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\begin{document}
\title{On the dimension of additive sets}
\author{P. Candela} \address{D\'epartement de math\'ematiques et applications\newline
\indent \'Ecole normale sup\'erieure,
Paris,
France} \email{pablo.candela@ens.fr} \author{H. A. Helfgott} \address{D\'epartement de math\'ematiques et applications\newline
\indent \'Ecole normale sup\'erieure,
Paris,
France}
\email{helfgott@dma.ens.fr} \thanks{Research supported by project ANR-12-BS01-0011 CAESAR and by a postdoctoral grant of the \'Ecole normale sup\'erieure, Paris.} \subjclass[2010]{Primary 11B30; Secondary 05D40} \keywords{Additive dimension, dissociated sets} \maketitle
\begin{abstract} We study the relations between several notions of dimension for an additive set, some of which are well-known and some of which are more recent, appearing for instance in work of Schoen and Shkredov. We obtain bounds for the ratios between these dimensions by improving an inequality of Lev and Yuster, and we show that these bounds are asymptotically sharp, using in particular the existence of large dissociated subsets of $\{0,1\}^n\subset \mathbb{Z}^n$. \end{abstract}
\section{Introduction} Let $A$ be an additive set, that is, a finite subset of an abelian group. A \emph{subset sum} of $A$ is a sum of the form $\sum_{a\in A'} a$ for some set $A'\subset A$. By a \emph{$[-1,1]$-combination of} $A$, we mean a sum $\sum_{a\in A} \varepsilon_a\, a$ with coefficients $\varepsilon_a$ lying in $ [-1,1]=\{-1,0,1\}$.
\begin{defn} A subset $D$ of an abelian group is said to be \emph{dissociated} if the subset sums of $D$ are pairwise distinct; equivalently, the only $[-1,1]$-combination of $D$ that equals 0 is the one with all coefficients equal to 0. We say that $D$ is a \emph{maximal} dissociated subset of $A$ if there is no dissociated set $D'\subset A$ such that $D'\supsetneq D$. \end{defn}
Dissociativity plays an important role in additive combinatorics and harmonic analysis; see \cite{TomNotes} and \cite[\S 4.5]{T-V}. In particular, it provides an analogue, in the setting of general abelian groups, of the concept of linear independence from linear algebra, and it is often used to define a notion of dimension for an additive set. For a recent instance, in the work of Schoen and Shkredov \cite{SS} the terminology `additive dimension of $A$' is used for the maximal cardinality of a dissociated subset of $A$. We shall call this quantity the dissociativity dimension.
\begin{defn} Let $A$ be an additive set. We define the \emph{dissociativity dimension} of $A$ to be the number
$d_d(A):=\max \{|D|: D\subset A,\; D\textrm{ is dissociated}\}$. We say that $D$ is a \emph{maximum} dissociated subset of $A$ if $|D|=d_d(A)$. We also define the \emph{lower dissociativity dimension} of $A$ to be the number
$d_d^-(A):=\min \{|D|: D\subset A \textrm{ is maximal dissociated}\}$. \end{defn}
The variant $d_d^-(A)$ is considered less often than $d_d(A)$ in the literature; it appears for instance in \cite[Section 8]{SS}, where it is denoted by $\tilde d(A)$.
In linear algebra, the concepts of linear independence and dimension are linked to that of a linear-span. The well-known basic result is that in a vector space the maximum cardinality of a linearly independent set, if finite, is equal to the minimum cardinality of a spanning set, the resulting number being by definition the dimension of the space. In the more general context of additive sets, there is an analogue of the linear span, related to dissociativity. We define it and give a corresponding notion of dimension, as follows.
\begin{defn} Given a subset $S$ of an abelian group $G$, the 1\emph{-span} of $S$, denoted $\langle S \rangle$, is the set of all $[-1,1]$-combinations of $S$. Given a subset $A\subset G$, we shall call a set $S\subset G$ satisfying $\langle S\rangle \supset A$ a \emph{1-spanning set} for $A$. We define the \emph{1-span dimension} of an additive set $A$ to be the number
$d_s(A):=\min \{|S|: S\subset A,\; \langle S\rangle \supset A\}$. \end{defn}
This quantity has also been considered in \cite[Section 8]{SS}, where it is denoted $d(A)$. A variant of this notion, which can be called the \emph{lower 1-span dimension} of $A$, is the number $d_s^-(A):=\min \{|S|: S\subset G,\; \langle S\rangle \supset A\}$; here $G$ is the ambient abelian group containing $A$ and the sets $S$ are allowed to have elements in $G\setminus A$. This variant also appears in \cite{SS}, where it is denoted $d_*(A)$. It had already appeared in previous works, notably as the number denoted $\ell(A)$ in \cite{Sch}.
Given the basic result from linear algebra recalled above, it is natural to compare the numbers $d_d(A),d_d^-(A)$ with $d_s(A),d_s^-(A)$. It follows promptly from the definitions that if $D$ is a maximal dissociated subset of $A$ then $\langle D\rangle \supset A$. We then deduce that \[ d_s^-(A)\leq d_s(A)\leq d_d^-(A)\leq d_d(A). \] In contrast to the linear-algebra setting, each of these inequalities can be a strict one. In this paper we study the extent to which these quantities can differ from each other.
Our first result is the following lower bound on the ratio $d_s^-(A) / d_d(A)$.
\begin{theorem}\label{thm:main} Let $A$ be an additive set. Then we have \begin{equation}\label{eq:mainineq} \frac{d_s^-(A)}{d_d(A)}\;\geq \; \frac{1}{\log_4 d_d(A)}\; \big(1+o(1)_{d_d(A)\to \infty}\big). \end{equation} \end{theorem}
We deduce this from an inequality relating the size of an arbitrary 1-spanning set for $A$ to the size of an arbitrary dissociated subset of $A$; see Proposition \ref{lem:dslb}. This inequality can be viewed as a refinement of an inequality of Lev and Yuster, namely inequality $(*)$ in \cite[Proof of Theorem 2]{LY}.
It is then natural to wonder whether there exist additive sets for which the ratio $d_s^-/d_d$ reaches the lower bound given by \eqref{eq:mainineq}, and more precisely whether each of the ratios of consecutive dimensions, i.e. $d_s^-/d_s,d_s/d_d^-,d_d^-/d_d$ can reach this lower bound.
For each positive integer $n$, let $Q_n$ denote the discrete cube $\{0,1\}^n$ viewed as an additive set in $\mathbb{Z}^n$. It follows from known results that $d_d(Q_n)=n\log_4 n\;(1+o(1))$ as $n\to \infty$. This was established independently by Lindstr\"om \cite{Lind} and by Cantor and Mills \cite{C&M}; the result is related to the \emph{coin weighing problem}, and similar results have been treated in other works (for a recent treatment, providing several references, see \cite{Bs}).
Let $D_n$ be a dissociated subset of $Q_n$ of cardinality $|D_n|=d_d(Q_n)$. Since the standard basis is itself a maximal dissociated subset of $Q_n$ of minimum size $n$, the set $Q_n$ shows that the ratio $d_d^-(A)/d_d(A)$ can be as small as $1/\log_4 d_d(A)$ asymptotically as $d_d(A)\to\infty$. Hence the lower bound in \eqref{eq:mainineq} is asymptotically sharp. Moreover, this set $D_n$ itself is an example showing that $d_s^-(A)/d_s(A)$ can also be as small as $1/\log_4 d_d(A)$, since for $D_n$ we have $d_s^-(D_n)=n$ yet $d_s(D_n)=|D_n|=d_d(D_n)$ (as $D_n$ is dissociated). Our second result completes the picture by showing that the remaining ratio $d_s(A)/d_d^-(A)$ can also be this small.
\begin{theorem}\label{thm:midratio} For each positive integer $n$ there exists a set $A_n\subset \{0,1,2\}^n$ satisfying $d_d(A_n)= n\log_4 n\, (1+o(1)_{n\to \infty})$ and such that \begin{equation}\label{eq:midratio} \frac{d_s(A_n)}{d_d^-(A_n)}\leq \frac{1}{\log_4 d_d(A_n)}\big(1+o(1)_{n\to\infty}\big). \end{equation} \end{theorem}
Theorems \ref{thm:main} and \ref{thm:midratio} are proved in Section \ref{section:main}.
In Section \ref{section:[N]} we consider sets of integers to examine whether, for at least some nice family of subsets of $\mathbb{Z}$, we have that for every set $A$ in the family the dissociativity dimensions $d_d(A)$, $d_d^-(A)$ lie closer to the spanning dimensions $d_s(A),d_s^-(A)$ than is guaranteed by \eqref{eq:mainineq}. The family of intervals $[N]=\{1,2,\ldots,N\}$ is a natural one to consider; let us recall for instance (see \cite[p. 59]{Erdos-Graham}) that it is one of the oldest problems of Erd\H os to prove that $d_d([N])=\log_2 N + O(1)$. We do not pursue that problem here, but we prove the following.
\begin{theorem}\label{thm:[N]} For any positive integer $N$ we have \[ d_s([N])=d_d^-([N])= \lfloor \log_3 N\rfloor + \big\lceil \log_3 2N -\lfloor \log_3 N\rfloor \big\rceil. \] \end{theorem} In the final section we briefly describe a relation between the dimension $d_s$ and a result of Schoen on maximal densities of subsets of $\Zmod{p}$ avoiding solutions to a linear equation with integer coefficients.
\section{On general additive sets: Theorems \ref{thm:main} and \ref{thm:midratio} }\label{section:main}
Given an additive set $A$, a 1-spanning set $S\subset A$ has size bounded below trivially by $\log_3(|A|)$, since $|\{-1,0,1\}^{S}|\geq |A|$. The argument leading to inequality $(*)$ in \cite[Proof of Theorem 2]{LY} is easily adapted to yield the following lower bound for $|S|$: we have $|S|\geq |D| / \log_2(2|D|+1)$ for every dissociated set $D\subset A$. This lower bound can be strengthened as follows.
\begin{proposition}\label{lem:dslb} Let $A$ be a finite subset of an abelian group $G$, let $D\subset A$ be dissociated, and let $S\subset G$ be a 1-spanning set for $A$. Then \begin{equation}\label{eq:dslb}
\frac{|D|}{\log_4|D|} \leq |S| \mathopen{}\mathclose\bgroup\originalleft(1+ \frac{4+\log_2 \log 4|S|}{\log_2 |D|}\aftergroup\egroup\originalright). \end{equation} \end{proposition} Theorem \ref{thm:main} follows from this, since $d_s^-(A)\leq d_d(A)$. \begin{proof}
Let $m=|S|, n=|D|$, and let us fix a labelling of the elements of $S$ and $D$, thus $S=\{s_1,s_2,\ldots, s_m\}$ and $D=\{d_1,d_2,\ldots,d_n\}$. Since $\langle S\rangle \supset A\supset D$, for each $j\in [n]$ we can fix a choice of a vector $(c_{i,j})_{i\in [m]}\in \{-1,0,1\}^m$ such that $d_j= \sum_{i\in [m]} c_{i,j} s_i$. Let $C$ be the $m\times n$ matrix with $(i,j)$ entry $c_{i,j}$.
The subset sums of $D$ are the combinations $\sum_{j=1}^{n} \lambda_j d_j$ with $\lambda=(\lambda_j)\in \{0,1\}^n$. We have \begin{equation}\label{eq:main1} \forall\, \lambda\in \{0,1\}^n,\qquad \sum_{j\in [n]} \lambda_jd_j = \sum_{i\in [m]} \Big(\sum_{j\in [n]} c_{i,j}\, \lambda_j \Big) s_i = \sum_{i\in [m]} (C\lambda)_i\,s_i . \end{equation}
We shall prove that, for some intervals of integers $\Lambda_1,\Lambda_2,\ldots, \Lambda_m$, each of width $O\Big(\sqrt{|D|\log |S|}\Big)$, for a large proportion of the elements $\lambda\in \{0,1\}^n$ we have $(C\lambda)_i \in \Lambda_i$ for every $i\in [m]$. To this end, fix any $i\in [m]$, and let us consider the terms $\lambda_1 c_{i,1},\ldots, \lambda_n c_{i,n}$ as independent random variables, the $j$th one taking value $c_{i,j}$ with probability $1/2$ and value 0 otherwise, for each $j\in [n]$. (Note that we are thus using the uniform probability on $\{0,1\}^n$.) Then letting $\mu_i=\frac{1}{2}\sum_{j\in [n]} c_{i,j}$, by Hoeffding's inequality \cite[Chapter 3, Theorem 1.3]{Gut} we have \[
\forall\, t>0,\qquad \mathbb{P}\mathopen{}\mathclose\bgroup\originalleft(\Big|\mu_i-\sum_{j\in [n]} \lambda_j\, c_{i,j} \Big| > t\Big(\sum_{j\in [n]} c_{i,j}^2\Big)^{1/2}\aftergroup\egroup\originalright) \leq 2 \exp \mathopen{}\mathclose\bgroup\originalleft(-2t^2 \aftergroup\egroup\originalright). \]
Since $\Big(\sum_{j\in [n]} c_{i,j}^2\Big)^{1/2}\leq |D|^{1/2}$, letting $t= \sqrt{ \log( 2 r |S|)/2}$, for $r>0$, we deduce that \[
\mathbb{P}\mathopen{}\mathclose\bgroup\originalleft(\Big|\mu_i - \sum_j \lambda_j\, c_{i,j} \Big| > |D|^{1/2} \sqrt{ \log( 2 r |S|)/2}\aftergroup\egroup\originalright) \leq (r|S|)^{-1} . \] By the union bound, the probability that the latter event holds for some $i\in [m]$ is thus at most $r^{-1}$. Hence \begin{equation}\label{eq:main3}
\mathbb{P}\Big(\Big|\mu_i - (C \lambda)_i \Big|\leq \sqrt{ |D| \log( 2 r |S|)/2}\,\,\textrm{ for all }i\in [m]\Big) \geq 1-r^{-1} . \end{equation}
Now let $\Lambda_i=\Big[\mu_i -\sqrt{ |D| \log( 2 r |S|)/2},\mu_i +\sqrt{ |D| \log( 2 r |S|)/2}\Big]$. Combining \eqref{eq:main1} and \eqref{eq:main3}, we obtain that for at least $(1-r^{-1})2^n$ values of $\lambda \in \{0,1\}^n$, the subset sum $\sum_{j\in [n]} \lambda_j d_j$ is an integer linear combination of the elements $s_1,\ldots, s_m$, with $i$th coefficient $(C\lambda)_i\in \Lambda_i$ for each $i\in [m]$. Since these subset sums are pairwise distinct (by dissociativity of $D$), we conclude that \[
\big(1-r^{-1}\big)\, 2^{|D|} \leq \prod_{j\in [m]} |\Lambda_j |\leq \big(2|D|\log(2r |S|)\big)^{|S|/2}. \] Choosing $r=2$, taking $\log_2$ of both sides and rearranging, we obtain \eqref{eq:dslb}. \end{proof}
We now turn to comparing $d_s$ and $d_d^-$, towards Theorem \ref{thm:midratio}.
We shall call a subset $S$ of an additive set $A$ satisfying $\langle S\rangle \supset A$ and $|S| = d_s(A)$ a \emph{minimum 1-spanning subset of} $A$.
The following small example shows that the dimensions $d_s$ and $d_d^-$ can indeed differ.
\begin{example}\label{lem:eg1} Let $\{x_1,x_2\}$ be the standard basis in $\mathbb{R}^2$, and let \[ A=\{x_1, x_2, x_1+x_2, 2x_1, 2x_2\}. \] This set has \textup{(}unique\textup{)} minimum 1-spanning subset $\{x_1,x_2,x_1+x_2\}$, while any maximal dissociated subset of $A$ has size $4$. \end{example}
The claims in this example are easily checked by inspection. In fact, this example is the simplest case of the following general construction, which is our main ingredient in our proof of Theorem \ref{thm:midratio}.
\begin{proposition}\label{lem:geneg} Let $B_n=\{x_1, x_2, \ldots, x_n\}$ be the standard basis of $\mathbb{R}^n$, let $s_n=\sum_{i\in [n]} x_i$, and let $D$ be a dissociated non-empty subset of $\{0,1\}^n$. Then the set \[ A_n= B_n\cup\{s_n\}\cup (2\cdot D) \]
satisfies $d_s(A_n)= n+1$ and $d_d^-(A_n)= d_d(A_n) = n+|D|$. \end{proposition} Here $2\cdot D$ denotes the set $\{2x:x\in D\}\subset \{0,2\}^n$. \begin{proof} To begin with, we claim that a 1-spanning subset $S\subset A_n$ must have at least $n+1$ elements. To show this, we distinguish two cases.
Case 1: $S$ does not contain $s_n$. Then, in order to be 1-spanning, $S$ must contain all other elements of $A_n$. Indeed, firstly, an element $x_i\in B_n$ must lie in $S$, for otherwise it cannot be in the 1-span of $S$, since every element of $A_n\setminus\{s_n, x_i\}$, modulo 2, has a zero $x_i$-component. An element of $2\cdot D$ must also lie in $S$, for it cannot be in the 1-span of other elements of $2\cdot D$ (since $D$ is dissociated), nor can it lie in $2\cdot D +\varepsilon_1 x_1+\dots+\varepsilon_n x_n$ with $\varepsilon_i\in [-1,1]$ not all zero, as it is congruent to 0 modulo 2. We have thus shown that $S$ must indeed contain $A_n\setminus \{s_n\}$, so our claim holds in this case, i.e. $|S|\geq n+1$.
Case 2: $S$ contains $s_n$, and does not contain some $x_j$. (If it contained $s_n$ and every $x_j$, then our claim would hold already.) In this case, in order to 1-span $x_j$ using $s_n$, the set $S$ must contain every $x_i$ with $i\neq j$. Moreover, $S$ must then also contain every element of $2\cdot D$. Indeed, an element of $2\cdot D$ equals either $2x_j$ or some combination $y$ involving some $2x_i$ with $i\neq j$. Now $2x_j$ must lie in $S$ in order to be 1-spanned by $S$, since $S$ does not contain $x_j$ and $D$ is dissociated. We claim that $S$ must also contain every other $y\in 2\cdot D$. Indeed, suppose that $y$ were not in $S$, and suppose that we had a $[-1,1]$-combination of elements of $S$ equal to $y$. This combination would then have to involve $s_n$, because otherwise it could only involve elements of $2\cdot D$ different from $y$, contradicting that $2\cdot D$ is dissociated. By involving $s_n$, this combination involves $x_j$. But the latter can then be neither cancelled nor increased to $2x_j$, since $S$ misses $x_j$, whence this combination could not equal $y$, a contradiction. We conclude that $S$ must be $A_n\setminus \{x_j\}$, so we have $|S|=n+|D|\geq n+1$ in this case.
The set $S_n:=B_n \cup \{s_n\}$, of size $n+1$, is 1-spanning for $A_n$ (and is not dissociated). We have thus shown that $d_s(A_n)= n+1$.
Now suppose that $S$ is a maximal dissociated subset of $A_n$. Then $S$ cannot contain $S_n$, so there exists some element $s\in S_n\setminus (S\cap S_n)$. Note also that, being maximal dissociated, $S$ must be 1-spanning for $A_n$. We can then distinguish the same two cases as above.
In the first case, we have $s=s_n$. Then, as in case 1 above, we must have $S= A_n\setminus \{s_n\}$, which is dissociated (as can be seen using that $B_n$ and $2\cdot D$ both are), clearly maximal, and of size $n+|D|$.
In the second case, we have $s = x_j$ for some $j\in [n]$. Then, $S$ must contain $s_n$ (it cannot 1-span it otherwise) and so we are in case 2 above, in which $S$ must be $A_n\setminus \{x_j\}$. Thus in this second case, either we get a contradiction (if $A_n\setminus \{x_j\}$ is not dissociated), or $S=A_n\setminus \{x_j\}$ is a maximal dissociated set of size $n+|D|$. \end{proof}
We now combine Proposition \ref{lem:geneg} with \cite[Theorem 1]{LY}. \begin{proof}[Proof of Theorem \ref{thm:midratio}]
As mentioned in the introduction, there exists a dissociated set $D_n\subset \{0,1\}^n$ of cardinality $|D_n|= n\log_4 n\, (1+o(1))$ as $n\to \infty$. Applying Proposition \ref{lem:geneg} with this set $D_n$, we obtain a set $A_n\subset \{0,1,2\}^n$ satisfying $d_s(A_n)=n+1$ and $d_d^-(A_n)=d_d(A_n)= n \log_4 n\,(1+o(1)_{n\to \infty})$, whence \eqref{eq:midratio} follows. \end{proof}
\section{Focusing on some sets of integers: Theorem \ref{thm:[N]}}\label{section:[N]}
So far, the examples that we have discussed of additive sets with small dimension-ratios have all been given by subsets of $\mathbb{Z}^n$ for large $n$. Note that by applying an appropriate Freiman isomorphism of sufficiently high order to such a set, we can obtain a subset of $\mathbb{Z}$ satisfying the same dimensional properties. For example, if for each $n$ we choose a Freiman isomorphism $\phi_n:\{0,1,2\}^n\to \mathbb{Z}$ of order $n^2$ (say) and satisfying\footnote{The existence of such Freiman isomorphisms is a standard result; see for instance \cite[Lemma 5.25]{T-V}.} $\phi_n(0)=0$, then applying $\phi_n$ to the set $A_n$ from Theorem \ref{thm:midratio} for each $n$ we obtain a family of sets $\phi_n(A_n)\subset \mathbb{Z}$ satisfying \eqref{eq:midratio}. One may wonder whether for some natural families of subsets of $\mathbb{Z}$ the dimensions $d_s^-, d_s,d_d^-,d_d$ lie closer to each other. In this section we show that this is the case for the family of intervals $[N]$, in the sense of Theorem \ref{thm:[N]}; thus we have $d_s([N])=d_d^-([N])$ for any positive integer $N$.
To prove Theorem \ref{thm:[N]}, we shall construct a maximal dissociated subset of $[N]$ of size $d_s([N])$, using the following simple fact concerning the powers of 3. \begin{lemma}\label{lem:P3} The set $P_3(k)=\{1,3,\ldots, 3^{k-1}\}$ satisfies $\langle P_3(k)\rangle = \mathopen{}\mathclose\bgroup\originalleft[-\frac{3^k-1}{2},\frac{3^k-1}{2}\aftergroup\egroup\originalright]$. \end{lemma}
\begin{proof} The claim holds for $k=1$. For $k>1$, we may suppose by induction that the claim holds for $k-1$, thus $\langle P_3(k-1)\rangle \supset \mathopen{}\mathclose\bgroup\originalleft[-\frac{3^{k-1}-1}{2},\frac{3^{k-1}-1}{2}\aftergroup\egroup\originalright]$. Then we have \begin{eqnarray*} \langle P_3(k)\rangle & = & \{-3^{k-1},0,3^{k-1}\} + \langle P_3(k-1)\rangle = \{-3^{k-1},0,3^{k-1}\} + \mathopen{}\mathclose\bgroup\originalleft[\frac{-3^{k-1}+1}{2},\frac{3^{k-1}-1}{2}\aftergroup\egroup\originalright] \\ & = & \mathopen{}\mathclose\bgroup\originalleft[-\frac{3^k-1}{2},\frac{3^k-1}{2}\aftergroup\egroup\originalright] . \end{eqnarray*} \end{proof} We shall also use the following. \begin{lemma}\label{lem:dissospan} Let $A$ be an additive set and let $S\subset A$ be dissociated and satisfy $\langle S \rangle \supset A$. Then $S$ is maximal dissociated. \end{lemma} \begin{proof} If there existed $a\in A\setminus S$ such that $S\cup \{a\}$ is dissociated, then $a$ could not lie in the 1-span of $S$, contradicting that $\langle S \rangle \supset A$. \end{proof}
To establish Theorem \ref{thm:[N]} we distinguish two cases, according to whether the fractional part $\{ \log_3 N \}:= \log_3 N - \lfloor \log_3 N \rfloor$ satisfies $\{ \log_3 N \}<1-\log_3 2$ or $\{ \log_3 N \}> 1-\log_3 2$.
\begin{proposition}\label{prop:case1} Let $N$ be a positive integer. The following statements are equivalent. \begin{enumerate}[leftmargin=30pt] \item We have $\{ \log_3 N \} <1-\log_3 2$.
\item The set $S_1:=\{1,3,3^2,\ldots, 3^{\lfloor \log_3 N\rfloor}\}$ is a minimum 1-spanning maximal dissociated subset of $[N]$. In particular $d_s([N])= d_d^-([N])= \lfloor\log_3 N\rfloor +1$. \end{enumerate} \end{proposition}
\begin{proof} It follows from Lemma \ref{lem:P3} that \[ \langle S_1\rangle = \langle P_3(\lfloor \log_3 N\rfloor+1) \rangle = \mathopen{}\mathclose\bgroup\originalleft[-\frac{3^{\lfloor \log_3 N\rfloor+1}-1}{2},\frac{3^{\lfloor \log_3 N\rfloor+1}-1}{2}\aftergroup\egroup\originalright]. \] Therefore $S_1$ is a 1-spanning subset of $[N]$ if and only if $\frac{3^{\lfloor \log_3 N\rfloor+1}-1}{2}\geq N$, that is if and only if $\{ \log_3 N \} <1-\log_3 2$. In particular, $(ii)$ implies $(i)$.
Now if $(i)$ holds, then we claim that $S_1$ is in fact \emph{minimum} 1-spanning for $[N]$. Indeed, any 1-spanning subset $S$ of $[N]$ must satisfy $N \leq (3^{|S|}-1)/2$, since to cover $[N]$ with $[-1,1]$-combinations of $S$ we only use the combinations with positive value. Hence $|S| \geq \log_3 (2N+1) >\log_3 N \geq \lfloor \log_3 N\rfloor$, so we have indeed that $|S| \geq \lfloor \log_3 N\rfloor +1=|S_1|$. Finally, note that $S_1$ is dissociated, so by Lemma \ref{lem:dissospan} it is maximal dissociated in $[N]$. We have thus shown that $(ii)$ holds. \end{proof}
We now treat the second case.
\begin{proposition}\label{prop:case2} Let $N$ be a positive integer, and let $t=1+ \sum_{i=0}^{ \lfloor \log_3 N\rfloor }3^i= \frac{3^{\lfloor \log_3 N \rfloor+1}+1}{2}$. The following statements are equivalent. \begin{enumerate}[leftmargin=30pt] \item We have $\{ \log_3 N \} > 1-\log_3 2$.
\item The set $S_2:=\{1,3,3^2,\ldots, 3^{\lfloor \log_3 N\rfloor}\}\cup \{t\}$ is a minimum 1-spanning maximal dissociated subset of $[N]$. In particular $d_s([N])= d_d^-([N])= \lfloor\log_3 N\rfloor +2$. \end{enumerate} \end{proposition} \begin{proof} By Lemma \ref{lem:P3} we have \[ \langle S_2\rangle = \langle P_3(\lfloor \log_3 N\rfloor+1) \rangle + \{-t,0,t\} = \mathopen{}\mathclose\bgroup\originalleft[-3^{\lfloor \log_3 N\rfloor+1},3^{\lfloor \log_3 N\rfloor+1}\aftergroup\egroup\originalright]. \] Thus $S_2$ is a 1-spanning set for $[N]$ which is dissociated. We have $S_2\subset [N]$ if and only if $t\leq N$, i.e. $\{ \log_3 N \} > 1-\log_3 2$. In particular, $(ii)$ implies $(i)$.
If $(i)$ holds, then we claim that $S_2$ is \emph{minimum} 1-spanning. Indeed, as shown at the end of the previous proof, if $S$ is 1-spanning for $[N]$ then we must have $|S|\geq \log_3(2N+1)$. If $|S|$ were less than $|S_2|$, i.e. if $|S|\leq \lfloor\log_3 N\rfloor +1$, then we would have $\lfloor \log_3 N \rfloor +1\geq \log_3(2N+1) > \log_3 2+ \log_3 N$, that is $\{ \log_3 N \}< 1-\log_3 2$, which contradicts $(i)$, so we must have $|S|\geq \lfloor\log_3 N\rfloor +2=|S_2|$. Note also that $S_2$ is dissociated, and therefore maximal dissociated in $[N]$ (by Lemma \ref{lem:dissospan} again). We have thus shown that $(ii)$ holds. \end{proof}
This completes the proof of Theorem \ref{thm:[N]}.
\section{Final remarks}
In \cite{Sch}, Schoen gave an interesting argument, using Chang's theorem, yielding an upper bound for the maximum density of a subset $A$ of $\Zmod{p}$ ($p$ prime) such that the Cartesian power $A^k$ contains no element $x$ solving a given integer linear equation $L(x)=c_1x_1+\cdots +c_k x_k=0$. We call such a set $A$ an \emph{$L$-free set}. Schoen's upper bound involves the dimension $d_s^-(C)$, where $C=\{c_1,\ldots,c_k\}$ is the set of coefficients of $L$ (in \cite{Sch} this dimension is denoted $\ell(C)$). It is a straightforward task to check that in Schoen's argument one can use $d_s(C)$ instead of $d_s^-(C)$. Thus one obtains the following version of Schoen's result.
\begin{theorem}\label{thm:circle-u-b}
Let $L(x)=c_1x_1+\cdots+c_k x_k$ be a linear form with coefficients $c_i \in \mathbb{Z}$, and let $m_L(\Zmod{p})=\max\{ |A|/p: A\subset \Zmod{p},\;A\textrm{ is }L\textrm{-free}\}$. Then \begin{equation}\label{eq:main-bounds} m_L(\Zmod{p})\leq e^{- d_s(C)/12}, \end{equation} where $C=\{c_1,c_2,\ldots,c_k\}$. \end{theorem}
As recalled in the introduction, there exists a dissociated set $D\subset \{0,1\}^n$ of size $\sim n\log_4 n$, and this has dimension $d_s(D)=|D|$, which is roughly $\log_4 n$ times $d_s^-(D)=n$. Applying an appropriate Freiman isomorphism $\phi: \{0,1\}^n \to \mathbb{Z}$, as in the previous section, we obtain a set $C=\phi(D) \subset \mathbb{Z}$ with the same properties (note that $d_s^-(C)\leq d_s^-(D)$ and $d_s(C)=d_s(D)$). For a linear form $L$ with coefficient-set $C$, the bound \eqref{eq:main-bounds} is thus stronger than the version with $d_s^-(C)$. It would be interesting to strengthen the upper bound on $m_L(\Zmod{p})$ further.
\textbf{Acknowledgements.} The first author is grateful to Jakob Vidmar for programming computer searches that shed light on problems treated in this paper. The authors are also very grateful to Vsevolod Lev for bringing to their attention the results on dissociated subsets of $\{0,1\}^n$ in \cite{Bs,C&M,Lind}.
\end{document} |
\begin{document}
\title{A Scalable Approach to Large Scale Risk-Averse Distribution Grid Expansion Planning}
\author{Alexandre Moreira, \IEEEmembership{Member,~IEEE,} Miguel Heleno, \IEEEmembership{Member,~IEEE,} Alan Valenzuela, Joseph H. Eto, \IEEEmembership{Senior Member,~IEEE,} Jaime Ortega, Cristina Botero.
\thanks{A. Moreira, A. Valenzuela, M. Heleno, J. Eto, are with the Lawrence Berkeley National Laboratory, Berkeley, CA, USA (e-mail: \mbox{\{AMoreira, AlanValenzuela, MiguelHeleno, JHeto\}@lbl.gov}). J. Ortega and C. Botero are with Commonwealth Edison, Chicago, IL, USA \mbox{\{Jaime.Ortega, Cristina.Botero\}@comed.com}.} }
\maketitle \begin{abstract}
Distribution grid reliability and resilience has become a major topic of concern for utilities and their regulators. In particular, with the increase in severity of extreme events, utilities are considering major investments in distribution grid assets to mitigate the damage of highly impactful outages. Communicating the overall economic and risk-mitigation benefits of these investments to regulators is an important element of the approval process. Today, industry reliability and resilience planning practices are based largely on methods that do not take explicit account of risk. This paper proposes a practical method for identifying optimal combinations of investments in new line segments and storage devices while considering the balance between the risk associated with high impact low probability events and the reliability related to routine failures. We show that this method can be scaled to address large scale networks and demonstrate its benefits using a Target Feeder from the Commonwealth Edison Reliability Program. \end{abstract}
\begin{IEEEkeywords} distribution expansion planning; large-scale distribution network; risk aversion; reliability. \end{IEEEkeywords}
\section*{Nomenclature}\label{Nomenclature}
The mathematical symbols used throughout this paper are classified below as follows.
\subsection*{Sets}
\begin{description} [\IEEEsetlabelwidth{5000000}\IEEEusemathlabelsep]
\item[${\Psi}^N$] Set of indexes of all nodes of the distribution grid.
\item[${\Psi}^{SS}$] Set of indexes of nodes that are substations of the distribution grid.
\item[${\Omega}$] Set of indexes of failure scenarios.
\item[${\Omega}^{resilience}$] Set of indexes of failure scenarios associated with resilience.
\item[${\Omega}^{routine}$] Set of indexes of routine failure scenarios.
\item[${\cal C}$] Set of indexes of failure states.
\item[${\cal D}$] Set of indexes of typical days.
\item[${\mathfrak{D}_{jec}}$] Set of indexes buses in each ``island'' $e$ when investment decision $j$ is taken for contingency state $c$.
\item[${E}_{jc}$] Set of indexes of islands if investment decision $j$ is taken under contingency state $c$.
\item[${H}$] Set of indexes of all storage devices (including existing and candidates).
\item[${H}^C$] Set of indexes of candidate storage devices.
\item[${\cal J}^{L,on}_j$] Set of indexes of candidate line segments that are build for the investment plan $j$.
\item[${\cal J}^{L,off}_j$] Set of indexes of candidate line segments that are not build for the investment plan $j$.
\item[${\cal L}$] Set of indexes of all lines (including existing and candidates).
\item[${\cal L}^{C}$] Set of indexes of existing transmission lines.
\item[${\cal L}^{E}$] Set of indexes of candidate transmission lines.
\item[${Rel}_c$] Set of indexes of relevant investments under contingency state $c$.
\item[${Rel}^{L,on}_{jc}$] Set of indexes of candidate line segments that are build for the investment plan $j$ that is relevant to failure state $c$.
\item[${Rel}^{L,off}_{jc}$] Set of indexes of candidate line segments that are not build for the investment plan $j$ that is relevant to failure state $c$.
\item[$T$] Set of indexes of operation periods during each typical day.
\end{description}
\subsection*{Indexes}
\begin{description} [\IEEEsetlabelwidth{5000000}\IEEEusemathlabelsep]
\item[$c$] Index of failure state.
\item[$d$] Index of typical days.
\item[$e$] Index of the islands that are formed under a contingency state $c$.
\item[${h}$] Index of storage devices.
\item[${j}$] Index of investment decision.
\item[$l$] Index of lines.
\item[$n$] Index of buses.
\item[$s$] Index of scenarios.
\item[$t$] Index of time periods.
\item[$t^0$] Index of the first time period of a day type $d$.
\end{description}
\subsection*{Parameters}
\begin{description} [\IEEEsetlabelwidth{5000000}\IEEEusemathlabelsep]
\item[${\alpha^{CVaR}}$] CVaR parameter.
\item[${\delta}$] Number of hours in a time period $t$.
\item[$\eta$] Round trip efficiency of batteries.
\item[${\lambda}$] Risk aversion user-defined parameter (between 0 and 1).
\item[${\rho}$] Probability of scenario $s$.
\item[$C^{Imb}$] Cost of imbalance.
\item[$C^{L,fix}_l$] Fixed investment cost of candidate line $l$.
\item[$C^{SD,fix}_h$] Fixed investment cost of candidate storage device $h$.
\item[$C^{SD,var}_h$] Variable investment cost of candidate storage device $h$.
\item[${D^{peak}_i}$] Peak demand of bus $i$.
\item[${D_{ntd}}$] Demand of bus $n$, at time period $t$ of typical day $d$.
\item[${{f}^{bat}_{h,t,d}}$] Percentage of state of charge of battery $h$ at time period $t$ of day type $d$.
\item[${{f}^{load}_{\tau,d}}$] Percentage of peak load at time period $\tau$ of day type $d$.
\item[${\overline{F}_l}$] Maximum capacity of line $l$.
\item[$\overline{G}^{Tr}_n$] Limit of injection in substation $n$.
\item[${k}_s$] Number of time periods of failure scenario $s$.
\item[${M}$] Sufficiently large number.
\item[$\overline{P}^{in}_h$] Maximum charging of storage device $h$ per time period.
\item[$\overline{P}^{out}_h$] Maximum discharging of storage device $h$ per time period.
\item[$pf$] Power factor.
\item[${r^{len}}$] Length of line $l$.
\item[$\overline{S}$] Number of hours to fully charge storage devices.
\item[${\underline{V}}$] Maximum voltage.
\item[${\overline{V}}$] Maximum voltage.
\item[${W_{d}}$] Number of days of type $d$ in one year.
\item[${x}^{state}_{cs}$] parameter that is equal to 1 if scenario $s$ implies in failure state $c$, being equal to 0 otherwise. Note that each scenario $s$ can only imply in one contingency state $c$.
\item[${Z^L_{l}}$] Impedance of line $l$.
\end{description}
\subsection*{Decision Variables}
\begin{description} [\IEEEsetlabelwidth{5000000}\IEEEusemathlabelsep]
\item[${\Delta^{+}_{ntd}}$] Positive imbalance in bus $n$ at time period $t$ of day type $d$.
\item[${\Delta^{-}_{ntd}}$] Negative imbalance in bus $n$ at time period $t$ of day type $d$.
\item[$\zeta_{td}$] CVaR auxiliary variable that represents the value at risk at time period $t$ of day type $d$.
\item[$\psi^{CVaR}_{tds}$] CVaR auxiliary variable.
\item[$f_{ltd}$] Flow in line $l$ at time period $t$ of day type $d$.
\item[$g^{Tr}_{ntd}$] Injection via substation $n$ at time period $t$ of day type $d$.
\item[$L^{\dagger}_{tds}$] Load shedding at time period $t$ of day type $d$ of scenario $s$.
\item[$L_{jec}$] Load shedding in island $e$ for relevant investment $j$ under failure state $c$.
\item[$p^{in}_{htd}$] Charging of storage device $h$ at time period $t$ of day type $d$.
\item[$p^{out}_{htd}$] Discharging of storage device $h$ at time period $t$ of day type $d$.
\item[$SOC_{htd}$] State of charge of storage device $h$ at time period $t$ of day type $d$.
\item[$SOC^{aux}_{hjec}$] State of charge of storage device $h$ that belongs to island $e$ for relevant investment $j$ under contingency state $c$.
\item[$SOC^{ref}_{h}$] Reference state of charge of storage device $h$.
\item[$v_{ntd}$] Voltage in bus $n$ at time period $t$ of day type $d$.
\item[$x^{ind}_{jc}$] Binary variable that indicates which relevant investment option $j$ has been taken under contingency state $c$.
\item[$x^{L,fix}_l$] Binary investment in line $l$.
\item[$x^{SD,fix}_{h}$] Binary investment in storage device $h$.
\item[$x^{SD,var}_{h}$] Continuous investment in storage device $h$.
\end{description}
\section{Introduction}\label{Introduction}
\IEEEPARstart{D}{distribution} grid assets represent a significant portion of the overall power system costs and, in the US, the highest share of capital investments of investor-owned utilities \cite{EEI2019}. Given this determinant role, utilities are periodically required to justify to regulators their proposed investments and the corresponding impact on consumer rates \cite{Cooke2018}. Typical reasons for those investments in the grid include expected load growth, hosting capacity and improvements in reliability performance.
In practice, grid investments driven by load growth can be justified using quantitative approaches, based on load flow simulations or, as done by Pacific Gas and Electric (PG\&E) in California, using more advanced methodologies including forecasting future feeder demands in different locations combined with consumer behavior under different meteorological seasons \cite{PGE2021_GNA}. Similarly, a hosting capacity analysis is often required to justify the corresponding grid investments, which can be a highly regulated process in some US jurisdictions, such as Minnesota, Hawaii, California, and New York \cite{Schwartz2020}.
In the reliability investments case, the process is slightly different. First, utilities are often evaluated by the reliability performance of their feeders and required to report reliability standardized metrics \cite{Cooke2018}, such as System Average Interruption Frequency Index (SAIFI), System Average Interruption Duration Index (SAIDI), Customer Average Interruption Frequency Index (CAIFI) and Customer Average Interruption Duration Index (CAIDI) \cite{IEEEStd1366}. Based on this ex-post reliability evaluation, utilities can suggest new investments to improve their performance. For example, in California, PG\&E publishes an annual report with reliability metrics in its service territory, including potential grid investments to improve them \cite{PGE2021_AnnualReliability}. In Illinois, utilities are requested to publish annual reliability performance reports and present a 3-year plan for reliability investments \cite{Illinois2020}, very similar to Ohio \cite{Ohio2021_1}, where utilities report metrics of their worse performing feeders \cite{Ohio2021_2}. Commonwealth Edison (ComEd) has a detailed process to propose grid investments \cite{ComEd2021_InvestmentsProposal}, being ``system performance'' (reliability) one among seven capital investment categories presented to the regulator. ``System performance'' includes investments that can improve the reliability of the system based on characteristics such as historical data of failures as well as material condition and age of system elements.
In short, the current practices of the industry show that distribution reliability investments are (1) based on an ex-post analysis of performance and (2) determined by empirical knowledge. Unlike other drivers of grid investments, such as load growth or hosting capacity, no forward-looking optimization nor simulation analysis is carried out. A forward-looking reliability assessment is already an usual practice in bulk power systems, in which forward-looking reliability indices, such of loss of load expectation (LOLE) and/or expected energy not served (EENS), are defined as requirements of the system \cite{NationalGrid2017_SecurityofSupply}.
Existing practices are even more limited when it comes to resilience investments. However, given the projected increase in frequency, intensity and duration of extreme weather hazards \cite{USGCRP_2017} and their consequences to the power supply and delivery \cite{DOE_2013}, resilience has become a central topic in the power systems community over the last few years. Despite the broader definition of resilience provided by FERC \cite{FERC2018_resilienceDef} - ``{\it the ability to withstand and reduce the magnitude and/or duration of disruptive events, which includes the capability to anticipate, absorb, adapt to, and/or rapidly recover from such an event}'' - resilience-related standards and metrics are still to be developed \cite{Vugrin2017}. In the absence of a consensus on resilience metrics, utilities remain relying on traditional reliability indices, conceived to capture routine failures instead of HILP events \cite{Schwartz2019_UtilityInvestmentsResilience} and to be used in ex-post evaluation. Therefore, the methods currently used by industry to plan the upgrade of distribution systems do not consider the risk associated with HILP events, which are much less predictable and much more impactful compared to routine events.
Thus, there is a need for analytical methodologies to support utilities' investment decisions, under reliability and resilience programs, that can capture forward-looking risk mitigation benefits and can demonstrate to regulators the added resilience value of different investment options. This paper presents a practical and scalable methodology to fill this gap and demonstrates it using Target Feeders from Commonwealth Edison (ComEd) Reliability Program.
\subsection{Literature Review}
Different metrics \cite{reliability_guide} and methods \cite{allan_billinton_1996} were developed in the past to perform reliability assessment in power systems, particularly in stochastic simulation environments, and later integrated into optimization methodologies addressing, for example, the expansion planning of distribution networks \cite{Munoz2016,Munoz2018}. However, recently, due to an increasing number of occurrences of natural disasters, a great deal of attention has been devoted to take resilience into consideration while planning and operating power systems. In this paper, we propose a methodology to plan the expansion of large-scale distribution systems while considering not only reliability but also resilience in the form of risk-aversion.
Several works have proposed approaches to tackle the distribution grid planning problem over the last years. In \cite{Moradijoz2018}, the authors propose a bilevel mixed-integer program that optimizes the distribution system expansion while taking into account the presence of Electric Vehicles (EVs). While the first level determines investments in the grid, the second level manages the strategies of charging and discharging of parked EVs so as to maximize the revenue of parking lots that provide grid services. In \cite{Amjady2018}, line reinforcement, distributed energy resources (DERs) and dispatchable units are candidate investments to be selected by the proposed methodology while facing uncertainty in DERs output and demand and neglecting reliability and resilience against failures of system elements. In \cite{Li2013}, a game-theoretical approach is presented to tackle the distribution planning problem. In \cite{Arasteh2019}, the distribution system expansion planning problem is addressed while considering the private investor (PI) who owns distributed generation, the distribution company (DISCO), and the demand response provider (DRP) as different players with different objectives. While the DISCO performs line reinforcements to improve reliability and to decrease costs by minimizing expected energy not served associated with line failures, DRP and PI aim to maximize the conditional value at risk (CVaR) of their profits under uncertainty in the availability of demand response and in renewable generation. In \cite{Ahmadian2019}, particle swarm optimization and tabu search are integrated into an algorithm that plans the expansion of distribution networks. In \cite{Zhao2020}, the distribution system planning is addressed by a stochastic optimization approach that determines investment in substations, feeders, and batteries while considering battery degradation and facing uncertainty in electricity prices and demand. In \cite{Troitzsch2020}, the flexibility to reduce peak demands provided by thermal building systems is considered while planning the distribution grid expansion. In \cite{Fan2020}, the distribution system expansion problem is addressed via a model that considers EVs and uncertainty in renewable energy sources.
Security under high impact and low probability (HILP) events has been a recent topic of concern in the context of expansion planning methodologies. At the transmission level, for example, a two-stage stochastic Mixed-Integer NonLinear Programming (MINLP) model is formulated in \cite{Romero2013} to determine the investment plan to increase resilience while considering seismic activity. Moreover, in \cite{Lagos2020}, an approach that leverages on simulation techniques and optimization is proposed to define the portfolio of investments needed to deal with potential events of earthquakes. In addition, relevant works have also considered resilience while planning investments at the distribution level. In \cite{Nazemi2020}, seismic hazards are considered in a model that decides sitting and sizing of storage devices. In \cite{Lin2018}, a trilevel model is proposed to select lines to be hardened to reduce the vulnerability of the distribution system to intentional or unintentional attacks. Finally, \cite{Barnes2019} proposes an approach to address the expansion planning (selecting network upgrades) of large scale distribution systems with a focus on preparing the grid to withstand extreme events specifically related to ice and wind storms.
\begin{comment} falar que metricas seriam bem-vindas e citar que menciona isso. falar que pratica atual visa mitigar o que ja aconteceu ao inves de projetar como quer que seja. falar que seria bom ter uma metodologia que otimiza investmentos com vistas a melhoria simultanea de metricas de confiabilidade e resiliencia e eh isso que propomos.
falar sobre o processo de selecao de investmentos da ComEd. \end{comment}
\subsection{Contributions}
In this paper, we propose a practical methodology to plan the expansion of large-scale distribution systems while minimizing the convex combination of the expected value and the CVaR of loss of load costs. Our results show that objective functions based on traditional risk-neutral metrics, e.g. the expected energy not served (EENS), produce expansion plans that neglect the consequences of HILP events. Consistent risk-aversion strategies can only be achieved through the inclusion of risk-based objectives. Unlike the previously mentioned works, we propose a methodology that can simultaneously (i) be general enough to consider routine (related to reliability) and extreme events (related to resilience) regardless of the cause while allowing the planner to place more importance on reliability or resilience according to their level of risk aversion, (ii) consider not only traditional investments in line segments but also in storage devices, and (iii) be scaled to realistic large scale distribution systems. Finally, we demonstrate our method using distribution planning information taken from a current US utility distribution system.
The contributions of this paper can be summarized as:
\begin{enumerate}
\item To propose a distribution system expansion planning model that accounts for reliability and resilience metrics while allowing the system planner to define their own level of risk-aversion. In this manner, the trade-off between focusing on reliability or on resilience can be evaluated so as to the determine the most appropriate portfolio of investments in new line segments and storage devices.
\item To reformulate the proposed model based on realistic assumptions in order to improve the scalability of the proposed methodology. As a result, our proposed model can be solved for real size large scale systems while considering several failure scenarios which can be based on historical data.
\end{enumerate}
The remainder of the paper is organized as follows. Section II presents a conventional scenario-based approach to formulate the problem under consideration in this paper. Section III describes the steps to alleviate the computational burden of the model presented in the previous section. Section IV presents case studies, and finally in Section VI we conclude.
\section{Conventional scenario-based approach }\label{sec.MathematicalFormulation}
Next, we present a methodology to select the optimal portfolio of investments to upgrade the distribution system with the objective of alleviating the impact of routine failures and the damage associated with HILP events. To achieve that, we consider not only the minimization of the expected value of the cost of loss of load, but also the CVaR of this cost for a range of failure scenarios (considering failures of line segments of the grid). In a conventional scenario-based approach, this problem can be formulated as follows.
\begin{align}
& \underset{{\substack{\Delta^-_{ntds},\Delta^+_{ntds},\zeta_{td},\psi^{CVaR}_{tds},\\f_{ltds},g^{Tr}_{ntds},p^{in}_{htds},p^{out}_{htds},\\SOC_{htds},v_{ntds},x^{L,fix}_{l}, x^{SD,fix}_{h},x^{SD,var}_{h}}}}{\text{Minimize}} \hspace{0.1cm} \sum_{l \in {\cal L}^C} C^{L,fix}_lx^{L,fix}_{l}
\notag\\
&\hspace{0pt} + \sum_{h \in H^C} \Bigl[ C^{SD,fix}_h x^{SD,fix}_{h} + C^{SD,var}_h x^{SD,var}_{h} {\color{black}\overline{S}}\overline{P}^{in}_h \Bigr] \notag \\
&\hspace{0pt}+ \sum_{d \in {\cal D}}W_d\sum_{t \in T}\Biggl[
pf C^{Imb} \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{n,t,d,1} + \Delta^+_{n,t,d,1} \Bigr] \Biggr ] \notag \\
&\hspace{0pt} + (1-\lambda) pf C^{Imb} \sum_{d \in {\cal D}} W_d \sum_{t \in T} \sum_{s \in \Omega \setminus{\{1\}}} \rho_s \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{ntds} \notag\\
&+ \Delta^+_{ntds} \Bigr]+ \lambda ~ pf ~ C^{Imb} \sum_{d \in {\cal D}} W_d \sum_{t \in T} \Bigl[ \zeta_{td}\notag\\
&\hspace{99pt} + \sum_{s \in \Omega \setminus{\{1\}}} \frac{\rho_s}{1-\alpha^{CVaR}} \psi^{CVaR}_{tds} \Bigr] \label{ScenarioBasedFormulation_1}\\
& \text{subject to:}\notag\\
& \psi^{CVaR}_{tds} + \zeta_{td} \geq \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{ntds} + \Delta^+_{ntds} \Bigr]; \forall d \in {\cal D}, \notag\\
&\hspace{153pt} t \in T, s \in \Omega \setminus \{1\} \label{ScenarioBasedFormulation_2}\\
& \psi^{CVaR}_{tds} \geq 0; \forall d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_3}\\
& x^{L,fix}_l \in \{0,1\}; \forall l \in {\cal L}^C \label{ScenarioBasedFormulation_4}\\
& x^{SD,fix}_h \in \{0,1\}; \forall h \in H^C \label{ScenarioBasedFormulation_5}\\
& 0 \leq x^{SD,var}_h \leq x^{SD,fix}_h \overline{x}^{SD}_h; \forall h \in H^C \label{ScenarioBasedFormulation_6}\\
& 0\leq g^{Tr}_{ntds} \leq \overline{G}^{Tr}_n; \forall n \in \Psi^{SS}, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_7}\\
& \underline{V} \leq v_{ntds}\leq \overline{V}; \forall n \in \Psi^N, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_8}\\
& -y_{ltds} \overline{F}_l \leq f_{ltds} \leq y_{ltds} \overline{F}_l; \forall l \in {\cal L}^E, d \in {\cal D}, t \in T, \notag\\
& \hspace{205pt} s \in \Omega \label{ScenarioBasedFormulation_9}\\
& -y_{ltds} x^{L,fix}_l \overline{F}_l \leq f_{ltds} \leq y_{ltds} x^{L,fix}_l \overline{F}_l; \forall l \in {\cal L}^C, \notag\\
& \hspace{144pt}d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_10}\\
& -M(1-y_{ltds}) \leq Z^L_l r^{len}_l f_{ltds} - \bigl( v_{fr(l),t,d,s} \notag\\
& \hspace{5pt} - v_{to(l),t,d,s} \bigl) \leq M(1-y_{ltds}); \forall l \in {\cal L}^{E}, d \in {\cal D}, t \in T, \notag\\
& \hspace{200pt} s \in \Omega \label{ScenarioBasedFormulation_11}\\
& - M(1-y_{ltds}) - M(1-x^{L,fix}_{l}) \leq Z^L_l r^{len}_l f_{ltds} \notag\\
& \hspace{5pt}- \bigl( v_{fr(l),t,d,s} - v_{to(l),t,d,s} \bigl) \leq M(1-y_{ltds}) \notag\\
& \hspace{26pt}+ M(1-x^{L,fix}_{l}); \forall l \in {\cal L}^{C}, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_12}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltds} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltds} + g^{Tr}_{ntds} = 0; \notag\\
& \hspace{97pt} \forall n \in {\Psi}^{SS}, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_13}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltds} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltds} = \sum_{h \in H_n} p^{in}_{htds} \notag\\
& \hspace{40pt} - \sum_{h \in H_n} p^{out}_{htds} - \Delta^-_{ntds} + \Delta^+_{ntds} + D_{ntd};\notag \\
& \hspace{72pt} \forall n \in {\Psi}^{N} \setminus {\Psi}^{SS}, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_14}\\
& SOC_{h|T|ds} = SOC_{ht^{0}ds}; \forall h \in H, d \in {\cal D}, s \in \Omega \label{ScenarioBasedFormulation_15}\\
& SOC_{htds} = SOC_{ht^{0}ds} + \eta \delta p^{in}_{htds} - \delta p^{out}_{htds}; \forall h \in H, \notag \\
& \hspace{143pt} d \in {\cal D}, t=1, s \in \Omega \label{ScenarioBasedFormulation_16}\\
& SOC_{htds} = SOC_{h,t-1,d,s} + \eta \delta p^{in}_{htds} - \delta p^{out}_{htds}; \notag \\
& \hspace{82pt} \forall h \in H, d \in {\cal D}, t \in T|t\geq2, s \in \Omega \label{ScenarioBasedFormulation_17}\\
& 0 \leq SOC_{htds} \leq \overline{S}\overline{P}^{in}_h; \forall h \in H \setminus H^C, s \in \Omega\label{ScenarioBasedFormulation_18}\\
& 0 \leq SOC_{htds} \leq \overline{S} x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C, s \in \Omega\label{ScenarioBasedFormulation_19}\\
& 0 \leq p^{in}_{htds} \leq \overline{P}^{in}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_20}\\
& 0 \leq p^{out}_{htds} \leq \overline{P}^{out}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T, s \in \Omega \label{ScenarioBasedFormulation_21}\\
& 0 \leq p^{in}_{htds} \leq x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C, d \in {\cal D}, t \in T, \notag\\
&\hspace{201pt} s \in \Omega \label{ScenarioBasedFormulation_22}\\
& 0 \leq p^{out}_{htds} \leq x^{SD,var}_h \overline{P}^{out}_h; \forall h \in H^C, d \in {\cal D}, t \in T, \notag\\
&\hspace{201pt} s \in \Omega \label{ScenarioBasedFormulation_23} \end{align}
The optimization problem \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_23} is a two-stage stochastic program formulated as a mixed-integer linear programming (MILP) model. The first-stage decision determines investment in new line segments and storage devices. The second-stage decision is associated with operation under a failure scenario.
The objective function to be minimized in \eqref{ScenarioBasedFormulation_1} comprises investment cost in new line segments and storage devices, cost of imbalance in the base case (scenario $s=1$), and a convex combination between expected value and CVaR of imbalance cost associated with a set of failure scenarios. Constraints \eqref{ScenarioBasedFormulation_2} and \eqref{ScenarioBasedFormulation_3} model the behavior of variables $\psi^{CVaR}_{tds}$ and $\zeta_{td}$ which are related to the CVaR of imbalance cost present in the objective function. Constraints \eqref{ScenarioBasedFormulation_4} and \eqref{ScenarioBasedFormulation_5} express the binary nature of investment variables $x^{L,fix}_l$ and $x^{SD,fix}_h$ that correspond to the installation of new line segments and storage devices, respectively. Constraints \eqref{ScenarioBasedFormulation_6} limit the continuous variable associated with the capacity of the candidate storage devices to a upper bound that depends on whether $x^{SD,fix}_h$ assumes value equal to one. Constraints \eqref{ScenarioBasedFormulation_7} limit the amount of power injected from the main transmission grid to the substations $n \in \Psi^{SS}$ of the distribution grid. Constraints \eqref{ScenarioBasedFormulation_8} impose voltage bounds for each bus of the distribution grid. Constraints \eqref{ScenarioBasedFormulation_9} and \eqref{ScenarioBasedFormulation_10} enforce transmission capacity limits to existing and candidate line segments, respectively, whereas constraints \eqref{ScenarioBasedFormulation_11} and \eqref{ScenarioBasedFormulation_12} relate power flows to voltages (also for existing and candidate lines) in a linear fashion as often done in distribution planning models (see \cite{Haffner2008} \cite{Munoz2016} for example). Constraints \eqref{ScenarioBasedFormulation_13} and \eqref{ScenarioBasedFormulation_14} ensure nodal power balance for substations and other buses, respectively. Constraints \eqref{ScenarioBasedFormulation_15}--\eqref{ScenarioBasedFormulation_17} model state of charge (SOC) variation along different periods. Constraints \eqref{ScenarioBasedFormulation_18} and \eqref{ScenarioBasedFormulation_19} impose SOC capacities for existing and candidate storage devices, respectively. Constraints \eqref{ScenarioBasedFormulation_20} and \eqref{ScenarioBasedFormulation_21} enforce limits to the charging and discharging of existing storage devices while \eqref{ScenarioBasedFormulation_22} and \eqref{ScenarioBasedFormulation_23} do the same to candidate storage devices.
\section{Scalability-oriented reformulation }\label{sec.Scalability}
The scenario-based formulation \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_23} can explicitly evaluate the cost of pre- and post-failure loss of load under a range of scenarios as it accounts for optimal power flow (OPF)-related constraints for both base case and each scenario of failure. However, for medium-sized systems and a reasonable number of scenarios, solving \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_23} is prohibitive due to large number of constraints, in particular the time coupling ones associated with the battery operation during outages. In this Section, we rewrite formulation \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_23} to address these scalablility issues by considering three assumptions that are based on industry practice.
\subsection{Assumptions} \textit{Assumption 1: Storage operation during outages}. Here we distinguish routine ($\Omega^{routine}$) from resilience ($\Omega^{resilience}$) outage events. The first correspond to spontaneous equipment failures that cannot be predicted nor anticipated by storage operation. Thus, we assume that storage is operated with other objectives (economic) and, when a routine failures occur, the existing storage SOC can be mobilized to mitigate it. The second are extreme events (e.g. storms, floods, wildfires) that can be predicted hours ahead. In this case, when the event occurs, it is assumed that the operators have preventively charged the batteries up to the maximum capacity.
\textit{Assumption 2: Power flow constraints during outages}. We consider that the loss of load associated with a particular state of failure can actually be modelled without writing the respective OPF-related constraints. This means that if a pre-outage state satisfies the steady-state load flow limits, any re-configuration of the network to mitigate an outage will also satisfy those limits. The realistic assumption behind it is that utilities only propose new ties as candidates after evaluating the peak condition of different topology realizations.
\textit{Assumption 3}: We assume that the number of candidate assets are very small in comparison with the number of outages and the grid size (utilities often evaluate a few investment options in grids with thousands of nodes).
\subsection{Scalability Approach} \textit{Assumption 1} allows to model storage operation during failure events exclusively as a function of (i) battery capacity and (ii) SOC at the time $t$ when the failure occurs. \textit{Assumption 2} allows to evaluate the loss of load as a function of those two variables and the duration $k$ of the outage when there is no possible reconfiguration to reconnect the portion of the grid that is disconnected by the failed line. With these two assumptions, an outage scenario $s$ can be represented as a state of failure of the grid $c$, starting at time $t$ with a duration $k_s$.
This separation between scenario and state of failure allows to reduce the dimensionality of the problem. Considering \textit{Assumption 3}, it is possible to say that for each state of failure $c$, there is only a small subset of relevant investments ($Rel_c$) that can mitigate the loss of load, regardless of the starting time $t$ and the duration $k_s$ of the outage. For example, investments in Zone A are irrelevant to mitigate the loss of load in Zone B when there is a failure in the line between Zones A and B.
\subsection{Model Formulation} Following this scalability approach, we considered the set of all states of failure of the grid ${\cal C}$ and we relate scenarios and states of failure using the binary parameter $x^{state}_{cs}$. For each $s \in {\cal S}$, this parameter is set to 1 just for one index $c$ within ${\cal C}$, so as to indicate the state of failure associated with each scenario. The parameter $k_s$ represents the duration of the state of failure $c$ in the outage scenario $s$. Following \textit{Assumption 1}, SOC at time $t$ is calculated separately, based on an economic objective (e.g. price signal), and modeled as a parameter $f^{bat}_{htd}$ both in the base case and failure scenarios. It is important to note that $f^{bat}_{htd}$ is used to determine the storage investment (which remains a variable). Still in \textit{Assumption 1}, the storage is modeled with a maximum SOC in response to extreme failure scenarios. Following \textit{Assumption 2}, the loss of load can be assessed by the energy balance within the multiple network islands that result from the states of failure. This assessment is similar to the expansion planning decision making framework provided in Section \ref{sec.MathematicalFormulation}, but defining the set of indexes of islanded buses $\mathfrak{D}_{jec}$ for each possible portfolio of investments $j$ and state of failure $c$, where $e \in E_c$ and $E_c$ is the set of indexes of islands created by the state of failure $c$. As mentioned in the scalability approach, we define the set relevant investments $Rel_c$ which contains the indexes $j$ of the investment combinations that are relevant to the state of failure $c$. In addition, we also create sets ${Rel}^{L,on}_{jc}$ and ${Rel}^{L,off}_{jc}$ which contain the indexes of line segments that are built and not built, respectively, under the relevant investment combination $j$ associated with failure state $c$. The model \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_23} is rewritten as follows.
\begin{align}
& \underset{{\substack{\Delta^+_{ntd},\Delta^-_{ntd},\zeta_{td},\psi^{CVaR}_{tds},\\f_{ltd},g^{Tr}_{ntd},L_{jec},L^{\dagger}_{tds},\\p^{in}_{htd},p^{out}_{htd}, SOC_{htd}, \\SOC^{aux}_{hjec}, SOC^{ref}_{h}, v_{ntd}\\mathbf{x}^{ind}_{jc}, x^{L,fix}_{l}, x^{SD,fix}_{l}, x^{SD,var}_{l}}}}{\text{Minimize}} \hspace{0.1cm} \sum_{l \in {\cal L}^C} \Bigl[ C^{L,fix}_lx^{L,fix}_{l} \Bigr] \notag\\
&\hspace{0pt} + \sum_{h \in H^C} \Bigl[ C^{SD,fix}_h x^{SD,fix}_{h} + C^{SD,var}_h x^{SD,var}_{h} {\color{black}\overline{S}}\overline{P}^{in}_h \Bigr] \notag \\
&\hspace{0pt}+ \sum_{d \in {\cal D}}W_d\sum_{t \in T}\Biggl[
pf C^{Imb} \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{ntd} + \Delta^+_{ntd} \Bigr] \Biggr ] \notag \\
&\hspace{0pt}+ (1-\lambda) pf C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \sum_{s \in \Omega} \rho_s L^{\dagger}_{tds}\notag\\
&\hspace{0pt}+ \lambda ~ pf ~ C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \Bigl[ \zeta_{td} \notag\\
&\hspace{110pt} + \sum_{s \in \Omega} \frac{\rho_s}{1-\alpha^{CVaR}} \psi^{CVaR}_{tds} \Bigr] \label{RepairV2_v7_1}\\
& \text{subject to:}\notag\\
& \psi^{CVaR}_{tds} + \zeta_{td} \geq L^{\dagger}_{tds}; \forall d \in {\cal D}, t \in T, s \in \Omega \label{RepairV2_v7_2}\\
& \psi^{CVaR}_{tds} \geq 0; \forall d \in {\cal D}, t \in T, s \in \Omega \label{RepairV2_v7_3}\\
& x^{ind}_{jc} \in \{0,1\}; \forall c \in {\cal C}, j \in {Rel}_c \label{RepairV2_v7_3_a}\\
& x^{L,fix}_{l} \in \{0,1\}; \forall l \in {\cal L}^C \label{RepairV2_v7_4}\\
& x^{SD,fix}_h \in \{0,1\}; \forall h \in H^C \label{RepairV2_v7_5}\\
& 0 \leq x^{SD,var}_h \leq x^{SD,fix}_h \overline{x}^{SD}_h; \forall h \in H^C\label{RepairV2_v7_6}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{5pt} - \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} f^{bat}_{htd} \Bigr ]; \forall t \in T, d \in {\cal D}, s \in \Omega^{routine}\label{RepairV2_v7_7}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{14pt}- \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} \Bigr ];\forall t \in T, d \in {\cal D}, s \in \Omega^{resilience}\label{RepairV2_v7_8}\\
& L^{\dagger}_{tds} \geq 0; \forall t \in T, d \in {\cal D}, s \in \Omega|s\geq2\label{RepairV2_v7_9}\\
& L^{\dagger}_{tds} = 0; \forall t \in T, d \in {\cal D}, s = 1\label{RepairV2_v7_10}\\
& \sum_{j \in Rel_c} x^{ind}_{jc} = 1; \forall c \in {\cal C}\label{RepairV2_v7_11}\\
&-M\sum_{l \in Rel^{L,on}_{jc}}(1-x^{L,fix}_l)
- M\sum_{l \in Rel^{L,off}_{jc}}x^{L,fix}_l
\notag\\
&\hspace{25pt}\leq x^{ind}_{jc} - 1 \leq M\sum_{l \in Rel^{L,on}_{jc}}(1-x^{L,fix}_l)
\notag\\
&\hspace{60pt}+ M\sum_{l \in Rel^{L,off}_{jc}}x^{L,fix}_l
;\forall c \in {\cal C}, j \in Rel_c\label{RepairV2_v7_12}\\
&-M (1-x^{ind}_{jc}) \leq SOC^{ref}_{h} - SOC^{aux}_{hjec} \notag\\
&\hspace{5pt} \leq M (1-x^{ind}_{jc});\forall c \in {\cal C}, j \in Rel_c, e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_13}\\
&-M x^{ind}_{jc} \leq SOC^{aux}_{hjec} \leq M x^{ind}_{jc}; \forall c \in {\cal C}, j \in Rel_c, \notag\\
&\hspace{150pt}e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_14}\\
&-M (1-x^{ind}_{jc}) \leq \Bigl[ \sum_{i \in \mathfrak{D}_{jec}} D_{i}^{peak} \Bigr] - L_{jec} \notag\\
&\hspace{48pt} \leq M (1-x^{ind}_{jc}); \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_15}\\
& L_{jec} \geq 0; \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_16}\\
& 0\leq g^{Tr}_{ntd} \leq \overline{G}^{Tr}_n; \forall n \in \Psi^{SS}, d \in {\cal D}, t \in T \label{RepairV2_v7_17}\\
& \underline{V} \leq v_{ntd}\leq \overline{V}; \forall n \in \Psi^N, d \in {\cal D}, t \in T \label{RepairV2_v7_18}\\
& -y_{ltd,0} \overline{F}_l \leq f_{ltd} \leq y_{ltd,0} \overline{F}_l; \forall l \in {\cal L}^E, d \in {\cal D}, t \in T \label{RepairV2_v7_19}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltd} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltd} + g^{Tr}_{ntd} = 0; \notag\\
&\hspace{125pt} \forall n \in {\Psi}^{SS}, d \in {\cal D}, t \in T \label{RepairV2_v7_20}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltd} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltd} = \sum_{h \in H_n} p^{in}_{htd} \notag\\
&\hspace{0pt} - \sum_{h \in H_n} p^{out}_{htd} - \Delta^-_{ntd} + \Delta^+_{ntd} + D_{ntd};\forall n \in {\Psi}^{N} \setminus {\Psi}^{SS},\notag\\
&\hspace{170pt} d \in {\cal D}, t \in T \label{RepairV2_v7_21}\\
& -M(1-y_{ltd,0}) \leq Z^L_l r^{len}_l f_{ltd} - \bigl( v_{fr(l),t,d} \notag\\
&\hspace{11pt}- v_{to(l),t,d} \bigl) \leq M(1-y_{ltd,0}); \forall l \in {\cal L}^{E}, d \in {\cal D}, t \in T \label{RepairV2_v7_22}\\
& SOC_{h|T|d} = SOC_{ht^{0}d}; \forall h \in H, d \in {\cal D}\label{RepairV2_v7_23}\\
& SOC_{htd} = SOC_{ht^{0}d} + \eta \delta p^{in}_{htd} - \delta p^{out}_{htd}; \forall h \in H, \notag\\
&\hspace{172pt}d \in {\cal D}, t=1 \label{RepairV2_v7_24}\\
& SOC_{htd} = SOC_{h,t-1,d} + \eta \delta p^{in}_{htd} - \delta p^{out}_{htd}; \forall h \in H,\notag\\
&\hspace{147pt} d \in {\cal D}, t \in T|t\geq2 \label{RepairV2_v7_25}\\
& 0\leq SOC^{ref}_{h} \leq \overline{S}\overline{P}^{in}_h; \forall h \in H \setminus H^C\label{RepairV2_v7_26}\\
& 0\leq SOC^{ref}_{h} \leq \overline{S} x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C\label{RepairV2_v7_27}\\
& SOC_{htd} = SOC^{ref}_{h} f^{bat}_{htd}; \forall h \in H, d \in {\cal D}, t \in T\label{RepairV2_v7_28}\\
& 0\leq p^{in}_{htd} \leq \overline{P}^{in}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_29}\\
& 0\leq p^{out}_{htd} \leq \overline{P}^{out}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_30}\\
& 0\leq p^{in}_{htd} \leq x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_31}\\
& 0\leq p^{out}_{htd} \leq x^{SD,var}_h \overline{P}^{out}_h; \forall h \in H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_32} \end{align}
\begin{comment}
\begin{align}
& \underset{{\substack{\Delta^+_{ntd},\Delta^-_{ntd},\zeta_{td},\psi^{CVaR}_{tds},\\f_{ltd},g^{Tr}_{ntd},L_{jec},L^{\dagger}_{tds},\\p^{in}_{htd},p^{out}_{htd}, SOC_{htd}, \\SOC^{aux}_{hjec}, SOC^{ref}_{h}, v_{ntd}\\mathbf{x}^{ind}_{jc}, x^{L,fix}_{l}, x^{SD,fix}_{l}, x^{SD,var}_{l}}}}{\text{Minimize}} \hspace{0.1cm} \sum_{l \in {\cal L}^C} \Bigl[ C^{L,fix}_lx^{L,fix}_{l} \Bigr] \notag\\
&\hspace{0pt} + \sum_{h \in H^C} \Bigl[ C^{SD,fix}_h x^{SD,fix}_{h} + C^{SD,var}_h x^{SD,var}_{h} \overline{P}^{in}_h \Bigr] \notag \\
&\hspace{0pt}+ \sum_{d \in {\cal D}}W_d\sum_{t \in T}\Biggl[
pf C^{Imb} \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{ntd} + \Delta^+_{ntd} \Bigr] \Biggr ] \notag \\
&\hspace{0pt}+ (1-\lambda) pf C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \sum_{s \in \Omega} \rho_s L^{\dagger}_{tds}\notag\\
&\hspace{0pt}+ \lambda ~ pf ~ C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \Bigl[ \zeta_{td} \notag\\
&\hspace{110pt} + \sum_{s \in \Omega} \frac{\rho_s}{1-\alpha^{CVaR}} \psi^{CVaR}_{tds} \Bigr] \label{RepairV2_v7_1}\\
& \text{subject to:}\notag\\
& \psi^{CVaR}_{tds} + \zeta_{td} \geq L^{\dagger}_{tds}; \forall d \in {\cal D}, t \in T, s \in \Omega \label{RepairV2_v7_2}\\
& \psi^{CVaR}_{tds} \geq 0; \forall d \in {\cal D}, t \in T, s \in \Omega \label{RepairV2_v7_3}\\
& x^{ind}_{jc} \in \{0,1\}; \forall c \in {\cal C}, j \in {Rel}_c \label{RepairV2_v7_3_a}\\
& x^{L,fix}_{l} \in \{0,1\}; \forall l \in {\cal L}^C \label{RepairV2_v7_4}\\
& x^{SD,fix}_h \in \{0,1\}; \forall h \in H^C \label{RepairV2_v7_5}\\
& x^{SD,var}_h \leq x^{SD,fix}_h \overline{x}^{SD}_h; \forall h \in H^C\label{RepairV2_v7_6}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{5pt} - \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} f^{bat}_{htd} \Bigr ]; \forall t \in T, d \in {\cal D}, s \in \Omega^{routine}\label{RepairV2_v7_7}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{14pt}- \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} \Bigr ];\forall t \in T, d \in {\cal D}, s \in \Omega^{resilience}\label{RepairV2_v7_8}\\
& L^{\dagger}_{tds} \geq 0; \forall t \in T, d \in {\cal D}, s \in \Omega|s\geq2\label{RepairV2_v7_9}\\
& L^{\dagger}_{tds} = 0; \forall t \in T, d \in {\cal D}, s = 1\label{RepairV2_v7_10}\\
& \sum_{j \in Rel_c} x^{ind}_{jc} = 1; \forall c \in {\cal C}\label{RepairV2_v7_11}\\
& \sum_{l \in Rel^{L,on}_{jc}}(1-x^{L,fix}_l) + \sum_{l \in Rel^{L,off}_{jc}}x^{L,fix}_l \notag\\
& \hspace{84pt} \leq M (1-x_{jc}^{ind}) ;\forall c \in {\cal C}, j \in Rel_c \label{RepairV2_v7_12}\\
&-M (1-x^{ind}_{jc}) \leq SOC^{ref}_{h} - SOC^{aux}_{hjec} \notag\\
&\hspace{5} \leq M (1-x^{ind}_{jc});\forall c \in {\cal C}, j \in Rel_c, e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_13}\\
&-M x^{ind}_{jc} \leq SOC^{aux}_{hjec} \leq M x^{ind}_{jc}; \forall c \in {\cal C}, j \in Rel_c, \notag\\
&\hspace{150}e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_14}\\
&-M (1-x^{ind}_{jc}) \leq \Bigl[ \sum_{i \in \mathfrak{D}_{jec}} D_{i}^{peak} \Bigr] - L_{jec} \notag\\
&\hspace{48} \leq M (1-x^{ind}_{jc}); \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_15}\\
& L_{jec} \geq 0; \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_16}\\
& 0\leq g^{Tr}_{ntd} \leq \overline{G}^{Tr}_n; \forall n \in \Psi^{SS}, d \in {\cal D}, t \in T \label{RepairV2_v7_17}\\
& \underline{V} \leq v_{ntd}\leq \overline{V}; \forall n \in \Psi^N, d \in {\cal D}, t \in T \label{RepairV2_v7_18}\\
& -y_{ltd,0} \overline{F}_l \leq f_{ltd} \leq y_{ltd,0} \overline{F}_l; \forall l \in {\cal L}^E, d \in {\cal D}, t \in T \label{RepairV2_v7_19}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltd} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltd} + g^{Tr}_{ntd} = 0; \notag\\
&\hspace{125} \forall n \in {\Psi}^{SS}, d \in {\cal D}, t \in T \label{RepairV2_v7_20}\\
& \sum_{l \in {\cal L}|to(l)=n} f_{ltd} - \sum_{l \in {\cal L}|fr(l)=n} f_{ltd} = \sum_{h \in H_n} p^{in}_{htd} \notag\\
&\hspace{0} - \sum_{h \in H_n} p^{out}_{htd} - \Delta^-_{ntd} + \Delta^+_{ntd} + D_{ntd};\forall n \in {\Psi}^{N} \setminus {\Psi}^{SS},\notag\\
&\hspace{170} d \in {\cal D}, t \in T \label{RepairV2_v7_21}\\
& -M(1-y_{ltd,0}) \leq Z^L_l r^{len}_l f_{ltd} - \bigl( v_{fr(l),t,d} \notag\\
&\hspace{11}- v_{to(l),t,d} \bigl) \leq M(1-y_{ltd,0}); \forall l \in {\cal L}^{E}, d \in {\cal D}, t \in T \label{RepairV2_v7_22}\\
& SOC_{h|T|d} = SOC_{ht^{0}d}; \forall h \in H, d \in {\cal D}\label{RepairV2_v7_23}\\
& SOC_{htd} = SOC_{ht^{0}d} + \eta \delta p^{in}_{htd} - \delta p^{out}_{htd}; \forall h \in H, \notag\\
&\hspace{172}d \in {\cal D}, t=1 \label{RepairV2_v7_24}\\
& SOC_{htd} = SOC_{h,t-1,d} + \eta \delta p^{in}_{htd} - \delta p^{out}_{htd}; \forall h \in H,\notag\\
&\hspace{147} d \in {\cal D}, t \in T|t\geq2 \label{RepairV2_v7_25}\\
& 0\leq SOC^{ref}_{h} \leq \overline{S}\overline{P}^{in}_h; \forall h \in H \setminus H^C\label{RepairV2_v7_26}\\
& 0\leq SOC^{ref}_{h} \leq \overline{S} x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C\label{RepairV2_v7_27}\\
& SOC_{htd} = SOC^{ref}_{h} f^{bat}_{htd}; \forall h \in H, d \in {\cal D}, t \in T\label{RepairV2_v7_28}\\
& 0\leq p^{in}_{htd} \leq \overline{P}^{in}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_29}\\
& 0\leq p^{out}_{htd} \leq \overline{P}^{out}_h; \forall h \in H \setminus H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_30}\\
& 0\leq p^{in}_{htd} \leq x^{SD,var}_h \overline{P}^{in}_h; \forall h \in H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_31}\\
& 0\leq p^{out}_{htd} \leq x^{SD,var}_h \overline{P}^{out}_h; \forall h \in H^C, d \in {\cal D}, t \in T\label{RepairV2_v7_32}
\end{align} \end{comment}
The objective function to be minimized \eqref{RepairV2_v7_1} and constraints \eqref{RepairV2_v7_2}--\eqref{RepairV2_v7_6} are similar to \eqref{ScenarioBasedFormulation_1}--\eqref{ScenarioBasedFormulation_6}. One difference is that, in \eqref{RepairV2_v7_1}, $\Delta^-_{ntd}$ and $\Delta^+_{ntd}$ correspond to imbalances only under base case condition where no failure takes place. Also, the loss of load for period $t$ of each typical day $d$ that belongs to each scenario $s$ is represented by $L^{\dagger}_{tds}$, which is bounded for routine failure scenarios in \eqref{RepairV2_v7_7} and for resilience failure scenarios in \eqref{RepairV2_v7_8}. Moreover, constraints \eqref{RepairV2_v7_3_a} enforce the binary nature of decision variables $x^{ind}_{jc}$ that indicate which portfolio of candidate assets will receive investments. For each scenario $s \in \Omega^{routine}$, the right-hand side of constraint \eqref{RepairV2_v7_7} corresponds to the loss of load under the respective failure state $c$, which is assigned to scenario $s$ via the only $x^{state}_{cs}$ equal to $1$ among all $c \in {\cal C}$. This loss of load is the result of the summation across all investment possibilities and islands created by line outages of the demand during the failure period minus the current SOC of batteries connected to the respective islands. Analogously, the right-hand side of constraints \eqref{RepairV2_v7_8} represent loss of load for resilience scenarios. The salient feature in \eqref{RepairV2_v7_8} is that the whole capacity of the storage device can be used under a resilience scenario. This assumption is realistic as extreme events (such as natural disasters) can be usually predicted with enough time in advance to charge batteries to their full potential and provision their capacities to respond to the adverse conditions. Constraints \eqref{RepairV2_v7_9} ensure the non-negativity of loss of variables $L^{\dagger}_{tds}$ while constraints \eqref{RepairV2_v7_10} enforce the loss of load to be zero for the most likely scenario where no element fails as in the base case condition. Constraints \eqref{RepairV2_v7_11} indicate that just one of the possible investment combinations in lines will be chosen and therefore have an impact for failure state $c$. Constraints \eqref{RepairV2_v7_12} associate the combination of lines that are installed (whose indexes are in $Rel^{L,on}_{jc}$) and not installed (whose indexes are in $Rel^{L,off}_{jc}$) with variable $x^{ind}_{jc}$. Constraints \eqref{RepairV2_v7_13} and \eqref{RepairV2_v7_14} indicate which storage devices will be associated with each island created after an outage according to the investment decision. Constraints \eqref{RepairV2_v7_15} associate the loss of load of each island (represented by variable $L_{jec}$) with the summation of the peak demand of the islanded buses according to the investment made. Note the peak demand of each island $L_{jec}$ is multiplied by a factor $f^{load}_{\tau,d}$ in \eqref{RepairV2_v7_7} and \eqref{RepairV2_v7_8} to be adjusted to the demand of time period $\tau$. Constraints \eqref{RepairV2_v7_16} ensure the non-negativity of variables $L_{jec}$. Constraints \eqref{RepairV2_v7_16}--\eqref{RepairV2_v7_32} represent the base case operating condition analogously to \eqref{ScenarioBasedFormulation_7}--\eqref{ScenarioBasedFormulation_23}. The salient features in \eqref{RepairV2_v7_16}--\eqref{RepairV2_v7_32} with respect to \eqref{ScenarioBasedFormulation_7}--\eqref{ScenarioBasedFormulation_23} are the inclusion of the decision variables $SOC^{ref}_h$ and constraints \eqref{RepairV2_v7_26} which enforce a predetermined hourly profile for each storage device that is dictated by parameters $f^{bat}_{htd}$. The values of $f^{bat}_{htd}$ are a priori determined by optimizing storage charging and discharging while only considering energy price variation within the different considered typical days. This assumption on fixed SOC hourly profiles makes sense as batteries are usually operated to avoid higher costs instead of capacity provision for potential routine failures. In the case of resilience failures, as aforementioned, the full capacity of the storage devices can be used.
\begin{comment}
\subsection{Reducing investment-related constraints}
Despite not having the burden of optimizing the power for each scenario, model \eqref{RepairV2_v7_1}--\eqref{RepairV2_v7_32} can become easily intractable due to considering the set of indexes of all possible combinations of candidate assets ${\cal J}$ in constraints \eqref{RepairV2_v7_3_a}, \eqref{RepairV2_v7_7}, \eqref{RepairV2_v7_8}, \eqref{RepairV2_v7_11}-\eqref{RepairV2_v7_16}. Nevertheless, for a particular failure state $c$, not all investments combinations $j \in {\cal J}$. Rather, just a few combinations of investments are relevant to failure state $c$. For example, a line investment in the north part of a distribution grid may not be able to prevent a loss of load associated with a failure in the south part of the grid. Hence, we create the set $Rel_c$ which contains the indexes $j$ of the investment combinations that relevant to the failure state $c$. In addition, we also create sets ${Rel}^{L,on}_{jc}$ and ${Rel}^{L,off}_{jc}$ which contain the indexes of line segments that are built and not built, respectively, under relevant investment combination $j$ associated with failure state $c$. Thus, the proposed model is formulated as follows.
\begin{align}
& \underset{{\substack{\Delta^+_{ntd},\Delta^-_{ntd},\zeta_{td},\psi^{CVaR}_{tds},\\f_{ltd},g^{Tr}_{ntd},L_{jec},L^{\dagger}_{tds},\\p^{in}_{htd},p^{out}_{htd}, SOC_{htd}, \\SOC^{aux}_{hjec}, SOC^{ref}_{h}, v_{ntd}\\mathbf{x}^{ind}_{jc}, x^{L,fix}_{l}, x^{SD,fix}_{l}, x^{SD,var}_{l}}}}{\text{Minimize}} \hspace{0.1cm} \sum_{l \in {\cal L}^C} \Bigl[ C^{L,fix}_lx^{L,fix}_{l} \Bigr] \notag\\
&\hspace{0pt} + \sum_{h \in H^C} \Bigl[ C^{SD,fix}_h x^{SD,fix}_{h} + C^{SD,var}_h x^{SD,var}_{h} {\color{blac}\overline{S}}\overline{P}^{in}_h \Bigr] \notag \\
&\hspace{0pt}+ \sum_{d \in {\cal D}}W_d\sum_{t \in T}\Biggl[
pf C^{Imb} \sum_{n \in \Psi^N \setminus \Psi^{SS}} \Bigl[ \Delta^-_{ntd} + \Delta^+_{ntd} \Bigr] \Biggr ] \notag \\
&\hspace{0pt}+ (1-\lambda) pf C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \sum_{s \in \Omega} \rho_s L^{\dagger}_{tds}\notag\\
&\hspace{0pt}+ \lambda ~ pf ~ C^{Imb} \sum_{d \in D} W_d \sum_{t \in T} \Bigl[ \zeta_{td} \notag\\
&\hspace{110pt} + \sum_{s \in \Omega} \frac{\rho_s}{1-\alpha^{CVaR}} \psi^{CVaR}_{tds} \Bigr] \label{RepairV2_v8_1}\\
& \text{subject to:}\notag\\
& \text{Constraints \eqref{RepairV2_v7_2}, \eqref{RepairV2_v7_3}, \eqref{RepairV2_v7_4}--\eqref{RepairV2_v7_6}, \eqref{RepairV2_v7_9}, \eqref{RepairV2_v7_10},}\notag\\
& \hspace{165pt} \text{and \eqref{RepairV2_v7_17}--\eqref{RepairV2_v7_32} } \\
& x^{ind}_{jc} \in \{0,1\}; \forall c \in {\cal C}, j \in {Rel}_c \label{RepairV2_v7_3_a_modified}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{5pt} - \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} f^{bat}_{htd} \Bigr ]; \forall t \in T, d \in {\cal D}, s \in \Omega^{routine}\label{RepairV2_v7_7_modified}\\
& L^{\dagger}_{tds} \geq \sum_{c \in {\cal C}} x^{state}_{cs} \sum_{j \in Rel_c} \sum_{e \in E_{jc}}\Bigl[ \Bigl [ \sum_{\tau=t}^{min\{t+k_s,|T|\}} L_{jec} f^{load}_{\tau,d} \Bigr ] \notag\\
& \hspace{14pt}- \sum_{h \in {\cal H}_{jec}} SOC^{aux}_{hjec} \Bigr ];\forall t \in T, d \in {\cal D}, s \in \Omega^{resilience}\label{RepairV2_v7_8_modified}\\
& \sum_{j \in Rel_c} x^{ind}_{jc} = 1; \forall c \in {\cal C}\label{RepairV2_v7_11_modified}\\
& \sum_{l \in Rel^{L,on}_{jc}}(1-x^{L,fix}_l) + \sum_{l \in Rel^{L,off}_{jc}}x^{L,fix}_l \notag\\
& \hspace{84pt} \leq M (1-x_{jc}^{ind}) ;\forall c \in {\cal C}, j \in Rel_c \label{RepairV2_v7_12_modified}\\
&-M (1-x^{ind}_{jc}) \leq SOC^{ref}_{h} - SOC^{aux}_{hjec} \notag\\
&\hspace{5} \leq M (1-x^{ind}_{jc});\forall c \in {\cal C}, j \in Rel_c, e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_13_modified}\\
&-M x^{ind}_{jc} \leq SOC^{aux}_{hjec} \leq M x^{ind}_{jc}; \forall c \in {\cal C}, j \in Rel_c, \notag\\
&\hspace{150}e \in E_{jc}, h \in {\cal H}_{jec}\label{RepairV2_v7_14_modified}\\
&-M (1-x^{ind}_{jc}) \leq \Bigl[ \sum_{i \in \mathfrak{D}_{jec}} D_{i}^{peak} \Bigr] - L_{jec} \notag\\
&\hspace{48} \leq M (1-x^{ind}_{jc}); \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_15_modified}\\
& L_{jec} \geq 0; \forall c \in {\cal C}, j \in Rel_c, e \in E_{jc} \label{RepairV2_v7_16_modified}. \end{align} \end{comment}
\section{Case study}
\begin{figure}
\caption{Distribution system map.}
\label{Fig.systemMap}
\end{figure}
The proposed methodology is illustrated in this section using a distribution network from the ComEd in Illinois, USA. This system (depicted in Fig. \ref{Fig.systemMap}) has 1435 customers, a peak load of 3.5MW and it is composed of 2055 nodes, 2062 existing lines, and 2 substations. In addition, we consider 13 candidate lines and 9 candidate nodes to receive storage investment. Each candidate line has an investment cost of \$158K per mile and each storage costs \$660/kWh. Our methodology {\color{black} was implemented on a Ubuntu-Linux server with two Intel\textsuperscript{\textregistered} Xeon\textsuperscript{\textregistered} E5-2680 processors @ 2.40GHz and 64 GB of RAM, using Python 3.8, Pyomo and solved via CPLEX 12.9.}
\begin{figure*}
\caption{Investment plans for different levels of risk aversion considering VoLL=\$1.50/kWh.}
\label{Fig.investmentPans_1e5Dollar/kWh}
\end{figure*}
To model the load, we considered 4 typical days, representing the electricity demand in different meteorological seasons. We combined the peak demand with the demand profile reported by the U.S Energy Information Administration in \cite{US_EIA_demandProfile} (considering Illinois in Zone 4 of MISO).
Routine failures of the network in Fig. \ref{Fig.systemMap} were modeled based on ComEd's historical outages from February 1998 to November 2020. Additional, we model three major events with a rate of failure of 0.0143 times/year (equivalent to once every 70 years). The first, involves a simultaneous failure of two line segments in the north par of the network that disconnects 46\% of consumers during 3 hours. The second, involves one of the substations and affects 55\% of consumers for 1 hour. The third, mimics a recent extreme event, caused by storm in Illinois in August 2020 (described in \cite{ComEd2021_InvestmentsProposal}), that, according to ComEd's data, simultaneously affected 5 line segments for 58 hours.
\begin{comment}
Illinois is in MISO Zone 4 \end{comment}
\begin{figure}
\caption{Extreme failure in August 2020 -- lines out-of-service and respective number of customers affected in the system under consideration.}
\label{Fig.extremeFailure}
\end{figure}
Considering these failures and the investment costs, we obtained investment plans for three levels of risk aversion: $\lambda=0$, $\lambda=0.5$, and $\lambda=1$. The first ($\lambda=0$) is a risk neutral plan, considering only the expected value of loss of load \eqref{RepairV2_v7_1}. The second ($\lambda=0.5$), has a medium level of risk aversion as it considers both expected value and CVaR of cost of loss of load with equal weight in \eqref{RepairV2_v7_1}, while the third plan (for $\lambda=1$) has the highest level of risk-aversion, exclusively minimizing the CVaR of cost of loss of load.
It is important to note that this cost is highly dependent on the user defined value of loss of load (VoLL), modeled by the parameter $C^{Imb}$. For routine outages, this economic value can be obtained by tools such as the Interruption Cost Estimate (ICE) Calculator \cite{ICE_calculator}. For the purpose of demonstrating our methodology, we obtain plans for VoLL=\$1.5/kWh and VoLL=\$5.0/kWh.
Table \ref{tab:investmentResults} presents the investments results associated with the different levels of risk aversion and values of loss of load and the respective values of annual expected value and CVaR of loss of load. In addition, Fig. \ref{Fig.investmentPans_1e5Dollar/kWh} illutrates the investments made for all considered values of $\lambda$ when considering the VoLL equal to 1.50/kWh. As expected, a larger cost of VoLL increases the values of expected value and CVaR of cost of loss of load and motivates investments to avoid a more expensive load shedding. In addition, higher levels of risk aversion ($\lambda=0.5$ and $\lambda=1$) substantially decrease the value of the annual costs associated with CVaR of loss of load.
\begin{table*}[htbp]
\footnotesize
\centering
\caption{Investments associated with each level of risk aversion and value of loss of load.}
\begin{tabular}{c c c c c c c c c }
\toprule
\multicolumn{1}{c}{\multirow{2}[4]{*}{Value of }} & \multicolumn{1}{c}{\multirow{5}[8]{*}{$\lambda$}} & \multicolumn{1}{c}{Annual } & \multicolumn{1}{c}{Annual } & \multicolumn{1}{c}{\multirow{2}[4]{*}{Total}} & \multicolumn{1}{c}{\multirow{2}[4]{*}{Total}} & \multicolumn{1}{c}{\multirow{2}[4]{*}{Number}} & \multicolumn{1}{c}{\multirow{2}[4]{*}{Installed}} & \multicolumn{1}{c}{\multirow{2}[4]{*}{Computing}} \\
\multicolumn{1}{c}{} & & \multicolumn{1}{c}{expected value } & \multicolumn{1}{c}{CVaR} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\
\multicolumn{1}{c}{loss of} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{(loss of load)} & \multicolumn{1}{c}{(loss of load)} & \multicolumn{1}{c}{investments} & \multicolumn{1}{c}{investments} & \multicolumn{1}{c}{of } & \multicolumn{1}{c}{storage} & \multicolumn{1}{c}{times} \\
\multicolumn{1}{c}{load} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{ costs } & \multicolumn{1}{c}{ costs } & \multicolumn{1}{c}{in lines } & \multicolumn{1}{c}{in storage } & \multicolumn{1}{c}{installed} & \multicolumn{1}{c}{capacity} & \multicolumn{1}{c}{\multirow{2}[2]{*}{(s)}} \\
\multicolumn{1}{c}{(\$/kWh) } & \multicolumn{1}{c}{} & \multicolumn{1}{c}{(\$k/year)} & \multicolumn{1}{c}{(\$k/year)} & \multicolumn{1}{c}{(\$k)} & \multicolumn{1}{c}{(\$k)} & \multicolumn{1}{c}{lines} & \multicolumn{1}{c}{(MWh)} & \multicolumn{1}{c}{} \\
\midrule
1.50 & 0 & {\color{white}0}71.31 & 11,388,684.38 & 256.80 & {\color{white}0,00}0.00 & {\color{white}0}6 & {\color{white}0}0.00 & {\color{white}0,}380.05 \\
1.50 & 0.5 & {\color{white}0}61.88 & {\color{white}00,00}1,237.58 & 572.80 & 1,038.20 & 11 & {\color{white}0}1.60 & 1,926.94 \\
1.50 & 1 & {\color{white}0}57.52 & {\color{white}00,00}1,150.49 & 572.80 & 4,609.60 & 11 & {\color{white}0}7.00 & 2,727.73 \\
\midrule
5.00 & 0 & 216.05 & 37,962,281.25 & 476.50 & {\color{white}0,00}0.00 & {\color{white}0}9 & {\color{white}0}0.00 & {\color{white}0,}445.73 \\
5.00 & 0.5 & 185.76 & {\color{white}00,00}3,715.13 & 824.20 & 5,942.40 & 13 & {\color{white}0}9.00 & 2,106.29 \\
5.00 & 1 & 183.65 & {\color{white}00,00}3,673.09 & 824.20 & 7,438.70 & 13 & 11.30 & 2,216.20 \\
\bottomrule
\end{tabular}
\label{tab:investmentResults} \end{table*}
\subsection{Simulation of system performance under an extreme failure}
For all obtained expansion plans, we have simulated the system performance under the extreme failure reported by ComEd in August 2020. For illustrative purposes, we have limited this failure to 12 hours in a summer day. In Fig. \ref{Fig.resilienceTrapezoids}, we depict how much of the demand was served for each plan considering VoLL = 1.50/kWh and VoLL = 5.00/kWh, respectively. Compared to the plan obtained for $\lambda=0$, the plan attained for $\lambda=1$ can serve up to 12\% more of the demand during the extreme event when considering VoLL = 1.50/kWh. This difference increases to 29\% for VoLL = 5.00/kWh. In fact, since the plan for $\lambda=0$ is risk neutral and therefore can only capture the effect of expected outages during normal operating conditions, the performance of this plan under this extreme failure is the same as not investing in anything. In Fig. \ref{Fig.totalLoadSheddingVersusStorage}, we compare the investment made in storage to the total load not served during the day simulated with an extreme event. As can be seen, higher levels of risk aversion and VoLL significantly decrease the total load not served.
\begin{comment}
\begin{figure}
\caption{Hourly served demand under extreme event for investments considering VoLL=\$1.50/kWh.}
\label{Fig.resilienceTrapezoid_1e5Dollar/kWh}
\end{figure}
\begin{figure}
\caption{Hourly served demand under extreme event for investments considering VoLL=\$5.00/kWh.}
\label{Fig.resilienceTrapezoid_5Dollars/kWh}
\end{figure} \end{comment}
\begin{figure}
\caption{Hourly served demand under extreme event for investments considering VoLL=\$1.50/kWh on the left and VoLL=\$5.00/kWh on the right.}
\label{Fig.resilienceTrapezoids}
\end{figure}
\begin{figure}
\caption{Total load shedding under extreme event versus investment in storage capacity.}
\label{Fig.totalLoadSheddingVersusStorage}
\end{figure}
\begin{figure}
\caption{Out-of-sample analysis---CVaR$_{1\%}$ of hourly energy not served for expansion plans obtained under different levels of risk aversion while considering VoLL=\$1.50/kWh.}
\label{Fig.CVaR_hourlyEnergyNotServed}
\end{figure}
\subsection{Out-of-sample simulation}
We have generated 1000 annual scenarios of operation to evaluate the performance of the six obtained expansion plans in an out-of-sample analysis. For each hour of each scenario, we generated Bernoulli trials for line states \textcolor{black}{(1~in service; 0 failure)} with probabilities according to the rates of failure used while attaining the expansion plans. The performance of the obtained expansion plans was then assessed under the realization of the generated scenarios and compared to a base case without investments. This assessment involved computing hourly and annual energy not served as well as SAIFI and SADI for each scenario. In Tables \ref{tab:outOfSampleEnergyNotServedMetrics} and \ref{tab:SAIFI_SAIDI_metrics}, we present the resulting metrics and, in Fig. \ref{Fig.CVaR_hourlyEnergyNotServed}, we present a histogram that shows the distributions of the CVaR of hourly energy not served for the plans obtained under different levels of risk aversion and the base case. Average metrics in Tables \ref{tab:outOfSampleEnergyNotServedMetrics} and \ref{tab:SAIFI_SAIDI_metrics} are related to reliability while CVaR and worst case metrics are associated with resilience. As can be seen, both reliability and resilience metrics significantly improve when the level of risk aversion and the VoLL increase. In addition, in Fig. \ref{Fig.CVaR_hourlyEnergyNotServed}, it is clearly demonstrated that higher levels of risk aversion when determining new investments result in less hours with higher levels of CVaR of energy not served.
\begin{table}[htbp]
\footnotesize
\centering
\caption{Out-of-sample analysis -- Metrics of annual energy not served for expansion plans obtained under different levels of risk aversion and values of loss of load.}
\begin{tabular}{cccccc}
\toprule
\textbf{VoLL} & \multirow{2}[2]{*}{\textbf{Metric}} & \textbf{No } & \multirow{2}[2]{*}{\textbf{$\lambda=0$}} & \multirow{2}[2]{*}{\textbf{$\lambda=0.5$}} & \multirow{2}[2]{*}{\textbf{$\lambda=1$}} \\
\textbf{(\$/kWh)} & & \textbf{Inv.} & & & \\
\midrule
\multirow{9}[6]{*}{1.50} & \textbf{Average annual } & \multirow{3}[2]{*}{20.95} & \multirow{3}[2]{*}{6.09} & \multirow{3}[2]{*}{3.47} & \multirow{3}[2]{*}{2.61} \\
& \textbf{energy not } & & & & \\
& \textbf{served (MWh)} & & & & \\ \cmidrule{2-6} & \textbf{CVaR$_{1\%}$ of } & \multirow{3}[2]{*}{39.03} & \multirow{3}[2]{*}{17.05} & \multirow{3}[2]{*}{13.20} & \multirow{3}[2]{*}{10.36} \\
& \textbf{annual energy} & & & & \\
& \textbf{not served (MWh)} & & & & \\ \cmidrule{2-6} & \textbf{Worst case} & \multirow{3}[2]{*}{44.17} & \multirow{3}[2]{*}{23.21} & \multirow{3}[2]{*}{21.57} & \multirow{3}[2]{*}{17.48} \\
& \textbf{annual energy} & & & & \\
& \textbf{not served (MWh)} & & & & \\
\midrule
\multirow{9}[6]{*}{5.00} & \textbf{Average annual } & \multirow{3}[2]{*}{20.95} & \multirow{3}[2]{*}{4.18} & \multirow{3}[2]{*}{2.36} & \multirow{3}[2]{*}{2.34} \\
& \textbf{energy not } & & & & \\
& \textbf{served (MWh)} & & & & \\ \cmidrule{2-6} & \textbf{CVaR$_{1\%}$ of } & \multirow{3}[2]{*}{39.03} & \multirow{3}[2]{*}{14.05} & \multirow{3}[2]{*}{8.81} & \multirow{3}[2]{*}{8.54} \\
& \textbf{annual energy} & & & & \\
& \textbf{not served (MWh)} & & & & \\ \cmidrule{2-6} & \textbf{Worst case} & \multirow{3}[2]{*}{44.17} & \multirow{3}[2]{*}{22.49} & \multirow{3}[2]{*}{16.08} & \multirow{3}[2]{*}{15.36} \\
& \textbf{annual energy} & & & & \\
& \textbf{not served (MWh)} & & & & \\
\bottomrule
\end{tabular}
\label{tab:outOfSampleEnergyNotServedMetrics} \end{table}
\begin{table}[htbp]
\footnotesize
\centering
\caption{Out-of-sample analysis -- Metrics of SAIFI and SAIDI for expansion plans obtained under different levels of risk aversion and values of loss of load.}
\begin{tabular}{cccccc}
\toprule
\textbf{VoLL} & \multirow{2}[2]{*}{\textbf{Metrics}} & \textbf{No} & \multirow{2}[2]{*}{\textbf{$\lambda=0$}} & \multirow{2}[2]{*}{\textbf{$\lambda=0.5$}} & \multirow{2}[2]{*}{\textbf{$\lambda=1$}} \\
\textbf{(\$/kWh)} & & \textbf{Inv.} & & & \\
\midrule
\multirow{8}[8]{*}{1.50} & \textbf{Average } & \multirow{2}[2]{*}{1.337} & \multirow{2}[2]{*}{0.432} & \multirow{2}[2]{*}{0.305} & \multirow{2}[2]{*}{0.265} \\
& \textbf{SAIFI} & & & & \\ \cmidrule{2-6} & \multicolumn{1}{l}{\textbf{CVaR$_{5\%}$}} & \multirow{2}[2]{*}{1.901} & \multirow{2}[2]{*}{0.720} & \multirow{2}[2]{*}{0.507} & \multirow{2}[2]{*}{0.439} \\
& \textbf{SAIFI} & & & & \\ \cmidrule{2-6} & \textbf{Average } & \multirow{2}[2]{*}{0.668} & \multirow{2}[2]{*}{0.360} & \multirow{2}[2]{*}{0.284} & \multirow{2}[2]{*}{0.252} \\
& \textbf{SAIDI (h)} & & & & \\ \cmidrule{2-6} & \multicolumn{1}{l}{\textbf{CVaR$_{5\%}$}} & \multirow{2}[2]{*}{0.827} & \multirow{2}[2]{*}{0.544} & \multirow{2}[2]{*}{0.469} & \multirow{2}[2]{*}{0.406} \\
& \textbf{SAIDI (h)} & & & & \\
\midrule
\multirow{8}[8]{*}{5.00} & \textbf{Average } & \multirow{2}[2]{*}{1.337} & \multirow{2}[2]{*}{0.336} & \multirow{2}[2]{*}{0.257} & \multirow{2}[2]{*}{0.253} \\
& \textbf{SAIFI} & & & & \\ \cmidrule{2-6} & \multicolumn{1}{l}{\textbf{CVaR$_{5\%}$}} & \multirow{2}[2]{*}{1.901} & \multirow{2}[2]{*}{0.573} & \multirow{2}[2]{*}{0.421} & \multirow{2}[2]{*}{0.421} \\
& \textbf{SAIFI} & & & & \\ \cmidrule{2-6} & \textbf{Average } & \multirow{2}[2]{*}{0.668} & \multirow{2}[2]{*}{0.302} & \multirow{2}[2]{*}{0.247} & \multirow{2}[2]{*}{0.245} \\
& \textbf{SAIDI (h)} & & & & \\ \cmidrule{2-6} & \multicolumn{1}{l}{\textbf{CVaR$_{5\%}$}} & \multirow{2}[2]{*}{0.827} & \multirow{2}[2]{*}{0.515} & \multirow{2}[2]{*}{0.398} & \multirow{2}[2]{*}{0.393} \\
& \textbf{SAIDI (h)} & & & & \\
\bottomrule
\end{tabular}
\label{tab:SAIFI_SAIDI_metrics} \end{table}
\section{Conclusions}\label{sec.Conclusions}
In this paper, we propose scalable risk-based method for reliability and resilience planning of distribution systems. Our results using a ComEd distribution network demonstrate that the proposed method is able to produce investment plans (for a real-scale feeder) that have been optimized according to the degree of risk aversion, considering both investment costs and outage frequency and severity.
The proposed method is intended to support ``cost vs risk'' discussions between utilities and regulators by providing an internally consistent framework for evaluating trade-offs and synergies between reliability and resilience investments.
\begin{comment}
\section*{Acknowledgment}
\begin{IEEEbiographynophoto}{Alexandre Moreira}
(S'12) received the Electrical Engineering and Industrial Engineering degrees from the Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, in 2011. He received his M.Sc. degree from the Electrical Engineering Department of PUC-Rio, in 2014. Currently, he is pursuing his Ph.D. degree at the Department of Electrical and Electronic Engineering of the Imperial College London, London, UK.
His current research interests include decision making under uncertainty as well as power system economics, operation, and planning.
\end{IEEEbiographynophoto}
\end{comment}
\end{document} |
\begin{document}
\baselineskip=18pt \title{f On subgroups in division rings of type 2}
\newcommand{\dpcm}{
\rule{3mm}{3mm}} \def\mathbb{Q}{\mathbb{Q}} \def\mathbb{F}{\mathbb{F}} \newcommand{\ts}[1]{\langle #1\rangle}
\begin{abstract} \baselineskip=18pt Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring. \end{abstract}
{\bf {\em Key words:}} division ring; type 2; finitely generated subgroups.
{\bf{\em Mathematics Subject Classification 2010}}: 16K20
\section{Introduction}
In the theory of division rings, one of the problems is to determine which groups can occur as multiplicative groups of non-commutative division rings. There are some interesting results relating to this problem. Among them we note the famous discovery of Wedderburn in 1905, which states that {\it if $D^*$ is a finite group, then $D$ is commutative}, where $D^*$ denotes the multiplicative group of $D$. Later, L. K. Hua (see, for example, in [12, p. 223]) proved that the multiplicative group of a non-commutative division ring cannot be solvable. Recently, in \cite{hai-thin2} it was shown that the group $D^*$ cannot even be locally nilpotent. Note also Kaplansky's Theorem (see [12,(15.15), p. 259]) which states that if the group $D^*/F^*$ is torsion, then $D$ is commutative, where $F$ is the center of $D$. Some other results of this kind can be found for example, in \cite{Ak1}-\cite{Ak3}, \cite{hai-huynh}-\cite{hai-thin2},...
In this paper we consider this question for division rings of type $2$. Recall that a division ring $D$ with center $F$ is said to be {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. This concept is an extension of that of locally finite division rings. By definition, a division ring $D$ is {\em centrally finite} if it is a finite dimensional vector space over its center $F$ and $D$ is {\em locally finite} if for every finite subset $S$ of $D$, the division subring $F(S)$ generated by $S\cup F$ in $D$ is a finite dimensional vector space over $F$. There exist locally finite division rings which are not centrally finite (it is not hard to give some examples). Of course, every locally finite division ring is a ring of type $2$. However, at present no example of a division ring of type $2$ is known which is not locally finite. The difficulties are related with the following famous longstanding conjecture known as the Kurosh Problem for division rings \cite{kha}. Recall that a division ring $D$ is {\em algebraic} over its center $F$ (briefly, $D$ is {\em algebraic}), if every element of $D$ is algebraic over $F$. Clearly, a locally finite division ring is algebraic. Kurosh conjectured that any algebraic division ring is locally finite. Unfortunately, this problem remains still unsolved in general, it is answered in the affirmative for the following special cases: for $F$ uncountable \cite{row}, $F$ finite \cite{lam}, and for $F$ having only finite algebraic field extensions (in particular for $F$ algebraically closed). The last case follows from the Levitzki-Shirshov Theorem which states that {\em any algebraic algebra of bounded degree is locally finite} (see e.g. \cite{dren}, \cite{kha}). The answer for the case of finite $F$ is due to Jacobson who proved that {\em an algebraic division ring $D$ is commutative provided its center is finite} (see, for example, \cite{lam}). Later, more general theorems of this kind (known as commutativity theorems) were proved by Jacobson and Herstein. For more information we refer to [9, Ch. 3]. Finally, we would like to note that the results obtained in this paper for division rings of type $2$ have not been proved elsewhere before for locally finite division rings. So, at least (in the fortunate case if the Kurosh Problem will be answered in the affirmative, as we would like to see) our results generalize previous results for the finite dimensional case.
Throughout this paper the following notation will be used consistently: $D$ denotes a division ring with center $F$ and $D^*$ is the multiplicative group of $D$. If $S$ is a nonempty subset of $D$, then we denote by $F[S]$ and $F(S)$ the subring and the division subring of $D$ generated by $S$ over $F$, respectively. The symbol $D'$ is used to denote the derived group $[D^*, D^*]$. An element $x$ in $D$ is said to be {\em radical} over a subring $K$ of $D$ if there exists some positive integer $n(x)$ depending on $x$ such that $x^{n(x)}\in K$. A nonempty subset $S$ of $D$ is {\em radical} over $K$ if every element of $S$ is radical over $K$. We denote by $N_{D/F}$ and $RN_{D/F}$ the norm and the reduced norm, respectively. Finally, if $G$ is any group then we always use the symbol $Z(G)$ to denote the center of $G$.
\section{Finitely generated subgroups}
The main purpose in this section is to prove that in any non-commutative division ring of type $2$ there are no finitely generated subgroups containing the center.
\begin{lemma}\label{lem:2.1} Let $D$ be a division ring with center $F$, $D_1$ be a division subring of $D$ containing $F$. Suppose that $D_1$ is a finite dimensional vector space over $F$ and $a\in D_1$. Then, $N_{D_1/F}(a)$ is periodic if and only if $N_{F(a)/F}(a)$ is periodic. \end{lemma}
\begin{proof} Let $F_1=Z(D_1)\supset F$, $m^2=[D_1:F_1]$ and $n=[F_1(a):F_1]$. By [4, Lemma 3, p.145] and [4, Corollary 4, p. 150], we have $$N_{D_1/F_1}(a)=[RN_{D_1/F_1}(a)]^m=[N_{F_1(a)/F_1}(a)]^{m^2/n}.$$ Now, using the Tower formulae for the norm (cf. \cite{dra}), from the equality above we get $$N_{D_1/F}(a)=[N_{F_1(a)/F}(a)]^{m^2/n}.$$ Since $a\in F(a), we have N_{F_1(a)/F(a)}(a)=a^k$, where $k=[F_1(a):F(a)]$. Therefore $$N_{F(a)/F}(a^{k})=N_{F(a)/F}(N_{F_1(a)/F(a)}(a))=N_{F_1(a)/F}(a).$$ It follows that $N_{D_1/F}(a)=[N_{F(a)/F}(a)]^{km^2/n}$, and the conclusion is now obvious. \end{proof}
The following proposition is useful. In particular, it is needed to prove the subsequent theorem.
\begin{proposition}\label{prop:2.2} Let $D$ be a division ring with center $F$. If $N$ is a subnormal subgroup of $D^*$ then $Z(N)=N\cap F$. \end{proposition}
\begin{proof} If $N$ is contained in $F$ then there is nothing to prove. Thus, suppose that $N$ is non-central. By [15, 14.4.2, p. 439], $C_D(N)=F$. Hence $Z(N)\subseteq N\cap F$. Since the inclusion $N\cap F\subseteq Z(N)$ is obvious, $Z(N)= N\cap F$. \end{proof}
\begin{theorem}\label{thm:2.3} Let $D$ be a division ring of type $2$. Then $Z(D')$ is a torsion group.
\end{theorem}
\begin{proof} By Proposition \ref{prop:2.2}, $Z(D')=D'\cap F^*$. Any element $a\in Z(D')$ can be written in the form $a=c_1 c_2\ldots c_r$, where $c_i=[x_i, y_i]$ with $x_i, y_i\in D^*$ for $i\in\{1, \ldots, r\}$. Put $D_1=D_2:= F(c_1, c_2)$, $D_3:= F(c_1 c_2, c_3)$, $\ldots$, $D_r:= F(c_1...c_{r-1}, c_r)$ and $F_i=Z(D_i)$ for $i\in\{1, \ldots, r\}$. Since $D$ is of type $2$, $[D_i:F]<\infty$.
Since $N_{F(x_i, y_i)/F}(c_i)=1$, by Lemma \ref{lem:2.1}, $N_{F(c_i)/F}(c_i)$ is periodic. Again by Lemma~\ref{lem:2.1}, $N_{D_i/F}(c_i)$ is periodic. Therefore, there exists some positive integer $n_i$ such that $N_{D_i/F}(c_i^{n_i})=1$. Recall that $D_2=D_1$. Hence we get $$N_{D_2/F}(c_1 c_2)^m=N_{D_2/F} (c_1)^m N_{D_2/F}(c_2)^m=1,$$ where $m=n_1 n_2$. Again by Lemma \ref{lem:2.1}, $N_{F(c_1 c_2)/F}(c_1 c_2)$ is periodic; hence $N_{D_3/F}(c_1 c_2)$ is periodic. By induction, $N_{D_r/F}(c_1... c_{r-1})$ is periodic. Suppose that $N_{D_r/F}(c_1... c_{r-1})^n=1$. Then $$N_{D_r/F}(a^n)=N_{D_r/F}(c_1... c_{r-1})^n N_{D_r/F}(c_r)^n=1.$$ Hence, $a^{n[D_r:F]}=1$. Therefore, $a$ is periodic. Thus $Z(D')$ is torsion. \end{proof}
\begin{corollary}\label{cor:2.4} Let $D$ be a non-commutative ring of type $2$ with center $F$. Then $D'\setminus Z(D')$ contains no elements purely inseparable over $F$. \end{corollary}
\begin{proof} Suppose that $a\in D'\setminus Z(D')$ is purely inseparable over $F$. Then, there exists some positive integer $m$ such that $a^{p^m}\in F$. Since $Z(D')=D'\cap F$ (by Proposition 2.2), $a^{p^m}\in Z(D')$. By Theorem \ref{thm:2.3}, there exists some positive integer $r$ such that $a^{rp^m}=1$. Denote by $k$ the order of $a$ in the group $D^*$. If $p$ divides $k$, then $k=pt$ and we have $$1=a^k=a^{pt}=(a^t)^p.$$
Consequently, $a^t=1$, which is impossible in view of the choice of $k$. Now, suppose that $p$ does not divide $k$. Then, $(k, p^m)=1$ and $\alpha k+\beta p^m=1$ for some integers $\alpha$ and $\beta$. Therefore, we have $$a=a^{\alpha k+\beta p^m}=(a^k)^\alpha.a^{\beta p^m}=(a^{p^m})^\beta\in F.$$
Consequently, $a\in F\cap D'=Z(D')$, a contradiction. \end{proof}
Note that in \cite{mah} the author proved that $Z(D')$ is finite if $D$ is centrally finite. In virtue of this fact, he expressed his ideas that $Z(D')$ is torsion for any division ring $D$ algebraic over its center, but he has not been able to prove this. Therefore, Theorem~\ref{thm:2.3} represents some progress in this direction. Moreover (and this is more important for our purpose), we need this theorem to establish the main result in the present section. In fact, we shall prove that in a division ring $D$ of type $2$ with center $F$, there are no finitely generated subgroups containing $F^*$. Consequently, if $D$ is of type $2$ and $D^*$ is finitely generated, then $D$ is a field. Note that if the multiplicative group of a field is finitely generated, then it is finite. So, if $D$ is of type $2$ and $D^*$ is finitely generated, then $D$ is even a finite field. Our next theorem strongly generalizes the result obtained in [2, Theorem 1] which states that, if $D$ is centrally finite and $D^*$ is finitely generated, then $D$ is commutative.
\begin{theorem}\label{thm:2.6} Let $D$ be a non-commutative division ring of type $2$ with center $F$ and suppose that $N$ is a subgroup of $D^*$ containing $F^*$. Then $N$ is not finitely generated. \end{theorem}
\begin{proof} Suppose that there is a finitely generated subgroup $N=\ts{x_1, x_2, \ldots, x_n}$ of $D^*$ containing $F^*$. Then, in virtue of [15, 5.5.8, p. 113], $F^*N'/N'$ is a finitely generated abelian group, where $N'$ denotes the derived subgroup of $N$.
\noindent {\em Case 1: $char(D)=0$.}
Then, $F$ contains the field $\mathbb{Q}$ of rational numbers and it follows that $\mathbb{Q}^*/(\mathbb{Q}^*\cap N')\simeq \mathbb{Q}^*N'/N'$. Since $F^*N'/N'$ is finitely generated, $\mathbb{Q}^*N'/N'$ is finitely generated and consequently $\mathbb{Q}^*/(\mathbb{Q}^*\cap N')$ is finitely generated. Consider an arbitrary element $a\in \mathbb{Q}^*\cap N'$. Then $a\in F^*\cap D'=Z(D')$. By Theorem \ref{thm:2.3}, $a$ is periodic. Since $a\in \mathbb{Q}$, we get $a=\pm{1}$. Thus, $\mathbb{Q}^*\cap N'$ is finite. Since $\mathbb{Q}^*/(\mathbb{Q}^*\cap N')$ is finitely generated, $\mathbb{Q}^*$ is finitely generated, which is impossible.
\noindent {\em Case 2: $char(D)=p > 0$.}
Denoting by $\mathbb{F}_p$ the prime subfield of $F$, we shall prove that $F$ is algebraic over $\mathbb{F}_p$. In fact, suppose that $u\in F$ and $u$ is transcendental over $\mathbb{F}_p$. Then, the group $\mathbb{F}_p(u)^*/(\mathbb{F}_p(u)^*\cap N')$ considered as a subgroup of $F^*N'/N'$ is finitely generated. Consider an arbitrary element $f(u)/g(u)\in \mathbb{F}_p(u)^*\cap N'$, where $f(X), g(X)\in \mathbb{F}_p[X], ((f(X), g(X))=1$ and $g(u)\neq 0$. As above, we have $f(u)^s/g(u)^s=1$ for some positive integer $s$. Since $u$ is transcendental over $\mathbb{F}_p$, it follows that $f(u)/g(u)\in \mathbb{F}_p$. Therefore, $\mathbb{F}_p(u)^*\cap N'$ is finite and consequently, $\mathbb{F}_p(u)^*$ is finitely generated, so $\mathbb{F}_p(u)$ is finite field, which is impossible. Hence $F$ is algebraic over $\mathbb{F}_p$ and it follows that $D$ is algebraic over $\mathbb{F}_p$. Now, in virtue of Jacobson's Theorem [12, (13.11), p. 219], $D$ is commutative, a contradiction. \end{proof}
\begin{corollary}\label{cor:2.7} Let $D$ be a division ring of type $2$. If $D^*$ is finitely generated, then $D$ is a finite field. \end{corollary}
If $M$ is a finitely generated maximal subgroup of $D^*$, then clearly $D^*$ is finitely generated. So, the next result follows immediately from Corollary \ref{cor:2.7}.
\begin{corollary}\label{cor:2.8} Assume that $D$ is a division ring of type $2$. If $D^*$ has a finitely generated maximal subgroup, then $D$ is a finite field. \end{corollary}
In the same way as in the proof of Theorem \ref{thm:2.6}, we obtain the following corollary.
\begin{corollary}\label{cor:2.9} Assume that $D$ is a non-commutative division ring of type $2$ with center $F$ and $S$ is a subgroup of $D^*$. If $N=SF^*$, then $N/N'$ is not finitely generated. \end{corollary}
\begin{proof} Suppose that $N/N'$ is finitely generated. Since $N'=S'$ and $F^*/(F^*\cap S') \simeq S'F^*/S'$, it follows that $F^*/(F^*\cap S')$ is a finitely generated abelian group. Now, in the same way as in the proof of Theorem \ref{thm:2.6}, we conclude that $D$ is commutative and this is a contradiction. \end{proof}
The following result follows immediately from Corollary \ref{cor:2.9}.
\begin{corollary}\label{cor:2.10}
If $D$ is a non-commutative division ring of type $2$, then $D^*/D'$ is not finitely generated. \end{corollary}
\section{The radicality of subgroups}
In this section we study subgroups of $D^*$ which are radical over some subring of $D$. To prove the next theorem we need the following useful property of division rings of type $2$.
\begin{lemma}\label{lem:3.1} Let $D$ be a division ring of type $2$ with center $F$ and let $N$ be a subnormal subgroup of $D^*$. If for every pair of elements $x, y\in N$, there exists some positive integer $n_{xy}$ such that $x^{n_{xy}}y=yx^{n_{xy}}$, then $N\subseteq F$. \end{lemma}
\begin{proof} Since $N$ is subnormal in $D^*$, there exists a series of subgroups $$N=N_1\triangleleft N_2\triangleleft\ldots\triangleleft N_r=D^*.$$
Suppose that $x, y\in N$ and $K:=F(x, y)$. By putting $M_i=K\cap N_i,\, \forall i\in\{1, \ldots, r\},$ we obtain the following series of subgroups: $$M_1\triangleleft M_2\triangleleft\ldots\triangleleft M_r=K^*.$$
For any $a\in M_1\leq N_1=N$, suppose that $n_{ax}$ and $n_{ay}$ are positive integers such that $$a^{n_{ax}}x=xa^{n_{ax}} \mbox{ and } a^{n_{ay}}y=ya^{n_{ay}}.$$
Then, for $n:=n_{ax}n_{ay}$ we have $$a^n=(a^{n_{ax}})^{n_{ay}}=(xa^{n_{ax}}x^{-1})^{n_{ay}}=xa^{n_{ax}n_{ay}}x^{-1}=xa^nx^{-1}$$ and $$a^n=(a^{n_{ay}})^{n_{ax}}=(ya^{n_{ay}}y^{-1})^{n_{ax}}=ya^{n_{ay}n_{ay}}y^{-1}=ya^ny^{-1}.$$
Therefore $a^n\in Z(K)$. Hence $M_1$ is radical over $Z(K)$. By [6, Theorem 1], $M_1\subseteq Z(K)$. In particular, $x$ and $y$ commute with each other. Consequently, $N$ is an abelian group. By [15, 14.4.4, p. 440], $N\subseteq F$. \end{proof}
\begin{theorem}\label{thm:3.2} Let $D$ be a division ring of type $2$ with center $F$, $K$ be a proper division subring of $D$, and suppose that $N$ is a normal subgroup of $D^*$. If $N$ is radical over $K$, then $N\subseteq F$. \end{theorem}
\begin{proof} Suppose that $N$ is not contained in the center $F$. If $N\setminus K=\emptyset$, then $N\subseteq K$. By [15, p. 433], either $K\subseteq F$ or $K=D$. Since $K\neq D$ by the assumption, it follows that $K\subseteq F$. Hence $N\subseteq F$, which contradicts the assumption. Thus, we have $N\setminus K\neq\emptyset$.
Now, to complete the proof of our theorem we shall show that the elements of $N$ satisfy the requirements of Lemma \ref{lem:3.1}. To this end, suppose that $a, b\in N$. We examine the following cases:
\noindent {\em Case 1:} $a\in K$.
{\em Subcase 1.1:} $b\not\in K$.
We shall prove that there exists some positive integer $n$ such that $a^nb=ba^n$. Suppose that $a^nb\neq ba^n, \forall n\in\mathbb{N}$. Then, $a+b\neq 0, a\neq \pm{1}$ and $b\neq \pm{1}$. So we have $$x=(a+b)a(a+b)^{-1}, y=(b+1)a(b+1)^{-1}\in N.$$
Since $N$ is radical over $K$, we can find positive integers $m_x$ and $m_y$ such that $$x^{m_x}=(a+b)a^{m_x}(a+b)^{-1}, y^{m_y}=(b+1)a^{m_y}(b+1)^{-1}\in K.$$
Putting $m=m_xm_y$, we have
$$x^m=(a+b)a^m(a+b)^{-1}, y^m=(b+1)a^m(b+1)^{-1}\in K.$$
Direct calculations give the equalities $$x^mb-y^mb+x^ma-y^m=x^m(a+b)-y^m(b+1)=(a+b)a^m-(b+1)a^m=a^m(a-1),$$ from which we get $$(x^m-y^m)b=a^m(a-1)+y^m-x^ma.$$
If $(x^m-y^m)\neq 0$, then $b=(x^m-y^m)^{-1}[a(a^m-1)+y^m-x^ma]\in K$, contrary to the choice of $b$. Therefore $(x^m-y^m)= 0$ and consequently, $a^m(a-1)=y^m(a-1)$. Since $a\neq 1,a^m=y^m=(b+1)a^m(b+1)^{-1}$ and it follows that $a^mb=ba^m$, a contradiction.
{\em Subcase 1.2:} $b\in K$.
Consider an element $x\in N\setminus K$. Since $xb\not\in K$, by Subcase 1.1, there exist positive integers $r, s$ such that $$a^rxb=xba^r \mbox{ and } a^sx=xa^s.$$
From these equalities it follows that $$a^{rs}=(xb)^{-1}a^{rs}(xb)=b^{-1}(x^{-1}a^{rs}x)b=b^{-1}a^{rs}b,$$
and consequently, $a^{rs}b=ba^{rs}.$
\noindent {\em Case 2:} $a\not\in K$.
Since $N$ is radical over $K$, there exists some positive integer $m$ such that $a^m\in K$. By Case 1, there exists a positive integer $n$ such that $a^{mn}b=ba^{mn}$. \end{proof}
Theorem \ref{thm:3.2} is closely related to the following conjecture of Herstein in \cite{her2}: {\em ``For a division ring $D$, given a subnormal subgroup $N$ of $D^*$. If $N$ is radical over the center $F$ of $D$, then $N$ is central, i. e. $N\subseteq F$."} In Theorem \ref{thm:3.2}, the subgroup $N$ is required to be radical over an arbitrary proper division subring $K$ of $D$, which does not necessarily coincide with the center $F$. Notice that $N$ is required to be normal in $D^*$. So, the following question seems to be interesting: {\em ``For a division ring $D$, given a subnormal subgroup $N$ of $D^*$. Is $N$ contained in the center $F$of $D$, provided it is radical over some proper division subring of $D$?"}
Finally, we consider the question of the existence of maximal subgroups in $D$ which are radical over $F$. Recall that if $D$ is centrally finite of index different from the characteristic of $F$, then $D^*$ contains no such subgroups (see [1, Theorem 5]). Here, we consider the case when $D$ is of type $2$ with $[D: F]=\infty$ and we prove that, if $char\,F=p > 0$, then $D^*$ contains no such subgroups.
\begin{theorem}\label{thm:3.3} Let $D$ be a division ring of type $2$ with center $F$ such that $[D:F]=\infty$ and $char\, F = p>0$. Then the group $D^*$ contains no maximal subgroups which are radical over $F$. \end{theorem}
\begin{proof} Suppose that $M$ is a maximal subgroup of $D^*$ which is radical over $F$. Put $G=D'\cap M$. For each $x\in G$, there exists a positive integer $n(x)$ such that $x^{n(x)}\in F$. It follows that $x^{n(x)}\in D'\cap F=Z(D')$. By Theorem \ref{thm:2.3}, $Z(D')$ is torsion, so $x$ is periodic. Thus, $G$ is a torsion group. Since $M'\leq G, M'$ is also torsion. For any $x,y\in M'$, put $H=\ts{x,y}$ and $D_1=F(x,y)$. Then $n:=[D_1:F]<\infty$ and $H$ is a torsion subgroup of $D_1^*\leq GL_n(F)$. By [12, (9.9'), p. 154], $H$ is finite. Since $char F=p>0$, by [12, (13.3), p. 215], $H$ is cyclic. In particular, $x$ and $y$ commute with each other, and consequently, $M'$ is abelian. It follows that $M$ is a solvable group. Thus $M$ is a solvable maximal subgroup of $D^*$. By [1, Corollary 2, p. 432] and [3, Theorem 6], $[D:F]<\infty$, a contradiction. \end{proof}
\end{document} |
\begin{document}
\title{A $2\ell k$ Kernel for $\ell$-Component Order Connectivity}
\begin{abstract}
In the $\ell$-\textsc{Component Order Connectivity} problem ($\ell \in \mathbb{N}$), we are given a graph $G$ on $n$ vertices, $m$ edges and a non-negative integer $k$ and asks whether there exists a set of vertices $S\subseteq V(G)$ such that $|S|\leq k$ and the size of the largest connected component in $G-S$ is at most $\ell$. In this paper, we give a kernel for $\ell$-\textsc{Component Order Connectivity} with at most $2\ell k$ vertices that takes $n^{\mathcal{O}(\ell)}$ time for every constant $\ell$. On the way to obtaining our kernel, we prove a generalization of the $q$-Expansion Lemma to weighted graphs. This generalization may be of independent interest.
\end{abstract}
\section{Introduction} In the classic {\sc Vertex Cover} problem, the input is a graph $G$ and integer $k$, and the task is to determine whether there exists a vertex set $S$ of size at most $k$ such that every edge in $G$ has at least one endpoint in $S$. Such a set is called a {\em vertex cover} of the input graph $G$. An equivalent definition of a vertex cover is that every connected component of $G - S$ has at most $1$ vertex. This view of the {\sc Vertex Cover} problem gives rise to a natural generalization: can we delete at most $k$ vertices from $G$ such that every connected component in the resulting graph has at most $\ell$ vertices? Here we study this generalization. Formally, for every integer $\ell \geq 1$, we consider the following problem, called \textsc{$\ell$-Component Order Connectivity} ($\ell$-\textsf{COC}).
\noindent \fbox{\parbox{\textwidth-\fboxsep}{ \textsc{$\ell$-Component Order Connectivity} ($\ell$-\textsf{COC})\\ \textbf{Input:} A graph $G$ on $n$ vertices and $m$ edges, and a positive integer $k$.\\
\textbf{Task:} determine whether there exists a set $S\subseteq V(G)$ such that $|S|\leq k$ and the maximum size of a component in $G-S$ is at most $\ell$. }}
The set $S$ is called an $\ell$-COC {\em solution}. For $\ell = 1$, $\ell$-\textsf{COC} is just the {\sc Vertex Cover} problem. Aside from being a natural generalization of {\sc Vertex Cover}, the family $\{\ell$-\textsf{COC} $: \ell \geq 1\}$ of problems can be thought of as a vulnerability measure of the graph $G$ - how many vertices of $G$ have to fail for the graph to break into small connected components? For a study of $\ell$-\textsf{COC} from this perspective see the survey of Gross et al.~\cite{gross}.
From the work of Lewis and Yannakakis~\cite{LewisY80} it immediately follows that $\ell$-\textsf{COC} is NP-complete for every $\ell \geq 1$. This motivates the study of $\ell$-\textsf{COC} within paradigms for coping with NP-hardness, such as approximation algorithms~\cite{approx_book}, exact exponential time algorithms~\cite{exact_book}, parameterized algorithms~\cite{pc_book,DowneyF13book} and kernelization~\cite{kernel_survey_kratsch, kernel_survey_loks}. The $\ell$-\textsf{COC} problems have (for some values of $\ell$) been studied within all four paradigms, see the related work section.
In this work we focus on $\ell$-\textsf{COC} from the perspective of parameterized complexity and kernelization. Our main result is an algorithm that given an instance $(G, k)$ of $\ell$-\textsf{COC}, runs in polynomial time, and outputs an equivalent instance $(G', k')$ such that $k' \leq k$ and $|V(G')| \leq 2\ell k$. This is called a {\em kernel} for $\ell$-\textsf{COC} with $2\ell k$ vertices. Our kernel significantly improves over the previously best known kernel with $O(\ell k(k+\ell))$ vertices by Drange et al.~\cite{DrangeDH14}. Indeed, for $\ell = 1$ our kernel matches the size of the smallest known kernel for {\sc Vertex Cover}~\cite{ChenKJ01} that is based on the classic theorem of Nemhauser and Trotter~\cite{NemhauserT74}.
\noindent {\bf Related Work.} $1$-\textsf{COC}, better known as {\sc Vertex Cover}, is extremely well studied from the perspective of approximation algorithms~\cite{approx_book,dinur2005hardness}, exact exponential time algorithms~\cite{FominGK09,Robson86,XiaoN13}, parameterized algorithms~\cite{pc_book,ChenKX10} and kernelization~\cite{ChenKJ01,NemhauserT74}. The kernel with $2k$ vertices for {\sc Vertex Cover} is considered one of the fundamental results in the field of kernelization. The $2$-\textsf{COC} problem is also well studied, and has been considered under several different names. The problem, or rather the dual problem of finding a largest possible set $S$ that induces a subgraph in which every connected component has order at most $2$, was first defined by Yannakakis~\cite{Yannakakis81a} under the name Dissociation Set. The problem has attracted attention in exact exponential time algorithms~\cite{KardosKS11,XiaoK15}, the fastest currently known algorithm~\cite{XiaoK15} has running time $O(1.3659^n)$. $2$-\textsf{COC} has also been studied from the perspective of parameterized algorithms~\cite{ChangCHRS16,Tu15} (under the name {\sc Vertex Cover} $P_3$) as well as approximation algorithms~\cite{Tu}. The fastest known parameterized algorithm, due to Chang et al.~\cite{ChangCHRS16} has running time $1.7485^kn^{O(1)}$, while the best approximation algorithm, due to Tu and Zhou~\cite{Tu} has factor $2$.
For the general case of $\ell$-\textsf{COC}, $\ell \geq 1$, Drange et al.~\cite{DrangeDH14} gave a simple parameterized algorithm with running time $(\ell + 1)^kn^{O(1)}$, and a kernel with $O(k\ell(\ell+k))$ vertices. The parameterized algorithm of Drange et al.~\cite{DrangeDH14} can be improved to $(\ell + 0.0755)^kn^{O(1)}$ by reducing to the $(\ell + 1)$-{\sc Hitting Set} problem, and applying the iterative compression based algorithm for $(\ell + 1)$-{\sc Hitting Set} due to Fomin et al.~\cite{FominGKLS10}. The reduction to $(\ell + 1)$-{\sc Hitting Set}, coupled with the simple factor $(\ell + 1)$-approximation algorithm for $(\ell + 1)$-{\sc Hitting Set}~\cite{approx_book} immediately also yields an $(\ell + 1)$-approximation algorithm for $\ell$-\textsf{COC}. There has also been some work on $\ell$-\textsf{COC} when the input graph is restricted to belong to a graph class, for a discussion of this work see~\cite{DrangeDH14}.
Comparing the existing results with our work, we see that our kernel improves over the kernel of Drange et al.~\cite{DrangeDH14} from at most $O(k\ell(\ell+k))$ vertices to at most $2k\ell$ vertices. Our kernel is also the first kernel with a linear number of vertices for every fixed $\ell \geq 2$.
\noindent
{\bf Our Methods.} Our kernel for $\ell$-\textsf{COC} hinges on the concept of a {\em reducible pair} of vertex sets. Essentially ({\em this is not the formal definition used in the paper!}), a reducible pair is a pair $(X,Y)$ of disjoint subsets of $V(G)$ such that $N(Y) \subseteq X$, every connected component of $G[Y]$ has size at most $\ell$, and every solution $S$ to $G$ has to contain at least $|X|$ vertices from $G[X \cup Y]$. If a reducible pair is identified, it is easy to see that one might just as well pick all of $X$ into the solution $S$, since any solution has to pay $|X|$ inside $G[X \cup Y]$, and after $X$ is deleted, $Y$ breaks down into components of size at most $\ell$ and is completely eliminated from the graph.
At this point there are several questions. (a) How does one argue that a reducible pair is in fact reducible? That is, how can we prove that any solution has to contain at least $|X|$ vertices from $X \cup Y$? (b) How big does $G$ have to be compared to $k$ before we can assert the existence of a reducible pair? Finally, (c) even if we can assert that $G$ contains a reducible pair, how can we find one in polynomial time?
To answer (a) we restrict ourselves to reducible pairs with the additional property that each connected component $C$ of $G[Y]$ can be assigned to a vertex $x \in N(C)$, such that for every $x \in X$ the total size of the components assigned to $x$ is at least $\ell$. Then $x$ together with the components assigned to it form a set of size at least $\ell+1$ and have to contain a vertex from the solution. Since we obtain such a connected set for each $x \in X$, the solution has to contain at least $|X|$ vertices from $X \cup Y$. Again we remark that this definition of a reducible pair is local to this section, and not the one we actually end up using.
To answer (b) we first try to use the $q$-Expansion Lemma (see~\cite{pc_book}), a tool that has found many uses in kernelization. Roughly speaking the Expansion Lemma says the following: if $q \geq 1$ is an integer and $H$ is a bipartite graph with bipartition $(A, B)$ and $B$ is at least $q$ times larger than $A$, then one can find a subset $X$ of $A$ and a subset $Y$ of $B$ such that $N(Y) \subseteq X$, and an assignment of each vertex $y \in Y$ to a neighbor $x$ of $y$, such that every vertex $x$ in $X$ has at least $q$ vertices in $Y$ assigned to it.
Suppose now that the graph does have an $\ell$-\textsf{COC} solution $S$ of size at most $k$, and that $V(G) \setminus S$ is sufficiently large compared to $S$. The idea is to apply the Expansion Lemma to the bipartite graph $H$, where the $A$ side of the bipartition is $S$ and the $B$ side has one vertex for each connected component of $G - S$. We put an edge in $H$ between a vertex $v$ in $S$ and a vertex corresponding to a component $C$ of $G - S$ if there is an edge between $v$ and $C$ in $G$. If $G - S$ has at least $|S| \cdot \ell$ connected components, we can apply the $\ell$-Expansion Lemma on $H$, and obtain a set $X \subseteq S$, and a collection ${\cal Y}$ of connected components of $G - X$ satisfying the following properties. Every component $C \in {\cal Y}$ satisfies $N(C) \subseteq X$ and $|C| \leq \ell$. Furthermore, there exists an assignment of each connected component $C$ to a vertex $x \in N(C)$, such that every $x \in X$ has at least $\ell$ components assigned to it. Since $x$ has at least $\ell$ components assigned to it, the total size of the components assigned to $x$ is at least $\ell$. But then, $X$ and $Y = \bigcup_{C \in {\cal Y}} C$ form a reducible pair, giving an answer to question (b). Indeed, this argument can be applied whenever the number of components of $G - S$ is at least $\ell \cdot |S|$. Since each component of $G - S$ has size at most $\ell$, this means that the argument can be applied whenever $|V(G) \setminus S| \geq \ell^2 \cdot |S| \geq \ell^2k$.
Clearly this argument fails to yield a kernel of size $2\ell k$, because it is only applicable when $|V(G)| = \Omega(\ell^2k)$. At this point we note that the argument above is extremely wasteful in one particular spot: we used the {\em number} of components assigned to $x$ to lower bound the {\em total size} of the components assigned to $x$. To avoid being wasteful, we prove a new variant of the Expansion Lemma, where the vertices on the $B$ side of the bipartite graph $H$ have non-negative integer weights. This new Weighted Expansion lemma states that if $q, W \geq 1$ are integers, $H$ is a bipartite graph with bipartition $(A, B)$, every vertex in $B$ has a non-negative integer weight which is at most $W$, and the total weight of $B$ is at least $(q+W-1) \cdot |A|$, then one can find a subset $X$ of $A$ and a subset $Y$ of $B$ such that $N(Y) \subseteq X$, and an assignment of each vertex $y \in Y$ to a neighbor $x$ of $y$, such that for every vertex in $X$, the total weight of the vertices assigned to it is at least $q$. The proof of the Weighted Expansion Lemma is based on a combination of the usual, unweighted Expansion Lemma with a variant of an argument by Bez\'{a}kov\'{a} and Dani~\cite{BezakovaD05} to round the linear program for Max-min Allocation of goods to customers.
Having the Weighted Expansion Lemma at hand we can now repeat the argument above for proving the existence of a reducible pair, but this time, when we build $H$, we can give the vertex corresponding to a component $C$ of $G - S$ weight $|C|$, and apply the Weighted Expansion Lemma with $q = \ell$ and $W = \ell$. Going through the argument again, it is easy to verify that this time the existence of a reducible pair is guaranteed whenever $|V(G) \setminus S| \leq (2\ell - 1)k$, that is when $|V(G)| \geq 2\ell k$.
We are now left with question (c) - the issue of how to {\em find} a reducible pair in polynomial time. Indeed, the proof of existence crucially relies on the knowledge of an (optimal) solution $S$. To find a reducible pair we use the linear programming relaxation of the $\ell$-\textsf{COC} problem. We prove that an optimal solution to the LP-relaxation has to highlight every reducible pair $(X,Y)$, essentially by always setting all the variables corresponding to $X$ to $1$ and the variables corresponding to $Y$ to $0$. For {\sc Vertex Cover} (i.e $1$-\textsf{COC}), the classic Nemhauser Trotter Theorem~\cite{NemhauserT74} implies that we may simply include all the vertices whose LP variable is set to $1$ into the solution $S$. For $\ell$-\textsf{COC} with $\ell \geq 2$ we are unable to prove the corresponding statement. We are however, able to prove that if a reducible pair $(X,Y)$ exists, then $X$ (essentially) has to be assigned $1$ and $Y$ (essentially) has to be assigned $0$. We then give a polynomial time algorithm that extracts $X$ and $Y$ from the vertices assigned $1$ and $0$ respectively by the optimal linear programming solution. Together, the arguments (b) and (c) yield the kernel with $2\ell k$ vertices. We remark that to the best of our knowledge, after the kernel for Vertex Cover~\cite{ChenKJ01} our kernel is the first example of a kernelization algorithm based on linear programming relaxations.
\noindent {\bf Overview of the paper.} In Section~\ref{sec:prelim} we recall basic definitions and set up notations. The kernel for $\ell$-\textsf{COC} is proved in Sections~\ref{sec:maxmin},~\ref{sec:expansion} and~\ref{sec:kernel}. In Section~\ref{sec:maxmin} we prove the necessary adjustment of the results on Max-Min allocation of Bez\'{a}kov\'{a} and Dani~\cite{BezakovaD05} that is suitable to our needs. In Section~\ref{sec:expansion} we state and prove our new Weighted Expansion Lemma, and in Section~\ref{sec:kernel} we combine all our results to obtain the kernel.
\section{Preliminaries}\label{sec:prelim}
Let $\mathbb{N}$ denote the set of positive integers $\{0,1,2,\dots\}$. For any non-zero $t\in \mathbb{N}$, $[t]:=\{1,2,\dots,t\}$. We denote a constant function $f:X\to \mathbb{N}$ such that for all $x\in X, f(x)=c$, by $f=c$. For any function $f:X\to \mathbb{N}$ and a constant $c\in \mathbb{N}$, we define the function $f+c:X\to \mathbb{N}$ such that for all $x\in X, (f+c)(x)=f(x)+c$. We use the same symbol $f$ to denote the restriction of $f$ over a subset of it's domain, $X$. For a set $\{v\}$ containing a single element, we simply write $v$. A vertex $u\in V(G)$ is said to be incident on an edge $e\in E(G)$ if $u$ is one of the endpoints of $e$. A pair of edges $e,e'\in E(G)$ are said to be adjacent if there is a vertex $u\in V(G)$ such that $u$ is incident on both $e$ and $e'$. For any vertex $u\in V(G)$, by $N(u)$ we denote the set of neighbors of $u$ i.e. $N(u):=\{v\in V(G)\mid uv\in E(G)\}$. For any subgraph $X\subseteq G$, by $N(X)$ we denote the set of neighbors of vertices in $X$ outside $X$, i.e. $N(X):=(\bigcup_{u\in X}N(u))\setminus X$. A pair of vertices $u,v\in V(G)$ are called \emph{twins} if $N(u)=N(v)$. An induced subgraph on $X\subseteq V(G)$ is denoted by $G[X]$.
A \emph{path} $P$ is a graph, denoted by a sequence of vertices $v_1v_2\dots v_t$ such that for any $i,j\in [t], v_iv_j\in E(P)$ if and only if $|i-j|=1$. A \emph{cycle} $C$ is a graph, denoted either by a sequence of vertices $v_1v_2\dots v_t$ or by a sequence of edges $e_1e_2\dots e_t$, such that for any $i,j\in [t]$ $u_iu_j\in E(C)$ if and only if $|i-j|=1\mod ~t$ or in terms of edges, for any $i,j\in [t]$, $e_i$ is adjacent to $e_j$ if and only if $|i-j|=1\mod t$. The \emph{length} of a path(cycle) is the number of edges in the path(cycle). A \emph{triangle} is a cycle of length $3$. In $G$, for any pair of vertices $u,v \in V(G)$ \textsf{dist}$(u,v)$ represents the length of a shortest path between $u$ and $v$. A \emph{tree} is a connected graph that does not contain any cycle. A \emph{rooted tree} $T$ is a tree with a special vertex $r$ called the root of $T$. With respect to $r$, for any edge $uv\in E(T)$ we say that $v$ is a child of $u$ (equivalently $u$ is parent of $v$) if \textsf{dist}$(u,r)<$\textsf{dist}$(v,r)$. A \emph{forest} is a collection of trees. A \emph{rooted forest} is a collection of rooted trees. A \emph{clique} is a graph that contains an edge between every pair of vertices. A vertex cover of a graph is a set of vertices whose removal makes the graph edgeless.
\noindent {\bf Fixed Parameter Tractability.} A {\em parameterized problem} $\Pi$ is a subset of $\Sigma^* \times \mathbb{N}$. A parameterized problem $\Pi$ is said to be \emph{fixed parameter tractable}(\textsc{FPT}) if there exists an algorithm that takes as input an instance $(I, k)$ and decides whether $(I, k) \in \Pi$ in time $f(k)\cdot n^c$, where $n$ is the length of the string $I$, $f(k)$ is a computable function depending only on $k$ and $c$ is a constant independent of $n$ and $k$.
A \emph{kernel} for a parameterized problem $\Pi$ is an algorithm that given an instance $(T,k)$ runs in time polynomial in $|T|$, and outputs an instance $(T',k')$ such that $|T'|,k' \leq g(k)$ for a computable function $g$ and $(T,k) \in \Pi$ if and only if $(T',k') \in \Pi$. For a comprehensive introduction to \textsc{FPT} algorithms and kernels, we refer to the book by Cygan et al.~\cite{pc_book}.
A \emph{data reduction rule}, or simply, reduction rule, for a parameterized problem $Q$ is a function $\phi:\Sigma^*\times\mathbb{N}\to \Sigma^*\times\mathbb{N}$ that maps an instance $(I,k)$ of $Q$ to an equivalent instance $(I',k')$ of $Q$ such that $\phi$ is computable in time polynomial in $|I|$ and $k$. We say that two instances of $Q$ are \emph{equivalent} if $(I,k)\in Q$ if and only if $(I',k')\in Q$; this property of the reduction rule $\phi$, that it translates an instance to an equivalent one, is referred as the \emph{safeness} of the reduction rule.
\section{Max-min Allocation}\label{sec:maxmin}
We will now view a bipartite graph $G:=((A,B),E)$ as a relationship between ``customers'' represented by the vertices in $A$ and ``items'' represented by the vertices in $B$. If the graph is supplied with two functions $w_a : A \to \mathbb{N}$ and $w_b : B \to \mathbb{N}$, we treat these functions as a ``demand function'' and a ``capacity'' function, respectively. That is, we consider each item $v \in B$ to have value $w_b(v)$, and every customer $u \in A$ wants to be assigned items worth at least $w_a(u)$. An edge between $u \in A$ and $v \in B$ means that the item $v$ can be given to $u$.
A weight function $f : E(G) \to \mathbb{N}$ describes an assignment of items to customers, provided that the items can be ``divided'' into pieces and the pieces can be distributed to different customers. However this ``division'' should not create more value than the original value of the items. Formally we say that the weight function {\em satisfies} the capacity constraint $w_b(v)$ of $v \in B$ if $\sum_{uv \in E(G)} f(uv) \leq w_b(v)$. The weight function satisfies the capacity constraints if it satisfies the capacity constraints of all items $v \in B$.
For each item $u \in A$, we say that $f$ {\em allocates} $\sum_{uv \in E(G)} f(uv)$ value to $u$. The weight function $f$ {\em satisfies} the demand $w_a(u)$ of $u \in A$ if it allocates at least $w_a(u)$ value to $u$,
and $f$ satisfies the demand constraints if it does so for all $u \in A$. In other words, the weight function satisfies the demands if every customer gets items worth at least her demand. The weight function $f$ {\em over-satisfies} a demand constraint $w_a(u)$ of $u$ if it allocates strictly more than $w_a(u)$ to $u$.
We will also be concerned with the case where items are indivisible. In particular we say that a weight function $f : E(G) \to \mathbb{N}$ is {\em unsplitting} if for every $v \in B$ there is at most one edge $uv \in E(G)$ such that $f(uv) > 0$. The essence of the next few lemmas is that if we have a (splitting) weight function $f$ of items whose value is at most $W$, and $f$ satisfies the capacity and demand constraints, then we can obtain in polynomial-time an unsplitting weight function $f'$ that satisfies the capacity constraints and violates the demand constraints by at most $(W-1)$. In other words we can make a splitting distribution of items unsplitting at the cost of making each customer lose approximately the value of the most expensive item.
Allocating items to customers in such a way as to maximize satisfaction is well studied in the literature. The lemmata~\ref{forrest} and~\ref{stars} are very similar, both in statement and proof, to the work of Bez\'{a}kov\'{a} and Dani~\cite{BezakovaD05}[Theorem 3.2], who themselves are inspired by Lenstra et al.~\cite{LenstraST90}. However we do not see a way to directly use the results of Bez\'{a}kov\'{a} and Dani~\cite{BezakovaD05}, because we need a slight strengthening of (a special case of) their statement.
\begin{lemma}\label{forrest} There exists a polynomial-time algorithm that given a bipartite graph $G$, a capacity function $w_b : B \to \mathbb{N}$, a demand function $w_a : A \to \mathbb{N}$ and a weight function $f: E(G)\to \mathbb{N}$ that satisfies the capacity and demand constraints, outputs a function $f': E(G)\to \mathbb{N}$ such that $f'$ satisfies the capacity and demand constraints and the graph $G_{f'} = (V(G), \{uv \in E(G) \mid f'(uv) > 0\}) $ induced on the non-zero weight edges of $G$ is a forest. \end{lemma}
\begin{proof} We start with $f$ and in polynomially many steps, change $f$ into the required function $f'$. If $G_{f} = (V(G), \{uv \in E(G) \mid f(uv) > 0\})$ is a forest, then we return $f'=f$. Otherwise, suppose that $G_{f}$ contains a cycle $C:=e_1e_2e_3\dots e_{2s}$. Proceed as follows. Without loss of generality, suppose $c = f(e_1) = min\{f(e) \mid e\in C\}$, and note that $c > 0$. Compute the edge weight function $f^\star:E\to \mathbb{R}$ defined as follows. For $e_i \in C$, we define $f^\star(e_i) = f(e)-c$ if $i$ is odd, and define $f^\star(e_i) = f(e)+c$ if $i$ is even. For $e \notin C$ we define $f^\star(e_i) = f(e)$.
Every vertex of $G$ is incident to either $0$ or exactly $2$ edges of $C$. If the vertex $v$ is incident to two edges of $C$ then one of these edges, say $e_{2i}$, has even index in $C$, and the other, $e_{2i+1}$ has odd. For the edge $e_{2i}$ we have $f^\star(e_{2i}) = f(e_{2i})+c$ and for $e_{2i + 1}$ we have $f^\star(e_{2i + 1}) = f(e_{2i + 1}) - c$. Thus we conclude that for all $v\in V(G)$, $\sum_{u\in N(v)}f^\star(uv)=\sum_{u\in N(v)}f(uv)$, and that therefore $f^\star$ satisfies the capacity and demand constraints. Furthermore at least one edge that is assigned non-zero weight by $f$ is assigned $0$ by $f^\star$ and $G_{f^\star} = (V(G), \{uv \in E(G) \mid f^\star(uv) > 0\})$ has one less cycle than $G_f$. For a polynomial-time algorithm, repeatedly apply the process described above to reduce the number of edges with non-zero weight, as long as $G_{f^\star}$ contains a cycle. \end{proof}
\begin{lemma}\label{stars} There exists a polynomial-time algorithm with the following specifications. It takes as input a bipartite graph $G:=((A,B),E)$, a demand function $w_a:A\to \mathbb{N}$, a capacity function $w_b:B\to \mathbb{N}$, an edge weight function $f:E(G)\to \mathbb{N}$ that satisfies both the capacity and demand constraints, and a vertex $r \in A$. The algorithm outputs an unsplitting edge weight function $h: E(G)\to \mathbb{N}$ that satisfies the capacity constraints, satisfies the demands $w_a' = w_a - (W-1)$ where $W=\max_{v\in B}w_b(v)$, and additionally satisfies the demand $w_a(r)$ of $r$. \end{lemma} \begin{proof} Without loss of generality the graph $G_{f} := (V(G), \{uv \in E(G) \mid f(uv) > 0\})$ is a forest. If it is not, we may apply Lemma~\ref{forrest} to $f$, and obtain a function $f'$ that satisfies the capacity and demand constraints, and such that $G_{f'} = (V(G), \{uv \in E(G) \mid f'(uv) > 0\})$ is a forest. We then rename $f'$ to $f$. By picking a root in each connected component of $G_f$ we may consider $G_f$ as a rooted forest. We pick the roots as follows, if the component contains the special vertex $r$, we pick $r$ as root. If the component does not contain $r$, but contains at least one vertex $u \in A$, we pick that vertex as the root. If the component does not contain any vertices of $A$ then it does not contain any edges and is therefore a single vertex in $B$, we pick that vertex as root. Thus, every item $v \in B$ that is incident to at least one edge in $G_f$ has a unique {\em parent} $u \in A$ in the forest $G_f$. We define the new weight function $h$. For every edge $uv \in E(G)$ with $u \in A$ and $v \in B$ we define $h(uv)$ as follows.
\begin{figure}
\caption{Proof of Lemma \ref{forrest} and \ref{stars}. Cyclically shift smallest weight in a non-zero weight cycle to obtain a forest. Root each tree in the forest at a vertex in $A$ such that each vertex in $B$ has a parent in $A$. Assign the value of $v\in B$ to its parent $u\in A$. In this new assignment, a non-root vertex $u\in A$ \emph{loses} its parent $v_0\in B$ and $f(v_0u)\leq W-1$ which explains the cost of making a splitting assignment into an unsplitting assignment.}
\end{figure}
$h(uv) = w_b(v)$ if $u$ is the parent of $v$ in $G_f$, and $h(uv) = 0$ otherwise.
Clearly $h$ is unsplitting and satisfies the capacity constraints. We now prove that $h$ also satisfies the demand constraints $w_a'$ and satisfies the demand constraint $w_a(r)$ of $r$. Consider the demand constraint $w_a'(u)$ for an arbitrary customer $u \in A$. There are two cases, either $u$ is the root of the component of $G_f$ or it is not. If $u$ is the root, then for every edge $uv \in E(G)$ such that $f(uv) > 0$ we have that $uv \in E(G_f)$ and consequently that $u$ is the parent of $v$. Hence $h(uv) = w_b(v) \geq f(uv)$, and therefore $h$ satisfies the demand $w_a(u)$ of $u$. Since $w_a(u) \geq w_a'(u)$, we have that $h$ satisfies the demand $w_a'(u)$. Furthermore, since $r$ is the root of its component this also proves that $h$ satisfies the demand $w_a(r)$.
Consider now the case that $u$ is not the root of its component in $G_f$. Then $u$ has a unique parent in $G_f$, call this vertex $v^\star \in B$. We first prove that $f(uv^\star) \leq w_b(v^\star) - 1$. Indeed, since $v^\star$ is incident to the edge $uv^\star$ we have that $v^\star$ has a parent $u^\star$ in $G_f$, and that $u^\star \neq u$ because $v^\star$ is the parent of $u$. We have that $f(u^\star v^\star) + f(uv^\star) \leq w_b(v^\star)$ and that $f(u^\star v^\star) \geq 1$, because $u^\star v^\star$ is an edge in $G_f$. It follows that $f(uv^\star) \leq w_b(v^\star) - 1$. We now proceed to proving that $h$ satisfies the demand $w_a'(u)$.
For every edge $uv \in E(G) \setminus \{uv^\star\}$ such that $uv \in E(G)$ such that $f(uv) > 0$ we have that $uv \in E(G_f)$ and consequently that $u$ is the parent of $v$. Hence we have that $h(uv) = w_b(v) \geq f(uv)$. Furthermore $h(uv^\star) = 0$ while $f(uv^\star) \leq w_b(v^\star) - 1 \leq W - 1$. Therefore $h$ satisfies the demand $w_a'(u)$.
\end{proof}
\section{The Weighted Expansion Lemma}\label{sec:expansion}
Our kernelization algorithm will use ``$q$-expansions'' in bipartite graphs, a well known tool in kernelization~\cite{pc_book}. We begin by stating the definition of a $q$-expansion and review the facts about them that we will use.
\begin{definition}[\textbf{$q$-expansion}]
Let $G:=((A,B),E)$ be a bipartite graph. We say that $A$ has $q$-expansion into $B$ if there is a family of sets $\{V_a\mid V_a\subseteq N(a),|V_a|\geq q,a\in A\}$ such that for any pair of vertices $a_i,a_j\in A$,$i\neq j$, $V_{a_i}\cap V_{a_j}=\emptyset$. \end{definition}
\begin{definition}[\textbf{Twin graph}]\label{def:twing}
For a bipartite graph $G:=((A,B),E)$ with a weight function $w_b:B\to \mathbb{N}$, the twin graph $T_{AB}:=(A,B')$ of $G$ is obtained as follows: $B'$ contains $|w_b(v)|$ twins of every vertex $v\in B$ i.e. $B':=\{v_1,v_2,\dots v_{w_b(v)}\mid v\in B\}$ and edges in $T_{AB}$ such that for all $v\in B$ and $i\in[w_b(v)], N(v_i)=N(v)$ i.e. $E(T_{AB}):=\{av_i|a\in A,v_i\in B',v\in B,av\in E(G)\}$. \end{definition}
\begin{lemma}\label{q}\cite{pc_book}
Let $G$ be a bipartite graph with bipartition $(A,B)$. Then there is a $q$-expansion from $A$ into $B$ if and only if $|N(X)|\geq q|X|$ for every $X\subseteq A$. Furthermore, if there is no $q$-expansion from $A$ into $B$, then a set $X\subseteq A$ with $|N(X)|<q|X|$ can be found in polynomial-time. \end{lemma}
\begin{lemma}[Expansion Lemma~\cite{pc_book}]\label{expansion}
Let $q\geq 1$ be a positive integer and $G$ be a bipartite graph with vertex bipartition $(A,B)$ such that $|B|\geq q|A|$, and there are no isolated vertices in $B$. Then there exist nonempty vertex sets $X\subseteq A$ and $Y\subseteq B$ such that there is a $q$-expansion of $X$ into $Y$, and no vertex in $Y$ has a neighbor outside $X$, i.e. $N(Y)\subseteq X$. Furthermore, the sets $X$ and $Y$ can be found in time polynomial in the size of $G$. \end{lemma}
\begin{lemma}[folklore]\label{qset}
There exists a polynomial-time algorithm that given a bipartite graph $G:=((A,B),E)$ and an integer $q$ decides (and outputs in case yes) if there exist sets $X\subseteq A, Y\subseteq B$ such that there is a $q$-expansion of $X$ into $Y$. \end{lemma} \begin{proof} We describe a recursive algorithm. If $A=\emptyset$ or $B=\emptyset$, then output \textsc{no} and terminate. Otherwise, construct the twin graph $T_{BA}$ with weight function $w:A\to \mathbb{N}$ where for all $u\in A,w(u)=q$ and let $M$ be a maximum matching in $T_{BA}$. Consider the graph $G':=(A,B)$ with edge set $E(G'):=\{uv,u\in A,v\in B\mid u_iv\in M \}$. Let $A'\subseteq A$ such that for all $u\in A', d_{G'}(u)\geq q$ and let $B'\subseteq B$ such that $B':= \bigcup_{u\in A'}N_{G'}(u)$. If $N(B')\subseteq A'$, then return $(A',B')$ and terminate. Otherwise, recurse on $G[A'\cup (B\setminus N_G(A\setminus A'))]$.
If there are no sets $X,Y$ such that there is a $q$-expansion of $X$ into $Y$, then for any pair of sets $A'\subseteq A,B'\subseteq B$ either $N(B')\setminus A'\neq \emptyset$ or $|B'|<q|A'|$. Since at each recursive step, the size of the graph with which the algorithm calls itself decreases, eventually either $A'$ becomes empty or $B\setminus N_G(A\setminus A')$ becomes empty. Hence, the algorithm outputs no. Now we need to show that if there exist sets $(A^*,B^*)$ such that there is a $q$-expansion of $A^*$ into $B^*$, then at each recursive call, we have that $A^*\subseteq A$ and $B^*\subseteq B$. At the start of the algorithm, $A^*\subseteq A$ and $B^*\subseteq B$. Since $N(B^*)\subseteq A^*$ and for all $u\in A^*$ $d_G(u)\geq q$, we have that $A^*\cup B^*\subseteq V(G')$. If $N(B')\subseteq A'$, then the algorithm of Lemma \ref{expansion} when run on $G',q$ will output $(A^*,B^*)$.
Note that $B^*\subseteq B'$. At the recursive step, $A^*\subseteq A'$ and since $B^*\cap N_G(A\setminus A')=\emptyset$, we have that $B^*\subseteq B'\setminus N_G(A\setminus A')$. Hence, $G[A^*\cup B^*]$ is a subgraph of $G[A'\cup (B\setminus N_G(A\setminus A'))]$ which concludes the correctness of the algorithm. Since at each recursive call the size of the graph decreases by at least 1, the total time taken by the above algorithm is polynomial in $n$. \end{proof}
One may think of a $q$-expansion in a bipartite graph with bipartition $(A,B)$ as an allocation of the items in $B$ to each customer in $A$ such that every customer gets at least $q$ items. For our kernel we will need a generalization of $q$-expansions to the setting where the items in $B$ have different values, and every customer gets items of total value at least $q$.
\begin{definition}[\textbf{Weighted $q$-expansion}] Let $G:=((A,B),E)$ be a bipartite graph with capacity function $w_b : B \to \mathbb{N}$. Then, a weighted $q$-expansion in $G$ is an edge weight function $f : E(G)\to \mathbb{N}$ that satisfies the capacity constraints $w_b$ and also satisfies the demand constraints $w_a = q$.
For an integer $W \in \mathbb{N}$, the $q$-expansion $f$ is called a $W$-{\em strict} $q$-expansion if $f$ allocates at least $q+W-1$ value to at least one vertex $r$ in $A$, and in this case we say that $f$ is $W$-strict at $r$. Further, a $q$-expansion $f$ is {\em strict} (at $r$) if it is $1$-strict (at $r$). If $f$ is unsplitting we call $f$ an {\em unsplitting} $q$-expansion. \end{definition}
\begin{lemma}\label{qexp} There exists a polynomial-time algorithm that given a bipartite graph $G:=((A,B),E)$, an integer $q$ and a capacity function $w_b : B\to \mathbb{N}$ outputs (if it exist) two sets $X\subseteq A$ and $Y\subseteq B$ along with a weighted $q$-expansion in $G[X\cup Y]$ such that $N(Y)\subseteq X$. \end{lemma} \begin{proof}
Construct the twin graph $T_{AB}:=(A,B')$ of $G$. Run the algorithm of Lemma \ref{qset} with input $T_{AB},q$ that outputs sets $X\subseteq A$ and $Y'\subseteq B'$ such that $X$ has $q$-expansion into $Y'$ and $N(Y')\subseteq X$. Consider the set $Y:=\{v\in B\mid v_i\in Y'\}$. Define a weight function $f:E(G[X\cup Y])\to \mathbb{N}$ as follows: for all $uv\in E(G[X\cup Y])$ $f(uv)=|\{v_i\in Y'|v_i \text{ matched to }u\}|$.
Clearly, $N(Y)\subseteq X$. Now we claim that $f$ is a weighted $q$-expansion in $G[X\cup Y]$ with capacity function $w_b$ and demand function $w_a=q$. For any vertex $u\in A$, there are at least $q$ vertices in $Y'$ are matched to $u$. Hence for all $u\in A$, we have that $\sum_{v\in N(u)}f(uv)\geq q=w_a$. At the same time, for any vertex $v\in B$, there are at most $w_b(v)$ copies of $v$ in $Y'$. Therefore, for all $v\in Y$ we have $\sum_{u\in N(v)}f(uv)\leq w_b(v)$. \end{proof}
\begin{lemma}\label{qstrict}
There exists a polynomial-time algorithm that given a weighted $q$-expansion $f:E(G)\to \mathbb{N}$ in $G:=((A,B),E)$, a capacity function $w_b:B\to \mathbb{N}$ and an integer $W$ such that $W=\max_{e\in E(G)}f(e)$ outputs an unsplitting $W$-strict weighted $(q-W+1)$-expansion in $G$. \end{lemma} \begin{proof} Run the algorithm of Lemma \ref{stars} with inputs $G,f,w_a=q,w_b,W$ and a vertex $u\in A$. In case $f$ is strict, $u$ is the vertex $r$ that makes $f$ strict. Let the function $h:E(G)\to \mathbb{N}$ be the output of Lemma \ref{stars}. Now $h$ is an unsplitting edge weight function that satisfies the capacity constraints, satisfies the demands $q - W + 1$, and additionally satisfies the demand $q$ of $u$. Hence, $h$ is the required unsplitting weighted $W$-strict $(q-W+1)$-expansion in $G$. \end{proof}
\begin{lemma}[Weighted Expansion Lemma]\label{wexpansion}
Let $q, W \geq 1$ be positive integers and $G$ be a bipartite graph with vertex bipartition $(A,B)$ and $w_b : B \to \{1,\ldots, W\}$ be a capacity function such that $\sum_{v \in B} w_b(v) \geq (q+W-1)\cdot|A|$, and there are no isolated vertices in $B$. Then there exist nonempty vertex sets $X\subseteq A$ and $Y\subseteq B$ such that $N(Y)\subseteq X$ and there is an unsplitting weighted $W$-strict $q$-expansion of $X$ into $Y$. Furthermore, the sets $X$ and $Y$ can be found in time polynomial in the size of $G$. \end{lemma}
\begin{proof} Construct the twin graph $T_{AB}$ from $G$ and $w_b$, the bipartition of $T_{AB}$ is $(A,B')$. Now, obtain using the Expansion Lemma~\ref{expansion} with $q' = q + W - 1$ on $T_{AB}$ sets $X \subseteq A$ and $Y' \subseteq B'$, such that $N(Y') \subseteq X$ and there is a $(q+W-1)$-expansion from $X$ to $Y'$ in $T_{AB}$.
Let $Y:=\{v\in B\mid v_i\in Y'\}$ (here the $v_i \in Y'$ are as in Definition~\ref{def:twing}). Then $N(Y) \subseteq X$ and the $(q+W-1)$-expansion from $X$ to $Y'$ in $T_{AB}$ immediately yields a weighted $(q+W-1)$-expansion $f$ from $X$ to $Y$ in $G$. Applying Lemma~\ref{qstrict} on $G[X \cup Y]$ using the weighted $(q+W-1)$-expansion $f$ proves the statement of the lemma. \end{proof}
\section{Obtaining the Linear Kernel}\label{sec:kernel}
\begin{definition} For a graph $G$ and a pair of vertex-disjoint sets $X,Y\subseteq V(G)$, we define the weighted graph $\tilde{G}_{XY}$ as follows:
$V(\tilde{G}_{XY}) := X\cup \tilde{Y}$ such that there is a bijection $h:cc(G[Y])\to \tilde{Y}$ where $cc(G[Y])$ is the set of connected components of $G[Y]$. $E(\tilde{G}_{XY}) :=\{xc\mid x\in X,c\in \tilde{Y}, c=h(C) \text{ and } x\in N_G(C)\}$. We also define a weight function $w:\tilde{Y}\to \mathbb{N}$ such that for all $c\in \tilde{Y}, w(c)=|h^{-1}(c)|$. \end{definition} \begin{definition}[\textbf{Reducible Pair}] For a graph $G$, a pair of vertex-disjoint sets $(X,Y)$ where $X,Y\subseteq V(G)$ is called a (strict) reducible pair if $N(Y)\subseteq X$, the size of every component in $G[Y]$ is at most $\ell$, and there exists a (strict) weighted $(2\ell-1)$-expansion in $\tilde{G}_{XY}$. \end{definition} \begin{definition}
A reducible pair $(X,Y)$ is called minimal if there is no reducible pair $(X',Y')$ such that $X'\subset X$ and $Y'\subseteq Y$. \end{definition}
\begin{lemma}\label{biker}
There exists a polynomial-time algorithm that given an $\ell$-COC instance $(G,k)$ together with a vertex-disjoint set pair $A,B\subseteq V(G)$ outputs (if it exists) a reducible pair $(X,Y)$ where $X\subseteq A$ and $Y\subseteq B$. \end{lemma} \begin{proof} Construct $\tilde{G}_{AB}:=(A,\tilde{B})$ and run the algorithm of Lemma \ref{qexp} with input $\tilde{G}_{AB},w,q=2\ell-1$ which outputs sets $X\subseteq A$ and $Y'\subseteq \tilde{B}$ (if it exists) along with a weighted $(2\ell-1)$-expansion of $X$ into $Y'$ such that $N(Y')\subseteq X$. Now from $Y'$ we obtain the set $Y:=\bigcup_{y\in Y'}h^{-1}(y)$. Clearly, $N(Y)\subseteq X$ and hence, $(X,Y)$ is the desired reducible pair. \end{proof}
\begin{lemma}\label{redexist}
Given an $\ell$-COC instance $(G,k)$, if $|V(G)|\geq 2\ell k$ and $(G,k)$ is a \textsc{yes}-instance, then there exists a reducible pair $(X,Y)$. \end{lemma} \begin{proof}
Without loss of generality, we can assume that $G$ is a connected graph. Let $S$ be an $\ell$-COC solution of size at most $k$. Clearly, $|V\setminus S|\geq (2\ell-1)k$. We define $A:=S$ and $B:=V\setminus S$ and construct $\tilde{G}_{AB}=(A,\tilde{B})$. We have the weight function $w_b:\tilde{B}\to \mathbb{N}$ such that for all $v\in \tilde{B},w_b(v)=|h^{-1}(v)|\leq \ell$, as the size of components in $G[V\setminus S]$ is at most $\ell$. We have that $\sum_{v\in \tilde{B}}w_b(v)\geq (2\ell-1)|A|$ and there are no isolated vertices in $\tilde{B}$. Hence, $(A,B)$ is the desired reducible pair. \end{proof}
\begin{lemma}\label{pack}
Let $(X,Y)$ be a reducible pair. Then, there exists a partition of $X\cup Y$ into $C_1,...,C_{|X|}$ such that (i) for all $u_i\in X$, we have $u_i\in C_j$ if and only if $i=j$, (ii) for all $i\in [|X|],~|C_i|\geq \ell+1$, (iii) for every component $C$ in $G[Y]$, there exists a unique $C_i$ such that $V(C)\subseteq C_i$ and $u_i\in N(C)$ and (iv) if $(X,Y)$ is a strict reducible pair, then there exists $C_j$ such that $|C_j|\geq 2\ell+1$. \end{lemma} \begin{proof}
Construct $\tilde{G}_{XY}:=(X,\tilde{Y})$. Run the algorithm of Lemma \ref{qstrict} with input $\tilde{G}_{XY},q=2\ell-1$, and $W=\ell$(as the capacity of any vertex in $\tilde{Y}$ is at most $\ell$) which outputs an unsplitting weighted $\ell$-expansion $f'$ in $\tilde{G}_{XY}$. In polynomial time, we modify $f'$ such that if there is a vertex $v\in \tilde{Y}$ such that $\forall u\in N(v), f'(uv)=0$, we choose a vertex $u\in N(v)$ and set $f'(uv)=w_b(v)$. For each $u_i\in X$ define $C_i:=u_i\bigcup_{f'(u_iv)\neq 0}h^{-1}(v)$. Since $f'$ is unsplitting, the collection $C_1,\dots,C_{|X|}$ forms a partition of $X\cup Y$. By the definition of $C_i$, we have that for any $u_i\in X$, $u_i\in C_j$ if and only if $i=j$. For any component $C$ in $G[Y]$, $h(C)$ is matched to a unique vertex $u_i\in X$ by $f'$, we have that $V(C)\subseteq C_i$. As $f'$ is a weighted $\ell$-expansion, $|C_i|=1+\sum_{f'(u_iv)\neq 0}|h^{-1}(v)|=1+\sum_{f'(u_iv)\neq 0}f'(u_iv)\geq 1+\ell$. Let $(X,Y)$ be strict at $u_j\in X$. Then, we can
use Lemma \ref{qstrict} to obtain the expansion $f'$ such that it is strict at $u_j$. Hence, $|C_j|=1+\sum_{f'(u_jv)\neq 0}|h^{-1}(v)|=1+\sum_{f'(u_jv)\neq 0}f'(u_jv)> 1+\ell+(\ell-1)$ which implies $|C_j|\geq 2\ell +1$. This concludes the proof of the lemma. \end{proof}
\begin{lemma}\label{X}
Let $(X,Y)$ be a reducible pair. If $(G,k)$ is a \textsc{yes}-instance for $\ell$-COC, then there exists an $\ell$-COC solution $S$ of size at most $k$ such that $X\subseteq S$ and $S\cap Y=\emptyset$. \end{lemma} \begin{proof}
By Lemma \ref{pack} we have that there are $C_1,\dots, C_{|X|}\subseteq X\cup Y$ vertex disjoint sets of size at least $\ell+1$ such that for all $i\in [|X|]$, $G[C_i]$ is a connected set. Let $S'$ be an arbitrary solution. Then, $S'$ must contain at least one vertex from each $C_i$. Let $S:=S'\setminus (X\cup Y)\cup X$. We have that $|S|\leq |S'|-|X|+|X|=|S'|$. As any connected set of size $\ell+1$ that contains a vertex in $Y$ also contains a vertex in $X$ and $X\subseteq S$, $S$ is also an $\ell$-COC solution. \end{proof}
Now we encode an $\ell$-COC instance $(G,k)$ as an \textsc{Integer Linear Programming} instance. We introduce $n=|V(G)|$ variables, one variable $x_v$ for each vertex $v\in V(G)$. Setting the variable $x_v$ to 1 means that $v$ is in $S$, while setting $x_v=0$ means that $v$ is not in $S$. To ensure that $S$ contains a vertex from every connected set of size $\ell+1$, we can introduce constraints $\sum_{v\in C}x_v\geq 1$ where $C$ is a connected set of size $\ell+1$. The size of $S$ is given by $\sum_{v\in V(G)}x_v$. This gives us the following ILP formulation:\\ $\begin{matrix} \mbox{minimize} & \sum_{v\in V(G)}x_v,\\ \mbox{subject to} & \sum_{v\in C}x_v\geq 1 & \mbox{for every connected set }C\mbox{ of size }\ell+1\\ & 0\leq x_v\leq 1 &\mbox{ for every }v\in V(G)\\ & x_v\in \mathbb{Z} & \mbox{ for every }v\in V(G). \end{matrix}$\\
Note that there are $n^{\mathcal{O}(\ell)}$ connected sets of size at most $\ell$ in a graph on $n$ vertices. Hence, providing an explicit ILP requires $n^{\mathcal{O}(\ell)}$ time which forms the bottleneck for the runtime of the kernelization algorithm that follows. We consider the Linear Programming relaxation of above ILP obtained by dropping the constraint that $x\in \mathbb{Z}$. By an optimal LP solution $S_L$ with weight $L$ we mean the set of values assigned to each variable,
and optimal value is $L$. For a set of vertices $X\in V(G)$, $X=1$ ($X=0$) denotes that every variable corresponding to vertices in $X$ is set to $1$ ($0$).
\begin{lemma}\label{helper}
Let $S_L$ be an optimal LP solution for $G$ such that $x_v=1$ for some $v\subseteq V(G)$. Then, $S_L-x_v$ is an optimal LP solution for $G-v$ of value $L-1$. \end{lemma} \begin{proof}
Clearly, $S_L-x_v$ is feasible solution for $G-v$ of value $L-1$. Suppose it is not optimal. Let $S_{L'}$ be an optimal LP solution for $G-v$ such that $L'<L-1$. Then, $S_{L'}\cup x_v$ with $x_v=1$ is an optimal LP solution for $G$ with value $<L-1+1=L$ contradicting that the optimal solution value of LP for $G$ is $L$. \end{proof} From now on by running LP after setting $x_v=1$ for some vertex $v$, we mean running the LP algorithm for $G-v$ and including $x_v=1$ in the obtained solution to get a solution for $G$.
\begin{lemma}\label{strictx}
Let $(X,Y)$ be a strict reducible pair. Then every optimal LP solution sets at least one variable corresponding to a vertex in $X$ to 1. \end{lemma} \begin{proof}
By Lemma \ref{X}, we have that every connected set of size $\ell+1$ in $G[X\cup Y]$ contains a vertex in $X$. Hence, from any LP solution $S_L$, a feasible LP solution can be obtained by setting $X=1$ and $Y=0$. Since, we have at least $|X|$ many vertex disjoint LP constraints, for each $v_i\in X$, we have $\sum_{u\in C_i}x_u=1$. By Lemma \ref{pack}, there is a set $C_j\subseteq X\cup Y$ such that $|C_j|\geq 2\ell+1$. If $x_{v_j}\neq 1$, then there is a vertex $w\in C_j$ such that $x_w>0$. Let $w\in C\subset C_j$ where $G[C]$ is a connected component in $G[Y]$. Since $|C|\leq \ell$, there is a connected set $C'$ of size at least $\ell+1$ in $G[C_j]-C$. But now $\sum_{u\in C'}x_u<1$ contradicting that $S_L$ is feasible. \end{proof}
\begin{lemma}\label{redpair}
Let $(X,Y)$ be a minimal reducible pair. If for any vertex $v\in X$, an optimal LP solution sets $x_v=1$, then it also sets $X=1$ and $Y=0$. \end{lemma} \begin{proof}
We prove the lemma by contradiction. Let $X'\subset X$ be the largest subset of $X$ such that $X'=1$. Consider $\tilde{G}_{XY}$. Let $Y'\subseteq \tilde{Y}$ be the set of vertices such that $N(Y')\subseteq X'$. Let $Z:=\bigcup_{v\in Y'}h^{-1}(v)$. By the minimality of $(X,Y)$, we have that $\sum_{v\in Y'}w(v) < (2\ell-1)|X'|$. Hence, $\sum_{v\in \tilde{Y}\setminus Y'}w(v)>(2\ell-1)|X\setminus X'|$. Clearly, the weighted $(2\ell-1)$-expansion in the reducible pair $(X,Y)$ when restricted to $(X\setminus X',Y\setminus Z)$ provides a weighted $(2\ell-1)$-expansion of $X\setminus X'$ into $Y\setminus Z$. This implies that $(X\setminus X',Y\setminus Z)$ is a strict reducible pair in $G-(X'\cup Z)$. By Lemma \ref{helper}, we have that the LP solution restricted to $G-(X'\cup Z)$ is optimal. Since $(X\setminus X',Y\setminus Z)$ is a strict reducible pair, by Lemma \ref{strictx}, there is a vertex $u\in X\setminus X'$ such that $x_u=1$, but this contradicts the maximality of $X'$. Therefore, if for any vertex $v\in X$, an LP solution sets $x_v=1$, then it sets $X=1$ and $Y=0$. \end{proof}
\begin{lemma}\label{ker}
There exists a polynomial time algorithm that given an integer $\ell$ and $\ell$-COC instance $(G,k)$ on at least $2\ell k$ vertices either finds a reducible pair $(X,Y)$ or concludes that $(G,k)$ is a \textsc{no}-instance. \end{lemma} \begin{proof} If $(G,k)$ is a \textsc{yes}-instance of $\ell$-COC, then by Lemma \ref{redexist}, there exists a reducible pair $(X,Y)$. We use the following algorithm to find one: \begin{description}
\item[Step 1] Run the LP algorithm. Let $A=1$ and $B=0$ in the LP solution.
\item[Step 2] If both $A$ and $B$ are non-empty, then run the algorithm of Lemma \ref{biker} with input $(G,k),A,B$. If it outputs a reducible pair $(X,Y)$, then return $(X,Y)$ and terminate. Otherwise, go to step 3.
\item[Step 3] Now we do a linear search for a vertex in $X$. For each vertex $v\in V(G)$, do the following: in the original LP introduce an additional constraint that sets the value of the variable $x_v$ to $1$ i.e. $x_v=1$ and run the LP algorithm. If the optimal value of the new LP is the same as the optimal value of the original LP, then let $A=1$ and $B=0$ be the sets of variables set to $1$ and $0$ respectively in the optimal solution of the new LP and go to step 2.
\item[Step 4] Output a trivial \textsc{no}-instance. \end{description} Step 1 identifies the set of variables set to 1 and 0 by the LP algorithm. By Lemma \ref{redpair}, we have that if there is a minimal reducible pair $(X,Y)$ in $G$, then $X\subseteq A$ and $Y\subseteq B$. So, in Step 2 if the algorithm succeeds in finding one, we return the reducible pair and terminate otherwise we look for a potential vertex in $X$ and set it to 1. If $(X,Y)$ exists, then for at least one vertex, setting $x_v=1$ would set $X=1$ and $Y=0$ (by Lemma~\ref{redpair}) without changing the LP value and we go to Step 2 to find it. If for each choice of $v\in V(G)$, the LP value changes when $x_v$ is set to 1, we can conclude that there is no reducible pair and output a trivial \textsc{no} instance. Since, we need to do this search at most $n$ times and each step takes only polynomial time, the total time taken by the algorithm is polynomial in the input size. \end{proof}
\begin{theorem}\label{LinearKernel}
For every constant $\ell\in \mathbb{N}$, $\ell$-\textsc{Component Order Connectivity} admits a kernel with at most $2\ell k$ vertices that takes $n^{\mathcal{O}(\ell)}$ time. \end{theorem}
\subparagraph*{Acknowledgements} The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 306992 and the Beating Hardness by Pre-processing grant funded by the Bergen Research Foundation.
{}
\end{document} |
\begin{document}
\title{Solving Totally Unimodular LPs with the\\ Shadow Vertex Algorithm\thanks{This research was supported by ERC Starting Grant 306465 (BeyondWorstCase).}}
\author{Tobias Brunsch \and Anna Gro{\ss}wendt \and Heiko R\"oglin}
\institute{ Department of Computer Science\\ University of Bonn, Germany\\ \email{\small\{brunsch,grosswen,roeglin\}@cs.uni-bonn.de}}
\maketitle
\begin{abstract} We show that the shadow vertex simplex algorithm can be used to solve linear programs in strongly polynomial time with respect to the number~$n$ of variables, the number~$m$ of constraints, and~$1/\delta$, where~$\delta$ is a parameter that measures the flatness of the vertices of the polyhedron. This extends our recent result that the shadow vertex algorithm finds paths of polynomial length (w.r.t.~$n$, $m$, and~$1/\delta$) between two given vertices of a polyhedron~\cite{BrunschR13}.
Our result also complements a recent result due to Eisenbrand and Vempala~\cite{GRE} who have shown that a certain version of the random edge pivot rule solves linear programs with a running time that is strongly polynomial in the number of variables~$n$ and~$1/\delta$, but independent of the number~$m$ of constraints. Even though the running time of our algorithm depends on~$m$, it is significantly faster for the important special case of totally unimodular linear programs, for which~$1/\delta\le n$ and which have only~$O(n^2)$ constraints. \end{abstract}
\section{Introduction}
The shadow vertex algorithm is a well-known pivoting rule for the simplex method that has gained attention in recent years because it was shown to have polynomial running time in the model of smoothed analysis~\cite{SpielmanTeng}. Recently we have observed that it can also be used to find short paths between given vertices of a polyhedron~\cite{BrunschR13}. Here short means that the path length is~$O(\frac{mn^2}{\delta^2})$, where~$n$ denotes the number of variables, $m$ denotes the number of constraints, and~$\delta$ is a parameter of the polyhedron that we will define shortly.
Our result left open the question whether or not it is also possible to solve linear programs in polynomial time with respect to~$n$,~$m$, and~$1/\delta$ by the shadow vertex simplex algorithm. In this article we resolve this question and introduce a variant of the shadow vertex simplex algorithm that solves linear programs in strongly polynomial time with respect to these parameters.
For a given matrix~$A = [a_1, \ldots, a_m]{^\textnormal{T}} \in \mathbb{R}^{m \times n}$ and vectors~$b \in \mathbb{R}^m$ and~$c_0\in \mathbb{R}^n$ our goal is to solve the linear program~$\max \lbrace c_0{^\textnormal{T}} x \,|\, Ax \leq b \rbrace$. We assume without loss of generality that~$\|c_0\|=1$ and~$\|a_i\|=1$ for every row~$a_i$ of the constraint matrix.
\begin{definition}\label{def:delta} The matrix~$A$ satisfies the \emph{$\delta$-distance property} if the following condition holds: For any~$I\subseteq\{1,\ldots,m\}$ and any~$j\in\{1,\ldots,m\}$, if $a_j\notin\SPAN{a_i \,|\, i\in I}$ then~$\mathrm{dist}(a_j,\SPAN{a_i \,|\, i\in I})\ge\delta$. In other words, if~$a_j$ does not lie in the subspace spanned by the~$a_i$, $i\in I$, then its distance to this subspace is at least~$\delta$. \end{definition}
We present a variant of the shadow vertex simplex algorithm that solves linear programs in strongly polynomial time with respect to~$n$, $m$, and~$1/\delta$, where~$\delta$ denotes the largest~$\delta'$ for which the constraint matrix of the linear program satisfies the $\delta'$-distance property. (In the following theorems, we assume~$m\ge n$. If this is not the case, we use the method from Section~\ref{dimension} to add irrelevant constraints so that~$A$ has rank~$n$. Hence, for instances that have fewer constraints than variables, the parameter~$m$ should be replaced by~$n$ in all bounds.)
\begin{theorem}\label{thm:MainNumberOfPivots} There exists a randomized variant of the shadow vertex simplex algorithm (described in Section~\ref{fullalgorithm}) that solves linear programs with $n$~variables and $m$~constraints satisfying the $\delta$-distance property using~$O\big(\frac{mn^3}{\delta^2}\cdot \log\big(\frac{1}{\delta}\big)\big)$ pivots in expectation if a basic feasible solution is given. A basic feasible solution can be found using~$O\big(\frac{m^5}{\delta^2}\cdot \log\big(\frac{1}{\delta}\big)\big)$ pivots in expectation. \end{theorem} We stress that the algorithm can be implemented without knowing the parameter~$\delta$. From the theorem it follows that the running time of the algorithm is strongly polynomial with respect to the number~$n$ of variables, the number~$m$ of constraints, and~$1/\delta$ because every pivot can be performed in time $O(mn)$ in the arithmetic model of computation (see Section~\ref{runtime}).\footnote{By \emph{strongly polynomial with respect to~$n$, $m$, and~$1/\delta$} we mean that the number of steps in the arithmetic model of computation is bounded polynomially in~$n$, $m$, and~$1/\delta$ and the size of the numbers occurring during the algorithm is polynomially bounded in the encoding size of the input.}
Let~$A\in\mathbb{Z}^{m\times n}$ be an integer matrix and let~$A'\in\mathbb{R}^{m\times n}$ be the matrix that arises from~$A$ by scaling each row such that its norm equals~$1$. If~$\Delta$ denotes an upper bound for the absolute value of any sub-determinant of~$A$, then~$A'$ satisfies the $\delta$-distance property for~$\delta = 1/(\Delta^2 n)$~\cite{BrunschR13}. For such matrices~$A$ Phase~1 of the simplex method can be implemented more efficiently and we obtain the following result.
\begin{theorem}\label{thm:MainNumberOfPivots2} For integer matrices~$A\in\mathbb{Z}^{m\times n}$, there exists a randomized variant of the shadow vertex simplex algorithm (described in Section~\ref{fullalgorithm}) that solves linear programs with $n$~variables and $m$~constraints using~$O\big(mn^5\Delta^4 \log(\Delta+1)\big)$ pivots in expectation if a basic feasible solution is given, where~$\Delta$ denotes an upper bound for the absolute value of any sub-determinant of~$A$. A basic feasible solution can be found using~$O\big(m^6\Delta^4 \log(\Delta+1)\big)$ pivots in expectation. \end{theorem}
Theorem~\ref{thm:MainNumberOfPivots2} implies in particular that totally unimodular linear programs can be solved by our algorithm with~$O\big(mn^5\big)$ pivots in expectation if a basic feasible solution is given and with~$O\big(m^6 \big)$ pivots in expectation otherwise.
Besides totally unimodular matrices there are also other classes of matrices for which~$1/\delta$ is polynomially bounded in~$n$. Eisenbrand and Vempala~\cite{GRE} observed, for example, that~$\delta=\Omega(1/\sqrt{n})$ for edge-node incidence matrices of undirected graphs with~$n$ vertices. One can also argue that~$\delta$ can be interpreted as a condition number of the matrix~$A$ in the following sense: If~$1/\delta$ is large then there must be an $(n\times n)$-submatrix of~$A$ of rank~$n$ that is almost singular.
\subsection{Related Work}
\paragraph{Shadow vertex simplex algorithm} We will briefly explain the geometric intuition behind the shadow vertex simplex algorithm. For a complete and more formal description, we refer the reader to~\cite{Borgwardt86} or~\cite{SpielmanTeng}. Let us consider the linear program~$\max \lbrace c_0{^\textnormal{T}} x \,|\, Ax \leq b \rbrace$ and let~$P= \SET{ x \in \mathbb{R}^n \,|\, Ax \leq b }$ denote the polyhedron of feasible solutions. Assume that an initial vertex~$x_1$ of~$P$ is known and assume, for the sake of simplicity, that there is a unique optimal vertex~$x^{\star}$ of~$P$ that maximizes the objective function~$c_0{^\textnormal{T}} x$. The shadow vertex pivot rule first computes a vector~$w\in\mathbb{R}^n$ such that the vertex~$x_1$ minimizes the objective function~$w{^\textnormal{T}} x$ subject to~$x\in P$. Again for the sake of simplicity, let us assume that the vectors~$c_0$ and~$w$ are linearly independent.
In the second step, the polyhedron~$P$ is projected onto the plane spanned by the vectors~$c_0$ and~$w$. The resulting projection is a (possibly open) polygon~$P'$ and one can show that the projections of both the initial vertex~$x_1$ and the optimal vertex~$x^{\star}$ are vertices of this polygon. Additionally, every edge between two vertices~$x$ and~$y$ of~$P'$ corresponds to an edge of~$P$ between two vertices that are projected onto~$x$ and~$y$, respectively. Due to these properties a path from the projection of~$x_1$ to the projection of~$x^{\star}$ along the edges of~$P'$ corresponds to a path from~$x_1$ to~$x^{\star}$ along the edges of~$P$.
This way, the problem of finding a path from~$x_1$ to~$x^{\star}$ on the polyhedron~$P$ is reduced to finding a path between two vertices of a polygon. There are at most two such paths and the shadow vertex pivot rule chooses the one along which the objective~$c_0{^\textnormal{T}} x$ improves.
\paragraph{Finding short paths} In~\cite{BrunschR13} we considered the problem of finding a short path between two given vertices~$x_1$ and~$x_2$ of the polyhedron~$P$ along the edges of~$P$. Our algorithm is the following variant of the shadow vertex algorithm: Choose two vectors~$w_1,w_2\in\mathbb{R}^n$ such that~$x_1$ uniquely minimizes~$w_1{^\textnormal{T}} x$ subject to~$x\in P$ and~$x_2$ uniquely maximizes~$w_2{^\textnormal{T}} x$ subject to~$x\in P$. Then project the polyhedron~$P$ onto the plane spanned by~$w_1$ and~$w_2$ in order to obtain a polygon~$P'$. Let us call the projection~$\pi$. By the same arguments as above, it follows that~$\pi(x_1)$ and~$\pi(x_2)$ are vertices of~$P'$ and that a path from~$\pi(x_1)$ to~$\pi(x_2)$ along the edges of~$P'$ can be translated into a path from~$x_1$ to~$x_2$ along the edges of~$P$. Hence, it suffices to compute such a path to solve the problem. Again computing such a path is easy because~$P'$ is a two-dimensional polygon.
The vectors~$w_1$ and~$w_2$ are not uniquely determined, but they can be chosen from cones that are determined by the vertices~$x_1$ and~$x_2$ and the polyhedron~$P$. We proved in~\cite{BrunschR13} that the expected path length is~$O(\frac{mn^2}{\delta^2})$ if~$w_1$ and~$w_2$ are chosen randomly from these cones. For totally unimodular matrices this implies that the diameter of the polyhedron is bounded by~$O(mn^4)$, which improved a previous result by Dyer and Frieze~\cite{DyerF94} who showed that for this special case paths of length~$O(m^3n^{16}\log(mn))$ can be computed efficiently.
Additionally, Bonifas et al.~\cite{BonifasDEHN12} proved that in a polyhedron defined by an integer matrix~$A$ between any pair of vertices there exists a path of length~$O(\Delta^2 n^4 \log (n\Delta))$ where~$\Delta$ is the largest absolute value of any sub-determinant of~$A$. For the special case that~$A$ is a totally unimodular matrix, this bound simplifies to~$O(n^4\log{n})$. Their proof is non-constructive, however.
\paragraph{Geometric random edge}
Eisenbrand and Vempala~\cite{GRE} have presented an algorithm that solves a linear program~$\max \lbrace c_0{^\textnormal{T}} x | Ax \leq b \rbrace$ in strongly polynomial time with respect to the parameters~$n$ and~$1/\delta$. Remarkably the running time of their algorithm does not depend on the number~$m$ of constraints. Their algorithm is based on a variant of the random edge pivoting rule. The algorithm performs a random walk on the vertices of the polyhedron whose transition probabilities are chosen such that it quickly attains a distribution close to its stationary distribution.
In the stationary distribution the random walk is likely at a vertex~$x_c$ that optimizes an objective function~$c{^\textnormal{T}} x$
with~$\|c_0-c\|<\frac{\delta}{2n}$. The $\delta$-distance property guarantees that~$x_c$ and the optimal vertex~$x^{\star}$ with respect to the objective function~$c_0{^\textnormal{T}} x$ lie on a common facet. This facet is then identified and the algorithm is run again in one dimension lower. This is repeated at most~$n$ times until all facets of the optimal vertex~$x^{\star}$ are identified. The number of pivots to identify one facet of~$x^{\star}$ is proven to be~$O(n^{10}/\delta^{8})$. A single pivot can be performed in polynomial time but determining the right transition probabilities is rather sophisticated and requires to approximately integrate a certain function over a convex body.
Let us point out that the number of pivots of our algorithm depends on the number~$m$ of constraints. However, Heller showed that for the important special case of totally unimodular linear programs~$m=O(n^2)$~\cite{Heller57}. Using this observation we also obtain a bound that depends polynomially only on~$n$ for totally unimodular matrices.
\paragraph{Combinatorial linear programs} {\'{E}}va Tardos has proved in 1986 that combinatorial linear programs can be solved in strongly polynomial time~\cite{Tardos1986}. Here combinatorial means that~$A$ is an integer matrix whose largest entry is polynomially bounded in~$n$. Her result implies in particular that totally unimodular linear programs can be solved in strongly polynomial time, which is also implied by Theorem~\ref{thm:MainNumberOfPivots2}. However, the proof and the techniques used to prove Theorem~\ref{thm:MainNumberOfPivots2} are completely different from those in~\cite{Tardos1986}.
\subsection{Our Contribution}\label{sub:contribution}
We replace the random walk in the algorithm of Eisenbrand and Vempala by the shadow vertex algorithm. Given a vertex~$x_0$ of the polyhedron~$P$ we choose an objective function~$w{^\textnormal{T}} x$ for which~$x_0$ is an optimal solution. As in~\cite{BrunschR13} we choose~$w$ uniformly at random from the cone determined by~$x_0$. Then we randomly perturb each coefficient in the given objective function~$c_0{^\textnormal{T}} x$ by a small amount. We denote by~$c{^\textnormal{T}} x$ the perturbed objective function. As in~\cite{BrunschR13} we prove that the projection of the polyhedron~$P$ onto the plane spanned by~$w$ and~$c$
has~$O\big(\frac{mn^2}{\delta^2}\big)$ edges in expectation. If the perturbation is so small that~$\|c_0-c\|<\frac{\delta}{2n}$, then the shadow vertex algorithm yields with~$O\big(\frac{mn^2}{\delta^2}\big)$ pivots a solution that has a common facet with the optimal solution~$x^{\star}$. We follow the same approach as Eisenbrand and Vempala and identify the facets of~$x^{\star}$ one by one with at most~$n$ calls of the shadow vertex algorithm.
The analysis in~\cite{BrunschR13} exploits that the two objective functions possess the same type of randomness (both are chosen uniformly at random from some cones). This is not the case anymore because every component of~$c$ is chosen independently uniformly at random from some interval. This changes the analysis significantly and introduces technical difficulties that we address in this article.
The problem when running the simplex method is that a feasible solution needs to be given upfront. Usually, such a solution is determined in Phase~1 by solving a modified linear program with a constraint matrix~$A'$ for which a feasible solution is known and whose optimal solution is feasible for the linear program one actually wants to solve. There are several common constructions for this modified linear program, it is, however, not clear how the parameter~$\delta$ is affected by modifying the linear program. To solve this problem, Eisenbrand and Vempala~\cite{GRE} have suggested a method for Phase~1 for which the modified constraint matrix~$A'$ satisfies the $\delta$-distance property for the same~$\delta$ as the matrix~$A$. However, their method is very different from usual textbook methods and needs to solve~$m$ different linear programs to find an initial feasible solution for the given linear program. We show that also one of the usual textbook methods can be applied. We argue that $1/\delta$ increases by a factor of at most~$\sqrt{m}$ and that~$\Delta$, the absolute value of any sub-determinant of~$A$, does not change at all in case one considers integer matrices. In this construction, the number of variables increases from~$n$ to~$n+m$.
\subsection{Outline and Notation}
In the following we assume that we are given a linear program $\max \lbrace c_0{^\textnormal{T}} x \,|\, Ax \leq b \rbrace$ with vectors $b \in \mathbb{R}^m$ and $c_0 \in \mathbb{R}^n$ and a matrix $A=[a_1,\dots,a_m]{^\textnormal{T}} \in \mathbb{R}^{m \times n}$.
Moreover, we assume that $\|c_0\|=\|a_i\|=1$ for all $ i \in [m]$, where $[m]:=\{1,\ldots,m\}$
and $\|\cdot\|$ denotes the Euclidean norm. This entails no loss of generality since any linear program can be brought into this form by scaling the objective function and the constraints appropriately. For a vector $x \in \mathbb{R}^n\setminus\{0^n\}$ we denote by $\mathcal{N}(x) = \frac{1}{\|x\|} \cdot x$ the normalization of vector~$x$.
For a vertex $v$ of the polyhedron $P=\SET{ x \in \mathbb{R}^n \,|\, Ax \leq b }$ we call the set of row indices $B_v=\lbrace i\in\{1,\ldots,m\} \,|\, a_i\cdot v=b_i \rbrace$ \textit{basis} of $v$. Then the \textit{normal cone} $C_v$ of $v$ is given by the set \[
C_v=\SET{ \sum \limits_{i \in B_v} \lambda_i a_i\,|\, \lambda_i\ge 0 }. \]
We will describe our algorithm in Section~\ref{sec:Algorithm} where we assume that the linear program in non-degenerate, that~$A$ has full rank~$n$, and that the polyhedron~$P$ is bounded. We have already described in Section~3 of~\cite{BrunschR13} that the linear program can be made non-degenerate by slightly perturbing the vector~$b$. This does not affect the parameter~$\delta$ because~$\delta$ depends only on the matrix~$A$. In Appendix~\ref{specialcases} we discuss why we can assume that~$A$ has full rank and why~$P$ is bounded. There are, of course, textbook methods to transform a linear program into this form. However, we need to be careful that this transformation does not change~$\delta$.
In Section~\ref{sec:analysis} we analyze our algorithm and prove Theorem~\ref{thm:MainNumberOfPivots}. In Section~\ref{initial} we discuss how Phase~1 of the simplex method can be implemented and in Appendix~\ref{sdelta} we give an alternative definition of~$\delta$ and discuss some properties of this parameter.
\section{Algorithm}\label{fullalgorithm}
Given a linear program $\max \lbrace c_0 {^\textnormal{T}} x \,|\, Ax \leq b \rbrace$ and a basic feasible solution~$x_0$, our algorithm randomly perturbs each coefficient of the vector~$c_0$ by at most~$1/\phi$ for some parameter~$\phi$ to be determined later. Let us call the resulting vector~$c$. The next step is then to use the shadow vertex algorithm to compute a path from $x_0$ to a vertex $x_c$ which maximizes the function $c{^\textnormal{T}} x$ for $x \in P$. For~$\phi>\frac{2n^{3/2}}{\delta}$ one can argue that the solution~$x$ has a facet in common with the optimal solution~$x^{\star}$ of the given linear program with objective function~$c_0{^\textnormal{T}} x$. Then the algorithm is run again on this facet one dimension lower until all facets that define~$x^{\star}$ are identified.
This section is organized as follows. In Section~\ref{reduct} we repeat a construction from~\cite{GRE} to project a facet of the polyhedron~$P$ into the space $\mathbb{R}^{n-1}$ without changing the parameter~$\delta$. This is crucial for being able to identify the facets that define~$x^{\star}$ one after another. In Section~\ref{identify} we also repeat an argument from~\cite{GRE} that shows how a common facet of~$x_c$ and~$x^{\star}$ can be identified if~$x_c$ is given. Section~\ref{sec:Algorithm} presents the shadow vertex algorithm, the main building block of our algorithm. Finally in Section~\ref{runtime} we discuss the running time of a single pivot step of the shadow vertex algorithm.
\input{Reduction}
\input{Identification}
\input{ShadowVertexMethod}
\section{Analysis of the Shadow Vertex Algorithm}\label{sec:analysis}
For given linear functions $L_1 \colon \mathbb{R}^n \to \mathbb{R}$ and $L_2 \colon \mathbb{R}^n \to \mathbb{R}$ we denote by $\pi = \pi_{L_1, L_2}$ the function $\pi \colon \mathbb{R}^n \to \mathbb{R}^2$, given by $\pi(x) = (L_1(x), L_2(x))$. Note that $n$-dimensional vectors can be treated as linear functions. By $P' = P'_{L_1, L_2}$ we denote the projection $\pi(P)$ of the polytope~$P$ onto the Euclidean plane, and by $R = R_{L_1, L_2}$ we denote the path from the bottommost vertex of~$P'$ to the rightmost vertex of~$P'$ along the edges of the lower envelope of~$P'$.
Our goal is to bound the expected number of edges of the path $R = R_{c, w}$, which is random since~$c$ and~$w$ are random. Each edge of~$R$ corresponds to a slope in $(0, \infty)$. These slopes are pairwise distinct with probability one (see Lemma~\ref{lemma:failure probability II}). Hence, the number of edges of~$R$ equals the number of distinct slopes of~$R$.
\begin{definition} \label{definition:failure event} For a real $\varepsilon > 0$ let~$\mathcal{F}_\varepsilon$ denote the event that there are three pairwise distinct vertices $z_1, z_2, z_3$ of~$P$ such that~$z_1$ and~$z_3$ are neighbors of~$z_2$ and such that \[
\left| \frac{w{^\textnormal{T}} \cdot (z_2-z_1)}{c{^\textnormal{T}} \cdot (z_2-z_1)} - \frac{w{^\textnormal{T}} \cdot (z_3-z_2)}{c{^\textnormal{T}} \cdot (z_3-z_2)} \right| \leq \varepsilon \,. \] \end{definition}
Note that if event~$\mathcal{F}_\varepsilon$ does not occur, then all slopes of~$R$ differ by more than~$\varepsilon$. Particularly, all slopes are pairwise distinct. First of all we show that event~$\mathcal{F}_\varepsilon$ is very unlikely to occur if~$\varepsilon$ is chosen sufficiently small. The proof of the following lemma is almost identical to the corresponding proof in~\cite{BrunschR13} except that we need to adapt it to the different random model of~$c$. The proof as well as the proofs of some other lemmas that are almost identical to their counterparts in~\cite{BrunschR13} can be found in Appendix~\ref{appendix:OmittedProofs} for the sake of completeness. Proofs that are completely identical to~\cite{BrunschR13} are omitted.
\begin{lemma} \label{lemma:failure probability II} The probability of event~$\mathcal{F}_\varepsilon$ tends to~$0$ for $\varepsilon \to 0$. \end{lemma}
Let~$p$ be a vertex of~$R$, but not the bottommost vertex $\pi(x_0)$. We call the slope~$s$ of the edge incident to~$p$ to the left of~$p$ \emph{the slope of~$p$}. As a convention, we set the slope of $\pi(x_0)$ to~$0$ which is smaller than the slope of any other vertex~$p$ of~$R$.
\begin{figure}
\caption{Slopes of the vertices of~$R$}
\label{fig:slopes}
\end{figure}
Let $t \geq 0$ be an arbitrary real, let~$p^\star$ be the rightmost vertex of~$R$ whose slope is at most~$t$, and let~$\hat{p}$ be the right neighbor of~$p^\star$, i.e., $\hat{p}$ is the leftmost vertex of~$R$ whose slope exceeds~$t$ (see Figure~\ref{fig:slopes}). Let~$x^\star$ and~$\hat{x}$ be the neighboring vertices of~$P$ with $\pi(x^\star) = p^\star$ and $\pi(\hat{x}) = \hat{p}$. Now let $i = i(x^\star, \hat{x}) \in [m]$ be the index for which $a_i{^\textnormal{T}} x^\star = b_i$ and for which~$\hat{x}$ is the (unique) neighbor~$x$ of~$x^\star$ for which $a_i{^\textnormal{T}} x < b_i$. This index is unique due to the non-degeneracy of the polytope~$P$. For an arbitrary real $\gamma \geq 0$ we consider the vector $\tilde{w} \mathbin{:=} w - \gamma \cdot a_i$.
\begin{lemma}[Lemma~9 of~\cite{BrunschR13}] \label{lemma:reconstruct} Let $\tilde{\pi} = \pi_{c, \tilde{w}}$ and let $\tilde{R} = R_{c, \tilde{w}}$ be the path from $\tilde{\pi}(x_0)$ to the rightmost vertex~$\tilde{p}_r$ of the projection $\tilde{\pi}(P)$ of polytope~$P$. Furthermore, let~$\tilde{p}^\star$ be the rightmost vertex of~$\tilde{R}$ whose slope does not exceed~$t$. Then $
\tilde{p}^\star = \tilde{\pi}(x^\star) $. \end{lemma}
Let us reformulate the statement of Lemma~\ref{lemma:reconstruct} as follows: The vertex~$\tilde{p}^\star$ is defined for the path~$\tilde{R}$ of polygon $\tilde{\pi}(R)$ with the same rules as used to define the vertex~$p^\star$ of the original path~$R$ of polygon $\pi(P)$. Even though~$R$ and~$\tilde{R}$ can be very different in shape, both vertices, $p^\star$ and~$\tilde{p}^\star$, correspond to the same solution~$x^\star$ in the polytope~$P$, that is, $p^\star = \pi(x^\star)$ and $\tilde{p}^\star = \tilde{\pi}(x^\star)$.
Lemma~\ref{lemma:reconstruct} holds for any vector~$\tilde{w}$ on the ray $\vec{r} = \SET{ w - \gamma \cdot a_i \,|\, \gamma \geq 0}$. As $\|w\| \leq n$ (see Section~\ref{sec:Algorithm}), we have $w \in [-n,n]^n$. Hence, ray~$\vec{r}$ intersects the boundary of $[-n,n]^n$ in a unique point~$z$. We choose $\tilde{w} = \tilde{w}(w, i) \mathbin{:=} z$ and obtain the following result.
\begin{corollary} \label{corollary:reconstruct} Let $\tilde{\pi} = \pi_{c, \tilde{w}(w, i)}$ and let~$\tilde{p}^\star$ be the rightmost vertex of path $\tilde{R} = R_{c, \tilde{w}(w, i)}$ whose slope does not exceed~$t$. Then $
\tilde{p}^\star = \tilde{\pi}(x^\star) $. \end{corollary}
Note that Corollary~\ref{corollary:reconstruct} only holds for the right choice of index $i = i(x^\star, \hat{x})$. However, the vector $\tilde{w}(w, i)$ can be defined for any vector $w \in [-n, n]^n$ and any index $i \in [m]$. In the remainder, index~$i$ is an arbitrary index from~$[m]$.
We can now define the following event that is parameterized in~$i$, $t$, and a real $\varepsilon > 0$ and that depends on~$c$ and~$w$.
\begin{definition} \label{definition:event E} For an index $i \in [m]$ and a real $t \geq 0$ let~$\tilde{p}^\star$ be the rightmost vertex of $\tilde{R} = R_{c, \tilde{w}(w, i)}$ whose slope does not exceed~$t$ and let~$y^\star$ be the corresponding vertex of~$P$. For a real $\varepsilon > 0$ we denote by~$E_{i, t, \varepsilon}$ the event that the conditions
\begin{itemize}[leftmargin=0.5cm]
\item[$\bullet$] $a_i{^\textnormal{T}} y^\star = b_i$ and
\item[$\bullet$] $\frac{w{^\textnormal{T}} (\hat{y} - y^\star)}{c{^\textnormal{T}} (\hat{y} - y^\star)} \in (t, t+\varepsilon]$, where~$\hat{y}$ is the neighbor~$y$ of~$y^\star$ for which $a_i{^\textnormal{T}} y < b_i$,
\end{itemize}
are met. Note that the vertex~$\hat{y}$ always exists and that it is unique since the polytope~$P$ is non-degenerate. \end{definition}
Let us remark that the vertices~$y^\star$ and~$\hat{y}$, which depend on the index~$i$, equal~$x^\star$ and~$\hat{x}$ if we choose $i = i(x^\star, \hat{x})$. For other choices of~$i$, this is, in general, not the case.
Observe that all possible realizations of~$w$ from the line $L \mathbin{:=} \SET{ w + x \cdot a_i \,|\, x \in \mathbb{R} }$ are mapped to the same vector $\tilde{w}(w, i)$. Consequently, if~$c$ is fixed and if we only consider realizations of~$\lambda$ for which $w \in L$, then vertex~$\tilde{p}^\star$ and, hence, vertex~$y^\star$ from Definition~\ref{definition:event E} are already determined. However, since~$w$ is not completely specified, we have some randomness left for event $E_{i, t, \varepsilon}$ to occur. This allows us to bound the probability of event $E_{i, t, \varepsilon}$ from above (see proof of Lemma~\ref{lemma:probability bound}). The next lemma shows why this probability matters.
\begin{lemma}[Lemma~12 from~\cite{BrunschR13}] \label{lemma:event covering} For any $t \geq 0$ and $\varepsilon > 0$ let $A_{t, \varepsilon}$ denote the event that the path $R = R_{c, w}$ has a slope in $(t, t+\varepsilon]$. Then, $A_{t, \varepsilon} \subseteq \bigcup_{i=1}^m E_{i, t, \varepsilon}$. \end{lemma}
With Lemma~\ref{lemma:event covering} we can now bound the probability of event $A_{t, \varepsilon}$. The proof of the next lemma is almost identical to the proof of Lemma~13 from~\cite{BrunschR13}. We include it in the appendix for the sake of completeness. The only differences to Lemma~13 from~\cite{BrunschR13}
are that we can now use the stronger upper bound $\|c\|\le 2$ instead of $\|c\|\le n$ and that we have more carefully analyzed the case of large~$t$.
\begin{lemma} \label{lemma:probability bound} For any~$\phi\ge\sqrt{n}$, any~$t \geq 0$, and any $\varepsilon > 0$ the probability of event $A_{t, \varepsilon}$ is bounded by \[
\Pr{A_{t, \varepsilon}} \leq \frac{2mn^2\varepsilon}{\max \SET{ \frac n2, t } \cdot \delta^2}
\leq \frac{4mn\varepsilon}{\delta^2}\,. \] \end{lemma}
\begin{lemma} \label{lemma:expectation bound} For any interval~$I$ let~$X_{I}$ denote the number of slopes of $R = R_{c,w}$ that lie in the interval~$I$. Then, for any~$\phi\ge\sqrt{n}$, \[
\Ex{X_{(0,n]}} \leq \frac{4mn^2}{\delta^2} \] \end{lemma}
\begin{proof} For a real $\varepsilon > 0$ let~$\mathcal{F}_\varepsilon$ denote the event from Definition~\ref{definition:failure event}. Recall that all slopes of~$R$ differ by more than~$\varepsilon$ if~$\mathcal{F}_\varepsilon$ does not occur. For~$t\in\mathbb{R}$ and~$\varepsilon>0$ let~$Z_{t,\varepsilon}$ be the random variable that indicates whether~$R$ has a slope in the interval $(t,t+\varepsilon]$ or not, i.e., $Z_{t,\varepsilon} = 1$ if~$X_{(t,t+\varepsilon]}>0$ and $Z_{t,\varepsilon} = 0$ if~$X_{(t,t+\varepsilon]}=0$.
Let~$k \geq 1$ be an arbitrary integer. We subdivide the interval~$(0,n]$ into~$k$ subintervals. If none of them contains more than one slope then the number~$X_{(0,n]}$ of slopes in the interval~$(0,n]$ equals the number of subintervals for which the corresponding $Z$-variable equals~1. Formally \[
X_{(0,n]} \leq \begin{cases}
\sum_{i=0}^{k-1} Z_{i\cdot\frac{n}{k},\frac{n}{k}} & \text{if $\mathcal{F}_{\frac{n}{k}}$ does not occur} \,, \cr
m^n & \text{otherwise} \,.
\end{cases} \] This is true because $\binom{m}{n-1} \leq m^n$ is a worst-case bound on the number of edges of~$P$ and, hence, of the number of slopes of~$R$. Consequently,
\begin{align*}
\Ex{X_{(0,n]}}
&\leq \sum_{i=0}^{k-1} \Ex{Z_{i \cdot \frac{n}{k}, \frac{n}{k}}} + \Pr{\mathcal{F}_{\frac{n}{k}}} \cdot m^n
= \sum_{i=0}^{k-1} \Pr{A_{i \cdot \frac{n}{k}, \frac{n}{k}}} + \Pr{\mathcal{F}_{\frac{n}{k}}} \cdot m^n \cr
&\leq \sum_{i=0}^{k-1} \frac{2mn^2 \cdot \frac{n}{k}}{\frac n2\delta^2} + \Pr{\mathcal{F}_{\frac{n}{k}}} \cdot m^n
= \frac{4mn^{2}}{\delta^2} + \Pr{\mathcal{F}_{\frac{n}{k}}} \cdot m^n \,. \end{align*}
The second inequality stems from Lemma~\ref{lemma:probability bound}. Now the lemma follows because the bound on $\Ex{X_{(0,n]}}$ holds for any integer $k \geq 1$ and since $\Pr{\mathcal{F}_\varepsilon} \to 0$ for $\varepsilon \to 0$ in accordance with Lemma~\ref{lemma:failure probability II}. \end{proof}
In~\cite{BrunschR13} we only computed an upper bound for the expected value of~$X_{(0,1]}$. Then we argued that the same upper bound also holds for the expected value of~$X_{(1,\infty)}$. In order to see this, we simply exchanged the order of the objective functions in the projection~$\pi$. Then any edge with a slope of~$s>1$ becomes an edge with slope~$\frac{1}{s}<1$. Hence the number of slopes in~$[1,\infty)$ equals the number of slopes in~$(0,1]$ in the scenario in which the objective functions are exchanged. Due to the symmetry in the choice of the objective functions in~\cite{BrunschR13} the same analysis as before applies also to that scenario.
We will now also exchange the order of the objective functions~$w{^\textnormal{T}} x$ and~$c{^\textnormal{T}} x$ in the projection. Since these objective functions are not anymore generated by the same random experiment, a simple argument as in~\cite{BrunschR13} is not possible anymore. Instead we have to go through the whole analysis again. We will use the superscript~$-1$ to indicate that we are referring to the scenario in which the order of the objective functions is exchanged. In particular, we consider the events~$\mathcal{F}_{\varepsilon}^{-1}$, $A_{t,\varepsilon}^{-1}$, and~$E_{i,t,\varepsilon}^{-1}$ that are defined analogously to their counterparts without superscript except that the order of the objective functions is exchanged. The proof of the following lemma is analogous to the proof of Lemma~\ref{lemma:failure probability II}. \begin{lemma} \label{lemma:failure probability II2} The probability of event~$\mathcal{F}^{-1}_\varepsilon$ tends to~$0$ for $\varepsilon \to 0$. \end{lemma}
\begin{lemma} \label{lemma:probability bound2} For any~$\phi\ge\sqrt{n}$, any~$t \geq 0$, and any $\varepsilon > 0$ the probability of event $A^{-1}_{t,\varepsilon}$ is bounded by \[
\Pr{A_{t,\varepsilon}^{-1}} \leq \frac{2mn^{3/2}\varepsilon\phi}{\max \SET{ 1,\frac {nt}2} \cdot \delta}
\le \frac{2mn^{3/2}\varepsilon\phi}{\delta} \,. \] \end{lemma} \begin{proof} Due to Lemma~\ref{lemma:event covering} (to be precise, due to its canonical adaption to the events with superscript~$-1$) it suffices to show that \[
\Pr{E_{i, t, \varepsilon}^{-1}}
\leq \frac{1}{m} \cdot \frac{2mn^{3/2}\varepsilon\phi}{\max \SET{ 1,\frac {nt}2} \cdot \delta}
= \frac{2n^{3/2}\varepsilon\phi}{\max \SET{ 1,\frac {nt}2} \cdot \delta} \] for any index $i \in [m]$.
We apply the principle of deferred decisions and assume that vector~$w$ is already fixed. Now we extend the normalized vector~$a_i$ to an orthonormal basis $\SET{ q_1, \ldots, q_{n-1}, a_i }$ of~$\mathbb{R}^n$ and consider the random vector $(Y_1, \ldots, Y_{n-1}, Z){^\textnormal{T}} = Q{^\textnormal{T}} c$ given by the matrix vector product of the transpose of the orthogonal matrix $Q = [q_1, \ldots, q_{n-1}, a_i]$ and the vector $c = (c_1,\ldots,c_n){^\textnormal{T}}$. For fixed values $y_1, \ldots, y_{n-1}$ let us consider all realizations of~$c$ such that $(Y_1, \ldots, Y_{n-1}) = (y_1, \ldots, y_{n-1})$. Then,~$c$ is fixed up to the ray \[
c(Z)
= Q \cdot (y_1, \ldots, y_{n-1}, Z){^\textnormal{T}}
= \sum_{j=1}^{n-1} y_j \cdot q_j + Z \cdot a_i
= v + Z \cdot a_i \] for $v = \sum_{j=1}^{n-1} y_j \cdot q_j$. All realizations of $c(Z)$ that are under consideration are mapped to the same value~$\tilde{c}$ by the function $c \mapsto \tilde{c}(c, i)$, i.e., $\tilde{c}(c(Z), i) = \tilde{c}$ for any possible realization of~$Z$. In other words, if $c = c(Z)$ is specified up to this ray, then the path $R_{\tilde{c}(c, i),w}$ and, hence, the vectors~$y^\star$ and~$\hat{y}$ from the definition of event $E_{i, t, \varepsilon}^{-1}$, are already determined.
Let us only consider the case that the first condition of event~$E_{i, t, \varepsilon}^{-1}$ is fulfilled. Otherwise, event~$E_{i, t, \varepsilon}$ cannot occur. Thus, event $E_{i, t, \varepsilon}^{-1}$ occurs iff \[
(t, t+\varepsilon]
\ni \frac{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)}
= \underbrace{\frac{v{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)}}_{\mathbin{=:} \alpha} + Z \cdot \underbrace{\frac{a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)}}_{\mathbin{=:} \beta} \,. \]
The next step in this proof will be to show that the inequality $|\beta| \geq \max \SET{ 1, \sqrt{n}\cdot t } \cdot \frac{\delta}{n}$ is necessary for event~$E_{i, t, \varepsilon}^{-1}$ to happen. For the sake of simplicity let us assume that $\|\hat{y} - y^\star\| = 1$ since~$\beta$ is invariant under scaling. If event~$E_{i, t, \varepsilon}^{-1}$ occurs, then $a_i{^\textnormal{T}} y^\star = b_i$, $\hat{y}$ is a neighbor of~$y^\star$, and $a_i{^\textnormal{T}} \hat{y} \neq b_i$. That is, by Lemma~\ref{lemma:delta properties}, Claim~\ref{delta properties:neighboring vertices} we obtain $|a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)| \geq \delta \cdot \|\hat{y} - y^\star\| = \delta$ and, hence, \[
|\beta|
= \left| \frac{a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)} \right|
\geq \frac{\delta}{|w{^\textnormal{T}} \cdot (\hat{y} - y^\star)|} \,. \]
On the one hand we have $|w{^\textnormal{T}} \cdot (\hat{y} - y^\star)| \leq \|w\| \cdot \|\hat{y}-y^\star\| \leq \Big(\sum_{i=1}^n\|u_i\|\Big) \cdot 1 \le n$. On the other hand, due to $\frac{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)} \geq t$ we have \[
|w{^\textnormal{T}} \cdot (\hat{y} - y^\star)|
\leq \frac{|c{^\textnormal{T}} \cdot (\hat{y} - y^\star)|}{t}
\leq \frac{\|c\| \cdot \|\hat{y} - y^\star\|}{t}
\leq \frac{\Big(1+\frac{\sqrt{n}}{\phi}\Big)}{t}
\leq \frac{2}{t} \,, \] where the third inequality is due to the choice of~$c$ as perturbation of the unit vector~$c_0$ and the fourth inequality is due to the assumption~$\phi\ge\sqrt{n}$. Consequently, \[
|\beta|
\geq \frac{\delta}{\min \SET{ n, \frac{2}{t} }}
= \max \SET{ 1,\frac{nt}2 } \cdot \frac{\delta}{n} \,. \]
Summarizing the previous observations we can state that if event~$E_{i, t, \varepsilon}^{-1}$ occurs, then $|\beta| \geq \max \SET{ 1,\frac {nt}2} \cdot \frac{\delta}{n}$ and $\alpha + Z \cdot \beta \in (t, t+\varepsilon]$. Hence, \[
Z \cdot \beta
\in (t, t+\varepsilon] - \alpha \,, \] i.e., $Z$ falls into an interval $I(y_1, \ldots, y_{n-1})$ of length at most $\varepsilon/(\max \SET{ 1,\frac {nt}2} \cdot \delta/n) = n \varepsilon/(\max \SET{ 1,\frac {nt}2} \cdot \delta)$ that only depends on the realizations $y_1, \ldots, y_{n-1}$ of $Y_1, \ldots, Y_{n-1}$. Let~$B_{i, t, \varepsilon}^{-1}$ denote the event that~$Z$ falls into the interval $I(Y_1, \ldots, Y_{n-1})$. We showed that $E_{i, t, \varepsilon}^{-1} \subseteq B_{i, t, \varepsilon}^{-1}$. Consequently, \[
\Pr{E_{i, t, \varepsilon}^{-1}}
\leq \Pr{B_{i, t, \varepsilon}^{-1}}
\leq \frac{2\sqrt{n} n \varepsilon\phi}{\max \SET{ 1,\frac {nt}2}}
\leq \frac{2n^{3/2}\varepsilon\phi}{\max \SET{ 1,\frac {nt}2} \cdot \delta} \,, \] where the second inequality is due to Theorem~\ref{theorem.Prob:enough randomness} for the orthogonal matrix~$Q$. \end{proof}
\begin{lemma} \label{lemma:expectation bound2} For any interval~$I$ let~$X_{I}^{-1}$ denote the number of slopes of~$R_{w,c}$ that lie in the interval~$I$. Then \[
\Ex{X^{-1}_{(0,1/n]}} \leq \frac{2m\sqrt{n}\phi}{\delta} \,. \] \end{lemma} \begin{proof} As in the proof of Lemma~\ref{lemma:expectation bound} we define for~$t\in\mathbb{R}$ and~$\varepsilon>0$ the random variable~$Z_{t,\varepsilon}^{-1}$ that indicates whether~$R_{w,c}$ has a slope in the interval $(t, t+\varepsilon]$ or not. For any integer~$k\ge 1$ we obtain \begin{align*}
\Ex{X^{-1}_{\big(0,\frac{1}{n}\big]}}
&\leq \sum_{i=0}^{k-1} \Ex{Z^{-1}_{i\cdot\frac{1}{kn},\frac{1}{kn}}} + \Pr{\mathcal{F}^{-1}_{\frac{1}{kn}}} \cdot m^n \cr
& = \sum_{i=0}^{k-1} \Pr{A^{-1}_{i\cdot\frac{1}{kn},\frac{1}{kn}}} + \Pr{\mathcal{F}^{-1}_{\frac{1}{kn}}} \cdot m^n \cr
&\leq \sum_{i=0}^{k-1} \frac{2mn^{3/2}\phi}{kn\delta} + \Pr{\mathcal{F}^{-1}_{\frac{1}{k2^{\ell}\sqrt{n}}}} \cdot m^n
= \frac{2m\sqrt{n}\phi}{\delta} + \Pr{\mathcal{F}^{-1}_{\frac{1}{k2^{\ell}\sqrt{n}}}} \cdot m^n \,. \end{align*} The second inequality stems from Lemma~\ref{lemma:probability bound2}. Now the lemma follows because the bound holds for any integer $k \geq 1$ and $\Pr{\mathcal{F}^{-1}_\varepsilon} \to 0$ for $\varepsilon \to 0$ in accordance with Lemma~\ref{lemma:failure probability II2}. \end{proof}
The following corollary directly implies Theorem~\ref{main}.
\begin{corollary} \label{corollary:expectation bound} The expected number of slopes of $R = R_{c,w}$ is \[
\Ex{X_{(0,\infty)}} = \frac{4mn^2}{\delta^2}+\frac{2m\sqrt{n}\phi}{\delta}\,. \] \end{corollary} \begin{proof} We divide the interval~$(0,\infty)$ into the subintervals~$(0,n]$ and~$(n,\infty)$. Using Lemma~\ref{lemma:expectation bound}, Lemma~\ref{lemma:expectation bound2}, and linearity of expectation we obtain \begin{align*}
\Ex{X_{(0,\infty)}} & = \Ex{X_{(0,n]}} + \Ex{X_{(n,\infty)}}
= \Ex{X_{(0,n]}} + \Ex{X^{-1}_{\big(0,\frac{1}{n}\big]}} \cr
& \le \frac{4mn^2}{\delta^2} + \frac{2m\sqrt{n}\phi}{\delta} \,. \end{align*} In the second step we have exploited that by definition~$X_{(a,b)}=X^{-1}_{(1/b,1/a)}$ for any interval~$(a,b)$. \end{proof}
\section{Finding a Basic Feasible Solution}\label{initial}
In this section we discuss how Phase~1 can be realized. In general there are, of course, several known textbook methods how Phase~1 can be implemented. However, for our purposes it is crucial that the parameter~$\delta$ (or~$\Delta$) is not too small (or too large) for the linear program that needs to be solved in Phase~1. Ideally we would like it to be identical with the parameter~$\delta$ (or~$\Delta$) of the matrix~$A$ of the original linear program. Eisenbrand and Vempala have addressed this problem and have presented a method to implement Phase~1. Their method is, however, very different from usual textbook methods and needs to solve~$m$ different linear programs to find an initial feasible solution for the given linear program.
In this section we will argue that also one of the usual textbook methods can be applied. We argue that $1/\delta$ increases by a factor of at most~$\sqrt{m}$ and that~$\Delta$ does not change at all in case one considers integer matrices (in particular, for totally unimodular matrices).
Let~$m$ and~$n$ be arbitrary positive integers, let $A \in \mathbb{R}^{m \times n}$ be an arbitrary matrix without zero-rows, and let $c \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$ be arbitrary vectors. For finding a basic feasible solution of the linear program
\begin{align*} \text{(LP)} \left\{ \begin{aligned}
\max &\ c{^\textnormal{T}} x \cr
\text{s.t.} &\ Ax \leq b \end{aligned} \right. \end{align*}
if one exists, or detecting that none exists, otherwise, we can solve the following linear program:
\begin{align*} \text{(LP')} \left\{ \begin{aligned}
\min &\sum_{i=1}^m y_i \cr
\text{s.t.} &\ Ax - y \leq b \cr
&\ y \geq 0 \end{aligned} \right. \end{align*}
In the remainder of this section let us assume that matrix~$A$ has full column rank, that is, $\textnormal{rank}(A) = n$. Otherwise, we can transform the linear program (LP) as stated in Section~\ref{dimension} before considering (LP'). Furthermore, let us assume that the matrix~$\bar{A}$, formed by the first~$n$ rows of matrix~$A$, is invertible. This entails no loss of generality as this can always be achieved by permuting the rows of matrix~$A$.
Let~$\bar{b}$ denote the vector given by the first~$n$ entries of vector~$b$ and let~$\bar{x}$ denote the vector for which $\bar{A} \bar{x} = \bar{b}$. The vector $(x',y') = (\bar{x}, \max \{A\bar{x}-b, \NULL\})$ is a feasible solution of (LP'), where the maximum is meant component-wise and~$\NULL$ denotes the $m$-dimensional null vector. This is true because $Ax'-y' \leq A\bar{x} - (A\bar{x}-b) = b$ and $y' \geq \NULL$. Moreover, $(x',y')$ is a basic solution: By the choice of~$\bar{x}$ the first~$n$ inequalities of $Ax-y \leq b$ are tight as well as the first~$n$ non-negativity constraints. For each $k > m$ the $k^\text{th}$ inequality of $Ax-y \leq b$ or the $k^\text{th}$ non-negativity constraint is tight. Hence, the number of tight constraints is at least $2n+(m-n) = m+n$, which equals the number of variables of (LP').
Finally, we observe that a vector $(x, \NULL)$ is a basic feasible solution of (LP') if and only if~$x$ is a basic feasible solution of (LP). Consequently, by solving the linear program (LP') we obtain a basic feasible solution of the linear program (LP) (if the optimal value is~$0$) or we detect that (LP) is infeasible (if the optimal value is larger than~$0$). The linear program (LP') can be solved as described in Section~\ref{sec:Algorithm}. However, the running time is now expressed in the parameters $m'=2m$, $n'=m+n$ and $\delta(B)$ (or $\Delta(B)$) of the matrix \[
B = \begin{bmatrix}
A & -\ID[m] \cr
\ZERO[m \times n] & -\ID[m]
\end{bmatrix}
\in \mathbb{R}^{(m+m) \times (n+m)} \,. \] Before analyzing the parameters $\delta(B)$ and $\Delta(B)$, let us show that matrix~$B$ has full column rank.
\begin{lemma} \label{lemma:Phase I full rank} The rank of matrix~$B$ is $m+n$. \end{lemma}
\begin{proof} Recall that we assumed that the matrix~$\bar{A}$ given by the first~$n$ rows of matrix~$A$ is invertible. Now consider the first~$n$ rows and the last~$m$ rows of matrix~$B$. These rows form a submatrix~$\bar{B}$ of~$B$ of the form \[
\bar{B}
= \begin{bmatrix}
\bar{A} & C \cr
\ZERO[m \times n] & -\ID[m]
\end{bmatrix} \] for $C = [-\ID[n \times n], \ZERO[n \times (m-n)]]$. As~$\bar{B}$ is a $2 \times 2$-block-triangular matrix, we obtain $\det(\bar{B}) = \det(\bar{A}) \cdot \det(-\ID[n]) \neq 0$, that is, the first~$n$ rows and the last~$m$ rows of matrix~$B$ are linearly independent. Hence, $\textnormal{rank}(B) = m+n$. \end{proof}
The remainder of this section is devoted to the analysis of $\delta(B)$ and $\Delta(B)$, respectively.
\subsection[A lower Bound for delta(B)]{A Lower Bound for $\delta(B)$} \label{Phase I delta}
Before we derive a bound for the value $\delta(B)$, let us give a characterization of $\delta(M)$ for a matrix~$M$ with full column rank.
\newcommand{\ATOP}[2]{\genfrac{}{}{0pt}{}{#1}{#2}} \begin{lemma} \label{lemma:delta characterization} Let $M \in \mathbb{R}^{m \times n}$ be a matrix with rank~$n$. Then \[
\frac{1}{\delta(M)} = \max_{k \in [n]} \max \SET{ \|z\| \,|\, \ATOP{r_1{^\textnormal{T}}, \ldots, r_n{^\textnormal{T}}\ \text{linear independent rows}}{\text{of~$M$ and}\ [\mathcal{N}(r_1), \ldots, \mathcal{N}(r_n)]{^\textnormal{T}} \cdot z = e_k } } \,, \] where~$e_k$ denotes the $k^\text{th}$ unit vector. \end{lemma}
\begin{proof} The correctness of the above statement follows from
\begin{align*}
\frac{1}{\delta(M)}
&= \max \SET{ \frac{1}{\delta(r_1, \ldots, r_n)} \,|\, r_1{^\textnormal{T}}, \ldots, r_n{^\textnormal{T}}\ \text{lin. indep. rows of~$M$} } \cr
&= \max \SET{ \frac{1}{\delta(\mathcal{N}(r_1), \ldots, \mathcal{N}(r_n))} \,|\, r_1{^\textnormal{T}}, \ldots, r_n{^\textnormal{T}}\ \text{lin. indep. rows of~$M$} } \cr
&= \max \SET{ \max_{k \in [n]} \|v_k\| \,|\, \ATOP{r_1{^\textnormal{T}}, \ldots, r_n{^\textnormal{T}}\ \text{lin. indep. rows of~$M$ and}}{[v_1, \ldots, v_n]^{-1} = [\mathcal{N}(r_1), \ldots, \mathcal{N}(r_n)]{^\textnormal{T}}} } \,. \end{align*}
The first equation is due to the definition of~$\delta$, the second equation holds as~$\delta$ is invariant under scaling of rows, and the third equation is due to Claim~\ref{delta properties:inverse} of Lemma~\ref{lemma:delta properties}. The vector~$v_k$ from the last line is exactly the vector~$z$ for which $[\mathcal{N}(r_1), \ldots, \mathcal{N}(r_n)]{^\textnormal{T}} \cdot z = e_k$. This finishes the proof. \end{proof}
For the following lemma let us without loss of generality assume that the rows of matrix~$A$ are normalized. This does neither change the rank of~$A$ nor the value $\delta(A)$.
\begin{lemma} \label{lemma:Phase I delta} Let~$A$ and~$B$ be matrices of the form described above. Then \[
\frac{1}{\delta(B)} \leq \frac{2\sqrt{m-n+1}}{\delta(A)} \,. \] \end{lemma}
\begin{proof} In accordance with Lemma~\ref{lemma:delta characterization}, it suffices to show that for any~$m+n$ linearly independent rows $r_1{^\textnormal{T}}, \ldots, r_{m+n}{^\textnormal{T}}$ of~$B$ and any $k = 1, \ldots, m+n$ the inequality \[
\|z\| \leq \frac{2\sqrt{m-n+1}}{\delta(A)} \] holds, where~$z$ is the vector for which $[\mathcal{N}(r_1), \ldots, \mathcal{N}(r_{m+n})]{^\textnormal{T}} \cdot z = e_k$.
Let $r_1{^\textnormal{T}}, \ldots, r_{m+n}{^\textnormal{T}}$ be arbitrary $m+n$ linearly independent rows of~$B$ and let $k \in [m+n]$ be an arbitrary integer. We consider the equation $\hat{B} \cdot z = e_k$, where $\hat{B} = [\mathcal{N}(r_1), \ldots, \mathcal{N}(r_{m+n})]{^\textnormal{T}}$. Each row~$r_\ell$ is of either one of the two following types: Type~1 rows correspond to a row from~$A$ and for these we have $\|r_\ell\| = 2$ as the rows of~$A$ are normalized. Type~2 rows correspond to a non-negativity constraint of a variable~$y_i$. For these we have $\|r_\ell\| = 1$. Observe that each row has exactly one ``$-1$''-entry within the last~$m$ columns.
We categorize type~1 and type~2 rows further depending on the other selected rows: Type~1a rows are type~1 rows for which a type~2 row exists among the rows $r_1, \ldots, r_{m+n}$ which has its ``$-1$''-entry in the same column. This type~2 row is then classified as a type~2a row. The remaining type~1 and type~2 rows are classified as type~1b and type~2b rows, respectively. Observe that we can permute the rows of matrix~$\hat{B}$ arbitrarily as we show the claim for all unit vectors~$e_k$. Furthermore, we can permute the columns of~$\hat{B}$ arbitrarily because this only permutes the rows of the solution vector~$z$. This does not influence its norm. Hence, without loss of generality, matrix~$\hat{B}$ contains normalizations of type~1a, of type~2a, of type~1b, and of type~2b rows in this order and the normalizations of the type~2a rows are ordered the same way as the normalizations of their corresponding type~1a rows.
Let~$m_1$, $m_2$, and~$m_3$ denote the number of type~1a, type~1b, and type~2b rows, respectively. Observe that the number of type~2a rows is also~$m_1$. As matrix~$\hat{B}$ is invertible, each column contains at least one non-zero entry. Hence, we can permute the columns of~$\hat{B}$ such that~$\hat{B}$ is of the form \[
\hat{B}
= \begin{bmatrix}
\frac{1}{2} A_1 & -\frac{1}{2} \ID[m_1] & \ZERO & \ZERO \cr
\ZERO & -\ID[m_1] & \ZERO & \ZERO \cr
\frac{1}{2} A_2 & \ZERO & -\frac{1}{2} \ID[m_2] & \ZERO \cr
\ZERO & \ZERO & \ZERO & -\ID[m_3]
\end{bmatrix}
\in \mathbb{R}^{(m+n) \times (m+n)} \,, \] where~$A_1$ and~$A_2$ are $m_1 \times n$- and $m_2 \times n$-submatrices of~$A$, respectively. The number of rows of~$\hat{B}$ is $2m_1+m_2+m_3 = m+n$, whereas the number of columns of~$\hat{B}$ is $n+m_1+m_2+m_3=m+n$. This implies $m_1 = n$ and $m_2 \leq m-n$. Particularly, $A_1$ is a square matrix. As matrix~$\hat{B}$ is a $2 \times 2$-block-triangular matrix and the top left and the bottom right block are $2 \times 2$-block-triangular matrices as well, we obtain \[
\det(\hat{B})
= \det \left( \frac{1}{2} A_1 \right) \cdot (-1)^{m_1} \cdot \left( -\frac{1}{2} \right)^{m_2} \cdot (-1)^{m_3}
= \pm\det(A_1) \cdot \frac{1}{2^{n+m_2}} \,. \] Due to the linear independence of the rows $r_1{^\textnormal{T}}, \ldots, r_{m+n}{^\textnormal{T}}$ we have $\det(\hat{B}) \neq 0$. Consequently, $\det(A_1) \neq 0$, that is, matrix~$A_1$ is invertible.
We partition vector~$z$ and vector~$e_k$ into four components $z_1, \ldots, z_4$ and $e_k^{(1)}, \ldots, e_k^{(4)}$, respectively, and rewrite the system $\hat{B} \cdot z = e_k$ of linear equations as follows:
\begin{align*}
\frac{1}{2} A_1 z_1 - \frac{1}{2} z_2 &= e_k^{(1)} \cr
-z_2 &= e_k^{(2)} \cr
\frac{1}{2} A_2 z_1 - \frac{1}{2} z_3 &= e_k^{(3)} \cr
-z_4 &= e_k^{(4)} \end{align*}
Now we distinguish between four pairwise distinct cases $e_k^{(i)} \neq \NULL$ for $i = 1, \ldots, 4$. In any case recall that the rows of~$A_1$ and~$A_2$ are rows of~$A$, which are normalized. Furthermore, recall that the rows of~$A_1$ are linearly independent.
\begin{itemize}[leftmargin=0.5cm]
\item \textbf{Case~1:} $e_k^{(1)} \neq \NULL$. In this case we obtain $z_2 = \NULL$ and $z_4 = \NULL$. This implies $z_1 = 2\hat{z}$, where~$\hat{z}$ is the solution of the equation $A_1 \hat{z} = e_k^{(1)} + \frac{1}{2} \cdot \NULL = e_k^{(1)}$. As the rows of matrix~$A_1$ are normalized, Lemma~\ref{lemma:delta characterization} yields $\|\hat{z}\| \leq 1/\delta(A)$ and, hence, $\|z_1\| \leq 2/\delta(A)$. Next, we obtain $z_3 = A_2 z_1 - 2 \cdot e_k^{(3)} = A_2 z_1 - \NULL = A_2 z_1$. Each entry of~$z_3$ is a dot product of a (normalized) row from~$A$ and~$z_1$. Hence, the absolute value of each entry is bounded by $\|z_1\| \leq 2/\delta(A)$. This yields the inequality
\begin{align*}
\|z\|
&= \sqrt{\|z_1\|^2 + \|z_2\|^2 + \|z_3\|^2 + \|z_4\|^2}
\leq \sqrt{(1+m_2) \cdot (2/\delta(A))^2} \cr
&\leq \frac{2\sqrt{m-n+1}}{\delta(A)} \,.
\end{align*}
For the last inequality we used the fact that $m_2 \leq m-n$.
\item \textbf{Case~2:} $e_k^{(2)} \neq \NULL$. Here we obtain $z_2 = -e_k^{(2)}$, $z_4 = \NULL$, and $A_1 z_1 = 2 \cdot e_k^{(1)} + z_2 = 2 \cdot \NULL - e_k^{(2)} = -e_k^{(2)}$, that is, $z_1 = -\hat{z}$, where~$\hat{z}$ is the solution of the equation $A_1 \hat{z} = e_k^{(2)}$. Analogously as in Case~1, we obtain $\|\hat{z}\| \leq 1/\delta(A)$ and, hence, $\|z_1\| \leq 1/\delta(A)$. Moreover, we obtain $z_3 = A_2 z_1 - 2 \cdot e_k^{(3)} = A_2 z_1 - \NULL = A_2 z_1$, that is, the absolute value of each entry of~$z_3$ is bounded by $\|z_1\| \leq 1/\delta(A)$. Consequently,
\begin{align*}
\|z\|
&\leq \sqrt{1+(1+m_2) \cdot (1/\delta(A))^2}
\leq \frac{\sqrt{m-n+2}}{\delta(A)}
\leq \frac{2\sqrt{m-n+1}}{\delta(A)} \,.
\end{align*}
For the second inequality we used $m_2 \leq m-n$ and $\delta(A) \leq 1$ by definition of~$\delta(A)$. In the last inequality we used the fact that $m-n+1 \geq 1$ and $\sqrt{x+1} \leq 2\sqrt{x}$ for all $x \geq 1/3$.
\item \textbf{Case~3:} $e_k^{(3)} \neq \NULL$. In this case we obtain $z_2 = \NULL$, $z_4 = \NULL$, and hence, $z_1 = \NULL$. This yields $z_3 = -2 \cdot e_k^{(3)}$ and
\[
\|z\|
= \|z_3\|
= 2
\leq \frac{2\sqrt{m-n+1}}{\delta(A)} \,,
\]
where we again used $\delta(A) \leq 1$.
\item \textbf{Case~4:} $e_k^{(4)} \neq \NULL$. Here we obtain $z_2 = \NULL$, $z_4 = -e_k^{(4)}$, and hence, $z_1 = \NULL$ and $z_3 = \NULL$. Consequently, we get
\[
\|z\|
= \|z_4\|
= 1
\leq \frac{2\sqrt{m-n+1}}{\delta(A)} \,,
\]
which completes this case distinction.
\end{itemize}
As we have seen, in any case the inequality $\|z\| \leq 2\sqrt{m-n+1}/\delta(A)$ holds, which finishes the proof.
\end{proof}
\subsection[An Upper Bound for Delta(B)]{An Upper Bound for $\Delta(B)$} \label{Phase I Delta}
Although parameter $\Delta(B)$ can be defined for arbitrary real-valued matrices, its meaning is limited to integer matrices when considering our analysis of the expected running time of the shadow vertex method. Hence, in this section we only deal with the case that matrix~$A$ is integral. Unlike in Section~\ref{Phase I delta}, we do not normalize the rows of matrix~$A$ before considering the linear program (LP'). As a consequence, matrix~$B$ is also integral.
The following lemma establishes a connection between $\Delta(A)$ and $\Delta(B)$.
\begin{lemma} \label{lemma:Phase I Delta} Let~$A$ and~$B$ be of the form described above. Then $\Delta(B) = \Delta(A)$. \end{lemma}
\begin{proof} It is clear that $\Delta(B) \geq \Delta(A)$ as matrix~$B$ contains matrix~$A$ as a submatrix. Thus, we can concentrate on proving that $\Delta(B) \leq \Delta(A)$. For this, consider an arbitrary $k \times k$-submatrix~$\hat{B}$ of~$B$. Matrix~$\hat{B}$ is of the form \[
\hat{B}
= \begin{bmatrix}
A' & -I_1 \cr
\ZERO[k_1 \times (k-k_2)] & -I_2 \cr
\end{bmatrix} \,, \]
where~$A'$ is a $(k-k_1) \times (k-k_2)$-submatrix of~$A$ and~$I_1$ and~$I_2$ are $(k-k_1) \times k_2$- and $k_1 \times k_2$-submatrices of $\ID[m]$, respectively. Our goal is to show that $|\det(\hat{B})| \leq \Delta(A)$. By analogy with the proof of Lemma~\ref{lemma:Phase I delta} we partition the rows of~$\hat{B}$ into classes. A row of~$\hat{B}$ is of type~1 if it contains a row from~$A'$. Otherwise, it is of type~2. Consequently, there are $k-k_1$ type~1 and~$k_1$ type~2 rows.
These type~1 and type~2 rows are further categorized into three subtypes depending on the ``$-1$''-entry (if exists) within the last~$k_2$ columns. Type~1 and type~2 rows that only have zeros in the last~$k_2$ entries are classified as type~1c and type~2c rows, respectively. The remaining type~1 and type~2 rows have exactly one ``$-1$''-entry within the last~$k_2$ columns. These are partitioned into subclasses as follows: If there are a type~1 row and a type~2 row that have their ``$-1$''-entry in the same column, then these rows are classified as type~1a and type~2a, respectively. The type~1 and type~2 rows that are neither type~1a nor type~1c nor type~2a nor type~2c are referred to as type~1b and type~2b rows, respectively.
Note that type~2c rows only contain zeros. If matrix~$\hat{B}$ contains such a row, then $|\det(\hat{B})| = 0 \leq \Delta(A)$. Hence, in the remainder we only consider the case that matrix~$\hat{B}$ does not contain type~2c rows. With the same argument we can assume, without loss of generality, that matrix~$\hat{B}$ does not contain a column with only zeros. As permuting the rows and columns of matrix~$\hat{B}$ does not change the absolute value of its determinant, we can assume that~$\hat{B}$ contains type~1a, type~1c, type~2a, type~1b, and type~2b rows in this order and that the type~2a rows are ordered the same ways as their corresponding type~1a rows. Furthermore, we can permute the columns of~$\hat{B}$ such that it has the following form: \[
\hat{B}
= \begin{bmatrix}
A_1 & -\ID & \ZERO & \ZERO \cr
A_2 & \ZERO & \ZERO & \ZERO \cr
\ZERO & -\ID & \ZERO & \ZERO \cr
A_3 & \ZERO & -\ID & \ZERO \cr
\ZERO & \ZERO & \ZERO & -\ID
\end{bmatrix} \,, \] where~$A_1$, $A_2$, and~$A_3$ are submatrices of~$A'$ and, hence, of~$A$. Iteratively decomposing matrix~$\hat{B}$ into blocks and exploiting the block-triangular form of the matrices obtained in each step yields
\begin{align*}
|\det(\hat{B})|
&= \left| \det \left( \begin{bmatrix}
A_1 & -\ID \cr
A_2 & \ZERO \cr
\ZERO & -\ID
\end{bmatrix} \right) \right|
\cdot
\left| \det \left( \begin{bmatrix}
-\ID & \ZERO \cr
\ZERO & -\ID
\end{bmatrix} \right) \right|
= \left| \det \left( \begin{bmatrix}
A_1 & -\ID \cr
A_2 & \ZERO \cr
\ZERO & -\ID
\end{bmatrix} \right) \right| \cr
&= \left| \det \left( \begin{bmatrix}
A_1 \cr
A_2
\end{bmatrix} \right) \right|
\cdot
|\det(-\ID)|
= \left| \det \left( \begin{bmatrix}
A_1 \cr
A_2
\end{bmatrix} \right) \right| \,. \end{align*}
The absolute value of the latter determinant is bounded from above by~$\Delta(A)$. This completes the proof. \end{proof} \section{Conclusions}
We have shown that the shadow vertex algorithm can be used to solve linear programs possessing the $\delta$-distance property in strongly polynomial time with respect to $n$, $m$, and~$1/\delta$. The bound we obtained in Theorem~\ref{thm:MainNumberOfPivots} depends quadratically on~$1/\delta$. Roughly speaking, one term~$1/\delta$ is due to the fact that the smaller~$\delta$ the less random is the objective function~$w{^\textnormal{T}} x$. This term could in fact be replaced by~$1/\delta(B)$ where~$B$ is the matrix that contains only the rows that are tight for~$x$. The other term~$1/\delta$ is due to our application of the principle of deferred decisions in the proof of Lemma~\ref{lemma:probability bound}. The smaller~$\delta$ the less random is~$w(Z)$.
For packing linear programs, in which all coefficients of~$A$ and~$b$ are non-negative and one has~$x\ge 0$ as additional constraint, it is, for example, clear that~$x=0^n$ is a basic feasible solution. That is, one does not need to run Phase~1. Furthermore as in this solution without loss of generality exactly the constraints~$x\ge 0$ are tight,~$\delta(B)=1$ and one occurrence of~$1/\delta$ in Theorem~\ref{thm:MainNumberOfPivots} can be removed.
\section*{Acknowledgments} The authors would like to thank Friedrich Eisenbrand and Santosh Vempala for providing detailed explanations of their paper and the anonymous reviewers for valuable suggestions how to improve the presentation.
\begin{appendix}
\section*{Appendix}
In Appendix~\ref{sdelta} we give an equivalent definition of~$\delta$ and state some important properties that are used later. Appendix~\ref{sec:protheory} contains some theorems from probability theory that will be used in Appendix~\ref{appendix:OmittedProofs}, which contains the omitted proofs from Section~\ref{sec:analysis}. In Appendix~\ref{specialcases} we argue how to cope with unbounded linear programs and linear programs without full column rank. We conclude with Appendix~\ref{sec:RandomBits} in which we analyze the number of random bits necessary to run the shadow vertex method.
\input{delta}
\input{ProbTheory}
\section{Proofs from Section~\ref{sec:analysis}}\label{appendix:OmittedProofs}
In this section we give the omitted proofs from Section~\ref{sec:analysis}. These are merely contained for the sake of completeness because they are very similar to the corresponding proofs in~\cite{BrunschR13}.
\subsection{Proof of Lemma~\ref{lemma:failure probability II}}
\begin{lemma} \label{lemma:failure probability I}
The probability that there are two neighboring vertices $z_1, z_2$ of~$P$ such that $|c{^\textnormal{T}} \cdot (z_2-z_1)| \leq \varepsilon \cdot \|z_2-z_1\|$ is bounded from above by $2m^n n\varepsilon\phi$. \end{lemma}
\begin{proof}
Let~$z_1$ and~$z_2$ be arbitrary points in $\mathbb{R}^n$, let $u = z_2 - z_1$, and let~$A_\varepsilon$ denote the event that $|c{^\textnormal{T}} \cdot u| \leq \varepsilon \cdot \|u\|$. As this inequality is invariant under scaling, we can assume that $\|u\| = 1$. Hence, there exists an index~$i$ for which $|u_i| \geq 1/\sqrt{n} \geq 1/n$. We apply the principle of deferred decisions and assume that the coefficients~$c_j$ for $j \neq i$ are already fixed arbitrarily. Then event~$A_\varepsilon$ occurs if and only if $c_i \cdot u_i \in [-\varepsilon, \varepsilon] - \sum_{j \neq i} c_j u_j$. Hence, for event~$A_\varepsilon$ to occur the random coefficient~$c_i$ must fall into an interval of length $2\varepsilon/|u_i| \leq 2n\varepsilon$. The probability for this is bounded from above by~$2n\varepsilon\phi$.
As we have to consider at most $\binom{m}{n-1} \leq m^n$ pairs of neighbors $(z_1, z_2)$, a union bound yields the additional factor of $m^n$. \end{proof}
\begin{proof}[Proof of Lemma~\ref{lemma:failure probability II}]
Let $z_1, z_2, z_3$ be pairwise distinct vertices of~$P$ such that~$z_1$ and~$z_3$ are neighbors of~$z_2$ and let $\Delta_z \mathbin{:=} z_2-z_1$ and $\Delta'_z \mathbin{:=} z_3-z_2$. We assume that $\|\Delta_z\| = \|\Delta'_z\| = 1$. This entails no loss of generality as the fractions in Definition~\ref{definition:failure event} are invariant under scaling. Let $i_1, \ldots, i_{n-1} \in [m]$ be the $n-1$ indices for which $a_{i_k}{^\textnormal{T}} z_1 = b_{i_k} = a_{i_k}{^\textnormal{T}} z_2$. For the ease of notation let us assume that $i_k = k$. The rows $a_1, \ldots, a_{n-1}$ are linearly independent because~$P$ is non-degenerate. Since $z_1, z_2, z_3$ are distinct vertices of~$P$ and since~$z_1$ and~$z_3$ are neighbors of~$z_2$, there is exactly one index~$\ell$ for which $a_\ell{^\textnormal{T}} z_3 < b_\ell$, i.e., $a_\ell{^\textnormal{T}} \Delta'_z \neq 0$. Otherwise, $z_1, z_2, z_3$ would be collinear which would contradict the fact that they are pairwise distinct vertices of~$P$. Without loss of generality assume that $\ell = n-1$. Since $a_k{^\textnormal{T}} \Delta_z = 0$ for each $k \in [n-1]$, the vectors $a_1, \ldots, a_{n-1}, \Delta_z$ are linearly independent.
We apply the principle of deferred decisions and assume that~$c$ is already fixed. Thus, $c{^\textnormal{T}} \Delta_z$ and $c{^\textnormal{T}} \Delta'_z$ are fixed as well. Moreover, we assume that $c{^\textnormal{T}} \Delta_z \neq 0$ and $c{^\textnormal{T}} \Delta'_z \neq 0$ since this happens almost surely due to Lemma~\ref{lemma:failure probability I}. Now consider the matrix $M = [a_1, \ldots, a_{n-2}, \Delta_z, a_{n-1}]$ and the random vector $
(Y_1, \ldots, Y_{n-1}, Z){^\textnormal{T}}
= M^{-1} \cdot w
= -M^{-1} \cdot [u_1, \ldots, u_n] \cdot \lambda $. For fixed values $y_1, \ldots, y_{n-1}$ let us consider all realizations of~$\lambda$ for which $(Y_1, \ldots, Y_{n-1}) = (y_1, \ldots, y_{n-1})$. Then
\begin{align*}
w{^\textnormal{T}} \Delta_z
&= \big( M \cdot (y_1, \ldots, y_{n-1}, Z){^\textnormal{T}} \big){^\textnormal{T}} \Delta_z \cr
&= \sum_{k=1}^{n-2} y_k \cdot a_k{^\textnormal{T}} \Delta_z + y_{n-1} \cdot \Delta_z{^\textnormal{T}} \Delta_z + Z \cdot a_{n-1}{^\textnormal{T}} \Delta_z \cr
&= y_{n-1} \,, \end{align*}
i.e., the value of $w{^\textnormal{T}} \Delta_z$ does not depend on the outcome of~$Z$ since~$\Delta_z$ is orthogonal to all~$a_k$. For~$\Delta'_z$ we obtain
\begin{align*}
w{^\textnormal{T}} \Delta'_z
&= \big( M \cdot (y_1, \ldots, y_{n-1}, Z){^\textnormal{T}} \big){^\textnormal{T}} \Delta'_z \cr
&= \sum_{k=1}^{n-2} y_k \cdot a_k{^\textnormal{T}} \Delta'_z + y_{n-1} \cdot \Delta_z{^\textnormal{T}} \Delta'_z + Z \cdot a_{n-1}{^\textnormal{T}} \Delta'_z \cr
&= y_{n-1} \cdot \Delta_z{^\textnormal{T}} \Delta'_z + Z \cdot a_{n-1}{^\textnormal{T}} \Delta'_z \end{align*}
as~$\Delta'_z$ is orthogonal to all~$a_k$ except for $k = \ell = n-1$. The chain of equivalences
\begin{align*}
&\left| \frac{w{^\textnormal{T}} \Delta_z}{c{^\textnormal{T}} \Delta_z} - \frac{w{^\textnormal{T}} \Delta'_z}{c{^\textnormal{T}} \Delta'_z} \right| \leq \varepsilon \cr
&\iff \frac{w{^\textnormal{T}} \Delta'_z}{c{^\textnormal{T}} \Delta'_z} \in [-\varepsilon, \varepsilon] + \frac{w{^\textnormal{T}} \Delta_z}{c{^\textnormal{T}} \Delta_z} \cr
&\iff w{^\textnormal{T}} \Delta'_z \in \Big[ -\varepsilon \cdot |c{^\textnormal{T}} \Delta'_z|, \varepsilon \cdot |c{^\textnormal{T}} \Delta'_z| \Big] + \frac{w{^\textnormal{T}} \Delta_z}{c{^\textnormal{T}} \Delta_z} \cdot c{^\textnormal{T}} \Delta'_z \cr
&\iff Z \cdot a_{n-1}{^\textnormal{T}} \Delta'_z \in \Big[ -\varepsilon \cdot |c{^\textnormal{T}} \Delta'_z|, \varepsilon \cdot |c{^\textnormal{T}} \Delta'_z| \Big] + \frac{w{^\textnormal{T}} \Delta_z}{c{^\textnormal{T}} \Delta_z} \cdot c{^\textnormal{T}} \Delta'_z - y_{n-1} \cdot \Delta_z{^\textnormal{T}} \Delta'_z \end{align*}
implies, that for event~$\mathcal{F}_\varepsilon$ to occur~$Z$ must fall into an interval $I = I(y_1, \ldots, y_{n-1})$ of length $2\varepsilon \cdot |c{^\textnormal{T}} \Delta'_z|/|a_{n-1}{^\textnormal{T}} \Delta'_z|$. The probability for this to happen is bounded from above by \[
\frac{2n \cdot 2\varepsilon \cdot \frac{|c{^\textnormal{T}} \Delta'_z|}{|a_{n-1}{^\textnormal{T}} \Delta'_z|}}{\delta(r_1, \ldots, r_n) \cdot \min_{k \in [n]} \|r_k\|}
= \underbrace{\frac{4n \cdot |c{^\textnormal{T}} \Delta'_z|}{\delta(r_1, \ldots, r_n) \cdot \min_{k \in [n]} \|r_k\| \cdot |a_{n-1}{^\textnormal{T}} \Delta'_z|}}_{\mathbin{=:} \gamma} \cdot \varepsilon \,, \]
where $[r_1, \ldots, r_n] = -M^{-1} \cdot [u_1, \ldots, u_n]$. This is due to $(Y_1, \ldots, Y_{n-1}, Z){^\textnormal{T}} = [r_1, \ldots, r_n] \cdot \lambda$ and Corollary~\ref{corollary.Prob:enough randomness} (applied with~$\phi=1$). Since the vectors $r_1, \ldots, r_n$ are linearly independent, $\delta(r_1, \ldots, r_n)$ is a well-defined positive value and $\min_{k \in [n]} \|r_k\| > 0$. Furthermore, $|a_{n-1}{^\textnormal{T}} \Delta'_z| > 0$ since~$i_{n-1}$ is the constraint which is not tight for~$z_3$, but for~$z_2$. Hence, $\gamma < \infty$, and thus $\Pr{\left| \frac{w{^\textnormal{T}} \Delta_z}{c{^\textnormal{T}} \Delta_z} - \frac{w{^\textnormal{T}} \Delta'_z}{c{^\textnormal{T}} \Delta'_z} \right| \leq \varepsilon} \to 0$ for $\varepsilon \to 0$.
As there are at most $m^{3n}$ triples $(z_1, z_2, z_3)$ we have to consider, the claim follows by applying a union bound. \end{proof}
\subsection{Proof of Lemma~\ref{lemma:reconstruct}}
\begin{proof}[Proof of Lemma~\ref{lemma:reconstruct}] We consider a linear auxiliary function $\bar{w} \colon \mathbb{R}^n \to \mathbb{R}$, given by $\bar{w}(x) \mathbin{:=} \tilde{w}{^\textnormal{T}} x + \gamma \cdot b_i$. The paths $\bar{R} = R_{c, \bar{w}}$ and~$\tilde{R}$ are identical except for a shift by $\gamma \cdot b_i$ in the second coordinate because for $\bar{\pi} = \pi_{c, \bar{w}}$ we obtain \[
\bar{\pi}(x)
= (c{^\textnormal{T}} x, \tilde{w}{^\textnormal{T}} x + \gamma \cdot b_i)
= (c{^\textnormal{T}} x, \tilde{w}{^\textnormal{T}} x) + (0, \gamma \cdot b_i)
= \tilde{\pi}(x) + (0, \gamma \cdot b_i) \] for all $x \in \mathbb{R}^n$. Consequently, the slopes of~$\bar{R}$ and~$\tilde{R}$ are exactly the same (see Figure~\ref{fig:reconstruct shift}).
\begin{figure}
\caption{Relation between $\bar{R}$ and $\tilde{R}$}
\caption{Relation between $\bar{R}$ an $R$}
\caption{Relations between $R$, $\tilde{R}$, and $\bar{R}$}
\label{fig:reconstruct shift}
\label{fig:reconstruct below}
\end{figure}
Let $x \in P$ be an arbitrary point from the polytope~$P$. Then, $
\tilde{w}{^\textnormal{T}} x
= w{^\textnormal{T}} x - \gamma \cdot a_i{^\textnormal{T}} x
\geq w{^\textnormal{T}} x - \gamma \cdot b_i $. The inequality is due to $\gamma \geq 0$ and $a_i{^\textnormal{T}} x \leq b_i$ for all $x \in P$. Equality holds, among others, for $x = x^\star$ due to the choice of~$a_i$. Hence, for all points $x \in P$ the two-dimensional points $\pi(x)$ and $\bar{\pi}(x)$ agree in the first coordinate while the second coordinate of $\pi(x)$ is at most the second coordinate of $\bar{\pi}(x)$ as $\bar{w}(x) = \tilde{w}{^\textnormal{T}} x + \gamma \cdot b_i \geq w{^\textnormal{T}} x$. Additionally, we have $\pi(x^\star) = \bar{\pi}(x^\star)$. Thus, path~$\bar{R}$ is above path~$R$ but they have point $p^\star = \pi(x^\star)$ in common. Hence, the slope of~$\bar{R}$ to the left (right) of~$p^\star$ is at most (at least) the slope of~$R$ to the left (right) of~$p^\star$ which is at most (greater than)~$t$ (see Figure~\ref{fig:reconstruct below}). Consequently, $p^\star$ is the rightmost vertex of~$\bar{R}$ whose slope does not exceed~$t$. Since~$\bar{R}$ and~$\tilde{R}$ are identical up to a shift of $(0, \gamma \cdot b_i)$, $\tilde{\pi}(x^\star)$ is the rightmost vertex of~$\tilde{R}$ whose slope does not exceed~$t$, i.e., $\tilde{\pi}(x^\star) = \tilde{p}^\star$. \end{proof}
\subsection{Proof of Lemma~\ref{lemma:probability bound}}
\begin{proof}[Proof of Lemma~\ref{lemma:probability bound}] Due to Lemma~\ref{lemma:event covering} it suffices to show that \[
\Pr{E_{i, t, \varepsilon}}
\leq \frac{1}{m} \cdot \frac{2mn^2\varepsilon}{\max \SET{ \frac n2, t } \cdot \delta^2}
= \frac{2n^2\varepsilon}{\max \SET{ \frac n2, t } \cdot \delta^2} \] for any index $i \in [m]$.
We apply the principle of deferred decisions and assume that vector~$c$ is already fixed. Now we extend the normalized vector~$a_i$ to an orthonormal basis $\SET{ q_1, \ldots, q_{n-1}, a_i }$ of~$\mathbb{R}^n$ and consider the random vector $(Y_1, \ldots, Y_{n-1}, Z){^\textnormal{T}} = Q{^\textnormal{T}} w$ given by the matrix vector product of the transpose of the orthogonal matrix $Q = [q_1, \ldots, q_{n-1}, a_i]$ and the vector $w = -[u_1, \ldots, u_n] \cdot \lambda$. For fixed values $y_1, \ldots, y_{n-1}$ let us consider all realizations of~$\lambda$ such that $(Y_1, \ldots, Y_{n-1}) = (y_1, \ldots, y_{n-1})$. Then,~$w$ is fixed up to the ray \[
w(Z)
= Q \cdot (y_1, \ldots, y_{n-1}, Z){^\textnormal{T}}
= \sum_{j=1}^{n-1} y_j \cdot q_j + Z \cdot a_i
= v + Z \cdot a_i \] for $v = \sum_{j=1}^{n-1} y_j \cdot q_j$. All realizations of $w(Z)$ that are under consideration are mapped to the same value~$\tilde{w}$ by the function $w \mapsto \tilde{w}(w, i)$, i.e., $\tilde{w}(w(Z), i) = \tilde{w}$ for any possible realization of~$Z$. In other words, if $w = w(Z)$ is specified up to this ray, then the path $R_{c, \tilde{w}(w, i)}$ and, hence, the vectors~$y^\star$ and~$\hat{y}$ from the definition of event $E_{i, t, \varepsilon}$, are already determined.
Let us only consider the case that the first condition of event~$E_{i, t, \varepsilon}$ is fulfilled. Otherwise, event~$E_{i, t, \varepsilon}$ cannot occur. Thus, event $E_{i, t, \varepsilon}$ occurs iff \[
(t, t+\varepsilon]
\ni \frac{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)}
= \underbrace{\frac{v{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)}}_{\mathbin{=:} \alpha} + Z \cdot \underbrace{\frac{a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)}}_{\mathbin{=:} \beta} \,. \]
The next step in this proof will be to show that the inequality $|\beta| \geq \max \SET{ \frac n2, t } \cdot \frac{\delta}{n}$ is necessary for event~$E_{i, t, \varepsilon}$ to happen. For the sake of simplicity let us assume that $\|\hat{y} - y^\star\| = 1$ since~$\beta$ is invariant under scaling. If event~$E_{i, t, \varepsilon}$ occurs, then $a_i{^\textnormal{T}} y^\star = b_i$, $\hat{y}$ is a neighbor of~$y^\star$, and $a_i{^\textnormal{T}} \hat{y} \neq b_i$. That is, by Lemma~\ref{lemma:delta properties}, Claim~\ref{delta properties:neighboring vertices} we obtain $|a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)| \geq \delta \cdot \|\hat{y} - y^\star\| = \delta$ and, hence, \[
|\beta|
= \left| \frac{a_i{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)} \right|
\geq \frac{\delta}{|c{^\textnormal{T}} \cdot (\hat{y} - y^\star)|} \,. \]
On the one hand we have $|c{^\textnormal{T}} \cdot (\hat{y} - y^\star)| \leq \|c\| \cdot \|\hat{y}-y^\star\| \leq \Big(1+\frac{\sqrt{n}}{\phi}\Big) \cdot 1 \le 2$, where the second inequality is due to the choice of~$c$ as perturbation of the unit vector~$c_0$ and the third inequality is due to the assumption~$\phi\ge\sqrt{n}$. On the other hand, due to $\frac{w{^\textnormal{T}} \cdot (\hat{y} - y^\star)}{c{^\textnormal{T}} \cdot (\hat{y} - y^\star)} \geq t$ we have \[
|c{^\textnormal{T}} \cdot (\hat{y} - y^\star)|
\leq \frac{|w{^\textnormal{T}} \cdot (\hat{y} - y^\star)|}{t}
\leq \frac{\|w\| \cdot \|\hat{y} - y^\star\|}{t}
\leq \frac{n}{t} \,. \] Consequently, \[
|\beta|
\geq \frac{\delta}{\min \SET{ 2, \frac{n}{t} }}
= \max \SET{ \frac n2, t } \cdot \frac{\delta}{n} \,. \]
Summarizing the previous observations we can state that if event~$E_{i, t, \varepsilon}$ occurs, then $|\beta| \geq \max \SET{ \frac n2, t } \cdot \frac{\delta}{n}$ and $\alpha + Z \cdot \beta \in (t, t+\varepsilon]$. Hence, \[
Z \cdot \beta
\in (t, t+\varepsilon] - \alpha \,, \] i.e., $Z$ falls into an interval $I(y_1, \ldots, y_{n-1})$ of length at most $\varepsilon/(\max \SET{ \frac n2, t } \cdot \delta/n) = n \varepsilon/(\max \SET{ \frac n2, t } \cdot \delta)$ that only depends on the realizations $y_1, \ldots, y_{n-1}$ of $Y_1, \ldots, Y_{n-1}$. Let~$B_{i, t, \varepsilon}$ denote the event that~$Z$ falls into the interval $I(Y_1, \ldots, Y_{n-1})$. We showed that $E_{i, t, \varepsilon} \subseteq B_{i, t, \varepsilon}$. Consequently, \[
\Pr{E_{i, t, \varepsilon}}
\leq \Pr{B_{i, t, \varepsilon}}
\leq \frac{2n \cdot \frac{n \varepsilon}{\max \sSET{ \frac n2, t } \cdot \delta}}{\delta(Q{^\textnormal{T}} u_1, \ldots, Q{^\textnormal{T}} u_n)}
\leq \frac{2n^2\varepsilon}{\max \sSET{ \frac n2, t } \cdot \delta^2} \,, \] where the second inequality is due to Corollary~\ref{corollary.Prob:enough randomness} (applied with~$\phi=1$): By definition, we have \[
(Y_1, \ldots, Y_{n-1}, Z){^\textnormal{T}}
= Q{^\textnormal{T}} w
= Q{^\textnormal{T}} \cdot -[u_1, \ldots, u_n] \cdot \lambda
= [-Q{^\textnormal{T}} u_1, \ldots, -Q{^\textnormal{T}} u_n] \cdot \lambda \,. \] The third inequality stems from the fact that $
\delta(-Q{^\textnormal{T}} u_1, \ldots, -Q{^\textnormal{T}} u_n)
= \delta(u_1, \ldots, u_n)
\geq \delta $, where the equality is due to the orthogonality of~$-Q$ (Claim~\ref{delta properties:orthogonal matrix} of Lemma~\ref{lemma:delta properties}). \end{proof}
\input{SpecialCases}
\input{randombits}
\end{appendix}
\end{document} |
\begin{document}
\title{Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation}
\begin{abstract} We prove that the rescaled one-point fluctuations of the boundary of the percolation cluster in the Bernoulli-Exponential first passage percolation around the diagonal converge to a new family of distributions. The limit law is indexed by the rescaled level of percolation $s\ge0$, it is Gaussian for $s=0$ and it converges to the Tracy--Widom distribution as $s\to\infty$. For a fixed level $s>0$ the width of the cluster in the limit as a function of a time parameter $t$ is of order $t^{2/3}$ with Tracy--Widom fluctuations as in the discrete model. \end{abstract}
\section{Introduction}
The Bernoulli-Exponential directed first passage percolation was introduced in~\cite{BC17} as follows. Let $a,b>0$ be fixed. Let $(E_e)$ be a family of independent random variables indexed by the edges of the lattice $\mathbb Z^2$ where the distribution of $E_e$ is exponential with parameter $a$ if $e$ is a vertical edge and exponential with parameter $b$ is $e$ is a horizontal edge. Let $(\xi_{i,j})$ be independent Bernoulli random variables with parameter $b/(a+b)$ which are also independent of $(E_e)$. The passage times of edges are given by \begin{equation} t_e=\left\{\begin{array}{cl} \xi_{i,j}E_e & \mbox{if $e$ is the vertical edge $(i,j)\to(i,j+1)$},\\ (1-\xi_{i,j})E_e & \mbox{if $e$ is the horizontal edge $(i,j)\to(i+1,j)$.}\end{array}\right. \end{equation} For non-negative integers $n$ and $m$ the point-to-point first passage time is given by \begin{equation} T^{\rm pp}(n,m)=\min_{\pi:(0,0)\to(n,m)}\sum_{e\in\pi}\,t_e \end{equation} where the minimum is over all up-right paths $\pi$ from $(0,0)$ to $(n,m)$.
We also introduce the point to half-line first passage time $T(n,m)$ between $(0,0)$ and the half-line \begin{equation} D_{n,m}=\{(i,n+m-i):0\le i\le n\} \end{equation} to be given by \begin{equation} T(n,m)=\min_{\pi:(0,0)\to D_{n,m}}\sum_{e\in\pi}\,t_e \end{equation} where the minimum is taken over all up-right paths $\pi$ from $(0,0)$ to $D_{n,m}$.
It was proved in~\cite{BC17} that for any slope $\kappa>a/b$, the fluctuations of the passage time $T(n,\kappa n)$ converges to the GUE Tracy--Widom distribution, but the behaviour around the slope $a/b$ was not considered. These results were extended in~\cite{BR19} with a theorem about the GUE Tracy--Widom fluctuations of $T(n,an/b+cn^{2/3})$ for any $c>0$.
In this note we investigate the asymptotic fluctuations of the passage time when approaching the diagonal of slope $a/b$ on the scale $\sqrt n$ on which a new family of distribution arises in the limit. The asymptotic fluctuations around the diagonal can be expressed in two equivalent ways. We state the main result in Theorem~\ref{thm:BEasymptotics} in terms of the shape of the percolation cluster. In Corollary~\ref{cor:FPP} we explicitly write the fluctuations of the first passage time value $T(n,an/b+cn^{1/2})$.
For any level $r\ge0$ the percolation cluster is defined by \begin{equation} C(r)=\{(n,m):T^{\rm pp}(n,m)\le r\}. \end{equation} It is natural to introduce the height function \begin{equation}\label{defH} H(n,r)=\max\{k\in\mathbb Z:T^{\rm pp}(bn-k,an+k)\le r\} \end{equation} where $n$ is a non-negative integer and $r\ge0$. Note that the maximum always exists on the right-hand side of \eqref{defH} for any $r\ge0$ because there is always a path from $(0,0)$ to $D_{(a+b)n,0}$ with zero first passage time value. We state the main result in terms of the height function $H(n,r)$ as follows.
\begin{theorem}\label{thm:BEasymptotics} Fix an $s>0$. Then \begin{equation}\label{BEasymptotics} \sqrt{\frac{a+b}{ab}}\frac1{\sqrt n}H\left(n,\frac s{\sqrt{ab(a+b)}}n^{-1/2}\right)\stackrel{\mathrm d}{\Longrightarrow}H_s \end{equation} in distribution as $n\to\infty$ where the distribution of $H_s$ is given as follows. For any $h\in\mathbb R$, \begin{equation}\label{Hsdistribution} \mathbf P(H_s<h)=\det(\mathbbm{1}-K_s)_{L^2((h,\infty))} \end{equation} with the kernel \begin{equation}\label{defKs} K_s(x,y)=\frac1{(2\pi i)^2}\int_{1+i\mathbb R}\mathrm d u\int_{\mathcal C_0}\mathrm d v\,\frac{e^{u^2/2-yu-s/u}}{e^{v^2/2-xv-s/v}}\frac uv\frac1{v-u} \end{equation} where the integration contour $\mathcal C_0$ is a small circle around $0$ with positive orientation such that it does not intersect $1+i\mathbb R$. \end{theorem}
\begin{remark} The formal substitution $s=0$ in \eqref{Hsdistribution}--\eqref{defKs} yields the standard Gaussian distribution. It can be seen by observing that the $v$-integral is equal to the residue at $v=0$ and by computing the $u$-integral directly to get that $K_0(x,y)=\frac1{\sqrt{2\pi}}e^{-y^2/2}$. This corresponds to taking the limit of $\sqrt{\frac{a+b}{ab}}\frac1{\sqrt n}H(n,0)$ which is not covered by the statement of Theorem~\ref{thm:BEasymptotics}, but this limit is known to be Gaussian since it is the scaling limit of a simple random walk with Bernoulli steps. \end{remark}
\begin{theorem}\label{thm:HstoTW} The rescaled random variables \begin{equation}\label{HstoTW} 2^{4/9}3^{-1/3}s^{1/9}\left(H_s-2^{-2/3}3s^{1/3}\right)\stackrel\mathrm d\Longrightarrow\xi \end{equation} as $s\to\infty$ where $\xi$ has GUE Tracy--Widom distribution. \end{theorem}
\begin{corollary}\label{cor:clusterwidth} For a fixed $s>0$ we introduce the height of the percolation cluster of level $s$ after time $t>0$ to be \begin{equation}\label{defHst} H_s(t)=\lim_{n\to\infty}\sqrt{\frac{a+b}{ab}}\frac1{\sqrt n}H\left(tn,\frac s{\sqrt{ab(a+b)}}n^{-1/2}\right). \end{equation} For any $s>0$ the rescaled cluster height converges, that is, \begin{equation}\label{Hstconv} \frac{H_s(t)-2^{-2/3}3s^{1/3}t^{2/3}}{2^{-4/9}3^{1/3}s^{-1/9}t^{4/9}}\stackrel\mathrm d\Longrightarrow\xi \end{equation} as $t\to\infty$ where $\xi$ has GUE Tracy--Widom distribution. \end{corollary}
\begin{remark} The limit in \eqref{defHst} exists by Theorem~\ref{thm:BEasymptotics} for any fixed $t>0$. Corollary~\ref{cor:clusterwidth} does not imply the existence of the time process $t\mapsto H_s(t)$. We expect that the limit process in \eqref{defHst} can be constructed as a function of $t$ based on the Brownian web, see~\cite{TW98,FINR04}.
By the results of~\cite{BR19}, the width of the percolation cluster of a fixed level in the Bernoulli-Exponential model along the diagonal $an/b$ is of order $n^{2/3}$ with Tracy--Widom fluctuations on the scale $n^{4/9}$. By Corollary~\ref{cor:clusterwidth}, the height of the cluster in the limit as a function of $t$ has the same limiting fluctuations under the same scaling as in the discrete model. \end{remark}
\begin{remark} The kernel $K_s$ in \eqref{defKs} is reminiscent of the correlation kernel of the hard-edge Pearcey process which arises in the neighbourhood of the cusp point of the limit shape in the situation when non-intersecting paths are pushed towards a hard wall. In the case of non-intersecting squared Bessel paths, the single-time kernel of the limit process was first described in~\cite{KMW11} and the multi-time kernel was given in~\cite{DV15}. We describe the connection of the two kernels below in more details. Let \begin{equation} L_s(x,y)=\frac1{(2\pi i)^2}\int_{\mathcal C_0}\mathrm d w\int_{1+i\mathbb R}\mathrm d z\,\frac1{wz(w-z)}\frac{e^{-w^2/2+sw+x/w}}{e^{-z^2/2+sz+y/z}} \end{equation} be the single-time kernel of the hard-edge Pearcey process. It was given in a slightly different form in Theorem 1.2 of~\cite{KMW11} with $\alpha=-1$ and more explicitly up to a conjugation in Proposition 2.21 of~\cite{DV15} with $t=s$, $\alpha=-1$ and $\sigma=0$. Here $\alpha$ denotes the index of the squared Bessel paths which is assumed to be $\alpha>-1$ in~\cite{KMW11,DV15}, hence the substitution $\alpha=-1$ is formal. \end{remark}
\begin{proposition} The derivative of the kernel $K_s$ and that of $L_s$ with respect to $s$ factorize as \begin{align} \frac{\mathrm d}{\mathrm d s}K_s(x,y)&=f(s,x)g(s,y),\\ -\frac{\mathrm d}{\mathrm d s}L_s(x,y)&=f(x,s)g(y,s) \end{align} where \begin{align} f(s,x)&=\frac1{2\pi i}\int_{\mathcal C_0}\frac{\mathrm d v}{v^2}\,e^{-v^2/2+xv+s/v},\\ g(s,y)&=\frac1{2\pi i}\int_{1+i \mathbb R}\mathrm d u\,e^{u^2/2-yu-s/u}. \end{align} \end{proposition}
The upper tail decay of the random variables $H_s$ is close to Gaussian.
\begin{proposition}\label{prop:Hstail} \begin{enumerate} \item There is a universal constant $C$ and a threshold $h_0>0$ such that we have \begin{equation}\label{Hstail} \mathbf P(H_s>h)\le C\frac{e^{-h^2/2+4\sqrt{sh}}}h \end{equation} for all $h\ge h_0$ if $0\le s\le h$ holds. \item If both $h,s\to\infty$ in a way that $s\ll h^3$, then the tail bound in \eqref{Hstail} remains valid with the factor $4$ in the exponent is replaced by $2+o(1)$ as $h\to\infty$. \item There is a $c^*\simeq0.0468$ such that if $s=ch^3$ with $c\in(0,c^*)$, then $\mathbf P(H_s>h)\le e^{-\delta(c)h^2}$ as $h\to\infty$ with some $\delta(c)>0$. \end{enumerate} \end{proposition}
Theorem~\ref{thm:BEasymptotics} can be translated into a fluctuation result on the passage times as follows. It is a direct consequence of the definition \eqref{defH} of the height function $H(n,r)$ that \begin{equation}\label{eqevents} \{T(bn-k,an+k)>r\}=\{H(n,r)<k\}. \end{equation} This equality of events yields the following result on the passage times.
\begin{corollary}\label{cor:FPP} Let $h\in\mathbb R$ be fixed. Then \begin{equation} \sqrt{ab(a+b)}\sqrt n\,T\left(bn-\sqrt{\frac{ab}{a+b}}h\sqrt n,an+\sqrt{\frac{ab}{a+b}}h\sqrt n\right)\stackrel{\mathrm d}{\Longrightarrow}T_h \end{equation} in distribution as $n\to\infty$. The distribution of $T_h$ has an atom at $0$ with weight \begin{equation} \mathbf P(T_h=0)=\int_h^\infty\frac1{\sqrt{2\pi}}e^{-y^2/2}\mathrm d y. \end{equation} The distribution function of $T_h$ for any $s>0$ is given by \begin{equation}\label{Thdistribution} \mathbf P(T_h>s)=\det(\mathbbm{1}-K_s)_{L^2((h,\infty))} \end{equation} where the kernel $K_s$ is defined in \eqref{defKs}. \end{corollary}
The Tracy--Widom limit of $H_s$ in Theorem~\ref{thm:HstoTW} implies a similar result for the limiting passage times.
\begin{corollary}\label{cor:passagetimetoTW} For the rescaled limiting passage time it holds that \begin{equation}\label{passagetimetoTW} \left(\frac32\right)^{4/3}h^{-5/3}\left(\frac{4h^3}{27}-T_h\right)\stackrel\mathrm d\Longrightarrow\xi \end{equation} as $h\to\infty$ where $\xi$ has GUE Tracy--Widom distribution. \end{corollary}
We expect that the Tracy--Widom limit of the passage time extends to the following convergence to the Airy process. The scaling of the space variable $x$ in \eqref{TtoAiry} below can be guessed based on the Taylor expansion of the limit shape in \eqref{passagetimetoTW}.
\begin{conjecture}\label{conj:TtoAiry} In the parameter $x\in\mathbb R$ we have that \begin{equation}\label{TtoAiry} \left(\frac32\right)^{4/3}h^{-5/3}\left(\frac{4h^3}{27}+\left(\frac23\right)^{5/3}h^{7/3}x-T_{h+(3/2)^{1/3}h^{1/3}x}\right)\stackrel\mathrm d\Longrightarrow\mathcal A(x)-x^2 \end{equation} as $h\to\infty$ where $\mathcal A(x)$ is the stationary Airy process. \end{conjecture}
The rest of this note is organized as follows. In Section~\ref{s:reformulation}, we reformulate the Fredholm determinant expression from~\cite{BC17} for the point to half-line first passage time in the Bernoulli-Exponential model. We prove Theorem~\ref{thm:BEasymptotics} the main result in this note in Section~\ref{s:asymptotics} which is based on some asymptotic statements proved in Section~\ref{s:asympproofs}. We prove the Tracy--Widom fluctuations in the $s\to\infty$ limit in Section~\ref{s:TW} and the decay bounds of Proposition~\ref{prop:Hstail} in Section~\ref{s:decaybounds}.
\paragraph{Acknowledgements:} We thank B\'alint Vir\'ag and Patrik Ferrari for discussions about polymer models and correlation kernels and Guillaume Barraquand for pointing out the scaling in Theorem~\ref{thm:HstoTW} and for his comments. The work of the author was supported by the NKFI (National Research, Development and Innovation Office) grants FK142124 and KKP144059 ``Fractal geometry and applications'', by the Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the \'UNKP--22--5--BME--250 New National Excellence Program of the Ministry for Innovation and Technology from the source of the NKFI.
\section{Reformulation of the passage time distribution} \label{s:reformulation}
The distribution of the point to half-line Bernoulli-Exponential first passage time is characterized by the following result which is based on Theorem 1.18 of~\cite{BC17} taking into account Remark 1.6 in~\cite{BR19} about a correct sign in \eqref{BEprobBC} below.
\begin{theorem}\label{thm:BEfinite} Let $r>0$ and let $n,m$ be non-negative integers. Then for the point to half-line Bernoulli-Exponential first passage time $T(n,m)$ with parameters $a,b>0$, we have \begin{equation}\label{BEprobBC} \mathbf P(T(n,m)>r)=\det(\mathbbm{1}-\widehat K_r)_{L^2(\mathcal C'_0)} \end{equation} where $\mathcal C'_0$ is a small positively oriented circle around $0$ not containing $-a-b$, and the kernel is given by \begin{equation}\label{defKrhat} \widehat K_r(u,u')=\frac1{2\pi i}\int_{\frac12+i\mathbb R}\frac{e^{rs}}s\frac{\widehat g(u)}{\widehat g(u+s)}\frac{\mathrm d s}{s+u-u'} \end{equation} with \begin{equation}\label{defghat} \widehat g(u)=\left(\frac{a+u}u\right)^n\left(\frac{a+u}{a+b+u}\right)^m\frac1u. \end{equation} \end{theorem}
We reformulate the statement of Theorem~\ref{thm:BEfinite} by a change of variables as follows.
\begin{proposition}\label{prop:BEalternative} Let $n$ be a non-negative integer and $r>0$. Then for any $k\in\mathbb Z$, \begin{equation}\label{BEprob} \mathbf P(H(n,r)<k)=\det(\mathbbm{1}+\widetilde K_r)_{L^2(\mathcal C_{-1/(a+b)})} \end{equation} where the kernel is given by \begin{equation} \widetilde K_r(u,u')=\frac1{2\pi i}\int_{\mathcal D_0}\frac{e^{r(1/v-1/u)}}{(v-u)(v-u')}\frac{g(u)}{g(v)}\,\mathrm d v \end{equation} and \begin{equation}\label{defg} g(u)=\frac{(1+au)^{(a+b)n}}{(1+(a+b)u)^{an+k}}u. \end{equation} The contour $\mathcal D_0$ is a circle around $0$ not containing $-1/(a+b)$ and $\mathcal C_{-1/(a+b)}$ is a large contour which encircles $\mathcal D_0$ and $-1/(a+b)$. \end{proposition}
\begin{proof} We use the statement of Theorem~\ref{thm:BEfinite} to derive \eqref{BEprob}. The left-hand side of \eqref{BEprobBC} and that of \eqref{BEprob} are equal due to the equality of the events \eqref{eqevents}. The equality of the right-hand sides follows in the steps given below. First note that the integration over $1/2+i\mathbb R$ is formal in \eqref{defKrhat} because of the oscillatory behaviour of the integrand. One way how it can be understood is to integrate over the contour \begin{equation} \mathcal D_R=\left\{1/2+iy:y\in[-R,R]\right\}\cup\left\{1/2+Re^{i\phi}:\phi\in\left[\pi/2,3\pi/2\right]\right\}. \end{equation}
Then we rewrite the integral in \eqref{defKrhat} in terms of the variables $v=u+s$ over the same integration contour $\mathcal D_R$ as follows \begin{equation} \widehat K_r(u,u')=\frac1{2\pi i}\int_{\mathcal D_R}\frac{e^{r(v-u)}}{v-u}\frac{\widehat g(u)}{\widehat g(v)}\frac{\mathrm d v}{v-u'}. \end{equation} The main step is the change of variables $u\to1/u$, $u'\to1/u'$, $v\to1/v$. It yields the equality of Fredholm determinants $\det(\mathbbm{1}-\widehat K_r)_{L^2(\mathcal C'_0)}=\det(\mathbbm{1}+\widetilde K_r)_{L^2(\mathcal C_{-1/(a+b)})}$ with \begin{equation} \widetilde K_r(u,u')=\frac1{uu'}\widehat K_r\left(\frac1u,\frac1{u'}\right) =\frac1{2\pi i}\int_{\mathcal D_0}\frac{e^{r(1/v-1/u)}}{v-u}\frac{\widehat g(1/u)}{\widehat g(1/v)}\frac{\mathrm d v}{v-u'} \end{equation} with the contours $\mathcal C_{-1/(a+b)}$ and $\mathcal D_0$ defined below \eqref{defg}. The sign change of the kernel is due to the orientation of the contours. Then \eqref{BEprob} follows by comparing the definition \eqref{defghat} with $(n,m)$ replaced by $(bn-k,an+k)$ and \eqref{defg}. \end{proof}
\section{Asymptotic analysis} \label{s:asymptotics}
This section is devoted to the proof of Theorem~\ref{thm:BEasymptotics} which is the main result in this note. The technical proofs of Propositions~\ref{prop:steep}, \ref{prop:localization}, \ref{prop:Taylor} and \ref{prop:convergence} about specific parts of the asymptotics are postponed to Section~\ref{s:asympproofs}.
With the notation \begin{equation} s_n=\frac s{\sqrt{ab(a+b)}}n^{-1/2},\qquad h_n=\sqrt{\frac{ab}{a+b}}hn^{1/2} \end{equation} the convergence result \eqref{BEasymptotics} can be written as \begin{equation} \lim_{n\to\infty}\mathbf P(H(n,s_n)<h_n)=\mathbf P(H_s<h). \end{equation}
By Proposition~\ref{prop:BEalternative}, we have that \begin{equation}\label{Hprobscaled} \mathbf P(H(n,s_n)<h_n)=\det(\mathbbm{1}+\widetilde K_{s_n})_{L^2(C_{-1/(a+b)})} \end{equation} where the kernel can be given as \begin{equation}\label{defKsntilde} \widetilde K_{s_n}(u,u')=\frac1{2\pi i}\int_{\mathcal D_0}e^{n(f_0(u)-f_0(v))+\sqrt n(f_1(u)-f_1(v))+s_n(\frac1v-\frac1u)}\,\frac uv\frac{\mathrm d v}{(v-u)(v-u')} \end{equation} with \begin{align} f_0(u)&=(a+b)\ln(1+au)-a\ln(1+(a+b)u),\\ f_1(u)&=-\sqrt{\frac{ab}{a+b}}h\ln(1+(a+b)u). \end{align} Hence the proof of Theorem~\ref{thm:BEasymptotics} boils down to show the convergence of the Fredholm determinants \begin{equation} \lim_{n\to\infty}\det(\mathbbm{1}+\widetilde K_{s_n})_{L^2(C_{-1/(a+b)})}=\det(\mathbbm{1}-K_s)_{L^2((h,\infty))}. \end{equation}
Since \begin{equation} f_0'(u)=\frac{ab(a+b)u}{(1+au)(1+(a+b)u)}, \end{equation} the function $f_0(u)$ has a unique critical point at $0$. Its Taylor expansion around this point is \begin{equation}\label{f0Taylor} f_0(u)=\frac12ab(a+b)u^2+\mathcal{O}(u^3) \end{equation} as $u\to0$. The first step of the asymptotic analysis is to find contours which enable us to localize the contour on which the Fredholm determinant is defined as well as the integration in \eqref{defKsntilde} to a neighbourhood of $0$. The existence of appropriate contours is ensured by Proposition~\ref{prop:steep} below. We introduce the V-shaped contour \begin{equation}
V_{\alpha,\varphi}^\delta=\{\alpha+e^{i\varphi\operatorname{sgn}(t)}|t|:t\in[-\delta,\delta]\} \end{equation} where $\alpha\in\mathcal C$ is the tip of the V, $\varphi\in(0,\pi)$ is its half-angle and $\delta\in\mathbb R_+\cup\{\infty\}$ is its length.
\begin{proposition}\label{prop:steep} There exist two bounded closed contours $\gamma_\pm$ such that $\operatorname{Re}(f_0(v))\ge0$ for $v\in\gamma_+$ and $\operatorname{Re}(f_0(u))\le0$ for $u\in\gamma_-$. Moreover, for a small $\delta>0$, \begin{equation}\label{gammapmdef} \gamma_+\cap B(0,\delta)=V_{0,5\pi/6}^\delta,\qquad\gamma_-\cap B(0,\delta)=V_{0,\pi/2}^\delta \end{equation} where $B(0,\delta)$ denotes the ball of radius $\delta$ around $0$. As a consequence, for any $\varepsilon>0$ small enough there is a $\delta'>0$ such that $\operatorname{Re}(f_0(u))<-\varepsilon$ for $u\in\gamma_-\setminus B(0,\delta')$ and $\operatorname{Re}(f_0(v))>\varepsilon$ for $v\in\gamma_+\setminus B(0,\delta')$. \end{proposition}
A possible choice of these contours is shown on Figure~\ref{fig:level_lines}. Let $\gamma_+^n$ be equal to $\gamma_+$ of Proposition~\ref{prop:steep} except for an $n^{-1/2}$ neighbourhood of $0$ where $\gamma_+^n$ is defined to be \begin{equation} \gamma_+^n\cap B(0,n^{-1/2})=\{n^{-1/2}e^{i\varphi}:\varphi\in[-5\pi/6,5\pi/6]\} \end{equation} for $n$ large enough. Let $\gamma_-^n$ be equal to $\gamma_-$ of Proposition~\ref{prop:steep} except for a $2n^{-1/2}$ neighbourhood of $0$ where $\gamma_-^n$ is defined to be \begin{equation} \gamma_-^n\cap B(0,2n^{-1/2})=\{2n^{-1/2}e^{i\varphi}:\varphi\in[-\pi/2,\pi/2]\} \end{equation} for $n$ large enough. Then the contours used on the right-hand side of \eqref{Hprobscaled} can be replaced by $\gamma_\pm$ as follows. By Cauchy's integral theorem, we can deform the contour $\mathcal C_{-1/(a+b)}$ to $\gamma_-^n$ on the right-hand side of \eqref{Hprobscaled}. The integration contour in the formula \eqref{defKsntilde} for the kernel $\widetilde K_{s_n}$ can also be deformed to $\gamma_+^n$ without changing the value of the Fredholm determinant. Note that there is no singularity in the variable $v$ at $-1/(a+b)$.
Next we localize the integration to a neighbourhood of $0$ on the right-hand side of \eqref{Hprobscaled}. For $\delta>0$, let \begin{equation} \gamma_\pm^{n,\delta}=\gamma_\pm^n\cap B(0,\delta) \end{equation} denote the contours $\gamma_\pm^n$ restricted to the $\delta$-neighbourhood of $0$. We define the kernel \begin{equation}\label{Krtildedelta} \widetilde K_{s_n}^\delta(u,u')=\frac1{2\pi i}\int_{\gamma_+^{n,\delta}}e^{n(f_0(u)-f_0(v))+\sqrt n(f_1(u)-f_1(v))+s_n(\frac1v-\frac1u)}\, \frac uv\frac{\mathrm d v}{(v-u)(v-u')} \end{equation} which differs from $\widetilde K_{s_n}$ given in \eqref{defKsntilde} only in the choice of the integration contour. The Fredholm determinant in \eqref{Hprobscaled} and that of \eqref{Krtildedelta} over the sequence of contours $\gamma_-^{n,\delta}$ have the same limit, that is, the localization does not change the $n\to\infty$ limit.
\begin{proposition}\label{prop:localization} For any $\delta>0$ small enough, we have that \begin{equation}\label{localization} \lim_{n\to\infty}\det(\mathbbm{1}+\widetilde K_{s_n})_{L^2(\gamma_-^n)}=\lim_{n\to\infty}\det(\mathbbm{1}+\widetilde K_{s_n}^\delta)_{L^2(\gamma_-^{n,\delta})}. \end{equation} \end{proposition}
The next statement is about the Taylor expansion of the localized Fredholm determinant.
\begin{proposition}\label{prop:Taylor} For $\delta>0$ small enough, the following limits are equal \begin{equation}\label{deteqTaylor} \lim_{n\to\infty}\det(\mathbbm{1}+\widetilde K_{s_n}^\delta)_{L^2(\gamma_-^{n,\delta})}=\lim_{n\to\infty}\det(\mathbbm{1}+K'_{s,n})_{L^2(\Gamma_n')} \end{equation} where \begin{equation}\label{defK'} K'_{s,n}(U,U')=\frac1{2\pi i}\int_{\Gamma_n}\frac{e^{U^2/2-hU-s/U}}{e^{V^2/2-hV-s/V}}\frac UV\frac{\mathrm d V}{(V-U)(V-U')}. \end{equation} The integration contour $\Gamma_n=\Gamma\cap B(0,\sqrt{ab(a+b)n}\delta)$ where $\Gamma$ is a path from $e^{-5\pi i/6}\infty$ to $e^{5\pi i/6}\infty$ so that it crosses the real axis between $0$ and $1$. The contour $\Gamma_n'$ is the vertical segment between $\pm i\sqrt{ab(a+b)n}\delta$ oriented upwards and modified around $0$ so that it does not intersect $\Gamma_n$. \end{proposition}
Finally, the proposition below yields the convergence of the localized Fredholm determinant to the right-hand side of \eqref{Hsdistribution}. That is, Theorem~\ref{thm:BEasymptotics} follows from Propositions~\ref{prop:localization}, \ref{prop:Taylor} and \ref{prop:convergence}.
\begin{proposition}\label{prop:convergence} Let $\delta>0$ be small. Then \begin{equation}\label{K'conv} \lim_{n\to\infty}\det(\mathbbm{1}+K'_{s,n})_{L^2(\Gamma_n')}=\det(\mathbbm{1}-K_s)_{L^2((h,\infty))}. \end{equation} \end{proposition}
\section{Proofs of the asymptotic statements} \label{s:asympproofs}
In this section we prove the asymptotic statements used in the proof of Theorem~\ref{thm:BEasymptotics}.
\begin{proof}[Proof of Proposition~\ref{prop:steep}]
Since $f_0$ is analytic away from its singularities, $\operatorname{Re}(f_0)$ is harmonic and its level lines of the form $\operatorname{Re}(f_0(u))=0$ can be described as follows. The level lines can only cross at singularities or critical points. There are two singularities of $f_0$ at $-1/a$ and at $-1/(a+b)$ and a critical point at $0$. It follows from the Taylor expansion \eqref{f0Taylor} that the branches of the level lines $\operatorname{Re}(f_0(u))=0$ cross at $0$ with angles $\pm\pi/4$ and $\pm3\pi/4$. As $|u|\to\infty$ in any direction, $\operatorname{Re}(f_0(u))\to\infty$, hence all level lines remain bounded. By the maximum principle, any closed path formed by portions of level lines must enclose a singularity. Around the singularity at $-1/a$, $\operatorname{Re}(f_0)$ is negative and around $-1/(a+b)$, $\operatorname{Re}(f_0)$ is positive.
Based on this information, the only possible configuration of the level lines $\operatorname{Re}(f_0)=0$ up to a continuous deformation of the lines which does not cross any singularity is shown on Figure~\ref{fig:level_lines}. Then the contours $\gamma_\pm$ are defined in two steps. We first choose a small $\delta>0$ and give $\gamma_\pm$ in $B(0,\delta)$ to be defined by \eqref{gammapmdef} and we let the value of $\operatorname{Re}(f_0)$ at the endpoints be denoted by $\varepsilon_+=\operatorname{Re}(f_0(e^{\pm5\pi i/6}\delta))>0$ and $\varepsilon_-=\operatorname{Re}(f_0(\pm i\delta))<0$. Then in the second step, we define $\gamma_+$ outside of $B(0,\delta)$ to coincide with that branch of the level line $\operatorname{Re}(f_0(v))=\varepsilon_+$ which connects the two points $e^{\pm5\pi i/6}\delta$. Similarly, we let $\gamma_-$ outside of $B(0,\delta)$ to be the same as the branch of the level line $\operatorname{Re}(f_0(u))=\varepsilon_-$ which connects the points $\pm i\delta$. Then the contours $\gamma_\pm$ satisfy the required properties by the Taylor expansion \eqref{f0Taylor}. \begin{figure}
\caption{The level lines $\operatorname{Re}(f_0)=0$ shown by dashed lines and a possible choice of the integration contours $\gamma_\pm$ shown by solid lines. }
\label{fig:level_lines}
\end{figure} \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:localization}] We fix a small $\delta>0$. The integrand in \eqref{Krtildedelta} can be upper bounded as \begin{equation}\label{Krtildeintegrand}
\left|e^{n(f_0(u)-f_0(v))+\sqrt n(f_1(u)-f_1(v))+s_n(\frac1v-\frac1u)}\,\frac uv\frac1{(v-u)(v-u')}\right|\le Cn^{3/2}e^{n(\operatorname{Re}(f_0(u)-f_0(v)))+K\sqrt n} \end{equation} for $u,u'\in\gamma_-^n$ and $v\in\gamma_+^n$ with some finite constants $C,K$.
By Proposition~\ref{prop:steep}, $\operatorname{Re}(f_0(v))>\varepsilon$ for all $v\in\gamma_+^n\setminus\gamma_+^{n,\delta}$ with some $\varepsilon>0$. Hence the integration over $\gamma_+^n\setminus\gamma_+^{n,\delta}$ can be upper bounded as \begin{equation}
\left|\widetilde K_{s_n}(u,u')-\widetilde K_{s_n}^\delta(u,u')\right|\le e^{-n\varepsilon/2} \end{equation} for $n$ large enough for all $u,u'\in\gamma_-^n$.
Next we consider the Fredholm expansion \begin{equation} \det(\mathbbm{1}+\widetilde K_{s_n})_{L^2(\gamma_-^n)} =\sum_{k=0}^\infty\frac1{k!}\int_{\gamma_-^n}\mathrm d u_1\dots\int_{\gamma_-^n}\mathrm d u_k\det\left(\widetilde K_{s_n}(u_i,u_j)\right)_{i,j=1}^k. \end{equation} The integration in the $k$th term of the expansion can be written as the sum of the integral over $(\gamma_-^{n,\delta})^k$ and the integral over $(\gamma_-^n)^k\setminus(\gamma_-^{n,\delta})^k$. By Proposition~\ref{prop:steep}, $\operatorname{Re}(f_0(u))<-\varepsilon$ for all $u\in\gamma_-^n\setminus\gamma_-^{n,\delta}$, hence we can use the bound in \eqref{Krtildeintegrand} to conclude that the total contribution of the integrals over $(\gamma_-^n)^k\setminus(\gamma_-^{n,\delta})^k$ for $k=0,1,2,\dots$ goes to $0$ as $n\to\infty$. In the integral over $(\gamma_-^{n,\delta})^k$, we can write the kernel $\widetilde K_{s_n}(u,u')$ as $\widetilde K_{s_n}^\delta(u,u')$ plus an error at most $e^{-n\varepsilon/2}$. Hence the difference of the contribution over $(\gamma_-^{n,\delta})^k$ in the $k$th term of the Fredholm expansion of $\det(\mathbbm{1}-\widetilde K_{s_n})_{L^2(\gamma_-^n)}$ and the $k$th term in the Fredholm expansion of $\det(\mathbbm{1}+\widetilde K_{s_n}^\delta)_{L^2(\gamma_-^{n,\delta})}$ is summable in $k$ and the sum goes to $0$ as $n\to\infty$ proving \eqref{localization}. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:Taylor}] By the Taylor expansion \eqref{f0Taylor} and \begin{equation} f_1(u)=-\sqrt{ab(a+b)}hu+\mathcal{O}(u^2) \end{equation} as $u\to0$, we can rewrite \eqref{Krtildedelta} as \begin{multline} \widetilde K_{s_n}^\delta(u,u')\\ =\frac1{2\pi i}\int_{\gamma_+^{n,\delta}}e^{n\frac12ab(a+b)(u^2-v^2)+\mathcal{O}(n(u^3+v^3))-\sqrt n\sqrt{ab(a+b)}h(u-v)+\mathcal{O}(\sqrt n(u^2+v^2))+s_n(\frac1v-\frac1u)}\\ \times\frac uv\frac{\mathrm d v}{(v-u)(v-u')}. \end{multline} By the change of variables $U=\sqrt n\sqrt{ab(a+b)}u$, $U'=\sqrt n\sqrt{ab(a+b)}u'$ and $V=\sqrt n\sqrt{ab(a+b)}v$, we get that the rescaled kernel is given by \begin{multline}\label{Ksndeltawitherror} \frac{n^{-1/2}}{\sqrt{ab(a+b)}}\widetilde K_{s_n}^\delta\left(\frac{n^{-1/2}}{\sqrt{ab(a+b)}}U,\frac{n^{-1/2}}{\sqrt{ab(a+b)}}V\right)\\ =\frac1{2\pi i}\int_{\Gamma_n}e^{U^2/2-V^2/2+\mathcal{O}(n^{-1/2}(U^3+V^3))-h(U-V)+\mathcal{O}(n^{-1/2}(U^2+V^2))+s/V-s/U}\\ \times\frac UV\frac{\mathrm d V}{(V-U)(V-U')}. \end{multline}
The difference between the rescaled kernel above and $K'_{s_n}(U,U')$ in \eqref{defK'} is the presence of the error terms $\mathcal{O}(n^{-1/2}(U^3+V^3))$ and $\mathcal{O}(n^{-1/2}(U^2+V^2))$ in the exponent. Hence the integrand of the rescaled kernel above converges to that of $K'_{s_n}(U,U')$ for any $U,U'\in\Gamma_n'$ and $V\in\Gamma_n$. In order to see the convergence of the kernels and that of the Fredholm determinants, we use dominated convergence. We observe that along the integration contours the error terms can be bounded by a fixed constant times $\delta(U^2+V^2)$. We bound the difference of the integrand with and without the error terms in the exponent by applying the inequality $|e^x-1|\le|x|e^{|x|}$. The decay of the integrand in \eqref{Ksndeltawitherror} comes from the main term $e^{U^2/2-V^2/2}$, hence in the presence of the error terms bounded by $e^{C\delta(U^2+V^2)}$, it remains integrable in both variables $U$ and $V$ if $\delta$ is small enough. Hence the difference of the Fredholm determinants goes to $0$ as $n\to\infty$ by dominated convergence. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:convergence}] The integrand in \eqref{defK'} has a Gaussian decay in both $U$ and $V$ due to the factors $e^{U^2/2-V^2/2}$. Hence by dominated convergence, the integration contours $\Gamma_n'$ and $\Gamma_n$ can be extended to infinity in the Fredholm determinant without changing the limit on the right-hand side of \eqref{deteqTaylor}. The integration contours for $U$ and $V$ can be deformed to $1+i\mathbb R$ and to $\mathcal C_0$ respectively by Cauchy's integral theorem.
Finally we reformulate the kernel as follows. Since $\operatorname{Re}(U-V)>0$ for $U\in1+i\mathbb R$ and $V\in\mathcal C_0$, we have that \begin{equation} \frac1{U-V}=\int_{\mathbb R_+}e^{-x(U-V)}\mathrm d x. \end{equation} Hence we can write the kernel with the contours extended to infinity as \begin{equation} K'_{s,\infty}(U,U')=-AB(U,U') \end{equation} where \begin{equation} A(U,x)=e^{U^2/2-(h+x)U-s/U},\qquad B(x,U)=\frac1{2\pi i}\int_{\mathcal C_0}e^{-V^2/2+(h+x)V+s/V}\frac{\mathrm d V}{V(V-U)}. \end{equation} Since $BA(x,y)=K_s(x,y)$, we conclude \eqref{K'conv} by using the fact that $\det(\mathbbm{1}-AB)_{L^2(1+i\mathbb R)}=\det(\mathbbm{1}-BA)_{L^2(\mathbb R_+)}$. \end{proof}
\section{Tracy--Widom limit} \label{s:TW}
In this section we prove Theorem~\ref{thm:HstoTW} and Corollaries~\ref{cor:clusterwidth} and \ref{cor:passagetimetoTW} about the Tracy--Widom limit of $H_s$ as well as its consequences on the height of the percolation cluster.
\begin{proof}[Proof of Theorem~\ref{thm:HstoTW}] We introduce the scaling of the space variables given by $x=2^{-2/3}3s^{1/3}+2^{-4/9}3^{1/3}s^{-1/9}X$ and $y=2^{-2/3}3s^{1/3}+2^{-4/9}3^{1/3}s^{-1/9}Y$ and we apply the change of variables $u=2^{1/3}s^{1/3}+2^{4/9}3^{-1/3}Us^{1/9}$ and $v=2^{1/3}s^{1/3}+2^{4/9}3^{-1/3}Vs^{1/9}$ in \eqref{defKs}. In the exponent after using the identity $1/(1+q)=1-q+q^2-q^3/(1+q)$ the linear and quadratic terms in $U$ and $V$ cancel and we get that \begin{equation}\label{exponentcalc} \frac{u^2}2-yu-\frac su=-2^{-1/3}3s^{2/3}-2^{-1/9}3^{1/3}s^{2/9}Y+\frac{U^3}3\frac1{1+2^{1/9}3^{-1/3}s^{-2/9}U}-UY \end{equation} and a similar identity in $v$ and $x$. This means that the rescaled kernel after a conjugation is equal to \begin{equation}\label{Airyconv}\begin{aligned} &e^{2^{-1/9}3^{1/3}s^{2/9}(X-Y)}\\ &\quad\times2^{-4/9}3^{1/3}K_s\left(2^{-2/3}3s^{1/3}+2^{-4/9}3^{1/3}s^{-1/9}X,2^{-2/3}3s^{1/3}+2^{-4/9}3^{1/3}s^{-1/9}Y\right)\\ &\qquad=\frac1{(2\pi i)^2}\int\mathrm d U\int\mathrm d V\frac{e^{\frac{U^3}3\frac1{1+2^{1/9}3^{-1/3}s^{-2/9}U}-UY-\frac{V^3}3\frac1{1+2^{1/9}3^{-1/3}s^{-2/9}V}+VX}}{V-U}+o(1). \end{aligned}\end{equation} The integration contours for $U$ and $V$ can be obtained as follows. We first deform the original contours for $u$ and $v$ in \eqref{defKs} so that they pass through $2^{1/3}s^{1/3}$. We may choose the contour for $u$ to be the $V$ shaped contour $V_{2^{1/3}s^{1/3},\pi/2-\varepsilon}^\infty$ and the contour for $v$ to be a circle of radius $2^{1/3}s^{1/3}$ which is deformed locally so that it coincides with $V_{2^{1/3}s^{1/3},\pi/2+\varepsilon}^\infty$ around $2^{1/3}s^{1/3}$ for some small fixed $\varepsilon>0$. We claim that the contour for $U$ on the right-hand side of \eqref{Airyconv} can be chosen to be the one which follows the semi-infinite straight lines from $e^{-i(\pi/2-\varepsilon)}\infty$ to $0$ and from $0$ to $e^{i(\pi/2+\varepsilon)}\infty$ and the contour for $V$ can be the one which goes from $e^{-i(\pi/2+\varepsilon)}\infty$ to $0$ and from $0$ to $e^{i(\pi/2+\varepsilon)}\infty$. The fact that the two contours intersect at $0$ does not cause divergence, alternatively it can be avoided by local deformation.
To validate the choice of contours described above we prove that the integrand has enough decay so that the integral in $U$ and $V$ can be localized to a small neighbourhood of $2^{1/3}s^{1/3}$. In order to justify the localization we first prove that if $U=e^{i(\pi-\varepsilon)}t$ and $t\ge0$ then for $s\ge2^{1/2}3^{-3/2}$ it holds that \begin{equation}\label{Rebound} \operatorname{Re}\left(\frac{U^3}3\frac1{1+2^{1/9}3^{-1/3}s^{-2/9}U}\right)\le-\frac{\sin(3\varepsilon)}3\frac{t^2}{1+t}. \end{equation} To see \eqref{Rebound} we observe that the argument $\arg(U^3/3)=3\pi/2-3\varepsilon$ and that
$\arg(1+2^{1/9}3^{-1/3}s^{-2/9}U)\in[0,\pi/2-\varepsilon]$ for all $t\ge0$. On the other hand $|U^3/3|=t^3/3$ and for $s\ge2^{1/2}3^{-3/2}$ we have that $|1+2^{1/9}3^{-1/3}s^{-2/9}U|\le1+t$. This shows that for the complex number $z=U^3/(3(1+2^{1/9}3^{-1/3}s^{-2/9}U))$ it holds that $\arg(z)\in[\pi-2\varepsilon,3\pi/2-3\varepsilon]$
and $|z|\ge t^3/(3(1+t))$ hence its real part satisfies $\operatorname{Re}(z)\le-\sin(3\varepsilon)t^3/(3(1+t))$ proving \eqref{Rebound}.
The bound on the real part of the exponent given in \eqref{Rebound} and its analogue for $V$ proves that the integrand on the right-hand side of \eqref{Airyconv} has at least Gaussian decay in $U$ and $V$ hence the error caused by changing the contours to be the ones given above causes an error going to $0$. The integrand on the right-hand side of \eqref{Airyconv} converges for any $U$ and $V$ so that the double integral formally goes to the Airy kernel. By the bound \eqref{Rebound} the Gaussian decay of the integrand is enough to conclude the convergence of the kernel.
For the convergence of the Fredholm determinants we can write \begin{equation}\label{lambdaint} \frac1{U-V}=\int_0^\infty\mathrm d\lambda e^{-\lambda(U-V)} \end{equation} because $\operatorname{Re}(U-V)>0$. Using \eqref{lambdaint} on the right-hand side of \eqref{Airyconv} factorizes the integrand into $U$ and $V$ dependent parts. Each of them has an Airy decay in $X$ and $Y$ which can be seen in the same way as for the Airy function. The contour for $U$ can be deformed to coincide with the vertical line at $\sqrt{Y+\lambda}$ around the real axis and to have $\operatorname{Re}(U)\ge\sqrt{Y+\lambda}$ along the whole contour. Then $\operatorname{Re}(U^3/3-U(Y+\lambda))\le-\frac23(Y+\lambda)^{3/2}$ which yields the Airy decay and the convergence of the Fredholm determinants. \end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:clusterwidth}] We can write the definition \eqref{defHst} as \begin{equation}\label{Hstcomp} H_s(t)=\lim_{n\to\infty}\sqrt t\sqrt{\frac{a+b}{ab}}\frac1{\sqrt{tn}}H\left(tn,\frac{\sqrt ts}{\sqrt{ab(a+b)}}(tn)^{-1/2}\right)\stackrel\mathrm d=\sqrt tH_{\sqrt ts} \end{equation} using Theorem~\ref{thm:BEasymptotics} in the second equality in distribution. For any fixed $s>0$ Theorem~\ref{thm:HstoTW} with $s$ replaced by $\sqrt ts$ implies that the right-hand side of \eqref{Hstcomp} can be written as \begin{equation} \sqrt tH_{\sqrt ts}=\sqrt t\left(2^{-2/3}3(\sqrt ts)^{1/3}+2^{-4/9}3^{1/3}(\sqrt ts)^{-1/9}\xi_t\right) \end{equation} where $\xi_t$ converges in law to the Tracy--Widom distribution which proves \eqref{Hstconv}. \end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:passagetimetoTW}] The statement of Theorem~\ref{thm:HstoTW} in terms of the Fredholm determinant in \eqref{Hsdistribution} means that by setting $h(s)=2^{-2/3}3s^{1/3}+2^{-4/9}3^{1/3}s^{-1/9}r$ for any $r\in\mathbb R$ we have $\det(\mathbbm{1}-K_s)_{L^2((h(s),\infty))}\to\mathbf P(\xi<r)$ as $s\to\infty$. We express the convergence of the Fredholm determinant using the variable $h$ as follows. We introduce $s(h)=4h^3/27-(2/3)^{4/3}h^{5/3}r$ which has the property that $h(s(h))=h+\mathcal{O}(h^{-5/3})$ where the error is of smaller order than the fluctuations. Hence we can write $\det(\mathbbm{1}-K_{s(h)})_{L^2((h,\infty))}\to\mathbf P(\xi<r)$ as $h\to\infty$. By \eqref{Thdistribution} this implies \eqref{passagetimetoTW}. \end{proof}
\section{Decay bounds} \label{s:decaybounds}
In this section we prove the decay bounds in Proposition~\ref{prop:Hstail}.
\begin{proof}[Proof of Proposition~\ref{prop:Hstail}] By Cauchy's integral theorem, the integration contours in the definition \eqref{defKs} of the kernel $K_s$ can be deformed as long as no singularity is crossed and the decay along the infinite contour is guaranteed during the deformation. Our choice is $K+i\mathbb R$ for the variable $u$ and the circle of radius $\varepsilon$ around $0$ for $v$ with the values of $K$ and $\varepsilon$ to be specified later so that $K>\varepsilon>0$. Writing $u=K+it$ with $t\in\mathbb R$ one observes that $\operatorname{Re}(u^2)=K^2-t^2$, hence by bounding the absolute value of each factor of the kernel, we have that \begin{equation}\label{Ksbound}
|K_s(x,y)|\le C\varepsilon\int_\mathbb R\mathrm d t\,e^{K^2/2-t^2/2-Ky+s/K+\varepsilon^2/2+\varepsilon x+s/\varepsilon}\frac{\sqrt{K^2+t^2}}\varepsilon\frac1{K-\varepsilon}. \end{equation} In the formula above and later in this proof $C$ denotes a finite positive constant which may change from line to line.
The integration in $t$ can be performed after using the inequality $\sqrt{K^2+t^2}\le K+|t|$ as \begin{equation}\label{tintbound} \int_\mathbb R\mathrm d t\,e^{-t^2/2}\sqrt{K^2+t^2}\le C(1+K). \end{equation} If $K\ge1$, then the integral above can be bounded by $CK$. Based on \eqref{Ksbound} and using \eqref{tintbound} we get that \begin{equation}\label{Ksbound2}
|K_s(x,y)|\le Ce^{K^2/2-Ky+\varepsilon^2/2+\varepsilon x+s/K+s/\varepsilon}. \end{equation}
By the Fredholm expansion on the right-hand side of \eqref{Hsdistribution}, we have that the tail probability of $H_s$ can be written as \begin{equation} \mathbf P(H_s>h)=\sum_{m=1}^\infty\frac{(-1)^{m+1}}{m!}\int_h^\infty\mathrm d x_1\dots\int_h^\infty\mathrm d x_m\det(K_s(x_i,x_j))_{i,j=1}^m. \end{equation} Using \eqref{Ksbound2} and Hadamard's inequality on the $m\times m$ determinant above, we get that \begin{equation}\label{Hstailbound}\begin{aligned} \mathbf P(H_s>h)&\le\sum_{m=1}^\infty\frac{m^{m/2}}{m!}C^me^{m(K^2/2+\varepsilon^2/2+s/K+s/\varepsilon)} \int_h^\infty\mathrm d x_1\dots\int_h^\infty\mathrm d x_m\,e^{-(K-\varepsilon)\sum_{i=1}^mx_i}\\ &=\sum_{m=1}^\infty\frac{m^{m/2}}{m!}\left(\frac{Ce^{K^2/2-(K-\varepsilon)h+\varepsilon^2/2+s/K+s/\varepsilon}}{K-\varepsilon}\right)^m. \end{aligned}\end{equation}
The values of $K$ and $\varepsilon$ are to be chosen in a way that we get the best bound in \eqref{Hstailbound}. With $K=h$, the expression $K^2/2-Kh$ in the exponent on the right-hand side of \eqref{Hstailbound} is minimized and its value is $-h^2/2$. If $0\le s\le h$, then the term $s/K$ in the exponent is bounded by $1$. In this case, we choose $\varepsilon=\sqrt{s/h}$ which minimizes the term $\varepsilon h+s/\varepsilon$ in the exponent with minimal value $2\sqrt{sh}$. With this choice of $K$ and $\varepsilon$ each of the terms $s/K$ and $\varepsilon^2/2$ in the exponent is upper bounded by $s/h$. If we choose $h_0>1$ which means together with the condition $s\le h$ that $s<h^3$ then $s/h<\sqrt{sh}$ holds. This also guarantees that $\varepsilon<K$ and the two integration contours do not cross. Hence we get that \begin{equation} \mathbf P(H_s>h)\le\sum_{m=1}^\infty\frac{m^{m/2}}{m!}\left(\frac{Ce^{-h^2/2+4\sqrt{sh}}}h\right)^m \end{equation} where the $m=1$ term gives the desired upper bound on the right-hand side of \eqref{Hstail} and further terms are negligible compared to it if $h$ is large enough. This proves the first part of the proposition.
If $s\to\infty$ with $h$ satisfying $s\ll h^3$, then we again choose $\varepsilon=\sqrt{s/h}$ and $K=h$. We have that $\varepsilon h+\varepsilon^2/2+s/K+s/\varepsilon=(2+o(1))\sqrt{sh}$ and the rest of the proof is the same as in the first case.
If $s=ch^3$, then we choose $K=\kappa(c)h$ and $\varepsilon=e(c)h$ where $\kappa(c)$ and $e(c)$ minimize the expressions $\kappa(c)^2/2-\kappa(c)+c/\kappa(c)$ and $e(c)^2/2+e(c)+c/e(c)$ which appear as the coefficients of the $h^2$ term in the exponent of \eqref{Hstailbound}. By taking the derivative we solve the equations $\kappa(c)-1-c/\kappa(c)^2=0$ and $e(c)+1-c/e(c)^2=0$ which have exactly one positive solution for $c>0$. These solutions denoted by $\kappa(c)$ and $e(c)$ satisfy $\lim_{c\to0}\kappa(c)=1$ and $\lim_{c\to0}e(c)=0$ with $\lim_{c\to0}e(c)/\sqrt c=1$. Hence the sum of $\kappa(c)^2/2-\kappa(c)+c/\kappa(c)$ and $e(c)^2/2+e(c)+c/e(c)$ is negative for all $c\in(0,c^*)$ with some $c^*>0$. Furthermore $e(c)<\kappa(c)$ also holds on this interval so that the two contours do not cross. Numerical approximation yields that $c^*\simeq0.0468$. \end{proof}
\end{document} |
\begin{document}
\title{Quantum simulation for three-dimensional chiral topological insulator}
\author {Wentao Ji$^{1,2,5\ast}$, Lin Zhang$^{3,4\ast}$, Mengqi Wang$^{1,2,5}$, Long Zhang$^{3,4}$, Yuhang Guo$^{1,2,5}$, Zihua Chai$^{1,2,5}$, Xing Rong$^{1,2,5}$, Fazhan Shi$^{1,2,5}$, Xiong-Jun Liu$^{3,4,6,7\dag}$, Ya Wang$^{1,2,5\dag}$, Jiangfeng Du $^{1,2,5\dag}$ \\ \normalsize{$^{1}$ Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China (USTC), Hefei, 230026, China.}\\
\normalsize{$^{2}$ CAS Key Laboratory of Microscale Magnetic Resonance, USTC, Hefei, 230026, China.}\\ \normalsize{$^{3}$ International Center for Quantum Materials, School of Physics,} \normalsize{Peking University, Beijing 100871, China.}\\ \normalsize{$^{4}$ Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.}\\ \normalsize{$^{5}$ Synergetic Innovation Center of Quantum Information and Quantum Physics,} \normalsize{USTC, Hefei, 230026, China.}\\ \normalsize{$^{6}$ Shenzhen Institute for Quantum Science and Engineering and Department of Physics,} \normalsize{Southern University of Science and Technology, Shenzhen 518055, China.}\\ \normalsize{$^{7}$ Beijing Academy of Quantum Information Science, Beijing 100193, China}\\ \normalsize{$^{\ast}$ These authors contributed equally to this work.}\\ \normalsize{$^\dag$ E-mail: xiongjunliu@pku.edu.cn, ywustc@ustc.edu.cn, djf@ustc.edu.cn } }
\date{\today}
\begin{abstract} Quantum simulation, as a state-of-art technique, provides the powerful way to explore topological quantum phases beyond natural limits. Nevertheless, a complete simulation of the bulk and surface topological physics, and their correspondence is usually hard to achieve in one single simulator. Here we build up a quantum simulator using nitrogen-vacancy center to investigate a previously-not-realized three-dimensional (3D) chiral topological insulator, and demonstrate by quantum quenches a complete study of both the bulk and surface topological physics. First, a dynamical bulk-surface correspondence in momentum space is observed, showing that the bulk topology of the 3D phase uniquely corresponds to the nontrivial quench dynamics emerging on 2D momentum hypersurfaces called band inversion surfaces (BISs), equivalent to the bulk-boundary correspondence in real space. Further, the symmetry protection of the 3D chiral phase is uncovered by measuring dynamical spin textures on BISs, which exhibit perfect (broken) topology when the chiral symmetry is preserved (broken). Finally we measure the topological charges to characterize directly the bulk topology, and identify an emergent dynamical topological transition when varying the quenches from deep to shallow regimes. This work opens a new avenue of quantum simulation towards the complete study of topological quantum phases. \end{abstract}
\maketitle
\emph{Introduction.}---The past over one decade has witnessed the explosive progress in the field of topological quantum phases~\cite{Hasan2010,Qi2011,Chiu2016}, with many exotic topological states having been discovered in the tabletop materials~\cite{Yan2012,Ando2013,Yan2017}. The most prominent classes of topological materials include the time-reversal invariant topological insulators~\cite{Konig2007,Hsieh2008,Xia2009,Knez2011}, quantum anomalous Hall insulators~\cite{Chang2013}, topological semimetals~\cite{Liu2014a,Lv2015,Xu2015}, and topological superconductors~\cite{Mourik2012,Sun2016,Zhang2018}. These topological phases are characterized by nontrivial topology in the bulk, and host topology- or symmetry-protected gapless boundary modes which are connected to the bulk through the bulk-boundary correspondence~\cite{Hasan2010,Qi2011,Chiu2016}. Such bulk-boundary correspondence has been the dominant mechanism for the observation of the topological states, with most topological materials having been uncovered in experiment by resolving the boundary physics~\cite{Hsieh2008,Xia2009,Lv2015,Xu2015}, while the bulk topology, however, is hard to be directly measured for solid systems.
\begin{figure*}\label{fig:1}
\end{figure*}
Despite the considerable achievements, only a small portion of the broad classes of topological phases predicted in theory have been observed in condensed matter physics~\cite{Zhang2019,Vergniory2019,Tang2019}. Quantum simulation~\cite{Feynman1982}, as a state-of-art technique, provides a powerful way to explore exotic quantum phases beyond natural limits~\cite{Georgescu2014}. A number of exotic quantum systems, such as the two-dimensional (2D) Haldane model~\cite{Jotzu2014} and 2D spin-orbit (SO) coupled minimal model for quantum anomalous Hall phase~\cite{Liu2014b,Wu2016,Sun2018}, 1D chiral topological phase \cite{Atala2013,Liu2013,Song2018,Xie2019}, and 3D semimetals~\cite{Song2019,Tan2019} have been successfully realized in a controllable fashion with various quantum simulators including the ultracold atoms~\cite{Bloch2012}, photonic crystals~\cite{Aspuru-Guzik2012,Lu2014}, and solid-state qubit systems~\cite{Houck2012}. Nevertheless, in these studies, either the bulk or only the boundary physics of the simulated topological states can be well explored. For example, in ultracold atoms, it is convenient to measure the bulk topology but hard to simulate the boundary~\cite{Jotzu2014,Wu2016,Sun2018,Aidelsburger2015,note-1}. Therefore, a complete study of both the bulk and boundary topological physics, and their correspondence is challenging for quantum simulators.
In this work, we build up a quantum simulator using nitrogen-vacancy (NV) center to investigate 3D chiral topological insulator which was not accessible in solid systems, and demonstrate a complete simulation of the bulk and surface topological physics of the simulated chiral phase. This study is based on the recently proposed dynamical bulk-surface correspondence in momentum space~\cite{Zhanglin2018,Zhanglong2018-1,Zhanglong2019a,Zhanglong2019b,LZhou2018,Sun2018b,Wang2019,Yi2019}, which bridges the bulk topology of a $d$D equilibrium phase and the nontrivial quench dynamics emerging on $(d-1)$D momentum hypersurfaces called band inversion surfaces (BISs). The dynamical bulk-surface correspondence resembles the bulk-boundary correspondence in real space, and is easier to emulate than the latter, since the momentum space can be readily engineered for quantum simulators. This facilitates complete study of the simulated topological phases. In demonstrating the correspondence between the bulk and surface topological physics, we show in experiment the chiral symmetry protection of the 3D topological phase, and further measure the topological charges to directly characterize the bulk topology, with an emergent dynamical topological transition being observed.
\emph{Simulation of the 3D model.}---The 3D chiral topological insulator simulated in the current experiment is described by the Bloch Hamiltonian ${\cal H}_{\rm 3D}({\bold k}) = \sum_{j=0}^4 h_j \gamma_j$ as \begin{align}
\mathcal H_{\rm 3D}(\bf k)= & \bigr[ m_z - t_0 ( \cos k_x + \cos k_y + \cos k_z )\bigr] \gamma_0\nonumber\\
& + t_{\rm so} ( \sin k_x \gamma_1 + \sin k_y \gamma_2 + \sin k_z \gamma_3 ), \end{align} where the Bloch momentum ${\bold k} = (k_x, k_y, k_z)$, the Dirac matrices $\gamma_0 = \sigma_z \otimes \tau_z$, $\gamma_1 = \sigma_x \otimes \textbf 1$, $\gamma_2 = \sigma_y \otimes \textbf 1$, and $\gamma_3 = \sigma_z \otimes \tau_x$, with the Pauli matrices $\sigma_{x,y,z}$ and $\tau_{x,y,z}$ in the present simulator corresponding to the electron and nuclear spins, respectively.
The $ h_0(\bold k)$-term with the parameters $m_z$ and $t_0$ characterizes the dispersion of four uncoupled bands. The remaining part, written as $ {\bf h}_{\rm so}( {\bf k} ) = (h_1,h_2,h_3) $, represents a spin-orbit field which couples the four different bands, with $t_{\rm so}$ simulating the spin-flipped hopping coefficient. The Hamiltonian has a chiral symmetry defined by $\gamma_4=\sigma_z \otimes \tau_y$, hence it belongs to AIII class according to the Altland-Zirnbauer ten-fold symmetry classification~\cite{Chiu2016,AZ1997} and is characterized by 3D winding numbers in the equilibrium theory. The topological phases include three nontrivial regions: (i) $t_0<m_0<3t_0$ with winding number $\nu_3=1$; (ii) $-t_0<m_0<t_0$ with $\nu_3=-2$; and (iii) $-3t_0<m_0<-t_0$ with $\nu_3=1$. Beyond these regions the phase is trivial, and across the phase transition points the bulk gap is closed.
We realize the Hamiltonian $\mathcal H_{\rm 3D}$ by a quantum simulator built from NV center in diamond~\cite{Doherty2013}. The electrons around the center form an effective electron spin with a triplet ground state ($S=1$). Together with the intrinsic nitrogen-14 nuclear spin ($I=1$), it forms a coupled system, as depicted in Fig.~\ref{fig:1}(a). The Hamiltonian of the NV center is \begin{equation}
\mathcal H_{\rm NV} = 2\pi (D S_z^2 + \omega_e S_z + Q I_z^2 + \omega_n I_z + A S_z I_z), \end{equation} where $S_z$ ($I_z$) denotes the electron (nuclear) spin operator, $D=2.87\mathrm{GHz}$ is the electronic zero-field splitting, $Q=-4.95\mathrm{MHz}$ is the nuclear quadrupolar interaction, and $A=-2.16\mathrm{MHz}$ is the hyperfine interaction. A magnetic field of $514\mathrm{G}$ is applied along the NV's symmetry axis, yielding an electron (nuclear) Zeeman splitting $\omega_e$ ($\omega_n$) of $1439\ \mathrm{MHz}$ ($154\ \mathrm{kHz}$). The subspace of $\{m_s=0,-1\} \otimes \{m_i=+1,0\}$ is utilized to form a two-qubit system,
relabeled as $\{\left|0\right> ,\left|1\right>\} \otimes \{\left|0\right> ,\left|1\right>\}$, on which the Pauli operators $\sigma_i$ and $\tau_i$ are defined. A microwave pulse is applied to produce an external driving field $\Omega_{mw}$. Under the rotating-wave approximation, the effective Hamiltonian capturing the couplings in the subspace reads $\mathcal H_{mw,{\rm RWA}} =2\pi\left( \frac{A}{4}\sigma_z\tau_z +\Omega_x\sigma_x +\Omega_y\sigma_y \right)$, with $\Omega_x=\Omega_{mw} \cos \phi$ and $\Omega_y=-\Omega_{mw} \sin \phi$, where $\phi$ is the phase of microwave pulse. Finally, the $\sigma_z\tau_x$ term can be generated via a unitary rotation about $\tau_y$ axis by $\theta$ angle, yielding \begin{eqnarray}
\mathcal H_{\rm eff} = \frac{A}{4}\cos \theta \sigma_z\tau_z +\Omega_x\sigma_x
+\Omega_y\sigma_y +\frac{A}{4}\sin \theta \sigma_z\tau_x, \end{eqnarray} where the factor $2\pi$ is neglected. This $\tau_y$-rotation of the Hamiltonian is realized by applying a radio-frequency pulse to rotate the nuclear spin. The experiment was performed on a home-built confocal setup at room temperature. We use a [111] oriented NV center with solid immersion lens. The MW and radio-frequency control of NV center are realized through an arbitrary wave generator.
The 3D chiral topological insulator model $\mathcal H_{\rm 3D}$ can emulated by $\mathcal H_{\rm eff}$ after mapping the parameter space to Bloch momentum space, i.e. $(\theta, \phi, \Omega_{mw})\rightarrow \bold k$. Any state evolving under $\mathcal H_{\rm eff}$ is then mapped to the one evolving under $\mathcal H_{\rm 3D}$~\cite{Supp}. Thus the $\bold k$-space of the 3D chiral phase can fully engineered, while the real space including the boundary cannot be simulated for the quantum simulator. The key observation is that, as studied below, the dynamical bulk-surface correspondence in momentum space provides the alternative full investigation of the bulk and surface topological physics.
\emph{Dynamical bulk-surface correspondence.}---We present the nontrivial quench dynamics emerging on BISs and connected to the bulk topology. The experimental procedure for quench along $\gamma_i$ axis consists of three steps [see Fig.~\ref{fig:1}(b)]. First, we initialize the state to the state $\left | 00 \right >$, which is then prepared to be fully (or incompletely) polarized along $\gamma_i$ axis by a unitary control. Then, the initialized state evolves by time $t$ under ${\cal H}_{\rm 3D}$, as simulated by $\mathcal H_{\rm eff}$, rendering the quench dynamics. Finally, we measure the spin polarization $\langle\gamma_j(t)\rangle$. The opposite unitary operations are respectively used to perform the quench and measurement with respect to all the spin components [Fig.~\ref{fig:1}(c)]. Following the measurement, we obtain the time-averaged spin polarizations $\overline{\left < \gamma_j \right >}_i=\lim_{T\rightarrow\infty}(1/T)\int_0^T\left < \gamma_j(t) \right >_idt\propto h_i h_j$, which are key ingredients to characterize the topology~\cite{Zhanglin2018}. Here the index $i$ ($j$) denotes the quench (measurement) axis.
The dynamical bulk-surface correspondence states that the bulk topology of the 3D chiral topological phase uniquely corresponds to the nontrivial quench dynamics emerging on the 2D BISs~\cite{Zhanglin2018}. For the initial state fully polarized in the axis $\gamma_0$, the 2D BISs are formed by all the momenta points where spin oscillations are {\em resonant} and easily measurable, giving the vanishing time-averaged spin-polarizations \begin{align}
\mathrm{BIS}=\{\mathbf{k}\vert\overline{\langle\gamma_{i}(\mathbf{k})\rangle}_{0}=0, \ i=0,1,2,3\}. \end{align} A dynamical invariant can be defined on the BISs as the winding of an emergent dynamical spin-texture field $\mathbf{g}(\mathbf{k})$, with the $i$-th component $\ g_i( \textbf k ) = \frac{1}{\mathcal N_k} \partial_{k_\perp} \overline{\left < \gamma_i \right >}_0\big\vert_{\scriptscriptstyle{\bf k}\in\rm BIS}$ describing the variation slope of $\overline{\left < \gamma_i \right >}_0$ along the local direction $k_\perp$ perpendicular to the BISs and normalized by ${\mathcal N_k}$, and \begin{align}
\mathcal{W} = \frac{1}{8\pi}\int_{\rm BIS}\mathrm{d}^{2}\mathbf{k}\,\mathbf{g}\cdot(\nabla\mathbf{g}\times\nabla\mathbf{g}), \end{align} Geometrically, the dynamical invariant describes the coverage of the dynamical field $\mathbf{g}(\mathbf{k})$ over a 2D spherical surface. This dynamical topological invariant equals the bulk topological invariant of the ground band of ${\cal H}_{\rm 3D}$, and provides the dynamical characterization of the 3D chiral phase~\cite{Zhanglin2018}.
We show the experimental measurements of the three different topological regimes in Fig.~\ref{fig:1}(d-f). The BISs are measured and exhibit very different shapes in different phases. The measured dynamical field $\mathbf{g}(\mathbf{k})$ is depicted as arrows, from which with the 2D dynamical invariant $\mathcal{W}$ can be computed, and is verified to characterize the 3D bulk topology.
\begin{figure}
\caption{
(color online). \textbf{Measuring the chiral symmetry protection}. The blue data points are experimental results of winding numbers obtained from the emergent dynamical spin-texture field, and the error-bars represent three standard deviation. The orange line is calculated from the theoretic model. The insets are the emergent dynamical spin-texture field $\mathbf{g}(\mathbf{k})=(g_1,g_2,g_3,g_4)$ correspond to each data point. The arrows denote $g_{1,2,3}$ components, and the color of the arrows denotes the component $g_4$.
}
\label{fig:sym_break}
\end{figure}
\begin{figure*}
\caption{
(color online). \textbf{Dynamical measurement of topological charges.}
(a) Experimental results of the dynamical field $\mathbf \Theta (\mathbf k)$ constructed from $\overline{\left < \gamma_0 (\textbf k) \right >}_{1,2,3}$. The color of the pixels indicates the norm of $\mathbf \Theta (\mathbf k)$, which vanishes at the location of BIS and topological charges. The positive (negative) charges are marked out with blue (green) circles.
(b)(c) Dynamical field $\mathbf \Theta (\mathbf k)$ at around the charges on $k_z=0$ and $k_z=-\pi$ plane.
}
\label{fig:charge}
\end{figure*}
\emph{Measuring the chiral symmetry protection}.---The dynamical bulk-surface correspondence is protected by the chiral symmetry. It is important to verify the symmetry protection by studying the symmetry-breaking effect on the quench dynamics on BISs, similar to the symmetry-breaking in the boundary states in real space. We create a constant term $h_{4}\gamma_4=h_4\sigma_z\tau_y$ into $\mathcal{H}_{\rm 3D}$ to break the chiral symmetry via an additional rotation in the $\tau_x$ axis. Then the dynamical spin-texture field on BISs becomes a 4D vector $\mathbf{g}(\mathbf{k})=(g_1,g_2,g_3,g_4)$ with $g_i=\partial_{k_{\perp}}\overline{\langle\gamma_{i}\rangle}_{0}/\mathcal{N}_{\mathbf{k}}$,
which locates on a 3D spherical surface $S^3$. To quantify the geometric property of $\mathbf{g}(\mathbf{k})$, we note that without symmetry-breaking, i.e. $h_4=0$, the dynamical field $\mathbf{g}(\mathbf{k})$ sits on the equator of $S^3$ (equivalent to $S^2$). The solid angle enclosed by $\mathbf{g}(\mathbf{k})$ is a multiple of the half 3-sphere area, characterizing the invariant which can be generalized to the symmetry breaking case as \begin{align}\label{W_CSB}
\mathcal{W}_{\rm SB}=\frac{1}{\pi^2} \int_{\mathcal{S}\vert_{\partial\mathcal{S}=\textbf g(\textbf k)}} \textrm d S^3, \end{align} where $\pi^2$ is the area of the half unit 3-sphere, $\textrm d S^3$ is the corresponding area element, and the integral is performed over the region $\mathcal{S}$ with boundary $\partial\mathcal{S}=\textbf g(\textbf k)$~\cite{Supp}.
The experimental measurement of the symmetry breaking effect is shown in Fig.~\ref{fig:sym_break}. It is observed that once $h_4\neq0$, the 4D dynamical field $\mathbf{g}(\mathbf{k})$ is shifted away from the equator of $S^3$, with a nonzero polarization in the $\gamma_{4}$ axis, for which the value $\mathcal{W}_{\rm SB}\neq\mathcal{W}$ is no longer quantized and decreases with the strength $h_4$. The results show that the symmetry protection of the bulk topological phase can be identified from the the dynamical spin textures on BISs, which exhibit perfect (broken) topology and zero (nonzero) $\gamma_{4}$-polarization when the chiral symmetry is preserved (broken), similar to the boundary modes in real space which can be gapped out and polarized by the symmetry-breaking term.
\emph{Topological charges and emergent dynamical transition}.---We proceed to detect topological charges and characterize directly the bulk topology by the total charges enclosed by BISs~\cite{Zhanglong2019a}, which further demonstrates the correspondence between the bulk and surface topological physics. In this case, instead of measuring all spin components after a single quench along $\gamma_{0}$, we perform a series of quantum quenches along different $\gamma_{i}$ ($i=0,1,2,3$) axes but measure only $\gamma_{0}$ component, i.e. $\overline{\langle\gamma_{0}(\mathbf{k})\rangle}_{i}$ after each quench~\cite{Zhanglong2019a,Zhanglong2019b}. To realize quenches in different axes $\gamma_{i}$, the quench process [see Fig.~\ref{fig:1}(c)] is modified to an appropriate combination of the nuclear and electron spin rotations, such that the resulting initial state is the eigenstate of the pre-quench Hamiltonian $\mathcal H_{\rm pre}=m_i \gamma_i + \mathcal H_{\rm 3D}$~\cite{Supp}. The BISs are again the collection of momenta on which time-averaged spin polarizations all vanish, namely $\mathrm{BIS}=\{\mathbf{k}\vert\overline{\left < \gamma_0( \textbf k ) \right >}_{i} =0,\forall i\}$, and the locations of topological charges are determined by $ \overline{\left < \gamma_0( \textbf k ) \right >}_{1,2,3} =0 $ while $\overline{\left < \gamma_0( \textbf k ) \right >}_0 \neq 0 $~\cite{Zhanglong2019a}. The topological charge is characterized by the dynamical field $\mathbf \Theta (\mathbf k)$ with components ($i=1,2,3$) \begin{align}
\Theta_i(\textbf{k})\equiv\frac{\mathrm{sgn}(h_0(\textbf{k}))}{\mathcal N_k} \overline{\left < \gamma_0( \textbf k ) \right >}_i. \end{align}
The norm of $\mathbf \Theta (\mathbf k)$ vanishes at a charge, and the charge value equals the winding of the dynamical field $\mathbf \Theta (\mathbf k)$ near the charge. In Fig.~\ref{fig:charge}(a), we measure the norm of $|\mathbf \Theta (\mathbf k)|$ on the $k_{z}=0,-\pi$ and $k_{x}=0,-\pi$ planes for the phase with $m_z=1.4t_{0}$ and $t_{\rm so}=t_{0}$, which shows eight topological charges $O_{i}$ in the bulk. The dynamical field $\mathbf \Theta (\mathbf k)$ near the charges is shown in Fig.~\ref{fig:charge}(b) and (c), showing that the topological charges $O_{1,3,5,7}$ have value $+1$ while the charges $O_{2,4,6,8}$ have value $-1$. The bulk topology is characterized by the total topological charge enclosed by the BIS [see Fig.~\ref{fig:charge}(a)], which is $O_1$, giving $\nu_{3}=+1$ for the bulk phase [Fig.~\ref{fig:1}(d)].
\begin{figure}
\caption{
(color online). \textbf{Identifying dynamical topological transition.}
(a) Experimental results of dynamical spin-texture field $\textbf f(\textbf k_0)$ projected onto $[\overline 1 \overline 1 \overline 1]$ direction versus quench field. Here $k_{0x}=k_{0y}=k_{0z} = -1.084$ is the momentum on BIS in the $[\overline 1 \overline 1 \overline 1]$ direction. The emergent topological transition point is around $m_c = 2.7 t_0$. Insets are $\textbf f(\textbf k)$ for a shallow quench (left inset, $m_i = 2 t_0$) and a deep quench (right inset, $m_i = \infty$). At $\textbf k_0$ point (blue circle) in the left inset, $\textbf f(\textbf k)$ points inward the BIS (highlighted in red).
(b) Locations of topological charges versus quench field. Dynamical field $\mathbf \Theta (\mathbf k)$ is measured along the line connecting charges $O_1$-$O_8$ (left) and $O_3$-$O_6$ (right), with the quench field $m_i$ being varied.
The charge $O_8$ cross the BIS at $m_c$ and merges with $O_1$.
}
\label{fig:charge_moving}
\end{figure}
An interesting observation of the dynamical characterization with topological charges is that an emergent dynamical topological transition occurs when the quenches are varied from deep to shallow regimes~\cite{Zhanglong2019b}. The deep (shallow) quench regime corresponds to large (small) $m_{i}$, and the initial state is fully (partially) polarized in the axis $\gamma_i$, independent of (dependent on) $\bold k$. For simplicity, in each set of quenches we take the same $m_{i}$ when quenching in different axes. The BISs are not affected by the quench depth $m_i$~\cite{Zhanglong2019b}. To characterize the emergent topological transition we measure the dynamical field $\mathbf{f}(\mathbf{k})$ on BISs, with components given by \begin{equation}
f_i( \textbf k ) = \frac{1}{\mathcal N_k} \partial_{k_\perp} \overline{\left < \gamma_0 \right >}_i\big\vert_{\scriptscriptstyle{\bf k}\in\rm BIS}, \end{equation} with $i=1,2,3$. For deep quenches, $\mathbf{f}(\mathbf{k})$ is equivalent to $\mathbf{g}(\textbf k)$, whose winding on BISs characterizes the post-quench topology [see the right insert in Fig.~\ref{fig:charge_moving}(a)]. In the left insert of Fig.~\ref{fig:charge_moving}(a), we show that the dynamical field $\mathbf{f}(\mathbf{k})$ for shallow quenches with $m_i=2$ is deformed and has zero winding, implying that between deep and shallow quenches an emergent topological transition occurs. To determine the critical quench depth $m_{c}$, we notice that the dynamical field changes most dramatically near the momentum $\mathbf{k}_{0}$ on the BIS in the $[\overline{111}]$ direction. In reducing $m_i$ across $m_c$, the direction of $\mathbf{f}(\mathbf{k}_0)$ changes from the outward to inward of BIS, and vanishes at $m_i=m_c$, where the winding on BIS is ill-defined. Fig.~\ref{fig:charge_moving}(a) displays the projection of $\mathbf{f}(\mathbf{k}_0)$ in the $[\overline{111}]$ direction. Our measurement determines the critical value $m_{c}\backsimeq2.7t_{0}$, which agrees on the theoretical prediction $m_c=2.653t_{0}$.
The dynamical topological transition corresponds to the movement of topological charges across BIS, as illustrated in Fig.~\ref{fig:charge_moving}(b). The dynamical field $\mathbf{\Theta}(\mathbf{k})$ is measured along the line connecting charges $O_1$-$O_8$ or $O_3$-$O_6$. We observe that the locations of charges depend on the quench depth $m_{i}$. Particularly, the charge $O_8$ passes through BIS when reducing $m_i$ across the critical value $m_{c}$. Then no topological charge is enclosed by the BIS, providing the alternative picture of the emergent topological transition. The topological charges $O_3$ and $O_6$ can also annihilate at certain $m_{i}$, but do not change the dynamical topology on the BIS.
{\em Conclusion.---}In summary, we have achieved by quantum quenches a complete study of bulk and surface topological physics for a novel 3D chiral topological insulator, using a quantum simulator built from solid-state spin system. We experimentally identified the dynamical bulk-surface correspondence in momentum space, as a momentum-space counterpart of the bulk-boundary correspondence in real space, which bridges the bulk topology of the 3D chiral phase and the nontrivial quench dynamics emerging on 2D band inversion surfaces. As the momentum space is more convenient to engineer for quantum simulators, the dynamical bulk-surface correspondence enables a complete study of the simulated topological phases, without the necessity of constructing real-space boundaries. The novel topological physics have been observed in experiment, including the chiral symmetry protection, the topological charges, and the dynamical topological transition emerging in the quench studies. The present work showed the insightful techniques of quantum simulation, which can be easily extended to other simulators, and opens a broad avenue to explore high dimensional topological phases beyond the limits of condensed matter physics.
\emph{Acknowledgement.-} This work is supported by the National Key R$\&$D Program of China (Grant No. 2018YFA0306600, 2017YFA0305000, 2016YFA0301604, 2016YFB0501603), the NNSFC (Grants No. 11775209, 11825401, 81788101, 11761161003, 11921005, 11761131011, 11722544), the CAS (Grants No. GJJSTD20170001, No. QYZDY-SSW-SLH004, No. QYZDB-SSW-SLH005), Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000), the Fundamental Research Funds for the Central Universities, the Thousand-Young-Talent Program of China.
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\section{\textbf{\large{Supplementary material}}}
\renewcommand{A\arabic{figure}}{A\arabic{figure}}
\setcounter{figure}{0}
\section{Generalized topological invariant}
As described in the main text, in the presence of the symmetry-breaking term $h_4 \gamma_4$, the generalized topological invariant is defined on a 3-sphere as \begin{equation} \mathcal{W}_{\rm SB}=\frac{1}{\pi^2} \int_{\mathcal{S}\vert_{\partial\mathcal{S}=\textbf g(\textbf k)}} \mathrm{d} S^3, \end{equation} where $\pi^2$ is the area of the half 3-sphere, $\mathrm{d} S^3$ is the area element of the 3-sphere, and the integral is taken over the region bounded by the curve $\partial\mathcal{S}=\textbf g(\textbf k)$.
When there is no symmetry-breaking term $\gamma_4$, the dynamical field $\mathbf g (\mathbf k)$ lies on the equator (a 2D object on $S^3$), and the integral $\int_{\mathcal{S}\vert_{\partial\mathcal{S}=\textbf g(\textbf k)}} \mathrm{d} S^3$ is taken over the upper-half sphere and equals $n \pi^2$ if the dynamical field $\mathbf g (\mathbf k)$ winds the equator $n$ times, giving the bulk topological invariant $\mathcal W = n$. When the symmetry is broken, the dynamical field $\mathbf g (\mathbf k)$ deviates from the equator, the integral $\int_{\mathcal{S}\vert_{\partial\mathcal{S}=\textbf g(\textbf k)}} \mathrm{d} S^3,$ generally is a fraction multiplying $\pi^2$, then $\mathcal W$ is a non-integer number as shown below.
In the sphere coordinates, we have \begin{equation} \mathrm{d} S^3 = \sin^2 \phi_3 \sin \phi_2 \mathrm{d} \phi_1 \mathrm{d} \phi_2 \mathrm{d} \phi_3, \end{equation} where \begin{equation} \begin{aligned} g_4 & = \cos \phi_3,\\ g_3 & = \sin \phi_3 \cos \phi_2,\\ g_2 & = \sin \phi_3 \sin \phi_2 \cos \phi_1,\\ g_1 & = \sin \phi_3 \sin \phi_2 \sin \phi_1 \end{aligned} \end{equation} with $0<\phi_1<2\pi$ and $0<\phi_{2,3}<\pi$. Inversely, we have \begin{equation} \begin{aligned} \tan \phi_1 & = g_1/g_2,\\ \tan \phi_2 & = \sqrt{g_1^2+g_2^2}/g_3,\\ \tan \phi_3 & = \sqrt{g_1^2+g_2^2+g_3^2}/g_4. \end{aligned} \end{equation} If the symmetry is not broken, then $g_4=0$ and $\phi_3=\pi/2$, the dynamical field lies on the equator of the 3-sphere. Here we consider the case where the dynamical field $\mathbf g (\mathbf k)$ winds the equator $n$ times, i.e., the bulk topological invariant is $n$, then we have \begin{equation} \begin{aligned} \mathcal W_{\rm SB} & = \frac{1}{\pi^2} \int_{ \mathcal{S} \vert _{\partial \mathcal{S} = \textbf g(\textbf k)}} \mathrm{d} S^3 \\ & = \frac{1}{\pi^2} \int_{0}^{2\pi n} \mathrm{d} \phi_1 \int_{0}^{\pi} \mathrm{d} \phi_2 \int_{0}^{\pi/2} \mathrm{d} \phi_3 \sin^2 \phi_3 \sin \phi_2 \\ & = n, \end{aligned} \end{equation}
Now we add the symmetry-breaking term, and we consider the following simple example, $g_4(\textbf k)=m$, where $|m|<1$ is a constant. Then the quantity $\mathcal W_{\rm SB}$ becomes \begin{equation}\label{eq:winding} \begin{aligned} \mathcal W_{\rm SB} & = \frac{1}{\pi^2} \int_{0}^{2\pi n} \mathrm{d} \phi_1 \int_{0}^{\pi} \mathrm{d} \phi_2 \int_{0}^{\arccos m} \mathrm{d} \phi_3 \sin^2 \phi_3 \sin \phi_2 \\ & = \frac{2n}{\pi}(\arccos m - m\sqrt{1-m^2}), \end{aligned} \end{equation} which is not an integer.
\begin{figure}
\caption{
\textbf{Illustration for the quantity $\mathcal W$ proportional to the area of the shadow region bounded by the dynamical field $\mathbf g (\mathbf k)$.}
(a) is the case with the chiral symmetry, the dynamical field $\mathbf g (\mathbf k)$ lies on the equator.
(b) is the case with the symmetry broken by the $\gamma_4$ term, and the dynamical field deviates from the equator. Note that the equator is now a 2D manifold on the 3-sphere.
}
\label{fig:winding}
\end{figure}
\section{Experimental simulation of the 3D model}
\subsection{Post-quench dynamics} \label{sec:sys.ham}
The target Hamiltonian $\mathcal H_{\rm 3D}$ can be written in a general form \begin{equation} \mathcal H_{\rm 3D}(\textbf k) = h_0 \sigma_z \tau_z + h_1 \sigma_x + h_2 \sigma_y + h_3 \sigma_z \tau_x. \end{equation} We realize this Hamiltonian with a diamond nitrogen-vacancy (NV) center system, whose Hamiltonian is \begin{equation} \mathcal H_{NV} = 2\pi (D S_z^2 + \omega_e S_z + Q I_z^2 + \omega_n I_z + A S_z I_z), \end{equation}
where $S_z$ ($I_z$) is the electron (nuclear) spin operator. A subspace of $\{m_s=0,-1\} \otimes \{m_i=+1,0\}$ is utilized to form a two-qubits system, which is relabeled as $\{\left|0\right> ,\left|1\right>\} \otimes \{\left|0\right> ,\left|1\right>\}$. The first (second) qubit corresponds to the Pauli operator $\sigma_i$ ($\tau_i$) in $\mathcal H_{\rm 3D}$. The subspace Hamiltonian can be rewritten as \begin{equation} H_0 = 2\pi \left( \begin{array}{cccc} \omega_1 & 0 & 0 & 0 \\ 0 & \omega_2 & 0 & 0 \\ 0 & 0 & \omega_3 & 0 \\ 0 & 0 & 0 & \omega_4 \end{array} \right), \end{equation} where $\omega_1=Q+\omega_n$, $\omega_2=0$, $\omega_3=D-\omega_e+Q+\omega_n-A$ and $\omega_4=D-\omega_e$.
In order to introduce $\sigma_x$ and $\sigma_y$ terms in $\mathcal H_{\rm 3D}$, we apply a microwave pulse of frequency $\omega_{mw}=(\omega_3-\omega_1+\omega_4-\omega_2)/2=D-\omega_e-A/2$, coupling both $\left|00\right> \leftrightarrow \left|10\right>$ and $\left|01\right> \leftrightarrow \left|11\right>$ transitions. The interaction Hamiltonian reads \begin{equation} V_{mw} = 2\pi \Omega_{mw} \cos(\omega_{mw} t+\phi) \left( \begin{array} {cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right)+h.c. \end{equation} After transforming the total Hamiltonian $H_0+V_{mw}$ to the rotating frame defined by the MW field, and applying proper rotating-wave approximation, the system Hamiltonian reads \begin{equation} \begin{aligned} \mathcal H_{mw,\rm RWA} &= 2\pi \left( \begin{array} {cccc} A/4 & 0 & \Omega_x-\mathrm i \Omega_y & 0 \\ 0 & -A/4 & 0 & \Omega_x-\mathrm i \Omega_y \\ \Omega_x+\mathrm i \Omega_y & 0 & -A/4 & 0 \\ 0 & \Omega_x+\mathrm i \Omega_y & 0 & A/4 \end{array} \right)\\ &=2\pi\left( \frac{A}{4}\sigma_z\tau_z +\Omega_x\sigma_x +\Omega_y\sigma_y \right), \end{aligned} \end{equation} where $\Omega_x=\Omega_{mw} \cos \phi$ and $\Omega_y=-\Omega_{mw} \sin \phi$. $\mathcal H_{mw,\rm RWA}$ is only a single $\tau_y$ rotation away from the target Hamiltonian form $\mathcal H_{\rm 3D}=h_0 \sigma_z \tau_z + h_1 \sigma_x + h_2 \sigma_y + h_3 \sigma_z \tau_x$. After applying the rotation $U_{\rm rot}=\exp(-\mathrm i \theta \tau_y)$ to the system Hamiltonian, we have our effective Hamiltonian \begin{equation} \mathcal H_{\rm eff}=2\pi\left(\frac{A}{4}\cos \theta\ \sigma_z\tau_z +\Omega_x\sigma_x +\Omega_y\sigma_y +\frac{A}{4}\sin \theta\ \sigma_z\tau_x\right). \end{equation} We can imply that \begin{equation}\label{eq:map1} \theta=\arctan(h_3/h_0). \end{equation} Note that $\mathcal H_{\rm eff}$ is subject to the limitation of having $h_0^2+h_3^2=\pi^2 A^2 /4$. In order to simulate $\mathcal H_{\rm 3D}$ with any $h_i$, we here define the effective time as a rescale of the simulation time $t$, i.e., $ t_{\rm eff} = \alpha t$. We only need to reproduce the same effect as $U_{3D}=\exp( -\mathrm i \mathcal H_{\rm 3D} t )$ with the simulated evolution $U_{\rm eff}=\exp( -\mathrm i \mathcal H_{\rm eff} t_{\rm eff} )$. Hence we have $\mathcal H_{\rm 3D} = \alpha \mathcal H_{\rm eff} $, giving \begin{equation}\label{eq:map2} \begin{aligned}
\alpha & =\frac{2}{\pi|A|} \sqrt{h_0^2+h_3^2}, \\ \Omega_{mw} & =\frac{1}{2\pi \alpha} \sqrt{h_1^2+h_2^2}, \\ \phi & =-\arctan(h_2/h_1). \end{aligned} \end{equation}
The corresponding experimental circuit of this post-quench evolution is depicted in Fig.~\ref{fig:evol}. We first rotate the nuclear spin along $-y$ axis for an angle $\theta$. Then the microwave with a driving strength of $\Omega_{mw}$ and a phase of $\phi$, for a time duration of $ t_{\rm eff}$ is applied. Finally we rotate back the nuclear spin along $y$ axis for the same angle $\theta$. The net effect of this whole process is identical to the evolution of the system under $\mathcal H_{\rm 3D}$ during an evolution time of $ t $.
For the symmetry breaking case with the additional $\gamma_4$ term, the rotation operation $U_{\rm rot}$ is modified to $U_{\rm rot,SB}=\exp [-\mathrm i \theta (\cos \phi_{\rm SB} \tau_y - \sin \phi_{\rm SB} \tau_x)] $, which will give the post-rotation Hamiltonian as \begin{equation} \begin{aligned} \mathcal H_{\rm sim,SB}= & 2\pi\left( \frac{A}{4}\cos \theta\ \sigma_z\tau_z +\Omega_x\sigma_x -\Omega_y\sigma_y \right. \\ &\hspace{0.35cm} \left. +\frac{A}{4}\sin \theta \cos \phi_{\rm SB}\ \sigma_z\tau_x +\frac{A}{4}\sin \theta \sin \phi_{\rm SB}\ \sigma_z\tau_y \right). \end{aligned} \end{equation} This modified rotation is realized by setting the phase of the RF pulse to \begin{equation} \phi_{\rm SB}=\arctan(h_4/h_3). \end{equation} $\theta$ and $\phi$ are also modified by substituting $h_3$ with $\sqrt{h_3^2+h_4^2}$, giving \begin{equation} \begin{aligned} \theta & =\arctan(\sqrt{h_3^2+h_4^2}/h_0),\\
\alpha & =\frac{2}{\pi|A|} \sqrt{h_0^2+h_3^2+h_4^2}. \end{aligned} \end{equation} Note that when setting $h_4=0$, we have $\phi_{\rm SB}=\pi$, $U_{\rm rot,SB}$, $\theta$ and $\alpha$ reduce to the non-symmetry-breaking case.
\begin{figure}
\caption{
\textbf{Post-quench dynamics.}
$R^\theta_{\pm Y}$ represents rotation of nuclear spin about axis $\pm y$ for an angle $\theta$.
The operation in the middle is evolution of the system under the microwave driving $\Omega_{mw}$, for a time duration of $ t_{\rm eff}$.
The net effect is equivalent to evolution under $\mathcal H_{\rm 3D}$ for a duration of $ t $.
}
\label{fig:evol}
\end{figure}
\subsection{Deep and shallow quench process} \label{sec:quench}
In the experiment the system is initially polarized by a green laser pulse to the state $\left| 00 \right>$, which is an eigenstate of $\gamma_0 =\sigma_z \tau_z$. This is equivalent to a deep quench along $\gamma_0$. The deep quenching along other axis is realized by either applying a microwave or radio-frequency pulse to prepare the system onto the eigenstate of $\gamma_{1,2,3}$.
For the shallow quench process, one need to initialize the state to the eigenstate of the quench Hamiltonian $\mathcal H_{\rm pre}=m_i \gamma_j + \mathcal H_{\rm 3D}$, with a finite quench field $m_i$ along quench axis $\gamma_j$. In general, the quench Hamiltonian can be rewritten as \begin{equation} \mathcal H_{\rm pre} = h_{\rm pre,0} \sigma_z \tau_z + h_{\rm pre,1} \sigma_x + h_{\rm pre,2} \sigma_y + h_{\rm pre,3} \sigma_z \tau_x. \end{equation} which denpends on both $m_i$ and $\textbf k$, and is no longer aligned with any of the $\gamma_i$ axes. To prepare an eigenstate of $\mathcal H_{\rm pre}$, we consider a rotation of electron spin \begin{equation} \begin{aligned}[t]
U_{\rm init,mw} &= \exp \left( -\mathrm i \frac{\theta_{\rm init,mw}}{2} (-\sin \phi_{\rm init,mw} \sigma_x +\cos \phi_{\rm init,mw} \sigma_y) \right),\\
\theta_{\rm init,mw} &= \arctan \left( \sqrt{h_{\rm pre,1}^2+h_{\rm pre,2}^2}/\sqrt{h_{\rm pre,0}^2+h_{\rm pre,3}^2} \right),\\
\phi_{\rm init,mw} &= \arctan \left( h_{\rm pre,2}/h_{\rm pre,1} \right). \end{aligned} \end{equation}
After the rotation, the state $U_{\rm init,mw} \left| 00 \right>$ becomes an eigenstate of $\sqrt{ h_{\rm pre,0}^2 + h_{\rm pre,3}^2 } \sigma_z + h_{\rm pre,1} \sigma_x + h_{\rm pre,2} \sigma_y$. Due to the fact that $U_{\rm init,mw}$ operates only on electron spin, it commutes with $\tau_z$. As a result, $U_{\rm init,mw} \left| 00 \right>$ is also an eigenstate of $\sqrt{ h_{\rm pre,0}^2 + h_{\rm pre,3}^2 } \sigma_z \tau_z + h_{\rm pre,1} \sigma_x + h_{\rm pre,2} \sigma_y$. We can further rotate the nuclear spin as \begin{equation} \begin{aligned}[t]
U_{\rm init,rf} &= \exp \left( -\mathrm i \frac{\theta_{\rm init,rf}}{2} \tau_y \right),\\
\theta_{\rm init,rf} &= \arctan \left( h_{\rm pre,3} / h_{\rm pre,0} \right).\\ \end{aligned} \end{equation} With this rotation, we end up with an eigenstate of $\mathcal H_{\rm pre}$. Note that $U_{\rm init,rf}$ commutes with $U_{\rm init,mw}$, the order of these two operations can be switched in the experiment.
\subsection{Readout time-averaged spin polarization}
\begin{figure}
\caption{
\textbf{Sequences to measure populations.}
The operation labeled $\gamma_i$ (yellow) corresponds to quench along $\gamma_i$ axis, which requires an operation to transform the state $\left| 00 \right>$ to an eigenstate of the quench Hamiltonian $\mathcal H_{\rm pre}$.
The operation $\mathcal H_{\rm 3D}$ (blue) corresponds to the post-quench dynamics as depicted in Fig.~\ref{fig:evol}.
The operation $\gamma_j$ (red) corresponds to readout $\gamma_j$ component, which requires an operation to transform $\gamma_j$ to the $z$ basis.
PL. (green) corresponds to a photoluminescence measurement, which is realized by applying a 532nm laser and counting the emitted photons.
Idle corresponds to a waiting time equal to the total time of the $\gamma_i$, $\mathcal H_{\rm 3D}$ and $\gamma_j$ steps.
}
\label{fig:tomo}
\end{figure}
\begin{figure}
\caption{
\textbf{A typical measurement result.}
(a) Photon number per readout for different readout sequences as shown in Fig.~\ref{fig:tomo} at different time. Error bars are estimated by photon shot noise.
(b) Population and spin polarization calculated from results in (a). The time-averaged spin polarization calculated from this result is $ \overline{\left < \gamma_0 \right >}_0 = 0.423 \pm 0.056 $, comparing to the theory value of 0.460.
For both (a) and (b), we set $m_z=1.4 t_0$, $t_{so}=0.2 t_0$, and the momentum point is $(k_x,k_y,k_z)=(0.1\pi,0.6\pi,0.1\pi)$, with deep quench along $\gamma_0$.
}
\label{fig:tomo_demo}
\end{figure}
The spin polarizations $\gamma_i$ of a given final state is measured by transforming the interested component to the $z$ basis of electron and nuclear spins, followed by a population measurement($P_{\left| i,j \right>}(i,j=0,1)$) through the optical readout. For the $\gamma_0=\sigma_z \tau_z$ readout, the spin polarization is essentially $P_{\left| 00 \right>} -P_{\left| 01 \right>} -P_{\left| 10 \right>} +P_{\left| 11 \right>}$, which is already in $z$ basis. For the case of $\gamma_{1,2} = \sigma_{x,y} \otimes \textbf 1$, a $\pi/2$ rotation on the electron spin about $-y$ or $x$ axis will map the $\sigma_{x,y}$ components to $\sigma_z$, of which the spin polarization is given by $P_{\left| 00 \right>} +P_{\left| 01 \right>} -P_{\left| 10 \right>} -P_{\left| 11 \right>}$. Similarly, for the $\gamma_{3,4} = \sigma_z \tau_{x,y}$ readout, a $\pi/2$ rotation on the nuclear spin about $-y$ or $x$ axis will transform the $\gamma_{3,4}$ readout to a $\gamma_0$ readout. These operations are depicted in Fig.~1(c) of the main text.
For the populations readout, the photoluminescence (PL) photon count of the spin state is recorded. Since the total PL count is the average of all four levels weighted by their populations, i.e., $N_{total}= N_1 P_{\left| 00 \right>} +N_2 P_{\left| 01 \right>} +N_3 P_{\left| 10 \right>} +N_4 P_{\left| 11 \right>}$, we apply RF and MW pulses in different ways to produce different linear combinations of the populations, and then combine all the equations to solve for the populations. The sequences are depicted in Fig.~\ref{fig:tomo}, and the system of equations for the populations is \begin{equation}
\left( \begin{array} {cccc}
N_1 & N_2 & N_3 & N_4 \\
N_3 & N_4 & N_1 & N_2 \\
N_2 & N_1 & N_3 & N_4 \\
N_3 & N_4 & N_2 & N_1
\end{array} \right)
\cdot \left( \begin{array} {c}
P_{\left| 00 \right>} \\
P_{\left| 01 \right>} \\
P_{\left| 10 \right>} \\
P_{\left| 11 \right>}
\end{array} \right)
= \left( \begin{array} {l}
N_f^0 \\
N_f^{mw} \\
N_f^{rf0} \\
N_f^{mw,rf0}
\end{array} \right). \end{equation} Note that the $N_{1,2,3,4}$ also need to be determined. The sequences are also depicted in Fig.~\ref{fig:tomo}.
The time-averaged spin polarization $\overline{ \left< \gamma_i( \textbf{k} ) \right> }$ is obtained by measuring and averaging spin polarization over a series of time. In order to maintain consistency, the time steps are chosen in such a way that the corresponding simulation time $ t $ are the same in all comparable measurements. Note that since the effective time $ t_{\rm eff}$ also depends on $\textbf{k}$, the same $ t $ may correspond to different $ t_{\rm eff}$. For experiments in which the effect of dephasing is ignored, the time range of $ t $ is chosen from 0 to \begin{equation}
t_{\rm max}=\frac{2}{\sqrt 3 \ t_{\rm so} \sin{ (\arccos{ (3 m_z/t_0) } )}}.\\ \end{equation}\label{eq:t_eff_max} For the experiments with dephasing, the time range is chosen from $2 t_{\rm max}$ to $3 t_{\rm max}$. A typical experimental result is shown in Fig.~\ref{fig:tomo_demo}, which corresponds to $m_z=1.4 t_0$, $t_{so}=0.2 t_0$, $(k_x,k_y,k_z)=(0.1,0.6,0.1)$, quenching $\gamma_0$ and measuring $\gamma_0$. The result correspond to a time-averaged spin polarization $ \overline{\left < \gamma_0 \right >}_0 $ of $ 0.423 \pm 0.056 $, and theory value is $0.460$.
\section{Data processing method}
\subsection{Reconstruction of the BIS}
\begin{figure}
\caption{
\textbf{Reconstruction of BIS.}
(a) The raw experimental results of $ \overline{\left < \gamma_0 \right >}_0 $ within the first octant. Note that, for clarity, only the $k_x=0$, $k_y=0$ and $k_z=0$ planes are plotted.
(b) The smoothed and interpolated $ \overline{\left < \gamma_0 \right >}_0 $ field and the initial triangular mesh.
(c,d,e) 2nd, 3rd and 5th iteration of triangular mesh.
(f) Flip the mesh in (e) and combine them to form the full BIS.
}
\label{fig:bis}
\end{figure}
To obtain the BIS, we quench along $\gamma_0$ and measure $\overline{ \left< \gamma_0 \right> }$ at different $\textbf k$. Since the Hamiltonian is symmetric under $k_x$-, $k_y$-, and $k_z$-reflections, the $\overline{ \left< \gamma_0( \textbf{k} ) \right> }_0$ result in the first octant of the Brillouin zone is sufficient to reconstruct the BIS. We measure $\overline{ \left< \gamma_0( \textbf{k} ) \right> }_0$ in a mesh grid with a $0.1\pi$ step size for $k_x$, $k_y$ and $k_z$, which is sufficient to reconstruct the BIS well in our interested case. As shown in Fig.~\ref{fig:bis}, the whole reconstruction process is based on data smoothing and iteratively interpolating a triangular mesh. Specifically, we first define an initial triangle, as a coarse representation of BIS, by finding the minimum of the smoothed $\overline{ \left< \gamma_0 \right> }_0$ field along all three axes. For each edge of the old mesh, we locate its center, and find the minimum of the $\overline{ \left< \gamma_0 \right> }_0$ field along the norm line of the old face at that location to define a new vertex. Combining the new vertices with the old ones, we can obtain a refined mesh and describe the BIS more accurately. With repeating of this process, we can reconstruct the BIS mesh in the first octant form the measurement result to any demanding accuracy. Finally, we flip the BIS mesh to other octants and combine them to obtain the full BIS mesh in the Brillouin zone.
\subsection{Measurement of g field winding number}
\begin{figure}\label{fig:gvec}
\end{figure}
Based on the previously obtained BIS mesh, we measure the emergent dynamical spin-texture field, i.e., the $\textbf g( \textbf k )$ field, of which the components are defined as \begin{equation}
g_i(\textbf k) = \frac{1}{\mathcal N_k} \partial_{\textbf k_\perp} \overline{ \left< \gamma_i( \textbf{k} ) \right> }. \end{equation} We measure the $\textbf g( \textbf k )$ field by sampling 6 points across the BIS, along the norm direction, with a step size of $0.02\pi$, and measuring the time-averaged spin polarization $\overline{ \left< \gamma_{1,2,3} \right> }_0$. The slopes fitted from the results, after normalization, give the $\textbf g( \textbf k )$ field. A typical experimental result is depicted in Fig.~\ref{fig:gvec}.
The integral calculating the winding number of the $\textbf g( \textbf k )$ field is discretized as a summation over all the triangular meshes, i.e., \begin{align} \mathcal{W} = \frac{1}{8\pi}\int_{\rm BIS}\mathrm{d}^{2}\mathbf{k}\,\mathbf{g}\cdot(\nabla\mathbf{g}\times\nabla\mathbf{g}) = \frac{1}{4\pi} \sum_i S_i, \end{align} where $i$ is the label of triangular element and $S_i$ is the solid angle formed by the three $\textbf g$ vectors on the vertices of the $i$-th triangular element. The solid angle is calculated by the sum of the three internal sphere angles subtracted by $\pi $.
For the symmetry-breaking case, the $g_4$ term is approximated by a constant within the same triangular element. From Eq.~\ref{eq:winding} we have \begin{equation} \begin{aligned} \mathcal W_{\rm SB} & = \frac{1}{\pi^2} \sum_{i} \int_{\mathcal S_i} \mathrm{d} S^2 \int_{0}^{\phi_{3,i}} \mathrm{d} \phi_3 \sin^2 \phi_3\\ & = \frac{1}{\pi^2} \sum_{i} S_i \frac{\phi_{3,i} - \sin \phi_{3,i} \cos \phi_{3,i}}{2} . \end{aligned} \end{equation} where $\mathrm{d} S^2$ is the area element of the 2-sphere, and the integral is taken within each triangular mesh, giving the solid angle formed by $(g_1,g_2,g_3)$. The $\mathcal W_{\rm SB}$ is calculated in the same way as $\mathcal W$, with each element multiplied by a factor depending on $g_4$.
\subsection{Error analysis}
The dominant error in our experiment comes from the shot noise in the the optical readout, which yields a normal distribution of the photon counts with a mean of $N$ and a standard deviation of $\sqrt{N}$, where $N$ is between 1000 to 2000 in the experiment for a fixed 10,000 repetitions of each sequence. This random distribution then introduces an uncertainty in obtaining the dynamical spin-texture field.
To estimate the error associated with a quantity, e.g. the winding number, we adopt the Monte Carlo method. First, we randomly generate photon counts of the same distribution with the measurements. Then we feed the generated counts to the algorithm for calculating the winding number. This process is repeated sufficient times, and we take the standard deviation of the results as the error of the winding number.
\section{Results with spin dephasing} \label{sec:deco}
\begin{figure}
\caption{
\textbf{Dephasing in the evolution.}
The results shown correspond to $m_z=1.4 t_0$, $t_{so}=t_0$, $ k_x = k_y = k_z = -0.6 \pi $, and deep quench along $\gamma_0$.
The yellow (red) area denotes the time range from $0$ to $t_{\rm max}$ (from $2 t_{\rm max}$ to $3 t_{\rm max}$).
}
\label{fig:deco_demo}
\end{figure}
\begin{figure}
\caption{
\textbf{Measurement of charge movement with dephasing.}
Repeat the measurement in main text Fig. 4(b), with the time range chosen from $2 t_{\rm max}$ to $3 t_{\rm max}$ to introduce dephasing. The results are almost identical, showing the robustness of the dynamical characterization methods against dephasing.
}
\label{fig:deco}
\end{figure}
We investigate the effect of dephasing in the simulation by extending the evolution time. All the previous time-averaged spin polarization measurements are averaged with evolution time $ t $ from $0$ to $t_{\rm max}$ as determined by Eq.~\ref{eq:t_eff_max}), while the results with dephasing are averaged with $ t $ from $2 t_{\rm max}$ to $3 t_{\rm max}$. As an example, we choose $m_z=1.4 t_0$, $t_{so}=t_0$, $ k_x = k_y = k_z = -0.6 \pi $, and deep quench along $\gamma_0$. $ \left < \gamma_{0,1,2,3,4} \right >_0 $ are measured for a series of evolution time, from $0$ to $3 t_{\rm max}$. The experimental results are shown in Fig.~\ref{fig:deco_demo}, with yellow (red) area denoting the time range from $0$ to $t_{\rm max}$ (from $2 t_{\rm max}$ to $3 t_{\rm max}$). One can easily see that although the amplitude of the oscillation damps, the average value maintains the same, which means that averaging over time range with or without dephasing will give the same result. The $ \left < \gamma_{1,2} \right >_0 $ results decay fast, while the $ \left < \gamma_{3,4} \right >_0 $ results decay only negligibly. This is due to the fact that $ \gamma_{1,2} $ correspond to $\sigma_{x,y}$, which depend on the short electron spin coherence time, while $ \gamma_{3,4} $ depend on the nuclear spin, of which the coherence time is much longer than this time scale.
We further demonstrate this robustness against dephasing by repeating the measurement in main text Fig. 4(b), with time range chosen from $2 t_{\rm max}$ to $3 t_{\rm max}$. The comparison of results with and without dephasing is shown in Fig.~\ref{fig:deco}.
\end{document} |
\begin{document}
\title{Derived non-archimedean analytic spaces}
\author{Mauro PORTA} \address{Mauro PORTA, University of Pennsylvania, David Rittenhouse Laboratory, 209 South 33rd Street, Philadelphia, PA 19104, United States} \email{maurop@math.upenn.edu}
\author{Tony Yue YU} \address{Tony Yue YU, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France} \email{yuyuetony@gmail.com} \date{January 5, 2016 (Revised on December 30, 2016)} \subjclass[2010]{Primary 14G22; Secondary 14A20, 18B25, 18F99} \keywords{derived geometry, rigid analytic geometry, non-archimedean geometry, Berkovich space, analytic stack, higher stack, pregeometry, structured topos}
\begin{abstract}
We propose a derived version of non-archimedean analytic geometry.
Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings.
Such a naive definition turns out to be insufficient.
In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie.
We prove the following three fundamental properties of derived non-archimedean analytic spaces:
(1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the $\infty$-category of derived non-archimedean analytic spaces.
(2) The $\infty$-category of derived non-archimedean analytic spaces admits fiber products.
(3) The $\infty$-category of higher non-archimedean analytic Deligne-Mumford stacks embeds fully faithfully into the $\infty$-category of derived non-archimedean analytic spaces.
The essential image of this embedding is spanned by $n$-localic discrete derived non-archimedean analytic spaces.
We will further develop the theory of derived non-archimedean analytic geometry in our subsequent works.
Our motivations mainly come from intersection theory, enumerative geometry and mirror symmetry. \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
\paragraph{\textbf{Motivations}} Derived algebraic geometry is a far-reaching enhancement of classical algebraic geometry. We refer to Toën-Vezzosi \cite{HAG-I,HAG-II} and Lurie \cite{Lurie_Thesis,DAG-V} for foundational works. The prototypical idea of derived algebraic geometry originated from intersection theory: Let $X$ be a smooth complex projective variety. Let $Y, Z$ be two smooth closed subvarieties of complementary dimension. We want to compute their intersection number. When $Y$ and $Z$ intersect transversally, it suffices to count the number of points in the set-theoretic intersection $Y\cap Z$. When $Y$ and $Z$ intersect non-transversally, we have two solutions: The first solution is to perturb $Y$ and $Z$ into transverse intersection; the second solution is to compute the Euler characteristic of the derived tensor product $\mathcal O_Y\otimes^\mathrm L_{\mathcal O_X}\mathcal O_Z$ of the structure sheaves. The second solution can be seen as doing perturbation in a more conceptual and algebraic way. It suggests us to consider spaces with a structure sheaf of derived rings instead of ordinary rings. This is one main idea of derived algebraic geometry.
Besides intersection theory, motivations for derived algebraic geometry also come from deformation theory, cotangent complexes, moduli problems, virtual fundamental classes, homotopy theory, etc.\ (see Toën \cite{Toen_Derived_2014} for an excellent introduction). All these motivations apply not only to algebraic geometry, but also to analytic geometry. Therefore, a theory of derived analytic geometry is as desirable as derived algebraic geometry.
The \emph{purpose} of this paper is to define a notion of derived space in non-archimedean analytic geometry and then study their basic properties. A non-archimedean field is a field with a complete nontrivial non-archimedean absolute value. By non-archimedean analytic geometry, we mean the theory of analytic geometry over a non-archimedean field $k$, initiated by Tate \cite{Tate_Rigid_1971}, then systematically developed by Raynaud \cite{Raynaud_Geometrie_analytique_rigide_1974}, Berkovich \cite{Berkovich_Spectral_1990,Berkovich_Etale_1993}, Huber \cite{Huber_Generalization_1994,Huber_Etale_1996} and other mathematicians with different levels of generalizations. The survey \cite{Conrad_Several_approaches_2008} by Conrad gives a friendly overview of the subject. We will restrict to the category of quasi-paracompact\footnote{A rigid $k$-analytic\xspace space is called quasi-paracompact if it has an admissible affinoid covering of finite type.} quasi-separated rigid $k$-analytic spaces, which is the common intersection of the various approaches to non-archimedean analytic geometry mentioned above. For readers more familiar with Berkovich spaces, we remark that this category is equivalent to the category of paracompact strictly $k$-analytic spaces in the sense of Berkovich (cf.\ \cite[\S 1.6]{Berkovich_Etale_1993}).
A more direct motivation of our study comes from mirror symmetry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds (cf.\ \cite{Yau_Essays_1992,Voisin_Symetrie_1996,Cox_Mirror_symmetry_1999,Hori_Mirror_symmetry_2003}). More precisely, mirror symmetry concerns degenerating families of Calabi-Yau manifolds instead of individual manifolds. An algebraic family of Calabi-Yau manifolds over a punctured disc gives rise naturally to a non-archimedean analytic space over the field $\mathbb C(\!( t)\!)$ of formal Laurent series. In \cite[\S 3.3]{Kontsevich_Homological_2001}, Kontsevich and Soibelman suggested that the theory of Berkovich spaces may shed new light on the study of mirror symmetry. Progresses along this direction are made by Kontsevich-Soibelman \cite{Kontsevich_Affine_2006} and by Tony Yue Yu \cite{Yu_Balancing_2013,Yu_Gromov_2014,Yu_Tropicalization_2014,Yu_Enumeration_cylinders_2015,Yu_Enumeration_cylinders_II_2016}. The works by Gross, Hacking, Keel, Siebert \cite{Gross_Real_Affine_2011,Gross_Mirror_Log_published,Gross_Tropical_2011} are in the same spirit.
More specifically, in \cite{Yu_Enumeration_cylinders_2015}, a new geometric invariant is constructed for log Calabi-Yau surfaces, via the enumeration of holomorphic cylinders in non-archimedean geometry. These invariants are essential to the reconstruction problem in mirror symmetry. In order to go beyond the case of log Calabi-Yau surfaces, a general theory of virtual fundamental classes in non-archimedean geometry must be developed. The situation here resembles very much the intersection theory discussed above, because moduli spaces in enumerative geometry can often be represented locally as intersections of smooth subvarieties in smooth ambient spaces. The virtual fundamental class is then supposed to be the set-theoretic intersection after perturbation into transverse situations. However, perturbations do not necessarily exist in analytic geometry. Consequently, we need a more general and more robust way of constructing the virtual fundamental class, whence the need for derived non-archimedean geometry.
\paragraph{\textbf{Basic ideas and main results}}
Our previous discussion on intersection numbers suggests the following definition of a derived scheme:
\begin{defin}[cf.\ \cite{Toen_Derived_2014}]\label{def:derived_scheme} A \emph{derived scheme} is a pair $(X,\mathcal O_X)$ consisting of a topological space $X$ and a sheaf $\mathcal O_X$ of commutative simplicial rings on $X$, satisfying the following conditions: \begin{enumerate}[(i)] \item The ringed space $(X,\pi_0(\mathcal O_X))$ is a scheme. \item For each $j\ge 0$, the sheaf $\pi_j(\mathcal O_X)$ is a quasi-coherent sheaf of $\pi_0(\mathcal O_X)$-modules. \end{enumerate} \end{defin}
In order to adapt \cref{def:derived_scheme} to analytic geometry, we need to impose certain analytic structures on the sheaf $\mathcal O_X$. For example, we would like to have a notion of norm on the sections of $\mathcal O_X$; moreover, we would like to be able to compose the sections of $\mathcal O_X$ with convergent power series. A practical way to organize such analytic structures is to use the notions of pregeometry and structured topos introduced by Lurie \cite{DAG-V}. We will review these notions in \cref{sec:definitions} (see also the introduction of \cite{Porta_DCAGI} for an expository account of these ideas).
We will define a pregeometry $\cT_{\mathrm{an}}(k)$ which will help us encode the theory of non-archimedean geometry responsible for our purposes.
After that, we are able to introduce our main object of study: derived $k$-analytic\xspace spaces. It is a pair $(\mathcal X,\mathcal O_\mathcal X)$ consisting of an $\infty$-topos\xspace $\mathcal X$ and a $\cT_{\mathrm{an}}(k)$-structure $\mathcal O_\mathcal X$, satisfying analogs of \cref{def:derived_scheme} Conditions (i)-(ii). We will explain more intuitions in \cref{rem:definition_intuition}.
The goal of this paper is to study the basic properties of derived $k$-analytic\xspace spaces and to compare them with ordinary $k$-analytic\xspace spaces. Here are our main results:
\begin{thm}[cf.\ \cref{thm:fully_faithfulness}] The category of quasi-paracompact quasi-separated rigid $k$-analytic\xspace spaces embeds fully faithfully into the $\infty$-category\xspace of derived $k$-analytic\xspace spaces. \end{thm}
\begin{thm}[cf.\ \cref{thm:fiber_products}] The $\infty$-category\xspace of derived $k$-analytic\xspace spaces admits fiber products. \end{thm}
Let $(\mathrm{An}_k,\tau_\mathrm{\acute{e}t})$ denote the étale site of rigid $k$-analytic\xspace spaces (cf.\ \cite[\S 8.2]{Fresnel_Rigid_2004}) and let $\mathbf P_\mathrm{\acute{e}t}$ denote the class of étale morphisms. The triple $(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ constitutes a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}. The associated geometric stacks are called \emph{higher $k$-analytic\xspace Deligne-Mumford stacks}.
\begin{thm}[cf.\ \cref{cor:underived_higher_kanal_stacks}]
The $\infty$-category\xspace of higher $k$-analytic\xspace Deligne-Mumford stacks embeds fully faithfully into the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
The essential image of this embedding is spanned by $n$-localic discrete derived $k$-analytic\xspace spaces. \end{thm}
\paragraph{\textbf{Outline of the paper}}
In \cref{sec:definitions}, we introduce the pregeometry $\cT_{\mathrm{an}}(k)$ and the notion of derived $k$-analytic\xspace space.
In \cref{sec:pregeometry}, we study the properties of the pregeometry $\cT_{\mathrm{an}}(k)$. We prove the unramifiedness conditions as well as the compatibility with truncations.
In \cref{sec:fullyfaithfulness}, we construct a functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ from the category of $k$-analytic\xspace spaces to the $\infty$-category\xspace of derived $k$-analytic\xspace spaces. We prove that $\Phi$ is a fully faithful embedding.
In \cref{sec:closed_etale}, we study closed immersions and étale morphisms under the embedding $\Phi$.
In \cref{sec:fiber_products}, we prove the existence of fiber products between derived $k$-analytic\xspace spaces.
In \cref{sec:essential_image}, we characterize the essential image of the embedding $\Phi$. Moreover, we compare derived $k$-analytic\xspace spaces with higher $k$-analytic\xspace stacks in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
\ifpersonal
\paragraph{\textbf{Personal note of outline: }} In \cref{sec:pregeometry}, we prove the unramifiedness of $\cT_{\mathrm{an}}(k)$ (\cref{cor:Tkan_unramified}) and the unramifiedness of the morphism $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$ (\cref{prop:unramified_transformation}) following \cite[\S 4]{DAG-IX}.
For the first, we first show that a closed immersion of $k$-analytic\xspace spaces induces a closed immersion of $\infty$-topoi\xspace. \cref{lem:descent_for_closed_subtopoi} is a gluing lemma which allows us to reason only for affinoid spaces.
\cref{lem:alg_conservative} and \cref{prop:alg_effective_epi} are two auxiliary results concerning the morphism $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$.
\cref{prop:closed_fiber_products_Top} is a corollary of unramifiedness, which shows the interest of the definition of unramifiedness. That is the property of unramifiedness which will be used later.
Unramifiedness of transformation of pregeometries implies that pullback of structured topoi along such morphism preserves closed immersions and pullbacks along closed immersions.
The main result of \cref{sec:fullyfaithfulness} is \cref{thm:fully_faithfulness}.
We define the functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ as follows. First we define the functor $\Phi$ on objects by \cref{lem:inclusion}. In order to define $\Phi$ on morphisms, we need to show that the mapping spaces are discrete (\cref{prop:discrete_mapping_spaces_I}), so we do not need to worry about higher homotopies. In order to prove \cref{prop:discrete_mapping_spaces_I}, we construct an auxiliary functor $\Upsilon\colon\mathrm{An}_k\to\mathrm{LRT}$ and prove that it is fully faithful (\cref{lem:first_fully_faithful}). Then \cref{prop:discrete_mapping_spaces_I} follows from \cref{lem:alg_faithful} and \cref{lem:alg_homotopy_monomorphism}.
Now the proof of \cref{thm:fully_faithfulness} is done as follows. The faithfullness of $\Phi$ is easy, which is the second paragraph of the proof of \cref{thm:fully_faithfulness}. The fullness is proved in the following way. By construction, $\Upsilon$ is $\Phi$ composed with truncation and algebraization. Given a morphism between $\varphi\colon\Phi(X)\to\Phi(Y)$, first we apply algebraization. Then we use \cref{lem:first_fully_faithful} to obtain a morphism $f\colon X\to Y$, which induces another morphism $\Phi(f)\colon\Phi(X)\to\Phi(Y)$, whose algebraization equals that of $\varphi$. \cref{lem:alg_faithful} says that for 0-truncated topoi, algebraization is faithful. Therefore, $f$ is what we want.
\cref{sec:fiber_products} shows the existence of fiber products. \cref{prop:closed_fiber_products_dAn} shows the existence of fiber products along a closed immersion. It is analog of \cite[Proposition 12.10]{DAG-IX}. Lurie deduces (v) from (iv). We will first prove (v) and then deduce (iv). In order to deduce (iv) from (v), we need \cref{lem:sheaves_coherent_modules}, which is analog of \cite[12.11]{DAG-IX}.
\cref{lem:closed_devissage} shows that a derived $k$-analytic\xspace space can locally be embedded into non-derived smooth $k$-analytic\xspace spaces. It is analog of \cite[12.13]{DAG-IX}. \cref{lem:products_dAn} shows the existence of products over a point. It is analog of \cite[12.14]{DAG-IX}. Finally, we are able to deduce \cref{thm:fiber_products} as in \cite[12.12]{DAG-IX}. Lurie's proof in the complex analytic case is a bit easier because the underlying topological space of the fiber product of complex analytic spaces is just the fiber product of topological spaces.
\cref{sec:essential_image}. \cite[12.8]{DAG-IX} cannot literally hold in $k$-analytic\xspace case. Because the category of $k$-analytic\xspace spaces is not closed under étale equivalence relations (cf.\ \cite{Conrad_Non-archimedean_analytification_2009}). Moreover, $k$-analytic\xspace spaces gives rise to $1$-localic topoi and not to $0$-localic ones. So we present a different statement and a different proof here. In fact we didn't understand Lurie's proof of \cite{Conrad_Non-archimedean_analytification_2009}, which involves loop spaces.
\fi
\paragraph{\bfseries Notations and terminology}
We refer to Bosch-Güntzer-Remmert \cite{Bosch_Non-archimedean_1984} and Fresnel-van der Put \cite{Fresnel_Rigid_2004} for the classical theory of non-archimedean analytic geometry, to Lurie \cite{HTT,Lurie_Higher_algebra} for the theory of $\infty$-categories\xspace, and to Lurie \cite{DAG-V} for the theory of structured spaces.
Throughout the paper, by $k$-analytic\xspace spaces, we mean quasi-paracompact quasi-separated rigid $k$-analytic\xspace spaces.
We denote by $\mathrm{Set}$ the category of sets and by $\mathcal S$ the $\infty$-category\xspace of spaces. For any small $\infty$-category\xspace $\mathcal C$ equipped with a Grothendieck topology $\tau$ and any presentable $\infty$-category\xspace $\mathcal D$, we denote by $\mathrm{PSh}_\mathcal D(\mathcal C)$ the $\infty$-category\xspace of $\mathcal D$-valued presheaves on $\mathcal C$ and by $\mathrm{Sh}_\mathcal D(\mathcal C,\tau)$ the $\infty$-category\xspace of $\mathcal D$-valued sheaves on the $\infty$-site\xspace $(\mathcal C,\tau)$. We will refer to $\mathcal S$-valued presheaves (resp.\ sheaves) simply as presheaves (resp.\ sheaves), and denote $\mathrm{PSh}(\mathcal C) \coloneqq \mathrm{PSh}_\mathcal S(\mathcal C)$, $\mathrm{Sh}(\mathcal C,\tau) \coloneqq \mathrm{Sh}_\mathcal S(\mathcal C,\tau)$. We denote the Yoneda embedding by \[ h\colon\mathcal C\to\mathrm{PSh}(\mathcal C),\qquad X\mapsto h_X.\]
\paragraph{\textbf{Related works and further developments}}
Our approach is very much based on the foundational works of Lurie \cite{DAG-V,DAG-VII,DAG-VIII,DAG-IX} on derived algebraic geometry and derived complex analytic geometry.
In \cite{Porta_DCAGI,Porta_DCAGII}, Mauro Porta studied the theories of analytification and deformation in derived complex analytic geometry, more specifically, the analytification functor, relative flatness, derived GAGA theorems, square-zero extensions, analytic modules and cotangent complexes.
The papers by Ben-Bassat and Kremnitzer \cite{Ben-Bassat_Non-archimedean_2013}, by Bambozzi and Ben-Bassat \cite{Bambozzi_Dagger_2015}, and by Paugam \cite{Paugam_Overconvergent_2014} suggest other approaches to derived analytic geometry.
In order to apply derived non-archimedean analytic geometry to enumerative geometry, mirror symmetry as well as other domains of mathematics, we must show that derived non-archimedean analytic spaces arise naturally in these contexts. The key to the construction of derived structures is to prove a representability theorem in derived non-archimedean geometry. This will be the main goal of our subsequent work \cite{Porta_Yu_Representability}.
\paragraph{\textbf{Acknowledgements}} We are grateful to Vladimir Berkovich, Antoine Chambert-Loir, Brian Conrad, Antoine Ducros, Bruno Klingler, Maxim Kontsevich, Jacob Lurie, Marco Robalo, Matthieu Romagny, Pierre Schapira, Carlos Simpson, Michael Temkin, Bertrand Toën and Gabriele Vezzosi for valuable discussions. The authors would also like to thank each other for the joint effort. This research was partially conducted during the period Mauro Porta was supported by Simons Foundation grant number 347070 and the group GNSAGA, and Tony Yue Yu served as a Clay Research Fellow.
\section{Basic definitions} \label{sec:definitions}
Intuitively, a derived non-archimedean analytic\xspace space is a ``topological space'' $\mathcal X$ equipped with a structure sheaf $\mathcal O_\mathcal X$ of ``derived non-archimedean analytic rings''. In order to give the precise definition, we introduce the notions of pregeometry and structured topos following \cite{DAG-V}.
\begin{defin}[{\cite[3.1.1]{DAG-V}}] A \emph{pregeometry} is an $\infty$-category\xspace $\mathcal T$ equipped with a class of \emph{admissible} morphisms and a Grothendieck topology generated by admissible morphisms, satisfying the following conditions: \begin{enumerate}[(i)] \item The $\infty$-category\xspace $\mathcal T$ admits finite products. \item The pullback of an admissible morphism along any morphism exists, and is again admissible. \item For morphisms $f,g$, if $g$ and $g\circ f$ are admissible, then $f$ is admissible. \item Every retract of an admissible morphism is admissible. \end{enumerate} \end{defin}
We now define two pregeometries responsible for derived non-archimedean geometry.
\begin{construction}
We define a pregeometry $\cT_{\mathrm{an}}(k)$ as follows:
\begin{enumerate}[(i)]
\item the underlying category of $\cT_{\mathrm{an}}(k)$ is the category of smooth $k$-analytic spaces;
\item a morphism in $\cT_{\mathrm{an}}(k)$ is admissible if and only if it is étale;
\item the topology on $\cT_{\mathrm{an}}(k)$ is the étale topology (cf.\ \cite[\S 8.2]{Fresnel_Rigid_2004}).
\end{enumerate} \end{construction}
\begin{construction}
We define a pregeometry $\cT_{\mathrm{disc}}(k)$ as follows:
\begin{enumerate}[(i)]
\item the underlying category of $\cT_{\mathrm{disc}}(k)$ is the full subcategory of the category of $k$-schemes spanned by affine spaces $\Spec(k[x_1, \ldots, x_n])$;
\item a morphism in $\cT_{\mathrm{disc}}(k)$ is admissible if and only if it is an isomorphism;
\item the topology on $\cT_{\mathrm{disc}}(k)$ is the trivial topology, i.e.\ a collection of admissible morphisms is a covering if and only if it is nonempty.
\end{enumerate} \end{construction}
\begin{defin}[{\cite[3.1.4]{DAG-V}}] \label{def:structure} Let $\mathcal T$ be a pregeometry, and let $\mathcal X$ be an $\infty$-topos\xspace. A \emph{$\mathcal T$-structure} on $\mathcal X$ is a functor $\mathcal O\colon\mathcal T\to\mathcal X$ with the following properties: \begin{enumerate}[(i)] \item The functor $\mathcal O$ preserves finite products. \item Suppose given a pullback diagram \[ \begin{tikzcd} U' \arrow{r} \arrow{d} & U \arrow{d}{f} \\ X' \arrow{r} & X \end{tikzcd} \] in $\mathcal T$, where $f$ is admissible. Then the induced diagram \[ \begin{tikzcd} \mathcal O(U') \arrow{r} \arrow{d} & \mathcal O(U) \arrow{d} \\ \mathcal O(X') \arrow{r} & \mathcal O(X) \end{tikzcd} \] is a pullback square in $\mathcal X$. \item Let $\{U_\alpha\to X\}$ be a covering in $\mathcal T$ consisting of admissible morphisms. Then the induced map \[\coprod_\alpha\mathcal O(U_\alpha)\to\mathcal O(X)\] is an effective epimorphism in $\mathcal X$. \end{enumerate} A morphism of $\mathcal T$-structures $\mathcal O\to\mathcal O'$ on $\mathcal X$ is \emph{local} if for every admissible morphism $U\to X$ in $\mathcal T$, the resulting diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \mathcal O'(U) \arrow{d} \\
\mathcal O(X) \arrow{r} & \mathcal O'(X)
\end{tikzcd} \] is a pullback square in $\mathcal X$. We denote by $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal X)$ the $\infty$-category\xspace of $\mathcal T$-structures on $\mathcal X$ with local morphisms.
A \emph{$\mathcal T$-structured $\infty$-topos\xspace} is a pair $(\mathcal X,\mathcal O_\mathcal X)$ consisting of an $\infty$-topos\xspace $\mathcal X$ and a $\mathcal T$-structure $\mathcal O_\mathcal X$ on $\mathcal X$. We denote by $\RTop(\mathcal T)$ the $\infty$-category\xspace of $\mathcal T$-structured $\infty$-topoi\xspace (cf.\ \cite[Definition 1.4.8]{DAG-V}). Note that a 1-morphism $f\colon (\mathcal X, \mathcal O_\mathcal X) \to (\mathcal Y, \mathcal O_\mathcal Y)$ in $\RTop(\mathcal T)$ consists of a geometric morphism of $\infty$-topoi\xspace $f_*\colon\mathcal X\rightleftarrows\mathcal Y\colon f^{-1}$ and a local morphism of $\mathcal T$-structures $f^\sharp \colon f^{-1} \mathcal O_\mathcal Y \to \mathcal O_\mathcal X$. \end{defin}
We have a natural functor $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ induced by analytification. Composing with this functor, we obtain an ``algebraization'' functor \[ (-)^\mathrm{alg} \colon \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X). \] In virtue of \cite[Example 3.1.6, Remark 4.1.2]{DAG-V}, we have an equivalence induced by the evaluation on the affine line \[\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X) \xrightarrow{\ \sim\ } \mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X), \] where $\mathrm{CAlg}_k$ denotes the $\infty$-category\xspace of simplicial commutative algebras over $k$.
We are now ready to introduce our main object of study: derived $k$-analytic\xspace spaces.
\begin{defin}\label{def:derived_space} A $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal X,\mathcal O_\mathcal X)$ is called a \emph{derived $k$-analytic\xspace space} if $\mathcal X$ is hypercomplete and there exists an effective epimorphism from $\coprod_i U_i$ to the final object of $\mathcal X$ satisfying the following conditions, for every index $i$: \begin{enumerate}[(i)]
\item The pair $(\mathcal X_{/U_i}, \pi_0(\mathcal O^\mathrm{alg}_\mathcal X | U_i))$ is equivalent to the ringed $\infty$-topos\xspace associated to the étale site on a $k$-analytic\xspace space $X_i$.
\item For each $j\ge 0$, $\pi_j(\mathcal O^\mathrm{alg}_\mathcal X | U_i)$ is a coherent sheaf of $\pi_0(\mathcal O^\mathrm{alg}_\mathcal X | U_i)$-modules on $X_i$. \end{enumerate} We denote by $\mathrm{dAn}_k$ the full subcategory of $\RTop(\cT_{\mathrm{an}}(k))$ spanned by derived $k$-analytic\xspace spaces. \end{defin}
\begin{rem}\label{rem:definition_intuition}
Let us explain the heuristic relation between \cref{def:derived_space} and \cref{def:derived_scheme} in the introduction.
Let $(\mathcal X,\mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space as in \cref{def:derived_space}.
Let $\mathbf A^1_k$ be the $k$-analytic\xspace affine line and let $O\coloneqq \mathcal O_\mathcal X (\mathbf A^1_k)\in\mathcal X$.
We have the sum operation $+\colon \mathbf A^1_k \times \mathbf A^1_k \to \mathbf A^1_k$ and the multiplication operation $\bullet\colon \mathbf A^1_k\times \mathbf A^1_k \to \mathbf A^1_k$.
By \cref{def:structure}(i), they induce respectively a sum operation $+\colon O\times O\to O$ and a multiplication operation $\bullet\colon O\times O\to O$ on $O$.
Therefore, intuitively, we can think of $O$ as a sheaf of commutative simplicial rings as in \cref{def:derived_scheme}.
Moreover, the sheaf $O$ is also equipped with analytic structures.
For example, let $\mathbf D^1_k\subset \mathbf A^1_k$ denote the closed unit disc.
By \cref{def:structure}(ii), we obtain a monomorphism $\mathcal O_\mathcal X(\mathbf D^1_k)\hookrightarrow O$.
We can think of $\mathcal O_\mathcal X(\mathbf D^1_k)$ as the subsheaf of $O$ consisting of functions of norm less than or equal to one.
Furthermore, any holomorphic function $f$ on $\mathbf D^1_k$ induces a morphism $f_O\colon \mathcal O_\mathcal X(\mathbf D^1_k)\to O$, which we think of as the composition with $f$.
(See also the discussion after \cref{def:derived_scheme}.) \end{rem}
\begin{rem}
The hypercompleteness assumption in \cref{def:derived_space} will ensure that the underlying $\infty$-topos\xspace of a derived $k$-analytic\xspace space has enough points (cf.\ \cref{rem:points_of_cX_X}). \end{rem}
The goal of this paper is to study the basic properties of derived $k$-analytic\xspace spaces and to compare them with ordinary $k$-analytic\xspace spaces as well as with the higher $k$-analytic\xspace stacks introduced in \cite{Porta_Yu_Higher_analytic_stacks_2014}.
Before moving on, we stress that the underlying $\infty$-topos\xspace of a derived $k$-analytic\xspace space is, by definition, hypercomplete. Therefore, using the notations of \cite[\S 2.2]{DAG-V}, for $X \in \cT_{\mathrm{an}}(k)$, the $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ is \emph{not} a derived $k$-analytic\xspace space. We remedy this problem by introducing the hypercomplete spectrum as follows:
Let $\RTop$ (resp.\ $\LTop$) denote the $\infty$-category\xspace of $\infty$-topoi\xspace where morphisms are right (resp.\ left) adjoint geometric morphisms. Denote by $\RHTop$ the full subcategory of $\RTop$ spanned by hypercomplete $\infty$-topoi\xspace. Denote by $\RHTop(\cT_{\mathrm{an}}(k))$ the full subcategory of $\RTop(\cT_{\mathrm{an}}(k))$ spanned by $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is a hypercomplete. It follows from \cite[6.5.2.13]{HTT} that the inclusion $\RHTop \to \RTop$ admits a right adjoint, given by hypercompletion. This induces a right adjoint to the inclusion $\RHTop(\cT_{\mathrm{an}}(k)) \hookrightarrow \RTop(\cT_{\mathrm{an}}(k))$, as the next lemma shows:
\begin{lem} \label{lem:hyp_right_adjoint}
The inclusion $\RHTop(\cT_{\mathrm{an}}(k)) \hookrightarrow \RTop(\cT_{\mathrm{an}}(k))$ admits a right adjoint, which we denote by $\mathrm{Hyp} \colon \RTop(\cT_{\mathrm{an}}(k)) \to \RHTop(\cT_{\mathrm{an}}(k))$. \end{lem}
\begin{proof}
Fix $X\coloneqq(\mathcal X, \mathcal O_\mathcal X) \in \RTop(\cT_{\mathrm{an}}(k))$.
Since the hypercompletion $L \colon \mathcal X \to \mathcal X^\wedge$ is left exact, we obtain a well defined functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X^\wedge)$ induced by composition with $L$.
Let $X^\wedge \coloneqq (\mathcal X^\wedge, L(\mathcal O_\mathcal X))$ be the resulting hypercomplete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
In $\RTop(\cT_{\mathrm{an}}(k))$ there is a natural morphism $\varphi \colon X^\wedge \to X$.
Using the dual of \cite[5.2.7.8]{HTT} it suffices to show that $\varphi$ exhibits $X^\wedge$ as a colocalization of $X$ relative to $\RHTop(\cT_{\mathrm{an}}(k))$. In order to prove this, let $Y \coloneqq (\mathcal Y, \mathcal O_\mathcal Y)$ be any hypercomplete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
Using \cite[6.5.2.13]{HTT} we obtain an equivalence
\[ \Map_{\RTop}(\mathcal Y, \mathcal X^\wedge) \to \Map_{\RTop}(\mathcal Y, \mathcal X). \]
Fix a geometric morphism $g_*\colon\mathcal Y\rightleftarrows\mathcal X^\wedge\colon g^{-1}$ and let $(f^{-1}, f_*)$ denote the induced geometric morphism $\mathcal Y \rightleftarrows \mathcal X$.
We remark that $f^{-1} \simeq g^{-1} \circ L$.
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we obtain a morphism of fiber sequences:
\[ \begin{tikzcd}[column sep=small]
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(g^{-1} L(\mathcal O_X), \mathcal O_\mathcal Y) \arrow{r} \arrow{d} & \Map_{\RTop(\cT_{\mathrm{an}}(k))}(Y, X^\wedge) \arrow{r} \arrow{d} & \Map_{\RTop}(\mathcal Y, \mathcal X^\wedge) \arrow{d} \\
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(f^{-1} \mathcal O_X, \mathcal O_\mathcal Y) \arrow{r} & \Map_{\RTop(\cT_{\mathrm{an}}(k))}(Y, X) \arrow{r} & \Map_{\RTop}(\mathcal Y, \mathcal X).
\end{tikzcd} \]
Since $f^{-1} \simeq g^{-1} \circ L$, we see that the left vertical morphism is an equivalence. Since this holds for all base points in $\Map_{\RTop}(\mathcal Y, \mathcal X)$, we conclude that the middle vertical morphism is an equivalence as well, completing the proof. \end{proof}
\begin{defin}
Given $X \in \cT_{\mathrm{an}}(k)$, we define its \emph{hypercomplete (absolute) spectrum} $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X)$ to be $\mathrm{Hyp}(\Spec^{\cT_{\mathrm{an}}(k)}(X))$. \end{defin}
\begin{lem} \label{lem:universal_property_HSpec}
Let $Y \coloneqq (\mathcal Y, \mathcal O_\mathcal Y)$ be a derived $k$-analytic\xspace space and let $X \in \cT_{\mathrm{an}}(k)$.
The natural morphism $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X) \to \Spec^{\cT_{\mathrm{an}}(k)}(X)$ induces an equivalence
\[ \Map_{\RHTop(\cT_{\mathrm{an}}(k))}(Y, \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(X)) \xrightarrow{\ \sim\ } \Map_{\RTop(\cT_{\mathrm{an}}(k))}(Y, \Spec^{\cT_{\mathrm{an}}(k)}(X)). \] \end{lem}
\begin{proof}
Since $Y$ belongs to $\RHTop(\cT_{\mathrm{an}}(k))$, the statement is an immediate consequence of \cref{lem:hyp_right_adjoint}. \end{proof}
\section{Properties of the pregeometry} \label{sec:pregeometry}
In this section, we study the properties of the pregeometry $\cT_{\mathrm{an}}(k)$ introduced in \cref{sec:definitions}. More specifically, we will prove the unramifiedness of $\cT_{\mathrm{an}}(k)$, the unramifiedness of the algebraization and the compatibility of $\cT_{\mathrm{an}}(k)$ with $n$-truncations.
\subsection{Unramifiedness}
In order that the collection of closed immersions behaves well with respect to fiber products, our pregeometry $\cT_{\mathrm{an}}(k)$ has to verify a condition of unramifiedness.
\begin{defin}[{\cite[1.3]{DAG-IX}}]\label{def:unramified_pregeometry}
A pregeometry $\mathcal T$ is said to be \emph{unramified} if for every morphism $f\colon X\to Y$ in $\mathcal T$ and every object $Z\in\mathcal T$, the diagram
\[ \begin{tikzcd}
X\times Z \arrow{r} \arrow{d} & X\times Y\times Z \arrow{d} \\
X \arrow{r} & X\times Y
\end{tikzcd} \]
induces a pullback square
\[ \begin{tikzcd}
\mathcal X_{X\times Z} \arrow{r} \arrow{d} & \mathcal X_{X\times Y\times Z} \arrow{d} \\
\mathcal X_X \arrow{r} & \mathcal X_{X\times Y}
\end{tikzcd} \]
in $\RTop$, where the symbol $\mathcal X_{(-)}$ denotes the associated $\infty$-topos\xspace. \end{defin}
Our first goal is to prove that the pregeometry $\cT_{\mathrm{an}}(k)$ is unramified (cf.\ \cref{cor:Tkan_unramified}). In order to do this, we need to describe explicitly the $\infty$-topos\xspace $\mathcal X_X$ associated to a $k$-analytic\xspace space $X$ and prove that the assignment $X \mapsto \mathcal X_X$ is well behaved with respect to closed immersions (cf.\ \cref{prop:preserve_closed_immersion}).
Let $\mathrm{An}_k$ denote the category of $k$-analytic\xspace spaces and let $\mathrm{Afd}_k$ denote the category of $k$-affinoid spaces. For $X\in\mathrm{An}_k$, let $(\mathrm{An}_X)_\mathrm{\acute{e}t}$ (resp.\ $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$) denote the category of étale morphisms from $k$-analytic\xspace spaces (resp.\ $k$-affinoid spaces) to $X$. We equip the categories $(\mathrm{An}_X)_\mathrm{\acute{e}t}$ and $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$ with the étale topology. By \cite[Proposition 2.24]{Porta_Yu_Higher_analytic_stacks_2014}, the inclusion $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}\hookrightarrow(\mathrm{An}_X)_\mathrm{\acute{e}t}$ induces an equivalence of $\infty$-topoi\xspace \begin{equation}\label{eq:afd_in_an} \mathrm{Sh}((\mathrm{Afd}_X)_\mathrm{\acute{e}t})\xrightarrow{\ \sim\ }\mathrm{Sh}((\mathrm{An}_X)_\mathrm{\acute{e}t}). \end{equation} We call the two equivalent $\infty$-topoi\xspace above the \emph{étale $\infty$-topos\xspace associated to $X$}, and denote it by $\mathcal X_X$. We will denote the site $(\mathrm{Afd}_X)_\mathrm{\acute{e}t}$ by $X_\mathrm{\acute{e}t}$ for simplicity.
\begin{rem}
The $\infty$-topos\xspace $\mathcal X_X$ is not hypercomplete in general.
In the subsequent sections we will also consider its hypercompletion $\mathcal X_X^\wedge$.
\end{rem}
\begin{rem}\label{rem:points_of_cX_X} Since the site $X_\mathrm{\acute{e}t}$ is a 1-category, the $\infty$-topos\xspace $\mathcal X_X$ is $1$-localic. It follows that for any $\infty$-topos\xspace $\mathcal Y$ one has an equivalence of $\infty$-categories \[ \Fun_*(\mathcal Y, \mathcal X_X) \simeq \Fun_*(\tau_{\le 0} \mathcal Y, \tau_{\le 0} \mathcal X_X), \] where $\Fun_*$ denotes the $\infty$-category of geometric morphisms (taken in $\RTop$). Put $\mathcal Y = \mathcal S$ and observe that $\tau_{\le 0} \mathcal X_X = \mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t})$ and $\tau_{\le 0}(\mathcal S) \simeq \mathrm{Set}$. We conclude that the points of $\mathcal X_X$ correspond bijectively to the points of the classical 1-topos associated to the site $X_\mathrm{\acute{e}t}$. The latter is classified by the geometric points of the adic space associated to $X$ in the sense of Huber (cf.\ \cite[Proposition 2.5.17]{Huber_Etale_1996}).
Since the site $X_\mathrm{\acute{e}t}$ is finitary, it follows from \cite[Corollary 3.22]{DAG-VII} that the hypercompletion $\mathcal X_X^\wedge$ is locally coherent. Therefore, by Theorem 4.1 in loc.\ cit., the $\infty$-topos\xspace $\mathcal X_X^\wedge$ has enough points. \end{rem}
\begin{rem}\label{rem:spectrum} As we already discussed in \cref{sec:definitions}, \cite[\S 2.2]{DAG-V} assigns to every $X \in \cT_{\mathrm{an}}(k)$ a $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $\Spec^{\cT_{\mathrm{an}}(k)}(X)$, called the \emph{spectrum} of $X$. It is characterized by the following universal property: for any $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal Y, \mathcal O_\mathcal Y)$ there is a natural equivalence \[ \Map_{\RTop(\cT_{\mathrm{an}}(k))}( (\mathcal Y, \mathcal O_\mathcal Y), \Spec^{\cT_{\mathrm{an}}(k)}(X) ) \simeq \Map_{\mathrm{Ind}(\mathcal G_\mathrm{an}(k)^{\mathrm{op}})}( X, \Gamma_{\mathcal G}(\mathcal Y, \mathcal O_\mathcal Y) ), \] where $\mathcal G_{\mathrm{an}}(k)$ denotes a geometric envelope of $\cT_{\mathrm{an}}(k)$ (cf.\ \cite[Theorem 2.2.12]{DAG-V}). We note that the underlying $\infty$-topos\xspace of $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ can be identified with $\mathcal X_X$. \end{rem}
We refer to \cite[7.3.2]{HTT} for the notion of closed immersion of $\infty$-topoi\xspace.
\begin{prop}\label{prop:preserve_closed_immersion} The functor \begin{align*} \mathrm{An}_k&\longrightarrow \mathrm{h}(\RTop) \\ X&\longmapsto\mathcal X_X \end{align*} preserves closed immersions, where $\mathrm h(\RTop)$ denotes the homotopy category of $\RTop$. \end{prop}
\begin{rem}
It will follow from the results of \cref{sec:fullyfaithfulness} (see in particular \cref{lem:rigidity} and the construction of $\Phi$) that the functor above can be promoted to an $\infty$-functor $\mathrm{An}_k \to \RTop$.
\end{rem}
\begin{lem} \label{lem:descent_for_closed_subtopoi} Let $\mathcal X, \mathcal Y$ be $\infty$-topoi\xspace and let $U \in \mathcal X$. Let $f^{-1} \colon \mathcal X /U \rightleftarrows \mathcal Y \colon f_*$ be a geometric morphism. Then $(f^{-1}, f_*)$ is an equivalence if and only if there exists an effective epimorphism $V \to 1_\mathcal X$ such that $\mathcal X_{/V} / (U \times V) \rightleftarrows \mathcal Y_{/ f^{-1}(V)}$ is an equivalence. \end{lem} \begin{proof} To see that the condition is necessary it is enough to take $V \to 1_\mathcal X$ to be the identity of $1_\mathcal X$. We now prove the sufficiency. Let us denote by $j^{-1} \colon \mathcal X \leftrightarrows \mathcal X / U \colon j_*$ (resp.\ $i^{-1} \colon \mathcal X \leftrightarrows \mathcal X_{/V} \colon i_*$) the given closed (resp.\ \'etale) morphism of $\infty$-topoi\xspace. We claim that \[\mathcal X_{/V} / (U \times V) \simeq (\mathcal X/U)_{/j^{-1}(V)}.\] Indeed, the left hand side can be identified with the pullback $\mathcal X_{/V} \times_{\mathcal X} \mathcal X/U$ in virtue of \cite[6.3.5.8]{HTT}. The right hand side can be identified with the same pullback in virtue of \cite[7.3.2.13]{HTT}. At this point, we obtain a commutative square of geometric morphisms in $\RTop$ \[ \begin{tikzcd} \mathcal Y_{/f^{-1}(V)} \arrow{r} \arrow{d} & (\mathcal X/U)_{/j^{-1}(V)} \arrow{d} \\ \mathcal Y \arrow{r}{f_*} & \mathcal X / U. \end{tikzcd} \] So the lemma follows from the descent property of $\infty$-topoi\xspace \cite[6.1.3.9(3)]{HTT}. \end{proof}
\begin{lem} \label{lem:qet_structure} Let $A \to B$ be a surjective morphism of $k$-affinoid algebras. Let $B \to B'$ be an \'etale morphism of $k$-affinoid $A$-algebras. Then there exists an \'etale $A$-algebra $A'$ and a pushout square: \[ \begin{tikzcd} A \arrow{r} \arrow{d} & B \arrow{d} \\ A' \arrow{r} & B'. \end{tikzcd} \] \end{lem}
\begin{proof} Since $B \to B'$ is \'etale, by \cite[Proposition 1.7.1]{Huber_Etale_1996}, we can write \[ B' = B \langle y_1, \ldots, y_m \rangle / (f_1, \ldots, f_m), \] such that the Jacobian $J\coloneqq\mathrm{Jac}(f_1, \ldots, f_m)$ is invertible in $B'$. So \[\rho\coloneqq\min_{x\in\Sp B'}\abs{J(x)}\] is positive. Since $A \to B$ is surjective, the induced morphism \[ A \langle y_1, \ldots, y_m \rangle \to B \langle y_1, \ldots, y_m \rangle \] is surjective as well. Therefore we can find elements $\overline{f_1}, \ldots, \overline{f_m} \in A \langle y_1, \ldots, y_m \rangle$ lifting $f_1, \ldots, f_m$. Set \[ A_0 \coloneqq A \langle y_1, \ldots, y_m \rangle / (\overline{f_1}, \ldots, \overline{f_m}). \] Let $\widebar J\coloneqq\mathrm{Jac}(\overline{f_1}, \ldots, \overline{f_m})$. Let $n$ be a positive integer such that $\rho^n\in\abs{k}$ and let $a$ be an element in $k$ such that $\abs{a}=\rho^n$. Set $A' \coloneqq A_0 \langle w \rangle / (w \widebar J^n - a)$. We see that the natural morphism $A_0\to B'$ factors as \[ A_0 \to A' \to B'. \] It follows from the construction that $A \to A'$ is \'etale, and moreover $B' \simeq A' \cotimes_A B$, completing the proof. \end{proof}
\begin{proof}[Proof of \cref{prop:preserve_closed_immersion}] Let $f \colon Y \to X$ be a closed immersion in $\mathrm{An}_k$. Let $U \colon X_\mathrm{\acute{e}t} \to \mathcal S$ be the functor defined by the formula \[ U(Z) = \begin{cases} \{*\} & \text{if } Z\times_X Y =\emptyset, \\ \emptyset & \text{otherwise.} \end{cases} \] This is a sheaf and therefore determines a closed subtopos $\mathcal X_X / U$. The morphism $f$ induces a geometric morphism \[f^{-1} \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f_* .\] We claim that $f_*$ factors through the closed subtopos $\mathcal X_X / U$. Indeed, it suffices to check that for every sheaf $G \in \mathcal X_Y$ and every representable sheaf $h_{Z}$ in $\mathcal X_X$ such that $\Map_{\mathcal X_X}(h_{Z}, U) \ne \emptyset$, the space $\Map_{\mathcal X_X}(h_{Z}, f_*(G))$ is contractible. This is true, because we have \[ \Map_{\mathcal X_X}(h_{Z}, f_*(G)) \simeq G(Z\times_X Y) = G(\emptyset) \simeq \{*\}. \] We denote by $(f^{-1},f_*)$ again the induced adjunction \begin{equation} \label{eq:induced_adjunction} f^{-1} \colon \mathcal X_X / U \rightleftarrows \mathcal X_Y \colon f_* . \end{equation} We conclude our proof by the following lemma. \end{proof}
\begin{lem} \label{lem:closed_immersion_closed_subtopos} The adjunction in \cref{eq:induced_adjunction} is an equivalence. \end{lem} \begin{proof} By \cref{lem:descent_for_closed_subtopoi}, we can assume that both $X$ and $Y$ are affinoid. Note that $\mathcal X_X / U$ and $\mathcal X_Y$ are $1$-localic $\infty$-topoi\xspace in virtue of \cite[7.5.4.2]{HTT} and \cite[Lemma 1.2.6]{DAG-VIII}. Therefore it suffices to show that the adjunction $(f^{-1}, f_*)$ induces an equivalence when restricted to $1$-truncated objects of $\mathcal X_X / U$ and $\mathcal X_Y$.
Let us prove that the functor $f_*$ is conservative. Let $\alpha \colon F \to F'$ be a morphism in $\mathcal X_Y$ and suppose that $f_*(\alpha)$ is an equivalence. By the equivalence \eqref{eq:afd_in_an}, it is enough to show that $\alpha$ induces equivalences $F(Y') \to F'(Y')$ for every étale morphism $Y' \to Y$. Using \cref{lem:qet_structure}, we can form a pullback diagram \[ \begin{tikzcd} Y' \arrow{r} \arrow{d} & X' \arrow{d} \\ Y \arrow{r} & X , \end{tikzcd} \] where $X' \to X$ is étale. It follows that \[ F(Y') = (f_* F)(X') \to (f_* F')(X') = F(Y') \] is an equivalence.
We are left to check that the unit of the adjunction $(f^{-1}, f_*)$ is an equivalence over $1$-truncated objects. For this, it suffices to check that for every $1$-truncated sheaf $F \in \mathcal X_X$, the unit $u \colon F \to f_* f^{-1} F$ induces an equivalence on sheaves of homotopy groups. Since both $F$ and $f_* f^{-1} F$ are $1$-truncated, they are hypercomplete objects. Therefore, it suffices to check that $\eta^{-1}(u)$ is an equivalence for every geometric morphism $\eta^{-1} \colon \mathcal X_X \to \mathcal S\colon\eta_*$. Such a geometric morphism corresponds to a geometric point $x$ of the adic space associated to $X$ (cf.\ \cref{rem:points_of_cX_X}). Let $\{V_\alpha\}$ be a system of étale neighborhoods of $x$. We have $\eta^{-1}(G)=\colim G(V_\alpha)$.
If $x$ does not meet $Y$, we see that $\eta^{-1}(G)$ is contractible whenever $G \in \mathcal X_X / U$. In particular $\eta^{-1}(u)$ is an equivalence for every $1$-truncated $F \in \mathcal X_X / U$.
Otherwise, $x$ lifts to a geometric morphism $\eta_1^{-1} \colon \mathcal X_Y \to \mathcal S$, satisfying $\eta^{-1} = \eta_1^{-1} \circ f^{-1}$. So we have \begin{align*} \eta^{-1}(f_* f^{-1} F) & \simeq \colim (f_* f^{-1} F)(V_\alpha) \\ & \simeq \colim (f^{-1} F)(V_\alpha \times_{X} Y) \\ & \simeq \eta_1^{-1} f^{-1} F \simeq \eta^{-1} F , \end{align*} completing the proof. \end{proof}
\begin{prop} \label{prop:closed_immersion_pullback_of_topoi}
Let
\[ \begin{tikzcd}
W \arrow{r} \arrow{d} & Y \arrow{d}{g} \\
X \arrow{r}{f} & Z
\end{tikzcd} \]
be a pullback square in $\mathrm{An}_k$ and assume that $f$ is a closed immersion. The induced square of $\infty$-topoi\xspace
\[ \begin{tikzcd}
\mathcal X_W \arrow{r} \arrow{d} & \mathcal X_Y \arrow{d}{g_*} \\
\mathcal X_X \arrow{r}{f_*} & \mathcal X_Z
\end{tikzcd} \]
is a pullback diagram in $\RTop$.
\end{prop}
\begin{proof}
Let $U_X$ be the sheaf on the étale site $Z_\mathrm{\acute{e}t}$ of $Z$ defined by
\[ U_X(T) \coloneqq \begin{cases} \{*\} & \text{if } T \times_Z X = \emptyset \\ \emptyset & \text{otherwise.} \end{cases} \]
Define $U_W$ to be the sheaf on the étale site $Y_\mathrm{\acute{e}t}$ of $Y$ in a similar way.
Using \cref{lem:closed_immersion_closed_subtopos} twice, we can rewrite the induced square of $\infty$-topoi\xspace as
\[ \begin{tikzcd}
\mathcal X_Y / U_W \arrow{r} \arrow{d} & \mathcal X_Y \arrow{d}{g_*} \\
\mathcal X_Z / U_X \arrow{r} & \mathcal X_Z .
\end{tikzcd} \]
In virtue of \cite[7.3.2.13]{HTT}, we only need to show that $g^{-1} U_X \simeq U_W$.
First of all, let us observe that there exists a map $U_X \to g_* U_W$: indeed, if $T \to X$ is étale with $T \to Z$ a smooth morphism such that $T \times_Z X = \emptyset$, then we also have $(T \times_Z Y) \times_Y W \simeq (T\times_Z W) \times_Z Y = \emptyset$, and therefore $g_*(U_W)(T) = U_W(T \times_Z Y) = \Delta^0$.
This allows to define the desired map, which induces by adjunction a morphism $g^{-1} U_X \to U_W$.
By construction, $U_W$ is $(-1)$-truncated and \cite[5.5.6.16]{HTT} shows that $g^{-1} U_X$ is $(-1)$-truncated too.
Therefore they are both hypercomplete.
So it suffices to check that $g^{-1} U_X \to U_W$ is an isomorphism on the stalks of $\mathcal X_Y$. This is true because a geometric point $\eta_* \colon \mathcal S \to \mathcal X_Y$ factors through $\mathcal X_W$ if and only if $g_* \circ \eta_*$ factors through $\mathcal X_X$ (cf.\ \cref{rem:points_of_cX_X}). \end{proof}
\begin{cor} \label{cor:Tkan_unramified}
The pregeometry $\cT_{\mathrm{an}}(k)$ is unramified. \end{cor}
\begin{proof}
We check that \cref{def:unramified_pregeometry} is satisfied.
Let $X, Y, Z \in \cT_{\mathrm{an}}(k)$ and let $f \colon Y \to X$ be any morphism.
The diagram
\[ \begin{tikzcd}
X \arrow{r} \arrow{d}{\mathrm{id}_X \times f} & Y \arrow{d}{\Delta} \\
X \times Y \arrow{r} & Y \times Y
\end{tikzcd} \]
is a pullback diagram. Since $Y$ is separated, $Y$ is a closed immersion, and therefore the same goes for $X \to X \times Y$.
We can therefore use \cref{prop:closed_immersion_pullback_of_topoi} to conclude that the induced square
\[ \begin{tikzcd}
\mathcal X_{X \times Z} \arrow{r} \arrow{d} & \mathcal X_X \arrow{d} \\
\mathcal X_{X \times Y \times Z} \arrow{r} & \mathcal X_{X \times Y}.
\end{tikzcd} \]
is a pullback diagram in $\RTop$. \end{proof}
\subsection{Algebraization}
The functor $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ induced by analytification is a transformation of pregeometries in the following sense:
\begin{defin}[{\cite[3.2.1]{DAG-IX}}]
A \emph{transformation of pregeometries} from $\mathcal T$ to $\mathcal T'$ is a functor $\theta\colon\mathcal T\to\mathcal T'$ such that
\begin{enumerate}[(i)]
\item it preserves finite products;
\item it sends admissible morphisms in $\mathcal T$ to admissible morphisms in $\mathcal T'$;
\item it sends coverings in $\mathcal T$ to coverings in $\mathcal T'$;
\item it sends any pullback in $\mathcal T$ along an admissible morphism to a pullback in $\mathcal T'$.
\end{enumerate} \end{defin}
In the following, we study some properties of the transformation of pregeometries $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$.
\begin{lem} \label{lem:alg_conservative}
Let $\mathcal X$ be an $\infty$-topos\xspace.
The algebraization functor
\[
(-)^\mathrm{alg} \colon \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X)
\]
induced by composition with the transformation $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ is conservative. \end{lem}
\begin{proof} Let $f \colon \mathcal O \to \mathcal O'$ be a local morphism of $\cT_{\mathrm{an}}(k)$-structures on $\mathcal X$ such that $f^\mathrm{alg} \colon \mathcal O^\mathrm{alg} \to \mathcal O^{\prime \mathrm{alg}}$ is an equivalence. We will show that for every $X\in\cT_{\mathrm{an}}(k)$, the induced morphism $\mathcal O(X) \to \mathcal O'(X)$ is an equivalence. Since $X$ is smooth, there exists an affinoid G-covering $\{\Sp B_i\to X\}$ such that every $\Sp B_i$ admits an étale morphism to a $k$-analytic\xspace affine space.
So we obtain a commutative square \[ \begin{tikzcd} \coprod \mathcal O(\Sp B_i) \arrow{d} \arrow{r} & \coprod \mathcal O'(\Sp B_i) \arrow{d} \\ \mathcal O(X) \arrow{r} & \mathcal O'(X) , \end{tikzcd} \] where the vertical morphisms are effective epimorphisms. Moreover, since admissible open immersions are étale and $f$ is a local morphism, we see that the above square is a pullback. We are therefore reduced to show that $\mathcal O(\Sp B) \to \mathcal O'(\Sp B)$ is an equivalence whenever $\Sp B$ admits an étale morphism to a $k$-analytic\xspace affine space $\mathbf A^n_k$. Since $f$ is a local morphism, we have in this case a pullback square \[ \begin{tikzcd} \mathcal O(\Sp B) \arrow{r} \arrow{d} & \mathcal O(\mathbf A^n_k) \arrow{d} \\ \mathcal O'(\Sp B) \arrow{r} & \mathcal O'(\mathbf A^n_k). \end{tikzcd} \] Let $\mathbb A^n_k$ denote the $n$-dimensional algebraic affine space over $k$. Since $\mathcal O(\mathbf A^n_k) = \mathcal O^\mathrm{alg}(\mathbb A^n_k) \to \mathcal O^{\prime \mathrm{alg}}(\mathbb A^n_k) = \mathcal O'(\mathbf A^n_k)$ is an equivalence by our assumption, we deduce that $\mathcal O(\Sp B) \to \mathcal O'(\Sp B)$ is an equivalence as well, completing the proof. \end{proof}
\begin{prop} \label{prop:alg_effective_epi}
Let $\mathcal X$ be an $\infty$-topos\xspace and let $f \colon \mathcal O \to \mathcal O'$ be a morphism in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
The following conditions are equivalent:
\begin{enumerate}[(i)]
\item The morphism $f$ is an effective epimorphism, i.e.\ for every $U \in \cT_{\mathrm{an}}(k)$ the morphism $\mathcal O(U) \to \mathcal O'(U)$ is an effective epimorphism in $\mathcal X$.
\item The morphism $f^\mathrm{alg} \colon \mathcal O^\mathrm{alg} \to \mathcal O^{\prime \mathrm{alg}}$ is an effective epimorphism.
\item The morphism $\mathcal O(\mathbf A^1_k) \to \mathcal O'(\mathbf A^1_k)$ is an effective epimorphism.
\end{enumerate} \end{prop}
\begin{proof}
It follows directly from the definition of effective epimorphism of $\cT_{\mathrm{an}}(k)$-structures that (i) implies (ii) and (ii) implies (iii).
Let us show that (iii) implies (i).
Let $X \in \cT_{\mathrm{an}}(k)$. Choose an étale covering $\{U_i \to X \}$ such that each $U_i$ admits an étale morphism to $\mathbf A^n_k$.
Since $f$ is a local morphism, we have the following pullback square:
\[ \begin{tikzcd}
\coprod \mathcal O(U_i) \arrow{r} \arrow{d} & \coprod \mathcal O'(U_i) \arrow{d} \\
\mathcal O(X) \arrow{r} & \mathcal O'(X) .
\end{tikzcd} \]
The vertical arrows are effective epimorphisms, and therefore it suffices to check that the upper horizontal map is an effective epimorphism.
Since $f$ is a local morphism, we see that the diagram
\[ \begin{tikzcd}
\mathcal O(U_i) \arrow{r} \arrow{d} & \mathcal O'(U_i) \arrow{d} \\
\mathcal O(\mathbf A^n_k) \arrow{r} & \mathcal O'(\mathbf A^n_k)
\end{tikzcd} \]
is a pullback diagram.
So it suffices to show that $\mathcal O(\mathbf A^n_k) \to \mathcal O'(\mathbf A^n_k)$ is an effective epimorphism. This follows from the hypothesis and the fact that both $\mathcal O$ and $\mathcal O'$ commute with products. \end{proof}
\begin{defin}[{\cite[10.1]{DAG-IX}}]
Let $\theta\colon\mathcal T'\to\mathcal T$ be a transformation of pregeometries, and $\Theta\colon\mathcal T\mathrm{op}(\mathcal T)\to\mathcal T\mathrm{op}(\mathcal T')$ the induced functor given by composition with $\theta$.
We say that $\theta$ is \emph{unramified} if the following conditions hold:
\begin{enumerate}[(i)]
\item The pregeometries $\mathcal T$ and $\mathcal T'$ are unramified.
\item For every morphism $f\colon X\to Y$ in $\mathcal T$ and every object $Z\in\mathcal T$, the diagram
\[ \begin{tikzcd}
\Theta\Spec^\mathcal T(X\times Z) \arrow{r} \arrow{d} & \Theta\Spec^\mathcal T(X) \arrow{d} \\
\Theta\Spec^\mathcal T(X\times Y\times Z) \arrow{r} & \Theta\Spec^\mathcal T(X\times Y)
\end{tikzcd} \]
is a pullback square in $\mathcal T\mathrm{op}(\mathcal T')$.
\end{enumerate} \end{defin}
\begin{prop} \label{prop:unramified_transformation} The transformation of pregeometries $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ is unramified. \end{prop}
\begin{proof}
For $X\in\cT_{\mathrm{an}}(k)$, we denote the spectrum $\Spec^{\cT_{\mathrm{an}}(k)}(X)$ by $(\mathcal X_X, \mathcal O_X)$. For a morphism $X \to Y$ in $\cT_{\mathrm{an}}(k)$, we denote by $\mathcal O_Y^\mathrm{alg} | X$ the image of $\mathcal O_Y^\mathrm{alg}$ under the pullback functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_Y) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_X)$. We have to show that for every morphism $f \colon X \to Y$ in $\cT_{\mathrm{an}}(k)$ and every $Z\in\cT_{\mathrm{an}}(k)$, the commutative square \begin{equation} \label{eq:desired_pushout} \begin{tikzcd}
\mathcal O_{X \times Y}^\mathrm{alg} | (X \times Z) \arrow{r} \arrow{d} & \mathcal O_X^\mathrm{alg} | (X \times Z) \arrow{d} \\
\mathcal O_{X \times Y \times Z}^\mathrm{alg} | (X \times Z) \arrow{r} & \mathcal O_{X \times Z}^\mathrm{alg} \end{tikzcd} \end{equation} is a pushout in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X_{X \times Z}) \simeq \mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X_{X \times Z})$.
Form the pushout \[ \begin{tikzcd}
\mathcal O_{X \times Y}^\mathrm{alg} | (X \times Z) \arrow{r} \arrow{d} & \mathcal O_X^\mathrm{alg} | (X \times Z) \arrow{d} \\
\mathcal O_{X \times Y \times Z}^\mathrm{alg} | (X \times Z) \arrow{r} & \mathcal A \end{tikzcd} \] in $\mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal X_{X \times Z})$. Let $\mathcal A^\wedge$ be the hypercompletion of $\mathcal A$. We will prove below that $\mathcal A^\wedge$ is equivalent to $\mathcal O_{X \times Z}^\mathrm{alg}$. Assuming this, we see that $\mathcal A^\wedge$ is discrete. It follows that $\mathcal A$ is discrete as well, and therefore it is hypercomplete. We thus conclude that the square \eqref{eq:desired_pushout} is a pushout.
So we are reduced to show that the map $\mathcal A^\wedge \to \mathcal O_{X \times Z}^\mathrm{alg}$ is an equivalence. Both sheaves are hypercomplete and \cref{rem:points_of_cX_X} shows that $\mathcal X_{X \times Z}^\wedge$ has enough points. Thus, it suffices to show that for every geometric point $(x,z)$ of the adic space associated to $X \times Z$ in the sense of Huber, the diagram \begin{equation}\label{eq:pushout_stalks} \begin{tikzcd} \mathcal O_{(x,y)}^\mathrm{alg} \arrow{r} \arrow{d} & \mathcal O_x^\mathrm{alg} \arrow{d} \\ \mathcal O_{(x,y,z)}^\mathrm{alg} \arrow{r} & \mathcal O_{(x,z)}^\mathrm{alg} \end{tikzcd} \end{equation} is a pushout square, where we set $y \coloneqq f(x)$. Choose a fundamental system of étale affinoid neighborhoods $\{V_\alpha\}$ of $(x,y)$ in $X \times Y$. Set $U_\alpha \coloneqq V_\alpha \times_{X \times Y} X$ and observe that $\{U_\alpha\}$ forms a fundamental system of étale affinoid neighborhoods of $x$ in $X$. Choose moreover a fundamental system $\{W_\beta\}$ of étale affinoid neighborhoods of $z$ in $Z$. We have pullback squares \begin{equation}\label{eq:neighborhood_systems} \begin{tikzcd} U_\alpha \times W_\beta \arrow{r} \arrow{d} & U_\alpha \arrow{d} \\ V_\alpha \times W_\beta \arrow{r} & V_\alpha. \end{tikzcd} \end{equation} Assume $U_\alpha = \Sp A_\alpha$, $V_\alpha = \Sp B_\alpha$ and $W_\beta = \Sp C_\beta$. Since $U_\alpha \to V_\alpha$ is a closed immersion, the pullback above corresponds to a pushout in the category of $k$-algebras\begin{equation}\label{eq:algebraic_neighborhood_systems} \begin{tikzcd} B_\alpha \arrow{r} \arrow{d} & B_\alpha \cotimes_k C_\beta \arrow{d} \\ A_\alpha \arrow{r} & A_\alpha \cotimes_k C_\beta \end{tikzcd} \end{equation} Taking limit in Diagram \ref{eq:neighborhood_systems} (or equivalently, taking colimit in Diagram \ref{eq:algebraic_neighborhood_systems}), we observe that Diagram \ref{eq:pushout_stalks} is a pushout diagram in the category of $k$-algebras. Since the projections $V_\alpha \times W_\beta \to V_\alpha$ are flat, we see that every morphism $B_\alpha \to B_\alpha \cotimes_k C_\beta$ is flat. As a consequence, $\mathcal O_{(x,y)}^\mathrm{alg} \to \mathcal O_{(x,y,z)}^\mathrm{alg}$ is flat. The pushout \eqref{eq:pushout_stalks} is therefore a derived pushout square, completing the proof. \end{proof}
Intuitively, the pregeometry $\cT_{\mathrm{an}}(k)$ enables us to consider structure sheaves with ``non-archimedean analytic structures'' in addition to the usual algebraic structures. The unramifiedness of the transformation $\cT_{\mathrm{disc}}(k)\to\cT_{\mathrm{an}}(k)$ in \cref{prop:unramified_transformation} will imply that for certain purposes, this additional analytic structure can be ignored. Here is a simple example illustrating this phenomenon: Consider the completed tensor product $A \cotimes_B C$ of three $k$-affinoid algebras. When $C$ is finitely presented as a $B$-module, we have an isomorphism $A \cotimes_B C\simeq A\otimes_B C$. That is, in this case, for the purpose of tensor product, the analytic structure on affinoid algebras can be ignored. The proposition below elaborates on this idea:
\begin{prop} \label{prop:closed_fiber_products_Top}
Let $f \colon (\mathcal Y, \mathcal O_{\mathcal Y}) \to (\mathcal X, \mathcal O_{\mathcal X})$ and $g \colon (\mathcal X', \mathcal O_{\mathcal X'}) \to (\mathcal X, \mathcal O_{\mathcal X})$ be morphisms in $\RTop(\cT_{\mathrm{an}}(k))$.
Assume that the induced map $\theta \colon f^{-1} \mathcal O_{\mathcal X}^\mathrm{alg} \to \mathcal O_{\mathcal Y}^\mathrm{alg}$ is an effective epimorphism.
Then:
\begin{enumerate}[(i)]
\item \label{item:pullback_structured_Top} There exists a pullback diagram
\[ \begin{tikzcd}
(\mathcal Y', \mathcal O_{\mathcal Y'}) \arrow{r}{f'} \arrow{d}{g'} & (\mathcal X', \mathcal O_{\mathcal X'}) \arrow{d}{g} \\
(\mathcal Y, \mathcal O_{\mathcal Y}) \arrow{r}{f} & (\mathcal X, \mathcal O_{\mathcal X})
\end{tikzcd} \]
in $\RTop(\cT_{\mathrm{an}}(k))$. If moreover $(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \RHTop(\cT_{\mathrm{an}}(k))$, then $\mathrm{Hyp}(\mathcal Y', \mathcal O_{\mathcal Y'})$ is equivalent to the pullback computed in $\RHTop(\cT_{\mathrm{an}}(k))$.
\item \label{item:pullback_Top} The underlying diagram of $\infty$-topoi\xspace
\[ \begin{tikzcd}
\mathcal Y' \arrow{r} \arrow{d} & \mathcal X' \arrow{d} \\
\mathcal Y \arrow{r} & \mathcal X
\end{tikzcd} \]
is a pullback square in $\RTop$.
If moreover
\[(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \RHTop(\cT_{\mathrm{an}}(k)),\] then $(\mathcal Y')^\wedge$ is equivalent to the pullback computed in $\RHTop$.
\item \label{item:pushout_analytic_algebras} The diagram
\[ \begin{tikzcd}
f^{\prime -1} g^{-1} \mathcal O_{\mathcal X}^\mathrm{alg} \arrow{r} \arrow{d} & f^{\prime -1} \mathcal O_\mathcal Y^\mathrm{alg} \arrow{d} \\
g^{\prime -1} \mathcal O_{\mathcal Y}^\mathrm{alg} \arrow{r} & \mathcal O_{\mathcal Y'}^\mathrm{alg}
\end{tikzcd} \]
is a pushout square in $\mathrm{Sh}_{\mathrm{CAlg}_k}(\mathcal Y')$. If moreover $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal X', \mathcal O_{\mathcal X'})$, $(\mathcal Y, \mathcal O_\mathcal Y) \in \RHTop(\cT_{\mathrm{an}}(k))$, the same holds after applying the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$.
\item \label{item:effective_epi} The map $\theta' \colon f^{\prime -1} \mathcal O_{\mathcal X'} \to \mathcal O_{\mathcal Y'}$ is an effective epimorphism. If moreover $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal X', \mathcal O_{\mathcal X'})$, $(\mathcal Y, \mathcal O_\mathcal Y) \in \RHTop(\cT_{\mathrm{an}}(k))$, the same holds after applying the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$
\end{enumerate} \end{prop}
\begin{proof}
We first deal with the non-hypercomplete case.
\cref{prop:alg_effective_epi} shows that the morphism $f^{-1} \mathcal O_{\mathcal X} \to \mathcal O_{\mathcal Y}$ is an effective epimorphism.
Moreover, $\cT_{\mathrm{an}}(k)$ is unramified in virtue of \cref{cor:Tkan_unramified}.
Therefore \cite[Theorem 1.6]{DAG-IX} implies the first two statements.
Combining \cref{prop:unramified_transformation}, \cref{prop:alg_effective_epi} and \cite[Proposition 10.3]{DAG-IX}, we deduce the other two statements.
We now assume that $(\mathcal X, \mathcal O_\mathcal X), (\mathcal X', \mathcal O_{\mathcal X'}), (\mathcal Y, \mathcal O_\mathcal Y) \in \RHTop(\cT_{\mathrm{an}}(k))$.
Then (\ref{item:pullback_structured_Top}) and (\ref{item:pullback_Top}) follow from what we already proved and the fact that $\mathrm{Hyp}$ commutes with limits, being a right adjoint by \cref{lem:hyp_right_adjoint}.
On the other side, (\ref{item:pushout_analytic_algebras}) and (\ref{item:effective_epi}) follow from the fact that the hypercompletion functor $L \colon \mathcal Y' \to (\mathcal Y')^\wedge$ commutes with colimits and finite limits. \end{proof}
\subsection{Truncations}
Now we discuss the compatibility of the pregeometry $\cT_{\mathrm{an}}(k)$ with $n$-truncations.
\begin{defin}[{\cite[3.3.2]{DAG-V}}]
Let $\mathcal T$ be a pregeometry and let $n \ge -1$ be an integer.
The pregeometry $\mathcal T$ is said to be \emph{compatible with $n$-truncations} if for every $\infty$-topos\xspace $\mathcal X$, every $\mathcal T$-structure $\mathcal O \colon \mathcal T \to \mathcal X$ and every admissible morphism $U \to V$ in $\mathcal T$, the induced square
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le n}(\mathcal O(U)) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le n}(\mathcal O(V))
\end{tikzcd} \]
is a pullback in $\mathcal X$. \end{defin}
This definition is equivalent to say that for every $\mathcal T$-structure $\mathcal O \colon \mathcal T \to \mathcal X$ the composition $\tau_{\le n} \circ \mathcal O$ is again a $\mathcal T$-structure and the canonical morphism $\mathcal O \to \tau_{\le n} \circ \mathcal O$ is a local morphism of $\mathcal T$-structures, where $\tau_{\le n} \colon \mathcal X \to \mathcal X$ denotes the truncation functor of the $\infty$-topos\xspace $\mathcal X$.
In order to prove that $\cT_{\mathrm{an}}(k)$ is compatible with $n$-truncations for every $n \ge 0$, it will be convenient to introduce a pregeometry slightly different from $\cT_{\mathrm{an}}(k)$.
\begin{construction}
We define a pregeometry $\cT_{\mathrm{an}}^G(k)$ as follows:
\begin{enumerate}
\item the underlying category of $\cT_{\mathrm{an}}^G(k)$ is the category of smooth $k$-analytic spaces;
\item a morphism in $\cT_{\mathrm{an}}^G(k)$ is admissible if and only if it is an admissible open embedding;
\item the topology on $\cT_{\mathrm{an}}^G(k)$ is the G-topology.
\end{enumerate} \end{construction}
\begin{lem} \label{lem:G_pregeometry_0_truncations}
The pregeometry $\cT_{\mathrm{an}}^G(k)$ is compatible with $n$-truncations for every $n \ge 0$. \end{lem}
\begin{proof}
Since admissible open immersions are monomorphisms, the lemma is a direct consequence of \cite[3.3.5]{DAG-V}. \end{proof}
\begin{lem} \label{lem:etale_analytic_domain}
Let $U \to V$ be an étale morphism in $\cT_{\mathrm{an}}(k)$.
There exists a G-covering $\{V_i\to V\}_{i\in I}$, G-coverings $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ for every $i\in I$, smooth algebraic $k$-varieties $Y_i$ and $X_{ij}$, étale morphisms $X_{ij}\to Y_i$, admissible open immersions $V_i\hookrightarrow Y_i^\mathrm{an}$ and $U_{ij}\hookrightarrow X_{ij}^\mathrm{an}$ such that the morphism $U_{ij}\to V_i$ equals the restriction of the morphism $X_{ij}^\mathrm{an}\to Y_i^\mathrm{an}$ to $V_i$ for every $i\in I$ and $j\in J_i$. \end{lem} \begin{proof}
Since $V$ is smooth, there exists an affinoid G-covering $\{V_i\to V\}_{i\in I}$ such that every $V_i$ admits an étale morphism to a polydisc $\mathbf D^{n_i}$.
By \cite[Proposition 1.7.1]{Huber_Etale_1996}, the affinoid algebra associated to $V_i$ has a presentation of the form
\[k\langle T_1,\dots,T_{n_i},T'_1,\dots,T'_{m_i}\rangle/(f_1,\dots,f_{m_i})\]
such that the determinant $\det\big(\frac{\partial f_\alpha}{\partial T'_\beta}\big)_{\alpha,\beta=1,\dots,m_i}$ is invertible in $k\langle T_1,\dots,T_{n_i}\rangle$.
By \cite[Chap.\ III Theorem 7 and Remark 2]{Elkik_Solutions_1973}, there exists a smooth affine scheme $Y_i$ and an étale morphism $Y_i\to\mathbb A^{n_i}_k$ such that $V_i$ is isomorphic to the fiber product $Y_i^\mathrm{an}\times_{(\mathbb A^{n_i}_k)^\mathrm{an}} \mathbf D^{n_i}$.
We now fix $i\in I$.
Since the morphism $U\to V$ is étale, by base change, the morphism $U\times_V V_i\to V_i$ is étale.
So the composition
\[ U\times_V V_i\to V_i\to\mathbf D^{n_i}\]
is étale.
Let $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ be an affinoid G-covering.
For every $j\in J_i$, by \cite[Proposition 1.7.1]{Huber_Etale_1996}, the affinoid algebra associated to $U_{ij}$ has a presentation of the form
\[k\langle T_1,\dots,T_{n_i},T'_1,\dots,T'_{m_{ij}}\rangle/(f_1,\dots,f_{m_{ij}})\]
such that the determinant $\det\big(\frac{\partial f_\alpha}{\partial T'_\beta}\big)_{\alpha,\beta=1,\dots,m_{ij}}$ is invertible in $k\langle T_1,\dots,T_{n_i}\rangle$.
By \cite[Chap.\ III Theorem 7]{Elkik_Solutions_1973} again, there exists a smooth affine scheme $Z_{ij}$ and an étale morphism $Z_{ij}\to\mathbb A^{n_i}_k$ such that $U_{ij}$ is isomorphic to the fiber product $Z_i^\mathrm{an}\times_{(\mathbb A^{n_i}_k)^\mathrm{an}} \mathbf D^{n_i}$.
Let $X_{ij}\coloneqq Y_i\times_{\mathbb A^{n_i}_k} Z_{ij}$.
By the universal property of the fiber product, there exists a unique map $r\colon U_{ij}\to X^\mathrm{an}_{ij}$ making the following diagram commutative:
\[\begin{tikzcd}
U_{ij} \arrow{d} \arrow[swap]{rd}{r} \arrow{rrd}{t} & &\\
U\times_V V_i \arrow{d} & X^\mathrm{an}_{ij}\arrow{d}\arrow{r}{s} & Z^\mathrm{an}_{ij}\arrow{d}\\
V_i \arrow[hookrightarrow]{r} & Y^\mathrm{an}_i \arrow{r} & (\mathbb A^{n_i}_k)^\mathrm{an}
\end{tikzcd}\]
The map $t$ is an admissible open immersion, so it is in particular étale.
The map $s$ is étale by base change, so it is étale.
Since $t=s\circ r$, we deduce that the map $r$ is étale.
Moreover, the map $t$ is a monomorphism, so the map $r$ is also a monomorphism.
Since the map $r$ is étale, we deduce that it is an admissible open immersion.
\end{proof}
\begin{lem} \label{lem:pullbacks_are_local}
Let $\mathcal X$ be an $\infty$-topos\xspace and let
\[ \begin{tikzcd}
U \arrow{r} \arrow{d} & W \arrow{r} \arrow{d} & Z \arrow{d} \\
V \arrow{r}{p} & Y \arrow{r} & X
\end{tikzcd} \]
be a diagram in $\mathcal X$.
Assume that the left and the outer squares are pullbacks and that $p$ is an effective epimorphism.
Then the right square is a pullback as well. \end{lem}
\begin{proof}
Let $W' \coloneqq Y \times_X Z$.
We obtain a commutative diagram
\[ \begin{tikzcd}
U \arrow{r} \arrow{d} & W' \arrow{r} \arrow{d} & Z \arrow{d} \\
V \arrow{r}{p} & Y \arrow{r} & X .
\end{tikzcd} \]
Since the outer square is a pullback by our assumption, the left square is a pullback as well.
The universal property of pullbacks induces a morphism $\alpha \colon W \to W'$.
By hypothesis, the induced map $\alpha \times_Y V \colon W \times_Y V \to W' \times_Y V$ is an equivalence.
Since $p$ is an effective epimorphism, the pullback functor $p^{-1} \colon \mathcal X_{/Y} \to \mathcal X_{/V}$ is conservative (cf.\ \cite[6.2.3.16]{HTT}).
We conclude that $\alpha$ is an equivalence, completing the proof. \end{proof}
\begin{thm} \label{thm:compatibility_truncations}
The pregeometry $\cT_{\mathrm{an}}(k)$ is compatible with $n$-truncations for every $n \ge 0$. \end{thm}
\begin{proof}
When $n \ge 1$, the statement is a direct consequence of \cite[3.3.5]{DAG-V}.
We now prove the case $n = 0$.
Let $\mathcal X$ be an $\infty$-topos\xspace and let $\mathcal O \in \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
For the purpose of this proof, we will say that a morphism $f \colon U \to V$ in $\cT_{\mathrm{an}}(k)$ is \emph{compatible} if the induced diagram
\begin{equation} \label{eq:compatibility_zero_truncations}
\begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(U) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V)
\end{tikzcd}
\end{equation}
is a pullback square.
We need to show that every étale morphism is compatible.
Let us start by observing the following properties of compatible morphisms:
\begin{enumerate}
\item \label{item:compatible_analytic_domain} Admissible open immersions are compatible.
This follows from \cref{lem:G_pregeometry_0_truncations}.
\item \label{item:compatible_analytification_etale} If $f \colon X \to Y$ is an \'etale morphism of smooth $k$-varieties, then the analytification $f^\mathrm{an} \colon X^\mathrm{an} \to Y^\mathrm{an}$ is compatible.
Indeed, let $\cT_{\mathrm{\acute{e}t}}(k)$ be the pregeometry of \cite[Definition 4.3.1]{DAG-V}.
The analytification functor induces a morphism of pregeometries $\varphi \colon \cT_{\mathrm{\acute{e}t}}(k) \to \cT_{\mathrm{an}}(k)$.
We have $\mathcal O(X^\mathrm{an}) = (\mathcal O \circ \varphi)(X)$ and $\mathcal O(Y^\mathrm{an}) = (\mathcal O \circ \varphi)(Y)$.
Since $\mathcal O \circ \varphi$ is a $\cT_{\mathrm{\acute{e}t}}(k)$-structure on $\mathcal X$, the statement follows from the fact that $\cT_{\mathrm{\acute{e}t}}(k)$ is compatible with $0$-truncations (cf.\ \cite[Proposition 4.3.28]{DAG-V}).
\item \label{item:compatible_composition} Compatible morphisms are stable under composition. This follows from the composition property of pullback squares (cf.\ \cite[4.4.2.1]{HTT}).
\item \label{item:compatible_pullback} Suppose given a pullback square
\[ \begin{tikzcd}
U \arrow{r}{g} \arrow{d}{f'} & V \arrow{d}{j} \\
X \arrow{r}{f} & Y
\end{tikzcd} \]
where $f$ is compatible and $j$ is an admissible open immersion.
Then $f'$ is compatible.
To see this, consider the commutative diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{d} \arrow{r} & \mathcal O(X) \arrow{d} \arrow{r} & \tau_{\le 0} \mathcal O(X) \arrow{d} \\
\mathcal O(V) \arrow{r} & \mathcal O(Y) \arrow{r} & \tau_{\le 0} \mathcal O(Y).
\end{tikzcd} \]
Since admissible open immersions are in particular étale morphisms and since $\mathcal O$ is a $\cT_{\mathrm{an}}(k)$-structure, we see that the left square is a pullback diagram.
On the other side, the right square is a pullback because $f$ is compatible by assumption.
We conclude that the outer square in the commutative diagram
\[ \begin{tikzcd}
\mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(U) \arrow{r} \arrow{d} & \tau_{\le 0} \mathcal O(X) \arrow{d} \\
\mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(Y)
\end{tikzcd} \]
is a pullback square.
We remark that $\tau_{\le 0} \circ \mathcal O$ is a $\cT_{\mathrm{an}}^G(k)$-structure in virtue of \cref{lem:G_pregeometry_0_truncations}.
So by \cite[Proposition 3.3.3]{DAG-V}, the right square is a pullback as well.
It follows that the left square is also a pullback, completing the proof of the claim.
\item \label{item:compatible_G_local_source} Being compatible is G-local on the source.
Indeed, let $f \colon X \to Y$ be a morphism in $\cT_{\mathrm{an}}(k)$ and assume there exists a G-covering $\{X_i\}$ of $X$ such that each composite $f_i \colon X_i \to X \to Y$ is compatible.
We want to prove that $f$ is compatible as well.
Consider the commutative diagram
\[ \begin{tikzcd}
\coprod \mathcal O(X_i) \arrow{r} \arrow{d} & \mathcal O(X) \arrow{r} \arrow{d} & \mathcal O(Y) \arrow{d} \\
\coprod \tau_{\le 0} \mathcal O(X_i) \arrow{r} & \tau_{\le 0} \mathcal O(X) \arrow{r} & \tau_{\le 0} \mathcal O(Y).
\end{tikzcd} \]
Since G-coverings are étale coverings, it follows from the properties of $\cT_{\mathrm{an}}(k)$-structures that the total morphism $\coprod \mathcal O(U_i) \to \mathcal O(U)$ is an effective epimorphism. Since $\tau_{\le 0}$ commutes with coproducts (being a left adjoint) and with effective epimorphisms (cf.\ \cite[7.2.1.14]{HTT}), we conclude that the total morphism $\coprod \tau_{\le 0} \mathcal O(U_i) \to \tau_{\le 0} \mathcal O(U)$ is an effective epimorphism as well.
Since each $X_i \to X$ is an admissible open immersion, Property (\ref{item:compatible_analytic_domain}) implies that the left square is a pullback.
Moreover, the outer square is a pullback by hypothesis.
Thus, \cref{lem:pullbacks_are_local} shows that the right square is a pullback as well, completing the proof of this property.
\end{enumerate}
Let now $f \colon U \to V$ be an étale morphism in $\cT_{\mathrm{an}}(k)$.
We will prove that $f$ is compatible.
Using \cref{lem:etale_analytic_domain} we obtain a G-covering $\{V_i\to V\}_{i\in I}$, G-coverings $\{U_{ij}\to U\times_V V_i\}_{j\in J_i}$ for every $i\in I$, smooth algebraic $k$-varieties $Y_i$ and $X_{ij}$, étale morphisms $X_{ij}\to Y_i$, admissible open immersions $V_i\hookrightarrow Y_i^\mathrm{an}$ and $U_{ij}\hookrightarrow X_{ij}^\mathrm{an}$ such that the morphism $U_{ij}\to V_i$ equals to restriction of the morphism $X_{ij}^\mathrm{an}\to Y_i^\mathrm{an}$.
In particular we can factor $U_{ij} \to V_i$ as the composition
\[ \begin{tikzcd}
U_{ij} \arrow{r} & X_{ij}^\mathrm{an} \times_{Y_i^\mathrm{an}} V_i \arrow{r} & V_i
\end{tikzcd} \]
where the first morphism is an admissible open immersion and the second is compatible by Property (\ref{item:compatible_pullback}) of compatible morphisms.
Therefore, Property (\ref{item:compatible_composition}) implies that $U_{ij} \to V_i$ is compatible.
Finally, using Property (\ref{item:compatible_G_local_source}) we conclude that the morphisms $U \times_V V_i \to V_i$ are compatible.
We are therefore reduced to prove the following statement: given a morphism $f \colon U \to V$, suppose that there exists a G-covering $\{v_i \colon V_i \to V\}$ such that each base change $f_i \colon U_i \coloneqq U \times_V V_i \to V_i$ is compatible, then $f$ is compatible.
We consider the commutative diagram
\[ \begin{tikzcd}
\coprod \mathcal O(U_i) \arrow{r} \arrow{d} & \mathcal O(U) \arrow{d} \arrow{r} & \tau_{\le 0} \mathcal O(U) \arrow{d} \\
\coprod \mathcal O(V_i) \arrow{r} & \mathcal O(V) \arrow{r} & \tau_{\le 0} \mathcal O(V).
\end{tikzcd} \]
Since $\mathcal O$ is a $\cT_{\mathrm{an}}(k)$-structure, the total morphism $\coprod \mathcal O(U_i) \to \mathcal O(U)$ is an effective epimorphism.
Moreover, since each $V_i \to V$ is an admissible open immersion, so in particular étale, we see that the left square is a pullback.
By hypothesis, the outer square is a pullback as well, so we conclude the proof using \cref{lem:pullbacks_are_local}. \end{proof}
\begin{cor} \label{cor:truncation_derived_kanal_spaces}
Let $(\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
Then $(\mathcal X, \pi_0( \mathcal O_\mathcal X ))$ is also a derived $k$-analytic\xspace space.
Moreover, we have $(\pi_0(\mathcal O_\mathcal X))^\mathrm{alg} \simeq \pi_0(\mathcal O_\mathcal X^\mathrm{alg})$. \end{cor}
\begin{proof}
It follows from \cref{thm:compatibility_truncations} that $\pi_0(\mathcal O_\mathcal X)$ is a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X$.
Let $\varphi \colon \cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$ be the transformation of pregeometries induced by the analytification functor.
Then we have by definition
\[ (\pi_0(\mathcal O_\mathcal X))^\mathrm{alg} = (\pi_0^{\mathcal X} \circ \mathcal O_\mathcal X ) \circ \varphi \simeq \pi_0^\mathcal X \circ ( \mathcal O_\mathcal X \circ \varphi) = \pi_0( \mathcal O_\mathcal X^\mathrm{alg}) , \]
where $\pi_0^\mathcal X$ denotes the truncation functor of the $\infty$-topos\xspace $\mathcal X$.
In particular, we see that $(\mathcal X, \pi_0(\mathcal O_\mathcal X)^\mathrm{alg})$ is a derived $k$-analytic\xspace space. \end{proof}
\begin{defin}
A $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos $(\mathcal X, \mathcal O_\mathcal X)$ is said to be \emph{discrete} if $\mathcal O_\mathcal X$ is a discrete object in $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)$.
We denote by $\RTop^0(\cT_{\mathrm{an}}(k))$ the full subcategory of $\RTop(\cT_{\mathrm{an}}(k))$ spanned by discrete $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace.
We say that a derived $k$-analytic\xspace space $(\mathcal X, \mathcal O_\mathcal X)$ is \emph{discrete} if it is discrete as a $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace.
We denote by $\mathrm{dAn}_k^0$ the full subcategory of $\mathrm{dAn}_k$ spanned by discrete derived $k$-analytic\xspace spaces.
\end{defin}
Choose a geometric envelope $\mathcal G_{\mathrm{an}}(k)$ for $\cT_{\mathrm{an}}(k)$ and let $\mathcal G_{\mathrm{an}}(k) \to \mathcal G_{\mathrm{an}}^{\le 0}(k)$ be a $0$-stub for $\mathcal G_{\mathrm{an}}(k)$ (cf.\ \cite[Definition 1.5.10]{DAG-V}). It follows from \cite[Proposition 1.5.14]{DAG-V} that \[ \RTop(\mathcal G_{\mathrm{an}}^{\le 0}(k)) \simeq \RTop^0(\cT_{\mathrm{an}}(k)) . \] The relative spectrum (cf.\ \cite[§ 2.1]{DAG-V}) is a functor \[ \Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}} \colon \RTop(\mathcal G_{\mathrm{an}}(k)) \to \RTop(\mathcal G_{\mathrm{an}}^{\le 0}(k)) \simeq \RTop^0(\cT_{\mathrm{an}}(k)) , \] which we refer to as the \emph{truncation functor}. Using \cref{thm:compatibility_truncations}, we can identify the action of this functor on objects with the assignment \[ (\mathcal X, \mathcal O_\mathcal X) \mapsto (\mathcal X, \pi_0(\mathcal O_\mathcal X)) . \]
The following proposition is an analogue of \cite[Proposition 3.13]{Porta_DCAGI} and of \cite[Proposition 2.2.4.4]{HAG-II}:
\begin{prop} \label{prop:truncation_and_finite_limits}
Let $i \colon \mathrm{dAn}_k^0 \to \mathrm{dAn}_k$ denote the natural inclusion functor. Then:
\begin{enumerate}[(i)]
\item \label{item:truncation_derived_analytic_spaces} The functor $\Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}} \colon \mathcal T\mathrm{op}(\cT_{\mathrm{an}}(k)) \to \mathcal T\mathrm{op}^0(\cT_{\mathrm{an}}(k))$ restricts to a functor $\mathrm{t}_0 \colon \mathrm{dAn}_k \to \mathrm{dAn}_k^0$.
\item \label{item:truncation_right_adjoint} The functor $i$ is left adjoint to the functor $\mathrm{t}_0$.
\item \label{item:truncated_spaces_embeds_fully_faithfully} The functor $i$ is fully faithful.
\end{enumerate} \end{prop}
\begin{proof}
The statement (\ref{item:truncated_spaces_embeds_fully_faithfully}) holds by definition of the functor $i$.
It follows from \cref{cor:truncation_derived_kanal_spaces} that the functor $\Spec^{\cG_{\mathrm{an}}^{\le 0}(k)}_{\cG_{\mathrm{an}(k)}}$ respects the $\infty$-category\xspace of derived $k$-analytic\xspace spaces.
Therefore the statements (\ref{item:truncation_derived_analytic_spaces}) and (\ref{item:truncation_right_adjoint}) follow immediately. \end{proof}
\section{Fully faithful embedding of $k$-analytic\xspace spaces} \label{sec:fullyfaithfulness}
In this section, we construct a functor $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ from the category of $k$-analytic\xspace spaces to the category of derived $k$-analytic\xspace spaces. We will prove that $\Phi$ is a fully faithful embedding.
First we will define the functor $\Phi$ on objects, then we will define it on morphisms.
Let $\mathrm{dAn}_k^{1, 0}$ be the full subcategory of $\mathrm{dAn}_k$ spanned by derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_{\mathcal X})$ such that $\mathcal X$ is $1$-localic and $\mathcal O_{\mathcal X}$ is $0$-truncated.
\begin{defin}
Let $X$ be a $k$-analytic\xspace space and let $\mathcal X_X$ be the étale $\infty$-topos\xspace associated to $X$.
We define a functor $\mathcal O_X\colon\cT_{\mathrm{an}}(k)\to\mathcal X_X$ by the formula
\[ \mathcal O_X (M) (U) = \Hom_{\mathrm{An}_k} ( U, M ).\] \end{defin}
\begin{lem} \label{lem:inclusion}
Let $X$ be a $k$-analytic\xspace space.
Then $\mathcal O_X$ is a 0-truncated $\cT_{\mathrm{an}}(k)$-structure on the $\infty$-topos\xspace $\mathcal X_X$.
Let $\mathcal X_X^\wedge$ denote the hypercompletion of $\mathcal X_X$ and let $\Phi(X)$ denote the pair $(\mathcal X_X^\wedge,\mathcal O_X)$.
Then $\Phi(X)$ is a derived $k$-analytic\xspace space. \end{lem} \begin{proof}
In order to prove that $\mathcal O_X$ is a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X_X$, it suffices to verify that if $\{ M_i \to M\}$ is an étale covering of $M\in\cT_{\mathrm{an}}(k)$, then the induced map $\coprod_i \mathcal O_X (M_i) \to \mathcal O_X (M)$ is an effective epimorphism in $\mathcal X_X$.
Observe that for any $U$ in the étale site on $X$ and any morphism $U \to M$, there exists an étale covering $\{U_j\to U\}$ such that the composite morphisms $U_j\to M$ factor though $\coprod M_i \to M$.
So we conclude using \cite[Corollary 2.9]{Porta_Yu_Higher_analytic_stacks_2014}.
Since $\mathcal O_X$ is $0$-truncated by construction, it is hypercomplete.
Therefore the second statement follows from the first. \end{proof}
In order to define the functor $\Phi$ on morphisms, our strategy is to prove that the mapping spaces $\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))$ are discrete for all $X, Y \in \mathrm{An}_k$ (cf.\ \cref{prop:discrete_mapping_spaces_I}). In this way we can promote $\Phi$ to an $\infty$-functor without the need to specify higher homotopies.
We begin by introducing an auxiliary functor $\Upsilon$. Let $\mathrm{LRT}$ denote the $2$-category of locally ringed 1-topoi and let $\Upsilon\colon\mathrm{An}_k\to\mathrm{LRT}$ be the functor sending every $k$-analytic\xspace space to the associated locally ringed étale 1-topos. For $X\in\mathrm{An}_k$, we denote by $\mathcal O_X^\mathrm{alg}$ the structure sheaf of $k$-algebras of $\Upsilon(X)$.
\begin{lem}\label{lem:affine_embedding}
Let $X$ be a $k$-affinoid space, $Y$ a $k$-analytic space and $\alpha\colon X\to Y$ a morphism.
Then there exists a positive integer $N$ and a monomorphism $\beta\colon X\hookrightarrow\mathbf D^N_Y$ over $Y$, where $\mathbf D^N_Y$ denotes the unit polydisc over $Y$. \end{lem} \begin{proof}
Let $A\coloneqq\Gamma(\mathcal O_X)$.
Write $A=k\langle x_1,\dots,x_n\rangle /I$ as a quotient of a Tate algebra.
Denote by $a_1,\dots,a_n$ the images of $x_1,\dots,x_n$ in $A$.
We cover $X$ by finitely many rational domains $U_i$ such that $\alpha(U_i)$ is contained in an affinoid domain $V_i\subset Y$.
Write \[\Gamma(\mathcal O_{U_i})=A\Big\langle\frac{b_{i1}}{b_{i0}},\dots,\frac{b_{in_i}}{b_{i0}}\Big\rangle,\]
where $b_{i0},\dots,b_{in_i}$ is a collection of elements in $A$ with no common zero.
Let $c_{i0},\dots,c_{in_i}$ be elements in $k$ such that $\abs{c_{ij}}\ge\rho(b_{ij})$ for $j=0,\dots,n_i$, where $\rho(\cdot)$ denotes the spectral radius.
Consider the morphism
\[\Gamma(\mathcal O_{V_i})\langle y_1,\dots,y_n, y_{i0},\dots,y_{in_i}\rangle\to\Gamma(\mathcal O_{U_i})\]
that sends $y_j$ to $a_j$ and $y_{ij}$ to $b_{ij}/c_{ij}$.
It induces a monomorphism $U_i\hookrightarrow\mathbf D^{n+n_i+1}_{V_i}$.
Let $N\coloneqq n+\sum_{i=1}^m(n_i+1)$.
Consider the unit polydisc $\mathbf D^N_Y$ over $Y$.
We denote by $y_i,y_{ij}$ for $i=1,\dots,m$, $j=0,\dots,n_i$ the coordinate functions on $\mathbf D^N_Y$.
Let $\beta\colon X\to\mathbf D^N_Y$ be the morphism that sends $y_i$ to $a_i$ and $y_{ij}$ to $b_{ij}/c_{ij}$ for all $i=1,\dots,m$, $j=0,\dots,n_i$.
Let $Z_i$ be the admissible open subset in $\mathbf D^N_Y$ given by the inequalities $\abs{c_{i0}\cdot y_{ij}}\le\abs{c_{i0}\cdot y_{i0}}$ for $j=1,\dots,n_i$.
Let $Z'_i\coloneqq Z_i\times_Y V_i$.
We see that $\beta^{-1}(Z'_i)$ is $U_i$.
By construction, $\beta|_{U_i}\colon U_i\to Z'_i$ is a monomorphism.
We conclude that $\beta\colon X\to\mathbf D^N_Y$ is a monomorphism. \end{proof}
\begin{lem}\label{lem:rigidity}
Let $f\colon X\to Y$ be a morphism of $k$-analytic\xspace spaces.
Let
\[(f,f^\#)\colon\big(\mathrm{Sh}_\mathrm{Set} (X_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_X\big) \to \big(\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_Y\big)\]
denote the induced morphism of locally ringed 1-topoi.
Let $t$ be a 2-morphism from $(f,f^\#)$ to itself.
Then $t$ equals the identity. \end{lem} \begin{proof}
Using \cref{lem:affine_embedding}, the same proof of \cite[Tag 04IJ]{stacks-project} applies. \end{proof}
\begin{lem}\label{lem:first_fully_faithful}
The functor
\begin{align*}
\Upsilon \colon \mathrm{An}_k &\longrightarrow \mathrm{LRT}\\
X &\longmapsto (\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X^\mathrm{alg})
\end{align*}
is fully faithful. \end{lem}
\begin{proof}
Let $X,Y$ be two $k$-analytic\xspace spaces.
Let \[(g,g^\#)\colon\big(\mathrm{Sh}_\mathrm{Set} (X_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_X\big) \to \big(\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}),\mathcal O^\mathrm{alg}_Y\big)\]
be a morphism of locally ringed 1-topoi.
We would like to show that there exists a unique morphism of $k$-analytic\xspace spaces $f\colon X\to Y$ which induces $(g,g^\#)$.
We proceed along the same lines as \cite[Tag 04JH]{stacks-project}.
Let $g^{-1}\colon\mathrm{Sh}_\mathrm{Set} (Y_\mathrm{\acute{e}t}) \leftrightarrows \mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}) \colon g_*$ denote the morphism of 1-topoi.
First, we assume that $X=\Sp A$, $Y=\Sp B$ for some $k$-affinoid algebras $A$ and $B$.
Since $B=\Gamma(Y_\mathrm{\acute{e}t},\mathcal O^\mathrm{alg}_Y)$ and $A=\Gamma(X_\mathrm{\acute{e}t},\mathcal O^\mathrm{alg}_X)$, we see that $g^\#$ induces a map of affinoid algebras $\varphi\colon B\to A$.
Let $f=\Sp\varphi\colon X\to Y$.
Let us show that $f$ induces $(g,g^\#)$.
Let $V\to Y$ be an affinoid space étale over $Y$.
Assume $V=\Sp C$.
By \cite[Proposition 1.7.1]{Huber_Etale_1996}, we can write
\[ C=B\langle x_1,\dots,x_n\rangle / (r_1,\dots,r_n), \]
where $r_1,\dots,r_n\in B\langle x_1,\dots,x_n\rangle$ and the determinant $\mathrm{Jac}(r_1,\dots,r_n)$ is invertible in $C$.
Now the sheaf $h_V$ on $Y_\mathrm{\acute{e}t}$ is the equalizer of the two maps
\[
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_Y \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_Y
}
\]
where $b=0$ and $a(h_1,\dots,h_n)=\big(r_1(h_1,\dots,h_n),\dots,r_n(h_1,\dots,h_n)\big)$.
We have the following commutative diagram
\begin{equation}\label{eq:equalizers}
\xymatrix{
g^{-1} h_V \ar@{.>}[d]^{\alpha} \ar[r] & \prod_{i=1}^n g^{-1} \mathcal O^\mathrm{alg}_Y \ar[d]^{\prod g^\#} \ar@<0.6ex>[r]^{g^{-1} a} \ar@<-0.6ex>[r]_{g^{-1} b} & \prod_{j=1}^n g^{-1}\mathcal O^\mathrm{alg}_Y \ar[d]^{\prod g^\#}\\
h_{X\times_Y V}\ar[r] & \prod_{i=1}^n \mathcal O^\mathrm{alg}_X \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_X,
}
\end{equation}
where $b'=0$, $a'(h_1,\dots,h_n)=\big(\varphi(r_1)(h_1,\dots,h_n),\dots,\varphi(r_n)(h_1,\dots,h_n)\big)$, the two horizontal lines are equalizer diagrams and the dotted arrow $\alpha$ is obtained by the universal property of equalizers.
We claim that the map $\alpha\colon g^{-1} h_V\to h_{X\times_Y V}$ is an isomorphism.
Let us check this on the stalks.
Let $\bar x$ be a geometric point of the adic space $X^\mathrm{ad}$ associated to $X$ in the sense of Huber.
Denote by $p$ the associated point of the 1-topos $\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t})$ (cf.\ \cref{rem:points_of_cX_X}).
Applying localization at $p$ to Diagram \eqref{eq:equalizers}, we would like to show that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Set $q\coloneqq g\circ p$.
This is a point of the 1-topos $\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t})$.
We denote by $\bar y$ the corresponding geometric point of the adic space $Y^\mathrm{ad}$ associated to $Y$.
Then the localization of the map $g^\#$ at $p$ has the following description
\[(g^\#)_p\colon\mathcal O^\mathrm{alg}_{Y,\bar y} = \mathcal O^\mathrm{alg}_{Y,q} = (g^{-1} \mathcal O^\mathrm{alg}_Y)_p \longrightarrow \mathcal O^\mathrm{alg}_{X,p} = \mathcal O^\mathrm{alg}_{X,\bar x}.\]
It suffices to treat the two cases: either $V\to Y$ is finite étale, or $V\to Y$ is an affinoid domain embedding.
In the first case, there exists an étale neighborhood $U$ of $\bar y$ in $Y^\mathrm{ad}$ such that the pullback morphism $V\times_Y U\to U$ splits.
Then the equalizer of
\begin{equation} \label{eq:O_Y_V}
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_Y(U) \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_Y(U)
}
\end{equation}
is isomorphic to the equalizer of
\begin{equation} \label{eq:ky}
\xymatrix{
\prod_{i=1}^n k(\bar y) \ar@<0.6ex>[r]^{a} \ar@<-0.6ex>[r]_{b} & \prod_{j=1}^n k(\bar y),
}
\end{equation}
where $k(\bar y)$ denotes the residue field of $\bar y$.
Similarly, there exists an étale neighborhood $U'$ of $\bar x$ in $X^\mathrm{ad}$ such that the pullback morphism $X\times_Y V\times_X U'\simeq V\times_Y U'\to U'$ splits.
Then the equalizer of
\begin{equation} \label{eq:O_X_V}
\xymatrix{
\prod_{i=1}^n \mathcal O^\mathrm{alg}_X(U') \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n \mathcal O^\mathrm{alg}_X(U')
}
\end{equation}
is isomorphic to the equalizer of
\begin{equation} \label{eq:kx}
\xymatrix{
\prod_{i=1}^n k(\bar x) \ar@<0.6ex>[r]^{a'} \ar@<-0.6ex>[r]_{b'} & \prod_{j=1}^n k(\bar x).
}
\end{equation}
Since the equalizer of \cref{eq:ky} and the equalizer of \cref{eq:kx} are isomorphic by construction,
we deduce that the equalizer of \cref{eq:O_Y_V} and the equalizer of \cref{eq:O_X_V} are isomorphic.
Taking colimits over all such étale neighborhoods, we conclude that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Then let us consider the second case where $V\to Y$ is an affinoid domain embedding.
If the geometric point $\bar y$ can be lifted to a geometric point in $V$, then for any étale neighborhood $U$ of $\bar y$ in $Y^\mathrm{ad}$ refining $V$, the equalizer of \cref{eq:O_Y_V} consists of a single element.
The same goes for the equalizer of \cref{eq:O_X_V}.
If the geometric point cannot be lifted to a geometric point in $V$, then the equalizer of \cref{eq:O_Y_V} is empty, so is the equalizer of \cref{eq:O_X_V}.
We conclude that $\alpha_p\colon (g^{-1} h_V)_p\to (h_{X\times_Y V})_p$ is an isomorphism.
Now the same argument in \cite[Tag 04I6]{stacks-project} shows that the isomorphisms $g^{-1} h_V\to h_{X\times_Y V}$ are functorial with respect to $V$ and that the map $f\colon X\to Y$ indeed induces the morphism of locally ringed 1-topoi $(g,g^\#)$ we started with.
Finally, the argument in \cite[Tag 04I7]{stacks-project} allows us to deduce the general case from the affinoid case considered above. \end{proof}
\begin{lem}\label{lem:alg_faithful}
Let $\mathcal X$ be an $\infty$-topos\xspace.
The induced functor
\[
\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X)
\]
is faithful. \end{lem}
\begin{proof}
We can factor the functor $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X)$ as
\[ \begin{tikzcd}
\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \arrow{r} & \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\tau_{\le 0} \mathcal X) \arrow{r}{U} & \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\tau_{\le 0} \mathcal X) ,
\end{tikzcd} \]
where $\cT_{\mathrm{\acute{e}t}}(k)$ is the pregeometry introduced in \cite[Definition 4.3.1]{DAG-V}.
Combining \cite[Propositions 4.3.16, 2.6.16 and Remark 2.5.13]{DAG-V} we see that the functor $U$ is faithful.
So we are reduced to prove the same statement for
\[ \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\tau_{\le 0} \mathcal X) \to \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\tau_{\le 0} \mathcal X) . \]
The mapping spaces of $\tau_{\le 0} \mathcal X$ are discrete by definition.
It follows from \cite[2.3.4.18]{HTT} that we can find a minimal $1$-category $\mathcal D$ and a categorical equivalence $\tau_{\le 0} \mathcal X \simeq \mathcal D$.
Let $F, G \in \mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal D)$. We want to show that the natural morphism
\[
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal D)}(F,G) \to \Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)}(F^\mathrm{alg}, G^\mathrm{alg})
\]
is a homotopy monomorphism.
Since $F$ and $G$ take values in the $1$-category $\mathcal D$, both mapping spaces above are sets.
We want to prove that the given map is a monomorphism.
Since $\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)$ is a $1$-category, two natural transformations $\varphi$ and $\psi$ represent the same object in $\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{\acute{e}t}}(k)}(\mathcal D)}(F^\mathrm{alg}, G^\mathrm{alg})$ if and only if they are equal, in the sense that
\[
\begin{tikzcd}
F^\mathrm{alg}(X) \arrow{r}{\varphi^\mathrm{alg}_{X}} \arrow{d}[swap]{\mathrm{id}_{F^\mathrm{alg}(X)}} & G^\mathrm{alg}(X) \arrow{d}{\mathrm{id}_{G^\mathrm{alg}(X)}} \\
F^\mathrm{alg}(X) \arrow{r}{\psi^\mathrm{alg}_{X}} & G^\mathrm{alg}(X)
\end{tikzcd}
\]
commutes for every $X \in \cT_{\mathrm{\acute{e}t}}(k)$.
Fix $U \in \cT_{\mathrm{an}}(k)$.
We first assume that $U$ is isomorphic to an affinoid domain in $X^\mathrm{an}$ for a smooth $k$-variety $X$.
Since $U \to X^\mathrm{an}$ is a monomorphism, we have a pullback square
\[\begin{tikzcd}
U \arrow[-, double equal sign distance]{r} \arrow[-, double equal sign distance]{d} & U \arrow[hook]{d} \\
U \arrow[hook]{r} & X^\mathrm{an}.
\end{tikzcd}\]
Since $U \to X$ is an affinoid embedding, it is étale, so it is an admissible morphism in $\cT_{\mathrm{an}}(k)$.
Applying the functor $F$, we obtain another pullback square
\[\begin{tikzcd}
F(U) \arrow[-, double equal sign distance]{r} \arrow[-, double equal sign distance]{d} & F(U) \arrow{d} \\
F(U) \arrow{r} & F(X^\mathrm{an}).
\end{tikzcd}\]
So $F(U) \to F(X^\mathrm{an})$ is a monomorphism in the category $\mathcal D$.
We have a commutative cube
\[
\begin{tikzcd}[column sep=small, row sep=small]
{} & F(U) \arrow{dl} \arrow{rr} \arrow[dotted]{dd} & & F(X^\mathrm{an}) \arrow{dl} \\
G(U) \arrow[crossing over]{rr} & & G(X^\mathrm{an}) \\
{} & F(U) \arrow{rr} \arrow{dl} & & F(X^\mathrm{an}) \arrow[-, double equal sign distance]{uu} \arrow{dl} \\
G(U) \arrow{rr} \arrow[-, double equal sign distance]{uu} & & G(X^\mathrm{an}) \arrow[-, double equal sign distance, crossing over]{uu}
\end{tikzcd}
\]
where the dotted arrow exists by the universal property of the pullbacks.
Since $F(U) \to F(X^\mathrm{an})$ is a monomorphism, the dotted arrow is in fact the identity of $F(U)$.
Let us now consider a general $U \in \cT_{\mathrm{an}}(k)$.
Choose a G-covering of $U$ by affinoid domains $\{U_i \to U\}$ such that each $U_i$ is isomorphic to an affinoid domain in $X_i^\mathrm{an}$ for some smooth $k$-variety $X_i$.
Set $U^0 \coloneqq \coprod U_i$ and consider the \v{C}ech nerve $U^\bullet \to U$. Observe that both $F(U^\bullet)$ and $G(U^\bullet)$ are groupoid objects in the 1-topos $\mathcal D$ and that their realizations are respectively $F(U)$ and $G(U)$. Since we have a commutative square of groupoids
\[
\begin{tikzcd}
F(U^\bullet) \arrow{r}{\varphi_{U^\bullet}} \arrow[-, double equal sign distance]{d} & G(U^\bullet) \arrow[-, double equal sign distance]{d} \\
F(U^\bullet) \arrow{r}{\psi_{U^\bullet}} & G(U^\bullet),
\end{tikzcd}
\]
the square
\[
\begin{tikzcd}
F(U) \arrow{r}{\varphi_U} \arrow[-, double equal sign distance]{d} & G(U) \arrow[-, double equal sign distance]{d} \\
F(U) \arrow{r}{\psi_U} & G(U)
\end{tikzcd}
\]
commutes as well.
Since the identity is functorial, the proof is now complete. \end{proof}
\begin{lem}\label{lem:1_truncated_mapping_space}
Let $\mathcal T$ be a pregeometry and let $(\mathcal X, \mathcal O_\mathcal X)$, $(\mathcal Y, \mathcal O_\mathcal Y)$ be $\mathcal T$-structured $\infty$-topoi\xspace such that $\mathcal X$ and $\mathcal Y$ are $1$-localic and the structure sheaves $\mathcal O_\mathcal X$, $\mathcal O_\mathcal Y$ are discrete.
Then $\Map_{\RTop(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y}))$ is $1$-truncated.
Moreover, the canonical morphism
\[ \Map_{\RTop(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y})) \to \Map_{\RTop_1(\mathcal T)}((\tau_{\le 0} \mathcal X, \mathcal O_{\mathcal X}), (\tau_{\le 0} \mathcal Y, \mathcal O_{\mathcal Y})) \]
is an equivalence, where $\RTop_1$ denotes the $\infty$-category of 1-topoi with morphisms being right adjoint geometric morphisms. \end{lem}
\begin{proof}
Consider the coCartesian fibration $\LTop(\mathcal T) \to \LTop$.
We know from \cite[Remark 1.4.10]{DAG-V} that the fiber over an $\infty$-topos\xspace $\mathcal X$ is equivalent to $\mathrm{Str}^\mathrm{loc}_{\mathcal T}(\mathcal X)$.
Let $f^{-1} \colon \mathcal X \rightleftarrows \mathcal Y \colon f_*$
be a geometric morphism between $\mathcal X$ and $\mathcal Y$.
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we obtain a fiber sequence
\[ \begin{tikzcd}
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_\mathcal X) \arrow{r} & \Map_{\RTop(\mathcal T)}((\mathcal X, \mathcal O_{\mathcal X}), (\mathcal Y, \mathcal O_{\mathcal Y})) \arrow{r} & \Fun_*(\mathcal X, \mathcal Y) ,
\end{tikzcd} \]
where the fiber is taken at the geometric morphism $(f^{-1}, f_*)$.
Since both $\mathcal X$ and $\mathcal Y$ are $1$-localic, there is an equivalence
\[ \Fun_*(\mathcal X, \mathcal Y) \simeq \Fun_*(\tau_{\le 0} \mathcal X, \tau_{\le 0} \mathcal Y). \]
Therefore $\Fun_*(\mathcal X, \mathcal Y)$ is $1$-truncated.
On the other side, $\mathcal O_\mathcal X$ is $0$-truncated, so $\Map_{\mathrm{Str}^\mathrm{loc}_{\mathcal T}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_\mathcal X)$ is discrete.
The second statement follows as well. \end{proof}
\begin{lem} \label{lem:alg_homotopy_monomorphism}
Let $X = (\mathcal X, \mathcal O_{\mathcal X})$ and $Y = (\mathcal Y, \mathcal O_{\mathcal Y})$ be two $\cT_{\mathrm{an}}(k)$-structured $\infty$-topoi\xspace.
Let $X^\mathrm{alg} \coloneqq (\mathcal X, \mathcal O_\mathcal X^\mathrm{alg})$ and $Y^\mathrm{alg} \coloneqq (\mathcal Y, \mathcal O_\mathcal Y^\mathrm{alg})$ be the underlying $\cT_{\mathrm{disc}}(k)$-structured $\infty$-topoi\xspace.
Assume that $\mathcal X$ and $\mathcal Y$ are $1$-localic and that $\mathcal O_{\mathcal X}$ and $\mathcal O_{\mathcal Y}$ are $0$-truncated.
Then the canonical map
\[ \Map_{\RTop(\cT_{\mathrm{an}}(k))}(X, Y) \to \Map_{\RTop(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) \]
induces monomorphisms on $\pi_0$ and on $\pi_1$ (for every choice of base point). \end{lem}
\begin{proof}
Let $f_* \colon \mathcal X \rightleftarrows \mathcal Y \colon f^{-1}$ be a geometric morphism in $\RTop$.
We have a commutative diagram in $\mathcal S$:
\[ \begin{tikzcd}[column sep=small]
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal X)}(f^{-1} \mathcal O_\mathcal Y, \mathcal O_{\mathcal X}) \arrow{r} \arrow{d} & \Map_{\RTop(\cT_{\mathrm{an}}(k))} (X, Y) \arrow{r} \arrow{d} & \Fun_*(\mathcal X, \mathcal Y) \arrow[-, double equal sign distance]{d} \\
\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{disc}}(k)}(\mathcal X)}(f^{-1} \mathcal O_{\mathcal Y}^\mathrm{alg}, \mathcal O_{\mathcal X}^\mathrm{alg}) \arrow{r} & \Map_{\RTop(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) \arrow{r} & \Fun_*(\mathcal X, \mathcal Y).
\end{tikzcd} \]
Using \cite[2.4.4.2]{HTT} and \cite[Remark 1.4.10]{DAG-V} we see that the two horizontal lines are fiber sequences.
Moreover, since $\mathcal O_\mathcal X$ and $\mathcal O_\mathcal Y$ are $0$-truncated, we can use \cref{lem:alg_faithful} to deduce that the first vertical map is a homotopy monomorphism.
Passing to the long exact sequences of homotopy groups and applying the five lemma, we obtain monomorphisms
\begin{gather*}
\pi_0 \Map_{\RTop(\cT_{\mathrm{an}}(k))}(X, Y) \to \pi_0 \Map_{\RTop(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) , \\
\pi_1 \Map_{\RTop(\cT_{\mathrm{an}}(k))}(X, Y) \to \pi_1 \Map_{\RTop(\cT_{\mathrm{disc}}(k))}(X^\mathrm{alg}, Y^\mathrm{alg}) ,
\end{gather*}
completing the proof. \end{proof}
\begin{lem} \label{lem:mapping_space_hypercompletion_localic}
Let $\mathcal Y$ be an $n$-localic $\infty$-topos\xspace and let $\mathcal X$ be any $\infty$-topos\xspace.
Then there is a canonical equivalence in the homotopy category of spaces $\mathcal H$:
\[ \Map_{\RTop}(\mathcal X^\wedge, \mathcal Y^\wedge) \simeq \Map_{\RTop}(\mathcal X, \mathcal Y) . \] \end{lem}
\begin{proof}
Using \cite[6.5.2.13]{HTT} we see that the canonical morphism
\[ \Map_{\RTop}(\mathcal X^\wedge, \mathcal Y^\wedge) \to \Map_{\RTop}(\mathcal X^\wedge, \mathcal Y) \]
is an equivalence.
Since $\mathcal Y$ is $n$-localic, the restriction
\[ \Map_{\RTop}(\mathcal X^\wedge, \mathcal Y) \to \Map_{\RTop_n}(\tau_{\le n - 1} (\mathcal X^\wedge), \tau_{\le n - 1} \mathcal Y) \]
is an equivalence as well.
On the other side, the restriction
\[ \Map_{\RTop}(\mathcal X, \mathcal Y) \to \Map_{\RTop_n}(\tau_{\le n - 1}\mathcal X, \tau_{\le n - 1} \mathcal Y) \]
is also an equivalence.
We now conclude by observing that $\tau_{\le n - 1} \mathcal X \simeq \tau_{\le n - 1}(\mathcal X^\wedge)$. \end{proof}
\begin{prop} \label{prop:discrete_mapping_spaces_I}
Let $X, Y \in \mathrm{An}_k$. Then $\Map_{\RTop(\cT_{\mathrm{an}}(k))}(\Phi(X), \Phi(Y))$ is discrete. \end{prop}
\begin{proof}
It follows from \cref{lem:mapping_space_hypercompletion_localic} that
\begin{equation} \label{eq:from_hypercomplete_to_non-hypercomplete}
\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y)) \simeq \Map_{\RTop(\cT_{\mathrm{an}}(k))}((\mathcal X_X, \mathcal O_X), (\mathcal X_Y, \mathcal O_Y) ) .
\end{equation}
On the other side, \cref{lem:1_truncated_mapping_space} shows that the right hand side is $1$-truncated and
\begin{equation} \label{eq:from_sheaves_of_spaces_to_sheaves_of_sets}
\Map_{\RTop(\cT_{\mathrm{an}}(k))}((\mathcal X_X, \mathcal O_X), (\mathcal X_Y, \mathcal O_Y) ) \simeq \Map_{\RTop_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y))
\end{equation}
We can now apply \cref{lem:alg_homotopy_monomorphism} to conclude that the canonical map
\begin{multline*}
\Map_{\RTop_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y)) \to\\
\Map_{\RTop_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0}\mathcal X_Y, \mathcal O_Y^\mathrm{alg}))
\end{multline*}
induces monomorphisms on $\pi_0$ and on $\pi_1$.
It follows from \cref{lem:first_fully_faithful} that the canonical map
\[ \Hom_{\mathrm{An}_k}(X,Y) \to \pi_0 \Map_{\RTop_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y^\mathrm{alg})) \]
is an isomorphism.
At this point, we can invoke \cref{lem:rigidity} to deduce that, for every choice of base point, we have
\[ \pi_1 \Map_{\RTop_1(\cT_{\mathrm{disc}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X^\mathrm{alg}), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y^\mathrm{alg})) = 0 . \]
Thus, we conclude that
\[ \pi_1 \Map_{\RTop_1(\cT_{\mathrm{an}}(k))}((\tau_{\le 0} \mathcal X_X, \mathcal O_X), (\tau_{\le 0} \mathcal X_Y, \mathcal O_Y)) = 0 \]
for every choice of base point.
It follows from the equivalences \eqref{eq:from_hypercomplete_to_non-hypercomplete} and \eqref{eq:from_sheaves_of_spaces_to_sheaves_of_sets} that $\Map_{\mathrm{dAn}_k}(\Phi(X),\allowbreak \Phi(Y))$ is discrete, completing the proof. \end{proof}
We can now promote $X \mapsto \Phi(X)$ to an $\infty$-functor.
\todo{Be careful with $X_\mathrm{\acute{e}t}$ below.} Let $\mathcal C$ temporarily denote the full subcategory of $\mathrm{dAn}_k$ spanned by the objects which are equivalent to $\Phi(X)$ for some $X \in \mathrm{An}_k$. \cref{prop:discrete_mapping_spaces_I} shows that mapping spaces in $\mathcal C$ are discrete, hence $\mathcal C$ is equivalent to a $1$-category. Fix a morphism $f \colon X \to Y$ in $\mathrm{An}_k$. It induces a morphism of sites \[ \varphi \colon Y_\mathrm{\acute{e}t} \to X_\mathrm{\acute{e}t} \] given by base change along $f$. Since all the morphisms in $X_\mathrm{\acute{e}t}$ and $Y_\mathrm{\acute{e}t}$ are étale, it follows that $X_\mathrm{\acute{e}t}$ and $Y_\mathrm{\acute{e}t}$ have fiber products. Moreover, $\varphi$ is left exact. Therefore, it follows from \cite[Lemma 2.16]{Porta_Yu_Higher_analytic_stacks_2014} that the induced adjunction \[ \varphi^s \colon \mathcal X_Y \rightleftarrows \colon \mathcal X_X \colon \varphi_s \] is a geometric morphism of $\infty$-topoi\xspace. In particular, we obtain an induced geometric morphism $\mathcal X_Y^\wedge \rightleftarrows \mathcal X_X^\wedge$, which we denote by \[ f^{-1} \colon \mathcal X^\wedge \rightleftarrows \mathcal X_X^\wedge \colon f_* . \] We obtain in this way a well defined morphism $(\mathcal X_X^\wedge, \mathcal O_X) \to (\mathcal X_Y^\wedge, \mathcal O_Y)$. Since mapping spaces in $\mathcal C$ are discrete, we see that this assignment is functorial. We denote the resulting $\infty$-functor by \[ \Phi \colon \mathrm{An}_k \to \mathrm{dAn}_k . \]
\begin{thm} \label{thm:fully_faithfulness} The functor $\Phi\colon \mathrm{An}_k \rightarrow \mathrm{dAn}_k$ is fully faithful. \end{thm}
\begin{proof}
Let $X, Y \in \mathrm{An}_k$. We want to show that
\[
\Hom_{\mathrm{An}_k}(X,Y) \to \Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))
\]
is an equivalence.
\cref{lem:1_truncated_mapping_space} allows us to identify $\Map_{\mathrm{dAn}_k}(\Phi(X), \Phi(Y))$ with
\[
\Map_{\RTop_1(\cT_{\mathrm{an}}(k))}\big((\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X), (\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t}), \mathcal O_Y)\big) .
\]
Let us first prove the faithfulness.
Let $f, g \colon X \to Y$ be two morphisms and assume that $\Phi(f) = \Phi(g)$.
Since the question of $f$ being equal to $g$ is local on both $X$ and $Y$, we can assume that both $X$ and $Y$ are affinoid.
In this case, $f$ (resp.\ $g$) can be recovered as global section of the natural transformation $\Phi(f)(\mathbf A^1_k)$ (resp.\ $\Phi(g)(\mathbf A^1_k)$), where $\mathbf A^1_k$ denote the $k$-analytic\xspace affine line.
Therefore we have $f = g$.
Let us now turn to the fullness.
Let
\[ (f, f^\sharp) \colon (\mathrm{Sh}_\mathrm{Set}(X_\mathrm{\acute{e}t}), \mathcal O_X) \to (\mathrm{Sh}_\mathrm{Set}(Y_\mathrm{\acute{e}t}), \mathcal O_Y) \]
be a morphism in $\RTop(\cT_{\mathrm{an}}(k))$.
After forgetting along the morphism $\cT_{\mathrm{disc}}(k) \to \cT_{\mathrm{an}}(k)$, we get a morphism of locally ringed 1-topoi.
\cref{lem:first_fully_faithful} implies that this morphism comes from a map $\varphi \colon X \to Y$.
This means that $\Phi(\varphi)^\mathrm{alg}$ and $(f,f^\sharp)^\mathrm{alg}$ coincide.
\cref{lem:alg_faithful} implies that $\Phi(\varphi)$ and $(f,f^\sharp)$ coincide as well, completing the proof. \end{proof}
\section{Closed immersions and étale morphisms} \label{sec:closed_etale}
In this section, we study closed immersions and étale morphisms under the fully faithful embedding $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$.
\begin{defin}[{\cite[1.1]{DAG-IX}},{\cite[2.3.1]{DAG-V}}]\label{def:closed_immersion_and_etale}
Let $\mathcal T$ be a pregeometry and $\mathcal X$, $\mathcal Y$ two $\infty$-topoi\xspace.
A morphism $\mathcal O\to\mathcal O'$ in $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal X)$ is said to be an \emph{effective epimorphism} if for every object $X\in\mathcal T$, the induced map $\mathcal O(X)\to\mathcal O'(X)$ is an effective epimorphism in $\mathcal X$.
A morphism $f\colon(\mathcal X, \mathcal O_\mathcal X)\to(\mathcal Y, \mathcal O_\mathcal Y)$ in $\RTop(\mathcal T)$ is called a \emph{closed immersion} (resp.\ an \emph{étale morphism}) if the following conditions are satisfied:
\begin{enumerate}[(i)]
\item the underlying geometric morphism $f_*\colon\mathcal X\to\mathcal Y$ is a closed immersion (resp.\ an étale morphism) of $\infty$-topoi\xspace;
\item the morphism of structure sheaves $f^{-1} \mathcal O_\mathcal Y \to \mathcal O_\mathcal X$ is an effective epimorphism (resp.\ an equivalence) in $\mathrm{Str}^\mathrm{loc}_\mathcal T(\mathcal Y)$.
\end{enumerate} \end{defin}
\begin{lem} \label{lem:hypercompletion_closed_immersions}
The hypercompletion functor $\RTop \to \RHTop$ preserves closed immersions. \end{lem}
\begin{proof}
Let $f_* \colon \mathcal X \rightleftarrows \mathcal Y \colon f^{-1}$ be a closed immersion of $\infty$-topoi\xspace.
By definition we can find a $(-1)$-truncated object $U \in \mathcal Y$ such that the geometric morphism $f_*$ is equivalent to the induced geometric morphism $j_* \colon \mathcal Y / U \rightleftarrows \mathcal Y \colon j^{-1}$.
Since $U$ is $(-1)$-truncated, it belongs to $\mathcal Y^\wedge$.
It is therefore enough to prove that $(\mathcal Y / U)^\wedge \simeq \mathcal Y^\wedge / U$.
The geometric morphism $\mathcal Y / U \to \mathcal Y$ induces by passing to hypercompletions a morphism $(\mathcal Y / U)^\wedge \to \mathcal Y^\wedge$ which by construction fits in the commutative diagram
\[ \begin{tikzcd}
\mathcal Y / U \arrow{r}{j_*} & \mathcal Y \\
(\mathcal Y / U)^\wedge \arrow{u}{i_{U*}} \arrow{r}{j_*^\wedge} & \mathcal Y^\wedge \arrow{u}{i_*} .
\end{tikzcd} \]
Since $j_*$, $i_*$ and $i_{U*}$ are fully faithful, the same goes for $j_*^\wedge$.
Observe that by \cite[7.3.2.5]{HTT}, an object $V \in \mathcal Y^\wedge$ belongs to $\mathcal Y^\wedge / U$ if and only if $V \times U \simeq U$.
Since both $i_*$ and $j_*$ commute with products, we conclude that $j_*^\wedge$ factors through $\mathcal Y^\wedge / U$.
This provides us a fully faithful functor $(\mathcal Y / U)^\wedge \to \mathcal Y^\wedge / U$.
In order to complete the proof, it is enough to prove that it is essentially surjective.
The canonical map $\mathcal Y^\wedge / U \to \mathcal Y^\wedge \to \mathcal Y$ factors through $\mathcal Y / U$.
Now it suffices to prove that this functor can be further factored through $(\mathcal Y / U)^\wedge$.
This follows from the fact that $j_*$ respects the collection of $\infty$-connected morphisms.
To see this, let $V \in \mathcal Y / U$. Since $U$ is $(-1)$-truncated, we see that for every $n \ge 0$ one has:
\[ \tau_{\le n}(V) \times U \simeq \tau_{\le n}(V) \times \tau_{\le n}(U) \simeq \tau_{\le n}(V \times U) \simeq \tau_{\le n}(U) \simeq U . \]
In particular, $\tau_{\le n}(V)$ belongs to $\mathcal Y / U$ as well.
It follows that $j_*$ commutes with truncations, and therefore with $\infty$-connected morphisms. \end{proof}
\begin{lem} \label{lem:different_closed_immersion}
Let $f^{-1} \colon \mathcal X \rightleftarrows \mathcal Y \colon f_*$ be a closed immersion of $\infty$-topoi\xspace.
Let $F \in \mathcal X$, $G \in \mathcal Y$ and let $f^{-1} F \to G$ be a morphism in $\mathcal Y$.
If the morphism $F \to f_* G$ is an effective epimorphism, then so is the morphism $f^{-1} F \to G$. \end{lem}
\begin{proof}
Since $f^{-1}$ is left exact, it commutes with effective epimorphisms.
Therefore, $f^{-1} F \to f^{-1} f_* G$ is an effective epimorphism.
Since $f_*$ is fully faithful, we see that $f^{-1} f_* G \simeq G$, hence completing the proof. \end{proof}
\begin{thm} \label{thm:Phi_classes_of_morphisms}
Let $f \colon X \to Y$ be a morphism in $\mathrm{An}_k$.
Then:
\begin{enumerate}[(i)]
\item The morphism $f$ is an étale morphism if and only if $\Phi(f)$ is an \'etale morphism.
\item The morphism $f$ is a closed immersion if and only if $\Phi(f)$ is a closed immersion.
\end{enumerate} \end{thm}
\begin{proof}
We start by dealing with étale morphisms.
Assume first that $f$ is an étale morphism.
If $X$ is affinoid, it determines an object in the site $Y_\mathrm{\acute{e}t}$.
Let us denote by $U$ this object. It follows from \cite[5.1.6.12]{HTT} that the adjunction $f_* \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f^{-1}$ induced by $f$ can be identified with the \'etale morphism $(\mathcal X_Y)_{/U} \rightleftarrows \mathcal X_Y$.
Since $X$ is an ordinary $k$-analytic\xspace space, $U$ is $0$-truncated and therefore it is hypercomplete.
It follows that we can identify the adjunction
\[ f_* \colon \mathcal X_X^\wedge \rightleftarrows \mathcal X_Y^\wedge \colon f^{-1} \]
with the \'etale morphism $j_* \colon (\mathcal X_Y^\wedge)_{/U} \rightleftarrows \mathcal X_Y^\wedge \colon j^{-1}$.
Moreover, since $f$ is étale, we see that $(f^{-1} \mathcal O_Y)(V) = \mathcal O_Y(V)$.
In particular, we deduce that $f^{-1} \mathcal O_Y = \mathcal O_X$.
In other words, $\Phi(f)$ is \'etale.
If now $X$ is arbitrary, we choose an étale covering $\{X_i \to X\}$ such that every $X_i$ is affinoid.
The above argument shows that the induced morphisms $\mathcal X_{X_i} \rightleftarrows \mathcal X_X$ and $\mathcal X_{X_i} \rightleftarrows \mathcal X_Y$ are \'etale.
It follows that $f_* \colon \mathcal X_X \rightleftarrows \mathcal X_Y \colon f^{-1}$ is \'etale as well.
Let us now assume that $\Phi(f)$ is \'etale.
We will prove that $f$ is étale.
The question being local on $X$ and $Y$, we can assume that they are affinoid, say $X = \Sp B$, $Y = \Sp A$.
By hypothesis, $f^{-1} \mathcal O_Y \to \mathcal O_X$ is an equivalence.
Since the morphism of $\infty$-topoi\xspace $f_* \colon \mathcal X_X^\wedge \rightleftarrows \mathcal X_Y^\wedge\colon f^{-1}$ is \'etale, we see that, for every $U \to X$ étale, one has
\[ f^{-1}(\mathcal O_Y)(U) = \mathcal O_Y(U) . \]
Consider the sheaf $\mathbb L_{\mathcal O_X / f^{-1} \mathcal O_Y}$ on $\mathcal X_X^\wedge$ defined by
\[ C \mapsto \mathbb L^\mathrm{an}_{\mathcal O_X(C) / f^{-1} \mathcal O_Y(C)} = \mathbb L^\mathrm{an}_{C / f^{-1} \mathcal O_Y(C)} , \]
where the symbol $\mathbb L^\mathrm{an}$ denotes the analytic cotangent complex (cf.\ \cite[\S 7.2]{Gabber_Almost_2003}).
Since $f^{-1} \mathcal O_Y \simeq \mathcal O_X$, this sheaf is identically zero.
On the other side, if $\eta^{-1} \colon \mathcal X_X^\wedge \to \mathcal S$ is a geometric point, then
\[ \eta^{-1}(\mathbb L^\mathrm{an}_{\mathcal O_A / f^{-1} \mathcal O_B}) \simeq \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / \eta^{-1} f^{-1} \mathcal O_B}. \]
We can identify $\eta^{-1} f^{-1} \mathcal O_B$ with a strictly henselian $B$-algebra $B'$.
Since the map $B \to B'$ is formally \'etale, we conclude that
\[ \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / \eta^{-1} f^{-1} \mathcal O_B} \simeq \mathbb L^\mathrm{an}_{\eta^{-1} \mathcal O_A / B}. \]
This is also the stalk of the sheaf on $\mathcal X_X^\wedge$ defined by
\[ C \mapsto \mathbb L^\mathrm{an}_{C / B}. \]
Therefore, this sheaf vanishes as well.
In particular, $\mathbb L^\mathrm{an}_{A / B} \simeq 0$, completing the proof.
\todo{A proof without cotangent complex maybe better.}
We now turn to closed immersions.
Assume first that $f$ is a closed immersion in $\mathrm{An}_k$.
\cref{prop:preserve_closed_immersion} and \cref{lem:hypercompletion_closed_immersions} show that the induced geometric morphism $f_* \colon \mathcal X_Y^\wedge \rightleftarrows \mathcal X_X^\wedge \colon f^{-1}$ is a closed immersion of $\infty$-topoi\xspace.
We are left to show that the morphism $f^{-1} \mathcal O_X \to \mathcal O_Y$ is an effective epimorphism.
In virtue of \cref{prop:alg_effective_epi}, it suffices to show that $(f^{-1} (\mathcal O_X))(\mathbf A^1_k) \to \mathcal O_Y(\mathbf A^1_k)$ is an effective epimorphism, where $\mathbf A^1_k$ denote the $k$-analytic\xspace affine line.
Observe that $(f^{-1} (\mathcal O_X))(\mathbf A^1_k) \simeq f^{-1} (\mathcal O_X(\mathbf A^1_k))$.
Since $(f^{-1}, f_*)$ is a closed immersion of $\infty$-topoi\xspace, \cref{lem:different_closed_immersion} shows that it is sufficient to check that
\begin{equation} \label{eq:closed_immersion_effective_epi}
\mathcal O_X(\mathbf A^1_k) \to f_* ( \mathcal O_Y(\mathbf A^1_k))
\end{equation}
is an effective epimorphism in $\mathcal X_X^\wedge$.
This question is local on $\mathcal X_X^\wedge$, so we can assume that $X$ is an affinoid space.
Observe now that $\mathcal O_X(\mathbf A^1_k)$ is the underlying sheaf of (discrete) spaces associated to the structure sheaf of $X$. In the same way, $f_*(\mathcal O_Y(\mathbf A^1_k))$ is the underlying sheaf of spaces associated to the pushforward of the structure sheaf of $Y$.
Both are coherent on $X$, and $f_*(\mathcal O_Y(\mathbf A^1_k))$ is the quotient of $\mathcal O_X(\mathbf A^1_k)$ by some coherent sheaf of ideals.
In particular, the map \eqref{eq:closed_immersion_effective_epi} is an effective epimorphism.
Assume now that $\Phi(f)$ is a closed immersion.
We want to prove that $f$ is a closed immersion as well.
The question is local both on the source and on the target, so we can assume that $X$ and $Y$ are affinoid, say $X = \Sp A$ and $Y = \Sp B$.
In this case, it follows from the proof of \cref{thm:fully_faithfulness} that $f$ corresponds to the morphism
\[ A = \mathcal O_X(\mathbf A^1_k)(X) \to B = \mathcal O_Y(\mathbf A^1_k)(Y) . \]
Therefore, we only have to show that this morphism is surjective.
Let $U = \Sp C \to X$ be an étale morphism.
Then it follows again from the proof of \cref{thm:fully_faithfulness} that
\begin{align*}
f_* \mathcal O_Y(\mathbf A^1_k)(U) & = \mathcal O_Y(\mathbf A^1_k)(Y \times_X U) = B {\cotimes}_A C \\
& = f_* \mathcal O_Y(\mathbf A^1_k)(X) {\cotimes}_{\mathcal O_X(\mathbf A^1_k)(X)} \mathcal O_X(\mathbf A^1_k)(U) .
\end{align*}
In particular, $f_* \mathcal O_Y(\mathbf A^1_k)$ is a coherent sheaf of $\mathcal O_X(\mathbf A^1_k)$-modules.
We can thus apply Tate's acyclicity theorem to conclude that $A \to B$ is surjective, completing the proof. \end{proof}
\section{Existence of fiber products} \label{sec:fiber_products}
The goal of this section is to prove the existence of fiber products of derived $k$-analytic\xspace spaces.
First we will prove the existence of fiber products along a closed immersion (\cref{prop:closed_fiber_products_dAn}). Then we will prove the existence of products over a point (\cref{lem:products_dAn}). We will deduce the existence of fiber products in the general case from the two special cases above, plus \cref{lem:closed_devissage}, which shows that any derived $k$-analytic\xspace space can locally be embedded into a non-derived smooth $k$-analytic\xspace space.
\begin{lem} \label{lem:sheaves_coherent_modules}
Let $f \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal Y, \mathcal O_{\mathcal Y})$ be a map of derived $k$-analytic\xspace spaces such that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ and $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \simeq \Phi(Y)$ for two $k$-analytic\xspace spaces $X, Y \in \mathrm{An}_k$.
Assume that $\mathcal F$ is a connective sheaf of $\mathcal O_{\mathcal Y}^\mathrm{alg}$-modules on $\mathcal Y$ and that each $\pi_n \mathcal F$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal Y}^\mathrm{alg}$-modules. Then the tensor product $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_{\mathcal Y}^\mathrm{alg}} \mathcal O_{\mathcal X}^\mathrm{alg}$ is connective, and each $\pi_n \mathcal F'$ is a coherent sheaf of $\pi_0(\mathcal O_{\mathcal X}^\mathrm{alg})$-modules. \end{lem}
\begin{proof}
The connectivity of $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y} \mathcal O_\mathcal X$ follows from the compatibility of the tensor product with the $t$-structure (cf.\ \cite[Proposition 2.1.3(6)]{DAG-VIII}.)
In order to prove that the homotopy groups $\pi_k \mathcal F'$ are coherent $\pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}$-modules, we first remark that the question is local both on $\mathcal X$ and on $\mathcal Y$.
so we can assume that $X$ and $Y$ are affinoid, say $X = \Sp A$ and $Y = \Sp B$.
We follow closely the proof of \cite[Lemma 12.11]{DAG-IX}.
Thus, we start by proving that for every integer $m \ge - 1$ there exists a sequence of morphisms
\[ 0 = \mathcal F(-1) \to \mathcal F(0) \to \mathcal F(1) \to \cdots \to \mathcal F(m) \to \mathcal F \]
of $\mathcal O_\mathcal Y^\mathrm{alg}$-modules with the following properties:
\begin{enumerate}[(i)]
\item For $0 \le i \le m$, the fiber of $\mathcal F(i-1) \to \mathcal F(i)$ is equivalent to a direct sum of finitely many copies of $\mathcal O_\mathcal Y^\mathrm{alg}[i]$.
\item For $0 \le i \le m$, the fiber of $\mathcal F(i) \to \mathcal F$ is $i$-connective.
\item For $-1 \le i \le m$, the homotopy groups $\pi_j \mathcal F(i)$ are coherent $\pi_0(\mathcal O_\mathcal Y^\mathrm{alg})$-modules, which vanish for $j < 0$.
\end{enumerate}
We proceed by induction on $m$.
If $m = -1$, we simply take $\mathcal F(-1) = 0$. The fiber of $\mathcal F(-1) \to \mathcal F$ is then $\mathcal F[1]$, which is $(-1)$-connective because $\mathcal F$ is connective.
Assume now that we are given a sequence
\[ 0 = \mathcal F(-1) \to \mathcal F(0) \to \cdots \to \mathcal F(m) \to \mathcal F \]
satisfying the conditions above.
Let $\mathcal G$ be the fiber of the map $\mathcal F(m) \to \mathcal F$, so $\mathcal G$ is $m$-connective.
We have an exact sequence
\[ \pi_{m+1} \mathcal F(m) \to \pi_{m+1} \mathcal F \to \pi_m \mathcal F' \to \pi_m \mathcal F(m) \to \pi_m \mathcal F , \]
from which we deduce that $\pi_m \mathcal F'$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
In particular, there exists a positive integer $l$ and a surjection $B^l \to \mathcal G(Y)$.
This induces an epimorphism $(\pi_0 \mathcal O_\mathcal Y^\mathrm{alg})^l \to \mathcal G[-m]$.
Composing with the canonical map $(\mathcal O_\mathcal Y^\mathrm{alg})^l \to (\pi_0 \mathcal O_\mathcal Y^\mathrm{alg})^l$, we obtain a map
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G . \]
Let $\mathcal F(m+1)$ be the cofiber of the composite map $(\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G \to \mathcal F(m)$.
Then the property (i) is satisfied by construction and the property (iii) follows from the long exact sequence associated to the cofiber sequence $(\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal F(m) \to \mathcal F(m+1)$.
Let $\mathcal G'$ denote the fiber of the map $\mathcal F(m+1) \to \mathcal F$, so we have a fiber sequence
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[m] \to \mathcal G \to \mathcal G' . \]
Passing to the long exact sequence, we deduce that $\mathcal G'$ is $(m+1)$-connective, proving the property (ii).
Let us now prove that the homotopy groups of $\mathcal F' \coloneqq f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg}$ are coherent sheaves of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
Fix an integer $n \ge 0$.
Choose a sequence
\[ 0 \to \mathcal F(-1) \to \mathcal F(0) \to \cdots \to \mathcal F(n+1) \to \mathcal F \]
satisfying the properties (i), (ii) and (iii) above.
In particular, the fiber of $\mathcal F(n+1) \to \mathcal F$ is $(n+1)$-connective and therefore the same goes for the map
\[ f^{-1} \mathcal F(n+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \to f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} . \]
So we obtain an isomorphism
\[ \pi_n \left( f^{-1} \mathcal F(n+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right) \to \pi_n \left( f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right) . \]
We can therefore replace $\mathcal F$ by $\mathcal F(n+1)$.
We will now prove that for $-1 \le i \le n+1$, $\pi_n \left( f^{-1} \mathcal F(i) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right)$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
We proceed by induction on $i$.
The case $i = -1$ is trivial.
To deal with the inductive step, we note that the property (i) implies the existence of a fiber sequence
\[ (\mathcal O_\mathcal Y^\mathrm{alg})^l[i] \to \mathcal F(i) \to \mathcal F(i+1) . \]
We therefore obtain a long exact sequence
\begin{multline*}
\cdots \to ( \pi_{n-i} \mathcal O_\mathcal X^\mathrm{alg} )^l \to \pi_n ( f^{-1} \mathcal F(i) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} ) \to\\
\pi_n( f^{-1} \mathcal F(i+1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} ) \to ( \pi_{n - i - 1} \mathcal O_\mathcal X^\mathrm{alg} )^l \to \cdots
\end{multline*}
We conclude that $\pi_n \left( f^{-1} \mathcal F(i + 1) \otimes_{f^{-1} \mathcal O_\mathcal Y^\mathrm{alg}} \mathcal O_\mathcal X^\mathrm{alg} \right)$ is a coherent sheaf of $\pi_0(\mathcal O_\mathcal X^\mathrm{alg})$-modules.
\end{proof}
\begin{prop} \label{prop:closed_fiber_products_dAn}
Assume we are given maps of derived $k$-analytic\xspace spaces $f \colon (\mathcal Y, \mathcal O_{\mathcal Y}) \allowbreak{}\to (\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal X', \mathcal O_{\mathcal X'}) \to (\mathcal Y, \mathcal O_{\mathcal Y})$.
Assume moreover that $f$ is a closed immersion.
Then we have the following statements:
\begin{enumerate}[(i)]
\item \label{item:dAn_fiber_products_Top} There exists a pullback diagram $\sigma$:
\[ \begin{tikzcd}
(\mathcal Y', \mathcal O_{\mathcal Y'}) \arrow{r}{f'} \arrow{d} & (\mathcal X', \mathcal O_{\mathcal X'}) \arrow{d} \\
(\mathcal Y, \mathcal O_{\mathcal Y}) \arrow{r}{f} & (\mathcal X, \mathcal O_{\mathcal X})
\end{tikzcd} \]
in the $\infty$-category $\RHTop(\cT_{\mathrm{an}}(k))$.
\item \label{item:dAn_fiber_products_topoi} The image of $\sigma$ in $\RHTop$ is a pullback diagram of hypercomplete $\infty$-topoi\xspace.
\item \label{item:dAn_closed_immersion} The map $f'$ is a closed immersion.
\item \label{item:dAn_fiber_product_dAn} The structured $\infty$-topos\xspace $(\mathcal Y', \mathcal O_{\mathcal Y'})$ is a derived $k$-analytic\xspace space.
\item \label{item:truncation_pullback} Assume that $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) = \Phi(Y)$, $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) = \Phi(X)$ and $(\mathcal X', \pi_0 \mathcal O_{\mathcal X'}) = \Phi(X')$. Then $(\mathcal Y', \pi_0 \mathcal O_{\mathcal Y'})$ can be identified with $\Phi(Y \times_X X')$.
\end{enumerate} \end{prop}
\begin{proof} The statements (\ref{item:dAn_fiber_products_Top}), (\ref{item:dAn_fiber_products_topoi}) and (\ref{item:dAn_closed_immersion}) follow from \cref{prop:closed_fiber_products_Top}.
We now prove (\ref{item:truncation_pullback}).
Observe that the map $f$ induces a closed immersion $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \to (\mathcal X, \pi_0 \mathcal O_{\mathcal X})$.
So by \cref{thm:Phi_classes_of_morphisms}, it corresponds to a closed immersion $\varphi \colon Y \to X$ of $k$-analytic\xspace spaces.
On the other side, the map $\Phi(X') \to \Phi(X)$ corresponds to a map $X' \to X$ by \cref{thm:fully_faithfulness}.
Let $Y' \coloneqq Y \times_X X'$ be the fiber product computed in $\mathrm{An}_k$.
Then \cref{prop:closed_immersion_pullback_of_topoi} allows us to identify $\mathcal X_{Y'}\coloneqq\mathrm{Sh}(Y'_\mathrm{\acute{e}t})^\wedge$ with $\mathcal Y'$.
It follows from the universal property of the fiber product that there exists a map in $\RHTop(\cT_{\mathrm{an}}(k))$
\[ (\mathcal Y', \mathcal O_{Y'}) \to (\mathcal Y', \mathcal O_{\mathcal Y'}) \]
Moreover, it follows from \cref{prop:closed_fiber_products_Top}(iii) that we have an identification
\[ \mathcal O_{\mathcal Y'}^\mathrm{alg} \simeq f^{\prime -1} \mathcal O_{X'}^\mathrm{alg} \otimes_{f^{\prime -1} g^{-1} \mathcal O_{X}} g^{\prime -1} \mathcal O_{Y} . \]
Using \cite[7.2.1.22]{Lurie_Higher_algebra}, we obtain an equivalence
\[ \pi_0(\mathcal O_{\mathcal Y'}^\mathrm{alg}) \simeq \mathrm{Tor}_0^{ f^{\prime -1} g^{-1}( \pi_0 \mathcal O_X^\mathrm{alg} )}(f^{\prime -1} \pi_0( \mathcal O_{X'}^\mathrm{alg} ), g^{\prime -1}( \pi_0 \mathcal O_Y^\mathrm{alg} )) . \]
As $\pi_0( \mathcal O_\mathcal X) \to f_* \pi_0( \mathcal O_\mathcal Y)$ is surjective, we see that the same formula can be used to describe $\mathcal O_{Y'}$.
Hence $\pi_0(\mathcal O_{\mathcal Y'}) \simeq \mathcal O_{Y'}$.
This proves (\ref{item:truncation_pullback}).
We are left to prove the statement (\ref{item:dAn_fiber_product_dAn}).
The assertion is local on $\mathcal Y'$, so we can assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) = \Phi(X)$, $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) = \Phi(Y)$ and $(\mathcal X', \pi_0 \mathcal O_{\mathcal Y}) = \Phi(X')$ for $k$-analytic\xspace spaces $X, X'$ and $Y$.
It follows from (\ref{item:truncation_pullback}) that $(\mathcal Y', \pi_0 \mathcal O_{\mathcal Y'}^\mathrm{alg})$ is a $k$-analytic\xspace space.
Moreover, since $f$ is a closed immersion, we see that for each $n \ge 0$ the pushforward $f_* \pi_n \mathcal O_{Y}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_X^\mathrm{alg}$-modules on $X$.
Using \cref{lem:sheaves_coherent_modules} and \cref{prop:closed_fiber_products_Top}, we conclude that for each $n \ge 0$, the pushforward $f'_* \pi_n \mathcal O_{\mathcal Y'}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal X'}^\mathrm{alg}$-modules.
Then each $\pi_n \mathcal O_{\mathcal Y'}^\mathrm{alg}$ is a coherent sheaf of $\pi_0 \mathcal O_{\mathcal Y'}^\mathrm{alg}$-modules.
This completes the proof. \end{proof}
\begin{lem} \label{lem:closed_devissage}
Let $(\mathcal X, \mathcal O_{\mathcal X})$ be a derived $k$-analytic\xspace space and let $\mathbf 1_\mathcal X$ be the final object of $\mathcal X$.
Then there exists an effective epimorphism $\coprod U_i \to \mathbf 1_\mathcal X$ and a collection of closed immersions $(\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$, where $V_i$ is a smooth $k$-analytic\xspace space. \end{lem}
\begin{proof}
We can assume without loss of generality that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ for a $k$-affinoid space $X$.
So we have a closed immersion into a $k$-analytic\xspace polydisc $X \hookrightarrow \mathbf D^n_k$. Composing with the affinoid domain embedding $\mathbf D^n_k \hookrightarrow \mathbf A^n_k$, we obtain an embedding $X \hookrightarrow \mathbf A^n_k$.
This embedding is given by $n$ global sections $f_1, \ldots, f_n \in \pi_0(\mathcal O_{\mathcal X}^\mathrm{alg})(X)$.
Let $\{u_i \colon U_i \to X\}_{i \in I}$ be an étale covering such that each restriction $f_j \circ u_i$ is represented by some $\widetilde{f}_{ij} \in \mathcal O_{\mathcal X}(\mathbf A^1_k)(U_i)$.
Combining \cref{lem:universal_property_HSpec} and \cite[Theorem 2.2.12]{DAG-V}, we deduce that these global sections determine a morphism of derived $k$-analytic\xspace spaces
\[ \varphi_i \colon (\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(\mathbf A^n_k). \]
Choose a factorization of $U_i \to X \to \mathbf D^n_k$ as $U_i \xrightarrow{p} V_i \xrightarrow{g} \mathbf D^n_k$, where $p$ is a closed immersion and $g$ is étale.
The composite map $V_i \to \mathbf D^n_k \to \mathbf A^n_k$ is étale and therefore by \cref{thm:Phi_classes_of_morphisms}(i) the induced morphism of derived $k$-analytic\xspace spaces $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(\mathbf A^n_k)$ is \'etale.
Then \cite[Remark 2.3.4]{DAG-V} shows that the map $\varphi_i$ factors through $\mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$ if and only if the underlying morphism of $\infty$-topoi\xspace factors through $\mathcal X_{V_i}$.
The latter holds by construction.
Moreover, the truncation of $\psi_i \colon (\mathcal X_{/U_i}, \mathcal O_{\mathcal X}|_{U_i}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V_i)$ corresponds to the map $U_i \to V_i$, which is a closed immersion. It follows that $\psi_i$ is a closed immersion as well, completing the proof.
\end{proof}
\begin{lem} \label{lem:products_dAn}
Let $(\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal Y, \mathcal O_{\mathcal Y})$ be derived $k$-analytic\xspace spaces.
We have the following statements:
\begin{enumerate}[(i)]
\item There exists a product $(\mathcal Z, \mathcal O_{\mathcal Z}) \simeq (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal Y, \mathcal O_{\mathcal Y})$ in $\RTop(\cT_{\mathrm{an}}(k))$.
\item The structured $\infty$-topos\xspace $(\mathcal Z, \mathcal O_{\mathcal Z})$ is a derived $k$-analytic\xspace space.
\item Assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ and $(\mathcal Y, \pi_0 \mathcal O_{\mathcal Y}) \simeq \Phi(Y)$. Then $(\mathcal Z, \pi_0\mathcal O_{\mathcal Z})$ is equivalent to $\Phi(X \times Y)$.
\item Assume that $(\mathcal X, \pi_0 \mathcal O_{\mathcal X}) \simeq \Phi(X)$ where $X$ is a separated $k$-analytic\xspace space.
Then the diagonal map $\delta \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X})$ is a closed immersion.
\end{enumerate} \end{lem}
\begin{proof}
The statements (i) and (ii) are local on $(\mathcal X, \mathcal O_{\mathcal X})$ and $(\mathcal Y, \mathcal O_{\mathcal Y})$, so we can assume in virtue of \cref{lem:closed_devissage} that there exists closed immersions $(\mathcal X, \mathcal O_{\mathcal X}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V)$ and $(\mathcal Y, \mathcal O_{\mathcal Y}) \to \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(W)$, where $V$ and $W$ are smooth $k$-analytic\xspace spaces.
\cref{prop:closed_fiber_products_dAn} allows us to reduce to the case $(\mathcal X, \mathcal O_{\mathcal X}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V)$ and $(\mathcal Y, \mathcal O_{\mathcal Y}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(W)$.
In this case, we have
\[(\mathcal Z, \mathcal O_{\mathcal Z}) \simeq \mathrm{HSpec}^{\cT_{\mathrm{an}}(k)}(V \times W).\]
The statement (iii) follows from the construction of $(\mathcal Z, \mathcal O_{\mathcal Z})$ we described and \cref{prop:closed_fiber_products_dAn}(\ref{item:truncation_pullback}).
We are left to prove the statement (iv).
The statement (ii) shows that the induced map
\[ \pi_0(\delta) \colon (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \to (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \times (\mathcal X, \pi_0 \mathcal O_{\mathcal X}^\mathrm{alg}) \]
corresponds to $\Phi(\Delta) \colon \Phi(X) \to \Phi(X \times X)$.
Since $X$ is separated, $\Delta \colon X \to X \times X$ is a closed immersion and therefore \cref{thm:Phi_classes_of_morphisms} implies that $\Phi(\Delta)$ is a closed immersion.
Now, the assertion follows from \cref{prop:alg_effective_epi}. \end{proof}
Now we can deduce the main result of this section:
\begin{thm} \label{thm:fiber_products}
The $\infty$-category\xspace $\mathrm{dAn}_k$ admits fiber products. \end{thm}
\begin{proof}
Let $(\mathcal Y,\mathcal O_\mathcal Y)\to(\mathcal X,\mathcal O_\mathcal X)\leftarrow(\mathcal X',\mathcal O_{\mathcal X'})$ be maps of derived $k$-analytic\xspace spaces.
We would like to construct the fiber product.
Working locally on $\mathcal X$, we can assume that $(\mathcal X,\pi_0\mathcal O_\mathcal X^\mathrm{alg})\simeq\upsilon(X)$ for a separated $k$-analytic\xspace space $X$.
Using \cref{lem:products_dAn}(i), we deduce the existence of two products $(\mathcal Z,\mathcal O_\mathcal Z)\coloneqq(\mathcal X',\mathcal O_\mathcal X')\times(\mathcal Y,\mathcal O_\mathcal Y)$ and $(\mathcal X,\mathcal O_\mathcal X)\times(\mathcal X,\mathcal O_\mathcal X)$ in $\mathcal T\mathrm{op}(\cT_{\mathrm{an}}(k))$.
By \cref{lem:products_dAn}(iv), the diagonal map $\delta \colon (\mathcal X, \mathcal O_{\mathcal X}) \to (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X})$ is a closed immersion.
We now apply \cref{prop:closed_fiber_products_dAn} to produce a fiber product
\[ \begin{tikzcd}
(\mathcal Y',\mathcal O_{\mathcal Y'}) \arrow{r} \arrow{d} & (\mathcal Z,\mathcal O_\mathcal Z) \arrow{d} \\
(\mathcal X,\mathcal O_\mathcal X) \arrow{r} & (\mathcal X, \mathcal O_{\mathcal X}) \times (\mathcal X, \mathcal O_{\mathcal X}).
\end{tikzcd} \]
Note that $(\mathcal Y',\mathcal O_{\mathcal Y'})$ is the fiber product of $(\mathcal Y,\mathcal O_\mathcal Y)\to(\mathcal X,\mathcal O_\mathcal X)\leftarrow(\mathcal X',\mathcal O_{\mathcal X'})$ , completing the proof. \end{proof}
\begin{comment} We record the following fact for later use: \todo{We never use it. It was needed in the old essential image. Shall we remove it?}
\begin{prop} \label{prop:Phi_etale_pullback}
Assume that the square $\sigma$
\[ \begin{tikzcd}
X' \arrow{r} \arrow{d} & Y' \arrow{d} \\
X \arrow{r}{j} & Y
\end{tikzcd} \]
is a pullback in $\mathrm{An}_k$ and that $j$ is an étale morphism.
Then $\Phi$ preserves this pullback. \end{prop}
\begin{proof}
Using \cite[6.3.5.8]{HTT} we deduce that $\Phi(\sigma)$ is a pullback square in $\RTop(\cT_{\mathrm{an}}(k))$.
In particular, it is a pullback also in $\mathrm{dAn}_k$. \end{proof} \end{comment}
\section{Comparison between derived spaces and non-derived stacks} \label{sec:essential_image}
In this section, we will characterize the essential image of the embedding $\Phi\colon\mathrm{An}_k\to\mathrm{dAn}_k$ constructed in \cref{sec:fullyfaithfulness}. Moreover, we will compare derived $k$-analytic\xspace spaces with higher $k$-analytic\xspace stacks in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
\subsection{Construction of the comparison functor}
On the $\infty$-category\xspace $\mathrm{dAn}_k$ of derived $k$-analytic\xspace spaces, we define the étale topology $\tau_\mathrm{\acute{e}t}$ to be the Grothendieck topology generated by collections of étale morphisms $\{U_i\to U\}$ such that $\coprod U_i\to U$ is an effective epimorphism (cf.\ \cref{def:closed_immersion_and_etale}).
\begin{rem}
The restriction of $\tau_\mathrm{\acute{e}t}$ to the full subcategory $\mathrm{An}_k$ of $\mathrm{dAn}_k$ coincides with the étale topology $\tau_\mathrm{\acute{e}t}$. \end{rem}
\begin{lem}
Every representable presheaf on $\mathrm{dAn}_k$ is a hypercomplete sheaf for the topology $\tau_\mathrm{\acute{e}t}$. \end{lem}
\begin{proof}
Let $X \coloneqq (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
The universal property of \'etale morphisms (cf.\ \cite[Remark 2.3.4]{DAG-V}) shows that a $\tau_\mathrm{\acute{e}t}$-hypercovering of $X$ can be identified with a hypercovering $U^\bullet$ of $\mathbf 1_\mathcal X$ in the $\infty$-topos\xspace $\mathcal X$.
Given such a hypercovering, the associated $\tau_\mathrm{\acute{e}t}$-hypercovering $X^\bullet$ of $X$ is described by $X^n \coloneqq (\mathcal X_{/U^n}, \mathcal O_\mathcal X |_{U^n})$.
Therefore, we have to prove that
\[ \colim_\Delta ( \mathcal X_{/U^\bullet}, \mathcal O_\mathcal X |_{U^\bullet} ) \simeq (\mathcal X, \mathcal O_\mathcal X) \]
in the $\infty$-category $\mathrm{dAn}_k$.
Using the statement (3') in the proof of \cite[Proposition 2.3.5]{DAG-V}, we see that it is enough to prove that $\mathcal X \simeq \colim \mathcal X_{/U^\bullet}$ in $\RTop$.
Since $\mathcal X$ is hypercomplete, this follows from the descent theory of $\infty$-topoi\xspace (cf.\ \cite[6.1.3.9]{HTT}) and from the fact that $|U^\bullet| \simeq \mathbf 1_\mathcal X$ (cf.\ \cite[6.5.3.12]{HTT}).
\end{proof}
\begin{defin} \label{def:derived_affinoid}
A \emph{derived $k$-affinoid space} is a derived $k$-analytic\xspace space $(\mathcal X, \mathcal O_\mathcal X)$ such that $(\mathcal X, \pi_0(\mathcal O_\mathcal X)) \simeq \Phi(X)$ for some $k$-affinoid space $X$.
We denote by $\mathrm{dAfd}_k$ the full subcategory of $\mathrm{dAn}_k$ spanned by derived $k$-affinoid spaces. \end{defin}
The Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ induces by restriction a Grothendieck topology on $\mathrm{dAfd}_k$ which we denote again by $\tau_\mathrm{\acute{e}t}$. We define the functor $\widetilde{\phi}$ as the composition \[ \begin{tikzcd}
\mathrm{dAn}_k \arrow{r} & \Fun(\mathrm{dAn}_k^{\mathrm{op}}, \mathcal S) \arrow{r} & \Fun( ( \mathrm{dAfd}_k )^{\mathrm{op}}, \mathcal S ) , \end{tikzcd} \] where the first functor is the Yoneda embedding and the second one is the restriction along $\mathrm{dAfd}_k \subset \mathrm{dAn}_k$. Since the Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ is subcanonical, the functor $\widetilde{\phi}$ factors through $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$. We denote by \[ \phi \colon \mathrm{dAn}_k \to \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \] the induced functor. Our first goal is to show that $\phi$ is fully faithful.
\begin{lem} \label{lem:affine_site_big_site}
Let $X = (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space and let $p \colon U \to \mathbf 1_\mathcal X$ be an effective epimorphism.
Let $U^\bullet$ be the \v{C}ech nerve of $p$ and put $X^n \coloneqq (\mathcal X_{/U^n}, \mathcal O_\mathcal X|_{U^n})$.
Then in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ we have
\[ \phi(X) \simeq \colim_{\Delta} \phi(X^\bullet) . \] \end{lem}
\begin{proof}
Let $j\colon\mathrm{dAfd}_k\hookrightarrow\mathrm{dAn}_k$ denote the inclusion functor.
It is continuous and cocontinuous in the sense of \cite[\S 2.4]{Porta_Yu_Higher_analytic_stacks_2014}.
It induces a pair of adjoint functors
\[ j_s\colon \mathrm{Sh}(\mathrm{dAn}_k,\tau_\mathrm{\acute{e}t})\leftrightarrows\mathrm{Sh}(\mathrm{dAfd}_k,\tau_\mathrm{\acute{e}t})\colon j^s.\]
Since the Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ is subcanonical, we can factor $\phi$ as
\[\mathrm{dAn}_k \xrightarrow{\ \psi\ } \mathrm{Sh}( \mathrm{dAn}_k, \tau_\mathrm{\acute{e}t} ) \xrightarrow{\ j_s\ } \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} ).\]
Moreover, we have
\[ \psi(X) \simeq \colim_{\Delta} \psi(X^\bullet) . \]
Since the functor $j_s$ is a left adjoint, it commutes with colimits, completing the proof. \end{proof}
\begin{lem} \label{lem:affine_hypercover}
Let $X = (\mathcal X, \mathcal O_\mathcal X)$ be a derived $k$-analytic\xspace space.
Then there exists a hypercovering $X^\bullet$ of $X$ in $\mathrm{dAn}_k$ such that each $X^n$ is a disjoint union of derived $k$-affinoid spaces. \end{lem} \begin{proof}
It follows directly from Definitions \ref{def:derived_space} and \ref{def:derived_affinoid}. \end{proof}
\begin{prop} \label{prop:phi_fully_faithful}
The functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ is fully faithful. \end{prop}
\begin{proof}
Let $X, Y \in \mathrm{dAn}_k$ and consider the natural map
\[ \psi_{X,Y} \colon \Map_{\mathrm{dAn}_k}(X,Y) \to \Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )}(\phi(X), \phi(Y)) . \]
Keeping $Y$ fixed, let $\mathcal C$ be the full subcategory of $\mathrm{dAn}_k$ spanned by those $X$ such that $\psi_{X,Y}$ is an equivalence.
Since $\mathcal C$ is stable under colimits, combining Lemmas \ref{lem:affine_site_big_site} and \ref{lem:affine_hypercover}, we are reduced to the case where $X$ belongs to $\mathrm{dAfd}_k$.
In this case, the statement follows entirely from the Yoneda lemma. \end{proof}
Our second goal is to identify the essential image of the functor $\phi$. For this, we need to introduce some notations.
\begin{defin} Let $\mathbf P_\mathrm{\acute{e}t}$ denote the class of étale morphisms in $\mathrm{dAn}_k$. The triple $(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ constitutes a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}. We call the associated geometric stacks \emph{derived $k$-analytic\xspace Deligne-Mumford stacks}. We denote by $\mathrm{DM}$ the $\infty$-category\xspace of derived $k$-analytic\xspace Deligne-Mumford stacks. \end{defin}
\begin{defin}
Let $F \in \mathrm{DM}$.
We say that $F$ is \emph{$n$-truncated} if $F(X)$ is $n$-truncated for every $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAfd}_k$ such that $\mathcal O_\mathcal X$ is discrete.
We denote by $\mathrm{DM}_n$ the full subcategory of $\mathrm{DM}$ spanned by $n$-truncated $k$-analytic\xspace Deligne-Mumford stacks. \end{defin}
We denote by $\mathrm{dAn}_k^{\le n}$ the full subcategory of $\mathrm{dAn}_k$ spanned by those derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is $n$-localic (cf.\ \cite[6.4.5.8]{HTT}).
With these notations we can now state our main comparison theorem, which is an analogue of \cite[Theorem 3.7]{Porta_DCAGI} and \cite[Theorem 1.7]{Porta_Comparison_2015}.
\begin{thm} \label{thm:functor_of_points_vs_dAnk}
For every integer $n \ge 1$, the functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ restricts to an equivalence of $\infty$-categories\xspace $\mathrm{dAn}_k^{\le n} \simeq \mathrm{DM}_n$. \end{thm}
The proof will occupy the rest of this section. Before plunging ourselves into the details, let us deduce from this theorem an important application.
Let $(\mathrm{An}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ be the geometric context consisting of the category of $k$-analytic spaces, the étale topology and the class of étale morphisms. The associated geometric stacks are called \emph{higher $k$-analytic\xspace Deligne-Mumford stacks}. They are in particular higher $k$-analytic\xspace stacks considered in \cite{Porta_Yu_Higher_analytic_stacks_2014}. So all the results in loc.\ cit.\ apply.
\begin{cor} \label{cor:underived_higher_kanal_stacks}
Let $\mathrm{Geom}(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ denote the $\infty$-category\xspace of higher $k$-analytic\xspace Deligne-Mumford stacks.
There is a fully faithful embedding $\mathrm{Geom}(\mathrm{An}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t}) \to \mathrm{dAn}_k$ whose essential image is spanned by those derived $k$-analytic\xspace spaces $(\mathcal X, \mathcal O_\mathcal X)$ such that $\mathcal X$ is $n$-localic for some $n$ and $\mathcal O_\mathcal X$ is discrete. \end{cor}
\begin{proof}
Let $(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$ be the geometric context consisting of the category of $k$-affinoid spaces, the étale topology and the class of étale morphisms.
Let $\mathrm{Geom}(\mathrm{Afd}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})$ denote the $\infty$-category\xspace of geometric stacks associated to this geometric context.
By \cite[\S 2.5]{Porta_Yu_Higher_analytic_stacks_2014}, we have an equivalence
\[\mathrm{Geom}(\mathrm{An}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t})\simeq\mathrm{Geom}(\mathrm{Afd}_k,\tau_\mathrm{\acute{e}t},\mathbf P_\mathrm{\acute{e}t}).\]
It follows from \cref{thm:fully_faithfulness} that the natural inclusion $j \colon \mathrm{Afd}_k \to \mathrm{dAfd}_k$ is fully faithful.
So the induced functor
\[ j_s \colon \mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}) \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \]
is fully faithful as well.
We know moreover that $j_s$ preserves geometric stacks.
Therefore $j_s$ factors through the full subcategory $\mathrm{DM} = \bigcup \mathrm{DM}_n$.
Applying \cref{thm:functor_of_points_vs_dAnk}, we obtain the desired fully faithful functor $\mathrm{Geom}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t}) \to \mathrm{dAn}_k$.
Now it suffices to observe that if a geometric stack $X \in \mathrm{dAn}_k^{\le n}$ is discrete, then $\phi(X)$ lies in the essential image of $j_s$.
Indeed, if $X$ is discrete, then
\[ \Map_{\mathrm{dAn}_k}( Y, X ) = \Map_{\mathrm{dAn}_k}(t_0(Y), X). \]
Therefore $\phi(X)$ coincides with the left Kan extension of its restriction along $j$, completing the proof. \end{proof}
\subsection{The case of algebraic spaces}
Given a derived $k$-analytic\xspace space $X$, we denote by $\mathrm{dAfd}_X$ the overcategory $(\mathrm{dAfd}_k)_{/X}$. The Grothendieck topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAn}_k$ induces a Grothendieck topology on $\mathrm{dAfd}_X$, which we still denote by $\tau_\mathrm{\acute{e}t}$. Let $X_\mathrm{big,\acute{e}t}$ denote the Grothendieck site $(\mathrm{dAfd}_X, \tau_\mathrm{\acute{e}t})$.
Let $(\mathrm{dAfd}_X)_\mathrm{\acute{e}t}$ be the full subcategory of the overcategory $\mathrm{dAfd}_X$ spanned by \'etale morphisms $Y \to X$. The \'etale topology $\tau_\mathrm{\acute{e}t}$ on $\mathrm{dAfd}_X$ restricts to a Grothendieck topology on $(\mathrm{dAfd}_X)_\mathrm{\acute{e}t}$, which we still denote by $\tau_\mathrm{\acute{e}t}$. Let $X_\mathrm{\acute{e}t}$ denote the Grothendieck site $((\mathrm{dAfd}_X)_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
\begin{rem}
Let $X$ be an ordinary $k$-analytic\xspace space. Let $f \colon (\mathcal Y, \mathcal O_\mathcal Y) \to \Phi(X)$ be an \'etale morphism in $\mathrm{dAn}_k$. Since the morphism $f^{-1} \mathcal O_X \to \mathcal O_\mathcal Y$ is an equivalence, we see that $\mathcal O_\mathcal Y$ is discrete. In particular, if $(\mathcal Y, \mathcal O_\mathcal Y)$ is a derived $k$-affinoid space, then it belongs to the essential image of $\Phi$.
This shows that there is a canonical equivalence $X_\mathrm{\acute{e}t} \simeq \Phi(X)_\mathrm{\acute{e}t}$. \end{rem}
We have continuous functors between the sites \[ \begin{tikzcd}
(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t}) \arrow{r}{u} & (X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \arrow{r}{v} & (\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) . \end{tikzcd} \] By \cite[\S 2.4]{Porta_Yu_Higher_analytic_stacks_2014}, they induce adjunctions on the $\infty$-categories\xspace of sheaves \begin{gather*}
u_s \colon \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t}) \rightleftarrows \mathrm{Sh}(X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \colon u^s, \\
v_s \colon \mathrm{Sh}(X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \rightleftarrows \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}) \colon v^s . \end{gather*} Moreover, since $u$ is left exact, $(u_s, u^s)$ is a geometric morphism of $\infty$-topoi\xspace. In particular, $u_s$ takes $n$-truncated objects to $n$-truncated objects. On the other side, we can identify the adjunction $(v_s, v^s)$ with the canonical adjunction \[ \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)} \rightleftarrows \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}), \] where the right arrow is the forgetful functor.
\begin{defin}
Let $X \in \mathrm{dAfd}_k$, $Y \in \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ and $\alpha \colon Y \to \phi(X)$ a natural transformation.
We say that \emph{$\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$} if there exists a $0$-truncated sheaf $F \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ and an equivalence $Y \simeq v_s(u_s(F))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$. \end{defin}
\begin{prop} \label{prop:etale_algebraic_spaces}
Let $X \in \mathrm{dAfd}_k$, $Y \in \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ and $\alpha \colon Y \to \phi(X)$ a natural transformation.
The following statements are equivalent:
\begin{enumerate}[(i)]
\item The natural transformation $\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$.
\item There exists a discrete object $U \in \mathcal X$ such that $\phi(j)$ is equivalent to $\alpha$, where $j \colon (\mathcal X_{/U}, \mathcal O_\mathcal X |_U) \to (\mathcal X, \mathcal O_\mathcal X)$ is the induced \'etale morphism.
\item The natural transformation $\alpha$ is $0$-truncated and $0$-representable by étale morphisms.
\end{enumerate} \end{prop}
\begin{proof}
We first prove the equivalence between (i) and (ii).
If $\alpha$ exhibits $Y$ as an \'etale derived algebraic space over $X$, we can find a $0$-truncated sheaf $U \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ and an equivalence $Y \simeq v_s(u_s(U))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Consider $X_U \coloneqq (\mathcal X_{/U}, \mathcal O_\mathcal X |_U)$ and let $j \colon X_U \to X$ be the induced \'etale map.
We want to prove that $\phi(j)$ is equivalent to $\alpha$.
For any $Z = (\mathcal Z, \mathcal O_\mathcal Z) \in \mathrm{dAfd}_k$ and any map $f \colon \phi(Z) \to \phi(X)$, we have a fiber sequence
\[ \begin{tikzcd}
\Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )_{/\phi(X)}}(\phi(Z)_f, u_s(U)) \arrow{r} \arrow{d} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), v_s(u_s(U))) \arrow{d} \\
\{f\} \arrow{r} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)),
\end{tikzcd} \]
where $\phi(Z)_f$ denotes the object $f \colon \phi(Z) \to \phi(X)$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Since $\phi$ is fully faithful by \cref{prop:phi_fully_faithful}, we can view $\phi(Z)_f$ as a representable object in $\mathrm{Sh}( X_\mathrm{big,\acute{e}t}, \tau_\mathrm{\acute{e}t}) \simeq \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})_{/\phi(X)}$.
Therefore, the Yoneda lemma combined with \cite[4.3.2.15]{HTT} implies that
\[ \Map_{\mathrm{Sh}( \mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t} )_{/\phi(X)}}(\phi(Z)_f, u_s(U)) \simeq \Gamma(\mathcal Z, f^{-1}(U)) . \]
In particular, taking $Z$ to be an atlas for $X_U$ and choosing $f$ to be $j$, we obtain a canonical map $\phi(X_U) \to v_s(u_s(U))$.
For any $Z \in \mathrm{dAfd}_k$, we obtain in this way a commutative square
\[ \begin{tikzcd}
\Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X_U)) \arrow{r} \arrow{d} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)) \arrow[-, double equal sign distance]{d} \\
\Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), v_s(u_s(U))) \arrow{r} & \Map_{\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})}(\phi(Z), \phi(X)) .
\end{tikzcd} \]
For any morphism $f \colon \phi(Z) \to \phi(X)$, we can combine the fully faithfulness of $\phi$ and \cite[Remark 2.3.4]{DAG-V} to identify the fiber of the top horizontal morphism with $\Gamma(\mathcal Z, f^{-1}(U))$.
The same holds for the lower horizontal morphism in virtue of the above discussion.
Therefore, there is a canonical identification of $\phi(X_U)$ with $Y = v_s(u_s(U))$ in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$, and a canonical identification of $\phi(j)$ with $\alpha$.
On the other side, if (ii) is satisfied, then $U$ defines an \'etale derived algebraic space $v_s(u_s(U))$ over $X$, which can be identified with $Y$ using the same argument as above.
Let us now prove the equivalence between (i) and (iii).
First, assume that (iii) is satisfied.
In this case, we can define a sheaf $U \colon X_\mathrm{\acute{e}t} \to \mathcal S$ by sending an \'etale map $f \colon Z \to X$ to the fiber product
\[ \begin{tikzcd}
U(Z) \arrow{r} \arrow{d} & Y(Z) \arrow{d}{\alpha_Z} \\
\{*\} \arrow{r}{f} & \phi(X)(Z).
\end{tikzcd} \]
Since $\alpha$ is $0$-truncated, we see that $U$ takes values in $\mathrm{Set}$.
Since both $\phi(X)$ and $Y$ are sheaves, the same goes for $U$.
It follows that $U$ defines a $0$-truncated object in $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Since $\alpha$ is $0$-representable by \'etale maps we obtain a canonical map $Y \to v_s(u_s(Y))$, and \cite[Remark 2.3.4]{DAG-V} shows that this map is an equivalence.
Finally, let us prove that (i) implies (iii).
Choose a $0$-truncated sheaf $U \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ such that $Y \simeq v_s(u_s(U))$.
We already remarked that in this case $\alpha$ is $0$-truncated.
Choose $V_i \in X_\mathrm{\acute{e}t}$ and sections $\eta_i \in U(V_i)$ which generate $U$, we obtain an effective epimorphism
\[ \coprod \phi(V_i) = \coprod v_s(u_s( V_i )) \to v_s(u_s(U)) \]
in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Suppose there is a $(-1)$-truncated morphism $v_s(u_s(U)) \to \phi(Z)$ for some $Z \in X_\mathrm{\acute{e}t}$.
In this case, we see that
\[ \phi(V_i) \times_{v_s(u_s(U))} \phi(V_j) \simeq \phi(V_i) \times_{\phi(Z)} \phi(V_j) \]
and therefore the maps $\phi(V_i) \to v_s(u_s(U)) \simeq Y$ is $(-1)$-representable by \'etale maps.
In the general case, the fiber product $Y_{i,j} \coloneqq \phi(V_i) \times_{v_s(u_s(U))} \phi(V_j)$ is again a derived algebraic space \'etale over $X$.
We claim that the canonical map $Y_{i,j} \to \phi(V_i \times_X V_j)$ is $(-1)$-truncated.
Indeed, we have a pullback diagram
\[ \begin{tikzcd}
Y_{i,j} \arrow{r} \arrow{d} & v_s(u_s(U)) \arrow{d} \\
\phi(V_i) \times_{\phi(X)} \phi(V_j) \arrow{r} & v_s(u_s(U)) \times_{\phi(X)} v_s(u_s(U)).
\end{tikzcd} \]
Since the map $\alpha \colon Y \to \phi(X)$ is $0$-truncated, we conclude the proof of the claim by \cite[5.5.6.15]{HTT}.
At this point, we deduce that $Y_{i,j} \to X$ is $(-1)$-representable by \'etale maps, and therefore that each $\phi(V_i) \to Y$ is $0$-representable by \'etale maps. \end{proof}
\subsection{Proof of \cref{thm:functor_of_points_vs_dAnk}}
We begin with the following analogue of \cite[Lemma 2.7]{Porta_Comparison_2015}
\begin{lem} \label{lem:decreasing_truncated_level}
Let $n \ge 0$ be an integer.
Fix $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAn}_k^{\le n+1}$ and let $V \in \mathcal X$ be an object such that $X_V \coloneqq (\mathcal X_{/V}, \mathcal O_\mathcal X |_V)$ is a derived $k$-affinoid space.
Then $V$ is $n$-truncated. \end{lem}
\begin{proof}
We have to prove that for every object $U \in \mathcal X$, the space
\[ \Map_\mathcal X(U, V) \simeq \Map_{\mathcal X_{/U}}(U, U \times V) \]
is $n$-truncated.
This property is local on $U$, so we can restrict ourselves to the situation where $X_U \coloneqq (\mathcal X_{/U}, \mathcal O_\mathcal X |_U)$ is a derived $k$-affinoid space.
Using \cite[Remark 2.3.4]{DAG-V}, we see that this space fits into a fiber sequence
\[ \Map_{\mathcal X}(U, V) \to \Map_{\mathrm{dAn}_k}( X_U, X_V ) \to \Map_{\mathrm{dAn}_k}( X_U, X ) . \]
Since a derived $k$-analytic\xspace space $(\mathcal Y, \mathcal O_\mathcal Y)$ belongs to $\mathrm{dAfd}_k$ if and only if its truncation $(\mathcal Y, \pi_0(\mathcal O_\mathcal Y))$ does, we can replace $X$ with its truncation.
Let us denote by $F_X \colon \mathrm{Afd}_k \to \mathcal S$ the functor of points associated to $X$ and by $F_V \colon \mathrm{Afd}_k \to \mathcal S$ the functor of points associated to $(\mathcal X_{/V}, \mathcal O_\mathcal X |_V)$.
The arguments above show that it is enough to prove that for every ordinary $k$-affinoid space $Z$, the fibers of $F_X(Z) \to F_V(Z)$ are $n$-truncated.
By hypothesis, $F_V$ is the functor of points associated to some $k$-affinoid space, so it takes values in $\mathrm{Set}$.
Since $F_V(Z)$ is discrete, it suffices to show that $F_X(Z)$ is $(n+1)$-truncated. This follows directly from \cite[Lemma 2.6.19]{DAG-V}.
\end{proof}
\begin{prop} \label{prop:geometricity}
Let $n \ge 1$ and let $X = (\mathcal X, \mathcal O_\mathcal X) \in \mathrm{dAn}_k$ be a derived $k$-analytic\xspace space such that $\mathcal X$ is $n$-localic.
Then $\phi(X)$ belongs to $\mathrm{DM}_n$. \end{prop}
\begin{proof}
Let $Y = (\mathcal Y, \mathcal O_\mathcal Y) \in \mathrm{dAfd}_k$ be a derived $k$-affinoid space such that $\mathcal O_\mathcal Y$ is discrete.
For every geometric morphism $f^{-1} \colon \mathcal X \leftrightarrows \mathcal Y \colon f_*$ we can use \cite[2.4.4.2]{HTT} to obtain a fiber sequence
\[ \Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}(f^{-1} \mathcal O_\mathcal X, \mathcal O_\mathcal Y ) \to \Map_{\mathrm{dAn}_k}( Y, X ) \to \Map_{\RTop}( \mathcal Y, \mathcal X ) , \]
where the fiber is taken at $(f^{-1}, f_*)$.
Since $\mathcal Y$ is $1$-localic and $n \ge 1$, it is also $n$-localic.
Therefore, \cite[Lemma 2.2]{Porta_Comparison_2015} shows that $\Map_{\RTop}( \mathcal Y, \mathcal X )$ is $n$-truncated.
Since $\mathcal O_\mathcal Y$ is discrete, we see that $\Map_{\mathrm{Str}^\mathrm{loc}_{\cT_{\mathrm{an}}(k)}(\mathcal Y)}( f^{-1} \mathcal O_\mathcal X, \mathcal O_\mathcal Y )$ is $0$-truncated, hence $n$-truncated.
So
\[\phi(X)(Y) = \Map_{\mathrm{dAn}_k}(Y,X)\]
is $n$-truncated as well.
Let us now prove that $\phi(X)$ is geometric. Combining \cref{cor:representability_diagonal} and \cref{cor:dAfdk_closed_under_tau}, we see that it is enough to prove that $\phi(X)$ admits an atlas.
Choose objects $U_i \in \mathcal X$ such that $(\mathcal X_{/U_i}, \mathcal O_\mathcal X |_{U_i})$ is a derived $k$-affinoid space and that the joint morphism $\coprod U_i \to \mathbf 1_\mathcal X$ is an effective epimorphism.
Put $X_i \coloneqq (\mathcal X_{/U_i}, \mathcal O_\mathcal X |_{U_i})$.
By functoriality we obtain maps $\phi(X_i) \to \phi(X)$.
It follows from \cref{lem:affine_site_big_site} that the total morphism $\coprod \phi(X_i) \to \phi(X)$ is an effective epimorphism.
We are therefore left to prove that $\phi(X_i) \to \phi(X)$ are $(n-1)$-representable by \'etale maps.
First of all, we remark that if $Z \in \mathrm{dAfd}_k$, then for any map $\phi(Z) \to \phi(X)$, using full faithfulness of $\phi$, we obtain
\[ \phi(Z) \times_{\phi(X)} \phi(X_i) \simeq \phi( Z \times_X X_i ) , \]
and $Z \times_X X_i$ is \'etale over $Z$. Therefore we are reduced to prove that the stacks $\phi(X) \times_{\phi(Z)} \phi(X_i)$ are $(n-1)$-geometric.
We prove this by induction on $n$.
If $n = 1$, \cref{lem:decreasing_truncated_level} shows that the objects $U_i$ are discrete.
It follows from \cref{prop:etale_algebraic_spaces} that $\phi(Z) \times_{\phi(X)} \phi(X_i)$ is $0$-geometric.
Now suppose that $\mathcal X$ is $n$-localic and $n > 1$.
\cref{lem:decreasing_truncated_level} shows again that the objects $U_i$ are $(n-1)$-truncated.
Therefore \cite[Lemma 2.3.16]{DAG-V} shows that the underlying $\infty$-topos\xspace of $Z \times_X X_i$ is $(n-1)$-localic.
We conclude by the inductive hypothesis. \end{proof}
As a consequence of \cref{prop:geometricity}, the functor $\phi \colon \mathrm{dAn}_k \to \mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ induces a well defined functor \[ \phi_n \colon \mathrm{dAn}_k^{\le n} \to \mathrm{DM}_n . \] In order to achieve the proof of \cref{thm:functor_of_points_vs_dAnk}, we are left to show that $\phi_n$ is essentially surjective.
We will need the following elementary observation:
\begin{lem} \label{lem:equivalence_etale_sites}
Let $X$ be a geometric stack for the geometric context $(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t}, \mathbf P_\mathrm{\acute{e}t})$.
The functor $\mathrm{t}_0 \colon X_\mathrm{\acute{e}t} \to (\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ is an equivalence of sites. \end{lem}
\begin{proof}
We prove this by induction on the geometric level of $X$.
If $X$ is $(-1)$-geometric we can find $Y = (\mathcal Y, \mathcal O_\mathcal Y) \in \mathrm{dAfd}_k$ such that $X \simeq \phi(Y)$.
Consider the chain of equivalences
\[ (\RTop(\cT_{\mathrm{an}}(k))_{/Y})_\mathrm{\acute{e}t} \simeq (\RTop_{/Y})_{_\mathrm{\acute{e}t}} \simeq (\RTop(\cT_{\mathrm{an}}(k))_{/\mathrm{t}_0(Y)})_\mathrm{\acute{e}t} . \]
We now remark that, if $X \to Y$ is an \'etale map in $\RTop(\cT_{\mathrm{an}}(k))$, then $X$ is a derived $k$-analytic\xspace space.
Moreover, a derived $k$-analytic\xspace space belongs to $\mathrm{dAfd}_k$ if and only if its truncation does.
These observations imply that the above equivalence restricts to an equivalence
\[ Y_\mathrm{\acute{e}t} \simeq (\mathrm{t}_0(Y))_\mathrm{\acute{e}t}, \]
thus achieving the proof of the base step of the induction.
Suppose now that $X$ is $n$-geometric and that the statement holds for $(n-1)$-geometric stacks.
Choose an \'etale $n$-groupoid presentation $U^\bullet$ for $X$.
This means that $U^\bullet$ is a groupoid object in the $\infty$-category $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$ such that each $U^m$ is $(n-1)$-geometric and that the map $U^0 \to X$ is $(n-1)$-representable by \'etale maps.
Since $\mathrm{t}_0$ commutes with products in virtue of \cref{prop:truncation_and_finite_limits} and it takes effective epimorphisms to effective epimorphisms by \cite[7.2.1.14]{HTT}, we see that $V^\bullet \coloneqq \mathrm{t}_0(U^\bullet)$ is a groupoid presentation for $\mathrm{t}_0(X)$.
Now, let $Y \to \mathrm{t}_0(X)$ be an \'etale map. We see that $Y \times_{\mathrm{t}_0(X)} V^\bullet \to V^\bullet$ is an \'etale map (i.e.\ it is a map of groupoids which is \'etale in each degree). By the inductive hypothesis, we obtain a map of simplicial objects $Z^\bullet \to U^\bullet$, such that
\[ \mathrm{t}_0(Z^\bullet) = Y \times_{\mathrm{t}_0(X)} V^\bullet . \]
Since $Y \times_{\mathrm{t}_0(X)} V^\bullet$ is a groupoid, so is $Z^\bullet$ (here we use again the equivalence guaranteed by the inductive hypothesis).
The geometric realization of $Z^\bullet$ provides us with an \'etale map $Z \to X$. Since $\mathrm{t}_0$ preserves effective epimorphisms, we conclude that $\mathrm{t}_0(Z) = Y$.
This construction is functorial in $Y$, and it provides the inverse to the functor $\mathrm{t}_0$. \end{proof}
\begin{prop} \label{prop:phi_essentially_surjective}
The functor $\phi_n \colon \mathrm{dAn}_k^{\le n} \to \mathrm{DM}_n$ is essentially surjective. \end{prop}
\begin{proof}
Let $X\in\mathrm{DM}_n$.
By \cref{lem:equivalence_etale_sites}, $X_\mathrm{\acute{e}t}$ is equivalent to $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$.
By hypothesis, $\mathrm{t}_0(X)$ is $n$-truncated.
Therefore, \cref{prop:over_n_category} shows that the mapping spaces in $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ are $(n-1)$-truncated.
In other words, $(\mathrm{t}_0(X))_\mathrm{\acute{e}t}$ is equivalent to an $n$-category (cf.\ \cite[2.3.4.18]{HTT}).
As a consequence, $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$ is $n$-localic.
Put $\mathcal X \coloneqq \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})^\wedge$.
Consider the composition
\[ \cT_{\mathrm{an}}(k) \times X_\mathrm{\acute{e}t}^{\mathrm{op}} \to \mathrm{dAfd}_k \times \mathrm{dAfd}_k^{\mathrm{op}} \xrightarrow{y} \mathcal S , \]
where the last functor classifies the Yoneda embedding (cf.\ \cite[\S 5.2.1]{Lurie_Higher_algebra}).
This induces a well defined functor
\[ \overline{\mathcal O_\mathcal X} \colon \cT_{\mathrm{an}}(k) \to \mathrm{PSh}(X_\mathrm{\acute{e}t}) , \]
which factors through $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Let $\mathcal O_\mathcal X$ be its hypercompletion.
Since the functor $\cT_{\mathrm{an}}(k) \to \mathrm{dAfd}_k$ preserves products and admissible pullbacks, the same holds for $\mathcal O_\mathcal X$.
Moreover, \cref{lem:affine_site_big_site} implies that $\mathcal O_\mathcal X$ takes $\tau_\mathrm{\acute{e}t}$-coverings to effective epimorphisms.
In other words, $\mathcal O_\mathcal X$ defines a $\cT_{\mathrm{an}}(k)$-structure on $\mathcal X$.
If $\{U_i \to X\}$ is an \'etale $n$-atlas of $X$, each $U_i$ defines an object $V_i$ in $\mathcal X$.
Unraveling the definitions, we see that the $\cT_{\mathrm{an}}(k)$-structured $\infty$-topos\xspace $(\mathcal X_{/V_i}, \mathcal O_X |_{V_i})$ is canonically isomorphic to $U_i \in \mathrm{dAn}_k$ itself.
Therefore $X' \coloneqq (\mathcal X, \mathcal O_X)$ is a derived $k$-analytic\xspace space.
We are left to prove that $\phi(X') \simeq X$.
We can proceed by induction on the geometric level $n$ of $X$.
If $n = -1$, $\phi(X')$ is the functor represented by $X'$, and the same holds for $X$.
Let now $n \ge 0$.
Choose an \'etale $n$-atlas $\{U_i \to X\}$ for $X$.
Set $U \coloneqq \coprod U_i$ and let $U^\bullet$ denote the \v{C}ech nerve of $U \to X$.
Every map $U^n \to X$ is \'etale.
In particular, the functor $X_\mathrm{\acute{e}t} \to \mathcal S$ sending $Y$ to $\Map_{X_\mathrm{\acute{e}t}}(Y, U^n)$ defines an element $V^n \in \mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$.
Using \cref{lem:equivalence_etale_sites}, we see that
\[ \Map_{X_\mathrm{\acute{e}t}}(Y, U^n) \simeq \Map_{\mathrm{t}_0(X)_\mathrm{\acute{e}t}}(\mathrm{t}_0(Y), \mathrm{t}_0(U^n)). \]
Since $t_0(U^n)$ is a geometric stack, we conclude that the above space is truncated.
In particular, the object $V^n$ is a truncated object in $\mathrm{Sh}(X_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})$, so it is hypercomplete.
In other words, $V^n$ belongs to $\mathcal X$.
We can therefore identify $\mathrm{Sh}(U^n_\mathrm{\acute{e}t}, \tau_\mathrm{\acute{e}t})^\wedge$ with $\mathcal X_{/V^n}$.
The universal property of \'etale morphisms (cf.\ \cite[Remark 2.3.4]{DAG-V}) shows that we can arrange the $V^n$s into a simplicial object $V^\bullet$ in $\mathcal X$, whose geometrical realization coincides with $\mathbf 1_\mathcal X$.
The inductive hypothesis shows that $\phi( \mathcal X_{/V^\bullet}, \mathcal O_X |_V^\bullet ) \simeq U^\bullet$ as simplicial objects in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Since $\phi$ commutes with \v{C}ech nerves of \'etale maps and their realizations (in virtue of \cref{lem:affine_site_big_site}), we conclude that $\phi(X')$ is equivalent to $X$ itself. \end{proof}
The proof of \cref{thm:functor_of_points_vs_dAnk} is now achieved.
\section{Appendices}
\subsection{Complements on overcategories}
The goal of this subsection is to provide a proof of the following basic result, for which we do not know a reference: if $(\mathcal C, \tau)$ is a Grothendieck site and $\mathcal C$ is a $1$-category, then for every $n$-truncated sheaf $X \in \mathrm{PSh}(\mathcal C)$, the overcategory $\mathcal C_{/X}$ is an $(n-1)$-category. The proof relies on the following lemma:
\begin{lem} \label{lem:fiber_sequence_for_over}
Let $\mathcal C$ be an $\infty$-category.
Let $X \in \mathcal C$ be an object and let $f \colon U \to X$, $g \colon V \to X$ be two $1$-morphisms of $\mathcal C$ viewed as objects of $\mathcal C_{/X}$.
For every morphism $h \colon U \to V$ in $\mathcal C$, choose a $2$-simplex $\sigma \colon \Delta^2 \to \mathcal C$ extending the morphism $\Lambda^2_1 \to \mathcal C$ classified by $h$ and $g$.
Put $f' \coloneqq d_1(\sigma)$.
Then we have a fiber sequence
\[ \mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \to \Map_{\mathcal C_{/X}}(f,g) \to \Map_{\mathcal C}(U,V). \] \end{lem}
\begin{proof}
It follows from \cite[Proposition 2.1.2.1]{HTT} that the canonical map $p \colon \mathcal C_{/X} \to \mathcal C$ is a right fibration. In particular, it is a Cartesian fibration where every edge of $\mathcal C_{/X}$ is $p$-Cartesian.
The $2$-simplex $\sigma \colon \Delta^2 \to \mathcal C$ can be viewed as an edge of $\mathcal C_{/X}$.
Reviewing the Kan complex $\Map_{\mathcal C}(U,X)$ as an $\infty$-category, we have a canonical equivalence $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \simeq \Map_{\Map_{\mathcal C}(U,X)}(f,f')$.
The conclusion follows at this point from \cite[Proposition 2.4.4.2]{HTT}. \end{proof}
\begin{prop} \label{prop:over_n_category}
Let $\mathcal C$ be an $\infty$-category.
Let $X \in \mathcal C$ be an $n$-truncated object.
Let $f \colon U \to X$ and $g \colon V \to X$ be two morphisms viewed as objects in $\mathcal C_{/X}$.
If $V$ is $m$-truncated with $m < n$, then $\Map_{\mathcal C_{/X}}(U, V)$ is $(n-1)$-truncated. \end{prop}
\begin{proof}
Choosing $f'$ as in \cref{lem:fiber_sequence_for_over}, we obtain a fiber sequence
\[ \mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \to \Map_{\mathcal C_{/X}}(f,g) \to \Map_{\mathcal C}(U,V). \]
Now, $\Map_{\mathcal C}(U,V)$ is $m$-truncated by hypothesis.
On the other hand, we have a pullback diagram
\[ \begin{tikzcd}
\mathrm{Path}_{\Map_{\mathcal C}(U,X)}(f,f') \arrow{r} \arrow{d} & \{*\} \arrow{d}{f'} \\
\{*\} \arrow{r}{f} & \Map_{\mathcal C}(U,X).
\end{tikzcd} \]
Therefore $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f')$ fits in the pullback diagram
\[ \begin{tikzcd}
\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f') \arrow{d} \arrow{r} & \Map_{\mathcal C}(U,X) \arrow{d}{\Delta} \\
\{*\} \arrow{r}{(f,f')} & \Map_{\mathcal C}(U,X) \times \Map_{\mathcal C}(U,X) .
\end{tikzcd} \]
Since $X$ is $n$-truncated, it follows that $\Map_{\mathcal C}(U,X)$ is $n$-truncated.
Therefore, \cite[5.5.6.15]{HTT} shows that $\Delta$ is $(n-1)$-truncated.
We deduce that $\mathrm{Path}_{\Map_\mathcal C(U,X)}(f,f')$ is $(n-1)$-truncated.
Thus the fiber sequence of \cref{lem:fiber_sequence_for_over} implies that $\Map_{\mathcal C_{/X}}(f,g)$ is $(n-1)$-truncated as well, completing the proof. \end{proof}
\subsection{Complements on geometric stacks}
\begin{defin}\label{def:C_closed_under_tau}
Let $(\mathcal C,\tau)$ be an $\infty$-site\xspace.
The $\infty$-category\xspace $\mathcal C$ is said to be \emph{closed under $\tau$-descent} if for any morphism from a sheaf $X$ to a representable sheaf $Y$, any $\tau$-covering $\{Y_i\to Y\}$, the representability of $X\times_Y Y_i$ for every $i$ implies the representability of $X$. \end{defin}
We need the following converse to \cite[Corollary 2.12]{Porta_Yu_Higher_analytic_stacks_2014}:
\begin{lem} \label{lem:factorizing_property_effective_epimorphisms}
Let $(\mathcal C, \tau)$ be a subcanonical $\infty$-site.
Let $F \to G$ be an effective epimorphism in $\mathrm{Sh}(\mathcal C, \tau)$.
For any object $X \in \mathcal C$ and any morphism $h_X \to G$, there exists a $\tau$-covering $\{U_i \to X\}$ such that the composite morphisms $h_{U_i} \to h_X \to G$ factor as
\[ \begin{tikzcd}
{} & {} & F \arrow{d} \\
h_{U_i} \arrow[dashed]{urr} \arrow{r} & h_X \arrow{r} & G .
\end{tikzcd} \] \end{lem}
\begin{proof}
Using \cite[Proposition 2.11]{Porta_Yu_Higher_analytic_stacks_2014} we see that the morphism $\pi_0(F) \to \pi_0(G)$ is an effective epimorphism of sheaves of sets.
In particular, there exists a covering $\{V_j \to X\}$ such that the composite morphisms $\pi_0( h_{V_j}) \to \pi_0(h_X) \to \pi_0(G)$ factor through $\pi_0(F)$.
Since $\pi_0(F)$ is by definition the sheafification of the presheaf $Y \mapsto \pi_0(F(Y))$ and since
\[ \Map_{\mathrm{Sh}(\mathcal C, \tau)}( \pi_0(h_{V_j}), \pi_0(F) ) \simeq \Map_{\mathrm{Sh}(\mathcal C, \tau)}(h_{V_j}, \pi_0(F)) \simeq \pi_0(F) (V_j) , \]
we can find a $\tau$-covering $\{U_{ij} \to V_j\}$ such that every composite morphism $h_{U_{ij}} \to h_{V_j} \to \pi_0(F)$ factors through $F \to \pi_0(F)$.
Finally, again since $\pi_0(G)$ is the sheafification of the presheaf $Y \mapsto \pi_0(G(Y))$, we can further refine the covering such that the morphisms $h_{U_{ij}} \to F$ are homotopic to the compositions $h_{U_{ij}} \to h_X \to G$. This completes the proof. \end{proof}
\begin{prop} \label{prop:geometric_stacks_closed_under_tau}
Let $(\mathcal C, \tau, \mathbf P)$ be a geometric context in the sense of \cite{Porta_Yu_Higher_analytic_stacks_2014}.
Assume that $\mathcal C$ is closed under $\tau$-descent.
Then the class of $n$-representable morphisms is closed under $\tau$-descent, in the sense that for any morphism $f \colon X \to Y$ with $Y$ a $n$-geometric stack, if there exists an $n$-atlas $\{U_i\}$ of $Y$ such that $X \times_Y U_i$ is $n$-geometric for every $i$, then $F$ is $n$-geometric as well. \end{prop}
\begin{proof}
The proof goes by induction on the geometric level $n$.
When $n = -1$, this holds because $\mathcal C$ is closed under $\tau$-descent.
Let now $n \ge 0$. Let $\{U_i\}$ be an $n$-atlas of $Y$ such that $X_i \coloneqq X \times_Y U_i$ is $n$-geometric for every $i$.
Choose an $n$-atlas $\{V_{ij}\}$ of $X \times_Y U_i$. The compositions $V_{ij} \to X_i \to X$ provide an $n$-atlas of $X$.
We are therefore left to prove that the diagonal of $X$ is $(n-1)$-representable.
Let $V \coloneqq \coprod V_{ij}$ be the $n$-atlas of $X$ introduced above. By construction, the map $V \to X$ is $(n-1)$-representable.
It follows that the induced map $V \times_X V \to V$ is $(n-1)$-representable as well.
Since $V$ is a disjoint union of $(-1)$-representable stacks, it follows that $V \times_X V$ is $(n-1)$-geometric.
Observe now that $V \times V \to X \times X$ is an effective epimorphism.
Therefore for every morphism $S \to X \times X$ from a $(-1)$-representable stack $S$, by \cref{lem:factorizing_property_effective_epimorphisms}, we can choose a $\tau$-covering $S_i \to S$ such that the composite map $S_i \to S \to X \times X$ factors as
\[ \begin{tikzcd}
{} & {} & V \times V \arrow{d} \\
S_i \arrow[dashed]{urr} \arrow{r} & S \arrow{r} & X \times X .
\end{tikzcd} \]
In order to prove that the diagonal $\Delta_X \colon X \to X \times X$ is $(n-1)$-representable, we have to show that $S \times_{X \times X} X$ is $(n-1)$-geometric.
Using the induction hypothesis, it suffices to show that each stack $S_i \times_{X \times X} X$ is $(n-1)$-geometric.
Note that this stack fits in the following diagram of cartesian squares:
\[ \begin{tikzcd}
S_i \times_{X \times X} X \arrow{r} \arrow{d} & V \times_X V \arrow{r} \arrow{d} & X \arrow{d} \\
S_i \arrow{r} & V \times V \arrow{r} & X \times X .
\end{tikzcd} \]
Since $V \times V$, $V \times_X V$ and $S_i$ are $(n-1)$-geometric, it follows that the same goes for $S_i \times_{X \times X} X$, thus completing the proof. \end{proof}
\begin{cor} \label{cor:representability_diagonal}
Let $(\mathcal C, \tau, \mathbf P)$ be a geometric context and assume that $\mathcal C$ is closed under $\tau$-descent.
If $X \in \mathrm{Sh}(\mathcal C, \tau)$ admits an $n$-atlas, then it is $n$-geometric. \end{cor}
\begin{proof}
We have to prove that the diagonal of $X$ is $(n-1)$-representable.
Let $V \to X$ be an $n$-atlas.
Then $V \times V \to X \times X$ is an $n$-atlas for $X\times X$.
By \cref{lem:factorizing_property_effective_epimorphisms}, for any map $S \to X \times X$, with $S$ being representable, we can find a $\tau$-covering $\{ S_i \to S\}$ such that the composite maps $S_i \to S \to X \times X$ factor through $V \times V$.
Using \cref{prop:geometric_stacks_closed_under_tau}, we are reduced to prove that each $S_i \times_{X \times X} X$ is $(n-1)$-geometric.
Consider the diagram
\[ \begin{tikzcd}
S_i \times_{X \times X} X \arrow{r} \arrow{d} & V \times_X V \arrow{r} \arrow{d} & X \arrow{d} \\
S_i \arrow{r} & V \times V \arrow{r} & X \times X .
\end{tikzcd} \]
The right and the outer squares are pullback diagrams by construction.
Therefore, so is the left square.
Now we conclude from the fact that $V \times_X V$ is $(n-1)$-geometric. \end{proof}
\begin{prop} \label{prop:Afdk_closed_under_tau}
The category $\mathrm{Afd}_k$ of $k$-affinoid spaces is closed under $\tau_\mathrm{\acute{e}t}$-descent. \end{prop} \begin{proof}
Let $Y$ be a $k$-affinoid space and let $f \colon F \to h_Y$ be a morphism in $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$.
Let $\{Y_i \to Y\}_{i \in I}$ be a finite étale covering in the category $\mathrm{Afd}_k$.
Assume that for every index $i$, the fiber product $h_{Y_i} \times_{h_Y} F$ is representable by $X_i \in \mathrm{Afd}_k$.
Put $Y^0 \coloneqq \coprod_{i \in I} Y_i$ and let $Y^\bullet$ be the \v{C}ech nerve of $Y^0 \to Y$.
By assumption, we see that for every integer $n$, $h_{Y^n} \times_{h_Y} F$ is representable.
Choose $X^n \in \mathrm{Afd}_k$ such that $h_{Y^n} \times_{h_Y} F \simeq h_{X^n}$.
Fully faithfulness of the Yoneda embedding implies that we can arrange the objects $X^n$ into a simplicial object $X^\bullet$ in $\mathrm{Afd}_k$.
Let $\bm{\Delta}_s$ be the semisimplicial category.
It follows from \cite[6.5.3.7]{HTT} that the inclusion $\bm{\Delta}_s^\mathrm{op} \subset \bm{\Delta}^\mathrm{op}$ is cofinal.
Let $\bm{\Delta}_{s, \le 2}$ be the full subcategory of $\bm{\Delta}_s$ spanned by the objects $[0]$, $[1]$ and $[2]$.
The inclusion $\bm{\Delta}_{s, \le 2}^\mathrm{op} \subset \bm{\Delta}_s^\mathrm{op}$ is $1$-cofinal, in the sense that for every $[n] \in \bm{\Delta}_s$, the undercategory $(\bm{\Delta}_{s,\le 2}^\mathrm{op})_{[n]/}$ is nonempty and connected.
Let $j \colon \bm{\Delta}_{s, \le 2}^\mathrm{op} \hookrightarrow \bm{\Delta}$ be the composite functor.
Since $\mathrm{Afd}_k$ is a $1$-category, we see that $X^\bullet$ admits a colimit if and only if $X^\bullet_{s, \le 2} \coloneqq X^\bullet \circ j$ does.
The latter statement is true because $\mathrm{Afd}_k$ admits finite colimits.
Let $X$ be the colimit of $X^\bullet$ and let $g \colon X \to Y$ be the canonical map.
We claim that $X^n \simeq Y^n \times_Y X$.
To prove this, it is enough to show that $X^0 \simeq Y^0 \times_Y X$.
We first remark that if the map $X^0 \to Y^0$ is a closed immersion, then the statement follows directly from the fpqc descent of coherent sheaves (cf.\ \cite{Conrad_Descent_for_coherent_2003}).
In the general case, we factor $X^0 \to Y^0$ as $X^0 \hookrightarrow \mathbf D^N_{Y^0} \to Y^0$, where $\mathbf D^N_{Y^0}$ denotes the $N$-dimensional unit polydisc over $Y^0$ and the first arrow is a closed immersion.
Observe that the colimit of $\mathbf D^N_{Y^\bullet}$ is $\mathbf D^N_Y$, and that $\mathbf D^N_{Y^0} \simeq Y^0 \times_Y \mathbf D^N_Y$.
Consider the following diagram:
\[ \begin{tikzcd}
X^0 \arrow{r} \arrow{d} & X \arrow{d} \\
\mathbf D^N_{Y^0} \arrow{r} \arrow{d} & \mathbf D^N_Y \arrow{d} \\
Y^0 \arrow{r} & Y .
\end{tikzcd} \]
Since $X_0 \hookrightarrow \mathbf D^N_{Y^0}$ is a closed immersion, we see that the top square is a pullback.
Moreover, we remarked that the bottom square is also a pullback.
Hence so is the outer square, completing the proof of the claim.
As a consequence, we see that $X^\bullet$ is the \v{C}ech nerve of the étale covering $X^0 \to X$.
In particular, in $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$ we have
\[ h_X \simeq | h_{X^\bullet} | ,\]
where $\abs{\cdot}$ denotes the geometric realization.
Finally, since $\mathrm{Sh}(\mathrm{Afd}_k, \tau_\mathrm{\acute{e}t})$ is an $\infty$-topos\xspace, we obtain:
\[ h_X \simeq | h_{X^\bullet} | \simeq |h_{Y^\bullet} \times_{h_Y} F| \simeq |h_{Y^\bullet}| \times_{h_Y} F \simeq F. \]
This shows that $F$ is representable, thus completing the proof.
\end{proof}
\begin{cor} \label{cor:dAfdk_closed_under_tau}
The category $\mathrm{dAfd}_k$ of derived $k$-affinoid spaces is closed under $\tau_\mathrm{\acute{e}t}$-descent. \end{cor}
\begin{proof}
Let $Y = (\mathcal Y, \mathcal O_\mathcal Y)$ be a derived $k$-affinoid space.
Let $F \to h_Y$ be a morphism in $\mathrm{Sh}(\mathrm{dAfd}_k, \tau_\mathrm{\acute{e}t})$.
Assume there exists an \'etale covering $Y_i \to Y$ such that each base change $h_{Y_i} \times_{h_Y} F$ is representable by a derived $k$-affinoid space $X_i$.
In particular, $\mathrm{t}_0(h_{Y_i} \times_{h_Y} F) \simeq \mathrm{t}_0(h_{Y_i}) \times_{\mathrm{t}_0(h_Y)} \mathrm{t}_0(F)$ is representable by an ordinary $k$-affinoid space $\mathrm{t}_0(X_i)$.
It follows from \cref{prop:Afdk_closed_under_tau} that $\mathrm{t}_0(F)$ is representable by an ordinary $k$-affinoid space $Z$.
Form the \v{C}ech nerve $G^\bullet$ of $\coprod h_{Y_i} \times_{h_Y} F \to F$.
By hypothesis, each $G^n$ is a disjoint union of derived $k$-affinoid spaces.
Since $\phi$ is fully faithful, we obtain in this way a simplicial object $X^\bullet$ in $\mathrm{dAn}_k$, such that all the face maps are \'etale morphisms.
It follows from \cite[Proposition 2.3.5]{DAG-V} that this simplicial object admits a colimit $Y$ in $\RTop(\cT_{\mathrm{an}}(k))$ and that the canonical maps $X^n \to X$ are \'etale.
This shows that we can cover $X$ with derived $k$-affinoid spaces.
In particular, $X$ is a derived $k$-analytic\xspace space.
We are left to prove that $X$ is a derived $k$-affinoid space.
Observe that the maps $\mathrm{t}_0(X^n) \to \mathrm{t}_0(X)$ are \'etale.
Since $X$ (resp.\ $X^n$) and $\mathrm{t}_0(X)$ (resp.\ $\mathrm{t}_0(X^n)$) share the same underlying $\infty$-topos\xspace, we can use the statement (3') in the proof of \cite[Proposition 2.3.5]{DAG-V} to conclude that the colimit of $\mathrm{t}_0(X^\bullet)$ in $\RTop(\cT_{\mathrm{an}}(k))$ is $\mathrm{t}_0(X)$.
On the other hand, since $\mathrm{t}_0$ commutes with limits, we can further identify $\phi(\mathrm{t}_0(X^\bullet))$ with the \v{C}ech nerve of the map $\coprod \mathrm{t}_0(h_{Y_i} \times_{h_Y} F) \to \mathrm{t}_0(F) \simeq \phi(Z)$.
It follows that $\mathrm{t}_0(X) \simeq Z$ in $\mathrm{dAn}_k$. This shows that $X$ is a derived $k$-affinoid space, and $\phi(X) \simeq F$.
The proof is thus complete. \end{proof}
\end{document} |
\begin{document}
\title{Continuous State-Space Models for Optimal Sepsis Treatment - a Deep Reinforcement Learning Approach}
\author{Aniruddh Raghu \email araghu@mit.edu \\
\AND
\name Matthieu Komorowski \email mkomo@mit.edu\\
\AND
\name Leo Anthony Celi \email lceli@mit.edu \\
\AND
\name Peter Szolovits \email psz@mit.edu \\
\AND
\name Marzyeh Ghassemi \email mghassem@mit.edu \\
\\
\addr Computer Science and Artificial Intelligence Lab, MIT\\
Cambridge, MA\\}
\maketitle
\begin{abstract} Sepsis is a leading cause of mortality in intensive care units (ICUs) and costs hospitals billions annually. Treating a septic patient is highly challenging, because individual patients respond very differently to medical interventions and there is no universally agreed-upon treatment for sepsis. Understanding more about a patient's physiological state at a given time could hold the key to effective treatment policies. In this work, we propose a new approach to deduce optimal treatment policies for septic patients by using continuous state-space models and deep reinforcement learning. Learning treatment policies over continuous spaces is important, because we retain more of the patient's physiological information. Our model is able to learn clinically interpretable treatment policies, similar in important aspects to the treatment policies of physicians. Evaluating our algorithm on past ICU patient data, we find that our model could reduce patient mortality in the hospital by up to 3.6\% over observed clinical policies, from a baseline mortality of 13.7\%. The learned treatment policies could be used to aid intensive care clinicians in medical decision making and improve the likelihood of patient survival. \end{abstract}
\section{Introduction} Sepsis (severe infections with organ failure) is a dangerous condition that costs hospitals billions of pounds in the UK alone \citep{sepsiscost}, and is a leading cause of patient mortality \citep{sepsismortality}. The clinicians' task of deciding treatment type and dosage for individual patients is highly challenging. Besides antibiotics and infection source control, a cornerstone of the management of severe infections is administration of intravenous fluids to correct hypovolemia. This may be followed by the administration of vasopressors to counteract sepsis-induced vasodilation. Various fluids and vasopressor treatment strategies have been shown to lead to extreme variations in patient mortality, which demonstrates how critical these decisions are \citep{waechter2014interaction}. While international efforts attempt to provide general guidance for treating sepsis, physicians at the bedside still lack efficient tools to provide individualized real-term decision support \citep{rhodes2017surviving}. As a consequence, individual clinicians vary treatment in many ways, e.g., the amount and type of fluids used, the timing of initiation and the dosing of vasopressors, which antibiotics are given, and whether to administer corticosteroids.
In this work, we propose a data-driven approach to discover optimal sepsis treatment strategies. We use deep reinforcement learning (RL) algorithms to identify how best to treat septic patients in the intensive care unit (ICU) to improve their chances of survival.
While RL has been used successfully in complex decision making tasks \citep{atari,go}, its application to clinical models has thus far been limited by data availability \citep{nemati} and the inherent difficulty of defining clinical state and action spaces \citep{prasad2017reinforcement,komorowski}.
Nevertheless, RL algorithms have many desired properties for the problem of deducing high-quality treatments. Their intrinsic design for sparse reward signals makes them well suited to overcome complexity from the stochasticity in patient responses to medical interventions, and delayed indications of efficacy of treatments. Importantly, RL algorithms also allow us to infer optimal strategies from suboptimal training examples.
In this work, we demonstrate how to surmount the modeling challenges present in the medical environment and use RL to successfully deduce optimal treatment policies for septic patients.\footnote{Either patients who develop sepsis in their ICU stay, or those who are already septic at the start of their stay.} We focus on continuous state-space modeling, represent a patient's physiological state at a point in time as a continuous vector (using either raw physiological data or sparse latent state representations), and find optimal actions with Deep-Q Learning \citep{atari}. Motivating this approach is the fact that physiological data collected from ICU patients provide very rich representations of a patient's physical state, allowing for the discovery of interpretable and high-quality policies.
In particular, we: \begin{enumerate}[nosep] \item Propose deep reinforcement learning models with continuous-state spaces, improving on earlier work with discretized models. \item Identify treatment policies that could improve patient outcomes, potentially reducing patient mortality in the hospital by 1.8 - 3.6\%, from a baseline mortality of 13.7\%. \item Investigate the learned policies for clinical interpretability and potential use as a clinical decision support tool. \end{enumerate}
\section{Background and Related Work} In this section we outline important reinforcement learning algorithms used in the paper and motivate our approach in comparison to prior work. \subsection{Reinforcement Learning}
Reinforcement learning (RL) models time-varying state spaces with a Markov Decision Process (MDP), in which at every timestep $t$ an agent observes the current state of the environment $s_t$, takes an action $a_t$ from the allowable set of actions $\cal A = \{$1$, \dots, M\}$, receives a reward $r_t$, and then transitions to a new state $s_{t+1}$. The agent selects actions at each timestep that maximize its expected discounted future reward, or \emph{return}, defined as $R_t = \sum_{t'=t}^{T} \gamma^{t'-t}r_{t'}$, where $\gamma$ captures the tradeoff between immediate and future rewards, and $T$ is the terminal timestep. The optimal action value function $Q^{*}(s,a)$ is the maximum discounted expected reward obtained after executing action $a$ in state $s$; that is, performing $a$ in state $s$ and proceeding optimally from this point onwards. More concretely, $Q^{*}(s,a) = \max_{\pi}\mathbb{E}[R_t | s_t = s, a_t = a, \pi]$, where $\pi$ --- also known as the \emph{policy} --- is a mapping from states to actions. The optimal value function is defined as $V^{*}(s)=\max_{\pi}\mathbb{E}[R_t|s_t = s,\pi]$, where we act according to $\pi$ throughout.
In Q-learning, the optimal action value function is estimated using the Bellman equation, \newline $Q^{*}(s,a) = \mathop{\mathbb{E}}_{s'\sim T(s'|s,a)}[r + \gamma \max_{a'} Q^{*}(s',a')| s_t = s, a_t = a]$, where $T(s'|s,a)$ refers to the state transition distribution. Learning proceeds either with value iteration \citep{sutton} or by directly approximating $Q^{*}(s,a)$ using a function approximator (such as a neural network) and learning via stochastic gradient descent. Note that Q-learning is an \emph{off-policy} algorithm, as the optimal action-value function is learned with samples $<s,a,r,s'>$ that are generated to explore the state space. An alternative to Q-learning is the SARSA algorithm \citep{sarsa}; an on-policy method to learn $Q^{\pi}(s,a)$, which is the action-value function when taking action $a$ in state $s$ at time $t$, and then proceeding according to policy $\pi$ afterwards.
In this work, the state $s_t$ is a patient's physiological state, either in raw form (as discussed in Section 3.2) or as a latent representation. The action space, $\mathcal{A}$, is of size 25 and is discretized over doses of vasopressors and IV fluids, two drugs commonly given to septic patients, detailed further in Section 3.3. The reward $r_t$ is $\pm R_\textit{max}$ at terminal timesteps and zero otherwise, with positive rewards being issued when a patient survives. At every timestep, the agent is trained to take an action $a_t$ with the highest Q-value, aiming to increase the chance of patient survival.
\subsection{Reinforcement Learning in Health} Much prior work in clinical machine learning has focused on supervised learning techniques for diagnosis \citep{dnn-skincancer} and risk stratification \citep{riskstrat-diabetes}. The incorporation of time in a supervised setting could be implicit within the feature space construction~\citep{hug2009icu,joshi2012prognostic}, or captured with multiple models for different timepoints ~\citep{fialho2013disease,ghassemi2014unfolding}. We prefer RL for sepsis treatment over supervised learning, because the ground truth of ``good'' treatment strategy is unclear in medical literature \citep{demise-egdt}. Importantly, RL algorithms also allow us to infer optimal strategies from training examples that do not represent optimal behavior. RL is well-suited to identifying ideal septic treatment strategies, because clinicians deal with a sparse, time-delayed reward signal in septic patients, and optimal treatment strategies may differ.
\cite{nemati} applied deep RL techniques to modeling ICU heparin dosing as a Partially Observed Markov Decision Process (POMDP), using both discriminative Hidden Markov Models and Q-networks to discover the optimal policy. Their investigation was made more challenging by the relatively small amount of available data. \cite{shortreed2011informing} learned optimal treatment policies for schizophrenic patients, and quantified the uncertainty around the expected outcome for patients who followed the policies. \cite{prasad2017reinforcement} use off-policy reinforcement learning algorithms to determine ICU strategies for mechanical ventilation administration and weaning, but focus on simpler learning algorithms and a heuristic action space. We experiment with using a sparse autoencoder to generate latent representations of the state of a patient, likely leading to an easier learning problem. We also propose neural network architectures that obtain more robust methods for optimal policy deduction.
Optimal sepsis treatment strategy was tackled most recently by \cite{komorowski}, using a discretized state and action-space to deduce optimal treatment policies for septic patients. Their work applied on-policy SARSA learning to fit an action-value function to the physician policy and value-iteration techniques to find an optimal policy \citep{sutton}. The optimal policy was then evaluated by comparing the Q-values that would have been obtained following chosen actions to the Q-values obtained by the physicians. We reproduce a similar model as our baseline, using related data pre-processing and clustering techniques. We additionally build on this approach by extending the results to the continuous domain, where policies are learned directly from the physiological state data, without discretization. We also propose a novel evaluation metric, different from ones used in \cite{komorowski}. We focus on in-hospital mortality instead of 90-day mortality (used in \cite{komorowski}) because of the other unobserved factors that could affect mortality in a 3-month timeframe.
\section{Data and Preprocessing} \subsection{Cohort} Data for these patients were obtained from the Multiparameter Intelligent Monitoring in Intensive Care (MIMIC-III v1.4) database \citep{mimic}, which is publicly available, and contains hospital admissions from approximately 38,600 adults (at least 15 years old). We extracted a cohort of patients fulfilling the Sepsis-3 criteria \citep{sepsis3}, and note that summary information about the populations is similar in sepsis survivors and mortalities (Table \ref{tab:cohort}). \begin{table}[htbp]
\centering
\begin{tabular}{|l|l||l|l||l|}\hline
& \% Female & Mean Age & Hours in ICU & Total Population \\ \hline
Survivors & 43.6 & 63.4 & 57.6 & 15,583 \\ \hline
Non-survivors & 47.0 & 69.9 & 58.8 & 2,315 \\ \hline
\end{tabular}
\caption{Comparison of cohort statistics for subjects that fulfilled the Sepsis-3 criteria.}
\label{tab:cohort} \end{table}
\subsection{Feature Preprocessing} For each patient, we extracted relevant physiological parameters including demographics, lab values, vital signs, and intake/output events. Data were aggregated into windows of 4 hours, with the mean or sum being recorded (as appropriate) when several data points were present in one window. Variables with excessive missingness were removed, and any remaining missing values were imputed with k-nearest neighbors, yielding a $47\times1$ feature vector for each patient at each timestep. Values exceeding clinical limits were capped, and capped data was normalized per-feature to zero mean and unit variance. See Appendix \ref{sec:appendix_features} for a full feature list.
\subsection{Action Discretization} We defined a $5\times5$ action space for the medical interventions covering the space of intravenous (IV) fluid (volume adjusted for fluid tonicity). and maximum vasopressor (VP) dosage in a given 4 hour window. The action space was restricted to these two interventions as both drugs are extremely important in the management of septic patients, but there is no agreement on when, and how much, of each drug to give \citep{demise-egdt}. We discretized the action space into per-drug quartiles based on all non-zero dosages of the two drugs, and converted each drug at every timestep into an integer representing its quartile bin. We included a special case of no drug given as bin 0. This created an action representation of interventions as tuples of (total IV in, max VP in) at each time.
\section{Methods} The challenge of applying RL to optimal medication dosing is that all available data are \emph{offline sampled}; that is, data are collected previously and models can only be fit to a retrospective dataset. In an RL context, this limits exploration of the state space in question, and makes learning the truly `optimal' policy difficult. This limitation motivates trying several different approaches, with varied modeling constraints, to determine the best medication strategy for patients.
We focus on off-policy RL algorithms that learn an optimal policy through data that is generated by following an alternative policy. This makes sense for our problem because the available data are generated from a policy followed by physicians, but our goal is to learn a different, optimal policy rather than to evaluate the physician's policy. We propose deep models with continuous state spaces and discretized action spaces to retain more of the underlying state representation.
\subsection {Discretized State-space and Discretized Action-space} Following \cite{komorowski}, we create a baseline model with discretized state and action spaces, aiming to capture the underlying representation while simplifying the learning procedure. We use this approach to evaluate the performance of other techniques, and to understand the significance of learned Q values. We use the SARSA algorithm \citep{sarsa} to learn $Q^{\pi}(s,a)$, and the action-value function for the physician policy (more detail in Appendix \ref{sec:appendix_train}).
\subsection {Continuous State-spaces} Continuous state-space models directly capture a patient's physiological state, and allow us to discover high-quality treatment policies. To learn an optimal policy with continuous state vectors, we use neural networks to approximate the optimal action-value function, $Q^{*}(s,a)$.
\subsubsection {Model Architecture} \label{sec:continuous_spaces} Our model is based on a variant of Deep Q Networks \citep{atari}. Deep Q Networks seek to minimize a squared error loss between the output of the network, $Q(s,a;\theta)$, and the desired target, $Q_{\textit{target}} = r + \gamma \max_{a'}Q(s',a';\theta)$, observing tuples of the form $<s,a,r,s'>$. The network has outputs for all the different actions that can be taken --- for all $a \in \cal A = \{$1$, \dots, M\}$. Concretely, the parameters $\theta^{*}$ are found such that: $$\theta^{*} = \argmin_{\theta}\mathbb{E}\left[\mathcal{L}(\theta)\right] = \argmin_{\theta}\mathbb{E}\left[\left( Q_{\textit{target}} - Q(s,a;\theta)\right)^{2}\right] $$
In practice, the expected loss is minimized via stochastic batch gradient descent. However, this method can be unstable due to non-stationarity of the target values, and using a separate network to determine the target Q values ($Q(s',a')$), which is periodically updated towards the main network (used to estimate $Q(s,a)$) helps to improve performance.
Simple Q-Networks have several shortcomings, so we made several important modifications to make our model suitable for this situation. Firstly, Q-values are frequently overestimated in practice, leading to incorrect predictions and poor policies. We solve this problem with a Double-Deep Q Network \citep{ddqn}, where the target Q values are determined using actions found through a feed-forward pass on the main network, as opposed to being determined directly from the target network.
In the context of finding optimal treatments, we want to separate the influence on Q-values of 1) a patient's \textit{underlying state} being good (e.g. near discharge), and 2) the correct action being taken at that timestep. To this end, we use a Dueling Q Network \citep{dueling}, where the action-value function for a given $(s,a)$ pair, $Q(s,a)$, is split into separate \emph{value} and \emph{advantage} streams, where the \emph{value} represents the quality of the current state, and the \emph{advantage} represents the quality of the chosen action.
Training such a model can be slow as reward signals are sparse and only available on terminal timesteps. We use Prioritized Experience Replay \citep{per} to accelerate learning by sampling a transition from the training set with probability proportional to the previous error observed.
Our final network architecture is a Dueling Double-Deep Q Network (Dueling DDQN), combining both of the above ideas. The network has two hidden layers of size 128, uses batch normalization \citep{batchnorm} after each, Leaky-ReLU activation functions, a split into equally sized advantage and value streams, and a projection onto the action space by combining these two streams. For more details, see Appendix \ref{sec:appendix_model}.
After training the Dueling DDQN, we can then obtain the optimal policy for a given patient state as: $\pi^{\ast}(s) = \argmax_{a} Q(s,a)$.
\subsection{Autoencoder Latent State Representation} Deep RL approaches for optimal medication are challenging to learn, because patient state is a high-dimensional continuous vector without clear structure. We examined both ordinary autoencoders \citep{autoencoder} and sparse autoencoders \citep{sparse-autoencoder} to produce latent state representations of the physiological state vectors and simplify the learning problem. Sparse autoencoders were trained with an additional term in the loss function to encourage sparsity. Our autoencoder models all had a single hidden layer, which was used as the latent state representation. These latent state representations were used as inputs to the Dueling DDQN (Section \ref{sec:continuous_spaces}).
\section{Evaluation} The evaluation of off-policy models is challenging, because it is difficult to estimate whether the rollout of a learned policy (using the learned policy to determine actions at each state) would eventually lead to lower patient mortality. Furthermore, directly comparing Q values on off-policy data, as done in prior applications of RL to healthcare \citep{komorowski} can provide incorrect performance estimates \citep{off-policy-eval}. Finding the average Q-value as in \cite{komorowski} is suboptimal because the $Q^{\pi}$ used for assessment represents the expected return when acting optimally at state $s_t$, but then proceeding according to $\pi_{\textit{physician}}$, the physician policy. In this work, we propose evaluating learned policies with several approaches.
\subsection{Discounted Returns vs. Mortality} \label{sec:ev_discounted} To understand how expected discounted returns relate to mortality, we bin Q-values obtained via SARSA on the test set into discrete buckets, and for each, if it is part of a trajectory where a patient died, we assign it a label of 1. If the patient survived, we assign a label of 0. These labels represent the ground truth, as we know the actual outcome of patients when the physician's policy is followed. We compute the average mortality in each bin, enabling us to produce an empirically derived function of proportion of mortality versus expected return (Figure \ref{fig:m_vs_r}). We expect to see an inverse relationship between mortality and expected return, and this function enables us to associate returns with mortality for the purpose of evaluation.
\subsection{Off-Policy Evaluation} \label{sec:ev_off} We use the method of Doubly Robust Off-policy Value Evaluation \citep{off-policy-eval} to obtain an unbiased estimate of the value of the learned optimal policy using off-policy sampled data (the trajectories in our training set). For each trajectory $H$ we compute an unbiased estimate of the value of the learned policy, $V_{\textit{DR}}^{H}$,
and average the results obtained across the observed trajectories. More details are provided in \cite{off-policy-eval}. We can also compute the mean discounted return of chosen actions under the physician policy. Using both these estimates, and the empirically learned proportion of mortality vs.\ expected return function, we can assess the potential improvement our policy could bring in terms of reduction in patient mortality. This method allows to accurately compare the returns obtained via different methodologies on off-policy data and estimate the mortality we would observe when following the learned policies. Directly comparing returns without the use of such an estimator is likely to give invalid results \citep{off-policy-eval}.
\subsection{Qualitative Examination of Treatment Policies} \label{sec:ev_qual} We examine the overall choice of treatments proposed by the optimal policy to derive more clinical understanding, and compare these choices to those made by physicians to understand how differences in the chosen actions contribute to patient mortality.
\section{Results} \subsection {Fully Discretized Models are Well-calibrated with Test Set Mortality} Figure \ref{fig:m_vs_r} shows the proportion of mortality versus the expected return for the physician policy on the held out test set. Note that $R_{\textit{max}} = 15$ is the reward issued at terminal timesteps. As shown, we observe high mortality with low returns, and low mortality with high returns. We also confirm that the empirically derived mortality for the physician's policy matches the actual proportion of mortality in the test set. For the empirically derived mortality, we average the expected return for the physician on the test set to obtain $13.9 \pm 0.5 \% $. This matches the actual proportion of mortality on the test set ($13.7\%$). \begin{figure}
\caption{The relationship between expected returns ---learned from observational data and actions taken by actual physicians --- and the risk of mortality in the test set of 3,580 patients (see Sec \ref{sec:ev_discounted}). The model appears to be well calibrated, with an inverse relationship between return and mortality. This function is not monotonically decreasing for low returns due to there being fewer training examples in this regime.}
\label{fig:m_vs_r}
\end{figure}
\subsection {Continuous State-space Models} We present the results for the two proposed networks: the Dueling Double-Deep Q Network (Dueling DDQN) and the Sparse Autoencoder Dueling DDQN. These are referred to as the \emph{normal Q-N} model and \emph{autoencode Q-N} model respectively for clarity.
\subsubsection{Quantitative Value Estimate of Learned Policies} Table \ref{tab:performance} demonstrates the relative performance of the three policies --- physician, \emph{normal Q-N}, and \emph{autoencode Q-N} --- on expected returns and estimated mortality. As described in Sec \ref{sec:ev_off}, we first obtain unbiased estimates of the value of our learned policies on the test data. The expected returns shown are $\bar{V}_{\textit{DR}}^{\textit{Physician}}$, $\bar{V}_{\textit{DR}}^{\textit{normal Q-N}}$, and $\bar{V}_{\textit{DR}}^{\textit{autoencode Q-N}}$. We estimate the mortality under each policy using Figure \ref{fig:m_vs_r}. As shown, the \emph{autoencode Q-N} policy has the lowest estimated mortality and could reduce patient mortality by up to 4\%. We examine a histogram of mortality counts against the first two principal components of the sparse representation (Figure \ref{fig:heatmap}) and observe a clear gradient of mortality counts, indicating how the autoencoder's hidden state may provide a rich representation of physiological state that leads to better policies.
\begin{table}[ht]
\centering
\begin{tabular}{l|l||l}
Policy & Expected Return & Estimated Mortality \\ \hline
Physician & 9.87 & $13.9 \pm 0.5 \%$ \\ \hline
Normal Q-N & 10.16 & $12.8 \pm 0.5 \%$ \\ \hline
Autoencode Q-N & 10.73 & $11.2 \pm 0.4 \%$
\end{tabular}
\caption{Comparison of expected return and estimated mortality under the physician's policy, Normal Q-N, and Autoencode Q-N.}
\label{tab:performance} \end{table}
\begin{figure}
\caption{Histogram of mortality counts against first two principal components of sparse autoencoder representation. Note the association between these values and the eventual outcome of the patient, potentially indicating why this model was able to learn a good quality policy.}
\label{fig:heatmap}
\end{figure}
\subsubsection{Qualitative Examination of Learned Policies} Figure \ref{fig:policies} demonstrates what the three policies --- physician, \emph{normal Q-N}, and \emph{autoencode Q-N} --- have learned as optimal policies. The action numbers index the different discrete actions selected at a given timestep, and the charts shown aggregate actions taken over all patient trajectories. Action 0 refers to no drugs given to the patient at that timestep, and increasing actions refer to higher drug dosages, where drug dosages are represented by quartiles. \begin{figure}
\caption{Policies learned by the different models, as a 2D histogram, where we aggregate all actions selected by the physician and models on the test set over all timesteps. The axes labels index the discretized action space, where 0 represents no drug given, and 4 the maximum of that particular drug. Both models learn to prescribe vasopressors sparingly, a key feature of the physician's policy. }
\label{fig:policies}
\end{figure}
As shown, physicians do not often prescribe vasopressors to patients (note the high density of actions corresponding to vasopressor dose = 0) and this behavior is strongly in the policy learned by the \emph{autoencode Q-N} model. This result is sensible; even though vasopressors are commonly used in the ICU to elevate mean arterial pressure, many patients with sepsis are not hypotensive and therefore do not need vasopressors. In addition, there have been few controlled clinical trials that have documented improved outcomes from their use~\citep{mullner2004vasopressors}. The \emph{normal Q-N} also learns a policy where vasopressors are not given in with high frequency, but that policy is less evident. There are interesting parallels between the two learned policies (\emph{normal Q-N}, and \emph{autoencode Q-N}). For example, both favor action (0,2) (corresponding to no IV fluids given and an intermediate dosage of vasopressor given), and action (2,3) (corresponding to a medium dosage of IV fluids and vasopressors).
\subsubsection{Quantifying Optimality of Learned Policies} Figure \ref{fig:diff_mort} shows the correlation between 1) the observed mortality, and 2) the difference between the optimal doses suggested by the policy, and the actual doses given by clinicians. The dosage differences at individual timesteps were binned, and mortality counts were aggregated. We observe consistently low mortalities when the optimal dosage and true dosage coincide, i.e. at a difference of 0, indicating the validity of the learned policy. The observed mortality proportion then increases as the difference between the optimal dosage and the true dosage increases. Results are less reliable when the optimal dose and physician dose differ by larger amounts.
\begin{figure}
\caption{Comparison of how observed mortality (y-axis) varies with the difference between the dosages recommended by the optimal policy and the dosages administered by clinicians (x-axis). For every timestep, this difference was calculated and associated with whether the patient survived or died in the hospital, allowing the computation of observed mortality. In general, we see low mortality for when the difference is zero, indicating that when the physician acts according to the optimal policy we observe more patient survival. }
\label{fig:diff_mort}
\end{figure}
Both models appear to learn useful policies for vasopressors, with a large increase in observed mortality seen in the \emph{autoencode Q-N} because of relatively few cases in the test set where the optimal dose and given dose differed positively by a large amount. For IV-fluids, \emph{normal Q-N} learns a policy that shows a clear improvement over that of the physician's, indicated by the significant drop in observed mortality at the 0 mark. The \emph{autoencode Q-N} model learns a weaker policy over IV fluids, shown by the observed mortality decreasing as the difference between dosages increases.
\section{Conclusion} In this work, we explored methods of applying deep reinforcement learning (RL) to the problem of deducing optimal medical treatments for patients with sepsis. There remain many interesting areas to be investigated. Firstly, the credit assignment in this model is quite sparse, with rewards/penalties only being issued at terminal states. There is scope for improvement here; one idea could be to use a clinically informed reward function based on patient blood counts to help learn better policies. Another approach could be to use inverse RL techniques \citep{Abbeel2010} to derive a suitable reward function based on the actions of experts (the physicians). As our dataset of patient trajectories is collected from recording the actions of many different physicians, this approach may allow us to infer a more appropriate reward function and in turn learn a better model.
Our contributions build on recent work by \cite{komorowski}, investigating a variety of techniques to find optimal treatment policies that improve patient outcome. We started by building a discretized state and action-space model, where the underlying states represent the physiological data averaged over four hour blocks and the action space is over two commonly administered drugs for septic patients --- IV fluids and vasopressors. Following this, we explored a fully continuous state-space/discretized action-space model, using Dueling Double-Deep Q Networks to learn an approximation for the optimal action-value function, $Q^{*}(s,a)$.
We demonstrated that using continuous state space modeling found policies that could reduce patient mortality in the hospital by 1.8--3.6\%, which is an exciting direction for identifying better medication strategies for treating patients with sepsis. Our policies learned that vasopressors may not be favored as a first response to sepsis, which is sensible given that vasopressors may be harmful in some populations~\citep{d2015blood}. Our learned policy of intermediate fuild dosages also fits well with recent clinical work finding that large fluid dosages on first ICU day are associated with increased hospital costs and risk of death~\citep{marik2017fluid}. The learned policies are also clinically interpretable, and could be used to provide clinical decision support in the ICU. To our knowledge, this is the first extensive application of novel deep reinforcement learning techniques to medical informatics, building significantly on the findings of \cite{nemati}.
\acks{This research was funded in part by the Intel Science and Technology Center for Big Data and the National Library of Medicine Biomedical Informatics Research Training grant (NIH/NLM 2T15 LM007092-22). }
\section{APPENDICES}
\subsection{Cohort definition}
Following the latest guidelines, sepsis was defined as a suspected infection (prescription of antibiotics and sampling of bodily fluids for microbiological culture) combined with evidence of organ dysfunction, defined by a Sequential Organ Failure Assessment (SOFA) score greater or equal to 2 \citep{sepsis3}. We assumed a baseline SOFA of zero for all patients. For cohort definition, we respected the temporal criteria for diagnosis of sepsis: when the microbiological sampling occurred first, the antibiotic must have been administered within 72 hours, and when the antibiotic was given first, the microbiological sample must have been collected within 24 hours \citep{sepsis3}. The earliest event defined the onset of sepsis. We excluded patients who received no intravenous fluid, and those with missing data for 8 or more out of the 47 variables. This method yield a cohort of 17,898 patients.
\subsection{Data extraction}
MIMIC-III was queried using pgAdmin 4. Raw data were extracted for all 47 features and processed in Matlab (version 2016b). Data was included from up to 24h preceding until 48h following the onset of sepsis, in order to capture the early phase of its management including initial resuscitation, which is the time period of interest. The features were converted into multidimensional time series with a time resolution of 4 hours. The outcome of interest was in-hospital mortality.
\subsection {Model Features}
\label{sec:appendix_features} The physiological features used in our model are presented below. \newline\newline
\textbf{Demographics/Static}\newline Shock Index, Elixhauser, SIRS, Gender, Re-admission, GCS - Glasgow Coma Scale, SOFA - Sequential Organ Failure Assessment, Age \newline \newline \textbf{Lab Values}\newline Albumin, Arterial pH, Calcium, Glucose, Haemoglobin, Magnesium, PTT - Partial Thromboplastin Time, Potassium, SGPT - Serum Glutamic-Pyruvic Transaminase, Arterial Blood Gas, BUN - Blood Urea Nitrogen, Chloride, Bicarbonate, INR - International Normalized Ratio, Sodium, Arterial Lactate, CO2, Creatinine, Ionised Calcium, PT - Prothrombin Time, Platelets Count, SGOT - Serum Glutamic-Oxaloacetic Transaminase, Total bilirubin, White Blood Cell Count \newline \newline \textbf{Vital Signs} \newline Diastolic Blood Pressure, Systolic Blood Pressure, Mean Blood Pressure, PaCO2, PaO2, FiO2, PaO/FiO2 ratio, Respiratory Rate, Temperature (Celsius), Weight (kg), Heart Rate, SpO2 \newline \newline \textbf{Intake and Output Events}\newline Fluid Output - 4 hourly period, Total Fluid Output, Mechanical Ventilation
\subsection{Discretized State and Action Space Model} \label{sec:appendix_train} We present here how the discretized model was built.
\subsubsection {State Discretization} As in the continuous case, the data are partitioned into a training set (80\%) and held-out test set (20\%) by selecting a proportionate number of patient trajectories for each set. These sets were checked to ensure they provide an accurate representation of the complete dataset, in terms of distribution of outcomes and some demographic features. We apply k-means clustering to the training set, discretizing the states into 1250 clusters. As in \cite{komorowski}, we use a simple, sparse reward function, issuing a reward of +15 at a timestep if a patient survives, -15 if they die, and 0 otherwise. Test set data points are discretized according to whichever training set cluster centroid they fall closest to.
\subsubsection{SARSA for Physician Policy} \label{sec:appendix_sarsa} To learn the action-value function associated with the model, we used an offline, SARSA approach with the Bellman optimality equation, randomly sampling trajectories from our training set, and using tuples of the form $<s,a,r,s',a'>$ to update the action-value function: \begin{center} $Q(s, a) \leftarrow Q(s, a) + \alpha \ast [r + \gamma Q(s', a')$ - $Q(s, a)]$ \end {center}
Here, $(s,a)$ is the current (state, action) tuple considered, $(s',a')$ is a tuple representing the next state and action, $\alpha$ is the learning rate and $\gamma$ the discount factor. As our state and action spaces are both finite in this model, we represent the Q-function using a table with rows for each $(s,a)$ tuple. This learned function was then used in model evaluation - after convergence, it represents $Q^{\pi}(s,a) = \mathop{\mathbb{E}}_{s'\sim T(s'|s,a)}[r + \gamma Q^{\pi}(s',a')| s_t = s, a_t = a, \pi]$, where $\pi$ is the physician policy.
\subsection {Continuous Model Architecture and Implementation Details} \label{sec:appendix_model} Our final network architecture had two hidden layers of size 128, using batch normalization \citep{batchnorm} after each, Leaky-ReLU activation functions, a split into equally sized advantage and value streams, and a projection onto the action space by combining these two streams together.
The activation function is mathematically described by: $f(z) = \max(z,0.5z)$, where z is the input to a neuron. This choice of activation function is motivated by the fact that Q-values can be positive or negative, and standard ReLU, tanh, and sigmoid activations appear to lead to saturation and `dead neurons' in the network. Appropriate feature scaling helped alleviate this problem, as did issuing rewards of $\pm15$ at terminal timesteps to help model stability.
We added a regularization term to the standard Q-network loss that penalized output Q-values which were outside of the allowed thresholds ($\pm15$), in order to encourage the network to learn a more appropriate Q function. Clipping the target network outputs to $\pm15$ was also found to be useful. The final loss function was:
$$\mathcal{L}(\theta) = \mathbb{E}\left[\left(Q_{\textit{double-target}} - Q\left(s,a;\theta\right)\right)^{2}\right] + \lambda \cdot \max\left(\left| Q(s,a;\theta)-R_{\max} \right|,0\right)$$ with $R_{\max}$ being the absolute value of the reward/penalty issued at a terminal timestep, and $$ Q_{\textit{double-target}} = r + \gamma Q(s’,\argmax_{a'}Q(s’,a';\theta);\theta ')$$ where $\theta$ are the weights used to parameterize the main network, and $\theta '$ are the weights used to parameterize the target network.
We use a train/test split of 80/20 and ensure that a proportionate number of patient outcomes are present in both sets. Batch normalization is used during training. All models were implemented in TensorFlow v1.0, with Adam being used for optimization \citep{adam}.
During training, we sample transitions of the form $<s,a,r,s'>$ from our training set, perform feed-forward passes on the main and target networks to evaluate the output and loss, and update the weights in the main network via backpropagation.
\subsection{Autoencoder Implementation Details}
For the autoencoder, a desired sparsity $\rho$ is chosen, and the weights of the autoencoder are adjusted to minimize ${\mathcal{L}_{\textit{sparse}}(\theta) = \mathcal{L}_{\textit{reconstruction}}(\theta) + \beta \sum_{j=1}^{n}\textit{KL}(\rho || \rho_{j})}$. Here, $n$ is the total number of hidden neurons in the network, $\rho_{j}$ is the actual output of neuron $j$, $\beta$ is a hyperparameter controlling the strength of the sparsity term, $\textit{KL}(\cdot || \cdot)$ is the KL divergence, and $\mathcal{L}_{\textit{reconstruction}}$ is the loss for a normal autoencoder.
\end{document} |
\begin{document}
\title[Norm inflation with infinite loss of regularity for gIBQ]{Norm inflation with infinite loss of regularity for the generalized improved Boussinesq equation}
\author{Pierre de Roubin}
\maketitle
\begin{abstract} In this paper, we study the ill-posedness issue for the generalized improved Boussinesq equation. In particular we prove there is norm inflation with infinite loss of regularity at general initial data in $\jb{\nabla}^{-s}\big(L^2 \cap L^\infty\big)(\mathbb{R})$ for any $s < 0$. This result is sharp in the $L^2$-based Sobolev scale in view of the well-posedness in $L^2(\mathbb{R}) \cap L^\infty(\mathbb{R})$. We also show that the same result applies to the multi-dimensional generalized improved Boussinesq equation. Finally, we extend our norm inflation result to Fourier-Lebesgue, modulation and Wiener amalgam spaces. \end{abstract}
\section{Introduction}
We consider the generalized improved Boussinesq equation (gIBq): \begin{equation} \begin{cases}\label{imBq} \partial_t^{2}u-\partial_x^2 u - \partial_t^2 \partial_x^2u = \partial_x^2 \left(f(u) \right) \\
(u,\partial_t u)|_{t = 0} = (u_0,u_1), \end{cases} \qquad ( t, x) \in \mathbb{R} \times \mathcal{M}, \end{equation}
\noindent where $\mathcal{M} = \mathbb{R}$ or $\mathbb{T}$ with $\mathbb{T} = \mathbb{R} / \mathbb{Z}$ and $f(u) = u^k$ for $k \geq 2$ an integer. The equation \eqref{imBq} appears in a wide variety of physical problems. Makhankov~\cite{Mak} derived the improved Boussinesq equation, namely with $f(u) = u^2$, in order to study ion-sound wave propagation and mentioned the case of the improved modified Boussinesq equation, with $f(u) = u^3$, as modeling nonlinear Alfv\'en waves. He also mentioned the possibility to use the linear term of \eqref{imBq} to describe waves propagating at right angles to the magnetic field. Besides, Clarkson-LeVeque-Saxton \cite{CLS} considered the cases $f(u) = u^p$, with $p = 3$ or $5$, as a model for the propagation of longitudinal deformation waves in an elastic rod. See also \cite{Tur} for further discussion on the physical aspect of \eqref{imBq}.
This equation has also attracted attention from a mathematical point of view. In particular, the well-posedness of this problem has been studied extensively and Constantin-Molinet~\cite{CM} proved local well-posedness for \eqref{imBq} in $L^2(\mathbb{R}) \cap L^{\infty}(\mathbb{R})$ (and global well-posedness under additional conditions). See also \cite{CO, GL, Zhi}. On the other hand, it is known that \eqref{imBq} possesses some undesirable behaviour in negative Sobolev spaces. Following the work of Bourgain~\cite{Bou97}, it was showed in \cite{GL} that the solution map $\Phi \colon H^s(\mathbb{R}) \times H^s (\mathbb{R})\to C([-T,T], H^s(\mathbb{R}))$ of \eqref{imBq}, for nonlinearity $f(u) = u^k$, fails to be $C^k$ for $s< 0$. This result in particular implies that we can not study the well-posedness of \eqref{imBq} via a contraction argument. Note however that their result does not imply failure of continuity for the data-to-soluton map in negative Sobolev spaces. Our main goal in this article is to complete the well-posedness issue of this problem by proving the discontinuity of the data-to-solution map of \eqref{imBq} in $\jb{\nabla}^{-s}\big(L^2 \cap L^\infty\big) (\mathcal{M})$ for any $s < 0$. For more clarity, let us denote $\mathcal{W}^{s, 2, \infty}(\mathcal{M}) \coloneqq W^{s, 2, \infty}(\mathcal{M}) \times W^{s, 2, \infty}(\mathcal{M})$ with $$ W^{s, 2, \infty}(\mathcal{M}) = \jb{\nabla}^{-s}\big(L^2 \cap L^\infty\big) (\mathcal{M}) = \{ f, \jb{\nabla}^sf \in L^2 (\mathcal{M}) \cap L^\infty (\mathcal{M})\}. $$ \noindent We exhibit the following norm inflation with infinite loss of regularity behaviour:
\begin{theorem} \label{THM:mainLinfty} Let $k \geq 2$, $\sigma \in \mathbb{R}$ and $s < 0$. Fix $(u_0 , u_1) \in \mathcal{W}^{s, 2, \infty} (\mathcal{M})$. Then, given any $\varepsilon > 0$, there exists a solution $u_\varepsilon$ to \eqref{imBq} with nonlinearity $f(u) = u^k$ on $\mathcal{M}$ and $t_\varepsilon \in (0, \varepsilon)$ such that \begin{equation} \label{EQ:NIwithILORLinfty}
\| (u_\varepsilon (0), \partial_t u_\varepsilon (0)) - (u_0 , u_1) \|_{ \mathcal{W}^{s, 2, \infty}} < \varepsilon \quad \text{ and } \quad \| u_\varepsilon (t_\varepsilon) \|_{W^{\sigma, 2, \infty}} > \varepsilon^{-1}. \end{equation} \end{theorem}
Note that, when $\sigma =s$ and $(u_0, u_1) = (0,0)$, \eqref{EQ:NIwithILORLinfty} is already called {\it norm inflation}. Since we have \eqref{EQ:NIwithILORLinfty} for any arbitrary initial data $(u_0, u_1) \in \mathcal{W}^{s, 2, \infty} (\mathcal{M})$, we say that we have norm inflation at {\it general initial data}. Besides, Theorem \ref{THM:main} also yields \eqref{EQ:NIwithILORLinfty} for any arbitrary $\sigma < s$, leading to the so-called {\it infinite loss of regularity}. Observe as well that, as a corollary, Theorem~\ref{THM:mainLinfty} gives the following discontinuity of the solution map.
\begin{corollary} \label{COR:DiscSolMap} Let $s<0$. Then, for any $T > 0$, the solution map $\Phi \colon (u_0, u_1) \in \mathcal{W}^{s, 2, \infty}(\mathcal{M}) \to (u, \partial_t u) \in C([-T, T], W^{s, 2, \infty}(\mathcal{M})) \times C([-T, T], W^{s, 2, \infty} (\mathcal{M})) $ to the generalized improved Boussinesq equation \eqref{imBq} is discontinuous everywhere in $\mathcal{W}^{s, 2, \infty} (\mathcal{M})$.
\end{corollary}
Throughout the rest of this paper, we prove in fact the following result of norm inflation at general initial data with infinite loss of regularity in negative Sobolev spaces:
\begin{theorem} \label{THM:main} Let $k \geq 2$, $\sigma \in \mathbb{R}$ and $s < 0$. Fix $(u_0 , u_1) \in \mathcal{H}^s (\mathcal{M}) \coloneqq H^s (\mathcal{M}) \times H^s (\mathcal{M})$. Then, given any $\varepsilon > 0$, there exists a solution $u_\varepsilon$ to \eqref{imBq} with nonlinearity $f(u) = u^k$ on $\mathcal{M}$ and $t_\varepsilon \in (0, \varepsilon)$ such that \begin{equation} \label{EQ:NIwithILOR}
\| (u_\varepsilon (0), \partial_t u_\varepsilon (0)) - (u_0 , u_1) \|_{\mathcal{H}^s} < \varepsilon \quad \text{ and } \quad \| u_\varepsilon (t_\varepsilon) \|_{H^\sigma} > \varepsilon^{-1}. \end{equation} \end{theorem}
Once Theorem \ref{THM:main} is proved, Theorem \ref{THM:mainLinfty} follows as a corollary. See Remark~\ref{REM:NIinWs2infty}. In fact, we first prove the usual norm inflation at general initial data, namely we prove \eqref{EQ:NIwithILOR} with $\sigma = s$. Then, the loss of regularity follows with a slight modification. See Remark~\ref{REM:ILOR}. Note also that, in a similar manner, Theorem \ref{THM:main} implies the discontinuity of the solution map in negative Sobolev spaces.
There are several ways to prove norm inflation results. Christ-Colliander-Tao \cite{CCT} first introduced norm inflation with a method based on a dispersionless ODE approach. See also \cite{BTz1, Xia}. On the other hand, Carles with his collaborators in \cite{AC, BC, CK} proved norm inflation with infinite loss of regularity results for nonlinear Schr\"odinger equations (NLS) by using geometric optics.
Our strategy is to follow a Fourier analytic argument that originated from the abstract work of Bejenaru-Tao \cite{BT} on quadratic nonlinear Schr\"odinger equation. Their idea was to expand the solution into a power series and to exploit the {\it high-to-low energy transfer} in one of the terms to prove discontinuity of the solution map. This approach was refined later on by Iwabuchi-Ogawa~\cite{IO} who used it to extend the ill-posedness result in \cite{BT} into a norm inflation result. Kishimoto~\cite{Kis} then further developed these methods to prove norm inflation for the nonlinear Schr\"odinger equation. Meanwhile, Oh \cite{Oh} refined the argument of \cite{IO} in the context of cubic NLS by introducing a way to index the power series by trees, and to estimate each term separately. Forlano-Okamoto \cite{FO} proved afterwards norm inflation with an approach inspired by \cite{Oh} for nonlinear wave equations (NLW) in Sobolev spaces of negative regularities, and we use the same reasoning in our proof. For other papers with similar argument, the interested reader might turn to \cite{BH2, COW, CP, OOT, Ok, WZ}. See also \cite{Chevyrev} for an implementation of this method in probabilistic settings.
One of the key ingredients to our proof is the use of the Wiener algebra $\mathcal{F}L^1 (\mathcal{M})$, which we define now. Given $\mathcal{M} = \mathbb{R}$ or $\mathbb{T}$, let $\widehat{\mathcal{M}}$ denote the Pontryagin dual of $\mathcal{M}$, i.e. \begin{equation} \label{DEF:PontryaginDual} \widehat{\mathcal{M}} = \begin{cases} \mathbb{R} & \text{if} \quad \mathcal{M} = \mathbb{R}, \\ \mathbb{Z} & \text{if} \quad \mathcal{M} = \mathbb{T}. \end{cases} \end{equation}
\noindent Note that, when $\widehat{\mathcal{M}} = \mathbb{Z}$, we endow it with the counting measure. We can then define the following Fourier-Lebesgue spaces:
\begin{definition}[Fourier-Lebesgue spaces] \rm \label{DEF:FLspaces} For $s \in \mathbb{R}$ and $p \geq 1$, we define the Fourier-Lebesgue space $\mathcal{F}L^{s,p} (\mathcal{M})$ as the completion of the Schwartz class of functions $\mathcal{S} ( \mathcal{M})$ with respect to the norm $$
\| f \|_{\mathcal{F}L^{s,p}(\mathcal{M})} = \| \jb{\xi}^s \widehat{f}(\xi) \|_{L^p_\xi (\widehat{\mathcal{M}})}, $$
\noindent where $\jb{\xi} \coloneqq (1 + \abs{\xi}^2 )^{\frac 12}$. \end{definition}
\noindent In particular, $\mathcal{F}L^{0,1}(\mathcal{M})$ corresponds to the Wiener algebra, and its algebra property allows us to prove easily that \eqref{imBq} is analytically locally well-posed in $\mathcal{F}L^{0,1}(\mathcal{M})$.
Another major point of our proof is the following power series expansion of a solution $u$ to \eqref{imBq} with $(u, \partial_t u ) |_{t=0} = \vec{u}_0$: $$ u = \sum^\infty_{j=0} \Xi_j (\vec{u}_0 ), $$
\noindent where $\Xi_j (\vec{u}_0 )$ denotes a multilinear term in $\vec{u}_0$ of degree $(k-1)j + 1$ (for nonlinearity $f(u) = u^k$). More precisely, these multilinear terms are exactly the successive terms of a Picard iteration expansion. See Section \ref{PowerSeries}. Then, by explicit computation we show that $\Xi_1(\vec{u}_0)$ grows rapidly in a short time, achieving the desired growth, while we control the other terms. See Section \ref{Sec3}. \begin{remark} \rm In \cite{WC, WC2}, Wang-Chen studied the {\it multi-dimensional} generalized improved Boussinesq equation \begin{equation} \begin{cases}\label{generalizedimBq} \partial_t^{2}u-\Delta u - \partial_t^2 \Delta u = \Delta \left(f(u) \right) \\
(u,\partial_t u)|_{t = 0} = (u_0,u_1), \end{cases} \qquad ( t, x) \in [0, +\infty) \times \mathbb{R}^d, \end{equation}
\noindent which corresponds essentially to the $d$-dimensional form of \eqref{imBq}, for $d \geq 1$. More precisely, they proved this problem is locally well-posed in $W^{2,p}\cap L^{\infty}$, for any $1 \leq p \leq \infty$, and in $H^s$ for $s \geq \frac d2$. They also showed global well-posedness under additional conditions. We claim that norm inflation with infinite loss of regularity also applies for this problem, for any dimension $d \geq 1$ and $s < 0$. See Remark \ref{RK:ProofGenIBq} for more details. \end{remark}
\begin{remark} \rm \label{REM:NIforFLMW} \cite{BH} used this method to extend the result of Forlano-Okamoto~\cite{FO} and prove infinite loss of regularity for the nonlinear wave equation in some Fourier-Lebesgue, modulation and Wiener amalgam spaces\footnote{However, we point out that, while Forlano-Okamoto did not state it, their argument implies infinite loss of regularity.}. We claim that we can prove the same result for equation \eqref{imBq}. See Appendix \ref{appendixA} for more details. \end{remark}
\section{Power series extension} \label{PowerSeries}
In this section, we prove \eqref{imBq} with nonlinearity $f(u) = u^k$ is well-posed in the Wiener algebra, and it can be expanded into a power series. First, let us fix some notations. We define $$ \overrightarrow{\mathcal{F}L}^{s,p}(\mathcal{M}) \coloneqq \mathcal{F}L^{s,p}(\mathcal{M}) \times \mathcal{F}L^{s,p}(\mathcal{M}) $$
\noindent and, for more clarity, we denote $\mathcal{F}L^{p}(\mathcal{M}) \coloneqq \mathcal{F}L^{0, p}(\mathcal{M})$ and $\overrightarrow{\mathcal{F}L}^{p}(\mathcal{M}) \coloneqq \overrightarrow{\mathcal{F}L}^{0,p}(\mathcal{M})$.
We denote $S(t)$ the linear propagator: \begin{equation} \label{EQ:linearOp} S(t)(u,v) = \cos\left(t P(D)\right) u + \frac{\sin\left(tP(D)\right)}{P(D)} v, \end{equation}
\noindent with $P(D) \coloneqq \frac{\abs{\nabla}}{\jb{\nabla}}$. Namely, for any $\xi \in \mathcal{M}$, $$ \mathcal{F} \left[P(D)f\right](\xi) = \frac{\abs{\xi}}{( 1 + \xi^2)^{1/2}} \widehat{f}(\xi) \eqqcolon \lambda(\xi) \widehat{f}(\xi). $$
\noindent We also denote $\mathcal{I}_k$ the multilinear Duhamel operator. \begin{equation} \label{Eq:DuhamelOp} \mathcal{I}_k(u_1, \dots, u_k)(t) = \int^{t}_0 \sin\left((t-t')P(D)\right) P(D) \prod^k_{j = 1} u_j (t') \mathrm{d}t'. \end{equation}
\noindent This gives the following Duhamel formulation for \eqref{imBq} \begin{equation} \label{Eq:duhamelP} u(t) = S(t) (u_0, u_1) + \mathcal{I}_k (u). \end{equation}
\noindent Note that, in the aforementioned formula, we used the short-hand notation $\mathcal{I}_k (u) \coloneqq \mathcal{I}_k(u, \dots, u)$.
To index the power series we intend to create, we need the following tree structure:
\begin{definition}[$k$-ary trees] \label{DEF:Trees} \rm \begin{enumerate}
\item Given a set $\mathcal{T}$ with partial order $\leq$, we say that $b \in \mathcal{T}$ with $b \leq a$ and $b \ne a$ is a child of $a \in \mathcal{T}$, if $b\leq c \leq a$ implies either $c = a$ or $c = b$. If the latter condition holds, we also say that $a$ is the parent of $b$.
\item A tree $\mathcal{T}$ is a finite partially ordered set, satisfying the following properties\footnote{We do not identify two trees even if there is an order-preserving bijection between them.}: \begin{itemize}
\item Let $a_1, a_2, a_3, a_4 \in \mathcal{T}$. If $a_4 \leq a_2 \leq a_1$ and $a_4 \leq a_3 \leq a_1$, then we have $a_2\leq a_3$ or $a_3 \leq a_2$,
\item A node $a\in \mathcal{T}$ is called terminal, if it has no child. A non-terminal node $a\in \mathcal{T}$, for $\mathcal{T}$ a $k$-ary tree, is a node with exactly $k$ children,
\item There exists a maximal element $r \in \mathcal{T}$, called the root node, such that $a \leq r$ for all $a \in \mathcal{T}$,
\item $\mathcal{T}$ consists of the disjoint union of $\mathcal{T}^0$ and $\mathcal{T}^\infty$, where $\mathcal{T}^0$ and $\mathcal{T}^\infty$ denote the collections of non-terminal nodes and terminal nodes, respectively. \end{itemize}
\item Let ${\bf T}(j)$ denote the set of all trees of $j$-th generation, namely trees with $j$ non-terminal nodes. \end{enumerate} \end{definition}
Note that a tree of $j$-th generation $\mathcal{T} \in {\bf T}(j)$ has $kj + 1$ nodes. Indeed, it has $j$ non-terminal nodes by definition and an induction argument shows it has $(k-1)j + 1$ terminal nodes. Besides, we have the following bound on the number of trees of $j$-th generation:
\begin{lemma} \label{LEM:NumberOfTrees} There exists a constant $C_0 > 0$, depending only on $k$, such that, for any $j \in \mathbb{N}$, we have \begin{equation} \label{EQ:NumberOfTrees} \abs{{\bf T} (j)} \leq \frac{C^j_0}{(1 + j)^2 } \leq C^j_0 \end{equation} \end{lemma}
The following proof is an adaptation of the one in \cite{Oh} for ternary trees, we include it for completeness.
\begin{proof} We prove \eqref{EQ:NumberOfTrees} by induction. Note that the right inequality is immediate, so we only have to prove $$ \abs{{\bf T} (j)} \leq \frac{C^j_0}{(1 + j)^2 } $$
\noindent for any $j \geq 0$. Observe first that $$ \abs{{\bf T} (0)} = \abs{{\bf T} (1)} =1. $$
\noindent Then, fix $j \geq 2$. Assume equation \eqref{EQ:NumberOfTrees} holds for any $0 \leq m \leq j-1$ and take $\mathcal{T} \in {\bf T}(j)$. By Definition \ref{DEF:Trees}, there exist $k$ trees $\mathcal{T}_1 \in {\bf T}(j_1), \dots, \mathcal{T}_k \in {\bf T}(j_k )$, with $j_1 + \cdots + j_k = j -1$, such that $\mathcal{T}$ is the tree consisting of a root node whose children are $\mathcal{T}_1, \dots, \mathcal{T}_k$. Thus, applying the induction hypothesis we get \begin{align*} \abs{{\bf T} (j)} & = \sum_{\substack{j_1 + \cdots + j_k = j-1, \\ j_1, \cdots, j_k \geq 0}} \abs{{\bf T} (j_1)} \times \cdots \times \abs{{\bf T} (j_k)} \\
& \leq \sum_{\substack{j_1 + \cdots + j_k = j-1, \\ j_1, \cdots, j_k \geq 0}} \frac{C^{j_1}_0}{(1 + j_1)^2 } \times \cdots \times \frac{C^{j_k}_0}{(1 + j_k )^2 } \times \frac{(1+j)^2}{(1+j)^2} \\
& \leq k^2 \left( \sum_{j_2, \cdots j_k \geq 0} \frac{1}{(1 + j_2)^2 \times \cdots \times (1 + j_k )^2 } \right) \frac{C^{j-1}_0}{(1 + j)^2 } \end{align*}
\noindent where we used $k \max \left( 1+ j_1 , \dots, 1 + j_k \right) \geq 1+j$ and rearranged the sum so the maximum is reached for $j_1$. Then, choosing $C_0 = k^2 \sum_{j_2, \cdots j_k \geq 0} \frac{1}{(1 + j_2)^2 \cdots (1 + j_k )^2 } < \infty$ ends the induction.
\end{proof}
From now on, for any $\vec{\phi} \in \overrightarrow{\mathcal{F}L}^1$, we will associate to any tree $\mathcal{T} \in {\bf T} (j)$, $j \geq 0$, a space-time distribution $\Psi (\mathcal{T}) (\vec{\phi}) \in \mathcal{D}' \left( (-T,T) \times \mathcal{M} \right)$ as follows: \begin{itemize}
\item replace a non-terminal node by Duhamel integral operator $\mathcal{I}_k$ with its $k$ arguments being the children of the node,
\item replace a terminal node by $S(t) \vec{\phi}$. \end{itemize}
\noindent For any $j \geq 0$ and $\vec{\phi} \in \overrightarrow{\mathcal{F}L}^1$, we also define $\Xi_j$ as follows:
\begin{equation} \label{DEF:Xi_j} \Xi_j (\vec{\phi}) = \sum_{\mathcal{T} \in {\bf T} (j)} \Psi(\mathcal{T})( \vec{\phi} ). \end{equation}
\noindent For instance, we have the two following terms: $$
\Xi_0 (\vec{\phi}) = S(t) \vec{\phi}, \quad \text{ and } \quad \Xi_1 (\vec{\phi}) = \mathcal{I}_k ( S(t) \vec{\phi}, \cdots, S(t) \vec{\phi}). $$
\noindent Let us now state some basic multilinear estimates that will be useful both for norm inflation and for local well-posedness in the Wiener algebra $\mathcal{F}L^1$.
\begin{lemma} \label{LEM:TreesEstFL} There exists $C > 0$ such that, for any $\vec{\phi} \in \overrightarrow{\mathcal{F}L}^1$, $j \in \mathbb{N}$ and $0 < T \leq 1$, we have \begin{equation} \label{EQ:TreesEst1}
\big\| \Xi_j (\vec{\phi}) (T) \big\|_{\mathcal{F}L^1} \leq C^j T^{2j} \| \vec{\phi} \|^{(k-1)j+1}_{\overrightarrow{\mathcal{F}L}^1} \end{equation}
\noindent Moreover, if $j \geq 1$ and $\vec{\psi} \in \overrightarrow{\mathcal{F}L}^1 \cap \mathcal{H}^0 \left(\mathbb{R}\right)$, \begin{equation} \label{EQ:TreesEstInf}
\left\| \Xi_j (\vec{\psi} ) (T) \right\|_{\mathcal{F}L^\infty} \leq C^j T^{2j} \| \vec{\psi} \|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^1} \| \vec{\psi} \|^{2}_{\mathcal{H}^0}. \end{equation} \end{lemma}
\begin{proof} Let $\mathcal{T} \in {\bf T} (j)$. Using the same tree structure argument as for Lemma \ref{LEM:NumberOfTrees}, there exist $k$ trees $\mathcal{T}_1 \in {\bf T}(j_1), \dots, \mathcal{T}_k \in{\bf T}(j_k)$, with $j_1 + \cdots + j_k = j-1$, such that the root nodes of $\mathcal{T}_1, \dots, \mathcal{T}_1$ are the children of the root node of $\mathcal{T}$. Thus, we can write $$ \Psi(\mathcal{T}) (\vec{\phi}) = \mathcal{I}_k (\Psi(\mathcal{T}_1) (\vec{\phi}), \cdots , \Psi(\mathcal{T}_k) (\vec{\phi}) ), $$
\noindent and $\Psi(\mathcal{T}) (\vec{\phi})$ consists essentially of $j = \abs{\mathcal{T}^0}$ iterations of the Duhamel operator $\mathcal{I}_k$ with $(k-1)j + 1$ times the term $S(t)\vec{\phi}$ as arguments.
Meanwhile, we deduce from \eqref{EQ:linearOp} that $S(t)$ is unitary in $\mathcal{F}L^1$ for any $ 0 < T \leq 1$, and, since $\abs{\sin y} \leq \abs{y}$ for any $y \in \mathbb{R}$ and $\lambda(\xi) \leq 1$ for any $\xi \in \mathbb{R}$, the algebra property of $\mathcal{F}L^1$ gives \begin{equation} \label{EQ:EstDuhamelOp}
\| \mathcal{I}_k [u_1, \dots, u_k] \|_{C_T \mathcal{F}L^1} \leq \int^T_0 \abs{T-t'} \abs{\lambda(\xi)}^2 \| u_1 \cdots u_k \|_{C_T \mathcal{F}L^1} \mathrm{d}t' \leq \frac 12 T^2 \prod^k_{j=1} \| u_j \|_{C_T \mathcal{F}L^1} \end{equation}
\noindent Hence, \eqref{EQ:TreesEst1} follows from an induction argument and Lemma \ref{LEM:NumberOfTrees}. Besides, Young's inequality and a similar argument give \eqref{EQ:TreesEstInf}.
\end{proof}
Let us now use Lemma \ref{LEM:TreesEstFL} to prove local well-posedness of \eqref{imBq} in the Wiener algebra and to justify the power series expansion.
\begin{lemma} \label{LEM:ExistenceOfSolution}
Let $M > 0$. Then, for any time $T$ such that $0 < T \ll \min(M^{-\frac{k-1}{2} }, 1)$ and $\vec{u}_0 \in \overrightarrow{\mathcal{F}L}^1$ with $\| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^1} \leq M$, \begin{enumerate}
\item there exists a unique solution $u \in \mathbb{C} ([0,T]; \mathcal{F}L^1 (\mathbb{R}))$ to \eqref{imBq} satisfying $(u, \partial_t u) = \vec{u}_0$.
\item Moreover, u can be expressed as \begin{equation} \label{EQ:PowerSeriesSol} u = \sum_{j = 0}^\infty \Xi_j (\vec{u}_0) = \sum_{j = 0}^\infty \sum_{\mathcal{T} \in {\bf T} (j)} \Psi(\mathcal{T})(\vec{u}_0) \end{equation} \end{enumerate}
\end{lemma}
\begin{proof}
Let us first prove our problem is locally well-posed in $\mathcal{F}L^1$. We define the functional $\Gamma$ by $$ \Gamma [u] (t) \coloneqq S(t)\vec{u}_0 + \mathcal{I}_k (u)(t) $$
\noindent for any $t \in [0,T]$. Then, using the unitarity of $S(t)$ and \eqref{EQ:EstDuhamelOp}, we have $$
\left\| \Gamma[u] \right\|_{C_T \mathcal{F}L^1} \leq \| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^1} + \frac 12 T^2 \| u \|_{\mathcal{F}L^1}^k $$
\noindent Using the multilinearity of $\mathcal{I}_k$, we ensure, for $0 < T \leq 1$ such that $\frac 12 T^2 M^{k-1} \ll 1$, that $\Gamma$ is a strict contraction on the ball
$$
B_{2M} \coloneqq \{ v \in C([0,T], \mathcal{F}L^1 (\mathcal{M})), \| v \|_{C_T \mathcal{F}L^1} \leq 2M\}. $$
\noindent Then, the contraction mapping theorem and an a posteriori continuity argument proves the local well-posedness. Let us move onto the power series expansion.
Fix $\varepsilon >0$. We choose $0 < T \leq 1$ such that $\frac 12 T^2 M^{k-1} \ll 1$. Then, from \eqref{EQ:TreesEst1}, the sum in \eqref{EQ:PowerSeriesSol} converges absolutely in $C([0,T], \mathcal{F}L^1 (\mathcal{M}))$. Let us denote $$ U_J = \sum_{j = 0}^J \Xi_j (\vec{u}_0) \qquad \text{ and } \qquad U = \sum_{j = 0}^\infty \Xi_j (\vec{u}_0). $$
\noindent There exists $J_1 \geq 0$ such that \begin{equation} \label{EQ:PfPowerSeries1}
\| U - U_J \|_{C_T \mathcal{F}L^1} < \frac \eps3 \end{equation}
\noindent for any $J \geq J_1$. In particular, this implies that $U$ and $U_J$ belong to the ball $B_{2M}$. Then, by continuity of $\Gamma$ as a map from $B_{2M}$ into itself, \eqref{EQ:PfPowerSeries1} implies there exists $J_2 \leq 0$ such that, for any $J \geq J_2$, \begin{equation} \label{EQ:PfPowerSeries2}
\| \Gamma[U] - \Gamma[U_J] \|_{C_T \mathcal{F}L^1} < \frac \eps3. \end{equation}
\noindent All that is left now is to estimate $U_J - \Gamma[U_J]$. Fix an integer $J \geq 1$. Then, from the tree structure argument we already used in the proof of Lemma \ref{LEM:TreesEstFL}, we get \begin{align*} U_J - \Gamma[U_J] & = \sum_{j = 1}^J \Xi_j (\vec{u}_0) - \sum_{0 \leq j_1, \dots, j_k \leq J} \mathcal{I}_k \left( \Xi_{j_1}(\vec{u}_0) , \cdots, \Xi_{j_k} (\vec{u}_0) \right) \\
& = - \sum^{kJ}_{l = J} \sum_{\substack{0 \leq j_1 , \dots, j_k \leq J, \\ j_1 + \cdots + j_k = l}} \mathcal{I}_k \left( \Xi_{j_1}(\vec{u}_0) , \cdots, \Xi_{j_k} (\vec{u}_0) \right). \end{align*}
\noindent Now a crude estimation of the sums, along with \eqref{EQ:EstDuhamelOp} and \eqref{EQ:TreesEst1}, give \begin{align*}
\| U_J - \Gamma[U_J] \|_{C_T \mathcal{F}L^1} & \leq \frac 12 T^2 \sum^{kJ}_{l = J} \ \sum_{\substack{0 \leq j_1 , \dots, j_k \leq J, \\ j_1 + \cdots + j_k = l}} \ \prod^k_{m=1} \| \Xi_{j_m}(\vec{u}_0) \|_{C_T \mathcal{F}L^1} \\
& \leq \frac 12 T^2 \sum^{kJ}_{l = J} \ \sum_{\substack{0 \leq j_1 , \dots, j_k \leq J, \\ j_1 + \cdots + j_k = l}} \ \prod^k_{m= 1} C^{j_m} T^{2j_m} \| \vec{u}_0 \|^{(k-1)j_m + 1}_{\overrightarrow{\mathcal{F}L}^1} \\
& \leq \frac 12 T^2 J^k M^k \sum^{\infty}_{l = J} (C T^2 M^{k-1})^l \end{align*}
\noindent Since we assumed $0 < T \ll \min(M^{-\frac{k-1}{2} }, 1)$, the sum converges and the right-hand side is bounded by $\frac 12 T^2 J^k M^k (C T^2 M^{k-1})^J$, which tends to $0$ as $J$ tends to infinity. Thus, there exists $J_3 \geq 1$ such that for every $J \geq J_3$, \begin{equation} \label{EQ:PfPowerSeries3}
\| U_J - \Gamma[U_J] \|_{C_T \mathcal{F}L^1} < \frac \eps3. \end{equation}
\noindent Now, for any $J \geq \max(J_1, J_2, J_3)$, \eqref{EQ:PfPowerSeries1}, \eqref{EQ:PfPowerSeries2} and \eqref{EQ:PfPowerSeries3} imply $$
\| U - \Gamma[U] \|_{C_T \mathcal{F}L^1} \leq \| U - U_J \|_{C_T \mathcal{F}L^1} + \| U_J - \Gamma[U_J] \|_{C_T \mathcal{F}L^1} + \| \Gamma[U] - \Gamma[U_J] \|_{C_T \mathcal{F}L^1} < \varepsilon. $$
\noindent Therefore, $U$ is a fixed point of $\Gamma$ and this ends the proof by uniqueness.
\end{proof}
\section{Norm inflation for IBq} \label{Sec3}
In this section, we present first the proof of Theorem \ref{THM:main}. Actually, our main goal is to prove the following proposition:
\begin{proposition} \label{THM:main2} Let $k \geq 2$ and $s < 0$. Fix $(u_0 , u_1) \in \mathcal{H}^s (\mathcal{M})$. Then, given any $\varepsilon > 0$, there exists a solution $u_\varepsilon$ to \eqref{imBq} with nonlinearity $f(u) = u^k$ on $\mathcal{M}$ and $t_\varepsilon \in (0, \varepsilon)$ such that $$
\| (u_\varepsilon (0), \partial_t u_\varepsilon (0)) - (u_0 , u_1) \|_{\mathcal{H}^s} < \varepsilon \quad \text{ and } \quad \| u_\varepsilon (t_\varepsilon) \|_{H^s} > \varepsilon^{-1}. $$ \end{proposition}
Indeed, once Proposition \ref{THM:main2} is proved, the proof of Theorem \ref{THM:main} follows in the same way, with a slight modification that we treat separately for more clarity. See Remark \ref{REM:ILOR}.
To do so, suppose first that we proved the following proposition:
\begin{proposition} \label{PROP:final2} Let $k \geq 2$ and $s < 0$. Fix $(u_0, u_1) \in \mathcal{S} ( \mathcal{M}) \times \mathcal{S}(\mathcal{M})$. Then, for any $n \in \mathbb{N}$, there exists a solution $u_n$ to \eqref{imBq} with nonlinearity $f(u) = u^k$ and $t_n \in (0, \frac 1n )$ such that \begin{equation} \label{EQ:propFinal}
\| (u_n (0), \partial_t u_n (0)) - (u_0 , u_1) \|_{\mathcal{H}^s} < \frac 1n \quad \text{ and } \quad \| u_n (t_n ) \|_{H^s} > n. \end{equation} \end{proposition}
\noindent Then, let us fix $\varepsilon > 0$ and choose $n \in \mathbb{N}$ such that $n > \varepsilon^{-1}$. According to Proposition \ref{PROP:final2}, there exists a solution $u_n$ to \eqref{imBq} and a time $t_n \in (0, \frac 1n ) \subset (0, \varepsilon)$ such that $$
\| (u_n (0), \partial_t u_n (0)) - (u_0 , u_1) \|_{\mathcal{H}^s} < \frac 1n < \varepsilon \quad \text{ and } \quad \| u_n (t_n ) \|_{H^s} > n > \varepsilon^{-1}. $$
\noindent Therefore, Proposition \ref{THM:main2} follows from Proposition \ref{PROP:final2} and the density of $\mathcal{S}(\mathcal{M})$ in $H^s (\mathcal{M})$. Similarly, Theorem \ref{THM:main} follows from the following proposition:
\begin{proposition} \label{PROP:final} Let $k \geq 2$, $\sigma \in \mathbb{R}$ and $s < 0$. Fix $(u_0 , u_1) \in \mathcal{S} ( \mathcal{M}) \times \mathcal{S}(\mathcal{M})$. Then, given any $n \in \mathbb{N}$, there exists a solution $u_n$ to \eqref{imBq} with nonlinearity $f(u) = u^k$ on $\mathcal{M}$ and $t_n \in (0, \frac 1n )$ such that $$
\| (u_n (0), \partial_t u_n (0)) - (u_0 , u_1) \|_{\mathcal{H}^s} < \frac 1n \quad \text{ and } \quad \| u_n (t_n) \|_{H^\sigma} > n. $$ \end{proposition}
Consequently, the rest of this paper is devoted to the proofs of Proposition \ref{PROP:final2} and Proposition \ref{PROP:final}. In the following, we fix some $\vec{u}_0 \in \mathcal{S} ( \mathcal{M}) \times \mathcal{S}(\mathcal{M})$.
\subsection{Multilinear estimates}
In this subsection, we establish some multilinear estimates that are essentials to our proof.
Given $n \in \mathbb{N}$, let us fix some $N = N(n) \gg 1$, $R = R(N) \gg 1$ and $ 1 \ll A = A(N) \ll N$ to be determined later. We define $\vec{\phi}_n \coloneqq (\phi_{n} , 0)$ by setting \begin{equation} \label{DEF:phi_n} \widehat{\phi_n} = R \chi_\Omega \end{equation}
\noindent where \begin{equation} \label{DEF:Omega} \Omega = \bigcup_{\eta \in \Sigma} (\eta + Q_A) \end{equation}
\noindent with $Q_A = [-\frac A2 , \frac A2]$ and \begin{equation} \label{DEF:Sigma} \Sigma = \{ -2N, -N, N, 2N\}. \end{equation}
\noindent Note that the condition $A \ll N$ ensures the union \eqref{DEF:Omega} is disjoint. Besides, observe that \eqref{DEF:phi_n}, \eqref{DEF:Omega} and \eqref{DEF:Sigma} imply for any $s \in \mathbb{R}$ \begin{equation} \label{EQ:EstPhi_n}
\| \vec{\phi}_n \|_{\overrightarrow{\mathcal{F}L}^1} \sim RA \quad \text{ and } \quad \| \vec{\phi}_n \|_{\mathcal{H}^s} \sim RN^s A^{1/2}. \end{equation}
\noindent We define finally $\vec{u}_{0,n} \coloneqq \vec{u}_0 + \vec{\phi}_n$. Suppose $N$, $R$ and $A$ satisfy \begin{equation} \label{EQ:CondFL1u}
\| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^1} \ll RA. \end{equation}
\noindent Therefore, for each $n \in \mathbb{N}$, provided \begin{equation} \label{EQ:CondOnT} 0 < T \ll \min( (RA)^{-\frac{k-1}{2}} , 1), \end{equation}
\noindent Lemma \ref{LEM:ExistenceOfSolution} implies there exists a unique solution $u_n \in C([0,T], \mathcal{F}L^1(\mathcal{M}))$ to \eqref{imBq} with $(u_n , \partial_t u_n) |_{t=0} = \vec{u}_{0,n}$ and admitting the power series expansion: \begin{equation} \label{EQ:PowerSeriesSolRankn} u_n = \sum_{j = 0}^\infty \Xi_j (\vec{u}_{0,n}) = \sum_{j = 0}^\infty \Xi_j (\vec{u}_{0} + \vec{\phi}_n) . \end{equation}
\noindent The purpose of this subsection is then to estimate the terms of the power series on the right-hand side of \eqref{EQ:PowerSeriesSolRankn}. But first, let us recall the following lemma:
\begin{lemma} \label{LEM:ConvolutionIneq} Let $a,b \in \mathbb{R}$ and $A > 0$, then we have $$ C A \chi_{a + b + Q_A } (\xi) \leq \chi_{a + Q_A } \ast \chi_{b + Q_A } (\xi) \leq \widetilde{C} A \chi_{a + b + Q_{2A}}(\xi ) $$
\noindent where $C, \widetilde{C} > 0$ are constants. \end{lemma}
The proof of the following lemma is essentially the same as in \cite[Lemma 3.2]{FO} and is included for completeness.
\begin{lemma} \label{LEM:MultilinearEst}
For any $s <0$, $t \in [0, T]$ and $j \in \mathbb{N}$, the following estimates hold: \begin{align}
\| \vec{u}_{0,n} - \vec{u}_0 \|_{\mathcal{H}^s} & \sim RN^s A^{1/2}, \label{EQ:MultilinearEst1} \\
\| \Xi_0 (\vec{u}_{0,n})(t) \|_{H^s} & \lesssim 1 + RA^{1/2} N^s, \label{EQ:MultilinearEst2} \\
\| \Xi_1 (\vec{u}_{0,n})(t) - \Xi_1 (\vec{\phi}_n) (t) \|_{H^s} & \lesssim t^2 R^{k-1}A^{k-1} \| \vec{u}_0 \|_{\mathcal{H}^0}, \label{EQ:MultilinearEst3} \\
\| \Xi_j (\vec{u}_{0,n})(t) \|_{H^s} & \lesssim C^j t^{2j} R^{(k-1)j}A^{(k-1)j} \left( \| \vec{u}_0 \|_{\mathcal{H}^0} + R g_s (A) \right), \label{EQ:MultilinearEst4} \end{align}
\noindent where $g_s (A)$ is defined by \begin{equation} \label{DEF:gs} g_s (A) \coloneqq \begin{cases}
1 & \textup{if } s<-\frac{1}{2}, \\
\left( \log A \right)^{\frac{1}{2}}& \textup{if } s=-\frac{1}{2}, \\
A^{\frac{1}{2}+s} & \textup{if } s>-\frac{1}{2} \end{cases} \end{equation}
\end{lemma}
\begin{proof}
The proofs of \eqref{EQ:MultilinearEst1} and \eqref{EQ:MultilinearEst2} follow directly from $\vec{u}_{0,n} = \vec{u}_0 + \vec{\phi}_n$, \eqref{EQ:EstPhi_n}, the unitarity of $S(t)$ for $t \leq 1$ and the fact that $\vec{u}_0$ is fixed, implying $\|\vec{u}_0\|_{\mathcal{H}^s} \lesssim 1$. Besides, the definition \eqref{DEF:Xi_j} of $\Xi_j$ and the multilinearity of $\mathcal{I}_k$ imply $$ \Xi_1 (\vec{u}_{0,n})(t) - \Xi_1 (\vec{\phi}_n) (t) = \sum\limits_{(v_1, \dots, v_k) \in E} \mathcal{I}_k (v_1, \cdots, v_k)(t) $$
\noindent where $E$ is the subset of $\{ S(t)\vec{u}_0 , S(t)\vec{\phi}_n \}^k$ such that at least one of the choice is $S(t)\vec{u}_0$. Thus, since $s<0$, Young's inequality, \eqref{EQ:EstDuhamelOp} and the unitarity of $S(t)$ imply \begin{align*}
\| \Xi_1 (\vec{u}_{0,n})(t) - \Xi_1 (\vec{\phi}_n) (t) \|_{H^s} & \lesssim t^2 \|\vec{u}_0 \|_{\mathcal{H}^0} \left( \| \vec{u}_0 \|^{k-1}_{\overrightarrow{\mathcal{F}L}^1} + \| \vec{\phi}_n \|^{k-1}_{\overrightarrow{\mathcal{F}L}^1} \right) \\
& \lesssim t^2 \|\vec{u}_0 \|_{\mathcal{H}^0} \left( \| \vec{u}_0 \|^{k-1}_{\overrightarrow{\mathcal{F}L}^1} + R^{k-1}A^{k-1} \right) \end{align*}
\noindent which, combined with assumption \eqref{EQ:CondFL1u}, proves \eqref{EQ:MultilinearEst3}.
For the last inequality, we split the left-hand side into two terms: \begin{equation} \label{EQ:SplitXi_j} \Xi_j (\vec{u}_{0,n})(t) = \left( \Xi_j (\vec{u}_{0,n})(t) - \Xi_j (\vec{\phi}_{n})(t) \right) + \Xi_j (\vec{\phi}_{n})(t) \end{equation}
\noindent which we will estimate separately.
{\bf Part 1:} $\mathcal{F}\big(\Xi_j (\vec{\phi}_{n})(t)\big)$ essentially consists of $(k-1)j + 1$ convolutions of terms of the form $\mathcal{F}(S(t) \vec{\phi}_n)$. According to \eqref{EQ:linearOp} and \eqref{DEF:phi_n}, the support of each of these terms is contained within at most $4$ disjoint cubes of volume approximately $A$. Thus, Lemma \ref{LEM:ConvolutionIneq} and a countability argument show the support of $ \mathcal{F}( \Xi_j (\vec{\phi}_{n})(t) )$ is contained within at most $4^{(k-1)j+1}$ cubes of volume approximately $A$. Therefore, \eqref{DEF:Xi_j} and Lemma \ref{LEM:NumberOfTrees} imply there exist $c,C > 0$ such that $$
|\supp \mathcal{F}( \Xi_j (\vec{\phi}_{n})(t) ) | \leq C^j A \leq c |C^j Q_A |. $$
\noindent Since $s < 0$, $\jb{\xi}^s$ is decreasing in $\abs{\xi}$ and Young's inequality, \eqref{EQ:TreesEstInf} and the unitarity of $S(t)$ yield \begin{align*}
\| \Xi_j (\vec{\phi}_{n})(t) \|_{H^s} & \leq \| \jb{\xi}^s \|_{L^2 (\supp \mathcal{F}( \Xi_j (\vec{\phi}_{n})(t) ) ) } \| \Xi_j (\vec{\phi}_{n})(t) \|_{\mathcal{F}L^\infty} \\
& \lesssim \| \jb{\xi}^s \|_{L^2 (c C^j Q_A) } C^j t^{2j} \| \vec{\phi}_n \|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^1} \| \vec{\phi}_n \|^2_{\mathcal{H}^0}. \end{align*}
\noindent Since $\| \jb{\xi}^s \|_{L^2 (c C^j Q_A) } \lesssim g_s(A)$ with $g_s$ defined as in \eqref{DEF:gs}, we get from \eqref{EQ:EstPhi_n} \begin{equation} \label{EQ:EstXi_jPhi}
\| \Xi_j (\vec{u}_{0,n})(t) \|_{H^s} \lesssim C^j t^{2j} R^{(k-1)j}A^{(k-1)j} R g_s (A). \end{equation}
{\bf Part 2:} On the other hand, since $\mathcal{F}\big(\Xi_j (\vec{u}_{0})(t) - \Xi_j (\vec{\phi}_{n})(t)\big)$ is essentially a sum of terms made of $j$ integrals in time and $(k-1)j + 1$ convolutions of terms of the form $\mathcal{F}(v)$, with $v \in \{ S(t)\vec{u}_0 , S(t)\vec{\phi}_n \}$, a similar argument as for \eqref{EQ:MultilinearEst3} along with $s < 0$ yield \begin{align*}
\| \Xi_j (\vec{u}_{0,n})(t) - \Xi_j (\vec{\phi}_{n})(t) \|_{H^s} & \leq \| \Xi_j (\vec{u}_{0,n})(t) - \Xi_j (\vec{\phi}_{n})(t) \|_{L^2} \\
& \lesssim C^j t^{2j} \| \vec{u}_0 \|_{\mathcal{H}^0} \left( \| S(t)\vec{u}_0\|_{\mathcal{F}L^1} + \| S(t)\vec{\phi}_n \|_{\mathcal{F}L^1} \right)^{(k-1)j}. \end{align*}
\noindent Thus, the unitarity of $S(t)$, \eqref{EQ:EstPhi_n} and \eqref{EQ:CondFL1u} give \begin{equation} \label{EQ:EstXi_jRem}
\| \Xi_j (\vec{u}_{0,n})(t) - \Xi_j (\vec{\phi}_{n})(t) \|_{H^s} \lesssim C^j t^{2j} \| \vec{u}_0 \|_{\mathcal{H}^0} (RA)^{(k-1)j}. \end{equation}
\noindent Now, \eqref{EQ:MultilinearEst4} follows from triangular inequality, \eqref{EQ:EstXi_jPhi} and \eqref{EQ:EstXi_jRem}.
\end{proof}
With the help of Lemma \ref{LEM:ConvolutionIneq}, we can also state the following crucial proposition. This proposition exploits the high-to-low energy transfer mentioned before to identify the first multilinear term in the Picard expansion as the one responsible for the instability in Proposition \ref{PROP:final2} and in Proposition \ref{PROP:final}.
\begin{proposition} \label{PROP:EstSecondPicard} Let $s < 0$, $\vec{\phi}_n$ defined as in equation \eqref{DEF:phi_n}, $T \ll 1$ and $A \ll N $. Then, there exists a constant $C > 0$ such that \begin{equation} \label{EQ:EstSecondPicard}
\| \Xi_1 (\vec{\phi}_n) (T) \|_{ H^s} \gtrsim C^{k-1} R^k T^2 A^{k - \frac 12 + s}. \end{equation}
\end{proposition}
\begin{proof} To simplify notation, we write $$ \Lambda \coloneqq \left\{ (\xi_1, \dots, \xi_k) \in \widehat{\mathcal{M}}^k \colon \sum^k_{j=1} \xi_j = \xi \right\} \quad \text{ and } \quad \mathrm{d}\xi_\Lambda \coloneqq \mathrm{d}\xi_1 \cdots \mathrm{d}\xi_{k-1}. $$ Using \eqref{DEF:phi_n} and a product-to-sum formula, we have \begin{align*} \mathcal{F}[ \Xi_1 (\vec{\phi}_n)] (T, \xi) = R^k \sum\limits_{(\eta_1, \dots, \eta_k) \in \Sigma^k} & \int^T_0 \sin((T - t') \lambda(\xi)) \lambda(\xi) \\
& \times\int_\Lambda \prod^k_{j=1} \cos(t' \lambda(\xi_j)) \chi_{\eta_j + Q_A} (\xi_j) \mathrm{d}\xi_\Lambda \mathrm{d}t'. \end{align*}
\noindent We split the set $\Sigma^k$, and hence the sum, into two parts $\Sigma_1$ and $\Sigma_2$ where $$ \Sigma_1 = \left\{ (\eta_1, \dots, \eta_k) \in \Sigma^k \colon \eta_1+ \cdots +\eta_k = 0 \right\} $$
\noindent and $\Sigma_2 = \Sigma^k \backslash \Sigma_1$. Note that, for any integer $k \geq 2$, $\Sigma_1$ and $\Sigma_2$ are both non-empty. We denote then \begin{equation} \label{EQ:ProofSecondPicardSplit} \mathcal{F}[ \Xi_1 (\vec{\phi}_n)] (T, \xi) = R^k \left( I_1 (T, \xi) + I_2 (T,\xi) \right) \end{equation}
\noindent with, for $r = 1,2$, $$ I_r (T, \xi) = \sum\limits_{(\eta_1, \dots, \eta_k) \in \Sigma_r} \int^T_0 \sin((T - t') \lambda(\xi)) \lambda(\xi) \int_\Lambda \prod^k_{j=1} \cos(t' \lambda(\xi_j)) \chi_{\eta_j + Q_A} (\xi_j) \mathrm{d}\xi_\Lambda \mathrm{d}t'. $$
Since $T \ll 1$, $\lambda(\xi) \leq 1$, $\sin(x) \sim x$ for $\abs{x} \leq 1$ and $\cos(x) \geq \frac 12$ for $\abs{x} \ll 1$, we have \begin{align*} I_1 (T, \xi) & \gtrsim \sum\limits_{(\eta_1, \dots, \eta_k) \in \Sigma_1} \frac 12 T^2 \lambda(\xi)^2 \int_\Lambda \frac{1}{2^k} \prod^k_{j=1} \chi_{\eta_j + Q_A}(\xi_j) \mathrm{d}\xi_\Lambda \\
& \gtrsim \frac{1}{2^{k+1}} T^2 C^{k-1}_1 A^{k-1} \lambda(\xi)^2 \chi_{Q_A} (\xi), \end{align*}
\noindent where the second inequality comes from $\abs{\Sigma_1} > 1$ and Lemma \ref{LEM:ConvolutionIneq}. Computing the $H^s$ norm, we get \begin{equation} \label{EQ:PicardSecondLowFreq}
\| I_1(T) \|_{H^s} \gtrsim C^{k-1}_1 T^2 A^{k - \frac 12 + s}. \end{equation}
Meanwhile, since $\sin(x) \sim x$ for $\abs{x} < 1$, $\abs{ \cos(x)}\leq 1$, and $\lambda(\xi) \leq 1$, we have $$ \abs{I_2 (T, \xi)} \lesssim \sum\limits_{(\eta_1, \dots, \eta_k) \in \Sigma_2} \frac 12 T^2 \int_\Lambda \prod^k_{j=1} \chi_{\eta_j + Q_A}(\xi_j) \mathrm{d}\xi_\Lambda. $$
\noindent Fix $(\eta_1 , \dots, \eta_k) \in \Sigma_2$. Since $\eta_1 + \cdots + \eta_k \neq 0$, \eqref{DEF:Sigma} implies there exists $m \in \mathbb{Z} \backslash \{ 0 \}$ such that $$ \eta_1 + \cdots + \eta_k = m N. $$
\noindent Lemma \ref{LEM:ConvolutionIneq} implies then \begin{align*}
\bigg\| \int_\Lambda \prod^k_{j=1} \chi_{\eta_j + Q_A}(\xi_j) \mathrm{d}\xi_\Lambda \bigg\|_{H^s} & \lesssim C^{k-1}_1 A^{k-1} \| \chi_{ mN + Q_{kA}} (\xi)\|_{H^s} \\
& \lesssim C^{k-1}_1 A^{k-\frac 12} N^s. \end{align*}
\noindent Since $\abs{\Sigma_2} \leq \abs{\Sigma^k} \leq 4^k$, we get \begin{equation} \label{EQ:PicardSecondHighFreq}
\| I_2 (T) \|_{H^s} \lesssim C^{k-1}_1 T^2 A^{k - \frac 12} N^s . \end{equation}
\noindent Combining \eqref{EQ:PicardSecondLowFreq} and \eqref{EQ:PicardSecondHighFreq} with triangular inequality, we have \begin{align*}
\| \Xi_1 (\vec{\phi}_n) (T) \|_{ H^s} & \geq R^k \big| \| I_1(T) \|_{H^s} - \| I_2 (T) \|_{H^s} \big| \\
& \gtrsim R^k C^{k-1}_1 T^2 A^{k - \frac 12} \big( A^{s} - N^s \big) \end{align*}
\noindent which, with the assumption $A \ll N$, proves our result.
\end{proof}
\subsection{Proof of Proposition \ref{PROP:final2}}
In this subsection we give the proof of Proposition \ref{PROP:final2}. We claim it suffices to show that, given $n \in \mathbb{N}$, the following inequalities hold: \begin{align*} \textup{(i)} & \quad RA^{\frac 12} N^s < \frac 1n, \\ \textup{(ii)} & \quad T^2 R^{k-1} A^{k-1} \ll 1, \\
\textup{(iii)} & \quad \| \vec{u}_0 \|_{\mathcal{H}^0} \ll R A^{\frac 12 + s} \text{ and } \| \vec{u}_0\|_{\overrightarrow{\mathcal{F}L}^1} \ll RA, \\ \textup{(iv)} & \quad T^4 (RA)^{2(k-1)} R g_s (A) \ll T^2 R^k A^{k - \frac 12 + s}, \\ \textup{(v)} & \quad T^2 R^k A^{k - \frac 12 + s} \gg n, \\ \textup{(vi)} & \quad A \ll N \end{align*}
\noindent for some particular $R$, $T$ and $N$, all depending on $n$.
Let us show conditions (i) through (vi) prove Proposition \ref{PROP:final2}. First, condition (i) along with \eqref{EQ:MultilinearEst1} verifies the first estimate in \eqref{EQ:propFinal}. Besides, conditions (ii) and the second of (iii), along with Lemma \ref{LEM:ExistenceOfSolution}, prove existence and uniqueness of a solution $u_n$ in $C([0,T], \mathcal{F}L^1)$ with $(u_n , \partial_t u_n) |_{t=0} = \vec{u}_{0,n}$, as well as the convergence of the power series \eqref{EQ:PowerSeriesSolRankn}. Furthermore, these conditions, along with \eqref{EQ:MultilinearEst4}, yield \begin{align*}
\sum^{\infty}_{j = 2} \| \Xi_j (\vec{u}_{0,n})(T) \|_{H^s} & \lesssim T^4 (RA)^{2(k-1)} \left(\sum^{\infty}_{j = 0} ( C T^2 R^{k-1}A^{k-1} )^j \right) \left( \| \vec{u}_0 \|_{\mathcal{H}^0} + Rg_s (A) \right) \\
& \lesssim T^4 (RA)^{2(k-1)} R g_s (A). \end{align*}
\noindent Then, using \eqref{EQ:PowerSeriesSolRankn}, Lemma \ref{LEM:MultilinearEst} and Proposition \ref{PROP:EstSecondPicard} - which is applicable by condition (vi) - conditions (i) through (v) give: \begin{align*}
\| u_n (T) \|_{H^s} & \geq \|\Xi_1 (\vec{\phi}_{n})(T) \|_{H^s} \\
& \qquad - \bigg\| \Xi_0 (\vec{u}_{0,n})(T) + \left( \Xi_1 (\vec{u}_{0,n})(T) - \Xi_1 (\vec{\phi}_{n})(T) \right) + \sum^{\infty}_{j = 2} \Xi_j (\vec{u}_{0,n})(T) \bigg\|_{H^s} \\
& \gtrsim R^k T^2 A^{k - \frac 12 + s} - 1 - RA^{1/2} N^s - T^2 R^{k-1}A^{k-1} \| \vec{u}_0 \|_{\mathcal{H}^0} - T^4 (RA)^{2(k-1)} R g_s (A) \\
& \sim R^k T^2 A^{k - \frac 12 + s} \gg n. \end{align*}
Thus, this verifies the second estimate in \eqref{EQ:propFinal} at time $t_n \coloneqq T$. Finally, a suitable choice of $T$ in terms of $N = N(n)$ ensures $t_n \in (0, \frac 1n)$, for $N(n)$ sufficiently large. See details below. So ends the proof of Proposition \ref{PROP:final2}.
Consequently, it only remains to verify the conditions (i) through (vi). To do so, we express $A$, $R$ and $T$ in terms of $N$. More precisely, let us choose $$ A = 10, \quad R = N^{-s-\delta} \quad \text{ and } \quad T = N^{\frac{k-1}{2}(s + \frac \dl2)} $$
\noindent where $\delta$ is sufficiently small, namely $0 < \delta < \min \big(1, -\frac{2}{k+1}s \big)$. Since $A$ is a constant, condition (vi) is trivially satisfied and, since $\vec{u}_0$ is fixed, condition (iii) reduces to $$ R \gg 1 $$
\noindent which is true from $-s - \delta > 0$, for $N$ sufficiently large. Besides, for $N$ sufficiently large, we get \begin{align*} & RA^{\frac 12} N^s \sim N^{-\delta} \ll \frac 1n \\ & T^2 R^{k-1} A^{k-1} \sim N^{- \frac{k-1}{2} \delta} \ll 1 \\ & T^2 R^k A^{k - \frac 12 + s} \sim N^{-s - \frac{k+1}{2}\delta} \gg 1 \end{align*}
\noindent which prove conditions (i), (ii) and (v). Lastly, condition (iv) is equivalent to $$
T^2 R^{k-1} A^{k - 2}g_s(A) \ll A^{-\frac 12 + s} $$
\noindent which, in our setting, is equivalent to condition (ii) since $A$ is constant. Finally, since $\delta < -s$, observe that $T$ goes to $0$ as $N \to \infty$, so $T \in (0, \frac 1n )$ for $N$ sufficiently large. Taking $$ N = n^{\frac 2 \delta} $$ completes the proof since $\delta < 1$.
\begin{remark} \rm \label{REM:ILOR}
In this remark, we use our previous arguments to prove infinite loss of regularity, namely Proposition \ref{PROP:final}. The idea is to use the same construction as before. Indeed, this construction allowed us to choose our parameter $A$ to be constant. In the following, we observe that the change of regularity between Proposition \ref{PROP:final2} and Proposition \ref{PROP:final} is only expressed in powers of $A$, so that this change has no major implications whatsoever.
Let $s < 0$, $\sigma \in \mathbb{R}$ and $\vec{u}_0 = ( u_0, u_1) \in \overrightarrow{\mathcal{F}L}^1$. Define $\vec{\phi}_n$ and $\vec{u}_{0,n}$ as before. There exists, for any given $n \in \mathbb{N}$, a unique solution $u_n$ to \eqref{imBq} in $C([0,T], \mathcal{F}L^1)$, with $T$ satisfying \eqref{EQ:CondOnT}, such that $(u_n , \partial_t u_n)|_{t=0} = \vec{u}_{0,n}$. Moreover, $u_n$ can be expressed as the power series in \eqref{EQ:PowerSeriesSolRankn}. First, we claim that it suffices to consider the case $\sigma < s$. Indeed, in the case $\sigma \geq s$ the estimate on the initial data remains the same as for Proposition \ref{PROP:final2} and, by Sobolev embedding, we have $$
\| u_n (t_n) \|_{H^\sigma} \geq \| u_n (t_n) \|_{H^s} > n. $$
Let us then assume $\sigma < s$. Lemma \ref{LEM:MultilinearEst}, \eqref{EQ:PowerSeriesSolRankn}, \eqref{EQ:CondOnT} and \eqref{EQ:CondFL1u} yield \begin{align*}
\| u_n(T) - \Xi_1(\vec{\phi}_n) (T) \|_{H^\sigma} & \leq \| u_n(T) - \Xi_1(\vec{\phi}_n) (T) \|_{H^s} \\
& \lesssim 1 + RA^{1/2} N^s + T^2 (RA)^{k-1} \| \vec{u}_0 \|_{\mathcal{H}^0} + T^4 (RA)^{2(k-1)} R g_s (A) \end{align*}
\noindent while Proposition \ref{PROP:EstSecondPicard} gives $$
\| \Xi_1 (\vec{\phi}_n) (T) \|_{ H^\sigma} \gtrsim R^k T^2 A^{k - \frac 12 + \sigma}. $$
\noindent Therefore, the same arguments as before allow us to say that, to prove Proposition \ref{PROP:final}, it suffices to verify the following hold: \begin{align*} \textup{(i)} & \quad RA^{\frac 12} N^s < \frac 1n, \\ \textup{(ii)} & \quad T^2 R^{k-1} A^{k-1} \ll 1, \\
\textup{(iii)} & \quad \| \vec{u}_0 \|_{\mathcal{H}^0} \ll R A^{\frac 12 + \sigma} \text{ and } \| \vec{u}_0\|_{\overrightarrow{\mathcal{F}L}^1} \ll RA, \\ \textup{(iv)} & \quad T^4 (RA)^{2(k-1)} R g_s (A) \ll T^2 R^k A^{k - \frac 12 + \sigma}, \\ \textup{(v)} & \quad T^2 R^k A^{k - \frac 12 + \sigma} \gg n, \\ \textup{(vi)} & \quad A \ll N. \end{align*}
\noindent Note that, compared to the conditions in the proof of Proposition \ref{PROP:final2}, the only changes are in conditions (iii), (iv) and (v), where the power of $A$ changes. Yet, if we choose $A$ to be constant again, the rest of the reasoning stays the same. Hence, the choices $$ A = 10, \quad R = N^{-s-\delta}, \quad T = N^{\frac{k-1}{2}(s + \frac \dl2)}, \quad \text{ and } \quad N = n^{\frac 2 \delta} $$
\noindent prove also Proposition \ref{PROP:final}. \end{remark}
\begin{remark} \rm \label{RK:ProofGenIBq} The proof we gave applies directly to the study of the multi-dimensional generalized improved Boussinesq equation \eqref{generalizedimBq}. The only major change to make is in the definition of the functions $\phi_n$. Indeed, let us denote $e_1 = (1, 0, \cdots, 0) \in \mathbb{R}^d$, we then define $\phi_n$ by $$ \widehat{\phi}_n (\xi) = R \chi_\Omega = R \sum_{\eta \in \Sigma} \chi_{\{ \eta e_1 + Q_A \}} $$
\noindent with $\Sigma$ as in \eqref{DEF:Sigma} and $Q_A = [-\frac A2 , \frac A2]^d$. Then, the same arguments are still applicable. In particular, observe that Lemma \ref{LEM:ConvolutionIneq} can be rephrased as:
\begin{lemma} \label{LEM:ConvolutionIneqInRd} Let $a,b \in \mathbb{R}^d$ and $A > 0$, then we have $$ C_d A^d \chi_{a + b + Q_A } (\xi) \leq \chi_{a + Q_A } \ast \chi_{b + Q_A } (\xi) \leq \widetilde{C}_d A^d \chi_{a + b + Q_{2A}}(\xi ) $$
\noindent where $C_d, \widetilde{C}_d > 0$ are constants depending only on the dimension $d$. \end{lemma}
Subsequently, the only changes caused from the change of dimension are expressed in powers of $A$. However, since $A$ is chosen to be constant, it does not change the argument, as for when we proved infinite loss of regularity from ``standard" norm inflation. Therefore, norm inflation with infinite loss of regularity for initial data $\vec{u}_0 \in \mathcal{H}^s(\mathbb{R}^d)$ with $s<0$ follows naturally from the same choice of parameters $A$, $R$, $T$ and $N$.
Similarly, our result would still apply on the same multidimensional problem on the torus $\mathbb{T}^d$, although such a problem has not been seen in the litterature yet, at least not to our knowledge. \end{remark}
\begin{remark}\rm \label{REM:NIinWs2infty}
In this remark, we show that our proof also implies Theorem~\ref{THM:mainLinfty}. The idea is to use exactly the same construction and to estimate the norm. Using the same notations, observe that we have $$
\| u_n (t_n) \|_{W^{\sigma, 2, \infty}} \geq \| u_n (t_n) \|_{H^\sigma} > n. $$
\noindent Therefore, we only need to show that $$
\| \vec{u}_0 - \vec{u}_{0,n} \|_{\mathcal{W}^{s,2,\infty}} < \frac 1n $$
\noindent which means that we want $\vec{u}_{0}$ and $\vec{u}_{0,n}$ to satisfy $$
\| \vec{u}_0 - \vec{u}_{0,n} \|_{\mathcal{H}^{s}} < \frac 1n \quad \text{ and } \quad \| \vec{u}_0 - \vec{u}_{0,n} \|_{W^{s,\infty} \times W^{s,\infty}} < \frac 1n. $$
\noindent We already proved that the first estimate is satisfied, so we are only interested in the second one. Observe that $\vec{u}_0 - \vec{u}_{0,n} = \vec{\phi}_n = (\phi_n , 0)$ and $\phi_n$ is frequently supported on $\Omega$, defined by \eqref{DEF:Omega}. Then, we get by Cauchy-Schwarz and inverse Fourier transform $$
| \jb{\nabla}^s \phi_n | \leq \int_{\widehat{\mathcal{M}}} | \jb{\xi}^s \widehat{\phi}_n (\xi)| \mathrm{d}\xi = \int_{\Omega} \jb{\xi}^s |\widehat{\phi}_n (\xi)| \mathrm{d}\xi \leq 2A^{\frac 12} \| \phi_n \|_{H^s} $$
\noindent Since $A = 10$ in our proof, and $\| \vec{\phi}_n \|_{\mathcal{W}^{s, 2, \infty}} = \| \phi_n \|_{W^{s, 2, \infty}}$, this shows the desired result and Theorem~\ref{THM:mainLinfty}. Furthermore, combining this argument with Remark~\ref{RK:ProofGenIBq}, this result still holds in higher dimension.
\end{remark}
\appendix
\section{Norm inflation for other spaces} \label{appendixA}
In this appendix, we come back to Remark \ref{REM:NIforFLMW} and show how to prove norm inflation with infinite loss of regularity at general initial data for equation \eqref{imBq} in Fourier-Lebesgue, modulation and Wiener amalgam spaces. However, we do not show the entire proof since it is mostly similar to the one we gave for Sobolev spaces. Indeed, we use the same construction and decomposition as before, so the first differences are in the estimates we find in Lemma \ref{LEM:MultilinearEst} and Proposition \ref{PROP:EstSecondPicard}. Fortunately we find that the new estimates are, under the condition $A = 10$, equivalent to the ones we had for Sobolev spaces, so the last part of our argument remains unchanged. Therefore, we just show quickly how to get these estimates. Again, a whole proof would be quite redundant with what was shown before, the arguments being essentially the same. Thus, we just point out the key differences. Before that, we introduce our spaces. Let us recall, at this point, that modulation and Wiener amalgam spaces were introduced by Feichtinger~\cite{Fei83}.
\subsection{New spaces and some preliminary results.}
First, let us recall that Fourier-Lebesgue spaces are defined in Definition \ref{DEF:FLspaces}. Now, we introduce the modulation and Wiener amalgam spaces.
For any $n \in \mathbb{Z}^d$, we define $Q_n = n + \left[- \frac 12, \frac 12 \right]^d$ so that $\mathbb{R}^d = \bigcup_{n \in \mathbb{Z}^d} Q_n$. Then, let $\rho \in \mathcal{S} (\mathbb{R}^d)$ such that $\rho \colon \mathbb{R}^d \to [0,1]$ and \begin{equation} \rho(\xi) = \begin{cases} 1 \quad \text{ if } \quad \abs{\xi} \leq \frac 12 \\ 0 \quad \text{ if } \quad \abs{\xi} \geq 1 \end{cases} \end{equation}
\noindent We also denote $\rho_n (\xi) = \rho (\xi - n)$ for any $n \in \mathbb{Z}^d$. Let us define $$ \sigma_n = \frac{\rho_n}{\sum_{l \in \mathbb{Z}} \rho_l } $$
\noindent and $P_n f = \mathcal{F}^{-1} ( \sigma_n \mathcal{F}(f))$ for any suitable $f$. Then, we define the modulation spaces in the following way:
\begin{definition}[Modulation spaces] \rm Let $s \in \mathbb{R}$ and $p,q \geq 1$. The Modulation space $M^{p,q}_s (\mathbb{R}^d)$ is the completion of the Schwartz class of functions $\mathcal{S} ( \mathbb{R}^d)$ with respect to the norm $$
\| f \|_{M^{p,q}_s (\mathbb{R}^d)} = \big\| (1 + \abs{n}^s) \|P_n f \|_{L^p_x (\mathbb{R}^d)} \big\|_{\ell^q_n (\mathbb{Z}^d)}. $$ \end{definition}
\noindent We also define Wiener amalgam spaces in the following way:
\begin{definition}[Wiener amalgam spaces] \rm Let $s \in \mathbb{R}$ and $p,q \geq 1$. The Wiener amalgam space $W^{p,q}_s (\mathbb{R}^d)$ is the completion of the Schwartz class of functions $\mathcal{S} ( \mathbb{R}^d)$ with respect to the norm $$
\| f \|_{W^{p,q}_s (\mathbb{R}^d)} = \big\| \|(1 + \abs{n}^s)P_n f \|_{\ell^q_n (\mathbb{Z}^d)} \big\|_{L^p_x (\mathbb{R}^d)}. $$ \end{definition}
\noindent We have the following relations between modulation and Wiener amalgam spaces:
\begin{lemma} \label{LEM:RelationsMW} Let $1 \leq p,q, p_1, q_1, p_2, q_2 \leq \infty$ and $d \geq 1$ be any finite dimension. Then: \begin{enumerate}
\item For $p_1 \leq p_2$ and $q_1 \leq q_2$, we have \begin{equation} \label{EQ:EmbeddingsM} M^{p_1, q_1}_s (\mathbb{R}^d) \hookrightarrow M^{p_2, q_2}_s (\mathbb{R}^d), \end{equation} \noindent and \begin{equation} \label{EQ:EmbeddingsW} W^{p, q_1}_s (\mathbb{R}^d) \hookrightarrow W^{p, q_2}_s (\mathbb{R}^d), \end{equation}
\item for $q\leq p$, we have \begin{equation} \label{EQ:EmbeddingMintoW} M^{p, q}_s (\mathbb{R}^d) \hookrightarrow W^{p, q}_s (\mathbb{R}^d), \end{equation}
\noindent and for $p \leq q$ we have \begin{equation} \label{EQ:EmbeddingWintoM} W^{p, q}_s (\mathbb{R}^d) \hookrightarrow M^{p, q}_s (\mathbb{R}^d), \end{equation}
\item and we finally have \begin{equation} \label{EQ:RelationsMandW_end} M^{p, \min(p,q)}_s (\mathbb{R}^d) \hookrightarrow W^{p, q}_s (\mathbb{R}^d) \hookrightarrow M^{p, \max(p,q)}_s (\mathbb{R}^d). \end{equation} \end{enumerate} \end{lemma}
\noindent The proof of this Lemma can be seen in \cite{BH}, but split into different proofs. For completeness and ease of reading, we include it here. However, we first need the following lemma, which proof can be seen in \cite[Lemma 6.1] {WHHG}:
\begin{lemma} \label{LEM:lem6.1} Let $\Omega$ be a compact subset of $\mathbb{R}^d$ with $d \geq 1$, such that $\mathrm{diam} \ \Omega < 2R$, with $R > 0$, and $1 \leq p \leq q \leq \infty$. Then, there exists a constant $C > 0$ depending only on $p$, $q$ and $R$ such that: $$
\| f \|_{L^q (\mathbb{R}^d)} \leq C \| f \|_{L^p (\mathbb{R}^d)} $$
\noindent for any function $f \in L^p (\mathbb{R}^d)$ such that its Fourier transform $\widehat{f}$ is compactly supported in $\Omega$. \end{lemma}
\begin{proof}[Proof of Lemma \ref{LEM:RelationsMW}] The embeddings \eqref{EQ:EmbeddingsM} and \eqref{EQ:EmbeddingsW} follow from Lemma \ref{LEM:lem6.1} and the fact that, for any $p \leq q$, $\ell^{p} (\mathbb{Z}^d) \hookrightarrow \ell^{q} (\mathbb{Z}^d)$.
Embeddings \eqref{EQ:EmbeddingMintoW} and \eqref{EQ:EmbeddingWintoM} follow from Minkowski's inequality.
Finally, \eqref{EQ:RelationsMandW_end} is a combination of \eqref{EQ:EmbeddingsM}, \eqref{EQ:EmbeddingsW}, \eqref{EQ:EmbeddingMintoW} and \eqref{EQ:EmbeddingWintoM}. \end{proof}
\noindent Besides, we also have the following algebra property:
\begin{lemma} \label{LEM:algebraM21} For any $d \geq 1$, the space $M^{2,1}_0 (\mathbb{R}^d)$ is a Banach algebra. \end{lemma}
\begin{proof} We want to show that, for any $u,v \in M^{2,1}_0 (\mathbb{R}^d)$, we have $$
\| uv \|_{M^{2,1}_0 (\mathbb{R}^d)} = \sum_{n \in \mathbb{Z}^d} \| P_n (uv) \|_{L^2 (\mathbb{R}^d)} \leq \| u \|_{M^{2,1}_0 (\mathbb{R}^d)} \| v \|_{M^{2,1}_0 (\mathbb{R}^d)}. $$
\noindent Note that $u = \sum_{m \in \mathbb{Z}^d} P_m u $ and $v = \sum_{k \in \mathbb{Z}^d} P_k v$. Therefore $$ P_n (uv) = \sum_{m,k \in \mathbb{Z}^d} P_n (P_m u P_k v ). $$
\noindent The idea then is to study the Fourier support of each of the terms $P_n (P_m u P_k v )$. Since, for any $\xi \in \mathbb{R}^d$, $$ \mathcal{F} [ P_n (P_m u P_k v ) ] (\xi) = \sigma_n (\xi) \int_{\xi = \xi_1 + \xi_2} \sigma_m (\xi_1) \widehat{u}(\xi_1) \sigma_k (\xi_2) \widehat{v}(\xi_2) \mathrm{d}\xi_1 $$
\noindent and, for any $N \in \mathbb{Z}^d$, $\supp \sigma_N \subset \{ \xi \in \mathbb{R}^d, \abs{\xi - N} \leq 1 \}$, we have then $$ \supp \big[ (\sigma_m \widehat{u}) \ast (\sigma_k \widehat{v}) \big] \subset \{ \xi \in \mathbb{R}^d, \abs{\xi - m - k} \leq 2 \} $$
\noindent and $$ \mathcal{F} [ P_n (P_m u P_k v ) ] \equiv 0 \quad \text{ if } \quad \abs{n - m - k} > 3. $$
\noindent Plancherel's identity, H\"older's inequality and \eqref{EQ:EmbeddingsM} yield then: \begin{align*}
\| P_n (P_m u P_k v ) \|_{L^2 ( \mathbb{R}^d)} & \leq \| P_m u P_k v \|_{L^2(\mathbb{R}^d)} \chi_{(\abs{n- k - m} \leq 3)}, \\
& \leq \| P_m u \|_{L^\infty (\mathbb{R}^d)} \| P_k v \|_{L^2 ( \mathbb{R}^d) } \chi_{(\abs{n- k - m} \leq 3)}, \\
& \lesssim \| P_m u \|_{L^2 (\mathbb{R}^d)} \| P_k v \|_{L^2 ( \mathbb{R}^d) } \chi_{(\abs{n- k - m} \leq 3)}. \end{align*}
\noindent Hence, \begin{align*}
\| uv \|_{M^{2,1}_0 (\mathbb{R}^d)} & \leq \sum_{n \in \mathbb{Z}^d} \sum_{m,k \in \mathbb{Z}^d} \| P_n (P_m u P_k v ) \|_{L^2 ( \mathbb{R}^d)}, \\
& \lesssim \sum_{n \in \mathbb{Z}^d} \sum_{m,k \in \mathbb{Z}^d} \| P_m u \|_{L^2 (\mathbb{R}^d)} \| P_k v \|_{L^2 ( \mathbb{R}^d) } \chi_{(\abs{n- k - m} \leq 3)}, \\
& \lesssim \sum_{m,k \in \mathbb{Z}^d} \| P_m u \|_{L^2 (\mathbb{R}^d)} \| P_k v \|_{L^2 ( \mathbb{R}^d) } \end{align*}
\noindent which ends the proof.
\end{proof}
\subsection{Proofs of norm inflation in other spaces}
In this subsection, we claim that the following result is true:
\begin{theorem} \label{THM:NIwithILORforFLMW} Assume that $s < 0$ and $1 \leq q \leq \infty$ and let \begin{equation} \label{EQ:DefZsq} Z_s^q \coloneqq \mathcal{F}L^{s,q}(\mathcal{M}) \quad \text{ or } \quad M_s^{2,q}(\mathbb{R}) \quad \text{ or } \quad W_s^{2,q}(\mathbb{R}). \end{equation}
\noindent Then, norm inflation with infinite loss of regularity occurs at the origin for \eqref{imBq} in $Z_s^q$.
\noindent Namely, let $\theta \in \mathbb{R}$, $s < 0$ and fix $\vec{u}_0 \in Z_s^q \times Z_s^q$. Given any $\varepsilon > 0$, there exists a solution $u_\varepsilon \in C([0,T], Z_s^q)$ to \eqref{imBq} with $T > 0$ and $t_\varepsilon \in (0, \varepsilon)$ such that $$
\| (u_\varepsilon (0), \partial_t u_\varepsilon (0) ) - \vec{u}_0 \|_{Z_s^q \times Z_s^q} < \varepsilon \qquad \text{ and } \qquad \| u_\varepsilon (t_\varepsilon ) \|_{Z_\theta^q} > \varepsilon^{-1}. $$
\end{theorem}
As explained before, the idea to prove our result is to use the same construction as in Section \ref{Sec3}. Actually, we show that, by keeping the same notations and choosing $A$ to be constant, we have estimates that are equivalent to the ones given in Lemma \ref{LEM:MultilinearEst} and Proposition \ref{PROP:EstSecondPicard}. Then, the proof of Theorem \ref{THM:NIwithILORforFLMW} follows with the same argument.
\subsubsection{The case of Fourier-Lebesgue spaces}
In the case of Fourier-Lebesgue spaces $\mathcal{F}L^{s,q} (\mathcal{M})$, we get from the same computations as for the Sobolev spaces $H^s (\mathcal{M})$:
\begin{proposition} \label{PROP:EstforFL} Let $s < 0$ and $1 \leq q \leq \infty$. We have then \begin{equation} \label{EQ:normXi0FL}
\| \Xi_0 [\vec{u}_{0, n}](T) \|_{\mathcal{F}L^{s,q}} \lesssim 1 + R N^s A^{\frac 1q}, \end{equation}
\noindent as well as \begin{equation} \label{EQ:EstXi1UpperFL}
\|\Xi_1 [\vec{u}_{0, n}] (T) - \Xi_1 [\vec{\phi}_{ n}] (T) \|_{\mathcal{F}L^{s,q}} \lesssim T^{2} (RA)^{(k-1)} \| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^{q}( \mathcal{M})}, \end{equation}
\noindent and, for any $j \geq 2$, \begin{equation} \label{EQ:EstXijFL}
\|\Xi_j [\vec{u}_{0, n}] (T) \|_{\mathcal{F}L^{s,q}} \lesssim T^{2j} (RA)^{(k-1)j} \big( \| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^{q}( \mathcal{M})} + Rf_{s,q} (A) \big) \end{equation}
\noindent where \begin{equation} \label{EQ:deffsqFL}
f_{s,q}(A) = \| \jb{\xi}^s \|_{L^q_\xi (Q_A) }. \end{equation}
\noindent Also, if $A \ll N$, we have the following lower bound: \begin{equation} \label{EQ:EstSecondPicardFL}
\| \Xi_1 (\vec{\phi}_n) (T) \|_{ \mathcal{F}L^{s,q}} \gtrsim R^k T^2 A^{k - 1 + \frac 1q + s} \end{equation} \end{proposition}
\noindent Under the condition $A = 10$, we observe that \eqref{EQ:normXi0FL}, \eqref{EQ:EstXi1UpperFL}, \eqref{EQ:EstXijFL} and \eqref{EQ:EstSecondPicardFL} are essentially equivalent respectively to \eqref{EQ:MultilinearEst2}, \eqref{EQ:MultilinearEst3}, \eqref{EQ:MultilinearEst4} and \eqref{EQ:EstSecondPicard}, so that the expected result follows from the same argument. There is one condition to change though, which is $\| \vec{u}_0 \|_{\overrightarrow{\mathcal{F}L}^{q}( \mathcal{M})} \lesssim 1$ instead of $\| \vec{u}_0 \|_{\mathcal{H}^0 ( \mathcal{M})} \lesssim 1$, but this is quite reasonable since our initial data $\vec{u}_0$ is fixed.
\subsubsection{The case of modulation spaces}
The computations in this section follow the same ideas as in the previous section, but the modulation spaces make them a bit trickier to apply. We first state our results:
\begin{proposition} \label{PROP:EstForMs2qspaces} Let $s < 0$ and $1 \leq q \leq \infty$. Keeping the same notations as before, and asssuming
$$
\| \vec{u}_0 \|_{M^{2,q}_s \times M^{2,q}_s} \lesssim 1 \quad \text{ and } \quad \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} \ll R \| (1 + \abs{n})^s \|_{\ell^q ( 1 \leq \abs{n} \leq A)}, $$
\noindent we have \begin{equation} \label{EQ:normXi0M}
\| \Xi_0 [\vec{u}_{0,n}](T) \|_{M_s^{2,q}} \lesssim 1 + R N^s A^{\frac 12}, \end{equation}
\noindent as well as \begin{equation} \label{EQ:EstXi1UpperM}
\|\Xi_1 [\vec{u}_{0, n}] (T) - \Xi_1 [\vec{\phi}_{ n}] (T) \|_{M_s^{2,q}} \lesssim T^{2} (RA)^{(k-1)} \| \vec{u}_0 \|_{M^{2,1}_0( \mathcal{M})}, \end{equation}
\noindent and, for any $j \geq 2$ \begin{equation} \label{EQ:EstXijFL}
\| \Xi_j [\vec{u}_{0,n}] (T) \|_{M_s^{2,q}} \lesssim T^{2j} (RA)^{(k-1)j} R \| (1 + \abs{n})^s \|_{\ell^q ( 1 \leq \abs{n} \leq A)}. \end{equation}
\noindent Also, if $A \ll N$, we have the following upper bound: \begin{equation} \label{EQ:EstSecondPicardFL}
\| \Xi_1 (\vec{\phi}_n) (T) \|_{M_s^{2,q}} \gtrsim R^k T^2 A^{k - 1}A^{ \frac 1q + s}. \end{equation} \end{proposition}
Again, we see that if we choose $A$ to be constant, all these estimates are equivalent to the estimates we had for the Sobolev spaces, and the norm inflation result becomes straightforward. We include the proof of Proposition \ref{PROP:EstForMs2qspaces} both for completeness and to put an emphasis on the differences between these spaces and Sobolev spaces.
\begin{proof}[Proof of Proposition \ref{PROP:EstForMs2qspaces}]
First, we have from $\abs{ \cos x } \leq 1$ and $\abs{\sin x} \lesssim \abs{x}$, \begin{equation} \label{EQ:EstLinearM}
\| S(t)(u_0 , u_1 ) \|_{M_s^{2,q}} \lesssim \| u_0 \|_{M_s^{2,q}} + \abs{t} \| u_1 \|_{M_s^{2,q}} \lesssim \| (u_0, u_1) \|_{M^{2,q}_s \times M^{2,q}_s} \end{equation}
\noindent for any $0 \leq t \leq 1$ and $u_0, u_1 \in M^{2,q}_s$. Besides, we also have \begin{align*}
\| \phi_n \|_{M_s^{2,q}} & = R \big\| (1 + \abs{n})^s \| P_n \mathcal{F}^{-1}(\sum_{\eta \in \Sigma} \chi_{\eta + Q_A}) \|_{L^2} \big\|_{\ell^q_n} \\
& = R \big\| (1 + \abs{n})^s \| \sigma_n \sum_{\eta \in \Sigma} \chi_{\eta + Q_A} \|_{L^2} \big\|_{\ell^q_n} \end{align*}
\noindent but since the sets $\eta + Q_A$ are disjoint for any two $\eta_1 \neq \eta_2$ in $\Sigma$, we get $$
\left\| \sigma_n \sum_{\eta \in \Sigma} \chi_{\eta + Q_A} \right\|^2_{L^2} \sim \chi_{N + Q_A} (n) \int_{\mathbb{R}} \sigma^2_n (\xi) \chi^2_{n + Q_A} (\xi) \mathrm{d}\xi \sim \chi_{N + Q_A} (n) $$
\noindent and $$
\| \phi_n \|_{M_s^{2,q}} \sim R \| (1 + \abs{n} )^s \chi_{N + Q_A} (n)\|_{\ell^q_n} \sim RN^s A^{\frac 1q} $$
\noindent which, combined with \eqref{EQ:EstLinearM}, proves our first estimate. Besides, \eqref{EQ:EmbeddingsM} and the algebra property of $M^{2,1}_0$ yield, in a similar argument as for \eqref{EQ:MultilinearEst3}: \begin{align*}
\|\Xi_1 [\vec{u}_{0, n}] (T) - \Xi_1 [\vec{\phi}_{ n}] (T) \|_{M_s^{2,q}} & \lesssim \|\Xi_1 [\vec{u}_{0, n}] (T) - \Xi_1 [\vec{\phi}_{ n}] (T) \|_{M_0^{2,1}}, \\
& \lesssim T^2 \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} \big( \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} + \| \vec{\phi}_n \|_{M^{2,1}_0 \times M^{2,1}_0} \big)^{k-1}, \\
& \lesssim T^2 \| (u_0, u_1) \|_{M^{2,1}_0 \times M^{2,1}_0} (RA^{\frac 12})^{k-1}. \end{align*}
\noindent Hence \eqref{EQ:EstXi1UpperM}.
Now, let $j \geq 2$. Using the same argument on the support as in the proof for \eqref{EQ:MultilinearEst4}, we have \begin{align*}
\| \sigma_n \mathcal{F}[\Xi_j (\vec{\phi}_n)](T) \|_{L^2_\xi} & \leq \| \sigma_n \|_{L^2_\xi (\supp \mathcal{F}[ \Xi_j (\vec{\phi}_n)](T))} \| \Xi_j (\vec{\phi}_n) (T) \|_{\mathcal{F}L^\infty} \\
& \lesssim \| \sigma_n\|_{L^2_\xi (Q_A \cap Q_n)} C^j T^{2j} \| \vec{\phi}_n \|^{(k-1)j-1}_{\overrightarrow{\mathcal{F}L}^1} \| \vec{\phi}_n \|^2_{\mathcal{H}^0} \\
& \lesssim \mathds{1}_{\{ n \in Q_A\}} T^{2j}(RA)^{(k-1)j} R \end{align*}
\noindent so that $$
\| \Xi_j [\vec{\phi}_n] (T) \|_{M_s^{2,q}} \lesssim T^{2j} (RA)^{(k-1)j} R \| (1 + \abs{n})^s \mathds{1}_{\{ n \in Q_A \}} \|_{\ell^q_n}. $$
\noindent Besides, a similar argument as for \eqref{EQ:EstXi1UpperM} yields \begin{align*}
\|\Xi_j [\vec{u}_{0, n}] (T) - \Xi_j [\vec{\phi}_{ n}] (T) \|_{M_s^{2,q}} & \lesssim \|\Xi_j [\vec{u}_{0, n}] (T) - \Xi_j [\vec{\phi}_{ n}] (T) \|_{M_0^{2,1}}, \\
& \lesssim T^{2j} \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} \big( \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} + \| \vec{\phi}_n \|_{M^{2,1}_0 \times M^{2,1}_0} \big)^{(k-1)j}, \\
& \lesssim T^{2j} \| \vec{u}_0 \|_{M^{2,1}_0 \times M^{2,1}_0} (RA^{\frac 12})^{(k-1)j}, \end{align*}
\noindent which gives our third estimate.
Finally, our fourth estimate follows from a straightforward adaptaton of the proof of Proposition \ref{PROP:EstSecondPicard}, the only difference being the norm used.
\end{proof}
\subsubsection{The case of Wiener amalgam spaces}
This section relies a lot on the interactions between modulation and Wiener amalgam spaces. Indeed, using \eqref{EQ:RelationsMandW_end} and Proposition \ref{PROP:EstForMs2qspaces}, we get respectively \begin{equation} \label{EQ:EstForWs2q_Xi0}
\| \Xi_0 [\vec{u}_{0,n}](T) \|_{W_s^{2,q}} \lesssim \| \Xi_0 [\vec{u}_{0,n}](T) \|_{M_s^{2,\min(2,q)}} \lesssim 1 + R N^s A^{\frac 12}, \end{equation}
\noindent as well as \begin{align} \label{EQ:EstForWs2q_Xi1Upper}
\| \Xi_1 [\vec{u}_{0,n}] (T) - \Xi_1 [\vec {\phi}_n](T) \|_{W_s^{2,q}} & \lesssim \| \Xi_1 [\vec{u}_{0,n}] (T) - \Xi_1 [\vec {\phi}_n](T) \|_{M_s^{2,\min(2,q)}}\\
& \lesssim T^{2} (RA)^{(k-1)} \| \vec{u}_0 \|_{M^{2,1}_0( \mathcal{M})}, \end{align}
\noindent and, for any $j \geq 2$, \begin{align} \label{EQ:EstForWs2q_Xij}
\| \Xi_j [\vec{u}_{0,n}] (T) \|_{W_s^{2,q}} & \lesssim \| \Xi_j [\vec{u}_{0,n}] (T) \|_{M_s^{2,\min(2,q)}}\\
& \lesssim T^{2j} (RA)^{(k-1)j} R \| (1 + \abs{n})^s \|_{\ell^{\min(2,q)} ( 1 \leq \abs{n} \leq A)}. \end{align}
\noindent Besides, we also get the following lower bound: \begin{equation} \label{EQ:EstForWs2q_Xi1}
\| \Xi_1 (\vec{\phi}_n) (T) \|_{W_s^{2,q}} \gtrsim \| \Xi_1 (\vec{\phi}_n) (T) \|_{W_s^{2,\max(2,q)}} \gtrsim R^k T^2 A^{k - 1}A^{ \frac 1q + s}. \end{equation}
\noindent Again, these estimates are equivalent to the ones we had for Sobolev spaces under the condition $A = 10$, so norm inflation with infinite loss of regularity follows.
\section*{Acknowledgments} The author would like to thank Tadahiro Oh for suggesting this problem and his continuous support throughout this work. The author acknowledges support from Tadahiro Oh's ERC grant (no. 864138 ``SingStochDispDyn"). The author would also like to thank Younes Zine for several helpful discussions.
\end{document} |
\begin{document}
\author{Alexander E Patkowski} \title{On asymptotic expansions for basic hypergeometric functions }
\maketitle \begin{abstract} This paper establishes new results concerning asymptotic expansions of $q$-series related to partial theta functions. We first establish a new method to obtain asymptotic expansions using a result of Ono and Lovejoy, and then build on these observations to obtain asymptotic expansions for related multi hypergeometric series \end{abstract}
\keywords{\it Keywords: \rm $L$-function; $q$-series, asymptotics}
\subjclass{ \it 2010 Mathematics Subject Classification Primary 33D90; Secondary 11M41}
\section{Introduction and Main Results}
\par In keeping with usual notation [6], put $(y)_n=(y;q)_{n}:=\prod_{0\le k\le n-1}(1-yq^{k}),$ and define $(y)_{\infty}=(y;q)_{\infty}:=\lim_{n\rightarrow\infty}(y;q)_{n}.$ Recall (see [11, pg.182]) the parabolic cylinder function $D_s(x),$ defined as the solution to the differential equation of Weber [7, pg.1031, eq.(9.255), \#1] $$\frac{\partial^2 f}{\partial x^2}+(s+\frac{1}{2}-\frac{x^2}{4})f=0.$$ This function also has a relationship with the confluent hypergeometric function ${}_1F_1(a;b; x)$ [11, pg.183], [7, pg.1028] $$D_s(x)=e^{-x^2/4}\sqrt{\pi}\left(\frac{2^{s/2}}{\Gamma(\frac{1}{2}-\frac{s}{2})}{}_1F_1(-\frac{s}{2};\frac{1}{2};\frac{x^2}{2})-\frac{x2^{s/2+1/2}}{\Gamma(-\frac{s}{2})}{}_1F_1(\frac{1}{2}-\frac{s}{2};\frac{3}{2};\frac{x^2}{2})\right).$$ In the last two decades considerable attention has been given to asymptotic expansions of basic hypergeometric series when $q=e^{-t}$ as $t\rightarrow0^{+}.$ Special examples related to negative values of $L$-functions have been offered in [3, 8, 9, 12, 13, 14]. Recall that an $L$-function is defined as the series $L(s)=\sum_{n\ge1}a_nn^{-s},$ for a suitable arithmetic function $a_n:\mathbb{N}\rightarrow\mathbb{C}.$ In the simple case $a_n=1,$ we have the Riemann zeta function $L(s)=\zeta(s),$ for $\Re(s)>1,$ and if $a_n=(-1)^n,$ we obtain $L(s)=(1-2^{1-s})\zeta(s).$ \par As it turns out, there are general expansions for basic hypergeometric series with $q=e^{-t}$ which include special functions as terms along with values of $L(s).$ To this end, we offer the first instance of a more general expansion in the literature for certain multi-dimensional basic hypergeometric series. To illustrate our approach we first recall the result due to Ono and Lovejoy [9, Theorem 1], which says that for $k\ge2,$ integers $0<m<l,$ if \begin{equation}F_k(z,q):=\sum_{r_{k-1}\ge r_{k-2}\dots\ge r_1\ge0}\frac{(q)_{n_{k-1}}(z)_{n_{k-1}}z^{n_{k-1}+2n_{k-2}+\dots+2n_1}q^{r_1^2+r_1+r_2^2+r_2+\dots+r_k^2+r_k}}{(q)_{r_{k-1}-r_{k-2}}(q)_{r_{k-2}-r_{k-3}}\dots(q)_{r_{2}-r_{1}}(q)_{r_1}(-z)_{r_1+1}},\end{equation} then as $t\rightarrow0^{+},$ $$e^{-(k-1)m^2t}F_k(e^{-lmt},e^{-l^2t})=\sum_{n\ge0}L_{l,m}(-2n)\frac{((1-k)t)^n}{n!}, $$ where $L_{l, m}(s)=(2l)^{-s}\left(\zeta(s,\frac{m}{2l})-\zeta(s,\frac{l+m}{2l})\right).$ Here the Hurwitz zeta function is $\zeta(s,x)=\sum_{n\ge0}(n+x)^{-s},$ a one parameter refinement of $\zeta(s).$ Their paper uses a clever specialization of Andrews' refinement of a transformation of Watson to obtain (1.1). By [9, Theorem 2.1] \begin{equation} F_k(z,q)=\sum_{n\ge0}(-1)^nz^{(2k-2)n}q^{(k-1)n^2}.\end{equation}
Putting $q=e^{-wt^2},$ and $z=e^{-vt},$ and taking the Mellin transform (see (2.10) or [11]) of (1.2), Lemma 2.4 of the next section tells us that for $\Re(s)>0,$ $\Re(w)>0,$
\begin{equation}\int_{0}^{\infty}t^{s-1} \left(F_k(e^{-vt},e^{-wt^2})-1\right)dt\end{equation} $$= (2w(k-1))^{-s/2}(1-2^{1-s})\zeta(s)\Gamma(s)e^{v^2(k-1)/(2w)}D_{-s}\left(\frac{v2(k-1)}{\sqrt{2(k-1)w}}\right).$$ Now by Mellin inversion and Cauchy's residue theorem, it can be shown that \begin{equation}F_k(e^{-vt},e^{-wt^2})-1\end{equation} $$\sim e^{v^2(k-1)/(2w)}\sum_{n\ge0}\frac{(2w(k-1))^{n/2}}{n!}(-t)^n(1-2^{1+n})\zeta(-n)D_{n}\left(\frac{v2(k-1)}{\sqrt{2(k-1)w}}\right),$$ as $t\rightarrow0^{+}.$ As a result of our new observation, we are able to produce general expansions of a similar type as (1.4) by appealing to Bailey chains [1]. We believe our observation is significant in its implications for further asymptotic expansions for basic hypergeometric series.
\begin{theorem}\label{thm:theorem1} Define for $k\ge1$ $$A_{n,k}(z,q):=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{a^{r_1+r_2+\dots+r_k}q^{r_1^2+r_2^2+\dots+r_k^2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\frac{(z)_{r_k+1}(q/z)_{r_k}}{(q)_{2r_k+1}}.$$ For any $v\in\mathbb{C},$ and $\Re(w)>0,$ $$-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}A_{n,k}(e^{-vt-wt^2(k+1)},e^{-wt^2})$$ $$\sim t^{-1}\sqrt{\frac{\pi}{4w(k+1)}}e^{v^2/(4(k+1)w)}\bigg(\erf\left(\frac{v}{2\sqrt{(k+1)w}}\right)-\erf\left(-\frac{v}{2\sqrt{(k+1)w}}\right)\bigg)$$ $$+e^{v^2/(8(k+1)w)}\sum_{n\ge0}\frac{(2w(k+1))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}\left(-\frac{v}{\sqrt{2(k+1)w}}\right)$$ $$- e^{v^2/(8(k+1)w)}\sum_{n\ge0}\frac{(2w(k+1))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}\left(\frac{v}{\sqrt{2(k+1)w}}\right).$$ as $t\rightarrow 0^{+}.$ \end{theorem}
We mention that the function in Theorem 1.1. should be compared to the $G(z,q)$ function contained in [9].
\begin{theorem}\label{thm:theorem2} Define for $k\ge1$ $$B_{n,k}(z,q):=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}P_{n,r_1,r_2,\dots,r_k}(q)q^{n-r_1+2(r_1-r_2)+\dots+2^{k-1}(r_{k-1}-r_k)},$$ where $$P_{n,r_1,r_2,\dots,r_k}(q):=\frac{(-q^2;q)_{2r_1}(-q^{2^2};q^{2})_{2r_2}\dots(-q^{2^{k}};q^{2^{k-1}})_{2r_k}}{(q^2;q^2)_{n-r_1}(q^{2^2};q^{2^2})_{r_1-r_2}\dots(q^{2^{k}};q^{2^{k}})_{r_{k-1}-r_{k}}}\frac{(z;q^{2^k})_{r_k+1}(q^{2^k}/z;q^{2^k})_{r_k}}{(q^{2^k};q^{2^k})_{2r_k+1}}.$$ For any $v\in\mathbb{C},$ and $\Re(w)>0,$ $$-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}B_{n,k}(e^{-vt-wt^2(2^{k}+1)},e^{-wt^2})$$ $$\sim t^{-1}\sqrt{\frac{\pi}{4w(2^k+1)}}e^{v^2/(4(2^k+1)w)}\bigg(\erf\left(\frac{v}{2\sqrt{(2^k+1)w}}\right)-\erf\left(-\frac{v}{2\sqrt{(2^k+1)w}}\right)\bigg)$$ $$+e^{v^2/(4(k+1)w)}\sum_{n\ge0}\frac{(w(2^{k}+1))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}(-\frac{v}{\sqrt{(2^{k}+1)w}})$$ $$- e^{v^2/(4(k+1)w)}\sum_{n\ge0}\frac{(w(2^{k}+1))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}(\frac{v}{\sqrt{(2^{k}+1)w}}).$$ \end{theorem}
\begin{theorem}\label{thm:theorem3} Define for $k\ge1$ $$C_{n.k}(z,q)=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{q^{r_1^2/2+r_1/2+r_2^2/2+r_2/2+\dots+r_k^2/2+r_k/2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\frac{(z)_{r_k+1}(q/z)_{r_k}}{(q)_{{r_k}}(q;q^2)_{r_k+1}}.$$ For any $v\in\mathbb{C},$ and $\Re(w)>0,$ $$-1+\sum_{n\ge0}\frac{(e^{-wt^2};e^{-wt^2})_n}{(-e^{-wt^2};e^{-wt^2})_n}(-1)^ne^{-wt^2n(n+1)/2}C_{n,k}(e^{-vt-wt^2(k+1)/2},e^{-wt^2})$$ $$\sim t^{-1}\sqrt{\frac{\pi}{2w(k+2)}}e^{v^2/(2(k+2)w)}\bigg(\erf\left(\frac{v}{2\sqrt{(k+2)w}}\right)-\erf\left(-\frac{v}{2\sqrt{(k+2)w}}\right)\bigg)$$ $$+e^{v^2/(4(k+2)w)}\sum_{n\ge0}\frac{(w(k+2))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}(-\frac{v}{\sqrt{(k+2)w}})$$ $$-e^{v^2/(4(k+2)w)} \sum_{n\ge0}\frac{(w(k+2))^{n/2}}{n!}(-t)^n\zeta(-n)D_{n}(\frac{v}{\sqrt{(k+2)w}}),$$ as $t\rightarrow0^{+}.$ \end{theorem} We note that it may be desirable to utilize the Hermite polynomial representation of the parabolic cylinder function [7, pg.1030, eq.(9.253)] $D_n(x)=2^{-n/2}e^{-x^2/4}H_n(\frac{x}{\sqrt{2}}),$ as an alternative. Due to the parity relationship $D_n(-x)=(-1)^nD_n(x),$ there is some collapsing that takes place for Theorems 1.1 through Theorem 1.3. \par Our last result is an apparently new asymptotic expansion for a partial theta function involving a nonprincipal Dirichlet character $\chi.$
\begin{theorem}\label{thm:theorem4} Let $\chi(n)$ be a real, primitive, nonprincipal Dirichlet character associated with $L(s,\chi).$ Let $v\in\mathbb{C},$ and $\Re(w)>0.$ Then if $\chi$ is an even character, $$\sum_{n\ge1}\chi(n)e^{-w n^2t^2-v nt}\sim e^{v^2/(8w)}\sum_{n\ge0}\frac{(2w)^{(2n+1)/2}(-t)^{2n+1}}{(2n+1)!}L(-2n-1,\chi)D_{2n+1}(\frac{v}{\sqrt{2w}}),$$ as $t\rightarrow0^{+},$ and if $\chi$ is an odd character, $$\sum_{n\ge1}\chi(n)e^{-w n^2t^2-v nt}\sim e^{v^2/(8w)}\sum_{n\ge0}\frac{(2w)^{n}(-t)^{2n}}{(2n)!}L(-2n,\chi)D_{2n}(\frac{v}{\sqrt{2w}}),$$ as $t\rightarrow0^{+}.$
\end{theorem}
\section{Proofs of results}
We recall that a pair $(\alpha_n(a, q),\beta_n(a,q))$ is referred to as a Bailey pair [4] with respect to $(a,q)$ if \begin{equation}\beta_n(a,q)=\sum_{0\le j\le n}\frac{\alpha_j(a,q)}{(q;q)_{n-j}(aq;q)_{n+j}}.\end{equation} Iterating [5, (S1)] $k$ times gives us the following lemma. \begin{lemma} For $k\ge1,$ \begin{equation} \beta'_n(a,q)=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{a^{r_1+r_2+\dots+r_k}q^{r_1^2+r_2^2+\dots+r_k^2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\beta_{r_k}(a,q)\end{equation} \begin{equation} \alpha'_n(a,q)=a^{kn}q^{kn^2}\alpha_n(a,q).\end{equation} \end{lemma} Iterating [5, (D1)] $k$ times gives us the following different lemma. \begin{lemma} For $k\ge1,$ \begin{equation} \beta'_n(a,q)=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}p_{n,r_1,r_2,\dots,r_k}(a,q)q^{n-r_1+2(r_1-r_2)+\dots+2^{k-1}(r_{k-1}-r_k)}\beta_{r_k}(a^{2^{k}},q^{2^{k}}),\end{equation} where $$p_{n,r_1,r_2,\dots,r_k}(a,q):=\frac{(-aq;q)_{2r_1}(-a^{2}q^{2};q^{2})_{2r_2}\dots(-a^{2^{k}}q^{2^{k-1}};q^{2^{k-1}})_{2r_k}}{(q^2;q^2)_{n-r_1}(q^{2^2};q^{2^2})_{r_1-r_2}\dots(q^{2^{k}};q^{2^{k}})_{r_{k-1}-r_{k}}},$$ and \begin{equation} \alpha'_n(a,q)=\alpha_n(a^{2^{k}},q^{2^{k}}).\end{equation} \end{lemma} Lastly, we iterate [5, (S2)] $k$ times. \begin{lemma} For $k\ge1,$ \begin{equation} \beta'_n(a,q)=\frac{1}{(-\sqrt{aq})_{n}}\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{(-\sqrt{aq})_{r_k}a^{r_1/2+r_2/2+\dots+r_k/2}q^{r_1^2/2+r_2^2/2+\dots+r_k^2/2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\beta_{r_k}(a,q)\end{equation} \begin{equation} \alpha'_n(a,q)=a^{kn/2}q^{kn^2/2}\alpha_n(a,q).\end{equation}
\end{lemma}
Now it is well-known [1. Lemma 6] that $(\alpha_n(q,q),\beta_n(q,q))$ form a Bailey pair where
\begin{equation}\alpha_n(q,q)=(-z)^{-n}\frac{q^{\binom{n+1}{2}}(1-z^{2n+1})}{1-q},\end{equation} \begin{equation}\beta_n(q,q)=\frac{(z)_{n+1}(q/z)_{n}}{(q)_{2n+1}}.\end{equation} The Mellin transform is defined as [11] (assuming $g$ satisfies certain growth conditions) \begin{equation}\mathfrak{M}(g)(s):=\int_{0}^{\infty}t^{s-1}g(t)dt.\end{equation} The main integral formula we will utilize is given in [7, pg.365, eq.(3.462), \#1]. \begin{lemma} For $\Re(s)>0,$ $\Re(w)>0,$ $$\int_{0}^{\infty}t^{s-1}e^{-w t^2-v t}dt=(2w)^{-s/2}\Gamma(s)e^{v^2/(8w)}D_{-s}(\frac{v}{\sqrt{2w}}).$$
\end{lemma} By the Legendre duplication formula $\Gamma(\frac{s}{2})\Gamma(\frac{s}{2}+\frac{1}{2})=2^{1-s}\sqrt{\pi}\Gamma(s),$ and the value $D_{-s}(0)\Gamma(\frac{1+s}{2})=2^{s/2}\sqrt{\pi},$ it is readily observed that the $v\rightarrow0$ case of this lemma reduces to the integral formula for $w^{-s/2}\Gamma(\frac{s}{2}).$ The parabolic cylinder function is an analytic function in $v$ and $x$ that enjoys the property that it has no singularities.
\begin{proof}[Proof of Theorem~\ref{thm:theorem1}] Inserting (2.8)--(2.9) into Lemma 2.1 gives us the following Bailey pair
\begin{equation}\bar{\beta}_n(q,q):=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{a^{r_1+r_2+\dots+r_k}q^{r_1^2+r_2^2+\dots+r_k^2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\frac{(z)_{r_k+1}(q/z)_{r_k}}{(q)_{2r_k+1}}\end{equation} \begin{equation}\bar{\alpha}_n(q,q)=(-z)^{-n}\frac{q^{(2k+1)n(n+1)/2}(1-z^{2n+1})}{1-q},\end{equation} A limiting case of Bailey's lemma [1, pg.270, eq.(2.4)] (with $a=q,$ $\rho_1=q,$ $\rho_2\rightarrow\infty,$ and $N\rightarrow\infty$) says that \begin{equation} \sum_{n\ge0}(q)_n(-1)^nq^{n(n+1)/2}\beta_n(q,q)=(1-q)\sum_{n\ge0}(-1)^n q^{n(n+1)/2}\alpha_n(q,q).\end{equation} Inserting (2.11)--(2.12) into (2.13) gives \begin{equation} \begin{aligned}&\sum_{n\ge0}(q)_n(-1)^nq^{n(n+1)/2}\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{a^{r_1+r_2+\dots+r_k}q^{r_1^2+r_2^2+\dots+r_k^2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\frac{(z)_{r_k+1}(q/z)_{r_k}}{(q)_{2r_k+1}}\\ &=\sum_{n\ge0}(-1)^n(-z)^{-n}q^{(k+1)n(n+1)}(1-z^{2n+1}).\end{aligned} \end{equation}
Putting $q=e^{-wt^2},$ and $z=e^{-vt-wt^2(k+1)},$ we have that (2.14) becomes \begin{equation} \begin{aligned} &\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_ne^{-wt^2n(n+1)/2}(-1)^nA_{n,k}(e^{-vt-wt^2(k+1)},e^{-wt^2})\\ &=\sum_{n\ge0}e^{nvt-wt^2(k+1)n^2}(1-e^{-vt(2n+1)-wt^2(k+1)(2n+1)})\\ &= \sum_{n\ge0}e^{nvt-wt^2(k+1)n^2}-\sum_{n\ge0}e^{nvt-wt^2(k+1)n^2-vt(2n+1)-wt^2(k+1)(2n+1)}\\ &= \sum_{n\ge0}e^{nvt-wt^2(k+1)n^2}-\sum_{n\ge1}e^{-wt^2(k+1)n^2-vtn} .\end{aligned}\end{equation}
Subtracting a $1$ from (2.15) and then taking the Mellin transform, we compute that $$\begin{aligned} &\mathfrak{M}\bigg\{ \sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_ne^{-wt^2n(n+1)/2}(-1)^nA_{n,k}(e^{-vt-wt^2(k+1)},e^{-wt^2})-1\bigg\}\\ &= \int_{0}^{\infty}t^{s-1}\left( \sum_{n\ge1}e^{nvt-wt^2(k+1)n^2}-\sum_{n\ge1}e^{-wt^2(k+1)n^2-vtn}\right)dt\\ &= \sum_{n\ge1}\int_{0}^{\infty}t^{s-1} \left(e^{nvt-wt^2(k+1)n^2} - e^{-wt^2(k+1)n^2-vtn}\right)dt\\ &=\sum_{n\ge1}\Bigg((2w(k+1)n^2)^{-s/2}\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}\left(-\frac{v}{\sqrt{2(k+1)w}}\right)\\ &- (2w(k+1)n^2)^{-s/2}\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}\left(\frac{v}{\sqrt{2(k+1)w}}\right)\Bigg)\\ &=(2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}\left(-\frac{v}{\sqrt{2(k+1)w}}\right)\\ &- (2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}\left(\frac{v}{\sqrt{2(k+1)w}}\right). \end{aligned}$$
Here we employed Lemma 2.4 with $v$ replaced by $nv,$ and $w$ replaced by $w(k+1)n^2,$ and the resulting formula is analytic for $\Re(s)>1.$ Now applying Mellin inversion, we compute that for $\Re(s)=c>1,$ \begin{equation}\begin{aligned} &-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}A_{n,k}(e^{-vt-wt^2(k+1)},e^{-wt^2})\\ &=\frac{1}{2\pi i}\int_{(c)}\bigg((2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}(-\frac{v}{\sqrt{2(k+1)w}})\\ &-(2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}(\frac{v}{\sqrt{2(k+1)w}})\bigg)t^{-s}ds.\end{aligned}\end{equation}
The modulus of the integrand can be seen to be estimated as follows (see [11, pg.398] for a similar example). Making the change of variable $s\rightarrow s+\frac{1}{2},$ we obtain an integral for $\Re(s)>\frac{1}{2}.$ For $\Re(s)=\sigma>\frac{1}{2},$ we have $\zeta(s+\frac{1}{2})\ll \zeta(\sigma+\frac{1}{2}).$ The growth of the integrand is then seen to be dominated by $\Gamma(s+\frac{1}{2})e^{v^2/(8(k+1)w)}D_{-s-\frac{1}{2}}(\frac{v}{\sqrt{2(k+1)w}}),$ due to Stirling's formula. Now an estimate of Paris [10, pg. 425, A(10)] for the parabolic cylinder function $D_{-s-\frac{1}{2}}(x)$ for fixed $x$ as $|s|\rightarrow\infty,$ says that \begin{equation}D_{-s-\frac{1}{2}}(x)=\frac{\sqrt{\pi}e^{-x\sqrt{s}}}{2^{s/2+1/4}\Gamma(\frac{s}{2}+\frac{3}{4})}\left(1-\frac{x^3}{24\sqrt{s}}+\frac{x^2}{24s}(\frac{x^2}{48}-\frac{3}{2})+O(s^{-3/2})\right),\end{equation}
uniformly for $|\arg(s)|\le \pi-\delta<\pi.$ By the asymptotic estimate (2.17) in conjunction with [11, pg.39, Lemma 2.2], we see the growth of the integrand is well controlled. Hence, we may apply Cauchy's residue theorem to obtain our expansion.
\par The integrand of (2.16) has simple poles at $s=1$ and the negative integers $s=-n$ due to $\Gamma(s).$ Using $\lim_{s\rightarrow1}(s-1)\zeta(s)=1,$ and [7, pg.1030, eq.(9.254),\#1] $D_{-1}(x)=\sqrt{\pi/2}e^{x^2/4}(1-\erf(\frac{x}{\sqrt{2}})),$ $$\begin{aligned} &\lim_{s\rightarrow1}(s-1)t^{-s}\bigg((2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}(-\frac{v}{\sqrt{2(k+1)w}})\\ &-(2w(k+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(8(k+1)w)}D_{-s}(\frac{v}{\sqrt{2(k+1)w}})\bigg)\\ &=t^{-1}(2w(k+1))^{-1/2}e^{v^2/(8(k+1)w)}\bigg(D_{-1}(-\frac{v}{\sqrt{2(k+1)w}})-D_{-1}(\frac{v}{\sqrt{2(k+1)w}})\bigg)\\ &=t^{-1}\sqrt{\frac{\pi}{4w(k+1)}}e^{v^2/(4(k+1)w)}\bigg(\left(1-\erf\left(-\frac{v}{\sqrt{4(k+1)w}})\right)\right) \\ &-\left(1-\erf\left(\frac{v}{\sqrt{4(k+1)w}})\right)\right)\bigg).\end{aligned}$$
Using standard properties of Mellin transforms, we compute the residues at the negative integers to see that as $t\rightarrow0^{+},$ $$-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}A_{n,k}(e^{-vt-wt^2(k+1)},e^{-wt^2})$$ $$\sim t^{-1}\sqrt{\frac{\pi}{4w(k+1)}}e^{v^2/(4(k+1)w)}\bigg(\erf\left(\frac{v}{2\sqrt{(k+1)w}}\right)-\erf\left(-\frac{v}{2\sqrt{(k+1)w}}\right)\bigg)$$ $$+\sum_{n\ge0}\frac{(2w(k+1))^{n/2}}{n!}(-t)^n\zeta(-n)e^{v^2/(8(k+1)w)}D_{n}\left(-\frac{v}{\sqrt{2(k+1)w}}\right)$$ $$- \sum_{n\ge0}\frac{(2w(k+1))^{n/2}}{n!}(-t)^n\zeta(-n)e^{v^2/(8(k+1)w)}D_{n}\left(\frac{v}{\sqrt{2(k+1)w}}\right).$$ \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:theorem2}] Inserting (2.8)--(2.9) (with $q$ replaced by $q^{2^k}$) into Lemma 2.2 gives us the following Bailey pair \begin{equation} \hat{\beta}_n(q,q)=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}P_{n,r_1,r_2,\dots,r_k}(q)q^{n-r_1+2(r_1-r_2)+\dots+2^{k-1}(r_{k-1}-r_k)},\end{equation} where $$P_{n,r_1,r_2,\dots,r_k}(q):=\frac{(-q^2;q)_{2r_1}(-q^{2^2};q^{2})_{2r_2}\dots(-q^{2^{k}};q^{2^{k-1}})_{2r_k}}{(q^2;q^2)_{n-r_1}(q^{2^2};q^{2^2})_{r_1-r_2}\dots(q^{2^{k}};q^{2^{k}})_{r_{k-1}-r_{k}}}\frac{(z;q^{2^k})_{r_k+1}(q^{2^k}/z;q^{2^k})_{r_k}}{(q^{2^k};q^{2^k})_{2r_k+1}},$$
\begin{equation}\hat{\alpha}_n(q,q)=(-z)^{-n}\frac{q^{2^{k}n(n+1)/2}(1-z^{2n+1})}{1-q^{2^{k}}},\end{equation} Inserting (2.18)--(2.19) into (2.13) gives \begin{equation} \sum_{n\ge0}(q)_n(-1)^nq^{n(n+1)/2}B_{n,k}(z,q)=\sum_{n\ge0}(-1)^n(-z)^{-n}q^{(2^{k}+1)n(n+1)/2}(1-z^{2n+1}).\end{equation} Putting $q=e^{-wt^2},$ and $z=e^{-vt-wt^2(2^{k}+1)/2},$ we have that (2.20) becomes \begin{equation} \begin{aligned} &\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}B_{n,k}(e^{-vt-wt^2(2^{k}+1)/2},e^{-wt^2})\\ &=\sum_{n\ge0}e^{nvt-wt^2(2^{k}+1)n^2/2}(1-e^{-vt(2n+1)-wt^2(2^{k}+1)(2n+1)/2})\\ &= \sum_{n\ge0}e^{nvt-wt^2(2^{k}+1)n^2/2}-\sum_{n\ge0}e^{-wt^2(2^{k}+1)(n+1)^2/2-vt(n+1)}\\ &= \sum_{n\ge0}e^{nvt-wt^2(2^{k}+1)n^2/2}-\sum_{n\ge1}e^{-wt^2(2^{k}+1)n^2/2-vtn} .\end{aligned}\end{equation}
Subtracting a $1$ from (2.21) and then taking the Mellin transform, we compute that $$\begin{aligned} &\mathfrak{M}\bigg\{ \sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_ne^{-wt^2n(n+1)/2}B_{n,k}(e^{-vt-wt^2(2^{k}+1)},e^{-wt^2})-1\bigg\}\\ &= \int_{0}^{\infty}t^{s-1}\left( \sum_{n\ge1}e^{nvt-wt^2(2^{k}+1)n^2/2}-\sum_{n\ge1}e^{-wt^2(2^{k}+1)n^2/2-vtn}\right)dt\\ &= \sum_{n\ge1}\int_{0}^{\infty}t^{s-1} \left(e^{nvt-wt^2(2^{k}+1)n^2/2} - e^{-wt^2(2^{k}+1)n^2/2-vtn}\right)dt\\ &=\sum_{n\ge1}\Bigg((w(2^{k}+1)n^2)^{-s/2}\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(-\frac{v}{\sqrt{(2^{k}+1)w}}\right)\\ &- (w(2^{k}+1)n^2)^{-s/2}\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(\frac{v}{\sqrt{2(2^{k}+1)w}}\right)\Bigg)\\ &=(w(2^{k}+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(-\frac{v}{\sqrt{(2^{k}+1)w}}\right)\\ &- (w(2^{k}+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(\frac{v}{\sqrt{2(2^{k}+1)w}}\right). \end{aligned}$$
Here we employed Lemma 2.4 with $v$ replaced by $nv,$ and $w$ replaced by $w(2^{k}+1)n^2/2,$ and again the resulting formula is analytic for $\Re(s)>1.$ By Mellin inversion, we compute for $\Re(s)=c>1,$ $$\begin{aligned} &-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}A_{n,k}(e^{-vt-wt^2(2^{k}+1)},e^{-wt^2})\\ &=\frac{1}{2\pi i}\int_{(c)}\bigg((w(2^{k}+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(-\frac{v}{\sqrt{(2^{k}+1)w}}\right)\\ &- (w(2^{k}+1))^{-s/2}\zeta(s)\Gamma(s)e^{v^2/(4(2^{k}+1)w)}D_{-s}\left(\frac{v}{\sqrt{(2^{k}+1)w}}\right)\bigg)t^{-s}ds.\end{aligned}$$ The integrand has a simple pole at $s=1$ and the negative integers $s=-n$ due to $\Gamma(s).$ We compute the residues at the negative integers to see that as $t\rightarrow0^{+},$ $$-1+\sum_{n\ge0}(e^{-wt^2};e^{-wt^2})_n(-1)^ne^{-wt^2n(n+1)/2}B_{n,k}(e^{-vt-wt^2(2^{k}+1)},e^{-wt^2})$$ $$\sim t^{-1}\sqrt{\frac{\pi}{4w(2^k+1)}}e^{v^2/(4(2^k+1)w)}\bigg(\erf\left(\frac{v}{2\sqrt{(2^k+1)w}}\right)-\erf\left(-\frac{v}{2\sqrt{(2^k+1)w}}\right)\bigg)$$ $$+\sum_{n\ge0}\frac{(w(2^{k}+1))^{n/2}}{n!}(-t)^n\zeta(-n)e^{v^2/(4(2^{k}+1)w)}D_{n}\left(-\frac{v}{\sqrt{(2^{k}+1)w}}\right)$$ $$- \sum_{n\ge0}\frac{(w(2^{k}+1))^{n/2}}{n!}(-t)^n\zeta(-n)e^{v^2/(4(2^{k}+1)w)}D_{n}\left(\frac{v}{\sqrt{(2^{k}+1)w}}\right).$$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:theorem3}] Since the proof is identical to the proof of our previous two theorems, we only outline some of the major details. Inserting the Bailey pair (2.8)--(2.9) into Lemma 2.3 and using (2.13) we obtain the identity \begin{equation} \sum_{n\ge0}\frac{(q)_n}{(-q)_{n}}(-1)^nq^{n(n+1)/2}C_{n,k}(z,q)\end{equation} $$=\sum_{n\ge0}z^{-n}q^{(k+2)n(n+1)/2}(1-z^{2n+1}),$$ where $$C_{n.k}(z,q)=\sum_{n\ge r_1\ge r_2\dots\ge r_k\ge0}\frac{q^{r_1^2/2+r_1/2+r_2^2/2+r_2/2+\dots+r_k^2/2+r_k/2}}{(q)_{n-r_1}(q)_{r_1-r_2}\dots(q)_{r_{k-1}-r_{k}}}\frac{(z)_{r_k+1}(q/z)_{r_k}}{(q)_{{r_k}}(q;q^2)_{r_k+1}}.$$ Putting $q=e^{-wt^2},$ and $z=e^{-vt-wt^2(k+2)/2},$ and proceeding as before the result follows. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:theorem4}] By Lemma 2.4, we have for $\Re(s)=c'>1,$ $\Re(w)>0,$ $$\sum_{n\ge1}\chi(n)e^{-w nt^2-v nt}=\frac{1}{2\pi i}\int_{(c')}(2w)^{-s/2}\Gamma(s)L(s,\chi)e^{v^2/(8w)}D_{-s}(\frac{v}{\sqrt{2w}})t^{-s}ds.$$ Since $L(s,\chi)$ is assumed to be nonprincipal, there is no pole at $s=1.$ Hence, computing the residues at the poles $s=-n$ from the gamma function, $$\sum_{n\ge1}\chi(n)e^{-w n^2t^2-v nt}\sim e^{v^2/(8w)}\sum_{n\ge0}\frac{(2w)^{n/2}(-t)^n}{n!}L(-n,\chi)D_{n}(\frac{v}{\sqrt{2w}}),$$ as $t\rightarrow0^{+}.$ After noting that if $\chi$ is even, then $L(-2n,\chi)=0,$ and if $\chi$ is odd, then $L(-2n-1,\chi)=0$ the result follows.
\end{proof} \section{Concluding Remarks}
Here we have of course limited ourselves with our choices of changing base of $q$ to three examples from [5]. Therefore, many more examples may be obtained by appealing to different Bailey chains from [5]. It would be desirable to obtain expansions for other $L$-functions, such as those contained in [3].
1390 Bumps River Rd. \\* Centerville, MA 02632 \\* USA \\* E-mail: alexpatk@hotmail.com, alexepatkowski@gmail.com
\end{document} |
\begin{document}
\begin{abstract} A special Danielewski surface is an affine surface which is the total space of a principal $(\C,+)$-bundle over an affine line with a multiple origin. Using a fiber product trick introduced by Danielewski, it is known that cylinders over two such surfaces are always isomorphic provided that both bases have the same number of origins. The goal of this note is to give an explicit method to find isomorphisms between cylinders over special Danielewski surfaces. The method is based on the construction of appropriate locally nilpotent derivations. \end{abstract}
\title{Isomorphisms between cylinders over Danielewski surfaces}
\section{Introduction}
In 1989, Danielewski exhibited a family of pairwise non-isomorphic complex affine rational surfaces $Y_n$, $n\geq1$, such that the cylinders $Y_n\times \A^1$ are all isomorphic. The surface $Y_n$ is defined to be the hypersurface in $\A^3$ defined by $x^ny=z^2-1$ for every positive integer $n$. Since this result, several authors have generalized Danielewski's construction and have introduced the notion of Danielewski surfaces. These are certain affine surfaces which can be realized as the total space of an $\A^1$-fibration over the affine line. Special Danielewski surfaces have the stronger property of being the total space of a principal $(\C,+)$-bundle over an affine line with a multiple origin. They were introduced in \cite {DuPo} and are those Danielewski surfaces for which Danielewski's original argument can be used to find isomorphic cylinders. However, the proof of these isomorphisms is not constructive.
The main result of this article is to give a method to find explicit isomorphisms of these cylinders. More precisely, the theorem~\ref{main-thm} produces, for every special Danielewski surface, an isomorphism between the cylinder over this surface and the cylinder over a classical Danielewski surface defined by an equation of the form $xy=P(z)$ in $\A^3$. This involves the construction of an appropriate $(\C,+)$-action on the cylinder of one surface whose quotient gives the other Danielewski surface.
As a corollary, one gets explicit embeddings of all special Danielewski surfaces as complete intersections in $\A^4$.
The paper is organized as follows.
In the section two, we recall the construction of Danielewski surfaces and some of their important properties, due to Fieseler and Dubouloz. Then in the following section we introduce three particular families of special Danielewski surfaces which are later used as examples for the main result in section 4. Two of these families are constructed as hypersurfaces, whereas for the last family, we do not know if they are realizable as hypersurfaces or not. In section four, we establish the theorem~\ref{main-thm}, which shows how to construct an isomorphism between the cylinders of any two special Danielewski surfaces. Finally, in section 5, we apply this result to the families of surfaces described in section 3. In particular, we obtain in the proposition~\ref{prop-classical-DS} a very simple explicit isomorphism between the cylinders of any two classical Danielewski surfaces whose respective equations are of the form $x^ny=P(z)$ and $x^my=Q(z)$.
{\bf Acknowledgments.} Part of this work was done during the first joint meeting Brazil-France in Mathematics. The second-named author gratefully acknowledges financial support from the R\'eseau Franco-Br\'esilien de Math\'ematiques (RFBM).
\section{Danielewski surfaces after Danielewski, Fieseler and Dubouloz}
In this section, we introduce some notations and summarize basic facts about Danielewski surfaces due to Fieseler \cite{Fi} and Dubouloz \cite{Du} (see also \cite{DuPo}).
\subsection{Construction of Danielewski surfaces}
\begin{definition} A Danielewski surface is a smooth complex affine surface $S$ equipped with an $\A^1$-fibration $\pi\colon S\to\A^1=\mathrm{Spec}(\C[x])$ that restricts to a trivial $\A^1$-bundle over $\A^1_*=\mathrm{Spec}(\C[x,x^{-1}])$ such that the exceptional fiber $\pi^{-1}(0)$ is reduced and consists of a disjoint union \[\pi^{-1}(0)=\coprod_{i=1}^d \ell_i\] of $d\geq2$ curves, $\ell_1,\ldots,\ell_d$, all isomorphic to the affine line. \end{definition}
For every $1\leq i\leq d$, we denote by $\Ucal_i\subset S$ the open subvariety of $S$ defined by \[\Ucal_i=S\smallsetminus\coprod_{j\neq i}\ell_j\subset S.\] Since every $\Ucal_i$ is isomorphic to the affine plane $\A^2$, every Danielewski surface can be constructed by gluing together $d\geq2$ copies of $\A^2$ along $\A^1_*\times\A^1$. More precisely, every Danielewski surface is isomorphic to a variety $S(d,\boldsymbol{\sigma})$ defined as follows.
\begin{definition} Let $d\geq2$ be an integer and let \[\boldsymbol{\sigma}=\big((n_1,\sigma_1(x)),\ldots,(n_d,\sigma_d(x))\big)\in(\Z_{>0}\times\C[x])^d\] be a sequence such that the polynomials $\sigma_i$ are distinct and satisfy that $\deg(\sigma_i(x))<n_i$ for all $1\leq i\leq d$. We denote by $S(d,\boldsymbol{\sigma})$ the surface obtained by gluing together $d$ copies $\Ucal_i=\mathrm{Spec}(\C[x,u_i])$ of $\A^2$ along the open subsets \[\Ucal_i^*=\mathrm{Spec}(\C[x,x^{-1},u_i])\simeq\C^*\times\C\] via the transition functions \begin{align*} \Ucal_i^* &\to\Ucal_j^* \\ (x,u_i)&\mapsto (x,x^{n_i-n_j}u_i+\frac{\sigma_i(x)-\sigma_j(x)}{x^{n_j}}). \end{align*} \end{definition} By \cite[Proposition 1.4]{Fi}, every such surface $S=S(d,\boldsymbol{\sigma})$ is affine. Moreover, the inclusion $\C[x]\hookrightarrow\C[S]$ defines an $\A^1$-fibration $\pi\colon S\to\A^1$ such that $\pi^{-1}(\A^1_*)\simeq\A^1_*\times\A^1$ and such that the unique special fiber $\pi^{-1}(0)$ consists of a disjoint union of $d$ reduced copies of $\A^1$. This shows that $S(d,\boldsymbol{\sigma})$ is indeed a Danielewski surface.
By construction, every Danielewski surface $S=S(d,\boldsymbol{\sigma})$ is canonically equipped with a regular function $u\in\C[S]$ whose restrictions to each of the open subsets $\Ucal_i$ are given by
\[u|_{\Ucal_i} = x^{n_i} u_i +\sigma_i(x) \in \C[x, u_i].\] Note that $u$ restricts to a coordinate function on every general fiber of $\pi=\textrm{pr}_x\colon S\to\A^1$, but not on the exceptional fiber $\pi^{-1}(0)$.
\subsection{Additive group actions and isomorphic cylinders.}
Every Danielewski surface $S=S(d,\boldsymbol{\sigma})$ is canonically equipped with a regular $(\C,+)$-action $\delta:\C\times S\to S$ defined on each chart $\Ucal_i$ by \[\delta(\lambda,(x,u_i))=(x,u_i+\lambda x^{n-n_i}),\] where $n=\max\{n_i\mid 1\leq i\leq d\}$. Algebraically, the action $\delta$ corresponds to the locally nilpotent derivation $D\in\textrm{LND}(\C[S])$ that is defined by $D(x)=0$ and $D(u_i)=x^{n-n_i}$. Note that $D(u)=x^n$.
An important property of Danielewski surfaces is the fact that the map $\pi=\textrm{pr}_x\colon S\to\A^1$ factors through a locally trivial fiber bundle $S\to Z(d)$ over the affine line $Z(d)$ with a $d$-fold origin, where the preimages of the $d$ origins are the $d$ affine lines $\ell_i$. In the case when the $(\C,+)$-action $\delta$ is free, we have moreover that $S$ is the total space of a $(\C,+)$-principal bundle over $Z(d)$. Recall (see \cite[Section 2.10]{DuPo}) that $\delta$ is free if and only if all $n_i$ are equal to each other, i.e.~ if and only if $n_i=n$ for all $1\leq i\leq d$. The latter condition is equivalent to the fact that the canonical class of $S$ is trivial. These Danielewski surfaces were called \emph{special} in \cite{DuPo}.
Danielewski's fiber product trick goes then as follows. Take two Danielewski surfaces, say $S$ and $S'$, that are $(\C,+)$-principal bundles over the same $Z(d)$ and consider their fiber product $S\times_{Z(d)}S'$. Since every $(\C,+)$-principal bundle over an affine base is trivial, we get at once that \[S\times\A^1\simeq S\times_{Z(d)}S'\simeq S'\times\A^1,\] hence that the cylinders over $S$ and $S'$ are isomorphic to each other.
\section{Examples of special Danielewski surfaces}
\subsection{Classical Danielewski surfaces.} These surfaces are the ones originally considered by Danielewski. They are defined as the hypersurfaces $W_{n,P}$ in $\A^3$ of equation \[W_{n,P}\colon x^ny=P(z),\] where $n\geq1$ is a positive integer and where $P(z)=\prod_{i=1}^d(z-r_i)\in\C[z]$ is a polynomial with $d\geq2$ simple roots.
Together with the restriction of the first projection $\pi=\textrm{pr}_x\colon W_{n,P}\to \A^1$, every such surface defines a Danielewski surface. The special fiber $\pi^{-1}(0)$ is the union of the lines $\ell_1,\ldots,\ell_d$ given by \[\A^1\simeq\ell_i=\{(0,y,r_i)\mid y\in\C\}\subset W_{n,P}.\]
Every open set $\Ucal_i=W_{n,P}\smallsetminus\coprod_{j\neq i}\ell_j$ is isomorphic to $\A^2$ and we have the isomorphisms \[\varphi_i\colon \Ucal_i\xrightarrow{\sim}\A^2, (x,y,z)\mapsto(x,u_i), \text{ where } u_i=\frac{z-r_i}{x^n}=\frac{y}{\prod_{j\neq i}(z-r_j)}\in\C[\Ucal_i].\]
\subsection{Danielewski hypersurfaces.} The hypersurfaces in $\A^3$ that are defined by an equation of the form \[H_{n,Q}\colon x^ny=Q(x,z),\] where $n\geq1$ and where $Q(x,z)\in\C[x,z]$ is such that $\deg(Q(0,z))\geq2$ are called \emph{Danielewski hypersurfaces}. If moreover the polynomial $Q(0,z)\in\C[z]$ has $d\geq2$ simple roots, say $r_1,\ldots,r_d$, then $\pi=\text{pr}_x\colon H_{n,Q}\to\A^1$ defines a Danielewski surface. Its special fiber is the union of the lines $\ell_1,\ldots,\ell_d$ given by \[\A^1\simeq\ell_i=\{(0,y,r_i)\mid y\in\C\}\subset H_{n,Q}.\]
Furthermore, there exist unique polynomials $\sigma_1(x),\ldots,\sigma_d(x)\in\C[x]$ of degree strictly smaller than $n$ such that $\sigma_i(0)=r_i$ and such that the congruences \[Q(x,\sigma_i(x))\equiv 0 \mod(x^n)\] hold for all $1\leq i\leq d$. Then, every open set $\Ucal_i=H_{n,Q}\smallsetminus\coprod_{j\neq i}\ell_j$ is isomorphic to $\A^2$ and we have the isomorphisms \[\varphi_i\colon \Ucal_i\xrightarrow{\sim}\A^2, (x,y,z)\mapsto(x,u_i), \text{ where } u_i=\frac{z-\sigma_i(x)}{x^n}.\] (See \cite{DuPo} for the details.)
\subsection{Iterated Danielewski hypersurfaces}\label{Section:construction-iterated-Danielewski}
Introduced by Alhajjar \cite{Al}, iterated Danielewski hypersurfaces are the hypersurfaces in $\A^3$ that are defined by an equation of the form \[H_{n,Q,m,R}\colon x^mz=R(x,x^ny-Q(x,z)),\] where $n,m\geq1$ and where $Q(x,t),R(x,t)\in\C[x,t]$. If the polynomial $R(0,-Q(0,t))$ in $\C[t]$ has only $d\geq2$ simple roots, then $H_{n,Q,m,R}$ is a Danielewski surface.
We discuss now a specific example in details. Consider the surface $H\subset\A^3$ defined by \[H\colon \{xz=(xy+z^2)^2-1\}.\] The special fiber of $\pi=\textrm{pr}_x\colon H\to\A^1$ consists of the four lines $\ell_1,\ldots,\ell_4$ given by \[\ell_i=\{(0,y,\varepsilon^i)\mid y\in\C\},\] where $\varepsilon=\boldsymbol{i}$ denotes a primitive fourth root of the unity.
Letting $u=xy+z^2$ and $\sigma_i(x)=\varepsilon^{2i}+\frac{\varepsilon^{-i}}{2}x$ for all $1\leq i\leq 4$, it follows that the open set $\Ucal_i=H\smallsetminus\coprod_{j\neq i}\ell_j$ is isomorphic to $\A^2$ and one claims that the map \[\varphi_i\colon \Ucal_i\xrightarrow{\sim}\A^2, (x,y,z)\mapsto(x,u_i), \text{ where } u_i=\frac{u-\sigma_i(x)}{x^2}\] is an isomorphism. \begin{proof} First, we remark that the rational functions \[\alpha_i=\frac{u-\varepsilon^{2i}}{x}=\frac{z}{u+\varepsilon^{2i}}\quad \text{ and }\quad \beta_i=\frac{z-\varepsilon^i}{x}=\frac{z-xy^2-2yz^2}{\prod_{j\in\{1,\ldots,4\}\smallsetminus\{i\}}(z-\varepsilon^j)}\] are regular on $\Ucal_i$. It follows that $u_i$ is also an element of $\C[\Ucal_i]$, since one easily checks that \[\beta_i-\left(\alpha_i\right)^2=2\varepsilon^{2i}u_i.\] Finally, the fact that $\varphi_i$ is an isomorphism follows from the following identities in $\C[\Ucal_i]$. \begin{align*} \alpha_i &= xu_i+\frac{\varepsilon^{-i}}{2}\\ \beta_i &=2\varepsilon^{2i}u_i+\left(\alpha_i\right)^2\\ z&=\varepsilon^i+x\beta_i\\ y&= \frac{u-z^2}{x}=\frac{u-(\varepsilon^i+x\beta_i)^2}{x}=\alpha_i-2\varepsilon^i\beta_i-x(\beta_i)^2. \end{align*} \end{proof}
\subsection{Double Danielewski surfaces}\label{Section:construction-double-Danielewski}
In \cite{GuSe}, Gupta and Sen studied some surfaces defined by two equations in $\A^4$ of the form \[S\colon \{x^ny=Q(x,z) \text{ and } x^mt=R(x,z,y)\},\] where $n,m\geq1$ and where $Q(x,z)\in\C[x,z]$ and $R(x,z,y)\in\C[x,z,y]$. They call them \emph{double Danielewski surfaces}. Indeed, if $Q(0,z)\in\C[z]$ has $d\geq2$ simple roots, say $r_1,\ldots,r_{d}$, and if every polynomial $R(0,r_i,y)\in\C[y]$ also has only simple roots, then $S$ is a Danielewski surface together with the first projection $\textrm{pr}_x\colon S\to\A^1$.
Let us study here a specific example in details, namely the surface $D\subset\A^4$ defined by \[D\colon \{xy=z^2-1 \text{ and } xt=y^2-1\}.\]
It is a Danielewski surface, its special fiber $\textrm{pr}_x^{-1}(0)$ consisting of the four lines given by $\{(0,\pm1,\pm1,t)\mid t\in\C\}\subset D$. Let us introduce the following notation. For every pair $(i,j)\in\{-1,1\}\times\{-1,1\}$, we let \[\ell_{i j}=\{(0,i,j,t)\mid t\in\C\}\] and \[\sigma_{i j}(x)=j+\frac{ij}{2}x.\] Then, every open set $\Ucal_{i j}=D\smallsetminus\coprod_{(i',j')\neq (i,j)}\ell_{i' j'}$ is isomorphic to $\A^2$ and one claims that the map \[\varphi_{i j}\colon \Ucal_{i j}\xrightarrow{\sim}\A^2, (x,y,z,t)\mapsto(x,u_{i j}) \text{ where } u_{i j}=\frac{z-\sigma_{i j}(x)}{x^2}\] is an isomorphism.
\begin{proof} First, remark that $\alpha_j=\frac{z-j}{x}=\frac{y}{z+j}$ and $\beta_i=\frac{y-i}{x}=\frac{t}{y+i}$ are regular functions on $\Ucal_{i j}$. Hence, it is straightforward to check that $u_{i j}$ is a regular function on $\Ucal_{i j}$, since \[\beta_i-\left(\alpha_j\right)^2=2ju_{i j}.\] The fact that $\varphi_{i j}$ is an isomorphism follows from the following identities in $\C[\Ucal_{i j}]$. \begin{align*} \alpha_j &= xu_{i j}+\frac{ij}{2}\\ \beta_i &=2ju_{i j}+\left(\alpha_j\right)^2\\ z&=j+x\alpha_j\\ y&=i+x\beta_i\\ t&= (y+i)\beta_i. \end{align*} \end{proof}
\section{Isomorphisms between cylinders}\label{Section:plan of the construction}
In this section, we fix an integer $d\geq 2$ and denote by $Z(d)$ the affine line with $d$ origins. Let
$P(z)=\prod_{i=1}^d(z-r_i)\in\C[z]$ be a polynomial with simple roots. We will explain how to construct, given a special Danielewski surface $S$ which is a principal bundle over $Z(d)$, an isomorphism between its cylinder $S\times\A^1$ and the cylinder $W\times\A^1$ over the classical Danielewski surface \[W=W_{1,P}\colon \{xy=P(z)=\prod_{i=1}^d(z-r_i)\} \text { in } \A^3.\]
Recall that a special Danielewski surface $S$ is constructed from a data set consisting of a positive integer $n\geq1$ and of distinct polynomial $\sigma_1(x),\ldots,\sigma_d(x)\in\C[x]$ of degree strictly smaller than $n$. More precisely, $S$ is obtained by gluing $d$ copies, $\Ucal_1,\ldots,\Ucal_d$, of $\A^2=\mathrm{Spec}(\C[x,u_i])$ along $\A^1_*\times\A^1$ by means of the transition functions \[(x,u_i)\mapsto (x,u_i+\frac{\sigma_i(x)-\sigma_j(x)}{x^{n}}).\]
We also recall that the inclusion $\C[x]\hookrightarrow S$ defines an $\A^1$-fibration $\pi\colon S\to\A^1$ with a unique special fiber $\pi^{-1}(0)=\coprod_{i=1}^d \ell_i$ consisting of $d$ disjoint reduced copies of $\A^1$, and that we can define, by considering the regular function $u\in\C[S]$ whose restrictions on the open sets $\Ucal_i$ are given by \[u|_{\Ucal_i} = x^{n} u_i +\sigma_i(x) \in \C[x, u_i],\]
the canonical locally nilpotent derivation $D\in\mathrm{LND}(\C[S])$ by setting $D(x)=0$ and $D(u)=x^n$.
Following Danielewski's original argument, we consider the fiber product $S\times_{Z(d)}W$, which we denote by $V$. We will use the following notations. We identify the ring of regular functions on $W$ with its canonical image in the ring of regular functions on $V$ and write \[\C[W]=\C[x,y,z]\subset\C[V], \quad\text{ where } xy=P(z).\]
Similarly, we identify $\C[S]$ as a subring of $\C[V]$. Then, $V$ can be naturally seen as being obtained by gluing $d$ copies $\Vcal_i=\mathrm{Spec}[x,u_i,z_i]$ of $\A^3$ where $z_i=(z-r_i)/x$. The open subvarieties $\Vcal_i$ are glued together along $\A^1_*\times\A^2$ via the transition functions \[(x,u_i,z_i)\mapsto(x,u_i+\frac{\sigma_i(x)-\sigma_j(x)}{x^{n}},z_i+\frac{r_i-r_j}{x}).\] In particular, we have that the regular functions $u\in\C[S]\subset\C[V]$ and $z\in\C[W]\subset\C[V]$ satisfy that
\[u|_{\Vcal_i} = x^{n} u_i +\sigma_i(x) \in \C[\Vcal_i]=\C[x,u_i,z_i]\] and
\[z|_{\Vcal_i} = x z_i +r_i \in \C[\Vcal_i]=\C[x,u_i,z_i]\] for all $1\leq i\leq d$.
{\bf Plan of the construction.} Our construction of an isomorphism between $S\times\A^1$ and $W\times\A^1$ consists of three steps. We first find a regular function $\alpha\in \C[V]$ on the fiber product $V=S\times_{Z(d)}W$ such that \[\C[V]=\C[S][\alpha]\simeq\C[S\times\A^1].\] We then use this equality to extend the canonical derivation on $\C[S]$ to a locally nilpotent derivation $\tilde{D}$ on $\C[V]$ in such a way that \[\mathrm{Ker}(\tilde{D})=\C[x,y,z]=\C[W]\subset\C[V].\] Finally, in the last step, we construct an element $s\in\C[V]$ which is a slice for $\tilde{D}$. This gives \[\C[S\times\A^1]\simeq\C[S][\alpha]=\C[V]=\mathrm{Ker}[\tilde{D}][s]=\C[W][s]\simeq\C[W\times\A^1]\] hence the desired isomorphism between $S\times\A^1$ and $W\times\A^1$.
{\bf Step 1.} Since $\ell_1,\ldots,\ell_d$ are disjoint closed subvarieties of the affine variety $S$, there exists a regular function $f\in\C[S]$ such that \begin{equation}\tag{$\star$}\label{equa*}
f|_{\ell_i} = r_i \quad\text{for all } 1\leq i\leq d. \end{equation} In other words, we can choose a function $f\in\C[V]$ such that
\[f|_{\Vcal_i}=r_i+x\tilde{f}_i \quad \text{for all } 1\leq i\leq d,\] where $\tilde{f}_i$ is some element in $\C[x,u_i]\subset\C[\Vcal_i]$. Therefore, the rational function \[\alpha=\frac{z-f}{x}\] is in fact a regular function on $V$, since
\[(z-f)|_{\Vcal_i}=xz_i+r_i-r_i-x\tilde{f}_i=x(z_i-\tilde{f}_i)\] is divisible by $x$ for all $i$.
It is then straightforward to check that $z=f+x\alpha$ and $y=x^{-1}P(f+x\alpha)$ are both elements of $\C[S][\alpha]$, hence \[\C[V]=\C[S][\alpha].\]
{\bf Step 2.} Note that the image $D(f)\in\C[S]$ of $f$ under the derivation $D$ is divisible by $x$. Therefore, we can extend $D$ to a locally nilpotent derivation $\tilde{D}$ on $\C[V]=\C[S][\alpha]$ by letting \[\tilde{D}(\alpha)=-\frac{D(f)}{x}.\] With this choice, we then have that $\tilde{D}(z)=\tilde{D}(f+x\alpha)=0$ and that $\tilde{D}(y)=\tilde{D}(\frac{P(z)}{x})=0$.
{\bf Step 3.} To find a slice $s$ for $\tilde{D}$, it suffices to take a polynomial $g(x,t)\in\C[x,t]$ such that the congruences
\begin{equation}\tag{$\star\star$}\label{equa**} g(x,r_i+xt)\equiv \sigma_i(x) \mod(x^n) \end{equation} hold in $\C[x,t]$ for all $1\leq i\leq d$, and to define \[s=\frac{u-g(x,z)}{x^n}.\] Indeed, since every restriction
\[(u-g(x,z))|_{\Vcal_i}=x^nu_i+\sigma_i(x)-g(x,r_i+xz_i)\] is divisible by $x^n$ in $\C[\Vcal_i]$, it follows that $s$ is a regular function on $V$. Moreover, it is clear that $\tilde{D}(s)=1$.
In order to construct a suitable polynomial $g(x,t)$, one can proceed as follows. If we denote \[\sigma_i(x)=\sum_{j=0}^{n-1}a_{ij}x^j\quad \text{ with } a_{ij}\in\C,\] then we can define \[g(x,t)=\sum_{j=0}^{n-1}g_j(t)x^j,\] where the $g_j(t)\in\C[t]$ are Hermite interpolation polynomials such that \[g_j(r_i)=a_{ij}\quad\text{ and }\quad g_j^{(k)}(r_i)=0\] for all $1\leq i\leq d$ and all $0\leq j\leq n-1$, $1\leq k\leq n-1-j$.
{\bf The isomorphism.} Finally, the above three steps have produced the desired isomorphism. We have therefore proven the following result. \begin{theorem}\label{main-thm} Let $S$ be a special Danielewski surface over $Z(d)$, and let $W$ be the hypersurface defined by the equation $XY=P(Z)$, where $P$ is a polynomial of degree $d$ whose roots are all simple. Suppose $f$ and $g$ are chosen to satisfy $($\ref{equa*}$)$ and $($\ref{equa**}$)$ above. Then the map \[\Phi\colon\C[W\times\A^1]=\C[X,Y,Z,W]/(XY-P(Z))=\C[x,y,z,w]\xrightarrow{\sim}\C[S\times\A^1]=\C[S][\alpha]\] defined by \begin{align*} \Phi(x)&=x\\ \Phi(z)&=f+x\alpha\\ \Phi(y)&=\frac{P(f+x\alpha)}{x}\\ \Phi(w)&=\frac{u-g(x,f+x\alpha)}{x^n} \end{align*} is an isomorphism. \end{theorem}
\begin{corollary} Keeping the same notation as in the previous theorem, it follows that the special Danielewski surface $S$ is isomorphic to the surface defined by the equations \[xy=P(z) \text{ and } \Phi^{-1}(\alpha)=\lambda\] in $\A^4$, where $\lambda\in\C$ is any constant. \end{corollary}
\section{Some explicit examples}
\subsection{Russell's isomorphism}
In \cite{SY}, the authors give an explicit isomorphism, which is due to Russell, between the cylinders over the Danielewski surfaces of respective equations $xy=z^2-1$ and $x^2y=z^2-1$. See also Theorem 10.1 in \cite{Fre}. With our method, we can recover this isomorphism easily.
In this section we will show how to apply the method of the previous section to treat a slightly more general case, and, in the end of the section, we will specialize to the case of the Russell isomorphism. We shall use the following notations. We denote by $P(z)=\prod_{i=1}^d(z-r_i)\in\C[z]$ a polynomial with $d\geq2$ simple roots and by $W_{n,P}$ the hypersurface in $\A^3$ defined by the equation $x^ny=P(z)$, where $n\geq1$ is a positive integer. Moreover, we let \[\C[W_{n,P}]=\C[X,Y,Z]/(X^nY-P(Z))=\C[x_n,y_n,z_n]\] and \[\C[W_{n,P}\times\A^1]=\C[X,Y,Z,W]/(X^nY-P(Z))=\C[x_n,y_n,z_n,w_n],\]
where $(x_n)^ny_n=P(z_n)$.
Accordingly with the previous section, we now proceed to construct an isomorphism between $\C[W_{1,P}\times\A^1]$ and $\C[W_{2,P}\times\A^1]$.
First, note that the regular function $f=z_2\in\C[W_{2,P}]$ is equal to $r_i$ on every point of the line $\{x=0, z=r_i\}\subset W_{2,P}$. Also, in this case, $u=z_2\in\C[W_{2,P}]$ restricts to a coordinate function on every general fiber of the projection $\textrm{pr}_{x}\colon W_{2,P}\to\C$.
Secondly, since $P$ has only simple roots, there exist two polynomials $U,V\in\C[z]$ such that $U(z)P'(z)+V(z)P(z)=1$ in $\C[z]$. Then, the polynomial \[g(z)=z-P(z)U(z)\in\C[z]\] satisfies that \[g(r_i)=r_i\] and \[g'(r_i)=1-P'(r_i)U(r_i)-P(r_i)U'(r_i)=0\] for all $1\leq i\leq d$. With these choices for $f$, $u$ and $g$, we get the isomorphism \[\Phi\colon\C[W_{1,P}\times\A^1]=\C[x_1,y_1,z_1,w_1]\xrightarrow{\sim}\C[W_{2,P}\times\A^1]=\C[x_2,y_2,z_2,w_2]\] defined by \begin{align*} \Phi(x_1)&=x_2\\ \Phi(z_1)&=z_2+x_2w_2\\ \Phi(y_1)&=\frac{P(z_2+x_2w_2)}{x_2}\\ \Phi(w_1)&=\frac{z_2-g(z_2+x_2w_2)}{x_2^2}, \end{align*} whose inverse isomorphism \[\Psi\colon\C[W_{2,P}\times\A^1]=\C[x_2,y_2,z_2,w_2]\xrightarrow{\sim}\C[W_{1,P}\times\A^1]=\C[x_1,y_1,z_1,w_1]\] is defined by \begin{align*} \Psi(x_2)&=x_1\\ \Psi(z_2)&=x_1^2w_1+g(z_1)\\ \Psi(y_2)&=\frac{P(x_1^2w_1+g(z_1))}{x_1^2}\\ \Psi(w_2)&=\frac{z_1-(x_1^2w_1+g(z_1))}{x_1}. \end{align*}
In the special case where $P(z)=z^2-1$, we have $g(z)=z-(z^2-1)\dfrac{z}{2}$ and we thus obtain the inverse isomorphisms \[\Phi_*\colon \{x^2y=z^2-1\}\times\A^1\to \{xy=z^2-1\}\times\A^1\] and \[\Psi_*\colon \{xy=z^2-1\}\times\A^1\to \{x^2y=z^2-1\}\times\A^1\] defined by \begin{align*} \Phi_*(x,y,z,w)&=\Big(x,\frac{P(z+xw)}{x},z+xw,\frac{z-g(z+xw)}{x^2}\Big)\\ &=\Big(x,\frac{(z+xw)^2-1}{x},z+xw,\\ &\qquad\qquad\frac{z-(z+xw)+\frac{1}{2}(z+xw)((z+xw)^2-1)}{x^2}\Big)\\ &=\Big(x,\frac{z^2-1}{x}+2zw+xw^2,z+xw,\\ &\qquad\qquad\frac{\frac{1}{2}(z+xw)(z^2-1+x^2w^2)+xw(z^2-1)+x^2w^2z}{x^2}\Big)\\ &=\Big(x,xy+2zw+xw^2,z+xw,\frac{1}{2}(z+xw)(y+w^2)+xyw+w^2z\Big)\\ &=\Big(x,xy+2zw+xw^2,z+xw,\frac{1}{2}(yz+3zw^2+3xyw+xw^3)\Big) \end{align*} and \begin{align*} \Psi_*(x,y,z,w)&=\Big(x,\frac{P(x^2w+g(z))}{x^2},x^2w+g(z),\frac{z-x^2w-g(z)}{x}\Big)\\ &=\Big(x,\frac{x^4w^2+2x^2wg(z)+(z^2-1)^2(\frac{1}{4}z^2-1)}{x^2},x^2w+g(z),\\ &\qquad\qquad-xw+\frac{1}{2}\cdot\frac{z(z^2-1)}{x}\Big)\\ &=\Big(x,x^2w^2+2wg(z)+y^2(\frac{1}{4}z^2-1),x^2w+g(z),-xw+\frac{1}{2}zy\Big). \end{align*}
\subsection{Classical Danielewski surfaces}
In light of the previous example, we obtain simple explicit isomorphisms between the cylinders over two classical Danielewski surfaces.
\begin{proposition}\label{prop-classical-DS} Let $d,n,m\geq1$ be positive integers and let $P(z)=\prod_{i=1}^d(z-a_i)$ and $Q(z)=\prod_{i=1}^d(z-b_i)$ be polynomials in $\C[z]$ with simple roots. Recall that $W_{n,P}$ and $W_{m,Q}$ denote the hypersurfaces in $\A^3=\mathrm{Spec}(\C[x,y,z])$ that are defined respectively by the equation \[W_{n,P}\colon x^ny=P(z)\] and \[W_{m,Q}\colon x^my=Q(z).\]
Let $f,g\in\C[z]$ be two Hermite interpolating polynomials such that \[f(b_i)=a_i\quad\text{ and }\quad f^{(k)}(b_i)=0\quad \text{ for all } 1\leq i\leq d, 1\leq k\leq n-1\] and \[g(a_i)=b_i\quad\text{ and }\quad g^{(k)}(a_i)=0\quad \text{ for all } 1\leq i\leq d, 1\leq k\leq m-1.\] Then, the maps \begin{align*} \varphi\colon &W_{n,P}\times\A^1\to W_{m,Q}\times\A^1\\ &(x,y,z,w)\mapsto(x,\frac{Q(g(z)+x^mw)}{x^m},g(z)+x^mw,\frac{z-f(g(z)+x^mw)}{x^n}) \end{align*} and \begin{align*} \psi\colon &W_{m,Q}\times\A^1\to W_{n,P}\times\A^1\\ &(x,y,z,w)\mapsto(x,\frac{P(f(z)+x^nw)}{x^n},f(z)+x^nw,\frac{z-g(f(z)+x^nw)}{x^m}) \end{align*} are regular and define inverse isomorphisms between the cylinders $W_{n,P}\times\A^1$ and $W_{m,Q}\times\A^1$. \end{proposition}
\begin{proof} On the one hand, we have that \[(P\circ f)(b_i)=(P\circ f)'(b_i)=\cdots=(P\circ f)^{(n-1)}(b_i)=0\quad\text{ for all } i.\] This shows that $P(f(z))$ is divisible by $(Q(z))^n$, hence that $P(f(z))/x^n$ is a regular function on $W_{m,Q}$. Similarly, $Q(g(z))/x^m$ is a regular function on $W_{n,P}$.
On the other hand, we have that \[z-g(f(z)+x^nw)=z-g(f(z))-\sum_{k=1}^{\infty}\frac{(x^nw)^k}{k!}g^{(k)}(f(z))\] is an element of the ideal $(Q(z),x^m)\C[x,z,w]$. Therefore, $x^{-m}(z-g(f(z)+x^nw))$ is a regular function on $W_{m,Q}\times\A^1=\mathrm{Spec}(\C[x,y,z,w]/(x^my-Q(z)))$. Similarly, $x^{-n}(z-f(g(z)+x^mw))$ is a regular function on $W_{n,P}\times\A^1$.
Thus, $\varphi$ and $\psi$ are regular maps. It is moreover straightforward to check that they are inverse of each other. \end{proof}
\subsection{An iterated Danielewski hypersurface}
Let us look again at the iterated Danielewski hypersurface $H=\{xz=(xy+z^2)^2-1\}$ in $\A^3$ that we studied at Section \ref{Section:construction-iterated-Danielewski}. Recall that the special fiber consists of the four lines \[\ell_i=\{(0,y,\varepsilon^i)\mid y\in\C\}, 1\leq i\leq 4,\] where $\varepsilon=\boldsymbol{i}\in\C$ denotes a primitive fourth root of the unity, and that the surface $H$ corresponds to the data $n=2$ and $\sigma_i(x)=\varepsilon^{2i}+\frac{\varepsilon^{-i}}{2}x$ for all $1\leq i\leq 4$.
We give now an isomorphism from $H\times\A^1$ to $\{xy=z^4-1\}\times\A^1$. Keeping the notations of Section~\ref{Section:plan of the construction}, we obtain the isomorphism \begin{align*} \{xz=(xy+z^2)^2-1\}\times\A^1&\to \{xy=z^4-1\}\times\A^1\\ (x,y,z,w)&\mapsto(x,\frac{(f+xw)^4-1}{x},f+xw,\frac{u-g(x,f+xw)}{x^2}), \end{align*} where \begin{align*} r_i&=\varepsilon^i\\ f&=z\\ u&=xy+z^2\\ g(x,z)&=z^2-\frac{1}{2}z^2(z^4-1)+x\frac{1}{2}z^3. \end{align*}
\subsection{A double Danielewski surface} We consider again the double Danielewski surface \[D= \{xy=z^2-1 \text{ and } xt=y^2-1\}\quad \text{ in } \A^4\] that we described at Section \ref{Section:construction-double-Danielewski}. Recall that the special fiber consists of the four lines \[\ell_{i j}=\{(0,i,j,t)\mid t\in\C\}\}\] and that the surface $D$ corresponds to the data $n=2$ and $\sigma_{i j}(x)=j+\frac{ij}{2}x$, where $(i,j)\in\{1,-1\}\times\{1,-1\}$.
Following the notations of Section~\ref{Section:plan of the construction}, we denote by $\varepsilon=\boldsymbol{i}\in\C$ a primitive fourth root of unity and define \begin{align*} f&=\frac{z+y}{2}+\varepsilon\frac{y-z}{2}\\ u&=z\\ g(x,z)&=\frac{1-\varepsilon}{2}z^3+\frac{1+\varepsilon}{2}z-(z^4-1)\frac{z}{4}(3\frac{1-\varepsilon}{2}z^2+\frac{1+\varepsilon}{2})+x\frac{1}{2}z^2. \end{align*} Then, we have that \begin{align*}
&f|_{\ell_{11}}=1, && g(x,1)=\sigma_{1 1}(x)\\
&f|_{\ell_{1 -1}}=\varepsilon, && g(x,\varepsilon)=\sigma_{1 -1}(x)\\
&f|_{\ell_{-1 1}}=-\varepsilon, && g(x,-\varepsilon)=\sigma_{-1 1}(x)\\
&f|_{\ell_{-1 -1}}=-1, && g(x,-1)=\sigma_{-1 -1}(x) \end{align*} and \[\frac{\partial g}{\partial z}(x,\varepsilon^i)\equiv0 \mod (x)\] for all $1\leq i\leq 4$. This produces the isomorphism \begin{align*} D\times\A^1&\to \{xy=z^4-1\}\times\A^1\\ (x,y,z,t,w)&\mapsto(x,\frac{(f+xw)^4-1}{x},f+xw,\frac{u-g(x,f+xw)}{x^2}). \end{align*}
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\begin{document}
\author{David Joyner\thanks{Dept Math, US Naval Academy, Annapolis, MD, wdj@usna.edu.}} \title{A primer on computational group homology and
cohomology using {\tt GAP} and \SAGE\thanks{Dedecated to my friend and colleague Tony Gaglione on the occasion of his $60^{th}$ birthday.}}
\maketitle
\vskip .4in
These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology. They were written for someone who has had a first course in graduate algebra but no background in cohomology. You should know the definition of a (left) module over a (non-commutative) ring, what $\mathbb{Z}[G]$ is (where $G$ is a group written multiplicatively and $\mathbb{Z}$ denotes the integers), and some ring theory and group theory. However, an attempt has been made to (a) keep the presentation as simple as possible, (b) either provide an explicit reference or proof of everything.
Several computer algebra packages are used to illustrate the computations, though for various reasons we have focused on the free, open source packages, such as {\tt GAP} \cite{Gap} and \SAGE \cite{St} (which includes {\tt GAP}). In particular, Graham Ellis generously allowed extensive use of his HAP \cite{Ehap} documentation (which is sometimes copied almost verbatim) in the presentation below. Some interesting work not included in this (incomplete) survey is (for example) that of Marcus Bishop \cite{Bi}, Jon Carlson \cite{C} (in MAGMA), David Green \cite{Gr} (in C), Pierre Guillot \cite{Gu} (in GAP, C++, and \SAGE), and Marc R\"oder \cite{Ro}.
Though Graham Ellis' {\tt HAP} package (and Marc R\"oder's add-on {\tt HAPcryst} \cite{Ro}) can compute comhomology and homology of some infinite groups, the computational examples given below are for finite groups only.
\section{Introduction}
First, some words of motivation.
Let $G$ be a group and $A$ a $G$-module\footnote{We call an abelian group $A$ (written additively) which is a left $\mathbb{Z}[G]$-module a {\bf $G$-module}. }. \index{$G$-module}
Let $A^G$ denote the largest submodule of $A$ on which $G$ acts trivially. Let us begin by asking ourselves the following natural question.
{\bf Question}: Suppose $A$ is a submodule of a $G$-module $B$ and $x$ is an arbitrary $G$-fixed element of $B/A$. Is there an element $b$ in $B$, also fixed by $G$, which maps onto $x$ under the quotient map?
The answer to this question can be formulated in terms of group cohomology. (``Yes'', if $H^1(G,A)=0$.) The details, given below, will help motivate the introduction of group cohomology.
Let $A_G$ is the largest quotient module of $A$ on which $G$ acts trivially. Next, we ask ourselves the following analogous question.
{\bf Question}: Suppose $A$ is a submodule of a $G$-module $B$ and $b$ is an arbitrary element of $B_G$ which maps to $0$ under the natural map $B_G\rightarrow (B/A)_G$. Is there an element $a$ in $a_G$ which maps onto $b$ under the inclusion map?
The answer to this question can be formulated in terms of group homology. (``Yes'', if $H_1(G,A)=0$.) The details, given below, will help motivate the introduction of group homology.
Group cohomology arises as the right higher derived functor for $A\longmapsto A^G$. The {\bf cohomology groups of $G$ with coefficients in $A$} are defined by \index{cohomology groups of $G$ with coefficients in $A$}
\[ H^n(G,A)={\rm Ext}\, _{\mathbb{Z} [G]}^n(\mathbb{Z} ,A). \] (See \S \ref{sec:H^n} below for more details.) These groups were first introduced in 1943 by S. Eilenberg and S. MacLane \cite{EM}. The functor $A\longmapsto A^G$ on the category of left $G$-modules is additive and left exact. This implies that if
\[ 0 \rightarrow A {\rightarrow} B {\rightarrow} C {\rightarrow} 0 \] is an exact sequence of $G$-modules then we have a {\bf long exact sequence of cohomology} \index{long exact sequence of cohomology}
\begin{equation} \label{eqn:LESC} \begin{array}{c} 0 \rightarrow A^G {\rightarrow} B^G \rightarrow C^G \rightarrow H^1(G,A) \rightarrow \\ H^1(G,B) \rightarrow H^1(G,C) \rightarrow H^2(G,A) \rightarrow \dots \end{array} \end{equation}
Similarly, group homology arises as the left higher derived functor for $A\longmapsto A_G$. The {\bf homology groups of $G$ with coefficients in $A$} are defined by \index{homology groups of $G$ with coefficients in $A$}
\[ H_n(G,A)={\rm Tor}\, _n^{\mathbb{Z} [G]} (\mathbb{Z} ,A). \] (See \S \ref{sec:H_n} below for more details.) The functor $A\longmapsto A_G$ on the category of left $G$-modules is additive and right exact. This implies that if
\[ 0 \rightarrow A {\rightarrow} B {\rightarrow} C {\rightarrow} 0 \] is an exact sequence of $G$-modules then we have a {\bf long exact sequence of homology} \index{long exact sequence of homology}
\begin{equation} \label{eqn:LESH} \begin{array}{c} \dots \rightarrow H_2(G,C) \rightarrow H_1(G,A) \rightarrow H_1(G,B) \rightarrow \\ H_1(G,C) \rightarrow A_G \rightarrow B_G \rightarrow C_G \rightarrow 0. \end{array} \end{equation}
Here we will define both cohomology $H^n(G,A)$ and homology $H_n(G,A)$ using projective resolutions and the higher derived functors ${\rm Ext}\, ^n$ and ${\rm Tor}\, _n$. We ``compute'' these when $G$ is a finite cyclic group. We also give various functorial properties, such as corestriction, inflation, restriction, and transfer. Since some of these cohomology groups can be computed with the help of computer algebra systems, we also include some discussion of how to use computers to compute them. We include several applications to group theory.
One can also define $H^1(G,A)$, $H^2(G,A)$, \dots , by explicitly constructing cocycles and coboundaries. Similarly, one can also define $H_1(G,A)$, $H_2(G,A)$, \dots , by explicitly constructing cycles and boundaries. For the proof that these constructions yield the same groups, see Rotman \cite{R}, chapter 10.
For the general outline, we follow \S 7 in chapter 10 of \cite{R} on homology. For some details, we follow Brown \cite{B}, Serre \cite{S} or Weiss \cite{W}.
For a recent expository account of this topic, see for example Adem \cite{A}. Another good reference is Brown \cite{B}.
\section{Differential groups}
In this section cohomology and homology are viewed in the same framework. This ``differential groups'' idea was introduced by Cartan and Eilenberg \cite{CE}, chapter IV, and developed in R. Godement \cite{G}, chapitre 1, \S 2. However, we shall follow Weiss \cite{W}, chapter 1.
\subsection{Definitions}
A {\bf differential group} is a pair $(L,d)$, $L$ an abelian group and $d:L\rightarrow L$ a homomorphism such that $d^2=0$. We call $d$ a {\bf differential operator}. \index{differential operator} \index{differential group} The group
\[ H(L)={\rm Kernel}\, (d)/{\rm Image}\, (d) \] is the {\bf derived group} of $(L,d)$. If \index{derived group}
\[ L=\oplus_{n=-\infty}^\infty L_n \] then we call $L$ {\bf graded}. Suppose $d$
(more precisely, $d|_{L_n}$) satisfies, in addition, for some fixed $r\not= 0$,
\[ d:L_n\rightarrow L_{n+r},\ \ \ \ n\in\mathbb{Z}. \] We say $d$ is {\bf compatible} with the grading provided $r=\pm 1$. In this case, we call $(L,d,r)$ a {\bf graded differential group}. As we shall see, \index{graded differential group} the case $r=1$ corresponds to cohomology and the the case $r=-1$ corresponds to homology. Indeed, if $r=1$ then we call $(L,d,r)$ a (differential) {\bf group of cohomology type} \index{differential group of cohomology type} and if $r=-1$ then we call $(L,d,r)$ a {\bf group of homology type}. \index{differential group of homology type} Note that if $L=\oplus_{n=-\infty}^\infty L_n$ is a group of cohomology type then $L'=\oplus_{n=-\infty}^\infty L'_n$ is a group of homology type, where $L'_n=L_{-n}$, for all $n\in\mathbb{Z}$.
\vskip .7in
{\bf For the impatient}: For {\it cohomology}, we shall eventually take $L=\oplus_n {\rm Hom}_G(X_n,A)$, where the $X_n$ form a chain complex (with $+1$ grading) determined by a certain type of resolution. The group $H(L)$ is an abbreviation for $\oplus_n {\rm Ext}\, _{\mathbb{Z} [G]}^n(\mathbb{Z},A)$. For {\it homology}, we shall eventually take $L=\oplus_n \mathbb{Z} \otimes_{\mathbb{Z} [G]}X_n$, where the $X_n$ form a chain complex (with $-1$ grading) determined by a certain type of resolution. The group $H(L)$ is an abbreviation for $\oplus_n {\rm Tor}\, ^{\mathbb{Z} [G]}_n(\mathbb{Z},A)$.
\vskip .7in
Let $(L,d)=(L,d_L)$ and $(M,d)=(M,d_M)$ be differential groups (to be more precise, we should use different symbols for the differential operators of $L$ and $M$ but, for notational simplicity, we use the same symbol and hope the context removes any ambiguity). A homomorphism $f:L\rightarrow M$ satisfying $d\circ f=f\circ d$ will be called {\bf admissible}. \index{admissible} For any $n\in \mathbb{Z}$, we define $nf:L\rightarrow M$ by $(nf)(x)=n\cdot f(x)=f(x)+\dots +f(x)$ ($n$ times). If $f$ is admissible then so is $nf$, for any $n\in \mathbb{Z}$. An admissible map $f$ gives rise to a map of derived groups: define the map $f_*:H(L)\rightarrow H(M)$, by $f_*(x +dL)=f(x)+dM$, for all $x\in L$.
\subsection{Properties} \label{sec:properties}
Let $f$ be an admissible map as above.
\begin{enumerate} \item The map $f_*:H(L)\rightarrow H(M)$ is a homomorphism.
\item If $f:L\rightarrow M$ and $g:L\rightarrow M$ are admissible, then so is $f+g$ and we have $(f+g)_*=f_*+g_*$.
\item If $f:L\rightarrow M$ and $g:M\rightarrow N$ are admissible then so is $g\circ f:L\rightarrow N$ and we have $(g\circ f)_*=g_*\circ f_*$.
\item If
\begin{equation} \label{eqn:LMN0} 0 \rightarrow L \stackrel{i}{\rightarrow} M \stackrel{j}{\rightarrow} N \rightarrow 0 \end{equation} is an exact sequence of differential groups with admissible maps $i,j$ then there is a homomorphism $d_*:H(N)\rightarrow H(L)$ for which the following triangle is exact:
{\footnotesize{ \begin{equation} \label{eqn:LMN1} {\footnotesize{ \begin{picture}(200.00,130.00)(-60.00,0.00) \thicklines \put(-30.00,50.00){$H(N)$} \put(5.00,65.00){\vector(1,1){50.00}} \put(55.00,-10.00){\vector(-1,1){50.00}} \put(60.00,120.00){$H(L)$} \put(60,-30.00){$H(M)$} \put(70.00,110.00){\vector(0,-1){115.00}} \put(80.00,60.00){$i_*$} \put(20.00,105.00){$d_*$} \put(20.00,-5.00){$j_*$} \end{picture} }} \end{equation} }}
\vskip .7in
\noindent This diagram\footnote{This is a special case of Th\'eor\`eme 2.1.1 in \cite{G}.} encodes both the long exact sequence of cohomology (\ref{eqn:LESC}) and the long exact sequence of homology (\ref{eqn:LESH}).
Here is the construction of $d_*$:
Recall $H(N)={\rm Kernel}\, (d)/{\rm Image}\, (d)$, so any $x\in H(N)$ is represented by an $n\in N$ with $dn=0$. Since $j$ is surjective, there is an $m\in M$ such that $j(m)=n$. Since $j$ is admissible and the sequence is exact, $j(dm)=d(j(m))=dn=0$, so $dm \in {\rm Kernel}\, (j)={\rm Image}\, (i)$. Therefore, there is an $\ell \in L$ such that $dm=i(\ell)$. Define $d_*(x)$ to be the class of $\ell$ in $H(L)$, i.e., $d_*(x)=\ell + dL$.
Here's the verification that $d_*$ is well-defined:
We must show that if we defined instead $d_*(x)=\ell' + dL$, some $\ell' \in L$, then $\ell'-\ell\in dL$. Pull back the above $n\in N$ with $dn=0$ to an $m\in M$ such that $j(m)=n$. As above, there is an $\ell \in L$ such that $dm=i(\ell)$. Represent $x\in H(N)$ by an $n'\in N$, so $x=n'+dN$ and $dn'=0$. Pull back this $n'$ to an $m'\in M$ such that $j(m')=n'$. As above, there is an $\ell' \in L$ such that $dm'=i(\ell')$. We know $n'-n\in dN$, so $n'-n=dn''$, some $n''\in N$. Let $j(m'')=n''$, some $m''\in M$, so $j(m'-m-dm'')=n'=n-j(dm'')=n'-n-dj(m'')=n'-n-dn''=0$. Since the sequence $L-M-N$ is exact, this implies there is an $\ell_0\in L$ such that $i(\ell_0)=m'-m-dm''$. But $di(\ell_0)=i(d\ell_0)=dm'-dm=i(\ell')-i(\ell)=i(\ell'-\ell)$, so $\ell'-\ell\in dL$.
\item If $M=L\oplus N$ then $H(M)=H(L)\oplus H(N)$.
{\bf proof}:\ To avoid ambiguity, for the moment, let $d_X$ denote the differential operator on $X$, where $X\in \{L,M,N\}$. In the notation of (\ref{eqn:LMN0}), $j$ is projection and $i$ is inclusion. Since both are admissible, we know that
$d_M|_L=d_L$ and $d_M|_N=d_N$. Note that $H(X)\subset X$, for any differential group $X$, so $H(M)=H(M)\cap L\oplus H(M)\cap N\subset H(L)\oplus H(N)$. It follows from this that that $d_*=0$. From the exactness of the triangle (\ref{eqn:LMN1}), it therefore follows that this inclusion is an equality.
$\Box$
\item Let $L$, $L'$, $M$, $M'$, $N$, $N'$ be differential groups. If
{\footnotesize{ \begin{equation} \label{eqn:LMN} \begin{CD} 0 @>>> L @>i>> M @>j>> N @>>> 0\\ @. @VfVV @VgVV @VhVV @. \\ 0 @>>> L' @>i'>> M' @>j'>> N' @>>> 0 \end{CD} \end{equation} }}
\vskip .7in \noindent is a commutative diagram of exact sequences with $i,i',j,j',f,g,h$ all admissible then
\[ \begin{CD} H(L) @>i_*>> H(M) \\ @Vf_*VV @Vg_*VV \\ H(L') @>i'_*>> H(M') \end{CD} \] commutes,
\[ \begin{CD} H(M) @>j_*>> H(N) \\ @Vg_*VV @Vh_*VV \\ H(M') @>i'_*>> H(N') \end{CD} \] commutes, and
\[ \begin{CD} H(N) @>d_*>> H(L) \\ @Vh_*VV @Vf_*VV \\ H(N') @>d_*>> H(L') \end{CD} \] commutes.
This is a case of Theorem 1.1.3 in \cite{W} and of Th\'eor\`eme 2.1.1 in \cite{G}.
The proofs that the first two squares commute are similar, so we only verify one and leave the other to the reader. By assumption, (\ref{eqn:LMN}) commutes and all the maps are admissible. Representing $x\in H(M)$ by $x=m+dM$, we have
\[ \begin{split} h_*j_*(x)&=h_*(j(m)+dN)=hj(m)+dN'=gi'(m)+dN'\\ &= g_*(i'(m)+dM')=g_*i'_*(m+dM)=g_*i'_*(x), \end{split} \] as desired.
The proof that the last square commutes is a little different than this, so we prove this too. Represent $x\in H(N)$ by $x=n+dN$ with $dn=0$ and recall that $d_*(x)=\ell+dL$, where $dm=i(\ell)$, $\ell \in L$, where $j(m)=n$, for $m\in M$. We have
\[ f_*d_*(x)=f_*(\ell+dL)=f(\ell)+dL'. \] On the other hand,
\[ d_*h_*(x)=d_*(h(n)+dN')=\ell'+dL', \] for some $\ell'\in L'$. Since $h(n)\in N'$, by the commutativity of (\ref{eqn:LMN}) and the definition of $d_*$, $\ell'\in L'$ is an element such that $i'(\ell')=gi(\ell)$. Since $i'$ is injective, this condition on $\ell'$ determines it uniquely mod $dL'$. By the commutativity of (\ref{eqn:LMN}), we may take $\ell'=f(\ell)$.
\item Let $L$, $L'$, $M$, $M'$, $N$, $N'$ be differential graded groups with grading $+1$ (i.e., of ``cohomology type''). Suppose that we have a commutative diagram, with all maps admissible and all rows exact as in (\ref{eqn:LMN}). Then the following diagram is commutative and has exact rows:
{\tiny{ \[ \begin{CD} \dots @>>> H_{n-1}(N) @>d_*>> H_n(L) @>i_*>> H_n(M) @>j_*>> H_n(N) @>d_*>> H_{n+1}(L) @>>> \dots \\ @. @Vh_*VV @Vf_*VV @Vg_*VV @Vh_*VV @Vf_*VV @. \\ \dots @>>> H_{n-1}(N') @>d_*>> H_n(L') @>i'_*>> H_n(M') @>j'_*>> H_n(N') @>d_*>> H_{n+1}(L') @>>> \dots \end{CD} \] }}
This is Proposition 1.1.4 in \cite{W}. As pointed out there, it is an immediate consequence of the properties, 1-6 above.
Compare this with Proposition 10.69 in \cite{R}.
\item Let $L$, $L'$, $M$, $M'$, $N$, $N'$ be differential graded groups with grading $-1$ (i.e., of ``homology type''). Suppose that we have a commutative diagram, with all maps admissible and all rows exact, as in (\ref{eqn:LMN}). Then the following diagram is commutative and has exact rows:
{\tiny{ \[ \begin{CD} \dots @>>> H_{n+1}(N) @>d_*>> H_n(L) @>i_*>> H_n(M) @>j_*>> H_n(N) @>d_*>> H_{n-1}(L) @>>> \dots \\ @. @Vh_*VV @Vf_*VV @Vg_*VV @Vh_*VV @Vf_*VV @. \\ \dots @>>> H_{n+1}(N') @>d_*>> H_n(L') @>i'_*>> H_n(M') @>j'_*>> H_n(N') @>d_*>> H_{n-1}(L') @>>> \dots \end{CD} \] }}
This is the analog of the previous property and is proven similarly.
Compare this with Proposition 10.58 in \cite{R}.
\item Let $(L,d)$ be a differential graded group with grading $r$.
If $d_n=d|_{L_n}$ then $d_{n+r}\circ d_n=0$ and
\begin{equation} \label{eqn:d_n} \dots \rightarrow L_{n-r} \stackrel{d_{n-r}}{\rightarrow} L_n \stackrel{d_{n}}{\rightarrow} L_{n+r} \stackrel{d_{n}}{\rightarrow} L_{n+2r} \rightarrow \dots \end{equation} is exact.
\item
If $\{L_n\ |\ n\in\mathbb{Z}\}$ is a sequence of abelian groups with homomorphisms $d_n$ satisfying (\ref{eqn:d_n}) then $(L,d)$ is a differential group, where $L=\oplus_n L_n$ and $d=\oplus_n d_n$.
\end{enumerate}
\subsection{Homology and cohomology}
When $r=1$, we call $L_n$ the {\bf group of $n$-cochains}, $Z_n=L_n\cap {\rm Kernel}\, (d_n)$ the group of {\bf $n$-cocycles}, and $B_n=L_n\cap d_{n-1}(L_{n-1})$ the group of {\bf $n$-coboundaries}. We call $H_n(L) =Z_n/B_n$ the {\bf $n^{th}$ cohomology group}. When $r=-1$, we call $L_n$ the {\bf group of $n$-chains}, $Z_n=L_n\cap {\rm Kernel}\, (d_n)$ the group of {\bf $n$-cycles}, and $B_n=L_n\cap d_{n+1}(L_{n+1})$ the group of {\bf $n$-boundaries}. We call $H_n(L) =Z_n/B_n$ the {\bf $n^{th}$ homology group}. \index{group of $n$-cochains} \index{group of $n$-cocycles} \index{group of $n$-cycles} \index{group of $n$-chains}
\section{Complexes}
We introduce complexes in order to define explicit differential groups which will then be used to construct group (co)homology.
\subsection{Definitions} \label{sec:complexes}
Let $R$ be a non-commutative ring, for example $R=\mathbb{Z} [G]$.
We shall define a ``finite free, acyclic, augmented chain complex'' of left $R$-modules.
A {\bf complex} (or chain complex or $R$-complex with a negative grading) is a sequence of maps \index{complex}
\begin{equation} \label{eqn:del_n} \dots \rightarrow X_{n+1} \stackrel{\partial_{n+1}}{\rightarrow} X_{n} \stackrel{\partial_{n}}{\rightarrow} X_{n-1} \stackrel{\partial_{n-1}}{\rightarrow} X_{n-2} \rightarrow \dots \end{equation} for which $\partial_n\partial_{n+1}=0$, for all $n$. If each $X_n$ is a free $R$-module with a finite basis over $R$ (so is $\cong R^k$, for some $k$) then the complex is called {\bf finite free}. If this sequence is exact then it is called an {\bf acyclic complex}. The complex is {\bf augmented} if there is a surjective $R$-module homomorphism $\epsilon : X_0\rightarrow \mathbb{Z}$ and an injective $R$-module homomorphism $\mu : \mathbb{Z}\rightarrow X_{-1}$ such that $\partial_0= \mu\circ \epsilon$, where (as usual) $\mathbb{Z}$ is regarded as a trivial $R$-module. \index{acyclic complex} \index{augmented complex}
The {\bf standard diagram} for such an $R$-complex is \index{standard diagram}
{\footnotesize{ \[ \begin{CD} \dots @>>> X_2 @>\partial_2>> X_1 @>\partial_1>> X_0 @>\partial_0>> X_{-1} @>\partial_{-1}>> X_{-2} @>>> \dots \\ @. @. @. @V\epsilon VV @AA\mu A @. \\ @. @. @. \mathbb{Z} @= \mathbb{Z} @. \\ @. @. @. @VVV @AAA @. \\ @. @. @. 0 @. 0 @. \end{CD} \] }} Such an acyclic augmented complex can be broken up into the {\bf positive part}
\[ \dots \rightarrow X_{2} \stackrel{\partial_{2}}{\rightarrow} X_{1} \stackrel{\partial_{1}}{\rightarrow} X_{0} \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] and the {\bf negative part}
\[ 0 \rightarrow \mathbb{Z} \stackrel{\mu}{\rightarrow} X_{-1} \stackrel{\partial_{-1}}{\rightarrow} X_{-2} \stackrel{\partial_{-2}}{\rightarrow} X_{-3} \rightarrow \dots \ . \] Conversely, given a positive part and a negative part, they can be combined into a standard diagram by taking $\partial_0=\mu\circ\epsilon$.
If $X$ is any left $R$-module, let $X^*={\rm Hom}_R(X,\mathbb{Z})$ be the dual $R$-module, where $\mathbb{Z}$ is regarded as a trivial $R$-module. Associated to any $f\in {\rm Hom}_R(X,Y)$ is the pull-back $f^*\in {\rm Hom}_R(Y^*,X^*)$. (If $y^*\in Y^*$ then define $f^*(y^*)$ to be $y^*\circ f:X\rightarrow \mathbb{Z}$.) Since ``dualizing'' reverses the direction of the maps, if you dualize the entire complex with a $-1$ grading, you will get a complex with a $+1$ grading. This is the {\bf dual complex}. \index{dual complex}
When $R=\mathbb{Z} [G]$ then we call a finite free, acyclic, augmented chain complex of left $R$-modules, a {\bf $G$-resolution}. \index{$G$-resolution} The maps $\partial_i:X_i\rightarrow X_{i-1}$ are sometimes called {\bf boundary maps}. \index{boundary maps}
\begin{remark} {\rm Using the command {\tt BoundaryMap} in the {\tt GAP} {\tt CRIME} package of Marcus Bishop, one can easily compute the boundary maps of a cohomology object associated to a $G$-module. However, $G$ must be a $p$-group. } \end{remark}
\begin{example} \label{ex:hap1} {\rm We use the package {\tt HAP} \cite{Ehap} to illustrate some of these concepts more concretely. Let $G$ be a finite group, whose elements we have ordered in some way: $G=\{g_1,...,g_n\}$.
Since a $G$-resolution $X_*$ determines a sequence of finitely generated free $\mathbb{Z}[G]$-modules, to concretely describe $X_*$ we must be able to concretely describe a finite free $\mathbb{Z}[G]$-module. In order to represent a word $w$ in a free $\mathbb{Z}[G]$-module $M$ of rank $n$, we use a list of integer pairs $w=[ [i_1,e_1], [i_2,e_2], ..., [i_k,e_k] ]$. The integers $i_j$ lie in the range $\{-n,..., n\}$ and correspond to the free $\mathbb{Z}[G]$-generators of $M$ and their additive inverses. The integers $e_j$ are positive (but not necessarily distinct) and correspond to the group element $g_{e_j}$.
Let's begin with a {\tt HAP} computation.
\vskip .2in
\begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label={\tt GAP}]
gap> LoadPackage("hap"); true gap> G:=Group([(1,2,3),(1,2)]);; gap> R:=ResolutionFiniteGroup(G, 4);;
\end{Verbatim}
\vskip .1in \noindent This computes the first $5$ terms of a $G$-resolution ($G=S_3$)
\[ X_4 \stackrel{\delta_4}{\rightarrow} X_3
\stackrel{\delta_3}{\rightarrow} X_2
\stackrel{\delta_2}{\rightarrow} X_1
\stackrel{\delta_1}{\rightarrow} X_0 \rightarrow \mathbb{Z} \rightarrow 0. \] The bounday maps $\delta_i$ are determined from the {\tt boundary} component of the {\tt GAP} record {\tt R}. This record has (among others) the following components:
\begin{itemize} \item {\tt R!.dimension(k)} -- the $\mathbb{Z}[G]$-rank of the module $X_k$, \item {\tt R!.boundary(k, j)} -- the image in $X_{k-1}$ of the $j$-th free generator of $X_k$, \item {\tt R!.elts} -- the elements in $G$, \item {\tt R!.group} is the group in question. \end{itemize}
Here is an illustration:
\vskip .2in
\begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label={\tt GAP}]
gap> R!.group;
Group([ (1,2), (1,2,3) ]) gap> R!.elts;
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ] gap> R!.dimension(3);
4 gap> R!.boundary(3,1);
[ [ 1, 2 ], [ -1, 1 ] ] gap> R!.boundary(3,2);
[ [ 2, 2 ], [ -2, 4 ] ] gap> R!.boundary(3,3);
[ [ 3, 4 ], [ 1, 3 ], [ -3, 1 ], [ -1, 1 ] ] gap> R!.boundary(3,4);
[ [ 2, 5 ], [ -3, 3 ], [ 2, 4 ], [ -1, 4 ], [ 2, 1 ], [ -3, 1 ] ]
\end{Verbatim}
\vskip .1in \noindent In other words, $X_3$ is rank $4$ as a $G$-module, with generators $\{f_1, f_2, f_3, f_4\}$ say, and
\[ \delta_3(f_1) = f_1g_2 - f_1g_1, \] \[ \delta_3(f_2) = f_2g_2 - f_2g_4, \] \[ \delta_3(f_3) = f_3g_4 - f_3g_1+f_1g_3-f_1g_1, \] \[ \delta_3(f_4) = f_2(g_1+g_3+g_5) - f_3g_3 + f_1g_4-f_3g_1. \]
Now, let us create another resolution and compute the equivariant chain map between them. Below is the complete {\tt GAP} session:
\vskip .2in
{\footnotesize{ \begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label={\tt GAP}]
gap> G1:=Group([(1,2,3),(1,2)]); Group([ (1,2,3), (1,2) ]) gap> G2:=Group([(1,2,3),(2,3)]); Group([ (1,2,3), (2,3) ]) gap> phi:=GroupHomomorphismByImages(G1,G2,[(1,2,3),(1,2)],[(1,2,3),(2,3)]); [ (1,2,3), (1,2) ] -> [ (1,2,3), (2,3) ] gap> R1:=ResolutionFiniteGroup(G1, 4); Resolution of length 4 in characteristic 0 for Group([ (1,2), (1,2,3) ]) .
gap> R2:=ResolutionFiniteGroup(G2, 4); Resolution of length 4 in characteristic 0 for Group([ (2,3), (1,2,3) ]) .
gap> ZP_map:=EquivariantChainMap(R1, R2, phi); Equivariant Chain Map between resolutions of length 4 .
gap> map := TensorWithIntegers( ZP_map); Chain Map between complexes of length 4 .
gap> Hphi := Homology( map, 3); [ f1, f2, f3 ] -> [ f2, f2*f3, f1*f2^2 ] gap> AbelianInvariants(Image(Hphi)); [ 2, 3 ] gap> gap> GroupHomology(G1,3); [ 6 ] gap> GroupHomology(G2,3); [ 6 ]
\end{Verbatim} }} \vskip .1in \noindent In other words, $H(\phi)$ is an isomorphism (as it should be, since the homology is independent of the resolution choosen).
} \end{example}
\subsection{Constructions}
Let $R=\mathbb{Z}[G]$.
\subsubsection{Bar resolution} \label{sec:bar_res}
This section follows \S 1.3 in \cite{W}.
Define a symbol $[.]$ and call it the {\bf empty cell}. Let $X_0=R[.]$, so $X_0$ is a finite free (left) $R$-module whose basis has only $1$ element. For $n>0$, let $g_1,\dots ,g_n\in G$ and define an {\bf $n$-cell} to be the symbol $[g_1,\dots ,g_n]$. \index{cell} Let
\[ X_n=\oplus_{(g_1,\dots ,g_n)\in G^n} R[g_1,\dots ,g_n], \] where the sum runs over all ordered $n$-tuples in $G^n$.
Define the differential operators $d_n$ and the augmentation $\epsilon$, as $G$-module maps, by
\[ \begin{split} \epsilon(g[.])&=1,\ \ \ \ \ \ g\in G\\ d_1([g])&=g[.]-[.],\\ d_2([g_1,g_2])&=g_1[g_2]-[g_1g_2]+[g_1],\\
& \vdots \\ d_n([g_1,\dots ,g_n])&=g_1[g_2,\dots ,g_n] +\sum_{i=1}^{n-1}(-1)^i[g_1,\dots ,g_{i-1},g_ig_{i+1},g_{i+2},\dots ,g_n]\\
& \ \ \ \ \ \ +(-1)^n[g_1,\dots ,g_{n-1}], \end{split} \] for $n\geq 1$. Note that the condition $\epsilon(g[.])=1$ for all $g\in G$ is equivalent to saying $\epsilon([.])=1$. This is because $\epsilon$ is a $G$-module homomorphism and $\mathbb{Z}$ is a trivial $G$-module, so $\epsilon(g[.])=g\epsilon([.])=g\cdot 1=1$, where the (trivial) $G$-action on $\mathbb{Z}$ is denoted by a $\cdot$.
The $X_n$ are finite free $G$-modules, with the set of all $n$-cells serving as a basis.
\begin{proposition} \label{prop:bar} With these definitions, the sequence
\[ \dots \rightarrow X_{2} \stackrel{d_{2}}{\rightarrow} X_{1} \stackrel{d_{1}}{\rightarrow} X_{0} \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] is a free $G$-resolution. \end{proposition}
Sometimes this resolution is called the {\bf bar resolution}\footnote{This resolution is not the same as the resolution computed by {\tt HAP} in Example \ref{ex:hap1}. For details on the resolution used by {\tt HAP}, please see Ellis \cite{E2}.}. \index{bar resolution} There are two other resolutions we shall consider. One is the closely related ``homogeneous resolution'' and the other is the ``normalized bar resolution''.
This simple-looking proposition is not so simple to prove. First, we shall show it is a complex, i.e., $d^2=0$. Then, and this is the most non-trivial part of the proof, we show that the sequence is exact.
First, we need some definitions and a lemma.
Let $f:L\rightarrow M$ and $g:L\rightarrow M$ be $+1$-graded admissible maps. We say $f$ is {\bf homotopic} to $g$ if there is a homomorphism $D:L\rightarrow M$, called a {\bf homotopy}, such that \index{homotopy}
\begin{itemize}
\item
$D_n=D|_{L_n}:L_n\rightarrow M_{n+1}$,
\item $f-g=Dd+dD$. \end{itemize} If $L=M$ and the identity map $1:L\rightarrow L$ is homotopic to the zero map $0:L\rightarrow L$ then the homotopy is called a {\bf contracting homotopy for $L$}. \index{contracting homotopy}
\begin{lemma} If $L$ has a contracting homotopy then $H(L)=0$. \end{lemma}
{\bf proof}:\ Represent $x\in H(L)$ by $\ell\in L$ with $d\ell =0$. But $\ell=1(\ell)-0(\ell)=dD(\ell)+Dd(\ell)=dD(\ell)$. Since $D:L\rightarrow L$, this shows $\ell\in dL$, so $x=0$ in $H(L)$. $\Box$
Next, we construct a contracting homotopy for the complex $X_*$ in Proposition \ref{prop:bar} with differential operator $d$. Actually, we shall {\it temporarily} let $X_{-1}=\mathbb{Z}$, $X_{-n}=0$ and $d_{-n}=0$ for $n>1$, so that that the complex is infinite in both directions. We must define $D:X\rightarrow X$ such that
\begin{itemize}
\item
$D_{-1}=D|_{\mathbb{Z}}:\mathbb{Z}\rightarrow X_0$,
\item
$D_{n}=D|_{X_n}:X_n\rightarrow X_{n+1}$,
\item $\epsilon D_{-1}=1$ on $\mathbb{Z}$,
\item $d_1D_0+D_{-1}\epsilon =1$ on $X_0$,
\item $d_{n+1}D_n+D_{n-1}d_n=1$ in $X_n$, for $n\geq 1$.
\end{itemize} Define
\[ \begin{split} D_{-n}&=0,\ \ \ \ \ \ n>1,\\ D_{-1}(1)&=[.],\\ D_0(g[.])&=[g],\\ D_n(g[g_1,\dots ,g_n])&=[g,g_1,\dots ,g_n],\ \ \ \ \ \ n>0, \end{split} \] and extend to a $\mathbb{Z}$-basis linearly.
Now we must verify the desired properties.
By definition, for $m\in \mathbb{Z}$, $\epsilon D_{-1}(m)=\epsilon (m[.])=m\epsilon ([.])=m$. Therefore, $\epsilon D_{-1}$ is the identity map on $\mathbb{Z}$.
Similarly, \[ \begin{split} (d_1 D_0+D_{-1}\epsilon )(g[.])= d_1 ([g])+D_{-1}(1) \\ =g[.]-[.]+D_{-1}(1)=g[.]-[.]+[.]=g[.]. \end{split} \]
For the last property, we compute
\[ \begin{split} d_{n+1} D_n(g[g_1,\dots ,g_n]) &=d_{n+1} ([g,g_1,\dots ,g_n])\\ &=g[g_1,\dots ,g_n]-[gg_1,\dots ,g_n]\\ & \ \ \ \ \ \ +\sum_{i=1}^{n-1}(-1)^{i-1}[g,g_1,\dots , g_{i-1},g_ig_{i+1},g_{i+2},\dots ,g_n]\\
& \ \ \ \ \ \ +(-1)^{n+1}[g,g_1,\dots ,g_{n-1}], \end{split} \] and
\[ \begin{split} D_{n-1}d_{n} (g[g_1,\dots ,g_n])\\ &=D_{n-1}(gd_{n} ([g_1,\dots ,g_n]))\\ &=D_{n-1}(gg_1[g_2,\dots ,g_n]\\ & \ \ \ \ \ \ +\sum_{i=1}^{n-1}(-1)^ig[g_1,\dots ,g_{i-1},g_ig_{i+1},g_{i+2},\dots ,g_n]\\
& \ \ \ \ \ \ +(-1)^{n}g[g_1,\dots ,g_{n-1}])\\ &=[gg_1,g_2,\dots ,g_n]\\ & \ \ \ \ \ \ +\sum_{i=1}^{n-1}(-1)^i[g,g_1,\dots ,g_{i-1},g_ig_{i+1},g_{i+2},\dots ,g_n]\\
& \ \ \ \ \ \ +(-1)^{n}[g,g_1,\dots ,g_{n-1}]. \end{split} \] All the terms but one cancels, verifying that $d_{n+1}D_n+D_{n-1}d_n=1$ in $X_n$, for $n\geq 1$.
Now we show $d^2=0$. One verifies $d_1d_2=0$ directly (which is left to the reader). Multiply $d_kD_{k-1}+D_{k-2}d_{k-1}=1$ on the right by $d_k$ and $d_{k+1}D_{k}+D_{k-1}d_{k}=1$ on the left by $d_k$:
\[ d_kD_{k-1}d_k + D_{k-2}d_{k-1}d_k=d_k= d_kd_{k+1}D_{k}+d_kD_{k-1}d_{k}. \] Cancelling like terms, the induction hypothesis $d_{k-1}d_k=0$ implies $d_{k}d_{k+1}=0$. This shows $d^2=0$ and hence that the sequence in Proposition \ref{prop:bar} is exact. This completes the proof of Proposition \ref{prop:bar}. $\Box$
\vskip .1in
The above complex can be ``dualized'' in the sense of \S \ref{sec:complexes}. This dualized complex is of the form
\[ 0 \rightarrow \mathbb{Z} \stackrel{\mu}{\rightarrow} X_{-1} \stackrel{d_{-1}}{\rightarrow} X_{-2} \stackrel{d_{-2}}{\rightarrow} X_{-3} \rightarrow \dots \ . \] The {\bf standard $G$-resolution} is obtained by splicing these together. \index{standard $G$-resolution}
\subsubsection{Normalized bar resolution}
Define the {\bf normalized cells} by
\[ [g_1,...,g_n]^*= \left\{ \begin{array}{cc} [g_1,...,g_n], &{\rm if \ all\ }g_i\not= 1,\\ 0, & {\rm if \ some\ }g_i= 1. \end{array} \right. \]
Let $X_0=R[.]$ and
\[ X_n=\oplus_{(g_1,\dots ,g_n)\in G^n} R[g_1,\dots ,g_n]^*,\ \ \ \ \ n\geq 1, \] where the sum runs over all ordered $n$-tuples in $G^n$. Define the differential operators $d_n$ and the augmentation map exactly as for the bar resolution.
\begin{proposition} \label{prop:nbar} With these definitions, the sequence
\[ \dots \rightarrow X_{2} \stackrel{d_{2}}{\rightarrow} X_{1} \stackrel{d_{1}}{\rightarrow} X_{0} \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] is a free $G$-resolution. \end{proposition}
Sometimes this resolution is called the {\bf normalized bar resolution}. \index{normalized bar resolution}
{\bf proof}:\ See Theorem 10.117 in \cite{R}. $\Box$
\subsubsection{Homogeneous resolution}
Let $X_0=R$, so $X_0$ is a finite free (left) $R$-module whose basis has only $1$ element. For $n>0$, let $X_n$ denote the $\mathbb{Z}$-module generated by all $(n+1)$-tuples $(g_0,\dots ,g_n)$. Make $X_i$ into a $G$-module by defining the action by $g:X_n\rightarrow X_n$ by
\[ g:(g_0,...,g_n)\longmapsto (gg_0,\dots ,gg_n),\ \ \ \ \ g\in G. \]
Define the differential operators $\partial_n$ and the augmentation $\epsilon$, as $G$-module maps, by
\[ \begin{split} \epsilon (g)&=1,\\ \partial_n(g_0,\dots ,g_n)&=\sum_{i=0}^{n-1}(-1)^i (g_0,\dots ,g_{i-1},\hat{g}_i,g_{i+1},\dots ,g_n), \end{split} \] for $n\geq 1$.
\begin{proposition} \label{prop:homog} With these definitions, the sequence
\[ \dots \rightarrow X_{2} \stackrel{\partial_{2}}{\rightarrow} X_{1} \stackrel{\partial_{1}}{\rightarrow} X_{0} \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] is a $G$-resolution. \end{proposition}
Sometimes this resolution is called the {\bf homogeneous resolution}. \index{homogeneous resolution}
Of the three resolutions presented here, this one is the most straightforward to deal with.
{\bf proof}:\ See Lemma 10.114, Proposition 10.115, and Proposition 10.116 in \cite{R}.
$\Box$
\section{Definition of $H^n(G,A)$} \label{sec:H^n}
For convenience, we briefly recall the definition of ${\rm Ext}\, ^n$. Let $A$ be a left $R$-module, where $R=\mathbb{Z} [G]$, and let $(X_i)$ be a $G$-resolution of $\mathbb{Z}$. We define
\[ {\rm {\rm Ext}\, }^n_{\mathbb{Z} [G]}(\mathbb{Z},A)= {\rm Kernel}\, (d_{n+1}^*)/{\rm Image}\, (d_n^*), \] where
\[ d_n^*:Hom(X_{n-1},A)\rightarrow Hom(X_n,A), \] is defined by sending $f:X_{n-1}\rightarrow A$ to $fd_n:X_{n}\rightarrow A$. It is known that this is, up to isomorphism, independent of the resolution choosen. Recall ${\rm {\rm Ext}\, }^*_{\mathbb{Z} [G]}(\mathbb{Z},A)$ is the right-derived functors of the right-exact functor $A\longmapsto A^G={\rm Hom}_G(\mathbb{Z},A)$ from the category of $G$-modules to the category of abelian groups. We define
\begin{equation} \label{eqn:H^ndef} H^n(G,A)={\rm {\rm Ext}\, }^n_{\mathbb{Z} [G]}(\mathbb{Z},A), \end{equation} When we wish to emphasize the dependence on the resolution choosen, we write $H^n(G,A,X_*)$.
For example, let $X_*$ denote the bar resolution in \S \ref{sec:bar_res} above. Call $C^n=C^n(G,A)={\rm Hom}_G(X_n,A)$ the {\bf group of $n$-cochains of $G$ in $A$}, \index{group of $n$-cochains of $G$ in $A$} $Z^n=Z^n(G,A)=C^n\cap {\rm Kernel}\, (\partial)$ the group of {\bf $n$-cocycles}, \index{group of $n$-cocycles} and $B^n=B^n(G,A)= \partial(C^{n-1})$ the group of {\bf $n$-coboundaries}. \index{group of $n$-coboundaries} We call $H^n(G,A) =Z^n/B^n$ the {\bf $n^{th}$ cohomology group of $G$ in $A$}. This is an abelian group.
We call also define the cohomology group using some other resolution, the normalized bar resolution or the homogeneous resolution for example. If we wish to express the dependence on the resolution $X_*$ used, we write $H^n(G,A,X_*)$. Later we shall see that, up to isomorphism, this abelian group is independent of the resolution.
The group $H_2(G,\mathbb{Z})$ (which is isomorphic to the algebraic dual group of $H^2(G,\mathbb{C}^\times)$) is sometimes called the {\bf Schur multiplier} of $G$. Here $\mathbb{C}$ denotes the field of complex numbers. \index{Schur multiplier}
We say that the group $G$ has {\bf cohomological dimension} $n$, \index{cohomological dimension} written $cd(G)=n$, if $H^{n+1}(H,A)=0$ for all $G$-modules $A$ and all subgroups $H$ of $G$, but $H^n(H,A)\not= 0$ for some such $A$ and $H$.
\begin{remark} \begin{itemize}
\item If $cd(G)<\infty$ then $G$ is torsion-free\footnote{This follows from the fact that if $G$ is a cyclic group then $H^n(G,\mathbb{Z})\not= 0$, discussed below.}.
\item If $G$ is a free abelian group of finite rank then $cd(G)=rank(G)$.
\item If $cd(G)=1$ then $G$ is free. This is a result of Stallings and Swan (see for example \cite{R}, page 885). \end{itemize} \end{remark}
\subsection{Computations}
We briefly discuss computer programs which compute cohomology and some examples of known computations.
\subsubsection{Computer computations of cohomology}
{\tt GAP} \cite{Gap} can compute some cohomology groups\footnote{See \S 37.22 of the {\tt GAP} manual, M. Bishop's package {\tt CRIME} for cohomology of $p$-groups, G. Ellis' package {\tt HAP} for group homology and cohomology of finite or (certain) infinite groups, and M. R\"oder's {\tt HAPCryst} package (an add-on to the {\tt HAP} package). \SAGE \cite{St} computes cohomology via it's {\tt GAP} interface.}.
All the \SAGE commands which compute group homology or cohomology require that the package {\tt HAP} be loaded. You can do this on the command line from the main \SAGE directory by typing\footnote{This is the current package name - change {\tt 4.4.10\_3} to whatever the latest version is on \url{http://www.sagemath.org/packages/optional/} at the time you read this. Also, this command assumes you are using \SAGE on a machine with an internet connection.}
\verb+sage -i gap_packages-4.4.10_3.spkg+
\begin{example} {\rm This example uses \SAGE, which wraps several of the {\tt HAP} functions.
\vskip .2in
\begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\SAGE]
sage: G = AlternatingGroup(5) sage: G.cohomology(1,7) Trivial Abelian Group sage: G.cohomology(2,7) Trivial Abelian Group
\end{Verbatim}
\vskip .1in \noindent This implies $H^1(A_5,GF(7))=H^2(A_5,GF(7))=0$. } \end{example}
\subsubsection{Examples}
Some example computations of a more theoretical nature.
\begin{enumerate}
\item $H^0(G,A)=A^G$.
This is by definition.
\item Let $L/K$ denote a Galois extension with finite Galois group $G$. We have $H^1(G,L^\times)=1$. This is often called Hilbert's Theorem 90.
See Theorem 1.5.4 in \cite{W} or Proposition 2 in \S X.1 of \cite{S}.
\item Let $G$ be a finite cyclic group and $A$ a trivial torsion-free $G$-module. Then $H^1(G,A)=0$.
This is a consequence of properties given in the next section.
\item If $G$ is a finite cyclic group of order $m$ and $A$ is a trivial $G$-module then
\[ H^2(G,A)=A/mA \]
This is a consequence of properties given below.
For example, $H^2(GF(q)^\times,\mathbb{C})=0$.
\item If $|G|=m$, $rA=0$ and $gcd(r,m)=1$, then $H^n(G,A)=0$, for all $n\geq 1$.
This is Corollary 3.1.7 in \cite{W}.
For example, $H^1(A_5,\mathbb{Z}/7\mathbb{Z})=0$. \end{enumerate}
\section{Definition of $H_n(G,A)$} \label{sec:H_n}
We say $A$ is {\bf projective} if the functor \index{projective $R$-module} $B\longmapsto {\rm Hom}_G(A,B)$ (from the category of $G$-modules to the category of abelian groups) is exact. Recall, if
\begin{equation} \label{eqn:P_Z} P_{\mathbb{Z}}= \dots \rightarrow P_2 \stackrel{d_2}{\rightarrow} P_1 \stackrel{d_1}{\rightarrow} P_0 \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0 \end{equation} is a projective resolution of $\mathbb{Z}$ then
\[ {\rm Tor}\, _n^{\mathbb{Z} [G]}(\mathbb{Z},A) ={\rm Kernel}\, (d_n\otimes 1_A)/{\rm Image}\, (d_{n+1}\otimes 1_A). \] It is known that this is, up to isomorphism, independent of the resolution choosen. Recall ${\rm Tor}\, _*^{\mathbb{Z} [G]}(\mathbb{Z},A)$ are the right-derived functors of the right-exact functor $A\longmapsto A_G=\mathbb{Z}\otimes_{\mathbb{Z} [G]}A$ from the category of $G$-modules to the category of abelian groups. We define
\begin{equation} \label{eqn:H_ndef} H_n(G,A)={\rm Tor}\, _n^{\mathbb{Z} [G]}(\mathbb{Z},A), \end{equation} When we wish to emphasize the dependence on the resolution, we write $H_n(G,A,P_\mathbb{Z})$.
\begin{remark} {\rm If $G$ is a $p$-group, then using the command {\tt ProjectiveResolution} in {\tt GAP}'s {\tt CRIME} package, one can easily compute the minimal projective resolution of a $G$-module, which can be either trivial or given as a {\tt MeatAxe}\footnote{See for example \url{http://www.math.rwth-aachen.de/~MTX/}.} module. } \end{remark}
Since we can identify the functor $A\longmapsto A_G$ with $A\longmapsto A\otimes_{\mathbb{Z}[G]} \mathbb{Z}$ (where $\mathbb{Z}$ is considered as a trivial $\mathbb{Z} [G]$-module), the following is another way to formulate this definition.
If $\mathbb{Z}$ is considered as a trivial $\mathbb{Z} [G]$-module, then a free $\mathbb{Z} [G]$-resolution of $\mathbb{Z}$ is a sequence of $\mathbb{Z} [G]$-module homomorphisms
\[ ... {\rightarrow} M_n {\rightarrow} M_{n-1} {\rightarrow} ... {\rightarrow} M_1 {\rightarrow} M_0 \] satisfying: \begin{itemize} \item
(Freeness) Each $M_n$ is a free $\mathbb{Z}[G]$-module. \item
(Exactness) The image of $M_{n+1} {\rightarrow} M_n$ equals the kernel of $M_n {\rightarrow} M_{n-1}$ for all $n>0$. \item
(Augmentation) The cokernel of $M_1 {\rightarrow} M_0$ is isomorphic to the trivial $\mathbb{Z}[G]$-module $\mathbb{Z}$. \end{itemize} The maps $M_n {\rightarrow} M_{n-1}$ are the boundary homomorphisms of the resolution. Setting $TM_n$ equal to the abelian group $M_n/G$ obtained from $M_n$ by killing the $G$-action, we get an induced sequence of abelian group homomorphisms
\[ ... {\rightarrow} TM_n {\rightarrow} TM_{n-1} {\rightarrow} ... {\rightarrow} TM_1 {\rightarrow} TM_0 \] This sequence will generally not satisfy the above exactness condition, and one defines the integral homology of $G$ to be
\[ H_n(G,\mathbb{Z}) = {\rm Kernel}\, (TM_n {\rightarrow} TM_{n-1}) / {\rm Image}\, (TM_{n+1} {\rightarrow} TM_n) \] for all $n>0$.
\subsection{Computations}
We briefly discuss computer programs which compute homology and some examples of known computations.
\subsubsection{Computer computations of homology}
\begin{example} {\rm {\tt GAP} will compute the Schur multiplier $H_2(G,\mathbb{Z})$ using the \newline {\tt AbelianInvariantsMultiplier} command. To find $H_2(A_5,\mathbb{Z})$, where $A_5$ is the alternating group on $5$ letters, type
\vskip .2in
\begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label={\tt GAP}]
gap> A5:=AlternatingGroup(5); Alt( [ 1 .. 5 ] ) gap> AbelianInvariantsMultiplier(A5); [ 2 ]
\end{Verbatim}
\vskip .1in \noindent So, $H_2(A_5,\mathbb{C})\cong \mathbb{Z}/2\mathbb{Z}$.
Here is the same computation in \SAGE: \vskip .2in
\begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\SAGE]
sage: G = AlternatingGroup(5) sage: G.homology(2) Multiplicative Abelian Group isomorphic to C2
\end{Verbatim}
} \end{example}
\begin{example} {\rm The \SAGE command {\tt poincare\_series} returns the Poincare series of $G \pmod p$ ($p$ must be a prime). In other words, if you input a (finite) permutation group $G$, a prime $p$, and a positive integer $n$, {\tt poincare\_series(G,p,n)} returns a quotient of polynomials $f(x)=P(x)/Q(x)$ whose coefficient of $x^k$ equals the rank of the vector space $H_k(G,ZZ/pZZ)$, for all $k$ in the range $ 1\leq k \leq n$ .
\vskip .2in
{\footnotesize{ \begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\SAGE]
sage: G = SymmetricGroup(5) sage: G.poincare_series(2,10)
(x^2 + 1)/(x^4 - x^3 - x + 1) sage: G = SymmetricGroup(3) sage: G.poincare_series(2,10)
1/(-x + 1)
\end{Verbatim} }}
\vskip .1in \noindent This last one implies
\[ \dim_{GF(2)}H_k(S_2,\mathbb{Z}/2\mathbb{Z})=1, \] for $1\leq k\leq 10$. } \end{example}
\begin{example} \label{ex:ppart} {\rm Here are some more examples using \SAGE's interface to {\tt HAP}:
\vskip .2in
{\footnotesize{ \begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\SAGE]
sage: G = SymmetricGroup(5) sage: G.homology(1) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(2) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(3) Multiplicative Abelian Group isomorphic to C2 x C4 x C3 sage: G.homology(4) Multiplicative Abelian Group isomorphic to C2 sage: G.homology(5) Multiplicative Abelian Group isomorphic to C2 x C2 x C2 sage: G.homology(6) Multiplicative Abelian Group isomorphic to C2 x C2 sage: G.homology(7) Multiplicative Abelian Group isomorphic to C2 x C2 x C4 x C3 x C5
\end{Verbatim} }}
\vskip .1in \noindent The last one means that
\[ H_7(S_5,Z) = (\mathbb{Z}/2\mathbb{Z})^2\times (\mathbb{Z}/3\mathbb{Z})\times (\mathbb{Z}/4\mathbb{Z})\times (\mathbb{Z}/5\mathbb{Z}). \]
\vskip .2in
{\footnotesize{ \begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\SAGE]
sage: G = AlternatingGroup(5) sage: G.homology(1) Trivial Abelian Group sage: G.homology(1,7) Trivial Abelian Group sage: G.homology(2,7) Trivial Abelian Group
\end{Verbatim} }}
\vskip .1in \noindent This implies $H_1(A_5,\mathbb{Z})=H_1(A_5,GF(7))=H_2(A_5,GF(7))=0$.
} \end{example}
\subsubsection{Examples} \label{sec:homprops}
Some example computations of a more theoretical nature.
\begin{enumerate}
\item If $A$ is a $G$-module then ${\rm Tor}\, _0^{\mathbb{Z}[G]}(\mathbb{Z},A)= H_0(G,A)=A_G\cong A/DA$.
{\bf proof}: We need some lemmas.
Let $\epsilon :\mathbb{Z} [G]\rightarrow \mathbb{Z}$ be the augmentation map. This is a ring homomorphism (but not a $G$-module homomorphism). Let $D={\rm Kernel}\, (\epsilon)$ denote its kernel, the {\bf augmentation ideal}. \index{augmentation ideal} This is a $G$-module.
\begin{lemma} \label{lemma:Disfree} As an abelian group, $D$ is free abelian generated by
$G-1=\{g-1\ |\ g\in G\}$. \end{lemma}
We write this as $D=\mathbb{Z} \langle G-1\rangle$.
{\bf proof of lemma}: If $d\in D$ then $d=\sum_{g\in G}m_gg$, where $m_g\in \mathbb{Z}$ and $\sum_{g\in G}m_g=0$. Thus, $d=\sum_{g\in G}m_g(g-1)$, so $D\subset \mathbb{Z} \langle G-1\rangle$. To show $D$ is free: If $\sum_{g\in G}m_g(g-1)=0$ then $\sum_{g\in G}m_g g - \sum_{g\in G}m_g=0$ in $\mathbb{Z}[G]$. But $\mathbb{Z}[G]$ is a free abelian group with basis $G$, so $m_g=0$ for all $g\in G$. $\Box$
\begin{lemma} $\mathbb{Z}\otimes_{\mathbb{Z} [G]}A= A/DA$, where $DA$ is generated by elements of the form $ga-a$, $g\in G$ and $a\in A$. \end{lemma}
Recall $A_G$ denotes the largest quotient of $A$ on which $G$ acts trivially\footnote{Implicit in the words ``largest quotient'' is a universal property which we leave to the reader for formulate precisely.}.
{\bf proof of lemma}: Consider the $G$-module map, $A\rightarrow \mathbb{Z}\otimes_{\mathbb{Z}[G]}A$, given by $a\longmapsto 1\otimes a$. Since $\mathbb{Z}\otimes_{\mathbb{Z}[G]}A$ is a trivial $G$-module, it must factor through $A_G$. The previous lemma implies $A_G\cong A/DA$. (In fact, the quotient map $q:A\rightarrow A_G$ satisfies $q(ga-a)=0$ for all $g\in G$ and $a\in A$, so $DA\subset {\rm Kernel}\, (q)$. By maximality of $A_G$, $DA={\rm Kernel}\, (q)$. QED) So, we have maps $A\rightarrow A_G \rightarrow \mathbb{Z}\otimes_{\mathbb{Z}[G]}A$. By the definition of tensor products, the map $\mathbb{Z}\times A\rightarrow A_G$, $1\times a\longmapsto 1\cdot aDA$, corresponds to a map $ \mathbb{Z}\otimes_{\mathbb{Z}[G]}A \rightarrow A_G $ for which the composition $A_G \rightarrow \mathbb{Z}\otimes_{\mathbb{Z}[G]}A \rightarrow A_G $ is the identity. This forces $A_G \cong \mathbb{Z}\otimes_{\mathbb{Z}[G]}A$. $\Box$
See also \# 11 in \S \ref{sec:basisprops}.
\item If $G$ is a finite group then $H_0(G,\mathbb{Z})=\mathbb{Z}$.
This is a special case of the example above (taking $A=\mathbb{Z}$, as a trivial $G$-module).
\item $H_1(G,\mathbb{Z})\cong G/[G,G]$, where $[G,G]$ is the commutator subgroup of $G$.
This is Proposition 10.110 in \cite{R}, \S 10.7.
{\bf proof}:\ First, we {\bf claim}: $D/D^2 \cong G/[G,G]$, where $D$ is as in Lemma \ref{lemma:Disfree}. To prove this, define $\theta: G\rightarrow D/D^2$ by $g\longmapsto (g-1)+D^2$. Since $gh-1-(g-1)-(h-1)=(g-1)(h-1)$, it follows that $\theta(gh)=\theta(g)\theta(h)$, so $\theta$ is a homomorphism. Since $D/D^2$ is abelian and $G/[G,G]$ is the maximal abelian quotient of $G$, we must have ${\rm Kernel}\, (\theta)\subset [G,G]$. Therefore, $\theta$ factors through $\theta': G/[G,G]\rightarrow D/D^2$, $g[G,G]\longmapsto (g-1)+D^2$. Now, we construct an inverse. Define $\tau:D\rightarrow G/[G,G]$ by $g-1 \longmapsto g[G,G]$. Since $\tau (g-1 + h-1) = g[G,G]\cdot h[G,G]=gh [G,G]$, it is not hard to see that this is a homomorphism. We would be essentially done (with the construction of the inverse of $\theta'$, hence the proof of the claim) if we knew $D^2\subset {\rm Kernel}\, (\tau)$. (The inverse would be the composition of the quotient $D/D^2\rightarrow D/{\rm Kernel}\, (\tau)$ with the map induced from $\tau$, $D/{\rm Kernel}\, (\tau)\rightarrow G/[G,G]$.) This follows from the fact that any $x\in D^2$ can be written as $x=(\sum_g m_g (g-1)) (\sum_h m'_h (h-1)) = (\sum_{g,h} m_gm'_h (g-1)(h-1))$, so $\tau(x)=\prod_{g,h} (ghg^{-1}h^{-1})^{m_gm'_h}[G,G]=[G,G]$.
QED (claim)
Next, we show $H_1(G,\mathbb{Z})\cong D/D^2$. From the short exact sequence
\[ 0\rightarrow D \rightarrow \mathbb{Z}[G] \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] we obtain the long exact sequence of homology
\begin{equation} \label{eqn:LESHG} \begin{array}{c} \dots \rightarrow H_1(G,D) \rightarrow H_1(G,\mathbb{Z}[G]) \rightarrow \\ H_1(G,\mathbb{Z}) \stackrel{\partial}{\rightarrow} H_0(G,D) \stackrel{f}{\rightarrow } H_0(G,\mathbb{Z}[G]) \stackrel{\epsilon_*}{\rightarrow} H_0(G,\mathbb{Z}) \rightarrow 0. \end{array} \end{equation} Since $\mathbb{Z}[G]$ is a free $\mathbb{Z}[G]$-module, $H_1(G,\mathbb{Z}[G])=0$. Therefore $\partial$ is injective. By item \# 1 above (i.e., $H_0(G,A)\cong A/DA\cong A_G$, we have $H_0(G,\mathbb{Z})\cong \mathbb{Z}_G=\mathbb{Z}$ and $H_0(G,\mathbb{Z}[G])\cong \mathbb{Z}[G]/D\cong \mathbb{Z}$. By (\ref{eqn:LESHG}), $\epsilon_*$ is surjective. Combining the last two statements, we find $\mathbb{Z}/{\rm Kernel}\, (\epsilon_*)\cong \mathbb{Z}$.This forces $\epsilon_*$ to be injective. This, and (\ref{eqn:LESHG}), together imply $f$ must be $0$. Since this forces $\partial$ to be an isomorphism, we are done. $\Box$
\item Let $G=F/R$ be a presentation of $G$, where $F$ is a free group and $R$ is a normal subgroup of relations. {\bf Hopf's formula} states: $H_2(G,\mathbb{Z})\cong (F\cap R)/[F,R]$, where $[F,R]$ is the commutator subgroup of $G$.
See \cite{R}, \S 10.7.
The group $H_2(G,\mathbb{Z})$ is sometimes called the {\bf Schur multiplier} of $G$. \index{Schur multiplier}
\end{enumerate}
\section{Basic properties of $H^n(G,A)$, $H_n(G,A)$} \label{sec:basisprops}
Let $R$ be a (possibly non-commutative) ring and $A$ be an $R$-module. We say $A$ is {\bf injective} if the functor \index{injective $R$-module} $B\longmapsto {\rm Hom}_G(B,A)$ (from the category of $G$-modules to the category of abelian groups) is exact. \index{injective $R$-module} (Recall $A$ is projective if the functor $B\longmapsto {\rm Hom}_G(A,B)$ is exact.) We say $A$ is {\bf co-induced} if it has the form \index{co-induced $R$-module} ${\rm Hom}_{\mathbb{Z}}(R,B)$ for some abelian group $B$. We say $A$ is {\bf relatively injective} if \index{relatively injective $R$-module} it is a direct factor of a co-induced $R$-module. We say $A$ is {\bf relatively projective} if
\[ \begin{array}{ccc} \pi : & \mathbb{Z} [G]\otimes_\mathbb{Z} A \rightarrow & A,\\
& x\otimes a \longmapsto & xa, \end{array} \] maps a direct factor of $\mathbb{Z} [G]\otimes_\mathbb{Z} A $ isomorphically onto $A$. These are the $G$-modules $A$ which are isomorphic to a direct factor of the induced module $\mathbb{Z} [G]\otimes_\mathbb{Z} A $. \index{relatively projective $R$-module} When $G$ is finite, the notions of relatively injective and relatively projective coincide\footnote{These notions were introduced by Hochschild \cite{Ho}.}.
\begin{enumerate}
\item The definition of $H^n(G,A)$ does not depend on the $G$-resolution $X_*$ of $\mathbb{Z}$ used.
\item If $A$ is an projective $\mathbb{Z} [G]$-module then $H^n(G,A)=0$, for all $n\geq 1$.
This follows immediately from the definitions.
\item If $A$ is an injective $\mathbb{Z} [G]$-module then $H_n(G,A)=0$, for all $n\geq 1$.
See also \cite{S}, \S VII.2.
\item If $A$ is a relatively injective $\mathbb{Z} [G]$-module then $H^n(G,A)=0$, for all $n\geq 1$.
This is Proposition 1 in \cite{S}, \S VII.2.
\item If $A$ is a relatively projective $\mathbb{Z} [G]$-module then $H^n(G,A)=0$, for all $n\geq 1$.
This is Proposition 2 in \cite{S}, \S VII.4.
\item If $A=A'\oplus A''$ then $H^n(G,A)=H^n(G,A')\oplus H^n(G,A'')$, for all $n\geq 0$. More generally, if $I$ is any indexing family and $A=\oplus_{i\in I} A_i$ then $H^n(G,A)=\oplus_{i\in I} H^n(G,A_i)$, for all $n\geq 0$.
This follows from Proposition 10.81 in \S 10.6 of Rotman \cite{R}.
\item If
\[ 0 \rightarrow A {\rightarrow} B {\rightarrow} C {\rightarrow} 0 \] is an exact sequence of $G$-modules then we have a long exact sequence of cohomology (\ref{eqn:LESC}). See \cite{S}, \S VII.2, and properties of the $ext$ functor \cite{R}, \S 10.6.
\item $A\longmapsto H^n(G,A)$ is the higher right derived functor associated to $A\longmapsto A^G={\rm Hom}_G(A,\mathbb{Z})$ from the category of $G$-modules to the category of abelian groups.
This is by definition. See \cite{S}, \S VII.2, or \cite{R}, \S 10.7.
\item If
\[ 0 \rightarrow A {\rightarrow} B {\rightarrow} C {\rightarrow} 0 \] is an exact sequence of $G$-modules then we have a long exact sequence of homology (\ref{eqn:LESH}). In the case of a finite group, see \cite{S}, \S VIII.1. In general, see \cite{S}, \S VII.4, and properties of the ${\rm Tor}\, $ functor in \cite{R}, \S 10.6.
\item $A\longmapsto H_n(G,A)$ is the higher left derived functor associated to $A\longmapsto A_G=\mathbb{Z} \otimes_{\mathbb{Z} [G]}A$ on the category of $G$-modules.
This is by definition. See \cite{S}, \S VII.4, or \cite{R}, \S 10.7.
\item If $G$ is a finite cyclic group then
\[ \begin{split} H_0(G,A) &= A_G,\\ H_{2n-1}(G,A) & = A^G/NA,\\ H_{2n}(G,A) &={\rm Kernel}\, (N)/DA , \end{split} \] for all $n\geq 1$.
To prove this, we need a lemma.
\begin{lemma} \label{lemma:les_cyclic} Let $G=\langle g\rangle$ be acyclic group of order $k$. Let $M=g-1$ and $N=1+g+g^2+...+g^{k-1}$. Then
\[ \dots \rightarrow \mathbb{Z} [G] \stackrel{N}{\rightarrow} \mathbb{Z} [G] \stackrel{M}{\rightarrow} \mathbb{Z} [G] \rightarrow \mathbb{Z} [G] \stackrel{N}{\rightarrow} \mathbb{Z} [G] \stackrel{M}{\rightarrow} \mathbb{Z} [G] \stackrel{\epsilon}{\rightarrow} \mathbb{Z} \rightarrow 0, \] is a free $G$-resolution.
\end{lemma}
{\bf proof of lemma}: It is clearly free. Since $MN=NM=(g-1)(1+g+g^2+...+g^{k-1})=g^k-1=0$, it is a complex. It remains to prove exactness. Since ${\rm Kernel}\, (\epsilon)=D={\rm Image}\, (M)$, by Lemma \ref{lemma:Disfree}, this stage is exact.
To show ${\rm Kernel}\, (M)={\rm Image}\, (N)$, let $x=\sum_{j=0}^{k-1}m_jg^j\in {\rm Kernel}\, (M)$. Since $(g-1)x=0$, we must have $m_0=m_1=...=m_{k-1}$. This forces $x=m_0N\in {\rm Image}\, (N)$. Thus ${\rm Kernel}\, (M)\subset {\rm Image}\, (N)$. Clearly $MN=0$ implies ${\rm Image}\, (N)\subset {\rm Kernel}\, (M)$, so ${\rm Kernel}\, (M)={\rm Image}\, (N)$.
To show ${\rm Kernel}\, (N)={\rm Image}\, (M)$, let $x=\sum_{j=0}^{k-1}m_jg^j\in {\rm Kernel}\, (N)$. Since $Nx=0$, we have $0=\epsilon(Nx)=\epsilon(N)\epsilon(x)=k\epsilon(x)$, so $\sum_{j=0}^{k-1}m_j=0$. Observe that
\[ \begin{array}{ll} x&=m_0\cdot 1+m_1g+m_2g^2+...+m_{k-1}g^{k-1}\\
&=(m_0-m_0g)+(m_0+m_1)g+m_2g^2+...+m_{k-1}g^{k-1}\\
&=(m_0-m_0g)+(m_0+m_1)g-(m_0+m_1)g^2\\
&+(m_0+m_1+m_2)g^2-(m_0+m_1+m_2)g^3+...\\
&+(m_0+..+m_{k-1})g^{k-1}-(m_0+..+m_{k-1})g^{k}. \end{array} \] where the last two terms are actually $0$. This implies $x=-M(m_0+(m_0+m_1)g+(m_0+m_1+m_2)g^2 +...+ (m_0+..+m_{k-1})g^{k-1}\in {\rm Image}\, (M)$. Thus ${\rm Kernel}\, (N)\subset {\rm Image}\, (M)$. Clearly $NM=0$ implies ${\rm Image}\, (M)\subset {\rm Kernel}\, (N)$, so ${\rm Kernel}\, (N)={\rm Image}\, (M)$.
This proves exactness at every stage.$\Box$
Now we can prove the claimed property. By property 1 in \S \ref{sec:homprops}, it suffices to assume $n>0$. Tensor the complex in Lemma \ref{lemma:les_cyclic} on the right with $A$:
{\footnotesize{ \[ \begin{array}{cc} \dots \rightarrow &\mathbb{Z} [G]\otimes_{\mathbb{Z} [G]}A \stackrel{N_*}{\rightarrow} \mathbb{Z} [G]\otimes_{\mathbb{Z} [G]}A \stackrel{M_*}{\rightarrow} \mathbb{Z} [G]\otimes_{\mathbb{Z} [G]}A \stackrel{N_*}{\rightarrow} \\ &\mathbb{Z} [G]\otimes_{\mathbb{Z} [G]}A \stackrel{M_*}{\rightarrow} \mathbb{Z} [G] \otimes_{\mathbb{Z} [G]}A \stackrel{\epsilon}{\rightarrow} \mathbb{Z}\otimes{\mathbb{Z} [G]}A \rightarrow 0, \end{array} \] }} where the new maps are distinguished from the old maps by adding an asterisk. By definition, $\mathbb{Z} [G]\otimes_{\mathbb{Z} [G]}A \cong A$, and by property 1 in \S \ref{sec:homprops}, $\mathbb{Z} \otimes_{\mathbb{Z} [G]}A \cong A/DA$. The above sequence becomes
\[ \dots \rightarrow A \stackrel{N_*}{\rightarrow} A \stackrel{M_*}{\rightarrow} A \stackrel{N_*}{\rightarrow} A \stackrel{M_*}{\rightarrow} A \stackrel{\epsilon}{\rightarrow} A/DA \rightarrow 0. \] This implies, by definition of ${\rm Tor}\, $,
\[ {\rm Tor}\, _{2n-1}^{\mathbb{Z}[G]}(\mathbb{Z},A)={\rm Kernel}\, (M_*)/{\rm Image}\, (N_*) =A^G/NA, \] and \[ {\rm Tor}\, _{2n}^{\mathbb{Z}[G]}(\mathbb{Z},A)={\rm Kernel}\, (N_*)/{\rm Image}\, (M_*) =A[N]/DA. \]
See also \cite{S}, \S VIII.4.1 and the Corollary in \S VIII.4.
\item The group $H^2(G,A)$ classifies group extensions of $A$ by $G$.
This is Theorem 5.1.2 in \cite{W}. See also \S 10.2 in \cite{R}.
\item If $G$ is a finite group of order $m=|G|$ then $mH^n(G,A)=0$, for all $n\geq 1$.
This is Proposition 10.119 in \cite{R}.
\item If $G$ is a finite group and $A$ is a finitely-generated $G$-module then $H^n(G,A)$ is finite, for all $n\geq 1$.
This is Proposition 3.1.9 in \cite{W} and Corollary 10.120 in \cite{R}.
\item The group $H^1(G,A)$ constructed using resolutions is the same as the group constructed using $1$-cocycles. The group $H^2(G,A)$ constructed using resolutions is the same as the group constructed using $2$-cocycles.
This is Corollary 10.118 in \cite{R}.
\item If $G$ is a finite cyclic group then
\[ \begin{split} H^0(G,A) &= A^G,\\ H^{2n-1}(G,A) &={\rm {\rm Kernel}\, }\, N/DA ,\\ H^{2n}(G,A) &= A^G/NA , \end{split} \] for all $n\geq 1$. Here $N:A\rightarrow A$ is the norm map $Na=\sum_{g\in G}ga$ and $DA$ is the augmentation ideal defined above (generated by elements of the form $ga-a$).
{\bf proof}:\ The case $n=0$: By definition, $H^0(G,A)={\rm Ext}\, ^0_{\mathbb{Z}[G]}(\mathbb{Z},A)={\rm Hom}_G(\mathbb{Z},A)$. Define $\tau : {\rm Hom}_G(\mathbb{Z},A)\rightarrow A^G$ by sending $f\longmapsto f(1)$. It is easy to see that this is well-defined and, in fact, injective. For each $a\in A^G$, define $f=f_a\in {\rm Hom}_G(\mathbb{Z},A)$ by $f(m)=ma$. This shows $\tau$ is surjective as well, so case $n=0$ is proven.
Case $n>0$: Applying the functor ${\rm Hom}_G(*,A)$ to the $G$-resolution in Lemma \ref{lemma:les_cyclic} to get
{\footnotesize{ \[ \begin{array}{cc} \dots \leftarrow &{\rm Hom}_G(\mathbb{Z} [G],A) \stackrel{N_*}{\leftarrow} {\rm Hom}_G(\mathbb{Z} [G],A) \stackrel{M_*}{\leftarrow} {\rm Hom}_G(\mathbb{Z} [G],A) \stackrel{\epsilon_*}{\leftarrow} {\rm Hom}_G(\mathbb{Z},A) \leftarrow 0. \end{array} \] }} It is known that ${\rm Hom}_G(\mathbb{Z}[G],A)\cong A$ (see Proposition 8.85 on page 583 of \cite{R}). It follows that
\[ \begin{array}{cc} \dots \leftarrow &A \stackrel{N_*}{\leftarrow} A \stackrel{M_*}{\leftarrow} A \stackrel{\epsilon_*}{\leftarrow} A^G \leftarrow 0. \end{array} \] By definition of ${\rm Ext}\, $, for $n>0$ we have
\[ {\rm Ext}\, _{\mathbb{Z}[G]}^{2n}(\mathbb{Z},A)={\rm Kernel}\, (M_*)/{\rm Image}\, (N_*)=A^G/NA, \] and
\[ {\rm Ext}\, _{\mathbb{Z}[G]}^{2n-1}(\mathbb{Z},A)={\rm Kernel}\, (N_*)/{\rm Image}\, (M_*)={\rm Kernel}\, (N)/(g-1)A, \] where $g$ is a generator of $G$ as in Lemma \ref{lemma:les_cyclic}. $\Box$
See also \cite{S}, \S VIII.4.1 and the Corollary in \S VIII.4.
\item If $G$ is a finite cyclic group of order $m$ and $A$ is a {\it trivial} $G$-module then
\[ \begin{split} H^0(G,A) &= A^G,\\ H^{2n-1}(G,A) &\cong A[m],\\ H^{2n}(G,A) &\cong A/mA, \end{split} \] for all $n\geq 1$.
This is a consequence of the previous property.
\end{enumerate}
\section{Functorial properties}
In this section, we investigate some of the ways in which $H^n(G,A)$ depends on $G$.
One way to construct all these in a common framework is to introduce the notion of a ``homomorphism of pairs''. Let $G,H$ be groups. Let $A$ be a $G$-module and $B$ an $H$-module. If $\alpha :H\rightarrow G$ is a homomorphism of groups and $\beta:A\rightarrow B$ is a homomorphism of $H$-modules (using $\alpha$ to regard $B$ as an $H$-module) then we call $(\alpha,\beta)$ a {\bf homomorphism of pairs}, written \index{homomorphism of pairs}
\[ (\alpha,\beta):(G,A)\rightarrow (H,B). \]
Let $G\subset H$ be groups and $A$ an $H$-module (so, by restriction, a $G$-module). We say a map
\[ f_{G,H}:H^n(G,A)\rightarrow H^n(H,A), \] is {\bf transitive} if $f_{G_2,G_3}f_{G_1,G_2}=f_{G_1,G_2}$, for all subgroups $G_1\subset G_2\subset G_3$.
\index{transitive}
Let $X_*$ be a $G$-resolution and $X'_*$ a $H$-resolution, each with a $-1$ grading. Associated to a homomorphism of groups $\alpha :H\rightarrow G$ is a sequence of $H$-homomorphisms
\begin{equation} \label{eqn:hom_Hom} A_n:X'_n\rightarrow X_n, \end{equation} $n\geq 0$, such that $d_{n+1}A_{n+1}=A_nd'_{n+1}$ and $\epsilon A_0=\epsilon'$.
\begin{theorem} \label{thrm:hom_Hom} \begin{enumerate}
\item If $(\alpha,\beta):(G,A)\rightarrow (G',A')$ and $(\alpha',\beta'):(G',A')\rightarrow (G'',A'')$ are homomorphisms of pairs then so is $(\alpha'\circ \alpha,\beta'\circ \beta):(G,A)\rightarrow (G'',A'')$.
\item Suppose $(\alpha,\beta):(G,A)\rightarrow (G',A')$ is homomorphism of pairs, $X_*$ is a $G$-resolution, and $X'_*$ is a $G'$-resolution (each infinite in both directions, with a $-1$ grading). Let $H^n(G,A,X_*)$ denote the derived groups associated to the differential groups ${\rm Hom}_G(X_*,A)$ with $+1$ grading. There is a homomorphism
\[ (\alpha,\beta)_{X_*,X'_*}:H^n(G,A,X_*)\rightarrow H^n(G',A',X'_*) \] satisfying the following properties.
\begin{enumerate}
\item If $G=G'$, $A=A'$, $X=X'$, $\alpha=1$ and $\beta=1$ then $(1,1)_{X_*,X'_*}=1$.
\item If $(\alpha',\beta'):(G',A')\rightarrow (G'',A'')$ is homomorphism of pairs, $X''_*$ is a $G''$-resolution then \[ (\alpha'\circ \alpha,\beta'\circ \beta)_{X_*,X''_*}= (\alpha',\beta')_{X'_*,X''_*}\circ (\alpha,\beta)_{X_*,X'_*}. \]
\item If $(\alpha,\gamma):(G,A)\rightarrow (G',A')$ is homomorphism of pairs then \[ (\alpha,\beta+ \gamma)_{X_*,X'_*}= (\alpha,\beta)_{X_*,X'_*}+ (\alpha,\gamma)_{X_*,X'_*}. \]
\end{enumerate} \end{enumerate} \end{theorem}
\begin{remark} For an analogous result for homology, see \S\S III.8 in Brown \cite{B}. \end{remark}
{\bf proof}:\ We sketch the proof, following Weiss, \cite{W}, Theorem 2.1.8, pp 52-53.
(1): This is ``obvious''.
(2): Let $(\alpha,\beta):(G,A)\rightarrow (G',A')$ be a homomorphism of pairs. Using (\ref{eqn:hom_Hom}), we have an associated chain map
\[ \alpha^*: {\rm Hom}_{G}(X_*,A)\rightarrow {\rm Hom}_{G'}(X'_*,A') \] of differential groups (Brown \S III.8 in \cite{B}). The homomorphism of cohomology groups induced by $\alpha^*$ is denoted
\[ \alpha^*_{n,X_*,X'_*}:H^n(G,A,X_*)\rightarrow H^n(G',A',X'_*). \] Properties (a)-(c) follow from \S \ref{sec:properties} and the corresponding properties of $\alpha^*$. $\Box$
As the cohomology groups are independent of the resolution used, the map $(\alpha,\beta)_{X_*,X'_*}:H^n(G,A,X_*)\rightarrow H^n(G',A',X'_*)$ is sometimes simply denoted by
\begin{equation} \label{eqn:hompair} (\alpha,\beta)_{*}:H^n(G,A)\rightarrow H^n(G',A'). \end{equation}
\subsection{Restriction}
Let $X_*=X_*(G)$ denote the bar resolution.
If $H$ is a subgroup of $G$ then the cycles on $G$, $C^n(G,A)={\rm Hom}_G(X_n(G),A)$, can be restricted to $H$: $C^n(H,A)={\rm Hom}_H(X_n(H),A)$. The restriction map $C^n(G,A)\rightarrow C^n(H,A)$ leads to a map of cohomology classes: \[ {\rm Res}\, :H^n(G,A)\rightarrow H^n(H,A). \]
In this case, the homomorphism of pairs is given by the inclusion map $\alpha:H\rightarrow G$ and the identity map $\beta:A\rightarrow A$. The map ${\rm Res}\, $ is the induced map defined by (\ref{eqn:hompair}). By the properties of this induced map, we see that ${\rm Res}\, _{H,G}$ is transitive: if $G\subset G'\subset G''$ then\footnote{There is an analog of the restriction for homology which also satisfies this transitive property (Proposition 9.5 in Brown \cite{B}).} \[ {\rm Res}\, _{G',G}\circ {\rm Res}\, _{G'',G'}={\rm Res}\, _{G'',G}. \]
A particularly nice feature of the restriction map is the following fact.
\begin{theorem} If $G$ is a finite group and $G_p$ is a $p$-Sylow subgroup and if $H^n(G,A)_p$ is the $p$-primary component of $H^n(G,A)$ then
(a) there is a canonical isomorphism $H^n(G,A)\cong \oplus_p H^n(G,A)_p$, and
(b) $Res:H^n(G,A)\rightarrow H^n(G_p,A)$ restricted to $H^n(G,A)_p$ (identified with a subgroup of $H^n(G,A)$ via (a)) is injective.
\end{theorem}
{\bf proof}:\ See Weiss, \cite{W}, Theorem 3.1.15. $\Box$
\begin{example} \label{ex:sylow} {\rm Homology is a functor. That is, for any $n > 0$ and group homomorphism $f : G \rightarrow G'$ there is an induced homomorphism $H_n(f) : H_n(G,\mathbb{Z}) \rightarrow H_n(G',\mathbb{Z})$ satisfying
\begin{itemize}
\item $H_n(gf) = H_n(g)H_n(f)$ for group homomorphisms $f : G \rightarrow G'$ $g : G' \rightarrow G''$, \item $H_n(f)$ is the identity homomorphism if $f$ is the identity. \end{itemize}
The following commands compute $H_3(f) : H_3(P,\mathbb{Z}) \rightarrow H_3(S_5,\mathbb{Z})$ for the inclusion $f : P \hookrightarrow S_5$ into the symmetric group $S_5$ of its Sylow $2$-subgroup. They also show that the image of the induced homomorphism $H_3(f)$ is precisely the Sylow $2$-subgroup of $H_3(S_5,\mathbb{Z})$.
\vskip .2in
{\footnotesize{ \begin{Verbatim}[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label={\tt GAP}]
gap> S_5:=SymmetricGroup(5);; P:=SylowSubgroup(S_5,2);; gap> f:=GroupHomomorphismByFunction(P,S_5, x->x);; gap> R:=ResolutionFiniteGroup(P,4);; gap> S:=ResolutionFiniteGroup(S_5,4);; gap> ZP_map:=EquivariantChainMap(R,S,f);; gap> map:=TensorWithIntegers(ZP_map);; gap> Hf:=Homology(map,3);; gap> AbelianInvariants(Image(Hf)); [2,4] gap> GroupHomology(S_5,3); [2,4,3]
\end{Verbatim} }}
} \end{example}
If $H$ is a subgroup of finite index in $G$ then there is an analogous restriction map in group homology (see for example Brown \cite{B}, \S III.9).
\subsection{Inflation}
Let $X_*$ denote the bar resolution of $G$. Recall \[ X_n=\oplus_{(g_1,\dots ,g_n)\in G^n} R[g_1,\dots ,g_n], \] where the sum runs over all ordered $n$-tuples in $G^n$. If $H$ is a subgroup of $G$, let $X^H_*$ denote the complex defined by \[ X^H_n=\oplus_{(g_1H,\dots ,g_nH)\in (G/H)^n} R[g_1H,\dots ,g_nH]. \] This is a resolution, and we have a chain map defined on $n$-cells by $[g_1,\dots ,g_n]\longmapsto [g_1H,\dots ,g_nH]$.
Suppose that $H$ is a normal subgroup of $G$ and $A$ is a $G$-module. We may view $A^H$ as a $G/H$-module. In this case, the homomorphism of pairs is given by the quotient map $\alpha:G\rightarrow G/H$ and the inclusion map $\beta:A^H\rightarrow A$. The {\bf inflation} map ${\rm Inf}\, $ is the induced map defined by (\ref{eqn:hompair}), denoted \index{inflation}
\[ {\rm Inf}\, :H^n(G/H,A^H)\rightarrow H^n(G,A). \]
The {\bf inflation-restriction sequence in dimension $n$} is \index{inflation-restriction sequence in dimension $n$}
\[ 0\rightarrow H^n(G/H,A^H)\stackrel{{\rm Inf}\, }{\rightarrow} H^n(G,A) \stackrel{{\rm Res}\, }{\rightarrow} H^n(H,A). \] For a proof, see Weiss, \cite{W}, \S 3.4.
There an analog of this inflation-restriction sequence for homology.
We omit any discussion of transfer and Shapiro's lemma, due to space limitations.
{\it Acknowledgements}: I thank G. Ellis, M. Mazur and J. Feldvoss, P. Guillot for correspondence which improved the content of these notes.
\end{document} |
\begin{document}
\date{} \address{\textrm{(Akram Aldroubi)} Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240-0001 USA} \email{aldroubi@math.vanderbilt.edu}
\address{\textrm{(Longxiu Huang)}
Department of Mathematics,
Vanderbilt University,
Nashville, Tennessee 37240-0001 USA} \email{longxiu.huang@vanderbilt.edu}
\address{\textrm{(Armenak Petrosyan)} Computational and Applied Mathematics Group, Oak Ridge National Laboratory, 1 Bethel Valley Rd, Oak Ridge, Tennessee 37830, USA} \email{petrosyana@ornl.gov}
\thanks{ The authors were supported in part by NSF Grant DMS- 1322099. }
\keywords{continuous frames, semi-continuous frames, dynamical sampling, discretization of continuous frames, frames induced by continuous powers of operators} \subjclass [2010] {46N99, 42C15, 94O20}
\title{Frames Induced by the Action of Continuous Powers of an Operator} \begin{abstract}
We investigate systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ where $A \in B(\mathcal{H})$ is a normal operator in a separable Hilbert space $\mathcal{H}$, $\mathcal{G}\subset \mathcal{H}$ is a countable set, and $L$ is a positive real number. Although the main goal of this work is to study the frame properties of $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$, as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science. \end{abstract} \section{Introduction} In a foundational paper, Duffin and Schaeffer introduced the theory of frames in the context of non-harmonic Fourier series \cite{DS52}. In this remarkable paper, the authors first gave conditions on a sequence of real numbers $\{\lambda_n\}_{n\in \mathbb{Z}}$ that induce a Riesz basis of exponentials $\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}$ for $L^2(- \frac 1 2, \frac 1 2)$. They then proposed the concept of frames which generalizes that of Riesz bases. Specifically, a frame $\{\phi_n\}_{n \in \mathbb{Z}}$ in a separable Hilbert space $\mathcal{H}$ is a sequence of vectors satisfying \begin{equation} \label {DSframes}
c\left\|f\right\|^2\le \sum\limits_{n\in \mathbb{Z} }\left\vert \la f,\phi_n\ra \right\vert^2\le C\|f\|^2, ~ \text {for all } f \in \mathcal{H}, \end {equation} for some positive constants $c,C>0$. They showed that, if \eqref {DSframes} holds, then (similar to a Riesz basis) any function $f\in \mathcal{H}$ can be represented by the series \[ f=\sum\limits_{n\in \mathbb{Z} } \la f,\phi_n\ra \tilde \phi_n, \] where $\{ \tilde \phi_n \}_{n\in \mathbb{Z}}$ is a dual frame and the convergence of the series is unconditional. Thus, every Riesz basis is a frame but a frame may have redundant vectors and hence need not be a basis. However, the relation between Riesz bases and redundant frames is not self-evident. For example, there are frames for $\mathcal{H}$ that have no subsequences that are Riesz bases for $\mathcal{H}$ (see, e.g., \cite {Christ16, Heil11} and the references therein).
The conditions on $\{\lambda_n\}_{n\in \mathbb{Z}}$ under which a system of exponentials $\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}$ becomes a frame for $L^2(- \frac 1 2, \frac 1 2)$ is also obtained in \cite{DS52}. Using the Fourier transform, a set $\{e^{i\lambda_n t}\}_{n\in\mathbb{Z}}$ is a frame for $L^2(- \frac 1 2, \frac 1 2)$ if and only if any function $f$ is in the Paley-Wiener space $PW_{1/2}=\{f \in L^2(\mathbb{R}): \hat f (\xi)=0 ~a.e.~\xi \notin (- \frac 1 2, \frac 1 2) \}$ can be recovered from its samples $\{f(\lambda_n)\}_{n\in \mathbb{Z}}$ in a {\em stable way}, i.e., there exists a bounded operator $R:\ell^2(\mathbb{Z}) \to \mathcal{H}$ such that $R(f(\lambda_n))=f$. This duality between reconstruction of functions from samples and frames has been used and extended in many directions, including for wavelet representations, time-frequency analyses, and sampling in shift-invariant spaces (see, e.g., \cite {ABK08, AG01, CCK13, DS16, Daubechies_1992, Han09, Mallat09, BM08, NO12, Sun14, ST17}).
\subsection{Dynamical sampling and frames induced by the action of continuous powers of an operator}
\subsubsection {Dynamical sampling} The general problem in sampling theory is to reconstruct a function $f$ in a separable Hilbert space $\mathcal{H}$ from its samples. A natural idea is to sample the function $f$ at many accessible positions and one expects that, with some {\it a priori} information, $f$ can be reconstructed from those samples. This idea is precisely the impetus of classical sampling theory. Related results can be found in, e.g., \cite{Adcock_2012,ABK08,AG01,BG12,Cand_s_2006,S01,Su06}. However, in real-world applications, there are many restrictions. For example, sampling may not be accessible at some required locations. Moreover, the spatial sampling density can be very limited, because sensors are often expensive and it is costly to achieve a high sampling density.
In many instances, the functions evolve over time by a known driving operator. A common example is provided by diffusion and modeled by the heat equation \cite{Lu_2009}. For such functions, a novel theory has been developed recently, and it is termed {\it dynamical sampling theory}. The general idea of dynamical sampling is to reduce the spatial sampling density by increasing the temporal sampling rate.
In dynamical sampling, the samples $\{(A^{t_j}f)(x_i): i\in \mathbb{Z}, j=0,\dots,J\}$ are taken repeatedly over time at some fixed spatial locations $X=\{x_i\}_{i\in \mathbb{Z}}$. Since the operator $A$ driving the evolution of $f$ can combine the information of $f$ from different locations, one may expect to recover the original function $f$ from $\{(A^{t_j}f)(x_i): i\in \mathbb{Z}, t_j\in T\}$, if the sampling locations are well chosen, the operator $A$ is well-behaved, and the time-set $T=\{t_0,\dots, t_J\}$ (or $T=[0,L]$) is large enough. The dynamical sampling problem is to derive necessary and sufficient conditions in terms of the operator $A$, the sampling set $X$, and the sampling time-set $T$ such that the samples from different time levels are adequate to recover the original signal.
\subsubsection {Frames induced by the action of powers of an operator} The mathematical formulation of dynamical sampling can be stated as follows. Let $A$ be a bounded linear operator on a separable Hilbert space $\mathcal{H}$, and let $f\in\mathcal{H}$ be the initial state of an evolution system. At time $t$, the initial signal $f$ evolves to become $$f_t=A^t f.$$ Given a countable (finite or countably infinite) set of vectors $\mathcal{G} \subset \mathcal{H}$, the task is to find conditions on $A\in B(\mathcal{H})$, $\mathcal{G}$, and $T\subset [0, \infty)$ that allow the recovery or stable recovery of any function $f\in\mathcal{H}$ from the set of samples \begin{equation} \left\{\langle A^tf,g\rangle: g\in\mathcal{G}, t\in T\right\}. \end{equation} By the recovery of $f$ we mean that there exists an operator $R$ from $\mathcal{G}\times T$ to $\mathcal{H}$ such that $R\big(\langle A^tf,g\rangle\big)=f$ for all $ f \in \mathcal{H}$. While by stable recovery of $f$ we mean that the operator $R$ is bounded. The problem above is equivalent to finding conditions on $A$, $\mathcal{G}$, and $T$ such that $\{A^{*t}g\}_{g\in\mathcal{G},t\in T}$ (where $A^*$ denotes the adjoint of $A$) is complete or a continuous frame for $\mathcal{H}$, where the notion of continuous frames generalizes that in \eqref {DSframes} \cite{Ali_1993,Ali_2000,Fornasier_2005,Gabardo_2003}. \begin{definition}
Let $\mathcal{H}$ be a complex Hilbert space and let $(\Omega, \mu)$ be a measure space with positive measure $\mu$. A mapping $F : \Omega \rightarrow \mathcal{H}$ is called a frame with respect to $(\Omega,\mu)$, if
\begin{enumerate}[label=(\roman*)]
\item F is weakly-measurable, i.e., $\omega \rightarrow \langle f, F(\omega)\rangle$ is a measurable
function on $\Omega$ for all $ f\in\mathcal{H}$;
\item there exist constants $c$ and $C > 0$ such that
\begin{equation}\label{ConFrame}
c\|f\|^2\leq \int_{\Omega}|\langle f,F(\omega)\rangle|^2d\mu(\omega)\leq C\|f\|^2,\text{ for all } f\in\mathcal{H}.
\end{equation}
\end{enumerate}
Here the constants $c$ and $C$ are called continuous frame (lower and upper) bounds. In addition, F is called a tight continuous frame if $c=C$. The mapping F is called Bessel if the second inequality in \eqref{ConFrame} holds.
In this case, C is called a Bessel constant. \end{definition} The frame operator $S=S_{F}$ on $\mathcal{H}$ associated with $F$ is defined in the weak sense by $$S_{F}f=\int_{\Omega}\langle f,F(\omega)\rangle F(\omega)d\mu(\omega).$$ According to \eqref{ConFrame}, $S_F$ is well defined, invertible with bounded inverse (see \cite{Fornasier_2005}). Thus every $f\in\mathcal{H}$ has the representations \[f=S_{F}^{-1}S_F f=\int_{\Omega}\langle f,F(\omega)\rangle S_{F}^{-1}F(\omega)d\mu(\omega), \] \[f=S_{F}S_F^{-1} f=\int_{\Omega}\langle f,S_F^{-1}F(\omega)\rangle F(\omega)d\mu(\omega). \]
If $\mu$ is the counting measure and $\Omega=\mathbb{N}$, then one gets back the Duffin-Schaffer frame in \eqref {DSframes}.
In the sequel, $\Omega=\mathcal{G}\times[0,L]$, and $\mu$ is the product of the counting measure on $\mathcal{G}$ and the Lebesgue measure on $[0,L]$. In this case, $F$ is called a {\em semi-continuous frame} and \eqref {ConFrame} becomes \begin {equation}\label {SCF}
c\|f\|^2\leq\sum\limits_{g\in\mathcal{G}}\int\limits_{0}^{L}|\la f,A^tg\ra|^2 dt \leq C\|f\|^2,\text{ for all } f\in\mathcal{H}. \end{equation} \subsection{Recent results on dynamical sampling and frames} Existing studies on various aspect of the dynamical sampling problem and related frame theory grew out of the work in \cite{AADP13, ACCMP17, ACMT17, ADK13, LuDV:11, RCLV11}, see, for example, \cite{AH17, CMPP17, CJS15,CH17, JT14, JD15, JD17, Phi17, ZLL17_2, ZLL17} and the references therein. However, except for a few, they all focus on uniform discrete time-sets $T\subseteq\{0,1,2,\ldots\}$, e.g., $T=\{1,\dots,N\}$ or $T=\mathbb{N}$ (see e.g., \cite {GRUV15}).
Even though the general dynamical sampling problem for discrete-time sets in finite dimensions (hence problems of systems and frames induced by iterations $\{A^ng:g \in \mathcal{G}, n \in T\}$) have been mostly resolved in \cite{ACMT17}, many problems and conjectures remain open for the infinite dimensional case. This state of affairs is not surprising because some of these problems take root in the deep theory of functional analysis and operator theory such as the Kadison Singer Theorem \cite {MSS15}, some open generalizations of the M\"untz-Sz\'asz Theorem \cite{Rudinrc}, and the famous invariant subspace conjecture.
When $T=\mathbb{N}$ and $A \in \mathcal B(\mathcal{H})$, it is not difficult to show that \begin{theorem}[\cite{AP17}]\label{noframeany}
If, for an operator $A\in B(\mathcal{H})$, there exists a countable set of vectors $\mathcal{G} $ in $ \mathcal{H} $ such that $\{A^n g\}_{g\in \mathcal{G}, \;n\geq 0}$ is a frame in $\mathcal{H}$, then for every $f\in \mathcal{H}$, $(A^*)^nf\to 0$ as $n\to \infty$. \end{theorem} Thus, in particular it is not possible to construct frames using non-negative iterations when $A$ is a unitary operator. For example, the right-shift operator $S$ on $\mathcal{H}=\ell^2(\mathbb{N})$ generates an orthonormal basis for $\ell^2(\mathbb{N})$ by iterations over $\mathcal{G}=\{(1,0,\dots,)\}$. Clearly, $(S^*)^nf\to 0$ as $n\to \infty$ for this case. However, if we change the space to $\mathcal{H}=\ell^2(\mathbb{Z})$, the right-shift operator $S$ becomes unitary, and there exists no subset $\mathcal{G}$ of $\ell^2(\mathbb{Z})$ such that $\{S^n g\}_{g\in \mathcal{G}, \;n\geq 0}$ is a frame for $\ell^2(\mathbb{Z})$.
On the other hand, for normal operators, it is possible to find frames of the form $\{A^n g\}_{g\in \mathcal{G}, \;n\geq 0}$; however, no such a frame can be a basis \cite{ACCMP17}.
Frames for $\mathcal{H}$ can be generated by the iterative action on a single vector $g$, i.e., there exist normal operators and associated cyclic vectors such that $\{A^n g\}_{\;n\geq 0}$ is a frame for $\mathcal{H}$ \cite{ACMT17}. Specifically, \begin{theorem} [\cite{ACCMP17}]\label {OnePointFrame}
Let $A$ be a bounded normal operator on an infinite dimensional Hilbert space $\mathcal{H}$. Then, $\{A^n g\}_{n\geq 0}$ is a frame for $\mathcal{H}$ if and only if the following five conditions are satisfied:
(i) $A=\sum_j\lambda_jP_j$, where $ P_j $ are rank one orthogonal projections;
(ii) $|\lambda_k| < 1$ for all $k$;
(iii) $|\lambda_k| \to 1$;
(iv) $\{\lambda_k\}$ satisfies Carleson's condition
$
\inf_{n} \prod_{k\neq n} \frac{|\lambda_n-\lambda_k|}{|1-\bar{\lambda}_n\lambda_k|}\geq \delta,
$
for some $\delta>0$;
and
(v) $0<c\le \frac {\|P_jg\|} {\sqrt{1-|\lambda_k|^2}} \le C< \infty$, for some constants $c, C$. \end{theorem}
It turns out that if $A$ is normal in an infinite dimensional Hilbert space $\mathcal{H}$, and $\{A^n g\}_{g\in \mathcal{G}, \;n\geq 0}$ is a frame for some $ \mathcal{G} \subset \mathcal{H}$ with $|\mathcal{G}| < \infty$, then $A$ is necessarily of the form described in Theorem \ref {OnePointFrame}: \begin{theorem}[\cite{AP17}]\ \label{cor53}
Let $A$ be a bounded normal operator in an infinite dimensional Hilbert space $\mathcal{H}$. If the system of vectors $\{A^n g\}_{g\in \mathcal{G}, \;n\geq 0}$ is a frame for some $ \mathcal{G} \subset \mathcal{H}$ with $|\mathcal{G}| < \infty$, then $A=\sum_j\lambda_jP_j$ where $ P_j $ are projections such that ${\rm rank\,}(P_j)\leq |\mathcal{G}|$ $\left(\text{i.e., the global multiplicity of $A$ is less than or equal to $ |\mathcal{G}| $}\right).$ In addition, (ii) and (iii) of Theorem \ref {OnePointFrame} are satisfied. \end{theorem}
The necessary and sufficient conditions generalizing Theorem \ref {OnePointFrame} for the case $1<|G|<\infty$ have been derived in \cite {CMPP17}.
Viewing Theorem \ref {OnePointFrame} from a different perspective, Christensen and Hasannasab ask whether a frame $\{h_n\}_{n \in I}$ has a representation of the form $h_n=A^nh_0$ for some operator $A$ when $I=\mathbb{N}\cup \{0\}$ or $I=\mathbb{Z}$. This question is partially answered in \cite{CH17OpRep} and gives rise to many new open problems and conjectures \cite{CH17}.
The set of self-adjoint operators is an important class of normal operators because it is often encountered in applications. For this class, one can rule out certain types of normalized frames. \begin{theorem} [\cite{ACMT17}] \label{normframe}
If $A$ is a self-adjoint operator on $\mathcal{H}$, then the system $\left\{\frac{A^n g}{\norm{A^n g}}\right\}_{g \in \mathcal{G},\;n\geq 0}$ is not a frame for $\mathcal{H}$. \end{theorem} However, for normal operators, the following conjecture remains open: \begin {conjecture} \label {normframeConj} The statement of Theorem \ref {normframe} holds for normal operators. \end {conjecture} Conjecture \ref {normframeConj} does not hold if the operator is not normal. For example, the shift-operator $S$ on $\ell^2(\mathbb{N})$ defined by $S(x_1,x_2,\dots)=(0,x_1,x_2,\dots)$, is not normal, and $\{S^ne_1\}$ is an orthonormal basis for $\ell^2(\mathbb{N})$, where $e_1=(1,0,\dots )$.
\subsection{Contributions and Organization} The present work concentrates on systems of the form $\{A^tg:g\in\mathcal{G},t\in[0,L]\} \subseteq \mathcal{H}$, where $A\in\mathcal B(\mathcal{H})$. The goal is to study the frame property of such systems. To this end, we need to derive some other properties in the intermediate steps. In particular, we study the completeness and Besselness of these systems.
For the completeness of $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$, necessary and sufficient conditions are derived in Section \ref{completeness section}. In light of the results in \cite{ACCMP17}, the form of the necessary and sufficient conditions are not surprising. However, the proofs and reductions to the known cases are appealing due to the use of certain techniques of complex analysis, and they are useful for the analysis of frames in the subsequent sections.
The Bessel property of the system $\{A^tg:g\in\mathcal{G},t\in[0,L]\}$ is investigated in Section \ref{bessel section}. Specifically, if $\mathcal{H}$ is a finite dimensional space (e.g., $\mathbb{C}^d$) and $A$ is a normal operator in $\mathcal{H}$, then the system $\{A^tg\}_{g\in\mathcal{G}, t\in[0,L]}$ being Bessel is equivalent to the Besselness of $\mathcal{G}$ in the space $Range(A)$. On the other hand, if $\mathcal{H}$ is an infinite dimensional separable Hilbert space and $A$ is a bounded invertible normal operator, then the only condition ensuring that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel is that $\mathcal{G}$ itself is a Bessel system in $\mathcal{H}$. In addition, an example is described to explain that the non-singularity of $A$ is necessary for the equivalence between the Besselness of $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ and that of $\mathcal{G}$.
Section 5 is devoted to the relations between a semi-continuous frame $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ generated by the action of an operator $A\in \mathcal{B}(\mathcal{H})$ and the discrete systems generated by its time discretization. Specifically, we show that under some mild conditions, $\{A^{t}g\}_{g\in\mathcal{G},t\in [0,L]}$ is a semi-continuous frame if and only if there exists $T=\{t_i:i=I\}\subseteq[0,L)$ with $|I|<\infty$ such that $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a frame system in $\mathcal{H}$. Additionally, Theorem \ref{SCFrSA} shows that under proper conditions, the property that $\{A^t g\}_{g\in\mathcal{G}, t\in [0,L]}$ is a semi-continuous frame is independent of $L$.
\section{Notation and preliminaries}\label{sec2}
\subsection{Normal operators} Let $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on a complex separable Hilbert space $\mathcal{H}$. In the sequel, all the operators are assumed to be normal. Normal operators have the following invertibility property (see \cite[Theorem 12.12]{rudinfa91}).
\begin{theorem}
If $A\in\mathcal{B}(\mathcal{H})$, then $A$ is invertible $\left(\text{i.e., $A$ has bounded inverse}\right)$ if and only if there exists $c>0$ such that $\|Af\|\geq c\|f\|$ for all $f\in\mathcal{H}$. \end{theorem}
For completeness, the spectral theorem with multiplicity is stated below, and the following notation is used in its statement.
For a non-negative regular Borel measure $\mu$ on $\mathbb{C}$, $N_{\mu}$ will denote the multiplication operator acting on $ L^2(\mu)$, i.e., for a $\mu$-measurable function $f:\mathbb{C}\to \mathbb{C}$ such that $\int_\mathbb{C} |f(z)|^2d\mu(z)< \infty$, $$N_{\mu}f(z)=zf(z).$$
We will use the notation $[\mu]=[\nu]$ to denote two mutually absolutely continuous measures $\mu$ and $\nu$.
The operator $N_{\mu}^{(k)}$ will denote the direct sum of $k$ copies of $N_{\mu}$, i.e., \begin{equation*} (N_\mu)^{(k)}=\oplus_{i=1}^{k}N_{\mu}. \end{equation*} Similarly, the space $(L^2(\mu))^{(k)}$ will denote the direct sum of $k$ copies of $L^2(\mu)$.
\begin{theorem}[\textbf{Spectral theorem with multiplicity}]\label{spectral theorem} For any normal operator $A$ on $\mathcal{H}$ there are mutually singular non-negative Borel measures $\mu_j, 1\leq j\leq\infty$, such that $A$ is equivalent to the operator
$$N_{\mu_{\infty}}^{(\infty)}\oplus N_{\mu_1}
\oplus N_{\mu_2}^{(2)}\oplus\ldots,$$
i.e., there exists a unitary transformation
$$U:\mathcal{H}\rightarrow(L^2(\mu_{\infty}))^{(\infty)}\oplus L^2(\mu_1)\oplus (L^2(\mu_2))^{(2)}\oplus\ldots$$ such that
\begin{equation}\label{representation of normal}
UAU^{-1}=N_{\mu_{\infty}}^{(\infty)}\oplus N_{\mu_1}\oplus N_{\mu_2}^{(2)}\oplus\ldots.
\end{equation}
Moreover, if $\tilde{A}$ is another normal operator with corresponding measures $\nu_{\infty},\nu_1,\nu_2,\ldots$, then $\tilde{A}$ is unitary equivalent to A if and only if $[\nu_j]=[\mu_j]$ for $j=1,\ldots,\infty.$ \end{theorem} A proof of the theorem can be found in \cite[Ch. IX, Theorem 10.16]{conway} and \cite[Theorem 9.14]{conway1}.
Since the measures $\mu_j$ are mutually singular, there are mutually disjoint Borel sets $\{\mathcal{E}_j\}_{j=1}^{\infty}\cup \{\mathcal{E}_\infty\}$ such that $\mu_j$ is supported on $\mathcal{E}_j$ for every $1\leq j\leq\infty$. The scalar-valued spectral measure $\mu$ associated with the normal operator $A$ is defined as \begin{equation}\label{scalar spectral measure} \mu=\sum_{1\leq j\leq\infty}\mu_j. \end{equation}
The Borel function $m_{A}:\mathbb{C}\rightarrow \mathbb{N}^*\cup\{0\}$ given by \begin{equation} m_{A}(z)=\infty\cdot\chi_{\mathcal{E}_{\infty}}(z)+\sum_{j=1}^{\infty}j\chi_{\mathcal{E}_j}(z) \end{equation} is called the multiplicity function of the operator $A$, where $\mathbb{N}$ is the set of natural numbers starting with $1$, $\mathbb{N}^*=\mathbb{N}\cup\{\infty\}$, $\chi_{E}(z)$ is the characteristic function on set $E$ defined by $\chi_{E}(z)=\begin{cases} 1,z\in E\\ 0, \text{otherwise} \end{cases}$ and $\infty\cdot 0=0.$
From Theorem \ref{spectral theorem}, every normal operator is uniquely determined, up to a unitary equivalence, by the pair $([\mu],m_A)$.
For $j\in \mathbb{N}$, let $\Omega_j$ be the set $ \{1,...,j\}$ and let $\Omega_\infty$ be the set $\mathbb{N}$. Then $\ell^2(\Omega_j) \cong \mathbb{C}^j$, for $j\in \mathbb{N}$, and $\ell^2(\Omega_\infty) = \ell^2(\mathbb{N}).$ For $j=0$, we use $\ell^2(\Omega_0)$ to represent the trivial space $\{0\}$.
Let $\mathcal{W}$ be the Hilbert space $$\mathcal{W} =(L^2(\mu_\infty))^{(\infty)}\oplus L^2(\mu_1)\oplus (L^2(\mu_2))^{(2)}\oplus\cdots$$ associated with the operator $A$ and let $U:\mathcal{H} \rightarrow \mathcal{W}$ be the unitary operator given by Theorem~\ref{spectral theorem}. If $g\in \mathcal{H}$, we denote by $\widetilde{g}$ the image of $g$ under $U$. Since $\widetilde{g} \in \mathcal{W}$, one has $\widetilde{g} = (\widetilde{g}_j)_{j\in \mathbb{N}^*}$, where $\widetilde{g}_j$ is the restriction of $\widetilde g$ to $(L^2(\mu_j))^{(j)}$. Thus, for any $j\in \mathbb{N}^*$, $ \widetilde{g}_j $ is a function from $ \mathbb{C} $ to $ \ell^2(\Omega_j) $ and
$$\sum_{j\in \mathbb{N}^*} \quad \int_\mathbb{C} \|\widetilde{g}_j(z)\|_{\ell^2(\Omega_j)}^2d\mu_j(z) =\|g\|^2<\infty .$$ Let $P_j$ be the projection defined for every $\widetilde g \in \mathcal W$ by $P_j\widetilde g=\widetilde f$, where $\widetilde f_j=\widetilde g_j$ and $\widetilde f_k=0$ for $k\ne j$.
Let $E$ be the spectral measure for the normal operator $A$. Then, for every $\mu$-measurable set $G\subseteq \mathbb{C}$ and vectors $f,g$ in $\mathcal{H}$, one has the following formula $$\la E(G)f,g\ra _{\mathcal{H}}\; = \int_G\left[\sum_{1\leq j\leq\infty} \chi_{\mathcal{E}_j}(z)\la \widetilde f_j(z),\widetilde g_j(z)\ra _{\ell^2(\Omega_j)}\right]d\mu(z),$$ which relates the spectral measure of $A$ to the scalar-valued spectral measure of $A$. \begin{definition}
Given a normal operator $A$, $A^t$ is defined as follows: $$A^t:\mathcal{H}\rightarrow\mathcal{H}$$ by
$$\langle A^tf_1, f_2\rangle=\int_{z\in\sigma(A)}z^t\langle \tilde{f}_1(z),\tilde{f}_2(z)\rangle d\mu(z), \text{ for all }f_1,f_2\in\mathcal{H},$$ where $z^t=\exp(t(\ln(|z|)+i\arg(z)))$ and $\arg(z)\in[-\pi,\pi).$
Using the fact that $\exp(i\arg(z)+i\arg(\bar z))=1,$ it follows that $(A^*)^t=(A^t)^*$ for $t\in\mathbb{R}$.
\end{definition} Section \ref{completeness section} will exploit the reductive operators which were introduced by P.Halmos and J.Wermer \cite{halmos,wermer}. For clarity, the definition is given as follows. \begin{definition}
A closed subspace $V\subseteq\mathcal{H}$ is called reducing for the operator $A$ if both $V$ and its orthogonal complement $V^{\perp}$ are invariant subspaces of $A$. \end{definition} \begin{definition}
An operator $A$ is called reductive if every invariant subspace of $A$ is reducing. \end{definition} \subsection{Holomorphic Function} The techniques of complex analysis, e.g., the properties of holomorphic functions (see \cite{Conway_1973,Rudinrc} and the references therein), are used extensively in the present work, including Montel's Theorem as stated below. \begin{definition}[\textbf{Normal family}]
A family $\mathfrak{F}$ of holomorphic functions in a region $X$ of the complex plane with values in $\mathbb{C}$
is called normal if every sequence in $\mathfrak{F}$ contains a subsequence which converges uniformly to a holomorphic function on compact subsets of X. \end{definition} \begin{theorem}[\textbf{Montel's Theorem}] \label{Montel's theorem} A uniformly bounded family of holomorphic functions defined on an open subset of the complex numbers is normal. \end{theorem}
\section{Completeness}\label{completeness section} In this section, we characterize the completeness of the system $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$, where $A$ is a (reductive) normal operator on a separable Hilbert space $\mathcal{H}$, $\mathcal{G}$ is a set of vectors in $\mathcal{H}$, and $L$ is a finite positive number. \begin{theorem}\label{theorem 3.1}
Let $A\in\mathcal{B}(\mathcal{H})$ be a normal operator,
and let $\mathcal{G}$ be a countable set of vectors in $\mathcal{H}$ such that $\{A^tg\}_{g\in\mathcal{G},t\in [0,L]}$ is complete in $\mathcal{H}$. Let $\mu_{\infty},\mu_1,\mu_2,\ldots$ be the measures in the representation \eqref{representation of normal} of the operator $A$. Then for every $1\leq j\leq\infty$ and $\mu_j$-a.e. $z$, the system of vectors $\{\tilde{g}_j(z)\}_{g\in\mathcal{G}}$ is complete in $\ell^2(\Omega_j)$.
If $A$ is also reductive, then $\{A^tg\}_{g\in\mathcal{G},t\in [0,L]}$ being complete in $\mathcal{H}$ is equivalent to $\{\tilde{g}_j(z)\}_{g\in\mathcal{G}}$ being complete in $\ell^2(\Omega_j)$ $\mu_j$-a.e. $z$ for every $1\leq j\leq\infty$. \end{theorem} Particularly, if the evolution operator belongs to the following class $\mathcal{A}$ of bounded self-adjoint operators: \begin{eqnarray} \mathcal{A}&=&\{A\in\mathcal{B}(\ell^2(\mathbb{N})): A=A^*, \nonumber\\ &&\text{and there exists a basis of $\ell^2(\mathbb{N})$ of eigenvectors of $A$}\},\label{equationA*} \end{eqnarray} then, for $A\in\mathcal{A}$, there exists a unitary operator $U$ such that $A=U^*DU$ with $D=\sum_{j}\lambda_j P_j$, where $\lambda_j$ are the spectrum of $A$ and $P_j$ is the orthogonal projection to the eigenspace $E_j$ of $D$ associated with the eigenvalue $\lambda_j$. Since the operators in $\mathcal{A}$ are also normal and reductive, the following corollary holds. \begin{corollary}\label{corollary 3.2}
Let $A\in\mathcal{A}$ with $A=U^*DU$, and let $\mathcal{G}$ be a countable set of vectors in $\ell^2(\mathbb{N})$. Then, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is complete in $\ell^2(\mathbb{N})$ if and only if $\{P_{j}(Ug)\}_{g\in\mathcal{G}}$ is complete in $E_j$. \end{corollary} The proof of Theorem \ref{theorem 3.1} below, also shows that, for normal reductive operators, completeness in $\mathcal{H}$ is equivalent to completeness of the system $\{N_{\mu_j}^t\tilde{g}_j\}_{g\in\mathcal{G},t\in[0,L]}$ in $(L^2(\mu_j))^{(j)}$ for every $1\leq j\leq \infty$. In other words, the completeness of $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is equivalent to the completeness of its projections onto the mutually orthogonal subspaces $U^*P_{j}U\mathcal{H}$ of $\mathcal{H}$. The following Theorem \ref{theorem 3.4} summarizes the discussion above. \begin{theorem}\label{theorem 3.4}
Let $A\in\mathcal{B}(\mathcal{H})$ be a normal reductive operator on the Hilbert space $\mathcal{H}$, and let $\mathcal{G}$ be a countable system of vectors in $\mathcal{H}$. Then, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is complete in $\mathcal{H}$ if only if the system $\{N^t_{\mu_j}\tilde{g}_j\}_{g\in\mathcal{G},t\in[0,L]}$ is complete in $(L^2(\mu_j))^{(j)}$ for every $j$, $1\leq j\leq\infty$. \end{theorem}
\subsection{Proofs} We begin this section by stating and proving a lemma used to prove Theorem \ref{theorem 3.1} as well as other results in later sections.\\
Let $A$ be a normal operator, $L$ be a positive number, $f\in\mathcal{H}$, $\tilde{f}=Uf=(\tilde f_j)$, and $\tilde{g}=Ug=(\tilde g_j)$ (as in the notation section). Define $F(t)$ by \[F(t)=\langle A^tg,f\rangle=\int_{\mathbb{C}}z^t\langle \tilde{g}(z),\tilde{f}(z)\rangle d\mu(z).\]
Then, the following lemma holds.
\begin{lemma}\label{lemma 3.6}
$F(t)$ is an analytic function of $t$ in the domain $\Omega=\{t:\Re(t)>L/2\}$, where $\Re(t)$ stands for the real part of $t$. \end{lemma} \begin{proof}
First, we aim to
prove that $F(t)$ is a continuous function in $\Omega$. Consider $t_0\in\Omega$. For $|z|\le M,$ where $M=\|A\|$, and for $t\in\Omega$ with $|t-t_0|<L/4$,
one has
\begin{eqnarray*}
|z^t\langle\tilde{g}(z),\tilde{f}(z)\rangle|&=& |e^{t\ln(z)}||\langle \tilde{g}(z),\tilde{f}(z)\rangle|\\
&\leq&e^{(|\ln (M)|+\pi)|t|}|\langle \tilde{g}(z),\tilde{f}(z)\rangle|\\
&\leq&e^{(|\ln (M)|+\pi)(|t_0|+\frac L 4)}|\langle \tilde{g}(z),\tilde{f}(z)\rangle|. \end{eqnarray*}
Since the right hand side of the last inequality is an $L^2(\mu)$ function, we can use the dominated convergence theorem for $\Re(t)>L/2>0$, and get that for $t_0\in\Omega$,
\begin{equation*}
\lim_{t\rightarrow t_0}F(t)=\lim_{t\rightarrow t_0}\int_\mathbb{C} z^t\langle \tilde{g}(z),\tilde{f}(z)\rangle d\mu(z)=\int_\mathbb{C}\lim_{t\rightarrow t_0}z^t\langle\tilde{g}(z),\tilde{f}(z)\rangle d\mu(z)=F(t_0).
\end{equation*}
Therefore, $F(t)$ is a continuous function in $\Omega$.
Next we show that for every closed piecewise $C^1$ curve $\gamma$ in $\Omega$,
\[\oint_{\gamma}F(t)dt=0.\]
For fixed $\gamma$, there exists finite $M_1>0$ such that $L/2<|t|<M_1$. Therefore, for $|z|\le M$,
\[|z^t\langle \tilde{g}(z),\tilde{f}(z)\rangle|\leq e^{\tilde{M}}|\langle \tilde{g}(z),\tilde{f}(z)\rangle|,\] with $\tilde{M}=M_1(|\ln M|+\pi)$.
Then $$\oint_{\gamma}\int_\mathbb{C}|z^t||\langle\tilde{g}(z),\tilde{f}(z)\rangle|d\mu(z)dt\leq e^{\tilde{M}}\|f\|_2\|g\|_2\cdot m_1(\gamma)<\infty, $$ where $m_1(\gamma)$ stands for the length of $\gamma.$
By Fubini's theorem,
\begin{eqnarray*}
\oint_{\gamma}\int_\mathbb{C} z^t\langle\tilde{g}(z),\tilde{f}(z)\rangle d\mu(z)dt&=&\int_\mathbb{C}\oint_{\gamma}z^t\langle\tilde{g}(z),\tilde{f}(z)\rangle dt d\mu(z)\\
&=&\int_\mathbb{C}\langle\tilde{g}(z),\tilde{f}(z)\rangle\oint_{\gamma}z^t dt d\mu(z)=0.
\end{eqnarray*}
where the last equality follows from the fact that for $z\in\mathbb{C}$, $h_z(t)=z^t$ is an analytic function of $t$ in $\Omega$ and hence $\oint_{\gamma}z^tdt=0$. Then, by Morera's Theorem \cite[pp 208]{Rudinrc}, $F(t)$ is analytic on $\Omega$. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{theorem 3.1}}]
Since $\{A^tg\}_{g\in\mathcal{G},t\in [0,L]}$ is complete in $\mathcal{H}$,
\[U\{A^tg:g\in\mathcal{G},t\in [0,L]\}=\{(N^t_{\mu_j}\tilde{g}_j)_{j\in\mathbb{N}^*}:g\in\mathcal{G},t\in [0,L]\}\] is complete in $\mathcal{W}=U\mathcal{H}$. Hence, for every $1\leq j\leq\infty$, the system $\widetilde{\mathcal{S}}_j=\{N^t_{\mu_j}\tilde{g}_j\}_{g\in\mathcal{G},t\in[0,L]}$ is complete in $(L^2(\mu_j))^{(j)}.$
To finish the proof of the first statement of Theorem \ref{theorem 3.1} we use the following lemma, which is an adaptation of \cite[Lemma 1]{kriete} (\cite[Lemma 3.5]{ACCMP17}).
\begin{lemma}\label{lemma 3.5}
Let $\mathfrak{S}$ be a complete countable set of vectors in $(L^2(\mu_j))^{(j)}$, then for $\mu_j$-almost every $z$, $\{h(z):h\in\mathfrak{S}\}$ is complete in $\ell^2(\Omega_j)$.
\end{lemma}
Since $\mathcal{H}$ is separable, there exists a countable set $T=\{t_i\}_{i=1}^{\infty}\subseteq [0,L]$ with $t_1=0$ such that $\overline{span}\{A^{t}g\}_{g\in\mathcal{G},t\in T}=\overline{span}\{A^{t}g\}_{g\in\mathcal{G},t\in [0,L]}$. Hence, the fact that $\widetilde{\mathcal{S}}_j=\{N^t_{\mu_j}\tilde{g}_j\}_{g\in\mathcal{G},t\in [0,L]}$ is complete in $(L^2(\mu_j))^{(j)}$ (together with Lemma \ref{lemma 3.5}) implies that $\{z^{t}\tilde{g}_j(z)\}_{g\in\mathcal{G},t\in T}$ is complete in $\ell^2(\Omega_j)$ for each $j\in\mathbb{N}^*$.
Let $f\in\mathcal{H}$ and $F(t)=\langle A^tg,f \rangle=0$ for all $g\in\mathcal{G},t\in [0,L]$. Since $F(t)=0$ for all $t\in[0,L]$, and $F$ is analytic for $t \in \Omega=\{t:\Re(t)>L/2\}$, it follows that $F(t)=0,$ for all $t\in\Omega$ (see \cite[Theorem 10.18]{Rudinrc}). Thus, $F(n)=0$ for all $n\in\mathbb{N}$, i.e., $\text{for all } n\in\mathbb{N}$,
\begin{equation}
\int_\mathbb{C} z^n\langle\tilde{g}(z),\tilde{f}(z)\rangle d\mu(z)= \int_{\mathbb{C}}z^n\left[\sum_{1\leq j\leq\infty}\chi_{\mathcal{E}_j}(z)\langle\tilde{g}_j(z),\tilde{f}_j(z)\rangle_{\ell^2(\Omega_j)} \right]d\mu(z)=0.
\end{equation}
To finish the proof, we need the following proposition from \cite{wermer}.
\begin{proposition}\label{proposition reductive}
Let $A$ be a normal operator on the Hilbert space $\mathcal{H}$ and let $\mu_j$ be
the measures in the representation \eqref{representation of normal} of $A$. Let $\mu$ be as in \eqref{scalar spectral measure}. Then, A is reductive if and only if, for any two vectors $f, g \in\mathcal{H}$,
$$\int_{\mathbb{C}}z^n\left[\sum_{1\leq j\leq\infty}\chi_{\mathcal{E}_j}(z)\langle\tilde{g}_j(z),\tilde{f}_j(z)\rangle_{\ell^2(\Omega_j)} \right]d\mu(z)=0$$
for every $n\geq 0$ implies $\mu_j$-a.e. $\langle \tilde{g}_j(z),\tilde{f}_j(z)\rangle_{\ell^2(\Omega_j)}=0$ for every $j\in\mathbb{N}^*$.
\end{proposition}
Since $A$ is reductive, it follows from Proposition \ref{proposition reductive} that $\langle \tilde{g}_j(z),\tilde{f}_j(z)\rangle_{\ell^2(\Omega_j)}=0$ for every $j\in\mathbb{N}^*$.
Finally, since $\{\tilde{g}_j(z)\}_{g\in\mathcal{G}}$ is complete in $\ell^2(\Omega_j)$ for $\mu_j$-a.e. $z$, we get that $\tilde{f}_j(z)=0, \mu_j\text{-a.e. z} \text{ for every } j\in\mathbb{N}^*$. Thus, $\tilde{f}=0$ $\mu$-a.e. $z$, and hence $f=0$. Therefore, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is complete in $\mathcal{H}$. \end{proof}
\section{Bessel system}\label{bessel section} The goal of this section is to study the conditions for which the system $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel in $\mathcal{H}$. There are two main theorems that correspond to the finite dimensional case and the infinite dimensional case, respectively. The proofs of the results are relegated to the last subsection. We begin with the following proposition which is valid for both finite and infinite dimensional spaces.
\begin{proposition}\label{BesselEqu}
Let $A \in \mathcal B(\mathcal{H})$ be normal, $\mathcal{G}\subset \mathcal{H}$ be a countable set of vectors, and let $L$ be a positive finite number. If $\mathcal{G}$ is a Bessel system in $\mathcal{H}$, then $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathcal{H}$. \end{proposition}
The fact that $\mathcal{G}$ is a Bessel system in $\mathcal{H}$ implies that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel in $\mathcal{H}$ is not too surprising. However, the converse implication is not obvious. The next result characterizes the finite dimensional case.
\begin{theorem}[Besselness in finite dimensional space]\label{FDBessel}
Let $A$ be a normal operator on $\mathbb{C}^d$ and $L$ be a positive finite number. Let $M=Range(A^*)$ and $P_M\mathcal{G}=\{P_Mg\}_{g \in \mathcal{G}}$, where $P_M$ is the orthogonal projection on $M$. Then, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathbb{C}^d$ if and only if $P_M\mathcal{G}$ is a Bessel system in $M$. \end{theorem}
Under the appropriate restrictions on the spectrum $\sigma (A)$ of $A$, one can obtain a result similar to Theorem \ref {FDBessel} for the infinite dimensional case. However, if $0 \notin \sigma (A)$, the main result for the infinite dimensional Hilbert space is stated in the following theorem. \begin{theorem} [Besselness in infinite dimensions] \label {MainBessel ID}
Let $A\in\mathcal{B}(\mathcal{H})$ be an invertible normal operator, and let $\mathcal{G}$ be a countable system of vectors in $\mathcal{H}$. Then, for any finite positive number $L$, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathcal{H}$
if and only if $\mathcal{G}$ is a Bessel system in $\mathcal{H}$. \end{theorem}
The condition that $A$ is invertible is necessary in Theorem \ref{MainBessel ID} as can be shown by the following example.
\begin{example}\label{example1}
Let $\mathcal{G}=\{ne_n\}_{n=1}^{\infty}$ with $\{e_n\}_{n=1}^{\infty}$ being the standard basis of $\ell^2(\mathbb{N})$, $f\in\ell^2(\mathbb{N})$ with $f(n)=1/n$, and let $D$ be the diagonal infinite matrix with diagonal entries $D_{n,n}=e^{-{n^2}}$. The operator $D$ is injective but not an invertible operator on $\ell^2(\mathbb{N})$.
Note that $$\sum_{g\in \mathcal{G}}|\langle f,g\rangle|^2=\infty.$$ Hence, $\mathcal{G}$ is not a Bessel system in $\ell^2(\mathbb{N})$.
On the other hand,
\begin{equation}
\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f,D^tg\rangle|dt= \sum_{n=1}^{\infty}\frac{1-e^{-2n^2}}{2}|f_n|^2\leq\|f\|^2/2.
\end{equation} Thus $\{D^{t}g\}_{g\in\mathcal{G},t\in[0,1]}$ is Bessel in $\ell^2(\mathbb{N})$. \end{example} \subsection {Proofs for Section \ref {bessel section}} \begin{proof}[\textbf{Proof of Proposition \ref{BesselEqu}}]
For all $ f\in\mathcal{H}$,
\begin{eqnarray}
\sum\limits_{g\in\mathcal{G}}\int\limits_{0}^{L}|\la f,A^tg\ra|^2 dt&=& \sum\limits_{g\in\mathcal{G}}\int\limits_{0}^{L}|\la A^{*t}f,g\ra|^2 dt\nonumber\\
&=&\int\limits_{0}^{L}\sum\limits_{g\in\mathcal{G}}|\la A^{*t} f,g\ra|^2 dt
\leq\int\limits_{0}^{L}C_\mathcal{G}\|A^{*t}f\|^2dt\nonumber\\ &\leq&\int\limits_{0}^{L}C_\mathcal{G}\|A\|^{2t}\|f\|^2dt=\begin{cases}
C_\mathcal{G}\frac{\|A\|^{2L}-1}{\ln\|A\|^2}\|f\|^2, \|A\|\neq 1\\
C_{\mathcal{G}}L\|f\|^2,\|A\|=1,\\
\end{cases}\nonumber
\end{eqnarray}
where $C_{\mathcal{G}}$ is a Bessel constant of the Bessel system $\mathcal{G}$.
Therefore, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel in $\mathcal{H}$. \end{proof}
In order to prove Theorem \ref {FDBessel}, we need the following lemma: \begin{lemma} \label{FDIG}
Let $\mathcal{G}=\{g_{j}\}_{j\in J}\subset \mathbb{C}^d$ where $J$ is a countable set. Then, $\mathcal{G}$ is a Bessel system if and only if $\sum_{j\in J}\|g_j\|^2<\infty$. \end{lemma}
\begin{proof}[\textbf{Proof of Lemma \ref {FDIG}}]
$(\Longrightarrow)$Let $\{u_i\}_{i=1}^d$ be an orthonormal basis in $\mathbb{C}^d$. If $\{g_j\}_{j\in J}$ is a Bessel system with Bessel constant $C$, then, for $i=1,\ldots,d$
\[\sum_{j\in J}|\langle u_i,g_j\rangle|^2 \leq C.\]
Since $\|g_j\|^2=\sum_{i=1}^{d}|\langle u_i,g_j\rangle|^2$ for $j\in J$, one obtains
$$\sum_{j\in J}\|g_j\|^2=\sum_{j\in J}\sum_{i=1}^d|\langle u_i,g_j\rangle|^2\leq Cd<\infty.$$
$(\Longleftarrow)$ For any $f\in\mathcal{H}$, one has
\begin{eqnarray*}
\sum_{j\in J}|\langle f,g_j\rangle|^2\leq \sum_{j\in J}\|f\|^2\|g_j\|^2=\|f\|^2(\sum_{j\in J}\|g_j\|^2).
\end{eqnarray*}
Therefore, $\{g_j\}_{j\in J}$ is Bessel in $\mathbb{C}^d$. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{FDBessel}}]
$(\Longleftarrow)$ Since $A$ is a normal operator on $\mathcal{H}=\mathbb{C}^d$, it is clear that $A=\sum_{i\in I}\lambda_iP_i$ where $P_iP_j=0$ for $i\ne j$, $I=\{i:\lambda_i\neq 0\}$, and $(\sum_{i\in I} P_i)(\mathbb{C}^d)=M$, where $M=Range(A^*)=Null^\perp (A)=Null^\perp (A^*)$.
For $f\in\mathbb{C}^d$, one has
\begin{eqnarray}
\sum_{g\in G}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt&=&\sum_{g\in G}\int_{0}^{L}\left|\langle A^{*t}f,g\rangle\right|^2dt
=\sum_{g\in G}\int_{0}^{L}\left|\sum_{i\in I}\overline{\lambda_i}^t\langle P_if,P_ig\rangle\right|^2dt\nonumber\\
&\le&\sum_{g\in G}\int_{0}^{L}\|A\|^{2t}\left(\sum_{i\in I}\lvert{\langle P_if,P_ig\rangle}\rvert\right)^2dt\nonumber\\
&\leq&\int_{0}^{L}\|A\|^{2t}\sum_{g\in G}\left(\sum_{i\in I}\|P_if\|^2\right)\left(\sum_{i\in I}\|P_ig\|^2\right)dt\nonumber\\
&\leq&\int_{0}^{L}\|A\|^{2t}\|P_Mf\|^2 \sum_{g\in G}\|P_Mg\|^2dt\leq C_1\cdot C_{P_M\mathcal{G}}\cdot \|f\|^2, \nonumber
\end{eqnarray}
where $ C_1=\begin{cases}
(\|A\|^{2L}-1)/\ln(\|A\|^2),\|A\|\neq 1\\
L,\|A\|=1\\
\end{cases}$ and $C_{P_M\mathcal{G}}=\sum_{g\in G}\|P_Mg\|^2$.
In addition, one can use Lemma \ref {FDIG} to conclude that $C_{P_M\mathcal{G}}=\sum_{g\in G}\|P_Mg\|^2< \infty$. Therefore, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel in $\mathbb{C}^d$.\\
($\Longrightarrow$)
Since $A$ is normal, $A$ can be written as $A=\sum_{i\in I}\lambda_iP_i$, with ${\rm rank\,}(P_i)=1$ (in this representation, we allow $\lambda_i=\lambda_j$ for $i\ne j$) and $I=\{i:\lambda_i\neq 0\}$, $P_iP_j=0$ for $i\ne j$, and $(\sum_{i\in I}P_i)(\mathbb{C}^d)=M$.
Specifically, by setting $f=u_i$, where $u_i$ is a unit vector in the one dimensional space $P_i(\mathbb{C}^d)$ with $i\in I$, one has
\begin{eqnarray*}
\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle u_i,A^tg\rangle|^2dt&=&\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle u_i,\lambda_i^tP_ig\rangle|^2dt \\
&=&\sum_{g\in\mathcal{G}}\int_{0}^{L}\lvert \lambda_i\rvert^{2t}\|P_ig\|^2dt \\
&=&\begin{cases}
L\sum_{g\in\mathcal{G}}\|P_ig\|^2,\quad |\lambda_i|=1\\
\frac{|\lambda_i|^{2L}-1}{2\ln|\lambda_i|}\sum_{g\in\mathcal{G}}\|P_ig\|^2,\text{ otherwise}.
\end{cases}.
\end{eqnarray*}
In addition, since by assumption $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathbb{C}^d$ with Bessel constant $C$, then
$\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle u_i,A^tg\rangle|^2dt\leq C\|u_i\|^2=C$. Hence, for each $i$,
\begin{equation*}
\sum_{g\in\mathcal{G}}|\\|P_ig\|^2<\infty.
\end{equation*}
Therefore, summing over (the finitely many) $i\in I$ we obtain
\begin{equation*}
\sum_{g\in\mathcal{G}}|\\|P_Mg\|^2<\infty.
\end{equation*}
If $f\in M=Range(A^*)$, then
\begin{eqnarray*}
\sum_{g\in\mathcal{G}}|\langle f,g\rangle|^2&=&\sum_{g\in\mathcal{G}}\left\lvert \sum_{i\in I} \langle P_if,P_ig\rangle\right\rvert^2\nonumber\\
&\le&\sum_{g\in\mathcal{G}}\left({\sum_{i\in I} \|P_if\|^2}\right)\left({\sum_{i\in I} \|P_ig\|^2}\right)\nonumber\\
&=&
\|f\|^2\sum_{g\in\mathcal{G}} \|P_Mg\|^2.
\end{eqnarray*}
Thus, $P_M\mathcal{G}$ is Bessel in $M$. \end{proof} Before proving Theorem \ref{MainBessel ID}, we first state and prove the following lemmas. \begin{lemma}\label{BT}
Let $A\in\mathcal{B}(\mathcal{H})$ be an invertible operator in $\mathcal{H}$, then a countable set $\mathcal{G} \subseteq \mathcal{H}$ is a Bessel system in $\mathcal{H}$ if and only if $\tilde{\mathcal{G}}=A\mathcal{G}$ is a Bessel system in $\mathcal{H}$. \end{lemma} \begin{proof}[\textbf{Proof of Lemma \ref{BT}}]
$(\Longrightarrow)$
For all $ f\in\mathcal{H}$,
\begin{eqnarray*}
\sum_{g\in\mathcal{G}}|\langle f,Ag\rangle|^2&=&\sum_{g\in\mathcal{G}}|\langle A^*f,g\rangle|^2\\
&\leq&C\|A^*f\|_2^2\leq C\|A\|_2^2\|f\|_2^2,
\end{eqnarray*}where $C$ is a Bessel constant of the Bessel system $\mathcal{G}$.
Therefore, $A\mathcal{G}$ is a Bessel system in $\mathcal{H}.$\\
$(\Longleftarrow)$ For all $f\in\mathcal{H}$,
\begin{eqnarray*}
\sum_{g\in\mathcal{G}}|\langle f,g\rangle|^2&=&\sum_{g\in\mathcal{G}}\left|\langle (A^*)^{-1}f,Ag\rangle\right|^2\\
&\leq&C_1\|(A^*)^{-1}f\|_2^2\leq C_1\|A^{-1}\|_2^2\|f\|_2^2,
\end{eqnarray*}
where $C_1$ is a Bessel constant of the Bessel system $A\mathcal{G}$.
Therefore, $\mathcal{G}$ is a Bessel system in $\mathcal{H}$. \end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{MainBessel ID}}]
$(\Longleftarrow)$ See Proposition \ref{BesselEqu}. \\
$(\Longrightarrow)$
Since $A$ is a normal operator in $\mathcal{H}$, by the Spectral Theorem, there exists a unitary operator $U$ such that $$UAU^{-1}=N_{\mu_{\infty}}^{(\infty)}\oplus N_{\mu_1}^{(1)}\oplus N_{\mu_2}^{(2)}\oplus\ldots$$ and $\mu$ is defined as by \eqref{scalar spectral measure}. Therefore, the task of proving that $\mathcal{G}$ is a Bessel system in $\mathcal{H}$ is equivalent to the task of showing that $U\mathcal{G}$ is a Bessel system in $\mathcal{W}=U\mathcal{H}$.
Let $T:\mathcal{W}\rightarrow \mathcal{W}$ be the operator defined by:
\begin {equation}\label {DefofT} T \tilde{f}(z):=\int_0^\ell z^tdt\tilde{f}(z),\text{ for all } \tilde{f}\in\mathcal{W} \text{ and }z\in\sigma(A) \text{ with }\ell=\min\{L,1/2\}. \end{equation} The condition that $\ell=\min\{L,1/2\}$ ensures that $T$ is an invertible operator as will be proved later.
By Lemma \ref{BT}, $U\mathcal{G}$ is a Bessel system in $\mathcal{W}$ if and only if $T(U\mathcal{G})$ is a Bessel system in $\mathcal{W}$ as long as $T$ is a bounded invertible normal operator. The fact that $T$ is a bounded invertible operator is stated in the following lemma whose proof is postponed till after the completion of the proof of this theorem. \begin{lemma}\label{boundedT}
$T$ is a bounded invertible operator in $\mathcal{W}$. \end{lemma} So, to finish the proof of Theorem \ref {MainBessel ID}, it only remains to show that $T(U\mathcal{G})$ is a Bessel system in $\mathcal{W}$ which we do next.
Since $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathcal{H}$, and $0<\ell\leq L$, one has that, for all $f\in\mathcal{H}$, \begin{equation*}\label{Besselell}
\sum_{g\in\mathcal{G}}\int_{0}^{\ell}|\langle f,A^tg\rangle|^2dt\leq C\|f\|^2. \end{equation*} Thus, using H\"older's inequality, we get
\begin{equation} \label{fTg}\sum_{g\in\mathcal{G}}\left|\int_{0}^{\ell}\langle f,A^tg\rangle dt\right|^2\leq \ell\cdot \sum_{g\in\mathcal{G}}\int_{0}^{\ell}|\langle f,A^tg\rangle|^2dt\leq\ell C\|f\|^2. \end{equation} In addition, \begin{eqnarray}
\sum_{g\in\mathcal{G}}\left|\int_{0}^{\ell}\langle f,A^tg\rangle dt\right|^2&=&\sum_{g\in\mathcal{G}}\left|\int_{0}^{\ell}\int_\mathbb{C} \overline{z}^t\langle \tilde{f}(z),\tilde{g}(z)\rangle d\mu(z)dt\right|^2\nonumber\\
&=&\sum_{g\in\mathcal{G}}\left|\int_\mathbb{C} \int_{0}^{\ell}\overline{z}^t dt\langle\tilde{f}(z),\tilde{g}(z)\rangle d\mu(z)\right|^2\nonumber\\
&=&\sum_{g\in\mathcal{G}}|\langle \tilde{f},T\tilde{g}\rangle|^2\label{inequation 18}. \end{eqnarray} Together, \eqref{fTg} and \eqref{inequation 18} induce the following inequality: \begin{equation*}
\sum_{g\in\mathcal{G}}|\langle\tilde{f}, T\tilde{g}\rangle|^2\leq\ell C\|f\|^2=\ell C\|\tilde{f}\|^2,\text{ for all } f\in\mathcal{H}. \end{equation*} This shows that $T(U\mathcal{G})$ is a Bessel system in $\mathcal{W}$.
In conclusion, by Lemma \ref{boundedT}, $T$ is bounded invertible. In addition, $T$ is normal. Hence, $U\mathcal{G}$ is a Bessel system in $\mathcal{W}$ by Lemma \ref{BT}. Consequently, $\mathcal{G}$ is a Bessel in $\mathcal{H}$. \end{proof}
\begin{proof}[\textbf{Proof of Lemma \ref{boundedT}}]
\begin{eqnarray*}
\|T\tilde{f}\|^2&=&\langle T\tilde{f},T\tilde{f}\rangle\\
&=&\left\langle \int_0^\ell z^tdt\tilde{f}(z),\int_0^\ell z^{\tau}d\tau\tilde{f}(z)\right\rangle_{L^2(\sigma(A))}\\
&=&\int_\mathbb{C} \int_0^\ell \int_{0}^{\ell} z^t \overline{z}^{\tau}\langle \tilde{f}(z),\tilde{f}(z)\rangle dtd\tau d\mu(z)\\
&=&\int_\mathbb{C} |\phi(z)|^2\|\tilde{f}(z)\|^2d\mu(z),
\end{eqnarray*}
where
\begin{equation} \label {defPhi}
\phi(z)= \begin{cases}
\ell,\quad z=1\\
0,\quad z=0\\
\frac{z^{\ell}-1}{\ln(z)},\text{ otherwise }
\end{cases}.
\end{equation}
Let $m=\inf\{|\phi(z)|:z\in\sigma(A)\}$ and $M=\sup\{|\phi(z)|:z\in\sigma(A)\}$. As shown below in claim \ref {lemma 4.11}, $m>0$ and $M<\infty$. Thus
\begin{eqnarray*}
\|T\tilde{f}\|^2&\leq&\int_\mathbb{C} M^2\|\tilde{f}(z)\|^2d\mu(z)=M^2\|\tilde{f}\|^2,\\
\|T\tilde{f}\|^2&\geq&\int_\mathbb{C} m^2\|\tilde{f}(z)\|^2d\mu(z)=m^2\|\tilde{f}\|^2,\text{ for all } \tilde{f}\in\mathcal{W}.
\end{eqnarray*}
Since $T$ is normal, it follows that $T$ is a bounded invertible operator (see \cite[Theorem 12.12]{rudinfa91}).
We finish by proving the following fact that was used in the proof of this lemma.
\begin{claim}\label{lemma 4.11} Let $\phi$ be the function defined in \eqref{defPhi}. Then
$M=\sup\{|\phi(z)|:z\in\sigma(A)\}<\infty$, and $m=\inf\{|\phi(z)|:z\in\sigma(A)\}>0$.
\end{claim}
\noindent {\bf Proof of Claim \ref{lemma 4.11}.}
Since $A$ is a bounded invertible normal operator, it follows that $\|A^{-1}\|^{-1}\leq|z|\leq\|A\|$ for $z\in\sigma(A)$. Let $S=\{z\in\mathbb{C}:\|A^{-1}\|^{-1}\leq|z|\leq\|A\|\}$. Since $\sigma(A)\subset S$, $M\leq\sup\{|\phi(z)|:z\in S \}$ and $m\geq\{|\phi(z)|:z\in S\}$. Therefore, in order to prove Claim \ref{lemma 4.11}, it is sufficient to show that
$\sup\{|\phi(z)|:z\in S\}<\infty$, $\inf\{|\phi(z)|:z\in S\}>0$.
To prove that $\sup\{|\phi(z)|:z\in S\}<\infty$, it is noteworthy that
\begin{eqnarray*}
|\phi(z)|&=&\left|\int_{0}^{\ell}z^{t}dt\right|\leq\int_{0}^{\ell}|z^t|dt =\int_{0}^{\ell}|z|^tdt=\begin{cases}
\ell,\quad \quad z\in S\text{ and } |z|=1\\
\frac{|z|^{\ell}-1}{\ln|z|},z\in S\text{ and }|z|\neq 1.
\end{cases}
\end{eqnarray*}
Let $$\psi(x)=\begin{cases}
\ell,x=1\\
\frac{x^{\ell}-1}{\ln x},x\in\mathbb{R}^+\setminus\{1\},
\end{cases}$$
and note that (since $\lim_{x\rightarrow 1}\frac{x^{\ell}-1}{\ln x}=\ell=\psi(1)$) $\psi$ is continuous at $x=1$. In addition, $\frac{x^{\ell}-1}{\ln x}$ is a continuous function on $\mathbb{R}^+\setminus\{1\}$. Hence, $\psi$ is continuous on $\mathbb{R}^+$. Particularly, $\psi$ is continuous on the closed interval $[\|A^{-1}\|^{-1},\|A\|]$. Therefore, $$\sup\{|\phi(z)|:z\in S\}=\max_{x\in[\|A^{-1}\|^{-1},\|A\|]}\psi(x)<\infty.$$
Finally, it remains to show that $\inf\{\left|\phi(z)\right|:z\in S\}>0$. First, we divide $S$ into two sets with $S_1=\{z\in S:\arg(z)\in[-\pi/2,\pi/2]\}$ and $S_2=S\setminus S_1$. Since $|\phi(z)|$ is a continuous function on $S_1$ and $S_1$ is compact, there exists $z_0\in S_1$ such that $|\phi(z_0)|=\inf\{|\phi(z)|:z\in S_1\}$. In addition, $|\phi(z)|$ has no root on $S_1$. Hence, $\inf\{|\phi(z)|:z\in S_1\}>0$.
For $z\in S_2$, $\pi/2\leq|\arg(z)|\leq\pi$. Therefore,
\begin{eqnarray*}
|z^{\ell}-1|&=&||z|^{\ell}e^{i\ell\arg(z)}-1|\\
&\geq&|z|^{\ell}|\sin(\ell\arg(z))|\\
&\geq&\min\{\|A^{-1}\|^{-\ell} \sin(\ell\pi),\|A^{-1}\|^{-\ell} \sin(\ell\pi/2)\}>0,
\end{eqnarray*}
where the last inequality follows from the fact that $0<\ell<1$ (in particular, we chose $\ell=\min\{L,1/2\}$ as in Definition \eqref {DefofT} for $T$).
In addition, for $z \in S_2$, one has
\begin{eqnarray*}
|\ln(z)|&\leq&|\ln(|z|)|+|\arg(z)|\leq\max\{|\ln(\|A^{-1}\|^{-1})|,|\ln(\|A\|)|\}+\pi<\infty.
\end{eqnarray*}
Hence, $\inf\{|\phi(z)|: z\in S_2 \}>0$.
Combining the estimates on $S_1$ and $S_2$, we conclude that
$\inf\{|\phi(z)|:z\in S\}>0$. \end{proof}
\section{Frames generated by the action of bounded normal operators.}\label{frames} In this section, we study some properties of a semi-continuous frame of the form $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ generated by the continuous action of a normal operator $A\in \mathcal{B}(\mathcal{H})$ and relate them to the properties of the discrete systems generated by its time discretization. We also show that, under the appropriate conditions, if $\{A^tg\}_{g\in\mathcal{G},t\in[0,L_1]}$ is a semi-continuous frame for some positive number $L_1$, then $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ a semi-continuous frame for all $0<L<\infty$. Before presenting the two main theorems, we first provide some necessary conditions for obtaining semi-continuous frames, and treat some special cases. The proofs are postponed to Subsection \ref {ProofFrame}.
The following proposition (whose proof is obtained by direct calculation) provides a necessary condition to ensure the lower bound of the semi-continuous frame generated by $A\in \mathcal{B}(\mathcal{H})$.
\begin{proposition}\label{FrLrBd}
Let $A\in\mathcal{B}(\mathcal{H})$ be an invertible normal operator,
$L$ be a finite positive number, and $\mathcal{G}\subseteq \mathcal{H}$ be a countable set of vectors. If, for all $f\in\mathcal{H}$,
\begin{equation}\label{equationlowerdis}
\sum_{g\in\mathcal{G}}|\langle f,g\rangle|^2\geq c\|f\|^2,
\end{equation}
where $c$ is a positive constant,
then there exists a finite positive constant $C$ such that
\begin{equation}\label{equationlowercont}
\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt\geq C\|f\|^2, \text{ for all } f\in\mathcal{H} .
\end{equation} \end{proposition} The converse of Proposition \ref{FrLrBd} is false, even in finite dimensional space as shown in Example \ref{FrLrBdEx}. For the special case that $A$ is equivalent to a diagonal operator on $\ell^2(\mathbb{N})$ we get: \begin{lemma}\label{DOPCase}
Let $A\in\mathcal{A}$, where $\mathcal{A}$ is defined in \eqref{equationA*}, and let $\mathcal{G}\subseteq \ell^2(\mathbb{N})$ be a countable set of vectors. If $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ satisfies \eqref {equationlowercont} in $\ell^2(\mathbb{N})$, then
$$\sum_{g\in\mathcal{G}}\|g\|^2=\infty. $$ \end{lemma} From Lemma \ref {DOPCase}, it follows that the cardinality of $\mathcal{G}$ must be infinite as stated in the following corollary. \begin{corollary}
If the assumptions of Lemma \ref {DOPCase} hold then $|\mathcal{G}|=+\infty.$ In particular, $|\mathcal{G}|=+\infty $ if $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a frame for $\ell^2(\mathbb{N})$. \end{corollary}
The discretization of continuous frames is a central question and has been studied extensively (see \cite {FR05, FS16} and the references therein). In particular, Freeman and Speegle have found necessary and sufficient conditions for the discretization of continuous frames \cite {FS16}. In our situation, the systems $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ can be viewed as continuous frames and the theory in \cite {FS16} may be applied to conclude that the system can be discretized. However, because of the particular structure of the systems $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$, we can say more and obtain finer results for their discretization, as stated in the following theorem.
\begin{theorem}\label{ScToDscr}
Let $A\in\mathcal{B}(\mathcal{H})$ be a normal operator on the Hilbert space $\mathcal{H}$ and let $\mathcal{G}$ be a Bessel system of vectors in $\mathcal{H}$. If $\{A^tg \}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$, then there exists $\delta>0$ such that for any finite set $T=\{t_i:i=1,\ldots,n\}$ with $0=t_1< t_2<\ldots<t_n<t_{n+1}=L$ and $|t_{i+1}-t_{i}|<\delta$, the system $\{A^{t}g\}_{g\in \mathcal{G},t\in T}$ is a frame for $\mathcal{H}$.
If, in addition, $A$ is invertible, then $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$ if and only if there exists a finite set $T=\{t_i:i=1,\ldots,n\}$ and $0= t_1< t_2<\ldots<t_n<L$, such that $\{A^{t}g\}_{g\in \mathcal{G}, t\in T}$ is a frame for $\mathcal{H}.$ \end{theorem}
Example \ref {example3} shows that the condition that $A$ is invertible is necessary for the second statement of Theorem \ref{ScToDscr}.
The next theorem shows that, under some appropriate conditions, if $\{A^tg\}_{g\in\mathcal{G},t\in[0,L_1]}$ is a semi-continuous frame for some finite positive number $L_1$, then $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for any finite positive number $L$. \begin{theorem}\label{SCFrSA}
Let $A\in\mathcal{B}(\mathcal{H})$ be an invertible self-adjoint operator and $\mathcal{G}$ be a countable set in $\mathcal{H}$.
Then, $\{A^tg\}_{g\in\mathcal{G},t\in[0,1]}$ is a semi-continuous frame in $\mathcal{H}$ if and only if $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame in $\mathcal{H}$ for all finite positive $L$. \end{theorem} We postulate the following conjecture: \begin {conjecture} Theorem \ref {SCFrSA} remains true if $A$ is a normal reductive operator. \end {conjecture}
This first example shows that the converse of Proposition \ref{FrLrBd} is false. \begin{example}\label{FrLrBdEx}
Let $A=\begin{bmatrix}
\epsilon& 0\\
0&1\\
\end{bmatrix}$ with $0<\epsilon<1$ and $g=\begin{bmatrix}
1\\
1\\
\end{bmatrix}$.
Note that for $L>0$, $$\mathcal{G}_1=\left\{g=\begin{bmatrix}
1\\
1\\
\end{bmatrix}, A^{L/2}g=\begin{bmatrix}
\epsilon^{L/2}\\
1\\
\end{bmatrix}\right\}$$ is complete in $\mathbb{R}^2$. In addition, $A$ is a bounded invertible normal operator in $\mathbb{R}^2$. Therefore, $\mathcal{G}_1$ is a frame in $\mathbb{R}^2$. By Theorem \ref{ScToDscr}, $\{A^tg\}_{t\in[0,L]}$ is a semi-continuous frame in $\mathbb{R}^2$. However, the lower bound of \eqref{equationlowerdis} does not hold for $\mathcal{G}=\{g\}$. For example, let $f=\begin{bmatrix}
-1\\
1\\
\end{bmatrix}$, then $ \langle f,g\rangle=0$. \end{example} This next example shows that the condition that $A$ is invertible is required for the second statement of Theorem \ref{ScToDscr}. \begin{example}\label{example3}
Let $\mathcal{G}=\{e_j\}_{j=1}^{\infty}$ be the standard basis of $\ell^2(\mathbb{N})$. Because $\mathcal{G}$ is an orthonormal basis, one has $\mathcal{G}\subseteq \{A^{t}g\}_{g\in \mathcal{G}, t\in T}$, for any bounded operator $A$, and for any time steps $T=\{t_i:i=1,\ldots,n\}$ with $0= t_1< t_2<\ldots<t_n<L$. Thus, $\mathcal{G}\subseteq \{A^{t}g\}_{g\in \mathcal{G},t\in T}$ is a frame for $\ell^2(\mathbb{N}).$
However, there exists a non-trivial bounded operator such that $\{A^te_j\}_{j\in\mathbb{N},t\in[0,L]}$ is not a semi-continuous frame. For example, if $D$ is a diagonal infinite matrix with diagonal entries $D_{j,j}=\frac{1}{j}$, then
\begin{equation}
\sum_{j=1}^{\infty}\int_{0}^{L}|\langle e_k,D^te_j\rangle|^2dt=\frac{1/k^{2L}-1}{\ln(1/k^{2})}.
\end{equation}
Since
$\lim\limits_{k\to\infty}\frac{1/k^{2L}-1}{\ln(1/k^{2})}=0, $
it follows that $\{D^te_j\}_{j\in\mathbb{N},t\in[0,L]}$ is not a semi-continuous frame for $\ell^2(\mathbb{N})$. \end{example}
Additionally, a number of examples are available to illustrate that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$ does not require $\mathcal{G}$ to be a frame or even complete in $\mathcal{H}$. In fact, this is precisely why space-time sampling trade-off is feasible. The next two examples are toy examples to show this fact. \begin{example} [$\mathcal{G}$ is not a frame for $\mathcal{H}$]
Let $\{e_n\}_{n=1}^{\infty}$ be the standard basis of $\ell^2(\mathbb{N})$ and $\mathcal{G}=\{g_n=e_n+e_{n+1}:n\in\mathbb{N}\}$, and let $D$ be a diagonal operator with
$D_{n,n}= \begin{cases}
1, n \text{ is odd}\\
3, n\text{ is even}
\end{cases}.$ \\
It can be shown that $\mathcal{G}$ is complete but that $\mathcal{G}$ is neither a basis nor a frame for $\ell^2(\mathbb{N})$ \cite{C08}. However, for all $f\in\ell^2(\mathbb{N})$, after a somewhat tedious computation, one gets
\[\frac{1}{2}\|f\|^2\leq\sum_{n=1}^{\infty}\int_{0}^{1}|\langle f,D^tg_n\rangle|^2dt\leq\frac{16}{\ln(3)}\|f\|^2, \] so that $\{D^tg_n\}_{n\in\mathbb{N},t\in[0,1]}$ is a semi-continuous frame for $\ell^2(\mathbb{N})$. \end{example} \begin{example} [$\mathcal{G}$ is not complete in $\mathcal{H}$]
Let $\{e_n\}_{n=1}^{\infty}$ be the standard basis of $\ell^2(\mathbb{N})$ and $\mathcal{G}=\{g_n=e_n+2e_{n+1}:n\in\mathbb{N}\}$. The set $\mathcal{G}$ is not complete in $\ell^2(\mathbb{N})$. For example $f=(f_k)$ with $f_k=(-1)^k\frac 1 {2^k}$ is orthogonal to $\overline {\text {span } }\mathcal{G}$. Thus, $\mathcal{G}$ is not a frame in $\ell^2(\mathbb{N})$. Let $D$ be the diagonal operator with
$$D_{n,n}= \begin{cases}
9, \quad \quad n=1\\
1-\frac{1}{n}, n\geq 2
\end{cases}.$$
A lengthy computation yields
\begin{eqnarray*}
\frac{1}{4}\|f\|^2\leq\sum_{n=1}^{\infty}|\langle f,g_n\rangle|^2+\sum_{n=1}^{\infty}|\langle f,Dg_n\rangle|^2\leq 164\|f\|^2.
\end{eqnarray*}
This implies that $\{D^tg\}_{g\in\mathcal{G},t\in\{0,1\}}$ is a frame in $\ell^2(\mathbb{N})$. In addition, since $D$ is a self-adjoint invertible operator, Theorem \ref{ScToDscr} implies that
$\{D^{t}g_n\}_{n\in\mathbb{N},t\in[0,2]}$ is a semi-continuous frame of $\ell^2(\mathbb{N})$. \end{example}
\subsection {Proofs of Section \ref {frames}} \label {ProofFrame} \begin{proof}[\textbf{Proof of Lemma \ref {DOPCase}}]
One can always assume that $A=\sum\limits_{i=1}^\infty\lambda_iP_i$ with ${\rm rank\,} (P_i)=1$, $P_iP_j=0$ and $\sum_iP_i=Id_{\ell^2(\mathbb{N})}$ as long as $\lambda_i=\lambda_j$ for $i\ne j$ in the representation of $A$ is allowed.
Let $e_i$ be a vector such that $\|e_i\|=1$ and $span \{e_i\}=P_i(\mathcal{H})$. Then
\begin{eqnarray*}
\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle e_i,A^tg\rangle|_2^2dt&=&\sum_{g\in\mathcal{G}}\int_{0}^{L}|\lambda_i|^{2t}\vert\langle e_i,P_i(g)\rangle \vert^2 dt.
\end{eqnarray*}
Since $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ satisfies \eqref {equationlowercont}, we have that $\lambda_i\neq 0$ for all $i \in \mathbb{N}$.
Moreover, if $\sum_{g\in\mathcal{G}}\|g\|^2_2=\sum_{i\in\mathbb{N}}\sum_{g\in\mathcal{G}}\|P_ig\|^2<\infty$, then $\lim\limits_{i\to\infty}\sum_{g\in\mathcal{G}}\|P_ig\|^2=0.$
In addition, since $ \frac{\|A\|^{2L}-1}{2\ln(\|A\|)}\geq\frac{|\lambda_i|^{2L}-1}{2\ln(|\lambda_i|)}>0,$ we get that $ \lim\limits_{i\to \infty}\frac{|\lambda_i|^{2L}-1}{2\ln(|\lambda_i|)}\sum_{g\in\mathcal{G}}\|P_ig\|^2= 0. $
This contradicts \eqref {equationlowercont}. Hence, $\sum_{g\in\mathcal{G}}\|g\|^2=\infty$. \end{proof}
\begin{proof}[\textbf{Proof of theorem \ref{ScToDscr}}]
From the assumption that $\mathcal{G}$ is a Bessel sequence in $\mathcal{H}$, there exists $K>0$ such that $\sum_{g\in\mathcal{G}}|\langle f,g\rangle|^2\leq K\|f\|^2,$ for all $f\in\mathcal{H}$. Since $A$ is a bounded normal operator, for any $0\leq t<\infty$, one has
\begin{equation}\label{AtgBessel}
\sum_{g\in\mathcal{G}}|\langle f,A^{t}g\rangle|^2=\sum_{g\in\mathcal{G}}|\langle A^{*t}f,g\rangle|^2\leq K\|A^{*t}f\|^2\leq K\|A\|^{2 t}\|f\|^2. \end{equation}
Summing the inequalities \eqref {AtgBessel} over $t\in T=\{t_i:i=1,\dots, n\}$, it immediately follows that $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a Bessel sequence in $\mathcal{H}$.
Using \eqref {AtgBessel}, it follows that
\begin{equation}\label{equation 25}
\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt\leq K\int_{0}^{L}\|A\|^{2t}dt\|f\|^2.
\end{equation}
Inequality \eqref{equation 25} implies that
for any $\epsilon>0$, there exists an $l$ with $L/2>l>0$, such that
\begin {equation}\label {epsIflSmall}
\sum_{g\in\mathcal{G}}\int_{0}^{l}|\langle f,A^tg\rangle|^2dt<\epsilon\|f\|^2. \end{equation}
Next, the goal is to find $\delta>0$ such that for any finite set $T=\{t_i:i=1,\ldots,n\}$ with $0=t_1< t_2<\ldots<t_n<t_{n+1}=L$ and $|t_{i+1}-t_{i}|<\delta$, the system $\{A^{t}g\}_{g\in \mathcal{G},t\in T}$ is a frame for $\mathcal{H}$, as long as $\{A^tg \}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$, i.e., \begin{equation}\label{equation 24}
c\|f\|^2\leq\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle |^2dt\leq C\|f\|^2, \quad \text {for all } f \in \mathcal{H}, \end{equation} for some $c, C>0$.
To finish the proof, we use the following lemma.
\begin{lemma}\label{ContApower}
Let $A\in \mathcal{B}(\mathcal{H})$ be a normal operator and $\ell,L$ be positive numbers with $0<\ell<L$. Then for any $\epsilon>0$, there exists $\delta>0$ such that whenever $s_1,s_2\in[\ell,L]$ with $|s_1-s_2|<\delta$, we have $\|A^{s_1}-A^{s_2}\|<\epsilon$. \end{lemma} \begin{proof}[\textbf{Proof of Lemma \ref{ContApower}}]
For $s_1,s_2\in[\ell,L]$,
\begin{eqnarray*}
|z^{s_1}-z^{s_2}|^2&=&|z|^{2s_1}-2|z|^{s_1}|z|^{s_2}\cos((s_1-s_2)arg(z))+|z|^{2s_2}\\
&=&||z|^{s_1}-|z|^{s_2}|^2+2|z|^{s_1}|z|^{s_2}(1-\cos((s_1-s_2)arg(z))).
\end{eqnarray*}
For all $z\in\sigma(A)$, one has $0\leq|z|\leq\|A\|$. Thus $|z|^{s}$ is uniformly bounded for all $s\in[\ell,L].$ In addition, the function $(t,r)\mapsto r^t$ is a continuous function on the compact set $ [ \ell,L]\times[0,\|A\|]$ and the function $t\mapsto\cos(t\cdot arg(z))$ is equicontinuous at $t=0$ for $arg(z)\in[-\pi,\pi)$. The lemma then follows from the spectral theorem (i.e., Theorem \ref {spectral theorem}).
\end{proof}
By Lemma \ref{ContApower}, there exists $\delta$ with $l/2>\delta>0$ such that whenever $|s_1-s_2|<2\cdot\delta$ for $s_1,s_2\in[l/2,L]$, then $\|A^{s_1}-A^{s_2}\|<\epsilon$. Assume that the set $T=\{t_i:i=1,\ldots,n\}$ satisfies $0=t_1< t_2<\ldots<t_n<t_{n+1}=L$ and $|t_{i+1}-t_{i}|<\delta$. Set $m=\min\{i:t_i>l/2\}$. Note that $l/2>\delta>0$. Therefore $t_m<l$. Then, using \eqref {epsIflSmall}, the difference
\begin {equation}\label {DiffContDisc}
\Delta=\left|{\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt-\sum_{g\in\mathcal{G}}\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\ |\langle f,A^{t_i}g\rangle|^2}dt\right|, \end {equation} can be estimated as follows. \begin{eqnarray*}
\Delta
&=&\left|{\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt-\sum_{g\in\mathcal{G}}\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\ |\langle f,A^{t_i}g\rangle|^2}dt\right| \\
&\leq&\left(\sum_{g\in\mathcal{G}}\int_{0}^{t_m}|\langle f,A^tg\rangle|^2dt\right)+\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\sum_{g\in\mathcal{G}}\vert {|\langle f,A^tg\rangle|^2-|\langle f,A^{t_i}g\rangle|^2} \vert dt\\
&=&\left(\int_{0}^{t_m}\sum_{g\in\mathcal{G}}|\langle f,A^tg\rangle|^2dt\right)+\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\sum_{g\in\mathcal{G}}(|\langle f,A^tg\rangle|+|\langle f,A^{t_i}g\rangle|)(\vert |\langle f,A^tg\rangle|-|\langle f,A^{t_i}g\rangle| \vert )dt\\
&\leq&\epsilon \|f\|^2+\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\sum_{g\in\mathcal{G}}(|\langle A^{*t}f,g\rangle|+|\langle A^{*t_i}f,g\rangle|)(|\langle A^{*t}f-A^{*t_i}f,g\rangle|)dt\\
&\leq&\epsilon\|f\|^2+\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\left(\sum_{g\in\mathcal{G}}(|\langle A^{*t}f,g\rangle|+|\langle A^{*t_i}f,g\rangle|)^2\right)^{1/2}\left(\sum_{g\in\mathcal{G}}(|\langle A^{*t}f-A^{*t_i}f,g\rangle|)^2\right)^{1/2}dt\\
&\leq&\epsilon\|f\|^2+\sum_{i=m}^{n}\int_{t_i}^{t_{i+1}}\left(2K(\|A^{*t}f\|^2+\|A^{*t_i}f\|^2)\right)^{1/2}(K\|A^{*t}f-A^{*t_i}f\|^2)^{1/2}dt\\
&\leq&(\epsilon+2C_1 KL\epsilon)\|f\|^2,\text{ where }C_1=\max\{1,\|A\|^{L}\}. \end{eqnarray*} Using \eqref {DiffContDisc} and choosing $\epsilon$ so small that $(1+2C_1 KL)\epsilon<c/2$, we find $\delta$ such that
$$\delta\sum_{g\in\mathcal{G}}\sum_{i=m}^{n}|\langle f,A^{t_i}g\rangle|^2\geq c\|f\|^2-c/2\|f\|^2=c/2\|f\|^2.$$ Therefore, for any finite set $T=\{t_i:i=1,\ldots,n\}$ with $0=t_1< t_2<\ldots<t_n<t_{n+1}=L$ and $|t_{i+1}-t_{i}|<\delta$, the system $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a frame in $\mathcal{H}$.
To prove the second statement, it is sufficient to prove that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame under the assumption that $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a frame in $\mathcal{H}$ and $A$ is an invertible normal operator. We already know by Theorem \ref{MainBessel ID} that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is Bessel since $\mathcal{G}$ is Bessel by assumption. Let $T=\{t_i:i=1,\ldots,n\}$ with $0=t_1<t_2<\ldots<t_n<L$ be such that $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a frame for $\mathcal{H}$ with frame constants $c,C$ i.e., for all $ f\in\mathcal{H}$,
\[c\|f\|^2\leq\sum_{g\in\mathcal{G}}\sum_{i=1}^{n}|\langle f,A^{t_i}g\rangle|\leq C\|f\|^2. \] Let $m=\min\{t_{i+1}-t_{i},1\leq i\leq n\}$ with $t_{n+1}=L$. Then, \begin{eqnarray*}
\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt&=&\sum_{g\in\mathcal{G}}\sum_{i=1}^{n}\int_{t_i}^{t_{i+1}}|\langle f,A^tg\rangle|^2dt\\
&=&\sum_{g\in\mathcal{G}}\sum_{i=1}^{n}\int_{0}^{t_{i+1}-t_{i}}|\langle (A^{*t}f,A^{t_i}g\rangle|^2dt\\
&\geq&\sum_{g\in\mathcal{G}}\sum_{i=1}^{n}\int_{0}^{m}|\langle A^{*t}f,A^{t_i}g\rangle|^2dt\\
&\geq&\int_{0}^{m}c\|A^{*t}f\|^2_2dt. \end{eqnarray*} Since $A$ is an invertible bounded normal operator, we have \begin{eqnarray*}
\int_{0}^{m}c\|A^{*t}f\|^2_2dt&\geq&c\cdot \frac{1-\|A^{-1}\|^{-2m}}{2\ln(\|A^{-1}\|)}\|f\|^2. \end{eqnarray*} This concludes the proof that $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$. \end{proof} To prove Theorem \ref{SCFrSA}, the following three lemmas, i.e., Lemmas \ref{SeriesConv}, \ref{AnalPosA} and \ref{ReImBessel} are needed. \begin{lemma}\label{SeriesConv}
Let $\mathcal{G}$ be a countable Bessel sequence in $\mathcal{H}$ and let $A\in\mathcal{B}(\mathcal{H})$ be a normal operator. Let $L$ be any positive real number, $\Omega_L=\{z:\Re(z)>L>0\}$, and let $\{g_i\}_{i\in I}$ be any indexing of $\mathcal{G}$. Then, for fixed $f\in\mathcal{H}$,
the partial sums $\sum\limits_{i=1}^n|\langle A^zg_i,f\rangle|^2$ converge uniformly on any compact subset of $ \Omega_L$. \end{lemma} \begin{proof}[\textbf{Proof of Lemma \ref{SeriesConv}}]
Let $\overline {D_r}$ denote the closed disk of radius $r$. Then using the fact that $\mathcal{G}$ is Bessel with Bessel constant $ C_\mathcal{G}$, for $z \in \overline {D_r}\cap \overline {\Omega_L}$, one gets,
$$\sum\limits_{i=1}^n|\langle A^zg_i,f\rangle|^2=\sum\limits_{i=1}^n|\langle f,A^zg_i\rangle|^2=\sum\limits_{i=1}^n |\langle (A^{z})^*f,g\rangle|^2\le C_\mathcal{G} \cdot e^{2\pi r}\cdot\|A\|^{2r}\|f\|^2,$$
from which the lemma follows. \end{proof} \begin{lemma}\label{AnalPosA}
Let $\mathcal{G}$ be a countable Bessel sequence in $\mathcal{H}$ and let $A\in\mathcal{B}(\mathcal{H})$ be a normal operator. Let $L$ be any positive real number and let $\Omega_L=\{z:\Re(z)>L>0\}$. Then, for fixed $f\in\mathcal{H}$,
$$F(z)=\sum_{g\in\mathcal{G}}(\langle A^zg,f\rangle)^2,$$ is an analytic function of $z$ in $\Omega_L$. \end{lemma}
\begin{proof}[\textbf{Proof of Lemma \ref{AnalPosA}}]
Since $A$ is a normal operator on $\mathcal{H}$, by Lemma \ref{lemma 3.6}, $\left(\langle A^zg,f\rangle\right)^2$ is analytic in $\Omega_L$.
Since $\left|\sum_{g\in\mathcal{G}}(\langle A^zg,f\rangle)^2\right|\leq\sum_{g\in\mathcal{G}}|\langle A^zg,f\rangle|^2$, by Lemma \ref{SeriesConv}, the series $\sum_{g\in\mathcal{G}}(\langle A^zg,f\rangle)^2$ converges absolutely and uniformly on any compact subset of $\Omega_{L}$, and the partial sums of $\sum_{g\in\mathcal{G}}(\langle A^zg,f\rangle)^2$ are analytic in $\Omega_{L}$ and converge uniformly on any compact subset of $\Omega_{L}$. It follows that the series $\sum_{g\in\mathcal{G}}(\langle A^zg,f\rangle)^2$ is an analytic function of $z$ in $\Omega_L$ \cite[Theorem 10.28]{Rudinrc}. \end{proof} Let $A\in\mathcal{B}(\mathcal{H})$ be a normal operator, by the spectral theorem, there exists a unitary operator $U$ such that $$UAU^{-1}=N_{\mu_{\infty}}^{(\infty)}\oplus N_{\mu_1}^{(1)}\oplus N_{\mu_2}^{(2)}\oplus\ldots.$$
For every $f\in\mathcal{H}$, we define $\tilde{f}=Uf\in U\mathcal{H}$. Note that $\tilde{f}:\sigma(A)\rightarrow \ell^2(\Omega_\infty)\oplus\ell^2(\Omega_1)\oplus\ell^2(\Omega_2)\oplus\ldots$ is a function and hence it makes sense to talk about its real and imaginary parts. Set $f^{\Re}=U^{-1}\Re(\tilde{f})$ and $f^{\Im}=U^{-1}\Im(\tilde{f})$.
\begin{lemma}\label{ReImBessel}
If $\mathcal{G}$ is a Bessel sequence in $\mathcal{H}$, then, $\{g^{\Re}\}_{g\in\mathcal{G}}$ and $\{g^{\Im}\}_{g\in\mathcal{G}}$ are also Bessel sequences in $\mathcal{H}$ for any given normal operator $A\in\mathcal{H}$. \end{lemma}
\begin{proof}[\textbf{Proof of Lemma \ref{ReImBessel}}]
Consider the subspace $S\subseteq \mathcal{H}$ defined by $S=\{f\in\mathcal{H}:Uf \text{ is real valued}\}$.
Then, for $f\in S$, using the following identity
$$\sum_{g\in\mathcal{G}}|\langle f,g\rangle|^2=\sum_{g\in\mathcal{G}}|\langle \tilde{f},\tilde{g}\rangle|=\sum_{g\in\mathcal{G}}|\langle \tilde{f},\Re(\tilde{g})\rangle|^2+|\langle \tilde{f},\Im (\tilde{g})\rangle|^2=\sum_{g\in\mathcal{G}}|\langle f,g^{\Re}\rangle|^2+|\langle f, g^{\Im}\rangle|^2, $$ it follows that $\{ g^{\Re}\}_{g\in\mathcal{G}}$ and $\{g^{\Im}\}_{g\in\mathcal{G}}$ are Bessel sequences in $S$.
For general $f\in\mathcal{H}$, we have $f^\Re\in S$, $f^{\Im}\in S$, and
$$ \sum_{g\in\mathcal{G}}|\langle f, g^\Re\rangle|^2= \sum_{g\in\mathcal{G}}|\langle f^\Re, g^\Re\rangle|^2+ \sum_{g\in\mathcal{G}}|\langle f^\Im, g^\Re\rangle|^2,$$
$$ \sum_{g\in\mathcal{G}}|\langle f,g^\Im \rangle|^2= \sum_{g\in\mathcal{G}}|\langle f^\Re, g^\Im\rangle|^2+ \sum_{g\in\mathcal{G}}|\langle f^\Im, g^\Im\rangle|^2.$$
It follows that $\{ g^\Re\}_{g \in\mathcal{G}}$ and $\{g^\Im\}_{g\in\mathcal{G}}$ are Bessel sequences for $\mathcal{H}$.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{SCFrSA}}]
Assume that $\{A^tg\}_{g\in\mathcal{G},t\in[0,1]}$ is a semi-continuous frame in $\mathcal{H}$ with frame bounds $c$, $C$.
By Theorem \ref{ScToDscr}, there exists a finite set $T$ such that $\{A^{t}g\}_{g\in\mathcal{G},t\in T}$ is a frame for $\mathcal{H}$. Therefore, for $L\geq 1$, $\{A^tg\}_{g\in\mathcal{G}, t\in[0,L]}$ is also a semi-continuous frame.
To prove that $\{A^tg\}_{g\in\mathcal{G}, t\in[0,L]}$ is a semi-continuous frame for $L<1$, we note that the inequality
$$\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f,A^tg\rangle|^2dt\leq\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f,A^tg\rangle|^2dt\leq C\|f\|_2^2$$ implies that
$\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a Bessel system in $\mathcal{H}$. Moreover, $A$ is an invertible bounded self-adjoint operator. Therefore, by Theorem \ref{MainBessel ID}, $\mathcal{G}$ is Bessel in $\mathcal{H}$ with Bessel constant $C_{\mathcal{G}}.$
Suppose that $\{A^tg\}_{g\in\mathcal{G}, t\in[0,L]}$ is not a frame. Then, there exists a sequence $\{f_n\}$ with $\|f_n\|=1$ such that $\sum_{g\in\mathcal{G}}\int_{0}^{L}|\langle f_n, A^tg\rangle|^2dt\rightarrow 0$. It follows that $\sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2\rightarrow 0$ in measure. Thus, there exists a subsequence $\{f_{n_k}\}$ of $\{f_n\}$ such that $\sum_{g\in\mathcal{G}}|\langle f_{n_k},A^tg\rangle|^2\rightarrow 0,$ for a.e. $t\in[0,L]$. By passing to a subsequence, assume that $\sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2\rightarrow 0,$ for a.e. $t\in[0,L]$.
To finish the proof, we next prove that there exists a subsequence $\{f_{n_k}\}$ of $\{f_{n}\}$ such that $$\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f_{n_k},A^tg\rangle|^2dt\rightarrow 0.$$
Since $A$ is a self-adjoint operator, by the spectral theorem, there exists a unitary operator $U$ such that $A$ can be represented as \eqref{representation of normal} and $\sigma(A)\subseteq\mathbb{R}$. In addition, $A$ is invertible.
Then there exist $m,M>0$ such that $m\leq|z|\leq M$ for all $z\in\sigma(A)$. Set $\tilde{f}=Uf$ and $\tilde{g}=Ug$.
{\bf Case 1.} {\em $A$ is a positive self-adjoint operator, and $\{Ug\}_{g\in\mathcal{G}}$ and $\{Uf_{n}\}$ are real-valued, i.e., $Ug=\Re(\tilde{g}) \text{ for all }g\in\mathcal{G}$ and $ Uf_n=\Re(\tilde f_n)$}: In this case, one has $ |\langle f_{n},A^tg\rangle|^2=(\langle A^tg,f_n\rangle)^2, \text{ for all } t\in\mathbb{R}^+.$ Therefore
$$\sum_{g\in\mathcal{G}}|\langle f_{n},A^tg\rangle|^2=\sum_{g\in\mathcal{G}}(\langle A^tg,f_n\rangle)^2, \text{ for all } t\in\mathbb{R}^+.$$
Moreover, since $\mathcal{G}$ is Bessel, by Lemma \ref{AnalPosA}, the functions $F_n(t)=\sum_{g\in\mathcal{G}}(\langle A^tg,f_n\rangle)^2$ are analytic for $ t \in \Omega_{L/4}\cap D_r\subseteq \mathbb{C}$ and satisfy
\begin{eqnarray*}
|F_n(t)|=\left|\sum_{g\in\mathcal{G}}(\langle A^tg,f_n\rangle)^2\right|&\leq& \sum_{g\in\mathcal{G}}\left|\langle g,(A^{t})^*f_n\rangle\right|^2\leq C_{\mathcal{G}}\|A\|^{2r}, \text{ for } t \in \Omega_{L/4}\cap D_r.
\end{eqnarray*}
Thus, by Montel's theorem, there exists a subsequence $\{F_{n_k}\}$ of $\{F_n\}$ such that $\{F_{n_k}\}$ converge to an analytic function $F$ on $ \Omega_{L/4}\cap D_r$. Let $D_r \subset \mathbb{C}$ be a disk of radius $r$ containing $[L/2,1]$. Since $F_n$ are analytic and $F_n(t)\rightarrow 0,\text{ for all } t\in[L/2,L]$, it follows that $F(t)=0$, for all $t\in[L/2,L]$. Moreover, since $F$ is analytic, we conclude that $F(t)=0$ for all $ t \in \Omega_{L/4}\cap D_r$, and hence also on $[L/2,1]$, i.e., $\lim_{n_k\rightarrow\infty}F_{n_k}(t)=0$ for all $t\in[L/2,1]$. Thus,
\begin{eqnarray*}
&&\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f_{n_k},A^tg\rangle|^2dt\\
&=&\sum_{g\in\mathcal{G}}\int_{0}^{L/2}|\langle f_{n_k},A^tg\rangle|^2dt+\sum_{g\in\mathcal{G}}\int_{L/2}^{1}|\langle f_{n_k},A^tg\rangle|^2dt.
\end{eqnarray*} Taking limits as $n_k$ tends to infinity, one sees that $\lim\limits_{n_k\rightarrow\infty}\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f_{n_k},A^tg\rangle|^2dt=0$. This contradicts the assumption that $\{A^tg\}_{g\in\mathcal{G},t\in[0,1]}$ is a semi-continuous frame. Therefore, $\{A^tg\}_{g\in\mathcal{G}, t\in[0,L]}$ is a semi-continuous frame.\\
{\bf Case 2.} {\em The general case}:
Let $\tilde {f}_n=\Re(\tilde{f}_{n}) +i\Im(\tilde{f}_{n}) $ and $\tilde {g}=\Re(\tilde{g}) +i\Im(\tilde{g}) $. Define $f_n^\Re=U^{-1}\Re(\tilde{f}_{n})$, $f_n^\Im=U^{-1}\Im(\tilde{f}_{n})$, $g^\Re=U^{-1}\Re(\tilde{g})$, and $g^\Im=U^{-1}\Im(\tilde{g})$. Define $A_+^t$ and $A_{-}^t$ as
\begin {align*}
\langle A_+^tg,f\rangle&=\int_{z\in\sigma(A),z>0}z^t\langle \tilde{g},\tilde{f}\rangle d\mu(z),\\
\langle A_{-}^tg,f\rangle&=\int_{z\in\sigma(A),z<0}(-z)^t\langle \tilde{g},\tilde{f}\rangle d\mu(z). \end{align*} Then $A_{-}$ and $A_+$ are positive operators, and $\langle A^tg,f\rangle= \langle A_{+}^tg,f\rangle+e^{i\pi t}\langle A_{-}^tg,f\rangle$.
For $t\in\mathbb{R}^+$, one has \begin{equation}
\sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2=F_n(t)+G_n(t), \end{equation} where \begin{eqnarray*}
F_n(t)&=&\sum_{g\in\mathcal{G}}(\langle A_{+}^tg^\Re,f_n^\Re\rangle+\langle A_{+}^tg^\Im,f_n^\Im\rangle+\cos(\pi t)\cdot(\langle A_{-}^tg^\Re,f_n^\Re\rangle+\langle A_{-}^tg^\Im,f_n^\Im\rangle)+\\
&&\sin(\pi t)\cdot
(\langle A_{-}^tg^\Re,f_n^\Im\rangle-\langle A_{-}^tg^\Im,f_n^\Re\rangle))^2, \end{eqnarray*} and \begin{eqnarray*}
G_n(t)&=&\sum_{g\in\mathcal{G}}(\langle A_{+}^tg^\Im,f_n^\Re\rangle-\langle A_{+}^tg^\Re,f_n^\Im\rangle+\sin(\pi t)\cdot(\langle A_{-}^tg^\Re,f_n^\Re\rangle+\langle A_{-}^tg^\Im,f_n^\Im\rangle)+\\
&&\cos(\pi t)\cdot(\langle A_{-}^tg^\Im,f_n^\Re\rangle-\langle A_{-}^tg^\Re,f_n^\Im\rangle))^2. \end{eqnarray*} Note that for $t \in \Omega_{L/4}\cap D_r$, by Lemma \ref{ReImBessel}, one has \begin{eqnarray*}
|F_n(t)|&\leq&6\cdot\left(\sum_{g\in\mathcal{G}}| \langle f_n^\Re,A_{+}^tg^\Re\rangle|^2+|\langle f_n^\Im,A_{+}^tg^\Im\rangle|^2+\frac{3+e^{2\pi r}}{4}\cdot(|\langle f_n^\Re,A_{-}^tg^\Re\rangle|^2+\right.\\
&&\left.|\langle f_n^\Im,A_{-}^tg^\Im\rangle|^2)+\frac{3+e^{2\pi r}}{4}\cdot(|\langle f_n^\Re,A_{-}^tg^\Im\rangle|^2+|\langle f_n^\Im,A_{-}^tg^\Re\rangle|^2)\right)\\
&\leq&6\cdot\left(C_\mathcal{G}\|A\|^{2r}+\frac{3+e^{2\pi r}}{4}\cdot C_\mathcal{G}\|A\|^{2r}+\frac{3+e^{2\pi r}}{4}\cdot C_{\mathcal{G}}\|A\|^{2r}\right)\\
&=&(15+3e^{2\pi r})\cdot C_\mathcal{G}\cdot\|A\|^{2r}, \end{eqnarray*} and \begin{eqnarray*}
|G_n(t)|&\leq& (15+3e^{2\pi r})\cdot C_\mathcal{G}\cdot\|A\|^{2r}. \end{eqnarray*}
Thus, (using a similar proof as in Lemma \ref{AnalPosA}) $F_n$ and $G_n$ are uniformly bounded analytic functions in $\Omega_{L/4}\cap D_r$.
As in Case 1, one can find two subsequences $\{F_{n_k}\}$ and $\{G_{n_k}\}$ converging to analytic functions $F$ and $G$, respectively. Moreover, since $G_n(t)\leq \sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2$, and $F_n(t)\leq \sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2$ for all $ t\in \mathbb{R}^+$ , and $\lim_{n\rightarrow\infty}\sum_{g\in\mathcal{G}}|\langle f_n,A^tg\rangle|^2=0, a.e.~ t\in[0,L],$ one can proceed as in the proof of Case 1 and get the contradiction that
$$\lim_{n_{k_j}\rightarrow\infty}\sum_{g\in\mathcal{G}}\int_{0}^{1}|\langle f_{n_{k_j}},A^tg\rangle|^2=0.$$ Thus, $\{A^tg\}_{g\in\mathcal{G},t\in[0,L]}$ is a semi-continuous frame for $\mathcal{H}$. \end{proof}
\end{document} |
\begin{document}
\title{Wide-field SU(1,1) interferometer}
Interferometers have been used for more than a century to measure physical quantities with high accuracy. Recently, the experimental realization of ${\rm SU(1,1)}$ interferometers has raised significant interest due to the loss-tolerant sub-shot-noise sensitivity~\cite{Li:14,Hudelist:14,Chekhova:16,Lemieux:16,Linneman:16,Lukens:16,Anderson:17,Gong:17,Manceau:17PRL,Du:18,Liu:18,Lukens:18,Liu:18fibers,Shaked:18}. The core idea is to use a series of two optical parametric amplifiers (OPAs) to probe phase shifts between them~\cite{Yurke:86,Sparaciari:16}. Possible applications are in remote sensing~\cite{Liu:18} and in quantum information processing~\cite{Shaked:18}, but such a scheme is especially attractive for quantum metrology with optical \cite{Hudelist:14,Anderson:17,Manceau:17PRL}, atom~\cite{Gross:10,Linneman:16} and hybrid~\cite{Chen:15} interferometers.
The two-mode squeezed state employed in an ${\rm SU(1,1)}$ interferometer is a quantum resource that helps to overcome the limitations encountered in a classical framework. In the optical phase measurement, the achievable sensitivity for an ${\rm SU(1,1)}$ interferometer beats the shot-noise limit, especially in the low photon-number regime~\cite{Manceau:17PRL,Anderson:17}. The $\mathrm{SU(1,1)}$ configuration is tolerant to detection losses, provided that the second OPA has a sufficiently large gain~\cite{Manceau:17,Giese:17,Shaked:18}. To satisfy this condition, one can use two nonlinear $\chi^{\left(2\right)}$ crystals as OPAs, since unbalancing the OPAs does not highly affect the mode composition, compared to the case of atoms \cite{Gross:10,Linneman:16} or four-wave mixers~\cite{Hudelist:14,Lukens:16}. All ${\rm SU(1,1)}$ interferometers realized so far are spatially single-mode and allow high sensitivity in one dimension only.
In this Letter, we report the first demonstration of a spatially multimode ${\rm SU(1,1)}$ interferometer, using high-gain parametric down-conversion (PDC) produced in a nonlinear crystal. This paves the way towards 2D phase profiling in the quantum regime, close to the common idea of an interferometer with a 2D fringe pattern. Furthermore, our configuration opens up the possibility to enhanced quantum imaging~\cite{Knyazev:19} and to the detection of a small disturbance in the orbital angular momentum (OAM)~\cite{Liu:18}. Finally, we report the measurement of quadrature squeezing for plane-wave modes inside the interferometer, as a prerequisite for achieving high sensitivity. We find that for all plane-wave modes, it is approximately the same quadrature that is squeezed.
\begin{figure}
\caption{(a) In a wide-field ${\rm SU(1,1)}$ interferometer, the degenerate PDC radiation from the first $\chi^{\left(2\right)}$ crystal is imaged into the second one. This enables multimode amplification/de-amplification, shown in the experimental far-field image. (b) In our experiment, the pump and the PDC are split with dichroic mirrors DM1-2. A spherical mirror M1 images the PDC radiation onto the crystal. Mirror M2 is mounted on the piezoelectric actuator PA to control the phase and the half-wave plate HWP and quarter-wave plate QWP control the polarisation. The amplified/de-amplified PDC radiation is observed in the Fourier plane of lens L on the CCD camera. Dichroic mirror DM3 rejects the pump, while bandpass filter BF and long pass filter LPF provide spectral filtering.}
\label{fig:setup}
\end{figure} The principle of our $\mathrm{SU(1,1)}$ interferometer is based on two nonlinear crystals with a focusing element in between, as shown in Fig. \ref{fig:setup} (a). The PDC radiation produced in the first crystal in the degenerate type-I regime (shown with a red filled cone) is amplified or de-amplified in the second crystal (shown as a brighter cone) depending on the optical phase between the pump, signal and idler fields $\phi=\phi_{p}-\phi_{s}-\phi_{i}$~\cite{Manceau:17}.
To exploit the full multimode structure of the radiation generated in the first crystal, the divergence of the PDC light is corrected with a lens. Provided that the PDC generation region in the first crystal is imaged into the second crystal, the amplification occurs for all angles of emission, intrinsically restricted only by the phase matching conditions. The configuration presented here offers good visibility over broad angles due to the mode matching. In previous experiments without the focusing element, spatial filtering of a single spatial mode around the pump direction was necessary to obtain high visibility and achieve sub-shot-noise phase sensitivity~\cite{Manceau:17PRL}.
To avoid the focusing of the pump beam, the paths of the pump and the PDC radiation are split and folded, see Fig. \ref{fig:setup} (b). The pump is the third harmonic of a pulsed ${\rm Nd:YAG}$ laser (wavelength $354.67$ nm, repetition rate $1$ kHz, pulse duration $18$ ps, average power 60 mW). Type-I collinear degenerate PDC is generated in a $\beta$-barium borate (BBO) crystal. The half-wave plate HWP misaligns the linear polarization of the pump by $27$ deg with respect to the optimal (horizontal) direction in order to reduce the parametric gain in the first pass.
Through the dichroic mirror DM2, the PDC radiation is sent to the focusing element, i. e. a spherical mirror M1 with curvature radius $R=100$ mm, and is imaged back onto the crystal. The pump transmitted by DM2 is sent to the quarter-wave plate QWP for polarization control and to the mirror M2 mounted on a piezoelectric actuator PA for phase control. The QWP on a double pass acts as an HWP and, by rotating the pump polarization, controls the parametric gain of the PDC generated in the second pass. Additionally, the pump has a beam size of FWHM $300\pm10$ $\mu{\rm m}$ in the first pass and, to provide a higher parametric gain, $180\pm10$ $\mu{\rm m}$ in the second pass. From the nonlinear dependence of the PDC intensity on the pump power, $I(P)\propto\sinh^2G,\,G\propto\sqrt{P}$, we measure the gain $G$~\cite{Iskhakov:12}. We obtain separately the gain from the first pass $G_{1}=2.1\pm0.3$ and from the second pass $G_{2}=3.3\pm0.3$. The gain unbalancing does not reduce much the interference visibility (see \href{link}{Supplement 1} for details) but it is crucial for the detection of the squeezing, as explained below.
If the path lengths from M1 and M2 to the crystal are such that the pump and PDC radiation pulses overlap on the way back, phase-sensitive amplification/de-amplification occurs depending on the phase shift controlled by the PA. The phase can be locked with the use of an additional beam injected at the unused port of DM2 and a feedback circuit (not shown). The dichroic mirror DM3 reflects the amplified/de-amplified PDC radiation to a charge-coupled device (CCD) camera in the Fourier plane of lens L with a focal length $f=40$ mm. The spectral filtering is achieved with a bandpass filter BF (central wavelength either $700$ or $710$ nm, bandwidth $10$ nm) and a long-pass filter LPF with the edge at $645$ nm, directly attached to the CCD.
\begin{figure}
\caption{(a) Intensity profile at the output of the interferometer for different phases. (b) Measured weights of OAM modes at the output of the interferometer show independence on the phase as predicted by theory (red). Relative shifts of the interferometric phase with respect to the dark fringe $\phi=\pi$ are $+0.68$ rad (blue), $+0.88$ rad (green), $+1.08$ rad (yellow).}
\label{fig:OAM}
\end{figure} The profile of the single-shot intensity distribution measured at the output is shown in Fig. \ref{fig:OAM} (a). As the phase is scanned, the amplitude of the characteristic flat-top intensity distribution for phase-matched PDC emission varies periodically, while the width stays constant. As shown in Fig. \ref{fig:OAM} (a), the visibility of the interference pattern for a single pixel is $98\%$, but even for the intensity integrated in two dimensions it is more than $95\%$.
Another remarkable property of the interferometer presented here is the stability of the spatial spectrum at the output, and, in particular, of the OAM spectrum, as the phase changes. Indeed, for any kind of application, the operation of the interferometer needs to be the same for all spatial modes. Further, the change of the OAM spectrum due to an azimuthal phase perturbation can be sensitively detected. To obtain the OAM spectrum at the output, we use a method based on the measurement of the covariance of intensities at different far-field points~\cite{Beltran:17}, specified by the transverse wave-vectors $\vec{q},\vec{q'}$, with modules $q,q'$ and azimuthal angles $\theta,\theta'$. Under the condition $q=q'=q_{0}$, the covariance, formally defined as
$\mathrm{Cov}_{\left|q=q'=q_{0}\right.}\left(\theta,\theta'\right)=\left<I\left(\theta\right)I\left(\theta'\right)\right>-\left<I\left(\theta\right)\right>\left<I\left(\theta'\right)\right>$, can be shown to depend only on the difference $\Delta\theta=\theta-\theta'$~\cite{Beltran:17}. Moreover, if the cross-correlations in the PDC radiation are removed by filtering a slightly non-degenerate wavelength, the covariance averaged over the variable $q_0$ is \begin{equation} \mathrm{Cov}\left(\Delta\theta\right)=\left[\sum_{l=-\infty}^{\infty}\Lambda_{l}e^{il\Delta\theta}\right]^{2}.\label{eq:azimcov} \end{equation} Here, $\Lambda_{l}$ are the weights of modes with OAM $l$, normalized as $\sum_{l=-\infty}^{\infty}\Lambda_{l}=1$. They can be extracted by performing a Fourier decomposition on the square root of Eq. \ref{eq:azimcov}. In experiment, we removed the cross-correlations with a $10$ nm bandpass filter centered at $700$ nm, which is shifted with respect to the degenerate frequency $709.33$ nm. The covariance was measured from $\sim500$ single-pulse intensity spectra acquired with the CCD camera.
Figure~\ref{fig:OAM} (b) shows the OAM weights of the radiation at the output of the interferometer for three different optical phases, namely $+0.68$ rad, $+0.88$ rad and $+1.08$ rad from the dark fringe respectively with blue, green and yellow bars. The weights remain the same as the phase is changed and this demonstrates that all OAM modes are uniformly amplified/de-amplified. The negative-OAM part is not shown here since it is symmetric with respect to $l=0$. The effective number of OAM modes, given by the formula $\left(\sum_{l=-\infty}^{\infty}\Lambda_{l}^{2}\right)^{-1}$, is $7.6\pm0.2$. For comparison, the number of {OAM modes for} the radiation in the first pass is theoretically estimated to be $13$. The double-pass configuration reduces the number of modes since the effective gain is larger than the gain of the first pass~\cite{Sharapova:15}. We obtain the theoretical distribution of the OAM weights through the Schmidt decomposition of the two-photon amplitude of PDC for the double-pass configuration \cite{Sharapova:15,Zakharov:18}. The theory (red bars in Fig. \ref{fig:OAM} (b)) confirms the experimental results in the same range of phases. For the case $\phi=\pi$, the theoretical distribution shows a slower decay and a $18\%$ increase in the effective number of modes.
Estimating the number of radial modes at the output of the interferometer to be $4.5\pm0.2$ from a similar experimental reconstruction in the radial degree of freedom, the total number of spatial modes is $35\pm3$. The multimode feature is attractive for the realization of high-dimensional quantum spaces~\cite{vandeNes:06,Molina:07}, but also for imaging \cite{Boyer:08Scie}. Indeed, the high-order modes are connected with fine details, because of their high spatial frequency, and represent a resource for the resolution in imaging experiments. We therefore characterize our interferometer as a `wide-field' one: it provides both a relatively broad angle ($20$ mrad) and a large number of angular modes within this range.
Finally, we show that the quadrature noise for the radiation inside the interferometer is below the shot noise. This is fundamental to achieve enhanced sensitivity with respect to a `classical' interferometer with the same number of photons. Homodyne detection is not suitable in our case since the PDC emission is highly multimode and it requires the appropriate shaping of the local oscillator field for the measurement of the squeezing in particular eigenmodes (one at a time)~\cite{Bennink:02, Embrey:15}. However, it was recently shown~\cite{Shaked:18} that the second amplifier in an ${\rm SU(1,1)}$ interferometer can be used for an `optical homodyne' measurement of quadrature squeezing at the output of the first amplifier. In this approach, the variances of the input quadratures $\hat{x}_{i},\hat{p}_{i}$ can be found by measuring the total intensity at the output: \begin{equation} I=C\cdot{\rm Var}\left(\hat{x}_{\psi}\right), \label{eq:parametric-amplif} \end{equation} with the calibration constant $C$ and the generic quadrature operator $\hat{x}_{\psi}=\hat{x}_{i}\cos\psi+\hat{p}_{i}\sin\psi$ and the phase of the interferometer $\phi=2\psi$. Eq. \ref{eq:parametric-amplif} is valid only under the assumption $e^{4G_{2}}{\rm Var}\left(\hat{x}_{i}\right)\gg {\rm Var}\left(\hat{p}_{i}\right)$, where $\hat{x}_{i}$ is the squeezed quadrature. This holds true in our case because of the unbalanced gains and because ${\rm Var}\left(\hat{p}_{i}\right)/{\rm Var}\left(\hat{x}_{i}\right)$ cannot exceed $e^{4G_{1}}$. The constant $C$ can be calibrated by removing the input to the second-pass OPA, leaving only vacuum fluctuations. Any loss induced at the detection stage will not contribute because it is already included in the constant $C$. This consideration is valid for single mode, but it can be generalized to the multimode scenario. In the measurement of the total intensity at the output, the contribution of each mode to the amplification/de-amplification depends on the relative phase between the modes and on the overlap of the modes of the input state with the modes generated in the second-pass OPA~\cite{Wasilewski:06}. The shapes of the modes for PDC radiation change very little as the gain increases~\cite{Sharapova:18}, therefore, the overlap can be reasonably high.
\begin{figure}
\caption{(a) Quadrature variance in dB changes with time for a triangle-wave scan of the piezo actuator. The traces around zero show the shot-noise level. The two different colors correspond to the measurement on the full frame (blue) and a single pixel (red), as shown on the intensity distribution at the output of the interferometer in panel (b).}
\label{fig:squeezing}
\end{figure} The level of amplification/de-amplification is measured by scanning the phase with a triangle-wave voltage applied to the PA and the result is shown in Fig.~\ref{fig:squeezing} (a). The measurement can be considered simultaneously for two different regions of interest (ROI) shown in panel (b): a single pixel (red) and the total frame (blue). The constant $C$ of Eq. \ref{eq:parametric-amplif} is obtained by blocking the radiation from the first pass at the curved mirror M1 (fluctuations are shown with the traces around zero). The small and large ROI give respectively the best squeezing level of $-4.3\pm0.7$ dB, $-2.6\pm0.3$ dB and the best anti-squeezing level of $14.8\pm0.9$ dB, $13.2\pm0.1$ dB. There is a good agreement between the anti-squeezing level (less sensitive to mode mismatch and losses) and the value expected from the independently measured gain $G_{1}$. The measurement is made with the bandpass filter centered at $710$ nm, but the result is similar for $700$ nm.
\begin{figure}
\caption{`Optical homodyne' measurement of quadrature squeezing (a) and anti-squeezing (b) for different plane-wave modes and its theoretical simulation (c,d).}
\label{fig:squeezing-map}
\end{figure} By considering every pixel of the CCD, we measure the quadrature variance for all plane-wave modes, as shown in Fig. \ref{fig:squeezing-map}. Panel (a) is the 2D distribution of the squeezing, while panel (b) shows the anti-squeezing. The sharp borders to zero are applied in both panels to remove artifacts caused by very low intensity. One should expect the distributions in Fig. \ref{fig:squeezing-map} to be flat, but in our case, the slight difference in the phase-matching of PDC in the first and second passes leads to squeezing slightly changing towards the center. Indeed, the intensity distribution from the second pass is $35\%$ broader than the one from the first pass, because a slight misalignment of the pump beam at the mirror M2 modifies the phase-matching conditions. This effect leads to mode mismatch, which affects mainly the squeezing distribution. The theoretical simulation of this behaviour, shown in Fig. \ref{fig:squeezing-map} (c) and (d) respectively for squeezing and anti-squeezing, is in good agreement with the experiment.
The wide-field ${\rm SU(1,1)}$ interferometer can be expanded to place inside the interferometer an object, whose absorption or image can be detected with enhanced sub-shot-noise accuracy and tolerance to detection losses~\cite{Knyazev:19}.
Further, the scheme presented here can be used in information processing as it provides a `quantum network', i.e. a multimode (multipartite) quantum system, similarly to other realizations in space~\cite{Armstrong:12} and frequency~\cite{Cai:17}. Indeed, the multipartite entanglement depends on the allocation of the spatial modes specified by the user and should be readily available with the selection in the detection process. The important advantage introduced here is that detection losses do not contribute to the measurement of the quadrature noise. In our case, one could use a particular mode combination without worrying about the effect of losses and simulate a linear optical network with the change of basis.
In conclusion, we have constructed a spatially multimode ${\rm SU(1,1)}$ interferometer by introducing a focusing component between the two OPAs. We have proved that the interferometer has the same multimode structure (around $35$ modes) as the phase changes. The quadrature noise for the radiation inside the interferometer has been proved to be below the shot noise (with the best squeezing level of $-4.3$ dB), as required for the highly sensitive detection of phase shifts. The measurement of the 2D distribution of the amplification/de-amplification level for the plane-wave modes reveals the uniform amplification phase for the quadrature variance of such modes. Possible applications of such an interferometer are the detection of the disturbance in the OAM imparted by an object, imaging with sub-shot-noise precision, quantum information processing in a multidimensional space and quantum metrology in two dimensions.
\section*{Funding Information} We acknowledge financial support of the Russian Science Foundation Project No.19-42-04105 and of the Japan Society for the Promotion of Science, Overseas Challenge Program for Young Researchers.
\noindent See \href{link}{Supplement 1} for supporting content.
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\begin{document}
\begin{abstract} In this paper we provide a local construction of a Sasakian manifold given a K\"ahler manifold. Obatined in this way manifold we call Sasakian lift of K\"ahler base. Almost contact metric structure is determined by the operation of the lift of vector fields - idea similar to lifts in Ehresmann connections. We show that Sasakian lift inherits geometry very close to its K\"ahler base. In some sense geometry of the lift is in analogy with geometry of hypersurface in K\"ahler manifold. There are obtained structure equations between corresponding Levi-Civita connections, curvatures and Ricci tensors of the lift and its base. We study lifts of symmetries different kind: of complex structure, of K\"hler metric, and K\"ahler structure automorphisms. In connection with $\eta$-Ricci solitons we introduce more general class of manifolds called twisted $\eta$-Ricci solitons. As we show class of $\alpha$-Sasakian twisted $\eta$-Ricci solitons is invariant under naturally defined group of structure deformations. As corollary it is proved
that orbit of Sasakian lift of steady or shrinking Ricci-K\"ahler soliton contains
$\alpha$-Sasakian Ricci soliton. In case of expanding Ricci-K\"ahler soliton existence of
$\alpha$-Sasakina Ricci solition is assured provided expansion coefficient is small enough. \end{abstract} \maketitle
\section{Introduction} The relations between Sasakian and K\"ahler manifolds is now quite well understood. In case structure is regular, characteristic vector field or Reeb vector field is regular, manifold can be viewed as line or circle bundle over K\"ahler manifold, where K\"ahler structure is determined by Sasakian structure. In the paper we consider in some sense reverse construction which allow to create Sasakian manifold given K\"ahler manifold. Construction is natural. Obtained in this way Sasakian manifolds share many properties of K\"ahler manifolds. Construction is of Sasakian lift is purely local - so strictly speaking we should rather consider our construction in terms of germs of structures.
There is analogy between our construction and idea of Ehresmann connection on fiber bundle $\pi:\mathcal P \rightarrow \mathcal B$, $\dim \mathcal B=n$, $\dim \mathcal P=n+k$. Ehresmann connection is some $n$-dimensional distribution $\mathcal D$ on total space and there is 1-1 operation operation between vector fields on base manifold and sections of $\mathcal D$.
In terms of structure equation here is analogy with theory of hypersurfaces in Riemann manifolds. Say $\iota: (\mathcal M,\bar g) \rightarrow (\mathcal{\bar M}, g)$, $\iota$ is inclusion, $\bar g=\iota^* g$. The first structure equation relates connection of manifold and connection of hypersurface $\nabla_XY = \bar\nabla_XY+h(X,Y)\xi$, $h$ being second fundamental form, $\xi$ normal vector field. Now the first structure equation for the lift $\pi: (\mathcal M , g) \rightarrow (\mathcal B, \bar g) $, $\mathcal M= \mathcal B^L$ is Sasakian lift, $\mathcal B$ its K\"ahler base, reads \begin{equation} \bar\nabla_{X^L}Y^L = (\nabla_XY)^L-\Phi(X^L,Y^L)\xi, \end{equation} $\xi$ being Reeb vector field. So covariant derivative of Sasakian lift is determined by the lift of covariant derivative of its K\"ahler base. The other possible point of view of our construction is theory of Riemann submersions with 1-dimensional fibers.
Geometry of Sasakian lift is very close to geometry of its base. For example for the Ricci tensor $\bar Ric$, of the lift, tensor field $\rho = Ric(X,\phi Y)$ is totally skew-symmetric, ie. some 2-form, moreover its related to the K\"ahler-Ricci form $\rho$, and K\"ahler form $\omega$ \begin{equation}
\bar\rho =\pi^*\rho - 2\pi^*\omega,
\end{equation} in particular this Sasakian-Ricci form is closed.
In geometric terms we study relations between infinitesimal symmetries of complex structure, Killing vector fields on K\"ahler manifold and some class of infinitesimal symmetries of Sasakian lift, as we call our construction. In particular there is local map between inifinitesimal symmetries of K\"ahler structure and almost contact structure of Sasakian manifold.
The other results of this kind are relations between holomorphic space forms and Sasakian manifolds of constant $\phi$-sectional curvatures, also K\"ahler Einstein manifolds and Sasakian $\eta$-Einstein manifolds. Lift of K\"ahler Einstein base is Sasakian $\eta$-Einstein manifold.
We provide relations between curvatures and Ricci tensors of K\"ahler manifold and its Sasakian lift. Obtained results allow us to show that Sasakian lift of holomorphic space form is Sasakian manifold of constant $\phi$-sectional curvature, also that lift of K\"ahler Einstein manifold is $\eta$-Einstein Sasakian manifold.
One of important subject of this paper is to study properties of Sasakian lift of K\"ahler-Ricci soliton. Our main result here is that Sasakian lift satisfies what we call equation of twisted $\eta$-Ricci soliton \begin{align*}
& Ric +\frac{1}{2}\mathcal L_X g = \lambda g +2C_1 \alpha_X\odot\eta +C_2 \eta\otimes\eta, \\ & \alpha_X = \mathcal L_X\eta, \end{align*} where $\lambda$, $C_1$, $C_2$ are some constants and $\odot$ denotes symmmetric tensor product.
In this paper almost contact metric manifold is called $\eta$-Ricci soliton if there is vector field $X$ and \begin{equation} Ric +\frac{1}{2}\mathcal L_X g = \lambda g + \mu \eta\otimes \eta, \end{equation} thus our definition is more general than that provided in \cite{ChoKimura}, where strictly speaking $\eta$-Ricci soliton is a metric which satisfies above equation, where $X=\xi$. In case of $\alpha$-Sasakian manifold Reeb vector field is Killing, therefore manifold is $\eta$-Ricci soliton only if it is $\eta$-Einstein manifold.
Condition for soliton vector field $X=\xi$ is rather restrictive. For example for very wide class of manifolds which satisfy $\mathcal L_\xi\eta=0$, and $d\Phi = 2f\eta\wedge \Phi$, for some local function $f$, the shape of Ricci tensor of strict $\eta$-Ricci soliton is completely determined. Namely Ricci tensor is necessary of the form, $h=\frac{1}{2}\mathcal L_\xi \phi$, \begin{equation} Ric(X,Y) = \alpha g(X,Y)+\beta g(X, h\phi Y)+ \gamma \eta(X)\eta(Y), \end{equation} where $ \alpha$, $\beta$, $\gamma$ are some functions. In case $h=0$, manifold is $\eta$-Einstein.
For sake of completness paper contains short exposition of geometry of class of deformations of Sasakian manifold. These deformations extend well-known $\mathcal D$-homotheties. Main result here is that kind of deformation, we call them $\mathcal D_{\alpha,\beta}$-homotheties \begin{equation}
g |_{\mathcal D}\mapsto g' |_{\mathcal D} = \alpha g|_{\mathcal D}, \quad g |_{\{\xi\}} \mapsto g'|_{\{\xi\}} = \beta^2 g|_{\{\xi\}}, \end{equation} $\alpha$, $\beta = const. > 0$, map $\alpha$-Sasakian manifold into another $\alpha'$-Sasakian manifold.
As we will see action of $\mathcal D_{\alpha,\beta}$-homotheties determines three invariant classes of twisted $\alpha$-Sasakian $\eta$-Ricci solitons where $C_1 < \frac{1}{2}$, $C_1 = \frac{1}{2}$ or $C_1> \frac{1}{2}$. Lift of K\"ahler-Ricci soliton provides example of Sasakian twisted $\eta$-Ricci soliton with $C_1 < \frac{1}{2}$. This raises existence question: do exist $\alpha$-Sasakian twisted $\eta$-Ricci solitons with $C_1 \geqslant \frac{1}{2}$? In some cases we have stronger result: there is $\mathcal D_{\alpha,\beta}$-homothety such that image is $\alpha$-Sasakian Ricci soliton. As we will see this is the case of lift over K\"ahler-Ricci steady or shrinking solitons. The lift of expanding K\"ahler Ricci soliton can always be deformed into $\alpha$-Sasakian $\eta$-Ricci soliton and into Ricci soliton if expansion coefficient is small enough.
Used notation can be confused for the reader. Particularly this concerns the how we use the notion of $\alpha$-Sasakian manifold. In some parts of paper this term is used in wider sense: manifold is called $\alpha$-Sasakian if there is real constant $c > 0$, and covariant derivative $\nabla\phi$ satisfies $(\nabla_X\phi)Y=c(g(X,Y)\xi-\eta(Y)X)$. However in expressions like $\frac{\beta}{\alpha}$-Sasakian, it is assumed that $c=\frac{\beta}{\alpha}$.
\section{Preliminaries} In this section we will recall some basic facts about almost contact metric manifolds, and in particular about Sasakian manifolds. \subsection{Almost contact metric manifolds}
Let $\mathcal{M}$ be a smooth connected odd-dimensional manifold, $\dim \mathcal{M} =2n+1 \geqslant 3$. An almost contact metric structure on $\mathcal{M}$, is a quadruple of tensor fields $(\phi,\xi,\eta)$, where $\phi$ is $(1,1)$-tensor field, $\xi$ a vector field, $\eta$ a 1-form, and $g$ a Riemnnian metric, which satisfy \cite{Blair} \begin{align} & \phi^2X = -X + \eta(X)\xi, \quad \eta(\xi) = 1, \\ & g(\phi X,\phi Y) = g(X,Y)-\eta(X)\eta(Y), \end{align} where $X$, $Y$ are arbitrary vector fields on $\mathcal{M}$. Triple $(\phi,\xi,\eta)$ is called an almost contact structure (on $\mathcal{M}$). From definition it follows that tensor field $\Phi(X,Y)=g(X,\phi Y)$, is totally skew-symmetric, a 2-form on $\mathcal{M}$, called a fundamental form. In the literature vector field $\xi$ is referred to as characteristic vector field, or Reeb vector field. In analogy the form $\eta$ is called characteristic form. Distribution $\mathcal{D} = \{\eta =0\}$, is called characteristic distribution, or simply kernel distribution, as its sections $X\in \Gamma^\infty(\mathcal{D})$, satisfy $\eta(X)=0$. As $\eta$ is non-zero everywhere $\dim \mathcal{D} = 2n$. Manifold equipped with fixed almost contact metric structure is called almost contact metric manifold.
Let for $(1,1)$-tensor field $S$, $N_S$ denote its Nijenhuis torsion, thus \begin{equation} N_S(X,Y)=S^2[X,Y]+[SX,SY]-S([SX,Y]+[X,SY]). \end{equation} Almost contact metric structure $(\phi,\xi,\eta,g)$ is said to be normal if tensor field $N^{(1)} = N_\phi +2d\eta\otimes \xi$, vanishes everywhere. Normality is the condition of integrability of naturally defined complex structure $J$ on a product of real line (a circle) and almost contact metric manifold.
There are three classes most widely studied almost contact metric manifolds \begin{itemize} \item[a)] Contact metric manifolds defined by condition \begin{equation}
d\eta = \Phi; \end{equation} \item[b)] Almost Kenmotsu manifolds \begin{equation} d\eta = 0,\quad d\Phi = 2\eta\wedge\Phi; \end{equation} \item[c)] Almost cosymplectic (or almost coK\"ahler) manifolds \begin{equation} d\eta =0, \quad d\Phi =0. \end{equation} \end{itemize} If additionaly manifold is normal we obtain following corresponding classes \begin{itemize} \item[an)] Sasakian manifolds - ie. contact metric and normal \begin{equation} (\nabla_X\phi)Y = g(X,Y)\xi - \eta(Y)X; \end{equation} \item[bn)] Kenmotsu manifolds \begin{equation} (\nabla_X\phi)Y = g(\phi X,Y)\xi - \eta(Y)\phi X; \end{equation} \item[cn)] Cosymplectic manifolds \begin{equation} \nabla\phi =0. \end{equation} \end{itemize} Above we have provided characterization of respective manifold in terms of covariant derivative - with resp. to the Levi-Civita connection of $g$ - of the structure tensor $\nabla\phi$.
For an almost contact metric manifold $\mathcal{D}$-homothety, with coefficient $\alpha$, is a deformation of an almost contact metric structure $(\phi,\xi,\eta,g) \rightarrow (\phi',\xi',\eta',g')$, defined by \begin{align*} & \phi'=\phi,\quad \xi'=\frac{1}{\alpha}\xi,\quad \eta' = \alpha\eta, \\ & g' = \alpha g +(\alpha^2-\alpha)\eta\otimes\eta. \end{align*} In this paper we will consider more general deformations of an almost contact metric structure, defined by real parameters $\alpha$, $\beta > 0$, given by \begin{align*} & \phi' = \phi, \quad \xi'=\frac{1}{\beta}\xi, \quad \eta' = \beta\eta, \\ & g'= \alpha g + (\beta^2-\alpha)\eta\otimes\eta. \end{align*} We call such deformations $\mathcal D_{\alpha,\beta}$-homotheties.
\subsection{Sasakian manifolds} Here we provide some very basic properties of Sasakian manifold. On Sasakian manifold \begin{equation} (\nabla_X\phi)Y = g(X,Y)\xi - \eta(Y)X, \end{equation} which implies that Reeb vector field $\xi$, is Killing vector field $\mathcal{L}_\xi g=0$, moreover \begin{equation} \mathcal{L}_\xi\phi =0, \quad \nabla_X\xi = -\phi X,\quad \nabla_\xi\xi=0, \end{equation} for curvature $R_{XY}\xi$, and Ricci tensor $Ric(X,\xi)$, we have \begin{align} \label{l:curv2} & R_{XY}\xi = \eta(Y)X-\eta(X)Y, \quad R_{X\xi}\xi = X-\eta(X)\xi, \\ & \label{l:ric:xi} Ric(X,\xi) = 2n\eta(X), \end{align} in particular $Ric(\xi,\xi)=2n$, and $R_{XY}\xi =0$, for $X$, $Y$ sections of characteristic distribution, $X$, $Y\in \Gamma^\infty(\mathcal{D})$, see \cite{Blair}.
More general almost contact metric manifold is called $\alpha$-Sasakian \begin{equation} (\nabla_X\phi)Y = \alpha(g(X,Y)\xi -\eta(Y)X), \end{equation} for some non-zero real constant $\alpha \neq 0$.
As we will proceed further we will obtain following equation for Ricci tensor $Ric$ of $\alpha$-Sasakian manifold \begin{equation} \label{e:e:sol} Ric +\frac{1}{2}(\mathcal{L}_Xg) = \lambda g +2C_1\alpha_X\odot\eta + C_2\eta\otimes \eta, \end{equation} where $\lambda$, $C_1$, $C_2$ are some real constants, vector field $X$, and 1-form $\alpha_X$, satisfy \begin{equation} \label{e:e:sol2} \eta(X) = 0, \quad (\mathcal{L}_X\eta)(Y)=\alpha_X(Y), \end{equation} in particular on $\alpha$-Sasakian manifold $\alpha_X(\xi)=0$, as by assumption $\eta(X)=0$, hence $(\mathcal{L}_X\eta)(\xi)=2d\eta(X,\xi)=0$.
\subsection{K\"ahler manifolds} Almost complex structure on manifold is a $(1,1)$-tensor
field $J$, such that $J^2X = -Id$. Structure is said
to be complex if any point admits a local chart,
such that local coefficient of $J$, in this chart,
are all constants. Necessary and sufficient condition,
is vanishing Nijenhuis torsion of $J$. If additionally
there is Riemannian metric $g$, with properties
$g(JX,JY)= g(X,Y)$, and complex structure is covariant
constant for Levi-Civita connection - manifold is called a K\"ahler manifold. Tensor field
$\omega(X,Y)=g(X,JY)$, is a maximal rank 2-form, called K\"ahler
form, as $J$ is parallel, K\"ahler form is always
closed. In particular K\"ahler form determines symplectic structure on K\"ahler manifold.
In dimensions $ > 2$, K\"ahler manifold of constant sectional curvature is always locally flat, thus more natural notion is holomorphic sectional curvature. This is sectional curvature of "complex" plane ie. a plane spanned by vectors $X$, and $JX$. If holomorphic curvature does not depend neither on point nor on complex plane section - K\"ahler manifold is said to have constant holomorphic curvature.
Infinitesimal automorphism of complex structure is a vector field $X$, which generates local 1-parameter flow $f_t$, of biholomorphisms, that is $f_{t*}J = Jf_{t*}$. The vector field $X$ is an infinitesimal automorphism if and only if $\mathcal{L}_XJ =0$. Note that if $X$ determines infinitesimal automorphism, then also vector field $JX$ is an infinitesimal automorphism. Moreover complex vector field $X^\mathbb{C}=X-\sqrt{-1}JX$, is holomorphic. Its local coordinates in complex chart are holomorphic functions.
We say that vector field $X$ is K\"ahler structure automorphism if the two of the three conditions \begin{equation} \mathcal{L}_XJ = 0,\quad \mathcal{L}_Xg =0, \quad \mathcal{L}_X\omega =0, \end{equation} are satisfied. Then the third condition is automatically satisfied. For example if $X$ is complex structure automorphism and Killing vector field, then it preserves K\"aler form. In particular is locally Hamiltonian with respect to symplectic structure determined by K\"ahler form, that is locally \begin{equation}
\omega(X,Y)= dH(Y), \end{equation} for some locally defined function $H$. cf. \cite{Bess},\cite{KoNo2}.
\subsection{Ricci solitons} Given geometric objects on manifold is it important to know do exist objects with some particular properties. One of possible way to find such particular entity is trough geometric flow. Of course here the point is how to create proper law of evolution. This kind of object is so-called Ricci flow introduced by Hamilton \cite{Hamilton} \begin{equation} \frac{\partial}{\partial t}g= -2Ric(g_t), \quad t\in [0,T), \quad T > 0, \end{equation} where we search for solution on some non-empty interval, with given initial condition $g_0 = g$.
In present time Ricci flow is one of most extensively studied subject. The goals are two-fold: analytical and geometrical. In terms of analysis there are studies considering problems of existence of the Ricci, on side of geometry plenty new manifolds which admits non-trivially Ricci flows \cite{ChowKnopf},\cite{ChowLuNi}.
Particular case of solutions are flows of the form \begin{equation} g_t = c(t)f_t^*g, \end{equation} where $f_t$ $1$-parameter group of diffeomorphisms. Such solutions are called Ricci solitons. In some sense they represent trivial solutions of Ricci flow, say $g_0$, and $g_t$, are always isometric up to homothety. In infinitesimal terms Ricci soliton is Riemannian manifold (Riemannian metric),
which admits a vector field $X$, such that \begin{equation} Ric +\frac{1}{2}\mathcal{L}_Xg = \lambda g, \end{equation} for some real constant $\lambda \in \mathbb{R}$, \cite{Cao1},\cite{Cao2},\cite{Ivey},\cite{SongWeink}.
Depending on sign of $\lambda$ there are expanding Ricci solitons: $\lambda >0$,
steady: $\lambda =0$, and shrinking: $\lambda < 0$. Of course in this classification only sign
of $\lambda$ counts as homothety $g \mapsto g' = cg$, $c > 0$, provides Ricci soliton with
soliton constant $\lambda' = \lambda c$ (and soliton vector field $X' = cX$).
In particular we can always normalize equation to $\lambda =1, 0, -1$.
Assuming $X$ is gradient $g(X,Y)=dH(Y)$, we have $(\mathcal{L}_Xg)(Y,Z) = 2Hess_H(Y,Z)$, where as usually $Hess_H$, stands for Hessian of the function, which is defined by \begin{equation} Hess_H(Y,Z)=(\nabla_Y dH)(Z), \end{equation} ie. it is covariant derivative of differential form of the function $H$. Therefore in case $X=grad H$, we obtain \begin{equation} Ric + Hess_H = \lambda g. \end{equation}
The solution to the Ricci flow equation in case of initial metric being K\"aheler, is family of K\"ahler metrics. Therefore in particular case of Ricci soliton we obtain that the vector field $X$, satisfies $\mathcal L_X J=0$. Equivalently metric of K\"ahler manifold is K\"ahler-Ricci soliton if \begin{equation} \rho +\frac{1}{2}\mathcal{L}_X\omega = \lambda\omega, \quad \mathcal L_X J=0, \end{equation} where $\rho$ denotes the Ricci form \cite{Bryant}.
\section{Sasakian lift of K\"ahler manifold} Let $\mathcal{N}$ be K\"ahler manifold with K\"ahler structure $(J,g)$, let $\omega$ be a K\"ahler form $\omega(X,Y) = g(X,JY)$. Assume there is globally defined 1-form $\tau$, such that $\omega = d\tau$, if there is no such form, so real cohomology class $[\omega] \neq 0$, we restrict structure to some open subset $\subset \mathcal{N}$, to assure existence of $\tau$. Then on product $\mathcal{M} = \mathbb{R}\times\mathcal{N}$, we introduce structure of Sasakian manifold in terms of a lift of vector field on $\mathcal{N}$. Set $$ \xi = \partial_t,\quad \eta = dt + \pi_2^*\tau, $$
for vector field $X$ on K\"ahler base we define its lift $X \mapsto X^L$ $$X^L= -\pi_2^*\tau(X)\xi+X, $$ where $\pi_1$, $\pi_2$ are projections $$ \pi_1: \mathcal{M} \rightarrow \mathbb{R},\quad
\pi_2: \mathcal{M}\rightarrow \mathcal{N},
$$
on the first and second
product components. If $f:\mathcal{N} \rightarrow \mathbb{R}$,
is a smooth function on K\"ahler base, we set $\bar{f}:\mathcal{M}\rightarrow \mathbb{R}$, $\bar{f}=f\circ\pi_2$, then $(fX)^L = \bar{f}X^L$. Note for $\bar{f}$, we have $\xi\bar{f}= d\bar{f}(\xi)=df(\pi_{2*}\xi)=0$. For functions $f_i$, and vector fields $X_i$, $i=1,2$, on $\mathcal{N}$ we have \begin{equation} (f_1 X_1+f_2X_2)^L = \bar{f}_1X_1^L+\bar{f}_2X_2^L, \end{equation} in particular $(\cdot)^L$ is $\mathbb{R}$-linear.
With help of the operation of the lift we now introduce on $\mathcal{M}$
a tensor field $\phi$, and metric $g^L$ , requiring \begin{equation} \phi X^L = (JX)^L, \quad \phi\xi =0, \quad g^L = \eta\otimes\eta + \pi_2^*g. \end{equation}
Note following formula for
commutator of lifts
\begin{equation}
\label{e:bra:lif}
[X^L,Y^L] = [X,Y]^L-2d\eta(X^L,Y^L)\xi.
\end{equation}
The following proposition
is just simple verification with help of provided definitions.
\begin{proposition}
Tensor fields $(\phi^L,\xi,\eta, g^L)$, determine almost contact
metric structure on $\mathcal{M}$.
\end{proposition}
\begin{proof}
If $(X_1,X_2,\ldots, X_{2n})$ is a local frame of vector fields on K\"ahler base vector
fields $(X_1^L,X_2^L,\ldots,X^L_{2n})$ is a local frame which spans characteristic
distribution $\mathcal D = \ker \eta = \{ \eta = 0 \}$, with Reeb vector field they create local
frame on the lift $\mathcal M$. Therefore any vector $\bar Y$ on the lift field can be given locally by
\begin{equation} \bar Y =a^0\xi + \sum_{i=1}^{2n}a^i X_i^L, \end{equation} $a^i$, $i=0,\ldots, 2n$ are some functions, therefore it is enough to verify that $\phi^2 X^L = -X^L$. By definition \begin{equation} \phi^2 X^L = \phi (JX)^L = (J^2 X)^L = -X^L. \end{equation} Similarly in case of metric given vector fields $\bar Y$, $ \bar Z$, \begin{equation} g^L(\phi\bar Y,\phi \bar Z) = \sum_{i,j=1}^{2n} a^i b^j g^L(\phi X_i^L,\phi X_j^L), \end{equation} from definition we have \begin{equation} g^L(\phi X_i^L,\phi X_j^L) = g(JX_i,JX_j)\circ \pi_2, \end{equation} as base manifold is K\"ahler $g(JX_i,JX_j) = g(X_i,X_j)$, from other hand \begin{equation} g^L(X_i^L, X_j^L ) = g(X_i,X_j)\circ\pi_2, \end{equation} hence $g^L(\phi \bar Y, \phi \bar Z) = g^L(\bar Y, \bar Z)-\eta(\bar Y)\eta(\bar Z)$.
\end{proof}
The proof o the above Proposition says little more.
\begin{corollary}
For $(t,q)\in \mathbb{R}\times \mathcal{N}$, map $X \mapsto X^L$, establishes
isometry $(T_q\mathcal{N}, g_q)\rightarrow (\mathcal{D}_{(t,q)}, g^L|_\mathcal{D})$.
In particular if $E_i$, $i=1,\ldots 2n$, is an orthonormal local frame
on K\"ahler base, lifts $E_i^L$, $i=1,\ldots 2n$, form an orthonormal frame
spanning contact distribution.
\end{corollary}
In future we we simplify notation and instead of eg. \begin{equation*} g^L(X^L,Y^L) = g(X,Y)\circ \pi_2, \end{equation*} we write $ g^L(X^L,Y^L) = g(X,Y), $ if it does not lead to a confusion. Similarly $d\eta(X^L,Y^L)=\Phi(X^L,Y^L)=\omega(X,Y)$. For function on K\"ahler base, or according to our simplified notation $X^L\bar{f}=\overline{Xf}$.
\begin{proposition} Manifold $\mathcal{M}$, equipped with structure $(\phi^L,\xi,\eta,g^L)$, is Sasakian manifold. \end{proposition} \begin{proof} The first we note that the fundamental form of $\mathcal{M}$ is just pullback of K\"ahler form $\omega$, $\Phi = \pi_2^*\omega$. From other hand $d\eta = d\pi_2^*\tau = \pi_2^*d\tau = \pi_2^*\omega = \Phi$, by assumption about $\tau$, and our above remark. Therefore $\mathcal{M}$ is contact metric manifold. To end the proof we directly verify that $\mathcal{M}$, is normal $N^{(1)} = 0$, it is enough to verify normality on vector fields of form $X^L$, $N^{(1)}(X^L,Y^L)= 0$, as they span the module $\Gamma^\infty(\mathcal{D})$, of all sections of contact distribution, and to verify directly that $N^{(1)}(\xi,X^L)=0$. As $N^{(1)} = N_\phi +2d\eta\otimes\xi$, with help of (\ref{e:bra:lif}) we obtain \begin{equation} N_\phi(X^L,Y^L) = (N_J(X,Y))^L -2 d\eta(X^L,Y^L)\xi = -2 d\eta(X^L,Y^L)\xi, \end{equation} as $J$ is complex structure, by Newlander-Nirenberg theorem this equivalent to vanishing its Nijenhuis torsion $N_J=0$. So $$ N^{(1)}(X^L,Y^L) = -2d\eta(X^L,Y^L)\xi +2d\eta(X^L,Y^L)\xi =0. $$ The case $N^{(1)}(\xi, X^L)$ is almost evident as for every vector field on K\"ahler base there is $[\xi, X^L]=0$, $d\eta(\xi, \cdot) =0$. \end{proof}
The almost contact metric structure constructed as above we call Sasakian lift of a K\"ahler structure. Consequently manifold itself we call Sasakian lift of K\"ahler manifold. If it is not explicitely stated what particular manifold, we just use a term Sasakian lift to emphasize that almost contact metric structure is obtained from some K\"ahler base manifold with help of the above described construction.
\subsection{Structure equations} Here we provide fundamental relations between Levi-Civita connections of K\"ahler base and its Saskian lift. Let $\bar\nabla$ denote the operator of the covariant derivative of Levi-Civita connection of Sasakian lift metric $\bar{\nabla} = LC(g^L)$, while $\nabla=LC(g)$, the Levi-Civita connection of K\"ahler base.
\begin{proposition} \label{p:streqs} For vector fields $X$, $Y$ on K\"ahler base we have \begin{align} \label{streqs1} & \bar{\nabla}_{X^L}\xi = - \phi X^L = -(JX)^L, \\ \label{streqs2} & \bar{\nabla}_{X^L}Y^L = (\nabla_XY)^L -\Phi(X^L,Y^L)\xi, \end{align} \end{proposition} \begin{proof} The first structure equation comes from property of any Sasakian manifold and the definition of the lift. The second structure equation is consequence of Koszul formula for Levi-Civita connection, applied to both K\"ahler base and its lift. Using Koszul formula we need to take into account that \begin{equation*} g^L(X^L,Y^L) = g(X,Y),\quad X^Lg^L(Y^L,Z^L)=Xg(Y,Z). \end{equation*} Therefore \begin{align*} & 2g^L(\bar \nabla_{X^L}Y^L, Z^L) = X^Lg^L(Y^L,Z^L)+Y^Lg^L(X^L,Z^L) - \\ & \qquad Z^Lg^L(X^L,Y^L) + g^L([X^L, Y^L], Z^L) + g^L([Z^L,X^L],Y^L) + \\ & \qquad g^L([Z^L,Y^L],X^L) = 2 g(\nabla_XY, Z)\circ\pi_2 = 2g^L((\nabla_XY)^L, Z^L), \end{align*} from other hand projection $\bar\nabla_{X^L}Y^L$ on $\xi$, is given by $g^L(\bar\nabla_{X^L}Y^L,\xi) = -g^L(\bar{\nabla}_{X^L}\xi,Y^L) = -\Phi(X^L,Y^L)\xi$. Here we use only the fact that on Sasakian manifold always $\nabla\xi = -\phi$. \end{proof}
Note above formula coincides with formula of commutator of the lifts \begin{align*} & [X^L,Y^L] = \bar\nabla_{X^L}Y^L-\bar\nabla_{Y^L}X^L = \\ & \qquad (\nabla_XY)^L-\Phi(X^L,Y^L)\xi - (\nabla_YX)^L +\Phi(Y^L,X^L)\xi = \\ & \qquad [X,Y]^L -2\Phi(X^L,Y^L)\xi, \end{align*}
In words orthogonal projection of covariant derivative of lifts $\bar\nabla_X^LY^L$, on characteristic distribution is equal exactly to the lift of covariant derivative on K\"ahler base $(\nabla_XY)^L$, while projection on direction of Reeb vector field is equal to $-\Phi(X^L,Y^L)\xi$, however note that $\Phi(X^L,Y^L) = \omega(X,Y)\circ \pi_2$, ie. pullback of K\"ahler form on these vector fileds. In symbolic terms we can describe this as \begin{equation} \bar \nabla = \nabla^L -(\pi_2^*\omega)\otimes\xi. \end{equation}
The structure equations in the Proposition {\bf \ref{p:streqs}.} remind structure equations for hypersurface in Riemannian manifold. However there is remarkable difference: in case of hypersurface its second fundamental form is symmetric tensor field, while in our case the tensor which supposedly plays a role of second fundamental form is skew-symmetric. Note that (\ref{streqs2}) {\it is not } a definition of connection. $\bar \nabla $ is just Levi-Civita connection of the metric $g^L$. But in particular case of lifts of vector fields from K\"ahler base, (\ref{streqs2}) holds true.
Having the structure equations as above we proceed to obtain relations between corresponding curvature tensors of K\"ahler manifold and its Sasakian lift.
\begin{proposition} \label{p:curv} Curvatures $R$ and $\bar{R}$ of K\"ahler base and its Sasakian lift are related by \begin{align} \label{l:curv1} & \bar{R}_{X^LY^L}Z^L = (R_{XY}Z)^L + \Phi(Y^L,Z^L)\phi X^L - \Phi(X^L,Z^L)\phi Y^L - \\ &\nonumber \qquad 2\Phi(X^L,Y^L)\phi Z^L, \end{align} \end{proposition} \begin{proof} By the structure equations (\ref{streqs1}), (\ref{streqs2}) \begin{align} \label{eq1} & \bar{\nabla}_{X^L}\bar{\nabla}_{Y^L}Z^L = \bar{\nabla}_{X^L}(\nabla_YZ)^L - X^L\Phi(Y^L,Z^L)\xi + \\ & \nonumber \qquad \Phi(Y^L,Z^L)\phi X^L = (\nabla_X\nabla_YZ)^L - (\Phi(X^L,(\nabla_YZ)^L)+ \\ & \nonumber \qquad X^L\Phi(Y^L,Z^L))\xi + \Phi(Y^L,Z^L)\phi X^L, \\ \label{eq2} & \bar{\nabla}_{[X^L,Y^L]}Z^L = \bar{\nabla}_{[X,Y]^L}Z^L - 2d\eta(X^L,Y^L)\bar{\nabla}_\xi Z^L = \\ & \nonumber \qquad (\nabla_{[X,Y]}Z)^L - \Phi([X,Y]^L,Z^L)\xi - 2d\eta(X^L,Y^L)\bar{\nabla}_\xi Z^L. \end{align} For the lift of vector field $Z^L$, $[\xi, Z^L] = 0$. Therefore as Levi-Civita connection has no torsion, we have \begin{equation} \label{eq3} \bar{\nabla}_\xi Z^L = \bar{\nabla}_{Z^L}\xi = -\phi Z^L. \end{equation} For curvature $$ \bar{R}_{X^LY^L}Z^L = \bar{\nabla}_{X^L}\bar{\nabla}_{Y^L}Z^L - \bar{\nabla}_{Y^L}\bar{\nabla}_{X^L}Z^L - \bar{\nabla}_{[X^L,Y^L]}Z^L, $$ in view of (\ref{eq1})-(\ref{eq3}), we obtain \begin{align*} & \bar{R}_{X^LY^L}Z^L = (R_{XY}Z)^L +\Phi(Y^L,Z^L)\phi X^L - \Phi(X^L,Z^L)\phi Y^L + \\ & \nonumber \qquad ( ( -\bar\nabla_{X^L}\Phi)(Y^L,Z^L) + (\bar\nabla_{Y^L}\Phi)(X^L,Z^L))\xi -
2\Phi(X^L,Y^L)\phi Z^L , \end{align*} as manifold is Sasakian $(\bar{\nabla}_{X^L}\Phi)(Y^L,Z^L)=(\bar{\nabla}_{Y^L}\Phi)(X^L,Z^L)=0$, vanish. \end{proof}
The following proposition describes relation between Ricci tensors of K\"ahler base and its lift \begin{proposition} \label{p:ric} Ricci tensors and scalar curvatures of K\"ahler base and its Sasakian lift are related by \begin{align} \label{r1} & \bar{R}ic(X^L,Y^L) = Ric(X,Y)-2g(X,Y), \end{align} in particular we have for scalar curvatures of K\"ahler base and Sasakian lift $s$, $\bar{s}$ \begin{align} \label{r2} & \bar{s}= s-2n. \end{align} \end{proposition} \begin{proof} In the proof we use adopted local orthonormal frame $(\xi, E_1^L,\ldots E_{2n}^L)$, where $(E_1,\ldots E_{2n})$, is local orthonormal frame on K\"ahler base. Then \begin{equation} \label{ric1} \bar{R}ic(X^L,Y^L) = \bar{R}(\xi, X^L,Y^L,\xi) + \sum\limits_{i=1}^{2n}\bar{R}(E_i^L,X^L,Y^L,E_i^L), \end{equation} by the Proposition {\bf \ref{p:curv}.}, (\ref{l:curv1}), and curvature identities for Sasakian manifold (\ref{l:curv2}), we obtain \begin{align} \label{tr:curv} & \sum\limits_{i=1}^{2n}\bar{R}(E_i^L,X^L,Y^L,E_i^L) = \sum\limits_{i=1}^{2n}R(E_i,X,Y,E_i) - \\ & \nonumber \qquad 3\sum\limits_{i=1}^{2n}\Phi(E_i^L,X^L)\Phi(E_i^L,Y^L) = Ric(X,Y)-3g(JX,JY)= \\ & \nonumber \qquad Ric(X,Y)-3g(X,Y), \\ \label{xi-sect} & \bar{R}(\xi,X^L,Y^L,\xi) = \bar{R}(X^L,\xi,\xi,Y^L) =g^L(X^L,Y^L)=g(X,Y), \end{align} now with help of (\ref{ric1})-(\ref{xi-sect}), we find \begin{equation*} \bar{R}ic(X^L,Y^L)=Ric(X,Y)-2g(X,Y). \end{equation*} For the scalar curvature of the lift $ \bar{s}= \bar{R}ic(\xi,\xi)+\sum_{i=1}^{2n}\bar{R}ic(E_i^L,E_i^L), $ and by (\ref{l:ric:xi}), (\ref{r1}), \begin{equation*} \bar{s}=2n+\sum\limits_{i=1}^{2n}(Ric(E_i,E_i)-2g(E_i,E_i)) =2n +s -4n = s-2n. \end{equation*} \end{proof}
\begin{proposition} On Sasakian lift tensor field $\bar{\rho}(\cdot,\cdot)=\bar{R}ic(\cdot,\phi \cdot)$, is a closed 2-form, moreover \begin{equation} \bar{\rho} = \pi_2^*\rho -2\pi_2^*\omega, \end{equation} that is $\bar{\rho}$ is a pullback of difference of Ricci form and twice of K\"ahler form. \end{proposition} \begin{proof} We have $ \bar{\rho}(X^L,Y^L) = \bar{R}ic(X^L,\phi Y^L) = \bar{R}ic(X^L,(JY)^L), $ and in virtue of the Proposition {\bf \ref{p:ric}.}, \begin{align*} & \bar{R}ic(X^L,(JY)^L) = Ric(X,JY) -2g(X,JY) = \\ & \qquad \rho(X,Y)-2\omega(X,Y), \end{align*} clearly $\bar{\rho}(X^L,\xi)=\bar{\rho}(\xi,X^L)=0$, therefore $\bar{\rho}$ is skew-symmetric and closed, as both Ricci and K\"ahler forms are closed. \end{proof}
Here are some corollaries of obtained results. \begin{theorem} \label{th:chc} If K\"ahler base has constant holomorphic sectional curvature $c=const.$, then its Sasakian lift is Sasakian manifold of constant $\phi$-sectional curvature $\bar{c} = c-3$. \end{theorem} \begin{proof} Let fix a point $(t,q)\in \mathcal{M}$, and let $v\in \mathcal{D}_{(t,q)}$, be unit vector, then $\phi$-sectional curvature $K_\phi(v)$, is a sectional curvature of plane $(v,\phi v)$. Hence \begin{equation*} K_\phi(v) = \bar{R}(v,\phi v,\phi v,v) = g^L(\bar{R}_{v\phi v}\phi v, v). \end{equation*} As $(\cdot)^L$ is point-wise linear isometry between $T_q\mathcal{N}$ and $\mathcal{D}_{(t,q)}$, there is local vector field $X$ on K\"ahler base, such that $X^L= v$ at the point $(t,q)$. We can assume that $X$ is normalized. In view of the Proposition {\bf \ref{p:curv}}, eq. (\ref{l:curv1}), having in mind that $g^L(X^L,X^L)=g(X,X)$, and \begin{equation*} g^L((R_{XJX}JX)^L,X^L) = g(R_{XJX}JX,X), \end{equation*} we obtain \begin{equation*} \bar{R}(X^L,\phi X^L,\phi X^L,X^L) = R(X,JX,JX,X)-3g^2(X,X)= c-3, \end{equation*} where $c=R(X,JX,JX,X)$ is holomorphic sectional curvature of K\"ahler base. By assumption $c=const$, in particular at the point $(t,q)$, \begin{equation*} \bar{R}(X^L,\phi X^L,\phi X^L,X^L) = K_\phi(v)= c-3. \end{equation*} As point and vector are arbitrary this shows that $\mathcal M$ has constant $\phi$-sectional curvature $c-3$. \end{proof}
\begin{theorem} \label{th:ein} If K\"ahler base is K\"ahler-Einstein manifold with Einstein constant $c=const.$, then its Sasakian lift is $\eta$-Einstein manifold \begin{equation} \bar{R}ic = (c-2) \bar{g}+(2n-c+2)\eta\otimes\eta. \end{equation} In particular is Einstein if and only if $c=2n+2$. \end{theorem}
The particular case is when base K\"aler manifold has constant holomorphic curvature $c=4$. \begin{theorem} If K\"ahler base is locally isometric to complex projective space $\mathbb{C}P^n$, equipped with Fubini-Study metric of constant holomorphic curvature $c=4$, then its lift is locally isometric to unit sphere $\mathbb{S}^{2n+1}\subset\mathbb{C}^{n+1}$, equipped with its canonical Sasakian structure of constant sectional curvature $\bar{c}=1$. \end{theorem} \begin{proof} By Theorem {\bf \ref{th:chc}}, lift of the K\"ahler base of constant holomorphic curvature is Sasakian manifold of constant $\phi$-sectional curvature $\bar{c}=c-3$. Note that curvature operator of manifold with constant $\phi$-sectional curvature is completely determined, cf. {\bf Theorem 7.19, p. 139} in (\cite{Blair}). \end{proof}
Note that the case of dimension three is exceptional. As every two-dimensional K\"ahler manifold is Einstein, its lift is $\eta$-Einstein, yet coefficient $c$ now is in general a some function - in fact determined by Gaussian curvature of 2-dimensional base.
\section{$\mathcal D_{\alpha,\beta}$-homothety of Sasakian manifolds and $\alpha$-Sasakian manifolds} In this section we provide detailed study of $\mathcal D_{\alpha,\beta}$-homothety Sasakian manifold. Fundamental result here is that image of Sasakian manifold by some $\mathcal D_{\alpha,\beta}$-homothety with parameters $\alpha$, $\beta > 0$ is $\frac{\beta}{\alpha}$-Sasakian manifold.
Let $\mathcal{M}$, be a Sasakian manifold with almost contact metric structure $(\phi,\xi,\eta,g)$. For real positive parameters $\alpha$, $\beta$, let consider new structure on $\mathcal{M}$, $(\phi'=\phi,\xi',\eta',g')$, given by \begin{align} & \xi' = \frac{1}{\beta}\xi, \quad \eta'=\beta\eta, \\ & g' = \alpha g +(\beta^2-\alpha)\eta\otimes\eta. \end{align} It is useful to have explicitly inverse map for metric \begin{align}
g = \frac{1}{\alpha}g'+(\frac{1}{\beta^2}-
\frac{1}{\alpha})\eta'\otimes\eta'. \end{align}
Here our main goal is to study relations between Levi-Civita connection $\nabla = LC(g)$, $\nabla' = LC(g')$, Riemann curvatures and in particular Ricci tensors. Note that in general deformed structure is not contact metric. It satisfies weaker condition \begin{equation} d\eta' = \frac{\beta}{\alpha}\Phi', \end{equation} where $\Phi'(X,Y)=g'(X,\phi Y) = \alpha \Phi(X,Y)$.
\begin{proposition} Let $\nabla$ be Levi-Civita connection of Sasakian manifold, and $\nabla'$ Levi-Civita connection of metric obtained by $\mathcal D_{\alpha,\beta}$-homotheties of Sasakian metric. Connections are related by the following formula \begin{equation} \label{e:txy} \nabla_XY= \nabla'_XY+\frac{\beta^2-\alpha}{\alpha\beta}(\eta'(X)\phi Y + \eta'(Y)\phi X). \end{equation} \end{proposition} \begin{proof} The proof is rather standard with help of Koszul formula for Levi-Civita connection. Let denote by $T_XY$, the difference tensor $\nabla_XY=\nabla'_XY+T_XY$. As connections are torsion-less $T_XY$ is symmetric $T_XY=T_YX$. Therefore \begin{align*} & -2g'(T_XY,Z)= 2g'(\nabla'_XY,Z) - 2g'(\nabla_XY,Z) = (\beta^2-\alpha)(X \eta(Y)\eta(Z)+ \\ & \qquad Y\eta(X)\eta(Z) - Z\eta(X)\eta(Y)) + (\beta^2-\alpha)(\eta([X,Y])\eta(Z) + \\ & \qquad \eta([Z,X])\eta(Y) + \eta([Z,Y])\eta(X)) - 2(\beta^2-\alpha)\eta(\nabla_XY)\eta(Z), \end{align*} note $X\eta(Y)\eta(Z) = (X\eta(Y))\eta(Z)+\eta(Y)(X\eta(Z))$, just well-known Leibniz rule, moreover $\eta([X,Y])= \eta(\nabla_XY)-\eta(\nabla_YX)$. Therefore using Leibniz rule after regrouping we find \begin{align*} & (\beta^2-\alpha)(X\eta(Z)-Z\eta(X)+\eta([Z,X]))\eta(Y) + \\ & \qquad (\beta^2-\alpha)(Y\eta(Z)-Z\eta(Y)+\eta([Z,Y]))\eta(X) + \\ & \qquad (\beta^2-\alpha)(X\eta(Y)-\eta(\nabla_XY))\eta(Z)) + \\ & \qquad (\beta^2-\alpha)(Y\eta(X)-\eta(\nabla_YX))\eta(Z)) = \\ & \qquad 2(\beta^2-\alpha)(d\eta(X,Z)\eta(Y)+d\eta(Y,Z)\eta(X)) + \\ & \qquad (\beta^2-\alpha)((\nabla_X\eta)(Y)+(\nabla_Y\eta)(X))\eta(Z), \end{align*} however on Sasakian manifold $(\nabla_X\eta)(Y)+(\nabla_Y\eta)(X)=0$, thus we obtain \begin{equation*} -g'(T_XY,Z)= (\beta^2-\alpha)(d\eta(X,Z)\eta(Y)+d\eta(Y,Z)\eta(X)). \end{equation*} Note $d\eta=\frac{1}{\beta}d\eta' = \frac{1}{\alpha}\Phi'$, in terms of deformed structure the above equation reads \begin{equation*} -g'(T_XY,Z)= \frac{\beta^2-\alpha}{\alpha\beta}( \Phi'(X,Z))\eta'(Y)+ \Phi'(Y,Z)\eta'(X)), \end{equation*} from $\Phi'(X,Y)=g'(X,\phi Y)= -g'(\phi X,Y)$, we finally obtain \begin{equation} T_XY = \frac{\beta^2-\alpha}{\alpha\beta}(\eta'(X)\phi Y +\eta'(Y)\phi X), \end{equation} ie. (\ref{e:txy}). \end{proof} Of course once we have explicit form of tensor $T_XY$ we can directly verify that $\nabla'g'=0$, and use the fact that Levi-Civita connection is unique connection which is both torsion-less and $\nabla'g'$. Such direct verification provides alternative proof of the above statement.
For further reference we set $c=c_{\alpha,\beta}=\frac{\beta^2-\alpha}{\alpha\beta}$. \begin{proposition} Covariant derivative $\nabla'\phi$ is given by \begin{equation} \label{e:nabpfi} (\nabla'_X\phi)Y = \frac{\beta}{\alpha}(g'(X,Y)\xi'-\eta'(Y)X). \end{equation} \end{proposition} \begin{proof}
We have \begin{align} \label{e:nab1} & (\nabla_X\phi )Y = (\nabla'_X\phi)Y +(T_X\phi)Y= (\nabla'_X\phi)Y + \\ & \nonumber\qquad T_X\phi Y - \phi T_XY = (\nabla'_X\phi)Y + c(\eta'(X)\phi^2Y - \\ & \nonumber\qquad \eta'(X)\phi^2Y-\eta'(Y)\phi^2X) = (\nabla'_X\phi)Y + c(\eta'(Y)X-\eta'(X)\eta'(Y)\xi'), \end{align} As $\mathcal{M}$ is Sasakian \begin{align} \label{e:nab2} & (\nabla_X\phi)Y=g(X,Y)\xi -\eta(Y)X = \frac{\beta}{\alpha}g'(X,Y)\xi' - \\ &\nonumber \qquad c\eta'(X)\eta'(Y)\xi' -
\frac{1}{\beta}\eta'(Y)X, \end{align} comparing (\ref{e:nab1}), (\ref{e:nab2}), we obtain (\ref{e:nabpfi}). \end{proof}
\begin{corollary} Image of Sasakian manifold by $\mathcal D_{\alpha,\beta}$-homothety with parameters $\alpha$, $\beta > 0$ is $\frac{\beta}{\alpha}$-Sasakian manifold.
\end{corollary}
To find relation between corresponding curvature operators we use following general formula \begin{equation} \label{e:r:rp} R_{XY}Z=R'_{XY} +(\nabla'_XT)_YZ-(\nabla'_YT)_XZ+[T_X,T_Y]Z, \end{equation} where $[T_X,T_Y]Z=T_XT_YZ-T_YT_XZ$.
For covariant derivative $\nabla'_XT$ on base of above Propositions we find \begin{align} \label{e:nabptxy} & (\nabla'_XT)_YZ =c ((\nabla'_X\eta')(Y)\phi Z + (\nabla'_X\eta')(Z)\phi Y +\\ & \nonumber\qquad \eta'(Y)(\nabla'_X\phi)Z+\eta'(Z)(\nabla'_X\phi)Y) = \\
& \nonumber\qquad \frac{c\beta}{\alpha}(\Phi'(X,Y)\phi Z+ \Phi'(X,Z)\phi Y )+ \\ & \nonumber\qquad \frac{c\beta}{\alpha}(g'(X,Z)\eta'(Y)+g'(X,Y)\eta'(Z))\xi' - \\ & \nonumber\qquad \frac{c\beta}{\alpha} 2\eta'(Y)\eta'(Z)X), \end{align} we have used $(\nabla_X\eta')(Y)=\frac{\beta}{\alpha}\Phi'(X,Y)$. For $T_XT_YZ$, we obtain \begin{align} \label{e:txtyz} & T_XT_YZ = c\eta'(X)\phi T_YZ = c^2(\eta'(X)\eta'(Y)\phi Z + \\ & \nonumber\qquad\eta'(X)\eta'(Z)\phi Y) \end{align} as $\eta'(T_XY)=0$, for every $X$, $Y$. In view of (\ref{e:r:rp}),(\ref{e:nabptxy}),(\ref{e:txtyz}), we can establish following result. \begin{proposition} Let $g'$ be a $\mathcal D_{\alpha,\beta}$-homothety of Sasakian metric. Then Riemann curvature operator $R$ and the curvature operator $R'$ of the deformed metric are related by following formula \begin{align} & R_{XY}Z = R'_{XY}Z + \\ & \nonumber\qquad \frac{c\beta}{\alpha}( \Phi'(X,Z)\phi Y -\Phi'(Y,Z)\phi X+2\Phi'(X,Y)\phi Z) + \\ & \nonumber\qquad \frac{c\beta}{\alpha}(g'(X,Z)\eta'(Y)-g'(Y,Z)\eta'(X))\xi' - \\ & \nonumber \qquad \frac{c\beta}{\alpha}\eta'(Z)(\eta'(Y)X-\eta'(X)Y)) - \\ & \nonumber \qquad \frac{c^3\beta}{\alpha}\eta'(Z)(\eta'(Y)\phi X-\eta'(X)\phi Y), \end{align} \end{proposition}
We have $Ric(Y,Z) = Tr \{ X\mapsto R_{XY}Z\}$, so as corollary from the above proposition we obtain \begin{corollary} Ricci tensors of Sasakian manifold and its $\frac{\beta}{\alpha}$-Sasakian deformation are related by \begin{equation} Ric(Y,Z)=Ric'(Y,Z)+2\frac{c\beta}{\alpha}(g'(Y,Z)-(n+1)\eta'(Y)\eta'(Z)). \end{equation} \end{corollary}
As we know already $\mathcal D_{\alpha,\beta}$-homothety of Sasakian manifold is an $\frac{\beta}{\alpha}$-Sasakian manifold. Providing two consecutive homotheties with parameters $(\alpha_i,\beta_i)$, $i=1,2$, we obtain that resulting manifold is $\frac{\beta_1\beta_2}{\alpha_1\alpha_2}$-Sasakian. Therefore as conclusion we obtain general statement that $\mathcal D_{\alpha,\beta}$-homothety with parameters $\alpha_1$, $\beta_1$ of some $\alpha$-Sasakian manifold is $(\frac{\beta_1}{\alpha_1}\alpha)$-Sasakian.
\subsection{$\mathcal D_{\alpha,\beta}$-homotheties of $\alpha$-Sasakian twisted $\eta$-Ricci soliton} In this part of the paper we will prove important result that equation which defines twisted $\eta$-Ricci soliton is on $\alpha$-Sasakian manifolds, invariant under $\mathcal D_{\alpha,\beta}$-homotheties.
\begin{theorem} Let assume $\alpha$-Sasakian manifold $(\mathcal{M},\phi,\xi,\eta,g)$ is twisted $\eta$-Ricci soliton \begin{align} \label{e:sas:esol} & Ric +\frac{1}{2}(\mathcal{L}_Xg) = \lambda g+2C_1 \alpha_X\odot\eta + C_2\eta\otimes\eta, \\ & (\mathcal{L}_X\eta)(Y) = \alpha_X(Y),\quad \eta(X)=0, \end{align} then its image by $\mathcal D_{\alpha,\beta}$-homothety $(\mathcal{M},\phi,\xi',\eta',g')$,
is also a twisted $\eta$-Ricci soliton soliton, \begin{align} & Ric' +\frac{1}{2}\mathcal{L}_{X'}g' = \lambda'g' + 2C_1'\alpha'_{X'}\odot \eta' +C_2'\eta'\otimes\eta', \\ & (\mathcal{L}_{X'}\eta')(Y) = \alpha'_{X'}, \quad
\quad \eta'(X')=0, \end{align} \end{theorem} \begin{proof} The proof is almost evident. Without loosing generality we may assume that manifold is Sasakian. We only need to find how each term in equation (\ref{e:sas:esol}) changes under $\mathcal D_{\alpha,\beta}$-homothety. Therefore \begin{align} & Ric = Ric'+2\frac{c\beta}{\alpha}g'-
2(n+1)\frac{c\beta}{\alpha}\eta'\otimes \eta', \\ & \mathcal{L}_Xg= \frac{1}{\alpha}\mathcal{L}_Xg' + 2(\frac{1}{\beta^2}-\frac{1}{\alpha})(\mathcal{L}_X\eta')\odot\eta' = \\ & \nonumber\qquad\mathcal{L}_{X'}g'+ 2(\frac{\alpha}{\beta^2}-1)(\mathcal{L}_{X'}\eta')\odot\eta', \\ & \alpha_X = \mathcal{L}_X\eta = \frac{\alpha}{\beta}\mathcal{L}_{X'}\eta' = \frac{\alpha}{\beta}\alpha'_{X'}, \\ & g = \frac{1}{\alpha}g'+(\frac{1}{\beta^2}-\frac{1}{\alpha})\eta'\otimes\eta', \quad \eta\otimes\eta = \frac{1}{\beta^2}\eta'\otimes\eta', \end{align} however we need to rescale vector field $X$ by $\frac{1}{\alpha}$, $X \mapsto X'=\frac{1}{\alpha}X$, after regrouping we obtain that $\frac{\beta}{\alpha}$-Sasakian manifold is twisted $\eta$-Ricci soliton with constants $\lambda'$, $C_1'$, $C_2'$, given by \begin{align*} & \lambda' = \frac{1}{\alpha}(\lambda-2c\beta), \quad C_1' = \frac{\alpha}{\beta^2}(C_1-\frac{1}{2})+\frac{1}{2}, \\ & C_2' = \lambda(\frac{1}{\beta^2}-\frac{1}{\alpha}) + \frac{C_2}{\beta^2}+2(n+1)c\frac{\beta}{\alpha} \end{align*} where $c=\frac{\beta^2-\alpha}{\alpha\beta}$. \end{proof}
On the base of above formulas we can answer question whether or not it is possible to remove the twist from the equation. That means does exist $D_{\alpha,\beta}$-homothety so $C_1=0$? From above formulas we see that necessary and sufficient condition is that the source structure satisfies $C_1 < \frac{1}{2}$. Behavior under deformations determines three classes of $\alpha$-Sasakian twisted $\eta$-Ricci solitons determined by value of twist coefficient $C_1$. The first class are those manifolds where $C_1 < \frac{1}{2}$, second say singular class are manifolds where $C_1=\frac{1}{2}$, and the third class are manifolds where $C_1 > \frac{1}{2}$. Later on studying lifts of Ricci-K\"ahler solitons as corollary we obtain that class $C_1 < \frac{1}{2}$ is always nonempty. Exactly lift of Ricci-K\"ahler soliton belongs to this class. So basically there is problem to solve about the other two classes: do exist $\alpha$-Sasakian twisted $\eta$-Ricci solitons with $C_1 \geqslant \frac{1}{2}$? Note that it is the case where it is not possible to remove twist by $\mathcal D_{\alpha,\beta}$-homothety.
\section{Lifts of Killing vector fields, inifinitesimal biholomorhpisms and automorhpisms}
In this we are interested in lifts of vector fields which satisfy some additional conditions. We just want to ask questions what is a lift of complex structure infinitesimal automorphism and similarly what is a lift of Killing vector field. \begin{proposition} \label{p:inaut} Let $V$ be an infinitesimal automorphism of complex structure on K\"ahler base. Then its lift $V^L$ satisfies \begin{equation} (\mathcal{L}_{V^L}\phi)X^L = 2g^L(V^L,X^L)\xi, \quad (\mathcal{L}_{V^L}\phi)\xi = 0. \end{equation} \end{proposition} \begin{proof} We have \begin{equation} \label{e:v:phi} [V^L,\phi X^L] = [V, JX]^L -2 \Phi(V^L,\phi X^L)\xi, \end{equation} \begin{equation} \label{e:phi:v} \phi[V^L,X^L] = \phi [V,X]^L = (J[V,X])^L, \end{equation} as $\Phi(V^L,\phi X^L) = -g^L(V^L,X^L)$, by (\ref{e:v:phi}),
(\ref{e:phi:v}) \begin{align*} & (\mathcal{L}_{V^L}\phi)X^L = [V^L,\phi X^L] -\phi[V^L,X^L]= ((\mathcal{L}_VJ)X)^L+ \\ & \nonumber \qquad 2g^L(V^L,X^L)\xi, \end{align*} and result follows by assumption that $\mathcal{L}_VJ=0$. \end{proof}
In similar way we can prove following statement considering lift of Killing vector fields from K\"ahler base. \begin{proposition} \label{p:kill} Let $V$ be a Killing vector field on K\"ahler base. Then its lift satisfies \begin{equation} (\mathcal{L}_{V^L}g^L)(X^L,Y^L) = 0, \quad (\mathcal{L}_{V^L}g^L)(\xi, X^L) = 2\Phi(V^L,X^L). \end{equation} \end{proposition}
\begin{proposition} Let $V$ be an inifinitesimal automorphism of K\"ahler form. Its lift satisfies \begin{equation} \label{e:lie:vl:phi} \mathcal{L}_{V^L}\Phi = 0, \end{equation} in particular one-form $\mathcal{L}_{V^L}\eta$ is closed. \end{proposition} \begin{proof} To proof (\ref{e:lie:vl:phi}), we show that $(\mathcal{L}_{V^L}\Phi)(X^L,Y^L)=0$, and $(\mathcal{L}_{V^L}\Phi)(X^L,\xi)=0$. As Lie derivative and exterior derivative commute we have \begin{equation} 0=\mathcal{L}_{V^L}\Phi = \mathcal{L}_{V^L}d\eta = d(\mathcal{L}_{V^L}\eta), \end{equation} hence $\mathcal{L}_{V^L}\eta$ is closed one-form. \end{proof}
Now we provide result establishing kind of relationship between local symmetries of K\"ahler base and some vector fields on its
Sasakian lift \begin{theorem} Let $V$ be inifinitesimal automorphism of K\"ahler structure of K\"ahler base. Its lift $V^L$, satisfies \begin{equation} \mathcal{L}_{V^L}\phi = 2\alpha\otimes\xi , \quad \mathcal{L}_{V^L}g^L = 4\alpha^\phi\odot\eta, \quad \mathcal{L}_{V^L}\Phi = 0, \end{equation} where $\alpha(\cdot)=g^L(V^L,\cdot)$, $\alpha^\phi(\cdot)=\alpha(\phi\cdot)=-g^L(\phi V^L,\cdot)$. and form $\alpha^\phi$ is closed, $d\alpha^\phi=0$. \end{theorem} \begin{proof} Only what requires is a proof that form $\alpha^\phi$ is closed. Hover note that we may identify $\alpha^\phi$, with a pullback of closed form $X\mapsto \omega(V,X)$. \end{proof}
Here we formulate main result of this section. \begin{theorem} Let $V$ be automorphism of K\"ahler structure of K\"ahler base. Then there is locally defined function $f$, $df(\xi)=0$, on Sasakian lift, such that vector field $U_V=V^L+f\xi$, is local infinitesimal automorphism of almost contact metric structure of Sasakian lift. \end{theorem} \begin{proof} From the properties of the Lie derivative \begin{equation*} (\mathcal{L}_{U_V}\phi)Y^L=(\mathcal{L}_{V^L}\phi)Y^L + f(\mathcal{L}_\xi\phi)Y^L-df(\phi Y^L)\xi, \end{equation*} on Sasakian manifold $\mathcal{L}_\xi\phi=0$, in the view of the Proposition {\bf\ref{p:inaut}}., we obtain \begin{equation*} (\mathcal{L}_{U_V}\phi)Y^L = (2g^L(V^L,Y^L)-df(\phi Y^L))\xi. \end{equation*} As $V$ is an automorphism of K\"ahler structure, is locally Hamiltonian, with respect to K\"ahler form $\omega(V,Y)=dH(Y)$, for locally defined function $H$. Set $f=-2\bar{H}=-2H\circ\pi_2$, then \begin{equation*} df(\phi Y^L) = df((JY)^L) = -2dH(JY)= -2\omega(V,JY)=2g(V,Y). \end{equation*}
Having $f$ determined we verify directly that \begin{equation*} (\mathcal{L}_{U_V}\phi)\xi = 0, \quad (\mathcal{L}_{U_V}g^L)(Y^L,Z^L)=0, \quad (\mathcal{L}_{U_V}g^L)(\xi,Y^L)=0. \end{equation*} \end{proof}
\section{Sasakian lift of K\"ahler-Ricci soliton and $\alpha$-Sasakian $\eta$-Ricci solitons } In this section we are interested particularly in lifts of K\"ahler-Ricci solitons. Let vector field $X$ satisfies Ricci soliton equation on K\"ahler base \begin{equation} \label{e:ric:sol} Ric+\dfrac{1}{2}\mathcal{L}_Xg = \lambda g, \end{equation} where $\lambda = const.$ is a real constant. By direct computations we find $(\mathcal{L}_{X^L}g^L)(Y^L,Z^L) = (\mathcal{L}_Xg)(Y,Z)$, then by (\ref{e:ric:sol}) \begin{equation*} \frac{1}{2}(\mathcal{L}_{X^L}g^L)(Y^L,Z^L) = \frac{1}{2}(\mathcal{L}_Xg)(Y,Z)= \lambda g(Y,Z)-Ric(Y,Z), \end{equation*} From other hand by the Proposition {\bf\ref{p:ric}.}, $$ \bar{R}ic(Y^L,Z^L)=Ric(Y,Z)-2g(Y,Z), $$ summing up we find \begin{align} \label{e:ric:sol2} & \bar{R}ic(Y^L,Z^L)+\frac{1}{2}(\mathcal{L}_{X^L}g^L)(Y^L,Z^L) = Ric(Y,Z)-2g(Y,Z) + \\ & \nonumber \lambda g(Y,Z)-Ric(Y,Z) = (\lambda-2)g^L(Y^L,Z^L), \end{align} in similar way \begin{equation} \label{e:ric:sol3} \bar{R}ic(\xi,Y^L)+\frac{1}{2}(\mathcal{L}_{X^L}g^L)(\xi,Y^L)= \Phi(X^L,Y^L), \end{equation} \begin{equation} \label{e:ric:sol4} \bar{R}ic(\xi,\xi)+\frac{1}{2}(\mathcal{L}_{X^L}g^L)(\xi,\xi) = 2n. \end{equation} The identities (\ref{e:ric:sol2})-(\ref{e:ric:sol4}), allow us to state the following result \begin{theorem} Let K\"ahler base be a K\"ahler-Ricci soliton. Sasakian lift is twisted $\eta$-Ricci soliton \begin{align} & \bar{R}ic+\frac{1}{2}\mathcal{L}_{X^L}g^L= (\lambda-2)g^L-2(\mathcal L_{X^L}\eta)\odot\eta + (2n-2+\lambda)\eta\otimes \eta, \end{align} \end{theorem}
\begin{corollary} Sasakian lift of K\"ahler-Ricci soliton admits $\mathcal D_{\alpha,\beta}$-homothety so $\alpha$-Sasakian image is $\eta$-Ricci soliton. \end{corollary} \begin{proof} Sasakian lift satisfies equation of twisted $\eta$-Ricci soliton. By {\bf Corollary \ref{c:tesol:esol} .} there is a $\mathcal D_{\alpha,\beta}$-homothety so image is $\alpha$-Sasakina $\eta$-Ricci soliton. \end{proof}
As Ricci tensor determines a 2-form on Sasakian lift above equation can be expressed in the following form \begin{proposition}
Sasakian lift of K\"ahler-Ricci soliton, satisfies following equations with resp. to the fundamental form and Ricci form of Sasakian lift \begin{equation} \bar{\rho}+\frac{1}{2}\mathcal{L}_{X^L}\Phi = (\lambda-2)\Phi. \end{equation} \end{proposition}
Above corollary give us plenty of examples of $\eta$-Ricci solitons.
\end{document} |
\begin{document}
\begin{abstract} The solutions to certain nested recursions, such as Conolly's $C(n) = C(n-C(n-1))+C(n-1-C(n-2))$, with initial conditions $C(1)=1, C(2)=2$, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a $k$-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the $k$-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite $k$-ary tree associated with the solution of the corresponding homogeneous $k$-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions. \end{abstract}
\title{Solving Non-homogeneous Nested Recursions Using Trees}
\section{Introduction} \label{sec1} In this paper all values for the variables and parameters are integers unless otherwise specified. For $k \geq 1$, $a_i$, and $b_i > 0,\; i = 1 \ldots k$, consider the nested (also called meta-Fibonacci) homogeneous recursion \begin{equation} \label{nested} A(n) = \sum_{i=1}^k A(n - a_i - A(n-b_i))\end{equation} which we abbreviate as $\ang{a_1;b_1: \cdots: a_k;b_k}$. We call the sequences that appear as solutions to nested recursions \emph{meta-Fibonacci sequences}.
Over the past twenty years, many special cases of (\ref{nested}), together with alternative sets of initial conditions, have been examined (see the references for specifics). Examples include Hofstadter's famous and mysterious $Q$-sequence \cite{GEB} given by $\ang{0;1:0;2}$ with $Q(1) = Q(2) = 1$, and Conolly's well-known sequence $C(n)$ \cite{Con} given by $\ang{0;1:1;2}$ with $C(1) =1, C(2) = 2$. Recently, fascinating and unexpected combinatorial connections have been discovered between the solutions to certain such nested recursions and infinite, labeled trees \cite{BLT, DR, JR, IRT}. For example, it is shown in \cite{JR} that the shifted Conolly sequence $C_s(n)$ determined by $\ang{s;1:s+1;2}$ for any fixed $s \geq 0$ and initial conditions $C_s(i) = 1$ for $1 \leq i \leq s+1$ and $C_s(s+2) = 2$ counts the number of leaves in a suitably constructed infinite labeled binary tree (with root at infinity) that have labels that are less than or equal to $n$ (the construction depends on the parameter $s$). In the labeling, each node of the infinite binary tree receives one label except for the so-called $s$-nodes along the top of the tree, each of which receives $s$ labels. In \cite{DR} an analogous combinatorial interpretation is derived for solutions to the $k$-term recursion (\ref{nested}) with $a_i = s+i-1$ and $b_i= i$, and with $k+s$ initial conditions that are determined by the leaf counts of the correspondingly constructed infinite, labeled $k$-ary tree. These initial conditions are said to ``follow the tree", in the sense that they are precisely the ones that force the solution to conform to the specified labeled tree.
Building on this work, Isgur et al \cite{IRT} vary the labeling scheme by inserting $j$ labels in each node of the $k$-ary tree rather than a single label, where $j \geq 1$ is a fixed parameter. In this way they derive a combinatorial interpretation for the solution to the generalized Conolly recurrence \begin{equation} \label{Conolly} R(n) = \sum_{i=1}^kR(n-s-(i-1)j-R(n-ij)) \end{equation} with the initial conditions: $R(i) = i$ for $1 \leq i \leq j$, $R(i) = j$ for $j+1 \leq i \leq j+s$, $R(i) = i-s$ for $j+s+1 \leq i \leq kj + s$, and $R(i) = kj$ for $kj+s+1 \leq i \leq (k+1)j+ 2s$. It is shown that $R(n)$ counts the number of \emph{labels} in the leaves of the labeled $k$-ary tree that are less than or equal to $n$.
Here we extend the tree-based correspondence described above to combinatorially interpret solutions to a non-homogeneous version of the Conolly nested recursion (\ref{Conolly}), namely, \begin{equation} \label{recursion} R(n) = \sum_{i=1}^kR(n-s-(i-1)j-R(n-ij))+ \nu \end{equation} where $\nu$ is any constant, and with specified initial conditions. Interest in such nested recursions is natural and longstanding; see, for example, \cite{Gol}, where the recursion $g(n) = g(n-g(n-1)) + 1$, with $g(1) = 1$, is shown to have a neat, closed form solution. Our focus on a constant for the non-homogeneous term in (\ref{recursion}) can be readily explained: if the non-homogeneous term is an integer valued function $\nu(n)$ with $|\nu(n)| \geq cn$ for all $n$, then the right side of (\ref{recursion}) grows at least linearly in $n$. Therefore $R(n)$ will grow at least linearly in $n$ and it seems plausible that eventually one of the arguments $n-s-(i-1)j-R(n-ij)$ in (\ref{recursion}) will be negative or will exceed $n$. At that point the recursion will cease to be well-defined. So a constant value for $\nu(n)$ is a natural choice. \footnote{Of course, one could consider non-constant, sub-linear $\nu(n)$. To date our empirical investigations suggest that only constant $\nu(n)$ lead to well-defined, infinite solution sequences. Considerable further work is needed in this area to confirm this contention, or to determine which non-constant $\nu(n)$, if any, lead to an infinite solution sequence to (\ref{recursion}) for some set of initial conditions.}
To solve (\ref{recursion}) combinatorially we ``graft" infinitely many copies of a finite, rooted tree $\mathcal T$ (or in some cases a portion of $\mathcal T$) onto the original $k$-ary tree that solves the related homogeneous recursion (\ref{Conolly}). As we explain below, it turns out that for any given value of $\nu$ in (\ref{recursion}), we can find infinitely many finite trees $\mathcal T$ that correspond to that choice of $\nu$. Each of these finite trees determines a set of initial conditions for the recursion, and these sets of initial conditions may differ. Thus, our tree grafting technique permits us to find combinatorial interpretations for different sets of initial conditions for the same recursion. In particular, we can use our technique for the case $\nu =0$, thereby solving the homogeneous nested recursion (\ref{Conolly}) with initial conditions determined by the choice of $\mathcal T$. Prior to this, the only combinatorial solution to (\ref{Conolly}) is the one associated with the initial conditions imposed by the leaf counts of the usual $k$-ary tree \cite{DR, IRT}.
The outline of the rest of this paper is as follows. In Section \ref{sec2} we describe precisely the procedure for grafting copies of an arbitrary finite, rooted tree $\mathcal T$ on the infinite $k$-ary tree in \cite{DR, IRT}, and for labeling the resulting infinite tree $\mathcal K$. This construction depends upon $\mathcal T$, as well as the three parameters $k, j$ and $s$ in (\ref{recursion}). In Section \ref{sec3} we establish that the infinite tree $\mathcal K$ constructed in Section \ref{sec2} provides a combinatorial interpretation to (\ref{recursion}): $R(n)$ is the number of labels up to $n$ on leaves of $\mathcal K$. Finally, in Section \ref{sec4} we discuss alternative labeling schemes for $\mathcal K$ that give rise to a variety of interesting results; in particular, we derive a new combinatorial interpretation for Golomb's recursive sequence $g(n)$.
\section{Constructing the infinite tree $\mathcal K$: the grafting technique} \label{sec2} Let $\mathcal T$ be any finite rooted tree with at least two nodes. The \textbf{height} $p$ of $\mathcal T$ is the length of the longest path from the root to any of its nodes. Fix $k \geq 1$ corresponding to the desired value of $k$ in ($\ref{recursion}$). We create a modified labeled $k$-ary tree $\mathcal K$ using $\mathcal T$.
The construction of $\mathcal K$ requires two steps. First we construct the nodes and edges, that is, the skeleton, of $\mathcal K$; this involves grafting copies of $\mathcal T$ on the infinite $k$-ary tree in \cite{DR, IRT}. Then we insert labels, which are successive positive integers, within the nodes of $\mathcal K$. To do this, first we specify the order in which the nodes of $\mathcal K$ are to be traversed one at a time; then we insert the appropriate number of labels, either $j$ or $s$ (the parameters in ($\ref{recursion})$) in each node. As we traverse $\mathcal K$, we keep count of the number of labels up to that point that are located in the leaves of $\mathcal K$. The ``leaf label" sequence generated by this enumeration satisfies a nested, non-homogeneous Conolly-type recursion of the form (\ref{recursion}), where the tree $\mathcal T$ determines $\nu$.
To help describe this construction, we illustrate our discussion using the rooted tree $\mathcal T$ of height 3 in Figure \ref{fig1}, together with $k=2$. We show how this results in an infinite, labeled binary tree with leaf label counting function $R(n)$ that satisfies the recursion $R(n) = R(n-R(n-2))+R(n-2-R(n-4))-2$.
\begin{figure}
\caption{Example of a rooted tree $\mathcal T$ of height $p=3$.}
\label{fig1}
\end{figure}
\subsection*{Constructing the skeleton of $\mathcal K$} The skeleton of $\mathcal K$ consists of an infinite sequence $\mathcal K_i$ of rooted, finite subtrees of $\mathcal K$ that we join together to form $\mathcal K$. For each $i$, the root of $\mathcal K_i$ is called the $i^{th}$ \textbf{supernode} of $\mathcal K$, while all other nodes are \textbf{regular nodes}. For $i=p>1$, where $p$ is the height of $\mathcal T$, the subtree $\mathcal K_p$ is isomorphic to $\mathcal T$. The subtree $\mathcal K_{p-1}$ is obtained by making a copy of $\mathcal K_p$ and then deleting all the leaves of the copy. See Figure \ref{fig3}, where we illustrate the subtrees $\mathcal K_3$ and $\mathcal K_2$ using the tree $\mathcal T$ of height $p=3$ in Figure \ref{fig1}; we draw the supernodes as squares and the regular nodes as circles. For $2<i\leq p$, we repeat this process successively, making $\mathcal K_{i-1}$ by copying $\mathcal K_i$ and deleting the leaves of the copy. The subtree $\mathcal K_1$ is a special case: after deleting the leaves of a copy of $\mathcal K_2$, we attach an extra regular node as a child of the first supernode (this extra child can be considered as the zeroth supernode - see Section \ref{sec4}).
\iffalse \begin{figure}
\caption{Initial segment of the infinite tree $\mathcal K$ constructed from the tree in Figure \ref{fig1}. Supernodes are drawn as squares. Note how $\mathcal K_2$ is obtained from $\mathcal K_3$ by pruning the leaves of $\mathcal K_3$, and the additional node in $\mathcal K_1$.}
\label{fig2}
\end{figure} \fi
For $i > p$ we construct the subtrees $\mathcal K_i$ by first making a copy of $\mathcal K_{i-1}$ and then attaching precisely $k$ nodes to each of its leaves. Thus, for $i > p$ the subtree $\mathcal K_i$ consists of the tree $\mathcal T$ with each of its leaves the root of a $k$-ary subtree of height $i-p$, so $\mathcal K_i$ has height $i$. Finally, for all $i>0$ we connect all the subtrees $\mathcal K_i$ to $\mathcal K_{i+1}$ by adding an edge from the $i^{th}$ supernode to the $(i+1)^{st}$. See Figure \ref{fig3}. Notice that the tree $\mathcal K$ is closely related to the infinite $k$-ary tree in \cite{DR, IRT}: take the former $k$-ary tree, insert the finite tree $\mathcal T$ (or a portion of it), with the root of $\mathcal T$ coinciding with each of the supernodes of the $k$-ary tree, and leave the rest of the original $k$-ary tree untouched.
\begin{figure}
\caption{First 5 subtrees of the skeleton of $\mathcal K$ derived from the tree in Figure \ref{fig1}, using $k=2$.}
\label{fig3}
\end{figure}
\subsection*{Labeling $\mathcal K$} Fix the values $j \geq 1$ and $s \geq 0$, corresponding to the parameters of the same name in ($\ref{recursion}$). Insert $j$ labels into each regular node of $\mathcal K$ and $s$ labels into each supernode; for convenience, we refer to the regular nodes and supernodes as $j$-nodes and $s$-nodes, respectively. The labels consist of successive integers starting at 1. Before we can insert these labels we must specify the traversal order of the nodes in $\mathcal K$. We recursively define a \textbf{pre-order} traversal as follows: $\mathcal K_1$ is traversed by beginning at the first child of the first supernode followed by the supernode itself and then its remaining children (note this is not the pre-order traversal of $\mathcal K_1$). Having traversed $\mathcal K_i$ for $i \geq 1$, the subtree $\mathcal K_{i+1}$ is traversed next in the usual pre-order way by beginning at its root, which is the $(i+1)^{th}$ supernode. See Figure \ref{fig4}, where we label the nodes of $\mathcal K$ in the order in which they are traversed, and then Figure \ref{fig5}, where we insert $j=2$ labels in each regular node and $s=0$ labels in each supernode.
\begin{figure}
\caption{The order in which the nodes of $\mathcal K$ from Figure \ref{fig3} are traversed.}
\label{fig4}
\end{figure}
\begin{figure}
\caption{First 5 subtrees of the completed infinite tree $\mathcal K$ from our example. The labeling parameters are $j=2$ and $s=0$.}
\label{fig5}
\end{figure}
We explain in the next section how the combinatorial interpretation for the solution of (\ref{recursion}) is derived from the ``leaf label" sequence generated from $\mathcal K$. Before doing so, we require additional terminology and notation with which we conclude this section.
Call $\mathcal K(n)$ the subtree of $\mathcal K$ containing all the labels between $1$ and $n$ and all the nodes in pre-order up to the node containing $n$. For $m \geq 1$, define $m$ to be a \textbf{leaf label} of $\mathcal K$ if $m$ is contained in a leaf of $\mathcal K$. Throughout the paper the \textbf{leaf labels of $\mathcal K(n)$} are defined to be all the labels in $\mathcal K(n)$ that are leaf labels in $\mathcal K$. It is very important to note that a node may be a leaf in $\mathcal K(n)$ and not be a leaf in $\mathcal K$. For example, in Figure \ref{fig5}, the node containing the label 21 is a leaf of $\mathcal K(21)$ but not of $\mathcal K$. Thus, 21 is \emph{not} a leaf label.
Let $R(n)$ be the number of leaf labels in $\mathcal K(n)$. In Figure \ref{fig5}, $R(7) =5$ and $R(20) = 10$.
A \textbf{penultimate node} of $\mathcal K$ is a non-leaf node in $\mathcal K$ such that all of its children are leaves (for example, nodes 2 and 18 in Figure \ref{fig4} but not node 7). Call the labels in a penultimate node \textbf{penultimate labels}. The penultimate nodes (respectively, labels) of $\mathcal K(n)$ are the penultimate nodes (respectively, labels) of $\mathcal K$ that are included in $\mathcal K(n)$.
Note that for $i \geq 2$ the leaves of $\mathcal K_i$ are the penultimate nodes of $\mathcal K_{i+1}$; the leaves of $\mathcal K_1$, other than the first leaf, are the penultimate nodes of $\mathcal K_2$, and $\mathcal K_1$ always has exactly one penultimate node (the first $s$-node of $\mathcal K$). Let $\ell_i$ be the number of leaves in $\mathcal K_i$. Define $\alpha$ (respectively, $\beta$) as the number of leaf labels (respectively, penultimate labels) occurring in $\mathcal K_1$ through $\mathcal K_p$. Then by the preceding observation $\alpha = j\sum_{i=1}^p \ell_i$, and $\beta = j(\sum_{i=1}^{p-1} \ell_i -1) + s$. Finally, let $N(i)$ be the largest label of $\mathcal K$ that occurs in $\mathcal K_i$.
We are now prepared to state and prove our key finding.
\section{Solving the non-homogeneous Conolly nested recursion} \label{sec3}
\begin{theorem} \label{thm1} Let $\mathcal T$ be a finite rooted tree of height $p$. Let $\mathcal K$ be the infinite tree constructed using $\mathcal T$ and fixed parameters $k \geq 1, j \geq 1$, and $s \geq 0$. Define $\nu = \alpha - k(\beta - s + j)$. Let $R(n)$ be the leaf label counting function of $\mathcal K$. Then for $n > N(p+1)$, $R(n)$ satisfies the non-homogeneous nested recursion (\ref{recursion}), that is, \[R(n) = \displaystyle \sum_{i=1}^k R(n-s-(i-1)j-R(n-ij)) + \nu\,.\] Equivalently, if any function $L(n)$ is defined by (\ref{recursion}) and the first $N(p+1)$ values of $L(n)$ agree with the corresponding values for the leaf label counting function $R(n)$, then $L(n)=R(n)$ for all $n$. \end{theorem}
In Figure \ref{fig5} $p=3, k=j=2, s=0$ and $\nu = 10 - 2( 4 - 0 + 2) = -2$. Then for $n>N(4)=32$, the leaf label counting function $R(n)$ satisfies $R(n) = R(n-R(n-2)) + R(n-2-R(n-4)) - 2$.
Before turning to the proof of Theorem \ref{thm1}, we provide several observations. From the formulas for $\alpha$ and $\beta$ in Section \ref{sec2} we have a computationally simpler expression for $\nu$: \begin{equation*} \nu = j\ell_p - j(k-1)(\ell_1 + \cdots + \ell_{p-1})\;\text{if}\; p \geq 2,\;\text{and}\; \nu = j(\ell_1-k)\;\text{if}\; p =1\,.\end{equation*}
In some cases fewer than $N(p+1)$ initial conditions will suffice. For our purposes, we are only interested in knowing that for some sufficiently large number of initial conditions the recursion (\ref{recursion}) will generate the leaf label counting sequence as its solution. Finally, note that different choices of $\mathcal T$ enable us to solve recursions of the form (\ref{recursion}) with diverse initial conditions. In particular, now we are able to solve (\ref{Conolly}) with many different sets of initial conditions.
We begin the proof of Theorem \ref{thm1} by defining the \emph{pruning operation} on the subtree $\mathcal K(n)$ for $n > N(1)$. This operation yields a new tree $\mathcal P\mathcal K(n)$ defined as follows: first, delete all leaf labels of $\mathcal K(n)$ along with the nodes containing them. Then convert the first $s$-node into a $j$-node. Finally, relabel the new tree in pre-order, keeping in mind that the first $s$-node is now a $j$-node so it receives $j$ labels rather than $s$. See Figure \ref{fig6} for the pruning of $\mathcal K(27)$ from the tree in Figure \ref{fig5}. Note that the node of $\mathcal K(27)$ that contains the label 27 is a leaf of $\mathcal K(27)$ but not of $\mathcal K$, and as such it is not deleted.
The significance of the pruning operation on the subtree $\mathcal K(n)$ is that it results in $\mathcal K(m)$ for some $m < n$. In this regard, we can view $\mathcal K$ as self-similar with respect to the pruning operation. Let $\mathcal P R(n)$ denote the number of leaf labels in $\mathcal P\mathcal K(n)$. We build to the proof of Theorem \ref{thm1} via a series of lemmas concerning $\mathcal P\mathcal K(n)$.
\begin{figure}
\caption{The pruning operation on $\mathcal K(27)$ results in $\mathcal K(15)$.}
\label{fig6}
\end{figure}
\begin{lemma}[Pruning] \label{lem1} For $n > N(1)$ the tree $\mathcal P K(n)$ has $n-s+j-R(n)$ labels and is isomorphic to the subtree $\mathcal K(n-s+j-R(n))$. Consequently, $\mathcal P R(n) = R(n-s+j-R(n))$.\end{lemma}
\begin{proof} Since $\mathcal K(n)$ contains $R(n)$ leaf labels, deleting the leaf labels of $\mathcal K(n)$ results in a loss of $R(n)$ labels. Also, replacing the first $s$-node with a $j$-node results in a net change of $j-s$ labels following the pruning operation. Thus, the total number of labels in $\mathcal P \mathcal K(n)$ is $n-R(n)-s+j$.
That $\mathcal P K(n)$ is isomorphic to the subtree $\mathcal K(n-s+j-R(n))$ follows directly from the definition of the pruning operation and the construction of the tree $\mathcal K$, since deleting all the leaves of $\mathcal K_q$ results in $\mathcal K_{q-1}$. More generally, if we delete the leaves of $\mathcal K$ from the subtree of $\mathcal K_q$ that consists of the first $m$ nodes of $\mathcal K_q$ (in pre-order), and then relabel in pre-order, the result is the subtree consisting of the first $m' $ nodes of $\mathcal K_{q-1}$ for some $m' < m$.
Finally, since $\mathcal P K(n)$ is isomorphic to $\mathcal K(n-s+j-R(n))$ and $\mathcal K(n-s+j-R(n))$ contains $R(n-s+j-R(n))$ leaf labels by definition, $\mathcal P R(n) = R(n-s+j-R(n))$.\end{proof}
The key to the proof of Theorem \ref{thm1} is that since most penultimate nodes have $k$ children, $k$ times the number of penultimate labels in $\mathcal K(n)$ is essentially the number of leaf labels in $\mathcal K(n)$, with the difference being given by the non-homogeneous term $\nu$. Call $\mathcal K(n)$ \emph{complete} if each of its penultimate nodes has all of its children from $\mathcal K$, and each of these children has $j$ labels. If $\mathcal K(n)$ is complete then the number leaf labels in $\mathcal P \mathcal K(n)$ is $j$ times the the number of penultimate level nodes in $\mathcal K(n)$.
\begin{lemma}[Completeness] \label{lem2} For $n \geq N(p)$, if $\mathcal K(n)$ is complete then \[\mathcal P R(n) = \dfrac{R(n) - \nu}{k}\,.\]\end{lemma}
\begin{proof} Recall that $\alpha$ (respectively, $\beta$) is the number of leaf labels (respectively, penultimate labels) occurring in $\mathcal K_1$ through $\mathcal K_p$. Since $n \geq N(p)$, the pruned tree $\mathcal P K(n)$ contains the subtrees $\mathcal K_1$ to $\mathcal K_{p-1}$, so has $\beta -s+j$ leaf labels in these subtrees. So $\mathcal P R(n) - (\beta - s+j)$ is the number of leaf labels in $\mathcal P K(n)$ occurring after $\mathcal K_{p-1}$. But by the self-similarity of $\mathcal K$ with respect to pruning, this is also the number of penultimate labels in $\mathcal K(n)$ after label $N(p)$, so after $\mathcal K_p$.
By the completeness of $\mathcal K(n)$, each penultimate node of $\mathcal K(n)$ occurring after $\mathcal K_p$ has $k$ children, these children are the only leaves of $\mathcal K$ included in $\mathcal K(n)$ that occur after $\mathcal K_p$, and all these children, as well as their penultimate node parents, have $j$ labels each. Group these children with their penultimate level parents (who also come after $\mathcal K_p$). We then get a $k:1$ correspondence between these children and their parents, which also extends to a correspondence between the labels situated in them.
Now, $R(n) - \alpha$ counts the number of leaf labels in $\mathcal K(n)$ after label $N(p)$. So it equals the number of labels in the children mentioned in the preceding paragraph. On the other hand $\mathcal P R(n) - (\beta - s+j)$, the number of penultimate labels in $\mathcal K(n)$ after label $N(p)$, is the number of labels in the parents mentioned above. Thus, we use the correspondence above to count the number of leaf labels in $\mathcal K(n)$ after label $N(p)$ in two ways: \[ k(\mathcal P R(n) - (\beta -s+j)) = R(n) - \alpha\,.\] Simplifying and substituting $\nu = \alpha - k(\beta-s+j)$ we get the desired result. Note that this also explains the definition of $\nu$. \end{proof}
Now observe that if $\mathcal K(n)$ and $\mathcal K(m)$ have the same penultimate labels, then $\mathcal P R(n) = \mathcal P R(m)$. We use this to compute the values of $\mathcal P R(n)$ based on the location of $n$ in $\mathcal K$. To this end, let $\Delta(n)$ denote the minimal non-negative integer such that $\mathcal K(n + \Delta(n))$ is complete.
\begin{lemma} \label{lem3} The following holds for $n > N(p+1)$.\begin{enumerate} \item If $n$ is neither a leaf label nor a penultimate label then $\mathcal K(n)$ is complete. Consequently for every $n > N(p+1)$, we have that $0 \leq \Delta(n) < (k+1)j$.\\ \item Suppose $\Delta(n) > 0$ so that $\mathcal K(n)$ is not complete. Then $0 < \Delta(n) < kj$ if and only if $n$ is a leaf label, and $\Delta(n) \geq kj$ if and only if $n$ is a penultimate label.\\ \item If $0 \leq \Delta(n) \leq kj$ then $\mathcal P R(n) = \dfrac{R(n) + \Delta(n) - \nu}{k}$.\\ \item If $\Delta(n) > kj$ then $\mathcal P R(n) = \dfrac{R(n) + kj - \nu}{k} - \Delta(n) + kj$. \end{enumerate} \end{lemma}
\begin{proof} (1) and (2): The first statement in (1) follows from the definition of completeness and the construction of $\mathcal K$. To prove the second part of (1) and assertion (2), note that if $\Delta(n) > 0$ then $n$ is a label on either a penultimate node or on one of the $k$ children of a penultimate node. In either case we can complete $\mathcal K(n)$ by adding any missing labels on the penultimate node, and nodes and labels for any missing children until the last label in the last child of said penultimate node. In either case we add up to (but excluding) $(k+1)j$ labels. This proves the second statement in (1). Further, the label $n$ is on a penultimate node if and only if $kj \leq \Delta(n) < (k+1)j$ (since in this case we must add the nodes and labels for all $k$ children); otherwise, $n$ is a label in some child on the bottom level, and $0 \leq \Delta(n) \leq kj$. This establishes (2).
(3): If $\Delta(n) = 0$ then this assertion is simply Completeness Lemma \ref{lem2}. If $0 < \Delta(n) \leq kj$ then from (2) we get that there exists a penultimate node $X$ such that either $n$ is a label in one of its $k$ children (when $0 < \Delta(n) < kj)$ or $n$ is the final label (in pre-order) of $X$ (when $\Delta(n) = kj$). In both cases all of the trees $\mathcal K(n), \ldots, K(n + \Delta(n))$ have the same penultimate labels, namely, all the penultimate labels from $1$ through to the final label in $X$. It follows, as we observed just prior to the statement of this lemma, that $\mathcal P R(n) = \mathcal P R(n+\Delta(n))$. Further, $R(n + \Delta(n)) = R(n) + \Delta(n)$, since the $\Delta(n)$ labels following $n$ are all leaf labels in $\mathcal K$. Since $\mathcal K(n + \Delta(n))$ is complete, we apply the Completeness Lemma \ref{lem2} to it to deduce that \begin{equation*} \mathcal P R(n) = \mathcal P R(n + \Delta(n)) = \frac{R(n + \Delta(n)) - \nu}{k} = \frac{R(n) + \Delta(n) - \nu}{k}\,.\end{equation*}
(4): If $\Delta(n) > kj$ then using the same notation as in the previous paragraph we see from (2) that $n$ is a label of the penultimate node $X$ but it is not the last label of $X$. Let $n'$ be the last label of $X$ so $n'-n = \Delta(n) - kj$. Also $\mathcal P R(n') = \mathcal P R(n) + (n'-n) = \mathcal P R(n) + \Delta(n) - kj$, and clearly $R(n) = R(n')$ and $\Delta(n') = kj$. It follows by (3), applied to $n'$, that \begin{eqnarray*} \mathcal P R(n) &=& \mathcal P R(n') - \Delta(n) + kj \\ &=& \frac{R(n') + \Delta(n') - \nu}{k} - \Delta(n) + kj\\ &=& \frac{R(n') + kj - \nu}{k} - \Delta(n) + kj \;.\end{eqnarray*} This proves (4) and completes the proof of the lemma.\end{proof}
To prove Theorem \ref{thm1} we demonstrate the following key relation: for $n > N(p+1)$, \begin{equation} \label{keyrelation} R(n) - \nu = \displaystyle \sum_{i=1}^k \mathcal P R(n-ij)\,.\end{equation} From (\ref{keyrelation}) and the Pruning Lemma \ref{lem1} our desired result is immediate, since for $n > N(p+1)$, \begin{eqnarray*} R(n) &=& \sum_{i=1}^k \mathcal P R(n-ij) + \nu \\ &=& \sum_{i=1}^k R(n- s - (i-1)j - R(n-ij)) + \nu \,.\end{eqnarray*}
To prove relation (\ref{keyrelation}) we have two cases.
\paragraph{\textbf{Case 1:}} Suppose $n$ is a leaf label. Then there exists $q$ and $r$ such that $1 \leq q \leq k$ and $1 \leq r \leq j$ and $n$ is the $r^{th}$ smallest label on the $q^{th}$ child (in pre-order) of its parent node $X$ (a penultimate node). The trees $\mathcal K(n), \mathcal K(n-j), \ldots, \mathcal K(n-(q-1)j)$ all have the same penultimate labels consisting of all such labels up to and including the penultimate labels in $X$. The tree $\mathcal K(n-qj)$ ends on the $r^{th}$ label in $X$, so its penultimate labels differ from those of the previously mentioned trees only at the last $j-r$ labels of $X$. The trees $\mathcal K(n-(q+1)j), \ldots, \mathcal K(n- kj)$ do not end on penultimate nodes, so they all have the same penultimate labels, namely, all penultimate labels occurring before the labels in $X$.
We now apply the remark we made just prior to Lemma \ref{lem3} that if two trees have the same penultimate labels, their pruned trees have the same number of leaf labels. So $\mathcal P R(n) = \mathcal P R(n-ij)$ for $1 \leq i \leq q-1$. In the same way, $\mathcal P R(n-qj) = \mathcal P R(n) - (j-r)$ and $\mathcal P R(n-ij) = \mathcal P R(n) - j$ for $q+1 \leq i \leq k$. But $\Delta(n) = (j-r) + j(k-q)$, so by (3) of Lemma \ref{lem3} we get $\mathcal P R(n) = \frac{R(n) + (j-r) + j(k-q) - \nu}{k}$. Thus we conclude that \[ \sum_{i=1}^k \mathcal P R(n-ij) = k \mathcal P R(n) -(j-r)-j(k-q) = R(n) - \nu\,.\]
\paragraph{\textbf{Case 2:}} Suppose $n$ is not a leaf label. In this case the subtrees $\mathcal K(n-j), \dots, \mathcal K(n-kj)$ all have the same penultimate labels, so $\mathcal P R(n-j) = \mathcal P R(n-ij)$ for $1 \leq i \leq k$. The subtrees $\mathcal K(n)$ and $\mathcal K(n-j)$ may differ on at most one penultimate node, which happens precisely when $n$ lies on a penultimate node $X$. So we can write $\mathcal P R(n-j) = \mathcal P R(n) - r'$ where $0 \leq r' \leq j$. Here $r'=0$ if and only if $K(n)$ is complete, and otherwise $n$ is the $r'$-th smallest label on the penultimate node $X$.
If $r' = 0$ then $\mathcal P R(n) = \frac{R(n) - \nu}{k}$ by the Completeness Lemma \ref{lem2}. If $r' > 0$ then $\Delta(n) = kj + (j-r)$, and (4) of Lemma \ref{lem3} implies that $\mathcal P R(n) = \frac{R(n) - \nu}{k} + r'$ after simplification. Therefore in all cases we conclude that \begin{eqnarray*} \sum_{i=1}^k \mathcal P R(n-ij) = k \mathcal P R(n-j) &=& k(\mathcal P R(n) - r')\\ &=& k \left ( \dfrac{R(n) - \nu}{k} + r' - r' \right ) \\ &=& R(n) - \nu\,.\end{eqnarray*} This completes the proof of Case 2, and the theorem.
\section{Further applications} \label{sec4}
In the construction of the tree $\mathcal K$, we created each subtree $\mathcal K_i$ by starting with a complete $k$-ary tree of height $i$ and inserting an arbitrary tree $\mathcal T$. Here we describe how slight modifications to this construction, such as to the labeling scheme or to the number of labels in various nodes, can still yield a tree $\mathcal K$ whose leaf label counting function satisfies a recursion with the form (\ref{recursion}). The key requirement to these modifications is that they preserve the self-similarity of $\mathcal K$ with respect to a suitably adapted pruning operation (or in other words, provided that removing the leaves of $\mathcal K$ results in a tree isomorphic to $\mathcal K$ up to some consistent finite correction).
In what follows, instead of stating a complicated theorem describing the most general possible modification that we can devise, we illustrate the flexibility of our methodology and its ability to produce interesting results via several examples.
\begin{example}[Solving (\ref{recursion}) with arbitrary values of $s$] \label{ex1} We begin by describing how to adjust the labeling of $\mathcal K$ to yield a combinatorial interpretation for solutions to (\ref{recursion}) with arbitrary values for the parameter $s$. Our approach turns out to be somewhat simpler than that of \cite{DR}, where this is accomplished for (\ref{Conolly}), the homogeneous version of (\ref{recursion}), by \emph{removing} labels from specific nodes in the tree when $s<0$.
We change the number of labels inserted within each supernode of $\mathcal K$: let the $m^{th}$ supernode receive $s_m \geq 0$ labels. Next, let the extra child of the first supernode receive $s_0 \geq 0$ labels (instead of $j$). We now derive the resulting recursion related to $\mathcal K$. To do so, we must identify the nature of the pruning operation associated with $\mathcal K$.
For any label $n$, suppose that $n$ lies in the subtree $\mathcal K_m$ of $\mathcal K$, where $m= m(n)$. Prune the subtree $\mathcal K(n)$ as follows: delete all the leaf labels and the nodes containing them. Replace the $s_i$ labels in the $i^{th}$ supernode by $s_{i-1}$ labels for each $1 \leq i \leq m$. Then relabel the new tree $\mathcal P K(n)$ in the usual way by pre-order. The tree $\mathcal P \mathcal K(n)$ contains $n - R(n) + (s_0-s_1) + \cdots + (s_m-s_{m-1}) = n - R(n) +s_0 - s_m$ labels, and it is isomorphic to the subtree $K(n-R(n)+s_0 - s_m)$. Analogous to the Pruning Lemma \ref{lem1}, we have that $\mathcal P R(n) = R(n-R(n) + s_0 - s_m)$.
Similarly, we have the analogue of the Completeness Lemma \ref{lem2} with the new value of $\nu = \alpha - k(\beta +s_0 - s_1)$ (where $\alpha$ and $\beta$ retain the same meaning as before). In the same way, Lemma \ref{lem3} and the key relation (\ref{keyrelation}) continue to hold as before. Thus, we conclude that the leaf label counting function $R(n)$ satisfies \begin{equation} \label{non-homo_modif1} R(n) = \sum_{i=1}^k R(n-(s_{m(n-ij)}-s_0) - ij - R(n-ij)) + \nu \quad \text{for}\; n > N(p+1)\,.\end{equation} Notice that as $i$ ranges from $1$ to $k$, $m(n-ij)$ can only take the values $m(n)-1$ or $m(n)$, since jumping back by $ij$ labels for $1 \leq i \leq k$ takes us at worst to the previous subtree $\mathcal K_{m(n)-1}$.
When $s_0 = t+j$ and $s_m = s$ for all $m \geq 1$ then $s_m(n-ij) = s$ for all $n > N(p+1)$, and we deduce after some simplification that \begin{equation} \label{non-homo_modif2} R(n) = \sum_{i=1}^k R(n-(s-t) - (i-1)j - R(n-ij)) + \nu \end{equation} for $n > N(p+1)$. The parameter $s-t$ can take any integer value, whereas the equivalent parameter $s$ in (\ref{recursion}) had to be nonnegative. \end{example}
For the next application of our methodology, we apply the idea in Example \ref{ex1}, together with a modified labeling scheme, to solve (\ref{Conolly}) with specified initial conditions. We illustrate our approach with $k=2$, so with the recursion \begin{equation} \label{2-ary} R(n) = R(n-s-R(n-j)) + R(n-s-j-R(n-2j))\,.\end{equation} As we discussed in Section \ref{sec1}, this recursion, together with initial conditions that follow the corresponding labeled binary tree, is solved in \cite{IRT}.
\begin{example} [Solving (\ref{2-ary}) with more general initial conditions] \label{ex2} We demonstrate how to solve (\ref{2-ary}) with initial conditions that begin with a string of $s_1 +1$ 1s for any given $s_1 \geq 0$. These are followed by an additional $s + 5j -1$ initial values determined by the tree $\mathcal K$ that we now construct.
\begin{figure}
\caption{The binary tree with label counts satisfying (\ref{2-ary}) and initial conditions beginning with $s_1+ 1$ 1s. The entries in each node indicate the number of labels. We require $j_1 = 2j-1, j_2=j$, and $s_2 = s$.}
\label{fig7}
\end{figure}
Here $\mathcal K$ is the infinite binary tree in \cite{JR} and $\mathcal T$ is $\mathcal K_2$ (see Figure \ref{fig7}). We traverse $\mathcal K$ in the usual way. We label $\mathcal K$ as follows: insert $s_1$ labels in the first supernode, and $s_2$ labels in all the other supernodes. Insert one label in the left child of the first supernode and $j_1$ labels in the right child of this supernode. The unique child of every other supernode contains $j_2$ labels. All other nodes in the tree get $j$ labels (see Figure \ref{fig7}).
Now we determine values for the parameters $j_1, j_2$ and $s_2$ so that the leaf label counting function for $\mathcal K$ satisfies (\ref{2-ary}). By pruning the subtree $\mathcal K(n)$ of $\mathcal K$ we mean deleting all the leaf labels of $\mathcal K(n)$ along with the nodes containing them, replacing the first supernode with a regular node containing 1 label, replacing the $s_2$ labels in second supernode with $s_1$ labels, and replacing the $j_2$ labels inside the child of the second supernode with $j_1$ labels. The resulting pruned tree is isomorphic to $\mathcal K(n-R(n) + 1-s_2 + j_1 - j_2)$, and so it contains $\mathcal P R(n) = R(n-(s_2-1+j_2-j_1)-R(n))$ leaf labels.
Once again we have the key relation $R(n) = \mathcal P R(n-j) + \mathcal P R(n-2j) + \nu$ where $\nu = \alpha - 2(\beta -s_1+1-j_2+j_1) = 2j-j_1-1$. The term $\nu$ is the difference between the number of leaf labels in $\mathcal K_1$ and $\mathcal K_2$ and twice the number of leaf labels in them after they have been pruned. Since the non-homogeneous term in (\ref{2-ary}) is 0 we must have $j_1 = 2j-1$ for $\nu$ to be 0. Furthermore, in order for $\mathcal P R(n-j) = R(n-s-R(n-j))$ and $\mathcal P R(n-2j) = R(n-s-j-R(n-2j))$ we require that $s = j + s_2 - 1 + j_2 - j_1$, which simplifies to $s_2 + j_2 = s + j$. Thus, we may take $s_2 = s$ and $j_2 = j$. Then for all $n > 5j+s+s_1$, the leaf label counting function $R(n)$ satisfies (\ref{2-ary}), and $R(n)$ begins with $s_1 + 1$ 1s.\end{example}
It is worth emphasizing that the tree-based solutions we derive here for (\ref{2-ary}) are not usually the ones produced when this recursion is given \emph{exactly} $s_1+1$ 1s as the initial conditions (indeed, it is not necessarily true that any solution exists when the initial conditions are precisely $s_1+1$ 1s).\footnote{It is shown in Theorem 6.4 of \cite{IRT} that the recursion (\ref{2-ary}), together with exactly $s_1+1$ 1s as the initial conditions, where $s_1 +1 \geq s+2j$, has a well-defined solution, although no tree-based combinatorial interpretation for it could be identified.} The intuition for this is as follows: any binary tree-based solution $R(n)$ for (\ref{2-ary}) has the property that periodically it will have increments of 1 for a stretch of $2j$ indices, corresponding to that portion of the tree where we successively count the $2j$ consecutive leaf labels in the pair of leaves of the tree. But such a regularity to the increments in the solution is not usually present when the initial conditions for (\ref{2-ary}) are exactly $s_1+1$ 1s.
We now prove a necessary condition for a solution $A(n)$ of (\ref{nested}) to be the leaf label counting function for some tree $\mathcal K$ as constructed in Section \ref{sec2}. Any such $A(n)$ has the property that $A(n+1) - A(n) \in \{0,1\}$. Therefore the sequence $A$ is completely determined by its \textbf{frequency sequence} $F$ defined by $F(m) = | A^{-1}(\{m\})|$. We show that $F$ reflects the self-similarity of $\mathcal K$, in the sense that we can partition $F$ into blocks such that each block can be obtained from the previous block by a suitable transformation.
To see this, assume for simplicity that $\mathcal K$ contains one label in each regular node. Consider all values $A(n)$ as $n$ ranges over the labels in the subtree $\mathcal K_i$. Define the frequency sequence $F_i$ for that segment of $A$ by $F_i(m) = | A^{-1}(\{m\}) \cap \{n:\, n \in \mathcal K_i\}|$. We only consider the non-zero values of $F_i$ so $F_i$ is a finite sequence. For $i \geq p$, recall that $\mathcal K_{i+1}$ is obtained from $\mathcal K_i$ by adding $k$ children to each leaf of $\mathcal K_i$. It follows from this that for $i \geq p$, $F_{i+1}$ is obtained from $F_i$ as follows: first increase every value of $F_i$ by 1 except its last value (which is a 1 corresponding to the last leaf label of $\mathcal K_i$); then insert $k-1$ 1s between each successive pair of these values.
For $2 \leq i < p$, the derivation of $F_{i+1}$ from $F_i$ follows the same general procedure. However, in this range the number of children of each penultimate node in $K_{i+1}$ is not necessarily $k$, so the number of 1s inserted between pairs of values is not necessarily $k-1$. Instead it is determined by the finite tree $\mathcal T$ that is used to construct $\mathcal K$. Finally, $F_2$ and $F_1$ are determined directly from their definitions.
The partition of $F$ we seek is not given by the $F_i$ but by the sequences $F_i^*$ that are determined by removing the last value in each $F_i$ and for $i \geq 2$, increasing the first value of $F_i$ by 1. In this way they correct the frequency of $A(N(i))$. Here's why: the last value in $F_i$, which is a 1, results from the sole occurrence of $A(N(i))$ in the sequence $\{A(n);\,n \in \mathcal K_i\}$. However, from the construction of $\mathcal K$, $F(A(N(i)) = 1 + F_{i+1}(A(N(i)))= F_{i+1}^*(A(N(i)))$. The sequence $F$ is the infinite word resulting from the concatenation of all the $F_i^*$, that is, $F = \prod_{i=1}^{\infty} F_i^*$.
We illustrate the above discussion using Conolly's original recursion \[C(n) = C(n-C(n-1)) + C(n-1-C(n-2)); \;\; C(1) = 1,\; C(2) = 2\,.\] $C(n)$ counts the number of leaf labels in the binary tree of Figure \ref{fig7} with one label per regular node and no labels in the supernodes. The frequency sequence is $F(m) = \nu_2(2m)$ where $\nu_2(m)$ is the 2-adic valuation of $m$. We can decompose $F$ as $F_1^* = 1$, $F_2^* = 2,1$, $F_3^* = 3,1,2,1\, \ldots$ It is precisely the decomposability of the frequency sequence as above that allows one to interpret solutions to recursions of the form (\ref{nested}) as counting leaves in some infinite tree. While it is straightforward to decompose the frequency sequence of $C(n)$ (the beginning of $F_i^*$ is the first occurrence of $i$), we do not have a criterion to determine decomposability of general meta-Fibonacci sequences arising as solutions to (\ref{nested}). The problem of determining whether any tree $\mathcal T$, and hence $\mathcal K$, corresponds to a given frequency sequence appears challenging.
Our final observation is that when the initial conditions are specified by a tree $\mathcal K$ we may change the first few initial conditions arbitrarily without affecting the resulting solution sequence. Notice that if $n > N(p+1)$, pruning the tree $\mathcal K(n-ij)$ for $1 \leq i \leq j$ results in a tree containing the first $p$ subtrees $\mathcal K_1$ to $\mathcal K_p$. Suppose that for $1 \leq n \leq N(p)-1$ we set $R(n)$ arbitrarily, and for $N(p) \leq n \leq N(p+1)$ we leave $R(n)$ as the number of leaf labels in $\mathcal K(n)$. Then the recurrence relations (\ref{recursion}) or (\ref{non-homo_modif1}) will be satisfied by $R(n)$ with the new initial conditions, because the pruned trees $\mathcal P \mathcal K(n-ij)$ for $n > N(p+1)$ and $1 \leq i \leq k$ will contain the first $N(p)$ labels. As such, all the arguments of the recursion will have value at least $N(p)$ and the proof proceeds as before. Thus, the first $N(p)-1$ values of the sequence $R(n)$ can be set arbitrarily and the recurrence relations (\ref{recursion}) or (\ref{non-homo_modif1}) still holds.
We conclude by deriving the solution for the Golomb recursion $g(n)$ \cite{Gol} discussed in Section \ref{sec1} using our tree-grafting methodology.
\begin{example}[Golomb's triangular sequence] \label{ex3} Golomb's sequence is defined by \[g(n) = g(n-g(n-1)) + 1 \;\text{and}\; g(1) = 1.\] Let $\mathcal T$ be be a rooted path of length 2. Take $s_0 =1$, $s_m=0$ for all $m \geq 1$, and $j=1$. Then we construct the unary tree whose leaf counts generate Golomb's sequence (see Figure \ref{fig8}). This shows that Golomb's sequence is a step function that increases by one at the indices $n = \binom{k+1}{2} + 1$ for every $k \geq 1$.\footnote{The step function property implies that $g(n)$ has a closed form, namely, $g(n) = \lfloor \frac{\lfloor \sqrt{8n} \rfloor + 1}{2} \rfloor$. See \cite{Gol}.} \end{example}
\begin{figure}
\caption{The unary tree $\mathcal K$ that generates Golomb's recursive sequence.}
\label{fig8}
\end{figure}
\end{document} |
\begin{document}
\title{$\eta$-Ricci solitons in $(\varepsilon)$-almost paracontact metric manifolds}
\begin{abstract} The object of this paper is to study $\eta $-Ricci solitons on $\left( \varepsilon \right) $-almost paracontact metric manifolds. We investigate $\eta $-Ricci solitons in the case when its potential vector field is exactly the characteristic vector field $\xi $ of the $\left( \varepsilon \right) $-almost paracontact metric manifold and when the potential vector field is torse-forming. We also study Einstein-like and $\left( \varepsilon \right) $-para Sasakian manifolds admitting $\eta $-Ricci solitons. Finally we obtain some results for $\eta $-Ricci solitons on $ \left( \varepsilon \right) $-almost paracontact metric manifolds with a special view towards parallel symmetric $\left( 0,2\right) $-tensor fields. \end{abstract}
\noindent {\bf Mathematics Subject Classification:}
53C15, 53C25, 53C40, 53C42, 53C50.
\noindent {\bf Keywords and phrases: }$\left( \varepsilon \right) $-almost paracontact metric manifold, $\left( \varepsilon \right) $-para Sasakian manifold, Einstein-like manifold, $\eta $-Ricci soliton.
\section{Introduction}
The notion of Ricci soliton which is a natural generalization of an Einstein metric (i.e. the Ricci tensor $S$ is a constant multiple of $g$) was introduced by Hamilton \cite{Hamilton-1982} in 1982. A pseudo-Riemannian manifold $(M,g)$ is called a \textit{Ricci soliton} if it admits a smooth vector field $V$ (potential vector field) on $M$ such that \begin{equation} \frac{1}{2}\left( \pounds _{V}\,g\right) \left( X,Y\right) +S(X,Y)+\lambda g(X,Y)=0, \label{int-1} \end{equation} where $\pounds _{V}$ denotes the Lie-derivative in the direction $V,$ $ \lambda $ is a constant and $X$, $Y$ are arbitrary vector fields on $M$. A Ricci soliton is said to be shrinking, steady or expanding according to $ \lambda $ being negative, zero or positive respectively. It is obvious that a trivial Ricci soliton is an Einstein manifold with $V$ zero or Killing vector field. Since Ricci solitons are the fixed points of the Ricci flow, they are important in understanding Hamilton's Ricci flow \cite{Hamilton-1988}: $ \frac{\partial }{\partial t}g_{ij}=-2S_{ij}$, viewed as a dynamical system, on the space of Riemannian metrics modulo diffeomorphisms and scalings. In differential geometry, the Ricci flow is an intrinsic geometric flow. It can be viewed as a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out the irregularities in the metric.
Geometric flows, especially Ricci flows, have become important tools in theoretical physics. Ricci soliton is known as quasi Einstein metric in physics literature \cite{Friedan-1985} and the solutions of the Einstein field equations correspond to Ricci solitons \cite{Akbar-Woolgar-2009}. Relation with the string theory and \ the fact that (\ref{int-1}) is a particular case of Einstein field equation makes the equation of Ricci soliton interesting in theoretical physics.
In spite of introducing and studying firstly in Riemannian geometry, the Ricci soliton equation has recently been investigated in pseudo-Riemannian context, especially in Lorentzian case \cite{Blaga3, Brozos-2012, Case-2010, Mat-89}.
The concept of $\eta $-Ricci soliton was initiated by Cho and Kimura \cite {Cho-Kimura}. An \textit{$\eta $-Ricci soliton} is a data $\left( g,V,\lambda ,\mu \right) $ on a pseudo-Riemannian manifold satisfying \begin{equation} \frac{1}{2}\left( \pounds _{V}\,g\right) \left( X,Y\right) +S(X,Y)+\lambda g(X,Y)+\mu \eta \otimes \eta \left( X,Y\right) =0, \label{int-2} \end{equation} where $\pounds _{V}$ denotes the Lie-derivative in the direction $V,$ $S$ stands for the Ricci tensor field, $\lambda $ and $\mu $ are constants and $ X $, $Y$ are arbitrary vector fields on $M$. In \cite{Calin-Crasma} the authors studied $\eta $-Ricci solitons on Hopf hypersurfaces in complex space forms. In the context of paracontact geometry $\eta $-Ricci solitons were investigated in \cite{Blaga1, Blaga2, Blaga3}.
In 1923, Eisenhart \cite{Eisenhart} proved that if a Riemannian manifold admits a second order parallel symmetric covariant tensor which is not a constant multiple of the metric tensor, then the manifold is reducible. In 1925, it was shown by Levy \cite{Levy} that a second order parallel symmetric non-degenerate tensor field in a space form is proportional to the metric tensor. Also if a (pseudo-)Riemannian manifold admits a parallel symmetric (0,2)-tensor field, then it is locally the direct product of a number of (pseudo-)Riemannian manifold \cite{Einshart36}. Sharma \cite {Einshart26} studied second order parallel tensors by using Ricci identities. Second order parallel tensors have been studied by various authors in different structures of manifolds \cite{Einshart6, Eisenhart, Levy, Einshart19, Einshart26, Einshart27, Einshart28}. \pagebreak
In $1976$, S\={a}to \cite{Sato-76} introduced the almost paracontact structure as a triple $(\varphi ,\xi ,\eta )$ of a (1,1)-tensor field $\varphi$, a vector field $\xi$ and a $1$-form $\eta$ satisfying $ \varphi ^{2}=I-\eta \otimes \xi $ and $\eta (\xi )=1$. The structure is an analogue of the almost contact structure \cite{Sasaki-60-Tohoku} and is closely related to almost product structure (in contrast to almost contact structure, which is related to almost complex structure). An almost contact manifold is always odd-dimensional but an almost paracontact manifold could be even-dimensional as well. In $1969$, Takahashi \cite {Takahashi-69-Tohoku-1} introduced almost contact manifolds equipped with an associated pseudo-Riemannian metric and, in particular, he studied Sasakian manifolds equipped with an associated pseudo-Riemannian metric. These indefinite almost contact metric manifolds and indefinite Sasakian manifolds are also known as $\left( \varepsilon \right) $-almost contact metric manifolds and $\left( \varepsilon \right) $-Sasakian manifolds, respectively \cite{Bej-Dug-93}. In 1989, Matsumoto \cite{Mat-89} replaced the structure vector field $\xi $ by $-\xi $ in an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact manifold.
An $(\varepsilon )$-Sasakian manifold is always odd-dimensional. On the other hand, in a Lorentzian almost paracontact manifold given by Matsumoto, the pseudo-Riemannian metric has only index $1$ and the structure vector field $\xi $ is always timelike. These circumstances motivated the authors of \cite{Tri-KYK-10} to associate a pseudo-Riemannian metric, not necessarily Lorentzian, with an almost paracontact structure, and this indefinite almost paracontact metric structure was called an $\left( \varepsilon \right) $-almost paracontact structure, where the structure vector field $\xi $ is spacelike or timelike according as $\varepsilon =1$ or $\varepsilon =-1$ \cite{Yuk-KTK-12}.
Motivated by these studies, in the present paper we investigate $\eta $-Ricci solitons in $\left( \varepsilon \right) $-almost paracontact metric manifolds. The paper is organized as follows. Section 2 is devoted to basic concepts on $\left( \varepsilon \right) $-almost paracontact metric manifolds. In Section 3, we study $\eta $-Ricci solitons in the case when its potential vector field is exactly the characteristic vector field $ \xi $ of the Einstein-like $( \varepsilon )$-almost paracontact metric manifold and when the potential vector field is torse-forming in an $\eta $-Einstein $ (\varepsilon )$-almost paracontact metric manifold. In Section 4, we prove that an $\left( \varepsilon \right) $-para Sasakian manifold admitting $\eta $-Ricci soliton with a potential vector field pointwise collinear to $ \xi $ is an Einstein-like manifold. In Section 5 we give some characterizations for $\eta $-Ricci solitons on $(\varepsilon )$ -almost paracontact metric manifolds concerning parallel symmetric (0,2)-tensor fields.
\section{Preliminaries}
Let $M$ be an $n$-dimensional manifold equipped with an \textit{almost paracontact structure} $(\varphi ,\xi ,\eta )$ \cite {Sato-76} consisting of a tensor field $\varphi $ of type $(1,1)$, a vector field $ \xi $ and a $1$-form $\eta $ satisfying \begin{equation} \varphi ^{2}=I-\eta \otimes \xi , \label{eq-phi-eta-xi} \end{equation} \begin{equation} \eta (\xi )=1, \label{eq-eta-xi} \end{equation} \begin{equation} \varphi \xi =0, \label{eq-phi-xi} \end{equation} \begin{equation} \eta \circ \varphi =0. \label{eq-eta-phi} \end{equation} It is easy to verify that (\ref{eq-phi-eta-xi}) and one of (\ref{eq-eta-xi} ), (\ref{eq-phi-xi}) and (\ref{eq-eta-phi}) imply the other two equations. If $g$ is a pseudo-Riemannian metric such that \begin{equation} g\left( \varphi X,\varphi Y\right) =g\left( X,Y\right) -\varepsilon \eta (X)\eta \left( Y\right), \qquad X,Y\in \Gamma (TM), \label{eq-metric-1} \end{equation} where $\varepsilon =\pm 1$, then $M$ is called $\left( \varepsilon \right) ${\it -almost paracontact metric manifold} equipped with an \textit{$\left( \varepsilon \right) ${\em -}almost paracontact metric structure} $(\varphi ,\xi ,\eta ,g,\varepsilon )$ \cite{Tri-KYK-10}. In particular, if ${\rm index }(g)=1$, that is when $g$ is a Lorentzian metric, then the $(\varepsilon )$ -almost paracontact metric manifold is called {\it Lorentzian almost paracontact manifold}. From (\ref{eq-metric-1}) we have \begin{equation} g\left( X,\xi \right) =\varepsilon \eta (X) \label{eq-metric-3} \end{equation} \begin{equation} g\left( X,\varphi Y\right) =g\left( \varphi X,Y\right), \label{eq-metric-2} \end{equation} for all $X,Y\in \Gamma (TM)$. From (\ref{eq-metric-3}) it follows that \begin{equation} g\left( \xi ,\xi \right) =\varepsilon, \label{eq-g(xi,xi)} \end{equation} that is, the structure vector field $\xi $ is never lightlike.
Let $(M,\varphi ,\xi ,\eta ,g,\varepsilon )$ be an $(\varepsilon )$-almost paracontact metric manifold (resp. a Lorentzian almost paracontact manifold). If $\varepsilon =1$, then $M$ is said to be a spacelike $ (\varepsilon )$-almost paracontact metric manifold (resp. a spacelike Lorentzian almost paracontact manifold). Similarly, if $\varepsilon =-\,1$, then $M$ is said to be a timelike $(\varepsilon )$-almost paracontact metric manifold (resp. a timelike Lorentzian almost paracontact manifold) \cite{Tri-KYK-10}.
An $\left( \varepsilon \right) $-almost paracontact metric structure $(\varphi ,\xi ,\eta ,g,\varepsilon )$ is called $\left( \varepsilon \right) ${\it -para Sasakian structure} if \begin{equation} (\nabla _{X}\varphi )Y=-\,g(\varphi X,\varphi Y)\xi -\varepsilon \eta \left( Y\right) \varphi ^{2}X,\qquad X,Y\in \Gamma (T\!M), \label{para2} \end{equation} where $\nabla $ is the Levi-Civita connection with respect to $g$. A manifold endowed with an $\left( \varepsilon \right) $-para Sasakian structure is called $\left( \varepsilon \right) ${\it -para Sasakian manifold} \cite{Tri-KYK-10}. In an $\left( \varepsilon \right) ${\em -}para Sasakian manifold, we have \begin{equation} \nabla \xi =\varepsilon \varphi \label{para3} \end{equation} and the Riemann curvature tensor $R$ and the Ricci tensor $S$ satisfy the following equations \cite{Tri-KYK-10}: \begin{equation} R\left( X,Y\right) \xi =\eta \left( X\right) Y-\eta \left( Y\right) X, \label{eq-eps-PS-R(X,Y)xi} \end{equation} \begin{equation} R\left( \xi ,X\right) Y=-\,\varepsilon g\left( X,Y\right) \xi +\eta \left( Y\right) X, \label{eq-eps-PS-R(xi,X)Y} \end{equation} \begin{equation} \eta \left( R\left( X,Y\right) Z\right) =-\,\varepsilon \eta \left( X\right) g\left( Y,Z\right) +\varepsilon \eta \left( Y\right) g\left( X,Z\right), \label{eq-eps-PS-eta(R(X,Y),Z)} \end{equation} \begin{equation} S(X,\xi )=-(n-1)\eta (X), \label{eq-eps-PS-S(X,xi)} \end{equation} for all $X,Y,Z\in \Gamma (TM)$.
\begin{example} \cite{Tri-KYK-10} Let ${\Bbb R}^{5}$\ be the $5$-dimensional real number space with a coordinate system $\left( x,y,z,t,s\right) $. Defining \[ \eta =ds-ydx-tdz\ ,\qquad \xi =\frac{\partial }{\partial s}\, \] \[ \varphi \left( \frac{\partial }{\partial x}\right) =-\,\frac{\partial }{ \partial x}-y\frac{\partial }{\partial s}\ ,\qquad \varphi \left( \frac{ \partial }{\partial y}\right) =-\,\frac{\partial }{\partial y}\, \] \[ \varphi \left( \frac{\partial }{\partial z}\right) =-\,\frac{\partial }{ \partial z}-t\frac{\partial }{\partial s}\ ,\qquad \varphi \left( \frac{ \partial }{\partial t}\right) =-\,\frac{\partial }{\partial t}\ ,\qquad \varphi \left( \frac{\partial }{\partial s}\right) =0\, \] \[ g_{1}=\left( dx\right) ^{2}+\left( dy\right) ^{2}+\left( dz\right) ^{2}+\left( dt\right) ^{2}-\eta \otimes \eta \, \] \begin{eqnarray*} g_{2} &=&-\,\left( dx\right) ^{2}-\left( dy\right) ^{2}+\left( dz\right) ^{2}+\left( dt\right) ^{2}+\left( ds\right) ^{2} \\ &&-\,t\left( dz\otimes ds+ds\otimes dz\right) -y\left( dx\otimes ds+ds\otimes dx\right), \end{eqnarray*} then $(\varphi ,\xi ,\eta ,g_{1})$ is a timelike Lorentzian almost paracontact structure in ${\Bbb R}^{5}$, while $(\varphi ,\xi ,\eta ,g_{2})$\ is a spacelike $\left( \varepsilon \right) $-almost paracontact structure. Note that {\rm index}$\left( g_{2}\right) =3$. \end{example}
\section{$\protect\eta $-Ricci solitons on Einstein-like $(\protect \varepsilon )$-almost paracontact metric manifolds}
We introduce the following definition analogous to Einstein-like para Sasakian manifolds \cite{Sharma-82}.
\begin{definition} An $\left( \varepsilon \right) $-almost paracontact metric manifold $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ is said to be {\it Einstein-like} if its Ricci tensor $S$ satisfies \begin{equation} S\left( X,Y\right) =a\,g\left( X,Y\right) +b\,g\left( \varphi X,Y\right) +c\,\eta \left( X\right) \eta \left( Y\right),\qquad X,Y\in \Gamma (TM) \label{1} \end{equation} for some real constants $a$, $b$ and $c$. \end{definition}
We deduce the following properties:
\begin{proposition} In an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold \linebreak $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ we have \begin{equation} S\left( \varphi X,Y\right) =S(X,\varphi Y), \label{2} \end{equation} \begin{equation} S\left( \varphi X,\varphi Y\right) =S(X,Y)-(\varepsilon a+c)\eta \left( X\right) \eta \left( Y\right), \label{3} \end{equation} \begin{equation} S\left( X,\xi \right) =(\varepsilon a+c)\eta \left( X\right), \label{4} \end{equation} \begin{equation} S\left( \xi ,\xi \right) =\varepsilon a+c, \label{5} \end{equation} \begin{equation} \left( \nabla _{X}\,S\right) (Y,Z)=bg((\nabla _{X}\varphi )Y,Z)+\varepsilon c\left\{ \eta (Y)g(\nabla _{X}\xi ,Z)+\eta (Z)g(\nabla _{X}\xi ,Y)\right\}, \label{5a} \end{equation} \begin{equation} \left( \nabla _{X}\,Q\right) Y=b(\nabla _{X}\varphi )Y+\varepsilon c\left\{ \eta (Y)\nabla _{X}\xi +\varepsilon g(\nabla _{X}\xi ,Y)\xi \right\}, \label{5b} \end{equation} where $Q$ is the Ricci operator defined by $g(QX,Y)=S(X,Y),$ $X,Y\in \Gamma (TM)$. Moreover, if the manifold is $\left( \varepsilon \right) $-para Sasakian, then \begin{equation} \varepsilon a+c=1-n, \label{6} \end{equation} \begin{equation} r=na+b\,{\rm trace}(\varphi )+\varepsilon c, \label{7} \end{equation} where $r$\ is the scalar curvature. \end{proposition}
Remark that the Ricci operator $Q$ of an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold is of the form \[ Q=aI+b\varphi +\varepsilon c\eta \otimes \xi \] and the structure vector field $\xi $ is an eigenvector of $Q$ with the corresponding eigenvalue $a+\varepsilon c.$
Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an\textit{ $\eta $-Ricci soliton}, that is, a tuple $\left( g,\xi ,\lambda ,\mu \right) $ satisfying \begin{equation} \frac{1}{2}\pounds _{\xi }g+S+\lambda g+\mu \eta \otimes \eta =0, \label{10} \end{equation} with $\lambda $ and $\mu $ real constants. Replacing (\ref{1}) in the last equation we get \begin{equation} g(\nabla _{X}\xi ,Y)+g(\nabla _{Y}\xi ,X)+2\left\{ \left( a+\lambda \right) g(X,Y)+bg(\varphi X,Y)+(c+\mu )\eta (X)\eta (Y)\right\} =0, \label{11} \end{equation} for all $X,Y\in \Gamma (TM).$ If we take $X=Y=\xi $ in (\ref{11}) we have \begin{equation} \varepsilon (a+\lambda )+c+\mu =0, \label{12} \end{equation} by virtue of (\ref{eq-eta-xi}) and (\ref{eq-g(xi,xi)}).
Using (\ref{12}) and taking $Y=\xi $ in (\ref{11}), we obtain \begin{equation} g(\nabla _{\xi }\xi ,X)=0, \label{13} \end{equation} which implies $\nabla _{\xi }\xi =0$. So we easily see that \begin{equation} \left( \nabla _{\xi }\varphi \right) \xi =0\text{ \ \ \ \ and \ \ \ \ } \nabla _{\xi }\eta =0. \label{14} \end{equation}
Also from (\ref{5a}), (\ref{5b}) and (\ref{14}) we get \begin{equation} \left( \nabla _{\xi }\,S\right) (Y,Z)=bg((\nabla _{\xi }\varphi )Y,Z) \label{15} \end{equation} and \begin{equation} \nabla _{\xi }\,Q=b\nabla _{\xi }\varphi. \label{16} \end{equation}
Hence we have the following result:
\begin{proposition} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $. Then
i) $\varepsilon (a+\lambda )+c+\mu =0,$
ii) $\xi $ is a geodesic vector field,
iii) $\left( \nabla _{\xi }\varphi \right) \xi =0$ \ and $\ \nabla _{\xi }\eta =0,$
iv) $\left( \nabla _{\xi }\,S\right) (Y,Z)=bg((\nabla _{\xi }\varphi )Y,Z)$ \ and \ $\nabla _{\xi }\,Q=b\nabla _{\xi }\varphi.$
Moreover, if the manifold is $\left( \varepsilon \right) $-para Sasakian, then \[ \nabla _{\xi }\,S=0\text{\ \ and \ \ }\nabla _{\xi }Q=0. \] \end{proposition}
A vector field $\xi $ is called \textit{torse-forming} if \begin{equation} \nabla _{X}\xi =fX+w(X)\xi, \label{17} \end{equation} is satisfied for some smooth function $f$ and a $1$-form $w$.
Taking the inner product with $\xi$ we have \[ 0=g(\nabla _{X}\xi ,\xi )=\varepsilon \left( f\eta (X)+w(X)\right), \] for all $X\in \Gamma (TM)$, which implies \begin{equation} w=-f\eta. \label{19} \end{equation} It follows \begin{equation} \nabla _{X}\xi =f\left( X-\eta (X)\xi \right) =f\varphi ^{2}X. \label{20} \end{equation}
Now assume that $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ is an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $ and that the potential vector field $\xi $ is torse-forming. Replacing (\ref{20}) in (\ref{11}) we obtain, for all $X,Y\in \Gamma (TM)$ \[ 0 =(f+a+\lambda )\left\{ g(X,Y)-\varepsilon \eta (X)\eta (Y)\right\} +bg(\varphi X,Y) \] and \[ 0 =g((f+a+\lambda )\varphi X+bX,\varphi Y), \] which implies \begin{equation} 0=(f+a+\lambda )\varphi ^{2}X+b\varphi X, \label{26} \end{equation} that is \begin{equation} b\varphi X=-(f+a+\lambda )X+(f+a+\lambda )\eta (X)\xi. \label{21a} \end{equation}
So we have:
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $ with torse-forming potential vector field. Then $M$ is an $\eta $ -Einstein manifold. \end{theorem}
In the remaining part of this section, we shall consider $M$ an \textit{$\eta $-Einstein manifold} (that is an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold with $b=0$), admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $ with torse-forming potential vector field $\xi $. Using (\ref{26}) we have \[ f=-a-\lambda. \] So we can write \[ \nabla _{X}\xi =-\left( a+\lambda \right) \left( X-\eta (X)\xi \right) =-\left( a+\lambda \right) \varphi ^{2}X. \] By using (\ref{20}) we obtain \begin{equation} R(X,Y)\xi =(a+\lambda )^{2}\{\eta (X)Y-\eta (Y)X\} \label{24} \end{equation} and \begin{equation} S(X,\xi )=(a+\lambda )^{2}(1-n)\eta (X), \label{25} \end{equation} for all $X,Y\in \Gamma (TM).$ From (\ref{4}) and (\ref{25}) we get \[ (\varepsilon a+c)\eta \left( X\right) =(a+\lambda )^{2}(1-n)\eta (X), \] which implies \begin{equation} c=-\varepsilon a+(a+\lambda )^{2}(1-n). \label{25a} \end{equation} Also by using (\ref{12}) in the last equation we get \begin{equation} \mu =-\varepsilon \left( \lambda +\varepsilon (a+\lambda )^{2}(1-n)\right). \label{27} \end{equation}
So we have:
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,c\right) $ be an $\eta $ -Einstein $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $ with torse-forming potential vector field. Then $f$ is a constant function and \[ c=-\varepsilon a+(a+\lambda )^{2}(1-n), \] \[ \mu =-\varepsilon \left( \lambda +\varepsilon (a+\lambda )^{2}(1-n)\right). \] \end{theorem}
\begin{proposition} In an $\eta $-Einstein $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton with torse-forming potential vector field, we have \begin{equation} (\nabla _{X}S)(Y,Z)=-c\varepsilon \left( a+\lambda \right) \left\{ \eta (Y)g(X,Z)+\eta (Z)g(X,Y)-2\varepsilon \eta (X)\eta (Y)\eta (Z)\right\} \end{equation} and \begin{equation} \left( \nabla _{X}Q\right) Y=-c\left( a+\lambda \right) \left\{ \eta (Y)X+\varepsilon g(X,Y)\xi -2\eta (X)\eta (Y)\xi \right\}. \label{38} \end{equation} \end{proposition}
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,c\right) $ be an $\eta $ -Einstein $\left( \varepsilon \right) $-almost paracontact metric manifold admitting an $\eta $-Ricci soliton $\left( g,\xi ,\lambda ,\mu \right) $ with torse-forming potential vector field. If $f\neq 0$ and the Ricci operator $Q$ is Codazzi, then $M$ is an Einstein manifold and $\xi $ is a Killing vector field. \end{theorem}
\begin{proof} From the condition \begin{equation} \left( \nabla _{X}Q\right) Y=\left( \nabla _{Y}Q\right) X, \label{60} \end{equation} for all $X,Y\in \Gamma (TM)$, using (\ref{38}) we get \[ c(a+\lambda)\{\eta(X)Y-\eta(Y)X\}=0, \] for all $X,Y\in \Gamma (TM)$. Since $a+\lambda=-f\neq 0$ we obtain $c=0$ and hence, $M$ is Einstein manifold.
Moreover, writing (\ref{5b}) for $Y=\xi $ we obtain \[ \nabla _{X}\xi =0. \] Therefore, $L_{\xi}g=0$, hence $\xi$ is Killing vector field. \end{proof}
Let us remark the following particular cases:
Case I: $f=-1.$ In this case, $\xi $ is an irrotational vector field and we have \[ \nabla \xi =-I+\eta \otimes \xi, \] \[ \lambda =1-a, \] \[ \mu =-\varepsilon \left( 1+\varepsilon c\right), \] \[ R(X,Y)\xi =\eta (X)Y-\eta (Y)X. \] Since $\eta \neq 0$, from (\ref{19}) it is easy to see that $\xi $ can not be a concurrent vector field.
Case II: $f=0.$ In this case, $\xi $ is a recurrent vector field and we have \[ \nabla \xi =0, \] \[ \lambda =-a, \] \[ \mu =\varepsilon a=-c, \] \[ R(X,Y)\xi =0. \] Furthermore, $S$ and $Q$ are $\nabla $-parallel.
\begin{theorem} On an $n$-dimensional $(n>1)$ non-Ricci flat $\eta $-Einstein $(\varepsilon ) $-almost paracontact manifold with $a=0$ admitting a torse-forming Ricci soliton $(g,\xi,\lambda)$ we have \[ \lambda =\frac{\varepsilon }{n-1}\quad \text{and \ \ }c=-\frac{1}{n-1}. \] \end{theorem}
\begin{corollary} A torse-forming Ricci soliton on an $n$-dimensional $(n>1)$ non-Ricci flat $ \eta $-Einstein spacelike (resp. timelike) $(\varepsilon )$-almost paracontact manifold with $a=0$ is expanding (resp. shrinking). \end{corollary}
\section{$\protect\eta $-Ricci solitons on $\left( \protect\varepsilon \right) $-para Sasakian manifolds}
Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ be an $\left( \varepsilon \right) $-para Sasakian manifold admitting an $\eta $-Ricci soliton $\left( g,V,\lambda ,\mu \right) $ and assume that the potential vector field $V$ is pointwise collinear with the structure vector field $\xi $, that is, $V=k\xi $, for $k$ a smooth function on $M.$ Then from (\ref{int-2} ) and (\ref{eq-metric-3}) we have \begin{equation} \varepsilon (Xk)\eta (Y)+\varepsilon (Yk)\eta (X)+2\varepsilon kg(\varphi X,Y)+2S(X,Y)+2\lambda g(X,Y)+2\mu \eta (X)\eta (Y)=0, \label{31} \end{equation} for all $X,Y\in \Gamma (TM).$ Taking $Y=\xi $ in (\ref{31}) and using (\ref {eq-eps-PS-S(X,xi)}) we get \begin{equation} \varepsilon \left( Xk\right) +\left\{ \varepsilon \left( \xi k\right) -2(n-1)+2\varepsilon \lambda +2\mu \right\} \eta (X)=0. \label{32} \end{equation} If we replace $X$ by $\xi $ in the last equation we obtain \begin{equation} \xi k=\varepsilon (n-1)-\lambda -\varepsilon \mu. \label{33} \end{equation} Using (\ref{33}) in (\ref{32}) gives \[ Xk=\left( \varepsilon (n-1)-\lambda -\varepsilon \mu \right) \eta (X). \] We conclude that $k$ is constant if $\varepsilon (n-1)=\lambda +\varepsilon \mu $ and in this case, from (\ref{31}) we get \[ S(X,Y)=-\lambda g(X,Y)-\varepsilon kg(\varphi X,Y)-\mu \eta (X)\eta (Y). \]
So we have:
\begin{theorem}\label{t1} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ be an $ (\varepsilon )$-para Sasakian manifold. If $M$ admits an $\eta $-Ricci soliton $\left( g,V,\lambda ,\mu \right) $ and $V$ is pointwise collinear with the structure vector field $\xi ,$ then $V$ is a constant multiple of $ \xi $ provided $\varepsilon (n-1)=\lambda +\varepsilon \mu $ and $M$ is an Einstein-like manifold. \end{theorem}
\begin{remark} Under the hypotheses of Theorem \ref{t1}, if $R(\xi,\cdot)\cdot S=0$, then $V$ is a constant multiple of $\xi$. Indeed, the condition on $S$ is \[ S(R(\xi,X)Y,Z)+S(Y, R(\xi,X)Z)=0, \] for all $X,Y,Z\in \Gamma (TM).$ Using (\ref{int-2}), (\ref{para3}) and (\ref{eq-eps-PS-R(xi,X)Y}) we obtain \[ (\varepsilon(n-1)-\lambda)\{\eta(Y)g(X,Z)+\eta(Z)g(X,Y)\}-\varepsilon \{\eta(Y)g(\varphi X,Z)+\eta(Z)g(\varphi X,Y)\}-$$$$-2\mu \eta(X)\eta(Y)\eta(Z)=0, \] for all $X,Y,Z\in \Gamma (TM)$ and taking $X=Y=Z=\xi$ we get \[ \varepsilon(n-1)-\lambda-\varepsilon \mu=0. \] \end{remark}
Assuming $V=\xi$ we get:
\begin{theorem} On an $n$-dimensional $(n>1)$ $ (\varepsilon )$-para Sasakian manifold admitting a Ricci soliton $(g,\xi,\lambda)$ we have \[ \lambda =\frac{\varepsilon }{n-1}. \] \end{theorem}
\begin{corollary} A Ricci soliton on an $n$-dimensional $(n>1)$ $ (\varepsilon )$-para Sasakian spacelike (resp. timelike) manifold is expanding (resp. shrinking). \end{corollary}
\begin{remark}If we assume that $M$ is an Einstein-like $(\varepsilon )$-para Sasakian manifold and $V=\xi $, we have \[ \frac{1}{2}\left( \pounds _{\xi }\,g\right) \left( X,Y\right) +S(X,Y)+\lambda g(X,Y)+\mu \eta (X)\eta (Y)=$$$$=(\varepsilon +b)g(\varphi X,Y)+(a+\lambda )g(X,Y)+(c+\mu )\eta (X)\eta (Y), \] which implies that if \[ \varepsilon +b=0, \ \ a+\lambda=0, \ \ c+\mu=0, \] then $\left( g,\xi ,-a,-c\right)$ is an $\eta $-Ricci soliton on $M$. \end{remark}
We end these considerations by giving two examples of $\eta$-Ricci solitons on the $(\varepsilon )$-para Sasakian manifold considered in Example 5.2. from \cite{Tri-KYK-10}.
\begin{example} Let $M={\Bbb R}^3$ and $(x,y,z)$ be the standard coordinates in ${\Bbb R}^3$. Set $$\varphi:=\frac{\partial}{\partial x}\otimes dx-\frac{\partial}{\partial y}\otimes dy, \ \ \xi:=\frac{\partial}{\partial z}, \ \ \eta:=dz,$$ $$g:=e^{2z}dx\otimes dx+e^{-2z}dy\otimes dy+dz\otimes dz$$ and consider the orthonormal system of vector fields $$E_1:=e^{-z}\frac{\partial}{\partial x}, \ \ E_2:=e^z\frac{\partial}{\partial y}, \ \ E_3:=\frac{\partial}{\partial z}.$$ Follows $$\nabla_{E_1}E_1=-E_3, \ \ \nabla_{E_1}E_2=0, \ \ \nabla_{E_1}E_3=E_1, \ \ \nabla_{E_2}E_1=0, \ \ \nabla_{E_2}E_2=E_3,$$$$\nabla_{E_2}E_3=-E_2, \ \ \nabla_{E_3}E_1=0, \ \ \nabla_{E_3}E_2=0, \ \ \nabla_{E_3}E_3=0.$$ Then the Riemann and the Ricci curvature tensor fields are given by: $$R(E_1,E_2)E_2=E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \ R(E_2,E_1)E_1=E_2,$$ $$R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=-E_3, \ \ R(E_3,E_2)E_2=-E_3,$$ $$S(E_1,E_1)=0, \ \ S(E_2,E_2)=0, \ \ S(E_3,E_3)=-2.$$ In this case, for $\lambda=0$ and $\mu=2$, the data $(g,\xi,\lambda,\mu)$ is an $\eta$-Ricci soliton on the para Sasakian manifold $({\Bbb R}^3, \varphi , \xi , \eta , g)$. \end{example}
\begin{example} Let $M={\Bbb R}^3$ and $(x,y,z)$ be the standard coordinates in ${\Bbb R}^3$. Set $$\varphi:=\frac{\partial}{\partial x}\otimes dx-\frac{\partial}{\partial y}\otimes dy, \ \ \xi:=\frac{\partial}{\partial z}, \ \ \eta:=dz,$$ $$g:=e^{-2z}dx\otimes dx+e^{2z}dy\otimes dy-dz\otimes dz$$ and consider the orthonormal system of vector fields $$E_1:=e^{z}\frac{\partial}{\partial x}, \ \ E_2:=e^{-z}\frac{\partial}{\partial y}, \ \ E_3:=\frac{\partial}{\partial z}.$$ Follows $$\nabla_{E_1}E_1=-E_3, \ \ \nabla_{E_1}E_2=0, \ \ \nabla_{E_1}E_3=-E_1, \ \ \nabla_{E_2}E_1=0, \ \ \nabla_{E_2}E_2=E_3,$$$$\nabla_{E_2}E_3=E_2, \ \ \nabla_{E_3}E_1=0, \ \ \nabla_{E_3}E_2=0, \ \ \nabla_{E_3}E_3=0.$$ Then the Riemann and the Ricci curvature tensor fields are given by: $$R(E_1,E_2)E_2=-E_1, \ \ R(E_1,E_3)E_3=-E_1, \ \ R(E_2,E_1)E_1=-E_2,$$ $$R(E_2,E_3)E_3=-E_2, \ \ R(E_3,E_1)E_1=E_3, \ \ R(E_3,E_2)E_2=E_3,$$ $$S(E_1,E_1)=-2, \ \ S(E_2,E_2)=-2, \ \ S(E_3,E_3)=-2.$$ In this case, for $\lambda=2$ and $\mu=4$, the data $(g,\xi,\lambda,\mu)$ is an $\eta$-Ricci soliton on the Lorentzian para Sasakian manifold $({\Bbb R}^3, \varphi , \xi , \eta , g)$. \end{example}
\section{Parallel symmetric $(0,2)$-tensor fields on $\left( \protect\varepsilon \right) $-almost paracontact metric manifolds}
Let $\alpha $ be a $(0,2)$-tensor field which is assumed to be parallel with respect to Levi-Civita connection $\nabla $, that is $\nabla \alpha =0$. Applying the Ricci identity \[ \nabla ^{2}\alpha (X,Y;Z,W)-\nabla ^{2}\alpha (X,Y;W,Z)=0, \] we have \cite{Einshart26}: \begin{equation} \alpha (R(X,Y)Z,W)+\alpha (R(X,Y)W,Z)=0. \label{50} \end{equation} Taking $Z=W=\xi $ and using the symmetry property of $\alpha ,$ we write \begin{equation} \alpha (R(X,Y)\xi ,\xi )=0. \label{51} \end{equation}
Assume that $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ is an $ \left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field. Then from (\ref{20}) we have \begin{equation} R(X,Y)\xi =f^{2}\{\eta (X)Y-\eta (Y)X\}+X(f)\varphi ^{2}Y-Y(f)\varphi ^{2}X. \label{52} \end{equation} Replacing (\ref{52}) in (\ref{51}) we get \begin{equation} f^{2}\{ \eta (X)\alpha \left( Y,\xi \right) -\eta (Y)\alpha \left( X,\xi \right) \} +X(f)\alpha \left( \varphi ^{2}Y,\xi \right) -Y(f)\alpha \left( \varphi ^{2}X,\xi \right) =0. \label{53} \end{equation} If we take $X=\xi $ in (\ref{53}) we obtain \begin{equation} \left( f^{2}+\xi (f)\right) \left\{ \alpha \left( Y,\xi \right) -\eta (Y)\alpha \left( \xi ,\xi \right) \right\} =0. \label{54} \end{equation} Let $f^{2}+\xi (f)\neq 0$; then we have \begin{equation} \alpha \left( Y,\xi \right) =\eta (Y)\alpha \left( \xi ,\xi \right). \label{55} \end{equation}
\begin{definition} An $\left( \varepsilon \right) $-almost paracontact metric manifold $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ with torse-forming characteristic vector field is called \textit{regular} if $f^{2}+\xi (f)\neq 0.$ \end{definition}
Since $\alpha $ is a parallel $(0,2)$-tensor field, then $\alpha \left( \xi ,\xi \right) $ is a constant. Taking the covariant derivative of (\ref{55}) with respect to $X$ we derive \begin{equation} \alpha (\nabla _{X}Y,\xi )+f\left\{ \alpha (X,Y)-\eta (X)\eta (Y)\alpha \left( \xi ,\xi \right) \right\} =X\left( \eta (Y)\right) \alpha \left( \xi ,\xi \right), \label{56} \end{equation} which implies \begin{eqnarray*} f\left\{ \alpha (X,Y)-\eta (X)\eta (Y)\alpha \left( \xi ,\xi \right) \right\} &=&\varepsilon \left\{ X(g(Y,\xi )-g\left( \nabla _{X}Y,\xi \right) \right\} \alpha \left( \xi ,\xi \right) \\ &=&\varepsilon g\left( Y,\nabla _{X}\xi \right) \alpha \left( \xi ,\xi \right) \\ &=&\varepsilon f\left\{ g(X,Y)-\varepsilon \eta (X)\eta (Y)\right\} \alpha \left( \xi ,\xi \right) \end{eqnarray*} and we obtain \begin{equation} \alpha (X,Y)=\varepsilon g(X,Y)\alpha \left( \xi ,\xi \right). \label{57} \end{equation}
Therefore:
\begin{theorem} A symmetric parallel second order covariant tensor in a regular $\left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field is a constant multiple of the metric tensor. \end{theorem}
Applying this result to solitons, we deduce:
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon \right) $ be a regular $\left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field. Then $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S+\mu \eta\otimes \eta$ (with $\mu$ a real constant) is parallel if and only if $(g,\xi,\lambda=-\varepsilon \alpha(\xi,\xi),\mu)$ is an $\eta$-Ricci soliton on $M$. \end{theorem}
Assume that $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ is an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field. Then \begin{equation} \frac{1}{2}\left( \pounds _{\xi}\,g\right)(X,Y)+S(X,Y)+\mu \eta(X) \eta(Y)=$$$$=(f+a)g(X,Y)+bg(\varphi X,Y)+(c+\mu-\varepsilon f)\eta(X) \eta(Y). \label{34} \end{equation}
\begin{theorem}\label{36} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be a regular Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field. Then $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S+\mu \eta\otimes \eta$ (with $\mu$ a real constant) is parallel if and only if $(g,\xi,\lambda=-(a+\varepsilon (c+\mu)),\mu)$ is an $\eta$-Ricci soliton on $M$. \end{theorem}
\begin{proof} From (\ref{34}) we get $\alpha(\xi,\xi)=\varepsilon (a+\varepsilon c)+\mu$, so $\lambda=-\varepsilon \alpha(\xi,\xi)=-(a+\varepsilon (c+\mu))$. \end{proof}
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be a regular Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold with torse-forming characteristic vector field. Then $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S$ is parallel if and only if $(g,\xi,\lambda=-(a+\varepsilon c))$ is an expanding (resp. shrinking) Ricci soliton on $M$ provided $a+\varepsilon c<0$ (resp. $a+\varepsilon c>0$). \end{theorem}
Assume that $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon\right) $ is an $(\varepsilon)$-para Sasakian manifold. From (\ref{eq-eps-PS-R(X,Y)xi}) and ( \ref{51}) we have \begin{equation} \eta (X)\alpha (Y,\xi )-\eta (Y)\alpha (X,\xi) =0. \label{58} \end{equation} Taking $X=\xi $ and $Y=\varphi ^{2}Z$ in the last equation we obtain \begin{equation} 0=\alpha (\varphi ^{2}Z,\xi )=\alpha (Z,\xi )-\eta (Z)\alpha (\xi ,\xi ), \label{39} \end{equation} for all $Z\in \Gamma (TM).$\pagebreak
Since $\alpha $ is a parallel $(0,2)$-tensor field, then $\alpha \left( \xi ,\xi \right) $ is a constant. Taking the covariant derivative of (\ref{39}) with respect to $X$ we derive \begin{equation} \alpha (\nabla _{X}Z,\xi )+\varepsilon \alpha(Z,\varphi X) =X(\eta (Z)) \alpha (\xi,\xi), \label{56} \end{equation} which implies \[ \alpha(\varphi X,Z)=\varepsilon g(\varphi X,Z)\alpha (\xi,\xi). \] Taking $X=\varphi Y$ in the last equation we get \begin{equation} \alpha (Y,Z)=\varepsilon g(Y,Z)\alpha \left( \xi ,\xi \right). \label{57} \end{equation}
Therefore:
\begin{theorem} On an $(\varepsilon)$-para Sasakian manifold, any parallel symmetric (0,2)-tensor field is a constant multiple of the metric. \end{theorem}
Applying this result to solitons, we deduce:
\begin{theorem} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon\right) $ be an $(\varepsilon)$-para Sasakian manifold. Then $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S+\mu \eta\otimes \eta$ (with $\mu$ a real constant) is parallel if and only if $(g,\xi,\lambda=-\varepsilon \alpha(\xi,\xi),\mu)$ is an $\eta$-Ricci soliton on $M$. \end{theorem}
Assume that $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon, a,b,c\right) $ is an Einstein-like $(\varepsilon)$-para Sasakian manifold. Then \begin{equation} \frac{1}{2}\left( \pounds _{\xi}\,g\right)(X,Y)+S(X,Y)+\mu \eta(X) \eta(Y)=$$$$=ag(X,Y)+(\varepsilon+b)g(\varphi X,Y)+(c+\mu)\eta(X) \eta(Y). \label{35} \end{equation}
\begin{theorem}\label{37} Let $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ be an Einstein-like $\left( \varepsilon \right) $-para Sasakian manifold. Then $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S+\mu \eta\otimes \eta$ (with $\mu$ a real constant) is parallel if and only if $(g,\xi,\lambda=-(a+\varepsilon (c+\mu)),\mu)$ is an $\eta$-Ricci soliton on $M$. \end{theorem}
\begin{proof} From (\ref{35}) we get $\alpha(\xi,\xi)=\varepsilon (a+\varepsilon c)+\mu$, so $\lambda=-\varepsilon \alpha(\xi,\xi)=-(a+\varepsilon (c+\mu))$. \end{proof}
\begin{remark} From Theorem \ref{36} and Theorem \ref{37} we notice that the parallelism of the symmetric (0,2)-tensor field $\alpha:=\frac{1}{2}\left( \pounds _{\xi}\,g\right)+S+\mu \eta\otimes \eta$ on an Einstein-like $\left( \varepsilon \right) $-almost paracontact metric manifold $\left( M,\varphi ,\xi ,\eta ,g,\varepsilon ,a,b,c\right) $ which either is regular with torse-forming characteristic vector field or is $\left( \varepsilon \right) $-para Sasakian, yields the same $\eta$-Ricci soliton (which depends only on the constants $a,c$ and $\mu$). \end{remark}
\small{
\noindent Adara Monica Blaga
\noindent Department of Mathematics, West University of Timi\c{s}oara
\noindent Bld. V. Parvan nr. 4, 300223, Timi\c{s}oara, Romania
\noindent Email: adarablaga@@yahoo.com
\noindent Selcen Y\"{u}ksel Perkta\c{s}
\noindent Department of Mathematics, Faculty of Arts and Sciences,
\noindent Ad\i yaman University
\noindent 02040, Ad\i yaman, Turkey
\noindent Email: sperktas@@adiyaman.edu.tr
\noindent Bilal Eftal Acet
\noindent Department of Mathematics, Faculty of Arts and Sciences,
\noindent Ad\i yaman University
\noindent 02040, Ad\i yaman, Turkey
\noindent Email: eacet@@adiyaman.edu.tr
\noindent Feyza Esra Erdo\u{g}an
\noindent Faculty of Education, Department of Elementary Education,
\noindent Ad\i yaman University
\noindent 02040, Ad\i yaman, Turkey
\noindent Email: ferdogan@@adiyaman.edu.tr
}
\end{document} |
\begin{document}
\title{On the Fourier asymptotics of absolutely continuous measures with power-law singularities}
\begin{abstract} We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon-Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian. \end{abstract}
\
\noindent{\bf Keywords}: Fourier Analysis, Quantum Dynamics and Spectral Theory.
\
\noindent{\bf AMS classification codes}: 28A80 (primary), 42A85 (secondary).
\renewcommand{\Alph{table}}{\Alph{table}}
\section{Introduction}\label{sectIntrod}
\subsection{Contextualization}
The study of the long time behavior of the Fourier transform of fractal (spectral) measures and of the modulus of continuity of the distribution of such measures play an important role in spectral theory and quantum dynamics (such behavior is related to good transport properties). Actually, most of these works are motivated by possible applications to Schr\"odinger operators (see \cite{AvilaUaH,Last,strichartz1990,Zhao,Zhao2} and references therein). In this context, one may highlight two classical results on finite Borel measures on~$\mathbb{R}$: the Riemann-Lebesgue's Lemma (around 1900) and the Wiener's Lemma (around 1935).
One may also highlight Strichartz's Theorem \cite{strichartz1990}, from 1990 (see Theorem \ref{Strichartztheorem} (i) below), that establishes (power-law) convergence rates for the time-average behavior of the squared absolute value of the Fourier transform of uniformly $\alpha$-H\"older continuous measures. We present some details.
Let $\mu$ be a finite positive Borel measure on~$\mathbb{R}$ and $\alpha \in [0,1]$. We recall that $\mu$ is uniformly $\alpha$-H\"older continuous (denoted {\rm U}$\alpha${\rm H}) if there exists a constant $C>0$ such that for each interval $I$ with $\ell(I) < 1$, $\mu(I) \le C\, \ell(I)^\alpha$, where $\ell(\cdot)$ denotes the Lebesgue measure on~$\mathbb{R}$. Namely, one has the following result.
\begin{theorem}[Theorems 2.5 and 3.1 in \cite{Last}]\label{Strichartztheorem} Let $\mu$ be a finite Borel measure on $\mathbb{R}$ and $\alpha \in [0,1]$.
\begin{enumerate}
\item[\rm{i)}] If $\mu$ is {\rm U}$\alpha${\rm H}, then there exists a constant $C_\mu> 0$, depending only on $\mu$, such that for every $f \in {\mathrm L}^2(\mathbb{R}, d\mu )$ and every $t>0$,
\[\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} f(x)\, d\mu(x) \bigg|^2 ds < C_{\mu} \|f\|_{{\mathrm L}^2(\mathbb{R}, d\mu )}^2 t^{-\alpha}. \]
\item[\rm{ii)}] If there exists $C_{\mu}>0$ such that for every $t>0$,
\[\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx}\, d\mu(x) \bigg|^2 ds < C_{\mu} t^{-\alpha},\] then $\mu$ is {\rm U}$\frac{\alpha}{2}${\rm H}.
\end{enumerate} \end{theorem}
\begin{remark}{\rm Note that Theorem \ref{Strichartztheorem}-i) is, indeed, a particular case of Strichartz's Theorem~\cite{strichartz1990}, which holds for $\sigma$-finite measures.} \end{remark}
Motivated by applications in spectral theory and quantum dynamics, we use in this work Fourier analysis to prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities. Our main goal here is to obtain initial states (for the Schr\"odinger equation) for which the respective spectral measures have a singularity with a power-law growth rate, and for which the asymptotic behavior of the respective (time-average) quantum return probabilities (see definition ahead) depends continuously on such singularities (see Theorem \ref{maintheorem} and Example \ref{ex2} ahead). To the best knowledge of the present authors, this phenomenon has never been discussed, although it may be natural to specialists.
In the next remark we discuss the fact that Theorem \ref{Strichartztheorem}-i), in general, is not sufficient to obtain sharp estimates on such Fourier transform averages of absolutely continuous measures with power-law singularities.
\begin{remark}{\rm For some important classes of measures in spectral theory and quantum dynamics, such as spectral measures of dynamically defined Schr{\"o}dinger operators, $1/2$-H\"older continuity is typically optimal (usually due to the fact that there are square root singularities associated with the boundary of the spectrum; see \cite{AvilaUaH,Damanik,Zhao,Zhao2} for additional comments); by Theorem~\ref{Strichartztheorem}-i), the time-average behavior of the squared absolute value of the Fourier transform of such measures decays at least as $1/\sqrt{t}$. This rate, in general, is far from optimal as, for instance, is the case of the discrete Laplacian: let $\ell^2({\mathbb{Z}})$ and $\delta_j = (\delta_{jk})_{k \in {\mathbb{Z}}}$, $j\in{\mathbb Z}$, be its canonical basis, and consider the Laplacian, whose action on $\psi\in\ell^2({\mathbb{Z}})$ is given by \[\triangle\psi(k) = \psi(k+1) + \psi(k-1);\] so, although the spectral measure $\mu_{\delta_0}^{\triangle}$ of the pair $(\triangle,\delta_0)$ is at most uniformly $1/2$-H\"older continuous, one has \begin{equation}\label{eq00}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\delta_0}^{\triangle}(x)\bigg|^2 ds=O(\log(t)/t); \end{equation} see the case $\beta = \frac{1}{2}$ in Example \ref{ex1} and, e.g., Section 12.3 in \cite{Oliveira} for details of the Radon-Nikodym derivative of such spectral measure (here, $h(t) = O(r(t))$ indicates that there is $C>0$ so that, for each $t>0$, $h(t)\leq Cr(t)$).
By Theorem~\ref{Strichartztheorem}-i), one may conclude that \begin{equation*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\delta_0}^{\triangle}(x)\bigg|^2 ds=O(1/\sqrt{t}), \end{equation*} which gives a worse bound than the one given by~\eqref{eq00}. Moreover, since $\mu_{\delta_0}^{\triangle}$ is at most uniformly $1/2$-H\"older continuous, by Theorem \ref{Strichartztheorem}-ii), the rate in~(\ref{eq00}) is (power-law) optimal. Namely, suppose that there exists $\varepsilon>0$ such that one can replace $t^{-1}$ by $t^{-1-\varepsilon}$ in~\eqref{eq00}; then, by Theorem \ref{Strichartztheorem}-ii), $\mu_{\delta_0}^{\triangle}$ is at least uniformly $(1/2+\varepsilon/4)$-H\"older continuous.} \end{remark}
In order to put our work into perspective, we present the following example. To each $0<\beta <1$, denote by \begin{equation}\label{eqMbeta}
M_\beta:= \max_{\eta > 0}\bigg|\displaystyle\int_0^\eta e^{-iu} u^{-\beta} du \bigg|^2; \end{equation}
for $0<\beta <1$, $\bigg|\displaystyle\int_0^\infty e^{-iu} u^{-\beta} du \bigg| = \Gamma(1-\beta)$, where $\Gamma$ stands for the Gamma Function; thus, $M_\beta < \infty$.
\begin{example}\label{ex1} {\rm Set, for each $\frac{1}{2} \leq \beta < 1$ and each $f \in {\mathrm L}^1(\mathbb{R})$, \begin{equation}\label{eqKB}
K_{\beta,f} := \{(|\cdot|^{-\beta} \chi_{(0,1]}) \ast f\}, \end{equation} so $K_{\beta,f} \in {\mathrm L}^1(\mathbb{R})$. By the Convolution Theorem, it follows that for each $s >0$, \begin{eqnarray*} \widehat{K_{\beta,f}}(s) &=& \biggl\{\int_0^1 e^{-2\pi i x s} x^{-\beta} dx \biggl\} \, \hat{f}(s) = \biggl\{\frac{1}{(2\pi)^{1-\beta}s^{1-\beta}} \int_0^1 e^{-2\pi i x s} (2 \pi x s)^{-\beta} (2\pi s)\,dx \biggl\} \, \hat{f}(s)\\ &=& \biggl\{\frac{1}{(2\pi)^{1-\beta}s^{1-\beta}} \int_0^{2\pi s} e^{- iu} u^{-\beta} du \biggl\} \, \hat{f}(s), \end{eqnarray*} and so, for each $t >0$ and $\frac{1}{2} < \beta < 1$, \begin{eqnarray}\label{eqex}
\nonumber\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f}(x)\bigg|^2 ds &\leq& {M_\beta}\, \frac{1}{t}\int_0^ts^{2(\beta-1)}|\hat{f}(s)|^2 ds\\
&\le& {M_\beta} \frac{\|f\|^2_{{\mathrm L^1}(\mathbb{R})}}{t}\int_0^ts^{2(\beta-1)}ds
= \frac{{M_\beta}}{2\beta-1} \, \frac{\|f\|^2_{{\mathrm L^1}(\mathbb{R})}}{t^{2(1-\beta)}}, \end{eqnarray} where $d\mu_{\beta,f}(x) = K_{\beta,f}(x)\, dx$. For $\beta = \frac{1}{2}$ and for every $t> 1,$ one has \begin{eqnarray}\label{eqex777777}
\nonumber\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f}(x)\bigg|^2 ds &=& \nonumber\frac{1}{t} \int_0^1 \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f}(x)\bigg|^2 ds + \nonumber \frac{1}{t} \int_1^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f}(x)\bigg|^2 ds\\ \nonumber &\leq& \frac{\|K_{\beta,f}\|_{{\mathrm L}^1(\mathbb{R})}^2}{t} + \, \frac{M_\beta\|f\|^2_{{\mathrm L^1}(\mathbb{R})}}{t}\int_1^t \frac{1}{s} \, ds\\ &=& \frac{\|K_{\beta,f}\|_{{\mathrm L}^1(\mathbb{R})}^2}{t} + \, \frac{M_\beta\|f\|^2_{{\mathrm L^1}(\mathbb{R})} \log(t)}{t}. \end{eqnarray}
We argue that the above power-law upper estimates cannot be improved; suppose, on the contrary, that there exists $0<\varepsilon<1-\beta$ such that one can replace $2(1-\beta)$ by $2(1-\beta)+\varepsilon$ in the estimate~\eqref{eqex} or \eqref{eqex777777} (recall that in \eqref{eqex777777} $\beta = 1/2$). Then, by Theorem \ref{Strichartztheorem}-ii), for each $f \in {\mathrm L}^1(\mathbb{R})$, $\mu_{\beta,f}$ is at least {\rm U}$(1-\beta+ \varepsilon/4)${\rm H}.
Now, set, for each $0<\delta <1-\beta$ and each $x \in \mathbb{R}\setminus\{0\}$, \[f_\delta(x) = \frac{1}{x^{1-\delta}} \; \chi_{(0,1]}(x),\] and $f_\delta \in {\mathrm L}^1(\mathbb{R})$. Note that for $0< x \leq 1$, \begin{eqnarray*}
K_{\beta,f_\delta}(x) &=& \int_0^1 \frac{1}{y^{\beta} |x-y|^{1-\delta}} \, dy \geq \int_0^x \frac{1}{y^{\beta} |x-y|^{1-\delta}} \, dy \geq \frac{1}{x^\beta} \int_0^x \frac{1}{|x-y|^{1-\delta}} \, dy \geq x^{- \beta+\delta}. \end{eqnarray*} Thus, for $0< \epsilon < 1$, \[ \int_0^\epsilon K_{\beta,f_\delta}(x) \, dx \geq \int_0^\epsilon x^{-\beta+\delta} dx = \frac{\epsilon^{(1- \beta+ \delta)} }{(1-\beta + \delta)},\] and therefore, $\mu_{\beta,f_\delta}$ is at most {\rm U}$(1-\beta+ \delta)${\rm H}.
Finally, let $0<\delta<\varepsilon/4$; then, since $\mu_{\beta,f_\delta}$ is at most {\rm U}$(1-\beta+ \delta)${\rm H}, one gets a contradiction with the fact that $\mu_{\beta,f_\delta}$ is also {\rm U}$(1-\beta+ \varepsilon/4)${\rm H}.
We emphasize that by applying Theorem~\ref{Strichartztheorem}-i), it may be obtained at most that
\[\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f}(x)\bigg|^2 ds=O(t^{-(1-\beta)+\delta}), \] which gives a worse bound than the one given by~\eqref{eqex} (respec.~\eqref{eqex777777} for $\beta = \frac{1}{2}$) for $0<\delta<1-\beta$.
We also note that what makes this example interesting is the fact that the measure has a power-law singularity, in the sense that its Radon-Nikodym derivative has a power-law divergence. } \end{example}
In this work we use Fourier analysis to extend the estimates in (\ref{eqex})-\eqref{eqex777777} to the measures \[ d\mu_{\beta,f,g}(x) = K_{\beta,f}(x) g(x)\, dx, \] with $f \in {\mathrm L}^1(\mathbb{R})$ and $g \in {\mathrm L}^\infty[0,1]$ (see Theorem~\ref{2Stheorem} ahead), and then we discuss an application of this result to spectral measures of finite-rank perturbations of the Laplacian. Namely, by taking into account Example~3.1 in~\cite{Last}, we use this class of measures to obtain initial states (for the Schr\"odinger equation) for which the respective spectral measures have power-law singularities, and for which the asymptotic behavior of the respective (time-average) quantum return probabilities (see definition ahead) depend continuously on such singularities (see Theorem~\ref{maintheorem} and Example~\ref{ex2}).
The organization of this work is as follows. In Subsection~\ref{subsectStrich} we discuss a Strichartz's-like Inequality (Theorem~\ref{2Stheorem}). In Section~\ref{sectFRPFL} we use some well-known results on the Radon-Nikodym derivative of spectral measures \cite{Germinet,Lasttransfermatrix} to present an application to finite-rank perturbations of the Laplacian. The proof of Theorem~\ref{2Stheorem} is left to Section~\ref{sectProofMain}.
Some words about the notation:~$\hat{f}$ will always denote the Fourier transform of a function $f \in {\mathrm L}^1(\mathbb{R})$. If $h,g : \mathbb{R} \longrightarrow \mathbb{R}$ are mensurable functions, then $h \ast g$ denotes the convolution product of $h$ and $g$; $\mu$ always indicates a finite positive Borel measure on $\mathbb{R}$. For each $x \in \mathbb{R}$ and each $\epsilon>0$, $B(x,\epsilon)$ denotes the open interval $(x-\epsilon,x+\epsilon)$. If $g$ is a complex-valued function, then $\mathfrak{Re}(g)$ and $\mathfrak{Im}(g)$ denote its real and the imaginary parts, respectively. If~$f$ is a real-valued function, then $f^+$ and $f^-$ denote its positive and the negative parts, respectively.
\subsection{A Strichartz's-like Inequality} \label{subsectStrich}
Let $K_{\beta,f}$ be as in Example \ref{ex1}, $g \in {\mathrm L}^\infty[0,1]$ and consider \begin{equation} d\mu_{\beta,f,g}(x) = K_{\beta,f}(x) g(x) \, dx. \end{equation} For simplicity, suppose that $f,g$ are nonnegative (measurable) real-valued functions. So, by well-known arguments \cite{Last,strichartz1990} (see (\ref{maineq1}) ahead), it is possible to show that for every $t>0$, \begin{eqnarray} \label{eq0}
\nonumber \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f,g}(x) \bigg|^2\nonumber ds &=& \nonumber \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 ds\\ &\le& \frac{e^{2\pi} }{2 \sqrt{\pi}} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x)g(x) K_{\beta,f}(y) g(y) e^{-\frac{t^2|x-y|^2}{4}} dx dy. \end{eqnarray} Moreover, for each $x \in \mathbb{R}$ and each $0<\epsilon<1$, one has
\[ \mu_{\beta,f}(B(x,\epsilon)) \leq \|f\|_{{\mathrm L}^1(\mathbb{R})} \epsilon^{1-\beta},\] where $d\mu_{\beta,f}(x) = K_{\beta,f}(x) dx$. So, by using (\ref{eq0}) and a Strichartz's-like argument (as in \cite{Last,strichartz1990}), it follows that for every $t>0$,
\begin{eqnarray} \nonumber \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f,g}(x) \bigg|^2 ds &\leq& \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{2 \sqrt{\pi}} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x) K_{\beta,f}(y) e^{-\frac{t^2|x-y|^2}{4}} dx dy\\ \nonumber &=& \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{2 \sqrt{\pi}} \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\frac{t^2|x-y|^2}{4}} d\mu_{\beta,f}(y) d\mu_{\beta,f}(x)\\ &=&\nonumber \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{ \sqrt{\pi}} \int_{\mathbb{R}} \sum_{n=0}^\infty \int_{\frac{n}{t}\leq |x-y|<\frac{n+1}{t}} e^{-\frac{t^2|x-y|^2}{4}} d\mu_{\beta,f}(y) d\mu_{\beta,f}(x) \\ \nonumber &\leq& \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{ \sqrt{\pi}} \int_{\mathbb{R}} \sum_{n=0}^\infty e^{-n^2/4} \|f\|_{{\mathrm L}^1(\mathbb{R})} t^{\beta-1} d\mu_{\beta,f}(x)\\ &=& \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{ \sqrt{\pi}} \left(\sum_{n=0}^\infty e^{-n^2/4}\right) \|f\|_{{\mathrm L}^1(\mathbb{R})} \|K_{\beta,f}\|_{{\mathrm L}^1(\mathbb{R})}\; t^{-(1-\beta)}. \end{eqnarray} Thus, by Young's Convolution Inequality, one gets, for every $t>0$,
\begin{eqnarray}\label{eq2} \nonumber \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f,g}(x) \bigg|^2 ds &\leq& \frac{e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2}{ \sqrt{\pi}} \left(\sum_{n=0}^\infty e^{-n^2/4}\right) \|f\|^2_{{\mathrm L}^1(\mathbb{R})} \|(|\cdot|^{-\beta} \chi_{(0,1]})\|_{{\mathrm L}^1(\mathbb{R})} t^{-(1-\beta)} \\ &=& \left(\sum_{n=0}^\infty e^{-n^2/4}\right) \frac{ e^{2\pi} \|g\|_{{\mathrm L}^\infty[0,1]}^2 \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 }{ \sqrt{\pi}(1-\beta)}\; t^{-(1-\beta)}. \end{eqnarray}
We remark that the uniform estimate over $x$ in the discussion above makes the decay in~\eqref{eq2} far from optimal (see the proof of Theorem \ref{2Stheorem} and compare~\eqref{eq2} with~(\ref{eq00lemma}) and~(\ref{eq1lemma})). By using Fourier analysis, we will explore this point of the argument to obtain the following result.
\begin{theorem}\label{2Stheorem} For $\frac{1}{2} \leq \beta < 1$, let $ K_{\beta,f}$ and~$M_\beta$ be as before. To every $g \in {\mathrm L}^\infty[0,1]$, consider $d\mu_{\beta,f,g}(x)= K_{\beta,f}(x) g(x) dx$. Then:
\begin{enumerate} \item[\rm {i)}] if $\frac{1}{2} < \beta < 1$, for every $t>0$, \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx}\, d\mu_{\beta,f,g}(x) \bigg|^2 ds &\leq& 2^{18} e^{2\pi} {M_\beta} \Gamma(\beta - 1/2) \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \|g\|_{{\mathrm L}^\infty[0,1]}^2\; t^{-2(1-\beta)}; \end{eqnarray*}
\item[\rm {ii)}] if $\beta = \frac{1}{2}$, for every $t>0$, \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx}\, d\mu_{\beta,f,g}(x) \bigg|^2 ds &\leq& e^{2\pi} \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \|g\|_{{\mathrm L}^\infty[0,1]}^2 \biggr[\biggr( \frac{1}{t} + M_{\frac{1}{2}} \frac{\Gamma(0, 4\pi^2/t^2)}{t}\biggr) \biggr], \end{eqnarray*} in particular, for sufficiently large $t$ , \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx}\, d\mu_{\beta,f,g}(x) \bigg|^2 ds &\leq& e^{2\pi} \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \|g\|_{{\mathrm L}^\infty[0,1]}^2 \biggr[\biggr( \frac{1}{t} + 3 M_{\frac{1}{2}} \frac{\log(t)}{t}\biggr) \biggr], \end{eqnarray*}
since
\[\lim_{t \to \infty} \frac{\Gamma(0, 4\pi^2/t^2)}{\log(t)} = 2,\] \end{enumerate} where $\Gamma(\cdot,\cdot)$ denotes the Incomplete Gamma Function. \end{theorem} \begin{remark} \end{remark}
\begin{enumerate}
\item [i)] As mentioned in Example~\ref{ex1}, in general, one cannot get a better power-law estimate than $O(t^{-2(1-\beta)})$ for all $f \in {\mathrm L}^1(\mathbb{R})$.
\item [ii)] If $0 \leq \beta < \frac{1}{2}$ then, by Young's Convolution Inequality, $K_{\beta,f} \cdot g \in {\mathrm L}^2(\mathbb{R})$. Hence, by applying Theorem \ref{Strichartztheorem}-i) to $K_{\beta,f} \cdot g$ and to $\chi_{[0,1]} \, dx$ (which is U$1$H), one gets \begin{equation*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} d\mu_{\beta,f,g}(x) \bigg|^2 ds = \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) \, g(x)dx \bigg|^2 ds = O(t^{-1}), \end{equation*} and $\beta=1/2$ is a transition point, so justifying its peculiar behavior as in Theorem~\ref{2Stheorem}-ii).
\item [iii)] For an arbitrary $h \in {\mathrm L}^1(\mathbb{R})$, it is well known that $\hat{h}(s)$ can decay arbitrarily slow (see, e.g.,~\cite{Muller}). In this context, for $\frac{1}{2} \leq \beta < 1$, it is particularly interesting that the power-law asymptotic behavior of \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) \, g(x)dx \bigg|^2 ds \end{eqnarray*}
is inherited from the asymptotic behavior of the Fourier transform of $|\cdot|^{-\beta}$, which depends continuously on $\beta$. \end{enumerate}
\section{Finite-rank perturbations of the Laplacian}\label{sectFRPFL}
Let $\delta_j = (\delta_{jk})_{k \in \mathbb{N}}$, $j=1,2...$, be the canonical basis of~$\ell^2(\mathbb{N})$. Consider the Laplacian with Dirichlet boundary condition, whose action on $\psi\in\ell^2(\mathbb{N})$ is \[\triangle\psi(k) = \psi(k+1) + \psi(k-1),\] with $\psi(0)=0$. Consider also the finite-rank perturbations of the Laplacian with Dirichlet boundary condition, acting in $\ell^2(\mathbb{N})$ by the law \[ \begin{cases} H_0 \psi \, \, =-\triangle\psi, \\ H_N \psi = -\triangle\psi + \displaystyle\sum_{j=1}^N v_j \langle \psi, \delta_j \rangle \delta_j,\qquad N \geq 1, \end{cases} \] with $\psi(0)=0$, where $(v_n)_{n \in {\mathbb{N}}}$ is a given real sequence.
The study of the dynamical and spectral properties of these operators is a classical subject in spectral theory for at least two reasons: it is a relatively simple model, and therefore it can be used to discuss results in quantum dynamics by avoiding technical complications; they can be used as approximations to some Schr\"odinger operators (see, for instance, Section~3 in~\cite{Germinet}). In this context, our main goal here is to study the quantum dynamics of these operators for some initial states whose spectral measures may have power-law singularities. Naturally, some preparation is required.
Let $\mu_\psi^{N}$ denote the spectral measure associated with the pair $(H_N,\psi)$ and $\displaystyle\frac{d\mu_\psi^{N}}{dx}$ the Radon-Nikodym derivative of $\mu_\psi^{N}$ with respect to the Lebesgue measure, and let $R_N(z) = (H_N - z)^{-1}$ be the corresponding resolvent operator. It is well known that for every $N \in \mathbb{N} \cup \{0\}$, $H_N$ has purely absolutely continuous spectrum (for details, see \cite{Lasttransfermatrix}). Thus, in this case, $\displaystyle\frac{d\mu_\psi^{N}}{dx}\in {\mathrm L}^1(\mathbb{R})$. Recall that for every $N \in \mathbb{N} \cup \{0\}$, the transfer matrix $T_N(E,n,n-1)$ between the sites $n-1$ and $n$, $n\in\mathbb{N}$, is given by \[T_N(E,n,n-1) =\left(\begin{array}{cc}
E-v_n\chi_{[1,N]}(n) & -1\\
1 & 0\\ \end{array}\right).\] Moreover, if $u_\theta(E,n)$, with $ \theta \in [0,\pi]$, denotes the solution to the eigenvalue equation $H_Nu = Eu$ at $E \in \mathbb{R}$ that satisfies \[u_\theta(E,m) = \sin(\theta), \, \, \, \, u_\theta(E,m+1) = \cos(\theta),\] then \[\left(\begin{array}{cc}
u_\theta(E,n+1)\\
u_\theta(E,n)\\ \end{array}\right)= T_N(E,n,m)\left(\begin{array}{cc}
u_\theta(E,m+1)\\
u_\theta(E,m)\\ \end{array}\right),\] with $T_N(E,n,m)=T_N(E,n,n-1)\cdots T_N(E,m+1,m)$. We need the following technical result.
\begin{lemma}\label{teclemma} Let $N \in \mathbb{N} \cup \{0\}$. Then, there exist constants $C_{1,N},C_{2,N}> 0$ such that, for every $E \in [0,1]$, \[C_{1,N}\, \leq\, \displaystyle\frac{d\mu_{\delta_1}^{N}}{dx}(E)\, \leq\, C_{2,N};\] in particular, $\displaystyle\frac{d\mu_{\delta_1}^{N}}{dx} \in {\mathrm L}^\infty[0,1]$. \end{lemma}
\begin{remark}{\rm The result stated in Lemma \ref{teclemma} is expected, since the boundary of the interval $[0,1]$ is far from $\pm 2$, which are the only points where $\displaystyle\frac{d\mu_{\delta_1}^{H_0}}{dx}$ diverges; see \cite{Oliveira} for details on the spectral measure $\mu_{\delta_1}^{H_0}$. Although natural to specialists, we present a proof of this result for the convenience of the reader.} \end{remark}
\begin{proof} [{Proof} {\rm (Lemma~\ref{teclemma})}] Let $N\in\mathbb{N}\cup\{0\}$. It follows from Lemma 3.1 in \cite{Germinet} that there exists $D> 0$ such that for every $E \in [0,1]$,
\[\displaystyle\frac{d\mu_{\delta_1}^{N}}{dx}(E) \geq \frac{D}{\|T_N(E,N,0)\|^2}.\] Note that there exists $F_N>0$ such that for each $E\in[0,1]$, $\Vert T_N(E,N,0)\Vert<F_N$ (since $T_N(E,N,0)$ is the product of $N$ matrices whose norms are bounded); thus,
\[0 < C_{1,N} := \frac{D}{F_N^2} \leq \inf_{E \in [0,1]}\displaystyle\frac{d\mu_{\delta_1}^{N}}{dx}(E).\]
Now, if $n \geq N$, then $T_N(E,n+1,n) = T_0(E,n+1,n)=A(E)$, where
\begin{eqnarray*} A(E)=\left(\begin{array}{cc} E&-1\\1&0\end{array}\right).
\end{eqnarray*}
It is straightforward to show that for $E\in[0,1]$, $A(E)$ is similar to a rotation matrix; thus, there exists $C_N> 0$ such that for each $E \in [0,1]$ and each $n \geq N$, $\|T_N(E,n,0)\|^2 \leq C_N$. Indeed,
\[\|T_N(E,n,0)\|=\Vert T_N(E,n,N)\cdot \cdot \cdot T_N(E,N,0)\Vert\le \Vert A(E)^{n-N}\Vert\cdot\Vert T_N(E,N,0)\Vert;\] since $A(E)^{n-N}$ is similar to a rotation and since for each $E\in[0,1]$, $\Vert T_N(E,N,0)\Vert$ is bounded, the result follows.
Now, by Proposition 3.9 in \cite{Lasttransfermatrix}, one has for every $E \in [0,1]$ and every $L\in\mathbb{N}$,
\begin{eqnarray*} \mathfrak{Im} \biggr(\int_{\mathbb{R}} \frac{1}{E+i/L - z} d\mu_{\delta_1}^{N}(z) \biggr) \leq (5 + \sqrt{24}) \frac{1}{L} \sum_{n=0}^{L+1} \|T_N(E,n,0)\|^2. \end{eqnarray*} Thus, for every $E \in [0,1]$ and every $L\ge N$,
\begin{eqnarray*} \mathfrak{Im} \biggr(\int_{\mathbb{R}} \frac{1}{E+i/L - z} d\mu_{\delta_1}^{N}(z) \biggr) &\leq& (5 + \sqrt{24}) \frac{1}{L} \sum_{n=0}^{L+1} \|T_N(E,n,0)\|^2\\ &=& (5 + \sqrt{24}) \biggr( \frac{1}{L} \sum_{n=0}^{N} \|T_N(E,n,0)\|^2 + \frac{1}{L} \sum_{n=N+1}^{L+1} \|T_N(E,n,0)\|^2 \bigg)\\ &\leq& (5 + \sqrt{24}) \biggr( \frac{1}{L} \sum_{n=0}^{N} \|T_N(E,n,0)\|^2 + \frac{L-N}{L} C_N \bigg). \end{eqnarray*} By Stone's Formula, it follows that for each $E \in [0,1]$ \begin{eqnarray*} \frac{d\mu_{\delta_1}^{N}}{dx}(E) &=& \frac{1}{\pi} \lim_{ L \to \infty} \mathfrak{Im} \biggr\langle \delta_1, R^N\biggr(E+i\frac{1}{L}\biggr)\delta_1 \biggr\rangle = \frac{1}{\pi} \lim_{ L \to \infty} \mathfrak{Im} \biggr(\int_{\mathbb{R}} \frac{1}{E+i/L - z} d\mu_{\delta_1}^{N}(z) \biggr). \end{eqnarray*}
Since $\sum_{n=0}^{N} \|T_N(E,n,0)\|^2$ does not depend on $L$, one gets \[C_{2,N}:= \frac{C_N}{\pi}(5+\sqrt{24}) \geq \displaystyle\sup_{E \in [0,1]} \frac{d\mu_{\delta_1}^{N}}{dx}(E).\] \end{proof}
\subsection{Power-law singularities and quantum dynamics}
Recall that, for every $N \in \mathbb{N} \cup \{0\}$, ${\mathbb{R}} \ni t \mapsto e^{-itH_N}$ is a one-parameter strongly continuous unitary evolution group and, for each $\psi\in \ell^2({\mathbb{N}})$, $(e^{-itH_N}\psi)_{t \in \mathbb{R}}$ is the unique solution to the Schr\"odinger equation \[ \begin{cases} \partial_t \psi = -iH_N\psi, \quad t \in {\mathbb{R}}, \\ \psi(0) = \psi\in\ell^2(\mathbb{N}). \end{cases} \]
Now, we present a dynamical quantity usually considered to probe the large time behavior of $e^{-itH_N}\psi$, the so-called (time-average) {\em quantum return probability}, which gives the (time-average) probability of finding the particle at time $t>0$ in its initial state $\psi$: \begin{equation*}
\frac{1}{t}\int_0^t |\langle \psi, e^{-isH_N} \psi \rangle|^2 \, ds. \end{equation*}
For $0 \leq \beta < 1$ and every $f \in {\mathrm L}^1(\mathbb{R})$, let
\[K_{\beta,f} = \{(|\cdot|^{-\beta} \chi_{(0,1]}) \ast f\}.\] Suppose that $f \geq 0$ and set \[\psi_{\beta,f} := \sqrt{K_{\beta,f}}(H_N)\delta_1,\] where each $\sqrt{K_{\beta,f}}(H_N): {\mathrm{dom\,}} (\sqrt{K_{\beta,f}}(H_N)) \subset \ell^2({\mathbb{N}})\rightarrow \ell^2({\mathbb{N}})$ is defined through the functional calculus: for every $\psi\in {\mathrm{dom\,}} (\sqrt{K_{\beta,f}}(H_N))$, one has
\[\langle\psi,\sqrt{K_{\beta,f}}(H_N)\psi\rangle=\int \sqrt{K_{\beta,f}(x)}\,d\mu^{H_N}_\psi(x).\]
Note that $\delta_1 \in {\mathrm{dom\,}} (\sqrt{K_{\beta,f}}(H_N))$, since $\sqrt{K_{\beta,f}} \in {\mathrm L}^2(\mathbb{R},d\mu^{H_N}_{\delta_1})$; if $\displaystyle\int_0^1 f(x) \, dx = 1$, then for every $x \in (0,1)$, $\displaystyle\lim_{\beta \downarrow 0} \sqrt{K_{\beta,f}}(x) = 1$, by dominated convergence; thus, \[\displaystyle\lim_{\beta \downarrow 0} \psi_{\beta,f} = \delta_1.\] Our next result describes the behavior of the (time-average) {\em quantum return probability} of the initial states $\psi_{\beta,f}$.
\begin{theorem}\label{maintheorem} Let $\psi_{\beta,f}$ be as above, with $0\le f \in {\mathrm L}^1(\mathbb{R})$. Then: \begin{enumerate}
\item[{\rm i)}] if $0\leq \beta < \frac{1}{2}$, for every $t>0$, \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-isH_N} \psi_{\beta,f} \rangle|^2 \, ds \leq \frac{20\pi \|f\|_{{\mathrm L}^1(\mathbb{R})}^2}{(1-2\beta)} \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2 t^{-1}; \end{eqnarray*}
\item[{\rm ii)}] if $\frac{1}{2}< \beta <1$, for every $t>0$, \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-isH_N} \psi_{\beta,f} \rangle|^2 \, ds &\leq& \frac{\Gamma(\beta - 1/2) 2^{18} e^{2\pi} {M_\beta} \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2 }{(2\pi)^{2(\beta-1)}} \; t^{-2(1-\beta)}; \end{eqnarray*}
\item[{\rm iii)}] if $\beta = \frac{1}{2}$, for every $t>0$, \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-isH_N} \psi_{\beta,f} \rangle|^2 \, ds &\leq& 4 \pi^2 e^{2\pi} \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2\biggr[ \frac{1}{t} + M_{\frac{1}{2}} \frac{\Gamma(0, 16\pi^4/t^2)}{t} \biggr], \end{eqnarray*} in particular, for sufficiently large $t$ \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-isH_N} \psi_{\beta,f} \rangle|^2 \, ds &\leq& 4 \pi^2 e^{2\pi} \|f\|_{{\mathrm L}^1(\mathbb{R})}^2 \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2\biggr[ \frac{1}{t} + 3M_{\frac{1}{2}} \frac{\log(t)}{t} \biggr]. \end{eqnarray*} \end{enumerate} \end{theorem}
\begin{remark}{\rm By applying Theorem \ref{2Stheorem}, in general, one can extend the above result to families of Schr\"odinger operators whose Radon-Nikodym derivatives of each spectral measure (with respect to Lebesgue measure) is bounded (see the proof of Theorem~\ref{maintheorem}).} \end{remark}
We revisit Example \ref{ex1}, but now taking into account Theorem~\ref{maintheorem}.
\begin{example}\label{ex2} {\rm Let $\frac{1}{2} \leq \beta<1$ and $0<\delta<1-\beta$. Let $f_\delta \in {\mathrm L}^1(\mathbb{R})$ with
$\displaystyle\int_0^1 f_\delta(x) dx =1$ and suppose that there exists $C>0$ so that, for every $w \in (0,1]$, \[f_\delta(w) \geq \frac{C}{w^{1-\delta}}.\] Hence, for every $0< w \leq 1$, \begin{eqnarray*}
K_{\beta,f_\delta}(w) &\geq& C\int_0^1 \frac{1}{y^\beta |w-y|^{1-\delta}} \, dy \geq C\int_0^w \frac{1}{y^\beta |w-y|^{1-\delta}} \, dy \geq \frac{C}{w^\beta} \int_0^w \frac{1}{|w-y|^{1-\delta}} \, dy \geq C w^{- \beta+\delta}. \end{eqnarray*} Thus, for every $0< \epsilon < 1$, \[ \mu_{\psi_{\beta,f_\delta}}((0,\epsilon))=\int_0^\epsilon K_{\beta,f}(w) \frac{d\mu_{\delta_1}^{N}}{dx}(w) \, dw \geq C C_{1,N} \int_0^\epsilon w^{-\beta+\delta} dw = C C_{1,N} \frac{\epsilon^{(1-\beta+ \delta)} }{(1-\beta+ \delta)},\] with $C_{1,N}$ given by Lemma \ref{teclemma}. Therefore, in this case, \[ d\mu_{\psi_{\beta,f_\delta}} = K_{\beta,f_\delta} \frac{d\mu_{\delta_1}^{N}}{dx} \, dx\] is at most {\rm U}$(1-{\beta}+ \delta)${\rm H}, and so, by Theorem~\ref{Strichartztheorem}-i), one can say at most that
\begin{eqnarray*}\frac{1}{t}\int_0^t |\langle \psi_{\beta,f_\delta}, e^{-isH_N} \psi_{\beta,f_\delta} \rangle|^2 \,ds = O(t^{-(1-\beta)-\delta}). \end{eqnarray*} Nonetheless, it follows from Theorem \ref{maintheorem} that, for every $\frac{1}{2} < \beta < 1$, \begin{equation}\label{eqex2}
\frac{1}{t}\int_0^t \vert\langle \psi_{\beta,f_\delta}, e^{-isH_N} \psi_{\beta,f_\delta} \rangle\vert^2 \, ds = O(t^{-2(1-\beta)}) \end{equation} and \begin{equation}\label{eq9999}
\frac{1}{t}\int_0^t \vert\langle \psi_{1/2,f_\delta}, e^{-isH_N} \psi_{1/2,f_\delta} \rangle\vert^2 \, ds = O(\log(t)/t)) \end{equation}
We observe that the above rates are power-law optimal. Namely, if there exists $\varepsilon>0$ so that one can replace $t^{-2(1-\beta)}$ by $t^{-2(1-\beta)+\varepsilon}$ in~\eqref{eqex2} or (\ref{eq9999}), then by Theorem \ref{Strichartztheorem}-ii), $\psi_{\beta,f_\delta}$ will be at least uniformly $(1-\beta+\varepsilon/4)$-H\"older; for $0<\delta< \varepsilon/4$, one gets a contradiction.} \end{example}
\begin{remark}{\rm For each $0<\delta<1-\beta$, Example \ref{ex2} presents a family of initial states, $\psi_{\beta,f_\delta}$, such that $\displaystyle\lim_{\beta \downarrow 0} \psi_{\beta,f_\delta} = \delta_1$ and for which the correspondent (time-average) quantum return probabilities depend continuously on the power-law growth rates of the singularities of the respective spectral measures.} \end{remark}
\begin{proof}[{Proof} {\rm (Theorem~\ref{maintheorem})}] i) It follows from the Spectral Theorem that for every $t>0$, \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_\beta, e^{-isH_N} \psi_\beta \rangle|^2 \, ds &=& \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-isy} K_{\beta,f}(y) d\mu_{\delta_1}^{N}(y) \bigg|^2 ds \\ &=& \frac{2\pi}{t} \int_0^{\frac{t}{2\pi}} \bigg|\int_{\mathbb{R}} e^{- 2\pi isy} K_{\beta,f}( y) \frac{d\mu_{\delta_1}^{N}}{dx}(y)dy \bigg|^2 ds. \end{eqnarray*} Since $0\leq \beta < \frac{1}{2}$, by Young's Convolution Inequality and Lemma \ref{teclemma}, $K_{\beta,f} \frac{d\mu_{\delta_1}^{N}}{dx} \in {\mathrm L}^2(\mathbb{R})$. Thus, by Theorem \ref{Strichartztheorem}-i) applied to $d\mu=\chi_{[0,1]} dy$ and to the function $K_{\beta,f}( y) \frac{d\mu_{\delta_1}^{N}}{dx}( y)$, one obtains, for every $t>0$, \begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-itH_N} \psi_{\beta,f} \rangle|^2 \, ds &=& \frac{2\pi}{t} \int_0^{\frac{t}{2\pi}} \bigg|\int_{\mathbb{R}} e^{- 2\pi isy} K_{\beta,f}( y) \frac{d\mu_{\delta_1}^{N}}{dx}(y)dy \bigg|^2 ds \\ &\leq& 10 \|K_{\beta,f}\|_{{\mathrm L}^2(\mathbb{R})}^2 \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2 2\pi t^{-1} \\
&\leq &\frac{20\pi \|f\|_{{\mathrm L}^1(\mathbb{R})}^2}{(1-2\beta)} \biggr\|\frac{d\mu_{\delta_1}^{N}}{dx}\biggr\|_{{\mathrm L}^\infty[0,1]}^2 t^{-1}. \end{eqnarray*} We remark that if $d\mu = \chi_{[0,1]} dx$, then one can choose $C_\mu =10$ in Theorem \ref{Strichartztheorem}-i); for details, see page~416 in \cite{Last}.
\noindent ii) and iii) Let $\frac{1}{2}\leq \beta < 1$; since for every $t>0$,
\begin{eqnarray*}
\frac{1}{t}\int_0^t |\langle \psi_{\beta,f}, e^{-isH_N} \psi_{\beta,f} \rangle|^2 \, ds = \frac{2\pi}{t} \int_0^{\frac{t}{2\pi}} \bigg|\int_{\mathbb{R}} e^{-2\pi isy} d\mu_{\beta,f,g}(y) \bigg|^2 ds, \end{eqnarray*} where $g = \displaystyle\frac{d\mu_{\delta_1}^{N}}{dx}$, ii) and iii) are direct consequences of Lemma \ref{teclemma} and Theorem \ref{2Stheorem}. \end{proof}
\section{Proof of Theorem \ref{2Stheorem}}\label{sectProofMain}
\begin{lemma}\label{mainlemma} Let $ K_{\beta,f}$ be as before.
\begin{enumerate} \item[{\rm i)}] If $\frac{1}{2} < \beta < 1$, then, for every $t>0$, one has
\begin{eqnarray*}
\bigg| \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \overline{K_{\beta,f}} (y) \, dx dy \bigg| \; \leq \; {M_\beta} \Gamma(\beta - 1/2) \|f\|_{{\mathrm L}^1(\mathbb{R})}^2\; t^{-2(1-\beta)}. \end{eqnarray*}
\item[{\rm ii)}] If $\beta = \frac{1}{2}$, then, for every $t>0$, one has
\begin{eqnarray*}
\bigg| \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \overline{K_{\beta,f}} (y) \, dx dy \bigg| \; \leq \; \frac{\pi\|f\|_{{\mathrm L^1}(\mathbb{R})}^2}{t} + M_{\frac{1}{2}} \|f\|_{{\mathrm L^1}(\mathbb{R})}^2 \frac{\Gamma(0, \pi/t^2)}{t}, \end{eqnarray*} and recall that $M_\beta$ is given by~\eqref{eqMbeta}. \end{enumerate} \end{lemma}
\begin{proof} Let $\frac{1}{2} \leq \beta <1$ and let $(K_n)_{n \in \mathbb{N}} \subset {\mathrm L}^1(\mathbb{R}) \cap {\mathrm L}^2(\mathbb{R})$ be so that $\displaystyle\lim_{n \to \infty}\|K_n - K_{\beta,f}\|_{{\mathrm L}^1(\mathbb{R})} = 0$. Then, by Theorem 4.9 in \cite{Brezis}, there exist a subsequence $(K_{n_k})$ and a function $h \in {\mathrm L}^1(\mathbb{R})$ such that $\displaystyle\lim_{k \to \infty} K_{n_k}(x) = K_{\beta,f}(x)$ for almost every $x \in \mathbb{R}$, and for every $k \geq 1$, $ |K_{n_k}(x)| \leq h(x)$ for almost every $x \in \mathbb{R}$. We note that for each $t >0$, each $k \geq 1$ and each $\xi \in \mathbb{R}$,
\[ e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{n_k}}(\xi)| \leq e^{- \frac{\pi |\xi|^2}{t^2}} \|K_{n_k}\|_{{\mathrm L}^1(\mathbb{R})} \leq e^{- \frac{\pi |\xi|^2}{t^2}} \|h\|_{{\mathrm L}^1(\mathbb{R})},\] and for each $t >0$,
\[\int_{\mathbb{R}} e^{- \frac{\pi |\xi|^2}{t^2}} d\xi = t.\]
This show that, for every $t>0$, the sequence $e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{n_k}}(\xi)|$ is dominated by a integrable function.
Set, for each $t>0$ and each $x \in \mathbb{R}$, $\Phi_t(x) := e^{-\pi t|x|^2}$. Then, for each $t>0$, \begin{equation}\label{eq0lemma}
\widehat{\Phi_t}(\xi) = \frac{1}{t} e^{- \frac{\pi |\xi|^2}{t^2}}, \quad \xi \in \mathbb{R}. \end{equation}
It follows from the identity in (\ref{eq0lemma}), some basic properties of the Fourier transform, dominated convergence and Plancherel's Theorem that for each $y \in \mathbb{R}$ and each $t >0$,
\begin{eqnarray}\label{eq00lemma}\nonumber \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \, dx &=& \lim_{k \to \infty} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{n_k} (x) \, dx \nonumber = \lim_{k \to \infty} \int_{\mathbb{R}} \overline{(\tau_y \Phi_t)(x)} K_{n_k} (x)\, dx \\ \nonumber &=& \lim_{k \to \infty} \int_{\mathbb{R}} \overline{\widehat{(\tau_y \Phi_t)}(\xi)} \widehat{K_{n_k}}(\xi) \, d\xi = \lim_{k \to \infty} \int_{\mathbb{R}} e^{2\pi i y \xi} \widehat{\Phi_t}(\xi) \widehat{K_{n_k}}(\xi) \, d\xi \\ \nonumber &=& \lim_{k \to \infty} \frac{1}{t} \int_{\mathbb{R}} e^{2\pi i y \xi} e^{- \frac{\pi |\xi|^2}{t^2}} \widehat{K_{n_k}}(\xi) \, d\xi\\ &=& \frac{1}{t} \int_{\mathbb{R}} e^{2\pi i y \xi} e^{- \frac{\pi |\xi|^2}{t^2}} \widehat{K_{\beta,f}}(\xi) \, d\xi, \end{eqnarray} where $\tau_yf(\cdot) = f(\cdot-y)$ stands for the translation by $y\in\mathbb{R}$. Thus, by Fubini's Theorem, one obtains for each $t >0$,
\begin{eqnarray}\label{eq1lemma} \nonumber \int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \overline{K_{\beta,f}} (y) \, dx dy &=& \frac{1}{t} \int_{\mathbb{R}} \int_{\mathbb{R}} e^{2\pi i y \xi} \overline{K_{\beta,f}} (y) \, dy \, e^{- \frac{\pi |\xi|^2}{t^2}} \widehat{K_{\beta,f}}(\xi) \, d\xi\\ &=& \frac{1}{t} \int_{\mathbb{R}} e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi. \end{eqnarray}
Now, by the Convolution Theorem, it follows that for each $\xi >0$, \begin{eqnarray}\label{eq88888} \nonumber \widehat{K_{\beta,f}}(\xi) &=& \biggl\{\int_0^1 e^{-2\pi i x \xi } x^{-\beta} dx \biggl\} \, \hat{f}(\xi) = \biggl\{\frac{1}{(2\pi)^{1-\beta}\xi^{1-\beta}} \int_0^1 e^{-2\pi i x \xi } (2 \pi x \xi)^{-\beta} (2\pi \xi)dx \biggl\} \, \hat{f}(\xi)\\ &=& \biggl\{\frac{1}{(2\pi)^{1-\beta}\xi^{1-\beta}} \int_0^{2\pi\xi} e^{- iu} u^{-\beta} du \biggl\} \, \hat{f}(\xi). \end{eqnarray} One also has, for each $\xi <0$, that \begin{eqnarray*} \widehat{K_{\beta,f}}(\xi) = \biggl\{\frac{1}{(2\pi)^{1-\beta}(-\xi)^{1-\beta}} \overline{\int_0^{-2\pi\xi} e^{-iu} u^{-\beta} du} \biggl\} \, \overline{\hat{f}(\xi)}. \end{eqnarray*} Thus, for $\xi \neq 0$,
\begin{eqnarray}\label{eq2lemma} |\widehat{K_{\beta,f}}(\xi)|^2 \leq {M_\beta} \, \frac{\|f\|^2_{{\mathrm L^1}(\mathbb{R})}}{|\xi|^{2(1-\beta)}}. \end{eqnarray}
Now a separate argument is necessary for each item. i) $1/2<\beta<1$. Since it follows from Cauchy's Residue Theorem that for every $t >0$ \begin{equation}\label{eq01}
\int_{\mathbb{R}} \frac{e^{- \frac{\pi |\xi|^2}{t^2}} }{|\xi|^{2(1-\beta)} } \, d\xi = \pi^{1/2-\beta}\Gamma\biggr(\beta - \frac{1}{2}\biggr) t^{2\beta-1} \leq \Gamma\biggr(\beta - \frac{1}{2}\biggr) t^{2\beta-1} , \end{equation} one gets from (\ref{eq1lemma}) and (\ref{eq2lemma}) that for every $t >0$, \begin{eqnarray*}
\int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \overline{K_{\beta,f}} (y) \, dx dy &=& \frac{1}{t} \int_{\mathbb{R}} e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi \\ &\leq& \frac{{M_\beta} \|f\|_{{\mathrm L^1}(\mathbb{R})}^2}{t} \int_{\mathbb{R}} \frac{e^{- \frac{\pi |\xi|^2}{t^2}} }{|\xi|^{2(1-\beta)} } \, d\xi\\ &\leq& \Gamma(\beta - 1/2) {M_\beta} \|f\|_{{\mathrm L^1}(\mathbb{R})}^2 t^{-2(1-\beta)}. \end{eqnarray*}
ii) $\beta=1/2$. By (\ref{eq88888}), for every $t> 0$
\begin{eqnarray*} \int_{-1}^1 e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi &\leq& 2\|f\|_{{\mathrm L^1}(\mathbb{R})}^2 \int_0^1 e^{- \frac{\pi |\xi|^2}{t^2}} \frac{1}{\xi} \biggr(\int_0^{2\pi\xi} \frac{1}{\sqrt{u}} \, du \biggr)^2 \, d\xi \\ &=& \pi \|f\|_{{\mathrm L^1}(\mathbb{R})}^2 \int_{0}^1 e^{- \frac{\pi |\xi|^2}{t^2}} \, d\xi \leq \pi\|f\|_{{\mathrm L^1}(\mathbb{R})}^2. \end{eqnarray*} Since, by Cauchy's Residue Theorem, for every $t >0$
\[\int_1^\infty \frac{e^{- \frac{\pi |\xi|^2}{t^2}} }{\xi} \, d\xi = \frac{\Gamma(0, \pi/t^2)}{2},\] one gets from (\ref{eq1lemma}) and (\ref{eq2lemma}) that for every $t >0$, \begin{eqnarray*}
\int_{\mathbb{R}} \int_{\mathbb{R}} e^{-\pi t^2|x-y|^2} K_{\beta,f}(x) \overline{K_{\beta,f}} (y) \, dx dy &=& \frac{1}{t} \int_{\mathbb{R}} e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi \\ &=& \frac{1}{t} \int_{-1}^1 e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi + \frac{1}{t} \int_{|x|> 1} e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi\\ &\leq& \frac{1}{t} \int_{-1}^1 e^{- \frac{\pi |\xi|^2}{t^2}} |\widehat{K_{\beta,f}}(\xi)|^2 \, d\xi + \frac{ 2M_{\frac{1}{2}} \|f\|_{{\mathrm L^1}(\mathbb{R})}^2}{t} \int_1^\infty \frac{e^{- \frac{\pi |\xi|^2}{t^2}} }{\xi} \, d\xi\\ &\leq& \frac{\pi\|f\|_{{\mathrm L^1}(\mathbb{R})}^2}{t} + M_{\frac{1}{2}} \|f\|_{{\mathrm L^1}(\mathbb{R})}^2 \frac{\Gamma(0, \pi/t^2)}{t}. \end{eqnarray*} \end{proof}
\begin{remark}{\rm Since the integral $\displaystyle\int_{\mathbb{R}} \frac{e^{- \frac{\pi |\xi|^2}{t^2}} }{|\xi| } \, d\xi$ does not converge, a separate argument is necessary therein for the case $\beta = \frac{1}{2}$.} \end{remark}
\begin{proof} [{Proof} {\rm (Theorem~\ref{2Stheorem})}] Note that the proof of Theorem~\ref{2Stheorem} is a consequence of Lemma~\ref{mainlemma} and Fubini's Theorem. We present details of the proof of item~i): by the linearity of the convolution product, we may assume without loss of generality that $\|f\|_{{\mathrm L^1}(\mathbb{R})} \leq 1 $ and $ \|g\|_{{\mathrm L}^\infty[0,1]} \leq 1$. We divide this proof into two cases.
\
\noindent {\bf Case 1:} $f,g$ are nonnegative real-valued functions. By Fubini's Theorem, one has, for every $t>0$,
\begin{eqnarray}\label{maineq1}
\nonumber \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 ds &\leq& \frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 \, e^{2\pi-(2\pi s)^2/t^2} \, ds\\ \nonumber &\leq& \frac{e^{2\pi}}{t} \int_{-\infty}^{\infty} \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 \, e^{-(2\pi s)^2/t^2} \, ds\\ \nonumber &=& \frac{e^{2\pi}}{t} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x) g(x) \overline{K_{\beta,f}(y)} \, \overline{g(y)}\\ \nonumber &\times& \biggl\{ \int_{-\infty}^{\infty} \, e^{-((2\pi s)^2/t^2)-2\pi is(x-y)} \, ds \biggl\} dxdy\\ \nonumber &=& \frac{e^{2\pi} \sqrt{\pi}}{2 \pi} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x) g(x)\, K_{\beta,f}(y) g(y) e^{-\frac{t^2|x-y|^2}{4}} dx dy \\ \nonumber &\leq& \frac{e^{2\pi} \sqrt{\pi}}{2 \pi} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x) K_{\beta,f}(y) e^{-\frac{t^2|x-y|^2}{4}} dx dy \\ &=& \frac{e^{2\pi} \sqrt{\pi}}{2 \pi} \int_{\mathbb{R}} \int_{\mathbb{R}} K_{\beta,f}(x) K_{\beta,f}(y) e^{-\pi (t/2\sqrt{\pi})^2|x-y|^2} dx dy . \end{eqnarray}
It then follows from (\ref{maineq1}) combined with Lemma \ref{mainlemma} i) that, for every $t>0$, \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 ds &\leq& \frac{\Gamma(\beta - 1/2) e^{2\pi} {M_\beta} \sqrt{\pi}}{2^{2\beta-1} \pi^\beta}\; t^{-2(1-\beta)}. \end{eqnarray*}
\noindent {\bf Case 2:} $f,g$ are complex valued. This case is a direct consequence of {\bf Case 1}. Namely, by the linearity of the convolution product, by the inequality $(a+b)^2 \leq 2(a^2+b^2)$, $a,b>0,$ and by the identity \begin{eqnarray*} K_{\beta,f} \cdot g&=& \biggr\{K_{\beta,\mathfrak{Re}(f)^+} - K_{\beta,\mathfrak{Re}(f)^-} + i\biggr(K_{\beta,\mathfrak{Im}(f)^+} - K_{\beta,\mathfrak{Im}(f)^-}\biggr)\biggr\}\\ &\times& \biggr\{\mathfrak{Re}(g)^+ - \mathfrak{Re}(g)^- + i\biggr(\mathfrak{Im}(g)^+ - \mathfrak{Im}(g)^-\biggr)\biggr\}, \end{eqnarray*} it follows that, for every $t>0$, \begin{eqnarray*}
\frac{1}{t} \int_0^t \bigg|\int_{\mathbb{R}} e^{-2\pi isx} K_{\beta,f}(x) g(x) \, dx \bigg|^2 ds &\leq& \frac{\Gamma(\beta - 1/2) 2^{16}e^{2\pi} {M_\beta} \sqrt{\pi}}{2^{2\beta-1} \pi^\beta} \; t^{-2(1-\beta)}. \end{eqnarray*} \end{proof}
\begin{center} \Large{Acknowledgments} \end{center} \addcontentsline{toc}{section}{Acknowledgments}
\noindent M. Aloisio thank the partial support by CAPES (a Brazilian government agency; Finance Code 001). S. L. Carvalho thanks the partial support by FAPEMIG (Minas Gerais state agency; under contract 001/17/CEX-APQ-00352-17) and C. R. de Oliveira thanks the partial support by CNPq (a Brazilian government agency, under contract 303689/2021-8).
\noindent Email: moacir@ufam.edu.br, Departamento de Matem\'atica, UFAM, Manaus, AM, 369067-005 Brazil
\noindent Email: silas@mat.ufmg.br, Departamento de Matem\'atica, UFMG, Belo Horizonte, MG, 30161-970 Brazil
\noindent Email: oliveira@ufscar.br, Departamento de Matem\'atica, UFSCar, S\~ao Carlos, SP, 13560-970 Brazil
\noindent Email: edsonilustre@yahoo.com.br, Departamento de Matem\'atica, UFAM \& UEA, Manaus, AM, 369067-005 Brazil
\end{document} |
\begin{document}
\title{Coalitions of pulse-interacting dynamical units}
\author{Eleonora Catsigeras\thanks{Instituto de Matem\'{a}tica y Estad\'{\i}stica Rafael Laguardia (IMERL),
Fac. Ingenier\'{\i}a, Universidad de la Rep\'{u}blica, Uruguay.
E-mail: eleonora@fing.edu.uy
EC was partially supported by CSIC of Universidad de la Rep\'{u}blica and ANII of Uruguay.} }
\date{\today}
\maketitle
\begin{abstract}
We prove that large global systems of interacting (non necessarily similar) dynamical units that are coupled by cooperative impulses, recurrently exhibit the so called \em grand coalition \em, for which all the units arrive to their respective goals simultaneously. We bound from above the waiting time until the first grand coalition appears. Finally, we prove that if besides the units are mutually similar, then the grand coalition is the unique subset of goal-synchronized units that is recurrently shown by the global dynamics. \end{abstract}
{\noindent \footnotesize {{\em MSC }2010: {\ Primary: 37NXX, 92B20; Secondary: 34D06, 05C82, 94A17, 92B25} \\ \noindent {\em Keywords:}
{Pulse-coupled networks, interacting dynamical units, coalitions, synchronization}}}
\section{Introduction}
We study the global dynamics of a network $N$ composed by a large number $m $ of dynamical units that mutually interact by cooperative (i.e. positive) instantaneous pulses.
One of the most cited examples of the type of phenomena that we are contributing to explain mathematically along this work, is the large scale synchronization of the flashes of the fireflies \lq\lq Pteroptyx malaccae\rq\rq: a large number of individuals flash periodically all together after a waiting time, when they meet together on trees, with neither an external clock nor privileged individuals mastering the global synchronization \cite{ErmentroutFireflies}.
We are motivated on the study of the dynamics of such global systems to obtain abstract and very general mathematical results, that are independent of the concrete formulae governing the dynamics, and require very few hypothesis. They are applicable in particular to models used in Neuroscience for which more or less concrete formulae and hypothesis governing the individual dynamics of the neurons are assumed (see for instance \cite{Dale2,ErmentroutTerman2010,Izhikevich2007,MassBishop2001,stamov2007}).
The mathematical study of the global dynamics of abstract and general networks composed by mutually interacting units has a large diversity of concrete applications to other sciences and technology. As said above, they are widely used in Neuroscience. They have also applications to
Engineering, for instance in the design and construction of some systems used in communications \cite{YangChua,GlobalSyncSecureCommun2011}; also to Physics, for instance in the study of systems of light controlled oscillators \cite{LCOrecomMartiCabeza2_2003,CabezaMartiRubidoKahan2011}, and in the research of the evolution of physical lattices of coupled dynamical units of different nature \cite{ChazottesFernandez2005,WangSlotine}. They have other important applications to Biology, for instance in the research of mathematical models of genetic regulatory networks \cite{arnaud}; to Ecology, in the study of the equilibria of some ecological systems evolving on time \cite{ecologiaMutualInterference3_2010,ecology2009}; to Economy and other social sciences, in the research of coupled networks of different agents, individuals or coalitions of individuals, for instance by means of evolutive Game Theory \cite{miltaich,elvio2009}.
While not interacting with the other units of the network, each unit $i \in \{1, 2, \ldots, m\}$, which we also call \lq\lq cell\rq\rq, evolves governed by two rules that determine the \lq\lq free dynamics of $i$\rq\rq: the \em relaxation rule \em and the \em update rule, \em which we will precisely define in Subsection \ref{subsectionModel}. While the units are not interacting, the dynamics of the network is the product dynamics of its $m$ units, which evolve independently one from the other. But at certain instants, at least one unit $i$ changes the dynamical rules that govern the other units $j \neq i$. The instants when each unit $i$ acts on the others are exclusively determined by the state $x_i$ of $i$. The pulsed coupling hypothesis assumes that any action from $i$ to $j \neq i$ is a discontinuity jump in the instantaneous state of the cell $j$ according to the \em interactions rules \em which we will precisely define in Subsection \ref{subsectionModel}.
The free dynamics rules and the instantaneous interactions rules, as well as the mathematical results that we obtain from them, generalize to a wide context the particular cases that were studied for instance in \cite{MirolloStrogatz, Bottani1996,Cessac,jimenez2013,YoPierre}.
The results that we prove along the paper deal with the spontaneous formation of \em coalitions \em (subsets) of dynamical units during the dynamical evolution of the network, provided that the interactions among the units are all \lq\lq cooperative\rq\rq \ (i.e. positively signed). Roughly speaking, each coalition is a subset of units that synchronize certain milestones of their respective individual dynamics, which we call goals, and do that spontaneously without any external clock or master unit, infinitely many times in the future. In particular the formation of the so called \em grand coalition \em (i.e. the simultaneous arrival to a certain goal of all the units of the network) is spontaneously and recurrently exhibited from any initial state (Theorem \ref{theorem1}). The synchronization in the grand coalition was initially proved in 1992 by Mirollo and Strogatz \cite{MirolloStrogatz}, under restrictive hypothesis requiring that the units were identical, the interactions were also identical, and that the free dynamics of the units were one-dimensional oscillators whose evolution were linear on time. Later, in 1996, Bottani \cite{Bottani1996} proved the synchronization of the grand coalition requiring that the units were similar (non necessarily identical), but still one dimensional oscillators although their evolution were not necessarily linear on time. In Theorem \ref{theorem1} we will generalize the result to any network of non necessarily similar units with cooperative interactions that depend on the pair of interacting cells, with general free dynamics of each unit $i$, on any finite dimension (depending on $i$), and such that the cells do not necessarily behave as oscillators. The price to pay for such a general result is that the network has to be large enough, and, unless the units were mutually similar (Theorem \ref{theorem2}), the grand coalition is not necessarily the unique coalition that is exhibited recurrently in the future.
Due to the fact that the units may be very different and that the grand coalition is not necessarily the unique coalition that is exhibited in the future, the word \lq\lq synchronization\rq\rq \ in Theorem \ref{theorem1}, if applied, it is not in its classical meaning (\cite{Pikovsky}). In fact, the orbits of each of the units that recurrently exhibit the grand coalition, are not synchronized in the strict sense since they do not show the same state for all the instants $t \geq 0$. The states of two or more units may sensibly differ one from the others, at some instants between two consecutive formations of the grand coalition.
On the one hand, the synchronization in the strict or wide sense, for models of pulsed coupled dynamical units, were up to now proved for particular examples in which the free dynamics of each cell is governed by a differential equation or a discrete time mapping with \em a concrete formulae. \em For instance, the free dynamics is governed by affine mapping in \cite{Cessac}, by linear differential equations in \cite{LCOrecomMartiCabeza2_2003,CabezaMartiRubidoKahan2011}, and by piecewise contracting maps in \cite{WangSlotine} \cite{jimenez2013},\cite{YoPierre} or using known results about piecewise contractions in \cite{bremont}. In this sense, the novelty of the results here is that their proofs work independently of the concrete formulation of the free dynamics of the cells. They have almost no hypothesis about the second term of the differential equation governing the free dynamics of each of the cells.
On the other hand, the results along this paper hold independently of the dimension of the space $X_i$ where the state of each unit evolves, and they do not require the free dynamics of each unit to make it an oscillator. This freedom allows the results to be applied for instance to multidimensional chaotic free dynamics of the cells that recurrently shear certain milestones in the global collective dynamics (\cite{LSYoung2008,coombes2013JournMathBio}).
The paper is organized as follows: in Section \ref{section2} we state the mathematical definitions and theorems to be proved. In Section \ref{sectionProofs} we write the proofs.
\section{Definitions and statements of the results} \label{section2}
\subsection{Definitions and hypothesis} \label{subsectionModel}
\noindent{\bf The relaxation rule of the free dynamics of $i$:}
The relaxation rule of the free dynamics of the cell $i$ determines the evolution on time $t \geq 0$ of the state $x_i$ on a compact finite-dimensional manifold $X_i$ (whose dimension may depend of $i$). It is defined as the solution of any differential equation: \begin{equation} \label{eqnFreeDynamics} \frac{d x_i}{dt} = f_i(x_i), \ \ \ x_i \in X_i\end{equation}
satisfying just one condition as follows:
There exists a Lyapunov real function $S_i: X_i \mapsto \mathbb{R}$, which we call \em the satisfaction level \em of $i$, such that: \begin{equation} \label{eqnSatisfaction} \frac{d S_i(x_i(t))}{dt} = \bigtriangledown S_i (x_i(t)) \cdot f_i (x_i(t)) >v_i >0 \ \ \forall \ t \mbox{ such that } S_i(x_i (t)) < \theta_i,\end{equation} where $\theta_i$ is a positive constant (for each unit $i$) which we call the \em goal \em of $i$. (In formula (\ref{eqnSatisfaction}) $\bigtriangledown S_i\cdot f_i$ denotes the inner product in the tangent bundle of the manifold $X_i$).
In other words, the free dynamics of $i$ holds at all the instants for which $i$ is uncoupled to the network and its state is unchanged by interferences that may come from outside $i$. It is described by a finite dimensional variable $x_i$ evolving on time $t$ in such a way that the satisfaction level $S_i(x_i)$, while it does not reach the goal value $\theta_i$, is strictly increasing with $t$ and its (positive) velocity is bounded away from zero.
\noindent{\bf The update rule of the free dynamics of $i$:}
The update rule is a discontinuity jump in the state $x_i$ of the cell $i$ that is produced whenever the satisfaction variable $S_i(x_i(t))$ reaches (or is larger than) the goal level $\theta_i$. This discontinuity jump instantaneously resets the satisfaction level $S_i(x_i(t))$ to a \lq\lq reset value\rq\rq, which is strictly smaller than $\theta_i$. With no loss of generality, we assume that the reset value is zero (see Figure \ref{figure1}). Precisely:
\begin{equation} \label{eqnResetRule} S_i(x_i(t_0^-)) \geq \theta_i \ \ \Rightarrow \ \ S_i(x_i(t_0)) = 0,\end{equation}
where $S_i(x_i(t_0^-))$ denotes $ \lim_{t \rightarrow t_0^-} S_i(x_i(t))$.
Note that the alternation between the relaxation and update rules of the free dynamics of $i$ will occur while no interferences come from outside $i$ forcing its satisfaction variable to decrease (see Figure \ref{figure1}). Nevertheless, the free evolution $S_i(x_i(t))$ is not necessarily periodic if $\mbox{dim}(X_i) \geq 2$. In fact, the set $S_i^{-1}(\{0\}) \subset X_i$ of states with constant null satisfaction may be for instance a curve: there may exist infinitely many points in $X_i$ for which $S_i= 0$. So, each state $x_i(t)$ obtained from the reset rule $S_i(x_i(t)) = 0$ from the goal $S_i(x_i(t^-)) = \theta_i$, does not necessarily repeat in the future to make the evolution $S_i(x_i(t))$ periodic with an exact time-period. On the contrary, if the set of all the possible reset states $x_i \in S_i^{-1}(\{0\}) $ were finite (this can occur even if $S_i^{-1}(\{0\})$ is infinite), then the free dynamics of $i$ would make it be periodic, i.e. an oscillator.
\begin{definition}
{\bf (Spikes)} \em
Taking the name from Neuroscience, we call \em spike \em of the cell $i$ to the discontinuity jump of its satisfaction state from the goal value $\theta_i$ (which in Neuroscience is called \lq\lq threshold level\rq\rq) to its reset value (which is assumed to be zero). Note that the instants when each cell $i$ spikes, while not interacting with the other units of the network, are defined just by the value of its own satisfaction variable. There is neither an external clock nor a master unit in the network to force a synchronization of the spikes of the many cells of the network.
\end{definition}
{\begin{figure}
\caption{ The evolution on time $t$ of the satisfaction variable $S(x(t))$ of a dynamical unit while not interacting with the other units of the network.}
\label{figure1}
\end{figure}}
\noindent{\bf The interactions rules among the units}
Now, let us define the rules that govern the mutual interactions among the units, to compose a global dynamical system which we call network $N$. Consider a system composed by $m \geq 2$ dynamical units with the free dynamics as described above.
\begin{definition} {\bf (Spiking instants and inter-spike intervals)}
\label{definitionSpiking instants} \em We denote by $\{t_n\}_{n \geq 0}$ the sequence of instants $0 \leq t_n < t_{n+1} $ for which at least one cell of the system spikes. We call $t_n$ the \em $n$-th. spiking instant \em of the global system.
We call $(t_{n+1}, t_n) $ the \em $n$-th. inter-spike interval of the global system. \em
\end{definition}
First, by hypothesis, the interactions among the units of the global system are produced only at the spiking instants. In other words, during the inter-spike intervals the cells evolve independently one from the others. Hence, the dynamics of the global system along the inter-spike time intervals is the product dynamics of those of its units.
Second, at each instant $t_n$ the possible action from a cell $i$ to $j \neq i$ is weighted by a real number $\Delta_{ij}$. The interactions in the network are represented by the edges of a finite graph, whose vertices are the cells $i \in \{1, \ldots, m\}$ and whose edges $(i,j)$ are oriented and weighted by $\Delta_{i,j}$ respectively (see Figure \ref{figure2}). We call $ \Delta_{i,j} $ the interaction weight. We say that the graph of interactions is \em complete \em if $\Delta_{i,j} \neq 0$ for all $i \neq j$.
Third and finally, the satisfaction value of any cell $j$, at any spiking instant $t_n$ is defined by the following rule:
\begin{equation} \label{eqnInteraction} S_j(x_j(t_n)) = S_j(x_j(t_n^-)) + \sum_{i \in I(t_n), i \neq j} \Delta_{ij} \ \ \ \ \mbox{ if } \ \ \ \ S_j(x_j(t_n^-)) + \sum_{i \in I(t_n), i \neq j} \Delta_{ij} < \theta_j, \end{equation} $$ S_j(x_j(t_n))= 0 \mbox{ otherwise, }$$
where $I(t_n)$ is the set of neurons that spike at instant $t_n$, and $\Delta_{i,j}$ are the interactions weights. \begin{definition}\label{definitioncoalition}{\bf (Coalition)}
\em We call the set $I(t_n)$ \em the coalition \em at the spiking instant $t_n$. A coalition $I$ is a singleton if $\#I = 1$. From the definition of the spiking instant, no coalition is empty. \end{definition}
{\begin{figure}
\caption{ The graph of interactions of a global system of instantaneously coupled units $1, 2, \ldots, 5$. The oriented and nonzero weighted edges are denoted by $\Delta_{ij}$.}
\label{figure2}
\end{figure}}
If the interactions weights $\Delta_{i,j} $ are all positive and large enough, the coalition $I(t_n)$ may be the result of an avalanche process that makes more and more cells spike at the same instant $t_n$ when at least one cell spikes. In fact, we can always decompose $I(t_n) $ as the following union of pairwise disjoint (maybe empty) subsets of cells:
$$I(t_n) = \bigcup_{p \geq 0} I_p(t_n),$$
where
$I_0(t_n)$ is the set of cells $i$ such that $x_i(t_n^-) = \theta_i$, and for all $p \geq 1$, the set $I_{p}(t_n)$ is composed by the cells $j \not \in \cup_{k= 0}^{p-1} I_k(t_n)$ such that $x_j(t_n^-) + \sum_{k= 0}^{k= p-1} \sum_{i \in I_k(t_n)} \Delta_{ij} \geq \theta_{j}. $
\begin{definition} {\bf (Cooperative and antagonist cells)}
\label{definitionCooperative} \em
A cell $i$ is called \em cooperative \em if $\Delta_{ij} \geq 0$ for all $j \neq i$. It is called \em antagonist \em if $\Delta_{ij} \leq 0$ for all $j \neq i$. It is called \em mixed \em if it is neither cooperative nor antagonist.
\end{definition}
In Figure \ref{figure3} we draw the evolution on time of the satisfaction variables of two interacting dynamical units: one of the units is cooperative and the other is antagonist.
{\begin{figure}
\caption{ Evolution on time $t$ of the satisfaction variable of two interacting units. One cell is cooperative and the other is antagonist.}
\label{figure3}
\end{figure}}
From the rule (\ref{eqnInteraction}), when a cooperative cells spikes, it helps the other cells to increase the values of their respective satisfaction variables, so it shortens the time that the others must wait to arrive to their respective goals. On the contrary, an antagonist cell diminishes the values of the satisfaction variables of the other cells, opposing to them and enlarging the time that the others must wait to arrive to their goals.
Experimentally in Neuroscience, the nervous system of animals rarely show the existence of mixed cells. This is a reason why one usually assumes the so called Dale's Principle \cite{Dale1, Dale2}: any cell in the network is either cooperative or antagonist. In \cite{yoDale} abstract mathematical reasons that support Dale's principle were proved: it is a necessary condition for a maximum dynamical richness in the network. Precisely, the amount of information that the network can exhibit along its temporal evolution in the future acquires its maximum restricted to a constant number of nonzero interactions, only if Dale's principle holds.
Along this work we focuss on the global dynamics of networks that are composed by cooperative cells and that have a complete graph of interactions.
\noindent{\bf The global state and the vectorial satisfaction variable}
We denote by $${\bf{x}}(t) = (x_1(t), \ldots, x_m(t)) \in \prod_{i= 1}^m X_i$$ the state of the global system at instant $t \geq 0$. We denote by $${\bf S}({\bf{x}}(t)) = (S_1(x_1(t)), \ldots, S_m(x_m(t))) \in \mathbb{R}^m$$ the vectorial satisfaction variable of the global system at instant $t$. We consider the cube $$Q := \prod_{i= 1}^{m} [0, \theta_i) \subset \mathbb{R}^m.$$ From the hypothesis of the free dynamics of the cells and of the mutual interactions, if all the cells are cooperative then $${\bf S}({\bf{x}}(t)) \in Q \ \ \forall \ t \geq 0$$ provided that \begin{equation} \label{eqnInitialInQ} {\bf{x}}(0) \in {\bf S} ^{-1}(Q).\end{equation} Along this paper we will assume condition (\ref{eqnInitialInQ}). This assumption is not a restriction for the study of all the orbits of the global autonomous system. In fact, if ${\bf S} ({\bf x}(0)) \not \in Q$, then, applying the inequality (\ref{eqnSatisfaction}) and the reset rule (\ref{eqnResetRule}, and taking into account that the the interactions are non negative, we deduce that there exists a minimum positive instant $t_0$ such that ${\bf S} ({\bf x}(t_0)) \in Q$. So, translating the origin of the time axis to $t_0$, we have reduced the problem to the case for which the vectorial satisfaction value initially belongs to $Q$.
\begin{definition} {\bf (Grand coalition)}
\label{definitionGrandCoalition} \em
We call $I(t_n)$, defined in \ref{definitioncoalition}, the \em grand coalition \em if all the cells of the system spike at instant $t_n$. Namely, the grand coalition is exhibited at instant $t_n$ if $I(t_n) = \{1, 2, \ldots, m\}$.
\end{definition}
\begin{definition} {\bf (Waiting time)}
\label{definitionWaitingTime} \em If from some initial state of the global system the grand coalition is exhibited at some spiking instant $t_n \geq 0$, we call the minimum such an instant \em the waiting time \em until the grand coalition occurs. Note that in general, if existing, the finite waiting time depends on the initial state.
\end{definition} \noindent {\bf Weak interactions:} We will not need to assume the following condition (\ref{eqnWeakInteractions}) as an hypothesis. So, it is not an assumption in any part of this paper. Nevertheless, we pose condition (\ref{eqnWeakInteractions}) just because some of the theorems that we will prove along the work become more interesting for networks that satisfy it: \begin{equation} \label{eqnWeakInteractions}
\max_{i \neq j} |\Delta_{ij}| \ll \min_i \theta_i,\end{equation} where $\ll$ denotes \lq\lq much smaller than\rq\rq. For instance, one may be interested in considering $a \ll b$ (where $0< a < b$) if $a/b <10^{-3} $. Condition (\ref{eqnWeakInteractions}) says that the interactions weights are relatively very weak.
\begin{definition}
\label{definitionLargeNetwork} \em {\bf (Large networks)}
Let $N$ be a network composed by $m$ cooperative units, as described above. We say that $N$ is \em large enough \em in relation to the cooperative interactions if the following inequality holds:
\begin{equation}
\label{eqnLargeInteractions}
\sqrt m \geq 1 +\frac{\max_i \theta_i}{\min_{i \neq j} \Delta_{ij}}.
\end{equation}
Note that, inequality (\ref{eqnLargeInteractions}) implies that the graph of interactions is complete. In fact $ \Delta_{ij} \geq 0$ for all $i \neq j$ because the cells are all cooperative, but $$ \Delta_{ij} \neq 0 \ \forall \ i \neq j $$ to make the minimum in formula (\ref{eqnLargeInteractions}) be nonzero and make this formula hold for a finite value of $m$.
\end{definition}
\subsection{Statements of the results} \label{subsectionStatements}
The purpose of this paper is to prove the following results:
\begin{theorem} \label{theorem1} If the network is cooperative and large enough, then from any initial state the grand coalition is exhibited infinitely many times in the future.
\end{theorem}
\begin{theorem} \label{theorem1waitingTime}
If the network is cooperative and large enough, then from any initial state in ${\bf S}^{-1}(Q)$ the waiting time $t_{n_0}$ before the grand coalition appears for the first time is upper bounded by: $$ t_{n_0} \leq \max_{i} \ \frac{\theta_i}{\min_{x_i \in S_i^{-1}[0, \theta_i]} \bigtriangledown S_i(x_i) \cdot f_i(x_i) }. $$
\end{theorem}
\begin{theorem} \label{theorem2} If the network is cooperative, large enough and if besides all the cells are mutually similar, i.e. \begin{equation} \label{eqnSimilarCells}
\frac{\min_i \big({\theta_i}/{\max_{x_i \in S_i^{-1}[0, \theta_i]}\bigtriangledown S_i(x_i) \cdot f_i(x_i) }\big)}{\max_i \big({\theta_i}/{\min_{x_i \in S_i^{-1}[0, \theta_i]}\bigtriangledown S_i(x_i) \cdot f_i(x_i) }\big)} \geq 1- \frac{\min_{i \neq j} \Delta_{ij}}{\max_i \theta_i}
\end{equation} then, from any initial state and after a waiting time the grand coalition appears at every spiking instant of the system. \end{theorem}
Inequality (\ref{eqnSimilarCells}) is satisfied for instance if the cells have mutually identical free dynamics and besides, for each cell $i$, the maximum and minimum velocities $\bigtriangledown S_i(x_i) \cdot f_i(x_i)$ - according to which the satisfaction variable $S_i$ increases - are not very different. Hypothesis (\ref{eqnSimilarCells}) also holds if the cells are not identical but their differences are small enough so the quotient at left in inequality (\ref{eqnSimilarCells}) - which is strictly smaller than 1 - differs from 1 less than $ \displaystyle \frac{\min_{i \neq j} \Delta_{ij}}{\max_i \theta_i}$. If besides the interactions weights $\Delta_{i,j}$ are much smaller than $\theta_i$ - cf. condition (\ref{eqnWeakInteractions}) -, then the similarity among the cells must be very notorious to satisfy the hypothesis of Theorem \ref{theorem2}.
Roughly speaking, Theorem \ref{theorem2} states that if the cells are similar enough then, after a waiting time which depends on the initial state of the global system, the spike of one cell makes all the other cells also spike at the same instant. In other words, the only recurrent coalition is the grand coalition.
\section{The proofs} \label{sectionProofs}
\subsection{Proof of Theorem \ref{theorem1}}
{\em Proof: } Let $\{t_n\}_{n \geq 0}$ the strictly increasing sequence of spiking instants, as defined in \ref{definitionSpiking instants}. Let $$ r := 1 + \mbox{int} \Big(\frac{\max_i \theta_i}{\min_{i \neq j} \Delta_{ij}} \Big ),$$ where int denotes the lower integer part. Since by hypothesis the network is large, from Definition \ref{definitionLargeNetwork} we obtain: $$r^2 \leq m,$$ where $m$ is the number of units in the system.
It said in Section \ref{section2}, it is not restrictive to assume that the initial state ${\bf x}(0)$ belongs to ${\bf S}^{-1} (Q)$. Thus $S_i(x_i(0)) \in [0, \theta_i)$ for any unit $i$. We state
\noindent{\bf Assertion (A) } \em During the time interval $[0, t_{r-1}]$ all the units of the system have spiked at least once. \em
To prove Assertion (A), let argue by contradiction. Assume that there is a cell, say $j$, such that $x_j(t) < \theta_j$ for all $t \in [0, t_{r-1}]$. By the interactions rule (\ref{eqnInteraction}), and since at least one cell spikes at instant $t_k$ for all $k= 0, \ldots, {r-1}$, we have: $$S_j(x_j(t_{r-1})) \geq S_j(x_j(0)) + r \, \min_{i \neq j} \Delta_{ij} \geq S_j(x_j(0)) + \theta_j \geq \theta_j,$$ contradicting the initial assumption. So Assertion (A) is proved.
Now, we state
\noindent{\bf Assertion (B) } \em If at some instant $t_n$ at least $r$ cells spike simultaneously, then all the cells spike simultaneously at $t_n$.\em
To prove Assertion (B) we have, by hypothesis, $\#I(t_n) \geq r$. Due to the interactions rule (\ref{eqnInteraction}), for any cell $j \not \in I(t_n) $ we obtain: $$S_j(x_j(t_n)) \geq S_j(x_j(t_n^-)) + r \, \min_{i \neq j} \Delta_{ij} \geq \theta_j, $$ contradicting the assumption that $j \not \in I(t_n)$. Therefore, all cells are in $I(t_n)$ proving Assertion (B).
Consider the $r$ coalitions $I(t_0), I(t_1), \ldots, I(t_{r-1})$. Assertion (A) states that each cell $i$ belongs to at least one of those coalitions. Since the number of different cells is $m \geq r^2$, and the number of coalitions in the above list is $r$, there exists at least one of such coalitions, say $I(t_k)$ having at least $r$ different cells. In other words, there exists a spiking instant $t_k$ such that at least $r$ cells spike simultaneously at $t_k$. Applying Assertion (B) we deduce that all the cells spike simultaneously at $t_k$. We have proved that the grand coalition $I(t_k) = \{1, \ldots, m\}$ is spontaneously formed at the instant $t^*_0 := t_k >0$. Since this assertion holds for any initial state, we now translate the origin of the time axis to $t^*_0$, adopting a new initial state from which the grand coalition will be formed again at some future instant $t^*_ 1 > t^*_0$. By induction on $n$, the grand coalition will be exhibited recurrently in the future at an increasing sequence of instants $t_n^*$, ending the proof of Theorem \ref{theorem1}.
$\Box$
\subsection{Proof of Theorem \ref{theorem1waitingTime}}
{\em Proof:}
From the proof of Theorem \ref{theorem1}, the waiting time $t^*_0 $ until the first grand coalition appears is not larger than the instant $t_{r-1}$ such that all the cells have spiked at least once during the time interval $[0, t_{r-1}$. Since all the interactions are positive, $t_{r-1}$ is not larger than the time $T_i$ that the slowest cell, say $i$, would take to arrive to its goal $\theta_i$ if it were not coupled to the network, i.e. under the free dynamics:
$$t_0^* \leq t_{r-1} \leq T_i.$$ From the relaxation rules (\ref{eqnFreeDynamics}) and (\ref{eqnSatisfaction}) we get
$$\theta_i = S_i(x_i(T_i^-)) = \int_0^{T_i} \bigtriangledown S_i(x_i(t)) \cdot f_i(x_i(t)) \, dt \geq \Big(\min_{x_i \in S_i^{-1} ([0, \theta_i])} \bigtriangledown S_i(x_i) \cdot f_i (x_i) \, \Big) \, T_i$$
Thus
$$t_0^* \leq T_i \leq \frac{\theta_i}{\min_{x_i \in S_i^{-1}[0, \theta_i]} \bigtriangledown S_i(x_i) \cdot f_i(x_i) } \leq \max_{i} \ \frac{\theta_i}{\min_{x_i \in S_i^{-1}[0, \theta_i]} \bigtriangledown S_i(x_i) \cdot f_i(x_i) } ,$$
ending the proof of Theorem \ref{theorem1waitingTime}.
$\Box$
\subsection{Proof of Theorem \ref{theorem2}}
{\em Proof: }
From Theorem \ref{theorem1}, there exists a first instant $t_0^*$ such that the grand coalition is exhibited. From the update rule (\ref{eqnResetRule}, the state ${\bf x}(t_0^*)$ of the global system is such that ${\bf S}({\bf x}(t_0^*)) = {\bf 0}$. We now translate the origin of the time axis to $t_0^*$. So, the initial state is now ${\bf x}(0)$ such that ${\bf S} ({\bf x}(0)) = {\bf 0}$.
Hence, to prove Theorem \ref{theorem2} it is enough to show that, if the hypothesis of inequality (\ref{eqnSimilarCells}) holds, then for any initial state ${\bf x}(0)$ such that ${\bf S} ({\bf x}(0)) = {\bf 0}$, all the cells spikes simultaneously at the minimum instant $t_1 >0$ such at least one cell, say $i$, spikes.
So, let us compute the values of the satisfaction variables of all the cells at the instant $t_1^-$.
Due to the relaxation rules (\ref{eqnFreeDynamics}) and (\ref{eqnSatisfaction}) we have
\begin{equation}
\label{eqn01}
S_j(x_j(t_1^-)) = \int _0 ^{t_1} \bigtriangledown S_j(x_j(t)) \cdot f_j(x_j(t)) \, dt \geq \Big (\min_{x_j \in S_j^{-1}([0, \theta_j]} \bigtriangledown S_j(x_j) \cdot f_j (x_j) \Big) \ t_1 \ \ \ \forall \ 1 \leq j \leq m . \end{equation}
In particular for the spiking cell $i$ we have
\begin{equation}
\label{eqn02}\theta_i = S_i(x_i(t_1^-)) = \int _0 ^{t_1} \bigtriangledown S_i(x_j(t)) \cdot f_i(x_j(t)) \, dt \leq \Big(\max_{x_i \in S_i^{-1}([0, \theta_i])} \bigtriangledown S_i(x_i) \cdot f_i (x_i) \Big) \ t_1.\end{equation}
Combining inequalities (\ref{eqn01}) and (\ref{eqn02}) we deduce:
$$S_j(x_j(t_1^-)) \geq {\theta_i} \ \frac{\min_{x_j \in S_j^{-1}([0, \theta_j])} \bigtriangledown S_j(x_j) \cdot f_j (x_j) }{\max_{x_i \in S_i^{-1}([0, \theta_i])} \bigtriangledown S_i(x_i) \cdot f_i (x_i)} \geq $$
$$\geq \theta_j \ \ \frac{\min_i \big({\theta_i}/{\max_{x_i \in S_i^{-1}[0, \theta_i]}\bigtriangledown S_i(x_i) \cdot f_i(x_i) }\big)}{\max_j \big({\theta_j}/{\min_{x_j \in S_j^{-1}[0, \theta_j]}\bigtriangledown S_j(x_j) \cdot f_j(x_j) }\big)} \ \ \ \ \forall \ j \neq i. $$ Using now the hypothesis of inequality (\ref{eqnSimilarCells}, we obtain:
$$S_j(x_j(t_1^-)) \geq \theta_j \ \Big (1- \frac{\min_{i \neq j} \Delta_{ij}}{\max_i \theta_i}\Big) \geq \theta_j - \min_{i \neq j} \Delta_{ij} \ \ \ \ \forall \ j \neq i.$$
Since at least the cell $i$ spikes at instant $t_1$ we have
$$S_j(x_j(t_1^-)) + \sum_{i \in I(t_1), \ i \neq j} \Delta_{ij} \geq S_j(x_j(t_1^-)) + \min_{i \neq j} \Delta_{ij} \geq \theta_j.$$
So, applying the interaction rule (\ref{eqnInteraction}) we deduce that the cell $j$ spikes at instant $t_1$. This result holds for all the cells $j \neq i$. Thus, all the cells spike when at least one spikes, ending the proof of Theorem \ref{theorem2}.
$\Box$
\noindent{\bf \large Acknowledgement }
\noindent We thank the scientific and organizing committees of the IV Coloquio Uruguayo de Mate\-m\'{a}\-tica, for the invitation to give a talk during the event on the subject of this paper.
\end{document} |
\begin{document}
\title[To Raise or Not To Raise: The Autonomous Learning Rate Question]{To Raise or Not To Raise: The Autonomous Learning Rate Question}
\author*[1,2]{\fnm{Xiaomeng} \sur{Dong}}\email{Xiaomeng.Dong@ge.com}
\author[1]{\fnm{Tao} \sur{Tan}}
\author[1]{\fnm{Michael} \sur{Potter}}
\author[1]{\fnm{Yun-Chan} \sur{Tsai}}
\author[1]{\fnm{Gaurav} \sur{Kumar}}
\author[1]{\fnm{V. Ratna} \sur{Saripalli}}
\author[2]{\fnm{Theodore} \sur{Trafalis}}
\affil[1]{\orgname{GE HealthCare},\country{USA}}
\affil[2]{\orgname{University of Oklahoma}, \country{USA}}
\abstract{There is a parameter ubiquitous throughout the deep learning world: learning rate. There is likewise a ubiquitous question: what should that learning rate be? The true answer to this question is often tedious and time consuming to obtain, and a great deal of arcane knowledge has accumulated in recent years over how to pick and modify learning rates to achieve optimal training performance. Moreover, the long hours spent carefully crafting the perfect learning rate can come to nothing the moment your network architecture, optimizer, dataset, or initial conditions change ever so slightly. But it need not be this way. We propose a new answer to the great learning rate question: the Autonomous Learning Rate Controller. Find it at \url{https://github.com/fastestimator/ARC/tree/v2.0}.}
\keywords{Deep Learning, AutoML, Optimization, Learning Rate}
\maketitle
\section{Introduction}\label{sec1}
Learning Rate (LR) is one of the most important hyperparameters in deep learning training, a parameter everyone interacts with for all tasks. In order to ensure model performance and convergence speed, LR needs to be carefully chosen. Overly large LRs will cause divergence whereas small LRs train slowly and may get trapped in a bad local minima. As training schemes have evolved over time they have begun to move away from a single static LR and into scheduled LRs, as can be seen in a variety of state-of-the-art AI applications \cite{sota-transformer,sota-yolov4,sota-cspnet,sota-roberta}. LR scheduling provides finer control of LRs by allowing different LRs to be used throughout the training. However, the extra flexibility comes at a cost: these schedules bring more parameters to tune. Given this tradeoff, there are broadly two ways of approaching LR scheduling within the AI community.
Experts with sufficient computational resources tend to hand-craft their own LR schedules, because a well-customized LR schedule can often lead to improvements over current state-of-the-art results. For example, entries in the Dawnbench \cite{dawnbench} are known for using carefully tuned LR schedules to achieve world-record convergence speeds. However, such LR schedules come with significant drawbacks. First, these LR schedules are often specifically tailored to an exact problem configuration (architecture, dataset, optimizer, etc.) such that they do not generalize to other tasks. Moreover, creating these schedules tends to require a good deal of intuition, heuristics, and manual observation of training trends. As a result, building a well-customized LR schedule often requires great expertise and significant computing resources.
In contrast, others favor existing task-independent LR schedules since they often provide decent performance gains with less tuning efforts. Some popular choices are cyclic cosine decay \cite{cosinedecay}, exponential decay, and warmup \cite{lrwarmup}. While these LR schedules can be used across different tasks, they are not specially optimized for any of them. As a result, these schedules do not guarantee performance improvements over a constant LR. On top of that, many of these schedules still require significant tuning to work well. For example, in cyclic cosine decay, parameters such as $l_{max}$, $l_{min}$, $T_0$, and $T_{multi}$ must all be tuned in order to function properly. Recently some methods have been proposed which either provide techniques to easily infer their parameters (\emph{e.g.} super-convergence \cite{superconvergence}), or are even completely parameter free (\emph{e.g.} stochastic line search \cite{vaswani2019painless,mahsereci2017probabilistic}). While promising, these techniques can prove finicky in practice, as we demonstrate in Section \ref{sec:experiments}.
Recent advancements in AutoML on architecture search \cite{nas,enas,darts} and update rule search \cite{updaterule} have proved that it is possible to create automated systems that perform equal or better than human experts in designing deep learning algorithms. These successes have inspired us to tackle the LR scheduling problem. We aim to create a system that learns how to change LR effectively. Several attempts have been previously made to dynamically learn to set LRs \cite{LARS,shu2020metalrschedulenet,daniel2016learning}. Unfortunately these have introduced step level dual optimization loops into the training procedure, something which can add significant complexity and training time to a problem. Moreover, this family of techniques has been shown to be vulnerable to short-horizon bias \cite{wu2018understanding}, sacrificing long-term performance for short-term gains. We believe that both of these drawbacks can be overcome.
To that end we introduce ARC: an Autonomous LR Controller. It takes training signals as inputs and is able to intelligently adjust LRs in a real-time generalizable fashion. ARC overcomes the challenges faced by prior LR schedulers by encoding experiences over a variety of different training tasks, different time horizons, and by dynamically responding to each new training situation so that no manual parameter tuning is required.
ARC is also fully complementary to modern adaptive optimizers such as Adagrad \cite{adagrad} and Adam \cite{adam}. Adaptive optimizers compute updates using a combination of LR and `adaptive' gradients. When gradients have inconsistent directions across steps, the scale of the adaptive gradient is reduced. Conversely, multiple updates in the same direction result in gradient upscaling. This is sometimes referred to as adaptive LR even though the LR term has not actually been modified. Our method is gradient agnostic and instead leverages information from various training signals to directly modify the optimizer LR. This allows it to detect patterns which are invisible to adaptive optimizers. Thus the two can be used in tandem for even better results.
The key contributions of this work are:
\begin{enumerate} \item An overall methodology for developing autonomous LR systems, including problem framing and dataset construction.
\item A comparison of ARC with popular LR schedules across multiple computer vision and language tasks.
\item An analysis of failure modes and unexpected behaviors from ARC, informing future directions for research. \end{enumerate}
The rest of the paper is organized as follows: Section \ref{sec:CC} outlines constraints which guide further solution development, Section \ref{sec:methods} explains the ARC methodology, Section \ref{sec:experiments} experimentally compares the performance of ARC against other common LR scheduling methods, and Section \ref{sec:limits} discusses open problems and opportunities.
\section{Challenges and Constraints} \label{sec:CC}
Before delving into our methodology, we will first highlight some of the key challenges in developing an autonomous LR controller. These constraints inform many of our subsequent design decisions.
\begin{enumerate}[label=\alph*)] \item \label{challenges:subjective} \textbf{Subjectivity.} Determining the superiority of one model over another (each trained with a different LR) is fraught with subjectivity. There are many different ways to measure model performance (training loss, validation loss, accuracy, etc.) and they may often be in conflict with one another.
\item \label{challenges:cumulative} \textbf{Cumulativeness.} Associating the current model performance with an LR decision at any particular step is challenging, since the current performance is the result of the cumulative effect of all previous LRs used during training. This is also related to short-horizon bias \cite{wu2018understanding}, where long-term effects are easily overshadowed by short-term wins.
\item \label{challenges:random} \textbf{Randomness.} Randomness during training makes it difficult to compare two alternative LRs. Some common sources of randomness are dataset shuffling, data augmentation, and random network layers such as dropout. Any performance differences due to the choice of LR need to be large enough to overshadow these random effects.
\item \label{challenges:scale} \textbf{Scale.} Different deep learning tasks use different metrics to monitor training. The most task-independent of these are training loss, validation loss, and LR. Unfortunately, the magnitude of these values can still vary greatly between tasks. For example, categorical cross entropy for 1000-class classification is usually between 0 and 10, but a pixel-level cross entropy for segmentation can easily reach a scale of several thousand. Moreover, a reasonable LR for a given task can vary greatly, from 1e-6 up to 10 or more.
\item \label{challenges:footprint} \textbf{Footprint.} The purpose of having an automated LR controller is to achieve faster convergence and better results. Any solution must therefore have a small enough footprint that using it does not adversely impact training speed and memory consumption.
\end{enumerate}
\section{Methods} \label{sec:methods}
\subsection{Framing LR Control as a Learning Problem}
We frame the development of ARC as a supervised learning problem: predicting the next LR given available training history. Due to challenge \ref{challenges:cumulative}, the model needs to observe the consequence of a specific LR for long enough to form a clear association between LR and performance. We therefore only modify the LR on an epoch timescale. This has a secondary benefit of dramatically reducing our computational overhead compared with competing methods.
Per challenges \ref{challenges:subjective} and \ref{challenges:scale}, as well as the desire to create a generalizable system, we cannot use any task-specific metrics. We also cannot rely on model parameters or gradient inspection since ARC would then become architecture dependent and would likely also fail constraint \ref{challenges:footprint}. We therefore leverage only the historical training loss, validation loss, and LR as input features.
Due to challenges \ref{challenges:random} and \ref{challenges:scale}, rather than generating a continuous prediction of what new LR values should be, we instead pose this as a 3-class classification problem. Given the input features, should the LR: increase ($LR * 1.618$), remain the same ($LR * 1.0$), or decrease ($LR * 0.618$)? We chose these specific values based on the following intuitions: Let $\alpha$ be your desired increase factor and $\beta$ be your desired decrease factor. Then we would like $\alpha\beta=1$ such that you can easily undo a decision if it later turns out to have been a mistake. We also would like $\alpha$ not to be too large, since a single large jump in LR could easily trigger model divergence. $\alpha$ should likewise not be too small, since then you might miss an opportunity to properly take advantage of smooth loss landscapes. What then is the largest `safe' value for $\alpha$? While we have no perfect way of knowing this in the general case, one reasonable heuristic is to combine previous LR values which are already known to be safe. For example, to get a new increased LR, you could sum together the two largest LRs which you've already tested. In other words: $\alpha^{i+1}LR_0=\alpha^{i}LR_0+\alpha^{i-1}LR_0$. Solving for these two constraints gives $\alpha\approx 1.618$ and $\beta \approx 0.618$.
\subsection{Generating the Dataset} \label{subsec:dataset}
Having framed the problem, we now need to generate a dataset on which we can train ARC. Given that this specific learning problem is newly proposed, there are no existing datasets nor data curation workflows which we could leverage. We therefore build one ourselves from real deep learning training tasks. For each task we used the following procedure to generate data:
\begin{enumerate} \item \label{step:train} Train $n$ epochs with $LR=r$, then save the current state as checkpoint $C$
\item \label{step:increase} Reload $C$, train for $n$ epochs with $LR=1.618*r$, then compute validation loss ($l_+$)
\item \label{step:constant} Reload $C$, train for $n$ epochs with $LR=1.0*r$, then compute validation loss ($l_1$)
\item \label{step:decrease} Reload $C$, train for $n$ epochs with $LR=0.618*r$, then compute validation loss ($l_-$)
\item \label{step:truth} Note the $LR$ which resulted in $min\left\{ l_+, l_1, l_- \right\}$
\item \label{step:repeat} Reload $C$, eliminate $max\left\{ l_+, l_1, l_- \right\}$ and its corresponding $LR$, replace $r$ with one of the two remaining $LRs$ at random, and return to step \ref{step:train} \end{enumerate}
By executing steps \ref{step:train} - \ref{step:truth} we can create one input/ground truth pair. The input features are the training loss, validation loss, and LR during step \ref{step:train}, concatenated with those same features from the previous $2n$ epochs of training. The label is the LR noted in step \ref{step:truth}. This process is depicted in Figure \ref{fig:data}. Steps \ref{step:train} - \ref{step:repeat} continue until training finishes. Assuming the total number of training epochs is $N$, then we get $N/n$ data points from each training procedure.
\begin{figure}
\caption{Generating and labeling a data point. In this hypothetical example, increasing the learning rate was found to result in the lowest validation loss after a further $n$ epochs of training. Therefore, the prior $3n$ epochs of training loss, validation loss, and learning rate are assigned a corresponding ground truth label of `increase.'}
\label{fig:data}
\end{figure}
We adopt this specific workflow based on the following intuitions: step \ref{step:train}, by providing an identical starting point, ensures a fair comparison between different subsequent LR decisions. Steps \ref{step:increase} - \ref{step:decrease} evaluate the impact of different LR decisions in actual training. We randomly choose different values for $n$ across different data generation runs to avoid bias towards a particular time horizon / dataset size. For step \ref{step:truth} we prefer decisions which lead to lower validation losses for consistency with our learning objective. The random selection process in step \ref{step:repeat} is used to explore a larger search space without causing the loss to diverge. While there could be many alterations to this overall workflow, this form was chosen to generate the data required by our specific problem statement due to its relative simplicity.
To help ensure generalization, we gathered 12 different computer vision and language tasks - each having a different configuration (dataset, architecture, initial LR ($LR_0$), etc.) as shown in Table \ref{tbl:tasks}. Note that these tasks capture a large variety of different losses, from sparse and per-pixel CE to hinge, adversarial, dice, and multi-task losses. Each of the 12 tasks were trained an average of 42 times. Each training randomly selected an optimizer from \{Adam, SGD, RMSprop\cite{rmsprop}\}, an $LR_0$ $r\in [\text{Min Init LR}, \text{Max Init LR}]$, a value of $n\in [1, 10]$, and then trained for a total of $10n$ epochs. Thus approximately 5050 sample points were collected in total.
\begin{table*} \begin{center}
\resizebox{\textwidth}{!}{\begin{tabular}{|c|c|c|c|c|c|} \hline Task & Task Description & Dataset & Architecture & Max $LR_0$ & Min $LR_0$ \\ \hline\hline 1 & Image Classification & SVHN Cropped \cite{svhn} & VGG19 \cite{vgg} + BatchNorm \cite{batchnorm} & 1e-3 & 1e-5\\ 2 & Image Classification & SVHN Cropped & VGG16 \cite{vgg} + ECC \cite{ecc} & 1e-4 & 1e-5\\ 3 & Adversarial Training \cite{fgsm} & SVHN Cropped & VGG19 & 1e-3 & 1e-5\\ 4 & Image Classification & Food101 \cite{food101} & Densenet121 \cite{densenet} & 1e-2 & 1e-5\\ 5 & Image Classification & Food101 & InceptionV3 \cite{inceptionv3} & 1e-2 & 1e-5\\ 6 & Multi-Task \cite{multitask} & CUB200 \cite{cub200} & ResNet50 \cite{resnet50} + UNet \cite{unet} & 1e-4 & 1e-5\\ 7 & Text Classification & IMDB \cite{imdb} & LSTM & 1e-3 & 1e-5\\ 8 & Named Entity Recognition & MIT Movie Corpus \cite{mitmovie} & BERT \cite{bert} & 1e-4 & 1e-5\\ 9 & One Shot Learning & omniglot \cite{omniglot} & Siamese Network \cite{siamese} & 1e-3 & 1e-5\\ 10 & Text Generation & Shakespear \cite{shakespear}& GRU \cite{gru} & 1e-3 & 1e-5\\ 11 & Semantic Segmentation & montgomery\cite{montgomery}& UNet& 1e-4 & 1e-5\\ 12 & Semantic Segmentation & CUB200 & UNet & 1e-3 & 1e-5\\ \hline \end{tabular}} \end{center} \caption{Training dataset task overview. Training tasks spanned a variety of domains, datasets, architectures, and learning rates in order to provide ARC with a wide sampling of experiences to learn from.} \label{tbl:tasks} \end{table*}
\subsection{Correcting Ground Truth} \label{subsec:corrections}
Suppose that during step \ref{step:truth} of the data generation process you find that $l_+$, $l_1$, and $l_-$ are 0.113, 0.112, and 0.111 respectively. Due to challenge \ref{challenges:random} it may not be appropriate to confidently claim that decreasing LR is the best course of action.
Luckily, there is one more datapoint we can use to reduce uncertainty. Suppose that during step \ref{step:repeat} we choose to decrease the LR. Then the subsequent step \ref{step:train} is repeating exactly the prior step \ref{step:decrease}. Let $l_-^*$ be the validation loss at the end of step \ref{step:train}. If the relative order of $l_+$, $l_1$, and $l_-$ is the same as the relative order of $l_+$, $l_1$, and $l_-^*$, then we consider our ground truth labeling to be correct (for example, if $l_-^*=0.109)$. On the other hand, if the relative ordering is different (for example, if $l_-^*=0.115$), then random noise is playing a greater role than the LR in determining performance. In that case we take a conservative approach and modify the ground truth label to be `constant LR'.
\subsection{Building the Model} \label{subsec:model}
For preprocessing, considering challenge \ref{challenges:scale}, we apply z-score normalization to the training and validation losses, and then perform nearest-neighbor resizing to length 100, 200, or 300 (depending on whether we have $n$, $2n$, or $3n$ epochs of history). If only $n$ or $2n$ epochs of prior data were available (very early during training), we zero-prepend the loss vectors to length 300. For LR we normalize by dividing by the first available value, then perform resizing followed, if necessary, by prepending ones to ensure a final length of 300.
\begin{figure*}
\caption{Network architecture used in ARC. The model receives historical measurements of training loss, validation loss, and learning rate. After parsing this information through 8 lightweight layers it produces one of three classes as output (increase, keep, or decrease the current LR).}
\label{fig:network}
\end{figure*}
The ARC model architecture is shown in Figure \ref{fig:network}. It consists of three components: a feature extractor, an LSTM \cite{lstm}, and a dense classifier. The feature extractor consists of two 1D convolution layers, the LSTM of two stacked memory sequences, and the classifier of two densely connected layers. Considering constraint \ref{challenges:footprint}, we chose layer parameters such that the total number of trainable model parameters is less than 80k. Compared to the millions of parameters which are common in current state-of-the-art models, this architecture should add relatively minimal overhead.
To train ARC we used a hybrid loss which averaged classical categorical cross-entropy and a specialized binary cross-entropy loss. For the binary cross-entropy loss we performed a one-vs-all binarization of the task focusing on the `increase LR’ class, since an over-aggressive increase can lead to divergence / NaN values.
The model was trained with the corrected dataset from Section \ref{subsec:corrections}. We leveraged an Adam optimizer with the following parameters: $LR=1e-4$, $\beta_1=0.9$, and $\beta_2=0.999$. Training proceeded with a batch size of 128 for 300 epochs. Once trained, the ARC model can be used to periodically adjust the LR for other models, as we demonstrate in Section \ref{sec:experiments}.
\subsection{Finding the Best ARC Model}
We found that our training procedure produces ARC models with noticeably different behaviors from one another. Ideally we would select the best model based on some metric, however we found that neither loss, accuracy, weighted accuracy, calibration error \cite{calibration}, nor MCC \cite{mcc} on held-out training data were strongly correlated with how well the model would perform on downstream tasks (results in Appendix \ref{secA1}). We believe this indicates that certain LR decisions are much more important than others (for example, perhaps early decision matter more than later ones). This is not captured in aggregating metrics such as accuracy. To circumvent this issue we used real training performance measurements on a separate problem to score the models.
We generated 10 different candidate models by repeating our Section \ref{subsec:model} procedure 10 times. We then used each candidate to train an auxillary task on multiple $LR_0$s, selecting the model leading to the best average task performance across $LR_0$s for further use. In particular, we trained a wide residual network \cite{wrn} on the SVHN Cropped dataset (Adam optimizer) over three different $LR_0$s: 1e-1, 1e-3, and 1e-5. Each candidate was used to perform 5 independent trainings at each $LR_0$. The proxy problem scores for each candidate network can be found in Appendix \ref{secA2}. Note that after an ARC model has been created in this way it can then be used on a variety of different downstream tasks without any further tuning, as we demonstrate in Section \ref{sec:experiments}.
\section{Experiments} \label{sec:experiments}
In this section, we test how well ARC can guide training tasks on previously unseen datasets and architectures. Specifically, we deploy ARC on two computer vision tasks and one NLP task. For each task, we compare ARC against 4 standard LR schedules: Baseline LR (BLR) - in which LR is held constant, one-cycle Cosine Decay (CD), Cyclic Cosine Decay (CCD), and Exponential Decay (ED). We also compare against 2 more sophisticated LR approaches: Superconvergence (SC) and SGD+Armijo Stochastic Line Search (SLS). For each task we use the same ARC model, invoked once every 3 epochs (which we found to be a good general rule, see Appendix \ref{secA5}).
When confronted with a new dataset or network architecture, it is unclear \textit{a priori} what an ideal $LR_0$ will be. Ideally an LR schedule would therefore be robust against a variety of different $LR_0$s. In order to gain a holistic view of the effectiveness of different schedulers, we use 3 different $LR_0$s for each task. We selected our maximum $LR_0$ such that larger values risked BLR divergence, and our minimum $LR_0$ such that smaller values led BLR to converge too slowly to be useful. Each training configuration is run 5 times, with median test metric performance (\emph{e.g.} median test accuracy) being reported.
SC has its own process for determining which $LR_0$ to use, leveraging an `LR range test' ahead of the primary training. We therefore run SC on the $LR_0$ value indicated by the range test for each task (range test results in Appendix \ref{secA3}).
SLS computes its LR at every step of training, and thus does not take $LR_0$ as a parameter. We run it 5 times per task and report the median score in comparison to the other methods for each of their $LR_0$ values. We used the official implementation of the SLS optimizer provided by \cite{vaswani2019painless}.
\subsection{Image Classification on CIFAR10} \label{subsec:cifar}
For our first experiment we trained a model to perform CIFAR10 image classification. We used the same architecture and preprocessing as proposed in \cite{fastcifar}. We trained for 30 epochs (rather than 24 in the original implementation) using an Adam optimizer and a batch size of 128. Three different $LR_0$s were used: 1e-2, 1e-3, and 1e-4. For each $LR_0$, we compare the performance of BLR with the performance of ARC (invoked every 3 epochs), as well as CD, CCD (using the settings proposed in \cite{enas} for CIFAR10), ED ($\gamma=0.9$), SC ($LR_{max}=0.198$ per Appendix \ref{secA3}), and SLS.
The results for all experiment runs are summarized in Table \ref{tbl:cifar10}. The median-run graphs of LR and validation accuracy over time for each method are given in Figure \ref{fig:cifar}. A summary of all runs can be found in Appendix \ref{secA4}.
When the $LR_0$ is sufficiently large (1e-2 and 1e-3), all decaying LR schedulers outperform the baseline LR. From Figure \ref{fig:cifar} (a) and (b), we can see that ARC also decided to decrease the LR. Amongst all LR schedulers, ARC performed third best with the large $LR_0$ (1e-2), and was the best performer with the medium $LR_0$ (1e-3). Interestingly, even when ARC was outperformed by CD, it very closely emulated the CD decay pattern as shown in Figure \ref{fig:cifar} (a).
\begin{figure*}
\caption{Median run performance (validation accuracy and LR vs training steps) on CIFAR10. For better LR visualization, SC and SLS are separated from the decay schedulers.}
\label{fig:cifar}
\end{figure*}
\begin{table}[b] \begin{center}
\begin{tabular}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{} & $LR_0$ = 0.01 & $LR_0$ = 0.001 & $LR_0$ = 0.0001\\ \hline BLR & 90.19 & 91.42 & 88.82\\ CD & \textbf{92.60}* & 92.89 & 88.22\\ CCD & 92.16& 92.90& 87.61\\ ED & 91.73& 92.52& 85.91\\ SC & 87.64 & -- & -- \\ SLS & 90.90 & 90.90 & 90.90 \\ ARC & 92.00& \textbf{93.09}*& \textbf{91.87}*\\ \hline \end{tabular} \end{center} \caption{CIFAR10 test accuracy. Median over 5 runs, with best (highest) values in bold. Note that SLS is duplicated across columns since it is independent of $LR_0$. Statistically significant ($p < 0.05$) improvements over runner-up values are indicated by an *.} \label{tbl:cifar10} \end{table}
When the $LR_0$ is small (1e-4), however, the drawback of statically decaying LR schedulers becomes evident: decaying an already small LR damages convergence. In this case CD, CCD, and ED are all beaten by the baseline LR. On the other hand, as shown in Figure \ref{fig:cifar} (c), ARC is able to sense that the LR is too small and increase it, achieving the best final accuracy. Furthermore, even given this difficult $LR_0$, ARC outperforms every BLR configuration (1e-2, 1e-3, and 1e-4). Given that the baseline learning rate in this case represents a raw Adam optimizer, this result demonstrates that ARC can be combined with adaptive optimizers to further improve their performance.
ARC also improves over SC and SLS for this problem, both of which increase LR far too aggressively, as can be seen in Figure \ref{fig:cifar} (d).
\subsection{Object Detection on MSCOCO} \label{subsec:mscoco}
Our second and most time-consuming task is object detection using the MSCOCO dataset. We downscale the longest side of each image to 256 pixels in order to complete the trainings within a more reasonable computational budget. The RetinaNet \cite{retinanet} architecture was selected for this task. We used a batch size of 32 and trained for a total of 45000 steps, with validation every 1500 steps. We used a momentum optimizer with 0.9 for its momentum value, but kept all other parameters consistent with the official implementation. The configuration for our LR schedulers is the same as in Section \ref{subsec:cifar}, but with $LR_0$ of 0.01, 0.005, and 0.001. The SC $LR_{max}$ was set to 0.01 (see Appendix \ref{secA3}). For this task we use mean average precision (mAP) to benchmark model performance. Note that this task uses localization and focal losses which were not present in the ARC training dataset.
The results for all experiment runs are summarized in Table \ref{tbl:mscoco}. Median-run graphs of LR and mAP over time for each method are given in Figure \ref{fig:mscoco}. A summary of all runs can be found in Appendix \ref{secA4}.
\begin{figure*}
\caption{Median run performance (validation mAP and LR vs training steps) on MSCOCO. For better LR visualization, SC and SLS are separated from the decay schedulers.}
\label{fig:mscoco}
\end{figure*}
\begin{table}[t] \begin{center}
\begin{tabular}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{} & $LR_0$ = 0.01 & $LR_0$ = 0.005 & $LR_0$ = 0.001\\ \hline BLR & 0.1634 & 0.1629 & 0.1343\\ CD & 0.1692 & 0.1580 & 0.1135\\ CCD & 0.1694 & 0.1584 & 0.1112\\ ED & 0.1563 & 0.1462 & 0.0967\\ SC & -- & -- & \textbf{0.1747}* \\ SLS & 0.0955 & 0.0955 & 0.0955 \\ ARC & \textbf{0.1757} & \textbf{0.1668} & 0.1709\\ \hline \end{tabular} \end{center} \caption{MSCOCO test mAP. Median over 5 runs, with best (highest) values in bold. Note that SLS is duplicated across columns since it is independent of $LR_0$. Statistically significant ($p < 0.05$) improvements over runner-up values are indicated by an *.} \label{tbl:mscoco} \end{table}
Interestingly, the largest $LR_0$ (1e-2) we used was not large enough to allow ED to outperform the baseline LR. Unfortunately, larger $LR_0$s were found to lead to training divergence. This exposes a critical limitation of exponential LR decay: the rate of decay needs to be carefully tuned, otherwise the LR can be either too large early on or too small later in training. On the other hand, CD, CCD, and ARC outperform the baseline LR, with ARC achieving the best mAP. Unfortunately SLS does very poorly on this problem despite decent performance on CIFAR.
For the other two smaller LRs (5e-3 and 1e-3), all of the decaying schedules are worse than the baseline LR because they have no mechanism to raise the LR when doing so would be useful. In contrast, ARC can notice this deficiency and increase the LR accordingly - allowing it to achieve strong mAP across the board. SC is able to outperform ARC on this problem, but the two methods are much closer to one another than they are to any runner-up candidates. In fact, from Figure \ref{fig:mscoco} (d) it appears that ARC has learned to naturally mimic SC when it sees it beneficial, although it is limited by the fact that it only executes once every 3 epochs.
\subsection{Language Modeling on PTB}
For our final experiment we move beyond computer vision to verify whether ARC can be useful in natural language processing tasks as well. We performed language modeling using the PTB dataset \cite{ptb} with a vocabulary size of 10000. Our network for this problem leveraged 600 LSTM units with 300 embedding dimensions, and a 50\% dropout applied before the final prediction. Training progressed for 98 epochs, with a batch size of 128 and a sequence length of 20. A Stochastic Gradient Descent (SGD) optimizer was selected, with $LR_0$ values of 1.0, 0.1, and 0.01. Our CCD scheduler used $T_0=14$ and $T_{multi}=2$ such that we could fit 3 LR cycles into the training window. The ED scheduler $\gamma$ value was set to 0.96, and the SC $LR_{max}$ was 23.2 (see Appendix \ref{secA3}). For this task we used perplexity to measure model performance (lower is better).
\begin{figure*}
\caption{Median run performance (validation perplexity and LR vs training steps) on PTB. For better LR visualization, SC and SLS are separated from the decay schedulers.}
\label{fig:ptb}
\end{figure*}
\begin{table}[t] \begin{center}
\begin{tabular}{|c|c|c|c|} \cline{2-4}
\multicolumn{1}{c|}{} & $LR_0$ = 1.0 & $LR_0$ = 0.1 & $LR_0$ = 0.01\\ \hline BLR & 118.3 & 136.7 & 313.5\\ CD & 119.2 & 157.8 & 438.8\\ CCD & 122.6 & 160.0 & 450.1\\ ED & 122.1 & 202.0 & 603.5\\ SC & 116.1 & -- & -- \\ SLS & 150.1 & 150.1 & 150.1 \\ ARC & \textbf{111.3}* & \textbf{116.1}* & \textbf{115.9}*\\ \hline \end{tabular} \end{center} \caption{PTB test perplexity. Median over 5 runs, with best (lowest) values in bold. Note that SLS is duplicated across columns since it is independent of $LR_0$. Statistically significant ($p < 0.05$) improvements over runner-up values are indicated by an *.} \label{tbl:ptb} \end{table}
The results for all experiment runs are summarized in Table \ref{tbl:ptb}. Median-run graphs of LR and perplexity over time for each method are given in Figure \ref{fig:ptb}. A summary of all runs can be found in Appendix \ref{secA4}.
ARC had very strong performance on this problem, achieving the best scores regardless of which $LR_0$ it was given. SC also performed well, and SLS once again struggled, though not to the same extent as for object detection.
\section{Open Problems and Opportunities} \label{sec:limits}
As Section \ref{sec:experiments} demonstrates, ARC can be successfully deployed over a range of tasks, architectures, optimizers, dataset and batch sizes, and $LR_0$s. It does, however, have some open problems along with opportunities for future work which bear mentioning.
One open problem with ARC is that it assumes a constant optimization objective. While this is often the case for real-world problem-solving tasks, it is not true of generative adversarial networks (GANs), where the loss of the generator is based on the performance of an ever-evolving discriminator. Thus ARC, while applicable to many problems, may not be appropriate for all genres of deep learning research. On the other hand, the design of ARC does not technically preclude its use in such cases. We see investigating this as an opportunity in the future.
Another open problem with the current ARC implementation is that it sometimes provides unreliable decisions if queried too frequently. We found that once every 3 epochs is the best frequency (see Appendix \ref{secA5}). The need for 3 epochs may be attributable to the way in which we train ARC using short-, mid-, and long-term data, or else perhaps a consequence of short-horizon bias \cite{wu2018understanding}. Notably, in the only instance where ARC performed worse than SC, it was essentially emulating SC but was unable to do so quickly enough. See Figure \ref{fig:mscoco} (d). It may be possible to avoid this limitation by measuring validation loss more frequently, rather than once at the end of each epoch. We see exploring this possibility as an interesting direction for future work.
\begin{figure*}
\caption{Failure modes and unexpected behaviors. (a) shows the best evaluation performance of 5 independent trainings for different schedulers. (b) shows 3 sample runs from (a) with the outlier run in orange. (c) highlights the median PTB example run from Figure \ref{fig:ptb} (a).}
\label{fig:failures}
\end{figure*}
As for failure modes, just like any other deep learning model, ARC can also make incorrect decisions. For example, when training MSCOCO at an $LR_0$ of 0.01, 4 out of 5 ARC trainings noticeably outperformed all competing schedules as shown in Figure \ref{fig:failures} (a). There was, however, an outlier which performed significantly worse. Its training plot is shown in Figure \ref{fig:failures} (b) alongside with two other ARC training runs. In all 3 of the visualized training runs, ARC raised the LR too aggressively (circled in red in the Figure). In the blue and green runs, ARC automatically detected that the LR was too high and decreased it aggressively, leading to strong final performance. In the orange (failure) case, ARC for some reason doubled down on the excessively large LR, leading to further degradation in performance. Although this is clearly undesirable behavior, it is quite rare - occurring only three times across our total of 45 different experimental configurations/runs (one time in each of the different PTB $LR_0$ configurations). Since real-world applications tend to train multiple models and keep the best one, we do not foresee this being usage-limiting.
Figure \ref{fig:failures} (c) shows an interesting phenomenon which we did not anticipate. After achieving an optimal performance around step 5000, ARC started to drop the LR as might normally be expected to improve performance. However, after step 25000 it changed course and dramatically increased the LR. Comparing ARC's performance with the other schedulers in Figure \ref{fig:ptb} (a), it seems that ARC may be attempting to prevent the model from overfitting. It's not clear whether this is actually a useful strategy, but it's something we hope to investigate more thoroughly in future research.
\section{Conclusion} In this work we proposed an autonomous learning rate controller that can guide deep learning training to reliably better results. ARC overcomes several challenges in LR scheduling and is complementary to modern adaptive optimizers. We experimentally demonstrated its superiority to popular schedulers across a variety of tasks, optimizers, batch sizes, and network architectures, as well as identifying several areas for future improvement. Not only that, ARC achieves its objectives without tangibly increasing the training budget, adding additional optimization loops, nor introducing complex RL workflows. This is in sharp contrast to prior work in this field, as well as other AutoML paradigms in general. The true test of any automation system is not whether it can outperform any possible hand-crafted custom solution, but rather whether it can provide a high quality output with great efficiency. Given that, the fact that ARC actually does outperform popular scheduling methods while requiring no effort nor extra computation budget on the part of the end user makes it a valuable addition to the AutoML domain.
\backmatter
\section*{Conflict of Interest}
The authors declare that they have no conflict of interest.
\section*{Data Availability Statement}
The datasets used in this study are publicly available for download in their corresponding websites. The source code to reproduce this work has been open-sourced and can be found at \url{https://github.com/fastestimator/ARC/tree/v2.0}.
\begin{appendices}
\section{Common Performance Metrics and ARC}\label{secA1}
Common performance metrics do not accurately predict an ARC model's downstream performance, making it difficult to determine which of several candidate models is the best. We considered 5 different candidate metrics, along with the final models from each run, and the proxy task from Section \ref{sec:proxy}. Our candidate metrics were accuracy, MCC, validation loss, calibration error, and weighted accuracy (specified in Table \ref{tbl:wacc}).
\begin{table}[h!] \begin{tabular}{cc} $w_{acc} = \frac{\sum_i{CM[i,i] * \lvert RPM[i,i] \lvert}}{\sum_{i,j}{CM[i,j] * \lvert RPM[i,j] \lvert}}\notag$ &
\begin{tabular}{|c||c|c|c|} \hline Predict \textbackslash Actual & Decrease & Constant & Increase \\ \hline\hline Decrease & $+3$ & $-1$ & $-3$ \\ Constant & $-1$ & $+1$ & $-1$ \\ Increase & $-3$ & $-1$ & $+3$ \\ \hline \end{tabular} \end{tabular} \caption{Weighted Accuracy ($w_{acc}$) formula and corresponding Reward Penalty Matrix. CM is the Confusion Matrix.} \label{tbl:wacc} \end{table}
To test whether any of these metrics were useful, we ran the ARC training procedure a total of 5 times. During those runs, we saved the `best' model according to each of the different metrics, such that in the end we had 5 ARC models for each metric, where models may or may not be the same across metrics (it could be that for one training run, the model with the best accuracy also had the highest MCC). We then used each ARC model to train a 9 layer residual network for 30 epochs on CIFAR10, recording the best evaluation accuracy during training. We tested three different values of $LR_0$: 0.01, 0.001, and 0.0001, and repeated each CIFAR10 training 5 times. The results are summarized in Figure \ref{fig:metrics}. Unfortunately, it seems that none of the simple metrics are strongly indicative of ARC model effectiveness. In fact, simply taking the model from the end of training (epoch 300) was just as effective as any of the metrics we tested. This motivated our use of a proxy problem.
\begin{figure*}
\caption{Evaluating models selected by different metrics. The `proxy' boxplots contain 5 data points each (1 ARC model * 5 training runs). All other boxplots contain 25 data points (5 ARC models * 5 training runs). Y-axis is test accuracy.}
\label{fig:metrics}
\end{figure*}
\section{Model Selection via Proxy Task}\label{secA2} \label{sec:proxy}
\begin{figure*}
\caption{ARC candidate performance on proxy problem. Each box contains 5 data points (1 ARC model * 5 runs). Y-axis is test accuracy.}
\label{fig:proxy}
\end{figure*}
Since we were unable to find a metric which predicts ARC performance reliably we turn to a proxy task instead. We first train 10 candidate ARC models. We then use each model to train a WideResnet28 architecture on the SVHN Cropped dataset for 30 epochs, repeating each training 5 times. We perform each set of trainings for 3 different values of $LR_0$: 0.1, 0.001, and 0.00001. We gather the best accuracies from each of these SVHN training runs, and select the candidate model with the best mean. Results for each candidate are given in Figure \ref{fig:proxy}. Notice that candidates 1 and 7, which tended to diverge at high LRs, are naturally eliminated by this process. Model 4 was selected for later use.
\section{Superconvergence LR Search}\label{secA3} \label{sec:range}
To use superconvergence LR scheduling, one first needs to run a search routine to determine the ideal maximum LR. For CIFAR10 we use the same LR search routine found in the paper \cite{superconvergence}: LR is increased linearly from 0.0 to 3.0 over 5000 iterations, and evaluation accuracy and LR are plotted accordingly. The max LR is the LR that leads to the maximum accuracy (Figure \ref{fig:lrsearch}). The LR search configuration for other tasks is the same, except that for instance detection LR is increased linearly from 0.0 to 1.0, and for language modeling LR is increased from 0.0 to 100. For instance detection, we use the evaluation loss rather than mAP because the magnitude of the latter is extremely small during early stages of training. As a consequence, we look for the minimum metric score rather than the maximum when searching for the appropriate max LR. We then set $LR_0$ to be $\frac{max LR}{f}$, where $f\in[5,40]$ (consistent with the original paper's experiment section). The $f$ chosen for image classification, instance detection, and language modeling are 19.8, 10.0, and 23.2 respectively, such that their $LR_0$ values match those used in other experiments.
\begin{figure*}
\caption{Superconvergence LR search for various evaluation tasks.}
\label{fig:lrsearch}
\end{figure*}
\section{Performance Summaries for all Experiments}\label{secA4}
Here we present full performance summaries for all experiments from the paper. Each box in Figures \ref{fig:results_ic}, \ref{fig:results_id}, and \ref{fig:results_lm} contain 5 data points from each of 5 independent runs. SC is only visualized for the $LR_0$ indicated by the LR range test for the particular task (Section \ref{sec:range}).
\begin{figure*}
\caption{Image classification summary for all runs. Y-axis is test accuracy (higher is better).}
\label{fig:results_ic}
\end{figure*}
\begin{figure*}
\caption{Instance detection summary for all runs. Y-axis is mAP (higher is better).}
\label{fig:results_id}
\end{figure*}
\begin{figure*}
\caption{Language modeling summary for all runs. Y-axis is perplexity (lower is better).}
\label{fig:results_lm}
\end{figure*}
\begin{figure*}
\caption{5 ARC runs for PTB Language Modeling with $LR_0=1.0$. The blue ($\text{arc}_0$) outlier run had a dramatically worse initialization than the other 4 runs, but nonetheless made good learning rate adjustments.}
\label{fig:ptbarc}
\end{figure*}
Note that there is an extreme outlier for ARC in PTB language modeling with $LR_0=1.0$ (Figure \ref{fig:results_lm}). As Figure \ref{fig:ptbarc} demonstrates, this is a result of an extremely unlucky initialization rather than any mistake on the part of ARC.
\section{ARC Invocation Frequency}\label{secA5}
ARC executes at the per-epoch rather than per-step timescale, but this leaves the question of how many epochs should elapse in between invocations. An easy answer would be `as often as possible', but invocations which are too frequent may lead to greater instability or vulnerability to short-horizon bias. To find an ideal frequency we trained 5 ARC models, and then deployed them on top of a 9 layer residual network to train on the CIFAR10 dataset at $LR_0$ values of 0.01, 0.001, and 0.0001. For each $LR_0$ we experimented with invoking ARC once every $n$ epochs for $n\in[1,6]$. Every experimental configuration was repeated 5 times, with maximum accuracies summarized in Figure \ref{fig:Frequency}. For $n\in[1,3]$ the training proceeded for 30 epochs. For $n\in[4,6]$ we trained for $10n$ epochs such that each configuration would have an equal opportunity to adjust the LR. From the results in Figure \ref{fig:Frequency} we conclude that once every 3 epochs is the most frequent invocation schedule we can currently support without harming performance.
\begin{figure*}
\caption{Effect of ARC frequency on final performance on CIFAR10. Each box contains 25 data points (5 ARC models * 5 runs). Y-axis is test accuracy.}
\label{fig:Frequency}
\end{figure*}
\end{appendices}
\end{document} |
\begin{document}
\preprint{EHU-FT/0104}
\def\langle{\langle} \def\rangle{\rangle} \def\omega{\omega} \def\Omega{\Omega} \def\varepsilon{\varepsilon} \def\widehat{\widehat} \newcommand{\begin{equation}}{\begin{equation}} \newcommand{\end{equation}}{\end{equation}} \newcommand{\begin{eqnarray}}{\begin{eqnarray}} \newcommand{\end{eqnarray}}{\end{eqnarray}} \newcommand{\int_{-\infty}^\infty}{\int_{-\infty}^\infty} \newcommand{\int_0^\infty}{\int_0^\infty} \def\wh{T}_{AB}{\widehat{T}_{AB}} \def\wh{a}_p{\widehat{a}_p} \def\wh{a}_{q}{\widehat{a}_{q}} \def\wh{a}_{r}{\widehat{a}_{r}} \def\wh{a}_p^\dag{\widehat{a}_p^\dag} \def\wh{a}_{q}^\dag{\widehat{a}_{q}^\dag} \def\wh{a}_{r}^\dag{\widehat{a}_{r}^\dag} \def\wh{\psi}(X){\widehat{\psi}(X)} \def\wh{\psi}^\dag(X){\widehat{\psi}^\dag(X)} \def\wh{\psi}^\dag(y){\widehat{\psi}^\dag(y)}
\def\wh{v}_\alpha(X){\widehat{v}_\alpha(X)} \def\wh{v}_\alpha^\dag(X){\widehat{v}_\alpha^\dag(X)}
\title{Quantum times of arrival for multiparticle states}
\author{A. D. Baute} \affiliation{Fisika Teorikoaren Saila, Euskal Herriko Unibertsitatea, 644 P.K., 48080 Bilbao, Spain} \affiliation{Departamento de Qu\'\i mica-F\'\i sica, Universidad del Pa\'\i s Vasco, Apdo. 644, Bilbao, Spain} \author{I. L. Egusquiza} \affiliation{Fisika Teorikoaren Saila, Euskal Herriko Unibertsitatea, 644 P.K., 48080 Bilbao, Spain} \author{J. G. Muga} \affiliation{Departamento de Qu\'\i mica-F\'\i sica, Universidad del Pa\'\i s Vasco, Apdo. 644, Bilbao, Spain}
\date{\today}
\begin{abstract} Using the concept of crossing state and the formalism of second quantization, we propose a prescription for computing the density of arrivals of particles for multiparticle states, both in the free and the interacting case. The densities thus computed are positive, covariant in time for time independent hamiltonians, normalized to the total number of arrivals, and related to the flux. We investigate the behaviour of this prescriptions for bosons and fermions, finding boson enhancement and fermion depletion of arrivals. \end{abstract}
\pacs{03.65.-w}
\maketitle
\section{Introduction}
A long standing issue in the theory and experiment of quantum mechanics has been that of measuring and formalizing time observables. In the last two decades a substantial body of work has been produced clarifying theoretically and measuring experimentally quantities such as dwell times \cite{Muga01}, tunneling times, or arrival times \cite{ML00}. In particular, many recent papers have challenged the classical work of Allcock, who denied the possibility of defining a quantum arrival-time concept \cite{Allcock69,Allcock69a,Allcock69b}. In fact, these theoretical efforts and difficulties concerning arrival times have been essentially decoupled from the daily practice of many laboratories, where time-of-flight (TOF) methods are routinely used. One reason for such a divorce is that, in most cases, a classical analysis of the translational motion and the associated arrival-time distribution is sufficient. It is now the case, however, that the development of laser cooling techniques is bringing the quantum nature of the atomic dynamics to the fore, thus approaching the conditions for testing several proposed time-of-arrival (TOA) theoretical distributions in a regime that differs from the classical approximation.
Yet another difficulty for a comparison and further interaction between experiment and theory is the absence, up to now, of a TOA theory for multiparticle systems. While the possibility of detecting individual atoms with nanosecond time resolution in specific TOF experiments is open \cite{RSBPNBWA01}, in the generic case the TOF spectra are produced by clouds of many particles that may interact with each other or/and with an external field. The aim of this paper xis to provide a quantum TOA theory which is applicable for the generic (one dimensional) multiparticle case using the formalism of second quantization (see for example \cite{Baym74}), together with the crossing states introduced in \cite{BEMS00} and developed further in \cite{BEM01b}. We shall also portray several numerical examples to illustrate the phenomena of boson enhancement and fermion depletion of common arrivals.
\section{Time of arrival of a single particle}
One of the major hindrances to the consideration of time observables in the framework of standard quantum mechanics was Pauli's theorem, which, simply put, states that no self-adjoint operator can exist that has canonical commutation relations with a self-adjoint bounded or semibounded hamiltonian, thus implying that the standard recipe associating self-adjoint operators to observables cannot work for time.
Nonetheless, Aharonov and Bohm considered the motion of free particles as a clock to measure time, and introduced a time operator by symmetrizing the classical expression for the time when a particle, initially at the origin and with momentum $p$, passes point $x$, that is, $t=mx/p$. With a sign change this becomes the time of arrival at the origin of a free particle that, at time $t=0$, is at position $x$ with momentum $p$ \cite{MSP98,MLP98}. If the arrival occurs at $X$ rather than at the origin, then the corresponding ``Aharonov-Bohm time-of-arrival operator'' takes the form
\begin{equation} \wh{T}_{AB}(X)=\frac{m}{2}\left[(X-\widehat{x})\frac{1}{\widehat{p}} +\frac{1}{\widehat{p}}(X-\widehat{x})\right]\,. \end{equation}
For all practical purposes, this expression fulfills all the properties one would expect of an operator associated with the observable quantity time-of-arrival for free particles on the line \cite{AB61} (for more details on this and the following topics, see \cite{ML00}). It cannot be applied onto states with non vanishing zero momentum, which has been at times regarded as a drawback \cite{GRT96,BR01}. In fact, this ``difficulty'' is perfectly physical, and mirrors the classical divergence of the time of arrival when the particle's momentum tends to zero. It also explains how Pauli's theorem can be circumvented: Aharonov and Bohm's time operator $\wh{T}_{AB}$ is a maximally symmetric operator, therefore not self-adjoint. Other steps had to be taken before maximally symmetric operators and their concomitant POVMs (positive operator valued measures or generalized non-orthogonal resolutions of the identity) were understood physically, however.
Although the faith in Pauli's theorem could have been slightly shaken by Aharonov and Bohm's proposal, the issue seemed to be settled after the important series of papers of Allcock \cite{Allcock69,Allcock69a,Allcock69b}, which apparently put to rest all hope to obtain a sensible prescription for the quantum prediction of times of arrival. Even so, some adventurous souls kept on searching for alternative formulations within quantum mechanics. Kijowski in 1974 \cite{Kijowski74} put forward a procedure to compute time-of-arrival probability densities for the free particle case in a purely axiomatic way (see also similar later work by Werner \cite{Werner86}).
With the advent of a better understanding of positive operator valued measures (also known as generalized decompositions of the identity or non-orthogonal measurements) \cite{SV81,Holevo82,Peres93,BGL95}, the force of Pauli's argument was strongly diminished. In fact, it has been possible to show the relation between Aharonov and Bohm's time-of-arrival operator and Kijowski's distribution: they follow naturally one from each other \cite{MPL99,EM00a}. $\wh{T}_{AB}$ is the first operator moment of a POVM whose distribution function over a given state is the corresponding Kijowski distribution. Kijowski's distribution $\Pi(t)$ of times of arrival at $X$ for particles in a state $|\psi(0)\rangle\equiv |\psi(t=0)\rangle$, in the free particle case, can be written as
\begin{equation}
\Pi(t,X)=\sum_{\alpha=\pm}|\langle\psi(0)|t,\alpha\rangle|^2\,,\label{pite} \end{equation}
where $|t,\alpha\rangle$ are the two generalized eigenvectors of
$\wh{T}_{AB}$ with eigenvalue $t$, $\wh{T}_{AB}|t,\alpha\rangle=t|t,\alpha\rangle$ for $\alpha=\pm$. Note that the above mentioned domain ``difficulty'' of $\wh{T}_{AB}$ does not apply to the bilinear functional $\Pi(t)$, which may be defined for arbitrary physical states regardless of their behaviour at $p=0$ \cite{MLP98}.
The fact that $\wh{T}_{AB}$ is not a self-adjoint operator is clearly identified from the non-orthogonality of the complete basis $\left\{|t,\alpha\rangle\right\}_{t,\alpha=\pm}$. The eigenvectors are related to each other by means of the relation
\begin{equation}
|t,\alpha\rangle=e^{i\widehat{H}(t-t')/\hbar}|t',\alpha\rangle \end{equation}
which assures the invariance of the distribution with respect to time traslations. In particular,
\begin{equation}\label{talpha}
|t,\alpha\rangle=e^{i\widehat{H}t/\hbar}|v_{\alpha}\rangle, \end{equation}
where we have used a special notation for the $t=0$ (generalized) eigenvectors or ``crossing states'',
$|v_{\alpha}\rangle=|t=0,\alpha\rangle$, where again $\alpha$ stands for either $+$ or $-$ (we will not be denoting explicitly the point of arrival $X$, which is part of the definition of these states, but it is always implied).
In terms of these states we may rewrite Kijowski's distribution of times of arrival at the point $X$ as
\begin{equation}
\Pi(t,X)=\sum_{\alpha=\pm}|\langle\psi(t)|v_{\alpha}\rangle|^2. \label{pive} \end{equation}
This also suggests a rewritting of the distribution in terms of an operator for the density of arrivals at point $X$,
$\widehat{\pi}(X)\equiv\sum_\alpha |v_{\alpha}\rangle
\langle v_{\alpha}|$,
\begin{equation}\label{pigen}
\Pi(t,X)=\langle\psi(t)|\widehat{\pi}(X)|\psi(t)\rangle. \end{equation}
Consider now the explicit form of the states $|v_{\alpha}\rangle$ in momentum representation,
\begin{equation}
\langle p|v_{\alpha}\rangle= \left(\frac{\alpha p}{h m}\right)^{1/2}\Theta(\alpha p) e^{-ipX/\hbar}\,,\label{vmomen} \end{equation}
where $\Theta(\cdot)$ is Heaviside's unit step function. The correct correspondence of $\Pi(t,X)$ with the classical case becomes now evident. If the non commutativity of position and momentum operators could be neglected, $\widehat{\pi}$ would correspond to the sum of the moduli of the fluxes that cross $X$ from both sides. Also important is the fact that in a classical setting the corresponding dynamical variable provides the arrival distribution irrespective of the dynamics and interaction potentials. In other words, Eq. (\ref{pigen}) generalizes the free motion case in a natural and simple way for arbitrary interaction potentials, a task that could not be carried out using the original axiomatic procedure of Kijowski or by quantizing the classical time of arrival for each particular potential (the expressions are not analytically known in general and pose formidable ordering problems).
Underlying this rewriting of the time-of-arrival distribution a change of emphasis is to be found: whereas in Eq. (\ref{pite}) the time-of-arrival distribution is obtained from the overlap of the \emph{initial} wavefunction with the states associated with arrival at the instant $t$, be it from the left ($\alpha=+1$) or the right ($\alpha=-1$), in Eq. (\ref{pive}) it is obtained as the overlap of the \emph{evolved} wavefunction with the \emph{constant} states
$|v_{\alpha}\rangle$ that measure arrivals.
The first point of view is, in a way, predictive: given the initial state of the particle, one can predict when the arrivals will occur.
In the general case, with interacting potentials, this view may also be adopted with $|t,\alpha\rangle$ given by Eq.(\ref{talpha}), where the appropriate Hamiltonian is put in each case.
{}From the second perspective, which could be termed ``unconditional'', the arrival or otherwise of a particle at $x=0$ is directly measured in physical space at every instant, using local definitions that are in no way conditioned by the different potentials in which the particles might be moving.
This point of view, inspired by Wigner's formalization of the time-energy uncertainty relation \cite{Wigner72}, was advocated in \cite{BEMS00,BEM01b}, where the properties of the crossing states
$|v_{\alpha}\rangle$ were examined, and Eq. (\ref{pive}) was put forward as an expression of density of arrivals also for the case of interaction. In \cite{BEM01b} we rewrote some other distributions that had been proposed in the literature for time-of-arrival distributions of particles in a potential (\cite{LJPU99}, later superseded by \cite{LJPU00}; see also \cite{Leon00}) in terms of crossing states, and showed that those defined in Eq. (\ref{vmomen}) were the only ones considered that led to classical correspondence with the properties expected of such distributions.
Another particularly relevant aspect of the change of emphasis is that it helps to understand that Eq. (\ref{pive}) need no longer be normalized to unity. In which case $\Pi(t,X)$ is to be understood as a density of arrivals of one particle: there might be a non zero probability for the particle never arriving at $X$, or, if the interacting potential were confining (such as the harmonic oscillator), recurrences would appear corresponding to many different arrivals. Notice that $\Pi(t,X)$ is a density of arrivals, not of first arrivals only.
\section{Second quantization and time of arrival}
Even though TOF experiments with single atoms might be available in not too distant a future, we need to understand better how to predict time-of-arrival distributions for multiparticle systems. Most suited for such a purpose is the formalism of second quantization. One must first realize that the distribution of arrivals is a property of the same nature as the current density, or the kinetic enery, namely, it is obtained as the sum of ``single particle'' contributions, irrespective of the external or internal interactions affecting the $N$-particle system. This is a key observation to discard outright, even for free motion, quantizations that would provide two-particle terms.
Let $\wh{a}_p$ and $\wh{a}_p^\dag$ represent the annihilation and creation operators that respectively eliminate and create a plane wave of momentum $p$. Similarly, $\widehat{\psi}(x)$ and $\widehat{\psi}^\dagger(x)$ act on the vacuum disposing of and creating a particle at point $x$. The canonical commutation relations read
\begin{equation} \left[\wh{a}_p,\wh{a}_{q}^\dag\right]_\pm=\delta(p-q)\,,\qquad{\rm and}\quad \left[\widehat{\psi}(x),\widehat{\psi}^\dagger(y)\right]_\pm=\delta(x-y)\,,\label{ccr} \end{equation}
where, as usual, $[,]_\pm$ stands for the commutator in the case of bosons and for the anticommutator when fermions are involved. The position operator is written as
\[\widehat{x}=\int_{-\infty}^{+\infty}dx\,x\, \widehat{\psi}(x)\widehat{\psi}^\dagger(x)\,,\]
and the inverse of the momentum operator as
\[\widehat{p}^{-1}= \int_{-\infty}^{+\infty}dp\,\frac1p\, \wh{a}_p^\dag\wh{a}_p\,,\]
{}from which the following form for a generalization of the time-of-arrival operator of Aharonov and Bohm might be inferred ($X=0$):
\begin{eqnarray} \wh{T}_{AB}^{(1)}&=&-\frac{m}2\left(\widehat{x}\widehat{p}^{-1}+\widehat{p}^{-1}\widehat{x}\right) =\nonumber\\ &=&-\frac{m}{4\pi\hbar}\int_{-\infty}^\infty dx\,dp\,dq\,dr\, \frac{x e^{i(r-q)x/\hbar}}p \left[\left(\delta(p-q)+\delta(p-r) \right)\wh{a}_{q}^\dag\wh{a}_{r} + 2 \wh{a}_p^\dag\wh{a}_{q}^\dag\wh{a}_{r}\wh{a}_p\right]\,. \label{absecq} \end{eqnarray}
In this expression one can recognize a one-particle component but also a two-particle one. As pointed out above, this leads us to discard this procedure, because of its unphysicality.
At any rate, $\wh{T}_{AB}$ is only valid for the free particle case, a further limitation of this route. The proper quantization procedure for the multiparticle case starts, as noted above, by recognizing the additive character of the time of arrival in terms of single particle contributions.
The basic trick is that for additive quantities taking the form of a sum of single particle operators,
\begin{equation} \widehat{G}=\widehat{g}_1+\widehat{g}_2+_....+\widehat{g}_N\,, \end{equation}
each of which has matrix elements $g_{ji}=\langle j|\widehat{g}|i\rangle$ in a complete (single particle) basis, the multiparticle operator in second quantized form is given by the simple expression
\begin{equation} \widehat{G}=\sum_{ij}g_{ji}a^\dagger_j a_i\,, \end{equation}
where $a_i$ and $a^\dag_j$ are the i-th annihilation and j-th creation operators. That is, they connect states $|i\rangle$ and $|j\rangle$
respectively with the vaccum state ($a_i|i\rangle=|0\rangle$ and
$|j\rangle=a^\dag_j|0\rangle$).
In the case of the arrival density operator we can directly apply this procedure in momentum representation. An even more compact expression is obtained by using the crossing states, to generate
\emph{crossing operators}, both annihilation and creation. Consider any (generalized) one particle state $|\varphi\rangle$. We can write the annihilation and creation operators associated with the state as
\[
\widehat{\varphi}=\int_{-\infty}^\infty dp\,\langle\varphi|p\rangle\wh{a}_p\,;\qquad
\widehat{\varphi}^\dag=\int_{-\infty}^\infty dp\,\langle p|\varphi\rangle\wh{a}_p^\dag\,. \]
On applying this procedure to the crossing states, we obtain the crossing operators
\begin{eqnarray}
\wh{v}_\alpha(X) & = & \int_{-\infty}^{+\infty}dp\,\langle v_{\alpha}|p\rangle\wh{a}_p= \int_{-\infty}^{+\infty}dp\,\left(\frac{\alpha p}{h m}\right)^{1/2} \Theta(\alpha p) e^{ipX/\hbar}\wh{a}_p\,;\label{crucea}\\
\wh{v}_\alpha^\dag(X) & = & \int_{-\infty}^{+\infty}dp\,\langle p|v_{\alpha}\rangle\wh{a}_p^\dag= \int_{-\infty}^{+\infty}dp\, \left(\frac{\alpha p}{h m}\right)^{1/2}\Theta(\alpha p) e^{-ipX/\hbar}\wh{a}_p^\dag\,.\label{crucec} \end{eqnarray}
Let us now put together Eqs. (\ref{crucea}) and (\ref{crucec}) with Eq. (\ref{pive}) to write the \emph{arrival density operator} $\widehat{\Pi}(X)$ for arrivals at $X$ in second quantized form,
\begin{equation} \widehat{\Pi}(X)=\sum_{\alpha=\pm}\wh{v}_\alpha^\dag(X)\wh{v}_\alpha(X)= \int_{-\infty}^\infty dp\,dq\,\frac{\sqrt{pq}}{h m}\Theta(pq) e^{i(q-p)X/\hbar}\wh{a}_p^\dag\wh{a}_{q}\,.\label{picero} \end{equation}
The left and right arrivals density operators $\widehat{\Pi}_+(X)$ and $\widehat{\Pi}_-(X)$ are similarly defined as
\[ \widehat{\Pi}_\alpha(X)=\wh{v}_\alpha^\dag(X)\wh{v}_\alpha(X)\,. \]
We may also write from Eq. (\ref{picero}) the corresponding operator in Heisenberg picture, whose expectation value over the initial state will give us the density of arrivals at point $X$ at instant $t$,
\begin{eqnarray} \widehat{\Pi}(t,X)&=&\widehat{U}^\dag(t)\widehat{\Pi}(X)\widehat{U}(t)= \sum_{\alpha=\pm} \widehat{v}_\alpha^\dagger(X,t) \widehat{v}_\alpha(X,t) \\ &=& \int_{-\infty}^\infty dp\,dq\,\frac{\sqrt{pq}}{h m}\Theta(pq) e^{i(q-p)X/\hbar}\wh{a}_p^\dag(t)\wh{a}_{q}(t)\,,\label{pit} \end{eqnarray}
where $\wh{a}_p^\dag(t)$ and $\wh{a}_{q}(t)$ are the time evolved creation and annihilation operators, with evolution operator $\widehat{U}(t)$, i.e. $\wh{a}_p^\dag(t)=\widehat{U}^\dag(t)\wh{a}_p^\dag \widehat{U}(t)$ and similarly $\wh{a}_p(t)=\widehat{U}^\dag(t)\wh{a}_p \widehat{U}(t)$.
The density of arrivals at instant $t$ at point $X$ for a generic state $|\psi\rangle$ may thus be written as
\[
\Pi(t,X;\psi)=\langle\psi (0)|\widehat{\Pi}(t,X)|\psi (0)\rangle=
\langle \psi(t)|\widehat{\Pi}(X)|\psi(t)\rangle\,. \]
This expression agrees with Eq. (\ref{pive}) whenever $|\psi\rangle$ is a one particle state. Even though it is not immediately apparent from expression (\ref{pit}) that we are obtaining positive semidefinite distributions, this is indeed the case by construction: $\widehat{\Pi}(t,X)$ is a positive operator because it is a sum of two terms of the form $\widehat{A}^\dag\widehat{A}$.
Furthermore, the one particle operator that is an extension of Aharonov and Bohm's time-of-arrival operator at position $X$ for many particles, even in the interacting case, is straightforwardly written as
\[\widehat{T}_X=\int_{-\infty}^{+\infty}dt\,t\,\widehat{\Pi}(t,X)= \int_{-\infty}^\infty dt\,dp\,dq\,\frac{\sqrt{pq}}{h m}\Theta(pq) e^{i(q-p)X/\hbar}\,t\, \wh{a}_p^\dag(t)\wh{a}_{q}(t)\,. \]
By construction this is simply a one particle operator, which coincides with $\wh{T}_{AB}$ over states whose content is just one free particle.
If the evolution of the system is governed by the free particle Hamiltonian it is easy to check that the integral over time of the arrival-density operator $\widehat{\Pi}^{\rm free}(t,X)$ sums to the total particle number operator,
\[ \int_{-\infty}^{+\infty}dt\,\widehat{\Pi}^{\rm free}(t,X) =\int_{-\infty}^{+\infty}dp\,\wh{a}_p^\dag\wh{a}_p=\widehat{N}\,.\]
This is no longer the case whenever the evolution operator is not the free one; anyhow, we deduce from this expression that the arrival density is normalized to the total number of arrivals. Notice that in the interacting case the total number of arrivals need not coincide with the total particle number, it may be smaller or bigger.
A particularly important property of the arrival-density operator is that the density of arrivals $\Pi(t,X;\psi)$ over any state is covariant in time if the Hamiltonian is independent of time (as has been assumed all along). Even though the properties of covariance, positivity, and correct classical correspondence do not, by themselves, completely fix the density of times of arrival, they are minimal requirements, the lack of which would seriously impair any proposal.
Even though in the presentation above we have restricted ourselves to pure states, there is no problem in extending our proposal to mixed states, as follows:
\[ \Pi(t,X;\widehat{\rho})={\rm Tr}\left(\widehat{\Pi}(t,X)\widehat{\rho}(0)\right)= {\rm Tr}\left(\widehat{\Pi}(X)\widehat{\rho}(t)\right)\,. \]
For the sake of completeness, let us note down the flux operator for many particles, in Schr\"odinger's picture,
\begin{eqnarray} \widehat{j}(X)&=&\frac{-i\hbar}{2m} \left\{\wh{\psi}^\dag(X)\partial_X\wh{\psi}(X)-\left[\partial_X\wh{\psi}^\dag(X)\right]\wh{\psi}(X)\right\}= \nonumber\\ &=&\frac1{2 h m}\int_{-\infty}^\infty dp\,dq\,e^{i(q-p)X/\hbar}(p+q)\wh{a}_p^\dag\wh{a}_{q}\,. \label{jotacerot} \end{eqnarray}
or in Heisenberg's picture as
\begin{equation} \widehat{j}(t,X)=\widehat{U}^\dag(t)\widehat{j}(X)\widehat{U}(t)=\frac1{2 h m} \int_{-\infty}^\infty dp\,dq\,e^{i(q-p)X/\hbar}(p+q)\wh{a}_p^\dag(t)\wh{a}_{q}(t)\,,\label{jotat} \end{equation}
(again assuming that the Hamiltonian is independent of time). Notice that the flux, defined in this standard manner, is a one-particle operator.
A straightforward comparison of Eqs. (\ref{pit}) and (\ref{jotat}) reveals the differences and similarities between $\widehat{\Pi}$ and the flux. In the former a geometric mean of the momenta takes the place of the arithmetic mean in the latter. Moreover, $\Pi$ counts the case when $p$ and $q$ are both negative as a positive contribution to the arrival density, whereas the same case counts as a negative flux contribution in (\ref{jotat}). This means that the quantity that tends classically to the flux is $\Pi_+-\Pi_-$ rather than $\Pi$ itself.
\section{Free particles: boson enhancement and fermion depletion}
We have already made out several properties of the proposed arrival-density operator, namely positivity, covariance, one-particle status, classical limit, and normalization to total number of arrivals. There is an obvious missing element yet, in that we have not investigated so far whether the fermionic or bosonic character of the particles involved is somehow reflected in the properties of the distributions of times of arrival, as is to be expected.
In fact, this distinction between fermions and bosons is already present in the proposed arrival-density distributions, as we will be showing in this section. In order to portray this new property it is enough to consider simply two-particle states, of generic form
\[
|\psi\rangle=\int_{-\infty}^\infty dp_1\,dp_2\,\psi(p_1,p_2)|p_1,p_2\rangle
=\frac1{\sqrt{2}}\int_{-\infty}^\infty dp_1\,dp_2\,\psi(p_1,p_2)\wh{a}_{q}^\dag\wh{a}_p^\dag|0\rangle\,,\]
both for bosons and fermions, where $|0\rangle$ is the vacuum state, and the normalization condition reads
\[ \frac12\int_{-\infty}^\infty dp_1\,dp_2\,\left[\overline{\psi(p_1,p_2)} \pm\overline{\psi(p_2,p_1)}\right]\psi(p_1,p_2)=1\,, \]
where the upper sign corresponds to bosons and the lower one to fermions.
Consider $\psi(p_1,p_2)$ given as
\begin{equation} \psi_\pm(p_1,p_2)=
\frac1{\sqrt{2\left(1\pm|\langle\chi_a|\chi_b\rangle|^2\right)}} \left[\chi_a(p_1)\chi_b(p_2)\pm\chi_a(p_2)\chi_b(p_1)\right]\,, \label{simanti} \end{equation}
which fulfills the normalization requirement if $\chi_a$ and $\chi_b$ are normalized one particle wavefunctions. Quite obviously,
$\langle\chi_a|\chi_b\rangle$ stands for $\int dp\,\overline{\chi_a(p)}\chi_b(p)$. In order to compare with the case of distinguishable particles, we shall also be using
\begin{equation} \psi_{\rm d}(p_1,p_2)=\chi_a(p_1)\chi_b(p_2)\,.\label{distin} \end{equation}
Since the arrival-density operator $\widehat{\Pi}(t,X)$ is a one-particle operator, the density of arrivals over the state $|\psi_+\rangle$, say, can be reorganized as
\[\langle\psi_+(0)|\widehat{\Pi}(t,X)|\psi_+(0)\rangle=\frac1{N_+^2}
\sum_{i,j=a,b}\langle\chi_j|\chi_i\rangle\Pi_{ij}(t,X)\,,\]
where $N_+=\sqrt{2\left(1+|\langle\chi_a|\chi_b\rangle|^2\right)}$, and
\[\Pi_{ij}(t,X)=\sum_{\alpha=\pm}\langle\chi_i|\widehat{v}_\alpha^\dag(X,t)
\widehat{v}_\alpha(X,t)|\chi_j\rangle\,. \]
Over the fermionic state $|\psi_-\rangle$ the cross terms carry a negative sign in front. The evolved crossing states are given by Eqs. (\ref{crucea}) and (\ref{crucec}) on substituting $\wh{a}_p$ and $\wh{a}_p^\dag$ by $\wh{a}_p(t)$ and $\wh{a}_p^\dag(t)$, respectively.
On the other hand, the evaluation of the expectation value of the evolved arrival density operator over the state $|\psi_{\rm d}\rangle$, which computes the density of arrivals for two distinguishable particles in such a state, produces just the two diagonal terms, i.e.
\[\langle\psi_{\rm d}(0)|\widehat{\Pi}(t,x)|\psi_{\rm d}(0)\rangle =\Pi_{aa}(t,x)+\Pi_{bb}(t,x)\,.\]
It should be observed that these computations are general in that they hold true for the case of interacting particles as well, as long as the states have the form given above. These results indicate that fermions and bosons (antisymmetric and symmetric states) present cross terms in the density of arrivals completely analogous to those that in spatial density signal the statistics of the particles. In fact, the formalism of second quantization carries in itself the fermionic or bosonic character of the particles concerned, through the commutation relations.
As a consistency check one may compute for the above states,
$|\psi_\pm\rangle$ and $|\psi_d\rangle$, the corresponding reduced one particle density operators $\widehat{\rho}^{(j)}$, $j=1,2$ ($\rho^{(1)}=\rho^{(2)}$ for
$|\psi_\pm\rangle$) and note that in all three cases $\Pi(t,X)=\sum_{j} \Pi^{(j)}(t,X)$, where
\[\Pi^{(j)}(t,X)={\rm Tr}_j\left[\widehat{\Pi}(t,X) \widehat{\rho}^{(j)}\right]\,. \]
in agreement with the one-particle character of the arrival-time distribution.
\begin{figure}
\caption{ The solid line corresponds to distinguishable particles
($|\psi_d\rangle$), the dots show the density of arrivals for bosons
(symmetric state $|\psi_+\rangle$), and the dashed lines that of fermions (antisymmetric state $|\psi_-\rangle$). The states are defined by Eqs. \ref{simanti} and \ref{distin}, with $\chi_a$ and
$\chi_b$ gaussian states with minimum uncertainty at $t=0$, their central positions being $\langle\chi_a|\widehat{x}|\chi_a\rangle=-3.5$ and
$\langle\chi_b|\widehat{x}|\chi_b\rangle=0$. In both cases, their central positions in momentum space are at 3, and their spatial widths $\Delta x=1$ (where $\Delta x$ is the square root of the spatial variance). The point of observation is $X=3$, and the mass $m=1$. All magnitudes are expressed in atomic units.}
\label{mapt2}
\end{figure}
The difference between bosons, fermions and distinguishable particles
($|\psi_d\rangle$) is quite apparent in Fig. \ref{mapt2}. The one particle states $\chi_a$ and $\chi_b$ are gaussians with a spatial separation between them (in atomic units, $3.5$), while their width is
$1$ (a.u.). Correspondingly, there are two main arrival times (maxima of the arrival densities) for all three cases. Even so, the two sets of principal arrivals for bosons (symmetric state) are much closer together and much less differentiated than for distinguishable particles ($|\psi_d\rangle$), which in turn present closer and less differentiated maxima when compared to the fermionic (antisymmetric) case. It should be noticed that in this situation of free motion the distributions are normalized to 2, as can be readily checked in this numerical simulation.
\section{Interacting particles}
Consider now a pair of interacting particles, be they distinguishable, bosonic or fermionic, moving in otherwise free space. The two-particle subspace of Fock space can be rewritten in center of mass and relative coordinates, and we shall consider for simplicity factorized states of the form
\[ \psi(p_1,p_2)=\chi(P)\phi(p)\,, \]
where $P=p_1+p_2$ is the center of mass momentum and $p=(p_1-p_2)/2$ the relative one. Under exchange of the particles $P$ is unchanged, while $p$ flips sign. So in order to ensure that the state is bosonic we are forced to use even functions $\phi_+(p)=\phi_+(-p)$, whereas the fermionic case demands odd functions $\phi_-(p)=-\phi_-(-p)$. The total mass is $2m$, while the reduced mass $\mu$ pertaining to the relative system is $m/2$. The normalization condition is translated into the requirement that $\phi_\pm$ and $\chi(P)$ be normalized to unity.
In what follows we shall assume that the center of mass function is gaussian with minimum uncertainty product at $t=0$. As to the internal states, they will be evolving in a harmonic oscillator potential. We shall consider stationary and coherent internal states.
\begin{figure}
\caption{ Arrival density of two identical particles in an internal stationary state of the harmonic oscillator. The center of mass state is gaussian, initially centered on $x=0$ and $p=4$ with width $\Delta x=0.5$. Circles correspond to the (internal) ground state, dashes to the first excited state, solid line to the second excited state, dot-dash to the third one. The internal frequency is $\omega=\sqrt{0.02}$ and the oscillation period is $T\approx 44.4$. The point of crossing is $x=3$. All magnitudes in atomic units.}
\label{mapt4}
\end{figure}
Figs. \ref{mapt4} and \ref{mapt5} represent two different sets of cases concerning internal stationary states. The ground state and the even excited states are symmetric (bosonic), whereas the odd numbered excited states are antisymmetric (fermionic). The differences between Figs. \ref{mapt4} and \ref{mapt5} are due to the different ratios between internal energy and that of the center of mass motion. In Fig. \ref{mapt4} the internal oscillations are much slower than the center of mass motion, so the humps of the internal spatial wavefunction appear, somewhat distorted for later times because of the spreading, in the arrival density. However those humps are smoothed over in Fig. \ref{mapt5} due to the much slower center of mass motion relative to the internal motion. Correspondingly, the integral of the curves in Fig. \ref{mapt4} is very nearly 2, whereas there is a significant increase of this number in Fig. \ref{mapt5} for the excited states. The higher the excitation the broader the state is, spatially, thus leading to more crossings.
\begin{figure}
\caption{ As in figure \ref{mapt4}, with the center of mass central momentum changed to 1, $\Delta x=1$ and the internal frequency to $\omega=\sqrt{2}$. $T\approx 4.4$}
\label{mapt5}
\end{figure}
We have also studied a case where the internal motion is time dependent. If it is fast enough with respect to the translational motion, a peak structure corresponding to several oscillations may be observed. Let us consider, at time $t=0$, symmetric and antisymmetric combinations of coherent states, of the form
(remember that the coherent state $|z\rangle$ is given by $\exp(-|z|^2/2)\sum_{n=0}^\infty(z^n/\sqrt{n!})|n\rangle$, where $|n\rangle$ is the n-th excited state of the harmonic oscillator hamiltonian)
\begin{equation}
|\phi_\pm\rangle=\frac1{\sqrt{2}}\left[|z\rangle\pm| \bar{z}\rangle\right]\,,\label{coherent} \end{equation}
where $\bar{z}$ is the complex conjugate of $z$. We shall take in particular $z=i$. In the relative motion space $|i\rangle$ is a minimum uncertainty product gaussian centered at the origin with average momentum $(2\mu\omega\hbar)^{1/2}$ and spatial variance $\hbar/(2\omega m)$. As time progresses it oscillates back and forth along the relative motion coordinate with period $T=2\pi/\omega$.
\begin{figure}
\caption{ Arrivals density (solid line for fermions and dashed line for bosons) and flux (triangles for fermions and circles for bosons) for the internal states defined in Eq. (\ref{coherent}), with $z=i$ and internal frequency $\omega=\sqrt{0.02}$. The initial center of mass state is a gaussian with central position at $x=0$ and central momentum $p=4$. The width is $\Delta x=0.5$. The point of arrival is $X=3$. All magnitudes in atomic units.}
\label{mapt7}
\end{figure}
In Fig. \ref{mapt7} the translational motion is faster than the oscillations so we see just one peak for the symmetric case and two maxima for the antisymmetric (fermionic) case. Since there is hardly any component of negative momentum, there is no distinction between flux and density of arrivals.
\begin{figure}
\caption{ Arrivals density (solid line for fermions and dashed line for bosons) for the internal states defined in Eq. (\ref{coherent}), with $z=i$ and internal frequency $\omega=\sqrt{2}$. The initial center of mass state is a gaussian with central position at $x=0$, central momentum $p=1$, and spatial width $\Delta x=1$. The point of arrival is $X=3$. All magnitudes in atomic units.}
\label{mapt8}
\end{figure} In contrast to Fig. \ref{mapt7}, the internal potential is much stronger in the situation depicted in Fig. \ref{mapt8}. As opposed to the case of stationary internal states, the tighter binding produces here oscillations that can be clearly seen.
\section{Discussion}
In this work we have proposed a general method for computing densities of arrivals (and related arrival density operators) for multiparticle states, that fulfill a number of quite sensible demands: positivity, covariance (if the evolution is homogeneous in time), related to a one particle operator, normalized to the total number of particles in the free case, related to the flux, and consistent with the classical arrival density. The analysis of the density of arrivals (i.e., the one point function of the arrival density operator) shows consistency with the results one would expect for bosons, fermions, and distinguishable particles. Numerical computations also show the behaviour expected, both for the free and the interacting case, and reveal a number of physical effects, hitherto unexplored. The proposed distribution is also applicable to the case of external interaction potentials as shown already for the single particle states.
The fact that the arrival-density operator is a one particle operator implies that in fact no distinction is made for the one point function (the density of arrivals) between the bosonic case and the symmetric states of distinguishable particles. The full difference will be seen in two-point and higher order functions, such as the arrival - arrival correlation function. In fact, on computing this two point correlation function, one sees immediately that the one particle component of the two point operator behaves in the natural way and can be substracted from the arrival - arrival correlation function to give the correlation function of pairs of arrivals,
$\langle\psi|:\widehat{\Pi}(t)\widehat{\Pi}(0):|\psi\rangle$, where $:\ :$ stands for normal ordering. The full analysis of this object we will leave for future work.
In this paper we have not been overly concerned with domain problems and the like. We know that the one particle time operator in the free case is only maximally symmetric and cannot be made self-adjoint, and we have no reason to expect that the interacting multiparticle case will be simpler in this respect. Nonetheless, this is not particulary relevant for our main interest, which lies in the computation of densities of arrival. Notice furthermore that the fact that Aharonov and Bohm's operator is not self-adjoint is really no hindrance to a full quantum mechanical analysis of its associated densities.
There might be many other alternative prescriptions for times of arrival. In the present state of knowledge, we do not feel able to discard those outright. However, by pushing to the multiparticle case the definitions used for one particle, this analysis becomes more amenable to experimental test.
\begin{acknowledgments} We acknowledge G. Hegerfeldt for useful comments. This work is supported by Ministerio de Educaci\'on y Cultura (AEN99-0315), The University of the Basque Country (grant UPV 063.310-EB187/98), and the Basque Government (PI-1999-28). A. D. Baute acknowledges an FPI fellowship by Ministerio de Educaci\'on y Cultura. \end{acknowledgments}
\end{document} |
\begin{document}
\title{{f On the semiadditivity of the capacities associated with signed vector valued Riesz kernels} \newtheorem{teo}{Theorem} \newtheorem{co}[teo]{Corollary} \newtheorem{lemma}[teo]{Lemma} \newtheorem{defi}[teo]{Definition} \newtheorem{note}[teo]{Note} \newtheorem{prop}[teo]{Proposition} \newcommand{{\cal H}^{\alpha}}{{\cal H}^{\alpha}} \newcommand{{\cal H}^1}{{\cal H}^1} \newcommand{{\mathbb R}^n}{{\mathbb R}^n} \newcommand{{\mathbb R}^2}{{\mathbb R}^2} \newcommand{\varepsilon}{\varepsilon} \newcommand{\mathbb{N}}{\mathbb{N}} \newcommand{\mathbb{Z}}{\mathbb{Z}} \newcommand{\mathbb{C}}{\mathbb{C}} \newcommand{{\cal C}}{{\cal C}} \newcommand{\gamma_\alpha}{\gamma_\alpha} \newcommand{\gamma_{\alpha}^i}{\gamma_{\alpha}^i} \newcommand{\gamma_{\alpha,+}}{\gamma_{\alpha,+}} \newcommand{\gamma_{\alpha,op}}{\gamma_{\alpha,op}} \newcommand{\alpha}{\alpha} \begin{abstract}
The aim of this paper is to show the semiadditivity of the capacities associated to the signed vector valued Riesz kernels of homogeneity $-\alpha$ in ${\mathbb R}^n$, $0<\alpha<n$. \end{abstract}
\section{Introduction}
In this paper we study the capacity $\gamma_\alpha$ related to the signed vector valued Riesz kernels $x/|x|^{1+\alpha}$ in ${\mathbb R}^n$, $0<\alpha<n$. If $E\subset{\mathbb R}^n$ is a compact set and $0<\alpha<n$, one sets \begin{equation}\label{ga}
\gamma_\alpha(E)=\sup|\langle T,1\rangle|, \end{equation} where the supremum is taken over all real distributions $T$ supported on
$E$ such that $x_i/|x|^{1+\alpha}*T$ is a function in $L^\infty({\mathbb R}^n)$
and $\|x_i/|x|^{1+\alpha}*T\|_\infty\le 1$, for $1\leq i\le n$.
When $n=2$ and $\alpha=1$, by the celebrated result of X. Tolsa \cite{semiad}, $\gamma_1$ is basically analytic capacity. Recall that the analytic capacity of a compact set $E\subset\mathbb{C}$ is defined as \begin{equation}\label{acapacity}
\gamma(E)=\sup|\langle T, 1 \rangle|, \end{equation} the supremum taken over all complex distributions $T$ supported on $E$ whose Cauchy potential $1/z*T$ is a bounded function and
$\|1/z*T\|_\infty\leq 1$.
The case $\alpha=n-1$, $n\ge 2$, is also particularly relevant, because $\gamma_{n-1}$ coincides with the Lipschitz harmonic capacity, introduced in \cite{paramonov} to study problems of ${\cal C}^1$-approximation by harmonic functions in ${\mathbb R}^n$ (see also \cite{mp} and \cite{verderacm}). Notice that the fact that, in the plane, analytic capacity and $\gamma_1$ (Lipschitz harmonic capacity) are comparable cannot be deduced just by an inspection of (\ref{ga}) and (\ref{acapacity}). The reason is that the distributions involved in the supremum in (\ref{acapacity}) are complex.\newline
In \cite{laura1} one discovered the fact that if $0<\alpha<1$, then a compact set of finite $\alpha$-dimensional Hausdorff measure has zero $\gamma_\alpha$- capacity. This is in strong contrast with the situation for integer $\alpha$, in which $\alpha$-dimensional smooth hypersurfaces have positive $\gamma_\alpha$ capacity (see \cite{mp}). The case of non-integer $\alpha > 1$ is not well understood, although it was shown in \cite{laura1} that for Ahlfors-David regular sets the above mentioned result (for $0<\alpha<1$) still holds in this case (see also \cite{laura3}).\newline
In \cite{mpv}, the surprising equivalence between $\gamma_\alpha$, $0 <\alpha < 1$, and one of the well-known Riesz capacities of non-linear potential theory (see \cite[ Chapter 1, p. 38]{adamshedberg}) was established. It was shown that for some positive constant $C$,
\begin{equation}\label{crelle} C^{-1}C_{\frac 2 3(n-\alpha),\frac 3 2}(E)\le \gamma_\alpha(E)\le CC_{\frac 2 3(n-\alpha),\frac 3 2}(E). \end{equation}
Recall that the Riesz capacity $C_{s,p}$ of a compact set $E\subset{\mathbb R}^n$, $1 < p < \infty$, $0 < sp\leq n$, is defined by
$$C_{s,p}(E)=\inf\{\|\varphi\|_p^p: \varphi*\frac 1{|x|^{n-s}}\geq 1\,\,\mbox{ on }\,E\}$$ where the infimum is taken over all compactly supported infinitely di�erentiable functions on ${\mathbb R}^n$. The capacity $C_{s,p}$ plays a central role in understanding the nature of Sobolev spaces (see \cite{adamshedberg}).
In \cite{env} it has been shown that the first inequality in (\ref{crelle}) holds for all indices $0<\alpha<n$. The opposite inequality is false when $\alpha\in\mathbb{Z}$, for example if one takes $E$ contained in a $\alpha$-plane with positive $\alpha$-dimensional Hausdorff measure, then $\gamma_\alpha(E)>0$ while $C_{\frac 2 3(n-\alpha),\frac 3 2}(E)=0$. It is an open problem to prove (or disprove) the second inequality in (\ref{crelle}) for non-integer $1<\alpha<n$.\newline
Since $C_{s,p}$ is a subadditive set function, as a direct consequence of (\ref{crelle}) one gets that $\gamma_\alpha$ is semiadditive, that is, given compact sets $E_1$ and $E_2$,
$$\gamma_\alpha(E_1\cup E_2)\leq C(\gamma_\alpha(E_1)+\gamma_\alpha(E_2)),$$ for some constant $C$ depending only on $n$ and $\alpha$. In fact $\gamma_\alpha$ is countably semiadditive. In this paper we will show that the semiadditivity of $\gamma_\alpha$ holds for all indices $0<\alpha<n$ (see corollary $2$ below). \newline
If we restrict the supremum in (\ref{ga}) to distributions $T$ given by positive Radon measures supported on $E$, we obtain the capacities $\gamma_{\alpha,+}$. Clearly, we have $$\gamma_{\alpha,+}(E)\leq\gamma_\alpha(E).$$
The arguments of this paper will prove that, in fact, these two quantities are comparable, namely
\begin{teo}\label{alfa} There exists an absolute constant $C>0$ such that for any compact set $E\subset{\mathbb R}^n$ and any $0<\alpha<n$, \begin{equation}\label{mainineq} \gamma_\alpha(E)\leq C\gamma_{\alpha,+}(E). \end{equation} \end{teo}
This was first shown for $\alpha=1$ and $n=2$ by X. Tolsa \cite{semiad}, and it was extended to the case $\alpha=n-1$ by Volberg \cite{volberg}. For values of $\alpha\in(0,1)$ the result appears in \cite{mpv}. In \cite{mt}, Theorem \ref{alfa} was proven for a certain class of Cantor sets in ${\mathbb R}^n$, (see also \cite{tolsacantor} where it is proven for a wider class of Cantor sets). Recently, \cite{eidvol} have proven that this comparability result also holds on some examples of random Cantor sets.\newline
As a corollary from Theorem \ref{alfa}, one deduces that $\gamma_\alpha$ is countably semiadditive for $0<\alpha<n$.\newline
\begin{co} \label{semiadit} Let $E\subset{\mathbb R}^n$ be a compact set. Let $E_i$, $i\ge 1$, be Borel sets such that $E = \bigcup_{i=1}^{\infty} Ei$. Then, $$\gamma_\alpha(\bigcup_{i=1}^\infty E_i)\leq C\sum_{i=1}^\infty \gamma_\alpha(E_i)$$ where C is an absolute constant. \end{co}
The paper is organized as follows. In Section $2$ we prove Corollary \ref{semiadit}. In Section $3$ we deal with one of the main ingredients for the proof of Theorem \ref{alfa}, a localization
$L^\infty$-estimate for the scalar kernels $x_i/|x|^{1+\alpha}$, $0<\alpha<n$, $1\leq i\leq n$. In Section $4$ we prove that the capacities $\gamma_\alpha$ satisfy a exterior regularity property that will be used for the proof of Theorem \ref{alfa}. Finally, in the last section, we present a sketch of the proof of Theorem \ref{alfa}. It becomes clear that the proof depends on the following three facts: a localization $L^\infty$ estimate for the $\alpha-$Riesz kernels, the exterior regularity property of $\gamma_\alpha$, $0<\alpha<n$, and Volberg's extension \cite{volberg} of Tolsa's proof of the semiadditivity of analytic capacity \cite{semiad}.\newline
Our notation and terminology are standard. For example, ${\cal C}_0^\infty(E)$ denotes the set of all infinitely differentiable functions with compact support contained in the set $E$. Cubes will always be supposed to have sides parallel to the coordinate axis,
$l(Q)$ is the side length of the cube $Q$ and $|Q|=l(Q)^n$ its volume.
Throughout all the paper, the letters $c,\,C$ will stand for absolute constants depending only on $n$ and $\alpha$ that may change at different occurrences.
\section{Proof of Corollary \ref{semiadit}.}
In this section we will deduce the semiadditivity of the $\gamma_\alpha$ capacity, $0<\alpha<n$, from Theorem \ref{alfa}. For this, we need to introduce the $\alpha-$Riesz transform with respect to an underlying positive Radon measure $\mu$ satisfying the $\alpha-$growth condition \begin{equation}\label{mesgrowth} \mu(B(x,r))\leq Cr^\alpha,\,\,x\in{\mathbb R}^n,\,\,r\ge 0. \end{equation}
\noindent Given $\varepsilon>0$ we define the truncated $\alpha-$Riesz transform at level $\varepsilon$ as
$$R_\varepsilon(f\mu)(x)=\int_{|y-x|>\varepsilon}\frac{x-y}{|x-y|^{1+\alpha}}f(y)d\mu(y),\,\,x\in{\mathbb R}^n,$$
for $f\in L^2(\mu)$. The growth condition on $\mu$ insures that each $R_\varepsilon$ is a bounded operator on $L^2(\mu)$ with operator norm $\|R_{\varepsilon}\|_{L^2(\mu)}$ possibly depending on $\varepsilon$. We say that the $\alpha-$Riesz transform is bounded on $L^2(\mu)$ when
$$\|R\|_{L^ 2(\mu)}=\sup_{\varepsilon}\|R_\varepsilon\|_{L^2(\mu)}<\infty,$$ or, in other words, when the truncated $\alpha-$Riesz transforms are uniformly bounded on $L^2(\mu)$. Call $L_\alpha(E)$ the set of positive Radon measures supported on $E$ which satisfy (\ref{mesgrowth}) with $C=1$. One defines $\gamma_{\alpha,op}(E)$ by,
\begin{equation}\label{comppositiu}
\gamma_{\alpha,op}(E)=\sup\{\mu(E):\,\mu\in L_\alpha(E)\,\mbox{ and }\,\|R\|_{L^2(\mu)}\leq 1\}. \end{equation}
As it is well known, the capacities $\gamma_{\alpha,+}(E)$ and $\gamma_{\alpha,op}(E)$ are comparable, that is, for some positive constant $C$ one has
\begin{equation}\label{gaopgam} C^{-1}\gamma_{\alpha,op}(E)\leq\gamma_{\alpha,+}(E)\leq C\gamma_{\alpha,op}(E), \end{equation} for each compact set $E\subset{\mathbb R}^n$ (see Lemma 3 in \cite{laura3}).
Hence, once Theorem \ref{alfa} is available, namely the fact that $\gamma_\alpha(E)$ is comparable to $\gamma_{\alpha,+}(E)$, the semiadditivity of $\gamma_\alpha$ holds because $\gamma_{\alpha,op}$ is obviously semiadditive.
\section{Localization of $\alpha$-Riesz potentials}
\subsection{A growth condition}
Let $T$ be a compactly supported distribution in ${\mathbb R}^n$ and $0<\alpha<n$. Write $\alpha=[\alpha]+\{\alpha\}$, with $[\alpha]\in\mathbb{Z}$ and $0\le\{\alpha\}<1$. We say that the distribution $T$ has growth $\alpha$ provided \begin{equation}\label{growthG}
G_\alpha(T) = \sup_{\varphi_Q} \frac{|\langle T,\varphi_Q\rangle|}{l(Q)^{\alpha}} < \infty \,, \end{equation} where the supremum is taken over all $\varphi_Q \in {\cal C}^\infty_0(Q)$ satisfying the normalization inequalities
\begin{equation}\label{normal}
\|\nabla^{n-\alpha}\varphi_Q\|\leq l(Q)^{\alpha}, \end{equation}
where $\|\nabla^{n-\alpha}\varphi_Q\|$ is defined as follows:
\begin{enumerate}
\item For $\alpha=[\alpha]\in\mathbb{Z}$, condition (\ref{normal}) means that \begin{equation}\label{normalizationinteger}
\|\nabla^{n-\alpha}\varphi_Q\|:=\sup_{|s|=n-\alpha}\|\partial^s\varphi_Q\|_{L^1(Q)} \leq C l(Q)^{\alpha}\,. \end{equation}
\item for $\{\alpha\}>0$, condition (\ref{normal}) means that \begin{equation}\label{normalization}
\|\nabla^{n-\alpha}\varphi_Q\|:=\sup_{|s|=n-[\alpha]}\|\partial^s\varphi_Q*\frac{1}{|x|^{n-\{\alpha\}}}\|_{L^1} \leq C l(Q)^{\alpha}\,. \end{equation}
\end{enumerate}
Here we are adopting the standard notation related to multi-indexes, that is, $s=(s_1,s_2,\cdots,s_n)$, where each coordinate $s_j$ is a non-negative integer and $|s|=s_1+\cdots+s_n$.
For a compact set $E$ in ${\mathbb R}^n$ we define $g_{\alpha}(E)$ as the set of all distributions $T$ supported on $E$ having growth $\alpha$ with constant $G_\alpha(T)$ at most $1$\,.\newline
We start by showing that the usual $\alpha$-growth condition for a positive Radon measure (see (\ref{mesgrowth})) is equivalent to the notion of growth $\alpha$ for distributions, as defined in (\ref{growthG}).
Given a positive Radon measure $\mu$ set
$$L_\alpha(\mu)=\sup_Q\frac{\mu(Q)}{l(Q)^\alpha}$$ where the supremum is taken over all cubes $Q$ with sides parallel to the coordinate axis.
If $\varphi\in{\cal C}^\infty_0(Q)$, then by an inequality of Maz'ya \cite[p. 15 and p.134 ]{mazya}.
$$|\langle\mu,\varphi\rangle|=|\int\varphi d\mu|\leq\int|\varphi|d\mu\leq CL_\alpha(\mu)\|\nabla^{n-\alpha}\varphi\|.$$ Thus, $G_\alpha(\mu)\leq CL_\alpha(\mu).$
For the reverse inequality, given a cube $Q$, let $\varphi_Q$ be a function in ${\cal C}^\infty_0(2Q)$ such that $1\leq\varphi_Q$ on Q and
$\|\partial^s\varphi_Q\|_\infty\leq C_sl(Q)^{-|s|}$, $|s|\geq 0$. Then (\ref{normalizationinteger}) clearly holds when $\alpha\in\mathbb{Z}$. For $\{\alpha\}>0$ and $|s|=n-[\alpha]$, write $|r|=n-[\alpha]-1$. Bringing one derivative from $\partial^s\varphi_Q$ to the kernel to get integrability in $(4Q)^c$ and using Fubini we obtain
\begin{equation*} \begin{split}
\|\partial^s\varphi_Q*\frac 1{|x|^{n-\{\alpha\}}}\|_{L^1}&= \int_{(4Q)^c}\left(\partial^s\varphi_Q*\frac 1{|x|^{n-\{\alpha\}}}\right)(y)dy +
\int_{4Q}\left(\partial^s\varphi_Q*\frac 1{|x|^{n-\{\alpha\}}}\right)(y)dy\\\\&
\leq \frac{C}{l(Q)^{|r|}}\int_{(4Q)^c}\int_Q\frac{dydx} {|y-x|^{n+1-\{\alpha\}}}+\frac{C}{l(Q)^{|s|}}\int_{4Q}\int_Q\frac{dydx}{|y-x|^{n-\{\alpha\}}}\\\\& \leq Cl(Q)^{\alpha}. \end{split} \end{equation*}
Thus, $$\|\nabla^{n-\alpha}\varphi_Q\|:=\sup_{n-[\alpha]}\|\partial^s\varphi_Q*\frac 1{|x|^{n-\{\alpha\}}}\|_{L^1}\leq Cl(Q)^\alpha.$$
Therefore,
$$\mu(Q)\leq\int\varphi_Q d\mu\le |\langle\mu,\varphi_Q\rangle|\leq C\,G_\alpha(\mu)\,l(Q)^{\alpha}.$$
Next lemma shows that all distributions admissible in the definition of $\gamma_\alpha(E)$ have growth $\alpha$.
\begin{lemma}\label{growthlemma} Let $T$ be a distribution supported on the compact set $E\subset{\mathbb R}^n$. Let $0<\alpha<n$ and suppose that $T$ has bounded
$\alpha$-Riesz potential $x/|x|^{1+\alpha}*T$. Then $T\in g_\alpha(E)$. \end{lemma}
{\em Proof.} Our proof uses a reproduction formula for test functions involving the kernel $x_i/|x|^{1+\alpha},$ which was first introduced in \cite[Lemma 3.1]{laura1}. There are many variants of this formula depending, for instance, on whether de dimension $n$ and the integer part of $\alpha$ are even or odd. We will consider in full detail only the case of odd dimension of the form $n=2k+1$. We will also assume that $\alpha$ is non-integer and that its integer part is even, of the form $[\alpha]=2d$. The argument for the other cases follows the same line of reasoning but using the different variants of the corresponding reproduction formula. In our present case, the reproduction formula we use reads as follows,
\begin{equation}\label{betterrepro}
\varphi(x)=c\sum_{i=1}^n\Delta^{k-d}\partial^i\varphi*\frac{x_i}{|x|^{1+\alpha}}*\frac 1{|x|^{n-\{\alpha\}}}. \end{equation}
Let $\varphi_Q$ be a ${\cal C}_0^{\infty}(Q)$ function satisfying the normalization inequalities (\ref{normal}). Then, by (\ref{betterrepro}), the boundedness of the potential
$x_i/|x|^{1+\alpha}*T$, $1\le i\le n$, and Fubini
\begin{eqnarray*}
|\langle T,\varphi_Q\rangle|&\leq&
\,\sum_{i=1}^n|\langle \frac{x_i}{|x|^{1+\alpha}}*T,\Delta^{k-d}\partial_i\varphi_Q*\frac 1{|x|^{n-\{\alpha\}}}\rangle|\\\\
&\leq& C\sum_{i=1}^n\int|\Delta^{k-d}\partial_i\varphi_Q*\frac{1}{|x|^{n-\{\alpha\}}}(y)|dy\\\\&\leq&C\sum_{i=1}^n\int_{2Q}\int_Q\frac{|\Delta^{k-d}\partial_i\varphi_Q(z)|}{|z-y|^{n-\{\alpha\}}}dzdy +C\int_{(2Q)^c}\int_Q\frac{|\Delta^{k-d}\varphi_Q(z)|}{|z-y|^{n+1-\{\alpha\}}}dzdy\\\\&\leq& Cl(Q)^\alpha. \end{eqnarray*}
The other cases (namely for odd $n$, even $[\alpha]$...) are proven in the same way by using analogous formulae (see \cite[Lemma 3.1]{laura1}).\qed
\subsection{Localization of Riesz potentials.}
When analyzing the argument for the proof of the semiadditivity of analytic capacity (see Theorem 1.1 in \cite{semiad}) one realizes that one of the technical tools used is the fact that the Cauchy kernel~$1/z$ localizes in the uniform norm. By this we mean that if $T$ is a compactly supported distribution such that $1/z*T$ is a bounded measurable function, then $1/z*(\varphi \, T)$ is also bounded measurable for each compactly supported ${\mathcal C}^1$ function~$\varphi$. This is an old result, which is simple to prove because $1/z$ is related to the differential operator~$\overline{\partial}$ (see~\cite[Chapter V]{garnett}). The same localization result can be proved easily in any dimension for the kernel~$x/|x|^n$, which is, modulo a multiplicative constant, the gradient of the fundamental solution of the Laplacian. Again the proof is reasonably straightforward because the kernel is related to a differential operator (see~\cite{paramonov} and~\cite{verderacm}).
In \cite[Lemma 3.1]{mpv} we were concerned with the localization of the vector valued $\alpha$-Riesz kernel $x/|x|^{1+\alpha}$, $0<\alpha<n$. For general values of $\alpha$ there is no differential operator in the background and consequently the corresponding localization result becomes far from obvious. We state now the general localization Lemma proved in \cite{mpv}.\newline
In what follows, given a cube $Q$, $\varphi_Q$ will denote and infinitely differentiable function supported on $Q$ and such that $\|\partial^s\varphi_Q\|_\infty\le l(Q)^{-|s|}$, $0\le |s|\le n-[\alpha]$.\newline
\begin{lemma}\label{localizationmpv}
Let $T$ be a compactly supported distribution in ${\mathbb R}^n$ and let $0<\alpha<n$. Suppose that $x_i/|x|^{1+\alpha}*T$ is a bounded measurable function for $1\le i\le n$. Then there exists some constant $C=C(n,\alpha)>0$ such that
$$\sup_{1\le i\le n}\|\frac{x_i}{|x|^{1+\alpha}}*\varphi_Q T\|_\infty\leq C\sup_{1\le i\le n}\|\frac{x_i}{|x|^{1+\alpha}}* T\|_\infty.$$ \end{lemma}
Although Lemma \ref{localizationmpv} is enough for our purposes, that is to prove Theorem \ref{alfa}, in this paper we will give a proof of a stronger localization result, with a shorter and less technically involved proof. The main difference between the localization lemma in \cite{mpv} and the one we prove here is that we localize one component of the vector potential
$\frac{x}{|x|^{1+\alpha}}*T$, only assuming $L^\infty$ estimates on the potential of the same component, instead of assuming
$\|\frac{x}{|x|^{1+\alpha}}*T\|_\infty\leq 1$ for the whole vector. Our new localization lemma reads as follows,
\begin{lemma}\label{localization1}
Let $T$ be a compactly supported distribution in ${\mathbb R}^n$ with $\alpha$-growth, $0<\alpha<n$,
such that $(x_i / |x|^{1+\alpha}) *T$ is in $L^\infty({\mathbb R}^n)$ for some $i$\,, $1\leq i \leq n$.
Then $(x_i / |x|^{1+\alpha}) * \varphi_Q T$ is in $L^\infty({\mathbb R}^n)$ and
$$\|\frac{x_i}{|x|^{1+\alpha}}*\varphi_Q T\|_\infty\leq C\,(\| \frac{x_i}{|x|^2}*T\|_\infty+G_{\alpha}(T))\,,$$ for some positive constant $C=C(n)$ depending only on $n\,.$ \end{lemma}
For $\alpha=1$ the proof of the above lemma can be found in \cite{mpv2}. We remark here that when one deals with indexes $\alpha\in\mathbb{Z}$, the proof of Lemma \ref{localization1}
is less technically involved, since the derivatives $\partial^s\varphi_Q$, $|s|=n-\alpha$, are ordinary derivatives and therefore supported on the cube $Q$ (compare (\ref{normalization}) with (\ref{normalizationinteger})). \newline
For the proof of Lemma \ref{localization1} we need the following result (see Lemma $7$ in \cite{mpv2} for the case $\alpha=1$), that will be proved after the proof of Lemma \ref{localization1}.
\begin{lemma}\label{prelocalization}
Let $T$ be a compactly supported distribution in ${\mathbb R}^n$ with $\alpha-$growth, $0<\alpha<n$.
Then, for each coordinate $i$, the distribution $(x_i / |x|^{1+\alpha}) * \varphi_Q T$ is a locally integrable function
and there exists a point $x_0 \in \frac{1}{4}Q$
such that $$\left|\left(\frac{x_i}{|x|^{1+\alpha}}*\varphi_QT\right)(x_0)\right|\leq C \, G_{\alpha}(T)\,,$$ where $C=C(n)$ is a positive constant depending only on $n\,.$ \end{lemma}
\noindent {\em Proof of Lemma \ref{localization1}.} Without loss of generality take $i=1$.
We distinguish two cases: \begin{enumerate}
\item $x\in (\frac 3 2 Q)^c$. Then $ |(k^1*\varphi_QT)(x)|=|<T,\varphi_Q(y)k^1(x-y)>|.$ Notice that, for an appropiate dimensional constant $C$, the function $$\psi_Q(y)=Cl(Q)^\alpha\varphi_Q(y)k^1(x-y),$$ satisfies the normalization inequalities in the definition of $G_\alpha(T)$, namely \begin{equation}\label{normaliz}
\|\nabla^{n-\alpha}\psi_Q\|\leq l(Q)^{\alpha}\,. \end{equation} Therefore,
$$|(k^1*\varphi_QT)(x)|=Cl(Q)^{-\alpha}|<T,\psi_Q>|\leq C.$$
To see (\ref{normaliz}), observe that if $[\alpha]$ denotes the integer part of $\alpha$ and we write $\alpha=[\alpha]+\{\alpha\}$, then by Leibniz formula, \begin{equation}\label{psiq}
\|\partial^s\psi_Q\|_{L^\infty(Q)}\leq Cl(Q)^\alpha\sum_{|r|=0}^{|s|}l(Q)^{-|r|}l(Q)^{-|s|-\alpha+|r|}\leq Cl(Q)^{-|s|}, \end{equation}
for any multiindex $s=(s_1,\cdots,s_n)$ with $|s|\ge 0$.
If $\{\alpha\}=0$, (\ref{psiq}) immediately implies that condition (\ref{normaliz}) holds. When
$\{\alpha\}>0$, let $s=(s_1,\cdots,s_n)$ be any multiindex with $|s|=n-[\alpha]$ and write
\begin{equation*} \begin{split}
\int|(\partial^s\psi_Q*\frac 1{|z|^{n-\{\alpha\}}})(y)|dy&=\int_{2Q}|(\partial^s\psi_Q*\frac{1}{|z|^{n-\{\alpha\}}})(y)|dy\\\\&+
\int_{(2Q)^c}|(\partial^s\psi_Q*\frac{1}{|z|^{n-\{\alpha\}}})(y)|dy=A+B. \end{split} \end{equation*} By (\ref{psiq}), we have
$$A\le C\|\partial^s\psi_Q\|_{\infty}\int_{2Q}\int_Q\frac{dzdy}{|z-y|^{n-\{\alpha\}}}\le Cl(Q)^\alpha.$$
And by bringing one derivative from $\partial^s\psi_Q$ to the kernel $|z|^{-n+\{\alpha\}}$ and using (\ref{psiq}) again, we get \begin{equation}
\begin{split}
B&\le C\int_{(2Q)^c}|(\partial^t\psi_Q*\frac{1}{|z|^{n+1-\{\alpha\}}})(y)|dy\\\\&\le C\|\partial^t\psi_Q\|_{\infty}\int_{(2Q)^c}\int_Q\frac{dzdy}{|z-y|^{n+1-\{\alpha\}}} \le Cl(Q)^\alpha,
\end{split} \end{equation}
for some multiindex $t$ with $|t|=n-[\alpha]-1$.
\item $x\in \frac 3 2 Q$. Since $k^1*T$ and $\varphi_Q$ are bounded functions, we can write \begin{eqnarray*}
|(k^1*\varphi_QT)(x)|\leq|(k^1*\varphi_QT)(x)-\varphi_Q(x)(k^1*T)(x)|+\|\varphi_Q\|_\infty\|k^1*T\|_\infty. \end{eqnarray*}
Let $\psi_Q\in{\cal C}_0^{\infty}({\mathbb R}^n)$ be such that $\psi_Q\equiv 1$ in $2Q$, $\psi_Q\equiv 0$ in $(4Q)^c$ and
$\|\partial^s\psi_Q\|_\infty\leq C_s\,l(Q)^{-|s|}\,,$ for each multi-index $s$ . Then one is tempted to write \begin{eqnarray*}
|(k^1*\varphi_QT)(x)-\varphi_Q(x)(k^1*T)(x)|&\leq&|<T,\psi_Q(y)(\varphi_Q(y)-\varphi_Q(x))k^1(x-y)>|\\\\&+&
\|\varphi_Q \|_{\infty}\,|<T,(1-\psi_Q(y))k^1(x-y)>|\,. \end{eqnarray*} The problem is that the first term in the right hand side above does not make any sense because $T$ is acting on a function of $y$ which is not necessarily differentiable at the point $x\,.$ To overcome this difficulty one needs to use a standard regularization process. Take $\chi \in {\cal C}^\infty(B(0,1))$ such that $\int \chi(x)\,dx = 1$ and set $\chi_\varepsilon(x)= \varepsilon^{-n}\,\chi(x/\varepsilon)\,.$ The plan is to estimate, uniformly on $x$ and $\epsilon\,,$ \begin{equation}\label{reg}
|(\chi_\varepsilon*k^1*\varphi_QT)(x)-\varphi_Q(x)(\chi_\varepsilon*k^1*T)(x)|\,. \end{equation}
Clearly \eqref{reg} tends, as $\varepsilon$ tends to zero,
to $$
|(k^1*\varphi_QT)(x)-\varphi_Q(x)(k^1*T)(x)|\,, $$ for almost all $x \in {\mathbb R}^n$\,, which allows the transfer of uniform estimates. We now have \begin{eqnarray*}\label{dif}
|(\chi_\varepsilon*k^1*\varphi_QT)(x)&-&\varphi_Q(x)(\chi_\varepsilon*k^1*T)(x)|\\\\&\le&
|<T,\psi_Q(y)(\varphi_Q(y)-\varphi_Q(x))
(\chi_\varepsilon*k^1)(x-y)>|\\\\&+&
\|\varphi_Q\|_{\infty}|<T,(1-\psi_Q(y))(\chi_\varepsilon*k^1)(x-y)>|\\\\&=&A_1+A_2. \end{eqnarray*} \noindent where the last identity is the definition of $A_1$ and $A_2$. \noindent To deal with term $A_1$ set $$k^{1,x}_\varepsilon(y)=(\chi_\varepsilon*k^1)(x-y).$$ We claim that, for an appropriate dimensional constant $C$, the test function $$f(y)=Cl(Q)^\alpha\psi_Q(y)(\varphi_Q(y)-\varphi_Q(x))k_\varepsilon^{1,x}(y),$$ satisfies the normalization inequalities \eqref{normal} in the definition of $G_\alpha(T)\,,$ with $\varphi_Q$ replaced by $f$. If this is the case, then
$$A_1\leq C l(Q)^{-\alpha}|<T,f>|\leq C\,G_\alpha(T).$$
To prove the claim we have to show that \begin{equation}\label{regdins}
\|\nabla^{n-\alpha}f\|\leq l(Q)^{\alpha}. \end{equation}
We first notice that the regularized kernel $\chi_\varepsilon*k^1$ satisfies the inequalities \begin{equation}\label{regkernel}
|(\chi_\varepsilon* \partial^s \,k^1)(x)| \le \frac{C}{|x|^{\alpha+|s|}}\,,
\quad x \in {\mathbb R}^n\setminus \{0\}\quad \text{and}\quad 0\le |s| \leq n-[\alpha]-1\,, \end{equation} where $C$ is a dimensional constant, which, in particular, is independent of $\epsilon$.
The estimate of the $L^1-$norm in (\ref{regdins}) requires the use of Leibniz formula \begin{equation}\label{Leibniz2} \partial^s \left(\psi_Q(\varphi_Q -\varphi_Q(x))k_{\varepsilon}^{1,x}\right) =
\sum_{|r|=0}^{|s|}c_{r,s}\, \partial^r(\psi_Q(\varphi_Q -\varphi_Q(x)))\;\partial^{s-r}\, k_\varepsilon^{1,x} \end{equation}
and of \eqref{regkernel}\,. Notice that for any multiindex $t=(t_1,\cdots,t_n)$ with $|t|=n-[\alpha]-1$, \begin{equation}\label{partials} \begin{split}
\|\partial^t f \|_{L^1(4Q)}&\le C l(Q)^\alpha\sum_{|r|=0}^{|t|}\frac 1{l(Q)^{|r|}}\int_{4Q}|\partial^{t-r}(k_\varepsilon^{1,x})(y)|\,dy \\\\
&\leq C l(Q)^{n-|t|}=Cl(Q)^{[\alpha]+1}. \end{split} \end{equation}
And if $s=(s_1,\cdots, s_n)$ is such that $|s|=n-[\alpha]$, using the mean value theorem to gain integrability when $|r|=0$ and (\ref{regkernel}), \begin{equation}\label{partialt} \begin{split}
\|\partial^s f \|_{L^1(4Q)}&\le Cl(Q)^{\alpha}\|\nabla\varphi_Q\|_\infty\int_{4Q}\frac{dy}{|y-x|^{\alpha+|s|-1}} \\\\&+C l(Q)^\alpha\sum_{|r|=1}^{|s|}\frac 1{l(Q)^{|r|}}\int_{4Q}|\partial^{s-r}(k_\varepsilon^{1,x})(y)|\,dy\\\\&\leq Cl(Q)^{[\alpha]}. \end{split} \end{equation}
Estimate (\ref{partialt}), immediately yields (\ref{regdins}) for $\{\alpha\}=0$,
$$\|\nabla^{n-\alpha}f\|:=\sup_{|s|=n-\alpha}\int_{4Q}|\partial^sf(y)|dy\leq Cl(Q)^{\alpha}.$$
If $\{\alpha\}>0$, then for any multiindex $s=(s_1,\cdots,s_n)$ with $|s|=n-[\alpha],$ \begin{eqnarray*}
\int|\,(\partial^s f*\frac{1}{|z|^{n-\{\alpha\}}})(y)|\,dy&\le&
\int_{5Q}|\,(\partial^s f*\frac{1}{|z|^{n-\{\alpha\}}})(y)|\,dy\\\\&+
&C\int_{(5Q)^c}|\,(\partial^t f*\frac{1}{|z|^{n+1-\{\alpha\}}})(y)|\,dy\\\\&=&B_1+B_2, \end{eqnarray*}
where $t$ is some multiindex with $|t|=n-[\alpha]-1$.
To estimate $B_2$, we use Fubini and (\ref{partials}). Then,
$$B_2\leq\int_{4Q}|\partial^tf(z)|\int_{(5Q)^c}\frac
{dydz}{|z-y|^{n+1-\{\alpha\}}}\leq Cl(Q)^{\alpha}.$$
We turn now to term $B_1$. By Fubini and (\ref{partialt}), we get
$$B_1\leq C\int_{5Q}|\partial^sf(z)|\int_{4Q}\frac{dydz}{|y-z|^{n-\{\alpha\}}}\leq Cl(Q)^{\alpha}.$$
This finishes the proof of (\ref{regdins}). We now turn to $A_2$. By Lemma \ref{prelocalization}, there exists a point $x_0\in Q$ such that $|(k^1*\psi_QT)(x_0)|\leq C\, G_\alpha(T)$.
Then
$$|(k^1*(1-\psi_Q)T)(x_0)|\leq C\,(\|k^1*T\|_\infty +G_\alpha(T)).$$ The analogous inequality holds as well for the regularized potentials appearing in $A_2\,,$ uniformly in $\epsilon\,,$ and therefore
$$A_2\leq C\,|<T,(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})|+C\,(\|k^1*T\|_\infty +G_\alpha(T)).$$
To estimate $|<T,(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|$, we decompose ${\mathbb R}^n \setminus \{x\}$ into a union of rings $$R_j=\{z\in
{\mathbb R}^n:\;2^j\,l(Q)\leq|z-x|\leq 2^{j+1}\,l(Q)\},\;\; j\in\mathbb{Z},$$ and consider functions $\varphi_j$ in ${\cal C}^\infty_0({\mathbb R}^n)$, with support contained in $\frac 3 2 R_j$\,, such that
$\|\partial^s\varphi_j\|_\infty\leq C \,(2^j\,l(Q))^{-|s|}\,,$ $|s| \geq 0$, and $\sum_j\varphi_j=1$ on ${\mathbb R}^n\setminus\{x\}$. Then, since $x\in\frac 3 2 Q$ and $1-\psi_Q\equiv 0$ in $2Q$, the smallest ring $R_j$ that may intersect $(2Q)^c$ is $R_{-2}$. Therefore we have
\begin{eqnarray*}
|<T,(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|&=&<T,\sum_{j\geq -2}\varphi_j(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|\\\\
&\leq&|<T,\sum_{j\in I}\varphi_{j}(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|\\\\&+&\sum_{j\in J}|<T,\varphi_{j}(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|, \end{eqnarray*} where $I$ denotes the set of indices $j\geq -2$ such that the support of $\varphi_j$ intersects $4Q$ and $J$ the remaining indices, namely those $j \geq -2 $ such that the support of $\varphi_j$ is contained in the complement of $4Q\,.$ Notice that the cardinality of $I$ is bounded by a dimensional constant. \newline
\noindent Set $$g =C\,l(Q)^\alpha\,\sum_{j\in I}\varphi_j(1-\psi_Q)\,(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})\,,$$ and for $j\in J$
$$g_j=C\,2^j\,(2^jl(Q))^\alpha\,\varphi_j\,(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0}).$$ We show now that the test functions $g$ and $g_j$, $j\in J\,,$ satisfy the normalization inequalities \eqref{normal} in the definition of $G_\alpha(T)\,.$ Once this is available, using the $\alpha$- growth condition of $T$ we obtain \begin{eqnarray*}
|<T,(1-\psi_Q)(k_\varepsilon^{1,x}-k_\varepsilon^{1,x_0})>|&\leq& C l(Q)^{-\alpha}|<T,g>|\\\\&+&
C \sum_{j\in J} 2^{-j}(2^jl(Q))^{-\alpha}|<T,g_j>|\\\\
&\leq& C\,G_\alpha(T) + C\sum_{j\geq 2}2^{-j}\,G_\alpha(T)\leq C\,G_\alpha(T)\,, \end{eqnarray*} which completes the proof of Lemma 7.
We check now the normalization inequalities for $g$ and $g_j$. For $g$ one argues as in the proof of (\ref{regdins}), using that $\|\partial^s(1-\psi_Q)\|_\infty\leq Cl(Q)^{-|s|},$ $\|\partial^s\varphi_j\|_\infty\leq C\, l(Q)^{-|s|}$, $j\in I$, \eqref{regkernel}, the fact that $x, x_0 \in \frac 3 2 Q$\,,\,$y\in (2Q)^c$\,, and a gradient estimate.
For $g_j$ we use in addition Leibniz formula and a gradient estimate to show that, for $j\in J$, and $n-[\alpha]-1\le|s|\le n-[\alpha]$,
\begin{equation}\label{gj} \begin{split}
\|\partial^s g_j\|_{\infty}&\leq C\, 2^j(2^{j}\,l(Q))^\alpha\sum_{|r|=0}^{|s|}\frac 1{(2^jl(Q))^{|r|}}\frac{l(Q)}{(2^j\,l(Q))^{1+\alpha+|s|-|r|}}
\\&\leq C\,(2^jl(Q))^{-|s|}, \end{split} \end{equation}
\noindent If $\{\alpha\}=0$, we use (\ref{gj}) to obtain
$$\|\nabla^{n-\alpha}g_j\|:=\sup_{|s|=n-\alpha}\int|\partial^s g_j(y)|dy\le C(2^jl(Q))^{\alpha}.$$
\noindent If $\{\alpha\}>0$, then for any multiindex $s=(s_1,\cdots,s_n))$ with $|s|=n-[\alpha]-1$,
using Fubini, (\ref{gj}) and arguing similar to the proof of (\ref{regdins}) we get, for some multiindex $t=(t_1,\cdots, t_n)$ with $|t|=n-[\alpha]-1$,
\begin{equation*} \begin{split}
\int|(\partial^sg_j*\frac 1{|z|^{n-\{\alpha\}}})(y)|dy&\le\int_{2R_j}|(\partial^sg_j*\frac 1{|z|^{n-\{\alpha\}}})(y)|dy\\\\&+
C\int_{(2R_j)^c}|(\partial^t g_j*\frac 1{|z|^{n+1-\{\alpha\}}})(y)|dy\\\\&\le C(2^jl(Q))^\alpha. \end{split} \end{equation*}
Therefore, we can conclude that $$\|\nabla^{n-\alpha}g_j\| \le C(2^jl(Q))^\alpha.\qed$$
\end{enumerate}
{\em Proof of Lemma \ref{prelocalization}.} Without loss of generality set $i=1$ and write
$k^1(x)=x_1/|x|^{1+\alpha}$. Since $k^1 * \varphi_Q T$ is infinitely differentiable off the closure of ${Q}\,,$ we only need to show that $k^1* \varphi_Q T$ is integrable on $2Q\,.$ We will actually prove a stronger statement, namely, that $k^1* \varphi_Q T$ is in $L^p(2Q)$ for each $p$ in the interval $1\leq p< n\,.$ Indeed, fix any $q$ satisfying $n/(n-1) <q < \infty$ and call $p$ the dual exponent, so that $1 < p< n\,.$ We need to estimate the action of $k^1 * \varphi_Q T$ on functions $\psi \in {\cal C}^\infty_0(2Q)$ in terms of
$\|\psi\|_q \,.$ We clearly have $$ < k^1 * \varphi_Q T, \psi> = <T, \varphi_Q\,(k^1 * \psi)> \,. $$
We claim that, for an appropriate dimensional constant $C \,,$ the test function
$$\frac{\varphi_Q\,(k^1 * \psi)}{C \,l(Q)^{\frac n p-\alpha} \,\|\psi\|_q} $$ satisfies the normalization inequalities \eqref{normal} in the definition of $G_\alpha(T)\,.$ Once this is proved, by the definition of $G_\alpha(T)$ we get \begin{equation*}\label{Lq}
|< k^1 * \varphi_Q T, \psi>| \le C\, l(Q)^\frac n p\,\|\psi\|_q \,G_\alpha(T)\,, \end{equation*} and so \begin{equation*}\label{Lp}
\|k^1 * \varphi_Q T \|_{L^p(2Q)} \le C\, l(Q)^\frac n p\,G_\alpha(T)\,. \end{equation*} Hence \begin{equation*} \begin{split}
\frac{1}{|\frac{1}{4}Q|}\,\int_{\frac{1}{4} Q} |(k^1 * \varphi_Q T)(x)|\,dx &\le 4^n\,\left(\frac{1}{|Q|}\,\int_Q |(k^1 * \varphi_Q T)(x)|^p\,dx\right)^{\frac 1 p}\\& \le C\,G_\alpha(T)\,, \end{split} \end{equation*} which completes the proof of Lemma 8.
To prove the claim we have to show that \begin{equation}\label{condition}
\| \nabla^{n-\alpha}(\varphi_Q\,(k^1 * \psi))\| \le C\, l(Q)^{\frac n p}\, \|\psi\|_q. \end{equation}
Write $\alpha=[\alpha]+\{\alpha\}$, with $\{\alpha\}\in[0,1)$ and $[\alpha]\in\mathbb{Z}$. We distinguish now two cases, $\{\alpha\}$=0 and $\{\alpha\}>0$.
\begin{enumerate} \item Case $\{\alpha\}$=0, i.e. $\alpha=[\alpha]\in\mathbb{Z}$. This is the easiest case, because the derivatives appearing in \eqref{condition} are ordinary derivatives (see also lemma $7$ in \cite{mpv2}).
Let $s=(s_1,s_2,\cdots,s_n)$ be any multiindex with $|s|=n-\alpha$. Using Leibniz formula, \begin{equation}\label{Leibniz}
\partial^s \left(\varphi_Q \,(k^1 * \psi)\right) = \sum_{|r|=0}^{|s|} c_{s,r}\,\partial^r\varphi_Q \;\partial^{s-r}(k^1*\psi)\,, \end{equation} we obtain
\begin{equation*}
\begin{split}
\int_{Q}|\partial^s(\varphi_Q(k^1*\psi))(y)|dy&\le C\int_{2Q}|(\partial^s k^1*\psi)(y)|dy\\\\&+C\sum_{|r|=1}^{|s|}\int_{Q}|\partial^r\varphi_Q(y)||\partial^{s-r}(k^1*\psi)(y)|dy=A+B.
\end{split} \end{equation*}
To estimate term $A$, we remark that for
$|s|=n-\alpha$, $$\partial^s k^1*\psi=c\psi+T(\psi),$$ where $T$ is a smooth homogeneous convolution Calder\'on-Zygmund operator and $c$ a constant depending on $s$. This can be seen by computing the Fourier transform of $\partial^s k^1$ and then using that each homogeneous polynomial can be decomposed in terms of homogeneous harmonic polynomials of lower degrees ( see \cite[p. 69]{St}). Since Calder\'on-Zygmund operators preserve $L^q({\mathbb R}^n)$, $1<q<\infty$, we get, using H\"older
$$A\le Cl(Q)^{\frac n p}\|\psi\|_q.$$
To estimate term $B$, we use $|\partial^{s-r}k^1(x)| \le C\, |x|^{-(\alpha+|s|-|r|)}\,,$ Fubini, the fact that $\|\partial^r\varphi_Q\|_\infty\leq C l(Q)^{-|r|}$ and H\"older to obtain $B\le Cl(Q)^{\frac n p}\|\psi\|_q$. Therefore we get,
$$\| \nabla^{n-\alpha}(\varphi_Q\,(k^1 * \psi))\|=\sup_{|s|=n-\alpha}\int_{Q}|\partial^s(\varphi_Q(k^1*\psi))(y)|dy\le C l(Q)^{\frac n p}\|\psi\|_q. $$
\item Case $\{\alpha\}>0$. Let $s=(s_1,s_2,\cdots,s_n)$ be any
multiindex with $|s|=n-\alpha$ and write
\begin{equation*} \begin{split}
\int|\partial^s(\varphi_Q(k^1*\psi)*\frac 1{|x|^{n-\{\alpha\}}})(x)|dx&=
\int_{(2Q)^c}|(\partial^s \varphi_Q(k^1*\psi)*\frac 1{|x|^{n-\{\alpha\}}})(x)|dx\\\\&+
\int_{2Q}|\partial^s(\varphi_Q(k^1*\psi)*\frac 1{|x|^{n-\{\alpha\}}})(x)|dx\\\\&=A+B. \end{split} \end{equation*}
\noindent We deal first with term $A$. Bringing one derivative from $\partial^s(\varphi_Q(k^1*\psi))$ to the kernel
$|x|^{-n+\{\alpha\}}$ and using Fubini, we obtain
$$A\le C\int_{(2Q)^c}|\partial^{t}(\varphi_Q(k^1*\psi))*\frac 1{|x|^{n+1-\{\alpha\}}})(x)|dx,$$
for some multiindex $t=(t_1,\cdots,t_n)$ with $|t|=n-[\alpha]-1$.
\noindent We will now use Leibniz formula \eqref{Leibniz} (with $s$ replaced by $t$) and the fact that
\begin{equation}\label{cota}
|\partial^{t-r}k^1(x)| \le C\, |x|^{-(\alpha+|t|-|r|)}\,. \end{equation}
Therefore, since $\alpha+|t|-|r|<n$, by Fubini and H\"older we obtain
\begin{equation}\label{lu} \begin{split}
\int_{Q}|\partial^t(\varphi_Q(k^1*\psi))(y)|dy&\leq
C\sum_{|r|=0}^{|t|}\int_{Q}|\partial^r\varphi_Q(y)||\partial^{t-r}(k^1*\psi)(y)|dy\\\\&\leq
C\sum_{|r|=0}^{|t|}\int_{2Q}|\psi(x)|\int_Q\frac{|\partial^r\varphi_Q(y)|}{|y-x|^{\alpha+|t|-|r|}}dydx\\\\
&\leq C\|\psi\|_q l(Q)^{\frac n p+1-\{\alpha\}}. \end{split} \end{equation}
\noindent Hence, if we apply Fubini again, we get
$$A\le C \int_{Q}|\partial^t(\varphi_Q(k^1*\psi))(y)|\int_{(2Q)^c}\frac{dxdy}{|y-x|^{n+1-\{\alpha\}}}
\leq Cl(Q)^{\frac n p}\, \|\psi\|_q.$$
To estimate term $B$, we will use (\ref{Leibniz}) and for each $0\le |r|\le |t|$ we will add and substract $\partial^r\varphi_Q(x)\partial^{t-r}(k^1*\psi)(y)$ in the integral to gain integrability, namely
\begin{equation*} \begin{split}
B&\le C\sum_{|r|= 0}^{|t|}\int_{2Q}|\int_Q\frac{(\partial^r\varphi_Q(y)-\partial^r\varphi_Q(x))\partial^{t-r}(k^1*\psi)(y)}{|y-x|^{n+1-\{\alpha\}}}dy|dx\\\\&+
C\sum_{|r|=0}^{|t|}\int_Q|\partial^r\varphi_Q(x)||\left(\Delta\partial^{t-r}k^1*\psi*\frac{1}{|y|^{n-1-\{\alpha\}}}\right)(x)|dx\\\\&
+C\sum_{|r|=0}^{|t|}\int_Q|\partial^r\varphi_Q(x)||\int_{Q^c}\frac{\partial^{t-r}(k^1*\psi)(y)}{|y-x|^{n+1-\{\alpha\}}}dy|dx\\\\&=B_1+B_2+B_3, \end{split} \end{equation*} the last identity being a definition for $B_1$, $B_2$ and $B_3$.\newline
\noindent Since arguing as in (\ref{lu}), (recall that
$|t|=n-[\alpha]-1$), we obtain \begin{equation*}
\int_Q|\partial^{t-r}(k^1*\psi)(y)|dy\le\|\psi\|_ql(Q)^{\frac n p+1-\{\alpha\}+|r|}, \end{equation*} by the mean value theorem and Fubini, we get that
\begin{equation}\label{b1} \begin{split}
B_1&\leq\sum_{|r|=0}^{|t|}\frac{C}{l(Q)^{|r|+1}}\int_{Q}|\partial^{t-r}(k^1*\psi)(y)|\int_{2Q}\frac{dxdy}{|y-x|^{n-\{\alpha\}}}\\\\
&\leq C\sum_{|r|=0}^{|t|}l(Q)^{-|r|-1}\|\psi\|_ql(Q)^{\frac n p+1-\{\alpha\}+|r|}l(Q)^{\{\alpha\}}\\\\&\leq Cl(Q)^{\frac n p}\,
\|\psi\|_q. \end{split} \end{equation}
We deal now with term $B_2$. By computing the Fourier transform of the convolution $\Delta\partial^{t-r}k^1*\psi*\frac{1}{|y|^{n-1-\{\alpha\}}}$, one can see that for $|r|=0$,
$$\left(\Delta\partial^t k^1*\psi*\frac{1}{|y|^{n-1-\{\alpha\}}}\right)(x)=c\psi+ cS_0(\psi)(x),$$
\noindent where $c$ is a constant, and $S_0$ is a smooth homogeneous convolution Calder\'on-Zygmund operator. For $|r|\geq 1$, we obtain
$$\left(\Delta\partial^{t-r} k^1*\psi*\frac{1}{|y|^{n-1-\{\alpha\}}}\right)(x)=cS_r(\psi)(x),$$
with $S_r$ a convolution operator with kernel of homogeneity $-(n-|r|)$. Since Calder\'on-Zygmund operators preserve $L^q({\mathbb R}^n)$, $1<q<\infty$, using Young's inequality to estimate the $L^q(Q)$-norm of the convolution $S_r(\psi)$ and H\"older, we obtain
\begin{equation*} \begin{split}
B_2&=\sum_{|r|=0}^{|t|}\int_Q|\partial^r\varphi_Q(x)||S_r(\psi)(x)|dx\\\\&
\le C\|\varphi_Q\|_p\|\psi\|_q+C\sum_{|r|=1}^{|t|}\|\partial^r\varphi_Q\|_p\|S_r(\psi)\|_{L^q(Q)}\\\\&
\le Cl(Q)^{\frac n p}\|\psi\|_q+\|\psi\|_q\sum_{|r|= 1}^{|t|}l(Q)^{-|r|}l(Q)^{\frac n p}l(Q)^{|r|}\leq Cl(Q)^{\frac n p}\, \|\psi\|_q. \end{split} \end{equation*}
Now we are only left with term $B_3$. Since $\partial^r\varphi_Q$ is supported on $Q$, we can substract
$\partial^r\varphi_Q(y)=0$ for $y\in 3Q\setminus Q$, $|r|\geq 0$. Then, \begin{equation*} \begin{split}
B_3&=\int_Q|\int_{Q^c}\partial^r\varphi_Q(x)\frac{\partial^{t-r}(k^1*\psi)(y)}{|y-x|^{n+1-\{\alpha\}}}dy|dx\\\\
&\le\sum_{|r|=0}^{|t|}\int_Q|\int_{3Q\setminus Q}\partial^{t-r}(k^1*\psi)(y)\frac{\partial^r\varphi_Q(x)-\partial\varphi_Q(y)}{|y-x|^{n+1-\{\alpha\}}}dy|dx\\\\
&+\sum_{|r|=0}^{|t|}\int_Q|\int_{(3Q)^c}\partial^r\varphi_Q(x)\frac{\partial^{t-r}(k^1*\psi)(y)}{|y-x|^{n+1-\{\alpha\}}}dy|dx= B_{31}+B_{32}. \end{split} \end{equation*}
Using the mean value theorem and proceeding as in (\ref{b1}), $$
B_{31}\le\sum_{|r|=0}^{|t|}\frac{C}{l(Q)^{|r|+1}}\int_{Q}\int_{3Q}\frac{|\partial^{t-r}(k^1*\psi)(y)|}{|y-x|^{n-\{\alpha\}}}dydx\le Cl(Q)^{\frac n p}\|\psi\|_q.$$
\noindent Notice that by (\ref{cota}), for $y\in (3Q)^c$,
$$|\partial^{t-r}(k^1*\psi)(y)|\leq\int_{2Q}\frac{\psi(z)}{|z-y|^{\alpha+|t|-|r|}}dz\le C \|\psi\|_q\,l(Q)^{\frac n p}\,l(Q)^{-\alpha-|t|+|r|}.$$
\noindent Therefore, \begin{equation*} \begin{split}
B_{32}&\le\sum_{|r|=0}^{|t|}\frac{C}{l(Q)^{|r|}}\int_{Q}\int_{(3Q)^c}\frac{|\partial^{t-r}(k^1*\psi)(y)|}{|y-x|^{n+1-\{\alpha\}}}dydx\\\\&\le c\|\psi\|_ql(Q)^{\frac n p}\sum_{|r|=0}^{|t|}\frac{l(Q)^{-\alpha-|t|+|r|}}{l(Q)^{|r|}}\int_Q\int_{(3Q)^c}\frac 1{|y-x|^{n+1-\{\alpha\}}}dydx\\\\&\le Cl(Q)^{\frac n p}\|\psi\|_q, \end{split} \end{equation*} which finishes the proof of (\ref{condition}) for the case $\{\alpha\}>0.\qed$ \end{enumerate}
\section{A continuity property for the capacity $\gamma_\alpha$}
In this section we prove a continuity property for the capacity $\gamma_\alpha$, $0<\alpha<n$, which will be used in the proof of Theorem \ref{alfa}.
\begin{lemma}\label{extregalfa} Let $\{E_j\}_j$ be a decreasing sequence of compact sets, with intersection the compact set $E\subset{\mathbb R}^n$ and let $0<\alpha<n$. Then $$\gamma_\alpha(E)=\lim_{j\to\infty}\gamma_\alpha(E_j).$$ \end{lemma}
{\em Proof.} Since, by definition, the set function $\gamma_\alpha$ in non-decreasing, $$\lim_{j\to\infty}\gamma_\alpha(E_j)\geq\gamma_\alpha(E)\,,$$
and the limit clearly exists. For each $j\geq 1$, let $T_j$ be a distribution such that the potential $x/|x|^{1+\alpha}*T_j$ is in the unit ball of $L^\infty({\mathbb R}^n)$\,, and
$$\gamma_\alpha(E_j)-\frac 1 j < |\langle T_j,1\rangle| \leq\gamma_\alpha(E_j).$$
\noindent We want to show that for each test function $\varphi$, \begin{equation}\label{test} \langle T_j,\varphi\rangle\underset{j\to\infty}{\longrightarrow}\langle T,\varphi\rangle, \end{equation}
for some distribution $T$ whose potential $x/|x|^{1+\alpha}*T$ is in the unit ball of $L^\infty({\mathbb R}^n)$. If (\ref{test}) holds and $\varphi$ is a test function satisfying $\varphi\equiv 1$ in a neighbourhood of $E$, then
$$\lim_{j\to\infty}\gamma_\alpha(E_j)=\lim_{j\to\infty}|\langle T_j,1\rangle |=\lim_{j\to\infty}|\langle T_j,\varphi\rangle|= |\langle T,\varphi\rangle|\leq\gamma_\alpha(E).$$
To show (\ref{test}), set $k_\alpha(x)=1/|x|^{n-\alpha}$ and $f_j^i=x_i/|x|^{1+\alpha}*T_j$, $1\leq i\leq n$. We will treat first the case $n$ odd and of the form $n=2k+1$. By (\ref{betterrepro}) \begin{eqnarray*} \langle T_j,\varphi\rangle&=&c\sum_{i=1}^n \langle f_j^i,\Delta^k\partial_i\varphi*k_\alpha\rangle\\\\&=&c\sum_{i=1}^n\int f_j^i(x)\left(\Delta^k\partial_i\varphi*k_\alpha\right)(x)dx \end{eqnarray*}
We mention here that if $n=2k$, then one argues in the same way, but one has to use another reproduction formula analogous to (\ref{betterrepro}) for this case (see \cite[Lemma 3.1]{laura1}).
Passing to a subsequence, we can assume that for each $1\leq i\leq n$, $f_j^i\longrightarrow f^i$ in the weak $*$ topology of $L^\infty({\mathbb R}^n)$. But then $f_j^i(x)(\Delta^k\partial_i\varphi*k_\alpha)(x) \longrightarrow f^i(x)(\Delta^k\partial_i\varphi*k_\alpha) (x)\,,\quad x \in {\mathbb R}^n \,.$ Since this pointwise convergence is bounded,
the dominated convergence theorem yields
\begin{equation*} \begin{split} \lim_{j\to\infty}<T_j,\varphi>&=c\,\lim_{j\to\infty}\sum_{i=1}^n\int f_j^i(x)\left(\Delta^k\partial_i\varphi*k_\alpha\right)(x)dx \\\\&=c\sum_{i=1}^n\int f^i(x)\left(\Delta^k\partial_i\varphi*k_\alpha\right)(x)dx. \end{split} \end{equation*}
Define the distribution $T$ by
$$<T,\varphi>=c\sum_{i=1}^n\int f^i(x)\left(\Delta^k\partial_i\varphi*k_\alpha\right)(x)dx.$$
Now we want to show that for $1\leq i\leq n$, $f^i=x_i/|x|^{1+\alpha}*T$. For that we regularize $f_j^i$ and $T_j$\,. Take $\chi \in{\cal C}_0^{\infty}(B(0,1))$ with $\int \chi(x)\,dx=1$ and set $\chi_\varepsilon(x)=\varepsilon^{-n}\chi(x/\varepsilon)$\,. Then we have , as $j \rightarrow \infty$\,,
$$ \left(\chi_\varepsilon*\frac{x_i}{|x|^{1+\alpha}}*T_j\right)(x)= \left(\chi_\varepsilon*f_j^i\right)(x) \longrightarrow \left(\chi_\varepsilon*f^i\right)(x)\,,\quad x \in {\mathbb R}^n\,,$$
because $f_j^i$ converges to $f^i$ weak $*$ in $L^\infty({\mathbb R}^n)\,.$ On the other hand, since $\chi_\varepsilon *\frac{x_i}{|x|^{1+\alpha}} \in {\cal C}^\infty({\mathbb R}^n)$ and $T_j$ tends to $T$ in the weak topology of distributions, with controlled supports, we have $$
\left(\chi_\varepsilon*\frac{x_i}{|x|^{1+\alpha}}*T_j\right)(x) \longrightarrow
\left(\chi_\varepsilon*\frac{x_i}{|x|^{1+\alpha}}*T\right)(x)\,, \quad x \in {\mathbb R}^n\,. $$ Hence $$
\chi_\varepsilon*\frac{x_i}{|x|^{1+\alpha}}*T = \chi_\varepsilon*f^i\,, \quad \varepsilon > 0\,, $$
and so, letting $\varepsilon \rightarrow 0$\,, $\frac{x_i}{|x|^{1+\alpha}}*T = f^i\,.$ \qed
\section{Sketch of the proof of Theorem \ref{alfa}.}
This section will be devoted to the proof of inequality (\ref{mainineq}), namely
$$\gamma_\alpha(E)\leq C\gamma_{\alpha,+}(E).$$
\noindent We will adapt the line of reasoning in \cite{semiad} and \cite{semiad2}, where Tolsa proves the semiadditivity of analytic capacity and continuous analytic capacity respectively. We will also use the modifications introduced in \cite{volberg}, where the semiadditivity of Lipschitz harmonic capacity is proven (see also \cite{tolsaaleix}).
In fact, when one analizes the proofs of \cite{semiad}, \cite{semiad2} and \cite{volberg} one realizes that they depend on two technical facts, the exterior regularity property of $\gamma_\alpha$ (see Lemma \ref{extregalfa}) and an $L^\infty-$localization result, which is Theorem \ref{localization1} in our setting. We must mention that the positivity properties of the symmetrization method for the Cauchy kernel discovered in \cite{me} and \cite{meve} are an important ingredient for the proofs of \cite{semiad} and \cite{semiad2}. In \cite{volberg} one has to circumvent this lack of positivity and modify Tolsa's idea.
We will explain now how each of the above mentioned main ingredients take part in the proof of (\ref{mainineq}): As we proved in Lemma \ref{extregalfa}, the capacities $\gamma_\alpha$, $0<\alpha<n$, enjoy the exterior regularity property. This is also true for the capacities $\gamma_{\alpha,+}$, $0<\alpha<n,$ just by the weak $\star$ compactness of the the set of positive measures having total variation not exceeding $1\,.$
We therefore can approximate a general compact set $E$ by sets which are finite unions of cubes of the same side length in such a way that the capacities $\gamma_\alpha$ and $\gamma_{\alpha,+}$ of the approximating sets are as close as we wish to those of $E\,.$ Thus we can assume, without loss of generality, that $E$ is a finite union of cubes of the same size.
This will allow to implement an induction argument on the size of certain ($n$-dimensional) rectangles. The first step involves rectangles of diameter comparable to the side length of the cubes whose union is $E$.
The starting point of the general inductive step in the proof of Tolsa's Theorem in~\cite{semiad} (and \cite{semiad2}) and in \cite{volberg} for the Lipschitz harmonic capacity case, consists in the construction of a positive Radon measure $\mu$ supported on a compact set $F$ which approximates $E$ in an appropriate sense. The construction of $F$ and $\mu$ gives readily that \begin{equation}\label{mu} \gamma_\alpha(E) \le C\, \mu(F), \end{equation} and \begin{equation}\label{mu2} \gamma_{\alpha,+}(F) \le C\,\gamma_{\alpha,+}(E), \end{equation} which tells us that $F$ is not too small but also not too big.
However, one cannot expect, in the context of \cite{semiad} and \cite{semiad2} the Cauchy singular integral to be
bounded on~$L^2(\mu)$. In our case one cannot expect the $\alpha-$Riesz operator $R(\mu)$ to be bounded on~$L^2(\mu)$. One has to carefully look for a compact subset $G$ of $F$ such that \begin{itemize} \item $\mu(F) \le C\,\mu(G)$. \item The restriction $\mu_G$ of $\mu$ to $G$ has $\alpha-$growth. \item The operator $R(\mu_G)$, is bounded on $L^2(\mu_G)$ with dimensional constants. \end{itemize}
\noindent Moreover, recall from (\ref{gaopgam}), that one has
$$C^{-1}\,\gamma_{\alpha,op}(E)\le\gamma_{\alpha,+}(E)\le C\,\gamma_{\alpha,op}(E).$$
\noindent This completes the proof because then \begin{equation*} \begin{split} \gamma_\alpha(E) &\le C\, \mu(F) \le C\, \mu(G) \le C\,\gamma_{\alpha,op}(G) \le C\,\gamma_{\alpha,op}(F) \\*[5pt] & \le C\,\gamma_{\alpha,+}(F) \le C\, \gamma_{\alpha,+}(E) \le C\,\gamma_{\alpha,op}(E) . \end{split} \end{equation*}
In \cite{semiad}, \cite{semiad2} and \cite{volberg} the set $F$ is defined as the union of a special family of cubes $\{Q_i\}_{i=1}^N$ that cover the set $E$ and approximate $E$ at an appropriate intermediate scale. One then sets $$F=\bigcup_{i=1}^NQ_i.$$
The construction of the set $F$ is different in the analytic capacity case and in the Lipschitz harmonic capacity case. In Tolsa's proof this construction is performed by using the positivity properties of the symmetrization of the Cauchy kernel discovered in
\cite{me} and \cite{meve}. In our setting, the symmetrization of the Riesz kernels $x/|x|^{1+\alpha}$ only gives a positive quantity for $0<\alpha\leq 1$, (see \cite[Lemma 4.2]{laura1}), therefore we have to circumvent the use of this positivity property and therefore modify Tolsa's idea. For this modification we will follow chapter 5 of \cite{volberg}, where this was done for the Lipschitz harmonic capacity case, namely for $\alpha=n-1$. In fact the arguments in chapter 5 of \cite{volberg} are written for more general Calder\'on-Zygmund kernels of homogeneity $-\alpha$ and so they also work in our setting. Therefore, the construction of the approximating set $F$ with properties (\ref{mu}) and (\ref{mu2}) can be done just as in \cite[ch. 5]{volberg}.\newline
By the definition of the capacity $\gamma_\alpha$ it follows that there exists a real distribution $T_0$ supported on the compact set $E$ such that \begin{enumerate}
\item $\displaystyle{|<T_0,1>|\geq \frac{\gamma_\alpha(E)}{2}.}$
\item $T_0$ has $\alpha-$growth and $G_\alpha(T_0) \le 1$\,.
\item $\displaystyle{\|\frac{x_j}{|x|^{1+\alpha}}* T_0\|_\infty\leq 1,}\,\,\,\,\,\,1\le j\le n$. \end{enumerate}
The construction of $\mu$ is performed simultaneously with that of a real measure $\nu$, which should be viewed as a model for $T_0$ adapted to the family of cubes $\{Q_i\}_{i=1}^N\,.$ For each cube $Q_i$ choose a ball $B_i$ concentric with $Q_i$ with radius $r_i$ comparable to $\gamma_\alpha(E\cap 2Q_i)$ and set
$$\mu=\sum_{i=1}^N\frac{r_i^\alpha}{{\cal L}^n(B_i)}{\cal L}^n_{|B_i}.$$
Consider now functions $\varphi_i\in{\cal C}_0^{\infty}(2Q_i)$,
$0\leq\varphi_i\leq 1$, $\|\partial^s \varphi_i\|_\infty\leq C\,l(Q_i)^{-|s|}\,,$ and $\sum_{i=1}^N\varphi_i=1$ on $\bigcup_iQ_i$. The measure $\nu$ is defined as
$$\nu=\sum_{i=1}^N\frac{<T_0,\varphi_i>}{{\cal L}^n(B_i)}{\cal L}^n_{|B_i},$$ with ${\cal L}^n$ being the $n$-dimensional Lebesgue measure.
Notice that supp$(\nu)\subset\mbox{supp}(\mu)\subset F$. Moreover we have $d\nu=bd\mu$, with
$\displaystyle{b=\frac{<T_0,\varphi_i>}{r_i^\alpha}}$ on $B_i$. At this point, we need to show that our function $b$ is bounded to apply later a suitable $T(b)$ Theorem. To estimate $\|b\|_{\infty}$ we need the localization Lemma \ref{localization1}, proved in section $3.2$, which gives us
$$\|\frac{x_j}{|x|^{1+\alpha}}*\varphi_i T_0\|_\infty\leq C\,,\,\,\,\,1\le j\le n\,. $$ We therefore obtain, by the definition of $\gamma_\alpha$,
\begin{equation}\label{capita}
|<T_0 , \varphi_i>|\leq C\gamma_\alpha(2Q_i\cap E),\,\,\,\mbox{ for }\,1\le i\le N\,. \end{equation}
Hence $\|b\|_\infty\leq C$. It is now easy to see why $\gamma_\alpha(E)\leq C\mu(F)$:
$$\gamma_\alpha(E)\leq 2|\langle T_0,1\rangle|\le 2\sum_{i=1}^N |\langle T_0,\varphi_i\rangle|\leq C\sum_{i=1}^N\gamma_\alpha(E\cap 2Q_i)=C\mu(F).$$
Now everything is ready to apply a suitable variant of the $T(b)$ Theorem (see \cite{ntv}). There is still one more difficulty, in applying the Nazarov, Treil and Volberg $T(b)-$type theorem, one needs finding a substitute for what they call the suppressed operators. It was already explained in \cite{laura1} that there are at least two versions of such operators for the Riesz kernels that work appropriately (see \cite[(2.7) and (2.13)]{laura1}).
\noindent Departament de Matem\`atiques, Universitat Aut\`onoma de Barcelona, 08193 Bellaterra (Barcelona), Catalunya.\newline\newline {\em E-mail:} {\tt laurapb@mat.uab.cat}
\end{document} |
\begin{document}
\openup 1.0\jot \date{}\title{{\Large \bf On the first Banhatti-Sombor index
\thanks{ Supported by the National Natural Science Foundation of China (Nos. 12071411 and 11771443).} \begin{abstract} Let $d_v$ be the degree of the vertex $v$ in a connected graph $G$. The first Banhatti-Sombor index of $G$ is defined as $BSO(G) =\sum_{uv\in E(G)}\sqrt{\frac{1}{d^2_u}+\frac{1}{d^2_v}}$, which is a new vertex-degree-based topological index introduced by Kulli. In this paper, the mathematical relations between the first Banhatti-Sombor index and some other well-known vertex-degree-based topological indices are established. In addition, the trees extremal with respect to the first Banhatti-Sombor index on trees and chemical trees are characterized, respectively.
\noindent {\bf MSC Classification:} 05C05, 05C07, 05C09, 05C92
\noindent {\bf Keywords:} The first Banhatti-Sombor index; Degree; Tree \end{abstract} \baselineskip 20pt
\section{\large Introduction} Let $G$ be a simple undirected connected graph with vertex set $V(G)$ and edge set $E(G)$. The number of vertices and edges of $G$ is called order and size, respectively. Denote by $\overline{G}$ the complement of $G$. For $v\in V(G)$, $d_v$ denotes the degree of vertex $v$ in $G$. The minimum and the maximum degree of $G$ are denoted by $\delta(G)$ and $\Delta(G)$, or simply $\delta$ and $\Delta$, respectively. A pendant vertex of $G$ is a vertex of degree $1$. A graph $G$ is called $(\Delta, \delta)$-semiregular if $\{d_u, d_v\} = \{\Delta, \delta\}$ holds for all edges $uv\in E(G)$. Denote by $K_n$, $C_n$, $P_n$ and $K_{1,\,n-1}$ the complete graph, cycle, path and star with $n$ vertices, respectively.
The study of topological indices of various graph structures has been of interest to chemists, mathematicians, and scientists from related fields due to the fact that the topological indices play a significant role in mathematical chemistry especially in the QSPR/QSAR modeling. In 1975, the Randi\'{c} index of a graph $G$ introduced by Randi\'{c} \cite{R} is the most important and widely applied. It is defined as $$R(G)=\sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_ud_v}}.$$
The modified second Zagreb index of a graph $G$, introduced by Nikoli\'{c} et al. \cite{NKMT}, is defined as $$M_2^{*}(G)=\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}$$
The harmonic index and the inverse degree index of a graph $G$ proposed by Fajtlowicz \cite{F} are two the older vertex-degree-based topological indices. They are respectively defined as $$H(G)=\sum\limits_{uv\in E(G)}\frac{2}{d_u+d_v}, \quad ID(G)=\sum\limits_{uv\in E(G)}\left(\frac{1}{d_u^2}+\frac{1}{d_v^2}\right).$$
The symmetric division deg index, the inverse sum indeg index and the geometric-arithmetic index of a graph $G$, introduced by Vuki\v{c}evi\'{c} \cite{V, VG, VF}, Ga\v{s}perov \cite{VG} and Furtula \cite{VF}, are respectively defined as $$SDD(G)=\sum\limits_{uv\in E(G)}\frac{d_u^2+d_v^2}{2d_ud_v}, \quad ISI(G)=\sum\limits_{uv\in E(G)}\frac{d_ud_v}{d_u+d_v}, \quad GA(G)=\sum\limits_{uv\in E(G)}\frac{2\sqrt{d_ud_v}}{d_u+d_v}.$$ The forgotten topological index, introduced by Furtula and Gutman \cite{FG}, is defined as $$F(G)=\sum\limits_{uv\in E(G)}\left(d_u^2+d_v^2\right)$$
In 2021, the Sombor index of a graph $G$ is defined as $$SO(G) =\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$$ which is a novel vertex-degree-based molecular structure descriptor proposed by Gutman \cite{G}. The investigation of the Sombor index of graphs has quickly received much attention. In particular, Red\v{z}epovi\'{c} \cite{R2} showed that the Sombor index may be used successfully on modeling thermodynamic properties of compounds due to the fact that the Sombor index has satisfactory prediction potential in modeling entropy and enthalpy of vaporization of alkanes. Das et al. \cite{DCC} and Wang et al. \cite{WMLF} gave the mathematical relations between the Sombor index and some other well-known vertex-degree-based topological indices. For other related results, one may refer to \cite{CGR, DTW, LMZ, KG, MMM, RDA} and the references therein.
Inspired by work on Sombor index, the first Banhatti-Sombor index of a connected graph $G$ was introduced by Kulli \cite{K} very recently and is defined as $$BSO(G) =\sum_{uv\in E(G)}\sqrt{\frac{1}{d^2_u}+\frac{1}{d^2_v}}.$$ We find that the new index has close contact with numerous well-known vertex-degree-based topological indices. Moreover, the trees with the maximum and minimum first Banhatti-Sombor index among the set of trees with $n$ vertices are determined, respectively. In particular, the extremal values of the first Banhatti-Sombor index for chemical trees are characterized.
\section{\large Preliminaries}
\begin{lemma}\label{le2,1} For any edge $uv\in E(G)$, $d_u^2+d_v^2$ or $\frac{1}{d_u^2}+\frac{1}{d_v^2}$ is a constant if and only if $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{lemma}
\begin{lemma}\label{le2,2} For any positive real number $a$ and $b$, we have $$\frac{2\sqrt{2}(a^2+b^2+ab)}{3(a+b)}\leq \sqrt{a^2+b^2}\leq\frac{\sqrt{2}(a^2+b^2)}{a+b}$$ with equality if and only if $a=b$. \end{lemma}
\begin{lemma}{\bf (\cite{R1})}\label{le2,3} If $a_i>0$, $b_i>0$, $p>0$, $i=1, 2, \ldots, n$, then the following inequality holds: $$\sum\limits_{i=1}^{n}\frac{a_k^{p+1}}{b_k^p}\geq \frac{\left(\sum\limits_{i=1}^{n}a_i\right)^{p+1}}{\left(\sum\limits_{i=1}^{n}b_i\right)^{p}}$$ with equality if and only if $\frac{a_1}{b_1}=\frac{a_2}{b_2}=\cdots=\frac{a_n}{b_n}$. \end{lemma}
\begin{lemma}{\bf (\cite{DM})}\label{le2,4} Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be real numbers such that $q\leq \frac{a_i}{b_i}\leq Q$ and $a_i\neq 0$ for $i=1, 2, \ldots, n$. Then there holds $$\sum\limits_{i=1}^{n}b_i^2+Qq\sum\limits_{i=1}^{n}a_i^2\leq (Q+q)\sum\limits_{i=1}^{n}a_ib_i$$ with equality if and only if $b_i=qa_i$ or $b_i=Qa_i$ for et least one $i$, $i=1, 2, \ldots, n$. \end{lemma}
\begin{lemma}{\bf (\cite{D})}\label{le2,5} If $a=(a_1, a_2, \ldots, a_n)$, $b=(b_1, b_2, \ldots, b_n)$ are sequences of real numbers and $c=(c_1, c_2, \ldots, c_n)$, $d=(d_1, d_2, \ldots, d_n)$ are nonnegative, then $$\sum\limits_{i=1}^{n}d_i\sum\limits_{i=1}^{n}c_ia_i^2+\sum\limits_{i=1}^{n}c_i\sum\limits_{i=1}^{n}d_ib_i^2\geq 2\sum\limits_{i=1}^{n}c_ia_i\sum\limits_{i=1}^{n}d_ib_i$$ with equality if and only if $a=b=(k, k, \ldots, k)$ is a constant sequence for positive $c_i$ and $d_i$, $i=1, 2, \ldots, n$. \end{lemma}
\begin{lemma}{\bf (\cite{LMR})}\label{le2,6} Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be real numbers such that $a\leq a_i\leq A$ and $b\leq b_i\leq B$ for $i=1, 2, \ldots, n$. Then there holds $$\left\lvert \frac{1}{n}\sum\limits_{i=1}^{n}a_ib_i-\frac{1}{n}\sum\limits_{i=1}^{n}a_i\frac{1}{n}\sum\limits_{i=1}^{n}b_i\right\rvert\leq \frac{1}{n}\left\lfloor\frac{n}{2}\right\rfloor\left(1-\frac{1}{n}\left\lfloor\frac{n}{2}\right\rfloor\right)(A-a)(B-b),$$ where $\lfloor x\rfloor$ denotes the integer part of $x$. \end{lemma}
\section{\large On relations between the first Banhatti-Sombor index and other degree-based indices}
\subsection{\large Bounds in terms of order, size and degree}
\begin{theorem}\label{th3,1} Let $G$ be a connected graph of order $n$ and size $m$ with the minimum degree $\delta$. Then $$\frac{n}{\sqrt{2}}\leq BSO(G)\leq \frac{\sqrt{2}m}{\delta}$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof} Note that $$BSO(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{\delta^2}+\frac{1}{\delta^2}}=\frac{\sqrt{2}m}{\delta}$$ with equality if and only if $d_u=\delta$ for any vertex $u$, that is, $G$ is a regular graph.
By the Cauchy-Schwarz inequality, we have $$BSO(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\geq \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{2}}\left(\frac{1}{d_u}+\frac{1}{d_v}\right)=\frac{1}{\sqrt{2}}n$$ with equality if and only if $d_u=d_v$ for any edge $uv$, that is, $G$ is a regular graph.
\end{proof}
\begin{corollary}\label{cor3,1} Let $G$ be a regular connected graph with $n$ vertices. Then $$BSO(G)= \frac{n}{\sqrt{2}}.$$ \end{corollary}
\begin{remark} This implies that $BSO(G)$ dose not increase with the increase of the number of edges of $G$. Clearly, $BSO(K_n)=BSO(C_n)$. \end{remark}
\begin{corollary}\label{cor3,2} Let $U_n$ be a unicyclic graph with $n$ vertices. Then $$BSO(U_n)\geq \frac{n}{\sqrt{2}}$$ with equality if and only if $G\cong C_n$. \end{corollary}
\begin{corollary}\label{cor3,3} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\sqrt{2} n\leq BSO(G)+BSO(\overline{G})\leq \sqrt{2}\left(\frac{m}{\delta}+\frac{n(n-1)-2m}{2(n-1-\Delta)}\right)$$ with equality if and only if $G$ is a regular graph. \end{corollary}
\begin{theorem}\label{th3,2} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$. Then $$BSO(G)\leq n-m(2-\sqrt{2})\frac{1}{\Delta}$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof}Without loss of generality, we suppose that $d_u\geq d_v$. Then we have \begin{eqnarray*} BSO(G) & = & \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}
\leq \sum\limits_{uv\in E(G)}\left(\frac{1}{d_v}+(\sqrt{2}-1)\frac{1}{d_u}\right)\\ & \leq & \sum\limits_{uv\in E(G)}\left(\frac{1}{d_v}+\frac{1}{d_u}\right)+m(\sqrt{2}-2)\frac{1}{\Delta}
= n-m(2-\sqrt{2})\frac{1}{\Delta} \end{eqnarray*} with equality if and only if $G$ is a regular graph. \end{proof}
\begin{corollary}\label{cor3,4} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\sqrt{2} n\leq BSO(G)+BSO(\overline{G})\leq 2n-(2-\sqrt{2})\left(\frac{m}{\Delta}+\frac{n(n-1)-2m}{2(n-1-\delta)}\right)$$ with equality if and only if $G$ is a regular graph. \end{corollary}
\subsection{\large Bounds in terms of the Randi\'{c} index, the modified second Zagreb index and the inverse degree index}
\begin{theorem}\label{th3,3} Let $G$ be a connected graph with the maximum degree $\Delta$. Then $$\sqrt{2}R(G)\leq BSO(G)\leq \sqrt{2}\Delta M_2^{*}(G)$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof} By the arithmetic geometric inequality, we have $$BSO(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\geq \sum\limits_{uv\in E(G)}\sqrt{\frac{2}{d_ud_v}}=\sqrt{2}R(G)$$ with equality if and only if $d_u=d_v$ for any edges, that is, $G$ is a regular graph. It easy to see that $$BSO(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \sum\limits_{uv\in E(G)}\frac{\sqrt{2\Delta^2}}{d_ud_v}=\sqrt{2}\Delta M_2^{*}(G)$$ with equality if and only if $d_u=d_v=\Delta$ for any edges, that is, $G$ is a regular graph. \end{proof}
\begin{theorem}\label{th3,4} Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \sqrt{mID(G)}$$ with equality if and only if $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{theorem}
\begin{proof}By the Cauchy-Schwarz inequality, we have $$BSO(G)= \sum\limits_{uv\in E(G)}1\cdot \sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \sqrt{\sum\limits_{uv\in E(G)}1^2\sum\limits_{uv\in E(G)}\left(\frac{1}{d_u^2}+\frac{1}{d_v^2}\right)}=\sqrt{mID(G)},$$ with equality if and only if $\frac{1}{d_u^2}+\frac{1}{d_v^2}$ is a constant for any edge $uv$ in a connected graph $G$. By Lemma \ref{le2,1}, $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{proof}
\subsection{\large Bounds in terms of the harmonic index, the symmetric division deg index and the modified second Zagreb index}
\begin{theorem}\label{th3,5} Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\sqrt{2}H(G)\leq BSO(G)\leq \frac{1}{\sqrt{2}}\left(\frac{\Delta}{\delta}+\frac{\delta}{\Delta}\right)H(G)$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof} By Lemma \ref{le2,2}, we have \begin{eqnarray*} BSO(G) & = & \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \sum\limits_{uv\in E(G)}\frac{\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right)}{d_u+d_v}\\ & \leq & \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{2}}\left(\frac{\Delta}{\delta}+\frac{\delta}{\Delta}\right)\frac{2}{d_u+d_v}
= \frac{1}{\sqrt{2}}\left(\frac{\Delta}{\delta}+\frac{\delta}{\Delta}\right)H(G). \end{eqnarray*} with equality if and only if $d_u=d_v$ for any edges, that is, $G$ is a regular graph.
By Lemma \ref{le2,2}, we have \begin{eqnarray*} BSO(G) & = & \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\geq \sum\limits_{uv\in E(G)}\frac{2\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}+1\right)}{3(d_u+d_v)}\\ & \geq & \sum\limits_{uv\in E(G)}\frac{2\sqrt{2}(2+1)}{3(d_u+d_v)}= \sqrt{2}H(G) \end{eqnarray*} with equality if and only if $d_u=d_v$ for any edges, that is, $G$ is a regular graph. \end{proof}
\begin{theorem}\label{th3,6} Let $G$ be a connected graph with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\frac{2\sqrt{2}}{3\Delta}SDD(G)+\frac{\sqrt{2}}{3}H(G)\leq BSO(G)\leq \frac{\sqrt{2}}{\delta}SDD(G)$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof}By Lemma \ref{le2,2}, we have \begin{eqnarray*} BSO(G) & = & \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \sum\limits_{uv\in E(G)}\frac{\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right)}{d_u+d_v}\\ & \leq & \sum\limits_{uv\in E(G)}\frac{\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right)}{\delta+\delta}
= \frac{\sqrt{2}}{\delta}SDD(G). \end{eqnarray*} with equality if and only if $d_u=d_v=\delta$ for any edges, that is, $G$ is a regular graph.
By Lemma \ref{le2,2}, we have \begin{eqnarray*} BSO(G) & = & \sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}} \geq \sum\limits_{uv\in E(G)}\frac{2\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}+1\right)}{3(d_u+d_v)}\\ & = & \sum\limits_{uv\in E(G)}\frac{2\sqrt{2}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right)}{3(d_u+d_v)}+\sum\limits_{uv\in E(G)}\frac{2\sqrt{2}}{3(d_u+d_v)}
= \frac{2\sqrt{2}}{3\Delta}SDD(G)+\frac{\sqrt{2}}{3}H(G). \end{eqnarray*} with equality if and only if $d_u=d_v=\Delta$ for any edges, that is, $G$ is a regular graph. \end{proof}
\begin{theorem}\label{th3,7} Let $G$ be a connected graph with $n$ vertices. Then $$BSO(G)\leq \sqrt{2M_2^{*}(G)SDD(G)}$$ with equality if and only if $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{theorem}
\begin{proof} Let $r=1$, $a_i=\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}$ and $b_i=\frac{1}{d_ud_v}$ in Lemma \ref{le2,3}. Then we have $$\frac{\left(\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\right)^2}{\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}}\leq \sum\limits_{uv\in E(G)}\frac{\frac{1}{d_u^2}+\frac{1}{d_v^2}}{\frac{1}{d_ud_v}}=\sum\limits_{uv\in E(G)}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right),$$ that is, $$BSO(G)\leq \sqrt{2M_2^{*}(G)SDD(G)}$$ with equality if and only if $\sqrt{d_u^2+d_v^2}$ is a constant for any edge $uv$ in $G$, by Lemma \ref{le2,1}, $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{proof}
\begin{corollary}\label{cor3,5} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \sqrt{mM_2^{*}(G)\left(\frac{\Delta}{\delta}+\frac{\delta}{\Delta}\right)}$$ with equality if and only if $G$ is a regular graph or a $(\Delta, \delta)$-semiregular bipartite graph. \end{corollary}
\begin{proof} Without loss of generality, we assume that $d_u\geq d_v$. By the proof of Theorem \ref{th3,7}, we have $$\frac{\left(\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\right)^2}{\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}}\leq \sum\limits_{uv\in E(G)}\left(\frac{d_v}{d_u}+\frac{d_u}{d_v}\right)\leq \left(\frac{\Delta}{\delta}+\frac{\delta}{\Delta}\right)m$$ with equality if and only if $d_u=\Delta$ and $d_v=\delta$ for any edge $uv$. This implies that $G$ is $G$ is a regular graph or a $(\Delta, \delta)$-semiregular bipartite graph. Conversely, it is easy to check that equality holds in Corollary \ref{cor3,4} when $G$ is a regular graph or a $(\Delta, \delta)$-semiregular bipartite graph. \end{proof}
\subsection{\large Bounds in terms of the forgotten index}
\begin{theorem}\label{th3,8} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\geq \frac{\sqrt{2}}{\Delta^3+\delta^3}\left(\frac{m\delta^3}{\Delta}+\frac{F}{2}\right)$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof}Let $a_i=\sqrt{d_u^2+d_v^2}$ and $b_i=\frac{1}{d_ud_v}$ in Lemma \ref{le2,4}. Then $q=\frac{1}{\sqrt{2}\Delta^3}$ and $Q=\frac{1}{\sqrt{2}\delta^3}$. By Lemma \ref{le2,4}, we have $$\sum\limits_{uv\in E(G)}\frac{1}{d_u^2d_v^2}+\frac{1}{2\Delta^3\delta^3}\sum\limits_{uv\in E(G)}(d_u^2+d_v^2)\leq \frac{1}{\sqrt{2}}\left(\frac{1}{\Delta^3}+\frac{1}{\delta^3}\right)\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}},$$ that is, $$\frac{m}{\Delta^4}+\frac{1}{2\Delta^3\delta^3}F(G)\leq \frac{1}{\sqrt{2}}\left(\frac{1}{\Delta^3}+\frac{1}{\delta^3}\right)BSO(G),$$ that is, $$BSO(G)\geq \frac{\sqrt{2}}{\Delta^3+\delta^3}\left(\frac{m\delta^3}{\Delta}+\frac{F}{2}\right)$$ with equality if and only if $d_u=d_v=\Delta$ for any edge $uv$, that is, $G$ is a regular graph. \end{proof}
\begin{theorem}\label{th3,9} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \frac{2mSDD(G)+M_2^{*}(G)F(G)}{2SO(G)}$$ with equality if and only if $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{theorem}
\begin{proof} Let $a_i=b_i=\sqrt{d_u^2+d_v^2}$, $c_i=\frac{1}{d_ud_v}$ and $d_i=1$ in Lemma \ref{le2,5}. Then we have $$m\sum\limits_{uv\in E(G)}\frac{d_u^2+d_v^2}{d_ud_v}+\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}\sum\limits_{uv\in E(G)}(d_u^2+d_v^2)\geq 2\sum\limits_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_ud_v}\sum\limits_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$$ that is, $$2mSDD(G)+M_2^{*}(G)F(G)\geq 2BSO(G)SO(G),$$ that is, $$BSO(G)\leq \frac{2mSDD(G)+M_2^{*}(G)F(G)}{2SO(G)}$$ with equality if and only if $a_i=b_i=\sqrt{d_u^2+d_v^2}$ for any edge $uv$ in $G$, that is, $d_u^2+d_v^2$ is a constant for any edge $uv$ in $G$, by Lemma \ref{le2,1}, $G$ is a regular graph (when $G$ is non-bipartite) or $G$ is a $(\Delta, \delta)$-semiregular bipartite graph (when $G$ is bipartite). \end{proof}
\begin{corollary}\label{cor3,6} Let $G$ be a connected graph of size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \frac{m(\Delta^2\delta+\delta^2)+\Delta F(G)}{2\sqrt{2}\Delta\delta^3}$$ with equality if and only if $G$ is a regular graph. \end{corollary}
\begin{corollary}\label{cor3,7} Let $G$ be a connected graph of size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \frac{m^2(2\Delta^3+\Delta^2\delta+\delta^3)}{2\Delta\delta^2SO(G)}$$ with equality if and only if $G$ is a regular graph. \end{corollary}
\subsection{\large Bounds in terms of the inverse sum indeg index and the geometric-arithmetic index}
\begin{theorem}\label{th3,10} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \frac{H(G)SDD(G)+2M_2^{*}(G)ISI(G)}{\sqrt{2}GA(G)}$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof} Let $a_i=\sqrt{d_u^2+d_v^2}$, $b_i=\sqrt{2d_ud_v}$, $c_i=\frac{1}{d_ud_v}$ and $d_i=\frac{1}{d_u+d_v}$ in Lemma \ref{le2,5}. Then we have \begin{align*}
& \sum\limits_{uv\in E(G)}\frac{1}{d_u+d_v}\sum\limits_{uv\in E(G)}\frac{d_u^2+d_v^2}{d_ud_v}+\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}\sum\limits_{uv\in E(G)}\frac{2d_ud_v}{d_u+d_v}\\ \geq {}& 2\sum\limits_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_ud_v}\sum\limits_{uv\in E(G)}\frac{\sqrt{2d_ud_v}}{d_u+d_v}, \end{align*} that is, $$H(G)SDD(G)+2M_2^{*}(G)ISI(G)\geq \sqrt{2}BSO(G)GA(G),$$ that is, $$BSO(G)\leq \frac{H(G)SDD(G)+2M_2^{*}(G)ISI(G)}{\sqrt{2}GA(G)}$$ with equality if and only if $\sqrt{d_u^2+d_v^2}=\sqrt{2d_ud_v}$ for any edge $uv$, that is, $G$ is a regular graph. \end{proof}
\begin{corollary}\label{cor3,8} Let $G$ be a connected graph of size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$BSO(G)\leq \frac{m^2\Delta^2+m^2\delta^2+4m\Delta ISI(G)}{2\sqrt{2}\Delta\delta^2GA(G)}$$ with equality if and only if $G$ is a regular graph. \end{corollary}
\subsection{\large Bounds in terms of the Sombor index and the modified second Zagreb index}
\begin{theorem}\label{th3,11} Let $G$ be a connected graph of size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\frac{2m^2}{SO(G)}\leq BSO(G)\leq \frac{1}{\delta^2}SO(G)$$ with equality if and only if $G$ is a regular graph. \end{theorem}
\begin{proof}It is easy to see that $$BSO(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\leq \frac{1}{\delta^2}\sum\limits_{uv\in E(G)}\sqrt{d_u^2+d_v^2}\leq\frac{1}{\delta^2}SO(G),$$ with equality if and only if $d_u=d_v=\Delta$ for any edges, that is, $G$ is a regular graph.
Let $a_i=b_i=\frac{1}{\sqrt{d_ud_v}}$ and $c_i=d_i=\sqrt{d_u^2+d_v^2}$ in Lemma \ref{le2,5}. Then $$2\sum\limits_{uv\in E(G)}\sqrt{d_u^2+d_v^2}\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}\geq 2\left(\sum\limits_{uv\in E(G)}\sqrt{\frac{d_u^2+d_v^2}{d_ud_v}}\right)^2\geq 4m^2,$$ that is, $$2SO(G)BSO(G)\geq 4m^2,$$ that is, $$BSO(G)\geq \frac{2m^2}{SO(G)}$$ with equality if and only if $G$ is a regular graph. \end{proof}
\begin{theorem}\label{th3,12} Let $G$ be a connected graph of order $n$ and size $m$ with the maximum degree $\Delta$ and the minimum degree $\delta$. Then $$\left\lvert \frac{1}{m}BSO(G)-\frac{1}{m^2}SO(G)M_2^{*}(G)\right\rvert\leq \xi(m)\frac{\sqrt{2}(\Delta+\delta) (\Delta-\delta)^2}{\Delta^2\delta^2},$$ where $$\xi(m)=\frac{1}{4}\left(1-\frac{1+(-1)^{m+1}}{2m^2}\right).$$ \end{theorem}
\begin{proof} Let $a_i=\sqrt{d_u^2+d_v^2}$ and $b_i=\frac{1}{d_ud_v}$ in Lemma \ref{le2,6}. Then $a=\sqrt{2}\delta$, $A=\sqrt{2}\Delta$, $b=\frac{1}{\Delta^2}$ and $B=\frac{1}{\delta^2}$. By Lemma \ref{le2,6}, we have \begin{align*}
& \left\lvert \frac{1}{m}\sum\limits_{uv\in E(G)}\sqrt{\frac{1}{d_u^2}+\frac{1}{d_v^2}}-\frac{1}{m^2}\sum\limits_{uv\in E(G)}\sqrt{d_u^2+d_v^2}\sum\limits_{uv\in E(G)}\frac{1}{d_ud_v}\right\rvert \\ \leq {}& \frac{1}{m}\left\lfloor\frac{m}{2}\right\rfloor\left(1-\frac{1}{m}\left\lfloor\frac{m}{2}\right\rfloor\right) \sqrt{2}(\Delta-\delta)(\frac{1}{\delta^2}-\frac{1}{\Delta^2}), \end{align*} that is, $$\left\lvert \frac{1}{m}BSO(G)-\frac{1}{m^2}SO(G)M_2^{*}(G)\right\rvert\leq \xi(m)\frac{\sqrt{2}(\Delta+\delta) (\Delta-\delta)^2}{\Delta^2\delta^2},$$ where $$\xi(m)=\frac{1}{m}\left\lfloor\frac{m}{2}\right\rfloor\left(1-\frac{1}{m}\left\lfloor\frac{m}{2}\right\rfloor\right) =\frac{1}{4}\left(1-\frac{1+(-1)^{m+1}}{2m^2}\right).$$ \end{proof}
\section{\large The first Banhatti-Sombor index of trees}
In this section, we determine the trees with the maximum and minimum first Banhatti-Sombor index among the set of trees of order $n$, respectively. For a tree $T_n$ of order $n$ with maximum degree $\Delta$, denote by $n_i$ the number of vertices with degree $i$ in $T_n$ for $1\leq i\leq\Delta$, and $m_{i,j}$ the number of edges of $T_n$ connecting vertices of degree $i$ and $j$, where $1\leq i\leq j\leq\Delta$. Note that $T_n$ is connected, so $m_{1,1}=0$ for $n\geq 3$. Let $N=\{(i,j)\in\mathbb{N}\times\mathbb{N}:1\leq i\leq j\leq\Delta\}$. Then clearly the following relations hold:
\begin{equation}\label{eq1}
|V(T_n)|=n=\sum_{i=1}^{\Delta}n_i, \end{equation}
\begin{equation}\label{eq2}
|E(T_n)|=n-1=\sum_{(i,j)\in N}m_{i,j}, \end{equation} and \begin{equation}\label{eq3} \left\{ \begin{aligned} &2m_{1,1}+m_{1,2}+\ldots+m_{1,\Delta}=n_1,\\ &m_{1,2}+2m_{2,2}+\ldots+m_{2,\Delta}=2n_2,\\ &\ldots\\ &m_{1,\Delta}+m_{2,\Delta}+\ldots+2m_{\Delta,\Delta}=\Delta n_\Delta. \end{aligned} \right. \end{equation} It follows easily from (\ref{eq1}) and (\ref{eq3}) that
\begin{equation}\label{eq4}
n=\sum_{(i,j)\in N}\frac{i+j}{ij}m_{i,j}.
\end{equation} And the definition of the first Banhatti-Sombor index is equivalent to
\begin{equation}\label{eq5}
SO(G)=\sum_{(i,j)\in P}\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}m_{i,j}.
\end{equation}
\begin{theorem}\label{th4,1} Let $T_n$ be a tree with $n$-vertex. Then $$\frac{\sqrt{2}(n-3)}{2}+\sqrt{5}\leq BSO(T_n)\leq \sqrt{1+(n-1)^2}.$$ The equality in the left hand side holds if and only if $T_n\cong P_n$, and the equality in the right hand side holds if and only if $T_n\cong K_{1,\,n-1}$. \end{theorem}
\begin{proof}
First, we consider the equality in the left side. Let $N_1=\Big\{(i,j)\in N:(i,j)\neq(1,1),(i,j)\neq(1,2),(i,j)\neq(2,2)\Big\}$. By equation (\ref{eq4}), we have $$3m_{1,2}+2m_{2,2}=2n-\sum_{(i,j)\in N_1}\frac{2(i+j)}{ij}m_{i,j},$$ and by equation (\ref{eq2}), we have $$m_{1,2}+m_{2,2}=n-1-\sum_{(i,j)\in N_1}m_{i,j}.$$ Then we obtain the following expression for $m_{1,2}$ and $m_{2,2}$: $$m_{1,2}=2+\sum_{(i,j)\in N_1}\Big[2-\frac{2(i+j)}{ij}\Big]m_{i,j},$$ $$m_{2,2}=n-3+\sum_{(i,j)\in N_1}\Big[\frac{2(i+j)}{ij}-3\Big]m_{i,j}.$$ According to the expression (\ref{eq5}), we have \begin{eqnarray*} BSO(T_n)& = & m_{1,2}\sqrt{\frac{1}{4}+1}+m_{2,2}\sqrt{\frac{1}{4}+\frac{1}{4}}+\sum_{(i,j)\in N_1}\!\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}m_{i,j}\\ & = & \sqrt{5}\Big[1+\sum_{(i,j)\in N_1}\Big(1-\frac{i+j}{ij}\Big)m_{i,j}\Big]+\frac{\sqrt{2}}{2}\Big\{n-3+\sum_{(i,j)\in N_1}\Big[\frac{2(i+j)}{ij}-3\Big]m_{i,j}\Big\}\\ & &+\sum_{(i,j)\in N_1}\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}m_{i,j}\\ & = & \frac{\sqrt{2}}{2}(n-3)+\sqrt{5}+\sum_{(i,j)\in N_1}\Big[\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}+({\sqrt{2}-\sqrt{5}})\frac{i+j}{ij}+\sqrt{5}-\frac{3\sqrt{2}}{2}\Big]m_{i,j}. \end{eqnarray*}
Let $f(x,y)=\sqrt{\frac{1}{x^2}+\frac{1}{y^2}}+({\sqrt{2}-\sqrt{5}})\frac{x+y}{xy}+\sqrt{5}-\frac{3\sqrt{2}}{2}$, where $(x,y)\in N$, it is easy to see that $f(1,2)=0$, $f(2,2)=0$ and $f(x,y)> 0$ for $(x,y)\in N_1$. Therefore, $BSO(T_n)=\frac{\sqrt{2}}{2}(n-3)+\sqrt{5}$ if and only if $m_{i,j}=0$ for all $(i,j)\in N_1$. And this occurs if and only if $T_n\cong P_n$. Conversely, if $T_n\cong P_n$, by (\ref{eq5}), we obtain $$BSO(P_n)=2\sqrt{\frac{1}{4}+1}+(n-3)\sqrt{\frac{1}{4}+\frac{1}{4}}=\frac{\sqrt{2}}{2}(n-3)+\sqrt{5}.$$ Thus, we have $BSO(T_n)\geq BSO(P_n)$ with equality if and only if $T_n\cong P_n$.
Now, we consider the equality on the right side. Let $N_2=\Big\{(i,j)\in N:(i,j)\neq(1,1),(i,j)\neq(1,\Delta),(i,j)\neq(\Delta,\Delta)\Big\}$. Similar to the proof of the above, by equation (\ref{eq4}), we have $$(\Delta+1)m_{1,\Delta}+2m_{\Delta,\Delta}=\Delta n-\sum_{(i,j)\in N_2}\Delta\frac{i+j}{ij}m_{i,j},$$
and by equation (\ref{eq2}), we have $$m_{1,\Delta}+m_{\Delta,\Delta}=n-1-\sum_{(i,j)\in N_2}m_{i,j}.$$ Then we obtain the following expression for $m_{1,\Delta}$ and $m_{\Delta,\Delta}$: $$(\Delta-1)m_{1,\Delta}=(\Delta-2)n+2-\sum_{(i,j)\in N_2}\Big(\Delta\frac{i+j}{ij}-2\Big)m_{i,j},$$ $$(\Delta-1)m_{\Delta,\Delta}=n-(\Delta+1)+\sum_{(i,j)\in N_2}\Big(\Delta\frac{i+j}{ij}-(\Delta+1)\Big)m_{i,j}.$$ According to the expression (\ref{eq4}), we have \begin{eqnarray*} BSO(T_n) & = & m_{1,\Delta}\sqrt{\frac{1}{\Delta^2}+1}+m_{\Delta,\Delta}\sqrt{\frac{1}{\Delta^2}+\frac{1}{\Delta^2}}+\sum_{(i,j)\in N_2}\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}m_{i,j}\\ & = & \frac{\sqrt{\Delta^2+1}}{\Delta(\Delta-1)}\Big[(\Delta\!-\!2)n+2-\sum_{(i,j)\in N_2}\Big(\Delta\frac{i+j}{ij}-2\Big)m_{i,j}\Big]\\ & & +\frac{\sqrt{1+\Delta^2}}{\Delta(\Delta-1)}\Big[n\!-\!(\Delta+1)+\!\sum_{(i,j)\in N_2}\!\Big(\Delta\frac{i+j}{ij}\!-\!(\Delta+1)\Big)m_{i,j}\Big]\\ & & +\!\sum_{(i,j)\in N_2}\!\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}m_{i,j}\\ & = & \frac{(\Delta-2)n\sqrt{\Delta^2+1}+\sqrt{2}(n-\Delta-1)+2\sqrt{\Delta^2+1}}{\Delta(\Delta-1)}\\ & & +\sum_{(i,j)\in N_2}\Big[\sqrt{\frac{1}{i^2}+\frac{1}{j^2}}+\frac{\sqrt{2}-\sqrt{\Delta^2+1}}{\Delta-1}\frac{i+j}{ij}+\frac{2\sqrt{\Delta^2+1}-\sqrt{2}(\Delta+1)} {\Delta(\Delta-1)}\Big]m_{i,j}. \end{eqnarray*}
Let $g(x,y)=\sqrt{\frac{1}{x^2}+\frac{1}{y^2}}+\frac{\sqrt{2}-\sqrt{\Delta^2+1}}{\Delta-1}\frac{x+y}{xy}+\frac{2\sqrt{\Delta^2+1}-\sqrt{2}(\Delta+1)}{\Delta(\Delta-1)}$, where $(x,y)\in N$, it is easy to see that $f(1,\Delta)=0$, $f(\Delta,\Delta)=0$ and $f(x,y)< 0$ for $(x,y)\in N_2$. Therefore, $BSO(T_n)=\frac{(\Delta-2)n\sqrt{\Delta^2+1}+\sqrt{2}(n-\Delta-1)+2\sqrt{\Delta^2+1}}{\Delta(\Delta-1)}$ if and only if $m_{i,j}=0$ for all $(i,j)\in N_2$. And this occurs if and only if $n_2=n_3=\ldots=n_{\Delta-1}=0$.
Let $h(x)=\frac{(x-2)n\sqrt{x^2+1}+\sqrt{2}(n-x-1)+2\sqrt{x^2+1}}{x(x-1)}$. By derivative, we know that $h(x)$ is an increasing function for $[2, +\infty)$. Thus $$h(\Delta)\leq h(n-1)=\sqrt{1+(n-1)^2}.$$ Conversely, $BSO(K_{1,\,n-1})=\sqrt{1+(n-1)^2}$. Thus, we have $BSO(T_n)\leq BSO(K_{1,\,n-1})$ with equality if and only if $T_n\cong K_{1,\,n-1}$.
\end{proof}
Similar to the method used in Theorem \ref{th4,1}, we now give an upper bound on chemical trees without its proof.
\begin{theorem}\label{th4,2} Let $T_n$ is a chemical tree with $n$ vertices. If $n-2=0(mod\ 3)$, then $$BSO(T_n)\leq \frac{2\sqrt{17}(n+1)+\sqrt{2}(n-5)}{12}$$ with equality if and only if $n_2=n_3=0$. \end{theorem}
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\begin{document}
\newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{prop}[thm]{Proposition} \newtheorem{rem}[thm]{Remark} \newtheorem{eg}[thm]{Example} \newtheorem{defn}[thm]{Definition} \newtheorem{assum}[thm]{Assumption}
\renewcommand {\theequation}{\arabic{equation}} \def\arabic{section}{\arabic{section}}
\title{\bf An Elementary Proof for the Structure of Wasserstein Derivatives}
\author{ Cong Wu \thanks{ \noindent Department of Mathematics, University of Southern California, Los Angeles, CA 90089. E-mail: congw@usc.edu.} ~ and ~{Jianfeng Zhang} \thanks{\noindent Department of Mathematics, University of Southern California, Los Angeles, CA 90089. E-mail: jianfenz@usc.edu. This author is supported in part by NSF grant \#1413717. } }
\date{} \maketitle
\begin{abstract} Let $F: \mathbb{L}^2(\Omega, \mathbb{R})\footnote{The space $\mathbb{R}$ can be replaced with general $\mathbb{R}^d$. We assume $d=1$ here for simplicity. } \to \mathbb{R}$ be a law invariant and continuously Fr\'echet differentiable mapping. Based on Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \begin{eqnarray} \label{Derivative} D F (\xi) = g(\xi), \end{eqnarray} where $g: \mathbb{R}\to \mathbb{R}$ is a deterministic function which depends only on the law of $\xi$. See also Carmona \& Delarue \cite{CD} Section 5.2 and Gangbo \& Tudorascu \cite{GT}. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \cite{WZ}, which deals with a more general situation. \end{abstract}
\noindent Let ${\cal P}_2(\mathbb{R})$ denote the set of square integrable probability measures on $\mathbb{R}$, and consider a mapping $f: {\cal P}_2(\mathbb{R}) \to \mathbb{R}$. As in standard literature, we lift $f$ to a function $F: \mathbb{L}^2(\Omega, \mathbb{R})\to \mathbb{R}$ by $F(\xi):=f({\cal L}_\xi)$, where $(\Omega,{\cal F},\mathbb{P})$ is an atomless Polish probability space and ${\cal L}_\xi$ denotes the law of $\xi$.
If $F$ is Frech\'et differentiable, then $DF(\xi)$ can be identified as an element of $\mathbb{L}^2(\Omega, \mathbb{R})$: \begin{eqnarray} \label{Frechet} \mathbb{E}\big[ DF(\xi) \eta\big] = \lim_{\varepsilon\to 0} {F(\xi + \varepsilon \eta) - F(\xi) \over \varepsilon},\quad \mbox{for all}~\eta \in \mathbb{L}^2(\Omega, \mathbb{R}). \end{eqnarray}
We start with the simple case that $\xi$ is discrete. Let $\delta_x$ denote the Dirac measure of $x$.
\begin{prop} \label{prop-discrete} Assume
$\xi$ is discrete: $\mathbb{P}(\xi = x_i) = p_i$, $i\ge 1$. If $F$ is Fr\'echet differentiable at $\xi$, then \reff{Derivative} holds with \begin{eqnarray} \label{formula1} g(x_i) := \lim_{\varepsilon\to 0} {f(\sum_{j\neq i} p_j \delta_{x_j} + p_i \delta_{x_i+\varepsilon}) - f(\sum_{j\ge 1} p_j \delta_{x_j})\over \varepsilon p_i}, \quad i\ge 1. \end{eqnarray} \end{prop}
To prove the proposition, we need the following result. \begin{lem} \label{lem-elem} Let $X\in \mathbb{L}^2(\Omega, \mathbb{R})$. Assume $A\in {\cal F}$ with $\mathbb{P}(A)>0$ satisfies \begin{eqnarray} \label{elem} \mathbb{E}[X {\bf 1}_{A_1}]=\mathbb{E}[X {\bf 1}_{A_2}], \quad\mbox{for all}~ A_1,A_2\subset A~ \mbox{such that}~\mathbb{P}(A_1) = \mathbb{P}(A_2). \end{eqnarray} Then $X$ is a constant, $\mathbb{P}$-a.s. in $A$. \end{lem} \proof This result is elementary, we nevertheless provide a proof for completeness.
Assume the result is not true. Denote $c := {\mathbb{E}[X{\bf 1}_A]\over \mathbb{P}(A)}$ and $A_1 := \{X <c\}\cap A$, $A_2:= \{X>c\}\cap A$. Then $\mathbb{P}(A_1)>0$, $\mathbb{P}(A_2) >0$. Assume without loss of generality that $\mathbb{P}(A_1) \le \mathbb{P}(A_2)$. Since $(\Omega, {\cal F}, \mathbb{P})$ is atomless, there is a random variable $U$ with uniform distribution on $[0, 1]$. Denote $A_{2,x} := A_2 \cap \{U \le x\}$, $x\in [0, 1]$. Clearly there exists $x_0$ such that $\mathbb{P}(A_{2,x_0}) = \mathbb{P}(A_1)$. Apply \reff{elem} on $A_1$ and $A_{2,x_0}$ we obtain the desired contradiction.
\vrule width.25cm height.25cm depth0cm
\begin{rem} \label{rem-elem} {\rm Lemma \ref{lem-elem} may not hold if $(\Omega, {\cal F}, \mathbb{P})$ has atoms. Indeed, consider $\Omega := \{\omega_1, \omega_2\}$ with $\mathbb{P}(\omega_1) = {1\over 3}, \mathbb{P}(\omega_2) = {2\over 3}$. Set $A:= \Omega$ and $X$ is an arbitrary random variable. The \reff{elem} holds true trivially because $\mathbb{P}(A_1) \neq \mathbb{P}(A_2)$ whenever $A_1\neq A_2$. However, $X$ may not be a constant.
\vrule width.25cm height.25cm depth0cm
} \end{rem}
\noindent{\bf Proof of Proposition \ref{prop-discrete}.} Fix an $i\ge 1$. For an arbitrary $A_1 \subset A:= \{\xi = x_i\}$, set $\eta := {\bf 1}_{A_1}$. Note that, for any $\varepsilon > 0$, we have \begin{eqnarray*} {\cal L}_{\xi + \varepsilon \eta} = \sum_{j\neq i} p_j \delta_{x_j} + \mathbb{P}(A_1) \delta_{x_i + \varepsilon} + [p_i - \mathbb{P}(A_1)] \delta_{x_i}, \end{eqnarray*} which depends only on ${\cal L}_\xi$ and $\mathbb{P}(A_1)$. By \reff{Frechet}, \begin{eqnarray} \label{formula0} \mathbb{E}\big[DF(\xi) {\bf 1}_{A_1}\big] = \lim_{\varepsilon\to 0} {f\big( \sum_{j\neq i} p_j \delta_{x_j} + \mathbb{P}(A_1) \delta_{x_i + \varepsilon} + [p_i - \mathbb{P}(A_1)] \delta_{x_i}\big) - f(\sum_{j\ge 1} p_j \delta_{x_j})\over \varepsilon}. \end{eqnarray} In particular, $\mathbb{E}\big[DF(\xi) {\bf 1}_{A_1}\big]$ depends only on $\mathbb{P}(A_1)$ for $A_1\subset \{\xi = x_i\}$. Applying Lemma \ref{lem-elem}, we see that $DF(\xi)$ is a constant, $\mathbb{P}$-a.s. on $\{\xi=x_i\}$. Now set $A_1 := \{\xi=x_i\}$ in \reff{formula0}, we obtain \reff{formula1} immediately.
\vrule width.25cm height.25cm depth0cm
We now consider the general case.
\begin{thm} \label{thm-general}
If $F$ is continuously Fr\'echet differentiable, then \reff{Derivative} holds with $g$ depending only on ${\cal L}_\xi$ but not on the particular choice of $\xi$.
\end{thm}
\proof For each $n\ge 1$, denote $x^n_i := i 2^{-n}, i \in \mathbb{Z}$, and $\xi_n := \sum_{i=-\infty}^\infty x^n_i {\bf 1}_{\{x^n_i \le \xi < x^n_{i+1}\}}$. Since $\xi_n$ is discrete, by Proposition \ref{prop-discrete} we have $DF(\xi_n) = g_n(\xi_n)= \tilde g_n(\xi)$, where $g_n$ is defined on $\{x^n_i, i\in \mathbb{Z}\}$ by \reff{formula1} (with $g_n(x^n_i) :=0$ when $\mathbb{P}(\xi_n = x^n_i)=0$) and $\tilde g_n(x) := g_n(x^n_i)$ for $x\in [x^n_i, x^n_{i+1})$. Clearly $\lim_{n\to \infty}\mathbb{E}[|\xi_n-\xi|^2] = 0$. Then by the continuous differentiability of $F$ we see that $\lim_{n\to\infty} \mathbb{E}[|\tilde g_n(\xi) - DF(\xi)|^2] = 0$. Thus, there exists a subsequence $\{n_k\}_{k\ge 1}$ such that $\tilde g_{n_k} (\xi) \to D F(\xi)$, $\mathbb{P}$-a.s. Denote $K := \{x: \mathop{\overline{\rm lim}}_{k\to\infty} \tilde g_{n_k}(x) = \mathop{\underline{\rm lim}}_{k\to\infty} \tilde g_{n_k}(x)\}$, and $g(x) := \lim_{k\to \infty} \tilde g_{n_k}(x){\bf 1}_K(x)$. Then $\mathbb{P}(\xi\in K)=1$ and $D F(\xi) = g(\xi)$, $\mathbb{P}$-a.s.
Moreover, let $\xi'$ be another random variable such that ${\cal L}_{\xi'}={\cal L}_\xi$. Define $\xi_n'$ similarly. Then $D F(\xi'_n) = \tilde g_n(\xi')$ for the same function $\tilde g_n$. Note that $\mathbb{P}(\xi'\in K) = \mathbb{P}(\xi\in K) =1$, then $\lim_{k\to\infty} \tilde g_{n_k}(\xi') = g(\xi')$, $\mathbb{P}$-a.s. On the other hand, $D F(\xi'_{n_k}) \to DF (\xi')$ in $\mathbb{L}^2$. So $D F(\xi') = g(\xi')$, and thus $g$ does not depend on the choice of $\xi$.
\vrule width.25cm height.25cm depth0cm
\begin{rem} \label{rem-joint} {\rm One may also write $D F(\xi) = g({\cal L}_\xi, \xi)$, where $g: {\cal P}_2(\mathbb{R}) \times \mathbb{R} \to \mathbb{R}$. When $DF$ is uniformly continuous, one may easily construct $g$ jointly measurable in $(\mu, x) \in {\cal P}_2(\mathbb{R}) \times \mathbb{R}$. One may also extend the result to the case that $F$ is a function of processes. We leave the details to \cite{WZ}.
\vrule width.25cm height.25cm depth0cm
} \end{rem}
\end{document} |
\begin{document}
\title{Machine Learning Algorithms in Design Optimization} \author{Daniele Peri} \affil{\small CNR-IAC -- National Research Council \\
Istituto per le Applicazioni del Calcolo "Mauro Picone" \\
Via dei Taurini 19, 00185 Rome, Italy \\ \texttt{d.peri@iac.cnr.it}}
\maketitle
\begin{abstract}
Numerical optimization of complex systems benefits from the technological development of computing platforms in the last twenty years. Unfortunately, this is still not enough, and a large computational time is necessary for the solution of optimization problems when mathematical models that implement rich (and therefore realistic) physical models are adopted.
In this paper, we show how the combination of optimization and Artificial Intelligence (AI), in particular Machine Learning algorithms, can help, strongly reducing the overall computational times, making also possible the use of complex simulation systems within the optimization cycle. Original approaches are proposed.
\end{abstract}
\section{Introduction}
The very first obstacle to the solution of an optimization problem is represented by the required time. In practice, the optimization task needs to be included inside the design activities schedule, and a specific time window is assigned. Consequently, the efficiency of the optimization algorithm is very important, and the number of attempts (computations of the objective function) before the optimal configuration is identified has to be minimized. But, although the efficiency of the optimization algorithm is high, the time required for a single evaluation of the objective function could be so large that some compromises regarding the quality of the physical model become unavoidable: consequently, only a simplified mathematical model can be applied in practice, and the final solution is vitiated by this assumption.
For this reason, the use of interpolation/approximation algorithms has been widely adopted, particularly in the field of Multidisciplinary Design Optimization (MDO), where a cascade of several solvers, one for each discipline involved, are adopted together, greatly increasing the overall calculation time \cite{Haftka1992}. The first group of examples is given in \cite{Toropov1992,Barthelemy1992}, more recently other applications can be found in \cite{Peri2009,Viana2014}. A review can be also found in \cite{Parnianifard2019}. In general, the base idea is the generation of a {\em meta-model}, that is, {\em a model of the model}, so that an estimate of the objective function can be obtained by using a simple closed-form expression. A {\em meta-model} can be represented by a polynomial model \cite{Myers2016,Peri2012b} or something more flexible and sophisticated \cite{Poggio1990,Matheron1963,Peri2018}.
A great debate about the feasibility of a global interpolation/approximation model for a function of several variables is still on the table, particularly in the case of a large number of parameters. A sequence of local approximations has been soon proposed in \cite{Alexandrov1998}, and other techniques in the same framework can be observed in \cite{Jin2001,Acar2009,Teixeira2019,Peri2012b}: the objective is the reduction of the spatial validity of the interpolation, localizing the estimate. On the other side, the ambitious goal of building a single interpolation/approximation model for simulating the state of a system is the typical objective of Artificial Intelligence (AI), and those techniques, suitable for the management of a great amount of data, can be also adopted for the description of the digital twin of our physical system.
In this paper, we are describing some techniques able to provide a global interpolation of the state of a system as a function of the influencing parameters. A further improvement of the {\em meta-model} is then obtained by increasing the number of samples of the objective function in some critical areas, where the disagreement between the {\em meta-model} and the true value of the objective function is hypothesized to be high. A regularization technique is also introduced. The usefulness of these techniques is finally demonstrated thru the application to the solution of realistic design optimization problems.
\section{Machine Learning for the {\em meta-model} improvement}
As previously noticed, a central point for the optimization of complex systems can be represented by the determination of a simplified surrogate of a detailed mathematical model of the full system. We are referring to this as {\em meta-model} because it represents, in practice, {\em a model of the model}. Here we are recalling some topic elements of the building of the {\em meta-model}.
The first step for the definition of a {\em meta-model} is the generation of a dataset from which we can extract the information on the optimizing system. This is classically referred to as {\em Design Of Experiments} (DOE). Since we have typically no information about the function to be fitted, the DOE could be homogeneously ditributed into the full Design Variable Space (DVS): for the generation of the DOE we can tap into the family of the so-called {\em Uniformly Distributed Sequences} (UDS). An equally-spaced DOE is also a guaranteee of regularity of the resulting {\em meta-model}. Examples are the Halton/Hammeresley sequences \cite{Halton1964}, Orthogonal Arrays \cite{Hedayat1999}, Latin Hypercube Sampling \cite{McKay1992}, Sobol sequence\cite{Bratley1988} and PT-Network \cite{Statnikov1995}.
Furthermore, one may be also interested in evaluating the credibility of the {\em meta-model} in off-design conditions, that is, the quality of the estimated output when a configuration different from the ones included in the DOE is checked. Classical theory of Artificial Neural Network (ANN) foresees the division of the DOE in two subsets, training set and validation set. The training set is used to produce the control parameters of the ANN, the validation set is used in order to measure the predictive qualities of the ANN.
Since we are considering numerical expensive models (in terms of computational time), the use of a set of data for validation purposes could appear as a waste of resources. It would be convenient to use all the available points for the determination of the parameters of the {\em meta-model}, without exceptions. By the way, if a UDS is adopted, the extraction o a single point is affecting the uniformity of the distribution.
Now we can add some new points for the determination of the performances of the system in some hypothetical configuration, not included in the training set: this way, we can obtain information about the degree of precision of the current version of the {\em meta-model}. Once used for their purposes, validation data should be then added to the training set, re-calibrating the {\em meta-model} with a richer quantity of information. This is what is commonly called, in AI {\em Machine Learning} (ML), since we are using the original system (numerical or not) to learn something new to add to our AI system. This could become a very important phase of the formation of the {\em meta-model} if we could identify a specific area where it would be useful to add new points to the current {\em meta-model}. This is not easy, since the DOE is already uniformly distributed on the DVS, so the identification of a new sample cannot be performed based on purely geometric considerations (i.e., a specific region is not well covered by the training set).
A possible approach to this problem is proposed in \cite{Peri2009}. If we compare different {\em meta-models} over the full DVS, we observe a different behavior among them, and we cannot determine {\em a priori} which {\em meta-model} is the best to apply. This situation is typical of a small DOE (undersampling). What we can do with a moderate computational cost is to compare systematically the outcome of different {\em meta-models} over the entire DVS, generating a denser UDS for this purpose, and then trace the disagreement between the prediction of the {\em meta-models}. We can interpret the discrepancy as a measure of the uncertainty in the prediction so that an additional training point where the disagreement is maximum will surely help in aligning locally the outputs of the {\em meta-models}. This is far different from adding a new point in an area with a low density of training points because this approach is explicitly considering the local quality of the approximation. Numerical experiments reported in \cite{Peri2009} underline improvements of about 10\% of the quality of the {\em meta-model} for the same number of training points: this is a demonstration that a uniform distribution of points in the training set is for sure a good start, but customization of the training set represents a more efficient solution.
\section{Further tuning of the AI model}
A regular distribution of the training points is a prerequisite for the determination of a response surface as regular. This means that loss of regularity in the distribution of training points can affect the regularity of the {\em meta-model} response surface, and the previously proposed algorithm is not preserving the regularity of the distribution: a regularization technique could be helpful in this context.
In the following, we are indicating a possible approach for {\em meta-models} whose construction implies the solution of a linear system. In particular, we are considering Kriging \cite{Matheron1963,Peri2009} and Multi-dimensional Spline \cite{Peri2018}.
\subsection{Kriging regularization}
The training of Kriging is performed by assembling and then factorizing the self-correlation matrix of the sample points \cite{Peri2009}: the spatial correlation between two points of the DVS is determined uniquely based on the distance between the two points thru the so-called {\em semi-variogram} $\gamma$. In the original formulation, $\gamma$ is computed experimentally, based on the available dataset. Once the experimental values of the {\em semi-variogram} are computed, a possible approach is to define $\gamma$ as an exponential function obtained by fitting the experimental data, whose behavior is typically far from being regular. In theory, there is no reason why we should not define a different {\em semi-variogram} for each DOE point, so that we are indicating the local {\em semi-variogram} as
\[ \gamma_i = e^{-(r/a_i)^2} \]
using different values of $a_i$, one for each point of the DOE. The coefficients $a_i$ are the result of the fitting. $\gamma_i$ are required for assembling the self-correlation matrix $\Gamma$, whose element $\Gamma_{i,j}$ represents the correlation between the $i^{th}$ and the $j^{th}$ sample point. If the coefficients $a_i$ and $a_j$ are different, $\Gamma$ becomes unsymmetric. In practice, also due to the small amount of DOE points, a single {\em semi-variogram} is adopted and $\Gamma$ is symmetric. $\Gamma$ is then inverted, and the interpolation of the $N$ DOE values $F(x_i)$ at the generic point $x$ is obtained as
\[
f(x) = \sum_{i=1}^N w_i F(x_i) \]
where the weights $w_i$ are obtained by solving
\[ \Gamma W = \Gamma_0 \]
where $\Gamma$ and $\Gamma_0$ are respectievely
\[ \Gamma = \begin{bmatrix} \gamma_1(0) & \gamma_1(r_{2-1}) & ... & \gamma_1(r_{N-1}) & 1 \\ \gamma_2(r_{1-2}) & \gamma_2(0) & ... & \gamma_2(r_{N-2}) & 1 \\
... & ... & ... & ... & 1 \\ \gamma_N(r_{1-N}) & \gamma_N(r_{2-N}) & ... & \gamma_N(0) & 1 \\
1 & 1 & 1 & 1 & 0 \\ \end{bmatrix} \Gamma_0 = \begin{bmatrix} \gamma_1(r_{1-0}) \\ \gamma_2(r_{2-0}) \\
... \\ \gamma_N(r_{N-0}) \\ 1 \\ \end{bmatrix} \]
and $(r_{a-b})$ is $||{\bf x_a}-{\bf x_b}||$. These formulas are used in the Best Linear Unbiased Prediction (BLUP) of random variables \cite{Robinson}. Under the assumption that the irregularities in the Kriging computation can be mainly addressed to the condition number of $\Gamma$, we can try to act on the coefficients of the {\em semi-variogram} $a_i$ in order to maximize the condition number: a compass-search algorithm \cite{Beale1958} is here applied, adjusting the coefficients $a_i$ maximizing the condition number of $\Gamma$. A maximum variation of $\pm50\%$ is allowed: at each step, a golden section search \cite{Kiefer1953} is iteratively performed, so that the search limits are easily enforced.
In figure \ref{evoK} we can see the effect of the regularization of $\Gamma$ on the overall reconstruction of the objective function for a 2-dimensional closed form expression (Sasena function \footnote{$f(x) = 2+0.01*(x_2-x_1^2)^2+(1-x_1)^2+2*(2-x_2)^2+7*sin (0.5*x_1)*sin(0.7*x_1*x_2) \\ x_1 \in [0, 5], x_2 \in [0, 5]$}). The use of a 2-dimensional function is here justified by the necessity of data visualization. In the first frame on left, we have the Kriging interpolation, where a single value of $a_i$, obtained by the standard fitting procedure, is applied; in the central frame, we can see the effect of the maximization of the condition number, with a different value of $a_i$ for each sample point. In the extreme right frame, we have the effect of a minimization of the condition number. The effect of the maximization of the condition number is evident: the response surface is much more regular, in particular in the region between the sampled points or in the extreme regions, where there is not a sampled value. On the contrary, a minimization of the condition number is producing an interpolation formed by several Dirac-type regions, one for each sample point. It is evident how the criteria appear to be helpful in the regularization of the response surface.
\begin{figure}
\caption{Effect of the tuning of Kriging on the base of the condition number of the self-correlation matrix $\Gamma$.
On the left, the first guess, on the center the effect of a maximization of the condition number (that is, the
proposed approach), and on right the effect of a minimization of the condition number (that is, the opposite of
the proposed approach). The test has been performed using the Sasena function in $\Re^2$. The function is sampled
using 16 random points.
}
\label{evoK}
\end{figure}
\subsection{Spline regularization}
Also for Multi-Dimensional Spline (MDS), the weights are obtained through the solution of a linear system. Here the interpolation is obtained as a sum of N compact support functions $R(\rho)$, one for each sample point: also in this case can, the function can be different from each DOE point:
\begin{equation}\label{eq:pesi}
f(x) = \sum_{i=1}^N w_i R_i(\rho(x_i,x)) \end{equation}
where $\rho(x_i,x)$ is a measure of the distance between the i$^{th}$ sample point and the computational point $x$, and $R_i(\rho)$ is a compact support function, decreasing to zero at a certain distance from its center (the sample point). A simple expression for $R$, linear in the distance between the points, is
\begin{eqnarray*}
R & = & 1 - b \, ||x_i - x|| for \,\,\, b \, ||x_i-x|| \le 1 \\
R & = & 0 \,\,\,\,\,\, otherwise \end{eqnarray*}
The weights of the kernel functions are determined by solving a linear system enforcing explicitly the equality between equation \ref{eq:pesi} and the sampled value at every point of the DOE. To compare with the previous case, while for Kriging the objective function was indirectly included through the {\em semi-variogram}, here it appears explicitly at the right end side of the linear system.
The parameter $b$ represents a measure of the amplitude of the compact support function, and also this parameter can be adjusted for the improvement of the condition number of the matrix of the linear system to be solved, using a procedure equivalent to the one previously described. The effect of the tuning is reported in figure \ref{evoN}. Compared with Kriging, MDS tends to produce a smoother response surface probably because the weights have a direct link with the local value of the objective function, and there is a progressive passage between the different influence areas of the DOE points when we move across the DVS. As observed in the left frame of figure \ref{evoN}, the case where all the $b_i$s are all equal was already regular. The differences between the first case and the regularized one are not so large as in the Kriging example, but they can still be observed looking at the contour levels reported at the bottom of the plot: a more regular behavior is evident in the neighborhood of the minimum of the function. As a further check, a comparison with the contour lines in figure \ref{evoK} and figure \ref{evoN} indicates that the effect of the regularization of MDS is also going in the direction of a stricter similitude with Kriging.
\begin{figure}
\caption{Effect of the tuning of MDS on the base of the condition number of the variance of the
coefficients. On the left, is the first guess, and on right is the effect of tuning. The
test has been performed using the Sasena function in $\Re^2$. The function is sampled using
16 random points.
}
\label{evoN}
\end{figure}
\section{Optimization algorithm}
{\em Meta-models} (MM) and ML can be used as base elements for the definition of an optimization algorithm where the recourse to the mathematical model providing the value of the objective function is very limited. This is a situation absolutely valuable when the computational cost of a single value of the objective function is very high. Furthermore, since we are now dealing with a computationally inexpensive surrogate of the objective function, we can also avoid the recourse to a sophisticated optimization algorithm, proceeding by using a {\em brute force} approach, where the DVS is sampled extensively by using the MM. The algorithm can be depicted sketchily by the pseudocode reported in {\bf Algorithm \ref{codice}}.
\begin{algorithm} \caption{PSI-AI algorithm}\label{codice} \begin{algorithmic}
\STATE Perform initial sampling (DOE)
\FOR {$N_E < N_E^{max}$}
\STATE Apply ML
\STATE Search the DVS using MM
\STATE Update DOE
\STATE Center the DVS on the current best solution
\STATE Reduce the amplitude of DVS
\ENDFOR \end{algorithmic} \end{algorithm}
$N_E$ is the current number of objective function evaluations, and $N_E^{max}$ is the maximum effort we want to put into the search, measured as the maximum number of evaluations of the objective function we are willing to perform. The name {\tt PSI-AI} comes from the original formulation of the so-called Parameter Space Investigation (PSI) originally proposed in \cite{Statnikov1995} but without the ML improvement phase.
The initial sampling of the DVS is performed by using a uniform distribution of points: this is in the logic of the uniform probability that every point of the DVS could host the optimal value of the objective function unless some information on the objective function is gained. The DOE is obtained using a P$\tau$-net distribution, extensively reported in \cite{Statnikov1995}.
A relevant parameter is represented by the constriction factor ($< 1$), that is, the amount of reduction of the DVS amplitude when we are passing from one iteration to another. A small value means a strong reduction: in this case, the convergence toward the more promising area is fast, but we can lose the location of the global minimum due to premature focalization. On the contrary, a large value, close to 1, is a guarantee of the completeness of the exploration, but it can require a very large number of iterations (and then a large number of evaluations of the objective function) to get the convergence. Here we are performing a comparative study about the role of this parameter. In figure \ref{xred} the effects of a systematic variation of the constriction parameter $\alpha$ are reported. A quadratic function of 12 variables is here minimized\footnote{$ f(x) = \sum_{i=1}^{12} (x_i-0.5)^2 \,\,\,\,\, x_i \in [-10:10]$}: the DOE is composed of 192 points ($16 \times$ the number of design variables), and at each iteration, 8 further points are added during the ML phase, while the best 8 points selected on the base of the evaluation of a regular sampling of the DVS using the MM are also added at each iteration. 10 iterations are performed. In the picture, to focus on the convergence of the algorithm, the representation of DOE evaluation is not included. We can observe from figure \ref{xred} that a value of $\alpha=0.5$ causes a premature convergence, and the reduction of the objective function is quite large but also far from the best value. A value of $\alpha$ in between 0.8 and 0.9 appears to be the most appropriate choice, with a slight preference for 0.9, since it looks like a further improvement of the objective function would be possible if a larger number of iterations were allowed. Larger values appear to slow down too much the convergence of the algorithm, although they are a guarantee (asymptotically) of a more accurate exploration of the DVS.
\begin{figure}
\caption{Effects of different values of the constriction factor on the convergence of the {\tt PSI-AI} algorithm. On the
left panel, the percentage difference between the current best value of the objective function and the optimal
value, and on the right panel the numerical value of the first (of 12) design parameter: different graphs are
for different values of the constriction factor, with colors in accordance with the left side picture. On the
horizontal axis, in every graph, the current number of objective function evaluations.
}
\label{xred}
\end{figure}
\section{Application to realistic problems}
To check the qualities of the algorithm on a realistic application, whose resulting objective function is possibly not showing a single clear global minimum as in the adopted algebraic test case, the problem of the optimization of a monohull ship has been considered. The parent hull form (PHF) for the ship design optimization has been taken from \cite{Zotti2002}: it represents the bare hull geometry of a {\em Vaporetto}, the water-bus performing public transport in Venice. Lines and main dimensions are reported in figure \ref{Vaporetto}.
\begin{figure}
\caption{Section lines of the {\em Vaproetto} as from \cite{Zotti2002}.
}
\label{Vaporetto}
\end{figure}
\subsection{Pitch motion reduction}
As a first case study, the {\tt PSI-AI} algorithm has been compared with an efficient multi-agent optimization algorithm, to understand and evaluate the performance we can expect from {\tt PSI-AI}. The selected optimization algorithm is the hybrid version of the Imperialist Competitive Algorithm ({\tt ICA}), originally proposed in \cite{Atashpaz2007} and then here adopted in the improved version \cite{Peri2019} as {\tt hICA}. The {\tt hICA} includes also a local search algorithm, used in conjunction and in cooperation with the original multi-agent algorithm. {\tt hICA} has been proved to be more efficient than other multi-agent algorithms, like the NSGA-II implementation \cite{Deb2002} of a Genetic Algorithm ({\tt GA})\cite{Peri2019}.
To limit the computational effort, the energy associated with the pitch response of the ship in rough seas has been considered as objective function. This choice allows the application of a very fast (but reliable) simulation tool, the open-source {\tt PDStrip} seakeeping code \cite{Bertram2006}. The objective function is represented by the area below the RAO pitch curve, at the speed of 10 knots. Its evaluation has a computational cost lower than 9 seconds on a 3.30GHz Intel\textregistered Core\texttrademark i7-5775R: as a consequence, we can easily perform extensive tests in a reasonable time.
The parameterization of the hull has been produced by using the Free Form Deformation (FFD) approach, proposed and described in \cite{Sederberg1986}. A patch with $5 \times 4 \times 6$ subdivisions (respectively along the {\bf X}, {\bf Y} and {\bf Z} Cartesian axes) has been adopted, but only 12 control points are active: the first and the last planes along the {\bf X} direction are kept fixed, as well as three first slices in the {\bf Y} direction together the bottom plane and the first two top planes in the {\bf Z} direction. Only the $4 \times 3$ control points on the last lateral slice parallel to the {\bf XZ} plane can freely move along the lateral direction, their movement is limited to 25 centimeters (ship scale). The patch is including the PHF geometry up to the plane $z$ = 1.5 meters, in order not to change the bridge geometry: the fixed control points of the FFD enforce the continuity between the modified and unmodified parts of the hull. An example of possible deformation obtained with this configuration is reported in figure \ref{FFD}.
\begin{figure}
\caption{Example of shape deformation of the {\em Vaporetto} geometry. The hull is partly embedded into a single FFD patch,
leaving unchanged the top part of the ship. Here the deformation producing the optimal shape is reported. Black
dots are the control points of the FFD, while the hull shape is represented in yellow. Bow points toward the
positive direction so that we are here observing the hull from the stern.
}
\label{FFD}
\end{figure}
Among the different possible choices, Kriging has been adopted as a substitute for the mathematical model, while the MDS has been adopted for the ML phase together with Kriging.
The results, in terms of the objective function and design parameters, are reported in figure \ref{bestRAO}. Here is clear how the number of objective function evaluations required to reach convergence for {\tt PSI-AI} is about one-third with respect to {\tt hICA}. On the other hand, we have some differences in the optimal values of the design parameters, whose numerical values are reported in table \ref{tab:para}.
Firstly we analyze the evolution of the design parameter values thru the optimization procedure. We can clearly observe how the optimal values of the design parameters in many cases are similar, but not in all the cases. Since most of the parameters find their optimum value on the border of the admissible portion of the DVS, we can argue that there is not a stationary point in the constrained DVS, so the selected optimum is the one producing the best possible value of the objective function, but the local derivatives of the objective function are not null. This situation is clear for 9 out of 12 parameters. For the remaining 3 parameters (\#4, 8, and 12), the difference is mainly addressable to their small sensitivity of the objective function: in fact, the final value of the objective function for the two optimal points is absolutely comparable. The two algorithms have different attitudes in the exploration of the DVS. In {\tt PSI-AI} the search has not a preferential area, and the focusing is progressive, and the initial search is mainly grounded on the center of the DVS. On the contrary, the {\tt hICA} generally expands towards the extreme regions of the admissible DVS, eventually resizing the group of the agents in a second moment. As a consequence, if the sensitivity of a parameter is small, {\tt PSI-AI} will preferentially return a value not close to the borders of the admissible DVS, while {\tt hICA} is implicitly encouraging the values at the extremes.
\begin{table}[!htb] \caption{Optimal values of parameters obtained using the two different optimization algorithms, {\tt PSI-AI}
and {\tt hICA}. The two sets of parameters are represented side-by-side, ordered by the
number of the design parameter.
}\label{tab:para}\centering\small
\begin{tabular}{l|l|l||l|l|l||l|l|l} \# & {\tt hICA} & {\tt PSI-AI} & \# & {\tt hICA} & {\tt PSI-AI} & \# & {\tt hICA} & {\tt PSI-AI} \\ \hline
1 & 0.2490 & 0.2432 & 2 & 0.2484 & 0.2498 & 3 & 0.2495 & 0.2499 \\ \hline \cellcolor{orange}{4} & \cellcolor{orange}{-0.2454} & \cellcolor{orange}{0.2045} &
5 & 0.2441 & 0.2498 & 6 & 0.2499 & 0.2500 \\ \hline
7 & 0.2486 & 0.2496 & \cellcolor{orange}{8} & \cellcolor{orange}{-0.2494} & \cellcolor{orange}{0.1592} &
9 & 0.2490 & 0.2484 \\ \hline 10 & 0.2455 & 0.2485 & 11 & 0.2493 & 0.2412 & \cellcolor{LightCyan}{12} & \cellcolor{LightCyan}{ 0.1957} & \cellcolor{LightCyan}{ 0.2105} \\ \end{tabular} \end{table}
\begin{figure}
\caption{Comparison between the {\tt PSI-AI} approach and the {\tt hICA} algorithm. On the top, is the time history of
the 12 design variables. On the bottom, the progressive reduction of the objective function value thru the
iterations. A dot is plotted only if an improvement of the objective function occurs.
}
\label{bestRAO}
\end{figure}
This hypothesis is substantially confirmed by the results of the sensitivity analysis reported in figure \ref{sensi}. In this picture, a single parameter is changed at a time, revealing the influence of each parameter on the objective function. Here we can observe how there is a small group of design variables with a reduced sensitivity with respect to the objective function: all the previously listed parameters belong to this group. As a consequence, the small influence of these parameters is the reason why they are much more prone to follow the tendency subtly suggested by the algorithm.
\begin{figure}
\caption{Sensitivity analysis of the pitch peak as a function of the design parameters. In the sub-picture, the detail
of the effect of smaller variations of the design parameters: here the horizontal scale is about one-tenth of
the larger picture.
}
\label{sensi}
\end{figure}
In figure \ref{MachineLearning} the progressive reduction of the discrepancies between the two adopted MMs during the course of the optimization process is reported. The full number of calls to the objective function is about 800: this means that only a quarter of them are devoted to the ML algorithm. After 80 evaluations, the difference is of the order of 1\%: this can be considered a relevant achievement, demonstrating the efficiency of the ML algorithm.
\begin{figure}
\caption{Effect of the Machine Learning algorithm on the relative precision of the two different {\em meta-models}.
On the horizontal axis, the number of calls to the objective function required by the ML algorithm during
the optimization problem, on the vertical axis the maximum value across the full DVS of the percentage
difference in the prediction of the two different {\em meta-models}. A reduction of the discrepancies is
a guarantee of the reliability of the interpolations.
}
\label{MachineLearning}
\end{figure}
\subsection{Total resistance}
A second test case has been produced to further check the capabilities of the {\tt PSI-AI} algorithm. The objective function is now represented by the effective power in calm water of the {\em Vaporetto} at the speed of 4.5 m/s, which is a little lower than the maximum speed fixed by the rules (20 km/h). Due to the motivations of the present work, the shape of the sea bottom and the side walls of the channels have been not considered, although they are peculiar elements for this kind of sea vehicle traveling in the Venice area. Hydrodynamic computations are performed using a single-layer potential flow solver, of the class of the Boundary Element Methods (BEM) \cite{Gadd1976}. These kinds of solvers are correctly modeling the wave pattern generated by the hull, but the viscous effects are not included in the formulation, so they are introduced {\em a posteriori} in a simplified way. The resulting estimate of the effective power has been proved to be reliable, at least from the engineering point of view. The computational effort of a fully-3D BEM is larger than a strip-theory method, also because computations need to be repeated iteratively to obtain the actual values of the sinkage and trim of the ship. The CPU time for a single value of the objective function is now about 40 seconds on the same computational platform. For this reason, the comparison with the {\tt hICA} algorithm has been not repeated.
The same parameterization as from the previous test case has been adopted, including also the constraints on the design variables.
At the end of the optimization problem, the effective power required by the {\em Vaporetto} is passing from 22.44 to 15.10 kW, with a reduction of 32.72\%. To have a second check of the improvements, the RANSE solver {\tt interFoam}, from the suite {\tt OpenFOAM}\cite{OpenFOAM} has been also applied: results are reported in table \ref{tab:ave}, where we can see the substantial equivalence of the estimated percentual reductions (34.1 vs. 32.7\%). In this case, the Reynolds-averaged Navier-Stokes equations, including explicitly the viscous terms neglected in the Laplace equation of the BEM method, have been solved. Also {\tt interFoam} is taking into account the real sinkage and trim of the hull.
\begin{table}[!htb] \caption{Optimal values of the wave resistance, frictional resistance, and total resistance for the original and
optimized hulls, computed by using the RANSE solver {\tt interFoam} from the suite
{\tt OpenFOAM}\textcopyright \cite{OpenFOAM}. The simulations are performed with the hull able to take the
dynamic sinkage and trim. Since the effective power differs by the total resistance by a constant, the
percentage differences of total resistance are the same as the ones of the effective power.
}\label{tab:ave}\centering\small
\begin{tabular}{l|c|c|c} Ship & R$_W$ [N] & R$_F$ [N] & R$_T$ [N] \\ \hline Original & 2411.0 & 430.9 & 2841.9 \\ \hline Optimal & 1638.9 & 234.7 & 1873.6 \\ \hline\hline \cellcolor{orange}{$\Delta$\%} & \cellcolor{orange}{-32.0} & \cellcolor{orange}{-45.5} & \cellcolor{orange}{-34.1} \\ \end{tabular} \end{table}
Figure \ref{geometry} is reporting the changes in the shape of the PHF. The beam is increased, and a part of the volume is shifted fore: since the displacement is fixed, the draught of the optimal hull is smaller in comparison with the PHF one. The topside of the hull is not changing: as a consequence, the beam increase is causing a kind of lateral bulb in the central part of the ship. If required, the hull lines of the out-of-the-water part of the hull, not directly influencing the performance, can be reassessed to have a more regular shape. The red color on the hull geometry is evidencing the wetted part of the hull at the optimizing speed.
\begin{figure}
\caption{Perspective view of the PHF (on left) and the optimal (on right) hull shapes. The red color is
indicating the wetted part of the hull (as computed by using the {\tt interFoam} RANSE solver.
}
\label{geometry}
\end{figure}
In figure \ref{correlation} we have reported a different way to stress the increasing similitude between the prediction of the two MMs as the ML iterations are going on. The two MMs have been computed on a large number of points laying into the DVS: on each frame, we have on the horizontal axis the estimate provided by Kriging, while the estimate by MDS is reported on the vertical axis. If the values estimated by two MMs were the same, all the points would be aligned along the $x=y$ line in the plot. If the points are not well aligned, there is still a difference between the prediction of the two MMs. We can see how, proceeding from left to right, top to bottom, the points are thickening on the line of full correlation: this is a sign of a progressive increase in the coherence between the two MMs. Since all the points are aligned, the similitude of the two MMs is certified on the full DVS.
\begin{figure}
\caption{Effect of the application of the ML algorithm on the overall precision
of the adopted {\em meta-models}. In each framework, the values
of the objective function as predicted by the two different {\em meta-models}
at a specific iteration are reported on the two axes. Iteration number (frame number)
is running from left to right, top to bottom.
}
\label{correlation}
\end{figure}
In figure \ref{etaBEM} and \ref{viste} we have a snapshot of the wave pattern produced by the PHF and the optimal hull. Figure \ref{etaBEM} reports the differences between the wave profile on the hull (and on the centerplane) as simulated by the BEM solver. In the picture, the hull is located in between $x=-0.5$ and $x=0.5$, with the bow at $x=-0.5$. The optimal hull shows a more regular wave pattern along the hull, and the hollow observed in the rear part of the PHF has disappeared. In the wake, the wave profile is reduced after optimization, which is typically beneficial.
\begin{figure}
\caption{Comparison between wave pattern generated by the PHF and optimal hulls,
computed with the BEM solver \cite{Gadd1976}, the one applied for the evaluation
of the objective function. The ship hull is placed in between $x=-0.5$ and $x=0.5$,
with bow at $x=-0.5$.
}
\label{etaBEM}
\end{figure}
The same conditions have been also simulated by using a RANSE mathematical model, with richer physical content. Results are substantially confirmed, as previously mentioned. Here we can compare the wave pattern predictions: in figure \ref{viste} we can see a top view of the wave pattern and a comparison of a longitudinal wave cut, for both the PHF and optimal configuration. The position of the longitudinal cut is reported as a black line in the upper frame of the same picture. The reduction in the wave elevation along the hull and in the region close to the stern is clearly evident. We cannot completely exclude that part of the great success of the optimization activities could be possibly connected with a slightly inaccurate reproduction of the linesplans of the real PHF shape (both in the drawings reported in \cite{Zotti2002} and/or in the digitalized ones), so that the performances of the PHF are lower than reality.
\begin{figure}
\caption{Comparison between wave pattern generated by the PHF and optimal hulls, computed by using the RANSE solver
{\tt interFoam} from the suite {\tt OpenFOAM}\textcopyright \cite{OpenFOAM}. On top, is the top view of the
wave patterns, where the optimal hull is presented in the lower part. On the bottom, is the comparison of a
longitudinal wave cut of the PHF (black) and optimal (red) hulls. The trace of the cut is reported as a black
line in the top view.
}
\label{viste}
\end{figure}
\section{Conclusions}
The paper is evidencing the connections between AI and optimization, demonstrating how some techniques classically adopted in AI can be easily and fruitfully applied as base elements of an optimization algorithm. There are some limitations, mainly connected with the space dimensionality of the problem: in fact, to consider a large number of design parameters may imply the requirement of a very large DOE, causing at the same time a huge computational cost for the training of the MM. This situation may become even harder if the objective function is multimodal so that the number of points required for the synchronization of the prediction of the MMs during the ML phase becomes also larger than in the present cases. In these conditions, the use of the classical optimization approach could still represent a more viable solution.
More experiences are needed to better establish the limits of the approach. Also, the use of more than a couple of MM, to further improve the tuning phase, could be investigated.
\section*{Conflict of interest}
The authors declare that they have no conflict of interest.
\end{document} |
\begin{document}
\title{On reducing the order of arm-passes bandit streaming algorithms under memory bottleneck}
\begin{abstract}
In this work we explore multi-arm bandit streaming model, especially in cases where the model faces resource bottleneck. We build over existing algorithms conditioned by limited arm memory at any instance of time. Specifically, we improve the amount of streaming passes it takes for a bandit algorithm to incur a $O(\sqrt{T\log(T)})$ regret by a logarithmic factor, and also provide 2-pass algorithms with some initial conditions to incur a similar order of regret. \end{abstract}
\section{Introduction}
In this paper we improve upon previous work done on regret minimization in Multi-Armed Bandit (MAB) \cite{Berry et al.} in constrained memory setting. In the multi-armed bandit setting, in general, the arms can be thought as handles in slot machines, with each handle giving a reward each time we pull it; the rewards for a particular arm follow an instance dependent but fixed distribution. The objective is to maximize the rewards accumulated in the end, or to put it in another way learn a policy to pull arms in a particular(regret minimizing or reward maximizing) manner. \\\\ With multi-arm bandits having abundant applications in healthcare, finance, dynamic-pricing models, recommender systems, etc. it makes sense to think about budgetary and resource constraints, especially in settings where the number of levers are large. In \cite{Shen et al.}, the authors proposed a bandit algorithm for making online portfolio choices via exploiting correlations among multiple arms, now given that high frequency trading firms operate on large volumes of data \textit{frequently} they can surely benefit from near optimal performance in resource constraint setting. Even in social good problems like drug-testing \cite{Armitage et al.}, and other practically relevant tasks like crowd-sourcing \cite{Tran et al.} that are modelled using multi armed bandits it makes economic sense to explore options that operate near optimally without storing entire statistics of the system \\\\ To that end, in \cite{Chaudhari et al.} the authors have established an instance independent $O(\sqrt{Tlog(T)})$ bound with $O(log(T))$ streaming passes and recently \cite{Maiti et al.} have established a lower bound of $O(T^{2/3})$ for any single pass algorithm. Our objective thus was to explore the domain between $log(T)$ and one pass, and try to minimise the regret accumulated while also reducing the number of passes to the extent that we can.\\\\ Our contributions: \begin{itemize}
\item We propose a variation of Algorithm-1 in \cite{Chaudhari et al.} which accrues $O(\sqrt{Tlog(T)})$ regret with $log(log(T))$ passes instead of $log(T)$ passes.
\item Using the analysis devised for above result we then propose a 2-pass-algorithm, with some instance dependent initial conditions, with $O(\sqrt{Tlog(T)})$ regret.
\item Simulations to corroborate the results. \end{itemize} We also include an instance dependent two-pass-hybrid algorithm with some prior information about the system which incurs a $log(T)$ regret. In the following sections we first start by describing the streaming model, RAM model, that we use, describe the algorithms and the key intuitions behind proving the result, and then doing same with 2-pass algorithms. \section{Related work} Right from the seminal work of \cite{Robbins et al.} the predominant body of literature in stochastic multi-armed bandit is dedicated to the regret minimisation task on finite and infinite bandit instances. Later, a number of salient algorithms like UCB1 (\cite{Auer et al.}), Thompson Sampling (\cite{Chapelle et al.}; \cite{Agarwal et al.}), have been shown to achieve the order optimal cumulative regret on the finite instances. The study on multi arm bandit algorithms under constraint resources however is still limited despite its myriad practical applications today. \cite{Liau et al.} where they provide an instance dependant optimal regret with O(1) storage of arms and \cite{Chaudhari et al.} providing instance independent $\sqrt{Tlog(T)}$ regret with $log(T)$ passes. These results can however be further improved as shown in the following sections. \\\\ Since early 60s and 50s finite memory hypothesis testing has been looked at by researchers (\cite{Robbins et al.}; \cite{Cover76 et al.}). In multi armed bandit setting \cite{Cover68 et al.} first presented a finite memory algorithm for two-armed Bernoulli instance, achieving an average reward which converges to optimality, with high probability. The approach consisted of a collection of interleaved test and trial blocks, where each test block is divided into several sub-blocks and the switching among these sub-blocks is governed by a finite state ma- chine. However, he considered only two-armed Bernoulli instances, and the approach guarantees only an asymptotic convergence of the empirical average reward. Hence, this setup is not very interesting, as our objective is to present a finite-time analysis of regret for general bandit instances. \\ To that end, we now present our bandit streaming setup and corresponding algorithms and analysis. \\\\ \section{Preliminaries} \subsection{RAM-Model} It should be noted that given any bandit instance $B = (A, D)$, as we are not considering any special structure in $A,D$, putting a restriction on an algorithm to use a bounded number of words of space, either restricts the horizon of pulls, or restricts the algorithm to store statistics of only bounded number of arms simultaneously.
In this paper, we consider the latter and assume $M$ to be that number. We adopt the word RAM model (\cite{Aho et al.}, \cite{Cormen et al.}), that considers a word as the unit of space. This model facilitates to consider that each of the in- put values and variables can be stored in $O(1)$ word space. For finite bandit instances $(|A|<\infty)$, we consider a word to be consisted of $O(log(T))$ bits. Therefore, our algorithm needs space-complexity of $O(Mlog(T)+log(|A|))$ bits. We call this set of arm indices whose statistics are stored as arm memory and its cardinality as arm memory size. Hence, an algorithm with arm memory size M can store the statistics of at most M arms. Also, it should be noted that an algorithm is allowed to pull an arm only if it is stored in the memory. Hence, before pulling a new arm (which is not currently in the arm memory), the algorithm should replace an arm in its arm memory with this new arm. It is interesting to note that the algorithms that work with M = 1, can only keep the stat of the arm it is currently pulling. There- fore, switching to a new arm costs such an algorithm to lose all the experience gained by sampling the previous arm. However, for a finite bandit instance, as the algorithms are allowed to remember all the arm indices, such an algorithm can store the gained experience by storing a bounded number of arm indices for possible further special treatment.
\subsection{Simple Regret} It is one of the popular problems in \textit{pure exploration} bandit setting which focuses on the design of strategies making the best possible use of available numerical resources (e.g., as cpu time) in order to optimize the performance of some decision-making task. If $b_{t} \in A$ is the arm recommended by the algorithm after $t-th$ pull, then the simple regret at $t$ is defined as, \begin{equation}
E[r^{*}_{t}]= \mu^{*}- E[\mu_{b_{t}}] \end{equation} \\\\
In particular \cite{Bubeck et al.} studies the relationship between simple regret and cumulative regret, with results showing that upper bounds on cumulative regret should also lead to upper bounds on simple regret. \\\\ Figure 1. from \cite{Bubeck et al.} shows the framework of a pure-exploration problem. Where the forecaster can be interpreted as the algorithm in consideration for the bandits, e.g. UCB1 or Thompson sampling. The forecaster then, at the end of its duration, recommends a particular arm based on some criteria, which can for example be a. Most played arm, b. Empirically best arm, or c. Arm with highest upper confidence and so forth. In this study we're concerned with most played arm(MPA) as our recommendation strategy. \begin{theorem}[Distribution-free upper bound on Simple-Regret of UCB-MPA by \cite{Bubeck et al.}] \textit{Given a K- sized set of arms $K$ as input, if UCB-MPA runs for a horizon of $T$ pulls such that $T \geq K(K + 2)$, then for some constant $C > 0$, it achieves the expected simple regret } $E[r^{*}_{T}] \leq C\sqrt{\dfrac{K\log(T)}{T}}$. \end{theorem} \begin{figure}
\caption{The pure exploration problem for multi-armed bandits (with a finite number of arms)}
\end{figure}
\textbf{From hereon \textit{ln}: logarithm with base $e$ and $\log$: logarithm with base 2.} \section{Multi pass algorithm} Algorithm 1, which we call UCB-LAM (UCB-limited arm memory), is presented in this section. We establish the upper-bound on cumulative regret and also the improvement in the order of passes over UCB-M proposed in \cite{Chaudhari et al.}. The allocation strategy in UCB-LAM is UCB1 \cite{Auer et al.} for now, although it can in principle be MOSS, Thompson Sampling, etc. After exploring this multi pass algorithm we then move on to studying constant pass algorithms.
\subsection{Intuition behind UCB-LAM} We know that main driving idea behind any multi-arm bandit algorithm focused on minimising regret is to sample rewards from the best arm with time. Although when faced with resource bottlenecks it's not even guaranteed that the best arm will be present in the arm memory, let alone consistently pulling it. Thus, intuitively, any algorithm that's minimizing regret under limited arm memory ought to take care that, a. The probability of the best arm being present in arm memory increases with time, b. Given (a) the best arm is sampled often. To that end we work towards UCB-LAM and also provide probabilistic analysis solidifying our intuitions, (a) and (b).
\subsection{Description of UCB-LAM} In algorithm 1, UCB-LAM, we're given $A:$ a set of $K$ arms, $K > M \geq 2:$ arm memory size. Now when $M \geq K$ we can simply run UCB1 \cite{Auer et al.} on the arms $A$ since the arm memory can accommodate all the arms. But when $M < K$ we sample arms in phases. There are a total of $x_{0}$ phases and each phase $w$ is divided into $h_{0} = \lceil \dfrac{K-1}{M-1} \rceil$ sub-phases to accommodate appropriate $M-$sized subset of arms $S^{w, j}$ that're allowed in the memory for phase $w$, and sub-phase $j$. Once the Allocation Strategy is applied on $S^{w,j}$ the sub-phase then recommends $\hat{a} \in S^{w,j}$ to the next sub-phase $S^{w, j+1} \text{or} S^{w+1, 1}$. \\\\ Each of the $h_{0}$ sub-phases present in phase $w$ are allotted a time-duration of $b_{w}$, with $b_{1}=M(M+2)$ \{change? Why?\} and with the time allotted to a sub-phase increasing as the phase increases with $b_{w}= b_{w-1}^{2}$. The increasing time allocation with phase indicates the algorithms growing confidence in the best arm being present in arm memory and it being sampled often.
\subsection{Preliminary arguments for regret analysis} We note that with $K$ arms, $M$ memory size, and horizon $T$, the number if sub-phases is $h_{0}= \lceil \frac{K-1}{M-1} \rceil$ and the total number of phases or arm-passes is upper bounded by $x_{0}= 1+ \lceil \log(\log_{M(M+2)}( \dfrac{T}{h_{0}} )) \rceil$ (Lemma A.1 in Appendix A). \\\\ As we'll further see in our regret bifurcation and analysis further ahead, we need to upper-bound the mean of the recommended arms between two consecutive sub-phases. Let $a^{y,j}_{*} \in S^{y,j}$ be the arm recommended by sub-phase $j$ to $j+1$, then we need to bound $r^{y, j}= \mu_{*}^{y,j-1}- \mu_{*}^{y, j}$. Since we know that, $\mu_{*}^{y, j-1} \leq \max_{a \in S^{y,j}} \mu_{a}$, we can write \[r^{y, j}= \mu_{*}^{y,j-1}- \mu_{*}^{y, j} = \max_{a \in S^{y,j}} \mu_{a}- \mu_{*}^{y, j} - \epsilon \]
We thus get: \begin{corollary} \textit{Using Theorem 3.1 in phase} $y$, \textit{at the end of each sub-phase} $j$, \textit{the approximate simple regret with respect to} $\mu_{a_{*}^{y,j}}$ \textit{is upper bounded as} $E[r^{y,j}] \leq C\sqrt{\dfrac{K\log(b_{y})}{b_{y}}}$.
\end{corollary} We also need to be cognizant of the fact that the arm recommended by a sub-phase might not be the optimal arm, because in the worst case scenario the allocation strategy will only be sampling sub-optimal arms and thus incurring a huge regret. It is thus imperative to bound the \textit{approximate} simple regret and to take care that optimal arm is included the arm-memory, and once included, it is also the arm that's recommended to the next sub-phase with high probability. We'll come across these arguments naturally while doing the regret analysis. \begin{lemma} Consider events, \begin{enumerate}
\item A: Best arm(universal) is in the current ($w,j$) instance.
\item B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).
\end{enumerate} Let $s=(w,j)$ be the current instance with phase $w$ and sub-phase $j$, and let $r$ be the arm recommended to the next instance $(w, j+1) or (w+1, j)$ and let $\mu_{*}^{w,j}$ be the mean of that arm. Then if \begin{itemize}
\item $T_{1}= P(A_{s}=1)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1)$
\item $T_{2}= P(A_{s}=0)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \}$ \end{itemize} We have, \\ $\dfrac{T_{1}+T_{2}}{b_{w}} \leq 2C\dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}}(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$ \\
\end{lemma} The proof is presented in the Appendix A. Next we birfurcate the regret appropriately and use the above results to find the upper bound the cumulative regret ($R_{T}^{*}$) \\\\ \textbf{Bifurcation of $R^{*}_{T}$}. For any given phase $w$, and a sub-phase $j$, let $\mu_{*}^{w,j}=$ mean of the most played arm $a$ for $a\in S^{w,j}$, and $R_{w,j}$ be the incurred regret. Then,
\begin{align*} R_{w,j} &= b_{w}\mu^{*} - \sum_{t=1}^{b_{w}} E[\mu_{a_{t}}] \\ &= b_{w}E[\mu^{*}- \mu_{*}^{w,j}] + \sum_{t=1}^{b_{w}}(E[\mu_{*}^{w,j}] -E[\mu_{a_{t}}]) \end{align*}
Now let $R^{(1)}_{w,j}= b_{w}(\mu^{*}- \mu_{*}^{w,j})$, and $R^{(2)}_{w,j} = \sum_{t=1}^{b_{w}} (E[\mu_{*}^{w,j}] - E[\mu_{a_{t}}])$, we write,
\begin{equation} R^{*}_{T}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}}R_{w,j}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} (R^{(1)}_{w,j} + R^{(2)}_{w,j}) \end{equation}
Having bifurcated cumulative regret $R^{*}_{T}$ in terms of $R^{1}, R^{2}$ we can see that minimizing $R^{1}$ would essentially mean that with high probability, as we go further in phases, the best arm in memory is actually the optimal arm. As can be seen from \textbf{Lemma A.2} for phase $w$, the P(best arm is recommended)= $P(J_{b_{w}}= i^{*}) \geq 1- \dfrac{K-1}{\alpha -1}(b_{w}/K -1)^{2(\alpha-1)}$, for $\alpha=2$, increases with phase, which is also the key argument that we use while proving \textbf{Lemma 4.2} below. \\\\
\begin{lemma}
For $2 \leq M < K$, and for $T \geq KM^{2}(M+2)$, let $R^{(1)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(1)}_{w,j}$, then, \[ R^{1} \leq C_{0}+ C_{2} \times \log(\log_{b_{1}}(\dfrac{T}{h_{0}})) \] where $C_{2}$ is a constant depending on $K,M$. \end{lemma} For the proof, refer Appendix A. For calculating $\sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(2)}_{w,j}$, can be upper-bounded using the problem independent upper-bound on the cumulative regret of UCB1 \cite{Auer et al.}, which we restate below. \\\\
And $R^{2}$ can be interpreted as a local cumulative regret, or the regret accrued with respect to the best arm present in a particular memory instance. In the worst case scenario we expect the arms to be present in increasing order of means, and thus the local regret then would thus increase linearly with instances and will be within a certain factor of the local cumulative of the instance containing best arm. This what \textbf{Lemma 4.4} puts mathematically. \begin{lemma} (Distribution-Free Upper Bound on Cumula- tive Regret of UCB1 \cite{Auer et al.}). Given a set of K- arms as the input, for any horizon T, the cumulative regret incurred by UCB1 $R^{*}_{T} \leq 12\sqrt{TK\log(T)}+6K$. Further, if $T \geq K/2$, then $R^{*}_{T} \leq 18\sqrt{TK\log(T)}$. \end{lemma} Next using \textit{Lemma 4.3} we upper bound $\sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(2)}_{w,j}$, proof is given in Appendix A. \\\\ \begin{lemma} For $2 \leq M < K$, and for $T \geq KM^{2}(M+2)$, let, $R^{(2)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(2)}_{w,j}$, then, \[ R^{(2)} \leq C_{0}+ C_{4}\sqrt{(\log_{b_{1}}(\dfrac{T}{h_{0}}))\cdot(\dfrac{T}{h_{0}})} \] Where $C_{0} , C_{4}$ are constants depending on K and M. \end{lemma}
\begin{theorem}[Main result] For $2 \leq M < K$, and for $T \geq KM^{2}(M+2)$, let $R^{(1)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(1)}_{w,j}$ and $R^{(2)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(2)}_{w,j}$, so for cumulative regret $R^{*}_{T}= R^{(1)}+R^{(2)}$ we have, \[ R^{*}_{T} \leq 2C_{0}+C_{2}\times (\log(\log_{b_{1}}(\dfrac{T}{h_{0}})))+C_{4}\sqrt{(\log_{b_{1}}(\dfrac{T}{h_{0}}))\cdot \dfrac{T}{h_{0}}} \] where $C_{0}, C_{1}, C_{4}$ are functions of $K,M$. \end{theorem} \begin{proof}
One can easily see how combining bounds from \textbf{Lemma 4.2} and \textbf{Lemma 4.4} will give us the required upper-bound on cumulative regret of \textbf{Algorithm 1}. \end{proof}
\begin{algorithm}[tb] \caption{UCB-LAM(limited arm memory)} \label{alg:algorithm_ucblam} \textbf{Input}: $A:$ the set of $K$ arms indexed by [K], $M(\geq 2)$: Arm memory size\\
\begin{algorithmic}[1]
\IF {$M \geq K$} \STATE Run UCB1 on $A$ until horizon \ELSE \STATE $b_{1}= M(M+2).$ \{Initial horizon per sub-phase\} \STATE $\hat{a}= 1$. \{ Initial arm recommendation\} \STATE $w= 1$. \{Counts the number of phases \} \STATE $h_{0}= \lceil \dfrac{K-1}{M-1} \rceil$. \{ The number of sub-phases in a phase\} \STATE No random shuffling for now \WHILE{ the horizon is not finished} \STATE $l= 0$ \FOR {$j= 1, \cdots, h_{0};$ if the horizon is not finished} \STATE $S^{w,j}= \{l+1, \cdots, \min\{l+1+(M-1), K\} \} $ \STATE $l=$ The highest arm index in $S^{w,j}$. \IF { $\hat{a} \notin S^{w,j}$ } \STATE $S^{w,j} = \{\hat{a}\} \cup S^{w,j}\setminus \{l\}$. \STATE $l= l-1$ \ENDIF \STATE \{ALLOCATION STRATEGY\} \STATE Run UCB1 on $S_{w,j}$ for horizon of $b_{w}$ pulls or the remaining horizon; whichever is smaller. \STATE \{RECOMMENDATION STRATEGY\} \STATE $\hat{a}=$ The most played arm in $S^{w,j}$ \ENDFOR \STATE $w= w+1$. \{Increment phase count\} \STATE $b_{w}= (b_{w-1})^{2}$. \{ Increment horizon per sub-phase\} \ENDWHILE \ENDIF
\end{algorithmic} \end{algorithm}
\section{Constant pass algorithms} Now that we've introduced \textbf{UCB-LAM} which gives us $O(\sqrt{T\log(T)})$ cumulative regret in $O(\log(\log(T)))$ passes, we further explore constant pass algorithms in similar spirit. Decreasing the number of passes is practically very desirable since it frees up the memory reservoir required to store the dormant arms not being considered by the allocation strategy at any instance\{come up with better motivation\}, and in extension having \textit{constant} number of passes will aid in determining when a section of memory space will be freed (after $c$ passes) and thereby enabling us to better plan the use of the memory as opposed to the case where the number of passes is dependant on the time horizon thereby restricting the scope of any prior planning for the memory in consideration.
\subsection{2-pass UCB-LAM} 2-pass UCB-LAM is an extension of UCB-LAM when restricted to two passes. The with the only difference being for us to know the time horizon beforehand -- which is obvious. Since when we say that a certain algorithm takes 2-passes, we ought to know \textit{when the first pass ends} and \textit{when the second pass begins}. \\\\
\begin{algorithm}[ht] \caption{2-pass UCB-LAM} \label{alg:algorithm_2passucblam} \textbf{Input}: $A:$ the set of $K$ arms indexed by [K], $M(\geq 2)$: Arm memory size, $T:$ Total time horizon, large enough such that $T \geq K*(1+\dfrac{4\alpha ln(T)}{M \Delta_{i}^{2}})$ \\
\begin{algorithmic}[1]
\IF {$M \geq K$} \STATE Run UCB1 on $A$ until horizon \ELSE \STATE $\hat{a}= 1$. \{ Initial arm recommendation\} \STATE $w= 1$. \{Counts the number of phases \} \STATE $h_{0}= \lceil \dfrac{K-1}{M-1} \rceil$. \{ The number of sub-phases in a phase\} \STATE $b_{1}= \dfrac{\sqrt{1+4\dfrac{T}{h_{0}}}-1}{2}$ \STATE $b_{2}= \dfrac{2\dfrac{T}{h_{0}}+1-\sqrt{1+\dfrac{4T}{h_{0}}}}{2}= b_{1}^{2}$
\WHILE{ the horizon is not finished} \STATE $l= 0$ \FOR {$j= 1, \cdots, h_{0};$ if the horizon is not finished} \STATE $S^{w,j}= \{l+1, \cdots, \min\{l+1+(M-1), K\} \} $ \STATE $l=$ The highest arm index in $S^{w,j}$.
\IF { $\hat{a} \notin S^{w,j}$ } \STATE $S^{w,j} = \{\hat{a}\} \cup S^{w,j}\setminus \{l\}$. \STATE $l= l-1$ \ENDIF \STATE \{ALLOCATION STRATEGY\} \STATE Run UCB1 on $S_{w,j}$ for horizon of $b_{w}$ pulls or the remaining horizon; whichever is smaller. \STATE \{RECOMMENDATION STRATEGY\} \STATE $\hat{a}=$ The most played arm in $S^{w,j}$ \ENDFOR \STATE $w= w+1$. \{Increment phase count\} \ENDWHILE \ENDIF \end{algorithmic} \end{algorithm}
From \textbf{Lemma B.1} we know for sure that the total number of passes required by \textbf{Algorithm-2} is two. Having said that we now put across the regret analysis and argument for \textbf{Algorithm-2}. \subsubsection{Bifurcation of regret} We know that the regret obtained from \textbf{Algorithm 2 $R^{*}_{T}$} can be bifurcated as, following steps as we did the regret bifurcation of \textbf{Algorithm-1}-
For any given phase $w$, and a sub-phase $j$, let $\mu_{*}^{w,j}=$ mean of the most played arm $a$ for $a\in S^{w,j}$, and $R_{w,j}$ be the incurred regret, Then,
\begin{align*} R_{w,j} &= b_{w}\mu^{*} - \sum_{t=1}^{b_{w}} E[\mu_{a_{t}}] \\ &= b_{w}E[\mu^{*}- \mu_{*}^{w,j}] + \sum_{t=1}^{b_{w}}(E[\mu_{*}^{w,j}] -E[\mu_{a_{t}}]). \end{align*}
Now let $R^{(1)}_{w,j}= b_{w}(\mu^{*}- \mu_{*}^{w,j})$, and \\ $R^{(2)}_{w,j} = \sum_{t=1}^{b_{w}} (E[\mu_{*}^{w,j}] - E[\mu_{a_{t}}])$, we write,
\[ R^{*}_{T}= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}R_{w,j}= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}} (R^{(1)}_{w,j} + R^{(2)}_{w,j}) \] We now deal with $R_{1}$ and $R_{2}$ separately as we did earlier.
\begin{lemma} For $2 \leq M < K$, and total time-horizon $T$, \\ we get that, for Algorithm-2, \[ R_{2} \leq C_{2}+C_{0}\sqrt{T+0.25h_{0}}+C_{1}\sqrt{T\log(T/h_{0})} \] Where $C_{0}, C_{1}, C_{2}$ are constants depending on $M,K$. \end{lemma}
As we can see the order of $R_{2}-regret$ accrued by \textbf{Algorithm-2} almost same as the order of $R_{2}-regret$ accrued by \textbf{Algorithm-1}. Essentially $R_{2}$ here can be interpreted as the summation of all the \textit{local} regrets, where local means being limited to one's memory instance without being aware of all the arms. We now move onto $R_{1}$.
\begin{lemma} For $2 \leq M < K$, and total time-horizon $T$, \\ we get that, for Algorithm-2,\\ $R_{1}= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}(R^{(1)}_{w,j}) \leq O(\sqrt{T})$ \end{lemma} \{refer Appendix for the proof\}. $R^{1}$ here can be physically interpreted as the penalty we accrue as the result of recommending sub-optimal arm instead of the best arm. While there can be several reasons for this, like a. Best arm is not present in the current memory instance, b. Best arm is present but isn't recommended for some reason, and all of these factors are analysed for in the proofs. \{comment on how $R_{1}$ here differs from $R_{1}$ for algorithm-1\}
\begin{theorem} Given a set of K arms A with $K \leq 3$, an arm memory of size M, and the total time horizon T Algorithm-2 will incur a cumulative regret $R^{*}_{T}= O(A_{1}\sqrt{T}+A_{2}\sqrt{T\log(T)})$, where $A_{1}, A_{2}$ are constants depending on $K,M$. \end{theorem} \begin{proof} For phase $w$, and sub-phase $j$, we know that, $R^{(1)}_{w,j}= b_{w}(\mu^{*}- \mu_{*}^{w,j})$, and $R^{(2)}_{w,j} = \sum_{t=1}^{b_{w}} (E[\mu_{*}^{w,j}] - E[\mu_{a_{t}}])$, and we write,
\[ R^{*}_{T}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}}R^{*}_{w,j}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} (R^{(1)}_{w,j} + R^{(2)}_{w,j}) \] And from \textbf{Lemma 5.1} we know that \[ R_{2} \leq C_{2}+C_{0}\sqrt{T+0.25h_{0}}+C_{1}\sqrt{T\log(T/h_{0})} \] ,and from \textbf{Lemma 5.2} we know that $R_{1} \leq O(\sqrt{T})$, thus we get, \begin{equation}
R^{*}_{T}=R_{1}+R_{2} \leq O(A_{1}\sqrt{T}+A_{2}\sqrt{T\log(T)}) \end{equation} where $A_{1}, A_{2}$ are constants depending on $K,M$. \end{proof} \subsection{Constant pass algorithm from a kind-of explore-exploit perspective} As we've noted in the sections above, to accrue lesser regret in resource bottleneck settings intuitively what any algorithm tries to do is maximize the probability of the best arm being in the arm memory and being recommended often. In this subsection we explore a relatively simple algorithm that performs somewhat better than \textbf{2-pass UCB-LAM} theoretically speaking, building up on exactly the points we mentioned. \\ We'll first start with a \textbf{pseudo algorithm} and then based on our analysis build up \textbf{Algorithm-4}.
\begin{algorithm}\caption{(\textbf{pseudo}) 2-pass-hybrid}\label{alg:algorithm_2passhybrid} \textbf{Input}: $A:$ the set of $K$ arms indexed by [K], $M(\geq 2)$: Arm memory size, $T:$ Total time horizon \\ \begin{algorithmic}[1] \STATE \textbf{1st Pass:} \STATE \text{Uniformly play all the arms (details later)} \STATE \text{Recommend the arm with highest} $\overline{\mu}$ \STATE \textbf{2nd Pass:} \STATE \text{Run UCB} \cite{Auer et al.} \text{for the current sub-phase for time} $b_{2}$ \STATE \text{Recommend the most played arm to the next sub-phase} \STATE \text{Keep doing it for the remaining time} \end{algorithmic} \end{algorithm}
\subsubsection{Description of \textbf{(pseudo) 2-pass-hybrid}} For the set of $K$ arms $A$, arm memory $M\geq 2$, and total time horizon $T$, a. In the first pass we sample all arms for an equal time-horizon, the main idea is to get an estimate of empirically best arm, b. Recommend the empirically best arm to the second pass, and in the second pass run the similar allocation strategy that we've used in the previous algorithms, i.e. bifurcate the memory appropriately and recommend the most played arm to the next memory instance. The underlying idea being that once the best arm is being captured, it'll be further recommended with high probability. And since the the probability of making a mistake on the best arm is very low the expected regret will have reasonable bound order. \\ We find the necessary conditions for our \textbf{(pseudo)} 2-pass-hybrid to perform optimally and build \textbf{Algorithm-4} out of it.
\\\\
\begin{lemma} For given set $A$ of $K$ arms, $M$ arm memory, $\Delta_{\min}= \min_{\substack{i \in A}}\{\mu^{*}-\mu^{i}\}$, total time horizon $T$, then \begin{itemize}
\item The optimal total time duration($h_{0}b_{0}$) spent in 1st pass by \textbf{Algorithm-3} is,
\[ h_{0}b_{1}= \dfrac{h_{0}}{\Delta_{\min}^{2}}\log(1+\dfrac{\Delta_{\min}^{2}}{K}T^{2}) \] where $h_{0}= \lceil \dfrac{K-1}{M-1} \rceil$.
\item And for the above optimal time duration spent in the first pass, the cumulative regret accrued ($R^{*}_{T}$)
\[ R_{T}^{*}= \dfrac{K}{\Delta_{\min}^{2}}\log(f(T))\times(1- \dfrac{1}{f(T)}) + \dfrac{T}{f(T)}
\]
where $f(T)= 1+\dfrac{\Delta_{\min}^{2}}{K}T^{2}$ \end{itemize}
\end{lemma} \begin{proof}
Let $X_{i}^{k}$ be the reward drawn from arm $i$ at time $k$, therefore $\overline{\mu_{i}}= \dfrac{\sum_{k=1}^{b_{0}} X^{k}_{i}}{b_{1}}$. We thus want a bound on $P_{i \neq i^{*}}(\overline{\mu_{i}} \geq \overline{\mu_{i^{*}}})$ or that sub-optimal arm is recommended. Let $E_{i}$ be the event such that
$\{\overline{\mu_{i}} \geq \overline{\mu_{i^{*}}}\}$
\begin{align*}
P(E_{i})&= P(\overline{\mu_{i^{*}}}-\overline{\mu_{i}}- (\Delta_{i}) \leq -\Delta_{i} ) \\
&= P(\dfrac{\sum{X^{j}_{i^{*}}}- \sum{X_{i}^{j}}}{b_{1}} - \Delta_{i} \leq -\Delta_{i}) \\
&= P(\sum{X^{j}_{i^{*}}}- \sum{X_{i}^{j}} - b_{0}\Delta_{i} \leq -b_{0}\Delta_{i}) \end{align*}
Let $S_{2n}$= $\sum{X^{j}_{i^{*}}}- \sum{X_{i}^{j}}$, $t= -b_{1}\Delta_{i}$, so the above inequality can be rewritten as $P(S_{2n}- E[S_{2n}] \leq -t$, and thus using Hoeffding's inequality we get:
\begin{align*}
&P(S_{2n}- E[S_{2n}] \leq -t) \leq exp(\dfrac{-2t^{2}}{2b_{1}}) \\
&P(S_{2n}- E[S_{2n}] \leq -t) \leq exp(\dfrac{-2(b_{1}^{2})(\Delta_{i}^{2})}{2b_{1}})\\
\implies &P(S_{2n}- E[S_{2n}] \leq -t) \leq exp(-\Delta_{i}^{2}b_{1}) \\
\implies &P(E_{i}) \leq \exp(-\Delta_{i}^{2}b_{1}) \end{align*}
From above we now know that $P(E_{i}) \leq exp(-\Delta_{i}^{2}b_{1})$. Now let $\overline{BA}$ be the event such that the recommended arm is \textbf{not} the best arm or P($\overline{BA}$)=1-P($BA$)= $1- P(\cap_{i \in [K]}\{\overline{E_{i}}\})$, where $E_{i}$ is the event that $ith$ arm is the recommended arm. Thus we get,
\begin{equation}
1- P(\cap_{i \in [K]}\{\overline{E_{i}}\})= 1- \Pi_{i=1}^{K}(1-P(E_{i})) \end{equation}
Since we know that $P(E_{i}) \leq exp(-\Delta_{i}^{2}b_{1})$
\begin{align*}
P(\overline{BA}) &\leq 1- \Pi_{i=1}^{K}( 1- exp(-b_{1}\Delta_{i})) \\
&\leq 1- \Pi_{i=1}^{K}(1- exp(-b_{1}\Delta_{\min})) \\
&\leq 1-(1- K\times exp(-b_{1}\Delta_{\min})) \\
&\leq K \times exp(-b_{1}\Delta_{\min})
\end{align*}
Thus, we get that the P(The best arm isn't recommended)= $P(\overline{BA}) \leq K \times exp(-b_{1}\Delta_{\min})$ or that the P(best arm is recommended)= $P(BA) \geq 1- K \times exp(-b_{1}\Delta_{\min})$ \\ \textbf{Calculating and optimizing the regret:}
We'll calculate the total regret by bifurcating it in two passes. We know that for the first pass the regret accrues linearly for the worst case, or: \begin{equation}
R_{1} \leq Kb_{1} \end{equation}
For the second pass however, it's a bit involved. Let $BA$ be the event that best arm is recommended by first pass at the beginning of the second pass, then we have:
\begin{equation}
R_{2} \leq (1-P(BA))\times (h_{0}b_{2})+ P(BA)\times O(h_{0}\sqrt{\dfrac{\log(b_{2})}{b_{2}}}) \end{equation} Which can be easily derived from using Lemma 4.4 in \cite{Chaudhari et al.} and arguing that in the worst case regret will be accrued linearly. For the sake of simplicity we ignore the second term in equation (3), and using the expression for $P(\overline{BA}) \leq K \times exp(-b_{1}\Delta_{\min})$ which we derived in the earlier section of the analysis, we convert our argument into a simple optimization problem where we want to find:
\begin{equation}
b_{1, \text{opt}}, b_{2, \text{opt}}= \argmin_{b_{1}, b_{2}} (K\times b_{1}+e^{-b_{1}\Delta_{\min}^{2}}\times(h_{0}b_{1})) \end{equation} Such that, $T= Kb_{1}+h_{0}b_{2}$.
Let $R= K\cdot b_{1}+e^{-b_{1}\Delta_{\min}^{2}}\cdot(h_{0}b_{2})$, substituting $b_{2}$ in terms of $b_{1}$ we get,
\begin{equation}
R= Kb_{1}+ e^{-b_{1}\Delta_{\min}^{2}}(T- Kb_{1}) \end{equation}
Differentiating wrt $b_{1}$ we get:
\begin{align}
\dfrac{dR}{db_{1}}= 0 \implies K + e^{-b_{1}\Delta_{\min}^{2}}(-K)+ \\
\hspace{10pt} \hspace{10pt} \hspace{10pt} \hspace{10pt} e^{-b_{1}\Delta_{\min}^{2}}(T-Kb_{1})(-b_{1}\Delta^{2}_{\min})= 0 \\
\implies K= e^{-b_{1}\Delta_{\min}^{2}}(K+b_{1}\Delta_{\min}^{2}(T-Kb_{1})) \\
\implies e^{b_{1}\Delta_{i}^{2}}= \dfrac{K}{K+b_{1}\Delta_{\min}^{2}(T-Kb_{1})} \\
\implies e^{b_{1}\Delta_{i}^{2}} \geq \dfrac{K}{K+b_{1}T\Delta_{\min}^{2}} \\
\implies e^{b_{1}\Delta_{i}^{2}} \geq \dfrac{K}{K+\Delta_{\min}^{2}T^{2}} \hspace{10pt} \hspace{10pt} \hspace{10pt} \{ b_{1} < T\}\\
\implies -b_{1}\Delta_{i}^{2} \geq \log(\dfrac{K}{K+\Delta^{2}_{\min}T^{2}}) \\
\implies b_{1} \leq \dfrac{1}{\Delta_{\min}^{2}}\log(1+\dfrac{\Delta_{\min}^{2}}{K}T^{2}) \end{align}
Thus we get \textbf{(a).} \[ b_{1}= \dfrac{1}{\Delta_{\min}^{2}}\log(1+\dfrac{\Delta_{\min}^{2}}{K}T^{2}) \] as the optimal time spent in the first pass. \\\\
Substituting the expression for $b_{1}= \dfrac{1}{\Delta_{\min}^{2}}\log(1+\dfrac{\Delta_{\min}^{2}}{K}T^{2})$ in equation (6) and letting $f(T)= 1+\dfrac{\Delta_{\min}^{2}}{K}T^{2}$ we get, \textbf{(b).} \[ R^{*}_{T}= \dfrac{K}{\Delta_{\min}^{2}}\log(f(T))\times(1- \dfrac{1}{f(T)}) + \dfrac{T}{f(T)} \]
\end{proof}
Using \textbf{Lemma 5.4} we now propose a 2-pass \textbf{Algorithm-4}.
\begin{algorithm}[ht] \caption{2-pass-hybrid} \label{alg:algorithm_real2passhybrid} \textbf{Input}: $A:$ the set of $K$ arms indexed by [K], $M(\geq 2)$: Arm memory size, $T:$ Total time horizon, $\Delta_{\min}= \min{i}\{\mu^{*}-\mu_{i}\}$ \\
\begin{algorithmic}[1]
\IF {$M \geq K$} \STATE Run UCB1 on $A$ until horizon \ELSE \STATE $\hat{a}= 1$. \{ Initial arm recommendation\} \STATE $w= 1$. \{Counts the number of phases \} \STATE $h_{0}= \lceil \dfrac{K-1}{M-1} \rceil$. \{ The number of sub-phases in a phase\} \STATE $b_{1}= \dfrac{1}{\Delta_{\min}^{2}}\log(1+\dfrac{\Delta_{\min}^{2}}{K}T^{2})$ \STATE $b_{2}= T- b_{1}$
\WHILE{ the horizon is not finished} \STATE $l= 0$ \FOR {$j= 1, \cdots, h_{0};$ if the horizon is not finished} \STATE $S^{w,j}= \{l+1, \cdots, min\{l+1+(M-1), K\} \} $ \STATE $l=$ The highest arm index in $S^{w,j}$. \IF { $\hat{a} \notin S^{w,j}$ } \STATE $S^{w,j} = \{\hat{a}\} \cup S^{w,j}\setminus \{l\}$. \STATE $l= l-1$ \ENDIF \IF {w==1} \STATE \{ALLOCATION STRATEGY\} \STATE Play each arm in $S^{w,j}$ $b_{1}$ times. \STATE \{RECOMMENDATION STRATEGY\} \STATE $\hat{a}=$ empirically best arm. Or $\hat{a}= max_{i}\mu_{i}$ \ELSE \STATE \{ALLOCATION STRATEGY\} \STATE Run UCB1 on $S_{w,j}$ for horizon of $b_{w}$ pulls or the remaining horizon; whichever is smaller. \STATE \{RECOMMENDATION STRATEGY\} \STATE $\hat{a}=$ The most played arm in $S^{w,j}$ \ENDIF \ENDFOR \STATE $w= w+1$. \{Increment phase count\} \ENDWHILE \ENDIF \end{algorithmic} \end{algorithm}
\section{Simulations} We performed simulations to compare the performances of UCB-M from \cite{Chaudhari et al.}, UCB-LAM above, and standard UCB1. We set parameters to be K=30, M=4, and the bandit arms had means varying as $\mu_{i}= 0.99-0.1*i; i\in [0,K]$ with each arm $i$ following a Bernoulli distribution with mean $\mu_{i}$. The order of arrival was randomised and simulation shows the average over 10 simulations. \begin{figure}
\caption{Aggregate regret vs. Time horizon compared between UCB-M, UCB-LAM, UCB1}
\end{figure}
We thus observe that the order of regret is comparable across the algorithms, with UCB-LAM having $O(log(log(T)))$ number of passes over $O(log(T))$ number of passes of UCB-M. \section{Conclusion} We've thus explored the gap we set out to explore, which is the regret bound behaviour between one-pass \cite{Maiti et al.} and $log(T)$ passes. We've shown that in instance independent setting with no prior information about the system, $O(\sqrt{Tlog(T)})$ regret can be achieved with simply $log(log(T))$ passes instead of $log(T)$ passes. We've now als know that under large enough time horizons it's possible to achieve $O(\sqrt{Tlog(T)})$ regret with simply two passes. It makes sense intuitively because with the number of passes limited to two we want to learn the best arm behaviour as accurately as possible. However the behaviour of regret in smaller time horizons with constant number of passes needs to be explored further and we defer that to future studies.
\section{Appendix}
\appendix
\section{Proofs for Section 4} \begin{lemma}
For a given $K$- sized set of arms $A$, and an arm memory size $M < K$, the number of phases \textbf{Algorithm 1} executes is upper bounded by $x_{0}= 1+ \lceil \log(\log_{M(M+2)}( \dfrac{T}{h_{0}} )) \rceil$.
\end{lemma} \begin{proof}
Firstly we'd like to make it clear that there exists\{ $K$,$M$ time horizon: $T$ \} such that the time horizon is complete while the algorithm is in between a particular phase. We have $b_{w}= (b_{w-1})^{2}$ or the length of each sub-phase in phase $w$ varies as $\{ b_{1}^{2^{1-1}}, b_{1}^{2^{2-1}}, b_{1}^{2^{3-1}}, \cdots, b_{1}^{2^{x_{0}-1}} \}$, where $x_{0}$ is the total number of phases or total number of passes. Now,
\begin{align} T= \sum_{w= 1}^{x_{0}}\sum_{j= 1}^{h_{0}} b_{w,j} \\ = \sum_{w= 1}^{x_{0}} h_{0}b_{w} \cdots \cdots \{b_{w}= b_{w,1}= \cdots= b_{w, h_{0}} \} \\ \Rightarrow T \geq h_{0}b_{x_{0}} \\ \Rightarrow T \geq h_{0}(b_{1}^{(2^{x_{0}-1})}) \\ \Rightarrow \log_{b_{1}}(\dfrac{T}{h_{0}}) \geq 2^{x_{0}-1}\\ \Rightarrow \log\log_{b_{1}}(\dfrac{T}{h_{0}}) \geq x_{0} -1 \end{align}
Since $b_{1}= M(M+2)$ \begin{equation} \Rightarrow x_{0} \leq \lceil \log\log_{M(M+2)}(\dfrac{T}{h_{0}}) \rceil +1 \end{equation} Hence proved. \end{proof}
\begin{lemma} Consider events,
\begin{enumerate}
\item A: Best arm(universal) is in the current ($w,j$) instance.
\item B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).
\end{enumerate}
And let $J_{b_{w}}$ be the arm recommended by the recommendation strategy for \textbf{UCB-LAM}. Then for a particular instance $s= (w,j)$, $M:$ arm-memory and $b_{w}:$ time-horizon, $P(B_{s}=1|A_{s}=1)= P(J_{b_{w}}= i^{*}) \geq 1-\dfrac{M-1}{(b_{w}/M -1)^{2}}$
\end{lemma}
\begin{proof}
Using a side-result from \textbf{\textit{Lemma 1} in \cite{Bubeck et al.}} we get that whenever the most played arm $J_{b_{w}}$ is different from the optimal arm $i^{*}$ then at least one of the suboptimal arms $i$ is such that $T_{i}(b_{w}) \geq a_{i}b_{w}$, where $a_{1}, \cdots, a_{K}$ are real numbers such that $a_{1}+a_{2}+\cdots+a_{K}= 1$ and $a_{i} \geq 0, \forall i$. \\ And that $P\{T_{i}(b_{w}) \geq a_{i}b_{w}\} \leq \dfrac{1}{\alpha -1}(a_{i}b_{w} -1)^{2(1-\alpha)}$, where $T_{i}:$ number of pulls for arm $i$ and $\alpha$ refers to $UCB(\alpha)$ algorithm used during allocation strategy.\\\\
Now, we know that:
\begin{align*}
P(J_{b_{w}}= i^{*})+ (\sum_{i \in A \setminus \{ i^{*}\}} P(J_{b_{w}}=i)) =1 \\
\implies P(J_{b_{w}}= i^{*})= 1- \sum_{i \in A \setminus \{ i^{*}\}} P(J_{b_{w}}=i) \end{align*}
When $i$ is the most played arm or $(J_{b_{w}}=i)$ we know that $T_{i} \geq b_{w}/K$ and from \textbf{Lemma A.5} which is derived from from \cite{Bubeck et al.} we know that
\begin{align*} P_{i \neq i^{*}}(J_{b_{w}}= i) \leq \dfrac{1}{\alpha-1}(b_{w}/K-1)^{2(1-\alpha)} \end{align*}
\begin{multline*}
\implies 1- \sum_{i \in A \setminus \{ i^{*}\}} P(J_{b_{w}}=i) \geq \\ (1-\sum_{i \in A \setminus \{ i^{*}\}}\dfrac{1}{\alpha-1}(b_{w}/K-1)^{2(\alpha-1)}) \end{multline*}
\begin{align*}
&\implies P(J_{b_{w}}= i^{*}) \geq 1- \dfrac{K-1}{\alpha -1}(b_{w}/K -1)^{2(\alpha-1)} \end{align*}
Substituting arm memory: $M$, $\alpha= 2$ for our case, we get $P(J_{b_{w}}= i^{*}) \geq 1- \dfrac{M-1}{(b_{w}/M -1)^{2}}$\\\\ \end{proof}
\begin{corollary} Consider events,
\begin{enumerate}
\item A: Best arm(universal) is in the current ($w,j$) instance.
\item B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).
\end{enumerate} For an instance $s= (w,j)$ the probability of the optimal arm not being present in the current arm memory, $P(A_{s}= 0) \leq \dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}} $ \end{corollary} \begin{proof} From \textbf{\textit{Lemma A.2}} above we know that (probability of recommending best-arm) \\
$P(B_{s}=1|A_{s}=1) \geq 1- \dfrac{M-1}{(b_{w}/M -1)^{2}}$.\\\\
Let $t_{w}= b_{w}/M$, and $m=(w-1, k^{'})$ be the memory instance in the previous phase $w-1$ containing the universal best arm $a^{*}$. \\\\
We know that $P(A_{s}= 1)= \Pi_{m=(w-1,k^{'})}^{(w,j-1)}P(B_{m}=1|A_{m}=1))$, because the probability that the current instance will have the best arm depends on the fact that previous instance had the best arm which it passed on -- all the way to the instance $(w-1, k)$ which actually had the best arm. Thus we get,
\begin{multline*}
\Pi_{m=(w-1,k^{'})}^{(w,j-1)}P(B_{m}=1|A_{m}=1))\geq\\\Pi_{m=(w-1,k^{'})}^{(w,j-1)} (1- \dfrac{M-1}{(t_{m}-1)^{2}}) \end{multline*}
We're considering the worst case scenario here when the best arm lies in the previous phase $w-1$. As going from $m= (w-1, k^{'})$ to $(w,j-1)$ has a phase change, $w-1$ to $w$, we bifurcate the expression in two terms, a. considering phase $w-1: k_{1}= h_{0}+1-k^{'}$ and considering phase $w: k_{2}= j-1$.
\begin{multline*}
\implies \Pi_{m=(w-1,k^{'})}^{(w,j-1)}P(B_{m}=1|A_{m}=1))\geq \\ (1- \dfrac{M-1}{(t_{w-1} -1)^{2}})^{k_{1}}\times(1-\dfrac{M-1}{(t_{w}-1)^{2}})^{k_{2}} \end{multline*}
Where: $k_{1}+k_{2} \leq h_{0}$. Thus, \begin{multline*}
1-\Pi_{m= (w-1, k^{'})}^{ (w, j-1)}P ( B_{m}= 1| A_{m}=1)) \leq \\ 1-(1- \dfrac{M-1}{(t_{w-1} -1)^{2}})^{k_{1}}\times(1-\dfrac{M-1}{(t_{w} -1)^{2}})^{k_{2}} \end{multline*}
\begin{multline*}
\implies P(A_{s}= 0) \leq \\ 1-(1-\dfrac{M-1}{(t_{w-1} -1)^{2}})^{k_{1}}\times(1-\dfrac{M-1}{(t_{w} -1)^{2}})^{k_{2}} \end{multline*}
\begin{align*}
\implies P(A_{s}= 0) \leq 1-(1- \dfrac{M-1}{(t_{w-1} -1)^{2}})^{h_{0}} \end{align*}
The above expression is valid even in the case where the best arm lies in phase $w$ in instance $(w, k^{'})$, instead of $w-1$, with $j> k^{'}$. Primarily because $b_{w-1} < b_{w}$.\\
Using binomial expansion and $t_{w}= b_{w}/M$ thus we get,
\[ P(A_{s}= 0) \leq \dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}} \] \end{proof}
The significance of above lemmas is that they show, theoretically, that as we move ahead in time the probability that the best arm will be contained in the memory increases drastically which is what we guess intuitively. \begin{lemma} Consider events, \begin{enumerate}
\item A: Best arm(universal) is in the current ($w,j$) instance.
\item B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).
\end{enumerate} Let $s=(w,j)$ be the current instance with phase $w$ and sub-phase $j$, and let $r$ be the arm recommended to the next instance $(w, j+1) or (w+1, j)$ and let $\mu_{*}^{w,j}$ be the mean of that arm. Then $E[\mu^{*}-\mu_{*}^{w,j}]$ can be expressed in terms of $T_{1}, T_{2}$ or $b_{w}E[\mu^{*}-\mu_{*}^{w,j}]= T_{1}+T_{2}$, where: \begin{itemize}
\item $T_{1}= P(A_{s}=1)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1)$
\item $T_{2}= P(A_{s}=0)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \}$ \end{itemize} \end{lemma}
\begin{proof} Consider given events: \begin{enumerate}
\item A: Best arm(universal) is in the current ($w,j$) instance.
\item B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).
\end{enumerate} We know that: \begin{itemize}
\item $P(B=1 | A= 0)= 0$
\item $P(B=0| A= 0)= 1$
\item $P(B=1| A= 1)= x (\text{unknown})$
\item $P(B=0| A= 1)= 1-x$ \end{itemize} Now we know that $R^{1}_{w,j}$= $b_{w}E[\mu^{*}- \mu_{*}^{w,j}]$. \\ Let $\textbf{s}:$ current memory instance ($w,j$). We know that
$E[E[X]]= E[X]$, thus we write, \\\\ \begin{align*}
R^{1}_{w,j}&= P(A_{s}=1)\cdot \{ b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=1|A_{s}=1)\}\\
&+ P(A_{s}=1)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1) \} \\
&+ P(A_{s}=0)\cdot\{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=1|A_{s}=0) \} \\
&+ P(A_{s}=0)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=0) \} \end{align*}
Using $P(B|A)$ values above, we get,\\ \begin{align*}
R^{1}_{w,j}&= P(A_{s}=1)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1) \}\\
&+ P(A_{s}=0)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \} \end{align*}
Now from above, let:
\begin{align*}
T_{1}= P(A_{s}=1)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1)\\
T_{2}= P(A_{s}=0)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \} \end{align*} Hence, $R^{1}_{w,j}= b_{w}E[\mu^{*}-\mu_{*}^{w,j}]= T_{1}+T_{2}$ \end{proof}
\textbf{Lemma 4.1} \textit{Consider events, } \begin{enumerate}
\item \textit{A: Best arm(universal) is in the current ($w,j$) instance.}
\item \textit{B: Best arm(universal) is recommended to next instance (($w, j+1$) or ($w+1, 1$)).}
\end{enumerate} \textit{Let $s=(w,j)$ be the current instance with phase $w$ and sub-phase $j$, and let $r$ be the arm recommended to the next instance $(w, j+1) or (w+1, j)$ and let $\mu_{*}^{w,j}$ be the mean of that arm. Then if } \begin{itemize}
\item $T_{1}= P(A_{s}=1)\cdot \{b_{w}(E[\mu^{*}-\mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1)$
\item $T_{2}= P(A_{s}=0)\cdot \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \}$ \end{itemize} \textit{We have, \\} $\dfrac{T_{1}+T_{2}}{b_{w}} \leq 2C\dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}}(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$ \\
\begin{proof}
We prove it in two steps, wherein first step we establish an upper-bound on $T_{1}$ and in the second step we establish an upper-bound on $T_{2}$. \\
\textbf{Step 1.}
We know that $P(A_{s}=1) \leq 1$, and from \textbf{Lemma A.2}, $P(B_{s}=1|A_{s}=1)= P(J_{n}= i^{*}) \geq 1-\dfrac{M-1}{(b_{w}/M -1)^{2}} \implies P(B_{s}=0|A_{s}=1)= 1-P(J_{n}= i^{*}) \leq \dfrac{M-1}{(b_{w}/M -1)^{2}}$ \\
Thus using the probabilistic bounds from above and \textbf{Theorem 3.1} we get that,
\begin{align*}
T_{1} \leq 1\times \dfrac{M-1}{(b_{w}/M -1)^{2}} \times b_{w}C\sqrt{\dfrac{M\log(b_{w})}{b_{w}}} \end{align*}
\textbf{Step 2.} Establishing bounds for $T_{2}$ isn't so straightforward and will thus be slightly more involved, since the best arm isn't present in the current instance $s=(w,j)$. Let $k$ be the minimum sub-phase in phase $w$ that has the best-arm $a^{*}$ or let $k= \min\{i\in[h_{0}]; a^{*}\in S^{w,i} \}$, we then analyse cases where a. $j \geq k$ and b. $j < k$ separately. let $\mu_{*}^{w,j}$ be the mean of that arm recommended by instance $(w,j)$ or $\mu_{*}^{w,j}$ \\ \textbf{Step 2.a} (for $j \geq k$) \\ We can write $E[\mu^{*}-\mu_{*}^{w,j}]= E[\mu^{*}-\mu_{*}^{w,k}]+\sum_{i=k}^{j-1}E[\mu_{*}^{w,i}-\mu_{*}^{w,i+1}]$ \\
From \textbf{Corollary 4.0.1} we know that approximate simple regret $E[r^{y}]= E[\mu_{*}^{w,i}-\mu_{*}^{w,i+1}]$ can be bounded by $C\sqrt{\dfrac{M\log(b_{w})}{b_{w}}}$. And using \textbf{Theorem 3.1} we can bound $E[\mu^{*}-\mu_{*}^{w,k}]$. We thus get,
\begin{align*} E[\mu^{*}-\mu_{*}^{w,j}] \leq Ch_{0}\sqrt{\dfrac{M\log(b_{w})}{b_{w}}} \end{align*}
\textbf{Step 2.b}(for step $j< k$)\\
Let $l= \min\{i\in[h_{0}]; a^{*}\in S^{w-1,i} \}$, or the sub-phase in the previous phase containing best arm $a^{*}$. Thus we can then write, \begin{align*}
E[\mu^{*}-\mu_{*}^{w,j}] &\leq E[\mu^{*}-\mu_{*}^{w-1,l}] \\
&+ \sum_{i=l}^{h_{0}-1}E[\mu^{w-1,i}_{*}-\mu_{*}^{w-1,i+1}] \\
&+ E[\mu_{*}^{w-1,h_{0}}-\mu_{*}^{w,1}] \\
&+ \sum_{i=1}^{j-1}E[\mu_{*}^{w,i}-\mu_{*}^{w,i+1}] \end{align*}
We'll now bound each of the above four terms, a. we know that $a^{*} \in S^{w-1,l}$ thus by \textbf{Theorem 3.1} we get that, $E[\mu^{*}- \mu_{*}^{w-1,l}] \leq C\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$, b. using \textbf{Corollary 4.0.1} we can bound $\sum_{i=l}^{h_{0}-1}E[\mu^{w-1,i}_{*}-\mu_{*}^{w-1,i+1}] \leq C(h_{0}-l)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$ as it's nothing but the summation of consecutive recommended arms, c. for the last two terms similar to point (b) above, only now the time horizon for a sub-phase is $b_{w}$, we can again invoke \textbf{Corollary 4.0.1} and get the following,
\begin{align*}
E[\mu^{*}-\mu_{*}^{w,j}] &\leq C(h_{0}-l+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}\\
&+Cj\sqrt{\dfrac{M\log(b_{w})}{b_{w}}} \end{align*}
Since, $b_{w-1}<b_{w}$ \\ $\implies \sqrt{\dfrac{M\log(b_{w})}{b_{w}}}<\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$. Thus we get that,
\begin{align*}
E[\mu^{*}-\mu_{*}^{w,j}] \leq C(h_{0}+j-l+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}} \end{align*} Because there has to be atleast one instance containing best-arm $a^{*}$ in $h_{0}$ consecutive instances, $j \leq l$ and thus we get,
\begin{align*}
E[\mu^{*}-\mu_{*}^{w,j}] \leq C(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}} \end{align*}
Using results from \textbf{Step 2.a} and \textbf{Step 2.b} we get that $E[\mu^{*}-\mu_{*}^{w,j}] \leq C(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$. Finally using \textbf{Corollary A.2.1} we get that,
\begin{align*}
T_{2} \leq \dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}}b_{w}C(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}} \end{align*}
\textbf{Step 3} Combining results from \textbf{Step 1}, \textbf{Step 2}, and using the fact that $b_{w-1}<b_{w}$ along with \\
$\sqrt{\dfrac{M\log(b_{w})}{b_{w}}} < \sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}}$ \\\\
we finally get that,
\begin{align*}
\dfrac{T_{1}+T_{2}}{b_{w}} \leq 2C\dfrac{(M-1)\cdot h_{0}}{(b_{w-1}/M -1)^{2}}(h_{0}+1)\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}} \end{align*} \end{proof}
\begin{lemma} If a sequence $S$ is such that $S_{w}=(b)^{m^{w}/2}m^{w/2}$, then if $\sum(S)=\sum_{w=1}^{\log_{m}\log_{b}(T)} (b)^{m^{w}/2}m^{w/2}$, we have, \begin{align*} \sum(S) \leq \sqrt{\dfrac{m}{m-1}(\log_{b}(T))\cdot(T+ \dfrac{T^{1/m-1}}{b^{m/m-1}})} \end{align*} \end{lemma} \begin{proof}
We use \textit{\textbf{Cauchy-Schwartz Inequality}} which states that $\sum_{i=1}^{k}a_{i}b_{i} \leq \sqrt{\sum_{i=1}^{k}a^{2}_{i}}\cdot \sqrt{\sum_{i=1}^{k}b^{2}_{i}} $, where $a_{i}, b_{i} \in C, \forall i \in [1,k]$. \\\\
Let $K= \log_{m}\log_{b}(T)$, $A = \sqrt{\sum_{w=1}^{K} b^{m^{w}}}$, $B= \sqrt{\sum_{w=1}^{K}m^{w}}$, applying \textit{\textbf{Cauchy-Schwartz Inequality}} on $\sum(S)$ we get that,
\begin{equation} \sum(S) \leq A\cdot B \end{equation}
\begin{align*} B &= \sqrt{\sum_{w=1}^{K}m^{w}}= \sqrt{\dfrac{m(m^{K}-1)}{m-1}} \\ &= \sqrt{\dfrac{m}{m-1}(\log_{b}(T) - 1) } \end{align*} \begin{equation} \implies B \leq (\sqrt{\dfrac{m}{m-1}(\log_{b}(T))}) \end{equation}
Now, to bound $A$ consider, \begin{align*} A^{2}&= \sum_{w=1}^{K} b^{m^{w}} \\ A^{2}&= \sum_{w=1}^{\log_{m}\log_{b}(T)- 1}b^{m^{w}} + b^{m^{\log_{m}\log_{b}(T)}} \\ A^{2}&= \sum_{w=1}^{\log_{m}\log_{b}(T)- 1}b^{m^{w}} + T \\ \intertext{We know that, $\sum_{w=1}^{K-1} m^{w}= (m^{K}-m)/(m-1)$ }. \intertext{Since $\sum a_{i} \leq \prod a_{i}, \forall a_{i} \geq 2$} \intertext{Therefore, $\sum_{w=1}^{K-1} b^{m^{w}} \leq b^{\sum_{w=1}^{K-1}m^{w}}$}\\ \Rightarrow &\sum_{w=1}^{K-1} b^{m^{w}} \leq b^{(m^{K} - m)/(m - 1)} \\ \Rightarrow &\sum_{w=1}^{K-1} b^{m^{w}} \leq \dfrac{T^{1/m-1}}{b^{m/m-1}} \\ \Rightarrow &A^{2} \leq T+ \dfrac{T^{1/m-1}}{b^{m/m-1}} \end{align*}
\begin{equation} \Rightarrow A \leq (\sqrt{T+ \dfrac{T^{1/m-1}}{b^{m/m-1}}}) \end{equation}
Thus from (7),(8),(9) we get that: \[ \sum(S) \leq \sqrt{\dfrac{m}{m-1}(\log_{b}(T))*(T+ \dfrac{T^{1/m-1}}{b^{m/m-1}})}. \] \end{proof}
\textbf{Lemma 4.2} For $2 \leq M < K$, and for $T \geq KM^{2}(M+2)$, let $R^{(1)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(1)}_{w,j}$, then, \[ R^{1} \leq C_{0}+ C_{2} \times \log(\log_{b_{1}}(\dfrac{T}{h_{0}})) \] where $C_{2}$ is a constant depending on $K,M$. \begin{proof} We know that, \begin{align*} R^{(1)}&= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}}R^{(1)}_{w,j} \\ &= \sum_{j=1}^{h_{0}}R^{(1)}_{1,j}+ \sum_{w=2}^{x_{0}}\sum_{j=1}^{h_{0}}R^{(1)}_{w,j} \\ &\leq h_{0}b_{1} + \sum_{w=2}^{x_{0}}\sum_{j=1}^{h_{0}}R^{(1)}_{w,j} \\ &\leq h_{0}b_{1}+ \sum_{w=2}^{x_{0}}\sum_{j=1}^{h_{0}} b_{w}E[\mu^{*}- \mu^{w,j}_{*}] \end{align*}
From \textbf{Lemma A.3} we know that $E[\mu^{*}- \mu^{w,j}_{*}]$ can be expressed in terms of $T_{1}, T_{2}$ and thus invoking \textbf{Lemma 4.1} we get,
\begin{align*} R^{1} & \leq h_{0}b_{1}\\ &+\sum_{w=2}^{x_{0}}h_{0}b_{w}(2C\dfrac{(M-1)\cdot h_{0}(h_{0}+1)}{(b_{w-1}/M -1)^{2}}\sqrt{\dfrac{M\log(b_{w-1})}{b_{w-1}}})\\ \intertext{Let $C_{1}= 2C\dfrac{(M-1)\cdot h_{0}^{2}(h_{0}+1)}{1}\sqrt{M}$, then}\\ R^{1} &\leq h_{0}b_{1}+C_{1}\sum_{w=2}^{x_{0}}\dfrac{b_{w}}{(b_{w-1}/M-1)^{2}}\sqrt{\dfrac{\log(b_{w-1})}{b_{w-1}}}\\ \intertext{Using the fact that $\dfrac{\log(x)}{x} \leq 1/2, \forall x > 1$ we get,}\\ R^{1} &\leq h_{0}b_{1}+\dfrac{C_{1}}{\sqrt{2}}\sum_{w=2}^{x_{0}} \dfrac{b_{w}}{(b_{w-1}/M -1)^{2}}\\ \intertext{Since $b_{w}=b_{w-1}^{2}$ we can write,} \\ R^{1} &\leq h_{0}b_{1}+\dfrac{C_{1}M^{2}}{\sqrt{2}}\sum_{w=2}^{x_{0}} \dfrac{1}{(1-\dfrac{M}{b_{w-1}})^{2}} \\
&\leq h_{0}b_{1}+ \dfrac{C_{1}M^{2}}{\sqrt{2}}\sum_{w=2}^{x_{0}} \dfrac{1}{(1-\dfrac{M}{b_{1}})^{2}}\\ &\leq h_{0}b_{1}+ \dfrac{C_{1}M^{2}}{\sqrt{2}(1-\dfrac{M}{b_{1}})^{2}}\times (\log(\log_{b_{1}}(\dfrac{T}{h_{0}}))) \end{align*}
Thus we get $ R^{1} \leq C_{0}+ C_{2} \times \log(\log_{b_{1}}(\dfrac{T}{h_{0}}))$, where $C_{2}= \dfrac{C_{1}M^{2}}{\sqrt{2}(1-\dfrac{M}{b_{1}})^{2}}$ is a constant depending on $K, M$. \end{proof} \textbf{Lemma 4.4} \textit{For $2 \leq M < K$, and for $T \geq KM^{2}(M+2)$, let, $R^{(2)}= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}} R^{(2)}_{w,j}$, then,}
\[ R^{(2)} \leq C_{0}+ C_{4}\sqrt{(\log_{b_{1}}(\dfrac{T}{h_{0}}))*(\dfrac{T}{h_{0}})} \] \textit{Where $C_{0} , C_{4}$ are constants depending on K and M.} \begin{proof}
For any sub-phase $j$ of any phase $w \geq 2$, due to Lemma 3, we know that there exists a constant $C$ such that $R^{(2)}_{w,j} \leq C \sqrt{b_{w}M\log(b_{w})}$. Therefore,
\begin{align*} R^{(2)}&= \sum_{w=1}^{x_{0}}\sum_{j=1}^{h_{0}}R^{(2)}_{w,j} \\ &= \sum_{j=1}^{h_{0}}R^{(2)}_{1,j}+ \sum_{w=2}^{x_{0}}\sum_{j=1}^{h_{0}}R^{(2)}_{w,j} \end{align*}
Now $R^{(2)}_{w,j} = \sum_{t=1}^{b_{w}} (E[\mu_{*}^{w,j}] - E[\mu_{a_{t}}])$, and we know that $\mu_{*}^{w,j} \leq \max_{a \in S^{w,j}}\mu_{a}$. Thus using \textbf{Lemma 4.3} we get that,
\begin{align*} R^{(2)} & \leq h_{0}b_{1}+ C\sum_{w=2}^{x_{0}}\sum_{j=1}^{h_{0}} \sqrt{b_{w}M\log(b_{w})} \\ &\leq h_{0}b_{1}+ Ch_{0}\sum_{w=2}^{x_{0}} \sqrt{(b_{1})^{2^{w-1}}2^{w-1} (M\log(b_{1}))} \\ &\leq h_{0}b_{1}+ Ch_{0}(M\log(b_{1}))\sum_{w=1}^{x_{0}-1} \sqrt{(b_{1})^{2^{w}}2^{w}} \\ &\leq h_{0}b_{1}+ Ch_{0}(M\log(b_{1}))\sum_{w=1}^{\log\log_{b1}(T/h_{0})} \sqrt{(b_{1})^{2^{w}}2^{w}} \\ \end{align*}
Let $C_{3}= C(\dfrac{K-1}{M-1})M\log(M(M+2))$, \\ and $C_{0}= \dfrac{(K-1)(M)(M+2)}{(M-1)}$. \\
\begin{align*} &\leq h_{0}b_{1}+ C_{3}\sum_{w=1}^{\log\log_{b1}(T/h_{0})} \sqrt{(b_{1})^{2^{w}}2^{w}} \\ \intertext{Invoking \textbf{Lemma A.4} and using $m=2$ we get} \\ &\leq h_{0}b_{1}+ C_{3}\sqrt{(\log_{b_{1}}(T/h_{0}))\cdot(T/h_{0}+ \dfrac{(T/h_{0})^{1/(1)}}{b_{1}^{2/(2-1)}})}\\ \intertext{Thus we get,} \\ R^{(2)} &\leq C_{0}+ C_{4}\sqrt{(\log_{b_{1}}(\dfrac{T}{h_{0}}))*(\dfrac{T}{h_{0}})}\\\\ \intertext{where $C_{4}= C(\dfrac{K-1}{M-1})M\log(M(M+2))\sqrt{1+\dfrac{1}{b_{1}}}$} \\ \intertext{and $C_{0}= \dfrac{(K-1)(M)(M+2)}{(M-1)}$} \\ \end{align*}
\end{proof}
\begin{lemma}[From Lemma 1 in \cite{Bubeck et al.}] Let $a_{1}, \cdots, a_{K}$ be real numbers such that $a_{1}+\cdots+a_{K}=1$ and $a_{i} \geq 0$ for all $i$, with the additional property that for all suboptimal arms $i$ and all optimal arms $i^{*}$, one has $a_{i} \leq a_{i^{*}}$. Let $n$: total time horizon, and $J_{n}$: max played arm recommended.Then for $\alpha > 1$, and sufficiently large time horizons $a_{i}n \geq 1+\dfrac{4\alpha ln(n)}{\Delta_{i}^{2}}$, the allocation strategy given by UCB($\alpha$) associated with the recommendation given by the most played arm ensures that the probability of the suboptimal arm being the max played arm:
\begin{align*}
P_{i \neq i^{*}}(J_{n}= i) \leq \dfrac{1}{\alpha-1}(n/K-1)^{2(1-\alpha)} \end{align*}
\end{lemma}
\begin{proof}
We first prove that whenever the most played arm $J_{n}$ is different from an optimal arm $i^{*}$, then at least one of the suboptimal arms i is such that $T_{i}(n) \geq a_{i}n$. To do so, we use a contrapositive method and assume that $T_{i}(n) < a_{i}n$ for all suboptimal arms. Then,
\begin{align*}
(\sum_{i=1}^{K}a_{i})n= n = \sum_{i=1}^{K}T_{i}(n) < \sum_{i_{*}}T_{i_{*}}(n)+ \sum_{i}a_{i}n \end{align*}
where, in the inequality, the first summation is over the optimal arms, the second one, over the suboptimal ones. Therefore, we get
\begin{align*}
\sum_{i^{*}}a_{i^{*}}n < \sum_{i^{*}}T_{i^{*}}(n) \end{align*}
and there exists at least one optmial arm $i^{*}$ such that $T_{i^{*}} > a_{i^{*}}n$. Since by definition of the vector $(a_{1},\cdots,a_{K})$, one has $a_{i} \leq a_{i^{*}}$ for all suboptimal arms, it comes that $T_{i} < a_{i}n \leq a_{i^{*}}n < T_{i^{*}}(n)$ for all suboptimal arms, and the most played arm $J_{n}$ is thus an optimal arm. \\\\
A side-result extracted from [\cite{Audibert et al.}, proof of Theorem 7], see also [\cite{Auer et al.}, proof of Theorem 1], states that for all suboptimal arms $i$ and rounds $t \geq K+1$,
\begin{align*}
P\{I_{t}=i, T_{i}(t-1) \geq l\} \leq 2t^{1-2\alpha} \hspace{30pt} (l\geq \dfrac{4\alpha ln(n)}{\Delta_{i}^{2}}) \end{align*}
We denote by $\lceil x \rceil$ the upper integer part of a real number $x$. For a suboptimal arm $i$ and since by the assumptions on $n$ and $a_{i}$, the choice $l= \lceil a_{i}n \rceil-1$ satisfies $l \geq K+1$ and $l \geq (4\alpha ln(n))/\Delta_{i}^{2}$,
\begin{align*}
P\{T_{i}(n) \geq a_{i}n\} &= P\{T_{i}(n) \geq \lceil a_{i}n \rceil\} \\
&\leq \sum_{t= \lceil a_{i}n \rceil}^{n}P\{T_{i}(t-1)= \lceil a_{i}n \rceil-1, I_{t}=i\}\\
&\leq \sum_{t= \lceil a_{i}n \rceil}^{n}2t^{1-2\alpha} \leq 2\int_{\lceil a_{i}n \rceil -1}^{\infty} v^{1-2\alpha} \,dv \\
&\leq \dfrac{1}{\alpha-1}(a_{i}n-1)^{2(1-\alpha)}, \end{align*} With the uniform choice of $a_{i}= 1/K$ we finally get,
\begin{align*}
P_{i \neq i^{*}}(J_{n}= i) \leq P(T_{i}(n) \geq n/K) \leq \dfrac{1}{\alpha-1}(n/K-1)^{2(1-\alpha)} \end{align*} \end{proof}
\section{Proofs for Section 5} \begin{lemma} Total number of passes in 2-pass UCB-LAM is two. \end{lemma}
\begin{proof}
From Algorithm-2 above we know that $b_{1}= \dfrac{\sqrt{1+4\dfrac{T}{h_{0}}}-1}{2}$ and $b_{2}= \dfrac{2\dfrac{T}{h_{0}}+1-\sqrt{1+\dfrac{4T}{h_{0}}}}{2}$. So the total time taken for the first two passes($T_{\text{two-pass}}$) is:
\begin{align*}
T_{\text{two-pass}} &= h_{0}\times b_{1}+ h_{0}\times b_{2} \\
&= \dfrac{h_{0}\sqrt{1+4\dfrac{T}{h_{0}}}-1}{2} + \dfrac{2T+h_{0}-h_{0}\sqrt{1+\dfrac{4T}{h_{0}}}}{2} \\
& \implies T_{\text{two-pass}}= T\\ \end{align*}
Thus we see that the total time horizon is depleted by the end of second pass, and thus the total number of passes in Algorithm-2 in two. \end{proof}
\textbf{Lemma 5.1} \textit{For $2 \leq M < K$, and total time-horizon $T$, \\ we get that, for Algorithm-2,} \[ R_{2} \leq C_{2}+C_{0}\sqrt{T+0.25h_{0}}+C_{1}\sqrt{T\log(T/h_{0})} \] \textit{Where $C_{0}, C_{1}, C_{2}$ are constants depending on $M,K$.} \begin{proof}
We know that $R_{2}$ can be written as,
\begin{align*}
R_{2}&= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}(R^{(2)}_{w,j}) \\
&= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}(\sum_{t=1}^{b_{w}} (E[\mu_{*}^{w,j}] - E[\mu_{a_{t}}])) \\
&\leq h_{0}b_{1}+ \sum_{j=1}^{h_{0}}(\sum_{t=1}^{b_{2}}(E[\mu_{*}^{2,j}] - E[\mu_{a_{t}}])) \\ \end{align*}
From \textbf{Lemma 4.3} we get that given a set of $K$ arms as the input for any horizon $T$, the cumulative regret incurred by $UCB1$ when $T \geq K/2$, is upper-bounded by $18\sqrt{TK\log(T)}$ or $R_{T}^{*} \leq 18\sqrt{TK\\log(T)}$. And we know that $\mu_{*}^{w,j} \leq \max_{a \in S^{w,j}} \mu_{a} \implies E[\mu^{w,j}_{*}-\mu_{a_{t}}] \leq E[\max_{a\in S^{w,j}}\mu_{a}- \mu_{a_{t}}]$. Thus,
\begin{align*}
R_{2} \leq h_{0}b_{1}+ h_{0}C\sqrt{b_{2}M\log(b_{2})} \end{align*}
Substituting $b_{1}= \dfrac{\sqrt{1+4\dfrac{T}{h_{0}}}-1}{2}$ and $b_{2}= \dfrac{2\dfrac{T}{h_{0}}+1-\sqrt{1+\dfrac{4T}{h_{0}}}}{2}$, we get,
\begin{align*}
R_{2} \leq 0.5(\sqrt{h_{0}^{2}+4h_{0}T}-h_{0})+C_{1}\sqrt{T\log(T/h_{0})} \end{align*} where $C_{1}=f(M,K), C_{0}= g(M,K), C_{2}= h(M, K)$\\ Thus we get that
\[ R_{2} \leq C_{2}+C_{0}\sqrt{T+0.25h_{0}}+C_{1}\sqrt{T\log(T/h_{0})}. \] \end{proof}
\textbf{Lemma 5.2} \textit{For $2 \leq M < K$, and total time-horizon $T$, \\ we get that, for Algorithm-2,\\ $R_{1}= \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}(R^{(1)}_{w,j}) \leq O(\sqrt{T})$}
\begin{proof} We know that $R_{1}$ can be written as,\\
$R_{1}$= $\sum_{w=1}^{2}\sum_{j=1}^{h_{0}}\sum_{t=1}^{b_{w}}(E[\mu^{*}-\mu_{*}^{w,j}])$
\begin{align*} \implies &R_{1} = \sum_{w=1}^{2}\sum_{j=1}^{h_{0}}b_{w}(E[\mu^{*}-\mu_{*}^{w,j}]) \\ & R_{1} \leq h_{0}b_{1}+ \sum_{j=1}^{h_{0}}b_{2}(E[\mu^{*}-\mu_{*}^{2,j}]) \end{align*}
From \textbf{Lemma A.3} we know that $b_{2}E[\mu^{*}-\mu_{*}^{2,j}]$ can be written in terms of $T_{1}$ and $T_{2}$, or $b_{2}E[\mu^{*}-\mu_{*}^{2,j}]= T_{1}+T_{2}$, where,
\begin{align*}
T_{1}&= P(A_{s}=1)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}])P(B_{s}=0|A_{s}=1)\\
T_{2}&= P(A_{s}=0)\times \{b_{w}(E[\mu^{*}- \mu_{*}^{w,j}]) \} \end{align*}
Where $s=(w,j)$ the current instance and $\mu_{*}^{w,j}$ is the mean of the arm recommended to the next instance. And invoking \textbf{Lemma 4.1} will give us that,
\begin{align*}
T_{1}+T_{2} \leq 2Cb_{2}\dfrac{(M-1)\cdot h_{0}}{(b_{1}/M -1)^{2}}(h_{0}+1)\sqrt{\dfrac{M\log(b_{1})}{b_{1}}} \end{align*} Thus we get that,
\begin{align*}
R_{1}& \leq h_{0}b_{1}+2Cb_{2}\dfrac{(M-1)\cdot h_{0}^{2}}{(b_{1}/M -1)^{2}}(h_{0}+1)\sqrt{\dfrac{M\log(b_{1})}{b_{1}}} \\
&\text{Substituting values for} b_{1}, b_{2} for \textbf{Algorithm-2} \\
b_{1}&= \dfrac{\sqrt{1+4\dfrac{T}{h_{0}}}-1}{2}\\
b_{2}&= \dfrac{2\dfrac{T}{h_{0}}+1-\sqrt{1+\dfrac{4T}{h_{0}}}}{2}= b_{1}^{2} \end{align*} in the above inequality we finally get that $R_{1} \leq O(\sqrt{T})$. \end{proof}
\end{document} |
\begin{document}
\baselineskip=17pt
\titlerunning{$p$-Laplace equations and geometric Sobolev inequalities}
\title{Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities}
\author{Daniele Castorina \and Manel Sanch\'on}
\date{}
\maketitle
\address{D. Castorina: Departament de Matem\`atiques, Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Spain; \email{castorina@mat.uab.cat} \and M. Sanch\'on: Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; \email{msanchon@maia.ub.es}}
\subjclass{Primary 35K57, 35B65 ; Secondary 35J60 }
\begin{abstract} In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \textit{a priori} estimates for semi-stable solutions of $-\Delta_p u= g(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ when the domain is strictly convex. More precisely, we prove that $u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.
\keywords{Geometric inequalities, mean curvature of level sets, Schwarz symmetri- zation, $p$-Laplace equations, regularity of stable solutions} \end{abstract}
\section{Introduction} The aim of this paper is to obtain \textit{a priori} estimates for semi-stable solutions of $p$-Laplace equations. We will accomplish this by proving some geometric type inequalities involving the functionals
\begin{equation}\label{Ipq} I_{p,q}(v;\Omega):=\left( \int_{\Omega}
\Big(\frac{1}{p'}|\nabla_{T,v} |\nabla v|^{p/q}|\Big)^{q}
+ |H_v|^q |\nabla v|^p \, dx \right)^{1/p},\quad p,q\geq 1 \end{equation}
where $\Omega$ is a smooth bounded domain of $\mathbb{R}^n$ with
$n\geq 2$ and $v\in C_0^\infty(\overline{\Omega})$. Here, and in the rest of the paper, $H_v (x)$ denotes the mean curvature at $x$ of the hypersurface $\{y\in\Omega:|v(y)|=|v(x)|\}$ (which is smooth at points $x\in\Omega$ satisfying $\nabla v(x)\neq 0$), and $\nabla_{T,v}$
is the tangential gradient along a level set of $|v|$. We will prove a Morrey's type inequality when $n<p+q$ and a Sobolev inequality when $n>p+q$ (see Theorem~\ref{Theorem:Sobolev} below).
Then, as an application of these inequalities, we establish $L^r$ and $W^{1,r}$ \textit{a priori} estimates for semi-stable solutions of the reaction-diffusion problem
\begin{equation}\label{problem} \left\{ \begin{array}{rcll} -\Delta_p u &=& g(u) &\textrm{in } \Omega, \\
u&>& 0 &\textrm{in } \Omega, \\
u &=& 0 &\textrm{on } \partial \Omega. \end{array} \right. \end{equation}
Here, the diffusion is modeled by the $p$-Laplace operator $\Delta_p$
(remember that $\Delta_p u:= {\rm div}(|\nabla u|^{p-2}\nabla u)$) with $p>1$, while the reaction term is driven by any positive $C^1$ nonlinearity $g$.
As we will see, these estimates will lead to new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star$ of \eqref{problem} when $g(u)=\lambda f(u)$ and the domain $\Omega$ is strictly convex. More precisely, we prove that $u^\star\in L^\infty(\Omega)$ if $n\leq p+2$ and $u^\star\in L^{\frac{np}{n-p-2}}(\Omega)\cap W^{1,p}_0(\Omega)$ if $n>p+2$.
\subsection{Geometric Sobolev inequalities}
Before we establish our Sobolev and Morrey type inequalities we will state that the functional $I_{p,q}$ defined in \eqref{Ipq} decreases (up to a universal multiplicative constant) by Schwarz symmetrization. Given a Lipschitz continuous function $v$ and its Schwarz symmetrization $v^*$ it is well known that
$$
\int_{B_R} |v^*|^r\ dx=\int_\Omega |v|^r\ dx\quad \textrm{for all }r\in[1,+\infty] $$ and $$
\int_{B_R} |\nabla v^*|^r\ dx \leq \int_\Omega |\nabla v|^r\ dx \quad \textrm{for all }r\in[1,\infty). $$
Our first result establishes that $I_{p,q}(v^*;B_R)\leq C I_{p,q}(v;\Omega)$ for some universal constant $C$ depending only on $n$, $p$, and $q$.
\begin{thm}\label{thm:Ipq} Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^n$ with $n\geq 2$ and
$B_R$ the ball centered at the origin and with radius $R=(|\Omega|/|B_1|)^{1/n}$. Let $v\in C^\infty_0(\overline{\Omega})$ and $v^*$ its Schwarz symmetrization. Let $I_{p,q}$ be the functional defined in \eqref{Ipq} with $p,q \geq 1$. If $n>q+1$ then there exists a universal constant $C$ depending only on $n$, $p$, and $q$, such that
\begin{equation}\label{comp_integrals}
\left( \int_{B_R}\frac{1}{|x|^q} |\nabla v^*|^p \, dx \right)^{1/p} = I_{p,q}(v^*;B_R) \leq C I_{p,q}(v;\Omega). \end{equation}
\end{thm}
Note that the Schwarz symmetrization of $v$ is a radial function, and hence, its level sets are spheres. In particular, the mean curvature $H_{v^*}(x)=1/|x|$ and the tangential gradient
$\nabla_{T,{v^*}} |\nabla v^*|^{p/q}=0$. This explains the equality in \eqref{comp_integrals}.
A related result was proved by Trudinger \cite{Trudinger97} when $q=1$ for the class of mean convex functions (\textit{i.e.}, functions for which the mean curvature of the level sets is nonnegative). More precisely, he proved Theorem~\ref{thm:Ipq} replacing the functional $I_{p,q}$ by
\begin{equation}\label{tilde:Ipq} \tilde{I}_{p,q}(v;\Omega):=\left( \int_{\Omega}
|H_v|^q |\nabla v|^p \, dx \right)^{1/p} \end{equation}
and considering the Schwarz symmetrization of $v$ with respect to the perimeter instead of the classical one like us (see Definition~\ref{Schwarz-symm}
below). In order to define this symmetrization (with respect to the perimeter) it is essential to know that the mean curvature $H_v$ of the level sets of $|v|$ is nonnegative. Then using an Aleksandrov-Fenchel inequality for mean convex hypersurfaces (see \cite{Trudinger94}) he proved Theorem~\ref{thm:Ipq} for this class of functions when $q=1$.
We prove Theorem~\ref{thm:Ipq} using two ingredients. The first one is the classical isoperimetric inequality:
\begin{equation}\label{isop:ineq}
n|B_1|^{1/n}|D|^{(n-1)/n}\leq |\partial D| \end{equation}
for any smooth bounded domain $D$ of $\mathbb{R}^n$. The second one is a geometric Sobolev inequality, due to Michael and Simon \cite{MS} and to Allard \cite{A}, on compact $(n-1)$-hypersurfaces $M$ without boundary which involves the mean curvature $H$ of $M$: for every $q\in [1, n-1)$, there exists a constant $A$ depending only on $n$ and $q$ such that \begin{equation}\label{Sob:mean}
\left( \int_M |\phi|^{q^\star} d\sigma \right)^{1/q^\star} \leq A
\left( \int_M |\nabla \phi|^q + |H\phi|^q \ d\sigma \right)^{1/q} \end{equation} for every $\phi\in C^\infty(M)$, where $q^\star = (n-1)q/(n-1-q)$ and
$d\sigma$ denotes the area element in $M$. Using the classical isoperimetric inequality \eqref{isop:ineq} and the geometric Sobolev inequality \eqref{Sob:mean} with $M=\{x\in \Omega:|v(x)|=t\}$
and $\phi=|\nabla v|^{(p-1)/q}$ we will prove Theorem~\ref{thm:Ipq}
with the explicit constant $C=A^\frac{q}{p} |\partial B_1|^\frac{q}{(n-1)p}$, being $A$ the universal constant in \eqref{Sob:mean}.
{F}rom Theorem~\ref{thm:Ipq} and well known 1-dimensional weighted Sobolev inequalities it is easy to prove Morrey and Sobolev geometric inequalities involving the functional $I_{p,q}$. Indeed, by Theorem~\ref{thm:Ipq} and since Schwarz symmetrization preserves the $L^r$ norm, it is sufficient to prove the existence of a positive constant $\overline{C}$ independent of $v^*$ such that
$$
\|v^*\|_{L^r(B_R)}\leq \overline{C} I_{p,q}(v^*;B_R). $$
Using this argument we prove the following geometric inequalities.
\begin{thm}\label{Theorem:Sobolev} Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^n$ with $n\geq 2$ and $v\in C^\infty_0(\overline{\Omega})$. Let $I_{p,q}$ be the functional defined in \eqref{Ipq} with $p,q \geq 1$ and
$$ p_{q}^\star:= \frac{n p}{n - (p+q)}. $$
Assume $n>q+1$. The following assertions hold: \begin{enumerate} \item[$(a)$] If $n<p+q$ then
\begin{equation}\label{Morrey}
\|v\|_{L^\infty(\Omega)}
\leq C_1|\Omega|^{\frac{p+q-n}{np}}I_{p,q}(v;\Omega) \end{equation}
for some constant $C_1$ depending only on $n$, $p$, and $q$.
\item[$(b)$] If $n>p+q$, then
\begin{equation}\label{Sobolev}
\|v\|_{L^r(\Omega)}
\leq C_2|\Omega|^{\frac{1}{r}-\frac{1}{p_q^\star}} I_{p,q}(v;\Omega)\quad \textrm{for every }1\leq r \leq p_{q}^\star, \end{equation}
where $C_2$ is a constant depending only on $n$, $p$, $q$, and $r$.
\item[$(c)$] If $n = p+q$, then
\begin{equation}\label{Moser-Trudinger}
\int_\Omega\exp\left\{\left(\frac{|v|}{C_3 I_{p,q}(v;\Omega)}\right)^{p'}\right\}\ dx
\leq \frac{n}{n-1}|\Omega|,\quad \textrm{where }p'=p/(p-1), \end{equation}
for some positive constant $C_3$ depending only on $n$ and $p$. \end{enumerate} \end{thm}
Cabr\'e and the second author \cite{CS} proved recently Theorem~\ref{Theorem:Sobolev} under the assumption $q\geq p$ using a different method (without the use of Schwarz symmetrization). More precisely, they proved the theorem replacing the functional $I_{p,q}(v;\Omega)$ by the one defined in \eqref{tilde:Ipq}, $\tilde{I}_{p,q}(v;\Omega)$. Therefore, our geometric inequalities are only new in the range $1\leq q<p$.
\begin{open} Is Theorem~\ref{Theorem:Sobolev} true for the range $1\leq q<p$ and replacing the functional $I_{p,q}(v;\Omega)$ by the one defined in \eqref{tilde:Ipq}, $\tilde{I}_{p,q}(v;\Omega)$? \end{open}
This question has a posive answer for the class of mean convex functions. Trudinger \cite{Trudinger97} proved this result for this class of functions when $q=1$ and can be easily extended for every $q\geq 1$. However, to our knowledge, for general functions (without mean convex level sets) it is an open problem.
\subsection{Regularity of semi-stable solutions}
The second part of the paper deals with \textit{a priori} estimates for semi-stable solutions of problem \eqref{problem}. Remember that a regular solution $u\in C_0^1(\overline{\Omega})$ of \eqref{problem} is said to be \textit{semi-stable} if the second variation of the associated energy functional at $u$ is nonnegative definite, \textit{i.e.},
\begin{equation}\label{semi-stab1}
\int_\Omega |\nabla u|^{p-2} \left\{|\nabla \phi|^2+(p-2)
\left(\nabla \phi\cdot\frac{\nabla u}{|\nabla u|}\right)^2\right\} - g'(u) \phi^2\ dx \geq 0 \end{equation}
for every $\phi \in H_0$, where $H_0$ denotes the space of admissible functions (see Definition~\ref{H0} below). The class of semi-stable solutions includes local minimizers of the energy functional as well as minimal and extremal solutions of \eqref{problem} when $g(u)=\lambda f(u)$.
Using an appropriate test function in \eqref{semi-stab1} we prove the following \textit{a priori} estimates for semi-stable solutions. This result extends the ones in \cite{Cabre09} and \cite{CS} for the Laplacian case ($p=2$) due to Cabr\'e and the second author.
\begin{thm}\label{Theorem} Let $g$ be any $C^\infty$ function and $\Omega\subset\mathbb{R}^n$ any smooth bounded domain. Let $u\in C^1_0(\overline{\Omega})$ be a semi-stable solution of \eqref{problem}, \textit{i.e.}, a solution satisfying \eqref{semi-stab1}. The following assertions hold:
$(a)$ If $n\leq p+2$ then there exists a constant $C$ depending only on $n$ and $p$ such that
\begin{equation}\label{L-infinty}
\|u\|_{L^\infty(\Omega)}\leq s+\frac{C}{s^{2/p}}|\Omega|^\frac{p+2-n}{np}
\left(\int_{\{u\leq s\}} |\nabla u|^{p+2}\, dx\right)^{1/p}\quad \textrm{for all }s>0. \end{equation}
$(b)$ If $n>p+2$ then there exists a constant $C$ depending only on $n$ and $p$ such that
\begin{equation}\label{Lq:estimate}
\left(\int_{\{u>s\}} \Big(|u|-s\Big)^{\frac{np}{n-(p+2)}}\ dx\right)^{\frac{n-(p+2)}{np}} \leq \frac{C}{s^{2/p}}
\left(\int_{\{u\leq s\}} |\nabla u|^{p+2} \ dx\right)^{1/p} \end{equation}
for all $s>0$. Moreover, there exists a constant $C$ depending only on $n$, $p$, and $r$ such that
\begin{equation}\label{grad:estimate}
\int_\Omega |\nabla u|^r\ dx\leq C\left(|\Omega|+\int_\Omega|u|^\frac{np}{n-(p+2)}\ dx
+\|g(u)\|_{L^1(\Omega)}\right) \end{equation}
for all $1\leq r<r_1:=\frac{np^2}{(1+p)n-p-2}$. \end{thm}
To prove \eqref{L-infinty} and \eqref{Lq:estimate} we use the semi-stability condition
\eqref{semi-stab1} with the test function $\phi=|\nabla u|\eta$ to obtain
\begin{equation}\label{ineq:key}
\int_{\Omega}\left( \frac{4}{p^2}|\nabla_{T,u} |\nabla u|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_u^2 |\nabla u|^{p} \right) \eta^2 \, dx
\leq \int_{\Omega} |\nabla u|^{p}|\nabla \eta|^2\, dx \end{equation}
for every Lipschitz function $\eta$ in $\overline{\Omega}$ with
$\eta|_{\partial\Omega}=0$. Then, taking $\eta=T_s u = \min\{s,u\}$, we obtain \eqref{L-infinty} and \eqref{Lq:estimate} when $n\neq p+2$ by using the Morrey and Sobolev inequalities established in Theorem~\ref{Theorem:Sobolev} with $q=2$.
The critical case $n=p+2$ is more involved. In order to get \eqref{L-infinty} in this case, we take another explicit test function $\eta=\eta(u)$ in \eqref{ineq:key} and use the geometric Sobolev inequality \eqref{Sob:mean}. The gradient estimate established in \eqref{grad:estimate} will follow by using a technique introduced by B\'enilan \textit{et al.} \cite{BBGGPV95} to get the regularity of entropy solutions for $p$-Laplace equations with $L^1$ data (see Proposition \ref{Prop:bootstrap}).
The rest of the introduction deals with the regularity of extremal solutions. Let us recall the problem and some known results in this topic. Consider
\stepcounter{equation}
$$ \left\{ \begin{array}{rcll} -\Delta_p u&=&\lambda f(u)&\textrm{in }\Omega,\\ u&=&0&\textrm{on }\partial \Omega, \end{array} \right. \eqno{(1.15)_{\lambda}} $$
where $\lambda$ is a positive parameter and $f$ is a $C^1$ positive increasing function satisfying
\begin{equation}\label{p-superlinear} \lim_{t\rightarrow+\infty}\frac{f(t)}{t^{p-1}}=+\infty. \end{equation}
Cabr\'e and the second author \cite{CS07} proved the existence of an extremal parameter $\lambda^\star\in(0,\infty)$ such that problem $(1.15)_{\lambda}$ admits a minimal regular solution $u_\lambda\in C^1_0(\overline{\Omega})$ for $\lambda\in(0,\lambda^\star)$ and admits no regular solution for $\lambda>\lambda^\star$. Moreover, every minimal solution $u_\lambda$ is a semi-stable for $\lambda\in (0,\lambda^\star)$.
For the Laplacian case ($p=2$), the limit of minimal solutions
$$ u^\star:=\lim_{\lambda\uparrow\lambda^\star}u_\lambda $$
is a weak solution of the extremal problem $(1.15)_{\lambda^\star}$ and it is known as extremal solution. Nedev \cite{Nedev} proved, in the case of convex nonlinearities, that $u^\star\in L^\infty(\Omega)$ if $n\leq 3$ and $u^\star\in L^r(\Omega)$ for all $1\leq r<n/(n-4)$ if $n\geq 4$. Recently, Cabr\'e~\cite{Cabre09}, Cabr\'e and the second author \cite{CS}, and Nedev \cite{Nedev01} proved, in the case of convex domains and general nonlinearities, that $u^\star\in L^\infty(\Omega)$ if $n\leq 4$ and $u^\star\in L^{\frac{2n}{n-4}}(\Omega)\cap H^1_0(\Omega)$ if $n\geq 5$.
For arbitrary $p>1$ it is unknown if the limit of minimal solutions $u^\star$ is a (weak or entropy) solution of $(1.15)_{\lambda^\star}$. In the affirmative case, it is called the \textit{extremal solution of $(1.15)_{\lambda^\star}$}. However, in \cite{S} it is proved that the limit of minimal solutions $u^\star$ is a weak solution (in the distributional sense) of $(1.15)_{\lambda^\star}$ whenever $p\geq 2$ and $f$ satisfies the additional condition:
\begin{equation}\label{convex:assump} \textrm{there exists }T\geq0 \textrm{ such that }(f(t)-f(0))^{1/(p-1)} \textrm{ is convex for all }t\geq T. \end{equation}
Moreover,
$$ u^\star\in L^\infty(\Omega)\qquad\textrm{ if }n<p+p' $$
and
$$ u^\star\in L^r(\Omega),\textrm{ for all }r<\tilde{r}_0:=(p-1)\frac{n}{n-(p+p')},\quad \textrm{if }n\geq p+p'. $$
This extends previous results of Nedev \cite{Nedev} for the Laplacian case ($p=2$) and convex nonlinearities.
Our next result improves the $L^q$ estimate in \cite{Nedev,S} for strictly convex domains. We also prove that $u^\star$ belongs to the energy class $W^{1,p}_0(\Omega)$ independently of the dimension extending an unpublished result of Nedev~\cite{Nedev01} for $p=2$ to every $p\geq 2$ (see also \cite{CS}).
\begin{thm}\label{Theorem2} Let $f$ be an increasing positive $C^1$ function satisfying \eqref{p-superlinear}. Assume that $\Omega$ is a smooth strictly convex domain of $\mathbb{R}^n$. Let $u_\lambda\in C^1_0(\overline{\Omega})$ be the minimal solution of $(1.15)_{\lambda}$. There exists a constant $C$ independent of $\lambda$ such that: \begin{enumerate} \item[$(a)$] If $n\leq p+2$ then $
\|u_\lambda\|_{L^\infty(\Omega)}\leq C \|f(u_\lambda)\|_{L^1(\Omega)}^{1/(p-1)}. $ \item[$(b)$] If $n> p+2$ then $
\|u_\lambda\|_{L^\frac{np}{n-p-2}(\Omega)}\leq C
\|f(u_\lambda)\|_{L^1(\Omega)}^{1/(p-1)}. $ Moreover
$
\|u_\lambda\|_{W^{1,p}_0(\Omega)}\leq C' $
where $C'$ is a constant depending only on $n$, $p$, $\Omega$, $f$ and
$\|f(u_\lambda)\|_{L^1(\Omega)}$. \end{enumerate}
Assume, in addition, $p\geq 2$ and that \eqref{convex:assump} holds. Then \begin{enumerate} \item[$(i)$] If $n\leq p+2$ then $u^\star\in L^\infty(\Omega)$. In particular, $u^\star \in C^1_0(\overline{\Omega})$. \item[$(ii)$] If $n>p+2$ then $u^\star\in L^\frac{np}{n-p-2}(\Omega)\cap W^{1,p}_0(\Omega)$. \end{enumerate} \end{thm}
\begin{rem} If $f(u_\lambda)$ is bounded in $L^1(\Omega)$ by a constant independent of $\lambda$, then parts $(a)$ and $(b)$ will lead automatically to the assertions $(i)$ and $(ii)$ stated in the theorem (without the requirement that $p\geq 2$ and \eqref{convex:assump} hold true). However, as we said before, the estimate $f(u^\star)\in L^1(\Omega)$ is unknown in the general case, \textit{i.e}, for arbitrary positive and increasing nonlinearities $f$ satisfying \eqref{p-superlinear} and arbitrary $p>1$. \end{rem}
\begin{open} Is it true that $f(u^\star)\in L^1(\Omega)$ for arbitrary positive and increasing nonlinearities $f$ satisfying \eqref{p-superlinear}? \end{open}
Under assumptions $p\geq 2$ and \eqref{convex:assump} it is proved in \cite{S} that $f(u^\star)\in L^r(\Omega)$ for all $1\leq r<n/(n-p')$ when $n\geq p'$ and $f(u^\star) \in L^\infty(\Omega)$ if $n<p'$. In particular, one has $f(u^\star)\in L^1(\Omega)$ independently of the dimension $n$ and the parameter $p>1$. As a consequence, assertions $(i)$ and $(ii)$ follow immediately from parts $(a)$ and $(b)$ of the theorem.
To prove the $L^r$ \textit{a priori} estimates stated in part $(a)$ and $(b)$ we make three steps. First, we use the strict convexity of the domain $\Omega$ to prove that
$$ \{x\in\Omega:{\rm dist}(x,\partial\Omega)<\varepsilon\} \subset \{x\in\Omega:u_\lambda(x)<s\} $$ for a suitable $s$. This is done using a moving plane procedure for $p$-Laplace equations (see Proposition~\ref{Prop:1} below). Then, we prove that the Morrey and Sobolev type inequalities stated in Theorem~\ref{Theorem:Sobolev} for smooth functions, also hold for regular solutions of \eqref{problem} when $1\leq q\leq 2$. Finally, taking a test function $\eta$ related to ${\rm dist}(\cdot,\partial\Omega)$ in \eqref{ineq:key} and proceeding as in the proof of Theorem~\ref{Theorem} we will obtain the $L^r$ \textit{a priori} estimates established in the theorem.
The energy estimate established in parts (ii) and (b) of Theorem~\ref{Theorem2} follows by extending the arguments of Nedev \cite{Nedev01} for the Laplacian case (see also Theorem~2.9 in \cite{CS}). First, using a Poho${\rm\check{z}}$aev identity we obtain
\begin{equation}\label{key:Poho}
\int_\Omega|\nabla u_\lambda|^p\ dx \leq
\frac{1}{p'}\int_{\partial\Omega}|\nabla u_\lambda|^p\ x\cdot\nu\ d\sigma, \qquad\textrm{for all }p>1\textrm{ and }\lambda\in(0,\lambda^\star), \end{equation}
where $d\sigma$ denotes the area element in $\partial\Omega$ and $\nu$ is the outward unit normal to $\Omega$. Then, using the strict convexity of the domain (as in the $L^r$ estimates) and standard regularity estimates for $-\Delta_p u=\lambda f(u_\lambda(x))$ in a neighborhood of the boundary, we are able to control the right hand side of \eqref{key:Poho} by a constant whose dependence on $\lambda$ is given by a function of
$\|f(u_\lambda)\|_{L^1(\Omega)}$.
\begin{rem} Let us compare our regularity results with the sharp ones proved by Cabr\'e, Capella, and the second author in \cite{CCS09} when $\Omega$ is the unit ball $B_1$ of $\mathbb{R}^n$. In the radial case, the extremal solution $u^\star$ of $(1.15)_{\lambda^\star}$ is bounded if the dimension $n< p+\frac{4p}{p-1}$. Moreover, if $n\geq p+\frac{4p}{p-1}$ then $u^\star\in W^{1,r}_0(B_1)$ for all $1\leq r<\bar{r}_1$, where $$ \bar{r}_1:=\frac{np}{n-2\sqrt{\frac{n-1}{p-1}}-2}. $$ In particular, $u^\star\in L^r(B_1)$ for all $1\leq r<\bar{r}_0$, where $$ \bar{r}_0:=\frac{np}{n-2\sqrt{\frac{n-1}{p-1}}-p-2}. $$ It can be shown that these regularity results are sharp by taking the exponential and power nonlinearities.
Note that the $L^r(\Omega)$-estimate established in Theorem~\ref{Theorem2} differs with the sharp exponent $\bar{r}_0$ defined above by the term $2\sqrt{\frac{n-1}{p-1}}$. Moreover, observe that $\bar{r}_1$ is larger than $p$ and tends to it as $n$ goes to infinity. In particular, the best expected regularity independent of the dimension $n$ for the extremal solution $u^\star$ is $W^{1,p}_0(\Omega)$, which is the one we obtain in Theorem~\ref{Theorem2}.
\end{rem}
\subsection{Outline of the paper} The paper is organized as follows. In section \ref{section2} we prove Theorem~\ref{thm:Ipq} and the geometric type inequalities stated in Theorem~\ref{Theorem:Sobolev}. In section~\ref{section3} we prove that Theorem~\ref{Theorem:Sobolev} holds for solutions of \eqref{problem} when $1\leq q\leq 2$. Moreover we give boundary estimates when the domain is strictly convex. In section~\ref{section5}, we present the semi-stability condition \eqref{semi-stab1} and the space of admissible functions $H_0$. The rest of the section deals with the regularity of semi-stable solutions proving Theorems~\ref{Theorem}~and~\ref{Theorem2}.
\section{Geometric Hardy-Sobolev type inequalities}\label{section2} In this section we prove Theorems \ref{thm:Ipq} and \ref{Theorem:Sobolev}. As we said in the introduction, the geometric inequalities established in Theorem~\ref{Theorem:Sobolev} are new for the range $1\leq q<p$ since the case $q\geq p$ was proved in \cite{CS}. However, we will give the proof in all cases using Schwarz symmetrization, giving an alternative proof for the known range of parameters $q\geq p$.
We start recalling the definition of Schwarz symmetrization of a compact set and of a Lipschitz continuous function. \begin{defin}\label{Schwarz-symm} We define the \textit{Schwarz symmetrization of a compact set $D\subset\mathbb{R}^n$} as $$ D^*:=\left\{ \begin{array}{lll}
B_R(0) \textrm{ with } R=(|D|/|B_1|)^{1/n}&\textrm{if}&D\neq \emptyset,\\ \emptyset&\textrm{if}&D= \emptyset. \end{array} \right. $$
Let $v$ be a Lipschitz continuous function in $\overline{\Omega}$ and
$\Omega_t:=\{x\in\Omega:|v(x)|\geq t\}$. We define the \textit{Schwarz symmetrization of $v$} as
$$ v^*(x):= \sup\{t\in \mathbb{R}: x\in \Omega_t^*\}. $$
Equivalently, we can define the Schwarz symmetrization of $v$ as
$$
v^*(x)=\inf\{t\geq 0:V(t)<|B_1||x|^n\}, $$
where $V(t):=|\Omega_t|=|\{x\in\Omega:|v(x)|> t\}|$ denotes the distribution function of $v$. \end{defin}
The first ingredient in the proof of Theorem \ref{thm:Ipq} is the isoperimetric inequality for functions $v$ in $W^{1,1}_0(\Omega)$:
\begin{equation}\label{talenti}
n|B_1|^{1/n} V(t)^{(n-1)/n}\leq P(t):=\frac{d}{dt}\int_{\{|v|\leq t\}}|\nabla v|\ dx \qquad\text{for a.e. } t>0, \end{equation}
where $P(t)$ stands for the perimeter in the sense of De Giorgi (the total variation of the characteristic function of
$\{x\in \Omega: |v(x)|>t\}$).
The second ingredient is the following Sobolev inequality on compact hypersurfaces without boundary due to Michael and Simon \cite{MS} and to Allard \cite{A}.
\begin{thm}[\cite{A,MS}]\label{ThmMS} Let $M\subset \mathbb{R}^{n}$ be a $C^\infty$ immersed $(n-1)$-dimensional compact hypersurface without boundary and $\phi\in C^\infty(M)$. If $q\in [1, n-1)$, then there exists a constant $A$ depending only on $n$ and $q$ such that
\begin{equation}\label{MS}
\left( \int_M |\phi|^{q^\star} d\sigma \right)^{1/q^\star} \leq A
\left( \int_M |\nabla \phi|^q + |H\phi|^q \ d\sigma \right)^{1/q}, \end{equation} where $H$ is the mean curvature of $M$, $d\sigma$ denotes the area element in $M$, and $q^\star = \frac{(n-1)q}{n-1-q}$. \end{thm}
As we said in the introduction it is well known that Schwarz symmetrization preserves the $L^r$-norm and decreases the $W^{1,r}$-norm. Let us prove that it also decreases (up to a multiplicative constant) the functional $I_{p,q}$ defined in \eqref{Ipq} using the isoperimetric inequality
\eqref{talenti} and the geometric inequality \eqref{MS} applied to $M=M_t=\{x\in\Omega: |v(x)|=t\}$
and $\phi=|\nabla v|^{(p-1)/q}$.
\begin{proof}[Proof of Theorem {\rm\ref{thm:Ipq}}]
Let $v\in C_0^\infty(\overline{\Omega})$, $p\geq 1$, and $1\leq q<n-1$. By Sard's theorem, almost every $t\in(0,\|v\|_{L^\infty(\Omega)})$ is a regular value of $|v|$. By definition, if $t$ is a regular value of $|v|$, then $\left|\nabla v(x)\right|>0$ for all $x\in\Omega$ such that
$|v(x)|=t$. Therefore, $M_t:=\{x\in\Omega: |v(x)|=t\}$ is a $C^{\infty}$ immersed $(n-1)-$dimensional compact hypersurface of $\mathbb{R}^n$ without boundary for every regular value $t$ . Applying inequality \eqref{MS} to $M=M_t$ and
$\phi=|\nabla v|^{(p-1)/q}$ we obtain
\begin{equation}\label{MSv}
\left( \int_{M_t} |\nabla v|^{(p-1)\frac{q^\star}{q}} \, d\sigma \right)^{q/q^\star}
\leq A^q \int_{M_t} \Big|\nabla_{T,v} |\nabla v|^{\frac{p-1}{q}}\Big|^q
+ |H_v|^q |\nabla v|^{p-1} \, d \sigma \end{equation}
for a.e. $t\in(0,\|v\|_{L^\infty(\Omega)})$, where $H_v$ denotes the mean curvature of $M_t$, $d\sigma$ is the area element in $M_t$, $A$ is the constant in \eqref{MS} which depends only on $n$ and $q$, and
$$ q^\star:=\frac{(n-1)q}{n-1-q}. $$
Recall that $V(t)$, being a nonincreasing function, is differentiable almost everywhere and, thanks to the coarea formula and that almost every $t\in(0,\|v\|_{L^\infty(\Omega)})$
is a regular value of $|v|$, we have
\begin{equation*}
- V'(t) = \int_{M_t} \frac{1}{|\nabla v|} \, d\sigma
\qquad\textrm{and}\qquad P(t) = \int_{M_t} d\sigma\qquad\textrm{for a.e. }t\in(0,\|v\|_{L^\infty(\Omega)}). \end{equation*}
Therefore applying Jensen inequality and then using the isoperimetric inequality \eqref{talenti}, we obtain
\begin{equation}\label{new1}
\left( \int_{M_t} |\nabla v|^{(p-1)\frac{q^\star}{q}+1} \,
\frac{d\sigma}{|\nabla v|} \right)^{\frac{q}{q^\star}} \geq \frac{P(t)^{p-1+\frac{q}{q^\star}}}{\left(- V'(t)\right)^{p-1}} \geq \frac{(A_1 V(t)^{\frac{n-1}{n}})^{p-1+\frac{q}{q^\star}}} {\left(- V'(t) \right)^{p-1}} \end{equation}
for a.e. $t\in(0,\|v\|_{L^\infty(\Omega)})$, where $A_1:=n|B_1|^{1/n}$.
Note that for radial functions the inequalities in \eqref{new1} are equalities. Therefore, since the Schwarz symmetrization $v^*$ of $v$
is a radial function and it satisfies \eqref{MSv}, with an equality and with constant $A=|\partial B_1|^{-1/(n-1)}$, we obtain
\begin{equation}\label{new2} \begin{array}{lll}
\displaystyle \left( \int_{\{|v^*|=t\}} |\nabla v^*|^{(p-1)\frac{q^\star}{q}} \, d\sigma \right)^{q/q^\star} &=&\displaystyle
|\partial B_1|^{-\frac{q}{n-1}}\int_{\{v^*=t\}} |H_{v^*}|^q |\nabla v^*|^{p-1} \, d \sigma\\ &=&\displaystyle \frac{(A_1 V(t)^{\frac{n-1}{n}})^{p-1+\frac{q}{q^\star}}} {\left(- V'(t) \right)^{p-1}}. \end{array} \end{equation}
for a.e. $t\in(0,\|v\|_{L^\infty(\Omega)})$. Here, we used that $V(t)=|\{|v|>t\}|=|\{|v^*|>t\}|$ for a.e. $t\in(0,\|v\|_{L^\infty(\Omega)})$.
Therefore, from \eqref{MSv}, \eqref{new1}, and \eqref{new2}, we obtain
$$
|\partial B_1|^{-\frac{q}{n-1}}\int_{\{v^*=t\}} |H_{v^*}|^q |\nabla v^*|^{p-1} \, d \sigma
\leq A^q \int_{M_t} \Big|\nabla_{T,v} |\nabla v|^{\frac{p-1}{q}}\Big|^q
+ |H_v|^q |\nabla v|^{p-1} \, d \sigma, $$
for a.e. $t\in(0,\|v\|_{L^\infty(\Omega)})$. Integrating the previous inequality with respect to $t$ on
\linebreak $(0,\|v\|_{L^\infty(\Omega)})$ and using the coarea formula we obtain inequality \eqref{comp_integrals}, with the explicit constant
$C=A^\frac{q}{p} |\partial B_1|^\frac{q}{(n-1)p}$, proving the result. \end{proof}
\begin{rem} We obtained the explicit admissible constant
$C=A^\frac{q}{p} |\partial B_1|^\frac{q}{(n-1)p}$ in \eqref{comp_integrals}, where $A$ is the universal constant appearing in \eqref{MS}. \end{rem}
We prove Theorem \ref{Theorem:Sobolev} using Theorem~\ref{thm:Ipq} and known results on one dimensional weighted Sobolev inequalities.
\begin{proof}[Proof of Theorem~{\rm\ref{Theorem:Sobolev}}]
Let $v\in C_0^\infty(\overline{\Omega})$ and $v^*$ its Schwarz symmetrization. Recall that $v^*$ is defined in $B_R$ with $R=(|\Omega|/|B_1|)^{1/n}$.
(a) Assume $1+q<n<p+q$. Using H\"older inequality we obtain
\begin{equation}\label{pointwise} \begin{array}{lll}
v^*(s)&=&\displaystyle \int_s^R |(v^*)'(\tau)|\ d\tau\\
&\leq&\displaystyle \left(\int_0^R |(v^*)'(\tau)|^p\tau^{-q}\tau^{n-1}\ d\tau\right)^{1/p} \left(\int_s^R \tau^\frac{1+q-n}{p-1}\ d\tau\right)^{1/p'} \end{array} \end{equation}
for a.e. $s\in(0,R)$. In particular,
$$
v^*(s)\leq |\partial B_1|^{-1/p} \left(\frac{p-1}{p+q-n}\right)^{1/p'} \left(\frac{|\Omega|}{|B_1|}\right)^\frac{p+q-n}{np}I_{p,q}(v^*;B_R) $$
for a.e. $s\in(0,R)$. We conclude this case, by Theorem \ref{thm:Ipq}, noting that $\|v\|_{L^\infty(\Omega)}=v^*(0)$.
(b) Assume $n>p+q$. We use the following 1-dimensional weighted Sobolev inequality:
\begin{equation}\label{radial:Sob}
\left(\int_0^R|\varphi(s)|^{p_q^\star} s^{n-1}\ ds\right)^{1/p_q^\star}
\leq C(n,p,q)\left(\int_0^R s^{-q}|\varphi'(s)|^p s^{n-1}\ ds\right)^{1/p} \end{equation}
with optimal constant
\begin{equation}\label{ctant:C} C(n,p,q):=\left(\frac{p-1}{n-(p+q)}\right)^{1/p'}n^{-1/p_q^\star} \left[\frac{\Gamma\left(\frac{np}{p+q}\right)} {\Gamma\left(\frac{n}{p+q}\right)\Gamma\left(1+\frac{n(p-1)}{p+q}\right)}\right]^\frac{p+q}{np} \end{equation}
stated in \cite{Trudinger97}. Applying inequality \eqref{radial:Sob} to $\varphi=v^*$ and noting that the $L^{p_q^\star}$-norm is preserved by Schwarz symmetrization, we obtain
$$
|\partial B_1|^{-1/p_q^\star}\left(\int_{\Omega}|v|^{p_q^\star}\ dx\right)^{1/p_q^\star}
\leq C(n,p,q)|\partial B_1|^{-1/p}\left(\int_{B_R} |x|^{-q}|\nabla v^*|^p \ dx\right)^{1/p}. $$
Using Theorem \ref{thm:Ipq} again we prove \eqref{Sobolev} for $r=p_q^\star$. The remaining cases, $1\leq r<p_q^\star$, now follow easily from H\"older inequality.
(c) Assume $n=p+q$. From \eqref{pointwise} and Theorem~\ref{thm:Ipq} we obtain
$$ \begin{array}{lll} v^*(s) &\leq&\displaystyle
\left(\int_0^R |(v^*)'(\tau)|^p\tau^{-q}\tau^{n-1}\ d\tau\right)^{1/p} \left(\int_s^R \tau^{-1}\ d\tau\right)^{1/p'}\\ &\leq&\displaystyle
|\partial B_1|^{-1/p} C I_{p,q}(v;\Omega) \left(\ln\left(\frac{R}{s}\right)\right)^{1/p'} \end{array} $$
for a.e. $s\in(0,R)$. Equivalently $$
\exp\left\{\left(\frac{v^*(s)}{|\partial B_1|^{-1/p}C I_{p,q}(v;\Omega)}\right)^{p'}\right\}
|\partial B_1|s^{n-1} \leq
\frac{R}{s}|\partial B_1|s^{n-1} $$ for a.e. $s\in(0,R)$. Integrating the previous inequality with respect to $s$ in $(0,R)$ we obtain
$$
\int_{B_R}\exp\left\{\left(\frac{v^*}{|\partial B_1|^{-1/p}C I_{p,q}(v;\Omega)}\right)^{p'}\right\}\ dx \leq
|\partial B_1|\frac{R^n}{n-1}=\frac{n}{n-1}|\Omega|. $$
We conclude the proof noting that the integral in inequality \eqref{Moser-Trudinger} is preserved under Schwarz symmetrization. \end{proof}
\begin{rem}\label{rmk:ctans} Note that we obtained explicit admissible constants $C_1$, $C_2$, and $C_3$ in inequalities of Theorem \ref{Theorem:Sobolev}. More precisely, we obtained
$$
C_1=|\partial B_1|^{-\frac{1}{p}} \left(\frac{p-1}{p+q-n}\right)^{\frac{1}{p'}}
\left(\frac{|\Omega|}{|B_1|}\right)^\frac{p+q-n}{np}
A^\frac{q}{p} |\partial B_1|^\frac{q}{(n-1)p}, $$
$$
C_2= C(n,p,q)|\partial B_1|^{\frac{1}{p_q^\star}-\frac{1}{p}}
A^\frac{q}{p} |\partial B_1|^\frac{q}{(n-1)p}, $$
and
$$
C_3=|\partial B_1|^{-\frac{1}{p}} A^{\frac{n-p}{p}}|\partial B_1|^{\frac{n-p}{(n-1)p}}, $$
where $A$ is the universal constant appearing in \eqref{MS} and $C(n,p,q)$ is defined in \eqref{ctant:C}.
All the constants $C_i$ depend only on $n$, $p$, and $q$. However, the best constant $A$ in \eqref{MS} is unknown (even for mean convex hypersurfaces). Behind this Sobolev inequality there is the following geometric isoperimetric inequality
\begin{equation}\label{isop:mean}
|M|^{\frac{n-2}{n-1}} \leq A_2\int_M |H(x)|\ d\sigma. \end{equation}
Here, $M\subset \mathbb{R}^{n}$ is a $C^\infty$ immersed $(n-1)$-dimensional compact hypersurface without boundary and $H$ is the mean curvature of $M$ as in Theorem \ref{ThmMS}. The best constant in \eqref{isop:mean} is also unknown even for mean convex hypersurfaces. \end{rem}
\section{Properties of solutions of $p$-Laplace equations}\label{section3}
In this section, we first establish an \textit{a priori} $L^\infty$ estimate in a neighborhood of the boundary $\partial\Omega$ for any regular solution $u$ of \eqref{problem} when the domain $\Omega$ is stricly convex. More precisely, we prove that there exists positive constants $\varepsilon$ and $\gamma$, depending only on the domain $\Omega$, such that
\begin{equation}\label{eqqq} \Vert u\Vert_{L^\infty(\Omega_\varepsilon)}\leq \frac{1}{\gamma} \Vert u\Vert_{L^1 (\Omega)}, \ \ \text{ where } \Omega_\varepsilon:=\{x\in\Omega\, :\, \text{\rm dist}(x,\partial\Omega) <\varepsilon\}. \end{equation}
Then, we establish that the geometric inequalities of Theorem~\ref{Theorem:Sobolev} still hold for solutions of \eqref{problem} in the smaller range $1\leq q\leq 2$. In the next section, these two ingredients will allow us to obtain \textit{a priori} estimates for semi-stable solutions.
Let $u \in W_{0}^{1,p}(\Omega)$ be a weak solution (\textit{i.e.}, a solution in the distributional sense) of the problem
\begin{equation}\label{prob} \left\{ \begin{array}{rcll} -\Delta_p u &=& g(u) &\textrm{in } \Omega, \\
u&>& 0 &\textrm{in } \Omega, \\
u &=& 0 &\textrm{on } \partial \Omega, \end{array} \right. \end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, with $n \geq 2$, and $g$ is any positive smooth nonlinearity.
We say that $u \in W_{0}^{1,p}(\Omega)$ is a \textit{regular solution} of \eqref{prob} if it satisfies the equation in the distributional sense and $g(u)\in L^\infty(\Omega)$. By well known regularity results for degenerate elliptic equations, one has that every regular solution $u$ belongs to $C^{1,\alpha} (\Omega)$ for some $\alpha\in(0,1]$ (see \cite{DB,T}). Moreover, $u\in C^1(\overline{\Omega})$ (see \cite{Lie}). This is the best regularity that one can hope for solutions of $p$-Laplace equations. Therefore, equation \eqref{prob} is always meant in a distributional sense.
We prove the boundary \textit{a priori} estimate \eqref{eqqq} through a moving plane procedure for the $p$-Laplacian which is developed in \cite{DS}.
\begin{proposition}\label{Prop:1} Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{n}$ and $g$ any positive smooth function. Let $u$ be any positive regular solution of \eqref{prob}.
If $\Omega$ is strictly convex, then there exist positive constants $\varepsilon$ and $\gamma$ depending only on the domain $\Omega$ such that for every $x\in\Omega$ with $\text{\rm dist}(x,\partial\Omega)<\varepsilon$, there exists a set $I_x\subset\Omega$ with the following properties:
$$
|I_x|\geq\gamma \qquad\text{and}\qquad u(x) \leq u(y) \ \text{ for all } y\in I_x. $$
As a consequence, \begin{equation}\label{L1boundary} \Vert u\Vert_{L^\infty(\Omega_\varepsilon)}\leq \frac{1}{\gamma} \Vert u\Vert_{L^1 (\Omega)}, \ \ \text{ where } \Omega_\varepsilon:=\{x\in\Omega\, :\, \text{\rm dist}(x,\partial\Omega) <\varepsilon\}. \end{equation} \end{proposition}
\begin{proof} First let us observe that from the regularity of the solution $u$ up to the boundary $\partial \Omega$ and the fact that $\Delta_p u \leq 0$, we can apply the generalized Hopf boundary lemma \cite{V} to see that the normal derivative $\frac{\partial u}{\partial \nu} < 0$ on $\partial \Omega$. Thus, if we let $Z_u := \{ x \in \Omega : \nabla u (x) = 0 \}$ be the critical set of $u$, we have that $Z_u \cap \partial \Omega = \emptyset$. By the compactness of both sets, there exists $\varepsilon_0 > 0$ such that $Z_u \cap \Omega_\varepsilon = \emptyset$ for any $\varepsilon \leq \varepsilon_0$.
We will now prove that this neighborhood of the boundary is in fact independent of the solution $u$. In order to begin a moving plane argument we need some notations: let $e \in S^{n-1}$ be any direction and for $\lambda \in \mathbb{R}$ let us consider the hyperplane
$$ T = T_{\lambda,e} = \{ x \in \mathbb{R}^n : x \cdot e = \lambda \} $$
and the corresponding cap
$$ \Sigma = \Sigma_{\lambda,e} = \{ x \in \Omega : x \cdot e < \lambda \}. $$
Set
$$ a(e) = \inf_{x \in \Omega} x \cdot e $$
and for any $x \in \Omega$, let $x' = x_{\lambda,e}$ be its reflection with respect to the hyperplane $T$, \textit{i.e.},
$$ x' = x + (\lambda - 2 x \cdot e)\ e. $$
For any $\lambda > a (e)$ the cap $$\Sigma' = \{ x \in \Omega : x' \in \Sigma \}$$ is the (non-empty) reflected cap of $\Sigma$ with respect to $T$.
Furthermore, consider the function $v(x) = u (x') = u (x_{\lambda,e})$, which is just the reflected of $u$ with respect to the same hyperplane. By the boundedness of $\Omega$, for $\lambda - a(e)$ small, we have that the corresponding reflected cap $\Sigma'$ is contained in $\Omega$. Moreover, by the strict convexity of $\Omega$, there exists $\lambda_0 = \lambda_0 (\Omega)$ (independent of $e$) such that $\Sigma'$ remains in $\Omega$ for any $\lambda \leq \lambda_0$.
Let us then compare the function $u$ and its reflection $v$ for such values of $\lambda$ in the cap $\Sigma$. First of all, both functions solve the same equation since $\Delta_p$ is invariant under reflection; secondly, on the hyperplane $T$ the functions coincide, whereas for any $x \in \partial \Sigma \cap \partial \Omega$ we have that $u (x) = 0$ and that $v(x) = u (x') > 0$, since the reflection $x' \in \Omega$. Hence we can see that:
\begin{equation*} \Delta_p (u) + f (u) = \Delta_p (v) + f (v) \text{ in } \Sigma, \quad u \leq v \text{ on } \partial \Sigma. \end{equation*}
Again by the boundedness of $\Omega$, if $\lambda - a(e)$ is small, the measure of the cap $\Sigma$ will be small. Therefore, from the Comparison Principle in small domains (see \cite{DS}) we have that $u \leq v \text{ in } \Sigma$. Moreover, by Strong Comparison Principle and Hopf Lemma, we see that $u \leq v \text{ in } \Sigma_{\lambda,e}$ for any $a(e) < \lambda \leq \lambda_0$. In particular, this spells that $u(x)$ is nondecreasing in the $e$ direction for all $x \in \Sigma$.
Now, fix $x_0 \in \partial \Omega$ and let $e=\nu (x_0)$ be the unit normal to $\partial \Omega$ at $x_0$. By the convexity assumption $T_{a(\nu(x_0)),\nu(x_0)} \cap \partial\Omega = \{ x_0 \}$. If we let $\theta \in S^{n-1}$ be another direction close to the outer normal $\nu (x_0)$, the reflection of the caps $\Sigma_{\lambda,\theta}$ with respect to the hyperplane $T_{\lambda,\theta}$ (which is close to the tangent one) would still be contained in $\Omega$ thanks to its strict convexity. So the above argument could be applied also to the new direction $\theta$. In particular, we see that we can get a neighborhood $\Theta$ of $\nu (x_0)$ in $S^{n-1}$ such that $u (x)$ is nondecreasing in every direction $\theta \in \Theta$ and for any $x$ such that $x \cdot \theta < \frac{\lambda_0}{2}$.
By eventually taking a smaller neighborhood $\Theta$, we may assume that $$|x \cdot (\theta - \nu (x_0))| < \lambda_0 /8$$ for any $x \in \Sigma_{\lambda_0,\theta}$ and $\theta \in \Theta$. Moreover, noticing that $$x \cdot \theta = x \cdot (\theta - \nu (x_0)) + x \cdot \nu (x_0)$$ and $$\frac{\lambda_0}{2} = \frac{\lambda_0}{8} + \frac{3 \lambda_0}{8} > x \cdot \theta > \frac{\lambda_0}{8} - \frac{\lambda_0}{8} = 0$$ it is then easy to see that $u$ is nondecreasing in any direction $\theta \in \Theta$ on $\Sigma_0 = \{ x \in \Omega : \frac{\lambda_0}{8} < x \cdot \nu (x_0) <\frac{3\lambda_0}{8} \}$.
Finally, let us choose $\varepsilon = \frac{\lambda_0}{8}$. Fix any point $x \in \Omega_\varepsilon$ and let $x_0$ be its projection onto $\partial \Omega$. {F}rom the above arguments we see that $$u(x) \leq u( x_0 - \varepsilon \nu (x_0)) \leq u(y)$$ for any $y \in I_x$, where $I_x \subset \Sigma_0$ is a truncated cone with vertex at $x_1$, opening angle $\Theta$, and height $\frac{\lambda_0}{4}$. Hence, we have obtained that there exists a positive constant $\gamma = \gamma
(\Omega, \varepsilon)$ such that $|I_x| \geq \gamma$ and $u(x) \leq u(y)$ for any $y \in I_x$. Finally, choosing $x_\varepsilon$ as the maximum of $u$ in $\Omega_\varepsilon$, we get
\begin{equation*} \Vert u \Vert_{L^\infty(\Omega_\varepsilon)} = u_\varepsilon (x_\varepsilon) \leq \frac{1}{\gamma} \int_{I_{x_{\varepsilon}}} u (y) \, dy \leq \frac{1}{\gamma} \Vert u\Vert_{L^1 (\Omega)} \end{equation*}
which proves \eqref{L1boundary}. \end{proof}
We will now prove that inequalities in Theorem \ref{Theorem:Sobolev} are also valid for a positive solution $u$ of \eqref{prob} in the smaller range $1\leq q \leq 2$. To do this, we will construct an approximation of $u$ through smooth functions and see that, thanks to strong uniform estimates on this approximation, we can pass to the limit in all of the inequalities.
\begin{proposition}\label{Prop:2} Let $\Omega$ be a smooth bounded domain of $\mathbb{R}^{n}$ and $g$ any positive smooth function. Let $u$ be any positive regular solution of \eqref{prob}. If $1\leq q \leq 2$, then inequalities in Theorem {\rm\ref{Theorem:Sobolev}} hold for $v=u$. Given $s>0$, the same holds true also for $v = u-s$ and $\Omega$ replaced by $\Omega_s := \{ x\in \Omega : u > s\}$. \end{proposition}
\begin{proof} Let $Z_u=\{x\in\Omega : \nabla u(x)=0\}$. Recall that by standard elliptic regularity $u \in C^\infty (\Omega
\setminus Z_u)$ and that $|Z_u| = 0$ by \cite{DS}. Therefore, $u$ is smooth almost everywhere in $\Omega$. Let $x \in \Omega \setminus Z_u$ and observe that for the mean curvature $H_u$ of the level set passing through $x$ we have the following explicit expression
\begin{equation}\label{exp1}
-(n-1) H_u = {\rm div} \left(\frac{\nabla u}{|\nabla u|}\right) =
\frac{\Delta u}{|\nabla u|} - \frac{\langle D^2 u \nabla u, \nabla u
\rangle }{|\nabla u|^3} \end{equation}
whereas for the tangential gradient term we have
\begin{equation}\label{exp2}
\nabla_{T,u} |\nabla u| = \frac{D^2 u \nabla u}{|\nabla u|} -
\frac{\langle D^2 u \nabla u, \nabla u \rangle \nabla u }{|\nabla u|^3}, \end{equation}
where all the terms in these expressions are evaluated at $x$. Hence, there exists a positive constant $C = C (n,p,q)$ such that
\begin{equation}\label{hsdequadro}
\left(\frac{1}{p'}|\nabla_{T,u} |\nabla u|^{\frac{p}{q}}|\right)^{q} +
|H_u|^q |\nabla u|^{p} \leq C |D^2 u|^{q} |\nabla u|^{p - q}\quad \textrm{for a.e. }x\in\Omega. \end{equation}
{F}rom \cite{DS} we recall the following important estimate: for any $1 \leq q \leq 2$ there holds
\begin{equation}\label{sciunzi}
\int_\Omega |D^2 u|^{q} |\nabla u|^{p - q} \, dx < \infty. \end{equation}
Thanks to \eqref{hsdequadro} and \eqref{sciunzi}, all of the integrals in the geometric Hardy-Sobolev inequalities are well defined for any $1 \leq q \leq 2$.
However, since the solution $u$ is not smooth around $Z_u$, we need to regularize $u$ in a neighborhood of the critical set in order to apply the inequalities of Theorem {\rm\ref{Theorem:Sobolev}}. We will now describe an approximation argument due to Canino, Le, and Sciunzi~\cite{CLS} for the $p(\cdot)$-Laplacian (in our case $p(x) \equiv p$ constant).
\begin{lem}[\cite{CLS}] Let $D\subset \Omega$ be an open set, $1\leq q\leq 2$, and $\varepsilon\in(0,1)$. Let $u\in C^1(\overline{\Omega})$ be a solution of \eqref{problem} and $h:=g(u)$. If $h_\varepsilon\in C^\infty(\overline{D})$ is any sequence converging to $h$ in $C^1(\overline{D})$ as $\varepsilon\downarrow 0$, then the unique solution $v_\varepsilon$ of the following regularized problem
\begin{equation}\label{aprox:problem} \left\{ \begin{array}{rcll}
-{\rm div} \left( (\varepsilon^2 + |\nabla v_\varepsilon|^2)^{\frac{p-2}{2}} \nabla v_\varepsilon \right) &=& h_\varepsilon(x) &\textrm{in } D, \\
v_\varepsilon &=& u &\textrm{on } \partial D. \end{array} \right. \end{equation}
tends to $u$ strongly in $W^{1,p}(B)$. Moreover, there exists a constant $C$ independent of $\varepsilon$ such that
$$
\int_D |D^2 v_\varepsilon|^{q} ( \varepsilon^2 + |\nabla v_\varepsilon|^2)^{\frac{p-q}{2}} \, dx \leq C $$
and
\begin{equation}\label{limve}
\lim_{\varepsilon \to 0} \int_D |D^2 v_\varepsilon|^{q} (
\varepsilon^2 + |\nabla v_\varepsilon|^2)^{\frac{p-q}{2}} \, dx =
\int_D |D^2 u|^{q} |\nabla u|^{p - q} \, dx. \end{equation}
\end{lem}
Let $v_\varepsilon\in C^{\infty} (D)$ be the unique solution of \eqref{aprox:problem} and let us consider a smooth cut-off function $\eta$ with compact support contained in $\Omega$ and such that $\eta \equiv 1$ on $D$. We can construct a smooth regularization $u_\varepsilon$ of $u$ defining $u_\varepsilon := (1-\eta) u + \eta v_\varepsilon$. We can then apply Theorem \ref{Theorem:Sobolev} to any $u_\varepsilon$ to get the appropriate inequality $(a)$, $(b)$, or $(c)$. From \cite{DB,Lie} and standard elliptic regularity we know that the regularization $u_\varepsilon$ will converge to $u$, as $\varepsilon \downarrow 0$, both in $C^1 (\overline{\Omega})$ and $C^2(\overline{\Omega} \setminus Z_u)$. Hence we can easily pass to the limit as $\varepsilon \downarrow 0$ in the left hand side of \eqref{Morrey} and \eqref{Sobolev}.
In order to see that also the remaining terms $I_{p,q} (u_\varepsilon; \Omega)$ which involve tangential gradient and mean curvature behave well under this approximation the argument is the following. Splitting the domain $\Omega$ and recalling that $u_\varepsilon \equiv v_\varepsilon$ in $D$ we have that: $$I_{p,q} (u_\varepsilon;\Omega) = I_{p,q} (u_\varepsilon; D) + I_{p,q} (u_\varepsilon; \Omega \setminus D) = I_{p,q} (v_\varepsilon; D) + I_{p,q} (u_\varepsilon; \Omega \setminus D).$$ Clearly, from the $C^2$ convergence we have that $I_{p,q} (u_\varepsilon; \Omega \setminus D) \to I_{p,q} (u; \Omega \setminus D)$ as $\varepsilon \downarrow 0$. Therefore we can concentrate on the convergence of $I_{p,q}(v_\varepsilon;D)$.
{F}rom \eqref{exp1}, \eqref{exp2}, and through a simple expansion of
$(\varepsilon^2 + |\nabla v_\varepsilon|^2)^{\frac{p - q}{2}}$ around $\varepsilon =0$, we see that for a sufficiently small $\varepsilon_0 > 0$ there exists a constant $K = K (n,p,q, \varepsilon_0) > 0$ such that for any $\varepsilon \leq \varepsilon_0$ we have
\begin{equation}\label{hsdequadrove}
\left(\frac{1}{p'} |\nabla_{T,v_\varepsilon}|\nabla v_\varepsilon|^{\frac{p}{q}}|
\right)^{q} + |H_{v_\varepsilon}|^q |\nabla v_\varepsilon|^{p} \leq K \, |D^2 v_\varepsilon|^{q} (\varepsilon^2 + |\nabla v_\varepsilon|^2)^{\frac{p - q}{2}}. \end{equation}
Moreover, by the fact that $v_\varepsilon \to u$ in $C^2 (D
\setminus Z_u)$ and $|Z_u|=0$, almost everywhere in $D$ we have
\begin{equation}\label{aeve}
\lim_{\varepsilon \to 0}\left(\frac{1}{p'} |\nabla_{T,v_\varepsilon} |\nabla v_\varepsilon|^{\frac{p}{q}}| \right)^{q} + |H_{v_\varepsilon}|^q
|\nabla v_\varepsilon|^{p} = \left(\frac{1}{p'} |\nabla_{T,u} |\nabla u|^{\frac{p}{q}}| \right)^{q} + |H_{u}|^q |\nabla u|^{p}. \end{equation}
Now, thanks to \eqref{limve}, \eqref{hsdequadrove}, and \eqref{aeve}, by dominated convergence theorem we see that:
$$
\lim_{\varepsilon \to 0} \int_D \left(\frac{1}{p'} |\nabla_{T,v_\varepsilon} |\nabla v_\varepsilon|^{\frac{p}{q}}| \right)^{q} + |H_{v_\varepsilon}|^q
|\nabla v_\varepsilon|^{p} \, dx $$ $$
= \int_D \left(\frac{1}{p'} |\nabla_{T,u} |\nabla u|^{\frac{p}{q}}| \right)^{q} + |H_{u}|^q |\nabla u|^{p} \, dx. $$
Thus, the assertions of Theorem~\ref{Theorem:Sobolev} hold for $v=u$.
To conclude the proof let us fix any $s >0$ and consider $v = u-s$ on $\Omega_s = \{x\in\Omega : u > s \}$. It is clear that the integrands in the inequalities remain unchanged in this case, so the only problem comes from the fact $\Omega_s$ might not be smooth. If this is the case, let us consider two sequences $\varepsilon_n \to 0$ and $s_n \to s$, with the corresponding regularizations of $v$ given by $v_n := v_{\varepsilon_n} = u_{\varepsilon_n} - s_n$. Thanks to the smoothness of any $v_n$ and Sard Lemma, we can choose each $s_n$ as a regular value of $v_n$, so that the level set $\{ v_n > 0 \} = \{ u_n > s_n \}$ is smooth. Moreover, from the $C^1$ convergence, it is clear that for the characteristic functions we have $\chi_{\{ u_n > s_n \}} \to \chi_{\{ u > s \}}$. Hence we can conclude the proof using the same dominated convergence argument as above.\end{proof}
\section{Regularity of stable solutions. Proof of Theorems \ref{Theorem} and \ref{Theorem2}}\label{section5}
We are now ready to establish $L^r$ and $W^{1,r}$ \textit{a priori} estimates of semi-stable solutions to $p$-Laplace equations proving Theorems~\ref{Theorem}~ and~\ref{Theorem2}.
Before the proof our regularity results let us recall some known facts on the linearized operator associated to \eqref{problem} and semi-stable solutions.
\subsection{Linearized operator and semi-stable solutions}\label{section4}
This subsection deals with the linearized operator at any regular semi-stable solution $u \in C_{0}^{1} (\overline{\Omega})$ of
\begin{equation}\label{prob2} \left\{ \begin{array}{rcll} -\Delta_p u &=& g(u) &\textrm{in } \Omega, \\
u&>& 0 &\textrm{in } \Omega, \\
u &=& 0 &\textrm{on } \partial \Omega, \end{array} \right. \end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, with $n \geq 2$, and $g$ is any positive $C^1$ nonlinearity.
The linearized operator $L_u$ associated to \eqref{prob2} at $u$ is defined by duality as
$$ \begin{array}{l}
L_u(v,\phi):=\displaystyle \hspace{-0.2cm}\int_\Omega |\nabla u|^{p-2}\left\{\nabla v\cdot\nabla\phi
+(p-2)\left(\nabla v\cdot\frac{\nabla u}{|\nabla u|}\right)
\left(\nabla\phi\cdot\frac{\nabla u}{|\nabla u|}\right)\right\} dx\\ \displaystyle \hspace{2cm}- \int_\Omega g'(u)v \phi\ dx \end{array} $$
for all $(v,\phi)\in H_0\times H_0$, where the Hilbert space $H_0$ is defined according to \cite{DS} as follows.
\begin{defin}\label{H0} Let $u \in C_{0}^{1} (\overline{\Omega})$ be a regular semi-stable solution of \eqref{prob2}. We introduce the following weighted $L^2$-norm of the gradient
$$
|\phi|:=\left(\int_\Omega \rho |\nabla \phi|^2\ dx\right)^{1/2}
\quad \textrm{where }\rho:=|\nabla u|^{p-2}. $$ According to \cite{DS}, the space
$$
H^1_\rho(\Omega):=\{\phi \in L^2(\Omega) \hbox{ weakly differentiable}:\:|\phi|<+\infty\} $$
is a Hilbert space and is the completion of $C^\infty(\Omega)$ with respect to the
$|\cdot|$-norm.
We define the Hilbert space $H_0$ of admissible test functions as
$$ H_0:= \left\{ \begin{array}{lll}
\{\phi \in H_0^1(\Omega):\: |\phi|<+\infty \}&\textrm{if}&1<p\leq 2\\ \\ \textrm{the closure of }C_0^\infty(\Omega) \textrm{ in }H^1_\rho(\Omega)&\textrm{if}&p>2. \end{array} \right. $$
\end{defin}
Note that for $1<p\leq 2$, $H_0$ is a subspace of $H_0^1(\Omega)$ and since
$$
\int_\Omega |\nabla \phi|^2 \leq \|\nabla u\|_{L^\infty(\Omega)}^{2-p}|\phi|^2, $$
we see that $(H_0,|\cdot|)$ is a Hilbert space. For $p>2$, the weight $\rho=|\nabla u|^{p-2}$ is in $L^\infty(\Omega)$ and satisfies $\rho^{-1} \in L^1(\Omega)$, as shown in \cite{DS}.
Now, thanks to the above definition, the operator $L_u$ is well defined for $\phi \in H_0$ and, therefore, the semistability of the solution $u$ reads as
\begin{equation}\label{semistab}
L_u (\phi,\phi) = \int_\Omega |\nabla u|^{p-2} \left\{|\nabla \phi|^2
+(p-2)\left(\nabla \phi\cdot\frac{\nabla u}{|\nabla u|}\right)^2\right\} - g'(u) \phi^2\ dx \geq 0, \end{equation}
for every $\phi \in H_0$.
On the one hand, considering $\phi = |\nabla u| \eta$ as a test function in the semistability condition \eqref{semistab} for $u$, we obtain
\begin{equation}\label{StZu}
\int_{\Omega}\left[ (p-1) |\nabla u|^{p-2} |\nabla_{T,u} |\nabla u||^{2}
+ B_u^2 |\nabla u|^{p} \right] \eta^2 \, dx
\leq (p-1) \int_{\Omega} |\nabla u|^{p} |\nabla \eta|^2 \, dx \end{equation}
for any Lipschitz continuous function $\eta$ with compact support. Here, $B_u^2$ denotes the $L^2$-norm of the second fundamental form of the level set of $|u|$ through $x$ (\textit{i.e.}, the sum of the squares of its principal curvatures). The fact that $\phi=\eta
|\nabla u|$ is an admissible test function derives from the estimate \eqref{sciunzi}, whereas the computations behind \eqref{StZu} are done in \cite{FSV} (see Theorem~2.5 \cite{FSV}).
On the other hand, noting that $(n-1) H_u^{2} \leq B_u^2$ and
$$
|\nabla u|^{p-2} |\nabla_{T,u} |\nabla u||^{2} = \frac{4}{p^2} |\nabla_{T,u} |\nabla u|^{\frac{p}{2}}|^{2}, $$
we obtain the key inequality to prove our regularity results for semi-stable solutions
\begin{equation}\label{StZu2}
\int_{\Omega}\left( \frac{4}{p^2}|\nabla_{T,u} |\nabla u|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_u^2 |\nabla u|^{p} \right) \eta^2 \, dx
\leq \int_{\Omega} |\nabla u|^{p} |\nabla \eta|^2 \, dx \end{equation}
for any Lipschitz continuous function $\eta$ with compact support.
\subsection{\textit{A priori} estimates of stable solutions. Proof of Theorem~\ref{Theorem}}\label{subsection5:1}
In order to prove the gradient estimate \eqref{grad:estimate} established in Theorem~{\rm \ref{Theorem}}~(b) we will use the following result. Its proof is based on a technique introduced by B\'enilan \textit{et al.} \cite{BBGGPV95} to obtain the regularity of entropy solutions for $p$-Laplace equations with $L^1$ data.
\begin{proposition}\label{Prop:bootstrap} Assume $n\geq 3$ and $h\in L^1(\Omega)$. Let $u$ be the entropy solution of
\begin{equation}\label{linear} \left\{ \begin{array}{rcll} -\Delta_p u&=&h(x)&\textrm{in }\Omega,\\ u&=&0&\textrm{on }\partial \Omega. \end{array} \right. \end{equation}
Let $r_0\geq (p-1)n/(n-p)$. If $\int_\Omega |u|^{r_0}\ dx<+\infty$, then the following \textit{a priori} estimate holds:
$$
\int_\Omega |\nabla u|^r\ dx
\leq r|\Omega|
+\left(\frac{r_1}{r}-1\right)^{-1}\left(\int_\Omega |u|^{r_0}\ dx+\|h\|_{L^1(\Omega)}\right) $$
for all $r< r_1:=pr_0/(r_0+1)$. \end{proposition}
\begin{rem}
B\'enilan \textit{et al.} \cite{BBGGPV95} proved the existence and uniqueness of entropy solutions to problem \eqref{linear}. Moreover, they proved that $|\nabla u|^{p-1}\in L^r(\Omega)$ for all $1\leq r<n/(n-1)$ and $|u|^{p-1}\in L^r(\Omega)$ for all $1\leq r<n/(n-p)$. Proposition~\ref{Prop:bootstrap} establishes an improvement of the previous gradient estimate knowing an \textit{a priori} estimate of
$\int_\Omega |u|^{r_0} dx$ for some $r_0>(p-1)n/(n-p)$. \end{rem}
\begin{proof}[Proof of Proposition {\rm\ref{Prop:bootstrap}}] Multiplying \eqref{linear} by $T_s u=\max\{-s,$ $\min\{s,u\}\}$ we obtain
$$
\int_{\{|u|\leq s\}}|\nabla u|^p\ dx=\int_\Omega h(x)T_su\ dx\leq s\|h\|_{L^1(\Omega)}. $$
Let $t=s^{(r_0+1)/p}$. {F}rom the previous inequality, recalling that
$V(s)=|\{x\in\Omega:|u|>s\}|$, we deduce
$$ \begin{array}{lll}
\displaystyle s^{r_0}|\{|\nabla u|>t\}|&\leq&\displaystyle s^{r_0}\int_{\{|\nabla u|>t\}\cap\{|u|\leq s\}}\left(\frac{|\nabla u|}{t}\right)^p dx
+s^{r_0}\int_{\{|u|>s\}}\ dx\\ \\
&\leq &\displaystyle\|h\|_{L^1(\Omega)} +s^{r_0}V(s)\quad\textrm{for a.e. }s>0. \end{array} $$
In particular
\begin{equation}\label{lalarito}
t^{\frac{pr_0}{r_0+1}}|\{|\nabla u|>t\}| \leq
\|h\|_{L^1(\Omega)}+\sup_{\tau>0}\Big\{\tau^{r_0}V(\tau)\Big\}\ \quad\textrm{for a.e. }t>0. \end{equation}
Moreover, since
$$ \tau^{r_0}V(\tau) \leq
\tau^{r_0}\int_{\{|u|>\tau\}}\left(\frac{|u|}{\tau}\right)^{r_0}\ dx
\leq\int_\Omega|u|^{r_0}\ dx \quad\textrm{for a.e. }\tau>0, $$
we have $\sup_{\tau>0}\Big\{\tau^{r_0}V(\tau)\Big\}\leq \int_\Omega|u|^{r_0}\ dx$.
Let $r< r_1:=pr_0/(r_0+1)$. From \eqref{lalarito} and the previous inequality, we have
$$ \begin{array}{lll}
\displaystyle \int_\Omega |\nabla u|^r\ dx &=&
\displaystyle r\int_0^\infty t^{r-1}|\{|\nabla u|>t\}|\ dt\\ &\leq&
\displaystyle r|\Omega|+r\left(\int_\Omega|u|^{r_0}\ dx +
\|h\|_{L^1(\Omega)}\right) \int_1^\infty t^{r-1}t^{-\frac{p{r_0}}{{r_0}+1}}\ dt \end{array} $$
proving the proposition. \end{proof}
Now, we have all the ingredients to prove the \textit{a priori} estimates established in Theorem~\ref{Theorem} for semi-stable solutions. It will follow from Theorem~\ref{Theorem:Sobolev} and Propositions~\ref{Prop:2} and \ref{Prop:bootstrap} choosing adequate test functions in the semistability condition \eqref{StZu2}.
First, we prove Theorem \ref{Theorem} when $n\neq p+2$. We will take $\eta=T_s u =\min\{s,u\}$ as a test function in \eqref{StZu2} and then, thanks to Proposition~\ref{Prop:2}, we apply our Morrey and Sobolev inequalities \eqref{Morrey} and \eqref{Sobolev} with $q=2$.
\begin{proof}[Proof of Theorem {\rm \ref{Theorem}} for $n\neq p+2$] Assume $n\neq p+2$. Let $u\in C^1_0(\overline{\Omega})$ be a semi-stable solution of \eqref{problem}. By taking $\eta=T_s u=\min\{s,u\}$ in the semistability condition \eqref{StZu2} we obtain
$$
\int_{\{u>s\}}\left( \frac{4}{p^2}|\nabla_{T,u} |\nabla u|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_u^2 |\nabla u|^{p} \right)\, dx
\leq \frac{1}{s^2}\int_{\{u<s\}} |\nabla u|^{p+2}\, dx $$
for a.e. $s>0$. In particular,
$$ \min\left(\frac{4}{(n-1)p},1\right) I_{p,2}(u-s;\{x\in\Omega:u>s\})^p \leq
\frac{p-1}{(n-1)s^2}\int_{\{u<s\}} |\nabla u|^{p+2}\, dx $$
for a.e. $s>0$, where $I_{p,2}$ is the functional defined in \eqref{Ipq} with $q=2$. By Proposition~\ref{Prop:2} we can apply Theorem~\ref{Theorem:Sobolev} with $\Omega$ replaced by $\{x\in\Omega: u > s\}$, $v=u-s$, and $q=2$. Then, the $L^r$ estimates established in parts (a) and (b) follow directly from the Morrey and Sobolev type inequalities \eqref{Morrey} and \eqref{Sobolev}.
Finally, the gradient estimate \eqref{grad:estimate} follows directly from Proposition~\ref{Prop:bootstrap} with $r_0=np/(n-p-2)$. \end{proof}
Now, we deal with the proof of Theorem {\rm \ref{Theorem}} $($a$)$ when $n=p+2$. This critical case follows from Theorem \ref{ThmMS} and the semistability condition \eqref{StZu2} with the test function $\eta=\eta(u)$ defined in \eqref{etaa} and \eqref{psii} below.
\begin{proof}[Proof of Theorem {\rm \ref{Theorem}} when $n=p+2$] Assume $n = p+2$ (and hence, $n>3$). Taking a Lipschitz function $\eta = \eta (u)$ (to be chosen later) in \eqref{StZu} and using the coarea formula we obtain
\begin{equation}\label{semi:n=p+2} \begin{array}{l} C\displaystyle \int_{0}^{\infty} \int_{\{ u = t \}}
\left\{\left|\nabla_{T,u} |\nabla u|^\frac{p-1}{2}\right|^{2}
+ \left|H_u|\nabla u|^\frac{p-1}{2}\right|^2\right\}\ \eta(t)^2 \, d\sigma dt \\ \displaystyle \hspace{3cm} \leq
\int_{0}^{\infty} \int_{\{ u = t \}} |\nabla u|^{p+1}\ \dot{\eta}(t)^2 \, d\sigma dt, \end{array} \end{equation}
where $d\sigma$ denotes the area element in $\{u=t\}$ and $C$, here and in the rest of the proof, is a constant depending only on $p$.
To apply the Sobolev inequality \eqref{MS} in the left hand side of the previous inequality we need to make an approximation argument. Consider the sequence $u_k$ of smooth regularizations of $u$ introduced in the proof of Proposition \ref{Prop:2} and note that $\{u_k=t\}$ is a smooth hypersurface for a.e. $t\geq0$. Then, from the Sobolev inequality \eqref{MS} with $\phi = |\nabla u_k|^{\frac{p-1}{2}}$, $q=2$, and $M=\{u_k=t\}$, and noting that
$$ (p-1) \frac{n-1}{n-3} = p+1\quad \textrm{ when }n = p+2, $$
we obtain
\begin{equation}\label{semi:u_k} \begin{array}{l} \displaystyle C \int_{0}^{\infty} \left(\int_{\{ u_k = t \}}
|\nabla u_k|^{p+1}\right)^\frac{n-3}{n-1} \eta(t)^2 \, d\sigma \ dt \\ \hspace{1.0cm} \leq \displaystyle \int_{0}^{\infty} \int_{\{ u_k = t \}}
\left\{\left|\nabla_{T,u_k} |\nabla u_k|^\frac{p-1}{2}\right|^{2}
+ \left|H_{u_k}|\nabla u_k|^\frac{p-1}{2}\right|^2\right\}\ \eta(t)^2 \, d\sigma dt. \end{array} \end{equation}
Now, we will pass to the limit in the previous inequality. Note that, if $\eta$ is bounded, through a dominated convergence argument as in Proposition \ref{Prop:2} we have
$$ \begin{array}{l} \displaystyle \lim_{k \to \infty} \int_{0}^{\infty} \int_{\{ u_k = t \}}
\left\{\left|\nabla_{T,u_k} |\nabla u_k|^\frac{p-1}{2}\right|^{2}
+ \left|H_{u_k}|\nabla u_k|^\frac{p-1}{2}\right|^2\right\}\ \eta(t)^2 \, d\sigma dt \\ \displaystyle \hspace{0.5cm} =\int_{0}^{\infty} \int_{\{ u = t \}}
\left\{\left|\nabla_{T,u} |\nabla u|^\frac{p-1}{2}\right|^{2}
+ \left|H_u|\nabla u|^\frac{p-1}{2}\right|^2\right\}\ \eta(t)^2 \, d\sigma dt. \end{array} $$ Moreover, from the $C^1$ convergence of $u_k$ to $u$ we obtain
$$ \lim_{k \to \infty} \int_{0}^{\infty} \left(\int_{\{ u_k = t \}}
|\nabla u_k|^{p+1}\right)^\frac{n-3}{n-1} \eta(t)^2\, d\sigma \ dt = \int_{0}^{\infty}\left(\int_{\{ u = t \}}
|\nabla u|^{p+1}\right)^\frac{n-3}{n-1} \eta(t)^2\, d\sigma \ dt. $$
Therefore, taking the limit as $k$ goes to infinity in \eqref{semi:u_k} and using \eqref{semi:n=p+2}, we get
\begin{equation}\label{p+2} C \int_{0}^{\infty} \psi(t)^{\frac{n-3}{n-1}} \, \eta (t)^2 \, dt \leq \int_{0}^{\infty} \psi(t) \, \dot{\eta} (t)^2 \, dt =
\int_{0}^{\infty} \int_{\{ u = t \}} |\nabla u|^{p+1} \, d\sigma\, \dot{\eta} (t)^{2}\, dt, \end{equation}
where \begin{equation}\label{psii}
\psi(t) := \int_{\{ u = t \}} |\nabla u|^{p+1} \, d\sigma. \end{equation}
Now, let $\bar{M}:={\| u \|_{L^\infty(\Omega)}}$. Given $s > 0$, choose
\begin{equation}\label{etaa} \eta(t)=\eta_s (t) :=\left\{ \begin{array}{lll} \displaystyle t/s&\textrm{ if }& 0\leq t \leq s,\\ \displaystyle \exp \left( \frac{1}{\sqrt{2}} \int_{s}^{t} \left(\frac{C \psi(\tau)^{\frac{n-3}{n-1}}}{ \psi(\tau)}\right)^{\frac12} \, d\tau \right)&\text{ if }& s < t \leq \bar{M}\\ \displaystyle \eta(M)&\textrm{ if }& t >\bar{M}. \end{array} \right. \end{equation}
It is then clear that
\begin{equation*}
\int_{0}^{\bar{M}} \int_{\{ u = t \}} |\nabla u|^{p+1} \, d\sigma \, \dot{\eta}_{s} (t)^{2}\, dt =
\frac{1}{s^2} \int_{\{ u \leq s \}} |\nabla u|^{p+2} \, dx + \frac{C}{2} \int_{s}^{\bar{M}} \psi(t)^{\frac{n-3}{n-1}} \, \eta_{s} (t)^{2} \, dt. \end{equation*}
Therefore, from \eqref{p+2} we obtain
\begin{equation}\label{p+2:bis} \frac{C}{2} \int_{s}^{\bar{M}} \psi(t)^{\frac{n-3}{n-1}} \, \eta_{s} (t)^{2} \, dt \leq
\frac{1}{s^2} \int_{\{ u \leq s \}} |\nabla u|^{p+2} \, dx. \end{equation}
Let us choose $\alpha = \frac{2}{n-2}$, $\beta = \frac{n-3}{(n-2)(n-1)}$, and $m = n-2$. Note that $\alpha,\beta>0$, $m>1$, and $\beta m'=1/(n-1)$. Moreover, using the definition of $\eta_s$ we have
\begin{equation}\label{asdf} \frac{1}{\psi(t)^{\beta m'} \eta_{s}(t)^{\alpha m'}} = \sqrt{\frac{2}{C}}\frac{\dot{\eta}_{s}(t)}{\eta_{s}(t)^{\alpha m' + 1}} \end{equation}
for all $t>s$. By \eqref{asdf}, H\"older inequality, and \eqref{p+2:bis}, we see that
$$ \begin{array}{lll} \displaystyle \bar{M} - s &=&\displaystyle \int_{s}^{\bar{M}} \frac{\psi(t)^{\beta}\eta_{s}(t)^{\alpha}}{\psi(t)^{\beta}\eta_{s}(t)^{\alpha}} \, dt\\ &\leq&\displaystyle \left( \int_{s}^{\bar{M}} \psi(t)^{\beta m}\eta_{s}(t)^{\alpha m} \, dt \right)^{\frac{1}{m}} \left( \int_{s}^{\bar{M}} \frac{dt}{\psi(t)^{\beta m'}\eta_{s}(t)^{\alpha m'}} \right)^{\frac{1}{m'}}\\ &\leq&\displaystyle \left( \int_{s}^{\bar{M}} \psi(t)^{\frac{n-3}{n-1}}\eta_{s}(t)^{2} \, dt \right)^{\frac{1}{n-2}} \left( \sqrt{\frac{2}{C}}\int_{s}^{\bar{M}} \frac{\dot{\eta}_{s} (t)} {\eta_{s}(t)^{m'\alpha + 1}} \, dt \right)^{\frac{n-3}{n-2}}\\ &\leq&\displaystyle
\left( \frac{2}{Cs^2} \int_{\{ u\leq s \}} |\nabla u|^{p+2} \, dx \right)^{\frac{1}{n-2}} \left(\sqrt{\frac{2}{C}}\frac{n-3}{2}\right)^{\frac{n-3}{n-2}} \end{array} $$ which is exactly \eqref{L-infinty} (note that $n-2=p$ and $\eta(\bar{M})\geq 1$). \end{proof}
\subsection{Regularity of the extremal solution. Proof of Theorem~\ref{Theorem2}}\label{subsection5:2}
In this subsection we will prove the \textit{a priori} estimates for minimal and extremal solutions of $(1.15)_{\lambda}$ stated in Theorem~\ref{Theorem2}. Let us remark that in the proof of Theorem~\ref{Theorem2} we will assume the nonlinearity $f$ to be smooth. However, if it is only $C^1$ we can proceed with an approximation argument as in the proof of Theorem 1.2 in \cite{Cabre09}.
The $W^{1,p}$-estimate established in Theorem~\ref{Theorem2} has as main ingredient the following result.
\begin{lem}\label{lemma:Poho} Let $f$ be an increasing positive $C^1$ function satisfying \eqref{p-superlinear} and $\lambda\in(0,\lambda^\star)$. Let $u=u_\lambda\in C^1_0(\overline{\Omega})$ be the minimal solution of $(1.15)_{\lambda}$. The following inequality holds:
\begin{equation}\label{Pohozaev}
\int_\Omega|\nabla u|^p\ dx \leq
\left(\max_{x\in\overline{\Omega}}|x|\right)
\frac{1}{p'}\int_{\partial\Omega}|\nabla u|^p\ d\sigma. \end{equation}
\end{lem}
\begin{proof} Let $G'(t)=g(t)=\lambda f(t)$. First, we note that
$$ x\cdot\nabla u\ g(u) =x\cdot\nabla G(u)={\rm div}\Big(G(u)x\Big)-nG(u) $$
and that almost everywhere on $\Omega$ we can evaluate
$$ \begin{array}{lll} \displaystyle x\cdot\nabla u\ \Delta_p u
-{\rm div}\Big(x\cdot\nabla u\ |\nabla u|^{p-2}\nabla u\Big)
&=&\displaystyle
-|\nabla u|^{p-2}\nabla u\cdot\nabla(x\cdot\nabla u)\\ &=& \displaystyle
-|\nabla u|^{p}-\frac{1}{p}\nabla|\nabla u|^p\cdot x\\ &=& \displaystyle
\frac{n-p}{p}|\nabla u|^{p}-\frac{1}{p}{\rm div} \Big(|\nabla u|^p x\Big). \end{array} $$ As a consequence, multiplying $(1.15)_{\lambda}$ by $x\cdot\nabla u$ and integrating on $\Omega$, we have
\begin{equation}\label{Pohozaev0}
n\int_\Omega G(u)\ dx-\frac{n-p}{p}\int_\Omega |\nabla u|^p\ dx=
\frac{1}{p'}\int_{\partial\Omega}|\nabla u|^p\ x\cdot\nu\ d\sigma, \end{equation}
where $\nu$ is the outward unit normal to $\Omega$.
Noting that $u$ is an absolute minimizer of the energy functional
$$
J(u)=\frac{1}{p}\int_\Omega|\nabla u|^p\ dx-\int_\Omega G(u)\ dx $$
in the convex set $\{v\in W^{1,p}_0(\Omega):0\leq v\leq u\}$ (see \cite{CS07}), we have that $J(u)\leq J(0)=0$. Therefore, from \eqref{Pohozaev0} we obtain
$$
\int_\Omega|\nabla u|^p\ dx = n J(u)
+\frac{1}{p'}\int_{\partial\Omega}|\nabla u|^p\ x\cdot\nu\ d\sigma \leq
\left(\max_{x\in\overline{\Omega}}|x|\right)
\frac{1}{p'}\int_{\partial\Omega}|\nabla u|^p\ d\sigma $$
proving the lemma. \end{proof}
Finally, we prove Theorem~\ref{Theorem2} (using the semistability condition \eqref{StZu2} with an appropriate test function), Theorem~\ref{Theorem:Sobolev}, and Lemma~\ref{lemma:Poho}.
\begin{proof}[Proof of Theorem {\rm \ref{Theorem2}}] Let $u_\lambda$ be the minimal solution of $(1.15)_{\lambda}$ for $\lambda\in(0,\lambda^\star)$. {F}rom \cite{CS07} we know that minimal solutions are semi-stable. In particular, $u_\lambda$ satisfies the semistability condition \eqref{StZu2} for all $\lambda\in(0,\lambda^\star)$.
Assume that $\Omega$ is strictly convex. Let $\delta(x) := {\rm dist}(x,\partial\Omega)$ be the distance to the boundary and $\Omega_\varepsilon:=\{x\in\Omega:\delta (x)<\varepsilon\}$. By Proposition~\ref{Prop:1} there exist positive constants $\varepsilon$ and $\gamma$ such that for every $x_0\in \Omega_\varepsilon$ there exists a set $I_{x_0}\subset \Omega$
satisfying $|I_{x_0}|>\gamma$ and
\begin{equation}\label{kkkkey} u_\lambda(x_0)^{p-1} \leq u_\lambda(y)^{p-1}\quad\textrm{for all }y\in I_{x_0}. \end{equation}
Let $x_\varepsilon\in\overline{\Omega}_\varepsilon$ be such that
$u_\lambda(x_\varepsilon)=\|u_\lambda\|_{L^\infty(\Omega_\varepsilon)}$. Integrating with respect to $y$ in $I_{x_\varepsilon}$ inequality \eqref{kkkkey} and using \eqref{p-superlinear}, we obtain
\begin{equation}\label{kkkkeyyyyy}
\|u_\lambda\|_{L^\infty(\Omega_\varepsilon)}^{p-1} \leq \frac{1}{\gamma}\int_{I_{x_\varepsilon}}u_\lambda^{p-1}\ dy \leq \frac{1}{\gamma}\int_{\Omega}u_\lambda^{p-1}\ dy
\leq \frac{C}{\gamma}\|f(u_\lambda)\|_{L^1(\Omega)}, \end{equation} where $C$, here and in the rest of the proof, is a constant independent of $\lambda$. Letting
$s=\left(\frac{C}{\gamma}\|f(u_\lambda)\|_{L^1(\Omega)}\right)^{1/(p-1)}$, we deduce
\begin{equation}\label{ghjklk} \Omega_\varepsilon\subset\{x\in\Omega:u_\lambda(x) \leq s\}. \end{equation}
Now, choose
$$ \eta (x) := \left\{ \begin{array}{lll} \delta (x)&\textrm{if}&\delta (x) < \varepsilon,\\ \varepsilon&\textrm{if}&\delta (x) \geq \varepsilon, \end{array} \right. $$
as a test function in \eqref{StZu2} and use \eqref{ghjklk} to obtain
$$ \varepsilon^2
\int_{\{u_\lambda>s\}}\left( \frac{4}{p^2}|\nabla_{T,u_\lambda} |\nabla u_\lambda|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_{u_\lambda}^2 |\nabla u_\lambda|^{p} \right) \, dx
\leq \int_{\{u_\lambda \leq s\}} |\nabla u_\lambda|^{p} \, dx. $$
Multiplying equation $(1.15)_{\lambda}$ by $T_su_\lambda=\min\{s,u_\lambda\}$ we have
\begin{equation}\label{umens}
\int_{\{u_\lambda<s\}}|\nabla u_\lambda|^p\ dx=\lambda\int_\Omega f(u_\lambda)T_su\ dx \leq\lambda^\star s\|f(u_\lambda)\|_{L^1(\Omega)}=C
\|f(u_\lambda)\|_{L^1(\Omega)}^{p'}. \end{equation}
Combining the previous two inequalities we obtain
$$
\int_{\{u_\lambda>s\}}\left( \frac{4}{p^2}|\nabla_{T,u_\lambda} |\nabla u_\lambda|^{p/2}|^{2}
+ \frac{n-1}{p-1}H_{u_\lambda}^2 |\nabla u_\lambda|^{p} \right) \, dx
\leq C \|f(u_\lambda)\|_{L^1(\Omega)}^{p'}. $$
At this point, proceeding exactly as in the proof of Theorem~\ref{Theorem}, we conclude the $L^r$ estimates established in parts $(a)$ and $(b)$.
In order to prove the $W^{1,p}$-estimate of part $(b)$, recall that by \eqref{Pohozaev0} we have $$
\int_\Omega|\nabla u_\lambda|^p\ dx \leq C
\int_{\partial\Omega}|\nabla u_\lambda|^p\ d\sigma. $$
Therefore, we need to control the right hand side of the previous inequality. Since the nonlinearity $f$ is increasing by hypothesis we obtain
$$
f(u_\lambda)\leq f\left(C\|f(u_\lambda)\|_{L^1(\Omega)}^\frac{1}{p-1}\right) \quad\textrm{in }\Omega_\varepsilon $$
by \eqref{kkkkeyyyyy}, where $C$ is a constant independent of $\lambda$.
Now, since $-\Delta_p u_\lambda = \lambda f(u_\lambda)\in L^\infty(\Omega_\varepsilon)$ in $\Omega_\varepsilon$, it holds $$
\| u_\lambda \|_{C^{1,\beta} (\overline{\Omega}_\varepsilon)} \leq C' $$ for some $\beta\in(0,1)$ by \cite{Lie}, where $C'$ is a constant depending only on
$n$, $p$, $\Omega$, $f$, and $\|f(u_\lambda)\|_{L^1(\Omega)}$ proving the assertion.
Finally, assume that $p\geq 2$ and \eqref{convex:assump} holds. From \cite{S} we know that $f(u^\star)\in L^r(\Omega)$ for all $1\leq r<n/(n-p')$. In particular, $f(u^\star)\in L^1(\Omega)$. Therefore, parts $(i)$ and $(ii)$ follow directly from $(a)$ and $(b)$. \end{proof}
\footnotesize \noindent\textit{Acknowledgments.}
The authors were supported by grant 2009SGR345(Catalunya) and MTM2011-27739-C04 (Spain). The second author was also supported by grant MTM2008-06349-C03-01 (Spain).
\end{document} |
\begin{document}
\title[]{Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms}
\author{Kosuke Mitarai} \email{mitarai@qc.ee.es.osaka-u.ac.jp} \affiliation{Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.} \affiliation{Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.} \affiliation{JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan.}
\author{Yasunari Suzuki} \affiliation{NTT Secure Platform Laboratories, NTT Corporation, Musashino 180-8585, Japan} \affiliation{JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan.}
\author{Wataru Mizukami} \affiliation{Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.} \affiliation{JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan.}
\author{Yuya O. Nakagawa} \affiliation{QunaSys Inc., Aqua Hakusan Building 9F, 1-13-7 Hakusan, Bunkyo, Tokyo 113-0001, Japan}
\author{Keisuke Fujii} \affiliation{Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.} \affiliation{Center for Quantum Information and Quantum Biology, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Japan.} \affiliation{Center for Emergent Matter Science, RIKEN, Wako Saitama 351-0198, Japan}
\date{\today}
\begin{abstract}
Variational quantum algorithms are considered to be appealing applications of near-term quantum computers. However, it has been unclear whether they can outperform classical algorithms or not. To reveal their limitations, we must seek a technique to benchmark them on large scale problems. Here, we propose a perturbative approach for efficient benchmarking of variational quantum algorithms. The proposed technique performs perturbative expansion of a circuit consisting of Clifford and Pauli rotation gates, which is enabled by exploiting the classical simulatability of Clifford circuits. Our method can be applied to a wide family of parameterized quantum circuits consisting of Clifford gates and single-qubit rotation gates. The approximate optimal parameter obtained by the method can also serve as an initial guess for further optimizations on a quantum device, which can potentially solve the so-called ``barren-plateau'' problem. As the first application of the method, we perform a benchmark of so-called hardware-efficient-type ansatzes when they are applied to the VQE of one-dimensional hydrogen chains up to $\mathrm{H}_{24}$, which corresponds to $48$-qubit system, using a standard workstation.
\end{abstract}
\pacs{Valid PACS appear here} \maketitle
\section{Introduction}
As promising candidates for possible applications of early-days quantum devices, variational quantum algorithms (VQAs) have been developed rapidly. VQAs utilize parameterized quantum circuits $U(\bm{\theta})$ with parameters $\bm{\theta}$ which are optimized with respect to some suitably defined cost function $\mathcal{L}(\bm{\theta})$ depending on specific tasks. Since there are quantum circuits that cannot be simulated classically \cite{Arute2019}, we might gain practical speedups from VQAs if we choose $U(\bm{\theta})$ from such ones. This choice of $U(\bm{\theta})$ guarantees classical inability to perform exactly the same tasks.
Target applications of VQAs range among quantum chemistry calculations \cite{Peruzzo2014,McClean2016,Kandala2017}, combinational optimization \cite{farhi2014quantum} and machine learning \cite{Mitarai2018, Schuld2019,Havlicek2019,farhi2018classification,benedetti2019generative}. Despite the vast amount of theoretical proposals, demonstrations and benchmarks of algorithms, whether they be experiments with actual quantum devices \cite{arute2020quantum, Arute2020hartree, Nam2020, Havlicek2019} or numerical simulations, are limited to relatively small scale problems, where classical simulations are still feasible. Efficient techniques for their benchmark in large scale problems are strongly demanded to understand the limitations of VQAs and to develop more sophisticated algorithms.
Here we aim to resolve the above problem with a perturbative expansion of the cost function. We assume that a parametric circuit utilized in an algorithm is made of Clifford gates and single-qubit rotation gates whose angles are the circuit parameters and initialized to zero. The above form of the circuit includes a wide family of parameterized circuits, which guarantees the wide applicability of the proposed method. With the assumption, we can efficiently compute first and second derivatives of a cost function by exploiting the classical simulatability \cite{10.5555/1972505, Aaronson2004} of Clifford circuits. This allows us to perform a simple minimization of a quadratic function to obtain an approximately optimal value of parameters and a cost function. In particular, for the variational quantum eigensolver (VQE) \cite{Peruzzo2014}, which is an algorithm to obtain an approximate ground state of a quantum system, the perturbative treatment can be justified because classically tractable variational solutions such as Hartree-Fock or mean-field states are expected to be close to true ones. Our method serves as a performance benchmark of a VQA in such cases.
The method can also be seen as an efficient initializer of the circuit parameters as the above procedure corresponds to the first step of Newton-Raphson optimization. The obtained Hessian together with the obtained parameter can be passed as an initial guess to quasi-Newton optimizer such as the BFGS method for further optimization on quantum devices. This method can potentially solve the so-called ``barren-plateau'' problem \cite{McClean2018} which states that the gradient of the cost function vanishes exponentially to the number of qubits when circuits and parameters are chosen randomly. The problem itself is not an obstacle to the proposed method, because the whole process is run on a classical computer where the accuracy of the computation can be improved exponentially with only polynomially scaling resource, which is in contrast to the case of using quantum devices.
In the following, we first give the concrete algorithm of the proposed method. Then, as an application of the method, we benchmark the so-called hardware-efficient-type ansatzes applied to the VQE up to 48 qubits using the hydrogen chain as a testbed. This benchmark can be performed because the proposed technique provides us an approximately optimal value of the cost function, which is the energy expectation value in this case. The benchmark itself can be seen as a demonstration of a ``quantum-inspired'' quantum chemistry calculation, where the ansatz wavefunction is constructed with the language of the quantum circuit. Finally, we also show the effectiveness of our perturbative initialization approach by another numerical experiment.
\section{Algorithms}
\subsection{Variational quantum algorithms}
VQA generally refers to a family of algorithms that involve use of parametrized quantum circuits $U(\bm{\theta})$ whose parameters $\bm{\theta}$ are optimized with respect to a suitable cost function $\mathcal{L}(\bm{\theta})$. $U(\bm{\theta})$ is used to generate a $n$-qubit parametrized state $\ket{\psi(\bm{\theta})}:=U(\bm{\theta})\ket{0}^{\otimes n}$. Hereafter we abbreviate $\ket{0}^{\otimes n}$ by $\ket{0}$ when it is clear from the context. A famous example of VQAs is the VQE \cite{Peruzzo2014}, where the cost function is defined as an energy expectation value $E(\bm{\theta})$ with respect to a Hamiltonian $H$, i.e. $\mathcal{L}(\bm{\theta})=E(\bm{\theta})=\bra{\psi(\bm{\theta})}H\ket{\psi(\bm{\theta})}$. The cost function is usually computed from expectation values of observables also in other examples such as machine learning \cite{Mitarai2018, Schuld2019,benedetti2019generative,farhi2018classification, Havlicek2019}, and combinational optimization \cite{farhi2014quantum}. A general form of $\mathcal{L}(\bm{\theta})$ can be written as $\mathcal{L}(\bm{\theta}) = L\left(\braket{O(\bm{\theta})}\right)$ where $O$ denote the measured observable and $\braket{O(\bm{\theta})}:= \bra{\psi(\bm{\theta})}O \ket{\psi(\bm{\theta})}$. $O$ is typically expressed as a sum of $N_o$ $n$-qubit Pauli operators $\{P_i\} \subset \{I,X,Y,Z\}^{\otimes n}$ as $O=\sum_{i=1}^{N_o} c_i P_i$ with coefficients $\{c_i\}$.
\subsection{Main result: Quadratic Clifford expansion}
We consider an ansatz in the form of \begin{align}
U(\bm{\theta}) = R_K(\theta_K)C_K \cdots R_2(\theta_2)C_2 R_1(\theta_1) C_1 \label{eq:ansatz}, \end{align} where $\bm{\theta}=\{\theta_k\}_{k=1}^K$, $R_k$ is a single-qubit rotation gate generated by a Pauli operator $P_k$, i.e. $R_k(\theta_k)=e^{i\theta_k P_k}$, and $C_k$ is a circuit consisting of Clifford gates. We furthermore assume $R_k(0)$ to be Clifford. Note that this form of the ansatz is quite general. When we wish to build a hardware-efficient ansatz \cite{Kandala2017}, it is frequently in the form of Eq. (\ref{eq:ansatz}) because the two-qubit gates which are tuned to give a high-fidelity on the hardware are usually Clifford gates such as controlled-NOT or controlled-Z gates. More sophisticated ansatz such as unitary coupled-cluster (UCC) \cite{Peruzzo2014} can also be written in this form.
This form of the ansatz allows us to efficiently compute the perturbative form of the cost function $\mathcal{L}(\bm{\theta})$. More concretely, a Taylor expansion of $\braket{O(\bm{\theta})}$ around $\bm{\theta}=0$ can be written as, \begin{align}
\braket{O(\bm{\theta})}
&=\braket{O(0)} + \sum_k g_k\theta_k + \frac{1}{2}\sum_{k,m} A_{km}\theta_k \theta_m + \mathcal{O}(\|\bm{\theta}\|^3),\label{eq:taylor} \end{align} where, \begin{align}
g_k &= 2\mathrm{Re}\left[\bra{0} U^\dagger(0) O \frac{\partial U(0)}{\partial \theta_k}\ket{0}\right], \\
\begin{split}
A_{km} &= 2\mathrm{Re}\left[\bra{0} \frac{\partial U^\dagger(0)}{\partial \theta_k} O \frac{\partial U(0)}{\partial \theta_m}\ket{0}\right] \\
&\quad+ 2\mathrm{Re}\left[\bra{0} U^\dagger(0) O \frac{\partial }{\partial \theta_k} \frac{\partial U(0)}{\partial \theta_m}\ket{0}\right],
\end{split} \end{align}
and $\frac{\partial U(0)}{\partial \theta_k}:=\left.\frac{\partial U(\bm{\theta})}{\partial \theta_k}\right|_{\bm{\theta}=0}$, which can then be used to expand $\mathcal{L}(\bm{\theta})=L(\braket{O(\bm{\theta})})$ itself. Since we assumed $U$ to be in the form of Eq. (\ref{eq:ansatz}), $\frac{\partial U(0)}{\partial \theta_i}$ can be efficiently computed. To see this, observe that, \begin{align}
\frac{\partial U(0)}{\partial \theta_k} = iR_K(0)C_K\cdots P_k R_k(0)C_k \cdots R_2(0)C_2 R_1(0) C_1. \end{align} Since we assume $R_k(0)$ to be Clifford for all $k$, $P_k$ can be passed through $R_K(0)C_K \cdots R_{k+1}(0)C_{k+1}$. Let \begin{align}
\begin{split}
&R_K(0)C_K \cdots R_{k+1}(0)C_{k+1} P_k \\
&= P_k'R_K(0)C_K \cdots R_{k+1}(0)C_{k+1}
\end{split} \end{align} for some Pauli operator $P_k'$.
$P_k'$ can be found in time $\mathcal{O}(nK)$ on a classical computer if $\{C_k\}$ are local. Using $P_k'$, the coefficients appearing in the second-order Taylor expansion (Eq. (\ref{eq:taylor})) can be written in the terms of the expectation values $\braket{\psi(0)|OP_k'|\psi(0)}$, $\braket{\psi(0)|P_k'OP_m'|\psi(0)}$ and $\braket{\psi(0)|OP_k'P_m'|\psi(0)}$. The decomposition of the operators $OP_k'$, $P_k'OP_m'$ and $OP_k'P_m'$ into a sum of Pauli operators can be computed in $\mathcal{O}(nN_o)$ on a classical computer. This can be performed simply by multiplying $P_k'$ and $P_m'$ to each Pauli operator in $O$. The expectation values of these operators can also be evaluated classically because $\ket{\psi(0)}$ is a stabilizer state under the assumption that $U(0)$ is Clifford. More concretely, we evaluate expectation values of each Pauli operator constituing $OP_k'$, $P_k'OP_m'$ and $OP_k'P_m'$, and then take the summation. This process can be performed in time $\mathcal{O}(n^2N_oK^2)$ using a standard simulation technique \cite{Aaronson2004}, which gives the leading order complexity of the perturbative expansion. We call this technique quadratic Clifford expansion. The technique itself might be useful for classical simulations of near-Clifford circuits.
The perturbative expansion given in Eq. (\ref{eq:taylor}) is justified especially for the VQE \cite{Peruzzo2014}, where we can obtain an approximate ground state classically by using techniques such as Hartree-Fock methods. If we construct $U(\bm{\theta})$ in such a way that $\ket{\psi(0)}$ becomes the Hartree-Fock ground state, $\ket{\psi(0)}$ is considered to be close to the true ground state. Therefore, in this case, we can presume the optimal value of $\bm{\theta}$ to be small, which justifies the perturbative treatment of the cost function $E(\bm{\theta})$. As long as the perturbation is accurate enough, we can obtain the optimal value of $\braket{O(\bm{\theta})}$ by simply minimizing the quadratic function obtained with the second-order expansion, which can be done in time $\mathcal{O}(K^3)$ and provides us an optimal parameter $\bm{\theta}^*=-A^{+}\bm{g}$, where $A^{+}$ is the Moore-Penrose pseudo-inverse of the Hessian $A$. The approximate, perturbative optimal value of $\braket{O(\bm{\theta})}$ can be calculated by substituting $\bm{\theta}^*$ into Eq. (\ref{eq:taylor}) and neglecting cubic error term, which we denote by $\braket{O}^{*}$.
It must be noted that whether the above perturbation is accurate or not cannot be determined classically in general, since we cannot obtain the expectation value $\braket{O(\bm{\theta})}$ for general $\bm{\theta}$ on a classical computer. This indicates the need for a quantum device for validating the result. Therefore, a possible strategy of using the proposed technique is to evaluate $\braket{O(\bm{\theta}^*)}$ on a quantum computer, and if it returns a value close to the perturbative one, then we just conclude that we have found an optimal approximate ground state; otherwise, we further optimize the parameters starting from $\bm{\theta}^*$. Note that $\bm{\theta}^*$ can also be seen as the parameter obtained by the first step of the Newton-Raphson method and provides a good starting point for further optimization. Moreover, the Hessian $A_{km}$ can be passed to quasi-Newton optimizers such as the BFGS method which are frequently utilized in the VQE.
We can also apply the proposed method to the machine learning algorithms \cite{Mitarai2018, Schuld2019,benedetti2019generative,farhi2018classification, Havlicek2019} In this direction, techniques for computing a ``good'' initial guess like mean-field solution are not yet developed to the best of our knowledge. However, for example, we might be able to obtain an initial guess by using the correspondence of the Boltzmann machine and the quantum circuit developed in \cite{PhysRevResearch.2.033125}. Also, one can use the gradient and Hessian of the cost function obtained by the above protocol for performing the first optimization step and also pass the latter to the quasi-Newton optimizers.
\section{Numerical experiment}
To demonstrate the effectiveness of our idea, we apply the method described in the previous section to the VQE to benchmark the performance of so-called hardware-efficient ansatz \cite{Kandala2017}. For this purpose, we use electronic Hamiltonians of evenly-spaced one-dimensional chains of hydrogen atoms $\mathrm{H}_m$, which are frequently used as a benchmark system for quantum chemistry calculations~\cite{PhysRevX.7.031059}. All benchmarks are performed on a workstation with two Intel Xeon Silver 4108 processors. For quantum circuit simulations, we utilized an NVIDIA Tesla-V100 GPU.
\subsection{Experimental details}
Electronic Hamiltonians of hydrogen chains are generated by OpenFermion \cite{McClean_2020} and PySCF \cite{sun2018pyscf,sun2020recent} using the STO-3G minimal basis set. The generated fermionic Hamiltonians are mapped to qubit ones by Jordan-Wigner transformation implemented in OpenFermion, which results in a $2m$-qubit Hamiltonian for an $m$-hydrogen chain $\mathrm{H}_m$. A thorough review of these procedures can be found at e.g. Ref. \cite{RevModPhys.92.015003}. All conventional quantum chemistry calculations are also performed with PySCF.
As for the ansatz, we use the one shown in Fig. \ref{fig:ansatz}, which can be regarded as a ``hardware-efficient'' ansatz constructed on a one-dimensional qubit array. It consists of alternating layers of two-qubit Clifford gates and single-qubit rotation gates. This form of the ansatz can generate sufficiently non-local evolutions that give non-zero gradients. In Fig. \ref{fig:ansatz}, the two-qubit Clifford gates in the region shaded by blue are randomly chosen as shown in the upper left of the figure. Those in the orange region are chosen so that the overall circuit becomes identity when $\bm{\theta}=0$. This allows us to easily guarantee $\ket{\psi(0)}$ to be the Hartree-Fock state $\ket{\psi_{HF}}$; we can just inject $\ket{\psi_{HF}}$ to the input of the circuit. Note that Hartree-Fock states are computational basis states under fermion-to-qubit mappings such as Jordan-Wigner transformation, and its evolution under Clifford gates can efficiently be simulated. The single-qubit rotations hold the parameter $\bm{\theta}$ to be optimized. Each of them has three parameters as $x$, $y$, and $z$-rotation angles. Note that, although the two-qubit Clifford gates are chosen to satisfy the above constraint, the parameters implemented in single-qubit rotations in each region are independent.
Using this ansatz, we calculate the gradient and Hessian based on the method described in the previous section. Then, we perform the minimization of the second-order perturbative energy to obtain approximately optimal energies and parameters. This provides us $\bm{\theta}^*$ and perturbative energies $E^*=\braket{H}^*$. Finally, when possible, we simulate an ansatz whose parameters are set to the perturbatively optimal ones to check if the perturbative treatment can be justified. This simulation is performed with Qulacs \cite{Qulacs}.
To make the benchmark systematic, we set the depth of the ansatzes equal to the number of hydrogen atoms, $m$. This scaling of the depth can be considered as the largest possible value for today's most advanced quantum computer \cite{Arute2019}. Note that this choice corresponds to $L=\mathcal{O}(m^2)$. Combining with the fact that $N_o = \mathcal{O}(m^4)$ and $n=2m$ in this case, the overall time complexity of computing Hessian for this system is $\mathcal{O}(m^8)$. Since the minimization of the quadratic function can be done in time $\mathcal{O}(L^3)=\mathcal{O}(m^6)$, the Hessian part contributes the most to the total time.
Finally, to somewhat relax the randomness of the ansatz, we first randomly generate 200 ansatzes in the form of Fig.~\ref{fig:ansatz} for each $m$ used in the experiment. For each generated ansatz, we calculate $g_l$ using the Hamiltonian with the spacing of 1.0 \AA. Then, the circuit with the largest $\sum_l |g_l|$ is chosen as the ansatz to be used for each hydrogen chain with different spacings. This is based on our expectation that an ansatz with large gradients would provide the highest performance.
\begin{figure}
\caption{Ansatz used in the numerical experiment. Grey boxes represent (fixed) 2-qubit Clifford gates in the form of the upper left, where white boxes are randomly chosen from 24 single-qubit Clifford gates \cite{Selinger_2015}. Green boxes represent parametrized single qubit rotations consisting of $x$, $y$ and $z$-axis rotations respectively.}
\label{fig:ansatz}
\end{figure}
\subsection{Results and discussion}
\subsubsection{Benchmark results of small-scale systems} Figure \ref{fig:SmallScaleResults} shows the result of the numerical experiment at $m=2$, $4$, and $6$ along with the energy obtained from standard quantum chemistry calculations as references. For $m=2$ and $4$, we can observe that the energies obtained from the circuit simulation and the one from the perturbative optimization match well at small spacings. Here, the Hartree-Fock method gives a relatively accurate description of the ground state, and the perturbative treatment works fine as expected. The effectiveness of the perturbation also means that we can achieve the optimal parameter with this technique. This implies that the hardware-efficient ansatzes considered in this work can only achieve the accuracy of second-order M{\o}ller-Plesset perturbation (MP2), which is a technique used widely in current quantum chemistry calculations as one of the easiest post-Hartree-Fock methods, for $\mathrm{H}_4$ as we can observe from Fig. \ref{fig:SmallScaleResults} (b). There is a possibility of improving the accuracy by optimizing from a randomly initialized $\bm{\theta}$ as the above discussion only considers the case where we take $\bm{\theta}=0$ as the initial parameter. However, such a strategy would not be generally scalable because of the barren plateau problem \cite{McClean2018}. On the other hand, the perturbative treatment breaks down at the larger spacings where the electronic correlation becomes larger. As mentioned in the previous section, one has to perform further optimization in such a case.
In the case of $m=6$, we cannot observe the clear break down of the perturbative treatment, i.e., the energies obtained from the perturbation match well with the ones from the circuit simulation. Again, it means that the optimal parameters and corresponding energies can be obtained with the proposed technique. We can see that the hardware-efficient circuit cannot even achieve the MP2 energy for $\mathrm{H}_6$. Note that MP2 considers up to double electron excitations and involves $\mathcal{O}(n^4)$ parameters in its construction. This scaling is considerably greater than the number of parameters implemented in the ansatz of Fig. \ref{fig:ansatz} with depth $n$. In this sense, the performance worse than MP2 is expected behavior. This trend of decreasing accuracy will also be certified with the result in the next subsection.
\begin{figure}
\caption{Results of the numerical experiments at (a) $m=2$, (b) $m=4$ and (c) $m=6$. Full-CI is the exact ground state energy, and MP2, CCSD, and HF respectively mean approximate ground state energy obtained with second-order M{\o}ller-Plesset perturbation, coupled cluster with single and double excitations, and Hartree-Fock. Circuit value and perturbative value represent $E(\bm{\theta}^*)$ and its perturbative approximation computed from Eq. (\ref{eq:taylor}), respectively.}
\label{fig:SmallScaleResults}
\end{figure}
\subsubsection{Benchmark results of large-scale systems}\label{sec:largescaleresults}
Although the complexity of $\mathcal{O}(m^8)$ is polynomial to $m$, the large exponent prohibits us from extending the analysis to larger scales. Due to this complexity, we need some modifications to the experimental settings. First, we modify the ansatz to only involve real numbers by generating an ansatz that has the same form as the one shown in Fig. \ref{fig:ansatz}, but the random single-qubit Clifford gates are randomly chosen from identity and Hadamard gates only, and the single-qubit rotations just contain a $y$-rotation. This modification reduces the number of parameters by a third. We find that this modification does not significantly alter the result as shown in Appendix, which can be explained by the fact that the eigenstates of non-relativistic quantum chemistry Hamiltonians can be described with states that are real in the computational basis. Second, to further reduce the number of parameters involved in the Hessian calculation, we ``drop out'' the parameters that give zero gradients to the energy, i.e., $y$-rotations that do not give the first-order contribution to the energy are removed from the ansatz after calculating the gradient. This is motivated by our observation in the preliminary experiment where we have found that the gradients with respect to most of the parameters are exactly zero. In the following experiment, the results are obtained by removing the rotation gates with $|g_l|< 10^{-6}$ Hartree. In Appendix, we show that this modification does not significantly alter the results either.
Figure \ref{fig:LargeScaleResults} shows the results for $m=14$ and $m=24$ cases. Note that $m=14$ corresponds to 28 qubits, which is the largest number of qubits that can be handled with the Qulacs-GPU simulator \cite{Qulacs} using an NVIDIA Tesla-V100 processor. At $m=16$, the required memory far exceeds its capacity. For this reason, we do not show the circuit value in the case of $m=24$ which corresponds to 48 qubits (Fig. \ref{fig:LargeScaleResults} (c), (d)). This is, to the best of our knowledge, the first benchmark of the VQE at this scale. Also, as the exact diagonalization at $m=24$ could not be performed under our environment, it is not shown in the figure. The energy of coupled-cluster with single and double excitations (CCSD) is not depicted in both cases because we experienced its numerical instability.
In Fig. \ref{fig:LargeScaleResults} (b) and (d), we show the correlation energy $E_{\mathrm{corr}}$ defined as the difference between the obtained energy and the Hartree-Fock reference energy to illustrate the performance of the proposed method and the hardware-efficient ansatz itself. From Fig. \ref{fig:LargeScaleResults} (b), we can observe that the perturbative energy and the circuit value match almost exactly, indicating that we can analyze the performance of the ansatz itself with this perturbative treatment. Therefore, Fig. \ref{fig:LargeScaleResults} (b) and (d) show that the correlation energy that can be achieved by the hardware-efficient ansatz used in this work is a tenth or a hundredth smaller than that of MP2. Note that the perturbative minimum obtained by this method is a local one that locates around $\bm{\theta}=0$, and the global optimal solution for this ansatz can perform better. Nevertheless, this result indicates the need for more structured ansatzes such as unitary coupled-cluster \cite{Peruzzo2014} to correctly capture the correlation of electrons in a molecule.
\begin{figure}
\caption{Results of the numerical experiments at (a), (b) $m=14$ and (c), (d) $m=24$. Graph legends follow those of Fig. \ref{fig:SmallScaleResults}. (a) and (c) show the total energy, while (b) and (d) show the correlation energy.}
\label{fig:LargeScaleResults}
\end{figure}
\subsubsection{Initialization performance}
Finally, we show that the approximate optimal parameter $\bm{\theta}^*=-A^+ \bm{g}$ together with the initial Hessian $A$ can indeed serve as a good initial guess of the VQE. To this end, we take Hamiltonians of $\mathrm{H}_4$ at different atom spacings as examples, and compare the convergence of the optimization procedure of the VQE when using different initial parameters $\bm{\theta}_{\mathrm{init}}$. The BFGS method implemented in SciPy \cite{virtanen2020scipy} which is a popular quasi-Newton technique is employed as the optimizer. We compare three cases: $\bm{\theta}_{\mathrm{init}}=0$, $\bm{\theta}_{\mathrm{init}}=-A^{+}\bm{g}$, and $\bm{\theta}_{\mathrm{init}}=-A^{+}\bm{g}$ with the initial Hessian provided to the optimizer.
Figure \ref{fig:VQE} shows the result of the numerical experiment. We can observe that the $\bm{\theta}_{\mathrm{init}}=-A^{+}\bm{g}$ cases exhibits the faster convergence than $\bm{\theta}_{\mathrm{init}}=0$ in all cases. Also, the optimizer with the initial Hessian performs equally well to or better than the case $\bm{\theta}_{\mathrm{init}}=-A^{+}\bm{g}$ without providing Hessian. This result demonstrates the effectiveness of our perturbative initialization approach.
\begin{figure}\label{fig:VQE}
\end{figure}
\section{Conclusion}
We proposed a technique to efficiently compute an approximate optimal parameter and the corresponding value of the cost function in the VQAs. It is based on the observation that we can efficiently compute the gradient and Hessian of the cost function if an ansatz is in the form of Eq. (\ref{eq:ansatz}) which includes a wide range of circuits. Since the method is based on a perturbative expansion, we can obtain an accurate solution when the initial guess of the parameter from which we perform the Taylor expansion of Eq. (\ref{eq:taylor}) is close to an optimal one. Even if we do not have such an initial guess, the gradient and Hessian can be used to perform the first step optimization, and those quantities can be passed to optimizers. The generality of the ansatz allows us to apply the proposed method to various VQAs such as VQE \cite{Peruzzo2014,McClean2016,Kandala2017}, quantum approximate optimization \cite{farhi2014quantum}, and variational machine learning algorithms \cite{Mitarai2018, Schuld2019,Havlicek2019,farhi2018classification,benedetti2019generative}.
We applied the method to the VQE of hydrogen chains with a one-dimensional hardware-efficient ansatz shown in Fig. \ref{fig:ansatz} for its benchmark. The numerical experiments showed that the performance of such a hardware-efficient ansatz in the VQE cannot even achieve that of classical MP2 calculation. To the best of our knowledge, the proposed method is the only one that enables us the benchmark of the VQAs beyond the scale that is classically simulatable. Although the benchmark results are pessimistic, it also motivates us to construct more structured ansatzes such as unitary coupled-cluster \cite{Peruzzo2014} and to make other initialization strategies such as the one presented in Ref. \cite{Grant2019initialization}. For example, one might be able to use genetic optimization to improve the ansatz in Fig. \ref{fig:ansatz} from the random choice of Clifford gates. One might also be able to improve the performance of this ansatz by using localized orbitals instead of the naive Hartree-Fock orbitals utilized in this work to express the Hamiltonian, which would make it easier for the ansatz to capture the electronic correlation. We believe that the proposed technique will be of use to a wide range of the VQAs.
\begin{acknowledgments}
KM is supported by JST PRESTO Grant No. JPMJPR2019 and JSPS KAKENHI Grant No. 20K22330.
YS is supported by JST PRESTO Grant No. JPMJPR1916.
WM is supported by JST PRESTO Grant No. JPMJPR191A.
KF is supported by JSPS KAKENHI Grant No. 16H02211, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.
This work is supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant Number JPMXS0118067394 and JPMXS0120319794. \end{acknowledgments}
\appendix*
\section{Reducing the number of parameters in the ansatz}
First, we demonstrate that the modification of the ansatz to be real in the computational basis (see Sec. \ref{sec:largescaleresults}) does not alter the results significantly. It is illustrated in Fig. \ref{fig:dropout-compare} (a), where we plot the difference of the energy obtained by the modified and original ansatz, respectively denoted as $E_{\mathrm{real}}$ and $E_{\mathrm{complex}}$. The energy difference does not exceed $10^{-2}$ Hartree in the figure when the spacing is less than 2.0 \AA, which is negligible compared to the correlation energy.
The effect of the ``drop-out'' utilized in Sec. \ref{sec:largescaleresults} does not alter results either. Figure \ref{fig:dropout-compare} (b) shows the comparison of the results with and without dropout at $m=6$ using the modified real ansatz. We can observe that the ``drop-out'' only slightly alters the result by about the same magnitude as Fig. \ref{fig:dropout-compare} (a).
\begin{figure}
\caption{(a) Difference of the results obtained with the original ansatz in Fig. \ref{fig:ansatz} and the one modified to generate only real-valued state vectors at $m=6$. (b) Difference of the result with and without dropout at $m=6$ using the modified ansatz similar to Fig. \ref{fig:ansatz}, which only generates real wavefunctions. }
\label{fig:dropout-compare}
\end{figure}
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\title{Graphs with 3-rainbow index $n-1$ and $n-2$\Large\bf \footnote{Supported by NSFC No.11071130.}} \author{\small Xueliang~Li, Kang Yang, Yan~Zhao\\ \small Center for Combinatorics and LPMC-TJKLC\\ \small Nankai University, Tianjin 300071, China\\ \small lxl@nankai.edu.cn;
yangkang@mail.nankai.edu.cn; zhaoyan2010@mail.nankai.edu.cn} \date{} \maketitle \begin{abstract}
Let $G$ be a nontrivial connected graph with an edge-coloring $c:E(G)\rightarrow \{1,2,\ldots,q\},$ $q\in \mathbb{N}$, where adjacent edges may be colored the same. A tree $T$ in $G$ is a $rainbow~tree$ if no two edges of $T$ receive the same color. For a vertex set $S\subseteq V(G)$, the tree connecting $S$ in $G$ is called an $S$-tree. The minimum number of colors that are needed in an edge-coloring of $G$ such that there is a rainbow $S$-tree for each $k$-set $S$ of $V(G)$ is called the $k$-rainbow index of $G$, denoted by $rx_k(G)$. In \cite{Zhang}, they got that the $k$-rainbow index of a tree is $n-1$ and the $k$-rainbow index of a unicyclic graph is $n-1$ or $n-2$. So there is an intriguing problem: Characterize graphs with the $k$-rainbow index $n-1$ and $n-2$. In this paper, we focus on $k=3$, and characterize the graphs whose 3-rainbow index is $n-1$ and $n-2$, respectively.
{\flushleft\bf Keywords}: rainbow $S$-tree, $k$-rainbow index.
{\flushleft\bf AMS subject classification 2010}: 05C05, 05C15, 05C75.
\end{abstract}
\section{Introduction} All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty \cite{Bondy}. Let $G$ be a nontrivial connected graph with an edge-coloring $c: E(G)\rightarrow \{1,2,\ldots,q\}$, $q\in \mathbb{N}$, where adjacent edges may be colored the same. A path of $G$ is a \emph{rainbow path} if every two edges of the path have distinct colors. The graph $G$ is \emph{rainbow connected} if for every two vertices $u$ and $v$ of $G$, there is a rainbow path connecting $u$ and $v$. The minimum number of colors for which there is an edge coloring of $G$ such that $G$ is rainbow connected is called the \emph{rainbow connection number}, denoted by $rc(G)$. Results on the rainbow connectivity can be found in \cite{Chartrand1,Caro, Chartrand,ChartrandZhang,LSS, LiSun}.
These concepts were introduced by Chartrand et al. in \cite{Chartrand1}. In \cite{Zhang}, they generalized the concept of rainbow path to rainbow tree. A tree $T$ in $G$ is a $rainbow~tree$ if no two edges of $T$ receive the same color. For $S\subseteq V (G)$, a $rainbow\ S$-$tree$ is a rainbow tree connecting $S$. Given a fixed integer $k$ with $2\leq k \leq n$, the edge-coloring $c$ of $G$ is called a $k$-$rainbow~coloring$ of $G$ if for every $k$-subset $S$ of $V(G)$, there exists a rainbow $S$-tree. In this case, $G$ is called $k$-$rainbow~connected$. The minimum number of colors that are needed in a $k$-$rainbow~coloring$ of $G$ is called the $k$-$rainbow~index$ of $G$, denoted by $rx_k(G)$. Clearly, when $k=2$, $rx_2(G)$ is nothing new but the rainbow connection number $rc(G)$ of $G$. For every connected graph $G$ of order $n$, it is easy to see that $rx_2(G)\leq rx_3(G)\leq \cdots \leq rx_n(G)$.
The $Steiner~distance$ $d_G(S)$ of a set $S$ of vertices in $G$ is the minimum size of a tree in $G$ connecting $S$. The $k$-$Steiner~diameter$ $sdiam_k(G)$ of $G$ is the maximum Steiner distance of $S$ among all sets $S$ with $k$ vertices in $G$. Then there is a simple upper bound and lower bound for $rx_k(G)$.
\begin{obs}[\cite{Zhang}]\label{obs1} For every connected graph $G$ of order $n\geq 3$ and each integer $k$ with $3\leq k\leq n$, $k-1\leq sdiam_k(G)\leq rx_k(G)\leq n-1$. \end{obs}
They showed that trees are composed of a class of graphs whose k-rainbow index attains the upper bound.
\begin{pro}[\cite{Zhang}]\label{pro1} Let $T$ be a tree of order $n\geq 3$. For each integer $k$ with $3\leq k\leq n$, $rx_k(T)=n-1$. \end{pro}
They also showed that the $k$-rainbow index of a unicyclic graph is $n-1$ or $n-2$.
\begin{thm}[\cite{Zhang}]\label{thm1} If $G$ is a unicyclic graph of order $n\geq 3$ and girth $g\geq 3$, then \begin{equation}
rx_k(G)=
\begin{cases}
n-2, & \text{$k=3$ and $g\geq4$}; \\
n-1, & \text{$g=3$ or $4\leq k\leq n$}.
\end{cases} \end{equation} \end{thm}
A natural thought is that which graph of order $n$ has the $k$-rainbow index $n-1$ except for a tree and a unicyclic graph of girth 3? Furthermore, which graph of order $n$ has the $k$-rainbow index $n-2$ except for a unicyclic graph of girth at least 4? In this paper, we focus on $k=3$. In addition, some known results are mentioned.
\begin{obs}[\cite{Zhang}]\label{obs2} Let $G$ be a connected graph of order $n$ containing two bridges $e$ and $f$. For each integer $k$ with $2\leq k\leq n$, every $k$-rainbow coloring of $G$ must assign distinct colors to $e$ and $f$. \end{obs}
\begin{lem}[\cite{CLYZ}]\label{lem3} If $G$ is a connected graph and $\{H_1, H_2,\cdots, H_k\}$ is a partition of $V(G)$ into connected subgraphs, then $rx_3(G)\leq k-1+\sum_{i=1}^krx_3(H_i)$. \end{lem}
\begin{thm}[\cite{CLYZ}]\label{thm3} Let $G$ be a connected graph of order $n$. Then $rx_3(G)=2$ if and only if $G=K_5$ or $G$ is a 2-connected graph of order 4 or $G$ is of order 3. \end{thm}
\begin{obs}[\cite{CLYZ}]\label{obs3} Let $G$ be a connected graph of order $n$, and $H$ be a connected spanning subgraph of $G$. Then $rx_3(G)\leq rx_3(H)$. \end{obs}
This paper is organized as follows. In section 2, some basic results and notations are presented. In section 3, we characterize the graphs whose 3-rainbow index is $n-1$ and $n-2$, respectively. And we take two steps for the latter case. First, we consider the bicyclic graphs. Second, we consider the tricyclic graphs. Finally, we characterize the graphs whose 3-rainbow index is $n-2$.
\section{Some basic results} First of all, we need some more terminology and notations.
\begin{definition} Let $G$ be a connected graph with $n$ vertices and $m$ edges. Define the $cyclomatic~number$ of $G$ as $c(G)=m-n+1$. A graph $G$ with $c(G)=k$ is called a $k$-$cyclic$ graph. According to this definition, if a graph $G$ meets $c(G)=0$, 1, 2 or 3, then the graph $G$ is called acyclic(or a tree), unicyclic, bicyclic, or tricyclic, respectively. \end{definition}
\begin{definition} For a subgraph $H$ of $G$ and $v\in V(G)$, let $d(v,H)=min\{d_G(v,x): x\in V(H)\}$. \end{definition}
Next we define some new notations.
\begin{definition} For a connected graph $G$ of order $n$, set $V(G)=\{v_1,v_2,\cdots,v_n\}$, we define a $lexicographic$ $ordering$ between any two edges of $G$ by $v_iv_j<v_sv_t$ if and only if $i<s$ or $i=s$, $j<t$. \end{definition}
Note that, the lexicographic ordering of a connected graph is unique. Given a coloring $c$ of a connected graph $G$, denote by $c_{\ell}(G)$ a sequence of colors of the edges which are ordered by the lexicographic ordering.
Let $G$ be a connected graph, to $contract$ an edge $e=xy$ is to delete $e$ and replace its ends by a single vertex incident to all the edges which were incident to either $x$ or $y$. Let $G^{'}$ be the graph obtained by contracting some edges of $G$. Given a rainbow coloring of $G^{'}$, when it comes back to $G$, we keep the colors of corresponding edges of $G^{'}$ in $G$ and assign a new color to a new edge, which makes $G$ 3-rainbow connected. Hence, the following lemma holds.
\begin{lem}\label{lem1}
Let $G$ be a connected graph, and $G^{'}$ be a connected graph by contracting some edges of $G$. Then $rx_3(G)\leq rx_3(G^{'})+|G|-|G^{'}|$. \end{lem}
\begin{definition} Let $G_0$ be the graph obtained by contracting all the cut edges of $G$, then $G_0$ is called the $basic$ $graph$ of $G$.
\end{definition}
\section{Main results}
\subsection{Characterize the graphs with $rx_3(G)\bf{=n-1}$} \begin{thm}\label{thm4} Let $G$ be a connected graph of order $n$. Then $rx_3(G)=n-1$ if and only if $G$ is a tree or $G$ is a unicyclic graph with girth 3. \end{thm} \begin{proof} If $G$ is a tree or a unicyclic graph with girth 3, by Proposition \ref{pro1} and Theorem \ref{thm1}, $rx_3(G)=n-1$. Conversely, suppose $G$ is a graph with $rx_3(G)=n-1$ but not a tree, then $G$ must contain cycles. Let $\{H_1, H_2,\cdots, H_k\}$ be a partition of $V(G)$ into connected subgraphs. If $G$ contains a cycle of length $r$ at least 4, then let $H_1$ be the $r$-cycle, and each other subgraph a single vertex. We color $H_1$ with $r-2$ dedicated colors, then by Lemma \ref{lem3}, $rx_3(G)\leq n-r+rx_3(H_1)=n-2$. Suppose then $G$ contains at least two triangles $C_1$ and $C_2$. If $C_1$ and $C_2$ have a vertex in common, then let $H_1$ be the union of $C_1$ and $C_2$, and each other subgraph a single vertex. We color both $C_1$ and $C_2$ with the same three dedicated colors, thus $rx_3(G)\leq n-5+rx_3(H_1)=n-2$. If $C_1$ and $C_2$ are vertex disjoint, then let $H_1=C_1$, $H_2=C_2$, and each other subgraph a single vertex. We color $H_1$ with three new colors and $H_2$ with the same three colors of $H_1$, thus $rx_3(G)\leq n-5+rx_3(H_1)+rx_3(H_2)=n-2$. Combining the above two cases, $G$ is a unicyclic graph with girth 3. Therefore, the result holds. \end{proof}
\subsection{Characterize the graphs with $rx_3(G)\bf{=n-2}$} Next, we characterize the graphs whose 3-rainbow index is $n-2$. We begin with a useful theorem from \cite{CLYZ}.
A 3-$sun$ is a graph constructed from a cycle $C_6=v_1v_2\cdots v_6v_1$ by adding three edges $v_2v_4$, $v_2v_6$ and $v_4v_6$.
\begin{thm}[\cite{CLYZ}]\label{thm8} Let $G$ be a 2-edge-connected graph of order $n~(n\geq 4)$. Then $rx_3(G)\leq n-2$, with equality if and only if $G=C_n$ or $G$ is a spanning subgraph of one of the following graphs: a 3-sun, $K_5-e$, $K_4$, $G_1$, $G_2$, $H_1$, $H_2$, $H_3$, where $G_1$, $G_2$ are defined in Figure 1 and $H_1$, $H_2$, $H_3$ are defined in Figure 2. \end{thm}
Since all the 2-edge-connected graphs with the 3-rainbow index $n-2$ have been characterized in Theorem \ref{thm8}, it remains to characterize the graphs with 3-rainbow index $n-2$ which have cut edges. Notice that the cut edges of a graph must be assigned with distinct colors, our main purpose is to check out how the addition of cut edges to $G$ affect the 3-rainbow index of a 2-connedted graph $G$ when $rx_3(G)=n-2$. In other words, share the colors of cut edges with the colors of the non-cut edges as many as possible.
Given a connected graph $G$ of order $n$, and a coloring $c$ of $G$, we always let $A_1$ be the set of colors assigned to the non-cut edges of $G$ and $A_2$ the set of colors assigned to the cut edges of $G$. For each positive integer $k$, let $N_k=\{1,2,\cdots,k\}$. We always set that $A_2=N_s$, where $s$ is the number of cut edges of $G$. Note that, $A_1$ and $A_2$ may intersect and suppose $|A_1\cap A_2|=p$. We can interchange the colors of cut edges suitably such that $A_1\cap A_2=\{1,2,\cdots,p\}$. Set $A_1\setminus A_2=\{a_1,\cdots, a_t\}$, $t\leq m-s$ and $a_j\in N_{|c|}$.
For a connected graph $G$, a $block$ is a maximal 2-connected subgraph. In this paper, we regard $K_2$ other than a block. An $internal~ cut~ edge$ is a cut edge which is on the unique path joining some two blocks. Denote the cut edges of $G$ by $e_1=x_1y_1,\cdots,e_p=x_py_p$ and the colors of these cut edges by $1,\cdots,p$, respectively. Moreover, if $x_iy_i$ is not an internal cut edge, we always set $d(x_i,B)\leq d(y_i,B)$ where $B$ is an arbitrary block.
Let $H$ be a connected subgraph of $G$, denote by $i\in H$ if the color $i$ appears in $H$. Given a graph $G$, let $G_0$ be its basic graph. Deleting the corresponding edges of $G_0$ in $G$, we obtain a forest. Each component corresponds to a vertex $v$ in $G_0$, denoted by $T(v)$. Denote by $U(v)$ the number of leaves of $T(v)$ in $G$ and $U(G)=\sum_{v\in V(G)}U(v)$. Let $W(v)$ be the number of edges of $T(v)$ whose colors are appeared in $A_1$, that is, $W(v)=|c(T(v))\cap A_1|$.
\subsubsection{Bicyclic graphs with $rx_3(G)\bf{=n-2}$} First, we introduce some graph classes. Let $G_i$ be the graphs shown in Figure 1, define by $\mathcal{G}^{*}_i$ the set of graphs whose basic graph is $G_i$, where $1\leq i\leq 6$. Set
$\mathcal{G}_1=\{G\in \mathcal{G}^{*}_1|U(v_3)\leq 1\}$,
$\mathcal{G}_2=\{G\in \mathcal{G}^{*}_2|U(v_3)+U(v_i)\leq 1, ~i=4,6\}$, $\mathcal{G}_3=\{G\in \mathcal{G}^{*}_3|U(v_i)+U(v_j)\leq 2,~v_iv_j\in E(G_3)\}$, $\mathcal{G}_4=\{G\in \mathcal{G}^{*}_4|U(v_i)\leq 2,~i=1,3\}$, $\mathcal{G}_5=\{G\in \mathcal{G}^{*}_5|U(v_2)+U(v_3)\leq 2,~U(v_4)+U(v_5)\leq 2\}$, $\mathcal{G}_6=\{G\in \mathcal{G}^{*}_6|U(v_2)=U(v_6)=0,~U(v_4)\leq 1,~U(v_4)+U(v_i)\leq 2,~i=3,5 \}$ and set $\mathcal{G}=\{\mathcal{G}_1,\mathcal{G}_2,\cdots,\mathcal{G}_6\}$.
\begin{lem}\label{lem8} Let $G$ be a connected bicyclic graph of order $n$. Then $rx_3(G)=n-2$ if and only if $G\in \mathcal{G}$. \end{lem}
\begin{proof}
Suppose that $G$ is a graph with $rx_3(G)=n-2$ but $G\notin \mathcal{G}$. Let $G_0$ be the basic graph of $G$, then $G_0$ is a 2-edge-connected bicyclic graph. If $G_0\neq G_i$, by Theorem \ref{thm8}, $rx_3(G_0)\leq |G_0|-3$. Moreover, by Lemma \ref{lem1}, we have $rx_3(G)\leq rx_3(G_0)+|G|-|G_0|\leq n-3$. Hence $G_0=G_i$. Next we show that if $G\in \mathcal{G}^{*}_i\setminus \mathcal{G}_i$, then $rx_3(G)\leq n-3$, where $1\leq i\leq 6$. As pointed out before, all the cut edges of $G$ are colored with $1,2,\cdots$. We only provide a coloring $c_{\ell}$ of $G_0$, namely, color the corresponding edges of $G$, with parts of colors used in cut edges, and the position of cut edges will be determined as following: $\{1,2,\cdots,q\}\subseteq T(v)$ means to assign colors $\{1,2,\cdots,q\}$ to $q$ leaves of $T(v)$ arbitrarily. If $G\in \mathcal{G}^{*}_1\setminus \mathcal{G}_1$, then $U(v_3)\geq 2$, set $c_{\ell}(G_1)=1a_1a_2a_2a_12$ and $\{1,2\}\subseteq T(v_3)$. If $G\in \mathcal{G}^{*}_2\setminus \mathcal{G}_2$, then $U(v_3)+U(v_4)\geq 2$ or $U(v_3)+U(v_6)\geq 2$. By contracting $v_3v_4$ or $v_3v_6$, we obtain a graph $G'$ belonging to $\mathcal{G}^{*}_1\setminus \mathcal{G}_1$. Then the coloring of $G$ can be obtained easily from $G'$ by Lemma \ref{lem1}. If $G\in \mathcal{G}^{*}_3\setminus \mathcal{G}_3$, then there is an edge $v_iv_j\in E(G_3)$ such that $U(v_i)+U(v_j)\geq 3$. By symmetry, there exist four cases for $G$: (1) $U(v_1)\geq3$; (2) $U(v_1)\geq2$, $U(v_2)\geq1$; (3) $U(v_1)\geq1$, $U(v_2)\geq2$; (4) $U(v_2)\geq3$. Set $c_{\ell}(G_3)=a_1a_2a_2123$ and set $\{1,2,3\}\subseteq T(v_1)$ for (1); $\{1\}\subseteq T(v_2)$, $\{2,3\}\subseteq T(v_1)$ for (2); $\{1\}\subseteq T(v_1)$, $\{2,3\}\subseteq T(v_2)$ for (3); $\{1,2,3\}\subseteq T(v_2)$ for (4). If $G\in \mathcal{G}^{*}_4\setminus \mathcal{G}_4$, then $U(v_1)\geq3$ or $U(v_3)\geq3$. By symmetry, suppose $U(v_1)\geq 3$ and set $c_{\ell}(G_4)=123a_1a_1$ and $\{1,2,3\}\subseteq T(v_1)$. If $G\in \mathcal{G}^{*}_5\setminus \mathcal{G}_5$, then by contracting $v_2v_3$ or $v_4v_5$, we obtain a graph $G'$ belonging to $\mathcal{G}^{*}_4\setminus \mathcal{G}_4$. Now consider $G\in \mathcal{G}^{*}_6\setminus \mathcal{G}_6$. Then $U(v_2)\geq 1$, or $U(v_6)\geq 1$, or $U(v_4)\geq 2$, or $U(v_4)+U(v_3)\geq 3$, or $U(v_4)+U(v_5)\geq 3$. For the last two cases, it belongs to $\mathcal{G}^{*}_5\setminus \mathcal{G}_5$ by contracting $v_3v_4$ or $v_4v_5$. If $U(v_2)\geq1$, set $c_{\ell}(G_6)=a_3a_2a_4a_4a_2a_31$ and $\{1\}\subseteq T(v_2)$. If $U(v_4)\geq2$, set $c_{\ell}(G_6)=a_31a_22a_1a_2a_1$ and $\{1,2\}\subseteq T(v_4)$. It is not hard to check that the colorings above make $G$ rainbow connected with $n-3$ colors, thus $rx_3(G)\leq n-3$.
Conversely, let $G$ be a bicyclic graph such that $G\in \mathcal{G}$. Assume, to the contrary, that $rx_3(G)\leq n-3$. Then there exists a rainbow coloring $c$ such that $A_1\cup A_2=N_{n-3}$. By Theorem \ref{thm8}, we focus on the graphs with cut edges and $|A_1\cap A_2|\geq 1$. We write $d_{G_i}(u,v,w)$ to mean that the number of edges of a $\{u,v,w\}$-tree in $G$ which correspond to the edges of $G_i$, the basic of $G$. We divide into three cases.
{\bf Case 1.}~~$G\in \mathcal{G}_1\cup \mathcal{G}_2$.
First assume that $G\in \mathcal{G}_2$ and we give the following claims. If there is a nontrivial path $P_{\ell}$ connecting $B_1$ and $B_2$ in $G$, then denote its ends by $v_3'(\in B_1)$ and $v_3''(\in B_2)$.
{\bf Claim 1.}~~Each block $B_i$ has at most one edge use the color from $A_2$, where $i\in \{1,2\}$. Moreover, if a color of $A_2$ appears in $B_i$, then the other edges of $B_i$ must be assigned with different colors in $A_1\setminus A_2$.
\emph{Proof.}~Suppose two edges of $B_1$ are colored with 1, 2, respectively. We also set $d(x_i,B_1)\leq d(y_i,B_1)$, where $x_iy_i$ belongs to $P_{\ell}$. Since the cut edges colored with 1 and 2 should be contained in the rainbow tree whose vertices contain $y_1$ and $y_2$, by deleting the edges assigned with 1 and 2 in $B_1$, $G$ is disconnected. Let $w$ be a vertex in the component that does not contain $y_1$, then there is no rainbow tree connecting $\{y_1,y_2,w\}$, a contradiction. We can take the similar argument for the other cases when two edges of $B_1$ $(B_2)$ are colored with 1 or two edges of $B_2$ are colored with 1, 2, respectively.
Now suppose $1\in B_i\cap A_2$ and two edges of $B_i$ have the same color $a_1$. Let $w_1$, $w_2$ be the end vertices of the edge assigned with 1, then $\{y_1,w_1,w_2\}$ has no rainbow tree. $\square$
{\bf Claim 2}~~The colors of the path $P_{\ell}$ can not appear in $A_1$.
\emph{Proof.}~ Assume $e$ is the edge of $P_{\ell}$ colored with 1. The color 1 can not appear in $B_1$. Otherwise suppose the three edges of $B_1$ are assigned with 1, $a_1$ and $a_2$, respectively. Consider $\{v_1,v_2,v_5\}$, then $c(v_3^{''}v_4),c(v_4v_5)\in \{2,a_3\}$ or $c(v_3^{''}v_6),c(v_5v_6)\in \{2,a_3\}$. Without loss of generality, suppose $c(v_3^{''}v_4),c(v_4v_5)\in \{2,a_3\}$, then by Claim 1, $c(v_3^{''}v_6),c(v_5v_6)\in \{a_1,a_2\}$, thus $\{v_1,v_2,v_6\}$ has no rainbow tree. On the other hand, 1 can not be in $B_2$. It is easy to see that neither $c(v_3^{''}v_4)$ nor $c(v_3^{''}v_6)$ can be 1 by considering $\{v_1,v_2,v_6\}$. If $c(v_5v_6)=1$, consider $\{v_1,v_5,v_6\}$, $\{v_2,v_5,v_6\}$, then $c(v_1v_3^{'}),c(v_2v_3^{'})\in A_2$, a contradiction to Claim 1. $\square$
By Claim 1, we have $1\leq|A_1\cap A_2|\leq2$ and only color 1 and 2 can exist in $A_1$. We should discuss all the situations according to which cut edges are colored with 1, 2. By the definition of $G$, $U(v_3)=1$ or $U(v_3)=0$. By similarity, we only deal with the former case. First assume $|A_1\cap A_2|=1$, then $A_1=\{1,a_1,a_2,a_3\}$. We consider the subcase when $1\in T(v_3)$. In this case we claim that the color $1$ appears in neither $B_1$ nor $B_2$. Indeed, if $c(v_3''v_6)=1$, since every tree whose vertices contain $y_1$ must contain the cut edge colored with 1, $d_{G_2}(y_1,v_1,v_6)=4$. Thus $\{y_1,v_1,v_6\}$ has no rainbow tree. If now $c(v_5v_6)=1$, then consider $\{y_1,v_5,v_6\}$, $\{y_1,v_5,v_1\}$, $\{y_1,v_5,v_2\}$ successively, we have $c(v_1v_3')=c(v_2v_3')=c(v_3''v_6)$, leading to a contradiction when considering $\{v_1,v_2,v_6\}$. Else if $c(v_1v_3')=1$, then $\{y_1,v_1,v_5\}$ has no rainbow tree. The last possibility is that $c(v_1v_2)=1$, we may set $c(v_1v_3)=a_1$, $c(v_2v_3)=a_2$. Consider $\{y_1,v_1,v_4\}$, $\{y_1,v_2,v_4\}$, $\{y_1,v_1,v_6\}$, $\{y_1,v_2,v_6\}$ successively, we have $c(v_3''v_4)=c(v_3''v_6)=a_3$ and 1 can not appear in $B_2$, hence $\{v_1,v_4,v_6\}$ has no rainbow tree. The other subcases are similar.
Thus $|A_1\cap A_2|=2$, $A_1=\{1,2,a_1,a_2,a_3\}$. By Claim 1, set $1\in B_1$, $2\in B_2$, and the other edges in each block have distinct colors. If $1,2\in T(v_3)$, assume that $d(y_1,T(v_3))>d(y_2,T(v_3))$, there always exist two vertices which come from different blocks such that there is no rainbow tree connecting them and $y_1$. If $1\in T(v_3)$, $2\in T(v_1)$, the most difficult case is that $c(v_1v_2)=1$, $c(v_5v_6)=2$. In this case, consider $\{y_2,v_5,v_6\}$, forcing that one of $v_1v_3'$, $v_3''v_4$, $v_3''v_6$, $v_4v_5$ is colored with 1, contradicting to Claim 1. With an analogous argument, we would get a contradiction if 1, 2 are in other cut edges of $G$.
For $G\in \mathcal{G}_1$, it can be obtained by contracting an edge of a graph in $\mathcal{G}_2$. Then by Lemma \ref{lem1}, $rx_3(G)\geq n-2$.
{\bf Case 2.}~~$G\in \mathcal{G}_3$.
First note that each path from $v_1$ to $v_5$ in $G_3$ can have at most one color in $A_2$. Thus $|A_1\cap A_2|\leq3$. On the other hand, noticing that $d_{G_3}(v_2,v_3,v_4)=3> 2$, all the cases satisfying $W(v_1)=W(v_5)=0$ and $W(v_2)$, $W(v_3)$, $W(v_4)\leq 1$ are easy to get a contradiction, so we omit them here.
First assume $|A_1\cap A_2|=1$, then $A_1=\{1,a_1,a_2\}$. If $1\in T(v_1)$, consider $\{y_1,v_2,v_3\}$, $\{y_1,v_2,v_4\}$ and $\{y_1,v_3,v_4\}$ successively, $v_1v_2$, $v_1v_3$, $v_1v_4$ must be colored with distinct colors from $A_1\setminus \{1\}$, which is impossible.
Assume now $|A_1\cap A_2|=2$, then $A_1=\{1,2,a_1,a_2\}$. If $1,2\in T(v_1)$, then consider $\{y_1,y_2,v_5\}$, without loss of generality, set $c(v_1v_2)=a_1$, $c(v_2v_5)=a_2$. Thus $c(v_1v_3)$ can be neither 1 nor 2, otherwise there is no rainbow $\{y_1,y_2,v_3\}$-tree. On the other hand, $c(v_1v_3)$ cannot be $a_1$, otherwise $c(v_3v_5)=i$($i=1,2$), then $\{y_i,v_2,v_3\}$ has no rainbow tree. Meanwhile, $v_1v_3$ cannot be colored with $a_2$, otherwise $c(v_3v_5)=i$($i=1,2$), then $\{y_i,v_3,v_5\}$ has no rainbow tree. If $1\in T(v_1)$, $2\in T(v_5)$, then every path from $v_1$ to $v_5$ must color $\{a_1,a_2\}$, a contradiction to $|A_1\cap A_2|=3$. If $1,2\in T(v_2)$, then by the same reason, we conclude that $c(v_1v_2)$, $c(v_2v_5)\notin \{1,2\}$ and we may set $c(v_1v_3)=1$. But now $\{y_1,v_1,v_3\}$ has no rainbow tree. If $1\in T(v_1)$, $2\in T(v_2)$. By considering $\{y_1,y_2,v_3\}$, $\{y_1,y_2,v_4\}$, $\{y_1,y_2,v_5\}$, we may set $c(v_1v_3)=c(v_1v_4)=c(v_2v_5)=a_1$, this force $c(v_3v_5)=i$($i=1,2$). However, there is no rainbow tree connecting $\{y_i,v_3,v_5\}$.
Thus $|A_1\cap A_2|=3$, $A_1=\{1,2,3,a_1,a_2\}$. If $1,2,3\in T(v_1)$, since $U(v_1)\leq 2$, we may assume that $y_2$ is on the unique path from $y_1$ to $v_1$. Thus one path from $v_1$ to $v_5$ must be colored with $\{a_1,a_2\}$, a contradiction to $|A_1\cap A_2|=3$. If $1,2\in T(v_2)$, $3\in T(v_5)$, and without loss of generality, $y_2$ is on the unique path from $y_1$ to $v_2$. Considering $\{y_1,v_3,y_3\}$ and $\{y_1,v_4,y_3\}$, we may set $c(v_1v_2)=a_1$, $c(v_1v_3)=c(v_1v_4)=a_2$. But there is no rainbow $\{y_1,v_3,v_4\}$-tree. Each other case is similar or easier.
{\bf Case 3.}~~$G\in \mathcal{G}_4\cup\mathcal{G}_5\cup\mathcal{G}_6$.
First let $G\in \mathcal{G}_6$. Similarly, each path from $v_2$ to $v_6$ in $G_6$ can have at most one color in $A_2$. Thus we have $1\leq|A_1\cap A_2|\leq3$. Assume first $|A_1\cap A_2|=1$, $A_1=\{1,a_1,a_2,a_3\}$.
We only focus on the case that $1\in T(v_4)$. To make sure there are rainbow trees connecting $\{y_1,v_1,v_3\}$ and $\{y_1,v_1,v_5\}$, only $c(v_2v_6)$ can be 1, but now $\{y_1,v_2,v_6\}$ has no rainbow tree.
Assume then $|A_1\cap A_2|=2$, $A_1=\{1,2,a_1,a_2,a_3\}$. If $1,2\in T(v_1)$, we may set $c(v_1v_2)=a_1$, $c(v_2v_3)=a_2$, $c(v_3v_4)=a_3$ by considering $\{y_1,y_2,v_4\}$. $c(v_5v_6)$ can be neither 1 nor 2, otherwise $\{y_1,y_2,v_5\}$ has no rainbow tree. Moreover, $c(v_4v_5)$ can be neither 1 nor 2, otherwise when $c(v_4v_5)=i$($i=1,2$), there is no rainbow $\{y_i,v_4,v_5\}$-tree. Thus $v_1v_6$, $v_2v_6$ must use colors $\{1,2\}$, but now $\{y_1,y_2,v_6\}$ has no rainbow tree. If $1,2\in T(v_3)$, first we claim that at most one edge of the triangle $v_1v_2v_6$ uses a color from $\{1,2\}$. Otherwise if $c(v_1v_2)$, $c(v_2v_6)\in \{1,2\}$, then $\{y_1,y_2,v_1\}$ has no rainbow tree. If $c(v_1v_6)$, $c(v_2v_6)\in \{1,2\}$, the rest non-cut edges must color $\{a_1,a_2,a_3\}$. It is easy to verify that either $\{y_1,v_1,v_5\}$ or $\{y_1,v_1,v_4\}$ has no rainbow tree. So the longest path from $v_2$ to $v_6$ has an edge colored with 1 or 2. However, we will show that it is impossible. It is easy to check that $c(v_2v_3),c(v_3v_4)\notin \{1,2\}$. If $c(v_5v_6)\in \{1,2\}$, then we may set $c(v_5v_6)=1$. Consider $\{y_1,v_1,v_5\}$ and $\{y_1,v_5,v_6\}$, then $c(v_1v_2)=c(v_2v_6)=2$, a contradiction. It is similar to check that $c(v_4v_5)$ can not be 1 or 2, a contradiction.
Now assume that $|A_1\cap A_2|=3$, $A_1=\{1,2,3,a_1,a_2,a_3\}$. If $1\in T(v_1)$, $2,3\in T(v_3)$. Again, we may set $c(v_1v_2)=a_1$, $c(v_2v_3)=a_2$. Thus $c(v_1v_6),c(v_2v_6)\in \{1,2,3\}$. If $c(v_1v_6),c(v_2v_6)\in \{1,i\}$, then there is a contradiction by considering $\{y_1,y_i,v_6\}$ ($i=2,3$). Thus we may set that $c(v_1v_6)=2$, $c(v_2v_6)=3$. By considering $\{y_1,y_3,v_4\}$ and $\{y_1,y_3,v_5\}$, we get that $c(v_3v_4)=c(v_5v_6)=a_3$, $c(v_4v_5)=1$, but now $\{y_1,v_4,v_5\}$ has no rainbow tree. If $1\in T(v_3)$, $2,3\in T(v_5)$, then we set $v_3v_4=a_1$, $v_4v_5=a_2$. If $c(v_2v_6)=i$, $c(v_5v_6)=j$, $i,j\in\{1,2,3\}$, then $\{y_i,y_j,v_6\}$ has no rainbow tree. The only possibility is $c(v_2v_3)=2$, $c(v_2v_6)=3$, $c(v_5v_6)=a_3$. However, $\{y_1,y_2,v_1\}$ has no rainbow tree.
For $G\in \mathcal{G}_5$, notice that $|A_1\cap A_2|\leq3$. If $U(v_2)=0$ or $U(v_5)=0$, then $G$ can be obtained by contracting an edge of a graph in $\mathcal{G}_6$. Then by Lemma \ref{lem1}, $rx_3(G)\geq n-2$. Thus we need to consider the case when $W(v_2)\geq 1$ and $W(v_5)\geq 1$. If $|A_1\cap A_2|=2$, then suppose $1\in T(v_2)$, $2\in T(v_5)$. Consider $\{y_1,y_2,v_3\}$, $\{y_1,y_2,v_4\}$, we have $c(v_2v_3)$, $c(v_2v_5)$, $c(v_4v_5)\in \{a_1,a_2\}$ and $c(v_2v_3)=c(v_4v_5)$. But now $c(v_3v_4)=i$ ($i=1,2$), then there is no rainbow $\{y_i,v_3,v_4\}$-tree. If $|A_1\cap A_2|=3$, then $A_1=\{1,2,3,a_1,a_2\}$. If $1\in T(v_1)$, $2\in T(v_2)$, $3\in T(v_5)$, then consider $\{y_1,y_2,y_3\}$, we have that two of $v_1v_2$, $v_1v_5$, $v_2v_5$ have colors outside $A_2$, contradicting to $|A_1\cap A_2|=3$. If $1\in T(v_2)$, $2\in T(v_5)$, $3\in T(v_2)$ and we may assume that $y_3$ is on the unique path from $y_1$ to $v_2$. Then consider $\{y_1,v_3,y_3\}$ and $\{y_1,v_4,y_3\}$, we have $c(v_2v_3)=c(v_4v_5)$, thus $c(v_3v_4)$ can not be in $A_2$, contradicting to $|A_1\cap A_2|=3$. If $1\in T(v_2)$, $2\in T(v_5)$, $3\in T(v_i)$ ($i=3,4$), then consider $\{y_1,y_2,y_3\}$, we have that $c(v_2v_5)$ is in $A_1\setminus A_2$, contradicting to $|A_1\cap A_2|=3$.
Finally, for a graph $G$ belonging to $\mathcal{G}_4$, it can be obtained by contracting an edge of a graph in $\mathcal{G}_3\cup \mathcal{G}_6$. Then by Lemma \ref{lem1}, $rx_3(G)\geq n-2$.
Combining all the cases above, we have $rx_3(G)\geq n-2$ for $G\in \mathcal{G}$. By Theorem \ref{thm4}, it follows that $rx_3(G)=n-2$. \end{proof}
\subsubsection{Tricyclic graphs with $rx_3(G)\bf{=n-2}$} Define by $\mathcal{H}^{*}_i$ the set of graphs whose basic graph is $H_i$, where $H_i$ is shown in Figure 2 and $1\leq i\leq 8$.
Now, we introduce another graph class $\mathcal{H}$. Set $\mathcal{H}=\{\mathcal{H}_1,\mathcal{H}_2,\cdots,\mathcal{H}_8\}$, where
$\mathcal{H}_1=\{G\in \mathcal{H}^{*}_1|U(G)=0\}$,
$\mathcal{H}_2=\{G\in \mathcal{H}^{*}_2|U(v_i)\leq 1, ~U(v_j)=0,~i=5,6,~j=1,3,4\}$,
$\mathcal{H}_3=\{G\in \mathcal{H}^{*}_3|U(v_2)\leq 1,~ U(v_5)+U(v_6)\leq 1,~ U(v_i)=0,~i=1,3,4\}$, $\mathcal{H}_4=\{G\in \mathcal{H}^{*}_4|U(v_i)\leq 1,~ U(v_j)\leq 2, ~U(v_i)+U(v_j)\leq 1,~U(v_j)+U(v_k)\leq 3,~ i=1,5,~ j,k=2,3,4\}$,
$\mathcal{H}_5=\{G\in \mathcal{H}^{*}_5|U(v_i)\leq 1,~U(v_j)=0,~i=1,3,5,~j=2,4,6\}$, $\mathcal{H}_6=\{G\in \mathcal{H}^{*}_6|U(v_3)=0,~ U(v_i)\leq 1, ~ U(v_1)+U(v_5)\leq 1,~ i=1,2,4,5\}$,
$\mathcal{H}_7=\{G\in \mathcal{H}^{*}_2|U(v_2)+U(v_4)\leq 1, ~U(v_3)+U(v_5)\leq 1,~ U(v_5)+U(v_1)\leq 1,~ U(v_j)+U(v_{j+1})\leq 1,~ j=1,2,4\}$,
$\mathcal{H}_8=\{G\in \mathcal{H}^{*}_8|U(v_i)\leq 2,~U(v_i)+U(v_j)+U(v_k)\leq 3, ~i,j,k=1,2,3,4\}$.
\begin{lem}\label{lem9} Let $G$ be a connected tricyclic graph of order $n$. Then $rx_3(G)=n-2$ if and only if $G\in \mathcal{H}$. \end{lem} \begin{proof} Suppose that $rx_3(G)=n-2$ but $G\notin \mathcal{H}$. Let $G_0$ be the basic graph of $G$. Similar to Lemma \ref{lem8}, we have $G_0=H_i$ and we can rainbow color $G$ with $n-3$ colors for $G\in \mathcal{H}_i^{*}\setminus \mathcal{H}_i$, $i=1,\cdots,8$.
If $G\in \mathcal{H}_1^{*}\setminus \mathcal{H}_1$, then if $U(v_2)\geq 1$, set $c_{\ell}(H_1)=a_41a_1a_2a_2a_4a_3a_3a_4$ and $\{1\}\subseteq T(v_2)$; if $U(v_3)\geq 1$, set $c_{\ell}(H_1)=a_4a_1a_11a_3a_4a_2a_2a_4$ and $\{1\}\subseteq T(v_3)$; if $U(v_4)\geq 1$, set $c_{\ell}(H_1)=a_3a_2a_2a_11a_3a_4a_4a_1$ and $\{1\}\subseteq T(v_4)$.
If $G\in \mathcal{H}_2^{*}\setminus \mathcal{H}_2$, then if $U(v_3)\geq 1$ ($U(v_1)\geq 1$ is similar), set $c_{\ell}(H_2)=a_3a_21a_2a_3a_1a_1a_2$ and $\{1\}\subseteq T(v_3)$; if $U(v_4)\geq 1$, set $c_{\ell}(H_2)=a_1a_3a_21a_2a_3a_3a_1$ and $\{1\}\subseteq T(v_4)$; if $U(v_6)\geq 2$ ($U(v_5)\geq 2$ is similar), set $c_{\ell}(H_2)=a_31a_22a_2a_3a_11$ and $\{1,2\}\subseteq T(v_3)$.
If $G\in \mathcal{H}_3^{*}\setminus \mathcal{H}_3$, then if $U(v_2)\geq 2$, set $c_{\ell}(H_3)=a_12a_2a_32a_2a_11$ and $\{1,2\}\subseteq T(v_2)$; if $U(v_5)\geq 2$, set $c_{\ell}(H_3)=2a_2a_31a_2a_11a_3$ and $\{1,2\}\subseteq T(v_5)$; if $U(v_5)+U(v_6)\geq 2$, set $c_{\ell}(H_3)=1a_12a_2a_3a_1a_2a_3$ and $\{1\}\subseteq T(v_6)$, $\{2\}\subseteq T(v_5)$; if $U(v_3)\geq 1$, set $c_{\ell}(H_3)=a_1a_2a_21a_1a_3a_3a_2$ and $\{1\}\subseteq T(v_3)$; if $U(v_4)\geq 1$, set $c_{\ell}(H_3)=a_2a_11a_1a_2a_3a_3a_2$ and $\{1\}\subseteq T(v_4)$.
If $G\in \mathcal{H}_4^{*}\setminus \mathcal{H}_4$, then there are four cases for the graph $G$: (1) $U(v_i)\geq 3$ ($i=2,3,4$); (2) $U(v_i)\geq 2$ ($i=1,5$); (3) $U(v_i)+U(v_j)\geq 2$ ($i\in\{1,5\}$, $j\in\{2,3,4\}$); (4) $U(v_i)\geq 2$ and $U(v_j)\geq 2$, $i,j\in\{2,3,4\}$. If $G$ is a graph in case (1), then there exists a graph in $\mathcal{G}_3^{*}\setminus \mathcal{G}_3$ which is a subgraph of $G$. Thus the result is obvious. If $U(v_1)\geq 2$, set $c_{\ell}(H_4)=a_2a_2a_1a_121a_2$ and $\{1,2\}\subseteq T(v_1)$; if $U(v_1)+U(v_2)\geq 2$, set $c_{\ell}(H_4)=a_1a_2a_22a_21a_1$ and $\{1\}\subseteq T(v_1)$, $\{2\}\subseteq T(v_2)$; if $U(v_2)\geq 2$ and $U(v_4)\geq 2$, set $c_{\ell}(H_4)=3421a_2a_2a_1$ and $\{1,2\}\subseteq T(v_2)$, $\{3,4\}\subseteq T(v_4)$.
If $G\in \mathcal{H}_5^{*}\setminus \mathcal{H}_5$, then $U(v_i)\geq 1$ ($i=2,4,6$) or $U(v_i)\geq 2$ ($i=1,3,5$). If $G$ is a graph in the former case, there exists a graph in $\mathcal{G}_6^{*}\setminus \mathcal{G}_6$ which is a subgraph of $G$. If $U(v_3)\geq 2$, set $c_{\ell}(H_5)=a_22a_1a_2a_31a_3a_2$ and $\{1,2\}\subseteq T(v_3)$; if $U(v_5)\geq 2$, set $c_{\ell}(H_5)=1a_22a_1a_3a_2a_3a_1$ and $\{1,2\}\subseteq T(v_5)$;
If $G\in \mathcal{H}_6^{*}\setminus \mathcal{H}_6$, then $U(v_3)\geq 1$ or $U(v_i)\geq 2$ ($i=1,2,4,5$) or $U(v_1)+U(v_5)\geq 2$. If $U(v_3)\geq 1$, set $c_{\ell}(H_6)=a_2a_11a_1a_2a_2a_1$ and $\{1\}\subseteq T(v_3)$; if $U(v_1)\geq 2$, set $c_{\ell}(H_6)=a_2a_1a_11a_212$ and $\{1,2\}\subseteq T(v_1)$; if $U(v_2)\geq 2$, set $c_{\ell}(H_6)=a_21a_1a_1a_212$ and $\{1,2\}\subseteq T(v_2)$; if $U(v_1)+U(v_5)\geq 2$, set $c_{\ell}(H_6)=a_1a_1a_2a_221a_1$ and $\{1\}\subseteq T(v_1)$, $\{2\}\subseteq T(v_5)$;
If $G\in \mathcal{H}_7^{*}\setminus \mathcal{H}_7$, then $U(v_i)\geq 2$ ($i=1,2,3,4,5$) or $U(v_1)+U(v_2)\geq 2$ or $U(v_2)+U(v_3)\geq 2$ or $U(v_4)+U(v_5)\geq 2$ or $U(v_2)+U(v_4)\geq 2$ or $U(v_3)+U(v_5)\geq 2$ or $U(v_1)+U(v_5)\geq 2$. If $U(v_1)\geq 2$, set $c_{\ell}(H_7)=a_1a_2a_212a_1a_1$ and $\{1,2\}\subseteq T(v_3)$; if $U(v_2)\geq 2$, set $c_{\ell}(H_7)=a_2a_1a_1a_1a_212$ and $\{1,2\}\subseteq T(v_2)$; if $U(v_3)\geq 2$, set $c_{\ell}(H_7)=a_2a_1a_12a_1a_21$ and $\{1,2\}\subseteq T(v_3)$; if $U(v_1)+U(v_2)\geq 2$, set $c_{\ell}(H_7)=a_2a_1a_1a_1a_221$ and $\{1\}\subseteq T(v_1)$, $\{2\}\subseteq T(v_2)$; if $U(v_2)+U(v_3)\geq 2$, set $c_{\ell}(H_7)=a_2a_1a_12a_2a_21$ and $\{1\}\subseteq T(v_2)$, $\{2\}\subseteq T(v_3)$; if $U(v_2)+U(v_4)\geq 2$, set $c_{\ell}(H_7)=a_212a_1a_2a_1a_2$ and $\{1\}\subseteq T(v_2)$, $\{2\}\subseteq T(v_4)$;
If $G\in \mathcal{H}_8^{*}\setminus \mathcal{H}_8$, then $U(v_i)\geq 3$ ($i=1,2,3,4$) or $U(v_i)+U(v_j)+U(v_k)\geq 4, ~i,j,k=1,2,3,4$. If $G$ is a graph in the former case, then a graph belonging to $\mathcal{G}_4^{*}\setminus \mathcal{G}_4$ is a subgraph of $G$. If $U(v_1)+U(v_2)+U(v_4)\geq 4$, set $c_{\ell}(H_8)=1a_1a_1423$ and $\{1,2\}\subseteq T(v_1)$, $\{3\}\subseteq T(v_2)$, $\{4\}\subseteq T(v_4)$; if $U(v_2)+U(v_3)\geq 4$, set $c_{\ell}(H_8)=12a_1a_134$ and $\{1,2\}\subseteq T(v_2)$, $\{3,4\}\subseteq T(v_3)$.
It is not hard to check that the colorings above make $G$ rainbow connected with $n-3$ colors, thus $rx_3(G)\leq n-3$.
Conversely, let $G$ be a tricyclic graph such that $G\in \mathcal{H}$. Similar to Lemma \ref{lem8}, we only need to consider the case that $G$ has cut edges and $|A_1\cap A_2|\geq 1$. Assume, to the contrary, that $rx_3(G)\leq n-3$. Then there exists a rainbow coloring $c$ of $G$ using colors in $N_{n-3}$.
For $G\in \mathcal{H}_1$, if there is a nontrivial path $P'$ connecting $B_1$ and $B_2$ in $G$, then denote its ends by $v_3'(\in B_1)$ and $v_3''(\in B_2)$ and if there is a nontrivial path $P''$ connecting $B_2$ and $B_3$ in $G$, then denote its ends by $v_5'(\in B_2)$ and $v_5''(\in B_3)$. Similar to Claim 2 in Lemma \ref{lem8}, the colors in the path $P^{'}$ and $P^{''}$ can not appear in $A_1$, which implies $|A_1\cap A_2|=0$, contradicting to $|A_1\cap A_2|\geq 1$. For $G\in \mathcal{H}_5$, notice that $d_{H_5}(v_1,v_3,v_5)= 4$ and $|A_1\setminus A_2|=3$, the result holds. The same argument applies to the case when $G\in \mathcal{H}_6$. Thus, we mainly discuss the rest cases for $G$ as follows.
{\bf Case 1.}~~$G\in \mathcal{H}_2$. we have $1\leq|A_1\cap A_2|\leq4$ and $|A_1\setminus A_2|=3$. If there is a nontrivial path $P'$ connecting $B_1$ and $B_2$ in $G$, then denote its ends by $v_4'(\in B_1)$ and $v_4''(\in B_2)$. We can also claim that $c(P')\cap A_1= \emptyset$. Noticing that $d_{H_2}(v_2,v_5,v_6)=4>3$, we only check the case when $W(v_2)\geq 2$. Since the case of $|A_1\cap A_2|=1$ or $|A_1\cap A_2|=4$ is easy to check, we consider the remaining two cases. Assume $|A_1\cap A_2|=2$, $A_1=\{1,2,a_1,a_2,a_3\}$ and $1,2\in T(v_2)$, consider $\{y_1,y_2,v_5\}$ and we may set $c(v_2v_3)=a_1$, $c(v_3v_4')=a_2$, $c(v_4''v_5)=a_3$. If 1 and 2 are in $B_1$, and 1 appears in $v_1v_2$ or $v_1v_4'$, then we have $c(v_4''v_6)=a_3$ by considering $\{y_1,y_2,v_6\}$, and thus $c(v_5v_6)\notin A_2$, but now $\{y_1,v_5,v_6\}$ has no rainbow tree. So one of 1, 2, say 1, is in $B_2$ and $c(v_5v_6)=1$. Now we have $c(v_4''v_6)\neq a_3$ and $c(v_1v_2),c(v_1v_4'),c(v_4''v_6)\in \{a_1,a_2,a_3\}$ by considering $\{y_1,y_2,v_6\}$. Then every $\{y_1,v_5,v_6\}$-tree of size 5 can not have the color 2. Thus there is no rainbow $\{y_1,v_5,v_6\}$-tree.
Assume then $|A_1\cap A_2|=3$ and $A_1=\{1,2,a_1,a_2,a_3\}$. If $1,2,3\in T(v_2)$, first we claim that $v_2v_3,v_3v_4'$ can not use colors from $A_2$ both. Otherwise assume $c(v_2v_3)=1$, $c(v_3v_4)=2$, $c(v_1v_2)=a_1$, $c(v_1v_4')=a_2$, and by considering $\{y_1,y_2,v_5\}$ and $\{y_1,y_2,v_6\}$, we have $c(v_4''v_5)=c(v_4''v_6)=a_3$, and $c(v_5v_6)$ can be $a_1$ or $a_2$. However, there is no rainbow $\{y_1,v_5,v_6\}$-tree or $\{y_2,v_5,v_6\}$-tree. With the same reason, we conclude that exactly one edge of the unique 4-cycle of $H_2$ can be colored with a color from $A_2$. Thus there are two cases by symmetry. If $c(v_1v_2)=1$, $c(v_1v_3)=2$, $c(v_5v_6)=3$. Consider $y_1$ and $y_2$, together with $v_5$, $v_6$ respectively, we have $c(v_4''v_5)=c(v_4''v_6)$, which is impossible. If $c(v_1v_4')=1$, $c(v_1v_3)=2$, $c(v_5v_6)=3$. Consider $y_1$ and $y_3$, together with $v_5$, $v_6$ respectively. suppose $c(v_4''v_5)=a_1$, $c(v_4''v_6)=a_2$ and $c(v_3v_4')=a_3$, $c(v_1v_2),c(v_2v_3)\in \{a_1,a_2\}$, but now there is no rainbow $\{y_3,v_5,v_6\}$-tree. If $1,2\in T(v_2)$, $3\in T(v_6)$, similarly set $c(v_1v_2)=a_1$, $c(v_1v_4')=a_2$, $c(v_4''v_6)=a_3$. First we can easily claim that the color 3 can not appear in $B_2$. Thus there are three possibilities for the color 3. If $c(v_3v_4')=3$, then consider $\{y_1,y_3,v_3\}$ and $\{y_2,y_3,v_3\}$, we have $c(v_1v_3),c(v_2v_3)\in \{1,2\}$. Consider $\{y_1,y_2,v_5\}$, one of $c(v_4''v_5)$ and $c(v_5v_6)$ is $a_3$, but now there is no rainbow $\{y_2,y_3,v_5\}$-tree. The case when $c(v_1v_3)=3$ is similar to the case of $c(v_3v_4')=3$. If $c(v_2v_3)=3$, then similarly we get $c(v_1v_3),c(v_2v_3)\in \{1,2\}$ and one of $c(v_4''v_5)$ and $c(v_5v_6)$ is $a_3$. Consider $\{y_1,y_3,v_5\}$, this forces one of $c(v_4''v_5)$ and $c(v_5v_6)$ is $2$, it is impossible.
{\bf Case 2.}~~$G\in \mathcal{H}_3$. Then $1\leq|A_1\cap A_2|\leq 4$ and $|A_1\setminus A_2|=3$. If there is a nontrivial path $P'$ connecting $B_1$ and $B_2$ in $G$, then denote its ends by $v_4'(\in B_1)$ and $v_4''(\in B_2)$. Similarly, it is easy to check that $c(P^{'})\cap A_1= \emptyset$. We only focus on the case that $|A_1\cap A_2|=1$, where $A_1=\{1,a_1,a_2,a_3\}$. If $1\in T(v_6)$, consider $\{y_1,v_1,v_3\}$, set $c(v_4''v_6)=a_1$, $c(v_1v_4')=a_2$, $c(v_3v_4')=a_3$. Then $c(v_5v_6)\neq 1$, otherwise there is no rainbow tree connecting $\{y_1,v_1,v_5\}$ or $\{y_1,v_3,v_5\}$ depending on $c(v_4''v_5)$. Similarly, $c(v_4''v_5)\neq 1$. Next $c(v_2v_4')\neq 1$ by considering $\{y_1,v_2,v_5\}$. Suppose $c(v_1v_2)=1$, to make sure there is a rainbow $\{y_1,v_1,v_2\}$-tree and $\{y_1,v_2,v_3\}$-tree, we have $c(v_2v_4')=a_3$ and $c(v_2v_3)=a_2$. But now $\{v_2,v_3,v_5\}$ has no rainbow tree. If $1\in T(v_2)$, then consider $\{y_1,v_5,v_6\}$, $\{y_1,v_1,v_5\}$, $\{y_1,v_1,v_6\}$, $\{y_1,v_3,v_5\}$, $\{y_1,v_3,v_6\}$ successively. Set $c(v_2v_4')=a_1$, then $c(v_1v_2)$, $c(v_1v_4')$, $c(v_2v_3)$, $c(v_3v_4')$, $c(v_4''v_5)$ and $c(v_4''v_6)$ can only be $a_2$ or $a_3$. It is easy to check that $\{v_2,v_3,v_5\}$ has no rainbow tree.
{\bf Case 3.}~~$G\in \mathcal{H}_4$. $1\leq |A_1\cap A_2|\leq 4$ and $|A_1\setminus A_2|=2$. First notice that $d_{H_4}(v_2,v_3,v_4)=3$, the case that $W(v_1)=W(v_5)=0$, $W(v_2)$, $W(v_3)$, $W(v_4)\leq1$ is evident. Assume $|A_1\cap A_2|=1$, then $1\in T(v_1)$, the case is similar with the case that $G\in \mathcal{G}_3$ in Lemma \ref{lem8}. Assume now $|A_1\cap A_2|=2$, if $1\in T(v_1)$, $2\in T(v_5)$, by considering all the trees containing $y_1$ and $y_2$, without loss of generality, set $c(v_1v_5)=a_1$, $c(v_1v_2)=1(2)$, $c(v_2v_5)=a_2$. Moreover, by considering $\{y_1(y_2),v_2,v_3\}$ and $\{y_1(y_2),v_2,v_4\}$, the remaining two paths of length 2 from $v_1$ to $v_5$ must be colored with 2(1), $a_2$, respectively. However, there is no rainbow $\{y_2(y_1),v_3,v_4\}$-tree. If $1,2\in T(v_2)$, by considering $\{y_1,y_2,v_4\}$, set $c(v_2v_5)=a_1$, $c(v_4v_5)=a_2$. Since the two possible rainbow trees connecting $\{y_1,v_3,v_4\}$ and $\{y_2,v_3,v_4\}$ are the same, we may set $c(v_3v_5)=1$. It is easy to see that $c(v_1v_2)$, $c(v_1v_3)$ cannot use colors from $A_2$ by considering $\{y_1,y_2,v_3\}$, and $c(v_1v_4)=2$ by considering $\{y_1,v_3,v_4\}$. But now if $c(v_1v_5)=1$ or $c(v_1v_5)=2$, there is no rainbow $\{y_1,v_2,v_5\}$-tree or $\{y_2,v_1,v_4\}$-tree, respectively.
Assume $|A_1\cap A_2|=3$, then $1,2\in T(v_2)$, $3\in T(v_4)$. Similarly as above, we may set $c(v_2v_5)=a_1$, $c(v_4v_5)=a_2$, $c(v_3v_5)=1$, $c(v_1v_5)=3$, one of $c(v_1v_2)$, $c(v_1v_4)$ is 2. However, there is no rainbow $\{y_2,y_3,v_1\}$-tree.
Finally assume $|A_1\cap A_2|=4$ and $1,2\in T(v_2)$, $3\in T(v_3)$, $4\in T(v_4)$, consider $\{y_1,y_3,y_4\}$ and $\{y_2,y_3,y_4\}$, at least four of the non-cut edges must be colored with $\{a_1,a_2\}$. This contradicts to $|A_1\cap A_2|=4$.
{\bf Case 4.}~~$G\in \mathcal{H}_7$. Since $d_{H_7}(v_1,v_3,v_4)=3$, we only focus on the case $1\in T(v_2)$. Consider all the three vertices containing $y_1$, it is not hard to obtain a contradiction.
{\bf Case 5.}~~$G\in \mathcal{H}_8$. First notice $d_{H_8}(v_1,v_2,v_3)=2$, the case that $W(v_1)$, $W(v_2)$, $W(v_3)$, $W(v_4)\leq1$ is evident. Assume $|A_1\cap A_2|=2$, then $1,2\in T(v_1)$. Consider $\{y_1,y_2,v_2\}$, $\{y_1,y_2,v_3\}$, $\{y_1,y_2,v_4\}$ successively, we have $c(v_1v_2)=c(v_1v_3)=c(v_1v_4)=a_1$. However, there is no rainbow tree connecting $\{y_1,v_2,v_3\}$ or $\{y_2,v_2,v_3\}$, a contradiction. Now focus on $|A_1\cap A_2|=3$, then $1,2\in T(v_1)$, $3\in T(v_2)$. Consider $\{y_1,y_3,v_3\}$, $\{y_1,y_3,v_4\}$, $\{y_2,y_3,v_3\}$ and $\{y_2,y_3,v_4\}$ successively, $c(v_1v_3)$, $c(v_1v_4)$, $c(v_2v_3)$, $c(v_2v_4)$ must be 1 or 2. Again, there is no rainbow $\{y_1,y_2,v_3\}$-tree.
By the detailed analysis above, we have $rx_3(G)\geq n-2$ for $G\in \mathcal{H}$. By Theorem \ref{thm4}, it follows that $rx_3(G)=n-2$. \end{proof}
\subsubsection{Characterize the graphs with $rx_3(G)\bf{=n-2}$} We begin with a lemma about a connected 5-cyclic graph.
\begin{lem}\label{lem11} Let $G$ be a connected 5-cyclic graph of order $n$. Then $rx_3(G)=n-2$ if and only if $G=K_5-e$. \end{lem}
\begin{proof} Let $G\neq K_5-e$ and $rx_3(G)=n-2$, by Lemma \ref{lem1} and Theorem \ref{thm8}, $rx_3(G)\leq n-3$, a contradiction. Conversely, suppose $G=K_5-e$, by Theorem \ref{thm3}, $rx_3(G)\geq3$, on the other hand, $rx_3(G)\leq rx_3(C_5)=3$. Thus $rx_3(G)=n-2$.
\end{proof}
For $n\geq 3$, the $wheel$ $W_n$ is a graph constructed by joining a vertex $v_0$ to every vertex of a cycle $C_{n}:v_1,v_2,\cdots,v_{n},v_{n+1}=v_1$.
A third graph class is defined as follows. Let $\mathcal{J}_1$ be a class of graphs such that every graph is obtained from a graph in $\mathcal{H}_5$ by adding an edge $v_4v_6$. Let $\mathcal{J}_2$ be a class of graphs such that every graph is obtained from a graph in $\mathcal{H}_7$ where $U(v_2)=0$ and $U(v_5)=0$ by adding an edge $v_2v_5$. Set $\mathcal{J}=\{\mathcal{J}_1,\mathcal{J}_2, W_4\}$.
Now we are ready to show our second main theorem of this paper.
\begin{thm}\label{thm2} Let $G$ be a connected graph of order $n$ $(n\geq 6)$. Then $rx_3(G)=n-2$ if and only if $G$ is unicyclic with the girth of $G$ at least 4 or $G\in \mathcal{G}\cup \mathcal{H}\cup \mathcal{J}$ or $G=K_5-e$. \end{thm}
\begin{proof} Let $G$ be a $t$-cyclic graph with $rx_3(G)=n-2$, but not a graph listed in the theorem. By Proposition \ref{pro1}, Theorem \ref{thm1}, Lemma \ref{lem8} and Lemma \ref{lem9}, we need to consider the cases $t\geq 4$. If $t=4$, by Theorem \ref{thm8}, the basic graph of $G$ should be a 3-sun or the basic graph of $\mathcal{J}_2$ or $W_4$. If $G\notin \mathcal{J}_1$ or $G\notin \mathcal{J}_2$, then by the similar arguments with Lemma \ref{lem8}, we have $rx_3(G)\leq n-3$, a contradiction. If the basic graph of $G$ is $W_4$ and there are some cut edges in $G$. If $U(v_0)\geq1$, then a graph belonging to $\mathcal{G}_6^{*}\setminus \mathcal{G}_6$ and satisfying $U(v_3)\geq1$ is a subgraph of $G$. If $U(v_1)\geq1$(other cases are similar), then set $c_{\ell}(W_4)=a_21a_1a_1a_1a_2a_2a_1$ and $\{1\}\subseteq T(v_1)$. If $t\geq 5$, by Theorem \ref{thm8}, the basic graph of $G$ should be $K_5-e$, since $n\geq 6$, by the similar argument with $t=4$, we have $rx_3(G)\leq n-3$, a contradiction.
Conversely, by Theorem \ref{thm1}, Theorem \ref{thm3}, Lemma \ref{lem8}, Lemma \ref{lem9} and Lemma \ref{lem11}, suppose $G$ is a graph such that $G\in \mathcal{J}_1$ or $G\in \mathcal{J}_2$. Assume, to the contrary, that $rx_3(G)\leq n-3$. Then there exists a rainbow coloring $c$ of $G$ using $n-3$ colors. Both cases can be considered similar to the case that $G\in \mathcal{H}_5$ or $G\in \mathcal{H}_7$ in Lemma \ref{lem9}, a contradiction. \end{proof}
\end{document} |
\begin{document}
\title{A description of the Thomas-Fermi ion with \ fast converging function series}
\begin{flushleft}
\begin{abstract} This article concerns the description of the electron sea of an atomic ion with the Thomas-Fermi model. The normalized ion radius $X$, ionization potential $b$ and electronic binding energy $B$ of the Thomas-Fermi ion are functions of the ratio $N$ of electrons to protons in the ion. A scheme is given to calculate the Taylor series of $X(N)$, $b(N)$ and $B(N)$. With this scheme, the Taylor coefficients are calculated up to 5th order. The obtained 0th to 3rd order coefficients agree with the values presently available in the literature. To the authors knowledge, the 4th and 5th order coefficients are new results. It is then argued that a different series description of these functions, based on the Taylor series of $c(N) \equiv b^{-1/3}X^{-4/3}$, leads to a significant improvement in convergence. \end{abstract}
\tableofcontents
\section{Introduction}
\subsection{Elementary quantities}
Throughout this article we (the author and the readers) use the normalized units and most of the notation that are standard in texts about the Thomas-Fermi ion \cite{Englert}. We abbreviate \textit{electron} by \textit{e}, and \textit{proton} by \textit{p}.\\ \quad \begin{multicols}{2}
Number of protons: \textit{Z}
Unit of Length: $a_B/1.1295 Z^{1/3}$
Radius from the nucleus: $x$
$e$:$p$-ratio inside radius $x$: $n(x) $
Numerical factor $1.1295 = \left( 128/9 \pi^2 \right)^{1/3}$
Unit of Energy: $2ZRy\cdot 1.1295 Z^{1/3}$
Ion radius: $X$
overall $e$:$p$-ratio: $n(X) \equiv N$ \end{multicols}
Potential energy of an $e$ due to the nucleus alone: $-\frac{1}{x}$
Overall potential energy of an $e$: $ \phi \quad (<0) $
Ionization potential: $b \equiv -\phi(X)$
$e$:$p$ number density: $\rho $
Fermi energy: $\rho^{2/3}$
Fermi surface: $\phi+\rho^{2/3} \quad (<0)$
Electronic binding energy \textit{per proton}: $ B$
\subsection{Outline}
Our goal is to calculate the ion radius $X$, the ionization potential $b$ and the binding energy $B$ as functions of the $e$:$p$-ratio $N$. The steps to this goal are the following.
In section 1 we collect the basic formulas describing the Thomas-Fermi (TF) ion.
In section 2 we express the Thomas-Fermi differential equation (TF DE) with the roles of $x$ and $\chi$ interchanged: the radius $x$ is the dependent variable, the screening number $\chi$ is the independent variable. We solve the new DE in a semi-analytical manner for $x(\chi;a)$, where $a$ is the initial slope of the screening function $\chi(x)$.
In section 3 we will then be able to express $X$, $b$, $B$ and $N$ as Taylor series in $a$.
In section 4 we express $X$, $b$, $B$ and $a$ as Taylor series in $N$.
In section 5 we derive new series for $X(N)$, $b(N)$, $B(N)$ and $a(N)$, with faster convergence.
In section 6 we sum up and plot the results.
\subsection{Description of the TF ion with the potential $\phi$}
The following basic equations of the Thomas-Fermi atom or ion can be found in most textbooks on classical quantum mechanics \cite{Englert}.
Let's start the description of the TF ion with some potential $\phi$, with the constraints \begin{equation} \phi \rightarrow -1/x \quad (x \rightarrow 0) \qquad \qquad \phi= -bX/x \quad (x \geq X) \label{e1.1} \end{equation} Outside the ion radius X, $\phi$ is a Coulomb potential. $b$ is the ionization potential. From electrostatics, we have \begin{equation} b =(1-N)/X \label{e1.2} \end{equation}
The $e$:$p$ number density is given by Poisson's equation: \begin{equation} \rho=-\Delta \left( \phi+\frac{1}{x} \right) \label{e1.3} \end{equation} with $\rho(x)=0$ for $x \geq X$.
By inserting this equation into \begin{equation*} \frac{dn}{dx} = - x^2 \rho(x) \end{equation*} and integrating we obtain the $e$:$p$-ratio inside radius $x$ \begin{equation} n=-x^2\frac{d}{dx} \left( \phi+\frac{1}{x} \right) \label{e1.4} \end{equation} with $n(0)=0$ and $n(x) \equiv N$ for $x \geq X$.
The equations given thus far allow us to construct physically reasonable state functions $\phi(x)$, $n(x)$ and $\rho(x)$, given the overall $e$:$p$-ratio $N$ and the ion radius $X$. We can then proceed to calculate the binding energy \textit{per proton} \begin{equation} B=\int_0^{X} dx x^2 \rho \left( \frac{1}{2x}-\frac{\phi}{2}-\frac{3}{5}\rho^{2/3} \right) \label{e1.5} \end{equation}
The actual state of the ion is the one with the highest binding energy $B$. To find this state, we vary the binding energy: \begin{equation*} \delta B=-\int_0^{X} dx x^2\delta \rho \left(\phi+\rho^{2/3}\right) \end{equation*} while keeping the overall $e$:$p$ ratio constant: \begin{equation*} \delta N=\int_0^{X} dx x^2\delta \rho =0 \end{equation*} Apparently, the maximal binding energy is achieved when the Fermi-surface is flat: \begin{equation*} \phi+\rho^{2/3} = -b' \end{equation*} The constant $b'$ on the RHS still depends on our choice of the ion radius $X$. However, it is intuitively clear that the ion radius is not a free parameter, but is determined by $N$. We therefore vary $X$ in the next step until the binding energy becomes minimal with respect to $X$. This will be achieved when the eletron density continuously approaches 0 as $x$ approaches $X$, i. e. we will have \begin{equation*} b'=b \end{equation*} In this way we obtain the relation \begin{equation} \rho = \left(-\phi-b \right)^{3/2} \quad \label{e1.6} \end{equation}
Incidentally, the last four unnumbered equations contain an important relation between the binding energy $B(N)$ and the ionization potential $b(N)$: \begin{equation*} \delta B=b \delta N \end{equation*} If we drop the restriction $\delta N= 0$, the physical interpretation is straightforward: addition of an electron ($\delta N = 1/Z$) to an ion with a level Fermi surface increases the binding energy $ZB$ by the ionization potential $b$! It follows that the binding energy (now again per proton) is the primitive of the ionization potential with respect to the $e$:$p$-ratio: \begin{equation} B(N)= \int_0 dN b(N) \quad \label{e1.7} \end{equation}
\subsection{Description of the TF ion with the screening number $\chi$}
For mathematical convenience, we now describe the potential with the screening number $\chi$, according to \begin{equation} \phi = -\frac{\chi}{x}-b \label{e1.8} \end{equation} The earlier statements about $\phi$ translate to \begin{equation} \chi(0)=1 \qquad \qquad \chi(X)=0 \qquad \qquad \chi(x)=-b(x-X) \quad (x>X) \label{e1.9} \end{equation}
The fraction of electrons inside radius \textit{x} is now \begin{equation} n(x)= 1-\chi -x\psi \quad \end{equation}
Here we have introduced the slope of the screening function \begin{equation} \psi \equiv -\frac{d\chi}{dx} \quad \label{e1.11} \end{equation} The initial slope is customarily designated by $a$, and the final slope equals the ionization potential $b$ (see eqn \eqref{e1.9}): \begin{equation} \psi(0) \equiv a \qquad \qquad \psi(x)=b \quad (x \geq X) \label{e1.12} \end{equation}
Poisson's equation for the electron density \eqref{e1.3} is now \begin{equation} \rho=-\frac{1}{x}\frac{d\psi}{dx}=\frac{1}{x}\frac{d^2\chi}{dx^2} \label{e1.13} \end{equation}
The binding energy \eqref{e1.5} becomes (after a short calculation, see appendix A) \begin{equation*} B=\frac{a-b(1-N)}{2} - \int_0^X dx x \rho \left( \frac{3}{5}x \rho^{2/3} - \frac{\chi}{2}\right) \end{equation*} According to eqn \eqref{e1.6} the maximum binding energy $B$ is obtained for \begin{equation} \rho = \left( \frac{\chi}{x} \right)^{3/2} \label{e1.14} \end{equation}
Inserting this density in Poisson's equation \eqref{e1.13}, we obtain the Thomas-Fermi differential equation (TF DE): \begin{equation} \frac{d^2\chi}{dx^2} = \frac{\chi ^{3/2}}{x^{1/2}} \label{e1.15} \end{equation}
The solution of the TF DE will maximise the binding energy to \begin{equation*} B= \frac{a-b(1-N)}{2}-\frac{1}{10}\int_0^X dx x^{-1/2} \chi^{5/2} \end{equation*} In Appendix A it is shown that the last integral is $\frac{5}{7}(a-b(1-N))$. Thus the binding energy finally becomes \begin{equation} B= \frac{3}{7}(a-b(1-N)) \label{e1.16} \end{equation}
\subsection{Mathematical relations between the quantities of interest}
In deriving the TF DE, we have come across three interesting relations between the initial slope of the screening function $a$, the ionization potential = final slope of the screening function $b$, the ion radius $X$, the overall $e$:$p$-ratio $N$ and the binding energy $B$: \begin{itemize} \item eqn. \eqref{e1.2}\enskip: $N=1-Xb$ \item eqn. \eqref{e1.7}\enskip: $B=\int_0 dN b$ \item eqn. \eqref{e1.16}: $B= \frac{3}{7}(a-b(1-N))$ \end{itemize} Any solution of the TF DE will give us a set of values $a$, $b$, $X$ from which we can calculate $B$ and $N$ by means of eqn.s \eqref{e1.2} and \eqref{e1.16}.
Furthermore we can eke out an interesting differential equation between $a$, $b$ and $X$. The differentials of eqn.s \eqref{e1.16} and \eqref{e1.7} are $dB= \frac{3}{7}(da-db(1-N)+bdN)$ and $dB=bdN$. Setting them equal and rearranging, we obtain $4bdN=3(da-db(1-N))$. Eliminating $N$ with eqn \eqref{e1.2}, we obtain after a little calculation: \begin{equation} bXdb+4b^2dX+3da=0 \label{e1.17} \end{equation} Some more calculation gives \begin{equation} (bX)^{7/3}dc=da \label{e1.18} \end{equation} where \begin{equation} c \equiv b^{-1/3}X^{-4/3} \label{e1.19} \end{equation} We will return to these equations later on.
\section{General solution of the TF differential equation}
The following general solution of the Thomas-Fermi differential equation is a Taylor-series in the parameter $a$ (initial slope of the screening function). The coefficient-functions are calculated numerically.
\subsection{Changing the independent variable}
The second order TF DE can be split into two first order differential equations:
\begin{minipage}{0.45\linewidth} \begin{equation} -\frac{d\chi}{dx} = \psi \end{equation} \end{minipage} \begin{minipage}{0.45\linewidth} \begin{equation} -\frac{d\psi}{dx}= \frac{\chi^{3/2}}{x^{1/2}} \end{equation} \end{minipage}
This system of DE's is non-linear. Historically, Riemann solved the non-linear DE for the plane sound wave by exchanging the roles of the dependent variable \textit{p} (pressure) and the independent variable \textit{x} (position) \cite{Sommerfeld}. We will use the same device here: $\chi$ becomes our independent variable, and \textit{x} and all other functions of interest become dependent variables.
The two equations then must be written in the form
\begin{minipage}[b]{0.45\linewidth} \begin{equation} \frac{dx}{d\chi} = -\frac{1}{\psi} \label{e2.9} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \psi\frac{d\psi}{d\chi}= -\frac{\chi^{3/2}}{x^{1/2}} \label{e2.10} \end{equation} \end{minipage}
Let us formally integrate these equations, starting at $\chi = 1$:
\begin{minipage}[b]{0.45\linewidth} \begin{equation} x=\int^1\frac{d\chi}{\psi} \label{e2.1} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \psi^2 = a^2-2\int^1 d\chi \frac{\chi^{3/2}}{x^{1/2}} \label{e2.2} \end{equation} \end{minipage}
The form of eqn.s \eqref{e2.1} and \eqref{e2.2} suggests that the functions $x(\chi)$ and $\psi(\chi)$ can be calculated iteratively: the output of one equation is the input to the other.
\subsection{Second normalization}
The two integral equations can be brought into a more symmetric form by substituting
\begin{minipage}[b]{0.45\linewidth} \begin{equation} \xi = \frac{ax}{1-\chi} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \eta =\left(\frac{\psi}{a}\right)^2 \\ \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} t=\sqrt{1-\chi} \\ \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} K = \frac{2}{a^{3/2}} \end{equation} \end{minipage}
The result is
\begin{minipage}[b]{0.45\linewidth} \begin{equation} \xi=\frac{2}{t^2} \int_0 dt \frac{t}{\eta^{1/2}} \label{e2.3} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \eta = 1-2K\int_0 dt \frac{(1-t^2)^{3/2}}{\xi^{1/2}} \label{e2.4} \end{equation} \end{minipage}
\pagebreak \subsection{Taylor development in $K$}
The coupled integral eqn.s \eqref{e2.3} and \eqref{e2.4} contain the parameter $K = 2/a^{3/2}$. $K$ varies from 0 for the ``empty ion'' ($N = 0$) to 0.999367 for the neutral atom ($N = 1$). The fact that $K$ doesn't exceed 1 suggests that it can be used as an expansion coefficient for the functions $\eta(\chi)$ and $\xi(\chi)$. For $K=0$, i. e. in lowest order, we have $\eta = 1$, or $\psi = a=constant$, which seems to be quite a reasonable starting point for the slope of the screening function, whatever the value of $N$. We therefore set \begin{gather} \eta=1+K\eta_1+K^2\eta_2+K^3\eta_3+K^4\eta_4+K^5\eta_5 \ldots \\ \xi=1+K\xi_1+K^2\xi_2+K^3\xi_3+K^4\xi_4+K^5\xi_5 \ldots \end{gather}
The integral eqn.s \eqref{e2.3} and \eqref{e2.4} also contain the powers $\eta^{-1/2}$ and $\xi^{-1/2}$. In fact, on several occasions we will have to calculate some power $\beta$ of a given power series. So let us deal with this problem first.
Let $f(K)$ some Taylor series in $K$, starting with 1: \begin{equation} f=1+Kf_1+K^2f_2+K^3f_3+K^4f_4+K^5f_5 \ldots \label{e2.8} \end{equation} Raising this series to the power of $\beta$ gives \begin{equation} f^{\beta}=1+Kf_1^{(\beta)}+K^2f_2^{(\beta)}+K^3f_3^{(\beta)}+K^4f_4^{(\beta)}+K^5f_5^{(\beta)} \ldots \end{equation} The coefficients $f_m^{(\beta)}$ can be calculated by repeated application of the binomial theorem. The coefficients $f_1 \ldots f_5$ are: \begin{align} f_1^{(\beta)}=&\beta f_1 \nonumber \\ f_2^{(\beta)}=&\beta f_2+\frac{\beta(\beta-1)}{2}f_1^2 \nonumber \\ f_3^{(\beta)}=&\beta f_3+\beta(\beta-1)f_1f_2+\frac{\beta(\beta-1)(\beta-2)}{6}f_1^2 \label{e2.5} \\ f_4^{(\beta)}=&\beta f_4+\frac{\beta(\beta-1)}{2}(f_2^2+2f_1f_3)+\frac{\beta(\beta-1)(\beta-2)}{2}f_1^2f_2 +\frac{\beta\ldots(\beta-3)}{24}f_1^4 \nonumber \\ f_5^{(\beta)}=&\beta f_5+\beta(\beta-1)(f_2f_3+f_1f_4)+ \frac{\beta(\beta-1)(\beta-2)}{2}(f_1^2f_3+f_1f_2^2) + \nonumber \\ &\frac{\beta\ldots(\beta-3)}{6}f_1^3f_2+ \frac{\beta\ldots(\beta-4)}{120}f_1^5 \nonumber \end{align}
The power series for $\eta^{-1/2}$ and $\xi^{-1/2}$ in eqn.s \eqref{e2.3} and \eqref{e2.4} are now formally \begin{gather} \eta^{-1/2}=1+K\eta_1^{(-1/2)}+K^2\eta_2^{(-1/2)}+K^3\eta_3^{(-1/2)}+ K^4\eta_4^{(-1/2)}+K^5\eta_5^{(-1/2)} \ldots \\ \xi^{-1/2}=1+K\xi_1^{(-1/2)}+K^2\xi_2^{(-1/2)}+K^3\xi_3^{(-1/2)}+ K^4\xi_4^{(-1/2)}+K^5\xi_5^{(-1/2)} \ldots \end{gather} The coefficients $\eta_m^{(-1/2)}$ and $\xi_m^{(-1/2)}$ can be calculated with eqn \eqref{e2.5}.
After ordering in powers of $K$, eqn.s \eqref{e2.3} and \eqref{e2.4} can be rewritten as
\begin{minipage}[b]{0.45\linewidth} \begin{equation} \xi_m=\frac{2}{t^2} \int_0 dt \: t \eta_m^{(-1/2)} \label{e2.6} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \eta_{m+1} = -2\int_0 dt \: (1-t^2)^{3/2}\xi_m^{(-1/2)} \label{e2.7} \end{equation} \end{minipage}
\subsection{Numerical integration}
The integration of eqn.s \eqref{e2.6} and \eqref{e2.7} can be performed sequentially on the computer. We begin with $\eta_0 = 1$ $\rightarrow \eta_0^{(-1/2)} = 1$ $\rightarrow \xi_0 = 1$ $\rightarrow \xi_0^{(-1/2)} = 1$. The first non-trivial term $\eta_1$ is obtained by inserting $\xi_0^{-1/2} = 1$ into eqn \eqref{e2.7} .
The result of the numerical integration of $\eta_m(t)$ and $\xi_m(t)$ is shown in figures (1) and (2). The color of the orders is red (1), yellow (2), green (3), cyan (4), blue (5), magenta (6). It is striking that the $\eta_m(t)$ get small very quickly with increasing $m$. On the other hand the $\xi_m(t)$ vanish only slowly, since the radius $\xi(1)=aX$ of the neutral TF atom ($a=1.588071$ and $K=0.999367$) is infinite. \begin{figure}
\caption{Taylor coefficients of \\ $\eta(t,K)=1+K\eta_1(t)+K^2\eta_2(t)+\ldots$}
\caption{Taylor coefficients of \\ $\xi(t,K)=1+K\xi_1(t)+K^2\xi_2(t)+\ldots$}
\end{figure}
The first impression is confirmed if we plot the partial sums of $x=\xi(1-\chi)/a$ and $\eta=(\psi/a)^2$ versus $\chi=1-t^2$ for the neutral atom and compare them to the known limiting functions. The approximation of $\eta(\chi)$ is fair, the approximation of $\chi(x)$ is poor. \begin{figure}
\caption{Successive approximations of $\eta(\chi,K)=1+K\eta_1(\chi)+K^2\eta_2(\chi)+\ldots$ for $K=0.999367$ (neutral atom)}
\caption{Successive approximations of $x(\chi,K)=(1+K\xi_1(\chi)+K^2\xi_2(\chi)+\ldots)\times \\ (1-\chi)/a$ for $K=0.999367$ (neutral atom)}
\end{figure}
\section{Taylor series for $X^{-1}(K)$, $b(K)$, $B(K)$ and $N(K)$}
\subsection{Formulas for Taylor coefficients}
The integration of eqn.s \eqref{e2.6} and \eqref{e2.7} provides two of the Taylor series in $K$ we are looking for (remember $K=2/a^{3/2}$): \begin{align} aX = \xi(1) &= 1+K\xi_1(1)+K^2\xi_2(1)+K^3\xi_3(1)+K^4\xi_4(1)+K^5\xi_5(1) \ldots \label{e3.3} \\ (b/a)^2 = \eta(1) &= 1+K\eta_1(1)+K^2\eta_2(1)+K^3\eta_3(1)+K^4\eta_4(1)+K^5\eta_5(1) \ldots \label{e3.4} \end{align} For physical reasons, we are interested in the power series for $b/a$ rather than $(b/a)^2$. From a mathematical point of view, the power series for $(aX)^{-1}$ has some advantages over the one for $aX$ ($(aX)^{-1} \rightarrow 0$ for $K \rightarrow 0.999367$, and eqn \eqref{e1.2}). These two series are \begin{align} b/a=\eta^{1/2}(1) & =1+K \eta_1^{(1/2)}+K^2 \eta_2^{(1/2)} +K^3 \eta_3^{(1/2)} +K^4 \eta_4^{(1/2)}+K^5 \eta_5^{(1/2)} \ldots \label{e3.1} \\ (aX)^{-1}=\xi^{-1}(1) & =1+K \eta^{(-1)}_1+K^2 \eta_2^{(-1)} +K^3 \eta_3^{(-1)} +K^4 \eta_4^{(-1)}+K^5 \eta_5^{(-1)} \ldots \label{e3.2} \end{align} The RHSs of eqn.s \eqref{e3.1} and \eqref{e3.2} must be evalated at $t=1$. The coefficients $\eta_m^{(1/2)}$ and $\xi_m^{(-1)}$ can be calculated with eqn \eqref{e2.5}.
The Taylor series for the other two quantities we seek, $B(K)$ and $N(K)$, are obtained as follows.
Eqn. \eqref{e1.16} for $B$ can be reformulated in terms of our new variables $\eta$ and $\xi$: \begin{equation} \frac{B}{a}=\frac{3}{7}(1-\eta(1)\xi(1)) \end{equation} Inserting the series \eqref{e3.1} and \eqref{e3.2}, we obtain \begin{align} \frac{B}{a}= -\frac{3}{7} & \left[ K(\eta_1+\xi_1)+K^2(\eta_2+\eta_1\xi_1+\xi_2)+K^3(\eta_3+\eta_2\xi_1+\eta_1\xi_2+\xi_3) + \right. \label{e3.5} \\ & \ \left. K^4(\eta_4+\eta_3\xi_1+\eta_2\xi_2+\eta_1\xi_3+\xi_4)+ K^5(\eta_5+\eta_4\xi_1+\eta_3\xi_2+\eta_2\xi_3+\eta_1\xi_4+\xi_5) \ldots \right] \nonumber \end{align} The RHSs of eqn \eqref{e3.5} must be evalated at $t=1$.
Eqn. \eqref{e1.2} for $N$ now becomes \begin{equation} N = 1-\xi(1)\eta^{1/2}(1) \end{equation} Inserting the power series \eqref{e3.3} and \eqref{e3.1}, we obtain \begin{align} N= - & \left[ K (\eta_1^{(1/2)}+\xi)+K^2 (\eta_2^{(1/2)}+\eta_1^{(1/2)}\xi_1+\xi_2) +K^3 (\eta_3^{(1/2)}+\eta_2^{(1/2)}\xi_1+\eta_1^{(1/2)}\xi_2+\xi_3)+\right. \nonumber \\ & \ \left. K^4 (\eta_4^{(1/2)}+\eta_3^{(1/2)}\xi_1+\eta_2^{(1/2)}\xi_2+\eta_1^{(1/2)}\xi_3+\xi_4 )+\right. \nonumber \\ & \ \left. K^5 (\eta_5^{(1/2)}+\eta_4^{(1/2)}\xi_1+\eta_3^{(1/2)}\xi_2+ \eta_2^{(1/2)}\xi_3+\eta_1^{(1/2)}\xi_4+\xi_5) \ldots \right] \label{e3.6} \end{align} Again, the RHSs of eqn \eqref{e3.6} must be evalated at $t=1$.
\subsection{Taylor coefficients }
Let us collect the Taylor coefficients of $X^{-1}(K)$, $b(K)$, $B(K)$, $N(K)$ (and of two more quantities) in a table:
\begin{table}[H]
\begin{tabular}{ |l|r|r|r|r|r|r|} \hline $f$ & $aX=\xi(1)$ & $(aX)^{-1}$ & $(b/a)^2=\eta(1)$ & $b/a$ & $B/a$ & $N$ \\ \hline $f_0$ & 1& 1 & 1& 1 & 0 & 0 \\ $f_1$ & 0.490873 & $-$0.490873 & $-$1.178097 & $-$0.589049 & 0.294524 & 0.098175 \\ $f_2$ & 0.339148 & $-$0.098191 & 0.122481 & $-$0.112248 & 0.050000 & 0.062248 \\ $f_3$ & 0.263353 & $-$0.048674 & 0.024990 & $-$0.053624 & 0.021892 & 0.045145 \\ $f_4$ & 0.217190 & $-$0.030722 & 0.010085 & $-$0.032845 & 0.012502 & 0.035173 \\ $f_5$ & 0.185856 & $-$0.021795 & 0.005303 & $-$0.022715 & 0.008155 & 0.028663 \\ $f_6$ & 0.163069 & $-$0.016574 & 0.003220 & $-$0.016894 & 0.005768 & 0.024093 \\ \hline \end{tabular} \caption{Coefficients of the Taylor series $f=f_0+Kf_1+K^2f_2+K^3f_3+\ldots $ for selected functions $f$, with $K=(2/a)^{3/2}$.} \end{table}
\subsection{Convergence}
We already know the values of all the quantities $f$ in table 1 for the neutral atom ($K=0.99936725$, the most difficult case to calculate). Let us see how quickly the series converge towards these values.
\begin{table}[H]
\begin{tabular}{|l|r|r|r|r|r|r|r|} \hline $f$ & $aX=\xi(1)$ & $(aX)^{-1}$ & $(b/a)^2=\eta(1)$ & $b/a$ & $B/a$ & $N$ \\ \hline $S_0$ & 1 & 1 & 1 & 1 & 0 & 0 \\ $\ldots$ & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ $S_3$ & 2.092136 & 0.362787 & $-$0.030081 & 0.245695 & 0.366125 & 0.205342 \\ $S_4$ & 2.308777 & 0.332142 & $-$0.020022 & 0.212933 & 0.378597 & 0.240426 \\ $S_5$ & 2.494047 & 0.310415 & $-$0.014735 & 0.190290 & 0.386726 & 0.269000 \\ $S_6$ & 2.656499 & 0.293903 & $-$0.011527 & 0.173460 & 0.392473 & 0.293002 \\ \hline $S_\infty$ & $\infty$ & 0 & 0 & 0 & 0.428571 & 1 \\ \hline \end{tabular} \caption{Partials sums $S_m = f_0+Kf_1+K^2f_2+\ldots+K^m f_m$ for $K=(2/a)^{3/2}=0.999367$ (neutral atom) and selected functions $f$.} \end{table}
The convergence is moderate for $b^2$, $b$ and $B$, slow for $X$ and $X^{-1}$, and very slow for $N$. This is a problem. There is no use calculating a precise value of, say, the binding energy $B$ for some value of $K$, if we don't know the precise $e$:$p$-ratio $N$ for this $K$.
Well, let's just proceed to the next point on our agenda, which is eliminiating the intermediate parameter $K$. When this is done, we will evaluate the convergence of the new series for $f(N)$.
\section{Taylor series for $X^{-1}(N)$, $b(N)$, $B(N)$ and $a(N)$}
\subsection{Eliminating the parameter $K$}
Our problem is as follows. We are given the Taylor developments of two functions $N(K)$, see table 1, and $f(K)$, which could be any of the other quantities in table 1: \begin{gather} N=N_1K+N_2K^2+N_3K^3+N_4K^4+N_5K^5 \ldots \label{e4.1} \\ f=(K/2)^\alpha(f_0+f_1K+f_2K^2+f_3K^3+f_4K^4+f_5K^5 \ldots) \label{e4.4} \end{gather} The initial factor $(K/2)^\alpha=a^{-3\alpha/2}$ has been absorbed in $f$ in the course of the second normalization ($aX$, $b/a$, $B/a$ \ldots). In \eqref{e4.4} we have reverted to the more familiar "first normalization" ($X$, $b$, $B$ \ldots).
We want to express $f$ as a power series in $N$. It is not difficult to see that this series must be of the following form: \begin{equation} f=N^\alpha ( \tilde f_0+ \tilde f_1 N+ \tilde f_2 N^2+ \tilde f_3 N^3+ \tilde f_4 N^4+ \tilde f_5 N^5 \ldots) \end{equation}
We want to express the new coefficients $\tilde f_n$ as linear combinatons of the old coefficients $f_m$. To this end we first rewrite eqn \eqref{e4.1} as \begin{equation} \frac{N}{N_1}=K(1+Kh_1+K^2h_2+K^3h_3+K^4h_4+K^5h_5 \ldots) \label{e4.2} \end{equation} where \begin{equation} h_1=\frac{N_2}{N_1} \qquad h_2=\frac{N_3}{N_1} \qquad \ldots \end{equation}
Elevating eqn \eqref{e4.2} to the power of $\beta$ gives \begin{equation} \left(\frac{N}{N_1}\right)^{\beta}=K^{\beta}(1+h_1^{(\beta)}K+h_2^{(\beta)}K^2+h_3^{(\beta)}K^3+h_4^{(\beta)}K^4+h_5^{(\beta)}K^5 \ldots) \label{e4.3} \end{equation} with the coefficients $h_n^{(\beta)}$ given earlier in eqn \eqref{e2.5}.
The $\tilde f_n$ are now obtained by repeatedly subtracting eqn \eqref{e4.3} from eqn \eqref{e4.1}, in such a way that the powers of $K$ on the RHS of \eqref{e4.1} are one by one eliminated. In the course of this elimination procedure, it appears that it is useful to introduce intermediate coefficients $g_n$ defined by \begin{equation} \tilde f_n=\frac{g_n}{2^{\alpha}N_1^{\alpha+n}} \quad (n \geq 0) \label{e4.6} \end{equation}
The Taylor series \eqref{e4.4} written with the $g_n$ is \begin{equation} f=\left(\frac{N}{2N_1}\right)^\alpha\left\{ g_0 +g_1\frac{N}{N_1} +g_2\left(\frac{N}{N_1}\right)^2 +g_3 \left(\frac{N}{N_1}\right)^3 +g_4 \left(\frac{N}{N_1}\right)^4 + \ldots \right\} \label{e4.5} \end{equation}
The elimination procedure can now be formalized by the following scheme. It shows how to calculate the coefficients $g_n$ out of the known $f_m$ and $h_m^{(\beta)}$: \begin{equation} \begin{array}{l l l l} G_{0,0} = f_0 & G_{0,1} = f_1 & G_{0,2}=f_2 & G_{0,3}=f_3\\ G_{1,0} = G_{0,1}-G_{0,0}h_1^{(\alpha)} & G_{1,1} = G_{0,2}-G_{0,0}h_2^{(\alpha)} & G_{1,2}=G_{0,3}-G_{0,0}h_3^{(\alpha)} & \ldots \\ G_{2,0} = G_{1,1}-G_{1,0}h_1^{(\alpha+1)} & G_{2,1} = G_{1,2}-G_{1,0}h_2^{(\alpha+1)} & \ldots & \\ G_{3,0} = G_{2,1}-G_{2,0}h_1^{(\alpha+2)} & \ldots & & \\ \ldots & & &\\ \end{array} \end{equation}
The general recursion relation in this scheme is \[G_{n,k} = G_{n-1,k+1}-G_{n-1,0}h_{k+1}^{(\alpha+n-1)} \]
Our $g_n$ are the first column of $G_{n,k} $: \begin{equation} g_n=G_{n,0} \end{equation}
\subsection{A linear transformation of Taylor coefficients}
The relation between the coefficients $f_m$ in the $K$-series \eqref{e2.8} and $g_n$ in the $N$-series \eqref{e4.5} is linear, i. e. it can be represented by a transformation matrix $T_{mn}(\alpha)$: \begin{equation} g_n=f_mT_{mn} \label{e4.7} \end{equation}
The $T_{mn}(\alpha)$ can be obtained with the scheme described above, by setting: \begin{equation} f_{m'}=\delta_{mm'} \Rightarrow T_{mn}=g_n \end{equation}
In the following subsection, we will mainly need the transformation matrix for $\alpha=-2/3$. A numerical evalation of the above scheme on the computer gives
\[\mathbf{T} \left( \alpha=-\frac{2}{3}\right) = \] \begin{equation} \left( \begin{array}{r r r r r r } 1 & 0.422703 & -0.006118 & 0.000023 & 0.000000 & 0.000000 \\ 0 & 1 & -0.211351 & 0.070063 & -0.025553 & 0.009775 \\ 0 & 0 & 1 & -0.845406 & 0.548271 & -0.317675 \\ 0 & 0 & 0 & 1 & -1.479461 & 1.428504 \\ 0 & 0 & 0 & 0 & 1 & -2.113515 \\ 0 & 0 & 0 & 0 & 0 & 1 \label{e4.8} \\ \end{array} \right) \end{equation}
The transformation of the Taylor series in $K$ into Taylor series in $N$ is now achieved by applying eqn.s \eqref{e4.6}, \eqref{e4.7} and \eqref{e4.8} to the coefficient vectors in table 1.
\subsection{Taylor coefficients }
Again we collect the Taylor coefficients, this time of $X^{-1}(N)$, $b(N)$, $B(N)$ and $a(N)$, in a table. For reasons that will soon become clear, we start the table with the coefficients of a function we have encountered in section 1.5: $c \equiv b^{-1/3}X^{-4/3}$.
\begin{table}[H]
\begin{tabular}{ |l|r|r|r|r|r|} \hline $f$ & $c$ & $X^{-1}$ & $b$ & $B$ & $a$ \\ \hline $\alpha$ & $-$2/3 & $-$2/3 & $-$2/3 & 1/3 & $-$2/3 \\ $\tilde f_0$ & 0.337821 & 0.337821 & 0.337821 & 1.013463 & 0.337821 \\ $\tilde f_1$ & $-$0.121969 & $-$0.234576 & $-$0.572397 & $-$0.429297 & 1.454528 \\ $\tilde f_2$ & $-$0.022859 & $-$0.019738 & 0.214837 & 0.092072 & $-$0.214459 \\ $\tilde f_3$ & $-$0.011826 & $-$0.011507 & 0.008230 & 0.002471 & 0.008229\\ $\tilde f_4$ & $-$0.007633 & $-$0.007524 & 0.003983 & 0.000907 & 0.001520\\ $\tilde f_5$ & $-$0.005504 & $-$0.005410 & 0.002114 & 0.000445 & 0.000249\\ \hline \end{tabular} \caption{Coefficients of the Taylor series $f=N^\alpha (\tilde f_0+\tilde f_1 N+\tilde f_2 N^2+\tilde f_3 N^3+ \ldots)$ for selected functions $f$.} \end{table}
The three relations collected in section 1.5 can be restated as recursive relations between the Taylor-coefficients of $X^{-1}(N)$, $b(N)$, $B(N)$ and $a(N)$: \begin{itemize} \item Eqn \eqref{e1.2}\enskip : $b=X^{-1}(1-N)$ \quad \qquad $\Rightarrow$ \quad $b_n = X^{(-1)}_n - X^{(-1)}_{n-1}$ \\ \item Eqn \eqref{e1.7}\enskip: $B = \int_0 dN b$ \quad \qquad \qquad $\Rightarrow$ \quad $B_n=b_n/(n+1/3)$ \\ \item Eqn \eqref{e1.16}: $a =7B/3+b(1-N)$ \enskip\;$\Rightarrow$ \quad $a_n = 7B_{n-1}/3+b_n-b_{n-1}$ \\ \end{itemize}
The reader may check that the coefficients in table 3 fulfill all these relations. In particular, the series for $X^{-1}$, $a$ and $b$ have the same first term $0.3378N^{-2/3}$.
\subsection{Convergence }
As we did for the $f(K)$-series, we check the convergence of the $f(N)$-series for the neutral atom, $N=1$.
\begin{table}[H]
\begin{tabular}{ |l|r|r|r|r|r|} \hline $f$ & $c$ & $X^{-1}$ & $b$ & $B$ & $a$ \\ \hline $S_0$ & 0.337821 & 0.337821 & 0.337821 & 1.013462 & 0.337821 \\ $\ldots$ & \ldots & \ldots & \ldots & \ldots & \ldots\\ $S_3$ & 0.181166 & 0.071998 & $-$0.011507 & 0.678709 & 1.586119 \\ $S_4$ & 0.173532 & 0.064474 & $-$0.007524 & 0.679617 & 1.587639 \\ $S_5$ & 0.168028 & 0.059063 & $-$0.005410 & 0.680063 & 1.587889 \\ \hline $S_\infty$ & 0.0977 & 0 & 0 & 0.680601 & 1.588071 \\ \hline \end{tabular} \caption{Partials sums $S_n=N^\alpha (\tilde f_0+ \tilde f_1 N+\tilde f_2 N^2+\ldots +\tilde f_n N^n )$ for $N=1$ (neutral atom) and selected functions $f$.} \end{table}
The convergence is rather slow for $c(N)$ and $X^{-1}(N)$, fair for $b(N)$, and excellent for $B(N)$ und $a(N)$. Eliminating the parameter $K$ brought about an unexpected improvement in convergence!
\section{Improved series for $X^{-1}(N)$, $b(N)$, $B(N)$ and $a(N)$}
When we want to describe a quantity $f(N)$ by a series, we actually have a lot of choices. We have encountered this situation in section 2, when we calculated Taylor series for $\eta$ and $\eta^{1/2}$, or for $\xi$ and $\xi^{-1}$. Both series contain the same information, and the question arises: which series converges faster? We leave this question unanswered for the $K$-series, and turn our attention to the (more important) $N$-series. In the last chapter, we had a loose hierarchy of the functions of interest: $X^{-1}(N) \rightarrow b(N) \rightarrow B(N) \rightarrow a(N)$. The Taylor coefficients of each function can be calculated from the Taylor coefficients of the previous functions, with $X^{-1}(N)$ being the fundamental function. We now declare \begin{equation} b^{-1/3}X^{-4/3} \equiv c(N)= \sum_{n=0}^\infty c_n N^{n-2/3} \label{e5.1} \end{equation} to be our fundamental function. Its Taylor coefficients $c_n$ are listed in table 3.
\begin{figure}
\caption{Successive approximations of the fundamental function $c(N)$ with the Taylor series \eqref{e5.1}. The color of the orders is red (0), yellow (1), green (2), cyan (3), blue (4), magenta (5). For $N=1$, the series converges towards 0.0977 (o).}
\end{figure}
Our quantities of interest $X(N)$, $b(N)$, $B(N)$ and $a(N)$ are obtained as follows:
The reciprocal ion radius is (see eqn.s \eqref{e1.2} and \eqref{e5.1}) \begin{equation} X^{-1}(N)=(1-N)^{1/3}c(N)=\sum_{n=0}^\infty c_n N^{n-2/3}(1-N)^{1/3} \label{e5.2} \end{equation}
The ionization potential is (see eqn.s \eqref{e1.2} and \eqref{e5.1}) \begin{equation} b(N)=(1-N)^{4/3}c(N)=\sum_{n=0}^\infty c_n N^{n-2/3}(1-N)^{4/3} \label{e5.3} \end{equation}
The binding energy $B(N)$ is the primitive of $b(N)$ (see eqn.s \eqref{e1.7} and \eqref{e5.3}): \begin{equation} B(N)=\int_0 dN (1-N)^{4/3}c(N)=\sum_{n=0}^\infty c_n\tilde B(N;n+1/3,7/3) \label{e5.4} \end{equation} where $\tilde B$ is Euler's incomplete Betafunction.
The initial slope of the screening function is (see eqn.s \eqref{e1.2}, \eqref{e1.18} and \eqref{e5.1}): \begin{align} a(N)=\int (1-N)^{7/3}dc =& c_0\left[ N^{-2/3}(1-N)^{7/3} +\tfrac{7}{3}\tilde{B}(N;1/3,7/3) \right]+ \nonumber \\ &\sum_{n=0}^\infty c_n (n-2/3)\tilde B(N;n-2/3,10/3) \label{e5.5} \end{align} Because of its divergence, the first term of $a(N)$ needs a separate analysis.
Why is $c(N)$ a smarter fundamental function than $X^{-1}(N)$? Well, the latter function tends towards $0$ for $N \rightarrow 1$. The Taylor series of $X^{-1}(N)$ cannot reproduce this behaviour well, in the sense that the relative error becomes infinite. On the other hand, the function $c(N)$ has a non-zero value for $N=1$: $c(N=1) = 0.0977$ (see Appendix B), and therefore can be reproduced with a Taylor series with finite relative error. As a consequence, the above formulas for $X(N)$, $b(N)$, $B(N)$ and $a(N)$ all have the correct behaviour for $N \rightarrow 1$ in-built!
By way of example, let's check this argument for the reciprocal ion radius $X^{-1}$. In fig.s (6) and (7), we compare the convergence of the Taylor series (see table 3) and of the improved series (see eqn \eqref{e5.2} and table 3). For $N \rightarrow 1$, the convergence of the improved series is indeed much better.
\begin{figure}
\caption{Successive approximations of the reciprocal ion radius $X^{-1}(N)$ with the Taylor series given in table 3. }
\caption{Successive approximations of the reciprocal ion radius $X^{-1}(N)$ with the improved series \eqref{e5.2}.}
\end{figure}
As a further test, let us compare the convergence of the Taylor series (see tables 3 and 4) \begin{equation*}
a(N=1)=\sum_{n=0}^\infty a_n \qquad (i) \end{equation*} and of the improved series (see eqn\eqref{e5.5} and table 3) \begin{equation*} a(N=1)=\tfrac{7}{3}\sum_0^\infty c_n\tilde B(n+1/3,7/3) \qquad (ii) \end{equation*} where $\tilde B$ is now Euler's ''complete'' Betafunction, and where we have used the identity $p\tilde B(p,q+1)=q\tilde B(p+1,q)$.
\begin{tabular}{ |l|rrrrrr|r|} \hline $a$ & $a_0$ & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_5$ & \\ \hline (i) & 0.337821 & 1.454528 & -0.214459 & 0.008229 & 0.001520 & 0.000249 & \\ (ii) & 1.671061 & -0.075416 & -0.005140 & -0.001330 & -0.000505 & -0.000237 & \\ \hline $a$ & $S_0$ & $S_1$ & $S_2$ & $S_3$ & $S_4$ & $S_5$ & $S_\infty$\\ \hline
(i) & 0.337821 & 1.792349 & 1.577890 & 1.586119 & 1.587639 & 1.587889 & 1.588071 \\ (ii) & 1.671061 & 1.595645 & 1.590505 & 1.589176 & 1.588671 & 1.588434 &1.588071 \\ \hline \end{tabular}
The outcome of the comparison is less clearcut here. Initially, the improved series $(ii)$ approaches the final value much faster than the Taylor series. However, as more terms are added, the convergence of the two series becomes similar, and series $(i)$ even seems to have the edge.
\section{Summing up and plots}
In this section we collect the formulas for the ion radius $X(N)$, the ionization potential $b(N)$, the electronic binding energy $B(N)$ and the initial slope of the screening function $a(N)$, where the independent variable $N$ is the electron to proton ratio of the ion. The fundamental quantity is $c \equiv b^{-1/3}X^{-4/3}$ with the Taylor series \begin{align*} c(N) =\sum_{n=0}^\infty c_n N^{n-2/3} &= 0.337821N^{-2/3}-0.121969N^{1/3}-0.022859N^{4/3} \\ & -0.011826N^{7/3}-0.007633N^{10/3} -0.005504N^{13/3} \end{align*} The ``improved series'' for $X^{-1}(N)$, $b(N)$, $B(N)$ and $a(N)$ are all expressed in terms of the Taylor series of $c(N)$. We plot the partial sums of the ``improved series'' of the four quantities. The color of the orders is red (0), yellow (1), green (2), cyan (3), blue (4), magenta (5). The convergence is excellent in all cases.
\subsection{The radius of the TF ion}
The ion radius $X$ in units of $a_B/1.1295 Z^{1/3}$ is given by $X^{-1}(N)=(1-N)^{1/3}c(N)$.
\begin{figure}
\caption{Successive approximations of the ion radius $X$.}
\end{figure}
\subsection{The ionization potential of the TF ion}
The ionization potential in units of $2.2590 Z^{4/3} Ry$ is $b(N)=(1-N)^{4/3}c(N)$.
\begin{figure}
\caption{Successive approximations of the ionization potental $b$.}
\end{figure}
\subsection{The electronic binding energy of the TF ion}
The electronic binding energy in units of $2.2590 Z^{7/3} Ry$ is $B(N)=\sum_0^\infty c_n\tilde B(N;n+1/3,7/3)$, where $\tilde B$ is Euler's incomplete Betafunction.
\begin{figure}
\caption{Successive approximations of the electronic binding energy $B$.}
\end{figure}
\subsection{The initial slope of the screening function}
The initial slope of the screening function is $a(N)=c_0\left[ N^{-2/3}(1-N)^{7/3} +\tfrac{7}{3}\tilde{B}(N;1/3,7/3) \right] +\sum_{n=1}^\infty c_n (n-2/3)\tilde B(N;n-2/3,10/3)$. In the following figure, the related quantity $K=2/a^{3/2}$ is plotted.
\begin{figure}
\caption{Successive approximations of the parameter $K=2/a^{3/2}$.}
\end{figure}
\section{A note about the screening function}
By means of the integration scheme described in section 2, we obtained a Taylor series for the ion radius $X(K)$. By applying the linear transformation described in section 4, we obtained the corresponding series for $X(N)$. Actually, in section 2 we obtained a series in $K$ for the whole inverse screening function $x(\chi;K)$; the ion radius $X$ is simply the value of the inverse screening function for $\chi=0$. So, why not apply the transformation from $K$- to $N$-series to the whole inverse screening function? When the author did this, he found to his surprise (and dismay) that the series \begin{equation} x(\chi;N)=N^{2/3}\left(x_0(\chi)+ x_1(\chi)N+ x_2(\chi)N^2+ x_3(\chi)N^3+ \ldots\right) \end{equation} diverges for all values of $\chi$ except for the border values $\chi=1$ (ion center) and $\chi=0$ (ion edge). When it comes to calculating the screening function of the TF ion, this article has therefore not much to contribute.
\appendix
\section{Appendix: Integral formulas}
Eqn \eqref{e1.5} for the binding energy $B$ contains integrals of the type \begin{equation} I_0(\beta)=\int_0^X dx x^{\beta-1/2} \chi^{3/2} =\int_a^b d\psi x^\beta \end{equation} and \begin{equation} I_1(\beta)=\int_0^X dx x^{\beta-1/2}\chi^{5/2} =\int_a^b d\psi x^\beta \chi \end{equation} where \begin{equation*}
\psi \equiv -\frac{d\chi}{dx} \end{equation*} and the TF differential eq. \begin{equation*} -\frac{d\psi}{dx}= \frac{\chi^{3/2}}{ x^{1/2}} \end{equation*} is assumed to hold.
\subsection*{Evaluation of the integrals}
The first integral is \begin{equation*} I_0(\beta)=\int_a^b d\psi x^\beta \end{equation*}
For $\beta=0$, integration by parts gives \begin{equation} I_0(0)=a-b \label{eA.1} \end{equation}
For $\beta>0$, integration by parts gives \begin{equation*} I_0(\beta)=\beta \int_0^1 d\chi x^{\beta-1}-bX^\beta \end{equation*} The case $\beta=1$ is readily evaluated: \begin{equation} I_0(1)=1-bX \label{eA.2} \end{equation}
The second integral is \begin{equation*} I_1(\beta)=\int_b^a d\psi x^\beta \chi \end{equation*}
For $\beta=0$, integration by parts gives \begin{equation*} I_1(0)=a-\int_0^1 d\chi \psi \end{equation*} Setting $d\chi=-\psi dx$ and integrating by parts gives \begin{equation*} I_1(0)=a-b^2X -2 \int_b^a d\psi \psi x \end{equation*} From \eqref{e2.10}, $\psi d \psi = x^{-1/2} \chi^{3/2} d\chi$, and therefore \begin{equation*} I_1(0)=a-b^2X -2 \int_0^1 d\chi x^{1/2} \chi^{3/2} \end{equation*} Integrating the last integral one more time by parts gives \begin{equation*} I_1(0)=a-b^2X -\frac{2}{5} \int_0^X dx x^{-1/2} \chi^{5/2} \end{equation*} But the last integral is again $I_1(0)$ ! Bringing this term to the LHS, we finally obtain \begin{equation} I_1(0)=\frac{5}{7}(a-b^2X) \label{eA.3} \end{equation}
For $\beta>0$, integration by parts gives \begin{equation*} I_1(\beta)=\beta \int_0^X dx \psi \chi x^{\beta-1} - \int_0^1 d\chi x^\beta \psi \end{equation*} The rest of the calculation is analogous to the case $\beta=0$. The final result is \begin{equation*} I_1(\beta)=\frac{5}{9\beta+7} \left( (\beta+1)\beta \int_0^1 d\chi \chi x^{\beta-1} - b^2 X^{\beta+1} \right) \end{equation*}
The case $\beta=1$ is readily evaluated: \begin{equation} I_1(1) =\frac{5}{16} \left( 1 - b^2 X^2 \right) \label{eA.4} \end{equation}
\subsection*{Summary and physical meaning}
We have evaluated four integrals.
The first integral \eqref{eA.1} is the electrostatic binding energy $B_{pe}$ between nucleus and electrons: \begin{equation*} B_{pe}=I_0(0)=\int_0^X dx x^2 \frac{\rho}{x} =a-b \end{equation*}
The second integral \eqref{eA.2} is the $e$:$p$-ratio $N$: \begin{equation*} N=I_0(1)=\int_0^X dx x^2 \rho = 1-bX \end{equation*}
The third integral \eqref{eA.3} is the kinetic energy of the electrons $T$ times a factor: \begin{equation*} \frac{5}{3}T=I_1(0)=\int_0^X dx x^2 \rho^{5/3}=\frac{5}{7}(a-b^2X) \end{equation*}
The binding energy $B$, eqn \eqref{e1.5}, contains contributions from these three integrals: \begin{equation*}
B=\frac{1}{2}I_0(0)+\frac{b}{2}I_0(1)-\frac{1}{10}I_1(0)=\frac{3}{7}(a-b^2 X) \end{equation*}
The last two equations are in agreement with the virial theorem: the kinetic energy $T$ of a system of electrons trapped in the Coulomb potential of a nucleus equals the electronic binding energy $B$.
The author can't recognise a physical meaning in the fourth integral \eqref{eA.4}. By combining it with eqn \eqref{e1.2}: $N=1-bX$, we obtain the curious relation \begin{equation*} I_1(1)=\int_0^X dx x^2 \rho \chi =\frac{5}{16} N(2-N) \end{equation*}
\section{Appendix: The limiting value of c}
The two quantities
\begin{minipage}[b]{0.45\linewidth} \begin{equation} C \equiv X^{4/5}b^{1/5} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} c \equiv X^{-4/3}b^{-1/3} = C^{-5/3} \end{equation} \end{minipage}
contain the factors $b$ and $X$ raised to a power. $X$ tends to $\infty$ for $N\rightarrow 1$, and $b$ tends to $0$ for $N\rightarrow 1$. The following calculation shows that $C$ and $c$ assume finite and non-zero values for $N\rightarrow 1$.
Let us formally integrate eqn.s \eqref{e2.9} and \eqref{e2.10}, this time starting at $\chi = 0$:
\begin{minipage}[b]{0.45\linewidth} \begin{equation*} x= X-\int_0 \frac{d\chi}{\psi} \end{equation*} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation*} \psi^2 = b^2+2\int_0 d\chi \frac{\chi^{3/2}}{x^{1/2}} \end{equation*} \end{minipage}
The two integral equations can be normalized by substituting
\begin{minipage}[b]{0.45\linewidth} \begin{equation*} \xi \equiv \frac{x}{X} \end{equation*} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation*} \eta \equiv \left(\frac{\psi}{b}\right)^2 \end{equation*} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation*} t \equiv \frac{\chi}{X^{1/5}b^{4/5}} \end{equation*} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation*} T \equiv \frac{1}{X^{1/5}b^{4/5}} \end{equation*} \end{minipage}
The result is
\begin{minipage}[b]{0.45\linewidth} \begin{equation} \xi =1-\frac{1}{C} \int_0 \frac{dt}{\eta^{1/2}} \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} \eta = 1+2 \int_0 dt \frac{t^{3/2}}{\xi^{1/2}} \end{equation} \end{minipage}
$\quad$ \\ The quantity $C$ is a function of the $e$:$p$-ratio $N$, and thus parametrizes the solutions $\xi(t)$ and $\eta(t)$. The equations $x(\chi=1)=0$ and $\psi^2(\chi=1)=a^2$ translate to $\xi(t=T)=0$ and $\eta(t=T)=(a/b)^2$. It follows that
\begin{minipage}[b]{0.45\linewidth} \begin{equation*} C = \int_0^T \frac{dt}{\eta^{1/2}} \end{equation*} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation*} \left( \frac{a}{b} \right) ^2 = 1+2 \int_0^T dt \frac{t^{3/2}}{\xi^{1/2}} \end{equation*} \end{minipage}
We are only interested in the limes $N \rightarrow 1$, where $T \rightarrow \infty$. The left eqn. then becomes. \begin{equation} C = \int_0^\infty \frac{dt}{\eta^{1/2}} \end{equation}
Eqn.s (B.3), (B.4), (B.5) can be solved iteratively on the computer, beginning with $C=\infty \rightarrow \xi=1$. The result is
\begin{minipage}[b]{0.45\linewidth} \begin{equation} C=4.03623 \end{equation} \end{minipage} \begin{minipage}[b]{0.45\linewidth} \begin{equation} c=0.097733 \end{equation} \end{minipage}
\end{flushleft}
\end{document} |
\begin{document}
\setcounter{page}{1}
\vspace*{1.0cm} \title[A New Penalty Dual-Primal Augmented Lagrangian Method] {A New Penalty Dual-Primal Augmented Lagrangian Method and Its Extensions} \author[J. Liu, X. Ou, J. Chen ]{ Jie Liu$^{1}$, Xiaoqing Ou$^{1,2}$, Jiawei Chen$^{1}$} \maketitle \vspace*{-0.6cm}
\begin{center} {\footnotesize {\it
$^1$School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;\\ $^2$College of Management, Chongqing College of Humanities, Science \& Technology, Chongqing 401524, China }}\end{center}
\vskip 4mm {\small\noindent {\bf Abstract.} In this paper, we propose a penalty dual-primal augmented lagrangian method for solving convex minimization problems under linear equality or inequality constraints. The proposed method combines a novel penalty technique with updates the new iterates in a dual-primal order, and then be extended to solve multiple-block separable convex programming problems with splitting version and partial splitting version. We establish the convergence analysis for all the introduced algorithm in the lens of variational analysis. Numerical results on the basic pursuit problem and the lasso model are presented to illustrate the efficiency of the proposed method.
\noindent {\bf Keywords.} Convex minimization; Augmented lagrangian method ; Novel penalty technique ; Dual-primal order ; Variational analysis. }
\renewcommand{\thefootnote}{} \footnotetext{ E-mail addresses: liujie66j@163.com (J. Liu), ouxiaoqing413@163.com (X. Ou), jwhen713@swu.edu.cn(J. Chen). \par Received May 7, 2023; Accepted xxx, xxx. \par The manuscript was first finished in August 2021}
\section{Introduction}
In this paper, we focus on the following convex minimization problem subject to linear equality or inequality constraints: \begin{eqnarray}\label{1.1}
\min \left \{\theta(x) \,| \,Ax=b\ (or \geq b),\ \ x\in\mathcal{X}\right\}, \end{eqnarray} where $\theta:x\rightarrow \mathbb{R}$ is a closed proper convex but not necessarily smooth function; $\mathcal{X}\subset \mathbb{R}^{n}$ is a nonempty closed convex sets; $A \in \mathbb{R}^{m\times n}$ is a given matrix and $b\in \mathbb{R}^{m}$ is a known vector. The problems (\ref{1.1}) is assumed to have solution throughout this paper.
There are a large number of algorithms that can be used to solve problem (\ref{1.1}), where a benchmark method is the Augmented Lagrangian Method (ALM) which was proposed in \cite{1hes,2pow}. It plays a significant role in both algorithmic design and practical application for various convex programming problems. We refer to \cite{3ber,4cha,5for,6roc,7bir,8glo,9ito} and the references therein.
In practice, for a given iterate $(x^k,\lambda^k)$, the iterative scheme of the classical ALM reads as
\begin{eqnarray}\label{1.2}
\left\{ \begin{array}{ll}
x^{k+1} = \arg \min \left\{ \mathcal{L}_{\beta} (x,\lambda^{k}):=\theta(x)-\lambda^{\top}(Ax-b)+\frac{\beta}{2}\|Ax-b\|^{2} \,|\, x\in\mathcal{X} \right\},\\ \lambda^{k+1}= \lambda^{k}- \beta (Ax^{k+1}-b), \end{array} \right. \end{eqnarray}
where $\beta >0$ denotes the penalty parameter, $\lambda \in \mathbb{R}^{m}$ is the associated Lagrange multiplier. Hereafter, $ x $ and $\lambda$ are referred to the primal and dual variables respectively, and $I$ and $\mathbf{0}$ are regarded as a identity matrix and a zero matrix with proper dimensions, respectively.
Ignoring some constant terms,the $x$-subproblem of (\ref{1.2}) can be rewritten as
$$x^{k+1} = \theta(x)+\frac{\beta}{2}\|Ax-b-\frac{1}{\beta}\lambda^{k}\|^{2}.$$
It is obvious that the objective function $\theta$, the coefficient matrix A, and the set $\mathcal{X}$ are all appear at the same time, so it is still difficult to be solved if without utilizing some linearization techniques or inner solvers. Some existing algorithm can be applied to decoupled the objection function $\theta$ and coefficient matrix A, so as to alleviate the $x$-subproblem substantially, such as the linearized ALM \cite{10he} and primal-dual method \cite{11cha}. In above-mentioned algorithm, the $x$-subproblem only depends on $\theta$ and $\mathcal{X}$, and the proximity operator of the objective function $\theta$, which is defined as
$$Prox_{\theta}^{\beta}(x):= \arg \min \left\{\theta(y)+\frac{\beta}{2} \|y-x\|^2\ |y \in\mathbb{R}^{n}\right\},\ \ \forall x \in \mathbb{R}^{n},\ \ \forall \beta >0, $$
has a closed-form representation . But in order to ensure convergence, there is an extra restriction on step-size , i.e., $\sigma>\beta\|A^\top A\|$, where $\sigma>0$ and $\|A^\top A\|$ represent the spectral norm of $A^\top A$. Hence the step-size for solving (\ref{1.2}) becomes small when $\|A^\top A\|$ is too large, and the convergence may not be guaranteed consequently \cite{12he}. Recently, a balanced version of ALM was proposed in \cite{12he}, which has no limitation on step-size and takes the following iterative scheme:
\begin{eqnarray}\label{1.3} {\rm (Balanced\ ALM)}\, \left\{ \begin{array}{ll}
x^{k+1}= \arg \min \left\{ \theta(x)+\frac{1}{\beta}{\|x-(x^k+\frac{1}{\beta})A^\top \lambda^{k}\|}^2 \,|\, x\in\mathcal{X} \right\},\\ \lambda^{k+1}= \lambda^{k}- \left(\frac{1}{\beta} AA^\top +\delta I_m\right)^{-1} \left\{A(2x^{k+1}-x^{k})-b \right\}, \end{array} \right. \end{eqnarray}
In which $\beta>0$ and $\delta>0$. It is clear that the $x$-subproblem of the balanced ALM decouples the objective function and the coefficient matrix without any extra condition. Namely, the parameter $\beta$ does not depend on $\|A^\top A\|$, and the $x$-subproblem have a closed-form solution since its solution can be expressed as a proximal mapping. However, balanced ALM will take much time to update $\lambda^{k+1}$, and in practice it will take an inner solver to tackle the dual subproblem or use the well-known Cholesky factorization to deal with an equivalent linear equation of dual problem. In this sence, a new Penalty ALM was proposed in \cite{13bai} to solve it, which reads:
\begin{eqnarray}\label{1.4}
\left\{ \begin{array}{ll}
x^{k+1}= \arg \min \left\{ \theta(x)-\langle \lambda^{k}, Ax-b \rangle +\frac{\beta}{2}{\|A(x-x^k)\|}^2+\frac{1}{2}\|x-x^k\|_Q^2 \,|\, x\in\mathcal{X} \right\},\\ \lambda^{k+1}= \lambda^{k}-\beta [A(2x^{k+1}-x^{k})-b], \end{array} \right. \end{eqnarray}
where $\beta>0$, $Q\succ0$ is an arbitrarily given positive-defined matrix; the quadratic terms $\frac{\beta}{2}{\|A(x-x^k)\|}^2$ can be treated as a penalty term, while the quadratic terms $\frac{1}{2}\|x-x^k\|_Q^2$ can be regarded as a matrix proximal term.
Both the balanced ALM and the new Penalty ALM update the new iterate by a primal-dual order. Exploiting the variational inequality structure of the balanced ALM, a dual-primal version of the balanced ALM was proposed in \cite{14xu}. The novel proposed method generates the new iterates by a dual-primal order and enjoys the same computational difficulty with the original primal-dual balanced ALM, which reads
\begin{eqnarray}\label{1.5}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\left(\frac{1}{\beta} AA^\top +\delta I_m\right)^{-1}(Ax^{k}-b),\\
x^{k+1}= \arg \min \left\{ \theta(x)+\frac{\beta}{2}{\left\|x-\left\{x^{k}+\frac{1}{\beta}A^\top(2\lambda^{k+1}-\lambda^{k})\right\}\right\|}^2 \,|\, x\in\mathcal{X} \right\}, \end{array} \right. \end{eqnarray} where $\beta>0$ and $\delta>0$.
It is clear that the original primal-dual balanced ALM also will take much time to update $\lambda^{k+1}$, and in practice it will take an inner solver or use the well-known Cholesky factorization to deal with an equivalent linear equation of dual problem the same as the balanced ALM. So our main purpose is to alleviate the difficulty for solving dual-subproblem of the original primal-dual balanced ALM (\ref{1.5}) by utilizing the novel penalty technique proposed in \cite{13bai}. Inspired by the works \cite{13bai,14xu}, this paper propose a penalty dual-primal ALM combines a novel penalty technique with updates the new iterates in a dual-primal order, as follows: \begin{center} \fbox{ \parbox{\textwidth} { {\bfseries Algorithm 1: the novel penalty dual-primal ALM }\\ Let $\beta>0$ and $Q\succ0$ be an arbitrarily given positive-defined matrix. Then the new iterate $(x^{k+1},\lambda^{k+1})$ is generated with $(x^k,\lambda^k)$ via the following steps:
\begin{eqnarray}\label{1.6}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta(Ax^{k}-b),\\
x^{k+1}= \arg \min \left\{ \theta(x)-\langle 2\lambda^{k+1}-\lambda^k,Ax-b \rangle +\frac{\beta}{2}{\|A(x-x^k)\|}^2+\frac{1}{2}\|x-x^k\|_Q^2 \,|\, x\in\mathcal{X} \right\}. \end{array} \right. \end{eqnarray} } } \end{center}
Where the quadratic terms $\frac{\beta}{2}{\|A(x-x^k)\|}^2$ can be treated as a penalty term, while the quadratic term $\frac{1}{2}{\|x-x^k\|_Q}^2$ can be regarded as a matrix proximal term.
It is clear that the $x$-subproblem can be written equivalently as
$$x^{k+1}= \arg \min \left\{ \theta(x)-\langle 2\lambda^{k+1}-\lambda^k,Ax-b \rangle +\frac{1}{2}\|x-x^k\|_{\beta A^\top A+Q}^2 \,|\, x\in\mathcal{X} \right\}.$$
In particular, \begin{enumerate}
\item when talking $Q=\tau I-\beta A^\top A$ with $\tau >\beta \|A^\top A\|$, could convert the $x$-update to
$$x^{k+1}=\arg \min \left\{ \theta(x) +\frac{\tau}{2}\|x-x^k-\frac{1}{\tau}A^\top(2\lambda^{k+1}-\lambda^k)\|^2 \,|\, x\in\mathcal{X} \right\};$$
\item when talking $Q=\beta(\tau I- A^\top A$) with $\tau > \|A^\top A\|$, could convert the $x$-update to
$$x^{k+1}=\arg \min \left\{ \theta(x) +\frac{\tau \beta}{2}\|x-x^k-\frac{1}{\tau}A^\top(2\lambda^{k+1}-\lambda^k)\|^2 \,|\, x\in\mathcal{X} \right\},$$ \end{enumerate} which have a closed-form solution by proximity operator of the objective function $\theta(x)$.
The dual update of (\ref{1.6}) is similar to \cite{11cha} and is comparatively much easier than that of the dual-primal balanced ALM. As said before, the global convergence of this penalty dual-primal ALM (\ref{1.6}), compared with some existing splitting algorithms, will no longer depend on the value of $\|A^\top A\|$. We also raise the extension versions of the proposed penalty dual-primal ALM (\ref{1.6}) to tackle the multi-block separable convex minimization problem with both linear equality and inequality constraints.
The paper is organized as follows. In Section 2, we recall some preliminaries which are used in the sequel. In Section 3, we establish the convergence analysis of the penalty dual-primal ALM. We extend the proposed method to solve the multiple-block separable convex problems and show its convergence analysis in Section 4. In Section 5, we further establish the partial splitting version and its convergence analysis. In Section 6, we present some computational experiments. Finally, we show our conclusions in Section 7.
\section{Preliminaries}
In this section, we recall some fundamental variational inequality characterization and fundamental lemma to simplify the convergence analysis of the proposed novel penalty dual-primal ALM (\ref{1.6}).
In what follows, $\mathbb{R}^{n}$ will stand for the $n$-dimensional Euclidean space, $\langle x, y \rangle=x^\top y=\sum_{i=1}^n x_iy_i$,\ \ $\|x\|=\sqrt{\langle x, x\rangle},$ where $x,y \in \mathbb{R}^{n}$ and $\top$ stands for the transpose operation.
We first derive the optimality condition of the model (\ref{1.1}) in the lens of variational inequality (more detailed introduction refer to, e.g. \cite{15gu,16he,17he} ). The Lagrangian function of model (\ref{1.1}) is defined as \begin{eqnarray}\label{2.1} \mathcal{L}(x,\lambda):=\theta(x)-\lambda^{\top}(Ax-b), \end{eqnarray}
with $\lambda \in \mathbb{R}^{m}$ is the Lagrange multiplier. To take into account both linear equality and inequality constraints in (\ref{1.1}),we define \begin{eqnarray*} \Omega:= \mathcal{X} \times \Lambda,\, \, \mbox{where}\, \Lambda:=
\left\{ \begin{array}{ll} \mathbb{R}^{m},\, \, \mbox{if} \, \, Ax=b,\\ \mathbb R_+^{m},\, \, \mbox{if} \, \, Ax\geq b. \end{array} \right. \end{eqnarray*}
The pair $(x^\ast,\lambda^\ast)\in \Omega$ is called a saddle point of Lagrangian function \eqref{2.1} which means it is a solution point of \eqref{1.1}, if it satisfies
\begin{eqnarray*} \begin{aligned} \mathcal{L}_{\lambda \in \Lambda}(x^\ast,\lambda) \leq \mathcal{L}(x^\ast,\lambda^\ast) \leq \mathcal{L}_{x \in \mathcal{X}}(x,\lambda^\ast). \end {aligned} \end{eqnarray*}
Which can be separately rewritten as the following variational inequalities:
\begin{equation*} \begin{aligned} \left\{ \begin{array}{ll} x^\ast \in \mathcal{X},\ \ \theta(x)-\theta(x^\ast)+(x-x^\ast)^\top(-A^\top\lambda^\ast)\geq 0,\ \forall x \in \mathcal{X},\\ \lambda^\ast \in \Lambda, \ \ \ (\lambda-\lambda^\ast)^\top(Ax^\ast-b)\geq 0,\ \forall \lambda \in \Lambda. \end{array} \right. \end {aligned} \end{equation*}
Which can be further written as the following compact format:
\begin{equation}\label{2.2} \begin{aligned} \omega^\ast \in \Omega,\ \ \ \theta(x)-\theta(x^\ast)+(\omega-\omega^\ast)^\top F(\omega^\ast) \geq 0,\ \ \forall \omega \in \Omega, \end {aligned} \end{equation}
with \begin{equation}\label{2.3} \omega=\left(\begin{matrix} x\\ \lambda \end{matrix}\right),\ \ F(\omega)=\left(\begin{matrix} -A^\top \lambda\\ Ax-b \end{matrix}\right)\ \ and\ \ \Omega=\mathcal{X} \times \Lambda. \end{equation}
Since the operator $F(\omega)$ defined in (\ref{2.3}) is affine with a skew-symmetric matrix and thus satisfies
\begin{equation}\label{2.4} (\omega-\widetilde{\omega})^\top(F(\omega)-F(\widetilde{\omega}))\equiv 0\ \ \ \ \forall \omega, \widetilde{\omega} \in \Omega. \end{equation}
We denote by $\Omega^\ast$ the solution set of the variational inequality (\ref{2.3},\ref{2.4}), which is also the saddle points of the Lagrangian function (\ref{2.1}) and the solution set of the model (\ref{1.1}).
The following basic lemma will be used frequently for our further discussions,which proof is elementary and can be found in, e.g. \cite{18bec}.
\begin{lemma}\label{lem2.1} Let $x\in\mathcal{X}$ be a closed convex set, $\theta(x)$ and $f(x)$ are convex functions. If $f$ is differentiable, and the solution set of the minimization problem
$$\min\left\{\theta(x)+f(x)|\ x\in\mathcal{X}\right\},$$
is nonempty, then it holds that
$$x^\ast \in \arg \min \left\{\theta(x)+f(x)|\ x\in\mathcal{X}\right\}$$
if and only if
$$x^\ast \in\mathcal{X},\ \ \theta(x)-\theta(x^\ast)+(x-x^\ast)^\top\nabla f(x^\ast) \geq 0,\ \ \forall x \in \mathcal{X}.$$ \end{lemma}
We recall the so called quasi-Fej\'{e}r convergence theorem, which will be used in our convergence analysis. \begin{definition}\label{plusdef1} A sequence $ \left\{ u^k \right\} \subset \mathbb{R}^{n} $ is called quasi-Fej\'{e}r convergent to a nonempty set $ U\subset\mathbb{R}^{n}$ if, for every $u \in U $, there exists a sequence $ \left\{\epsilon_k \right\} \subset \mathbb{R}_{+} $ such that
$$\|u^{k+1}-u\|^2 \leq \|u^{k}-u\|^2+\epsilon_k, $$ with $\sum_{k=0}^\infty < \infty. $ \end{definition}
\begin{lemma}\label{pluslem1}\cite[Theorem 1]{19plus} If $ \left\{ u^k \right\} \subset \mathbb{R}^{n} $ is a quasi-Fej\'{e}r convergent sequence to a nonempty set $ U$, then $ \left\{ u^k \right\} $ is bounded. Furthermore, if a cluster point $\bar{u}$ of $ \left\{ u^k \right\} $ belongs to $U$, then $\lim \limits_{k\to \infty} u^k=\bar{u}$. \end{lemma}
\section{Convergence analysis}
In this section, we establish the convergence analysis of the introduced penalty dual-primal ALM, following the analogous analysis method in \cite{12he}. We prove the first lemma which plays a key role in analysis of the convergence as follows. \begin{lemma}\label{lem3.1} Let $\left\{\omega^{k}=(x^{k},\lambda^{k})\right\}$ be the sequence generated by the penalty dual-primal balanced ALM (\ref{1.6}). Then we have
\begin{equation}\label{3.1} \begin{aligned}
\omega^{k+1}\in\Omega, \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H(\omega^k-\omega^{k+1}),\ \forall\omega \in \Omega, \end {aligned} \end{equation}
where
\begin{equation}\label{3.2} \begin{aligned}
H=\left(\begin{matrix}\beta A^\top A+Q & -A^\top \\ -A & \frac{1}{\beta}I \end{matrix}\right). \end{aligned} \end{equation} \end{lemma}
\begin{proof} For the $x$-subproblem in (\ref{1.6}), it follows from Lemma \ref{lem2.1} that $x^{k+1}\in \mathcal{X},$ \begin{equation*} \begin{aligned}
\theta(x)-\theta(x^{k+1})+(x-x^{k+1})^\top\left\{-A^\top(2\lambda^{k+1}-\lambda^{k})+(\beta A^\top A+Q)(x^{k+1}-x^{k})\right\} \geq 0, \end{aligned} \end{equation*} for all $x\in\mathcal{X}$, which can be rewritten as \begin{equation}\label{3.3} \begin{aligned} &x^{k+1}\in \mathcal{X},\ \ \theta(x)-\theta(x^{k+1})+(x-x^{k+1})^\top(-A^\top\lambda^{k+1}) \\ &\geq (x-x^{k+1})^\top\left\{-A^\top(\lambda^{k}-\lambda^{k+1})+(\beta A^\top A+Q)(x^{k}-x^{k+1})\right\}, \ \ \forall x\in\mathcal{X}. \end{aligned} \end{equation}
For the $\lambda$-subproblem in (\ref{1.6}), we have
$$ Ax^{k}-b+\frac{1}{\beta}(\lambda^{k+1}-\lambda^{k})=0 , $$
which implies that \begin{equation*} \begin{aligned} (\lambda-\lambda^{k+1})^\top \left\{Ax^{k+1}-b-A(x^{k+1}-x^{k})+\frac{1}{\beta}(\lambda^{k+1}-\lambda^{k}) \right\} \geq 0,\ \ \ \forall \lambda \in \Lambda, \end{aligned} \end{equation*}
which leads to
\begin{equation}\label{3.4} \begin{aligned} (\lambda-\lambda^{k+1})^\top(Ax^{k+1}-b) \geq (\lambda-\lambda^{k+1})^\top \left\{-A(x^{k}-x^{k+1})+\frac{1}{\beta}(\lambda^{k}-\lambda^{k+1})\right\},\ \ \ \forall \lambda \in \Lambda. \end{aligned} \end{equation}
Combining(\ref{3.3}) and(\ref{3.4}), we have
\begin{equation*} \begin{aligned} \theta(x)-\theta(x^{k+1})+\left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} \left(\begin{matrix}-A^\top \lambda^{k+1}\\ Ax^{k+1}-b \end{matrix}\right) \geq \left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} \left(\begin{matrix}\beta A^\top A+Q & -A^\top \\ -A & \frac{1}{\beta}I \end{matrix}\right) \left(\begin{matrix}x^{k}-x^{k+1}\\ \lambda^{k}-\lambda^{k+1} \end{matrix}\right), \end{aligned} \end{equation*} namely, \begin{equation*} \begin{aligned} \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H (\omega^{k}-\omega^{k+1}). \end{aligned} \end{equation*} \end{proof}
Convergence of the penalty dual-primal ALM depends on the positive definiteness of the matrix $H$, and following proposition gives the prove.
\begin{proposition}\label{pro3.1} The matrix $H$ defined in (\ref{3.2}) is positive definite. \end{proposition}
\begin{proof} Note that \begin{equation*} \begin{aligned} H&=\left(\begin{matrix}\beta A^\top A+Q & -A^\top \\ -A & \frac{1}{\beta}I \end{matrix}\right)\\
&=\left(\begin{matrix}\beta A^\top A & -A^\top \\ -A & \frac{1}{\beta}I \end{matrix}\right)+\left(\begin{matrix}Q & 0 \\ 0 & 0 \end{matrix}\right) \\
&=\left(\begin{matrix}-\sqrt{\beta} A^\top \\ \frac{1}{\sqrt{\beta}}I \end{matrix}\right)
\left(\begin{matrix}-\sqrt{\beta} A & \frac{1}{\sqrt{\beta}}I \end{matrix}\right) +
\left(\begin{matrix}Q & 0 \\ 0 & 0 \end{matrix}\right). \end {aligned} \end{equation*}
Then, for arbitrary $\omega=(x,\lambda)\neq 0$, we have
\begin{equation*} \begin{aligned}
\omega^\top H \omega =\left\|\frac{1}{\sqrt{\beta}} \lambda-\sqrt{\beta}Ax \right\|^2+\|x\|_Q^2 >0, \end {aligned} \end{equation*}
and hence the matrix $H$ is positive definite.
\end{proof}
The following lemma is also the basis of convergence analysis of the proposed novel penalty dual-primal ALM (\ref{1.6}).
\begin{lemma}\label{lem3.2} Let $ \left\{\omega^{k}=(x^{k},\lambda^{k})\right\} $ be the sequence generated by the proposed penalty dual-primal balanced ALM (\ref{1.6}). Then, we obtain
\begin{eqnarray}\label{3.5} \begin{aligned}
\theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega) &\geq \frac{1}{2}(\|\omega-\omega^{k+1}\|_H^2-\|\omega-\omega^{k}\|_H^2)\\
&+\frac{1}{2}\|\omega^{k}-\omega^{k+1}\|_H^2, \ \forall \omega \in \Omega. \end {aligned} \end{eqnarray} \end{lemma}
\begin{proof}
Since $F(\omega)$ is affine with a skew-symmetric matrix, like (\ref{2.4}) we have
\begin{equation*} \begin{aligned} (\omega-\omega^{k+1})F(\omega^{k+1})=(\omega-\omega^{k+1})F(\omega). \end {aligned} \end{equation*}
According to (\ref{3.1}), we have
\begin{equation}\label{3.6} \begin{aligned} \omega^{k+1}\in\Omega, \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega) \geq (\omega-\omega^{k+1})^\top H(\omega^k-\omega^{k+1}),\ \forall\omega \in \Omega. \end {aligned} \end{equation}
Applying the identity
\begin{equation*} \begin{aligned}
(a-b)^\top H (c-d)=\frac{1}{2}\left\{\|a-d\|_H^2-\|a-c\|_H^2\right\}+\frac{1}{2}\left\{\|c-b\|_H^2-\|d-b\|_H^2\right\}, \end{aligned} \end{equation*} to the right-hand side of (\ref{3.6}) with $a=\omega$,\ \ $b=d=\omega^{k+1}$,\ \ $c=\omega^{k}$, then we obtain
\begin{equation}\label{3.7} \begin{aligned} (\omega-\omega^{k+1})^\top H(\omega^k-\omega^{k+1})
=\frac{1}{2}\left\{\|\omega-\omega^{k+1}\|_H^2-\|\omega-\omega^{k}\|_H^2\right\}+\frac{1}{2}\|\omega^{k}-\omega^{k+1}\|_H^2. \end {aligned} \end{equation}
Combining with (\ref{3.6}) and (\ref{3.7}), then we obtain the conclusion
\begin{equation*} \begin{aligned}
\theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega) & \geq \frac{1}{2}(\|\omega-\omega^{k+1}\|_H^2-\|\omega-\omega^{k}\|_H^2) \\
& +\frac{1}{2}\|\omega^{k}-\omega^{k+1}\|_H^2, \ \forall \omega \in \Omega. \end {aligned} \end{equation*} \end{proof}
Combining the two lemmas given above, we can directly obtain the following key theorem.
\begin{lemma}\label{thm3.1} Let$ \left\{\omega^{k}=(x^{k},\lambda^{k})\right\} $ be the sequence generated by the proposed penalty dual-primal balanced ALM (\ref{1.6}). Then, we have
\begin{equation}\label{3.8} \begin{aligned}
\|\omega^{k+1}-\omega^\ast\|_H^2 \leq \|\omega^{k}-\omega^{\ast}\|_H^2-\|\omega^{k}-\omega^{k+1}\|_H^2,\ \ \forall \omega^\ast \in \Omega^\ast. \end {aligned} \end{equation} \end{lemma}
\begin{proof} Setting $\omega$ in (\ref{3.5}) as any fixed $\omega^\ast \in \Omega^\ast$, then we get \begin{equation}\label{3.9} \begin{aligned}
&\|\omega^{k}-\omega^{\ast}\|_H^2-\|\omega^{k+1}-\omega^\ast\|_H^2-\|\omega^{k}-\omega^{k+1}\|_H^2 \\ & \geq 2\left\{\theta(x^{k+1})-\theta(x^\ast)+(\omega^{k+1}-\omega^\ast)^\top F(\omega^\ast) \right\},\ \forall \omega^\ast \in \Omega^\ast. \end{aligned} \end{equation}
Since $\omega^\ast \in \Omega^\ast$ and $\omega^{k+1} \in \Omega^\ast$, according to (\ref{2.2},\ref{2.3}), the right-hand of the inequality (\ref{3.9}) is non-negative, i.e.
\begin{equation*} \begin{aligned} \theta(x^{k+1})-\theta(x^\ast)+(\omega^{k+1}-\omega^\ast)^\top F(\omega^\ast) \geq 0. \end{aligned} \end{equation*}
This leads to the assertion of this theorem immediately.
\end{proof}
\begin{theorem}\label{plusthe1} The sequence $ \left\{\omega^{k}=(x^{k},\lambda^{k})\right\} $ generated by the proposed penalty dual-primal balanced ALM (\ref{1.6}) is quasi-Fej\'{e}r convergent to $\Omega^\ast$. \end{theorem}
\begin{proof} It follows from (\ref{3.8}) that \begin{equation}\label{plus1} \begin{aligned}
\|\omega^{k+1}-\omega^\ast\|_H^2 \leq \|\omega^{k}-\omega^{\ast}\|_H^2,\ \ \forall \omega^\ast \in \Omega^\ast, \end {aligned} \end{equation} by considering (\ref{plus1}) and Definition \ref{plusdef1} with $\varepsilon_k =0 ,\ \ \ k=1, 2, \ldots $, we conclude the proof. \end{proof}
Base on the establishment of the key contraction in Theorem (\ref{plusthe1}), we can easily prove the convergence of the proposed novel penalty dual-primal ALM (\ref{1.6}) through the following theorem.
\begin{theorem}\label{thm3.2} Let$ \left\{\omega^{k}=(x^{k},\lambda^{k})\right\} $ be the sequence generated by the proposed penalty dual-primal balanced ALM (\ref{1.6}) and $H$ be defined in (\ref{3.2}). Then, the sequence $\left\{\omega^{k}\right\} $ converges to some $\bar{\omega} \in \Omega^\ast$. \end{theorem}
\begin{proof} According to Lemma \ref{pluslem1}, we know that the sequence $\left\{\omega^{k}\right\} $ is bounded. According to (\ref{3.8}), we have
\begin{equation}\label{3.10} \begin{aligned}
\lim_{k\to \infty}\|\omega^{k}-\omega^{k+1}\|_H^2=0 \end{aligned} \end{equation}
Let $\bar{\omega}$ be a cluster point of $\left\{\omega^{k}\right\} $, and $\left\{\omega^{k_j}\right\} $ be a subsequence converging to $\bar{\omega}$. It follows from (\ref{3.1}) that
\begin{equation*} \begin{aligned} \omega^{k_j} \in \Omega, \theta(x)-\theta(x^{k_j})+(\omega-\omega^{k_j})^\top F(\omega^{k_j}) \geq (\omega-\omega^{k_j})^\top H(\omega^{k_j-1}-\omega^{k_j}),\ \forall\omega \in \Omega. \end{aligned} \end{equation*}
Since the matrix $H$ is positive definite, it follows from (\ref{3.10}) and the continuity of $\theta(x)$ and $F(\omega)$ that
\begin{equation*} \begin{aligned} \bar{\omega} \in \Omega, \ \ \ \theta(x)-\theta(\bar{x})+(\omega-\bar{\omega})^\top F(\bar{\omega}) \geq 0, \ \ \ \forall \omega \in \Omega. \end{aligned} \end{equation*}
This implies $ \bar{\omega} \in \Omega^\ast$. Then it follows from Lemma \ref{pluslem1} that $\lim \limits_{k\to \infty}\omega^{k}=\bar{\omega} \in \Omega^\ast $, then the proof is complete.
\end{proof}
\begin{theorem}\cite[Theorem 3.5]{12he} Let$ \left\{\omega^{k}=(x^{k},\lambda^{k})\right\} $ be the sequence generated by the proposed penalty dual-primal balanced ALM (\ref{1.6}) and $H$ be defined in (\ref{3.2}). For any integer number $t>O$, if we define
\begin{eqnarray}\label{3.12} \tilde{\omega}_t :=\frac{1}{t+1}\sum_{k=0}^t \omega^{k+1}, \end{eqnarray} then we have
$$\tilde{\omega}_t \ \ \in \Omega, \theta(\tilde{x_t})-\theta(x)+(\tilde{\omega_t}-\omega)^\top F(\omega) \leq
\frac{1}{2(t+1)} \|\omega-\omega^0\|_H^2, \ \ \forall \omega \in \Omega. $$ \end{theorem}
\begin{proof} It follows from (\ref{3.5}) that $\omega^{k+1} \in \Omega$, \begin{eqnarray}\label{3.13} \begin{aligned}
\theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega)+\frac{1}{2}\|\omega-\omega^{k}\|_H^2 \geq \frac{1}{2}\|\omega-\omega^{k+1}\|_H^2, \ \forall \omega \in \Omega. \end {aligned} \end{eqnarray}
Summarizing the inequalities (\ref{3.13}) over $k=0,1,\cdots,t$, we obtain
\begin{eqnarray*} \begin{aligned}
(t+1)\theta(x)-\sum_{k=0}^t\theta(x^{k+1})+((t+1)\omega-\sum_{k=0}^t\omega^{k+1})^\top F(\omega)+\frac{1}{2}\|\omega-\omega^{0}\|_H^2 \geq 0, \ \forall \omega \in \Omega. \end {aligned} \end{eqnarray*}
It follows from (\ref{3.12}) that
\begin{eqnarray}\label{3.14} \begin{aligned}
\frac{1}{t+1}\sum_{k=0}^t\theta(x^{k+1})-\theta(x)+(\tilde{\omega}_t-\omega)^\top F(\omega) \leq \frac{1}{2(t+1)}\|\omega-\omega^{0}\|_H^2, \ \forall \omega \in \Omega. \end {aligned} \end{eqnarray}
Note that $\tilde{\omega}_t$ define in (\ref{3.12}) is a convex combination of all iterates $\omega^k$ for $k=0,\cdots,t$, and $\theta(x)$ is convex. We thus have
$$\tilde{x}_t=\frac{1}{t+1}\sum_{k=0}^t x^{k+1},$$
and also
$$\theta(\tilde{x}_t) \leq \frac{1}{t+1}\sum_{k=0}^t \theta(x^{k+1}).$$
Substituting it into (\ref{3.14}), the theorem follows directly. \end{proof}
The above theorem shows the worst-case $O(1/t)$ convergence rate of the proposed penalty dual-primal balanced ALM \eqref{1.6}, where $t$ denotes the total iteration counter.
\section{Splitting version}
In this section, following the same extension technique in \cite{12he}, we also design the splitting version for the dual-primal balanced ALM (\ref{1.6}) to solve the following multi-block separable convex optimization problem with both linear equality and inequality constraints:
\begin{equation}\label{4.1}
\min\left\{\sum_{i=1}^{p}\theta_i(x_i)|\ \sum_{i=1}^{p}A_{i}x_{i}=b\ (or \geq b),\ x_{i}\in\mathcal{X}_{i}\right\}, \end{equation} where $p\geq 1$ is the number of subfunctions, and $\sum_{i=1}^{p}n_{i}=n$; $\theta_{i},\ i=1, \ldots, p,$ are closed proper convex but not necessarily smooth functions; $\mathcal{X}_{i}\subset \mathbb{R}^{n_{i}},\ i=1, \ldots, p,$ are nonempty closed convex sets; $A_{i} \in \mathbb{R}^{m\times n_{i}},\ i=1, \ldots, p,$ are given matrixs and $b\in \mathbb{R}^{m}$ is a known vector. The model (\ref{4.1}) can be applied to various domains, such as, e.g. \cite{20boy,21sun,22yua}. It is clear that the model (\ref{4.1}) is an extension of the model (\ref{1.1}) from $p=1$ to $p\geq 1$.
\subsection{Splitting version of the penalty dual-primal ALM}
In this section, we extend the penalty dual-primal ALM to solve the multi-block separable convex optimization problem (\ref{4.1}) and proposed a splitting version of (\ref{1.6}) below.
\begin{center} \fbox{ \parbox{\textwidth} { {\bfseries Algorithm 2: the spitting penalty dual-primal balanced ALM }\\ Let $i=1,\cdots, p,$ and $x_{i}\in\mathcal{X}_{i}$, $\beta_{i}>0$; $Q_{i}\succ0$ are arbitrarily given positive-defined matrixes. Then the new iterate $(x^{k+1},\lambda^{k+1})$ is generated with $(x^k,\lambda^k)$ via the following steps:
\begin{eqnarray}\label{4.2}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\sum_{i=1}^{p}\beta_{i}(\sum_{i=1}^{p}A_{i}x_{i}^{k}-b),\\
x_{i}^{k+1}= \arg \min \left\{ \theta(x_{i})-\langle 2\lambda^{k+1}-\lambda^k,A_{i}x_{i}-b \rangle +\frac{\beta_{i}}{2}{\|A_{i}(x_{i}-x_{i}^k)\|}^2+\frac{1}{2}\|x_{i}-x_{i}^k\|_{Q_{i}}^2 \right\}. \end{array} \right. \end{eqnarray} } } \end{center}
Where the quadratic terms $\frac{\beta_{i}}{2}{\|A_{i}(x_{i}-x_{i}^k)\|}^2$ can be treated as the penalty terms, while the quadratic terms $\frac{1}{2}\|x-x^k\|_Q^2$ can be regarded as the matrix proximal terms.
\subsection{Variational inequality characterization of the splitting version }
In order to analyze convergence, we also need to give the optimality condition of the model (\ref{4.2}) in the variational inequality context. Firstly, we reuse the letters and set
\begin{equation*} \Omega:= \mathcal{X}_{1} \times \ldots \times \mathcal{X}_{p} \times \Lambda\ \ where\ \ \Lambda:=
\left\{ \begin{array}{ll} \mathbb{R}^{m},\ \ \ if \ \ \sum_{i=1}^{p}A_{i}x_{i}=b,\\ \mathbb R_+^{m},\ \ \ if \ \ \sum_{i=1}^{p}A_{i}x_{i}\geq b. \end{array} \right. \end{equation*}
Analogous to the analysis in Section 2, it is clear that the optimal condition of (\ref{4.2}) is equivalent to finding a saddle point of the Lagrangian function of model (\ref{4.2}), which satisfy
\begin{equation}\label{4.3} \begin{aligned} \omega^\ast \in \Omega,\ \ \ \theta(x)-\theta(x^\ast)+(\omega-\omega^\ast)^\top F(\omega^\ast) \geq 0,\ \ \forall \omega \in \Omega, \end {aligned} \end{equation}
where \begin{equation}\label{4.4} \theta=\sum_{i=1}^{p}\theta_{i},\ \ \omega=\left(\begin{matrix} x\\ \lambda \end{matrix}\right),\ \ x=\left(\begin{matrix} x_{1}\\ \vdots\\ x_{p}\end{matrix}\right),\ \ F(\omega)=\left(\begin{matrix} -A_{1}^\top \lambda\\ \vdots\\ -A_{p}^\top \lambda\\ \sum_{i=1}^{p}A_{i}x_{i}-b \end{matrix}\right)\ \ and\ \ \Omega=\mathcal{X} \times \Lambda. \end{equation}
Similarly, since the operator $F(\omega)$ defined in (\ref{4.4}) is affine with a skew-symmetric matrix and thus satisfies
\begin{equation}\label{4.5} (\omega-\widetilde{\omega})^\top(F(\omega)-F(\widetilde{\omega}))\equiv 0\ \ \ \forall \omega, \widetilde{\omega} \in \Omega. \end{equation}
\subsection{Convergence analysis of the splitting version}
According to the same analysis route in Section 3, we next give the essential lemma and the key proposition below.
\begin{lemma}\label{lem4.1} Let $\left\{\omega^{k}=(x^{k},\lambda^{k})\right\}$ be the sequence generated by the spitting penalty dual-primal balanced ALM (\ref{4.2}). Then we have \begin{equation}\label{4.6} \begin{aligned}
\omega^{k+1}\in\Omega, \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H(\omega^k-\omega^{k+1}),\ \forall\omega \in \Omega, \end {aligned} \end{equation}
where
\begin{equation}\label{4.7} \begin{aligned}
H=\left(\begin{matrix}\beta_1 A_1^\top {A_1}+Q_1 & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots& \vdots\\
\mathbf{0} & \ldots & \beta_P A_p^\top {A_P}+Q_P & -A_P^\top \\ -A_1 & \ldots & -A_P & \sum_{i=1}^p \frac{1}{\beta_i}I
\end{matrix}\right). \end{aligned} \end{equation} \end{lemma}
\begin{proof} According to Lemma \ref{lem2.1}, for $i=1,\ldots, p, x_i^{k+1}\in \mathcal{X}_i$,
\begin{equation*} \begin{aligned}
\theta_i(x_i)-\theta_i(x_i^{k+1})+(x-x_i^{k+1})^\top\left\{-A_i^\top(2\lambda^{k+1}-\lambda^{k})+ (\beta_i A_i^\top A_i+Q_i)(x_i^{k+1}-x_i^{k})\right\} \geq 0, \end{aligned} \end{equation*}
for all ${x_i} \in\mathcal{X}_i$. It can be written as \begin{equation}\label{4.8} \begin{aligned} &x_i^{k+1}\in \mathcal{X}_i,\ \ \theta({x_i})-\theta(x_i^{k+1})+({x_i}-x_i^{k+1})^\top(-A_i^\top\lambda^{k+1})\\ & \geq ({x_i}-x_i^{k+1})^\top\left\{-A_i^\top(\lambda^{k}-\lambda^{k+1})+(\beta_i A_i^\top A_i+Q_i)(x_i^{k}-x_i^{k+1})\right\}, \ \ \forall {x_i}\in\mathcal{X}_i. \end{aligned} \end{equation}
For the $\lambda$-subproblem in (\ref{4.2}), we have $$ \sum_{i=1}^{p}A_ix_i^{k}-b+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k+1}-\lambda^{k})=0,$$ which implies that \begin{equation*} \begin{aligned} (\lambda-\lambda^{k+1})^\top \left\{\sum_{i=1}^{p}A_ix_i^{k+1}-b-\sum_{i=1}^{p}A_i(x_i^{k+1}-x_i^{k})+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k+1}-\lambda^{k}) \right\} \geq 0,\ \ \forall \lambda \in \Lambda, \end{aligned} \end{equation*}
which leads to
\begin{equation}\label{4.9} \begin{aligned} (\lambda-\lambda^{k+1})^\top(\sum_{i=1}^{p}A_ix_i^{k+1}-b) \geq (\lambda-\lambda^{k+1})^\top (-\sum_{i=1}^{p}A_i(x_i^{k}-x_i^{k+1})+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k}-\lambda^{k+1})), \end{aligned} \end{equation} for all $\lambda \in \Lambda$. Combining (\ref{4.8}) and (\ref{4.9}), we have \begin{equation*} \begin{aligned} &\theta(x)-\theta(x^{k+1})+\left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} \left(\begin{matrix}-A_1^\top \lambda^{k+1}\\ \cdots \\ -A_p^\top \lambda^{k+1} \\ \sum_{i=1}^{p}A_ix_i^{k+1}-b \end{matrix}\right)\\ &\geq\left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} \left(\begin{matrix}\beta_1 A_1^\top {A_1}+Q_1 & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots& \vdots\\
\mathbf{0} & \ldots & \beta_P A_p^\top {A_P}+Q_P & -A_P^\top \\ -A_1 & \ldots & -A_P & \sum_{i=1}^p \frac{1}{\beta_i}I
\end{matrix}\right) \left(\begin{matrix}x^{k}-x^{k+1}\\ \lambda^{k}-\lambda^{k+1} \end{matrix}\right), \end{aligned} \end{equation*}
namely
\begin{equation*} \begin{aligned} \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H (\omega^{k}-\omega^{k+1}). \end{aligned} \end{equation*} \end{proof}
\begin{proposition}\label{pro4.1} The matrix $H$ defined in (\ref{4.7}) is positive definite. \end{proposition}
\begin{proof} Note that
\begin{equation*} \begin{aligned} H=\bar{H}+\left(\begin{matrix}Q_1 & \cdots & \mathbf{0} & \mathbf{0} \\ \vdots & \cdots & \vdots & \vdots \\
\mathbf{0} & \ddots & Q_P & \mathbf{0} \\ \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} \end{matrix}\right) \end {aligned} \end{equation*}
and \begin{equation*} \begin{aligned} \bar{H}&=\left(\begin{matrix}\beta_1 A_1^\top {A_1} & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots& \vdots\\ \mathbf{0} & \ldots & \beta_P A_p^\top {A_P} & -A_P^\top \\ -A_1 & \ldots & -A_P & \sum_{i=1}^p \frac{1}{\beta_i}I \end{matrix}\right)\\
&=\left(\begin{matrix}\beta_1 A_1^\top {A_1} & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots& \vdots\\ \mathbf{0} & \ldots & \mathbf{0} & \mathbf{0} \\ -A_1 & \ldots & \cdots & \frac{1}{\beta_1}I \end{matrix}\right)+\cdots +\left(\begin{matrix} \mathbf{0} & \ldots & \mathbf{0} & \mathbf{0} \\ \vdots & \ddots & \vdots& \vdots\\ \mathbf{0} & \ldots & \beta_P A_p^\top {A_P} & -A_P^\top \\ \mathbf{0} & \ldots & -A_p & \frac{1}{\beta_p}I \end{matrix}\right)\\ &= \left(\begin{matrix}-\sqrt{\beta_1} A_1^\top \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ \frac{1}{\sqrt{\beta_1}}I \end{matrix}\right) \left(\begin{matrix} -\sqrt{\beta_1} {A_1} & \mathbf{0} &\cdots & \mathbf{0} & \frac{1}{\sqrt{\beta_1}}I \end{matrix}\right)
+\cdots+\left(\begin{matrix} \mathbf{0} \\ \vdots \\ \mathbf{0} \\ -\sqrt{\beta_P} A_P^\top \\ \frac{1}{\sqrt{\beta_P}}I \end{matrix}\right)
\left(\begin{matrix} \mathbf{0} & \cdots & \mathbf{0} & -\sqrt{\beta_P} {A_P} & \frac{1}{\sqrt{\beta_P}}I\end{matrix}\right) \end{aligned} \end{equation*} Then, for arbitrary $\omega=(x,\lambda)\neq 0$, we have
\begin{equation*} \begin{aligned}
\omega^\top H \omega =\sum_{i=1}^p \left\|\frac{1}{\sqrt{\beta_i}} \lambda-\sqrt{\beta_i}A_ix_i \right\|^2+\sum_{i=1}^p \|x_i\|_{Q_i}^2 >0 \end {aligned} \end{equation*} and hence the matrix $H$ is positive definite. \end{proof}
\begin{remark} Based on the Lemma \ref{lem4.1} and Proposition \ref{pro4.1}, similar lemmas and theorems for convergence and convergence rate as those in Section 3 can be obtained for the splitting version of the penalty dual-primal ALM (\ref{4.2}). Hence it is easy to prove the worst-case $O(1/t)$ convergence rate of the spitting penalty dual-primal balanced ALM (\ref{4.2}), where $t$ denotes the total iteration counter. For brevity, we omit the details proof. \end{remark}
\section{Partial proximal strategy for the penalty dual-primal ALM}
The penalty dual-primal ALM (\ref{1.6}) can be generalized to the splitting version (\ref{4.2}) when the background issue changes from the one-block (\ref{1.1}) to the multiple-block case (\ref{4.1}). We can also proposed another way for the generalization, which only adds proximal matrix terms to partial $x_i$-subproblem of the splitting version (\ref{4.2}). In section 4, it is clear that each of the $x_i$-subproblem involve the matrix proximal terms $\frac{1}{2}\|x_i-x_i^k\|_{Q_i}^2$ , so that it has two quadratic terms in each $x_i$-subproblem. In this sense, we think about partial proximity to the $x_i$-subproblem, so we proposed the Partial proximal penalty dual-primal ALM to solve the same multi-block separable convex optimization problem as in Section 4 (\ref{4.1}).
\subsection{Partial proximal penalty dual-primal ALM}
Without ambiguity, we add the proximal matrix terms to the former $p_1$ $x_i$-subproblem, while no to the latter $p_2$ $x_i$-subproblem, where $1\leq p_1, p_2 \leq p$, and $p_1 +p_2 =p$. For succinctness of notation, we reused the notation in Section 4.
A partial proximal penalty dual-primal ALM for the multiple-block model (\ref{4.1}) can be read as:
\begin{center} \fbox{ \parbox{\textwidth} { {\bfseries Algorithm 3: the partial proximal penalty dual-primal ALM }\\ Let$\beta_{i}>0$ and $Q_{i}\succ0$ are arbitrarily given positive-defined matrixes, $i=1,\cdots, p,$ and $x_{i}\in\mathcal{X}_{i}$. Then the new iterate $(x^{k+1},\lambda^{k+1})$ is generated with $(x^k,\lambda^k)$ via the following steps:
\begin{eqnarray}\label{5.1}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\sum_{i=1}^{p}\beta_{i}(\sum_{i=1}^{p}A_{i}x_{i}^{k}-b),\\
x_{i}^{k+1}= \arg \min \left\{ \theta(x_{i})-\langle 2\lambda^{k+1}-\lambda^k,A_{i}x_{i}-b \rangle +\frac{1}{2}\|x_{i}-x_{i}^k\|_{\beta_i A_i^\top A_i +Q_{i}}^2 \right\}, i=1, \cdots, p_{1},\\
x_{i}^{k+1}= \arg \min \left\{ \theta(x_{i})-\langle 2\lambda^{k+1}-\lambda^k,A_{i}x_{i}-b \rangle +\frac{\beta_{i}}{2}{\|A_{i}(x_{i}-x_{i}^k)\|}^2\right\}, i= p_{1} +1, \cdots, p. \end{array} \right. \end{eqnarray} } } \end{center}
\subsection{Convergence analysis of the partial proximal penalty dual-primal ALM (\ref{5.1})}
According to the same variational inequality characterization and the same analysis route in Section 4, we next give the essential lemma and the key proposition below.
\begin{lemma}\label{lem5.1} Let $\left\{\omega^{k}=(x^{k},\lambda^{k})\right\}$ be the sequence generated by the partial proximal penalty dual-primal ALM (\ref{5.1}). Then we have \begin{equation}\label{5.2} \begin{aligned}
\omega^{k+1}\in\Omega, \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H(\omega^k-\omega^{k+1}),\ \forall\omega \in \Omega, \end {aligned} \end{equation}
where $H$ defined as
\begin{equation}\label{5.3} \begin{aligned}
\left(\begin{matrix}
\beta_1 A_1^\top {A_1}+Q_1 & \ldots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & -A_1^\top \\
\vdots & \ddots & \vdots& \vdots& \ddots & \vdots & \vdots& \\
\mathbf{0} & \ldots & \beta_{P_1} A_{P_1}^\top {A_{P_1}}+Q_{P_1} & \mathbf{0} & \ldots & \mathbf{0} & -A_{P_1}^\top \\
\mathbf{0}& \ldots & \mathbf{0} & \beta_{P_1+1} A_{P_1+1}^\top {A_{P_1+1}} & \ldots & \mathbf{0} & -A_{P_1+1}^\top\\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots& \vdots \\
\mathbf{0} & \ldots & \mathbf{0} & \mathbf{0} & \ldots & \beta_{P} A_{P}^\top {A_{P}} & -A_{P}^\top \\
-A_1 & \ldots & -A_{P_1} & -A_{P_1+1} & \ldots & -A_{P} & \sum_{i=1}^p \frac{1}{\beta_i}I
\end{matrix}\right). \end{aligned} \end{equation} \end{lemma}
\begin{proof} According to Lemma \ref{lem2.1}, for $i=1,\ldots, p_1, x_i^{k+1}\in \mathcal{X}_i$,
\begin{equation*} \begin{aligned}
\theta_i(x_i)-\theta_i(x_i^{k+1})+(x-x_i^{k+1})^\top\left\{-A_i^\top(2\lambda^{k+1}-\lambda^{k})+ (\beta_i A_i^\top A_i+Q_i)(x_i^{k+1}-x_i^{k})\right\} \geq 0, \end{aligned} \end{equation*} for all ${x_i} \in\mathcal{X}_i$. Which can be written as \begin{equation}\label{5.4} \begin{aligned} &x_i^{k+1}\in \mathcal{X}_i,\ \ \theta({x_i})-\theta(x_i^{k+1})+({x_i}-x_i^{k+1})^\top(-A_i^\top\lambda^{k+1})\\ & \geq ({x_i}-x_i^{k+1})^\top\left\{-A_i^\top(\lambda^{k}-\lambda^{k+1})+(\beta_i A_i^\top A_i+Q_i)(x_i^{k}-x_i^{k+1})\right\}, \ \ \forall {x_i}\in\mathcal{X}_i. \end{aligned} \end{equation}
Similarly, for $i=p_1 +1,\ldots, p$, according to Lemma \ref{lem2.1} we have $x_i^{k+1}\in \mathcal{X}_i$, \begin{equation*} \begin{aligned}
\theta_i(x_i)-\theta_i(x_i^{k+1})+(x-x_i^{k+1})^\top\left\{-A_i^\top(2\lambda^{k+1}-\lambda^{k})+ \beta_i A_i^\top A_i(x_i^{k+1}-x_i^{k})\right\} \geq 0, \end{aligned} \end{equation*} for all ${x_i} \in\mathcal{X}_i$. Which can be written as \begin{equation}\label{5.5} \begin{aligned} &x_i^{k+1}\in \mathcal{X}_i,\ \ \theta({x_i})-\theta(x_i^{k+1})+({x_i}-x_i^{k+1})^\top(-A_i^\top\lambda^{k+1})\\ & \geq ({x_i}-x_i^{k+1})^\top\left\{-A_i^\top(\lambda^{k}-\lambda^{k+1})+\beta_i A_i^\top A_i (x_i^{k}-x_i^{k+1})\right\}, \ \ \forall {x_i}\in\mathcal{X}_i. \end{aligned} \end{equation} For the $\lambda$-subproblem in (\ref{5.1}), we have $$ \sum_{i=1}^{p}A_ix_i^{k}-b+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k+1}-\lambda^{k})=0,$$ which implies that \begin{equation*} \begin{aligned} (\lambda-\lambda^{k+1})^\top \left\{\sum_{i=1}^{p}A_ix_i^{k+1}-b-\sum_{i=1}^{p}A_i(x_i^{k+1}-x_i^{k})+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k+1}-\lambda^{k}) \right\} \geq 0,\ \ \forall \lambda \in \Lambda, \end{aligned} \end{equation*} which leads to
\begin{equation}\label{5.6} \begin{aligned} (\lambda-\lambda^{k+1})^\top(\sum_{i=1}^{p}A_ix_i^{k+1}-b) \geq (\lambda-\lambda^{k+1})^\top (-\sum_{i=1}^{p}A_i(x_i^{k}-x_i^{k+1})+\sum_{i=1}^{p}\frac{1}{\beta_i}(\lambda^{k}-\lambda^{k+1})),\ \ \ \forall \lambda \in \Lambda. \end{aligned} \end{equation} Combining(\ref{5.4}), (\ref{5.5}) and(\ref{5.6}), we obtain
\begin{equation*} \begin{aligned} \theta(x)-\theta(x^{k+1})+\left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} \left(\begin{matrix}-A_1^\top \lambda^{k+1}\\ \cdots \\ -A_p^\top \lambda^{k+1} \\ \sum_{i=1}^{p}A_ix_i^{k+1}-b \end{matrix}\right) \geq\left(\begin{matrix}x-x^{k+1}\\ \lambda-\lambda^{k+1}\end{matrix}\right)^{\top} H \left(\begin{matrix}x^{k}-x^{k+1}\\ \lambda^{k}-\lambda^{k+1} \end{matrix}\right), \end{aligned} \end{equation*} namely,
\begin{equation*} \begin{aligned} \theta(x)-\theta(x^{k+1})+(\omega-\omega^{k+1})^\top F(\omega^{k+1}) \geq (\omega-\omega^{k+1})^\top H (\omega^{k}-\omega^{k+1}). \end{aligned} \end{equation*} \end{proof}
\begin{proposition}\label{pro5.1} The matrix $H$ defined in (\ref{5.3}) is positive definite. \end{proposition}
\begin{proof} Note that
\begin{equation*} \begin{aligned} H=\bar{H}+ \left(\begin{matrix} Q_1 & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
\mathbf{0} & \cdots & Q_{P_1} & \mathbf{0} & \cdots & \mathbf{0} \\
\mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\
\mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} \\
\mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0}
\end{matrix}\right) \end {aligned} \end{equation*}
and
\begin{equation*} \begin{aligned} \bar{H}&=\left(\begin{matrix} \beta_1 A_1^\top {A_1} & \ldots & \mathbf{0} & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots & \ddots & \vdots & \vdots \\ \mathbf{0} & \ldots & \beta_{P_1} A_{P_1}^\top {A_{P_1}} & \ldots & \mathbf{0} & -A_{P_1}^\top \\ \vdots & \ddots & \vdots & \ddots & \vdots & \vdots\\ \mathbf{0} & \ldots & \mathbf{0} & \ldots & \beta_{P} A_{P}^\top {A_{P}} & -A_{P}^\top\\ -A_1 & \ldots & -A_{P_1} & \ldots & -A_{P} & \sum_{i=1}^p \frac{1}{\beta_i}I \end{matrix}\right)\\
&=\left(\begin{matrix}\beta_1 A_1^\top {A_1} & \ldots & \mathbf{0} & -A_1^\top \\ \vdots & \ddots & \vdots& \vdots\\ \mathbf{0} & \ldots & \mathbf{0} & \mathbf{0} \\ -A_1 & \ldots & \cdots & \frac{1}{\beta_1}I \end{matrix}\right)+\cdots +\left(\begin{matrix} \mathbf{0} & \ldots & \mathbf{0} & \mathbf{0} \\ \vdots & \ddots & \vdots& \vdots\\ \mathbf{0} & \ldots & \beta_P A_p^\top {A_P} & -A_P^\top \\ \mathbf{0} & \ldots & -A_p & \frac{1}{\beta_p}I \end{matrix}\right)\\ &= \left(\begin{matrix}-\sqrt{\beta_1} A_1^ \top \\ \mathbf{0} \\ \vdots \\ \mathbf{0} \\ \frac{1}{\sqrt{\beta_1}}I \end{matrix}\right) \left(\begin{matrix} -\sqrt{\beta_1} {A_1} & \mathbf{0} &\cdots & \mathbf{0} & \frac{1}{\sqrt{\beta_1}}I \end{matrix}\right)
+\cdots+\left(\begin{matrix} \mathbf{0} \\ \vdots \\ \mathbf{0} \\ -\sqrt{\beta_P} A_P^\top \\ \frac{1}{\sqrt{\beta_P}}I \end{matrix}\right)
\left(\begin{matrix} \mathbf{0} & \cdots & \mathbf{0} & -\sqrt{\beta_P} {A_P} & \frac{1}{\sqrt{\beta_P}}I\end{matrix}\right) \end{aligned} \end{equation*}
Then, for arbitrary $\omega=(x,\lambda)\neq 0$, we have
\begin{equation*} \begin{aligned}
\omega^\top H \omega =\sum_{i=1}^p \left\|\frac{1}{\sqrt{\beta_i}} \lambda-\sqrt{\beta_i}A_ix_i \right\|^2+\sum_{i=1}^{p_1} \|x_i\|_{Q_i}^2 >0 \end {aligned} \end{equation*}
and hence the matrix $H$ is positive definite.
\end{proof}
\begin{remark} Based on the Lemma \ref{lem5.1} and Proposition \ref{pro5.1}, similar lemmas and theorems for convergence and convergence rate as those in Section 3 can be obtained for the splitting version of the penalty dual-primal ALM (\ref{5.1}). Hence it is easy to prove the worst-case $O(1/t)$ convergence rate of the spitting penalty dual-primal balanced ALM (\ref{5.1}), where $t$ denotes the total iteration counter. For brevity, we omit the details proof. \end{remark}
\section{Numerical experiments}
In this section, two numerical tests and a practical application are used for the purpose of demonstrate the proposed algorithm performance. All code are written in Matlab and all experiments are performed in Matlab R2015b on a workstation with an Intel(R) Core(TM) i7-8550U CPU(1.80GHz) and 8GB RAM.
We firstly show the comparison among the penalty dual-primal balanced ALM (\ref{1.6}), the primal-dual balanced ALM proposed in \cite{14xu} and the balanced ALM proposed in \cite{12he} and for soling the basic pursuit problem (equality-constrained $l_1$ minimization problem). The preliminary numerical result shows that the proposed method has a better performance. \begin{example} The basic pursuit problem is: \begin{eqnarray}\label{6.1}
\min\{\|x\|_1\,|\,Ax=b, x\in \mathbb{R}^{n}\}, \end{eqnarray}
with $\|x\|_1=\sum_{i=1}^{n}|x_i|$ denotes the $l_1$-norm of a vector, $A\in \mathbb{R}^{m\times n}(m<n)$ is a given matrix and $b \in \mathbb{R}^{m}$ is a given vector.
Apply the proposed method (\ref{1.6}) to the problem (\ref{6.1}), the iterative scheme is as follows:
\begin{eqnarray*}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta(Ax^{k}-b),\\
x^{k+1}= \arg \min \left\{ \|x\|_1-\langle 2\lambda^{k+1}-\lambda^k,Ax-b \rangle
+\frac{1}{2}\|x-x^k\|_{\beta A^\top A +Q}^2 \right\}. \end{array} \right. \end{eqnarray*}
In particular, take $Q=\tau I-\beta A^\top A$ with $\tau >\beta \|A^\top A\|$, the iterate scheme could converted to:
\begin{eqnarray}\label{p6.2}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta(Ax^{k}-b),\\
x^{k+1}= \arg \min \left\{\|x\|_{1}+\frac{\tau}{2}\|x-x^k-\frac{1}{\tau}A^\top(2\lambda^{k+1}-\lambda^k)\|^2 \right\}. \end{array} \right. \end{eqnarray}
Then the solutions of the problem (\ref{p6.2}) are given respectively by the following explicit form: \begin{eqnarray*}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta(Ax^{k}-b),\\ x^{k+1}=S_{1/\tau}[x^{k}+\frac{1}{\tau}A^\top(2\lambda^{k+1}-\lambda^k)], \end{array} \right. \end{eqnarray*} where $S_{\delta}(t)$ is the soft thresholding operator \cite{24che} defined as
\begin{eqnarray}\label{p6.3}
(S_{\delta}(t))_{i}:=(1-\delta/|t_{i}|)_{+}\cdot t_{i},i=1,2,\cdot\cdot\cdot,m. \end{eqnarray}
Follow the same rules, apply the primal-dual balanced ALM \cite{14xu} for (\ref{6.1}), the iterative scheme is read as:
\begin{eqnarray*} {\rm (DP-BALM)} \left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}- (\frac{1}{\tau}AA^\top+\delta I_m)^{-1} (Ax^{k}-b),\\
x^{k+1} = \arg \min \left\{\|x\|_{1}+\frac{\tau}{2}\|x-x^k-\frac{1}{\tau}A^\top(2\lambda^{k+1}-\lambda^k)\|^2 \right\}. \end{array} \right. \end{eqnarray*}
And the balanced ALM \cite{12he} for (\ref{6.1}) reads as : \begin{eqnarray*} {\rm (Balanced \ ALM)} \left\{ \begin{array}{ll}
x^{k+1} = \arg \min \left\{\|x\|_{1}+\frac{\tau}{2}\|x-x^k-\frac{1}{\tau}A^\top\lambda^k\|^2 \right\},\\ \lambda^{k+1}= \lambda^{k}- (\frac{1}{\tau}AA^\top+\delta I_m)^{-1} (A(2x^{k+1}-x^{k})-b). \end{array} \right. \end{eqnarray*}
We generate the data by the same way in \cite{14xu}: the matrixes $A$ are generated from independently normal distribution $\mathcal{N}(0,1)$; for all tested algorithm the initial point $(x^0,\lambda^0)$ is randomly generated, and we take the following toned values of parameters for the mentioned experiments: \begin{enumerate} \item the penalty dual-primal balanced ALM: $\beta:=0.001$ and $\tau=2.5$;
\item the primal-dual balanced ALM and the balanced ALM :$\delta=1000$ and $\tau:2.5$. \end{enumerate}
The termination criteria is defined by \begin{eqnarray*}
R(k)=\max\left\{\|x^{k+1}-x^{k}\|, \|\lambda^{k+1}-\lambda^{k}\| \right\} < 10^{-7} \end{eqnarray*}
Table \ref{tab1} lists the number of iterations and runtime in seconds respectively of the the PDP-ALM, the DP-BALM and the balanced ALM for solving the basic pursuit problem with different dimension $m \times n$ of $A$, and $CR=\|Ax-b\|^2$ stands for constrained residual. From the numerical experimental result, it is clearly that compared with the the PDP-ALM and the DP-BALM under different dimension of $A$, the proposed method has much better performs both in the number of iterations and runtime. To further visualize the numerical results, we also plot the convergence curves versus iteration numbers of some representative examples in Figure \ref{fig1}.
\begin{table}\label{tab1} \centering\caption{The number of iterations and runtime of the PL-ADMM and the proposed method for solving LASSO model } \begin{tabular}{cccccccccc}\hline
\multicolumn{1}{l}{}&\multicolumn{3}{l}{PDP-ALM}&\multicolumn{3}{l}{DP-ALM}&\multicolumn{3}{l}{B-ALM}\\ $m \times n$ &Iter. &Time &$CR$ &Iter. &Time &$CR$ &Iter. &Time &$CR$\\ \hline $300\times500$ &465 &0.11 &2.86e-4 &562 &0.16 &3.64e-4 &564 &0.18 &3.64e-4\\ $400\times600$ &503 &0.15 &1.49e-4 &669 &0.28 &1.82e-4 &671 &0.28 &1.82e-4\\ $450\times750$ &453 &0.18 &6.60e-5 &610 &0.39 &1.01e-4 &612 &0.35 &1.01e-4\\ $500\times900$ &373 &0.20 &4.63e-5 &511 &0.39 &5.98e-5 &513 &0.39 &5.98e-5\\ $500\times1000$ &1294 &0.73 &7.25e-5 &1969 &1.50 &2.29e-4 &1971 &1.47 &2.29e-4\\ $600\times1150$ &843 &0.70 &1.77e-4 &1263 &1.40 &1.65e-4 &1266 &1.34 &1.70e-4\\ $700\times1300$ &672 &0.75 &8.90e-5 &1073 &1.51 &1.15e-4 &1074 &1.50 &1.15e-4\\ $800\times1450$ &455 &0.65 &4.63e-5 &704 &1.26 &1.53e-4 &706 &1.31 &1.48e-4\\ $900\times1600$ &547 &1.03 &5.99e-5 &943 &2.21 &8.24e-5 &945 &2.27 &8.27e-5\\ $1000\times1750$ &876 &1.95 &4.32e-5 &1523 &4.14 &6.37e-5 &1525 &4.22 &6.40e-5\\ $1100\times1900$ &725 &1.83 &4.78e-5 &1179 &3.81 &8.53e-5 &1181 &3.89 &8.58e-5\\ $1100\times2000$ &694 &1.85 &4.47e-5 &1263 &4.24 &9.12e-5 &1265 &4.30 &9.13e-5\\ $1200\times2150$ &406 &1.31 &3.85e-5 &602 &3.65 &5.38e-5 &896 &3.88 &1.10e-4\\ $1300\times2300$ &420 &1.52 &6.66e-5 &762 &3.53 &6.27e-5 &760 &3.58 &7.01e-5\\ \hline \end{tabular} \end{table}
\begin{figure}
\caption{Convergence curves of the DPD-ALM, DP-ALM and the balanced ALM compared with iteration number under various dimension of $A$}
\label{fig1}
\end{figure}
\end{example}
Then we show the comparison among the spitting penalty dual-primal balanced ALM (\ref{4.2}) , the PL-ADMM \cite{23yan} and the PIPL-ADMM \cite{24che} for solving the well-known LASSO model, and the preliminary numerical result shows that the proposed method has a better performance. \begin{example} The LASSO model is: \begin{eqnarray}\label{6.2}
\min_{y} \frac{1}{2}\|Ay-b\|^{2}+\sigma\|y\|_{1}, \end{eqnarray}
where $\|y\|_{1}:=\Sigma^{n}_{i=1}|y_{i}|$, $A\in \mathbb{R}^{m\times n}$ is a design matrix usually with $m\ll n$, $m$ is the number of date point, $n$ is the number of features, $b\in \mathbb{R}^{m}$ is the response vector and $\sigma>0$ is a regularization parameter.
By a new auxiliary variable $x$, (\ref{6.2}) can be rewritten as the form:
\begin{eqnarray}\label{6.3}
\min\{\frac{1}{2}\|x-b\|^{2}+\sigma\|y\|_{1}\,|\,x-Ay=0, \ \ \ x\in \mathbb{R}^{m}, y\in \mathbb{R}^{n}\}. \end{eqnarray}
which is a special case of (\ref{4.1}). Then apply the spitting penalty dual-primal balanced ALM (\ref{4.2}) to (\ref{6.3}), we have:
\begin{eqnarray*}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta_{1}(x-Ay)-\beta_{2}(x-Ay),\\
x^{k+1}= \arg \min \left\{ \frac{1}{2}\|x-b\|^2-\langle 2\lambda^{k+1}-\lambda^k,x-b \rangle
+\frac{1}{2}\|x-x^k\|_{\beta_1 I^\top I +Q_{1}}^2 \right\} \\
y^{k+1}= \arg \min \left\{ \sigma\|y\|_{1}-\langle 2\lambda^{k+1}-\lambda^k,-Ay-b \rangle
+\frac{1}{2}\|y-y^k\|_{\beta_2 A^\top A +Q_{2}}^2 \right\}. \end{array} \right. \end{eqnarray*}
In particular, take $Q_1=\tau_1 I-\beta_1 I$ with $\tau_1 >\beta_1 \|I^\top I\|$, $Q_2=\tau_2 I-\beta_2 A^\top A$ with $\tau_2 >\beta_2 \|A^\top A\|$, the iterate could converted to:
\begin{eqnarray}
\left\{ \begin{array}{ll}{\label{6.4}} \lambda^{k+1}= \lambda^{k}-\beta_{1}(x-Ay)-\beta_{2}(x-Ay),\\
x^{k+1}= \arg \min \left\{ \frac{1}{2}\|x-b\|^2+\frac{\tau_1}{2}\|x-x^k-\frac{1}{\tau_1}(2\lambda^{k+1}-\lambda^k)\|^2 \right\} \\
y^{k+1}= \arg \min \left\{ \sigma\|y\|_{1}+\frac{\tau_2}{2}\|y-y^k-\frac{1}{\tau_2}A^\top(2\lambda^{k+1}-\lambda^k)\|^2 \right\}. \end{array} \right. \end{eqnarray}
\begin{table}[htbp] \centering\caption{The number of iterations and runtime of the PL-ADMM and the proposed method for solving LASSO model }\label{tab2} \begin{tabular}{ccccccccccc}\hline
\multicolumn{1}{l}{}&\multicolumn{2}{l}{PL-ADMM}&\multicolumn{2}{l}{PIPL-ADMM}&\multicolumn{2}{l}{PDP-ALM}\\ $\gamma$ &Iter.1&Time.1&Iter.2&Time.2&Iter.3&Time.3& $\frac{Iter.3}{Iter.1}$& $\frac{Time3}{Time1}$& $\frac{Iter.3}{Iter.2}$& $\frac{Time3}{Time2}$\\\hline $0.05$ &376 &29.35 &204 &16.16 &101 &3.53 &0.27 &0.12 &0.50 &0.22\\ $0.10$ &380 &29.63 &417 &33.98 &111 &3.92 &0.29 &0.13 &0.27 &0.12\\ $0.15$ &384 &30.13 &397 &31.66 &118 &4.15 &0.31 &0.14 &0.30 &0.13\\ $0.20$ &399 &32.97 &410 &34.58 &123 &4.34 &0.31 &0.13 &0.30 &0.13\\ $0.25$ &413 &32.18 &409 &33.57 &127 &4.54 &0.31 &0.14 &0.31 &0.14\\ $0.30$ &409 &31.92 &406 &34.08 &130 &4.56 &0.32 &0.14 &0.32 &0.13\\ $0.35$ &420 &33.02 &404 &32.64 &132 &4.65 &0.31 &0.14 &0.33 &0.14\\ $0.40$ &420 &32.87 &397 &32.32 &134 &4.76 &0.32 &0.14 &0.34 &0.15\\ $0.45$ &431 &33.82 &392 &31.34 &136 &4.78 &0.32 &0.14 &0.35 &0.15\\ $0.50$ &438 &34.67 &392 &35.86 &137 &4.83 &0.31 &0.14 &0.35 &0.13\\ $0.55$ &443 &34.69 &395 &31.34 &138 &4.85 &0.31 &0.14 &0.35 &0.15\\ $0.60$ &452 &35.34 &395 &31.78 &138 &4.87 &0.31 &0.14 &0.35 &0.15\\ $0.65$ &407 &23.41 &394 &31.38 &138 &3.53 &0.34 &0.15 &0.35 &0.11\\ $0.70$ &413 &23.74 &400 &32.67 &137 &3.53 &0.33 &0.15 &0.34 &0.11\\ \hline \end{tabular} \end{table}
\begin{figure}
\caption{The number of iterations and average runtime of splitting PDP-ALM and PL-ADMM in different $\tau_2$}
\label{fig2}
\end{figure}
Then the solutions of the problem (\ref{6.4}) are given respectively by the following explicit form:
\begin{eqnarray*}
\left\{ \begin{array}{ll} \lambda^{k+1}= \lambda^{k}-\beta_{1}(x-Ay)-\beta_{2}(x-Ay),\\ x^{k+1}= \frac{1}{\tau_1} [ \tau_1x^k+(2\lambda^{k+1}-\lambda^k) ] \\ y^{k+1}=S_{\sigma/\tau_2}[y^{k}+\frac{1}{\tau_2}A^\top(2\lambda^{k+1}-\lambda^k)], \end{array} \right. \end{eqnarray*} where $S_{\delta}(t)$ is the soft thresholding operator \cite{24che} defined as (\ref{p6.3}).
We generate the data by the same way in \cite{24che}: we first choose $A_{ij}\sim\mathcal{N}(0,1)$ and then scaled the columns to have unit norm. We use the script \lq sprandn\rq \, to generate a sparse vector $y^{*}$ which have approximately density $=100/n$ non-zeros entries taken from the normal distribution with zero mean and unit variance. We generate $b$ via $b:=Ay^{*}+e$, where $e$ is a small white noise taken from $e\sim\mathcal{N}(0,10^{-3}I)$. We choose the dimension of $A$ is $1050\times3500$. We set the regularization parameter $\sigma$ to $0.1$, and for all tested algorithm the initial point $(x^0,y^0,\lambda^0)$ is randomly generated, and we take the following toned values of parameters for the mentioned experiments: \begin{enumerate}
\item The spitting penalty dual-primal balanced ALM: $\beta_1 =\beta_2:=\frac{2-\tau_2}{\tau_2|\tau_2-1|}$ and $\tau_1 :=\frac{|\tau_2-1|}{5\beta_1\tau_2}+\frac{4}{5}$;
\item PL-ADMML:$\beta=\frac{2-\tau_2}{\tau_2|\tau_2-1|}$ and $\tau:=\frac{|\tau_2-1|}{5\beta_1\tau_2}+\frac{4}{5}$, \end{enumerate} and we all set $\tau_2=0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7. $
The termination criteria is defined by \begin{eqnarray*}
\max\left\{\|x^{k+1}-x^{k}\|, \|y^{k+1}-y^{k}\|, \|\lambda^{k+1}-\lambda^{k}\| \right\} < 10^{-10} \end{eqnarray*}
Table \ref{tab2} lists the number of iterations and runtime in seconds respectively of the the PL-ADMM and the spitting penalty dual-primal balanced ALM for solving the LASSO model with different parameter $\tau_2$. From the numerical experimental result, it is clearly that compared with the PL-ADMM under different parameter $\tau_2$, the proposed method has much better performs both in the number of iterations and runtime. To further visualize the numerical results, we also plot the iterations results in terms of the various parameters $\tau_2$ in Figure \ref{fig2}. \end{example}
\section{Conclusions}
This paper have proposed a penalty dual-primal augmented lagrangian method for solving convex minimization problems under linear equality or inequality constraints, and two extensions to solve the multiple-block separable convex programming problems. The global convergence and sub-linear convergence rate of the proposed methods have been establish in the lens of variational analysis. Furthermore, the numerical test on the basic pursuit problem and the lasso model demonstrate that the proposed method has better performance compared with the dual-primal ALM and the balanced ALM. This work may enhance the rich literature of the most recent balanced ALM.
\section*{ Acknowledgments.}
This paper was partially supported by the Youth Project of Science and Technology Research Program of Chongqing Education Commission of China (No. KJQN202201802), the the Natural Science Foundation of China (No. 12071379),
the Natural Science Foundation of Chongqing (No. cstc2021jcyj-msxmX0925, cstc2022ycjh-bgzxm0097).
\end{document} |
\begin{document}
\title{f Domination of the \ rectangular queen's graph}
\begin{abstract}
The queens graph $Q_{m \times n}$ has the squares of the $m \times n$ chessboard as its vertices;
two squares are adjacent if they are in
the same row, column, or diagonal of the board. A set $D$ of squares of $Q_{m \times n}$ is a \emph{dominating set}
for $Q_{m \times n}$ if every square of $Q_{m \times n}$ is either in $D$
or adjacent to a square in $D$. The minimum size of a dominating set
of $Q_{m \times n}$ is the \emph{domination number}, denoted by $\gamma(Q_{m \times n})$.
Values of $\gamma(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$ are given here, in each case with a f{\kern0pt}ile of minimum dominating sets (often all of them, up to symmetry) in an online \href{https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML}{appendix}. In these ranges for $m$ and $n$, monotonicity fails once: $\gamma(Q_{8 \times 11}) = 6 > 5 = \gamma(Q_{9 \times 11})
= \gamma(Q_{10 \times 11}) = \gamma(Q_{11 \times 11})$.
Let $g(m)$ [respectively $g^{*}(m)$] be the largest integer such that $m$ queens suf{\kern0pt}f{\kern0pt}ice to dominate the $(m+1) \times g(m)$ board [respectively, to dominate the $(m+1) \times g^{*}(m)$
board with no two queens in a row]. Starting from the elementary bound $g(m) \leq 3m$, domination when the board is far from square is investigated. It is shown (Theorem \ref{by3k}) that $g(m) = 3m$ can only occur when $m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod 9)}$, with an online appendix showing that this does occur for $m \leq 40, m \neq 3$. Also (Theorem \ref{upperbounds}), if $m \equiv 5, 6, \mbox{or } 7 \mbox{ (mod 9)}$ then $g^{*}(m) \leq 3m-2$, and if $m \equiv 8 \mbox{ (mod 9)}$ then $g^{*}(m) \leq 3m-4$. It is shown that equality holds in these bounds for $m \leq 40 $.
Lower bounds on $\gamma(Q_{m \times n})$ are given. In particular, if $m \leq n$ then $\gamma(Q_{m \times n}) \geq \min \{ m, \lceil (m+n-2)/4 \rceil \}$.
Two types of dominating sets (orthodox covers and centrally strong sets) are developed; each type is shown to give good upper bounds of $\gamma(Q_{m \times n})$ in several cases.
Three questions are posed: whether monotonicity of $\gamma(\q{m}{n})$ holds (other than from $(m, n) = (8, 11)$ to $(9, 11)$), whether $\gamma(\q{m}{n}) = (m+n-2)/4$ occurs with $m \leq n < 3m+2$ (other than for $(m, n) = (3, 3)$ and $(11, 11)$), and whether the lower bound given above can be improved.
A set of squares is \emph{independent} if no two of its squares are adjacent.
The minimum size of an independent dominating set of $Q_{m \times n}$ is the \emph{independent domination number}, denoted by $i(Q_{m \times n})$. Values of $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18, \,$
are given here, in each case with some minimum dominating sets. In these ranges for $m$ and $n$, monotonicity fails twice: $i(Q_{8 \times 11}) = 6 > 5 = i(Q_{9 \times 11}) = i(Q_{10 \times 11}) = i(Q_{11 \times 11})$, and $i(Q_{11 \times 18}) = 9 > 8 = i(Q_{12 \times 18})$.
\noindent \textbf{Keywords:} chessboard combinatorics, queen's graph,
domination, covering problems \\
\small {\textbf{Mathematics Subject Classifications:} 05C69, 05C99} \end{abstract}
\section{Introduction}
Let $m$ and $n$ be positive integers. We will identify the $m \times n$ chessboard with a rectangle in the Cartesian plane, having sides parallel to the coordinate axes. We place the board so that the center of every square has integer coordinates, and refer to each square by the coordinates \mbox{($x$,$y$)}\ of its center. Unless otherwise noted, squares have edge length one, and the board is placed so that the lower left corner has center $(1, 1)$; sometimes it is more convenient to use squares of edge length two or to place the board with its center at the origin of the coordinate system. By symmetry it suf{\kern0pt}f{\kern0pt}ices to consider the case $m \leq n$, which we will assume throughout: the board has at least as many columns as rows.
The square \mbox{($x$,$y$)}\ is in {\em column} $x$ and {\em row} $y$. Columns and rows will be referred to collectively as {\em orthogonals}. The {\em dif{\kern0pt}ference diagonal\/} (respectively {\em sum diagonal\/}) through square \mbox{($x$,$y$)}\ is the set of all board squares with centers on the line of slope +1 (respectively $-1$) through the point \mbox{($x$,$y$)}. The value of $y-x$ is the same for each square \mbox{($x$,$y$)}\ on a dif{\kern0pt}ference diagonal, and we will refer to the diagonal by this value. Similarly, the value of $y+x$ is the same for each square on a sum diagonal, and we associate this value to the diagonal. Orthogonals and diagonals are collectively referred to as \!\!\emph{ lines} of the board.
The queens graph \mbox{$Q_{m \times n}$}\ has the squares of the $m \times n$ chessboard as its vertices; two squares are adjacent if they are in some line of \mbox{$Q_{m \times n}$}. A set $D$ of squares of \mbox{$Q_{m \times n}$}\ is a {\em dominating set} for \mbox{$Q_{m \times n}$}\ if every square of \mbox{$Q_{m \times n}$}\ is either in $D$ or adjacent to a square in $D$. The minimum size of a dominating set is the \emph{domination number}, denoted by $\gamma(\mbox{$Q_{m \times n}$})$. A set of squares is {\em independent} if no two squares in the set are adjacent.
Almost all previous work on queen domination has concerned square boards. The problem of f{\kern0pt}inding values of $\gamma(\q{n}{n})$ has interested mathematicians for over 150 years. The f{\kern0pt}irst published work is that of De Jaenisch \cite{JA} in 1862, who gave minimum dominating sets and minimum independent dominating sets of $Q_{n \times n}$ for $n \leq 8$. His work was brief{\kern0pt}ly summarized by Rouse Ball \cite{RB} in 1892, who considered several other questions about queen domination. In 1901, W. Ahrens \cite[Chapter X]{AH1} gave minimum dominating sets for \q{9}{9}, and in 1902-3, K. von Szily \cite{V1, V2} gave minimum dominating sets of \q{n}{n} for $10 \leq n \leq 13$ and $n = 17$. Proof that these sets were minimum had to wait for later work, described below. De Jaenisch, Ahrens, and von Szily also worked extensively to f{\kern0pt}ind the number of dif{\kern0pt}ferent minimum dominating sets for each $n$, often giving lists with one set from each symmetry class. Many of these results were collected by Ahrens in the 1910 edition \cite{AH} of his book, and can also be found in its later editions.
More detail and some examples from recent work on domination of $Q_{n \times n}$ can be found in \cite{Weakley2018}.
The f{\kern0pt}irst published work on nonsquare boards of which we are aware is in Watkins \cite{Watkins2004}: the values $\gamma(Q_{5 \times 12}) = 4$ and $\gamma(Q_{6 \times 10}) = 4$ (see Problem 8.4 on p.~132 and Figure 8.19 on p.~137), found by D.~C.~Fisher.
Say that two minimum dominating set s of \mbox{$\gamma(\qq)$}\ are
\emph{equivalent} if there is an isometry of the $m \times n$ chessboard that carries one to the other.
We have computed \mbox{$\gamma(\qq)$}\ for rectangular chessboards with $4 \leq m \leq n \leq 18$. \linebreak Results are given in Table 1; for most $m$ and $n$ we give a f{\kern0pt}ile of minimum dominating \linebreak sets with one from every equivalence class, unless the number of equivalence classes is large. The online appendix at \\ \url{https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML} \linebreak includes the computational results. For each set, we describe its symmetry and say whether it can be obtained by one of the constructions in Section \ref{construct}.
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
$n \diagdown m$ & \hspace*{0.1mm} 4 \hspace*{0.1mm}
& \hspace*{0.1mm} 5 \hspace*{0.1mm}
& \hspace*{0.1mm} 6 \hspace*{0.1mm}
& \hspace*{0.1mm} 7 \hspace*{0.1mm}
& \hspace*{0.1mm} 8 \hspace*{0.1mm}
& \hspace*{0.1mm} 9 \hspace*{0.1mm} & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \hline
4 & \href{https://www.combinatorics.org/files/v26i4p45/04x04_2Q.html}{\emph{2}} & & & & & & & & & & & & & & \\ \hline
5 & \href{https://www.combinatorics.org/files/v26i4p45/04x05_2Q.html}{2} &
\href{https://www.combinatorics.org/files/v26i4p45/05x05_3Q.html}{\emph{3}} & & & & & & & & & & & & & \\ \hline
6 & \href{https://www.combinatorics.org/files/v26i4p45/04x06_3Q.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/05x06_3Q.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/06x06_3Q.html}{\emph{3}} & & & & & & & & & & & & \\ \hline
7 & \href{https://www.combinatorics.org/files/v26i4p45/04x07_3Q.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/05x07_3Q.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/06x07_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x07_4Q.html}{\emph{4}} & & & & & & & & & & & \\ \hline
8 & \href{https://www.combinatorics.org/files/v26i4p45/04x08_3Q.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/05x08_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x08_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x08_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x08_5Q.html}{\emph{5}} & & & & & & & & & & \\ \hline
9 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x09_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x09_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x09_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x09_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/09x09_5Q.html}{\emph{5}} & & & & & & & & & \\ \hline
10 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x10_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x10_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x10_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x10_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/09x10_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/10x10_5Q.html}{\emph{5}} & & & & & & & & \\ \hline
11 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x11_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x11_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x11_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x11_6Q.html}{\textbf{6}} &
\href{https://www.combinatorics.org/files/v26i4p45/09x11_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/10x11_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/11x11_5Q.html}{\emph{5}} & & & & & & &\\ \hline
12 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x12_4Q.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x12_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x12_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x12_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x12_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x12_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/11x12_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/12x12_6Q.html}{\emph{6}} & & & & & & \\ \hline
13 & 4 &
{5} &
\href{https://www.combinatorics.org/files/v26i4p45/06x13_5Q.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x13_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x13_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x13_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x13_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x13_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x13_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/13x13_7Q.html}{\emph{7}} & & & & & \\ \hline
14 & 4 &
{5} &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x14_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x14_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x14_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x14_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x14_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x14_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x14_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/14x14_8Q.html}{\emph{8}} & & & & \\ \hline
15 & 4 &
{5} &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x15_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x15_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x15_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x15_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x15_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x15_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x15_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/14x15_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/15x15_9Q.html}{\emph{9}} & & & \\ \hline
16 & 4 &
5 &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x16_6Q.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x16_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x16_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x16_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x16_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/12x16_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x16_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/14x16_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x16_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/16x16_9Q.html}{\emph{9}} & & \\ \hline
17 & 4 &
5 &
6 &
{7} &
\href{https://www.combinatorics.org/files/v26i4p45/08x17_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x17_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x17_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/12x17_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x17_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/14x17_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x17_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/16x17_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/17x17_9Q.html}{\emph{9}} & \\ \hline
18 & 4 &
5 &
6 &
{7} &
\href{https://www.combinatorics.org/files/v26i4p45/08x18_7Q.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x18_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/10x18_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/11x18_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/12x18_8Q.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x18_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/14x18_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x18_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/16x18_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/17x18_9Q.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/18x18_9Q.html}{\emph{9}} \\
\hline \end{tabular} \\[3mm] Table 1: Values of $\gamma(Q_{m \times n}), \, 4 \leq m \leq n \leq 18 $ (\href{http://oeis.org/A274138}{OEIS A274138}) \end{center}
The computation was done with a backtracking algorithm. The backtrack condition minimizes the number of queens placed. If a solution is found with $k$ queens, then the remaining search space is limited to at most $k-1$ queens. The algorithm places a single queen in a position covering the top left cell and does a recursive call to cover all remaining cells. Some heuristics are used also to {f}{i}nd the {f}{i}rst solution faster: the {f}{i}rst queen is placed in the middle of the board (actually in the closest to middle position attacking the top left cell); other possible attacking positions are only tried later. Frequently this position is part of a minimal solution.
Once it is shown that there is no solution with $k-1$ queens, a search for other solutions with $k$ queens is made.
Cockayne \cite[Problem 1]{CO} introduced monotonicity \[ \gamma(Q_{n \times n}) \stackrel{?}{\leq} \gamma(Q_{(n+1) \times (n+1)}) \] as an open problem (see also in Chartrand, Haynes, Henning and Zhang \cite[Conjecture 1.2.1 on page 7]{ChartrandHaynesHenningZhang2019}.
A remarkable observation about \q{8}{11}: six queens (with bold typeface in Table 1) are necessary to dominate it, though f{\kern0pt}ive queens are suf{\kern0pt}f{\kern0pt}icient (and necessary) to dominate each of \q{9}{11}, \q{10}{11}, \q{11}{11}. A possible explanation for this is given later. We note that f{\kern0pt}ive queens can cover all but one square of {\q{8}{11}}. One of the \href{https://www.combinatorics.org/files/v26i4p45/08x11_5Qalmost.html}{8} placements is in Figure 1.
\begin{figure}
\caption{Five queens dominate $Q_{8 \times 11}$ except for one square (\hspace*{-0.2mm}$\bullet$)}
\label{fig8x11almost}
\end{figure}
We extend Cockayne's question to the rectangular case.
\begin{ques} \label{cockayne} Column-wise monotonicity: Does $\mbox{$\gamma(\qq)$} \leq \gamma(\q{m}{(n+1)})$ hold for $m \leq n$?
Row-wise monotonicity: Does $\mbox{$\gamma(\qq)$} \leq \gamma(\q{(m+1)}{n})$ hold for $m \leq n, (m, n) \neq (8, 11)$? \end{ques}
We discuss one type of internal symmetry of minimum dominating set s that frequently occurs. A \emph{foursome} is a set of four squares $(x+a, y+b)$, $(x-a, y-b)$, $(x-b, y+a)$, $(x+b, y-a)$, where either each of $x, y, a, b$ is an integer or each is half an odd integer, and $a$ and $b$ are unequal and nonzero. The \emph{center} of the foursome is the point $(x, y)$, which need not be a square center. For examples, see Figure 1 above, the f{\kern0pt}irst minimum dominating set s given for $Q_{9 \times 9}$ and $Q_{11 \times 11}$, as well as the f{\kern0pt}irst four minimum dominating set s given for $Q_{11 \times 12}$.
If a foursome $F$ is f{\kern0pt}lipped across any of the four lines through its center, the result is another foursome $F'$ that occupies the same lines as $F$; this is illustrated in Figure 2.
\begin{figure}
\caption{The four squares with white queens are a foursome; the lines through its squares are shown. Ref{\kern0pt}lecting this foursome across any of the four lines through its center gives another foursome (the squares with black queens) that occupies the same lines.}
\label{fig4some}
\end{figure}
Thus if a dominating set $D$ of $Q_{m \times n}$ contains $F$, we may replace $F$ in $D$ with $F'$ and obtain a dominating set $D'$ of the same size as $D$. Usually $D$ and $D'$ are not equivalent.
As an example, we analyze the minimum dominating set s of $Q_{11 \times 17}$, which have size 8. Up to equivalence there are 131 solutions, shown in the \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html}{f{\kern0pt}ile}. Of these, 85 have no foursomes, 41 have exactly one foursome, four (\#\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution69}{69}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution70}{70}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution62}{62}) have exactly two foursomes, and one (\#\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution76}{76}) has 3 foursomes.
We may def{\kern0pt}ine a relation on the set ${\cal S}(11, 17)$ of minimum dominating sets of \q{11}{17} by saying that two sets are related if either they are equivalent, or f{\kern0pt}lipping a foursome of the f{\kern0pt}irst set yields a set equivalent to the second set. This relation is ref{\kern0pt}lexive and symmetric, and its transitive closure gives a partition ${\cal P}(11, 17)$ of ${\cal S}(11, 17)$, which may also be regarded as a partition of the set of (isometric) equivalence classes, as we will do.
For example, solution \#\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125} has two foursomes: one centered at $(12,6)$ with $(a, b) = (4, 2)$, and one centered at $(9, 7)$ with $(a, b) = (3, -1)$. Flipping the f{\kern0pt}irst gives solution \#\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution124}{124}. If instead we f{\kern0pt}lip the second, we get the ref{\kern0pt}lection of \#\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125} across its vertical line of symmetry. This implies that one part of the partition ${\cal P}(11, 17)$ contains just the equivalence classes of \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125} and \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution124}{124}, and we denote this part by \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution124}{124}\}. It is then straightforward to see that ${\cal P}(11, 17)$ has 85 parts with one member and 20 parts with two members: \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution8}{8}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution9}{9}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution20}{20}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution24}{24}\} \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution19}{19}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution23}{23}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution21}{21}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution25}{25}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution22}{22}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution26}{26}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution15}{15}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution27}{27}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution13}{13}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution6}{6}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution12}{12}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution5}{5}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution128}{128}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution129}{129}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution73}{73}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution68}{68}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution127}{127}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution130}{130}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution126}{126}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution131}{131}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution103}{103}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution97}{97}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution125}{125}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution124}{124}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution39}{39}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution44}{44}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution96}{96}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution95}{95}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution63}{63}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution62}{62}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution72}{72}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution70}{70}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution101}{101}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution77}{77}\}, \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution100}{100}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution78}{78}\}. There are also two parts with three members: \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution80}{80}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution71}{71}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution69}{69}\} and \{\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution79}{79}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution76}{76}, \href{https://www.combinatorics.org/files/v26i4p45/11x17_8Q.html\#Solution75}{75}\}.
It would be possible to reduce the size of the \href{https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML}{appendix} by giving for each $(m, n)$ one solution from each part of the partition ${\cal P}(m, n)$ rather than one solution from each isometry equivalence class. But when two solutions dif{\kern0pt}fer by the f{\kern0pt}lip of a foursome, it is not clear which is most useful to see, so we have not done this.
\section{Lower bounds on queen domination numbers}
We begin by looking at what happens when the board is far from square.
\begin{prop} \label{prop1} If $n \geq 3m-2$, then $ \gamma(\q{m}{n}) = m.$ \end{prop}
\begin{proof} Each queen attacks all squares in her own row, but at most three squares in any other row. Thus $m-1$ queens occupy at most $m-1$ rows and cover at most $3(m-1)$ squares in any row that does not contain a queen. On the other hand, $m$ queens are certainly suf{\kern0pt}f{\kern0pt}icient. \end{proof}
Note that $\gamma(\q{3}{6}) = 2$ (see the set given immediately after Theorem \ref{by3k}) and $\gamma(\q{5}{12}) = 4$ (see the database), but as shown by our computations, for $m = 4, 6, 7$, \mbox{$\gamma(\qq)$}\ reaches $m$ before $n$ reaches $3m-2$.
We change viewpoint slightly, focusing on the size of the dominating set rather than the dimensions of the board. For each positive integer $m$, let $g(m)$ be the largest integer such that $m$ queens can cover the $(m+1) \times g(m)$ board. (Proposition \ref{prop1} asserts $g(m) \leq 3m$.) Let $g^{*}(m)$ be the largest integer such that $m$ queens, no two in a row, can cover the $(m+1) \times g^{*}(m)$ board.
\begin{thm} \label{by3k} If $g(m) = 3m$ then $m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod $9$)}$.
Take the board to be a rectangle in the Cartesian plane with sides parallel to the axes, board center at the origin, and squares of edge length two.
Assume that $D$ is a dominating set of size $m$ for the $(m+1) \times 3m$ board. One of two cases occurs:
(1) there is only one empty row, which without loss of generality has index $h$, $0 \leq h \leq m$, and each other row contains exactly one square of $D$, or
(2) there are exactly two empty rows; in this case $m$ is even, row 0 contains two squares of $D$, the empty rows are indexed $\pm h$ for some $h$, $0 < h \leq m$, and each other row contains exactly one square of $D$.
In either case:\\ if $m \equiv 0 \mbox{ or } 4 \mbox{ (mod $9$)}$, then $h \equiv 0 \mbox{(mod $3$)}$; \\ if $m \equiv 1 \mbox{ or } 3 \mbox{ (mod $9$)}$, then $h \not \equiv 0 \mbox{ (mod $3$)}$. \end{thm}
\begin{proof}
The $3m$ column indices are $x = 1-3m, 3-3m, \ldots, 3m-3, 3m-1$ and the $m+1$ row indices are $y = -m, 2-m, \ldots, m-2, m$.
Let $D = \{ (x_{i}, y_{i})\mbox{ : }1 \leq i \leq m\}$ be a dominating set of the $(m+1) \times 3m$-board. Since $|D| = m$ and there are $m+1$ rows, at least one row contains no square of $D$. Let $h$ be the index of such a row. Since $|D| = m$, and a queen covers at most three squares of any line she does not occupy, the $3m$ squares of row $h$ are each covered exactly once by $D$. For each $(x, y) \in D$, the squares of row $h$ covered by $(x, y)$ are $(x-(y-h), h), (x, h), (x+(y-h), h)$. We can add up the squares of the column indices of the squares in row $h$ in two ways, giving the equation \[ (1-3m)^{2} + (3-3m)^{2} + \ldots +
(3m-1)^{2} = \sum_{i=1}^{m} \left[ (x_{i}-(y_{i}-h))^{2} + x_{i}^{2} + (x_{i}+(y_{i}-h))^{2} \right]. \] This reduces to \begin{equation} \label{longeq2} 2 \binom{3m+1}{3} = 3 \sum_{i=1}^{m} x_{i}^{2} + 2 \sum_{i=1}^{m} y_{i}^{2} - 4h \sum_{i=1}^{m} y_{i} + 2mh^{2}. \end{equation} For a particular dominating set $D$ we may regard this as a quadratic equation in $h$, so there are at most two empty rows.
If there are two empty rows $h_{1}, h_{2}$, then (\ref{longeq2}) implies $2mh_{1}^{2} - 4h_{1}\sum_{i=1}^{m} y_{i} = 2mh_{2}^{2} - 4h_{2}\sum_{i=1}^{m} y_{i}$ and then \begin{equation} \label{h1ph2}
\sum_{i=1}^{m} y_{i} = \frac{m(h_{1}+h_{2})}{2}. \end{equation}
As $|D| = m$, there is exactly one row, say $l$, with two queens, and all rows except $h_{1}, h_{2}, l$ have just one queen. Thus $\sum_{i=1}^{m} y_{i} = -h_{1}-h_{2}+l$. With (\ref{h1ph2}) this implies $-h_{1}-h_{2}+l = m(h_{1}+h_{2})/2$ and then $l = (m+2)(h_{1}+h_{2})/2$. From $-m \leq l \leq m$ we have $-2m \leq (m+2)(h_{1}+h_{2}) \leq 2m$ so $-2 < h_{1}+h_{2} < 2$. But $h_{1}+h_{2}$ is even since all row indices have the same parity. Thus $h_{1}+h_{2} = 0$, so $l = 0$ and all row indices are even, which implies $m$ is even. So there is $h$, $0 < h \leq m$, such that the empty rows are $\pm h$. Here $\sum_{i=1}^{m} y_{i} = 0$ and $\sum_{i=1}^{m} y_{i}^{2} = 2 \binom{m+2}{3} - 2h^{2}$, and (\ref{longeq2}) becomes \begin{equation} \label{2empty}
2 \binom{3m+1}{3} - 4 \binom{m+2}{3} = 3 \sum_{i=1}^{m} x_{i}^{2} + 2(m-2)h^{2}. \end{equation} If instead there is only one empty row $h$, then we may assume $0 \leq h \leq m$ by f{\kern0pt}lipping across the $x$-axis if necessary. Then $\sum_{i=1}^{m} y_{i} = -h$ and $\sum_{i=1}^{m} y_{i}^{2} = 2 \binom{m+2}{3} - h^{2}$, and (\ref{longeq2}) gives \begin{equation} \label{1empty}
2 \binom{3m+1}{3} - 4 \binom{m+2}{3} = 3 \sum_{i=1}^{m} x_{i}^{2} + 2(m+1)h^{2}. \end{equation}
The left sides of (\ref{2empty}) and (\ref{1empty}) reduce to $m(25m^{2}-6m-7)/3$. Multiplying either of (\ref{2empty}) and (\ref{1empty}) by 3 and reducing modulo 9 gives the congruence \begin{equation} \label{maincong}
m(25m^{2}-6m-7) \equiv -3(m-2)h^{2} {\mbox{ (mod 9)}}. \end{equation} For $m \equiv {5, 6, 7} \mbox{ or } {8} \mbox{ (mod 9)}$, (\ref{maincong}) leads to $h^{2} \equiv -1 \mbox{ (mod 3)}$ or another impossibility. For $m \equiv 0 \mbox{ or } 4 \mbox{ (mod 9)}$, (\ref{maincong}) implies $h^{2} \equiv 0 \mbox{ (mod 3)}$ and thus $h \equiv 0 \mbox{ (mod 3)}$. For $m \equiv 1 \mbox{ or } 3 \mbox{ (mod 9)}$, (\ref{maincong}) gives $h^{2} \equiv 1 \mbox{ (mod 3)}$, so $h \equiv \pm 1 \mbox{ (mod 3)}$. (For $m \equiv 2 \mbox{ (mod 9)}$, (\ref{maincong}) is satisf{\kern0pt}ied for any $h$.) \end{proof}
Only one example of the second case of Theorem \ref{by3k} is known: for $m=2$, the set $D = \{ (-3, 0), (3, 0)\}$ covers the $3 \times 6$ board. A computer search shows that there is no other example with $m \leq 40$. It seems likely that no other example exists; if for some $m$ there is such a set $D = \{ (x_{i}, y_{i}) \mbox{ : }1 \leq i \leq m\}$, we have been able to show that $0 = \sum x_{i} = \sum x_{i}y_{i} = \sum x_{i}^{2}y_{i} = \sum (x_{i}^{3}y_{i}+2x_{i}y_{i}^{2})$.
By computer search (see the \href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem2.html}{f{\kern0pt}ile}) we have shown that for all $m$ such that $m \leq 40$, $m \neq 3$, and $m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod 9)}$, there exist sets of $m$ queen squares dominating $Q_{(m+1)\times 3m}$, with $h$ taking all values not ruled out by Theorem \ref{by3k}. We believe $g(m) = g^{*}(m) = 3m$ holds for all $m \neq 3$ with $m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod 9)}$.
The following is immediate from Theorem \ref{by3k}.
\begin{cor}
If $m \equiv 5, 6, 7, \mbox{or } 8 \mbox{ (mod $9$)}$ then $\gamma(Q_{(m+1) \times 3m}) = m+1$. \end{cor}
We now examine the cases $m \equiv 5, 6, 7, 8 \mbox{ (mod $9$)}$.
\begin{thm} \label{upperbounds}
If $m \equiv 5, 6, \mbox{or } 7 \mbox{ (mod $9$)}$ then $g^{*}(m) \leq 3m-2$.\\
If $m \equiv 8 \mbox{ (mod $9$)}$ then $g^{*}(m) \leq 3m-4$.
Take the board to be a rectangle in the Cartesian plane with sides parallel to
the axes, board center at the origin, and squares of edge length two.
For $m \equiv 6 \mbox{ or } 7 \mbox{ (mod $9$)}$, assume that $D$ is a set of $m$ squares that dominates
the $(m+1) \times (3m-2)$ board, occupying all but row $h$. If $m \equiv 6 \mbox{ (mod $9$)}$ then
$h \not \equiv 0 \mbox{ (mod $3$)}$.
If $m \equiv 7 \mbox{ (mod $9$)}$ then $h \equiv 0 \mbox{ (mod $3$)}$. \end{thm}
\begin{proof}
Let $m$ be a positive integer and $j$ an integer with $0 \leq j \leq 4$. Let $D$ be a dominating set of $m$ queens for the $(m+1) \times (3m-j)$-board, with only row $h$ empty.
The $3m-j$ column indices form the set $S_{\rm col} = \{ j+1-3m, j+3-3m, \ldots, 3m-j-3, 3m-j-1\}$ and the $m+1$ row indices form the set $S_{\rm row} = \{-m, 2-m, \ldots, m-2, m \}$.
It will be useful to consider row $h$ extended beyond the board, and to look at the congruence classes modulo 3 of the column indices. To this end, for each integer $i$, let $C'_{i} = \{x \in \mathbb{Z} \mbox{ : } x \equiv i \mbox{ (mod 3)} \}$. The restriction of $C'_{i}$ to column indices of the $(m+1) \times (3m-j)$-board is $C_{i} = C'_{i} \cap S_{\rm col}$. We write $c_{i}$ for the size of $C_{i}$. By symmetry, $C_{-1} = \{-x \mbox{ : } x \in C_{1}\} = -C_{1}$ so $c_{-1} = c_{1}$.
For each integer $i$ let $R_{i} = \{y \in S_{\rm row} \mbox{ : } y \equiv i \mbox{ (mod 3)} \}$. Write $r_{i}$ for the size of $R_{i}$. By symmetry, $R_{-1} = - R_{1}$ so $r_{-1} = r_{1}$.
\begin{lemma} \label{techlemma}
Choose $s, b \in \{ -1, 0, 1 \}$ such that $s \equiv j$ and $b \equiv k+1 \mbox{ (mod 3)}$.
Then $c_{-1} = c_{1} = c_{0} + s$ and $r_{-1} = r_{1} = r_{0} - b$.
\end{lemma}
\begin{proof} The facts that $c_{-1} = c_{1}$, $c_{-1} + c_{0} + c_{1} = 3m-j$ and the $c_{i}$'s dif{\kern0pt}fer by at most one
imply the f{\kern0pt}irst equation, and the second is similar.
\end{proof}
Let $p$ be the number of squares $(x, y)$ of $D$ with $y \not \equiv h \mbox{ (mod 3)}$. For each $i \in \{ -1, 0, 1 \}$, let
$t_{i}$ be the number of squares $(x, y)$ of $D$ such that $x \equiv i \mbox{ (mod 3)}$ and $y \equiv h \mbox{ (mod 3)}$. Let $t = t_{-1} + t_{0} + t_{1}$. Thus $|D| = p + t_{-1} + t_{0} + t_{1} = p + t$.
Let $(x, y)$ be a square of $D$. The squares in the extension of row $h$ covered by $(x, y)$ are $(x-(y-h), h), (x, h), (x+(y-h), h)$, some of which may be of{\kern0pt}f the $(m+1) \times (3m-j)$-board. Their $x$-coordinates $x - (y-h), x, x+(y-h)$ are an arithmetic progression with dif{\kern0pt}ference $y-h$. If $y \equiv h \mbox{ (mod 3)}$ then all of $x - (y-h), x, x+(y-h)$ are in $C'_{x}$. If $y \not \equiv h \mbox{ (mod 3)}$ then the three values $x - (y-h), x, x+(y-h)$ are dif{\kern0pt}ferent modulo 3, so they contribute one member to each of $C'_{-1}, C'_{0}, C'_{1}$. Thus for each $i \in \{ -1, 0, 1\}$, $p + 3t_{i}$ is the number of covers (with multiplicity) of squares $(x, h)$ with $x \in C'_{i}$.
For each $i \in \{ -1, 0, 1 \}$ let $a_{i}$ be the number of ``wasted covers'' by $D$ of squares $(x, h)$ with $x \in C'_{i}$. That is, $a_{i}$ counts every cover of any square $(x, h)$
that is of{\kern0pt}f the board ($|x| > 3m-j-1$) and all but one cover of each multiply covered square $(x,h)$ on the board. Thus each $a_{i}$ is nonnegative, and since each of the $m$ squares of $D$ covers 3 squares of the extended row $h$, $a_{-1} + a_{0} + a_{1} = j$.
Since each square in row $h$ of the board is covered by $D$, we get a system of equations: \begin{eqnarray}
p + 3t_{-1} & = & c_{-1} + a_{-1} \label{one} \\
p + 3t_{0} & = & c_{0} + a_{0} \label{two} \\
p + 3t_{1} & = & c_{1} + a_{1} \label{three} \\
a_{-1} + a_{0} + a_{1} & = & j \label{four}. \end{eqnarray}
As $c_{-1} = c_{1}$, subtracting (\ref{one}) from (\ref{three}) shows \begin{equation} \label{cong1}
a_{-1} \equiv a_{1} \mbox{ (mod 3)}. \end{equation}
For a dominating set $D$ as hypothesized to exist, it is necessary that the total number $t$ of squares $(x,y)$ in $D$ with $y \equiv h \mbox{ (mod 3)}$ is one less than the number $r_{h}$ of rows in $R_{h}$. When $t = r_{h} - 1$, we will say $h$ is \emph{eligible} for $t$.
Theorem \ref{by3k} covers the case $j = 0$, so we pass to less wide boards, only considering $m \equiv 5, 6, 7, 8 \beem{9}$.
\noindent Let $j=1$. From (\ref{cong1}) we have $a_{-1}=a_{1}=0$ and then $a_{0}=1$. Here the $s$ of Lemma \ref{techlemma} is 1 so $c_{-1} = c_{1} = m, c_{0} = m-1$. Then equations (\ref{one}-\ref{three}) imply $t_{-1} = t_{0} = t_{1}$, so $t \equiv 0 \mbox{ (mod 3)}$, and the following analysis shows that no $h$ is eligible for any $t$ for $m \equiv 5, 6, 7, 8 \beem{9}$.
\indent If $m \equiv 5 \beem{9}$ then $r_{-1} = r_{1} = r_{0} = (m+1)/3 \equiv -1 \beem{3}$.\\ \indent If $m \equiv 6 \beem{9}$ then $r_{-1} = r_{1} = m/3 \equiv -1 \beem{3}, r_{0} = (m/3)+1 \equiv 0 \beem{3}$.\\ \indent If $m \equiv 7 \beem{9}$ then $r_{-1} = r_{1} = (m+2)/3 \equiv 0 \beem{3}, r_{0} = (m-1)/3 \equiv -1 \beem{3}$.\\ \indent If $m \equiv 8 \beem{9}$ then $r_{-1} = r_{1} = r_{0} = (m+1)/3 \equiv 0 \beem{3}$.
\noindent Let $j=2$. Since (\ref{four}) here implies all $a_{i} \leq 2$, (\ref{cong1}) gives $a_{-1} = a_{1}$, and then (\ref{one}-\ref{three}) imply $t_{-1} = t_{1}$. The $s$ of Lemma \ref{techlemma} is $-1$ so $c_{-1} = c_{1} = m, c_{0} = m+1$. There are two possibilities:
$(a_{-1}, a_{0}, a_{1}) = (0, 2, 0)$, when equations (\ref{one}-\ref{three}) imply $t_{0} = t_{1}+1$, so $t \equiv 1 \beem{3}$, or
$(a_{-1}, a_{0}, a_{1}) = (1, 0, 1)$, when equations (\ref{one}-\ref{three}) imply $t_{0} = t_{1}$, so $t \equiv 0 \beem{3}$.
Then examining the values of the $r_{i}$'s found above for $m \equiv 5, 6, 7, 8 \beem{9}$, we see:\\ For $m \equiv 5 \beem{9}$, all $h$ are eligible for $t \equiv 1 \beem{3}$ and none for $t \equiv 0 \beem{3}$;\\ For $m \equiv 6 \beem{9}$, $h \not \equiv 0 \beem{3}$ are eligible for $t \equiv 1 \beem{3}$ and none for $t \equiv 0 \beem{3}$;\\ For $m \equiv 7 \beem{9}$, $h \equiv 0 \beem{3}$ is eligible for $t \equiv 1 \beem{3}$ and none for $t \equiv 0 \beem{3}$;\\ For $m \equiv 8 \beem{9}$, no $h$ is eligible for either $t$.
We continue with $m \equiv 8 \beem{9}$.
\noindent Let $j=3$. Here $c_{-1} = c_{0} = c_{1} = m-1$ so from equations (\ref{one}-\ref{three}) we see all $a_{i}$'s are congruent modulo 3. Either $(a_{-1}, a_{0}, a_{1}) = (1, 1, 1)$, and then $t_{-1} = t_{0}= t_{1}$, so $t \equiv 0 \beem{3}$, or one of the $a_{i}$'s is 3 and the other two are zero, which gives $t \equiv 1 \beem{3}$. For $m \equiv 8 \beem{9}$, neither of these gives an eligible $h$, as before.
\noindent Let $j=4$. Here there are more possibilities for $(a_{-1}, a_{0}, a_{1})$, but the only helpful one for $m \equiv 8 \beem{9}$ is $(2, 0, 2)$, which gives $t \equiv -1 \beem{3}$, with all $h$ eligible. \end{proof}
Computer search reveals (see the \href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem4.html}{f{\kern0pt}ile}) that for $m \leq 40$ and $ m \equiv 5,6,7 \beem{9},$ all minimum dominating sets of \q{m+1}{3m-2} have just one empty row, and all eligible values of $h$ actually occur. For $ m \equiv 8 \beem{9},$ Theorem 4 does not say any $h$ are ineligible; indeed, our search has found solutions with one empty row for all the $h$ values. The only board size in the $3 \leq m \leq 40$ range where the minimum dominating sets found have two empty rows is \q{9}{20}. Those dominating sets demonstrate numerous patterns of pairs of empty rows, as shown in this \href{https://www.combinatorics.org/files/v26i4p45/appendix-9x20Spec.html}{f{\kern0pt}ile}.\\
We next show that by ``pasting together'' dominating sets of a certain type, we can extend the range of values for which the bounds of Theorem \ref{upperbounds} are known to be exact.
Say that a \emph{topless} dominating set for the $(m+1)\times n$ board is a dominating set of size $m$ having one square in each row except the top row, which is empty.
\begin{prop} \label{pasting}
(A) Suppose that for $i = 1, 2$ there is a topless dominating set
of $m_{i}$
queens for the $(m_{i}+1) \times n_{i}$ board. Then there is a dominating set
of $m_{1} + m_{2}$ queens for the $(m_{1}+m_{2}+1) \times (n_{1}+n_{2})$
board.
(B) Let $k$ be a positive integer. Suppose that for each $l$, $1 \leq l \leq k$, and for $i = 0, 1, 2$, there exist topless
dominating sets of size $9l+i$ for the $(9l+i+1) \times (27l + 3i)$ board.
Then for each $m \neq 3$, $1 \leq m \leq 9k+8$, there is a dominating set of
size $m$ for the $(m+1) \times (3m-j)$ board, where $j=0$ if
$m \equiv 0, 1, 2, 3, \mbox{or } 4 \mbox{ (mod $9$)}$,
$j=2$ if $m \equiv 5, 6, \mbox{or } 7 \mbox{ (mod $9$)}$, and $j=4$ if
$m \equiv 8 \mbox{ (mod $9$)}$.
\end{prop}
\begin{proof} We return to our usual scheme of indexing columns and
rows from the bottom left board corner.
For (A): For $i = 1, 2$, let $S_{i}$ be a topless dominating set of $m_{i}$
queens for the $(m_{i}+1) \times n_{i}$ board.
On the $(m_{1}+m_{2}+1) \times (n_{1}+n_{2})$ board, the squares in which the columns
indexed $1$ to $n_{1}$ and the rows indexed $1$ to $m_{1}+1$ meet
form a copy of the $(m_{1}+1)\times n_{1}$ board. Place a copy $S_{1}'$ of $S_{1}$
on that. The squares in which the columns
indexed $n_{1}+1$ to $n_{1}+n_{2}$ and the rows indexed $m_{1}+1$ to $m_{1}+m_{2}+1$ meet
form a copy of the $(m_{2}+1)\times n_{2}$ board. Place a copy $S_{2}'$ of $S_{2}$, rotated
by a half turn, on that. The union of the two copies is a set $S_{1} \oplus
S_{2}$ of $m_{1}+m_{2}$ squares that leaves only row $m_{2}+1$ empty on the $(m_{1}+m_{2}+1) \times (n_{1}+n_{2})$
board. As $S_{1}'$ covers the left $n_{1}$ squares of that row and $S_{2}'$
covers the remainder, $S_{1} \oplus S_{2}$ dominates the $(m_{1}+m_{2}+1) \times (n_{1}+n_{2})$
board.
For (B): We will use topless dominating sets of the $(m+1) \times (3m-j)$ board
for $(m,j)$ =
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem2.html#m1h1}{(1,0)},
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem4.html#m5h5}{(5,2)},
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem4.html#m8h8}{(8,4)}, \mbox{and} \hspace{.03in}
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem2.html#m11h11}{(11,0)}.
Assume the hypotheses; we then need to prove the existence of (minimum) dominating sets of size
$m$ for the $(m+1) \times (3m-j)$ boards in the ranges claimed.
For $m < 18$ the database (see the f{\kern0pt}iles
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem2.html}{here}
and
\href{https://www.combinatorics.org/files/v26i4p45/appendix-Theorem4.html}{here})
contain the claimed dominating sets. For other board sizes of $m = 9l, 9l + 1, 9l + 2, \, 2 \leq l \leq k$,
we are assuming there are such sets (in fact, topless).
For $m = 9l + 3$, we use part (A) to ``paste together" topless sets for $m = 9l + 2$ and $m = 1$.
For $m = 9l + 4$, we paste together topless sets for $m = 9(l - 1) + 2$ and $m = 11$.
Pasting together a topless set for each of $m = 9l, 9l + 1, 9l + 2$ with a topless set for $m = 5$ gives
the result for $m = 9l + 5, 9l + 6, 9l + 7$.
Finally, pasting together topless sets for $m = 9l$ and $m = 8$ gives the result for $m = 9l + 8$. \end{proof}
As mentioned after Theorem \ref{by3k}, computer search gave topless dominating sets as
required in Proposition \ref{pasting}(B)
for $k = 4$.
It follows that $g^{*}(m)$ equals the bound of Theorem \ref{upperbounds}
for $m \leq 44, m \neq 3$.
We next develop a lower bound for $\gamma(Q_{m \times n})$ for more general $m, n$.
Raghavan and Venketesan \cite{RV} and Spencer \cite{CO,WE} independently proved that \begin{equation} \label{rvs} \gamma(\q{n}{n}) \geq \left \lceil \frac{n-1}{2} \right \rceil. \end{equation}
It has been shown \cite{FW} that $\gamma(\q{n}{n}) = (n-1)/2$ only for $n = 3, 11$. Both of these values are signif{\kern0pt}icant for our work here, as we now discuss.
A central queen on \q{3}{3}\ shows $\gamma(\q{3}{3}) = 1$. This simple fact has a useful generalization: if $C$ is a central sub-board of \mbox{$Q_{m \times n}$}\ such that every square of \mbox{$Q_{m \times n}$}\ has a line meeting $C$, then a subset of $C$ that occupies all those lines is a dominating set of \mbox{$Q_{m \times n}$}. More than a hundred years ago, Szily \cite{V1, V2} gave dominating sets of this type for \q{13}{13}\ and \q{17}{17}, which were later shown to be minimum. We found that \q{13}{16} has a minimum dominating set (solution \href{https://www.combinatorics.org/files/v26i4p45/13x16_8Q.html\#Solution23}{\#23} in the database) of this \emph{centrally strong} form and have also used this idea to produce good upper bounds of \mbox{$Q_{m \times n}$}\ for some $m, n$, as shown below.
It follows from \cite{LB} that there are exactly two minimum dominating sets for \q{11}{11}. Placing the origin of our coordinate system at board center, these sets are ${\cal D} = \{ (0, 0)$, $\pm(2, 4)$, $\pm(4, -2) \}$ (see Figure 3) and the ref{\kern0pt}lection of ${\cal D}$ across the column $x = 0$. \begin{figure}
\caption{The minimum dominating set ${\cal D}$ of $Q_{11 \times 11}$ is shown, with the unique cover of each edge square (see Corollary \ref{coro}) indicated.}
\label{figq11}
\end{figure} So up to equivalence ${\cal D}$ is the unique minimum dominating set of \href{https://www.combinatorics.org/files/v26i4p45/11x11_5Q.html}{\q{11}{11}}, consisting of a foursome and a queen at its center. This amazing set has an inf{\kern0pt}luence on many other values of \mbox{$\gamma(\qq)$}.
First, since ${\cal D}$ f{\kern0pt}its on \q{9}{9}, by omitting edge rows and columns of \q{11}{11}\ we get dominating sets of \mbox{$Q_{m \times n}$}\ for $(m, n) = (10, 11), (10, 10), (9, 11), (9, 10), (9, 9)$, and these turn out to be minimum dominating sets. In a sense, the observed failure of monotonicity, $\gamma(\q{8}{11}) = 6 > 5 = \gamma(\q{9}{11})$, occurs simply because ${\cal D}$ does not f{\kern0pt}it on \q{8}{11}.
Also, by adding edge rows or columns to \q{11}{11}\ and adding corner squares to ${\cal D}$, we obtain minimum dominating sets for the values $\gamma(\q{11}{12}) = 6$, $\gamma(\q{11}{13}) = 7$, $\gamma(\q{12}{12}) = 6$, $\gamma(\q{12}{13}) = 7$, $\gamma(\q{12}{14}) = 8$, $\gamma(\q{13}{13}) = 7$, $\gamma(\q{13}{14}) = 8$, $\gamma(\q{14}{14}) = 8$, and $\gamma(\q{15}{15}) = 9$. Finally, it is shown in \cite{UB} that ${\cal D}$ gives a set implying $\gamma(\q{53}{53}) = 27$.
It was observed by Eisenstein et al. \cite{EI} that if a dominating set $D$ of $\q{n}{n}$
contains no edge squares, the facts that there are $4(n-1)$ edge squares and every queen covers eight edge squares imply $|D| \geq \lceil (n-1)/2 \rceil$. This suggests the bound (\ref{rvs}).
A similar approach leads one to guess the bound of our next theorem, but some care is needed to handle the general case.
\begin{thm} \label{main} Let $m, n$ be positive integers with $m \leq n$. Then
\begin{equation} \label{rvs2} \gamma(Q_{m \times n}) \geq \min \left\{m, \left\lceil \frac{m+n-2}{4} \right\rceil\right\}. \end{equation} \end{thm}
\begin{proof} It suf{\kern0pt}f{\kern0pt}ices to show that if $\gamma(Q_{m \times n}) \leq m-1$ then $\gamma(Q_{m \times n}) \geq (m+n-2)/4 $. So we assume that $\gamma(Q_{m \times n}) \leq m-1$.
First, suppose $\gamma(\q{m}{n})=m-1$. Then by Proposition \ref{prop1} we have $n < 3m-2$, which implies $m-1 > (m+n-2)/4$ as needed.
Thus we may take $\gamma(\q{m}{n}) \leq m-2$ and let $D$ be a minimum dominating set of $Q_{m \times n}$. Since $m \leq n$, there are at least two rows and at least two columns that do not contain squares of $D$. Let $a$ be the index of the leftmost empty column, $b$ the index of the rightmost empty column, $c$ the index of the lowest empty row, $d$ the index of the highest empty row. The board has a rectangular sub-board $U$ with corner squares $(a, c), (a, d), (b, c), \mbox{ and } (b, d)$. Let $E$ be the set of edge squares of this sub-board. We say that $U$ is the \emph{box} of $D$ and $E$ is the \emph{box border} of $D$; these sets are def{\kern0pt}ined for any square set $D$ with
$|D| \leq m-2$. Here $|E| = 2(d-c) + 2(b-a)$.
Removing columns $a$ and $b$ and rows $c$ and $d$ divides the board into nine regions (some possibly empty). Let $C$ be the set of squares of $D$ inside $U$; that is, $C = \{ (x, y) \in D \mbox{ : } a<x<b \mbox{ and } c<y<d \}$. Let $T_{nw}$ be the set of squares of $D$ in the ``northwest'' region of the $m \times n$ board; that is, $T_{nw} = \{ (x, y) \in D \mbox{ : } x < a \mbox{ and } y > d \}$. Similarly we label seven more subsets of $D$ by their ``geographic direction'' from the central region: $T_{n}, T_{ne}, T_{e}, T_{se}, T_{s}, T_{sw}, \mbox{ and } T_{w}$. Let $R = T_{nw} \cup T_{ne} \cup T_{sw} \cup T_{se}$, the set of those squares of $D$ whose orthogonals do not meet $U$. Let $S = T_{n} \cup T_{e} \cup T_{s} \cup T_{w}$, the set of those squares of $D$ having exactly one orthogonal that meets $U$. Then $D$ is the disjoint union of $R$, $S$, and $C$.
Since each column to the left of column $a$ contains at least one square of $D$, \begin{equation} \label{in1}
|T_{sw}| + |T_{w}| + |T_{nw}| \geq a-1. \end{equation} Similarly, \begin{eqnarray}
\label{in2} |T_{se}| + |T_{e}| + |T_{ne}| & \geq & n-b, \\
\label{in3}|T_{sw}| + |T_{s}| + |T_{se}| & \geq & c-1, \\
\label{in4}|T_{nw}| + |T_{n}| + |T_{ne}| & \geq & m-d. \end{eqnarray} Adding inequalities (\ref{in1})-(\ref{in4}) and using the def{\kern0pt}initions of $R$ and $S$ gives \begin{equation} \label{in5}
2 |R| + |S| \geq m + n - 2 - (d-c) - (b-a). \end{equation} Each square in $R$ covers at most two squares of $E$, as the square's orthogonals and one of its diagonals miss $E$. Each square in $S$ covers at most six squares of $E$, as one of the square's orthogonals misses $E$. Each square in $C$ covers eight squares of $E$. Since $D$ is a dominating set, $D$ covers all squares of $E$, so \begin{equation} \label{in6}
2 |R| + 6 |S| + 8 |C| \geq 2(d-c) + 2(b-a). \end{equation} Adding two times (\ref{in5}) to (\ref{in6}) gives \begin{equation} \label{in7}
6 |R| + 8 |S| + 8 |C| \geq 2(m+n-2). \end{equation}
Since $|D| = |R| + |S| + |C|$, adding $2 |R|$ to both sides of (\ref{in7}) gives \[
8 |D| \geq 2(m+n-2+|R|). \] Thus \begin{equation} \label{ineq}
\gamma(Q_{m \times n}) = |D| \geq (m+n-2+|R|)/4, \end{equation} which implies the desired conclusion. \end{proof}
A diagram illustrating the proof for $Q_{10 \times 17}$ is given in Figure 4.
\begin{figure}
\caption{Illustration of the proof of Theorem \ref{main} with $Q_{10 \times 17}$: $R, S, C$ are the sets of queen squares having respectively $0, 1, \mbox{or } 2$ orthogonals meeting the box of $D$, and $E$ is the box border of $D$.}
\label{fig10x17sol8482}
\end{figure}
There are 120 pairs $(m, n)$ satisfying $4 \leq m \leq n \leq 18$. Of these, the bound (\ref{rvs2}) is achieved for 40 pairs (28 with $m \leq 6$), for 76 pairs the bound is exceeded by one and for the pairs $(12, 14), (13, 17), (14, 16)$, and $(15, 15)$ the bound is exceeded by two.
We next explore when $\gamma(\q{m}{n}) = (m+n-2)/4$. From Proposition \ref{prop1} it follows that for any positive integer $m$, if $n = 3m+2$ then $\gamma(\q{m}{n}) = m$, and here $m = (m+n-2)/4$. So we restrict to $n < 3m+2$.
\begin{cor} \label{coro} Suppose $m \leq n < 3m+2$ and $\gamma(\q{m}{n}) = (m+n-2)/4$. Let $D$
be a minimum dominating set of \q{m}{n}. Then $|D| \leq m-2$, each box border square is covered exactly once by $D$, and $D$ is independent. \end{cor}
\begin{proof}
From $n < 3m+2$ we have $|D| = (m+n-2)/4 \leq m-2$, so the box of $D$ is def{\kern0pt}ined. Since $|D| = (m+n-2)/4$, in this setting we have equality in inequalities (\ref{in1})\mbox{--}(\ref{ineq}),
so each square of the box border $E$ is covered exactly once by $D$. Thus any line meeting $E$ contains at most one square of $D$. Every square of $D$ must be diagonally adjacent to four squares of $E$, so if any line containing a square of $D$ does not meet $E$, it is an orthogonal. From (\ref{ineq}) we see that here the set $R$ of ``corner squares'' in $D$ is empty, so every square of $D$ has at least one orthogonal meeting $E$; then since (\ref{in1})--(\ref{in4}) are equations here, each orthogonal that misses $E$ contains exactly one square of $D$. Thus $D$ is independent. \end{proof}
Rarely does a minimum dominating set cover each of its box border squares uniquely; see Figure 3 for an example. We also note that the minimum dominating sets \#1-4 for \q{11}{12} have this property. Each of these sets consists of a foursome centered at $(13/2, 13/2)$ plus the corner squares $(1, 1)$ and $(12, 1)$, so is not independent.
As mentioned earlier, $\gamma(\q{n}{n}) = (n-1)/2$ is achieved only for $n = 3, 11$. Considering Corollary \ref{coro}, we suspect that the answer to the following question is no.
\begin{ques} \label{wonderful} Does $\mbox{$\gamma(\qq)$} = (m+n-2)/4$ with $m \leq n < 3m + 2$ occur, other than for $(m, n) = (3, 3)$ and $(11, 11)$? \end{ques}
We next extend the method of proof used in \cite{CO, RV, WE} for the lower bound (\ref{rvs}) to show that the dimensions of the box of $D$ give a lower bound for $|D|$.
\begin{prop} \label{morebound} Let $m \leq n$ and let $D$ be a dominating set of \mbox{$Q_{m \times n}$}\ of size at most $m-2$. Let $m'$ be the number of rows and $n'$ the number of columns of the box of $D$. Then:
If $m' > n'$ then $|D| \geq \lceil \frac{n}{2} \rceil$;
If $m' \leq n'$ then $|D| \geq \left \lceil \frac{n - 1 - (n' - m')}{2} \right \rceil$. \end{prop}
\begin{proof} Let $a, b, c, d$ be def{\kern0pt}ined as in the proof of Theorem \ref{main}. (Then $m' = d-c + 1$ and $n' = b - a + 1$.) Since $m' \leq m \leq n$, we may choose an integer $e$ such that $e$ through $e + m' - 2$ are indices of columns of the board. Set $S = \{ (x, c), (x, d) \mbox{ : } e \leq x \leq e + m' -2 \}$ and
$P = \{ (x, y) \in D \mbox{ : } x < e \mbox{ or } x > e + m' - 2 \}$. Then no square is diagonally adjacent to more than two squares of $S$ and no square of $P$ is orthogonally adjacent to any square of $S$. As the $2(m'-1)$ squares of $S$ are covered by $D$, $2(m'-1) \leq 2|P| + 4(|D| - |P|)$, which implies \begin{equation} \label{bounder}
|D| \geq \left \lceil \frac{m'-1+|P|}{2} \right \rceil. \end{equation}
If $m' > n'$ then we can choose $e$ so that all columns that do not meet $S$ are occupied, so $|P| \geq n - (m'-1)$. If $m' \leq n'$ we can choose $e$ so that $S$ is contained in the top and bottom edges of $U$, and then
$|P| \geq n - n'$. In both cases, (\ref{bounder}) implies the conclusion. \end{proof}
As $\gamma(\q{n}{n}) = (n-1)/2$ only for $n = 3, 11$, we have $\gamma(\q{n}{n}) \geq \lceil n/2 \rceil$ for all other positive integers $n$. There is much evidence that this lower bound is quite good. Work from \cite{BM1,BMC,GW, KG, OW, WE, LB} reported in \cite{OW} shows that for $n$ from 1 to 120, excluding 3 and 11, we have $\lceil n/2 \rceil \leq \gamma(\q{n}{n}) \leq \lceil n/2 \rceil + 1$. In this range, $\gamma(\q{n}{n}) = \lceil n/2 \rceil$ is known for 46 values of $n$ and $\gamma(\q{n}{n}) = \lceil n/2 \rceil +1$ is known for $n = 8, 14, 15, 16$. Also, $\gamma(\q{(4k+1)}{(4k+1)}) = 2k+1$ is known for $1 \leq k \leq 32$.
For $m < n$, we have little evidence that the bound (\ref{rvs2}) is good. We were not able to use the
methods of the proofs of Theorem \ref{main} and Proposition \ref{morebound} to improve on this bound. Also, a computer search using a greedy algorithm for some larger $m, n$ did not supply evidence about lower bounds for $\gamma(\mbox{$Q_{m \times n}$})$.
The statement of Proposition \ref{morebound} leads one to consider the quantity $n/2$. We have checked that when $4 \leq m \leq n \leq 18$, $\gamma(\q{m}{n}) \geq \min\{m-1, \lfloor n/2 \rfloor -1 \}$. This bound and the bound (\ref{rvs2}) are close only when $m$ and $n$ are close. So we ask the following.
\begin{ques} \label{whatlowerbound} For $m, n$ with $m \leq n$, what is a good general lower bound for $\gamma(\mbox{$Q_{m \times n}$})$? In particular, is it true that $\gamma(\mbox{$Q_{m \times n}$}) \geq \min \{ m-1, \lfloor n/2 \rfloor - 1 \}$? \end{ques}
\section{Construction of dominating sets} \label{construct}
Given dimensions $m$ and $n$, we would like a general approach that would allow us to construct minimum dominating sets of \q{m}{n}, or at least reasonably small dominating sets. We have two dif{\kern0pt}f{\kern0pt}iculties to consider.
The f{\kern0pt}irst dif{\kern0pt}f{\kern0pt}iculty was just discussed: in general we know the value of $\gamma(\mbox{$Q_{m \times n}$})$ only approximately.
The second dif{\kern0pt}f{\kern0pt}iculty is that construction of a dominating set of \mbox{$Q_{m \times n}$}\ generally means specifying most or all of the lines that the set is to occupy. There are some restrictions that the indices of the lines must satisfy, as we now describe.
Let $D = \{ (x_{i}, y_{i}) \mbox{ : } 1 \leq i \leq l \}$ be a set of $l$ squares of \mbox{$Q_{m \times n}$}\ that occupies dif{\kern0pt}ference diagonals $(d_{i})_{i=1}^{l}$ and sum diagonals $(s_{i})_{i=1}^{l}$. Since the square \mbox{($x$,$y$)}\ is on the difference diagonal\ with index $y-x$ and the sum diagonal\ with index $y+x$, summing over $D$ gives \begin{equation} \label{linearcon} \sum_{i=1}^{l} d_{i} = \sum_{i=1}^{l} y_{i} - \sum_{i=1}^{l} x_{i} \mbox{ and } \sum_{i=1}^{l} s_{i} = \sum_{i=1}^{l} y_{i} + \sum_{i=1}^{l} x_{i}. \end{equation}
The Parallelogram Law $2x^{2} + 2y^{2} = (y-x)^{2} + (y+x)^{2}$ gives a quadratic constraint \begin{equation} \label{plleleq} 2 \sum_{i=1}^{l} x_{i}^2 + 2 \sum_{i=1}^{l} y_{i}^2 = \sum_{i=1}^{l} d_{i}^2 + \sum_{i=1}^{l} s_{i}^2 \end{equation} on the line indices.
In each of the two constructions given below, we will refer to lines that must be occupied for domination as \emph{required} lines and other lines as \emph{auxiliary} lines.
Both constructions produce a number of minimum dominating sets, but neither can produce a dominating set of \q{m}{n}\ of size less than $\lfloor n/2 \rfloor$. This is a little evidence for the possible bound mentioned in Question \ref{whatlowerbound}.
\subsection{Domination by orthodox covers}
This idea generalizes \cite[Section 2]{UB}. Let $D$ be a set of squares of \mbox{$Q_{m \times n}$}. If it is possible to place the origin of the coordinate system so that every even column and every even row contains a square of $D$, we will say $D$ is an \emph{orthodox} set. That is, an orthodox set is one that occupies at least every other column and every other row of \mbox{$Q_{m \times n}$}.
Say that square \mbox{($x$,$y$)}\ of \mbox{$Q_{m \times n}$}\ is \emph{even} if $x+y$ is even, \emph{odd} if $x+y$ is odd. We divide the even squares of \mbox{$Q_{m \times n}$}\ into two classes: \mbox{($x$,$y$)}\ is {\em even-even\/} if both $x$ and $y$ are even, {\em odd-odd\/} if both are odd. If $D$ is an orthodox set and each odd-odd square of \mbox{$Q_{m \times n}$}\
is covered by some square of $D$, we say $D$ is an \emph{orthodox cover}. For example, solution \href{https://www.combinatorics.org/files/v26i4p45/07x11_5Q.html\#Solution10}{\#10} for \q{7}{11} given in Table 1 is an orthodox cover; take the origin at $(6, 3)$ to see this.
It is clear from the def{\kern0pt}inition that an orthodox cover dominates every even square of \mbox{$Q_{m \times n}$}, and since every odd square of \mbox{$Q_{m \times n}$}\ is on one even-indexed orthogonal, all odd squares are also dominated: an orthodox cover is a dominating set of \mbox{$Q_{m \times n}$}. Many orthodox covers appear in the \href{https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML}{appendix}, and are labeled there as such.
Since \mbox{$Q_{m \times n}$}\ has at least $\lfloor n/2 \rfloor$ even-indexed columns, an orthodox set on \mbox{$Q_{m \times n}$}\ has at least $\lfloor n/2 \rfloor$ members. Generally, we expect that most of the squares of $D$ will be even-even, to help dominate the odd-odd squares diagonally. When $n$ is considerably larger than $m$, there are more possibilities of placing queens on odd squares that occupy even columns. Also, it is possible sometimes to achieve a dominating set of size less than $\lfloor n/2 \rfloor$ by a minor modif{\kern0pt}ication, as is shown in solution \href{https://www.combinatorics.org/files/v26i4p45/07x12_5Q.html\#Solution1}{\#1} for \q{7}{12} in Table 1. If the center of the square there labeled $(6, 3)$ is taken to be the origin of the coordinate system, the dominating set shown misses being an orthodox set only by not occupying the rightmost column. Thus the three odd-odd squares in that column are not covered along their column, as they would be by an orthodox set. But the queen on a dark square covers those three squares, and the odd squares of its column, and thus completes a dominating set of size 5.
Minimum dominating set \href{https://www.combinatorics.org/files/v26i4p45/12x16_8Q.html\#Solution147}{\#147} for \q{12}{16}\ is an orthodox cover with a single queen on a dark square, at $(6, 10)$. The squares covered only by this queen are the dark squares in its column and $(1, 10)$. Replacing $(6, 10)$ with the white square $(6, 5)$ covers those squares and, adding a row 0 to the board, also the square $(1, 0)$. In fact, the full set now covers all of row 0 and is thus a minimum dominating set of \q{13}{16}; it is solution \href{https://www.combinatorics.org/files/v26i4p45/13x16_8Q.html\#Solution15}{\#15} for \q{13}{16}, rotated by a half-turn.
There are many ways to create orthodox covers, and we will only give one example. An approach is to regard \q{m}{n}\ as the union of overlapping copies of \q{m}{m}; for odd $m$, this allows us to use \cite[Theorem 1]{UB}, which gives suf{\kern0pt}f{\kern0pt}icient conditions for an orthodox set on \q{m}{m} to be an orthodox cover.
\begin{exam} \label{orthexam} An orthodox cover implying $\gamma(\q{13}{19}) \leq 10$.\\ We take the origin of the coordinate system to be the center of \q{13}{19}, and regard \q{13}{19}\ as the union of two copies of \q{13}{13}, centered at $(\pm 3, 0)$. From \cite[Theorem 1]{UB}, if we regard the center of \q{13}{13}\ as the origin, an orthodox set on \q{13}{13}\ dominates if the set occupies the sum and difference diagonal s with indices in $\{-6, -2, 0, 2, 6 \}$. Asking this on both copies of \q{13}{13}, we wish to have our orthodox set occupy the sum and difference diagonal s which (on \q{13}{19}) have indices in $\{-6, -2, 0, 2, 6 \} \pm 3$, which is $\{ \pm 1, \pm 3, \pm 5, \pm 9 \}$, so there will be two auxiliary difference diagonal\ indices $d_{1}, d_{2}$ and two auxiliary sum diagonal\ indices $s_{1}, s_{2}$. The required column indices are $\pm 1, \pm 3, \pm 5, \pm 7, \pm 9$, so there will be no auxiliary column indices. The required row indices are $0, \pm 2, \pm 4, \pm 6$, so there will be three auxiliary row indices $r_{1}, r_{2}, r_{3}$.
From (\ref{linearcon}) we have $d_{1} + d_{2} = r_{1} + r_{2} + r_{3} = s_{1} + s_{2}$ and (\ref{plleleq}) gives $d_{1}^{2} + d_{2}^{2} + s_{1}^{2} + s_{2}^{2} = 420 + 2(r_{1}^{2} + r_{2}^{2} + r_{3}^{2})$. We attempt to f{\kern0pt}ind a solution with symmetry by a half-turn about the board center: this means $r_{1} = 0, r_{2} = - r_{3}, d_{1} = -d_{2}, \mbox{ and } s_{1} = -s_{2}$. Then the quadratic constraint simplif{\kern0pt}ies to $d_{1}^{2} + s_{1}^{2} = 210 + 2r_{2}^{2}$, of which one solution is $d_{1} = 13, s_{1} = 7, r_{2} = 2$. Now all lines are specif{\kern0pt}ied, and it is not dif{\kern0pt}f{\kern0pt}icult to f{\kern0pt}ind the solution $D = \{ \pm(9, 0), \pm(7, -6), \pm(5, 2), \pm(3, 2), \pm(1, -4) \}$. \end{exam}
\subsection{Domination by centrally strong sets}
We begin by considering a board $C$ which is to be a central sub-board of a larger board $B$. Say that $C$ has $m_{1}$ rows and $n_{1}$ columns, with $m_{1} \geq n_{1}$ and $m_{1}, n_{1}$ not both even. It is convenient here to have the board squares of side length two, and place $C$ with its center at the origin. Thus if, for example, $m_{1}$ is odd and $n_{1}$ is even, then each square has center \mbox{($x$,$y$)}\ with $x$ an odd integer and $y$ an even one.
We then wish to choose a nonnegative integer $k$ and a set $D$ of squares of $B$ (actually, all or almost all in $C$) such that $D$ contains at least one square from the extension of each orthogonal of $C$ to $B$, and $D$ contains exactly one square from the extension of each difference diagonal\ of $C$, except none from the highest $k$ and the lowest $k$ extended difference diagonal s; similarly for sum diagonal s. Let \begin{equation} \label{mnk} m = m_{1} + 2n_{1} - 2k, \hspace{.3in} n = 2m_{1} + n_{1} - 2k, \hspace{.3in} g = m_{1} + n_{1} - 2k - 1. \end{equation}
Then $m \leq n$, and it is straightforward to verify that if $C$ is taken to be the central $m_{1} \times n_{1}$ sub-board of the $m \times n$ board $B$, then $D$ is a dominating set of \mbox{$Q_{m \times n}$}\ and $|D| = g$. Such a $D$ will be called a \emph{centrally strong} set, as it generalizes the idea discussed for square boards in \cite[page 234]{UB}. We note that our def{\kern0pt}inition requires each square of $D$ to have both diagonals among the required ones, and thus both have indices of absolute value at most $m_{1} + n_{1} - 2k - 2$, but this does not imply $D \subseteq C$. If in fact $D \subseteq C$, we say that $D$ is a \emph{strict} centrally strong set.
A number of strict centrally strong sets occur in the \href{https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p45/HTML}{appendix}, and are labeled there as such. We note that these sets can only occur when $m \leq n < 2m$; this follows from (\ref{mnk}) and the fact that since there will be $n_{1} - 2k - 1$ auxiliary row indices, this quantity is nonnegative.
One merit of this construction is that a single centrally strong $D$ gives an upper bound for \mbox{$\gamma(\qq)$}\ for several pairs $(m, n)$ since $D$ is conf{\kern0pt}ined to a small central region of the $m \times n$ board, especially if $D$ is strict. For example, there is a strict centrally strong set $D = \{ \pm(-5, 0)$, $\pm(-3, 4)$, $\pm(-1, 6)$, $\pm(1, 2)$, $\pm(3, 6) \}$
with $m_{1} = 9, n_{1} = 6, \mbox{ and } k=2$, and $|D| = 10$, which shows that $\mbox{$\gamma(\qq)$} \leq 10$ when $9 \leq m \leq 17 \mbox{ and } 6 \leq n \leq 20$. For some of these pairs $(m, n)$, this bound is poor, but for the six pairs with $m + n \geq 35$, combining with the bound (\ref{rvs2}) gives $9 \leq \mbox{$\gamma(\qq)$} \leq 10$, and 10 is a useful upper bound for some of the smaller boards also.
The simplest centrally strong sets occur with $m_{1} \geq 1$, $n_{1} = 1$ and $k = 0$, where we get $m_{1}$ queens occupying all squares of the $m_{1} \times 1$ board $C$, and the following bound (which we have stated in terms of $m = m_{1} + 2$). For $3 \leq m \leq 10$ at least, this bound gives the exact value of $\gamma(\q{m}{(2m-3)})$.
\begin{prop} \label{n1equal1} For $m \geq 3$, $\gamma(\q{m}{(2m-3)}) \leq m-2$. \end{prop}
We next consider the ef{\kern0pt}fect of (\ref{linearcon}) and (\ref{plleleq}) on the search for centrally strong sets. Symmetry and the requirement that each difference diagonal\ contains exactly one square of $D$ imply that the sum of the difference diagonal\ indices of $D$ is zero. Similarly the sum of the sum diagonal\ indices of $D$ is zero, and then (\ref{linearcon}) implies that $\sum_{(x, y) \in D} x = 0$ and $\sum_{(x, y) \in D} y = 0$. As we require a centrally strong set to occupy all (extended) columns of the sub-board, we regard the $n_{1}$ indices of these columns as required column indices; by symmetry their sum is zero. As $C$ has $n_{1}$ columns and $g$ occupied squares, there will be $g - n_{1} = m_{1}-2k-1$ auxiliary column indices, each having parity opposite to that of $n_{1}$. Since $\sum_{(x, y) \in D} x = 0$ and all required column indices sum to zero, so do the auxiliary column indices. Similarly there will be $m_{1}$ required row indices and $g - m_{1} = n_{1} - 2k - 1$ auxiliary row indices, with sum zero, each having parity opposite to that of $m_{1}$. If $D$ is strict, then all indices of occupied columns have absolute value at most $n_{1} - 1$ and all indices of occupied rows have absolute value at most $m_{1} - 1$. (We have required that $m_{1}, n_{1}$ not both be even because if they were, there would be an odd number of auxiliary row indices, each odd, so their sum could not be even, thus not zero.)
Using the identities $\sum_{i=1}^{j} (2i-1)^{2} = \binom{2j+1}{3}$ and $\sum_{i=1}^{j} (2i)^{2} = \binom{2j+2}{3}$, we see that the sum of the squares of the indices of all occupied diagonals of $C$ is $4 \binom{g+1}{3}$, the sum of the squares of the required column indices is $2 \binom{n_{1}+1}{3}$ and the sum of the squares of the required row indices is $2 \binom{m_{1}+1}{3}$. Letting $\sum_{orth}$ denote the sum of the squares of the auxiliary column indices and auxiliary row indices, the quadratic constraint (\ref{plleleq}) gives \begin{equation} \label{quadcstrong} \sum_{orth} = 2\left[ \binom{g+1}{3} - \binom{m_{1} + 1}{3} - \binom{n_{1} + 1}{3} \right]. \end{equation}
Combined with Proposition \ref{n1equal1}, part (a) of the following proposition shows how small a centrally strong set can be. In parts (b) and (c), we limit the values of $m_{1}, n_{1}, k$ that need be considered when constructing centrally strong sets.
We say that a value of $k$ for which there exists a centrally strong set on \q{m_{1}}{n_{1}}\ is \emph{feasible} for $(m_{1}, n_{1})$.
\begin{prop} \label{atbesthalfn}
(a) For any centrally strong set $D$ with $n_{1} > 1$, $|D| \geq n/2$.\\ (b) For any $(m_{1}, n_{1})$, it is only necessary to use the largest feasible $k$ to determine all upper bounds for \mbox{$\gamma(\qq)$}\ implied by centrally strong sets from $(m_{1}, n_{1})$.\\ (c) If $k$ is feasible for $(m_{1}, n_{1})$ and $k+1$ is feasible for $(m_{1}, n_{1}+2)$, the latter gives the more useful result. \end{prop}
\begin{proof} (a): As the number $n_{1}-2k-1$ of auxiliary row indices is nonnegative, $n_{1} \geq 2k+1$. If $n_{1} = 2k+1$ then $g = m_{1}$ by (\ref{mnk}), and then the fact that the right side of (\ref{quadcstrong}) is nonnegative implies $n_{1}=1$ and $k=0$, the situation of Proposition \ref{n1equal1}. Thus for $k \geq 1$
we have $n_{1} \geq 2k+2$, which by (\ref{mnk}) is equivalent to $|D| \geq n/2$.
(b): Suppose for some integer $h>0$ that both $k$ and $k-h$ are feasible for $(m_{1}, n_{1})$. Then the triple $m_{1}, n_{1}, k$ gives a dominating set $D$ of size $g$ on \q{m}{n}, where $m, n, g$ are determined by (\ref{mnk}), and similarly the triple $m_{1}, n_{1}, k-h$ gives a dominating set $D'$ of size $g+2h$ on \q{(m+2h)}{(n+2h)}. However, by repeating $2h$ times the process of adding an edge row
and edge column to the board and the new corner square to the dominating set, we can construct from $D$ a dominating set of \q{(m+2h)}{(n+2h)}\ of the same size as $D'$.
(c): Using (\ref{mnk}), if $m_{1}, n_{1}, k$ gives a dominating set of size $g$ for \mbox{$Q_{m \times n}$}, then $m_{1}, n_{1}+2, k+1$ gives a dominating set of size $g$ for \q{(m+2)}{n}. \end{proof}
\begin{exam} \label{13by16} A centrally strong set implying $\gamma(\q{13}{16}) \leq 8$.
Let $m_{1} = 7$ and $n_{1} = 4$, and $k=1$. Then a strict centrally strong set $D$ is to have one auxiliary row index, which from $\sum_{(x, y) \in D} y = 0$ must be zero, and two auxiliary column indices, say $c_{1}, c_{2}$, each in $\{ -3, -1, 1, 3 \}$. From $\sum_{(x, y) \in D} x = 0$ we see $c_{2} = -c_{1}$ and from (\ref{quadcstrong}) we have $c_{1}^{2} + c_{2}^{2} = 18$, so we can take $c_{1} = 3$ and $c_{2} = -3$. We then easily obtain $D = \{ \pm(1, -6), \pm(3, 4), \pm(3, 0), \pm(3, -2) \}$; see solution \href{https://www.combinatorics.org/files/v26i4p45/13x16_8Q.html\#Solution23}{\#23} for $\q{13}{16}$. (Recall that board squares have edge length two, column indices are even integers, and row indices are odd integers here.) Using (\ref{mnk}) this gives $\gamma(\q{13}{16}) \leq 8$ (and equality holds by our computer search). \end{exam}
We give two inf{\kern0pt}inite families of strict centrally strong sets, each including a minimum dominating set found by von Szily \cite{V1,V2}.
\begin{exam} Strict centrally strong sets for $n_{1} = 5, k = 1$ and odd $m_{1} \geq 5$, and for $n_{1} = 7, k=2$ and odd $m_{1} \geq 7$.
In our approach described above, all orthogonal indices would be even here; we have divided by two, thus returning to a board with squares of edge length one.
For $n_{1} = 5, k = 1, \mbox{ and } m_{1} \equiv 1 \beem{4}$, $D$ consists of $\pm(-1, \frac{m_{1}-1}{2})$, $(0, 0)$, and $\pm(0, 2i)$ and $\pm(2, \frac{m_{1} + 5}{2} - 4i)$ for $1 \leq i \leq \frac{m_{1} - 1}{4}$. With $m_{1} = 5$, this gives a minimum dominating set of \q{13}{13}\ found by von Szily \cite{V1}; see also solution \href{https://www.combinatorics.org/files/v26i4p45/13x13_7Q.html\#Solution41}{\#41}.
For $n_{1} = 5, k = 1, \mbox{ and } m_{1} \equiv -1 \beem{4}$, $D$ consists of $\pm(\pm 1, \frac{m_{1}-1}{2})$, $\pm(-1, \frac{m_{1}-3}{2})$, $(0, 0)$, $\pm(0, 2i)$ for $1 \leq i \leq \frac{m_{1} - 7}{4}$, and $\pm(2, \frac{m_{1} + 3}{2} - 4i)$ for $1 \leq i \leq \frac{m_{1} - 3}{4}$.
These sets show that for $i \geq 3$, if $2i-1 \leq m \leq 2i+7$ and $5 \leq n \leq 4i+1$, then $\gamma(\q{m}{n}) \leq 2i+1$.
Now let $n_{1} = 7$.
For $m_{1} = 7$, let $D = \{ i(1, 2) + j(2, -1) \mbox{ : } -1 \leq i, j \leq 1 \}$. This gives a minimum dominating set of \q{17}{17}\ found by von Szily \cite{V2}; see also solution
\href{https://www.combinatorics.org/files/v26i4p45/17x17_9Q.html\#Solution21}{\#21}.
For $m_{1} = 9$, let $D = \{ (0, 0), \pm(1, 4), \pm(2, -3), \pm(1, 2) + j(2, -1) \mbox{ : } -1 \leq j \leq 1 \}$. This gives $\gamma(\q{19}{21}) \leq 11$, which is the best we know.
The following complicated description of a placement is the result of unifying four cases depending on the residue of $m_{1}$ modulo 8. Any odd $m_{1} \geq 11$ has a unique expression $m_{1} = 11 + 2(l_{1} + l_{2})$ with $l_{1}$ an integer and either $l_{2} = l_{1}$ or $l_{2} = l_{1} + 1$. (Here $l_{1} = \lfloor (m_{1} - 11)/4 \rfloor$ and $l_{2} = \lceil (m_{1} - 11)/4 \rceil$.)
Start with $(0, 0), \pm(1, 2), \pm(2, -3), \pm(3, -1), \pm((-1)^{l_{1}}, -2l_{1}-5), \pm((-1)^{l_{2}+1}, -2l_{2}-4)$. Add $\pm(2, 4j)$ and $\pm(2, 4j+1)$ for $1 \leq j \leq \lceil l_{2}/2 \rceil$, and add $\pm(2, -4j-2)$ and $\pm(2, -4j-3)$ for $1 \leq j \leq \lfloor l_{2}/2 \rfloor$. If $l_{2} = l_{1}$ then add $\pm(2, (-1)^{l_{1}}(2l_{1}+4))$.
These sets show that for $i \geq 4$, if $2i-1 \leq m \leq 2i+9$ and $7 \leq n \leq 4i+1$, then $\gamma(\q{m}{n}) \leq 2i+1$. \end{exam}
Above we have described two approaches to the construction of dominating sets. In both, once a set of lines to be occupied by the dominating set is specif{\kern0pt}ied, it is necessary to see whether one can f{\kern0pt}ind such a dominating set. A fast backtrack search idea of Hitotumatu and Noshita \cite{HN}, explained and amplif{\kern0pt}ied by Knuth \cite{KN}, was used by \"{O}sterg{\aa}rd and Weakley \cite{OW} to f{\kern0pt}ind values and bounds of $\gamma(\q{n}{n})$ up to $n = 120$. This approach and also the algorithm of Neuhaus \cite{NE} can be applied to rectangular boards as well. But as mentioned, neither of our constructions can produce a dominating set of size less than $\lfloor n/2 \rfloor$ for $Q_{m \times n}$ (with $m \leq n$). Thus a resolution of Question \ref{whatlowerbound} would be needed to determine whether extensive search based on these constructions is useful.
The complexity of computing minimum dominating set of queens is another open question \cite[Section 5]{FE}. Backtracking algorithms, dynamic programming, and treewidth technique are analyzed extensively by Fernau \cite[Sections 2-4]{FE}.
Applications of backtracking algorithms to a variety of domination problems are studied in the doctoral dissertation of Bird \cite{Bird}; in particular, he examines how recursive backtracking search can be split among multiple processes by partitioning the search tree. We give some of his results on queens at the end of the next section.
\section{Independent domination}
We have calculated the independent domination number $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18$,
as shown in the table. Each table entry is linked to some minimum independent
dominating sets.
In these ranges for $m$ and $n$, monotonicity fails twice: $i(Q_{8 \times 11}) = 6 > 5 = i(Q_{9 \times 11}) = i(Q_{10 \times 11}) = i(Q_{11 \times 11})$, and $i(Q_{11 \times 18}) = 9 > 8 = i(Q_{12 \times 18})$. The f{\kern0pt}irst instance is essentially the same failure as for $\gamma(Q_{m \times n})$. The second is similar in that the only (up to symmetry) independent dominating set of size 8 for $Q_{12 \times 18}$ does not f{\kern0pt}it on $Q_{11 \times 18}$.
From the def{\kern0pt}initions it is clear that $\gamma(Q_{m \times n}) \leq i(Q_{m \times n})$, and this appears to be an excellent lower bound for $i(Q_{m \times n})$. In the table, we have highlighted the entries where these two numbers are unequal. We know of no case where $\gamma(Q_{m \times n}) + 1 < i(Q_{m \times n})$.
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
$n \diagdown m$ & \hspace*{0.1mm} 4 \hspace*{0.1mm}
& \hspace*{0.1mm} 5 \hspace*{0.1mm}
& \hspace*{0.1mm} 6 \hspace*{0.1mm}
& \hspace*{0.1mm} 7 \hspace*{0.1mm}
& \hspace*{0.1mm} 8 \hspace*{0.1mm}
& \hspace*{0.1mm} 9 \hspace*{0.1mm} & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \hline
4 & \cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/04x04_3Qi.html}{\emph{3}} & & & & & & & & & & & & & & \\ \hline
5 & \cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/04x05_3Qi.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/05x05_3Qi.html}{\emph{3}} & & & & & & & & & & & & & \\ \hline
6 & \href{https://www.combinatorics.org/files/v26i4p45/04x06_3Qi.html}{3} &
\href{https://www.combinatorics.org/files/v26i4p45/05x06_3Qi.html}{3} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/06x06_4Qi.html}{\emph{4}} & & & & & & & & & & & & \\ \hline
7 & \href{https://www.combinatorics.org/files/v26i4p45/04x07_3Qi.html}{3} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/05x07_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x07_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x07_4Qi.html}{\emph{4}} & & & & & & & & & & & \\ \hline
8 & \cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/04x08_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x08_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x08_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x08_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x08_5Qi.html}{\emph{5}} & & & & & & & & & & \\ \hline
9 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x09_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x09_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x09_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x09_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/09x09_5Qi.html}{\emph{5}} & & & & & & & & & \\ \hline
10 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x10_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x10_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/07x10_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x10_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/09x10_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/10x10_5Qi.html}{\emph{5}} & & & & & & & & \\ \hline
11 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x11_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x11_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x11_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x11_6Qi.html}{\textbf{6}} &
\href{https://www.combinatorics.org/files/v26i4p45/09x11_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/10x11_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/11x11_5Qi.html}{\emph{5}} & & & & & & &\\ \hline
12 & {4} &
\href{https://www.combinatorics.org/files/v26i4p45/05x12_4Qi.html}{4} &
\href{https://www.combinatorics.org/files/v26i4p45/06x12_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x12_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/08x12_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x12_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x12_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/11x12_6Qi.html}{6} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/12x12_7Qi.html}{\emph{7}} & & & & & & \\ \hline
13 & 4 &
{5} &
\href{https://www.combinatorics.org/files/v26i4p45/06x13_5Qi.html}{5} &
\href{https://www.combinatorics.org/files/v26i4p45/07x13_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x13_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x13_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x13_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x13_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x13_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/13x13_7Qi.html}{\emph{7}} & & & & & \\ \hline
14 & 4 &
{5} &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x14_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x14_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/09x14_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/10x14_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x14_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x14_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x14_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/14x14_8Qi.html}{\emph{8}} & & & & \\ \hline
15 & 4 &
{5} &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x15_6Qi.html}{6} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/08x15_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x15_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x15_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x15_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/12x15_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x15_8Qi.html}{8} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/14x15_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x15_9Qi.html}{\emph{9}} & & & \\ \hline
16 & 4 &
5 &
{6} &
\href{https://www.combinatorics.org/files/v26i4p45/07x16_6Qi.html}{6} &
\href{https://www.combinatorics.org/files/v26i4p45/08x16_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x16_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x16_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/11x16_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/12x16_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x16_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/14x16_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x16_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/16x16_9Qi.html}{\emph{9}} & & \\ \hline
17 & 4 &
5 &
6 &
{7} &
\href{https://www.combinatorics.org/files/v26i4p45/08x17_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x17_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/10x17_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/11x17_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/12x17_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x17_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/14x17_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x17_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/16x17_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/17x17_9Qi.html}{\emph{9}} & \\ \hline
18 & 4 &
5 &
6 &
{7} &
\href{https://www.combinatorics.org/files/v26i4p45/08x18_7Qi.html}{7} &
\href{https://www.combinatorics.org/files/v26i4p45/09x18_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/10x18_8Qi.html}{8} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/11x18_9Qi.html}{\textbf{9}} &
\href{https://www.combinatorics.org/files/v26i4p45/12x18_8Qi.html}{8} &
\href{https://www.combinatorics.org/files/v26i4p45/13x18_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/14x18_9Qi.html}{9} &
\href{https://www.combinatorics.org/files/v26i4p45/15x18_9Qi.html}{9} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/16x18_10Qi.html}{10} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/17x18_10Qi.html}{10} &
\cellcolor{lightgray} \href{https://www.combinatorics.org/files/v26i4p45/18x18_10Qi.html}{\emph{10}} \\
\hline \end{tabular} \\[1mm] Table 2: Values of independent domination number $i(Q_{m \times n}), \, 4 \leq m \leq n \leq 18$ (\href{http://oeis.org/A299029}{OEIS A299029}). Highlighted cells indicate where $\gamma \neq i$. \end{center}
Bird \cite[Chapter 5]{Bird} reports the new values $i(Q_{n \times n}) = \gamma(Q_{n \times n}) = (n/2) + 1$ for $n = 20, 22, 24$, $i(Q_{19 \times 19}) = i(Q_{21 \times 21}) = 11$, and $i(Q_{23 \times 23}) = 13$. He also gives the number of minimum dominating sets and the number of minimum independent dominating sets, up to equivalence, for $Q_{n \times n}$ up to $n = 18$.
\end{document} |
\begin{document}
\begin{abstract} \noindent We demonstrate that Joyal's category $\Theta_n$, which is central to numerous definitions of $(\infty,n)$-categories, naturally encodes the homotopy type of configuration spaces of marked points in $\mathbb{R}^n$. This article is largely self-contained and uses only elementary techniques in combinatorics and homotopy theory. \end{abstract}
\title{Configuration spaces and $\Theta_n$}
\section{Introduction} \label{Introduction:Section}
Among the many approaches to a theory of $(\infty,1)$-categories, two of the more developed are Joyal's theory of quasi-categories \cite{JoyalQuasi} and Rezk's theory of \emph{complete Segal spaces} \cite{RezkCSS}. In the former approach, an $(\infty,1)$-category is a contravariant functor from the simplicial category $\Delta$ into sets, while in the latter it is a contravariant functor from $\Delta$ into spaces; in each case this functor is required to satisfy certain conditions.
Numerous deep and natural questions involving higher category theory, for instance the cobordism hypothesis of Baez and Dolan \cite{BaezDolan, LurieTFT}, require a developed theory of $(\infty,n)$-categories for $n\geqslant 0$. It was in order to initiate such a theory of $(\infty,n)$-categories that Joyal introduced the categories $\Theta_n$ with $\Theta_1 = \Delta$. Indeed, he defined an $(\infty,n)$-category to be a contravariant functor from $\Theta_n$ into sets, directly generalising the notion of a quasi-category~\cite{JoyalDiscs}. Thereafter, Rezk formulated a different notion of $(\infty,n)$-category as a contravariant functor from $\Theta_n$ into spaces, directly generalising the theory of complete Segal spaces~\cite{RezkCartesian}.
At present the categories $\Theta_n$ appear in several places in the higher category theory literature, and likewise admit various definitions. We will use Berger's definition of $\Theta_n$ as the \emph{$n$-fold wreath product} of $\Delta$ with itself \cite{BergerTwo}. This notion of wreath product is important in Lurie's theory of $\infty$-operads~\cite{LurieHA}.
The purpose of this paper is to demonstrate that the category $\Theta_n$ very naturally encodes properties of an important class of topological spaces, namely the spaces of configurations of $r$ marked points in Euclidean space $\mathbb{R}^n$. Such configuration spaces arise in various situations throughout algebraic and geometric topology. For one, when $n=2$ this configuration space is precisely the classifying space of the pure braid group on $r$ strands. Secondly, keeping $r$ fixed and taking the limit as $n\to\infty$ and forgetting the markings one obtains the classifying space of the symmetric group on $r$ letters. Lastly, the configuration space of $r$ marked points in $\mathbb{R}^n$ is homotopy equivalent to the space of $r$-ary operations of the $E_n$-operad.
Let us introduce some terminology before stating our main result. Fix a finite set $A$. Recall that there is a natural assembly functor $\gamma_n\colon\Theta_n\to\Gamma$ taking values in Segal's category of finite sets.
\begin{definition*} Let $\Theta_n(A)$ denote the following category. The objects are pairs $(S,f)$ where $S$ is an object of $\Theta_n$ and $\sigma\colon\gamma_n(S)\to A$ is an isomorphism. A morphism $(S,\sigma)\to (T,\tau)$ is a morphism $\lambda\colon S\to T$ in $\Theta_n$ for which $\tau\circ\gamma_n(\lambda)=\sigma$. \end{definition*}
\begin{definition*} Let $\mathrm{Conf}_A(\mathbb{R}^n)$ denote the space of all injective functions $A\to\mathbb{R}^n$. When $A=\{1,\ldots,r\}$ this is simply the space of configurations of $r$ marked points in $\mathbb{R}^n$. \end{definition*}
\begin{theorem*} There is a homotopy equivalence \[B(\Theta_n(A))\simeq \mathrm{Conf}_A(\mathbb{R}^n)\] between the classifying space of $\Theta_n(A)$ and the configuration space $\mathrm{Conf}_A(\mathbb{R}^n)$. \end{theorem*}
It is our hope that this paper will be accessible to category theorists new to the configuration space $\mathrm{Conf}_A(\mathbb{R}^n)$, and also to topologists new to the category $\Theta_n$. In particular we do not rely on any of the literature on configuration spaces, and we do not assume any prior knowledge of the category $\Theta_n$.
Both $B(\Theta_n(A))$ and $\mathrm{Conf}_A(\mathbb{R}^n)$ admit evident free actions of the permutation group $\Sigma_A$, and the equivalence $B(\Theta_n(A))\simeq \mathrm{Conf}_A(\mathbb{R}^n)$ is in fact a $\Sigma_A$-equivariant homotopy equivalence. It is our expectation that this equivalence extends to a more general statement in which the set $A$ is allowed to vary not just by bijections, but by surjections. More concretely, we expect to show in future work that (a simplicial localisation of) $\Theta_n$ is equivalent to (the exit-path category of a non-compact version of) the Ran space of $\mathbb{R}^n$. A consequence of such a result would be an explicit comparison between $\Theta_n$-spaces (that is, contravariant functors from $\Theta_n$ to spaces) and $E_n$-algebras which would make use of `factorisation algebras' in the sense of Lurie~(\cite{LurieHA}). Such a comparison is to be expected. For instance, Berger (\cite{BergerTwo}) has shown that group-like reduced $\Theta_n$-spaces are a `model' for $n$-fold loop spaces. \\
Let us sketch the proof of the theorem. We will make use of an elementary combinatorial object which we call the \emph{poset of $n$-orderings of $A$} and write as $n\mathrm{Ord}(A)$. The elements of $n\mathrm{Ord}(A)$ are certain trees of height $n$ with leaves labelled by $A$, and the partial order is determined by a simple criterion that we call the \emph{branching condition}. When $n=1$ the poset $n\mathrm{Ord}(A)$ is simply the set of linear orderings of $A$, and the partial ordering is the trivial one. The poset of $n$ orderings appears elsewhere in other other guises: in \cite{Batanin}
it is the poset of total complementary $n$-orders on $A$, while in \cite{BFSV} it is the poset of $|A|$-ary operations in the Milgram preoperad. The relevance of $n\mathrm{Ord}(A)$ is that it mediates between $\Theta_n(A)$ and $\mathrm{Conf}_A(\mathbb{R}^n)$:
\begin{TheoremA} There is a homotopy equivalence $B(n\mathrm{Ord}(A))\simeq \mathrm{Conf}_A(\mathbb{R}^n)$. \end{TheoremA}
\begin{TheoremB} There is a full embedding $n\mathrm{Ord}(A)\hookrightarrow \Theta_n(A)$ which induces a homotopy equivalence on geometric realisations. \end{TheoremB}
These two theorems together imply the main result above.
Theorem~A is related to the \emph{Fox-Neuwirth cell decomposition} of $\mathrm{Conf}_A(\mathbb{R}^n)$. Fox and Neuwirth exhibited in~\cite{FoxNeuwirth} a decomposition of $\mathrm{Conf}_A(\mathbb{R}^2)$ into finitely many open cells. This is not a CW-decomposition, but still the topological boundary of each cell meets only cells of lower dimension. Consequently the cells themselves form the elements of a poset. The generalisation to $n>2$ is discussed in \cite[Section~5.4]{GetzlerJones}, \cite[section~6]{Batanin} and \cite{GiustiSinha}. From this point of view, Theorem~A identifies the poset of Fox-Neuwirth cells with $n\mathrm{Ord}(A)$, and shows that its realisation is homotopy-equivalent to $\mathrm{Conf}_A(\mathbb{R}^n)$ itself.
Theorem~A is contained in a theorem of Balteanu, Fiedorowicz, Schw\"anzl and Vogt \cite[Theorem~3.14]{BFSV}. However, we present our own proof of the result, both in order to give a self-contained account of our main theorem, and because our proof is significantly simpler. (This is not surprising: Theorem~A appears in \cite{BFSV} as just one part of a more elaborate result.)
Theorem~B amounts to a careful study of the morphisms of $\Theta_n$. For it is well-known that the objects of $\Theta_n$ admit a simple description as the \emph{planar level trees of height $n$}. Thus it is simple to construct the claimed embedding $n\mathrm{Ord}(A)\hookrightarrow\Theta_n(A)$ on the level of objects. However, given objects $S$ and $T$ of $\Theta_n$, described as planar level trees, it is difficult to describe the collection of all morphisms $S\to T$ in a similarly combinatorial way. (This can be seen as one of the causes for the profusion of definitions of $\Theta_n$ itself.) Nevertheless, we are able to prove a theorem that gives a simple combinatorial description of the set of all \emph{active} morphisms $S\to T$ when $T$ is \emph{healthy}. (Here \emph{active} and \emph{healthy} are appropriate restricted classes of morphisms and objects.) This is sufficient to prove Theorem~B.
The paper is arranged as follows. In section~\ref{nOrd:Section} we introduce the poset $n\mathrm{Ord}(A)$, then in section~\ref{nOrdHomotopy:Section} we compute its homotopy type and prove Theorem~A. Section~\ref{Definitions:Section} recalls the definition of $\Theta_n$ in detail. Then section~\ref{Morphisms:Section} proves our characterisation of the active morphisms $S\to T$ when $T$ is healthy. This result is then used in section~\ref{ThetanA:Section} to prove Theorem~B.
\subsection*{Acknowledgments}
This work began in the Topology Reading Seminar in the mathematics department of Copenhagen University. Together with the other members of the topology group, we spent several weeks in 2010 studying Clemens Berger's papers~\cite{BergerOne} and~\cite{BergerTwo}, which inspired the results presented here. We would like to thank Berger for his work, and we would like to thank the other members of Copenhagen's topology group for their friendship and support.
\section{The poset of $n$-orderings on $A$}\label{nOrd:Section}
As in the introduction, we fix a finite set $A$ and an integer $n\geqslant 1$. This section will first define the notion of an \emph{$n$-ordering} on $A$, and then make the set of $n$-orderings on $A$ into a poset. (For us, a \emph{poset} is a category in which for each pair of objects $c,d$ there is at most one morphism $c\to d$, and in which the only isomorphisms are the identity morphisms.) As explained in the introduction, this poset appears elsewhere in the literature, in particular in \cite{Batanin} and \cite{BFSV}. Here we will introduce the poset from scratch.
\begin{definition}\label{PlanarLevelTrees:Definition} A \emph{level tree} is a tree equipped with a preferred vertex called the \emph{root}. The root gives a preferred direction to all edges of the tree, and there is a unique directed path from any vertex to the root. The \emph{incoming} edges at a vertex are those edges directed toward the vertex. A \emph{planar level tree} is a level tree equipped with a linear ordering on the {incoming} edges at each vertex. The \emph{level} of a vertex is the length of the directed path from that vertex to the root. A tree has \emph{height $n$} if the maximum of the levels of its vertices is at most $n$. Notice then that a tree of height $n$ is a tree of height $n+k$ for any $k\geqslant 0$. A vertex $v$ is called a \textit{leaf} if it has no incoming edges. We depict planar level trees as a diagrams as follows, with the root at the bottom and with the linear orderings read from left to right. \[ \xy (0,0)*{\bullet}="A"; (-5,5)*{\bullet}="B1"; (-0,5)*{\bullet}="B2"; (5,5)*{\bullet}="B4"; (-10,10)*{\bullet}="C1"; (-5,10)*{\bullet}="C2"; (0,10)*{\bullet}="C3"; (5,10)*{\bullet}="C4"; (10,10)*{\bullet}="C6"; "A";"B1" **\dir{-}; "A";"B2" **\dir{-}; "A";"B4" **\dir{-}; "B1";"C1" **\dir{-}; "B1";"C2" **\dir{-}; "B1";"C2" **\dir{-}; "B2";"C3" **\dir{-}; "B2";"C4" **\dir{-}; "B4";"C6" **\dir{-}; \endxy \quad\quad \xy (0,0)*{\bullet}="A"; (-15,5)*{\bullet}="B1"; (-5,5)*{\bullet}="B2"; (5,5)*{\bullet}="B3"; (15,5)*{\bullet}="B4"; (-10,10)*{\bullet}="C3"; (-5,10)*{\bullet}="C4"; (0,10)*{\bullet}="C5"; (15,10)*{\bullet}="C6"; "A";"B1" **\dir{-}; "A";"B2" **\dir{-}; "A";"B3" **\dir{-}; "A";"B4" **\dir{-}; "B2";"C3" **\dir{-}; "B2";"C4" **\dir{-}; "B2";"C5" **\dir{-}; "B4";"C6" **\dir{-}; \endxy \] \end{definition}
\begin{definition} A planar level tree of height $n$ is \emph{healthy} if it has no leaves at levels $1,\ldots,n-1$. In the illustration above the first tree is healthy of height $2$, whereas the second tree is not healthy. \end{definition}
Note that the question of whether a planar level tree of height $n$ is healthy is dependent on $n$. The tree of height $0$, which consists of the root and nothing more, is healthy regardless of the value of $n$. This might seem anomalous, but it will allow the empty set to admit a (unique) $n$-ordering for each $n$, as we see now.
\begin{definition}\label{nOrderings:Definition} An \emph{$n$-ordering} on $A$ is a pair $(S,\sigma)$ consisting of a healthy planar level tree $S$ of height $n$, together with a bijection $\sigma$ between $A$ and the level-$n$ leaves of $S$. We will usually denote an $n$-ordering $(S,\sigma)$ by $S$ alone, leaving the bijection $\sigma$ implicit. \end{definition}
\begin{example}\label{nOrderings:ExampleOne} A $1$-ordering on $A$ is precisely a linear ordering on $A$. \end{example}
\begin{example}\label{nOrderings:ExampleTwo} There are exactly {four} $2$-orderings on the set $A=\{a,b\}$, and they are depicted below. \[\xy (-60,0)*{\bullet}="T1"; (-60,-5)*{S}; (-65,5)*{\bullet}="T2"; (-55,5)*{\bullet}="T3"; (-65,10)*{\bullet}="T4"; (-65,13)*{a}; (-55,10)*{\bullet}="T5"; (-55,13.5)*{b}; "T1";"T2" **\dir{-}; "T1";"T3" **\dir{-}; "T2";"T4" **\dir{-}; "T3";"T5" **\dir{-};
(-30,0)*{\bullet}="T1"; (-30,-5)*{T}; (-35,5)*{\bullet}="T2"; (-25,5)*{\bullet}="T3"; (-35,10)*{\bullet}="T4"; (-35,13.5)*{b}; (-25,10)*{\bullet}="T5"; (-25,13)*{a}; "T1";"T2" **\dir{-}; "T1";"T3" **\dir{-}; "T2";"T4" **\dir{-}; "T3";"T5" **\dir{-};
(0,0)*{\bullet}="T1"; (0,-5)*{U}; (0,5)*{\bullet}="T2"; (-5,10)*{\bullet}="T3"; (-5,13)*{a}; (5,10)*{\bullet}="T4"; (5,13.5)*{b}; "T1";"T2" **\dir{-}; "T2";"T3" **\dir{-}; "T2";"T4" **\dir{-};
(30,0)*{\bullet}="T1"; (30,-5)*{V}; (30,5)*{\bullet}="T2"; (25,10)*{\bullet}="T3"; (25,13.5)*{b}; (35,10)*{\bullet}="T4"; (35,13)*{a}; "T1";"T2" **\dir{-}; "T2";"T3" **\dir{-}; "T2";"T4" **\dir{-}; \endxy\] \end{example}
Next, we wish to define a notion of \emph{morphism} between different $n$-orderings. In order to do so we introduce a little more notation.
\begin{definition}\label{OrderBranchingLevels:DefinitionOne} Let $S$ be an $n$-ordering on $A$. The leaves of $S$ inherit a canonical linear order, and this induces a linear ordering on $A$ that we denote by $<_S$. Given $a,b\in A$, the \emph{branching level} \[b_S(a,b)\] is defined to be the level of the vertex at which the directed paths from $a$ and $b$ to the root first meet. \end{definition}
\begin{example}\label{nOrderings:ExampleThree} Let us return to the four elements of $2\mathrm{Ord}(\{a,b\})$, which were listed in Example~\ref{nOrderings:ExampleTwo}. For the orderings we have \[a<_S b,\qquad b<_T a,\qquad a<_U b,\qquad b<_V a\] and for the branching levels we have \[b_S(a,b)=0,\qquad b_T(a,b)=0,\qquad b_U(a,b)=1,\qquad b_V(a,b)=1.\] \end{example}
\begin{definition}\label{BranchingCondition:DefinitionOne} Let $S$ and $T$ be $n$-orderings on $A$. The \emph{branching condition} for a morphism $S\to T$ states that the following criterion holds for all $a,b\in A$. \begin{quotation} We have $b_T(a,b)\leqslant b_S(a,b)$, with equality only if the ordering of $a$ and $b$ under $<_T$ agrees with the ordering of $a$ and $b$ under $<_S$. \end{quotation} The condition on orderings means that ($a<_T b$ and $a<_S b$) or ($b<_T a$ and $b<_S a$). \end{definition}
\begin{definition}\label{nOrd:Definition} The \emph{poset of $n$-orderings on $A$}, denoted $n\mathrm{Ord}(A)$, is the poset whose objects are the $n$-orderings on $A$, and in which there is a morphism $S\to T$ if and only if the branching condition holds. \end{definition}
It is trivial to verify that $n\mathrm{Ord}(A)$ is indeed a poset. In other words \begin{itemize} \item for every $n$-ordering $S$ there is a morphism $S\to S$, \item if there are morphisms $S\to T$ and $T\to U$, then there is a morphism $S\to U$, \item if there are morphisms $S\to T$ and $T\to S$ then $S=T$. \end{itemize}
\begin{example}\label{nOrderings:ExampleFour} Let us return again to $2\mathrm{Ord}(\{a,b\})$, as in Examples~\ref{nOrderings:ExampleTwo} and \ref{nOrderings:ExampleThree}. There are exactly four objects, and there are also exactly four non-identity morphisms, which we depict below. \[\xy (0,20)*{\bullet}="T1"; (0,15)*{U}; (0,25)*{\bullet}="T2"; (-5,30)*{\bullet}="T3"; (-5,33)*{a}; (5,30)*{\bullet}="T4"; (5,33.5)*{b}; "T1";"T2" **\dir{-}; "T2";"T3" **\dir{-}; "T2";"T4" **\dir{-};
(0,-20)*{\bullet}="T1"; (0,-25)*{V}; (0,-15)*{\bullet}="T2"; (-5,-10)*{\bullet}="T3"; (-5,-6.5)*{b}; (5,-10)*{\bullet}="T4"; (5,-7)*{a}; "T1";"T2" **\dir{-}; "T2";"T3" **\dir{-}; "T2";"T4" **\dir{-};
(-30,0)*{\bullet}="T1"; (-30,-5)*{S}; (-35,5)*{\bullet}="T2"; (-25,5)*{\bullet}="T3"; (-35,10)*{\bullet}="T4"; (-35,13)*{a}; (-25,10)*{\bullet}="T5"; (-25,13.5)*{b}; "T1";"T2" **\dir{-}; "T1";"T3" **\dir{-}; "T2";"T4" **\dir{-}; "T3";"T5" **\dir{-};
(30,0)*{\bullet}="T1"; (30,-5)*{T}; (25,5)*{\bullet}="T2"; (35,5)*{\bullet}="T3"; (25,10)*{\bullet}="T4"; (25,13.5)*{b}; (35,10)*{\bullet}="T5"; (35,13)*{a}; "T1";"T2" **\dir{-}; "T1";"T3" **\dir{-}; "T2";"T4" **\dir{-}; "T3";"T5" **\dir{-};
{\ar@/_1 ex/(-5,25)*{};(-20,10)*{}}; {\ar@/^1 ex/(5,25)*{};(20,10)*{}}; {\ar@/^1 ex/(-5,-15)*{};(-20,0)*{}}; {\ar@/_1 ex/(5,-15)*{};(20,0)*{}}; \endxy\] For example there is a morphism $U\to T$ since $b_T(a,b)<b_U(a,b)$, and there is no morphism $U\to V$ since $b_U(a,b)=b_V(a,b)$ while $a<_Ub$ and $a>_V b$.
Observe that the classifying space $B(2\mathrm{Ord}(\{a,b\})$ is homeomorphic to $S^1$, which is homotopy equivalent to the configuration space $\mathrm{Conf}_2(A)$ of two labelled points in $\mathbb{R}^2$. \end{example}
\section{The homotopy type of $n\mathrm{Ord}(A)$} \label{nOrdHomotopy:Section}
Now we turn to the proof of Theorem~A, which states that there is a homotopy equivalence $B(n\mathrm{Ord}(A))\simeq \mathrm{Conf}_A(\mathbb{R}^n)$. As explained in the introduction, the key to the proof is that $n\mathrm{Ord}(A)$ is the poset indexing the `cells' in the Fox-Neuwirth decomposition of $\mathrm{Conf}_A(\mathbb{R}^n)$ \cite{FoxNeuwirth,GetzlerJones,Batanin,GiustiSinha}, and the theorem itself is contained in \cite[Theorem~3.14]{BFSV}. The proof we give here does not depend on any of this literature.
\begin{definition}\label{Cells:Definition} Let $S$ be an object of $n\mathrm{Ord}(A)$. Define $C(S)\subset\mathrm{Conf}_A(\mathbb{R}^n)$ to be the space of injections $\phi\colon A\hookrightarrow\mathbb{R}^n$ such that for each pair $a,b\in A$ with $a<_S b$, we have \begin{enumerate} \item $\phi(a)_i=\phi(b)_i$ for $i=1,\ldots,b_S(a,b)$; \item \label{Second} $\phi(a)_i\leqslant \phi(b)_i$ for $i=b_S(a,b)+1$. \end{enumerate} \end{definition}
Inspecting the branching condition, observe that $S\to T$ implies $C(S) \subset C(T)$. In this way, the assignment $S\mapsto C(S)$ defines a functor \[ C\colon n\mathrm{Ord}(A) \to \mathrm{Top}. \]
\begin{lemma}\label{Sphi:Lemma} Let $\phi\in\mathrm{Conf}_A(\mathbb{R}^n)$. Then there is an object $S_\phi$ in $n\mathrm{Ord}(A)$ with the property that $\phi\in C(S)$ if and only if there is a morphism $S_\phi\to S$. \end{lemma} \begin{proof} The lexicographic ordering on $\phi(A)\subset\mathbb{R}^n$ induces an ordering on $A$ itself that we denote by $<$. For $a,b\in A$ we define $b(a,b)$ to be the largest integer $i$ for which $\phi(a)_i=\phi(b)_i$. There is a unique object $S_\phi$ in $n\mathrm{Ord}(A)$ with ordering $<$ and branching levels $b(a,b)$. It is now trivial to check that $S_\phi$ has the required property. \end{proof}
\begin{lemma} \label{Colimit:Lemma} The colimit of $C\colonn\mathrm{Ord}(A)\to\mathrm{Top}$ is homeomorphic to $\mathrm{Conf}_A(\mathbb{R}^n)$. \end{lemma} \begin{proof}
The inclusions $C(S)\hookrightarrow\mathrm{Conf}_A(\mathbb{R}^n)$ determine a natural transformation from $C$ to the constant functor with value $\mathrm{Conf}_A(\mathbb{R}^n)$. This induces a map $f\colon\colim(C)\to\mathrm{Conf}_A(\mathbb{R}^n)$. Define a function $g\colon \mathrm{Conf}_A(\mathbb{R}^n)\to\colim(C)$ by specifying that each $g|_{C(S)}$ is the tautological map $C(S)\to\colim(C)$. By Lemma~\ref{Sphi:Lemma} it is well-defined. Since $n\mathrm{Ord}(A)$ is finite and each $C(S)\subset\mathrm{Conf}_A(\mathbb{R}^n)$ is closed, $g$ is continuous. Finally note that $g$ is inverse to $f$. This completes the proof. \end{proof}
\begin{lemma} The natural map $\hocolim(C)\to\colim(C)$ is a homotopy equivalence. \end{lemma} \begin{proof} We will prove this using the method of Proposition~13.4 of \cite{Dugger}. Define the \emph{degree} of an object of $n\mathrm{Ord}(A)$ to be the number of edges of the corresponding tree. Non-identity morphisms raise the degree, so this makes $n\mathrm{Ord}(A)$ into a \emph{directed Reedy category}. Then it will suffice to show that for each object $S$ the natural map \[ L_S(C) \to C(S) \] from the \emph{latching object} \[L_S(C)= \colim_{T\to S,\deg(T)<\deg(S)}(C(T))\] is a cofibration. By adapting the proof of Lemma~\ref{Colimit:Lemma}, one can see that this latching object is the subspace of $C(S)$ consisting of all $\phi\colon A\hookrightarrow\mathbb{R}^n$ that satisfy the conditions of Definition~\ref{Cells:Definition}, and for which at least one of the inequalities \eqref{Second} is an equality. Thus $L_S(C) \to C(S)$ is the inclusion into an (unbounded) convex polyhedron of its boundary, and in particular is a cofibration. This completes the proof. \end{proof}
\begin{lemma} The spaces $C(S)$ are all contractible. \end{lemma} \begin{proof} These spaces are naturally contained in $(\mathbb{R}^n)^A$, with respect to whose linear structure they are convex. \end{proof}
\begin{proof}[Proof of Theorem~A] The last three lemmas give us the first three homotopy equivalences in the computation \[ \mathrm{Conf}_A(\mathbb{R}^n) \simeq \colim(C) \simeq \hocolim(C) \simeq \hocolim(\ast) \simeq B(n\mathrm{Ord}(A)),\] where $\ast\colonn\mathrm{Ord}(A)\to\mathrm{Top}$ denotes the constant functor with value a point. The last homotopy equivalence holds by definition. \end{proof}
\section{The categories $\Theta_n$} \label{Definitions:Section}
In this section we will recall Berger's inductive definition of the categories $\Theta_n$, and the description of the objects of $\Theta_n$ in terms of trees. Except where noted, the material of this section is due to Berger~\cite{BergerTwo}.
\subsection{Segal's category of finite sets} For $Z$ a finite set, denote its set of subsets by $\mathcal{P}(Z)$. Recall from \cite{Segal} that $\Gamma$ is the category whose objects are the finite sets, and in which a morphism $\theta\colon X\to Y$ is a function $\theta\colon X\to\mathcal{P}(Y)$ with the property that $\theta({x_1})$ and $\theta({x_2})$ are disjoint when $x_1\neq x_2$. Composition is defined by $(\phi\circ\theta) (s)=\bigcup_{t\in\theta(s)}\phi(t)$. There is a natural functor \[\gamma\colon\Delta\longrightarrow\Gamma.\] It sends $[n]=\{0<\dots<n\}$ to $\mathbf{n}=\{1,\ldots,n\}$, and sends a morphism $f$ to the morphism $\gamma(f)$ defined by \[i\longmapsto\{j \mid f(i-1)<j\leqslant f(i) \}.\]
\subsection{Wreath products} The \emph{wreath product} $\Gamma\wr\mathcal{D}$ of $\Gamma$ with an arbitrary category $\mathcal{D}$ is defined as follows. An object of $\Gamma\wr\mathcal{D}$ is a symbol \[X(D_x)\] where $X$ is a finite set and $(D_x)_{x\in X}$ is a tuple of objects of $\mathcal{D}$ indexed by $X$. A morphism \[\theta\colon X(D_x)\longrightarrow Y(E_y)\] consists of a morphism $\theta_\Gamma\colon X\to Y$ in $\Gamma$ and a morphism $\theta_{xy}\colon D_x\to E_y$ in $\mathcal{D}$ whenever $y\in\theta_\Gamma(x)$. Composition in $\Gamma\wr\mathcal{D}$ is given by composition in $\Gamma$ and in $\mathcal{D}$. There is an apparent forgetful functor $\Gamma\wr \mathcal{D} \to \Gamma$ given by $X(D_x) \mapsto X$.
If $\mathcal{C}$ is a category over $\Gamma$ then the \emph{wreath product} $\mathcal{C}\wr\mathcal{D}$ is defined to be the pullback $\mathcal{C}\times_\Gamma (\Gamma\wr\mathcal{D})$. This wreath product is functorial in the arguments $\mathcal{C} \to \Gamma$ and $\mathcal{D}$.
The \emph{assembly} functor $\alpha\colon\Gamma\wr\Gamma\longrightarrow\Gamma$ is obtained by taking unions. To be precise, an object $X(A_x)$ is sent to the disjoint union $\bigsqcup_{x\in X} A_x$ and a morphism $\theta\colon X(A_x)\to Y(B_y)$ is sent to the morphism $\bigsqcup_{x\in X}A_x \to \bigsqcup_{y\in Y}B_y$ which assigns to $a\in A_x$ the subset $\bigsqcup_{y\in\theta_\Gamma(x)}(\theta_{xy})(a)$.
\subsection{Wreath product with $\Delta$}\label{DeltaWreath:Subsection} The Segal functor $\gamma\colon\Delta\to\Gamma$ allows us to define the wreath product $\Delta\wr\mathcal{D}$ for any category $\mathcal{D}$. Unravelling the definitions above, we see that the objects of $\Delta\wr\mathcal{D}$ are symbols of the form \[[s](D_1,\ldots,D_s)\] where $s\geqslant 0$ and $D_1,\ldots,D_s$ are objects of $\mathcal{D}$. A morphism in $\Delta\wr\mathcal{D}$ \[f\colon [s](D_1,\ldots,D_s)\longrightarrow [t](E_1,\ldots,E_t)\] consists of a morphism in $\Delta$ \[f_\Delta\colon [s]\longrightarrow[t]\] and morphisms in $\mathcal{D}$ \[f_{ij}\colon S_i\to T_j\] for every pair $i,j$ satisfying $f(i-1)<j\leqslant f(i)$.
\subsection{The categories $\Theta_n$}
\begin{definition}\label{ThetaN:Definition} The categories $\Theta_n$ are defined inductively by setting \[\Theta_1=\Delta\qquad\mathrm{and}\qquad \Theta_{n}=\Delta\wr\Theta_{n-1}.\] The \emph{assembly functors} $\gamma_n\colon\Theta_n\to\Gamma$ are defined inductively by setting $\gamma_1=\gamma$ and $\gamma_{n}=\alpha\circ(\gamma\wr\gamma_{n-1})$. (The categories $\Theta_n$ were first introduced by Joyal \cite{JoyalDiscs}. However, the above definition in terms of wreath products is due to Berger.) \end{definition}
\subsection{The objects of $\Theta_n$}\label{Objects:Section} \label{Objects:Subsection} The objects of $\Theta_n$ are naturally identified with the planar level trees of height $n$. (See Definition~\ref{PlanarLevelTrees:Definition}.) When $n=1$ this identification sends the object $[s]$ to the tree with exactly $s$ non-root vertices, all with level $1$. For $n>1$ the object $[s](S_1,\ldots,S_s)$ of $\Theta_n$ is identified with the planar level tree that has exactly $s$ vertices with level $1$, and in which the tree associated to $S_i$ appears as the subtree spanned by those vertices for which there is a directed path to the $i$-th such level-$1$ vertex.
The value of the assembly functor $\gamma_n\colon\Theta_n\to\Gamma$ on a tree $T$ is naturally identified with the set of leaves of that tree.
\begin{example} The object $[4]([2],[3],[0],[1])$ of $\Theta_2$ corresponds to the following planar level tree. \[ \xy (0,0)*{\bullet}="A"; (-15,10)*{\bullet}="B1"; (-5,10)*{\bullet}="B2"; (5,10)*{\bullet}="B3"; (15,10)*{\bullet}="B4"; (-17.5,20)*{\bullet}="C1"; (-12.5,20)*{\bullet}="C2"; (-10,20)*{\bullet}="C3"; (-5,20)*{\bullet}="C4"; (0,20)*{\bullet}="C5"; (15,20)*{\bullet}="C6"; "A";"B1" **\dir{-}; "A";"B2" **\dir{-}; "A";"B3" **\dir{-}; "A";"B4" **\dir{-}; "B1";"C1" **\dir{-}; "B1";"C2" **\dir{-}; "B1";"C2" **\dir{-}; "B2";"C3" **\dir{-}; "B2";"C4" **\dir{-}; "B2";"C5" **\dir{-}; "B4";"C6" **\dir{-}; \endxy \] \end{example}
\section{Morphisms in $\Theta_n$} \label{Morphisms:Section}
From the category-theoretic point of view this section is the technical heart of the paper.
We have just seen that the objects of $\Theta_n$ admit a simple description as the planar level trees of height $n$. The morphisms in $\Theta_n$ are much less easy to describe from this point of view. This is indicated by the assembly functor $\gamma_n\colon\Theta_n\to\Gamma$, which sends a tree to its set of leaves with level $n$. Just as a morphism in $\Gamma$ is not a map of sets, so a morphism in $\Theta_n$ is not a map of trees in any obvious sense.
In this section we will define \emph{healthy} objects and \emph{active} morphisms in $\Theta_n$, and we will give a simple, combinatorial description of the set of active morphisms $S\to T$ when $T$ is healthy. In the next section this description will be applied to the category $\Theta_n(A)$.
\subsection{Active morphisms and healthy trees}
\begin{definition} An object of $\Theta_n$ is \emph{healthy} if the corresponding planar level tree is healthy, or in other words, has no leaves at level $1,\ldots,n-1$. \end{definition}
\begin{definition} A morphism $\theta\colon X\to Y$ in $\Gamma$ is \emph{active} if $\bigcup_{x\in X}\theta(x)=Y$. A morphism in $\Theta_n$ is \emph{active} if its image under $\gamma_n$ is active. (Our definition of active morphisms in $\Gamma$ corresponds to Lurie's notion of active morphism in the category of based finite sets, which is the opposite of $\Gamma$ \cite{LurieHA}.) \end{definition}
By way of section~\textsection\ref{Objects:Section}, from now on we will not distinguish between an object of $\Theta_n$ and a planar level tree of height $n$.
\subsection{The branching condition}
Now we introduce some notation regarding objects and morphisms in $\Theta_n$. These are closely related to the definitions introduced in section~\ref{nOrd:Section}.
\begin{definition}\label{OrderBranchingLevels:DefinitionTwo} Let $T$ be an object of $\Theta_n$. Then the planar structure of $T$ endows the set of level-$n$ leaves $\gamma_n(T)$ with a linear ordering that we denote $<_T$. Given $a,b\in\gamma_n(T)$ we define the \emph{branching level} \[ b_T(a,b)\] to be the level of the vertex at which the directed paths from $a$ and $b$ to the root meet. Note that the branching levels between consecutive elements of $\gamma_n(T)$ determine all the other branching levels. \end{definition}
\begin{definition}\label{BranchingCondition:DefinitionTwo} Let $S$, $T$ be objects of $\Theta_n$, with $T$ healthy. An active morphism $\bar f\colon \gamma_n(S)\to \gamma_n(T)$ satisfies the \emph{branching condition} if, for all quadruples $a,b,c,d$ with $a,b\in\gamma_n(S)$ and $c\in\bar f(a)$, $d\in\bar f(b)$, the following condition holds: \begin{quotation} $b_T(c,d)\leqslant b_S(a,b)$, with equality only if the order of $c,d$ in $T$ is the same as the order of $a,b$ in $S$. \end{quotation} The condition on orderings means that ($c<_T d$ and $a<_S b$) or ($d<_T c$ and $b<_S a$). \end{definition}
This branching condition is closely related to the one that describes when there is a morphism in $n\mathrm{Ord}(A)$. Here, however, there is no need for the map $\bar f$ to be an isomorphism. So for fixed $a,b\in\gamma_n(S)$ there may be several choices of elements $c\in\bar f(a)$, $d\in\bar f(b)$.
\subsection{Active morphisms into healthy trees}
The next theorem fully characterises active morphisms into healthy objects in terms of the branching condition.
\begin{theorem}\label{Morphisms:Theorem} Let $S$ and $T$ be trees in $\Theta_n$ with $T$ healthy. Then the assignment \[f\longmapsto \gamma_n(f)\] determines a bijection between active morphisms $f\colon S\to T$ and active morphisms $\bar f\colon\gamma_n(S)\to\gamma_n(T)$ satisfying the branching condition. \end{theorem}
The theorem will be applied in the next section to give us a description of the category $\Theta_n(A)$. For that application we only need the case when $\gamma_n(f)$ is an isomorphism. However, the theorem proceeds by induction on $n$, and at the induction step we are forced to pass from isomorphisms to general active morphisms.
\subsection{Proof of Theorem~\ref{Morphisms:Theorem}} Our proof of the theorem consists of the next three lemmas, all of which are proved by induction on $n$. We use the notation of section~\ref{DeltaWreath:Subsection} throughout.
\begin{lemma} Let $f,g\colon S\to T$ be active morphisms in $\Theta_n$ with $T$ healthy and $\gamma_n(f)=\gamma_n(g)$. Then $f=g$. \end{lemma} \begin{proof} In the case $n=1$ it is trivial to check that an active morphism in $\Delta$ is determined by its image in $\Gamma$.
In the general case, if $T$ is the trivial tree then there is nothing to prove. So we assume that all leaves of $T$ are at level $n$. Writing \[ \gamma_n(S)=\gamma_{n-1}(S_1)\cup\cdots\cup\gamma_{n-1}(S_s) \qquad\mathrm{and}\qquad \gamma_n(T)=\gamma_{n-1}(T_1)\cup\cdots\cup\gamma_{n-1}(T_t), \] we find that if $j\in \gamma(f_\Delta)(i)$, then $\gamma_{n-1}(T_j)$ must lie in $\gamma_{n}(f)(\gamma_{n-1}(S_i))$. Since each $\gamma_{n-1}(T_j)$ is nonempty, this means that $\gamma_n(f)$ determines $\gamma(f_\Delta)$. The same reasoning holds for $g$, so that we have $\gamma(f_\Delta)=\gamma(g_\Delta)$ and so $f_\Delta=g_\Delta$.
Now $f$ and $g$ are determined by \emph{active} morphisms $f_{ij},g_{ij}\colon S_i\to T_j$ in $\Theta_{n-1}$ for which each $T_j$ is healthy and that satisfy $\gamma_{n-1}(f_{ij})=\gamma_{n-1}(g_{ij})$. By the induction hypothesis we have $f_{ij}=g_{ij}$, and the proof is complete. \end{proof}
\begin{lemma} Let $f\colon S\to T$ be an active morphism to a healthy tree. Then $\gamma_n(f)$ satisfies the branching condition. \end{lemma} \begin{proof} Let $a,b,c,d$ be as in the statement of the branching condition. Suppose that $a\in\gamma_{n-1}(S_i)$ and $b\in\gamma_{n-1}(S_{j})$ and $c\in\gamma_{n-1}(T_{i'})$ and $d\in\gamma_{n-1}(T_{j'})$. Without loss $i\leqslant j$. There are now three possibilities, in each of which we will verify that $a,b,c,d$ satisfy the necessary condition.
First, $i< j$, so that $b_S(a,b)=0$ and $a<_Sb$. Since $i'\in \gamma(f_\Delta)(i)$ and $j'\in \gamma(f_\Delta)(j)$, we must have $i'<j'$, so that $b_T(c,d)=0$ and $c<_Td$. So the condition holds in this case.
Second, $i=j$ but $i'\neq j'$. Then $b_S(a,b)\geqslant 1$ while $b_T(c,d)=0$, so the branching condition holds.
Third, $i=j$ and $i'=j'$. Then $b_S(a,b)=b_{S_i}(a,b)+1$ and $b_T(c,d)=b_{T_{i'}}(c,d)+1$, and $f_{ii'}\colon S_i\to T_{i'}$ is an active morphism for which $c\in\gamma_{n-1}(f_{ii'})(a)$ and $d\in\gamma_{n-1}(f_{ii'})(b)$. So it suffices to check that the required condition holds when $f$ is replaced by $f_{ii'}$, but this follows from the induction hypothesis. \end{proof}
\begin{lemma} Let $S$ and $T$ be objects of $\Theta_n$ with $T$ healthy, and let $\bar f\colon \gamma_n(S)\to \gamma_n(T)$ be an active morphism in $\Gamma$ that satisfies the branching condition. Then there is $f\colon S\to T$ in $\Theta_n$ such that $\gamma_n(f)=\bar f$. \end{lemma}
\begin{proof} Again, we proceed by induction on $n$, the case $n=1$ being a trivial observation about Segal's functor $\gamma\colon\Delta\to\Gamma$.
If $t=0$ then there is nothing to prove, so we assume $t>0$. Writing \[\gamma_n(S)=\gamma_{n-1}(S_1)\cup\cdots\cup\gamma_{n-1}(S_s) \quad \mathrm{and} \quad \gamma_n(T)=\gamma_{n-1}(T_1)\cup\cdots\cup\gamma_{n-1}(T_t),\] we see first that the branching condition means first that each $\bar f (\gamma_{n-1}(S_i))$ is the union of certain of the $\gamma_{n-1}(T_j)$, and second that this union in fact has the form $\gamma_{n-1}(T_{j})\cup\cdots\cup\gamma_{n-1}(T_{j'})$ for some $j\leqslant j'$. We may therefore find $0= r_0\leqslant \cdots \leqslant r_{s}=t$ such that $\bar f (\gamma_{n-1}(S_i))=\gamma_{n-1}(T_{r_{i-1}+1})\cup\cdots\cup\gamma_{n-1}(T_{r_{i}})$.
Now we can write down a morphism $f_\Delta\colon[s]\to[t]$ and, by the inductive hypothesis, morphisms $\bar f_{ij}\colon\gamma_{n-1}(S_i)\to\gamma_{n-1}(T_j)$ whenever $j\in \gamma(f_\Delta)(i)$, with the property that for $x\in\gamma_{n-1}(S_i)$ we have \[\bar f(x)=\bigcup_{j\in\gamma(f_\Delta)(i)}\bar f_{ij}(x).\] It is simple to check that each $\bar f_{ij}$ is active and satisfies the branching condition for a morphism $S_i\to T_j$, and thus by the induction hypothesis there is a morphism $f_{ij}\colon S_i\to T_j$ with $\gamma_{n-1}f_{ij}=\bar f_{ij}$. Now $f_\Delta$ and the $f_{ij}$ define the required morphism $f$. \end{proof}
\section{The category $\Theta_n(A)$} \label{ThetanA:Section}
We now use the results of the last section to study the categories $\Theta_n(A)$, and prove Theorem~B.
\subsection{Morphisms in $\Theta_n(A)$} Recall from the introduction that for a finite set $A$, the category $\Theta_n(A)$ consists of pairs $(S,\sigma)$ where $S$ is an object of $\Theta_n$ and $\sigma\colon\gamma_n(S)\to A$ is an isomorphism. A morphism $f\colon (S,\sigma)\to (T,\tau)$ in $\Theta_n(A)$ is a morphism $f\colon S\to T$ in $\Theta_n$ for which $\tau\circ\gamma_n(f)=\sigma$. From this point we will suppress the isomorphism $\sigma$ from the notation.
In light of Section~\ref{Objects:Subsection}, the objects of $\Theta_n(A)$ may be described as planar level trees of height $n$, whose leaves at level $n$ are labelled in bijection with $A$. So an object $S$ of $\Theta_n(A)$ determines an {ordering} $<_S$ on $A$ and branching levels $b_S(a,b)$ for all $a,b\in A$, exactly as in Definition~\ref{OrderBranchingLevels:DefinitionOne}. Moreover, it makes sense to ask whether the branching condition holds for a morphism $S\to T$ in $\Theta_n(A)$, exactly as in Definition~\ref{BranchingCondition:DefinitionOne}.
Comparing with Definitions~\ref{OrderBranchingLevels:DefinitionTwo} and~\ref{BranchingCondition:DefinitionTwo} and Theorem~\ref{Morphisms:Theorem}, we immediately obtain the following.
\begin{corollary}\label{ThetaNAMorphisms:Corollary} Let $S$ and $T$ be objects of $\Theta_n(A)$ with $T$ healthy. Then there is at most one morphism $S\to T$, and it exists if and only if the branching condition holds. \end{corollary}
\subsection{Proof of Theorem~B}
Recall that Theorem~B states that there is a full embedding $n\mathrm{Ord}(A)\hookrightarrow\Theta_n(A)$ that induces a homotopy equivalence on geometric realisations.
The objects of $n\mathrm{Ord}(A)$ are the healthy planar level trees of height $n$ equipped with a labelling of their leaves in bijection with $A$. The objects of $\Theta_n(A)$ have exactly the same description, except that the tree need not be healthy. This gives us an inclusion $i\colon n\mathrm{Ord}(A)\hookrightarrow\Theta_n(A)$ on objects, and by comparing Corollary~\ref{ThetaNAMorphisms:Corollary} with Definition~\ref{BranchingCondition:DefinitionOne} we see that it is a full functor.
Let $S$ be an object of $\Theta_n(A)$. Denote by $S^h$ the healthy subtree spanned by the level-$n$ vertices. Then it is easily seen that the branching condition for a morphism $S\to T$ is identical to the branching condition for a morphism $S^h\to T$. It follows that there is a morphism $S\to S^h$ in $\Theta_n(A)$, with target in $n\mathrm{Ord}(A)$, and which is initial among all morphisms from $S$ to an object of $n\mathrm{Ord}(A)$.
Now the assignment $S\mapsto S^h$ determines a functor $r\colon \Theta_n(A)\ton\mathrm{Ord}(A)$ for which $r\circ i=1$ and for which there is a natural transformation $1\Rightarrow i\circ r$. It follows that the maps $B(n\mathrm{Ord}(A))\to B(\Theta_n(A))$ and $B(\Theta_n(A))\to B(n\mathrm{Ord}(A))$ induced by $i$ and $r$ respectively are homotopy-inverse to one another. This completes the proof of Theorem~B.
\end{document} |
\begin{document}
\title{Analytical study of bound states in graphene nano-ribbons and carbon nanotubes: the variable phase method and the relativistic Levinson theorem}
\date{\today}
\author{D.~S.~Miserev$^{1, 2}$} \email{d.miserev@student.unsw.edu.au} \affiliation{$^{1}$School of Physics, University of New South Wales, Sydney, Australia} \affiliation{$^{2}$Rzhanov Institute of Semiconductor Physics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 13, Novosibirsk, 630090 Russia}
\begin{abstract} The problem of localized states in 1D systems with the relativistic spectrum, namely, graphene stripes and carbon nanotubes, has been analytically studied. The bound state as a superposition of two chiral states is completely described by their relative phase which is the foundation of the variable phase method (VPM) developed herein. Basing on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound state can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincare indices theorem for these closed trajectories. The reduction of the VPM equation to the non-relativistic and semi-classical limits has been done. The limit of the small momentum $p_y$ of the transverse quantization is applicable to arbitrary integrable potential. In this case the only confined mode is predicted. \end{abstract}
\maketitle \section{Introduction} Graphene, carbon nanotubes and topological insulators have attracted keen attention for intensive theoretical and experimental research in recent years. The uniqueness of these quantum materials with respect to fundamental physics lies in the opportunity to observe QED effects with a significantly larger coupling constant $g = e^2/s \hbar \varepsilon \sim 1$, where $s \approx c/300$ is the Fermi velocity, $\varepsilon$ is an average dielectric constant of environment (for instance, for graphene sheet on the substrate with the dielectric constant $\varepsilon_s$ one obtains $\varepsilon = (1 + \varepsilon_s)/2$). Effects such as the atomic collapse and pair production in the super-critical potentials~\cite{Popov}--\cite{Milstein}, the Adler-Bell-Jackiw anomaly (the chiral anomaly)~\cite{Nielsen}--\cite{Landsteiner} have been intensively studied. The Klein tunnelling of electrons in the gated graphene~\cite{Ando}--\cite{Reijnders} reveals the complete suppression of the backscattering.
The present work is related to the general theoretical study of the confined electronic states in graphene nano-ribbons or single-walled carbon nanotubes affected by a longitudinal electric field. Omitting inter-valley scattering, we consider electron behavior near one of two independent Dirac points where electrons are well-described by the Dirac-Weyl hamiltonian~(\ref{hamiltonian}) in the single-particle approach.
We propose a convenient technique to analyse bound states analytically for the 2D Dirac-Weyl equation with a 1D potential $U(x)$. It refers to the variable phase method (VPM) developed generally by P. M. Morse and W. P. Allis~\cite{Morse}, V. V. Babikov~\cite{Babikov}, F. Calogero~\cite{Calogero} and others~\cite{Sobel}--\cite{Ouerdane}. The wave function is expressed as a linear combination of two Weyl fermions and the phase between them is considered as a desired phase function for the VPM to be applied. Following this, we demonstrate the reduction to the non-relativistic and semi-classical limits. Furthermore, we consider one more limiting case of the $\delta$-potential which is applicable to any integrable potentials at sufficiently small transverse momentum $p_y$. Physically, this limit contains both the shallow quantum well limit and the opposite limit of a strongly supercritical potential.
Our VPM allows one to formulate the relativistic analogue of the Levinson theorem~\cite{Levinson}. The relativistic Levinson theorem for the Dirac equation was formulated in 3D by M. Klaus~\cite{Klaus} for central potentials, K. Hayashi~\cite{Hayashi} and R. L. Warnock~\cite{Warnock} as a relation between zeroes of the vertex function and particle poles of the total amplitude. This problem has been considered in two dimensions with the compact supported central potential~\cite{Dong}. D. P. Clemence~\cite{Clemence} thoroughly investigated the Levinson theorem for the Dirac equation with a 1D potential which satisfies the condition $\int_{-\infty}^{\infty} U(x) (1 + |x|) \, dx < \infty$ via the scattering matrix approach taking into account the half-bound states. The particular case of the relativistic Levinson theorem for symmetric 1D potentials has been studied by Q. Lin~\cite{Lin} with additional restriction for the potential to be a compact supported function, A. Calogeracos and N. Dombey~\cite{Calogeracos} for potentials of definite sign, Z. Ma {\it et al.}~\cite{Ma} with the similar condition as in~\cite{Clemence}. The developed herein method permits one to prove the Levinson theorem with the minimal restriction $\int_{-\infty}^\infty U(x) \, dx < \infty$ which significantly broadens the result obtained by D. P. Clemence. For example, our results are applicable to so-called top-gate potential~(\ref{top-gate}) for which asymptotics are expected to be realistic for the gated graphene structures~\cite{Hartmann}. Afterwards, a geometrical interpretation of the Levinson theorem together with the corresponding numerical method of integral curves analysis of some vector field are considered.
\section{Theoretical model} Near the conic points, electrons in graphene with the gated potential $U(x)$ are approximately described by the Dirac-Weyl Hamiltonian: \begin{equation} \hat H = s { \mbox{\boldmath{$\sigma$}} \hat{\bf p}} + U(x)= s \sigma_x \hat p_x +s \sigma_y \hat p_y + U(x) \label{hamiltonian} \end{equation} where $s$ is the Fermi velocity, $ \mbox{\boldmath{$\sigma$}} = (\sigma_x, \sigma_y)$ are Pauli matrices, ${\bf p} = - i \hbar \bm{\nabla}$. Henceforth, it is assumed that the potential decays at infinity. Further calculations are executed in the dimensionless variables: $\hbar = s = 1$. It is also assumed $p_y > 0$ where $p_y$ is the quantized transverse momentum of quasi-1D systems such as graphene nano-ribbons and single-walled carbon nanotubes where $y = r \phi$, $r$ is the radius, $\phi$ is the cyclic variable. The spectrum of the free-particle Hamiltonian is linear on the momentum: $E = \pm \sqrt{p_x^2+p_y^2}$. The negative-energy states correspond to the hole's description according to the conventional views.
The stationary wave function can be represented in a symmetric form: \begin{equation} \label{psi} \Psi = \frac{ e^{i p_y y}}{\sqrt{4 W}}{g (x)+p_y^{-1} g'(x)\choose g (x)-p_y^{-1} g'(x)}e^{i \int \limits^x\left(E-U(\zeta)\right)\, d\zeta} \end{equation} via the axillary function $g\left( x \right)$ which is introduced in~\cite{Miserev}: \begin{equation} \label{eq} g''\left(x\right)+2i\left(E-U\left(x\right)\right)g'\left(x\right)-p_y^2 g\left(x\right)=0 \end{equation} where $E$ is the electron energy and $W$ is the normalization coefficient. Eq.~(\ref{eq}) represents an equivalent statement of the problem described by the Hamiltonian~(\ref{hamiltonian}). Further we deal with electronic states of zero current along $x$-direction.
We now apply this condition to the analysis of confined states. Zero flow $j_x=\Psi^\dag(x) \sigma_x \Psi(x) = 0$ along $x$-direction yields the restriction on the function $g(x)$: \begin{equation} \label{feature1}
|g\left(x\right)|=|p_y^{-1} g'\left(x\right)|. \end{equation}
The first consequence is that $g(x)$ and hence the electron density of confined states $\rho(x) = \Psi^\dag(x) \Psi(x) = |g(x)|^2/W$ vanishes only at infinity. Otherwise, we have from~(\ref{feature1}): $g(x_0) = g'(x_0) = 0$, $|x_0| < \infty$, which yields $g(x) \equiv 0$.
Separating modulus and phase $g(x)=R e^{i\Phi}$, we arrive at the condition: \begin{equation} \label{pifagor} \left(\Phi'\right)^2+\left(R'/R\right)^2=p_y^2, \end{equation} which allows for the following substitution: \begin{equation} \label{newfunc} \left\{ \begin{array}{rcl} \Phi'(x)&=&p_y \sin {\Omega(x)}\\ R'/R&=&p_y \cos {\Omega(x)} \end{array} \right. \end{equation} where the function $\Omega(x)$ is the solution of the first-order differential equation: \begin{equation} \label{supereq} \Omega'(x)=2 \left(U(x)- E\right) -2 p_y \sin {\Omega(x)}. \end{equation} Thereby, we arrived at the desired VPM equation. We emphasize here that Eq.~(\ref{supereq}) is valid for any quantum state with zero flow, not only for bound states.
Considering bound states, we have to set the boundary conditions for the function $\Omega(x)$: \begin{equation} \label{asymptotes} \left\{ \begin{array}{ll} \Omega(x \to + \infty)=\pi +\arcsin {\frac{E}{p_y}}+2 \pi n\\ \Omega(x \to - \infty)=-\arcsin {\frac{E}{p_y}}. \end{array} \right. \end{equation} At $E \in(-p_y, p_y)$ these conditions provide the exponential decay of the density $\rho(x) \sim R^2(x)$ at infinity as it follows from~(\ref{newfunc}), $n$ being an integer.
To reveal the physical meaning of the function $\Omega(x)$, we use the following representation of the wave function: \begin{equation} \label{psidiscrete} \Psi(x, y)= \frac{ e^{i p_y y}}{\sqrt{4 W}}\left({1 \choose 1} + e^{i \Omega} {1 \choose -1}\right) R(x) e^{-i \Omega/2}. \end{equation} Hence, confined state appears as a linear combination of two chiral (Weyl) states and is completely described by the phase between them. Another form of Eq.~(\ref{psidiscrete}) refers to the spin with the polar angle $\Omega$ and the azimuthal angle $-\pi/2$: \begin{equation} \label {psidiscrete2} \Psi (x, y)= \frac{R(x) e^{i p_y y}}{\sqrt{W}}{\phantom{-i}\cos \frac{\Omega}{2}\choose -i \sin \frac{\Omega}{2}}. \end{equation}
\section{Non-relativistic limit} Let us show that Eq.~(\ref{supereq}) can be reduced to a non-relativistic equation. To be more specific, consider the non-relativistic limit for electrons: \begin{eqnarray*} E = p_y + \varepsilon, \\ \varepsilon = - k^2 /2 p_y, \end{eqnarray*} where we imply that all energy scales are small as compared with $p_y$: $k, U(x), 1/d \ll p_y$, $d$ is the characteristic width of the confinement. Boundary conditions~(\ref{asymptotes}) for $\Omega(x)$ take the form: $\Omega(-\infty) = -\pi/2 + k/p_y$, $\Omega(+\infty) = -\pi/2 - k/p_y +2 \pi n$, $n$ being an integer.
Suppose $\Omega(x) = -\pi/2 + \delta \Omega$, where $\delta \Omega \ll 1$ almost everywhere. This assumption is violated only when $\Omega' \sim p_y$ which corresponds to $\delta \Omega \sim 1$. The behaviour of the phase function $\Omega(x)$ in this region does not depend on the potential because $U(x) \ll p_y$. Notice that the width of this region $\delta x \sim 1/p_y \ll d$ is small in the non-relativistic limit. Hence, the expansion of the initial equation~(\ref{supereq}) results in the Riccati equation: \begin{equation} \label{NonRel} \delta \Omega' = 2(U(x) - \varepsilon) - p_y \delta \Omega^2, \end{equation} where $\psi(x) = \exp \left( p_y \int \delta \Omega(x) \, dx \right)$ satisfies the 1D Schrodinger equation for a non-relativistic particle with mass $p_y$. The function $\delta \Omega(x)$ tends to the infinity in zeroes of the wave function $\psi(x)$.
\section{Semi-classical limit} Let us rewrite Eq.~(\ref{supereq}) in the dimensional quantities: \begin{equation} \hbar \Omega' = \frac{2}{s} \left( U(x) - E\right) - 2 p_y \sin \Omega, \end{equation} where $s$ is the Fermi velocity. In the semi-classical limit $\hbar \to 0$ the elimination of the left-hand part of this equation yields: \begin{equation} \label{quasiw} \sin \Omega = \frac{U(x) - E}{s p_y}. \end{equation} Let us show that Eq.~(\ref{quasiw}) represents the usual quasi-classical approach.
This approximation is solvable in the real-valued functions when $|U(x) - E| < s p_y$, which conforms to the case of non-classical motion where the wave function decays. At breakpoints $x_i$, when $U(x_i) - E = -\mu \cdot s p_y$ we define $\Omega(x_i) = - \mu \pi/2$, $\mu = \pm 1$ is definite for each region of motion.
In the regions of classical motion where the wave function is oscillatory, $\Omega(x)$ is a complex function, namely, $\Omega(x) = -\mu \pi/2 + i \delta \Omega$: \begin{equation} \label{domega}
\cosh \delta \Omega(x) = -\mu \frac{U(x) - E}{s p_y} = \left| \frac{U(x) - E}{s p_y} \right|. \end{equation} Eq.~(\ref{domega}) has two solutions $\pm \delta \Omega$ (for definiteness, we set the first solution $\delta\Omega \ge 0$). The corresponding amplitude of the wave function $R_\pm(x)$ is determined from Eq.~(\ref{newfunc}): $$ R_\pm (x) \sim \exp\left(\pm i \frac{p_y}{\hbar} \int \sinh \delta \Omega(x) \,dx \right). $$ According to the definition, it is required that the function $R(x)$ is real-valued. It means that we have to consider a linear combination of corresponding functions $g_\pm (x) = R_\pm (x) e^{i \Phi_\pm(x)}$ where $$
\Phi_\pm (x) = -\mu \displaystyle \int \left| \frac{U(x) - E}{s}\right| \, \frac{dx}{\hbar} = \displaystyle \int \frac{U(x) - E}{s} \, \frac{dx}{\hbar}, $$ which follows from Eq.~(\ref{newfunc}) and $\Phi$ is the same for the two different solutions of Eq.~(\ref{domega}). Finally, the semi-classical amplitude reads: \begin{equation} R(x) \sim \cos\left(\int p_x \, \frac{dx}{\hbar} + \phi_0 \right) \end{equation} where the semi-classical momentum $p_x= p_y \sinh \delta \Omega(x)= \sqrt{\left( E - U(x) \right)^2/s^2 - p_y^2}$ is introduced. The phase $\phi_0$ is defined by the matching conditions.
Hence, Bohr-Sommerfeld quantization takes the usual form: \begin{equation} \oint p_x dx = 2 \pi \hbar (n + \gamma) \end{equation} where $n \gg 1$ is an integer, $\gamma \sim 1$ is defined from the matching conditions in the turning points; for example, $\gamma = 1/2$ for smooth potentials. The semi-classical approximation is valid when $\hbar p_y U'(x) \ll s p_x^3$.
\section{Delta-potential limit} \label{Deltap} Before we start, we emphasize that we do not require from the confinement $U(x)$ to be $\delta$-like. The reason why we name this limit as the delta-potential limit is that at some conditions the discrete spectrum and corresponding wave functions of any integrable potential are of the same analytical form as for the actual $\delta$-potential which is considered in Appendix A.
In this section we are interested in all possible cases when we are entitled to neglect the non-linear term in Eq.~(\ref{supereq}). It allows to find the spectrum and corresponding wave functions exactly. Let us formulate the following
\begin{SpectrumT} Let the potential $U(x)$ be an integrable function, $d$ is the characteristic width of $U(x)$, $p_y > 0$ is transverse momentum. Introduce the integral
\begin{equation}
G = \int_{-\infty}^\infty U(x) \, dx = \pi (n_G + \delta n_G),
\label{gexp}
\end{equation} where $n_G$ is integer and $\delta n_G \in [0, 1)$ Assume $\delta n_G \ne 0$.
Let the condition be met: \begin{equation} p_y d \ll \min \{\delta n_G, 1 - \delta n_G\}. \label{pycond} \end{equation} Then: \begin{itemize}
\item[{\bf a}]
The discrete spectrum contains the only one level with energy $E \in (-p_y, p_y)$:
\begin{equation}
\label{Deltalimit}
E = (-1)^{n_G+1} p_y \cos G,
\end{equation}
\item[{\bf b}]
If additionally $\int\limits_{x_0}^{x} U(x') x' \, dx'$ converges at $x \to \pm \infty$ at some $|x_0| < \infty$, the corresponding wave function takes the form~(\ref{psidiscrete2}) with the phase function:
\begin{equation}
\Omega(x) = -\arcsin \frac{E}{p_y} + 2 \int \limits_{-\infty}^x U(x') \, dx'.
\label{wavef}
\end{equation}
\end{itemize} \end{SpectrumT}
\begin{proof}[{\bf Proof.}]
We mean here that $U(x)$ is an integrable function in a sense that the primitive integral
$$
f_{x_0}(x) = \int_{x_0}^x U(x') \, dx'
$$
for some $|x_0| < \infty$ is defined for any $x \in (-\infty, +\infty)$ except maybe some finite set of points, and $f_{x_0}(x)$ is bounded function. We set parameter $E \in(-p_y, p_y)$.
\begin{itemize}
\item
Let $\Omega(x)$ is a physical solution with boundary conditions~(\ref{asymptotes}).
Then the total variance of the phase function $\Delta\Omega = \Omega(+\infty) - \Omega(-\infty)$ is straightforward from~(\ref{asymptotes}):
\begin{equation}
\Delta \Omega = 2 \arcsin \frac{E}{p_y} + 2 \pi \left(n+\frac{1}{2}\right).
\label{omvar}
\end{equation}
On the other hand, the integration of Eq.~(\ref{supereq}) yields:
\begin{equation}
\label{DeltaOmega}
\Delta \Omega = 2 G + \mathfrak{K},
\end{equation}
where $n$ is the integer. We introduced the integral:
\begin{equation}
\mathfrak{K} = \int \limits_{-\infty}^\infty 2 (E + p_y \sin \Omega(x))\, dx.
\end{equation}
{\it Convergence of $\mathfrak{K}$.}
Let us use {\bf Lemma 2} about the properties of solutions of Eq.~(\ref{supereq}) and rewrite $\mathfrak{K}$:
$$
\mathfrak{K} = 2 p_y \int\limits_{-\infty}^\infty(\sin \Omega(x) - \sin \Omega_\pm)\, dx.
$$
From {\bf Lemma 2} we know that the physical solution corresponds to the degeneration of two separatrix families of Eq.~(\ref{supereq}). Let us consider the behavior of this physical solution at $x \to -\infty$ where we can represent it in the form:
$$
\Omega(x) = \Omega_- + \delta\Omega(x).
$$
At $x \to -\infty$, $\delta\Omega(x)$ satisfies the approximate equation which follows directly from Eq.~(\ref{supereq}):
$$
\delta\Omega'(x) \approx 2 U(x) - 2 k \cdot \delta\Omega(x),
$$
where we accounted for that $p_y \cos\Omega_- = k > 0$, $k = \sqrt{p_y^2 - E^2}$. The solution which meets the initial condition $\delta\Omega(-\infty) = 0$ reads:
\begin{equation}
\delta\Omega(x) = 2 \int\limits_{-\infty}^{x} U(x') e^{-2k (x-x')} \, dx'.
\label{domex}
\end{equation}
Apply it to analyze the convergence of $\mathfrak{K}$ at $-\infty$. If $x \to -\infty$ we can use the expansion $p_y (\sin\Omega(x) - \sin\Omega_-) \approx k \cdot \delta\Omega(x)$. Then we get:
\begin{eqnarray*}
&& 2 p_y \int\limits_{-\infty}^x (\sin\Omega(x') -\sin\Omega_-) \, dx' \approx \\
&& \approx 2 k \int\limits_{-\infty}^x \delta\Omega(x') \, dx' = 2 \int\limits_{-\infty}^x U(x') \, dx' - \delta \Omega(x).
\end{eqnarray*}
It proves the convergence of $\mathfrak{K}$ at $-\infty$ once $U(x)$ is an integrable function. One can prove by analogy the convergence at $+\infty$. Hence, $\mathfrak{K}$ converges.
{\it Estimation of $\mathfrak{K}$.}
The convergence allows us to introduce some characteristic scale $D(\varepsilon)$ which is a diameter of the convergence domain of $\mathfrak{K}$. Mathematically, for any $\varepsilon > 0$ the number $0 < D(\varepsilon) < \infty$ exists that
$$
\left|\mathfrak{K} - 2 p_y \int\limits_{-D(\varepsilon)/2}^{D(\varepsilon)/2} (\sin \Omega(x) - \sin \Omega_\pm)\, dx \right| < \varepsilon.
$$
We will consider only those cases when we can omit $\mathfrak{K}$ in Eq.~(\ref{DeltaOmega}). Then, let us estimate the order of magnitude. As we can see from the convergence proof, integrals $\mathfrak{K}$ and $G$ converge simultaneously. Then:
\begin{equation}
\mathfrak{K} \sim O(p_y \cdot d),
\label{Kestim}
\end{equation}
where $d$ is the characteristic convergence length of the integral $G$ or, alternatively, the characteristic length of the confinement.
We are ready now to prove the theorem.
\item[{\bf a}]
Combining Eq.~(\ref{omvar}) and Eq.~(\ref{DeltaOmega}) we get:
\begin{equation}
\label{spec}
\arcsin \frac{E}{p_y} = \pi\left( \delta n_G + \frac{\mathfrak{K}}{2\pi} - \frac{1}{2} + n_G-n\right).
\end{equation}
If the condition~(\ref{pycond}) is met, we can omit $\mathfrak{K}$ in Eq.~(\ref{spec}). After that we can set $n = n_G$ because $\arcsin x \in [-\pi/2, \pi/2]$ which finally gives:
$$
\arcsin \frac{E}{p_y} = \pi \left(\delta n_G - \frac{1}{2}\right)
$$
that is equivalent to Eq.~(\ref{Deltalimit}).
\item[{\bf b}]
In order to obtain the wave function, we can naively neglect the influence of the non-linear term of Eq.~(\ref{supereq}) and, hence, the approximate solution reads:
$$
\Omega_0(x) = \Omega_- + 2\int\limits_{-\infty}^x U(x') \, dx'
$$
which coincides with~(\ref{wavef}).
However, this approximation is valid when there is no divergence in the following correction of order of $p_y d$. This correction can be estimated as follows:
\begin{eqnarray*}
&&\Omega_1(x) = \\
&&= -2 p_y \int\limits_{-\infty}^x (\sin\Omega_0(x) - \sin\Omega_-) \, dx' + \Omega_1(-\infty),
\end{eqnarray*}
where we imply that the integral converges. Checking the convergence at $x \to -\infty$:
\begin{eqnarray*}
&&\Omega_1(x) \approx -2 k \int\limits_{-\infty}^x \int\limits_{-\infty}^{x'} U(x'') \, dx'\, dx'' + \Omega_1(-\infty),
\end{eqnarray*}
where this double integral reduces to $\int\limits_{-\infty}^x U(x') x' \, dx'$, which means that we can use the approximate wave function~(\ref{wavef}) only when $x U(x)$ is integrable.
This is unsurprising because for the convergence of $\mathfrak{K}$ at the condition of integrability of $U(x)$ we required the exponential decay of $\Omega(x)$ to $\Omega_-$ at $x \to -\infty$ as it is shown by Eq.~(\ref{domex}). It means that we cannot neglect the dependence of wave function on $k$ and thus, we are not allowed to use the approximate wave function~(\ref{wavef}) if $U(x)$ is integrable but not $x U(x)$. However, the spectrum~(\ref{Deltalimit}) is valid even if $x U(x)$ is non-integrable once $U(x)$ is integrable and the condition~(\ref{pycond}) is met.
Physically, this limit can be understood as a supercritical regime for the confinement $U(x)$. If we consider the case where $U(x)$ is a quantum well with the characteristic depth $U_0$ and width $d$, then, $\pi \delta n_G \lesssim G \sim U_0 \cdot d$ and the condition~(\ref{pycond}) gives $U_0 \gg p_y$ which corresponds to the strong supercritical regime.
Hence, once the condition~(\ref{pycond}) is valid, we get for any integrable potential:
\begin{equation}
\label{DeltaLimit}
\arcsin \frac{E}{p_y} \approx G - \pi \left(n+\frac{1}{2}\right).
\end{equation} \end{itemize} \end{proof}
We did not consider the cases $G = \pi n_G$, $n_G$ is an integer because it requires more fine analysis than represented above.
{\it Zero-energy states}
We are going to compare our results with some recent analytical works on graphene states. As an example, let us consider the condition for the existence of confined modes with zero energy (exactly in Dirac point). Zero-energy confined states and their importance in possible construction of 1D gated structures (waveguides) were discussed thoroughly in~\cite{Hartmann}.
According to Eq.~(\ref{DeltaLimit}), we arrive at the desirable restriction, if Eq.~(\ref{pycond}) is valid: \begin{equation} \label{Zeromodes} G =\pi \left( n + \frac{1}{2} \right), \end{equation} where $n$ is an integer. This constriction means that we cannot have zero-energy confined states at arbitrarily small potential strength $G$. However, at any $G \ne \pi n$ we have at least one bound state.
In~\cite{Hartmann} the analytical solution for zero-energy modes in the gate potential $V(x) = - U_0 /\cosh(x/d)$, $U_0 > 0$, is provided. Taking into account that for this case $G = - \pi U_0 d$ we arrive at the condition for zero-energy mode existence in the limit of small $p_y$: $$ U_0 d = n + \frac{1}{2} $$
where $n$ is a non-negative integer. Hence, we cannot have a confined zero-energy modes once $|U_0 d| < 1/2$ which coincides exactly with the condition obtained analytically in~\cite{Hartmann}.
Thorough analytical study of bound states in the potential \begin{equation}
V(x) = - U_0 /\cosh(x/d)
\label{secant} \end{equation} for non-zero energies has been done in the recent paper~\cite{Quasi}. The authors claim that there is a threshold value of the potential strength $G = \pi U_0 d > \pi/2$ for the first confined state to appear. We suppose that something essential is missing in the work~\cite{Quasi} since this strong statement immediately contradicts the non-relativistic limit and the limit of $\delta$-potential that are developed herein.
Let us now compare our VPM method with one developed by D. A. Stone {\it et al.}~\cite{Portnoi}. They considered another phase function which satisfies a more complex equation. One of the substantial points of their paper is that zero-energy mode exists for arbitrarily small power-law decaying (faster than $1/x$) potentials. And again this statement strongly contradicts with Eq.~(\ref{Zeromodes}). Moreover, their asymptotic analysis resulted in no bound states for the potential~(\ref{secant}) if $p_y < 1/d$. It apparently contradicts with our $\delta$-limit.
Finally, consider the potential $V(x) = U_0 \exp(- |x|/d)$. Zero-energy mode condition was found analytically in~\cite{Portnoi} where the minimal potential strength is stated as $(U_0 d)_{min} = \pi/4$. Our model predicts zero-energy modes when $2 U_0 d = \pi (n+1/2)$ in excellent agreement with analytical solution.
Due to the simplicity of our method, let us calculate the condition of zero-energy mode existence for so-called top-gate potential $V_{t}(x)$ (see reference~\cite{Hartmann}): \begin{equation} \label{top-gate} V_t(x) = \displaystyle \frac{U_0}{2} \ln \left(\frac{x^2 + (h_2 - h_1)^2}{x^2 + (h_2 + h_1)^2}\right) \end{equation} where parameters $h_1 < h_2$ depend on geometry of the gate electrodes. Namely, $h_1$ is a width of the insulator between the graphene plane and so-called back-gate electrode, $h_2$ is a distance between top and back electrodes. Applying Eq.~(\ref{Zeromodes}) one receives the condition of zero mode existence: $$ U_0 h_1 = \frac{1}{2} \left( n + \frac{1}{2}\right) \ge \frac{1}{4}. $$ Notice that this condition does not depend on the bigger parameter $h_2$ which in our case determines the distance between electrodes.
Hence, the $\delta$-potential limit is a simple and powerful tool to study one-particle confined states in arbitrary integrable 1D gate potentials in graphene stripes and it should be included in the analysis of bound states for concrete configuration of the gate potential to avoid possible misconceptions.
\section{Relativistic Levinson theorem}
In this section, we formulate the oscillation theorem in terms of the phase function $\Omega(x)$ as it has been done for the case of massive non-relativistic particles through the analysis of the scattering phase function~\cite{Morse}.
Before we set out the main theorem, we give some properties of the solutions to Eq.~(\ref{supereq}).
\begin{lemma}[of continuity]
Define the following function: $f_{x_0}(x) =\int_{x_0}^{x} U(x') \, dx'$, $|x_0| < \infty$ is some constant. Let $f_{x_0}(x) \in C^k$, where $k$ is a non-negative integer, $C^k$ is the $k$-th class of differentiability. Then every solution of Eq.~(\ref{supereq}) belongs to $C^k$. \end{lemma} \begin{proof}[{\bf Proof}]
We prove this by induction.
\begin{itemize}
\item[{\bf a}] If $k=0$ then $f_{x_0}(x)$ is a continuous function. It is equivalent to the condition:
$ \int_{x}^{x+\epsilon} U(x') \, dx' \to 0$ if $\epsilon \to 0$ at arbitrary $x \in (-\infty, \infty)$.
Then, integrate Eq.~(\ref{supereq}) from $x$ to $x+\epsilon$:
\begin{eqnarray*}
|\Omega(x+\epsilon) - \Omega(x)| = \left| 2\int_{x}^{x+\epsilon} U(x') \, dx' - \right.\\
\left.-2 \int_{x}^{x+\epsilon} (E + p_y \sin \Omega(x')) \, dx'\right| \le \\
2 \left|\int_{x}^{x+\epsilon} U(x') \, dx' \right| + 2 \epsilon (p_y +|E|) \to 0,
\end{eqnarray*}
which confirms the continuity of any solution of Eq.~(\ref{supereq}).
\item[{\bf b}] Assume that the statement of the lemma is true at all $k<n$, where $n$ is positive integer. Let $f_{x_0}(x) \in C^n$. Then prove the Lemma at $k=n$. Differentiate Eq.~(\ref{supereq}) $n-1$ times:
$$
\Omega^{(n)}(x) = f_{x_0}^{(n)}(x) - 2(E+ p_y \sin\Omega(x))^{(n-1)},
$$
where $f_{x_0}^{(n)}(x)$ is continuous by the condition of the lemma. $2(E+ p_y \sin\Omega(x))^{(n-1)}$ is continuous by inductive assumption because it contains derivatives of $\Omega(x)$ not higher than $n-1$. Then $\Omega^{(n)}(x)$ is continuous function, or $\Omega(x) \in C^{(n)}$.
\end{itemize} \end{proof}
We need to make one additional comment. If $f_{x_0}(x)$ is a piecewise-continuous function (this means that $U(x)$ has $\delta$-like singularities at discontinuity points), all solutions of Eq.~(\ref{supereq}) are piecewise-continuous with the same discontinuity points as $f_{x_0}(x)$. In other words, the statement of the {\bf Lemma 1} is valid even if $f_{x_0}(x)$ is a piecewise-continuous function.
\begin{lemma}[of attractors and repellors]
Let $U(x) \to 0$ at $x \to \infty$, $E\in (-p_y, p_y)$. Then:
\begin{itemize}
\item[{\bf a}] All solutions of Eq.~(\ref{supereq}) at infinity come to stationary points of the free motion equation (i.e. with zero potential).
\item[{\bf b}] There are two families of stationary points:
\begin{equation}
\label{attract}
\left\{
\begin{array}{ll}
\Omega_- = -\arcsin (E/p_y) + 2 \pi n\\
\Omega_+ = \arcsin (E/p_y) + 2 \pi \left(n + 1/2\right).
\end{array}
\right.
\end{equation}
\item[{\bf c}] $\Omega_+$ ($\Omega_-$) is an attractor (repellor) at $x\to -\infty$; \\
$\Omega_+$ ($\Omega_-$) is a repellor (attractor) at $x\to +\infty$.
\item[{\bf d}] There are two types of separatrix solutions which are defined by following Cauchy problems:
\begin{equation}
\label{separatrix}
\left\{
\begin{array}{ll}
\Omega_l(x \to -\infty)=\Omega_-\\
\Omega_r(x \to +\infty)=\Omega_+.
\end{array}
\right.
\end{equation}
We call $\Omega_l(x)$ ($\Omega_r(x)$) the left (right) separatrix.
\item[{\bf e}] The bound state problem is equivalent to the degeneracy of two separatrix families $\Omega_l$ and $\Omega_r$.
\end{itemize} \end{lemma}
\begin{proof}[{\bf Proof}]
\begin{itemize}
\item[{\bf a}]
Consider the free motion equation:
\begin{equation}
\Omega'(x) = -2 p_y \left(\sin \Omega(x) + \frac{E}{p_y} \right).
\label{freemotion}
\end{equation}
This equation has stationary points $\Omega(x) \equiv const$ when $\sin \Omega = -E/p_y$. Every solution of Eq.~(\ref{freemotion}) comes to $\Omega_+$ ($\Omega_-$) at $x \to -\infty$ ($x \to +\infty$), where $\Omega_\pm$ are defined according to~(\ref{attract}). Moreover, $\Omega_\pm$ are solutions by itself. However, there are no physical solutions amid the solutions of the free motion equation because it is impossible to satisfy physical boundary conditions~(\ref{asymptotes}).
If we have $U(x) \to 0$, $x \to \infty$, asymptotics of solutions at infinity resemble those of the free motion equation. Thus, {\bf a} is proven.
\item[{\bf b}]
Two families of stationary points of the free motion equation (which present the whole set of attractors and repellors of Eq.~(\ref{supereq})) obviously arise from the equation $\sin \Omega_\pm = -E/p_y$.
\item[{\bf c}]
Let us demonstrate that $\Omega_+$ are repellors at $x \to +\infty$ and attractors at $x \to - \infty$. Consider the solution which comes closely to $\Omega_+$ at some point $x^*$. Represent it in the form $\Omega(x) = \Omega_+ -\epsilon + \delta \Omega(x)$, $\delta\Omega(x^*) = 0$, where $\epsilon$ is a small deviation from $\Omega_+$ at $x = x^*$. Substitute it into Eq.~(\ref{supereq}) and expand $\sin \Omega(x)$ via smallness of $\delta\Omega(x)$ at the vicinity of $x^*$:
\begin{equation}
\delta\Omega'(x) \approx 2 U(x) + 2 k \cdot (\delta \Omega(x) - \epsilon),
\end{equation}
where we accounted that $p_y \cos \Omega_+ = -k$, $k=\sqrt{p_y^2 - E^2}>0$.
The solution with the appropriate boundary condition is:
\begin{eqnarray}
\delta\Omega(x) = 2 \int \limits_{x^*}^x U(x') e^{2 k (x - x')} \, dx' + \nonumber \\
+\epsilon \cdot (1-e^{2k(x-x^*)}).
\label{variations}
\end{eqnarray}
In the region $x > x^*$ both terms in~(\ref{variations}) give exponential divergence at $x \to + \infty$ ($x-x' \ge 0$ under the integral). So, the solution which approaches $\Omega_+$ (up to some arbitrarily small value $\epsilon$) runs away exponentially. It proves the statement that $\Omega_+$ are repellors at $x \to +\infty$.
In the region $x < x^*$, $\delta \Omega(x) \to \epsilon$ exponentially fast ($x-x' \le 0$ under the integral) when $x\to -\infty$ and hence $\Omega(x) \to \Omega_+$. It proves that $\Omega_+$ are attractors at $x \to - \infty$.
We can prove the statement for $\Omega_-$ in {\bf c} by analogy. For this, we just notice the change of sign in exponents because $p_y \cos \Omega_- = k$.
We have to remark that we can finely adjust the constant $\epsilon$ to cancel out the exponential divergence from the integral part of~(\ref{variations}) at $x \to +\infty$. As we can see below, such solutions indeed exist!
\item[{\bf d}]
As it follows from {\bf c}, asymptotes $\Omega_+$ ($\Omega_-$) are unstable at $x \to +\infty$ ($x \to -\infty$). However, we require the solutions to satisfy one of the initial conditions~(\ref{separatrix}). We call such solutions left and right separatrices because they separate all solutions by regions. For example, the separatrix $\Omega_r$ separates solutions which are above and below its value $\Omega_+$ at $+\infty$ according the fact that $\Omega_+$ is a repellor at $+\infty$.
Let us demonstrate that once we fixed one of the conditions~(\ref{separatrix}) it defines the only solution. To be more specific, consider $\Omega_r(x)$. To demonstrate the existence of such solution we need to set $x^* = +\infty$ and $\epsilon = 0$ in the previous item. Then $\Omega_r(x) = \Omega_+ + \delta\Omega_r(x)$ where at $x\to +\infty$ we can write by analogy with~(\ref{variations})
$$
\delta\Omega_r(x) = 2 \int \limits_{+\infty}^x U(x') e^{2 k (x - x')} \, dx',
$$
where $\delta\Omega_r(x) \to 0$ at $x\to +\infty$ which proves the existence of the solution. To show its uniqueness, we suppose two solutions with the same condition $\Omega_{1, 2}(x) \to \Omega_+$ at $x \to +\infty$ and consider its difference $\delta\Omega = \Omega_2 - \Omega_1$ which continuously tends to zero at $x\to +\infty$. While $\delta\Omega$ is small it satisfies the equation:
$$
\delta \Omega' = -2 p_y \cos\Omega_1(x) \cdot \delta\Omega
$$
with solution:
$$
\delta\Omega(x) = \delta\Omega(x_0)\cdot e^{-2 p_y \int\limits_{x_0}^x \cos \Omega_1(x') \, dx'},
$$
where $x \le x_0 \to +\infty$. While $x_0$ is fixed we use the limit relation $p_y \cos\Omega_1(x) \to -k$ at $x \to +\infty$ which exposes the exponential divergence at any non-zero $\delta\Omega(x_0)$, ergo $\delta\Omega(x) \equiv 0$.
It should be emphasized that the uniqueness of solutions with the conditions~(\ref{separatrix}) is not valid if $E = \pm p_y$ since $k = 0$.
\item[{\bf e}] Compare now the boundary conditions~(\ref{asymptotes}) for solutions that correspond to physical states with initial conditions~(\ref{separatrix}) for two families of separatrices. The physical solution must fulfill both conditions which is possible only when two separatrix families merge. Thence, the bound state problem is equivalent to the degeneracy of separatrices of Eq.~(\ref{supereq}).
Notice that the physical solutions are stated by degenerated separatrices, and the corresponding parameter $E$ when the degeneracy occurs is the discrete energy level in a given potential $U(x)$.
\end{itemize} \end{proof}
Remark that we denote as $\Omega_l$, $\Omega_r$ the whole families of separatrices. If we need some particular function from a family, we indicate the dependence from $x$: $\Omega_l(x)$, $\Omega_r(x)$. Again, we use notations $\Omega_+$, $\Omega_-$ to describe the whole families of attractors and repellors if we do not indicate explicitly some particular point from these families.
\begin{lemma}[of boundedness]
Let $U(x)\to 0$ at $x \to \pm \infty$. Let the primitive integral $f_{x_0}(x) = \int_{x_0}^{x} U(x') \, dx'$ of the potential $U(x)$ be a continuous function and the limit $\lim\limits_{x\to \pm \infty} f_{x_0}(x)$ exists (maybe, infinite). Then:
\begin{itemize}
\item[{\bf a}]
Any solution of Eq.~(\ref{supereq}) is a bounded function for any parameter $E \in (-p_y, p_y)$.
\item[{\bf b}]
If $|\lim\limits_{x\to \pm \infty} f_{x_0}(x)| < \infty$, then all solutions of Eq.~(\ref{supereq}) are bounded functions for any parameter $E \in [-p_y, p_y]$. \end{itemize} \end{lemma} \begin{proof}[{\bf Proof}]
\begin{itemize}
\item[{\bf a}]
First, consider the situation when $k \ne 0$ or $E \in (-p_y, p_y)$.
Continuity of $f_{x_0}(x)$ results in $\Omega(x)$ being a continuous function as to {\bf Lemma 1}.
Suppose that $\Omega(x)$ diverges at $+\infty$. From continuity, we always can find an arbitrarily large positive $x_0$ where $p_y \cos \Omega(x_0) = k >0$. We expand $\Omega(x)$ at the vicinity of $x_0$: $\Omega(x) = \Omega(x_0) + \delta \Omega(x)$.
Up to the first order of $\delta\Omega$ we have:
\begin{equation}
\delta\Omega'(x) = 2 U(x) - 2 k \cdot \delta\Omega(x),
\label{lemma3pr}
\end{equation}
which yields the solution:
\begin{equation}
\delta\Omega(x) = 2\int \limits_{x_0}^x U(x') e^{-2 k (x - x')} \, dx'.
\label{sollem3}
\end{equation}
We clearly see that $\delta\Omega(x)$ converges at $x \to +\infty$ even at arbitrarily small $k > 0$. Hence, we arrived at the contradiction with our initial assumption of the unboundedness of $\Omega(x)$ at $+\infty$.
By analogy, one can prove the boundedness of any solution of Eq.~(\ref{supereq}) at $x \to -\infty$. Here we will choose an arbitrary large negative $x_0$ where $p_y \cos \Omega(x_0) = -k$.
Notice that $\delta\Omega(+\infty) = 0$; we integrate Eq.~(\ref{lemma3pr}) and substitute~(\ref{sollem3}) into the right-hand side. It yields:
\begin{eqnarray*}
&&\int\limits_{x_0}^{+\infty} \delta\Omega(x) \, dx = \\
&&= 2 \int\limits_{x_0}^{+\infty} \int \limits_{x_0}^x U(x') e^{-2k(x-x')} \, dx\, dx'=\frac{f_{x_0}(+\infty)}{k}.
\end{eqnarray*}
On the other hand, the direct integration of Eq.~(\ref{lemma3pr}) results in:
$$
\delta\Omega(+\infty) = 2 f_{x_0}(+\infty) - 2k \int\limits_{x_0}^{+\infty} \delta\Omega(x) \, dx.
$$
Hence, $\delta\Omega(+\infty) = 0$ or $\Omega(+\infty) = \Omega(x_0)$. This result is not surprising because we intentionally chose $x_0$ in that way to satisfy $\Omega(x_0) = \Omega_-$ which is attractor at $x \to +\infty$.
\item[{\bf b}]
If $f_{x_0}(x)$ has finite limits at $x \to \pm \infty$, one can show that solutions of Eq.~(\ref{supereq}) are bound on the closed interval $E \in [-p_y, p_y]$. To show this, we need to check what happens on the boundaries of the continuum when $E= \mu p_y$, $\mu = \pm 1$, $k=0$.
As in item {\bf a}, we assume that $\Omega(x)$ diverges at $x \to +\infty$, thus, we can write $\Omega(x) = \Omega(x_0) + \delta\Omega(x)$, $\sin\Omega(x_0) = \mu$ where $x_0$ can be an arbitrarily large positive number. In Eq.~(\ref{lemma3pr}) we omitted summands of order $\delta\Omega^2$ and higher because $k\ne0$. In this case we have to account for the first non-zero term that is quadratic in $\delta\Omega$:
\begin{equation*}
\delta\Omega'(x) = 2 U(x) - \mu p_y \delta\Omega^2(x).
\end{equation*}
This equation resembles that of a non-relativistic limit with zero non-relativistic energy.
There are three possible scenarios of the behavior at $+\infty$. The first one, $\delta \Omega^2(x) \sim U(x)$, $x \to +\infty$, gives explicit convergence of $\delta\Omega$ since $U(x) \to 0$, $x \to +\infty$. The second one corresponds to $\delta\Omega^2(x) \sim \delta\Omega'(x)$ which provides the convergence $\delta\Omega \sim 1/x$. The last situation is $\delta\Omega'(x) \sim U(x)$ which gives the convergence if and only if $f_{x_0}(x)$ converges at infinity.
Hence, any solution of Eq.~(\ref{supereq}) is bounded at any parameter $E \in [-p_y, p_y]$ as soon as $f_{x_0}(x)$ is continuous and converges at infinity.
\end{itemize} \end{proof}
As it can be seen from {\bf Lemma 2}, we are interested in the separatrix solutions because only these solutions are related to physical ones. For all further discussions we choose the family of left separatrices $\Omega_l$. We are going to show that the total variance: $$ \Delta\Omega_l(E) = \Omega_l(+\infty) - \Omega_l(-\infty) $$ as a function of energy contains the full information of the discrete spectrum. It is stated in the following
\begin{Lev}[Levinson] Let $f_{x_0}(x)$ be a continuous function which converges at infinity, $E \in [-p_y, p_y]$. Then:
\begin{itemize}
\item[{\bf a}]
$\Delta \Omega_l(E)$ is a bounded function on the interval $E \in [-p_y, p_y]$.
\item[{\bf b}]
$\Delta \Omega_l(E)$ is a multiple of $2 \pi$ if $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$, $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is a discrete specter of $U(x)$ at given $p_y$.
\item[{\bf c}]
Any $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is a point of continuity of $\Delta \Omega_l(E)$.
\item[{\bf d}]
$\Delta \Omega_l(E)$ has finite jumps of $-2 \pi$ at every point $E_d \in \mathop{\mathrm{Spec}}\nolimits(U, p_y)$:
\begin{equation}
\Delta\Omega_l(E_d+0)-\Delta\Omega_l(E_d -0) = -2\pi.
\label{djump}
\end{equation}
\item[{\bf e}]
The total number $N_d(p_y)$ of discrete levels of $U(x)$ at any given $p_y > 0$ is defined by:
\begin{equation}
\label{Totallevels}
N_d(p_y)=\frac{\Delta\Omega_l(-p_y)-\Delta\Omega_l(p_y)}{2 \pi}.
\end{equation}
\end{itemize} \end{Lev}
\begin{proof}[{\bf Proof}] \begin{itemize}
\item[{\bf a}] We know from {\bf Lemma 3} that, under conditions of the theorem, $\Omega_l(x)$ is a bounded function on $x \in (-\infty, \infty)$ at any parameter $E \in [-p_y, p_y]$. In other words $\Delta \Omega_l(E)$ is finite for any $E \in [-p_y, p_y]$ or $\Delta \Omega_l(E)$ is bounded function of $E$.
\item[{\bf b}] According to {\bf Lemma 2, e)}, two families $\Omega_l$, $\Omega_r$ of separatrices merge if and only if the parameter $E$ corresponds to some discrete energy level. Let $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$. Therefore $\Omega_l$ and $\Omega_r$ are disjoint families; $\Omega_l(x)$ starts from some $\Omega_-$ at $x=-\infty$ and comes to, perhaps, some other $\Omega_-$ from the family at $x=+\infty$. Otherwise $\Omega_l(x)$ must tend to $\Omega_+$ at $+\infty$ resulting in $\Omega_l(x) = \Omega_r(x)$ which violates our assumption that $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$. Hence, $\Delta\Omega_l(E)$ is a multiple of $2 \pi$.
\item[{\bf c}] Let $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$ where it is natural to assume that $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is a discrete set. Then some $\delta$-vicinity of $E$ is disjoint with $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$, $\delta > 0$. Let us consider how $\Omega_l(x, E)$ changes with small variation of the parameter $E$: $$ \delta\Omega_l(x, E, \epsilon) = \Omega_l(x, E+\epsilon) - \Omega_l(x, E), $$
where small $0 <|\epsilon| < \delta$. In contrast with the previous consideration where $E$ was fixed, we indicate here $E$ among variables of functions. Subtracting Eq.~(\ref{supereq}) for $\Omega_l(x, E+\epsilon)$ and $\Omega_l(x, E)$, we arrive at the equation for the variation function: \begin{equation} \delta\Omega_l' \approx - 2 \epsilon - 2 p_y \cdot \cos \Omega_l(x, E) \cdot \delta\Omega_l. \label{leftsep} \end{equation} Remark that the initial condition depends on $\epsilon$ because: \begin{equation} \delta \Omega_l(-\infty, E, \epsilon) \! = \! \Omega_-(E+\epsilon) \! - \! \Omega_-(E) \!\approx \!-\frac{\epsilon}{k}. \label{incond} \end{equation} The solution reads: \begin{eqnarray} &&\delta \Omega_l(x, E, \epsilon) = \nonumber\\ && = - 2 \epsilon \int \limits_{-\infty}^x e^{2 p_y \int_x^y \cos \Omega_l(y', E) \, dy'} \, dy. \label{varepsilon} \end{eqnarray}
First, let's demonstrate that~(\ref{varepsilon}) meets the initial condition~(\ref{incond}). According to~(\ref{separatrix}), we may approximate $p_y \cos\Omega_l(y', E) \to p_y \cos\Omega_- = k$ at $x \to -\infty$ because $y \le y' \le x$. Hence, at $x \to -\infty$ we see that: $$ \delta \Omega_l(-\infty, E, \epsilon) \to - 2 \epsilon \int \limits_{-\infty}^x e^{2 k (y -x)} \, dy = - \epsilon/k. $$
Now we are ready to show the convergence of~(\ref{varepsilon}) at $+\infty$ and that $\delta\Omega_l(+\infty, E, \epsilon) = -\epsilon/k$. First, divide~(\ref{varepsilon}) into two parts: the first part is the $y$-integral where $-\infty < y < x_0$, the second part is the $y$-integral where $x_0 < y < x$. $x_0 < x$ is big positive number such that we can use the approximation $p_y \cos \Omega_l(y', E) \approx p_y \cos \Omega_- = k$ while $y' > x_0$. The first part can be estimated at $x \to +\infty$ as follows: \begin{eqnarray*} &&-2 \epsilon \int\limits_{-\infty}^{x_0} e^{2 p_y (\int_x^{x_0} +\int_{x_0}^y)\cos \Omega_l(y', E) \, dy'} \, dy \approx \\ &&-2 \epsilon \int\limits_{-\infty}^{x_0} e^{2 p_y \int_{x_0}^y\cos \Omega_l(y', E) \, dy'} \, dy \cdot e^{-2 k (x - x_0)} = \\ && = \delta\Omega_l(x_0, E, \epsilon) \cdot e^{-2 k (x - x_0)} \to 0. \end{eqnarray*} The second part gives the desirable limit $\delta\Omega_l(+\infty, E, \epsilon)$: \begin{eqnarray*}
&&-2 \epsilon \int\limits_{x_0}^x e^{2 p_y \int_x^y\cos \Omega_l(y', E) \, dy'} \, dy \approx \\
&&\approx -2 \epsilon \int\limits_{x_0}^x e^{2 k (y - x)} \, dy \to -\frac{\epsilon}{k}. \end{eqnarray*} Hence, $\delta\Omega_l(+\infty, E, \epsilon) = \delta\Omega_l(-\infty, E, \epsilon) = -\epsilon/k + O(\epsilon^2)$. We remark the equality of values of $\delta\Omega_l$ at $\pm \infty$ not just up to order of $\epsilon^2$ because we have proven here that the difference tends to zero with $\epsilon$. But according to item {\bf b} of this theorem, the difference must be a multiple of $2 \pi$ whence the only one opportunity is possible. Finally, we conclude that: \begin{eqnarray*} &&\Delta\Omega_l(E+\epsilon) - \Delta\Omega_l(E) = \\ &&= \delta\Omega_l(+\infty, E, \epsilon) - \delta\Omega_l(-\infty, E, \epsilon) = 0. \end{eqnarray*}
Hence, we proved that any $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is the point of continuity of the function $\Delta\Omega_l(E)$. {\bf We also proved that } $\Delta\Omega_l(E)$ {\bf is a piecewise-constant function with only possible discontinuity points from} $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$.
We emphasize that the statement of this item is true even for the boundaries of continuum where $E = \pm p_y$ since $E = \pm p_y$ are not limit points of $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ (see the {\bf Remark 1}). For example, for $E = p_y$ we take $$ \delta \Omega_l(x, E=p_y, \epsilon) = \Omega_l(x, p_y - \epsilon) - \Omega_l(x, p_y), $$ where $\epsilon \approx k^2/(2 p_y) \to +0$. Then the condition~(\ref{incond}) is valid because $\epsilon/k \approx k/(2 p_y) \to 0$.
\item[{\bf d}] Now we understand the behavior of $\Delta\Omega_l(E)$ when $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$. In this item we consider the situation when $E = E_d \in \mathop{\mathrm{Spec}}\nolimits(U, p_y)$ where we assume that $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is a discrete set or each element is an isolated point. As it follows from {\bf Lemma 2, e)}, two separatrix families merge when $E=E_d$. We call these merged separatrices as $\Omega_d$ family.
$E_d$ is an isolated point of $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$. Then $\delta > 0$ exists such that $\delta$-vicinity of $E_d$ does not contain any other points from $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ except $E_d$. Let us consider the variation function: $$ \delta\Omega_l(x, E_d, \epsilon) = \Omega_l(x, E_d+\epsilon) - \Omega_d(x, E_d), $$
where $\epsilon$ can be arbitrarily small, $0 <|\epsilon| < \delta$ . Afterwards, we repeat the procedure from item {\bf c} of the theorem which gives exactly the same initial condition~(\ref{incond}) and in Eq.~(\ref{leftsep}) we need to substitute $\Omega_l(y', E) \to \Omega_d(y', E_d)$. Thence the approximate solution for $\delta\Omega_l(x, E_d, \epsilon)$ reads: \begin{eqnarray} &&\delta \Omega_l(x, E_d, \epsilon) = \nonumber\\ && = - 2 \epsilon \int \limits_{-\infty}^x e^{2 p_y \int_x^y \cos \Omega_d(y', E_d) \, dy'} \, dy. \label{vardepsilon} \end{eqnarray}
But analysis of Eq.~(\ref{vardepsilon}) at $x \to +\infty$ gives different result from those of Eq.~(\ref{varepsilon}). The reason is that $\Omega_d(x, E_d)$ comes to $\Omega_+$ at $x \to +\infty$ as per the conditions~(\ref{asymptotes}). This gives $p_y \cos \Omega_d(+\infty, E_d) = p_y \cos\Omega_+ = -k$d which results in the exponential divergence of $\delta\Omega_l(x, E_d, \epsilon)$ at $x \to +\infty$ for any $|\epsilon| > 0$. Formally, this divergence indicates instability of the solution $\Omega_d(x, E_d)$ towards infinitely small variations from the parameter $E_d$. This conclusion is already obvious because we know that at $E = E_d + \epsilon$ we have two disjoint families of separatrices and our separatrix $\Omega_l$ tends to $\Omega_-$ at $x \to +\infty$.
The non-trivial conclusion which can be drawn from~(\ref{vardepsilon}) is that: \begin{equation} \mbox{sign}\left( \delta \Omega_l \right)= - \mbox{sign}( \varepsilon). \label{signs} \end{equation} We are going to show that it leads to~(\ref{djump}).
We can use the approximate solution~(\ref{vardepsilon}) at the region $x < R$ if the condition $\delta\Omega_l(x<R, E_d, \epsilon) \ll 1$ is met. Fix some small value of $\delta\Omega_l$: $$ \delta\Omega_l(R, E_d, \epsilon) \equiv \alpha. $$ It means that $R$ is a function of two parameters $\alpha$ and $\epsilon$ and $R(\alpha,\epsilon) \to +\infty$ at fixed $\alpha$ and $\epsilon \to 0$. Introduce the following variance: $$
\delta\Omega_d=\Omega_d(R(\alpha, \epsilon), E_d) - \Omega_+, $$ where $\delta\Omega_d \to 0$ at $R \to +\infty$. Finally, we have for the left separatrix: \begin{eqnarray*} &&\Omega_l(R(\alpha,\epsilon), E_d+\epsilon) = \Omega_+ + \delta\Omega_d + \alpha, \end{eqnarray*} where $\alpha$ is fixed and $\delta\Omega_d \to 0$ at $\epsilon \to 0$ or equivalently: $$ \Omega_l(R(\alpha,\epsilon), E_d+\epsilon) \to \Omega_+ + \alpha $$ at $\epsilon \to 0$ and arbitrarily small but fixed $\alpha$. According to the definition of $\alpha$ and Eq.~(\ref{signs}), we get $$ \mbox{sign}(\alpha) = - \mbox{sign}(\epsilon). $$ It means that at $\epsilon > 0$ ($\epsilon < 0$) the left separatrix $\Omega_l(R, E_d+\epsilon) < \Omega_+$ ($\Omega_l(R, E_d+\epsilon) > \Omega_+$) at $R \to +\infty$ and ergo $\Omega_l(x, E_d+\epsilon)$ falls onto the asymptote $\Omega_-$ which is right under (above) the asymptote $\Omega_+ = \Omega_d(+\infty, E_d)$. Thence: $$ \Omega_l(+\infty, E_d + 0) - \Omega_l(+\infty, E_d - 0) = - 2 \pi $$ or equivalently: $$ \Delta\Omega_l(E_d+0)-\Delta\Omega_l(E_d -0) = -2\pi. $$ We used the fact that here $\Omega_l(-\infty, E_d+0)= \Omega_l(-\infty, E_d-0)$.
One can show by analogy that the right separatrix experiences jumps with the same sign: $$ \Delta\Omega_r(E_d+0)-\Delta\Omega_r(E_d -0) = -2\pi. $$ In this sense, the right separatrix does not give any additional information about the discrete spectrum.
\item[{\bf e}] We proved that the function $\Delta\Omega_l(E)$ is a bounded piecewise-constant function which experiences final jumps of $-2 \pi$ at every point $E_d$ of discrete spectrum of the confinement $U(x)$. $\Delta\Omega_l(E)$ is continuous at any other points where $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$.
It allows us to calculate the total number of discrete levels as the difference of $\Delta\Omega_l(E)$ on the ends of the interval $[-p_y, p_y]$ which immediately gives Eq.~(\ref{Totallevels}).
However, we understand $\Delta\Omega_l(\pm p_y)$ only in the sense of the limit relation $\Delta\Omega_l(\pm p_y) = \lim \limits_{\epsilon \to +0} \Delta\Omega_l(\pm (p_y- \epsilon))$ because separatrices are not well defined at the boundaries of the continuum as to {\bf Lemma 2}.
\end{itemize} \end{proof}
\begin{remark1}[for the Levinson Theorem] We need to remark that assumptions made in the head of the Levinson theorem provide that $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ is discrete set. Indeed, assume that $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ has one limit point $E_0 \in [-p_y, p_y]$. It means that infinitesimal vicinity of this point contains an infinite number of isolated points from $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$. But for any isolated point, the item {\bf d} of the theorem is valid which leads to $\Delta\Omega_l(E \to E_0) \to \infty$; this contradicts with the item {\bf a} of the theorem of boundedness of this function for any $E \in [-p_y, p_y]$. Hence, $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ does not contain limit points. \end{remark1}
\begin{remark1}[for the Levinson Theorem]
Even if $|\lim \limits_{x\to\pm \infty} f_{x_0}(x)| = \infty$, all proofs and statements of the Theorem are valid for open interval $E \in (-p_y, p_y)$ because $k = \sqrt{p_y^2 - E^2} > 0$. However, at least one of the points $E = \pm p_y$ is limit point of $\mathop{\mathrm{Spec}}\nolimits(U, p_y)$ which makes $\Delta\Omega_l(E)$ unbound on the closed interval $E\in [-p_y, p_y]$. \end{remark1}
\begin{remark1}[for the Levinson Theorem]
One can get the number of discrete levels between any two given energies $|E_{1, 2}| \le p_y$, $E_{1, 2} \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$:
\begin{equation}
\label{Numlevels}
N_d(p_y, E_1, E_2)=\left|\frac{\Delta\Omega_l(E_2)-\Delta\Omega_l(E_1)}{2 \pi} \right|.
\end{equation} \end{remark1}
Hence, the function $\Delta \Omega_l(E)$ plays the same role as the scattering phase in the non-relativistic theory. In other words, the theorem represents the relativistic Levinson theorem for the 2D Dirac equation with the 1D potential. \\
{\it Example for $\delta$-potential}
Finally, we give an example for the simple case of the $\delta$-potential $U(x) = G \cdot \delta(x)$. Let us demonstrate that the total number of discrete levels $N_d(p_y) = 1$ at any $p_y \ne 0$ and $G \ne \pi n$, $n$ is integer, $N_d$ is defined by Eq.~(\ref{Totallevels}). We need to consider Eq.~(\ref{supereq}) only at $E= \pm p_y$.
All solutions of Eq.~(\ref{supereq}) are constructed from solutions of the free motion equation~(\ref{freemotion}) separately at $x<0$ and $x>0$ with the matching condition \begin{equation}
\Omega(+0) = \Omega(-0) + 2 G. \end{equation}
We first analyze the solutions of Eq.~(\ref{freemotion}). If $E = p_y$, then we have $\Omega'(x) = -2 p_y (1 + \sin \Omega) \le 0$ and $\Omega'(x) = 0$ only for the case of stationary points $\Omega_0 \equiv \Omega_\pm = -\pi/2 + 2 \pi n$. Hence, all non-stationary solutions of Eq.~(\ref{freemotion}) decrease strictly monotonically from some stationary point $\Omega_0 + 2\pi$ at $x = -\infty$ to $\Omega_0$ at $x = +\infty$. Notice that two families of stationary points merge at $E = \pm p_y$.
In the case $E = - p_y$ all non-stationary solutions of Eq.~(\ref{freemotion}) increase strictly monotonically from some stationary point $\Omega_0 - 2\pi$ at $x = -\infty$ to $\Omega_0$ at $x = +\infty$.
Represent the confinement strength in the following form: $$ G = \pi (n_G + \delta n_G), $$ where $n_G$ is integer and $\delta n_G \in(0, 1)$. Then: $$ \Omega_l(x < 0, \pm p_y) = \Omega_-(\pm p_y) $$ and $$\Omega_l(+0, \pm p_y) = \Omega_-(\pm p_y) + 2 \pi n_G + 2 \pi \delta n_G, $$ where $\Omega_0 = \Omega_- + 2 \pi n_G$ is stationary point and $2 \pi \cdot \delta n_G \in (0, 2\pi)$ which means that $\Omega_l(x, \pm p_y)$ at $x>0$ comes along some non-stationary solution which decreases (increases) at $E=p_y$ ($E = -p_y$), ergo $\Omega_l(+\infty, p_y) = \Omega_0$ ($\Omega_l(+\infty, -p_y) = \Omega_0 + 2\pi$) at $E=p_y$ ($E = -p_y$). Equivalently, $\Delta\Omega_l(p_y) = 2 \pi n_G$ and $\Delta\Omega_l(-p_y) = 2 \pi n_G + 2 \pi$. Hence, $N_d(p_y) = 1$.
\section{Geometrical interpretation of the relativistic Levinson theorem}
The problem of bound states in graphene stripes can be analyzed similarly to what happens in mechanical autonomous systems. Let us consider the following system of equations: \begin{equation} \label{Autosys} \left\{ \begin{array}{ll} U'(x) = G(U) \\ \Omega'(x)=2 \left(U(x)- E\right) -2 p_y \sin {\Omega(x)}, \end{array} \right. \end{equation} where the second equation here is just Eq.~(\ref{supereq}). We may consider that Eq.~(\ref{Autosys}) represents integral curves of some vector field $$ {\bf F}(U, \Omega) = { G(U) \choose 2 \left(U - E\right) -2 p_y \sin {\Omega} }, $$ whereas the coordinate $x$ is just some parametrization of these curves. Though the system~(\ref{Autosys}) is not Hamiltonian as in usual mechanics, it is still an autonomous system of differential equations and, therefore, it can be analyzed in terms of the phase trajectories in so-called phase space $\mathfrak{D}$. In our case, the phase space $\mathfrak{D}$ is the $(U, \Omega)$-stripe: $$
\mathfrak{D} = \{(U, \Omega)| U \in [\inf\limits_{x \in \mathbb R} U(x), \sup\limits_{x \in \mathbb R} U(x)], \Omega\in \mathbb R \}, $$ where $\mathbb R = (-\infty, +\infty)$.
However, our system~(\ref{Autosys}) is more complicated than usual autonomous systems. To see this, notice that the function $G(U)$ is different for each interval of monotonicity $I_j = [x_{j-1}, x_j]$ of $U(x)$. It means that we have different maps for each $I_j$ and we need to match these maps continuously. In other words, instead of one autonomous system we have the whole chain of systems: \begin{equation} {\bf F}_j(U, \Omega) = {U'(x) \choose \Omega'(x)} = {G_j(U) \choose 2 \left(U - E\right) -2 p_y \sin {\Omega}} \label{jtheq} \end{equation} which are autonomous on the corresponding intervals $I_j$, $x\in I_j$ is some parametrization, and ${\bf F}_j(x_j) = {\bf F}_{j+1}(x_j)$. All trajectories of the field ${\bf F}_j$ fill the whole stripe: $$
\mathfrak{D}_j = \{(\Omega , U)| U \in [\inf\limits_{x \in I_j} U(x), \sup\limits_{x \in I_j} U(x)], \Omega\in \mathbb R \}. $$
Let us formulate the following \begin{lemma}[of stationary points]
Let $U(x) \in C^1$ have a finite number $N$ of monotonicity intervals $I_j = [x_{j-1}, x_j]$, $x_0 = -\infty < x_1 < \dots < x_{N-1 }< x_N = +\infty$. Let $U(x)$ be a strictly monotonic function on each $I_j$. Let $U(x) \to 0$ at $x \to \pm \infty$. Then:
\begin{itemize}
\item[{\bf a}]
$U'(x) \to 0$ at $x \to \pm \infty$.
\item[{\bf b}]
Functions $G_j(U)$ are definite on corresponding intervals $I_j$, $j = 1, \dots, N$ and $G_1(0) = G_N(0) = 0$.
\item[{\bf c}]
The number of stationary points of $j$-th Eq.~(\ref{jtheq}) is exhausted by the following series:
$$
\left( U_\sigma, \arcsin \left(\frac{U_\sigma - E}{p_y} \right) + 2\pi n \right)
$$
or
$$
\left( U_\sigma, \pi -\arcsin \left(\frac{U_\sigma - E}{p_y} \right) + 2 \pi n\right)
$$
where $n$ is integer, $|U_\sigma - E| \le p_y$ and $G_j(U_\sigma)=0$.
\end{itemize} \end{lemma}
\begin{proof}[{\bf Proof}]
\begin{itemize}
\item[{\bf a}]
It is straightforward from the monotonic behavior of $U(x)$ at infinity and $U(x) \to 0$ at $x \to \infty$.
\item[{\bf b}]
$U(x)$ is strictly monotonic on each $I_j$, therefore an inverse function exists: $x_j(U)$. Thereby we get $G_j(U) = U'(x_j(U))$.
We know that $I_1 = (-\infty, x_1]$, $I_N = [x_{N-1}, +\infty)$ and $U'(x) \to 0$ at $x \to \pm \infty$ where $U(x) \to 0$. It immediately yields: $G_1(0) = \lim\limits_{x\to - \infty} U'(x) = 0$ and $G_N(0) = \lim\limits_{x\to +\infty} U'(x) = 0$.
\item[{\bf c}]
This statement follows from the solution of the equation: $$ {\bf F}_j (U, \Omega) = 0.$$
\end{itemize} \end{proof}
Further we call the whole chain of connected maps for ${\bf F}_j(U, \Omega)$ as ${\bf F}(U, \Omega)$ where each trajectory from $\mathfrak{D}$ corresponds to some solution of Eq.~(\ref{Autosys}). The properties of these trajectories are formulated in the \begin{Poincare}[of Poincare indeces]
Let all restrictions of {\bf Lemma 4} be valid. Let us consider the following mapping
$\mathfrak{D} \to \mathfrak{R}$ by the rule:
\begin{equation}
\left\{
\begin{aligned}
X(U, \Omega) = (U + a \cdot p_y) \cos\Omega, \\
Y(U, \Omega) = (U + a \cdot p_y) \sin\Omega,
\end{aligned}
\right.
\label{map}
\end{equation}
where $+\infty > a \cdot p_y > -\inf \limits_{x \in \mathbb R} U(x)$ is some parameter, $E \in (-p_y, p_y)$, $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$. Then:
\begin{itemize}
\item[{\bf a}]
All stable trajectories of the vector field ${\bf P}(X, Y) = {\bf F}(U(X, Y), \Omega(X, Y))$, $(X, Y) \in \mathfrak{R}$
are open.
All unstable trajectories (separatrices) are closed.
\item[{\bf b}]
In the previous section we introduced the total variance $\Delta\Omega_s(E)$, $s$ indicates left or right separatrix. The relation $\Delta\Omega_s(E)/(2 \pi)$ equals to integer number $\mathfrak{p}$ of full rotations of corresponding closed trajectory in the phase space $\mathfrak{R}$:
$$
\Delta\Omega_s(E) = 2 \pi \mathfrak{p}_s.
$$
$\mathfrak{p}_s$ is the Poincare index of closed trajectory.
\end{itemize} \end{Poincare} \begin{proof}[{\bf Proof}]
\begin{itemize}
\item[{\bf a}]
The mapping~(\ref{map}) is the mapping of stripe $\mathfrak{D}$ to the ring $\mathfrak{R}$ where all points $(U, \Omega + 2 \pi n)$, $n$ is integer, are identified.
The asymptotic behavior of stable trajectories of the field ${\bf P}(X, Y)$ is referred to stable solutions of Eq.~(\ref{supereq}) which start from attractor $\Omega_+$ at $x \to -\infty$ and finish to attractor $\Omega_-$ at $x \to +\infty$ as to {\bf Lemma 2}. Accounting that $U(x) \to 0$ at $x \to \pm \infty$, we conclude that stable trajectories in $\mathfrak{R}$ space start from the point
$$
P_i = (-a \cdot k, -a \cdot E)
$$
because $X_i = a \cdot p_y \cos\Omega_+$, $Y_i = a \cdot p_y \sin \Omega_+$;
and finish by another point
$$
P_f = (a \cdot k, -a \cdot E)
$$
because $X_f = a \cdot p_y \cos\Omega_-$, $Y_f = a \cdot p_y \sin \Omega_-$. If $E\in (-p_y, p_y)$ then $k > 0 $ and $P_f \ne P_i$. This means that stable trajectories are open.
According to~(\ref{separatrix}), if $E \notin \mathop{\mathrm{Spec}}\nolimits(U, p_y)$, $\Omega_l$ ($\Omega_r$) starts and finishes on the asymptotes from the same family: $\Omega_-$ for $\Omega_l$ and $\Omega_+$ for $\Omega_r$. Then, $P_i$ and $P_f$ are identical for them or, equivalently, their trajectories in $\mathfrak{R}$ space are closed.
\item[{\bf b}]
It follows from the Levinson Theorem that $\Delta \Omega_l(E) = 2 \pi \mathfrak{p}_l$ where $\mathfrak{p}_l$ is integer. But from the continuity of $\Omega_l(x)$ we conclude that $\mathfrak{p}_l$ is the number of full rotations of the closed trajectory corresponding to the separatrix $\Omega_l$ in $\mathfrak{R}$ space. In other words, $\mathfrak{p}_l$ is the Poincare index of this closed trajectory~\cite{Poincare}.
\end{itemize} \end{proof}
\begin{figure}
\caption{The vector field ${\bf F}(E = p_y)$, $p_y = 0.1$, $U_0 = 1$ on the interval $I_1 = (-\infty, 0)$. The trajectory $(U, \Omega_l(x_1(U)))$ corresponding to the separatrix $\Omega_l(x)$ (red streamline) starts from the initial (red) point $(U = 0, \Omega = -\pi/2)$ and ends when $U = -U_0 = -1$ (black point). The distance between red points is equal to $2 \pi$.}
\label{Vec1}
\end{figure}
\begin{figure}
\caption{The vector field ${\bf F}(E = p_y)$, $p_y = 0.1$, $U_0 = 1$ on the interval $I_2 = (0, +\infty)$. The trajectory $(U, \Omega_l(x_2(U)))$ corresponding to the separatrix $\Omega_l(x)$ (red streamline) starts from the black point which provides the continuity of $\Omega_l(x)$ at $x = 0$ and ends at the red point $(U = 0, \Omega = -9 \pi/2)$. The distance between red points is equal to $2 \pi$.
}
\label{Vec2}
\end{figure}
Here we present a simple example of the spectral analysis for the Lorentzian shaped confinement $$ U(x) = - U_0/(x^2+1). $$ We are going to plot the vector field ${\bf F}(U, \Omega)$ and calculate the number of bound states at some particular $p_y$ and $U_0$.
First, we need to find $G_j(U)$ for each interval of monotonicity $I_1 = (-\infty, 0)$ and $I_2 = (0, + \infty)$: $$ G_{n}(U) = (-1)^n \frac{2U^2}{U_0} \sqrt{-\frac{U_0}{U} - 1} $$ for the interval $I_n$, $n = \{1, 2\}$, $U \in [-U_0, 0]$.
Then we set the parameters $p_y = 0.1$, $U_0 = 1$. In order to find the total number of confined modes, we apply Eq.~(\ref{Totallevels}). We need to plot the phase portrait only for two energies $E = \pm p_y$. Pictures (Fig.~\ref{Vec1}--\ref{Vec2}) of the vector field ${\bf F}(E = p_y)$ show the approximate trajectory $(U, \Omega_l(x(U)))$ (red line) for two intervals $I_{1, 2}$. We chose the point $(U = -10^{-6}, \Omega = - \pi/2 + 0.05)$ as the initial condition for the trajectory $(U, \Omega_l(x_1(U)))$ on the interval $I_1$. Matching trajectories corresponding to the intervals $I_1$ and $I_2$ (black points on Fig.~\ref{Vec1}--\ref{Vec2}) we finally obtain the variance $\Delta \Omega_l(p_y) = -4 \pi$. Analogically, drawing such pictures for $E = -p_y$ we get $\Delta \Omega_l(p_y) = 0$. Eq.~(\ref{Totallevels}) yields $N_d(p_y) = 2$ confined energy levels for $p_y = 0.1$.
We have to remark that initial condition for $\Omega_l$ must be perturbed from ideal point $(U=0, \Omega = \Omega_-)$ because it is stationary point of Eq.~(\ref{Autosys}) according to {\bf Lemma 4}. However, the result is stable towards little shaking of initial conditions because of the stability of the Poincare index or so-called topological charge.
\section{Conclusions} The variable phase method has been developed herein for the electrostatically confined 2D massless Dirac-Weyl particles such as electrons in graphene devices. The desirable phase function $\Omega(x)$ appears as the phase between two chiral states whose superposition yields the wave function of the confined state. Besides the well-known non-relativistic and semi-classical limits, it has been shown that confined states with small $p_y$ (see the condition~(\ref{pycond})) are successfully described in the so-called $\delta$-potential limit that is valid for every integrable potential $U(x)$. The relativistic Levinson theorem has then been formulated and proved for the variance $\Delta \Omega_l(E)$ of the separatrix $\Omega_l(x)$ of Eq.~(\ref{supereq}). As a consequence of the theorem, the number of confined modes with given $p_y$ has been derived. Finally, the geometrical approach to find the function $\Delta \Omega_l(E)$ has been suggested.
We note that this paper is dedicated exceptionally to the discrete part of the specter. The developed approach can be extended to analyze half-bound and quasi-bound states where the last ones are important for better understanding of supercriticality.
\section{Appendix A: unambiguous solution of the $\delta$-potential}
One can find in the literature that $U(x) = G \delta(x)$ does not have definite solutions for Dirac-Weyl equation~\cite{Calkin}--\cite{McKellar}. This problem arises from the fact that the wave function is discontinuous at $x = 0$ and it results in the ambiguous integral of the type $$ \int\limits_{-\epsilon}^\epsilon \delta(x) \theta(x) \, dx $$ which takes an arbitrary value from the segment $[0, 1]$, $\theta(x)$ is the Heaviside step function, $\epsilon \to +0$. This problem is bypassed by A. Calogeracos {\it et al.}~\cite{Imagawa}. They represented the wave function $\Psi(x)$ as the $x$-ordered exponent (the analogue of the evolution operator) acting on the wave function in the initial point $x_0$. We cite herein the exact solution of Eq.~(\ref{eq}) in order to demonstrate explicitly the absence of any ambiguities.
Let us start from Eq.~(\ref{eq}): \begin{equation} \label{eqDelta} g''\left(x\right)+2i\left(E-G \delta(x)\right)g'\left(x\right)-p_y^2 g\left(x\right)=0. \end{equation} The function $g(x)$ appears to be continuous, $g'(x)$ is discontinuous at $x = 0$. Assume that $g'(\pm 0) \ne 0$ and divide this equation over the function $g'(x)$, $x \in I_\epsilon = (-\epsilon, \epsilon)$. Integrating then this equation over the interval $I_\epsilon$ and taking the limit $\epsilon \to +0$ we arrive at the correct matching condition: \begin{equation} \label{MatchDelta} \frac{g'(+0)}{g'(-0)} = e^{2 i G}. \end{equation}
If one is interested in the discrete spectrum of this problem one has to apply the condition~(\ref{MatchDelta}) to the function $g(x) = g_0 e^{-i E x} e^{- k |x|}$ which represents the common form of the continuous at $x = 0$ bounded solution of Eq.~(\ref{eqDelta}), $k = \sqrt{p_y^2 - E^2}$. This yields explicitly the spectrum~(\ref{Deltalimit}). The initial assumption $g'(\pm 0) \ne 0$ is obviously valid for such functions $g(x)$.
If we consider the scattering problem with definite $|E| > p_y$, the continuous function $g(x)$ has the following form: $$ g(x) = \left\{ \begin{array}{ll} A e^{ix (k - E)} + B e^{-ix (k + E)}, x < 0\\ (A + B) e^{ix (k - E)}, x > 0, \end{array} \right. $$ $k = \sqrt{E^2 - p_y^2}$. Applying the condition~(\ref{MatchDelta}) one can receive the transmission coefficient: $$
T = \left|1 + \frac{B}{A} \right|^2 = \frac{k^2}{k^2 + p_y^2 \sin^2 G}. $$ Finally, we have to check that the initial assumption $g'(\pm 0) \ne 0$ is not violated. $g'(+0) \ne 0$ as far as $E \ne k$ when $p_y \ne 0$. Suppose then that $g'(-0) = 0$ which leads to $A (k - E) = B (k + E)$ or equivalently $T = 4 k^2 / (k + E)^2$. This makes no physical sense because the transmission coefficient $T$ is not dependent on the parameter $G$ in this case. Hence, the unambiguous solution for the case of the $\delta$-potential is provided.
We can suggest an easier way to get the discrete spectrum for this potential. By integrating Eq.~(\ref{supereq}) and applying boundary conditions~(\ref{asymptotes}) we finally get: \begin{equation} \Delta \Omega = \Omega_+ - \Omega_- = 2 G \end{equation} which gives explicitly the spectrum~(\ref{Deltalimit}).
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\end{document} |
\begin{document}
\author{Michel J.\,G. WEBER} \address{IRMA, UMR 7501, Universit\'e Louis-Pasteur et C.N.R.S., 7 rue Ren\'e Descartes, 67084 Strasbourg Cedex, France.
E-mail: {\tt michel.weber@math.unistra.fr}}
\keywords{Local limit theorem, asymptotic uniform distribution, Rozanov's Theorem, divisors, Bernoulli random variables, i.i.d. sums, Theta functions. \vskip 1pt 2010 \emph{Mathematics Subject Classification}: {Primary: 60F15, 60G50 ; Secondary: 60F05}.}
\begin{abstract} For sums $S_n=\sum_{k=1}^n X_k$, $n\ge 1$ of independent random variables $ X_k $ taking values in ${\mathbb Z}$
we prove, as a consequence of a more general result, that if (i) For some function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, and some constant $C$, we have for all $n$ and $\nu\in {\mathbb Z}$,
\begin{equation*}\label{abstract1}
\big|B_n{\mathbb P}\big\{ S_n=\nu\big\}- {1\over \sqrt{ 2\pi } }\ e^{-
{(\nu-M_n)^2\over 2 B_n^2} }\big|\,\le \, {C\over \,\phi(B_n)},
\end{equation*}
then (ii) There exists a numerical constant $C_1$, such that for all $n $ such that $B_n\ge 6$, all $h\ge 2$,
and $\m=0,1,\ldots, h-1$, \begin{align*}\label{abstract1}
\Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}- \frac{1}{h}\Big|
\le {1\over \sqrt{2\pi}\, B_n }+\frac{1+ 2 {C}/{h} }{ \phi(B_n)^{2/3} } + C_1 \,e^{-(1/ 16 )\phi(B_n)^{2/3}}. \end{align*} Assumption (i) holds if a local limit theorem in the usual form is applicable, and (ii) yields a strenghtening of Rozanov's necessary condition.
Assume in place of (i) that $\t_j =\sum_{k\in {\mathbb Z}}{\mathbb P}\{X_j= k\}\wedge{\mathbb P}\{X_j= k+1 \} >0$, for each $j$ and that
$\nu_n =\sum_{j=1}^n \t_j\uparrow \infty$.
We prove strenghtened forms of the asymptotic uniform distribution property. (iii) Let $\a\!>\!\a'\!>\!0$, $0\!<\!\e\!<\!1$. Then for each $n$ such that
$$|x|\le\frac12 \big( \frac{ 2\a\log (1-\epsilon)\nu_n}{ (1-\epsilon)\nu_n }\big)^{1/2}\qq \Rightarrow \qq{\sin x\over x}\ge (\a^\prime/\a)^{1/2},$$
we have
\begin{eqnarray*} \sup_{u\ge 0}\,\sup_{d< \pi
( {(1-\epsilon)\nu_n \over 2\a\log (1-\epsilon)\nu_n})^{1/2}
} \ \big| {\mathbb P} \{d|S_n+u \} - {1\over d} \big|
\,\le \,2 \,e^{- \frac{\epsilon^2 }{2}\nu_n}+
\,\big( (1-\epsilon)\nu_n\big)^{-\a'} . \end{eqnarray*}
(iv) Let $0<\rho<1 $ and $0<\e<1$. The sharper uniform bound $2 e^{- \frac{\epsilon^2 }{2}\nu_n}+e^{- ( (1-\epsilon)\nu_n)^\rho}$ is also proved (for a corresponding $d$-region of divisors), for each $n$ such that $$|x|\le\frac12 \,\big( \frac{ 2 }{ ((1-\epsilon)\nu_n)^{1-\rho} }\big)^{1/2}\qq \Rightarrow \qq{\sin x\over x}\ge \sqrt{1-\e}.$$ \end{abstract}
\maketitle
\section{\bf Local limit theorem and asymptotic uniform distribution.}\label{s1}
Let $X=\{X_i , i\ge 1\}$ be a sequence of independent variables taking values in ${\mathbb Z}$,
and let $S_n=\sum_{k=1}^n X_k$, for each $n$.
\vskip 3 pt The sequence $X$
is said to be {asymptotically uniformly distributed with respect to lattices of span $d$}, in short a.u.d.($d$), if for $m = 0,1,\ldots,d-1$, we have
\begin{equation} \label{aud1}\lim_{n\to \infty}\ {\mathbb P}\{S_n \equiv m \,{\rm (mod)}\,d\}=\frac1d.
\end{equation} Equivalenty for $m = 0,1,\ldots,d-1$, we have
\begin{equation}\label{uad.lim1}\lim_{n\to \infty}\ {\mathbb P}\{d|S_n-m\}=\frac1d.
\end{equation} The sequence $X$ is {asymptotically uniformly distributed}, in short a.u.d., if \eqref{aud1} holds true for any $d\ge 2$ and $m = 0,1,\ldots,d-1$.
\vskip 5 pt \vskip 3 pt
Dvoretsky and Wolfowitz \cite{DW} proved the following characterization. Assume that $X$ is composed with independent random variables taking only the values $$ 0, 1,\ldots, h-1.$$
In order that the partial sums $\{S_n, n\ge 1\}$ be a.u.d.($h$), it is necessary and sufficient that \begin{equation} \label{aud.dw.ns} \prod_{k=1}^\infty\bigg( \sum_{m=0}^{h-1}{\mathbb P}\{X_k=m\}\,e^{\frac{2i\pi }{h}rm}\bigg) \,=\, 0, \qq \quad (r=1,\ldots, h-1).
\end{equation} Equivalently, \begin{equation} \label{aud.dw.ns.} \prod_{k=1}^\infty\big({\mathbb E \,} e^{\frac{2i\pi }{h}rX_k}\big) \,=\, \lim_{N\to \infty} \big({\mathbb E \,} e^{\frac{2i\pi }{h}rS_N}\big) \,=\, 0, \qq \quad (r=1,\ldots, h-1).
\end{equation}
This notion plays an important role in the study of the local limit theorem. Let us assume that the random variables $X_k$ take values in a common lattice $\mathcal L(v_{0},D )$, namely defined by the
sequence $v_{ k}=v_{ 0}+D k$, $k\in {\mathbb Z}$, $v_{0} $ and $D >0$ being reals, and are square integrable,
and let
\begin{equation} \label{MnBn}M_n= {\mathbb E\,} S_n , \qq B_n^2={\rm Var}(S_n)\to \infty.
\end{equation}
\vskip 20pt We say that the local limit theorem (in the usual form) is applicable to $X$ if
\begin{equation}\label{def.llt.indep} \sup_{N=v_0n+Dk }\Big|B_n\, {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi } }e^{-
{(N-M_n)^2\over 2 B_n^2} }\Big| = o(1), \qq \quad n\to\infty. \end{equation}
When the random variables $X_i$ are identically distributed, \eqref{def.llt.indep} reduces to \begin{equation}\label{llt.iid} \sup_{N=v_0n+Dk }\Big| \s \sqrt{n}\, {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi } }e^{-
{(N-n\m)^2\over 2 n\s^2} }\Big| = o(1), \end{equation} where $\m={\mathbb E\,} X_1$, $\s^2={\rm Var}( X_1)$. By Gnedenko's Theorem \cite{G}, see also \cite{P}, p.\,187, \cite{SW}, Th.\,1.4, \eqref{llt.iid} holds if and only if the span $D$ is maximal (there are no other real numbers $v'_{0} $ and $D' >D$ for which ${\mathbb P}\{X \in\mathcal L(v'_0,D')\}=1$).
Note that the transformation \begin{equation}\label{llt.transf.}
X'_j= \frac{X_j-v_0}{D},
\end{equation} allows one to reduce to the case $v_0=0$, $D=1$.
\begin{remark}Note that the series (in $k$)
\begin{equation}\label{def.llt.indep.sum} \sum_{N=v_0n+Dk } \Big( {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi } B_n}e^{- {(N-M_n)^2\over 2 B_n^2} } \Big), \end{equation} is obviously convergent, whereas
nothing can be deduced concerning its order from the very definition of the local limit theorem. Further by using Poisson summation formula the series associated to the second summand verifies
\begin{equation}
\label{def.llt.indep.poisson}
\sum_{N=v_0n+Dk } {D\over \sqrt{ 2\pi } B_n}e^{- {(N-M_n)^2\over 2 B_n^2} }\,=\,\sum_{\ell \in{\mathbb Z}} e^{2i\pi \ell \{\frac{v_0n-M_n}{D}\}-\frac{2\pi^2\ell^2 B_n^2}{D^2}}, \end{equation} and so is $ 1+\mathcal O(D/B_n)$, whereas the one associated to the first is 1. Therefore
\begin{equation}\label{def.llt.indep.sum.} \sum_{N=v_0n+Dk }
\Big( {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi } B_n}e^{- {(N-M_n)^2\over 2 B_n^2} }\Big)\,=\, \mathcal O(D/B_n). \end{equation}
\end{remark}
When a strong local limit theorem with convergence in variation holds we have the more informative result
\begin{equation} \label{sllt1}
\lim_{n\to\infty}
\sum_{N=v_0n+Dk }\Big| {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi }B_n }e^{-
{(N-M_n)^2\over 2 B_n^2} }\Big| =0.
\end{equation}
\vskip 20pt The following result is well-known. \begin{theorem}[Rozanov] \label{l1} Let $X=\{X_i , i\ge 1\}$ be a sequence of independent variables taking values in ${\mathbb Z}$, and let $S_n=\sum_{k=1}^n X_k$, for each $n$. The local limit theorem is applicable to $X$ only if $X$ satisfies the a.u.d. property.
\end{theorem}
\begin{remark} In Petrov \cite{P}, Lemma 1,\,p.\,194, also in Rozanov's \cite{Ro} Lemma 1,\,p.\,261, Theorem \ref{l1} is stated under the assumption that a local limit theorem in the strong form holds, which is not necessary.
\end{remark} We will in fact prove the following stronger result providing an explicit link between the local limit theorem and the a.u.d. property, through a quantitative estimate of the difference ${\mathbb P} \{ S_n\equiv\, m\! \hbox{\rm{ (mod $h$)}} \}- {1}/{h}$.
\begin{theorem} \label{l1a} Let $X=\{X_i , i\ge 1\}$ be a sequence of independent variables taking values in ${\mathbb Z}$, and let $S_n=\sum_{k=1}^n X_k$, for each $n$. Assume that for some function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, and some constant $C$, we have for all $n$
\begin{equation}\label{phi.cond}
\sup_{m\in {\mathbb Z}}\Big|B_n{\mathbb P}\big\{ S_n=m\big\}- {1\over \sqrt{ 2\pi } }\ e^{-
{(m-M_n)^2\over 2 B_n^2} }\Big|\,\le \, {C\over \,\phi(B_n)}.
\end{equation}
\vskip 3 pt \noindent Then there exists a numerical constant $C_1$, such that for all $0<\e \le 1$, all $n $ such that $B_n\ge 6$, and all $h\ge 2$, \begin{align*}
\sup_{\m=0,1,\ldots, h-1} \,&\Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}- \frac{1}{h}\Big| \cr &\le {1\over \sqrt{2\pi}\, B_n }+\frac{2C}{h\,\sqrt{\e}\,\phi(B_n)}
+ {\mathbb P}\Big\{ \frac{|S_n -M_n |}{B_n}> \frac{1}{\sqrt \e}\Big\}+C_1 \,e^{-1/(16\e)}. \end{align*} \end{theorem}
\vskip 8 pt
\begin{remark} \label{rem.thl1a} It follows from the proof that $C_1=2e\sqrt{\pi}$ is suitable. \end{remark}
Choosing $\e= \phi(B_n)^{-2/3}$ and using Tchebycheff's inequality, we get the following
\begin{corollary}\label{cor}For all $n $ such that $B_n\ge 6$, and all $h\ge 2$, we have \begin{align}\label{eps.phi}
\sup_{\m=0,1,\ldots, h-1} \, \Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}- \frac{1}{h}\Big|
\le H_n , \end{align} with \begin{align}\label{eps.phi.Hn}
H_n= {1\over \sqrt{2\pi}\, B_n }+\frac{1+ 2 {C}/{h} }{ \phi(B_n)^{2/3} } + C_1 \,e^{-(1/ 16 )\phi(B_n)^{2/3}}. \end{align} \end{corollary}
Theorem \ref{l1a} contains Theorem \ref{l1}, since by definition such a function $\phi$ exists if the local limit theorem is applicable to $X$. Further condition \eqref{phi.cond} implies that the local limit theorem is applicable to $X$.
\begin{remark} Examples of LLT's with speed of convergence are given in Appendix. \end{remark}
\begin{proof} By assumption, \begin{equation*}
\Big|B_n{\mathbb P}\big\{ S_n=m\big\}- {1\over \sqrt{ 2\pi } }\ e^{-
{(m-M_n)^2\over 2 B_n^2} }\Big|\,\le \, {C\over \phi(B_n) },
\end{equation*}
for all $m$ and $n$. Let $\e>0$.
We have \begin{eqnarray*}
\Big|{\mathbb P}\big\{ S_n\equiv\, m\ \hbox{\rm (mod $h$)}\big\}- \sum_{|k-M_n|\le B_n/\sqrt \e
\atop k\equiv m\, (h)} {\mathbb P}\big\{S_n=k\}\Big|&\le &
{\mathbb P}\Big\{ \frac{|S_n -M_n |}{B_n}> \frac{1}{\sqrt \e}\Big\} , \end{eqnarray*} \begin{align*}
\Big| \sum_{|k-M_n|\le B_n/\sqrt \e
\atop k\equiv m\, (h)} {\mathbb P}\big\{S_n=k\}- & {1\over \sqrt{ 2\pi }B_n } \sum_{|k-M_n|\le B_n/\sqrt \e \atop k \equiv m\, (h)} e^{-
{(k-M_n)^2\over 2 B^2_n} } \Big| \cr &\le \, {C\over B_n\phi(B_n) }\,\sum_{|k-M_n|\le B_n/\sqrt \e \atop k\equiv m\, (h)}1 \ \le \, \frac{2C}{h\,\sqrt{\e}\,\phi(B_n)} . \end{align*}
Letting $z_n= \lfloor M_n\rfloor$, we have
\begin{eqnarray*} \sum_{k\in{\mathbb Z}\atop|k-M_n|> B_n/\sqrt \e } e^{- {(k-M_n)^2\over 2 B^2_n} }
&\le & \sum_{Z\in {\mathbb Z} \atop |Z-z_n |> B_n/ \sqrt \e } e^{- {(Z-z_n)^2\over 2 B^2_n} }. \end{eqnarray*}
Now using the elementary inequality $(a+b)^2\le 2(a^2+b^2)$ for reals $a$, $b$, we have $|Z-z_n |\le\sqrt 2( |Z |+|z_n|) $ and $|Z-z_n |^2\ge |Z |^2/2-z_n^2$. We can thus continue as follows
\begin{eqnarray*}\,\le\, \sum_{Z\in {\mathbb Z} \atop \sqrt 2( |Z |+|z_n|) > B_n/ \sqrt \e } e^{- {(Z-z_n)^2\over 2 B^2_n} } &\le& e^{
{1\over 2 B^2_n} }\,\sum_{ Z\in {\mathbb Z} \atop |Z | > (B_n/ \sqrt{2 \e}) -1} e^{- { Z ^2\over 4 B^2_n} } .\qq \end{eqnarray*}
Assume that $B_n\ge \max( 1/\sqrt 2,4\sqrt{2 \e})$, then ${B_n\over \sqrt{2 \e}}-2\ge {B_n\over 2\sqrt{2 \e}}$.
In particular $|Z|\ge 1$ in the previous series, and so we have the estimates
\begin{eqnarray*}\ \le\ 2\,e^{ {1\over 2 B^2_n} }\,\sum_{ Z > (B_n/ 2\sqrt{2 \e}) +1} e^{- { Z ^2\over 4 B^2_n} }&\le & 2\,e
\sum_{ Z > (B_n/ 2\sqrt{2 \e}) +1} \int_{Z-1}^Z e^{-{t^2\over 4 B^2_n}} {\rm d} t \cr &\le & 2\,e
\int_{B_n/ 2\sqrt{2 \e} }^\infty e^{-{t^2\over 4 B^2_n}} {\rm d} t \cr( t= \sqrt 2B_n u)\quad&=& 2 \sqrt{2 }e
B_n\int_{1/4\sqrt{ \e}}^\infty e^{-{u^2\over 2 }} {\rm d} u \cr &\le & 2\sqrt{2 }e
B_n\sqrt{{\pi\over 2}}\, e^{-1/(16\e)} \cr &= & 2e\sqrt{\pi}
B_n \, e^{-1/(16\e)},\end{eqnarray*} since $ e^{x^2/2}\int_x^\infty e^{-t^2/2}{\rm d}t \le \sqrt{{\pi\over 2}}$, for any $x\ge 0$.
Therefore \begin{eqnarray}\label{est.1}
& &\Big|{\mathbb P}\big\{ S_n\equiv\, m\ \hbox{\rm (mod $h$)}\big\}
- {1\over \sqrt{ 2\pi }B_n } \sum_{ k \equiv m\, (h)} e^{-
{(k-M_n)^2\over 2 B^2_n} }\Big|\cr &\le & {\mathbb P}\Big\{ \frac{|S_n -M_n |}{B_n}> \frac{1}{\sqrt \e}\Big\} + \frac{2C}{h\,\sqrt{\e}\,\phi(B_n)} +C_1 \, e^{-1/(16\e)} , \end{eqnarray} with $C_1=2e\sqrt{\pi}$.
\vskip 5 pt Recall Poisson summation formula: for $x\in {\mathbb R},\ 0\le \d\le 1 $, \begin{equation}\label{poisson}\sum_{\ell\in {\mathbb Z}} e^{-(\ell+\d)^2\pi x^{-1}}=x^{1/2} \sum_{\ell\in {\mathbb Z}} e^{2i\pi \ell\d -\ell^2\pi x}.
\end{equation}
Write $k=m+l h$, $M'_n=M_n-m$, \begin{equation}{(k-M_n)^2\over 2 B^2_n}={( l h-M'_n)^2\over 2 B^2_n}={( l -\lceil M'_n/h\rceil+\{ M'_n/h\})^2\over 2 B^2_n/h^2}={( \ell +\{ M'_n/h\})^2\over 2 B^2_n/h^2},
\end{equation} letting $ \ell=l -\lceil M'_n/h\rceil$. \vskip 3 pt By applying it with $x=2 B^2_n\pi /h^2$, $\d=\{ M'_n/h\}$, we get \begin{equation} \sum_{ k \equiv m\, (h)} e^{- {(k-M_n)^2\over 2 B^2_n} }\,=\, \sum_{\ell \in {\mathbb Z}}e^{-{( \ell -\{ M'_n/h\})^2\over 2 B^2_n/h^2}}\,=\, {\sqrt{2 \pi}B_n\over h}\,\sum_{\ell \in {\mathbb Z}}e^{ -2i\pi \ell \{ M'_n/h\} -2\pi^2B_n^2\ell^2/h^2}. \end{equation}
Whence
\begin{equation} \Big|{ h\over \sqrt{2 \pi}B_n} \sum_{ k \equiv m\, (h)} e^{-
{(k-M_n)^2\over 2 B^2_n} } -1\Big|\le \sum_{|\ell |\ge 1}e^{ -2\pi^2B_n^2\ell^2/h^2}. \end{equation} But for any positive real $a$,
\begin{equation}\label{aux.est1}\sum_{H=1}^\infty e^{-aH^2}\le {\sqrt \pi\over 2 }\min({1\over \sqrt a}, {1\over a}).
\end{equation} Therefore with $a= 2\pi^2B_n^2/h^2$,
\begin{equation*} \Big|{ h\over \sqrt{2 \pi}B_n} \sum_{ k \equiv m\, (h)}
e^{-{(k-M_n)^2\over 2 B^2_n} } -1\Big|\le {\sqrt \pi }\min({h\over \sqrt{2}\pi B_n }, {h^2\over 2\pi^2B_n^2 })\le {h\over \sqrt{2\pi}\, B_n }. \end{equation*} We have thus obtained the explicit bound
\begin{equation}\label{est2} \Big|{ 1\over \sqrt{2 \pi}B_n} \sum_{ k \equiv m\, (h)} e^{-
{(k-M_n)^2\over 2 B^2_n} } -\frac1h\Big| \le {1\over \sqrt{2\pi}\, B_n }. \end{equation}
By carrying it back to \eqref{est.1}, we get for any $\e>0$, all $n $ such that $B_n\ge \max( 1/\sqrt 2,4\sqrt{2 \e})$, and all $h\ge 2$,
\begin{align}\label{est.3}\sup_{\m=0,1,\ldots, h-1}
\Big|{\mathbb P}\big\{ S_n\equiv\, \m\ \hbox{\rm{ (mod $h$)}}\big\}-\frac{1}{h}\Big|
&\le {1\over \sqrt{2\pi}\, B_n }+ \frac{2C}{h\,\sqrt{\e}\,\phi(B_n)}
\cr &\quad+ {\mathbb P}\Big\{ |S_n -M_n |> {B_n\over \sqrt \e}\Big\}+ C_1\, e^{-1/(16\e)}. \end{align} This is fulfilled if we choose $0<\e \le 1$, and $n$ such that $B_n\ge 6$, whence the claimed estimate. \end{proof}
\section{Local limit theorem in the strong form}\label{s2} There are easy examples of sequences $X$ for which the fulfilment of the local limit theorem depends on the behavior of the first members of $X$.
Hence it is reasonable to introduce the following definition due to Prohorov \cite{Pr}. A local limit theorem in the {\it strong form} (or {\it in a strengthened form})
is said to be applicable to $ X$, if a local limit theorem in the usual form is applicable to any subsequence extracted from $ X$, which differs from $ X$ only in a finite number of members. \vskip 3 pt This definition can be made a bit more convenient, see Gamkrelidze \cite{Gam3}. Let \begin{equation}S_{k,n}=X_{k+1}+ \ldots +X_{k+n},\qq A_{k,n}={\mathbb E \,} S_{k,n}, \qq B^2_{k,n}={\rm Var} (S_{k,n}). \end{equation} The local limit theorem in the strong form holds if and only if \begin{equation}\label{lltsf.ref} {\mathbb P}\big\{ S_{k,n}=m\big\}= {D\over B_{k,n} \sqrt{ 2\pi } }\ e^{- {(m-A_{k,n})^2\over 2 B_{k,n}^2} }+o\Big({1\over B_{k,n} }\Big), \end{equation} uniformly in $m$ and every finite $k$, $k=0,1,2, \ldots$, as $n\to \infty$ and $B_{k,n}\to \infty$. \vskip 8 pt Rozanov's necessary condition states as follows.
\begin{theorem}[\cite{Ro},\,Th.\,I]\label{rozanov.I} Let $ X= \{ X_j , j\ge 1\}$ be a sequence of independent, square integrable random variables taking values in ${\mathbb Z}$. Let $b_k^2= {\rm Var}(X_k)$, $B_n^2 =b_1^2+\ldots+ b_n^2$. Assume that \begin{equation}\label{Ro.A}B_n \to \infty\qq\qq {\rm as} \ n\to \infty. \end{equation} The following condition is necessary for the applicability of a local limit theorem in the strong form to the sequence $X$, \begin{equation}\label{rozn}\prod_{k=1}^\infty \big[ \max_{0\le m< h} {\mathbb P}\big\{X_k\equiv m\, {\rm (mod {\it \, h })} \big\}\big]=0\qq {\it for\ any\ } h\ge 2 . \end{equation} \end{theorem} \vskip 2 pt Condition \eqref{rozn} is also sufficient in some important examples, in particular if $X_j$ have stable limit distribution, see Mitalauskas \cite{Mit}.
We briefly indicate how Theorem \ref{rozanov.I} is proved.
If the local limit theorem in the strong form is applicable to the sequence $X$, then \begin{equation} \label{roz.cond} \sum_{k=1}^\infty \ {\mathbb P}\big\{ X_k\not\equiv 0\, ({\rm mod} \ h) \big\}= \infty, \qq {\it for\ any} \ h\ge 2. \end{equation} Indeed, otherwise given $h\ge 2$, by the Borel--Cantelli lemma, on a set of measure greater than $3/4$, $X_k\equiv 0\, ({\rm mod} \ h)$ for all $k\ge k_0$, say. The new sequence $X'$ defined by $X'_k=0$ if $k< k_0$, $X'_k=X_k$ unless, with partial sums $S'_n$, verifies ${\mathbb P}\{ S'_n\equiv 0\, ({\rm mod} \ h) \}>3/4$ for all $n$ large enough, and this can be used to bring a contradiction with the fact
that ${\mathbb P}\{ S'_n\equiv 0\, ({\rm mod} \ h) \}$ should converge to $1/h$. \vskip 2 pt
\vskip 3 pt
The arithmetical quantity $$\max_{0\le m< h} {\mathbb P}\big\{X_k\equiv m\, {\rm (mod({\it h})} \big\}$$
also appears in the study of
local limit theorems with arithmetical sufficient conditions. The approaches used (Freiman, Moskvin and Yudin \cite{FMY}, Mitalauskas \cite{Mit1}, Raudelyunas \cite{Rau} and later Fomin \cite{Fo}, for instance) require the random variables to do not overly much concentrate in a particular residue class $m$ (mod $h$) of ${\mathbb Z}$, and impose arithmetical conditions of type: For all $h\ge 2$ \begin{equation}\label{llt.arithm.cond.}
\max_{0\le m<h}{\mathbb P}\{X_k\equiv m \ {\rm (mod\, {\it h})}\}\le 1-\a_k,
\end{equation} for all $k$, where $\a_k$ is some specific sequence of reals decreasing to $0$. In addition, one generally have that $\sum _k \a_k = \infty$. Although the simple form of local limit theorem is here considered, for obvious reasons, condition \eqref{rozn} brings nothing more in this context.
\vskip 3 pt
\vskip 3 pt As a consequence of the quantitative formulation of the a.u.d. property obtained in Theorem \ref{l1a}, we have the following result.
\begin{theorem} Under the assumptions of Theorem \ref{rozanov.I}, assume further that the local limit theorem is applicable to a sequence $X$. Then \vskip 3 pt
{\rm(i)}
\begin{eqnarray*} \limsup_{h\to \infty}\ \prod_{k=1}^\infty \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}\,=\ 0. \end{eqnarray*}
{\rm(ii)} There exists a function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, such that \begin{eqnarray*} \sum_{k=1}^n\frac{ \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}}{1- \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}}
&\ge & -\log\big(\frac{1}{h}+H_n\big), \end{eqnarray*} where $H_n={ 1\over \sqrt{2\pi}\, B_n }+\frac{1+ 2 {C}/{h}}{ \phi(B_n)^{2/3} } + C_1\,e^{-(1/ 16 )\phi(B_n)^{2/3}} $, and $C,C_1$ are absolute constants.\end{theorem} \begin{proof} We purpose a direct argument.
Consider a
sequence $Y$ where $Y_k=X_k-m_k$, $m_k$ are integers, for all $k\ge 1$. Let $h\ge 2$ be fixed. Choose $m_k$ so that
$$\max_{0\le m< h} {\mathbb P}\big\{X_k\equiv m\, {\rm mod({\it h})} \big\}= {\mathbb P}\big\{X_k\equiv m_k\, {\rm mod({\it h})} \big\}
= {\mathbb P}\big\{Y_k\equiv 0\, {\rm mod({\it h})} \big\}
, $$ and let $\m_n=\sum_{k=1}^nm_k$. Note that $\sum_{k=1}^n Y_k=S_n -\m_n$, ${\rm Var}(\sum_{k=1}^nY_k)={\rm Var}(S_n)=B_n^2$.
\vskip 2 pt \vskip 2 pt As the local limit theorem is applicable to the sequence $X$,
condition \eqref{phi.cond} is satisfied for some function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, namely we have for all $n$,
\begin{equation*}
\sup_{\nu\in {\mathbb Z}}\Big|B_n{\mathbb P}\big\{ S_n=\nu\big\}- {1\over \sqrt{ 2\pi } }\ e^{-
{(\nu-M_n)^2\over 2 B_n^2} }\Big|\,\le \, {C\over \,\phi(B_n)}.
\end{equation*}
Given $n$, letting $\nu=m+\m_n$ and observing that ${\mathbb P} \{ \sum_{k=1}^n Y_k =m\}={\mathbb P} \{ S_n -\m_n =m\}$, we get for $m\in {\mathbb Z}$, $n\ge 1$,
\begin{equation*}
\Big|B_n{\mathbb P}\Big\{ \sum_{k=1}^n Y_k =m \Big\}- {1\over \sqrt{ 2\pi } }\ e^{-
{(m+\m_n-M_n)^2\over 2 B_n^2} }\Big|\,\le \, {C\over \,\phi(B_n)}.
\end{equation*} Thus $Y$ satisfies condition \eqref{phi.cond} with the same function $\phi(n) $.
\vskip 2 pt \vskip 2 pt Applying Remark \ref{rem.thl1a} to the sequence $Y$, it follows that,
\begin{eqnarray}\label{kb} \prod_{k=1}^n \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}&=&\prod_{k=1}^n {\mathbb P}\{Y_k\equiv\, 0\ \hbox{\rm (mod $h$)}\} \cr &\le & {\mathbb P}\big\{ \sum_{k=1}^n Y_k\equiv\, 0\ \hbox{\rm (mod $h$)}\big\}\le \frac{1}{h}+H_n, \end{eqnarray} where $H_n$ has the form given in the statement, and $H_n\to 0$ as $n\to \infty$.
\vskip 2 pt Letting $n$ tend to infinity in \eqref{kb} implies,
\begin{eqnarray}\label{kbh} \prod_{k=1}^\infty \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}&\le& \frac{1}{h}. \end{eqnarray} This being true for each $h$, $h\ge 2$, letting now $h$ tend to infinity in \eqref{kbh} yields,
\begin{eqnarray} \limsup_{h\to \infty}\ \prod_{k=1}^\infty \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}&=& 0. \end{eqnarray}
\vskip 2 pt \vskip 2 pt We also have by using the elementary inequality $\log( 1-x)\ge -x/(1-x)$, $0\le x<1$, \begin{eqnarray*} \prod_{k=1}^n {\mathbb P}\{Y_k\equiv\, m\ \hbox{\rm (mod $h$)}\} &=&\prod_{k=1}^n\big(1- {\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\}\big) \cr&=&e^{\sum_{k=1}^n \log(1-{\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\})}\cr&\ge &e^{-\sum_{k=1}^n {\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\}/(1-{\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\})}
. \end{eqnarray*}
Thus by Remark \ref{rem.thl1a}, \begin{eqnarray*} \sum_{k=1}^n\frac{ \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}}{1- \max_{0\le m< h}{\mathbb P}\{X_k\equiv\, m\ \hbox{\rm (mod $h$)}\}} &\,=\,&\sum_{k=1}^n\frac{ {\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\}}{1-{\mathbb P}\{Y_k\not\equiv\, m\ \hbox{\rm (mod $h$)}\}} \cr&\ge & -\log\big(\frac{1}{h}+H_n\big). \end{eqnarray*} \end{proof} \begin{remark}\label{kb.rem} (i) Note that the bound used in \eqref{kb} is very weak since \begin{eqnarray*} \prod_{k=1}^n {\mathbb P}\{Y_k\equiv\, m\ \hbox{\rm (mod $h$)}\}
\ =\ {\mathbb P}\big\{ \forall J\subset [1,n],\ \sum_{k\in J} Y_k\equiv\, m\ \hbox{\rm (mod $h$)}\big\} . \end{eqnarray*} One can replace individuals $Y_k$ by sums over blocks according to any partition of $\{1,\ldots,n\}$. \vskip 2 pt \noindent (ii) Sets of multiples serve as good test sets for the applicability of the local limit theorem because addition is a closed operation. What can be derived when testing the applicability of the local limit theorem with other remarkable sets of integers (squarefree numbers, primes numbers, power numbers, geometric growing sequences, \ldots) is unknown.
Concerning the squarefree integers, namely having no squared prime factors, we note the bound \begin{equation}\label{squarefree} \Big|2^{-n}
\sum_{j\, { \rm squarefree}} C_n^j - \frac{6}{\pi^2}\Big|\le C_1e^{-C_2 {(\log n^{{3/5}}}/{(\log\log n) ^{1/5}}}. \end{equation}
We refer to \cite{DS}. \end{remark}
\section{Random sequences satisfying the a.u.d. property}\label{s3}
It has some interest to relate the a.u.d. property for Bernoulli sums to the one of sets having Euler density, in this particular case here, arithmetic progressions. A subset $A$ of $ {\mathbb N}$ is said to have Euler density $\l$ with parameter $\varrho $ (in short ${ E}_\varrho$ density $\l$) if \begin{equation*} \lim_{n\to \infty}\sum_{j\in A} C_n^j \varrho^j (1-\varrho)^{n-j}= \l. \end{equation*}
By a result due to Diaconis and Stein, we have the following characterization.
\begin{theorem}[\cite{DS},\,Th.\,1]\label{ds}For any $A\subset {\mathbb N}$, and $\varrho\in ]0,1[$ the following assertions are equivalent:
\begin{eqnarray*} \label{dschar} ({\rm i}) & &\qq \hbox{\it $A$ has ${E}_\varrho$ density $\l$},
\cr ({\rm ii}) & &\qq\lim_{t\to \infty} e^{-t}\sum_{j\in A} \frac{t^j}{j!}=\l,
\cr ({\rm iii}) & &\qq \hbox{\it for all $\e>0$}, \quad \lim_{n\to \infty} \frac{\#\{j\in A : n\le j< n+\e\sqrt n\} }{\e\sqrt n}
=\l .
\end{eqnarray*} \end{theorem} Applying (iii) with $\rho=\frac12$, to \begin{equation}\label{setA}A= \{ u+kd,\ k\ge 1\}, \end{equation} straightforwardly implies
\begin{lemma}\label{ber.uad} Let $\B_n=\b_1+\ldots+\b_n$, where $ \b_i $ are i.i.d. Bernoulli random variables. Then $\{\B_n, n\ge 1\}$ is a.u.d.($d$) for any $d \ge 2$.
\end{lemma} \vskip 10 pt
Now consider the independent case and introduce the following characteristic.
Let $Y$ be
a random variable with values in ${\mathbb Z}$. Put
\begin{eqnarray}\label{vartheta} \t_Y =\sum_{k\in {\mathbb Z}}{\mathbb P}\{Y= k\}\wedge{\mathbb P}\{Y= k+1 \} , \end{eqnarray} where $a\wedge b=\min(a,b)$. Note that $0\le \t_Y<1$.
\begin{theorem} \label{t3} Let $ X= \{ X_j , j\ge 1\}$ be a sequence of independent random variables taking values in ${\mathbb Z}$. Assume that $\t_{X_j}>0$ for each $j$. Further assume that the series
$\sum_{j=1}^\infty \t_{X_j}$ diverges. Then $X$ is a.u.d.,
the conclusion holds in particular if the $X_j$ are i.i.d. and $\t_{X_1}>0$.
\end{theorem}
Note that no integrability condition is required, whereas square integrability is required in order that the local limit theorem be applicable.
We prove in the next section that if the series $\sum_{j=1}^\infty \t_{X_j}$ diverges, much more is in fact true.
Under the assumption made, each $X_j$ admits a Bernoulli component. This is the principle of a coupling method (the Bernoulli part extraction) introduced by McDonald \cite{M}, Davis and McDonald \cite{MD} in the study of the local limit theorem. See Weber \cite{W} for an application of this method to almost sure local limit theorem, and Giuliano and Weber \cite{GW3} where this method is used to obtain approximate local limit theorems with effective rate.
\vskip 3 pt Before passing to the proof, we briefly recall some facts and state an auxiliary Lemma. Let $\mathcal L(v_0,D)$ be a lattice defined by the
sequence $v_{ k}=v_{ 0}+D k$, $k\in {\mathbb Z}$,
$v_{0} $ and $D >0$ being real numbers. Let $X$ be
a random variable such that ${\mathbb P}\{X \in\mathcal L(v_0,D)\}=1$, and assume that $\t_X>0$. Let $ f(k)= {\mathbb P}\{X= v_k\}$, $k\in {\mathbb Z}$. Let also $0<\t\le\t_X$. Associate to $\t$ and $X$ a sequence $ \{ \tau_k, k\in {\mathbb Z}\}$ of non-negative reals such that \begin{equation}\label{basber0} \tau_{k-1}+\tau_k\le 2f(k), \qq \qq\sum_{k\in {\mathbb Z}} \tau_k =\t. \end{equation} For instance $\tau_k= \frac{\t}{\nu_X} \, (f(k)\wedge f(k+1)) $ is suitable. Next define a pair of random variables $(V,\e)$ as follows:
\begin{eqnarray}\label{ve} \qq\qq\begin{cases} {\mathbb P}\{ (V,\e)=( v_k,1)\}=\tau_k, \cr
{\mathbb P}\{ (V,\e)=( v_k,0)\}=f(k) -{\tau_{k-1}+\tau_k\over 2} . \end{cases}\qq (\forall k\in {\mathbb Z}) \end{eqnarray}
\begin{lemma} \label{lemd} Let $L$ be a Bernoulli random variable which is independent of $(V,\e)$, and let $Z= V+ \e DL$. Then $Z\buildrel{\mathcal D}\over{ =}X$. \end{lemma}
\begin{proof}[Proof of Theorem \ref{t3}]
We apply Lemma \ref{lemd} with $D=1$ to each $X_j$, and choose $0<\t_j\le\t_{X_j}$ so that the series $\sum_{j=1}^\infty \t_j$ diverges. One can associate to them a sequence of independent vectors $ (V_j,\e_j, L_j) $, $j=1,\ldots,n$ such that
\begin{eqnarray}\label{dec0} \big\{V_j+\e_j L_j,j=1,\ldots,n\big\}&\buildrel{\mathcal D}\over{ =}&\big\{X_j, j=1,\ldots,n\big\} . \end{eqnarray}
Further the sequences $\{(V_j,\e_j),j=1,\ldots,n\}
$ and $\{L_j, j=1,\ldots,n\}$ are independent. For each $j=1,\ldots,n$, the law of $(V_j,\e_j)$ is defined according to (\ref{ve}) with $\t=\t_j$. And $\{L_j, j=1,\ldots,n\}$ is a sequence of independent Bernoulli random variables. Set \begin{equation}\label{dec}
W_n =\sum_{j=1}^n V_j,\qq M_n=\sum_{j=1}^n \e_jL_j, \quad B_n=\sum_{j=1}^n
\e_j . \end{equation}
Denoting again $X_j= V_j+ \e_jL_j$,
$j\ge 1$, we have
\begin{eqnarray}\label{dep} {\mathbb P} \{d|S_n +u\} &=& {\mathbb E \,}_{(V,\e)} \, {\mathbb P}_{\!L}
\Big\{d|\big( \sum_{j= 1}^n \e_jL_j+W_n \big)+u \Big\}
. \end{eqnarray} As $\sum_{j= 1}^n \e_jL_j\buildrel{\mathcal D}\over{ =}\sum_{j=1}^{B_n } L_j$, we have
\begin{eqnarray*} {\mathbb P}_{\!L}
\Big\{d|\big( \sum_{j= 1}^n \e_jL_j+W_n \big)+u \Big\} &=& {\mathbb P}_{\!L}
\Big\{d| \, \sum_{j=1}^{B_n } L_j+\big(W_n +u\big) \Big\}. \end{eqnarray*}
In view of the dominated convergence theorem, it suffices to prove that for each $d\ge 2$,
\begin{eqnarray*} {\mathbb P}_{\!L}
\Big\{d| \, \sum_{j=1}^{B_n } L_j+ (W_n +u) \Big\}\ \to \frac{1}{d}, \end{eqnarray*} as $n\to \infty$, ${\mathbb P}_{(V,\e)}$ almost surely. But the set (compare with \eqref{setA}) \begin{equation*} A= \{ (W_n +u)+kd,\ k\ge 1\}, \end{equation*} now depends on $W_n$, thus on $n$, which is complicating things. However we can write
\begin{equation*}
\chi\Big({d\, \big|\,\sum_{j=1}^{B_n } L_j+ (W_n +u)}\Big)=\frac1d\,\sum_{j=0}^{d-1} e^{2i\pi {j \over d} (W_n +u)}e^{2i\pi {j \over d}\sum_{j=1}^{B_n } L_j}. \end{equation*}
By integrating with respect to $ {\mathbb P}_{\!L}$ we get,
$$ {\mathbb P}_{\!L}
\Big\{d| \, \sum_{j=1}^{B_n } L_j+\big(W_n +u\big) \Big\}={1\over d}+{1\over d}\sum_{j=1}^{d-1} e^{2i\pi {j \over d} (W_n +u)}\big(\cos { \pi j\over d}\big)^{B_n}
.$$ By the assumption made,
$B_n$ tends to infinity ${\mathbb P}_{(V,\e)}$ almost surely, ((8.3.5) in \cite{W2} for instance). Thus the latter sum tends to 0 as $n\to \infty$, ${\mathbb P}_{(V,\e)}$ almost surely. Therefore by the convergence argument invoked before, $ {\mathbb P} \{d|S_n +u\}$ tends to ${1\over d}$ as $n$ tends to infinity, for any $d\ge 2$ and $u\in {\mathbb N}$. Whence it follows that the sequence $\{S_n, n\ge 1\}$ is {\rm a.u.d.}\,. \end{proof}
\section{Random sequences satisfying a strenghtened a.u.d. property.}\label{s4}
For Bernoulli sums, the a.u.d. property is only a rough aspect of the value distribution of divisors of $\B_{ n}+u$, $u\ge 0$ integer. Much more is known. \begin{theorem}[\cite{W3},\, Th.\,2.1]\label{estPdlBn.u}
We have the uniform estimate \begin{equation*}
\sup_{u\ge 0}\,\sup_{2\le d\le n}\Big|{\mathbb P}\big\{ d| \B_{ n}+u \big\}- {1\over d}\sum_{ 0\le |j|< d } e^{i\pi (2u+n){j\over d}}\ e^{ -n
{\pi^2j^2\over 2d^2}}\Big|= {\mathcal O}\big((\log n)^{5/2}n^{-3/2}\big). \end{equation*} \end{theorem}
The special case $u=0$ was proved in \cite[Th.\,II]{W1}. Introduce the Theta function
\begin{equation}\label{theta.u.} \Theta_u(d,n) = \sum_{\ell\in {\mathbb Z}} e^{i\pi (2u+n){\ell\over d}}\ e^{ -n {\pi^2\ell^2\over 2d^2}}.
\end{equation}
By Poisson summation formula
\begin{equation}\label{theta.u..} \Theta_u(d,n) = \Big(d\sqrt{\frac{2}{\pi n}}\Big)\ \sum_{\ell \in {\mathbb Z}} e^{-(\ell+\{\frac{u+n/2}{d}\})^2\frac{2d^2}{n}}.
\end{equation}
As a consequence of Theorem \ref{estPdlBn.u}, we get\begin{corollary}\label{cor.estPdlBn.u}
We have the uniform estimate
\begin{equation*} \sup_{u\ge 0}\, \sup_{2\le d\le n}\Big|{\mathbb P}\big\{ d| \B_{ n}+u \big\}-{\Theta_u(d,n)\over d} \Big| \le C \,(\log n)^{5/2}n^{-3/2} . \end{equation*}
\end{corollary}
Apart from this important but specific case, it seems that the speed of convergence in the limit \eqref{aud1} was not investigated, in particular when $d$ and $n$ are varying simultaneously.
\vskip 3 pt
Consider the independent case and assume as in Theorem \ref{t3}, that $\nu_n =\sum_{j=1}^n \t_j\uparrow \infty $. The speed of uniform convergence over regions (in $d$ and $n$) presents a singularity when $d$ is getting too close to $\sqrt {\nu_n}$. That quantity already appears in Davis and McDonald \cite{MD}. On the other hand when $d$ is not close to $\sqrt {\nu_n}$, in a sense that we shall make precise, we show that an explicit speed of convergence can be assigned, this under the {\it sole} divergence assumption of the series $\sum_{j=1}^\infty \t_j$. So, for this important class of independent sequences, the well-known a.u.d. necessary condition turns up to be a particularly weak requirement. Further one can show by using Poisson summation formula that in the Bernoulli case, the local limit theorem implies a weaker speed of convergence than the one obtained in Theorem \ref{estPdlBn.u}.
\vskip 3 pt The speed of uniform convergence problem for {\it all} $d$ and $n$, $n\ge d\ge 2$, $n\to\infty$, is more complicated and one must restrict to the i.i.d. case. In place of the limiting term ${1}/{d}$ appears a more complicated Theta elliptic function. See \cite{W3}. For the independent case, the approach used becomes inoperant, due to appearance of integral products with interlaced integrants.
In fact, what will make possible to handle the independent case, is not just that $d$ and $\sqrt {\nu_n}$ are not too close, but also that in background, symmetries properties of the Bernoulli model permitted to effect the necessary calculations
in the first quadrant and {\it not} in the half-circle. This point is crucial for getting the uniform speed of convergence in Theorem \ref{estPdlBn.u}. This is explained in \cite{W3}, see reduction Lemma 2.3. In short, when the Bernoulli extraction part applies, these symmetry properties allow one to get a speed of convergence. The proof in the Bernoulli case is transposable to
other systems of random variables when such symmetries exist. This is not the case for the Hwang and Tsai model of the Dickman function \cite{HT}, \cite{GSW}, neither for the Cram\'er model of primes \cite{W4}.
\vskip 3 pt We prove the following result.
\begin{theorem}\label{saud1} Assume that $D=1$, $\t_{X_j}>0$ for each $j$, and that the series
$\sum_{j=1}^\infty \t_{X_j}$ diverges.
Let $\a\!>\!\a'\!>\!0$, $0\!<\!\e\!<\!1$. Then for each $n$ such that
$$|x|\le\frac12 \sqrt{ \frac{ 2\a\log (1-\epsilon)\nu_n}{ (1-\epsilon)\nu_n }}\qq \Rightarrow \qq{\sin x\over x}\ge (\a^\prime/\a)^{1/2},$$
recalling that $\nu_n =\sum_{j=1}^n \t_j$, we have
\begin{eqnarray*} \sup_{u\ge 0}\,\sup_{d< \pi \sqrt{ (1-\epsilon)\nu_n \over 2\a\log (1-\epsilon)\nu_n}} \ \Big| {\mathbb P} \{d|S_n+u \} - {1\over d} \Big|
&\le &2 \,e^{- \frac{\epsilon^2 }{2}\nu_n}+
\,\big( (1-\epsilon)\nu_n\big)^{-\a'} . \end{eqnarray*} \end{theorem} \ For the proof we use the following Lemma. \begin{lemma}[\cite{di}, Theorem 2.3] \label{di.1} Let $X_1, \dots, X_n$ be independent random variables, with $0 \le X_k \le 1$ for each $k$. Let $S_n = \sum_{k=1}^n X_k$ and $\mu = {\mathbb E \,} S_n$. Then for any $\epsilon >0$,
\begin{eqnarray*} {\rm (a)} &&
{\mathbb P}\big\{S_n \ge (1+\epsilon)\mu\big\}
\le e^{- \frac{\epsilon^2\mu}{2(1+ \epsilon/3) } } . \cr {\rm (b)} & &{\mathbb P}\big\{S_n \le (1-\epsilon)\mu\big\}\le e^{- \frac{\epsilon^2\mu}{2}}.
\end{eqnarray*} \end{lemma} We also need the following result. \begin{proposition}[\cite{W3}, Corollary 2.4]
\label{special.cases}{\rm (i)} For each $\a\!>\!\a'\!>\!0$ and $n$ such that $ \tau_n\ge (\a^\prime/\a)^{1/2}$, where
\begin{equation*}
\tau_n= {\sin\p_n/2\over \p_n /2}, \qquad\qquad \p_n= \big( {2\a\log n \over n}\big)^{1/2},
\end{equation*} we have \begin{equation*}
\sup_{u\ge 0}\,\sup_{d< \pi \sqrt{ n \over 2\a\log n}}\Big|{\mathbb P}\big\{ d| \mathcal B_{ n} +u\big\}-{1\over d}
\Big|\,\le\, n^{-\a'}. \end{equation*} {\rm (ii)} Let $0<\rho<1 $. Let also $0<\eta<1$, and suppose $n$ sufficiently large so that $\widetilde\tau_n\ge \sqrt{1-\eta}$, where $$ \widetilde\tau_n= {\sin\psi_n/2\over \psi_n /2}\qq \qq \psi_n= \big({2n^\rho \over n}\big)^{1/2}.$$ Then,
\begin{equation*} \sup_{u\ge 0}\,\sup_{d< (\pi/\sqrt 2) n^{(1-\rho)/2} }\Big|{\mathbb P}\big\{ d| \mathcal B_{ n} +u\big\}-{1\over d}
\Big|\,\le\, e^{-(1-\eta)\, n^\rho}. \end{equation*}
\end{proposition}
\begin{proof}[Proof of Theorem \ref{saud1}] We use the Bernoulli part extraction displayed at Lemma \ref{lemd}, \eqref{dec0}, \eqref{dec} as well as the notation introduced. Let
\begin{eqnarray}\label{dep0}A_n=\big\{B_n\le (1-\e)\nu_n \big\} . \end{eqnarray}
We deduce from Lemma \ref{di.1} that ${\mathbb P}\{A_n \} \, \le e^{- \frac{\epsilon^2\nu_n}{2}}$ for all positive $n$. We write
\begin{equation}\label{dep..} {\mathbb P} \{d|S_n \} -{1\over d} \,=\, {\mathbb E \,}_{(V,\e)}
\, \big( \chi(A_n)+\chi(A_n^c)\big)
\, \, \Big({\mathbb P}_{\!L}
\big\{d|\big( \sum_{j= 1}^n \e_jL_j+W_n \big) \big\}-{1\over d}\Big)
. \end{equation}
On the one hand,
\begin{eqnarray}\label{proof.th.saud1}& & {\mathbb E \,}_{(V,\e)}
\chi(A_n)
\, \Big| {\mathbb P}_{\!L}
\big\{d|\big( \sum_{j= 1}^n \e_jL_j+W_n \big) \big\}
-{1\over d}
\Big|\ \le \,2 {\mathbb P}\{A_n \} \, \le 2 e^{- \frac{\epsilon^2 }{2}\nu_n}. \end{eqnarray}
So that
\begin{equation}\label{dep1} \big|{\mathbb P} \{d|S_n \} -{1\over d} \big| \,\le \,2 e^{- \frac{\epsilon^2 }{2}\nu_n}+ {\mathbb E \,}_{(V,\e)}
\,
\chi(A_n^c) \, \cdot \, \Big|{\mathbb P}_{\!L}
\big\{d|\big( \sum_{j= 1}^n \e_jL_j+W_n \big)
\big\}-{1\over d}\Big|
. \end{equation}
Now on $A_n^c$, $B_n\ge (1-\epsilon)\nu_n $, and since $ \sqrt{ x / \log x}$ is increasing on $[e,\infty)$, we have \begin{equation}\label{phintaun1}
\sqrt{{ (1-\epsilon)\nu_n \over 2\a\log (1-\epsilon)\nu_n}}\le \sqrt{{ B_n \over 2\a\log B_n}}.
\end{equation} Also \begin{equation}\label{phintaun2} \p_n=\sqrt{\frac {2\a\log B_n}{ B_n}}\le \sqrt{ \frac{ 2\a\log (1-\epsilon)\nu_n}{ (1-\epsilon)\nu_n }} \quad \hbox{\rm and \ thus} \quad {\sin\p_n/2\over \p_n /2}\ge (\a^\prime/\a)^{1/2}, \end{equation}
by the assumption made.
\vskip 2 pt By applying Proposition \ref{special.cases}, we have ${\mathbb P}_{(V,\e)}$ almost surely, \begin{equation*}
\sup_{u\ge 0}\,\sup_{d< \pi \sqrt{ B_n \over 2\a\log B_n}}\Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big|\,\le\, B_n^{-\a'}.\end{equation*} Whence on $A_n^c$,
\begin{eqnarray}\label{proof.th.saud2}& & \sup_{u\ge 0}\,\sup_{d< \pi \sqrt{ (1-\epsilon)\nu_n \over 2\a\log (1-\epsilon)\nu_n}}\Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big|
\cr &\le& \sup_{u\ge 0}\,\sup_{d< \pi \sqrt{ B_n \over 2\a\log B_n}}\ \Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big|\cr &\le& B_n^{-\a'}\, \le \,\big( (1-\epsilon)\nu_n\big)^{-\a'} . \end{eqnarray}
In view of \eqref{dep1} and \eqref{proof.th.saud2}, we get for all $u\ge 0$ and $d< \pi \sqrt{{ (1-\epsilon)\nu_n \over 2\a\log (1-\epsilon)\nu_n}}$,
\begin{eqnarray}
\big| {\mathbb P} \{d|S_n+u \} - {1\over d} \big|
&\le & 2 e^{- \frac{\epsilon^2 }{2}\nu_n}+ \,\big( (1-\epsilon)\nu_n\big)^{-\a'} {\mathbb E \,}_{(V,\e)}
\, \chi(A_n^c)
\cr &\le &2 e^{- \frac{\epsilon^2 }{2}\nu_n}+
\,\big( (1-\epsilon)\nu_n\big)^{-\a'} . \end{eqnarray} \end{proof}
\vskip 3 pt \vskip 3 pt The next result shows a considerable variation of the speed of convergence when $d$ is less close to $\sqrt{\nu_n}$.
\begin{theorem}\label{saud2} Let $0<\rho<1 $ and $0<\e<1$.Then for each $n$ such that
$$|x|\le\frac12 \,\sqrt{ \frac{ 2 }{ ((1-\epsilon)\nu_n)^{1-\rho} }}\qq \Rightarrow \qq{\sin x\over x}\ge \sqrt{1-\e}$$
we have\begin{eqnarray*}
\sup_{u\ge 0}\,\sup_{d< (\pi/\sqrt 2) ((1-\e)\nu_n)^{(1-\rho)/2} }\ \big| {\mathbb P} \{d|S_n+u \} - {1\over d} \big|
&\le & 2 e^{- \frac{\epsilon^2 }{2}\nu_n}+e^{- ( (1-\epsilon)\nu_n)^\rho}
. \end{eqnarray*} \end{theorem}
\begin{proof} The proof is similar. We operate with the same set $A_n$ as in \eqref{dep0}, and use the decomposition \eqref{dep}. Let $0<\rho<1 $ and $0<\e<1$.
By applying Proposition \ref{special.cases} with $\eta=\e$, we have ${\mathbb P}_{(V,\e)}$ almost surely, for
$n$ such that $\widetilde\tau_n\ge \sqrt{1-\e}$,
where here $$ \widetilde\tau_n= {\sin\psi_n/2\over \psi_n /2}\qq {\rm with}\qq \psi_n= \big({2B_n^\rho \over B_n}\big)^{1/2},$$
\begin{equation*} \sup_{u\ge 0}\,\sup_{d< (\pi/\sqrt 2) B_n^{(1-\rho)/2} }\Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big|\,\le\, e^{-(1-\e) B_n^\rho}. \end{equation*}
By using corresponding estimates to \eqref{phintaun1}, \eqref{phintaun2}, namely that on $A_n^c$, $$\psi_n=\Big(\frac{2}{B_n^{1-\rho}}\Big)^{1/2}\le \Big(\frac{2}{((1-\e)\nu_n)^{1-\rho}}\Big)^{1/2}, $$ so that $\widetilde\tau_n\ge \sqrt{1-\e}$, we deduce that on
$A_n^c$,$$ \sup_{u\ge 0}\,\sup_{d< (\pi/\sqrt 2) ((1-\e)\nu_n)^{(1-\rho)/2} }\ \Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big|
$$ $$\,\le\, \sup_{u\ge 0}\,\,\sup_{d< (\pi/\sqrt 2) B_n^{(1-\rho)/2} }\ \Big|{\mathbb P}_{\!L}
\Big\{d\,\big|\Big( \sum_{j=1}^{B_n } L_j+W_n +u \Big) \Big\}-{1\over d}
\Big| \,\le\, e^{-(1-\e) B_n^\rho}.$$
Therefore
\begin{eqnarray} && \sup_{u\ge 0}\,\sup_{d< (\pi/\sqrt 2) ((1-\e)\nu_n)^{(1-\rho)/2} }\ \big| {\mathbb P} \{d|S_n+u \} - {1\over d} \big| \cr &\le & 2 e^{- \frac{\epsilon^2 }{2}\nu_n}+ {\mathbb E \,}_{(V,\e)}
\, \chi(A_n^c)
\,e^{-(1-\e) B_n^\rho} \,\le \, 2 e^{- \frac{\epsilon^2 }{2}\nu_n}+e^{-(1-\e)^{1+\rho} \nu_n ^\rho}
. \end{eqnarray}
\end{proof}
\begin{remark} So far we only have considered necessary conditions for the validity of the local limit theorem, which are formulated in terms of a.u.d. property, as well as strenghtenings of this property yielding effective speed of convergence bounds. It is important to mention in that context, that in 1984, Mukhin found a remarkable necessary and sufficient condition for the validity of the local limit theorem. Let $\{S_n,n\ge 1\}$ be a sequence of ${\mathbb Z}$--valued random variables such that an integral limit theorem holds: there exist $a_n\in {\mathbb R}$ and real $b_n\to \infty$ such that the sequence of distributions of $(S_n-a_n)/b_n$ converges weakly to an absolutely continuous distribution $G$ with density $g(x)$, which is uniformly continuous in ${\mathbb R}$.
The local limit theorem is valid if \begin{equation}\label{ilt.llt} {\mathbb P}\{S_n=m\}=B_n^{-1} g\Big(\frac{m-A_n}{B_n}\Big) + o(B_n^{-1}), \end{equation} uniformly in $m\in {\mathbb Z}$. Muhkin showed that the validity of the local limit theorem is equivalent to the existence
of a sequence of integers $v_n=o(b_n)$ such that
\begin{equation}\label{ilt.llt.diff} \sup_{m}\Big|{\mathbb P}\{S_n=m+v_n\big\}-{\mathbb P}\{S_n=m \big\}\Big|\,=\,\,o\Big(\frac{1}{b_n}\Big). \end{equation}
Revisiting the succint proof given in \cite{Mu2}, we however could only prove rigorously a weaker necessary and sufficient condition, with a significantly different formulation, namely that a necessary and sufficient condition for the local limit theorem in the usual form to hold is
\begin{equation} \sup_{m, k\in{\mathbb Z}\atop |m-k|\le \max\{1, [\sqrt \e_n b_n]\}}\Big|{\mathbb P}\{S_n=m\big\}-{\mathbb P}\{S_n=k \big\}\Big|\,=\,\,o\Big(\frac{1}{b_n}\Big),
\end{equation} where \begin{equation} \label{en}\e_n:=\sup_{x\in {\mathbb R}}\Big|{\mathbb P}\Big\{\frac{S_n-a_n}{b_n}<x\Big\}-G(x)\Big|\ \to \ 0, \end{equation} by the integral limit theorem. This is the object of the Note \cite{W5}, with remarks and references on general relations of type \eqref{ilt.llt.diff} therein. Mukhin wrote at this regard in \cite{Mu2}: \lq\lq ... getting from here more general sufficient conditions turns out to be difficult in view of the lack of good criteria. Working with asymptotic equidistribution properties are more convenient in this respect\,\rq\rq.
\end{remark}
\appendix
\vskip 6pt \section{LLT's with speed of convergence.} Let $S_n=X_1+\ldots +X_n$, $n\ge 1$, where $X_j$ are independent random variables such that
${\mathbb P}\{X_j \in\mathcal L(v_{ 0},D )\}=1$. \vskip 3 pt Assume first that the random variables $X_j$ are identically distributed. Then we have the following characterization result. \begin{theorem} \label{r} Let $F$ denote the distribution function of $X_1$.
{\rm (i) (\cite{IBLIN}, Theorem 4.5.3)} In order that the property \begin{equation} \label{alfa}
\sup_{N=an+Dk}\Big|
{ \frac{\s \sqrt n}{D} }{\mathbb P}\{S_n=N\}-{1 \over \sqrt{ 2\pi}\s}e^{-
{(N-n\m )^2\over 2 n \s^2} }\Big| ={\mathcal O}\big(n^{-\alpha{/2}} \big) ,
\end{equation}
{ where $0<\a<1$},
it is necessary and sufficient that the following conditions be satisfied:
\begin{eqnarray*} (1) \ D \ \hbox{is maximal}, \ \qq\qq
(2) \ \ \int_{|x|\ge u} x^2 F(dx) = \mathcal O(u^{-\a})\quad \hbox{as $u\to \infty$.} \end{eqnarray*}
{\rm (ii) (\cite{P} Theorem 6 p.\,197)} If ${\mathbb E\,} |X_1|^3<\infty$, then \eqref{alfa} holds with $\a =1/2$.
\end{theorem}
\vskip 6 pt Now consider the non-identically distributed case. Assume that (see \eqref{vartheta}) \begin{equation}\label{basber.pos} \t_{X_j}>0, \qq \quad j=1,\ldots, n. \end{equation} Let $\nu_n =\sum_{j=1}^n \t_j$. Let $\psi:{\mathbb R}\to {\mathbb R}^+$ be even, convex and such that $\frac {\psi(x)}{x^2}$ and $\frac{x^3}{\psi(x)}$ are non-decreasing on ${\mathbb R}^+$. We further assume that
\begin{equation}\label{did} {\mathbb E \,} \psi( X_j )<\infty .
\end{equation} Put $$L_n=\frac{ \sum_{j=1}^n{\mathbb E \,} \psi (X_j) } { \psi (\sqrt { {\rm Var}(S_n )})} .$$
The following result is Corollary 1.7 in Giuliano-Weber in \cite{GW3}.
\begin{theorem}\label{ger3} Assume that $\frac{ \log \nu_n }{\nu_n}\le {1}/{14} $. Then, for all $\k\in \mathcal L( v_{ 0}n,D )$ such that $$\frac{(\k- {\mathbb E \,} S_n)^2}{ {\rm Var}(S_n) } \le \sqrt{\frac{7 \log \nu_n} {2\nu_n}},$$ we have
\begin{eqnarray*} \Big| {\mathbb P} \{S_n =\kappa \} -{ D e^{- \frac{(\k- {\mathbb E \,} S_n)^2}{ 2 {\rm Var}(S_n) } } \over \sqrt{2\pi {\rm Var}(S_n) }} \Big| & \le & C_3\Big\{ D\big({ { \log \nu_n } \over
{ {\rm Var}(S_n) \nu_n} } \big)^{1/2} + { L_n
+ \nu_n^{-1} \over \sqrt{ \nu_n} } \Big\} .
\end{eqnarray*} And $C_3=\max (C_2, 2^{ 3/2}C_{{\rm E}}) $, $C_{{\rm E}}$ being an absolute constant arising from Berry-Esseen's inequality. \end{theorem}
\vskip 8 pt We pass to another speed of convergence result due to Mukhin. Consider the structural characteristic of a random variable $X$, introduced and studied by Mukhin in \cite{Mu1} and \cite{Mu} for instance, $$ H(X ,d) = {\mathbb E \,} \langle X^*d\rangle^2,$$ where $\langle \a \rangle$ denotes the distance from $\a$ to the nearest integer, and $X^*$ is a symmetrization of $X$. Let $\p_X$ be the characteristic function $X$.
The two-sided inequality
\begin{eqnarray}\label{fih} 1-2\pi^2 H(X ,\frac{t }{2\pi}) \le |\p_X(t)|\le 1-4 H(X ,\frac{t }{2\pi}) , \end{eqnarray} is established in the above references. See also Szewczak and Weber \cite{SW} for more.
The following is the one-dimensional version of Theorem 5 in \cite{Mu}, see also \cite{SW} and is stated without proof, however.
\begin{theorem}[Mukhin]\label{Mukhin.th.Hn} Let $X_1,\ldots, X_n$ have zero mean and finite third moments. Let
$$ B_n^2= \sum_{j=1}^n{\mathbb E\,} |X_j|^2 ,\qq H_n= \inf_{1/4\le d\le 1/2}\sum_{j=1}^n H(X_j
,d), \qq L_n= \frac{\sum_{j=1}^n{\mathbb E\,} |X_j|^3}{(B_n)^{3/2}} .$$ Then
\begin{equation}\label{llt} \sup_{N=v_0n+Dk }\Big|B_n {\mathbb P}\{S_n=N\}-{D\over \sqrt{ 2\pi } }e^{-
{(N-M_n)^2\over 2 B_n^2} }\Big|\,\le CL_n\, \big( {B_n }/{ H_n}\big) . \end{equation} \end{theorem} \vskip 8 pt
\end{document} |
\begin{document}
\title{Cherlin's conjecture for almost simple groups of Lie rank $1$}
\author{Nick Gill} \address{ Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K.} \email{nick.gill@southwales.ac.uk}
\author{Francis Hunt} \address{ Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K.} \email{francis.hunt@southwales.ac.uk}
\author{Pablo Spiga} \address{Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy} \email{pablo.spiga@unimib.it}
\begin{abstract}
We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to ${\rm PSL}_2(q)$, $\mathop{{^2\mathrm{B}_2}(q)}$, $\mathop{{^2\mathrm{G}_2}(q)}$ or $\mathrm{PSU}_3(q)$. Our method uses the notion of a ``strongly non-binary action''. \end{abstract}
\maketitle
\section{Introduction}
All groups in this paper are finite. In this note our main result is the following.
\begin{thm}\label{t: psl2}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to a linear group ${\rm PSL}_2(q)$, or to a Suzuki group $\mathop{{^2\mathrm{B}_2}(q)}$, or to a Ree group $\mathop{{^2\mathrm{G}_2}(q)}$, or to a unitary group $\mathrm{PSU}_3(q)$. Then, either $G$ is not binary, or $G=\mathop{\mathrm{Sym}}(\Omega)\cong \mathop{\mathrm{Sym}}(5)\cong \mathrm{P\Gamma L}_2(4)\cong \mathrm{PGL}_2(5)$, or $G=\mathop{\mathrm{Sym}}(\Omega)\cong \mathop{\mathrm{Sym}}(6)\cong \mathrm{P}\Sigma\mathrm{L}_2(9)$. \end{thm}
Theorem~\ref{t: psl2} is a contribution towards a proof of a conjecture of Cherlin \cite{cherlin1}. This conjecture asserts that a primitive binary permutation group lies on a short explicit list of known actions.
The precise definition of ``binary'' and ``binary action'' is given in Section~\ref{s: bin back} below. An equivalent definition, couched in terms of ``relational structures'', can be found in \cite{cherlin2}; the connection between this conjecture and Lachlan's theory of sporadic structures can be found in \cite{cherlin1}. It is this connection that really enlivens the study of binary permutation groups, and provides motivation to work towards a proof of Cherlin's conjecture.
Let us briefly describe the status of this conjecture. By work of Cherlin \cite{cherlin2} and Wiscons \cite{wiscons}, this very general conjecture has been reduced to the following statement concerning almost simple groups.
\begin{conj}\label{conj: cherlin}
If $G$ is a binary almost simple primitive permutation group on the set $\Omega$, then $G=\mathrm{Sym}(\Omega)$. \end{conj}
One sees immediately that Theorem~\ref{t: psl2} settles Conjecture~\ref{conj: cherlin} for almost simple primitive permutation groups with socle isomorphic to ${\rm PSL}_2(q)$, or $\mathop{{^2\mathrm{B}_2}(q)}$, or $\mathop{{^2\mathrm{G}_2}(q)}$, or $\mathrm{PSU}_3(q)$, that is, for each Lie type group of twisted Lie rank $1$. Theorem~\ref{t: psl2} is the third recent result of this type; in recent work, the first and third authors settled Conjecture~\ref{conj: cherlin} for groups with alternating socle, and for the $\mathcal{C}_1$ primitive actions of groups with classical socle \cite{gs_binary}.
A brief word about our methods: the aforementioned work on groups with alternating or classical socle was based on the study of so-called ``beautiful subsets''. These objects are defined below, and their usefulness is explained by Lemma~\ref{l: forbidden} and Example~\ref{ex: snba1} below, which together imply that whenever an action admits a beautiful subset the action is not binary.
In the current note our approach is different for the reason that the family of actions under consideration -- the primitive actions of almost simple groups with socle a Lie group of Lie rank $1$-- very often do not have beautiful subsets.
To deal with this situation we need to develop a more general theory: Suppose that we have a group $G$ acting on a set $\Omega$, and we want to show that this action is non-binary. The key property of beautiful subsets that makes them useful is that they allow us to argue ``inductively'', in the sense that if we can find a subset $\Lambda$ of $\Omega$ that is ``beautiful'', then the full action of $G$ on $\Omega$ is non-binary. In order to deal with the absence of beautiful subsets, we have studied this inductive property more formally via the notion of a ``strongly non-binary subset''. The theory of such subsets is developed in \S\ref{s: bin back} and allows us to apply an inductive argument in a more general setting.
The advantages of Theorem~\ref{t: psl2} and of this theory are several: firstly, Theorem~\ref{t: psl2} is a material advance towards a proof of Conjecture~\ref{conj: cherlin}; secondly, it demonstrates the possibility of obtaining results in situations where one cannot use the notion of a beautiful subset, as in~\cite{gs_binary}; thirdly, it turns out that the rank $1$ groups tend to be a sticking point when making general arguments concerning binary groups. We hope, therefore, that by disposing of this case here, we will be able to deal more easily with the remaining cases required for a proof of Cherlin's conjecture. Investigation in this direction is in progress, see~\cite{gls_binary}.
\subsection{Structure of the paper}
The proof of Theorem~\ref{t: psl2} is split into several parts. First, in \S\ref{s: bin back}, after giving a number of definitions, we prove some general results about binary actions; in particular Lemma~\ref{l: forbidden} is vital.
In \S\ref{s: structure} we give some basic information concerning groups with socle isomorphic to $\mathrm{PSL}_2(q)$; then in \S\ref{s: fp} we calculate the size of the fixed set for various elements of $\mathrm{P\Gamma L}_2(q)$ in various primitive actions; these results are then used to prove Lemmas~\ref{l: handy q odd} and \ref{l: handy q even}; it is worth remarking that these fixed point calculations yield the required conclusions almost immediately for the groups ${\rm PSL}_2(q)$ and $\mathrm{PGL}_2(q)$, however a finer analysis is required to deal with those almost simple groups that contain field automorphisms. The three lemmas just mentioned -- Lemmas~\ref{l: forbidden}, \ref{l: handy q odd} and \ref{l: handy q even} -- directly imply Theorem~\ref{t: psl2} for $\mathrm{PSL}_2(q)$ when $q\geq 9$. The remaining small cases, when $q\in\{4,5,7,8\}$, can be verified directly using GAP \cite{GAP} or by referencing the calculations of Wiscons \cite{wiscons2}.
In \S\ref{s: suzuki}, \S\ref{s: ree} and \S\ref{s: psu}, we give a proof of Theorem~\ref{t: psl2} for groups with socle $\mathop{{^2\mathrm{B}_2}(q)}$, $\mathop{{^2\mathrm{G}_2}(q)}$ and $\mathrm{PSU}_3(q)$, respectively. In the first two cases the theorems are easy consequences of propositions asserting that the primitive actions in question admit strongly non-binary subsets (see \S\ref{s: bin back} for the definition of a strongly non-binary subset). The final case -- socle $\mathrm{PSU}_3(q)$ -- is dealt with somewhat differently.
\section{Binary actions and strongly non-binary actions}\label{s: bin back}
Throughout this section $G$ is a finite group acting (not necessarily faithfully) on a set $\Omega$ of cardinality $t$. Here, our job is to give a definition of ``binary action'', and of ``strongly non-binary action'', and to connect these definitions to earlier work on ``beautiful sets''. Given a subset $\Lambda$ of $\Omega$, we write $G_\Lambda:=\{g\in G\mid \lambda^g\in\Lambda,\forall \lambda\in \Lambda\}$ for the set-wise stabilizer of $\Lambda$, $G_{(\Lambda)}:=\{g\in G\mid \lambda^g=\lambda, \forall\lambda\in \Lambda\}$ for the point-wise stabilizer of $\Lambda$, and $G^\Lambda$ for the permutation group induced on $\Lambda$ by the action of $G_\Lambda$. In particular, $G^\Lambda\cong G_\Lambda/G_{(\Lambda)}$.
Given a positive integer $r$, the group $G$ is called \textit{$r$-subtuple complete} with respect to the pair of $n$-tuples $I, J \in \Omega^n$, if it contains elements that map every subtuple of size $r$ in $I$ to the corresponding subtuple in $J$ i.e. $$\textrm{for every } k_1, k_2, \dots, k_r\in\{ 1, \ldots, n\}, \textrm{ there exists } h \in G \textrm{ with }I_{k_i}^h=J_{k_i}, \textrm{ for every }i \in\{ 1, \ldots, r\}.$$ Here $I_k$ denotes the $k^{\text{th}}$ element of tuple $I$ and $I^g$ denotes the image of $I$ under the action of $g$. Note that $n$-subtuple completeness simply requires the existence of an element of $G$ mapping $I$ to $J$.
The group $G$ is said to be of {\it arity $r$} if, for all $n\in\mathbb{N}$ with $n\geq r$ and for all $n$-tuples $I, J \in \Omega^n$, $r$-subtuple completeness (with respect to $I$ and $J$) implies $n$-subtuple completeness (with respect to $I$ and $J$). When $G$ has arity 2, we say that $G$ is {\it binary}. A pair $(I,J)$ of $n$-tuples of $\Omega$ is called a {\it non-binary witness for the action of $G$ on $\Omega$}, if $G$ is $2$-subtuple complete with respect to $I$ and $J$, but not $n$-subtuple complete, that is, $I$ and $J$ are not $G$-conjugate. To show that the action of $G$ on $\Omega$ is non-binary it is sufficient to find a non-binary witness $(I,J)$.
We say that the action of $G$ on $\Omega$ is \emph{strongly non-binary} if there exists a non-binary witness $(I,J)$ such that \begin{itemize}
\item $I$ and $J$ are $t$-tuples where $|\Omega|=t$;
\item the entries of $I$ (resp. $J$) are distinct entries of $\Omega$. \end{itemize}
\begin{example}\label{ex: snba1}{\rm If $G$ acts $2$-transitively on $\Omega$ with kernel $K$ and $G/K\cong G^\Omega\not\cong\mathop{\mathrm{Sym}}(\Omega)$, then $G$ is strongly non-binary.
Indeed, by $2$-transitivity, any pair $(I,J)$ of $t$-tuples of distinct elements from $\Omega$ is $2$-subtuple complete. Since $G/K\cong G^\Omega\not\cong\mathop{\mathrm{Sym}}(\Omega)$, we can choose $I$ and $J$ in distinct $G$-orbits. Thus $(I,J)$ is a non-binary witness.} \end{example}
\begin{example}\label{ex: snba2}{\rm Let $G$ be a subgroup of $\mathop{\mathrm{Sym}}(\Omega)$, let $g_1, g_2,\ldots,g_r$ be elements of $G$, and let $\tau,\eta_1,\ldots,\eta_r$ be elements of $\mathop{\mathrm{Sym}}(\Omega)$ with \[
g_1=\tau\eta_1,\,\,g_2=\tau\eta_2,\,\,\ldots,\,\,g_r=\tau\eta_r. \] Suppose that, for every $i\in \{1,\ldots,r\}$, the support of $\tau$ is disjoint from the support of $\eta_i$; moreover, suppose that, for each $\omega\in\Omega$, there exists $i\in\{1,\ldots,r\}$ (which may depend upon $\omega$) with $\omega^{\eta_i}=\omega$. Suppose, in addition, $\tau\notin G$. Now, writing $\Omega=\{\omega_1,\dots, \omega_t\}$, observe that
\[
((\omega_1,\omega_2,\dots, \omega_t), (\omega_1^{\tau},\omega_2^{\tau}, \ldots,\omega_t^{\tau}))
\] is a non-binary witness. Thus the action of $G$ on $\Omega$ is strongly non-binary.} \end{example}
The notion of a strongly non-binary action allows us to ``argue inductively'' using suitably chosen set-stabilizers. The following lemma (which was first stated in~\cite{gs_binary} and which, in any case, is virtually self-evident) clarifies what we mean by this.
\begin{lem}\label{l: forbidden} Suppose that there exists a subset $\Lambda \subseteq \Omega$ such that $G^\Lambda$ is strongly non-binary. Then $G$ is not binary. \end{lem}
In what follows a \emph{strongly non-binary subset} is a subset $\Lambda$ of $\Omega$ such that $G^\Lambda$ is strongly non-binary.
We are ready for the third main concept of this section, that of a ``beautiful subset''; this is closely related to the example of a strongly non-binary action given in Example~\ref{ex: snba1}. Specifically, we say that a subset $\Lambda\subseteq \Omega$ is a \emph{$G$-beautiful subset} if $G^\Lambda$ is a $2$-transitive subgroup of $\mathrm{Sym}(\Lambda)$ which is neither $\mathrm{Alt}(\Lambda)$ nor $\mathrm{Sym}(\Lambda)$. Note that we will tend to drop the ``$G$'' in $G$-beautiful, so long as the context is clear (for instance, when $G$ is a permutation group on $\Omega$, that is, $G\le\mathop{\mathrm{Sym}}(\Omega)$).
In the light of Example~\ref{ex: snba1}, the curious reader may be wondering why the definition of a beautiful subset excludes also the possibility that $G^\Lambda=\mathrm{Alt}(\Lambda)$. This exclusion is explained by the following lemma, which is~\cite[Corollary 2.3]{gs_binary}.
\begin{lem}\label{l: beautiful}
Suppose that $G$ is almost simple with socle $S$. If $\Omega$ contains an $S$-beautiful subset, then $G$ is not binary. \end{lem}
In what follows we will see a number of examples of strongly non-binary actions of the types given in Examples~\ref{ex: snba1} and \ref{ex: snba2}, as well as examples of beautiful subsets. To study these examples we will make use of the fact that the finite faithful 2-transitive actions are all known thanks to the Classification of Finite Simple Groups.
One naturally wonders whether other examples of strongly non-binary witnesses exist. This is indeed the case and the existence of a strongly non-binary witness is related to the classic concept of $2$-closure introduced by Wielandt~\cite{Wielandt}. Given a permutation group $G$ on $\Omega$, the \emph{$2$-closure of $G$} is the set $$G^{(2)}:=\{\sigma\in \mathop{\mathrm{Sym}}(\Omega)\mid \forall (\omega_1,\omega_2)\in \Omega\times \Omega, \textrm{there exists }g_{\omega_1\omega_2}\in G \textrm{ with }\omega_1^\sigma=\omega_1^{g_{\omega_1\omega_2}}, \omega_2^\sigma=\omega_2^{g_{\omega_1\omega_2}}\},$$ that is, $G^{(2)}$ is the largest subgroup of $\mathop{\mathrm{Sym}}(\Omega)$ having the same orbitals as $G$. The group $G$ is said to be $2$-closed if and only if $G=G^{(2)}$. We claim that $G$ is not $2$-closed if and only if $G$ has a strongly non-binary witness. Write $\Omega:=\{\omega_1,\ldots,\omega_t\}$. If $G$ is not $2$-closed, then there exists $\sigma\in G^{(2)}\setminus G$. Now, it is easy to verify that $I:=(\omega_1,\ldots,\omega_t)$ and $J:=I^\sigma=(\omega_1^\sigma,\ldots,\omega_t^\sigma)$ are $2$-subtuple complete (because $\sigma\in G^{(2)}$) and are not $G$-conjugate (because $g\notin G$). Thus $(I,J)$ is a strongly non-binary witness. The converse is similar.
\section{Groups with socle isomorphic to \texorpdfstring{$\mathrm{PSL}_2(q)$}{PSL2(q)}}\label{s: structure}
In this section we start by studying some of the basic properties of involutions and Klein $4$-subgroups of the almost simple groups $G$ with socle $\mathrm{PSL}_2(q)$. (In particular, $\mathrm{PSL}_2(q)\le G\le \mathrm{P\Gamma L}_2(q)$.) All of these properties are well-known and/or easy to verify by direct calculation. We also set up some basic notation for what follows.
For a group $J$, write $m_2(J)$ for the \emph{$2$-rank} of $J$, i.e.\ the maximum rank of an abelian $2$-subgroup of $J$. If $q$ is odd and $J$ is a section of $G$ (i.e.\ a quotient of a subgroup of $G$), then $m_2(J)\leq 3$. What is more, $m_2(J)\leq 2$ unless $q$ is a square and $G$ contains a field automorphism of order $2$.
\begin{lem}\label{l: quotients split}
Let $L$ be a subgroup of $\mathrm{PGL}_2(q)$ with $q$ odd, and let $K$ be a subgroup of $\nor {\mathrm{PGL}_2(q)} L $ with $K$ isomorphic to a Klein $4$-group and with $K\cap L=1$. Then $|L|$ is odd. \end{lem} \begin{proof} Let $P$ be a Sylow $2$-subgroup of $\langle K,L\rangle=K\ltimes L$ containing $K$. Then $P=K\ltimes Q$, for some Sylow $2$-subgroup $Q$ of $L$. If $Q\ne 1$, then $K$ centralises a non-identity element of $Q$ and hence $m_2(\mathrm{PGL}_2(q))\ge m_2(P)=m_2(K\ltimes Q)\ge m_2(K)+m_2(\cent Q K)\ge 2+1=3$, a contradiction. \end{proof}
Suppose that $q$ is odd. There is exactly one $\mathrm{PGL}_2(q)$-conjugacy class of Klein $4$-subgroups of $\mathrm{PSL}_2(q)$, and one can check directly that $\cent{\mathrm{PGL}_2(q)} K =K$ for each Klein $4$-subgroup of $\mathrm{PSL}_2(q)$. When $q\equiv \pm 3\pmod 8$, a Sylow $2$-subgroup of $\mathrm{PSL}_2(q)$ is a Klein $4$-subgroup and, by Sylow's theorems, there is exactly one $\mathrm{PSL}_2(q)$-conjugacy class of Klein $4$-subgroups of $\mathrm{PSL}_2(q)$; in this case $\nor {\mathrm{PSL}_2(q)} K \cong \mathop{\mathrm{Alt}}(4)$. When $q\equiv \pm 1\pmod 8$, there are two $\mathrm{PSL}_2(q)$-conjugacy classes of Klein $4$-subgroups of $\mathrm{PSL}_2(q)$ and these are fused in $\mathrm{PGL}_2(q)$; in this case $\nor {\mathrm{PSL}_2(q)} K \cong\mathop{\mathrm{Sym}}(4)$.
We need information concerning involutions in $\mathrm{P\Gamma L}_2(q)\setminus\mathrm{PGL}_2(q)$ -- such involutions must be field automorphisms, as defined in~\cite{gls3}. The following result is a special case of \cite[Prop. 4.9.1]{gls3}.
\begin{lem}\label{l: fields}
Let $f_1,f_2\in\mathrm{P\Gamma L}_2(q)\setminus\mathrm{PGL}_2(q)$ be of order $t$ for some prime $t$, and suppose that $f_1\mathrm{PGL}_2(q)=f_2\mathrm{PGL}_2(q)$. Then $f_1$ and $f_2$ are $\mathrm{PGL}_2(q)$-conjugate. \end{lem}
\subsection{Fixed point calculations}\label{s: fp}
We let $G$ be a group having socle $S$ with $S\cong \mathrm{PSL}_2(q)$. Using the classification of the maximal subgroups of $G$ (see for example~\cite{bhr}), it is important to observe that, for every maximal subgroup $M$ of $G$ there exists a maximal subgroup $H$ of $S$ with $M=\nor G H$; in particular, this allows us to identify (up to permutation isomorphism) each primitive $G$-set $\Omega$ with the set of $G$-conjugates of some maximal subgroup $H$ of $S$. Therefore, we let $H$ be a maximal subgroup of $S$ with $\nor G H $ maximal in $G$, and set $\Omega$ to be $H^G:=\{H^g\mid g\in G\}$, the set of all conjugates of $H$ in $G$. All possibilities for $H$ and $|\Omega|$ are given in the first and in the third column of Tables~\ref{t: inv q odd} and \ref{t: inv q even}, where in Table~\ref{t: inv q odd} the symbol $\zeta$ is defined by \begin{equation}\label{e: zeta}
\zeta:=\begin{cases}
2 & \textrm{if }G\not\le \mathrm{P}\Sigma L_2(q) \textrm{ and }q \textrm{ is odd}, \textrm{ or }q\textrm{ is even},\\
1 & \textrm{if }G\le \mathrm{P}\Sigma L_2(q) \textrm{ and }q \textrm{ is odd}.
\end{cases} \end{equation} (See \cite{bhr} to verify this. The conditions that are listed in Table~\ref{t: inv q odd} are necessary for the action of $G$ on $\Omega$ to be primitive, but they are not necessarily sufficient.) Finally, we write $\mathcal{P}(H)$ for the power set of $H$.
In what follows, we calculate the number of fixed points of an involution $g\in S$, and (when $q$ is odd) of a Klein $4$-subgroup $K\leq S$, for the action of $G$ on $\Omega$. (Given a subset $Y$ of a permutation group $X$ on $\Omega$, we write ${\rm Fix}_\Omega(Y):=\{\omega\in \Omega\mid \omega^y=\omega,\forall y\in Y\}$ and simply ${\rm Fix}_\Omega(y)$ when the set $Y$ consists of the single element $y$.)
To calculate the number of fixed points of $g$ and of $K$, we make use of the well-known formulas (see for instance~\cite[Lemma~$2.5$]{LiebeckSaxl}) \begin{equation}\label{e: fora}
|{\rm Fix}_\Omega(g)| = \frac{|\Omega|\cdot |H\cap g^G|}{|g^G|},\qquad
|{\rm Fix}_\Omega(K)| = \frac{|\Omega|\cdot |\mathcal{P}(H)\cap K^G|}{|K^G|}. \end{equation}
Given an involution $g\in S$, from~\cite{gls3} we obtain \[
|g^G|=\begin{cases}
\frac12q(q-1), & \textrm{if } q\equiv 3\pmod 4,\\
\frac12q(q+1), & \textrm{if }q\equiv 1\pmod 4, \\
q^2-1, &\textrm{if } q \textrm{ even}.
\end{cases} \] Using this information, Eq.~\eqref{e: fora} and the fact that $\mathrm{PSL}_2(q)$ has a unique conjugacy class of involutions, it is a straightforward computation to verify the fourth and fifth column in Table~\ref{t: inv q odd} and the third and fourth column in Table~\ref{t: inv q even}. \begin{table} \begin{adjustbox}{angle=90}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
$H$ & Conditions & $|\Omega|$ & $|H\cap g^G|$ & $|{\rm Fix}_\Omega(g)|$&$|\mathcal{P}(H)\cap K^G|$&$|{\rm Fix}_\Omega(K)|$\\ \hline $[q]:(\frac{q-1}{2})$ & None & $q+1$ & $\begin{cases}
0, & q\equiv 3(4) \\
q, & q\equiv 1(4)
\end{cases}$ & $\begin{cases}
0, & q\equiv 3(4) \\
2, & q\equiv 1(4)
\end{cases}$ &$0$&$0$\\ $D_{q-1}$ & None & $\frac{q(q+1)}{2}$ & $\begin{cases}
\frac{q-1}{2}, & q\equiv 3(4) \\
\frac{q+1}{2}, & q\equiv 1(4)
\end{cases}$ & $\frac{q+1}{2}$&$\begin{cases}0,&q\equiv 3(8)\\ \frac{q-1}{4},&q\equiv 5(8)\\\frac{\zeta(q+1)}{8},&q\equiv 1(8)\\0,&q\equiv 7(8)\end{cases}$&$\begin{cases}0,&q\equiv 3(8)\\ 3,&q\equiv 5(8)\\3,&q\equiv 1(8)\\0,&q\equiv 7(8)\end{cases}$\\ $D_{q+1}$ & None & $\frac{q(q-1)}{2}$ & $\begin{cases}
\frac{q+3}{2}, & q\equiv 3(4) \\
\frac{q+1}{2}, & q\equiv 1(4)
\end{cases}$ & $\begin{cases}
\frac{q+3}{2}, & q\equiv 3(4) \\
\frac{q-1}{2}, & q\equiv 1(4)
\end{cases}$ &$\begin{cases}\frac{q+1}{4},&q\equiv 3(8)\\0,&q\equiv 5(8)\\0,&q\equiv 1(8)\\\frac{\zeta(q+1)}{8},&q\equiv 7(8)\end{cases}$&$\begin{cases}3,&q\equiv 3(8)\\ 0,&q\equiv 5(8)\\0,&q\equiv 1(8)\\3,&q\equiv 7(8)\end{cases}$\\ $\mathrm{PSL}_2(q_0)$ & $q=q_0^a$, $a$\textrm{ odd } & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\begin{cases}
\frac{q_0(q_0-1)}{2}, & q_0\equiv 3(4)\\
\frac{q_0(q_0+1)}{2}, & q_0\equiv 1(4) \\
\end{cases}$ & $\begin{cases}
\frac{q+1}{q_0+1}, & q_0\equiv 3(4)\\
\frac{q-1}{q_0-1}, & q_0\equiv 1(4) \\
\end{cases}$ &$\begin{cases}\frac{q_0(q_0^2-1)}{24},&q\equiv \pm 3(8)\\\frac{\zeta q_0(q_0^2-1)}{48},&q\equiv \pm 1(8)\end{cases}$&$1$\\ $\mathrm{PGL}_2(q_0)$ & $q=q_0^2$, $\zeta=1$ & $\frac{\sqrt{q}(q+1)}{2}$ & $q$ & $\sqrt{q}$ &$\frac{q_0(q_0^2-1)}{24}$ or $\frac{q_0(q_0^2-1)}{8}$&$1$ or $3$\\ $\mathop{\mathrm{Alt}}(4)$ & $q=p\equiv \pm 3(8)$ & $\frac{q(q^2-1)}{24}$ & $3$ & $\begin{cases}
\frac{q+1}{4}, & q\equiv 3(8) \\
\frac{q-1}{4}, & q\equiv 5(8)
\end{cases}$ &$1$&$1$\\ $\mathop{\mathrm{Sym}}(4)$ & $q=p\equiv \pm 1(8)$, $\zeta=1$ & $\frac{q(q^2-1)}{48}$ & $9$ & $\begin{cases}
\frac{3(q+1)}{8}, & q\equiv 7(8) \\
\frac{3(q-1)}{8}, & q\equiv 1(8)
\end{cases}$ &$1$ or $3$&$1$ or $3$\\ $\mathop{\mathrm{Alt}}(5)$ & $\begin{array}{l}q=p, q\equiv \pm 1 (10), \textrm{ or}\\ q=p^2, p\equiv \pm 3(10) \end{array}$ & $\frac{\zeta q(q^2-1)}{120}$ & $15$ & $\begin{cases}
\frac{\zeta(q+1)}{4}, & q\equiv 3(4) \\
\frac{\zeta(q-1)}{4}, & q\equiv 1(4)
\end{cases}$&$5$&$\begin{cases}\zeta,&q\equiv \pm 3(8)\\2,&q\equiv \pm 1(8)\end{cases}$ \\
\hline \end{tabular} \end{adjustbox} \caption{Fixed points of involutions in $S$ and of a Klein $4$-subgroup of $S$, for $q$ odd. The symbol $\zeta$ is defined in~\eqref{e: zeta}.}\label{t: inv q odd} \end{table} \begin{table}
\begin{tabular}{|c|c|c|c|} \hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|{\rm Fix}_\Omega(g)|$\\ \hline $[q]:(q-1)$ & $q+1$ & $q-1$ & $1$ \\ $D_{2(q-1)}$ & $\frac12q(q+1)$ & $q-1$ & $\frac12q$ \\ $D_{2(q+1)}$ & $\frac12q(q-1)$ & $q+1$ & $\frac12q$ \\
$\mathrm{SL}_2(q_0)$ & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $q_0^2-1$ & $\frac{q}{q_0}$ \\
\hline \end{tabular} \caption{Fixed points of involutions in $S$ for $q$ even.}\label{t: inv q even} \end{table}
Suppose that $q\equiv \pm 3\pmod 8$ and let $K$ be a Klein $4$-subgroup of $S$. As we mentioned above, $K$ is a Sylow $2$-subgroup of $S$, all Klein $4$-subgroups of $S$ are conjugate and $\nor {\mathrm{PSL}_2(q)}K\cong \mathop{\mathrm{Alt}}(4)$. Therefore $|K^G|=\frac{1}{24}q(q^2-1)$. Using this and Eq.~\eqref{e: fora}, it is easy to confirm (when $q\equiv \pm 3\pmod 8$) the veracity of the sixth and seventh column in Table~\ref{t: inv q odd}. (Note that the $\mathrm{PGL}_2(q_0)$ and $\mathop{\mathrm{Sym}}(4)$ rows do not apply when $q\equiv \pm3\pmod 8$.)
Suppose now that $q\equiv \pm 1\pmod 8$ and let $K$ be a Klein $4$-subgroup of $S$. In this case, there are two $S$-conjugacy classes of Klein $4$-subgroups and, regardless of the $S$-conjugacy class on which $K$ lies, we have $\nor G K\cong \mathop{\mathrm{Sym}}(4)$. In particular, \[
|K^G|=\frac{1}{48}\zeta q(q^2-1), \] where $\zeta$ is the parameter that was defined in \eqref{e: zeta}. As above, using this and Eq.~\eqref{e: fora}, it is easy to confirm (when $q\equiv \pm 1\pmod 8$) the veracity of the sixth and seventh column in Table~\ref{t: inv q odd}. (Note that the $\mathop{\mathrm{Alt}}(4)$ row does not apply when $q\equiv \pm1\pmod 8$.)
For the proof of Theorem~\ref{t: psl2}, we also need to compute the number of fixed points of field involutions of $G$ only for certain primitive actions when $q$ is odd: this information is tabulated in Table~\ref{t: f field}. Of course, here we assume that $q$ is a square and that $G$ does contain a field automorphism of order $2$. Now observe that \[
|f^G|= \begin{cases}\frac{\zeta}{2}\sqrt{q}(q+1),&\textrm{if }q \textrm{ is odd},\\ \sqrt{q}(q+1),&\textrm{if }q \textrm{ is even}. \end{cases} \] From this and~\eqref{e: fora}, the veracity of Table~\ref{t: f field} follows from easy calculations (which we omit).
Note that Lemma~\ref{l: fields} means that it is convenient to assume that $G\geq \mathrm{PGL}_2(q)$ where this makes no difference; however for the final action in Table~\ref{t: f field}, we must assume that $G$ does {\bf not} contain $\mathrm{PGL}_2(q)$ since otherwise the action is not primitive. To make this clear we state the assumed value of $\zeta$ in the ``Conditions'' column in each case.
\begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline
$H$ & Conditions & $|\Omega|$ & $|\nor GH\cap f^G|$ & $|{\rm Fix}_\Omega(f)|$\\ \hline
$D_{q-1}$ & $\zeta=2$ & $\frac12q(q+1)$ & $2\sqrt{q}$ & $q$ \\ $D_{q+1}$ & $\zeta=2$ & $\frac12q(q-1)$ & 0 & 0 \\ $\mathrm{PSL}_2(q_0)$ & $q=q_0^a$, $a$\textrm{ odd }, $\zeta=2$, & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\sqrt{q_0}(q_0+1)$ & $\frac{\sqrt{q}(q-1)}{\sqrt{q_0}(q_0-1)}$ \\
$\mathop{\mathrm{Alt}}(5)$ & $q=p^2\equiv \pm1\pmod{10}$, $\zeta=1$ & $\frac1{120}q(q^2-1)$ & $10$ & $\frac{1}{6} \sqrt{q}(q-1)$ \\
\hline \end{tabular} \caption{Fixed points of field automorphisms of order $2$ for selected primitive actions of $G$ with $q$ odd.}\label{t: f field} \end{table}
We are now ready to prove the two lemmas that together yield Theorem~\ref{t: psl2} for groups with socle $\mathrm{PSL}_2(q)$.
\begin{lem}\label{l: handy q odd} Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to ${\rm PSL}_2(q)$ with $q$ odd. If $q>9$, then $\Omega$ contains a strongly non-binary subset. \end{lem} \begin{proof} Our notation here is consistent with that established above. For instance, we identify $\Omega$ with the set of $G$-conjugates of $H$. We must consider the actions corresponding to the first column of Table~\ref{t: inv q odd}.
\noindent\textsc{Line 1: $H$ is a Borel subgroup of $S$}. In this case $G$ acts $2$-transitively on $\Omega$, but $ \mathrm{Alt}(\Omega)\nleq G$. Thus $\Omega$ itself is a beautiful subset and hence strongly non-binary (of the type given in Example~\ref{ex: snba1}).
\noindent\textsc{Line 5: $H\cong \mathrm{PGL}_2(q_0)$ where $q=q_0^2$}. We regard $S$ as the projective image of those elements of $\mathrm{GL}_2(q)$ that have square determinant, and we may assume that $H$ consists of the projective image of those elements in $\mathrm{GL}_2(q_0)$ whose entries are all in $\mathbb{F}_{q_0}$. Let $T$ be the set of diagonal elements in $S$; let $T_0:=H\cap T$, a maximal split torus in $H$; let $\alpha$ be an element of $\mathbb{F}_q$ that does not lie in any proper subfield of $\mathbb{F}_q$; and define \[
N_0:=\left\{\begin{pmatrix}
1 & \alpha b \\ 0 & 1
\end{pmatrix} \mid b\in\mathbb{F}_{q_0}\right\}. \] Clearly, $N_0$ is a subgroup of $G$, $T_0$ normalizes $N_0$ and $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X:=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$. Let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0$ on which $X$ acts $2$-transitively. If $q_0>5$, then $G$ does not contain a section isomorphic to $\mathrm{Alt}(q_0)$ and we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$. If $q_0=5$, then $q=5^2$ and we can check the result directly using \texttt{magma}~\cite{magma}.
\noindent\textsc{Line 4: $H\cong \mathrm{PSL}_2(q_0)$ where $q=q_0^a$ for some odd prime $a$}. We consider first the special situation where $q$ is a square. We consider $S$ as before, with $H$ the projective image of those elements in $\mathrm{GL}_2(q_0)$ whose entries are all in $\mathbb{F}_{q_0}$ and which have square determinant in $\mathbb{F}_{q_0}$; finally, we know that $H$ has a subgroup $H_1$ isomorphic to $\mathrm{PGL}_2(\sqrt{q_0})$ (since $q_0$ is a square by assumption). We take $H_1$ to be the projective image of those elements in $\mathrm{GL}_2(\sqrt{q_0})$ whose entries are all in $\mathbb{F}_{\sqrt{q_0}}$. Let $T$ be the set of diagonal elements in $S$; let $T_0:=H_1\cap T$, a maximal split torus in $H_1$; let $\alpha$ be an element of $\mathbb{F}_q$ that does not lie in any proper subfield of $\mathbb{F}_q$; and define \[
N_0:=\left\{\begin{pmatrix}
1 & \alpha b \\ 0 & 1
\end{pmatrix} \mid b\in\mathbb{F}_{\sqrt{q_0}}\right\}. \] As above, $N_0$ is a subgroup of $G$, $T_0$ normalizes $N_0$ and $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X:=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$. Let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $\sqrt{q_0}$ on which $X$ acts $2$-transitively. If $\sqrt{q_0}>5$, then $G$ does not contain a section isomorphic to $\mathrm{Alt}(q_0)$ and we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$. The outstanding cases (that is, $\sqrt{q_0}\le 5$ or $q$ is not a square) will be dealt with below.
\noindent\textsc{Lines 2,3,4,6,7,8}. Here we will show that in every case we can find a strongly non-binary subset $\Lambda$ for which $G^\Lambda$ is as in Example~\ref{ex: snba2}. We let $g$ be an involution in $S$ and $h\in g^G$ with $K:=\langle g,h\rangle$ a Klein $4$-subgroup of $S$ and we let \[
\Lambda={\rm Fix}(g)\cup{\rm Fix}(h)\cup{\rm Fix}(gh). \] Observe that $\Lambda$, ${\rm Fix}(g)$, ${\rm Fix}(h)$ and ${\rm Fix}(gh)$ are $g$-invariant and $h$-invariant. Write $\tau_1$ for the permutation induced by $g$ on ${\rm Fix}(gh)$ and $\tau_2$ for the permutation induced by $g$ on ${\rm Fix}(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$. Observe, furthermore, that $h$ induces the permutation $\tau_1$ on ${\rm Fix}(gh)$; now write $\tau_3$ for the involution induced by $h$ on ${\rm Fix}(g)$, and observe that the supports of $\tau_1$ and $\tau_3$ are disjoint, and that $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$. Observe, finally, that the supports of $\tau_2$ and $\tau_3$ are disjoint and that, since $g,h$ and $gh$ are conjugate, the permutations $\tau_1,\tau_2$ and $\tau_3$ all have support of equal size.
Comparing the entries in the fifth and seventh column of Table~\ref{t: inv q odd}, we see that $|{\rm Fix}(g)|\geq |{\rm Fix}(K)|+2$. (Here we are using our assumption that $q>9$.) This implies, in particular, that $\tau_1,\tau_2$ and $\tau_3$ are non-trivial permutations of order $2$. Observe that either there exists $f\in G_\Lambda$ inducing the permutation $\tau_1$ on $\Lambda$ or else $\Lambda$ is a strongly non-binary subset of $\Omega$ (it corresponds to Example~\ref{ex: snba2}).
Suppose that $G$ does not contain a field automorphism of order $2$, and suppose that $f\in G_\Lambda$ induces the permutation $\tau_1$ on $\Lambda$. This would imply that $G_{\Lambda}$ contained an elementary-abelian subgroup of order $8$. But, as we observed earlier, $m_2(Q)\leq 2$ for any section $Q$ in $G$, which is a contradiction. We conclude that $\Lambda$ is a strongly non-binary subset of $\Omega$.
Note that this argument disposes of Lines 6 and 7 of Table~\ref{t: inv q odd}. It also deals with one of the outstanding cases for Line 4, namely the situation where $q$ is not a square.
Suppose from here on that $G$ contains a field automorphism of order $2$. In particular $q$ is a square and $q\equiv 1\pmod 8$. Now, the previous argument implies that $G^\Lambda$ is strongly non-binary unless $G_\Lambda$ contains a field automorphism that induces the element $\tau_1$, so assume that this is the case. There are two possibilities:
\begin{enumerate}
\item[(a)] there is a field automorphism $f$ of order $2$ that induces the element $\tau_1$ on $\Lambda$;
\item[(b)] there is a field automorphism $f$ of order $2$ that fixes $\Lambda$ point-wise (and some element of $G_\Lambda\cap (G\setminus \mathrm{PGL}_2(q))$ of order divisible by $4$ induces the element $\tau_1$ on $\Lambda$).
\end{enumerate}
Note first that Line 3 of Table~\ref{t: inv q odd} is immediately excluded since field automorphisms of order $2$ have no fixed points in this action (see Table~\ref{t: f field}). We are left only with Lines 2 and 8, as well as Line 4 with $q_0\in\{9,25\}$.
Assume that Case~(a) holds. Observe that, the action on $\Lambda$ gives a natural homomorphism $\langle S_{(\Lambda)},f,g,h\rangle \to \mathop{\mathrm{Sym}}(\Lambda)$ whose image is elementary abelian of order $8$, and whose kernel is $S_{(\Lambda)}$. What is more, by Lemma~\ref{l: quotients split}, $S_{(\Lambda)}$ has odd order, and we conclude that $\langle f,g,h\rangle$ is elementary abelian of order $8$.
Since $f$ centralizes $\langle g,h\rangle$ we may consider the action of $\langle g,h\rangle$ on ${\rm Fix}(f)$. Observe that if $\gamma\not\in{\rm Fix}(g)\cup {\rm Fix}(h)\cup{\rm Fix}(gh)$, then $\gamma^g\neq \gamma^h$, and so $\langle g, h\rangle$ acts semi-regularly on ${\rm Fix}(f)\setminus({\rm Fix}(f)\cap\Lambda)$ and so \[
|{\rm Fix}(f)\setminus({\rm Fix}(f)\cap\Lambda)|\equiv 0 \pmod 4. \] Now in this case ${\rm Fix}(f)\cap \Lambda = {\rm Fix}(g)\cup{\rm Fix}(h)$ and we conclude that \begin{equation}\label{eq: f}
|{\rm Fix}(f)|-2|{\rm Fix}(g)|+|{\rm Fix}(K)|\equiv 0\pmod 4. \end{equation} Let us consider the remaining actions, one by one.
\noindent\textsc{Line 2: $H\cong D_{q-1}$}. In this case \eqref{eq: f} implies that
\[
|{\rm Fix}(f)|-2|{\rm Fix}(g)|+|{\rm Fix}(K)|=q-(q+1)+3\equiv 0\pmod 4
\] which is a contradiction.
\noindent\textsc{Line 4: $H\cong \mathrm{PSL}_2(q_0)$ with $q=q_0^a$ and $a$ an odd prime}. Note first that we may assume that $q_0\in\{9,25\}$, with $p=\sqrt{q_0}$. Choose $g\in S$ to be an element of order $p$; an easy calculation using \eqref{e: fora} confirms that $g$ fixes $\frac{q}{q_0}$ points of $\Omega$. Now choose $h\in S$ to be an element of order $p$ (hence also fixing the same number of points of $\Omega$) such that $\langle g,h\rangle$ is an elementary-abelian group of order $q_0$. We require, moreover, that $\langle g,h\rangle$ fixes no points of $\Omega$: for this we just make sure that $\langle g,h\rangle$ is not conjugate to a Sylow $p$-subgroup of $H$. As usual we set $\Lambda={\rm Fix}(g)\cup{\rm Fix}(h)\cup {\rm Fix}(gh)$. We define $\tau_1, \tau_2, \tau_3$ exactly as in the argument for \textsc{Lines 2,3,4,6,7,8}.
Now if $f$ is an element inducing the permutation $\tau_1$, then $f$ has order divisible by $p$, and $f$ fixes at least $\frac{2q}{q_0}$ elements of $\Omega$. This implies immediately that $f\not\in S$, and we conclude that $a=p$. Now, referring to Lemma~\ref{l: fields}, we see that $f$ must be a field automorphism of degree $a=p$ and an easy calculation with \eqref{e: fora} implies that such an element fixes $\frac12p(p^2+1)$ points of $\Omega$ and so cannot induce the permutation $\tau_1$. Now, referring to Example~\ref{ex: snba2}, we conclude that $\Lambda$ is a strongly non-binary subset.
\noindent\textsc{Line 8: $H\cong A_5$.} In this case we assume that $G\leq {\rm P\Sigma L}_2(q)$, otherwise the action on $\Omega$ is not primitive. In particular $\zeta=1$ and \eqref{eq: f} implies that \[
|{\rm Fix}(f)|-2|{\rm Fix}(g)|+|{\rm Fix}(K)|=\frac{1}{6}\sqrt{q}(q-1)-\frac{1}{2}(q-1)+2\equiv 0\pmod 4.
\] which is a contradiction.
We are left with Case~(b). Note in this case that $q=p^a$ where $a$ is divisible by $4$. This immediately excludes Line 8 of the table (since $q=p^2$ here) as well as the remaining cases for Line $4$ (since here $q=9^a$ or $25^a$ where $a$ is an odd prime). Thus the only line left to consider is Line 2. But note that, for Case (b) to hold, ${\rm Fix}(f)$ must contain $\Lambda$ and so \[
|{\rm Fix}(f)|\geq 3|{\rm Fix}(g)|-2|{\rm Fix}(K)|. \] But Tables~\ref{t: inv q odd} and \ref{t: f field} then give that \[
q\geq \frac{3}{2}(q+1)-6. \] This is a contradiction for $q>9$ and we are done. \end{proof}
\begin{lem}\label{l: handy q even} Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle isomorphic to ${\rm PSL}_2(q)$ with $q=2^a$. If $a>3$, then $\Omega$ contains a strongly non-binary subset. \end{lem} \begin{proof} Our notation here is consistent with that established above. We must consider the actions corresponding to the first column of Table~\ref{t: inv q even}.
\noindent\textsc{Line 1: $H$ is a Borel subgroup of $S$}. In this case $G$ acts 2-transitively on $\Omega$, but $G\not\cong \mathrm{Alt}(\Omega)$ or $\mathrm{Sym}(\Omega)$. Thus $\Omega$ itself is a beautiful subset (and hence strongly non-binary).
\noindent\textsc{Line 2: $H\cong D_{2(q-1)}$}. We may assume that $H$ contains $T$, the set of diagonal elements in $S$. We define \[
N:=\left\{\begin{pmatrix}
1 & \alpha \\ 0 & 1
\end{pmatrix} \mid \alpha\in\mathbb{F}_{q}\right\}. \] Now it is clear that $T$ normalizes $N$ and that $T\cap N=\{1\}$. Thus we can form the semidirect product $X=N\rtimes T_0$ and we observe that $X\cap H=T$.
Using the identification of $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q\geq 8$ on which $X$ acts 2-transitively. Since $G$ does not contain a section isomorphic to $\mathrm{Alt}(q)$ we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$ and we are done.
\noindent\textsc{Line 3: $H\cong D_{2(q+1)}$}. We proceed similarly to the case where $q$ is odd in Lemma~\ref{l: handy q odd}: let $g$ be an involution in $S$ and $h\in g^G$ with $K:=\langle g,h\rangle$ a Klein $4$-subgroup of $S$ and we let \[
\Lambda={\rm Fix}(g)\cup{\rm Fix}(h)\cup{\rm Fix}(gh). \] Observe that $\Lambda$, ${\rm Fix}(g)$, ${\rm Fix}(h)$ and ${\rm Fix}(gh)$ are $g$-invariant and $h$-invariant. Observe, furthermore, that ${\rm Fix}(g)$, ${\rm Fix}(h)$ and ${\rm Fix}(gh)$ are all disjoint and, by Table~\ref{t: inv q even}, are of size $\frac12 q$. Write $\tau_1$ for the permutation induced by $g$ on ${\rm Fix}(gh)$, $\tau_2$ for the permutation induced by $g$ on ${\rm Fix}(h)$, and $\tau_3$ for the permutation induced by $g$ on ${\rm Fix}(gh)$.
Then $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$, while $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$. Then $\Lambda$ is a strongly non-binary subset provided there is no element $f\in G_\Lambda$ that induces the permutation $\tau_1$. Such an element must have even order and must fix at least $q$ elements of $\Omega$. Now Table~\ref{t: inv q even} implies that $f\not\in S$. On the other hand, if $f$ is a field-automorphism of order $2^c$, then it does not fix any elements of $\Omega$. We conclude that $\Lambda$ is a strongly non-binary subset and we are done.
\noindent\textsc{Line 4: $H\cong \mathrm{SL}_2(q_0)$ where $q=q_0^b$ for some prime $b$.} Note that, using \cite{bhr}, we can exclude the possibility that $q_0=2$. Suppose first that $q_0>4$, and take $\beta \in \mathbb{F}_q\setminus \mathbb{F}_{q_0}$.
We may assume that $H$ consists of those elements in $S=\mathrm{SL}_2(q)$ whose entries are all in $\mathbb{F}_{q_0}$. Let $T$ be the set of diagonal elements in $S$; let $T_0=S\cap T$, a maximal split torus in $S$; and define \[
N_0:=\left\{\begin{pmatrix}
1 & \beta \alpha \\ 0 & 1
\end{pmatrix} \mid \alpha\in\mathbb{F}_{q_0}\right\}. \] Now it is clear that $T_0$ normalizes $N_0$ and that $T_0\cap N_0=\{1\}$. Thus we can form the semidirect product $X=N_0\rtimes T_0$ and we observe that $X\cap H=T_0$.
Using the identification of $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0\geq 8$ on which $X$ acts 2-transitively. Since $G$ does not contain a section isomorphic to $\mathrm{Alt}(q_0)$ we conclude that $\Lambda$ is a beautiful subset for the action of $G$ on $\Omega$ and we are done.
The only remaining case is when $q_0=4$. As $q=2^a$, we have $a=2b$. In this case we make use of the fact that the number of $S$-conjugacy classes of subgroups of $S$ isomorphic to a Klein 4-subgroup is $(q+2)/6$. Since $H$ contains a unique conjugacy class of Klein 4-subgroups, there exists a Klein $4$-subgroup $K:=\langle g,h\rangle$ of $S$ with $K\nleq H^g$, for every $g\in S$, that is, ${\rm Fix}(K)=\emptyset$ for the action on cosets of $H$.
Observe that $q$ is a square. We choose $K$ so that, not only does it not lie in a conjugate of $H$, it also doesn't lie in a conjugate of $\mathrm{SL}_2(\sqrt{q})=\mathrm{SL}_2(2^b)$, the centralizer of a field automorphism of order $2$. Define \[
\Lambda={\rm Fix}(g)\cup{\rm Fix}(h)\cup{\rm Fix}(gh). \] Observe that $g$ acts on $\Lambda$, and on ${\rm Fix}(h)$, and on ${\rm Fix}(gh)$. Write $\tau_1$ for the involution induced by $g$ on ${\rm Fix}(gh)$ and $\tau_2$ for the permutation induced by $g$ on ${\rm Fix}(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$.
Exactly as in the case when $H=D_{2(q-1)}$, $\Lambda$ is either strongly non-binary (and we are done), or else there exists $f\in G^\Lambda$ such that $f$ induces the permutation $\tau_1$ on $\Lambda$. Suppose that this latter possibility occurs, and observe that ${\rm Fix}(f)$ contains ${\rm Fix}(g)\cup{\rm Fix}(h)$ and so $|{\rm Fix}(f)|\geq \frac{q}{2}$. If $f\in S$, then $f$ is conjugate to $g$ and $|{\rm Fix}(g)|=\frac{q}{4}$, so we have a contradiction.
Suppose that $f\not\in S$. The subgroup structure of $\mathrm{SL}_2(q)$ implies that if a subgroup $X$ is normalized by a Klein $4$-group, then $X$ is elementary abelian of even order. Thus $S_{(\Lambda)}$ is elementary abelian of even order. But if $S_{(\Lambda)}$ is non-trivial, then an involution fixes at least $\frac{3q}{4}$ points which is a contradiction. Thus $S_{(\Lambda)}$ is trivial.
Now, note that since $q=4^b$, where $b$ is prime, either $q=16$, or else we may assume that $f$ is a field automorphism of order $2$. Thus $\langle K, f\rangle$ is elementary-abelian. But, since $K$ does not lie in a conjugate of $\mathrm{SL}_2(\sqrt{q})$ we have a contradiction here. In the case $q=16$, a moment's thought shows that either $f$ is a field automorphism of order $2$, or else there is a field automorphism of order $2$ that fixes $\Lambda$ point-wise. Either way one concludes that there is a field automorphism $f_1$ such that $\langle K, f_1\rangle$ is elementary-abelian and, again, we have a contradiction. Thus in all cases we have a strongly non-binary subset of the type given in Example~\ref{ex: snba2} and we are done. \end{proof}
We remark again that Theorem~\ref{t: psl2} for groups with socle $\mathrm{PSL}_2(q)$ is an immediate consequence of Lemmas~\ref{l: forbidden}, \ref{l: handy q odd} and \ref{l: handy q even}.
\section{Groups with socle isomorphic to \texorpdfstring{$^{2}B_2(q)$}{2B2(q)}}\label{s: suzuki}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\mathop{{^2\mathrm{B}_2}(q)}$. This theorem follows, {\it \`a la} the other main results, from Lemma~\ref{l: suzuki} combined with Lemma~\ref{l: forbidden}. In what follows $G$ is an almost simple group with socle $S\cong\mathop{{^2\mathrm{B}_2}(q)}$, where $q=2^a$ and $a$ is an odd integer with $a\geq 3$. We write $r:=2^{\frac{a+1}{2}}$ and define $\theta$ to be the following field automorphism of $\mathbb{F}_q$: \[
\theta: \mathbb{F}_q\to \mathbb{F}_q, \,\,\, x \mapsto x^r. \]
We need some basic facts, all of which can be found in \cite{suzuki}. First, ${\rm Out}(S)$ is a cyclic group of odd order $a$. Second, $G$ contains a single conjugacy class of involutions; writing $g$ for one of these involutions we note that \[
|g^G|=(q^2+1)(q-1). \] Third, the maximal subgroups of $S$ fall into three families: Borel subgroups, normalizers of maximal tori, and subfield subgroups. For the second of these families, we need some fixed point calculations, and these are given in Table~\ref{t: suz} (making use of \eqref{e: fora}). Each line of this table corresponds to a conjugacy class of maximal tori in $S$; we write $H$ for a maximal subgroup of $S$ and $\Omega$ for the set of right cosets of $H$ in $S$; in the final column we write $K$ for a Klein $4$-subgroup of $S$. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|} \hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|{\rm Fix}_\Omega(g)|$ & $|\mathcal{P}(H)\cap K^S|$&$|{\rm Fix}_\Omega(K)|$\\ \hline $D_{2(q-1)}$ & $\frac12q^2(q^2+1)$ & $q-1$ & $\frac12q^2$&$0$ & $0$\\ $(q+r+1)\rtimes 4$ & $\frac14q^2(q-1)(q-r+1)$ & $q+r+1$ & $\frac14q^2$ &$0$ & $0$\\ $(q-r+1)\rtimes 4$ & $\frac14q^2(q-1)(q+r+1)$ & $q-r+1$ & $\frac14q^2$ &$0$ &$0$\\
\hline \end{tabular} \caption{Fixed points of involutions and Klein $4$-subgroups for selected primitive actions of almost simple Suzuki groups.}\label{t: suz} \end{table}
\begin{lem}\label{l: suzuki}
Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle $S\cong\mathop{{^2\mathrm{B}_2}(q)}$. Then $\Omega$ contains a strongly non-binary subset. \end{lem}
\begin{proof}
Note that $|S|$ is not divisible by $3$ and hence $G$ does not contain a section isomorphic to an alternating group $\mathop{\mathrm{Alt}}(n)$ with $n\geq 3$. Referring to \cite{suzuki}, we see that a maximal subgroup of $G$ is necessarily the normalizer in $G$ of a maximal subgroup $H$ of $S$. Thus we can identify $\Omega$ with the set of right cosets of $H$ in $S$. We split into three families, as per the discussion above.
First, if $H$ is a Borel subgroup, then the action of $G$ on $\Omega$ is $2$-transitive and, since $G$ contains no alternating sections, we obtain immediately that $\Omega$ itself is a beautiful subset.
Second, if $H$ is the normalizer in $S$ of a maximal torus, then we set $K$ to be a Klein 4-subgroup of $S$, and we let $g,h$ be distinct involutions in $K$. Referring to Table~\ref{t: suz}, we see that $g$ and $h$ fix at least $16$ points of $\Omega$, while $K$ fixes none. We set $\lambda_3$ to be one of the fixed points of $g$ and write $\lambda_4$ for the point $\lambda_3^h$. Similarly $\lambda_5\in{\rm Fix}(h)$ and $\lambda_6=\lambda_5^g$. Finally pick $\lambda_1\in{\rm Fix}(gh)$ and let $\lambda_2=\lambda_1^g$; observe that $\lambda_2=\lambda_1^h$. Now let $\Lambda=\{\lambda_1,\lambda_2,\lambda_3,\lambda_4,\lambda_5,\lambda_6\}$ and observe that $K$ acts on this set with the element $g$ inducing the permutation $(\lambda_1,\lambda_2)(\lambda_5,\lambda_6)$ while the element $h$ induces the permutation $(\lambda_1,\lambda_2)(\lambda_3,\lambda_4)$. Suppose that $f\in G_\Lambda$ induces the permutation $(\lambda_1,\lambda_2)$ on $\Lambda$. This would imply that $f$ and $g$ fix the point $\lambda_3$ and so the stabilizer of $\lambda_3$ must contain a section isomorphic to a Klein 4-subgroup. This is impossible: the Sylow $2$-subgroups of $H$ are cyclic of order $2$ or $4$ and, since $|{\rm Out}(S)|$ is odd, this is true of the stabilizer in $G$ of $\lambda_3$. Therefore $\Lambda$ is a strongly non-binary subset of $\Omega$: it corresponds to Example~\ref{ex: snba2}.
Third, suppose that $H$ is a subfield subgroup of $S$. It is convenient to take $S$ to be the set of $4\times 4$ matrices over $\mathbb{F}_q$ described on \cite[p.133]{suzuki}; then we take $H$ to be the subgroup of $S$ consisting of matrices with entries over $\mathbb{F}_{q_0}$ with $q=q_0^b$ for some prime $b$, and $q_0>2$. The following set forms a Sylow $2$-subgroup of $S$:
\[
P_2(q):= \left\{\begin{pmatrix}
1 & 0 & 0 & 0 \\ \alpha & 1 & 0 & 0 \\ \alpha^{1+\theta}+\beta & \alpha^\theta & 1 & 0 \\ \alpha^{2+\theta}+\alpha\beta+\beta^\theta & \beta&\alpha & 1
\end{pmatrix} \mid
\alpha, \beta \in \mathbb{F}_q \right\}.
\] The subgroup $P_2(q)$ is normalized by the following subgroup of $S$, \[
K(q):= \left\{\begin{pmatrix}
\zeta_1 & 0 & 0 & 0 \\ 0 & \zeta_2 & 0 & 0 \\ 0 & 0 & \zeta_3 & 0 \\ 0 & 0 & 0 & \zeta_4
\end{pmatrix} \mid
\exists\kappa \in \mathbb{F}_q\setminus\{0\}, \zeta_1^\theta = \kappa^{1+\theta}, \zeta_2^\theta=\kappa, \zeta_3=\zeta_2^{-1}, \zeta_4=\zeta_1^{-1} \right\}. \] The group $P_2(q)\rtimes K(q)$ is a maximal Borel subgroup of $S$, while $P_2(q_0)\rtimes K(q_0)$ is a maximal Borel subgroup of $H$. Observe that the center $\Zent {P_2(q)}$ of $P_2(q)$ consists of those matrices for which $\alpha=0$.
Let $\zeta\in\mathbb{F}_q\setminus\mathbb{F}_{q_0}$ and consider the group \[
ZP_2(\zeta,q_0):= \left\{\begin{pmatrix}
1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \zeta\beta & 0 & 1 & 0 \\ (\zeta\beta)^\theta & \zeta\beta & 0 & 1
\end{pmatrix} \mid
\beta \in \mathbb{F}_{q_0} \right\}. \] Observe that $K(q_0)$ normalizes $ZP_2(\zeta,q_0)$, that $K(q_0)<H$, that $ZP_2(\zeta,q_0)\cap H=\{1\}$ and that $K(q_0)$ acts fixed-point-freely on $ZP_2(\zeta,q_0)$. Let $X:=ZP_2(\zeta,q_0)\rtimes K(q_0)$; identifying $\Omega$ with the set of $G$-conjugates of $H$, we let $\Lambda$ be the orbit of $H$ under the group $X$. One obtains immediately that $\Lambda$ is a set of size $q_0\geq 8$ on which $X$ acts 2-transitively. The absence of alternating sections implies that $\Lambda$ is a beautiful subset. \end{proof}
\section{Groups with socle isomorphic to \texorpdfstring{$^{2}G_2(q)$}{2G2(q)}}\label{s: ree}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\mathop{{^2\mathrm{G}_2}(q)}$. This theorem follows, {\it \`a la} the other main results, from Lemma~\ref{l: ree} combined with Lemma~\ref{l: forbidden}. In what follows $G$ is an almost simple group with socle $S\cong\mathop{{^2\mathrm{G}_2}(q)}$, where $q=3^a$ and $a$ is an odd integer with $a\geq 3$. We write $r:=3^{\frac{a+1}{2}}$ and define $\theta$ to be the following field automorphism of $\mathbb{F}_q$: \[
\theta: \mathbb{F}_q\to \mathbb{F}_q, \,\,\, x \mapsto x^r. \]
We need some basic facts, all of which can be found in \cite{kleidman}. First, ${\rm Out}(S)$ is a cyclic group of odd order $a$. Second, $G$ contains a single conjugacy class of involutions; writing $g$ for one of these involutions we note that \[
|g^G|=q^2(q^2-q+1). \] Third, the order of $S$ is not divisible by $5$, and so $G$ does not contain a section isomorphic to $\mathrm{Alt}(n)$ with $n\geq 5$. Fourth, the maximal subgroups of $G$ fall into four families: Borel subgroups, normalizers of maximal tori, involution centralizers, and subfield subgroups. For all but the first of these families, we need some fixed point calculations, and these are given in Table~\ref{t: ree} (making use of \eqref{e: fora}). In each line of this table we write $H$ for a maximal subgroup of $S$ and $\Omega$ for the set of right cosets of $H$ in $S$; in the final column we write $K$ for a Klein $4$-subgroup of $S$. \begin{table}
\begin{tabular}{|c|c|c|c|c|c|} \hline
$H$ & $|\Omega|$ & $|H\cap g^G|$ & $|{\rm Fix}_\Omega(g)|$ & $|K^G\cap \mathcal{P}(H)|$ & $|{\rm Fix}_\Omega(K)|$\\ \hline $(2^2\times D_{\frac{q+1}{2}})\rtimes 3$ & $\frac{q^3(q^2-q+1)(q-1)}{6}$ & $q+4$ & $\frac{q(q-1)(q+4)}{6}$ & $\frac{3q+5}{2}$ & $\frac{3q+5}{2}$\\ $(q+r+1)\rtimes 6$ & $\frac{q^3(q^2-1)(q-r+1)}{6}$ & $q+r+1$ & $\frac{q(q^2-1)}{6}$ & $0$ & $0$ \\ $(q-r+1)\rtimes 6$ & $\frac{q^3(q^2-1)(q+r+1)}{6}$ & $q-r+1$ & $\frac{q(q^2-1)}{6}$ & $0$ & $0$ \\ $2\times \mathrm{PSL}_2(q)$ & $q^2(q^2-q+1)$ & $q^2-q+1$ & $q^2-q+1$ & $\frac{(q+4)q(q-1)}{6}$ & $q+4$\\ ${^2G_2}(q_0)$ & $\frac{q^3(q^3+1)(q-1)}{q_0^3(q_0^3+1)(q_0-1)}$ & $q_0^2(q_0^2-q_0+1)$ & $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ & $\frac16q_0^3(q_0^2-q_0+1)(q_0-1)$ & $\frac{q+1}{q_0+1}$ \\ \hline \end{tabular} \caption{Fixed points of involutions and Klein $4$-subgroups for selected primitive actions of almost simple Ree groups.}\label{t: ree} \end{table}
The calculations given in Table~\ref{t: ree} make use of the fact there is a unique class of involutions and a unique class of Klein $4$-subgroups in $S$; their normalizers are maximal subgroups. In particular the normalizer of a Klein $4$-subgroup in $S$ is the group $H$ in the first line of Table~\ref{t: ree}; combined with the fact that a Sylow $2$-subgroup of $S$ is elementary abelian of order $8$, we are able to complete the final entry in that first row. The other entries in the table follow from easy calculations.
\begin{lem}\label{l: ree} Let $G$ be an almost simple primitive permutation group on the set $\Omega$ with socle $S\cong\mathop{{^2\mathrm{G}_2}(q)}$. Then $\Omega$ contains a strongly non-binary subset. \end{lem} \begin{proof}
Referring to \cite{kleidman}, we see that a maximal subgroup of $G$ is necessarily the normalizer in $G$ of a maximal subgroup $H$ of $S$. Thus we can identify $\Omega$ with the set of right cosets of $H$ in $S$. We split into two cases.
First, if $H$ is a Borel subgroup, then the action of $G$ on $\Omega$ is $2$-transitive and, since $G$ contains no sections isomorphic to $\mathrm{Alt}(n)$ with $n\geq 5$, we obtain immediately that $\Omega$ itself is a beautiful subset.
Second, if $H$ is not a Borel subgroup, then we set $K$ to be a Klein 4-subgroup of $S$, we let $g,h$ be distinct involutions in $K$, and we let \[
\Lambda={\rm Fix}(g)\cup{\rm Fix}(h)\cup{\rm Fix}(gh). \] Observe that $\Lambda$, ${\rm Fix}(g)$, ${\rm Fix}(h)$ and ${\rm Fix}(gh)$ are $g$-invariant and $h$-invariant. Write $\tau_1$ for the involution induced by $g$ on ${\rm Fix}(gh)$ and $\tau_2$ for the permutation induced by $g$ on ${\rm Fix}(h)$, and observe that the supports of $\tau_1$ and $\tau_2$ are disjoint, and that $g$ induces the permutation $\tau_1\tau_2$ on $\Lambda$. Observe, furthermore, that $h$ induces the permutation $\tau_1$ on ${\rm Fix}(gh)$; now write $\tau_3$ for the involution induced by $h$ on ${\rm Fix}(g)$, and observe that the supports of $\tau_1$ and $\tau_3$ are disjoint, and that $h$ induces the permutation $\tau_1\tau_3$ on $\Lambda$.
Observe, finally, that the supports of $\tau_2$ and $\tau_3$ are disjoint. Now, suppose that $f\in G_\Lambda$ induces the permutation $\tau_1$ on $\Lambda$. This would imply that $f$ fixes more points than $g$. Since $f$ has even order and all involutions in $G$ are conjugate, some odd power of $f$ is a conjugate of $g$, which is a contradiction. Thus $\Lambda$ is a strongly non-binary subset of $\Omega$ (it corresponds to Example~\ref{ex: snba2}). \end{proof}
\section{Groups with socle isomorphic to \texorpdfstring{$\mathrm{PSU}_3(q)$}{PSU(3,q)}}\label{s: psu}
In this section we prove Theorem~\ref{t: psl2} for groups with socle $\mathrm{PSU}_3(q)$. Strictly speaking, this theorem does not follow {\it \`a la} the other main results. Firstly, we do not prove the existence of beautiful subsets or of strongly non-binary subsets: we simply prove that the primitive groups under consideration are not binary. Second, for some primitive actions we make use of computer aided computations. The basic ideas for these computations are inspired from a deeper analysis in~\cite{DV_NG_PS}, where Conjecture~\ref{conj: cherlin} is proved for most almost simple groups with socle a sporadic simple group.
The following lemmas are taken from~\cite{DV_NG_PS} and are stated in a form tailored to our needs in this paper. \begin{lem}\label{l: again0}Let $G$ be a transitive group on a set $\Omega$, let $\alpha$ be a point of $\Omega$ and let $\Lambda\subseteq \Omega$ be a $G_\alpha$-orbit. If $G$ is binary, then $G_\alpha^\Lambda$ is binary. \end{lem} \begin{proof}Assume that $G$ is binary. Let $\ell\in\mathbb{N}$ and let $I:=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ and $J:=(\lambda_1',\lambda_2',\ldots,\lambda_\ell')$ be two tuples in $\Lambda^\ell$ which are $2$-subtuple complete for the action of $G_\alpha$ on $\Lambda$. Clearly, $I_0:=(\alpha,\lambda_1,\lambda_2,\ldots,\lambda_\ell)$ and $J_0:=(\alpha,\lambda_1',\lambda_2',\ldots,\lambda'_\ell)$ are $2$-subtuple complete for the action of $G$ on $\Omega$; as $G$ is binary, $I_0$ and $J_0$ are in the same $G$-orbit; hence $I$ and $J$ are in the same $G_\alpha$-orbit. From this we deduce that $G_\alpha^\Lambda$ is binary. \end{proof}
We caution the reader that in the next lemma when we write $\Lambda$ we \emph{are not} referring to a subset of $\Omega$ -- here the set $\Lambda$ is allowed to be any set whatsoever that satisfies the listed suppositions.
\begin{lem}\label{l: again} Let $G$ be a primitive group on a set $\Omega$, let $\alpha$ be a point of $\Omega$, let $M$ be the stabilizer of $\alpha$ in $G$ and let $d$ be an integer. Suppose $M\ne 1$ and, for each transitive action of $M$ on a set $\Lambda$ satisfying: \begin{enumerate}
\item $|\Lambda|>1$, and \item every composition factor of $M$ is isomorphic to some section of $M^\Lambda$, and \item $M$ is binary in its action on $\Lambda$, \end{enumerate}
we have that $d$ divides $|\Lambda|$. Then either $d$ divides $|\Omega|-1$ or $G$ is not binary. \end{lem} \begin{proof}
Suppose that $G$ is binary. Since $\{\beta\in\Omega\mid \beta^m=\beta,\forall m\in M\}$ is a block of imprimitivity for $G$ and since $G$ is primitive, we obtain that either $M$ fixes each point of $\Omega$ or $\alpha$ is the only point fixed by $M$. The former possibility is excluded because $M\neq 1$ by hypothesis. Therefore $\alpha$ is the only point fixed by $M$. Let $\Lambda\subseteq\Omega\setminus\{\alpha\}$ be an $M$-orbit. Thus $|\Lambda|>1$ and (1) holds. Since $G$ is a primitive group on $\Omega$, from~\cite[Theorem~3.2C]{dixon_mortimer}, we obtain that every composition factor of $M$ is isomorphic to some section of $M^\Lambda$ and hence (2) holds. From Lemma~\ref{l: again0}, the action of $M$ on $\Lambda$ is binary and hence (3) also holds. Therefore, $d$ divides $|\Lambda|$ and hence each orbit of $M$ on $\Omega\setminus\{\alpha\}$ has cardinality divisible by $d$. Thus $|\Omega|-1$ is divisible by $d$. \end{proof}
\begin{proof}[Proof of Theorem~$\ref{t: psl2}$ for almost simple groups with socle $\mathrm{PSU}_3(q)$.]
Let $G$ be an almost simple primitive group on the set $\Omega$ with socle $S$ isomorphic to $\mathrm{PSU}_3(q)$. Observe that $q\ge 3$ because $\mathrm{PSU}_3(2)$ is soluble. When $q\le 9$, we can check directly with \texttt{magma} the veracity of our statement by constructing all the primitive actions under consideration and checking one-by-one that none is binary (in each case we are able to exhibit a non-binary witness). For the rest of the proof we assume that $q>9$, that is, $q\ge 11$: among other things, this will allow us to exclude some ``novelties'' in dealing with the maximal subgroups of $G$. Moreover, we let $V:=\mathbb{F}_{q^2}^3$ be the natural $3$-dimensional Hermitian space over the field $\mathbb{F}_{q^2}$ of cardinality $q^2$ for the appropriate covering group of $G$.
Let $\alpha\in \Omega$ and let $M:=G_\alpha$ be the stabilizer in $G$ of the point $\alpha$. We subdivide the proof according to the structure of $M$ as described in~\cite[Section~8, Tables~8.5,~8.6]{bhr}. In this proof we use~\cite{bhr} as a crib.
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_1$. }This case is completely settled in~\cite{gs_binary}, where the authors have proved Cherlin's conjecture for almost simple classical groups acting on the cosets of a maximal subgroup in the Aschbacher class $\mathcal{C}_1$.
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_2$.} From~\cite{bhr}, we get that the action of $G$ on $\Omega$ is permutation equivalent to the natural action of $G$ on \[ \{\{V_1,V_2,V_3\}\mid \dim_{\mathbb{F}_{q^2}}(V_1)= \dim_{\mathbb{F}_{q^2}}(V_2)=\dim_{\mathbb{F}_{q^2}}(V_3)=1, V=V_1\perp V_2\perp V_3, V_1,V_2,V_3 \textrm{ non isotropic}\}. \]
Therefore we identify $\Omega$ with the latter set. Let $e_1,e_2,e_3$ be the canonical basis of $V$ and, replacing $G$ by a suitable conjugate, we may assume that the matrix associated to the Hermitian form on $V$ with respect to the basis $e_1,e_2,e_3$ is the identity matrix. Thus $\omega_0:=\{\langle e_1\rangle,\langle e_2\rangle,\langle e_3\rangle\}\in\Omega$. Consider $\Omega_0:=\{\{V_1,V_2,V_3\}\in \Omega\mid V_1=\langle e_1\rangle\}$. Clearly, $G_{\Omega_0}=G_{\langle e_1\rangle}$, $G_{\langle e_1\rangle}$ is a classical group, $G_{\Omega_0}/\Zent {G_{\Omega_0}}$ is almost simple with socle isomorphic to $\mathrm{PSL}_2(q)$ (here we are using $q>3$), and the action of $G_{\Omega_0}$ on $\Omega_0$ is permutation equivalent to the action of $G_{\langle e_1\rangle}$ on $\Omega_0':=\{\{W_1,W_2\}\mid \dim(W_1)=\dim(W_2), \langle e_1\rangle^\perp=W_1\perp W_2, W_1,W_2 \textrm{ non degenerate}\}$. Therefore $G^{\Omega_0}$ is an almost simple primitive group with socle isomorphic to $\mathrm{PSL}_2(q)$ and having degree $|\Omega_0|=q(q-1)/2$. Applying Theorem~\ref{t: psl2} to $G^{\Omega_0}$, we obtain that $G^{\Omega_0}$ is not binary and hence there exist two $\ell$-tuples $(\{W_{1,1},W_{1,2}\},\ldots,\{W_{\ell,1},W_{\ell,2}\})$ and $(\{W'_{1,1},W'_{1,2}\},\ldots,\{W'_{\ell,1},W'_{\ell,2}\})$ in $\Omega_0^\ell$ which are $2$-subtuple complete for the action of $G_{\Omega_0}$ but not in the same $G_{\Omega_0}$-orbit. By construction the two $\ell$-tuples
\begin{align*} I& :=(\{\langle e_1\rangle,W_{1,1},W_{1,2}\},\{\langle e_1\rangle,W_{2,1},W_{2,2}\},\ldots,\{\langle e_1\rangle,W_{\ell,1},W_{\ell,2}\}), \\ J&:=(\{\langle e_1\rangle,W'_{1,1},W'_{1,2}\},\{\langle e_1\rangle,W'_{2,1},W'_{2,2}\},\ldots,\{\langle e_1\rangle,W'_{\ell,1},W'_{\ell,2}\})
\end{align*}
are in $\Omega^\ell$ and are $2$-subtuple complete. Moreover, a moment's thought yields that $I$ and $J$ are not in the same $G$-orbit. Thus $G$ is not binary.
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_3$. }Here $M$ is the normalizer in $G$ of a maximal non-split torus $T$ of $S$ of order $(q^2-q+1)/\gcd(q+1,3)$. From~\cite{bhr}, we infer that $\nor S T$ is a split extension of $T$ by a cyclic group $\langle x\rangle$ of order $3$ (arising from an element of order $3$ in the Weyl group of $S$), thus $M=C\rtimes K$, with $C=\langle c\rangle$ cyclic such that $C\cap S=T$ and with $K$ abelian. (The group $K$ is the direct product of a cyclic group of order $3$ and a cyclic group of order $|G:G\cap\mathrm{PGU}_3(q)|$.) An inspection of the maximal subgroups of $\mathrm{PSU}_3(q)$ reveals that there exists $g\in \nor S K\setminus M$. Set $\beta:=\alpha^g$. Since $g\notin M$, we get $\beta\neq \alpha$ and, since $g\in \nor S K$, we get $G_\alpha\cap G_\beta=M\cap M^g\ge K$. Therefore $G_\alpha\cap G_\beta=C'\rtimes K$, for some cyclic subgroup $C'$ of $C$.
Set $\Lambda:=\beta^M$. Now, the action induced by $M$ on the $M$-orbit $\Lambda$ is permutation isomorphic to the action of $M=C\rtimes K$ on the right cosets of $M\cap M^g=C'\rtimes K$. We use the ``bar'' notation and denote by $\bar{M}$ the group $M^\Lambda$. Thus $\bar{M}=\langle\bar{c}\rangle\rtimes \bar{K}$ and the action of $\bar{M}$ on $\Lambda$ is permutation isomorphic to the natural action of $\langle\bar{c}\rangle\rtimes\bar{K}$ on $\langle\bar{c}\rangle$: with $\langle\bar{c}\rangle$ acting on $\langle\bar{c}\rangle$ via its regular representation and with $\bar{K}\cong K$ acting on $\langle\bar{c}\rangle$ via conjugation. Now, $\bar{c}^{\bar{x}}=\bar{c}^\kappa$, for some $\kappa\in\mathbb{Z}$ with $\kappa^3\equiv 1\pmod {|\bar{c}|}$ and $\kappa\not\equiv 1\pmod {|\bar{c}|}$. Consider the two triples $I:=(1,\bar{c},\bar{c}^{1+\kappa^2})$ and $J:=(1,\bar{c},\bar{c}^{1+\kappa})$. Now $(1,\bar{c})^{id_{\bar{M}}}=(1,\bar{c})$, $(1,\bar{c}^{1+\kappa^2})^{\bar{x}}=(1,\bar{c}^{\kappa+\kappa^3})=(1,\bar{c}^{\kappa+1})$ and $$(\bar{c},\bar{c}^{1+\kappa^2})^{\bar{c}^{-1}\bar{x}^2\bar{c}}=(\bar{c}^{\bar{c}^{-1}\bar{x}^2\bar{c}},(\bar{c}^{1+\kappa^2})^{\bar{c}^{-1}\bar{x}^2\bar{c}})=(\bar{c},\bar{c}^{\kappa^4+1})=(\bar{c},\bar{c}^{\kappa+1}).$$
Thus $I$ and $J$ are $2$-subtuple complete for the action of $\bar{M}$ on $\langle\bar{c}\rangle$. Observe that $I$ and $J$ are not in the same $\bar{M}$-orbit because the only element of $\bar{M}$ fixing $1$ and the generator $\bar{c}$ of $\langle\bar{c}\rangle$ is the identity, but $\bar{c}^{1+\kappa^2}\ne \bar{c}^{1+\kappa}$ because $\kappa\not\equiv 1\pmod {|\bar{c}|}$. Therefore $\bar{M}$ is not binary. From Lemma~\ref{l: again0}, we deduce that $G$ is not binary.
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_5$.} Let $H$ be the stabilizer in $M$ of a non-isotropic $1$-dimensional subspace $\langle v\rangle$ of $V$ and let $K$ be the stabilizer of $\langle v\rangle$ in $G$. Thus $K$ is a maximal subgroup of $G$ in the Aschbacher class $\mathcal{C}_1$; moreover, using~\cite{bhr} and $q>8$, we see that there exists $g\in \Zent K\setminus M$. Set $\beta:=\alpha^g$. Since $g\notin M$, we get $\beta\neq \alpha$ and, since $g\in \Zent K$, we get $G_\alpha\cap G_\beta=M\cap M^g\ge M\cap K=H$. Since $H$ is maximal in $M$, we obtain $G_\alpha\cap G_\beta=H$ and hence the action induced by $G_\alpha=M$ on the $G_\alpha$-orbit $\beta^{G_\alpha}$ is permutation isomorphic to the action of $M$ on the right cosets of $H$. By construction, this latter action is the natural action of the classical group $M$ on the cosets of a maximal subgroup in its $\mathcal{C}_1$-Aschbacher class. From~\cite[Theorem~B]{gs_binary}, this action is not binary. Therefore, $G$ is not binary by Lemma~\ref{l: again0}.
\noindent\textsc{The group $M$ is in the Aschbacher class $\mathcal{C}_6$ or in the Aschbacher class $\mathcal{S}$.} Now the isomorphism class of $M$ is explicitly given in~\cite{bhr}. Since $|M|$ is very small (actually $|M|\le 720$), with the invaluable help of the computer algebra system \texttt{magma}, we compute all the transitive $M$-sets and we select the $M$-sets $\Lambda$ with $|\Lambda|>1$, with every composition factor of $M$ isomorphic to some section of $M^\Lambda$ and with $M^\Lambda$ binary. In all cases, we see that $|\Lambda|$ is even. Therefore, applying Lemma~\ref{l: again}, we obtain that either $G$ is not binary or $|\Omega|-1$ is even. In the
latter case, $|\Omega|$ is odd and hence $M$ contains a Sylow $2$-subgroup of $G$. From~\cite{bhr}, we see that a Sylow $2$-subgroup of $M\cap S$ has size $8$, but this is a contradiction because $|\mathrm{PSU}_3(q)|$ is always divisible by $16$.
\end{proof}
\end{document} |
\begin{document}
\title[]{A useful lemma for calculating the Hausdorff dimension of certain sets in Engel expansions}
\author {Lei Shang}
\subjclass[2010]{Primary 11K55; Secondary 28A80} \keywords{Engel expansions, Growth rate of digits, Hausdorff dimension}
\begin{abstract} Let $\{s_n\}$ and $\{t_n\}$ be two sequences of positive real numbers. Under some mild conditions on $\{s_n\}$ and $\{t_n\}$, we give the precise formula of the Hausdorff dimension of the set \[ \mathbb{E}(\{s_n\},\{t_n\}):=\Big\{x\in(0,1): s_{n}<d_{n}(x)\leq s_n+t_n, \forall n\geq1\Big\}, \] where $d_n(x)$ denotes the digit of the Engel expansion of $x$. This result improves the Lemma 2.6 of Shang and Wu (2021JNT), and is very useful for calculating the Hausdorff dimension of certain sets in Engel expansions. \end{abstract}
\maketitle
\section{Introduction}
Let $T:[0,1)\to [0,1)$ be the \emph{Engel expansion map} defined by $T(0):=0$ and \[ T(x) := x\left\lceil\frac{1}{x}\right\rceil -1,\ \ \ \forall x \in (0,1), \] where $\lceil y\rceil$ denotes the least integer not less than $y$. Denote by $T^k$ the $k$th iteration of $T$. For $x\in (0,1)$, if $x$ is rational, then there exists $n \in \mathbb{N}$ such that $T^n(x)=0$; if $x$ is irrational, then $T^n(x)>0$ for all $n \geq 1$. Then every irrational number $x\in (0,1)$ admits an infinite series with the form \begin{equation}\label{EE} x= \frac{1}{d_1(x)} + \frac{1}{d_1(x)d_2(x)}+\cdots+\frac{1}{d_1(x)\cdots d_n(x)}+\cdots \end{equation} by letting $d_1(x) := \lceil 1/x\rceil$ and $d_{n+1}(x) := d_1(T^n(x))$ for all $n \geq 1$. The expression \eqref{EE} is called the \emph{Engel expansion} of $x$ and the integers $d_n(x)$ are called the \emph{digits} of the Engel expansion of $x$. It was remarked in \cite[p.\,7]{ERS58} that $d_{n+1}(x) \geq d_n(x) \geq 2$ for all $n \geq 1$ and $d_n(x) \to \infty$ as $n \to \infty$. We refer the reader to Galambos \cite{Gal76} for more information of Engel expansions.
Let $\{s_n\}$ and $\{t_n\}$ be sequences of positive real numbers. Write \[ \mathbb{E}(\{s_n\},\{t_n\}):=\Big\{x\in(0,1): s_{n}<d_{n}(x)\leq s_n+t_n, \forall n\geq1\Big\}. \] We will prove the following result.
\begin{lemma}\label{A} Assume that: (1)\,$s_n \geq t_n \geq 2$; (2)\,$s_{n+1} \geq s_n+t_n$; (3)\,$\lim_{n \to \infty}s_n=\infty$. Then \begin{align}\label{Formula} \dim_\mathrm{H}\mathbb{E}(\{s_n\},\{t_n\})=\liminf_{n \to\infty}\frac{\sum_{k=1}^n \log t_k}{\sum_{k=1}^{n+1} \log s_k+\log s_{n+1}-\log t_{n+1}}. \end{align} \end{lemma}
We remark that a similar result was obtained by Liao and Rams \cite{LR} for a class of infinite iterated function systems, while the Engel expansion system does not belong to their setting, see \cite{JR, LR}. We also point out that the result of Lemma A would be very useful for calculating the Hausdorff dimension of certain sets in Engel expansions, see for example \cite{FWnon, LW03, LL, SWjmaa}.
The paper is organized as follows. Section 2 is devoted to several definitions and basic properties of the Engel expansion. The proof of Lemma A will be given in Section 3.
\section{Preliminaries}
\begin{definition} A finite sequence $(\sigma_1, \cdots, \sigma_n) \in \mathbb{N}^n$ is said to be \emph{admissible} if there exists $x \in (0,1)$ such that $d_k(x) = \sigma_k$ for all $1\leq k\leq n$. An infinite sequence $(\sigma_1, \cdots, \sigma_k, \cdots) \in \mathbb{N}^\mathbb{N}$ is said to be \emph{admissible} if there exists $x \in (0,1)$ such that for all $n \geq 1$, $d_k(x) = \sigma_k, \forall 1 \leq k \leq n$.
\end{definition}
Denote by $\Sigma_n$ the collection of all admissible sequences with length $n$ and by $\Sigma$ that of all infinite admissible sequences. The following result gives a characterisation of admissible sequences.
\begin{proposition}[\cite{Gal76}]\label{AD} $(\sigma_1, \cdots, \sigma_n) \in \Sigma_n$ if and only if $2 \leq \sigma_1 \leq \cdots \leq \sigma_n$; $(\sigma_1, \cdots, \sigma_n, \cdots) \in \Sigma$ if and only if \[ \sigma_{n+1} \geq \sigma_n \geq 2, \ \forall n \geq 1 \ \ \ \ \ \text{and} \ \ \ \ \ \lim_{n \to \infty}\sigma_n = \infty. \] \end{proposition}
\begin{definition} Let $(\sigma_1, \cdots, \sigma_n) \in \Sigma_n$. We call \begin{equation*} I_n(\sigma_1, \cdots, \sigma_n) := \big\{x \in (0,1): d_1(x)=\sigma_1, \cdots,d_n(x)=\sigma_n\big\} \end{equation*} the \emph{cylinder} of order $n$ associated to $(\sigma_1,\dots,\sigma_n)$. \end{definition}
We use $|I|$ to denote the diameter of an interval $I$.
\begin{proposition}[{\cite{Gal76}}]\label{cylinder} Let $(\sigma_1, \cdots, \sigma_n) \in \Sigma_n$. Then $I_n(\sigma_1, \cdots, \sigma_n) =[A_n, B_n)$, where \[ A_n:=\frac{1}{\sigma_1}+\cdots+\frac{1}{\sigma_1\sigma_2\cdots \sigma_{n-1}}+ \frac{1}{\sigma_1\sigma_2\cdots \sigma_{n-1}\sigma_n} \] and \[ B:=\frac{1}{\sigma_1}+\cdots+ \frac{1}{\sigma_1\sigma_2\cdots \sigma_{n-1}}+ \frac{1}{\sigma_1\sigma_2\cdots \sigma_{n-1}(\sigma_n-1)}. \] Moreover, \begin{equation*}\label{cylinder length}
\left|I_n(\sigma_1, \sigma_2, \cdots, \sigma_n)\right| = \frac{1}{\sigma_1\sigma_2\cdots \sigma_{n-1}\sigma_n(\sigma_n-1)}. \end{equation*} \end{proposition}
\begin{proposition}[{\cite[Proposition 4.1]{Fal90}}]\label{upp} Suppose $\mathbb{F}$ can be covered by $\mathcal{N}_n$ sets of diameter at most $\delta_n$ with $\delta_n \to 0$ as $n \to \infty$. Then \[ \dim_{\rm H}\mathbb{F} \leq \liminf_{n \to \infty} \frac{\log \mathcal{N}_n}{-\log \delta_n}. \] \end{proposition}
\begin{proposition}[{\cite[Example 4.6]{Fal90}}]\label{low} Let $[0,1] = \mathbb{E}_0 \supset \mathbb{E}_1 \supset \cdots$ be a decreasing sequence of sets and $\mathbb{E} = \bigcap_{n \geq 0} \mathbb{E}_n$. Assume that each $\mathbb{E}_n$ is a union of a finite number of disjoint closed intervals (called basic intervals of order $n$) and each basic interval in $\mathbb{E}_{n-1}$ contains $m_n$ intervals of $\mathbb{E}_n$ which are separated by gaps of lengths at least $\varepsilon_n$. If $m_n \geq 2$ and $\varepsilon_{n-1}> \varepsilon_n >0$, then \[ \dim_\mathrm{H} \mathbb{E} \geq \liminf_{n \to \infty} \frac{\log(m_1m_2 \cdots m_{n-1})}{-\log (m_{n}\varepsilon_n)}. \] \end{proposition}
\section{Proof of Lemma A}
In this section, we will give the proof of Lemma A. Assume that: (1)\,$s_n \geq t_n \geq 2$; (2)\,$s_{n+1} \geq s_n+t_n$; (3)\,$\lim_{n \to \infty}s_n=\infty$. Let \begin{equation*} \mathcal{D}_n:=\big\{(\sigma_1,\cdots \sigma_n)\in \mathbb{N}^n:s_k<\sigma_k\leq s_k+t_k,\forall 1\leq k\leq n\big\}. \end{equation*} Note that $\sigma_1>s_1\geq 2$ and $\sigma_{k+1}>s_{k+1} \geq s_k+t_k >\sigma_k$, so $\sigma_n \geq \cdots\geq \sigma_1\geq 2$. That is to say, $(\sigma_1,\cdots \sigma_n)$ is admissible. For $(\sigma_1,\cdots \sigma_n) \in \mathcal{D}_n$, let \begin{equation*} J_n(\sigma_1,\cdots,\sigma_n):=\bigcup_{s_{n+1}<j\leq s_{n+1}+t_{n+1}} cl(I_{n+1}(\sigma_1,\cdots,\sigma_n,j)), \end{equation*} where $cl(\cdot)$ denotes the closure of a set. By Proposition \ref{cylinder}, we know that $J_n(\sigma_1,\cdots,\sigma_n)$ is a closed interval, which is called the \emph{basic interval} of order $n$. Write \begin{equation}\label{En} \mathbb{E}_n:=\bigcup_{(\sigma_1,\cdots,\sigma_n)\in \mathcal{D}_n} J_n(\sigma_1,\cdots,\sigma_n) \end{equation} with the convention $\mathbb{E}_0:=[0,1]$. Then \begin{equation*} \mathbb{E}(\{s_n\},\{t_n\})=\bigcap_{n=0}^\infty\mathbb{E}_n. \end{equation*}
For the upper bound of $\dim_{\rm H}\mathbb{E}(\{s_n\},\{t_n\})$, for any $n \geq 1$, we derive from \eqref{En} that $\{J_n(\sigma_1,\cdots,\sigma_n):(\sigma_1,\cdots,\sigma_n) \in \mathcal{D}_n\}$ is a cover of $\mathbb{E}(\{s_n\},\{t_n\})$. Then \begin{align*} \mathcal{N}_n:=\#\mathcal{D}_n=(\lfloor s_1+t_1\rfloor-\lfloor s_1\rfloor)\cdots(\lfloor s_n+t_n\rfloor-\lfloor s_n\rfloor)\leq 2^nt_1\cdots t_n. \end{align*} For any $(\sigma_1,\cdots \sigma_n) \in \mathcal{D}_n$, by Proposition \ref{cylinder}, we see that \begin{align*}
|J_{n}(\sigma_1,\cdots,\sigma_n)|&=\sum_{j=\lfloor s_{n+1}\rfloor+1}^{\lfloor s_{n+1}+t_{n+1}\rfloor}|I(\sigma_1,\cdots,\sigma_n,j)|
=\frac{1}{\sigma_1\cdots \sigma_n}\sum_{\lfloor s_{n+1}\rfloor+1}^{\lfloor s_{n+1}+t_{n+1}\rfloor}\frac{1}{j(j-1)}. \end{align*} Note that $s_{n+1}-1\geq s_{n+1}/2$ and $s_{n+1}+t_{n+1}\geq s_{n+1}$, so \begin{align*} \sum_{j=\lfloor s_{n+1}\rfloor+1}^{\lfloor s_{n+1}+t_{n+1}\rfloor}\frac{1}{j(j-1)} = \frac{1}{\lfloor s_{n+1}\rfloor} - \frac{1}{\lfloor s_{n+1}+t_{n+1}\rfloor}
\leq \frac{1}{s_{n+1}-1}-\frac{1}{s_{n+1}+t_{n+1}} \leq\frac{4t_{n+1}}{s_{n+1}^2}. \end{align*} Since $\sigma_k\geq s_k$, we have \begin{align*}
|J_n(\sigma_1,\cdots,\sigma_n)|\leq\frac{1}{s_1\cdots s_n}\cdot\frac{4t_{n+1}}{s_{n+1}^2}=:\delta_n. \end{align*} From Proposition \ref{upp}, we conclude that \begin{align*} \dim_\mathrm{H}\mathbb{E}(\{s_n\},\{t_n\})&\leq \liminf_{n\to \infty}\frac{\log\mathcal{N}_n}{-\log \delta_n}\\ &\leq\liminf_{n\to \infty}\frac{n\log2+\log(t_1\cdots t_n)}{\log(s_1\cdots s_{n+1})+\log s_{n+1}-\log t_{n+1}-\log 4}\\ &=\liminf_{n\to \infty}\frac{\sum_{k=1}^n\log t_{k}}{\sum_{k=1}^{n+1}\log s_{k}+\log s_{n+1}-\log t_{n+1}}. \end{align*}
For the lower bound of $\dim_{\rm H}\mathbb{E}(\{s_n\},\{t_n\})$, by the structure of basic intervals, we deduce that each basic interval of order $n-1$ contains \[ \frac{t_n}{2}<\lfloor t_n\rfloor\leq m_n:=\lfloor s_n+t_n\rfloor - \lfloor s_n\rfloor < t_n+1 <2t_n \] basic intervals of order $n$. Next we will estimate the gaps between two basic intervals with the same order. For two adjacent sequences $(\sigma_1,\cdots,\sigma_n)$ and $(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$ in $\mathcal{D}_n$, we deduce that the cylinders $J_n(\sigma_1,\cdots,\sigma_n)$ and $J_n(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$ are different and then $J_n(\sigma_1,\cdots,\sigma_n)$ is on the left-hand or right-hand side of $J_n(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$. Without loss of generality, we assume that $J_n(\sigma_1,\cdots,\sigma_n)$ is on the left-hand side of $J_n(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$. Then \[ \Pi_1:=\bigcup_{\sigma_n\leq j\leq \lfloor s_{n+1}\rfloor}I_{n+1}(\sigma_1,\cdots,\sigma_n,j) \ \text{and}\ \Pi_2:=\bigcup_{j\geq \lfloor s_{n+1}+t_{n+1}\rfloor+1}I_{n+1}(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n,j) \] are the gap of $J_n(\sigma_1,\cdots,\sigma_n)$ and $J_n(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$. For $\Pi_1$, we have \begin{align*}
|\Pi_1|&=\sum_{\sigma_n\leq j\leq \lfloor s_{n+1}\rfloor}|I_{n+1}(\sigma_1,\cdots,\sigma_n,j)|\\ &=\frac{1}{\sigma_1\cdots\sigma_n}\sum_{\sigma_n\leq j\leq \lfloor s_{n+1}\rfloor}\frac{1}{j(j-1)}\\ &=\frac{1}{\sigma_1\cdots\sigma_n}\left(\frac{1}{\sigma_n-1}-\frac{1}{\lfloor s_{n+1}\rfloor}\right)\\ &\geq\frac{1}{2s_1\cdots 2s_n}\left(\frac{1}{s_n+t_n-1}-\frac{1}{s_{n+1}-1}\right). \end{align*} For $\Pi_2$, we obtain \begin{align*}
|\Pi_2|&=\sum_{j\geq \lfloor s_{n+1}+t_{n+1}\rfloor+1}|I_{n+1}(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n,j)|\\ &=\frac{1}{\sigma^{\prime}_1\cdots\sigma^{\prime}_n}\sum_{j\geq \lfloor s_{n+1}+t_{n+1}\rfloor+1}\frac{1}{j(j-1)}\\ &\geq\frac{1}{2s_1\cdots 2s_n}\cdot\frac{1}{s_{n+1}+t_{n+1}}. \end{align*} Note that $s_{n+1} \geq s_n+t_n$, so \begin{align*} \frac{1}{s_n+t_n-1}-\frac{1}{s_{n+1}-1} +\frac{1}{s_{n+1}+t_{n+1}} &= \frac{1}{s_n+t_n-1}- \frac{t_{n+1}+1}{(s_{n+1}-1)(s_{n+1}+t_{n+1})}\\ &\geq \frac{1}{s_n+t_n-1}\cdot \left(1- \frac{t_{n+1}+1}{s_{n+1}+t_{n+1}}\right)\\ &=\frac{1}{s_n+t_n-1}\cdot\frac{s_{n+1}-1}{s_{n+1}+t_{n+1}}. \end{align*} Since $s_n+t_n-1 < 2s_n$, $s_{n+1}-1 \geq s_{n+1}/2$ and $s_{n+1}+t_{n+1} \leq 2s_{n+1}$, we see that the length of the gap of $J(\sigma_1,\cdots,\sigma_n)$ and $J(\sigma^{\prime}_1,\cdots,\sigma^{\prime}_n)$ is at least \begin{align*}
|\Pi_1|+|\Pi_2| &\geq\frac{1}{2s_1\cdots2s_n}\cdot\frac{1}{s_n+t_n-1}\cdot\frac{s_{n+1}-1}{s_{n+1}+t_{n+1}}\\ &\geq \frac{1}{2s_1\cdots2s_n}\cdot\frac{1}{8s_n}\\ &=\frac{1}{2^{n+3}}\cdot\frac{1}{s_1\cdots s_ns_n}:=\varepsilon_n. \end{align*} It follows from Proposition \ref{low} that \begin{align*} \dim_\mathrm{H}\mathbb{E}(\{s_n\},\{t_n\})&\geq \liminf_{n\to \infty}\frac{\log(m_1m_2\cdots m_{n-1})}{-\log(m_n\varepsilon_n)}\\ &\geq\liminf_{n\to \infty}\frac{-n\log2+\sum _{k=1}^n \log t_k }{(n+5)\log2+\sum_{k=1}^{n+1}\log s_k+\log s_{n+1}-\log t_{n+1}}\\ &=\liminf_{n\to \infty}\frac{\sum _{k=1}^n \log t_k}{\sum_{k=1}^{n+1}\log s_k+\log s_{n+1}-\log t_{n+1}}. \end{align*}
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\title{The Randers metrics of weakly isotropic scalar curvature} \author{Xinyue Cheng\footnote{supported by the National Natural Science Foundation of China (No.11871126) and Chongqing Normal University Science Research Fund (No. 17XLB022)}, Yannian Gong}
\maketitle
\begin{abstract} In this paper, we study the Randers metrics of weakly isotropic scalar curvature. We prove that a Randers metric of weakly isotropic scalar curvature must be of isotropic $S$-curvature. Further, we prove that a conformally flat Randers metric of weakly isotropic scalar curvature is either Minkowskian or Riemannian.\\ {\bf Keywords:} Finsler geometry, Randers metric, Ricci curvature tensor, scalar curvature, $S$-curvature. \end{abstract}
\section{Introduction}
Randers metrics form a special and important class of metrics in Finsler geometry. A Randers metric on a manifold $M$ is a Finsler metric in the following form: \[ F = \alpha +\beta, \]
where $\alpha =\sqrt{a_{ij}(x)y^iy^j}$ is a Riemannian metric and $\beta = b_i(x) y^i$ is a $1$-form satisfying $\|\beta_x\|_{\alpha} < 1$ on $M$. Randers metrics were first introduced by physicist G. Randers in 1941 from the standpoint of general relativity, here the Riemannian metric $\alpha$ denotes the gravitation field and $\beta$ denotes a electromagnetic field. Later on, these metrics were applied to the theory of the electron microscope by R. S. Ingarden in 1957, who first named them Randers metrics. An interesting fundamental fact about Randers metrics is as follows: any Randers metric can be expressed as the solution of Zermelo navigation problem with navigation data $(h, W)$, where $h$ is a Riemannian metric and $W$ is a vector field with $h(x, -W)<1$ on $M$ (\cite{CS}).
Finsler geometry is just the Riemannian geometry without quadratic restriction. Ricci curvature in Finsler geometry is the natural extension of that in Riemannian geometry. However, there is no unified definition of Ricci curvature tensor in Finsler geometry. Hence, we can find several different versions of the definition of scalar curvature in Finsler geometry. Here, we adopt the definitions introduced by H. Akbar-Zadeh for Ricci curvature tensor and scalar curvature (\cite{AZ}). For a Finsler metric $F$ on an $n$-dimensional manifold $M$, let $\bf Ric$ be the Ricci curvature of $F$. Then the scalar curvature of $F$ is defined as folows \begin{equation} {\bf r}:=g^{ij}{\bf Ric}_{ij},\label{eqb0} \end{equation} where \begin{eqnarray*} {\bf Ric}_{ij}:=\frac {1}{2}{\bf Ric}_{y^{i}y^{j}} \end{eqnarray*} denote the Ricci curvature tensor and $(g^{ij}):=(g_{ij})^{-1}, \ g_{ij}:=\frac {1}{2}[F^{2}]_{y^{i}y^{j}}$. We say that $F$ is of weakly isotropic scalar curvature if there exists a 1-form $\theta:=\theta_{i}(x)y^{i}$ and a scalar function $\mu (x)$ on $M$ such that \begin{equation} {\bf r}=n(n-1)\left[\frac{\theta}{F}+\mu(x)\right]. \label{wisc} \end{equation} In particular, when $\theta=0$, that is, ${\bf r}=n(n-1)\mu(x)$, we say that $F$ is of isotropic scalar curvature. We can find many Finsler metrics of weakly isotropic scalar curvature which are not of isotropic scalar curvature (see Example \ref{ex1} below).
The $S$-curvature ${\bf S} = {\bf S}(x, y)$ is an important non-Riemannian quantity in Finsler geometry which was first introduced by Z. Shen when he studied volume comparison in Riemann-Finsler geometry (\cite{Sh}). Shen proved that the Bishop-Gromov volume comparison holds for Finsler manifolds with vanishing $S$-curvature. The recent studies show that $S$-curvature plays a very important role in Finsler geometry. In 2014, the first author and M. Yuan verified that a Randers metric of isotropic scalar curvature must be of isotropic $S$-curvature (see {\cite{CY}}). In this paper, we mainly study the Randers metrics of weakly isotropic scalar curvature. Firstly, we obtain the following theorem which generalizes the related result in \cite{CY} mentioned above.
\begin{thm} \label{SCS} \ Let $F = \alpha +\beta$ be a Randers metric on an $n$-dimensional manifold $M$. If $F$ is of weakly isotropic scalar curvature, ${\bf r}=n(n-1)\left[\frac{\theta}{F}+\mu(x)\right]$, then $F$ is of isotropic $S$-curvature. \end{thm}
The following is an example about Randers metrics of weakly isotropic scalar curvature which arises from \cite{CS1}. \begin{ex}{\rm (\cite{CS1})}\label{ex1} Let us consider the following Randers metric
\begin{eqnarray*} F&=& \frac{\sqrt{\left(1-|a|^{2}|x|^{4}\right)|y|^{2}+\left(|x|^{2}\langle a, y\rangle- 2\langle a, x\rangle\langle x, y\rangle\right)^{2}}}{1-|a|^{2}|x|^{4}} \\
&& -\frac{|x|^{2}\langle a, y\rangle- 2\langle a, x\rangle\langle x, y\rangle}{1-|a|^{2}|x|^{4}}, \end{eqnarray*} where $a$ is a constant vector in ${\bf R}^{n}$ and $\langle ~, \rangle$ denotes the standard inner product in ${\bf R}^{n}$. By direct computation, one can easily verify that $F$ is of weakly isotropic scalar curvature. Precisely, we have \begin{equation} {\bf r}=n(n-1)\left(\frac{\theta}{F}+\mu(x)\right), \ \ \ \theta =\frac{3(n+1)c_{m}y^{m}}{2n}, \label{ex1}
\end{equation} where $c=\langle a, x\rangle$ and $\mu = 3\langle a, x\rangle^{2}-2|a|^{2}|x|^{2}$, \ $c_{m}:=c_{x^{m}}$. Further, we can also prove that $F$ is of isotropic $S$-curvature, \[ {\bf S}=(n+1) c F. \] Actually, in this case, $F$ is of weakly isotropic flag curvature, \[ \mathbf{K}=\frac{3 c_{m} y^{m}}{F}+ \mu (x). \] Then we can get (\ref{ex1}) by Lemma \ref{wEisc}. \end{ex}
The study on conformal geometry has a long and venerable history. From the beginning, conformal geometry has played an important role in differential geometry and physical theories. The Weyl theorem shows that the conformal and projective properties of a Finsler space determine the properties of metric completely (see{\cite{Kn}}, {\cite{Ru}} and \cite{BC}). Undoubtedly, Finsler conformal geometry is an important part of Finsler geometry. We say two Finsler metrics $F$ and $\bar{F}$ are conformally related if there is a scalar function $\sigma (x)$ on the manifold such that $F=e^{\sigma(x)}\bar{F}$. Further, if $\bar{F}$ is a Minkowskian, the Finsler metric $F$ is called the conformally flat Finsler metric. It is an important topic in Finsler geometry to reveal and characterize deeply geometric structures and properties of conformally flat Finsler metrics. G. Chen and the first author have proved that a conformally flat weak Einstein $(\alpha, \beta)$-metric must be either a locally Minkowski metric or a Riemannian metric on a manifold M with the dimension $n \geq 3$ ({\cite {CC}}). Chen-He-Shen proved that a conformally flat $(\alpha, \beta)$-metric with constant flag curvature is either a locally Minkowskian or Riemannian metric (\cite{GQZ}). On the other hand, Cheng-Yuan proved that a conformally flat non-Riemannian Randers metric of isotropic scalar curvature must be locally Minkowskian when the dimension $n \geq 3$ (\cite{CY}). Further, B. Chen and K. Xia studied a class of conformally flat polynomial $(\alpha, \beta)$-metrics in the form $F=\alpha\left(1+\Sigma^m_{j=2}a_{j}(\frac {\beta}{\alpha})^{j}\right)$ with $m\geq 2$. They proved that, if such a conformally flat $(\alpha, \beta)$-metric F is of weaklly isotropic scalar curvature, then it must have zero salar curvature. Moreover, if $a_{m-1}a_{m}\neq 0$, then $F$ must be either locally Minkowkian or Riemannian when the dimension $n\geq 3$ (\cite{BK}). When $m=1$, that is, when $F$ is a Randers metric, Chen-Xia have not confirmed that the same conclusion still holds. Therefore, it is a natural problem to characterize conformally flat Randers metrics of weakly isotropic scalar curvature. We have got the following theorem.
\begin{thm} \label{CWS}\ Let $F = \alpha +\beta$ be a conformally flat non-Riemannian Randers metric on an n-dimensional manifold $M$ with $n\geq 2$. If $F$ is of weakly isotropic scalar curvature, that is, ${\bf r}=n(n-1)[\frac{\theta}{F}+\mu(x)]$, then $F$ must be locally Minkowkian. \end{thm}
\section{Preliminaries}
Let $M$ be an $n$-dimensional smooth manifold and $(x^{i}, y^{i})$ denote the local coordinates of point $(x,y)$ on the tangent boudle $TM$ with $y=y^{i}\frac{{\partial}}{{\partial} x^{i}}\in T_{x}M$. Let $F$ be a Finsler metric on $M$ and $g_{y}=g_{kl}(x, y)dx^{k}\otimes dx^{l}$ be the fundamental tensor of $F$, where $g_{kl}:=\frac {1}{2}[F]^{2}_{y^{k}y^{l}}$. The geodesic coefficients of $F$ are given by
\begin{equation}
G^{k}=\frac {1}{4}g^{kl}\{[F]^{2}_{x^{m}y^{l}}y^{m}-[F^{2}]_{x^{l}}\},
\end{equation} where $(g^{kl}):=(g_{kl})^{-1}$. For any $x\in M$ and $y\in T_xM\backslash \left\{0\right\}$, the Riemann curvature ${\bf R}_{y}:=R^{i}_{\ k}(x, y)\frac {\partial}{\partial x^{i}}$ $\otimes dx^{k}$ is defined by
\begin{equation}
R^{i}_{\ k}(x, y):=2G^{i}_{x^{k}}-G^{i}_{x^jy^k}y^{j}+2G^{j}G^{i}_{y^{j}y^{k}}-G^{i}_{y^{j}}G^{j}_{y^{k}}.
\end{equation} The Ricci curvature of Finsler metric $F$ is defined as the trace of Riemann curvature , that is
\begin{equation} {\bf Ric}(x, y):=R^{m}_{\ m}(x, y).
\end{equation}
It is not difficult to see that Ricci curvature is a positive homogeneous function of degree two in y. Further, the Ricci curvature tensor is given by
\begin{equation}
{\bf Ric}_{ij}:=\frac {1}{2}{\bf Ric}_{y^{i}y^{j}}.\label{Ricci def}
\end{equation}
One can get ${\bf Ric}(x, y)={\bf Ric}_{ij}y^{i}y^{j}$ by the homogeneity of ${\bf Ric}$. A Finsler metric $F$ is called a weak Einstien metric, if there exists a 1-form $\xi= \xi_{i}(x)y^{i}$ and a scalar function $\mu=\mu(x)$ on $M$ such that
\begin{equation}
{\bf Ric}=(n-1)\left(\frac {3 \xi}{F}+\mu\right)F^{2} . \label{wE}
\end{equation}
In particular, if $\xi =0$, that is, ${\bf Ric}=(n-1)\mu F^{2}$, $F$ is called an Einstein metric.
The scalar curvature of a Finsler metric $F$ introduced by Akbar-Zadeh is defined by (1.1), that is, ${\bf r}:=g^{ij}{\bf Ric}_{ij}$. The following lemma is natural and important.
\begin{lem}\label{wEisc}
Assume that $F$ is a weak Einstein Finsler metric satisfying (\ref{wE}). Then $F$ must be of weakly isotropic scalar curvature satisfying
\begin{equation}
{\bf r}= n(n-1)\left(\frac{\theta}{F}+\mu\right), \ \ \theta = \frac{3(n+1)}{2n}\xi. \label{wEWIS}
\end{equation}
\end{lem}
The distortion $\tau$ of a Finsler metric $F$ is defined by \[ \tau(x, y):=\ln\frac {\sqrt{det(g_{ij}(x, y))}}{\sigma_{BH}(x)}, \]
where
\[
\sigma_{BH}:=\frac{Vol({\bf B}^{n}(1))}{Vol\{(y^{i})\in R^{n}|F(x, y^{i}\frac{\partial}{\partial x^{i}})<1\}}
\]
is Busemann-Hausdorff volume coefficient. $\tau=0$ if and only if Finsler metric $F$ is Riemannian (\cite{SSZ}). $S$-curvature ${\bf S}$ of $F$ characterizes the change rate of distortion $\tau$ along geodesics, that is,
\[
{\bf S}(x,y):=\tau_{|m}(x, y)y^{m}.
\] In local coordinate system, the $S$-curvature of $F$ can be expressed as
\[
{\bf S}(x, y)=\frac{{\partial} G^{m}}{{\partial} y^{m}}-y^{m}\frac{{\partial}}{{\partial} x^{m}}\left[\ln\sigma_{BH}(x)\right].
\]
We say that a Finsler metric $F$ is of isotropic $S$-curvature, if there is a scalar function $c(x)$ on $M$ such that
\begin{equation}
{\bf S}(x, y)=(n-1)c(x)F(x, y).
\end{equation}
The mean Cartan torsion ${\bf I}_{y}=I_{i}(x, y) d x^{i}: T_{x} M \rightarrow {\bf R}$ is defined by $$ I_{i}:=g^{j k} C_{i j k}, $$ where $C_{ijk}$ denote the Cartan torsion of $F$. It is easy to check that $ I_{i}=\tau_{y^{i}}$.
\vskip 2mm
For a Randers metric $F=\alpha+\beta$, we have the following \begin{equation}\label{gij} g^{ij}=\frac {\alpha}{F}a^{ij}-\frac {\alpha}{F^{2}}(b^{i}y^{j}+b^{j}y^{i})+\frac {b^{2}\alpha+\beta}{F^{3}}y^{i}y^{j},
\end{equation} where $b:=\|\beta\|_{\alpha}$ denotes the norm of $\beta$ with respect to $\alpha$ (\cite{SSZ}). Let \[ r_{ij}:=\frac {1}{2}(b_{i;j}+b_{j;i}), \ \ \ \ s_{ij}:=\frac {1}{2}(b_{i;j}-b_{j;i}),\ \ \ \ \ \ \ \ \ \\ \] \[ e_{ij}:=r_{ij}+s_{i}b_{j}+s_{j}b_{i},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \] \[ w_{ij}:=r_{im}r^{m}\!_{j},\ \ t_{ij}:=s_{im}s^{m}\!_{j},\ \ q_{ij}:=r_{im}s^{m}\!_{j}, \ \ \ \ \ \ \ \\ \] \[ r^{i}_{\ j}:=a^{im}r_{mj},\ s^{i}_{\ j}:=a^{im}s_{mj}, \ t^{i}_{\ j}:=a^{im}t_{mj},\ q^{i}_{\ j}:=a^{im}q_{mj},\\ \] \[ r_{i}:=b^{m}r_{mi},\ s_{i}:=b^{m}s_{mi},\ t_{i}:=b^{m}t_{mi},\ q_{i}:=b^{m}q_{mi},\ \ \\ \] \[ r:=b^{i}b^{j}r_{ij},\ \ \ t:=b^{i}t_{i},\ \ \ p_{i}:=r_{im}s^{m}, \] where `` ; " denotes the covariant derivative with respect to $\alpha$. Besides, put $ r_{00}:=r_{ij}y^{i}y^{j},\ e_{00}:=e_{ij}y^{i}y^{j},\ q_{00}:=q_{ij}y^{i}y^{j},\ s_{0}:=s_{i}y^{i}$, etc..
Further, the Ricci curvature of Randers metric $F=\alpha +\beta$ is given by \begin{equation} {\bf Ric}={}^{\alpha}{\bf Ric}+(2\alpha s^{m}\!_{0;m}-2t_{00}-\alpha^{2}t^{m}_{\ m})+(n-1)\Xi,\label{Ricci curvature} \end{equation} where ${}^{\alpha}{\bf Ric}$ denotes the Ricci curvature of $\alpha$ and \begin{equation} \Xi:=\frac {2\alpha}{F}(q_{00}-\alpha t_{0})+\frac {3}{4F^{2}}(r_{00}-2\alpha s_{0})^{2}-\frac
{1}{2F}(r_{00;0}-2\alpha s_{0;0}). \label{Xi} \end{equation} Besides, the mean Cartan tensor ${\bf I}=I_{i}dx^{i}$ of $F=\alpha +\beta$ is given by \begin{equation}\label{Ii} I_{i}=\frac {n+1}{2F}(b_{i}-\frac {\beta y_{i}}{\alpha^{2}}), \end{equation} where $y_{i}:=a_{ij}y^{j}$. For related details, see \cite{CS} or \cite{SSZ}.
The following lemma is very important for the proof of Theorem \ref{SCS}. \begin{lem}{\rm (\cite{CS})}\label{e00} Let $F = \alpha +\beta$ be a Randers metric on an $n$-dimensional manifold $M$. Then $F$ is of isotropic $S$-curvature, ${\bf S}=(n+1)c(x)F$ if and only if \begin{equation} e_{00}=2c(x)(\alpha^{2}-\beta^{2}), \end{equation} where $c=c(x)$ is a scalar function on $M$. \end{lem}
\section{The scalar curvature of Randers metric}\label{section3}
In \cite{CY}, the first author and M. Yuan have obtained the formula of the scalar curvature of Randers metrics. In this section, we will further improve and optimize this formula. For readers' convenience, we will derive the formula of the scalar curvature of Randers metrics step by step.
By (\ref{gij}), (\ref{Ricci curvature}) and the definition of scalar curvature, we can get \begin{equation} {\bf r}:=g^{ij}{\bf Ric}_{ij}= {}^{\alpha}{\bf Ric}_{ij}g^{ij}+\frac {1}{2}E_{ij}g^{ij}+\frac {1}{2}(n-1)\Xi_{ij}g^{ij},\label{scRanders} \end{equation} where ${}^{\alpha}{\bf Ric}_{ij}$ denote Ricci curvature tensor of $\alpha$ and \begin{eqnarray*} && E:= 2\alpha s^{m}_{\ 0;m}-2t_{00}-\alpha^{2}t^{m}_{\ m},\\ && E_{ij}:= E_{y^{i}y^{j}},\ \ \ \ \Xi_{ij}:=\Xi_{y^{i}y^{j}}. \end{eqnarray*} Next, we will compute each term on the right side of (\ref {scRanders}). Firstly, we get the following \begin{equation}\label{term 1} {}^{\alpha}{\bf Ric}_{ij}g^{ij}=\frac {\alpha}{F}{\bf r}_{\alpha}-\frac{2\alpha}{F^{2}}\ {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+\frac {b^{2}\alpha+\beta}{F^{3}}\ {}^{\alpha}{\bf Ric}, \end{equation} where ${\bf r}_{\alpha}$ denotes the scalar curvature of $\alpha$. Moreover, we obtain the following \begin{eqnarray} E_{ij}g^{ij}&=& \frac{2\alpha}{F}\left[\frac{n+1}{\alpha}s^{m}_{\ 0;m}-(n+2)t^{m}_{\ m}\right] \nonumber\\
&&-\frac{4\alpha}{F^{2}}\big[s^{m}_{\ 0;m}s+\alpha b^{i} s^{m}_{\ i;m}-2t_{0}-\beta t^{m}_{\ m}\big]+2\frac {b^{2}\alpha+\beta}{F^{3}}E, \label{Eij} \end{eqnarray} where $s:=\beta / \alpha$.
In order to obtian $\Xi_{ij}g^{ij}$, we rewrite (\ref{Xi}) as the following \begin{equation}
\Xi:=\Xi^{1}+\Xi^{2}+\Xi^{3}, \label{reXi} \end{equation} where, \[ \Xi^{1}:=2\left(\frac {D}{F}\right), \ \ \Xi^{2}:=\frac {3}{4}\left(\frac {A^{2}}{F^{2}}\right), \ \ \Xi^{3}:= -\frac {1}{2}\left(\frac {B}{F}\right), \] and \[ A:=r_{00}-2\alpha s_{0},\ \ \ B:=r_{00;0}-2\alpha s_{0;0}, \] \[
D:=\alpha D_{1}=\alpha(q_{00}-\alpha t_{0}), \ \ \ D_{1}:= q_{00}-\alpha t_{0}. \]
From (\ref{reXi}), one can write \begin{equation}
\Xi_{ij}:=\Xi^{1}_{ij}+\Xi^{2}_{ij}+\Xi^{3}_{ij},\label{XXi} \end{equation} where \begin{eqnarray*} \Xi^{1}_{ij}&:=& \Big[2\frac{\alpha}{F}(q_{00}-\alpha t_{0})\Big]_{y^{i}y^{j}}=2\Big(\frac{D}{F}\Big)_{y^{i}y^{j}},\\ \Xi^{2}_{ij}&:=& \Big[\frac{3}{4F^{2}}(r_{00}-2\alpha s_{0})^{2}\Big]_{y^{i}y^{j}}=\Big(\frac{3}{4}\Big)\Big(\frac{A^{2}}{F^{2}}\Big)_{y^{i}y^{j}},\\ \Xi^{3}_{ij}&:=& \Big[-\frac{1}{2F}(r_{00;0}-2\alpha s_{0;0})\Big]_{y^{i}y^{j}}=-\Big(\frac{1}{2}\Big)\Big(\frac{B}{F}\Big)_{y^{i}y^{j}}. \end{eqnarray*} By a series direct computations, we can get the following \begin{eqnarray} \Xi^{1}_{ij}g^{ij}&=& \frac{2}{F}\left \{\frac{\alpha}{F}\Big[(n+3)\frac{D_{1}}{\alpha}+2\alpha q^{m}\!_{m}-(n+1)t_{0}\Big]\right. \nonumber \\ && \left.-\frac{4\alpha}{F^{2}}\big[sD_{1}+\alpha (q_{00\cdot i}b^{i}-st_{0}-\alpha t)\big]+6D\frac{b^{2}\alpha+\beta}{F^{3}}\right\} \nonumber\\ && -\frac{4}{F^{2}}\left\{\frac{\alpha}{F}\big[(3+s)D_{1}+\alpha (q_{00\cdot i}b^{i}-st_{0}-\alpha t)\big]\right. \nonumber\\ && \left. -\frac{\alpha}{F^{2}}\big[ F(sD_{1}+\alpha (q_{00\cdot i}b^{i}-st_{0}-\alpha t))+3D(b^{2}+s)\big]+ 3D\frac{b^{2}\alpha+\beta}{F^{2}}\right\} \nonumber\\ && -2(n-1)\frac{D}{F^{3}}+4\frac{D}{F^{3}}\Big[\frac{\alpha}{F}(1-b^{2})+\frac{b^{2}\alpha+\beta}{F}\Big] \nonumber\\ & =& \frac{1}{2F^{5}}\left\{4F^{3}\big[(n+3)D_{1}-(n+1)\alpha t_{0}+2\alpha^{2}q^{m}_{\ m}\big]\right. \nonumber\\ && -16F^{2}\big[\beta D_{1}+\alpha^{2}q_{00\cdot i}b^{i}-\alpha\beta t_{0}-\alpha^{3}t\big]- 4F^{2}(n+3)D \nonumber\\ && \left.+24F(b^{2}\alpha+\beta)D\right\}, \label{Xieq1} \end{eqnarray} \begin{eqnarray} \Xi^{2}_{ij}g^{ij}&=&\frac{6}{F^{2}}\left\{\frac{\alpha}{F}\Big[w_{00}-2\frac{r_{00}s_{0}}{\alpha}-2\alpha p_{0}+3s^{2}_{0}-\alpha^{2}t\Big]\right.\nonumber\\ && \left.-\frac{2\alpha}{F^{2}}A(r_{0}-ss_{0})+\frac{b^{2}\alpha+\beta}{F^{3}}A^{2}\right\}+\frac{3A}{F^{2}}\left\{\frac{\alpha}{F}\big[r^{m}_{\ m}-(n+1)\frac{s_{0}}{\alpha}\big]\right. \nonumber\\ && \left.-\frac{2\alpha}{F^{2}}(r_{0}-ss_{0})+\frac{b^{2}\alpha+\beta}{F^{3}}A\right\}-\frac{12A}{F^{3}}\left\{\frac{\alpha}{F}\Big[\frac{r_{00}}{\alpha}+r_{0}-(2+s)s_{0}\Big] \right. \nonumber\\ && \left.-\frac{\alpha}{F^{2}}\Big[F(r_{0}-ss_{0})+A(s+b^{2})\Big]+\frac{b^{2}\alpha+\beta}{F^{2}}A\right\}-\frac{3(n-1)A^{2}}{2F^{4}} \nonumber\\ && +\frac{9A^{2}}{2F^{4}}\Big[\frac{\alpha}{F}(1-b^{2})+\frac{b^{2}\alpha+\beta}{F}\Big] \nonumber\\ &=& \frac{1}{2F^{5}}\left\{12F^{2}(\alpha w_{00}-2r_{00}s_{0}-2\alpha^{2}p_{0}+3\alpha s^{2}_{0}-\alpha^{3}t)\right.\nonumber\\ && +6F^{2}A\big[\alpha r^{m}\!_{m}-(n+1)s_{0}\big]-12FA(\alpha r_{0}-\beta s_{0}) \nonumber\\ && -24FA\big[r_{00}+\alpha r_{0}-(2\alpha+\beta)s_{0}\big]-3(n-4)FA^{2} \nonumber\\ && \left.+18A^{2}(b^{2}\alpha+\beta)\right\} \label{Xieq2} \end{eqnarray} and \begin{eqnarray} \Xi^{3}_{ij}g^{ij}&=&\frac{1}{2F}\left\{-\frac{2\alpha}{F}\big[2r^{m}_{\ 0;m}+r^{m}_{\ m;0}-(n+3)\frac{s_{0;0}}{\alpha}-2\alpha s^{m}_{\ ;m}\big]\right. \nonumber\\ && \left.+\frac{4\alpha}{F^{2}}\big[r_{00;0\cdot i}b^{i}-2ss_{0;0}-2\alpha s_{0;0\cdot i}b^{i}\big]-2\frac{b^{2}\alpha+\beta}{F^{3}}B\right\} \nonumber\\ && +\frac{1}{F^{2}}\left\{\frac{\alpha}{F}\big[\frac{3r_{00;0}}{\alpha}-2(3+s)s_{0;0}+r_{00;0\cdot i}b^{i}-2\alpha s_{0;0\cdot i}b^{i}\big]\right. \nonumber\\ && \left.-\frac{\alpha}{F^{2}}\big[F(r_{00;0\cdot i}b^{i}-2ss_{0;0}-2\alpha s_{0;0\cdot i}b^{i})+3B(s+b^{2})\big]\right\} \nonumber\\ && +\frac{n-1}{2F^{3}}B-\frac{ B}{F^{4}}\alpha(1-b^{2})\nonumber\\ &=&\frac{1}{2F^{5}}\left\{-2F^{3}\big[\alpha r^{m}_{\ m;0}+2\alpha r^{m}_{\ 0;m}-2\alpha^{2}s^{m}_{\ ;m}-(3+n)s_{0;0}\big]\right.\nonumber\\ && +F^{2}\big[4\alpha r_{00;0\cdot i}b^{i}-8\alpha^{2}s_{0;0\cdot i}b^{i}-8\beta s_{0;0}+(n+3)B \big] \nonumber\\ && \left.-6F(b^{2}\alpha+\beta)B\right\}, \label{Xieq3} \end{eqnarray} where `` $\cdot i$ " denotes the partial derivative with respect to $y^{i}$.
Now, plugging (\ref{term 1}), (\ref{Eij}) and (\ref{Xieq1})-(\ref{Xieq3}) into (\ref{scRanders}), one obtains the formula of the scalar curvature for Randers metric $F=\alpha +\beta$ as follows \begin{equation} {\bf r}=\frac {\alpha}{F}{\bf r}_{\alpha}+\frac {1}{4F^{5}}\Big\{\Sigma_{1}+\alpha\Sigma_{2}\Big\}, \label{r1} \end{equation} where $\Sigma_{1}$ and $\Sigma_{2}$ are both polynomials in $y$. Concretely, we have the following expressions: \[
\begin{aligned} \Sigma_{1}:=&\left\{\big[-4(2b^{2}+4n+7)t^{m}_{\ m}-24b^{i}s^{m}_{\ i;m}+4(n-1)(6q^{m}_{\ m}+3s^{m}_{\ ;m}+2t)\big]\beta\right.\\ &-8\ {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+4(2b^{2}+n+1)s^{m}_{\ 0;m}-4(6b^{2}n-6b^{2}+n^{2}-5)t_{0}\\ &\left.-2(n-1)(6r^{m}_{\ m}s_{0}+8q_{00\cdot i}b^{i}+r^{m}_{\ m;0}+2r^{m}_{\ 0;m}+4s_{0;0\cdot i}b^{i}+12p_{0})\right\}\alpha^{4}\\ &+\left\{-4\big[(3+4n)t^{m}_{\ m}+2b^{i}s^{m}_{\ i;m}-(n-1)(2q^{m}_{\ m}+s^{m}_{\ ;m})\big]\beta^{3}\right.\\ &+\big[-24 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+8(b^{2}+3n+2)s^{m}_{\ 0;m}-4(n+1)(5n-11)t_{0}\\ &-2(n-1)(6r^{m}_{\ m}s_{0}+8q_{00\cdot i}b^{i}+3r^{m}_{\ m;0}+6r^{m}_{\ 0;m}+4s_{0;0\cdot i}b^{i}+12p_{0})\big]\beta^{2}\\ &+\big[4(2b^{2}+1)~ {}^{\alpha}{\bf Ric}-8(2b^{2}+1)t_{00}+4(n-1)(6b^{2}+n+5)q_{00}\\ &+12(n-1)(3n-4)s_{0}^{2}+2(n-1)(6b^{2}+n+5)s_{0;0}+4(n-1)(3r^{m}_{\ m}r_{00}\\ &+18r_{0}s_{0}+2r_{00;0\cdot i}b^{i}+6w_{00})\big]\beta-6(n-1)\big[(12b^{2}+3n-19)s_{0}+6r_{0}\big] r_{00} \\ & \left.-(n-1)(6b^{2}-n-3)r_{00;0} \right\}\alpha^{2}+4(n-1)s^{m}_{\ 0;m}\beta^{4}+\big[4 {}^{\alpha}{\bf Ric}-8t_{00} \\ &+2(n-1)^{2}(2q_{00}+s_{0;0})\big]\beta^{3}+(n-1)\big[-6(n-1)s_{0}r_{00}+(n-3)r_{00;0}\big]\beta^{2}\\ &+3(n-1)(n-6)r_{00}^{2}\beta
\end{aligned} \] and \[
\begin{aligned} \Sigma_{2}:=&\big[-4(b^{2}+n+2)t^{m}_{\ m}-8b^{i}s^{m}_{\ i;m}+4(n-1)(2q^{m}_{\ m}+s^{m}_{\ ;m}+t)\big]\alpha^{4}\\ &+\left\{[-4(b^{2}+6n+8)t^{m}_{\ m}-24b^{i}s^{m}_{\ i;m}+4(n-1)(6q^{m}_{\ m}+3s^{m}_{\ ;m}+t)]\beta^{2}\right.\\ &+[-24 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+16(b^{2}+n+1)s^{m}_{\ 0;m}-2(n-1)(12r^{m}_{\ m}s_{0}+24p_{0}\\ &+16q_{00\cdot i}b^{i}+3r^{m}_{\ m;0}+6r^{m}_{\ 0;m}+8s_{0;0\cdot i}b^{i})-8(3b^{2}n-3b^{2}+2n^{2}-8)t_{0}\big]\beta\\ &+4\ {}^{\alpha}{\bf Ric}b^{2}-8b^{2}t_{00}+2(n-1)\big[12(3b^{2}+n-4)s_{0}^{2}+36s_{0}r_{0}+6b^{2}s_{0;0}\\ &\left.+12b^{2}q_{00}+3r^{m}_{\ m}r_{00}+2r_{00;0\cdot i}b^{i}+6w_{00}]\right\}\alpha^{2}-4nt^{m}_{\ m}\beta^{4}+2\big[8ns^{m}_{\ 0;m}\\ &+(n-1)(2r^{m}_{\ 0;m}+r^{m}_{\ m;0})-4 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}-4n(n-3)t_{0}\big]\beta^{3}\\ &+\left\{4(b^{2}+2)({}^{\alpha}{\bf Ric}-2t_{00})+2(n-1)\big[6(n-2)s_{0}^{2}+2(n+2)s_{0;0}+6w_{00} \right. \\ &\left.+3r^{m}_{\ m}r_{00}+4(n+4)q_{00}+2r_{00;0\cdot i}b^{i}\big]\right\}\beta^{2}-2(n-1)\left\{3b^{2}r_{00;0}-nr_{00;0}\right.\\ &\left.+6[2(n-2)s_{0}-3r_{0}]r_{00}\right\}\beta+3(n-1)(6b^2+n-12)r_{00}^{2}.
\end{aligned} \] It is obvious that $\Sigma_{1}$ and $\Sigma_{2}$ are homogeneous polynomials of degree 5 and 4 in $y$, respectively.
\begin{rem} We have totally corrected some errors occurred in the formulas of $\Sigma_{1}$ and $\Sigma_{2}$ in \cite{CY}. We must point out that, in the proofs of main theorems in \cite{CY}, the authors just used the facts that $\Sigma_{1}$ and $\Sigma_{2}$ are homogeneous polynomials of dgree 5 and 4 in $y$, respectively. Hence, the main results in \cite{CY} are still true. \end{rem}
\section{Proof of Theorems}
In this section, we are going to prove Theorem \ref{SCS} and Theorem \ref{CWS}. \vskip 2mm
{\bf Proof of Theorem \ref{SCS}.} \ Let $F=\alpha+\beta$ be a Randers metric on an $n$-dimentional manifold $M$ with weakly isotropic scalar curvature.
Firstly, note that \[
e_{ij}:=r_{ij}+b_{i}s_{j}+b_{j}s_{i}. \] We have \begin{equation} r_{00}=e_{00}-2\beta s_{0} \label{r00} \end{equation} and \begin{equation} r_{00;0}=e_{00;0}-2(\beta s_{0;0}+s_{0}e_{00}-2\beta s_{0}^{2}). \label{r00;0} \end{equation} Then plugging (\ref{r00}), (\ref{r00;0}) into (\ref{r1}) and multiplying ($\ref{r1}$) by $4F^{5}$, one gets \begin{equation} 4F^{5}{\bf r}=\Gamma_{1}+\alpha\Gamma_{2},\label{r2} \end{equation} where $\Gamma_{1}$ and $\Gamma_{2}$ are homogeneous polynomials of degree 5 and 4 in $y$, respectively, which have the following expressions:
\begin{eqnarray} \Gamma_{1}&:=&\left\{\big[16{\bf r}_{\alpha}-4(2b^{2}+4n+7)t^{m}_{\ m}-24b^{i}s^{m}_{\ i;m}+4(n-1)(6q^{m}_{\ m}+3s^{m}_{\ m}+2t)\big]\beta\right.\nonumber\\ && -8 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+4(2b^{2}+n+1)s^{m}_{\ 0;m}-2(n-1)\big[6r^{m}_{\ m}s_{0}+8q_{00\cdot i}b^{i} \nonumber\\ && \left. +r^{m}_{\ m;0}+2r^{m}_{\ 0;m}+4s_{0;0\cdot i}b^{i}+12p_{0}\big]-4(6b^{2}n-6b^{2}+n^{2}-5)t_{0}\right\}\alpha^4 \nonumber\\ && + \left\{\big[16{\bf r}_{\alpha}-4(4n+3)t^{m}_{\ m}-8b^{i}s^{m}_{\ i;m}+4(n-1)(2q^{m}_{\ m}+s^{m}_{\ m})\big]\beta^3\right. \nonumber\\ && +\big[-2(n-1)(18r^{m}_{\ m}s_{0}+ 8q_{00\cdot i}b^{i}+3r^{m}_{\ m;0}+6r^{m}_{\ 0;m}+4s_{0;0\cdot i}b^{i}+12 p_{0}) \nonumber\\ && -24 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+8(b^{2}+3n+2)s^{m}_{\ 0;m}-4(n+1)(5n-11)t_{0}\big]\beta^2 \nonumber \\ && +\Big[4(2b^2+1)({}^{\alpha}{\bf Ric}-2t_{00})+4(n-1)\big[(6b^{2}+n+5)q_{00}+2r_{00;0\cdot i}b^{i} \nonumber\\ && +(6b^{2}+1)s_{0;0}+3r^{m}_{\ m}e_{00}+36s_{0}r_{0}+(30b^{2}+19n-66)s_{0}^{2}+6w_{00}\big]\Big]\beta \nonumber\\ && \left. -(n-1)(6b^{2}-n-3)e_{00;0}-4(n-1)\big[9r_{0}+(15b^{2}+5n-27)s_{0}\big]e_{00}\right\}\alpha^{2} \nonumber\\ && +4(n-1)s^{m}_{\ 0;m}\beta^{4}+\left\{4 {}^{\alpha}{\bf Ric}-8t_{00}+4(n-1)\big[(n-1)q_{00}+s_{0;0}\right. \nonumber\\ && \left.+(7n-24)s_{0}^{2}\big]\right\}\beta^{3}+(n-1)\big[(n-3)e_{00;0}-4(5n-21)s_{0}e_{00}\big]\beta^{2} \nonumber\\ && +3(n-1)(n-6)e_{00}^{2}\beta \label{Gma1} \end{eqnarray} and \begin{eqnarray} \Gamma_{2}&:=&\left\{4{\bf r}_{\alpha}-4(b^{2}+n+2)t^{m}_{\ m}-8b^{i}s^{m}_{\ i;m}+4(n-1)(2q^{m}_{\ m}+s^{m}_{\ m}+t)\right\}\alpha^{4} \nonumber\\ && +\left\{\big[24{\bf r}_{\alpha} -4(b^{2}+6n+8)t^{m}_{\ m}-24b^{i}s^{m}_{\ i;m} +4(n-1)(6q^{m}_{\ m}+t +3s^{m}_{\ m})\big]\beta^2 \right. \nonumber\\ &&+\big[-24 {}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+16(b^{2}+n+1)s^{m}_{\ 0;m} \nonumber\\ && -2(n-1)(18r^{m}_{\ m}s_{0}+16q_{00\cdot i}b^{i}+3r^{m}_{\ m;0}+6r^{m}_{\ 0;m}+8s_{0;0\cdot i}b^{i}+24p_{0}) \nonumber\\ && -8(3b^{2}n-3b^{2}+2n^{2}-8)t_{0}\big]\beta+4 b^{2}\ {}^{\alpha}{\bf Ric}- 8b^{2}t_{00}+2(n-1)\big[12b^{2}q_{00} \nonumber\\ && \left. +2r_{00;0\cdot i}b^{i}+6b^{2}s_{0;0}+3r^{m}_{\ m}e_{00}+36s_{0}r_{0}+12(3b^{2}+n-4)s_{0}^{2} +6w_{00}\big]\right\}\alpha^{2} \nonumber\\ && -4(nt^{m}_{\ m}-{\bf r}_{\alpha})\beta^{4}+\big[-2(n-1)(6r^{m}_{\ m}s_{0}+r^{m}_{\ m;0}+2r^{m}_{\ 0;m}) \nonumber\\ && -8{}^{\alpha}{\bf Ric}_{ij}b^{i}y^{j}+16ns^{m}_{\ 0;m}-8n(n-3)t_{0}\big]\beta^{3}+\left\{4(b^{2}+2)({}^{\alpha}{\bf Ric}-2t_{00})\right. \nonumber\\ && +2(n-1)\big[4(n+2)q_{00}+2r_{00;0\cdot i}b^{i}+2(3b^{2}+2)s_{0;0}+3r^{m}_{\ m}e_{00}+3s_{0}r_{0} \nonumber\\ && \left.+4(6b^{2}+10n-33)s_{0}^{2}+6w_{00}\big]\right\}\beta^{2}- 2(n-1)\left\{(3b^{2}-n)e_{00;0}\right. \nonumber\\ && \left.+\big[18r_{0}+2(15b^{2}+10n-48)s_{0}\big] e_{00}\right\}\beta+3(n-1)(6b^{2}+n-12)e_{00}^{2}.\label{Gma2} \end{eqnarray} By (\ref{Gma1}) and (\ref{Gma2}), we can find that \begin{equation} \Gamma_{2}\beta-\Gamma_{1}=-18(1-b^{2})\beta e_{00}^{2}+(\alpha^{2}-\beta^{2})K_{000},\label{G2-G1} \end{equation} where $K_{000}$ is a homogeneous polynomial of degree 3 in $y$.
On the other hand, by the assumption, Randers metric of $F=\alpha +\beta$ is of weakly isotropic scalar curvature, that is, \begin{equation} {\bf r}=n(n-1)\left[\frac{\theta}{F}+\mu(x)\right]. \label{r22} \end{equation} Then, multipying ($\ref{r22}$) by $4F^{5}$, one has \begin{equation} \label{r3} 4F^{5}{\bf r}=4n(n-1)(\Pi_{1}+\alpha\Pi_{2}), \end{equation} where, $\Pi_{1}$ and $\Pi_{2}$ are homogeneous polynomials of degree 5 and 4 in $y$, respectively, which are expressed as \begin{eqnarray*} \Pi_{1}&:=&(5\mu\beta+\theta)\alpha^{4}+(10\mu\beta^{3}+6\theta\beta^{2})\alpha^{2}+\mu\beta^{5}+\theta\beta^{4},\\ \Pi_{2}&:=& \mu\alpha^{4}+(10\mu\beta^{2}+4\theta\beta^{2})\alpha^{2}+5\mu\beta^{4}+4\theta\beta^{3}. \end{eqnarray*} Further, we can get \begin{equation} \Pi_{2}\beta-\Pi_{1}=-(\alpha^{2}-\beta^{2})(4\mu\beta\alpha^{2}+4\mu\beta^{3}+\theta\alpha^{2}+3\theta\beta^{2}). \label{r5} \end{equation} Comparing (${\ref {r2}}$) and (${\ref {r3}}$), we obtian the following \begin{equation} \label{r4} \Gamma_{1}=4n(n-1)\Pi_{1}, \ \ \Gamma_{2}=4n(n-1)\Pi_{2}. \end{equation} Therefore, $\Gamma _{2}\beta - \Gamma _{1}= 4n(n-1)(\Pi_{2}\beta-\Pi_{1})$. From (${\ref {G2-G1}}$) and (${\ref {r5}}$), one obtains \begin{eqnarray*} && -18(1-b^{2})\beta e_{00}^{2}+(\alpha^{2}-\beta^{2})K_{000}\\ &&= -4n(n-1)(\alpha^{2}-\beta^{2})(4\mu\beta\alpha^{2}+4\mu\beta^{3}+\theta\alpha^{2}+3\theta\beta^{2}), \end{eqnarray*} which is equivalent to \begin{equation} \-18(1-b^{2})\beta e_{00}^{2}=(\alpha^{2}-\beta^{2})[K_{000}+4n(n-1)(4\mu\beta\alpha^{2}+4\mu\beta^{3}+\theta\alpha^{2}+3\theta\beta^{2})]. \end{equation} Because $\alpha^{2}-\beta^{2}$ is an irreducible polynomial in $y$ and $1-b^{2}>0$, we know that $e_{00}$ must be divided by $\alpha^{2}-\beta^{2}$. That is, there exists a scalar function $c(x)$ on $M$ such that \begin{equation} e_{00}=2c(x)(\alpha^{2}-\beta^{2}). \end{equation} By Lemma {\ref {e00}}, $F$ is of isotropic $S$-curvature. \hspace*{\fill}Q.E.D.
\vskip 2mm
In order to prove Theorem \ref{CWS}, we prove the following lemma firstly.
\begin{lem}\label{lmS} Let $F = \alpha +\beta$ be a non-Riemannian Randers metric with isotropic $S$-curvature on an n-dimensional ($n\geq 2$) manifold $M$. If there is scalar function $\sigma =\sigma (x)$ on $M$ such that $\bar {F}:=e^{\sigma (x)}F$ is of weakly isotropic scalar curvature, then $\sigma$ must be a constant. \end{lem}
{\bf Proof.} \ By the assumption, $\bar{F}$ is conformally related to $F$, $\bar{F} =e^{\sigma(x)}F$. Then we have the following equality ({\cite {BC}}). \begin{equation} \bar{\bf S} ={\bf S}+F^{2}\sigma^{r}I_{r}, \label{S1} \end{equation} where $\sigma^{r}:=g^{rm}\sigma_{x^{m}}$ and $I_{r}$ is the mean Cartan tensor of $F$.
Also, by the assumption, Randers metric $F$ is of isotropic $S$-curvature, that is, there is a scalar function $\lambda(x)$ on $M$ such that \begin{equation} {\bf S}=(n+1)\lambda(x)F. \label{S2} \end{equation} On the other hand, by the assumption that Randers metric $\bar{F}:=e^{\sigma(x)}F$ is of weakly isotropic scalar curvature and by Theorem \ref{SCS}, $\bar{F}$ must be of isotropic $S$-curvature , \begin{equation} {\bar{\bf S}}=(n+1){\bar\lambda}(x){\bar F}, \label{S3} \end{equation} where $\bar{\lambda}(x)$ is a scalar function on $M$. Plugging ($\ref{gij}$), ($\ref{Ii}$), ($\ref{S2}$) and ($\ref{S3}$) into ($\ref{S1}$), and then, similar to the proof of Theorem 1.2 in {\cite{CY}}, we can conclude that $\sigma(x)$ is a constant. \hspace*{\fill}Q.E.D.
\vskip 2mm
Lemma \ref{lmS} shows that, if $F$ is a non-Riemannian Randers metric with isotropic $S$-curvature on an $n$-dimensional manifold $M (n\geq 3),$ then there is no non-constant scalar function $\sigma=\sigma(x)$ such that $\bar{F}:=e^{\sigma} F$ is of weakly isotropic scalar curvature.
Now, we are in the position to prove Theorem \ref{CWS}.
{\bf Proof of Theorem \ref{CWS}.} \ By the assumption, $F=\alpha+\beta$ is a conformally flat metric, that is, there exists a scalar function $\kappa(x)$ on $M$, such that \begin{equation} F=e^\kappa(x){\bar{F}}, \end{equation} where ${\bar{F}}$ is a Minkowski metric.
Obviously, $\bar{F}$ is of isotropic S-curvature, $\bar{\bf S}=0$. Further, since $F=e^\kappa(x){\bar{F}}$ is of weakly isotropic scalar curvature, by Lemma \ref{lmS}, $\kappa(x)$ must be a constant. Hence, $F$ is a Minkowski metric. This completes the proof. \hspace*{\fill}Q.E.D.
\vskip 8mm
\vskip 10mm
\noindent Xinyue Cheng \\ School of Mathematical Sciences \\ Chongqing Normal University \\ Chongqing 401331, P. R. China \\ chengxy@cqnu.edu.cn
\vskip 5mm
\noindent Yannian Gong \\ School of Sciences \\ Chongqing University of Technology \\ Chongqing 400054, P. R. China \\ gyn@2017.cqut.edu.cn
\end{document} |
\begin{document}
\title{A secure deterministic remote state preparation via a seven-qubit entangled channel of an arbitrary two-qubit state under the impact of quantum noise}
\author{Deepak Singh \orcidA{}}
\email{dsingh@ma.iitr.ac.in} \affiliation{Department of Mathematics,\\ Indian Institute of Technology Roorkee 247667, Uttarakhand, India}
\author{Sanjeev Kumar \orcidB{}}
\email{sanjeev.kumar@ma.iitr.ac.in} \affiliation{Department of Mathematics, Mehta Family School of Data Science and Artificial Intelligence\\ Indian Institute of Technology Roorkee 247667, Uttarakhand, India}
\author{Bikash K. Behera \orcidC{}} \email{bikas.riki@gmail.com} \affiliation{Bikash's Quantum (OPC) Pvt. Ltd., Balindi, Mohanpur 741246, Nadia, West Bengal, India}
\begin{abstract} As one of the most prominent subfields of quantum communication research, remote state preparation (RSP) plays a crucial role in quantum networks. Here we present a deterministic remote state preparation scheme to prepare an arbitrary two-qubit state via a seven-qubit entangled channel created from Borras \emph{et al.} state. Quantum noises are inherent to each and every protocol for quantum communication that is currently in use, putting the integrity of quantum communication systems and their dependability at risk. The initial state of the system was a pure quantum state, but as soon as there was any noise injected into the system, it transitioned into a mixed state. In this article, we discuss the six different types of noise models namely bit-flip noise, phase-flip noise, bit-phase-flip noise, amplitude damping, phase damping and depolarizing noise. The impact these noises had on the entangled channel may be seen by analysing the density matrices that have been altered as a result of the noise. For the purpose of analysing the impact of noise on the scheme, the fidelity between the original quantum state and the remotely prepared state has been assessed and graphically represented. In addition, a comprehensive security analysis is performed, demonstrating that the suggested protocol is safe against internal and external attacks. \end{abstract}
\begin{keywords} {Quantum Communication, Remote State Preparation, Entanglement, Borras \emph{et al.} state} \end{keywords}
\maketitle \onecolumngrid \section{Introduction}\label{sec:introduction}\label{Section-1} Quantum communication is a very essential field in quantum mechanics, quantum computation, and quantum information theory. Entanglement is the backbone of quantum communication; it is considered the essential asset of overall quantum information processing \cite{nielsen2001quantum}. It is well known that there are some novel phenomena in the applications of quantum communication and information theory, which include quantum teleportation \cite{bennett1993teleporting, boschi1998experimental}, quantum secure direct communication (QSDC)\cite{long2007quantum}, quantum key distribution \cite{scarani2009security} and quantum dense coding \cite{mattle1996dense}, quantum secret sharing \cite{gisin2002quantum, zhang2005multiparty}, quantum data hiding \cite{verstraete2003quantum}, quantum private comparison (QPC) \cite{yang2013corrigendum}, quantum remote state preparation (RSP) \cite{bennett2001remote, bennett2005remote, zhang2016deterministic} to name a few. Using an entangled channel to send the quantum state securely in all of these quantum communication protocols is an important area of research in quantum communication. At least in theory, quantum protocols have the potential to attain a better degree of security than their conventional counterparts.
In a remote state preparation (RSP) protocol, the sender uses a shared entangled channel and the right measurements to set up a known quantum state for the faraway receiver. RSP is believed to be a more efficient method of teleporting to a known state than the traditional method since it uses fewer classical bits \cite{pati2000minimum}. Since then, several distinct RSP algorithms \cite{berry2003optimal, devetak2001low} were put up as potential solutions. Later on, a number of research that were based on RSP were carried out. These included joint remote state preparation (JRSP) \cite{chen2011joint, hou2009joint, yang2012joint} and controlled remote state preparation (CRSP) [\cite{chen2012controlled}. On the other hand, the vast majority of RSP algorithms made in the past have a chance of success that is less than 1. To raise the success probability of RSP to 1, a new RSP algorithm called the deterministic remote state preparation (DRSP) algorithm \cite{zhang2017deterministic, ma2017deterministic} was proposed. This algorithm can set up the needed quantum state with a one-in-one chance of success, which saves a lot of quantum resources. In recent years, several fascinating studies about RSP have been carried out. A unique technique for the implementation of RSP of a generic m-qubit entangled state was suggested by Wang et al. \cite{wang2015generalized} by employing the GHZ-type states as quantum channels with a high success probability, reducibility, and generalizability. This strategy was developed by using the GHZ-type states. Wang et al.\cite{wang2015efficient} suggested two successful measurement-based ways for executing the RSP techniques for generic W-class entangled states for three and four particles, employing GHZ-type states as the quantum channels in the same year. In comparison to the previous schemes, these approaches stand a better chance of being implemented successfully with a higher success probability compared to the existing schemes. In addition, the suggested methods may be successfully implemented with a total success probability of one when the utilised channels are condensed into the most maximally entangled versions of themselves. Wang et al. \cite{wang2016practical} proposed two optics-based implementations for RSP and JRSP of an arbitrary single-photon pure state. The protocols for these implementations may be achieved with a specific success probability with the assistance of appropriate LOCC.
Every quantum protocol is practically not error-free. It has some ambiguities in the system. These ambiguities are considered quantum noise. Quantum noise is inevitable in implementing a quantum communication system under realistic conditions. Quantum noise will hurt the safety and reliability of the quantum communication system in a big way. Research on RSP \cite{wang2017effect, liang2015effects} in noisy environments is now being conducted and explored. The JRSP method was computed in amplitude-damping noise, and phase-damping noise by Guan et al. \cite{guan2014joint}, who also conducted an in-depth analysis of the impact that noise has on the output state. Ma et al. \cite{ma2017deterministic} examined the influence of amplitude-damping and phase-damping noise on the DJRSP method and presented a deterministic approach for constructing distant states using a Brown et al. state as the underlying channel. In \cite{dash2020deterministic} the noisy environment is studied for deterministic joint remote state preparation of arbitrary two-qubit state. A complexity analysis of the teleportation scheme under the influence of noise is studied in \cite{singh2021complexity}.
Motivated by these schemes, we have developed a protocol for the deterministic remote state preparation of a two-qubit qubit quantum state via a maximally entangled seven-qubit state that is derived from Borras \emph{et al.} state. The use of highly entangled Borras \emph{et al.} \cite{borras2007multiqubit} state in the remote state preparation has not been done in any of the schemes proposed till now. This scheme involves four participants in preparation for the remote state. Alice, Bob, Charlie, and David share a predetermined seven-qubit entangled channel. Alice has the first qubit, Bob has the second and third, Charlie has the fourth and sixth, and David has the last two qubits, which are the fifth and seventh. The first action that has to be performed is to factorize the entangled channel into the sum of the Bell basis states that have the same coefficients as the two-qubit state that needs to be prepared remotely. Then Alice measures her qubit and conveys the result to Bob through a classical channel. Charlie and David likewise communicate their outcomes to Bob via classical channels. Finally, in order to remotely prepare the desired two qubit-state, Bob needs to perform certain unitary operation on his qubit based on the collapse state of other participants. All the recovery operations are discussed in details in table \ref{table:Operation}. Next, we will study the impact of six types of noise on the entangled channel. These six kinds of noise are named bit-flip noise, phase-flip noise, bit-phase-flip noise, amplitude damping, phase damping and depolarizing noise. These noise models are studied with the help of the action of the Kraus operator on the qubit. When the Kraus operator acts on a quantum state, it becomes a mixed state. Upon evaluation of the density matrices, the fidelity is calculated between the initial state and the remotely prepared state. The variation in fidelity is represented with the help of a graph. In \cite{chen2012controlled}, the security analysis of the remote state preparation protocol is performed. This study analyses the security attack from the outside and the inside participants.
The rest of the article is structured as follows: Section \ref{Section-2} covers introductory terminology about the underlying entangled channel and remote state preparation procedures. In Section \ref{Section-3}, the noise analysis of the entangled teleportation channel is described. Section \ref{Section-5} consists of the security analysis of the protocol, and \ref{Section-5} concludes the report with a discussion of the study's results, followed by suggestions for further research.
\section{Remote state preparation of an arbitrary two-qubit state}\label{Section-2}
Suppose Alice, Bob, Charlie, and David are the participants in this system. Alice, Charlie and David combining together, want to remotely prepare a known quantum state at Bob's end. The two-qubit quantum state can be parameterized as $\ket{\xi} = \alpha \ket{00} + \beta \ket{11}$, where $ \alpha,\beta \in \mathbb{C}$ such that $|\alpha|+|\beta|^2 = 1$ takes care of the normalization of the state $\Ket{\phi}$. The seven-qubit entangled channel shared among the participants can be prepared by using the Borras \emph{et al.} state and an ancilla qubit in the $\Ket{0}$ state. The Borras \emph{et al.} state is given by \begin{eqnarray}
\ket{\psi} &=& \frac{1}{4}\Big(\ket{000}(\ket{0}\ket{\psi^{+}} + \ket{1}\ket{\phi^{+}}) +\ket{001}(\ket{0}\ket{\phi^{-}} - \ket{1}\ket{\psi^{-}})
+\ket{010}(\ket{0}\ket{\phi^{+}} - \ket{1}\ket{\psi^{+}}) + \ket{011}(\ket{0}\ket{\psi^{-}} + \ket{1}\ket{\phi^{-}}) \nonumber\\
&-& \ket{100}(\ket{0}\ket{\phi^{-}} + \ket{1}\ket{\psi^{-}}) +\ket{101}(-\ket{0}\ket{\psi^{+}} + \ket{1}\ket{\phi^{+}}) + \ket{110}(\ket{0}\ket{\psi^{-}} - \ket{1}\ket{\phi^{-}}) + \ket{111}(\ket{0}\ket{\phi^{+}} + \ket{1}\ket{\psi^{+}}) \Big) \nonumber \\ \label{eq-Borras} \end{eqnarray}
where $\ket{\phi^{\pm}} = \frac{1}{\sqrt{2}} (\ket{01} \pm \ket{10})$ and $\ket{\psi^{\pm}} = \frac{1}{\sqrt{2}} (\ket{00} \pm \ket{11})$. The ancilla qubit is entangled with the Borras \emph{et al.} state and the final entangled channel is given by the following equation \begin{eqnarray}
\ket{\Psi} &=& \Big( \ket{\psi}_{123456} \otimes \ket{0}_{7} \Big)_{CX(6,7)} \nonumber \\
&=& \frac{1}{4\sqrt{2}}\Big( \ket{0000000} +\ket{0000111} +\ket{0001011} +\ket{0001100} +\ket{0010011} -\ket{0010100} -\ket{0011000} +\ket{0011111} \nonumber \\ &+& \ket{0100011} +\ket{0100100} -\ket{0101000} -\ket{0101111} +\ket{0110000} -\ket{0110111} +\ket{0111011} -\ket{0111100} -\ket{1000011} \nonumber \\ &+& \ket{1000100} -\ket{1001000} +\ket{1001111} -\ket{1010000} -\ket{1010111} +\ket{1011011} +\ket{1011100} +\ket{1100000} -\ket{1100111} \nonumber \\ &-& \ket{1101011} +\ket{1101100} +\ket{1110011} +\ket{1110100} +\ket{1111000} +\ket{1111111}
\Big)_{AB_1B_2C_1D_1C_2D_2} \label{Borras_Eq1} \end{eqnarray} The Borras \emph{et al.} state is a six-qubit highly entangled state, which included two-qubit maximally entangled Bell states, $\ket{\phi^{\pm}}$ and $\ket{\psi^{\pm}}$ \cite{borras2007multiqubit}. Because the prepared entangled state $\ket{\Psi}$ was generated using the C-NOT operation on a highly entangled six-qubit Borras et al. state, which is considered to add entanglement to a quantum state, we believe that it is indeed extremely entangled, if not maximally entangled. Once the entangled channel is obtained, Alice reserves the first qubit for herself and distributes the other qubits as follows: Bob receives the second and third qubits, Charlie receives the fourth and sixth qubits, and David receives the fifth and seventh qubits. Alice owns the qubit $A$, Bob owns the qubits $B_1$ and $B_2$, Charlie owns the qubits $C_1$ and $C_2$, and David owns the qubits $D_1$ and $D_2$. In order to remotely prepare the quantum state at Bob's end, the factorization of the quantum state plays a crucial role. So, the known state $\ket{\Psi}$ is primarily factorized in a certain way that only the sender utilises a basis created out of the known parameters $\alpha$ and $\beta$. Here the factorization of $\ket{\Psi}$ can be done in the following basis -
\begin{figure}
\caption{ (a) Diagram representation of the remote state preparation protocol (b) Quantum Circuit representation of the RSP scheme. The }
\label{fig:Graph Plot 1}
\label{fig:Graph Plot 2}
\label{fig:Figure}
\end{figure}
\begin{eqnarray}
\ket{\Psi} &=& \dfrac{1}{4} \Big[ \ket{\Upsilon_1}\ket{\zeta_1} + \ket{\Upsilon_2}\ket{\zeta_2} \Big]_{AB_1B_2B_2C_1C_2D_1D_2} \label{Eq2} \end{eqnarray} where the quantum state $\ket{\Upsilon_1}$ and $\ket{\Upsilon_1}$ are given by the Eq.\eqref{Eq3} \begin{eqnarray}
\ket{\Upsilon_1}_A &=& (\alpha \Ket{0} + \beta \Ket{1})_{A} \nonumber \\
\ket{\Upsilon_2}_A &=& (\alpha \Ket{1} - \beta \Ket{0})_{A} \label{Eq3} \end{eqnarray}
And the quantum states $\Ket{\zeta_1}$ and $\Ket{\zeta_2}$ are given by the following expressions
\begin{eqnarray}
\ket{\zeta_1} = \dfrac{1}{\sqrt{2}} &\Big[& \big( \alpha \Ket{00} + \beta \Ket{10} + \alpha \Ket{11}-\beta \Ket{01} \big)_{B_1B_2} \Ket{00}_{C_1C_2}\Ket{00}_{D_1D_2} \nonumber\\
&+& \big(\alpha \Ket{01} + \beta \Ket{11} + \alpha \Ket{10}-\beta \Ket{00}\big)_{B_1B_2} \Ket{01}_{C_1C_2} \Ket{01}_{D_1D_2} \nonumber \\
&+& \big(-\alpha \Ket{01} + \beta \Ket{11} - \alpha \Ket{10}-\beta \Ket{00}\big)_{B_1B_2} \Ket{10}_{C_1C_2} \Ket{00}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{00} - \beta \Ket{10} + \alpha \Ket{11}+\beta \Ket{01}\big)_{B_1B_2} \Ket{11}_{C_1C_2} \Ket{01}_{D_1D_2} \nonumber \\
&+& \big(-\alpha \Ket{01} + \beta \Ket{11} + \alpha \Ket{10}+\beta \Ket{00}\big)_{B_1B_2} \Ket{00}_{C_1C_2} \Ket{10}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{00} + \beta \Ket{10} - \alpha \Ket{11} + \beta \Ket{01}\big)_{B_1B_2} \Ket{10}_{C_1C_2} \Ket{10}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{00} - \beta \Ket{10} - \alpha \Ket{11} - \beta \Ket{01}\big)_{B_1B_2} \Ket{01}_{C_1C_2} \Ket{11}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{01} + \beta \Ket{11} - \alpha \Ket{10} + \beta \Ket{00}\big)_{B_1B_2} \Ket{11}_{C_1C_2} \Ket{11}_{D_1D_2} \Big]
\label{Eq01} \end{eqnarray}
\begin{eqnarray}
\ket{\zeta_2} = \dfrac{1}{\sqrt{2}} &\Big[& \big(\alpha \Ket{10} - \beta \Ket{00} - \alpha \Ket{01}-\beta \Ket{11}\big)_{B_1B_2} \Ket{00}_{C_1C_2}\Ket{00}_{D_1D_2} \nonumber\\
&+& \big(\alpha \Ket{11} - \beta \Ket{01} - \alpha \Ket{00}-\beta \Ket{10}\big)_{B_1B_2} \Ket{01}_{C_1C_2} \Ket{01}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{11} + \beta \Ket{01} - \alpha \Ket{00}+\beta \Ket{10}\big)_{B_1B_2} \Ket{10}_{C_1C_2} \Ket{00}_{D_1D_2} \nonumber \\
&+& \big(-\alpha \Ket{10} - \beta \Ket{00} + \alpha \Ket{01}-\beta \Ket{11}\big)_{B_1B_2} \Ket{11}_{C_1C_2} \Ket{01}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{11} + \beta \Ket{01} + \alpha \Ket{00} - \beta \Ket{10}\big)_{B_1B_2} \Ket{00}_{C_1C_2} \Ket{10}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{10} - \beta \Ket{00} + \alpha \Ket{01} + \beta \Ket{11}\big)_{B_1B_2} \Ket{10}_{C_1C_2} \Ket{10}_{D_1D_2} \nonumber \\
&+& \big(- \alpha \Ket{10} - \beta \Ket{00} - \alpha \Ket{01} + \beta \Ket{11}\big)_{B_1B_2} \Ket{01}_{C_1C_2} \Ket{11}_{D_1D_2} \nonumber \\
&+& \big(\alpha \Ket{11} - \beta \Ket{01} + \alpha \Ket{00} + \beta \Ket{10}\big)_{B_1B_2} \Ket{11}_{C_1C_2} \Ket{11}_{D_1D_2} \Big] \label{Eq02} \end{eqnarray} Now Alice measure her qubit in the basis $\{ \ket{\Upsilon_1}, \ket{\Upsilon_1} \}$, Charlie and David measure their qubits in $\{\Ket{00}, \Ket{01}, \Ket{10}, \Ket{11} \}$ basis then Bob can easily remotely prepare the state $\Ket{\xi}$ knowing the outcome of Alice, Charlie and David. For instance, if Alice's measurement collapse in the state $\Ket{\Upsilon_1}_A$, Charlie and David's measurement collapse to the state $\Ket{01}_{C_1C_2}\Ket{01}_{D_1D_2}$ after then, each participant will use a conventional channel to communicate their results to Bob. Bob will next be required to do the necessary unitary operations on his qubit in order to remotely prepare the state at his end., in this case, Bob will apply a controlled-not gate form ``his first qubit" to the second qubit ($B_1$ to $B_2$), denoted as $CX_{1-2}$. A Hadamard gate on his first qubit, denoted as $H(1)$, again a not operation on his second qubit, denoted as $X_2$ and finally a $Z_1$ gate on his first qubit to flip the phase of the first qubit. At this stage of the protocol, Bob has successfully prepared the quantum state $\Ket{\xi}$ at his end. All the other possible cases are given in table \ref{table:Operation}.
\begin{table*}[!htb] \centering \renewcommand{1.65}{1.65}
\begin{tabular}{|l|l|l|c|} \hline \textbf{Alice's Outcome} & \textbf{Charlie \& David's Outcome} & \textbf{Bob's Collapse State} & \textbf{Bob's Operations} \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{00}_{C_1C_2}\Ket{00}_{D_1D_2}$ & $\frac{1}{\sqrt{2}}(\alpha \Ket{00} + \beta \Ket{10} + \alpha \Ket{11} - \beta \Ket{01})$ & $CX_{1-2}~H_1~Z_{1}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{01}_{C_1C_2}\Ket{01}_{D_1D_2}$ & $\frac{1}{\sqrt{2}}(\alpha \Ket{01} + \beta \Ket{11} + \alpha \Ket{10}-\beta \Ket{00})$ & $CX_{1-2}~H_1~X_2~Z_{1}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{10}_{C_1C_2}\Ket{00}_{D_1D_2}$ & $\frac{1}{\sqrt{2}}(-\alpha \Ket{01} + \beta \Ket{11} - \alpha \Ket{10}-\beta \Ket{00})$ & $CX_{1-2}~H_1~Z_{1}~Z_{2}~X_{2}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{11}_{C_1C_2}\Ket{01}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{00} - \beta \Ket{10} + \alpha \Ket{11}+\beta \Ket{01} )$ & $CX_{1-2}~H_1$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{00}_{C_1C_2}\Ket{10}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (-\alpha \Ket{01} + \beta \Ket{11} + \alpha \Ket{10}+\beta \Ket{00} )$ & $CX_{1-2}~H_1~Z_{1}~X_{1}~X_{2}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{10}_{C_1C_2}\Ket{10}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{00} + \beta \Ket{10} - \alpha \Ket{11}+\beta \Ket{01})$ & $CX_{1-2}~H_1~Z_{2}~X_{1}~CX_{2-1}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{10}_{C_1C_2}\Ket{11}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} ( \alpha \Ket{00} - \beta \Ket{10} - \alpha \Ket{11} - \beta \Ket{01} )$ & $CX_{1-2}~H_1~Z_{2}~X_{1}$ \\ \hline $\ket{\Upsilon_1}_A$ & $\Ket{11}_{C_1C_2}\Ket{11}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{01} + \beta \Ket{11} - \alpha \Ket{10} + \beta \Ket{00} )$ & $CX_{1-2}~H_1~X_{2}~X_{1}$ \\ \hline \hline $\ket{\Upsilon_2}_A$ & $\Ket{00}_{C_1C_2}\Ket{00}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{10} - \beta \Ket{00} - \alpha \Ket{01} - \beta \Ket{11})$ & $CX_{1-2}~H_1~Z_{1}~X_{1}~X_{2}~Z_{1}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{01}_{C_1C_2}\Ket{01}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{11} - \beta \Ket{01} - \alpha \Ket{00} - \beta \Ket{10} )$ & $CX_{1-2}~H_1~Z_{2}~Z_{1}~X_{1}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{10}_{C_1C_2}\Ket{00}_{D_1D_2}$ & $\frac{1}{\sqrt{2}}(\alpha \Ket{11} + \beta \Ket{01} - \alpha \Ket{00} + \beta \Ket{10})$ & $CX_{1-2}~H_1~Z_{1}~X_{1}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{11}_{C_1C_2}\Ket{01}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (-\alpha \Ket{10} - \beta \Ket{00} + \alpha \Ket{01} - \beta \Ket{11} )$ & $CX_{1-2}~H_1~X_{1}~X_{2}~Z_{1}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{00}_{C_1C_2}\Ket{10}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{11} + \beta \Ket{01} + \alpha \Ket{00} - \beta \Ket{10} )$ & $CX_{1-2}~H_1$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{10}_{C_1C_2}\Ket{10}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{10} - \beta \Ket{00} + \alpha \Ket{01} + \beta \Ket{11} )$ & $CX_{1-2}~H_1~Z_{1}~X_{2}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{10}_{C_1C_2}\Ket{11}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (-\alpha \Ket{10} - \beta \Ket{00} - \alpha \Ket{01} + \beta \Ket{11} )$ & $CX_{1-2}~H_1~Z_{1}~Z_{2}~X_{2}$ \\ \hline $\ket{\Upsilon_2}_A$ & $\Ket{11}_{C_1C_2}\Ket{11}_{D_1D_2}$ & $\frac{1}{\sqrt{2}} (\alpha \Ket{11} - \beta \Ket{01} + \alpha \Ket{00} + \beta \Ket{10} )$ & $CX_{1-2}~H_1~Z_{1}$ \\ \hline \end{tabular} \caption{Here are the operations that Bob must do on his qubits in order to obtain a state based on the results of Alice and Charlie/David. Here $CX_{1-2}$ represents the CNOT gate from qubit 1 to 2, $H_1$ represents the hadamard gate on qubit 1, $Z_2$ represents the Z gate on qubit 2, and $X_2$ represents the not operation on qubit 2.} \label{table:Operation} \end{table*}
\section{Effect of noisy environment on the quantum channel }\label{Section-3} The influence that noise has on the quantum channel will be the topic of our next discussion. Ambiguities are produced in the quantum system as a result of the fact that a real quantum system does not operate in perfect circumstances and that it interacts with its surrounding environment. This kind of ambiguity is described by the phrase quantum noise. Quantum noise must be taken into account in order to undertake precise research on a quantum communication method.In any actual quantum experiment conducted in the real world, noise will play a crucial role in deciding the success of the scheme. These noises may be studied and classified into one of six categories. Bit flip, phase flip, bit-phase flip, amplitude damping, phase damping, and depolarizing noise are the six fundamental forms of quantum noise that may occur in quantum channels. In this section, these six forms of quantum noise are examined. Therefore, the most productive course of action would be to investigate the causes and consequences of the system's noises and to seek to reduce their impact wherever feasible. We can analyze this by studying the evolution of the density matrix $\rho=\ket{\psi}\bra{\psi}$ with the help of Kraus operators. The impact of noise on the quantum channel can be done using the operator sum representation. The action of Kraus operators $E_k$ on a particular qubit $k$ described by the density matrix $\rho_k$ is given by the Eq. \eqref {eq:Xi}
\begin{equation}
\mathcal{G}^r(\rho_k) = \sum_{j=1}^{n} \big({E_j}\big)\ \rho_k\ \big({E_{j}}\big)^{\dagger} \label{eq:Xi} \end{equation} where $r \in \{b,w,f,a,p,d\}$ for bit-flip, phase-flip, bit-phase-flip, amplitude damping, phase damping and depolarizing damping respectively, $j \in \{0,1\}$ for $r=b,w,f,a$, $j \in \{0,1,2\}$ for $r=p$ and $j \in \{0,1,2,3\}$ for $r=d$.
Alice retains the first qubit and transfers the remaining qubits to Bob, Charlie, and David. After the distribution of qubits in a noisy environment, the shared entangled state would transform into a mixed quantum state. Bob must perform the proper unitary operations on his qubits to remotely construct the quantum state, therefore the final state $\rho_{out} ^r$ may be written as the density matrix indicated in the following Eq. \eqref{eq:rho_out}.
\begin{equation}
\rho_{out}^r = Tr_{i_1i_2...i_{n-1}} \{ \mathcal{U} [\rho_k \otimes \xi^r(\rho_l)] \mathcal{U}^{\dagger} \}
\label{eq:rho_out} \end{equation} where $Tr_{i_1i_2...i_{n-1}}$ is the partial trace over the qubits $i_1,i_2,...,i_{n-1}$ and $\mathcal{U}$ is the unitary operations, which Bob will apply on his qubit to remotely prepare the quantum state, the operation $\mathcal{U}$ is given by the following Eq. \eqref{eq:U}, $\mathcal{U}$
\begin{eqnarray} \mathcal{U} &=& \{\mathbb{I}_1 \otimes \mathbb{I}_2 \otimes . . .\mathbb{I}_{n-1} \otimes \sigma_{n} ^{i_1 i_2 ... i_{n-1}} \} \{\ket{\phi}_{12} \bra{\phi}_{12} \otimes \mathbb{I}_3 ... \otimes \mathbb{I}_n \} \{\mathbb{I}_1 \otimes \mathbb{I}_2 ... U_{j_1 j_2} ... \otimes \mathbb{I}_n \} \nonumber \\ && \{\mathbb{I}_1 \otimes \mathbb{I}_2 ... U_{k_1} ... \otimes \mathbb{I}_n \} \label{eq:U} \end{eqnarray}
where $\sigma_{n} ^{i_1 i_2 ... i_{n-1}}$ is the the Bob's recovery unitary operations after Alice has measured her qubit and communicated her result to Bob via a classical channel, $\ket{\phi}_{12} \bra{\phi}_{12}$ is the Bell basis measurement on the first two qubits, $U_{j_1 j_2}$ represents the C-NOT gate from qubit $j_1$ to $j_2$ and $U_{k_1}$ represents the unitary gate on qubit $k_1$. The unitary operations vary according to the collapse states of Alice, Charlie, and David, as given in the table \ref{table:Operation}. By computing the fidelity between the initial two-qubit state $ket{\xi}$ and the density matrix $\rho_{out}^r$, the influence of noise in the entangled channel can now be illustrated. Fidelity reflects the proximity between two quantum states and provides a mathematical formula for quantifying the degree of resemblance between quantum states. The mathematical expression for nose-effected fidelity given by Eq. \eqref{eq:Fid} \cite{liang2019quantum}.
\begin{equation}
\mathcal{F} = \bra{\Psi} \rho_{out}^r \ket{\Psi}
\label{eq:Fid}
\end{equation}
Due to the fact that this communication is occurring under the influence of noise, some quantum information may be lost during the remote state preparation. Fidelity is the ideal statistic for measuring how much data is lost. When the fidelity $\mathcal{F}=1$, this indicates the ideal case where the noise hasn't affected the communication, and no information has been lost. Meanwhile, $\mathcal{F}=0$ implies that the noise has very badly affected the communication and the state being communicated has been changed completely to a different state resulting in all the information being lost. Thus, the fidelity ranges between $0$ and $1$, i.e., $0\leq \mathcal{F} \leq 1$. We now discuss the effect of six types of noises (bit-flip, phase-flip, bit-phase-flip, amplitude damping, phase damping and depolarizing noise) in the next section. The entangle channel given in Eq. \eqref{Borras_Eq1} can be factorized in the following manner \eqref{eq-f}
\begin{eqnarray}
\ket{\Psi} &=& \frac{1}{32}\Big(
\ket{000}(\ket{0}\ket{\mu^{+}} + \ket{1}\ket{\lambda^{+}})
+\ket{001}(\ket{0}\ket{\lambda^{-}} - \ket{1}\ket{\mu^{-}})
+\ket{010}(\ket{0}\ket{\lambda^{+}} - \ket{1}\ket{\mu^{+}}) + \ket{011}(\ket{0}\ket{\mu^{-}} + \ket{1}\ket{\lambda^{-}})
\nonumber\\ &+&
\ket{100}(-\ket{0}\ket{\lambda^{-}} - \ket{1}\ket{\mu^{-}})
+\ket{101}(\ket{1}\ket{\lambda^{+}}-\ket{0}\ket{\mu^{+}})
+\ket{110}(\ket{0}\ket{\mu^{-}} - \ket{1}\ket{\lambda^{-}})
+\ket{111}(\ket{0}\ket{\lambda^{+}} + \ket{1}\ket{\mu^{+}}) \Big) \nonumber \\ \label{eq-f} \end{eqnarray} where $\ket{\lambda^{\pm}} = \frac{1}{\sqrt{2}} (\ket{011} \pm \ket{100})$ and $\ket{\mu^{\pm}} = \frac{1}{\sqrt{2}} (\ket{000} \pm \ket{111})$
\subsection{Bit-flip noisy environment} The bit-flip noise changes the state of computational qubit $\ket{0}$ to $\ket{1}$ and vice-versa with probability $\eta_b$ and the qubits remain unchanged with the probability $(1-\eta_b)$\cite{fortes2015fighting, oh2002fidelity}. Its operations on a qubit can be described by Kraus operators given by the following matrices in Eq.\eqref{eqn_bit_noise}
\begin{eqnarray}
&& E_0^b= \sqrt{1-\eta_b} \mathbb{I} =
\sqrt{1-\eta_b} \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}, ~~~~E_1^b= \sqrt{\eta_b}\mathbb{X} = \sqrt{\eta_b}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix} \label{eqn_bit_noise} \end{eqnarray}
Where $ \eta_b \in [0,1]$ represents the probability parameter for the bit-flip error in the quantum system. The impact of bit-flip noise on the entangled channel may be analyzed with the help of density matrices of the noisy channel. The affected density matrix under the bit-flip noise is denoted by $\mathcal{G}^b (\rho)$, given in Eq. \eqref{eq:bit2}.
\begin{eqnarray}
\mathcal{G}^b (\rho) &=& \frac{1}{32}
\Big\{(1-\eta_b)^{7} \Big[\Ket{\Psi}\bra{\Psi}\Big] + (\eta_b)^7
\Big[\ket{111}(\ket{1}\ket{\mu^{+}} + \ket{0}\ket{\lambda^{+}})
+\ket{110}(\ket{0}\ket{\mu^{-}} -\ket{1}\ket{\lambda^{-}} )
+\ket{101}(\ket{1}\ket{\lambda^{+}} - \ket{0}\ket{\mu^{+}}) \nonumber \\&-&
\ket{100}(\ket{1}\ket{\mu^{-}} + \ket{0}\ket{\lambda^{-}}) + \ket{011}(\ket{1}\ket{\lambda^{-}} - \ket{0}\ket{\mu^{-}}) +\ket{010}(\ket{0}\ket{\lambda^{+}}-\ket{1}\ket{\mu^{+}}) + \ket{001}(\ket{0}\ket{\lambda^{-}} -\ket{1}\ket{\mu^{-}} ) \nonumber \\ &+&
\ket{000}(\ket{1}\ket{\lambda^{+}} + \ket{0}\ket{\mu^{+}}) \Big]
\times
\Big[\bra{111}(\bra{1}\bra{\mu^{+}} + \bra{0}\bra{\lambda^{+}})
+\bra{110}(\bra{0}\bra{\mu^{-}} -\bra{1}\bra{\lambda^{-}} )
+\bra{101}(\bra{1}\bra{\lambda^{+}} - \bra{0}\bra{\mu^{+}}) \nonumber \\ &-&
\bra{100}(\bra{1}\bra{\mu^{-}} + \bra{0}\bra{\lambda^{-}})
+\bra{011}(\bra{1}\bra{\lambda^{-}} - \bra{0}\bra{\mu^{-}}) +\bra{010}(\bra{0}\bra{\lambda^{+}}-\bra{1}\bra{\mu^{+}})
+\bra{001}(\bra{0}\bra{\lambda^{-}} -\bra{1}\bra{\mu^{-}} ) \nonumber \\ &+&
\bra{000}(\bra{1}\bra{\lambda^{+}} + \bra{0}\bra{\mu^{+}}) \Big] \Big\}
\label{eq:bit2} \end{eqnarray}
\subsection{Phase-flip noisy environment} The phase-flip noise affects the phase of the computational qubit. If the system has a phase-flip noise then the computational qubit changes from $\ket{1}$ to $-\ket{1}$, whereas it does not alter the qubit $\ket{0}$. It's Kraus operators \cite{fortes2015fighting, oh2002fidelity} are given by Eq. \eqref{eqn_phase_noise},
\begin{eqnarray}
&& E_0^w= \sqrt{1-\eta_w} \mathbb{I} =
\sqrt{1-\eta_w} \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}~, ~~~~E_1^w= \sqrt{\eta_w}\mathbb{Z} = \sqrt{\eta_w}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} \label{eqn_phase_noise} \end{eqnarray} where $\eta_{w} \in [0,1]$ represents the probability parameter for the phase-flip error in the quantum system. The effect of phase-flip noise on the entangled channel can be studied by using the density matrices of the noisy phase-affected channel, which is given by $\mathcal{G}^w (\rho)$ \begin{eqnarray} \mathcal{G}^w (\rho) &=& \frac{1}{32} \Big\{(1-\eta_w)^{7} \Big[\Ket{\Psi}\bra{\Psi}\Big] + (\eta_w)^7 \Big[\ket{000}(\ket{0}\ket{\mu^{-}} - \ket{1}\ket{\lambda^{-}}) - \ket{001} (\ket{0}\ket{\lambda^{+}} +\ket{1}\ket{\mu^{+}} ) - \ket{010}(\ket{0}\ket{\lambda^{-}} + \ket{1} \ket{\mu^{-}}) \nonumber \\ &-& \ket{011}(\ket{0}\ket{\mu^{+}} - \ket{1}\ket{\lambda^{+}}) - \ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}}) - \ket{101} (+\ket{0}\ket{\mu^{-}}+\ket{1}\ket{\lambda^{-}}) + \ket{110}(\ket{0}\ket{\mu^{+}} +\ket{1}\ket{\lambda^{+}} ) \nonumber \\ &-& \ket{111}(\ket{0}\ket{\lambda^{-}} - \ket{1}\ket{\mu^{-}}) \Big] \times \Big[\bra{000}(\bra{0}\bra{\mu^{-}} - \bra{1}\bra{\lambda^{-}}) - \bra{001} (\bra{0}\bra{\lambda^{+}} +\bra{1}\bra{\mu^{+}} ) - \bra{010}(\bra{0}\bra{\lambda^{-}} + \bra{1} \bra{\mu^{-}}) \nonumber \\ &-& \bra{011}(\bra{0}\bra{\mu^{+}} - \bra{1}\bra{\lambda^{+}}) - \bra{100}(\bra{1}\bra{\mu^{+}} - \bra{0}\bra{\lambda^{+}} ) - \bra{101} (+\bra{0}\bra{\mu^{-}}+\bra{1}\bra{\lambda^{-}}) + \bra{110}(\bra{0}\bra{\mu^{+}} + \bra{1}\bra{\lambda^{+}} ) \nonumber \\ &-& \bra{111}(\bra{0}\bra{\lambda^{-}} - \bra{1}\bra{\mu^{-}}) \Big] \Big\} \label{eq:phase} \end{eqnarray}
\subsection{Bit-phase-flip noisy environment} The combination of a phase flip and a bit flip is referred to as a bit–phase flip, and it is represented by the Kraus operators given in Eq. \eqref{eqn_bitphase_noise} \cite{fortes2015fighting, oh2002fidelity},
\begin{eqnarray}
&& E^f= \sqrt{1-\eta_f} \mathbb{I} =
\sqrt{1-\eta_f} \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}~, ~~~~ E^f = \sqrt{\eta_f}\mathbb{Y} = \sqrt{\eta_f}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix} \label{eqn_bitphase_noise} \end{eqnarray} where $ \eta_{f} \in [0,1]$ denotes the bit-phase-flip noise parameter probability, which defines the likelihood of an error in the quantum state arising owing to a computational qubit. The impact of bit-phase flip noise on the entangled channel can be determined by calculating the noise affected density matrix. The impacted density matrix under bit-phase flip noise is denoted by $\mathcal{G}^f (\rho)$, which is provided by Eq. \eqref{eq:flip}
\begin{eqnarray} \mathcal{G}^f (\rho) &=& \frac{1}{32} \Big\{(1-\eta_f)^{7} \Big[\Ket{\Psi}\bra{\Psi}\Big] -(\eta_f)^7 \Big[\ket{111}(-\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}}) -\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}}) +\ket{101}(\ket{1}\ket{\lambda^{-}} + \ket{0}\ket{\mu^{-}}) \nonumber \\ &+& \ket{100}(\ket{1}\ket{\mu^{+}} - \ket{0}\ket{\lambda^{+}}) - \ket{011}(\ket{0}\ket{\mu^{+}} - \ket{1}\ket{\lambda^{+}}) +\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}}) +\ket{001}(\ket{0}\ket{\lambda^{+}} + \ket{1}\ket{\mu^{+}}) \nonumber \\ &-& \ket{000}(\ket{0}\ket{\lambda^{-}} + \ket{0}\ket{\mu^{-}}) \Big] \times \Big[\bra{111}(-\bra{1}\bra{\mu^{-}}+ \bra{0}\bra{\lambda^{-}}) -\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}}) +\bra{101}(\bra{1}\bra{\lambda^{-}} + \bra{0}\bra{\mu^{-}}) \nonumber \\ &+& \bra{100}(\bra{1}\bra{\mu^{+}} - \bra{0}\bra{\lambda^{+}}) -\bra{011}(\bra{0}\bra{\mu^{+}} - \bra{1}\bra{\lambda^{+}}) +\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}}) +\bra{001}(\bra{0}\bra{\lambda^{+}} + \bra{1}\bra{\mu^{+}}) \nonumber \\ &-& \bra{000}(\bra{0}\bra{\lambda^{-}} + \bra{0}\bra{\mu^{-}}) \Big] \Big\} \label{eq:flip} \end{eqnarray}
\subsection{Effect of amplitude damping (AD) noise} The amplitude damping plays a vital role since it is responsible for characterising the energy loss of a system, which is the effect that occurs the most frequently in open systems. The idea of amplitude damping is crucial to the modelling of energy dissipation in a variety of quantum systems, and the matrices that follow supply the Kraus operators for this process \cite{fortes2015fighting, oh2002fidelity}. \begin{eqnarray}
E_0^a=
\begin{pmatrix}
1 & 0 \\
0 & \sqrt{1-\eta_{a}}
\end{pmatrix}~, \quad
E_1^a=
\begin{pmatrix}
0 & \sqrt{\eta_{a}} \\
0 & 0
\end{pmatrix} \label{eqn_AD_noise} \end{eqnarray} Where $\eta_{a} \in [0,1]$ signifies the decoherence rate of amplitude damping, which specifies the likelihood of quantum state inaccuracy associated with computational qubits. The influence of amplitude damping on the entangled channel may be detected by analysing the channel's noise-affected density matrix. This matrix is represented by $\mathcal{G}^a$, given in Eq. \eqref{eq:AP}
\begin{eqnarray} \mathcal{G}^a (\rho) &=& \frac{1}{32} \Big\{ \Big[\ket{000}(\ket{0}\ket{\Xi^{+}}+\sqrt{1-\eta_a}\ket{1} \ket{\Omega^{+}}) +\sqrt{1-\eta_a}\ket{001}(\ket{0}\ket{\Omega^{-}}-\sqrt{1-\eta_a}\ket{1}\ket{\Xi^{-}}) \nonumber \nonumber \\ &+& \sqrt{1-\eta_a}\ket{010}(\ket{0}\ket{\Omega^{+}} - \sqrt{1-\eta_a}\ket{1}\ket{\Xi^{+}}) + (1-\eta_a) \ket{011}(\ket{0}\ket{\Xi^{-}} + \sqrt{1-\eta_a}\ket{1}\ket{\Omega^{-}}) \nonumber \\ &-& \sqrt{1-\eta_a}\ket{100}(\ket{0}\ket{\Omega^{-}} + \sqrt{1-\eta_a} \ket{1}\ket{\Xi^{-}}) +(1-\eta_a)\ket{101}(\sqrt{1-\eta_a}\ket{1}\ket{\Omega^{+}}-\ket{0}\ket{\Xi^{+}}) \nonumber \\ &+& (1-\eta_a)\ket{110}(\ket{0}\ket{\Xi^{-}} - \sqrt{1-\eta_a}\ket{1}\ket{\Omega^{-}}) + \sqrt{(1-\eta_a)^3}\ket{111}(\ket{0}\ket{\Omega^{+}} + \sqrt{1-\eta_a}\ket{1}\ket{\Xi^{+}}) \Big] \nonumber \\ &\times& \Big[\bra{000}(\bra{0}\bra{\Xi^{+}}+\sqrt{1-\eta_a}\bra{1} \bra{\Omega^{+}}) +\sqrt{1-\eta_a}\bra{001}(\bra{0}\bra{\Omega^{-}}-\sqrt{1-\eta_a}\bra{1}\bra{\Xi^{-}}) \nonumber \\ &+& \sqrt{1-\eta_a}\bra{010}(\bra{0}\bra{\Omega^{+}} - \sqrt{1-\eta_a}\bra{1}\bra{\Xi^{+}}) \nonumber + (1-\eta_a) \bra{011}(\bra{0}\bra{\Xi^{-}} + \sqrt{1-\eta_a}\bra{1}\bra{\Omega^{-}}) \nonumber \\ &-& \sqrt{1-\eta_a}\bra{100}(\bra{0}\bra{\Omega^{-}} + \sqrt{1-\eta_a} \bra{1}\bra{\Xi^{-}}) +(1-\eta_a)\bra{101}(\sqrt{1-\eta_a}\bra{1}\bra{\Omega^{+}}-\bra{0}\bra{\Xi^{+}}) \nonumber \\ &+& (1-\eta_a)\bra{110}(\bra{0}\bra{\Xi^{-}} - \sqrt{1-\eta_a}\bra{1}\bra{\Omega^{-}}) + \sqrt{(1-\eta_a)^3}\bra{111}(\bra{0}\bra{\Omega^{+}} + \sqrt{1-\eta_a}\bra{1}\bra{\Xi^{+}}) \Big] \nonumber \\ &+& (\eta_a)^7 \Ket{1111111}\bra{1111111} \Big\} \label{eq:AP} \end{eqnarray}
where the quantum states $\ket{\Omega^{\pm}} = \dfrac{\sqrt{1-\eta_a}\big(\sqrt{1-\eta_a}\Ket{011} \pm \Ket{100} \big)}{\sqrt{2}}$ and $\ket{\Xi^{\pm}} = \dfrac{\big(\Ket{000} \pm \sqrt{(1-\eta_a)^3}\Ket{111} \big)}{\sqrt{2}}$
\subsection{Phase damping noisy environment} The phenomenon of phase-damping results in the loss of information regarding the relative phases of a quantum state. During phase damping, the fundamental quantum system becomes entangled with the surrounding environment \cite{mcmahon2007quantum}. The Kraus operators \{$E_{0}^p, E_{1}^p, E_{2}^p$\} for phase-damping noise are given in Eq. \eqref{eqn_PD_noise} \cite{fortes2015fighting, oh2002fidelity}. \begin{eqnarray}
&& E_0^p = \sqrt{1-\eta_{p}}
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}~, \quad
E_1^p=
\begin{pmatrix}
\sqrt{\eta_{p}} & 0 \\
0 & 0
\end{pmatrix}~, \quad ~~~ E_2^p=
\begin{pmatrix}
0 & 0 \\
0 & \sqrt{\eta_{p}}
\end{pmatrix} \label{eqn_PD_noise} \end{eqnarray} where $\eta_{p} \in [0,1]$ denotes the phase-damping decoherence rate, which specifies the likelihood of an error occurring in the quantum state associated with the computational qubit. After the noise has been introduced into the channel, the affected density matrix can be used to figure out how phase damping affected the entangled channels. The affected density matrix under the phase damping noise is denoted by $\mathcal{G}^p$.
\begin{eqnarray}
\mathcal{G}^p (\rho) &=& \frac{1}{32}
\Big\{(1-\eta_b)^{7} \Big[\Ket{\Psi}\bra{\Psi}\Big] + (\eta_p)^7
\Big[\ket{0000000}\bra{0000000} + \Ket{1111111}\bra{1111111}\Big] \Big\}
\label{eq:PD} \end{eqnarray}
\subsection{Depolarizing Noisy Environment} When exposed to a depolarizing noisy environment, the quantum state's qubits are depolarized with a probability of $\eta_d$, and the qubits are left with an invariant probability of $(1-\eta_d)$. The following matrices give the Kraus operators for depolarizing noise \cite{fortes2015fighting, oh2002fidelity}.
\begin{eqnarray}
&& E_0^d = \sqrt{1-\eta_{d}}
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}~, \quad
E_1^d = \sqrt{\frac{\eta_{d}}{3}}
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}~, ~~~ E_2^d = \sqrt{\frac{\eta_{d}}{3}}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}~, \quad
E_3^d = \sqrt{\frac{\eta_{d}}{3}}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} \label{eqn_DN_noise} \end{eqnarray}
After introducing noise into a channel, the impact of depolarizing noise on the entangled channel may be determined by analysing the affected density matrix, the noise affected density matrix under the depolarizing noise is denoted by $\mathcal{G}^D$ \begin{eqnarray} \mathcal{G}^d (\rho) &=& \frac{1}{32} \Big\{(1-\eta_d)^{7} \Big[\Ket{\Psi}\bra{\Psi}\Big] + (\eta_b)^7 \Big[\ket{111}(\ket{1}\ket{\mu^{+}} + \ket{0}\ket{\lambda^{+}}) +\ket{110}(\ket{0}\ket{\mu^{-}} -\ket{1}\ket{\lambda^{-}} ) +\ket{101}(\ket{1}\ket{\lambda^{+}} - \ket{0}\ket{\mu^{+}}) \nonumber \\&-& \ket{100}(\ket{1}\ket{\mu^{-}} + \ket{0}\ket{\lambda^{-}}) + \ket{011}(\ket{1}\ket{\lambda^{-}} - \ket{0}\ket{\mu^{-}}) +\ket{010}(\ket{0}\ket{\lambda^{+}}-\ket{1}\ket{\mu^{+}}) + \ket{001}(\ket{0}\ket{\lambda^{-}} -\ket{1}\ket{\mu^{-}} ) \nonumber \\ &+& \ket{000}(\ket{1}\ket{\lambda^{+}} + \ket{0}\ket{\mu^{+}}) \Big] \times \Big[\bra{111}(\bra{1}\bra{\mu^{+}} + \bra{0}\bra{\lambda^{+}}) +\bra{110}(\bra{0}\bra{\mu^{-}} -\bra{1}\bra{\lambda^{-}} ) +\bra{101}(\bra{1}\bra{\lambda^{+}} - \bra{0}\bra{\mu^{+}}) \nonumber \\ &-& \bra{100}(\bra{1}\bra{\mu^{-}} + \bra{0}\bra{\lambda^{-}}) +\bra{011}(\bra{1}\bra{\lambda^{-}} - \bra{0}\bra{\mu^{-}}) +\bra{010}(\bra{0}\bra{\lambda^{+}}-\bra{1}\bra{\mu^{+}}) +\bra{001}(\bra{0}\bra{\lambda^{-}} -\bra{1}\bra{\mu^{-}} ) \nonumber \\ &+& \bra{000}(\bra{1}\bra{\lambda^{+}} + \bra{0}\bra{\mu^{+}}) \Big]-(\eta_f)^7 \Big[\ket{111}(-\ket{1}\ket{\mu^{-}}+\ket{0}\ket{\lambda^{-}}) -\ket{110}(\ket{0}\ket{\mu^{+}}+\ket{1}\ket{\lambda^{+}}) +\ket{101}(\ket{1}\ket{\lambda^{-}} + \ket{0}\ket{\mu^{-}}) \nonumber \\ &+& \ket{100}(\ket{1}\ket{\mu^{+}} - \ket{0}\ket{\lambda^{+}}) - \ket{011}(\ket{0}\ket{\mu^{+}} - \ket{1}\ket{\lambda^{+}}) +\ket{010}(\ket{0}\ket{\lambda^{-}}+\ket{1}\ket{\mu^{-}}) +\ket{001}(\ket{0}\ket{\lambda^{+}} + \ket{1}\ket{\mu^{+}}) \nonumber \\ &-& \ket{000}(\ket{0}\ket{\lambda^{-}} + \ket{0}\ket{\mu^{-}}) \Big] \times \Big[\bra{111}(-\bra{1}\bra{\mu^{-}}+ \bra{0}\bra{\lambda^{-}}) -\bra{110}(\bra{0}\bra{\mu^{+}}+\bra{1}\bra{\lambda^{+}}) +\bra{101}(\bra{1}\bra{\lambda^{-}} + \bra{0}\bra{\mu^{-}}) \nonumber \\ &+& \bra{100}(\bra{1}\bra{\mu^{+}} - \bra{0}\bra{\lambda^{+}}) -\bra{011}(\bra{0}\bra{\mu^{+}} - \bra{1}\bra{\lambda^{+}}) +\bra{010}(\bra{0}\bra{\lambda^{-}}+\bra{1}\bra{\mu^{-}}) +\bra{001}(\bra{0}\bra{\lambda^{+}} + \bra{1}\bra{\mu^{+}}) \nonumber \\ &-& \bra{000}(\bra{0}\bra{\lambda^{-}} + \bra{0}\bra{\mu^{-}}) \Big] + (\eta_w)^7 \Big[\ket{000}(\ket{0}\ket{\mu^{-}} - \ket{1}\ket{\lambda^{-}}) - \ket{001} (\ket{0}\ket{\lambda^{+}} +\ket{1}\ket{\mu^{+}} ) - \ket{010}(\ket{0}\ket{\lambda^{-}} + \ket{1} \ket{\mu^{-}}) \nonumber \\&-& \ket{011}(\ket{0}\ket{\mu^{+}} - \ket{1}\ket{\lambda^{+}}) - \ket{100}(\ket{1}\ket{\mu^{+}}-\ket{0}\ket{\lambda^{+}}) - \ket{101} (+\ket{0}\ket{\mu^{-}}+\ket{1}\ket{\lambda^{-}}) + \ket{110}(\ket{0}\ket{\mu^{+}} + \ket{1}\ket{\lambda^{+}} ) \nonumber \\ &-& \ket{111}(\ket{0}\ket{\lambda^{-}} - \ket{1}\ket{\mu^{-}}) \Big] \times \Big[\bra{000}(\bra{0}\bra{\mu^{-}} - \bra{1}\bra{\lambda^{-}}) - \bra{001} (\bra{0}\bra{\lambda^{+}} +\bra{1}\bra{\mu^{+}} ) - \bra{010}(\bra{0}\bra{\lambda^{-}} + \bra{1} \bra{\mu^{-}}) \nonumber \\ &-& \bra{011}(\bra{0}\bra{\mu^{+}} - \bra{1}\bra{\lambda^{+}}) - \bra{100}(\bra{1}\bra{\mu^{+}} - \bra{0}\bra{\lambda^{+}} ) - \bra{101} (+\bra{0}\bra{\mu^{-}}+\bra{1}\bra{\lambda^{-}}) + \bra{110}(\bra{0}\bra{\mu^{+}} + \bra{1}\bra{\lambda^{+}} ) \nonumber \\ &-& \bra{111}(\bra{0}\bra{\lambda^{-}} - \bra{1}\bra{\mu^{-}}) \Big]\Big\} \label{eq:Depo} \end{eqnarray}
\begin{figure}
\caption{The plot of fidelity against the noise parameter $\eta$ for all the six kinds of noise models.}
\label{fig:RSPFig}
\end{figure}
\section{Security analysis}\label{Section-4} Quantum communication protocols, in comparison to their classical counterparts, often have the characteristic of a superior level of security. Our protocols are protected not just from attacks coming from the outside, but also from those coming from inside the system from the dishonest participant. Here, we provide two different types of security analysis about the process of remotely preparing a state. The first of these is an outside attack from an eavesdropper, attempting to learn the state that is being remotely prepared. And the second threat is an attempt from a non-authorized participant who is interested in finding out the quantum state that is being prepared remotely.
\subsection{Outside attack} Before the remote state preparation scheme is put into action, the seven-quit entangled channel is to be created from the six-qubit Borras \emph{et al.} state and then distributed among the participants. Without loss of generality, suppose Alice prepares the state $\Ket{\Psi}_{1234567}$ and send the qubits $(2,3)$ to Bob, qubits $(4,6)$ to Charlie and $(5,7)$ to David. Alice includes a predetermined number of decoy-state particles in the transmission which are randomly distributed in one of the following four quantum states $\{ \Ket{0}, \Ket{1}, \Ket{+}, \Ket{-} \}$. After it has been determined that all three participants have received the particles, Alice will then proceed to make the statement about the placements of the decoy particles as well as the measurement basis. Then, participants Bob, Charlie, and David measure their qubits in accordance with the provided basis and declare their results. Next, Alice makes a comparison between the results of the measurement and the initial states of the decoy particles. The fact that any eavesdropping leaves a trace in the outcomes of the decoy sampling photons \cite{wang2008efficient}, enables the security checking process to detect multiple types of attacks coming from an outside attacker Eve. These attacks include several attacks mentioned in \cite{chen2012controlled} named as an intercept-resend attack, a measurement-resend attack, an entanglement-measure attack, and a denial-of-service attack. During the security checking, the impact of these attacks will be detected with a probability larger than zero. This verification approach is based on the notion of the BB84 QKD protocol \cite{bennett2020quantum}, which has been shown to be completely safe by a number of different researchers \cite{shor2000simple}. Moreover, the particles used to construct the quantum channel do not transmit any concealed information. As a result, if an eavesdropper is present, she is not only identifiable but also incapable of gathering any relevant information during the security screening process. After the completion of the security checks, the entangled channel that is sufficiently safe will be distributed among the participants. If there is evidence of eavesdropping, the participants will abandon this procedure and begin over. After three different parties have validated the safety of the quantum channel, an eavesdropper from the outside can no longer attack the protocol since no qubits are being sent at this point. During the implementation of the protocol, only classical information is sent, which has no relevance to the secrets. Therefore, our remote state preparation protocol is robust against an attack from an outsider.
\subsection{Inside attack} It is possible for a participant to carry out his attack by entangling an auxiliary particle with his own particle. Let's suppose that Alice is the participant interested in finding out about Bob's state, being remotely prepared in an unethical way. She can prepare the auxiliary state $\Ket{\varepsilon}$ and entangle it using the local unitary operation $\mathcal{\hat{U}}$, which is defined as follows:
\begin{eqnarray}
\mathcal{\hat{U}} (\Ket{0}_1\Ket{\varepsilon}_E) &=& \Ket{0}_1 \Ket{\varepsilon_{00}} + \Ket{1}_1 \Ket{\varepsilon_{01}} \nonumber \\
\mathcal{\hat{U}} (\Ket{1}_1\Ket{\varepsilon}_E) &=& \Ket{0}_1 \Ket{\varepsilon_{10}} + \Ket{1}_1 \Ket{\varepsilon_{11}}
\label{eq:EntQ} \end{eqnarray}
Where $\braket{\varepsilon_{00}|\varepsilon_{00}} + \braket{\varepsilon_{01}|\varepsilon_{01}} = 1 $ and
$\braket{\varepsilon_{10}|\varepsilon_{10}} + \braket{\varepsilon_{11}|\varepsilon_{11}} = 1$. Suppose Charlie and David simultaneously measure their qubits and broadcast their results to Bob through a classical channel. Without compromising generality, assume that the Charlie and David's measurement results are $\Ket{00}_{C_1C_2}\Ket{00}_{D_1D_2}$. From table \ref{table:Operation}, Alice's measurement result is $\ket{\Upsilon_1}_A$ or $\ket{\Upsilon_2}_A$. After completing the unitary operations \eqref{eq:EntQ} on her state, Alice entangled an auxiliary qubit on it, and the resulting state becomes
\begin{eqnarray}
\ket{\Psi}_{AEB_1B_2} &=& \frac{1}{4} \Big( \mathcal{\hat{U}} \Ket{\Upsilon_1}_A \Ket{\varepsilon}_E\Big) \otimes \Big[ \frac{1}{\sqrt{2}}(\alpha \Ket{00} + \beta \Ket{10} + \alpha \Ket{11} - \beta \Ket{01}) \Big] \nonumber \\
&+& \frac{1}{4} \Big( \mathcal{\hat{U}} \Ket{\Upsilon_2}_A \Ket{\varepsilon}_E \Big) \otimes \Big[ \frac{1}{\sqrt{2}}(\alpha \Ket{10} - \beta \Ket{00} - \alpha \Ket{01} - \beta \Ket{11}) \Big] \nonumber \\
&=& \frac{1}{4} \Big[ \alpha (\Ket{0}\Ket{\varepsilon_{00}} + \Ket{1}\Ket{\varepsilon_{01}}) + \beta (\Ket{0} \Ket{\varepsilon_{10}} +\Ket{1} \ket{\varepsilon_{11}}) \Big] \Big[ \alpha \Ket{\psi^+} - \beta \Ket{\phi^-} \Big]
\nonumber \\ &+&
\frac{1}{4} \Big[ \alpha (\Ket{0}\Ket{\varepsilon_{10}} + \Ket{1}\Ket{\varepsilon_{11}}) - \beta (\Ket{0} \Ket{\varepsilon_{00}} +\Ket{1} \ket{\varepsilon_{01}}) \Big] \Big[-\beta \Ket{\psi^+} - \alpha\Ket{\phi^-} \Big]
\label{eq:EntQ2} \end{eqnarray}
At this point, Alice must follow the rules for preparing the remote state and tell Bob the result of her measurement. Bob will then do the unitary operations on his qubit. Now, we will concentrate on the quantum system of particles $AE$, which represents a portion of the entire system.
\begin{eqnarray}
\rho_{AE} &=& tr_{B_1B_2} \big( \rho_{AEB_1B_2} \big) \nonumber \\
&=& \dfrac{(\alpha^4 + \beta^4)}{16} \Big[ (\Ket{0}\Ket{\varepsilon_{00}} + \Ket{1}\Ket{\varepsilon_{01}}) (\bra{0}\bra{\varepsilon_{00}} + \bra{1}\bra{\varepsilon_{01}}) + (\Ket{0}\Ket{\varepsilon_{10}} + \Ket{1}\Ket{\varepsilon_{11}}) (\bra{0}\bra{\varepsilon_{10}} + \bra{1}\bra{\varepsilon_{11}}) \Big] \nonumber \\
&=& \dfrac{(\alpha^4 + \beta^4)}{16} \Big( (\Ket{0\varepsilon_{00}} + \Ket{1\varepsilon_{01}}) (\bra{0\varepsilon_{00}} + \bra{1\varepsilon_{01}}) + (\Ket{0\varepsilon_{10}} + \Ket{1\varepsilon_{11}}) (\bra{0\varepsilon_{10}} + \bra{1\varepsilon_{11}}) \Big)
\label{eq:EntQ3} \end{eqnarray}
Now, we will check the purity of the quantum state by evaluating the trace of the square of the density matrix, given by
\begin{eqnarray}
tr(\rho^2_{AE}) &=& \Big(\dfrac{\alpha^4 + \beta^4}{16}\Big)^2 + \Big(\dfrac{\alpha^4 + \beta^4}{16}\Big)^2 + 2\Big( \dfrac{\alpha^4 + \beta^4}{16} \Big) \Big( \dfrac{\alpha^4 + \beta^4}{16} \Big) |\braket{\varepsilon_{00}|\varepsilon_{10}} + \braket{\varepsilon_{01}|\varepsilon_{11}}|^2 \nonumber \\
&\leq& 2 \Big(\dfrac{\alpha^4 + \beta^4}{16}\Big)^2 \Big[1+(|\varepsilon_{00}|\varepsilon_{10}| + |\varepsilon_{10}|\varepsilon_{11}|)^2] \nonumber \\
&\leq& 2 \Big(\dfrac{\alpha^4 + \beta^4}{16}\Big)^2 \Big[1+(\sqrt{\braket{\varepsilon_{00}|\varepsilon_{00}} \braket{\varepsilon_{10}|\varepsilon_{10}}} + \sqrt{\braket{\varepsilon_{01}|\varepsilon_{01}} \braket{\varepsilon_{11}|\varepsilon_{11}}})^2 \Big] \nonumber \\
&\leq& 2 \Big(\dfrac{\alpha^4 + \beta^4}{16}\Big)^2 \Big[1+\Big( \dfrac{\braket{\varepsilon_{00}|\varepsilon_{00}} +\braket{\varepsilon_{10}|\varepsilon_{10}}}{2} + \dfrac{\braket{\varepsilon_{01}|\varepsilon_{01}}
+\braket{\varepsilon_{11}|\varepsilon_{11}}}{2} \Big)^2 \Big]
\label{eq:EntQ4} \end{eqnarray}
Simplifying the expression using the Cauchy–Schwarz inequality and the triangular inequality, we get \begin{eqnarray}
tr(\rho^2_{AE}) < 1
\label{eq:EntQ6} \end{eqnarray} If this is the case, the quantum state of the particle $AE$ will be in a mixed state, and Alice will not be able to extract any information from the system. That is to say, the inner attack initiated by Alice cannot be considered valid and Alice will not be able to steal any information from the attack.
\section{Discussion and Conclusion }\label{Section-5}
In this study, the deterministic remote state preparation is studied via the seven-qubit entangled channel, created from the maximally entangled Borras \emph{et al.} state under the effect of six different kinds of noises. First, the seven-qubit entangled channel was constructed from the highly entangled Borras state by taking an ancillary qubit and then performing a C-NOT operation from the terminal qubit of the Borras state to the ancillary qubit. This was done in order to establish the highly entangled Borras state. It generates an entangled channel in order to facilitate the remote state preparation of a two-qubit quantum state. The qubits that make up the entangled channels are split up among the four participants. Alice is the one who is sending the information, while Bob is the one who is participating and preparing the state at his end remotely. After Alice and the other two players, Charlie and David, have sent their measurement results to Bob using a classical channel, Bob then performs the relevant operation to his qubit in order to remotely prepare the state. The operations in the different scenarios are presented in table \ref{table:Operation}.
In addition, the impacts of six distinct types of noise are investigated during the course of this study. The noise may be modelled using the Kraus operators by applying them to the entangled channel. We have evaluated the density matrices for all the noise channels and calculated the fidelity of the remotely prepared state with respect to the noise-affected channel.
For various noise channels, it is possible to discern how the system loses information due to decoherence when it interacts with its environment. A fidelity metric has been used to calculate the amount of information that is lost due to each type of noise, for which variation of fidelity with respect to the noise parameter $\eta$ is plotted in figure \ref{fig:RSPFig}. The plot shows that as the noise parameter increases in the range $[0,1]$, the fidelity shows different trends in different types of noise. The graph reveals that depolarizing noise has the most negative effect on the channel, hence reducing its fidelity and causing information loss. In addition, the phase flip noise loses the least amount of information compared to all others. Moreover, the security analysis is conducted for a protocol intruder. In the event of an external attack, an eavesdropper may attempt to attack the protocol, tamper with the process, and determine the prepared quantum state. However, it appears that our protocol is secure against these types of threats, as outsiders cannot steal information. Again, in the event of an inside attack, a dishonest player may attempt to steal information by entangling an auxiliary qubit $\Ket{\varepsilon}$ with his qubit in order to determine the state being prepared. Attempting to do so, however, will result in the quantum state transforming into a mixed state, resulting in the loss of information.
\end{document} |
\begin{document}
\title{Minimal cylinders in the three-dimensional Heisenberg group} \dedicatory{}
\author[S.-P.~Kobayashi]{Shimpei Kobayashi}
\address{Department of Mathematics, Hokkaido University,
Sapporo, 060-0810, Japan}
\email{shimpei@math.sci.hokudai.ac.jp}
\thanks{The author is partially supported by Kakenhi 18K03265 and 22K03304.}
\subjclass[2020]{Primary~53A10, 58E20, Secondary~53C42}
\keywords{Minimal surfaces; Heisenberg group; cylinders;
generalized Weierstrass type representation}
\date{\today} \pagestyle{plain} \begin{abstract} We study minimal cylinders in the three-dimensional
Heisenberg group ${\rm Nil}_3$ using the generalized Weierstrass
type representation, the so-called loop group method.
We characterize all non-vertical minimal cylinders in terms of
pairs of two closed plane curves which have the same signed area.
Moreover, as a byproduct of the construction,
spacelike CMC cylinders can also be obtained.
\end{abstract} \maketitle
\section{Introduction and the main result}
\textbf{(A)} The construction of surfaces with special properties is an important and
standard task of classical differential geometry. A particularly important class of surfaces in the
three-dimensional Heisenberg group is formed by the minimal surfaces.
They have been investigated by many authors for many years, e.g.,
\cite{CPR, Figueroa, FMP, IKOS, Fer-Mira2, Daniel:GaussHeisenbergw}.
In \cite{DIKAsian} the basic loop group construction of all such surfaces with contractible domain
of definition has been presented. In \cite{DIK;Nil3-sym} this was generalized to minimal surfaces with non-trivial topology. More generally minimal surfaces with symmetries have been investigated.
As an illustration of our technique we have discussed equivariant surfaces and helicoidal surfaces in some detail.
The present paper continues the discussion of special surfaces in the
three-dimensional Heisenberg group ${\rm Nil}_3$ by investigating minimal cylinders.
Before we can describe the present work we would like to recall
the generalized Weierstrass type representation for non-vertical minimal surfaces in ${\rm Nil}_3$. For details and notation we refer to \cite{DIKAsian, DIK;Nil3-sym}.
In this paper we consider exclusively Riemann surfaces $M$ and their universal cover $\widetilde{M} = \mathbb D$, a contractible open subset of $\mathbb C$, since there is no minimal $S^2$ in ${\rm Nil}_3$. \begin{steps}
\item Choose any holomorphic potential $\eta = A (z, \lambda) dz$
so that $A(z, \lambda) = \sum_{k=-1}^\infty A_k(z) \lambda^k$ is defined for $\lambda \in \mathbb C^*$ and all $z \in \mathbb D$ and
the diagonal entries are even, and the off-diagonal entries are odd in $\lambda$.
Moreover, assume that the $(1,2)$-entry of $A_{-1}$ never vanishes.
\item Solve the ordinary differential equation $d C = C \eta$
on $\mathbb D$ with initial condition identity at some base point $z_0$.
Here we consider a holomorphic potential and therefore do not have any problems to solve the ode (Below we extend what is explained here to certain meromorphic $\mathfrak{su}_{1, 1}$-potentials).
\item
Consider the connected component $\mathcal{I}_{z_0}$,
containing $z_0$, of all points in $\mathbb D$ such that
$C(z, \lambda) $ has for all $\lambda \in S^1$ an ${\rm SU}_{1, 1}$-Iwasawa (twisted) loop group decomposition on
$\mathcal{I}_{z_0}$,
that is, $C = F V_+$, where $F : \mathcal{I}_{z_0}\rightarrow
\Lambda {\rm SU}_{1, 1 \sigma}$ and
$V_+ : \mathcal{I}_{z_0} \rightarrow \Lambda^+ {\rm SL}_{2} \mathbb C_{\sigma}$.
Then $F$ is on $\mathbb D(z_0)$ the extended frame of some minimal surface
in ${\rm Nil}_3$, that is, $F$ takes values in $\Lambda {\rm SU}_{1, 1 \sigma}$ and
its Maurer-Cartan form
has the particular $\lambda$-distribution characteristic for the Maurer-Cartan forms of extended frames.
Also note, the leading term $V_0$ of $V_+$ is a diagonal matrix and
we can assume w.l.g. that it has positive diagonal elements. In this case the decomposition $C = F V_+$
is unique.
\item Next we apply Sym-type formulas to the extended frame $F$ as follows: First we compute the spacelike CMC surface in Minkowski space $f_{\mathbb L_3}$ and its
Gauss map $N_{\mathbb L_3}$ respectively by the formulas
\begin{equation}\label{eq:SymMin}
f_{\mathbb L_3}=-i \lambda (\partial_{\lambda} F) F^{-1}
- \frac{i}{2} \operatorname{Ad} (F) \sigma_3,\quad \mbox{and} \quad
N_{\mathbb L_3}= \frac{i}{2} \operatorname{Ad} (F) \sigma_3
\end{equation}
with
$\sigma_3 = \left(
\begin{smallmatrix}
1 &0 \\
0 & -1
\end{smallmatrix}
\right)$.
Then we finally compute $f^{\lambda}$ by the formula \begin{equation}\label{eq:symNil}
f^{\lambda}:=\Xi_{\mathrm{nil}}\circ \hat{f}\quad\mbox{with} \quad
\hat f =
(f_{\mathbb L_3})^o -\frac{i}{2} \lambda (\partial_{\lambda} f_{\mathbb L_3})^d, \end{equation}
where the superscripts ``$o$'' and ``$d$'' denote the off-diagonal and
diagonal part,
respectively. Moreover, $\Xi_{\mathrm{nil}}$ denotes the exponential map from $\mathfrak{nil}_3$ to
${\rm Nil}_3$. Then $f^{\lambda}, {\lambda \in S^1}$ defines a
minimal surface in ${\rm Nil}_3$ for each $\lambda \in S^1$. \end{steps}
The Sym-type formula (\ref{eq:symNil}) first was stated in \cite{Cartier}, and the form of \eqref{eq:symNil} has been given in \cite{DIKAsian}.
\textbf{(B)} It is clear that the choice of holomorphic potentials $\eta$ is
the most important task.
Since the extended frames of a minimal surface in ${\rm Nil}_3$
and a spacelike CMC surfaces in the Minkowski $3$-space $\mathbb L_3$
are the same, see \eqref{eq:SymMin}
and \eqref{eq:symNil},
the construction technique of spacelike CMC surfaces can be
applied. Moreover, it has been known that there exists
a natural choice of holomorphic potentials for CMC-cylinders in
the Euclidean $3$-space, see \cite{DK:cyl}. Note that the extended frames
of a CMC surface in $\mathbb E^3$ and a spacelike CMC surface in
$\mathbb L_3$ just take values in different real forms of the loop group of ${\rm SL}_2 \mathbb C$.
We thus follow the strategy from Section 5.2 in \cite{DK:cyl} for
the construction of all non-vertical minimal cylinders in ${\rm Nil}_3$.
By the appendix we can assume without loss of generality that one can start from a
holomorphic and periodic matrix $C_0(z)$ of period $p>0$ of the form
\begin{equation}\label{eq:frameperiodic1}
C_0(z) = \begin{pmatrix} a_0(z) & b_0(z) \\
\overline {b_0(\bar z)} & \overline{ a_0(\bar z)} \end{pmatrix}\;\;,
\end{equation}
where $a_0(z), b_0(z)$ are periodic functions of period $p >0$ satisfying
$\det C_0(z) = a_0(z) \overline{a_0(\bar z)} - b_0(z) \overline{b_0(\bar z)} =1$.
In particular $C_0(z)$ is in ${\rm SU}_{1, 1}$ for $z \in \mathbb R$. Moreover, we assume in addition $C_0(z =0) = \operatorname{id}$.
\begin{Remark}
If we also want to admit the case
$C_0(z + p) = -C_0(z),$ then we can change in two ways to the periodic case:
\begin{enumerate}
\item[(a)] We change $C_0$ by a gauge $k$, where $k$ is a diagonal matrix with entries $\exp(i \pi z/ p)$ and $\exp(-i\pi z/ p)$. This does not change anything of geometric substance. However, strictly speaking it changes the original surface to another one by diagonal dressing.
\item[(b)] Or we consider the period $2p$.
\end{enumerate}
\end{Remark}
Set
\begin{align}\label{eq:C0}
\zeta_0 (z) = C_0^{-1} \partial_z C_0 dz
=
\begin{pmatrix} \nu(z) & \kappa (z) \\
\overline { \kappa (\bar z)} & -\nu(z) \end{pmatrix}dz, \end{align}
then $\nu$ and $\kappa$ are periodic functions of period $p >0$
and $\overline{\nu (\bar z)} = - \nu (z)$.
It is easy to see that \begin{equation}\label{eq:kappa}
\kappa (z) = \overline{a_0(\bar z)}\partial_z b_0 (z)
- b_0 (z) \partial_z \overline{a_0(\bar z)} \end{equation}
and $\nu (z) = \overline{a_0(\bar z)}\partial_z a_0 (z)
- b_0 (z) \partial_z \overline{b_0(\bar z)}$ hold.
We take a meromorphic function $h$ on $\mathbb D$, which is periodic of period $p$,
and set \begin{equation}\label{eq:frameperiodicPot}
\zeta (z, \lambda) =
\begin{pmatrix}
\nu (z) & \lambda^{-1}h(z) + \lambda(\kappa(z) - h(z)) \\
\lambda^{-1}(\overline{\kappa (\bar z)}- \overline{h(\bar z)} )
+ \lambda \overline{ h(\bar z)} & -\nu (z)
\end{pmatrix} dz.
\end{equation}
Then $\zeta (z, \lambda =1) = \zeta_0 (z)$ is meromorphic for
$z \in \mathbb D$ and in $\mathfrak{su}_{1, 1}$ for $z \in \mathbb R \cap \mathbb D$, that is,
$\zeta_0$ is $\mathfrak{su}_{1, 1}$ on $\mathbb R \cap \mathbb D$.
The potential $\zeta$ satisfies the
assumption stated in Proposition 5.1 of \cite{DK:cyl} after replacing ``skew-hermitian''
by \textquote{$\mathfrak{su}_{1, 1}$}, and will be called an \textit{$\mathfrak{su}_{1, 1}$-potential}
.
Using the $\mathfrak{su}_{1, 1}$-potential $\zeta$ as the input data for
Step $1$ above, one can construct a minimal surface in ${\rm Nil}_3$ with
particularly nice properties. We will explain them in Section \ref{sc:cylinder} below.
\begin{Remark}
The $\mathfrak{su}_{1, 1}$-potential
$\zeta$ is meromorphic and its poles can be off from the real line. In general poles of $\zeta$
give special properties of the resulting minimal surface, for examples,
ends, branch points or smooth points etc.
\begin{enumerate}
\item[(a)] Branch points: At a point $z_1$ where the product of the two (off-diagonal)
terms of $A_{-1}$ vanishes, considering the partial derivatives of (the Sym formula for)
the minimal surface in ${\rm Nil}_3$ immersion derived from the potential.
It is easy to see that the differential of the \textquote{immersion} is $1$ or $0$.
Thus we obtain a branch point.
\item[(b)] Ends: From \cite{DIK;Nil3-sym} one infers that sometimes poles of specific structure
characterize some also characterize \textquote{ends}. \end{enumerate}
\end{Remark}
\textbf{(C)} To state the main result of this paper, we introduce
two complex functions $\ell$ and $m$ defined by \begin{align}\label{eq:ell1} \ell (t) &=\int_{0}^{t} \left\{a_0 (s)^2 (2 h(s) - \kappa (s) )
+ b_0 (s)^2 (2 \overline{h(s)} - \overline{\kappa (s)}) \right\}ds \\
m(t)&= a_0(t) b_0(t), \label{eq:m} \end{align}
for $t \in \mathbb R \cap \mathbb D$.
Note that $\ell$ and $m$ give plane curves in $\mathbb C$, and
the curve $m$ is determined by the periodic matrix $C_0$
only, and in particular, the curve $m(t)$ is automatically periodic and thus closed.
Now we can formulate the main result of this paper. \begin{Theorem*}\label{thm:main}
The following statements hold$:$
{\rm (i)} Let $C_0$ and $\zeta$ be a periodic matrix and (after choosing a function $h$) the corresponding $\mathfrak{su}_{1, 1}$-potential as defined in \eqref{eq:frameperiodic1} and \eqref{eq:frameperiodicPot},
respectively,
and let $\ell$ and $m$ be the plane curves defined in \eqref{eq:ell1} and \eqref{eq:m} respectively.
Then $\zeta$ gives, via the generalized Weierstrass representation explained in part {\rm (A)} above,
a non-vertical minimal cylinder in ${\rm Nil}_3$ with possibly singular points
if the curves $\ell$ and $m$ are closed and their
signed enclosed areas are equal.
{\rm (ii)} Let $\tilde \ell$ and $\tilde m$ be analytic closed plane curves
whose signed enclosed areas are equal. Then there exist a periodic matrix
$C_0$ and an $\mathfrak{su}_{1, 1}$-potential $\zeta$ such that the plane curves
$\ell$ and $m$ in \eqref{eq:ell1} and \eqref{eq:m}
are $\tilde \ell$ and $\tilde m$ respectively, and $\zeta$ gives a non-vertical
minimal cylinder in ${\rm Nil}_3$ via the generalized Weierstrass representation explained in part {\rm (A)} above.
{\rm (iii)} All non-vertical minimal cylinders in ${\rm Nil}_3$ can be constructed from
two analytic curves which satisfy the conditions in {\rm (ii)}. \end{Theorem*}
The proof will be given in Section \ref{subsc:proof} below.
\begin{Remark}
One of the closed curves $\ell$ or $m$ can be degenerate, i.e. for example in Figure \ref{fig:0}, $m=0$ while $\ell$
is
a regular curve. On the one hand, in Example \ref{ex:cyl4}
\end{Remark}
\begin{figure}
\caption{The closed plane curve $\ell$ (left, the lemniscate) and two views of the corresponding
non-vertical minimal cylinder in ${\rm Nil}_3$ (right, bottom) from
the
potential $\zeta$ stated in \eqref{eq:zetapotential} with
$h$ defined by first one in \eqref{eq:choiceh0and1} and initial frame $C_0(z) \equiv \operatorname{id}$.
Note, we have $T = 0$ in this case, and the curve $m$ is degenerate to a point $0$.
The figures were drawn by using \cite{Br:Matlab} developed by David Brander.}
\label{fig:0}
\end{figure}
\section{Minimal cylinders in ${\rm Nil}_3$}\label{sc:cylinder} \subsection{Constructing $C_0$ and $\zeta$ for non-vertical minimal surfaces in ${\rm Nil}_3$} We show first the last part of the theorem, i.e. that each non-vertical minimal cylinder in ${\rm Nil}_3$ can be obtained from a potential $\zeta$ defined by using a matrix $C_0$ and a function $h$ as discussed in the introduction.
We follow the scheme outlined in \cite{DIK;Nil3-sym} and change \textquote{Mutatis Mutandis}.
First we recall that we have shown in Section 2.3.1 loc.cit. that
after multiplication by some diagonal matrix $L$ the
extended frame $F$ has a meromorphic extension $U = FL$.
In Section 2.3.2 loc.cit. we have discussed the extension of symmetries
to the meromorphic extension.
The closing condition is stated in Theorem 2.11. loc.cit.
We thus have proven the results of Section 3 of \cite{DK:cyl}.
\subsection{The construction of frame periodic minimal immersions in ${\rm Nil}_3$ from $C_0$ and $\zeta$ and $h$} We have already discussed the construction of a minimal surface in ${\rm Nil}_3$ from the initial data $C_0$ and $h$ in (B) and (C) of the introduction.
Recall, if $\zeta$ is the potential in \eqref{eq:frameperiodicPot} and
let $C = C(z, \lambda)$ denote the solution to $d C = C \zeta$,
$C(0, \lambda) =\operatorname{id}$.
Then it is easy to see that $C$ takes values in $\Lambda {\rm SU}_{1, 1 \sigma}$ for $z \in \mathbb R$,
since $\zeta$ takes values in $\Lambda \mathfrak{su}_{1, 1 \sigma}$ for $z \in \mathbb R$.
Our construction implies $C(z, 1) = C_0 (z)$, whence the first closing condition in Theorem 2.11 of \cite{DIK;Nil3-sym}
is satisfied, that is, \begin{equation}\label{eq:firstclosing}
M(\lambda)\in \Lambda {\rm SU}_{1, 1 \sigma}\quad\mbox{and}\quad M(\lambda = 1) = \pm \operatorname{id} \end{equation}
for the monodromy matrix $M$ defined by $C(z+p, \lambda) = M(\lambda) C(z, \lambda)$.
Moreover, $C(z, \lambda)$ is the extended frame along $z \in \mathbb R$ up to
a ${\rm U}_1$ factor, that is $F= C k$ for some $k \in {\rm U}_1$ on $z \in \mathbb R$.
Then the extended frame $F$ of a minimal surface in ${\rm Nil}_3$
satisfies $F(z+ p, \lambda) = M(\lambda) F(z, \lambda) k(z)$ and \eqref{eq:firstclosing}.
Such a surface will be called a \textit{frame
periodic minimal immersion in ${\rm Nil}_3$.}
The following theorem is an exact analogue of
Theorem 5.4 in \cite{DK:cyl} for $\mathfrak{su}_{1, 1}$-potentials
and the proof is as well.
\begin{Theorem}\label{thm:framePeriodic}
Let $C_0$ be given by \eqref{eq:frameperiodic1}.
Set $\zeta_0 = C_0^{-1} d C_0$ and define a $\mathfrak{su}_{1, 1}$-potential $\zeta$ by
\eqref{eq:frameperiodicPot}.
Then $\zeta$ defines a frame
periodic minimal immersion into ${\rm Nil}_3$.
Moreover, every frame periodic minimal immersion in ${\rm Nil}_3$ can be obtained this way.
\end{Theorem}
\subsection{The remaining two closing conditions for minimal cylinders in ${\rm Nil}_3$}\label{subsc:secondthird}
From Theorem 2.11 of \cite{DIK;Nil3-sym} we derive that in addition to the condition of
frame periodic we also need to satisfy two more closing conditions
to obtain a minimal \textquote{cylinder} in ${\rm Nil}_3$:
\begin{equation}\label{eq:closing}
X^o(\lambda =1) = O \quad \mbox{and} \quad Y^d(\lambda =1)= O, \end{equation}
where $ X = -i \lambda (\partial_{\lambda} M) M^{-1}$ and
$Y = - \frac{1}{2} \lambda \partial_{\lambda} (\lambda (\partial_{\lambda} M) M^{-1})$,
considered as functions of $\lambda$, and where the superscript ``$d$'' and ``$o$'' means ``diagonal part''
and ``off-diagonal part'' respectively.
It is natural to call $M(\lambda)= \pm \operatorname{id}$ the \textit{first} closing condition.
Accordingly, the condition $X^o(\lambda =1) = O $ and
the condition $Y^d(\lambda =1) = O$ will be called the \textit{second} and
the \textit{third} closing condition, respectively.
To consider the second and third closing conditions, we next
prove \cite[Theorem 5.1.2]{Kil} in our setting.
\begin{Proposition} \label{Kiliansformula}
Let $C$ be the solution of $d C = C \zeta$ with $C(0, \lambda) =\operatorname{id}$
where the $1$-form
$\zeta$ is given as before by \eqref{eq:frameperiodicPot}. Then the following formula holds$:$
\[
(\partial_{\lambda} C(z)) C(z)^{-1} = \int_0^z C(t, \lambda) (\partial_{\lambda} \zeta (t,\lambda) ) C(t, \lambda)^{-1}
\]
for all real $z$ and integration along the real axis. \end{Proposition} \begin{proof}
Consider the $z$-derivative of $(\partial_{\lambda} C) C^{-1}$: \begin{align*}
\partial_{z} ((\partial_{\lambda} C) C^{-1}) dz&= ( \partial_{\lambda}( \partial_{z} C) ) C^{-1}dz -
(\partial_{\lambda} C) C^{-1} (\partial_{z}C) C^{-1}dz \\& = ( \partial_{\lambda}( C \zeta) ) C^{-1} -
(\partial_{\lambda} C) C^{-1} (C \zeta) C^{-1}\\&= C (\partial_{\lambda} \zeta) C^{-1}. \end{align*}
Now an integration from $0$ to $z$ along the real axis yields \[
(\partial_{\lambda} C(z, \lambda)) C(z, \lambda)^{-1} = \int_0^z C(t, \lambda) (\partial_{\lambda} \zeta (t,\lambda) ) C(t, \lambda)^{-1} + A(\lambda). \]
But since $C(0,\lambda) = \operatorname{id}$, substituting $z = 0$ above yields
$A(\lambda) = 0$ for all $\lambda$. This completes the proof. \end{proof}
Since $C(z+p, \lambda) = M(\lambda) C(z,\lambda)$ implies $M(\lambda) = C(p, \lambda)$, we infer \begin{Corollary} \label{closingcondition}
Let
$X(\lambda) = -i \lambda (\partial_{\lambda} M(\lambda )) M(\lambda)^{-1}$ and
$Y(\lambda) = - \frac{1}{2} \lambda \partial_{\lambda} (\lambda (\partial_{\lambda} M(\lambda ))
M(\lambda )^{-1})$ respectively. Then the
following formulas hold$:$ \begin{align}
X(\lambda) &= -i \int_0^p C(t, \lambda) (\lambda \partial_{\lambda} \zeta (t,\lambda) ) C(t, \lambda)^{-1}, \\
Y(\lambda ) &=- \frac{i}{2} \lambda \partial_{\lambda} X(\lambda)
= - \frac12 \lambda \partial_{\lambda} \left(\int_0^p C(t, \lambda) (\lambda \partial_{\lambda} \zeta (t,\lambda) ) C(t, \lambda)^{-1}\right). \end{align} \end{Corollary}
Using Proposition \ref{Kiliansformula} and Corollary \ref{closingcondition},
it is straightforward to rewrite the second and third closing conditions for $\lambda = 1$ in a simple way. \begin{Proposition}
Let $\zeta$ be as before a $\mathfrak{su}_{1, 1}$-potential as
given in \eqref{eq:frameperiodicPot}.
Moreover, let $\alpha$ be the integrand in \eqref{eq:ell1},
\begin{equation}\label{eq:alpha}
\alpha(s)= a_0 (s)^2 (2 h(s) - \kappa (s) )
+ b_0 (s)^2 (2 \overline{h(s)} - \overline{\kappa (s)}).
\end{equation}
Then the second closing condition $X^o = 0$ is for
$\lambda = 1$ equivalent to
\begin{equation}\label{eq:secondclosing} \int_0^{p}
\alpha(t) dt =0.
\end{equation}
Moreover, the third closing condition $Y^d = 0$ is for $\lambda = 1$
equivalent to \begin{equation}\label{eq:thirdclosing}
\int_0^p \operatorname {Im} \left(
\alpha(t) \left\{\int_0^t \overline{\alpha(s)} ds \right\}\right) dt = \int_0^p \operatorname {Im} \left(a_0(t) \overline{b_0(t)} \kappa (t)\right) dt. \end{equation} \end{Proposition} \begin{proof}
Equation \eqref{eq:secondclosing} follows from
Corollary \ref{closingcondition},
since the integrand of $X(\lambda = 1)$ is
\[
C_0(t) \left\{\lambda \partial_{\lambda} \zeta (t,\lambda)
\right\}|_{\lambda = 1} C_0(t)^{-1}
\]
with $(1,2)$-entry equal to $-i\alpha(t)$.
To evaluate the third (and last) closing condition we need to
differentiate first of all $X(\lambda)$ for $\lambda$,
and then evaluate at $\lambda = 1$. \begin{align*} 2 Y(\lambda) &= -i \lambda \partial{_\lambda}X(\lambda)
\\ &= - \int_0^p \left\{\lambda \partial{_\lambda}C(t, \lambda)\right\}
\left\{\lambda \partial_{\lambda} \zeta (t,\lambda)\right\} C(t, \lambda)^{-1} \\
& \quad
- \int_0^p C(t, \lambda) [\lambda \partial{_\lambda} \left\{\lambda \partial_{\lambda} \zeta (t,\lambda) \right\} ] C(t, \lambda)^{-1}\\ & \quad +
\int_0^p C(t, \lambda) \left\{\lambda \partial_{\lambda} \zeta (t,\lambda) \right\} C(t, \lambda)^{-1}
\left\{\lambda \partial{_\lambda}C(t,\lambda)\right\} C(t, \lambda)^{-1} \end{align*}
A straightforward computation shows \[
\lambda \partial{_\lambda} \left(\lambda \partial_{\lambda} \zeta (t,\lambda) \right)
|_{\lambda =1}
= \begin{pmatrix} 0 &\kappa(t) \\ \overline{\kappa(t)} & 0 \end{pmatrix} dt. \]
Recall that $C(t, \lambda = 1) = C_0(t) = \left( \begin{smallmatrix}
a_0(t) & b_0(t) \\ \overline{ b_0(t)} & \overline{a_0(t)}
\end{smallmatrix}\right)$ holds.
Therefore the $(1, 1)$-entry of the second summand can be computed at $\lambda = 1$ as \begin{equation}\label{eq:secondsummand}
\int_0^p \left( a_0(t) \overline{b_0(t)} \kappa (t) -\overline{a_0 (t)} b_0(t) \overline{\kappa (t)} \right) dt. \end{equation}
Moreover, the first and third summands can be
computed at $\lambda = 1$ by using Proposition
\ref{Kiliansformula} as \begin{align*}
& - \int_0^p \left\{\int_0^t C_0(s) (\partial_{\lambda} \zeta (s,\lambda) ){|_{\lambda = 1}} C_0(s)^{-1} \right\}
\left\{C_0(t) (\lambda \partial_{\lambda} \zeta (t,\lambda) ){|_{\lambda = 1}} C_0(t)^{-1}\right\} \\ &+
\int_0^p \left\{C_0(t) (\lambda \partial_{\lambda} \zeta (t,\lambda) ){|_{\lambda = 1}} C_0(t)^{-1} \right\}
\left\{ \int_0^t C_0(s) (\partial_{\lambda} \zeta (s,\lambda) ){|_{\lambda = 1}} C_0(s)^{-1} \right\}. \end{align*}
Putting $A(t) = C_0(t) (\partial_{\lambda} \zeta (t,\lambda) ){|_{\lambda = 1}} C_0(t)^{-1}$
this is equivalent to
\begin{equation}\label{eq:thridclosing2}
\int_0^p \left[A(t), \int_0^t A(s)\right]. \end{equation}
where $[P, Q] = P Q -Q P$ denotes the matrix commutator.
It is easy to compute the $(1,1)$-entry of \eqref{eq:thridclosing2} as \begin{align} \label{eq:thirdsummand} - \int_0^p \left(\alpha (t)
\int_0^t\overline{\alpha(s)} ds
- \overline{\alpha(t)}
\int_0^t\alpha (s)ds \right)dt. \end{align}
Putting together \eqref{eq:secondsummand} and
\eqref{eq:thirdsummand}, the condition \eqref{eq:thirdclosing}
for $Y(\lambda = 1)^d = 0$ follows. This completes the proof. \end{proof} \begin{Remark}
In general the closing conditions in \eqref{eq:closing} can be
relaxed as follows: \begin{equation}\label{eq:nclosing}
\tilde X^o (\lambda =1)=0 \quad \mbox{and} \quad
\tilde Y^d (\lambda =1)=0, \end{equation}
where $\tilde X =- i \lambda (\partial_{\lambda} \tilde M) \tilde M^{-1}$
and $\tilde X =-\tfrac12 \partial_\lambda (\lambda (\partial_{\lambda} \tilde M) \tilde
M^{-1})$ with $\tilde M = \left\{M(\lambda)\right\}^n$
for some $n \in \mathbb N$. Then the second and the third closing condition in
\eqref{eq:secondclosing} and \eqref{eq:thirdclosing} respectively can be
stated by replacing the period \textquote{$p$}
by \textquote{$np$}, for some
$n \in \mathbb N$. \end{Remark}
\subsection{The proof of Theorem \ref{thm:main}}\label{subsc:proof}
\begin{proof}
We assume that the two curves $l$ and $m$ are closed and have the same signed area.
We want to show that the closing conditions for a minimal cylinder in ${\rm Nil}_3$ are satisfied.
Recall that the first closing condition is satisfied by construction.
(i) We look at the closing conditions \eqref{eq:secondclosing} and
\eqref{eq:thirdclosing} more closely.
It is evident that the second closing condition, i.e., \eqref{eq:secondclosing}
is equivalent to
that $\ell$ is a closed curve with period $p>0$.
It thus remains to consider the third closing condition.
But by using
the relation (\ref{eq:kappa}) between $\kappa$, $a_0$ and $b_0$ and
integration by parts, the third closing condition, \eqref{eq:thirdclosing},
is easily seen to be equivalent to \[ \int_0^{p}\operatorname {Im} \left( \overline{\ell(t)} \ell^{\prime}(t) \right) dt = \int_0^{p}\operatorname {Im} \left( \overline{m(t)} m^{\prime}(t) \right) dt. \]
On the left-hand side let $\ell(t) = x(t) + i y (t)$ be
written in terms of its real
and its imaginary part.
Then a straightforward computation shows that \begin{align*}
\int_0^{p} \overline{\ell(t)} \ell^{\prime}(t) dt&= \frac12 \int_0^{p} (x(t)^2 + y(t)^2)^{\prime} dt -i \int_0^{p} (x(t)y(t))^{\prime} dt + 2 i \int_0^{p}x(t)y^{\prime}(t) dt \end{align*} showing that the original integral expresses (up to the factor $2i$) the signed area of $\ell$.
The analogous argument applies to the curve $m$. Since we have assumed that the signed areas of $\ell$ and $m$ are equal, the two integrals above are equal and thus the third closing condition is satisfied.
(ii) First we define periodic functions $a_0$ and $b_0$
by the equations:
\begin{equation}\label{eq:a0b0}
\tilde m(z) = a_0(z) b_0(z), \quad a_0(z) \overline{a_0(\bar z)} - b_0(z) \overline{b_0(\bar z)} =1.
\end{equation}
Note, we first solve for functions defined on $\mathbb R$ and extend holomorpically, since our original functions were assumed to be real-analytic.
Note that the functions $a_0$ and $b_0$ are determined up to some phase factors, i.e.,
$a_0 e^{i \theta}$ and $b_0 e^{-i \theta}$ for some real function $\theta$.
Then the periodic matrix $C_0$ is defined by
\[
C_0(z) =\begin{pmatrix}
a_0(z) & b_0(z) \\
\overline{b_0(\bar z)} & \overline{a_0(\bar z)}
\end{pmatrix} k, \quad k =
\begin{pmatrix}
e^{i \theta(z)} & 0 \\
0 & e^{-i \theta(z)}
\end{pmatrix}.
\]
The diagonal matrix $k$ is just a gauge
and thus the functions $a_0$ and $b_0$
are uniquely determined by \eqref{eq:a0b0}.
Actually, different $C_0$ may yield the same cylinder: If we have $\tilde C_0 = C_0 k,$ with
$k$ diagonal unitary, then the Iwasawa decomposition yields a new $F$, namely $Fk$. But the Sym-formula
ignores $k$.
Next the plane curve $\tilde \ell$ is real analytic
and also $\tilde \ell^{\prime}(t)$ is.
Thus $\tilde \ell^{\prime}(t)$ can be extended holomorphically
around $t \in \mathbb R$ and we denote this curve by $\alpha$.
Then we define a function $\mu$
by \begin{equation}\label{eq:mu}
\mu (z) = \frac1{a_0 (z) \overline{a_0(\bar z)} + b_0 (z) \overline{b_0(\bar z)}}
\left(\overline{a_0(\bar z)}^2 \alpha(z) - b_0(z)^2 \overline{\alpha(\bar z)}
\right), \end{equation}
where $a_0, b_0$ are entries of $C_0$. From the construction it is easy to see that \[
\alpha(z) = a_0(z)^2 \mu(z) + b_0(z)^2 \overline{\mu(\bar z)} \]
holds.
Then we define $h$ to be \[
h(z) = \frac{\kappa(z) + \mu(z)}{2}, \]
where the function $\kappa(z)= \overline{a_0(\bar z)}\partial_z b_0 (z)
- \partial_z \overline{a_0(\bar z)} b_0 (z)$ is given by
\eqref{eq:kappa}, and define the potential $\zeta$ in \eqref{eq:frameperiodicPot} by the data $C_0$ and $h$.
Then the plane curve $\tilde \ell$ is exactly $\ell$ in \eqref{eq:ell1}
and $\zeta$ clearly produces a non-vertical minimal cylinder in ${\rm Nil}_3$.
\textrm{(iii)} To prove the final statement, we consider a non-vertical minimal cylinder in ${\rm Nil}_3$.
Then without loss of generality, there exists the extended frame $F \in \Lambda {\rm SU}_{1, 1 \sigma}$
such that $F$ is periodic on $\mathbb R$
with period $p>0$ and $F(z+p, \lambda) = M(\lambda) F(z)$
and the monodromy $M(\lambda)$ satisfies all closing conditions, that is, $M|_{\lambda=1}
=\pm \operatorname{id}$,
$X^o|_{\lambda=1 }=O$ and $Y^d|_{\lambda=1}=O$. We have shown in Section 2.3.1
in \cite{DIK;Nil3-sym} that there exists some diagonal
matrix $L$ such that $U= FL$ has a meromorphic extension, see e.g., Theorem 3.2 in \cite{DK:cyl}. Then the Maurer-Cartan
form of $U$ is a potential $\zeta = U^{-1} d U$
of the desired type, and the two plane curves $\ell$ and $m$
given by $C_0$ and $\zeta$ satisfy the properties in (ii).
\end{proof}
\subsection{Examples of non-vertical minimal cylinders in ${\rm Nil}_3$ constructed from matrices $C_0$}\label{subsc:diagonal}
In this subsection, we illustrate the theory presented in the previous section by constructing
non-vertical minimal cylinders in terms of simple specific periodic matrices $C_0$ and specific functions $h$.
Thus we start by considering a diagonal matrix \begin{equation}\label{eq:diagC0} C_0(z) =
\begin{pmatrix}
e^{icz} & 0 \\
0 & e^{-icz}
\end{pmatrix}, \quad c \in \mathbb R. \end{equation}
Note that $a_0(z) = e^{icz}$ and $b_0(z) =0$ in $C_0(z)$.
If $c \neq 0$, then we have the minimum period $p$ of $C_0(z)$ given by
$\pi/ |c|>0$, and if $c=0$, then we have any period. We compute \begin{align}
\zeta_0 (z)& = C_0 (z)^{-1} \partial_z C_0(z) dz=
\begin{pmatrix}
ic & 0 \\
0 & -ic
\end{pmatrix} dz. \end{align}
We note that in our case we have $\kappa(z) \equiv 0$ (the off-diagonal part of $\zeta_0$ is
identically zero).
For the construction of $\zeta$ we still need to choose a periodic function $h(z)$ of period $p$, which must be a positive integer
multiple of the minimum period of $C_0$. Then we obtain
\begin{equation}\label{eq:zetapotential}
\zeta (z, \lambda) =
\begin{pmatrix}
i c & \lambda^{-1} h(z) - \lambda h(z) \\
-\lambda^{-1} \overline{h(\bar z)}+ \lambda \overline{h(\bar z)} & -ic
\end{pmatrix} dz.
\end{equation}
From Corollary \ref{closingcondition} we derive that $X(\lambda = 1)$ is off-diagonal and
the integrals of both entries are complex conjugate.
Since the first closing condition is always satisfied by our construction we only need to consider the second and the third closing condition.
For this we compute the functions $\alpha$ and $m$ in \eqref{eq:alpha}
and \eqref{eq:m}.
We obtain \begin{equation}\label{eq:alphabeta}
\alpha(t) = 2 e^{2 i c t} h(t) \quad \mbox{and} \quad
m(t) = 0. \end{equation} \begin{Example}[The case $c = 0$]\label{ex:cyl1} We construct non-vertical minimal cylinders from the constant frame $C_0(z) \equiv \operatorname{id}$.
For this we only need to choose
a periodic holomorphic function $h$.
We present two examples, by choosing \begin{align} \label{eq:choiceh0and1}
h(z) = \frac{1+ i \sin z}{(i + \sin z)^2}, \quad \mbox{or}\quad
h(z) = \cos(z) - i \sin (3 z). \end{align}
Note that we choose the period $p$
for both choices of $h$ as $p = 2 \pi$.
For our first choice of $h$
the curve $\ell(t)= \int_0^t 2h(s) ds$
is the lemniscate, that is, \[
\ell(t) = - \frac{2 \cos z}{i + \sin z}, \]
and it is clear
that the signed area of $\ell$ is zero.
Moreover, the resulting non-vertical
minimal cylinder does not have any branch point on $\mathbb R$, since $h^2(z)+$\textquote{positive function} is the conformal factor of the metric
and $h$ in \eqref{eq:choiceh0and1} does vanish on $\mathbb R$.
This example is shown in Figure \ref{fig:0}.
For our second choice of $h$ the curve $\ell (t)$ can be computed as
\[
\ell(t) = \frac{2i}3 \cos 3 t + 2 \sin t,
\]
and the signed area of $\ell$ is zero.
The plane curve $\ell$ and the minimal cylinder given by our second choice of $h$ as stated in \eqref{eq:choiceh0and1}
are shown in Figure \ref{fig:1}. \end{Example}
\begin{figure}
\caption{A closed plane curve $\ell$ (left) and the corresponding
non-vertical minimal cylinder in ${\rm Nil}_3$ (right) from
the $\mathfrak{su}_{1, 1}$-potential $\zeta$ in \eqref{eq:zetapotential} with
the second choice of $h$ in \eqref{eq:choiceh0and1} and identity initial condition. The signed
area enclosed by $\ell$ is zero in this case.
The figure is constructed by \cite{Br:Matlab}.}
\label{fig:1}
\end{figure}
\begin{Example}[The case $c = 1$] \label{ex:cyl2}
We construct a on-vertical minimal cylinder in ${\rm Nil}_3$ from the simple periodic frame $C_0(z)$ as stated in (\ref{eq:diagC0}) for $c = 1$.
We choose
the periodic holomorphic function $h$ \begin{equation}\label{eq:choiceh2}
h(z) = \exp(-i \pi/4)+\sqrt{6} \cos (4z). \end{equation}
Note that the minimal period is $\frac{\pi}{2}$, but
we choose the period $p = \pi$,
and the plane curve $\ell$ is given by \[ \ell(t) = \frac16 e^{-2 i t}\left(3 \sqrt 6 i - \sqrt 6 i e^{8 i t}
- 6 \sqrt i e^{4 i t} \right). \]
It is easy to see that $\ell$ is closed curve whose signed area is
zero for the period $p$.
Therefore the resulting minimal surface in ${\rm Nil}_3$ is a cylinder.
Moreover, the resulting non-vertical
minimal cylinder does not have any branch point on $\mathbb R$, since $h^2(z)+$\textquote{positive function} is the conformal factor of the metric
and $h$ in \eqref{eq:choiceh2} does vanish on $\mathbb R$.
The corresponding curve $\ell$ and the corresponding surface are shown in Figure \ref{fig:2}
\begin{figure}
\caption{The closed plane curve $\ell$ (left) and the corresponding
non-vertical minimal cylinder in ${\rm Nil}_3$ (right) constructed from
the $\mathfrak{su}_{1, 1}$-potential $\zeta$ stated in \eqref{eq:zetapotential} with
$h$ as defined in \eqref{eq:choiceh2} and
$C_0(z)$ defined as in (\ref{eq:diagC0}) for $c = 1$. The signed
area enclosed by $\ell$ is zero in this case.
The right figure is constructed by \cite{Br:Matlab}.}
\label{fig:2}
\end{figure} \end{Example}
\subsection{An example of non-vertical minimal cylinders in ${\rm Nil}_3$ constructed from matrices $C_0$ with full entries} \begin{Example}\label{ex:cyl3}
We consider a periodic matrix
$C_0$ with full entries$:$ \begin{equation}\label{eq:diagC03} C_0(z) =
\begin{pmatrix}
\cosh (\sin z) & \sinh (\sin z) \\
\sinh (\sin z) & \cosh (\sin z)
\end{pmatrix}. \end{equation}
The minimum period $p$ of $C_0(z)$ is obviously $p = 2 \pi$.
Then we compute \begin{align}
\zeta_0 (z)& = C_0 (z)^{-1} \partial_z C_0(z)dz =
\begin{pmatrix}
0 & \cos z \\
\cos z & 0
\end{pmatrix} dz, \end{align}
thus $\kappa (z) = \cos z$. It is also easy to see that the
signed are of $m$ in \eqref{eq:ell1}
vanishes since $a_0$ and $b_0$ are real on $\mathbb R$.
Let us choose the complex function $h$ defined by \begin{equation}\label{eq:choiceh3}
h(z) = \frac12 \cos z + (\cos z) \operatorname{sech} (2 \sin z) - i \sin (3 z), \end{equation}
and define an $\mathfrak{su}_{1, 1}$-potential $\zeta$ as
\begin{equation}\label{eq:zetapotential3}
\zeta (z, \lambda) =
\begin{pmatrix}
0 & \lambda^{-1} h(z) + \lambda(\kappa (z)- h(z)) \\
\lambda^{-1}(\overline{\kappa (\bar z)}- \overline{h(\bar z)}) + \lambda \overline{h(\bar z)} & 0
\end{pmatrix} dz.
\end{equation}
Since the first closing condition is always satisfied by our construction we only need to consider the second and the third closing condition.
For this we compute the function $\alpha$ and the signed area of $m$ in \eqref{eq:alpha} and \eqref{eq:m}. It is easy to see that the signed area of $m$ vanishes, since $m$ is real on $\mathbb R$.
On the one hand, the function $\alpha$ in \eqref{eq:alpha} can be computed as \begin{equation}\label{eq:alpha3}
\alpha(t) = 2(\cos t - i \sin 3 t), \end{equation}
which is the same function in Example \ref{ex:cyl1}.
Thus the second and third closing conditions
can be satisfied.
\begin{figure}
\caption{Two views of the same non-vertical minimal cylinders in ${\rm Nil}_3$
by the $\mathfrak{su}_{1, 1}$-potential in \eqref{eq:zetapotential3}
with $h$ in \eqref{eq:choiceh3} and identity initial condition.
The figures are constructed by \cite{Br:Matlab}.}
\label{fig:3}
\end{figure}
\end{Example} Finally we will give an example of non-vertical minimal cylinders in ${\rm Nil}_3$ constructed from the non-diagonal matrix $C_0$. \begin{Example}\label{ex:cyl4}
We consider a periodic matrix
$C_0$ with full entries$:$ \begin{equation}\label{eq:diagC04} C_0(z) =
\begin{pmatrix}
e^{-i z}\cosh (\sin z) & \sinh (\sin z) \\
\sinh (\sin z) & e^{i z}\cosh (\sin z)
\end{pmatrix}. \end{equation}
The minimum period $p$ of $C_0(z)$ is obviously $p = 2 \pi$.
Then we compute \begin{align*}
\zeta_0 (z)& = C_0 (z)^{-1} \partial_z C_0(z)dz \\
&=
\begin{pmatrix}
-i \left\{\cosh (\sin z)\right\}^2 & e^{i z}
\left\{\cos z - \frac{i}2\sinh (2\sin z ) \right\} \\
e^{-i z} \left\{\cos z + \frac{i}2\sinh (2\sin z) \right\} & i \left\{\cosh (\sin z)\right\}^2
\end{pmatrix} dz, \end{align*}
thus
\[
\nu(z) = -i \left\{\cosh (\sin z)\right\}^2, \quad \kappa (z) = e^{i z}
\left\{\cos z - \frac{i}2\sinh (2\sin z ) \right\}.
\]
It is also easy to see that the signed area $T$ of $m(t)$
in \eqref{eq:m} can be computed as
\[
T = -\frac18\pi(I_{0}(4)-1) \fallingdotseq -4.04556,
\]
where $I_{\nu}(x)$ is the modified Bessel function of the first kind,
see Chapter III \cite{Bowman} for definition. Let us choose the complex function $h$ defined by \begin{equation}\label{eq:choiceh4}
h(z)=-\frac{i}4 \left\{2 c_1 + 2 i \cos z+ \sinh(2 \sin z)\right\},
\quad c_1 = \sqrt{|T|/\pi} \end{equation}
and define an $\mathfrak{su}_{1, 1}$-potential $\zeta$ as
\begin{equation}\label{eq:zetapotential4}
\zeta (z, \lambda) =
\begin{pmatrix}
\nu(z) & \lambda^{-1} h(z) + \lambda(\kappa (z)- h(z)) \\
\lambda^{-1}(\overline{\kappa (\bar z)}- \overline{h(\bar z)}) + \lambda \overline{h(\bar z)} & - \nu (z)
\end{pmatrix} dz.
\end{equation}
Since the first closing condition is always satisfied by our construction we only need to consider the second and the third closing condition.
For this we compute the function $\alpha$ \eqref{eq:alpha}.
As we have shown that the signed are of $m$ is zero. On the one hand,
the function $\alpha$ in \eqref{eq:alpha} can be computed as \begin{equation}\label{eq:alpha4}
\alpha(t) = -i c_1 e^{-i t}, \end{equation}
which is the derivative of the round circle about origin
$\ell(t) = c_1 e^{-it}$ with radius $c_1$.
Then the signed area of $\ell$ and $m$ are the same,
and thus the second and third conditions can be satisfied. \begin{figure}
\caption{
The plane curve $\ell$ (left, the round circle) and
the corresponding non-vertical minimal cylinder in ${\rm Nil}_3$ (right)
by the $\mathfrak{su}_{1, 1}$-potential in \eqref{eq:zetapotential4}
with $h$ in \eqref{eq:choiceh4} and identity initial condition.
The bottom figure shows a close-up view of the right one.
The figures are constructed by \cite{Br:Matlab}.
}
\label{fig:4}
\end{figure} \end{Example}
\begin{Remark}
The Abresch-Rosenberg differential of these examples are
$B(z) = h(z) \overline{h(\bar z)} dz^2$, and $h (z)$ is given in \eqref{eq:choiceh0and1}, \eqref{eq:choiceh2} or \eqref{eq:choiceh3}. \end{Remark}
\section{Spacelike CMC cylinders in $\mathbb L_3$} The loop group technique used in \cite{DIKAsian}, \cite{DIK;Nil3-sym} and in this paper for the construction of minimal surfaces in ${\rm Nil}_3$ uses the close relationship with spacelike CMC surfaces in Minkowski three-space $\mathbb L_3$ as manifestly expressed by the closely related, but different, Sym formulas.
Spacelike CMC surfaces in $\mathbb L_3$ have been investigated in some detail in \cite{BRS:Min}. We trust that the present reader will be able to translate without problems notation and results from there into our setting. We therefore do not include any special notation nor any special results about this surface class into this note.
In Theorem \ref{thm:framePeriodic}, we have shown that
the $\mathfrak{su}_{1, 1}$-potentials considered there produce frame periodic minimal immersions in ${\rm Nil}_3$ and vice versa.
It is obvious that the periodic matrix $C_0$
stated in \eqref{eq:frameperiodic1}
also produces
a spacelike CMC-immersion in $\mathbb L_3$ by the formula defining $f_{\mathbb L_3}$ in \eqref{eq:SymMin} inserting the
extended frame $F$,
and thus we have the following.
\begin{Theorem}\label{thm:framePeriodicMin}
Let $C_0$ be given by \eqref{eq:frameperiodic1}.
Set $\zeta_0 = C_0^{-1} d C_0$ and define
a $\mathfrak{su}_{1, 1}$-potential $\zeta$ by
\eqref{eq:frameperiodicPot}, where $h$ is an arbitrary holomorphic function along $\mathbb R \cap \mathbb D$.
Then $\zeta$ defines a frame
periodic spacelike CMC-immersion into $\mathbb L_3$.
Moreover, every frame periodic CMC-immersion in $\mathbb L_3$
can be obtained this way.
\end{Theorem}
It is known (see e.g. \cite{BRS:Min}) that the closing conditions for $f_{\mathbb L_3}$ at $\lambda =1$
can be phrased as follows: \begin{equation}\label{eq:Minclosings}
M(\lambda=1) = \pm \operatorname{id} \quad\mbox{and}\quad
(\partial_{\lambda}M)(\lambda=1) = 0. \end{equation}
Since $X(\lambda)$ is defined by $- i \lambda (\partial_{\lambda} M) M^{-1}$, the above
closing conditions can be rephrased as \begin{equation}\label{eq:Minclosings2} M(\lambda=1) = \pm \operatorname{id} \quad\mbox{and}\quad
X(\lambda=1)=0. \end{equation}
Since for the frame periodic spacelike CMC-immersions
we already have $M(\lambda) \in \Lambda {\rm SU}_{1, 1 \sigma}$ and
$M(\lambda=1)=\pm \operatorname{id}$, and
thus only the second closing condition $X(\lambda=1)=0$ remains.
Therefore we have the following corollary. \begin{Corollary}
Let $\zeta$ be an $\mathfrak{su}_{1, 1}$-potential as stated in \eqref{eq:frameperiodicPot}.
Define functions $\alpha$ and $\beta$ by the equations \begin{align}\label{eq:alphaMin}
\alpha(t) &= a_0(t)^2 (2h(t) -\kappa(t)) + b_0 (t)^2(2\overline{h(t)} -\overline{k(t)}), \\ \label{eq:betaMin} \beta (t) &= \operatorname {Im} \left\{a_0(t)\overline{b_0(t)} (2h(t) -\kappa(t))\right\}, \end{align}
where $a_0$, $b_0$, $h$ and $\kappa$ are the functions occurring in \eqref{eq:frameperiodic1}
and \eqref{eq:kappa}, respectively.
Then $\zeta$ defines a spacelike CMC-cylinder in $\mathbb L_3$ if and only if \begin{equation}\label{eq:closingMin}
\int_0^p\alpha(t) dt = \int_0^p\beta(t) dt =0 \end{equation}
hold. \end{Corollary} \begin{Remark}
Note that the function $\alpha$ stated in \eqref{eq:alphaMin} above is the same function as defined in \ref{eq:alpha}. \end{Remark}
It is easy to see that for the potentials $\zeta$ occurring in
Examples \ref{ex:cyl1} and \ref{ex:cyl2}, the closing conditions stated in
\eqref{eq:closingMin} are satisfied.
Let us rephrase the closing conditions
in \eqref{eq:closingMin} more geometrically.
First let us consider the plane curve $\ell$ in \eqref{eq:ell1} as before:
then $\ell(t) = \int_0^t \alpha(s) ds$ and \[
2 h(z)- \kappa (z) = \frac{1 }{a_0 (z) \overline{a_0(\bar z)} + b_0 (z) \overline{b_0(\bar z)}}\left(
\overline{a_0(\bar z)}^2 \alpha(z) - b_0(z)^2 \overline{\alpha(\bar z)}\right), \]
hold.
We now simplify the condition $\int_0^p \beta (t) dt =0$. After plugging $2 h - \kappa$ into
$\beta$ a straightforward computation using
the relation $a_0 (z) \overline{a_0(\bar z)} - b_0 (z) \overline{b_0(\bar z)}=1$,
yields \[ \beta (z) = \operatorname {Im} \left(\overline{m(\bar z)} \alpha(z) \right), \]
where $m(z) = a_0(z) b_0(z)$.
Therefore $\int_0^p \beta (t) dt =0$ is equivalent to that
\[
\operatorname {Im} \langle {\ell}^\prime, m \rangle = 0,
\]
i.e., the imaginary part of the $L^2$-inner product of $m$ with $\alpha$ vanishes. Similar to Theorem \ref{thm:main}, we have the following. \begin{Theorem}\label{thm:mainspacelike}
Let $\zeta$ be an $\mathfrak{su}_{1, 1}$-potential as stated in \eqref{eq:frameperiodicPot},
and let $\ell$ and $m$ be the curves in $\mathbb C$ as defined by \eqref{eq:ell1}
and $m=a_0 b_0$ with the entries of $C_0$ in \eqref{eq:frameperiodic1}, respectively.
Then $\zeta$, via the generalized Weierstrass representation,
gives a spacelike CMC-cylinder in $\mathbb L_3$
if $\ell$ is closed and
the imaginary part of the $L^2$-inner product of $\ell^{\prime}$ and
$m$ vanishes.
Conversely for closed analytic plane curves $\tilde \ell$ and $\tilde m$
such that the imaginary part of the $L^2$-inner product of $\tilde \ell^{\prime}$ and $\tilde m$ vanishes, there exist
a periodic matrix $C_0$ and an $\mathfrak{su}_{1, 1}$-potential $\zeta$
such that the curves $\ell$ in
\eqref{eq:ell1} and $m=a_0 b_0$ are $\tilde \ell$ and $\tilde m$,
respectively and the generalized Weierstrass
representation yields a spacelike CMC-cylinder in $\mathbb L_3.$
Moreover, all spacelike CMC-cylinders in $\mathbb L_3$ can be constructed in this way. \end{Theorem}
\begin{Example}[Two spacelike CMC cylinders -Two minimal cylinders in ${\rm Nil}_3$ revisited] We will illustrate our results for spacelike CMC surfaces by reusing the data we had chosen in the ${\rm Nil}_3$ case, namely two potentials for a diagonal periodic matrix $C_0$, with diagonal entries $e^{icz}$ and $e^{-icz}$, and discuss the cases $c=1$ and $c = 0$.
Let $\zeta$ be an $\mathfrak{su}_{1, 1}$-potential as stated in \eqref{eq:zetapotential} with $C_0$ as just above.
As we have already seen that in Section
\ref{subsc:diagonal}, from Corollary \ref{closingcondition},
$X(\lambda=1)$ is off-diagonal, and thus
the second condition
$\int_0^p \beta(t) dt = 0$ in \eqref{eq:closingMin} holds anyway.
As before we choose a periodic holomorphic function $h$ as \begin{align} \label{eq:cmch1} h(z) &= \cos(z)-i \sin (3 z)\quad \mbox{for} \quad c=0, \\ \label{eq:cmch2} h(z) & = \exp(-i \pi/4)+\sqrt{6} \cos (4z) \quad \mbox{for} \quad c=1.
\end{align}
Then the first condition $\int_0^p \alpha(t) dt = 0$
in \eqref{eq:closingMin} also holds since we have
shown them in Examples \ref{ex:cyl1} and \ref{ex:cyl2}. \begin{figure}
\caption{A spacelike CMC-cylinders in $\mathbb L_3$ from
the $\mathfrak{su}_{1, 1}$-potentials $\zeta$ in \eqref{eq:zetapotential}
with $h$ in \eqref{eq:cmch1} (the left figure) or
\eqref{eq:cmch1} (the right figure) with identity initial condition.
The figure is constructed by \cite{Br:Matlab}. }
\label{fig:sCMC}
\end{figure} \end{Example}
\appendix \section{Basic results} In this appendix we will collect first basic definitions and will then present some results enabling us to use holomorphic potentials in place of meromorphic potentials.
\subsection{Notation and definitions}
We first define the twisted ${\rm SL}_2 \mathbb C$ loop group as a space of continuous maps from $\mathbb{S}^1$ to the Lie group ${\rm SL}_2 \mathbb C$, that is,
$\Lambda {\rm SL}_2 \mathbb C_{\sigma} =\{g : \mathbb{S}^1 \to {\rm SL}_2 \mathbb C \;|\; g(-\lambda) = \sigma g(\lambda) \}$, where $\sigma =\operatorname{Ad} (\sigma_3)$. We restrict
our attention to loops in $\Lambda {\rm SL}_2 \mathbb C_{\sigma}$ such that
the associate Fourier series of the loops are absolutely convergent.
Such loops determine a Banach algebra, the so-called
\textit{Wiener algebra}, and
it induces a topology on $\Lambda {\rm SL}_2 \mathbb C_{\sigma}$,
the so-called \textit{Wiener topology}.
From now on, we consider only $\Lambda {\rm SL}_2 \mathbb C_{\sigma}$ equipped with the Wiener topology.
Let $D^{\pm}$ denote respective the inside of unit disk and
the union of outside of the unit disk and infinity.
We define \textit{plus} and \textit{minus} loop subgroups of $\Lambda {\rm SL}_2 \mathbb C_{\sigma}$;
$\Lambda^{\pm} {\rm SL}_{2} \mathbb C_{\sigma}=\{ g \in \Lambda {\rm SL}_2 \mathbb C_{\sigma} \;|\; \mbox{$g$ can be extended holomorphically to $D^{\pm}$} \}$.
By $\Lambda_*^+ {\rm SL}_{2} \mathbb C_{\sigma}$ we denote the subgroup of
elements of $\Lambda^+ {\rm SL}_{2} \mathbb C_{\sigma}$ which take the value identity at zero.
Similarly, by $\Lambda_*^{-} {\rm SL}_{2} \mathbb C_{\sigma}$ we denote the subgroup of
elements of $\Lambda^{-} {\rm SL}_{2} \mathbb C_{\sigma}$ which take the value identity at infinity.
We also define the ${\rm SU}_{1, 1}$-loop group as follows:
\[\Lambda {\rm SU}_{1, 1 \sigma} =\left\{ g \in \Lambda {\rm SL}_2 \mathbb C_{\sigma} \;|\; \sigma_3 \overline{g(1/\bar \lambda)}^{t-1} \sigma_3
= g(\lambda)\right\}.\]
It will be convenient to use $\tau (g)(\lambda) = \sigma_3 \overline{g(1/\bar \lambda)}^{t-1} \sigma_3 $
for $\lambda \in S^1$.
For all geometric quantities, we can assume $\lambda \in \mathbb C^*$.
For potentials $\zeta = \hat \zeta dz$ with Lie algebra elements $\hat \zeta \in \Lambda \mathfrak{sl}_2 \mathbb C_{\sigma} $ we say that $\zeta$ is \textit{an $\mathfrak{su}_{1, 1}$-potential} if $\tau(\hat \zeta)(\lambda) = - \hat \zeta(\lambda)$ holds.
\subsection{Holomorphic $C(z,\lambda)$} We should probably be a bit more careful about where from we choose our variables: at one hand it is convenient to choose $\mathbb D$ as complex plane or unit disk, since they are invariant under complex conjugation. But since we consider already very early periodic functions with real period, we actually are interested in strips $\mathbb S$. There are three types of strips: the whole complex plane, the upper half-plane and a strip of finite width. In the first and the last case we can assume w.l.g. that the strip contains the real line \textquote{in the middle}. Meaning the real line for the case of $\mathbb S = \mathbb C$, and the real line for a strip of type $-c_0 < \operatorname {Im} (z) < c_0$. For the third case, the upper half-plane $\mathbb{H} =\{ z \mid \operatorname {Im}(z) > 0\}$, it is perhaps best to let $\mathbb{Z}$ translate parallel to the real axis and to choose the base point to be $i$, instead of moving the upper-half plane down so that the new domain covers the actual real line.
\begin{Proposition}
Let $f : \mathcal{C} \rightarrow {\rm Nil}_3$ be a minimal (immersed) cylinder in ${\rm Nil}_3$
with universal cover $\mathbb S$.
Then there exists a maximal minimal (immersed) cylinder in ${\rm Nil}_3$ prolonging $\mathbb S$. \end{Proposition} For the notion of \textquote{prolongable} see \cite[p.207]{Kra}.
\begin{proof} Clearly, if $\mathbb S = \mathbb C$, no extension is possible and the given cylinder already is maximal. If $\mathbb S$ is a finite strip with $\mathbb R$ in the middle, we consider all extensions, realized by finite strips with $\mathbb R$ in the middle. Then by Zorn's Lemma there exists a maximal object. If $\mathbb S$ is the upper half plane, then any extension has a universal cover which can be realized as a \textquote{shifted down} upper half plane. Now a procedure as in the last case yields a maximal element. \end{proof}
\subsection{Holomorphic $\mathfrak{su}_{1, 1}$-potentials}
For this we recall that the extended coordinate frame
has a meromorphic extension.
Let's have a closer look:
The right upper corner of the Maurer-Cartan form of the coordinate
frame $F$ is of the form
$-\lambda^{-1}e^{u/2} -\lambda \bar B e^{-u/2}$,
where $u$ is the metric exponent and $B$ the Hopf differential
(see \cite{DIKAsian}).
Since $F$ has a meromorphic extension to $\mathbb D \times \mathbb D$, also the expression above has such an extension. Since $F$ is real-analytic, also $e^{u/2}$ is real analytic. Therefore there exists an open subset ${\mathfrak D}_0$ of $\mathbb D \times \mathbb D$ containing $\mathbb D^\sharp = \{(z, \bar z); z \in \mathbb D \}$ to which $F$ extends holomorphically. We thus obtain:
\begin{Proposition}
There exists an open subset $\mathfrak D_0$ of $\mathbb D \times \mathbb D$ containing $\mathbb D^\sharp = \{(z, \bar z); z \in \mathbb D \}$ to which $F$ extends holomorphically in $(z,w)$.
In particular, also $\zeta$ extends holomorphically in $(z,w)$ to $\mathfrak D_0$. \end{Proposition} \begin{Remark} \mbox{} \begin{enumerate} \item If one starts from some potential like $\zeta$, then the maximal minimally immersed cylinder in ${\rm Nil}_3$ is defined on the largest strip on which the ${\rm SU}_{1, 1}$-Iwasawa decomposition is real analytic.
\item Singularities along the boundary of such a maximal strip indicate several different geometric features of the minimal cylinder. \end{enumerate} \end{Remark}
\def$'${$'$}
\end{document} |
\begin{document}
\title{ Generation of random chordal graphs using subtrees of a tree\thanks{A preliminary version of this paper has appeared in the Proceedings of the 10th International Conference on Algorithms and Complexity, CIAC 2017 \cite{CIAC2017}.} \thanks{This work is supported by the Research Council of Norway and Bo\u{g}azi\c{c}i University Research Fund (grant 11765); and T. Ekim is supported by Turkish Academy of Sciences GEBIP award.}}
\author{
Oylum \c{S}eker\inst{1}
\and
Pinar Heggernes\inst{2}
\and
T{\i}naz Ekim\inst{1}
\and Z.~Caner Ta\c{s}k{\i}n\inst{1} }
\institute{
Department of Industrial Engineering, Bo\u{g}azi\c{c}i University, Istanbul, Turkey.
\texttt{\{oylum.seker,tinaz.ekim,caner.taskin\}@boun.edu.tr}
\and
Department of Informatics, University of Bergen, Norway.
\texttt{pinar.heggernes@uib.no} }
\maketitle
\begin{abstract} Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the ``intersection of subtrees of a tree'' characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in a ``intersection of subtrees of a tree'' representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.
\end{abstract}
\textit{Keywords}: Random chordal graph generation; Intersection of subtrees of a tree
\section{Introduction}\label{chapter:intro}
Algorithms particularly tailored to exploit properties of various graph classes have formed an increasingly important area of graph algorithms during the last five decades. With the introduction of relatively new theories for coping with NP-hard problems, like parameterized algorithms, algorithmic research on graph classes has become even more popular recently, and the number of results in this area appearing at international conferences and journals is now higher than ever. One of the most studied graph classes in this context is the class of chordal graphs, i.e., graphs that contain no induced cycle of length 4 or more. Chordal graphs arise in practical applications from a wide variety of unrelated fields, like sparse matrix computations, database management, perfect phylogeny, VLSI, computer vision, knowledge based systems, and Bayesian networks \cite{BLS99,Gol04,pearl2014probabilistic,rose1972graph,Spinrad03}. This graph class that first appeared in the literature as early as 1958 \cite{Hajnal}, has steadily increased its popularity, and there are now more than 20 thousand references on chordal graphs according to Google Scholar.
With a large number of existing algorithms specially tailored for chordal graphs, it is interesting to note that not much has been done to test these algorithms in practice. Very few such tests are available as published articles \cite{andreou2005generating,markenzon2008two,pemmaraju2005approximating}. In particular, there seems to be no efficient chordal graph generator available that is capable of producing every chordal graph. Most of the work in this direction involves generating chordal graphs tailored to test a particular algorithm or result \cite{andreou2005generating,pemmaraju2005approximating}. This is a clear shortcoming for the field, and it was even mentioned as an important open task at a Dagstuhl Seminar \cite{Dagstuhl}. Until some years ago, most of the algorithms tailored for chordal graphs had polynomial running time, and testing was perhaps not crucial. Now, however, many parameterized and exponential-time algorithms exist for chordal graphs, for problems that remain hard on this graph class, see e.g., \cite{Sofsem2014,Pinar,DMarx,Neel}. The proven running times of such algorithms might often be too high compared to the practical running time. Just to give some examples from the field of enumeration, there are now several algorithms and upper bounds on the maximum number of various objects in chordal graphs \cite{Pinar1,Pinar2,Pinar}. However, the lower bound examples at hand usually do not match these upper bounds. Tests on random chordal graphs is a good way of getting better insight about whether the known upper bounds are too high or tight.
In this paper, we present an algorithm for generating random chordal graphs. Our algorithm is based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Surprisingly, this characterization does not seem to have been directly used for random chordal graph generation earlier. Starting from a random host tree, we propose three different methods for generating subtrees of the host tree to give different neighborhood and density properties. Our algorithm, with two of these methods, can be implemented in such a way that the overall running time is best possible for an algorithm producing the entire node set of subtrees in a ``intersection of subtrees '' representation of a chordal graph. One of these fast subtree generation methods, which we call GrowingSubtree, is also the method that turns out to generate the largest variety of chordal graphs. We measure the variety using the characteristics of the maximal cliques of the generated graph, as it has been done in previous work \cite{pemmaraju2005approximating}. After proving the correctness, we give extensive computational tests to demonstrate the kind of chordal graphs that our algorithm generates using each of the different subtree generation methods. We compare our methods with one another and with existing test results; we also implement one of the earlier proposed methods and include this in our tests. Note that {\sc Graph Isomorphism} is as hard on chordal graphs as on general graphs \cite{lueker1979linear}, which adds to the difficulty of producing chordal graphs uniformly at random. Our algorithm is able to generate every chordal graph with positive probability.
\section{Background, terminology and existing algorithms} \label{chapter:literature} In this section we give the necessary background on chordal graphs, as well as a short review of the existing algorithms for chordal graph generation. We work with simple and undirected graphs, and we use standard graph terminology.
In particular, for a given graph $G$, we denote its vertex set by $V(G)$, and edge set by $E(G)$. We let $n=|V(G)|$ and $m=|E(G)|$. The set of {\it neighbors}, or the {\it neighborhood}, of a vertex is the set of vertices adjacent to it. The size of the neighborhood of a vertex $x$ is the {\it degree} of $x$, denoted by $d(x)$. The neighborhood of a set $X$ of vertices is the union of the neighborhoods of the vertices in $X$, excluding $X$ itself.
Let $F=\{S_1, S_2, \ldots, S_n\}$ be a family of sets from the same universe. A graph $G$ is called an {\it intersection graph of} $F$ if there is a bijection between the set of vertices $\{v_1, v_2, \ldots, v_n\}$ of $G$ and the sets in $F$ such that $v_i$ and $v_j$ are adjacent if and only if $S_i \cap S_j \neq \emptyset$, for $1 \leq i,j \leq n$. In the special case where there is a tree $T$ such that each set in $F$ corresponds to the vertex set of a subtree of $T$, then $G$ is called the {\it intersection graph of subtrees of a tree}.
A {\it clique} is a set of vertices that are pairwise adjacent. An ordering $(v_1, v_2, \ldots, v_n)$ of the vertices of a graph is a {\it perfect elimination order (peo)} if the set of higher numbered neighbors of each vertex forms a clique. A {\it maximal clique} is a clique $C$ such that no set of vertices properly containing $C$ is a clique. Let $K$ be the set of maximal cliques of a graph $G$. A tree $T$ with a bijection between its vertex set and the cliques in $K$ is called a {\it clique tree} of $G$ if for every vertex $v$ of $G$, the set of vertices of $T$ that correspond to the cliques containing $v$ induce a connected subtree of $T$.
A graph is {\it chordal} if it contains no induced cycle of length 4 or more. A chordal graph on $n$ vertices has at most $n$ maximal cliques \cite{Dirac}. Chordal graphs have many different characterizations. For our purposes, the following will be sufficient.
\begin{theorem} [\cite{buneman1974characterisation,fulkerson1965incidence,gavril1972algorithms,gavril1974intersection}] \label{chordal-big} Let $G$ be a graph. The following are equivalent. \begin{itemize}
\item $G$ is chordal.
\item $G$ has a perfect elimination order.
\item $G$ is the intersection graph of subtrees of a tree.
\item $G$ has a clique tree. \end{itemize} \end{theorem}
Especially the last two points of Theorem \ref{chordal-big} are crucial for our algorithm and its implementation. To make sure that there is no confusion between the vertices of $G$ and the vertices of a tree or a clique tree, we will from now on refer to vertices of a tree as {\it nodes}.
Rose, Tarjan, and Lueker \cite{rose1976algorithmic} gave an algorithm called Maximal Cardinality Search (MSC) that creates a perfect elimination order of a chordal graph in time $O(n+m)$. Blair and Peyton \cite{BP} gave a modification of MCS to list all the maximal cliques of a chordal graph in time $O(n+m)$. Implicit in their proofs is the following well-known fact, which can be characterized as folklore.
\begin{lemma}[\cite{BP,rose1976algorithmic}] \label{chordal-cliquesum} The sum of the sizes of the maximal cliques of a chordal graph is $O(n+m)$. \end{lemma}
Next, we briefly mention the algorithms for generating chordal graphs from the works of Andreou, Papadopoulou, Spirakis, Theodorides, and Xeros \cite{andreou2005generating}; Pemmaraju, Penumatcha, and Raman \cite{pemmaraju2005approximating}; and Markenzon, Vernet, and Araujo \cite{markenzon2008two}. Some of these algorithms create very limited chordal graphs, which is either mentioned by the authors or clear from the algorithm. Thus, in the following we only mention the algorithms that are general enough to be interesting in our context.
It should also be noted that the purpose of Andreou et al.~\cite{andreou2005generating} is not to obtain general chordal graphs, but rather chordal graphs with a known bound on some parameter. One of the algorithms that they propose starts from an arbitrary graph and adds edges to obtain a chordal graph. How the edges are added is not given in detail, however it should be noted that there are many algorithms for generating a chordal graph from a given graph by adding a minimal set of edges and their running time is usually $O(nm)$ \cite{heggernes2006minimal}. Andreou et al.~\cite{andreou2005generating} do not report on the quality of chordal graphs obtained by this method.
We highlight below the algorithms that are the most promising with respect to generating random chordal graphs. In addition to these, there is an $O(n^2)$-time algorithm by Markenzon et al.~\cite{markenzon2008two} that generates a random tree and adds edges to this tree until a chordal graph with desired edge density is obtained. However, no test results about the quality of the generated graphs is given.
\noindent {\bf Alg 1 \cite{andreou2005generating}.} The algorithm constructs a chordal graph by using a peo. At every iteration, a new vertex is added and made adjacent to a random selection of already existing vertices. Then necessary edges are added to turn the neighborhood of the new vertex into a clique such that a given maximum degree is not exceeded. No test results are given in the paper about the quality of the chordal graphs this algorithm produces. As we found the algorithm interesting, we have implemented it, and we compare the resulting graphs to those generated by our algorithm in Section \ref{chapter:experiments}.
\noindent {\bf Alg 2 \cite{markenzon2008two,pemmaraju2005approximating}.} The algorithm starts from a single vertex. At each subsequent step, a clique $C$ in the existing graph is chosen at random, and a new vertex is added adjacent to exactly the vertices of $C$. The inverse of the order in which the vertices are added is a peo of the final graph. It is observed by the authors of both papers that this procedure results in chordal graphs with approximately $2n$ edges experimentally. They propose the following changes:
{\bf Alg 2a \cite{markenzon2008two}} modifies the above generated graph by randomly choosing maximal cliques that are adjacent according to the clique tree and merging these until desired edge density is obtained. Some test results about graphs generated by Alg 2a are provided in \cite{markenzon2008two}. Although these tests are not as comprehensive as the ones we give on our algorithms in Section \ref{chapter:experiments}, we compare our results to those of \cite{markenzon2008two} as best we can. The running time of Alg 2a is $O(m+n\alpha(2n,n))$.
{\bf Alg 2b \cite{pemmaraju2005approximating}} is a modification of Alg 2 in a different way: instead of randomly choosing a clique, a maximum clique is chosen and a random subset of it is made adjacent to the new vertex. Although test results for Alg 2b are provided in \cite{pemmaraju2005approximating}, the authors acknowledge that the produced graphs are still very particular with very few large maximal cliques and many very small maximal cliques. For this reason, we do not include Alg 2b in our comparisons.
\section{Generating chordal graphs using subtrees of a tree}\label{chapter:algo}
We find it surprising that the intersection graph of subtrees of a tree characterization of chordal graphs has not been used directly for their generation. Of course, since all characterizations of a chordal graph are equivalent, even the existing algorithms mentioned above could be interpreted as based on any of these characterization. Especially the algorithms based on clique trees can easily be translated to generate subtrees of a tree. However, none of these algorithms generate random subtrees of a randomly generated tree to produce the resulting chordal graph. One reason could be that this does not give a direct way to decide the number of edges in the generated graph. We will see that edge density can be regulated by adjusting the sizes of the generated subtrees. Let us first observe the following.
\begin{lemma} \label{treetree} For every chordal graph $G$ on $n$ vertices, there is a tree $T$ on $n$ nodes, and a set of $n$ subtrees of $T$, such that $G$ is the intersection graph of these subtrees. \end{lemma}
\begin{proof}
Let $G$ be a chordal graph on $n$ vertices and let $T'$ be a clique tree of $G$. Let us call the vertices of $G$: $v_1, v_2, \ldots, v_n$. Define subtree $T'_i$ to be the subtree of $T'$ that corresponds to the nodes (maximal cliques) that contain vertex $v_i$, for $1 \le i \le n$. By the definition of a clique tree, $T'$ has at most $n$ nodes and each $T'_i$ is a connected subgraph of $T'$. If $T'$ has fewer than $n$ nodes, we can add new nodes adjacent to arbitrary nodes of $T'$ until we get a new tree $T$ with $n$ nodes. The subtrees stay the same. As two vertices are adjacent in $G$ if and only if they appear together in a clique, $G$ is the intersection graph of subtrees $T'_1, \ldots, T'_n$ of $T$. \qed \end{proof}
Based on Lemma \ref{treetree}, we are ready to present our main algorithm for generating chordal graphs on $n$ vertices:
{\bf Algorithm ChordalGen}
{\bf Input:} $n$ and one or two other numbers to guide the number of resulting edges.
{\bf Output:} A chordal graph $G$ on $n$ vertices and $m$ edges.
1. Generate a tree $T$ on $n$ nodes uniformly at random.
2. Generate $n$ non-empty subtrees of $T: \{T_1, \ldots, T_n\}$.
3. Output as $G$ the intersection graph of $\{V(T_1), \ldots, V(T_n)\}$.
By Theorem \ref{chordal-big}, the graph generated by Algorithm ChordalGen is chordal. By Lemma \ref{treetree}, our algorithm can create any chordal graph. Later in this section we will describe three different methods for generating the $n$ subtrees in Step 2. Each method will take one or two parameters to guide the average size of the generated subtrees, with the purpose of controlling the resulting number of edges in $G$. Our algorithm is flexible in the sense that additional ways to generate the subtrees can be suggested and tested later.
We need to evaluate the order of $\sum_{i=1 }^{n} |V(T_i)|$ to analyze the running time of Algorithm ChordalGen. Let us explain at this point that in the preliminary version of this paper \cite{CIAC2017}, $\sum_{i=1 }^{n} |V(T_i)|$ was mistakenly claimed to be linear in the size of the generated chordal graph, that is $O(n+m)$. However, it turns out that $\sum_{i=1 }^{n} |V(T_i)|$ is $\Omega(mn^{1/4})$ as shown in what follows.
\begin{lemma}\label{lem:key}
Let $T$ be a tree on $n$ nodes, and let $T_1,\ldots ,T_n$ be $n$ subtrees of $T$ whose intersection graph is a chordal graph on $n$ vertices and $m$ edges. Then $\sum_{i=1 }^{n} |V(T_i)|$is $\Omega(mn^{1/4})$. \end{lemma}
\begin{proof} Let $G$ be a chordal graph that is the intersection graph of $\{V(T_1),\ldots ,V(T_n)\}$, such that every subtree $T_i$ is associated with a vertex $v_i$ of $G$. Choose an integer $p\geq 2$. Let $n = p^4, n' = p^2 + 1, s = p^2 - p$, and $s_{n'} = p^3$. Note that $n > n'$. Let $T$ be a path on $n$ nodes $x_1, \ldots x_n$. Let $T_i$ be the subtree of $T$ consisting of the single node $x_i$, for $i= 1, \ldots , n'-1$. Let $T_{n'}$ be the subpath of $T$ consisting of the nodes from $n'$ to $n$. The representation contains $s$ copies of $T_i$ for every $i=1, \ldots ,n'-1$ and also $s_{n'}$ copies of $T_{n'}$. Therefore, the number of subtrees is $$ s (n' - 1) + s_{n'} = (p^2 - p) p^2 + p^3 = p^4 = n.$$ The corresponding graph is the disjoint union of a $K_{s_{n'}}$ and $n' - 1$ copies of $K_s$. The number $m$ of edges of this graph is $$ m = (n' - 1)\frac{s(s - 1)}{2} + \frac{s_{n'}(s_{n'} - 1)}{2} = p^2 \frac{(p^2 - p)(p^2 - p - 1)}{2} + \frac{p^3(p^3 - 1)}{2} = p^6 - p^5.$$ On the other hand, the total size of the subtrees is $$s (n' - 1) + s_{n'}(n - n' + 1) = (p^2 - p)p^2 + p^3(p^4 - p^2) = p^7 - p^5 + p^4 - p^3 \geq p^7 - p^5.$$ Therefore,
$$\frac{\sum_{i=1 }^{n} |V(T_i)|}{m} \geq \frac{p^7 - p^5}{p^6 - p^5} = p + 1 > n^{1/4}.$$ Since the inequality holds for infinitely many values of $p$, we conclude the proof. \qed \end{proof}
We are now ready to state our main result about Algorithm ChordalGen.
\begin{theorem} \label{runningtime}
Algorithm ChordalGen generates a chordal graph with $n$ vertices in time $O(S+\sum_{i=1 }^{n} |V(T_i)|)$, where $O(S)$ is the time required to generate $n$ subtrees of a host tree on $n$ nodes. \end{theorem}
\begin{proof} By Theorem \ref{chordal-big}, graph $G$ generated by Algorithm ChordalGen is chordal. Clearly $G$ has $n$ vertices. Let $m$ be the number of edges of $G$. Let us go through the steps of the algorithm to argue for the running time.
Step 1. For the generation of a random host tree $T$ on $n$ vertices, we use the following method by Rodionov and Choo \cite{Rodionov}, which can easily be implemented in $O(n)$ time: start with a tree $T$ that contains only one node. Then repeat $n-1$ times the following: pick a random vertex $x$ of $T$ and add a new vertex adjacent to it.
Step 2. We generate $n$ subtrees $T_1,\ldots ,T_n$ of $T$ using a subtree generator. According to the premises of the theorem, this adds $O(S)$ to the overall time.
Step 3. The sum of the sizes of the generated subtrees is $\sum_{i=1 }^{n} |V(T_i)|$. Let us now explain how we can obtain in time $O(n+m)$ the chordal graph which is the intersection graph of these subtrees. For every node $x$ of $T$, let us define the following list: $C_x = \{v_j \mid T_j$ contains $x \}$, i.e., vertices of $G$ whose corresponding subtrees contain node $x$ of $T$. Let ${\cal C}= \{C_x \mid x \in V(T)\}$. Observe that every set in $\cal{C}$ is a clique of $G$, and $\cal{C}$ contains all maximal cliques of $G$. However, some of the cliques in $\cal{C}$ may not be maximal. If we blow up every node $x$ of $T$ to the set $C_x$, we get a tree which is almost a clique-tree of $G$; this procedure can clearly be done in $O(n+m)$ time. In the resulting tree, non-maximal cliques simply need to be deleted or merged into the maximal ones. By the methods described by Blair and Peyton \cite{BP} it is possible to turn $T$ into a proper clique tree for $G$ in time $O(n+m)$. Thus, in total $O(\sum_{i=1 }^{n} |V(T_i)|+n+m)$ time we both have a representation of our output graph $G$ and a list of maximal cliques of it. It could, however, be desirable to output an adjacency list representation for $G$. Markenzon et al.~\cite{markenzon2008two}, using the methods of Blair and Peyton \cite{BP}, explain how this can be done in $O(n+m)$ time. By Lemma \ref{lem:key}, the overall running time of this step is $O(\sum_{i=1 }^{n} |V(T_i)|)$. \qed \end{proof}
Now we are ready to give the details of how the subtrees of Step 2 are generated. In the subsections below, we present three methods for generating $n$ subtrees of $T$. Then, in Section \ref{chapter:experiments}, we will test our algorithm with each of these methods and compare the results with each other, as well as with Alg 1 and Alg 2a.
\subsection{Algorithm GrowingSubtree} \label{section:blooming_subtree} This algorithm takes as input a tree $T$ on $n$ nodes, and an integer $k$, and generates $n$ subtrees of $T$ of average size $ \frac{k+1}{2}$. In our test results, we give both $k$ and the resulting number of edges, $m$, to give an indication of how $k$ affects the density of the generated graph.
For each subtree $T_i$, the algorithm picks a size $k_i$ randomly from $[1, k]$. Then a random node of $T$ is chosen as the single node of $T_i$ to start with. In each of the subsequent $k_i-1$ iterations, we pick a random node of $T$ in the neighborhood of $T_i$ and add it to $T_i$.
{\bf Algorithm GrowingSubtree}
{\bf Input:} A tree $T$ on $n$ nodes and a positive integer $k \le n$
{\bf Output:} A set of $n$ non-empty subtrees of $T$ of average size $\frac{k+1}{2}$
{\bf for} $i=1$ {\bf to} $n$ {\bf do}
\quad Select a random node $x$ of $T$ and set $T_i=\{x\}$
\quad Select a random integer $k_i$ between 1 and $k$
\quad {\bf for} $j=1$ {\bf to} $k_i-1$ {\bf do}
\quad \quad Select a random node $y$ of $T_i$ that has neighbors in $T$ outside of $T_i$
\quad \quad Select a random neighbor $z$ of $y$ outside of $T_i$ and add $z$ to $T_i$
Output $\{T_1, T_2, \ldots, T_n\}$ \\
\begin{lemma} \label{growingsubtree-runtime}
The running time of Algorithm GrowingSubtree is $O(\sum_{i=1 }^{n} |V(T_i)|)$. \end{lemma}
\begin{proof} Observe first that each subtree $T_i$ can be represented simply a list of nodes of $T$.
We show that after an initial $O(n)$ preprocessing time, each subtree $T_i$ can be generated in time $O(|V(T_i)|)$. For this, we need to be able to add a new node to $T_i$ in constant time, at each of the $k_i - 1$ steps.
As selecting random elements in constant time is easier when accessing the elements of an array directly by indices, we start with copying the nodes of $T$ into an array $A$ of size $n$, and copying the adjacency list of each node $x$ into an array $A_x$ of size $d(x)$. This can clearly be done in total time $O(n)$ since $T$ is a tree.
In general, selecting an unselected element of a set at random can be done easily in constant time if the set is represented with an array. Let us say we have an array $S$ of $t$ elements. We keep a separation index $s$ that separates the selected elements from the not selected ones. At the beginning $s$ is 1. At each step, we generate a random integer $r$ between $s$ and $t$. $S[r]$ is our randomly selected element. Then we swap the elements $S[s]$ and $S[r]$ and increase $s$ by 1.
We can use this method both for selecting a node $y$ of $T_i$ that still has neighbors outside and for selecting a neighbor $z$ of $y$ that has not yet been selected. For the latter, whenever we select a neighbor $z$ of $y$, we move $z$ to the first part of the array $A_x$ using swap. When the separation index reaches the degree of $y$ then we know that $y$ should not be selected to grow the subtree $T_i$ at later steps. Representing $T_i$ with an array of size $k_i$, we can use the same trick to move $y$ to a part of the array that we will not select from. Also, when $z$ is added, we can check whether it is a leaf in $T$ in constant time, and immediately move it to the irrelevant part of the array for $T_i$ if so, since $z$ can then not be used for growing $T_i$ at later steps. It is sufficient to check that $z$ is a leaf of $T$, because otherwise it must have neighbors outside of $T_i$, since $T$ is a tree and we cannot have cycles. When the generation of $T_i$ is finished, the separation indices of each
of its nodes should be reset before we start generating $T_{i+1}$. The adjacency arrays need not be reorganized, as we will anyway be selecting neighbors at random.
Note that we do not need this trick to select an initial node $x$ of each subtree $T_i$, because we should indeed be able to select the same node several times (and grow another subtree from it perhaps in a different way).
With the described method, each step of Algorithm GrowingSubtree takes $O(1)$ time, in addition to initial $O(n)$ time to copy the information into appropriate arrays. Thus the total running time is $O(\sum_{i=1 }^{n} |V(T_i)|)$. \qed \end{proof}
Lemma \ref{growingsubtree-runtime} together with Theorem \ref{runningtime} gives the following.
\begin{corollary}
Algorithm ChordalGen, using the GrowingSubtree method, runs in time $O(\sum_{i=1 }^{n} |V(T_i)|)$. \end{corollary}
We will see in Section \ref{chapter:experiments} that this method generates chordal graphs with the most even distribution of maximal clique sizes.
\subsection{Algorithm ConnectingNodes} \label{section:thres_prob_vertices}
This algorithm takes as input a tree $T$ on $n$ nodes, and a positive real number $\lambda$. The purpose of the parameter $\lambda$ is to guide the desired number of resulting edges in the graph. To generate each subtree $T_i$, we first select $k_i$ nodes of $T$, where $k_i$ is a random integer generated by making use of $\lambda$. $T_i$ is then generated to be the minimal subtree that contains the selected $k_i$ nodes. This implies that a subtree will most likely have many more nodes than those selected initially, and this must be taken into consideration when choosing $\lambda$. In our test results, we give both $\lambda$ and the resulting number of edges, $m$, to give an indication of how $\lambda$ affects the density of the generated graph.
In the following algorithm, we will make use of the standard Breadth First Search (BFS) algorithm from an arbitrary node $r$ of $T$. We will then treat $T$ as a rooted tree with root $r$, and speak about parent-child relation in the standard way, with respect to root $r$. The BFS level of a vertex is simply the distance of that vertex from $r$.
{\bf Algorithm ConnectingNodes}
{\bf Input:} A tree $T$ on $n$ nodes and a positive real number $\lambda$
{\bf Output:} A set of $n$ subtrees of $T$
Let $r$ be an arbitrary node of $T$
Perform BFS from $r$, and identify the parent $P(x)$ and the BFS level $l(x)$ of each node $x$
$L \gets \emptyset$
{\bf for} $i=1$ {\bf to} $n$ {\bf do}
\quad $T_i \gets \emptyset$
\quad Select a random integer $k_i$ from Poisson distribution with mean $\lambda$
\quad {\bf if} $k_i=0$ {\bf then} $k_i \gets 1$
\quad {\bf else if} $k_i > n$ {\bf then} $k_i \gets n$
\quad Select $k_i$ random nodes from $T$ to form $T_i = \{x_1,\ldots,x_{k_i}\}$ and $L = \{x_1,\ldots,x_{k_i}\}$
\quad $d \gets \max_{x \in T_i} l(x)$
\quad {\bf while} $|L| > 1$
\qquad {\bf for all} $x \in L$ such that $l(x) = d$ {\bf do}
\qquad \quad $T_i \gets T_i \cup P(x) $, $L \gets (L \setminus \{x\}) \cup P(x) $
\qquad $d \gets d-1 $
Output $\{T_1, T_2, \ldots, T_n\}$ \\
For each subtree $T_i$, we first generate a random integer $k_i$ by making use of Poisson distribution with mean $\lambda$. Poisson distribution is a discrete probability distribution widely used to model number of occurrences of an event over a specified domain such as time, space etc. In our case, the domain is the host tree, and an event is selection of a node from the host tree. The parameter of this distribution is $\lambda$ and it is the average rate of event occurrences, which implies that the initial $k_i$ values will tend to increase on the average as $\lambda$ increases. The set of possible values a Poisson random variable can take is nonnegative integers, regardless of the value of $\lambda$. However, the minimum and maximum number of nodes that a subtree of an $n$-node host tree can contain are 1 and $n$ respectively. Therefore, we equate $k_i$ to 1 if it is zero, and to $n$ if it is greater than $n$. In this method, the only reason why we use Poisson distribution is that we were not able to achieve a good precision for edge density by picking a random integer uniformly at random from a given interval. To generate the minimal subtree that contains the $k_i$ selected nodes, we make use of the nodes' parent and level information retrieved during BFS. Our key observation is that the minimal subtree must contain the parents of all selected nodes at some level $d$ if there are other selected nodes at levels less than or equal to $d$. Using this idea, we add a node to $T_i$ only when an edge incident to it has to be in $T_i$ to join a node to the others. We start from the highest level (highest distance from $r$) and proceed by moving toward the root until all nodes in the selection become connected. The set $L$ keeps the unprocessed nodes yet to be connected to form the subtree $T_i$. At each step, we consider the nodes in $L$ that are at the same level, which are going to be joined as we move through the levels. Once the parents of those nodes are identified, we are done with level $d$ and there is no need to reconsider nodes at level $d$ any further. Therefore, we replace those nodes with their parents in $L$, which are to be considered at the next step. Parent nodes are also added to $T_i$ because they lie on the paths that connect the node selection. Afterwards, we move to the next level and apply the same procedure. This process continues until a single node is left in $L$, which simply means that we have a node set that has been linked at a single node already and that $T_i$ includes all nodes of the subtree that minimally connects the randomly selected node set.
\begin{lemma}
\label{connectingnodes-runtime}
The running time of Algorithm ConnectingNodes is $O(\sum_{i=1}^{n} |V(T_i)|)$.
\end{lemma}
\begin{proof}
Rooting $T$ from an arbitrary node, and determining the parent $P(x)$ of each node $x$ in $T$ as well as its level $l(x)$ with respect to the root node, takes $O(n)$ time in total for all nodes using BFS. The set $L$ is represented by an array of lists with length equal to $\max_{x\in T}{l(x)}$. Each index $d$ of $L$ represents a list of unprocessed nodes having $l(x) = d$. The lists in $L$ will be empty initially. If a node $x$ at level $d$ is selected, we add $x$ to the list at index $d$ of $L$ in constant time. We also need to keep an $n$-dimensional boolean array $B$, which will be comprised of zeros at first, in order to check whether a node already exists in $L$. If a node $x$ is chosen, the element at index $x$ of $B$ will be set to one in constant time. Note that $B$ is a different representation of the set of nodes in $L$. Both $L$ and $B$ are initialized only once at the start of the algorithm after performing the BFS. Since the number of levels is at most $n$, initialization of $L$ and $B$ can be done in time $O(n)$. We represent $T_i$ as a list of nodes, which can be initialized in $O(1)$. For each subtree, we generate a random integer $k_i$ by making use of Poisson distribution with mean $\lambda$. Generation of some random integer $x$ from Poisson distribution normally takes $O(x)$ time; starting from zero, the value of $x$ is incremented one by one until the stopping condition is met \cite{knuth1969art}. However, since we need $k_i$, which is the number nodes to be selected, to lie between 1 and $n$, we terminate the process if we reach $n$ before stopping condition is met, and we set $k_i$ to 1 if the process returns a zero value. This way, we only spend $O(k_i)$ time to generate $k_i$.
Then, at each iteration, $k_i$ random nodes are chosen. To do this in time $O(k_i)$, we can copy the nodes of $T$ into an array, which is done at the beginning only once and hence takes $O(n)$ time in total, and keep a separation index $s$ that separates the selected elements from the ones that have not been selected yet, as explained in the proof of Lemma \ref{growingsubtree-runtime}. Adding a chosen node to $T_i$, $L$ and $B$ can be done in $O(1)$. At the end of $n$ such iterations, a total of $\sum_{i =1 }^{n} k_i$ random choices are made and this is clearly less than or equal to $\sum_{i=1}^{n} |V(T_i)|$.
Now, it remains to show that generation of $n$ subtrees can also be done in time $O(\sum_{i=1}^{n} |V(T_i)|)$ once $T_i$, $L$ and $B$ are populated with initial randomly selected nodes. Our aim is to construct the minimal subtree of $T$ connecting all the nodes in $\{x_1,\ldots, x_{k_i}\}$. To this end, at every iteration, we add the parents of all nodes of highest level to subtree $T_i$ and replace these nodes by their parents in $L$. The way we store the nodes in $L$ enables us to access unprocessed nodes at a given level in constant time. However, to be able to start with the highest level in constant time initially, we need to know the highest level of the randomly selected $k_i$ nodes, which can be found in $O(k_i)$. While processing some node $x$ at level $d$, we first investigate whether its parent node $P(x)$ has already been included in $L$ by checking index $P(x)$ of $B$ in constant time. If it is one, it means that the parent node has already been included in $L$ and $T_i$, and we do not do anything. Otherwise, we append $P(x)$ to the list at index $(d-1)$ of $L$ and set the corresponding index of $B$ to one. When done with $x$, we remove it from $L$ in $O(1)$ since $x$ is an element of a list, and set index $x$ of $B$ to zero, which is again $O(1)$. Recall that since $T_i$ is represented as a list, adding an element to $T_i$ can be done in $O(1)$. Thus, we perform constant-time operations for each node under consideration in the inner for loop.
At the beginning of the while loop $L$ has $k_i$ isolated nodes. Whenever two nodes in $L$ have a common parent, the cardinality of $L$ decreases by one at the next step. Noting that $|L|$ indicates the number of currently existing connected components, which are to be attached together to reveal the subtree, the while loop to add new nodes to $T_i$ terminates when $|L|=1$; that is, as soon as the minimal subtree has been found. Now, it is enough to notice that during the generation of each subtree $T_i$ using this method, we only consider and add the nodes of $T$ which are in $T_i$, and iterate only through the levels that are contained in $T_i$. In other words, $|L|$ becomes 1 and the while loop stops after exactly when $|V(T_i)|$ nodes are considered. Because we spend constant time for each of the $|V(T_i)|$ nodes, the overall complexity of the operations within the while loop becomes $O(|V(T_i)|)$ for each subtree $T_i$. In order to obtain $O(|V(T_i)|)$ for the entire loop, we need to ensure that termination condition of the while loop can be checked in constant
time. For this purpose, we keep the number of nodes in $L$ as a variable, incrementing whenever a new node is added and decrementing upon removal of a node, which takes at most $O(|V(T_i)|)$ time. Finally, the arrays $L$ and $B$ should be reset before being passed to the next subtree. We know that $L$ will contain a single node at the end of the while loop, and equivalently a single nonzero element will exist in $B$. When the loop terminates, we will know at which index (level) of $L$ we were finally at. So, we can simply access the final node, set the element in $B$ corresponding to that node to zero, and delete it from $L$, all in constant time. This way, the algorithm from while loop on to the next subtree completes in time $O(|V(T_i)|)$. In total, these operations add to the running time of Algorithm ConnectingNodes a term of $O(\sum_{i=1}^{n} |V(T_i)|)$. \qed
\end{proof}
Lemma \ref{connectingnodes-runtime}, together with Theorem \ref{runningtime}, gives the following:
\begin{corollary}
Algorithm ChordalGen, using the ConnectingNode method, runs in time $O(\sum_{i=1}^{n} |V(T_i)|)$. \end{corollary}
We have thus presented two different methods for generating subtrees of a given tree, both of which result in an algorithm for generating random chordal graphs in time$O(\sum_{i=1}^{n} |V(T_i)|)$. In the next subsection we present yet another subtree generation method, having running time $O(n^2+\sum_{i=1}^{n} |V(T_i)|)$. We include this algorithm for the sake of completeness and better comparison basis in our tests in the next section. This algorithm can for example be used when one is interested in generating chordal graphs with predominantly large maximal cliques as the density grows.
\subsection{Algorithm PrunedTree} \label{section:removal_of_edges}
The input to this algorithm consists of a tree $T$ on $n$ nodes, an edge deletion fraction $f$, which is a rational number between 0 and 1, and a selection barrier $s$, which is a real number between 0 and 1. To generate subtree $T_i$, we randomly select a fraction $f$ of the edges on the tree and remove them. The number of edges to delete, say $l$, is calculated as $\lfloor (n-1)f \rfloor$, which will leave $l+1$ subtrees in total. We then determine the sizes of the $l+1$ subtrees and store the distinct values. We pick a random size $k_i$ from the set of largest $100(1-s) \%$ of distinct values, and randomly choose a subtree with size $k_i$. We repeat this $n$ times to generate all the subtrees. One should note that we could simply select one connected component (subtree) at random without any preferential treatment; however, parameter $s$ makes it easier to increase the density of the chordal graph by favoring larger components as the value of $s$ advises. So, parameter $s$ is an additional means to tune the edge density of the chordal graph; as its value increases, the size of the subtree to be selected tends to increase too. Increasing the edge deletion fraction $f$, however, tends to decrease the average size of subtrees emerging from deletion of edges.
{\bf Algorithm PrunedTree}
{\bf Input:} A tree $T$ on $n$ nodes, edge deletion fraction $f$, and selection barrier $s$
{\bf Output:} A set of $n$ non-empty subtrees of $T$
{\bf for} $i=1$ {\bf to} $n$ {\bf do}
\quad Create a copy $T'$ of $T$
\quad Select randomly $\lfloor (n-1)f \rfloor$ edges of $T'$ and delete them from $T'$
\quad Determine connected components of $T'$ and their sizes
\quad Select randomly a subtree size $k_i$ from the highest $ 100(1-s) \% $ subtree sizes.
\quad Select a random component of size $k_i$ and choose it as $T_i$
Output $\{T_1, T_2, \ldots, T_n\}$ \\
\begin{lemma}
\label{prunedtree-runtime}
The running time of Algorithm PrunedTree is $O(n^2)$. \end{lemma}
\begin{proof}
Creating a copy of $T$, deleting a subset of its edges, and computing the resulting connected components takes $O(n)$ time by standard BFS. Now, we create an array $A$ of size $n$, where each element in $A$ is a linked list. For each connected component of $T'$ of size $t$, we add this component at the end of the list in $A[t]$. Clearly, initializing $A$, and adding all subtrees to appropriate lists takes $O(n)$ time. We also make an additional array $B$ which simply stores the sizes of all subtrees, in sorted order. $B$ can be created in time $O(n)$, using $A$. We use $B$ to find the highest $ 100(1-s) \% $ subtree sizes, by simply using the corresponding last portion of $B$. Random selection of a subtree of size $k_i$ is simply done by picking a subtree from the list $A[k_i]$ in $O(1)$ time. Thus every subtree requires $O(n)$ time to generate.
Repeating this $n$ times, the overall complexity of PrunedTree algorithm amounts to $O(n^2)$.
\qed \end{proof}
We now obtain the following result using Theorem \ref{runningtime} together with Lemma \ref{prunedtree-runtime}.
\begin{corollary}
Algorithm ChordalGen, using the PrunedTree method, runs in time $O(n^2+\sum_{i=1}^{n} |V(T_i)|)$. \end{corollary}
Let us conclude this section with a remark which applies to all of the three subtree generation methods. Algorithm ChordalGen does not guarantee the connectedness of its output graphs, as also revealed by our experimental results in Section \ref{chapter:experiments}. If connectedness is of particular importance and must be achieved, a possible modification to our algorithms can guarantee it without increasing the overall time complexity.
To this end, we randomly choose one vertex from each connected component of the resulting graph. From each subtree corresponding to the set of vertices selected from the components, we pick one arbitrary node, and we form the last subtree with the union of paths on the host tree that connect these nodes. This way, lastly added vertex $v$ is guaranteed to be linked to at least one vertex from each connected component of $G - v$, and so we ensure the connectedness of the output graph. This process is very similar to a single iteration of ConnectingNodes subroutine, thus by the proof of Lemma \ref{connectingnodes-runtime}, it takes $O(\sum_{i=1}^{n} |V(T_i)|)$ time and does not affect the overall complexity of Algorithm ChordalGen.
\section{Experimental Results}\label{chapter:experiments}
In this section, we give extensive test results to show what kind of chordal graphs are generated by Algorithm ChordalGen. In Tables \ref{tab:GrowingSubtree}-\ref{tab:PrunedTree} we give the experimental results of our presented methods GrowingSubtree, ConnectingNodes and PrunedTree, respectively. We show how the selection of the input parameters affects the number of resulting edges $m$ and connected components (``\# conn. comp.s"). We also present the number of maximal cliques (``\# maximal cliques"), and the minimum, maximum, and mean size for the maximal cliques (``min clique size", ``max clique size", ``mean clique size"), along with their standard deviation (``sd of clique sizes"). For each parameter combination, we performed ten independent runs and report the average values across those ten runs. For each $n$, we tuned the parameter values in order to approximately achieve some selected average edge density values of 0.01, 0.1, 0.5, and 0.8, where edge density is defined as $\frac{m}{n(n-1)/2}$. We made all instances that we present here available at http://www.ie.boun.edu.tr/$\sim$taskin/data/chordal/ where we also offer a broad collection of relatively small-sized chordal graphs on 50 to 500 vertices with varying edge densities.
Algorithm ChordalGen together with GrowingSubtree is able to output connected chordal graphs unless density is too low, as the results in Table \ref{tab:GrowingSubtree} show. In fact, for average edge density of 0.01, as $n$ increases, the average number of connected components converge to one. If we examine the ``min clique size" column, we see that it is usually one for cases where the average number of connected components is greater than one, suggesting that the reason for obtaining disconnected chordal graphs is largely due to a few isolated vertices and that the dominating rest of the graph is comprised of a connected body. The fact that the starting point of the subtrees is selected uniformly at random and we can directly control the maximum size of them leaves little chance for a set of subtrees not to intersect with any other and so lead to a separate connected component, unless the maximum subtree size $k$ is very small.
\begin{table}[H]
\centering
\caption{Experimental results of Algorithm ChordalGen with GrowingSubtree method}
\scalebox{0.8}{ \begin{tabular}{ m{1.3cm} m{1.5cm} m{1.5cm} m{2cm} m{1.5cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} }
\toprule
$n$ & \parbox[t]{2cm}{max\\subtree\\size ($k$)} & \parbox[t]{2cm}{density} & \parbox[t]{2cm}{$m$} & \parbox[t]{2cm}{\#\\ conn.\\comp.s} & \parbox[t]{2cm}{ \# \\maximal\\cliques} & \parbox[t]{2cm}{ min \\clique \\size} & \parbox[t]{2cm}{max\\ clique \\size} & \parbox[t]{2cm}{mean \\clique \\size} & \parbox[t]{2cm}{sd of\\clique \\sizes} \\
\midrule
\multirow{4}[2]{*}{1000}
& 7 & 0.011 & 5551.4 & 16.7 & 357.1 & 1.0 & 21.6 & 6.1 & 3.4 \\
& 33 & 0.104 & 51768.5 & 1.0 & 173.0 & 4.8 & 141.5 & 30.7 & 20.4 \\
& 139 & 0.497 & 248033.5 & 1.0 & 81.3 & 30.6 & 474.3 & 137.9 & 89.2 \\
& 324 & 0.803 & 400918.7 & 1.0 & 47.5 & 66.8 & 717.4 & 312.2 & 159.5 \\
\midrule
\multirow{4}[2]{*}{2500}
& 13 & 0.011 & 34605.4 & 2.7 & 678.3 & 1.2 & 54.5 & 11.5 & 6.8 \\
& 63 & 0.104 & 326287.0 & 1.0 & 300.4 & 9.7 & 349.9 & 61.7 & 45.0 \\
& 269 & 0.505 & 1577474.1 & 1.0 & 134.3 & 50.2 & 1177.4 & 292.8 & 207.7 \\
& 635 & 0.806 & 2518595.5 & 1.0 & 83.3 & 132.6 & 1861.8 & 673.1 & 397.4 \\
\midrule
\multirow{4}[2]{*}{5000}
& 20 & 0.010 & 129763.8 & 1.6 & 1104.6 & 1.6 & 96.5 & 18.1 & 11.4 \\
& 100 & 0.104 & 1296493.4 & 1.0 & 474.0 & 15.9 & 695.0 & 103.0 & 80.0 \\
& 450 & 0.498 & 6226843.9 & 1.0 & 205.7 & 74.8 & 2390.8 & 501.6 & 374.2 \\
& 1097 & 0.804 & 10053952.1 & 1.0 & 124.1 & 202.0 & 3656.7 & 1220.6 & 741.9 \\
\midrule
\multirow{4}[2]{*}{10000}
& 31 & 0.010 & 502155.4 & 1.0 & 1754.4 & 3.4 & 199.2 & 29.1 & 19.8 \\
& 169 & 0.107 & 5362219.2 & 1.0 & 709.8 & 22.2 & 1376.0 & 181.6 & 149.6 \\
& 751 & 0.506 & 25298684.2 & 1.0 & 304.9 & 104.9 & 4687.5 & 894.0 & 711.5 \\
& 1855 & 0.802 & 40103196.8 & 1.0 & 184.1 & 278.3 & 7445.0 & 2141.8 & 1459.7 \\
\bottomrule
\end{tabular}
\label{tab:GrowingSubtree} } \end{table}
Table \ref{tab:ConnectingNodes} reports the outputs of Algorithm ChordalGen using ConnectingNodes for generating subtrees. As the experimental results given in Table \ref{tab:ConnectingNodes} reveal, Algorithm ConnectingNodes should be input very small $\lambda$ values in order to achieve even quite dense graphs. Since even few number of selected nodes may result in large subtrees, which increases the chances of potential intersections with other subtrees and hence the number of edges in the output graph, the number of selected nodes has to be restricted via low values for $\lambda$, which is the main ingredient in setting the cardinality of node selection. Because of this, the selected node set, and hence the subtree, commonly be comprised of a single node, especially in graphs with low density. Therefore, when there are many single-node subtrees, intersections are not very likely. Thus, we observe many isolated vertices in the generated chordal graphs, as implied by the high number of connected components and minimum size of one in maximal cliques (see ``min clique size" column) in ConnectingNodes method.
\begin{table}[H]
\centering
\caption{Experimental results of Algorithm ChordalGen with ConnectingNodes method}
\scalebox{0.8}{
\begin{tabular}{ m{1.3cm} m{1.4cm} m{1.5cm} m{2.2cm} m{1.5cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} }
\toprule
$n$ & \parbox[t]{2cm}{$\lambda$} & \parbox[t]{2cm}{density} & \parbox[t]{2cm}{$m$} & \parbox[t]{2cm}{\#\\ conn.\\comp.s} & \parbox[t]{2cm}{ \# \\maximal\\cliques} & \parbox[t]{2cm}{ min \\clique \\size} & \parbox[t]{2cm}{max\\ clique \\size} & \parbox[t]{2cm}{mean \\clique \\size} & \parbox[t]{2cm}{sd of\\clique \\sizes} \\
\midrule
\multirow{4}[2]{*}{1000}
& 0.5 & 0.011 & 5455.4 & 349.0 & 597.0 & 1.0 & 75.8 & 3.0 & 5.5 \\
& 1.2 & 0.100 & 49805.1 & 121.4 & 495.3 & 1.0 & 266.5 & 8.0 & 23.1 \\
& 2.7 & 0.507 & 253074.6 & 8.6 & 238.7 & 1.0 & 627.0 & 30.7 & 87.4 \\
& 4.1 & 0.804 & 401708.6 & 1.8 & 96.3 & 1.6 & 835.4 & 81.5 & 183.9 \\
\midrule
\multirow{4}[2]{*}{2500}
& 0.6 & 0.010 & 31559.8 & 849.2 & 1475.8 & 1.0 & 194.6 & 3.5 & 10.1 \\
& 1.2 & 0.101 & 314818.0 & 298.5 & 1215.3 & 1.0 & 657.5 & 9.6 & 40.7 \\
& 2.7 & 0.503 & 1571946.8 & 27.9 & 594.0 & 1.0 & 1620.6 & 36.0 & 142.1 \\
& 4.1 & 0.800 & 2498034.2 & 3.1 & 226.5 & 1.2 & 2074.8 & 102.8 & 315.3 \\
\midrule
\multirow{4}[2]{*}{5000}
& 0.6 & 0.010 & 127700.5 & 1693.5 & 2960.1 & 1.0 & 395.8 & 3.8 & 15.2 \\
& 1.2 & 0.103 & 1290089.2 & 578.3 & 2409.8 & 1.0 & 1396.8 & 10.6 & 57.9 \\
& 2.7 & 0.505 & 6308093.4 & 44.4 & 1148.6 & 1.0 & 3217.5 & 41.0 & 211.0 \\
& 4.1 & 0.805 & 10060406.5 & 4.7 & 435.7 & 1.0 & 4261.4 & 119.4 & 460.6 \\
\midrule
\multirow{4}[2]{*}{10000}
& 0.6 & 0.010 & 501760.2 & 3365.6 & 5901.9 & 1.0 & 806.2 & 4.0 & 21.8 \\
& 1.2 & 0.100 & 5022899.1 & 1180.4 & 4794.0 & 1.0 & 2703.4 & 11.7 & 83.8 \\
& 2.7 & 0.502 & 25114409.8 & 97.3 & 2300.6 & 1.0 & 6355.1 & 44.0 & 291.5 \\
& 4.1 & 0.803 & 40154270.6 & 9.3 & 852.9 & 1.0 & 8484.9 & 136.2 & 682.4 \\
\bottomrule
\end{tabular}
\label{tab:ConnectingNodes} } \end{table}
Table \ref{tab:PrunedTree} presents the experimental results of Algorithm ChordalGen when used with PrunedTree method. The two columns ``edge del. fr. ($f$)" and ``selection barrier ($s$)" correspond to input parameters that PrunedTree takes as input, whose role are explained in Section \ref{section:removal_of_edges}. Here, we observe that for density values of 0.1, 0.5, and 0.8, the output graphs are predominantly connected. As in the previous two methods, minimum size of maximal cliques in case of 0.01 density is one, implying that the main cause of the number of connected components is probably a small group of isolated vertices.
\begin{table}[H]
\caption{Experimental results of Algorithm ChordalGen with PrunedTree method}
\scalebox{0.8}{
\begin{tabular}{ m{1.3cm} m{1.5cm} m{1.5cm} m{1.5cm} m{2cm} m{1.5cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} }
\toprule
$n$ & \parbox[t]{2cm}{edge\\ del. fr.\\($f$)} & \parbox[t]{2cm}{selection\\ barrier\\($s$)} & \parbox[t]{2cm}{density} & \parbox[t]{2cm}{$m$} & \parbox[t]{2cm}{\#\\ conn.\\comp.s} & \parbox[t]{2cm}{ \# \\maximal\\cliques} & \parbox[t]{2cm}{ min \\clique \\size} & \parbox[t]{2cm}{max\\ clique \\size} & \parbox[t]{2cm}{mean \\clique \\size} & \parbox[t]{2cm}{sd of\\clique \\sizes} \\
\midrule
\multirow{4}[2]{*}{1000}
& 0.950 & 0.35 & 0.011 & 5619.3 & 45.8 & 324.5 & 1.0 & 30.4 & 5.5 & 4.1 \\
& 0.700 & 0.60 & 0.104 & 51765.9 & 1.0 & 99.9 & 4.2 & 133.6 & 35.9 & 25.2 \\
& 0.140 & 0.85 & 0.497 & 248172.1 & 1.0 & 50.2 & 193.0 & 337.1 & 278.3 & 35.2 \\
& 0.100 & 0.93 & 0.806 & 402349.8 & 1.0 & 36.5 & 397.7 & 621.6 & 530.8 & 53.6 \\
\midrule
\multirow{4}[2]{*}{2500}
& 0.950 & 0.70 & 0.011 & 34013.9 & 28.3 & 542.5 & 1.0 & 72.3 & 9.5 & 8.8 \\
& 0.700 & 0.70 & 0.101 & 316270.6 & 1.0 & 150.4 & 5.5 & 335.0 & 70.8 & 58.0 \\
& 0.120 & 0.90 & 0.507 & 1584225.2 & 1.0 & 66.2 & 492.0 & 844.4 & 703.2 & 84.4 \\
& 0.077 & 0.95 & 0.801 & 2500840.1 & 1.0 & 56.5 & 996.2 & 1530.1 & 1304.7 & 123.0 \\
\midrule
\multirow{4}[2]{*}{5000}
& 0.950 & 0.77 & 0.010 & 130970.2 & 21.7 & 833.8 & 1.0 & 177.7 & 14.0 & 15.2 \\
& 0.700 & 0.75 & 0.097 & 1216527.9 & 1.0 & 202.6 & 4.8 & 655.2 & 117.0 & 106.5 \\
& 0.080 & 0.91 & 0.495 & 6182739.6 & 1.0 & 101.0 & 1083.4 & 1571.7 & 1395.7 & 109.0 \\
& 0.045 & 0.96 & 0.801 & 10004264.5 & 1.0 & 99.9 & 2269.3 & 2955.8 & 2672.9 & 146.1 \\
\midrule
\multirow{4}[2]{*}{10000}
& 0.900 & 0.50 & 0.010 & 479501.2 & 22.5 & 1394.8 & 1.0 & 286.6 & 20.6 & 24.5 \\
& 0.700 & 0.81 & 0.102 & 5076707.1 & 1.0 & 260.0 & 7.2 & 1415.8 & 204.5 & 206.0 \\
& 0.060 & 0.93 & 0.507 & 25357868.2 & 1.0 & 143.5 & 2359.9 & 3176.8 & 2882.4 & 177.7 \\
& 0.031 & 0.96 & 0.793 & 39653114.8 & 1.0 & 157.7 & 4705.0 & 5709.5 & 5319.0 & 198.9 \\
\bottomrule
\end{tabular}
\label{tab:PrunedTree} } \end{table}
We want to compare our results to the results showing the kind of chordal graphs that are generated by Alg 2a \cite{markenzon2008two}. Note, however that, the results given by \cite{markenzon2008two} only contain graphs on 10000 vertices, with varying number of edges. Most metrics presented in \cite{markenzon2008two} are about the number of edges. When it comes to the maximal cliques, they present only the average maximum clique size over the generated graphs for each edge density. Comparing these to our numbers we see that graphs corresponding to edge densities 0.01, 0.1, 0.5, and 0.8 of Alg 2a have average maximum clique sizes 727, 2847, 6875, and 8760, respectively. As can be seen from Tables \ref{tab:GrowingSubtree}-\ref{tab:PrunedTree}, these numbers are quite higher than the corresponding numbers for the graphs generated by Algorithm ChordalGen. In fact, studying the numbers more carefully, we can conclude that the maximum clique of a graph generated by Alg 2a contains almost all the edges of the graph. In the case of density 0.01, such a clique contains more than half of the edges, whereas in the case of higher densities, the largest clique contains more than 80, 94, and 95 percent of the edges, respectively. Thus, there does not seem to be an even distribution of the sizes of maximal cliques of graphs generated by Alg 2a.
As we mentioned in Section \ref{chapter:literature}, we also implemented Alg 1 \cite{andreou2005generating}, but without imposing a limit on the maximum degree of the output graph, because no detail was given about how the method avoids exceeding a given maximum degree in \cite{andreou2005generating}. In Table \ref{tab:Alg1} we give results of Alg 1 analogous to Tables \ref{tab:GrowingSubtree}--\ref{tab:PrunedTree} for 1000, 2500, and 5000 vertices. In order to obtain results for Table \ref{tab:Alg1} comparable to those given in Tables \ref{tab:GrowingSubtree}-\ref{tab:PrunedTree}, we aimed to have approximately the same edge density values. For this purpose, when determining the number of new neighbors of a vertex at each step in Alg 1, we multiplied the total number of candidate vertices with a coefficient between 0 and 1, which we call \textit{upper bound coefficient}. A running time analysis for this algorithm has not been given in \cite{andreou2005generating}. With our implementation, this algorithm turned out to be too slow to allow testing graphs on 10000 vertices in a reasonable amount of time. However, already from the obtained numbers, we can reach a conclusion for Alg 1 similar to that on Alg 2a. Observe that the maximum clique sizes obtained for 5000 vertices by Alg 1, are comparable to the maximum clique sizes obtained for 10000 vertices by Algorithm ChordalGen. Hence, like Alg 2a, also Alg 1 seems to generate graphs with few big maximal cliques.
As can be seen in Table \ref{tab:Alg1}, Alg 1 outputs connected chordal graphs for the selected set of average edge density values and number of vertices. The minimum size of the maximal cliques did not show much variation throughout our experiments and almost always turned out to be two. The consistency in this measure may be an indication of the lack of potential to produce a diverse range of maximal clique sizes.
\begin{table}[H]
\centering
\caption{Experimental results of our implementation of Alg 1 \cite{andreou2005generating}}
\scalebox{0.8}{
\begin{tabular}{ m{1.3cm} m{1.7cm} m{1.5cm} m{2cm} m{1.5cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} m{1.6cm} }
\toprule
$n$ & \parbox[t]{2cm}{upper\\ bound\\coef.} & \parbox[t]{2cm}{density} & \parbox[t]{2cm}{$m$} & \parbox[t]{2cm}{\#\\conn.\\comp.s} & \parbox[t]{2cm}{ \# \\maximal \\cliques} & \parbox[t]{2cm}{min \\clique \\size} & \parbox[t]{2cm}{max\\clique\\size} & \parbox[t]{2cm}{mean\\clique\\size} & \parbox[t]{2cm}{sd of\\clique\\sizes} \\
\midrule
\multirow{4}[2]{*}{1000}
& 0.00130 & 0.011 & 5659.3 & 1.0 & 933.3 & 2.0 & 58.0 & 4.9 & 9.4 \\
& 0.00300 & 0.100 & 49717.1 & 1.0 & 753.3 & 2.0 & 219.3 & 28.4 & 60.0 \\
& 0.01100 & 0.506 & 252864.4 & 1.0 & 401.5 & 2.0 & 562.5 & 190.1 & 233.6 \\
& 0.03500 & 0.805 & 401945.6 & 1.0 & 191.6 & 2.4 & 788.6 & 399.4 & 342.6 \\
\midrule
\multirow{4}[2]{*}{2500}
& 0.00053 & 0.011 & 33201.8 & 1.0 & 2320.8 & 2.0 & 154.5 & 8.8 & 26.1 \\
& 0.00120 & 0.100 & 313001.8 & 1.0 & 1882.3 & 2.0 & 548.8 & 69.3 & 159.3 \\
& 0.00440 & 0.502 & 1568857.4 & 1.0 & 1006.5 & 2.0 & 1400.6 & 462.7 & 593.3 \\
& 0.01400 & 0.799 & 2495447.1 & 1.0 & 469.8 & 2.0 & 1975.9 & 936.8 & 888.2 \\
\midrule
\multirow{4}[2]{*}{5000}
& 0.00027 & 0.011 & 133829.0 & 1.0 & 4629.0 & 2.0 & 313.9 & 15.9 & 56.2 \\
& 0.00062 & 0.107 & 1339169.7 & 1.0 & 3717.9 & 2.0 & 1136.5 & 147.0 & 342.7 \\
& 0.00220 & 0.494 & 6179872.9 & 1.0 & 2032.5 & 2.0 & 2774.2 & 897.7 & 1180.8 \\
& 0.00700 & 0.801 & 10011146.3 & 1.0 & 939.0 & 2.0 & 3950.0 & 1901.6 & 1794.5 \\
\bottomrule
\end{tabular}
\label{tab:Alg1} } \end{table}
\begin{figure}
\caption{Results from Algorithm ChordalGen with GrowingSubtree method}
\label{fig:kabakcicegi_1000_4lu}
\caption{Results from Algorithm ChordalGen with ConnectingNodes method}
\label{fig:nodeDeletion_1000_4lu}
\caption{ Results from Algorithm ChordalGen with PrunedTree method}
\label{fig:edgeDeletion_1000_4lu}
\caption{Results from our implementation of Alg 1 \cite{andreou2005generating}}
\label{fig:peoBased_1000_4lu}
\caption{Histograms of maximal clique sizes for $ n = 1000 $ and average edge densities 0.01, 0.1, 0.5, and 0.8 (from left to right)}
\label{fig:histograms_1000}
\end{figure}
In our next set of experimental results we investigate how the sizes of the maximal cliques are distributed. Figures \ref{fig:histograms_1000}-\ref{fig:histograms_10000} show the average number of maximal cliques across ten independent runs in intervals of width five, for 1000, 2500, 5000, and 10000 vertices and varying edge densities. These figures consist of four subfigures, except Figure \ref{fig:histograms_10000} which contains only the first three, and each subfigure is comprised of four histograms corresponding to four different average edge density values. The first three sub-figures on the top row show the results from Algorithm ChordalGen combined with each one of the three subtree generation methods presented, and the bottom row shows results of our implementation of Alg 1 \cite{andreou2005generating}. For a given $n$ and average density value, the ranges of $ x $-axes are kept the same in order to render the histograms comparable. The $y$-axes, however, have different ranges because maximum frequencies in histograms may vary drastically.
\begin{figure}
\caption{Results from Algorithm ChordalGen with GrowingSubtree method}
\label{fig:kabakcicegi_2500_4lu}
\caption{Results from Algorithm ChordalGen with ConnectingNodes method}
\label{fig:nodeDeletion_2500_4lu}
\caption{Results from Algorithm ChordalGen with PrunedTree method}
\label{fig:edgeDeletion_2500_4lu}
\caption{Results from our implementation of Alg 1 \cite{andreou2005generating}}
\label{fig:peoBased_2500_4lu}
\caption{Histograms of maximal clique sizes for $ n = 2500 $ and average edge densities 0.01, 0.1, 0.5, and 0.8 (from left to right)}
\label{fig:histograms_2500}
\end{figure}
The sizes of maximal cliques of graphs produced by GrowingSubtree method are dispersed fairly over the range, which becomes more noticeable with the increase in edge densities (as we proceed to the right). Frequencies do not show any obvious bias toward some portion of its domain, which may be considered as an indicator of the diversity of the chordal graphs produced, which is a desired characteristic of a random chordal graph generator. In ConnectingNodes method, however, the vast majority of cliques have size ten or less. The frequencies of larger cliques are barely noticeable compared to the dominant small-sized set. As the graphs become denser, frequencies of relatively larger cliques start to become visible too, but general behaviour remains the same. So, if chordal graphs with predominantly very small clique sizes are sought, ConnectingNodes method can be preferred. In PrunedTree method, the mode of the distribution shifts with the increase in edge densities and the sizes of cliques become clustered around some moderate value over the given range. As for Alg 1, the vast majority of maximal cliques of its output graphs have sizes of 2 to 15 for graphs with low densities of 0.01 and 0.1. With the increase in edge densities, frequencies of large-size maximal cliques become visible relative to the dominant small clique frequencies; however, all but the extremes of the range is barely used regardless of selection of $n$ and edge density.
\begin{figure}
\caption{Results from Algorithm ChordalGen with GrowingSubtree method}
\label{fig:kabakcicegi_5000_4lu}
\caption{Results from Algorithm ChordalGen with ConnectingNodes method}
\label{fig:nodeDeletion_5000_4lu}
\caption{Results from Algorithm ChordalGen with PrunedTree method}
\label{fig:edgeDeletion_5000_4lu}
\caption{Results from our implementation of Alg 1 \cite{andreou2005generating}}
\label{fig:peoBased_5000_4lu}
\caption{Histograms of maximal clique sizes for $ n = 5000 $ and average edge densities 0.01, 0.1, 0.5, and 0.8 (from left to right)}
\label{fig:histograms_5000}
\end{figure}
\begin{figure}
\caption{Results from Algorithm ChordalGen with GrowingSubtree method}
\label{fig:kabakcicegi_10000_4lu}
\caption{Results from Algorithm ChordalGen with ConnectingNodes method}
\label{fig:nodeDeletion_10000_4lu}
\caption{Results from Algorithm ChordalGen with PrunedTree method}
\label{fig:edgeDeletion_10000_4lu}
\caption{Histograms of maximal clique sizes for $ n = 10000 $ and average edge densities 0.01, 0.1, 0.5, and 0.8 (from left to right)}
\label{fig:histograms_10000}
\end{figure}
\section{Conclusion}\label{chapter:conc}
To the best of our knowledge, Algorithm ChordalGen is the first algorithm for random chordal graph generation based directly on subtree intersection characterization. It is very general and flexible in the sense that many different methods for subtree generation can be plugged in.
The three different subtree generation methods presented here each offer output graphs of different structures. As far as the distribution of maximal clique sizes are concerned, ConnectingNodes and PrunedTree methods yield graphs of somewhat more specific structure in the sense that the sizes of maximal cliques are always clustered in the very initial portion in ConnectingNodes method, and in PrunedTree method in the initial part of the range for low densities, in middle portions for moderate to high densities. GrowingSubtree method, though, in addition to its advantageous time complexity, generates the most varied chordal graphs compared to both existing methods and to other two of our suggested methods. Depending on the context and structural needs for the output graph, Algorithm ChordalGen can be used with one of the three subroutines chosen suitably in order to produce chordal graphs of varying size and density. For those who prefer to test various algorithms on chordal graphs without focusing on the generation procedure, we offer all instances used in this paper as well as a broad collection of relatively small-sized chordal graphs on 50 to 500 vertices with varying edge densities at http://www.ie.boun.edu.tr/$\sim$taskin/data/chordal/.
Last but not least, our work gives rise to an open question about chordal graphs which has not been addressed to date to the best of our knowledge: what is the worst case time complexity of a chordal graph generation algorithm which produces the entire set of nodes of all subtrees in a subtree intersection model? In the current paper, we give a lower bound, namely $\Omega(mn^{1/4})$, on the time complexity. However, the exact worst case time complexity remains unknown.
\end{document} |
\begin{document}
\title[Fisher-KPP equation with time delay] {Propagation dynamics of Fisher-KPP equation with time delay and free boundaries}
\begin{abstract} Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed. \end{abstract}
\maketitle
\section{Introduction}\label{sec:intr} In the pioneer work of Fisher \cite{Fisher}, and Kolmogorov, Petrovski and Piskunov \cite{KPP}, it was shown that \begin{equation}\label{eq:KPP} u_t =u_{xx}+f(u), \quad x\in\mathbb{R} \end{equation} with \begin{equation}\label{eq:KPP-cond} f\in C^1(\mathbb{R},\mathbb{R}),\quad f(0)=0=f(1),\quad f(s)\leqslant f'(0)s,\ s\geqslant 0, \end{equation} admits traveling waves solutions of the form $u(t,x)=\phi(x-ct)$ satisfying $\phi(-\infty)=1$ and $\phi(+\infty)=0$ if and only if $c\geqslant c_0:=2\sqrt{f'(0)}$. In 1970s', Aronson and Weinberger \cite{AW2} proved that the minimal wave speed $c_0$ is also the asymptotic speed of spread (spreading speed for short) in the sense that \begin{equation}
\lim_{t\to\infty} \sup_{|x|\geqslant (c_0+\epsilon)t}u(t,x)=0,\quad \lim_{t\to\infty} \inf_{|x|\leqslant (c_0-\epsilon)t}u(t,x)=1 \end{equation} for any small $\epsilon>0$ provided that the initial function $u(0,x)$ is compactly supported. These works have stimulated volumes of studies for the propagation dynamics of various types of evolution systems. Among others, of particular interest to the Fisher-KPP equation \eqref{eq:KPP}-\eqref{eq:KPP-cond} with time delay or free boundary are two typical ones.
Schaaf \cite{Sc} studied the following delayed reaction-diffusion equation \begin{equation}\label{eq:Schaaf} u_t(t,x)=u_{xx}(t,x)+f(u(t,x),u(t-\tau,x)),\quad x\in\mathbb{R},\ t>0, \end{equation} where $\tau>0$ is the time delay. With the Fisher-KPP condition on $\tilde{f}(s):=f(s,s)$ and the quasi-monotone condition $\partial_2 f\geqslant 0$, it was shown that the minimal wave speed $c_0=c_0(\tau)$ exists and it is determined by the system of two transcendental equations \begin{equation}\label{eq:speed} F(c,\lambda)=0, \quad \frac{\partial F}{\lambda}(c,\lambda)=0, \end{equation} where \begin{equation} F(c,\lambda)=\lambda^2+c\lambda+\partial_1 f(0,0)+\partial_2 f(0,0)e^{-\lambda\tau}. \end{equation} The delay-induced spatial non-locality was brought to attention by So, Wu and Zou \cite{SWZ}, where they derived the following time-delayed reaction-diffusion model equation with nonlocal response for the study of age-structured population \begin{equation}\label{eq:SWZ} u_t=u_{xx}-d u+\gamma \int_\mathbb{R} b(u(t-\tau,x-y))k(y)dy,\quad x\in\mathbb{R},\ t>0, \end{equation} where $u$ represents the density of mature population, $\tau>0$ is the maturation age, $d$ is the death rate, $b$ is the birth rate function, $\gamma$ is the survival rate from newborn to being mature, and $k$ is the redistribution kernel during the maturation period. As such, introducing time delay into diffusive equation usually gives rises to spatial non-locality due to the interaction of time lag (for maturation) and diffusion of immature population. In the extreme case where the immature population does not diffuse, the kernel $k$ becomes the Dirac measure, and hence \eqref{eq:SWZ} reduces to \eqref{eq:Schaaf}. We refer to the survey article \cite{GW} for the delay-induced nonlocal reaction-diffusion problems. In \cite{SWZ}, the authors obtained the minimal wave speed $c_0(\tau)$ that is determined by a similar system to \eqref{eq:speed} provided that $b$ is nondecreasing and $f(s):=-ds+b(s)$ is of Fisher-KPP type. Wang, Li and Ruan \cite{WLR} proved that $c_0(\tau)$ is decreasing in $\tau$. Liang and Zhao \cite{LZ} showed that $c_0(\tau)$ is also the spreading speed for the solutions satisfying the following initial condition \begin{equation} \text{$u(\theta,x)$ is continuous and compactly supported in $\theta\in [-\tau,0]$ and $x\in\mathbb{R}$.} \end{equation} Similar to the classical Fisher-KPP equation, the spreading speed $c_0(\tau)$ for delayed reaction-diffusion equation is still linearly determined for both local and nonlocal problems thanks to the Fisher-KPP type condition.
We refer to \cite{MS} for more properties that are induced by time delay in reaction-diffusion equations, including the well-posedness of initial value problems as well as the role of the quasi-monotone condition on the comparison principle, and \cite{FangZhao14, FangZhao15} for the delay-induced weak compactness of time-$t$ solution maps when $t\in(0,\tau]$ as well as its role in the study of wave propagation.
Recently, Du and Lin \cite{DuLin} proposed a Stefan type free boundary to the Fisher-KPP equation \begin{equation}\label{freeb} \left\{ \begin{array}{ll}
u_t =u_{xx}+u(1-u), & g(t)< x<h(t),\; t>0,\\
u(t,g(t))=0,\ \ g'(t)=-\mu u_x(t, g(t)), & t>0,\\
u(t,h(t))=0,\ \ h'(t)=-\mu u_x (t, h(t)) , & t>0, \end{array} \right. \end{equation} where the free boundaries $x=g(t)$ and $x=h(t)$ represent the spreading fronts, which are determined jointly by the gradient at the fronts and the coefficient $\mu$ in the Stefan condition. For more background of proposing such free boundary conditions, we refer to \cite{DuLin, BDK}. It was proved in \cite{DuLin} that the unique global solution $(u,g,h)$ has a spreading-vanishing dichotomy property as $t\to\infty$: either $(g(t),h(t))\to\mathbb{R}$ and $u\to1$ (spreading case), or $g(t)\to g_\infty$, $h(t)\to h_\infty$ with $h_\infty-g_\infty\leqslant \pi$, and $u\to 0$ (vanishing case). Moreover, it was also proved that when spreading happens, there is a constant $k_0>0$ such that $-g(t)$ and $h(t)$ behave like a straight line $k_0t$ for large time, where $k_0$ is called the asymptotic speed of spread (spreading speed for short). Different from the classical Fisher-KPP speed, $k_0$ is the unique value of $c$ such that the following nonlinear semi-line problem is solvable: \begin{equation}\label{k0} \left\{
\begin{array}{ll}
q'' - cq'+q(1-q)=0, & z>0,\\
q(\infty)=1, \ \ \mu q_+'(0)=c,\ \ q(z)>0, & z\leqslant 0,\\
q(z)=0, & z\leqslant 0,
\end{array}
\right. \end{equation} where $q_+'(0)$ is the right derivative of $q(z)$ at $0$. In particular, as $\mu$ increases to infinity, $k_0$ increases to the classical Fisher-KPP speed $2\sqrt{f'(0)}$. Later on, Du and Lou \cite{DuLou} obtained a rather complete characterization on the asymptotic behavior of solutions for \eqref{freeb} with some general nonlinear terms. For further related work on free boundary problems, we refer to \cite{DuGuo, DGP, DMZ} and the references therein.
In this paper, we aim to explore how to incorporate time delay and free boundary into the Fisher-KPP equation \eqref{eq:KPP}-\eqref{eq:KPP-cond} so that the problem is well-posed, and then study their joint influence on the propagation dynamics.
Keeping a smooth flow for the organizations of the paper, we write down here the problem of interest while leaving in the next section the derivation details, including the emergence of the compatible condition \eqref{CC} for the well-posedness of the initial value problem. \begin{equation}\label{p} \left\{ \begin{array}{ll}
u_t(t,x) =u_{xx}(t,x)- d u(t,x) +f(u(t-\tau,x)), & x\in(g(t),h(t)),\; t>0,\\
u(t,g(t))=0,\ \ g'(t)=-\mu u_x(t, g(t)), & t>0,\\
u(t,h(t))=0,\ \ h'(t)=-\mu u_x (t, h(t)) , & t>0,\\
u(\theta,x) =\phi (\theta,x),& g(\theta) \leqslant x \leqslant h(\theta),\; \theta\in[-\tau,0], \end{array} \right. \tag{$P$} \end{equation} where $d$ and $\tau$ are two positive constants, the nonlinear function $f$ satisfies \[ \bf{(H)}\ \hskip 16mm \left\{
\begin{array}{l}
f(s)\in C^{1+\tilde{\nu}}([0,\infty))\ \mbox{ for some } \tilde{\nu}\in(0,1),\ \ f(0)=0,\ \ f'(0)-d>0;\\
f(s)-d s=0 \mbox{ has a unique positive constant root } u^*;\\
f(s) \mbox{ is monotonically increasing in } s \in[0,u^*];\\
\frac{f(s)}{s} \mbox{ is monotonically decreasing in } s \in[0,u^*]
\end{array}
\right. \ \hskip 15mm
\] and the initial data $(\phi(\theta,x), g(\theta), h(\theta))$ satisfies \begin{equation}\label{def:X} \left\{ \begin{array}{ll} \phi(\theta,x) \in C^{1,2} ([-\tau,0]\times[g(\theta),h(\theta)]),\\ 0<\phi(\theta,x)\leqslant u^*\ \mbox{ for } (\theta,x)\in[-\tau,0]\times(g(\theta), h(\theta)),\\ \phi(\theta,x) \equiv 0\ \ \mbox{ for } \theta\in[-\tau,0],\; x\not\in(g(\theta),h(\theta)) \end{array} \right. \end{equation} as well as the compatible condition \begin{equation}\label{CC} [g(\theta), h(\theta)]\subset [g(0), h(0)]\ \ \ \mbox{ for }\ \theta\in[-\tau,0]. \end{equation}
Assumption {\bf (H)} ensures the Fisher-KPP structure as well as the comparison principle. Due to the nature of delay differential equations, the initial value, including the initial domain, has to be imposed over the history period $[-\tau,0]$, as in \eqref{def:X}. The interaction of time delay and free boundary gives rise to the compatible condition \eqref{CC} that is essential for the well-posedness of the problem. If $\tau=0$, then the compatible condition (1.12) becomes trivial and problem (P) reduces to \eqref{freeb}.
\begin{thm}\label{wellposedness} {\rm(Well-posedness)} For an initial data $(\phi(\theta,x), g(\theta), h(\theta))$ satisfying \eqref{def:X} and \eqref{CC}, there exists a unique triple $(u, g, h)$ solving \eqref{p} with $u\in C^{1,2}((0,\infty) \times[g(t),h(t)])$ and $g,\, h\in C^1([0,\infty))$. \end{thm} With the compatible condition \eqref{CC} we can cast the problem into a fixed boundary problem and then apply the Schauder fixed point theorem to establish the local existence of solutions. The extension to all positive time is based on some a priori estimates\footnote{We sincerely thank Professor Avner Friedman for his valuable comments and suggestions on the proof of the well-posedness.}.
From the maximum principle and {\bf (H)}, it follows that when $t>0$ the solution $u>0$ as $x\in (g(t),h(t))$, $u_x(t,g(t))>0$ and $u_x(t,h(t))<0$, and hence, $g'(t)<0<h'(t)$ for all $t>0$. Therefore, we can denote $$ g_{\infty}:=\lim_{t\to\infty}g(t)\ \ \ \mbox{and }\quad h_{\infty}:= \lim_{t\to\infty}h(t). $$
\begin{thm}{\rm (Spreading-vanishing dichotomy)}\label{thm:asy be} Let $(u,g,h)$ be the solution of \eqref{p}. Then the following alternative holds:
Either
{\rm (i) Spreading:} $(g_\infty, h_\infty)=\mathbb{R}$ and \[ \lim_{t\to\infty}u(t,x)=u^* \mbox{ locally uniformly in $\mathbb{R}$}, \]
or
{\rm (ii) Vanishing:} $(g_\infty, h_\infty)$ is a finite interval with length no bigger than $\frac{\pi}{\sqrt{f'(0)-d}}$ and \[ \lim_{t\to\infty}\max_{g(t)\leqslant x\leqslant h(t)} u(t,x)=0. \] \end{thm}
When spreading happens, we characterize the spreading speed and profile of the solutions. The nonlinear and nonlocal semi-wave problem \begin{equation}\label{sw11} \left\{
\begin{array}{ll}
q'' - cq'-d q+ f( q(z-c\tau))=0, & z>0,\\
q(\infty)=u^*, \ \ \mu q_+'(0)=c,\ \ q(z)>0, & z\leqslant 0,\\
q(z)=0, & z\leqslant 0
\end{array}
\right. \end{equation} will play an important role. If $\tau=0$ then \eqref{sw11} reduces to the local form \eqref{k0}.
\begin{thm}\label{thm:semiwave} Problem \eqref{sw11} admits a unique solution $(c^*, q_{c^*})$ and $c^*=c^*(\tau)$ is decreasing in delay $\tau\geqslant 0$. \end{thm} Due to the presence of time delay, the proof of Theorem \ref{thm:semiwave} highly relies on the distribution of complex solutions of the following transcendental equation \begin{equation} \lambda^2-c\lambda-d+f'(0)e^{-\lambda c\tau}=0. \end{equation} We refer to Lemma \ref {lem:eigen} and Proposition \ref{prop:qoan1}, which are independently of interest.
With the semi-wave established above, we can construct various super- and subsolutions to estimate the spreading fronts $h(t),g(t)$ and the spreading profile as $t\to\infty$.
\begin{thm}\label{thm:profile of spreading sol} {\rm(Spreading profile)} Let $u$ be a solution satisfying Theorem \ref{thm:asy be}(i). Then there exist two constants $H_1$ and $G_1$ such that \[ \lim\limits_{t\to\infty}[h(t)- c^*t] = H_1 ,\quad \ \lim\limits_{t\to\infty} h'(t)=c^*, \] \[ \lim\limits_{t\to\infty}[g(t) + c^*t] = G_1 ,\quad\ \lim\limits_{t\to\infty} g'(t)=-c^*, \] \begin{equation}\label{profile convergence 1}
\lim\limits_{t\to\infty} \left\| u(t,\cdot)- q_{c^*}(c^*t+ H_1-\cdot) \right\|_{L^\infty ( [0, h(t)])}=0, \end{equation} \begin{equation}\label{profile convergence 1-left}
\lim\limits_{t\to\infty} \left\| u(t,\cdot)- q_{c^*}(c^*t- G_1+\cdot )\right\|_{L^\infty ([g(t), 0])} =0, \end{equation} where $(c^*,q_{c^*})$ is the unique solution of \eqref{sw11}. \end{thm}
The rest of this paper is organized as follows. In Section 2 we derive the compatible condition \eqref{CC}, with which we formulate problem (P) and then establish the well-posedness as well as the comparison principle. Section 3 is devoted to the study of the semi-wave problem \eqref{sw11}. In section 4, we establish the spreading-vanishing dichotomy result. Finally in Section 5, we characterize the spreading speed and profile of spreading solutions of \eqref{p}.
\section{The compatible condition, well-posedness and comparison principle}\label{sec:basic}
\subsection{The compatible condition} To formulate problem \eqref{p}, we start from the age-structured population growth law \begin{equation}\label{ase} p_t+p_a=D(a)p_{xx}-d(a)p, \end{equation} where $p=p(t,x;a)$ denotes the density of species of age $a$ at time $t$ and location $x$, $D(a)$ and $d(a)$ denote the diffusion rate and death rate of species of age $a$, respectively.
Next we consider the scenario that the species has the following biological characteristics. \begin{itemize} \item[(A1)] The species can be classified into two stages by age: mature and immature. An individual at time $t$ belongs to the mature class if and only if its age exceeds the maturation time $\tau>0$. Within each stage, all individuals share the same behavior. \item[(A2)] Immature population does not move in space. \end{itemize} The total mature population $u$ at time $t$ and location $x$ can be represented by the integral \begin{equation}\label{mimuv} u(t,x)=\int_\tau^\infty p(t,x;a)da. \end{equation} We assume that the mature population $u$ lives in the habitat $[g(t),h(t)]$, vanishes in the boundary \begin{equation}\label{vc} u(t,g(t))=0=u(t,h(t)),\quad t>0 \end{equation} and extends the habitat by obeying the Stefan type moving boundary conditions: \begin{equation}\label{fbc} h'(t)=-\mu u_x(t,h(t)),\ \ g'(t)=-\mu u_x(t,g(t)), \quad t>0, \end{equation} where $\mu$ is a given positive constant. Note that the immature population does not contribute to the extension of habitat due to their immobility, as assumed in (A2).
According to (A1) we may assume that \[ D(a)=\left\{ \begin{array}{ll} 1,& a\geqslant \tau ,\\ 0, & 0\leqslant a<\tau, \end{array} \right. \ \ \ \ d(a)=\left\{ \begin{array}{ll} d ,& a\geqslant \tau ,\\ d_I, & 0\leqslant a<\tau, \end{array} \right.
\]
where $d$ and $d_I$ are two positive constants. Differentiating the both sides of the equation \eqref{mimuv} in time yields \begin{eqnarray}\label{diff-u} u_t & = &\int_\tau^\infty p_t da = \int_\tau^\infty [-p_a+ p_{xx}-d p]da\nonumber\\ & = & u_{xx}-d u+p(t,x;\tau) -p(t,x;\infty).\label{du1} \end{eqnarray} Since no individual lives forever, it is nature to assume that \begin{equation}\label{infinity} p(t,x;\infty)=0. \end{equation} To obtain a closed form of the model, one then needs to express $p(t,x;\tau)$ by $u$ in a certain way. Indeed, $p(t,x;\tau)$ denotes the newly matured population at time $t$, and it is the evolution result of newborns at $t-\tau$. In other words, there is an evolution relation between the quantities $p(t,x;\tau)$ and $p(t-\tau,x;0)$. Such a relation is obeyed by the growth law \eqref{ase} for $0<a<\tau$, and hence it is the time-$\tau$ solution map of the following equation \begin{equation}\label{ast} \left\{
\begin{array}{ll}
q_s=-d_Iq, & x\in\mathbb{R},\ 0\leqslant s\leqslant \tau,\\
q(0,x)=p(t-\tau,x;0), & x\in\mathbb{R}.
\end{array}
\right. \end{equation} Therefore, $p(t,x;\tau)=q(\tau,x)=e^{-d_I\tau}p(t-\tau,x,0)$. Further, the newborns $p(t-\tau,x;0)$ is given by the birth $b(u(t-\tau,x))$, where $b$ is the birth rate function with $b(0)=0$. Consequently, \begin{equation}\label{pptst} p(t,x;\tau)=e^{-d_I\tau}b(u(t-\tau,x)). \end{equation} Combining \eqref{vc}-\eqref{infinity} and \eqref{pptst}, we are led to the following system:
\begin{equation}\label{p-nonsim} \left\{ \begin{array}{lll} u_t(t,x) =u_{xx}(t,x)- d u(t,x) +e^{-d_I\tau}b(u(t-\tau,x)), & t>0, x\in[g(t-\tau),h(t-\tau)]\\ u_t(t,x) =u_{xx}(t,x)- d u(t,x), & t>0, x\in[g(t),h(t)]\setminus [g(t-\tau),h(t-\tau)]\\ u(t,g(t))=0=u(t,h(t)), &t>0\\ h'(t)=-\mu u_x(t,h(t)),\ \ g'(t)=-\mu u_x(t,g(t)), & t>0. \end{array} \right. \end{equation} For $t>0$, outside the habitat $(g(t),h(t))$ the mature population does not exist, that is, \begin{equation}\label{uhhgg} u(t,x)\equiv0 \ \ \ \mbox{ for } \ t>0,\; x\not\in(g(t),h(t)). \end{equation} Clearly, since the habitat is expanding for $t>0$, we have \begin{equation}\label{habitat} [g(t-\tau),h(t-\tau)]\subset [g(t),h(t)],\quad t\geqslant \tau. \end{equation} Hence, the first two equations in \eqref{p-nonsim} can be written as the following single one \begin{equation} u_t(t,x) =u_{xx}(t,x)- d u(t,x) +e^{-d_I\tau}b(u(t-\tau,x)), \quad t>0, x\in[g(t),h(t)] \end{equation} provided that \eqref{habitat} holds for $t\geqslant 0$. As such, in view of \eqref{habitat} we need an additional condition \begin{equation}\label{AC} [g(t-\tau),h(t-\tau)]\subset [g(t),h(t)], \quad t\in[0,\tau). \end{equation} Note that $[g(0),h(0)]\subset[g(t),h(t)]$ for $t>0$. And as the coefficient $\mu\to+\infty$ we have $[g(t),h(t)]\to [g(0,h(0))]$ uniformly for $t\in [0,\tau]$. Therefore, regardless of the influence of $\mu$, \eqref{AC} is strengthened to be \[ [g(\theta), h(\theta)]\subset [g(0), h(0)]\ \ \ \mbox{ for }\ \theta\in[-\tau,0], \] which is the aforementioned compatible condition \eqref{CC}.
Setting $f(s):=e^{-d_I\tau}b(s)$ in \eqref{p-nonsim}, we obtain problem \eqref{p}.
\subsection{Well-posedness} We employ the Schauder fixed point theorem to establish the local existence of solutions to \eqref{p}, and prove the uniqueness,
then extend the solutions to all time by an estimate on the free boundary.
\begin{thm}\label{thm:local} Suppose {\bf(H)} holds. For any $\alpha\in (0,1)$, there is a $T>0$ such that problem \eqref{p} admits a solution $$(u, g, h)\in C^{(1+\alpha)/2, 1+\alpha}([0,T]\times[g(t),h(t)])\times C^{1+\alpha/2}([0,T])\times C^{1+\alpha/2}([0,T]).$$ \end{thm}
\begin{proof} We divide the proof into three steps.
$Step\ 1$. We use a change of variable argument to transform problem \eqref{p} into a fixed boundary problem with a more complicated equation which is used in \cite{CF, DuLin}. Denote $l_1=g(0)$ and $l_2=h(0)$ for convenience, and set $h_0=\frac{1}{2}(l_2-l_1)$. Let $\xi_{1}(y)$ and $\xi_{2}(y)$ be two nonnegative functions in $C^{3}(\mathbb{R})$ such that \[
\xi_{1}(y)=1\ \mbox{ if}\ | y-l_2|< \frac{h_{0}}{4},\ \xi_{1}(y)=0\ \mbox{ if} \ |y-l_2| > \frac{h_{0}}{2},\
|\xi_{1}'(y)|<\frac{6}{h_{0}}\ \mbox{for}\ y\in \mathbb{R}; \] \[
\xi_{2}(y)=1\ \mbox{ if}\ | y-l_1| < \frac{h_{0}}{4},\ \xi_{2}(y)=0\ \mbox{ if}\ | y-l_1| > \frac{h_{0}}{2},\
|\xi_{2}'(y)| < \frac{6}{h_{0}}\ \mbox{for}\ y\in \mathbb{R}. \] Define $y= y(t,x)$ through the identity \begin{align*} &x=y+\xi_{1}(y)(h(t)-l_2)+\xi_{2}(y)(g(t)-l_1)\ \ \ \mbox{ for } t>0,\\ &x\equiv y\ \ \ \mbox{ for } -\tau\leqslant t\leqslant 0. \end{align*} and set \begin{align*} &w(t,y):=u(t,y+\xi_{1}(y)(h(t)-l_2)+\xi_{2}(y)(g(t)-l_1))=u(t,x)\ \ \ \mbox{ for } t>0,\\ &w(\theta,y):=\phi(\theta,y)\ \ \ \mbox{ for } -\tau\leqslant \theta\leqslant 0. \end{align*} Then the free boundary problem \eqref{p} becomes \begin{equation}\label{lin1} \left\{ \begin{array}{ll}
w_t -A(g,h,y)w_{yy} + B(g,h,y)w_{y}=f(w(t-\tau,y))- d w, & y\in(l_1, l_2),\ t>0,\\
w(t,l_i)=0, & t>0,\ i=1, 2,\\ w(\theta,y) =\phi(\theta,y),& y\in[l_1,l_2],\ \theta\in[-\tau,0], \end{array} \right. \end{equation} and \begin{equation}\label{lghin1} g'(t)=-\mu\, w_y(t,l_1), \ \ h'(t) = -\mu w_{y}(t,l_2),\ \ \ t>0, \end{equation} with $f(w(t-\tau,y))=f(u(t-\tau,y))$ and $A(g,h,y)=[1+\xi_1'(y)(h(t)-l_2)+\xi_2'(y)(g(t)-l_1)]^{-2}$, \[ B(g,h,y)=[\xi_1''(y)(h(t)-l_2)+\xi_2''(y)(g(t)-l_1)]A(g,h,y)^{\frac {3}{2}}-[\xi_1(y)h'(t)+\xi_2(y)g'(t)]A(g,h,y)^{\frac {1}{2}}. \]
Denote $ h_{1}=-\mu (u_{0})_y(0,l_2)$, and $h_{2}=\mu (u_{0})_y(0,l_1)$. For $ 0<T\leqslant \min\big\{\frac{h_{0}}{16(1+ h_{1} +h_{2})},\ \tau\big\}$, we define $\Omega_{T}:=[0,T]\times[l_1,l_2]$, \begin{align*}
&\mathcal{D}^{h}_{T}=\{h\in C^{1}([0,T]):\ h(0)=l_2,\ h'(0)=h_{1},\ \| h'-h_{1}\|_{C([0,T])} \leqslant 1\},\\
&\mathcal{D}^{g}_{T}=\{g\in C^{1}([0,T]):\ g(0)=l_1,\ g'(0)=-h_{2},\ \| g'+h_{2}\|_{C([0,T])} \leqslant 1\}. \end{align*} Clearly, $\mathcal{D}:=\mathcal{D}^{g}_{T}\times\mathcal{D}^{h}_{T}$ is a bounded and closed convex set of $C^1([0,T])\times C^1([0,T])$.
Noting that the restriction on $T$, it is easy to see that the transformation $(t,y)\rightarrow(t,x)$ is well defined. By a similar argument as in \cite{W}, applying standard $L^p$ theory and the Sobolev embedding theorem, we can deduce that for any given $(g,h)\in \mathcal{D}$, problem \eqref{lin1} admits a unique $w(t,y;g,h)\in W^{1,2}_p(\Omega_{T})\hookrightarrow C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})$, which satisfies \begin{equation}\label{eq1}
\|w\| _{W^{1,2}_p(\Omega_{T})}+\|w\| _{C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})}\leqslant C_{1}, \end{equation}
where $p>1$ and $C_{1}$ is a constant dependent on $g(\theta)$, $h(\theta)$, $\alpha$, $p$ and $\| \phi\|_{C^{1,2}([-\tau,0]\times[g(\theta),h(\theta)])}$.
Defining $\hat{h}$ and $\hat{g}$ by $\hat{h}(t)=l_2-\int_0^t \mu w_{y}(s, l_2)ds$ and $\hat{g}(t)=l_1-\int_0^t \mu w_{y}(s, l_1)ds$, respectively, then we have \[ \hat{h}'(t)=-\mu w_{y}(t, l_2),\ \hat{h}(0)=l_2,\ \hat{h}'(0)=-\mu w_{y}(0, l_2)=h_1, \] and thus $\hat{h}'\in C^{\frac{\alpha}{2}}([0,T])$, which satisfies \begin{equation}\label{eq2}
\|\hat{h}'\| _{C^{\frac{\alpha}{2}}([0,T])}\leqslant \mu C_{1}=:C_{2}. \end{equation} Similarly $\hat{g}'\in C^{\frac{\alpha}{2}}([0,T])$, which satisfies \begin{equation}\label{eq3}
\|\hat{g}'\| _{C^{\frac{\alpha}{2}}([0,T])}\leqslant \mu C_{1}=:C_{2}. \end{equation}
$ Step\ 2$. For any given triple $(g,h)\in \mathcal{D}$, we define an operator $ \mathcal{F}$ by \[
\mathcal{F}(g,h)=(\hat{g},\hat{h}). \] Clearly, $\mathcal{F}$ is continuous in $\mathcal{D}$, and $(g,h)\in \mathcal{D}$ is a fixed point of $\mathcal{F}$ if and only if $(w,g,h)$ solves \eqref{lin1} and \eqref{lghin1}. We will show that if $ T>0$ is small enough, then $\mathcal{F}$ has a fixed point by using the Schauder fixed point theorem.
Firstly, it follows from \eqref{eq2} and \eqref{eq3} that \[
\|\hat{h}'-h_{1}\|_{C([0,T])}\leqslant C_{2}T^{\frac{\alpha}{2}},\ \|\hat{g}'+h_{2}\|_{C([0,T])}\leqslant C_{2}T^{\frac{\alpha}{2}}. \] Thus if we choose $T\leqslant \min\big\{\frac{h_{0}}{16(1+ h_{1} +h_{2})},\ \tau, \ C^{-\frac{2}{\alpha}}_{2}\big\}$, then $\mathcal{F}$ maps $\mathcal{D}$ into itself. Consequently, $\mathcal{F}$ has at least one fixed point by using the Schauder fixed point theorem, which implies that \eqref{lin1} and \eqref{lghin1} have at least one solution $(w,g,h)$ defined in $[0,T]$. Moreover, by the Schauder estimates, we have additional regularity for $(w, g, h)$ as a solution of \eqref{lin1} and \eqref{lghin1}, namely, \[ (w,g,h)\in C^{1+\alpha/2,2+\alpha}((0,T]\times[l_1,l_2])\times C^{1+\alpha/2}((0,T]) \times C^{1+\alpha/2}((0,T]) \] and for any given $0<\varepsilon<T$, there holds \[
\|w\|_{C^{1+\alpha/2,2+\alpha}([\varepsilon,T]\times[l_1,l_2])}\leqslant C_3, \]
where $C_3$ is a constant dependent on $\varepsilon$, $ g(\theta)$, $h(\theta)$, $\alpha$ and $\| \phi\|_{C^{1,2}}$. Thus we deduce a local classical solution $(u,g,h)$ of \eqref{p} by $(w,g,h)$, and $u\in C^{1+\alpha/2,2+\alpha}((0,T]\times[g(t),h(t)])$ satisfies \[
\|u\|_{C^{1+\alpha/2,2+\alpha}([\varepsilon,T]\times[g(t),h(t)])}\leqslant C_3. \]
$ Step\ 3$. We will prove the uniqueness of solutions of \eqref{p}. Let $(u_i,g_i,h_i)$, $i=1,2$, be two solutions of \eqref{p} and set \[ w_i(t,y):=u_i(t,y+\xi_{1}(y)(h_i(t)-l_2)+\xi_{2}(y)(g_i(t)-l_1)). \] Then it follows from \eqref{eq1}, \eqref{eq2} and \eqref{eq3} that \[
\|w_i\| _{W^{1,2}_p(\Omega_{T})}+\|w_i\| _{C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})}\leqslant C_{1},\ \
\|h'_i\| _{C^{\frac{\alpha}{2}}([0,T])}\leqslant C_{2},
\ \ \|g'_i\| _{C^{\frac{\alpha}{2}}([0,T])}\leqslant C_{2}. \] Set \[ \tilde{w}(t,y):=w_{1}(t,y)-w_{2}(t,y), \ \ \tilde{g}(t):=g_1(t)-g_2(t),\ \mbox{ and }\ \tilde{h}(t):=h_1(t)-h_2(t), \] then we find that $\tilde{w}(t,y)$ satisfies that \begin{equation}\label{p1} \left\{ \begin{array}{ll}
\tilde{w}_{t} -A_{2}(t,y)\tilde{w}_{yy} + B_{2}(t,y)\tilde{w}_{y}=\tilde{f}(t,y), & y\in(l_1, l_2),\ t\in(0, T),\\
\tilde{w}(t,l_1)=\tilde{w}(t,l_2)= 0, & t\in(0, T),\\ \tilde{w}(\theta,y) =0 ,& y\in[l_1, l_2],\ \theta\in[-\tau, 0], \end{array} \right. \end{equation} where \[ \tilde{f}(t,y)=(A_{1}-A_{2})(w_{1})_{yy}-(B_{1}-B_{2})(w _{1})_{y}+f(w_1(t-\tau,y))-f(w_2(t-\tau,y))- d \tilde{w}, \] and $A_i$ and $B_i$ are the coefficients of problem \eqref{lin1} with $(w_i,g_i,h_i)$ instead of $(w,g,h)$.
Recalling that $T\leqslant \tau$, then $f(w_1(t-\tau,y))-f(w_2(t-\tau,y))=0$ for all $(t,y)\in \Omega_{T}$, thus \[ \tilde{f}(t,y)=(A_{1}-A_{2})(w_{1})_{yy}-(B_{1}-B_{2})(w_{1})_{y}- d \tilde{w}. \] Thanks to this, we can apply the $L^p$ estimates for parabolic equations to deduce that \begin{equation}\label{sobem}
\|\tilde{w}\|_{W^{1,2}_p(\Omega_{T})}\leqslant C_4 (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}) \end{equation} with $C_4$ depending on $C_1$ and $C_2$. By a similar argument as in \cite{W}, we obtain that \[
\| \tilde{w}\|_{C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})}\leqslant C \|\tilde{w}\|_{W^{1,2}_p(\Omega_{T})} \] for some positive constant $C$ independent of $T^{-1}$. Thus \begin{equation}\label{sobem}
\| \tilde{w}\|_{C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})}\leqslant C C_4 (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}) \end{equation} Since $\tilde{h}'(0)=h'_{1}(0)-h'_{2}(0)=0$, then \[
\| \tilde{h}'\|_{C^{\frac{\alpha}{2}}([0,T])}=\mu \| \tilde{w}_{y}\|_{C^{\frac{\alpha}{2},0}(\Omega_{T})}\leqslant
\mu \| \tilde{w}\|_{C^{\frac{1+\alpha}{2},{1+\alpha}}(\Omega_{T})}. \] This, together with \eqref{sobem}, implies that \[
\| \tilde{h}\|_{C^1([0,T])}\leqslant 2T^{\frac{\alpha}{2}} \| \tilde{h}'\|_{C^{\frac{\alpha}{2}}([0,T])}\leqslant C_5T^{\frac{\alpha}{2}} (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}), \] where $C_5=2\mu C C_4$. Similarly, we have
\[
\| \tilde{g}\|_{C^1([0,T])}\leqslant C_5T^{\frac{\alpha}{2}} (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}), \] As a consequence, we deduce that \[
\| \tilde{g}\|_{C^1([0,T])}\|+\| \tilde{h}\|_{C^1([0,T])}
\leqslant 2C_5 T^{\frac{\alpha}{2}} (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}). \] Hence for \[ T:=\min\Big\{\frac{h_{0}}{ 16(1+h_{1}+h_{2})},\ \tau,\ C^{-\frac{2}{\alpha}}_{2},\ (4C_5)^{-\frac{2}{\alpha}}\Big\}, \] we have \[
\| \tilde{g}\|_{C^1([0,T])}\|+\| \tilde{h}\|_{C^1([0,T])}
\leqslant \frac{1}{2} (\| \tilde{g}\|_{C^1([0,T])}+\| \tilde{h}\|_{C^1([0,T])}). \] This shows that $\tilde{g}\equiv 0 \equiv \tilde{h}$ for $0\leqslant t\leqslant T$, thus $\tilde{w}\equiv0$ in $[0,T]\times[l_1,l_2]$. Consequently, the uniqueness of solution of \eqref{p} is established, which ends the proof of this theorem. \end{proof}
\begin{lem}\label{lem:global} Assume that {\bf(H)} holds. Then every positive solution $(u, g, h)$ of problem \eqref{p} exists and is unique for all $t\in(0, \infty)$. \end{lem}
\begin{proof} Let $[0, T_{max})$ be the maximal time interval in which the solution exists. In view of Theorem \ref{thm:local}, it remains to show that $T_{max}=\infty$. We proceed by a contradiction argument and assume that $T_{max}<\infty$. Thanks to the choice of the initial data, the comparison principle implies that $u(t,x)\leqslant u^*$ for $(t,x)\in(0,T_{max})\times[g(t),h(t)]$. Construct the auxiliary function \[ \bar{u}(t,x)=u^*\big[2M(h(t)-x)-M^{2}(h(t)-x)^{2}\big],\ \ \ t\in[-\tau,T_{max}),\ x\in[h(t)-M^{-1},h(t)] \] where \[
M:=\max\Big\{\sqrt{d},\ \frac{2}{h(-\tau)-g(-\tau)},\ \frac{4}{3u^*}\max_{-\tau\leqslant \theta\leqslant 0}\|\phi(\theta,\cdot)\|_{C^1([g(\theta),h(\theta)])}\Big\}. \] It follows the proof of \cite[Lemma 2.2]{DuLin} to prove that there is a constant $C_0$ independent on $T_{max}$ such that $h'(t)\leqslant C_0$ for $t\in (0, T_{max})$. The proof for $-g'(t)\leqslant C_0$ for $t\in (0, T_{max})$ is parallel.
Let us now fix $\epsilon\in(0,T_{max})$. Similar to the proof of Theorem \ref{thm:local}, by standard $L^p$ estimate, the Sobolev embedding theorem and the H\"{o}lder estimates for parabolic equation, we can find $C_1>0$ depending only on $\epsilon$, $T_{max}$,
$u^*$, $ h_0$, $\| \phi\|_{C^{1,2}([-\tau,0]\times[g(\theta),h(\theta)])}$ and $C_0$ such that \[
||u||_{C^{1+\alpha/2,2+\alpha}([\varepsilon,T_{max}]\times[g(t), h(t)])}\leqslant C_1. \] This implies that $(u,g,h)$ exists on $[0,T_{max}]$. Choosing $t_n\in(0,T_{max})$ with $t_n\nearrow T_{max}$, and regarding $(u(t_n-\theta, x), h), g(t_n-\theta), h(t_n-\theta))$ for $\theta\in[0,\tau]$ as the initial function, it then follows from the proof of Theorem \ref{thm:local} that there exists $s_0>0$ depending on $C_0$, $C_1$ and $u^*$ independent of $n$ such that problem \eqref{p} has a unique solution $(u, g, h)$ in $[t_n, t_n+s_0]$. This yields that the solution $(u,g,h)$ of \eqref{p} can be extended uniquely to $[0,t_n+s_0)$. Hence $t_n+s_0>T_{max}$ when $n$ is large. But this contradicts the assumption, which ends the proof of this lemma. \end{proof}
\noindent
{\bf Proof of Theorem \ref{wellposedness}:} Combining Theorem \ref{thm:local} and Lemma \ref{lem:global}, we complete the proof.{
$\Box$} \subsection{Comparison Principle}\label{subsec:cp} In this subsection, we establish the comparison principle, which will be used in the rest of this paper. Let us
start with the following result. \begin{lem} \label{lem:comp1} Suppose that {\bf{(H)}} holds, $T\in (0,\infty)$, $\overline g,\ \overline h\in C^1([-\tau,T])$, $\overline u\in C(\overline D_T) \cap C^{1,2}(D_T)$ satisfies $\overline u \leqslant u^*$ in $\overline D_T$ with $D_T=\{(t,x)\in\mathbb{R}^2: -\tau<t\leqslant T,\ \overline g(t)<x<\overline h(t)\}$, and
\begin{eqnarray*} \left\{ \begin{array}{lll} \overline u_{t} \geqslant \overline u_{xx} -d \overline u+f(\overline u(t-\tau, x)),\; & 0<t \leqslant T,\ \overline g(t)<x<\overline h(t), \\ \overline u= 0,\quad \overline g'(t)\leqslant -\mu \overline u_x,\quad & 0<t \leqslant T, \ x=\overline g(t),\\ \overline u= 0,\quad \overline h'(t)\geqslant -\mu \overline u_x,\quad &0<t \leqslant T, \ x=\overline h(t). \end{array} \right. \end{eqnarray*} If $[g(\theta), h(\theta)]\subseteq [\overline g(\theta), \overline h(\theta)]$ for $\theta\in[-\tau,0]$ and $\overline u(\theta,x)\in C^{1,2}([-\tau,0]\times[\overline g(\theta), \overline h(\theta)])$ satisfies \[ \phi(\theta,x)\leqslant \overline u(\theta,x) \leqslant u^*\ \ \mbox{ in } [-\tau,0]\times[g(\theta),h(\theta)], \] then the solution $(u,g, h)$ of problem \eqref{p} satisfies $g(t)\geqslant \overline g(t)$, $h(t)\leqslant \overline h(t)$ in $(0,T]$, and \begin{align*} u(t,x)\leqslant \overline u(t,x)\ \ \mbox{ for } (t,x)\in (0, T]\times(g(t), h(t)). \end{align*} \end{lem} \begin{proof} We integrate the ideas of \cite[Lemma 5.7]{DuLin} and \cite[Corollary 5]{MS} to deal with free boundary and time delay.
Firstly, for small $\epsilon>0$, let $(u_\epsilon,g_\epsilon,h_\epsilon)$ denote the unique solution of \eqref{p} with $g(\theta)$ and $h(\theta)$ replaced by $g_\epsilon(\theta):=g(\theta)(1-\epsilon)$ and $h_\epsilon(\theta):=h(\theta)(1-\epsilon)$ for $\theta\in[-\tau,0]$, respectively, with $\mu$ replaced by $\mu_\epsilon:=\mu(1-\epsilon)$, and with $\phi(\theta,x)$ replaced by some $\phi_\epsilon(\theta,x)\in C^{1,2}([-\tau,0]\times[g_\epsilon(\theta),h_\epsilon(\theta)])$, satisfying \[ 0<\phi_\epsilon(\theta,x)\leqslant \phi(\theta,x),\ \ \ \phi_\epsilon(\theta,g_\epsilon(\theta))=\phi_\epsilon(\theta,h_\epsilon(\theta))=0 \ \ \mbox{ for }\ \theta\in[-\tau,0],\ x\in[g_\epsilon(\theta),h_\epsilon(\theta)], \] and for any fixed $\theta\in[-\tau,0]$ as $\epsilon\to 0$, $\phi_\epsilon (\theta,x )\to \phi(\theta,x)$ in the $C^2([g(\theta),h(\theta)])$ norm.
We claim that $h_\epsilon(t)<\overline h(t)$, $g_\epsilon(t)> \overline g(t)$ and $u_\epsilon(t,x)<\overline u(t,x)$ for all $t\in[0,T]$ and $x\in[g_\epsilon(t),h_\epsilon(t)]$. Obviously, this is true for all small $t>0$. Now, let us use an indirect argument and suppose that the claim does not hold, then there exists a first $t^*\in(0,T]$ such that \begin{align*}
u_\epsilon(t,x)< \overline{u}(t,x)\ \ \mbox{ for }\ t\in [0, t^*),\ x\in [g_\epsilon(t), h_\epsilon(t)]\subset (\overline {g}(t), \overline {h}(t)), \end{align*} and there is some $x^*\in[g_\epsilon(t^*),h_\epsilon(t^*)]$ such that $u_\epsilon(t^*,x^*)=\overline{u}(t^*,x^*)$.
Later, let us compare $u_\epsilon$ and $\overline u$ over the region \[ \Omega_{t^*}:=\{(t,x)\in\mathbb{R}^2: 0<t\leqslant t^*,\ g_\epsilon(t)< x < h_\epsilon(t)\}. \] An direct computation shows that for $(t,x)\in \Omega_{t^*}$, \[ (\overline{u}-u_\epsilon)_t- (\overline{u}-u_\epsilon)_{xx}+d (\overline{u}-u_\epsilon)\geqslant f(\overline{u}(t-\tau,x))-f(u_\epsilon(t-\tau,x))\geqslant 0, \] it then follows from the strong maximum principle that \begin{equation}\label{mao1} u_\epsilon(t,x)<\overline{u}(t,x)\ \ \mbox{ in } \Omega_{t^*}. \end{equation} Thus either $x^*=h_\epsilon(t^*)$ or $x^*=g_\epsilon(t^*)$. Without loss of generality we may assume that $x^*=h_\epsilon(t^*)$, then $\overline{u}(t^*,h_\epsilon(t^*))=u_\epsilon(t^*,h_\epsilon(t^*))=0$. This, together with \eqref{mao1}, implies that $\overline{u}_x(t^*,h_\epsilon(t^*))\leqslant (u_\epsilon)_x(t^*,h_\epsilon(t^*))$, from which we obtain that \begin{equation}\label{uhq} h'_\epsilon(t^*)=-\mu_\epsilon(u_\epsilon)_x(t^*,h_\epsilon(t^*))<-\mu \overline{u}_x(t^*,h_\epsilon(t^*))= \overline{h}'(t^*). \end{equation} As $h_\epsilon(t)< \overline{h}(t)$ for $t\in[0,t^*)$ and $h_\epsilon(t^*)=\overline{h}(t^*)$, then $h'_\epsilon(t^*)\geqslant \overline{h}'(t^*)$, which contradicts \eqref{uhq}. This proves our claim.
Finally, thanks to the unique solution of \eqref{p} depending continuously on the parameters in \eqref{p}, as $\epsilon \to 0$, $(u_\epsilon,g_\epsilon,h_\epsilon)$ converges to $(u,g,h)$, the unique of solution of \eqref{p}. The desired result then follows by letting $\epsilon\to 0$ in the inequalities $u_\epsilon< \overline{u},\ g_\epsilon> \overline{g}$ and $h_\epsilon< \overline{h}$. \end{proof}
By slightly modifying the proof of Lemma \ref{lem:comp1}, we obtain a variant of Lemma \ref{lem:comp1}. \begin{lem} \label{lem:comp2} Suppose that {\bf {(H)}} holds, $T\in (0,\infty)$, $\overline g,\, \overline h\in C^1([-\tau,T])$, $\overline u\in C(\overline D_T)\cap C^{1,2}(D_T)$ satisfies $\overline u \leqslant u^*$ in $\overline D_T$ with $D_T=\{(t,x)\in\mathbb{R}^2: -\tau<t\leqslant T,\ \overline g(t)<x<\overline h(t)\}$, and
\begin{eqnarray*} \left\{ \begin{array}{lll} \overline u_{t} \geqslant \overline u_{xx} -d \overline u+f(\overline u(t-\tau, x)),\; &0<t \leqslant T,\ \overline g(t)<x<\overline h(t), \\ \overline u\geqslant u, &0<t \leqslant T, \ x= \overline g(t),\\ \overline u= 0,\quad \overline h'(t)\geqslant -\mu \overline u_x,\quad &0<t \leqslant T, \ x=\overline h(t), \end{array} \right. \end{eqnarray*} with $\overline g(t)\geqslant g(t)$ in $[0,T]$, $h(\theta)\leqslant \overline h(\theta)$, $\phi(\theta,x)\leqslant \overline u(\theta,x)$ for $\theta\in[-\tau,0]$ and $x\in[\overline g(\theta),h(\theta)]$, where $(u,g, h)$ is a solution to \eqref{p}. Then \[ \mbox{ $h(t)\leqslant \overline h(t)$ in $(0, T]$,\quad $u(x,t)\leqslant \overline u(x,t)$ for $(t,x)\in (0, T]\times(g(t),h(t))$.} \] \end{lem}
\begin{remark} \label{rem5.8}\rm The function $\overline u$, or the triple $(\overline u,\overline g,\overline h)$, in Lemmas \ref{lem:comp1} and \ref{lem:comp2} is often called a supersolution to \eqref{p}. A subsolution can be defined analogously by reversing all the inequalities. There is a symmetric version of Lemma~\ref{lem:comp2}, where the conditions on the left and right boundaries are interchanged. We also have corresponding comparison results for lower solutions in each case. \end{remark}
\section{Semi-waves}\label{subsec:semiwave}
This section is devoted to proving the existence and uniqueness of a semi-wave $q(z)$ of \eqref{sw11}, which will be used to construct some suitable sub- and supersolutions to study the asymptotic profiles of spreading solutions of \eqref{p}. Let us consider the following nonlocal elliptic problem \begin{equation}\label{semiwave} \left\{
\begin{array}{ll}
q'' - cq'-d q+ f( q(z-c\tau))=0, & z>0,\\
q(z)=0, & z\leqslant 0,\\
\end{array}
\right. \end{equation} where $c\geqslant 0$ is a constant.
If $z$ is understood as the time variable, then we may regard problem \eqref{semiwave} as a time-delayed dynamical system in the phase space $C([-c\tau,0],\mathbb{R}^2)$. When $c\tau=0$, the phase space reduces to $\mathbb{R}^2$ and it follows from the phase plane analysis that \eqref{semiwave} admits a unique positive solution $q_0(z)$, which is increasing in $z$ and $q_0(z)\to u^*$ as $z\to \infty$. When $c\tau>0$, the phase space is of infinite dimension and the positivity and boundedness of the unique solution are not clear.
\begin{prop}\label{prop:semiwave} Suppose {\bf{(H)}} holds. For any given constant $c> 0$, problem \eqref{semiwave} has a maximal nonnegative solution $q_c$. Moreover, either $q_c(z)\equiv 0$ or $q_c(z)> 0$ in $(0,\infty)$. Furthermore, if $q_c>0$, then it is the unique positive solution of \eqref{semiwave}, $q_c'(z)>0$ in $(0,\infty)$ and $q_c(z)\to u^*$ as $z\to\infty$, in addition, for any given constant $c_1<c$, one has $q_c(z)<q_{c_1}(z)$ for $z\in(0,\infty)$, and $q'_c(0)<q'_{c_1}(0)$. \end{prop} \begin{proof} We divide the proof into four steps.
$ Step \ 1$. Problem \eqref{semiwave} always has a maximal nonnegative solution $\overline{q}$ and it satisfies \[ \overline{q}\leqslant u^*\ \ \mbox{ for } z\in[0,\infty). \] Clearly, $0$ is a nonnegative solution of \eqref{semiwave}. For any $l>0$, consider the following problem: \begin{equation}\label{semiwavel} \left\{
\begin{array}{ll}
w'' - cw'-d w+ f( w(z-c\tau))=0, & 0<z<l,\\ w(l)=u^*,\ \ \ w(z)=0, \ \ z\leqslant 0.
\end{array}
\right. \end{equation} It is well known problem \eqref{semiwavel} admits a unique solution $w^l(z)>0$ for $z\in(0,l]$. Applying the maximal principle, we can deduce that $w^l(z)\leqslant u^*$ for $z\in[0,l]$. Moreover, it is easy to check that $w^l(z)$ is decreasing in $l>0$ and increasing in $z\in[0,l]$ and \[ w^l(z)\to W(z)\ \ \mbox{ as } l\to\infty, \] where $W(z)$ is a nonnegative solution of problem \eqref{semiwave} and it satisfies $W(z)\leqslant u^*$ for $z\in[0,\infty)$.
In what follows, we want to prove that $W$ is the maximal nonnegative solution of \eqref{semiwave}. Let $q$ be an arbitrary nonnegative solution of \eqref{semiwave}, then $q(z)\leqslant u^*$ for $z\in[0,\infty)$. If $q\equiv 0$, then $q\leqslant W$. Suppose now $q\geqslant, \not\equiv 0$, then $q>0$ in $(0,\infty)$. Let us show $q(z)\leqslant W(z)$ for $z\in[0,\infty)$.
Firstly, for any fixed $l>0$ we can find $M>0$ large such that $Mw^l(z)\geqslant q(z)$ for $z\in[0,l]$. We claim that the above inequality holds for $M=1$. On the contrary, define \[ M_0:=\inf\{M>0:\ Mw^l(z)\geqslant q(z)\ \ \mbox{ for } z\in[0,l]\}, \] then $M_0>1$ and $M_0w^l(z)\geqslant,\not\equiv q(z)$ for $z\in[0,l]$. Thanks to the monotonicity of $w^l(z)$ in $z\in[0,l]$, then there is $z_0\in(0,l)$ such that $M_0w^l(z_0)=u^*$ and $M_0w^l(z)<u^*$ for $z\in[0,z_0)$. It is easy to check that $q(z_0)<u^*$. Then the strong maximal principle yields that $M_0 (w^l)'(0)>q'(0)$ and $M_0w^l(z)>q(z)$ for $z\in(0,z_0]$. Thus we can find a constant $0<\epsilon\ll1$ such that \begin{equation}\label{Mqw} M_1:=M_0(1+\epsilon)^{-1}>1, \ \ M_1w^l(z)>q(z)\ \ \mbox{ for } z\in(0,z_0], \end{equation} and$ M_1w^l(z_0+\tilde{z})> u^*$ for $\tilde{z}=\min\{c\tau,\ l-z_0\}$. So there is $z_1\in(0,\tilde{z}]$ such that $M_1w^l(z_0+z_1)= u^*$ and $M_1w^l(z_0+z)> u^*$ for $z\in (z_1,l-z_0]$.
Later, we want to prove that $M_1 w^l(z)>q(z)$ for all $z\in(z_0,l]$. Combining the definition of $z_1$, we only need to prove $M_1 w^l(z)\geqslant q(z)$ for all $z\in(z_0,z_0+z_1]$. Since $M_1 w^l(z)\geqslant q(z)$ for $z=z_0+z_1$ and $z=z_0$, and for $z\in(z_0,z_0+z_1)$, \begin{eqnarray*} && \big(M_1w^l-q\big)''-c\big(M_1w^l-q\big)'-d \big(M_1 w^l-q\big)\\ &=& f(q(z-c\tau))-M_1f\big( w^l(z-c\tau)\big)\\ & \leqslant& f(q(z-c\tau))-f\big(M_1 w^l(z-c\tau)\big) \leqslant 0, \end{eqnarray*} where the monotonicity of $f(v)$ in $v\in[0,u^*]$ and the fact where $M_1w^l(z-c\tau)\geqslant q(z-c\tau)$ for $z\leqslant z_0+z_1$ are used. The comparison principle yields that $M_1 w^l(z)\geqslant q(z)$ for all $z\in[z_0,z_0+z_1]$. This, together with the definition of $z_1$ and \eqref{Mqw}, yields that $M_1 w^l(z)\geqslant q(z)$ for all $z\in(0,l]$, which contradicts the definition of $M_0$. Thus we have proved that $w^l(z)\geqslant q(z)$ for $z\in[0,l]$.
Finally, letting $l\to\infty$, we deduce that \[ W(z)\geqslant q(z)\ \ \mbox{ for } z\in[0,\infty), \] as we wanted. Thus Step 1 is proved.
$ Step \ 2$. For any $c\geqslant 0$, if $q$ is a positive solution of \eqref{semiwave}, then $q_+'(0)>0$, $q'(z)>0$ for $z\in(0,\infty)$, and $q(z)\to u^*$ as $z\to\infty$.
Since $q>0$ for $z>0$, then the Hopf lemma can be used to deduce $q_+'(0)>0$, it follows that $q'(z)>0$ for all small $z>0$. Setting \[ \gamma^*:=\sup\{\gamma>0:\ q(2\gamma-z)>q(z)\ \mbox{ for } z\in[0,\gamma),\ \ q'(z)>0\ \mbox{ for } z\in(0,\gamma]\}. \] In the following, we shall show $\gamma^*=\infty$. Suppose by way of contradiction that $\gamma^*\in(0,\infty)$, then \[ q(2\gamma^*-z)\geqslant q(z), \ \mbox{ and }\ q'(z)\geqslant 0\ \ \mbox{ for } z\in[0,\gamma^*]. \] Define $\tilde{q}(z)=q(2\gamma^*-z)$ for $z\in[\gamma^*,2\gamma^*]$, then \[
\tilde{q}''-c\tilde{q}'-d\tilde{q}+f(\tilde{q}(z-c\tau))=-2cq_\xi,\ \ \ \xi=2\gamma^*-z\in[0,\gamma^*]. \] Let us set \[ Q(z;\gamma^*)=Q(z)=\tilde{q}(z)-q(z)=q(\xi)-q(2\gamma^*-\xi). \] Then $Q\leqslant 0$ for $z\in[\gamma^*,2\gamma^*]$ and it satisfies \begin{equation}\label{Qqq} \left\{
\begin{array}{ll}
Q'' - cQ'-d Q=f(q(z-c\tau))-f(\tilde{q}(z-c\tau))-2cq_\xi\leqslant 0, & \gamma^*\leqslant z\leqslant 2\gamma^*,\\ Q(\gamma^*)=0,\ \ \ Q(2\gamma^*)=-q(2\gamma^*)<0.
\end{array}
\right. \end{equation} The strong maximal principle and the Hopf lemma imply that \[ Q(z)<0,\ \ \ z\in(\gamma^*,2\gamma^*],\ \ \ Q'(\gamma^*)<0. \] It follows the continuity that for all small $\varepsilon\geqslant 0$, \[ Q'(\gamma^*+\varepsilon;\gamma^*+\varepsilon)<0,\ \ \ Q(z;\gamma^*+\varepsilon)<0\ \ \mbox{ for } z\in(\gamma^*+\varepsilon,2\gamma^*+2\varepsilon], \] which implies that $q(2\gamma^*+2\varepsilon-\xi)>q(\xi)$ for $\xi\in[0,\gamma^*+\varepsilon)$. Moreover, since $Q'(\gamma^*+\varepsilon;\gamma^*+\varepsilon)=-2q'(\gamma^*+\varepsilon)$, it then follows that $q'(\gamma^*+\varepsilon)>0$. But these facts contradict the definition of $\gamma^*$. Thus the monotonicity of positive solutions of \eqref{semiwave} is established.
Next, we consider the asymptotic behavior of positive solution $q$ of \eqref{semiwave}. The monotonicity of $q$ implies that there is a constant $a>0$ such that $\lim_{z\to\infty} q(z)=a$. We claim that $a=u^*$. For any sequence $\{z_n\}$ with $z_n\to\infty$ as $n\to\infty$, define $q_n(z)=q(z+z_n)$. Then $q_n$ solves the same equation as $q$ but over $(-z_n,\infty)$. Since $q_n\leqslant u^*$, it follows that there is a subsequence of $\{q_n\}$ (still denoted by $\{q_n\}$) such that $q_n\to \hat{q}$ locally in $C^2(\mathbb{R})$ as $n\to\infty$, and $\hat{q}$ is a solution of \[
v''-cv'-d v+f(v(z-c\tau))=0,\ \ \ z\in\mathbb{R}. \]
On the other hand, it follows from $\lim_{z\to\infty}q(z)=a$ that $ \hat{q}\equiv a$, which implies that $a=u^*$, as we wanted. Thus this completes the proof of Step 2.
$ Step \ 3$. We show that problem \eqref{semiwave} has at most one positive solution.
Suppose problem \eqref{semiwave} has two positive solutions $q_1$ and $q_2$, then $0<q_i<u^*$ in $(0,\infty)$, and $q_i(z)\to u^*$ as $z\to\infty$ for $i=1,\ 2$. Define \[ \rho^*:=\inf\left\{\frac{q_1(z)}{q_2(z)}:z>0\right\}. \] From Step 2 we have $(q_i)_+'(0)>0$, $i=1, 2$. Then by L'H\^{o}pital's rule we obtain $\lim_{z\downarrow 0}\frac{q_1(z)}{q_2(z)}>0$, which together with $\lim_{z\to+\infty}\frac{q_1(z)}{q_2(z)}=1$ implies that $\rho^*\in (0,1]$. Next we show $\rho^*=1$. Indeed, assume for the sake of contraction that $\rho^*\in (0,1)$. Define \[ w(z):=q_1(z)-\rho^*q_2(z). \] Then $w(z)\geqslant 0$ for $z\geqslant 0$, $w(0)=0$, $w(+\infty)=(1-\rho^*)u^*>0$ and \[ w''-cw'-dw=-f(q_1(z-c\tau))+\rho^*f(q_2(z-c\tau))\leqslant 0, \] where the sub-linearity and monotonicity of $f(z)$ for $z\in(0,u^*)$ are used. By Hopf's lemma, we see that $0<w'(0)=(q_1)_+'(0)-\rho^* (q_2)_+'(0)$, which implies that $\lim_{z\downarrow 0}\frac{q_1(z)}{q_2(z)}>\rho^*$. Thus, in view of the definition of $\rho^*$, we have an $z_0\in (0,+\infty)$ such that $w(z_0)=0$. By the elliptic strong maximum principle, we infer that $w(z)\equiv 0$ for $z>0$, a contradiction to $w(+\infty)>0$. Therefore, $\rho^*=1$, and hence, $q_1(z)\geqslant q_2(z)$. Changing the role of $q_1$ and $q_2$ and repeating the above arguments, we obtain $q_2(z)\geqslant q_1(z)$. The uniqueness is proved.
$ Step \ 4$. Let us consider the monotonicity of positive solutions in $c$.
Assume that $q_c$ is a positive solution of \eqref{semiwave}. Choose $c_1<c$ and let $q_{c_1}$ be the maximal nonnegative solution of \eqref{semiwave} with $c=c_1$. Since $u^*$ is a supersolution of \eqref{semiwave}, and by Step 2 we know that $q_c$ is a subsolution of \eqref{semiwave} with $c=c_1$, in view of the uniqueness of positive solution of this problem, then we see that $q_{c_1}(z)\geqslant q_c(z)$ for $z\in[0,\infty)$. It thus follows from the maximum principle and the Hopf lemma that \[ q_{c_1}(z)>q_c(z)\ \ \mbox{ for } z\in(0,\infty), \ \ \mbox{ and }\ \ q'_{c_1}(0)>q'_c(0). \]
The proof of this proposition is complete now. \end{proof}
Next we give a necessary and sufficient condition for the existence of a positive solution of \eqref{semiwave}. For this purpose, we need the following property on the distribution of complex solutions to a transcendental equation. \begin{lem}\label{lem:eigen} Let $c>0$ and $\tau>0$. Define \begin{equation} \Delta_c(\lambda,\tau)=\lambda^2-c\lambda-d+f'(0)e^{-\lambda c\tau}. \end{equation} Then there exists $c_0(\tau)\in (0,2\sqrt{f'(0)-d})$ such that the following statements hold: \begin{enumerate} \item[(i)] $\Delta_c(\lambda,\tau)=0$ has a positive solution if and only if $c\geqslant c_0(\tau)$; \item[(ii)] $\Delta_c(\lambda,\tau)=0$ has a complex solution in the domain \begin{equation}\label{def-Omega} \Omega:=\left\{\lambda\in \mathbb{C}: Re \lambda>0, Im \lambda \in \left(0,\frac{\pi}{c\tau}\right) \right\} \end{equation} provided that $c\in (0,c_0(\tau))$. \end{enumerate} \end{lem} Before the proof, we note that if $\tau=0$ then $\Delta_c(\lambda,\tau)=0$ reduces to a polynomial equation of order $2$. It admits at least one positive solution if and only if $c\geqslant 2\sqrt{f'(0)-d}$ and exactly a pair of complex eigenvalues in $\Omega$ when $c\in (0,2\sqrt{f'(0)-d})$.
\begin{proof} (i) Note that $\Delta_c(\lambda,\tau)$ is convex in $\lambda$, decreasing in $c>0$ when $\lambda>0$, $\Delta_0(\lambda,\tau)>0$ and $\Delta_c(\lambda,\tau)=0$ is negative for some $\lambda>0$ when $c$ is sufficiently large. Therefore, such $c_0(\tau)$ exists.
(ii) We employ a continuation method with $\tau$ being the parameter. From the proof of \cite[Theorem 2.1]{RuanWei2003}, we can infer that the solutions of $\Delta_c(\lambda,\tau)=0$ is continuous in $\tau>0$. We write $\lambda=\alpha(\tau)+i\beta(\tau)$, where $\alpha(\tau)$ and $\beta(\tau)$ are continuous in $\tau>0$. Separating the real and imaginary parts of $\Delta_c(\lambda,\tau)=0$ yields \begin{equation}\label{s1} \begin{cases} F_1(\alpha,\beta,\tau):=\alpha^2-\beta^2-c\alpha-d+f'(0)e^{-c\tau \alpha}\cos c\tau\beta=0\\ F_2(\alpha,\beta,\tau):=2\alpha\beta-c\beta-f'(0)e^{-c\tau \alpha}\sin c\tau\beta=0. \end{cases} \end{equation}
We proceed with four steps.
$ Step \ 1$. If $\tau$ is small enough, then there is a solution in $\Omega$. Indeed, At $\tau=0$, \eqref{s1} admits a solution $(\alpha,\beta)=\left(\frac{c}{2}, \frac{ \sqrt{|c^2-(f'(0)-d)^2}|}{2}\right)$. Note that \begin{equation} \det \left( \begin{matrix} \partial_\alpha F_1 & \partial_\beta F_1\\ \partial_\alpha F_2 & \partial_\beta F_2
\end{matrix}\right)|_{\tau=0 } =\det \left( \begin{matrix} 2\alpha-c &-2\beta\\ 2\beta & 2\alpha+c \end{matrix}\right) >0. \end{equation}
It then follows from the implicit function theorem that for small $\tau$, $\Delta_c(\lambda,\tau)$ admits a complex solution near $\frac{c}{2}+i\frac{ \sqrt{|c^2-(f'(0)-d)^2}|}{2}$, and hence, in the open domain $\Omega$.
$ Step \ 2$. For any $\tau>0$, $\Delta_c(\lambda,\tau)$ admits no solution with $\beta=0$ or $\beta=\frac{\pi}{c\tau}$ when $c\tau>0$. It follows from statement (i) that there is no solution with $\beta=0$ when $c<c_0(\tau)$. If $\beta$ equals $\frac{\pi}{c\tau}$, then from the second equation of \eqref{s1} we can infer that $\alpha=\frac{c}{2}$. Substituting $\alpha=\frac{c}{2}$ and $\beta=\frac{\pi}{c\tau}$ into the first equation of \eqref{s1}, we obtain $0=-\frac{1}{4}c^2-\left(\frac{\pi}{c\tau}\right)^2-d-f'(0)e^{-c^2\tau/2}$, a contradiction.
$ Step \ 3$. If a solution $\alpha(\tau)+i\beta(\tau)$ touches pure imaginary axis at some $\tau=\tau^*>0$, then $\alpha'(\tau^*)>0$. We use the implicit function theorem. By direct computations, we have \begin{eqnarray*} &&\det \left( \begin{matrix} \partial_\alpha F_1 & \partial_\beta F_1\\ \partial_\alpha F_2 & \partial_\beta F_2
\end{matrix}\right)|_{\tau=\tau^*}\\ =&&\det \left( \begin{matrix} -c-c\tau f'(0)\cos c\tau\beta &-2\beta-c\tau f'(0)\sin c\tau\beta\\ 2\beta+c\tau f'(0)\sin c\tau\beta& -c-c\tau f'(0)\cos c\tau\beta \end{matrix}\right)\\ =&& [-c-c\tau f'(0)\cos c\tau\beta]^2+ [2\beta+c\tau f'(0)\sin c\tau\beta]^2 \geqslant 0, \end{eqnarray*} where the equality holds if and only if $-c-c\tau f'(0)\cos c\tau\beta=0$ and $2\beta+c\tau f'(0)\sin c\tau\beta=0$. Taking these two relations into \eqref{s1} with $\alpha=0$, we obtain \begin{equation} \begin{cases} -\beta^2-d-\frac{1}{\tau}=0\\ -c\beta+\frac{2\beta}{c\tau}=0, \end{cases} \end{equation} which is not solvable for $\beta$. Therefore, \[ \det \left( \begin{matrix} \partial_\alpha F_1 & \partial_\beta F_1\\ \partial_\alpha F_2 & \partial_\beta F_2
\end{matrix}\right)|_{\tau=\tau^*}>0. \] On the other hand, \[ \left( \begin{matrix} \partial_\tau F_1\\ \partial_\tau F_2
\end{matrix}\right)|_{\tau=\tau^*} =-c\beta f'(0)\left( \begin{matrix} \sin c\tau\beta \\ \cos c\tau\beta \end{matrix}\right) \] Consequently, by the implicit function theorem we have \[ \left( \begin{matrix} \alpha'(\tau^*)\\ \beta'(\tau^*)
\end{matrix}\right)|_{\tau=\tau^*} =-\left( \begin{matrix} \partial_\alpha F_1 & \partial_\beta F_1\\ \partial_\alpha F_2 & \partial_\beta F_2
\end{matrix}\right)^{-1}|_{\tau=\tau^*}\left( \begin{matrix} \partial_\tau F_1\\ \partial_\tau F_2
\end{matrix}\right)|_{\tau=\tau^*}, \] from which we compute to have \begin{equation} \alpha'(\tau^*)=\frac{(2\beta^4+2d\beta^2+c^2)c}{\det \left( \begin{matrix} \partial_\alpha F_1 & \partial_\beta F_1\\ \partial_\alpha F_2 & \partial_\beta F_2
\end{matrix}\right)|_{\tau=\tau^*}}>0. \end{equation}
$ Step \ 4$. Completion of the proof. In Steps 2 and 3, we have verified that the perturbed solution at Step 1 can not escape $\Omega$ continuously as $\tau$ increases from $0$ to $\infty$. Therefore, it always stays in $\Omega$. \end{proof}
Based on the above results, we are ready to give the following necessary and sufficient condition for \eqref{semiwave} to have a unique positive solution.
\begin{prop}\label{prop:qoan1} Suppose {\bf{(H)}} holds. Problem \eqref{semiwave} has a unique positive solution $q\in C^2([0,\infty))$ if and only if $c\in[0,c_0(\tau))$, where $c_0(\tau)$ is given in Lemma \ref{lem:eigen}. \end{prop} \begin{proof} Firstly, let us show that problem \eqref{semiwave} admits a unique positive solution when $c\in[0,c_0(\tau))$. We employ the super- and subsolution method. The case where $c\tau=0$ is trivial and the proof is omitted. Fix $c\in (0,c_0(\tau))$. By Lemma \ref{lem:eigen} we can infer that there exists $\gamma>0$ such that \begin{equation} \widetilde{\Delta}_c(\lambda)=\lambda^2-c\lambda-d+(1-\gamma) f'(0)e^{-\lambda c\tau}=0 \end{equation} has a solution $\lambda=\alpha+i\beta$ in $\Omega$.
{\bf Claim.} The function \begin{equation} \underline{v}(x):= \begin{cases} \delta e^{\alpha x} cos \beta x, & \beta x\in (\frac{3\pi}{2}, \frac{5\pi}{2}),\\ 0, & \text{elsewhere}, \end{cases} \end{equation} is a subsolution provided that $\delta$ is small enough.
Indeed, for $\beta x\in (\frac{3\pi}{2}, \frac{5\pi}{2})$, we have \begin{eqnarray*} L[\underline{v}](x):=&&\underline{v}''(x)-c\underline{v}'(x)-d\underline{v}(x)+f(\underline{v}(x-c\tau))\\ =&& \underline{v}(x) \left[ \alpha^2-\beta^2-c\alpha -d- [2\alpha\beta -c\beta]\tan\beta x \right] +f(\underline{v}(x-c\tau))\\ =&& -\underline{v}(x) \frac{1}{\cos \beta x}(1-\gamma)f'(0)e^{-c\tau\alpha}\cos(\beta(x-c\tau))+f(\underline{v}(x-c\tau))\\ =&& -(1-\gamma) f'(0)\delta e^{\alpha(x-c\tau)}\cos\beta(x-c\tau)+f(\underline{v}(x-c\tau)). \end{eqnarray*} Choose $\delta>0$ sufficiently small such that \[ f(\underline{v}(x-c\tau)) \geqslant (1-\gamma) f'(0) \underline{v}(x-c\tau), \] with which we obtain \[ L[\underline{v}](x)\geqslant (1-\gamma)f'(0) [\underline{v}(x-c\tau)-\delta e^{\alpha(x-c\tau)}\cos\beta(x-c\tau)],\quad \beta x\in \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right). \] Clearly, if $\beta (x-c\tau) \in \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right)$, then $\underline{v}(x-c\tau)=\delta e^{\alpha(x-c\tau)}\cos\beta(x-c\tau)$, and hence, $L[\underline{v}](x)\geqslant 0$. If $\beta (x-c\tau) \not\in \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right)$, then $\underline{v}(x-c\tau)=0$, and hence, \[ L[\underline{v}](x)\geqslant -(1-\gamma)f'(0)\delta e^{\alpha(x-c\tau)}\cos\beta(x-c\tau) \] with $\beta (x-c\tau)\in \left(\frac{3\pi}{2}-\beta c\tau, \frac{5\pi}{2}-\beta c\tau\right)\setminus \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right)$. Since $\beta c\tau\leqslant \pi$ (as proved in Lemma \ref{lem:eigen}), we obtain $\cos\beta(x-c\tau)\leqslant 0$ when $\beta (x-c\tau)\in \left(\frac{3\pi}{2}-\beta c\tau, \frac{5\pi}{2}-\beta c\tau\right)\setminus \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right)$. To summarize, $L[\underline{v}](x)\geqslant 0$ for $\beta x\in \left(\frac{3\pi}{2}, \frac{5\pi}{2}\right)$ and $L[\underline{v}](x)= 0$ for $\beta x\not \in \left[\frac{3\pi}{2}, \frac{5\pi}{2}\right]$. The claim is proved.
Having such a subsolution, we can infer that \eqref{semiwave} admits a positive solution. The proof of uniqueness of the solution of \eqref{semiwave} follows from Proposition \ref{prop:semiwave}.
Next we show that \eqref{semiwave} does not admit a positive solution when $c\geqslant c_0(\tau)$. We employ a sliding argument. Assume for the sake of contradiction that there is a solution $q(z)$. Since $c\geqslant c_0(\tau)$, $\Delta_c(\lambda,\tau)=0$ admits a positive solution $\lambda_1$. Define $w(z)=le^{\lambda_1 z}-q(z), l>0$. Since $q(0)=0$ and $q(+\infty)=u^*$, we may choose $l$ such that $w(z)\geqslant 0$ for $z\geqslant 0$ and $w(z)$ vanishes at some $z\in (0,+\infty)$. Note that $f(u)\leqslant f'(0) u$. It then follows that \begin{equation} w''(z)-cw'(z)-dw(z)=-f'(0)w(z-c\tau)+[f(q(z-c\tau))-f'(0)q(z-c\tau)]\leqslant 0, \quad z \geqslant 0. \end{equation} By the elliptic strong maximum principle, we obtain $w(z)=0$ for $z\geqslant 0$, a contradiction. The nonexistence is proved. \end{proof}
Based on the above results, we obtain the solvability of \eqref{sw11}.
\begin{thm}\label{waves} For any given $\tau>0$, let $c_0(\tau)$ be given in Lemma \ref{lem:eigen}. For each $\mu>0$, there exists a unique $c^*=c^*_\mu(\tau)\in (0, c_0(\tau))$ such that $(q_{c^*})'_+(0)=\frac{c^*}{\mu}$, where $q_{c^*}(z)$ is the unique positive solution of \eqref{semiwave} with $c$ replaced by $c^*$. Moreover, $c^*_\mu(\tau)$ is increasing in $\mu$ with \[ \lim_{\mu\to\infty}c^*_\mu(\tau)=c_0(\tau). \] \end{thm} \begin{proof} From Propositions \ref{prop:semiwave} and \ref{prop:qoan1}, it is known that for each $c\in [0,c_0(\tau))$, problem \eqref{semiwave} admits a unique solution $q_c(z)>0$ for $z>0$, and for any $0\leqslant c_1<c_2\leqslant c_0(\tau)$, $q_{c_1}(z)>q_{c_2}(z)$ in $(0,\infty)$. Define \begin{equation}\label{def-P} P(0;c,\tau):=(q_c)_+'(0). \end{equation} Then $P(0;c,\tau)>0$ for all $c\in[0,c_0(\tau))$ and it decreases continuously in $c\in [0, c_0(\tau))$. Let $c_n\uparrow c_0(\tau)$. For each $c_n$ problem \eqref{semiwave} admits a unique solution $q_{c_n}(z)$. Clearly, $q_{c_n}$ converges to some $q^*$ and $(q_{c_n})'$ converges to $(q^*)'$ locally uniformly in $z\in[0,+\infty)$, and $q^*$ solves \eqref{semiwave} with $c=c_0(\tau)$. By the nonexistence established in Proposition \ref{prop:qoan1} we obtain $q^*\equiv 0$. In particular, \begin{equation} \lim_{c\uparrow c_0(\tau)} (q_c)_+'(0)=(q^*)_+'(0)=0. \end{equation} We now consider the continuous function \[ \eta(c;\tau)=\eta_\mu(c;\tau):=P(0;c,\tau)-\frac{c}{\mu}\ \ \mbox{ for }\ c\in [0,c_0(\tau)). \] By the above discussion we know that $\eta(c;\tau)$ is strictly decreasing in $c\in[0, c_0(\tau))$. Moreover, $\eta(0;\tau)=P(0;0,\tau)>0$ and $\lim_{c\uparrow c_0(\tau)}\eta(c;\tau)=-c_0(\tau)/\mu<0$. Thus there exists a unique $c^*=c^*_\mu(\tau)\in (0, c_0(\tau))$ such that $\eta(c^*;\tau)=0$, which means that \[ (q_{c^*})_+'(0)=\frac{c^*}{\mu}. \]
Next, let us view $(c^*_\mu, c_\mu^*/\mu)$ as the unique intersection point of the decreasing curve $y=P(0;c,\tau)$ with the increasing line $y=c/\mu$ in the $cy$-plane, then it is clear that $c^*_\mu(\tau)$ increases to $c_0(\tau)$ as $\mu$ increases to $\infty$. The proof is complete. \end{proof} \begin{remark}\label{tau0}\rm In \cite{DuLou}, the authors considered the case $\tau=0$. They obtained that for each $\mu>0$, there is a unique $c^*=c^*_\mu(0)\in (0, c_0(0))$ such that $(q_{c^*})'_+(0)=\frac{c^*}{\mu}$, where $q_{c^*}(z)$ is the unique of \eqref{semiwave} with $\tau=0$ and $c=c^*$, and $c_0(0)=2\sqrt{f'(0)-d}$. Moreover, $c^*_\mu(0)$ is increasing in $\mu$ with \[ \lim_{\mu\to\infty}c^*_\mu(0)=c_0(0). \]
\end{remark}
In the rest of this part, we study the monotonicity of $c^*_\mu(\tau)$ in $\tau$. For any given $\tau\geqslant 0$, the unique positive solution of \eqref{semiwave} with $c\in[0,c_0(\tau))$ may be denoted by $q_c(z;\tau)$. Now we give the proof of Theorem \ref{thm:semiwave}.
\noindent
{\bf Proof of Theorem \ref{thm:semiwave}:} For $\tau\geqslant 0$ and $\mu>0$, let $c^*_\mu(\tau)$ be given in Theorem \ref{waves} and Remark \ref{tau0} for $\tau>0$ and $\tau=0$, respectively. By Propositions \ref{prop:semiwave} and \ref{prop:qoan1}, we see that for $\tau\geqslant 0$ and $c\in (0,c_0(\tau))$, problem \eqref{semiwave} admits a unique positive solution $q_c(z;\tau)$. Moreover, $q_c(z;\tau)$ is increasing in $z>0$ and decreasing in $c\in (0,c_0(\tau))$. Let $P(0;c,\tau)$ be defined as in \eqref{def-P}.
{\bf Claim.} For $0\leqslant \tau_1<\tau_2$ , $P(0;c,\tau_1)>P(0;c,\tau_2)$ when $c\in (0,c_0(\tau_2))$.
We postpone the proof of the claim and reach the conclusion in a few lines. Note that $c^*_\mu(\tau)$ is the unique positive solution of $P(0;c,\tau)-\frac{c}{\mu}=0$. In view of $\lim_{c\uparrow c_0(\tau_2)} P(0;c,\tau_2)=0$, we have $c^*_\mu(\tau_2)\in (0,c_0(\tau_2))$. If $c^*_\mu(\tau_1)\geqslant c_0(\tau_2)$, then we are done. Otherwise, $c^*_\mu(\tau_1)\in (0,c_0(\tau_2))$, which, together with the claim, implies that \[ \frac{c^*_\mu(\tau_1)}{\mu}=P(0;c^*_\mu(\tau_1),\tau_1)>P(0;c^*_\mu(\tau_1),\tau_2). \] This further implies that $c^*_\mu(\tau_1)>c^*_\mu(\tau_2)$, due to the monotonicity of $P(0;c,\tau_2)-\frac{c}{\mu}$ in $c\in (0,c_0(\tau_2))$. Thus, $c^*_\mu(\tau)$ is decreasing in $\tau\geqslant 0$.
{\it Proof of the claim.} Since $c_0(\tau)$ is decreasing in $\tau\geqslant 0$, we see that $P(0;c,\tau_1)$ is well-defined when $c\in (0,c_0(\tau_2))$. By the monotonicity of $q_c(z;\tau_2)$ in $z>0$, we have $q_c(z-c\tau_2;\tau_2)<q_c(z-c\tau_1;\tau_2)$. This, together with the monotonicity of $f(v)$ in $v$, implies that $f(q_c(z-c\tau_2;\tau_2))< f(q_c(z-c\tau_1;\tau_2))$. Consequently, \[ q_c''(z;\tau_2)-cq_c'(z;\tau_2)-dq_c(z;\tau_2)+f(q_c(z-c\tau_1;\tau_2))> 0,\quad z>0. \] Consider the initial value problem \begin{equation} \begin{cases} v_t=v_{zz}-cv_z-dv+f(v(t,z-c\tau_1)),& t>0,\ z>0\\ v(t,z)=0, &t>0,\ z\leqslant 0\\ v(0,z)=q_c(z;\tau_2) \end{cases} \end{equation} By the maximum principle we know that $v(t,z)$ is nondecreasing in $t\geqslant 0$ and its limit $v^*(z)$ as $t\to\infty$ satisfies \eqref{semiwave} with $\tau=\tau_1$. By the uniqueness established in Proposition \ref{prop:semiwave}, we obtain $v^*(z)=q_c(z;\tau_1)$. Therefore, \begin{equation} q_c(z;\tau_2)=v(0,z)\leqslant v(t,z)\leqslant v(+\infty,z)=v^*(z)=q_c(z;\tau_1). \end{equation} The claim is proved. {
$\Box$}
\section{Long time behavior of the solutions}\label{seclo} In this section we study the asymptotic behavior of solutions of \eqref{p}. Firstly, we give some sufficient conditions for vanishing and spreading. Next, based on these results, we prove the spreading-vanishing dichotomy result of \eqref{p}. Let us start this section with the following equivalent conditions for vanishing.
\begin{lem}\label{lemvansmall} Assume that {\bf(H)} holds. Let $(u,g,h)$ be a solution of \eqref{p}. Then the following three assertions are equivalent: $$ {\rm (i)}\ h_\infty \mbox{ or } g_\infty \mbox{ is finite};\qquad
{\rm (ii)}\ h_\infty-g_\infty\leqslant \pi/\sqrt{f'(0)- d }; \qquad {\rm (iii)}\ \lim_{t\to\infty}\|u(t,\cdot)\|_{L^\infty ([g(t),h(t)])}= 0. $$ \end{lem}
\begin{proof} ``(i)$\Rightarrow$ (ii)". Without loss of generality we assume $h_\infty < -\infty$ and prove (ii) by contradiction. Assume that $h_\infty-g_\infty > \pi/\sqrt{f'(0)- d }$, then there exists $t_1 \gg 1$ such that \[ h(t_1) - g(t_1) > \frac{\pi}{\sqrt{f'(0)- d }}. \]
Let us consider the following auxiliary problem: \begin{equation}\label{subso} \left\{ \begin{array}{ll} v_t = v_{xx} - d v +f(v(t-\tau,x)), & t> t_1,\ x\in (g(t_1), \xi(t)),\\ v(t, \xi(t)) = 0,\quad \xi'(t)= -\mu v_x(t, \xi(t)),& t>t_1,\\ v (t,g(t_1))=0, & t> t_1,\\ \xi(t_1) = h(t_1),\ \ v(s, x)= u(s, x), & s\in[t_1-\tau, t_1],\ x\in [g(s), h(s)].
\end{array}
\right.
\end{equation} It is easy to check that $v$ is a subsolution of \eqref{p}, then $\xi(t)\leqslant h(t)$ and $\xi(\infty)<\infty$ by our assumption. Using a similar argument as in \cite[Lemma 3.3]{DGP} one can show that $$
\|v(t,\cdot)- V(\cdot)\|_{C^2([g(t_1),\xi(t)])} \to 0,\quad \mbox{as } t\to\infty, $$ where $V(x)$ is the unique positive solution of the problem \[ V''- d V +f(V)=0\ \ \mbox{ for}\ \ x\in(g(t_1),\xi(\infty)),\ \ \ \ V(g(t_1))=V(\xi(\infty))=0. \] Thus, \[ \lim_{t\to\infty} \xi'(t)=-\mu \lim_{t\to\infty}v_x (t,\xi(t)) =-\mu V'(\xi(\infty)) = \delta, \] for some $\delta>0$, which contradicts the fact that $\xi(\infty) < \infty$.
``(ii)$\Rightarrow$(iii)". It follows from the assumption and \cite[Proposition 2.9]{YCW} that the unique positive solution of the following problem \begin{equation}\label{upbsoper} \left\{ \begin{array}{ll} v_t=v_{xx}- d v+f(v(t-\tau,x)), & t>0,\ x\in[g_\infty,h_\infty],\\
v(t,g_\infty)= v(t,h_\infty)=0, & t>0,\\
v(\theta,x)\geqslant 0, & \theta\in[-\tau,0],\ x\in[g_\infty,h_\infty],
\end{array}
\right.
\end{equation} with $v(\theta,x)\geqslant \phi(\theta,x)$ in $[-\tau,0]\times[g(\theta),h(\theta)]$, satisfies $v\to0$ uniformly for $x\in[g_\infty,h_\infty]$ as $t\to\infty$. Then the conclusion (iii) follows easily from the comparison principle.
``(iii)$\Rightarrow$(ii)": Suppose by way of contraction argument that for some small $\varepsilon>0$ there exists $t_2\gg 1$ such that $h(t)-g(t)>\frac{\pi}{\sqrt{f'(0)- d }}+ 3\varepsilon$ for all $t>t_2-\tau$. Let $l_1:=\pi/\sqrt{f'(0)- d }+ \varepsilon$, it is well known that the following eigenvalue problem $$ \left\{
\begin{array}{ll} -\varphi_{xx} + d \varphi- f'(0)\varphi=\lambda_1\varphi, & 0<x<l_1,\\
\varphi(0)=\varphi(l_1)=0,
\end{array}
\right. $$ has a negative principal eigenvalue, denoted by $\lambda_1$, whose corresponding positive eigenfunction, denoted by
$\varphi$, can be chosen positive and normalized by $\|\varphi\|_{L^{\infty}}=1$. Set \[ w(t,x) :=\epsilon\varphi(x)\ \mbox{ for } x\in[0,l_1], \] with $\epsilon>0$ small such that \[ f(\epsilon\varphi)\geqslant f'(0)\epsilon\varphi+ \frac{1}{2}\lambda_1\epsilon\varphi\ \ \mbox{ in }[0, l_1]. \] It is easy to compute that for $x\in[0, l_1]$, $$ w_t-w_{xx}+ d w-f(w(t-\tau,x))= \epsilon\varphi [f'(0)+\lambda_1 ] -f(\epsilon\varphi) \leqslant 0. $$ Moreover one can see that \[ 0\leqslant w(x) = \epsilon \varphi(x) < u(t_2+s, x +g(t_2+s)+\varepsilon),\quad
x\in [0, l_1],\ s\in[-\tau,0] \] provided that $\epsilon$ is sufficiently small. Then we can apply the comparison principle to deduce $$ u(t+t_2,x +g(t_2) +\varepsilon) \geqslant w(x)>0,\quad (t,x)\in[0,\infty)\times(0, l_1), $$ contradicting (iii).
``(ii)$\Rightarrow$(i)". When (ii) holds, (i) is obvious. This proves the lemma. \end{proof}
Next, we give a sufficient condition for vanishing, which indicates that if the initial domain and initial function are both small, then the species dies out eventually in the environment.
\begin{lem}\label{vfsma} Assume that {\bf(H)} holds. Let $(u,g,h)$ be a solution of \eqref{p}. Then vanishing happens provided that
$h(0)-g(0)<\frac{\pi}{\sqrt{f'(0)- d }}$ and $\|\phi\|_{L^\infty([-\tau,0]\times[g(\theta),h(\theta)])}$ is sufficient small. \end{lem}
\begin{proof} Set \[ h_0=\frac{h(0)-g(0)}{2}, \] then $h_0<\pi/(2\sqrt{f'(0)- d })$, so there exists a small $\varepsilon >0$ such that \begin{equation}\label{choice of delta} \frac{\pi^2}{4 (1+\varepsilon)^2 h^2_0} - (f'(0)+\varepsilon)e^{\varepsilon\tau} + d \geqslant \varepsilon. \end{equation} For such $\varepsilon$, we can find a small positive constant $\delta$ such that $$ \pi \mu \delta \leqslant \varepsilon^2 h^2_0, \qquad f(v) \leqslant (f'(0) + \varepsilon) v \quad \mbox{for } v\in [0,\delta]. $$ Define \begin{align*}
& k(t) := h_0 \Big( 1+\varepsilon - \frac{\varepsilon}{2} e^{-\varepsilon t} \Big), \quad w(t,x):= \delta e^{-\varepsilon t} \cos\Big( \frac{\pi x}{2 k (t)}\Big),\ \ t>0,\ x\in[-k(t),k(t)],\\
& k(\theta)\equiv k_0 := h_0 \Big( 1+\frac{\varepsilon}{2} \Big), \quad w(\theta,x)\equiv w_0(x):= \delta \cos\Big(\frac{\pi x}{h_0(2+\varepsilon)}\Big),\ \ \theta\in[-\tau,0],\ x\in[-k_0,k_0]. \end{align*} and extend $w(t,x)$ by $0$ for $t\in[-\tau,\infty)$, $x\in(-\infty, -k(t)]\cup [k(t),\infty)$.
A direct calculation shows that for $t>0$, $x\in(-k(t),k(t))$ \begin{eqnarray*} && w_t - w_{xx} + d w - f(w(t-\tau,x))\\ &=& \left[ \frac{\pi^2}{4k^2(t)}-\varepsilon + d -(f'(0)+\varepsilon)\frac{w(t-\tau,x)}{w(t,x)} +\frac{\pi x k'(t)}{2k^2(t)}\tan \Big( \frac{\pi x}{2 k (t)}\Big)\right] w\\ & \geqslant& \left[ -\varepsilon + \frac{\pi^2}{4k^2(t)}+ d -(f'(0)+\varepsilon)\frac{w(t-\tau,x)}{w(t,x)} \right] w, \end{eqnarray*} where we have used $k'(t)>0$, $k(t)>0$ for $t>0$ and $y\tan y\geqslant 0$ for $y\in(-\frac{\pi}{2},\frac{\pi}{2})$.
When $t\geqslant \tau$ and $x\in(-k(t),k(t))$, it is easy to check that \begin{eqnarray*} \mathcal{A} & := & -\varepsilon + \frac{\pi^2}{4k^2(t)}+ d -(f'(0)+\varepsilon)\frac{w(t-\tau,x)}{w(t,x)} \\ & \geqslant & -\varepsilon + \frac{\pi^2}{4h_0^2(1+\varepsilon)^2}+ d -(f'(0)+\varepsilon)e^{\varepsilon\tau} \geqslant 0, \end{eqnarray*} where the fact that $\cos\Big(\frac{\pi x}{2 k (t-\tau)}\Big)\leqslant \cos\Big(\frac{\pi x}{2 k (t)}\Big)$ for $(t,x)\in[\tau,\infty)\times[-k(t),k(t)]$ and the monotonicity of $k(t)$ in $t\in[0,\infty)$ are used. If $t\in[0, \tau)$ and $x\in(-k(t),k(t))$, we have that \begin{eqnarray*} \mathcal{A} & \geqslant & -\varepsilon + \frac{\pi^2}{4h_0^2(1+\varepsilon)^2}+ d -(f'(0)+\varepsilon)e^{\varepsilon t}\frac{\cos\Big(\frac{\pi x}{h_0(2+\varepsilon)}\Big)}{\cos\Big(\frac{\pi x}{2k(t)}\Big)} \\ & \geqslant & -\varepsilon + \frac{\pi^2}{4h_0^2(1+\varepsilon)^2}+ d -(f'(0)+\varepsilon)e^{\varepsilon\tau} \geqslant 0. \end{eqnarray*} Thus we have $$ w_t - w_{xx} + d w - f(w(t-\tau,x)) \geqslant 0\ \ \mbox{ in }\ (0,\infty)\times(-k(t),k(t)). $$
On the other hand, $$
k'(t)=\frac{\varepsilon^2 h_0}{2} e^{-\varepsilon t}\geqslant \frac{\pi \mu \delta}{2h_0 } e^{-\varepsilon t}\geqslant
\frac{\pi \mu \delta}{2k(t)} e^{-\varepsilon t} \geqslant - \mu w_x(t, k(t)) =\mu w_x(t, -k(t)). $$
As a consequence, $(w(t,x), -k(t), k(t))$ will be a supersolution of \eqref{p} if $w(\theta,x)\geqslant \phi (\theta,x)$ in $[-\tau,0]\times[g(\theta),h(\theta)]$. Indeed, choose $\sigma_1 := \delta\cos \frac{\pi}{2+\varepsilon}$, which depends only on $\mu, h_0, d $ and $f$. Then when $\|\phi\|_{L^\infty([-\tau,0]\times[g(\theta),h(\theta)])} \leqslant \sigma_1$ we have $\phi(\theta,x)\leqslant \sigma_1 \leqslant w(\theta,x)$ in $[-\tau,0]\times[g(\theta), h(\theta)]$, since $h_0 < k(0)= h_0 (1+\frac{\varepsilon}{2})$. It follows from the comparison principle that $$ h(t)\leqslant k(t) \leqslant h_0 (1+\varepsilon),\; h_\infty<\infty. $$ This, together with the previous lemma, implies that vanishing happens. \end{proof} \begin{remark}\rm When $\tau=0$, the proof of Lemma \ref{vfsma} reduces to that of \cite[Theorem 3.2(i)]{DuLou}. \end{remark}
We now present a sufficient condition for spreading, which reads as follows.
\begin{lem}\label{lemuto1} Assume that {\bf(H)} holds. If $h(0)-g(0)\geqslant \pi/\sqrt{f'(0)- d }$, then spreading happens for every positive solution $(u, g, h)$ of \eqref{p}. \end{lem}
\begin{proof} Since $g'(t)<0<h'(t)$ for $t>0$, we have $h(t)-g(t)>\pi/\sqrt{f'(0)- d }$ for any $t>0$. So the conclusion $-g_\infty = h_\infty =\infty$ follows from Lemma \ref{lemvansmall}. In what follows we prove \begin{equation}\label{utoPt} \lim_{t\to\infty}u(t,x)=u^* \mbox{ locally uniformly in $\mathbb{R}$}. \end{equation}
First, it is well known that for any $L>\pi/(2\sqrt{f'(0)- d })$, the following problem \[ W_{xx}- d W+f(W)=0,\ \ \ x\in(-L,L),\ \ \ W(\pm L)= 0, \] admits a unique positive solution $W_L$, which is increasing in $L$ and satisfies \begin{equation}\label{WL1} \lim_{L\to\infty}W_L(x)=u^* \mbox{ locally uniformly in $\mathbb{R}$}. \end{equation} Moreover we can find an increasing sequence of positive numbers $L_n$ with $L_n\to\infty$ as $n\to\infty$ such that $L_n>\pi/\sqrt{f'(0)- d }$ for all $n\geqslant1$. Since $W_{L_n}$ converges to $u^*$ locally uniformly in $\mathbb{R}$, we can choose $t_n$ such that $h(t)\geqslant L_n$ and $g(t)\leqslant-L_n$ for $t\geqslant t_n$. It then follows from \cite{YCW} the following problem \[ \left\{ \begin{array}{ll}
w_t =w_{xx}- d w +f(w(t-\tau,x)), & t\geqslant t_n+\tau,\ x\in[-L_n,L_n],\\
w(t,\pm L_n)= 0, & t\geqslant t_n+\tau,\\
w(s,x)=u(s,x), & s\in[t_n, t_n+\tau],\ x\in[-L_n,L_n], \end{array} \right. \] has a unique positive solution $w_n(t,x)$, which satisfies that \[ w_n(t,x)\to W_{L_n}(x) \ \mbox{ uniformly for } x\in[-L_n,L_n]\ \mbox{ as } t\to\infty. \] Applying the comparison principle we have $w_n(t,x)\leqslant u(t,x)$ for all $t\geqslant t_n+\tau$, $x\in [-L_n,L_n]$. This, together with \eqref{WL1}, yields that \begin{equation}\label{uin1} \liminf_{t\to\infty} u(t ,x) \geqslant u^*\ \mbox{ locally uniformly for } x\in\mathbb{R}. \end{equation} Later, since the initial data $u_0(s,x)$ satisfies $0\leqslant u_0(s,x)\leqslant u^*$ for $(s,x)\in[-\tau,0]\times[g(s),h(s)]$, it thus follows from the comparison principle that \[ \limsup_{t\to\infty} u(t ,x) \leqslant u^*\ \mbox{ locally uniformly for } x\in\mathbb{R}. \]
Combining with \eqref{uin1}, one can easily obtain \eqref{utoPt}, which ends the proof of this lemma. \end{proof}
Now we are ready to give the proof of Theorem \ref{thm:asy be}.
\noindent
{\bf Proof of Theorem \ref{thm:asy be}}. It is easy to see that there are two possibilities: (i) $h_\infty-g_\infty\leqslant \pi/\sqrt{f'(0)- d }$;
(ii) $h_\infty-g_\infty>\pi/\sqrt{f'(0)- d }$. In case (i), it follows from Lemma \ref{lemvansmall} that $\lim_{t\to\infty}
\|u(t,\cdot)\|_{L^\infty([g(t),h(t)])}=0$. For case (ii), it follows from Lemma \ref{lemuto1} and its proof that $(g_\infty, h_\infty)=\mathbb{R}$ and $u(t,x)\to u^*$ as $t\to\infty$ locally uniformly in $\mathbb{R}$, which ends the proof.
$\square$
\section{Asymptotic profiles of spreading solutions}\label{sec:asybeh} Throughout this section we assume that {\bf(H)} holds and $(u,g,h)$ is a solution of \eqref{p} for which spreading happens. In order to determine the spreading speed, we will construct some suitable sub- and supersolutions based on semi-waves. Let $c^*$ and
$q_{c^*}(z)$ be given in Theorem \ref{waves}. The first subsection covers the proof of the boundedness for $|h(t)-c^*t|$ and $|g(t)+c^*t|$. Based on these results, we prove Theorem \ref{thm:profile of spreading sol} in the second subsection.
\subsection{Boundedness for $|h(t)-c^*t|$ and $|g(t)+c^*t|$.}\label{sub51} Let us begin this subsection with the following estimate. \begin{lem}\label{lem:u-to-1} Let $(u, g, h)$ be a solution of \eqref{p} for which spreading happens. Then for any $c\in (0,c^*)$, there exist small $\beta^*\in (0, d -f'(u^*))$ , $T>0$ and $ M>0$ such that for $t\geqslant T$, \begin{itemize} \item[\rm (i)] $ [g(t), h(t)]\supset [-ct, ct]; $ \item[\rm (ii)] $ u(t,x)\geqslant u^*\big(1-M e^{-\beta^* t}\big)\quad \mbox{for } x\in [-ct, ct]; $ \item[\rm (iii)] $ u(t,x) \leqslant u^*\big(1+M e^{-\beta^* t}\big) \quad \mbox{for } x \in [g(t), h(t)]. $ \end{itemize} \end{lem} \begin{proof} In order to prove conclusions (i) and (ii), inspired by \cite{FM}, we will use the semi-wave $q_{c^*}$ to construct the suitable subsolution. Here we mainly use the the monotonicity and exponentially convergent of $q_{c^*}$.
(i)\ Since $q_{c^*}(z)$ is the unique positive solution of
\begin{equation}\label{semiwave112} \left\{
\begin{array}{ll}
q_{c^*}'' - c^*q_{c^*}'- d q_{c^*}+ f( q_{c^*}(z-c^*\tau))=0,\ \ \ q_{c^*}'(z)>0, & z>0,\\
q_{c^*}(z)=0, & z\leqslant 0,\\
\mu q_{c^*}'(0)=c^*,\ \ q_{c^*}(\infty)=u^*,
\end{array}
\right. \end{equation} then it is easy to check that $q_{c^*}''(0)> 0$. Since $q_{c^*}'(z)> 0$ for $z\geqslant0$ and $q_{c^*}(z)\to u^*$ as $z\to\infty$, thus there is $z_0\gg 1$ such that $q_{c^*}''(z)<0$ for $z\geqslant z_0$. Thus there exists $\hat{z} \in (0,\infty)$ such that $q_{c^*}''(\hat{z})=0$ and $q_{c^*}''(z)>0 $ for $z\in[0,\hat{z})$. This means that $q_{c^*}'(z)$ is increasing in $z\in[0,\hat{z})$. Let $\hat{p}_0 \in (0,q_{c^*}(\hat{z}))$ be small. Define \[ G(u,p)=\left\{ \begin{array}{ll}
d +[f(u-p)-f(u)]/p ,& p>0 ,\\
d -f'(u), & p=0, \end{array} \right. \] for $p>0$ and $u>p$. Then $G(u,p)$ is a continuous function for $0 \leqslant p \leqslant \hat{p}_0$ and $G(u^*,p)>0$, $G(u^*,0)= d -f'(u^*)>0$, thus there exists $0<\gamma\ll d $ such that $G(u^*,p) \geqslant 2\gamma$ for $0\leqslant p\leqslant \hat{p}_0 $. By continuity, there exists $\rho>0$ small such that $G(u,p) \geqslant \gamma $ for $u^*-\rho \leqslant u\leqslant u^*$, $0\leqslant p\leqslant \hat{p}_0$. Furthermore, as $f(u^*)= d u^*$, then there is a constant $b>0$ such that \begin{equation}\label{fub1} f(v)- d v\leqslant b(u^*-v)\ \ \mbox{ for }\ v\in[u^*-\rho, u^*]. \end{equation} Inspired by \cite{FM}, let us construct the following function: \[ \underline{u}(t,x):= \max\{0,\ q_{c^*}(x+c^*t+\xi(t))+q_{c^*}(c^*t-x+\xi(t))-u^*-p(t)\},\ \ t>0, \] and denote $\underline{g}(t)$ and $\underline{h}(t)$ be the zero points of $\underline{u}(t,x)$ with $t>0$, that is \[ \underline{u}(t,\underline{g}(t))=\underline{u}(t,\underline{h}(t))=0. \]
In the following, we will show that $(\underline{u},\underline{g}, \underline{h})$ is a subsolution of problem \eqref{p}. We only prove the case where $x\geqslant 0$, since the other is analogous. For any function $J$ depended on $t$, we write $J_{\tau}(t):=J(t-\tau)$ if no confusion arises. For simplicity of notations, we will write \[ \zeta^-(t):=-x+c^*t+\xi(t), \ \zeta^+(t):=x+c^*t+\xi(t),\ \ \zeta^-_\tau:=\zeta^-(t-\tau), \ \zeta^+_\tau:=\zeta^+(t-\tau). \]
Firstly, a direct calculation shows that for $(t,x)\in(\tau,\infty)\times[0, \underline{h}(t)]$, \begin{align*} \mathcal{N}[\underline{u}]:&=\underline{u}_t-\underline{u}_{xx}+ d \underline{u}-f(\underline{u}(t-\tau,x))\\ &=\xi'[q'_{c^*}(\zeta^-)+q'_{c^*}(\zeta^+)]+f(q_{c^*}(\zeta^-_\tau))+f(q_{c^*}(\zeta^+_\tau))\\ &\ \ \ -f(q_{c^*}(\zeta^-_\tau)+q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau})- d (u^*+p) -p'. \end{align*} Assume that $\xi'(t) \leqslant 0$, and choose $\xi$ large such that $u^*-\frac{\rho}{2}\leqslant q_{c^*}(\zeta^+_\tau)\leqslant u^*$ in $(\tau,\infty) \times[0, \underline{h}(t)]$. The monotonicity of $q_{c^*}$ and its exponential rate of convergence to $u^*$ at $\infty$ imply that if we choose $\xi$ sufficiently large, then there exist positive constants $\nu$, $K_0$ and $K$ such that \[ u^*-q_{c^*}(\zeta^+_\tau)\leqslant K_0e^{-\nu \zeta^+_\tau}\leqslant Ke^{-\nu(\xi(t)+c^*t)}. \] Set $p(t)=p_0e^{-\beta t}$ with $p_0:=\frac{1}{2}\min\{\hat{p}_0,\ \frac{\rho}{2}\}$ and $\beta:=\frac{1}{2}\min\{\nu c^*,\ \alpha_0\}$, where $\alpha_0$ is the unique zero point of \[
d (e^{\tau y}-1)-\gamma e^{\tau y}+y=0. \]
Thus, when $q_{c^*}(\zeta^-_\tau)\in[u^*-\rho, u^*]$ and $(t,x)\in(\tau,\infty)\times[0, \underline{h}(t)]$, since $q_{c^*}'(z) \geqslant 0$, then \begin{align*} \mathcal{N}[\underline{u}]&=\xi'[q'_{c^*}(\zeta^-)+q'_{c^*}(\zeta^+)]+f(q_{c^*}(\zeta^-_\tau))+f(q_{c^*}(\zeta^+_\tau))\\ &\ \ \ -f(q_{c^*}(\zeta^-_\tau)+q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau})- d (u^*+p) -p'\\ &\leqslant \gamma [q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau}]+b[u^*-q_{c^*}(\zeta^+_\tau)]+ d (p_{\tau}-p)-p'\\ &\leqslant b[u^*-q_{c^*}(\zeta^+_\tau)]+ d (p_{\tau}-p)-p'-\gamma p_{\tau}\\ &\leqslant Kbe^{-\nu(\xi(t)+c^*t)}+p_0e^{-\beta t}\big[ d \big(e^{\beta \tau}-1\big)-\gamma e^{\beta \tau}+\beta\big]\leqslant 0, \end{align*} provided that $\xi$ is sufficiently large.
For the part $ q_{c^*}(\zeta^-_\tau)\in[0,u^*-\rho]$, then for $(t,x)\in(\tau,\infty)\times[0, \underline{h}(t)]$ and sufficiently large $\xi$, there are two positive constants $d_1$ and $d_2$ where $d_1<1$ such that $q'_{c^*}(\zeta^-)+q'_{c^*}(\zeta^+)\geqslant d_1$, and \[ f\big(q_{c^*}(\zeta^-_\tau)\big)-f\big(q_{c^*}(\zeta^-_\tau)+q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau}\big)+ d [q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau}]\leqslant d_2[u^*+p_{\tau}-q_{c^*}(\zeta^+_\tau)], \] thus we have \begin{align*} \mathcal{N}[\underline{u}]&=\xi'[q'_{c^*}(\zeta^-)+q'_{c^*}(\zeta^+)]+f(q_{c^*}(\zeta^-_\tau))+f(q_{c^*}(\zeta^+_\tau))\\ &\ \ \ -f(q_{c^*}(\zeta^-_\tau)+q_{c^*}(\zeta^+_\tau)-u^*-p_{\tau})- d (u^*+p) -p'\\ &\leqslant d_1\xi'+d_2 [u^*+p_{\tau}-q_{c^*}(\zeta^+_\tau)]+b[u^*-q_{c^*}(\zeta^+_\tau)+ d (p_{\tau}-p)-p'\\ &\leqslant d_1\xi'+(d_2+b)Ke^{-\nu(\xi+c^*t)} +p_0e^{-\beta t}\big[d_2e^{\beta \tau}+ d \big(e^{\beta \tau}-1\big)+\beta\big]\\ &\leqslant d_1\xi'+p_0e^{-\beta t}\big[d_2e^{\beta \tau}+ d (e^{\beta \tau}-1)+2\beta\big]. \end{align*} Now let us choose $\xi$ satisfies \[ d_1\xi'+\kappa p_0e^{-\beta t}=0 \] with $\xi(0)=\xi_0$ sufficiently large, and $\kappa:=d_2e^{\beta \tau}+ d \big(e^{\beta \tau}-1\big)+2\beta$, then $\xi'(t)\leqslant 0$. Hence from the above we obtain that $\mathcal{N}[\underline{u}]\leqslant 0$ in this part.
Next, let us check the free boundary condition. When $x=\underline{h}(t)$, we set $\zeta_1(t)=-\underline{h}(t)+c^*t+\xi(t)$ and $\zeta_2(t)=\underline{h}(t)+c^*t+\xi(t)$, then \begin{equation}\label{qq1} q_{c^*}(\zeta_1(t))+q_{c^*}(\zeta_2(t))=u^*+p(t). \end{equation} We differentiate \eqref{qq1} with respect to $t$ to obtain \begin{equation}\label{hf1} \big[q_{c^*}'(\zeta_2)-q_{c^*}'(\zeta_1)\big]\big(\underline{h}'(t)-c^*\big)= p'-2c^*q_{c^*}'(\zeta_2)-\big[q_{c^*}'(\zeta_2)+q_{c^*}'(\zeta_1)\big]\xi'. \end{equation} By shrinking $p_0$ and enlarge $\xi_0$ if necessary, then we can see that $\zeta_2(t)\gg1$, and $q_{c^*}(\zeta_2(t))\approx u^*$. This, together with \eqref{qq1}, yields that $q_{c^*}(\zeta_1(t))\approx p(t)$. Since $q''_{c^*}(z)>0>q''_{c^*}(y)$ for $0\leqslant z\ll 1$ and $y\gg 1$ and $q'_{c^*}(z)\searrow 0$ as $z\to\infty$, thus we have \begin{equation}\label{q1q21} 0<q_{c^*}'(\zeta_2)< q_{c^*}'(0)< q_{c^*}'(\zeta_1). \end{equation} Thanks to the choice of $\xi(t)$, we can compute that \begin{equation}\label{q1q22} p'-2c^*q_{c^*}'(\zeta_2)-[q_{c^*}'(\zeta_2)+q_{c^*}'(\zeta_1)]\xi'\geqslant \big(\frac{\kappa q_{c^*}'(0)}{d_1}-\beta\big) p_0e^{-\beta t}-2c^*K_1e^{-\nu(\xi(t)+c^*t)}\geqslant 0, \end{equation} where $K_1$ is a positive constant, $\kappa:=d_2e^{\beta \tau}+ d \big(e^{\beta \tau}-1\big)+2\beta>2\beta$ and we have used that by shrinking $d_1$ if necessary, then $\kappa q_{c^*}'(0)>\beta d_1$.
It follows from \eqref{hf1}, \eqref{q1q21}, \eqref{q1q22} and the monotonicity of $q_{c^*}'(z)$ in $z$ that \[
\underline{h}'(t)\leqslant c^*=\mu q_{c^*}'(0)\leqslant \mu[q_{c^*}'(\zeta_1)-q_{c^*}'(\zeta_2)]=-\mu \underline{u}_x(t,\underline{h}(t)). \]
Using \eqref{qq1} again, it is easy to see that $\zeta_1(t)$ is decreasing in $t\geqslant T_1$, thus for all $t\geqslant T_1$, \begin{equation}\label{huh} \underline{h}(t)-c^*t\geqslant \tilde{C}_0:=\underline{h}(T_1)-c^*T_1+\xi(\infty)-\xi(0). \end{equation} Since $(u,g,h)$ is a spreading solution of \eqref{p}, then there exists $T_2>0$ such that \begin{align*}
& u(T_1+T_2+\tilde{s},x)\geqslant \underline{u}(T_1+\tau,x)\ \mbox{ for }\ \tilde{s}\in[0,\tau],\ x\in[\underline{g}(\tau),\underline{h}(\tau)],\\
& g(T_1+T_2)\leqslant \underline{g}(T_1+\tau)\ \ \mbox{and }\ h(T_1+T_2)\geqslant \underline{h}(T_1+\tau). \end{align*}
Consequently, $(\underline{u},\underline{g}, \underline{h})$ is a subsolution of problem \eqref{p}, then we can apply the comparison principle to conclude that $u(t+T_1+T_2,x)\geqslant \underline{u}(t+T_1,x)$, $h(t+T_1+T_2)\geqslant \underline{h}(t+T_1)$ for $t>0$, $x\in[0,\underline{h}(t)]$. This, together with \eqref{huh}, implies that \[ h(t)-c^*t\geqslant -C_1\ \ \ \mbox{ for } t>0, \]
with $C_1:=-|\tilde{C}_0|-h(T_1+T_2+\tau)-c^*(T_1+T_2+\tau)$. Similarly, by enlarging $C_1$ if necessary, we can have $g(t)+c^*t\leqslant C_1$ for $t>0$. Thus result (i) holds for large $T$.
(ii)\ From the proof of (i), it is easy to see that $u(t+T_2)\geqslant \underline{u}(t,x)$ for $t>T_1$. The monotonicity of $q_{c^*}$ and its exponential rate of convergence to $u^*$ at $\infty$ can be used again to conclude that for any $c\in(0,c^*)$ there exist constants $\nu$, $K>0$ such that for any $x\in[0,ct]$ and $t>0$, \begin{align*}
& u^*-q_{c^*}(x+c^*t+\xi(t))\leqslant u^*-q_{c^*}(c^*t+\xi(t))\leqslant K e^{-\nu(c^*t+\xi(t))},\\
& q_{c^*}(-x+c^*t+\xi(t))\geqslant q_{c^*}((c^*-c)t+\xi(t))\geqslant u^*-K e^{-\nu[(c^*-c)t+\xi(t)]}. \end{align*} Based on above results, we can find $T_3>T_1+T_2$ large such that for $t>T_3$ and $x\in[0,ct]$, \begin{align*} u(t,x)&\geqslant q_{c^*}(x+c^*(t-T_2)+\xi(t-T_2))+q_{c^*}(-x+c^*(t-T_2)+\xi(t-T_2))-u^*-p_0e^{\beta (t-T_2)}\\
& \geqslant u^* -2K e^{-\nu\big[(c^*-c)(t-T_2)+\xi(t-T_2)\big]}-p_0e^{\beta (t-T_2)} \geqslant u^*-M u^*e^{-\beta^* t}, \end{align*} where $M>0$ is sufficiently large and $\beta^*:=\frac{1}{2}\min\big\{\nu(c^*-c),\ \beta,\ d -f'(u^*)\big\}$. The case where $x\in[-ct,0]$ can be proved by a similar argument as above. The proof of (ii) is now complete.
(iii)\ Thanks to the choice of the initial data, we know that for any given $\beta^*>0$ and $M>0$, \[ u(t,x) \leqslant u^*+ Mu^* e^{-\beta^* t}\ \ \ \mbox{ for }\ (t,x)\in[0,\infty)\times[g(t), h(t)]. \] This completes the proof. \end{proof}
Next we prove the boundedness of $h(t)-c^*t$ and show that $u(t,\cdot) \approx u^*$ in the domain $[0, h(t)-Z]$, where $Z>0$ is a large number.
\begin{prop}\label{pro:sigma01} Assume that spreading happens for the solution $(u,g,h)$. Then \begin{itemize} \item[(i)] there exists $C>0$ such that \begin{equation}\label{hghg1}
|h(t)-c^*t |\leqslant C \ \ \mbox{ for all } t\geqslant0 ; \end{equation}
\item[(ii)] for any small $\varepsilon>0$, there exists $Z_\varepsilon>0$ and $T_\epsilon >0$ such that \begin{equation}\label{ughu1}
\|u(t,\cdot ) - u^* \|_{L^\infty ([0, h(t) -Z_\varepsilon])} \leqslant u^*\varepsilon \ \ \mbox{ for } t> T_\varepsilon. \end{equation} \end{itemize} \end{prop}
\begin{proof} In order to prove conclusions in this proposition, inspired by \cite{DMZ}, we will use the semi-wave $q_{c^*}$ to construct the suitable sub- and supersolution. Compared with \cite{DMZ}, our problem deal with the case where $\tau>0$. Due to $\tau>0$, there will be some space-translation of the semi-wave $q_{c^*}$, which make our problem difficult to deal with. To overcome this difficulty, we mainly use the the monotonicity and exponentially convergent of $q_{c^*}$. Moreover, this idea also be used in Lemma \ref{limn21}. For clarity we divide the proof into several steps.
$Step\ 1$. To give some upper bounds for $h(t)$ and $u(t,x)$.
Fix $c\in(0,c^*)$. It follows from Lemma \ref{lem:u-to-1} that there exist $\beta^*\in(0, d -f'(u^*))$, $M >0$, and $T> 0$ such that for $t \geqslant T$, (i), (ii) and (iii) in Lemma \ref{lem:u-to-1} hold. Thanks to {\bf (H)}, by shrinking $\beta^*$ if necessary, we can find $\rho>0$ small such that \begin{equation}\label{vu1}
d -f'(v)e^{\beta^* \tau}\geqslant \beta^*\ \ \ \mbox{ for } v\in[u^*-\rho,u^*+\rho]. \end{equation}
For any $T_*>T+\tau$ large satisfying $Mu^* e^{-\beta^* (T_*-\tau)}<\frac{\rho}{2}$, there is $M' > M$ such that $M'u^*e^{-\beta^* (T_*-\tau)}< \rho$. Since $q_{c^*}(z)\to u^*$ as $z\to\infty$, we can find $Z_0 >0$ such that \begin{equation}\label{U1a1} \big(1+M'e^{-\beta^* (T_*+\tau) }\big)q_{c^*}(Z_0 )\geqslant u^*. \end{equation}
Now we construct a supersolution $(\bar{u} ,g, \bar{h})$ to \eqref{p} as follows: \begin{align*}
& \bar{h} (t): =c^*(t - T_*)+ h (T_*+\tau )+ K M'\big(e^{-\beta^* T_* }-e^{-\beta^* t}\big)+Z_0\ \ \ \mbox{ for }\ t\geqslant T_* ,\\
& \bar{u}(t,x):=\min\big\{\big(1+M'e^{-\beta^* t}\big)q_{c^*}\big(\bar{h} (t)-x\big),\ u^*\big\}\ \ \ \mbox{ for }\ t\geqslant T_* ,\ x\leqslant \bar{h} (t), \end{align*} where $K$ is a positive constant to be determined below.
Clearly, for all $t\geqslant T_*$, $\bar{u} (t, g(t))>0= u(t, g(t))$, $\bar{u}\big(t, \bar{h} (t)\big)=0$, and \begin{eqnarray*} -\mu \bar{u} _x(t,\bar{h}(t))& = & \mu \big(1+M'e^{-\beta^* t}\big)q_{c^*}'(0)=\big(1+M'e^{-\beta^* t}\big)c^*, \\ & < & c^*+M' K \beta^* e^{-\beta^* t} = \bar{h}'(t), \end{eqnarray*} if we choose $K$ with $K\beta^* > c^*$. By the definition of $\bar{h}$ we have $h (T_*+s )<\bar{h}(T_* +s)$ for $s\in[0,\tau]$. It then follows from \eqref{U1a1} that for $(s,x)\in[0,\tau]\times[ g (T_*+s ),h (T_* +s)]$, \[ \big(1+M'e^{-\beta^* (T_*+s) }\big)q_{c^*}\big(\bar{h} (T_*+s)-x\big) \geqslant\big(1+M'e^{-\beta^* (T_*+\tau) }\big)q_{c^*}(Z_0)\geqslant u^*, \] which yields that $\bar{u}(T_*+s,x)=u^*\geqslant u(T_*+s,x)$ for $(s,x)\in[0,\tau]\times[g(T_*+s),h(T_*+s)]$.
We now show that \begin{equation}\label{u+ upper}
\mathcal{N} [\bar{u}] := \bar{u}_t - \bar{u}_{xx} + d \bar{u}-f(\bar{u}(t-\tau,x)) \geqslant 0,\quad x\in [g(t), \bar{h}(t)],\ t> T_*+\tau . \end{equation} Thanks to the definition of $\bar{u}(t,x)$ and the monotonicity of $q_{c^*}(z)$ in $z$, we can find a decreasing function $\eta(t)<\bar{h}(t)$ for $t>T_*$, such that \[ \big(1+M'e^{-\beta^* t}\big)q_{c^*}\big(\bar{h}(t)-x\big)\left\{\begin{array}{ll} > u^*, & x<\eta(t),\\ = u^*, & x=\eta(t),\\ < u^*, & x\in\big(\eta(t),\bar{h}(t)\big], \end{array} \right. \] which implies that \[ \bar{u}(t,x)=u^*\ \mbox{ for } x \leqslant \eta(t), \ \mbox{ and }\ \bar{u}(t,x)=\big(1+M'e^{-\beta^* t}\big)q_{c^*}\big(\bar{h}(t)-x\big)\ \mbox{ for } x\in\big[\eta(t),\bar{h}(t)\big]. \] As $\mathcal{N} u^*=0$, thus in what follows, we only consider the case $ x\in\big[\eta(t),\bar{h}(t)\big]$. Set $q_\tau:=q_{c^*}\big(\bar{h}_\tau-x\big)$ for convenience. A direct calculation shows that, for $t>T_*+\tau$, \begin{align*} \mathcal{N} [\bar{u}] :& = \bar{u}_t - \bar{u}_{xx} + d \bar{u}-f(\bar{u}(t-\tau,x))\\
& = -\beta^* M'e^{-\beta^* t} q_{c^*}+\big(1+M'e^{-\beta^* t}\big) \{K\beta^* M'e^{-\beta^* t} q_{c^*}'+f(q_{\tau})\} -f\big((1+M'e^{-\beta^* (t-\tau)})q_{\tau}\big)\\
& = M'e^{-\beta^* t}\Big\{f(q_{\tau})+K \beta^*\big(1+M'e^{-\beta^* t}\big)q_{c^*}' -\beta^*q_{c^*} \Big\} + f(q_\tau)- f\big((1+M'e^{-\beta^* (t-\tau)})q_\tau\big)\\
& \geqslant M'e^{-\beta^* t}\Big\{K \beta^* \big(1+M'e^{-\beta^* t}\big)q_{c^*}'-\big[\big(f'\big((1+ \theta M'e^{-\beta^* (t-\tau)})q_\tau\big)e^{\beta^* \tau}- d \big)q_\tau-\beta^*q_{c^*}\big]\Big\}, \end{align*} for some $\theta \in (0,1)$. Since \begin{equation}\label{qcto1} q_{c^*}(z)\to u^*\ \mbox{ and } \frac{(q_{c^*}(z)-u^*)'}{q_{c^*}(z)-u^*}\to k^*\ \ \mbox{ as } z\to \infty \end{equation} where $k^*:=c^*-\sqrt{(c^*)^2+4( d -f'(u^*))}<0$, there are $z_0>0$ and $k_1>0$ such that \begin{equation}\label{qqq1} q_{c^*}''(z)<0,\ \ \ q_{c^*}(z)\geqslant u^*-\rho\ \ \mbox{ and }\ \ q_{c^*}'(z-2c^*\tau) \leqslant k_1 q_{c^*}'(z) \ \mbox{ for } \ z>z_0, \end{equation} Moreover, we can compute that \begin{align*} \triangle \bar{h}(t) := \bar{h}(t)-\bar{h}_\tau(t)= c^*\tau+KM'e^{-\beta^*t}(e^{\beta^*\tau}-1). \end{align*} For any given $K>0$, by enlarging $T_*$ if necessary, we have that \begin{equation}\label{deh} \triangle \bar{h}(t)\in[c^*\tau,2c^*\tau]\ \ \mbox{ for }\ t \geqslant T_*. \end{equation}
When $\bar{h}_\tau-x>z_0$ and $t> T_*+\tau$, it then follows that \begin{align*} \mathcal{B} :& = K \beta^* \big(1+M'e^{-\beta^* t}\big)q_{c^*}'-\big[\big(f'\big((1+ \theta M'e^{-\beta^* (t-\tau)})q_\tau\big)e^{\beta^* \tau}- d \big)q_\tau-\beta^*q_{c^*}\big]\\
& \geqslant \big[ d -f'\big(\big(1+ \theta M'e^{-\beta^* (t-\tau)}\big)q_\tau\big)e^{\beta^* \tau}-\beta^*\big]q_\tau+ K\beta^* q_{c^*}'+\beta^* (q_\tau-q_{c^*})\\ & \geqslant K\beta^* q_{c^*}'(\bar{h}(t)-x)-\beta^* q_{c^*}'(\bar{h}(t)-x-\tilde{\theta}\triangle\bar{h}(t))\triangle\bar{h}(t)\ \ \ (\mbox{with } \tilde{\theta}\in(0,1))\\ & \geqslant (K-2k_1c^*\tau) \beta^*q_{c^*}'(\bar{h}(t)-x)\geqslant 0 \end{align*} provided that $K$ is sufficiently large, and we have used $M'e^{-\beta^* (t-\tau)} u^*\leqslant\rho$ for $t> T_* $, $q_{c^*}'(z)>0$ for $z>0$, \eqref{vu1}, \eqref{qqq1} and \eqref{deh}. Thus $\mathcal{N} [\bar{u}]\geqslant 0$ in this case.
When $0\leqslant \bar{h}_\tau-x\leqslant z_0$ and $t> T_*+\tau $, for sufficiently large $K$, we have $$ \mathcal{N}[\bar{u}] \geqslant M'e^{-\beta^* t}\big[K \beta^* D_1 - D_2u^* e^{\beta^* \tau}-\beta^*u^*\big]\geqslant 0, $$ where $D_1:=\min_{z\in[0,z_0+2c^*\tau]}q_{c^*}'(z)>0$, $D_2:=\max_{v\in[0,2u^*]}f'( v)$, and \eqref{deh} are used.
Summarizing the above results we see that $(\bar{u}, g, \bar{h})$ is a supersolution of \eqref{p}. Thus we can apply the comparison principle to deduce $$ h(t) \leqslant \bar{h}(t) \quad \mbox{and} \quad u(t,x)\leqslant \bar{u}(t,x) \leqslant u^*+M' u^*e^{-\beta^* t}\quad \mbox{ for } x\in [g(t), h(t)],\ t>T_*. $$ By the definition of $\bar{h}$ we see that, for $C_r := h(T_*+\tau)+Z_0 +KM'$, we have \begin{equation}\label{hbd} h(t)< c^*t +C_r\ \ \ \mbox{ for all } t\geqslant 0. \end{equation} For any $\varepsilon>0$, if we choose $T_1(\varepsilon) >T_*$ large such that $M' e^{-\beta^* T_1(\varepsilon)} < \varepsilon$, then we have \begin{equation}\label{v<1+epsilon/P} u(t,x)\leqslant \bar{u}(t,x) \leqslant u^*(1 +\varepsilon) ,\quad x\in [g(t), h(t)],\ t> T_1(\varepsilon), \end{equation} which ends the proof of Step 1.
$Step\ 2$. To give some lower bounds for $h(t)$ and $u(t,x)$.
Let $c$, $M$, $T$ and $\beta^*$ be as before. By shrinking $c$ if necessary, we can find $T^*>T+\tau$ large such that \begin{equation}\label{vu2} M u^* e^{-\beta^* (t-\tau)}\leqslant \frac{\rho}{2}\ \ \ \ \mbox{ for }\ t\geqslant T^*\ \mbox{ and }\ \ \ h(T^*)-cT^*\geqslant c^*\tau. \end{equation} We will define the following functions \begin{align*}
& \underline{g}(t)=ct,\ \ \underline{h}(t)=c^*(t-T^*)+cT^*-\sigma M(e^{-\beta^*T^*}-e^{-\beta^*t}),\ \ \ t\geqslant T^*,\\
& \underline{u}(t,x)=\big(1-Me^{-\beta^* t}\big)q_{c^*}(\underline{h}(t)-x),\ \ \ t\geqslant T^*,\ \ x\in[\underline{g}(t),\underline{h}(t)], \end{align*} where $\sigma$ is a positive constant to be determined later.
We will prove that $(\underline{u},\underline{g},\underline{h})$ is a subsolution to \eqref{p} for $t>T^*$. Firstly, for $t\geqslant T^*$, \[ \underline{u}\big(t,\underline{g}(t)\big)=\underline{u}(t,-ct)\leqslant u^*-M u^* e^{-\beta^* t}\leqslant u(t,-ct)=u\big(t,\underline{g}(t)\big). \] Next, we check that $\underline{h}$ and $\underline{u}$ satisfy the required conditions at $x=\underline{h}(t)$. It is obvious that $\underline{u}(t,\underline{h}(t))=0$. If we choose $\sigma$ with $\sigma\beta^*\geqslant c^*$, then \begin{eqnarray*} -\mu \underline{u}_x(t,\underline{h}(t))& = & \mu\big(1-Me^{-\beta^* t}\big)q_{c^*}'(0)=c^*\big(1-Me^{-\beta^* t}\big), \\ & > & c^*-\sigma M \beta^* e^{-\beta^* t} = \underline{h}'(t). \end{eqnarray*}
Later, let us check the initial conditions. From Lemma \ref{lem:u-to-1}, it is easy to see that \begin{align*} &\underline{h}(T^*+s)\leqslant cT^*+c^*\tau \leqslant h(T^*+s),\\ &\underline{u}(T^*+s,x)\leqslant u^*\big(1-Me^{-\beta^* (T^*+s)}\big)\leqslant u(T^*+s,x), \end{align*} for $s\in[0,\tau]$ and $x\in [\underline{g}(T^*+s),\underline{h}(T^*+s)]$.
Finally we will prove that $\underline{u}_t-\underline{u}_{xx}+ d \underline{u}-f(\underline{u}(t-\tau,x))\leqslant 0$ for $t\geqslant T^*+\tau$. Put $z=\underline{h}(t)-x$ and $q_\tau=q_{c^*}(\underline{h}(t-\tau)-x)$. It is easy to check that \begin{align*} \mathcal{N}[\underline{u}]:&=\underline{u}_t-\underline{u}_{xx}+ d \underline{u}-f(\underline{u}(t-\tau,x))\\ &\leqslant M e^{-\beta^* t}\Big\{\beta^*q_{c^*}-\sigma \beta^*\big(1-M e^{-\beta^* t}\big)q_{c^*}'+\big[f'\big(\big(1-\theta_1M e^{-\beta^* (t-\tau)}\big)q_\tau\big)e^{\beta^*\tau}- d \big]q_\tau\Big\}. \end{align*} for some $\theta_1\in(0,1)$. It follows from \eqref{qcto1} that there are two constants $z_1>0$, $k_2>0$ such that \begin{equation}\label{qqq12} q_{c^*}''(z)<0,\ \ \ q_{c^*}(z)\geqslant u^*-\frac{\rho}{2}\ \ \mbox{ and }\ \ q_{c^*}'(z-c^*\tau) \leqslant k_2 q_{c^*}'(z) \ \mbox{ for } \ z>z_1, \end{equation}
Moreover, we can compute that \begin{align*} \triangle \underline{h}(t) : = \underline{h}(t)-\underline{h}_\tau(t) = c^*\tau-\sigma M e^{-\beta^*t}(e^{\beta^*\tau}-1). \end{align*} For any given $\sigma>0$, by enlarging $T^*$ if necessary, we have that \begin{equation}\label{deh2} \triangle \underline{h}(t)\in[0,c^*\tau]\ \ \mbox{ for }\ t \geqslant T^*. \end{equation} When $\underline{h}_\tau-x>z_1$ and $t\geqslant T^*+\tau$, it then follows that \begin{align*} \mathcal{C} :& = \beta^*q_{c^*}-\sigma \beta^*\big(1-M e^{-\beta^* t}\big)q_{c^*}'+\big[f'\big(\big(1-\theta_1M e^{-\beta^* (t-\tau)}\big)q_\tau\big)e^{\beta^*\tau}- d \big]q_\tau\\
& \leqslant \big[f'\big(\big(1-\theta_1M e^{-\beta^* (t-\tau)}\big)q_\tau\big)e^{\beta^*\tau}- d +\beta^*\big]q_\tau -\sigma\beta^* q_{c^*}'+\beta^*(q_{c^*}-q_{\tau})\\ & \leqslant -\sigma\beta^* q_{c^*}'(\underline{h}(t)-x)+\beta^* q_{c^*}'(\underline{h}(t)-x-\tilde{\theta}_1\triangle \underline{h}(t))\triangle \underline{h}(t) \ \ \ (\mbox{with } \tilde{\theta}_1\in(0,1))\\ & \leqslant (k_2c^*\tau-\sigma) \beta^*q_{c^*}'(\underline{h}(t)-x)\leqslant 0 \end{align*} provided that $\sigma$ is sufficiently large, and we have used $\big(1-\theta_1 M e^{-\beta^* (t-\tau)}\big)q_\tau\in[u^*-\rho,u^*]$ and \eqref{vu2} for $t\geqslant T^*$, and \eqref{vu1}, \eqref{qqq12}, \eqref{deh2}. Thus $\mathcal{N} [\underline{u}]\leqslant 0$ in this case.
When $0\leqslant \underline{h}_\tau-x\leqslant z_1$ and $t\geqslant T^*+\tau $, for sufficiently large $\sigma$, we have $$ \mathcal{N} [\underline{u}] \leqslant M e^{-\beta^*t} \Big[\beta^*u^*-\sigma \beta^* \Big(1-\frac{\rho}{2u^*} e^{-\beta^*\tau}\Big)D'_1 + D'_2u^* e^{\beta^* \tau}\Big]\leqslant 0, $$ where $D'_1:=\min_{z\in[0,z_1+c^*\tau]}q_{c^*}'(z)>0$, $D'_2:=\max_{v\in[0,2u^*]}f'( v)$ and \eqref{deh2} are used.
Consequently, $(\underline{u},\underline{g}, \underline{h})$ is a subsolution to \eqref{p}, then the comparison principle implies that \[ \underline{h}(t)\leqslant h(t),\ \ \underline{u}(t,x)\leqslant u(t,x)\ \ \mbox{ for }\ t\geqslant T^*,\ x\in[\underline{g}(t), \underline{h}(t)], \] which yields that \begin{equation}\label{hbd2}
h(t)\geqslant \underline{h}(t) - \max_{t\in[0,T^*]}|h(t)-\underline{h}(t)| \geqslant c^*t -C_l \ \ \mbox{ for all } t\geqslant 0, \end{equation}
where $C_l = \max_{t\in[0,T^*]}|h(t)-\underline{h}(t)|+c^*T^* +\sigma M$. Combining with \eqref{hbd} we obtain \eqref{hghg1}.
On the other hand, for any $\varepsilon>0$, since $q_{c^*}(\infty) =u^*$, there exists $Z_1(\varepsilon)>0$ such that $$ q_{c^*}(z)> u^*\Big(1- \frac{\varepsilon}{2}\Big)\ \ \mbox{ for } z\geqslant Z_1(\varepsilon). $$ It follows from \eqref{hbd2} and \eqref{hbd} that $$ \underline{h}(t) -x \geqslant c^*t -C_l -x \geqslant h(t) - C_r -C_l -x \geqslant Z_1(\varepsilon)\ \mbox{ for }\ t>T^*, $$ which yields that for $(t,x)\in \Phi_1 := \{ (t,x) : ct\leqslant x\leqslant h(t) -C_r -C_l -Z_1(\varepsilon),\ t>T^*\}$, $$ u(t,x) \geqslant \underline{u} (t,x) \geqslant \big(1-M e^{-\beta^* t} \big) q_{c^*}\big(Z_1(\varepsilon)\big) \geqslant u^*\big(1-M e^{-\beta^* t} \big) \Big( 1- \frac{\varepsilon}{2} \Big). $$ Moreover, if we choose $T_2(\epsilon) >T^*$ such that $2 M e^{-\beta^* T_2(\varepsilon)} <\varepsilon$, then \begin{equation}\label{v>1-epsilonP} u(t,x)\geqslant u^*\Big( 1- \frac{\varepsilon}{2}\Big)^2 > u^*(1- \varepsilon)\ \ \mbox{ for } (t,x)\in \Phi_1\ \mbox{ and }\ t>T_2(\varepsilon), \end{equation} which completes the proof of Step 2.
$ Step\ 3$. Completion of the proof of \eqref{ughu1}. Denote $T_\varepsilon :=T_1(\varepsilon)+T_2(\varepsilon)$ and $Z_\varepsilon := C_r + C_l +Z_1(\varepsilon)$, then by \eqref{v<1+epsilon/P} and \eqref{v>1-epsilonP} we have $$
|u(t,x)-u^*| \leqslant u^*\varepsilon\ \ \mbox{ for } 0\leqslant x\leqslant h(t) -Z_\varepsilon,\ t>T_\varepsilon. $$ This yields the estimate in \eqref{ughu1}, which completes the proof of this proposition. \end{proof}
Using a similar argument as above we can obtain the following result. \begin{prop}\label{pro:sigma12} Assume that spreading happens for the solution $(u,g,h)$. Then \begin{itemize} \item[(i)] there exists $C'>0$ such that \begin{equation}\label{hhgg1}
|g(t)+c^*t |\leqslant C' \ \ \mbox{for all } t\geqslant0 ; \end{equation}
\item[(ii)] for any small $\varepsilon>0$, there exists $Z'_\varepsilon>0$ and $T'_\epsilon >0$ such that \begin{equation}\label{uugghh1}
\|u(t,\cdot ) - u^* \|_{L^\infty ([g(t)+Z'_\varepsilon,0])} \leqslant u^*\varepsilon \ \ \mbox{ for } t> T'_\varepsilon. \end{equation} \end{itemize} \end{prop}
\subsection{Asymptotic profiles of the spreading solutions}\label{sub52} This subsection is devoted to the proof of Theorem \ref{thm:profile of spreading sol}. We will prove this theorem by a series of results. Firstly, it follows from Proposition \ref{pro:sigma01} that there exist positive constant $C$ such that \[ -C\leqslant h(t)-c^*t\leqslant C\ \ \mbox{ for } t\geqslant 0. \] Let us use the moving coordinate $y :=x-c^*t+2C$ and set $$ \begin{array}{l} h_1(t):= h(t)-c^*t+2C, \quad g_1(t) :=g(t)-c^*t+2C\ \ \mbox{ for } t\geqslant 0,\\ \mbox{and } u_1(t,y):= u (t, y+c^*t-2C)\ \ \mbox{ for } y\in[g_1(t),h_1(t)],\ t\geqslant 0. \end{array} $$ Then $({u_1},{g_1},{h_1})$ solves \begin{equation}\label{pWH} \left\{ \begin{array}{ll}
(u_1)_t =(u_1)_{yy}+c^*(u_1)_y- d u_1+ f(u_1(t-\tau,y+c^*\tau)), & {g_1}(t)<y<{h_1}(t),\ t>0,\\
{u_1}(t, y)= 0,\ {g'_1}(t)=-\mu (u_1)_y(t,y)-c^*, & y={g_1}(t),\ t>0,\\
{u_1}(t, y)= 0,\ {h'_1}(t)=-\mu (u_1)_y(t,y)-c^*, & y={h_1}(t),\ t>0. \end{array} \right. \end{equation} Let $t_n\to\infty$ be an arbitrary sequence satisfying $t_n>\tau$ for $n\geqslant1$. Define \[ v_n(t,y)=u_1(t+t_n,y),\ \ H_n(t)=h_1(t+t_n), \ \ k_n(t)=g_1(t+t_n). \] \begin{lem}\label{limn1} Subject to a subsequence, \begin{equation}\label{vhgt1}
H_n(t)\to H\ \ in \ C^{1+\frac{\nu}{2}}_{loc}(\mathbb{R})\ \ \ and \ \ \ \|v_n-V\|_{C^{\frac{1+\nu}{2},1+ \nu}_{loc}(\Omega_n)}\to 0, \end{equation} where $\nu\in(0,1)$, $\Omega_n=\{(t,y)\in\Omega: \ y\leqslant H_n(t)\}$, $\Omega=\{(t,y): \ -\infty<y\leqslant H(t), t\in\mathbb{R}\}$, and $(V(t,y),H(t))$ satisfies \begin{equation}\label{VGHQ} \left\{ \begin{array}{ll}
V_t =V_{yy}+c^*V_{y}- d V+ f(V(t-\tau,y+c^*\tau)), & (t,y)\in\Omega,\\
V (t, H(t))= 0,\ H'(t)=-\mu V_{y} (t,H(t))-c^*, & t\in\mathbb{R}. \end{array} \right. \end{equation} \end{lem} \begin{proof} It follows from the proof of Lemma \ref{lem:global} that there is $C_0>0$ such that $0<h'(t)\leqslant C_0$ for all $t>0$. One can deduce that \[ -c^*<H_n'(t)\leqslant C_0 \ \ \mbox{ for } t+t_n\ \mbox{ large and every } n\geqslant1. \] Define \[ z=\frac{y}{H_n(t)},\ \ \ w_n(t,z)=v_n(t,y), \] and direct computations yield that \[(w_n)_t =\frac{1}{H^2_n(t)}(w_n)_{zz}+\frac{c^*+zH'_n(t)}{H_n(t)}(w_n)_z- d w_n+ f\Big(w_n\Big(t-\tau,\frac{H_n(t)z+c^*\tau}{H_n(t-\tau)}\Big)\Big)\] for $\frac{k_n(t)}{H_n(t)} <z<1$, $t>\tau-t_n$, and \[ w_n(t,1)=0,\ \ H_n'(t)=-\mu \frac{(w_n)_z(t,1)}{H_n(t)}-c^*,\ \ t>\tau-t_n. \] Since $w_n\leqslant u^*$, then $f\Big(w_n\Big(t-\tau,\frac{H_n(t)z+c^*\tau}{H_n(t-\tau)}\Big)\Big)$ is bounded. For any given $Z>0$ and $T_0\in\mathbb{R}$, using the partial interior-boundary $L^p$ estimates and the Sobolev embedding theorem (see \cite{DMZ2, Fr}), for any $\nu'\in(0,1)$, we obtain \[
\|w_n\|_{C^{\frac{1+\nu'}{2},1+ \nu'}([T_0,\infty)\times[-Z,1])}\leqslant C_Z\ \ \mbox{ for all large } n, \] where $C_Z$ is a positive constant depending on $Z$ and $\nu'$ but independent of $n$ and $T_0$. Thanks to this, we have \[
\|H_n\|_{C^{1+\frac{\nu'}{2}}([T_0,\infty))}\leqslant C_1\ \ \mbox{ for all large } n, \] with $C_1$ is a positive constant independent of $n$ and $T_0$. Hence by passing to a subsequence we may assume that as $n\to\infty$, \[ w_n\to W\ \ \mbox{ in } C_{loc}^{\frac{1+\nu}{2},1+ \nu}(\mathbb{R}\times(-\infty,1]),\ \ \ H_n\to H\ \ \mbox{ in } C_{loc}^{1+\frac{\nu}{2}}(\mathbb{R}), \] where $\nu\in(0,\nu')$. Based on above results, we can see that $(W,H)$ satisfies that \[ \left\{ \begin{array}{ll}
W_t =\frac{W_{zz}}{H^2(t)}+\frac{c^*+zH'(t)}{H(t)}W_{z}- d W+ f(W(t-\tau,H(t)z+c^*\tau)), & (t,z)\in(-\infty,1]\times\mathbb{R},\\
W (t, 1)= 0,\ \ \ \ H'(t)=-\mu \frac{W_{z} (t,1)}{H(t)}-c^*, & t\in\mathbb{R}. \end{array} \right. \] Define $V(t,y)=W\big(t, \frac{y}{H(t)}\big)$. It is easy to check that $(V,H)$ satisfies \eqref{VGHQ} and \eqref{vhgt1} holds. \end{proof}
Later, we show by a sequence of lemmas that $H(t)\equiv H_0$ is a constant and hence \[ V(t,y)=q_{c^*}(H_0-y). \]
Since $C\leqslant h(t)-c^*t+2C\leqslant 3C$ for all $t\geqslant0$, then $C\leqslant H(t)\leqslant 3C$ for $t\in\mathbb{R}$. Denote \[ \phi(z):=q_{c^*}(-z)\ \ \mbox{ for } z\in\mathbb{R}, \] it follows from the proof of Proposition \ref{pro:sigma01} that for $x\in[(c-c^*)(t+t_n),H_n(t)]$ and $t+t_n$ large, \[
\big(1-Me^{-\beta^* (t+t_n)}\big)\phi(y-C)\leqslant v_n(t,y)\leqslant \min\Big\{\big(1+M'e^{-\beta^* (t+t_n)}\big)\phi(y-3C),\ u^*\Big\}. \] Letting $n\to\infty$ we have \[ \phi(y-C)\leqslant V(t,y)\leqslant \phi(y-3C) \ \ \ \mbox{for all } t\in\mathbb{R},\ y<H(t). \] Define \[ X^*:=\inf\{X:\ V(t,y)\leqslant \phi(y-X)\ \ \mbox{ for all } (t,y)\in D\} \] and \[ X_*:=\sup\{X:\ V(t,y)\geqslant \phi(y-X)\ \ \mbox{ for all } (t,y)\in D\} \] Then \[\phi(y-X_*)\leqslant V(t,y)\leqslant \phi(y-X^*)\ \ \mbox{ for all } (t,y)\in D,\] and \[ C\leqslant X_*\leqslant\inf_{t\in\mathbb{R}} H(t)\leqslant \sup_{t\in\mathbb{R}} H(t)\leqslant X^*\leqslant 3C. \] By a similar argument as in \cite{DMZ2}, we have the following result. \begin{lem}\label{limn2} $X^*=\sup_{t\in\mathbb{R}} H(t)$, $X_*=\inf_{t\in\mathbb{R}} H(t)$, and there exist two sequences $\{s_n\}$, $\{\tilde{s}_n\}\subset \mathbb{R}$ such that \[ H(t+s_n)\to X^*,\ \ V(t+s_n,y)\to \phi(y-X^*)\ \ \mbox{ as } n\to\infty \] uniformly for $(t,y)$ in compact subsets of $\mathbb{R}\times(-\infty,X^*]$, and \[ H(t+\tilde{s}_n)\to X_*,\ \ V(t+\tilde{s}_n,y)\to \phi(y-X_*)\ \ \mbox{ as } n\to\infty \] uniformly for $(t,y)$ in compact subsets of $\mathbb{R}\times(-\infty, X_*]$. \end{lem} Based on Lemma \ref{limn2}, we have the following lemma. \begin{lem}\label{limn21} $X^*=X_*$, and hence $H(t)\equiv H_0$ is a constant, which yields $V(t,y)=\phi (y-H_0)$. \end{lem} \begin{proof} Argue indirectly we may assume that $X_*<X^*$. Choose $\epsilon=(X^*-X_*)/4$. We will show next that there is $T_\epsilon>0$ such that \begin{equation}\label{HGH1} H(t)-X^*\geqslant -\epsilon\ \ \mbox{ and }\ \ H(t)-X_*\leqslant \epsilon\ \ \mbox{ for } t\geqslant T_\epsilon, \end{equation} which implies that $X^*-X_*\leqslant 2\epsilon$. This contraction would complete the proof.
To complete the proof, we need to prove that for given $\epsilon=(X^*-X_*)/4$, there exist $n_1(\epsilon)$ and $n_2(\epsilon)$ such that \[ H(t)-X^*\geqslant -\epsilon\ \ (\forall t\geqslant s_{n_1}),\ \ \ H(t)-X_*\leqslant \epsilon\ \ (\forall t\geqslant \tilde{s}_{n_2}). \] It follows from $\phi(y-X_*)\leqslant V(t,y)\leqslant \phi(y-X^*)$ that there exist $C_1>0$ and $\beta_1>0$ such that \[
|u^*-V(t,y)|\leqslant C_1e^{\beta_1y}. \] By Lemma \ref{limn2}, for any $\varepsilon>0$, there exist $K>0$, $T>0$ such that for $\tilde{s}_n>T+\tau$ and $s\in[0,\tau]$, \begin{equation}\label{UHG1}
\sup_{y\in(-\infty,K]}|V(\tilde{s}_n+s,y)-\phi(y-X^*)|<\varepsilon \end{equation}
Set $G(t)=H(t)+c^*t$ and $U(t,y)=V(t,y-c^*t)$, then $(W,G)$ satisfies \begin{equation}\label{UGu} \left\{ \begin{array}{ll}
U_t =U_{yy}- d U+ f(U(t-\tau,y)), & t\in\mathbb{R},\ y\leqslant G(t),\\
U (t, G(t))= 0,\ \ \ \ G'(t)=-\mu U_y (t,G(t)), & t\in\mathbb{R}. \end{array} \right. \end{equation} It follows from Lemma \ref{limn2} and \eqref{UHG1} that there is $n_1=n_1(\varepsilon)$ such that for $n\geqslant n_1$, \begin{eqnarray} & H(\tilde{s}_n+s)\leqslant X_*+\varepsilon\ \ \mbox{ for } s\in[0,\tau], \label{GR1}\\ & V(\tilde{s}_n+s,y)\leqslant \phi(y-X_*-\varepsilon)+\varepsilon\ \ \mbox{ for } s\in[0,\tau],\ y\leqslant X_*. \label{UR1} \end{eqnarray}
Thanks to {\bf (H)}, for $\beta_0\in(0,\beta^*)$ small with $\beta^*$ is given in the proof of Proposition \ref{pro:sigma01}, there is $\eta>0$ small such that \begin{equation}\label{vuf1}
d -f'(v)e^{\beta_0 \tau}\geqslant \beta_0\ \ \ \mbox{ for } v\in[u^*-\eta,u^*+\eta], \end{equation} and we can find $N>1$ independent of $\varepsilon$ satisfies \[
\phi(y-X_*-\varepsilon)+\varepsilon \leqslant \big(1+N\varepsilon e^{-\beta_0\tau}\big)\phi(y-X_*-N\varepsilon)\ \
\mbox{ for } y\leqslant X_*+\varepsilon, \]
Let us construct the following supersolution of problem \eqref{UGu}: $$ \begin{array}{l} \bar{G}(t):= X_*+N\varepsilon+c^*t+N\sigma\varepsilon\big(1-e^{-\beta_0(t-\tilde{s}_n)}\big),\\ \bar{U}(t,y):=\min\big\{\big(1+N\varepsilon e^{-\beta_0(t-\tilde{s}_n)}\big)\phi\big(y-\bar{G}(t)\big),\ u^*\big\}. \end{array} $$ Since $\lim_{y\to-\infty}\big(1+N\varepsilon e^{-\beta_0(t-\tilde{s}_n)}\big)\phi\big(y-\bar{G}(t)\big)>u^*$, then there is a smooth function $\bar{K}(t)$ of $t\geqslant \tilde{s}_n$ such that $\bar{K}(t)\to-\infty$ as $t\to\infty$ and $\big(1+N\varepsilon e^{-\beta_0(t-\tilde{s}_n)}\big)\phi\big(\bar{K}(t)-\bar{G}(t)\big)=u^*$. We will check that $(\bar{U},\bar{K},\bar{G})$ is a supersolution for $t\geqslant \tilde{s}_n+\tau$ and $y\in[\bar{K}(t),\bar{G}(t)]$. We note that \[ \bar{U}(t,y)=\big(1+N\varepsilon e^{-\beta_0(t-\tilde{s}_n)}\big)\phi\big(y-\bar{G}(t)\big)\ \mbox{ when }\ y\in[\bar{K}(t),\bar{G}(t)]. \] Firstly, it follows from \eqref{GR1} that for $s\in[0,\tau]$, \[ G(\tilde{s}_n+s)\leqslant X_*+\varepsilon+c^*(\tilde{s}_n+s)\leqslant X_*+N\varepsilon+c^*(\tilde{s}_n+s)\leqslant \bar{G}(\tilde{s}_n+s). \] In view of \eqref{UR1}, we have \begin{align*} \bar{U}(\tilde{s}_n+s,y)&=\big(1+N\varepsilon e^{-\beta_0s}\big)\phi\big(y-\bar{G}(\tilde{s}_n+s)\big)\\ &\geqslant\big(1+N\varepsilon e^{-\beta_0\tau}\big)\phi\big(y-X_*-N\varepsilon-c^*(\tilde{s}_n+s)\big)\\ &\geqslant\phi\big(y-X_*-\varepsilon-c^*(\tilde{s}_n+s)\big)+\varepsilon\\ &\geqslant V\big(\tilde{s}_n+s,y-c^*(\tilde{s}_n+s)\big)=U(\tilde{s}_n+s,y ). \end{align*} for $s\in[0,\tau]$ and $y\leqslant G(\tilde{s}_n+s)$. By definition $\bar{U}(t,\bar{G}(t))=0$ and direct computation yields \begin{eqnarray*} -\mu \bar{U} _y(t,\bar{G}(t))& = & c^*\big(1+N\varepsilon e^{-\beta_0( t-\tilde{s}_n)}\big), \\ & < & c^*+N\varepsilon \sigma \beta_0 e^{-\beta_0 ( t-\tilde{s}_n)} = \bar{G}'(t), \end{eqnarray*} if we choose $\sigma$ with $\sigma\beta_0 > c^*$. Since $U\leqslant u^*$, it then follows from the definition of $\bar{K}(t)$ that $\bar{U}(t,\bar{K}(t))=u^*\geqslant U(t,\bar{K}(t))$.
Finally, let us show \begin{equation}\label{uhupper}
\mathcal{N} [\bar{U}] := \bar{U}_t - \bar{U}_{yy} + d \bar{U}-f(\bar{U}(t-\tau,y)) \geqslant 0,\quad y\in [\bar{K}(t), \bar{G}(t)],\ t>\tilde{s}_n+\tau. \end{equation} Put $z:=y-\bar{G}(t)$, $\zeta(t):=N\varepsilon e^{-\beta_0( t-\tilde{s}_n)}$ and $\phi_\tau:=\phi\big(y-\bar{G}(t-\tau)\big)$. It is easy to compute that \begin{align*} \mathcal{N} [\bar{U}]&=\zeta\Big\{f(\phi_\tau)-\beta_0\phi-\sigma\beta_0(1+\zeta)\phi'-f'\big(\big(1+\theta_2\zeta e^{\beta_0\tau}\big)\phi_\tau\big)e^{\beta_0\tau}\phi_\tau\Big\}\ \ \ (\mbox{with } \theta_2\in(0,1))\\ &\geqslant\zeta\Big\{-\sigma\beta_0(1+\zeta)\phi'-\big[f'\big(\big(1+\theta_2\zeta e^{\beta_0\tau}\big)\phi_\tau\big)e^{\beta_0\tau}- d \big]\phi_\tau-\beta_0\phi\Big\}. \end{align*}
Since \[ \phi(z)\to u^*\ \mbox{ and } \frac{(\phi(z)-u^*)'}{\phi(z)-u^*}\to k^*\ \ \mbox{ as } z\to -\infty \] where $k^*:=c^*-\sqrt{(c^*)^2+4( d -f'(u^*))}<0$, there are two constants $z_\eta<0$ and $k_0$ such that \begin{equation}\label{qqq122} \phi''(z)>0,\ \ \ \phi(z)\geqslant u^*-\eta\ \ \mbox{ and }\ \ \phi'(z-2c^*\tau) \geqslant k_0 \phi'(z) \ \mbox{ for } \ z<z_\eta, \end{equation} Moreover, we can compute that \begin{align*} \triangle \bar{G}(t) :& = \bar{G}(t)-\bar{G}(t-\tau) = c^*\tau+N\sigma \varepsilon e^{-\beta_0(t-\tilde{s}_n)}(e^{\beta_0\tau}-1). \end{align*} For any given $\sigma>0$, by shrinking $\varepsilon$ if necessary, we have that \begin{equation}\label{deh22} \triangle \bar{G}(t)\in[c^*\tau,2c^*\tau]\ \ \mbox{ for }\ t > \tilde{s}_n+\tau. \end{equation}
For $y-\bar{G}(t-\tau)\leqslant z_\eta$ and $t>\tilde{s}_n+\tau$, direct calculation implies \begin{align*} \mathcal{N} [\bar{U}] &\geqslant\zeta\big\{-\sigma\beta_0(1+\zeta)\phi'-\big[f'\big(\big(1+\theta_2\zeta e^{\beta_0\tau}\big)\phi_\tau\big)e^{\beta_0\tau}- d \big]\phi_\tau-\beta_0\phi\big\}\\
& \geqslant \zeta\big\{\big[d-f'\big(\big(1+\theta_2\zeta e^{\beta_0\tau}\big)\phi_\tau\big)e^{\beta_0\tau}-\beta_0\big]\phi_\tau-\sigma\beta_0\phi'+\beta_0(\phi_\tau-\phi)\big\}\\ &\geqslant \zeta\big[\beta_0 \phi'(y-\bar{G}(t)+\tilde{\theta}_2\triangle\bar{G}(t))\triangle\bar{G}(t)-\sigma\beta_0\phi'(y-\bar{G}(t))\big]\ \ \ (\mbox{with } \tilde{\theta}_2\in(0,1))\\ & \geqslant \zeta (2k_0c^*\tau-\sigma) \beta_0\phi'(y-\bar{G}(t)) \geqslant 0 \end{align*} provided that $\sigma$ is sufficiently large, and we have used $\big(1+ \theta_2 \zeta e^{\beta_0\tau}\big)\phi_\tau\in[u^*-\eta,u^*+\eta]$ for $t>\tilde{s}_n+\tau$, \eqref{vuf1}, $\phi'(z)\leqslant0$ for $z\leqslant z_\eta$, \eqref{qqq122} and \eqref{deh22}.
When $z_\eta\leqslant y-\bar{G}(t-\tau)\leqslant 0$ and $t>\tilde{s}_n+\tau$, for sufficiently large $\sigma$, we have \[ \mathcal{N} [\bar{U}]\geqslant\zeta \big[-\sigma\beta_0 C_z-u^*e^{\beta_0\tau}C_f -\beta_0u^* \big]\geqslant0. \] where $C_z:=\max_{z\in[0,z_\eta+2c^*\tau]}\phi'(z)<0$, $C_f:=\max_{v\in[0,2u^*]}f'(v)$, and \eqref{deh22} are used.
Thus \eqref{uhupper} holds, then we can apply the comparison principle to conclude that \[ U(t,y)\leqslant \bar{U}(t,y),\ \ \ G(t)\leqslant \bar{G}(t)\ \ \mbox{ for }\ y\in[\bar{K}(t),\bar{G}(t)]\ \mbox{and } t>\tilde{s}_n+\tau. \] This, together with the definition of $H(t)$, yields that $H(t)\leqslant X_*+N\varepsilon(1+\sigma)$ for $t>\tilde{s}_n+\tau$. By shrinking $\varepsilon$ if necessary, we obtain \begin{equation}\label{Hsu1} H(t)\leqslant X_*+\epsilon\ \ \ \mbox{ for }\ t>\tilde{s}_n+\tau\ \mbox{and } n>n_1. \end{equation}
In the following, we show $H(t)\geqslant X^*-\epsilon$ for all large $t$. As in the construction of supersolution, for any $\varepsilon>0$, there exists $n_2=n_2(\varepsilon)$ such that, for $n\geqslant n_2$, \begin{eqnarray} & H(s_n+s)\geqslant X^*-\varepsilon\ \ \mbox{ for } s\in[0,\tau], \label{GRH1}\\ & V(s_n+s,y)\geqslant \phi(y-X^*+\varepsilon)-\varepsilon\ \ \mbox{ for } s\in[0,\tau],\ y\leqslant X^*-\varepsilon. \label{URH1} \end{eqnarray} We also can find $N_0>1$ independent of $\varepsilon$ such that \[
\phi(y-X^*+\varepsilon)-\varepsilon \geqslant (1-N_0\varepsilon e^{-\beta_0\tau})\phi(y-X^*+N_0\varepsilon)\ \ \mbox{ for } y\leqslant X^*-\varepsilon, \] We can define a subsolution as follows: $$ \begin{array}{l} \underline{G}(t):= X^*-N_0\varepsilon+c^*t-N_0\sigma\varepsilon\big(1-e^{-\beta_0(t-s_n)}\big),\\ \underline{U}(t,y):=\big(1-N_0\varepsilon e^{-\beta_0(t-s_n)}\big)\phi\big(y-\underline{G}(t)\big). \end{array} $$ Since $U(t,y)\geqslant\phi(y-X_*)$, there are $C_0$ and $\alpha>0$ such that $V(t,y)\geqslant u^*-C_0e^{\alpha y}$ for all $y\leqslant0$, which implies that $U(t,y)\geqslant u^*-C_0e^{\alpha (y-c^*t)}$. Let us fix $c\in(0,c^*)$ such that $\beta_0\leqslant \alpha(c+c^*)$. By enlarging $n$ if necessary we may assume that $C_0\leqslant u^*N_0\varepsilon e^{\beta_0 s_n}$. Denote $\underline{K}(t)\equiv-ct$.
By a similar argument as above and in Step 2 of Proposition \ref{pro:sigma01}, we can show that $(\underline{U},\underline{G},\underline{K})$ is a subsolution of problem \eqref{UGu} by taking $\sigma>0$ sufficiently large. The comparison principle can be used to conclude that \[ \underline{U}(t,y)\leqslant U(t,y),\ \ \ \ \underline{G}\leqslant G(t)\ \ \ \mbox{ for } t\geqslant s_n+\tau,\ y\in[-ct,\underline{G}(t)], \] which implies that $G(t)\geqslant X^*-N_0\varepsilon(1+\sigma)$ for $t\geqslant s_n+\tau$. By shrinking $\varepsilon$ if necessary, we have \[ X^*-\epsilon\leqslant G(t)\ \ \ \mbox{ for } t\geqslant s_n+\tau\ \mbox{ and }\ n\geqslant n_2. \] This completes the proof of this lemma. \end{proof} \begin{thm}\label{thm:WHG} Assume that {\bf (H)} and spreading happens. Then there exists $H_1\in\mathbb{R}$ such that \begin{equation}\label{HWt1} \lim_{t\to\infty}[h(t) - c^*t ]= H_1,\ \ \ \ \ \lim_{t\to\infty} h'(t) =c^*, \end{equation} \begin{equation}\label{WHt1}
\lim\limits_{t\to\infty} \| u(t,\cdot)- q_{c^*}(c^*t+H_1-\cdot)\| _{L^\infty ( [0, h(t)])}=0, \end{equation} where $(c^*, q_{c^*})$ be the unique solution of \eqref{sw11}. \end{thm} \begin{proof} It follows from Lemmas \ref{limn1} and \ref{limn21} that for any $t_n\to\infty$, by passing to a subsequence, $h(t+t_n)-c^*(t+t_n)\to H_1:=H_0-2C$ in $C_{loc}^{1+\frac{\nu}{2}}(\mathbb{R})$. The arbitrariness of $\{t_n\}$ implies that $h(t)-c^*t\to H_1$ and $h'(t)\to c^*$ as $t\to\infty$, which proves \eqref{HWt1}.
In what follows, we use the moving coordinate $z:= x-h(t)$ to prove \eqref{WHt1}. Set $$ g_2(t) := g(t)-h(t), \ \ \ \ \ \ u_2(t,z) := u(t, z+h(t))\ \ \mbox{ for }\ z\in [g_2 (t), 0],\ t\geqslant \tau, $$ \[ \tilde{g}_n(t)=g(t+t_n)-h(t+t_n),\ \ \ \tilde{h}_n(t)=h(t+t_n),\ \ \ \ \tilde{u}_n(t,z)=u_2(t+t_n,z), \] then the pair $(\tilde{u}_n, \tilde{g}_n,\tilde{h}_n)$ solves \begin{equation}\label{p u2} \left\{ \begin{array}{ll}
(\tilde{u}_n)_t =(\tilde{u}_n)_{zz}+\tilde{h}_n'(\tilde{u}_n)_z+ f(\tilde{u}_n(t-\tau,z+\tilde{h}_n(t)-\tilde{h}_n(t-\tau))- d \tilde{u}_n, & z\in(\tilde{g}_n(t),0),\ t>\tau,\\
{\tilde{u}_n}(t, z)= 0,\ \ {\tilde{g}'_n}(t)=- \mu (\tilde{u}_n)_z (t,z)-\tilde{h}_n'(t), & z={\tilde{g}_n}(t),\ t>\tau,\\
{\tilde{u}_n}(t, 0)= 0,\ \ \tilde{h}_n'(t) = -\mu (\tilde{u}_n)_z(t,0), & t>\tau.
\end{array} \right. \end{equation} By the same reasoning as in the proof of Lemma \ref{limn1}, the parabolic regularity to \eqref{p u2} plus the Sobolev embedding theorem can be used to conclude that, by passing to a further subsequence if necessary, as $n\to\infty$, $\tilde{u}_n\to W$ in $\ C_{loc}^{\frac{1+\nu}{2},1+\nu}(\mathbb{R}\times(-\infty,0])$, and $W$ satisfies, in view of $\tilde{h}'_n(t)\to c^*$, \[ \left\{ \begin{array}{ll}
W_t =W_{zz}+c^*W_z- d W+ f(W(t-\tau,z+c^*\tau)), & -\infty<z<0,\ t\in \mathbb{R},\\
W (t, 0)= 0, \ c^*=-\mu W_z (t,0), & t\in \mathbb{R}. \end{array} \right. \] This is equivalent to \eqref{VGHQ} with $V=W$ and $H=0$. Hence we can conclude \[ W(t,z)\equiv \phi(z)\ \ \ \mbox{ for } (t,z)\in\mathbb{R}\times(-\infty,0]. \] Thus we have proved that, as $n\to\infty$, \[ u(t+t_n,z+h(t+t_n))-q_{c^*}(-z)\to0 \ \ \ \mbox{ in } C_{loc}^{\frac{1+\nu}{2},1+\nu}(R\times(-\infty,0]). \] This, together with the arbitrariness of $\{t_n\}$, yields that \[ \lim_{t\to\infty}[u(t,z+h(t))-q_{c^*}(-z)]=0\ \ \mbox{ uniformly for } z \mbox{ in compact subsets of } (-\infty,0]. \] Then, for any $L>0$, \[
\|u(t,\cdot)-q_{c^*}(h(t)-\cdot)\|_{L^\infty ([h(t)-L,h(t)])}\to0 \ \quad \mbox{ as } t\to \infty. \] Using the limit $h(t)-c^*t\to H_1$ as $t\to\infty$ we obtain \begin{equation}\label{u to U near h(t)}
\|u(t, \cdot) - q_{c^*}(c^*t+H_1 -\cdot)\|_{L^\infty ([h(t)-L,h(t)])} \to 0\ \quad \mbox{ as } t\to \infty. \end{equation}
Finally we prove \eqref{WHt1}. For any given small $\varepsilon >0$, it follows from \eqref{ughu1} in Proposition \ref{pro:sigma01} that there exist two positive constants $Z_\varepsilon$ and $T_\varepsilon$ such that $$
|u(t,x) - u^*| \leqslant u^*\varepsilon \ \quad \mbox{ for }\ 0\leqslant x\leqslant h(t) - Z_\varepsilon,\ t>T_\varepsilon. $$ Since $q_{c^*}(z)\to u^*$ as $z\to\infty$, there exists $Z^*_\varepsilon > Z_\varepsilon$ such that $$
|q_{c^*}(c^*t +H_1 -x) -u^*| \leqslant u^*\varepsilon\ \quad \mbox{ for }\ x\leqslant c^*t+ 2H_1 -Z^*_\varepsilon. $$ Taking $T^*_\varepsilon >T_\varepsilon$ large such that $h(t) <c^*t+2H_1$ for $t>T^*_\varepsilon$, then we obtain $$
|u(t,x) - q_{c^*}(c^*t +H_1 -x) | \leqslant 2u^*\varepsilon\ \quad \mbox{ for }\ 0\leqslant x\leqslant h(t)-Z^*_\varepsilon, \ t> T^*_\varepsilon. $$ Taking $L= Z^*_\varepsilon$ in \eqref{u to U near h(t)} we see that for some $T^{**}_\varepsilon >T^*_\varepsilon$, we have $$
|u(t, x) - q_{c^*}(c^*t +H_1 -x)| \leqslant u^*\varepsilon\ \quad \mbox{ for }\ h(t) -Z^*_\varepsilon \leqslant x \leqslant h(t),\ t>T^{**}_\varepsilon. $$ This completes the proof of \eqref{WHt1}. \end{proof}
Taking use of a similar argument as above one can obtain the following result.
\begin{thm}\label{thm:WGH} Assume that {\bf (H)} and spreading happens. Then there exists $G_1\in\mathbb{R}$ such that \begin{equation}\label{HWgt1} \lim_{t\to\infty}[g(t) + c^*t ]= G_1,\ \ \ \ \ \lim_{t\to\infty} g'(t) =-c^*, \end{equation} \begin{equation}\label{WHtg1}
\lim\limits_{t\to\infty} \| u(t,\cdot)- q_{c^*}(c^*t-G_1+\cdot)\| _{L^\infty ( [g(t), 0])}=0, \end{equation} where $(c^*, q_{c^*})$ be the unique solution of \eqref{sw11}. \end{thm}
\noindent {\bf Proof of Theorem \ref{thm:profile of spreading sol}}. The results in Theorem \ref{thm:profile of spreading sol} follow from Theorems \ref{thm:WHG} and \ref{thm:WGH}.
$\square$
\end{document} |
\begin{document}
\maketitle
\begin{abstract} Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations.
{\noindent \textsc{R\'esum\'e}.} Grosso modo, en utilisant le programme des mod\`eles minimaux semi-stables, nous montrons que la partie modulaire d'une fibration lc-triviale co\"incide avec celle d'une fibration klt-triviale induite par adjonction apr\'es changement de base par un morphisme g\'en\'eriquement fini. Comme application, eu utilisant le r\'esultat de Ambro sur fibrations klt-triviales, on obtient que la partie modulaire d'une fibration lc-triviale est b-nef et abondante. \end{abstract}
\tableofcontents \section{Introduction}
In this paper, we prove the following theorem. More precisely, we reduce Theorem \ref{main} to Ambro's result (see \cite[Theorem 3.3]{ambro2}) by using the semi-stable minimal model program (see, for example, \cite{fujino3}). For a related result, see \cite[Theorem 1.4]{floris}.
\begin{thm}[{cf.~\cite[Theorem 3.3]{ambro2}}]\label{main} Let $f:X\to Y$ be a projective surjective morphism between normal projective varieties with connected fibers. Assume that $(X, B)$ is log canonical and $K_X+B\sim _{\mathbb Q, Y}0$. Then the moduli $\mathbb Q$-b-divisor $\mathbf M$ is b-nef and abundant. \end{thm}
Let us recall the definition of {\em{b-nef and abundant}} $\mathbb Q$-b-divisors.
\begin{defn}[{\cite[Definition 3.2]{ambro2}}] A $\mathbb Q$-b-divisor $\mathbf M$ of a normal complete algebraic variety $Y$ is called {\em{b-nef and abundant}} if there exists a proper birational morphism $Y'\to Y$ from a normal variety $Y'$, endowed with a proper surjective morphism $h:Y'\to Z$ onto a normal variety $Z$ with connected fibers, such that: \begin{itemize} \item[(1)] $\mathbf M_{Y'}\sim _\mathbb Q h^*H$, for some nef and big $\mathbb Q$-divisor $H$ of $Z$; \item[(2)] $\mathbf M=\overline {\mathbf M_{Y'}}$. \end{itemize} \end{defn}
Let us quickly explain the idea of the proof of Theorem \ref{main}. We assume that the pair $(X, B)$ in Theorem \ref{main} is dlt for simplicity. Let $W$ be a log canonical center of $(X, B)$ which is dominant onto $Y$ and is minimal over the generic point of $Y$.
We set $K_W+B_W=(K_X+B)|_W$ by adjunction. Then we have $K_W+B_W\sim _{\mathbb Q, Y}0$.
Let $h:W\to Y'$ be the Stein factorization of $f|_W:W\to Y$. Note that $(W, B_W)$ is klt over the generic point of $Y'$. We prove that the moduli part $\mathbf M$ of $f:(X, B)\to Y$ coincides with the moduli part $\mathbf M^{\min}$ of $h:(W, B_W)\to Y'$ after taking a suitable generically finite base change by using the semi-stable minimal model program. We do not need the {\em{mixed}} period map nor the infinitesimal {\em{mixed}} Torelli theorem (see \cite[Section 2]{ambro2} and \cite{ssu}) for the proof of Theorem \ref{main}. We just reduce the problem on lc-trivial fibrations to Ambro's result on klt-trivial fibrations, which follows from the theory of period maps. Our proof of Theorem \ref{main} partially answers the questions in \cite[8.3.8 (Open problems)]{kollar}.
It is conjectured that $\mathbf M$ is b-semi-ample (see, for example, \cite[0.~Introduction]{ambro1}, \cite[Conjecture 7.13.3]{ps}, \cite{floris}, \cite{birkar-chen}, and \cite[Section 3]{fujino10}). The b-semi-ampleness of the moduli part has been proved only for some special cases (see, for example, \cite{kawamata}, \cite{fujino1}, and \cite[Section 8]{ps}). See also Remark \ref{41} below.
\end{ack}
We will work over $\mathbb C$, the complex number field, throughout this paper. We will make use of the standard notation as in \cite{fujino-funda}.
\section{Preliminaries}
Throughout this paper, we do not use $\mathbb R$-divisors. We only use $\mathbb Q$-divisors.
\begin{say}[Pairs] A pair $(X, B)$ consists of a normal variety $X$ over $\mathbb C$ and a $\mathbb Q$-divisor $B$ on $X$ such that $K_X+B$ is $\mathbb Q$-Cartier. A pair $(X, B)$ is called {\em{subklt}} (resp.~{\em{sublc}}) if for any projective birational morphism $g:Z\to X$ from a normal variety $Z$, every coefficient of $B_Z$ is $<1$ (resp.~$\leq 1$) where $K_Z+B_Z:=g^*(K_X+B)$. A pair $(X, B)$ is called {\em{klt}} (resp.~{\em{lc}}) if $(X, B)$ is subklt (resp.~sublc) and $B$ is effective. Let $(X, B)$ be an lc pair. If there is a log resolution $g:Z\to X$ of $(X, B)$ such that $\mathrm{Exc}(g)$ is a divisor and that the coefficients of the $g$-exceptional part of $B_Z$ are $<1$, then the pair $(X, B)$ is called {\em{divisorial log terminal}} ({\em{dlt}}, for short). Let $(X, B)$ be a sublc pair and let $W$ be a closed subset of $X$. Then $W$ is called a {\em{log canonical center}} of $(X, B)$ if there are a projective birational morphism $g:Z\to X$ from a normal variety $Z$ and a prime divisor $E$ on $Z$ such that $\mult _EB_Z=1$ and that $g(E)=W$. Moreover we say that $W$ is {\em{minimal}} if it is minimal with respect to inclusion. \end{say}
In this paper, we use the notion of {\em{b-divisors}} introduced by Shokurov. For details, we refer to \cite[2.3.2]{corti} and \cite[Section 3]{fujino4}.
\begin{say} [Canonical b-divisors] Let $X$ be a normal variety and let $\omega$ be a top rational differential form of $X$. Then $(\omega)$ defines a b-divisor $\mathbf K$. We call $\mathbf K$ the {\em{canonical b-divisor}} of $X$. \end{say}
\begin{say}[$\mathbf A(X, B)$ and $\mathbf A^*(X, B)$] The {\em{discrepancy b-divisor}} $\mathbf A=\mathbf A(X, B)$ of a pair $(X, B)$ is the $\mathbb Q$-b-divisor of $X$ with the trace $\mathbf A_Y$ defined by the formula $$ K_Y=f^*(K_X+B)+\mathbf A_Y, $$ where $f:Y\to X$ is a proper birational morphism of normal varieties. Similarly, we define $\mathbf A^*=\mathbf A^*(X, B)$ by $$ \mathbf A_Y^*=\sum _{a_i>-1}a_i A_i $$ for $$ K_Y=f^*(K_X+B)+\sum a_i A_i, $$ where $f:Y\to X$ is a proper birational morphism of normal varieties. Note that $\mathbf A(X, B)=\mathbf A^*(X, B)$ when $(X, B)$ is subklt.
By the definition, we have $\mathcal O_X(\lceil \mathbf A^*(X, B)\rceil)\simeq \mathcal O_X$ if $(X, B)$ is lc (see \cite[Lemma 3.19]{fujino4}). We also have $\mathcal O_X(\lceil \mathbf A(X, B)\rceil) \simeq \mathcal O_X$ when $(X, B)$ is klt. \end{say}
\begin{say}[b-nef and b-semi-ample $\mathbb Q$-b-divisors] Let $X$ be a normal variety and let $X\to S$ be a proper surjective morphism onto a variety $S$. A $\mathbb Q$-b-divisor $\mathbf D$ of $X$ is {\em{b-nef over $S$}} (resp.~{\em{b-semi-ample over $S$}}) if there exists a proper birational morphism $X'\to X$ from a normal variety $X'$ such that $\mathbf D=\overline {\mathbf D_{X'}}$ and $\mathbf D_{X'}$ is nef (resp.~semi-ample) relative to the induced morphism $X'\to S$. \end{say}
\begin{say} Let $D=\sum _i d_iD_i$ be a $\mathbb Q$-divisor on a normal variety, where $D_i$ is a prime divisor for every $i$, $D_i\ne D_j$ for $i\ne j$, and $d_i\in \mathbb Q$ for every $i$. Then we set $$ D^{\geq 0}=\sum _{d_i\geq 0}d_iD_i \quad \text{and} \quad D^{\leq 0}=\sum _{d_i\leq 0}d_iD_i. $$ \end{say}
\section{A quick review of lc-trivial fibrations}
In this section, we quickly recall some basic definitions and results on {\em{klt-trivial fibrations}} and {\em{lc-trivial fibrations}} (see also \cite[Section 3]{fujino10}).
\begin{defn}[Klt-trivial fibrations]\label{def-klt} A {\em{klt-trivial fibration}} $f:(X, B)\to Y$ consists of a proper surjective morphism $f:X\to Y$ between normal varieties with connected fibers and a pair $(X, B)$ satisfying the following properties: \begin{itemize} \item[(1)] $(X, B)$ is subklt over the generic point of $Y$; \item[(2)] $\rank f_*\mathcal O_X(\lceil \mathbf A(X, B)\rceil)=1$; \item[(3)] There exists a $\mathbb Q$-Cartier $\mathbb Q$-divisor $D$ on $Y$ such that $$ K_X+B\sim _{\mathbb Q}f^*D. $$ \end{itemize} \end{defn}
Note that Definition \ref{def-klt} is nothing but \cite[Definition 2.1]{ambro1}, where a klt-trivial fibration is called an lc-trivial fibration. So, our definition of lc-trivial fibrations in Definition \ref{def-lc} is different from the original one in \cite[Definition 2.1]{ambro1}.
\begin{defn}[Lc-trivial fibrations]\label{def-lc} An {\em{lc-trivial fibration}} $f:(X, B)\to Y$ consists of a proper surjective morphism $f:X\to Y$ between normal varieties with connected fibers and a pair $(X, B)$ satisfying the following properties: \begin{itemize} \item[(1)] $(X, B)$ is sublc over the generic point of $Y$; \item[(2)] $\rank f_*\mathcal O_X(\lceil \mathbf A^*(X, B)\rceil)=1$; \item[(3)] There exists a $\mathbb Q$-Cartier $\mathbb Q$-divisor $D$ on $Y$ such that $$ K_X+B\sim _{\mathbb Q}f^*D. $$ \end{itemize} \end{defn}
In Section \ref{sec4}, we sometimes take various base changes and construct the induced lc-trivial fibrations and klt-trivial fibrations. For the details, see \cite[Section 2]{ambro1}.
\begin{say}[Induced lc-trivial fibrations by base changes]\label{33} Let $f:(X, B)\to Y$ be a klt-trivial (resp.~an lc-tirivial) fibration and let $\sigma:Y'\to Y$ be a generically finite morphism. Then we have an induced klt-trivial (resp.~lc-trivial) fibration $f':(X', B_{X'})\to Y'$, where $B_{X'}$ is defined by $\mu^*(K_X+B)=K_{X'}+B_{X'}$: $$ \xymatrix{
(X', B_{X'}) \ar[r]^{\mu} \ar[d]_{f'} & (X, B)\ar[d]^{f} \\
Y' \ar[r]_{\sigma} & Y, } $$ Note that $X'$ is the normalization of the main component of $X\times _{Y}Y'$. We sometimes replace $X'$ with $X''$ where $X''$ is a normal variety such that there is a proper birational morphism $\varphi:X''\to X'$. In this case, we set $K_{X''}+B_{X''}=\varphi^*(K_{X'}+B_{X'})$. \end{say}
Let us explain the definitions of the {\em{discriminant}} and {\em{moduli}} $\mathbb Q$-b-divisors.
\begin{say} [Discriminant $\mathbb Q$-b-divisors and moduli $\mathbb Q$-b-divisors] Let $f:(X, B)\to Y$ be an lc-trivial fibration as in Definition \ref{def-lc}. Let $P$ be a prime divisor on $Y$. By shrinking $Y$ around the generic point of $P$, we assume that $P$ is Cartier. We set $$
b_P=\max \left\{t \in \mathbb Q\, \left|\, \begin{array}{l} {\text{$(X, B+tf^*P)$ is sublc over}}\\ {\text{the generic point of $P$}} \end{array}\right. \right\} $$ and set $$ B_Y=\sum _P (1-b_P)P, $$ where $P$ runs over prime divisors on $Y$. Then it is easy to see that $B_Y$ is a well-defined $\mathbb Q$-divisor on $Y$ and is called the {\em{discriminant $\mathbb Q$-divisor}} of $f:(X, B)\to Y$. We set $$ M_Y=D-K_Y-B_Y $$ and call $M_Y$ the {\em{moduli $\mathbb Q$-divisor}} of $f:(X, B)\to Y$. Let $\sigma:Y'\to Y$ be a proper birational morphism from a normal variety $Y'$ and let $f':(X', B_{X'})\to Y'$ be the induced lc-trivial fibration by $\sigma:Y'\to Y$ (see \ref{33}). We can define $B_{Y'}$, $K_{Y'}$ and $M_{Y'}$ such that $\sigma^*D=K_{Y'}+B_{Y'}+M_{Y'}$, $\sigma_*B_{Y'}=B_Y$, $\sigma _*K_{Y'}=K_Y$ and $\sigma_*M_{Y'}=M_Y$. Hence there exist a unique $\mathbb Q$-b-divisor $\mathbf B$ such that $\mathbf B_{Y'}=B_{Y'}$ for every $\sigma:Y'\to Y$ and a unique $\mathbb Q$-b-divisor $\mathbf M$ such that $\mathbf M_{Y'}=M_{Y'}$ for every $\sigma:Y'\to Y$. Note that $\mathbf B$ is called the {\em{discriminant $\mathbb Q$-b-divisor}} and that $\mathbf M$ is called the {\em{moduli $\mathbb Q$-b-divisor}} associated to $f:(X, B)\to Y$. We sometimes simply say that $\mathbf M$ is the {\em{moduli part}} of $f:(X, B)\to Y$. \end{say}
For the basic properties of the discriminant and moduli $\mathbb Q$-b-divisors, see \cite[Section 2]{ambro1}.
Let us recall the main theorem of \cite{ambro1}. Note that a klt-trivial fibration in the sense of Definition \ref{def-klt} is called an lc-trivial fibration in \cite{ambro1}.
\begin{thm}[{see \cite[Theorem 2.7]{ambro1}}]\label{thm-klt-tri} Let $f:(X, B)\to Y$ be a klt-trivial fibration and let $\pi:Y\to S$ be a proper morphism. Let $\mathbf B$ and $\mathbf M$ be the induced discriminant and moduli $\mathbb Q$-b-divisors of $f$. Then, \begin{itemize} \item[(1)] $\mathbf K+\mathbf B$ is $\mathbb Q$-b-Cartier, that is, there exists a proper birational morphism $Y'\to Y$ from a normal variety $Y'$ such that $\mathbf {K}+\mathbf {B}=\overline{K_{Y'}+\mathbf{B}_{Y'}}$, \item[(2)] $\mathbf M$ is b-nef over $S$. \end{itemize} \end{thm}
Theorem \ref{thm-klt-tri} has some important applications, see, for example, \cite[Proof of Theorem 1.1]{fujino-kawa} and \cite[The proof of Theorem 1.1]{fujino4}.
By modifying the arguments in \cite[Section 5]{ambro1} suitably with the aid of \cite[Theorems 3.1, 3.4, and 3.9]{fujino2} (see also \cite{fujino-fujisawa}), we can generalize Theorem \ref{thm-klt-tri} as follows.
\begin{thm}\label{thm-lc-tri} Let $f:(X, B)\to Y$ be an lc-trivial fibration and let $\pi:Y\to S$ be a proper morphism. Let $\mathbf B$ and $\mathbf M$ be the induced discriminant and moduli $\mathbb Q$-b-divisors of $f$. Then, \begin{itemize} \item[(1)] $\mathbf K+\mathbf B$ is $\mathbb Q$-b-Cartier, \item[(2)] $\mathbf M$ is b-nef over $S$. \end{itemize} \end{thm}
Theorem \ref{thm-klt-tri} is proved by using the theory of variations of Hodge structure. On the other hand, Theorem \ref{thm-lc-tri} follows from the theory of variations of {\em{mixed}} Hodge structure. We do not adopt the formulation in \cite[Section 4]{fujino-pre} (see also \cite[8.5]{kollar}) because the argument in \cite{ambro1} suits our purposes better. For the reader's convenience, we include the main ingredient of the proof of Theorem \ref{thm-lc-tri}, which easily follows from \cite[Theorems 3.1, 3.4, and 3.9]{fujino2} (see also \cite{fujino-fujisawa}).
\begin{thm}[{cf.~\cite[Theorem 4.4]{ambro1}}]\label{thm-mhs} Let $f:X\to Y$ be a projective morphism between algebraic varieties. Let $\Sigma_X$ {\em{(}}resp.~$\Sigma_Y${\em{)}} be a simple normal crossing divisor on $X$ {\em{(}}resp.~$Y${\em{)}} such that $f$ is smooth over $Y\setminus \Sigma_Y$, $\Sigma_X$ is relatively normal crossing over $Y\setminus \Sigma_Y$, and $f^{-1}(\Sigma_Y)\subset \Sigma _X$. Assume that $f$ is semi-stable in codimension one. Let $D$ be a simple normal crossing divisor on $X$ such that $\Supp D\subset \Sigma_X$ and that every irreducible component of $D$ is dominant onto $Y$. Then the following properties hold. \begin{itemize} \item[(1)] $R^pf_*\omega_{X/Y}(D)$ is a locally free sheaf on $Y$ for every $p$. \item[(2)] $R^pf_*\omega_{X/Y}(D)$ is semi-positive for every $p$. \item[(3)] Let $\rho :Y'\to Y$ be a projective morphism from a smooth variety $Y'$ such that $\Sigma_{Y'}=\rho^{-1}(\Sigma_Y)$ is a simple normal crossing divisor on $Y'$. Let $\pi:X'\to X\times _YY'$ be a resolution of the main component of $X\times _YY'$ such that $\pi$ is an isomorphism over $Y'\setminus \Sigma_{Y'}$. Then we obtain the following commutative diagram{\em{:}} $$ \xymatrix{
X' \ar[r] \ar[d]_{f'} & X\ar[d]^{f} \\
Y' \ar[r]_{\rho} & Y. } $$ Assume that $f'$ is projective, $D'$ is a simple normal crossing divisor on $X'$ such that $D'$ coincides with $D\times _YY'$ over $Y'\setminus \Sigma_{Y'}$, and every stratum of $D'$ is dominant onto $Y'$. Then there exists a natural isomorphism $$ \rho^*(R^pf_*\omega_{X/Y}(D))\simeq R^pf'_*\omega_{X'/Y'}(D') $$ which extends the base change isomorphism over $Y\setminus \Sigma_Y$ for every $p$. \end{itemize} \end{thm}
\begin{rem} For the proof of Theorem \ref{thm-lc-tri}, Theorem \ref{thm-mhs} for $p=0$ is sufficient. Note that all the local monodromies on $R^q(f_{0})_*\mathbb C_{X_0\setminus D_0}$ around $\Sigma_Y$ are unipotent for every $q$ because $f$ is semi-stable in codimension one, where
$X_0=f^{-1}(Y\setminus \Sigma_Y)$, $D_0=D|_{X_0}$, and $f_0=f|_{X_0\setminus D_0}$. More precisely, let $C_0^{[d]}$ be the disjoint union of all the codimension $d$ log canonical centers of $(X_0, D_0)$. If $d=0$, then we put $C_0^{[0]}=X_0$. In this case, we have the following weight spectral sequence $$
_W\!E_1^{-d, q+d}=R^{q-d}(f|_{C_0^{[d]}})_* \mathbb C_{C_0^{[d]}}\Longrightarrow R^q(f_0)_*\mathbb C_{X_0\setminus D_0} $$ which degenerates at $E_2$ (see, for example, \cite[Corollaire (3.2.13)]{deligne}). Since $f$ is semi-stable in codimension one, all the local monodromies
on $R^{q-d}(f|_{C_0^{[d]}})_*\mathbb C_{C_0^{[d]}}$ around $\Sigma_Y$ are unipotent for every $q$ and $d$ (see, for example, \cite[VII.~The Monodromy theorem]{katz}). By the above spectral sequence, we obtain that all the local monodromies on $R^q(f_0)_*\mathbb C_{X_0\setminus D_0}$ around $\Sigma_Y$ are unipotent. \end{rem}
We add a remark on the proof of Theorem \ref{thm-lc-tri}. In Remark \ref{rem39}, we explain how to modify the arguments in the proof of \cite[Lemma 5.2]{ambro1} in order to treat lc-trivial fibrations. It will help the reader to understand the main difference between klt-trivial fibrations and lc-trivial fibrations and the reason why we need Theorem \ref{thm-mhs}.
\begin{rem}\label{rem39} We use the notation in \cite[Lemma 5.2]{ambro1}. We only assume that $(X, B)$ is sublc over the generic point of $Y$ in \cite[Lemma 5.2]{ambro1}. We write $$ B=\sum _{i\in I}d_iB_i $$ where $B_i$ is a prime divisor for every $i$ and $B_i\ne B_j$ for $i\ne j$. We set $$
J=\left\{i\in I \, \left|\, {\text{$B_i$ is dominant onto $Y$ and $d_i=1$}} \right. \right\} $$ and set $$ D=\sum _{i\in J}B_i. $$ In Ambro's original setting in \cite[Lemma 5.2]{ambro1}, we have $D=0$ because $(X, B)$ is subklt over the generic point of $Y$. In the proof of \cite[Lemma 5.2 (4)]{ambro1}, we have to replace $$ \widetilde f_*\omega_{\widetilde {X}/Y}= \bigoplus _{i=0}^{b-1}f_*\mathcal O_X(\lceil (1-i)K_{X/Y}-iB+if^*B_Y+if^*M_Y\rceil)\cdot \psi^i. $$ with $$ \widetilde f_*\omega_{\widetilde {X}/Y}(\pi^*D)= \bigoplus _{i=0}^{b-1}f_*\mathcal O_X(\lceil (1-i)K_{X/Y}-iB+D+if^*B_Y+if^*M_Y\rceil)\cdot \psi^i $$ in order to treat lc-trivial fibrations. We leave the details as exercises for the reader. \end{rem}
The following theorem is a part of \cite[Theorem 3.3]{ambro2}.
\begin{thm}[{see \cite[Theorem 3.3]{ambro2}}]\label{thm-moduli} Let $f:(X, B)\to Y$ be a klt-trivial fibration such that $Y$ is complete, the geometric generic fiber $X_{\overline \eta}=X\times \Spec \overline {\mathbb C(\eta)}$ is a projective variety, and
$B_{\overline \eta}=B|_{X_{\overline {\eta}}}$ is effective, where $\eta$ is the generic point of $Y$. Then the moduli $\mathbb Q$-b-divisor $\mathbf M$ is b-nef and abundant. \end{thm}
\section{Proof of Theorem \ref{main}}\label{sec4}
Let us give a proof of Theorem \ref{main}.
\begin{proof}[Proof of Theorem \ref{main}] By taking a dlt blow-up, we may assume that the pair $(X, B)$ is $\mathbb Q$-factorial and dlt (see, for example, \cite[Section 4]{fujino3}). If $(X, B)$ is klt over the generic point of $Y$, then Theorem \ref{main} follows from \cite[Theorem 3.3]{ambro2} (see Theorem \ref{thm-moduli}). Therefore, we may also assume that $(X, B)$ is not klt over the generic point of $Y$. Let $\sigma_1:Y_1\to Y$ be a suitable projective birational morphism such that $\mathbf M=\overline {\mathbf M_{Y_1}}$ and $\mathbf M_{Y_1}$ is nef by Theorem \ref{thm-lc-tri}. Let $W$ be an arbitrary log canonical center of $(X, B)$ which is dominant onto $Y$ and is minimal over the generic point of $Y$. We set
$$K_W+B_W=(K_X+B)|_W$$ by adjunction (see, for example, \cite[3.9]{fujino-what}). By the construction, we have $K_W+B_W \sim _{\mathbb Q, Y}0$.
We consider the Stein factorization of $f|_W:W\to Y$ and denote it by $h:W\to Y'$. Then $K_W+B_W\sim _{\mathbb Q, Y'}0$.
We see that $h:(W, B_W)\to Y'$ is a klt-trivial fibration since the general fibers of $f|_{W}$ are klt pairs. Let $Y_2$ be a suitable resolution of $Y'$ which factors through $\sigma_1:Y_1\to Y$. By taking the base change by $\sigma_2:Y_2\to Y_1$, we obtain $\mathbf M_{Y_2}=\sigma_2^*\mathbf M_{Y_1}$ (see \cite[Proposition 5.5]{ambro1}). Note that the proof of \cite[Proposition 5.5]{ambro1} works for lc-trivial fibrations by some suitable modifications. By the construction, on the induced lc-trivial fibration $f_2:(X_2, B_{X_2})\to Y_2$, where $X_2$ is the normalization of the main component of $X\times _YY_2$, there is a log canonical
center $W_2$ of $(X_2, B_{X_2})$ such that $f_2|_{W_2^\nu}: (W_2^\nu, B_{W_2^\nu})\to Y_2$ is a klt-trivial fibration, which is birationally equivalent to $h:(W, B_W)\to Y'$. Note that $\nu: W_2^\nu\to W_2$
is the normalization, $K_{W_2^\nu}+B_{W_2^\nu}=\nu^*(K_{X_2}+B_{X_2})|_{W_2}$,
and $f_2|_{W_2^\nu}=f_2|_{W_2}\circ \nu$. It is easy to see that $$ K_{Y_2}+\mathbf M_{Y_2}+\mathbf B_{Y_2}\sim _{\mathbb Q} K_{Y_2}+\mathbf M^{\min}_{Y_2}+\mathbf B^{\min}_{Y_2} $$ where $\mathbf M^{\min}$ and $\mathbf B^{\min}$ are the induced moduli and discriminant $\mathbb Q$-b-divisors
of $f_2|_{W^\nu_2}: (W^\nu_2, B_{W^\nu_2})\to Y_2$ such that $$
K_{W^\nu_2}+B_{W^\nu_2}\sim _{\mathbb Q}(f_2|_{W^\nu_2}) ^*(K_{Y_2}+\mathbf M_{Y_2}^{\min}+\mathbf B_{Y_2}^{\min}). $$ By replacing $Y_2$ birationally, we may further assume that $\mathbf M^{\min}=\overline {\mathbf M_{Y_2}^{\min}}$ by Theorem \ref{thm-klt-tri}. By Theorem \ref{thm-moduli}, we see that $\mathbf M_{Y_2}^{\min}$ is nef and abundant. Let $\sigma_3:Y_3\to Y_2$ be a suitable generically finite morphism such that the induced lc-trivial fibration $f_3:(X_3, B_{X_3})\to Y_3$ has a semi-stable resolution in codimension one (see, for example, \cite{kkms}, \cite[(9.1) Theorem]{ssu}, and \cite[Theorem 4.3]{ambro1}). Note that $X_3$ is the normalization of the main component of $X\times _YY_3$. Here we draw the following big diagram for the reader's convenience. $$ \xymatrix{ (V, B_V) \ar[dr]^{\textrm{log-res.}} & & & & & &\\ & (X_3,B_3) \ar[rrr] \ar[dd]^{f_3} & & & (X_2,B_2)\ar[rr]\ar[dd]^{f_2} & & (X, B)\ar[dd]^{f} \\(W_3,B_{W_3}) \ar[dr]_{g_3} \ar[ur] & & W_2^\nu
\ar[drr]_{{f_2}|_{W_2^\nu}} \ar[r]_\nu^{\textrm {norm.}} & W_2 \ar@{^{(}->}[ur]
\ar[dr]^{{f_2}|_{W_2}} & & W \ar@{^{(}->}[ur] \ar@{->>}[d]^h\ar@{->>}[dr] &\\ &Y_3 \ar@/_1pc/[rrr]_{\textrm {semistab.}} &&& Y_2\ar[r]^{\textrm {desing.}}\ar[dr]_{\sigma_2} & Y'\ar[r]^{\textrm {Stein}} & Y \\
& & & & & Y_1\ar[ur]_{\sigma_1} & } $$ Note that $g_3:(W_3, B_{W_3})\to Y_3$ is the induced klt-trivial
fibration from $f_2|_{W_2^\nu}: W_2^\nu\to Y_2$ by $\sigma_3:Y_3\to Y_2$. On $Y_3$, we will see the following claim by using the semi-stable minimal model program. \begin{claim} The following equality $$ \mathbf B_{Y_3}=\mathbf B_{Y_3}^{\min} $$ holds. \end{claim} \begin{proof}[Proof of Claim] By taking general hyperplane cuts, we may assume that $Y_3$ is a curve. We write $$ \mathbf B_{Y_3}=\sum _P(1-b_P)P \quad {\text{and}}\quad \mathbf B_{Y_3}^{\min}=\sum _P(1-b_P^{\min})P. $$ Let $\varphi: (V, B_V)\to (X_3, B_{X_3})$ be a resolution of $(X_3, B_{X_3})$ with the following properties: \begin{itemize} \item $K_V+B_V=\varphi^*(K_{X_3}+B_{X_3})$; \item $\pi^*Q$ is a reduced simple normal crossing divisor on $V$ for every $Q\in Y_3$, where $\pi: V\to X_3\to Y_3$; \item $\Supp \pi^*Q\cup \Supp B_V$ is a simple normal crossing divisor on $V$ for every $Q\in Y_3$; \item $\pi$ is projective. \end{itemize} Let $\Sigma$ be a reduced divisor on $Y_3$ such that $\pi$ is smooth over $Y_3\setminus \Sigma$ and that $\Supp B_V$ is relatively normal crossing over $Y_3\setminus \Sigma$. We consider the set of prime divisors $\{E_i\}$ where $E_i$ is a prime divisor on $V$ such that $\pi(E_i)\in \Sigma$ and $$\mult _{E_i} (B_V+\sum _{P\in\Sigma}b_P\pi^*P)^{\geq 0}<1. $$ We run the minimal model programs with ample scaling with respect to $$ K_V+(B_V+\sum _{P\in \Sigma}b_P\pi^*P)^{\geq 0}+\varepsilon \sum_i E_i $$ over $X_3$ and $Y_3$ for some small positive rational number $\varepsilon$. Note that $$ (V, (B_V+\sum _P b_P\pi^*P)^{\geq 0}+\varepsilon \sum _i E_i) $$ is a $\mathbb Q$-factorial dlt pair because $0<\varepsilon \ll 1$. We set $$E=-(B_V+\sum _P b_P\pi^*P)^{\leq0}+\varepsilon \sum_i E_i. $$ Then it holds that $$ K_V+(B_V+\sum _Pb_P\pi^*P)^{\geq 0}+\varepsilon \sum_i E_i \sim _{\mathbb Q, Y_3}E\geq 0. $$ First we run a minimal model program with ample scaling with respect to $$ K_V+(B_V+\sum _Pb_P\pi^*P)^{\geq 0}+\varepsilon \sum_i E_i \sim _{\mathbb Q, X_3}E\geq 0 $$ over $X_3$. Note that every irreducible component of $E$ which is dominant onto $Y_3$ is exceptional over $X_3$ by the construction. Thus, if $E$ is dominant onto $Y_3$, then it is not contained in the relative movable cone over $X_3$. Therefore, after finitely many steps, we may assume that every irreducible component of $E$ is contained in a fiber over $Y_3$ (see, for example, \cite[Theorem 2.2]{fujino3}). Next we run a minimal model program with ample scaling with respect to $$ K_V+(B_V+\sum _Pb_P\pi^*P)^{\geq 0}+\varepsilon \sum_i E_i \sim _{\mathbb Q, Y_3}E\geq 0 $$ over $Y_3$. Then the minimal model program terminates at $V'$ (see, for example, \cite[Theorem 2.2]{fujino3}). Note that all the components of $E+\sum _i E_i$ are contracted by the above minimal model programs. Thus, we have $$ K_{V'}+(B_{V'}+\sum _P b_P{\pi'}^*P)^{\geq 0}\sim _{\mathbb Q, Y_3}0, $$ where $\pi':V'\to Y_3$ and $B_{V'}$ is the pushforward of $B_V$ on $V'$. Note that $B_{V'}+\sum _Pb_P\pi'^*P$ is effective since $\Supp(E+\sum _i E_i)$ is contracted by the above minimal model programs. Of course, we see that $$ (V', (B_{V'}+\sum _P b_P\pi'^*P)^{\geq 0})=(V', B_{V'}+\sum _P b_P\pi'^*P) $$ is a $\mathbb Q$-factorial dlt pair. By the construction, the induced proper birational map $$ (V, B_V+\sum _Pb_P\pi^*P)\dashrightarrow (V', B_{V'}+\sum _P b_P \pi'^*P) $$ over $Y_3$ is $B$-birational (see \cite[Definition 1.5]{fujino-abun}), that is, we have a common resolution \begin{equation*} \xymatrix{ & Z\ar[dl]_{a} \ar[dr]^{b}\\
V \ar@{-->}[rr] & & V'} \end{equation*} over $Y_3$ such that $$ a^*(K_V+B_V+\sum _{P\in \Sigma}b_P\pi^*P)=b^*(K_{V'}+B_{V'}+\sum _{P\in \Sigma}b_P\pi'^*P). $$ Let $S$ be any log canonical center of $(V', B_{V'}+\sum _Pb_P{\pi'}^*P)$ which is dominant onto $Y_3$ and is minimal over the generic point of $Y_3$. Then $(S, B_{S})$, where $$
K_{S}+B_{S}=(K_{V'}+B_{V'}+\sum _Pb_P{\pi'}^*P)|_{S}, $$ is not klt but lc over every $P\in \Sigma$ since it holds that \begin{align}\label{shiki1} B_{V'}+\sum _{P\in \Sigma}b_P\pi'^*P\geq \sum _{P\in \Sigma}\pi'^*P. \tag{$\spadesuit$} \end{align} Note that \eqref{shiki1} follows from the fact that all the components of $\sum _i E_i$ are contracted in the minimal model programs. Let $g_3:(W_3, B_{W_3})\to Y_3$ be the induced klt-trivial fibration from $(W^\nu_2, B_{W^\nu_2})\to Y_2$ by $\sigma_2:Y_3\to Y_2$. By \cite[Claims $(A_n)$ and $(B_n)$ in the proof of Lemma 4.9]{fujino-abun}, there is a log canonical center $S_0$ of $(V', B_{V'}+\sum _Pb_P\pi'^*P)$ which is dominant onto $Y_3$ and is minimal over the generic point of $Y_3$ such that there is a $B$-birational map $$ (W_3, B_{W_3}+\sum _{P\in\Sigma}b_Pg_3^*P)\dashrightarrow (S_0, B_{S_0}) $$ over $Y_3$, where $$
K_{S_0}+B_{S_0}=(K_{V'}+B_{V'}+\sum _{P\in \Sigma}b_P\pi'^*P)|_{S_0}. $$ This means that there is a common resolution \begin{equation*} \xymatrix{ & T\ar[dl]_{\alpha} \ar[dr]^{\beta}\\
W_3 \ar@{-->}[rr] & & S_0} \end{equation*} over $Y_3$ such that $$ \alpha^*(K_{W_3}+B_{W_3}+\sum _Pb_P g_3^*P)=\beta^*(K_{S_0}+B_{S_0}). $$ This implies that $b_P=b_P^{\min}$ for every $P\in \Sigma$. Therefore, we have $\mathbf B_{Y_3}=\mathbf B_{Y_3}^{\min}$. \end{proof} Then we obtain $$\mathbf M_{Y_3}\sim _{\mathbb Q}\mathbf M_{Y_3}^{\min}=\sigma_3^*\mathbf M_{Y_2}^{\min} $$ because $$ K_{Y_3}+\mathbf M_{Y_3}+\mathbf B_{Y_3}\sim _{\mathbb Q} K_{Y_3} +\mathbf M_{Y_3}^{\min} +\mathbf B_{Y_3}^{\min}. $$ Thus, $\mathbf M_{Y_3}$ is nef and abundant. Since $$ \mathbf M_{Y_3}=\sigma_3^*\mathbf M_{Y_2}=\sigma_3^*\sigma_2^*\mathbf M_{Y_1}, $$ $\mathbf M$ is b-nef and abundant. Moreover, by replacing $Y_3$ with a suitable generically finite cover, we have that $\mathbf M_{Y_3}$ and $\mathbf M_{Y_3}^{\min}$ are both Cartier (see \cite[Lemma 5.2 (5), Proposition 5.4, and Proposition 5.5]{ambro1}) and $\mathbf M_{Y_3}\sim \mathbf M_{Y_3}^{\min}$. \end{proof}
We close this paper with a remark on the b-semi-ampleness of $\mathbf M$. For some related topics, see \cite[Section 3]{fujino10}.
\begin{rem}[b-semi-ampleness]\label{41} Let $f:X\to Y$ be a projective surjective morphism between normal projective varieties with connected fibers. Assume that $(X, B)$ is log canonical and $K_X+B\sim _{\mathbb Q, Y}0$. Without loss of generality, we may assume that $(X, B)$ is dlt by taking a dlt blow-up. We set $$
d_f(X, B)=\left\{\dim W-\dim Y \left| \begin{array}{l} {\text{$W$ is a log canonical center of}}\\ {\text{$(X, B)$ which is dominant onto $Y$}} \end{array}\right. \right\}. $$ If $d_f(X, B)\in \{0, 1\}$, then the b-semi-ampleness of the moduli part $\mathbf M$ follows from \cite{kawamata} and \cite{ps} by the proof of Theorem \ref{main}. Moreover, it is obvious that $\mathbf M\sim _\mathbb Q 0$ when $d_f(X, B)=0$. \end{rem}
\end{document} |
\begin{document}
\title{Generalised Fibonacci sequences constructed from balanced words}
\begin{abstract} We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference of the two preceding terms where the pluses and minuses follow a certain pattern. In 2012, McLellan proved that if the pluses and minuses follow a periodic pattern and $G_n$ is the $n$th term of the resulting generalised Fibonacci sequence, then \begin{equation*}
\lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation*} exists. We extend her results to recurrences of the form $G_{m+2} = \alpha_m G_{m+1} \pm G_{m}$ if the choices of pluses and minuses, and of the $\alpha_m$ follow a balancing word type pattern. \end{abstract}
\keywords{Fibonacci sequences; Balancing words; matrices}
\section{Introduction} The Fibonacci sequence, recursively defined by $f_1=f_2=1$ and $f_n=f_{n-1}+f_{n-2}$ for all $n\geq 3$, has been generalised in several ways. In 2000, Divakar Viswanath studied random Fibonacci sequences given by $t_1=t_2=1$ and $t_n=\pm t_{n-1}\pm t_{n-2}$ for all $n\geq 3$. Here each $\pm$ is chosen to be $+$ or $-$ with probability $1/2$, and
are chosen independently. Viswanath proved that \begin{equation*}
\lim_{n\rightarrow\infty}\sqrt[n]{|t_n|}=1.13198824\dots \end{equation*} with probability $1$ where the exact value of the above limit is given as the exponential function on an integral expression involving a measure defined on Stern–Brocot intervals \cite{viswanath}. Almost nothing is known about this constant, however, not even if it is irrational. One of the key ideas in his proof was to study random Fibonacci sequences by using products of matrices. More specifically, let \begin{equation*} A:=\begin{bmatrix} 0 & 1\\ 1 & 1 \end{bmatrix} \text{ and } B:=\begin{bmatrix} 0 & 1\\ 1 & -1 \end{bmatrix}. \end{equation*} Then for all $n\in\mathbb{N}$ the $(n+1)$th and $(n+2)$th terms of a random Fibonacci sequence satisfy \begin{equation*} [1,1]Q_n=[G_{n+1},G_{n+2}] \end{equation*} where $Q_n$ is a matrix product consisting of $n$ $A$s and $B$s as factors where the pattern of $A$s and $B$s reflect the pattern of pluses and minuses generating the random Fibonacci sequence in question.
In 2006, Jeffrey McGowan and Eran Makover used the formalism of trees to evaluate the growth of the average value of the $n$th term of a random Fibonacci sequence \cite{eran}, which by Jensen's inequality is larger than Viswanath's constant. More precisely, they proved that \begin{equation*}
1.12095\leq\sqrt[n]{E(|t_n|)}\leq 1.23375 \end{equation*}
where $E(|t_n|)$ is the expected value of the $n$th term of the sequence. In 2007, Rittaud \cite{rittaud} improved this result and obtained \begin{equation*}
\lim_{n\rightarrow\infty}\sqrt[n]{E(|t_n|)}=\alpha=1\approx 1.20556943\ldots \end{equation*} where $\alpha$ is the only real root of $f(x)=x^3-2x^2-1$. In 2010, Janvresse, Rittaud, and De La Rue generalised Viswanath's result to random Fibonacci sequences involving different coefficients and probabilities of the choices of pluses and minuses \cite{janvresse}. In 2012, Karyn McLellan used Viswanath's idea of representing random Fibonacci sequences as matrix products with the matrices $A$ and $B$ as factors to study sequences that begin with two real numbers with the next term always being either the sum or the difference of the two preceding terms, but where the pattern of the pluses and minuses was periodic instead of random \cite{mclellan}. McLellan determined the growth rate of any such sequence, showing that Viswanath's limit still exists, albeit evaluating to different values, depending on the particular sequence in question. She used these growth rates to provide an alternative method of calculation for Viswanath's constant in the random case.
In 2018, the authors, in \cite{hare}, extended Rittaud's results and determined the probability that a random infinite walk down the tree contains no $(1,1)$ pairs after the initial root. We also determined tight upper and lower bounds on the number of coprime $(a,b)$ pairs at any given depth in the tree for any coprime pair $(a,b)$.
In this paper we consider a more general model. Starting with $G_1$ and $G_2$ as any real numbers, consider the recurrence
$G_{m+2} = \alpha_m G_{m+1} \pm G_{m}$ where the $\alpha_m$ are taken from a finite
set. This can be modeled by matrix multiplication as \[ [G_{m+1},G_{m+2}] = [G_{m} , G_{m+1}] \begin{bmatrix} 0 & \pm 1 \\ 1 & \alpha_m
\end{bmatrix}. \] Here we extend McLellan's results and show that Viswanath's limit exits if the
pattern of matrix multiplications generating the sequence $\left(G_n\right)_n$ follows certain balancing word patterns. Recall that \begin{definition} A \textit{balanced word} or \textit{Sturmian word} $w$ is an infinite word over a two letter alphabet $\{a,b\}$ such that, for any two subwords from $w$ of the same length, the number of letters that are $a$ in each of these two subwords will differ by at most $1$. \end{definition} We consider the following construction of a balanced or Sturmian word throughout this paper. There exists a sequence of nonnegative integers $q_0,q_1,a_2,\ldots$ with $q_i>0$ for all $i>0$ such that the infinite set of words $\{s_n\}_{n\geq 0}$ constructed from $s_0=b$, $s_1=a$, and $s_{n+1}=s_n^{q_{n-1}}s_{n-1}$ for all $n\in\mathbb{N}$ converges to $w$. Strumian or balanced words have been great studied, and two relevant references are Allouche and Shallit \cite{allouche} and de Luca \cite{deLuca}. Instead of having the pluses and minuses follow a periodic pattern studied by McLellan, we will have the pluses and minuses follow the pattern in an infinite balanced word. \begin{notation}\label{P_1P_2} Let $v\geq 1$, and let \[ A_i := \begin{bmatrix} 0 & \epsilon_i \\ 1 & \alpha_i \end{bmatrix} \] for $i = 1, 2, \dots, v$, where $\epsilon_i \in \{-1, 1\}$ and $\alpha_i \in \mathbb{Z}$. We note that each $A_i$ has determinant $\pm 1$. Let $P_1$ and $P_2$ be products of these matrices of length $k_1$ and $k_2$ respectively, allowing multiplicity. That is \begin{equation*} P_j:=\begin{bmatrix} a_j & b_j\\ c_j & d_j \end{bmatrix} = A_{j,1} A_{j,2} \dots, A_{j,k_j}. \end{equation*} We further require for $j = 1, 2$ that \begin{equation}\label{inequalitiesentries}
b_j,c_j\neq 0,\left|d_j\right|\geq 2, \left|a_j\right|\leq\left|b_j\right|\leq\left|d_j\right|\text{ and }\left|a_j\right|\leq\left|c_j\right|\leq\left|d_j\right|. \end{equation} For a sequence of positive integers $(q_m)_{m\in\mathbb{N}}$ define $P_m$, $k_m$ and $A_{m, j_1}, \dots A_{m, j_{k_m}}$,
inductively as \begin{align*} P_{m+2} & := P_{m+1}^{q_m} P_m \\
& := \underbrace{\underbrace{A_{m+1, j_1} \dots A_{m+1, j_{k_{m+1}}}}_{P_{m+1}} \dots
\underbrace{A_{m+1, j_1} \dots A_{m+1, j_{k_{m+1}}}}_{P_{m+1}}}_{q_m \text{times}}
\underbrace{A_{m, j_1} \dots A_{m+1, j_{k_{m}}}}_{P_{m}} \\
& := A_{m+2, j_1} \dots A_{m+2, j_{k_{m+2}}} \\
& := \begin{bmatrix}
a_{m+2}& b_{m+2}\\
c_{m+2} & d_{m+2}
\end{bmatrix}. \end{align*} Notice that the sequence $\left(P_m\right)_m$ is a standard Sturmian word (or balanced word) on the alphabet $\{P_1,P_2\}$. We observe that for all $m\geq 3$ and $\ell\geq 1$ that $P_{m+\ell}$ always starts by $P_m$, which is a product of $k_m$ matrices. Thus, whenever $n\leq k_m$, the first $n$ matrices of $P_{m+\ell}$ is independent of the choice of $\ell\in\mathbb{N}$. With this observation, we define \begin{equation*} Q_n:= A_{m, 1} \dots A_{m, n}
:= \begin{bmatrix}
e_n & f_n\\
g_n & h_n \end{bmatrix} \end{equation*} for $k_m \geq n$. Notice that so long as $k_m\geq n$, $Q_n$ is independent of the choice of $m$. Finally, let $G_1$ and $G_2$ be any two real numbers and for every $n\in\mathbb{N}$ let $G_{n+2}=\alpha_{m,n}G_{n+1}+\epsilon_{m,n}G_n$ where \begin{equation*} A_{m,n}
:= \begin{bmatrix}
0 & \epsilon_{m,n}\\
1 & \alpha_{m,n} \end{bmatrix} \end{equation*} Notice that for all $n\in\mathbb{N}$ we have \begin{equation*} [G_1,G_2]Q_n=[G_{n+1},G_{n+2}]. \end{equation*} \end{notation} We prove the following: \begin{theorem}\label{bigthm} Let $q_m$, $P_m$, $a_m$, $c_m$, $Q_n$, and $G_n$ be as defined in Notation \ref{P_1P_2}. Then $\lim_{m\rightarrow\infty}\frac{a_m}{c_m}=\lim_{m\rightarrow\infty}\frac{b_m}{d_m}$ exists, and is positive, and is either $1$ or irrational. Let this limit be denoted by $M$. \begin{enumerate}
\item If $M=1$, then $|G_n|$ grows at most linearly, i.e. there exists $C>0$ such that \begin{equation*}
|G_n|<Cn \end{equation*} for all $n\in\mathbb{N}$. \item If $M$ is irrational and $G_1\neq\frac{-G_2}{M}$, then \begin{equation}\label{Gnlimit}
\lim_{n\rightarrow\infty}|G_n|^{1/n} \end{equation} exists with this limit being greater than $1$. \end{enumerate} \end{theorem}
The proof of Theorem \ref{bigthm} is divided up in the subsequent sections of the paper as follows. In Section \ref{sec2} we prove some necessary technical lemmas on the entries in the matrices $\left(P_m\right)_{m\in\mathbb{N}}$. We then divide the proof up into two cases. If $\left|b_i\right|=\left|c_i\right|=\left|d_i\right|-1=\left|a_i\right|+1$ for all sufficiently large $i$, then it turns out that $M=1$ and we get the case that $\left|G_n\right|$ grows at most linearly. This is proved in Section \ref{sec3}. Otherwise we get $M$ is irrational and the exponential growth of $\left|G_n\right|$, which is dealt in Section \ref{sec4}. For the rest of the present section though we include some remarks on Theorem \ref{bigthm} and illustrate them with an example. \begin{remark} If we restrict $G_1,G_2\in\mathbb{Z}$ or even to just $G_1,G_2\in\mathbb{Q}$, then we can ignore the condition that $G_1\neq\frac{-G_2}{M}$ in Theorem \ref{bigthm} since $M$ is irrational. \end{remark} \begin{remark}\label{alphal} Let $0<\alpha<1$ be irrational. If in Notation \ref{P_1P_2} we pick the sequence of positive integers $q_1,q_2,q_3,\ldots $ so that the continued fraction expansion of $\alpha$ can be represented as $[0;q_1,q_2,q_3,\ldots]$, then we have \begin{equation}\label{alphalimit} \lim_{m\rightarrow\infty}\frac{\text{number of }P_2\text{s in }P_m}{\text{number of }P_1\text{s and }P_2\text{s in }P_m}=\alpha. \end{equation} See, for example, \cite{deLuca}. \end{remark} We give some examples of $A_1$, $A_2$, $P_1$, and $P_2$ that satisfy Notation \ref{P_1P_2} with $v=2$. \begin{remark} For all $a,b,c,d\in\mathbb{R}$ observe that \begin{equation*} \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} a & b\\ c & d \end{bmatrix} = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} d & -b\\ -c & a \end{bmatrix}. \end{equation*} It follows that if we replace the inequalities in \eqref{inequalitiesentries} with the inequalities \begin{equation*}
0\leq\frac{|d_i|}{|b_i|},\frac{|c_i|}{|a_i|},\frac{|d_i|}{|c_i|},\frac{|b_i|}{|a_i|}\leq 1, \end{equation*} then Theorem \ref{bigthm} still holds. \end{remark} \begin{example}\label{entriessize} Let \begin{equation*} A:=\begin{bmatrix} 0 & 1\\ 1 & 1 \end{bmatrix} \text{ and } B:=\begin{bmatrix} 0 & 1\\ 1 & -1 \end{bmatrix}. \end{equation*} Let $P_1$ and $P_2$ be a product matrices of matrices of the form $A^j$ and $B^k$ where $j,k\geq 2$. Then the matrices $A$, $B$, $P_1$, and $P_2$ satisfy the matrices $A_1$, $A_2$, $P_1$, and $P_2$ respectively in Notation \ref{P_1P_2} with $v=2$ with $P_1$ and $P_2$ satisfying \eqref{inequalitiesentries}. \end{example} \begin{remark} In Theorem \ref{bigthm} it is possible that if the sequence $\left(G_n\right)_n$ grows at most linearly, then it could contain a bounded infinite subsequence of terms. For example, let the matrices $A$ and $B$ be as in Example \ref{entriessize} and let $P_1=P_2=A^3B^3$. Then for all $k\in\mathbb{N}$ we can verify that \begin{equation*} Q_{6k+1}=(A^3B^3)^kA=(-1)^k\begin{bmatrix} 4k & 1\\ 4k+1 & 1 \end{bmatrix}. \end{equation*}
We can thus deduce that $|G_{6k+3}|=|G_1+G_2|$ for all $k\in\mathbb{N}$. It is routine to check that the entries in $Q_n$ in Example \ref{entriessize} grow at most linearly and so any corresponding Fibonacci sequence will grow at most linearly. \end{remark} We prove that the matrices in Example \ref{entriessize} satisfy Notation \ref{P_1P_2} and Theorem \ref{bigthm} in the next section. We give a concrete example though of matrices $P_1$ and $P_2$ and a sequence of positive integers $(q_m)_{m\in\mathbb{N}}$ for illustration. \begin{example} Let $P_1=A^2$ and $P_2=B^2$. Consider the number $1/\pi$, which has continued fraction expansion $[0;3,7,15,1,\ldots]$. Let our sequence of positive integers $(q_m)_{m\in\mathbb{N}}$ be these convergents so that $q_1=3$, $q_2=7$, $q_3=15$, $q_4=1,\ldots$ Then $P_1$ and $P_2$ satisfy Notation \ref{P_1P_2} and \eqref{inequalitiesentries}. Then we have $P_3=B^6A^2$, $P_4=(B^6A^2)^7B^2,\ldots$ Let $G_1=G_2=1$ in Theorem \ref{bigthm}. Then the corresponding Fibonacci sequence starts out as follows. \begin{align*} G_3&=G_1-G_2=1-1=0\\ G_4&=G_2-G_3=1-0=1\\ G_5&=G_3-G_4=0-1=-1\\ G_6&=G_4-G_5=1-(-1)=2\\ G_7&=G_5-G_6=-1-2=-3\\ G_8&=G_6-G_7=2-(-3)=5\\ G_9&=G_7+G_8=-3+5=2\\ G_{10}&=G_8+G_9=5+2=7\\ G_{11}&=G_9-G_{10}=2-7=-5\\ G_{12}&=G_{10}-G_{11}=7-(-5)=12\\ \ldots& \end{align*} Also, we have \begin{equation*} P_3=\begin{bmatrix} -3 & -11\\ 5 & 18 \end{bmatrix} \text{ and } P_4=\begin{bmatrix} 88364872 & -21089221\\ -144059117 & 343812479 \end{bmatrix} \end{equation*} and notice that \begin{equation*} \frac{1}{2}<\frac{3}{5},\frac{11}{18},\frac{88364872}{144059117},\frac{21089221}{343812479}<\frac{2}{3}. \end{equation*} Then by induction $n\in\mathbb{N}$ using Lemma \ref{positive} we have that if \begin{equation*} P_n=\begin{bmatrix} a_n & b_n\\ c_n & d_n \end{bmatrix}, \end{equation*} then \begin{equation*}
\frac{1}{2}<\frac{\left|a_n\right|}{\left|c_n\right|},\frac{\left|b_n\right|}{\left|d_n\right|}<\frac{2}{3} \end{equation*}
for all $n\geq 3$. Using Lemma \ref{limitsinfinity}, we therefore have that $|b_m|=|c_m|=|d_m|-1=|a_m|+1$ cannot hold for sufficiently large $m$ so that from the work in Section \ref{sec4} the sequence $\left(G_n\right)_n$ grows exponentially with the limit in \eqref{Gnlimit} existing and greater than $1$. As well, we can deduce from Remark \ref{alphal} that the fraction of $+$'s creating the Fibonacci sequence tends to $1/\pi$. \end{example} \section{Preliminary Results}\label{sec2} To prove Theorem \ref{bigthm} we first require some preliminary lemmas. \begin{lemma}\label{positive} Suppose we have \begin{equation*} \begin{bmatrix} a_1 & b_1\\ c_1 & d_1 \end{bmatrix} \cdot \begin{bmatrix} a_2 & b_2\\ c_2 & d_2 \end{bmatrix} = \begin{bmatrix} a_3 & b_3\\ c_3 & d_3 \end{bmatrix} \end{equation*}
where $a_i,b_i,c_i,d_i\in\mathbb{Z}$ for $i=1,2,3$ with $c_1,d_1,b_2,d_2$ nonzero and that the determinants of all the matrices are $1$. Suppose that $|d_1|\geq|c_1|$, $|b_1|\geq|a_1|$, $|d_2|\geq|b_2|$, $|c_2|\geq|a_2|$ and that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_1|}{|c_1|},\frac{|b_1|}{|d_1|}\leq\frac{r_3}{r_4} \end{equation*}
where $r_i\in\mathbb{Z}$ for $i=1,2,3,4$ with $|d_1|>r_2$ and $|d_1|>r_4$. Then \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_3|}{|c_3|},\frac{|b_3|}{|d_3|}\leq\frac{r_3}{r_4}. \end{equation*} Also, suppose that \begin{equation*}
\frac{r_5}{r_6}\leq\frac{|a_2|}{|b_2|},\frac{|c_2|}{|d_2|}\leq\frac{r_7}{r_8} \end{equation*}
where $r_i\in\mathbb{Z}\backslash\{0\}$ for $i=5,6,7,8$ with $|d_2|>r_6$ and $|d_2|>r_8$. Then \begin{equation*}
\frac{r_5}{r_6}\leq\frac{|a_3|}{|b_3|},\frac{|c_3|}{|d_3|}\leq\frac{r_7}{r_8}. \end{equation*} \end{lemma} \begin{proof} We will assume that $\frac{c_1b_2}{d_1d_2}<0$. The case of $\frac{c_1b_2}{d_1d_2}>0$ follows similarly.
Since $\frac{c_1b_2}{d_1d_2}<0$, $c_1b_2$ and $d_1d_2$ have opposite signs. Thus $|d_3|=|d_1||d_2|-|c_1||b_2|$ since $|d_1|\geq |c_1|$ and $|d_2|\geq |b_2|$. Similarly, since the determinants of the matrices is $1$, we can argue similarly that $|b_3|=|b_1||d_2|-|a_1||b_2|$, $|c_3|=|d_1||c_2|-|c_1||a_2|$, and $|a_3|=|b_1||c_2|-|a_1||a_2|$. We have \begin{align*}
r_2|a_3|-r_1|c_3|&=r_2|b_1||c_2|-r_2|a_1||a_2|-r_1|d_1||c_2|+r_1|c_1||a_2|\\
&=(r_2|b_1|-r_1|d_1|)|c_2|-(r_2|a_1|-r_1|c_1|)|a_2|. \end{align*}
Since the determinants of the matrices is $1$, we have $|b_1||c_1|\geq |a_1||d_1|-1$. Thus we have \begin{equation*}
\frac{|b_1|}{|d_1|} \geq\frac{|a_1|}{|c_1|}-\frac{1}{|c_1||d_1|} \text{\ \ \ and\ \ \ }
\frac{|b_1|}{|d_1|}-\frac{r_1}{r_2} \geq\frac{|a_1|}{|c_1|}-\frac{r_1}{r_2}-\frac{1}{|c_1||d_1|} \end{equation*}
Since $|d_1|\geq|c_1|$ and $|d_1|>r_2$, we have \begin{equation*}
r_2|b_1|-r_1|d_1|\geq r_2|a_1|-r_1|c_1|-\frac{r_2}{|d_1|}>r_2|a_1|-r_1|c_1|-1. \end{equation*}
Since $r_2|b_1|-r_1|b_1|,r_2|a_1|-r_1|c_1|\in\mathbb{Z}$, we have $r_2|b_1|-r_1|d_1|\geq r_2|a_1|-r_1|c_1|\geq 0$. Since $|c_2|\geq|a_2|$, we thus have $r_2|a_3|-r_1|c_3|\geq 0$. Thus $\frac{r_1}{r_2}\leq\frac{|a_3|}{|c_3|}$. Next, we have \begin{align*}
r_3|c_3|-r_4|a_3|&=r_3(|d_1||c_2|-|c_1||a_2|)-r_4(|b_1||c_2|-|a_1||a_2|)\\
&=(r_3|d_1|-r_4|b_1|)|c_2|-(r_3|c_1|-r_4|a_1|)|a_2|. \end{align*}
Since the determinants of the matrices is $1$, we have $|a_1||d_1|\leq |b_1||c_1|+1$. Thus we have \begin{equation*}
\frac{|b_1|}{|d_1|} \leq\frac{|a_1|}{|c_1|}+\frac{1}{|c_1||d_1|} \text{\ \ \ and\ \ \ }
\frac{r_3}{r_4}-\frac{|b_1|}{|d_1|} \geq\frac{r_3}{r_4}-\frac{|a_1|}{|c_1|}-\frac{1}{|c_1||d_1|}. \end{equation*}
Since $|d_1|\geq|c_1|$ and $|d_1|>r_4$, we have \begin{equation*}
r_3|d_1|-r_4|b_1|\geq r_3|c_1|-r_4|a_1|-\frac{r_4}{|d_1|}>r_3|c_1|-r_4|a_1|-1. \end{equation*}
Since $r_3|d_1|-r_4|b_1|,r_3|c_1|-r_4|a_1|\in\mathbb{Z}$, we have $r_3|d_1|-r_4|b_1|\geq r_3|c_1|-r_4|a_1|\geq 0$. Since $|c_2|\geq |a_2|$, we thus have $r_3|c_3|-r_4|a_3|\geq 0$. Thus $\frac{|a_3|}{|c_3|}\leq\frac{r_3}{r_4}$. The rest of the inequalities follow similarly. \end{proof} We prove that the matrices in Example \ref{entriessize} satisfies Notation \ref{P_1P_2} and Theorem \ref{bigthm}. \begin{proof} First, $A$ and $B$ consist of integer entries and both matrices have determinate $-1$. We can prove by induction on $j,k\in\mathbb{N}$ that \begin{equation*} A^j= \begin{bmatrix} F_{j-1} & F_j\\ F_j & F_{j+1} \end{bmatrix} \text{\ \ \ and \ \ \ } B^k=(-1)^k \begin{bmatrix} F_{k-1} & -F_k\\ -F_k & F_{k+1} \end{bmatrix} \end{equation*} where $F_k$ is the $k$th Fibonacci number where $F_0=0$, and $F_1=F_2=1$.
Let $P_1$ and $P_2$ be product matrices of $A$s and $B$ satisfying the example. Without loss of generality, it is enough to prove that $P_1$ satisfies the matrix $P_1$ in Notation \ref{P_1P_2}. We prove this by induction on the number of matrices of the form $A^j$ and $B^k$ there are in the product. For the base cases of $P_1=A^k$ and $P_1=B^k$, we have \begin{equation*} 0\leq\frac{F_{k-1}}{F_k},\frac{F_k}{F_{k+1}}\leq 1, \end{equation*} $F_{k+1}\geq 2$, and $F_k\neq 0$.
Suppose the case holds for some product matrix $P$ \begin{equation*} P= \begin{bmatrix} a & b\\ c & d \end{bmatrix}. \end{equation*}
First we prove it holds for $PA^j$ where $j\geq 2$. By induction, we have $|d|\geq 2$ and $c\neq 0$. Thus $\frac{cF_{j-1}}{dF_j}\neq 0$.
First assume that $\frac{cF_{j-1}}{dF_j}>0$. By induction, we have all of the inequalities holding in Lemma \ref{positive} with $r_1=0$ and $r_2=r_3=r_4=1$. Lemma \ref{positive} thus gives us all of the desired inequalities holding for $PA^j$ with the observations that $|c||F_j|+|d||F_{j+1}|\geq 2$, $|c|F_{k-1}+|d|F_k\neq 0$, and $|a|F_k+|b|F_{k+1}\neq 0$.
Now assume that $\frac{cF_{j-1}}{dF_j}<0$. By induction, we have all of the inequalities holding in Lemma \ref{positive} with $r_1=r_5=0$ and $r_2=r_3=r_4=r_6=r_7=r_8=1$. Lemma \ref{positive} thus gives us all of the desired inequalities holding for $PA^j$ with the observations that $|d|F_{j+1}-|c|F_j\geq |d|\geq 2$, $d|F_k|-c|F_{k-1}|\neq 0$, and $|d|F_{k+1}-|c|F_k\neq 0$.
The case of $PB^k$ is similar. \end{proof} \begin{lemma}\label{ratio1} Consider a matrix $P$ \begin{equation*} P= \begin{bmatrix} a & b\\ c & d \end{bmatrix}. \end{equation*}
where $1\leq |a|<|c|<|d|$, $|a|<|b|<|d|$, and $|\det P|=|ad-bc|=1$. \begin{enumerate} \item Suppose there doesn't exist positive integers $r_1,r_2,r_3,r_4$ with $r_3<r_4$ such that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a|}{|c|},\frac{|b|}{|d|}\leq\frac{r_3}{r_4} \end{equation*}
and $|d|>r_2,r_4$. Then $|b|=|c|=|d|-1=|a|+1$. \label{case1} \item Suppose there doesn't exist positive integers $r_1,r_2,r_3,r_4$ with $r_3<r_4$ such that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a|}{|b|},\frac{|c|}{|d|}\leq\frac{r_3}{r_4} \end{equation*}
and $|d|>r_2,r_4$. Then $|b|=|c|=|d|-1=|a|+1$. \label{case2} \end{enumerate} \end{lemma} \begin{proof} We prove \ref{case1}. Case \ref{case2} follows by taking the transpose of the matrix $P$ and using \ref{case1}.
Suppose $|c|\geq |a|+2$. Then \begin{equation*}
\frac{1}{|d|-1}\leq\frac{1}{|c|}\leq\frac{|a|}{|c|}\leq\frac{|c|-2}{|c|}<\frac{|c|-1}{|c|}. \end{equation*} Also, we have \begin{equation*}
\frac{|b|}{|d|}\leq\left|\frac{b}{d}-\frac{a}{c}\right|+\frac{|a|}{|c|}
=\frac{1}{|cd|}+\frac{|a|}{|c|}
\leq\frac{1}{|cd|}+\frac{|c|-2}{|c|}
\leq\frac{|c|-2+\frac{1}{|d|}}{|c|}
<\frac{|c|-1}{|c|} \end{equation*}
with the last inequality following from $|d|\geq 3$. Also, we have \begin{equation*}
\frac{|b|}{|d|}\geq\frac{2}{|d|}>\frac{1}{|d|-1} \end{equation*}
since $|b|\geq 2$ and $|d|\geq 3$. Thus letting $r_1=1$, $r_2=|d|-1$, $r_3=|c|-1$, and $r_4=|c|$, we obtain the existence of four positive integers with the properties as stated in the theorem. Thus we may assume that $|c|=|a|+1$. By similar reasoning, if $|d|\geq |b|+2$, then we can see that $r_1=1$, $r_2=|d|-1$, $r_3=|d|-2$, and $r_4=|d|-1$ also satisfies the properties as stated in the theorem. Thus we may also assume that $|d|=|b|+1$.
We can deduce that $|a||c|-|b||d|=\pm 1$ from $|ac-bd|=1$. Thus we have \begin{equation*}
\pm 1=|a|(|b|+1)-|b|(|a|+1)=|a|-|b|. \end{equation*}
We know, however, that $|a|<|b|$. We therefore have that $|b|=|a|+1$ and so we must also have that $|b|=|c|$. \end{proof} \begin{lemma}\label{samesigns} Consider a matrix $P$ \begin{equation*} P= \begin{bmatrix} a & b\\ c & d \end{bmatrix}, \end{equation*}
where $|\det P|=|ad-bc|=1$, $a,b,c,d\in\mathbb{Z}\backslash\{0\}$. We have $\frac{a}{c}>0$ if and only if $\frac{b}{d}>0$. \end{lemma} \begin{proof}
It suffices to prove that $\frac{ad}{bc}>0$. Suppose for a contradiction that $\frac{ad}{bc}<0$. Then $ad$ and $bc$ have opposite signs. Also notice that $|ad|\geq 1$ and $|bc|\geq 1$. Then we have $|ad-bc|\geq 2$, a contradiction. The result follows. \end{proof} \begin{lemma}\label{power} Consider a matrix $P$ \begin{equation*} P= \begin{bmatrix} a & b\\ c & d \end{bmatrix}, \end{equation*}
where $a,b,c,d\in\mathbb{Z}$, $|c|,|d|\geq 2$, $b\neq 0$, $\det(P)=\pm 1$, and \begin{equation*}
0\leq\frac{|a|}{|b|},\frac{|c|}{|d|},\frac{|a|}{|c|},\frac{|b|}{|d|}\leq 1. \end{equation*} For all $i\in\mathbb{N}$, let \begin{equation*} P^i= \begin{bmatrix} a_i & b_i\\ c_i & d_i \end{bmatrix}. \end{equation*}
For all $i\geq 2$, we have $|d_i|-|b_i|\geq|d_{i-1}|-|b_{i-1}|$, $|d_i|-|c_i|\geq|d_{i-1}|-|c_{i-1}|$, $|b_i|-|a_i|\geq(|b|-|a|)(|b_{i-1}|-|a_{i-1}|)$, and $|d_i|>|d_{i-1}|$. \end{lemma} \begin{proof}
We can deduce that either $|d_i|=|d||d_{i-1}|-|c||b_{i-1}|$ and $|b_i|=|b||d_{i-1}|-|a||b_{i-1}|$ or $|d_i|=|d||d_{i-1}|+|c||b_{i-1}|$ and $|b_i|=|b||d_{i-1}|+|a||b_{i-1}|$. In the first case, we have \begin{align*}
|d_i|-|b_i|&=|d||d_{i-1}|-|c||b_{i-1}|-(|b||d_{i-1}|-|a||b_{i-1}|)\\
&=(|d|-|b|)|d_{i-1}|-(|c|-|a|)|b_{i-1}|. \end{align*}
Since $\det(P)=\pm 1$, we can deduce that $|a||d|\geq |b||c|-1$. Thus, we have $|a||d|-|a||c|\geq |b||c|-|a||c|-1$. Since $\det(P)=\pm 1$, we have $\gcd(|a|,|c|)=1$ and so since $|c|\geq 2$ and $|c|\geq |a|$, we have $|c|>|a|$. Since $|a||d|-|a||c|\geq |b||c|\geq 0$, we have \begin{equation*}
|d|-|c|\geq |b|-|a|-\frac{1}{|c|}>|b|-|a|-1. \end{equation*}
It follows that $|d|-|b|\geq |c|-|a|$ since $a,b,c,d\in\mathbb{Z}$. Thus \begin{equation*}
|d_i|-|b_i|\geq(|c|-|a|)(|d_{i-1}|-|b_{i-1}|)\geq |d_{i-1}|-|b_{i-1}|. \end{equation*} The other inequalities follows similarly. \end{proof} \begin{lemma}\label{limitsinfinity} For all $m\in\mathbb{N}$ let $q_m$, $P_m$, $a_m$, $b_m$, $c_m$, and $d_m$ be defined as in Notation \ref{P_1P_2}. We have \begin{equation*}
\lim_{m\rightarrow\infty}|a_m|=\lim_{m\rightarrow\infty}|b_m|=\lim_{m\rightarrow\infty}|c_m|=\lim_{m\rightarrow\infty}|d_m|=\infty. \end{equation*} \end{lemma} \begin{proof}
By Lemma \ref{positive}, we have for all $m\in\mathbb{N}$ that $\min\{|a_m|,|b_m|,|c_m|,|d_m|\}=|a_m|$, $\max\{|a_m|,|b_m|,|c_m|,|d_m|\}=|d_m|$, and $|d_m|\geq 2$. By induction for all $m\in\mathbb{N}$ we have $\det(P_m)=|a_md_m-b_mc_m|=\pm 1$. Thus for all $m\in\mathbb{N}$, we have $\gcd(|b_m|,|d_m|)=\gcd(|c_m|,|d_m|)=1$ so that for all $m\in\mathbb{N}$ $|c_m|<|d_m|$ and $|b_m|<|d_m|$. Also, for all $m\in\mathbb{N}$, let \begin{equation*} P_{m+1}^{q_m}= \begin{bmatrix} a_{m+1,q_m} & b_{m+1,q_m}\\ c_{m+1,q_m} & d_{m+1,q_m} \end{bmatrix}. \end{equation*}
From Lemma \ref{power}, we can deduce that $|c_{m,q}|<|d_{m,q}|$ and $|b_{m,q}|<|d_{m,q}|$. Thus, for all $m\in\mathbb{N}$, we have by Lemma \ref{power} \begin{equation*}
|d_{m+2}|\geq |d_{m+1,q_m}||d_{m}|-|c_{m+1,q_m}||b_{m}|>|d_m|(|d_{m}|-|c_{m}|)\geq |d_m|. \end{equation*}
It follows that $\lim_{m\rightarrow\infty}|d_m|=\infty$. Also, we have \begin{equation*}
|b_{m+2}|\geq |b_{m+1,q_m}||d_{m}|-|a_{m+1,q_m}||b_{m}|\geq|b_{m+1,q_m}|(|d_{m}|-|b_{m}|)\geq |b_{m+1,q_m}|. \end{equation*}
By similar reasoning, we have $|b_{m+1,q_m}|\geq |b_{m+1}|$ and so for all $m\in\mathbb{N}$ we have $|b_{m+2}|\geq |b_{m+1}|$ with equality only if $|a_{m+1,q_m}|=|b_{m+1,q_m}|=1$ and $|d_{m}|=|b_{m}|+1$. If $|b_m|$ is bounded for all $m\in\mathbb{N}$, then for sufficiently large $m$, we have $|d_{m}|=|b_{m}|+1$ and so $|d_m|$ is bounded, a contradiction. Thus $|b_m|$ isn't bounded and we have $\lim_{m\rightarrow\infty}|b_m|=\infty$.
Finally, choose $M\in\mathbb{N}$ such that for all $m\geq M$ $|b_m|\geq 2$. Then for all $m\geq M$, we have $|b_m|>|a_m|$. By Lemma \ref{power}, we have $|b_{m+1,q_m}|-|a_{m+1,q_m}|\geq(|b_{m+1}|-|a_{m+1}|)^{q_m}\geq 1$. So for all $m\geq M$, we have \begin{align*}
|a_{m+2}|&=|a_{m+1,q_m}a_{m}+b_{m+1,q_m}c_{m}|\\
&\geq |b_{m+1,q_m}||c_{m}|-|a_{m+1,q_m}||a_{m}|\\
&>(|b_{m+1,q_m}|-|a_{m+1,q_m}|)|a_{m}|\\
&\geq |a_m|. \end{align*}
Thus $\lim_{m\rightarrow\infty}|a_m|=\infty$. Since $|a_m|<|c_m|$ for all $m\in\mathbb{N}$, we also have $\lim_{m\rightarrow\infty}|c_m|=\infty$. \end{proof}
Letting $a_m$, $b_m$, $c_m$, and $d_m$ be as defined in Notation \ref{P_1P_2}, Lemma \ref{limitsinfinity} implies that for sufficiently large $m$, we have $|d_m|>|b_m|>|a_m|>0$. \begin{remark} In Notation \ref{P_1P_2}, since $P_3=(P_2)^{q_1}P_1$, $P_2$ is $P_3$ truncated after a certain point. Thus, by reindexing the matrices $P_i,{i\in\mathbb{N}}$, we will assume for the rest of the paper that $P_1$ is $P_2$ truncated after a certain point. \end{remark} \begin{lemma}\label{Q_nproductP_n} Let $q_m$, $P_m$, and $Q_n$ be as defined in Notation \ref{P_1P_2}. Then there exists uniquely $2\leq m_1<m_2<...<m_l$ and $n_1,\ldots,n_l\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_i\leq q_{m_i-1}$, \item for all $2\leq i\leq l$, if $n_i=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$, \item $Q_n=(P_{m_l})^{n_l}(P_{m_{l-1}})^{n_{l-1}}...(P_{m_1})^{n_1}M_n$ where the matrix $M_n$ is a product of the string of $A_1$s, $A_2$s, $\ldots$, and $A_v$s in $P_2$ truncated after a certain point unless $m_1=2$ and $n_1=q_1$ in which case $M_n$ is a product of the string of $A_1$s, $A_2$s, $\ldots$, and $A_v$s in $P_1$ truncated after a certain point. \end{enumerate} \end{lemma} \begin{proof} We prove by strong induction on $n\in\mathbb{N}$. If $n<k_2$, then we have $Q_n=M_n$. Suppose $n\geq k_2$ and that the lemma holds true for all values less than $n$. Choose $m\in\mathbb{N}$ such that $k_{m+1}>n\geq k_m$ where $m\geq 2$. Then $P_m$ is $Q_n$ truncated after a certain point and $Q_n$ is $P_{m+1}$ truncated after a certain point. Since $P_{m+1}=(P_m)^{q_{m-1}}P_{m-1}$, it follows that $Q_n=(P_m)^iR$ where $i$ and $R$ satisfy the following: \begin{enumerate} \item $i\leq q_{m-1}$ \item $R$ is a product of the string of $A_1$s, $A_2$s, $\ldots$, and $A_v$s in $P_m$ truncated after a certain point \item if $i=q_{m-1}$ and $m\geq 3$, then $R$ is a product of the string of $A_1$s, $A_2$s, $\ldots$, and $A_v$s in $P_{m-1}$ truncated after a certain point. \end{enumerate} Notice that if we replace $m$ by any other positive integer, say $m'$ and have $Q_n=\left(P_{m'}\right)^iR$ instead, then it follows that $m'<m$. But then if $i\leq q_{m-1}$, then we have that $P_{m'}$ is $R$ truncated after a certain point. Thus any expression for $Q_n$ in (3) in the lemma must begin with $m_l=m$. Similarly the value of $i$ satisfying the above must be unique.
If $m=2$ and $i<q_1$, then $R$ is $P_2$ truncated after a certain point and $R=M_n$ and the result follows. If $m=2$ and $i=q_1$, then $R$ is $P_1$ truncated after a certain point and $R=M_n$ and again the result follows. So assume that $m>2$. Then $R=Q_{n-ik_m}$ and so $Q_n=P_m^iQ_{n-ik_m}$ with $n-ik_m<k_m$ and if $i=q_{n-1}$, then $n-ik_m<k_{m-1}$. By induction, the result follows. \end{proof}
We divide into two cases. Case $1$ assumes that for sufficiently large $m$, we have $|b_m|=|c_m|=|d_m|-1=|a_m|+1$. Case $2$ deals with all other cases. \section{The Linear Growth Case}\label{sec3}
For this case, we may assume without loss of generality that $|b_m|=|c_m|=|d_m|-1=|a_m|+1$ for all $m\in\mathbb{N}$. By Lemma \ref{limitsinfinity}, we may also assume without loss of generality that $|a_m|\geq 1$ for all $m\in\mathbb{N}$. \begin{lemma}\label{nklinear} Let $P_m$ and $Q_n$ be as defined in Notation \ref{P_1P_2}. Let $n_1,n_2,\ldots,n_j,\ldots,$ be the list of natural numbers such that for each $n_j$, there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_{n_j}=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{n_{j,1}}$. \end{enumerate} We have \begin{equation*}
|g_{n_j}|\leq n_j\max\{|c_1|,|c_2|\} \end{equation*} for all $j\in\mathbb{N}$. \end{lemma} \begin{proof}
We prove by induction on $k\in\mathbb{N}$. First we observe that Lemma \ref{positive} implies that $|e_{n_k}|\leq|f_{n_k}|\leq |h_{n_k}|$ and $|e_{n_k}|\leq|g_{n_k}|\leq |h_{n_k}|$ for all $k\in\mathbb{N}$. Let $k\in\mathbb{N}$. If there exists $r_1,r_2,r_3,r_4\in\mathbb{N}$ with $r_3<r_4$ such that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|e_{n_k}|}{|g_{n_k}|},\frac{|f_{n_k}|}{|h_{n_k}|}\leq\frac{r_3}{r_4} \end{equation*}
and $|h_{n_k}|>r_2,r_4$, then we can use Lemma \ref{positive} to deduce that for all sufficiently large $m\in\mathbb{N}$, we have \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_m|}{|c_m|},\frac{|b_m|}{|d_m|}\leq\frac{r_3}{r_4}. \end{equation*} But this cannot be because \begin{equation*}
\lim_{m\rightarrow\infty}\frac{|a_m|}{|c_m|}=\lim_{m\rightarrow\infty}\frac{|b_m|}{|d_m|}=1. \end{equation*}
We can therefore see with Lemma \ref{ratio1} that for all $k\in\mathbb{N}$, we have $|f_{k_m}|=|g_{k_m}|=|h_{k_m}|-1=|e_{k_m}|+1$. Suppose the desired inequality holds for $k$. We will prove it also holds for $k+1$. Notice that Lemma \ref{Q_nproductP_n} implies that for all $n\in\mathbb{N}$ $Q_n$ is a product of the matrices $P_1$ and $P_2$ if and only if $n$ is in the sequence $(n_j)_j$. It follows that $Q_{n_{k+1}}=Q_{n_k}P_1$ or $Q_{n_{k+1}}=Q_{n_k}P_2$. Then we have that either $|g_{n_{k+1}}|=|h_{n_k}||c_i|+|g_{n_k}||a_i|$ or $|g_{n_{k+1}}|=|h_{n_k}||c_i|-|g_{n_k}||a_i|$ where $i=1$ or $2$. Suppose the first equality holds. Then we also have $|e_{n_{k+1}}|=|e_{n_k}||a_i|+|f_{n_k}||c_i|$ so that \begin{align*}
|g_{n_{k+1}}|-|e_{n_{k+1}}|&=|h_{n_k}||c_i|+|g_{n_k}||a_i|-|e_{n_k}||a_i|-|f_{n_k}||c_i|\\
&=(|h_{n_k}|-|f_{n_k}|)|c_i|+(|g_{n_k}|-|e_{n_k}|)|a_i|\\
&=|c_i|+|a_i|\\ &>1, \end{align*} a contradiction. Thus we have \begin{align*}
|g_{n_{k+1}}|&=|h_{n_k}||c_i|-|g_{n_k}||a_i|\\
&=(|g_{n_k}|+1)|c_i|-|g_{n_k}||c_i|+|g_{n_k}|\\
&=|g_{n_k}|+|c_i|\\
&\leq n_k\max\{|c_1|,|c_2|\}+|c_i|\\
&\leq(n_k+1)\max\{|c_1|,|c_2|\}\\
&\leq(n_{k+1})\max\{|c_1|,|c_2|\}. \end{align*} \end{proof} \begin{proposition}\label{entrieslimit1} Let $e_n$, $f_n$, $g_n$, and $h_n$ be as defined in Notation \ref{P_1P_2}. Then there exists $D>0$ such that for all $n\in\mathbb{N}$, we have \begin{equation*}
|e_n|,|f_n|,|g_n|,|h_n|<Dn. \end{equation*} \end{proposition} \begin{proof} Consider all product matrices constructed as the string of $A$s and $B$s in $P_1$ truncated after a certain point and in $P_2$ truncated after a certain point. Let $M$ be the largest entry in absolute value of all such matrices. Let $n\in\mathbb{N}$. By Lemma \ref{Q_nproductP_n}, we have there exists $2\leq m_1<m_2<...<m_l$ and $ n_1,\ldots,n_l\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_i\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_i=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$. \item $Q_n=(P_{m_l})^{n_l}(P_{m_{l-1}})^{n_{l-1}}...(P_{m_1})^{n_1}M_n$ where the matrix $M_n$ is a product of the string of $A_1$s, $A_2$s, $\ldots$, and $A_v$s in either $P_1$ or $P_2$ truncated after a certain point. \end{enumerate}
Let \begin{equation*} Q_{n'}=(P_{m_l})^{n_l}(P_{m_{l-1}})^{n_{l-1}}...(P_{m_1})^{n_1} \end{equation*} so that \begin{equation*} Q_n=Q_{n'}M_n. \end{equation*}
Note that $n'<n+k_2$. By Lemma \ref{nklinear}, we have $|g_{n'}|\leq n'\max\{|c_1|,|c_2|\}$. Let $C=\max\{|c_1|,|c_2|\}$. Then we have \begin{align*}
\max\{|e_n|,|f_n|,|g_n|,|h_n|\}& \leq 2M(|g_{n'}|+1)\leq 2M(Cn'+1)<2M(C(n+k_2)+1)\\ & \leq(2MC+2MCk_2+2M)n. \end{align*} Since none of $M,C$, or $k_2$ depend on $n$, letting $D=2MC+2MCk_2+2M$, we obtain our result. \end{proof} \begin{proof}[Proof of Theorem \ref{bigthm} for Case $1$] From Proposition \ref{entrieslimit1}, we get there exists $C>0$ such that for all $n\in\mathbb{N}$ \begin{equation*}
|G_n|<Cn. \end{equation*} \end{proof} \section{The Exponential Growth Case}\label{sec4} Case $2$ covers all other cases. By Lemma \ref{ratio1}, there exists $m\in\mathbb{N}$, $m\geq 2$, with the following properties. $a_m,b_m,c_m,d_m$ are all nonzero and there exists positive integers $r_1,r_2,r_3,r_4,r_5,r_6,r_7,r_8$ such that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_m|}{|c_m|},\frac{|b_m|}{|d_m|}\leq\frac{r_3}{r_4},
\text{\ \ \ and\ \ \ }
\frac{r_5}{r_6}\leq\frac{|a_m|}{|b_m|},\frac{|c_m|}{|d_m|}\leq\frac{r_7}{r_8}, \end{equation*}
with $r_1<r_2$, $r_3<r_4$, $r_5<r_6$, $r_7<r_8$, and $|d_m|>r_2,r_4,r_6,r_8$. Without loss of generality, we may assume that the inequalities hold for $m=2$. Also, by taking the minimum of $\frac{r_1}{r_2}$ and $\frac{r_5}{r_6}$ and the maximum of $\frac{r_3}{r_4}$ and $\frac{r_7}{r_8}$, we can say there exists positive integers $r_1,r_2,r_3,r_4$ such that \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_2|}{|c_2|},\frac{|b_2|}{|d_2|},\frac{|a_2|}{|b_2|},\frac{|c_2|}{|d_2|}\leq\frac{r_3}{r_4} \end{equation*}
with $r_1<r_2$, $r_3<r_4$, and $|d_2|>r_2,r_4$. \begin{lemma}\label{ratios1} Let $a_m$, $b_m$, $c_m$, and $d_m$ be as defined in Notation \ref{P_1P_2}. For all $m\geq 2$, we have \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_m|}{|c_m|},\frac{|b_m|}{|d_m|}\leq\frac{r_3}{r_4} \end{equation*}
and $|d_m|>r_2,r_4$. \end{lemma} \begin{proof}
We prove our result by induction on $m\in\mathbb{N}$. We have already established it for $m=2$. Suppose the case holds for some $m$ where $m\geq 2$. We prove it holds for $P_{m+1}=(P_m)^{q_{m-1}}P_{m-1}$.\par First assume that $\frac{c_{m,q_{m-1}}b_{m-1}}{d_{m,q_{m-1}}d_{m-1}}>0$. By Lemma \ref{power}, we have $|d_{m,q_{m+1}}|\geq |c_{m,q_{m+1}}|+1$. By Lemma \ref{positive}, we have the desired inequalities holding for $P_{m+1}$ with the observation by Lemma \ref{power} that \begin{align*}
|d_{m+1}|&\geq |d_{m,q_{m-1}}||d_{m-1}|-|c_{m,q_{m-1}}||b_{m-1}|\\
&\geq |d_{m,q_{m-1}}|(|b_{m-1}|+1)-|c_{m,q_{m-1}}||b_{m-1}|\\
&\geq(|d_{m,q_{m-1}}|-|c_{m,q_{m-1}}|)|b_{m-1}|+|d_m,q_{m-1}|\\
&>|d_m,q_{m-1}|\\
&>|d_m|\\ &>r_2,r_4. \end{align*}
Now assume that $\frac{c_{m,q_{m-1}}b_{m-1}}{d_{m,q_{m-1}}d_{m-1}}<0$. By induction, we have all of the inequalities holding in Lemma \ref{positive}. Lemma \ref{positive} thus gives us all of the desired inequalities holding for $P_{m+1}$ again with the observation that $|d_{m+1}|>|d_m|>r_2,r_4$. \end{proof} Also, with the help of Lemma \ref{positive}, we can obtain the following. \begin{lemma}\label{ratios3} Let $a_m$, $b_m$, $c_m$, and $d_m$ be as defined in Notation \ref{P_1P_2}. For all even $m\geq 2$, we have \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_m|}{|b_m|},\frac{|c_m|}{|d_m|}\leq\frac{r_3}{r_4} \end{equation*}
and $|d_m|>r_2,r_4$. \end{lemma} By Lemmas \ref{ratio1}, we can also obtain that there exists positive integers $r_9,r_{10},r_{11},r_{12}$ such that \begin{equation*}
\frac{r_9}{r_{10}}\leq\frac{|a_3|}{|b_3|},\frac{|c_3|}{|d_3|}\leq\frac{r_{11}}{r_{12}} \end{equation*}
with $r_9<r_{10}$, $r_{11}<r_{12}$, and $|d_2|>r_{10},r_{12}$ and so also with the help of Lemma \ref{positive}, we can obtain the following. \begin{lemma}\label{ratios2} Let $a_m$, $b_m$, $c_m$, and $d_m$ be as defined in Notation \ref{P_1P_2}. For all odd $m\geq 3$, we have \begin{equation*}
\frac{r_9}{r_{10}}\leq\frac{|a_m|}{|b_m|},\frac{|c_m|}{|d_m|}\leq\frac{r_{11}}{r_{12}} \end{equation*}
and $|d_m|>r_2,r_4$. \end{lemma} \begin{remark} Without loss of generality, we will assume that $r_9=r_1$, $r_{10}=r_2$, $r_{11}=r_3$, and $r_{12}=r_4$ for the rest of this section. \end{remark} \begin{lemma}\label{qratios} Let $a_m$, $b_m$, $c_m$, and $d_m$ be as defined in Notation \ref{P_1P_2}. For all $m\geq 2$ and $i\in\mathbb{N}$, let \begin{equation*} P_m^i:=\begin{bmatrix} a_{m,i} & b_{m,i}\\ c_{m,i} & d_{m,i} \end{bmatrix}. \end{equation*} Then we have \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|a_{m,i}|}{|c_{m,i}|},\frac{|b_{m,i}|}{|d_{m,i}|},\frac{|a_{m,i}|}{|b_{m,i}|},\frac{|c_{m,i}|}{|d_{m,i}|}\leq\frac{r_3}{r_4}. \end{equation*} \end{lemma} \begin{remark} For the rest of the section, we will let $t_1=\frac{(r_4-r_3)}{r_3r_4}$ and $t_2=\frac{(r_2r_4+r_1r_3)}{r_1r_4}$. \end{remark} \begin{lemma}\label{c_mrelations} Let $c_m$ be as defined in Notation \ref{P_1P_2}. We have \begin{equation*}
(t_1|c_{m-1}|)^{q_{m-2}}|c_{m-2}|\leq|c_m|\leq(t_2|c_{m-1}|)^{q_{m-2}}|c_{m-2}| \end{equation*} for all $m\geq 4$. \end{lemma} \begin{proof} Let $m\geq 4$. By Lemmas \ref{ratios3}, \ref{ratios2}, and \ref{qratios}, we have \begin{align*}
|c_m| &\geq |d_{m-1,q_{m-2}}||c_{m-2}|- |c_{m-1,q_{m-2}}||a_{m-2}|\\
&\geq\frac{r_4}{r_3}|c_{m-1,q_{m-2}}||c_{m-2}|-\frac{r_3}{r_4}|c_{m-1,q_{m-2}}||c_{m-2}|\\
&=\frac{(r_4-r_3)}{r_3r_4}|c_{m-1,q_{m-2}}||c_{m-2}|. \end{align*}
Through induction on $q_{m-2}$, we can similarly derive that $|c_{m-1,q_{m-2}}|\geq t_1^{q_{m-2}-1}|c_{m-1}|^{q_{m-2}}$. Also, we have \begin{align*}
|c_m| &\leq |d_{m-1,q_{m-2}}||c_{m-2}|+|c_{m-1,q_{m-2}}||a_{m-2}|\\
&\leq\frac{r_2}{r_1}|c_{m-1,q_{m-2}}||c_{m-2}|+\frac{r_3}{r_4}|c_{m-1,q_{m-2}}||c_{m-2}|\\
&=\frac{(r_2r_4+r_1r_3)}{r_1r_4}|c_{m-1,q_{m-2}}||c_{m-2}|. \end{align*}
Again, through induction on $q_{m-2}$, we can similarly derive that $|c_{m-1,q_{m-2}}|\leq t_2^{q_{m-2}-1}|c_{m-1}|^{q_{m-2}}$. Thus we have our result. \end{proof} \begin{proposition}\label{limc_mexists} Let $q_m$, $c_m$, and $k_m$ be as defined in Notation \ref{P_1P_2}. We have \begin{equation*}
\lim_{m\rightarrow\infty}|c_m|^{1/k_m} \end{equation*} exists, is finite, and is greater than $1$. \end{proposition} \begin{proof} It suffices to show that \begin{equation*}
\lim_{m\rightarrow\infty}\frac{\log|c_m|}{k_m} \end{equation*}
exists, is finite, and is positive. Let $u_m=\log |c_m|$ and $s_m:=\frac{u_m}{k_m}$. By Lemma \ref{c_mrelations}, we have that $q_{m-1}(u_m+\log t_1)+u_{m-1}\leq u_{m+1}\leq q_{m-1}(u_m+\log t_2)+u_{m-1}$ for all $m\in\mathbb{N}$, $m\geq 3$. Let $m\in\mathbb{N}$, $m\geq 3$. We have \begin{align} s_{m+1}-s_m&=\frac{u_m}{k_{m+1}}\left(\frac{u_{m+1}}{u_m}-\frac{k_{m+1}}{k_m}\right)\nonumber\\ &\leq\frac{u_m}{k_{m+1}}\left(\frac{q_{m-1}(u_m+\log t_2)+u_{m-1}}{u_m}-\frac{k_{m+1}}{k_m}\right)\nonumber\\ &=\frac{u_m}{k_{m+1}}\left(\frac{u_{m-1}+q_{m-1}\log t_2}{u_m}-\frac{k_{m-1}}{k_m}\right)\nonumber\\ &=\frac{u_{m-1}}{k_{m+1}}-\frac{u_mk_{m-1}}{k_{m+1}k_m}+\frac{q_{m-1}\log t_2}{k_{m+1}}\nonumber\\ &=\frac{\left(u_{m-1}-\dfrac{u_mk_{m-1}}{k_m}\right)}{k_{m+1}}+\frac{q_{m-1}\log t_2}{k_{m+1}}\nonumber\\ &=\frac{k_{m-1}(s_{m-1}-s_m)}{k_{m+1}}+\frac{q_{m-1}\log t_2}{k_{m+1}}.\label{t2ineq} \end{align} Similarly, we have \begin{equation} s_m-s_{m+1}\leq\frac{k_{m-1}(s_{m-1}-s_m)}{k_{m+1}}+\frac{q_{m-1}\log t_1}{k_{m+1}}.\label{t1ineq} \end{equation} Therefore \begin{equation*}
|s_m-s_{m+1}|\leq\frac{k_{m-1}|s_m-s_{m-1}|}{k_{m+1}}+\frac{q_{m-1}\log t}{k_{m+1}}, \end{equation*} where $t=\max\{t_2,t_1^{-1}\}$. Note that \begin{equation*} \frac{k_{m+1}}{k_{m-1}}=\frac{k_m+q_{m+1}k_{m-1}}{k_{m-1}}>1+q_{m+1}\geq 2. \end{equation*} Thus \begin{equation*}
|s_{m+1}-s_m|\le\frac{|s_m-s_{m-1}|}{2}+\frac{\log t}{k_m}. \end{equation*} Consider the Fibonacci sequence $F_1=1, F_2=1$, and $F_m=F_{m-1}+F_{m-2}$ for all $m\geq 3$. Then we have \begin{equation} k_m\geq F_m=\left\lfloor\frac{\varphi^m}{\sqrt{5}}\right\rfloor\geq\frac{\varphi^m}{5}.\label{kngrowth} \end{equation} Thus \begin{equation*}
|s_{m+1}-s_m|\le\frac{|s_m-s_{m-1}|}{2}+\frac{5\log t}{\varphi^m}. \end{equation*} We can prove by induction on $l\in\mathbb{N}$ that for all $m\geq 3$ and $l\geq 1$, we have \begin{align*}
|s_{m+l}-s_{m+l-1}|&\leq\frac{|s_m-s_{m-1}|}{2^l}+\left(\frac{1}{2^{l-1}\cdot\varphi^m}+\frac{1}{2^{l-2}\varphi^{m+1}}+\ldots+\frac{1}{\varphi^{m+l-1}}\right)\log t\\
&<\frac{|s_m-s_{m-1}|}{2^l}+\frac{1}{\varphi^{m+l-1}}\left(1+\frac{\varphi}{2}+\frac{\varphi^2}{4}+\ldots\right)\\
&=\frac{|s_m-s_{m-1}|}{2^l}+\frac{1}{\varphi^{m+l-1}}\left(\frac{2}{2-\varphi}\right). \end{align*}
By a geometric series argument, the limit exists and is finite. It remains to show the limit is positive. By Lemma \ref{c_mrelations}, we have $q_{m-2}(\log t_1+\log|c_{m-1}|)+\log|c_{m-2}|\leq\log|c_m|$ for all $m\geq 3$. We thus have \begin{equation*}
q_{m-2}(\log|c_{m-1}|+\log t_1)+(\log|c_{m-2}|+\log t_1)\leq\log|c_m|+\log t_1 \end{equation*}
for all $m\geq 3$. By Lemma \ref{limitsinfinity}, we have $\lim_{m\rightarrow\infty}\log|c_m|=\infty$. We can therefore deduce that there exists $C_2>0$ such that for all sufficiently large $m\in\mathbb{N}$, we have $\log|c_m|+\log t_1>C_2k_m$. It follows that the limit is positive. \end{proof} \begin{remark}
Let $q_m$, $c_m$, and $q_m$ be as in Proposition \ref{limc_mexists}. Let $L:=\lim_{m\rightarrow\infty}|c_m|^{1/k_m}$. \end{remark} \begin{lemma}\label{prodrelc_m} Let $P_m$, $c_m$, $Q_n$, $e_n$, $f_n$, $g_n$, and $h_n$ be as defined in Notation \ref{P_1P_2}. Let $n\in\mathbb{N}$ such that there exists $2\leq m_1<m_2<...<m_l$ and $n_1,\ldots,n_l\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_i\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_i=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$. \item $Q_n=(P_{m_l})^{n_l}(P_{m_{l-1}})^{n_{l-1}}...(P_{m_1})^{n_1}$ \end{enumerate} Then \begin{equation*}
\frac{r_1}{r_2}\leq\frac{|e_n|}{|g_n|},\frac{|f_n|}{|h_n|},\frac{|e_n|}{|f_n|},\frac{|g_n|}{|h_n|}\leq\frac{r_3}{r_4}. \end{equation*} Also, we have \begin{equation*}
t_1^{l+n_1+\ldots+n_l}|c_{m_l}|^{n_l}|c_{m_{l-1}}|^{n_{l-1}}...|c_{m_1}|^{n_1}
\leq|g_n|
\leq t_2^{l+n_1+\ldots+n_l}|c_{m_l}|^{n_l}|c_{m_{l-1}}|^{n_{l-1}}...|c_{m_1}|^{n_1}. \end{equation*} \end{lemma} \begin{proof} The first pair of inequalities follows by similar reasoning as in the proof of Lemma \ref{positive}. The second pair of inequalities can by proved by induction on $l\in\mathbb{N}$ with the base case and induction step proved as in the proof of Lemma \ref{c_mrelations}. \end{proof} \begin{proposition}\label{rootgnj} Let $P_m$, $k_m$, $Q_n$, and $g_n$ be as defined in Notation \ref{P_1P_2}. Let $n_1,n_2,\ldots,n_j,\ldots,$ be the list of natural numbers such that for each $n_j$, there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_{n_j}=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{j,n_1}$. \end{enumerate}
We have \begin{equation*}
\lim_{j\rightarrow\infty}|g_{n_j}|^{1/n_j}=L. \end{equation*} \end{proposition} \begin{proof} It suffices to prove that \begin{equation*}
\lim_{j\rightarrow\infty}\frac{\log|g_{n_j}|}{n_j}=\log L. \end{equation*} We have \begin{equation*}
\lim_{m\rightarrow\infty}\frac{\log|c_m|}{k_m}=\log L. \end{equation*} Let $\epsilon>0$. Pick $\frac{\log L}{2}>\delta_1>0$ and $\delta_2,\delta_3,\delta_4>0$ such that $(\log L+\delta_1)(1+\delta_2)<\log L+\epsilon$ and $\frac{(1-\delta_3)(\log L-\delta_1)}{(1+\delta_4)}>\log L-\epsilon$. Choose $M\in\mathbb{N}$ such that for all $m\geq M$, we have \begin{equation}
\left|\frac{\log|c_m|}{k_m}-\log L\right|<\delta_1\label{epsiloncond1}, \end{equation} \begin{equation} \frac{10}{\varphi^{M-1}(\varphi-1)\log L}+\frac{2}{k_M\log L}<\frac{\delta_2}{2}\label{epsiloncond5}, \end{equation} and \begin{equation} \frac{10\log(t_1^{-1})}{\varphi^{M-1}(\varphi-1)\log L}+\frac{2\log(t_1^{-1})}{k_M\log L}<\frac{\delta_3}{2}\label{epsiloncond6}. \end{equation} By \eqref{kngrowth}, we have $\lim_{m\rightarrow\infty}\frac{m}{k_m}=0$. Thus we can choose $N>M$ such that for all $m\geq N$, we have \begin{equation} \frac{2m\widehat{q_{M-1}}}{k_m\log L}+\frac{3\widehat{q_{M-1}}Mk_M}{k_m}<\frac{\delta_2}{2}\label{epsiloncond2}, \end{equation} \begin{equation} \frac{2m\widehat{q_{M-1}}\log(t_1^{-1})}{k_m\log L}<\frac{\delta_3}{2}\label{epsiloncond3}, \end{equation} and \begin{equation} \frac{\widehat{q_{M-1}}Mk_M}{k_{m}}<\delta_4\label{epsiloncond4} \end{equation} where \begin{equation*} \widehat{q_{M-1}}:=\max\{q_1,q_1,\ldots,q_{M-1}\}. \end{equation*} Let $n_j\geq k_{N}$. Then there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_n=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{j,n_1}$. \end{enumerate}
By Lemma \ref{prodrelc_m}, we have \begin{align}
& (l+n_{j,1}+\ldots+n_{j,l})\log(t_1)+n_{j,l}\log|c_{m_l}|+...+n_{j,1}\log|c_{m_1}|\nonumber\\
& \qquad \qquad \leq\log|g_{n_j}|
\leq(l+n_{j,1}+\ldots+n_{j,l})\log(t_2)+n_{j,l}\log|c_{m_l}|+...+n_{j,1}\log|c_{m_1}|\label{gbounds}. \end{align} Pick $1\leq y\leq l$ such that $m_y\geq M>m_{y-1}$. Thus $y<M$. By \eqref{epsiloncond1}, we have \begin{equation}
\log L-\delta_1<\frac{n_{j,l}\log|c_{m_l}|+...+n_{j,y}\log|c_{m_y}|}{n_{j,l}k_{m_l}+...+n_{j,y}k_{m_y}}<\log L+\delta_1\label{epsilon1bounds}. \end{equation} Also observe the following. \begin{align}
\frac{l+n_{j,1}+\ldots+n_{j,l-1}}{\log|c_{m_l}|}&<\frac{(l+q_{m_1-1}+\ldots+q_{m_{l-1}-1})}{\log|c_{m_l}|}\nonumber\\
&<\frac{(l+q_{m_1-1}+\ldots+q_{m_{y-1}-1})}{\log|c_{m_l}|}+\frac{q_{m_y-1}+\ldots+q_{m_{l-1}-1}}{\log |c_{m_l}|}\nonumber\\
&<\frac{l\widehat{q_{M-1}}}{\log |c_{m_l}|}+\frac{\dfrac{k_{m_y+1}}{k_{m_y}}+\ldots+\dfrac{k_{m_{l-1}+1}}{k_{m_l-1}}}{\log |c_{m_l}|}\nonumber\\ &<\frac{l\widehat{q_{M-1}}}{k_{m_l}(\log L-\delta_1)}+\frac{\dfrac{k_{m_y+1}}{k_{m_y}}+\ldots+\dfrac{k_{m_{l-1}+1}}{k_{m_l-1}}}{k_{m_l}(\log L-\delta_1)}\nonumber\\ &<\frac{l\widehat{q_{M-1}}}{k_{m_l}(\log L-\delta_1)}+\frac{1}{(\log L-\delta_1)}\sum_{j=m_y}^{\infty}\frac{1}{k_j}\nonumber\\ &<\frac{l\widehat{q_{M-1}}}{k_{m_l}(\log L-\delta_1)}+\frac{1}{(\log L-\delta_1)}\sum_{j=m_y}^{\infty}\frac{5}{\varphi^y}\nonumber\\ &=\frac{l\widehat{q_{M-1}}}{k_{m_l}(\log L-\delta_1)}+\frac{5}{\varphi^{m_y-1}(\varphi-1)(\log L-\delta_1)}\nonumber\\ &<\frac{2m_l\widehat{q_{M-1}}}{k_{m_l}\log L}+\frac{10}{\varphi^{M-1}(\varphi-1)\log L}.\label{nlog} \end{align} Thus, by \eqref{epsiloncond1}, \eqref{epsiloncond5}, and \eqref{epsiloncond2}, we have \begin{align}
&\frac{(l+n_{j,1}+\ldots+n_{j,l})\log(t_2)+n_{j,l}\log|c_{m_l}|+...+n_{j,1}\log|c_{m_1}|}{n_{j,l}\log|c_{m_l}|+...+n_{j,y}\log|c_{m_y}|}\nonumber\\
&\qquad \qquad < 1+\frac{(l+n_{j,1}+\ldots+n_{j,l})\log(t_2)+n_{j,y-1}\log|c_{m_l}|+...+n_{j,1}\log|c_{m_1}|}{n_{j,l}\log|c_{m_l}|}\nonumber\\
&\qquad \qquad <1+\frac{(l+n_{j,1}+\ldots+n_{j,l})\log(t_2)+\widehat{q_{M-1}}y\log|c_{m_{y-1}}|}{n_{j,l}\log|c_{m_l}|}\nonumber\\
&\qquad \qquad <1+\frac{(l+n_{j,1}+\ldots+n_{j,l-1})\log(t_2)+\widehat{q_{M-1}}M\log|c_M|}{\log|c_{m_l}|}+\frac{1}{\log |c_{m_l}|}\nonumber\\
&\qquad \qquad <1+\frac{(l+n_{j,1}+\ldots+n_{j,l-1})\log(t_2)}{\log |c_{m_l}|}+\frac{\widehat{q_{M-1}}Mk_M(\log L+\delta_1)}{k_{m_l}(\log L-\delta_1)}+\frac{1}{\log |c_M|}\nonumber\\
&\qquad \qquad <1+\frac{2m_l\widehat{q_{M-1}}}{k_{m_l}\log L}+\frac{10}{\varphi^{M-1}(\varphi-1)\log L}+\frac{\widehat{q_{M-1}}Mk_M(\log L+\delta_1)}{k_{m_l}(\log L-\delta_1)}+\frac{1}{\log |c_M|}\nonumber\\ &\qquad \qquad <1+\frac{2m_l\widehat{q_{M-1}}}{k_{m_l}\log L}+\frac{10}{\varphi^{M-1}(\varphi-1)\log L}+\frac{3\widehat{q_{M-1}}Mk_M}{k_{m_l}}+\frac{2}{k_M\log L}\nonumber\\ &\qquad \qquad <1+\delta_2\label{epsilon2bound}. \end{align} Combining \eqref{gbounds}, \eqref{epsilon1bounds}, and \eqref{epsilon2bound}, we thus have \begin{align*}
\frac{\log |g_{n_j}|}{n_j}&<\frac{(1+\delta_2)(n_{j,l}\log|c_{m_l}|+...+n_{j,y}\log|c_{m_y}|)}{n_{j,l}k_{m_l}+...+n_{j,y}k_{m_y}}\\ &<(\log L+\delta_1)(1+\delta_2)\\ &<\log L+\epsilon. \end{align*} Also, since $t_1\leq 1$, by \eqref{epsiloncond6}, \eqref{epsiloncond3}, and \eqref{nlog}, we have \begin{align}
&\frac{(l+n_{j,1}+\ldots+n_{j,l})\log(t_1)+n_{j,l}\log|c_{m_l}|+...+n_{j,1}\log|c_{m_1}|}{n_{j,l}\log|c_{m_l}|+...+n_{j,y}\log|c_{m_y}|}\nonumber\\
&\qquad \qquad >1-\frac{(l+n_{j,1}+\ldots+n_{j,l})\log(t_1^{-1})}{n_{j,l}\log|c_{m_l}|}\nonumber\\
&\qquad \qquad >1-\frac{(l+n_{j,1}+\ldots+n_{j,l-1})\log(t_1^{-1})}{\log|c_{m_l}|}-\frac{2\log(t_1^{-1})}{k_M\log L}\nonumber\\ &\qquad \qquad >1-\frac{2m_l\widehat{q_{M-1}}\log(t_1^{-1})}{k_{m_l}\log L}-\frac{10\log(t_1^{-1})}{\varphi^{M-1}(\varphi-1)\log L}-\frac{2\log(t_1^{-1})}{k_M\log L}\nonumber\\ &\qquad \qquad >1-\delta_3.\label{epsilon3bound} \end{align} Also \begin{align} \frac{n_j}{n_{j,l}k_{m_l}+...+n_{j,y}k_{m_y}}&=1+\frac{n_{j,y-1}k_{m_{y-1}}+\ldots+n_{j,1}k_{m_1}}{n_{j,l}k_{m_l}+...+n_{j,y}k_{m_y}}\nonumber\\ &<1+\frac{\widehat{q_{M-1}}yk_{m_{y-1}}}{k_{m_l}}\nonumber\\ &<1+\frac{\widehat{q_{M-1}}Mk_M}{k_{m_l}}\nonumber\\ &<1+\delta_4\label{epsilon4bound} \end{align} by \eqref{epsiloncond4}. Combining \eqref{gbounds}, \eqref{epsilon1bounds}, \eqref{epsilon3bound}, and \eqref{epsilon4bound}, we have \begin{align*}
\frac{\log |g_{n_j}|}{n_j}&>\frac{(1-\delta_3)(n_{j,l}\log|c_{m_l}|+...+n_{j,y}\log|c_{m_y}|)}{(1+\delta_4)(n_{j,l}k_{m_l}+...+n_{j,y}k_{m_y})}\\ &>\frac{(1-\delta_3)(\log L-\delta_1)}{(1+\delta_4)}\\ &>\log L-\epsilon. \end{align*} \end{proof} \begin{lemma}\label{limitratios1} Let $q_m$, $a_m$ and $c_m$ be as defined in Notation \ref{P_1P_2}. We have $\lim_{m\rightarrow\infty}\frac{a_m}{c_m}$ exists, is between $-1$ and $1$, and is irrational. \end{lemma} \begin{proof} We have \begin{equation*}
\lim_{m\rightarrow\infty}\frac{\log|c_m|}{k_m}=\log L>0. \end{equation*} Thus there exists $L'>1$ such that for all sufficiently large $m\in\mathbb{N}$, we have \begin{equation*}
|c_{m-1}|>L'^{k_{m-1}}. \end{equation*} Let \begin{equation*} P_{m-1}^{q_{m-2}-1}P_{m-2}=: \begin{bmatrix} a_m' & b_m'\\ c_m' & d_m' \end{bmatrix}. \end{equation*} Then for $m\in\mathbb{N}$ sufficiently large, we have \begin{align*}
\left|\frac{a_m}{c_m}-\frac{a_{m-1}}{c_{m-1}}\right|&=\left|\frac{a_{m-1}a_{m-2}'+b_{m-1}c_{m-2}'}{c_{m-1}a_{m-2}'+d_{m-1}c_{m-2}'}-\frac{a_{m-1}}{c_{m-1}}\right|\\
&=\left|\frac{\left(b_{m-1}-\dfrac{a_{m-1}d_{m-1}}{c_{m-1}}\right)c_m'}{c_{m-1}a_{m-2}'+d_{m-1}c_{m-2}'}\right|
=\frac{|c_m'|}{|c_m||c_{m-1}|}\\
&\leq\frac{|c_m'|}{t_1|c_m'||c_{m-1}|^2} \leq\frac{1}{t_1L'^{2k_{m-1}}}. \end{align*} By a geometric series argument, using \eqref{kngrowth}, the sequence $\frac{a_m}{c_m}$ is Cauchy and so converges. The fact that the limit is between $-1$ and $1$ follows from Lemma \ref{ratios1}. It remains to show the limit is irrational. Suppose for a contradiction that it is rational and let it be $\frac{a}{b}$ where $a,b\in\mathbb{N}$. Let $N\in\mathbb{N}$ be sufficiently large so that for all $m\geq N$, the above inequality holds and $L'^{-2k_{m-1}}<\frac{1}{2}$. Then for all $m\geq N$, we have \begin{align*}
\left|\frac{a}{b}-\frac{a_m}{c_m}\right|&\leq\sum_{i=m}^{\infty}\left|\frac{a_{i+1}}{c_{i+1}}-\frac{a_i}{c_i}\right| <\sum_{i=m}^{\infty}\frac{1}{t_1L'^{2k_i}} =\frac{1}{t_1L'^{2k_m}}\sum_{i=0}^{\infty}L'^{2k_m-2k_{m+i}}\\ &=\frac{1}{t_1L'^{2k_m}}\sum_{i=0}^{\infty}\prod_{j=0}^{i-1}L'^{2k_{m+j}-2k_{m+j+1}}\\ &<\frac{1}{t_1L'^{2k_m}}\sum_{i=0}^{\infty}\prod_{j=0}^{i-1}L'^{-2k_{m+j-1}}\\ &<\frac{1}{t_1L'^{2k_m}}\sum_{i=0}^{\infty}\left(\frac{1}{2}\right)^i\\ &=\frac{2}{t_1L'^{2k_m}}. \end{align*} Thus \begin{equation*}
\frac{|ac_m-ba_m|}{|bc_m|}<\frac{2}{t_1L'^{2k_m}}. \end{equation*} Suppose that $\frac{a}{b}\neq\frac{a_m}{c_m}$. Then we have \begin{equation*}
\frac{1}{|bc_m|}<\frac{2}{t_1L'^{2k_m}}. \end{equation*} Thus \begin{equation*}
\frac{t_1L'^{2k_m}}{2|b|}<|c_m|
\text{\ \ \ or\ \ \ }
\left(\frac{t_1}{2|b|}\right)^{1/k_m}L'^2<|c_m|^{1/k_m}. \end{equation*}
Thus if there are infinitely many $m\in\mathbb{N}$ such that $\frac{a}{b}\neq\frac{a_m}{c_m}$, then we have $L'^2\leq L'$, a contradiction since $L>1$. So for sufficiently large $m\in\mathbb{N}$, we have $\frac{a}{b}=\frac{a_m}{c_m}$. But for all $m\in\mathbb{N}$, we have $\gcd(a_m,c_m)=1$ and $\lim_{m\rightarrow\infty}|a_m|=\lim_{m\rightarrow\infty}|c_m|=\infty$ and so this cannot be the case either. Thus the limit must be irrational. \end{proof} \begin{remark} Let $q_m$, $a_m$ and $c_m$ as defined in Notation \ref{P_1P_2}. We will denote \begin{equation*} M:=\lim_{m\rightarrow\infty}\frac{a_m}{c_m}. \end{equation*} \end{remark} \begin{lemma}\label{n_jratiolimit} Let $q_m$, $P_m$, $Q_n$, $e_n$, $f_n$, $g_n$, and $h_n$ be as defined in Notation \ref{P_1P_2}. Let $n_1,n_2,\ldots,n_j,\ldots,$ be the list of natural numbers such that for each $n_j$, there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_n=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{j,n_1}$. \end{enumerate}
We have $\lim_{j\rightarrow\infty}\frac{e_{n_j}}{g_{n_j}}$ and $\lim_{j\rightarrow\infty}\frac{f_{n_j}}{h_{n_j}}$ both exist and are equal to $M$. \end{lemma} \begin{proof} By Lemma \ref{limitratios1}, we have that \begin{equation*} \lim_{m\rightarrow\infty}\frac{e_{k_m}}{g_{k_m}} \end{equation*} exists and is equal to $M$. We will prove that the desired limit is $M$. Let $\epsilon>0$. Choose $N\in\mathbb{N}$ such that for all $m\geq N$, we have \begin{equation*}
\left|\frac{e_{k_m}}{g_{k_m}}-M\right|<\frac{\epsilon}{2}
\text{\ \ \ and\ \ \ }
\frac{r_4^2}{(r_4^2-r_3^2)|g_{k_m}|}<\frac{\epsilon}{2}. \end{equation*} Let $n_j\geq k_N$. Then $k_{m+1}>n_j\geq k_m$ for some $m\geq N$. We have $Q_{n_j}=Q_{k_m}Q_{n_j-k_m}$. Thus we have the following using Lemma \ref{prodrelc_m}: \begin{align*}
\left|\frac{e_{n_j}}{g_{n_j}}-\frac{e_{k_m}}{g_{k_m}}\right|&=\left|\frac{e_{k_m}e_{n_j-k_m}+f_{k_m}g_{n_j-k_m}}{g_{k_m}e_{n_j-k_m}+h_{k_m}g_{n_j-k_m}}-\frac{e_{k_m}}{g_{k_m}}\right|\\
&=\left|\frac{g_{n_j-k_m}(f_{k_m}g_{k_m}-e_{k_m}h_{k_m})}{g_{k_m}(g_{k_m}e_{n_j-k_m}+h_{k_m}g_{n_j-k_m})}\right|\\
&=\frac{|g_{n_j-k_m}|}{|g_{k_m}||g_{k_m}e_{n_j-k_m}+h_{k_m}g_{n_j-k_m}|}\\
&\leq\frac{|g_{n_j-k_m}|}{|g_{k_m}|(|h_{k_m}g_{n_j-k_m}|-|g_{k_m}e_{n_j-k_m}|)}\\
&<\frac{|g_{n_j-k_m}|}{|g_{k_m}|\left(\left|h_{k_m}g_{n_j-k_m}\right|-\dfrac{r_3^2}{r_4^2}\left|h_{k_m}g_{n_j-k_m}\right|\right)}\\
&=\frac{r_4^2}{(r_4^2-r_3^2)|g_{k_m}h_{k_m}|}\\
&\leq\frac{r_4^2}{(r_4^2-r_3^2)|g_{k_m}|}\\ &<\frac{\epsilon}{2}. \end{align*} Thus \begin{equation*}
\left|\frac{e_{n_j}}{g_{n_j}}-M\right|\leq\left|\frac{e_{n_j}}{g_{n_j}}-\frac{e_{k_m}}{g_{k_m}}\right|+\left|\frac{e_{k_m}}{g_{k_m}}-M\right|<\epsilon. \end{equation*} Thus \begin{equation*} \lim_{j\rightarrow\infty}\frac{e_{n_j}}{g_{n_j}}=M.
\text{\ \ \ and\ \ \ }
\left|\frac{e_{n_j}}{g_{n_j}}-\frac{f_{n_j}}{h_{n_j}}\right|=\frac{1}{|g_{n_j}h_{n_j}|}, \end{equation*} from which the rest follow using Lemma \ref{prodrelc_m}. \end{proof} \begin{remark} For the rest of the paper, we will assume that $G_1\neq\frac{-G_2}{M}$. \end{remark} \begin{proposition}\label{partiallimit} Let $q_m$, $P_m$ and $Q_n$ be defined as in Notation \ref{P_1P_2}. Let $n_1,n_2,\ldots,n_j,\ldots,$ be the list of natural numbers such that for each $n_j$, there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_{n_j}=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{j,n_1}$. \end{enumerate}
Let \begin{equation*} [G_1,G_2]Q_n=[G_{n+1},G_{n+2}] \end{equation*} for all $n\in\mathbb{N}$. We have \begin{equation*}
\lim_{j\rightarrow\infty}|G_{n_j+1}|^{1/(n_j+1)}=\lim_{j\rightarrow\infty}|G_{n_j+2}|^{1/(n_j+2)}=L.
\text{\ \ \ and\ \ \ } \lim_{j\rightarrow\infty}\frac{G_{n_j+1}}{G_{n_j+2}}=M. \end{equation*} \end{proposition} \begin{proof} For all $n_j$, we have $G_1e_{n_j}+G_2g_{n_j}=G_{n_j+1}$. By Lemma \ref{prodrelc_m}, $g_{n_j}\neq 0$. Thus $\frac{G_1e_{n_j}}{g_{n_j}}+G_2=\frac{G_{n_j+1}}{g_{n_j}}$. By Lemma \ref{n_jratiolimit}, we have \begin{equation*} \lim_{j\rightarrow\infty}\frac{G_{n_j+1}}{g_{n_j}}=MG_1+G_2. \end{equation*} Since $G_1\neq\frac{-G_2}{M}$, we have the limit is nonzero and so \begin{equation*}
\lim_{j\rightarrow\infty}|G_{n_j+1}|^{1/(n_j+1)}=L. \end{equation*} follows from Proposition \ref{rootgnj}. The limit \begin{equation*}
\lim_{j\rightarrow\infty}|G_{n_j+2}|^{1/(n_j+2)}=L \end{equation*} follows similarly. Also, for all $j\in\mathbb{N}$, we have \begin{align*} \frac{G_{n_j+1}}{G_{n_j+2}}&=\frac{G_1e_{n_j}+G_2g_{n_j}}{G_1f_{n_j}+G_2h_{n_j}}\\ &=\frac{e_{n_j}}{f_{n_j}}\cdot\frac{G_1+\dfrac{G_2g_{n_j}}{e_{n_j}}}{G_1+\dfrac{G_2h_{n_j}}{f_{n_j}}}. \end{align*} Applying Lemma \ref{n_jratiolimit} gives us the third limit with the observation that \begin{equation*} \lim_{j\rightarrow\infty}G_1+\frac{G_2h_{n_j}}{f_{n_j}}=G_1+\frac{G_2}{M}\neq 0. \end{equation*} \end{proof} \begin{proof}[Proof of Theorem \ref{bigthm} for Case $2$] Let $n_1,n_2,\ldots,n_j,\ldots,$ be the list of natural numbers such that for each $n_j$, there exists $2\leq m_1<m_2<...<m_l$ and $n_{j,1},\ldots,n_{j,l}\in\mathbb{N}$ with the following properties: \begin{enumerate} \item for all $1\leq i\leq l$, we have $n_{j,i}\leq q_{m_i-1}$ \item for all $2\leq i\leq l$, if $n_{j,i}=q_{m_i-1}$, then $m_{i-1}+2\leq m_i$ \item $Q_n=(P_{m_l})^{n_{j,l}}(P_{m_{l-1}})^{n_{j,l-1}}...(P_{m_1})^{j,n_1}$. \end{enumerate} For all $j\in\mathbb{N}$, we have $n_{j+1}-n_j<k_2$ by Lemma \ref{Q_nproductP_n}. We can deduce that there exists $C>0$ such that for all $n\in\mathbb{N}$ there exists $n_j<n$ and integers $C_1,C_2<C$ such that $G_n=C_1G_{n_j+1}+C_2G_{n_j+2}$. By Proposition \ref{partiallimit}, we can deduce that \begin{equation*}
\limsup_{n\rightarrow\infty}|G_n|^{1/n}=L. \end{equation*} Also, out of all of the finite possibilities for $C_1$ and $C_2$, we observe that $MC_1+C_2\neq 0$ since $M$ is irrational. Let $M'$ denote the minimal possible value of $C_1+MC_2$ in absolute value. Then $M'>0$. By Proposition \ref{partiallimit}, for all $1\leq t<k_2$, we have \begin{equation*}
\liminf_{j\rightarrow\infty}\frac{|G_{n_j+t}|}{|G_{n_j+1}|}\geq M'. \end{equation*} It follows that \begin{equation*}
\liminf_{n\rightarrow\infty}|G_n|^{1/n}=L. \end{equation*} \end{proof} \section{Future Work} There are a couple of different directions this research can go in. The first direction involves studying the growth rates of generalised Fibonacci sequences produced by other patterns of words. Here we have examined Sturmian words, but there are other types of words as well. Some examples are words that follow a Thue-Morse pattern, as well as other morphisms. We could even try removing the condition $j,k\geq 2$ in Example \ref{entriessize}.
The other direction involves trying to calculate the exact growth rate of certain random Fibonacci sequences produced from words following such patterns and seeing how close to Viswanath's constant we can get. McLellan \cite{mclellan} used words following a periodic pattern to create a new way of calculating Viswanath's constant. By adding in new patterns into her method, we may be able to calculate Viswanath's constant even more accurately. We even might be able to calculate its exact value or at least shed some light on its nature (for example, if it's irrational, transcendental, etc.).
\section{Competing Interest Statement} The authors have no competing interests to declare.
\end{document} |
\begin{document}
\title{Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox}
\author{E. G. Cavalcanti}
\affiliation{Centre for Quantum Dynamics, Griffith University, Brisbane QLD 4111, Australia}
\affiliation{ARC Centre of Excellence for Quantum-Atom Optics, The University of Queensland, Brisbane, QLD 4072, Australia}
\author{S. J. Jones}
\affiliation{Centre for Quantum Dynamics, Griffith University, Brisbane QLD 4111, Australia}
\author{H. M. Wiseman}
\affiliation{Centre for Quantum Dynamics, Griffith University, Brisbane QLD 4111, Australia}
\author{M. D. Reid}
\affiliation{ARC Centre of Excellence for Quantum-Atom Optics, The University of Queensland, Brisbane, QLD 4072, Australia}
\date{\today{}} \begin{abstract} We formally link the concept of steering (a concept created by Schrödinger but only recently formalised by Wiseman, Jones and Doherty {[}Phys. Rev. Lett. 98, 140402 (2007){]} and the criteria for demonstrations of Einstein-Podolsky-Rosen (EPR) paradox introduced by Reid {[}Phys. Rev. A, 40, 913 (1989){]}. We develop a general theory of experimental EPR-steering criteria, derive a number of criteria applicable to discrete as well as continuous-variables observables, and study their efficacy in detecting that form of nonlocality in some classes of quantum states. We show that previous versions of EPR-type criteria can be rederived within this formalism, thus unifying these efforts from a modern quantum-information perspective and clarifying their conceptual and formal origin. The theory follows in close analogy with criteria for other forms of quantum nonlocality (Bell-nonlocality and entanglement), and because it is a hybrid of those two, it may lead to insights into the relationship between the different forms of nonlocality and the criteria that are able to detect them. \end{abstract} \maketitle
\section{Introduction}
In their seminal 1935 paper \citep{Einstein1935}, Einstein, Podolsky and Rosen (EPR) presented an argument which demonstrates the incompatibility between the concepts of \emph{local causality} \footnote{This is Bell's terminology \citep{Bell1971}. It is also commonly called local realism \citep{Reid1989}, which is arguably closer to EPR's terminology. See however Ref.~\citep{Wiseman2006} for a discussion of Einstein's later writings on locality and realism. } and the \emph{completeness} of quantum mechanics. Apart from the foundational importance of that work, it had long-reaching consequences \citep{Vedral2006}: it was the first time that physicists clearly noticed the strange phenomena associated with \emph{entanglement} --- the resource at the basis of modern quantum information science.
The situation depicted by EPR is often referred to as the ``EPR paradox''. The authors themselves did not intend to point out a true paradox; instead they argued that quantum mechanics was an incomplete theory, that is, that it did not give a complete description of reality. Schrödinger \citep{SchPCP35} seems to have been the first to name the situation a `paradox', as he could not believe with EPR that quantum mechanics was indeed incomplete but neither could he see a flaw in the argument. In hindsight, we now know (since Bell \citep{Bell1964}) that, while the argument is sound, one of the premises --- local causality --- is false. However, we will retain the historically prevalent term `paradox', if only because we still do not have a fully satisfactory understanding of the nature of quantum nonlocality.
The original EPR paradox involved an example of an idealized bipartite entangled state of continuous variables measured at the two subsystems. Later, Bohm \citep{Bohm1951} extended the EPR paradox to a scenario involving discrete (spin) observables. The essence of both of these arguments involved perfect correlations, and therefore neither the original EPR paradox nor Bohm's version could be directly tested in the laboratory without additional assumptions. Criteria for the experimental demonstration of the EPR paradox, which can be used in situations with non-ideal states, have been derived for the continuous-variables scenario by Reid in 1989 \citep{Reid1989} and more recently for discrete systems by Cavalcanti and Reid \citep{Cavalcanti2007b} and Cavalcanti\emph{ et al. }\citep{Cavalcanti2009b}.
In another recent development, Wiseman, Jones and Doherty \citep{Wiseman2007} have introduced a new classification of quantum nonlocality, a formalisation of the concept of \emph{steering} introduced by Schrödinger in 1935 \citep{Schroedinger1935} in a response to the EPR paper. In that Letter, the authors claimed that any demonstration of the EPR paradox, as proposed by Reid, is also a demonstration of steering. While that claim was essentially correct, the proof proposed there was incomplete, as we will see later in this paper. We will provide the missing proof and further show that the converse is also true: any demonstration of steering is also a demonstration of the EPR paradox. In other words, the EPR paradox and steering are equivalent notions of nonlocality.
In Ref. \citep{Wiseman2007} Wiseman, Jones and Doherty showed that EPR-steering constitutes a different class of nonlocality intermediate between the classes of quantum non-separability and Bell-nonlocality, with the distinction between these being explainable as a matter of trust between different parties. Therefore, besides its foundational interest, this classification could prove important in the context of quantum communication and information. It would be thus desirable to devise criteria to determine to which classes a given state (or a set of observed correlations) belongs. For that purpose we will formulate and develop the theory of \emph{EPR-steering criteria}, defined as any criteria which are sufficient to demonstrate EPR-steering experimentally. The theory will proceed in close analogy to the theories of entanglement criteria \citep{Duan2000,Simon2000,Hofmann2003a,Guhne2004a} and of Bell inequalities\emph{ }(or Bell-nonlocality criteria)\emph{ }\citep{Bell1964,Clauser1969,Mermin1980,Fine1982,Pitowsky1989,Ardehali1992,Belinskii1993,Peres1999,Werner2001,Collins2002a,Zukowski2002a,Cavalcanti2007a}\emph{.}
The structure of the paper is as follows: In Sec.~\ref{sec:History-and-concepts} we will review some of the history and concepts surrounding the EPR paradox and steering. The main purposes of this section are to review the conceptual motivation for the new formulation and to put the steering criteria proposed here in context with the relevant literature. In Sec.~\ref{sec:Locality-models} we will review the three classes of nonlocality, including Wiseman and coworkers' \citep{Wiseman2007} steering, and argue in more detail than in previous papers \citep{Jones2007} as to why it provides the correct formalization of Schrödinger's concept. In Sec.~\ref{sec:Experimental-criteria-for} we will introduce the formalism for derivation of general EPR-steering criteria. We develop two broad classes of EPR-steering criteria: the \emph{multiplicative variance criteri}a, and the \emph{additive convex criteria} (which includes linear EPR-steering inequalities as a special case). We show how the criteria in the existing literature can be rederived as special cases within this modern unifying approach. In Sec.~\ref{sec:Applications-to-classes} we will apply the criteria derived in Sec.~\ref{sec:Experimental-criteria-for} to some classes of quantum states, comparing their effectiveness in experimentally demonstrating EPR-steering. We consider both continuous variables (as in the original EPR paradox) and spin-half systems (as in Bohm's version).
\section{History and concepts\label{sec:History-and-concepts}}
\subsection{The Einstein-Podolsky-Rosen argument\label{sec:The-Einstein-Podolsky-Rosen-argument}}
The EPR argument has been exhaustively commented in the literature. However, since in this paper we will discuss a new mathematical formulation of it, it will be important to review it in detail.
The essence of Einstein and coworkers' \citep{Einstein1935} 1935 argument is a demonstration of the incompatibility between the premises of \emph{local causality} and the \emph{completeness} of quantum mechanics. EPR started the paper by making a distinction between \emph{reality} and the \emph{concepts} of a theory, followed by a critique of the operationalist position, clearly aimed at the views advocated by Bohr, Heisenberg and the other proponents of the Copenhagen interpretation. \begin{quote} ``Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.
In attempting to judge the success of a physical theory, we may ask ourselves two questions: (1) `Is the theory correct?' and (2) `Is the description given by the theory complete?' It is only in the case in which positive answers may be given to both of these questions, that the concepts of the theory may be said to be satisfactory.'' \citep{Einstein1935} \end{quote} Any theory will have some concepts which will be used to aid in the description and prediction of the phenomena which are their subject matter. In quantum theory, Schrödinger introduced the concept of the wave function and Heisenberg described the same phenomena with the more abstract matrix mechanics. EPR argued that we must distinguish those concepts from the reality they attempt to describe. One can see the physical concepts of the theory as mere calculational tools if one wishes, but it was those authors' opinion that one must be careful to avoid falling back into a pure operationalist position; the theory must strive to furnish a complete picture of reality.
EPR follow the previous considerations with a \emph{necessary condition for completeness:} \begin{quote} \textbf{EPR's necessary condition for completeness: }``Whatever the meaning assigned to the term \emph{complete, }the following requirement for a complete theory seems to be a necessary one: \emph{every element of the physical reality must have a counterpart in the physical theory}.'' \citep{Einstein1935} \end{quote} Soon afterward they note that this condition only makes sense if one is able to decide what are the elements of the physical reality. They did not attempt to \emph{define} `element of physical reality', saying ``The elements of the physical reality cannot be determined by \emph{a priori }philosophical considerations, but must be found by an appeal to results of experiments and measurements. A comprehensive definition of reality is, however, unnecessary for our purpose''. Instead they provide a \emph{sufficient condition}: \begin{quote} \textbf{EPR's sufficient condition for reality:} We shall be satisfied with the following criterion, which we regard as reasonable. \emph{If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity}.'' \citep{Einstein1935} \end{quote} Later in the same paragraph it is made explicit that this criterion is ``regarded not as a necessary, but merely as a sufficient, condition of reality''. This is followed by a discussion to the effect that, in quantum mechanics, if a system is in an eigenstate of an operator $A$ with eigenvalue $a$, by this criterion, there must be an element of physical reality corresponding to the physical quantity $A$. ``On the other hand'', they continue, if the state of the system is a superposition of eigenstates of $A$, ``we can no longer speak of the physical quantity $A$ having a particular value''. After a few more considerations, they state that ``the usual conclusion from this in quantum mechanics is that \emph{when the momentum of a particle is known, its coordinate has no physical reality}''\emph{. }We are left therefore, according to EPR, with two alternatives: \begin{quote} \textbf{EPR's dilemma: }``From this follows that either (1) \emph{the quantum-mechanical description of reality given by the wave function is not complete or }(2) \emph{when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality.}'' \citep{Einstein1935} \end{quote} They justify this by reasoning that ``if both of them had simultaneous reality --- and thus definite values --- these values would enter into the complete description, according to the condition for completeness''. And in the crucial step of the reasoning: ``If then the wave function provided such a complete description of reality it would contain these values; \emph{these would then be predictable \label{quo:EPR-predictable}} {[}our emphasis{]}. This not being the case, we are left with the alternatives stated''. Brassard and Méthot \citep{Brassard2006} (correctly) pointed out that strictly speaking EPR should conclude that (1) \emph{or }(2), instead of \emph{either} (1) or (2), since they could not exclude the possibility that (1) and (2) could be both correct. However, this does not affect EPR's conclusion. It was enough for them to show that (1) and (2) could not both be wrong, and therefore if one can find a reason for (2) to be false, (1) must be true \footnote{Brassard and Méthot's further conclusion that the EPR argument is logically unsound is not based on this mistake, which they acknowledge as irrelevant. Their conclusion is, in the present authors' opinion, based on a misinterpretation of EPR's paper. They read the quote ``In quantum mechanics it is usually assumed that the wave function \emph{does} contain a complete description of the physical reality {[}...{]}. We shall show however, that this assumption, together with the criterion of reality given above, leads to a contradiction'', as stating that $\neg(1)\wedge(2)\rightarrow false$. If that was the correct formalisation of the argument we would agree with their conclusion. However, by ``criterion of reality given above'' EPR clearly mean their \textquotedbl{}sufficient condition for reality\textquotedbl{}, not statement $(2)$. }.
The next section in EPR's paper intends to find a reason for (2) to be false, that is, to find a circumstance in which one can say that there are simultaneous elements of reality associated to two non-commuting operators. They consider a composite system composed of two spatially separated subsystems $S_{A}$ and $S_{B}$ which is prepared, by way of a suitable initial interaction, in an entangled state of the type\begin{equation}
|\Psi\rangle=\sum_{n}c_{n}|\psi_{n}\rangle_{A}\otimes|u_{n}\rangle_{B},\label{eq:entangled1}\end{equation}
where the $|\psi_{n}\rangle_{A}$ denote a basis of eigenstates of an operator, say $\hat{O}_{1}$, of subsystem $S_{A}$ and $|u_{n}\rangle_{B}$ denote some (normalised but not necessarily orthogonal) states of
$S_{B}$. If one measures the quantity $\hat{O}_{1}$ at $S_{A}$, and obtains an outcome corresponding to eigenstate $|\psi_{k}\rangle_{A}$
the global state is reduced to $|\psi_{k}\rangle_{A}\otimes|u_{k}\rangle_{B}$. If, on the other hand, one chooses to measure a non-commuting observable
$\hat{O}_{2}$, with eigenstates $|\phi_{s}\rangle_{A}$, one should instead use the expansion\begin{equation}
|\Psi\rangle=\sum_{s}c'_{s}|\phi_{s}\rangle_{A}\otimes|v_{s}\rangle_{B},\label{eq:entangled2}\end{equation}
where $|v_{s}\rangle_{B}$ represent, in general, another set of states of $S_{B}$. Now if the outcome of this measurement is, say, the one corresponding to $|\phi_{r}\rangle_{A}$, the global state is thereby reduced to $|\phi_{r}\rangle_{A}\otimes|v_{r}\rangle_{B}$. Therefore, ``as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions''. This is just what Schrödinger later termed \emph{steering, }and we will return to that later. Now enters the crucial assumption of \emph{locality}, justified by the fact that the systems are spatially separated and thus no longer interacting. \begin{quote} \textbf{EPR's necessary condition for locality: }``No real change can take place in the second system in consequence of anything that may be done to the first system.'' \citep{Einstein1935} \end{quote} Einstein \emph{et al.} never explicitly used the term `locality', but took this assumption for granted. Because of this we call this a ``necessary condition for locality'', as this is the most conservative reading of EPR's reasoning: if they had explicitly defined some assumption of locality, this would certainly be an implication of it, but there is no reason (and no need) to take it as a definition.
``\emph{Thus}'', conclude EPR, ``\emph{it is possible to assign two different wave functions to the same} \emph{reality}''. EPR could have now simply concluded by noting that two different (pure) states can in general assign unit probability (and thus an element of reality, according to the locality assumption and the sufficient condition for reality) to each of two non-commuting quantities, in contradiction of statement (2); this would imply, by way of EPR's dilemma, that quantum mechanics is incomplete. Instead, they consider a specific example, depicted in Fig.~\ref{fig:EPR}, where those different wave functions are respective eigenstates of position and momentum. Because they are canonically conjugate, this guarantees that $|u_{n}\rangle$
is different from $|v_{s}\rangle$ for \emph{every} possible outcome \emph{n} or \emph{s}. The paradox is thus guaranteed to be realised --- one cannot attempt to hide behind statistics. If the initial state was of type\begin{equation} \Psi(x_{A},x_{B})=\int_{-\infty}^{\infty}e^{ix_{A}p/\hbar}e^{-ix_{B}p/\hbar}dp,\label{eq:EPRstate}\end{equation} then if one measures momentum $\hat{p}^{A}$ at $S_{A}$ and finds outcome $p$, the reduced state of subsystem $S_{B}$ will be the one associated with outcome $-p$ of $\hat{p}^{B}$. On the other hand, if one measures position $\hat{x}^{A}$ and finds outcome $x$, the reduced state of $S_{B}$ will be the one corresponding to outcome $x$ of $\hat{x}^{B}$. By measuring position or momentum at $S_{A}$, one can predict with certainty the outcome of the same measurement on $S_{B}$. But $\hat{p}^{B}$ and $\hat{x}^{B}$ correspond to non-commuting operators. EPR conclude from this that \begin{quote} ``In accordance with our criterion of reality, in the first case we must consider the quantity {[}$\hat{p}^{B}${]} as being an element of reality, in the second case the quantity {[}$\hat{x}^{B}${]} is an element of reality. But, as we have seen, both wave functions {[}corresponding to $-p$ and $x${]} belong to the same reality.'' \citep{Einstein1935} \end{quote}
\begin{figure}
\caption{ The EPR scenario. Alice and Bob are two spatially separated observers who can perform one of two (position or momentum) measurements available to each of them.}
\label{fig:EPR}
\end{figure} In other words, by using the sufficient condition for reality, the necessary condition for locality and the predictions for the entangled state under consideration, EPR conclude that there must be elements of reality associated to a pair of non-commuting operators. So horn (2) of EPR's dilemma is closed, leaving as the only alternative option (1), namely, that the quantum mechanical description of physical reality is incomplete.
In more modern terminology, the conclusion of EPR was to infer the existence of a set of local hidden variables (LHVs) underlying quantum systems which should be able to reproduce the statistics. It is trivial to reproduce the statistics of EPR's example with LHVs, even though that is not possible with some entangled states, as later proved by Bell \citep{Bell1964}. Schrödinger arrived at a different conclusion from an analysis of the paradox raised by EPR, as we will see in the next section.
In hindsight, as we now know that the premise of locality is not justified, we can read EPR's argument as demonstrating the\emph{ incompatibility} between the premises of locality, the completeness of quantum mechanics and some of its predictions.
\subsection{Schrödinger's response: The concept of steering\label{sub:Schr=0000F6dinger's-response}}
EPR's argument prompted an interesting response from Schrödinger \citep{SchPCP35,Schroedinger1935}. He also considered nonfactorizable pure states describable by the wave function given by Eq. (\ref{eq:entangled1}). Schrödinger, however, had of course developed the wave function for atoms and believed that it gave a complete description of a quantum system. So while he was not prepared to accept EPR's conclusion that quantum mechanics was incomplete, neither could he see a flaw with their argument. For this reason he termed the situation described by EPR a \emph{paradox}.
Clearly Schrödinger was also interested in implications arising from composite quantum systems described by nonfactorizable pure states. He described this situation, coining a famous term, as follows: {}``If two separated bodies, each by itself known maximally, enter a situation in which they influence each other, and separate again, then there occurs regularly ... {[}an{]} entanglement of our knowledge of the two bodies.'' \citep{SchPCP35}
Having defined entanglement, Schrödinger then defined the process of \emph{disentanglement }which occurs when a non-degenerate observable is measured on one body: {}``\emph{After establishing one representative by observation, the other one can be inferred simultaneously ... this procedure will be called} \emph{the} disentanglement''. This leads us directly to the EPR paradox, as Schrödinger describes it: \begin{quote} {}``{[}EPR called attention{]} to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to \emph{one} system, the representative obtained for the \emph{other} system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are \emph{entirely} arbitrary.''~\citep{SchPCP35} \end{quote} Schrödinger describes this ability to affect the state of the remote subsystem as \emph{steering}: \begin{quote} {}``It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter's mercy in spite of his having no access to it.'' \citep{SchPCP35} \end{quote} EPR's example concerning position and momentum was recast in the context of steering as \begin{quote} {}``Since I can predict \emph{either} $x_{1}$ or $p_{1}$ without interfering with system No.~1 and since system No.~1, like a scholar in examination, cannot possibly know which of the two questions I am going to ask it first: it so seems that our scholar is prepared to give the right answer to the \emph{first} question he is asked \emph{anyhow}. He must know both answers; which is an amazing knowledge.'' \citep{SchPCP35} \end{quote} The remainder of Schrödinger's paper is a generalisation of steering to more than two measurements: \begin{quote} ``{[}System No.~1{]} does not only know these two answers but a vast number of others, and that with no mnemotechnical help whatsoever, at least none that we know of.'' \citep{SchPCP35} \end{quote} By {}``mnemotechnical help'' Schrödinger presumably means a cheat-sheet (to use his scholar analogy). That is, a set of local hidden variables (LHVs) that determine the measurement results. Thus, unlike EPR, Schrödinger explicitly rejected LHVs as an explanation of steering. Perhaps because he had performed explicit calculations generalizing EPR's example (which can be explained trivially using LHVs), he recognized steering as {}``a necessary and indispensable feature'' \citep{SchPCP36} of quantum mechanics. We now know, thanks to Bell's theorem, that Schrödinger's intuition was correct: there is no possible local hidden variable model (or local mnemotechnical help) to explain the correlations between measurement outcomes for certain entangled states \citep{Gisin1991}.
Like EPR, Schrödinger was troubled by the implications of steerability of entangled states for quantum theory. Unlike EPR, however, he saw the resolution of the paradox lying in the \emph{incorrectness} of the predictions of quantum mechanics. That is, he was {}``\emph{not satisfied about there being} \emph{sufficient experimental evidence for}'' steering in nature \citep{SchPCP36}. This raises the obvious question: what evidence would have convinced Schrödinger? The pure entangled states he discussed are an idealization, so we cannot expect ever to observe precisely the phenomenon he introduced. On the other hand, Schrödinger was quite explicit that a separable but classically correlated state which allows ``\emph{determining the state of the first system by }suitable\emph{ measurement of the second or }vice versa'' \citep{SchPCP36} could never exhibit steering. For this situation, he says that ``\emph{it would utterly eliminate the experimenter's influence on the state of that system which he does not touch}.'' \citep{SchPCP36}. Thus it is apparent that by steering Schrödinger meant something that could not be explained by Alice simply finding out which state Bob's system is in, out of some predefined ensemble of states. Following this reasoning leads to the general definition of steering as presented in Ref.~\citep{Wiseman2007}. We return to this concept in Sec.~\ref{sec:Locality-models}.
\subsection{Bohm's version \label{sub:Bohm's-version}}
Although making reference to a general entangled state, the original EPR argument used the specific case of a continuous-variable state for its final (and crucial) part. In his 1951 textbook \citep{Bohm1951}, Bohm presented a discussion of the EPR paradox in a modified scenario involving two entangled spin-1/2 particles. Although trivial in hindsight, this extension had a fundamental importance. It was the scenario used by Bell in the proof of his now famous theorem \citep{Bell1964} and for most of the subsequent discussions of Bell inequalities (a Bell-type inequality directly applicable to continuous-variables has only recently been derived \citep{Cavalcanti2007a}), and was instrumental for our present understanding of entanglement, and particularly for its applications in quantum information processing.
In Bohm's version the system of interest is a molecule containing two spin-1/2 atoms in a singlet state, in which the total spin is zero:\begin{equation}
|\Psi_{s}\rangle=|z_{+}\rangle_{A}\otimes|z_{-}\rangle_{B}-|z_{-}\rangle_{A}\otimes|z_{+}\rangle_{B}.\label{eq:Bohm_singlet_z}\end{equation}
Here $|z_{\pm}\rangle$ represent the $\pm1/2$ eigenstate of the spin projection operator along the z direction, $S_{z}$. Compare this state with Eq.~\eqref{eq:entangled1} used in the EPR argument. If $S_{z}$ is measured on system \emph{A}, and the outcome corresponding to $|z_{+}\rangle_{A}$ (or $|z_{-}\rangle_{A}$) is obtained, the state of subsystem \emph{B} is projected into $|z_{-}\rangle_{B}$
(or $|z_{+}\rangle_{B}$). Thus, one predicts an element of reality for the $z$ component of the spin of the second atom. But the same state can be written, in the basis of eigenstates of another spin projection, say $S_{x},$\begin{equation}
|\Psi_{s}\rangle=|x_{+}\rangle_{A}\otimes|x_{-}\rangle_{B}-|x_{-}\rangle_{A}\otimes|x_{+}\rangle_{B}.\label{eq:Bohm_singlet_x}\end{equation} Similarly, the $x$ component of the spin of the first atom could be measured instead, allowing inference of an element of reality associated with the $x$ component of spin for the second atom. With this mapping, the rest of the argument follows in analogy with EPR's.
Bohm's version of the EPR paradox is conceptually appealing, but (in his 1951 textbook at least) he did not present it as an argument for the incompleteness of quantum theory (as did EPR). Instead, he used it to argue that a complete description of nature need not contain a one-to-one correspondence between elements of reality and the mathematical description provided by the theory. Bohm defended, in 1951, the interpretation that the quantum state represents only ``potentialities'' of measurement results, which actually occur only when a system interacts with an appropriate apparatus. It is curious to find that already in 1952 Bohm must have found this interpretation wanting, since he then developed his famous non-local hidden-variable interpretation of quantum mechanics \citep{Bohm1952a,Bohm1952b}, where there \emph{is} such a one-to-one correspondence.
As the original continuous-variable example remained unrealizable for decades, several early experiments followed Bohm's proposal, such as Bleuler and Bradt (1948) \citep{Bleuler1948},\textbf{ }Wu and Shaknov (1950) \citep{Wu1950} and Kocher and Commins (1967) \citep{Kocher1967}. All of these suffered from low detection efficiencies and had no concern with causal separation, however, making their interpretation debatable.
\subsection{The EPR-Reid criterion\label{sec:The-EPR-Reid-criterion}}
While the EPR argument was logically sound, one could block its conclusion by rejecting those statistical predictions required to formulate it. As we have discussed in Sec.~\ref{sub:Schr=0000F6dinger's-response}, Schrödinger seems to have found this an appealing solution. This move is particularly easy to make since the necessary predictions are of perfect correlations, unobtainable in practice due to unavoidable inefficiency in preparation and detection of real physical systems. This problem was considered by Furry already in 1936 \citep{Furry1936} but experimentally useful criteria for the EPR paradox were only proposed in 1989 by Reid \citep{Reid1989}, which we will discuss in detail later in this section. The notation and terminology will closely follow that of a recent review on the EPR paradox \citep{Reid2008tb}. The essential difference in the derivation of the EPR-Reid criteria and the original EPR argument is in a modification of the sufficient condition for reality \footnote{Reid's original paper did not explicitly include this assumption, which was implicit in the logic. }. This could be stated as the following: \begin{quote} \textbf{Reid's extension of EPR's sufficient condition of reality:} If, without in any way disturbing a system, we can predict with \emph{some specified uncertainty} the value of a physical quantity, then there exists a \emph{stochastic} element of physical reality which determines this physical quantity with at most that specific uncertainty. \end{quote} The scenario considered is the same as the one for the EPR paradox above, as depicted in Fig.~\ref{fig:EPR}, but one does not need a state which predicts the perfect correlations considered by EPR. Instead, the two experimenters, Alice and Bob, can measure the conditional probabilities of Bob finding outcome $x_{B}$ in a measurement of
$\hat{x}_{B}$ given that Alice finds outcome $x_{A}$ in a measurement of $\hat{x}_{A}$, i.e., $P(x_{B}|x_{A})$. Similarly they can measure the conditional probabilities $P(p_{B}|p_{A})$ and the unconditional probabilities $P(x_{A})$, $P(p_{A})$. We denote by $\Delta^{2}(x_{B}|x_{A})$,
$\Delta^{2}(p_{B}|p_{A})$ the variances of the conditional distributions
$P(x_{B}|x_{A})$, $P(p_{B}|p_{A})$, respectively. Based on a result $x_{A},$ Alice can make an estimate of the result for Bob's outcome $x_{B}.$ Denote this estimate $x_{B}^{{\rm est}}(x_{A}).$ The\emph{ average inference variance} of $x_{B}$ given estimate $x_{B}^{{\rm est}}(x_{A})$ is defined as
\begin{multline} \Delta_{\mathrm{inf}}^{2}x_{B}\equiv\langle(x_{B}-x_{B}^{{\rm est}}(x_{A}))^{2}\rangle\\ =\int dx_{A}dx_{B}P(x_{A},x_{B})(x-x_{B}^{{\rm est}}(x_{A}))^{2}.\label{eq:Inf_var_def}\end{multline}
Note that this average inference variance is minimized when the estimate is just the expectation value of $x_{B}$ given $x_{A},$ i.e., the mean of the distribution $P(x_{B}|x_{A})$ \citep{Reid2008tb}.\emph{ }Therefore the \emph{optimal (or minimum) inference variance} of $x_{B}$ ($p_{B}$) given a measurement $\hat{x}_{A}$ ($\hat{p}_{A}$) is given by \emph{\begin{eqnarray} \Delta_{\mathrm{min}}^{2}x_{B} & = & \mathrm{min}_{x_{B}^{{\rm est}}}\{\Delta_{\mathrm{inf}}^{2}x_{B}\}\nonumber \\
& = & \int dx_{A}dx_{B}P(x_{A})\Delta^{2}(x_{B}|x_{A});\label{eq:Reid_min_inf_var_x}\\ \Delta_{\mathrm{min}}^{2}p_{B} & = & \mathrm{min}_{p_{B}^{{\rm est}}}\{\Delta_{\mathrm{inf}}^{2}p_{B}\}\nonumber \\
& = & \int dp_{A}dp_{B}P(p_{A})\Delta^{2}(p_{B}|p_{A}).\label{eq:Reid_min_inf_var_p}\end{eqnarray} }Reid showed, by use of the sufficient condition of reality above, that since Alice can, by measuring either position $\hat{x}_{A}$ or momentum $\hat{p}_{B}$, infer with some uncertainty $\Delta_{\mathrm{inf}}x_{B}=\sqrt{\Delta_{\mathrm{inf}}^{2}x_{B}}$ or $\Delta_{\mathrm{inf}}p_{B}=\sqrt{\Delta_{\mathrm{inf}}^{2}p_{B}}$ the outcomes of the corresponding experiments performed by Bob, and since by the locality condition of EPR her choice cannot affect the elements of reality of Bob, then there must be simultaneous stochastic elements of reality which determine $\hat{x}_{B}$ and $\hat{p}_{B}$ with at most those uncertainties. Now by Heisenberg's Uncertainty Principle (HUP), quantum mechanics imposes a limit to the precision with which one can assign values to observables corresponding to non-commuting operators such as $\hat{x}$ and $\hat{p}$. In appropriately rescaled units the relevant HUP reads $\Delta x\Delta p\geq1$. Therefore, if quantum mechanics is complete and the locality condition holds, by use of the extended sufficient condition of reality and EPR's necessary condition for completeness, the limit with which one could determine the average inference variances above is\begin{equation} \Delta_{\mathrm{inf}}x_{B}\Delta_{\mathrm{inf}}p_{B}\geq1.\label{eq:EPR-Reid}\end{equation} This is the \emph{EPR-Reid criterion}. Violation of that criterion signifies the EPR paradox, and has been experimentally demonstrated in continuous-variables quantum optics experiments with quadratures \citep{Ou1992,Zhang2000,Silberhorn2001,Schori2002,Bowen2003} and actual position-momentum measurements \citep{Howell2004}. While these were performed with high detection efficiency, none of these experimental demonstrations have been able to achieve causal separation between the measurements. For a detailed review see \citep{Reid2008tb}.
\subsection{Recent developments\label{sub:Recent-developments}}
Cavalcanti and Reid \citep{Cavalcanti2007b}\textbf{ }recently showed that a larger class of quantum uncertainty relations can be used to derive EPR inequalities. For example, from the uncertainty relation $\Delta^{2}x+\Delta^{2}p\geq2,$ which follows from $\Delta x\Delta p\geq1,$ one can derive, in analogy with the previous section, the EPR criterion\begin{equation} \Delta_{\mathrm{inf}}^{2}x_{B}+\Delta_{\mathrm{inf}}^{2}p_{B}\geq2.\label{eq:EPR_sum_Cav_Reid}\end{equation}
Using instead the spin uncertainty relation $\Delta J_{x}\Delta J_{y}\geq\frac{1}{2}|\langle J_{z}\rangle|,$ one can obtain the EPR criterion\begin{equation}
\Delta_{\mathrm{inf}}J_{x}^{B}\Delta_{\mathrm{inf}}J_{y}^{B}\geq\frac{1}{2}\sum_{J_{z}^{A}}P(J_{z}^{A})|\langle J_{z}^{B}\rangle_{J_{z}^{A}}|,\label{eq:EPR-Bohm_Cav_Reid}\end{equation} useful for demonstration of Bohm's version of the EPR paradox. Here
$\langle J_{z}^{B}\rangle_{J_{z}^{A}}$ is the mean of the conditional probability distribution $P(J_{z}^{B}|J_{z}^{A}).$ A weaker version of Eq.~\eqref{eq:EPR-Bohm_Cav_Reid},\begin{equation}
\Delta_{\mathrm{inf}}J_{x}^{B}\Delta_{\mathrm{inf}}J_{y}^{B}\geq\frac{1}{2}|\langle J_{z}^{B}\rangle|,\label{eq:Bowen_EPR_criterion}\end{equation} was used by Bowen \emph{et al. }\citep{Bowen2003} to demonstrate an EPR paradox in the continuum limit for optical systems, with Stokes operators playing the role of spin operators, in states where $\langle J_{z}^{B}\rangle\neq0.$
An inequality for demonstration of an EPR-Bohm paradox has also been derived using an uncertainty relation based on sums of observables. The uncertainty relation $\Delta^{2}J_{x}+\Delta^{2}J_{y}+\Delta^{2}J_{z}\geq\langle j\rangle,$ where $\langle j\rangle$ is the average total spin, has been used in \citep{Hofmann2003a} for derivation of separability criteria, and recently by \citep{Cavalcanti2009b} to derive the following EPR criterion \footnote{More precisely, inequality \eqref{eq:sum_inf_var_criterion_J} was presented in that work. The following follows with the substitution explained below \eqref{eq:sum_inf_var_criterion_J}. }
\begin{equation} \Delta_{\mathrm{inf}}^{2}J_{x}^{B}+\Delta_{\mathrm{inf}}^{2}J_{y}^{B}+\Delta_{\mathrm{inf}}^{2}J_{z}^{B}\geq\langle j^{B}\rangle.\label{eq:EPR-Bohm_sum_criterion}\end{equation}
All of the above EPR criteria will be rederived from an unifying perspective in Section \ref{sec:Experimental-criteria-for}, and shown to be special cases of broader classes of EPR-steering criteria.
\section{Locality models; EPR-steering\label{sec:Locality-models}}
In \citep{Wiseman2007}, a distinction was made between three locality models, the failure of each corresponding to three strictly distinct forms of nonlocality. To define those we will first establish some notation.
Let $a\in\mathfrak{M_{\alpha}}$ and $b\in\mathfrak{M}_{\beta}$ represent possible choices of measurements for two spatially separated observers Alice and Bob, with respective outcomes denoted by the upper-case variables $A\in\mathfrak{O}_{a}$ and $B\in\mathfrak{O}_{b}$, respectively. Here we follow the case convention introduced by Bell \citep{Bell1964}. Alice and Bob perform measurements on pairs of systems prepared by a reproducible preparation procedure $c$. We denote the set of ordered pairs $\mathfrak{M}\equiv\{(a,b):a\in\mathfrak{M}_{\alpha},b\in\mathfrak{M}_{\beta}\}$ a\emph{ measurement strategy}. The joint probability of obtaining outcomes $A$ and $B$ upon measuring $a$ and $b$ after preparation $c$ is denoted by\begin{equation}
P(A,B|a,b,c).\label{eq:jointProb}\end{equation}
The preparation procedure $c$ represents all those variables which are explicitly known in the experimental situation. The joint probabilities for all outcomes of all pairs of observables in a measurement strategy given a preparation procedure define a \emph{phenomenon. }Following Bell \citep{Bell1987}, we represent by $\lambda\in\Lambda$ any variables associated with events in the union of the past light cones of $a,\, A,\, b,\, B$ which are\emph{ }relevant\emph{ }to the experimental situation but are not explicitly known, and therefore not included in $c$. In this sense they may be deemed hidden variables, but our usage will not imply that they are necessarily hidden in principle (although in particular theories they may be).
\subsection{Bell-nonlocality}
Given that notation, it is said that a phenomenon has a \emph{local hidden variable} (\emph{LHV} or \emph{Bell-local }or \emph{locally causal}) model if and only if for all $a\in\mathfrak{M_{\alpha}},\, A\in\mathfrak{O}_{a},\, b\in\mathfrak{M_{\beta}},\, B\in\mathfrak{O}_{b},$
there exist (i) a probability distribution $P(\lambda|c)$ over the hidden variables, conditional on the information about the preparation procedure $c$ \footnote{In general one could have a continuum of hidden variables, and Eq. \eqref{eq:LHV} can be modified in the obvious way. No generality is gained with that procedure, though, so we use the sum notation for simplicity.
} and (ii) arbitrary probability distributions $P(A|a,c,\lambda)$
and $P(B|b,c,\lambda)$, which reproduce the phenomenon in the form: \begin{equation}
P(A,B|a,b,c)=\sum_{\lambda}P(\lambda|c)P(A|a,c,\lambda)P(B|b,c,\lambda).\label{eq:LHV}\end{equation}
Any constraint on the set of possible phenomena that can be derived from \eqref{eq:LHV} is called\emph{ a Bell inequali}ty. A state for which all phenomena can be given a LHV model, when the sets $\mathfrak{M_{\alpha}}$ and $\mathfrak{M_{\beta}}$ include all observables on the Hilbert spaces of each corresponding subsystems, is called\emph{ a Bell-local state}. If a state is not Bell-local it is called \emph{Bell-nonlocal}.
\subsection{Entanglement}
Similarly, it is said that a phenomenon has a \emph{quantum separable} model, or \emph{separable} model for simplicity, if and only if for all $a\in\mathfrak{M_{\alpha}},\, A\in\mathfrak{O}_{a},\, b\in\mathfrak{M_{\beta}},\, B\in\mathfrak{O}_{b},$
there exist $P(\lambda|c)$ as above and probability distributions
$P_{Q}(A|a,c,\lambda)$ and $P_{Q}(B|b,c,\lambda)$ such that
\begin{equation}
P(A,B|a,b,c)=\sum_{\lambda}P(\lambda|c)P_{Q}(A|a,c,\lambda)P_{Q}(B|b,c,\lambda),\label{eq:separable}\end{equation}
where now $P_{Q}(A|a,c,\lambda)$ represent probability distributions for outcomes $A$ which are compatible with a quantum state. That is, given a projector $\Pi_{a}^{A}$ associated to outcome $A$ of measurement $a,$ and given a quantum density operator $\rho_{\alpha}(c,\lambda)$ for Alice's subsystem (as a function of $c$ and $\lambda$), these probabilities are determined by \[
P_{Q}(A|a,c,\lambda)=\mathrm{Tr}\{\Pi_{a}^{A}\rho_{\alpha}(c,\lambda)\}.\] Similar definitions apply for Bob's subsystem.
Any constraint on the set of possible phenomena that can be derived from assumption \eqref{eq:separable} is called a\emph{ separability criterion} or\emph{ entanglement criterion}. A state for which all phenomena can be given a separable model, when the sets $\mathfrak{M_{\alpha}}$ and $\mathfrak{M_{\beta}}$ include all observables on the Hilbert spaces of each corresponding subsystems, is called a\emph{ separable state}. A state which is not separable is called\emph{ non-separabl}e o\emph{r entangled}. This definition is of course equivalent to the usual definition involving product states, since if there is a separable model for all possible measurement settings, then the joint state can be given as a convex combination of product states \begin{equation}
\rho=\sum_{\lambda}P(\lambda|c)\rho_{\alpha}(c,\lambda)\otimes\rho_{\beta}(c,\lambda).\label{eq:separable_state}\end{equation} Conversely, if the state is given as a convex combination of product states of form \eqref{eq:separable_state}, the joint probabilities for each pair of measurements are given straightforwardly by Eq.~\eqref{eq:separable}.
\subsection{EPR-steering}
Strictly intermediate between the LHV and separable models is the \emph{local hidden-state (LHS) model for Bob.} This was argued in \citep{Wiseman2007} to be the correct formalisation of non-steering correlations. That is, violation of a LHS model for Bob is a demonstration of EPR-steering, the concept introduced by Schrödinger to refer to the situation depicted in the EPR paradox. Following the previous notations, we say that a phenomenon has a \emph{no-Bob-steering model }or a \emph{LHS model for Bob }(or\emph{ LHS model }for short) \emph{} \footnote{It would perhaps be more logical to use the term \emph{LHV/LHS model} to denote no-steering, and the other types of nonlocality by LHV and LHS models respectively, but we will use the simpler terminology introduced in Ref.\citep{Wiseman2007}, as we believe there is no risk of confusion. } if and only if\emph{ }for all $a\in\mathfrak{M_{\alpha}},\, A\in\mathfrak{O}_{a},\, b\mathfrak{\in M_{\beta}},\, B\in\mathfrak{O}_{b},$
there exist $P(\lambda|c),$ $P(A|a,c,\lambda)$ and $P_{Q}(B|b,c,\lambda)$ defined as before such that
\begin{equation}
P(A,B|a,b,c)=\sum_{\lambda}P(\lambda|c)P(A|a,c,\lambda)P_{Q}(B|b,c,\lambda).\label{eq:LHS_model}\end{equation}
In other words, in a LHS model Bob's outcomes are described by some quantum state, but Alice's outcomes are free to be arbitrarily determined by the variables $\lambda.$ We call any constraint on the set of possible phenomena that can be derived from \eqref{eq:LHS_model} \emph{an EPR-steering criterion} \emph{or EPR-steering inequality}. A state for which all phenomena can be given a LHS model, when the sets $\mathfrak{M_{\alpha}}$ and $\mathfrak{M_{\beta}}$ include all observables on the Hilbert spaces of each corresponding subsystems, is called an\emph{ EPR-steerable state}. A state which is not steerable is called\emph{ non-EPR-steerable}.
\subsection{Foundational relevance of EPR-steering}
As we have seen in Section \eqref{sub:Schr=0000F6dinger's-response}, Schrödinger was {}``discomforted'' with the possibility of Alice being able to {}``steer'' Bob's system {}``in spite of {[}her{]} having no access to it''. In other words, the strange phenomenon revealed by the EPR paradox which he termed {}``steering'' was the possibility that Alice could prepare, simply by different choices of measurement on her own system, different ensembles of states for Bob which are incompatible with a LHS model, that is, which cannot be explained as arising from a coarse-graining from a pre-existing ensemble of local quantum states for Bob. This is an inherently asymmetric concept, thus the asymmetry in the formalization given by Eq.~\eqref{eq:LHS_model}.
For each choice of measurement $a,$ Alice will prepare for Bob one state out of an ensemble $E^{a}\equiv\{\tilde{\rho}_{a}^{A}:A\in\mathfrak{O}_{a}\}.$ If the state of the global system is $\mathrm{W}_{c},$ the (unnormalized) reduced state for Bob's subsystem corresponding to outcome $A$ will be \begin{equation} \tilde{\rho}_{a}^{A}\equiv\mathrm{Tr}_{\alpha}[\mathrm{W}_{c}(\Pi_{a}^{A}\otimes\mathbf{I})].\label{eq:Bob_red_state}\end{equation} Evidently, the reduced density matrix for Bob is independent of Alice's choice: $\rho_{\beta}=\mathrm{Tr}_{\alpha}[\mathrm{W}_{c}]=\sum_{A}\tilde{\rho}_{a}^{A}$ for all $a$ --- otherwise Alice could send faster-than-light signals to Bob.
In Ref. \citep{Wiseman2007} it was shown that for pure states $\mathrm{W}_{c},$ entangled states, steerable states and Bell-nonlocal states are all equivalent classes. The difficulty (and interest) comes when talking about mixed states. In this case, one certainly does not want to consider it as an example of steering when the ensembles prepared by Alice are just different coarse-grainings of some underlying ensemble of states. After all, these ensembles can be reproduced if Bob's local state is simply classically correlated with some variables available to Alice. These correlations would hardly constitute a puzzle for Schrödinger, as we have argued in Section \eqref{sub:Schr=0000F6dinger's-response}.
Thus, Wiseman and co-workers \citep{Wiseman2007} considered EPR-steering to occur iff it is not the case that there exists a decomposition of Bob's reduced state, $\rho_{\beta}=\sum_{\lambda}P(\lambda|c)\rho_{\beta}(c,\lambda)$ such that for all $a\in\mathfrak{M_{\alpha}},\, A\in\mathfrak{O}_{a}$
there exists a stochastic map $P(A|a,c,\lambda)$ which allows all states in the ensembles $E^{a}$ to be reproduced as \begin{equation}
\tilde{\rho}_{a}^{A}=\sum_{\lambda}P(A|a,c,\lambda)P(\lambda|c)\rho_{\beta}(c,\lambda).\label{eq:coarse-graining}\end{equation}
This definition leads directly to the formulation of a no-steering model, Eq.~\eqref{eq:LHS_model}. According to the reduced state \eqref{eq:coarse-graining}, the probability for outcome $B$ of Bob's measurement $b,$ given an outcome $A$ of Alice's measurement $a,$
is given by $P(B|A,a,b,c)=\mathrm{Tr}[\Pi_{b}^{B}\tilde{\rho}_{a}^{A}]/P(A|a,b,c),$ where the denominator is introduced for normalization. Therefore the joint probability becomes \begin{align}
P(A,B|a,b,c) & =\mathrm{Tr}[\Pi_{b}^{B}\tilde{\rho}_{a}^{A}]\nonumber \\
& =\sum_{\lambda}P(A|a,c,\lambda)P(\lambda|c)\mathrm{Tr}[\Pi_{b}^{B}\rho_{\beta}(c,\lambda)]\nonumber \\
& =\sum_{\lambda}P(\lambda|c)P(A|a,c,\lambda)P_{Q}(B|b,c,\lambda),\label{eq:LHS_model_again}\end{align} as in Eq.~\eqref{eq:LHS_model}. The converse can also be trivially shown.
One could propose that the definition of EPR-steering should take into account the fact that Alice's state is also describable by quantum mechanics. It can indeed be argued \citep{Cavalcanti2010tb} that the conjunction of the assumptions of local causality and the completeness of quantum mechanics (for both Alice and Bob) leads directly to a quantum separable model, and in that sense EPR's conclusion that quantum mechanics is incomplete (assuming local causality) could have been reached by simply pointing out the predictions from any entangled state. However, we are interested in capturing the phenomenon which is central to EPR's \emph{actual} argument, and in Schrödinger's generalization of this phenomenon, and hence we are led to the asymmetry in the definition. This is the phenomenon that Einstein famously described as \textquotedbl{}spooky action at a distance\textquotedbl{} \citep{Einstein1947}.
As we will see, this formalization also leads precisely to existing EPR criteria, putting in a modern context the phenomena that have already been discussed in the literature as generalizations of the EPR paradox. Following Einstein's informal turn of phrase, we could even call them tests of spooky action at a distance.
\subsection{EPR-steering as a quantum information task}
Wiseman and co-workers \citep{Wiseman2007,Jones2007} showed that the distinction between the three forms of nonlocality above can be formulated in a modern quantum information perspective, as a\emph{ task}. Suppose a third party, Charlie, wants proof that Alice and Bob share an entangled state. Alice and Bob are not allowed to communicate, but they can share any amount of classical randomness. If Charlie trusts both Alice and Bob, he would be convinced iff Alice and Bob are able to demonstrate entanglement, via violation of a separable model, Eq.~\eqref{eq:separable}. If Charlie trusts Bob but not Alice, he would be convinced they share entanglement iff they are able to demonstrate EPR-steering by violating the local hidden state model for Bob, Eq.~\eqref{eq:LHS_model}. If, on the other hand, Charlie trusts neither of them, Alice and Bob would have to demonstrate Bell-nonlocality, violating a local hidden variable model, Eq.~\eqref{eq:LHV}. The reason is that, in the absence of trust, it is possible for the weaker forms of nonlocality to be reproduced with the use of classical resources.
\section{Experimental criteria for EPR-steering\label{sec:Experimental-criteria-for}}
The above definition of EPR-steering invites the question: what are the analogues for EPR-steering of Bell inequalities or entanglement criteria, i.e., how can one derive what we have termed \emph{EPR-steering criteria} above? In Refs. \citep{Wiseman2007} and \citep{Jones2007} the emphasis was on the EPR-steering capabilities as a property of states, and an analysis was made of how the steerability of some families of quantum states depends on parameters which specify the states within those families. This was necessary and useful for proving the strict distinction between entangled, EPR-steerable and Bell-nonlocal states. In an experimental situation, however, this kind of analysis is insufficient. Quantum state tomography could be used to determine those parameters, but what if the prepared state is only approximately a member of the studied family? What about states which are not even approximately members of any useful class? An experimental EPR-steering criterion should not depend on any assumption about the type of state being prepared, but only on the measured data. Compare this situation with that of Bell inequalities, where a violation represents failure of a LHV model, independently of any assumption about the state being measured.
Another important issue is the relation between the EPR-type criteria existing in the literature and the above formalization of EPR-steering. In \citep{Wiseman2007} the authors provided a partial answer by showing that for a class of Gaussian states the EPR-Reid criterion is violated if and only if the state is steerable by Gaussian measurements. However, the EPR-Reid criterion is valid for arbitrary states, and therefore their conclusion that it is merely a special case of EPR-steering was not entirely justified. Furthermore, the relation between this formalization of EPR-steering and the other existing EPR-type criteria cited in Sec.~\ref{sub:Recent-developments} was not discussed. Here we will show that not only the EPR-Reid criterion but other existing EPR-type criteria are indeed special cases of EPR-steering. We will rederive those inequalities within this modern approach, and also derive a number of new criteria for EPR-steering.
There is an important difference between Bell inequalities and EPR-steering criteria. Since the LHV model \eqref{eq:LHV} does not depend on the Hilbert space structure of quantum mechanics, Bell inequalities are independent of the actual measurements being performed. To be clear, the \emph{violation} of the inequality will certainly depend on which measurements are performed (as well as the state being prepared), but the derivation of the inequality itself is independent of that information. In a Bell inequality the measurements are treated as {}``black boxes'', where the only important feature is (usually, but see \citep{Cavalcanti2007a}) their number of outcomes. In a LHS model, on the other hand, Bob's subsytem is treated as a quantum state, and therefore it is important in general to specify the actual quantum operators corresponding to Bob's measurement choices, just as in an entanglement criterion this information is in general required for both Alice and Bob \footnote{The qualification 'in general' here is needed because a Bell inequality \emph{is} an EPR-steering and an entanglement criterion. The failure of a LHV model implies the failure of a LHS model and of a separable model. However, in general a Bell inequality is inefficient as a criterion for these weaker forms of nonlocality. }.
The fact that in a no-steering model Bob's probabilities are constrained to be compatible with a quantum state suggests the use of quantum uncertainty relations as ingredients in the derivation of criteria for EPR-steering. A connection between uncertainty relations and EPR criteria has been pointed out by two of the present authors in \citep{Cavalcanti2007b} (although using the logic of the EPR-Reid criteria, not the present formalization of EPR-steering), and that between uncertainty relations and separability criteria has been shown by \citep{Hofmann2003a}, among others.
We identify two main types of EPR-steering criteria: the \emph{multiplicative variance} criteria, which include the EPR-Reid criteria and are based on product uncertainty relations involving variances of observables; and the \emph{additive convex} criteria, based on uncertainty relations which are sums of convex functions.
\subsection{Existence of linear EPR-Steering criteria}
An interesting special case of additive convex criteria will be the \emph{linear }criteria, based on linear functions of expectation values of observables, and which can therefore be written as the expectation value of a single Hermitian \emph{EPR-steering operator} $S$.
In general, for any (finite-dimensional) quantum state $W$, if the state in question is steerable, then there exists a linear criterion that would demonstrate EPR-steering for that phenomenon.
The proof is as follows. If the state is steerable, then by definition there exists a measurement strategy which can demonstrate steering with that state. Let $\mathfrak{M}$ be that measurement strategy. Consider the set $\mathfrak{P(M)}$ of all possible phenomena for $\mathfrak{M}$, i.e., the set of all possible sets of joint probabilities
$P(A,B|a,b)$ for all pair of outcomes $(A,B)$ of each pair of measurements $(a,b)\in\mathfrak{M}$. Let $M$ be the number of possible settings for the pair of measurements performed by Alice and Bob (i.e., the number of elements in $\mathfrak{M}$) and let $O$ be the number of possible pairs of outcomes $(A,B)$ for each pair of measurements.
A phenomenon is defined by specifying the $MO$ probabilities for all possible outcomes of all measurements in the measurement strategy. We represent those probabilities as an ordered set, and thus an element $\mathbf{P}$ of $\mathfrak{P(M)}$ is associated to a point in $\mathbb{R}^{MO},$ where the joint probability for each $(A,B,a,b)$ is associated to a coordinate $x_{ab}^{AB}$ of $\mathbb{R}^{MO}.$ For example, in a phenomenon with 2 measurements per site with 2 outcomes each, $M=O=4,$
and the number of probabilities to be specified is $MO=16.$ Denoting those measurements by $a\in\{a_{1},a_{2}\}$ and the outcomes of each measurement by $A\in\{0,1\}$ (and similarly for Bob), these probabilities would be represented by the vector $\mathbf{P}=(P(0,0|a_{1},b_{1}),P(0,1|a_{1},b_{1}),...,P(1,1|a_{2},b_{2})).$
Now consider two phenomena associated to $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$, and take a convex combination of the two vectors, i.e., \begin{equation} \mathbf{P}_{3}=p\mathbf{P}_{1}+(1-p)\mathbf{P}_{2},\label{eq:convex_comb}\end{equation} where $0\leq p\leq1$. If $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ have a no-steering model, then $\mathbf{P}_{3}$ also does. The proof is simple: by assumption we can write the joint probabilities given by $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ in form \eqref{eq:LHS_model}. Simple manipulation shows that Eq.~\eqref{eq:convex_comb} can also be written in form \eqref{eq:LHS_model}, with $P_{3}(\lambda)=pP_{1}(\lambda)+(1-p)P_{2}(\lambda)$. In other words, the set of phenomena $\mathfrak{NS}\mathfrak{(M)}\subset\mathfrak{P(M)}$ which do not demonstrate EPR-steering is a convex set. (The same is also true, of course, for the other forms of nonlocality.)
Now consider a phenomenon $\mathbf{P}_{s}\in\mathfrak{P(M)}$ which \emph{does} demonstrate EPR-steering. By definition it is not in $\mathfrak{NS}\mathfrak{(M)}$. Since, as shown above, that is a convex set, we can invoke a well known result from convex analysis: there exists a plane in $\mathbb{R}^{MO}$ separating $\mathbf{P}_{s}$ from points in $\mathfrak{NS}\mathfrak{(M)}.$ Denote by $\hat{n}$ an unit vector normal to this plane pointing away from $\mathfrak{NS}\mathfrak{(M)}$ and by $\mathbf{P}_{0}$ an arbitrary point on the plane. Then all points $\mathbf{P}_{\bar{s}}\in\mathfrak{NS}\mathfrak{(M)}$ satisfy \begin{equation} \hat{n}\cdot(\mathbf{P}_{\bar{s}}-\mathbf{P}_{0})\leq0.\label{eq:steering_ineq_plane}\end{equation} Inequality \eqref{eq:steering_ineq_plane} is an EPR-steering criterion. If for an arbitrary point $\mathbf{P}_{c}\in\mathfrak{P(M)},$ $\hat{\mathbf{n}}\cdot(\mathbf{P}_{c}-\mathbf{P}_{0})>0,$ then $\mathbf{P}_{c}\notin\mathfrak{NS}$ and so this phenomenon demonstrates EPR-steering. We can decompose $\mathbf{P}_{c}=\sum_{A,B,a,b}\langle\Pi_{a}^{A}\Pi_{b}^{B}\rangle_{c}\hat{\mathbf{e}}_{ab}^{AB},$
where $\langle\Pi_{a}^{A}\Pi_{b}^{B}\rangle_{c}\equiv P(A,B|a,b,c)=\mathrm{Tr}[W_{c}\,(\Pi_{a}^{A}\otimes\Pi_{b}^{B})]$ and $\{\hat{\mathbf{e}}_{ab}^{AB}\}$ is an orthonormal basis of $\mathbb{R}^{MO}$. Decomposing $\hat{\mathbf{n}}=\sum_{A,B,a,b}n_{ab}^{AB}\hat{\mathbf{e}}_{ab}^{AB}$ and denoting $d\equiv-\hat{\mathbf{n}}\cdot\mathbf{P}_{0},$ \eqref{eq:steering_ineq_plane} becomes $\sum_{A,B,a,b}n_{ab}^{AB}\langle\Pi_{a}^{A}\Pi_{b}^{B}\rangle_{c}+d\leq0.$ Defining a Hermitian operator $S\equiv\sum_{A,B,a,b}n_{ab}^{AB}\Pi_{a}^{A}\Pi_{b}^{B}+d\mathrm{I}$ we can rewrite the EPR-steering criterion \eqref{eq:steering_ineq_plane} as \begin{equation} \mathrm{Tr}[W_{c}S]\leq0,\label{eq:steer_operator_ineq}\end{equation} which completes the proof.
However, this is merely an existence proof. It is quite a different matter to produce the EPR-steering operator $S$ which will demonstrate EPR-steering for a given state $W_{c}$. This is analogous to the situation with Bell inequalities and entanglement, where one can prove the existence of a Bell operator or entanglement witness for states which can demonstrate the corresponding form of nonlocality, but cannot easily produce such operators beyond some simple cases.
Furthermore, in the case of EPR-steering (and also of entanglement) the matter is even more complicated: there is an infinite (and continuous) number of extreme points in the convex set of phenomena which allow a LHS model (or a separable model) --- the set is not a polytope. Therefore even for a finite measurement strategy, an infinite number of linear inequalities are needed to fully specify the set. So in general nonlinear criteria may be more useful, and we will consider that general case in this paper.
In the following subsections we will first derive the class of multiplicative variance criteria, which will reduce to the well-known EPR-Reid criterion as a special case. Then we will introduce the quite general class of additive convex criteria, a special case of which will be the linear criteria.
\subsection{Multiplicative variance criteria\label{sub:Multiplicative-variance-criteria}}
Following \citep{Reid1989}, we consider a situation where Alice tries to infer the outcomes of Bob's measurements through measurements on her subsystem. We denote by $B_{\mathrm{est}}(A)$ Alice's estimate of the value of Bob's measurement $b$ as a function of the outcomes of her measurement $a.$ As in Section \ref{sec:The-EPR-Reid-criterion}, the average inference variance of $B$ given estimate $B_{\mathrm{est}}(A)$ is defined by \begin{equation} \Delta_{\mathrm{inf}}^{2}B=\langle(B-B_{\mathrm{est}}(A))^{2}\rangle.\label{eq:inf_var}\end{equation} Here the average is over all outcomes $B$, $A$. Since for a given $A$, the estimate that minimizes $\langle(B-B_{\mathrm{est}}(A))^{2}\rangle$
is just the mean $\langle B\rangle_{A}$ of \textcolor{black}{the conditional probability $P(B|A),$} the optimal estimate for each $A$ is just $B_{\mathrm{est}}(A)=\langle B\rangle_{A}.$ We denote thus the \emph{optimal inference variance} of $B$ by measurement of $a$ as \begin{eqnarray} {\normalcolor \Delta_{\mathrm{min}}^{2}B} & = & \sum_{A,B}P(A,B)(B-\langle B\rangle_{A})^{2}\nonumber \\
& = & \sum_{A}P(A)\sum_{B}P(B|A)(B-\langle B\rangle_{A})^{2}\nonumber \\
& = & \sum_{A}P(A)\Delta^{2}(B|A)\label{eq:min_inf_var}\end{eqnarray}
where $\Delta^{2}(B|A)$ is the variance of $B$ calculated from the conditional probability distribution $P(B|A).$ As explained above, \begin{equation} \Delta_{\mathrm{inf}}^{2}B\geq\Delta_{\mathrm{min}}^{2}B\label{eq:min<inf}\end{equation}
for all choices of $B_{\mathrm{est}}(A).$ This minimum is optimal, but not always experimentally accessible, in EPR experiments, since it requires one to be able to measure conditional probability distributions.
We assume that the statistics of Alice's and Bob's experimental outcomes can be described by a LHS model, i.e., by a model of form \eqref{eq:LHS_model} {[}omitting henceforth, for notational simplicity, the preparation $c$ and the measurement choices $a,\, b$ from the conditional probabilities
$P(A,B|a,b,c),$ etc.{]}, \begin{equation}
P(A,B)=\sum_{\lambda}\, P(\lambda)\, P(A|\lambda)P_{Q}(B|\lambda).\label{eq:LHS model_2}\end{equation} Assuming this model, the conditional probability of $B$ given $A$ is\begin{eqnarray}
P(B|A) & = & \sum_{\lambda}\frac{P(\lambda)P(A|\lambda)}{P(A)}P_{Q}(B|\lambda)\nonumber \\
& = & \sum_{\lambda}P(\lambda|A)P_{Q}(B|\lambda).\label{eq:P(B|A)}\end{eqnarray}
As in Section \ref{sec:Locality-models}, $P_{Q}(B|\lambda)=\mathrm{Tr}[\Pi_{b}^{B}\rho_{\lambda}]$ represents the probability for $B$ predicted by a quantum state $\rho_{\lambda}.$
It is a general result that if a probability distribution has a convex decomposition of the type $P(x)=\sum_{y}P(y)P(x|y),$ then the variance
$\Delta^{2}x$ over the distribution $P(x)$ cannot be smaller than the average of the variances over the component distributions $P(x|y),$
\emph{i.e.}, $\Delta^{2}x\geq\sum_{y}P(y)\Delta^{2}(x|y).$ Therefore, by \eqref{eq:P(B|A)}, the variance $\Delta^{2}(B|A)$ satisfies\begin{equation}
\Delta^{2}(B|A)\geq\sum_{\lambda}P(\lambda|A)\Delta_{Q}^{2}(B|\lambda),\label{eq:bound_var(B|A)}\end{equation}
where $\Delta_{Q}^{2}(B|\lambda)$ is the variance of $P_{Q}(B|\lambda).$ Using this result, we can derive a bound for Eq. \eqref{eq:min_inf_var}, \begin{equation}
\Delta_{\mathrm{min}}^{2}B\geq\sum_{A,\lambda}P(A,\lambda)\Delta_{Q}^{2}(B|\lambda)=\sum_{\lambda}P(\lambda)\Delta_{Q}^{2}(B|\lambda).\label{eq:bound_inf_var}\end{equation}
Suppose Bob's set of measurements consists of $\mathfrak{M}_{\beta}=\{b_{1},b_{2},b_{3}\},$ with respective outcomes labeled by $B_{1},\, B_{2},\, B_{3}.$ Alice measures $\mathfrak{M}_{\alpha}=\{a_{1},a_{2},a_{3}\}.$ Suppose the corresponding quantum observables for Bob, $\{\hat{b}_{1},\hat{b}_{2},\hat{b}_{3}\},$ obey the commutation relation $[\hat{b}_{1},\hat{b}_{2}]=i\hat{b}_{3}.$ The outcomes must then satisfy the product uncertainty relation \begin{equation}
\Delta_{Q}(B_{1}|\rho)\Delta_{Q}(B_{2}|\rho)\geq\frac{1}{2}|\langle B_{3}\rangle_{\rho}|,\label{eq:mult_UR}\end{equation}
where $\Delta_{Q}(B_{i}|\rho)$ and $\langle B_{i}\rangle_{\rho}$ are respectively the standard deviation and the average of $B_{i}$ in the quantum state $\rho.$
We will use the uncertainty relation above and the Cauchy-Schwarz
(C-S) inequality to obtain an EPR-steering criterion. The C-S inequality states that, for two vectors $u$ and $v,$ $|u||v|\geq|u\cdot v|.$
Define $u=(\sqrt{P(\lambda_{1})}\Delta_{Q}(B_{1}|\lambda_{1})),\,\sqrt{P(\lambda_{2})}\Delta_{Q}(B_{1}|\lambda_{2}),\,\ldots)$
and $v=(\sqrt{P(\lambda_{1})}\Delta_{Q}(B_{2}|\lambda_{1}),\,\sqrt{P(\lambda_{2})}\Delta_{Q}(B_{2}|\lambda_{2}),\,\ldots).$ Then by \eqref{eq:bound_inf_var}\begin{eqnarray}
\Delta_{\mathrm{min}}B_{1}=\sqrt{\Delta_{\mathrm{min}}^{2}B_{1}} & \geq & |u|,\nonumber \\
\Delta_{\mathrm{min}}B_{2}=\sqrt{\Delta_{\mathrm{min}}^{2}B_{2}} & \geq & |v|.\label{eq:mult_var_u,v}\end{eqnarray} We thus obtain, from \eqref{eq:mult_var_u,v}, the C-S inequality and the uncertainty relation \eqref{eq:mult_UR}, \begin{eqnarray}
\Delta_{\mathrm{min}}B_{1}\Delta_{\mathrm{min}}B_{2} & \geq & |u||v|\nonumber \\
& \geq & |u\cdot v|\nonumber \\
& = & \sum_{\lambda}P(\lambda)\Delta_{Q}(B_{1}|\lambda)\Delta_{Q}(B_{2}|\lambda)\nonumber \\
& \geq & \frac{1}{2}\sum_{\lambda}P(\lambda)|\langle B_{3}\rangle_{\lambda}|.\label{eq:mult_var_step2}\end{eqnarray}
Here we denote by $\langle B\rangle_{\lambda}$ the expectation value of $B$ calculated from $P_{Q}(B|\lambda)$. Using again Eq.~\eqref{eq:P(B|A)}
and the fact that $f(x)=|x|$ is a convex function, that is, that
$\sum_{x}P(x)|x|\geq|\sum_{x}P(x)\, x|,$ we obtain a bound for the last term: \begin{eqnarray}
\sum_{\lambda}P(\lambda)|\langle B_{3}\rangle_{\lambda}| & = & \sum_{A_{3},\lambda}P(A_{3},\lambda)|\langle B_{3}\rangle_{\lambda}|\nonumber \\
& \geq & \sum_{A_{3}}P(A_{3})\left|\sum_{\lambda}P(\lambda|A_{3})\langle B_{3}\rangle_{\lambda}\right|\nonumber \\
& = & \sum_{A_{3}}P(A_{3})|\langle B_{3}\rangle_{A_{3}}|\nonumber \\
& \equiv & |\langle B_{i}\rangle|_{\mathrm{inf}}\label{eq:inf_ave}\end{eqnarray} Using now \eqref{eq:min<inf}, we obtain, from \eqref{eq:mult_var_step2} and \eqref{eq:inf_ave}, the EPR-steering criterion\begin{equation}
\Delta_{\mathrm{inf}}B_{1}\Delta_{\mathrm{inf}}B_{2}\geq\frac{1}{2}|\langle B_{3}\rangle|_{\mathrm{inf}}.\label{eq:mult_var_final}\end{equation} This inequality was introduced in \citep{Cavalcanti2007b}, but its derivation was based on the conceptual scheme of the EPR-Reid criterion. Here we have shown that it follows directly from the LHS model \eqref{eq:LHS model_2}. Its experimental violation implies the failure of the LHS model to represent the measurement statistics, that is, it is an experimental demonstration of EPR-steering. It is important to note that the choices of measurement $a_{1},\, a_{2},\, a_{3}$ used by Alice to infer the values of the corresponding measurements of Bob are arbitrary in this derivation; the specific quantum observables $\hat{a}_{i}$ played no role in the above because in a LHS model Alice's probabilities are allowed to depend arbitrarily on the variables $\lambda$. In an experimental situation, one should choose, of course, those which can maximise the violation of \eqref{eq:mult_var_final}.
One can also derive criteria involving collective variances such as $\Delta^{2}(g_{k}A_{k}+B_{k}),$ where $g_{k}$ is a real number. These measurements are often simpler to be realised as they do not require the full conditional distributions. These are just the average inference variances $\Delta_{\mathrm{inf}}^{2}B_{k}=\langle[B_{k}-B_{\mathrm{est}}(A_{k})]^{2}\rangle$ with a linear estimate $B_{\mathrm{est}}(A_{k})=-g_{k}A_{k}+\langle B_{k}+g_{k}A_{k}\rangle,$ as shown in \citep{Reid2008tb}. We can therefore straightforwardly derive, from \eqref{eq:mult_var_final}:
\begin{equation}
\Delta(g_{1}A_{1}+B_{1})\Delta(g_{2}A_{2}+B_{2})\geq\frac{1}{2}|\langle B_{3}\rangle|_{\mathrm{inf}},\label{eq:mult_collective_var}\end{equation} keeping in mind that the measurements for Alice and the values of $g_{k}$ are arbitrary, and should be chosen so as to optimize the violation of the inequality.
\subsubsection{Examples}
The first example of a multiplicative variance criterion is the original EPR-Reid criterion \citep{Reid1989}, reviewed in Section \ref{sec:The-EPR-Reid-criterion}. It was developed for continuous variables observables $\hat{x}^{B}$
and $\hat{p}^{B},$ which obey an uncertainty relation $\Delta_{Q}(x^{B}|\rho)\Delta_{Q}(p^{B}|\rho)\geq1,$ arising from the commutation relation (in appropriate units) $[\hat{x}^{B},\hat{p}^{B}]=2i$. Substituting $B_{1}=x^{B},$ $B_{2}=p^{B}$ and $B_{3}=2$ in \eqref{eq:mult_var_final} we obtain the EPR-Reid criterion \eqref{eq:EPR-Reid},
\begin{equation} \Delta_{\mathrm{inf}}x^{B}\Delta_{\mathrm{inf}}p^{B}\geq1.\label{eq:EPR-Reid_rederived}\end{equation}
This provides a formal proof of the incomplete conjecture put forth in \citep{Wiseman2007}, that the EPR-Reid criterion is a special case of EPR-steering. It is a direct consequence of the assumption of a LHS model; in particular this derivation does not require Reid's extension of EPR's necessary condition for reality.
For angular momentum observables, obeying a commutation relation $[\hat{J}_{x}^{B},\hat{J}_{y}^{B}]=i\hat{J}_{z}^{B}$
(and its cyclical permutations) the corresponding quantum uncertainty relation is $\Delta_{Q}(J_{x}^{B}|\rho)\Delta_{Q}(J_{y}^{B}|\rho)\geq\frac{1}{2}|\langle J_{z}^{B}\rangle_{\rho}|$ (and permutations). Substituting these in \eqref{eq:mult_var_final}, with $B_{1}=J_{x}^{B},$ $B_{2}=J_{y}^{B}$ and $B_{3}=J_{z}^{B},$ we obtain the criterion \eqref{eq:EPR-Bohm_Cav_Reid} reviewed in Section \ref{sub:Recent-developments}:\begin{equation}
\Delta_{\mathrm{inf}}J_{x}^{B}\Delta_{\mathrm{inf}}J_{y}^{B}\geq\frac{1}{2}|\langle J_{z}^{B}\rangle|_{\mathrm{inf}},\label{eq:EPR-Bohm_criterion}\end{equation} and of course, its permutations. Violation of one of these inequalities corresponds to a demonstration of the EPR-Bohm paradox discussed in Sec. \ref{sub:Bohm's-version}. Bowen \emph{et al.}'s \citep{Bowen2003}
inequality \eqref{eq:Bowen_EPR_criterion} is the special case in which Alice's choice of measurement used to infer $|\langle J_{z}^{B}\rangle|_{\mathrm{inf}}$
is the identity. We can see that it is a weaker criterion than the above by noting that the convexity of the function $f(x)=|x|$ implies
$|\langle J_{z}^{B}\rangle|_{\mathrm{inf}}\equiv\sum_{J_{z}^{A}}P(J_{z}^{A})|\langle J_{z}^{B}\rangle_{J_{z}^{A}}|\geq|\langle J_{z}^{B}\rangle|.$ Inequality \eqref{eq:Bowen_EPR_criterion} therefore will be violated only if \eqref{eq:EPR-Bohm_criterion} also is. In particular, \eqref{eq:EPR-Bohm_criterion} can detect EPR-steering for states in which the expectation value of $J_{z}^{B}$ is zero, such as the symmetric state originally considered by Bohm \citep{Bohm1951}. Applications of these criteria to specific classes of quantum states will be given in Sec. \ref{sec:Applications-to-classes}.
\subsection{Additive convex criteria}
We now present the derivation of the class of additive convex criteria. Suppose one has an uncertainty relation in the broadest sense --- a general constraint which must be obeyed by all quantum states of Bob's subsystem --- of form
\begin{equation} \sum_{j}f_{j}(\langle B_{j}\rangle_{\rho},\alpha_{j})\leq0,\label{eq:add_conv_constraint}\end{equation} where $j$ indexes observables on Bob's subsystem, $\langle B_{j}\rangle_{\rho}$ denotes the expectation value of observable $b_{j}$ on a quantum state $\rho,$ $\alpha_{j}\in\mathbb{R}$ are parameters of the constraint which can take any values in some set $\mathfrak{O}_{a_{j}}$ (the significance of which should be clear soon), and the functions $f_{j}$ are convex on the interval containing the possible values of the first argument (i.e., the possible expectation values $\langle B_{j}\rangle_{\rho}$, which is the convex hull $H_{\mathrm{convex}}\{\mathfrak{O}_{b_{j}}\}$ of the set of possible outcomes of $b_{j}$). This last requirement means that for all $x,\, y\in H_{\mathrm{convex}}\{\mathfrak{O}_{b_{j}}\},$ for all $z\in\mathfrak{O}_{a_{j}}$ and for all $p\in[0,1]$, \begin{equation} f_{j}(px+(1-p)y,z)\leq pf_{j}(x,z)+(1-p)f_{j}(y,z).\end{equation}
Although the product uncertainty relations considered in the previous section are not of form \eqref{eq:add_conv_constraint}, since they include terms like $\langle B_{1}^{2}\rangle\langle B_{2}^{2}\rangle,$ a large class of uncertainty relations can be written in this form. The negative of the variance of a variable $B$, that is, $-\Delta^{2}B=\langle B\rangle^{2}-\langle B^{2}\rangle,$ is a sum of two convex functions $f_{1}(\langle B\rangle)+f_{2}(\langle B^{2}\rangle),$
{[}with $f_{1}(x)=x^{2}$ and $f_{2}(x)=-x${]} and thus we can obtain EPR-steering criteria from uncertainty relations that involve sums of variances of observables. For example, the relation $\Delta^{2}B_{1}+\Delta^{2}B_{2}\geq|\langle B_{3}\rangle|$ \citep{Cohen-Tannoudji1977} can be rewritten as \begin{equation}
|\langle B_{3}\rangle|-\langle B_{1}^{2}\rangle+\langle B_{3}\rangle^{2}-\langle B_{3}^{2}\rangle+\langle B_{3}\rangle^{2}\leq0,\label{eq:UR_add_convex}\end{equation} which is of form \eqref{eq:add_conv_constraint}, with 5 terms in the sum. All terms are convex, since the coefficients of the square terms and absolute-value terms are positive. Any term linear on the expectation values $\langle B_{j}\rangle_{\rho}$ is clearly also of that form. As in the previous section, the assumption that the statistics of Alice and Bob can be described by a LHS model of form \eqref{eq:LHS model_2} implies that the conditional probability of outcome $B$ given outcome $A$ can be written as\begin{equation}
P(B|A)=\sum_{\lambda}P(\lambda|A)P_{Q}(B|\lambda).\label{eq:P(B|A)_again}\end{equation}
The average of this conditional probability,$\langle B\rangle_{A},$ can be thus written as\begin{equation}
\langle B\rangle_{A}=\sum_{\lambda}P(\lambda|A)\langle B\rangle_{\lambda},\label{eq:ave_B_A}\end{equation}
and we remind the reader that $\langle B\rangle_{\lambda}\equiv\sum_{B}P_{Q}(B|\lambda)\, B=\mathrm{Tr}\{\hat{b}\,\rho_{\lambda}\}.$
If $f$ is a convex function, \eqref{eq:ave_B_A} then implies, for all $A$,\begin{eqnarray}
f\left(\langle B\rangle_{A},A\right) & = & f\left(\sum_{\lambda}P(\lambda|A)\langle B\rangle_{\lambda},A\right)\nonumber \\
& \leq & \sum_{\lambda}P(\lambda|A)\, f\left(\langle B\rangle_{\lambda},A\right).\label{eq:conv_f_step_1}\end{eqnarray} Taking the average over $A$ we obtain\begin{equation} \sum_{A}P(A)\, f\left(\langle B\rangle_{A},A\right)\leq\sum_{A,\lambda}P(A,\lambda)\, f\left(\langle B\rangle_{\lambda},A\right).\label{eq:ave_over_A}\end{equation}
We now introduce the subscripts $j,$ sum both sides of \eqref{eq:ave_over_A} over $j$ and apply the quantum constraint \eqref{eq:add_conv_constraint} to obtain \begin{multline} \sum_{j,A_{j}}P(A_{j})\, f_{j}\left(\langle B_{j}\rangle_{A_{j}},A_{j}\right)\\ \leq\sum_{A_{j},\lambda}P(A_{j},\lambda)\,\sum_{j}f_{j}\left(\langle B_{j}\rangle_{\lambda},A_{j}\right)\leq0\;.\end{multline}
Introducing the simplifying notation $E_{b|a}[f_{j}]\equiv\sum_{A_{j}}P(A_{j})\, f_{j}\left(\langle B_{j}\rangle_{A_{j}},A_{j}\right),$ we write the general EPR-steering criterion\begin{equation}
\sum_{j}E_{b|a}[f_{j}]\leq0\;.\label{eq:add_convex_criteria}\end{equation} A weaker version of the inequality (i.e., one that detects steerability less efficiently) can be obtained by using the following bound, which is a consequence of the convexity of $f_{j},$ when $f_{j}$ does not depend explicitly on $A_{j}$:\begin{equation}
f_{j}(\langle B_{j}\rangle)\leq E_{b|a}[f_{j}].\label{eq:meas_bound}\end{equation}
One can therefore substitute $E_{b|a}[f_{j}]$ by $f_{j}(\langle B_{j}\rangle)$ for some $j$ in \eqref{eq:add_convex_criteria} and the inequality still holds.
\subsubsection{Examples: criteria from inference variances}
We will now give some examples of criteria that can be obtained with the general form of \eqref{eq:add_convex_criteria}.We note, to make contact with the previous notation, that when the $f_{j}$'s involve variances, the corresponding expressions on the left-hand side of \eqref{eq:add_convex_criteria} are just \begin{equation} \sum_{A}P(A)\,\left(\langle B\rangle_{A}^{2}-\langle B^{2}\rangle_{A}\right)=-\Delta_{\mathrm{min}}^{2}B,\label{eq:fj_inf_var}\end{equation} as defined on \eqref{eq:inf_var}. As before, the bound \begin{equation} \Delta_{\mathrm{inf}}^{2}B\geq\Delta_{\mathrm{min}}^{2}B\label{eq:inf>min_again}\end{equation}
can be used in the derivation of the inequalities.
We start considering arbitrary observables obeying commutation relation
$[\hat{b}_{1},\hat{b}_{2}]=i\hat{b}_{3},$ and use the uncertainty relation $\Delta^{2}(B_{1}|\rho)+\Delta^{2}(B_{2}|\rho)\geq|\langle B_{3}\rangle_{\rho}|,$ which is of form \eqref{eq:add_conv_constraint} as shown above. Expanding this in terms of the $f_{j}$'s, substituting on \eqref{eq:add_convex_criteria} and using \eqref{eq:fj_inf_var} and \eqref{eq:inf>min_again} we obtain the EPR-steering inequality\begin{equation}
\Delta_{\mathrm{inf}}^{2}B_{1}+\Delta_{\mathrm{inf}}^{2}B_{2}\geq|\langle B_{3}\rangle|_{\mathrm{inf}},\label{eq:sum_inf_var_criterion_arb}\end{equation}
where as before $|\langle B_{3}\rangle|_{\mathrm{inf}}\equiv\sum_{A_{3}}P(A_{3})|\langle B_{3}\rangle_{A_{3}}|,$
and the bound $|\langle B_{3}\rangle|_{\mathrm{inf}}\geq|\langle B_{3}\rangle|$ can be used if needed.
For continuous variables observables $[\hat{x}^{B},\hat{p}^{B}]=2i,$ \eqref{eq:sum_inf_var_criterion_arb} becomes inequality \eqref{eq:EPR_sum_Cav_Reid}, \begin{equation} \Delta_{\mathrm{inf}}^{2}x^{B}+\Delta_{\mathrm{inf}}^{2}p^{B}\geq2,\label{eq:sum_inf_var_criterion_x}\end{equation} and for angular momentum observables inequality \eqref{eq:sum_inf_var_criterion_arb} reads \begin{equation}
\Delta_{\mathrm{inf}}^{2}J_{x}^{B}+\Delta_{\mathrm{inf}}^{2}J_{y}^{B}\geq|\langle J_{z}^{B}\rangle|_{\mathrm{inf}}.\label{eq:sum_inf_var_criterion_|J|}\end{equation} Inequality \eqref{eq:sum_inf_var_criterion_x} has been derived (within the EPR-Reid formalism) in \citep{Cavalcanti2007b}. However, these inequalities are weaker than the corresponding multiplicative variance criteria: since for any pair of real numbers $x^{2}+y^{2}\geq2xy,$ inequality \eqref{eq:mult_var_final} directly implies \eqref{eq:sum_inf_var_criterion_arb} and thus the latter can be violated only if the former is.
Another special case of additive convex criterion has been recently derived in \citep{Cavalcanti2009b}.\textcolor{black}{{} Consider Schwinger spin operators defined as \begin{eqnarray} \hat{J}_{x}^{B} & = & \frac{1}{2}\left(\hat{b}_{-}\hat{b}_{+}^{\dagger}+\hat{b}_{-}^{\dagger}\hat{b}_{+}\right),\nonumber \\ \hat{J}_{y}^{B} & = & \frac{1}{2i}\left(\hat{b}_{-}\hat{b}_{+}^{\dagger}-\hat{b}_{-}^{\dagger}\hat{b}_{+}\right),\nonumber \\ \hat{J}_{z}^{B} & = & \frac{1}{2}\left(\hat{b}_{+}^{\dagger}\hat{b}_{+}-\hat{b}_{-}^{\dagger}\hat{b}_{-}\right),\nonumber \\ \hat{N}^{B} & = & \left(\hat{b}_{+}^{\dagger}\hat{b}_{+}+\hat{b}_{-}^{\dagger}\hat{b}_{-}\right),\label{eq:Schwinger}\end{eqnarray} where $\hat{b}_{\pm}$ ar}e boson operators for two field modes of Bob's subsystem, obeying commutation relations $[\hat{b}_{\pm},\hat{b}_{\pm}^{\dagger}]=1.$ Similar operators are defined for Alice\textcolor{black}{. The situation of the EPR-Bohm setup is therefore extended with number measurements.} We now use the quantum uncertainty relation \citep{Hofmann2003a} \begin{equation}
\Delta^{2}(J_{x}^{B}|\rho)+\Delta^{2}(J_{y}^{B}|\rho)+\Delta^{2}(J_{z}^{B}|\rho)\geq\frac{1}{4}\Delta^{2}(N^{B}|\rho)+\frac{1}{2}\langle N^{B}\rangle_{\rho},\label{eq:HUPHofmann}\end{equation}
and rewrite it in the form of \eqref{eq:add_conv_constraint}, $-\Delta^{2}(J_{x}^{B}|\rho)-\Delta^{2}(J_{y}^{B}|\rho)-\Delta^{2}(J_{z}^{B}|\rho)+\langle N^{B}\rangle_{\rho}/2\leq0,$ dropping the positive but non-convex term $\Delta^{2}N^{B}/4.$ Substituting this in \eqref{eq:add_convex_criteria}, and using \eqref{eq:fj_inf_var} and \eqref{eq:inf>min_again}, we obtain: \begin{equation} \Delta_{\mathrm{inf}}^{2}J_{x}^{B}+\Delta_{\mathrm{inf}}^{2}J_{y}^{B}+\Delta_{\mathrm{inf}}^{2}J_{z}^{B}\geq\frac{\langle N^{B}\rangle}{2}.\label{eq:sum_inf_var_criterion_J}\end{equation}
In the angular momentum basis $\{|j,m\rangle\},$ where $j(j+1)$ are the eigenvalues of $\hat{J}{}^{2}=(\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2})$ and $m$ are the eigenvalues of $\hat{J}_{z},$ the operator $\hat{N}/2$
corresponds to the {}``total angular momentum'' operator $\hat{J}_{T}=\sum_{j}j\sum_{m}|j,m\rangle\langle j,m|,$ i.e., the operator which has a spectral decomposition in terms of projectors onto each subspace of constant $j,$ with corresponding eigenvalues $j.$ \footnote{Note that the angular momentum-square operator $J{}^{2}$ is not the square of this operator. Although they have the same eigenvectors, the eigenvalues of $J{}^{2}$ are $j(j+1)$ and not $j^{2}.$ } Any criteria in which $\langle N^{B}\rangle$ occurs can therefore be modified by substituting $\langle N^{B}\rangle/2=\langle J_{T}^{B}\rangle$. For a spin-$j$ particle, this is just $\langle J_{T}^{B}\rangle=j.$ With this substitution we obtain inequality \eqref{eq:EPR-Bohm_sum_criterion}.
Using again the linear inferences $B_{\mathrm{est}}(A_{k})=-g_{k}A_{k}+\langle B_{k}+g_{k}A_{k}\rangle$ as discussed above Eq. \eqref{eq:mult_collective_var}, we can derive directly from \eqref{eq:sum_inf_var_criterion_J}, \eqref{eq:sum_inf_var_criterion_x} and \eqref{eq:sum_inf_var_criterion_arb} the respective criteria \begin{equation} \Delta^{2}(g_{x}J_{x}^{A}+J_{x}^{B})+\Delta^{2}(g_{y}J_{y}^{A}+J_{y}^{B})+\Delta^{2}(g_{z}J_{z}^{A}+J_{z}^{B})\geq\frac{\langle N^{B}\rangle}{2},\label{eq:collective_spin_criterion}\end{equation} \begin{equation} \Delta^{2}(g_{x}x^{A}+x^{B})+\Delta^{2}(g_{p}p^{A}+p^{B})\geq2,\label{eq:collective_CV_criterion}\end{equation} and\begin{equation}
\Delta^{2}(g_{1}A_{1}+B_{1})+\Delta^{2}(g_{2}A_{2}+B_{2})\geq|\langle B_{3}\rangle|_{inf}.\label{eq:collective_arb_criterion}\end{equation} Again we should keep in mind that the corresponding operators for Alice, and the values of $g_{k}$, are arbitrary, and therefore should be chosen so as to optimize the violation of the criteria. Inequality \eqref{eq:collective_CV_criterion}, which was introduced in \citep{Reid2008tb}, is the analogue for EPR-steering of the entanglement criteria of Duan \emph{et al.} \citep{Duan2000} and Simon \citep{Simon2000}. Note that the bound is half that of those authors (making it harder to violate), a consequence of the fact that EPR-steering is a stronger form of nonlocality than entanglement. Inequality \eqref{eq:collective_spin_criterion} is the analogue of the separability criteria of \citet{Hofmann2003a}.
The inference variance criteria have an immediate interpretation as a demonstration of the situation described by EPR, as they are based on an apparent violation of the uncertainty principle by inference of the variances of the distant subsystem. However, in general any constraint that can be derived from the LHS model is an EPR-steering criterion, and by the arguments of Sections \ref{sec:History-and-concepts} and \ref{sec:Locality-models}, a demonstration of the EPR paradox. We present below examples of such more general criteria which can be derived as special cases of the additive convex criterion \eqref{eq:add_convex_criteria}.
\subsubsection{Examples: linear criteria}
We first illustrate this approach by deriving a simple criteria for the case of two qubits. We start with a quantum constraint on expectation values of spin-1/2 observables:\begin{equation} \langle J_{x}\rangle_{\rho}+\langle J_{y}\rangle_{\rho}\leq\frac{\sqrt{2}}{2}.\label{eq:constraint_Jx_Jy}\end{equation} This must be satisfied by any quantum state of a qubit: $\frac{1}{\sqrt{2}}(\hat{J}_{x}+\hat{J}_{y})\equiv\hat{J}_{\theta}$ is simply the observable corresponding to the spin projection on a direction at $\theta=45^{o}$ between $\mathbf{x}$ and $\mathbf{y}$, and so for any quantum state $\rho$, $\langle\hat{J}_{\theta}\rangle_{\rho}\leq\frac{1}{2}$.
Now it must then also be the case that, for a pair of observables $\hat{J}_{x}^{B},\,\hat{J}_{y}^{B}$ for Bob and $\hat{J}_{x}^{A},\,\hat{J}_{y}^{A}$ for Alice, and where $\alpha_{i}\in\{-\frac{1}{2},\frac{1}{2}\}$ represent possible values for the outcomes of observable $\hat{J}_{i}^{A},$\begin{equation} \alpha_{x}\langle J_{x}^{B}\rangle_{\rho}+\alpha_{y}\langle J_{y}^{B}\rangle_{\rho}\leq\frac{\sqrt{2}}{4},\label{eq:bound_spin_1/2}\end{equation} for all values of $\alpha_{x},\,\alpha_{y}.$ This is easy to see by noting that the different values of $(\alpha_{x},\,\alpha_{y})$ lead to one of $\mp\frac{1}{2}\langle J_{x}^{B}\pm J_{y}^{B}\rangle,$ and for each of these the argument of the previous paragraph leads to \eqref{eq:bound_spin_1/2}. This is of the form \eqref{eq:add_conv_constraint}, and therefore, by substituting on \eqref{eq:add_convex_criteria} and noting that $\sum_{A}P(A)\, J_{i}^{A}\langle J_{i}^{B}\rangle_{A}=\langle J_{i}^{A}J_{i}^{B}\rangle,$ it leads to the EPR-steering criterion\begin{equation} \langle J_{x}^{A}J_{x}^{B}\rangle+\langle J_{y}^{A}J_{y}^{B}\rangle\leq\frac{\sqrt{2}}{4}.\label{eq:two_qubit_criterion_0}\end{equation} Following a similar procedure, and using the quantum constraint $\alpha_{x}\langle J_{x}^{B}\rangle_{\rho}+\alpha_{y}\langle J_{y}^{B}\rangle_{\rho}\geq-\frac{\sqrt{2}}{4},$ which is valid for the same reason as \eqref{eq:bound_spin_1/2}, we can derive the inequality $\langle J_{x}^{A}J_{x}^{B}\rangle+\langle J_{y}^{A}J_{y}^{B}\rangle\geq-\frac{\sqrt{2}}{4}.$ These two inequalities can be summarised in the EPR-steering criterion \begin{equation}
\left|\langle J_{x}^{A}J_{x}^{B}\rangle+\langle J_{y}^{A}J_{y}^{B}\rangle\right|\leq\frac{\sqrt{2}}{4}.\label{eq:two_qubit_criterion_1}\end{equation}
A similar, more powerful inequality can be derived from the analogous constraint on three observables \begin{equation} -\frac{\sqrt{3}}{2}\leq\alpha_{x}\langle J_{x}\rangle_{\rho}+\alpha_{y}\langle J_{y}\rangle_{\rho}+\alpha_{z}\langle J_{z}\rangle_{\rho}\leq\frac{\sqrt{3}}{2},\label{eq:constraint_Jx_Jy_Jz}\end{equation} which follows, as \eqref{eq:bound_spin_1/2}, from the fact that $\hat{J}_{\phi}\equiv\frac{1}{\sqrt{3}}(\hat{J}_{x}+\hat{J}_{y}+\hat{J}_{z})$ is another observable corresponding to a spin projection. From \eqref{eq:constraint_Jx_Jy_Jz} we can derive, following similar steps as above, the EPR-steering criterion\begin{equation}
\left|\langle J_{x}^{A}J_{x}^{B}\rangle+\langle J_{y}^{A}J_{y}^{B}\rangle+\langle J_{z}^{A}J_{z}^{B}\rangle\right|\leq\frac{\sqrt{3}}{4}.\label{eq:two_qubit_criterion_2}\end{equation}
We can now generalize this to an arbitrary total spin. For a spin-$j$
particle, the quantum constraint $\left|\alpha_{x}\langle J_{x}\rangle_{\rho}+\alpha_{y}\langle J_{y}\rangle_{\rho}+\alpha_{z}\langle J_{z}\rangle_{\rho}\right|\leq\sqrt{3}j^{2}$ holds. To see this, note that $\hat{J}_{\phi}\equiv(\alpha_{x}\hat{J}_{x}+\alpha_{y}\hat{J}_{y}+\alpha_{z}\hat{J}_{z})/\sqrt{\alpha_{x}^{2}+\alpha_{y}^{2}+\alpha_{z}^{2}}$ is again a spin projection operator, and that $\sqrt{\alpha_{x}^{2}+\alpha_{y}^{2}+\alpha_{z}^{2}}\leq\sqrt{3}j.$ Following the same steps as for the derivation of \eqref{eq:two_qubit_criterion_1} this leads to the EPR-steering inequality\begin{equation}
\left|\langle J_{x}^{A}J_{x}^{B}\rangle+\langle J_{y}^{A}J_{y}^{B}\rangle+\langle J_{z}^{A}J_{z}^{B}\rangle\right|\leq\sqrt{3}j^{2}.\label{eq:two_qudit_criterion}\end{equation}
\subsubsection{Generalisation for positive operator valued measures (POVMs)}
In all of the above we have assumed that the measurements on Bob's system can be described by observables, with projection operators associated to eigenvalues. There is no loss of generality in this assumption if we allow Bob's system to be supplemented by an ancilla system, uncorrelated with any other system \citep{Helstrom}. However it is often convenient to consider generalized measurements, described by a POVM, that is, a set of positive operators $F_{\mu}$ associated to measurement outcomes $\mu$, which sum to unity. In terms of finding appropriate EPR-steering criteria, the additive convex criteria are the ones most naturally generalizable to this case. We replace the $f_{j}(\langle B_{j}\rangle,\alpha_{j})$ in Eq. \eqref{eq:add_conv_constraint} by\[ f_{j}(\{\langle F_{\mu}^{j}\rangle_{\rho}:\mu\},\alpha_{j}),\]
where for all $j$ and $\mu$, $F_{\mu}^{j}\geq0$, and for all $j$, $\sum_{\mu}F_{\mu}^{j}=1$.
The convexity requirement in $\langle B_{j}\rangle_{\rho}$ would be replaced by a more general convexity requirement, that for all $j$ and $\alpha_{j}$, all $\rho$ and $\rho'$, and $0\leq p\leq1$,
\begin{multline} f_{j}(\{\langle F_{\mu}^{j}\rangle_{\rho''}:\mu\},\alpha_{j})\\ \leq pf_{j}(\{\langle F_{\mu}^{j}\rangle_{\rho}:\mu\},\alpha_{j})+(1-p)f_{j}(\{\langle F_{\mu}^{j}\rangle_{\rho'}:\mu\},\alpha_{j}),\label{eq:POVM_convexity}\end{multline} where $\rho''=p\rho+(1-p)\rho'$. The derivation of Eq. \eqref{eq:add_convex_criteria} then follows exactly as before.
\section{Applications to classes of quantum states \label{sec:Applications-to-classes}}
We now apply the criteria derived in the previous section to some classes of quantum states of experimental interest. Violations of those inequalities amount to demonstrations of the effect termed {}``steering'' by Schrödinger in his response to EPR, reviewed in Sec.~\ref{sub:Schr=0000F6dinger's-response}. In the continuous variables case, this provides a more modern and unifying approach to the demonstration of the correlations considered by EPR in their original example, discussed in Sec.~\ref{sec:The-Einstein-Podolsky-Rosen-argument}. In the discrete variables case this represents a modern approach to the demonstration of EPR-Bohm correlations discussed in Sec.~\ref{sub:Bohm's-version}. We consider each case in turn.
\subsection{Continuous variables}
We consider as a continuous variable example the case of two-mode Gaussian states prepared by optical parametric amplifiers \citep{Bowen2004}. Such states include the original EPR state as a special case with zero entropy and infinite energy. We define $\hat{x}^{A}=\hat{a}+\hat{a}^{\dagger}$ and $\hat{p}^{A}=-i(\hat{a}-\hat{a}^{\dagger})$ as the position and momentum observables to be measured by Alice, where $\hat{a}$ and $\hat{a}^{\dagger}$ are the annihilation and creation operators for a bosonic field mode at Alice's subsystem. We define $\hat{x}^{B},\,\hat{p}^{B}$ analogously for Bob's subsystem in terms of the annihilation and creation operators $\hat{b}$ and $\hat{b}^{\dagger}$ for his field mode. When the entanglement is symmetric between the two modes the covariance matrix describing such states has a particularly simple form. The continuous variable entanglement properties of such a state have recently been characterized experimentally \citep{Bowen2004}.. In this case the covariance matrix of the state $W$ has just two parameters, $\mu$ and $\bar{n}$:\begin{equation} {\rm CM}[W_{\bar{n}}^{\mu}]=V_{2}^{\alpha\beta}=\left(\begin{array}{cccc} \gamma & 0 & \delta & 0\\ 0 & \gamma & 0 & -\delta\\ \delta & 0 & \gamma & 0\\ 0 & -\delta & 0 & \gamma\end{array}\right),\label{eq:GaussCovariance}\end{equation} where $\gamma=1+2\bar{n}$ and $\delta=2\eta\sqrt{\bar{n}(1+\bar{n})}$. Here $\bar{n}$ is the mean photon number for each party, and $\mu$ is a mixing parameter defined such that the covariance matrix is linear in $\mu$ and that $0\leq\mu\leq1$, such that $\mu=0$ corresponds to an uncorrelated state and $\mu=1$ corresponds to a pure state \citep{Jones2007}. It has been shown by Duan \emph{et al.} \citep{Duan2000} and Simon \citep{Simon2000} that if a quantum state such as $W_{\bar{n}}^{\mu}$ is separable it must satisfy\begin{equation} \Delta^{2}(x^{A}-x^{B})+\Delta^{2}(p^{A}+p^{B})\geq4.\label{eq:Simon}\end{equation} It is straightforward to show that for states defined by Eq.~\eqref{eq:GaussCovariance} this leads to the condition that\begin{equation} \mu>\frac{\bar{n}}{\sqrt{\bar{n}(1+\bar{n})}}\end{equation} indicates entanglement. This condition is plotted in Fig.~\ref{fig:CVbounds}, where states above the line are entangled.
\begin{figure}
\caption{ (Color on-line.) Boundaries between different classes of symmetric two-mode Gaussian states. The lower line (green, dotted) is an entanglement boundary given by Eq.~\eqref{eq:Simon}: states above the line are entangled. The central (blue, dashed) line is a steerability (lower) boundary based on Eq.~\eqref{eq:EPRtwomodesteer} for the EPR paradox: states above this line are steerable. The upper line (red, full) is a second steerability (lower) boundary based on a generalisation of the entanglement criterion of Duan \emph{et al.} \citep{Duan2000} and Simon \citep{Simon2000}: states above this line are steerable.}
\label{fig:CVbounds}
\end{figure}
As discussed in Sec.~\ref{sec:Experimental-criteria-for}, the generalization of Duan \emph{et al.} and Simon's entanglement criterion to EPR-steering is given by inequality \eqref{eq:collective_CV_criterion}. For states of the form of Eq.~\eqref{eq:GaussCovariance}, the relevant criterion becomes, using the optimal scale factors $g_{x}=-1$ and $g_{p}=1$,\begin{equation} \Delta^{2}(x^{A}-x^{B})+\Delta^{2}(p^{A}+p^{B})\geq2.\label{eq:SimonLike}\end{equation} For the two-mode symmetric states we find\begin{equation} \Delta^{2}(x^{A}-x^{B})=\Delta^{2}(p^{A}+p^{B})=2\gamma-2\delta.\end{equation} Substituting into \eqref{eq:SimonLike} and rearranging we find that\begin{equation} \mu>\frac{1+4\bar{n}}{4\sqrt{\bar{n}(1+\bar{n})}}\label{eq:SimonSteer}\end{equation} indicates EPR-steering. This condition is plotted in Fig.~\ref{fig:CVbounds}, where states above the line are steerable. For this particular state the additive convex criterion \eqref{eq:SimonLike} and the corresponding multiplicative criterion\begin{equation} \Delta^{2}(x^{A}-x^{B})\Delta^{2}(p^{A}+p^{B})\geq1,\end{equation} derived from \eqref{eq:mult_collective_var}, give the same results, since both variances are identical in this case.
For comparison, recall the EPR-Reid criterion, \eqref{eq:EPR-Reid_rederived}, which tells us that the violation of\begin{equation} \Delta_{\mathrm{inf}}x^{B}\Delta_{\mathrm{inf}}p^{B}\geq1\label{eq:EPRsteer}\end{equation} indicates EPR-steering. Evaluating the left hand side of \eqref{eq:EPRsteer} for two-mode symmetric Gaussian states, using the optimal inference variances $\Delta_{\mathrm{min}}x^{B}$ as defined in Eq. \eqref{eq:min_inf_var}, we thus obtain\begin{equation} \mu>\sqrt{\frac{1+2\bar{n}}{2(1+\bar{n})}}\label{eq:EPRtwomodesteer}\end{equation} as a condition indicating the demonstration of EPR-steering. Also in this case inequality \eqref{eq:EPRsteer} detects EPR-steering just as well as the analogous additive criterion \eqref{eq:sum_inf_var_criterion_x}, since both inference variances for $x^{B}$ and $p^{B}$ have the same value. In Fig.~\ref{fig:CVbounds} we see that \eqref{eq:EPRsteer} provides a lower bound on steerability than that provided by \eqref{eq:SimonLike} (although for $\bar{n}\gg1$ the two bounds become arbitrarily close). This is not surprising when one remembers, as discussed in Sec.~\ref{sub:Multiplicative-variance-criteria}, that the optimal conditional variances \eqref{eq:EPRsteer} are lower bounds for the linear-estimate inference of the form $\Delta^{2}(g_{x}x^{A}+x^{B})$. In other words, as pointed out in Sec.~\ref{sec:Experimental-criteria-for}, the EPR criterion is a more sensitive witness to EPR-steering than inequality \eqref{eq:SimonLike}, derived as the steerability generalisation of the entanglement criterion of Duan \emph{et al.} and Simon.
\subsection{Discrete variables}
To illustrate the use of EPR-steering criteria in the discrete variable case we will make use of the Werner states \citep{Werner1989}. For the case of a two-dimensional subsystems, these are a natural mixed-state generalization of the singlet state considered by Bohm, and can be written as follows
\begin{equation}
\rho_{W}=\mu|\psi_{S}\rangle\langle\psi_{S}|+(1-\mu)\frac{\mathbf{I}}{4},\label{eq:WernerState}\end{equation}
where $|\psi_{S}\rangle=\frac{1}{\sqrt{2}}(|\frac{1}{2}\rangle|-\frac{1}{2}\rangle-|-\frac{1}{2}\rangle|\frac{1}{2}\rangle)$, $\mathbf{I}$ is the identity over both subsystems, and $\mu$ is a mixing parameter that can take values $\mu\leq1$, with $\mu=0$ again corresponding to a product state \citep{Wiseman2007}.
It was shown in Ref. \citep{Wiseman2007} that the Werner state is steerable in theory with an infinite number of measurements whenever $\mu>1/2$. In order to demonstrate EPR-steering in a realistic experimental setup it is sufficient to instead test a suitable EPR-steering criterion.
We will first evaluate the criterion given by inequality \eqref{eq:EPR-Bohm_criterion}. Calculation shows that for the Werner state \eqref{eq:WernerState},\[ \Delta_{\mathrm{inf}}^{2}J_{z}^{B}=\frac{1}{4}(1-\mu^{2})\] and\[
|\langle J_{z}^{B}\rangle|_{\mathrm{inf}}=\frac{\mu}{2}.\] The Werner state is rotationally symmetric, and thus $\Delta_{\mathrm{inf}}J_{x}^{B}=\Delta_{\mathrm{inf}}J_{y}^{B}=\Delta_{\mathrm{inf}}^{2}J_{z}^{B}$. We therefore find that inequality \eqref{eq:EPR-Bohm_criterion} will be violated (demonstrating EPR-steering) for $\mu>(\sqrt{5}-1)/2\approx0.62$. This inequality cannot therefore detect all steerable states.
For inequality \eqref{eq:sum_inf_var_criterion_J} we make the substitution (as explained below Eq. \eqref{eq:sum_inf_var_criterion_J}) $\langle N^{B}\rangle/2=j=1/2$, and with the values for $\Delta_{\mathrm{inf}}^{2}J_{z}^{B}$ a simple calculation reveals violation whenever $\mu>1/\sqrt{3}\approx0.58$, This inequality, more symmetric between the different measurements, thus detects more steerable states (within the class of Werner states) than the less symmetric \eqref{eq:EPR-Bohm_criterion}.
We now proceed to evaluating the linear inequalities \eqref{eq:two_qubit_criterion_1} and \eqref{eq:two_qubit_criterion_2}. The expectation value of the products of observables required for those inequalities, given the Werner state, is\[ \langle J_{i}^{A}J_{i}^{B}\rangle=-\frac{\mu}{4},\] where again by symmetry those expectation values are the same for all $i\in\{x,y,z\}$. Substituting in \eqref{eq:two_qubit_criterion_1} we obtain a violation for $\mu>1/\sqrt{2}\approx0.71$ and in \eqref{eq:two_qubit_criterion_2}, violation for $\mu>1/\sqrt{3}\approx0.58$. The first inequality, with only two measurements per site, performs worse (detects less steerable Werner states) than \eqref{eq:EPR-Bohm_criterion}, but the second, with three measurements, detects a larger range. Note that the range of states for which violation is predicted using \eqref{eq:sum_inf_var_criterion_J} is the same as that detected with \eqref{eq:two_qubit_criterion_2}. The latter, however, offers the advantage of being simpler to measure and calculate.
\section{Conclusion\label{sec:Discussion-and-conclusions}}
We have developed a general theory of EPR-steering criteria. These criteria are the experimental consequences of a LHS model for one party (Bob), just as Bell inequalities are the experimental consequence of a LHV model and entanglement criteria are consequences of a quantum separable model. The essential ingredients in the derivation of the criteria are the convexity of the set of correlations that allow a LHS model and (generalized) uncertainty relations which define bounds on how Bob's outcomes can be described by quantum states.
Analysing the different forms of nonlocality, we see that they differ only in how they treat the states of Alice and/or Bob, but they are all convex combinations of separable probability distributions. Some of the criteria derived here were therefore similar to known entanglement criteria, but with a more restrictive bound due to the fact that Alice's subsystem is treated as an arbitrary hidden-variable state. However others, in particular the \emph{linear EPR-steering criteria, }are entirely new. These criteria open the possibility to new experimental demonstrations of the EPR-steering phenomenon, with close links to topics in quantum information including entanglement witnesses and quantum cryptography. \begin{acknowledgments} We would like to acknowledge support from the Griffith University Postdoctoral Fellowship scheme, Australian Research Council grants FF0458313, DP0984863, the ARC Centre of Excellence for Quantum Computing Technology and the ARC Centre of Excellence for Quantum-Atom Optics. \end{acknowledgments}
\end{document} |
\begin{document}
\title{Towards Resistance Sparsifiers}
\begin{abstract} We study \emph{resistance sparsification} of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph.
Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers. \end{abstract}
\section{Introduction}
Compact representations of discrete structures are of fundamental importance, both from an applications point of view and from a purely mathematical perspective. Graph sparsification is perhaps one of the simplest examples: given a graph $G(V, E)$, is there a subgraph that represents $G$ truthfully, say up to a small approximation? This notion has had different names in different contexts, depending on the property that is being preserved: preserving distances is known as a \emph{graph spanner}~\cite{PS89}, preserving the size of cuts is known as a \emph{cut sparsifier}~\cite{BK96}, while preserving spectral properties is known as a \emph{spectral sparsifier}~\cite{ST04a}. These concepts are known to be related, for example, every spectral sparsifier is clearly also a cut sparsifier, and spectral sparsifiers can be constructed by an appropriate sample of spanners \cite{KP12}.
Our work is concerned with sparsification that preserves \emph{effective resistances}. We define this in Section~\ref{sec:results}, but informally the effective resistance between two nodes $u$ and $v$ is the voltage differential between them when we regard the graph as an electrical network of resistors with one unit of current injected at $u$ and extracted at $v$. Effective resistances are very useful in many applications that seek to cluster nodes in a network (see \cite{vLRH14} and references therein for a comprehensive list), and are also of fundamental mathematical interest. For example, they have deep connections to random walks on graphs (see~\cite{Lov96} for an excellent overview of this connection). Most famously, the commute time between two nodes $u$ and $v$ (the expected time for a random walk starting at $u$ to hit $v$ plus the expected time for a random walk starting at $v$ to hit $u$) is exactly $2m$ times the effective resistance between $u$ and $v$, where throughout $n\eqdef\card V$ and $m\eqdef\card E$. Hence, we are concerned with sparsification which preserves commute times.
We ask whether graphs admit a good \emph{resistance sparsifier}: a reweighted subgraph $G'(V,E',w')$ in which the effective resistances are equal, up to a $(1+\epsilon)$-factor, to those in the original graph. The short answer is yes, because every $(1+\epsilon)$-spectral sparsifier is also a $(1+\epsilon)$-resistance sparsifier. Using the spectral-sparsifiers of~\cite{BSS12}, we immediately conclude that every graph admits a $(1+\epsilon)$-resistance sparsifier with $O({n}/{\epsilon^2})$ edges.
Interestingly, the same $1/\epsilon^2$ factor loss appears even when we interpret ``sparsification'' far more broadly. For example, a natural approach to compressing the effective resistances is to use a metric embedding (instead of looking for a subgraph): map the nodes into some metric, and use the metric's distances as our resistance estimates. This approach is particularly attractive since it is well-known that effective resistances form a metric space which embeds isometrically into $\ell_2$-squared (i.e., the metric is of negative type, see e.g.~\cite{DL97}). Hence, using the Johnson-Lindenstrauss dimension reduction lemma, we can represent effective resistances up to a distortion of $(1+\epsilon)$ using vectors of dimension $O(\epsilon^{-2} \log n)$, i.e., using total space $\tO({n}/{\epsilon^2})$. In fact, this very approach was used by~\cite{SS11} to quickly compute effective resistance estimates, which were then used to construct a spectral sparsifier.
Since a ${1}/{\epsilon^2}$ term appears in both of these natural ways to compactly represent effective resistances, an obvious question is whether this is \emph{necessary}. For the stronger requirement of spectral sparsification, we know the answer is yes -- every spectral sparsifier of the complete graph requires $\Omega(n/\epsilon^2)$ edges~\cite[Section 4]{BSS12} (see also \cite{AKW14}). However, it is currently unknown whether such a bound holds also for resistance sparsifiers, and the starting point of our work is the observation (based on~\cite{vLRH14}) that for the complete graph, every $O({1}/{\epsilon})$-regular expander is a $(1+\epsilon)$-resistance sparsifier, despite not being a $(1+\epsilon)$-spectral sparsifier! We thus put forward the following conjecture.
\begin{conjecture} \label{conj:resistance} Every graph admits a $(1+\epsilon)$-resistance sparsifier with $\tO(n/\epsilon)$ edges. \end{conjecture}
We make the first step in this direction by proving the special case of dense regular expanders (which directly generalize the complete graph). Even this very special case turns out to be nontrivial, and in fact leads us to another beautiful problem which is interesting in its own right.
\begin{question} \label{q:expander} Does every dense regular expander contain a sparse regular expander as a subgraph? \end{question}
Our positive answer to this question (for a certain definition of expanders) forms the bulk of our technical work (Sections \ref{sec:subgraphs} and~\ref{sec:krv_extended_proof}), and is then used to find good resistance sparsifiers for dense regular expanders (Section~\ref{sec:resistance}).
\subsection{Results and Techniques} \label{sec:results}
Throughout, we consider undirected graphs, and they are unweighted unless stated otherwise. In a weighted graph, i.e., when edges have nonnegative weights, the \emph{weighted degree} of a vertex is the sum of weights on incident edges, and the graph is considered regular if all of its weighted degrees are equal. Typically, a sparsifying subgraph must be weighted even when the host graph is unweighted, in order to exhibit comparable parameters with far fewer edges.
Before we can state our results we first need to recall some basic definitions from spectral graph theory. Given a weighted graph $G$, let $D$ be the diagonal $n \times n$ matrix of weighted degrees, and let $A$ be the weighted adjacency matrix. The \emph{Laplacian} of $G$ is defined as $L\eqdef D-A$, and the \emph{normalized Laplacian} is the matrix $\hat L\eqdef D^{-1/2} L D^{-1/2}$.
\begin{definition}[Effective Resistance] \label{def:resistance_distance} Let $G(V,E,w)$ be a weighted graph, and let $P$ the Moore-Penrose pseudo-inverse of its Laplacian matrix. The \emph{effective resistance} (also called \emph{resistance distance}) between two nodes $u,v\in V$ is \begin{equation*}
R_G(u,v) \eqdef (e_u-e_v)^TP(e_u-e_v) , \end{equation*} where $e_u$ and $e_v$ denote the standard basis vectors in $\mathbb R^V$ that correspond to $u$ and $v$ respectively. \end{definition} When the graph $G$ is clear from context we will omit it and write $R(u,v)$. We can now define the main objects that we study.
\begin{definition}[Resistance Sparsifier] Let $G(V,E,w)$ be a weighted graph, and let $\epsilon\in(0,1)$. A \emph{$(1+\epsilon)$-resistance sparsifier} for $G$ is a subgraph $H(V,E',w')$ with reweighted edges such that $ (1-\epsilon) R_H(u,v) \leq R_G(u,v) \leq (1 + \epsilon)R_H(u,v)$, for all $u,v \in V$. \end{definition}
It will turn out that in order to understand resistance sparsifiers, we need to use expansion properties.
\begin{definition}[Graph Expansion] The \emph{edge-expansion} (also known as the \emph{Cheeger constant}) of a weighted graph $G(V,E,w)$ is \[
\phi(G) \eqdef
\min \left\{ \frac{w(S, \bar S)}{|S|} :
S\subset V,\ 0<|S|\leq|V|/2 \right\} , \] where $w(S, \bar S)$ denotes the total weight of edges with exactly one endpoint in $S\subset V$. The \emph{spectral expansion} of $G$, denoted $\lambda_2(G)$, is the second-smallest eigenvalue of the graph's normalized Laplacian. \end{definition}
Our main result is the following. Throughout this paper, ``efficiently'' means in randomized polynomial time.
\begin{theorem}\label{thm:main_resistance} Fix $\beta,\gamma>0$, let $n$ be sufficiently large, and ${1}/{n^{0.99}}<\epsilon<1$. Every $D$-regular graph $G$ on $n$ nodes with $D\geq\beta n$ and $\phi(G)\geq\gamma D$ contains (as a subgraph) a $(1+\epsilon)$-resistance sparsifier with at most $\epsilon^{-1}n(\log n)^{O(1/\beta\gamma^2)}$ edges, and it can be found efficiently. \end{theorem}
While dense regular expanders may seem like a simple case, even this special case requires significant technical work. The most obvious idea, of sparsifying through random sampling, does not work --- selecting each edge of $G$ uniformly at random with probability $\tO(1/(D\epsilon))$ (the right probability for achieving a subgraph with $\tO(n/\epsilon)$ edges) need not yield a $(1+\epsilon)$-resistance sparsifier. Intuitively, this is because the variance of independent random sampling is too large (see Theorem~\ref{thm:von_luxburg} for the precise effect), and the easiest setting to see this is the case of sparsifying the complete graph. If we sparsify through independent random sampling, then to get a $(1+\epsilon)$-resistance sparsifier requires picking each edge independently with probability at least $1/(\epsilon^2 n)$, and we end up with $n/\epsilon^2$ edges. To beat this, we need to use correlated sampling. More specifically, it turns out that a random $O(1/\epsilon)$-regular graph is a $(1+\epsilon)$-resistance sparsifier of the complete graph, despite not being a $(1+\epsilon)$-spectral sparsifier. So instead of sampling edges independently (the natural approach, and in fact the approach used to construct spectral sparsifiers by Spielman and Srivastava~\cite{SS11}), we need to sample a random regular graph.
In order to prove Theorem~\ref{thm:main_resistance}, we actually need to generalize this approach beyond the complete graph. But what is the natural generalization of a random regular graph when the graph we start with is not the complete graph? It turns out that what we need is an expander, which is sparse but maintains regularity of its degrees. This motivates our main structural result, that every dense regular expander contains a sparse regular expander (as a subgraph). This can be seen as a type of sparsification result that retains regularity.
\begin{theorem}\label{thm:main} Fix $\beta,\gamma>0$ and let $n$ be sufficiently large. Every $D$-regular graph $G$ on $n$ nodes with $D\geq\beta n$ and $\phi(G)\geq\gamma D$ contains a weighted $d$-regular subgraph $H$ with $d=(\log n)^{O(1/\beta\gamma^2)}$ and $\phi(H)\geq\frac{1}{3}$. All edge weights in $H$ are in $\{1,2\}$, and $H$ can be found efficiently. \end{theorem}
To prove this theorem, we analyze a modified version of the cut-matching game of Khandekar, Rao, and Vazirani~\cite{KRV09}. This game has been used in the past to construct expander graphs, but in order to use it for Theorem~\ref{thm:main} we need to generalize beyond matchings, and also show how to turn the graphs it creates (which are not necessarily subgraphs of $G$) into subgraphs of $G$.
The expansion requirement for $G$ in \Cref{thm:main} is equivalent to $\lambda_2(G)=\Omega(1)$, when $\beta$ and $\gamma$ are viewed as absolute constants. We note that $H$ is a much weaker expander, satisfying only $\lambda_2(H)=\Omega(1/\mathrm{polylog}(n))$, but this is nonetheless sufficient for \Cref{thm:main_resistance}. Also, $H$ is regular in weighted degrees. For completeness we give a variant of \Cref{thm:main} that achieves an unweighted $H$ by requiring stronger expansion from $G$, but this is not necessary for our application to resistance sparsifiers, which anyway involves reweighting the edges.
\begin{theorem}\label{thm:unweighted} For every $\beta>0$ there is $0<\gamma<1$ such the following holds for sufficiently large $n$. Every $D$-regular graph $G$ on $n$ nodes with $D\geq\beta n$ and $\phi(G)\geq\gamma D$ contains an (unweighted) $d$-regular subgraph $H$ with $d=(\log n)^{O(1/\beta\gamma)}$ and $\phi(H)\geq\frac{1}{3}$, and it can be found efficiently. \end{theorem}
The algorithm underlying Theorems~\ref{thm:main_resistance},~\ref{thm:main} and~\ref{thm:unweighted} turns out to be quite straightforward: decompose the host graph into disjoint perfect matchings or Hamiltonian cycles (which are ``atomic'' regular components), and subsample a random subset of them of size $d$ to form the target subgraph. However, since the decomposition leads to large dependencies between inclusion of different edges in the subgraph, it is unclear how to approach this algorithm with direct probabilistic analysis. Instead, our analysis uses the adaptive framework of~\cite{KRV09} to quantify the effect of gradually adding random matching/cycles from the decomposition to the subgraph.
\subsection{Related Work} \label{sec:related}
The line of work most directly related to resistance sparsifiers is the construction of spectral sparsifiers. This was initiated by Spielman and Teng~\cite{ST04a}, and was later pushed to its limits by Spielman and Teng~\cite{ST11}, Spielman and Srivastava~\cite{SS11}, and Batson, Spielman, and Srivastava~\cite{BSS12}, who finally proved that every graph has a $(1+\epsilon)$-spectral sparsifier with $O(n/\epsilon^2)$ edges and that this bound is tight (see also \cite{AKW14}).
The approach by Spielman and Srivastava~\cite{SS11} is particularly closely related to our work. They construct almost-optimal spectral sparsifiers (a logarithmic factor worse than~\cite{BSS12}) by sampling each edge independently with probability proportional to the effective resistance between the endpoints. This method naturally leads us to try the same thing for resistance sparsification, but as discussed, independent random sampling (even based on the effective resistances) cannot give improved resistance sparsifiers. Interestingly, in order to make their algorithm extremely efficient they needed a way to estimate effective resistances very quickly, so along the way they showed how to create a sketch of size $O(n\log n/\epsilon^2)$ from which every resistance distance can be read off in $O(\log n)$ time (essentially through an $\ell_2$-squared embedding and a Johnson-Lindenstrauss dimension reduction).
\section{Sparse Regular Expanding Subgraphs} \label{sec:subgraphs}
In this section we prove \Cref{thm:main}, building towards it in stages. Our starting point is the Cut-Matching game of Khandekar, Rao and Vazirani (KRV) \cite{KRV09}, which is a framework to constructing sparse expanders by iteratively adding perfect matchings across adaptively chosen bisections of the vertex set. The resulting graph $H$ is regular, as it is the union of perfect matchings, and if the matchings are contained in the input graph $G$ then $H$ is furthermore a subgraph of $G$, as desired. In \Cref{sec:cut-matching}, we employ this approach to prove \Cref{thm:main} in the case $D/n = \frac{3}{4}+\Omega(1)$.
To handle smaller $D$, we observe that the perfect matchings in the KRV game can be replaced with a more general structure that we call a \emph{weave}, defined as a set of edges where for every vertex at least one incident edge crosses the given bisection. To ensure that $H$ is regular (all vertices have the same degree), we would like the weaves to be regular. We thus decompose the input graph to disjoint regular elements -- either perfect matchings or Hamiltonian cycles -- and use them as building blocks to construct regular weaves. Leveraging the fact that for some bisections, $G$ contains no perfect matching but does contain a weave, we use this extension in \Cref{sec:Cut-Weave} to handle the case $D/n = \frac{1}{2}+\Omega(1)$.
Finally, for the general case $D/n=\Omega(1)$, we need to handle a graph $G$ that contains no weave on some bisections. The main portion of our proof constructs a weave that is not contained in $G$, but rather embeds in $G$ with small (polylogarithmic) congestion. Repeating this step sufficiently many times as required by the KRV game, yields a subgraph $H$ as desired.
\paragraph*{Notation and terminology.} For a regular graph $G$, we denote $\mathrm{deg}(G)$ the degree of each vertex. We say that a graph $H$ is an \emph{edge-expander} if $\phi(H)>\frac{1}{3}$. A \emph{bisection} of a vertex set of size $n$ is a partition $(S,\bar S)$ with equal sizes $\frac{1}{2}n$ if $n$ is even, or with sizes $\lfloor\frac{1}{2}n\rfloor$ and $\lceil\frac{1}{2}n\rceil$ if $n$ is odd.
\subsection{The Cut-Matching Game} \label{sec:cut-matching} Khandekar, Rao and Vazirani \cite{KRV09} described the following game between two players. Start with an empty graph (no edges) $H$ on a vertex set of even size $n$. In each round, the \emph{cut player} chooses a bisection, and the \emph{matching player} answers with a perfect matching across the bisection. The game ends when $H$ is an edge-expander. Informally, the goal of the cut player is to reach this as soon as possible, and that of the matching player is to delay the game's ending.
\begin{theorem}[\cite{KRV09,KKOV07}]\label{thm:krv} The cut player has an efficiently computable strategy that wins (i.e., is guaranteed to end the game) within $O(\log^2n)$ rounds, and a non-efficient strategy that wins within $O(\log n)$ rounds. \end{theorem}
The following result illustrates the use of the KRV framework in our setting. \begin{theorem}\label{thm:d34n} Let $\delta>0$ and let $n$ be even and sufficiently large ($n\ge n_0(\delta)$). Then every $n$-vertex graph $G(V,E)$ with minimum degree $D\geq(\frac{3}{4}+\delta)n$ contains an edge-expander $H$ that is $d$-regular for $d=O(\log n)$, and also an efficiently computable edge-expander $H'$ that is a $d'$-regular for $d'=O(\log^2n)$. \end{theorem}
\begin{proof} Apply the Cut-Matching game on $V$ with the following player strategies. For the cut player, execute the efficient strategy from \Cref{thm:krv} that wins within $O(\log^2n)$ rounds. For the matching player, given a bisection $(S,\bar S)$, consider the bipartite subgraph $G[S,\bar S]$ of $G$ induced by $(S,\bar S)$. Each vertex in $S$ has in $G$ at least $D\geq\frac{3}{4}n$ neighbors, but at most $\frac{1}{2}n-1$ of them are in $S$, and the rest must be in $\bar S$, which implies that $G[S,\bar S]$ has minimum degree $\geq\frac{1}{4}n$. Hence, as a simple consequence of Hall's theorem (see \Cref{prp:hall_consequence}), it contains a perfect matching that can be efficiently found. The matching player returns this matching as his answer. We then remove this matching from $G$ before proceeding to the next round, to ensure that different iterations find disjoint matchings. The slackness parameter $\delta$ (and $n$ being sufficiently large) ensure that the minimum degree of $G$ does not fall below $\frac{3}{4}n$ during the $O(\log^2n)$ iterations, so the above argument holds in all rounds.
The game ends with an edge-expander $H'$ which is a disjoint union of $d'=O(\log^2n)$ perfect matchings contained in $G$, and hence is a $d'$-regular subgraph of $G$, as required. To obtain the graph $H$, apply the same reasoning but using the non-efficient strategy from \Cref{thm:krv} that wins within $O(\log n)$ rounds. \end{proof}
\subsection{The Cut-Weave Game} \label{sec:Cut-Weave}
For values of $D$ below $\frac{3}{4}n$, we can no longer guarantee that every bisection in $G$ admits a perfect matching. However, we observe that one can allow the matching player a wider range of strategies while retaining the ability of the cut player to win within a small number of rounds.
\begin{definition}[weave] Given a bisection $(S,\bar S)$ of a vertex set $V$, a \emph{weave} on $(S,\bar S)$ is a subgraph in which every node has an incident edge crossing $(S,\bar S)$. \end{definition}
\begin{definition}[Cut-Weave Game] The \emph{Cut-Weave game} with parameter $r$ is the following game of two players. Start with a graph $H$ on a vertex set of size $n$ and no edges. In each round, the \emph{cut player} chooses a bisection of the vertex set, and the \emph{weave player} answers with an $r$-regular weave on the bisection. The edges of the weave are added to $H$. \end{definition} Note that the $r=1$ case is the original Cut-Matching game (when $n$ is even). The following theorem is an extension of \Cref{thm:krv}. For clarity of presentation, its proof is deferred to \Cref{sec:krv_extended_proof}.
\begin{theorem}\label{thm:krv_extended} In the Cut-Weave game with parameter $r$, the cut player has an efficient strategy that wins within $O(r\log^2n)$ rounds, and furthermore ensures $\phi(H)\geq\frac{1}{2}r$. \end{theorem} In order to construct regular weaves, we employ a decomposition of $G$ into disjoint Hamiltonian cycles. The following theorem was proven by Perkovic and Reed \cite{PR97}, and recently extended by Csaba, K\"{u}hn, Lo, Osthus and Treglown~\cite{CKLOT14}.
\begin{theorem}\label{thm:Hamiltonian_decomposition} Let $\delta>0$. Every $D$-regular graph $G$ on $n$ nodes with $D\geq(\frac{1}{2}+\delta)n$, admits a decomposition of its edges into $\lfloor\frac{1}{2}D\rfloor$ Hamiltonian cycles and possibly one perfect matching (if $D$ is odd). Furthermore, the decomposition can be found efficiently. \end{theorem}
Now we can use the Cut-Weave framework to make another step towards \Cref{thm:main}. \begin{theorem}\label{thm:d12n} Let $\delta>0$ and let $n$ be sufficiently large. Then every $n$-vertex graph $G(V,E)$ with minimum degree $D\geq(\frac{1}{2}+\delta)n$ contains a $d$-regular edge-expander $H$ with $d=O(\log^3n)$, which furthermore can be efficiently found. \end{theorem}
\begin{proof} We simulate the Cut-Weave game with $r=16\delta^{-1}\log n$. The proof is the same as \Cref{thm:d34n}, only instead of a perfect matching we need to construct an $r$-regular weave across a given bisection $(S,\bar S)$. We apply \Cref{thm:Hamiltonian_decomposition} to obtain a Hamiltonian decomposition of $G$. For simplicity, if $D$ is odd we discard the one perfect matching from \Cref{thm:Hamiltonian_decomposition}. Let $\mathcal C$ be the collection of Hamiltonian cycles in the decomposition.
Suppose w.l.o.g.~$|S|=\lceil\frac{1}{2}n\rceil$. Every $v\in S$ has at most $|S|-1\leq\frac{1}{2}n$ neighbors in $S$, and hence at least $\delta n$ incident edges crossing to $\bar S$. We set up a Set-Cover instance of the cycles $\mathcal C$ against the nodes in $S$, where a node $v$ is considered covered by a cycle $C$ is $v$ has an incident edge crossing to $\bar S$, that belongs to $C$. This is a dense instance: since each cycle visits $v$ only twice, $v$ can be covered by $\frac{1}{2}\delta n$ cycles. Therefore, $4\delta^{-1}\log n$ randomly chosen cycles form a cover with high probability (see \Cref{prp:dense_set_cover} for details). We then repeat the same procedure to cover the nodes on side $\bar S$. The result is a collection of $8\delta^{-1}\log n=\frac{1}{2}r$ disjoint Hamiltonian cycles, whose union forms an $r$-regular weave on $(S,\bar S)$, which we return as the answer of the weave player. Applying \Cref{thm:krv_extended} with $r=O(\log n)$ concludes the proof of \Cref{thm:d12n}. \end{proof}
Observe that in the proof of Theorem~\ref{thm:d12n}, the weave player is in fact oblivious to the queries of the cut player: all she does is sample random cycles from $\mathcal C$, and the output subgraph $H$ is the union of those cycles. Therefore, in order to construct $H$, it is sufficient to decompose $G$ into disjoint Hamiltonian cycles, and choose a random subset of size $O(\log^3n)$ of them. There is no need to actually simulate the cut player, and in particular, the proof does not require her strategy (from Theorem~\ref{thm:krv_extended}) to be efficient.
\subsection{Reduction to Double Cover}\label{sec:double_cover} We now begin to address the full range of parameters stated in \Cref{thm:main}. In this range there is no Hamiltonian decomposition theorem (or a result of similar flavor) that we are aware of, so we replace it with a basic argument which incurs edge weights $w:V\times V\rightarrow\{0,1,2\}$ in the target subgraph $H$, as well as a loss in its degree.
Given the input graph $G(V,E)$, we construct its \emph{double cover}, which is the bipartite graph $G''(V'',E'')$ defined by $V''=V\times\{0,1\}$ and $E'' = \{((v,0)(u,1)) : vu\in E \}$. It is easily seen that if $G$ is $D$-regular then so is $G''$, and since $|V''|=2|V|$ we have $D\geq\frac{1}{2}\beta|V''|$. It also well known that $\lambda_2(G)=\lambda_2(G'')$, and therefore by the discrete Cheeger inequalities, \begin{equation*}
\phi(G'') \geq
\tfrac{1}{2}\lambda_2(G'')D =
\tfrac{1}{2}\lambda_2(G)D \geq
\tfrac{1}{2}\gamma^2(G)D. \end{equation*}
$G''$ satisfies the requirements of \Cref{thm:main} with $\beta''=\frac{1}{2}\beta$ and $\gamma''=\tfrac{1}{2}\gamma^2$. Suppose we find in $G''$ a $d$-regular edge-expander $H''$ with $d = (\log n)^{O(1/\beta''\gamma'')} = (\log n)^{O(1/\beta\gamma^2)}$. We carry it over to a subgraph $H$ of $G$, by including each edge $uv\in E$ in $H$ with weight $\left|\{(v,0)(u,1),(u,0)(v,1)\} \cap E(H'')\right|$, where $E(H'')$ denotes the edge set of $H''$. Each edge then appears in $H$ with weight either $1$ or $2$ (or $0$, which means it is not present in $H$). It can be easily checked that $H$ is $d$-regular in weighted degrees, and $\phi(H)\geq\frac{1}{2}\phi(H'')$. Therefore $H$ is a suitable target subgraph for \Cref{thm:main}.
The above reduction allows us to restrict our attention to regular bipartite graphs $G$, but on the other hand we are forced to look for a subgraph $H$ which is unweighted and $d$-regular with $d=(\log n)^{O(1/\beta\gamma)}$ (which is tighter than stated in \Cref{thm:main}). We take this approach in the remainder of the proof. The gain is that such $G$ admits a decomposition into disjoint perfect matchings, which can be efficiently found, as a direct consequence of Hall's theorem. We will use this fact where we have previously used \Cref{thm:Hamiltonian_decomposition}.
\subsection{Constructing an Embedded Weave} We now get to the main technical part of the proof. Given a bisection $(S,\bar S)$ queried by the cut player, we need to construct an $r$-regular weave on the bisection, where this time we choose $r=(\log n)^{O(1/\beta\gamma)}$. Unlike the proof of \Cref{thm:d12n}, we cannot hope to find a weave which is a subgraph of $G$, since if $D<\frac{1}{2}n$, any bisection in which one side contains some vertex and all its neighbors would not admit a weave in $G$. Instead, we aim for a weave which embeds into $G$ with polylogarithmic congestion.
We will use two types of graph operations: The \emph{union} of two graphs on the same vertex set $V$ is obtained by simply taking the set union of their edge sets, whereas the \emph{sum} of the two graphs is given by keeping parallel edges if they appear in both graphs. We now construct the weave in 4 steps.
\paragraph*{Step 1.} Fix $\mu=\frac{\beta\gamma^2}{4}$. We partition the entire vertex set $V$ into subsets $S_0,S_1,\ldots,S_t$ by the following process: \begin{enumerate}
\item Set $S_0 \leftarrow \bar S$ and $T \leftarrow S$.
\item While $T\neq\emptyset$, take $S_i\subseteq T$ to be the subset of nodes with at least $\mu D$ neighbors in $S_{i-1}$, and set $T \rightarrow T\setminus S_i$. \end{enumerate} \begin{mylemma} The process terminates after $t\leq\frac{2}{\beta\gamma}$ iterations. \end{mylemma} \begin{proof}
Consider an iteration $i\leq\frac{2}{\beta\gamma}$ that ends with $T\neq\emptyset$. Denote $\bar T=V\setminus T=\cup_{j=0}^iS_j$. By the hypothesis $\phi(G)\geq\gamma D$ we have at least $\gamma D|T|$ edges crossing from $T$ to $\bar T$, so by averaging over the nodes in $T$, there is $v\in T$ with $\gamma D$ neighbors in $\bar T$. For every $j<i$, $v$ must have less than $\mu D$ neighbors in $S_j$, or it would already belong to $S_{j+1}\subseteq\bar T$. Summing over $j=0,\ldots,i-1$, we see that $v$ has less than $i\mu D\leq\frac{1}{2}\gamma D$ neighbors in $\bar T\setminus S_i$, so at least $\frac{1}{2}\gamma D$ neighbors in $S_i$. This implies $|S_i|\geq\frac{1}{2}\gamma D$. We have shown that each of the first $\frac{2}{\beta\gamma}$ iterations either terminates the process or removes $\frac{1}{2}\gamma D\geq\frac{1}{2}\gamma\beta n$ nodes from $T$, so after $\frac{2}{\beta\gamma}$ iterations we must have $T=\emptyset$. \end{proof}
\paragraph*{Step 2.} By \Cref{sec:double_cover} we have a decomposition of all the edges in $G$ into a collection $\mathcal M$ of $D$ disjoint perfect matchings. For every $i=1,\ldots,t$, we now cover the nodes in $S_i$ with perfect matchings, similar to the proof of \Cref{thm:d12n}. A node $v\in S_i$ is considered covered by a matching if $v$ has an incident edge with the other endpoint in $S_{i-1}$, and that edge lies on the matching. Since $v$ has $\mu D$ incident edges crossing to $S_{i-1}$, and each matching touches $v$ with at most one edge, we have $\mu D$ matchings that can cover $v$. Therefore $k=\frac{1}{\mu}\log n$ randomly chosen matchings from $\mathcal M$ form a cover of $S_i$ (see \Cref{prp:dense_set_cover}), which we denote as $K_i$. Thus, for each $i$ we have a subgraph $K_i$ which is $k$-regular, such that each node in $S_i$ has an incident edge in $K_i$ with the other endpoint in $S_{i-1}$. Denote henceforth \[ K = \cup_{i=1}^t K_i. \] Note that $K$ is a regular subgraph of $G$, since it is a union of disjoint perfect matchings from $\mathcal M$, and $\mathrm{deg}(K)\leq kt$.
\paragraph*{Step 3.} In this step we construct a graph $K^*$ from the subgraph $K$. As discussed, $K^*$ will not be a subgraph of $G$ but will embed into it with reasonable congestion. Let us formally define the notion of graph embedding that we will be using. \begin{definition}[Graph embedding with congestion] Let $G(V,E)$ and $G'(V,E')$ be graphs on the same vertex set. Denote by $\mathcal{P}_G$ the set of simple paths in $G$. An \emph{embedding} of $G'$ into $G$ is a map $f:E'\rightarrow\mathcal{P}_G$ such that every edge in $G'$ is mapped to a path in $G$ with the same endpoints.
The \emph{congestion} of $f$ on an edge $e\in E$ is $\mathrm{cng}_f(e):=|e'\in E':e\in f(e')|$. The congestion of $f$ is $\mathrm{cng}(f):=\max_{e\in E}\mathrm{cng}_f(e)$. We say that $G'$ embeds into $G$ with congestion $c$ if there is an embedding $f$ with $\mathrm{cng}(f)=c$. \end{definition} The following claim is a simple observation and we omit its proof. \begin{claim} If $G'$ embeds into $G$ with congestion $c$, then $\phi(G)\geq\frac{1}{c}\phi(G')$. \end{claim}
We generate $K^*$ with the following inductive construction.
\begin{mylemma}\label{clm:main_construction} Let $\rho_0=c_0=0$. We can efficiently construct subgraphs $K_1^*,\ldots,K_t^*$ (which may have parallel edges and self-loops), such that for every $i=1,\ldots,t$, \begin{enumerate}
\item $K_i^*$ is $\rho_i$-regular, where $\rho_i=k(1+\rho_{i-1})$.
\item $K_i^*$ embeds into $K$ with congestion $c_i$, where $c_i=1+kc_{i-1}$.
\item Every $v\in S_i$ has an incident edge in $K_i^*$ with the other endpoint in $S_0$. \end{enumerate} \end{mylemma} \begin{proof} We go by induction on $i$. For the base case $i=1$ we simply set $K_1^*=K_1$. The claim holds as we recall that \begin{enumerate}
\item $K_1$ is $k$-regular.
\item $K_1$ is a subgraph of $K$, hence it embeds into $K$ with congestion $1=1+kc_0$.
\item By Step 2, every $v\in S_1$ has an incident edge in $K_1$ crossing to $S_0$. \end{enumerate}
We turn to the inductive step $i>1$. Start with a graph $K'$ which is a fresh copy of $K_{i-1}^*$, with each edge duplicated into $k$ parallel edges. By induction, $K'$ is $(k\rho_{i-1})$-regular. Now sum $K_i$ into $K'$; recall this means keeping parallel edges instead of unifying them. Since $K_i$ is $k$-regular, $K'$ is $\rho_i$-regular.
Let $v\in S_i$. By Step 2, there is an edge $vw\in K_i$ such that $w\in S_{i-1}$. By induction, there is an edge $wu\in K_{i-1}^*$ such that $u\in S_0$. Note that both edges $vw$ and $wu$ are present in $K'$. Perform the following crossing operation on $K'$: Remove the edges $vw$ and $wu$, and add an edge $vu$ and a self-loop on $w$.
Perform this on every $v\in S_i$. The resulting graph is $K_i^*$. We need to show that it is well defined in the following sense: we might be using the same edge $wu$ for several $v$'s, and we need to make sure each $wu$ appears sufficiently many times, to be removed in all the crossing operations in which it is needed. Indeed, we recall that $K_i$ is the union of $k$ disjoint perfect matchings, and therefore each $w\in S_{i-1}$ has at most $k$ edges in $K_i$ incoming from $S_i$. Since $K'$ contains $k$ copies of each edge $wu$, we have enough copies to be removed in all necessary crossing operations.
Lastly we show that $K_i^*$ satisfies all the required properties. \begin{enumerate}
\item Since $K'$ was $\rho_i$-regular, and the switching operations do not effect vertex degrees, we see that $K_i^*$ is $\rho_i$-regular.
\item Each edge $vu$ in $K_i^*$ which is not original from $K'$, corresponds to a path (of length 2) in $K'$ that was removed upon adding that edge; hence $K_i^*$ embeds into $K'$ with congestion $1$. $K'$ is the sum of $K_i$, which is a subgraph of $K$, and $k$ copies of $K_{i-1}^*$, which by induction embeds into $K$ with congestion $c_{i-1}$. Hence $K'$ embeds into $K$ with congestion $1+kc_{i-1}=c_i$. Therefore, $K_i^*$ embeds into $K$ with congestion $c_i$.
\item For every $v\in S_i$, we added to $K_i^*$ an edge $vu$ such that $u\in S_0$. \end{enumerate} \end{proof}
We now take $K^* = \sum_{i=1}^tK_i^*$. By \Cref{clm:main_construction}, $K^*$ is $(\sum_{i=1}^t\rho_i)$-regular, embeds into $K$ with congestion $\sum_{i=1}^tc_i$, and every $v\in S$ has an incident edge $vu\in K^*$ such that $u\in \bar S$. (To see why the latter point holds, recall that we put $\bar S=S_0$.)
\paragraph*{Step 4.} In this final step we repeat Steps 1--3, only with the roles of $S$ and $\bar S$ interchanged. This results in a subgraph $\bar K$ of $G$ which is $kt$-regular, and a graph $\bar K^*$ which is $(\sum_{i=1}^t\rho_i)$-regular, embeds into $\bar K$ with congestion $\sum_{i=1}^tc_i$, and every $v\in\bar S$ has an incident edge $vu\in \bar K^*$ such that $u\in S$.
Our final weave is $K^*+\bar K^*$. By the above it is clearly a weave, and moreover it is $r$-regular and embeds into $K\cup\bar K$ (and hence into $G$, which contains $K\cup\bar K$) with congestion $c$, where $r = 2\sum_{i=1}^t\rho_i$ and $c = 2\sum_{i=1}^tc_i$. By inspecting the recurrence formulas from \Cref{clm:main_construction}, in which $\rho_i$ and $c_i$ were defined, we can bound $\rho_i,c_i \leq (2k)^i \leq (2k)^t$ for every $i$, and hence $r,c \leq 2t(2k)^t$. Recalling that $t\leq\frac{2}{\beta\gamma}+1$ and $k=\frac{1}{\mu}\log n=O(\log n)$, we find $r,c\leq(\log n)^{O(1/\beta\gamma)}$.
\subsection{Completing the Proof of \Cref{thm:main}} We play the Cut-Weave game for $L$ rounds, where $L=O(r\log^2n)$ is the number of rounds required by the efficient strategy in \Cref{thm:krv_extended}. For each round $\ell=1,\ldots,L$, we constructed above an $r$-regular weave $W^*_\ell=K^*+\bar K^*$, that embeds into a subgraph $W_\ell=K\cup\bar K$ of $G$ with congestion $c$. Let $H=\cup_{\ell=1}^L W_\ell$ and $H^*=\sum_{1=\ell}^L W^*_\ell$. Then $H$ is a union of disjoint perfect matchings from $\mathcal M$, and hence regular. Moreover $\mathrm{deg}(H)\leq 2ktL$, since $H$ is the union of $L$ subgraphs $\{W_\ell\}_{\ell=1}^L$, where each $W_\ell$ is a union $W_\ell$ of two $kt$-regular graphs $K,\bar K$.
Now consider $H^*$. Since each $W^*_\ell$ embeds into $W_\ell$ with congestion $c$, we see that $H^*$ embeds into $H$ with congestion (at most) $cL$. By \Cref{thm:krv_extended} we have $\phi(H^*)\geq\frac{1}{2}r$, and this now implies $\phi(H)\geq\frac{r}{2cL}$.
Recalling the parameters: \[ t = O(1) \;\; ; \;\; k = O(\log n) \;\; ; \;\; r,c=O(\log^{O(1/\beta\gamma)}n) \;\; ; \;\; L=O(r\log^2n), \] we see that $H$ is a $d$-regular subgraph of $d=(\log n)^{O(1/\beta\gamma)}$ and $\phi(H)\geq1/(\log n)^{O(1/\beta\gamma)}$. We can now repeat this Cut-Weave game $(\log n)^{O(1/\beta\gamma)}$ disjoint times, because if each time we remove the graph $H$ we have found, we decrease the degree $D=\beta n$ of each node by only $\polylog{n}$. By repeating the game this many times and taking the union of the disjoint resulting subgraphs, we find a regular subgraph $H$ of $G$ with $\mathrm{deg}(H)=(\log n)^{O(1/\beta\gamma)}$ and $\phi(H)\geq1$. Lastly recall that unfolding the reduction from \Cref{sec:double_cover} puts on $H$ edge weight in $\{1,2\}$, and weakens the degree bound to $\mathrm{deg}(H)=(\log n)^{O(1/\beta\gamma^2)}$. This completes the proof of \Cref{thm:main}.
Regarding the algorithm to construct $H$, the observation made after Theorem~\ref{thm:d12n} applies here as well. The weave player's strategy is oblivious to the queries of the cut player, since she just samples random matchings from $\mathcal M$ to form $H$. The cut player strategy does not actually need to be simulated, nor the graphs $K^*$ need to actually be constructed. The algorithm to construct $H$ then amounts to the following: Construct the double cover graph $G"$ of $G$; decompose $G"$ into disjoint perfect matchings; choose a random subset of $(\log n)^{O(1/\beta\gamma^2)}$ of them to form a subgraph $H"$ of $G"$; and unfold the double cover construction to obtain the final subgraph $H$ from $H"$.
\subsection{Proof of \Cref{thm:unweighted}} The theorem follows from replacing the reduction to the double cover in \Cref{sec:double_cover} by a Hamiltonian decomposition result that holds for this stronger expansion requirement, due to K\"{u}hn and Osthus \cite[Theorem 1.11]{KO14}. The trade-off between $\beta$ and $\gamma$ is inherited from their theorem (in which it is unspecified). Circumventing \Cref{sec:double_cover} also improves the dependence of $d$ on $\gamma$. The proof of \Cref{thm:unweighted} is otherwise identical to the proof of \Cref{thm:main}.
\section{Proof of the Cut-Weave Theorem}\label{sec:krv_extended_proof} Recall the setting of the Cut-Weave game with parameter $r$: The game starts with a graph $G_0$ on $n$ vertices and without edges. In each round $t=1,2,\ldots$, the weave player queries a bisection of the vertex set, and the weave player answers with an $r$-regular weave $H_t$ on that bisection. The weave is then unified into the graph, putting $G_t=G_{t-1}\cup H_t$.
We now prove \Cref{thm:krv_extended} by an adaptation of the analysis from \cite{KRV09}. The main change is in \Cref{lmm:krv_potential_reduction}.
For each step $t$, let $M_t$ be the matrix describing one step of the natural lazy random walk on $H_t$: W.p.~$\frac{1}{2}$ stay in the current vertex, and with probability $\frac{1}{2r}$ move to a neighbor. The cut player strategy is as follows: \begin{itemize}
\item Choose a random unit vector $z\perp\mathbf1$ in $\mathbb R^n$.
\item Compute $u=M_tM_{t-1}\ldots M_1z$.
\item Output the bisection $(S,\ldots S)$ where $S$ is the $\lfloor n/2 \rfloor$ vertices with smallest values in $u$. \end{itemize}
Let us analyze the game with this strategy. In the graph $G_t$ (which equals $\cup_{t'=1}^tH_{t'}$), we consider the following $t$-steps random walk: Take one (lazy) step on $H_1$, then on $H_2$, and so on until $H_t$. In other words, the walk is given by applying sequentially $M_1$, then $M_2$, and so on.
Let $P_{ij}(t)$ denote the probability to go from node $j$ to node $i$ within $t$ steps. Let $P_i$ denote the vector $(P_{i1},P_{i2},\ldots,P_{ji})$. We use the following potential function: \[ \Psi(t) = \sum_{i,j\in V}(P_{ij}-1/n)^2 = \sum_{i=1}^n\norm{P_i-\mathbf1/n}_2^2. \]
\begin{mylemma} For every $t$ and every $i\in V$, we have $\sum_{j\in V}P_{ij}(t)=1$. \end{mylemma} \begin{proof} By induction on $t$: It holds initially, and in each step $t$, vertex $i$ trades exactly half of its total present probability with its neighbors in $H_t$. (Note that this relies on the fact that $H_t$ is regular.) \end{proof}
\begin{mylemma}\label{lmm:krv_potential_to_expansion} If $\Psi(t)<1/4n^2$ then $G=G_t$ has edge-expansion at least $\frac{1}{2}r$. \end{mylemma} \begin{proof} If $\Psi(t)<1/4n^2$ then $P_{ji}(t)\geq\frac{1}{2n}$ for all $i,j\in V$. Hence the graph $K_t$ on $V$, in which each edge $ij$ has weight $P_{ji}(t)+P_{ij}(t)$, has edge-expansion $\frac{1}{2}$. We finish by showing that $K_t$ embeds into $G_t$ with congestion $1/r$. Proof by induction: Consider the transition from $G_{t-1}$ to $G_t$, which is unifying $H_t$ into $G_{t-1}$. Let $i,j\in V$ be connected with an edge in $H_t$, and let $k$ be any vertex. In the transition from $K_{t-1}$ to $K_t$, we need to ship $\frac{1}{2r}$ of the type-$k$ probability in $i$ (namely $\frac{1}{2r}P_{ik}$) to $j$, and similarly, ship $\frac{1}{2r}P_{jk}$ probability from $j$ to $i$. (The ``type-$k$'' probabiility is probability mass that was originally located in $k$.) In total, we need to ship $\frac{1}{2r}\sum_{k\in V}P_{ik}=\frac{1}{2r}$ from $i$ to $j$ and a similar amount from $j$ to $i$. In total the edge $ij$ in $H_t$ needs to support $\frac{1}{r}$ flow (of probability) in the transition, so the claim follows. \end{proof}
We turn to analyzing the change in potential in a single fixed round $t$. To simplify notation we let \[ P_{ji} = P_{ji}(t) \;\;\;\; ; \;\;\;\; Q_{ji} = P_{ji}(t+1) . \] Moreover recall we have a vector $u$ generated by the cut player in the current round: \[ u=M_tM_{t-1}\ldots M_1z. \] Denote its entries by $u_1,\ldots,u_n$. We are now adding the graph $H_{t+1}$ to $G_t$ to produce $G_{t+1}$.
\begin{mylemma} For every $i$, $u_i$ is the projection of $P_i$ on $r$, i.e.~$u_i=P_i^Tz$. \end{mylemma} \begin{proof} Fix $i$. Abbreviate $M=M_tM_{t-1}\ldots M_1\mathbf(\frac{1}{n}\mathbf1)$. If $\phi$ is any distribution on the vertices then $P_i^T\phi$ is the probability that the random walk lands in vertex $i$ after $t$ steps, meaning \begin{equation}\label{eq:krv_projection} (M\phi)_i=P_i^T\phi. \end{equation} Let $z'=\frac{1}{n\norm{z}_\infty}z$. Applying \Cref{eq:krv_projection} with $\phi=z'+\frac{1}{n}\mathbf1$ gives $(M(z'+\frac{1}{n}\mathbf1))_i=P_i^T(z'+\frac{1}{n}\mathbf1)$. Applying \Cref{eq:krv_projection} again with $\phi=\frac{1}{n}\mathbf1$ gives $(M\frac{1}{n}\mathbf1)_i=P_i^T(\frac{1}{n}\mathbf1)$ and together we get $(Mz')_i=P_i^Tz'$, which implies $u_i=(Mz)_i=P_i^Tz$. \end{proof}
\begin{mylemma}\label{lmm:krv_random_projection} With probability $1-1/n^{\Omega(1)}$ over the choice of $z$, for all pairs $i,j\in V$,
\[ \norm{P_i-P_j}_2^2 \geq \frac{n-1}{C\log n}|u_i-u_j|^2 . \] \end{mylemma} \begin{proof} Similar to \cite[Lemma 3.4]{KRV09}. \end{proof}
\begin{mylemma}\label{lmm:krv_cut_gain} Let $E(S,\bar S)$ denote the set of edges in $H_{t+1}$ that cross the bisection $(S,\bar S)$ produced by the cut player (from the vector $u$). Then,
\[ (n-1)\mathbb E\left[\sum_{ij\in E(S,\bar S)}|u_i-u_j|^2\right] \geq \Psi(t) . \] \end{mylemma} \begin{proof} Denote by $\text{deg}_{(S,\bar S)}(i)$ the number of edges in $E(S,\bar S)$ incident to vertex $i$. Note that $\text{deg}_{(S,\bar S)}(i)\geq 1$ for every $i\in V$, since $H_{t+1}$ is a weave on $(S,\bar S)$. Recall that $S$ contains the vertices with smallest entries in $u$. Hence there is a number $\eta\in\mathbb R$ such that $i\leq\eta\leq j$ for each edge $ij\in E(S,\bar S)$. Hence, \begin{align*}
\sum_{ij\in E(S,\bar S)}|u_i-u_j|^2 & \geq \sum_{ij\in E(S,\bar S)}((u_i-\eta)^2+(\eta-u_j)^2) \\ & = \sum_{i\in V}\text{deg}_{(S,\bar S)}(i)(u_i-\eta)^2 \\ & \geq \sum_{i\in V}(u_i-\eta)^2 \\ & = \sum_{i\in V}u_i^2- 2\eta\sum_{i\in V}u_i+n\eta^2 \\ & \geq \sum_{i\in V}u_i^2, \end{align*} where the last equality is by noting that $z\perp\mathbf1$, hence $u\perp\mathbf1$, hence $\sum_{i}u_i=0$.
Next, since $u_i=P_i^Tz$ and $z\perp\mathbf1$ we have $u_i=(P_i-\mathbf1/n)^Tz$. Hence $u_i$ is the projection of $P_i-\mathbf1/n$ on $z$. By properties of random projections we have $\mathbb E[u_i^2]=\frac{1}{n-1}\norm{P_i-\mathbf1/n}_2^2$ (see details in \cite{KRV09}), hence \[
\mathbb E\left[\sum_{i\in V}u_i^2\right] =
\frac{1}{n-1}\sum_{i\in V}\norm{P_i-\mathbf1/n}_2^2 =
\frac{1}{n-1}\Psi(t), \] and the lemma follows from combining this with the above. \end{proof}
\begin{mylemma}\label{lmm:krv_potential_reduction} Let $E_{t+1}$ denote the edge set of $H_{t+1}$. The potential reduction is \[ \Psi(t)-\Psi(t+1) = \frac{1}{r}\sum_{ij\in E_{t+1}}\norm{P_i-P_j}_2^2 . \] \end{mylemma} \begin{proof} We construct from $G$ a graph $G'$ by splitting each vertex $i$ into $r$ copies $i_1,\ldots,i_r$, assigning arbitrarily one edge from the $r$ edges incident to $i$ in $E_{t+1}$ to the copies, and distributing the type-$j$ probability in $i$, for each $j$, evenly among the copies. We denote by $P_{ji_k}$ the amount of type-$j$ probability on $i_k$ before adding $E_{t+1}$ to $G'$, and by $Q_{ji_k}$ the type-$j$ probability in $i$ after adding $E_{t+1}$. Note that we have defined $P_{ji_k}=\frac{1}{r}P_{ji}$ for all $i,j\in V$ and $k\in[r]$, but for the $Q_{ji_k}$'s all we know is that $\sum_{k=1}^rQ_{ji_k}=Q_{ji}$, so $Q_{ji}$ may be distributed arbitrarily among the $Q_{ji_k}$'s. As usual $P_{i_k}$ denotes the vector with entries $P_{ji_k}$, and $Q_{i_k}$ is defined similarly.
Define the potential of $G'$ as: \[ \Psi'(t) = \sum_{i\in V}\sum_{k=1}^r\norm{P_{i_k}-\mathbf1/nr}_2^2. \] We thus have \[ \Psi(t) =
\sum_{i\in V}\norm{P_i-\mathbf1/n}_2^2 =
r\sum_{k=1}^r\sum_{i\in V}\norm{\frac{1}{r}P_i-\mathbf1/nr}_2^2 =
r\sum_{k=1}^r\sum_{i\in V}\norm{P_{i_k}-\mathbf1/nr}_2^2 =
r\Psi'(t). \] To relate $\Psi(t+1)$ to $\Psi'(t+1)$, we use the general fact that for any constants $c$ and $X$, the solution to $\min\norm{x-c\mathbf1}$ s.t.~$x\in\mathbb R^r$, $\sum_ix_i=X$ is attained on $x=\frac{X}{r}\mathbf1$. Since we have $\sum_{k=1}^rQ_{ji_k}=Q_{ji}$ for all $i,j$, we infer \begin{align*} \Psi(t+1) &= \sum_{i\in V}\norm{Q_i-\mathbf1/n}_2^2 \\ & = \sum_{i,j\in V}(Q_{ji}-1/n)^2 & \\ & = \sum_{i,j\in V}r\sum_{k=1}^r(\frac{1}{r}Q_{ji}-1/nr)^2 &\\ & \leq \sum_{i,j\in V}r\sum_{k=1}^r(Q_{ji_k}-1/nr)^2 \\ & = r\sum_{i\in V}\sum_{k=1}^r\norm{Q_{i_k}-\mathbf1/nr}_2^2 \\ & = r\Psi'(t+1). \end{align*} We have thus proven, \[ \Psi(t)-\Psi(t+1) \geq r(\Psi'(t)-\Psi'(t+1)). \] Now observe that $E_{t+1}$ is, by construction, a perfect matching on $G'$. Therefore by \cite[Lemma 3.3]{KRV09} (which the current lemma generalizes), \begin{align*}
\Psi'(t)-\Psi'(t+1) & \geq \sum_{i_k,j_{k'}\in E_{t+1}}\norm{P_{i_k}-P_{j_{k'}}}_2^2 \\ & = \sum_{i_k,j_{k'}\in E_{t+1}}\norm{\frac{1}{r}P_i-\frac{1}{r}P_j}_2^2 &\\ & = \frac{1}{r^2}\sum_{i,j\in E_{t+1}}\norm{P_i-P_j}_2^2, \end{align*} and the lemma follows. \end{proof}
\begin{proof}[Proof of \Cref{thm:krv_extended}] The initial potential is $\Psi(0)=n-1$, and by \Cref{lmm:krv_potential_to_expansion} we need to get it below $1/4n^2$. Putting \Cref{lmm:krv_random_projection,lmm:krv_cut_gain,lmm:krv_potential_reduction} together, we see that in each step we have in expectation $\Psi(t+1)\leq(1-\frac{1}{Cr\log n})\Psi(t)$. Hence, in expectation, it is enough to play for $O(r\log^2n)$ rounds. \end{proof}
\section{Resistance Sparsification} \label{sec:resistance}
We prove \Cref{thm:main_resistance} by combining \Cref{thm:main} with the following known result. \begin{theorem}[von Luxburg, Radl and Hein \cite{vLRH14}] \label{thm:von_luxburg} Let $G$ be a non-bipartite weighted graph with maximum edge weight $w_{\max}$ and minimum weighted degree $d_{\min}$. Let $u,v$ be nodes in $G$ with weighted degrees $d_u,d_v$ respectively. Then \[
\left|R_G(u,v) - \left(\frac{1}{d_u} + \frac{1}{d_v}\right)\right| \leq
2 \left(\frac{1}{\lambda_2(G)} + 2\right) \frac{w_{\max}}{d_{\min}^2}. \] \end{theorem}
Qualitatively, the theorem asserts that in a sufficiently regular expander, the resistance distance is essentially determined by vertex degrees. Therefore an expanding subgraph $H$ of $G$ with the \emph{same} weighted degrees can serve as a resistance sparsifier. In particular, in order to resistance-sparsify a regular expander, all we need is a regular expanding subgraph, as we have by \Cref{thm:main}. Since \Cref{thm:von_luxburg} does not apply to bipartite graphs, we will use the following variant that holds also for bipartite graphs as long as they are regular. Its proof appears in \Cref{sec:von_luxburg_bipartite}.
\begin{theorem}\label{thm:von_luxburg_bipartite} Let $G$ be a weighted graph which is $d$-regular in weighted degrees, with maximum edge weight $w_{\max}$. Let $u,v$ be nodes in $G$. Then \[
\left|R_G(u,v) - \frac{2}{d}\right| \leq
12 \left(\frac{1}{\lambda_2(G)} + 2\right) \frac{w_{\max}}{d^2}. \] \end{theorem}
\begin{proof}[Proof of \Cref{thm:main_resistance}] Using \Cref{thm:main} we obtain a $d$-regular subgraph $H$ of $G$ with $\phi(H)>\frac{1}{3}$. By removing the obtained subgraph $H$ from $G$ and iterating, we can apply the theorem $3d/\epsilon$ times and obtain disjoint subgraphs $H$. Since $d=(\log n)^{O(1)}$ and $D=\Omega(n)$, the degree of $G$ does not significantly change in the process, and the requirements of \Cref{thm:main} continue to hold throughout the iterations (with a loss only in constants). Taking the union of the disjoint subgraphs produced in this process, we obtain a subgraph $H$ of $G$ which is $(3d^2/\epsilon)$-regular with $\phi(H)\geq d/\epsilon$. By the discrete Cheeger inequality, \[
\lambda_2(H) \geq
\frac{1}{2}\left(\frac{\phi(H)}{\mathrm{deg(H)}}\right)^2 \geq
\frac{1}{18d^2}. \]
Recall that $H$ has edge weights in $\{1,2\}$. We now multiply each weight by $\epsilon D/(3d^2)$, rendering it $D$-regular in weighted degrees. This does not affect $\lambda_2(H)$ since it is an eigenvalue of the \emph{normalized} Laplacian.
Let $u,v\in V$. Apply \Cref{thm:von_luxburg_bipartite} on both $G$ and $H$. As $G$ is $D$-regular with $w_{\max}=1$ and $\lambda_2(G)=\Omega(1)$, we know that $R_G(u,v) = \tfrac{2}{D} \pm O\left(\frac{1}{D^2}\right)$. And as $H$ is $D$-regular with $w_{\max}=O(\frac{\epsilon D}{d^2})$ and $\lambda_2(H)=\Omega(1/d^2)$, we know that $R_H(u,v) = \tfrac{2}{D} \pm O\left(\frac{\epsilon}{D}\right)$. Putting these together, we get $
\frac{R_H(u,v)}{R_G(u,v)}
= 1\pm O\left(\epsilon+\frac{1}{D}\right)
= 1\pm O\left(\epsilon\right), $ where the last equality holds for sufficiently large $n$ since $D=\Omega(n)$. Scaling $\epsilon$ down by the constant hidden in the last $O(\epsilon)$ notation yields the theorem. \end{proof}
\else
\fi
\appendix
\section{Appendix: Omitted Proofs} \subsection{Proof of \Cref{thm:von_luxburg_bipartite}} \label{sec:von_luxburg_bipartite}
In the non-bipartite case, \Cref{thm:von_luxburg_bipartite} follows from \Cref{thm:von_luxburg}. We henceforth assume that $G=(V,E,w)$ is bipartite with bipartition $V=V_1\cup V_2$. Note that since it is regular, we must have $|V_1|=|V_2|=\frac{1}{2}|V|$. Furthermore, as a weighted regular bipartite graph, $G$ is a convex combination of perfect matchings and hence is regular also in unweighed degrees. Let $d'$ denote the unweighted degree of each vertex in $G$. If $d'\leq2$ then it is easy to verify that the theorem holds (due to poor expansion), so we henceforth assume $d'\geq3$.
For brevity we denote the error term in \Cref{thm:von_luxburg} as \[
\mathrm{err}
\eqdef 2 \left(\frac{1}{\lambda_2(G)} + 2\right) \frac{w_{\max}}{d^2}. \] We will use the notion of \emph{hitting time}: For a pair of vertices $u,v$, the hitting time $H_G(u,v)$ is defined as the expected time it takes a random walk in $G$ that starts at $u$, to hit $v$. Define the \emph{normalized hitting time} $h_G(u,v)=\frac{1}{2W}H_G(u,v)$, where $W$ is the sum of all edge weights in $G$. We then have, \begin{equation}\label{eq:resistance_hitting_time} R_G(u,v) = h_G(u,v) + h_G(v,u). \end{equation} We will use the following bound on the normalized hitting time, which is given in the same theorem by von Luxburg, Radl and Hein~\cite{vLRH14}. \begin{theorem}\label{thm:von_luxburg_hitting_time} In the same setting of \Cref{thm:von_luxburg}, \[
\forall u\neq v\in V,
\qquad h_G(u,v) = \frac{1}{d_v}\pm \mathrm{err} . \] \end{theorem} (Like \Cref{thm:von_luxburg}, this theorem does not apply to bipartite graphs, and this is the obstacle we are now trying to circumvent.)
We begin by handling pairs of vertices contained within the same partition side, say $V_1$. We construct from $G$ a weighted graph $G_1$ on the vertex set $V_1$, with weights $w_1$, by putting \[ \forall i\neq j\in V_1,\;\;\;\; w_1(i,j) = \frac{1}{d}\sum_{k\in V_2}w(i,k)w(j,k). \]
We argue that $H_{G_1}(u,v)=\frac{1}{2}H_G(u,v)$. This follows by observing that we set the weights $w_1$ such that for any $i,j\in V_1$, the probability to walk in one step from $i$ to $j$ in $G_1$ equals the probability to walk in two steps from $i$ to $j$ in $G$ via an intermediate node in $V_2$. Furthermore, we have normalized the weights $w_1$ such that $G_1$ is $d$-regular in weighted degrees. Recalling that $|V_1|=\frac{1}{2}|V|$, we have
\[ h_{G_1}(u,v)=\frac{1}{d|V_1|}H_{G_1}(u,v) = \frac{2}{d|V|}\cdot\frac{1}{2}H_G(u,v) = h_G(u,v). \] Recalling that the unweighted degree in $G$ is $d'\geq3$, we see that by construction, $G_1$ contains a triangle and hence is non-bipartite. Hence we can apply to it \Cref{thm:von_luxburg_hitting_time} and obtain $h_{G_1}(u,v)=\frac{1}{d}\pm\mathrm{err}_1$, where $\mathrm{err}_1$ is the error term of $G_1$. Note that for every $i\neq j\in V_1$ we have $w_1(i,j)\leq\frac{w_{\mathrm{max}}}{d}\sum_{k\in V_2}w(i,k)=w_{\mathrm{max}}$, so the maximum edge weight in $G_1$ is bounded by $w_{\mathrm{max}}$, and $\lambda_2(G_1)\geq\lambda_2(G)$ (easy to verify by construction), so $\mathrm{err}_1\leq\mathrm{err}$, and we have $h_{G_1}(u,v)=\frac{1}{d}\pm\mathrm{err}$. Hence, \[ h_G(u,v)=\frac{1}{d}\pm\mathrm{err} . \] Recalling that $R_G(u,v) = h_G(u,v) + h_G(v,u)$, we have established that \[ R_G(u,v) = \frac{2}{d}\pm2\mathrm{err} \] for every pair $u,v\in V_1$. The same arguments hold for every pair $u,v\in V_2$ as well. We are left to handle the case $u\in V_1$, $v\in V_2$. Recalling the definition of hitting time, we have \begin{align*} H_G(u,v) &= 1+\frac{w(u,v)}{d}\cdot0+\sum_{x\in V_2\setminus\{v\}}\frac{w(u,x)}{d}H_G(x,v) & \text{(factoring out the first step)} &\\ &= 1+\frac{w(u,v)}{d}\cdot0+\sum_{x\in V_2\setminus\{v\}}\frac{w(u,x)}{d}\cdot 2W\cdot h_G(x,v) & &\\ &= 1+2W\sum_{x\in V_2\setminus\{v\}}\frac{w(u,x)}{d}\left(\frac{1}{d}\pm\mathrm{err}\right) & \text{(since $v,x\in V_2$)} &\\ &= 1+2W\left(1-\frac{w(u,v)}{d}\right)\left(\frac{1}{d}\pm\mathrm{err}\right). \end{align*} Therefore \[ h_G(u,v) = \frac{1}{2W}+\left(1-\frac{w(u,v)}{d}\right)\left(\frac{1}{d}\pm\mathrm{err}\right), \] which implies \[ h_G \leq \frac{1}{2W}+\frac{1}{d}\pm\mathrm{err} \] and \[
h_G(u,v) \geq
\frac{1}{2W}+\left(1-\frac{w_{\max}}{d}\right)\left(\frac{1}{d}\pm\mathrm{err}\right) =
\frac{1}{2W}+\frac{1}{d}\pm2\mathrm{err}. \] Together, $h_G(u,v)=\frac{1}{d}+\frac{1}{2W}\pm2\mathrm{err}$. Now, since for an arbitrary vertex $i$ we have \[ d = \mathrm{deg}(i) = \sum_{j\in V}w(i,j) \leq nw_{\max}, \] we see that $\frac{1}{2W}=\frac{1}{nd}\leq\frac{w_{\max}}{d^2} \leq \mathrm{err}$ and hence \[ h_G(u,v)=\frac{1}{d}\pm3\mathrm{err}. \] Plugging this into $R_G(u,v) = h_G(u,v) + h_G(v,u)$, we find \[ R_G(u,v)=\frac{2}{d}\pm6\mathrm{err}, \] which completes the proof of \Cref{thm:von_luxburg_bipartite}. \qed
\subsection{Further Omitted Proofs}\label{sec:omitted_proofs} \begin{proposition}\label{prp:hall_consequence}
Let $G(V,U;E)$ be a bipartite graph on $n$ nodes with $|V|=|U|=\frac{1}{2}n$, and minimum degree $\geq\frac{1}{4}n$. Then $G$ contains a perfect matching. \end{proposition} \begin{proof}
Let $S\subset V$ be non-empty, and denote $N(S)\subset U$ the set of nodes with a neighbor in $S$. If $|S|\leq\frac{1}{4}n$ then since any $v\in S$ has $\frac{1}{4}n$ neighbors in $U$, we have $|N(S)|\geq N(\{v\})\geq\frac{1}{4}n\geq|S|$. If $|S|>\frac{1}{4}n$ then by the minimum degree condition on side $U$, every $u\in U$ must have a neighbor in $S$, and hence $|N(S)|=|U|=|V|\geq|S|$. The same arguments apply for $S\subset U$, so the condition of Hall's Marriage Theorem is verified, and it implies that $G$ contains a perfect matching. \end{proof}
\begin{proposition}\label{prp:dense_set_cover}
Consider an instance of Set Cover with a set $S$ of $n$ elements, and a family $\mathcal M$ of subsets of $S$. Suppose each $x\in S$ belongs to at least a $\mu$-fraction of the subsets in $\mathcal M$. Then for sufficiently large $n$, we can efficiently find a cover $M\subset\mathcal M$ with $|M|\leq\frac{1.1}{\mu}\log n$. \end{proposition} \begin{proof} Pick $q$ uniformly random sets (with replacement) from $\mathcal M$ to form $M$. The probability that a given element in $S$ is not covered by $M$ is upper-bounded by $(1-\mu)^q$. Taking a union bound over the element, we need to ensure that $n(1-\mu)^q<1$ in order to ensure that with constant probability, $M$ is a solution to the given Set Cover instance. This can be achieved by $q\leq\frac{1.1}{\mu}\log n$. \end{proof}
\end{document} |
\begin{document}
\title[Diagrammatic unknotting of knots and links in $\mathbb RP^3$]{Diagrammatic unknotting of knots and links in the projective space} \author{Maciej Mroczkowski}
\keywords{descending link diagram, unknot, unlink, projective space} \subjclass{Primary 57M99; Secondary 57N35, 57M27} \address{Department of Mathematics,
Uppsala University, 751 06 Uppsala, Sweden} \email{mroczkow@math.uu.se} \begin{abstract} In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in $\mathbb R^3$ is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links. \end{abstract} \maketitle
\section{introduction} Embeddings are dense in the space of immersions of a curve to a 3-manifold. Hence any immersion of a collection of circles to a 3-manifold can be turned by a small regular homotopy into a differentiable embedding. Usually, the ambient isotopy type of an embedding, which can be obtained from a given immersion by an arbitrarily small (in $C^1$-topology) regular homotopy, is not entirely determined by the immersion. On the other hand, for any immersion there are ambient isotopy types which cannot be obtained from it.
The main results of this paper imply the following theorem.
\begin{theorem} Let $S$ be a smooth closed 1-manifold. For any immersion $f:S\to\mathbb RP^2$, its composition with the standard inclusion $in: \mathbb RP^2\to\mathbb RP^3$ is regularly homotopic via an arbitrarily small regular homotopy to an embedding $g:S\to\mathbb RP^3$, which depends, up to ambient isotopy and composition with a self-diffeomorphism of $S$, only on the homotopy class of $f$. In other words, $g(S)$ is ambiently isotopic to a standard nonoriented unlink $L_{p,q}\subset\mathbb RP^3$ which depends, up to ambient isotopy, only on the number $p$ of its components contractible in $\mathbb RP^3$ and the number $q$ of its components non contractible in $\mathbb RP^3$. \end{theorem}
From this theorem one cannot eliminate self-diffeomorphisms of $S$. This is also proven below (see Section \ref{counter}).
This paper presents a way to unknot knots and unlink links in the real projective 3-space $\mathbb RP^3$ and the results formulated above appear as straightforward corollaries. It is shown how to obtain a diagram of the unknot or unlink, starting from an arbitrary diagram, and performing some crossing changes on it. This is done through an extension of the classical notion of descending diagram to diagrams of knots and links in $\mathbb RP^3$.
In the case of knots in $\mathbb R^3$, the notion of descending diagram was used to study some knot invariants such as Jones polynomial, Homfly polynomial or finite type invariants. One can calculate these invariants using appropriate skein relations and the fact that it is possible to make any link diagram descending, which would turn it to a diagram of the unlink. Descending diagrams were used to define the Homfly polynomial (see for instance \cite{LM}).
The notion of descending diagram for knots in $\mathbb RP^3$ can be used to study some invariants of these knots. It can be useful when considering the Homflypt module of $\mathbb RP^3$. The Jones polynomial was extended in \cite{JD} by J. V. Drobotukhina to knots in $\mathbb RP^3$. An algorithm that makes a diagram descending gives an alternative way to calculate this polynomial.
I wish to thank Oleg Viro for stimulating conversations and for his help.
\section{Unknotting knot and link diagrams in $\mathbb R^3$}\label{r3descending}
In the case of knots in $\mathbb R^3$ there is a well-known way to obtain the unknot by doing some crossing changes on a diagram. We choose a starting point and proceed from this point according to some orientation of the knot. When we meet a crossing for the first time and arrive at the lower branch of the crossing, we change it in order to make this branch upper. If we arrive at a crossing for a second time, we leave it unchanged. Finally we get back at the initial point. When we have done all these changes the diagram takes a special form: it is descending.
We can imagine that the result is a diagram of a knot in which we descend from the initial point all the way (a $z$ coordinate is decreased if $z$ is the axis along which we project the knot to the diagram). When we get back at a point which has the same projection as the initial point (but is below it) we join these two points with a segment. The resulting knot is the unknot.
More generally, a link diagram can be unlinked with appropriate crossing changes, by putting different components one above an other (choosing some order) and making each component descending as above.
\section {knot and link diagrams in $\mathbb RP^3$} \subsection{Link diagrams} A link diagram in $\mathbb RP^3$ is a disk with a collection of generically immersed arcs. An arc is a compact connected 1-manifold with or without boundary. The endpoints of arcs with boundary are on the boundary of the disk, divided into pairs of antipodal points. Each double point of the immersions or {\it crossing} of the diagram is endowed with information of over- and undercrossing. An example of a knot diagram in $\mathbb RP^3$ is shown in Figure \ref{knot1}.
\begin{figure}\label{knot1}
\end{figure}
A link diagram is constructed from a link $L$ in the following way:
$\mathbb RP^3$ is represented as a ball $D^3$ with antipodal points of the bounding sphere identified. The link $L$ in $\mathbb RP^3$ is lifted to $L'$ in $D^3$. We can suppose that the poles of the ball are not in $L'$. Let $p$ be the projection of $L'$ to the equatorial disk $D^2$ where a point in $L'$ is projected along the metric circle in $D^3$ passing through this point and the poles of the ball.
We assume that $L$ satisfies the following conditions of general position: $p(L')$ contains no cusps, points of tangency, triple points; $L'$ intersects transversally the boundary of the ball; no two points in $L'$ lie on the same arc of the great circle joining the poles of the ball in the boundary of the ball.
The information of over- and undercrossings comes from some orientation of the circles along which $L'$ is projected to $D^2$ (for instance from north to south).
If the link $L$ is oriented, we get naturally an {\it oriented} link diagram for which each arc is oriented. An orientation of a link diagram gives rise to a cyclic ordering of arcs (when we travel on $L$ according to the orientation, we meet the arcs in this order).
As for diagrams of links in $\mathbb R^3$, there are Reidemeister moves for diagrams of links in $\mathbb RP^3$. The five of them are pictured in Figure \ref{reidemeister}. These moves appeared in \cite{JD}.
\begin{figure}
\caption{Reidemeister moves}
\label{reidemeister}
\end{figure}
\subsection{Nets, diagrammatic components, arc distance and first pass} A {\it net} is the projective plane $\mathbb RP^2$ with a distinguished line, called {\it the line at infinity}, and a collection of generically immersed circles together with information of over- and undercrossing for each double point. We can associate to each diagram $D$ of a link {\it its net} obtained from $D$ by identifying the antipodal points of the boundary circle of $D$, with the line at infinity coming from this boundary circle.
Let $D$ be a diagram of a link and $b$ the part of $D$ coming from a component, $L_b$, of the link. We will say that $b$ is a {\it diagrammatic component} of $D$. Suppose that $L_b$ is oriented, then $b$ is also oriented. Let $P$ and $Q$ be two points in the interior of some arcs of $b$. Then the {\it arc distance} between $P$ and $Q$ is defined to be the number of times the line at infinity is crossed in the net of $D$, if one travels from the image of $P$ to the image of $Q$ in the net, according to the orientation of the image of $b$ in the net. Suppose that $X$ is a crossing of $D$ such that at least one of its branches is in $b$. Then the {\it first pass} of $X$ from $P$ is, by definition, the branch of $X$ whose image in the net of $D$ is passed first, if one travels from the image of $P$ in the net, according to the orientation of the image of $b$ in the net.
\subsection{Unknots in $\mathbb RP^3$} The fundamental group of $\mathbb RP^3$ has two elements. In each of them there is a simple loop that is naturally called {\it unknot}. A planar circle and a projective line are two unknots, up to isotopy. They are the only knots for which there are diagrams without crossings. A knot in $\mathbb RP^3$ is homotopic to one of the unknots depending on the element of the fundamental group it realizes. Thus we can deform it to an unknot by a sequence of isotopies, which correspond to planar isotopies and Reidemeister moves on the level of diagrams, and some homotopies, namely the ones which correspond to crossing changes on the level of diagrams.
In the next section we will see that for any diagram of a knot we can obtain a diagram of an unknot solely by some crossing changes.
\section{Unknotting knot diagrams in $\mathbb RP^3$}
A natural question is whether, for knots in $\mathbb RP^3$, there is a way to obtain a diagram of the unknot, by changing some crossings on an arbitrary diagram. The goal of this section is to give a positive answer to this question.
A {\it basepoint} is a distinguished point of a diagram, distinct from crossings and endpoints of arcs. A diagram with a basepoint is called {\it based diagram}.
\begin{definition} A based oriented diagram $D$ is called {\it descending} provided that for every crossing $X$ of $D$, the first pass of $X$ from the basepoint is an overpass (resp. underpass), if the arc distance between the basepoint and this first pass is even (resp. odd). \end{definition}
\begin{theorem}\label{theorem_knots} Let $D$ be a based oriented diagram of a knot in $\mathbb RP^3$. If $D$ is descending, then $D$ is a diagram of an unknot. \begin{proof} Suppose that $D$ is descending.
First, note that if $D$ consists of a single arc without boundary, then $D$ is a diagram of 0-homologous unknot, because it is descending in the classical sense.
Now suppose that $D$ is not of that type. Denote the arcs of $D$ by $a_1, ..., a_n$, $n\ge 1$, where $a_1$ contains the basepoint and $a_2, ..., a_n$ are ordered according to the orientation of $D$.
The arc $a_1$ can be divided in two parts: one that comes after the basepoint (according to the orientation) and the other one that comes before the basepoint. Denote the first one by $a_1^a$ and the second one by $a_1^b$.
An arc, or a part of it is said to be {\it below} another one if at each crossing between the two of them the branch of the first one is below the branch of the second one. If $a$ is below $b$, we write $a\le b$. It is easy to see that in the descending diagram $D$ the following relation holds:
$$a_2\le a_4\le a_6\; ...\le a_1^b\; ...\le a_5\le a_3\le a_1^a$$
Observe that each arc is descending or ascending. Also $a_1^a$ and $a_1^b$ are descending or ascending.
We will show that $D$ is a diagram of an unknot by constructing a sequence of Reidemeister moves from $D$ to a diagram without crossings. As each arc is descending or ascending we kill all the crossings between an arc and itself with some $\Omega_1-\Omega_3$ moves. We do the same with $a_1^a$ and $a_1^b$.
Now consider Figure \ref{isotopy}. We want to reduce the number of arcs by eliminating $a_2$. One of its boundary points, say $P$, is antipodal to a boundary point of $a_1^a$. Denote the other boundary point of $a_2$ by $Q$. Denote by $P'$, resp. $Q'$ the antipodal points of $P$ resp. $Q$.
\begin{figure}\label{isotopy}
\end{figure}
In order to eliminate $a_2$ by $\Omega_4$ move, we want to have no endpoints between the two endpoints of $a_2$, on the boundary of the disk. For this purpose we move $P$ towards $Q$, below some endpoints. At the same time $P'$ moves towards $Q'$ above some other endpoints. This corresponds to several applications of $\Omega_5$ move. Secondly, we perform several $\Omega_1-\Omega_3$ moves to kill all the crossings between $a_2$ and other arcs. It is possible because $a_2$ is below all other arcs. Finally, we can perform $\Omega_4$ move on $a_2$. Now instead of $a_1^a$, $a_2$ and $a_3$ we have a single part of arc, call it $a_3^a$. It is unknotted and above all other arcs. There may be some self-crossings of $a_3^a$ and in that case we kill them with $\Omega_1-\Omega_3$ moves. $a_1^b$ is renamed $a_3^b$. The arcs are now positioned in the following way: $$a_4\le a_6\; ...\le a_3^b\; ...\le a_5\le a_3^a$$
We can repeat the process with $a_3^a$, $a_4$ and $a_5$. As long as there are at least three arcs or parts of arcs left this is possible. Finally we will end up with a single arc. It will be unknotted and we can kill all crossings again. Thus it will be 0-homologous or non zero-homologous unknot. \end{proof} \end{theorem}
\begin{corolary} Let $D$ be a diagram of a knot in $\mathbb RP^3$. By making some crossing changes on it, we can obtain a diagram of the unknot. \begin{proof} Choose orientation and basepoint on $D$. Let $D'$ be the diagram obtained from $D$ by making the necessary crossing changes that make $D'$ descending. Then $D'$ is a diagram of an unknot by Theorem \ref{theorem_knots}. \end{proof} \end{corolary}
\begin{remark} In the proof of Theorem \ref{theorem_knots}, there are two types of simplifications of the diagram. The first one is to make the arcs unknotted. In this case we use only $\Omega_1$, $\Omega_2$ and $\Omega_3$ moves. The second is to eliminate some arcs. In this case we use only $\Omega_2$ and $\Omega_3$ moves that involve at least two different arcs as well as $\Omega_4$ and $\Omega_5$ moves. In the proof we alternated these two types of simplifications.
Another proof is possible in which any descending diagram is changed to a diagram with no crossings by a sequence of simplifications of the second type (the number of arcs will be reduced to one), followed at the end by a simplification of the first type. \end{remark}
\begin{remark} In the case of a diagram that is descending and which represents 0-homologous unknot, consider the diagram with a single arc obtained after a sequence of simplifications of the second type defined in the previous remark. This diagram will be descending in the classical sense (when we move an ascending arc through the boundary it becomes descending and vice versa). In this way we see that the notion of descending diagram in $\mathbb RP^3$ is similar to the same notion in $\mathbb R^3$ even for some diagrams of non affine knots. \end {remark}
\section{unlinks in $\mathbb RP^3$}
There is no natural notion of unlink in $\mathbb RP^3$. A link has two types of components: the 0-homologous and the non 0-homologous ones (they will be called {\it 1-homologous}). For the first type there is no problem to see how they should look in an unlink: like in $\mathbb R^3$ they should be unknots and each of them should be in a ball which does not intersect the other components. For the second type, different definitions of unlink are possible. Note that two 1-homologous components will always intersect in a diagram. Even if we require that an unlink should have a diagram in which any couple of 1-homologous components has a single common crossing and there are no other crossings, there are still many choices for an unlink. This is related to the configurations of skew lines in \cite{OYV} by O. Ya. Viro and J. V. Drobotukhina.
In Figure \ref{unlinks}, the two links could be taken as unlink with three 1-homologous components and four 0-homologous. But in fact they are not isotopic. This suggests that one may define a {\it standard unlink}. We consider the following definition of standard unlink: Take a projective line. Add the next below the first slightly rotating it in the counterclockwise direction. The third will be under the two firsts, also rotated in the same direction. We continue in this way with all projective lines.
An equivalent way to obtain standard unlink is the following: Take several complex lines in $\mathbb C^2$. They represent projective lines in $\mathbb RP^3$ which form the standard unlink.
For instance in Figure \ref{unlinks} the first is standard unlink, but not the second.
\begin{figure}
\caption{unlinks in $\mathbb RP^3$}
\label{unlinks}
\end{figure}
\section{Unlinking link diagrams} Now the question arises: is it possible to unlink a link diagram (obtain a diagram of a standard unlink) by crossing changes ?
The approach that was used in the case of knots does not extend directly. The problems arise from the 1-homologous components.
Recall that a diagrammatic component of $D$ is a part of $D$ coming from a component of the link. We will say that a diagrammatic component is {\it 0-homologous} (resp. {\it 1-homologous}), if it comes from a 0-homologous (resp. 1-homologous) component of the link.
\subsection{0-homologous components} Suppose that we have a diagram of a link, $D$, with $k$ 0-homologous diagrammatic components, $a^1, ..., a^k$, and some 1-homologous diagrammatic components. Choose some basepoint and orientation for each 0-homologous diagrammatic component.
If we make $a^1$ descending, in the same way as it was done for knots (its arcs will alternate below and above the other components), it will become diagram of an unknot, unlinked to the other components. We can continue with $a^2, ..., a^n$ in the same way.
The crossings between the 1-homologous diagrammatic components of $D$ were not changed, so the problem of unlinking a link diagram reduces to the problem of unlinking a diagram with only 1-homologous diagrammatic components.
\subsection{1-homologous components} If we try to make the arcs of 1-homologous diagrammatic components of $D$ alternating in the similar way as for 0-homologous diagrammatic components, there will be a problem: after unknotting the first 1-homologous diagrammatic component it will lie below or above everything else and we will not be able to continue with the next one. For instance in Figure \ref{bad_1} we can try to put the component with a single arc below or above the other component. But in either case the resulting diagram will not be a diagram of standard unlink.
\begin{figure}\label{bad_1}
\end{figure}
Nonetheless, as was stated in the introduction, we will define a notion of descending diagram for arbitrary link diagrams. In order to do this, we need to enhance the data which is used in the definition of descending diagram.
In the next section we will construct the data needed for the 1-homologous diagrammatic components (different from the data needed for the 0-homologous diagrammatic components). For each such component a set of self-crossings will be specified. This set will determine some part of the diagram which will be called dashed part. We will also specify two antipodal endpoints of arcs on some part of the diagram that is not dashed.
With this data we define the notion of descending diagram. We prove that a descending diagram is a diagram of the standard unlink in two steps. In the first step the dashed part of the diagram is eliminated by a sequence of Reidemeister moves. The remaining diagram has a simpler form and, in the second step, it is transformed into a canonical diagram of standard unlink by another sequence of Reidemeister moves.
\subsection{Data for descending diagrams: simplifying sets.}
Let $D$ be a diagram. Consider a 1-homologous diagrammatic component of $D$, say $b$. By the set of {\it self-crossings} of $b$, we will mean the set of those crossings of $D$ for which both branches are in $b$.
Let $X$ be a self-crossing of $b$. We can associate to it an orientation of $b$ defined in such a way that, with this orientation, the arc distance between the upper branch of $X$ and the lower one is even. This gives a unique choice of orientation because $b$ is 1-homologous: if we reverse the orientation, the arc distance will be odd. We call the associated orientation {\it the orientation determined by $X$}.
We will also dash the part of the diagram along which we travel from the upper branch to the lower branch of $X$ according to the orientation determined by $X$. To be precise, we notice that we travel on the net of the diagram, but we can lift it to the diagram itself and dash in this way some part of it. We call it the {\it dashed part determined by $X$}. An example is shown in Figure \ref{dash}.
\begin{figure}\label{dash}
\end{figure}
A subset $M$ of the set of self-crossings of a 1-homologous diagrammatic component $b$ is called {\it simplifying set} if:
1) Any self-crossing of $b$ is either in $M$ or is such that at least one of its branches is in a dashed part determined by a crossing in $M$.
2) For any two crossings $X, Y$ in $M$, the intersection between the dashed part determined by $X$ and the one determined by $Y$ is empty or consists of some crossings (the dashed parts are disjoint except possibly for some crossings).
To see that it is always possible to construct a simplifying set $M$, we first define a partial order on the set of self-crossings of $b$ in a diagram $D$:
$Y\le X$ if both branches of $Y$ are in the dashed part determined by $X$. Also $X\le X$ for any $X$. It can be checked easily that this indeed gives a partial order.
A simplifying set $M$ is constructed in the following way: Choose any self-crossing of $b$, say $X_1$, that is maximal with respect to the partial order defined above. It is possible if the set of self-crossings of $b$ is non empty. Put $X_1$ in $M$. Now suppose that $X_1, ..., X_k$ are already in $M$. Consider the diagram $D'$ obtained from $D$ by removing the dashed parts determined by $X_1, ..., X_k$ (and joining the remaining part of the upper branch to the remaining part of the lower branch for each of the crossings $X_1, ..., X_k$). Choose any self-crossing of $b$ in $D'$, say $X_{k+1}$, that is maximal. It can be viewed as a crossing in $D$ because the crossings of $D'$ form naturally a subset of the set of crossings of $D$. Put $X_{k+1}$ in $M$. At some point the set of self-crossings of $b$ in $D'$ is empty and $M$ is constructed.
Such set $M$ is indeed a simplifying set:
1) Each crossing is either in $M$ or in a dashed part determined by a crossing in $M$ (otherwise more crossings could be put in $M$).
2) The intersection between the dashed part determined by $X_1$ and the dashed part determined by any other crossing $Y$ in $M$ is empty or consists of some crossings. This is the case because $X_1$ is maximal and none of the branches of $Y$ is in the dashed part determined by $X_1$. Similarly, the intersection between the dashed part determined by $X_k$ and the one determined by any crossing $X_l$, with $l>k$, is empty or consists of some crossings. This can be seen by considering the diagram $D'$ obtained from $D$ by removing the dashed parts determined by $X_1, ..., X_{k-1}$. Thus for any crossings $X, Y$ in $M$, the intersection between the dashed part determined by $X$ and the one determined by $Y$ is empty or consists of some crossings.
\subsection{Simple diagrams} Diagrams which have only 1-homologous diagrammatic components and which have crossings only between different diagrammatic components will be called {\it simple diagrams}.
Let $D$ be a diagram. Suppose that for each 1-homologous diagrammatic component of $D$, a simplifying set is chosen. Dash the parts of $D$ determined by all crossings in the simplifying sets. Then the diagram obtained from $D$ by removing the dashed parts and all the 0-homologous diagrammatic components, is called the {\it simple diagram} of the data consisting of $D$ and the simplifying sets. This diagram is indeed simple in the sense defined in the preceding paragraph. At the crossings which are in the simplifying sets, we join the remaining part of the upper branch to the remaining part of the lower branch.
\subsection{Simple descending diagrams}\label{simple_desc_dgms} We will first define the notion of descending diagram for a simple diagram $D$ with a couple of antipodal endpoints of some arcs specified, say $P^1$ and $Q^1$.
Denote by $b^1$ the diagrammatic component to which $P^1$ and $Q^1$ belong. Now travel along the boundary of the disk in counterclockwise direction, starting from these two antipodal endpoints. Each time a couple of antipodal endpoints is encountered consider whether it belongs to a new diagrammatic component. If this is the case, denote it by $b^2$ and denote this couple of endpoints by $P^2$ and $Q^2$. Continuing in this way, call the subsequent diagrammatic components from $b^3$ to $b^n$ and, for each one of them, call the couple of antipodal points encountered $P^3$, $Q^3$ to $P^n$, $Q^n$.
Orient each $b^i$ in such a way that $P^i$ becomes the initial point of the arc to which it belongs. In the definition below, we will use the arc distance between two points $P$ and $X$ where $P$ is an endpoint of an arc. The original definition of arc distance is extended to this case in the following way: consider a point $P'$ in the interior of the arc to which $P$ belongs, which is such that $X$ is not between $P$ and $P'$. Then the arc distance between $P$ and $X$ is by definition the arc distance between $P'$ and $X$.
With these conventions the definition is:
\begin{definition}\label{simple_desc} The simple diagram $D$ is called {\it descending} with respect to the couple of antipodal points $P^1$ and $Q^1$, provided that for each crossing between $b^i$ and $b^j$, $i<j$, the branch in $b^i$ is over (resp. under) the branch in $b^j$, if the arc distance between $P^i$ and the branch in $b^i$ is even (resp. odd). \end{definition}
\subsection{General descending diagrams.}\label{gendescdgms} Consider a diagram $D$. Suppose that $D$ has oriented based 0-homologous diagrammatic components, $a^1, ..., a^m$ and that for each 1-homologous diagrammatic component an {\it ordered} simplifying set is fixed. Moreover, suppose that a couple of endpoints $P^1$ and $Q^1$ is specified in $D$, that are not in the dashed part of $D$ determined by the crossings in the simplifying sets. With this data we define below the notion of descending diagram.
First we introduce some notations. Let $D'$ be the simple diagram of this data. We will consider $D'$ as a subset of $D$. Note that $P^1$ and $Q^1$ are in $D'$.
Denote by $b^1$ the diagrammatic component to which $P^1$ and $Q^1$ belong. Now travel along the boundary of the disk in counterclockwise direction, starting from these two antipodal endpoints, and consider the successive endpoints encountered. If they belong to a new diagrammatic component {\it and are in the simple diagram $D'$}, denote it by $b^2$ and denote this couple of endpoints by $P^2$ and $Q^2$. Continuing in this way, call the subsequent diagrammatic components from $b^3$ to $b^n$ and, for each one of them, call the couple of antipodal points encountered $P^3$, $Q^3$ to $P^n$, $Q^n$.
Finally for each $b^i$ call its ordered simplifying set $M^i$.
With these conventions the definition is:
\begin{definition}\label{main_def} $D$ is called {\it descending} with respect to the preceding data, provided that its simple diagram $D'$ is descending with respect to $P^1$ and $Q^1$ and the crossings that are not crossings of $D'$ satisfy the following:
Case 1: Suppose that a crossing $X$ has one branch in $a^i$ and the other either in $a^j$ with $i\le j$ or in $b^l$ for any $l$. Then, the first pass of $X$ from the basepoint of $a^i$ is an overpass (resp. underpass), if the arc distance between the basepoint and this first pass is even (resp. odd).
Case 2: Suppose that a crossing $Y$ between two 1-homologous diagrammatic components, which is not in a simplifying set, has one branch in a dashed part determined by a crossing $X$ in $M^i$ and for the other branch of $Y$ one of the following holds:
\begin{enumerate} \item it is in $b^j$ with $j>i$,
\item it is in $b^j$ with $j\le i$, but not in a dashed part of it,
\item it is in $b^i$, in the dashed part determined by a crossing in $M^i$ that is greater or equal to $X$ ($M^i$ is ordered). \end{enumerate}
Give $b^i$ the orientation determined by $X$. Then, the first pass of $Y$ from the upper branch of $X$ is an overpass (resp. underpass), if the arc distance between the upper branch of $X$ and this first pass of $Y$ is even (resp. odd). \end{definition}
\subsection{Unknottedness of simple descending diagrams} Suppose that in a simple diagram there is a diagrammatic component $a$ which has a unique arc and another diagrammatic component $b$. In a simple diagram all diagrammatic components are 1-homologous, so $a$ divides the disk in two halves. An arc of $b$ can have its endpoints in different halves (in which case we say that it {\it crosses} $a$) or in the same half (in which case we say that it {\it does not cross} $a$). In order to prove that simple descending diagrams are diagrams of standard unlinks, we need the following:
\begin{lemma}\label{cross} Let $D$ be a simple diagram. Let $a$ and $b$ be two of its diagrammatic components. Suppose that $a$ has a single arc and that $b$ has $n$ arcs, $n\ge 3$. Then at least two of the arcs of $b$ do not cross $a$. \begin{proof} First we prove that there is at least one such arc.
Suppose that all arcs of $b$ cross $a$. Consider the net of $D$, say $N$. Let $P$ be the image in the net of the endpoints of $a$. If we remove from $N$ the image of $a$ and a small disk centered at $P$ we get the Mobius strip $M$ together with some closed curves on it. By assumption the image of $b$ is a simple closed curve in $M$ representing $\pm n$ in the fundamental group of $M$ (the infinite cyclic group). This is impossible if $n\ge 3$ by \cite{DRJC} (p.145). Thus at least one of the arcs of $b$ does not cross $a$.
Furthermore, the number of crossings between $a$ and $b$, two 1-homologous diagrammatic components, is odd. We can count the crossings coming from $b$-arcs that do not cross $a$ (even number) and the other ones (odd number). If there was just one $b$-arc that does not cross $a$, the number of the remaining arcs being even ($b$ is 1-homologous) the total number of crossing between $b$ and $a$ would be even. This is impossible, so there has to be another $b$-arc that does not cross $a$. \end{proof} \end{lemma}
An arc of a diagrammatic component $b$ will be called {\it most nested} if it is possible to travel on the boundary circle of the diagram from one endpoint of the arc to the other without encountering endpoints that belong to other arcs of $b$.
If $P$, $Q$ is a couple of antipodal points on the boundary circle of the diagram and $b^1$ and $b^2$ are two arcs of $b$, we say that $b^1$ is {\it nested} in $b^2$ {\it with respect to} $P$, $Q$ if it is possible to travel on the boundary circle from $P$ to $Q$ and meet successively: an endpoint of $b^2$, the endpoints of $b^1$ and finally the other endpoint of $b^2$.
\begin{theorem}\label{main_lemma} Suppose that $D$ is a simple diagram with a couple of antipodal endpoints $P^1$ and $Q^1$ specified. If $D$ is descending with respect to $P^1$ and $Q^1$, then $D$ is a diagram of standard unlink.
\begin{proof} Suppose that $D$ is descending with respect to $P^1$ and $Q^1$.
We will consider successively the diagrammatic components $b^1, ..., b^n$ and the couples $P^1$, $Q^1$ to $P^n$, $Q^n$ defined in the same way as in section \ref{simple_desc_dgms}.
First suppose that $b^1$ has a single arc which has $P^1$ and $Q^1$ as endpoints and lies above everything else. Suppose that $b^2$ has $k$ arcs, $k\ge 3$. Denote them by $b^2_1$, ..., $b^2_k$ in such a way that $P^2$ is an endpoint of $b^2_1$, $Q^2$ is an endpoint of $b^2_k$ and the order comes from some orientation of $b^2$.
We will reduce the number of arcs of $b^2$. We choose the first arc to eliminate in such a way that it is most nested, does not cross $b^1$ and does not have $P^2$ or $Q^2$ as endpoint.
To see that it is always possible to find such arc we consider different cases, depending on whether $b^2_1$ or $b^2_k$ crosses $b^1$ or not. These cases are pictured in Figure \ref{3cases}.
Cases 1 and 2: If at least one of $b^2_1$ and $b^2_k$ crosses $b^1$, we know that there is another arc, say $b^2_i$, that does not cross $b^1$ by Lemma \ref{cross}. If it is most nested we will eliminate it. Otherwise it is easy to see that there is a most nested arc, say $b^2_j$, nested in $b^2_i$ with respect to the endpoints of $b^1$. Then $b^2_j$ does not cross $b^1$, it does not have $P^2$ or $Q^2$ as endpoint and we will eliminate it.
Case 3: From Figure \ref{3cases} it is clear that there is an arc nested in $b^2_k$ with respect to the endpoints of $b^1$. It does not cross $b^1$. As in cases 1 and 2, we will eliminate it or eliminate another most nested arc, nested in it with respect to the endpoints of $b^1$.
We consider also the situation where the roles of $b^2_1$ and $b^2_k$ are reversed in Case 3. We get a suitable arc to eliminate in that case, too.
\begin{figure}\label{3cases}
\end{figure}
To eliminate an arc, consider its position with respect to the arcs of $b^3$, ..., $b^k$. If it lies above these arcs, one of its endpoints is moved towards the other above all endpoints of other arcs. This is done by a sequence of $\Omega_5$ moves. In the next step all crossings between this arc and other arcs are killed with some $\Omega_1-\Omega_3$ moves. Finally it is eliminated with $\Omega_4$.
If an arc lies below the arcs of $b^3$, ..., $b^k$, the same moves are used except that the endpoint of the arc is moved below other endpoints.
Note that it is crucial that the chosen arc does not cross $b^1$. Otherwise, after some application of $\Omega_5$ move between the arc and $b^1$, there would be a crossing between $b^1$ and $b^2$ in which the upper branch would belong to $b^2$. Thus $b^1$ would no more lie above everything else.
After the elimination, a new arc lies above or below $b^3, ..., b^n$, and the number of arcs of $b^2$ is decreased by two.
In this way, the number of arcs of $b^2$ can be reduced to one. At the end $b^1$ and $b^2$, each with a single arc, lie above all other arcs. If we travel from $P^1$, $Q^1$ in counterclockwise direction, the first couple of endpoints encountered is $P^2$ and $Q^2$. This is shown in Figure \ref{2unlink}.
\begin{figure}\label{2unlink}
\end{figure}
We continue similarly with $b^3$. Its arcs can cross both $b^1$ and $b^2$ or none of them. So the reduction of arcs that was done for $b^2$ works for $b^3$ as well.
In this way we can continue for all arcs and at the end we get standard unlink.
It remains to prove that at the beginning of the isotopy, the number of arcs of $b^1$ can be reduced to one arc which has $P^1$ and $Q^1$ as endpoints and lies above everything else. To see this we add a new diagrammatic component $b^0$ which has a single arc lying above everything else, with $P^0$ and $Q^0$ as endpoints, which are such that if we travel from $P^0$, $Q^0$ in counterclockwise direction, the first couple of endpoints encountered is $P^1$ and $Q^1$. Now the number of arcs of $b^1$ is reduced to one in the same way as it was done for $b^2$. Finally we remove $b^0$. \end{proof} \end{theorem}
\subsection{Unknottedness of general descending diagrams}
\begin{theorem}\label{main_theorem}
Let $D$ be a diagram of a link in $\mathbb RP^3$. Suppose that its 0-homologous diagrammatic components are ordered and each of them is oriented and based. Moreover, suppose that for each 1-homologous diagrammatic component of $D$, an ordered simplifying set is chosen. Finally let $P$ and $Q$ be a couple of antipodal endpoints in the simple diagram of $D$. If $D$ is descending with respect to this data, then $D$ is a diagram of standard unlink.
\begin{proof} Suppose that $D$ is descending.
Denote the 0-homologous diagrammatic components of $D$ by $a^1, ..., a^m$, according to the order. Denote $P$ and $Q$ by $P^1$ and $Q^1$ respectively. In the proof, we use the same notation as in section \ref{gendescdgms} for the simple diagram of $D$, 1-homologous diagrammatic components, endpoints and simplifying sets.
First, the 0-homologous diagrammatic components are unlinked as in the case of knots, starting with $a^1$ and ending with $a^m$. At the end of this step, we get a diagram in which there are crossings only between 1-homologous diagrammatic components.
Next, consider the first crossing in the ordered simplifying set $M^1$, say $X$. Consider the dashed part determined by $X$. Denote by $b^X_1$, ..., $b^X_k$ the arcs and part of arcs encountered successively when traveling on the net according to the orientation determined by $X$, from the upper branch to the lower branch of $X$. Note that $k$ is odd because of the definition of orientation determined by a self-crossing. As $D$ is descending we have:
$$b^X_2\le b^X_4\; ...\le b^X_{k-1}\le {everything\; else}\le b^X_k\; ...\le b^X_3\le b^X_1$$
Now, as in the proof of Theorem \ref{theorem_knots}, we can eliminate $b^X_2$, $b^X_4$, etc. Finally, we get a single descending part of arc from the upper branch to the lower branch of $X$ that is eliminated at the end. In this way, the dashed part determined by $X$ is erased.
Similarly, we erase all dashed parts determined by crossings in $M^1, ..., M^n$.
Thus, we get the simple diagram of $D$ together with some 0-homologous diagrammatic components. But the simple diagram of $D$ is a diagram of standard unlink because of Theorem \ref{main_lemma}. Thus $D$ is a diagram of standard unlink. \end{proof} \end{theorem}
As in the case of knots the last result implies:
\begin{corolary}\label{nonoriented} Let $D$ be a diagram of a link. By making some crossing changes on it, we can obtain a diagram of standard unlink. \end{corolary}
\section{The oriented case}\label{counter} The next natural step is to try to unlink an oriented link. Now, for standard oriented unlink we can take a standard unlink and orient each 1-homologous component in such a way that at each crossing the local writhe is +1. For instance, standard oriented unlink for four 1-homologous components is pictured in Figure \ref{4unlink} (a).
\begin{figure}\label{4unlink}
\end{figure}
The notion of descending diagram cannot be extended to the oriented case in a satisfactory way. In other words, there is no way to obtain the equivalent of Corollary \ref{nonoriented} for oriented links: there is a diagram such that no matter what crossing changes are done on it, it will never become a standard oriented unlink.
A counterexample is presented in Figure \ref{4unlink} (b). This link with four 1-homologous components has a local writhe +1 at each crossing. Each couple of components has a unique common crossing so, if we want to obtain the standard oriented unlink, we cannot change any crossing because we would get at this crossing a local writhe of -1. But the link in Figure \ref{4unlink} (b) is not the standard oriented unlink. It is the mirror image of the link $6_3^4$, that appears in \cite{JD2}, whereas the standard oriented unlink is the mirror image of the link $6_2^4$, and these two links are not isotopic.
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Faster Reductions from Straight Skeletons to Motorcycle Graphs}
\author{John Bowers\fnref{Research supported by an NSF graduate fellowship under Grant No. S121000000211.}}
\address{Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA.}
\begin{abstract} We give an algorithm that reduces the straight skeleton to the motorcycle graph in $O(n\log n)$ time for (weakly) simple polygons and $O(n(\log n)\log m)$ time for a planar straight line graph with $m$ connected components. The current fastest algorithms for computing motorcycle graphs are an $O(n^{4/3 + \epsilon})$ time algorithm for non-degenerate cases and $O(n^{17/11 + \epsilon})$ for degenerate cases. Together with our algorithm this results in an algorithm computing the straight skeleton of a non-degenerate (weakly) simple polygon with $r$ reflex vertices in $O(n\log n + r^{4/3 + \epsilon})$ time and of a non-degenerate planar straight line graph with $m$ connected components in $O(n(\log n)\log m + r^{4/3 + \epsilon})$ time. For degenerate cases the algorithm takes $O(n\log n + r^{17/11 + \epsilon})$ and $O(n(\log n)\log m + r^{17/11 + \epsilon})$ time respectively. \end{abstract}
\begin{keyword} Computational geometry\sep Straight skeletons\sep Motorcycle graphs\sep Roof construction \end{keyword}
\end{frontmatter}
\section{Introduction} \label{sec:introduction}
The straight skeleton of a simple polygon (Fig.~\ref{fig:sskel0}b) is a tree-like structure that subdivides its interior into regions. It was first defined by Aichholzer et al.\ in \cite{aaag-ntsp-95} by tracing the vertices of the polygon during a wavefront process in which the sides of the polygon are moved inwards in parallel at constant speed. It was later generalized to planar straight line graphs (PSLGs) \cite{aa-ssgpf-96}. The trace of the vertices during the wavefront process forms the straight-skeleton. It has a wide array of applications including polygon interpolation \cite{Barequet:2003:SBC:644108.644129}, procedural modeling of urban environments \cite{Vanegas:2012:PGP:2322116.2322132}, biomedical imaging \cite{Cloppet:2000:ABN:331097.331118}, and polygon decomposition \cite{Tanase:2003:PDB:777792.777802}, to name a few. For convex polygons, the straight skeleton is identical to the medial axis and is linear time computable, but for general simple polygons and PSLGs the computational complexity is still an open problem. In the polygon case, the fastest algorithms for computing it first compute a structure called the induced motorcycle graph, which was introduced by Eppstein and Erickson \cite{EppEri-SCG-98}, and then compute the straight skeleton as a post-processing step.
In the case of PSLGs, however, no sub-quadratic reduction of the straight skeleton to the motorcycle graph is known. This results of this paper are summarized by:
\begin{theorem}[Main Results] \label{theorem:main}The straight skeleton problem can be reduced to the motorcycle graph problem in $O(n\log n)$ time and $O(n)$ space for simple polygons or $O(n(\log n)\log m)$ time and $O(n\log m)$ space for a PSLG with $m$ connected components.\end{theorem}
The current fastest algorithms for computing the induced motorcycle graph of a polygon or PSLG with $r$ reflex vertices are the $O(r^{4/3 + \epsilon})$ algorithm from \cite{Vigneron:2013vw} for non-degenerate input and the $O(r^{17/11 + \epsilon})$ time algorithm from \cite{EppEri-SCG-98} for degenerate input. Together with our results this gives an algorithm computing the straight skeleton of a non-degenerate (weakly) simple polygon with $r$ reflex vertices in $O(n\log n + r^{4/3 + \epsilon})$ time and of a non-degenerate planar straight line graph with $m$ connected components in $O(n(\log n)\log m + r^{4/3 + \epsilon})$ time. For degenerate cases the algorithm takes $O(n\log n + r^{17/11 + \epsilon})$ and $O(n(\log n)\log m + r^{17/11 + \epsilon})$ time respectively.
\paraskip{Overview.} In addition to the wavefront process, the straight skeleton has alternatively been defined as a terrain, sometimes called a {\em roof}, which can be characterized as a lower envelope of certain infinite strips in ${\bf R}\xspace^3$ \cite{aaag-ntsp-95, Cheng:2007go, HuberH12}. Our general approach for polygons is a divide and conquer algorithm which computes this roof. Given a polygon (or a sub-chain of the polygon) our algorithm subdivides the polygon into two sub-chains, recursively computes an intermediate structure we call a {\em partial roof} for each sub-chain, and then merges the result. The partial roof captures certain properties of the final roof which allow us to reconstruct the roof in the final merge operation. A key insight of this paper is that although the final roof is a terrain, which in particular implies no self-intersections, this restriction is unnecessary for the intermediate partial roofs. Indeed, our partial roofs are {\em intrinsically} topological disks, but their particular {\em realizations} in ${\bf R}\xspace^3$ may be more topologically complicated. This distinction may be made more clear by a familiar analog: a Klein bottle is intrinsically a 2D manifold, meaning locally it always looks like a small patch of the Euclidean plane, but any realization in ${\bf R}\xspace^3$ exhibits self-intersections and so their are points on the realization that are more complicated. However, if one is intrinsically walking along the surface (say, as a flatlander), one would never encounter such a self-intersection. The intersection is an extrinsic property of the particular realization chosen, but is not intrinsic to the underlying surface. Our result is extended to PSLGs by using a modification of the vertical subdivision procedure from \cite{chengESA2014}. This allows us to subdivide the PSLG into polygons and then employ our divide and conquer approach to compute the part of the straight skeleton roof which lies above each polygon.
\begin{figure}\label{fig:sskel0}
\end{figure}
\paraskip{Related Work.} Our algorithm for polygons has been circulating since Nov. 2013 and was posted to the ArXiV in May 2014 \cite{bowersArxiv2014}. Subsequently, two related papers have appeared. The first gives a reduction from the straight skeleton problem to the motorcycle graph problem taking $O(n(\log n)\log r)$ time for a polygon with $r$ reflex vertices with or without holes \cite{chengESA2014}. We adapt a technique from that paper--the vertical decomposition algorithm--to extend our polygon algorithm to PSLGs. The second, which appeared in EuroCG \cite{eurocg2014}, studies the problem of computing the straight skeleton for the special case of monotone polygons without holes. The main idea is similar to ours: subdivide the polygon into its two monotone chains, compute an intermediate terrain for the each chain, and merge the result. Their result makes use of the fact that the two chains are monotone, which allows them to efficiently compute a terrain for each chain and merge the result. This is different than our method, since we cannot assume our input polygons are monotone and thus drop the restriction that the intermediate results be terrains. The first sub-quadratic straight skeleton algorithm is due to Eppstein and Erickson \cite{EppEri-SCG-98} and takes $O(n^{1+\epsilon} + n^{8/11 + \epsilon}r^{9/11+\epsilon})$ time. Prior to the present work and that of \cite{chengArxiv2014}, this was the fastest for PSLGs and the fastest {\em deterministic} algorithm for polygons (with or without holes). They introduced {\em motorcycle graphs} as an abstraction of the main difficulty, but did not give an algorithm for straight skeletons that uses motorcycle graph as input. The first such algorithm was described by Cheng and Vigneron \cite{Cheng:2007go}. They give an algorithm computing a motorcycle graph in $O(n^{3/2}\log^2 n)$ time and a post-processing step computing the straight skeleton of a polygon with $h$ holes from its motorcycle graph in {\em expected} $O(n\sqrt{h}\log^2 n)$ time. The first step was recently improved to $O(n^{4/3 + \epsilon})$ time by Vigneron and Yan \cite{Vigneron:2013vw} for non-degenerate inputs. The best known lower bounds for straight skeletons are $\Omega(n\log n)$ for PSLGs \cite{EppEri-SCG-98} and polygons with holes \cite{Hub11}, and $\Omega(n)$ for simple polygons. A parallel thread of research focuses on algorithms which perform better in practice than their theoretical upper bounds. Huber and Held \cite{Huber:2011kr}, describe an $O(n^2\log n)$ time algorithm for computing the straight skeleton of planar straight-line graphs that uses the motorcycle graph which behaves like $O(n\log n)$ in practice--though worst case examples can be constructed. Similarly, Palfrader et al., \cite{phh-2012} investigate the algorithm from \cite{aa-ssgpf-96} and show that it behaves like $O(n\log n)$ in practice, though examples requiring $O(n^2\log n)$ are known. It remains open to close the gap between theoretical upper and lower bounds and experimental observation.
\section{Preliminary Terms} \label{sec:preliminaries}
\paraskip{Straight Skeletons.} Historically, the straight skeleton of a polygon $P$ has been defined by a wavefront process: move the edges of $P$ towards its interior at unit speed while keeping each edge parallel to its original position. Each edge grows or shrinks to maintain incidence with its neighboring edges. An edge may shrink to zero-length, in which case it is replaced by a vertex in the wavefront, or may hit some other edge of the wavefront, in which case the wavefront polygon is split into two, and the wavefront continues independently in each. The trace of the vertices during this process is the {\em straight skeleton}, denoted $SS(P)$. For a more thorough treatment of the wavefront definition of the straight skeleton, see \cite{aaag-ntsp-95}. The wavefront model is extended to PSLGs in \cite{aa-ssgpf-96}.
\paraskip{Motorcycle graphs.} Place ``motorcycles'' at points $p_1,\dots,p_n$ in the plane with velocity vectors $v_1,\dots,v_n$. A motorcycle $M_i$ begins at $p_i$ and moves along the ray $p_i + t v_i$, leaving a track behind it. It crashes if it encounters another motorcycle's track. The {\em motorcycle graph} is given by vertices for the initial positions $p_1, \dots, p_n$ and the crash sites $c_1,\dots,c_n$ for each motorcycle, and an edge for each track. The {\em motorcycle graph induced by a polygon $P$ (or PSLG $G$)}, denoted $MG(P)$, is given by creating a motorcycle for each reflex vertex $v$ of the polygon, with speed equal to $1/\sin{(\theta/2)}$, where $\theta$ is the interior angle at $v$ in $P$. In a PSLG a degree 1 vertex induces two motorcycles, each making an angle of $3\pi/2$ on either side with the incident edge. We show how to handle this more generically below. The speed of each motorcycle is the same as the speed a vertex moves in the wavefront algorithms for straight skeleton computation. In addition to the tracks, the polygon/PSLG edges are treated as obstacles and a motorcycle crashes if it encounters either an edge or a track. See Fig.~\ref{fig:sskel0}c. \looseness=-1
\paraskip{The roof model of the straight skeleton.} An alternative view of the straight skeleton to the wavefront model is the {\em roof model} \cite{aaag-ntsp-95}. In the roof model the straight skeleton is a polygonal ``roof'' of faces in ${\bf R}^3$ each lying in the upper half space $z\geq 0$ with the boundary edges embedded in the $xy$-plane. The roof model is given by lifting each vertex $v$ of the straight skeleton by augmenting its position with a $z$-coordinate equal to the time $t$ at which the wavefront reaches $v$. We call this the {\em straight skeleton roof}, denoted $R(P)$. The non-boundary edges of the roof is the (lifted) straight-skeleton, denoted $SS(P)$. See Fig.~\ref{fig:sskel0}d. Each face of the roof lies in a plane through its base edge making a dihedral angle of $\pi/4$ with the $xy$-plane. \looseness=-1
\paraskip{Edge and motorcycle slabs.} An alternative characterization of $R(P)$ is given in \cite{Cheng:2007go}. There $R(P)$ is defined as the lower envelope of a set of partially infinite strips in ${\bf R}^3$ called slabs defined with respect to the edges of the polygon $P$ and the edges of the motorcycle graph $MG(P)$. For each edge $e$ of $P$ they define an {\em edge slab} and for each reflex vertex $v$ of $P$ they define two {\em motorcycle slabs}, one for each edge incident $v$. Before defining the slabs, let us attach a coordinate frame to each edge of $P$.
Define three unit 3-vectors along $e$: an {\em edge vector} $\vec{E}_e$, a {\em slope vector} $\vec{S}_e$, and a {\em normal vector} $\vec{N}_e$. Given an edge $e$ of $P$, $\vec{E}_e$ is the unit vector pointing along $e$ in counter-clockwise direction around $P$; $\vec{S}_e$ is the unit vector orthogonal to $\vec{E}_e$ lying above the interior of $P$ and making an angle of $\pi/4$ with the $xy$-plane; and $\vec{N}_e = \vec{E}_e\times \vec{S}_e$. The {\em edge slab} of an edge $e$ is defined by $\{p + t \vec{S}_e\,|\,p\in e, t\geq 0\}$. Let $u$ be a reflex vertex of $P$ and $M_u$ be its motorcycle in $MG(P)$, $c_u$ be the crash site of $M_u$, and $t_u$ be the crash time. Lift $c_u$ into ${\bf R}^3$ to obtain $\bar{c}_u$ by augmenting $t_u$ as its $z$-coordinate. Let $e$ be an edge of $P$ incident $u$. Then the {\em motorcycle slab for $u$ with respect to $e$} are the points $\{p + t\vec{S}_e\,|\,p\in(u, \bar{c}_u), t\geq 0\}$ where $p$ is on the line segment $(u, \bar{c}_u)$. We call $(u, \bar{c}_u)$ the {\em lifted motorcycle track}. See Fig.~\ref{fig:sskel0}e, f. Each point on a slab can be written as a linear combination of the slab's slope and edge vectors. In other words if $p$ is a point on a slab $s$ with base edge $e$, then $p$ can be written as $a \vec{E}_e + b \vec{S}_e$ for some $a, b\in {\bf R}\xspace$. We call $(a, b)$ the {\em local coordinates} of $p$ in $s$. As a shorthand we treat $\vec{S}_e$ as the {\em (local) vertical axis} for a slab and $\vec{E}_e$ as the {\em (local) horizontal axis}.
\paraskip{The structure $\operatorname{slabs}(P)$.} Each edge $e$ has one edge slab and for both of its endpoints it has a motorcycle slab if the endpoint is reflex. All slabs for $e$ are contained in the plane through $e$ with normal $\vec{N}_e$. As in \cite{Huber:2011kr} we simplify the notation by referring to the union of the edge slab and any motorcycle slabs for an edge $e$ as {\em the slab for $e$}, denoted $\operatorname{slab}(e)$. See Fig.~\ref{fig:sskel0}g. We denote the set of slabs for all edges of the polygon by $\operatorname{slabs}(P)$ (i.e. $\operatorname{slabs}(P) = \{\operatorname{slab}(e)\,|\,e\in P\}$). The lower envelope of $\operatorname{slabs}(P)$ is given by keeping the part of each slab which is lower (in terms of $z$-coordinate) than all other slabs. In \cite{Cheng:2007go} it is shown that (1) $R(P)$ is equivalent to the part of the lower envelope of $\operatorname{slabs}(P)$ which projects orthogonally onto the interior of $P$ and (2) the face with base edge $e$ can be defined as the lower envelope in the direction of $\vec{S}_e$ in the plane supporting $\operatorname{slab}(e)$ of the line segments given by intersecting all other slabs with $\operatorname{slab}(e)$. We call (2) the {\em local (2D) definition} for a face of the straight skeleton roof and use these two characterizations in the remainder of the paper. Each face of the straight skeleton roof is monotone with respect to the base edge of its supporting slab, and its boundary is the union of two monotone chains. Furthermore, the lower monotone chain, which includes only the base edge, is convex. \looseness=-1
\paraskip{Planar straight line graphs.} The slab based roof definition was extended to PSLGs in \cite{HuberH12}. Vertices may now have degree different from 2. It is convenient to apply the following operation so that the boundary components of each face are combinatorially simple polygons (all vertices of degree 2): for each face compute a walk of the edges of each connected boundary component of the face. Each time a vertex or edge is visited by such a walk it is duplicated so that the walk is combinatorially simple. For any degree 1 vertex encountered on the walk, add a small zero-length edge, and for the purposes of the induced motorcycle graph, consider that it makes an angle of $\pi/2$ with its two incident edges. This gives rise to the same induced motorcycle graph as before, and each reflex vertex is incident to a single motorcycle edge on the interior of the face. The slab for such a zero-length edge is made up of the union of its two motorcycle slabs.
\paraskip{Vertical Slab.} Given a line, ray, or line segment $l$ in the $xy$-plane, we define its {\em vertical slab} $H(l)$ to be the set of points $(x, y, z)$ in ${\bf R}^3$ where $(x, y)\in l$ and $z\geq 0$. This concept is used in our algorithm for PSLGs.
\paraskip{Assumptions.} We assume real-RAM computation. For now we also assume that the input polygon or PSLG is {\em non-degenerate}, meaning no two motorcycles crash simultaneously and in {\em general position}, meaning that the intersection of any two slabs is either empty or a line segment and no four slabs meet at a point. In Sec.~\ref{sec:generalposition} we show how to remove these assumptions while maintaining the same time bounds.
\section{Partial roofs for subchains of simple polygons}\label{sec:partialroofs}
We now show how to compute the straight skeleton roof $R(P)$ for a (weakly) simple polygon $P$. By ``weakly simple'' we mean that the interior of $P$ is topologically a disk and each part of the boundary of $P$ is incident to the interior. Figure~\ref{fig:simple} shows several examples of weakly simple polygons. This allows, for instance, two boundary edges to coincide as long as the interior of $P$ is on opposite sides of the two edges. See Fig.~\ref{fig:simple}a. We consider such edges and their corresponding slabs disjoint, meaning in particular that when we test for the intersection of two edges of $P$, the result is either a vertex of $P$, if the two edges are consecutive along $P$, or is {\em empty}. We assume that $P$ is given by a walk along the boundary of its interior, meaning in particular that we have no vertices of degree 1. If the polygon has sharp turns, meaning a vertex of degree $2\pi$, then we replace the vertex with a zero-length edge which, for the purposes of the induced motorcycle graph, makes right angles with its incident faces. Such an edge produces two motorcycles in the induced motorcycle graph, and the slab for such an edge is the union of its two motorcycle slabs. See Fig.~\ref{fig:simple}c. This allows us to treat such cases generically, rather than as special cases.
\begin{figure}
\caption{\small{Weakly simple polygons. (a) The two edges $e_1$ and $e_2$ overlap, but we consider them disjoint. (b) The left polygon has two edges which have an interior angle of $0$. We handle this by restricting $P$ only to the parts incident to the interior (right). (c) Since we treat the polygon as a walk, the apparent single edge on the interior is represented by two edges $e_1$ and $e_2$ making a sharp turn of $2\pi$ (left). We handle this by adding a little zero length edge $e$ between the two (shown here with positive length for visualization). The induced motorcycle graph is shown on the right.}}
\label{fig:simple}
\end{figure}
\paraskip{Overview.} A straightforward divide and conquer approach for computing $R(P)$ is to subdivide $P$ into equal length chains $C_1$ and $C_2$, recursively compute the lower envelopes of their defining slab sets and merge the result. However, the combinatorial complexity of the lower envelope of the slabs in a chain may be $\Omega(n^2\alpha(n))$\footnote{Where $\alpha(n)$ denotes the inverse Ackermann function.} \cite{ed1989} and finding all intersections between two lower envelopes is non-trivial. But, not all intersections between the lower envelopes of $\operatorname{slabs}(C_1)$ and $\operatorname{slabs}(C_2)$ appear in the final roof $R(P)$. For instance, an edge of $C_1$ is associated with only one face of $R(P)$ but its slab may appear as multiple faces in the lower envelope of $\operatorname{slabs}(C_1)$. This motivates our definition of a {\em partial roof}: the main idea is to define an intrinsic surface which has edges along all intersections that will eventually be part of the final roof, {\em but may not have edges for all intersections}. We then merge partial roofs of subchains by computing a path of local intersection between the two, reminiscent of Shamos and Hoey's Voronoi diagram algorithm \cite{Shamos:1975:CP:1382429.1382488}.
\paraskip{Extending ${\bf R}\xspace^3$.} We extend ${\bf R}\xspace^3$ with points at infinity, each of which is given by an equivalence class of vectors with the same unit vector. This allows us to conveniently represent slabs and parts of slabs as polygons. For example, a slab bounded by a single edge and two rays becomes a triangle with one vertex at the point at infinity equivalent to the slab's slope vector\footnote{In geometric group theory this is known as extending ${\bf R}\xspace^3$ by the {\em visual boundary}.}. We call a polygon with a point at infinity {\em unbounded}.\looseness=-1
\paraskip{Partial roof: definition.} A {\em partial roof} $R$ for a $k$-length subchain $C$ of a polygon $P$ is a piecewise linear surface, topologically a disk, with $k$ faces (one for each slab of $\operatorname{slabs}(C)$) and satisfies four properties (defined below): face monotonicity, face containment, edge containment, and the boundary property. The geometry of each face is defined by a simple (possibly unbounded) polygon on its supporting slab and two faces may be glued together along an edge which lies on the intersection of the two supporting slabs. We denote the boundary by $\partial R$.\looseness=-1
\begin{wrapfigure}{l}{0.6\textwidth}
\centering \subfloat[]{
\begin{minipage}[c][1\width]{0.13\textwidth}
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\includegraphics[width=\textwidth]{concepts1}
\end{minipage} }
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\begin{minipage}[c][1\width]{0.15\textwidth}
\centering
\includegraphics[width=\textwidth]{concepts2}
\end{minipage} }
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\includegraphics[width=\textwidth]{concepts4}
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\includegraphics[width=\textwidth]{concepts5}
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\caption{\small{(a) A polygon, induced motorcycle graph (dotted), and a 4 edge subchain (bold). (b) The straight skeleton with the faces incident to the subchain shaded. (c) A partial roof for the subchain from (a). The vertices $7$ and $11$ correspond to points at infinity. (d) The combinatorially representation of the underlying surface as a disk. {\em Note:} the dotted line in (c) shows an intersection between the realizations of the left and right-most faces that is not part of the underlying surface.}}
\label{fig:sskel}
\end{wrapfigure}
\paraskip{Intrinsic and extrinsic properties.} An important distinction is between the underlying {\em intrinsic surface}, which is given by the local geometry of each face and how the faces are glued together along edges, and the {\em extrinsic realization} of the surface, which is what the surface ``looks like'' in 3D. It is helpful to think of a partial roof as defined by cutting out each face from its supporting slab independently of the other faces and then gluing the faces back together along certain edges. In doing this we temporarily forget where each face sits in ${\bf R}\xspace^3$: each face simply has its own local geometry and the neighboring faces it is glued to. A realization is given by mapping back the vertices, edges, and faces into ${\bf R}\xspace^3$ in such a way that the local geometry of each face and its incidence with faces it is glued to is respected. Since the geometry of each face is defined on the surface of a slab residing in ${\bf R}\xspace^3$, we define the canonical realization of the surface, which we refer to as {\em the (canonical) realization} and denote by vertical bars $|\cdot|$, to be the one mapping each vertex, edge, and face back onto the original position on the slab that it was cut out from. It is important to note, however, that other realizations exist. (Think of ``folding'' the surface along its edges. This changes how the surface is situated in ${\bf R}\xspace^3$ and possibly introduces self-intersections, but does not change the underlying surface.) Note that a surface may exhibit self-intersections in a particular realization that are not present in the underlying intrinsic surface\footnote{A familiar example is that of a Klein bottle, which always locally looks (to any observer sitting on the surface) like a patch of the Euclidean plane, even though in any realization of the surface in ${\bf R}\xspace^3$ there is a self-intersection.}. A partial roof is intrinsically a disk, even though it may exhibit self-intersections in ${\bf R}\xspace^3$ which make the realization (if we forget the underlying intrinsic surface) something more topologically complicated. See Fig.~\ref{fig:sskel}c, d.
\paraskip{Properties.} The {\em face monotonicity property} is that each face is a (simple, possibly unbounded) polygon that is monotone with respect to the base edge of its supporting slab. The {\em face containment property} is that the realization of each face geometrically contains the final face for that slab in the final straight skeleton roof. The {\em edge containment property} is that if there is an edge $e$ of the final straight skeleton roof between slabs $s_1$ and $s_2$, and $s_1, s_2\in \operatorname{slabs}(C)$, then there exists an edge between the faces of $R$ supported by $s_1$ and $s_2$ whose realization geometrically contains $|e|$.
\begin{figure}
\caption{\small{Examples of possible faces (dark gray) defined on their supporting slabs (light gray). The slabs are oriented so that the base edge is the horizontal edge at the bottom. Thick lines denote the slab border chain; thin lines denote the interior chains; and dotted lines denote the slope edges.}}
\label{fig:facetypes}
\end{figure}
\paraskip{Faces and the boundary property.} Each face is a (possibly unbounded) polygon defined on the surface of its defining slab by at most two chains we call its {\em interior chains} which are monotone with respect to the slab's base edge. There are six distinct types of faces that are illustrated in Fig.~\ref{fig:facetypes}. Each interior chain starts at a {\em base point} and ends at a {\em terminal point}. The base point starts either on the left (right) motorcycle edge, if it exists, or left (right) endpoint of the base edge otherwise. Each interior chain is a monotone chain moving rightwards (leftwards). If there are two interior chains, then one starts on the left motorcycle edge or base edge endpoint and moves rightwards, and the other starts on the right motorcycle edge or base edge endpoint and moves leftwards. The two chains may only overlap vertically at their terminal points (i.e. the union of the two chains is monotone except possibly at the terminal points). The face is defined by shooting a ray from the terminal point of each interior chain upwards along the slab's slope vector. This subdivides the slab into (possibly unbounded) polygons, and the face is the polygon containing the base edge. In Figure~\ref{fig:facetypes} (a) illustrates a face defined by no interior chains, in this case the face is simply the entire slab. (b)-(d) illustrate faces defined by one interior chain, and (e) and (f) illustrate faces defined by two interior chains. In each case the thick black lines denote the parts of the face lying along the base and motorcycle edges, the thing black lines denote the interior chains, and the dotted lines denote the edges lying along the rays extended from each terminal point. In (b) the terminal point of the chain is somewhere on the interior of the face, in (c) the terminal point lies on the boundary of the slab, in (d) the terminal point lies on the opposite motorcycle edge. Note that in (c) the chain begins at the left endpoint of the base edge, since there is no left motorcycle edge. (e) and (f) illustrate faces defined by two interior chains. In (f) the terminal point of the right chain lies directly above the terminal point from the left chain.
Each face is made up of at most four chains: the (at most) two interior chains, the chain of edges lying along the base and motorcycle edges of the slab, which we call its {\em slab border chain} (drawn as thick lines in Fig.~\ref{fig:facetypes}), and the (at most) two {\em slope edges} of the {\em slope chain} (drawn as the dotted lines in Fig.~\ref{fig:facetypes}). The {\em boundary property} is (1) that the boundary edges of each partial roof are formed by two distinct chains: a {\em defining chain} containing the base edges of each face, and a {\em fringe chain} containing the remaining edges and (2) for a given face each edge on its interior chain is internal in the partial roof, each slope edge lies on fringe of the partial roof, and each motorcycle edge of the slab border chain are on the fringe of the partial roof if and only if the defining chain of the partial roof ends at the base edge endpoint incident to the motorcycle edge.
\paraskip{Discussion.} The face containment property ensures that every face is large enough that it contains the final face for its slab, so that the merge operation can ``cut down'' each face until it eventually becomes equal to the final face. The edge containment property (we will see below) ensures that when two slabs intersect {\em and we need to know about the intersection}, the intersection is represented by an edge of the partial roof. This is crucial, and is the reason we can forget about the other intersections. The basic idea is this: if an intersection between two faces exists geometrically on the realization of a partial roof but is not an actual edge of the underlying intrinsic surface, then by the edge containment property {\em no edge} along that intersection exists in the final roof. Thus, if we are merging two partial roofs along a path that hits one of these non-edge intersections, at that point we no longer care exactly how the partial roofs are represented--we are now on parts of faces that will eventually be cut away and are not part of the final roof. The face monotonicity property is used to bound the complexity of the merge operation. Finally, the boundary property allows us to bound the combinatorial complexity of the partial roof.
\begin{lemma}[Linear complexity of partial roofs]\label{lem:partialroofcomplexity} The combinatorial complexity of a partial roof for an $k$-length subchain of a simple polygon is $O(k)$.
\end{lemma}
\noindent{\em Proof.\xspace}\xspace The internal edges of $R$ form a forest (otherwise there would be a cycle of internal edges, contradicting that each face is incident to $\partial R$ along its base edge). We are going to ensure that any vertex of the forest lying on the boundary is a leaf node, which guarantees that all non-leaf nodes of the forest have degree at least 3. This is violated when a face $f$ is incident to the boundary at a vertex $v$ but the two edges of $f$ incident to $v$, $e_1$ and $e_2$ are internal to $R$. Conceptually split the tree at each such a vertex: replace any such vertex $v$ with a zero-length dummy edge between two vertices $v'$ and $v''$ such that $e_1$ is incident to $v'$ and $e_2$ is incident to $v''$. Having done this for all such vertices in every incident face, any vertex of the forest incident to $\partial R$ is now a leaf node and all leaf nodes of the forest lie on $\partial R$ (since each face of $R$ is simple). We now show that the number of leaves in this forest is $O(k)$, which bounds the number of internal edges and vertices of $R$ by $O(k)$. The result then follows from the fact that each face has at most $O(1)$ edges on $\partial R$ (by the boundary property) and $R$ has exactly $k$ faces. Since each face has at most $O(1)$ edges incident to $\partial R$, aside from the dummy vertices added above, there are $O(k)$ vertices on $\partial R$. We now bound the number of added dummy vertices. Assume that a face has two vertices $v_1$ and $v_2$ that are replaced by dummy vertices. First: neither $v_1$ nor $v_2$ are incident to the defining chain, since this would imply that they are endpoints of the base edge, and the base edge is incident to the defining chain, a contradiction. Now, since both $v_1$ and $v_2$ lie on the fringe and the face $f$, there must be a chain of interior edges incident to $f$ between $v_1$ and $v_2$. But this implies that there exists a face not incident to the base chain, a contradiction. Thus any face has at most one vertex that is replaced with a zero-length dummy edge. After replacing all such vertices, we are still left with $O(k)$ vertices on the boundary, completing the proof. \qed
We now show that the only object which meets the definition of a partial roof for the entire polygon is the final roof $R(P)$:
\begin{lemma}[A partial roof of the entire polygon is the straight skeleton]\label{lem:partialrooftoroof}Let $P$ be a simple polygon, $R(P)$ be its straight skeleton roof, and $R$ be a partial roof for $P$. Then $R(P)=R$.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Let $e_1$ be an edge of $P$. Then $e_1$ is the base edge of a face $f_1$ in $R$ and a face $f_1'$ in $R(P)$.
We first claim that for each edge of $f_1'$ there is a corresponding edge of $f_1$ which is equal to it (in ${\bf R}^3$). Let $e'$ be any edge of $f_1'$ which is not the base edge. Then there is a second face $f_2'$ of $R(P)$ incident to $e'$. Denote its base edge by $e_2$ and let $f_2$ denote the corresponding face in $R$. By the edge containment property there must exist an edge $e$ in $R$ which is incident to both $f_1$ and $f_2$ such that $|e|$ contains $|e'|$. Further, if $|e|$ strictly contains $|e'|$, then $f_1$ is not simple because the edges incident $e'$ also have corresponding edges in $f_1$ that contain them and one must be crossed by $|e|$, a contradiction. Thus $|e| = |e'|$. It follows that the faces $f$ and $f'$ are identical.\looseness=-1 \qed
\section{Merging partial roofs for a simple polygon}\label{sec:mergeop}
\begin{figure}\label{fig:mergeop}
\end{figure}
\paraskip{Procedure.} The merge operation takes as input two partial roofs $R_1$ and $R_2$ for co-incident subchains of a simple polygon and produces a partial roof $R$ for the the combined subchain. The basic idea is to start at the {\em gluing vertex} common to both chains and compute a walk of each surface which {\em locally} lies on the intersection of $|R_1|$ and $|R_2|$. The purpose of this walk is to detect all intersections between $|R_1|$ and $|R_2|$ which must exist as edges in $R$ to satisfy the edge containment property. {\em This may detect other intersections between $|R_1|$ and $|R_2|$ but will not necessarily detect all intersections}. We then cut each surface along the path discarding some of the subdivided faces and glue the two surfaces together along the path to form $R$.
More specifically, we (1) compute the {\em splicing path} on each surface by starting at the vertex $\hat{v}$ (which is incident to exactly one face $f_1$ of $R_1$ and one face $f_2$ of $R_2$). We then compute the intersection of $|f_1|$ and $|f_2|$ and walk along this intersection until we hit an edge of either face (say $f_2$). If the edge is not a boundary edge of the surface, we traverse across it to the next edge, say $f_3$, and continue along the intersection of $|f_1|$ and $|f_3|$. It should be noted that this computation is {\em local}, meaning that it ignores self-intersections which may intersect the path in the realization in ${\bf R}\xspace^3$ that are not represented in the underlying intrinsic surface. The walk stops if we hit a boundary edge of either surface, or we detect that the next edge of the walk is an edge that is provably not required to satisfy the edge containment invariant: i.e. the edge is on an intersection of two slabs that we can prove does not appear in the final straight skeleton roof (more on this below). Once we have computed the splicing path, then (2) we cut each face it traverses along the path which subdivides the face into two. One of the subdivided faces will be incident to the defining chain, and we discard the other. This makes the path a chain of boundary edges, and (3) we glue the two surfaces together along the two corresponding boundary chains. Finally (4) we perform a ``clean-up'' operation to ensure that the boundary property is maintained. Figure~\ref{fig:mergeop} illustrates a single merge. We now give more details on steps (1), (2), and (4). Step (3) is a common operation on piecewise linear surfaces.
\begin{wrapfigure}{r}{0.5\textwidth}
\centering \includegraphics[width=0.4\textwidth]{subdividestep}
\caption{\small{The subdivision of two faces along the splicing path (dotted). Left: before the subdivision. Right: afterwards. The side containing the base edge (bold) is retained.}}
\label{fig:subdividestep} \end{wrapfigure}
\para{Subdividing the faces.} For most faces the splicing path traverses the entire face, and subdividing the face along the splicing path is well defined. The only special case is the last face encountered. If the splicing path does not simultaneously encounter a boundary edge in both partial roofs, then in one of the roofs, say $R_2$, the last face $f$ encountered by the path is not completely cut into two. Let $x$ be the endpoint of the splicing path in $f$. Intrinsically on f, start at $x$ and trace the ray emanating from $x$ along the slope vector of $\operatorname{slab}(f)$. This either hits an edge of $f$ or escapes to infinity. In the first case, split the hit edge at the hit point by adding a vertex $y$ and subdivide $f$ by $p$ and an edge from $x$ to $y$. Otherwise, $f$ must have an infinite vertex, say $v_\infty$. Split $f$ by cutting along $p$, and then adding an edge from $x$ to $v_\infty$. See Fig.~\ref{fig:subdividestep}. \looseness=-1
\paraskip{Stopping the walk.} The basic idea behind the stopping conditions is that we only need the splicing path to find edges along the intersection of the two roofs that are necessary to satisfy the edge containment property. In Lemma~\ref{lem:mergeproducespartialroof} we prove by induction that all such edges constitute the first edges along the splicing path and correspond to a simple path of internal edges on the final straight skeleton roof. Because of this, if at any point in the computation of the splicing path we detect that the an edge (or the next possible edge) of the splicing path is provably not needed to satisfy the edge containment invariant, we can stop the splicing path walk. We use properties of the final straight skeleton roof $R(P)$ to detect when we arrive at an edge that provably cannot be an edge of $R(P)$.
In the following we use properties of a simple path $p$ of interior edges of the final roof $R(P)$ that starts at a boundary vertex $\hat{v}$ to determine the stopping conditions for the splicing path walk. First: since interior edges of the straight skeleton form a tree, it is not possible that two edges of $p$ be incident to the same face such that there is an intermediate edge between the two that is not incident to the face. Thus we stop the splicing path walk if the path re-enters a face it has already traversed.
Second: the edges of $p$ incident to a face $f$ that are not motorcycle edges form the upper monotone chain of $f$. Thus, if we detect that adding the next potential edge of the splicing path walk makes the splicing path non-monotone with respect to the base edge of the face it is currently in, we stop.
Third: let $f_1$ and $f_2$ be the faces incident to the left and right (resp.) of the last edge of $p$. The base edges of $f_1$ and $f_2$ split $P$ into two subchains. Let $C$ denote the subchain containing $\hat{v}$, $C_1$ denote the part of $C$ from $\hat{v}$ to $\operatorname{base}(f_1)$ and $C_2$ denote the part from $\hat{v}$ to $\operatorname{base}(f_2)$. Since $R(P)$ is a disk, all faces to the left of $p$ have their base edges on $C_1$. Similarly all faces to the right of $p$ have their base edges on $C_2$. Furthermore the interior edges of a straight skeleton roof are valleys if and only if they lie along lifted motorcycle tracks (cf. \cite{Cheng:2007go}). Thus we require (1) that the non-discarded parts of all faces of $R_1$ traversed by splicing path lie to the same side of the path (similarly for $R_2$ on the opposite side) and (2) that a splicing path edge becomes a valley in $R$ if and only if the edge lies along a lifted motorcycle track. If we detect that the next edge added along the splicing path will violate this condition, we stop.
\paraskip{Fringe simplification.} Discarding split faces in Step (2) may result in edges from the interior chain of a face becoming boundary edges on $\partial R$, thus violating the boundary property. Let $e_1, \dots, e_m$ denote the edges of a face $f$ that lie on $\partial R$ but are not part of the slab border chain. These edges form a connected chain along the boundary (we prove this in Lemma~\ref{lem:mergeproducespartialroof}). We use this property to perform the following clean-up which ensures that the boundary property is maintained. Let $e_1,\dots,e_m$ be the chain of edges of a face $f$ on the boundary of $R$ that are not slab border edges (i.e. lie along the base edge and motorcycle edges) and $u$ and $v$ be the endpoints of the chain. Each interior vertex of the chain has degree 2. We replace the chain by adding a new vertex $w$ at the point at infinity equivalent to the slope vector of the slab supporting $f$ and swapping out the chain with two edges $\edge{u}{w}$ and $\edge{w}{v}$. Note that this step may introduce self-intersections to the realization of the surface, but all operations are performed on the underlying intrinsic surface, and are simply contractions of boundary chains to shorter boundary chains. Thus, though the realization may not be a disk, the underlying surface remains one. See Fig.~\ref{fig:mergeop}e, f.\looseness=-1
We now investigate several properties used to prove the correctness of the merge operation.
\begin{lemma}\label{lem:splicingpathdefiningchain}The splicing path does not intersect the defining chain of either $R_1$ or $R_2$.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Assume that it does, then for some edge of the polygon, there exists a slab that intersects the edge. But each slab is incident to the $xy$-plane only along an edge of $P$, so $P$ is not (weakly) simple, a contradiction.\qed
\begin{lemma}\label{lem:disk}The (intrinsic) surface produced by a merge operation is topologically a disk.\end{lemma}
\noindent{\em Proof.\xspace}\xspace By definition the splicing path is a simple path on the interior of the (intrinsic) surface $R_1$ (resp. $R_2$) which is incident to the boundary at the gluing vertex (and possibly along a motorcycle edge). Lemma~\ref{lem:splicingpathdefiningchain} implies that if the splicing path traverses all the way to a boundary edge of $R_1$ (which splits $R_1$ into two disks) only one of them will contain the base edges. Thus all faces in the other are discarded. (Similarly for $R_2$.) We now show that discarding the remaining faces maintains that $R_1$ (resp. $R_2$) is a disk. Assume not. Then either the remaining faces form at least two separate connected components, or they form at least two topological disks that are incident only at a vertex. In the first case, since each face is incident to a base edge, the base edges from one component are disconnected from the base edges of the other, so the splicing path must cut the defining chain contradicting Lemma~\ref{lem:splicingpathdefiningchain}. In the second case, if the vertex is on the defining chain, we contradict Lemma~\ref{lem:splicingpathdefiningchain}; if it is not on the defining chain, then again we have that the base edges are dicsonnected. Finally, by the stopping conditions every discarded face lies on the same side of the splicing path, which is opposite to the side that is glued to the other surface. Removing these faces, then, removes a topological disk from $R_1$ which is incident to the boundary and does not touch the defining path. Thus $R_1$ (resp. $R_2$) is a disk after the faces are discarded and the gluing path remains an intact series of edges along the boundary, so $R$ must be (intrinsically) a disk. \looseness=-1 \qed
\paraskip{Correctness.}
We now prove correctness by showing that $R$ satisfies the properties of a partial roof (Sec.~\ref{sec:partialroofs}):
\begin{lemma}\label{lem:mergeproducespartialroof} The merge operation correctly computes a partial roof. \end{lemma}
\noindent{\em Proof.\xspace}\xspace Let $R$ denote the output surface, $R_1$ and $R_2$ denote the input partial roofs, $R(P)$ denote the final straight skeleton roof, and $C$, $C_1$, and $C_2$ denote the defining chains for $R$, $R_1$ and $R_2$. By Lemma~\ref{lem:disk} $R$ is topologically a disk. The face monotonicity property follows directly from the second stopping condition.
\paraskip{Face containment.} Suppose the face containment property does not hold in $R$. Then there exists some slab $s$ such that (without loss of generality) $s\in \operatorname{slabs}(R_1)$ and the face $f$ corresponding to $s$ in $R$ violates the face containment property. Let $f'$ denote the corresponding face of $R(P)$ and $f''$ denote the corresponding face of $R_1$. In particular, this means that $|f|$ does not contain $|f'|$. However, since $R_1$ is a partial roof $|f''|$ contains $|f'|$. Thus the splicing path must have cut through $f$. But the splicing path can only cut along intersections between the slab $s$ and other slabs in $\operatorname{slabs}(R)$. This means that there is an intersection between $|f'|$ and a slab in $\operatorname{slabs}(R)$ which contradicts that $f'$ is a face of $R(P)$. \looseness=-1
\paraskip{Edge containment.} This property has two pieces: an existence claim and a geometric containment claim. The basic idea of the proof of existence is to use induction along the splicing path to show that it contains all of the edges required to satisfy the property. Once we have that, the geometric containment follows the same line of reasoning as face containment above.
Let $e'$ be an edge of $R(P)$ incident to faces supported by slabs $s_1$ and $s_2$ such that $s_1$ and $s_2$ are slabs in $\operatorname{slabs}(C)$. Without loss of generality, there are two cases, either $s_1, s_2\in \operatorname{slabs}(C_1)$ or $s_1\in\operatorname{slabs}(C_1)$ and $s_2\in\operatorname{slabs}(C_2)$.
\paraskip{Case 1:} Since $R_1$ is a partial roof, there is an edge $e$ in $R_1$ incident to the faces $f_1$ and $f_2$ of $R_1$ that are supported by $s_1$ and $s_2$ such that $|e|$ geometrically contains $e'$. For contradiction, suppose that no such edge exists in $R$. Then the splicing path must cut $f_1$ and $f_2$ below $e$ (otherwise $e$ cannot have been discarded) or in such a way that $|e|$ only partially covers $|e'|$. In either case the after cutting $f_1$ and $f_2$ along the splicing path, they no longer maintain the face containment property, a contradiction.\looseness=-1
\paraskip{Case 2:} We claim that the faces $f_1$ and $f_2$ in $R$ supported by $s_1$ and $s_2$ are incident along some edge $e$ and $|e|$ contains $|e'|$. Since the edges of the straight skeleton form a tree, there exists a unique path $p'$ along the interior edges of the straight skeleton roof $R(P)$ from $\hat{v}$ to $e'$. We claim that $p'$ corresponds to the first part of the splicing path $p$. Let $k$ be the length of $p'$. The proof is by induction for $i$ from 1 to $k$.
\paraskip{Base step:} By definition, the first edge of both $p'$ and $p$ is along the intersection of the slabs of the base edges incident to $\hat{v}$. Geometric containment follows the argument as above.\looseness=-1
\begin{wrapfigure}{r}{0.41\textwidth}
\centering \includegraphics[width=0.40\textwidth]{prop8}
\caption{\small{The setup in $R(P)$ for the inductive step of the edge containment property.}} \label{fig:prop7}
\end{wrapfigure}
\paraskip{Inductive step:} Now assume the claim is true for the first $i<k$ edges of $p'$. Denote the edges of $p$ and $p'$ in order from $\hat{v}$ by $p_1, p_2, \dots$ and $p'_1, p'_2, \dots$, resp. Let $\bar{f}'_1$ and $\bar{f}'_2$ denote the faces incident to $p'_i$ and $\bar{f}_1$ and $\bar{f}_2$ be the faces of $R$ with the same base edges. Then $\bar{f}_1$ and $\bar{f}_2$ are incident along $p_i$ and $|p_i|$ contains $|p'_i|$. We prove that this holds for $i+1$. We first need to prove that $p$ contains an edge with index $i+1$. Let $v$ be the vertex between $p'_i$ and $p'_{i+1}$. By genericity there is one other internal edge, say $d'$, that is also incident to $v$. Denote the faces incident to $p'_{i+1}$ by $\bar{f}'_1$ and $\bar{f}'_2$ such that the base edges are on $C_1$ and $C_2$ (resp.). Without loss of generality assume that $d'$ is incident to $\bar{f}'_2$. Let $\bar{f}'_3$ be the other face incident to $d'$. Since $\bar{f}'_2$ and $\bar{f}'_3$ lie on the same side of $p'$ (by the stopping condition) the base edge of $\bar{f}'_3$ is on $C_2$. By the edge containment property on $R_2$, there is an edge $d$ in $R_2$ which is incident to two faces with base edges equal to the base edges of $\bar{f}'_2$ and $\bar{f}'_3$. Let $\bar{f}_1$, $\bar{f}_2$, and $\bar{f}_3$ be the faces of $R$ with base edges corresponding to $\bar{f}'_1$, $\bar{f}'_2$, and $\bar{f}'_3$ (resp.). By Case 1 above, $d$ is an edge of $R$ between $\bar{f}_2$ and $\bar{f}_3$. Since $d$ borders $\bar{f}_3$ and $|d|$ contains $|d'|$, then the splicing path between $\bar{f}_1$ and $\bar{f}_3$ must hit $d$ at $f_2$. Since $d$ is not a boundary edge, the splicing path continues along the intersection of $\bar{f}_1$ and $\bar{f}_2$. This edge is $p_{i+1}$. Geometric containment then follows by the same proof by contradiction as above. \looseness=-1
\paraskip{Boundary property.} Since the path enters a face along an interior edge, the new edges along the path are part of the interior chain defining the face. However, the cutting and discarding step may have introduced edges to the boundary of $R$ that were previously part of the interior chains.
We claim that before the fringe simplification step, such edges form a connected chain. Assume not, then there are at least two distinct subchains of such edges incident to the same face $f$ of $R$ that lie on $\partial R$. Since these chains are disconnected, there must be some interior edge of $R$ between them. But that edge must be incident to a face that is incident to a base edge. Since $R$ is a disk, however, this implies that the defining chain is disconnected, which can only happen if the splicing path intersects the defining chain, contradicting Lemma~\ref{lem:splicingpathdefiningchain}. Thus the fringe simplification step is able to find a single connected chain containing all the violating edges and replaces them with a single slope chain, restoring the boundary property. \qed
\paraskip{Existence.} Existence of a partial roof follows from the observation that for any edge $e$ of a simple polygon $P$, $\operatorname{slab}(e)$ is itself a partial roof for $e$. That a partial roof exists for any subchain $C$ now follows by induction.
\para{Running time.} We store each partial roof as a doubly-connected edge list (\cite{guibas:stolfi:85}) which handles most of the operations we need efficiently out-of-the-box. The only non-trivial part is finding the splicing path.
The basic idea is that to compute the walk we need to use ray shooting across each face, which is in general an expensive operation. However, due to the monotonicity of each face and the walk we can quickly compute a trapezoidal decomposition of each face in order to accelerate the ray shooting. This gives us:
\begin{lemma}\label{lem:splicingpathcomplexity}The splicing path can be computed in linear time and space.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Suppose we merge a partial roof $R_1$ for a subchain of length $k_1$ to a partial roof $R_2$ for a subchain of length $k_2$ and let $k=k_1 + k_2$. To prove the lemma, we show that the splicing path has at most $O(k)$ edges, that each potential next edge can be found in $O(1)$ time, and that the stopping conditions can be checked in $O(1)$ time per potential edge.
At each iteration, the walk lies on one face of $R_1$ and one face of $R_2$. The basic idea is to use the intersection of the realization of the two faces to compute a direction and shoot a ray (intrinsically) across each face to find the first edge of either face hit, then advance the splicing path in both faces along this ray to the closer hit-point. This adds an edge to the splicing path and in one of the partial roofs we cross an edge into a new face. Since we only continue until we hit a boundary edge or a face we have already traversed, the length of the final splicing path is at most $k$ and requires shooting $O(k)$ rays.
To compute the ray shooting, we exploit the monotonicity of the splicing path across each face (second stopping condition). Subdivide the the faces of both partial roofs into trapezoids by extending chords from each vertex on the interior of each face perpendicular to its base edge. Each internal vertex of a partial roof has degree 3 and thus is incident to at most 6 trapezoids. This gives us a bounds of $O(k)$ on the number of trapezoids generated. We now perform the same ray-shooting/walking scheme as above except in the trapezoids. The path now traverses trapezoids, but still cannot cross the same trapezoid twice since by the stopping conditions it must remain monotone to the base edge and cannot re-enter a face. Thus it has at most one edge on any trapezoid for a total length of $O(k)$. Shooting a ray in a trapezoid takes $O(1)$ time.
The first stopping condition is handled by marking each face when the splicing path enters it. When the splicing path enters a new face, we simply check whether the face has already been marked. This requires an $O(k)$ time preprocessing step to initialize each marker, and then an $O(1)$ check each time we compute an edge of the splicing path. The second stopping condition requires us to check whether the next potential edge of the splicing path is non-monotone to the previous edge and can be checked in constant time. The final check requires us to check for each edge if we stop the splicing path at that edge not along a motorcycle track, which side of the splicing path the discarded face lies on. If we stop the splicing path before it traverses the entire face, then we complete the subdivision of the face by shooting a ray upwards along the slope vector. From this it follows that the subdivided face that includes the base edge slopes downwards away from the splicing path. The only way this changes is if we exit a face before the splicing path has traversed above the base edge. In other words, the splicing path enters and exits the same motorcycle edge. In this case, the subdivided face containing the base edge lies on the upper side of the splicing path. To check this property we check whether the bottom edge of each trapezoid lies along a base edge or a motorcycle edge. We can then check, if the next edge is going to exit a face, whether that edge flips the base edge onto the wrong side of the splicing path in constant time.
By Lemma~\ref{lem:partialroofcomplexity} each partial roof can be represented by a DCEL using $O(k)$ storage, and the additional storage requirements are only a representation of the splicing path and storing the additional $O(k)$ trapezoids, each of which takes $O(k)$ space in total. \qed
Then we have as a direct corollary:
\begin{corollary}[Merging partial roofs in linear time]\label{lem:merge} Given two partial roofs with defining chains that are co-incident subchains of a simple polygon $P$, there is an algorithm for computing a partial roof of the concatenated defining chains in $O(k)$ time. \looseness=-2 \end{corollary}
\para{Proof of Main Theorem for simple polygons.} Given the merge operation the procedure for computing the straight-skeleton is surprisingly straightforward: subdivide the polygon into equal length subchains, recursively compute a partial roof for each, and merge the results to produce the straight-skeleton roof. Taking Lemmas~\ref{lem:partialroofcomplexity} and \ref{lem:partialrooftoroof} and Corollary \ref{lem:merge} together we have:
\begin{theorem}\label{thm:polygon}The straight skeleton of a simple polygon can be computed from its induced motorcycle graph in $O(n\log n)$ time and $O(n)$ space.\end{theorem}
\section{Extending to planar straight line graphs}\label{sec:extension}
\paraskip{Overview.} We now show how to compute the straight skeleton roof of a planar straight line graph. The straight skeleton on the interior of each face of a PSLG is independent of the other faces. For this reason we focus here on the straight skeleton of the outer face. This is the most general case because its straight skeleton may include both bounded and unbounded faces. Let $R(G)$ denote the straight skeleton roof for a PSLG $G$ and $F$ denote its outer face. We will call the part of $R(G)$ that orthogonally projects onto $F$ in the $xy$-plane the {\em restriction of $R(G)$ to the region above $F$} and denote this $R(F)$. In fact, for any planar region $C$ we will denote by $R(C)$ the restriction of $R(G)$ to the region above $C$. The main idea is to subdivide the interior of $F$ into a set of cells $\{ C_i\}$, each of which captures the essential properties that allow the divide and conquer algorithm for simple polygons to work. Each cell $C_i$ has a defined slab set $\operatorname{slabs}(C_i)$ called a {\em subdivided slab set}, and the restriction of the lower envelope of $\operatorname{slabs}(C_i)$ to the area above $C_i$ is equal to $R(C_i)$. As before we define a notion of a {\em partial roof} to subchains of the boundary of each $C_i$ and then use our divide and conquer approach for polygons to compute $R(C_i)$.
\paraskip{Essential properties.} We now informally examine the essential properties used by our polygon algorithm that each cell $C_i$ captures. The first essential property is that the edges of the polygon $P$ represent known parts of the roof $R(P)$, namely $P$ is equal to the intersection of $R(P)$ with the $xy$-plane. For each cell $C_i$ we compute a lifting of its boundary, denoted $\partial C_i$, onto the final roof $R(F)$. This lifting can be computed efficiently for all cells without having to compute the entire $R(F)$. The lifted edges $\partial C_i$ take the place of the edges of $P$ in the polygon algorithm. The second essential property is that each slab in $\operatorname{slabs}(P)$ is incident to $P$ along its base edge. For a cell $C_i$, we require that each slab in $\operatorname{slabs}(C_i)$ be incident to $\partial C_i$ along a connected chain of edges. In particular these first two properties mean that we can employ the same divide and conquer approach: given two subchains of $\partial C_i$ incident to the same vertex, the vertex represents a known starting point along an intersection of two slabs that must appear on any partial roof for the merged chains. The third essential property is that the straight skeleton of a polygon $P$ is a tree (and thus acyclic), which is used in the proof of correctness to prove the edge containment property. For a cell $C_i$ we require that the interior edges of $R(C_i)$ (which are the lifted straight skeleton edges on the interior of $C_i$) be acyclic.
\paraskip{The subdivided roof.} Our algorithm computes a subdivision of the final roof $R(F)$. Let $C_1, C_2, \dots$ be a subdivision of $F$ into cells. Denote by $\partial C_i$ the lifting of the boundary of the cell $C_i$ onto $R(F)$. The lifting of the boundary of all cells induces a subdivision on $R(F)$ we call the {\em subdivided roof} and denote by $\hat{R}(F)$. Figure~\ref{fig:pslg14}a depicts the projection of the final straight skeleton roof $R(G)$ of a PSLG $G$ onto the $xy$-plane. Figure~\ref{fig:pslg58}b shows a subdivided roof $\hat{R}(F)$ for the outerface $F$ of $G$ produced by a particular cellular subdivision (the blue dashed and solid lines are the added edges used to form the subdivision). To perform the subdivision we use a modified version of the vertical subdivision procedure from \cite{chengArxiv2014} to divide $F$ into cells. We then use our divide and conquer approach for polygons to compute $R(C_i)$ for each cell $C_i$. The output of our algorithm is the subdivided roof $\hat{R}(F)$. From this the final roof $F$ can be constructed in a linear time by merging subdivided faces across cell boundary edges.
\paraskip{The subdivided slab set.} The subdivided roof presents one technical problem. A slab $s\in\operatorname{slabs}(F)$ supports only one face of $R(F)$, but may support more than one face of $\hat{R}(F)$. To handle this we extend the subdivision procedure from \cite{chengArxiv2014} to subdivide each slab $s$ resulting in a set of {\em subdivided slabs} for each cell $C_i$, denoted $\operatorname{slabs}(C_i)$. It is the lower envelope of these subdivided slabs that forms the roof $R(C_i)$.
\paraskip{Notation and assumptions.} As before we extend ${\bf R}\xspace^2$ and ${\bf R}\xspace^3$ with points at infinity which are equivalence classes over vectors that point in the same direction. We let $n$ denote the number of edges in $F$ and $m$ denote the number of boundary components. To simplify the presentation we assume that: (1) no edge of $F$ is parallel to the $x$ or $y$-axes and (2) no angle bisector line of the lines supporting any two edges of $\partial F$ is parallel to either axes\footnote{This can be enforced by the following $O(n\log n)$ preprocessing step which finds a small rotation to apply to $G$ to ensure the property holds. Let $L_1, \dots, L_n$ denote the lines through the origin parallel to the $n$ edges of $G$. Sort these by angle made with the $x$-axis. Let $\epsilon_x$ and $\epsilon_y$ be the smallest non-zero angles between any line and the $x$ and $y$ axes (resp.). Clearly, applying a small rotation by any angle $\alpha < \min\{\epsilon_x, \epsilon_y\}$ is sufficient to ensure that now edges are parallel to the $x$ or $y$ axes, since the rotation will make any edges currently parallel to either axes non-parallel but is too small to make any edge not yet parallel into a parallel edge. Now, for each line $L$, find the line $L'$ such that the slope $M$ of $L$ and the negative slope $M'$ of $L$ are closest but not equal (this can be done easily in $O(n\log n)$ time using a modified 1D closest pair algorithm). Given an $L$, this is the same as finding the line $L'$ such that the bisector lines of $L$ and $L'$ are closer to parallel to the $x$ and $y$ axes than $L$ with any other line $L''$. Let $\theta$ be the smallest angle of rotation that aligns the bisector lines of $L$ and $L'$ with the axes. Let $\theta'$ be the smallest non-zero angle over all such $\theta$. Then any rotation $\alpha < \theta'$ will ensure that the bisector lines of all pairs of lines $(L, L')$ are not the axes. Thus we apply a rotation of $\alpha = (\min\{\epsilon_x, \epsilon_y, \theta'\})/2$ to $G$. }. We also assume that each connected boundary component of $F$ is combinatorially a simple polygon. This can be ensured by ``walking around'' the part of each component incident to $F$. Sharp turns in the walk (of angle $2\pi$) are handled as in the polygon case by adding a zero-length line segment which we think of as making right angles with its neighboring edges. See Fig.~\ref{fig:pslg14}c.
\subsection{The subdivision procedure}
We use a modification of the vertical subdivision procedure from \cite{chengArxiv2014}. The basic idea is to use vertical lines (in the $xy$-plane) to partition the interior of $F$ into cells. The lines are defined with respect to a particular set of subdivision points. In \cite{chengArxiv2014} these points are chosen so that the part of the final roof on the interior of each cell is convex, which allows them to use existing techniques to efficiently compute the lower envelope of the supporting planes of slabs for a particular cell rather than the lower envelope of the slabs. However, to achieve this they require $O(r)$ subdivision points where $r$ is the number of reflex vertices. We use strictly fewer subdivision points: exactly one for each connected component of $F$. We also note that though they employ their subdivision procedure only for polygons with holes, the subdivision algorithm extends naturally to PSLGs. {\em Note the the reader:} there is one difference in terminology: in order to treat edge and motorcycle slabs in a unified manner we have defined a single slab for each edge of $F$ to be the union of the edge and motorcycle slabs (see Sec.~\ref{sec:preliminaries}). As was noted previously, this simplification was also employed by \cite{Huber:2011kr}. In \cite{chengArxiv2014}, instead of merging the edge and motorcycle slabs into a single slab, they subdivide each face of the final straight skeleton into the components that lie entirely on the edge and motorcycle slabs using what they call {\em flat edges} that lie along the common boundary of an base edge's edge and motorcycle slabs. Both of these methods are essentially equivalent and are semantic tools to allow us to simplify the discussion.
\begin{figure}\label{fig:pslg14}
\end{figure}
\begin{figure}\label{fig:pslg58}
\end{figure}
\paraskip{Preliminaries.} To understand the vertical subdivision procedure of \cite{chengArxiv2014}, we need several concepts. Given a point $p$ on the final roof $R(F)$, a {\em descent path} is the path of steepest descent from that point $p$ down until it hits an edge of $F$. If $p$ is on the interior of a face $f$ of $R(F)$, there is exactly one descent path from $p$. Otherwise there is one descent path for each face incident to $p$. Each descent path first follows a segment parallel to the slope vector of $f$ downwards until it hits either the base edge of $f$, or a motorcycle edge of $f$. We call this segment the {\em descent edge} of $p$ in $f$. Each motorcycle edge forms a valley in $R(F)$, and so the descent path then travels the rest of the way down the motorcycle edge until it hits the base. If $p$ lies on an edge or vertex of $R(F)$, then it will have a descent path for each face it is incident to. Each descent path lies entirely on a single face, and given a point $p$ which is known to reside on a face $f$ of $R(F)$, the descent path from $p$ can be found knowing only the slab $s$ supporting $f$. In other words, to compute a descent path in a face it suffices to know a single point on the face and the slab containing the face. This property is useful in that it allows us to compute descent paths without first computing the entire roof. See \cite{Cheng:2007go} for more details.
The subdivision procedure also makes use of {\em vertical lines} and {\em vertical planes}. A vertical line is a line $l$ in the $xy$-plane parallel to the $y$-axis and a vertical plane is plane through a vertical line that is orthogonal to the $xy$-plane. Each point $v$ in the $xy$-plane has a unique vertical line and vertical plane through it.
\paraskip{Intersecting a vertical plane with $R(C_i)$.} The subdivision procedure makes use of the following subroutine: given a vertical plane $X$ and a cell $C_i$, the intersection of $X$ with $R(C_i)$ can be found in $O(k\log k)$ time {\em without first computing the roof $R(C_i)$} (where $k = |\operatorname{slabs}(C_i)|$). This is done by intersecting each slab $s\in \operatorname{slabs}(C_i)$ with $X$. The intersection of each slab with $X$ is a line segment, and the intersection of $R(C_i)$ with $X$ is the lower envelope (in $X$) of these line segments. The lower envelope of the segments can be found in $O(n\log n)$ using the algorithm from \cite{Hershberger:1989:FUE:79765.79766}. For more information we refer the reader to \cite{Cheng:2007go}. Each of these line segments lies along the roof in ${\bf R}\xspace^3$, but also has a projection onto the $xy$-plane. For each such edge we will refer to its {\em lifting} into ${\bf R}\xspace^3$ and its {\em projection} onto the $xy$-plane. The projected edges are used to subdivide $F$ into cells, but we simultaneously keep track of the lifting of the boundary of each cell into ${\bf R}\xspace^3$.
\begin{wrapfigure}{r}{0.25\textwidth}
\centering \includegraphics[width=0.24\textwidth]{subdivcell.pdf}
\caption{\small{The subdivision from Fig.~\ref{fig:pslg58} without the final straight skeleton edges depicted. Each connected shaded region is a cell. }}
\label{fig:celldiv} \end{wrapfigure}
\paraskip{Subdividing $F$ into cells.} The subdivision is divide and conquer. At the beginning, we select a set of {\em subdivision points} $V$. At each point in the algorithm we have a division of $F$ into some number of cells $C_1, C_2, \dots$. Each cell $C_i$ is such that the restriction of the lower envelope of its slab set, $\operatorname{slabs}(C_i)$, to the region above $C_i$ is equal to $R(C_i)$. Each cell also maintains a {\em conflict list} $V_i$ of the points of $V$ on its interior.
\begin{wrapfigure}{r}{0.63\textwidth}
\centering \subfloat[]{\includegraphics[width=0.13\textwidth]{subdiv1}} \subfloat[]{\includegraphics[width=0.117\textwidth]{subdiv2}} \subfloat[]{\includegraphics[width=0.115\textwidth]{subdiv3}} \subfloat[]{\includegraphics[width=0.12\textwidth]{subdiv4}} \subfloat[]{\includegraphics[width=0.1\textwidth]{subdiv5}}
\caption{\small{The subdivision induced by a line $l$ on a face of $R(F)$ and its supporting slab. (a) The face $f$, its slab $s$, and the lifted subdivision line $\hat{l}$. (b) the edges induced by $\hat{l}$, $e_1$ and $e_2$ lie on $f$, $e'_0$ and $e'_1$ lie on $s$ but not on $f$. (c) The slab-columns and face-columns induced by $\hat{l}$. (d) The final subdivision slabs. (e) The final subdivision faces.}}
\label{fig:facesubdiv}
\end{wrapfigure}
The initial cell $C_1=F$. The boundary of $C_1$ is given by the connected boundary components of $F$, $\operatorname{slabs}(C_1) = \operatorname{slabs}(F)$, and $V_1 = V$. Each recursive step takes a cell $C_i$ for which $|V_i|\neq 0$ and $|\operatorname{slabs}(C_i)|>2$. It selects the point $v\in V_i$ with median $x$-coordinate, and then finds the intersection of the vertical plane through $v$ with $R(C_i)$ using the subroutine outlined above. We call the segments along this intersection {\em vertical edges}. The endpoints of each vertical edge lie on edges of $R(F)$ and represent intersection points between slabs on the final roof $R(F)$. From each intersection point we trace the descent paths in each of the incident faces of $R(F)$. The projection of the vertical edges and descent paths onto the $xy$-plane further subdivides the cell $C_i$. See Figs.~\ref{fig:pslg14} and \ref{fig:pslg58}. In \cite{chengArxiv2014}, they distinguish between three types of cells: {\em empty cells}, which have a single slab in $\operatorname{slabs}(C_i)$, {\em wedge cells}, which have two slabs in $\operatorname{slabs}(C_i)$ and the part of the straight skeleton on their interior is just the part of the projection of the intersection line of the two slabs that lies on the interior of $C_i$ (a single edge), and the remaining {\em general cells}, which have more than two slabs in $\operatorname{slabs}(C_i)$. The empty cells and wedge cells are base cases for the divide and conquer and are not further subdivided.
\paraskip{Subdividing faces and slabs.} The subdivision procedure above induces a subdivision on the faces of $R(F)$. When we subdivide $F$ into cells, we do so along vertical edges and descent paths. In both cases the subdivision edges lie on a particular face of $R(F)$ and the subdivision of $R(F)$ induced by the lifting of these edges is the subdivided roof $\hat{R}(F)$. But, a single slab of $\operatorname{slabs}(F)$, which supports only one face of $R(F)$, may support multiple faces of $\hat{R}(F)$. For this reason we also subdivide each slab in order to maintain that each slab supports exactly one face.
Let $f$ be a face of $R(C_i)$ which is subdivided during the subdivision of $C_i$. Let $X$ denote the vertical subdivision plane, $l$ denote the vertical line contained in $X$, $s$ be the supporting slab of $f$ in $\operatorname{slabs}(C_i)$, and $\hat{l}$ be the line supporting the intersection of $X$ with $s$. Note that because we assume no edge of $F$ is parallel to the $x$ axis (and thus perpendicular to $l$), $\hat{l}$ is not parallel to the slab's slope vector. Let $e_1, \dots, e_p$ be the vertical edges on $f$ (i.e. the connected components of $f\cap X$). These subdivide $\hat{l}$ into a series of segments: $e'_0, e_1, e'_1, e_2, e'_2, \dots, e'_{p-1}, e_p, e'_p$ where each $e'_i$ lies on $s$ but not on $f$ (Fig.~\ref{fig:facesubdiv}(b)). Let $l_p$ denote the line through a point $p$ on $s$ parallel to the slope vector of $s$. The induced subdivision on both $f$ and $s$ is defined by first subdividing $f$ and $s$ along $l_p$ for all points $p$ that are an endpoint of one of the edges $e_i$ not on the boundary of $s$ (Fig.~\ref{fig:facesubdiv}(c)). This divides $f$ into a series of {\em face-columns}, each of which is bounded below by some part of the boundary of $s$, and above some part of the upper monotone chain of $f$. This divides $s$ into a series of unbounded {\em slab-columns}, each of which contains exactly one face-column. Each slab-column either contains a vertical edge $e_i$ or contains one of the edges $e'_i$. In the first case, we further subdivide both the slab-column and face-column along $e_i$. In the second case we subdivide only the slab-column by $e'_i$ and keep only the part containing the corresponding face-column. We call these the {\em subdivided slabs and faces} (Fig.~\ref{fig:facesubdiv}(d,e)). We show an example of a face split by three lines in Fig.~\ref{fig:threesplit}. This procedure inductively maintains the following properties:
\begin{figure}\label{fig:threesplit}
\end{figure}
\begin{lemma}[Properties of subdivided slabs.]\label{lem:subdivprops} (1) Each subdivided slab supports a single subdivided face. (2) The edges of a subdivided face that lie along boundary edges of the slab are also edges of one of the cells produced by the subdivision, and these form a connected chain of edges on the boundary of the cell. (3) The combinatorial complexity of each slab is $O(1)$. (4) each subdivided slab and face is monotone with respect to the base edge of the original slab. (5) Letting $\operatorname{slabs}(C_i)$ denote the subdivided slabs incident to the edges in $\partial C_i$, the restriction of the lower envelope of $\operatorname{slabs}(C_i)$ to the region above $C_i$ is the roof $R(C_i)$. (6) Each vertex of $\partial C_i$ is incident to at most two slabs of $\operatorname{slabs}(C_i)$. \end{lemma}
\noindent{\em Proof.\xspace}\xspace (1) follows by induction from the definition. (2) follows from induction and the fact that the face is subdivided by the the vertical edges and descent paths, which are also the edges used to subdivide the cell that originally contained the face. (3) The original slabs have $O(1)$ complexity. A subdivided slab is given by ``clipping'' the slab between at most two lifted vertical subdivision lines and at most two descent edges parallel to the slab's slope vector. Thus each subdivided slab contains some connected portion of the original slab plus $O(1)$ additional edges. (4) The subdivision into slab-columns and face-columns maintains this property since the cuts are along the direction of monotonicity. The final cut along a vertical edge cuts clear across a column, by definition, since its endpoints define the sides of the column, which necessarily maintains monotonicity. (5) Suppose not. Since the subdivided slabs in $\operatorname{slabs}(C_i)$ cover the faces of $R(C_i)$, this means that one of the slabs in $\operatorname{slabs}(C_i)$ must appear in the lower envelope below some face $f$ of $R(C_i)$ that it does not support. However, since each subdivided slab is a subset of its original slab and each face of $R(C_i)$ is a subset of its original face, and the subdivided slabs cover the faces, then this means some part of $f$ cannot be part of the final roof $R(F)$, a contradiction. (6) This is true initially, since each vertex of $\partial F$ is incident only to the two slabs for its incident edges. Now assume it is true for a cell $C_i$ which is subdivided by a line $l$. Let $p$ be an intersection point along $l$. Then each face of $R(C_i)$ that is incident has a descent path traced from $p$. The descent paths form the new cell boundaries and separate the subdivided faces and slabs so that only two of each end up in any subdivided cell.\qed
\begin{figure}\label{fig:cellcompose}
\end{figure}
\paraskip{Properties of the subdivision.} Let $C_1, C_2, \dots$ denote the cells computed during the subdivision (including intermediate cells). We note the following properties, which are proven in \cite{chengArxiv2014}. Let $\kappa_i$ denote the number of edges of $R(F)$ intersecting the interior of $R(C_i)$. Then $\sum_i \kappa_i = O(n\log |V|)$, the number of edges in $\partial C_i$ is $O(\kappa_i)$, and computing the subdivision takes $O(n(\log n)\log |V|)$ time. Note that they use a different (and larger) set of points $V$ than we do, namely their $V$ is the set of vertices of the induced motorcycle graph, but the proofs for the properties above generalize both to unbounded straight skeleton edges, and point sets $V$ in which each point $v\in V$ lies on the boundary of $F$.
\begin{figure}\label{fig:cellcompose}
\end{figure}
\paraskip{The subdivision points.} In \cite{chengArxiv2014}, they choose $V$ to be the set of vertices of the induced motorcycle graph, so that $|V| = \Theta(r)$.
For our point set $V$ we arbitrarily select a point (which need not be a vertex) from each connected boundary component of $F$. We could, for instance, choose one reflex vertex from each connected component, making our choice of $V$ a strict subset of the one in \cite{chengArxiv2014}. The purpose of our choice of $V$ is to ensure that each cell is (weakly) simple, meaning:\looseness=-1
\begin{lemma}\label{lem:noholes}If $C_i$ is a final cell of the subdivision, then its interior contains no connected component of $F$.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Assume not. Then there must be a connected component on the interior of $C_i$. We first argue that $C_i$ has at least three slabs. Assume not. Since our connected components are combinatorial polygons, each has at least two edges, and thus $C_i$ has at least two slabs. The boundary of $C_i$ must lie along each of these slabs, but then there exists some intersection point on $C_i$ from which we would have traced a descent path downwards to the base edge of the slab, which is an edge of the connected component, contradicting that the connected component is not part of $\partial C_i$. Now, the connected component on the interior of $C_i$ necessarily has a subdivision point in $V$. Thus $V_i$ is non-empty and $|\operatorname{slabs}(C_i)| \geq 3$, which contradicts that $C_i$ is a final cell of the subdivision.\qed
\begin{lemma}\label{lem:contree} If $C_i$ is a final cell of the subdivision, then the interior edges of $R(C_i)$ are acyclic.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Assume there is such a cycle. Vertical and descent edges always become cell boundary edges, so all the edges of the cycle are edges of the final $R(F)$. There must be a face on the interior of the cycle, and since it is not sub-divided by any cell boundary it must contain a base edge which is not connected to $C_i$ by cell boundary edges. This implies that $C_i$ contains a connected component of $F$ on its interior, contradicting Lemma~\ref{lem:noholes}. \qed
We now have:
\begin{lemma}\label{lem:slabcount}The total number of subdivided slabs and the total number of edges in all cells is $O(n\log m)$.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Each subdivided slab is incident to at least one edge of its cell $C_i$ along its defining chain. The number of edges of $C_i$ is $O(\kappa_i)$ and the total number of subdivided slabs is less than the total number of edges over all cells, which is $O(\sum_i\kappa_i) = O(n\log m)$.\qed
\subsection{The partial roof of a cell}
\paraskip{Filling in each cell.} We now have that each final cell $C_i$ has a set of subdivided slabs, each of which is incident to it along a chain of edges, called the slab's {\em defining edges}. This is slightly different than in the polygon case, where each slab is incident to the polygon only along a single base edge. When we split $C_i$ into subchains, we keep all edges incident to the same slab together (in other words we split $C_i$ into two subchains with equal size slab sets). The definition of partial roofs extends naturally to subchains of $C_i$.
\paraskip{Partial roof of a subchain of a cell.} The partial roof $R$ of a $k$-length subchain $C$ of a cell $C_i$ is a piecewise linear surface, topologically a disk, which has exactly one face supported by each slab $s\in\operatorname{slabs}(C_i)$. The partial roof satisfies the (modified) edge containment, face containment, face monotonicity, and boundary properties.
\paraskip{(Modified) Properties.} The face and edge containment properties are defined the same way as for simple polygons, except in reference to the faces and edges in $R(C_i)$. The face monotonicity property is that each face $f$ of $R$ is monotonic with respect to the base edge of the original slab from which $f$'s supporting subdivided slab came from. The boundary property is essentially the same. Instead of each face being incident to $\partial R$ along a single base edge, it is now incident to $\partial R$ along the entire defining chain of its slab. The face again has a slab border chain, which includes the defining edges of the slab. Again we allow one or two interior chains of edges that are incident to other faces of $R$. Now the slope chain is defined as before, if the subdivided slab is unbounded. However, the subdivided slab may be bounded above by a lifted subdivision line, in which case we ``cap'' the unbounded edges with an edge along the subdividing line. In other words the slope chain may not either be two unbounded edges along the slope of the slab as before, or two bounded edges and a cap edge. Figure~\ref{fig:labeledchains} illustrates these slight variations. The boundary of $R$, denoted $\partial R$ again is made up of two chains: a defining chain which includes the defining edges of each slab in $\operatorname{slabs}(C)$ and a fringe chain, containing the remaining slab border edges and slope edges of each face.
\begin{lemma}\label{lem:partialroofcomplexitypslg}The combinatorial complexity of a partial roof $R$ for a $k$-length subchain $C$ of a cell $C_i$ is $O(k)$\end{lemma}
\noindent{\em Proof.\xspace}\xspace Each face as $O(1)$ vertices on $\partial R$ that are not part of its defining chain. There are $O(k)$ vertices on the defining chain. Each face is incident to $\partial R$ along its defining chain, which contains at least one edge. Thus the same argument as in in Lemma~\ref{lem:partialroofcomplexity} applies directly to prove the result.\qed
And from the face and edge containment properties we have:
\begin{lemma}\label{lem:partialrooftoroofpslg} Let $R$ be a partial roof for the entire cell $C_i$. Then $R = R(C_i)$. \end{lemma}
\noindent{\em Proof.\xspace}\xspace The proof follows the same reasoning as Lemma~\ref{lem:partialrooftoroof}, which uses only the edge and face containment properties and so applies generally to bounded and unbounded cells $C_i$.\qed
\paraskip{Merging partial roofs.} The merge operation is exactly the same except that vertices of the cell may be at infinity in the positive or negative $y$-direction, and the fringe simplification step has to take into account if the subdivided slab is bounded or unbounded. If a gluing vertex were at infinity, this could present a problem, since merging at this vertex would no longer have the property of starting at a known intersection point. However, since our preprocessing step ensures that no lines of the straight skeleton are vertical in the $xy$-plane, this cannot occur.
\begin{lemma}\label{lem:finitegluing}No gluing vertex $\hat{v}$ is an infinite vertex.\end{lemma}
\noindent{\em Proof.\xspace}\xspace Suppose for contradiction that a cell $C_i$ has an infinite vertex $v$ such that the two edges $e_1$ and $e_2$ of $\partial C_i$ incident to $v$ come from different slabs. Recall that the edges of $\partial C_i$ are of three types: edges of the PSLG, vertical edges created by the subdivision step, and descent edges. Of these only the vertical edges can possibly be unbounded, so $e_1$ and $e_2$ are both vertical edges. Furthermore, since they are incident to the same vertex they must appear on different subdivision planes. There must, then, be some unbounded face $f$ incident to $e_1$ that is not incident to $e_2$. Then $f$ must have some unbounded edge projecting onto the interior of $C_i$. This means that the projection of this edge onto the $xy$-plane is parallel to the $y$-axis (otherwise it would have been subdivided by both the subdividing plane through $e_1$ and through $e_2$ and would not be unbounded), a contradiction.\qed
We also note that the cells to the left of the left-most subdivision line and to the right of the rightmost subdivision line are not polygons, but rather polygonal chains unbounded at each end by an infinite vertex. This also cannot be a gluing vertex, since it is an endpoint of the chain, and is therefore not between two edges. By the containment properties, the partial roof for the chain is equivalent to restricted roof for the cell, and so the algorithm works correctly on the cell. When we apply the fringe simplification step to a bounded subdivided slab, instead of extending out to infinity, the two endpoints are extended up to the lifted subdivision line which bounds the slab.
\begin{figure}
\caption{\small{Left: a slab, a face, and a possible induced subdivision of the slab and face if it is part of a PSLG. Middle: an illustration of the parts of a face of a partial roof that appear in the polygon case. The slab chain lies along the boundary of the slab, which is made up of the base edge and motorcycle edges, has two interior chains which are made up of edges incident to other faces of the partial roof, and a single slope chain, which in this case is given by two unbounded edges. Right: a face of a partial roof (built on the same slab) in the case of a PSLG. In this case, the subdivided slab does not contain the base or motorcycle edges, so the slab chain includes edges along the lifted subdividing line and descent path that define the subdivided slab. The face also has an interior chain of edges incident to other face of the partial roof, and a slope chain, which includes a single edge parallel to the slope vector of the slab, and a cap edge along the upper lifted subdivision line.}}
\label{fig:labeledchains}
\end{figure}
Finally, letting $c_i = |\operatorname{slabs}(C_i)|$ we have:
\begin{lemma}The roof $R(C_i)$ for a final cell $C_i$ can be computed in $O(c_i\log c_i)$ time and $O(c_i)$ space\end{lemma}
\noindent{\em Proof.\xspace}\xspace We prove that the merge operation is correct. The lemma then follows from Lemmas~\ref{lem:partialroofcomplexitypslg} and \ref{lem:partialrooftoroofpslg}. Lemma~\ref{lem:finitegluing} ensures that when we start each merge step, the gluing vertex $\hat{v}$ is a known point on $R(C_i)$. This point is, by Lemma~\ref{lem:subdivprops} property (6), incident to two faces, $f_1$ and $f_2$ of $R(C_i)$ that are incident along an edge with $\hat{v}$ as an endpoint. By the face containment property, the two faces intersect at $\hat{v}$, and so the merge operation is able to get started, as in the polygon case.
The face containment property follows the same argument as for the polygon case: since $R(C_i)$ is the lower envelope of $\operatorname{slabs}(C_i)$ and we only cut along intersections between slabs, it is not possible that we cut a face so that it fails to satisfy the property.
For the edge containment property, the main difference is that before we had a tree and now we have a forest of interior edges on the final $R(C_i)$ (Lemma~\ref{lem:contree}). We need to show that given an edge $e'$ of $R(C_i)$ supported by subdivided slabs $s_1$ and $s_2$, such that $s_1$ is on the defining chain of one of the merging roofs and $s_2$ is on the defining chain of the other, we can get a path of interior edges in $R(C_i)$ back to the gluing vertex $\hat{v}$. Suppose not. Then the tree containing the gluing vertex is disconnected from the tree containing $e'$. Then there must be a face $f$ that separates the tree containing $e'$ from the tree containing $\hat{v}$. But this necessarily means that $f$ must be incident to the defining chain along an edge between the defining chain of $s_1$ and $\hat{v}$ and along an edge between the defining chain of $s_2$ and $\hat{v}$. But this is a contradiction, since each slab is incident to the defining chain along a connected chain of edges. Face monotonicity follows by the monotonicity of the splicing path, which is maintained by the stopping conditions. The boundary condition follows the same argument as before and the definition of the fringe simplification step. \qed
Finally, filling in a cell takes $O(c_i\log c_i)$ time and $O(c_i)$ space, $\sum_i c_i = O(n\log m)$ (by Lemma~\ref{lem:slabcount}), and computing the subdivision requires $O(n(\log n)\log m)$ time. Thus we have:
\begin{theorem}\label{thm:pslg}The straight skeleton of a PSLG with $m$ connected components can be computed from its induced motorcycle graph in $O(n(\log n)\log m)$ time and $O(n\log m)$ space.\end{theorem}
\section{Handling degeneracies} \label{sec:generalposition}
So far we have assumed that the polygon is in general position and non-degenerate. By {\em non-degenerate} we mean that no two motorcycles crash simultaneously. By {\em general position} we mean that no four slabs intersect at a point and no two slabs are coplanar.
We now show how to remove these assumptions.
\paraskip{Degenerate polygons.} In a degenerate polygon multiple motorcycles may collide simultaneously. Huber and Held showed how to handle this by launching a new motorcycle in such cases and extended the definition of slabs to include multiple motorcycle edges along the boundary \cite{HuberH12}. The straight skeleton of such polygons is the lower envelope of the (extended) slabs. We follow their approach and extend the definition of slabs in the same way. This requires extending the boundary property for partial roofs. In particular, we allow the {\em slab border chain} of each face to contain an edge on each of the motorcycle edges incident to a face. A single slab (or face in partial roof) may now be incident to $O(r)$ motorcycle edges, rather than just two, and thus a single face may have up to $O(r)$ edges on the boundary of a partial roof. However, the sum of all such edges is still $O(r)$ (cf. \cite{HuberH12}). But each slab appears in $O(\log k)$ merge steps, and thus the amortized cost of a merge operation remains unchanged. We also note that the vertical subdivision algorithm of \cite{chengArxiv2014} works within the same time bound using the extended motorcycle graph and slabs of \cite{HuberH12}.
\paraskip{Removing general position.} Let us first assume that the intersection of any two slabs is either empty or a line segment. We now show how to deal with the case where more than three slabs intersect at a point. The proof of the linear complexity of partial roofs explicitly allows for vertices of degree higher than 3, so the proof holds without modification. The main difficulty lies in what to do if the splicing path hits a vertex rather than an edge.
\paraskip{Computing the splicing path.} The splicing path may now cut through a vertex $v$ rather than an edge of the input roofs, and the hit point no longer tells us {\em a priori} which face the splicing path should traverse next as it does when the splicing path traverses an edge. There are two cases we need to deal with, the first is when the splicing path hits a vertex in one of the partial roofs (say $R_1$), but is still on the interior of a face (say $f$) in the other partial roof. The second is when the splicing path simultaneously hits vertices in both partial roofs.
In the first case, let $e$ denote the incoming edge of the splicing path into $v$ and assume without loss of generality that $e$ is oriented so that the downward slope of the supporting slab $s$ of $f$ is to its right. We use the fact that the final face supported by $s$ is the lower envelope of the line segments given by intersecting all other slabs with $s$. Intersect the slabs supporting the other faces incident to $v$ with $s$ to get a list of line segments $s_1, s_2, \dots$. If $e$ is an edge required by the edge containment property and for one of the segments $s_i$, to satisfy the edge containment property we need an edge along $s_i$, then $e$ and $s_i$ will be part of the lower envelope of $e$ and $s_1, s_2, \dots$ in $s$. This is equivalent to saying that $s_i$ will be the segment making the sharpest right hand turn from $e$ at $v$, and thus it can be found in $O(\operatorname{deg}(v))$ time, where $\operatorname{deg}(v)$ denotes the degree of $v$.
In the second case, let $c_r$ denote a cylinder centered at $v$ with radius $r$ with rotational axis parallel to the $z$-axis and let $F_1 = (f_1, f_2, \dots, f_i)$ and $F_2 = (f'_1, f'_2, \dots, f_j)$ denote the fans of faces incident to $v$ in $R_1$ and $R_2$ (resp.). Choose the radius $r$ small enough that no edge of either fan incident to $v$ lies entirely on the interior of $c_r$ (for instance half the length of the shortest edge of either fan incident to $v$. We compute a walk of each fan starting at the intersection of the splicing edge we just computed with $c_r$, and walk along the local intersection between each fan and $c_r$. As with the merge operation, this walk is guided only by the local intersection between the current intrinsic point on which it lies on a fan, and the intersection of a small neighborhood of that point with $c_r$. Each walk traces out a path of curved segments along cylindric sections of $c_r$. Note that two (curved) segments on $c_r$ may intersect at most at two points (rather than just one) and each segment is monotone in $c_r$ with respect to the $z$-axis. Each walk traces a (curved) polygonal chain on $c_r$. We stop the walk if either we hit a boundary edge of the fan (i.e. an edge such that the face on the other side is not incident to $v$), have traveled one complete turn around the cylinder (i.e. the projection of the walk into the $xy$-plane subtends an angle greater than $2\pi$), or the next edge of the walk is non-monotone with the previously computed segment with respect to the $z$ direction.
The intuition behind this is: suppose there is an edge $e$ in the final straight skeleton roof $R(P)$ along an intersection that we must detect in order to satisfy the edge containment property. Intersect $c_r$ with the final straight skeleton roof $R(P)$, then because $R(P)$ is a terrain, we obtain a polygon on the surface of $c_r$ that is monotone with respect to the $z$ direction. Thus in the walk of either fan, if we arrive at a point violating monotonicity, we know that that the rest of the walk cannot possibly be part of the lower envelope. The same general inductive argument as was used in the proof of Lemma~\ref{lem:mergeproducespartialroof} shows that the beginning of the walk of each fan lies along the intersection between $c_r$ and the final roof $R(P)$ and will only diverge at the edge of $R(P)$ that needs to be detected by the splicing path. This is necessarily at the first intersection between the two polygonal chains produced by the two walks. To find this point we compute the lower envelope in $c_r$ of the two chains with respect to the $z$ direction.
\begin{wrapfigure}{r}{0.5\textwidth}
\centering \includegraphics[width=0.5\textwidth]{overlap}
\caption{\small{An example where coplanar slabs overlap in a degenerate manner. (a) A polygon and its motorcycle graph. (b, c) the slabs for parallel edges $e_1$ and $e_2$. (d) the intersection of the two slabs is the darkly shaded region.}}
\label{fig:overlap} \end{wrapfigure}
Choosing the radius takes $O(\operatorname{deg}(v))$ time (where $\operatorname{deg}(v)$ denotes the larger degree of $v$ in either fan). On the walks we traverse each face once, and the stopping conditions are checked in constant time, so computing the walks requires $O(\operatorname{deg}(v))$ time. We then compute the lower envelope of the two monotone chains using standard sweep techniques in $O(\operatorname{deg}(v))$ time. Therefore computing the next edge of the splicing path takes $O(\operatorname{deg}(v))$ time instead of $O(1)$ time, and $\operatorname{deg}(v) = k$. However, we bound the amortized cost of a single merge operation by observing that (by the handshaking lemma), the sum of the degrees over all vertices is twice the number of edges in the partial roof, which is $O(k)$. This similarly bounds the number of trapezoids created by the ray-shooting sub-routine: the number of trapezoids incident to any vertex is at most twice the degree of that vertex, so the total number of trapezoids required for a merge step is $O(k)$. Thus the merge operation takes amortized $O(k)$ time.
\paraskip{Coplanar slabs.} The remaining ambiguity that arises when we remove the general position assumption is that when two base edges have coplanar slabs the intersection of the two slabs may be a region of the plane supporting the two, rather than a simple line segment. See Fig.~\ref{fig:overlap}. However, if the faces of the final straight skeleton roof supported by these two slabs are incident along an edge, then using the wavefront definition of the straight skeleton it is easy to show that their motorcycles crash simultaneously. (Their base edges are parallel, and the wavefront moves outwards at unit speed in parallel, so they can only crash if their endpoints are reflex and these reflex vertices collide during the wavefront propagation. It was shown in \cite{HuberH12} that the motorcycle edges cover the traces of the reflex vertices and so this implies the motorcycles crash simultaneously.) However, following \cite{HuberH12}, such a simultaneous crash necessitates the creation of a new motorcycle which becomes a boundary edge of both the slabs, negating that the slabs intersect. Therefore if the two slabs intersect in a non-degenerate way, their faces in the final roof are not co-incident along an edge. For this reason, we can add one more stopping condition to the splicing path computation: if we get to a point where the splicing path should traverse across two faces which are coplanar and do not share a common motorcycle edge (i.e. the ambiguous situation above), we stop. This constitutes a proof that the splicing path has already computed all of the edges necessary to satisfy the edge containment property.
\section{Conclusion.} The main theorem is proven by Theorems~\ref{thm:polygon} and \ref{thm:pslg}. This gives us faster algorithms for computing the straight skeleton of polygons and PSLGs by first computing the motorcycle graph and then using our reductions. However, there still exists a theoretical gap between the known lower bounds of $\Omega(n)$ for polygons and $\Omega(n\log n)$ for PSLGs. This remains an intriguing open problem.
\end{document} |
\begin{document}
\title{Quantized fluctuational electrodynamics for three-dimensional plasmonic structures} \date{January 30, 2017} \author{Mikko Partanen} \affiliation{Engineered Nanosystems group, School of Science, Aalto University, P.O. Box 12200, 00076 Aalto, Finland} \author{Teppo H\"ayrynen} \affiliation{DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, \O rsteds Plads, Building 343, DK-2800 Kongens Lyngby, Denmark} \author{Jukka Tulkki} \affiliation{Engineered Nanosystems group, School of Science, Aalto University, P.O. Box 12200, 00076 Aalto, Finland} \author{Jani Oksanen} \affiliation{Engineered Nanosystems group, School of Science, Aalto University, P.O. Box 12200, 00076 Aalto, Finland}
\begin{abstract} We recently introduced a quantized fluctuational electrodynamics (QFED) formalism that provides a physically insightful definition of an effective position-dependent photon-number operator and the associated ladder operators. However, this far the formalism has been applicable only for the normal incidence of the electromagnetic field in planar structures. In this work, we overcome the main limitation of the one-dimensional QFED formalism by extending the model to three dimensions, allowing us to use the QFED method to study, e.g., plasmonic structures. To demonstrate the benefits of the developed formalism, we apply it to study the local steady-state photon numbers and field temperatures in a light-emitting near-surface InGaN quantum-well structure with a metallic coating supporting surface plasmons. \end{abstract}
\maketitle
\section{Introduction}
The quantum optical processes in lossy and lossless material systems have been widely studied during the last few decades. This has led to advances, e.g., in nanoplasmonics \cite{Sorger2011,Oulton2009,Huang2013,Sadi2013a}, near-field microscopy \cite{Taubner2006,Hillenbrand2002}, thin-film light-emitting diodes \cite{Nakamura2013,Heikkila2013}, photonic crystals \cite{Russell2003,Akahane2003}, and metamaterials \cite{Tanaka2010,Mattiucci2013}. For describing spatial field evolution in resonant structures, one of the most widely used quantization approaches has been the input-output formalism of the photon creation and annihilation operators. The formalism was originally developed for describing lossless and dispersionless dielectrics \cite{Knoll1987} and was later extended for lossy and dispersive media \cite{Knoll1991,Allen1992,Huttner1992,Barnett1995,Matloob1995,Matloob1996}. The early studies clearly highlighted that the noise and field operators in nonuniform systems are position dependent and that the vector potential and electric-field operators obey the well-known canonical commutation relation as expected \cite{Barnett1995,Matloob1995}. However, the canonical commutation relations did not extend to the photon creation and annihilation operators, which were found to exhibit anomalies in resonant structures \cite{Ueda1994,Raymer2013,Barnett1996,Aiello2000,Stefano2000}. It was first concluded that these anomalies have no physical significance. Although the formalism was later successfully used to study, e.g., amplifying media and spontaneous decay in left-handed media \cite{Dung2003,Khanbekyan2005,Raabe2007,Raabe2008}, it was also recently shown that the anomalous commutation relations should, e.g., lead to the existence of a threshold for second-harmonic generation inside microcavities \cite{Gauvin2014,Collette2013}. The anomalous commutation relations have also been found to prevent a systematic description of local thermal balance between the field and interacting media \cite{Partanen2014a,Partanen2015a}.
We recently solved the cavity commutation relation anomaly and photon-number problem by introducing a quantized fluctuational electrodynamics (QFED) model to describe photon number and showed that the expectation values of the properly normalized annihilation and creation operators result in a meaningful photon-number model and thermal balance conditions \cite{Partanen2014a,Partanen2014c,Partanen2015a,Partanen2016b}. This far, our models have been strictly one-dimensional and limited to normal incidence in planar structures, which has provided an adequate framework for describing the fundamental properties of cavity fields. However, considering the associated transparent description of the photon number and field temperature, it becomes reasonable to ask how the description can be expanded to more complex systems involving, e.g., plasmons that have been of great topical interest \cite{Barnes2003,Okamoto2004,Yeh2008,Bonnand2006a,Bonnand2006b,Tanaka2010,Pitarke2007} and whose description could benefit from the new methodology clearly separating the local density of states (LDOS) and the photon number. Here we therefore present a generalized QFED model to account for fully three-dimensional propagation as well as the associated spectral expansion for planar structures. We also demonstrate the benefits of the formalism by applying it to study the local steady-state field properties and plasmonic interactions in a light-emitting near-surface InGaN quantum-well (QW) structure with a metallic coating supporting surface plasmons (SPs).
This paper is organized as follows: The theory of the QFED method is presented in Sec.~\ref{sec:theory}. As a background for QFED, we first review the general three-dimensional noise operator formalism and the use of Green's functions to obtain the solutions of the electromagnetic (EM) fields. This is followed by a presentation of the new contribution to the theory: the properly normalized position- and frequency-dependent photon ladder operators, the related photon-number presentation, and the generalized forms of the densities of states in the QFED method. After introducing the ladder and number operators, we also briefly review how the operators can be used to present the associated models for the field fluctuations, Poynting vector, and absorption and emission operators. Note that the expectation values of these macroscopic field quantities are equivalent to the values obtained by using the conventional fluctuational electrodynamics. In Sec.~\ref{sec:results}, we demonstrate the applicability and study the physical implications of the introduced QFED method in an example InGaN QW geometry.
\section{\label{sec:theory}Quantized fluctuational electrodynamics method}
In this section, we outline the derivation of the three-dimensional QFED theory. Detailed derivations are given in the appendixes. We start by introducing the fundamental equations of the conventional fluctuational electrodynamics theory and its quantization in Sec.~\ref{sec:noiseoperatorformalism} and the solution of fields using the dyadic Green's functions in Sec.~\ref{sec:green}. Then, in Sec.~\ref{sec:photonnumbers}, we present the properly normalized photon ladder operators and the related photon-number and density-of-states concepts that expand the classical and previously used quantized versions of fluctuational electrodynamics to enable an unambiguous photon-level description of the three-dimensional system. In Secs.~\ref{sec:poynting} and \ref{sec:balance}, we focus on calculating the Poynting vector operator and the thermal balance predicted by the quantized theory using the newly established operators.
\subsection{\label{sec:noiseoperatorformalism}Noise operator formalism} Maxwell's equations describe electric and magnetic fields generated by currents and charges in matter. They relate the electric field strength $\mathbf{E}$, the magnetic field strength $\mathbf{H}$, the electric flux density $\mathbf{D}$, and the magnetic flux density $\mathbf{B}$ to the free electric charge density $\rho_\mathrm{f}$ and the free electric current density $\mathbf{J}_\mathrm{f}$. In the frequency domain, Maxwell's equations are written for positive frequencies as \cite{Partanen2016b} \begin{align}
\nabla\cdot\mathbf{D} &=\rho_\mathrm{f},\label{eq:maxwell1}\\
\nabla\cdot\mathbf{B} &=0,\label{eq:maxwell2}\\
\nabla\times\mathbf{E} &=i\omega\mathbf{B}=i\omega\mu_0(\mu\mathbf{H}+\delta\mathbf{M}),\label{eq:maxwell3}\\
\nabla\times\mathbf{H} &=\mathbf{J}_\mathrm{f}-i\omega\mathbf{D}=\mathbf{J}_\mathrm{f}-i\omega(\varepsilon_0\varepsilon\mathbf{E}+\delta\mathbf{P}).\label{eq:maxwell4} \end{align} Here we have additionally used the constitutive relations $\mathbf{D}=\varepsilon_0\varepsilon\mathbf{E}+\delta\mathbf{P}$ and $\mathbf{B}=\mu_0(\mu\mathbf{H}+\delta\mathbf{M})$, where $\varepsilon_0$ and $\mu_0$ are the permittivity and permeability of vacuum, $\varepsilon=\varepsilon_\mathrm{r}+i\varepsilon_\mathrm{i}$ and $\mu=\mu_\mathrm{r}+i\mu_\mathrm{i}$ are the relative permittivity and permeability of the medium with real and imaginary parts denoted by subscripts $\mathrm{r}$ and $\mathrm{i}$, and the polarization and magnetization fields $\delta\mathbf{P}$ and $\delta\mathbf{M}$ denote the polarization and magnetization that are not linearly proportional to the respective field strengths \cite{Sipe1987}. In the context of the fluctuational electrodynamics and the present work, $\delta\mathbf{P}$ and $\delta\mathbf{M}$ describe the small thermal fluctuations of the linear polarization and magnetization fields \cite{Partanen2016b}. For the remainder of this work, the current density of free charges $\mathbf{J}_\mathrm{f}$ is also included in the electric permittivity for notational simplicity.
From Maxwell's equations in Eqs.~\eqref{eq:maxwell1}--\eqref{eq:maxwell4} it follows that the electric field obeys the well-known equation \cite{Partanen2016b} \begin{align}
\nabla\times\Big(\frac{\nabla\times\mathbf{E}}{\mu_0\mu}\Big)-\omega^2\varepsilon_0\varepsilon\mathbf{E} &=i\omega\mathbf{J}_\mathrm{e}
-\nabla\times\Big(\frac{\mathbf{J}_\mathrm{m}}{\mu_0\mu}\Big),\label{eq:HelmholtzE1} \end{align} where the terms $\mathbf{J}_\mathrm{e}=-i\omega\delta\mathbf{P}$ and $\mathbf{J}_\mathrm{m}=-i\omega\mu_0\delta\mathbf{M}$ represent the polarization and magnetization currents that act as field sources in the noise operator theory \cite{Partanen2016b} and in the classical fluctuational electrodynamics \cite{Narayanaswamy2014,Polimeridis2015}. The electric term $\mathbf{J}_\mathrm{e}$ includes contributions from both the electric currents due to free charges and polarization terms associated with dipole currents and thermal dipole fluctuations. For the magnetic current term $\mathbf{J}_\mathrm{m}$, the only contribution arises from the magnetic dipoles. Note that, after solving the electric field from Eq.~\eqref{eq:HelmholtzE1}, the calculation of the magnetic field is straightforward using Faraday's law in Eq.~\eqref{eq:maxwell3}.
In the previously known noise operator framework, we use the canonical quantization of the above classical equations where the classical field quantities are replaced by corresponding quantum operators \cite{Dung2003,Matloob1995,Partanen2014a,Partanen2016b}. The electric and magnetic noise current operators $\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r},\omega)$ and $\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r},\omega)$ are written in terms of bosonic source field operators $\hat f_\mathrm{e}(\mathbf{r},\omega)$ and $\hat f_\mathrm{m}(\mathbf{r},\omega)$ as $\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r},\omega)=\sum_\alpha j_\mathrm{0,e}(\mathbf{r},\omega)\hat{\mathbf{e}}_\alpha\hat f_\mathrm{e}(\mathbf{r},\omega)$ and $\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r},\omega)=\sum_\alpha j_\mathrm{0,m}(\mathbf{r},\omega)\hat{\mathbf{e}}_\alpha\hat f_\mathrm{m}(\mathbf{r},\omega)$, where $\hat{\mathbf{e}}_\alpha$ are unit vectors for the three coordinate directions $\alpha\in\{x,y,z\}$ \cite{Partanen2016b,Partanen2014a}. The operators $\hat f_\mathrm{e}(\mathbf{r},\omega)$ and $\hat f_\mathrm{m}(\mathbf{r},\omega)$ obey the canonical commutation relation $[\hat f_{j}(\mathbf{r},\omega),\hat f_k^\dag(\mathbf{r}',\omega')]=\delta_{jk}\delta(\mathbf{r}-\mathbf{r}')\delta(\omega-\omega')$, with $j,k\in\{\mathrm{e,m}\}$. The normalization factors $j_\mathrm{0,e}(\mathbf{r},\omega)$ and $j_\mathrm{0,m}(\mathbf{r},\omega)$ have been determined to be $j_\mathrm{0,e}(\mathbf{r},\omega)=\sqrt{4\pi\hbar\omega^2\varepsilon_0\varepsilon_\mathrm{i}(\mathbf{r},\omega)}$ \cite{Partanen2014a,Khanbekyan2005} and $j_\mathrm{0,m}(\mathbf{r},\omega)=\sqrt{4\pi\hbar\omega^2\mu_0\mu_\mathrm{i}(\mathbf{r},\omega)}$ \cite{Partanen2016b,Dung2003}.
\subsection{\label{sec:green}Green's functions}
In order to write the solution of Eq.~\eqref{eq:HelmholtzE1} in a general form, we apply the conventional dyadic Green's function formalism \cite{Eckhardt1984,Paulus2000}, where the field solutions are written as \begin{align}
\hat{\mathbf{E}}^+(\mathbf{r},\omega)
&=i\omega\mu_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&\hspace{0.5cm}+k_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)d^3r',
\label{eq:efield} \end{align} \begin{align}
\hat{\mathbf{H}}^+(\mathbf{r},\omega)
&=k_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&\hspace{0.5cm}+i\omega\varepsilon_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)d^3r'.
\label{eq:hfield} \end{align} Here $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ is the electric Green's function, $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')$ is the magnetic Green's function, and $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')$ are the exchange Green's functions. For completeness, the relations between these Green's functions are shown in Appendix \ref{apx:greengeneral}, and the Green's functions are explicitly presented for stratified media in Appendix \ref{apx:greenstratified}.
\subsection{\label{sec:photonnumbers}Photon numbers and densities of states}
In analogy with the one-dimensional QFED formalism \cite{Partanen2016b}, we can define the position- and frequency-dependent effective photon ladder operators $\hat a_j(\mathbf{r},\omega)$, which obey the canonical commutation relation $[\hat a_j(\mathbf{r},\omega),\hat a_j^\dag(\mathbf{r},\omega)]=\delta(\omega-\omega')$, for the electric, magnetic, and total EM fields, $j\in\{\mathrm{e,m,tot}\}$. These operators and the corresponding effective photon-number expectation values $\langle\hat n_\mathrm{j}(\mathbf{r},\omega')\rangle$ are given by \begin{align}
\hat a_j(\mathbf{r},\omega)&=\frac{1}{\sqrt{\int\rho_{\mathrm{NL},j}(\mathbf{r},\omega,\mathbf{r}')d^3r'}}\nonumber\\
&\hspace{0.5cm}\times\int\Big[\sqrt{\rho_{\mathrm{NL},j,\mathrm{e}}(\mathbf{r},\omega,\mathbf{r}')}\hat f_\mathrm{e}(\mathbf{r}',\omega)\nonumber\\
&\hspace{0.5cm}+\sqrt{\rho_{\mathrm{NL},j,\mathrm{m}}(\mathbf{r},\omega,\mathbf{r}')}\hat f_\mathrm{m}(\mathbf{r}',\omega)\Big]d^3r',
\label{eq:photonladder} \end{align} \begin{equation}
\langle\hat n_{j}(\mathbf{r},\omega)\rangle=\frac{\int\rho_{\mathrm{NL},j}(\mathbf{r},\omega,\mathbf{r}')\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'}{\int\rho_{\mathrm{NL},j}(\mathbf{r},\omega,\mathbf{r}')d^3r'},
\label{eq:photonnumbers} \end{equation} where $\langle\hat\eta(\mathbf{r}',\omega)\rangle$ is the source-field photon-number expectation value related to the bosonic noise operators as $\langle\hat\eta(\mathbf{r},\omega)\rangle =\int\langle\hat f_\mathrm{e}^\dag(\mathbf{r},\omega)\hat f_\mathrm{e}(\mathbf{r}',\omega')\rangle d^3r'd\omega' =\int\langle\hat f_\mathrm{m}^\dag(\mathbf{r},\omega)\hat f_\mathrm{m}(\mathbf{r}',\omega')\rangle d^3r'd\omega'$ and $\rho_{\mathrm{NL},j}(\mathbf{r},\omega,\mathbf{r}')$ are the nonlocal densities of states (NLDOSs) for the electric, magnetic, and total EM fields, written as \begin{align}
&\rho_\mathrm{NL,e}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3}{\pi c^4}\Big(\varepsilon_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big) \label{eq:enldos},\\
&\rho_\mathrm{NL,m}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3}{\pi c^4}\Big(\varepsilon_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big) \label{eq:hnldos},\\
&\rho_\mathrm{NL,tot}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{|\varepsilon(\mathbf{r},\omega)|}{2}\rho_\mathrm{NL,e}(\mathbf{r},\omega,\mathbf{r}')
+\frac{|\mu(\mathbf{r},\omega)|}{2}\rho_\mathrm{NL,m}(\mathbf{r},\omega,\mathbf{r}').
\label{eq:unldos} \end{align} The NLDOS components $\rho_{\mathrm{NL},j,\mathrm{e}}(\mathbf{r},\omega,\mathbf{r}')$ and $\rho_{\mathrm{NL},j,\mathrm{m}}(\mathbf{r},\omega,\mathbf{r}')$, with $j\in\{\mathrm{e,m}\}$, in Eq.~\eqref{eq:photonladder} denote, respectively, the first and the second terms of Eqs.~\eqref{eq:enldos} and \eqref{eq:hnldos}. The total NLDOS terms $\rho_{\mathrm{NL,tot,e}}(\mathbf{r},\omega,\mathbf{r}')$ and $\rho_{\mathrm{NL,j,m}}(\mathbf{r},\omega,\mathbf{r}')$ are calculated by using Eq.~\eqref{eq:unldos} with the corresponding terms in the electric and magnetic NLDOSs.
Note that the expressions for the photon ladder operators and the photon numbers in Eqs.~\eqref{eq:photonladder} and \eqref{eq:photonnumbers} are the same as the expression in the one-dimensional formalism \cite{Partanen2016b}, but the NLDOSs are different. The derivation of these NLDOSs is presented in Appendix \ref{apx:densities}, and for general stratified media, the densities of states are presented in Appendix \ref{apx:stratifieddos}. The LDOSs $\rho_j(\mathbf{r},\omega)$ are given in terms of the NLDOSs by \begin{equation}
\rho_j(\mathbf{r},\omega)=\int\rho_{\mathrm{NL},j}(\mathbf{r},\omega,\mathbf{r}')d^3r'.
\label{eq:ldos} \end{equation}
It is well-known that, in vacuum, the imaginary parts of the traces of the dyadic Green's functions give the electric and magnetic LDOSs $\rho_\mathrm{e}(\mathbf{r},\omega)$ and $\rho_\mathrm{m}(\mathbf{r},\omega)$ as \cite{Joulain2003,Joulain2005} \begin{align}
\rho_{j}(\mathbf{r},\omega) &=\frac{2\omega}{\pi c^2}\mathrm{Im}\{\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jj}(\mathbf{r},\omega,\mathbf{r})]\},
\label{eq:traceldos} \end{align} where $j\in\{\mathrm{e,m}\}$. A similar relation also applies for the normal components of the Fourier-transformed quantities in layered media \cite{Partanen2014c,Partanen2016b}, and typically, also, the spatially resolved form in Eq.~\eqref{eq:traceldos} is expected to be valid inside lossy media. However, in lossy media, these LDOSs are generally known to become infinite due to the contribution of evanescent waves \cite{Joulain2003,Joulain2005}.
In terms of the photon-number expectation values in Eq.~\eqref{eq:photonnumbers} and the LDOSs in Eq.~\eqref{eq:ldos}, the spectral electric and magnetic field fluctuations and the energy density are given by \cite{Partanen2014c} \begin{align}
\langle\hat{E}(\mathbf{r},t)^2\rangle_\omega & =\frac{\hbar\omega}{\varepsilon_0}\rho_\mathrm{e}(\mathbf{r},\omega)\Big(\langle\hat n_\mathrm{e}(\mathbf{r},\omega)\rangle+\frac{1}{2}\Big)\label{eq:efluct},\\[8pt]
\langle\hat{H}(\mathbf{r},t)^2\rangle_\omega & =\frac{\hbar\omega}{\mu_0}\rho_\mathrm{m}(\mathbf{r},\omega)\Big(\langle\hat n_\mathrm{m}(\mathbf{r},\omega)\rangle+\frac{1}{2}\Big)\label{eq:bfluct},\\[8pt]
\langle\hat u(\mathbf{r},t)\rangle_\omega & = \hbar\omega\rho_\mathrm{tot}(\mathbf{r},\omega)\Big(\langle\hat n_\mathrm{tot}(\mathbf{r},\omega)\rangle+\frac{1}{2}\Big)\label{eq:edensity}. \end{align} Here the subscript $\omega$ denotes the contribution of $\omega$ to the total quantities which are obtained as integrals over positive frequencies.
\subsection{\label{sec:poynting}Quantized Poynting vector operator}
To conform with our earlier works and to enable describing energy flow in detail, we also find the three-dimensional generalized expression for the Poynting vector. For an optical mode, the quantum optical Poynting vector is defined as a normal-ordered operator in terms of the positive- and negative-frequency parts of the electric and magnetic field operators as $\hat{\mathbf{S}}(\mathbf{r},t)=:\!\hat{\mathbf{E}}(\mathbf{r},t)\times\hat{\mathbf{H}}(\mathbf{r},t)\!:=\hat{\mathbf{E}}^-(\mathbf{r},t)\times\hat{\mathbf{H}}^+(\mathbf{r},t)-\hat{\mathbf{H}}^-(\mathbf{r},t)\times\hat{\mathbf{E}}^+(\mathbf{r},t)$ \cite{Loudon2000}. As detailed in Appendix \ref{apx:densities} and in analogy with the one-dimensional QFED formalism \cite{Partanen2014a,Partanen2016b}, we obtain the Poynting-vector-related interference density of states (IFDOS) as \begin{align}
&\boldsymbol{\rho}_\mathrm{IF}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3n_\mathrm{r}(\mathbf{r},\omega)}{\pi c^4}\nonumber\\
&\hspace{0.5cm}\times\Big(\mu_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\nonumber\\
&\hspace{0.5cm}-\varepsilon_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\Big), \end{align} where $n_\mathrm{r}(\mathbf{r},\omega)$ is the real part of the refractive index and we have used the short-hand notation $\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{1}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{2}^\dag(\mathbf{r},\omega,\mathbf{r}')]
=\sum_\alpha[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{1}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{e}}_\alpha]\times
[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{2}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{e}}_\alpha]^\dag$, which is a vector, in contrast to the conventional trace of a matrix. Using the IFDOS, the Poynting vector is given by \begin{equation}
\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle_\omega =
\hbar\omega v(\mathbf{r},\omega)\int\boldsymbol{\rho}_\mathrm{IF}(\mathbf{r},\omega,\mathbf{r}')\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r',
\label{eq:poynting} \end{equation} where $v(\mathbf{r},\omega)=c/n_\mathrm{r}(\mathbf{r},\omega)$ is the propagation velocity of the field in the direction of the wave vector. The integral of the IFDOS with respect to $\mathbf{r}'$ is always zero, i.e., $\int\boldsymbol{\rho}_\mathrm{IF}(\mathbf{r},\omega,\mathbf{r}')d^3r'=0$, which is required by the fact that, in a medium in thermal equilibrium, there is no net energy flow. For stratified media, the IFDOS is presented in Appendix \ref{apx:stratifieddos}.
\subsection{\label{sec:balance}Field-matter interaction operators and thermal balance}
A particularly insightful view of the effective photon numbers in the QFED framework is provided by their connection to local thermal balance between the field and matter \cite{Partanen2014a}. First, we define the normal-ordered emission and absorption operators $\hat Q_\mathrm{em}(\mathbf{r},t)$ and $\hat Q_\mathrm{abs}(\mathbf{r},t)$ as \begin{align}
\hat Q_\mathrm{em}(\mathbf{r},t)\!
&=\!-\!:\!\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r},t)\cdot\hat{\mathbf{E}}(\mathbf{r},t)\!:
\!-\!:\!\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r},t)\cdot\hat{\mathbf{H}}(\mathbf{r},t)\!:,
\label{eq:emissionop}\\
\hat Q_\mathrm{abs}(\mathbf{r},t)\!
&=:\!\hat{\mathbf{J}}_\mathrm{e,abs}(\mathbf{r},t)\cdot\hat{\mathbf{E}}(\mathbf{r},t)\!:
\!+\!:\!\hat{\mathbf{J}}_\mathrm{m,abs}(\mathbf{r},t)\cdot\hat{\mathbf{H}}(\mathbf{r},t)\!:,
\label{eq:absorptionop} \end{align} where the electric and magnetic absorption current operators $\hat{\mathbf{J}}_\mathrm{e,abs}(\mathbf{r},t)$ and $\hat{\mathbf{J}}_\mathrm{m,abs}(\mathbf{r},t)$ are written in the spectral domain as $\hat{\mathbf{J}}_\mathrm{e,abs}^+(\mathbf{r},\omega)=-i\omega\varepsilon_0\chi_\mathrm{e}(\mathbf{r},\omega)\hat{\mathbf{E}}^+(\mathbf{r},\omega)$ and $\hat{\mathbf{J}}_\mathrm{m,abs}^+(\mathbf{r},\omega)=-i\omega\mu_0\chi_\mathrm{m}(\mathbf{r},\omega)\hat{\mathbf{H}}^+(\mathbf{r},\omega)$, where $\chi_\mathrm{e}(\mathbf{r},\omega)=\varepsilon(\mathbf{r},\omega)-1$ and $\chi_\mathrm{m}(\mathbf{r},\omega)=\mu(\mathbf{r},\omega)-1$ are the electric and magnetic susceptibilities of the medium.
The net emission operator $\hat Q(\mathbf{r},t)=\hat Q_\mathrm{em}(\mathbf{r},t)-\hat Q_\mathrm{abs}(\mathbf{r},t)$, which describes the energy transfer between the EM field and the local medium, is given by \begin{align}
\hat Q(\mathbf{r},t)
=&:\!\hat{\mathbf{J}}_\mathrm{e,tot}(\mathbf{r},t)\cdot\hat{\mathbf{E}}(\mathbf{r},t)\!:+:\!\hat{\mathbf{J}}_\mathrm{m,tot}(\mathbf{r},t)\cdot\hat{\mathbf{H}}(\mathbf{r},t)\!:,
\label{eq:netemissionop} \end{align} where $\hat{\mathbf{J}}_\mathrm{e,tot}(\mathbf{r},t)=\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r},t)+\hat{\mathbf{J}}_\mathrm{e,abs}(\mathbf{r},t)$ and $\hat{\mathbf{J}}_\mathrm{m,tot}(\mathbf{r},t)=\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r},t)+\hat{\mathbf{J}}_\mathrm{m,abs}(\mathbf{r},t)$ correspond to the classical total current densities, which are sums of free and bound current densities. The spectral component of the expectation value of the net emission operator in Eq.~\eqref{eq:netemissionop} can be written in terms of the LDOSs and the electric- and magnetic-field photon numbers as \begin{align}
&\langle\hat Q(\mathbf{r},t)\rangle_\omega\nonumber\\
& =\hbar\omega^2\varepsilon_\mathrm{i}(\mathbf{r},\omega)\rho_\mathrm{e}(\mathbf{r},\omega)[\langle\hat\eta(\mathbf{r},\omega)\rangle-\langle\hat n_\mathrm{e}(\mathbf{r},\omega)\rangle]\nonumber\\ &\hspace{0.5cm}+\hbar\omega^2\mu_\mathrm{i}(\mathbf{r},\omega)\rho_\mathrm{m}(\mathbf{r},\omega)[\langle\hat\eta(\mathbf{r},\omega)\rangle-\langle\hat n_\mathrm{m}(\mathbf{r},\omega)\rangle].
\label{eq:divP} \end{align} This shows that local thermal balance [$\langle\hat Q(\mathbf{r},t)\rangle_\omega=0$] is generally reached when the source-field photon numbers coincide with the field photon numbers as defined in Eq.~\eqref{eq:divP}. In addition, the net emission operator satisfies $\langle\hat Q(\mathbf{r},t)\rangle_\omega=\nabla\cdot\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle_\omega$. In resonant systems where the energy exchange is dominated by a narrow frequency band, the condition $\langle\hat Q(\mathbf{r},t)\rangle_\omega=0$ can be used to approximately determine the steady-state temperature of a weakly interacting resonant particle \cite{Bohren1998}.
\section{\label{sec:results}Results}
\begin{figure}
\caption{(Color online) The studied structure formed by a Ag/GaN/In$_{0.15}$Ga$_{0.85}$N/GaN/Al$_2$O$_3$ heterostructure. The background temperature is $T=300$ K, the band gap of the light emitting In$_{0.15}$Ga$_{0.85}$N QW is $E_\mathnormal{g}=2.76$ eV, and the QW excitation corresponds to an applied voltage of $U=2.6$ V. Note that the figure is not to scale.}
\label{fig:structure}
\end{figure}
\begin{figure*}
\caption{(Color online) (a) The base-10 logarithm of the total EM LDOS, (b) the effective temperature of the total EM field in the case of a thermally excited QW, and (c) the effective-field temperature in the case of an electrically or optically excited QW corresponding to the bias voltage $U=2.6$ V for photon energy $\hbar\omega=E_\mathnormal{g}+k_\mathrm{B}T=2.786$ eV as a function of position and the in-plane component of the wave vector. The position $z=0$ is fixed to the Ag/air interface. The white dashed lines represent the light cones of GaN, sapphire, and air.}
\label{fig:pos}
\end{figure*}
We apply the QFED formalism presented in Sec.~\ref{sec:theory} to the study of an example plasmonic multilayer structure, which has recently been of experimental and theoretical interest \cite{Homeyer2013,Sadi2013b}. In contrast to the previous QFED models, the generalized model can, among other oblique-angle problems, describe the optical properties of plasmonic devices. Here we study the contribution of the evanescent SP modes to the position-dependent LDOSs and the effective-field temperatures in the vicinity of a light-emitting Ag/GaN/In$_{0.15}$Ga$_{0.85}$N/GaN/Al$_2$O$_3$ multilayer structure illustrated in Fig.~\ref{fig:structure}. The 2-nm In$_{0.15}$Ga$_{0.85}$N QW has a band gap of 2.76 eV ($\lambda=450$ nm), and it acts as the emitter layer. It is deposited 20 nm below the 20-nm silver layer which supports SP modes. The refractive indices of GaN and InN are taken from Refs.~\citenum{Leung1998,Barker1973,Ambacher1996,Djurisic1999,Trainor1974,Tansley1986}, and the refractive index of In$_{0.15}$Ga$_{0.85}$N is deduced by using Vegard's law; the refractive index of silver is calculated by using the Drude model plasma frequency $\omega_\mathrm{p}=9.04$ eV/$\hbar$ and damping frequency $\omega_\tau=0.02125$ eV/$\hbar$ taken from Ref.~\citenum{Zeman1987}, and the refractive index of sapphire is taken from Ref.~\citenum{Malitson1962}. For example, in the case of the photon energy $\hbar\omega=2.76$ eV corresponding to the QW band gap, the refractive indices of air, silver, GaN, In$_{0.15}$Ga$_{0.85}$N, and sapphire are 1.00, $0.013+3.119i$, $2.51+0.0029i$, $2.51+0.094i$, and 1.78, respectively.
The background temperature of the materials is $T=300$ K. We compare the emission of the structure in two cases: (1) the QW is thermally excited to temperature $T_\mathrm{ex}=350$ K, and (2) the QW is electrically or optically excited to a state corresponding to direct excitation by a $U=2.6$ V voltage source. In the first case, the QW source-field photon-number expectation value is modeled using the Bose-Einstein distribution $\langle\hat\eta_\text{\tiny QW}\rangle=1/(e^{\hbar\omega/(k_\mathrm{B}T_\mathrm{ex})}-1)$. In other words, we apply the local thermal equilibrium (LTE) approximation. The LTE approximation is justified when the gradients in the temperature are expected to be small compared to a material-dependent current-current correlation length scale, which is of the order of atomic scale or the phonon mean free path \cite{Polimeridis2015}. In the second case, the source-field photon number of the QW is modeled using $\langle\hat\eta_\text{\tiny QW}\rangle=1/(e^{(\hbar\omega-eU)/(k_\mathrm{B}T)}-1)$ for photon energies above the band gap $\hbar\omega\ge E_\mathnormal{g}$ and the background value $\langle\hat\eta_\text{\tiny BG}\rangle=1/(e^{\hbar\omega/(k_\mathrm{B}T)}-1)$ for photon energies below the band gap $\hbar\omega\le E_\mathnormal{g}$ corresponding to the interactions with the free carriers. For example, in the case of the photon energy $\hbar\omega=E_\mathnormal{g}+k_\mathrm{B}T=2.786$ eV, the source-field photon number of the electrically or optically excited QW is $\langle\hat\eta_\text{\tiny QW}\rangle=7.51\times 10^{-4}$, which is very large in comparison with the photon number of the thermal 300 K background $\langle\hat\eta_\text{\tiny BG}\rangle=1.57\times 10^{-47}$. As the photon-number expectation values are relatively small and depend strongly on the frequency, it is convenient to illustrate the results by using the effective field temperature that is defined in terms of the photon-number expectation value as $T_\mathrm{eff}(z,K,\omega)=\hbar\omega/\{(k_\mathrm{B}\ln[1+1/\langle\hat n(z,K,\omega)\rangle]\}$ \cite{Partanen2014c,Partanen2014b}. The corresponding effective-source-field temperature of the electrically or optically excited QW ranges from 5175 K (compare with $\sim$6000 K of solar radiation on earth) to 625 K as the photon energy ranges from 2.76 to 5 eV.
Figure \ref{fig:pos}(a) shows the base-10 logarithm of the total EM LDOS for photon energy $\hbar\omega=E_\mathnormal{g}+k_\mathrm{B}T=2.786$ eV as a function of position and the in-plane component of the wave vector. The sapphire substrate lies on the left and air on the right. The light cones for sapphire, GaN, and air are defined by the in-plane wave vector component values $K<nk_0$, where $n$ is the real part of the refractive index of the respective material. The light cones of the different material layers are clearly visible in the figure. Due to the evanescent fields, the LDOSs are slightly elevated also beyond the material interfaces. One can also see the very large LDOS associated with the GaN/Ag SP resonance near the position $z=0$ and $K/k_0=5.0$. The less visible air/Ag SP resonance is near the position $z=0$ and $K/k_0=1.0$. The GaN guided modes and the associated interference patterns can be seen between the GaN light cone and the sapphire light cone with $1.78<K/k_0<2.51$.
\begin{figure*}
\caption{(Color online) The base-10 logarithm of the total EM LDOS as a function of photon energy and the in-plane component of the wave vector (a) in the QW, (b) in air at 1 nm above the surface, and (c) in air at 1 $\mu$m above the surface. (d), (e), and (f) The effective temperature of the total EM field at the corresponding positions in the case of a thermally excited QW. (g), (h), and (i) The effective-field temperature at the corresponding positions in the case of an electrically or optically excited QW corresponding to the bias voltage $U=2.6$ V.}
\label{fig:ene}
\end{figure*}
Figure \ref{fig:pos}(b) shows the effective-field temperature of the total EM field corresponding to the LDOS in Fig.~\ref{fig:pos}(a) in the case of a thermally excited QW. For the narrow In$_{0.15}$Ga$_{0.85}$N layer located slightly left from the position $z=0$ $\mu$m, the source-field temperature is 350 K; for other material layers it is 300 K. It can be seen that the light cones of each material are visible also in the effective-field temperature. The evanescent fields near the material interfaces are even more pronounced when compared to the LDOS in Fig.~\ref{fig:pos}(a). At high values of $K$, the effective-field temperatures approach the source-field temperature in each layer, whereas for K within the light cones of air and sapphire, the effective temperature is reduced due to the strong coupling to the semi-infinite air and sapphire layers. Figure \ref{fig:pos}(c) shows the corresponding effective-field temperature of the total EM field in the case of an electrically or optically excited QW. The figure clearly resembles the case of thermal excitation in Fig.~\ref{fig:pos}(b), but the values of the effective-field temperature are significantly higher, as expected.
Figure \ref{fig:ene}(a) presents the base-10 logarithm of the total EM LDOS in the QW as a function of the photon energy and the in-plane component of the wave vector. The figure clearly shows the GaN/Ag SP resonance, as well as the GaN guided modes corresponding to the Fabry-P\'erot resonances of the cavity. At photon energy slightly above 3 eV, the GaN becomes absorptive, and therefore, there are no resonances visible above this energy. If, instead of the total EM LDOS, we were to plot the LDOS parts corresponding to the TE and TM polarizations, then the SP modes would be visible only in the TM case, as previously discussed, e.g., in Ref.~\cite{Sadi2013b}. Otherwise, the LDOSs of the TE and TM polarizations are qualitatively very similar. Figure \ref{fig:ene}(b) shows the corresponding base-10 logarithm of the total EM LDOS in air at 1 nm above the structure. In addition to the resonances visible in Fig.~\ref{fig:ene}(a), in Fig.~\ref{fig:ene}(b), one can also see the Ag/air SP mode just above the light cone of air. Figure \ref{fig:ene}(c) presents the base-10 logarithm of the EM LDOS in air at 1 $\mu$m above the structure. One can clearly see that there is only a small contribution of the evanescent fields remaining, especially at high frequencies, and the only significant contribution to the EM LDOS arises from the propagating modes in the light cone of air.
Figure \ref{fig:ene}(d) shows the effective-field temperature of the total EM field in the middle of the QW as a function of energy and $K/k_0$ for the case of a thermally excited QW. The effective temperature is essentially above the background temperature of 300 K when the imaginary part of the refractive index of the InGaN QW significantly deviates from zero either due to band-to-band or other absorption and emission mechanisms. At low frequencies, the thin InGaN becomes nearly transparent, and therefore, the effective temperature reaches the background temperature. The emissivity peak near photon energy $\hbar\omega=0.5$ eV follows from the peak in the infrared absorption coefficient of the QW \cite{Tansley1986}. The corresponding effective-field temperature in the case of an electrically or optically excited QW is shown in Fig.~\ref{fig:ene}(g). The emission begins at the photon energy corresponding to the band gap, where the resulting effective-field temperature also obtains its highest values, as expected. At high energies well above the band gap, the effective-field temperature again reaches the source-field temperature of 300 K. The effective temperature of the field generally increases as the optical confinement of the mode increases: In the case of an electrically or optically excited QW and photon energy $\hbar\omega=E_\mathnormal{g}+k_\mathrm{B}T=2.786$ eV, the modes extending into the light cone of air have $T_\mathrm{eff}\approx 2200$ K, whereas the modes bound in the light cone of GaN reach $T_\mathrm{eff}\approx 2700$ K, while the evanescent InGaN modes reach values as high as $T_\mathrm{eff}\approx 3500$ K. For all these cases, however, $T_\mathrm{eff}$ remains well below the source-field temperature of the QW due to the losses caused by the surrounding lossy materials.
Figure \ref{fig:ene}(e) presents the effective-field temperature of the total EM field in air at 1 nm above the structure, corresponding to the LDOS in Fig.~\ref{fig:ene}(b) in the case of a thermally excited QW. The values of the effective-field temperature are somewhat lower than the values of the effective-field temperature in the QW in Fig.~\ref{fig:ene}(d). This is mainly due to the attenuation related to the increased distance to the excited QW. The effective-field temperature in Fig.~\ref{fig:ene}(e), however, resembles the effective-field temperature in the QW. Also, the effective-field temperatures in the case of an electrically or optically excited QW at the two positions presented in Figs.~\ref{fig:ene}(g) and Fig.~\ref{fig:ene}(h) are quite similar. In the case of a thermally excited QW at low frequencies, the infrared emission of the QW is not visible in air as the silver layer between air and the QW becomes very lossy at low frequencies.
Figures \ref{fig:ene}(f) and \ref{fig:ene}(i) show the effective-field temperatures of the total EM field in air at 1 $\mu$m above the structure corresponding to the LDOS in Fig.~\ref{fig:ene}(c) in the cases of thermally and electrically or optically excited QWs. The contribution of the evanescent fields is reduced as in the case of the EM LDOS. Due to the longer distance to the structure and reflections at the interfaces, the values of the effective-field temperatures are also consequently lower compared to the values of the effective-field temperatures in the QW in Figs.~\ref{fig:ene}(d) and \ref{fig:ene}(g).
\section{\label{sec:conclusions}Conclusions}
We have developed a three-dimensional QFED method to describe the photon-number quantization and thermal balance in general lossy and lossless geometries. By appropriately defining the photon ladder operators and the densities of states, we were able to present the ladder operators and the photon-number expectation values using formulas that are equivalent to the forms previously obtained by using a one-dimensional formalism. The resulting generalized QFED method allows studying, e.g., plasmonic structures and defining an effective-field temperature that realistically describes the excitation of the optical field.
To demonstrate the applicability and physical implications of the presented QFED method, we have used the model to study the energy and position dependencies of the EM LDOSs and effective-field temperatures in a light-emitting InGaN QW structure, which has recently been of experimental and theoretical interest. The results show that the developed method is well suited for analyzing the emission of electrically, optically, or thermally excited QWs. The effective temperatures were studied both as a function of position and as a function of photon energy. Electrical and optical excitations of the QW produce high effective-field temperatures, whose energy spectrum is quite narrow, whereas the effective-field temperature of a thermally excited QW has a significantly broader emission spectrum, as expected.
In addition to providing further insight into the classical fluctuational electrodynamics theory, the QFED method enables interesting further studies as it bridges the classical propagating wave picture of the EM field and the fluctuational electrodynamics, which is widely used to model near-field effects. Therefore, we expect that using the QFED, method one could, for instance, find a radiative transfer equation that allows describing interference effects, thus widening the applicability of the conventional radiative transfer equation beyond its main limitation in describing interference effects. This would make it possible to use the radiative transfer equation to describe also near-field effects in resonant structures.
\begin{acknowledgments} This work has been funded in part by the Academy of Finland and the Aalto Energy Efficiency Research Programme. \end{acknowledgments}
\appendix
\section{\label{apx:greengeneral}Green's functions}
Here we briefly review the known relations between the electric, magnetic, and exchange Green's functions. We first define the electric Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ that satisfies \cite{Paulus2000} \begin{align}
&\nabla_\mathbf{r}\times\Big(\frac{\nabla_\mathbf{r}\times\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')}{\mu(\mathbf{r},\omega)}\Big)-k_0^2\varepsilon(\mathbf{r},\omega)\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{I}}\delta(\mathbf{r}-\mathbf{r}'),
\label{eq:greenee} \end{align} where $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{I}}$ is the unit dyadic and $k_0=\omega/c$ is the wavenumber in vacuum with the vacuum velocity of light $c$. The subscript $\mathbf{r}$ in $\nabla_\mathbf{r}$ highlights that the differentiation is here performed with respect to $\mathbf{r}$ instead of $\mathbf{r}'$. The solution of Eq.~\eqref{eq:HelmholtzE1} is then written in terms of the electric Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ by integrating the product of the Green's function and the source terms over all the source points $\mathbf{r}'$ as \begin{align}
\hat{\mathbf{E}}^+(\mathbf{r},\omega)
&= \mu_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot\Big[i\omega\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)\nonumber\\
&\hspace{0.5cm}-\nabla_{\mathbf{r}'}\times\Big(\frac{\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)}{\mu_0\mu(\mathbf{r}',\omega)}\Big)\Big]d^3r'\nonumber\\
&=i\omega\mu_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&\hspace{0.5cm}+k_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)d^3r',
\label{eq:efield2} \end{align} where the subscript $\mathbf{r}'$ in $\nabla_{\mathbf{r}'}$ inside the integral indicates that the differentiation is performed with respect to the source point $\mathbf{r}'$. In the case of the second term, we have applied the Stokes' theorem resulting in the integration by parts formula $\int_V\mathbf{G}^\alpha\cdot(\nabla_{\mathbf{r}'}\times\mathbf{J})d^3r'=\int_V(\nabla_{\mathbf{r}'}\times\mathbf{G}^\alpha)\cdot\mathbf{J}d^3r'-\int_{\partial V}(\mathbf{G}^\alpha\times\mathbf{J})\cdot d\mathbf{S}'$ separately for each row vector $\mathbf{G}^\alpha$ of the matrix representation of $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ with the boundary condition that the Green's functions go to zero when the separation between the source point $\mathbf{r}'$ and the field point $\mathbf{r}$ tends to infinity. Using the short hand notation $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times\nabla_{\mathbf{r}'} =-[\nabla_{\mathbf{r}'}\times\mathbf{G}^1,\nabla_{\mathbf{r}'}\times\mathbf{G}^2,\nabla_{\mathbf{r}'}\times\mathbf{G}^3]^T$, where $T$ denotes transpose, we then define the exchange Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')$ as \begin{equation}
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')
=\frac{\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times\nabla_{\mathbf{r}'}}{k_0\mu(\mathbf{r}',\omega)}.
\label{eq:greenem} \end{equation}
Solving for the magnetic field by using Faraday's law in Eq.~\eqref{eq:maxwell3} and substituting the electric field operator in terms of the Green's functions in Eq.~\eqref{eq:efield2} give \begin{align}
\hat{\mathbf{H}}^+(\mathbf{r},\omega) &=\frac{1}{i\omega\mu_0\mu(\mathbf{r},\omega)}\Big(\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r},\omega)+\nabla_\mathbf{r}\times\hat{\mathbf{E}}^+(\mathbf{r},\omega)\Big)\nonumber\\
&=k_0\int\frac{\nabla_\mathbf{r}\times\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')}{k_0\mu(\mathbf{r},\omega)}\cdot\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&\hspace{0.5cm}-\frac{ik_0^2}{\omega\mu_0}\int\Big[\frac{\nabla_\mathbf{r}\times\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')}{k_0\mu(\mathbf{r},\omega)}\nonumber\\
&\hspace{0.5cm}+\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{I}}\frac{\delta(\mathbf{r}-\mathbf{r}')}{k_0^2\mu(\mathbf{r},\omega)}\Big]\cdot\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&=k_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{e}^+(\mathbf{r}',\omega)d^3r'\nonumber\\
&\hspace{0.5cm}+i\omega\varepsilon_0\int\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r}',\omega)d^3r',
\label{eq:hfield2} \end{align} where we have first substituted the expression for $\hat{\mathbf{E}}^+(\mathbf{r},\omega)$ from Eq.~\eqref{eq:efield2} and incorporated the separate $\hat{\mathbf{J}}_\mathrm{m}^+(\mathbf{r},\omega)$ term into the integral using a suitable $\delta$-function presentation and then defined the exchange Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')$ and the magnetic Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')$ as \begin{equation} \overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')
=\frac{\nabla_\mathbf{r}\times\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')}{k_0\mu(\mathbf{r},\omega)},
\label{eq:greenme} \end{equation} \begin{equation} \overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')
=-\frac{\nabla_\mathbf{r}\times\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')}{k_0\mu(\mathbf{r},\omega)}
-\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{I}}\frac{\delta(\mathbf{r}-\mathbf{r}')}{k_0^2\mu(\mathbf{r},\omega)}.
\label{eq:greenmm} \end{equation} By using Eqs.~\eqref{eq:greenem} and \eqref{eq:greenmm}, one also obtains an expression of the magnetic Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')$ directly in terms of the electric Green's function $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ as \begin{equation}
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')
=-\frac{\nabla_\mathbf{r}\times[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')
\times\nabla_{\mathbf{r}'}]}{k_0^2\mu(\mathbf{r},\omega)\mu(\mathbf{r}',\omega)}
-\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{I}}\frac{\delta (\mathbf{r}-\mathbf{r}')}{k_0^2\mu(\mathbf{r},\omega)}.
\label{eq:greenmm2} \end{equation}
\section{\label{apx:greenstratified}Green's functions for stratified media}
To gain more insight and analytical formulas directly applicable to common planar geometries, and to partly lift the divergences associated with absorbing media, we apply the formalism developed above for stratified media. In the case of stratified media, it is convenient to use the plane wave representation for the dyadic Green's functions: A point in space is denoted in the Cartesian basis $(\hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}})$ by $\mathbf{r}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}+z\hat{\mathbf{z}}=\mathbf{R}+z\hat{\mathbf{z}}$, where $\mathbf{R}=x\hat{\mathbf{x}}+y\hat{\mathbf{y}}$ is the in-plane coordinate and the surface normals are along the $z$ coordinate. Similarly, a wave vector of a plane wave is denoted by $\mathbf{k}=\mathbf{K}+k_z\,\mathrm{sgn}(z-z')\hat{\mathbf{z}}$ where the component $\mathbf{K}$ is in the $x$-$y$ plane and $k_z$ is given by $k_z=\sqrt{k_0^2n^2-K^2}$, with $\mathrm{Im}(k_z)\ge 0$. For convenience, we also define the unit vector $\hat{\mathbf{K}}=\mathbf{K}/K$.
The above notation is convenient since at the $x$-$y$ plane, the dyadic Green's functions of stratified media depend only on the relative in-plane coordinate $\mathbf{R}-\mathbf{R}'$. Therefore, in the plane-wave representation, the dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}(\mathbf{r},\omega,\mathbf{r}')$, $j,k\in\{\mathrm{e,m}\}$, can be written as \cite{Tomas1995,Intravaia2015} \begin{align}
&\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{1}{4\pi^2}\int
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}^\mathrm{T}
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}
e^{i\mathbf{K}\cdot(\mathbf{R}-\mathbf{R}')}d^2K,\label{eq:greenfourier} \end{align} where the terms $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}^\mathrm{T} \overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z') \overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}$ are the Fourier transforms of $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}(\mathbf{r},\omega,\mathbf{r}')$ that have been obtained by rotating the dyadic plane-wave Greens functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')$ calculated using the standard techniques to evaluate the fields in layered structures as presented below. More specifically, $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')$ have been evaluated in a coordinate system where the in-plane noise components are taken to be perpendicular and parallel to $\mathbf{K}$, and the rotation with the rotation matrix $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}$ is used to return this convention of direction back to the coordinate system where the direction of the dipoles does not depend on $\mathbf{K}$.
Due to the symmetry properties of the Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}(\mathbf{r},\omega,\mathbf{r}')$, also $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')$ obey symmetry relations. When the field and source positions $z$ and $z'$ are interchanged, the values of the Green's functions are changed according to the reciprocity relations as follows. The spectral dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')$ obey the reciprocity relation $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{kj}(z',K,\omega,z) =\mathrm{diag}(-1,1,-1)\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')^T\mathrm{diag}(-1,1,-1)$, where $\mathrm{diag}(-1,1,-1)$ is a diagonal matrix with diagonal elements $-1$, $1$, and $-1$. In addition, they obey the complex conjugation relation $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}^*(z,K,\omega,z') =\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,-\omega,z')$.
\subsection{Multi-interface reflection and transmission coefficients} In order to write the spectral dyadic Green's functions for a multi-interface geometry in a compact form, we first define the multi-interface reflection and transmission coefficients that take into account all the reflections in the geometry. First, the single-interface reflection and transmission coefficients following from the boundary conditions of the tangential and normal polarizations ($\sigma\in\{\parallel,\perp\}$) requiring the tangential components of the electric and magnetic fields to be continuous at interfaces are given for the electric and magnetic fields by \begin{align}
r_{e,\parallel} & =\frac{\mu_2k_{z,1}-\mu_1k_{z,2}}{\mu_2k_{z,1}+\mu_1k_{z,2}},
\hspace{1cm} t_{e,\parallel}=\frac{2\mu_2k_{z,1}}{\mu_2k_{z,1}+\mu_1k_{z,2}},\nonumber\\
r_{e,\perp} & =\frac{\varepsilon_1k_{z,2}-\varepsilon_2k_{z,1}}{\varepsilon_2k_{z,1}+\varepsilon_1k_{z,2}},
\hspace{1.1cm} t_{e,\perp}=\frac{2\sqrt{\varepsilon_1/\mu_1}\,n_2k_{z,1}}{\varepsilon_2k_{z,1}+\varepsilon_1k_{z,2}},\nonumber\\
r_{m,\parallel} & =\frac{\varepsilon_2k_{z,1}-\varepsilon_1k_{z,2}}{\varepsilon_2k_{z,1}+\varepsilon_1k_{z,2}},
\hspace{1.1cm} t_{m,\parallel}=\frac{2\varepsilon_2k_{z,1}}{\varepsilon_2k_{z,1}+\varepsilon_1k_{z,2}},\nonumber\\
r_{m,\perp} & =\frac{\mu_1k_{z,2}-\mu_2k_{z,1}}{\mu_2k_{z,1}+\mu_1k_{z,2}},
\hspace{1cm} t_{m,\perp}=\frac{2\sqrt{\mu_1/\varepsilon_1}\,n_2k_{z,1}}{\mu_2k_{z,1}+\mu_1k_{z,2}},
\label{eq:fresnel} \end{align} where $\varepsilon_l$, $\mu_l$, $n_l$, $k_{z,l}$, $l=1,2$, are the relative permittivities, permeabilities, refractive indices, and the $z$ components of the wave vectors in the two materials. The single-interface coefficients in Eq.~\eqref{eq:fresnel} equal the conventional reflection and transmission coefficients used, e.g., in Ref.~\cite{Paulus2000}. In the following, with primed reflection and transmission coefficients we denote the reflection and transmission coefficients for the incidence from medium 2 to medium 1, and they are obtained by switching indices 1 and 2 in Eq.~\eqref{eq:fresnel}.
The multi-interface geometry is defined by interface positions $z_l$, $l=1,2,...,N$, separating material layers with relative permittivities and permeabilities $\varepsilon_l$ and $\mu_l$, $l=1,2,...,N+1$. The layer thicknesses are denoted by $d_l=z_l-z_{l-1}$, where $l=2,...,N$. The multi-interface reflection and transmission coefficients $\mathcal{R}_{l,j}$ and $\mathcal{T}_{l,j}$, which account for the multiple reflections in different medium layers, are recursively given in terms of the single-interface reflection and transmission coefficients as \begin{align}
\mathcal{R}_{l,j,\sigma} & =\frac{r_{l,j,\sigma}+\mathcal{R}_{l+1,j,\sigma}e^{2ik_{z,l+1}d_{l+1}}}{1+r_{l,j,\sigma}\mathcal{R}_{l+1,j,\sigma}e^{2ik_{z,l+1}d_{l+1}}}\label{eq:Rl},\\
\mathcal{T}_{l,j,\sigma} & =\frac{t_{l,j,\sigma}\nu_{l+1,j,\sigma}}{\nu_{l,j,\sigma}(1-\mathcal{R}_{l-1,j,\sigma}'r_{l,j,\sigma} e^{2ik_{z,l}d_l})}\label{eq:Tl}, \end{align} where $l=1,2,...,N$, $j\in\{\mathrm{e},\mathrm{m}\}$, $\sigma\in\{\parallel,\perp\}$ $\nu_{l,j,\sigma}=1/(1-\mathcal{R}_{l-1,j,\sigma}'\mathcal{R}_{l,j,\sigma} e^{2ik_{z,l}d_l})$, and $\mathcal{R}_{0,j,\sigma}'=\mathcal{R}_{N+1,j,\sigma}=0$. As in the case of single-interface coefficients in Eq.~\eqref{eq:fresnel} the primed coefficients denote the coefficients for right incidence. The layers are indexed such that $\mathcal{R}_{l,j,\sigma}'$ corresponds to the same interface as $\mathcal{R}_{l,j,\sigma}$. The propagation coefficient for a certain material layer of thickness $d_l$ is given as $\mathcal{P}_l=e^{ik_{z,l}d_l}$ when the transmission coefficient from layer $l'$ to layer $l>l'+1$ is recursively given by $\mathcal{T}_{l',l,j,\sigma}=\mathcal{T}_{l',l-1,j,\sigma}\mathcal{T}_{l-1,j,\sigma}e^{ik_{z,l-1}d_{l-1}}$, with $\mathcal{T}_{l',l'+1,j,\sigma}=\mathcal{T}_{l',j,\sigma}$, and that from layer $l'$ to layer $l<l'-1$ by $\mathcal{T}_{l',l,j,\sigma}'=\mathcal{T}_{l',l+1,j,\sigma}'\mathcal{T}_{l,j,\sigma}'e^{ik_{z,l+1}d_{l+1}}$, with $\mathcal{T}_{l',l'-1,j,\sigma}'=\mathcal{T}_{l'-1,j,\sigma}'$.
\subsection{Spectral dyadic Green's functions}
Here we give a compact componentwise representation of the spectral dyadic Green's functions for general stratified media. The presentation adapts the dyadic Green's functions given, e.g., in Ref.~\citenum{Paulus2000} or, in the case of purely dielectric structures, in Appendix A of Ref.~\citenum{Joulain2005}. However, the chosen presentation has a few differences: (1) We use the orthonormal basis $(\hat{\mathbf{K}}\times\hat{\mathbf{z}},\hat{\mathbf{K}},\hat{\mathbf{z}})$, where the in-plane noise components are taken to be perpendicular and parallel to $\mathbf{K}$. Then, the rotation matrix $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{R}}$ in Eq.~\eqref{eq:greenfourier} is used to return this convention of direction back to the coordinate system where the direction of the dipoles does not depend on $\mathbf{K}$. (2) Instead of using the orthonormal system of complex-valued unit dyads of Refs.~\citenum{Paulus2000} and \citenum{Joulain2005}, we write the dyadic Green's functions as matrices. (3) We use the scaled forms of the dyadic Green's functions; for example, $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')$ is obtained as a solution to the differential equation in Eq.~\eqref{eq:greenee} instead of the corresponding equation in Ref.~\citenum{Paulus2000}, whose right-hand side contains an additional factor $1/\mu(\mathbf{r},\omega)$. Thus, our notation corresponds to the notation used in the case of normal incidence in Ref.~\citenum{Partanen2016b}.
The spectral dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{ee}(z,K,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{mm}(z,K,\omega,z')$, in our notation, are given in terms of the scaled dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{e}}(z,K,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{m}}(z,K,\omega,z')$ as \begin{align}
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{ee}(z,K,\omega,z') &=
\mu(z',\omega)\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{e}}(z,K,\omega,z')
-\frac{\delta(z-z')}{k_0^2\varepsilon(z,\omega)}\hat{\mathbf{z}}\hat{\mathbf{z}},
\label{eq:greendefe1}\\
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{mm}(z,K,\omega,z') &=
\varepsilon(z',\omega)\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{m}}(z,K,\omega,z')
-\frac{\delta(z-z')}{k_0^2\mu(z,\omega)}\hat{\mathbf{z}}\hat{\mathbf{z}}.
\label{eq:greendefm1} \end{align} The scaled dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_\mathrm{e}(z,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_\mathrm{m}(z,\omega,z')$ in Eqs.~\eqref{eq:greendefe1} and \eqref{eq:greendefm1} are given in the orthonormal basis $(\hat{\mathbf{K}}\times\hat{\mathbf{z}},\hat{\mathbf{K}},\hat{\mathbf{z}})$ by \begin{align}
&\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{j}(z,K,\omega,z')\nonumber\\
&=\!\!\footnotesize\left(\begin{array}{ccc}
\!\!\!\xi_{j,\parallel}^{+}(z,\omega,z')\!\! & 0 & 0\\
0 & \frac{k_zk_z'}{kk'}\xi_{j,\perp}^{+}(z,\omega,z') & \!i\frac{k_z K}{kk'}\frac{\partial}{k_z\partial z}\xi_{j,\perp}^{-}(z,\omega,z')\!\!\!\!\\
0 & i\frac{K k_z'}{kk'}\frac{\partial}{k_z\partial z}\xi_{j,\perp}^{+}(z,\omega,z') & \frac{K^2}{kk'}\xi_{j,\perp}^{-}(z,\omega,z')
\end{array}\right)\!\!, \end{align} where the primed and unprimed quantities correspond to the quantities at positions $z'$ and $z$, respectively, and $\xi_{j,\sigma}^\pm(z,\omega,z')$ are the scaled scalar Green's functions, which are presented in the case of normal incidence in Ref.~\citenum{Partanen2016b}. For non-normal incidence, the generalization of the scaled scalar Green's functions of Ref.~\citenum{Partanen2016b} is obtained by substituting the wave number $k$ with its $z$ component $k_z$ and the reflection and transmission coefficients of normal incidence with the corresponding quantities for non-normal incidence, given in Eqs.~\eqref{eq:Rl} and \eqref{eq:Tl}. Assuming that the source point $z'$ is located in layer $l'$ ($z_{l'-1}<z'<z_{l'}$) and the field point $z$ is located in layer $l$ ($z_{l-1}<z<z_{l}$), in the three cases $l=l'$, $l>l'$, and $l<l'$, the scaled scalar Green's functions are compactly given by \begin{align}
&\xi_{l=l',j,\sigma}^\pm(z,\omega,z')\nonumber\\
&=\frac{i}{2k_{z,l'}}\Big(e^{ik_{z,l'}|z-z'|} \pm \nu_{l',j}\mathcal{R}_{l',j,\sigma}
[e^{-ik_{z,l'}(z+z'-2z_{l'})}\nonumber\\
&\hspace{0.5cm}\pm \mathcal{R}_{l'-1,j,\sigma}'e^{-ik_{z,l'}(z-z'-2d_{l'})}]\pm \nu_{l',j,\sigma}\mathcal{R}_{l'-1,j,\sigma}'\nonumber\\
&\hspace{0.5cm}\times[e^{ik_{z,l'}(z+z'-2z_{l'-1})} \pm \mathcal{R}_{l',j,\sigma}e^{ik_{z,l'}(z-z'+2d_{l'})}]\Big),
\label{eq:greenscalar1} \end{align} \begin{align}
&\xi_{l>l',j,\sigma}^\pm(z,\omega,z')\nonumber\\
&=\frac{i}{2k_{z,l'}}\mathcal{T}_{l',l,j,\sigma}\Big(e^{ik_{z,l'}(z_{l'}-z')} \pm \nu_{l',j,\sigma}\mathcal{R}_{l'-1,j,\sigma}'\nonumber\\
&\hspace{0.5cm}\times[e^{ik_{z,l'}(z'+d_{l'}-z_{l'-1})}\pm \mathcal{R}_{l',j,\sigma}e^{ik_{z,l'}(2d_{l'}-z'+z_{l'})}]\Big)\nonumber\\
&\hspace{0.5cm}\times\Big(e^{ik_{z,l}(z-z_{l-1})} \pm \mathcal{R}_{l,j,\sigma}e^{-ik_{z,l}(z-z_{l-1}-2d_{l})}\Big),
\label{eq:greenscalar2} \end{align} \begin{align}
&\xi_{l<l',j,\sigma}^\pm(z,\omega,z')\nonumber\\
&=\frac{i}{2k_{z,l'}}\mathcal{T}_{l',l,j,\sigma}'\Big(e^{ik_{z,l'}(z'-z_{l'-1})} \pm \nu_{l',j,\sigma}\mathcal{R}_{l',j,\sigma}\nonumber\\
&\hspace{0.4cm}\times[e^{-ik_{z,l'}(z'-2d_{l'}-z_{l'-1})}\!\pm\!\mathcal{R}_{l'-1,j,\sigma}'e^{ik_{z,l'}(z'+2d_{l'}-z_{l'-1})}]\Big)\nonumber\\
&\hspace{0.4cm}\times\Big(e^{-ik_{z,l}(z-z_{l})} \pm \mathcal{R}_{l-1,j,\sigma}'e^{ik_{z,l}(z+d_{l}-z_{l-1})}\Big).
\label{eq:greenscalar3} \end{align}
The spectral dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{me}(z,K,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{em}(z,K,\omega,z')$ following from Eqs.~\eqref{eq:greenem}, \eqref{eq:greenme}, and \eqref{eq:greenfourier} are, respectively, presented in terms of the scaled dyadic exchange Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,e}}(z,K,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,m}}(z,K,\omega,z')$ as \begin{align}
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{me}(z,K,\omega,z') &=\frac{\mu(z',\omega)}{\mu(z,\omega)} \overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,e}}(z,K,\omega,z'),
\label{eq:greendefe2}\\
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{em}(z,K,\omega,z') &=-\frac{\varepsilon(z',\omega)}{\varepsilon(z,\omega)}
\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,m}}(z,K,\omega,z').
\label{eq:greendefm2} \end{align} The scaled dyadic exchange Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,e}}(z,K,\omega,z')$ and $\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex,m}}(z,K,\omega,z')$ are given in terms of the scaled scalar Green's functions in Eqs.~\eqref{eq:greenscalar1}--\eqref{eq:greenscalar3} by \begin{align}
&\overset{\text{\tiny$\leftrightarrow$}}{\boldsymbol{\xi}}_{\mathrm{ex},j}(z,K,\omega,z')\nonumber\\
&\!=\!\!\footnotesize\left(\begin{array}{ccc}
0 & \!\!\!\!-\frac{k_z'k}{k_z k'}\frac{\partial}{k_0\partial z}\xi_{j,\perp}^{+}(z,\omega,z') & \!i\frac{Kk}{k_0k'}\xi_{j,\perp}^{-}(z,\omega,z')\!\!\\
\!\!\!\frac{\partial}{k_0\partial z}\xi_{j,\parallel}^{+}(z,\omega,z') & 0 & 0\\
\!\!\!-i\frac{K}{k_0}\xi_{j,\parallel}^{+}(z,\omega,z') & 0 & 0
\end{array}\right)\!\!. \end{align}
\section{\label{apx:densities}Derivation of the densities of states}
\subsection{\label{apx:nldos}Nonlocal densities of states}
The time-domain field operators are obtained from the frequency-domain operators by Fourier transforms. For example, the time-domain electric field operator is given by \begin{equation}
\hat{\mathbf{E}}(\mathbf{r},t)\!=\!\frac{1}{2\pi}\!\int_0^\infty\!\!\hat{\mathbf{E}}^+(\mathbf{r},\omega)e^{-i\omega t}d\omega
+\frac{1}{2\pi}\!\int_0^\infty\!\!\hat{\mathbf{E}}^-(\mathbf{r},\omega)e^{i\omega t}d\omega, \end{equation} where $\hat{\mathbf{E}}^-(\mathbf{r},\omega)$ is the negative-frequency part obtained by a Hermitian conjugate of the positive-frequency part $\hat{\mathbf{E}}^+(\mathbf{r},\omega)$ in Eq.~\eqref{eq:efield}.
The frequency-space correlation functions are given by \begin{align}
&\langle\hat{\mathbf{E}}^{-}(\mathbf{r},\omega)\cdot\hat{\mathbf{E}}^+(\mathbf{r},\omega')\rangle\nonumber\\
&=\mu_0^2\omega\omega'\int\langle\hat{\mathbf{J}}_\mathrm{e}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&\hspace{0.5cm}\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r}'',\omega')\rangle d^3r'd^3r''\nonumber\\ &\hspace{0.5cm}+k_0^2\int\langle\hat{\mathbf{J}}_\mathrm{m}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')\nonumber\\ &\hspace{0.5cm}\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r}'',\omega')\rangle d^3r'd^3r''\nonumber\\
&=\delta(\omega-\omega')\mu_0^2\omega^2\int|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'\nonumber\\
&\hspace{0.5cm}+\delta(\omega-\omega')k_0^2\int|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r', \end{align} \begin{align}
&\langle\hat{\mathbf{E}}^{+}(\mathbf{r},\omega)\cdot\hat{\mathbf{E}}^-(\mathbf{r},\omega')\rangle\nonumber\\
&=\mu_0^2\omega\omega'\int\langle\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&\hspace{0.5cm}\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{e}^\dag(\mathbf{r}'',\omega')\rangle d^3r'd^3r''\nonumber\\ &\hspace{0.5cm}+k_0^2\int\langle\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\ &\hspace{0.5cm}\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{m}^\dag(\mathbf{r}'',\omega')\rangle d^3r'd^3r''\nonumber\\
&=\delta(\omega-\omega')\mu_0^2\omega^2\int|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')][\langle\hat\eta(\mathbf{r}',\omega)\rangle+1]d^3r'\nonumber\\
&\hspace{0.5cm}+\delta(\omega-\omega')k_0^2\int|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')][\langle\hat\eta(\mathbf{r}',\omega)\rangle+1]d^3r'. \end{align}
In the time domain, we have \begin{align}
&\langle\hat{\mathbf{E}}(\mathbf{r},t)^2\rangle\nonumber\\
&=\frac{1}{4\pi^2}\int_0^\infty\!\int_0^\infty\langle\hat{\mathbf{E}}^-(\mathbf{r},\omega)\cdot\hat{\mathbf{E}}^+(\mathbf{r},\omega')\rangle e^{i(\omega-\omega')t}d\omega d\omega'\nonumber\\
&+\frac{1}{4\pi^2}\int_0^\infty\!\int_0^\infty\langle\hat{\mathbf{E}}^+(\mathbf{r},\omega)\cdot\hat{\mathbf{E}}^-(\mathbf{r},\omega')\rangle e^{i(\omega'-\omega)t}d\omega d\omega', \end{align} which then becomes \begin{align}
&\langle\hat{\mathbf{E}}(\mathbf{r},t)^2\rangle\nonumber\\
&=\int_0^\infty\!\!\!\int\!\frac{\mu_0^2\omega^2}{2\pi^2}|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2 \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}\times\Big(\langle\hat\eta(\mathbf{r}',\omega)\rangle+\frac{1}{2}\Big)d^3r'd\omega\nonumber\\
&\hspace{0.4cm}+\!\int_0^\infty\!\!\!\int\!\!\frac{k_0^2}{2\pi^2}|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2 \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}\times\Big(\langle\hat\eta(\mathbf{r}',\omega)\rangle+\frac{1}{2}\Big)d^3r'd\omega. \end{align}
Using $|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2=4\pi\hbar\omega^2\varepsilon_0\varepsilon_\mathrm{i}(\mathbf{r}',\omega)$ and
$|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2=4\pi\hbar\omega^2\mu_0\mu_\mathrm{i}(\mathbf{r}',\omega)$ gives \begin{align}
&\langle\hat{\mathbf{E}}(\mathbf{r},t)^2\rangle\nonumber\\
&=\int_0^\infty\!\!\int\frac{2\hbar\omega^4\mu_0}{\pi c^2}\Big(\varepsilon_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big)\nonumber\\ &\hspace{0.5cm}\times\Big(\langle\hat\eta(\mathbf{r}',\omega)\rangle+\frac{1}{2}\Big)d^3r'd\omega. \label{eq:apxefluct} \end{align} This allows defining the NLDOS for the electric field as \begin{align}
&\rho_\mathrm{NL,e}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3}{\pi c^4}\Big(\varepsilon_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big). \label{eq:apxenldos} \end{align} Corresponding equations can be written for the magnetic field. The NLDOS of the magnetic field is then given by \begin{align}
&\rho_\mathrm{NL,m}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3}{\pi c^4}\Big(\varepsilon_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(\mathbf{r}',\omega) \mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big). \label{eq:apxhnldos} \end{align}
\subsection{\label{apx:ifdos}Interference density of states}
For an optical mode the quantum optical Poynting vector is defined as a normal ordered operator in terms of the positive- and negative-frequency parts of the electric and magnetic field operators as $\hat{\mathbf{S}}(\mathbf{r},t)=:\!\hat{\mathbf{E}}(\mathbf{r},t)\times\hat{\mathbf{H}}(\mathbf{r},t)\!:=\hat{\mathbf{E}}^-(\mathbf{r},t)\times\hat{\mathbf{H}}^+(\mathbf{r},t)-\hat{\mathbf{H}}^-(\mathbf{r},t)\times\hat{\mathbf{E}}^+(\mathbf{r},t)$ \cite{Loudon2000}. Substituting the time-space forms of the electric and magnetic field operators in Eqs.~\eqref{eq:efield} and \eqref{eq:hfield} gives \begin{align}
&\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle\nonumber\\
&=\frac{1}{4\pi^2}\int_0^\infty\int_0^\infty\langle\hat{\mathbf{E}}^-(\mathbf{r},\omega)\times\hat{\mathbf{H}}^+(\mathbf{r},\omega')\rangle e^{i(\omega-\omega')t}d\omega d\omega'\nonumber\\
&-\frac{1}{4\pi^2}\int_0^\infty\int_0^\infty\langle\hat{\mathbf{H}}^-(\mathbf{r},\omega)\times\hat{\mathbf{E}}^+(\mathbf{r},\omega')\rangle e^{i(\omega-\omega')t}d\omega d\omega' \end{align}
The frequency-space correlation functions are given by \begin{align}
&\langle\hat{\mathbf{E}}^{-}(\mathbf{r},\omega)\times\hat{\mathbf{H}}^+(\mathbf{r},\omega')\rangle\nonumber\\
&=-i\omega\mu_0k_0\int\langle[\hat{\mathbf{J}}_\mathrm{e}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\
&\hspace{0.5cm}\times[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r}'',\omega')]\rangle d^3r'd^3r''\nonumber\\ &\hspace{0.5cm}+i\omega'\varepsilon_0k_0\int\langle[\hat{\mathbf{J}}_\mathrm{m}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}\times[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r}'',\omega')]\rangle d^3r'd^3r''\nonumber\\
&=-\delta(\omega-\omega')i\omega\mu_0k_0\int|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'\nonumber\\
&\hspace{0.5cm}+\delta(\omega-\omega')i\omega\varepsilon_0k_0\int|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r', \end{align} \begin{align}
&\langle\hat{\mathbf{H}}^{-}(\mathbf{r},\omega)\times\hat{\mathbf{E}}^+(\mathbf{r},\omega')\rangle\nonumber\\
&=-i\omega\varepsilon_0k_0\int\langle[\hat{\mathbf{J}}_\mathrm{m}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\
&\hspace{0.5cm}\times[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{m}(\mathbf{r}'',\omega')]\rangle d^3r'd^3r''\nonumber\\ &\hspace{0.5cm}+i\omega'\mu_0k_0\int\langle[\hat{\mathbf{J}}_\mathrm{e}^\dag(\mathbf{r}',\omega)\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\nonumber\\ &\hspace{0.5cm}\times[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega',\mathbf{r}'')\cdot\hat{\mathbf{J}}_\mathrm{e}(\mathbf{r}'',\omega')]\rangle d^3r'd^3r''\nonumber\\
&=-\delta(\omega-\omega')i\omega\varepsilon_0k_0\int|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'\nonumber\\
&\hspace{0.5cm}+\delta(\omega-\omega')i\omega\mu_0k_0\int|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')]\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'. \end{align} Here $\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jj}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{kj}(\mathbf{r},\omega,\mathbf{r}')]
=\sum_\sigma[\hat{\mathbf{e}}_\sigma\cdot\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jj}^\dag(\mathbf{r},\omega,\mathbf{r}')]\times
[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{kj}(\mathbf{r},\omega,\mathbf{r}')\cdot\hat{\mathbf{e}}_\sigma]$, which is a vector, in contrast to the conventional trace of a matrix. The Poynting vector then becomes \begin{align}
&\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle\nonumber\\
&=\frac{1}{4\pi^2}\int_0^\infty\int\Big(
-i\omega\mu_0k_0|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\
&\hspace{0.5cm}\times\mathrm{Tr}\Big[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&\hspace{0.5cm}+\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\Big]\nonumber\\
&\hspace{0.5cm}+i\omega\varepsilon_0k_0|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\ &\hspace{0.5cm}\times\mathrm{Tr}\Big[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&\hspace{0.5cm}+\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}^\dag(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}(\mathbf{r},\omega,\mathbf{r}')\Big]
\Big)\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'd\omega\nonumber\\
&=\frac{1}{2\pi^2}\int_0^\infty\int\Big(
-\omega\mu_0k_0|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\nonumber\\
&\hspace{0.5cm}+\omega\varepsilon_0k_0|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\Big)\nonumber\\
&\hspace{0.5cm}\times\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'd\omega. \end{align}
Using $|j_\mathrm{0,e}(\mathbf{r}',\omega)|^2=4\pi\hbar\omega^2\varepsilon_0\varepsilon_\mathrm{i}(\mathbf{r}',\omega)$ and
$|j_\mathrm{0,m}(\mathbf{r}',\omega)|^2=4\pi\hbar\omega^2\mu_0\mu_\mathrm{i}(\mathbf{r}',\omega)$ gives \begin{align}
&\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle\nonumber\\
&=\int_0^\infty\int\frac{2\hbar\omega^4}{\pi c^3}\nonumber\\
&\hspace{0.5cm}\times\Big(\mu_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\nonumber\\
&\hspace{0.5cm}-\varepsilon_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\Big)\nonumber\\
&\hspace{0.5cm}\times\langle\hat\eta(\mathbf{r}',\omega)\rangle d^3r'd\omega.
\label{eq:apxpoynting} \end{align} This allows defining the IFDOS as \begin{align}
&\boldsymbol{\rho}_\mathrm{IF}(\mathbf{r},\omega,\mathbf{r}')\nonumber\\
&=\frac{2\omega^3n_\mathrm{r}(\mathbf{r},\omega)}{\pi c^4}\nonumber\\
&\hspace{0.5cm}\times\Big(\mu_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\nonumber\\
&\hspace{0.5cm}-\varepsilon_\mathrm{i}(\mathbf{r}',\omega)
\mathrm{Im}\Big[\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]\Big]\Big), \end{align} where $n_\mathrm{r}(\mathbf{r},\omega)$ is the real part of the refractive index.
\section{\label{apx:stratifieddos}Densities of states for stratified media}
Here we present the densities of states for stratified media by using the components $g_{jk}^{\alpha\beta}$, $\alpha,\beta\in\{1,2,3\}$, of the matrix representations of the spectral dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}$.
\subsection{Nonlocal densities of states}
Using Eq.~\eqref{eq:apxefluct} with $\varepsilon(\mathbf{r}',\omega)=\varepsilon(z',\omega)$, $\mu(\mathbf{r}',\omega)=\mu(z',\omega)$, $\langle\hat\eta(\mathbf{r}',\omega)\rangle=\langle\hat\eta(z',\omega)\rangle$, and \begin{align}
&\int\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}^\dag(\mathbf{r},\omega,\mathbf{r}')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_{jk}(\mathbf{r},\omega,\mathbf{r}')]d^2R'\nonumber\\
&=\frac{1}{4\pi^2}
\int\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{jk}(z,K,\omega,z')]d^2K, \end{align} where $j,k\in\{\mathrm{e,m}\}$, gives \begin{align}
&\langle\hat{\mathbf{E}}(\mathbf{r},t)^2\rangle\nonumber\\
&=\int\int_0^\infty\int_{-\infty}^\infty\frac{\hbar\omega^4\mu_0}{2\pi^3c^2}\nonumber\\
&\hspace{0.5cm}\times\Big(\varepsilon_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}(z,K,\omega,z')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}(z,K,\omega,z')]\Big)\nonumber\\
&\hspace{0.5cm}\times\Big(\langle\hat\eta(z',\omega)\rangle+\frac{1}{2}\Big)dz'd\omega d^2K. \end{align} Then, the NLDOS for the electric field can be written as \begin{align}
&\rho_\mathrm{NL,e}(z,K,\omega,z')\nonumber\\
&=\frac{\omega^3}{2\pi^3c^4} \Big(\varepsilon_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}(z,K,\omega,z')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}(z,K,\omega,z')]\Big)\nonumber\\ &=\frac{\omega^3}{2\pi^3c^4}\sum_{\alpha,\beta} \Big(\varepsilon_\mathrm{i}(z',\omega)
|g_{\mathrm{ee}}^{\alpha\beta}(z,K,\omega,z')|^2\nonumber\\
&\hspace{0.5cm}+\mu_\mathrm{i}(z',\omega)
|g_{\mathrm{em}}^{\alpha\beta}(z,K,\omega,z')|^2\Big), \label{eq:enldosstrat} \end{align} where $g_\mathrm{ee}^{\alpha\beta}$ and $g_\mathrm{em}^{\alpha\beta}$, with $\alpha,\beta\in\{1,2,3\}$, are components of the matrix representations of the spectral dyadic Green's functions $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}$ and $\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}$. The NLDOS of the magnetic field is given by \begin{align}
&\rho_\mathrm{NL,m}(z,K,\omega,z')\nonumber\\
&=\frac{\omega^3}{2\pi^3c^4} \Big(\varepsilon_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{me}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{me}}(z,K,\omega,z')]\nonumber\\ &\hspace{0.5cm}+\mu_\mathrm{i}(z',\omega)
\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{mm}}^\dag(z,K,\omega,z')\cdot
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{mm}}(z,K,\omega,z')]\Big)\nonumber\\ &=\frac{\omega^3}{2\pi^3c^4}\sum_{\alpha,\beta} \Big(\varepsilon_\mathrm{i}(z',\omega)
|g_{\mathrm{me}}^{\alpha\beta}(z,K,\omega,z')|^2\nonumber\\
&\hspace{0.5cm}+\mu_\mathrm{i}(z',\omega)
|g_{\mathrm{mm}}^{\alpha\beta}(z,K,\omega,z')|^2\Big). \label{eq:hnldosstrat} \end{align}
\subsection{Local densities of states}
As integrals of the electric and magnetic NLDOSs in Eqs.~\eqref{eq:enldosstrat} and \eqref{eq:hnldosstrat}, the electric and magnetic LDOSs are given by \begin{align}
\rho_\mathrm{e}(z,K,\omega) &=\frac{\omega}{2\pi^3 c^2}\mathrm{Im}\Big[g_\mathrm{ee}^{11}+g_\mathrm{ee}^{22}+\frac{\varepsilon(z,\omega)^2}{|\varepsilon(z,\omega)|^2}g_\mathrm{ee}^{33}\Big]
\label{eq:eldos},\\
\rho_\mathrm{m}(z,K,\omega) &=\frac{\omega}{2\pi^3 c^2}\mathrm{Im}\Big[g_\mathrm{mm}^{11}+g_\mathrm{mm}^{22}+\frac{\mu(z,\omega)^2}{|\mu(z,\omega)|^2}g_\mathrm{mm}^{33}\Big]
\label{eq:hldos}. \end{align}
\subsection{Interference density of states}
Using Eq.~\eqref{eq:apxpoynting} with $\varepsilon(\mathbf{r}',\omega)=\varepsilon(z',\omega)$, $\mu(\mathbf{r}',\omega)=\mu(z',\omega)$, $\langle\hat\eta(\mathbf{r}',\omega)\rangle=\langle\hat\eta(z',\omega)\rangle$, and \begin{align}
&\int\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{ee}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{me}^\dag(\mathbf{r},\omega,\mathbf{r}')]d^2R'\nonumber\\
&=\frac{1}{4\pi^2}
\int\hat{\mathbf{z}}\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{ee}}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{me}}^\dag(z,K,\omega,z')]d^2K, \end{align} \begin{align}
&\int\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{mm}(\mathbf{r},\omega,\mathbf{r}')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{G}}_\mathrm{em}^\dag(\mathbf{r},\omega,\mathbf{r}')]d^2R'\nonumber\\
&=\frac{1}{4\pi^2}
\int\hat{\mathbf{z}}\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{mm}}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_{\mathrm{em}}^\dag(z,K,\omega,z')]d^2K \end{align} gives \begin{align}
&\langle\hat{\mathbf{S}}(\mathbf{r},t)\rangle\nonumber\\
&=\int\int_0^\infty
\int_{-\infty}^\infty\frac{\hbar\omega^4}{2\pi^3c^3}\Big(\mu_\mathrm{i}(z',\omega)\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\hat{\mathbf{z}}\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{mm}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{em}^\dag(z,K,\omega,z')]\Big]\nonumber\\
&\hspace{0.4cm}-\varepsilon_\mathrm{i}(z',\omega)\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\hat{\mathbf{z}}\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{ee}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{me}^\dag(z,K,\omega,z')]\Big]\Big)\nonumber\\
&\hspace{0.5cm}\times\langle\hat\eta(z',\omega)\rangle dz'd\omega d^2K. \end{align} Note that the Poynting vector points purely in the $z$ direction, which is natural due to the symmetry with respect to the $z$ axis. Hence, the IFDOS can be written as $\boldsymbol{\rho}_\mathrm{IF}(z,K,\omega,z')=\hat{\mathbf{z}}\rho_\mathrm{IF}(z,K,\omega,z')$, where the scalar IFDOS $\rho_\mathrm{IF}(z,K,\omega,z')$ is given by \begin{align}
&\rho_\mathrm{IF}(z,K,\omega,z')\nonumber\\
&=
\frac{\omega^3n_\mathrm{r}(z,\omega)}{2\pi^3c^4}\Big(\mu_\mathrm{i}(z',\omega)\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{mm}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{em}^\dag(z,K,\omega,z')]\Big]\nonumber\\
&\hspace{0.5cm}-\varepsilon_\mathrm{i}(z',\omega)\nonumber\\
&\hspace{0.5cm}\times\mathrm{Im}\Big[\hat{\mathbf{z}}\cdot\mathrm{Tr}[\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{ee}(z,K,\omega,z')\times
\overset{\text{\tiny$\leftrightarrow$}}{\mathbf{g}}_\mathrm{me}^\dag(z,K,\omega,z')]\Big]
\Big)\nonumber\\
&=\frac{\omega^3n_\mathrm{r}(z,\omega)}{2\pi^3c^4}\nonumber\\
&\hspace{0.5cm}\times\Big(\mu_\mathrm{i}(z',\omega)
\mathrm{Im}\Big[g_\mathrm{mm}^{11}g_\mathrm{em}^{21*}-g_\mathrm{mm}^{22}g_\mathrm{em}^{12*}-g_\mathrm{mm}^{23}g_\mathrm{em}^{13*}\Big]\nonumber\\
&\hspace{0.5cm}-\varepsilon_\mathrm{i}(z',\omega)
\mathrm{Im}\Big[g_\mathrm{ee}^{11}g_\mathrm{me}^{21*}-g_\mathrm{ee}^{22}g_\mathrm{me}^{12*}-g_\mathrm{ee}^{23}g_\mathrm{me}^{13*}\Big]\Big).
\label{eq:ifdosstrat} \end{align}
\end{document} |
\begin{document}
\title[Rational torsion subgroups and Eisenstein cochains] {The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains}
\author{Mihran Papikian} \address{Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A.} \email{papikian@psu.edu} \author{Fu-Tsun Wei} \address{Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan} \email{ftwei@math.sinica.edu.tw}
\thanks{The first author's research was partially supported by grants from the Simons Foundation (245676) and the National Security Agency (H98230-15-1-0008).}
\subjclass[2010]{11G09, 11G18, 11F12}
\keywords{Drinfeld modular curves; Cuspidal divisor group; Eisenstein ideal; Pseudo-harmonic cochains}
\begin{abstract} Let $\mathfrak n$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\mathfrak n)$ of the Drinfeld modular curve $X_0(\mathfrak n)$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$. \end{abstract}
\maketitle
\section{Introduction}
\subsection{Rational torsion of classical modular Jacobians} Let $N\geq 1$ be a positive integer. Let $X_0(N)$ be the modular curve over $\mathbb{Q}$ parametrizing the isomorphism classes of (generalized) elliptic curves with $\Gamma_0(N)$-structures. The rational torsion subgroup $\mathcal{T}(N):=J_0(N)(\mathbb{Q})_\mathrm{tor}$ of the Jacobian variety $J_0(N)$ of $X_0(N)$ is a finite group by the Mordell-Weil theorem. The cuspidal divisor class group $\mathcal{C}(N)$ of $J_0(N)$, i.e., the group generated by the classes of differences of two cusps of $X_0(N)$, is also finite by a theorem of Manin and Drinfeld. In the early 1970s, for $N=p$ prime, Ogg computed that $\mathcal{C}(p)\cong \mathbb{Z}\big/\frac{p-1}{(p-1,12)}\mathbb{Z}$ and conjectured that $\mathcal{T}(p)=\mathcal{C}(p)$; see \cite{OggBAMS}. In his seminal paper \cite{Mazur}, Mazur proved this conjecture by a detailed study of the Eisenstein ideal of the Hecke ring of level $p$.
When $N$ is square-free, all cusps of $X_0(N)$ are rational over $\mathbb{Q}$. Therefore, $\mathcal{C}(N)\subseteq \mathcal{T}(N)$. A natural generalization of Ogg's conjecture then would predict an equality $\mathcal{C}(N)=\mathcal{T}(N)$. Recently, Ohta \cite{Ohta} proved the following theorem toward this conjecture (given a finite abelian group $G$, we denote by $G_\ell$ the maximal $\ell$-primary subgroup of $G$): \begin{thm}\label{thmMOhta} Let $N$ be a square-free positive integer. We have $\mathcal{C}(N)_\ell=\mathcal{T}(N)_\ell$ for all prime numbers $\ell\geq 3$ when $N$ is not divisible by $3$; and for all prime numbers $\ell\geq 5$ when $N$ is divisible by $3$. \end{thm}
The proof of this theorem again relies on the study of Eisenstein ideals. Ohta computes the index of the Eisentein ideal in an appropriate Hecke algebra, up to a power of $2$, which turns out to be the order of $\mathcal{C}(N)$ by a formula of Takagi \cite{Takagi}, again up to a power of $2$. One important observation that Ohta makes is that the Hecke algebra best suited for the purpose of proving $\mathcal{T}(N)=\mathcal{C}(N)$ in the square-free case is the algebra generated by the Hecke operators $T_p$, for prime numbers $p$ not dividing $N$, and the Atkin-Lehner involutions $W_p$ for $p\mid N$ (instead of $U_p$ operators). The restriction $3\nmid N$ in Theorem \ref{thmMOhta} in the case $\ell=3$ is of technical nature, partly arising from the existence of constant modular forms over $\mathbb{F}_3$ of level $1$ and weight $2$.
On the other hand, the proof of $\mathcal{C}(N)_2=\mathcal{T}(N)_2$ is beyond the reach of the method used by Ohta. Even in the prime level case considered by Mazur \cite{Mazur}, proving $\mathcal{C}(p)_2=\mathcal{T}(p)_2$ requires deeper techniques related to the ring-theoretic properties of the Hecke algebra.
\subsection{Rational torsion of Drinfeld modular Jacobians} Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is a power of a prime number $p$. Let $A=\mathbb{F}_q[T]$ be the ring of polynomials in indeterminate $T$ with coefficients in $\mathbb{F}_q$, and $F=\mathbb{F}_q(T)$ the field of fractions of $A$. The degree map $\deg: F\to \mathbb{Z}\cup \{-\infty\}$, which associates to a non-zero polynomial its degree in $T$ and $\deg(0)=-\infty$, defines
a norm on $F$ by $|a|:=q^{\deg(a)}$; the corresponding place of $F$ is usually called the \textit{place at infinity} and is denoted by $\infty$.
We also define a norm and degree on the ideals of $A$ by $|\mathfrak n|:=\#(A/\mathfrak n)$ and $\deg(\mathfrak n):=\log_q|\mathfrak n|$. Let $F_\infty$ denote the completion of $F$ at $\infty$, and $\mathbb{C}_\infty$ denote the completion of an algebraic closure of $F_\infty$. The \textit{Drinfeld half-plane} $\Omega:=\mathbb{C}_\infty - F_\infty$ has a natural structure of a smooth connected rigid-analytic space over $F_\infty$; see \cite[$\S$1]{GR}.
Let $\mathfrak n\lhd A$ be a non-zero ideal. The level-$\mathfrak n$ \textit{Hecke congruence subgroup} of $\mathrm{GL}_2(A)$ $$
\Gamma_0(\mathfrak n):=\left\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in \mathrm{GL}_2(A)\ \bigg|\ c\in \mathfrak n \right\} $$ plays a central role in this paper. This group acts on $\Omega$ via linear fractional transformations.
Drinfeld proved in \cite{Drinfeld} that the quotient $\Gamma_0(\mathfrak n)\setminus \Omega$ is the space of $\mathbb{C}_\infty$-points of an affine curve $Y_0(\mathfrak n)$ defined over $F$, which is a moduli space of rank-$2$ Drinfeld modules. The unique smooth projective curve over $F$ containing $Y_0(\mathfrak n)$ as an open subvariety is denoted by $X_0(\mathfrak n)$. The \textit{cusps} of $X_0(\mathfrak n)$ are the finitely many geometric points of the complement of $Y_0(\mathfrak n)$ in $X_0(\mathfrak n)$. Let $J_0(\mathfrak n)$ be the Jacobian variety of $X_0(\mathfrak n)$. The \textit{cuspidal divisor group} $\mathcal{C}(\mathfrak n)$ is the subgroup of $J_0(\mathfrak n)$ generated by the classes of divisors $c-c'$, where $c, c'$ run through the set of cusps of $X_0(\mathfrak n)$. It is known that $\mathcal{C}(\mathfrak n)$ is a finite group; see \cite{GekelerIJM}. By the Lang-N\'eron theorem, the group of $F$-rational points of $J_0(\mathfrak n)$ is finitely generated, in particular, its torsion subgroup $\mathcal{T}(\mathfrak n):=J_0(\mathfrak n)(F)_\mathrm{tor}$ is finite. Finally, it is known that, when $\mathfrak n$ is square-free, all cusps of $X_0(\mathfrak n)$ are rational over $F$; see \cite[Prop. 6.7]{Invariants}. Therefore, $\mathcal{C}(\mathfrak n)\subseteq \mathcal{T}(\mathfrak n)$. In analogy with generalized Ogg conjecture that we discussed earlier, one can propose the following:
\begin{conj}\label{conjGOC} For a square-free ideal $\mathfrak n\lhd A$, we have $\mathcal{C}(\mathfrak n)= \mathcal{T}(\mathfrak n)$. \end{conj}
This statement for $\mathfrak n=\mathfrak p$ prime was proved by P\'al \cite{Pal}, following Mazur's method.
Now let $\mathfrak n=\prod_{i=1}^s\mathfrak p_i$ be a square-free ideal in $A$ with the given prime decomposition. Let $$
\mathbb{E}=\{(\varepsilon_1, \dots, \varepsilon_s)\ |\ \varepsilon_i=\pm 1,\ 1\leq i\leq s\}. $$ We single out two (not necessarily distinct) elements of $\mathbb{E}$: \begin{align*} {\bm{\epsilon}}_{H} &:=((-1)^{\deg(\mathfrak p_1)}, \dots, (-1)^{\deg(\mathfrak p_s)}), \text{ and} \\ {\bm{1}} &:=(1,1,\dots, 1). \end{align*} For ${\bm{\epsilon}}=(\varepsilon_1, \dots, \varepsilon_s)\in \mathbb{E}$, define $$ N({\bm{\epsilon}})= \begin{cases} 1 & \text{if ${\bm{\epsilon}}={\bm{1}}$};\\
\prod_{i=1}^s(1+\varepsilon_i|\mathfrak p_i|) & \text{if ${\bm{\epsilon}}={\bm{\epsilon}}_{H}$ and ${\bm{\epsilon}}\neq {\bm{1}}$}; \\
\frac{1}{q+1}\prod_{i=1}^s(1+\varepsilon_i|\mathfrak p_i|) & \text{if ${\bm{\epsilon}}\neq{\bm{\epsilon}}_{H}$ and ${\bm{\epsilon}}\neq {\bm{1}}$}. \end{cases} $$ The main result of this paper is the following: \begin{thm}\label{thmMainT} Let $\mathfrak n=\prod_{i=1}^s \mathfrak p_i$ be a square-free ideal in $A$. Let $\ell$ be a prime number not dividing $q(q-1)$. Then $$ \mathcal{T}(\mathfrak n)_\ell=\mathcal{C}(\mathfrak n)_\ell \cong \bigoplus_{{\bm{\epsilon}}\in \mathbb{E}} \mathbb{Z}_\ell/N({\bm{\epsilon}})\mathbb{Z}_\ell. $$ \end{thm}
The method that we use falls short of proving $\mathcal{T}(\mathfrak n)_\ell=\mathcal{C}(\mathfrak n)_\ell$ for primes $\ell$ dividing $(q-1)$; these primes are the analogues of $\ell=2$ over $\mathbb{Q}$. The prime $\ell=p$, being the characteristic of $F$, has peculiar properties, as far as Conjecture \ref{conjGOC} is concerned. On the one hand, it is not hard to prove the following: \begin{prop} Assume $\mathfrak n\lhd A$ is square-free and there is a prime divisor $\mathfrak p\mid \mathfrak n$ such that $\deg(\mathfrak n/\mathfrak p)\leq 2$. Then $\mathcal{T}(\mathfrak n)_p=0$. \end{prop} \begin{proof} If $\deg(\mathfrak n/\mathfrak p)\leq 2$, then $J_0(\mathfrak n)$ has purely toric reduction at $\mathfrak p$. Moreover, the component group $\Phi_\mathfrak p$ of the N\'eron model of $J_0(\mathfrak n)$ at $\mathfrak p$ has order coprime to $p$; see \cite[Thm. 5.3]{PW1}. On the other hand, because the reduction is purely toric, $\mathcal{T}(\mathfrak n)_p$ injects into $\Phi_\mathfrak p$; cf. \cite[Lem. 7.13]{Pal}.
Hence $\mathcal{T}(\mathfrak n)_p=0$. \end{proof}
On the other hand, when there is no prime dividing $\mathfrak n$ at which $J_0(\mathfrak n)$ has purely toric reduction, it is not clear to us how to analyse $\mathcal{T}(\mathfrak n)_p$. More precisely, it not clear whether the theory of the Eisenstein ideal is applicable at all to the study of $\mathcal{T}(\mathfrak n)_p$. (When $\ell\neq p$, the Eichler-Shimura
congruence relation implies that $T_\mathfrak p-(|\mathfrak p|+1)$, $\mathfrak p\nmid \mathfrak n$, annihilates $\mathcal{T}(\mathfrak n)_\ell$, hence one can use the Eisenstein ideal to study $\mathcal{T}(\mathfrak n)_\ell$.) In any case, we prove the following (see Proposition \ref{propCp=0}):
\begin{prop} If $\mathfrak n$ is square-free, then $\mathcal{C}(\mathfrak n)_p=0$. Therefore, if Conjecture \ref{conjGOC} is valid, then $\mathcal{T}(\mathfrak n)_p=0$. \end{prop}
\begin{rem} We proved in \cite{PW2} that when $\mathfrak n$ is not square-free, and is not equal to a square of a prime, $\mathcal{T}(\mathfrak n)_p\neq 0$. Note that in \cite{PW2} we constructed explicit elements in $\mathcal{T}(\mathfrak n)_p$ using cuspidal divisors, which can be shown to be annihilated by the Eisenstein ideal. \end{rem}
Now we give an outline of the proof of Theorem \ref{thmMainT}. The Atkin-Lehner involutions form a group $\mathbb{W}\cong (\mathbb{Z}/2\mathbb{Z})^s$ which acts on $\mathcal{T}(\mathfrak n)$ and $\mathcal{C}(\mathfrak n)$. Away from the $2$-primary components, one can decompose $\mathcal{T}(\mathfrak n)$ and $\mathcal{C}(\mathfrak n)$ into direct sums of $\mathbb{W}$-eigenspaces, each eigenspace corresponding to some ${\bm{\epsilon}}\in \mathbb{E}$. Hence, it is enough to show that $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}=\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}\cong \mathbb{Z}_\ell/N({\bm{\epsilon}})\mathbb{Z}_\ell$ for all $\ell\nmid q(q-1)$ and ${\bm{\epsilon}}=(\varepsilon_1, \dots, \varepsilon_s)\in \mathbb{E}$, where $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}$ denotes the largest direct summand of $\mathcal{T}(\mathfrak n)_\ell$ on which $W_{\mathfrak p_i}$ acts by $\varepsilon_i$, $1\leq i\leq s$, and similarly for $\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}$. Since $J_0(\mathfrak n)$ has split toric reduction at $\infty$, the $\ell$-primary subgroup $\mathcal{T}(\mathfrak n)_\ell$, $\ell\nmid q(q-1)$, maps injectively into the component group $\Phi_\infty$ of the N\'eron model of $J_0(\mathfrak n)$ at $\infty$. Then, using the rigid-analytic uniformization of $J_0(\mathfrak n)$ at $\infty$ and the Eichler-Shimura relations, one shows that the image of $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}$ in $\Phi_\infty$ can be identified with a subspace of $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})^{\bm{\epsilon}}$ for any sufficiently large $n$. Here $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})^{\bm{\epsilon}}$ denotes the module of $\Gamma_0(\mathfrak n)$-invariant $\mathbb{Z}/\ell^n\mathbb{Z}$-valued
cuspidal harmonic cochains on which $T_\mathfrak p$ ($\mathfrak p\nmid \mathfrak n$) acts by multiplication by $|\mathfrak p|+1$, and $\mathbb{W}$ acts by ${\bm{\epsilon}}$. We study the space $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})^{\bm{\epsilon}}$ in Section \ref{sEPHC}, where we show that it is generated by the reduction modulo $\ell^n$ of certain Eisenstein series with constant Fourier coefficient $qN({\bm{\epsilon}})$. We then use this to identify $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}$ with a subgroup of $\mathbb{Z}_\ell/N({\bm{\epsilon}})\mathbb{Z}_\ell$. On the other hand, in Section \ref{sCDG}, we construct an explicit element in $\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}$, and use it to show that $\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}$ contains a subgroup isomorphic to $\mathbb{Z}_\ell/N({\bm{\epsilon}})\mathbb{Z}_\ell$. Now Theorem \ref{thmMainT} immediately follows by comparing the orders of $\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}$ and $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}$. We note that the crucial idea of using the Atkin-Lehner involutions instead of $U_\mathfrak p$ operators to analyse the Eisenstein harmonic cochains was inspired by Ohta's paper \cite{Ohta}. Also, the trick with mapping $\mathcal{T}(\mathfrak n)_\ell$ into $\Phi_\infty$ was first used by P\'al in \cite{Pal} in the prime level case.
\begin{rem} In \cite{PW1}, we proved a result toward Conjecture \ref{conjGOC} when $\mathfrak n$ is a product of two distinct primes. In that paper, which was mostly written in 2012, we used $U_\mathfrak p$ operators instead of Atkin-Lehner involutions to analyse $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})$. That approach is technically more complicated and leads to a weaker result than Theorem \ref{thmMainT} specialized to $s=2$. On the other hand, in \cite{PW1}, we determined the structure of $\mathcal{C}(\mathfrak p_1\mathfrak p_2)_\ell$ for all $\ell$, including $\ell\mid (q-1)$. \end{rem}
The Eisenstein ideal does not explicitly appear in this paper, although it is in the background of the analysis of $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})^{\bm{\epsilon}}$. The most technical part of the paper is the analysis of $\mathcal{E}_0(\mathfrak n, \mathbb{Z}/\ell^n\mathbb{Z})^{{\bm{\epsilon}}_H}$ for $\ell \mid (q+1)$, which occupies a large portion of Section \ref{sEPHC}. The odd primes dividing $q+1$ are somewhat similar to $\ell=3$ in Ohta's setting; for example, there is a non-trivial $\mathrm{GL}_2(A)$-invariant $\mathbb{Z}/(q+1)\mathbb{Z}$-valued harmonic cochain, although there are no such cochains with values in rings where $q+1$ is invertible.
For our purposes we found it convenient to generalize the notion of harmonic cochain. In Section \ref{sAL}, we introduce what we call \textit{pseudo-harmonic cochains}, which are functions on the edges of the Bruhat-Tits tree of $\mathrm{PGL}_2(F_\infty)$ satisfying the flow condition of harmonic cochains but which are not necessarily alternating. It turns out that there is a $\mathrm{GL}_2(A)$-invariant $\mathbb{Z}$-valued pseudo-harmonic cochain $\widetilde{E}$, which in our setting plays a role of the classical Eisenstein series $E_2(z)=\sum_{c,d\in \mathbb{Z}}'(cz+d)^{-2}$. (Another function field analogue of $E_2$ was introduced by Gekeler in \cite{Improper} under the name of \textit{improper Eisenstein series}; we explain the relationship between our $\widetilde{E}$ and Gekeler's Eisenstein series in Remark \ref{remH}.)
\section{Pseudo-harmonic cochains and Hecke operators}\label{sAL}
\subsection{Notation} Besides $\infty$, the other places of $F$ are in bijection with the non-zero prime ideals of $A$. Given a place $v$ of $F$, we denote by $F_v$ the completion of $F$ at $v$, by $\mathcal{O}_v$ the ring of integers of $F_v$, and by $\mathbb{F}_v$ the residue field of $\mathcal{O}_v$. We fix $\pi_\infty:=T^{-1}$ as a uniformizer of $\mathcal{O}_\infty$.
Let $R$ be a commutative ring with identity. We denote by $R^\times$ the group of multiplicative units of $R$. Let
$\mathrm{GL}_n(R)$ be the group of $n\times n$ matrices over $R$ whose determinant is in $R^\times$, and $Z(R)\cong R^\times$ the subgroup of $\mathrm{GL}_n(R)$ consisting of scalar matrices.
Given an abelian group $H$ and an integer $n$, $H[n]$ is the kernel of multiplication by $n$ in $G$. For a prime number $\ell$, $H_\ell$ denotes the $\ell$-primary component of $H$.
Given an ideal $\mathfrak n\lhd A$, by abuse of notation, we denote by the same symbol the unique monic polynomial in $A$ generating $\mathfrak n$. It will always be clear from the context in which capacity $\mathfrak n$ is used; for example, if $\mathfrak n$ appears in a matrix, column vector, or a polynomial equation, then the monic polynomial is implied. The prime ideals $\mathfrak p\lhd A$ are always assumed to be non-zero. Given two ideals $\mathfrak n, \mathfrak m$ of $A$, $(\mathfrak n, \mathfrak m)$ stands for the greatest common divisor of $\mathfrak n$ and $\mathfrak m$, and $\mathfrak m\parallel \mathfrak n$ means that $\mathfrak m$ divides $\mathfrak n$ and $(\mathfrak m, \mathfrak n/\mathfrak m)=1$.
\subsection{Pseudo-harmonic cochains} \label{sec2.1}
Let $G$ be an oriented connected graph in the sense of Definition 1 of $\S$2.1 in \cite{SerreT}. We denote by $V(G)$ and $E(G)$ its set of vertices and edges, respectively. For an edge $e\in E(G)$, let $o(e)$, $t(e)\in V(G)$ and $\bar{e}\in E(G)$ be its origin, terminus and inversely oriented edge, respectively. \begin{comment} In particular, $t(\bar{e})=o(e)$ and $o(\bar{e})=t(e)$. We will assume that for any $v\in V(G)$ the number of edges with $t(e)=v$ is finite, and $\bar{e}\neq e$ for any $e\in E(G)$. A \textit{path} in $G$ is a sequence of edges $\{e_i\}_{i\in I}$ indexed by a set $I$ where $I=\mathbb{Z}$, $I=\mathbb{N}$ or $I=\{1,\dots, m\}$ for some $m\in \mathbb{N}$ such that $t(e_i)=o(e_{i+1})$ for every $i, i+1\in I$. We say that the path is \textit{without backtracking} if $e_i\neq \bar{e}_{i+1}$ for every $i, i+1\in I$. We say that the path without backtracking $\{e_i\}_{i\in \mathbb{N}}$ is a \textit{half-line} if for every vertex $v$ of $G$ there is at most one index $n\in \mathbb{N}$ such that $v=o(e_n)$. Let $\Gamma$ be a group acting on $G$. We say that $\Gamma$ acts with \textit{inversion} if there is $\gamma\in \Gamma$ and $e\in E(G)$ such that $\gamma e=\bar{e}$. If $\Gamma$ acts without inversion, then we have a natural quotient graph $\Gamma\setminus G$ such that $V(\Gamma\setminus G)=\Gamma\setminus V(G)$ and $E(\Gamma\setminus G)=\Gamma\setminus E(G)$, cf. \cite[p. 25]{SerreT}. \end{comment}
\begin{defn}\label{defnHarmG} Let $R$ be a commutative ring with identity. An $R$-valued \textit{pseudo-harmonic cochain} on $G$ is a function $f: E(G)\to R$ that satisfies \begin{equation}\label{eq-pharm} \sum_{\substack{e'\in E(G)\\ t(e')=o(e),\ e'\neq \bar{e}}} f(e')=f(e)\quad \text{for all $e\in E(G)$}. \end{equation} A pseudo-harmonic cochain is called \textit{harmonic} if it is alternating: \begin{equation}\label{eq-alt} f(e)+f(\bar{e})=0\quad \text{for all $e\in E(G)$}. \end{equation} Note that (\ref{eq-alt}) makes (\ref{eq-pharm}) equivalent to the following $$ \sum_{\substack{e\in E(G)\\ t(e)=v}} f(e)=0\quad \text{for all $v\in V(G)$}. $$ Denote by $\widetilde{\mathcal{H}}(G, R)$ (resp. $\mathcal{H}(G, R)$) the $R$-module of pseudo-harmonic (resp. harmonic) cochains on $G$.
\end{defn}
\begin{lem}\label{lem 2.1FT} Assume $G$ is connected. Given $f\in \widetilde{\mathcal{H}}(G, R)$, there is a constant $c\in R$ such that $$f(e) + f(\bar{e}) = c, \quad \forall e \in E(G).$$ \end{lem} \begin{proof} Since $G$ is connected, it suffices to show that for every $e_1,e_2 \in E(G)$ with $t(e_1) = o(e_2)$ and $e_1 \neq \bar{e}_2$, $$f(e_1)+f(\bar{e}_1) = f(e_2) + f(\bar{e}_2).$$ By definition $$f(e_2) = f(e_1) + \sum_{\substack{e' \in E(G) \\ t(e') = o(e_2),\ e' \neq \bar{e}_2, e_1}} f(e')$$ and $$ f(\bar{e}_1) = f(\bar{e}_2) + \sum_{\substack{e' \in E(G)\\ t(e') = o(\bar{e}_1),\ e' \neq e_1, \bar{e}_2}} f(e') = f(\bar{e}_2)+ \sum_{\substack{e' \in E(G)\\ t(e') = o(e_2),\ e' \neq \bar{e}_2, e_1}} f(e'). $$ Therefore, $f(e_2) - f(e_1) = f(\bar{e}_1) - f(\bar{e}_2)$, and the result follows. \end{proof}
\begin{rem} The pseudo-harmonic cochains on the Bruhat-Tits tree $\mathscr{T}$ are a special case of metaplectic forms over function fields whose general theory is developed in \cite{WeiMA}. In the terminology of \cite{WeiMA}, pseudo-harmonic cochains are the ``weight-2'' forms.
\end{rem}
The graphs that we consider in this paper are the Bruhat-Tits tree $\mathscr{T}$ of $\mathrm{PGL}_2(F_\infty)$ and the quotients of $\mathscr{T}$. We recall the definition and introduce some notation for later use. Fix a uniformizer $\pi_\infty$ of $F_\infty$. The sets of vertices $V(\mathscr{T})$ and edges $E(\mathscr{T})$ are the cosets $\mathrm{GL}_2(F_\infty)/Z(F_\infty)\mathrm{GL}_2(\mathcal{O}_\infty)$ and $\mathrm{GL}_2(F_\infty)/Z(F_\infty)\mathcal{I}_\infty$, respectively, where $\mathcal{I}_\infty$ is the Iwahori group: $$
\mathcal{I}_\infty=\left\{\begin{pmatrix} a & b\\ c & d\end{pmatrix}\in \mathrm{GL}_2(\mathcal{O}_\infty)\ \bigg|\ c\in \pi_\infty\mathcal{O}_\infty\right\}. $$ The matrix $\begin{pmatrix} 0 & 1\\ \pi_\infty & 0\end{pmatrix}$ normalizes $\mathcal{I}_\infty$, so the multiplication from the right by this matrix on $\mathrm{GL}_2(F_\infty)$ induces an involution on $E(\mathscr{T})$; this involution is $e\mapsto \bar{e}$. The matrices \begin{equation}\label{eq-setM}
E(\mathscr{T})^+=\left\{\begin{pmatrix} \pi_\infty^k & u \\ 0 & 1\end{pmatrix}\ \bigg|\ \begin{matrix} k\in \mathbb{Z}\\ u\in F_\infty,\ u\ \mathrm{mod}\ \pi_\infty^k\mathcal{O}_\infty\end{matrix}\right\} \end{equation} are in distinct left cosets of $\mathcal{I}_\infty Z(F_\infty)$, and there is a disjoint decomposition (cf.\ \cite[(1.6)]{Improper}) \begin{equation}\label{eq-edge+} E(\mathscr{T})=E(\mathscr{T})^+\bigsqcup E(\mathscr{T})^+\begin{pmatrix} 0 & 1\\ \pi_\infty & 0\end{pmatrix}. \end{equation} We call the edges in $E(\mathscr{T})^+$ \textit{positively oriented}.
The group $\mathrm{GL}_2(F_\infty)$ naturally acts on $E(\mathscr{T})$ by left multiplication. This induces an action on the group of $R$-valued functions on $E(\mathscr{T})$:
for a function $f$ on $E(\mathscr{T})$ and $\gamma\in \mathrm{GL}_2(F_\infty)$ we define the function $f|\gamma$ on $E(\mathscr{T})$ by
$(f|\gamma)(e)=f(\gamma e)$.
It is clear from the definition that $f|\gamma$ is pseudo-harmonic (resp. harmonic) if $f$ is pseudo-harmonic (resp. harmonic).
\begin{defn} Let $\Gamma$ be a subgroup of $\mathrm{GL}_2(F_\infty)$. Denote by $\widetilde{\mathcal{H}}(\mathscr{T}, R)^\Gamma$ (resp. $\mathcal{H}(\mathscr{T}, R)^\Gamma$) the $R$-submodule of $\Gamma$-invariant pseudo-harmonic (resp. harmonic) cochains,
i.e., $f|\gamma=f$ for all $\gamma\in \Gamma$. The module of $R$-valued \textit{cuspidal} harmonic cochains for $\Gamma$, denoted $\mathcal{H}_0(\mathscr{T}, R)^\Gamma$, is the submodule of $\mathcal{H}(\mathscr{T}, R)^\Gamma$ consisting of functions which have compact support modulo $\Gamma$.
Let $\mathcal{H}_{00}(\mathscr{T}, R)^\Gamma$ denote the image of $\mathcal{H}_0(\mathscr{T}, \mathbb{Z})^\Gamma\otimes R$ in $\mathcal{H}_0(\mathscr{T}, R)^\Gamma$. To simplify the notation, we denote the $R$-module of pseudo-harmonic (resp. harmonic, cuspidal) $\Gamma_0(\mathfrak n)$-invariant cochains by $$ \widetilde{\mathcal{H}}(\mathfrak n, R)\supset \mathcal{H}(\mathfrak n, R)\supset \mathcal{H}_0(\mathfrak n, R)\supset \mathcal{H}_{00}(\mathfrak n, R). $$ In general, all these inclusions are strict; in particular $\mathcal{H}_0(\mathfrak n, R)\neq \mathcal{H}_{00}(\mathfrak n, R)$, although one can show that $\mathcal{H}_0(\mathfrak n, R)= \mathcal{H}_{00}(\mathfrak n, R)$ if $R$ is flat over $\mathbb{Z}$ or $q(q^2-1)\in R^\times$; see \cite[$\S$2.1.]{PW1}. (We could have also defined cuspidal pseudo-harmonic $\Gamma_0(\mathfrak n)$-invariant cochains, but such cochains are necessarily harmonic by Lemma \ref{lem 2.1FT}, i.e., $\widetilde{\mathcal{H}}_0(\mathfrak n, R)=\mathcal{H}_0(\mathfrak n, R)$.) \end{defn}
It is known that the quotient graph $\Gamma_0(\mathfrak n)\setminus \mathscr{T}$ is the edge disjoint union $$ \Gamma_0(\mathfrak n)\setminus \mathscr{T} = (\Gamma_0(\mathfrak n)\setminus \mathscr{T})^0\cup \bigcup_{s\in \Gamma_0(\mathfrak n)\setminus \mathbb{P}^1(F)} h_s $$ of a finite graph $(\Gamma_0(\mathfrak n)\setminus \mathscr{T})^0$ with a finite number of half-lines $h_s$, called \textit{cusps}; cf. Theorem 2 on page 106 of \cite{SerreT}. (A half-line is a graph as in Figure \ref{Fig1}.) The cusps are in bijection with the orbits of the natural action of $\Gamma_0(\mathfrak n)$ on $\mathbb{P}^1(F)$; cf. Remark 2 on page 110 of \cite{SerreT}. It is clear that $f\in \mathcal{H}(\mathfrak n, R)$ is cuspidal if and only if it eventually vanishes on each $h_s$.
\begin{example}\label{example1} \begin{figure}
\caption{$\mathrm{GL}_2(A)\setminus \mathscr{T}$}
\label{Fig1}
\end{figure} The quotient graph $\mathrm{GL}_2(A)\setminus \mathscr{T}$ is a half-line depicted in Figure \ref{Fig1}, where the vertex $v_i$ ($i\geq 0$) is the image of $\begin{pmatrix} \pi_\infty^{-i} & 0\\ 0 & 1\end{pmatrix}\in V(\mathscr{T})$. Denote the edge with origin $v_i$ and terminus $v_{i+1}$ by $e_i$. It is clear that $f\in \widetilde{\mathcal{H}}(1, R)$ defines a function on $\mathrm{GL}_2(A)\setminus \mathscr{T}$, and $f$ itself can be uniquely recover from that function. Hence we consider $f\in \widetilde{\mathcal{H}}(1, R)$ as a function on the quotient graph. The stabilizers of vertices and edges of $\mathrm{GL}_2(A)\setminus \mathscr{T}$ are well-known, cf. \cite[p. 691]{GN}. Using this one easily computes that (\ref{eq-pharm}) is equivalent to the following relations (cf. \cite[Example 2.4]{PW1}): \begin{align*}
q\cdot f(\bar{e}_0)&=f(e_0), \\ q\cdot f(e_{i+1})&=f(e_i), \qquad \forall i\geq 0,
\\ f(\bar{e}_{i+1})+(q-1)f(e_i)&=f(\bar{e}_i), \qquad \forall i\geq 0. \end{align*} Therefore, if we put $f(\bar{e}_0)=\alpha$, then $f\in \widetilde{\mathcal{H}}(1, R)$ if and only if for all $i\geq 0$ \begin{align*} f(e_i) &=q^{i+1}\alpha,\\ f(\bar{e}_i) &=(1+q-q^{i+1})\alpha. \end{align*} In particular, $f(e_i)+f(\bar{e}_i) = (q+1)\alpha$ for all $i\geq 0$. We conclude that $\widetilde{\mathcal{H}}(1, R)\cong R$ is spanned by the function $\widetilde{E}$ with $\widetilde{E}(\bar{e}_0)=1$, $\mathcal{H}(1, R)\cong R[q+1]$, and $\mathcal{H}_0(1,R)=\mathcal{H}_{00}(1,R)=0$. \end{example}
\begin{rem} When $(q+1)$ is invertible in $R$, it is easy to see from Lemma \ref{lem 2.1FT} and the previous example that $$ \widetilde{\mathcal{H}}(\mathscr{T}, R)=\mathcal{H}(\mathscr{T}, R)\oplus R \widetilde{E}. $$ \end{rem}
\subsection{Fourier expansion}
The theory of Fourier expansions of automorphic forms over function fields was developed by Weil in \cite{Weil}. As was observed by P\'al in \cite{Pal}, Weil's theory works over more general rings than $\mathbb{C}$. Here we follow Gekeler's reinterpretation \cite{Improper} of Weil's adelic approach as analysis on the Bruhat-Tits tree, but we will extend \cite{Improper} to the setting of these more general rings.
\begin{defn} Following \cite{Pal} we say that $R$ is a \textit{coefficient ring} if $p\in R^\times$ and $R$ is a quotient of a discrete valuation ring $\tilde{R}$ which contains $p$-th roots of unity. Note that the image of the $p$-th roots of unity of $\tilde{R}$ in $R$ is exactly the set of $p$-th roots of unity of $R$. For example, any algebraically closed field of characteristic different from $p$ is a coefficient ring. \end{defn}
In this subsection $R$ is assumed to be a coefficient ring. Let \begin{align*} \eta: F_\infty &\to R^\times \\ \sum a_i\pi_\infty^i &\mapsto \eta_0\Big(\text{Trace}_{\mathbb{F}_q/\mathbb{F}_p}(a_1)\Big) \end{align*} where $\eta_0: \mathbb{F}_p\to R^\times$ is a non-trivial additive character fixed once and for all.
The group $$ \Gamma_\infty:=\left\{\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}\in \mathrm{GL}_2(A)\right\} $$ acts orientation preserving on $\mathscr{T}$, so $E(\Gamma_\infty\setminus \mathscr{T})=\Gamma_\infty\setminus E(\mathscr{T})$, and the orientation $E(\mathscr{T})^+$ on $\mathscr{T}$ induces an orientation $E(\Gamma_\infty\setminus \mathscr{T})^+$. For an $R$-valued $\Gamma_\infty$-invariant function $f$ on $E(\mathscr{T})$, its \textit{constant Fourier coefficient} is the $R$-valued function $f^0$ on $\pi_\infty^\mathbb{Z}$ defined by $$ f^0(\pi_\infty^k)= \begin{cases} q^{1-k} \sum_{\substack{u\in (\pi_\infty)/(\pi_\infty^k)}}f\left(\begin{pmatrix}\pi_\infty^k & u \\ 0 &1\end{pmatrix}\right) & \text{if $k\geq 1$}, \\ f\left(\begin{pmatrix}\pi_\infty^k & 0 \\ 0 &1\end{pmatrix}\right)& \text{if $k\leq 1$}. \end{cases} $$ For a divisor $\mathfrak m$ on $F$, the \textit{$\mathfrak m$-th Fourier coefficient} $f^\ast(\mathfrak m)$ of $f$ is $$ f^\ast(\mathfrak m) = q^{-1-\deg(\mathfrak m)}\sum_{u\in (\pi_\infty)/(\pi_\infty^{2+\deg(\mathfrak m)})} f\left(\begin{pmatrix}\pi_\infty^{2+\deg(\mathfrak m)} & u \\ 0 &1\end{pmatrix}\right)\eta(-m u), $$ if $\mathfrak m$ is non-negative, and $f^\ast(\mathfrak m)=0$, otherwise; here $m\in A$ is the monic polynomial such that $\mathfrak m=\mathrm{div}(m)\cdot \infty^{\deg(\mathfrak m)}$. Then $f$ has a \textit{Fourier expansion} \begin{equation}\label{eq-Fexp} f \left(\begin{pmatrix} \pi_\infty^k & y \\ 0 &1\end{pmatrix}\right) = f^0(\pi_\infty^k)+ \sum_{\substack{0\neq m\in A \\ \deg(m)\leq k-2}} f^\ast(\mathrm{div}(m)\cdot \infty^{k-2})\cdot \eta(my). \end{equation}
We refer to \cite[$\S$2]{Pal} and \cite[$\S$2]{Improper} for the proofs.
\begin{lem}\label{lemFH} Assume $f\in \widetilde{\mathcal{H}}(\mathscr{T}, R)^{\Gamma_\infty}$. Then \begin{itemize} \item[(i)] $f^0(\pi_\infty^k)=f^0(1)\cdot q^{-k}$ for any $k\in \mathbb{Z}$; \item[(ii)] $ f^\ast(\mathfrak m \infty^k)=f^\ast(\mathfrak m)\cdot q^{-k}$ for any non-negative divisor $\mathfrak m$ and $k \in \mathbb{Z}_{\geq 0}$. \end{itemize} In particular, when $f$ is harmonic, the Fourier coefficients $f^0(1)$ and $f^*(\mathfrak m)$ for $\mathfrak m \lhd A$ uniquely determine $f$. \end{lem} \begin{proof} Note that the pseudo-harmonicity of $f$ says: $$
f|T_\infty (g) := \sum_{u \in \mathbb{F}_q} f\left(g \begin{pmatrix} \pi_\infty &u\\ 0&1\end{pmatrix}\right) = f(g), \quad \forall g \in \mathrm{GL}_2(F_\infty). $$ It is then straightforward to check that for any non-negative divisor $\mathfrak m$ and $k \in \mathbb{Z}$,
$$(f|T_\infty)^0(\pi_\infty^k) = q f^0(\pi_\infty^{k+1}) \quad \text{ and } \quad
(f|T_\infty)^*(\mathfrak m) = q f^*(\mathfrak m \infty).$$ Hence we obtain (i) and (ii). The last statement follows from the Fourier expansion of $f$. \end{proof}
\begin{lem}\label{lem2.3FT} Let $\mathfrak n \lhd A$ be a non-zero ideal. Then $f \in \widetilde{\mathcal{H}}(\mathfrak n,R)$ is uniquely determined by its Fourier coefficients $f^0(1)$ and $f^*(\mathfrak m)$ for $\mathfrak m \lhd A$. \end{lem}
\begin{proof} The Fourier coefficients of $f$ determine the values of $f$ on positively oriented edges $e \in E(\mathscr{T})^+$. Therefore it suffices to show that for every edge $e \in E(\mathscr{T})^-$, there exists $\gamma \in \Gamma_0(\mathfrak n)$ such that $\gamma e \in E(\mathscr{T})^+$. By (\ref{eq-edge+}), after identifying $E(\mathscr{T})$ with $\mathrm{GL}_2(F_\infty)/Z(F_\infty) \mathcal{I}_\infty$, every edge $e \in E(\mathscr{T})^-$ is expressed by $$\begin{pmatrix} \pi_\infty^r & u \\ 0 & 1\end{pmatrix}\begin{pmatrix} 0 & 1 \\ \pi_\infty & 0 \end{pmatrix}, \quad r \in \mathbb{Z},\ u \in F_\infty.$$ By the approximation theorem, there exists $\alpha \in F^{\times}$ such that \begin{align*} &\mathrm{ord}_\infty(u-\alpha) \geq r \quad \text{ and }\\ &\mathrm{ord}_\mathfrak p\left(\mathfrak p^{-\mathrm{ord}_\mathfrak p(\mathfrak n)} - \alpha\right) \geq 0\ \text{ for all prime } \mathfrak p \mid \mathfrak n. \end{align*} Writing $\alpha = d/c$ with $c,d \in A$ and $(c,d) = 1$, the above inequalities imply that $$c = c_1 \mathfrak n \text{ with $c_1 \in A$,} \quad \text{ and } \quad \mathrm{ord}_\infty \left(u-\frac{d}{c_1 \mathfrak n}\right) \geq r.$$ Take $a,b \in A$ with $ad - b c_1 \mathfrak n =1$. Then $$\gamma:= \begin{pmatrix} a&b\\c_1 \mathfrak n &d \end{pmatrix} \in \Gamma_0(\mathfrak n) \quad \text{ and }\quad \mathrm{ord}_\infty \left(\frac{ c_1 \mathfrak n u + d}{ c_1 \mathfrak n \pi_\infty^r}\right) \geq 0.$$ Therefore $\gamma e \in E(\mathscr{T})^+$ as $$\begin{pmatrix} a&b \\ c_1 \mathfrak n & d\end{pmatrix} \cdot \begin{pmatrix} \pi_\infty^r & u \\ 0 & 1\end{pmatrix}\begin{pmatrix} 0 & 1 \\ \pi_\infty & 0 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ - \left(\frac{ c_1 \mathfrak n u + d}{c_1 \mathfrak n u \pi_\infty ^r}\right) \pi_\infty & 1\end{pmatrix} = \begin{pmatrix} -\frac{\pi_\infty}{c_1 \mathfrak n} & u \pi_\infty^r \\ 0 & c_1 \mathfrak n \pi_\infty^r \end{pmatrix}.$$ \end{proof}
\subsection{Hecke operators and Atkin-Lehner involutions}
Given a non-zero ideal $\mathfrak m\lhd A$, let \begin{equation} B_\mathfrak m=\begin{pmatrix} \mathfrak m & 0 \\ 0 & 1\end{pmatrix}, \end{equation} which we consider as a linear operator on $R$-valued functions on $E(\mathscr{T})$. Note that the action of $B_\mathfrak m^{-1}$ on functions on $E(\mathscr{T})$ is the same as the action of the matrix $\begin{pmatrix} 1 & 0 \\ 0 & \mathfrak m\end{pmatrix}$ (since the diagonal matrices act trivially). For $b\in A$, let $S_b:=\begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix}$. For a prime ideal $\mathfrak p\lhd A$, define the \textit{Hecke operators} $U_\mathfrak p$ and $T_\mathfrak p$ acting on the space of $R$-valued functions on $E(\mathscr{T})$ by \begin{align*}
f|U_\mathfrak p &=\sum_{\substack{b\in A\\ \deg(b)<\deg(\mathfrak p)}} f|B_\mathfrak p^{-1}S_b, \\ f|T_\mathfrak p &=f|U_\mathfrak p + f|B_\mathfrak p. \end{align*}
For an ideal $\mathfrak m \parallel \mathfrak n$, let $W_\mathfrak m$ be any matrix of the form \begin{equation}\label{ALmatrix} \begin{pmatrix} a\mathfrak m & b \\ c\mathfrak n & d\mathfrak m \end{pmatrix} \end{equation} such that $a,b,c,d, \in A$, and the ideal generated by $\det(W_\mathfrak m)$ in $A$ is $\mathfrak m$.
It is not hard to check that for $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$, $f|W_\mathfrak m$ does not depend on the choice of the matrix for $W_\mathfrak m$. Moreover, as $R$-linear endomorphisms of $\widetilde{\mathcal{H}}(\mathfrak n, R)$, $W_\mathfrak m$'s satisfy \begin{equation}\label{eqWs} W_{\mathfrak m_1}W_{\mathfrak m_2}=W_{\mathfrak m_3}, \quad \text{where} \quad \mathfrak m_3= \frac{\mathfrak m_1\mathfrak m_2}{(\mathfrak m_1, \mathfrak m_2)^2}. \end{equation} Therefore, the matrices $W_\mathfrak m$ acting on $\widetilde{\mathcal{H}}(\mathfrak n, R)$ generate an abelian group $\mathbb{W}\cong (\mathbb{Z}/2\mathbb{Z})^s$, called the group of \textit{Atkin-Lehner involutions}, where $s$ is the number of prime divisors of $\mathfrak n$.
\begin{lem}\label{lem-TU} Given a non-zero ideal $\mathfrak n\lhd A$, the operators $T_\mathfrak p$ $(\mathfrak p\nmid \mathfrak n)$, $U_\mathfrak p$ $(\mathfrak p\mid\mathfrak n)$, and $W_\mathfrak m$ $(\mathfrak m\parallel \mathfrak n)$ preserve the $R$-modules $\widetilde{\mathcal{H}}(\mathfrak n, R)$, $\mathcal{H}(\mathfrak n, R)$, $\mathcal{H}_0(\mathfrak n, R)$. Moreover, $T_\mathfrak p U_{\mathfrak p'}=U_{\mathfrak p'}T_\mathfrak p$, $T_\mathfrak p W_\mathfrak m=W_\mathfrak m T_\mathfrak p$, and if $\mathfrak p\nmid \mathfrak m$ then $U_\mathfrak p W_\mathfrak m=W_\mathfrak m U_\mathfrak p$. \end{lem} \begin{proof} The group-theoretic proofs of the analogous statement for operators acting on classical modular forms work also in this setting; cf. \cite[$\S$4.5]{Miyake}. \end{proof}
Given ideals $\mathfrak n, \mathfrak m\lhd A$, denote $$
\Gamma_0(\mathfrak n,\mathfrak m)=\left\{\begin{pmatrix} a & b \\ c & d\end{pmatrix}\in \mathrm{GL}_2(A)\ \big|\ c\in \mathfrak n, b\in \mathfrak m \right\}. $$
\begin{lem}\label{lemAL2} We have: \begin{enumerate}
\item If $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$, then $f|B_\mathfrak m$ is $\Gamma_0(\mathfrak n\mathfrak m)$-invariant and
$f|B_\mathfrak m^{-1}$ is $\Gamma_0(\mathfrak n/(\mathfrak n, \mathfrak m), \mathfrak m)$-invariant. \item If $\mathfrak n$ is coprime to $\mathfrak m$ and $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$, then
$$(f|B_\mathfrak m)|W_\mathfrak m=f,$$ where $W_\mathfrak m$ is the Atkin-Lehner involution acting on $\widetilde{\mathcal{H}}(\mathfrak n\mathfrak m, R)$. \item If $\mathfrak m\parallel \mathfrak n$, $(\mathfrak b, \mathfrak m)=1$, and $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$, then $$
(f|B_\mathfrak b)|W_\mathfrak m=(f|W_\mathfrak m)|B_\mathfrak b, $$ where on the left hand-side $W_\mathfrak m$ denotes the Atkin-Lehner involution acting on $\widetilde{\mathcal{H}}(\mathfrak n\mathfrak b, R)$ and on the right hand-side $W_\mathfrak m$ denotes the involution acting on $\widetilde{\mathcal{H}}(\mathfrak n, R)$. \item If $f\in \widetilde{\mathcal{H}}(\mathscr{T}, R)^{\Gamma_\infty}$, then $$
(f|B_\mathfrak p)|U_\mathfrak p=|\mathfrak p|\cdot f. $$ \end{enumerate} \end{lem} \begin{proof} This follows from straightforward manipulations with matrices; cf. \cite[$\S$2]{AL}. \end{proof}
\begin{comment} \begin{lem}\label{lem2.15} Let $\mathfrak p$ and $\mathfrak q$ be two distinct prime ideals of $A$. If $f\in\widetilde{\mathcal{H}}(\mathscr{T}, R)^{\Gamma_\infty}$, then \begin{align*}
(f|B_\mathfrak p)|U_\mathfrak p &=|\mathfrak p|\cdot f,\\
(f|B_\mathfrak p)|U_\mathfrak q &= (f|U_\mathfrak q)|B_\mathfrak p. \end{align*} \end{lem} \begin{proof} See Lemma 2.23 in \cite{PW1}. \end{proof} \end{comment}
\begin{lem}\label{lem_new14} Assume $R$ is a coefficient ring. For any non-zero ideal $\mathfrak m\lhd A$ and $f\in\widetilde{\mathcal{H}}(\mathscr{T}, R)^{\Gamma_\infty}$ we have \begin{align*}
(f|B_\mathfrak m)^0(\pi_\infty^k) &=f^0(\pi_\infty^{k-\deg(\mathfrak m)})\\ (f|B_\mathfrak m)^\ast(\mathfrak n) &=f^\ast(\mathfrak n/\mathfrak m). \end{align*} \begin{comment} \item For $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ and arbitrary non-zero $\mathfrak r\lhd A$, we have \begin{align*}
(f|T_\mathfrak p)^\ast(\mathfrak r) &=|\mathfrak p|f^\ast(\mathfrak r\mathfrak p) \\ (f|U_\mathfrak p)^\ast(\mathfrak r) &=|\mathfrak p|f^\ast(\mathfrak r\mathfrak p). \end{align*} \end{comment} \end{lem} \begin{proof} See Proposition 2.10 in \cite{Improper}. \end{proof}
\begin{lem}\label{lem1.15fm} Assume $R$ is a coefficient ring and $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ is an eigenfunction of all $T_\mathfrak p$, $\mathfrak p\nmid \mathfrak n$; that is,
$f|T_\mathfrak p=\lambda_\mathfrak p f$ for some $\lambda_\mathfrak p\in R$. Then the Fourier coefficients $f^\ast(\mathfrak m)$, with $\mathfrak m$ coprime to $\mathfrak n$, are uniquely determined by $f^\ast(1)$ and the eigenvalues $\lambda_\mathfrak p$, $\mathfrak p\nmid \mathfrak n$. \end{lem} \begin{proof} This follows from two well-known facts: (1) every Hecke operator $T_\mathfrak m$, with $\mathfrak m$ coprime to $\mathfrak n$, can be expressed as a polynomial in $T_\mathfrak p$'s with integer coefficients, where $\mathfrak p$'s are the prime divisors of $\mathfrak m$;
(2) $(f|T_\mathfrak m)^\ast(1)=|\mathfrak m|f^\ast(\mathfrak m)$. For more details, see the discussion in \cite[$\S$3]{Analytical}. \end{proof}
As we mentioned, the cusps of $\Gamma_0(\mathfrak n)$ are in bijection with the orbits of the action of $\Gamma_0(\mathfrak n)$ on $$
\mathbb{P}^1(F)=\mathbb{P}^1(A)=\left\{\begin{pmatrix} a \\ b\end{pmatrix}\ \big|\ a, b\in A, \ (a, b)=1,\ a \text{ is monic}\right\}, $$ where $\Gamma_0(\mathfrak n)$ acts on $\mathbb{P}^1(F)$ from the left as on column vectors. We leave the proof of the following lemma to the reader.
\begin{lem}\label{lemCusps} Assume $\mathfrak n$ is square-free. \begin{enumerate} \item
For a monic $\mathfrak m|\mathfrak n$, let $[\mathfrak m]$ be the orbit of $\begin{pmatrix} 1 \\ \mathfrak m \end{pmatrix}$ under the action of $\Gamma_0(\mathfrak n)$.
Then $[\mathfrak m]\neq [\mathfrak m']$ if $\mathfrak m\neq \mathfrak m'$, and the set $\{[\mathfrak m]\ |\ \mathfrak m|\mathfrak n\}$ is the set of cusps of $\Gamma_0(\mathfrak n)$. In particular, there are $2^s$ cusps, where $s$ is the number of prime divisors of $\mathfrak n$. \item Since $W_\mathfrak d$ normalizes $\Gamma_0(\mathfrak n)$, it acts on the set of cusps of $\Gamma_0(\mathfrak n)$. We have $$ W_\mathfrak d [\mathfrak m] = \left[\frac{\mathfrak m\mathfrak d}{(\mathfrak m, \mathfrak d)^2}\right]. $$ \end{enumerate} \end{lem}
\begin{defn} Let $\mathfrak n=\mathfrak p_1\cdots \mathfrak p_s$ be a square-free ideal in $A$ with the given prime decomposition. Let $$
\mathbb{E}=\{(\varepsilon_1, \dots, \varepsilon_s)\ |\ \varepsilon_i=\pm 1,\ 1\leq i\leq s\}. $$ Let $R$ be a $\mathbb{Z}[1/2]$-algebra and $M$ be an $R[\mathbb{W}]$-module. For each ${\bm{\epsilon}}\in \mathbb{E}$, we let $M^{\bm{\epsilon}}$ be the maximum direct summand of $M$ on which $W_{\mathfrak p_i}$ acts by multiplication by $\varepsilon_i$ ($1\leq i\leq s$), so that $$ M=\bigoplus_{{\bm{\epsilon}}\in \mathbb{E}} M^{\bm{\epsilon}}. $$ \end{defn}
\subsection{Atkin-Lehner type result} The results in this subsection are the analogues for $R$-valued pseudo-harmonic cochains of some of the results of Atkin and Lehner \cite{AL}. In our proofs we also use ideas of Ohta from \cite[$\S$2.1]{Ohta}.
\begin{lem}\label{lemLevelLow} Assume $R$ is a coefficient ring. Let $\mathfrak p\lhd A$ be a prime such that $\mathfrak p\parallel \mathfrak n$. Suppose $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ is such that $f^\ast(\mathfrak m)=0$ if $\mathfrak p\nmid \mathfrak m$. Then there is $g\in \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p, R)$ such that
$$f=g|B_\mathfrak p\quad \text{ and }\quad f|W_\mathfrak p= g.$$ Moreover, if $f$ is harmonic, then so is $g$. \end{lem}
\begin{proof} Let $g:=f|B_\mathfrak p^{-1}$. By Lemma \ref{lemAL2} (1), $g$ is $\Gamma_0(\mathfrak n/\mathfrak p, \mathfrak p)$-invariant and pseudo-harmonic. Moreover, given $a \in A$ and $e = \begin{pmatrix} \pi_\infty^k & u \\ 0 & 1 \end{pmatrix} \in E(\mathscr{T})^+$, \begin{eqnarray} g\left(\begin{pmatrix}1&a\\0&1\end{pmatrix} e\right) & = & f \left(\begin{pmatrix} \pi_\infty^{k + \deg \mathfrak p} & u + \frac{a}{\mathfrak p} \\ 0 & 1 \end{pmatrix}\right) \nonumber \\ & = & f^0 (\pi_\infty^{k + \deg \mathfrak p}) + \sum_{\mathfrak m \lhd A\atop \deg \mathfrak m \leq k+\deg \mathfrak p-2 } f^*(\text{div}(m) \infty^{k+\deg \mathfrak p - 2}) \eta(mu + \frac{ma}{\mathfrak p}) \nonumber \\ & = & f^0 (\pi_\infty^{k + \deg \mathfrak p}) + \sum_{\mathfrak m \lhd A\atop \deg \mathfrak m \leq k+\deg \mathfrak p-2 } f^*(\text{div}(m) \infty^{k+\deg \mathfrak p - 2}) \eta(mu) \nonumber \\ & = & g(e). \nonumber \end{eqnarray} The third equality in the above follows from the assumption that $f^\ast(\mathfrak m) = 0$ unless $\mathfrak p \mid \mathfrak m$ and $\eta(\alpha) = 1$ for every $\alpha \in A$. Moreover, by Lemma~\ref{lem 2.1FT}, there exists a constant $c \in R$ such that $$g\left(\begin{pmatrix}1&a\\0&1\end{pmatrix} \bar{e}\right) = c+g\left(\begin{pmatrix}1&a\\0&1\end{pmatrix} e\right) = c + g(e) = g(\bar{e}).$$ Thus $g$ is also $\Gamma_\infty$-invariant. Since the subgroup of $\mathrm{GL}_2(A)$ generated by $\Gamma_0(\mathfrak n/\mathfrak p, \mathfrak p)$ and $\Gamma_\infty$ is $\Gamma_0(\mathfrak n/\mathfrak p)$, we conclude that $g\in \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p, R)$ (resp. $g\in \mathcal{H}(\mathfrak n/\mathfrak p, R)$ if $f$ is harmonic).
Now choose some matrix $\begin{pmatrix} a\mathfrak p & b \\ \mathfrak n & d\mathfrak p\end{pmatrix}$ representing $W_\mathfrak p$. Since $$ B_\mathfrak p W_\mathfrak p =\begin{pmatrix} a\mathfrak p^2 & b\mathfrak p \\ \mathfrak n & d\mathfrak p\end{pmatrix}= \begin{pmatrix} \mathfrak p & 0 \\ 0 &\mathfrak p\end{pmatrix} \begin{pmatrix} a\mathfrak p & b \\ \mathfrak n/\mathfrak p & d\end{pmatrix} \in Z(F_\infty)\Gamma_0(\mathfrak n/\mathfrak p), $$
we have $f|W_\mathfrak p=g|B_\mathfrak p W_\mathfrak p =g$. \end{proof}
\begin{prop}\label{propALOhta} Let $\mathfrak n=\mathfrak p_1\cdots \mathfrak p_s\lhd A$ be square-free, and $R$ a coefficient ring. Assume $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ is such that $f^\ast(\mathfrak m)=0$
unless $\mathfrak m$ is divisible by some $\mathfrak p_i$, $1\leq i\leq s$. Assume $f|W_{\mathfrak p_s}=\pm f$. Then $f^\ast(\mathfrak m)=0$ unless $\mathfrak m$ is divisible by some $\mathfrak p_i$, $1\leq i\leq s-1$, when $s\geq 2$, and $f^\ast(\mathfrak m)=0$ for all $\mathfrak m$ when $s=1$. \end{prop} \begin{proof} Let $\mathfrak p \mid \mathfrak n$. Define the analogue of the ``annihilator'' operator of Atkin and Lehner \cite{AL}: $$
K_\mathfrak p:=1-\frac{1}{|\mathfrak p|}U_\mathfrak p B_\mathfrak p. $$
Since $f|U_\mathfrak p\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ satisfies $
(f|U_\mathfrak p)^\ast(\mathfrak m)=|\mathfrak p|f^\ast(\mathfrak m\mathfrak p)$
and $f|B_\mathfrak p\in \widetilde{\mathcal{H}}(\mathfrak n\mathfrak p, R)$ satisfies $(f|B_\mathfrak p)^\ast(\mathfrak m)=f^\ast(\mathfrak m/\mathfrak p)$, $f|K_\mathfrak p$ is in $\widetilde{\mathcal{H}}(\mathfrak n\mathfrak p, R)$ and has Fourier coefficients $$
(f|K_\mathfrak p)^\ast(\mathfrak m) = \begin{cases} f^\ast(\mathfrak m) & \text{if $\mathfrak p\nmid \mathfrak m$}; \\ 0 & \text{if $\mathfrak p \mid \mathfrak m$}. \end{cases} $$
Given two distinct primes $\mathfrak p, \mathfrak p'$ dividing $\mathfrak n$, $K_\mathfrak p$ commutes with $K_{\mathfrak p'}$ and $W_{\mathfrak p'}$; cf. \cite[Lem. 2.23]{PW1}.
Set $h:=f|\prod_{i=1}^{s-1}K_{\mathfrak p_i} \in \widetilde{\mathcal{H}}(\mathfrak n\mathfrak p_1\cdots\mathfrak p_{s-1}, R)$. Then $h^\ast(\mathfrak m)=0$ unless $\mathfrak p_s \mid \mathfrak m$, so Lemma \ref{lemLevelLow} implies that there is $h'\in \widetilde{\mathcal{H}}(\mathfrak n\mathfrak p_1\cdots\mathfrak p_{s-1}/\mathfrak p_s, R)$
such that $h=h'|B_{\mathfrak p_s}$ and $h|W_{\mathfrak p_s}=h'$. The equality $h=h'|B_{\mathfrak p_s}$ implies that $h^\ast(\mathfrak p_s\mathfrak m)={h'}^\ast(\mathfrak m)$ for all $\mathfrak m$.
On the other hand, $h|W_{\mathfrak p_s}=h'$ implies $$
h'=h|W_{\mathfrak p_s}=f|\prod_{i=1}^{s-1}K_{\mathfrak p_i} W_{\mathfrak p_s} = f|W_{\mathfrak p_s}\prod_{i=1}^{s-1}K_{\mathfrak p_i} =\pm f|\prod_{i=1}^{s-1}K_{\mathfrak p_i} = \pm h. $$ Thus $h^\ast(\mathfrak p_s\mathfrak m)={h'}^\ast(\mathfrak m)=\pm h^\ast(\mathfrak m)$ for all $\mathfrak m$. Now it is easy to see that $h^\ast(\mathfrak m)=0$ for all $\mathfrak m$, and therefore $f^\ast(\mathfrak m)=0$ unless $\mathfrak m$ is divisible by some $\mathfrak p_i$, $1\leq i\leq s-1$. \end{proof}
\section{Eisenstein pseudo-harmonic cochains}\label{sEPHC}
Let $\mathfrak n\lhd A$ be a non-zero ideal. We say that $f \in \widetilde{\mathcal{H}}(\mathfrak n,R)$ is \textit{Eisenstein}
if $f |T_{\mathfrak p} = (|\mathfrak p|+1)f$ for every prime ideal $\mathfrak p \lhd A$ not dividing $\mathfrak n$. It is clear that Eisenstein pseudo-harmonic cochains form an $R$-submodule of $\widetilde{\mathcal{H}}(\mathfrak n, R)$ which we denote by $\widetilde{\mathcal{E}}(\mathfrak n, R)$. Let $\mathcal{E}(\mathfrak n, R)$ (resp. $\mathcal{E}_0(\mathfrak n, R)$, $\mathcal{E}_{00}(\mathfrak n, R)$) be the intersection of $ \widetilde{\mathcal{E}}(\mathfrak n, R)$ with $\mathcal{H}(\mathfrak n, R)$ (resp. $\mathcal{H}_0(\mathfrak n, R)$, $\mathcal{H}_{00}(\mathfrak n, R)$). We have $$ \widetilde{\mathcal{E}}(\mathfrak n, R)\supset \mathcal{E}(\mathfrak n, R) \supset \mathcal{E}_0(\mathfrak n, R) \supset \mathcal{E}_{00}(\mathfrak n, R). $$ The main goal of this section is to determine $\mathcal{E}_0(\mathfrak n, R)$ for square-free $\mathfrak n$. For that purpose, using $\widetilde{E}$ from Example \ref{example1}, we will construct certain explicit Eisenstein pseudo-harmonic cochains.
\begin{lem}\label{lemtEEis}
For any prime $\mathfrak p\lhd A$, we have $\widetilde{E}|T_\mathfrak p=(1+|\mathfrak p|)\widetilde{E}$. Therefore, $\widetilde{\mathcal{E}}(1, R)=\widetilde{\mathcal{H}}(1, R)$. \end{lem} \begin{proof}
By Lemma \ref{lem-TU}, $\widetilde{E}|T_\mathfrak p\in \widetilde{\mathcal{H}}(1, R)$, and since $\widetilde{E}$ spans $\widetilde{\mathcal{H}}(1, R)$,
we have $\widetilde{E}|T_\mathfrak p = c \widetilde{E}$ for some $c\in R$. To find $c$, it is enough to compute the value $\widetilde{E}|T_\mathfrak p$ on some $e_i$; it is convenient to take $i=\deg(\mathfrak p)$: $$
\widetilde{E}|T_\mathfrak p (e_{\deg(\mathfrak p)})= \widetilde{E}\left(\begin{pmatrix} \mathfrak p^2 & 0 \\ 0 & 1\end{pmatrix}\right) +\sum_{\deg(b)<\deg(\mathfrak p)} \widetilde{E}\left(\begin{pmatrix} 1 & b \\ 0 & \mathfrak p\end{pmatrix}\begin{pmatrix} \mathfrak p & 0 \\ 0 & 1\end{pmatrix}\right) $$ $$
=\widetilde{E}(e_{2\deg(\mathfrak p)}) + |\mathfrak p| \widetilde{E}(e_{0})= q^{2\deg(\mathfrak p)+1}+q^{\deg(\mathfrak p)+1}=(|\mathfrak p|+1)\widetilde{E}(e_{\deg(\mathfrak p)}). $$ \end{proof}
We will need to know the Fourier coefficient of $\widetilde{E}$, which can be computed as follows. First, $$\widetilde{E}^0(1)=\widetilde{E}(e_0)=q.$$ Next, using (\ref{eq-Fexp}) and Lemma \ref{lemFH}, we compute \begin{align*} \widetilde{E}^\ast(1) & =-\left(\widetilde{E}\left(\begin{pmatrix} \pi_\infty^2 & \pi_\infty \\ 0 & 1\end{pmatrix}\right)- \widetilde{E}^0(\pi_\infty^2)\right)\\ &= -\left(\widetilde{E}\left(\begin{pmatrix} \pi_\infty^2 & \pi_\infty \\ 0 & 1\end{pmatrix}\right) - q^{-2}\widetilde{E}\left(\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\right)\right) \\ &= -\left(\widetilde{E}(e_0)- q^{-2}\widetilde{E}(e_0)\right) \\ &= \frac{1-q^2}{q}. \end{align*}
Finally, since $\widetilde{E}$ is an eigenfunction of $T_\mathfrak p$ with eigenvalue $|\mathfrak p|+1$ for all prime $\mathfrak p$, any $f^\ast(\mathfrak m)$ can be computed from $f^\ast(1)$; see Lemma \ref{lem1.15fm}. We get for $k \in \mathbb{Z}$ and $u \in F_\infty$, $$\widetilde{E}\begin{pmatrix} \pi_\infty^k & u \\ 0 & 1\end{pmatrix} = q^{-k+1}\left(1 + (1-q^2)\sum_{m \in A, \atop \deg m + 2 \leq k} \sigma(m) \eta(m u)\right),$$
where $$\sigma(m) := \sum_{\text{monic } m' \in A, \atop m' \mid m} |m'|.$$
\begin{rem}\label{remH} In \cite{Improper}, Gekeler introduced the so-called improper Eisenstein series $H$ on $\mathscr{T}$, as the analogue of the classical Eisenstein series $E_2(z)=\sum_{c,d\in \mathbb{Z}}'(cz+d)^{-2}$. \begin{comment} The function $H$ is defined in \cite{Improper} as $$ H(e)=\sum_{\Gamma_\infty\setminus \mathrm{GL}_2(A)} \mathrm{sgn}(e)q^{-\kappa(e)}, $$ where the summation is taken in a specific order (since the series fails to converge), $\kappa\left(\begin{pmatrix} \pi_\infty^k & u \\ 0 & 1\end{pmatrix}\right)=k$, and $\mathrm{sgn}(e)=1$ if $e\in E(\mathscr{T})^{+}$, $\mathrm{sgn}(e)=-1$ if $e\in E(\mathscr{T})^{-}$. \end{comment} This function $H$ is a $\Gamma_\infty$-invariant harmonic cochain on $E(\mathscr{T})$, which is ``half'' invariant under $\mathrm{GL}_2(A)$ in the sense that $H(\gamma e)=H(e)$ if and only if both $e$ and $\gamma e$ are positively or negatively oriented. It can be seen from \cite[p. 383]{Improper} that $\widetilde{E}(e)=q\cdot H(e)$ for all $e\in E(\mathscr{T})^+$, although these functions are distinct (for example, $\widetilde{E}$ is $\mathrm{GL}_2(A)$-invariant and is not alternating). \end{rem}
\begin{notn}\label{notn2.3} From now on $\mathfrak n=\mathfrak p_1\cdots \mathfrak p_s$ is a square-free ideal with the given prime decomposition. Let \begin{align*} {\bm{\epsilon}}_{H(\mathfrak n)} &:=((-1)^{\deg(\mathfrak p_1)}, \dots, (-1)^{\deg(\mathfrak p_s)}) \in \mathbb{E}, \\ {\bm{1}} &:=(1,1,\dots, 1)\in \mathbb{E}. \end{align*} Also, for a given ${\bm{\epsilon}}=(\varepsilon_1, \dots, \varepsilon_n)\in \mathbb{E}$ and a divisor $\mathfrak d \mid \mathfrak n$, put $$ \epsilon_\mathfrak d = \prod_{\mathfrak p_i\mid \mathfrak d}\varepsilon_i\in \{\pm 1\}. $$ (In particular, $\epsilon_{\mathfrak p_i}=\varepsilon_i$ and $\epsilon_1=1$.) \end{notn}
\begin{prop}\label{prop-cor2.4} Let $R$ be a coefficient ring with $(q-1)\in R^\times$. Given ${\bm{\epsilon}}\in \mathbb{E}$, we have $$
\{f\in \widetilde{\mathcal{H}}(\mathfrak n, R)^{\bm{\epsilon}}\ |\ f^\ast(\mathfrak m)=0 \text{ when } (\mathfrak m, \mathfrak n)=1\}= \begin{cases} \mathcal{H}(1, R) & \text{if }{\bm{\epsilon}}={\bm{\epsilon}}_{H(\mathfrak n)},\\ 0 & \text{otherwise}. \end{cases} $$ \end{prop} \begin{proof} Let $f\in \widetilde{\mathcal{H}}(\mathfrak n, R)^{\bm{\epsilon}}$ and assume $f^\ast(\mathfrak m)=0$ for all $\mathfrak m$ coprime to $\mathfrak n$. Then Proposition \ref{propALOhta} implies that $f^\ast(\mathfrak m)=0$ for all $\mathfrak m$. In particular, $f^\ast(1)=0$. Applying Lemma \ref{lemLevelLow} successively we get that there is $g\in \widetilde{\mathcal{H}}(1, R)$
such that $f=g|\prod_{i=1}^s B_{\mathfrak p_i}$ and $\epsilon_\mathfrak n f=f|W_\mathfrak n = g\in \widetilde{\mathcal{H}}(1, R)$. Therefore, there exists $a\in R$ such that $f=a\widetilde{E}$. This implies $$ f^\ast(1)=a\widetilde{E}^\ast(1)=\frac{a(1-q^2)}{q}=0. $$ Since $(q-1)$ is invertible in $R$, we must have $a\in R[q+1]$. From Example \ref{example1} we conclude that $f\in \mathcal{H}(1, R)$ is harmonic.
Now it is enough to show that $a\widetilde{E}|W_{\mathfrak p_i}=(-1)^{\deg(\mathfrak p_i)}a\widetilde{E}$ for all $1\leq i\leq s$ if $a\in R[q+1]$.
For that, let $\mathfrak p|\mathfrak n$ be a prime. By Lemma \ref{lemAL2} (2), $a\widetilde{E}|B_\mathfrak p W_\mathfrak p=a\widetilde{E}$. On the other hand, \begin{align*}
a\widetilde{E}|B_\mathfrak p\left(\begin{pmatrix}\pi_\infty^k & u \\ 0 &1\end{pmatrix}\right) & = a\widetilde{E} \left(\begin{pmatrix}\pi_\infty^{k-\deg(\mathfrak p)} & u \mathfrak p \\ 0 &1\end{pmatrix}\right)=aq^{k-\deg(\mathfrak p)+1}=a(-1)^{k-\deg(\mathfrak p)+1} \\ &= (-1)^{\deg(\mathfrak p)} a\widetilde{E} \left(\begin{pmatrix}\pi_\infty^k & u \\ 0 &1\end{pmatrix}\right). \end{align*}
Hence $a\widetilde{E}|W_\mathfrak p= (-1)^{\deg(\mathfrak p)}a\widetilde{E}|B_\mathfrak p W_\mathfrak p=(-1)^{\deg(\mathfrak p)}a\widetilde{E}$, as was required to be shown. \end{proof}
\begin{notn} For ${\bm{\epsilon}} \in \mathbb{E}$, let \begin{align*}
E^{\bm{\epsilon}} &:=\frac{1}{\nu({\bm{\epsilon}})}\widetilde{E}|\prod_{i=1}^s(1+\varepsilon_i W_{\mathfrak p_i})=\frac{1}{\nu({\bm{\epsilon}})}\widetilde{E}|\prod_{i=1}^s(1+\varepsilon_i B_{\mathfrak p_i})\\
&=\frac{1}{\nu({\bm{\epsilon}})}\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d \widetilde{E}|B_\mathfrak d \in \widetilde{\mathcal{E}}(\mathfrak n, \mathbb{Q}), \end{align*} where $$\nu ({\bm{\epsilon}}) = \begin{cases} 1 & \text{ if ${\bm{\epsilon}} = {\bm{\epsilon}}_{H(\mathfrak n)}$, } \\ q+1 & \text{ otherwise.} \end{cases}$$ (The fact that $E^{\bm{\epsilon}}$ is Eisenstein follows from Lemma \ref{lemtEEis} and the commutativity $B_\mathfrak d T_\mathfrak p=T_\mathfrak p B_\mathfrak d$, $\mathfrak p\nmid \mathfrak d$.) Let $$
N(\mathfrak n, {\bm{\epsilon}}):=\prod_{i=1}^s(1+\varepsilon_i|\mathfrak p_i|). $$ \end{notn} \begin{rem}\label{remNnu} Note that $q+1$ divides $N(\mathfrak n, {\bm{\epsilon}})$ if ${\bm{\epsilon}}\neq {\bm{\epsilon}}_{H(\mathfrak n)}$. In particular, $N(\mathfrak n, {\bm{\epsilon}})/\nu({\bm{\epsilon}})\in \mathbb{Z}$. Also note that $(q+1, N(\mathfrak n, {\bm{\epsilon}}_{H(\mathfrak n)})) = (q+1,2^s) = 2^b$ for some $0\leq b\leq s$, since
\begin{align*}
N(\mathfrak n, {\bm{\epsilon}}_{H(\mathfrak n)}) &= \prod_{i=1}^s(1+(-1)^{\deg(\mathfrak p_i)}|\mathfrak p_i|)\\ & \equiv \prod_{i=1}^s(1+(-1)^{\deg(\mathfrak p_i)}(-1)^{\deg(\mathfrak p_i)})=2^s\ (\mathrm{mod}\ q+1). \end{align*}
\end{rem}
\begin{lem}\label{lemEeps} We have $E^{\bm{\epsilon}}\in \widetilde{\mathcal{E}}(\mathfrak n, \mathbb{Z})^{\bm{\epsilon}}$. Moreover, $E^{\bm{\epsilon}}$ is harmonic if and only if ${\bm{\epsilon}}\neq {\bm{1}}$. \end{lem} \begin{proof} Since $\widetilde{E}$ is $\mathbb{Z}$-valued, $E^{\bm{\epsilon}}$ takes values in $\mathbb{Z}[\nu({\bm{\epsilon}})^{-1}]$. On the other hand, from the Fourier expansion of $\widetilde{E}$ and Lemma \ref{lem_new14}, one computes \begin{equation}\label{eqE(1)} (E^{{\bm{\epsilon}}})^0(1) = q \frac{N(\mathfrak n,{\bm{\epsilon}})}{\nu({\bm{\epsilon}})} \quad \text{ and } \quad (E^{{\bm{\epsilon}}})^*(1) = \frac{1-q^2}{q\nu({\bm{\epsilon}})}, \end{equation} and also that $(E^{{\bm{\epsilon}}})^*(\mathfrak m)\in \mathbb{Z}[p^{-1}]$ for all $\mathfrak m$. The Fourier expansion (\ref{eq-Fexp}) then implies that $E^{\bm{\epsilon}}$ on $E(\mathscr{T})^+$ takes values in $\mathbb{Z}[p^{-1}]$; cf. \cite[Cor. 3.15]{Analytical}.
Next, using the definition of $E^{\bm{\epsilon}}$ and Example \ref{example1}, we have \begin{align*}
E^{\bm{\epsilon}}(e_0) &=\frac{1}{\nu({\bm{\epsilon}})}\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d \widetilde{E}|B_\mathfrak d(e_0) \\ &= \frac{1}{\nu({\bm{\epsilon}})}\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d \widetilde{E}(e_{\deg(\mathfrak d)})
= \frac{q}{\nu(\mathfrak n)}\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d |\mathfrak d|=\frac{q}{\nu(\mathfrak n)}N(\mathfrak n, {\bm{\epsilon}}), \end{align*} and similarly $$
E^{\bm{\epsilon}}(\bar{e}_0)=\frac{q+1}{\nu(\mathfrak n)}\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d - \frac{q}{\nu(\mathfrak n)}N(\mathfrak n, {\bm{\epsilon}}) = \frac{q+1}{\nu({\bm{\epsilon}})}\prod_{i=1}^s(1+\varepsilon_{i}) - \frac{q}{\nu(\mathfrak n)}N(\mathfrak n, {\bm{\epsilon}}). $$ Therefore, by Lemma \ref{lem 2.1FT}, for any $e\in E(\mathscr{T})$ we have \begin{equation}\label{eqE+E} E^{{\bm{\epsilon}}}(e) + E^{{\bm{\epsilon}}}(\bar{e})= E^{{\bm{\epsilon}}}(e_0) + E^{{\bm{\epsilon}}}(\bar{e}_0) = \frac{q+1}{\nu({\bm{\epsilon}})}\prod_{i=1}^s(1+\varepsilon_{i}). \end{equation} This shows that $E^{\bm{\epsilon}}$ is harmonic if and only if ${\bm{\epsilon}}\neq {\bm{1}}$, and that $E^{\bm{\epsilon}}(e)\in \mathbb{Z}[p^{-1}]$ for all $e\in E(\mathscr{T})$. Since $p$ and $\nu({\bm{\epsilon}})$ are coprime, we conclude that $E^{\bm{\epsilon}}$ takes its values in $\mathbb{Z}$.
It remains to show that $E^{\bm{\epsilon}}|W_{\mathfrak p_i}=\varepsilon_i E^{\bm{\epsilon}}$, $1\leq i\leq s$. This follows from the definition of $E^{\bm{\epsilon}}$ and the following observation: $$ \prod_{j=1}^s(1+\varepsilon_j W_{\mathfrak p_j})W_{\mathfrak p_i} = (W_{\mathfrak p_i}+\varepsilon_i)\prod_{j\neq i}(1+\varepsilon_j W_{\mathfrak p_j})=\varepsilon_i \prod_{j=1}^s(1+\varepsilon_j W_{\mathfrak p_j}). $$ \end{proof}
The previous lemma allows us to consider $E^{\bm{\epsilon}}$ as an element of $\widetilde{E}(\mathfrak n, R)^{\bm{\epsilon}}$ for any ring $R$.
\begin{prop}\label{prop 3.2FT} Let $R$ be a coefficient ring with $(q-1) \in R^\times$. Assume ${\bm{\epsilon}} \neq {\bm{\epsilon}}_{H(\mathfrak n)}$. Then $$\widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}} = R\cdot E^{{\bm{\epsilon}}},$$ and $$\mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}} = \begin{cases} R\left[\frac{N(\mathfrak n,{\bm{\epsilon}})}{q+1}\right] \cdot E^{{\bm{\epsilon}}}, & \text{if ${\bm{\epsilon}}\neq {\bm{1}}$};\\ 0, & \text{if ${\bm{\epsilon}}={\bm{1}}$}. \end{cases} $$ \end{prop}
\begin{proof} Since ${\bm{\epsilon}} \neq {\bm{\epsilon}}_{H(\mathfrak n)}$, from (\ref{eqE(1)}) we get $(E^{{\bm{\epsilon}}})^*(1) = q^{-1}(1-q) \in R^{\times}$. Given $f \in \widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}}$, let $$ \tilde{f}:= f - \frac{qf^*(1)}{1-q}E^{{\bm{\epsilon}}}.$$ Then $\tilde{f} \in \widetilde{\mathcal{H}}(\mathfrak n,R)^{{\bm{\epsilon}}}$ has $\tilde{f}^*(\mathfrak m) = 0$ for every $\mathfrak m$ coprime to $\mathfrak n$. From Proposition \ref{prop-cor2.4} we get $\tilde{f} = 0$, i.e., \begin{equation}\label{eqSpanE} f = \frac{q f^*(1)}{1-q}E^{{\bm{\epsilon}}}. \end{equation} Hence $\widetilde{\mathcal{E}}(\mathfrak n, R)^{\bm{\epsilon}}$ is spanned by $E^{\bm{\epsilon}}$.
To prove the second claim, assume $f\in \mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}}$. In particular, $f$ is alternating, so from (\ref{eqSpanE}) and (\ref{eqE+E}) we get $$ 0=f(\bar{e}) + f(e) = \frac{q}{1-q}\left(\prod_{i=1}^s(1+\epsilon_{\mathfrak p_i})\right) \cdot f^*(1). $$ Thus, when ${\bm{\epsilon}} ={\bm{1}}$, we must have $f^*(1) = 0$; thus, by Proposition \ref{prop-cor2.4}, $f = 0$. (Note that $\prod_{i=1}^s(1+\varepsilon_i)=2^s\in R^\times$, since $p$ and $q-1$ are invertible in $R$, and one of these numbers is necessarily even.)
Finally, assume ${\bm{\epsilon}}\neq {\bm{1}}$.
According to \cite[Lem. 2.19]{PW1}, $f\in \mathcal{H}(\mathfrak n,R)$ is cuspidal if and only if $(f|W)^0(1)=0$ for all $W\in \mathbb{W}$. Since our $f$ is an eigenfunction of all Atkin-Lehner involutions, we conclude that $f$ is cuspidal if and only if $f^0(1)=0$. On the other hand, from (\ref{eqE(1)}) and (\ref{eqSpanE}) we have $$f^0(1) = \frac{q^2}{1-q} \cdot \frac{N(\mathfrak n,{\bm{\epsilon}})}{q+1} \cdot f^*(1).$$ Since $q(q-1)\in R^\times$, we must have $f^*(1) \in R\left[\frac{N(\mathfrak n,{\bm{\epsilon}})}{q+1}\right]$; thus $\mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}} = R\left[\frac{N(\mathfrak n,{\bm{\epsilon}})}{q+1}\right] \cdot E^{{\bm{\epsilon}}}$. \end{proof}
\begin{rem} \label{rem E_00=E_0} Assume $R$ is a coefficient ring with $(q-1)\in R^\times$. \begin{itemize} \item[(1)] Take ${\bm{\epsilon}} \in \mathbb{E}$ with ${\bm{\epsilon}}\neq {\bm{\epsilon}}_{H(\mathfrak n)}$. By (\ref{eqE+E}), $E^{\bm{\epsilon}}\in \widetilde{\mathcal{H}}(\mathfrak n, R)$ is harmonic if and only if ${\bm{\epsilon}}\neq {\bm{1}}$. Thus, Proposition \ref{prop 3.2FT} implies that $\widetilde{\mathcal{E}}(\mathfrak n, R)^{\bm{\epsilon}}=\mathcal{E}(\mathfrak n, R)^{\bm{\epsilon}}$ when ${\bm{\epsilon}}\neq {\bm{1}}$, and $\mathcal{E}(\mathfrak n, R)^{\bm{\epsilon}}=0$ when ${\bm{\epsilon}}={\bm{1}}$. \item[(2)] Suppose ${\bm{\epsilon}}_{H(\mathfrak n)} \neq {\bm{1}}$, i.e., there is a prime factor of $\mathfrak n$ with odd degree. Then by \cite[Lemma 2.7 (1)]{PW1} we have $\mathcal{E}_0(\mathfrak n, R)^{\bm{\epsilon}}=\mathcal{E}_{00}(\mathfrak n, R)^{\bm{\epsilon}}$ for every ${\bm{\epsilon}} \neq {\bm{\epsilon}}_{H(\mathfrak n)}$. \end{itemize} \end{rem}
When ${\bm{\epsilon}} = {\bm{\epsilon}}_{H(\mathfrak n)}$, $(E^{{\bm{\epsilon}}})^*(1)$ is not invertible in $R$, so the argument in the proof of Proposition~\ref{prop 3.2FT} does not work. We will deal with this case in three separate propositions, after proving the following lemma:
\begin{lem}\label{lem 3.8} Define ${\bm{\epsilon}}_{H(\mathfrak n),s} = (\varepsilon_1,\dots ,\varepsilon_s) \in \mathbb{E}$ by $$ \varepsilon_i =\begin{cases} (-1)^{\deg \mathfrak p_i} & 1\leq i<s;\\
-(-1)^{\deg \mathfrak p_s} & i=s. \end{cases} $$ \begin{itemize}
\item[(1)] Suppose $\deg \mathfrak p_s$ is even. Then $E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} | U_{\mathfrak p_s} = E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}$. \item[(2)] Suppose $\deg \mathfrak p_s$ is odd. Let $n$ be an integer dividing both $q+1$ and $\deg \mathfrak p_s$. Let $R$ be a coefficient ring. Then for an arbitrary $a \in R[n]$, we have
$$a \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} | U_{\mathfrak p_s} = -a \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}.$$ \end{itemize} \end{lem}
\begin{proof} We can write
$$E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} = \frac{1}{q+1} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} | (1 -(-1)^{\deg \mathfrak p_s} B_{\mathfrak p_s}).$$
By Lemma \ref{lemAL2} (4), $E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} | B_{\mathfrak p_s} U_{\mathfrak p_s} = |\mathfrak p_s| \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}$ and
$$(|\mathfrak p_s|+1) E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} = E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} | T_{\mathfrak p_s} = E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} | (U_{\mathfrak p_s} + B_{\mathfrak p_s}).$$ One then gets: \begin{align}
E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} | U_{\mathfrak p_s} &= \frac{1}{q+1} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}|(U_{\mathfrak p_s} -(-1)^{\deg \mathfrak p_s} B_{\mathfrak p_s} U_{\mathfrak p_s}) \notag \\
&= \frac{1}{q+1} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} | (T_{\mathfrak p_s}- B_{\mathfrak p_s}-(-1)^{\deg \mathfrak p_s} |\mathfrak p_s|) \notag \\
&= \left((1-(-1)^{\deg \mathfrak p_s}) \cdot \frac{|\mathfrak p_s|+1}{q+1}\right) \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}} +(-1)^{\deg \mathfrak p_s}\cdot E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}. \notag \end{align} Part (1) immediately follows from this; part (2) follows from the observation that if $q\equiv -1\ (\mathrm{mod}\ n)$ and $\deg(\mathfrak p_s)$ is odd, then $$
\frac{|\mathfrak p_s|+1}{q+1} =\sum_{i=0}^{\deg(\mathfrak p_s)-1}(-1)^i q^i\equiv \deg(\mathfrak p_s)\ (\mathrm{mod}\ n). $$ \end{proof}
\begin{prop}\label{prop 3.4} Let $R$ be a coefficient ring with $(q-1) \in R^\times$. Assume there exists at most one prime divisor $\mathfrak p_i$ of $\mathfrak n$ with $\deg \mathfrak p_i$ odd. Then $$\widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}} = R \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}},$$ and $$\mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}} = \begin{cases} R\left[N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)})\right]\cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}, & \text{ if ${\bm{\epsilon}}_{H(\mathfrak n)} \neq \mathbf{1}$}; \\ 0, & \text{ if ${\bm{\epsilon}}_{H(\mathfrak n)} = \mathbf{1}$.} \end{cases} $$ \end{prop}
\begin{proof} When $s =1$, i.e., $\mathfrak n = \mathfrak p$ is a prime, take ${\bm{\epsilon}}' = (-(-1)^{\deg \mathfrak p}) \in \mathbb{E}$. Then $(E^{{\bm{\epsilon}}'})^*(1) = q^{-1}(1-q) \in R^{\times}$. For $f \in \widetilde{\mathcal{E}}(\mathfrak p,R)^{{\bm{\epsilon}}_{H(\mathfrak p)}}$, put $$\tilde{f} := f - \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}'}.$$ Then by Proposition~\ref{propALOhta}, $\tilde{f}^*(\mathfrak m) = 0$ unless $\mathfrak p \mid \mathfrak m$. Hence by Lemma~\ref{lemLevelLow}, there exists $g \in \widetilde{\mathcal{H}}(1,R) = R \widetilde{E}$ such that
$\tilde{f} = g|B_\mathfrak p$ and
$$\tilde{f}|W_\mathfrak p = (-1)^{\deg \mathfrak p} \left(f+\frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}'}\right) = g.$$ In particular,
$$g+(-1)^{\deg \mathfrak p} g|B_\mathfrak p = g+(-1)^{\deg \mathfrak p}\tilde{f} = 2(-1)^{\deg \mathfrak p} f.$$ Writing $g = a \widetilde{E}$ for $a \in R$, we get
$$g+(-1)^{\deg \mathfrak p} g|B_\mathfrak p = a E^{{\bm{\epsilon}}_{H(\mathfrak p)}}$$ and so $f = (2^{-1} (-1)^{\deg \mathfrak p} a) \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p)}}$. This shows $$\widetilde{\mathcal{E}}(\mathfrak p,R)^{{\bm{\epsilon}}_{H(\mathfrak p)}} = R \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p)}}.$$
Now suppose $s>1$. Without loss of generality, we assume that $\deg \mathfrak p_i$ is even for $2\leq i \leq s$. Let ${\bm{\epsilon}}_{H(\mathfrak n),s} = (\varepsilon_1,\dots ,\varepsilon_s)\in \mathbb{E}$ be the element defined in Lemma~\ref{lem 3.8}.
In particular, ${\bm{\epsilon}}_{H(\mathfrak n),s} \neq \mathbf{1}$ and ${\bm{\epsilon}}_{H(\mathfrak n),s} \neq{\bm{\epsilon}}_{H(\mathfrak n)}$. Given $f \in \widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, put $$\tilde{f} := f - \frac{qf^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}.$$
Then, by Proposition \ref{propALOhta}, $\tilde{f}^*(\mathfrak m) = 0$ unless $\mathfrak p_{s} | \mathfrak m$. By Lemma~\ref{lemLevelLow}, there exists $g \in \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p_{s},R)$ such that
$\tilde{f} = g|B_{\mathfrak p_{s}}$ and
$$\tilde{f}|W_{\mathfrak p_{s}} = f + \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} = g.$$ In particular,
$$g+g|B_{\mathfrak p_{s}} = 2 f \quad \text{ and } \quad
g- g|B_{\mathfrak p_{s}} = g- g|W_{\mathfrak p_{s}} = \frac{2q f^*(1)}{1-q} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p),s}}.$$
Consider the trace map : $\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}: \widetilde{\mathcal{H}}(\mathfrak n,R) \rightarrow \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p_s,R)$ defined by:
$$\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(h):= h + h | W_{\mathfrak p_s} U_{\mathfrak p_s}.$$ Then Lemma~\ref{lem 3.8} (1) implies that
$$\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}) = E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} - E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} | U_{\mathfrak p_s} = 0.$$ Therefore
$$\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(g) = \text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(g |W_{\mathfrak p_s}).$$
The left hand side is equal to $(|\mathfrak p_s|+1)g$ as $g$ is of level $\mathfrak n/\mathfrak p_s$, and the right hand side is nothing but $g| T_{\mathfrak p_s}$. This implies that $g \in \widetilde{\mathcal{E}}(\mathfrak n/\mathfrak p_s,R)^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}$. By induction, there exists $a \in R$ so that $g = a E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}$, and hence
$$f = 2^{-1}\left(g+ g|B_{\mathfrak p_s}\right) = 2^{-1} a \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ Therefore $\widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}} = R \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, which proves the first statement of the proposition. The second statement now can be deduced by an argument similar to the argument in the proof of Proposition \ref{prop 3.2FT}. \end{proof}
\begin{prop}\label{prop 3.5} Let $R$ be a coefficient ring with $(q-1) \in R^\times$. Assume there are at least two prime factors of $\mathfrak n$ with odd degree, and $(q+1, \deg \mathfrak p_1,...,\deg \mathfrak p_s)$ is invertible in $R$. Then $$\mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}} = R\left[N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)})\right]\cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ \end{prop}
\begin{proof} For $f \in \mathcal{E}_0(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, put $$\tilde{f} := (q+1) f - \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n)}} \in \widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ Then $\tilde{f}^*(\mathfrak m) = 0$ for every $\mathfrak m$ coprime to $\mathfrak n$. By Proposition \ref{prop-cor2.4}, we get $\tilde{f} \in \mathcal{H}(1,R) = R[q+1] \cdot \widetilde{E}$. In particular, $$(q+1)^2f = \frac{q (q+1) f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ As $f$ is cuspidal, we get \begin{align}\label{eqn 3.1} (q+1)^2 f^0(1) &= \frac{q (q+1) f^*(1)}{1-q} (E^{{\bm{\epsilon}}_{H(\mathfrak n)}})^0(1) \\ \nonumber &= \frac{q}{1-q} \cdot (q+1)N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)}) f^*(1) = 0. \end{align}
For $1\leq i_0 \leq s$, define ${\bm{\epsilon}}_{H(\mathfrak n),i_0} = (\varepsilon_1,\dots ,\varepsilon_s)\in \mathbb{E}$ by $$\varepsilon_i := \begin{cases} (-1)^{\deg \mathfrak p_i} & \text{ if $i \neq i_0$,}\\ -(-1)^{\deg \mathfrak p_i} & \text{ if $i = i_0$.} \end{cases}$$ Since $s>1$ and there are at least two prime factors of $\mathfrak n$ with odd degree, ${\bm{\epsilon}}_{H(\mathfrak n),i_0} \neq \mathbf{1}$, ${\bm{\epsilon}}_{H(\mathfrak n),i_0} \neq {\bm{\epsilon}}_{H(\mathfrak n)}$ for $1\leq i_0\leq s$. Put $$\tilde{f}_{i_0} := f - \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),i_0}}.$$ Then $\tilde{f}_{i_0}^*(\mathfrak m) = 0$ unless $\mathfrak p_{i_0} \mid \mathfrak m$. Hence by Lemma~\ref{lemLevelLow} there exists $g \in \mathcal{H}(\mathfrak n/\mathfrak p_{i_0},R)$ such that
$\tilde{f}_{i_0} = g|B_{\mathfrak p_{i_0}}$ and
$$\tilde{f}_{i_0}|W_{\mathfrak p_{i_0}} = (-1)^{\deg \mathfrak p_{i_0}} \left(f + \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),i_0}}\right) = g.$$ In particular, we get $$
(-1)^{ \deg \mathfrak p_{i_0}}g+g|B_{\mathfrak p_{i_0}} = 2 f$$ and
$$(-1)^{ \deg \mathfrak p_{i_0}}g- g|B_{\mathfrak p_{i_0}} = (-1)^{ \deg \mathfrak p_{i_0}}g- g|W_{\mathfrak p_{i_0}} = \frac{ 2 q f^*(1) }{1-q} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p),i_0}}.$$ Adding this equations, we get $g=(-1)^{ \deg \mathfrak p_{i_0}}\left(f+\frac{q f^*(1) }{1-q} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p),i_0}}\right)$. Since $f$ is cuspidal, \begin{align*} g^0(1) &= (-1)^{ \deg \mathfrak p_{i_0}} \frac{q f^*(1)}{1-q} \cdot (E^{{\bm{\epsilon}}_{H(\mathfrak n),i_0}})^0(1) \\
&= \frac{ q^2f^*(1) }{q-1} \cdot N(\mathfrak n/\mathfrak p_{i_0},{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_{i_0})}) \cdot \frac{|\mathfrak p_{i_0}|-(-1)^{ \deg \mathfrak p_{i_0}}}{q+1}. \end{align*} Therefore, using Lemma \ref{lem_new14}, we obtain \begin{align}\label{eqn 3.2}
0 &= (-1)^{ \deg \mathfrak p_{i_0}}g^0(1)+ (g|B_{\mathfrak p_{i_0}})^0(1) \\
\nonumber & = \frac{q^2}{1-q} \cdot \left(N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)}) \cdot \frac{1-(-1)^{ \deg \mathfrak p_{i_0}}|\mathfrak p_{i_0}|}{q+1}\right) \cdot f^*(1). \end{align}
By assumption
$$\left(q+1, \frac{1-(-1)^{ \deg \mathfrak p_{1}}|\mathfrak p_{1}|}{q+1},\dots ,\frac{1-(-1)^{ \deg \mathfrak p_{s}}|\mathfrak p_{s}|}{q+1}\right) = (q+1,\deg \mathfrak p_1,...,\deg \mathfrak p_s) \in R^{\times},$$ so the equations~(\ref{eqn 3.1}) and (\ref{eqn 3.2}) force $f^*(1) \in R[N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)})]$. From the Eisenstein property of $f$, we get $N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)})f^*(\mathfrak m) = 0$ for every $\mathfrak m$ coprime to $\mathfrak n$. Therefore, by Proposition~\ref{prop-cor2.4}, we obtain $N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)}) f \in \mathcal{H}_0(1,R) = 0$. Finally, from the fact that $(q+1,N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)}))$ is a power of $2$, thus invertible in $R$, we conclude that there exists $\alpha \in R$ such that $$f = \alpha (q+1)^2 f = \frac{q(q+1)\alpha f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ Thus, $f \in R[N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)})] \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}$. \end{proof}
Given a prime $\ell$ and a positive integer $r$, let $R_{\ell}^{(r)} := \mathbb{Z}_\ell[\zeta_p]/ \ell^r \mathbb{Z}_\ell[\zeta_p]$. For our purposes, it suffices to focus on this particular coefficient ring for the remaining case:
\begin{prop}\label{prop 3.9}
Let $\ell$ be a prime such that $\ell \nmid q(q-1)$, $\ell \mid q+1$, and $\ell \mid \deg \mathfrak p_i$ for $1\leq i \leq s$. Then $$\widetilde{\mathcal{E}}(\mathfrak n,R_{\ell}^{(r)})^{{\bm{\epsilon}}_{H(\mathfrak n)}} = R_\ell^{(r)} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}, \quad \forall r \geq 1,$$ and $${\mathcal{E}}_0(\mathfrak n,R_\ell^{(r)})^{{\bm{\epsilon}}_{H(\mathfrak n)}} = R_\ell^{(r)}\left[N(\mathfrak n,{\bm{\epsilon}}_{H(\mathfrak n)}\right] \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}} = 0. $$ \end{prop}
\begin{proof} Note that the statement about ${\mathcal{E}}_0(\mathfrak n,R_\ell^{(r)})^{{\bm{\epsilon}}_{H(\mathfrak n)}}$ follows from the first statement
by an argument similar to the argument in the proof of Proposition \ref{prop 3.2FT} combined with Remark \ref{remNnu}. Thus, it suffices to prove the first statement. Let $R = R_\ell^{(r)}$, which says that $R = R[\ell^r]$.\\
First, assume $r=1$. By Proposition~\ref{prop 3.4} the case when $s=1$ holds. Suppose $s>1$. Let ${\bm{\epsilon}}_{H(\mathfrak n),s}$ be defined as in Lemma \ref{lem 3.8}.
In particular, ${\bm{\epsilon}}_{H(\mathfrak n),s} \neq \mathbf{1}, {\bm{\epsilon}}_{H(\mathfrak n)}$. Given $f \in \widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, put $$\tilde{f} := f - \frac{qf^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}.$$ By Proposition~\ref{propALOhta}, $\tilde{f}^*(\mathfrak m) = 0$ unless $\mathfrak p_{s} \mid \mathfrak m$; Lemma~\ref{lemLevelLow} then says that there exists $g \in \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p_{s},R)$ such that
$\tilde{f} = g|B_{\mathfrak p_{s}}$ and
$$\tilde{f}|W_{\mathfrak p_{s}} = (-1)^{\deg \mathfrak p_s} \left(f + \frac{q f^*(1)}{1-q} E^{{\bm{\epsilon}}_{H(\mathfrak n),s}}\right) = g.$$ In particular, we have \begin{itemize}
\item[(i)] $$g+(-1)^{\deg \mathfrak p_s} g|B_{\mathfrak p_{s}} = 2(-1)^{\deg \mathfrak p_s} f,$$ \item[(ii)]
$$g- (-1)^{\deg \mathfrak p_s} g|B_{\mathfrak p_{s}} = g- (-1)^{\deg \mathfrak p_s} g|W_{\mathfrak p_{s}} = \frac{2(-1)^{\deg \mathfrak p_s} q f^*(1)}{1-q} \cdot E^{{\bm{\epsilon}}_{H(\mathfrak p),s}}.$$ \end{itemize}
Consider the trace map : $\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}: \widetilde{\mathcal{H}}(\mathfrak n,R) \rightarrow \widetilde{\mathcal{H}}(\mathfrak n/\mathfrak p_s,R)$ defined by:
$$\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(h):= h + h | W_{\mathfrak p_s} U_{\mathfrak p_s}.$$ Since $\ell \cdot f^*(1)= 0$, by Lemma~\ref{lem 3.8} we get $\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(f^*(1) \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n),s}} ) = 0$ and
$$\text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(g) = (-1)^{\deg \mathfrak p_s} \text{Tr}^{\mathfrak n}_{\mathfrak n/\mathfrak p_s}(g |W_{\mathfrak p_s}).$$
The left hand side is equal to $(|\mathfrak p_s|+1)g$ as $g$ is of level $\mathfrak n/\mathfrak p_s$, and the right hand side is nothing but $(-1)^{\deg \mathfrak p_s} g\big| T_{\mathfrak p_s}$. This implies that $g \in \widetilde{\mathcal{E}}(\mathfrak n/\mathfrak p_s,R)^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}$ (when $\deg \mathfrak p_s$ is odd, we have $(|\mathfrak p_s|+1)g = -(|\mathfrak p_s|+1)g = 0$). By induction on $s$, there exists $a \in R$ so that $g = a E^{{\bm{\epsilon}}_{H(\mathfrak n/\mathfrak p_s)}}$, and hence
$$f = 2^{-1}\left(g+ g|B_{\mathfrak p_s}\right) = 2^{-1} a \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ ${}$\\
Now assume $r>1$. Given $f \in \widetilde{\mathcal{E}}(\mathfrak n,R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, consider $f \bmod \ell \in \widetilde{\mathcal{E}}(\mathfrak n,R/\ell R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}$. Since this latter module is spanned by $E^{{\bm{\epsilon}}_{H(\mathfrak n)}}$, there exists $a \in R$ such that $$ f \equiv a \cdot E^{{\bm{\epsilon}}_{H(\mathfrak n)}} \bmod \ell,$$ or equivalently $$f - a E^{{\bm{\epsilon}}_{H(\mathfrak n)}} \in \widetilde{\mathcal{E}}(\mathfrak n,\ell R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ Note that $\ell R$ is the maximal ideal in $R$, and the isomorphism $\iota_\ell : R/\ell^{r-1}R \cong \ell R$ (multiplication by $\ell$) induces the following (group) isomorphism: $$\widetilde{\mathcal{E}}(\mathfrak n,R/\ell^{r-1}R)^{{\bm{\epsilon}}_{H(\mathfrak n)}} \cong \widetilde{\mathcal{E}}(\mathfrak n,\ell R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ By the induction on $r$, there exists $b \in R/\ell^{r-1}R$ such that $$ f - a E^{{\bm{\epsilon}}_{H(\mathfrak n)}} = \iota_\ell(b) E^{{\bm{\epsilon}}_{H(\mathfrak n)}} \in \widetilde{\mathcal{E}}(\mathfrak n,\ell R)^{{\bm{\epsilon}}_{H(\mathfrak n)}}.$$ This completes the proof. \end{proof}
The results of this section imply the following:
\begin{thm}\label{thm 3.6FT} Let $\mathfrak n = \prod_{i=1}^s \mathfrak p_i \lhd A$ be a square-free ideal. Given a prime number $\ell$ not dividing $q(q-1)$ and a positive integer $r$, we have that for ${\bm{\epsilon}} \in \mathbb{E}$, $$\mathcal{E}_0(\mathfrak n,\mathbb{Z}/\ell^r\mathbb{Z})^{\bm{\epsilon}} \cong \begin{cases} \mathbb{Z}/(\ell^r,\frac{N(\mathfrak n,{\bm{\epsilon}})}{\nu({\bm{\epsilon}})})\mathbb{Z}, & \text{ if ${\bm{\epsilon}} \neq \mathbf{1}$};\\ 0, & \text{ if ${\bm{\epsilon}} = \mathbf{1}$.} \end{cases}$$
\end{thm}
\begin{proof}\label{rem3.13}
This follows from the observation that the coefficient ring $\mathbb{Z}_\ell[\zeta_p]/\ell^r\mathbb{Z}_\ell[\zeta_p]$ is a free $\mathbb{Z} /\ell^r\mathbb{Z}$-module; cf.\ the proof of Corollary 6.9 in \cite{Pal}.
\end{proof}
\section{Proof of the main theorem}\label{sCDG}
Let $\Omega$, $X_0(\mathfrak n)$, $J_0(\mathfrak n)$, $\mathcal{C}(\mathfrak n)$, $\mathcal{T}(\mathfrak n)$ be as in the introduction. The Hecke operators $T_\mathfrak p$ may also be defined as a correspondences on $X_0(\mathfrak n)$, so $T_\mathfrak p$ induce endomorphisms of $J_0(\mathfrak n)$. Similarly, the Atkin-Lehner involutions induce automorphisms of $J_0(\mathfrak n)$. Let $\mathcal{O}(\Omega)^\times$ be the group of non-vanishing holomorphic rigid-analytic functions on $\Omega$.
The group $\mathrm{GL}_2(F_\infty)$ act on $\mathcal{O}(\Omega)^\times$ via $(f|\gamma)(z)=f(\gamma z)$. To each $f\in \mathcal{O}(\Omega)^\times$ van der Put associated a harmonic cochain $r(f)\in \mathcal{H}(\mathscr{T}, \mathbb{Z})$ so that the sequence \begin{equation}\label{eqvdPut} 0\to \mathbb{C}_\infty^\times \to \mathcal{O}(\Omega)^\times\xrightarrow{r} \mathcal{H}(\mathscr{T}, \mathbb{Z})\to 0 \end{equation} is exact and $\mathrm{GL}_2(F_\infty)$-equivariant; cf. \cite[Thm. 2.1]{vdPut}, \cite[(1.7.2)]{GR}.
Let $\Delta(z)\in \mathcal{O}(\Omega)^\times$ be the Drinfeld discriminant function defined on page 183 of \cite[p. 183]{Discriminant}.
For a non-zero ideal $\mathfrak m\lhd A$, let
$$\Delta_\mathfrak m(z):=\Delta|B_\mathfrak m(z)=\Delta(\mathfrak m z).$$
From now on we assume $\mathfrak n=\mathfrak p_1\cdots \mathfrak p_s$ is square-free with the given prime decomposition. Given ${\bm{\epsilon}}\in \mathbb{E}$, we set $$
\Delta^{\bm{\epsilon}}:=\prod_{\mathfrak d|\mathfrak n}\Delta_\mathfrak d^{\epsilon_\mathfrak d} \in \mathcal{O}(\Omega)^\times. $$
(Recall that $\epsilon_\mathfrak d=\prod_{\mathfrak p_i|\mathfrak d}\varepsilon_i$; see Notation \ref{notn2.3}.)
The harmonic cochain $r(\Delta)$ has been extensively studied by Gekeler in \cite{Improper}, \cite{Discriminant}, where he shows that $r(\Delta)=q(1-q)H$; see Remark \ref{remH}. Therefore, by the same remark, for $e\in E(\mathscr{T})^+$, $$ r(\Delta)(e)=(1-q)\widetilde{E}(e) \quad \text{and}\quad r(\Delta)(\bar{e})=(q^2-1)+ (1-q)\widetilde{E}(\bar{e}). $$ More importantly for us, we also get $$ E^{\bm{\epsilon}}(e)=\frac{1}{\nu({\bm{\epsilon}})(1-q)} r(\Delta^{\bm{\epsilon}})(e) $$ for all $e\in E(\mathscr{T})^+$. The above equality holds for all $e\in E(\mathscr{T})$ if and only if ${\bm{\epsilon}}\neq {\bm{1}}$, since $E^{\bm{\epsilon}}$ is harmonic if and only if ${\bm{\epsilon}}\neq {\bm{1}}$; see Lemma \ref{lemEeps}.
\begin{lem}\label{lemrootDelta} Let $\ell$ be a prime number not dividing $q(q-1)$. Let $\ell^n$ be the largest power of $\ell$ such that there exists an $\ell^n$-th root of $\Delta^{\bm{\epsilon}}$ in $\mathcal{O}(\Omega)^\times$. Then $\ell^n$ divides $\nu({\bm{\epsilon}})$. \end{lem} \begin{proof}The following argument is essentially due to Gekeler; cf. \cite[Cor. 3.5]{Discriminant}. Let $f\in \mathcal{O}(\Omega)^\times$ be such that $f^{\ell^n}=\Delta^{\bm{\epsilon}}$. Since $r(f)\in \mathcal{H}(\mathscr{T}, \mathbb{Z})$ is $\Gamma_\infty$-invariant, it has Fourier expansion. Moreover, the Fourier coefficients are in $\mathbb{Z}[p^{-1}]$. On the other hand, $$ r(f)^0(1)=\ell^{-n}r(\Delta^{\bm{\epsilon}})^0(1)=\frac{\nu({\bm{\epsilon}})(1-q)}{\ell^n}(E^{\bm{\epsilon}})^0(1) =\frac{q N(\mathfrak n, {\bm{\epsilon}})(1-q)}{\ell^n}, $$ $$ r(f)^\ast(1)=\ell^{-n}r(\Delta^{\bm{\epsilon}})^\ast(1)=\frac{\nu({\bm{\epsilon}})(1-q)}{\ell^n}(E^{\bm{\epsilon}})^\ast(1) = \frac{(1-q)^2(1+q)}{q\ell^n}. $$ Thus, $\ell^n$ divides $N(\mathfrak n, {\bm{\epsilon}})$ and $(q+1)$; thus $\ell^n$ divides $\nu({\bm{\epsilon}})$; cf. Remark \ref{remNnu}. \end{proof}
The cusps of $X_0(\mathfrak n)$ are in natural bijection with the cusps of $\Gamma_0(\mathfrak n)\setminus \mathscr{T}$, and this bijection is compatible with the action of Hecke operators and Atkin-Lehner involutions; cf. \cite[(2.6)]{GR}. We will use the notation introduced in Lemma \ref{lemCusps} for the cusps of $X_0(\mathfrak n)$. Define a cuspidal divisor $D^{{\bm{\epsilon}}}$ on $X_0(\mathfrak n)$ by $$
D^{\bm{\epsilon}}=\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d [\mathfrak d]. $$ Note that $$
\deg(D^{\bm{\epsilon}})=\sum_{\mathfrak d|\mathfrak n}\epsilon_\mathfrak d = \prod_{i=1}^n(1+\varepsilon_i). $$ Hence $\deg(D^{\bm{\epsilon}})=0$ if ${\bm{\epsilon}}\neq {\bm{1}}$, and we can consider the class of $D^{\bm{\epsilon}}$ in $\mathcal{C}(\mathfrak n)$, which by abuse of notation we denote by the same symbol. Let $\langle D^{\bm{\epsilon}} \rangle$ be the finite cyclic subgroup generated by $D^{\bm{\epsilon}}$ in $\mathcal{C}(\mathfrak n)$.
\begin{prop}\label{thmCDG} Assume ${\bm{\epsilon}}=(\varepsilon_1, \dots, \varepsilon_s)\neq {\bm{1}}$. \begin{enumerate}
\item We have $W_{\mathfrak p_i}(D^{\bm{\epsilon}})=\varepsilon_i D^{\bm{\epsilon}}$, $1\leq i\leq s$.
\item Let $\ell$ be a prime number not dividing $q(q-1)$. Let $N$ be the order of $D^{\bm{\epsilon}}$ in $\mathcal{C}(\mathfrak n)$. Then $\mathrm{ord}_\ell(N)\geq \mathrm{ord}_\ell(N(\mathfrak n, {\bm{\epsilon}})/\nu({\bm{\epsilon}}))$. \end{enumerate} \end{prop} \begin{proof} Using Lemma \ref{lemCusps}, we compute $$
W_{\mathfrak p_i}(D^{\bm{\epsilon}})=W_{\mathfrak p_i}\sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i| \mathfrak d} \epsilon_\mathfrak d [\mathfrak d] +
W_{\mathfrak p_i}\sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i\nmid \mathfrak d} \epsilon_\mathfrak d [\mathfrak d]
= \sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i| \mathfrak d} \epsilon_\mathfrak d [\mathfrak d/\mathfrak p_i] + \sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i\nmid \mathfrak d} \epsilon_\mathfrak d [\mathfrak d \mathfrak p_i] $$ $$
= \varepsilon_i\sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i| \mathfrak d} \epsilon_{\mathfrak d/\mathfrak p_i} [\mathfrak d/\mathfrak p_i] + \varepsilon_i\sum_{\mathfrak d|\mathfrak n,\ \mathfrak p_i\nmid \mathfrak d} \epsilon_{\mathfrak d \mathfrak p_i} [\mathfrak d \mathfrak p_i] =\varepsilon_i D^{\bm{\epsilon}}, $$ which proves part (1).
By formulas (3.10) and (3.11) in \cite{Discriminant}, for divisors $\mathfrak m$ and $\mathfrak d$ of $\mathfrak n$ we have $$
\mathrm{ord}_{[\mathfrak m]}\Delta_\mathfrak d = \mathrm{ord}_{W_\mathfrak d [\mathfrak m]}\Delta = \frac{|\mathfrak n|\cdot |(\mathfrak m, \mathfrak d)^2|}{|\mathfrak m|\cdot |\mathfrak d|}. $$ Therefore $$
\mathrm{ord}_{[\mathfrak m]}\Delta^{\bm{\epsilon}} = \frac{|\mathfrak n|}{|\mathfrak m|}\sum_{\mathfrak d|\mathfrak n} \epsilon_\mathfrak d \frac{(\mathfrak m, \mathfrak d)^2}{|\mathfrak d|}
= \frac{|\mathfrak n|}{|\mathfrak m|}\prod_{\mathfrak p_i|\mathfrak m}(1+\varepsilon_i|\mathfrak p_i|)\prod_{\mathfrak p_i\nmid\mathfrak m}(1+\varepsilon_i|\mathfrak p_i|^{-1}) $$ $$
= \prod_{\mathfrak p_i|\mathfrak m}(1+\varepsilon_i|\mathfrak p_i|)\prod_{\mathfrak p_i\nmid\mathfrak m}(|\mathfrak p_i|+\varepsilon_i)=\epsilon_{\mathfrak n/\mathfrak m}N(\mathfrak n, {\bm{\epsilon}})=\epsilon_\mathfrak m\epsilon_\mathfrak n N(\mathfrak n, {\bm{\epsilon}}). $$ This implies $$
\mathrm{div}(\Delta^{\bm{\epsilon}})=\sum_{\mathfrak m|\mathfrak n}(\mathrm{ord}_{[\mathfrak m]}\Delta^{\bm{\epsilon}})[\mathfrak m]= \epsilon_\mathfrak n N(\mathfrak n, {\bm{\epsilon}}) \sum_{\mathfrak m|\mathfrak n}\epsilon_\mathfrak m[\mathfrak m] = \epsilon_\mathfrak n N(\mathfrak n, {\bm{\epsilon}}) D^{\bm{\epsilon}}. $$ It is easy to see from this equality that if $\mathrm{ord}_\ell(N)< \mathrm{ord}_\ell(N(\mathfrak n, {\bm{\epsilon}})/\nu({\bm{\epsilon}}))$, then $\Delta^{\bm{\epsilon}}$ has an $\ell^n$-th root in $\mathcal{O}(\Omega)^\times$ with $n>\mathrm{ord}_\ell(\nu({\bm{\epsilon}}))$. But this would contradict Lemma \ref{lemrootDelta}. \end{proof}
\begin{thm}\label{thmLast} Let $\ell$ be a prime number not dividing $q(q-1)$. Then $$ \mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}=\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}\cong \begin{cases} \mathbb{Z}_\ell\big/\frac{N(\mathfrak n, {\bm{\epsilon}})}{\nu({\bm{\epsilon}})}\mathbb{Z}_\ell, & \text{if ${\bm{\epsilon}}\neq {\bm{1}}$};\\ 0, & \text{if ${\bm{\epsilon}}= {\bm{1}}$}. \end{cases} $$ Moreover, if ${\bm{\epsilon}}\neq {\bm{1}}$, then $D^{\bm{\epsilon}}$ generates $\mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}}$. \end{thm} \begin{proof} As is explained in \cite[$\S$7.1]{PW1}, the canonical specialization of $\mathcal{T}(\mathfrak n)$ into the component group $\Phi_\infty$ at $\infty$ of the N\'eron model of $J_0(\mathfrak n)$ induces an injective homomorphism $$ \mathcal{T}(\mathfrak n)_\ell\hookrightarrow \mathcal{E}_{00}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell) $$ for any $n\geq \mathrm{ord}_\ell(\# \Phi_\infty)$. Moreover, as follows from the discussion in \cite[$\S$2.6]{PW2}, the above homomorphism is compatible with the action of $\mathbb{W}$. Therefore, we have \begin{equation}\label{eq-inclusions} \langle D^{\bm{\epsilon}} \rangle_\ell \subseteq \mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}}\subseteq \mathcal{T}(\mathfrak n)_\ell^{\bm{\epsilon}} \subseteq \mathcal{E}_{00}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell)^{\bm{\epsilon}}\subseteq \mathcal{E}_{0}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell)^{\bm{\epsilon}}. \end{equation}
By Theorem \ref{thm 3.6FT}, if ${\bm{\epsilon}}={\bm{1}}$ then $\mathcal{E}_{0}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell)^{\bm{\epsilon}}=0$, and the claim of the theorem follows. Next, assume ${\bm{\epsilon}}\neq {\bm{1}}$. By Theorem \ref{thm 3.6FT}, $\mathcal{E}_{0}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell)^{\bm{\epsilon}} \subseteq \mathbb{Z}_\ell\big/\frac{N(\mathfrak n, {\bm{\epsilon}})}{\nu({\bm{\epsilon}})}\mathbb{Z}_\ell$. On the other hand, by Proposition \ref{thmCDG}, we have $\mathbb{Z}_\ell\big/\frac{N(\mathfrak n, {\bm{\epsilon}})}{\nu({\bm{\epsilon}})}\mathbb{Z}_\ell\subseteq \langle D^{\bm{\epsilon}}\rangle_\ell$. Comparing the orders of the groups, one immediately deduces that the inclusions in (\ref{eq-inclusions}) are equalities, which implies the claim of the theorem in this case.
\end{proof}
\begin{rem}
The Jacobian variety $J_0(\mathfrak n)$ has bad reduction at $\infty$ and at the primes dividing $\mathfrak n$. A crucial role in the previous proof plays the fact that the canonical specialization $\wp_\infty: \mathcal{T}(\mathfrak n)_\ell\to \Phi_\infty$ is injective if $\ell\nmid (q-1)$. One can also consider the specializations $\wp_{\mathfrak p_i}: \mathcal{T}(\mathfrak n)\to \Phi_{\mathfrak p_i}$ into the component groups of $J_0(\mathfrak n)$ at finite places. The behaviour of these maps is somewhat different from $\wp_\infty$. Without loss of generality, we consider $\wp_{\mathfrak p_1}$. The reduction $X_0(\mathfrak n)_{\mathbb{F}_{\mathfrak p_1}}$ consists of two irreducible components $Z$ and $Z'$, both isomorphic to $X_0(\mathfrak n/\mathfrak p_1)_{\mathbb{F}_{\mathfrak p_1}}$, intersecting transversally at certain points. The Atkin-Lehner involution $W_{\mathfrak p_1}$ interchanges $Z$ and $Z'$. The reductions of the cusps lie in the smooth locus, and no two cusps reduce to the same point. Moreover, the reductions of all $[\mathfrak m]$, with $\mathfrak p_1\nmid \mathfrak m$, lie on one component, say $Z$, and the reductions
of $[\mathfrak m]$, with $\mathfrak p_1|\mathfrak m$, lie on the other component $Z'$. This implies that $$ \wp(D^{\bm{\epsilon}}) = \prod_{i=2}^s(1+\varepsilon_i)Z+\varepsilon_1\prod_{i=2}^s(1+\varepsilon_i)Z'=(1-\varepsilon_1)\prod_{i=2}^s(1+\varepsilon_i)(Z-Z'). $$ Let $z:=Z-Z'$. We conclude that $\wp(D^{\bm{\epsilon}})=2^s z$ if ${\bm{\epsilon}}=(-1, 1,\dots, 1)$, and $\wp(D^{\bm{\epsilon}})=0$ otherwise. By the argument in the proof of Lemma 8.3 in \cite{PW2}, if $\ell\nmid (q+1)$, then $$ (\Phi_{\mathfrak p_1})_\ell = \langle z\rangle_\ell. $$ Moreover, by \cite[Thm. 5.3]{PW1}, if $\ell\nmid (q^2-1)$, then $$ (\Phi_{\mathfrak p_1})_\ell \cong \mathbb{Z}_\ell/N(\mathfrak n, {\bm{\epsilon}})\mathbb{Z}_\ell, \text{ where ${\bm{\epsilon}}=(-1, 1,\dots, 1)$}. $$ Overall, we see that for $\ell\nmid (q^2-1)$, the map $\wp_{\mathfrak p_1}:\mathcal{C}(\mathfrak n)_\ell^{\bm{\epsilon}} \to (\Phi_{\mathfrak p_1})_\ell$ is an isomorphism if ${\bm{\epsilon}} =(-1, 1,\dots, 1)$, and is $0$, otherwise. \end{rem}
\begin{comment} \begin{rem} In this paper we did not explicitly define the Hecke algebra or its Eisenstein ideal. Nevertheless, the Eisenstein ideal is in the background of our main results. To make this more explicit, let $\ell$ be a prime not dividing $q(q^2-1)$. Let $\mathbb{T}_\ell(\mathfrak n)$ be the commutative subalgebra of $\mathrm{End}_{\mathbb{Z}_\ell}(\mathcal{H}_0(\mathfrak n, \mathbb{Z}_\ell))$ generated by the Atkin-Lehner involutions $\mathbb{W}$ and the Hecke operators $T_\mathfrak m$ with $\mathfrak m$ coprime to $\mathfrak n$. The \textit{Eisenstein ideal} $\mathfrak E_\ell(\mathfrak n)\lhd \mathbb{T}_\ell(\mathfrak n)$
is the ideal generated by $\{T_\mathfrak p-(|\mathfrak p|+1)\ |\ \mathfrak p\lhd A \text{ prime}, \mathfrak p\nmid \mathfrak n\}$.
One can deduce from Theorem \ref{thmLast} and its proof that $$ \mathbb{T}_\ell(\mathfrak n)^{\bm{\epsilon}}/\mathfrak E_\ell(\mathfrak n)^{\bm{\epsilon}}\cong \begin{cases} \mathbb{Z}_\ell/{N(\mathfrak n, {\bm{\epsilon}})}\mathbb{Z}_\ell & \text{if ${\bm{\epsilon}}\neq {\bm{1}}$},\\ 0 & \text{if ${\bm{\epsilon}}= {\bm{1}}$} \end{cases} $$ cf. \cite[Cor. 3.18]{PW1}.
It is easy to see that for any ${\bm{\epsilon}}\in \mathbb{E}$ the quotient $\mathbb{T}_\ell(\mathfrak n)^{\bm{\epsilon}}/\mathfrak E_\ell(\mathfrak n)^{\bm{\epsilon}}$ is a finite cyclic group $\mathbb{Z}_\ell/M({\bm{\epsilon}})\mathbb{Z}_\ell$, $M({\bm{\epsilon}})\geq 1$. Combining the argument in the proof of Theorem 3.17 in \cite{Analytical} with Proposition \ref{prop-cor2.4}, one can show that there is a perfect pairing $\mathbb{T}_\ell(\mathfrak n)^{\bm{\epsilon}}\times \mathcal{H}_{00}(\mathfrak n, \mathbb{Z}_\ell)^{\bm{\epsilon}}\to \mathbb{Z}_\ell$. Note that the proof of Theorem \ref{thmLast} implies that $\mathcal{E}_{00}(\mathfrak n, \mathbb{Z}_\ell/\ell^n\mathbb{Z}_\ell)^{\bm{\epsilon}}\cong \mathbb{Z}_\ell/N(\mathfrak n, {\bm{\epsilon}})\mathbb{Z}_\ell$ if ${\bm{\epsilon}}\neq {\bm{1}}$, and is $0$ otherwise. Now one can use the argument in the proof of Corollary 3.18 in \cite{PW1} to deduce that $\mathrm{ord}_\ell M({\bm{\epsilon}})\leq \mathrm{ord}_\ell N(\mathfrak n, {\bm{\epsilon}})$ if ${\bm{\epsilon}}\neq {\bm{1}}$, and $\mathrm{ord}_\ell M({\bm{\epsilon}})=0$ if ${\bm{\epsilon}}= {\bm{1}}$. On the other hand, since $\mathfrak E_\ell(\mathfrak n)^{\bm{\epsilon}}$ annihilates $\mathcal{C}_\ell^{\bm{\epsilon}}$, Theorem \ref{thmLast} implies that $\mathrm{ord}_\ell N(\mathfrak n, {\bm{\epsilon}})\leq \mathrm{ord}_\ell M({\bm{\epsilon}})$. Overall, we get $$ \mathbb{T}_\ell(\mathfrak n)^{\bm{\epsilon}}/\mathfrak E_\ell(\mathfrak n)^{\bm{\epsilon}}\cong \begin{cases} \mathbb{Z}_\ell/{N(\mathfrak n, {\bm{\epsilon}})}\mathbb{Z}_\ell & \text{if ${\bm{\epsilon}}\neq {\bm{1}}$}\\ 0 & \text{if ${\bm{\epsilon}}= {\bm{1}}$}. \end{cases} $$
\end{rem} \end{comment}
\begin{prop}\label{propCp=0} Assume $\mathfrak n=\mathfrak p_1\cdots \mathfrak p_s$ is square-free. Then the exponent of $\mathcal{C}(\mathfrak n)$ divides $$
\rho(\mathfrak n):= \prod_{i=1}^s\left((|\mathfrak p_i|-1)(|\mathfrak p_i|+1)^{s-1}\right).$$ In particular, $\mathcal{C}(\mathfrak n)_p=0$. \end{prop}
\begin{proof} When $\mathfrak n$ is prime (i.e., $s=1$), this follows from a result of Gekeler; cf. \cite[Cor. 3.23]{Discriminant}. Now suppose $s>1$. Take $\mathfrak p = \mathfrak p_1$ and $\mathfrak n':= \mathfrak n/\mathfrak p$. Let $\pi_\mathfrak p: X_0(\mathfrak n) \rightarrow X_0(\mathfrak n')$ be the canonical projection map, and $\widetilde{\pi}_\mathfrak p:= \pi_\mathfrak p \circ W_\mathfrak p$. For each cusp $[\mathfrak m]_{\mathfrak n'}$ of $X_0(\mathfrak n')$ ($\mathfrak m \mid \mathfrak n'$), we have \begin{align*}
\pi_\mathfrak p^*([\mathfrak m]_{\mathfrak n'}) &= |\mathfrak p|[\mathfrak m]_{\mathfrak n} + [\mathfrak m\mathfrak p]_{\mathfrak n} \in \mathrm{Div}(X_0(\mathfrak n)) \\
\widetilde{\pi}_\mathfrak p^*([\mathfrak m]_{\mathfrak n'}) &= |\mathfrak p|[\mathfrak m\mathfrak p]_{\mathfrak n} + [\mathfrak m]_{\mathfrak n} \in \mathrm{Div}(X_0(\mathfrak n)). \end{align*} Thus \begin{align*}
|\mathfrak p| \cdot \pi_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'}) - \widetilde{\pi}_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'})
&= (|\mathfrak p|^2-1)([\mathfrak m_1]_\mathfrak n - [\mathfrak m_2]_\mathfrak n),\\
|\mathfrak p| \cdot \widetilde{\pi}_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'}) - \pi_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'})
&= (|\mathfrak p|^2-1)([\mathfrak m_1\mathfrak p]_\mathfrak n - [\mathfrak m_2\mathfrak p]_\mathfrak n), \end{align*} and $$\pi_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'}) - \widetilde{\pi}_\mathfrak p^*([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'})
= (|\mathfrak p|-1)\big(([\mathfrak m_1]_\mathfrak n - [\mathfrak m_1 \mathfrak p]_\mathfrak n)-([\mathfrak m_2]_\mathfrak n-[\mathfrak m_2\mathfrak p]_\mathfrak n)\big).$$ By induction hypothesis we have $\rho(\mathfrak n')\cdot \big([\mathfrak m_1]_{\mathfrak n'}-[\mathfrak m_2]_{\mathfrak n'}\big) = 0 \in \mathcal{C}(\mathfrak n')$ for $\mathfrak m_1,\mathfrak m_2 \mid \mathfrak n'$. Therefore
$$\rho(\mathfrak n')(|\mathfrak p|^2-1)([\mathfrak m_1]_\mathfrak n-[\mathfrak m_2]_\mathfrak n) = \rho(\mathfrak n')(|\mathfrak p|^2-1)([\mathfrak m_1\mathfrak p]_\mathfrak n-[\mathfrak m_2\mathfrak p]_\mathfrak n) = 0 \in \mathcal{C}(\mathfrak n)$$ and
$$\rho(\mathfrak n')(|\mathfrak p|-1)\Big(([\mathfrak m_1]_\mathfrak n - [\mathfrak m_1 \mathfrak p]_\mathfrak n)-([\mathfrak m_2]_\mathfrak n-[\mathfrak m_2\mathfrak p]_\mathfrak n)\Big) = 0 \in \mathcal{C}(\mathfrak n)$$ for every $\mathfrak m_1,\mathfrak m_2 \mid \mathfrak n'$. Moreover, we have
$$\text{div}\left(\frac{\Delta}{\Delta_\mathfrak p}\right) = (|\mathfrak p|-1)\sum_{\mathfrak m \mid \mathfrak n'} \frac{|\mathfrak n'|}{|\mathfrak m|}([\mathfrak m]_\mathfrak n - [\mathfrak m\mathfrak p]_\mathfrak n) \in \mathrm{Div}^0(X_0(\mathfrak n)).$$ Thus for every $\mathfrak m_1 \mid \mathfrak n'$, \begin{eqnarray} \rho(\mathfrak n') \cdot \text{div}\left(\frac{\Delta}{\Delta_\mathfrak p}\right)
&=& \rho(\mathfrak n')(|\mathfrak p|-1)\left(\sum_{\mathfrak m \mid \mathfrak n'} \frac{|\mathfrak n'|}{|\mathfrak m|}\right) \cdot ([\mathfrak m_1]_\mathfrak n-[\mathfrak m_1\mathfrak p]_\mathfrak n) \nonumber\\
&=& \rho(\mathfrak n')(|\mathfrak p|-1)\left(\prod_{\mathfrak p' \mid \mathfrak n'}(|\mathfrak p'|+1)\right) \cdot ([\mathfrak m_1]_\mathfrak n-[\mathfrak m_1\mathfrak p]_\mathfrak n). \nonumber \end{eqnarray} Hence we conclude that for every $\mathfrak m_1,\mathfrak m_2$ dividing $\mathfrak n$,
$$\left(\rho(\mathfrak n')\cdot(|\mathfrak p|^2-1)\prod_{\mathfrak p' \mid \mathfrak n'}(|\mathfrak p'|+1)\right) \cdot ([\mathfrak m_1]_\mathfrak n - [\mathfrak m_2]_\mathfrak n) = \rho(\mathfrak n)\cdot ([\mathfrak m_1]_\mathfrak n - [\mathfrak m_2]_\mathfrak n) = 0 \in \mathcal{C}(\mathfrak n).$$ \end{proof}
\begin{comment} \begin{rem} If $\mathfrak n$ is divisible by a square of a prime $\mathfrak p^2$, but $\mathfrak n\neq \mathfrak p^2$, then $\mathcal{C}(\mathfrak n)$ always has an element of order divisible by $p$. Moreover, this element can be chosen to be rational over $F$. Thus, $\mathcal{C}(\mathfrak n)(F)_p\neq 0$; see \cite[Cor. 8.6]{PW2}. (When $\mathfrak n$ is not square-free, not every element of $\mathcal{C}(\mathfrak n)$ is rational over $F$.) If $\mathfrak n=\mathfrak p^2$, then $\mathcal{C}(\mathfrak p^2)(F)_p=0$, but $\mathcal{C}(\mathfrak p^2)(\mathbb{C}_\infty)_p$ is not necessarily trivial; see Lemma 8.7 and Example 8.8 in \cite{PW2}. \end{rem} \end{comment}
\renewcommand{\normalsize}{\normalsize}
\end{document} |
\begin{document}
\title{Spectral gap global solutions for degenerate Kirchhoff equations} \begin{abstract}
We consider the second order Cauchy problem
$$u''+\m{u}Au=0,
\hspace{2em}
u(0)=u_{0},\quad
u'(0)=u_{1},$$
where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and
$A$ is a self-adjoint nonnegative operator with dense domain on a
Hilbert space.
It is well known that this problem admits local-in-time solutions
provided that $u_{0}$ and $u_{1}$ are regular enough, depending on
the continuity modulus of $m$, and on the strict/weak
hyperbolicity of the equation.
We prove that for such initial data $(u_{0},u_{1})$ there exist
two pairs of initial data $(\overline{u}_{0},\overline{u}_{1})$,
$(\widehat{u}_{0},\widehat{u}_{1})$ for which the solution is
global, and such that $u_{0}=\overline{u}_{0}+\widehat{u}_{0}$,
$u_{1}=\overline{u}_{1}+\widehat{u}_{1}$.
This is a byproduct of a global existence result for initial data
with a suitable spectral gap, which extends previous results
obtained in the strictly hyperbolic case with a smooth
nonlinearity $m$.
\noindent{\bf Mathematics Subject Classification 2000 (MSC2000):} 35L70, 35L80, 35L90.
\noindent{\bf Key words:} uniqueness, integro-differential hyperbolic equation, degenerate hyperbolic equation, continuity modulus, Kirchhoff equations, Gevrey spaces. \end{abstract}
\section{Introduction}
Let $H$ be a real Hilbert space. For every $x$ and $y$ in $H$, let
$|x|$ denote the norm of $x$, and let $\langle x,y\rangle$ denote the scalar product of $x$ and $y$. Let $A$ be an unbounded linear operator on $H$ with dense domain $D(A)$. We always assume that $A$ is self-adjoint and nonnegative, so that for every $\alpha\geq 0$ the power $A^{\alpha}$ is defined in a suitable domain $D(A^{\alpha})$.
Given a continuous function $m:[0,+\infty)\to[0,+\infty)$ we consider the Cauchy problem \begin{equation}
u''(t)+\m{u(t)}Au(t)=0,
\hspace{2em}\forall t\in[0,T),
\label{pbm:h-eq} \end{equation} \begin{equation}
u(0)=u_0,\hspace{3em}u'(0)=u_1.
\label{pbm:h-data} \end{equation}
It is well known that (\ref{pbm:h-eq}), (\ref{pbm:h-data}) is the abstract setting of the Cauchy-boundary value problem for the quasilinear hyperbolic integro-differential partial differential equation \begin{equation}
u_{tt}(t,x)-
m{\left(\int_{\Omega}\left|\nabla u(t,x)\right|^2\,dx\right)}
\Delta u(t,x)=0
\hspace{2em}
\forall(x,t)\in\Omega\times[0,T),
\label{eq:k} \end{equation} where $\Omega\subseteq{\mathbb{R}}^{n}$ is an open set, and $\nabla u$ and $\Delta u$ denote the gradient and the Laplacian of $u$ with respect to the space variables.
Equation (\ref{pbm:h-eq}) is called strictly hyperbolic if \begin{equation}
m(\sigma)\geq\nu>0
\quad\quad
\forall\sigma\geq 0.
\label{hp:s-h} \end{equation}
Equation (\ref{pbm:h-eq}) is called weakly (or degenerate) hyperbolic if $$m(\sigma)\geq 0 \quad\quad \forall\sigma\geq 0.$$
Existence of local/global solutions to (\ref{pbm:h-eq}), (\ref{pbm:h-data}) has long been investigated in the last century. The theory is well established in the case of local solutions, which are known to exist in the following situations. \begin{enumerate}
\item[(L1)] When equation (\ref{pbm:h-eq}) is strictly hyperbolic,
$m$ is Lipschitz continuous, and initial data $(u_{0},u_{1})\in
D(A^{3/4})\times D(A^{1/4})$ (see \cite{ap} and the references
quoted therein).
\item[(L2)] When equation (\ref{pbm:h-eq}) is weakly hyperbolic,
$m$ is continuous, and initial data are analytic. In this case
solutions are actually global (see \cite{as}, \cite{das-an-1},
\cite{das-an-2}).
\item[(L3)] More generally, when initial data belong to suitable
intermediate spaces, depending on the continuity modulus of $m$,
and on the strict/weak hyperbolicity of (\ref{pbm:h-eq}) (see
\cite{hirosawa-main} and \cite{gg:k-derloss}). This is a sort of
interpolation between (L1) and (L2). We refer to
section~\ref{sec:prelim} for precise definitions of the functional
spaces in the abstract framework and a formal local existence
statement (Theorem~\ref{thm:hirosawa}). \end{enumerate}
Existence of global solutions is a much more difficult problem, and it is still widely open. A positive answer has been given in the case (L2), and in some special situations: quasi-analytic initial data (see \cite{nishihara}), or Sobolev-type data but special nonlinearities $m$ (see \cite{poho-m}), or dispersive operators and small data (see \cite{gh}, \cite{das}). But for (L2) all these results assume the strict hyperbolicity and the Lipschitz continuity of $m$.
Recently \textsc{R.\ Manfrin}~\cite{manfrin1,manfrin2} (see also~\cite{hirosawa2}) considered once again the strictly hyperbolic case with a smooth nonlinearity. He proved global existence in a special class of nonanalytic initial data. Manfrin's spaces are not vector spaces and do not contain any Gevrey space $\mathcal{G}_{s}$ with $s>1$. However they have the following astonishing property: \begin{enumerate}
\item[(M)] every pair of initial conditions $(u_{0},u_{1})\in
D(A)\times D(A^{1/2})$ is the sum of two pairs of initial
conditions in Manfrin's spaces, i.e., the sum of two initial
conditions for which the solution is global! \end{enumerate}
This theory requires the strict hyperbolicity and some smoothness of $m$, which is assumed to be of class $C^{2}$ both in \cite{manfrin1} and \cite{manfrin2}.
In this paper we extend Manfrin's theory to the general situation of (L3). We consider indeed both the strictly hyperbolic and the weakly hyperbolic case, and a nonlinearity $m$ with a given continuity modulus. In Theorem~\ref{thm:main} we prove global existence for initial data in a suitable subset of the spaces involved in (L3). In analogy with Manfrin's spaces, the definition (\ref{defn:m-space}) of our subset is made in terms of the spectral resolution of initial data. Of course our subset is not a vector space and it doesn't even contain all analytic functions. Nevertheless in Proposition~\ref{prop:sum} we show that this subset satisfies property (M) in the spaces involved in (L3).
From the point of view of property (M) our result extends Manfrin's one also in the framework (L1). In this case we obtain indeed property (M) for initial data in $D(A^{3/4})\times D(A^{1/4})$ and a locally Lipschitz continuous nonlinearity $m$, instead of initial data in $D(A)\times D(A^{1/2})$ and $m\in C^{2}$.
This paper is organized as follows. In section~\ref{sec:prelim} we recall the definition of continuity modulus and Gevrey-type functional spaces, and we state the local existence result for the case (L3). In section~\ref{sec:statements} we introduce our spaces and we state our main results. In section~\ref{sec:proofs} we prove these results.
\setcounter{equation}{0} \section{Preliminaries}\label{sec:prelim}
For the sake of simplicity we assume that $H$ admits a countable complete orthonormal system $\{e_{k}\}_{k\geq 1}$ made by eigenvectors of $A$. We denote the corresponding eigenvalues by $\lambda_{k}^{2}$ (with $\lambda_{k}\geq 0$), so that $Ae_{k}=\lambda_{k}^{2}e_{k}$ for every $k\geq 1$.
Under this assumption we can work with Fourier series. However, any definition or statement of this section can be easily extended to the general setting just by using the spectral decomposition instead of Fourier series. The interested reader is referred to \cite{ap} for further details.
By means of the orthonormal system every $u\in H$ can be written in a unique way in the form $u=\sum_{k=1}^{\infty}u_{k}e_{k}$, where $u_{k}=\langle u,e_{k}\rangle$ are the Fourier components of $u$. With these notations for every $\alpha\geq 0$ we have that $$D(A^{\alpha}):=\left\{u\in H:\sum_{k=1}^{\infty} \lambda_{k}^{4\alpha}u_{k}^{2}<+\infty\right\}.$$
Let now $\varphi:[0,+\infty)\to(0,+\infty)$ be any function. Then for every $\alpha\geq 0$ and $r>0$ one can set \begin{equation}
\trebar{u}_{\varphi,r,\alpha}^{2}:=\sum_{k=1}^{\infty}\lambda_{k}^{4\alpha}
u_{k}^{2} \exp\left(\strut r\varphi(\lambda_{k})\right),
\label{defn:trebar} \end{equation} and then define the spaces $$\mathcal{G}_{\varphi,r,\alpha}(A):= \left\{u\in H:\trebar{u}_{\varphi,r,\alpha}^{2}<+\infty\right\}.$$
These spaces are a generalization of the usual spaces of Sobolev, Gevrey or analytic functions. They are Hilbert spaces with norm
$(|u|^{2}+\trebar{u}_{\varphi,r,\alpha}^{2})^{1/2}$. We also set $$\mathcal{G}_{\varphi,\infty,\alpha}(A):=\bigcap_{r>0}\mathcal{G}_{\varphi,r,\alpha}(A).$$
A \emph{continuity modulus} is a continuous increasing function $\omega:[0,+\infty)\to[0,+\infty)$ such that $\omega(0)=0$, and $\omega(a+b)\leq\omega(a)+\omega(b)$ for every $a\geq 0$ and $b\geq 0$.
The function $m$ is said to be $\omega$-continuous if there exists a constant $L\in{\mathbb{R}}$ such that \begin{equation}
|m(a)-m(b)|\leq
L\,\omega(|a-b|)
\hspace{3em}
\forall a\geq 0,\ \forall b\geq 0.
\label{hp:m-ocont} \end{equation}
The following result sums up the state of the art concerning existence of local solutions. We refer to Theorem~2.1 and Theorem~2.2 in~\cite{hirosawa-main} for the existence part, to \cite{gg:k-derloss} for some counterexamples, and to \cite{gg:k-uniq} for uniqueness issues.
\begin{thmbibl}\label{thm:hirosawa}
Let $\omega$ be a continuity modulus, let
$m:[0,+\infty)\to[0,+\infty)$ be a (locally) $\omega$-continuous\ function, and
let $\varphi:[0,+\infty)\to(0,+\infty)$.
Let us assume that there exists a constant $\Lambda$ such that
\begin{equation}
\sigma
\omega\left(\frac{1}{\sigma}\right)\leq\Lambda\varphi(\sigma)
\quad\quad
\forall\sigma> 0
\label{hp:phi-ndg}
\end{equation}
in the strictly hyperbolic case, and
\begin{equation}
\sigma\leq\Lambda\varphi\left(\frac{\sigma}{
\sqrt{\omega(1/\sigma)}}\right)
\quad\quad
\forall\sigma> 0
\label{hp:phi-dg}
\end{equation}
in the weakly hyperbolic case.
Let
\begin{equation}
(u_{0},u_{1})\in
\mathcal{G}_{\varphi,r_{0},3/4}(A)\times\mathcal{G}_{\varphi,r_{0},1/4}(A)
\label{hp:hiro-data}
\end{equation}
for some $r_{0}>0$.
Then there exists $T>0$, and a nonincreasing function
$r:[0,T]\to(0,r_{0}]$ such that problem (\ref{pbm:h-eq}),
(\ref{pbm:h-data}) admits at least one local solution
\begin{equation}
u\in C^{1}\left([0,T];\mathcal{G}_{\varphi,r(t),1/4}(A)\right)\cap
C^{0}\left([0,T];\mathcal{G}_{\varphi,r(t),3/4}(A)\right).
\label{th:reg-sol} \end{equation} \end{thmbibl}
\setcounter{equation}{0} \section{Main result}\label{sec:statements}
Let $\mathcal{L}$ denote the set of all sequences $\{\rho_{n}\}$ of positive real numbers such that $\rho_{n}\to +\infty$ as $n\to +\infty$. Given $\varphi:[0,+\infty)\to(0,+\infty)$, $\{\rho_{n}\}\in\mathcal{L}$, $\alpha\geq 0$, and $\beta\geq 0$ we set \begin{equation}
\mathcal{GM}_{\varphi,\{\rho_{n}\},\alpha}^{(\beta)}(A):=\left\{ u\in
H:\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}^{4\alpha}u_{k}^{2}
\exp\left(\rho_{n}^{\beta}\varphi(\lambda_{k})\right)\leq\rho_{n}
\quad\forall n\in{\mathbb{N}}\right\},
\label{defn:m-space} \end{equation} and then $$\mathcal{GM}_{\varphi,\alpha}^{(\beta)}(A):= \bigcup_{\{\rho_{n}\}\in\mathcal{L}} \mathcal{GM}_{\varphi,\{\rho_{n}\},\alpha}^{(\beta)}(A).$$
These spaces are a generalization of Manfrin's spaces.
The following global existence result is the main result of this paper.
\begin{thm}\label{thm:main}
Let $\omega$ be a continuity modulus, let
$m:[0,+\infty)\to[0,+\infty)$ be a function satisfying
(\ref{hp:m-ocont}), let $\varphi:[0,+\infty)\to(0,+\infty)$, and
let $\{\rho_{n}\}\in\mathcal{L}$.
Let us assume that
\begin{itemize}
\item in the strictly hyperbolic case (\ref{hp:phi-ndg})
holds true for a suitable $\Lambda$, and
\begin{equation}
(u_{0},u_{1})\in
\mathcal{GM}_{\varphi,\{\rho_{n}\},3/4}^{(2)}(A)\times
\mathcal{GM}_{\varphi,\{\rho_{n}\},1/4}^{(2)}(A),
\label{hp:data-ndg}
\end{equation}
\item in the weakly hyperbolic case (\ref{hp:phi-dg})
holds true for a suitable $\Lambda$, and
\begin{equation}
(u_{0},u_{1})\in
\mathcal{GM}_{\varphi,\{\rho_{n}\},3/4}^{(3)}(A)\times
\mathcal{GM}_{\varphi,\{\rho_{n}\},1/4}^{(3)}(A).
\label{hp:data-dg}
\end{equation}
\end{itemize}
Then problem (\ref{pbm:h-eq}), (\ref{pbm:h-data})
admits at least one global solution $u(t)$ with
\begin{equation}
u\in C^{1}\left([0,+\infty);\mathcal{G}_{\varphi,r,3/4}(A)\right)\cap
C^{0}\left([0,+\infty);\mathcal{G}_{\varphi,r,1/4}(A)\right)
\label{hp:reg-sol}
\end{equation}
for every $r>0$. \end{thm}
We conclude by speculating on these spaces. First of all it is easy to prove that \begin{equation}
\mathcal{GM}_{\varphi,\alpha}^{(\beta)}(A)\subseteq
\mathcal{G}_{\varphi,\infty,\alpha}(A)
\label{eq:inclusion} \end{equation} for every admissible values of the parameters. On one hand this inclusion is ``very strict''. Roughly speaking indeed the inequalities in definition (\ref{defn:m-space}) require that the spectrum of $u$ ``has a big hole after each $\rho_{n}$''. For this heuristic reason we used ``spectral gap solutions'' to denote the solutions produced by Theorem~\ref{thm:main}.
On the other hand inclusion (\ref{eq:inclusion}) is ``not so strict'' in the sense that $$\mathcal{GM}_{\varphi,\alpha}^{(\beta)}(A)+\mathcal{GM}_{\varphi,\alpha}^{(\beta)}(A)= \mathcal{G}_{\varphi,\infty,\alpha}(A)$$ for any admissible values of the parameters. We state this property more precisely in the case of pairs of initial data.
\begin{prop}\label{prop:sum}
Let $\varphi:[0,+\infty)\to(0,+\infty)$, and let
\begin{equation}
(u_{0},u_{1})\in
\mathcal{G}_{\varphi,\infty,3/4}(A)\times\mathcal{G}_{\varphi,\infty,1/4}(A).
\label{hp:prop}
\end{equation}
Then for every $\beta\geq 0$ there exist $\{\overline{\rho}_{n}\}$ and
$\{\widehat{\rho}_{n}\}$ in $\mathcal{L}$, and
\begin{equation}
(\overline{u}_{0},\overline{u}_{1})\in
\mathcal{GM}_{\varphi,\{\overline{\rho}_{n}\},3/4}^{(\beta)}(A)\times
\mathcal{GM}_{\varphi,\{\overline{\rho}_{n}\},1/4}^{(\beta)}(A),
\label{th:prop-1} \end{equation}
\begin{equation}
(\widehat{u}_{0},\widehat{u}_{1})\in
\mathcal{GM}_{\varphi,\{\widehat{\rho}_{n}\},3/4}^{(\beta)}(A)\times
\mathcal{GM}_{\varphi,\{\widehat{\rho}_{n}\},1/4}^{(\beta)}(A),
\label{th:prop-2}
\end{equation}
such that $u_{0}=\overline{u}_{0}+\widehat{u}_{0}$ and
$u_{1}=\overline{u}_{1}+\widehat{u}_{1}$. \end{prop}
\begin{rmk}
\begin{em}
Combining Theorem~\ref{thm:main} and
Proposition~\ref{prop:sum} we obtain the following statement:
every pair of initial conditions satisfying
(\ref{hp:hiro-data}) with $r_{0}=\infty$ is the sum of two
pairs of initial conditions for which the solution is global.
We have thus extended to the general case the astonishing
aspect of Manfrin's result.
The extra requirement that $r_{0}=\infty$ is hardly
surprising. It is indeed a necessary condition for existence
of global solutions even in the theory of linear equations
with nonsmooth time dependent coefficients.
\end{em} \end{rmk}
\begin{rmk}
\begin{em}
The $\omega$-continuity assumption on $m$ can be easily
relaxed to local $\omega$-continuity in all the cases where
there is a uniform-in-time estimate of $|A^{1/2}u(t)|$ in
terms of the initial data. We refer to the paragraph ``Energy
conservation'' in section~\ref{sec:proof-prelim} for further
details.
\end{em} \end{rmk}
\begin{rmk}\label{rmk:reg-sol}
\begin{em}
It is possible to extend the result of Theorem~\ref{thm:main}
to larger spaces. A careful inspection of the proof reveals
that in the strictly hyperbolic case one can replace $\beta=2$
with any $\beta>1$, in the weakly hyperbolic case one can
replace $\beta=3$ with any $\beta>2$. It should also be
possible to enlarge these spaces in order to contain all
analytic functions, for which a global solution was already
known to exist.
Our choice (\ref{defn:m-space}) is optimized in order to
obtain both Theorem~\ref{thm:main} and
Proposition~\ref{prop:sum} under the more general assumptions
on $m$, and with a simple proof.
\end{em} \end{rmk}
\setcounter{equation}{0} \section{Proofs}\label{sec:proofs}
\subsection{Preliminaries}\label{sec:proof-prelim}
\paragraph{Estimates for a continuity modulus}
The following estimates are crucial in the proof of our main result (see also Lemma~3.1 in \cite{gg:k-uniq}).
\begin{lemma}
Let $\omega:[0,+\infty)\to[0,+\infty)$ be a continuity modulus.
Then
\begin{eqnarray}
& \omega(\lambda x)\leq(1+\lambda)\omega(x)
\quad\quad\forall\lambda\geq 0,\ \forall x\geq 0; &
\label{th:omega-lambda} \\
\noalign{
}
& \displaystyle{\omega(x)\geq\omega(1)\frac{x}{x+1}}
\quad\quad \forall x\geq 0; &
\label{th:omega-est} \\
\noalign{
}
& \displaystyle{1+\frac{1}{\omega(x)}\leq\left(1+\frac{1}{\omega(1)}\right)
\left(1+\frac{1}{x}\right)}
\quad\quad\forall x>0.&
\label{th:omega-3}
\end{eqnarray} \end{lemma}
{\sc Proof.}\ Inequality (\ref{th:omega-lambda}) can be easily proved by induction on the integer part of $\lambda$ using the monotonicity and the subadditivity of $\omega$. Inequality (\ref{th:omega-est}) follows from (\ref{th:omega-lambda}) applied with $\lambda=1/x$. Inequality (\ref{th:omega-3}) follows from (\ref{th:omega-est}). {\penalty 10000\mbox{$\quad\Box$}}
\paragraph{Energy conservation}
Let $u$ be any solution of (\ref{pbm:h-eq}) defined in an interval $[0,T)$. Let us set $$M(\sigma):=\int_{0}^{\sigma}m(s)\,ds \quad\quad \forall\sigma\geq 0,$$ and let us consider the usual Hamiltonian
$$\mathcal{H}(t):=|u'(t)|^{2}+M(|A^{1/2}u(t)|^{2}).$$
It is well known that $\mathcal{H}(t)$ is constant. In particular \begin{equation}
|u'(t)|^{2}\leq\mathcal{H}(0)
\quad\quad
\forall t\in[0,T).
\label{est:u'} \end{equation}
In the strictly hyperbolic case we have also that $M(\sigma)\geq\nu\sigma$, hence \begin{equation}
|A^{1/2}u(t)|^{2}\leq\frac{\mathcal{H}(0)}{\nu}
\quad\quad
\forall t\in[0,T).
\label{est:au} \end{equation}
This provides an estimate of $|A^{1/2}u(t)|$ in terms of the initial conditions. This type of estimate can be obtained also without the strict hyperbolicity provided that the limit of $M(\sigma)$ as $\sigma\to +\infty$ is $+\infty$ or at least larger than $\mathcal{H}(0)$.
\paragraph{Convolutions}
In the next result we recall the properties of convolutions which are needed in the sequel (we omit the standard proof).
\begin{lemma}\label{lemma:conv}
Let $\rho:{\mathbb{R}}\to [0,+\infty)$ be a function of class $C^{\infty}$,
with support contained in $[-1,1]$, and integral equal to 1.
Let $a>0$, and let $f:[0,a]\to{\mathbb{R}}$ be a continuous function. Let
us extend $f(x)$ to the whole real line by setting $f(x)=f(0)$ for
every $x\leq 0$, and $f(x)=f(a)$ for every $x\geq a$.
For every $\varepsilon>0$ let us set
$$f_{\varepsilon}(x):=\int_{{\mathbb{R}}}^{}f(x+\varepsilon
s)\rho(s)\,ds \quad\quad
\forall x\in{\mathbb{R}}.$$
Then $f_{\varepsilon}(x)$ has the following properties.
\begin{enumerate}
\renewcommand{(\arabic{enumi})}{(\arabic{enumi})}
\item $f_{\varepsilon}\in C^{\infty}({\mathbb{R}})$.
\item If $\mu_{1}\leq f(x)\leq\mu_{2}$ for every $x\in[0,a]$,
then $\mu_{1}\leq f_{\varepsilon}(x)\leq\mu_{2}$ for every
$x\in{\mathbb{R}}$ and every $\varepsilon>0$.
\item $|f_{\varepsilon}(0)|\leq\max\{|f(x)|:0\leq x\leq\varepsilon\}$ for
every $\varepsilon> 0$.
\item Let $\omega$ be a continuity modulus. Let us assume that
\begin{equation}
|f(x)-f(y)|\leq H\omega(|x-y|) \quad\quad \forall x\in[0,a],\
\forall y\in[0,a],
\label{hp:o-cont}
\end{equation}
for some $H\geq 0$. Then there exists a constant
$\gamma_{0}$ (independent on $\varepsilon$, $H$, and on the function
$f(t)$) such that
$$|f_{\varepsilon}(x)-f(x)|\leq\gamma_{0}H\omega(\varepsilon)
\quad\quad
\forall x\in{\mathbb{R}},\ \forall\varepsilon>0,$$
$$|f_{\varepsilon}'(x)|\leq\gamma_{0}H\,\displaystyle{\frac{\omega(\varepsilon)}{\varepsilon}}
\quad\quad
\forall x\in{\mathbb{R}},\ \forall\varepsilon>0.$$
\end{enumerate} \end{lemma}
\paragraph{Maximal local solutions}
By (\ref{eq:inclusion}) assumptions (\ref{hp:data-ndg}) and (\ref{hp:data-dg}) imply that $(u_{0},u_{1})\in\mathcal{G}_{\varphi,\infty,3/4}(A)\times \mathcal{G}_{\varphi,\infty,1/4}(A)$. Therefore the existence of a local solution to (\ref{pbm:h-eq}), (\ref{pbm:h-data}) follows from Theorem~\ref{thm:hirosawa} both in the strictly hyperbolic and in the weakly hyperbolic case. Since initial data satisfy (\ref{hp:hiro-data}) for every $r_{0}$, from the linear theory it easily follows that the local solution satisfies (\ref{th:reg-sol}) for every $r(t)$.
By a standard argument any local solution can be continued to a solution defined in a maximal interval $[0,T)$. If $T=+\infty$ there is nothing to prove. In order to exclude that $T<+\infty$ we prove that the time derivative of $|A^{1/2}u(t)|^{2}$ cannot blow-up in a finite time. The proof of this a priori estimate, which is the basic tool in all global existence results, is different in the strictly hyperbolic and in the weakly hyperbolic case.
\subsection{The strictly hyperbolic case}
Let us introduce some constants. From the strict hyperbolicity (\ref{hp:s-h}) and estimate (\ref{est:au}) we have that $$\nu\leq\m{u(t)}\leq \max\left\{m(\sigma):0\leq\sigma\leq \frac{\mathcal{H}(0)}{\nu}\right\}=:\mu \quad\quad \forall t\geq 0.$$
Let $L$, $\Lambda$, $\gamma_{0}$ be the constants appearing in (\ref{hp:m-ocont}), (\ref{hp:phi-ndg}), and in Lemma~\ref{lemma:conv}, and let $$\gamma_{1}:=\max\{1,\mu\}\cdot\max\left\{1,\nu^{-1}\right\},$$
$$H_{1}:=\max\left\{\left|\langle A^{3/4}u_{0},A^{1/4}u_{1}
\rangle\right|+1,\left(1+\nu^{-1}\right) \mathcal{H}(0)+2\gamma_{1}+1\right\},$$ $$\gamma_{2}:=\gamma_{0}L\Lambda(2H_{1}+1)\left( \frac{1}{\nu}+\frac{1}{\sqrt{\nu}}\right).$$
Since $\rho_{n}\to +\infty$ we can choose $n\in{\mathbb{N}}$ such that \begin{equation}
\rho_{n}\geq\max\{\gamma_{2}T,1\}.
\label{defn:rhon} \end{equation}
Let us set
$$S:=\sup\left\{\tau\leq T:\left|\langle A^{3/4}u(t),A^{1/4}u'(t)
\rangle\right|\leq H_{1}\rho_{n}\;\;\forall t\in[0,\tau]\right\}.$$
We remark that $S>0$ because
$\left|\langle A^{3/4}u_{0},A^{1/4}u_{1}
\rangle\right|<H_{1}\leq H_{1}\rho_{n}$.
Now we distinguish the case $S=T$ and $S<T$.
\subparagraph{\textmd{\emph{Case}} $S=T$}
The argument is quite standard. In the interval $[0,T)$ the function $u(t)$ is the solution of the linear problem \begin{equation}
v''(t)+c(t)Av(t)=0
\label{pbm:lin-eq} \end{equation} \begin{equation}
v(0)=u_{0},
\quad\quad
v'(0)=u_{1},
\label{pbm:lin-data} \end{equation} where \begin{equation}
c(t):=\m{u(t)}.
\label{defn:c} \end{equation}
Since $S=T$ in this case we have that \begin{equation}
\left|\frac{\mathrm{d}}{\mathrm{d}t}|A^{1/2}u(t)|^{2}\right|=
2\left|\langle A^{3/4}u(t),A^{1/4}u'(t)
\rangle\right|\leq 2H_{1}\rho_{n}
\label{eq:c'} \end{equation}
for every $t\in[0,T)$. It follows that $|A^{1/2}u(t)|^{2}$ is Lipschitz continuous in $[0,T)$, hence $c(t)$ can be extended to an $\omega$-continuous function defined in the closed interval $[0,T]$. By the linear theory (see \cite{dgcs} and \cite{hirosawa-main}) problem (\ref{pbm:lin-eq}), (\ref{pbm:lin-data}) has a solution $$v\in C^{0}\left([0,T];\mathcal{G}_{\varphi,r,3/4}(A)\right)\cap C^{1}\left([0,T];\mathcal{G}_{\varphi,r,1/4}(A)\right)$$ for every $r>0$. Since the solution of the linear problem is unique, this implies that there exist $$\widehat{u}_{0}:=\lim_{t\to T^{-}}u(t)\in\mathcal{G}_{\varphi,\infty,3/4}(A), \quad\quad \widehat{u}_{1}:=\lim_{t\to T^{-}}u'(t)\in\mathcal{G}_{\varphi,\infty,1/4}(A).$$
Applying Theorem~\ref{thm:hirosawa} with initial data $(\widehat{u}_{0},\widehat{u}_{1})$ one can therefore continue $u(t)$ on an interval $[0,T_{1})$ with $T_{1}>T$, which contradicts the maximality of $T$.
\subparagraph{\textmd{\emph{Case}} $S<T$}
By the maximality of $S$ we have that necessarily \begin{equation}
\left|\langle A^{3/4}u(S),A^{1/4}u'(S) \rangle\right|=
H_{1}\rho_{n}.
\label{eq:S-nec} \end{equation}
Let us consider the function $c(t)$ defined according to (\ref{defn:c}). In this case (\ref{eq:c'}) holds true for every $t\in[0,S]$, hence by (\ref{hp:m-ocont}) and (\ref{th:omega-lambda}) we have that \begin{eqnarray*}
\left|c(t)-c(s)\right| & = & \left|\m{u(t)}-\m{u(s)}\right| \\
& \leq & L\,\omega\left(\left|
|A^{1/2}u(t)|^{2}-|A^{1/2}u(s)|^{2}\right|\right) \\
& \leq & L\,\omega(2H_{1}\rho_{n}|t-s|) \\
& \leq & L(2H_{1}\rho_{n}+1)\,\omega(|t-s|) \\
& \leq & L(2H_{1}+1)\rho_{n}\,\omega(|t-s|) \end{eqnarray*} for every $t$ and $s$ in $[0,S]$. Let us extend $c(t)$ outside the interval $[0,S]$ as in Lemma~\ref{lemma:conv}, and let us set \begin{equation}
c_{\ep}(t):=\int_{{\mathbb{R}}}^{}c(t+\varepsilon s)\rho(s)\,ds \quad\quad \forall
t\in{\mathbb{R}}.
\label{defn:cep} \end{equation}
Since estimate (\ref{hp:o-cont}) holds true with $H:=L(2H_{1}+1)\rho_{n}$, from statements (2) and (4) of Lemma~\ref{lemma:conv} we deduce that \begin{equation}
\nu\leqc_{\ep}(t)\leq\mu
\quad\quad
\forall t\in{\mathbb{R}},\ \forall\varepsilon>0,
\label{est:cep-nu-mu} \end{equation} \begin{equation}
|c_{\ep}(t)-c(t)|\leq\gamma_{0}L(2H_{1}+1)\rho_{n}\omega(\varepsilon)
\quad\quad
\forall t\in{\mathbb{R}},\ \forall\varepsilon>0,
\label{est:cep-c} \end{equation} \begin{equation}
|c_{\ep}'(t)|\leq\gamma_{0}L(2H_{1}+1)\rho_{n}\,
\displaystyle{\frac{\omega(\varepsilon)}{\varepsilon}}
\quad\quad
\forall t\in{\mathbb{R}},\ \forall\varepsilon>0.
\label{est:cep'} \end{equation}
Let us consider the Fourier components $u_{k}(t)$ of $u(t)$, and let us set \begin{equation}
E_{k,\varepsilon}(t):=|u_{k}'(t)|^{2}+\lambda_{k}^{2}c_{\ep}(t)|u_{k}(t)|^{2}.
\label{defn:ekep} \end{equation}
An easy computation shows that \begin{eqnarray*}
E_{k,\varepsilon}'(t) & = & c_{\ep}'(t)\lambda_{k}^{2}|u_{k}(t)|^{2}+
2\lambda_{k}^{2}(c_{\ep}(t)-c(t))u_{k}(t)u_{k}'(t)\\
& \leq & \frac{|c_{\ep}'(t)|}{c_{\ep}(t)}c_{\ep}(t)\lambda_{k}^{2}|u_{k}(t)|^{2}+
\lambda_{k}\frac{|c_{\ep}(t)-c(t)|}{\sqrt{c_{\ep}(t)}}2|u_{k}'(t)|\cdot
\lambda_{k}\sqrt{c_{\ep}(t)}|u_{k}(t)|\\
& \leq & \frac{|c_{\ep}'(t)|}{c_{\ep}(t)}E_{k,\varepsilon}(t)+
\lambda_{k}\frac{|c_{\ep}(t)-c(t)|}{\sqrt{c_{\ep}(t)}}E_{k,\varepsilon}(t), \end{eqnarray*} hence by (\ref{est:cep-nu-mu}), (\ref{est:cep-c}), and (\ref{est:cep'}) we obtain that \begin{equation}
E_{k,\varepsilon}'(t)\leq \gamma_{0}L(2H_{1}+1)\rho_{n}\left(
\frac{1}{\nu}\frac{\omega(\varepsilon)}{\varepsilon}+
\frac{1}{\sqrt{\nu}}\lambda_{k}\omega(\varepsilon)\right)E_{k,\varepsilon}(t)
\quad\quad\forall t\in[0,S].
\label{est:ekep} \end{equation}
Let us consider now the eigenvalues $\lambda_{k}>\rho_{n}$, which are clearly positive, and let us set $\varepsilon_{k}:=\lambda_{k}^{-1}$. By (\ref{hp:phi-ndg}) we have that $$\frac{\omega(\varepsilon_{k})}{\varepsilon_{k}}=\lambda_{k}\omega(\varepsilon_{k})= \lambda_{k}\omega\left(\frac{1}{\lambda_{k}}\right)\leq\Lambda\varphi(\lambda_{k}).$$
Using these estimates in (\ref{est:ekep}) we obtain that $$E_{k,\varepsilon_{k}}'(t)\leq \gamma_{0}L(2H_{1}+1)\rho_{n}\left( \frac{1}{\nu}+\frac{1}{\sqrt{\nu}}\right) \Lambda\varphi(\lambda_{k}) E_{k,\varepsilon_{k}}(t)= \gamma_{2}\rho_{n}\varphi(\lambda_{k})E_{k,\varepsilon_{k}}(t).$$
Integrating this differential inequality and using (\ref{defn:rhon}) we find that $$E_{k,\varepsilon_{k}}(t)\leq E_{k,\varepsilon_{k}}(0) \exp\left(\gamma_{2}\rho_{n}\varphi(\lambda_{k})T\right)\leq E_{k,\varepsilon_{k}}(0) \exp\left(\rho_{n}^{2}\varphi(\lambda_{k})\right)$$ for every $t\in[0,S]$. Thanks to (\ref{est:cep-nu-mu}) we obtain that \begin{eqnarray*}
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2} & \leq &
\max\left\{1,\nu^{-1}\right\}E_{k,\varepsilon_{k}}(t)\\
& \leq & \max\left\{1,\nu^{-1}\right\}
\left(|u_{1k}|^{2}+\lambda_{k}^{2} c_{\varepsilon_{k}}(0)|u_{0k}|^{2}\right)
\exp\left(\rho_{n}^{2}\varphi(\lambda_{k})\right)\\
& \leq & \max\left\{1,\nu^{-1}\right\}\cdot\max\{1,\mu\}
\left(|u_{1k}|^{2}+\lambda_{k}^{2} |u_{0k}|^{2}\right)
\exp\left(\rho_{n}^{2}\varphi(\lambda_{k})\right)\\
& = & \gamma_{1}\left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right)
\exp\left(\rho_{n}^{2}\varphi(\lambda_{k})\right), \end{eqnarray*} where $u_{0k}$ and $u_{1k}$ denote the Fourier components of $u_{0}$ and $u_{1}$, respectively.
By assumption (\ref{hp:data-ndg}) we have therefore that $$\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}\left(
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2}\right)\leq \gamma_{1}
\sum_{\lambda_{k}>\rho_{n}}\lambda_{k} \left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right) \exp\left(\rho_{n}^{2}\varphi(\lambda_{k})\right)\leq 2\gamma_{1}\rho_{n}$$ for every $t\in[0,S]$. On the other hand, by (\ref{est:u'}) and (\ref{est:au}) we have that \begin{eqnarray*}
\sum_{\lambda_{k}\leq\rho_{n}}\lambda_{k}\left(
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2}\right) & \leq &
\rho_{n}\sum_{\lambda_{k}\leq\rho_{n}}\left(
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2}\right)\\
& \leq & \rho_{n}\left(|u'(t)|^{2}+|A^{1/2}u(t)|^{2}\right) \\
& \leq & \rho_{n}\left(\mathcal{H}(0)+
\frac{\mathcal{H}(0)}{\nu}\right) \end{eqnarray*} for every $t\in[0,S]$. In particular for $t=S$ we have that \begin{eqnarray*}
\lefteqn{\hspace{-2em}\left|\langle A^{3/4}u(S),A^{1/4}u'(S) \rangle\right| \leq
|A^{3/4}u(S)|^{2}+|A^{1/4}u'(S)|^{2}} \\
\noalign{
}
\hspace{2em} & = & \sum_{\lambda_{k}\leq\rho_{n}}\lambda_{k}\left(
|u_{k}'(S)|^{2}+\lambda_{k}^{2}|u_{k}(S)|^{2}\right)+
\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}\left(
|u_{k}'(S)|^{2}+\lambda_{k}^{2}|u_{k}(S)|^{2}\right) \\
& \leq & \rho_{n}\left(\mathcal{H}(0)+
\frac{\mathcal{H}(0)}{\nu}+2\gamma_{1}\right) \\
& < & H_{1}\rho_{n}. \end{eqnarray*}
This contradicts (\ref{eq:S-nec}).
\subsection{The weakly hyperbolic case}
Let us introduce some constants. Let $L$, $\Lambda$, $\gamma_{0}$ be the constants appearing in (\ref{hp:m-ocont}), (\ref{hp:phi-dg}), and in Lemma~\ref{lemma:conv}, and let $$\gamma_{3}:=1+\frac{1}{\omega(1)},$$ $$\gamma_{4}:=\max\left\{\m{u(t)}:t\in[0,T/2]\right\}+ \max\left\{\omega(\sigma):0\leq \sigma\sqrt{\omega(\sigma)}\leq 1\right\},$$ $$\gamma_{5}:=\gamma_{3}(1+\gamma_{4})(\Lambda+1)$$
$$H_{2}:=\max\left\{\left|\langle A^{3/4}u_{0},A^{1/4}u_{1}
\rangle\right|+1,(|u_{0}|+1)\sqrt{\mathcal{H}(0)}+ \gamma_{5}+1)\right\},$$ $$\gamma_{6}:=1+\gamma_{0}L(2H_{2}+1).$$
Since $\rho_{n}\to +\infty$ we can choose $n\in{\mathbb{N}}$ such that $\rho_{n}\geq 1$, and \begin{equation}
\rho_{n}^{1/2}\geq T\sqrt{\mathcal{H}(0)},
\hspace{3em}
\rho_{n}^{1/2}\geq 4\gamma_{6}\Lambda T,
\hspace{3em}
\rho_{n}\geq\frac{2}{T\sqrt{\omega(T/2)}}.
\label{defn:rhon-w} \end{equation}
Let us set
$$S:=\sup\left\{\tau\leq T:\left|\langle A^{3/4}u(t),A^{1/4}u'(t)
\rangle\right|\leq H_{2}\rho_{n}^{5/2}\ \ \forall t\in[0,\tau]\right\}.$$
We remark that $S>0$ because $\left|\langle A^{3/4}u_{0},A^{1/4}u_{1} \rangle\right|<H_{2}\leq H_{2}\rho_{n}^{5/2}$.
If $S=T$ we can conclude as in the strictly hyperbolic case (using the linear theory for the weakly hyperbolic case, for which we refer to \cite{cjs}). So let us assume that $S<T$. By the maximality of $S$ we have that necessarily \begin{equation}
\left|\langle A^{3/4}u(S),A^{1/4}u'(S) \rangle\right|=
H_{2}\rho_{n}^{5/2}.
\label{eq:S-nec-w} \end{equation}
Let us consider the function $c(t)$ defined according to (\ref{defn:c}), let us extend it outside the interval $[0,S]$ as in Lemma~\ref{lemma:conv}, and let us set $$c_{\ep}(t):=\omega(\varepsilon)+\int_{{\mathbb{R}}}^{}c(t+\varepsilon s)\rho(s)\,ds \quad\quad\forall t\in{\mathbb{R}}.$$
Arguing as in the strictly hyperbolic case we find that
$$\left|c(t)-c(s)\right| \leq L(2H_{2}+1)\rho_{n}^{5/2}\omega(|t-s|)$$ for every $t$ and $s$ in $[0,S]$. Therefore from statement (4) of Lemma~\ref{lemma:conv} we deduce that \begin{equation}
|c_{\ep}(t)-c(t)|\leq\left(1+\gamma_{0}L(2H_{2}+1)
\rho_{n}^{5/2}\right)\omega(\varepsilon)=
\gamma_{6}\rho_{n}^{5/2}\,\omega(\varepsilon),
\label{est:cep-c-w} \end{equation} \begin{equation}
|c_{\ep}'(t)|\leq\gamma_{0}L(2H_{2}+1)\rho_{n}^{5/2}
\displaystyle{\frac{\omega(\varepsilon)}{\varepsilon}}\leq
\gamma_{6}\rho_{n}^{5/2}\,
\displaystyle{\frac{\omega(\varepsilon)}{\varepsilon}}.
\label{est:cep'-w} \end{equation}
Let us consider the Fourier components $u_{k}(t)$ of $u(t)$, and let us define $E_{k,\varepsilon}(t)$ as in (\ref{defn:ekep}). Computing the time derivative as in the strictly hyperbolic case, and using (\ref{est:cep-c-w}), (\ref{est:cep'-w}), and the fact that $c_{\ep}(t)\geq\omega(\varepsilon)$ we find that $$E_{k,\varepsilon}'(t)\leq\gamma_{6}\rho_{n}^{5/2}\left( \frac{1}{\varepsilon}+\lambda_{k}\sqrt{\omega(\varepsilon)}\right)E_{k,\varepsilon}(t) \quad\quad\forall t\in[0,S].$$
Now we choose $\varepsilon$ as a function of $k$. The function $h(\sigma)=\sigma\sqrt{\omega(\sigma)}$ is invertible. Let us consider the eigenvalues $\lambda_{k}>\rho_{n}$, which are clearly positive, and let us set $\varepsilon_{k}:=h^{-1}(1/\lambda_{k})$. By (\ref{hp:phi-dg}) we have that \begin{equation}
\lambda_{k}\sqrt{\omega(\varepsilon_{k})}=\frac{1}{\varepsilon_{k}}\leq\Lambda
\varphi\left(\frac{1}{h(\varepsilon_{k})}\right)=
\Lambda\varphi(\lambda_{k}),
\label{est:Lambda} \end{equation} hence $$E_{k,\varepsilon_{k}}'(t)\leq 2\gamma_{6}\rho_{n}^{5/2}\Lambda \varphi(\lambda_{k})E_{k,\varepsilon_{k}}(t).$$
Integrating this differential inequality, and exploiting the second condition in (\ref{defn:rhon-w}) we thus obtain that $$E_{k,\varepsilon_{k}}(t)\leq E_{k,\varepsilon_{k}}(0) \exp\left(2\rho_{n}^{5/2}\gamma_{6}\Lambda \varphi(\lambda_{k})T\right)\leq E_{k,\varepsilon_{k}}(0) \exp\left(\frac{1}{2}\rho_{n}^{3}\varphi(\lambda_{k})\right)$$ for every $t\in[0,S]$. In order to estimate $E_{k,\varepsilon_{k}}(0)$ we need an estimate on $c_{\varepsilon_{k}}(0)$. To this end we first observe that $h(\varepsilon_{k})=1/\lambda_{k}<1$, hence \begin{equation}
\omega(\varepsilon_{k})\leq\max\{\omega(\sigma):0\leq h(\sigma)\leq 1\}.
\label{est:cep0-1} \end{equation}
Moreover the last condition in (\ref{defn:rhon-w}) is equivalent to $1/\rho_{n}\leq h(T/2)$. Therefore from the monotonicity of $h$ it follows that $$\varepsilon_{k}=h^{-1}\left(\frac{1}{\lambda_{k}}\right)\leq h^{-1}\left(\frac{1}{\rho_{n}}\right)\leq h^{-1}\left(h \left(\frac{T}{2}\right)\right)=\frac{T}{2},$$ hence from statement (3) of Lemma~\ref{lemma:conv} we deduce that \begin{equation}
\int_{{\mathbb{R}}}c(\varepsilon_{k}s)\rho(s)\,ds\leq
\max\{c(t):0\leq t\leq\varepsilon_{k}\}\leq
\max\{c(t):0\leq t\leq T/2\}.
\label{est:cep0-2} \end{equation}
From (\ref{est:cep0-1}) and (\ref{est:cep0-2}) it follows that $c_{\varepsilon_{k}}(0)\leq\gamma_{4}$, hence $$E_{k,\varepsilon_{k}}(0)\leq\max\left\{1,c_{\ep}(0)\right\}
\left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right)\leq
(1+\gamma_{4})\left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right).$$
Moreover from (\ref{th:omega-3}) and (\ref{est:Lambda}) it follows that $$\max\left\{1,\frac{1}{\omega(\varepsilon_{k})}\right\}\leq 1+\frac{1}{\omega(\varepsilon_{k})}\leq \gamma_{3}\left(1+\frac{1}{\varepsilon_{k}}\right)\leq \gamma_{3}(1+\Lambda\varphi(\lambda_{k})).$$
Since $(1+\Lambda x)\leq(\Lambda +1)e^{x/2}$ for every $\Lambda\geq 0$ and every $x\geq 0$, we have in particular that $$\max\left\{1,\frac{1}{\omega(\varepsilon_{k})}\right\}\leq \gamma_{3}(1+\Lambda\varphi(\lambda_{k}))\leq \gamma_{3}(1+\Lambda)\exp\left(\frac{1}{2}\varphi(\lambda_{k})\right)\leq$$ $$\leq\gamma_{3}(1+\Lambda)\exp\left(\frac{1}{2}\rho_{n}^{3} \varphi(\lambda_{k})\right).$$
From all these estimates it follows that \begin{eqnarray*}
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2} & \leq &
\max\left\{1,\frac{1}{\omega(\varepsilon_{k})}\right\}E_{k,\varepsilon_{k}}(t)\\
& \leq & \gamma_{3}(1+\Lambda)E_{k,\varepsilon_{k}}(0)
\exp\left(\rho_{n}^{3}\varphi(\lambda_{k})\right)\\
& \leq & \gamma_{3}(1+\Lambda)(1+\gamma_{4})
\left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right)
\exp\left(\rho_{n}^{3}\varphi(\lambda_{k})\right) \\
& = & \gamma_{5} \left(|u_{1k}|^{2}+\lambda_{k}^{2}|u_{0k}|^{2}\right)
\exp\left(\rho_{n}^{3}\varphi(\lambda_{k})\right). \end{eqnarray*}
By assumption (\ref{hp:data-dg}) we have therefore that $$\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}\left(
|u_{k}'(t)|^{2}+\lambda_{k}^{2}|u_{k}(t)|^{2}\right)\leq \gamma_{5}
\sum_{\lambda_{k}>\rho_{n}}\lambda_{k} \left(|u_{1k}|^{2}+
\lambda_{k}^{2}|u_{0k}|^{2}\right) \exp\left(\rho_{n}^{3}\varphi(\lambda_{k})\right)\leq 2\gamma_{5}\rho_{n}$$ for every $t\in[0,S]$, and in particular \begin{eqnarray*}
\left|\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}^{2}u_{k}'(S)\cdot u_{k}(S)\right|
& \leq & \sum_{\lambda_{k}>\rho_{n}}\lambda_{k}^{2}|u_{k}'(S)|\cdot|u_{k}(S)|
\\
& \leq & \frac{1}{2} \sum_{\lambda_{k}>\rho_{n}}\left(
\lambda_{k}|u_{k}'(S)|^{2}+\lambda_{k}^{3}|u_{k}(S)|^{2}\right) \\
& \leq & \gamma_{5}\rho_{n}. \end{eqnarray*}
On the other hand, by (\ref{est:u'}) and the first condition in (\ref{defn:rhon-w}) we have that
$$|u(t)|\leq|u_{0}|+S\cdot\max\{|u'(t)|:t\in[0,S]\}
\leq|u_{0}|+T\cdot\sqrt{\mathcal{H}(0)}
\leq \left(|u_{0}|+1 \right)\rho_{n}^{1/2}$$ for every $t\in[0,S]$, hence
$$\left|\sum_{\lambda_{k}\leq\rho_{n}}\lambda_{k}^{2}u_{k}'(t) u_{k}(t)\right|
\leq\rho_{n}^{2}\left|\langle u(t),u'(t)\rangle\right|\leq
\rho_{n}^{2}|u(t)|\cdot|u'(t)| \leq \rho_{n}^{5/2}\left(|u_{0}|+1 \right)\sqrt{\mathcal{H}(0)}$$ for every $t\in[0,S]$. In particular for $t=S$ we have that \begin{eqnarray*}
\left|\langle A^{3/4}u(S),A^{1/4}u'(S) \rangle\right| & \leq &
\left|\sum_{\lambda_{k}\leq\rho_{n}}\lambda_{k}^{2}u_{k}'(S)\cdot u_{k}(S)\right|+
\left|\sum_{\lambda_{k}>\rho_{n}}\lambda_{k}^{2}u_{k}'(S)\cdot u_{k}(S)\right| \\
\noalign{
}
& \leq & \rho_{n}^{5/2}\left(|u_{0}|+1
\right)\sqrt{\mathcal{H}(0)}+\gamma_{5}\rho_{n} \\
\noalign{
}
& < & H_{2}\rho_{n}^{5/2}. \end{eqnarray*}
This contradicts (\ref{eq:S-nec-w}).
\subsection{Proof of Proposition~\ref{prop:sum}}
Let us recursively define a sequence $\rho_{n}$ as follows. First of all we set $\rho_{0}=0$. Let us assume that a term $\rho_{n}$ has been defined. Assumption (\ref{hp:prop}) implies in particular that $$ (u_{0},u_{1})\in \mathcal{G}_{\varphi,r,3/4}(A)\times\mathcal{G}_{\varphi,r,1/4}(A)$$ with $r=\rho_{n}^{\beta}$, hence $$\sum_{k=1}^{\infty}u_{0k}^{2}\lambda_{k}^{3}\exp\left( \rho_{n}^{\beta}\varphi(\lambda_{k})\right)<+\infty, \hspace{3em} \sum_{k=1}^{\infty}u_{1k}^{2}\lambda_{k}\exp\left( \rho_{n}^{\beta}\varphi(\lambda_{k})\right)<+\infty.$$
We can therefore choose $\rho_{n+1}$ big enough in such a way that $\rho_{n+1}\geq\rho_{n}+1$, and $$\sum_{\lambda_{k}\geq\rho_{n+1}}u_{0k}^{2}\lambda_{k}^{3}\exp\left( \rho_{n}^{\beta}\varphi(\lambda_{k})\right)\leq\rho_{n}, \hspace{3em} \sum_{\lambda_{k}\geq\rho_{n+1}}^{\infty}u_{1k}^{2}\lambda_{k}\exp\left( \rho_{n}^{\beta}\varphi(\lambda_{k})\right)\leq\rho_{n}.$$
Let $\overline{u}_{0}$ and $\overline{u}_{1}$ be the elements of $H$ whose Fourier components are given by $$\overline{u}_{0k}:=\left\{ \begin{array}{ll}
0 & \mbox{if }\rho_{2k}\leq\lambda_{k}<\rho_{2k+1}, \\
u_{0k} & \mbox{if }\rho_{2k+1}\leq\lambda_{k}<\rho_{2k+2}, \end{array} \right. \hspace{1em} \overline{u}_{1k}:=\left\{\begin{array}{ll}
0 & \mbox{if }\rho_{2k}\leq\lambda_{k}<\rho_{2k+1}, \\
u_{1k} & \mbox{if }\rho_{2k+1}\leq\lambda_{k}<\rho_{2k+2}, \end{array} \right.$$ and let $\overline{\rho}_{n}:=\rho_{2n}$. We claim that (\ref{th:prop-1}) holds true. Indeed for every $n\in{\mathbb{N}}$ we have that \begin{eqnarray*}
\sum_{\lambda_{k}>\overline{\rho}_{n}}^{\infty}
\overline{u}_{0k}^{2}\lambda_{k}^{3}\exp\left(
\overline{\rho}_{n}^{\beta}\varphi(\lambda_{k})\right) & = &
\sum_{\lambda_{k}>\rho_{2n}}^{\infty}\overline{u}_{0k}^{2}\lambda_{k}^{3}
\exp\left( \rho_{2n}^{\beta}\varphi(\lambda_{k})\right)\\
& = & \sum_{\lambda_{k}\geq\rho_{2n+1}}^{\infty}\overline{u}_{0k}^{2}\lambda_{k}^{3}
\exp\left( \rho_{2n}^{\beta}\varphi(\lambda_{k})\right) \\
& \leq & \sum_{\lambda_{k}\geq\rho_{2n+1}}^{\infty}u_{0k}^{2}\lambda_{k}^{3}
\exp\left( \rho_{2n}^{\beta}\varphi(\lambda_{k})\right) \\
\noalign{
}
& \leq & \rho_{2n}=\overline{\rho}_{n}, \end{eqnarray*} and similarly for $\overline{u}_{1}$. Note that in the second equality we exploited the spectral gap of $\overline{u}_{0}$, whose components are equal to zero in the range $(\rho_{2n},\rho_{2n+1})$.
In the same way we can show that $\widehat{u}_{0}:=u_{0}-\overline{u}_{0}$ and $\widehat{u}_{1}:=u_{1}-\overline{u}_{1}$ satisfy (\ref{th:prop-2}) with $\widehat{\rho}_{n}:=\rho_{2n+1}$. {\penalty 10000\mbox{$\quad\Box$}}
\label{NumeroPagine}
\end{document} |
\begin{document}
\title{
\huge
Stochastic analysis of Bernoulli processes
}
\begin{abstract}
These notes survey some aspects of discrete-time chaotic calculus
and its applications, based on the chaos representation property
for i.i.d. sequences of random variables.
The topics covered include the Clark formula and predictable
representation, anticipating calculus, covariance identities and
functional inequalities (such as deviation and logarithmic Sobolev
inequalities), and an application to option hedging in discrete time. \end{abstract} \noindent
Keywords: Malliavin calculus,
Bernoulli processes, discrete time, chaotic calculus, functional inequalities,
option hedging. \\
\noindent
Classification: 60G42, 60G50, 60G51, 60H30, 60H07.
\baselineskip0.7cm
\section{Introduction}
Stochastic analysis can be viewed as an infinite-dimensional
version of classical analysis, developed in relation to stochastic
processes.
In this survey we present a construction of the basic operators
of stochastic analysis (gradient and divergence)
in discrete time for Bernoulli processes.
Our presentation is based on the chaos representation property
and discrete multiple stochastic integrals with respect to
i.i.d. sequences of random variables.
The main applications presented are to functional inequalities
(deviation inequalities, logarithmic Sobolev inequalities)
in discrete settings,
cf. \cite{gaopri}, \cite{hp}, \cite{prisch2}, and to option
pricing and hedging in discrete time mathematical finance.
Other approaches to discrete-time stochastic analysis can be found in Holden et al. \cite{holden1} (1992), \cite{holden2} (1993), Leitz-Martini \cite{leitz} (2000), and also in Attal \cite{attal1} (2003) in the framework of quantum stochastic calculus, see also the recent paper \cite{gzyl}
by H. Gzyl (2005).
This survey can be roughly divided into a first part (Sections~\ref{s2} to
\ref{s8}) in which we present the main basic results
and analytic tools, and a second part (Sections~\ref{devsec} to \ref{hdg})
which is devoted to applications.
We proceed as follows.
In Section~\ref{s2} we consider a family of discrete-time
normal martingales.
The next section is devoted to the construction of
the stochastic integral of predictable square-integrable
processes with respect to such martingales.
In Section~\ref{s3} we construct the associated multiple
stochastic integrals of symmetric functions on ${\mathord{\mathbb N}}^n$,
$n\geq 1$.
Starting with Section~\ref{s3.1} we focus on a particular
class of normal martingales satisfying a structure equation.
The chaos representation property is studied in Section~\ref{s3.2}
in the case of discrete time random walks with independent
increments.
A gradient operator $D$ acting by finite differences is
introduced in Section~\ref{s4} in connection with multiple
stochastic integrals, and used in Section~\ref{s5}
to state a Clark predictable representation formula.
The divergence operator $\delta$, adjoint of $D$, is
presented in Section~\ref{s6} as an extension of the
discrete-time stochastic integral.
It is also used in Section~\ref{s7} to express
the generator of the Ornstein-Uhlenbeck process.
Covariance identities are stated in Section~\ref{s8},
both from the Clark representation formula and by
use of the Ornstein-Uhlenbeck semigroup.
Functional inequalities on Bernoulli space are presented
as an application in Sections~\ref{devsec} and \ref{lsid}.
On the one hand, in Section~\ref{devsec}
we prove several deviation inequalities
for functionals of an infinite number of i.i.d. Bernoulli
random variables.
Then in Section~\ref{lsid} we state different versions of
the logarithmic Sobolev inequality in discrete settings
(modified, $L^1$, sharp) which allow one to control the
entropy of random variables.
In particular we recover and extend some results
of \cite{bobkov}, using the method of \cite{gaopri}.
Our approach is based on the intrinsic tools
(gradient, divergence, Laplacian) of
infinite-dimensional stochastic analysis.
We refer to \cite{bht2}, \cite{bobkovsyk}, \cite{ht2},
\cite{ledouxesaim}, for other versions of logarithmic
Sobolev inequalities in discrete settings,
and to \cite{daipra}, \cite{wuls2} for the Poisson case.
Section~\ref{s11} contains a change of variable formula
in discrete time, which is applied
with the Clark formula in Section~\ref{hdg}
to a derivation of the Black-Scholes formula
in discrete time, i.e. in the Cox-Ross-Rubinstein model, see e.g.
\cite{lamberton}, $\S$15-1 of \cite{williams}, or \cite{ruiz},
for other approaches.
\section{Discrete-Time Normal Martingales}\index{normal martingale!discrete time} \label{s2}
Consider a sequence $(Y_k)_{k\in {\mathord{\mathbb N}}}$
of (not necessarily independent) random
variables on a probability space $(\Omega , {\cal F} , \mathbb{P})$.
Let $({\cal F}_n)_{n\geq -1}$ denote the filtration generated by
$(Y_n)_{n\in {\mathord{\mathbb N}}}$, i.e. $$
{\cal F}_{-1}=\{\emptyset , \Omega \} , $$
and $${\cal F}_n = \sigma (Y_0,\ldots ,Y_n) , \qquad n\geq 0 . $$
Recall that a random variable $F$ is said to be
${\cal F}_n$-measurable if it can be written as a function $$F=f_n (Y_0,\ldots,Y_n)$$
of $Y_0,\ldots , Y_n$, where
$f_n : {\mathord{\mathbb R}}^{n+1} \to {\mathord{\mathbb R}}$.
\begin{assumption}
We make the following assumptions on the sequence
$(Y_n)_{n\in{\mathord{\mathbb N}}}$: \begin{description} \item{a)}
it is conditionally centered: \begin{equation} \label{dse0} \mathbb{E} [Y_n \mid {\cal F}_{n-1}]=0, \qquad n\geq 0, \end{equation} \item{b)} its conditional quadratic variation satisfies: $$\mathbb{E}[Y_n^2 \mid {\cal F}_{n-1} ]=1, \qquad
n\geq 0. $$ \end{description} \end{assumption}
Condition \eqref{dse0} implies that the process
$(Y_0+\cdots +Y_n)_{n\geq 0}$ is an ${\cal F}_n$-martingale.
More precisely, the sequence $(Y_n)_{n\in{\mathord{\mathbb N}}}$ and the process
$(Y_0+\cdots +Y_n)_{n\geq 0}$ can be viewed
respectively as a (correlated) noise and as a normal martingale
in discrete time. \section{Discrete Stochastic Integrals}\index{stochastic integral!discrete time}
In this section we construct the discrete stochastic integral
of predictable square-summable processes with respect to a
discrete-time normal martingale. \begin{definition}
Let $(u_k)_{k\in {\mathord{\mathbb N}}}$ be a uniformly bounded sequence of
random variables with finite support in ${\mathord{\mathbb N}}$, i.e. there
exists $N\geq 0$ such that $u_k=0$ for all $k\geq N$.
The stochastic integral $J(u)$ of
$(u_n)_{n\in{\mathord{\mathbb N}}}$ is defined as $$J( u) = \sum_{k=0}^\infty u_kY_k . $$ \end{definition}
The next proposition states a version of
the It\^o isometry in discrete time.
A sequence $( u_n)_{n\in{\mathord{\mathbb N}}}$ of random variables
is said to be ${\cal F}_n$-predictable\index{predictable process}
if $u_n$ is ${\cal F}_{n-1}$-measurable
for all $n\in {\mathord{\mathbb N}}$, in particular $u_0$ is constant
in this case. \begin{prop} \label{is}
The stochastic integral operator $J(u)$
extends to square-integrable predictable processes
$(u_n)_{n\in{\mathord{\mathbb N}}} \in L^2 (\Omega \times {\mathord{\mathbb N}} )$
via the (conditional) isometry formula\index{It\^o!isometry} \begin{equation} \label{iiiso}
\mathbb{E} [ | J( {\bf 1}_{[n,\infty )} u) |^2 | \mid {\cal F}_{n-1} ]
= \mathbb{E}[\Vert {\bf 1}_{[n,\infty )} u \Vert_{\ell^2({\mathord{\mathbb N}})}^2 \mid {\cal F}_{n-1} ],
\qquad n\in{\mathord{\mathbb N}}. \end{equation} \end{prop} \begin{Proof}
Let $(u_n)_{n\in{\mathord{\mathbb N}}}$ and $(v_n)_{n\in{\mathord{\mathbb N}}}$ be bounded
predictable processes with finite support in ${\mathord{\mathbb N}}$.
The product $u_kY_kv_l$, $0\leq k < l$, is ${\cal F}_{l-1}$-measurable,
and $u_kY_lv_l$ is ${\cal F}_{k-1}$-measurable, $0\leq l < k$.
Hence \begin{eqnarray*} \lefteqn{
\mathbb{E}\left[ \sum_{k=n}^\infty u_kY_k \sum_{l=0}^\infty v_lY_l
\Big| {\cal F}_{n-1} \right]
= \mathbb{E}\left[ \sum_{k,l=n}^\infty u_kY_k v_l Y_l
\Big| {\cal F}_{n-1} \right] } \\ & = & \mathbb{E}\left[
\sum_{k=n}^\infty u_kv_k Y_k^2
+ \sum_{n\leq k<l } u_kY_k v_lY_l
+ \sum_{n\leq l < k} u_kY_k v_lY_l
\Big| {\cal F}_{n-1} \right] \\ & = &
\sum_{k=n}^\infty \mathbb{E}[
\mathbb{E}[u_kv_k Y_k^2 \mid {\cal F}_{k-1} ]
\mid {\cal F}_{n-1} ] + \sum_{n\leq k<l }
\mathbb{E}[
\mathbb{E}[
u_kY_k v_lY_l \mid {\cal F}_{l-1} ]
\mid {\cal F}_{n-1}
] \\
& & +
\sum_{n\leq l < k}
\mathbb{E}[
\mathbb{E}[
u_kY_k v_lY_l \mid {\cal F}_{k-1} ]
\mid {\cal F}_{n-1}
] \\ & = &
\sum_{k=0}^\infty \mathbb{E}[
u_kv_k
\mathbb{E}[ Y_k^2 \mid {\cal F}_{k-1} ]
\mid {\cal F}_{n-1}
]
+
2 \sum_{n\leq k<l }
\mathbb{E}[
u_kY_k v_l
\mathbb{E}[
Y_l \mid {\cal F}_{l-1} ]
\mid {\cal F}_{n-1}
] \\
& = &
\sum_{k=n}^\infty \mathbb{E}[ u_kv_k \mid {\cal F}_{n-1} ] \\
& = &
\mathbb{E}\left[\sum_{k=n}^\infty u_kv_k \Big| {\cal F}_{n-1} \right]. \end{eqnarray*}
This proves the isometry property \eqref{iiiso} for $J$.
The extension to $L^2 (\Omega \times {\mathord{\mathbb N}} )$ follows then
from a Cauchy sequence argument.
Consider a sequence of bounded predictable
processes with finite support converging to $u$ in
$L^2(\Omega \times {\mathord{\mathbb N}} )$, for example the sequence
$(u^n)_{n\in{\mathord{\mathbb N}}}$ defined as $$
u^n
=
( u^n_k )_{k\in{\mathord{\mathbb N}}}
=
( u_k {\bf 1}_{\{ 0 \leq k \leq n\}} {\bf 1}_{\{ |u_k|\leq n \}} )_{k\in{\mathord{\mathbb N}}},
\qquad
n\in{\mathord{\mathbb N}} . $$
Then the sequence $(J(u^n))_{n\in{\mathord{\mathbb N}}}$ is Cauchy and
converges in $L^2(\Omega )$, hence we may define $$
J(u) : = \lim_{k \to \infty} J(u^k ). $$
From the isometry property \eqref{iiiso} applied with $n=0$,
the limit is clearly independent of the choice of the approximating sequence
$(u^k)_{k\in{\mathord{\mathbb N}}}$. \end{Proof}
Note that by bilinearity, \eqref{iiiso} can also be written as $$
\mathbb{E} [ J( {\bf 1}_{[n,\infty )} u) J( {\bf 1}_{[n,\infty )} v ) |
{\cal F}_{n-1} ]
=
\mathbb{E}[ \langle
{\bf 1}_{[n,\infty )} u
,
{\bf 1}_{[n,\infty )} v
\rangle_{\ell^2({\mathord{\mathbb N}})}
\mid {\cal F}_{n-1} ],
\qquad n\in{\mathord{\mathbb N}}, $$
and that for $n=0$ we get \begin{equation} \label{lkoi}
\mathbb{E} [ J( u) J( v ) ]
=
\mathbb{E}[ \langle
u
,
v
\rangle_{\ell^2({\mathord{\mathbb N}})} ], \end{equation}
for all square-integrable predictable processes
$u = (u_k)_{k\in {\mathord{\mathbb N}}}$ and $v = (v_k)_{k\in {\mathord{\mathbb N}}}$. \begin{prop} \label{isomp15}
Let $(u_k)_{k\in{\mathord{\mathbb N}}} \in L^2 (\Omega \times {\mathord{\mathbb N}} )$
be a predictable square-integrable process.
We have $$
\mathbb{E}[J (u ) \mid {\cal F}_k] = J (u{\bf 1}_{[0,k]}), \quad
k\in {\mathord{\mathbb N}} . $$ \end{prop} \begin{Proof}
It is sufficient to note that \begin{eqnarray*}
\mathbb{E}[J (u ) \mid {\cal F}_k] & = &
\mathbb{E}\left[ \sum_{i=0}^k u_iY_i \Big| {\cal F}_k \right]
+
\sum_{i=k+1}^\infty \mathbb{E}\left[ u_iY_i \mid {\cal F}_k \right] \\
& = &
\sum_{i=0}^k
u_iY_i
+
\sum_{i=k+1}^\infty
\mathbb{E}\left[ \mathbb{E}\left[
u_iY_i \mid {\cal F}_{i-1} \right] \mid {\cal F}_k \right] \\
& = &
\sum_{i=0}^k
u_iY_i
+
\sum_{i=k+1}^\infty
\mathbb{E}\left[
u_i
\mathbb{E}\left[
Y_i \mid {\cal F}_{i-1}
\right] \mid {\cal F}_k
\right] \\
& = &
\sum_{i=0}^k
u_iY_i \\
& = &
J (u{\bf 1}_{[0,k]}) . \end{eqnarray*} \end{Proof} \begin{corollary}
The indefinite stochastic integral
$(J (u{\bf 1}_{[0,k]}))_{k\in{\mathord{\mathbb N}}}$ is a discrete time martingale with
respect to $({\cal F}_n)_{n\geq -1}$. \end{corollary} \begin{Proof}
We have \begin{eqnarray*}
\mathbb{E}[J (u {\bf 1}_{[0,k+1]} ) \mid {\cal F}_k]
& = &
\mathbb{E}[ \mathbb{E}[
J (u{\bf 1}_{[0, k + 1 ]}) \mid {\cal F}_{k+1}
\mid {\cal F}_k] \\
& = &
\mathbb{E}[ \mathbb{E}[
J (u ) \mid {\cal F}_{k+1}
\mid {\cal F}_k] \\
& = &
\mathbb{E}[ J (u ) \mid {\cal F}_k] \\
& = &
J (u {\bf 1}_{[0,k]} ) . \end{eqnarray*} \end{Proof} \section{Discrete Multiple Stochastic Integrals}\index{multiple stochastic integrals!discrete time} \label{s3} \noindent
The role of multiple stochastic integrals
in the orthogonal expansions of random variables
is similar to that of polynomials
in the series expansions of functions of a real variable.
In some cases, multiple stochastic integrals can be expressed
using polynomials, for example Krawtchouk polynomials in
the symmetric discrete case with $p_n=q_n=1/2$, $n\in{\mathord{\mathbb N}}$,
see Relation~\eqref{krw} below. \begin{definition}
Let $\ell^2({\mathord{\mathbb N}})^{\circ n}$ denote the subspace
of $\ell^2({\mathord{\mathbb N}})^{\otimes n} = \ell^2 ({\mathord{\mathbb N}}^n)$ made of
functions $f_n$ that are symmetric in $n$ variables, i.e. such that
for every permutation $\sigma$ of $\{1,\ldots , n\}$, $$f_n (k_{\sigma (1)},\ldots ,k_{\sigma (n)})
= f_n (k_1,\ldots ,k_n), \quad k_1,\ldots ,k_n\in{\mathord{\mathbb N}}. $$ \end{definition}
Given $f_1\in l^2 ({\mathord{\mathbb N}})$ we let $$J_1(f_1 ) = J(f_1 ) = \sum_{k=0}^\infty f_1 (k) Y_k . $$
As a convention we identify $\ell^2({\mathord{\mathbb N}}^0)$ to ${\mathord{\mathbb R}}$ and let $J_0(f_0)=f_0$,
$f_0\in {\mathord{\mathbb R}}$.
Let $$
\Delta_{n}=\{(k_1,\ldots ,k_n)\in {\mathord{\mathbb N}}^n \ :
\ k_i\not= k_j, \ 1\leq i < j \leq n \},
\qquad n\geq 1.$$
The following proposition gives the definition of multiple stochastic
integrals by iterated
stochastic integration of predictable processes in the sense of Proposition~\ref{is}. \begin{prop} \label{prop1.18}
The multiple stochastic integral $J_n (f_n)$ of $f_n\in \ell^2({\mathord{\mathbb N}})^{\circ n}$,
$n\geq 1$, is defined as $$ J_n (f_n)
= \sum_{(i_1,\ldots ,i_n)\in \Delta_n}
f_n(i_1,\ldots ,i_n) Y_{i_1}\cdots Y_{i_n} . $$
It satisfies the recurrence relation \begin{equation} \label{recrel2}
J_{n}(f_{n}) =
n\sum_{k=1}^{\infty }
Y_k J_{n-1}(f_{n}( * ,k) {\bf 1}_{[0,k-1]^{n-1}}( * )) \end{equation}
and the isometry formula \begin{equation} \label{isois} \mathbb{E}[J_n(f_n)J_m(g_m)]
= \left\{ \begin{array}{ll}
n! \langle {\bf 1}_{ \Delta_n} f_n , g_m \rangle_{\ell^2({\mathord{\mathbb N}})^{\otimes n}}
& \mbox{if } n=m, \\
0 & \mbox{if } n\not= m. \end{array} \right. \end{equation} \end{prop} \begin{Proof}
Note that we have \begin{eqnarray} \nonumber
J_n (f_n)
& = & n! \sum_{0\leq i_1 < \cdots < i_n}
f_n(i_1,\ldots ,i_n) Y_{i_1}\cdots Y_{i_n} \\ \label{eg1}
& = &
n! \sum_{i_n = 0 ~}^\infty
\sum_{0\leq i_{n-1} < i_n}
\cdots
\sum_{0\leq i_1 < i_2}
f_n(i_1,\ldots ,i_n) Y_{i_1}\cdots Y_{i_n} . \end{eqnarray}
Note that since $0\leq i_1 < i_2 < \cdots < i_n$ and
$0 \leq j_1 < j_2 < \cdots < j_n$ we have $$
\mathbb{E}[Y_{i_1}\cdots Y_{i_n} Y_{j_1}\cdots Y_{j_n}] =
{\bf 1}_{\{ i_1=j_1, \ldots , i_n=j_n\}} . $$
Hence \begin{eqnarray*} \lefteqn{ \mathbb{E}[J_n(f_n)J_n(g_n)] } \\
& = &
(n!)^2
\mathbb{E}\left[ \sum_{0\leq i_1 < \cdots < i_n}
f_n(i_1,\ldots ,i_n) Y_{i_1}\cdots Y_{i_n} \sum_{0\leq j_1 < \cdots < j_n}
g_n(j_1,\ldots ,j_n) Y_{j_1}\cdots Y_{j_n} \right] \\
& = &
(n!)^2
\sum_{0 \leq i_1 < \cdots < i_n, \
0\leq j_1 < \cdots < j_n}
f_n(i_1,\ldots ,i_n)
g_n(j_1,\ldots ,j_n)
\mathbb{E}[Y_{i_1}\cdots Y_{i_n} Y_{j_1}\cdots Y_{j_n}] \\
& = &
(n!)^2 \sum_{0 \leq i_1 < \cdots < i_n}
f_n(i_1,\ldots ,i_n)
g_n(i_1,\ldots ,i_n) \\
& = &
n! \sum_{(i_1,\ldots ,i_n)\in \Delta_n }
f_n(i_1,\ldots ,i_n)
g_n(i_1,\ldots ,i_n) \\
& = &
n! \langle {\bf 1}_{\Delta_n} f_n , g_m \rangle_{\ell^2({\mathord{\mathbb N}})^{\otimes n}} . \end{eqnarray*}
When $n < m$ and $(i_1,\ldots, i_n)\in \Delta_n$ and
$(j_1,\ldots, j_m)\in\Delta_m$ are two
sets of indices, there necessarily exists $k\in \{1,\ldots, m \}$
such that $j_k\notin \{ i_1,\ldots, i_n \}$, hence $$\mathbb{E}[Y_{i_1}\cdots Y_{i_n} Y_{j_1}\cdots Y_{j_m}] = 0 , $$
and this implies the orthogonality of $J_n (f_n)$ and $J_m (g_m)$.
The recurrence relation \eqref{recrel2} is a direct
consequence of \eqref{eg1}.
The isometry property \eqref{isois} of $J_n$ also follows
by induction from \eqref{iiiso} and the recurrence relation. \end{Proof}
If $f_n \in \ell^2({\mathord{\mathbb N}}^n)$ is not symmetric
we let $J_n(f_n)=J_n(\tilde{f}_n)$, where
$\tilde{f}_n$ is the symmetrization of $f_n$, defined as $$\tilde{f}_n (i_1,\ldots ,i_n)
= \frac{1}{n!}
\sum_{\sigma \in \Sigma_n} f(i_{\sigma (1)}, \ldots ,i_{\sigma_n}),
\qquad
i_1,\ldots ,i_n \in {\mathord{\mathbb N}}^n, $$
and $\Sigma_n$ is the set of all permutations of $\{1,\ldots ,n\}$.
In particular,
if $(k_1, \ldots , k_n )\in \Delta_n$,
the symmetrization $\tilde{{\bf 1}}_{\{(k_1,\ldots ,k_n)\}}$
of ${\bf 1}_{\{ (k_1,\ldots ,k_n) \}}$ in $n$ variables is given by $$
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_n)\}}
(i_1,\ldots ,i_n)
= \frac{1}{n!} {\bf 1}_{\{
\{i_1,\ldots ,i_n\}
= \{k_1,\ldots ,k_n\}\}}, \quad
i_1,\ldots , i_n \in {\mathord{\mathbb N}}, $$
and $$
J_n(
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_n)\}})
= Y_{k_1}\cdots Y_{k_n}. $$ \begin{lemma} \label{lea2}
For all $n\geq 1$ we have $$\mathbb{E}[J_n(f_n) \mid {\cal F}_k] = J_n (f_n {\bf 1}_{[0,k]^n}),
\qquad k\in{\mathord{\mathbb N}}, \quad
f_n\in \ell^2 ({\mathord{\mathbb N}} )^{\circ n}. $$ \end{lemma} \begin{Proof}
This lemma can be proved in two ways, either as a consequence of
Proposition~\ref{isomp15} and Proposition~\ref{prop1.18} or via the
following direct argument, noting that for all
$m=0,\ldots ,n$ and $g_m \in \ell^2({\mathord{\mathbb N}} )^{\circ m}$ we have: \begin{eqnarray*}
\mathbb{E}[(J_n(f_n) -J_n (f_n {\bf 1}_{[0,k]^n})) J_m (g_m {\bf 1}_{[0,k]^m}) ]
& = &
{\bf 1}_{\{n=m\}}
n!\langle
f_n(1-{\bf 1}_{[0,k]^n}) , g_m {\bf 1}_{[0,k]^m}\rangle_{\ell^2 ({\mathord{\mathbb N}}^n)} \\
& = & 0 , \end{eqnarray*}
hence $J_n (f_n {\bf 1}_{[0,k]^n})\in L^2(\Omega , {\cal F}_k)$,
and $J_n(f_n) -J_n (f_n {\bf 1}_{[0,k]^n})$ is orthogonal to
$L^2(\Omega , {\cal F}_k)$. \end{Proof}
In other terms we have $$\mathbb{E}[ J_n(f_n)]=0, \qquad
f_n \in \ell^2({\mathord{\mathbb N}})^{\circ n},
\qquad
n\geq 1, $$
the process $(J_n(f_n {\bf 1}_{[0,k]^n}))_{k\in{\mathord{\mathbb N}}}$ is a discrete-time
martingale, and $J_n(f_n)$ is ${\cal F}_k$-measurable
if and only if $f_n {\bf 1}_{[0,k]^n}=f_n$, $0\leq k \leq n$. \section{Discrete structure equations}\index{structure equation!discrete time} \label{s3.1}
Assume now that the sequence $(Y_n)_{n\in{\mathord{\mathbb N}}}$
satisfies the discrete structure equation: \begin{equation} \label{dse} Y_n^{2}=1+\varphi_n Y_n, \qquad n\in {\mathord{\mathbb N}}, \end{equation}
where $(\varphi_n)_{n\in{\mathord{\mathbb N}}}$ is an ${\cal F}_n$-predictable
process.
Condition \eqref{dse0} implies that $$\mathbb{E}[Y_n^2 \mid {\cal F}_{n-1} ]=1, \qquad
n\in {\mathord{\mathbb N}} , $$
hence the hypotheses of the preceding sections are satisfied.
Since \eqref{dse} is a second order equation, there exists an
${\cal F}_n$-adapted process $(X_n)_{n\in {\mathord{\mathbb N}}}$
of Bernoulli $\{-1,1\}$-valued random variables such that \begin{equation} \label{yk} Y_n=\frac{\varphi_n}{2} + X_n \sqrt{1+\left(
\frac{\varphi _n}{2}\right)^2},\qquad n\in{\mathord{\mathbb N}}. \end{equation}
Consider the conditional probabilities \begin{equation} \label{*p13}
p_n = \mathbb{P}(X_n=1 \mid {\cal F}_{n-1} ) \quad \mbox{and} \quad
q_n = \mathbb{P}(X_n=-1 \mid {\cal F}_{n-1} ),
\qquad n\in {\mathord{\mathbb N}}. \end{equation}
From the relation $\mathbb{E}[Y_n \mid {\cal F}_{n-1} ]=0$, rewritten as $$p_n
\left(
\frac{\varphi_n}{2} + \sqrt{1+\left(
\frac{\varphi _n}{2}\right)^2}
\right)
+
q_n
\left(
\frac{\varphi_n}{2} - \sqrt{1+\left(
\frac{\varphi _n}{2}\right)^2}
\right)
= 0,
\qquad
n\in {\mathord{\mathbb N}}, $$
we get \begin{equation} \label{yk01}
p_n = \frac{1}{2}
\left(
1 -{\frac{\varphi _n}{\sqrt{ 4 + \varphi_n^{2}}}}
\right) ,
\qquad
q_n = {\frac{1}{2}} \left(
1 +{\frac{\varphi_n}{\sqrt{ 4 + \varphi_n^{2} }}}
\right) , \end{equation}
and $$
\varphi_n = \sqrt{\frac{q_n}{p_n}} - \sqrt{\frac{p_n}{q_n}}
= \frac{q_n-p_n}{\sqrt{p_nq_n}},
\qquad n\in {\mathord{\mathbb N}}, $$
hence $$
Y_n
= {\bf 1}_{\{X_n=1\}} \sqrt{\frac{q_n}{p_n}}
- {\bf 1}_{\{X_n=-1\}} \sqrt{\frac{p_n}{q_n}},
\qquad n\in {\mathord{\mathbb N}} . $$
Letting $$Z_n = \frac{X_n+1}{2} \in \{ 0,1\}, \qquad n\in{\mathord{\mathbb N}} , $$
we also have the relations \begin{equation} \label{ch1} Y_n = \frac{q_n-p_n+X_n}{2\sqrt{p_nq_n}}
= \frac{Z_n-p_n}{\sqrt{p_nq_n}},
\quad n\in{\mathord{\mathbb N}} , \end{equation}
which yield $${\cal F}_n = \sigma (X_0,\ldots ,X_n)
= \sigma (Z_0,\ldots ,Z_n) , \qquad
n\in{\mathord{\mathbb N}} . $$ \begin{remark} \label{rk01}
In particular, one can take $\Omega = \{-1,1\}^{\mathord{\mathbb N}}$ and
construct the Bernoulli process $(X_n)_{n\in{\mathord{\mathbb N}}}$
as the sequence of canonical projections on
$\Omega = \{-1,1\}^{\mathord{\mathbb N}}$
under a countable product $\mathbb{P}$ of Bernoulli measures on $\{-1,1\}$.
In this case the sequence $(X_n)_{n\in{\mathord{\mathbb N}}}$ can be viewed as the
dyadic expansion of $X(\omega ) \in [0,1]$ defined as: $$ X(\omega ) = \sum_{n=0}^\infty
\frac{1}{2^{n+1}}
X_n (\omega ) . $$
In the symmetric case $p_k=q_k=1/2$, $k\in {\mathord{\mathbb N}}$,
the image measure of $\mathbb{P}$ by the mapping $\omega \mapsto X(\omega )$
is the Lebesgue measure on $[0,1]$,
see \cite{MR1764269} for the non-symmetric case. \end{remark} \section{Chaos representation}\index{chaos representation!discrete time} \label{s3.2}
From now on we assume that the sequence $(p_k)_{k\in{\mathord{\mathbb N}}}$ defined in
\eqref{*p13} is deterministic, which implies that the random variables
$(X_n)_{n\in{\mathord{\mathbb N}}}$ are independent.
Precisely, $X_n$ will be
constructed as the canonical projection
$X_n:\Omega \to \{-1,1\}$ on $\Omega = \{-1,1\}^{{\mathord{\mathbb N}}}$
under the measure $\mathbb{P}$ given on cylinder sets by $$\mathbb{P}(\{\epsilon_0, \ldots, \epsilon_n \} \times \{ -1 , 1 \}^{{\mathord{\mathbb N}}})
= \prod_{k=0}^n
p_k^{(1+\varepsilon_k)/2}
q_k^{(1-\varepsilon_k)/2} ,
\qquad \{ \epsilon_0, \ldots, \epsilon_n \} \in \{-1,1\}^{n+1} . $$
The sequence $(Y_k)_{k\in{\mathord{\mathbb N}}}$ can be constructed as
a family of independent random variables given by $$
Y_n
=
\frac{\varphi_n}{2} + X_n \sqrt{1+\left(
\frac{\varphi _n}{2}\right)^2},\qquad n\in{\mathord{\mathbb N}}, $$
where the sequence $(\varphi_n)_{n\in{\mathord{\mathbb N}}}$ is deterministic.
In this case, all spaces
$L^r (\Omega, {\cal F}_n)$, $r\geq 1$,
have finite dimension $2^{n+1}$, with basis \begin{eqnarray*} \lefteqn{ \left\{ {\bf 1}_{\{
Y_0 = \epsilon_0, \ldots,
Y_n = \epsilon_n
\}}
\ : \ (\epsilon_0,\ldots, \epsilon_n) \in
\prod_{k=0}^n
\left\{
\sqrt{\frac{q_k}{p_k}}
, -\sqrt{\frac{p_k}{q_k}} \right\} \right\} } \\
& = & \left\{ {\bf 1}_{\{
X_0 = \epsilon_0, \ldots,
X_n = \epsilon_n
\}}
\ : \ (\epsilon_0,\ldots, \epsilon_n) \in
\prod_{k=0}^n
\left\{
-1 , 1
\right\} \right\} . \end{eqnarray*}
An orthogonal basis of $L^r (\Omega, {\cal F}_n)$ is given by $$\left\{ Y_{k_1}\cdots Y_{k_l}
= J_l (
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_l)\}})
\ : \ 0\leq k_1 < \cdots < k_l \leq n,
\ l=0,\ldots , n+1
\right\} . $$
Let \begin{equation} \label{defsn}
S_n = \sum_{k=0}^n \frac{1+X_k}{2}
= \sum_{k=0}^n Z_k, \quad n\in{\mathord{\mathbb N}}, \end{equation}
denote the random walk associated to $(X_k)_{k\in {\mathord{\mathbb N}}}$.
If $p_k=p$, $k\in {\mathord{\mathbb N}}$, then \begin{equation} \label{krw} J_n( {\bf 1}_{[0,N]}^{\circ n}) = K_n(S_N;N+1,p) \end{equation}
coincides with the Krawtchouk polynomial $K_n (\cdot ;N+1,p)$ of order $n$
and parameter $(N+1,p)$, evaluated at $S_N$, cf. \cite{prisch2}.
\noindent
Let now ${\cal H}_0={\mathord{\mathbb R}}$ and let
${\cal H}_n$ denote the subspace of $L^2(\Omega )$
made of integrals of order $n\geq 1$, and called chaos
of order $n$: $${\cal H}_n = \{ J_n (f_n) \ : \
f_n \in \ell^2({\mathord{\mathbb N}})^{\circ n}\} . $$
The space of ${\cal F}_n$-measurable random variables is denoted by $L^0 (\Omega , {\cal F}_n)$. \begin{lemma} \label{ll1.1}
For all $n\in {\mathord{\mathbb N}}$ we have \begin{equation} \label{ch}
L^0 (\Omega , {\cal F}_n) \subset {\cal H}_0\oplus \cdots \oplus {\cal H}_{n+1}. \end{equation} \end{lemma} \begin{Proof}
It suffices to note that ${\cal H}_l \cap L^0 (\Omega ,{\cal F}_n)$
has dimension ${n+1\choose l}$,
$1 \leq l\leq n+1$.
More precisely it is generated by the orthonormal basis $$\left\{ Y_{k_1}\cdots Y_{k_l} =
J_l (
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_l)\}})
\ : \ 0\leq k_1 < \cdots < k_l \leq n
\right\} , $$
since any element $F$ of ${\cal H}_l \cap L^0 (\Omega ,{\cal F}_n)$
can be written as $F = J_l (f_l {\bf 1}_{[0,n]^l} )$, hence $$
L^0 (\Omega , {\cal F}_n) = (
{\cal H}_0\oplus \cdots \oplus {\cal H}_{n+1} )
\bigcap L^0 (\Omega , {\cal F}_n) . $$ \end{Proof} \noindent
Alternatively, Lemma~\ref{ll1.1} can be proved by noting that $$J_n (f_n {\bf 1}_{[0,N]^n}) = 0,
\qquad
n>N+1,
\quad
f_n \in \ell^2({\mathord{\mathbb N}})^{\circ n} , $$
and as a consequence, any $F\in L^0 (\Omega , {\cal F}_N)$
can be expressed as $$
F = \mathbb{E}[F]
+
\sum_{n=1}^{N+1}
J_n (f_n {\bf 1}_{[0,N]^n}) . $$ \begin{definition}
Let ${\cal S}$ denote the linear space
spanned by multiple stochastic integrals, i.e. \begin{equation} \label{defs} {\cal S} = {\mathrm{{\rm Vect \ \!}}} \left\{
\bigcup_{n=0}^\infty {\cal H}_n
\right\}
= \left\{
\sum_{k=0}^n
J_k (f_k) \ : \
f_k\in \ell^2({\mathord{\mathbb N}} )^{\circ k}
, \ k=0, \ldots , n,\ n\in{\mathord{\mathbb N}}\right\}. \end{equation} \end{definition}
The completion of ${\cal S}$ in $L^2(\Omega )$
is denoted by the direct sum $$\bigoplus_{n=0}^\infty
{\cal H}_n . $$
The next result is the chaos representation property for
Bernoulli processes, which is analogous to the Walsh decomposition,
cf. \cite{leitz}.
This property is obtained under the assumption that the sequence
$(X_n)_{n\in{\mathord{\mathbb N}}}$ is i.i.d.
See \cite{emerycrp} for other instances of the chaos representation
property without this independence assumption. \begin{prop} \label{chaos}
We have the identity $$L^2(\Omega ) = \bigoplus_{n=0}^\infty
{\cal H}_n.$$ \end{prop} \begin{Proof}
It suffices to show that ${\cal S}$ is dense in $L^2(\Omega )$.
Let $F$ be a bounded random variable.
Relation \eqref{ch} of Lemma~\ref{ll1.1} shows
that $\mathbb{E}[F \mid {\cal F}_n] \in {\cal S}$.
The martingale convergence theorem,
cf. e.g. Theorem~27.1 in \cite{jacodprotterbk},
implies that $(\mathbb{E}[F \mid {\cal F}_n])_{n\in{\mathord{\mathbb N}}}$ converges
to $F$ a.s., hence every bounded
$F$ is the $L^2 (\Omega )$-limit of a sequence in ${\cal S}$.
If $F\in L^2(\Omega )$ is not bounded, $F$ is the limit
in $L^2(\Omega )$ of the sequence
$( {\bf 1}_{\{\vert F\vert\leq n\}} F)_{n\in{\mathord{\mathbb N}}}$
of bounded random variables. \end{Proof} \noindent
As a consequence of Proposition~\ref{chaos},
any $F\in L^{2}(\Omega ,\mathbb{P})$ has a unique decomposition \[ F= \mathbb{E}[F]
+ \sum_{n=1}^{\infty }J_{n}(f_{n}),\qquad f_{n}\in l^{2}({\mathord{\mathbb N}})^{\circ n},\ n\in {\mathord{\mathbb N}}, \]
as a series of multiple stochastic integrals.
Note also that the statement of Lemma~\ref{ll1.1} is sufficient
for the chaos representation property to hold. \section{Gradient Operator}\index{finite difference gradient!discrete time} \label{s4} \begin{definition}
We densely define the linear gradient operator $$D: {\cal S} \longrightarrow L^{2}(\Omega \times {\mathord{\mathbb N}}) $$
by \[ D_{k}J_n(f_n)
= n J_{n-1} (f_n(*,k) {\bf 1}_{\Delta_n}(*,k)), \]
$k\in {\mathord{\mathbb N}}$,
$f_n\in \ell^2({\mathord{\mathbb N}})^{\circ n}$
$n\in{\mathord{\mathbb N}}$. \end{definition}
Note that for all
$k_1,\ldots ,k_{n-1} , k \in {\mathord{\mathbb N}}$
we have $$ {\bf 1}_{\Delta_n} ( k_1,\ldots ,k_{n-1} , k)
= {\bf 1}_{\{ k \notin ( k_1,\ldots , k_{n-1} ) \}} {\bf 1}_{\Delta_{n-1}} ( k_1,\ldots ,k_{n-1}) , $$
hence we can write \[
D_{k}J_n(f_n)
= n J_{n-1} (f_n(*,k) {\bf 1}_{\{ k \notin * \}}), \quad
k\in {\mathord{\mathbb N}}, \]
where in the above relation, ``$*$'' denotes the
first $k-1$ variables
$( k_1,\ldots , k_{n-1} )$ of $f_n( k_1,\ldots , k_{n-1} , k)$.
We also have
$D_k F = 0$ whenever $F\in {\cal S}$ is ${\cal F}_{k-1}$-measurable. \\
On the other hand, $D_k$ is a continuous operator on the chaos
${\cal H}_n$ since \begin{eqnarray} \nonumber
\Vert D_k J_n (f_n) \Vert_{L^2(\Omega )}^2
& = &
n^2
\Vert J_{n-1} (f_n(*,k)) \Vert_{L^2(\Omega )}^2 \\ \label{frm11}
& = &
n n!
\Vert f_n(*,k) \Vert_{\ell^2({\mathord{\mathbb N}}^{\otimes (n-1)} )}^2 ,
\qquad
f_n \in \ell^2({\mathord{\mathbb N}}^{\otimes n} ),
\quad
k\in{\mathord{\mathbb N}}. \end{eqnarray}
The following result gives the probabilistic interpretation of $D_k$
as a finite difference operator. Given $$\omega = (\omega_0,\omega_1 , \ldots ) \in \{-1,1\}^{\mathord{\mathbb N}}, $$
let $$\omega_+^k = (\omega_0,\omega_1,\ldots ,\omega_{k-1}, +1,\omega_{k+1}, \ldots )$$
and $$\omega_-^k = (\omega_0,\omega_1,\ldots ,\omega_{k-1}, -1,\omega_{k+1}, \ldots ) . $$ \begin{prop} \label{fimdtl}
We have for any $F\in {\cal S}$: \begin{equation} \label{natdef}
D_k F (\omega ) = \sqrt{p_kq_k}
(F (\omega_+^k) - F (\omega_-^k ) ) , \quad k\in {\mathord{\mathbb N}} . \end{equation} \end{prop} \begin{Proof}
We start by proving the above statement for an ${\cal F}_n$-measurable
$F\in {\cal S}$.
Since $L^0(\Omega , {\cal F}_n)$ is finite dimensional it suffices to consider $$F= Y_{k_1}\cdots Y_{k_l} = f(X_0,\ldots ,X_{k_l}), $$
with from \eqref{ch1}: $$f(x_0,\ldots ,x_{k_l})
= \frac{1}{2^l}
\prod_{i=1}^{l}
\frac{q_{k_i}-p_{k_i}+x_{k_i}}{\sqrt{p_{k_i}q_{k_i}}}. $$
First we note that from \eqref{ch} we have for
$(k_1,\ldots , k_n)\in \Delta_n$: \begin{eqnarray} \nonumber
D_{k}\left( Y_{k_1}\cdots Y_{k_n} \right)
& = &
D_{k} J_n(
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_n)\}}) \\ \nonumber
& = &
n J_{n-1} (
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_n)\}}(*,k)) \\ \nonumber
& = &
\frac{1}{(n-1)!}
\sum_{i=1}^{n}
{\bf 1}_{\{k_i\}}(k)
\sum_{(i_1,\ldots , i_{n-1})\in \Delta_{n-1}}
{\bf \tilde{1}}_{\{\{i_1,\ldots ,i_{n-1}\} =
\{ k_1,\ldots ,k_{i-1},k_{i+1}, \ldots , k_n
\} \}
} \\ \nonumber
& = & \sum_{i=1}^{n}
{\bf 1}_{\{k_i\}}(k)
J_{n-1} (
{\bf \tilde{1}}_{\{(k_1,\ldots ,k_{i-1},k_{i+1}, \ldots , k_n)\}} ) \\ \label{opl}
& = &
{\bf 1}_{\{k_1,\ldots ,k_n\}}(k)
\prod_{i=1 \atop k_i\not= k}^{n} Y_{k_i}. \end{eqnarray}
If $k \notin \{k_1,\ldots ,k_l \}$
we clearly have $F(\omega_+^k ) =F(\omega_-^k)=F(\omega )$,
hence $$\sqrt{p_kq_k}
(F (\omega_+^k ) - F (\omega_-^k ) ) =0
=
D_kF (\omega ) . $$
On the other hand if $ k \in \{k_1,\ldots ,k_l \}$
we have $$F (\omega_+^k ) =
\sqrt{\frac{q_k}{p_k}}
\prod_{i=1 \atop k_i\not= k}^{l}
\frac{q_{k_i}-p_{k_i}+\omega_{k_i}}{2 \sqrt{p_{k_i}q_{k_i}}}, $$ $$
F (\omega_-^k )
=
- \sqrt{\frac{p_k}{q_k}}
\prod_{i=1 \atop k_i\not= k}^{l}
\frac{q_{k_i}-p_{k_i}+\omega_{k_i}}{2\sqrt{p_{k_i}q_{k_i}}}, $$
hence from \eqref{opl} we get \begin{eqnarray*} \sqrt{p_kq_k}(F(\omega_+^k ) -F (\omega_-^k ) ) & = &
\frac{1}{2^{l-1}}
\prod_{i=1 \atop k_i\not= k}^{l}
\frac{q_{k_i}-p_{k_i}+\omega_{k_i}}{\sqrt{p_{k_i}q_{k_i}}} \\ & = &
\prod_{i=1 \atop k_i\not= k}^{l} Y_{k_i} (\omega ) \\
& = &
D_{k}\left( Y_{k_1}\cdots Y_{k_l} \right) (\omega ) \\
& = & D_{k} F (\omega ) . \end{eqnarray*}
In the general case, $J_l(f_l )$ is the $L^2$-limit
of the sequence $\mathbb{E}[J_l(f_l ) \mid {\cal F}_n] = J_l ( f_l {\bf 1}_{[0,n]^l})$
as $n$ goes to infinity, and since from
\eqref{frm11} the operator $D_k$ is continuous
on all chaoses ${\cal H}_n$, $n\geq 1$, we have \begin{eqnarray*}
D_k F
& = &
\lim_{n\to \infty} D_k \mathbb{E}[F \mid {\cal F}_n]
\\
& = &
\lim_{n\to \infty}
(\mathbb{E}[F \mid {\cal F}_n] (\omega_+^k) - \mathbb{E}[F \mid {\cal F}_n] (\omega_-^k ) ) \\
& = &
\sqrt{p_kq_k}
(F (\omega_+^k) - F (\omega_-^k ) ) , \quad k\in {\mathord{\mathbb N}} . \end{eqnarray*} \end{Proof}
The next property follows immediately from Proposition~\ref{fimdtl} . \begin{corollary} \label{cormes}
A random variable $F:\Omega \to {\mathord{\mathbb R}}$
is ${\cal F}_n$-measurable if and only if $$D_k F = 0$$
for all $k>n$. \end{corollary}
If $F$ has the form $F = f(X_0,\ldots ,X_n)$,
we may also write $$ D_k F = \sqrt{p_kq_k}
(F^+_k - F^-_k ) , \qquad k\in {\mathord{\mathbb N}}, $$
with $$F_k^+ = f(X_0,\ldots ,X_{k-1},+1,X_{k+1}, \ldots ,X_n), $$
and $$ F_k^- =
f(X_0,\ldots ,X_{k-1},-1,X_{k+1}, \ldots ,X_n). $$
The gradient $D$ can also be expressed as $$
D_k F (S_\cdot ) = \sqrt{p_kq_k}
\left(
F
\left(
S_\cdot + {\bf 1}_{\{X_k=-1\}} {\bf 1}_{\{k\leq \cdot \}}
\right)
-
F
\left(
S_\cdot - {\bf 1}_{\{X_k=1\}} {\bf 1}_{\{k\leq \cdot \}}
\right)
\right)
, $$
where $F (S_\cdot )$ is an informal notation for
the random variable $F$ estimated on a given path
of $(S_n)_{n\in{\mathord{\mathbb N}}}$ defined in \eqref{defsn}
and $S_\cdot + {\bf 1}_{\{X_k=\mp 1 \}} {\bf 1}_{\{k\leq \cdot \}}$
denotes the path of $(S_n)_{n\in{\mathord{\mathbb N}}}$ perturbed by
forcing $X_k$ to be equal to $\pm 1$. \\
We will also use the gradient $\nabla_k$ defined as \begin{equation} \label{mod2} \nabla_k F =
X_k
\left(
f(X_0,\ldots ,X_{k-1}, -1,X_{k+1}, \ldots ,X_n)
- f(X_0,\ldots ,X_{k-1},1,X_{k+1}, \ldots ,X_n)
\right), \end{equation}
$k\in {\mathord{\mathbb N}}$,
with the relation $$
D_k = - X_k \sqrt{p_kq_k} \nabla_k ,
\qquad
k\in {\mathord{\mathbb N}} , $$
hence $\nabla_kF$ coincides with
$D_kF$ after squaring and multiplication by $p_kq_k$.
From now on, $D_k$ denotes the finite difference
operator which is extended to any $F:\Omega \to {\mathord{\mathbb R}}$
using Relation~\eqref{natdef}.
The $L^2$ domain of $D$ is naturally defined as
the space of functionals $F$ such that $\mathbb{E}[\Vert DF\Vert^2_{\ell^2 ({\mathord{\mathbb N}})}]
< \infty$, or equivalently $$\sum_{n=0}^\infty n n! \Vert f_n \Vert_{\ell^2 ({\mathord{\mathbb N}}^n)}^2 < \infty , $$
if $F = \sum_{n=0}^\infty J_n ( f_n )$.
The following is the product rule for the operator $D$. \begin{prop} \label{chnrle}
Let $F,G:\Omega \rightarrow {\mathord{\mathbb R}}$.
We have \begin{eqnarray*} D_{k}(FG)& =& FD_{k}G+GD_{k}F-\frac{X_k}{\sqrt{p_kq_k}} D_{k}FD_{k}G,
\qquad k\in {\mathord{\mathbb N}}. \end{eqnarray*} \end{prop} \begin{Proof}
Let $F_+^k (\omega ) = F (\omega_+^k)$, $F_-^k (\omega ) = F (\omega_-^k)$, $k\geq 0$.
We have \begin{eqnarray*} D_k(FG) & = & \sqrt{p_kq_k} (F_+^k G_+^k - F_-^k G_-^k ) \\ & = & {\bf 1}_{\{X_k=-1\}}
\sqrt{p_kq_k}
\left(
F(G_+^k -G)+
G(F_+^k -F)
+(F_+^k -F)(G_+^k -G )\right) \\ & & + {\bf 1}_{\{X_k=1\}} \sqrt{p_kq_k}
\left( F(G-G_-^k )+
G(F-F_-^k )
-(F-F_-^k )(G-G_-^k )\right) \\ & = &
{\bf 1}_{\{X_k=-1\}} \left( FD_kG+GD_kF+\frac{1}{\sqrt{p_kq_k}}D_kFD_kG\right) \\ & & + {\bf 1}_{\{X_k=1\}} \left(
FD_kG+GD_kF-\frac{1}{\sqrt{p_kq_k}}D_kFD_kG\right). \end{eqnarray*} \end{Proof} \section{Clark Formula and Predictable Representation}\index{Clark formula!discrete time} \index{predictable representation!discrete time} \label{s5}
In this section we prove a
predictable representation formula for the functionals of $(S_n)_{n\geq 0}$
defined in \eqref{defsn}. \begin{prop} \label{propclk}
For all $F \in {\cal S}$ we have \begin{eqnarray} \label{clk} F & = & \mathbb{E}[F]+\sum_{k=0}^{\infty }\mathbb{E}[D_{k}F \mid {\cal F}_{k-1}]Y_{k} \\ \nonumber
& = &
\mathbb{E}[F]+\sum_{k=0}^{\infty } Y_{k} D_{k} \mathbb{E}[F \mid {\cal F}_k ] . \end{eqnarray} \end{prop} \begin{Proof}
The formula is obviously true for $F = J_0(f_0)$.
Given $n\geq 1$, as a consequence of Proposition~\ref{prop1.18} above and
Lemma~\ref{lea2} we have: \begin{eqnarray*}
J_n (f_n) & = & n\sum_{k=0}^{\infty }J_{n-1}(f_{n}( * ,k) {\bf 1}_{[0,k-1]^{n-1}}( * ))Y_{k} \\ & = & n\sum_{k=0}^{\infty }J_{n-1}(f_{n}( * ,k) {\bf 1}_{\Delta_n}(*,k) {\bf 1}_{[0,k-1]^{n-1}}( * ))Y_{k} \\ & = & n\sum_{k=0}^{\infty }\mathbb{E}[J_{n-1}(f_{n}( * ,k) {\bf 1}_{\Delta_n}(*,k) ) \mid {\cal F}_{k-1}]
Y_{k} \\ & = & \sum_{k=0}^{\infty }\mathbb{E}[D_kJ_n(f_{n} ) \mid {\cal F}_{k-1}]
Y_{k}, \end{eqnarray*}
which yields \eqref{clk} for $F = J_n(f_n)$, since $\mathbb{E}[ J_n(f_n)]=0$.
By linearity the formula is established for $F\in {\cal S}$. \end{Proof}
Although the operator
$D$ is unbounded we have the following result,
which states the boundedness of the operator that maps
a random variable to the unique process involved in its predictable
representation. \begin{lemma} \label{lbdd}
The operator \begin{align*} L^{2}(\Omega ) & \longrightarrow L^{2}(\Omega \times {\mathord{\mathbb N}}) \\
F & \mapsto (\mathbb{E}[D_k F \mid {\cal F}_{k -1}])_{k\in{\mathord{\mathbb N}}} \end{align*}
is bounded with norm equal to one. \end{lemma} \begin{Proof}
Let $F\in {\cal S}$.
From Relation~\eqref{clk} and
the isometry formula \eqref{lkoi} for the stochastic
integral operator $J$ we get \begin{eqnarray} \label{bndd} \Vert \mathbb{E}[D_{\cdot }F \mid {\cal F}_{\cdot -1}]\Vert _{L^{2}(\Omega \times {\mathord{\mathbb N}})}^{2}
& = &
\Vert F-\mathbb{E}[F]\Vert_{L^{2}(\Omega )}^{2} \\ \nonumber
& \leq &
\Vert F-\mathbb{E}[F]\Vert _{L^{2}(\Omega )}^{2} + (\mathbb{E}[F])^2 \\ \nonumber
& = & \Vert F\Vert _{L^{2}(\Omega )}^{2}, \end{eqnarray}
with equality in case $F = J_1 (f_1)$. \end{Proof}
As a consequence of Lemma~\ref{lbdd} we have
the following corollary. \begin{corollary}
The Clark formula of
Proposition~\ref{propclk}
extends to any $F\in L^2 (\Omega )$. \end{corollary} \begin{Proof}
Since $F\mapsto \mathbb{E}[D_{\cdot }F \mid {\cal F}_{\cdot -1}]$ is bounded
from Lemma~\ref{lbdd}, the Clark formula extends to $F\in L^{2}(\Omega )$
by a standard Cauchy sequence argument.
For the second identity we use the relation $$\mathbb{E}[D_k F \mid {\cal F}_{k-1} ] = D_k \mathbb{E}[F \mid {\cal F}_k ] $$
which clearly holds since $D_k F$ is independent
of $X_k$, $k\in {\mathord{\mathbb N}}$. \end{Proof}
Let us give a first elementary application
of the above construction to the proof of a Poincar\'e inequality
on Bernoulli space. We have \begin{eqnarray*}
{\mathrm{{\rm var \ \!}}} (F) & = & \mathbb{E}[ | F-\mathbb{E}[F] |^2] \\
& = & \mathbb{E}\left[
\left(
\sum_{k=0}^\infty
\mathbb{E}[D_k F \mid {\cal F}_{k-1} ] Y_k \right)^2
\right] \\
& = & \mathbb{E}\left[
\sum_{k=0}^\infty
(\mathbb{E}[D_k F \mid {\cal F}_{k-1} ])^2
\right] \\
& \leq & \mathbb{E}\left[
\sum_{k=0}^\infty
\mathbb{E}[ | D_k F |^2 \mid {\cal F}_{k-1} ]
\right] \\
& = & \mathbb{E}\left[
\sum_{k=0}^\infty
| D_k F |^2
\right], \end{eqnarray*}
hence $${\mathrm{{\rm var \ \!}}} (F) \leq \Vert DF\Vert_{L^2(\Omega \times {\mathord{\mathbb N}} )}^2. $$
More generally the Clark formula implies the following. \begin{corollary} \label{lemmaa}
Let $a\in {\mathord{\mathbb N}}$ and $F\in L^2 (\Omega )$.
We have \begin{equation} \label{clark2a} F= \mathbb{E}[F \mid {\cal F}_a]
+ \sum_{k=a+1}^\infty \mathbb{E}[D_k F \mid {\cal F}_{k-1} ] Y_k, \end{equation}
and \begin{equation} \label{clark3}
\mathbb{E}[F^2] = \mathbb{E}[(\mathbb{E}[F \mid {\cal F}_a])^2]
+ \mathbb{E}\left[ \sum_{k=a+1}^\infty (\mathbb{E}[D_k F \mid {\cal F}_{k-1}])^2 \right]. \end{equation} \end{corollary} \begin{Proof}
From Proposition~\ref{isomp15} and the Clark formula
\eqref{clk} of Proposition~\ref{propclk} we have $$ \mathbb{E}[F \mid {\cal F}_a ] =
\mathbb{E}[F]+\sum_{k=0}^a \mathbb{E}[D_k F \mid {\cal F}_{k-1} ]Y_k , $$
which implies (\ref{clark2a}). Relation (\ref{clark3}) is an immediate consequence of
(\ref{clark2a}) and the isometry property of $J$. \end{Proof}
As an application of the Clark formula of Corollary~\ref{lemmaa}
we obtain the following predictable representation property for
discrete-time martingales. \begin{prop} \label{martrepr}
Let $(M_n)_{n\in {\mathord{\mathbb N}}}$ be a martingale in $L^2(\Omega)$ with respect
to $({\cal F}_n)_{n\in {\mathord{\mathbb N}}}$.
There exists a predictable process $(u_k)_{k\in {\mathord{\mathbb N}}}$ locally in $L^2(\Omega \times {\mathord{\mathbb N}})$,
(i.e. $u(\cdot ) {\bf 1}_{[0,N]} (\cdot ) \in L^2(\Omega \times {\mathord{\mathbb N}})$
for all $N>0$)
such that \begin{equation} \label{ghl0}
M_n = M_{-1} + \sum_{k=0}^n
u_k Y_k, \qquad n\in {\mathord{\mathbb N}}. \end{equation} \end{prop} \begin{Proof}
Let $k\geq 1$. From
Corollaries~\ref{cormes} and \ref{lemmaa}
we have: \begin{eqnarray*}
M_k & = &
\mathbb{E}[ M_k \mid {\cal F}_{k-1}]
+
\mathbb{E}[D_k M_k \mid {\cal F}_{k-1} ] Y_k \\ & = &
M_{k-1}
+
\mathbb{E}[D_k M_k \mid {\cal F}_{k-1} ] Y_k , \end{eqnarray*}
hence it suffices to let $$u_k = \mathbb{E}[D_k M_k \mid {\cal F}_{k-1}],
\qquad k\geq 0,$$
to obtain $$
M_n = M_{-1} + \sum_{k=0}^n M_k-M_{k-1}
= M_{-1} + \sum_{k=0}^n u_k Y_k. $$ \end{Proof} \section{Divergence Operator}\index{divergence operator!discrete time} \label{s6}
The divergence operator $\delta$ is introduced as the adjoint of $D$.
Let ${\cal U}\subset L^2 (\Omega \times {\mathord{\mathbb N}} )$
be the space of processes defined as $${\cal U} = \left\{
\sum_{k=0}^n
J_k (f_{k+1} (*,\cdot ) ),
\quad
f_{k+1} \in \ell^2({\mathord{\mathbb N}} )^{\circ k}\otimes \ell^2 ({\mathord{\mathbb N}}), \
\ k=, n \in{\mathord{\mathbb N}} \right\} . $$ \begin{definition}
Let $\delta : {\cal U}
\to L^2(\Omega )$ be the linear mapping defined on ${\cal U}$
as \[
\delta ( u )
=
\delta (J_{n}(f_{n+1}( * ,\cdot )))=J_{n+1}(\tilde{f}_{n+1}), \quad
f_{n+1}\in l^{2}({\mathord{\mathbb N}})^{\circ n}\otimes l^{2}({\mathord{\mathbb N}}), \]
for $(u_k)_{k\in{\mathord{\mathbb N}}}$ of the form $$
u_k =
J_{n}(f_{n+1}( * , k )),
\qquad
k\in{\mathord{\mathbb N}} , $$
where $\tilde{f}_{n+1}$ denotes the symmetrization of $f_{n+1}$
in $n+1$ variables, i.e. $$\tilde{f}_{n+1} (k_1,\ldots ,k_{n+1})
= \frac{1}{n+1}
\sum_{i=1}^{n+1}
f_{n+1} (k_1,\ldots ,k_{k-1},k_{k+1}, \ldots ,k_{n+1},
k_i ) . $$ \end{definition}
From Proposition~\ref{chaos}, ${\cal S}$ is dense in $L^2 (\Omega )$,
hence ${\cal U}$ is dense in $L^2 (\Omega \times {\mathord{\mathbb N}} )$. \begin{prop} \label{prdual}
The operator $\delta$ is adjoint to $D$: $$\mathbb{E}[\langle DF,u\rangle_{\ell^2 ({\mathord{\mathbb N}} )} ]
= \mathbb{E}[F\delta (u) ], \quad F\in {\cal S}, \ u\in {\cal U}. $$ \end{prop} \begin{Proof}
We consider $F= J_n(f_n)$ and $u_k = J_m(g_{m+1}(*,k))$, $k\in {\mathord{\mathbb N}}$,
where $f_n\in \ell^2({\mathord{\mathbb N}})^{\circ n}$ and
$g_{m+1}\in \ell^2({\mathord{\mathbb N}})^{\circ m}\otimes \ell^2({\mathord{\mathbb N}} )$.
We have \begin{eqnarray*} \lefteqn{
\mathbb{E}[\langle D_\cdot
J_n(f_n), J_m (g_{m+1}(*,\cdot )) \rangle_{\ell^2 ({\mathord{\mathbb N}} )}]
=
n \mathbb{E} [ \langle
J_{n-1} (f_n ( * , \cdot ) )
,
J_m (g_m ( * , \cdot ) )
\rangle_{l^2 ({\mathord{\mathbb N}} )}
]
} \\
& = &
n {\bf 1}_{\{n-1=m \}} \sum_{k=0}^\infty \mathbb{E}[J_{n-1}( f_n (*,k) {\bf 1}_{\Delta_n }(*,k) )
J_m(g_{m+1}(*,k) ) ] \\
& = &
n! {\bf 1}_{\{n-1=m \}} \sum_{k=0}^\infty
\langle
{\bf 1}_{\Delta_n } (*,k)
f_n (*,k), g_{m+1} (*,k)
\rangle_{\ell^2 ({\mathord{\mathbb N}}^{n-1})} \\
& = &
n! {\bf 1}_{\{n=m+1\}}
\langle
{\bf 1}_{\Delta_n }
f_n, g_{m+1} \rangle_{\ell^2 ({\mathord{\mathbb N}}^n)} \\
& = &
n! {\bf 1}_{\{n=m+1\}}
\langle {\bf 1}_{\Delta_n}
f_n, \tilde{g}_{m+1} \rangle_{\ell^2 ({\mathord{\mathbb N}}^n)} \\
& = & \mathbb{E}[J_n(f_n)J_m( \tilde{g}_{m+1} ) ]
\\
& = & \mathbb{E} [ \delta (u) F ]. \end{eqnarray*} \end{Proof}
The next proposition shows in particular that $\delta$ coincides with
the stochastic integral operator $J$ on the square-summable
predictable processes. \begin{prop} \label{pkl}
The operator $\delta$ can be extended to $u\in L^2(\Omega \times {\mathord{\mathbb N}})$
with \begin{equation} \label{dlta} \delta (u) = \sum_{k=0}^\infty u_k Y_k - \sum_{k=0}^\infty D_k u_k - \delta ( \varphi D u ), \end{equation}
provided that all series converges in $L^2 (\Omega )$,
where $(\varphi_k)_{k\in{\mathord{\mathbb N}}}$ appears in the
structure equation \eqref{dse}.
We also have for all $u \in {\cal U}$: \begin{equation}
\label{skois}
\mathbb{E}[ | \delta (u)| ^2 ] = \mathbb{E}[\Vert u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ]
+ \mathbb{E}\left[ \sum_{k,l=0\atop k \not= l}^\infty D_ku_l D_lu_k
- \sum_{k=0}^\infty ( D_ku_k)^2
\right]. \end{equation} \end{prop} \begin{Proof}
Using the expression \eqref{eg1} of
$u_k = J_n (f_{n+1} (*,k))$ we have \begin{eqnarray*} \delta (u)
& = &
J_{n+1}(\tilde{f}_{n+1} ) \\
& = &
\sum_{(i_1 , \ldots , i_{n+1} )\in \Delta_{n+1} }
\tilde{f}_{n+1} (i_1,\ldots ,i_{n+1} ) Y_{i_1}\cdots Y_{i_{n+1}} \\
& = &
\sum_{k=0}^\infty
\sum_{(i_1 , \ldots , i_n )\in \Delta_n }
\tilde{f}_{n+1} (i_1,\ldots ,i_n , k ) Y_{i_1}\cdots Y_{i_n} Y_k \\
& &
-
n
\sum_{k=0}^\infty
\sum_{(i_1 , \ldots , i_{n-1} )\in \Delta_{n-1} }
\tilde{f}_{n+1} (i_1,\ldots ,i_{n-1} , k , k ) Y_{i_1}\cdots Y_{i_{n-1}} | Y_k |^2 \\
& = & \sum_{k=0}^\infty
u_k Y_k
-
\sum_{k=0}^\infty D_k u_k | Y_k |^2 \\
& = & \sum_{k=0}^\infty
u_k Y_k
-
\sum_{k=0}^\infty D_k u_k
-
\sum_{k=0}^\infty \varphi_k D_k u_k Y_k. \end{eqnarray*}
Next, we note the commutation relation\footnote{See A.~Mantei, Masterarbeit ``Stochastisches Kalk\"ul in diskreter Zeit'', Satz 6.7, 2015.} \begin{eqnarray*}
D_k \delta (u)
& = &
D_k \left(
\sum_{l=0}^\infty
u_l Y_l
-
\sum_{l=0}^\infty | Y_l |^2 D_l u_l
\right)
\\
& = &
\sum_{l=0}^\infty
\left( Y_l D_k u_l + u_l D_k Y_l - \frac{X_k}{\sqrt{p_kq_k}} D_k u_l D_k Y_l
\right)
\\
& &
-
\sum_{l=0}^\infty
\left(
| Y_l |^2 D_k D_l u_l
+ D_l u_l D_k | Y_l |^2
- \frac{X_k}{\sqrt{p_kq_k}} D_k | Y_l |^2 D_k D_l u_l
\right)
\\
& = &
\delta ( D_k u)
+
u_k D_k Y_k - \frac{X_k}{\sqrt{p_kq_k}} D_k u_k D_k Y_k
- D_k u_k D_k | Y_k |^2
\\
& = &
\delta ( D_k u)
+ u_k -
\left( \frac{X_k}{\sqrt{p_kq_k}} + 2 Y_k D_k Y_k -\frac{X_k}{\sqrt{p_kq_k}}
D_k Y_k D_k Y_k
\right) D_k u_k
\\
& = &
\delta ( D_k u) + u_k - 2 Y_k D_k u_k. \end{eqnarray*}
On the other hand, we have \begin{eqnarray*}
\delta ( {\bf 1}_{\{ k \} } D_k u_k )
& = &
\sum_{l=0}^\infty
Y_l {\bf 1}_{\{ k \} } (l) D_k u_k
-
\sum_{l=0}^\infty | Y_l |^2 D_l ( {\bf 1}_{\{ k \} } (l) D_k u_k )
\\
& = & Y_k D_k u_k - | Y_k |^2 D_k D_k u_k
\\
& = & Y_k D_k u_k, \end{eqnarray*} hence \begin{eqnarray*} \Vert \delta (u) \Vert^2_{L^2(\Omega ) } & = & \mathbb{E}[
\langle u , D \delta (u) \rangle_{\ell^2({\mathord{\mathbb N}} )}
] \\
& = & \mathbb{E} \left[
\sum_{k=0}^\infty
u_k ( u_k + \delta ( D_k u ) - 2 Y_k D_k u_k )
\right] \\
& = & \mathbb{E}[\Vert
u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ]
+ \mathbb{E}\left[
\sum_{k,l=0}^\infty D_ku_l D_lu_k \right]
- 2 \mathbb{E} \left[ \sum_{k=0}^\infty u_k Y_k D_k u_k \right] \\
& = &
\mathbb{E}[\Vert
u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ]
+ \mathbb{E}\left[
\sum_{k,l=0}^\infty D_ku_l D_lu_k
- 2 \sum_{k=0}^\infty ( D_ku_k )^2
\right], \end{eqnarray*}
where we used the equality
\begin{eqnarray*}
\mathbb{E} \left[ u_k Y_k D_k u_k \right]
& = &
\mathbb{E} \left[ p_k {\bf 1}_{\{ X_k = 1 \} } u_k (\omega_+^k)
Y_k (\omega_+^k) D_k u_k
+
q_k {\bf 1}_{\{ X_k = - 1 \} } u_k (\omega_-^k)
Y_k (\omega_-^k) D_k u_k \right]
\\
& = &
\sqrt{p_kq_k} \mathbb{E} \left[
( {\bf 1}_{\{ X_k = 1 \} } u_k (\omega_+^k)
- {\bf 1}_{\{ X_k = - 1 \} } u_k (\omega_-^k)
)
D_k u_k \right]
\\
& = &
\mathbb{E}\left[ ( D_ku_k )^2 \right], \qquad k \in {\mathord{\mathbb N}}. \end{eqnarray*} \end{Proof}
In the symmetric case $p_k=q_k=1/2$ we have $\varphi_k=0$, $k\in{\mathord{\mathbb N}}$, and
$$ \delta (u) = \sum_{k=0}^\infty u_k Y_k - \sum_{k=0}^\infty D_k u_k. $$
The last two terms in the right hand side of
\eqref{dlta} vanish when $(u_k)_{k\in{\mathord{\mathbb N}}}$ is predictable,
and in this case the Skorohod isometry \eqref{skois} becomes
the It\^o isometry as in the next proposition. \begin{corollary}
If $(u_k )_{k\in{\mathord{\mathbb N}}}$ satisfies $D_ku_k = 0$,
i.e. $u_k$ does not depend on $X_k$, $k\in{\mathord{\mathbb N}}$, then $\delta (u)$
coincides with the (discrete time) stochastic integral \begin{equation} \label{iisoo} \delta (u)= \sum_{k=0}^{\infty }Y_{k}u_k , \end{equation}
provided that the series converges in $L^2(\Omega )$.
If moreover $(u_k)_{k\in{\mathord{\mathbb N}}}$ is predictable and square-summable
we have the isometry \begin{equation} \label{iiso}
\mathbb{E}[\delta (u)^{2}]=\mathbb{E}\left[\Vert u\Vert _{\ell^{2}({\mathord{\mathbb N}} )}^{2}\right], \end{equation}
and $\delta (u)$ coincides with $J (u)$
on the space of predictable square-summable processes. \end{corollary} \section{Ornstein-Uhlenbeck Semi-Group and Process}\index{Ornstein-Uhlenbeck!semigroup}\index{Ornstein-Uhlenbeck!process} \label{s7}
The Ornstein-Uhlenbeck operator $L$ is defined as $L = \delta D$, i.e. $L$ satisfies $$LJ_n ( f_n) = n J_n (f_n) , \qquad f_n \in \ell^2 ({\mathord{\mathbb N}})^{\circ n} . $$ \begin{prop}
For any $F\in {\cal S}$ we have $$LF = \delta D F = \sum_{k=0}^\infty Y_k (D_kF)
= \sum_{k=0}^\infty \sqrt{p_kq_k} Y_k (F_k^+ - F_k^- ) , $$ \end{prop} \begin{Proof}
Note that $D_kD_kF=0$, $k\in{\mathord{\mathbb N}}$, and
use Relation \eqref{dlta} of Proposition~\ref{pkl}. \end{Proof}
Note that $L$ can be expressed in other forms, for example $$
LF =
\sum_{k=0}^\infty
\Delta_k F , $$
where \begin{eqnarray*}
\Delta_k F
& = &
(
{\bf 1}_{\{X_k =1\}}
q_k
(F (\omega ) - F (\omega_-^k ) )
-
{\bf 1}_{\{X_k=-1\}}
p_k
(F (\omega_+^k) - F (\omega ) )
) \\
& = &
F
-
(
{\bf 1}_{\{X_k =1\}}
q_k
F (\omega_-^k )
+
{\bf 1}_{\{X_k=-1\}}
p_k
F (\omega_+^k)
) \\
& = &
F
-
\mathbb{E}
[
F
\mid
{\cal F}_k^c
] , \quad k\in {\mathord{\mathbb N}} , \end{eqnarray*}
and
${\cal F}_k^c$ is the $\sigma$-algebra generated by $$
\{ X_l \ : \ l\not=k, \ l \in {\mathord{\mathbb N}} \}. $$
Let now $(P_t)_{t\in {\mathord{\mathbb R}}_+} = (e^{tL})_{t\in {\mathord{\mathbb R}}_+}$
denote the semi-group associated to $L$ and defined as $$P_t F = \sum_{n=0}^\infty e^{-nt} J_n(f_n),
\qquad t\in {\mathord{\mathbb R}}_+,$$
on
$\displaystyle F = \sum_{n=0}^\infty J_n(f_n) \in L^2(\Omega )$.
The next result shows that $(P_t)_{t\in {\mathord{\mathbb R}}_+}$
admits an integral representation by a probability kernel.
Let $q^N_t : \Omega \times \Omega \to {\mathord{\mathbb R}}_+$ be defined by $$
q^N_t(\tilde{\omega} , \omega ) =
\prod_{i=0}^{N}
(1+e^{-t} Y_i( \omega ) Y_i( \tilde{\omega} ) ),
\qquad \omega, \tilde{\omega} \in \Omega,
\quad t\in {\mathord{\mathbb R}}_+. $$ \begin{lemma} \label{p11}
Let the probability kernel $Q_t (\tilde{\omega} , d\omega )$ be defined by $$\mathbb{E}\left[
\frac{dQ_t (\tilde{\omega} , \cdot )}{d\mathbb{P}} \Big|
{\cal F}_N \right] (\omega )
=
q^N_t(\tilde{\omega} , \omega ) , \qquad N\geq 1,
\quad t\in {\mathord{\mathbb R}}_+. $$
For $F\in L^2(\Omega, {\cal F}_N)$ we have \begin{equation} \label{repr}
P_t F ( \tilde{\omega} ) =
\int_\Omega F( \omega )
Q_t ( \tilde{\omega} , d \omega ),
\qquad \tilde{\omega} \in \Omega, \quad n\geq N. \end{equation} \end{lemma} \begin{Proof}
Since $L^2(\Omega ,{\cal F}_N)$ has finite
dimension $2^{N+1}$,
it suffices to consider functionals of the form
$F = Y_{k_1}\cdots Y_{k_n}$ with $0\leq k_1<\cdots < k_n\leq N$.
We have for $\omega \in \Omega$, $k\in {\mathord{\mathbb N}}$: \begin{eqnarray*} \lefteqn{ \mathbb{E}\left[Y_k(\cdot) (1+e^{-t} Y_k(\cdot ) Y_k(\omega ) )\right] } \\ & = &
p_k
\sqrt{\frac{q_k}{p_k}}
\left(1+e^{-t} \sqrt{\frac{q_k}{p_k}} Y_k (\omega )\right)
- q_k
\sqrt{\frac{p_k}{q_k}}
\left(1-e^{-t} \sqrt{\frac{p_k}{q_k}} Y_k (\omega )\right) \\
& = &
e^{-t} Y_k(\omega ), \end{eqnarray*}
which implies, by independence of the sequence $(X_k)_{k\in{\mathord{\mathbb N}}}$, \begin{eqnarray*} \mathbb{E}[Y_{k_1}\cdots Y_{k_n} q^N_t(\omega , \cdot )] & = & \mathbb{E}\left[
Y_{k_1}\cdots Y_{k_n}
\prod_{i=1}^{N}
(1+e^{-t} Y_{k_i}(\omega ) Y_{k_i} (\cdot ) )
\right] \\ & = &
\prod_{i=1}^{N} \mathbb{E}\left[
Y_{k_i} (\cdot )
(1+e^{-t} Y_{k_i} (\omega ) Y_{k_i} (\cdot ) )
\right] \\ & = &
e^{-nt} Y_{k_1}(\omega )\cdots Y_{k_n}(\omega ) \\ & = &
e^{-nt} J_n( {\bf \tilde{1}}_{\{(k_1, \ldots , k_n )\}} ) (\omega ) \\ & = &
P_t J_n(
{\bf \tilde{1}}_{\{(k_1, \ldots , k_n )\}} ) (\omega ) \\ & = & P_t(Y_{k_1}\cdots Y_{k_n}) (\omega ). \end{eqnarray*} \end{Proof}
Consider the $\Omega$-valued stationary process $(X(t))_{t\in {\mathord{\mathbb R}}_+}
= ((X_k(t) )_{k\in{\mathord{\mathbb N}}})_{t\in {\mathord{\mathbb R}}_+}$
with independent components and distribution given by \begin{align} \label{a1.x} & \mathbb{P}(X_k (t) = 1 \mid X_k (0) = 1) = p_k+e^{-t}q_k, \\ \nonumber \\ \label{a2.x} & \mathbb{P}(X_k (t) = -1 \mid X_k (0) = 1) = q_k-e^{-t}q_k, \\ \nonumber \\ \label{a3.x} & \mathbb{P}(X_k (t) = 1 \mid X_k (0) = -1) = p_k-e^{-t}p_k, \\ \nonumber \\ \label{a4.x} & \mathbb{P} (X_k (t) = -1 \mid X_k (0) = -1) = q_k+e^{-t}p_k, \end{align}
$k\in{\mathord{\mathbb N}}$, $t\in {\mathord{\mathbb R}}_+$. \begin{prop}
The process $(X (t))_{t\in {\mathord{\mathbb R}}_+} = ((X_k (t))_{k\in{\mathord{\mathbb N}}})_{t\in {\mathord{\mathbb R}}_+}$
is the Ornstein-Uhlenbeck process associated to $(P_t)_{t\in {\mathord{\mathbb R}}_+}$,
i.e. we have \begin{equation} \label{istheou}
P_t F = \mathbb{E}[F(X (t) ) \mid X (0) ],
\qquad t\in {\mathord{\mathbb R}}_+ . \end{equation} \end{prop} \begin{Proof}
By construction of $(X (t))_{t\in {\mathord{\mathbb R}}_+}$
in Relations~\eqref{a1.x}-\eqref{a4.x}
we have $$ \mathbb{P}(X_k (t) = 1 \mid X_k (0) )
= p_k\left(
1 + e^{-t}Y_k \sqrt{\frac{q_k}{p_k}}\right) , $$ \\ $$
\mathbb{P}(X_k (t) = -1 \mid X_k (0) )
= q_k\left( 1-e^{-t}Y_k \sqrt{\frac{p_k}{q_k}}\right) , $$ \\ $$ \mathbb{P}(X_k (t) = 1 \mid X_k (0) )
= p_k\left(
1 + e^{-t}Y_k \sqrt{\frac{q_k}{p_k}}\right) , $$ \\ $$
\mathbb{P}(X_k (t) = -1 \mid X_k (0) )
= q_k\left( 1-e^{-t}Y_k \sqrt{\frac{p_k}{q_k}}\right) , $$
thus $$
d \mathbb{P}(X_k (t)(\tilde{\omega} ) = \epsilon \mid X (0)) (\omega )
=
\left( 1 + e^{-t}Y_k (\omega ) Y_k(\tilde{\omega} )
\right)
d\mathbb{P}(X_k (\tilde{\omega} ) = \epsilon ) , $$
$\varepsilon = \pm 1$.
Since the components of $(X_k (t))_{k\in{\mathord{\mathbb N}}}$ are
independent, this shows that
the law of $(X_0 (t),\ldots ,X_n (t))$ conditionally
to $X (0)$ has the density $q_t^n(\tilde{\omega},\cdot )$
with respect to $\mathbb{P}$: \begin{eqnarray*} \lefteqn{ \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
d\mathbb{P}(X_0 (t)(\tilde{\omega} ) = \epsilon_0,
\ldots, X_n (t)(\tilde{\omega} ) =\epsilon_n \mid X (0) ) (\tilde{\omega} ) } \\
& = & q^n_t(\tilde{\omega},\omega )
d\mathbb{P}(X_0 (\tilde{\omega} ) = \epsilon_0, \ldots,
X_n (\tilde{\omega} ) = \epsilon_n ). \end{eqnarray*}
Consequently we have \begin{equation} \label{rprf}
\mathbb{E}[F ( X (t) ) \mid X (0) = \tilde{\omega} ]
= \int_\Omega F(\omega ) q^N_t (\tilde{\omega} , \omega )
\mathbb{P} (d\omega) , \end{equation}
hence from \eqref{repr}, Relation~\eqref{istheou} holds
for $F\in L^2(\Omega , {\cal F}_N)$, $N\geq 0$. \end{Proof}
The independent components $X_k (t)$, $k\in {\mathord{\mathbb N}}$,
can be constructed from the data of
$X_k (0) =\epsilon$ and an independent exponential random variable $\tau_k$
via the following procedure.
If $\tau_k <t$, let $X_k (t) =X_k (0) =\epsilon$,
otherwise if $\tau_k >t$, take $X_k (t)$ to be an independent copy of
$X_k$.
This procedure is illustrated in the following equalities: \begin{eqnarray} \label{r1r}
\mathbb{P}(X_k (t) = 1 \mid X_k (0) = 1)
& = & \mathbb{E}[ {\bf 1}_{\{\tau_k >t\}} ]
+ \mathbb{E}[ {\bf 1}_{\{\tau_k <t\}} {\bf 1}_{\{X_k=1\}}] \\ \nonumber
& = & e^{-t} + p_k (1-e^{-t}) , \\ \nonumber \\ \label{r2r}
\mathbb{P}(X_k (t) = -1 \mid X_k (0) = 1)
& = & \mathbb{E}[ {\bf 1}_{\{\tau_k <t\}} {\bf 1}_{\{X_k=-1\}}] \\ \nonumber
& = & q_k (1-e^{-t}) , \\ \nonumber \\ \label{r3r}
\mathbb{P}(X_k (t) = -1 \mid X_k (0) = -1)
& = & \mathbb{E}[ {\bf 1}_{\{\tau_k >t\}} ]
+ \mathbb{E}[ {\bf 1}_{\{\tau_k <t\}} {\bf 1}_{\{X_k=-1\}}] \\ \nonumber & = & e^{-t} + q_k (1-e^{-t}) , \\ \nonumber \\ \label{r4r}
\mathbb{P}(X_k (t) = 1 \mid X_k (0) = -1)
& = & \mathbb{E}[ {\bf 1}_{\{\tau_k <t\}} {\bf 1}_{\{X_k=1\}}] \\ \nonumber
& = & p_k (1-e^{-t}) . \end{eqnarray}
The operator $L^2 (\Omega \times {\mathord{\mathbb N}} ) \to
L^2 (\Omega \times {\mathord{\mathbb N}} )$
which maps $(u_k)_{k\in{\mathord{\mathbb N}}}$ to $(P_tu_k)_{k\in{\mathord{\mathbb N}}}$
is also denoted by $P_t$.
As a consequence of the representation of $P_t$ given
in Lemma~\ref{p11} we obtain the following bound. \begin{lemma} \label{bbdd}
For $F\in {\mathrm{{\rm Dom \ \!}}} (D)$ we have $$ \Vert P_t u\Vert_{L^\infty (\Omega , \ell^2({\mathord{\mathbb N}} ))}
\leq \Vert u\Vert_{L^\infty (\Omega , \ell^2({\mathord{\mathbb N}} ))},
\qquad
t\in {\mathord{\mathbb R}}_+,
\quad
u\in L^2 (\Omega \times {\mathord{\mathbb N}} ) . $$ \end{lemma} \begin{Proof}
As a consequence of the representation formula \eqref{rprf} we have
$\mathbb{P}(d \tilde{\omega} )$-a.s.: \begin{eqnarray*} \Vert P_t u \Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ( \tilde{\omega} )
& = & \sum_{k=0}^\infty
| P_t u_k ( \tilde{\omega} ) |^2 \\ & = & \sum_{k=0}^\infty
\left( \int_{\Omega}
u_k ( \omega ) Q_t ( \tilde{\omega} , d\omega ) \right)^2 \\ & \leq &
\sum_{k=0}^\infty
\int_{\Omega}
| u_k ( \omega )|^2 Q_t ( \tilde{\omega} , d\omega ) \\ & = &
\int_{\Omega}
\Vert u\Vert_{\ell^2({\mathord{\mathbb N}} )}^2 ( \omega )
Q_t ( \tilde{\omega} , d\omega ) \\ & \leq & \Vert u\Vert_{L^\infty (\Omega , \ell^2({\mathord{\mathbb N}} ))}^2 . \end{eqnarray*} \end{Proof} \section{Covariance Identities}\index{covariance identities!discrete time} \label{s8}
In this section we state the covariance identities
which will be used for the proof of deviation inequalities
in the next section.
The covariance
$\mathop{\hbox{\rm Cov}}\nolimits (F,G)$ of $F,G\in L^2 (\Omega )$
is defined as $$\mathop{\hbox{\rm Cov}}\nolimits (F,G) = \mathbb{E}[(F-\mathbb{E}[F])(G-\mathbb{E}[G])]
= \mathbb{E}[FG] - \mathbb{E}[F]\mathbb{E}[G] . $$ \begin{prop} \label{covclark0}
We have for
$F,G\in L^2 (\Omega )$ such that $\mathbb{E}[\Vert DF \Vert_{\ell^2({\mathord{\mathbb N}} )}^2]< \infty$: \begin{equation} \label{covclark} \mathop{\hbox{\rm Cov}}\nolimits (F,G) =
\mathbb{E}\left[ \sum_{k=0}^\infty
\mathbb{E}\left[ D_{k} G \mid {\cal F}_{k-1} \right] D_k F
\right]. \end{equation} \end{prop} \begin{Proof}
This identity is a consequence of the Clark formula \eqref{clk}: \begin{eqnarray*} \mathop{\hbox{\rm Cov}}\nolimits (F,G) & = & \mathbb{E}[(F-\mathbb{E}[F])(G-\mathbb{E}[G])] \\ & = & \mathbb{E}\left[ \left( \sum_{k=0}^\infty \mathbb{E}[D_kF \mid {\cal F}_{k-1}] Y_k \right) \left( \sum_{l=0}^\infty \mathbb{E}[D_lG \mid {\cal F}_{l-1}] Y_l \right) \right] \\ & = & \mathbb{E}\left[ \sum_{k=0}^\infty \mathbb{E}[D_kF \mid {\cal F}_{k-1}]
\mathbb{E}[D_kG\mid {\cal F}_{k-1}] \right] \\ & = & \sum_{k=0}^\infty
\mathbb{E}\left[
\mathbb{E}[\mathbb{E}[D_kG \mid {\cal F}_{k-1}]
D_kF\mid {\cal F}_{k-1}] \right] \\ & = &
\mathbb{E}\left[ \sum_{k=0}^\infty
\mathbb{E}[D_kG \mid {\cal F}_{k-1}] D_kF
\right]. \end{eqnarray*} \end{Proof}
A covariance identity can also be obtained
using the semi-group $(P_t)_{t\in {\mathord{\mathbb R}}_+}$. \begin{prop} \label{crsg}
For any $F,G\in L^2 (\Omega )$ such that $$\mathbb{E}[\Vert DF\Vert_{\ell^2({\mathord{\mathbb N}} )}^2] < \infty
\qquad
\mbox{and}
\qquad
\mathbb{E}[\Vert DG\Vert_{\ell^2({\mathord{\mathbb N}} )}^2] < \infty , $$
we have \begin{equation} \label{coov} \mathop{\hbox{\rm Cov}}\nolimits (F,G) = \mathbb{E}\left[
\sum_{k=0}^\infty
\int_0^\infty
e^{-t} ( D_k F ) P_t D_k G dt
\right]. \end{equation} \end{prop} \begin{Proof}
Consider $F = J_n (f_n)$ and $G= J_m(g_m )$.
We have \begin{eqnarray*} \lefteqn{ \mathop{\hbox{\rm Cov}}\nolimits (J_n(f_n),J_m(g_m))
= \mathbb{E}\left[ J_n(f_n) J_m(g_m) \right] } \\ & = &
{\bf 1}_{\{n=m\}} n! \langle f_n
, g_n {\bf 1}_{\Delta_n}
\rangle_{\ell^2({\mathord{\mathbb N}}^n)} \\
& =&
{\bf 1}_{\{n=m\}}
n! n
\int_0^\infty e^{-nt} dt
\langle f_n
, g_n {\bf 1}_{\Delta_n}
\rangle_{\ell^2({\mathord{\mathbb N}}^n)} \\ & = &
{\bf 1}_{\{n-1=m-1\}}
n! n
\int_0^\infty e^{-t} \sum_{k=0}^{\infty }
\langle f_n (*,k)
, e^{-(n-1)t}
g_n (*,k)
{\bf 1}_{\Delta_n} (*,k)
\rangle_{\ell^2({\mathord{\mathbb N}}^{n-1})} dt \\ & = &
n m
\mathbb{E}\left[ \int_0^\infty e^{- t} \sum_{k=0}^{\infty }
J_{n-1}( f_n(*,k) {\bf 1}_{\Delta_n} (*,k) )
e^{- ( m - 1 ) t} J_{m-1}( g_m(*,k) {\bf 1}_{\Delta_m} (*,k) ) dt \right] \\ & = &
n m
\mathbb{E}\left[ \int_0^\infty
e^{-t}
\sum_{k=0}^{\infty }
J_{n-1}( f_n(*,k) {\bf 1}_{\Delta_n}(*,k) )
P_t J_{m-1}( g_m(*,k) {\bf 1}_{\Delta_m}(*,k) ) dt \right] \\
& = & \mathbb{E}\left[ \int_0^\infty e^{-t} \sum_{k=0}^{\infty } D_k J_n(f_n) P_t D_k J_m(g_m) dt \right] . \end{eqnarray*} \end{Proof} \noindent
From \eqref{r1r}-\eqref{r4r}
the covariance identity \eqref{coov} shows that \begin{eqnarray} \nonumber \lefteqn{ \mathop{\hbox{\rm Cov}}\nolimits (F,G) = \mathbb{E}\left[
\sum_{k=0}^\infty
\int_0^\infty
e^{-t} D_k F P_t D_k G dt
\right] } \\ \nonumber & = &
\mathbb{E}\left[
\int_0^1
\sum_{k=0}^\infty
D_k F P_{-\log \alpha} D_k G d\alpha
\right] \\ \nonumber
& = & \int_0^1
\int_{\Omega \times \Omega}
\sum_{k=0}^\infty
D_k F (\omega )
D_k G ((\omega_i {\bf 1}_{\{\tau_i < -\log \alpha\}}
+ \omega'_i {\bf 1}_{\{\tau_i < -\log \alpha\}} )_{i\in{\mathord{\mathbb N}}} )
d\alpha \mathbb{P}(d\omega ) \mathbb{P}(d\omega') \\ \nonumber & = &
\int_0^1
\int_{\Omega \times \Omega}
\sum_{k=0}^\infty
D_k F (\omega )
D_k G ((\omega_i {\bf 1}_{\{\xi_i < \alpha\}}
+ \omega'_i {\bf 1}_{\{\xi_i > \alpha\}})_{i\in{\mathord{\mathbb N}}} )
\mathbb{P}(d\omega ) \mathbb{P}(d\omega') d\alpha , \\ & & \label{gghhh} \end{eqnarray}
where $(\xi_i)_{i\in{\mathord{\mathbb N}}}$ is a family of i.i.d. random variables,
uniformly distributed on $[0,1]$.
Note that the marginals of
$(X_k, X_k {\bf 1}_{\{\xi_k < \alpha\}}
+ X_k' {\bf 1}_{\{\xi_i > \alpha\}})$
are identical when $X_k'$ is an independent copy of
$X_k$.
Let $$\phi_\alpha (s,t) = \mathbb{E}[e^{is X_k}e^{it (X_k+{\bf 1}_{\{
\xi_k< \alpha \}}) + it (X_k'+{\bf 1}_{\{
\xi_k > \alpha \}})}]. $$
Then we have the relation $$\phi_\alpha (s,t) =
\alpha \phi (s+t) + (1-\alpha ) \phi (s) \phi (t),
\quad \alpha\in [0,1]. $$
Note that $$
\mathop{\hbox{\rm Cov}}\nolimits (e^{isX_k},e^{itX_k})
= \phi_1(s,t) - \phi_0 (s,t)
= \int_0^1 \frac{d \phi_\alpha }{d\alpha} (s,t) d\alpha
= \phi(s+t) - \phi(s)\phi(t). $$
Next we prove an iterated version of the covariance identity
in discrete time, which is an analog of a result proved
in \cite{houdre} for the Wiener and Poisson processes. \begin{theorem} \label{th1a}
Let $n\in {\mathord{\mathbb N}}$ and $F,G\in L^2 (\Omega )$.
We have \begin{eqnarray} \label{*a} \lefteqn{ \! \! \! \! \! \! \! \!
\mathop{\hbox{\rm Cov}}\nolimits (F,G) =
\sum_{d=1}^{d=n}
(-1)^{d+1} \mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_d\}}
(D_{k_d}\cdots D_{k_1} F)
(D_{k_d}\cdots D_{k_1} G)
\right] } \\ \nonumber
& &
+
(-1)^n
\mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_{n+1} \}}
( D_{k_{n+1}}\cdots D_{k_1} F )
\mathbb{E}\left[ D_{k_{n+1}}\cdots D_{k_1} G\mid {\cal F}_{k_{n+1}-1} \right]
\right]. \end{eqnarray} \end{theorem} \begin{Proof}
Take $F=G$.
For $n=0$, (\ref{*a}) is a consequence of the Clark formula.
Let $n\geq 1$.
Applying Lemma~\ref{lemmaa} to
$D_{k_n}\cdots D_{k_1} F$ with $a=k_n$ and
$b=k_{n+1}$,
and summing on
$(k_1, \ldots ,k_n)\in \Delta_n$,
we obtain \begin{eqnarray*} \lefteqn{ \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_n \}}
\left(
\mathbb{E}[D_{k_n}\cdots D_{k_1} F \mid {\cal F}_{k_n-1} ]\right)^2
\right]
=
\mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_n \}} \mid D_{k_n}\cdots D_{k_1} F |^2
\right] } \\ & &
-
\mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_{n+1} \}}
\left(
\mathbb{E}\left[ D_{k_{n+1}}\cdots D_{k_1} F\mid {\cal F}_{k_{n+1}-1}
\right]\right)^2
\right], \end{eqnarray*}
which concludes the proof by induction and bilinearity. \end{Proof}
As a consequence of Theorem~\ref{th1a},
letting $F=G$ we get the variance
inequality $$\sum_{k=1}^{2n} \frac{(-1)^{k+1}}{k!}
\mathbb{E} \left[
\Vert D^k F\Vert_{\ell^2(\Delta_k)}^2
\right]
\leq \mathop{\hbox{\rm Var}}\nolimits (F) \leq
\sum_{k=1}^{2n-1}
\frac{(-1)^{k+1}}{k!}
\mathbb{E} \left[
\Vert D^k F\Vert_{\ell^2(\Delta_k)}^2
\right] , $$
since \begin{eqnarray*} \lefteqn{ \mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_{n+1} \}}
( D_{k_{n+1}}\cdots D_{k_1} F )
\mathbb{E}\left[ D_{k_{n+1}}\cdots D_{k_1} G\mid {\cal F}_{k_{n+1}-1} \right]
\right] } \\ & = & \mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_{n+1} \}}
\mathbb{E}\left[
( D_{k_{n+1}}\cdots D_{k_1} F )
\mathbb{E}\left[ D_{k_{n+1}}\cdots D_{k_1} G\mid {\cal F}_{k_{n+1}-1} \right]
\mid {\cal F}_{k_{n+1}-1} \right]
\right] \\ & = & \mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_{n+1} \}}
(
\mathbb{E}\left[ D_{k_{n+1}}\cdots D_{k_1} G\mid {\cal F}_{k_{n+1}-1} \right]
)^2 \right] \\ & \geq & 0 , \end{eqnarray*}
see Relation~(2.15) in \cite{houdre} in continuous time.
In a similar way, another iterated covariance identity can be
obtained from Proposition~\ref{crsg}. \begin{corollary}
Let $n\in {\mathord{\mathbb N}}$ and $F,G \in L^2(\Omega, {\cal F}_N)$.
We have \begin{eqnarray} \nonumber \mathop{\hbox{\rm Cov}}\nolimits (F,G) & = &
\sum_{d=1}^{d=n}
(-1)^{d+1} \mathbb{E}\left[
\sum_{\{1\leq k_1< \cdots < k_d\leq N\}}
(D_{k_d}\cdots D_{k_1} F)
(D_{k_d}\cdots D_{k_1} G)
\right] \\ \nonumber
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
+
(-1)^n
\int_{\Omega \times \Omega}
\sum_{\{1\leq k_1< \cdots < k_{n+1} \leq N\}}
D_{k_{n+1}}\cdots D_{k_1} F(\omega )
D_{k_{n+1}}\cdots D_{k_1} G(\omega') \\ \label{****a}
& &
q^N_t(\omega,\omega')\mathbb{P}(d\omega )\mathbb{P}(d\omega' ). \end{eqnarray} \end{corollary}
The covariance and variance
have the tensorization property: $$\mathop{\hbox{\rm Var}}\nolimits (FG) = \mathbb{E}[F\mathop{\hbox{\rm Var}}\nolimits G] + \mathbb{E}[G\mathop{\hbox{\rm Var}}\nolimits F]$$
if $F,G$ are independent, hence most of the identities in this
section can be obtained by tensorization of a one dimensional
elementary covariance identity.
An elementary consequence of the covariance identities is
the following lemma. \begin{lemma} \label{m2}
Let $F, G\in L^2 (\Omega )$ such that
$$\mathbb{E} [D_k F | {\cal F}_{k-1} ]
\cdot
\mathbb{E} [D_k G | {\cal F}_{k-1} ]\ge 0,
\qquad
k\in{\mathord{\mathbb N}}. $$
Then $F$ and $G$ are non-negatively correlated: $$\mathop{\hbox{\rm Cov}}\nolimits (F, G)\ge 0. $$ \end{lemma} \noindent
According to the next definition, a non-decreasing functional $F$
satisfies $D_kF\ge 0$ for all $k\in{\mathord{\mathbb N}}$. \begin{definition}
A random variable $F : \Omega \to {\mathord{\mathbb R}}$ is said to be non-decreasing if
for all $\omega_1,\omega_2\in \Omega$ we have $$\omega_1 (k) \le \omega_2 (k), \qquad
k\in{\mathord{\mathbb N}},
\qquad \Rightarrow
\
F(\omega_1 ) \leq F(\omega_2) . $$ \end{definition} \noindent
The following result is then immediate from
Proposition~\ref{fimdtl} and Lemma~\ref{m2},
and shows that the FKG inequality holds on $\Omega$.
It can also be obtained from from Proposition~\ref{crsg}. \begin{prop}
If $F, G\in L^2 (\Omega )$ are non-decreasing
then $F$ and $G$ are non-negatively correlated: $$ \mathop{\hbox{\rm Cov}}\nolimits (F, G)\ge 0. $$ \end{prop} \noindent
Note however that the assumptions of
Lemma~\ref{m2} are actually weaker as they
do not require $F$ and $G$ to be non-decreasing. \section{Deviation Inequalities}\index{deviation inequalities!discrete time} \label{devsec}
In this section, which is based on \cite{hp},
we recover a deviation inequality of \cite{bobkov} in the case
of Bernoulli measures, using covariance representations instead
of the logarithmic Sobolev inequalities to be presented
in Section~\ref{lsid}.
The method relies on a bound on the Laplace transform
$L(t) = \mathbb{E} [ e^{t F} ]$ obtained via a differential inequality
and Chebychev's inequality. \begin{prop} \label{c1.02}
Let $F : \Omega \to {\mathord{\mathbb R}}$ be such that
$\vert F_k^+ - F_k^- \vert \leq K$, $k\in{\mathord{\mathbb N}}$, for
some $K\geq 0$, and
$\Vert DF \Vert_{L^\infty (\Omega , \ell^2({\mathord{\mathbb N}} ))} <\infty$.
Then \begin{eqnarray*} \mathbb{P}(F-\mathbb{E}[F]\geq x) & \leq & \exp \left(
- \frac{\Vert DF\Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2}{K^2}
g\left( \frac{x K}{\Vert DF\Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2}\right)
\right) \\ & \leq & \exp \left(
- \frac{x}{2 K}
\log \left(
1 + \frac{x K}{\Vert DF\Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2}
\right)\right), \end{eqnarray*}
with $g(u) = (1+u)\log (1+u)-u$, $u\geq 0$. \end{prop} \begin{Proof}
Although $D_k$ does not satisfy a derivation rule for products, from
Proposition~\ref{chnrle} we have \begin{eqnarray*} D_k e^F & = &
{\bf 1}_{\{X_k=1\}}
\sqrt{p_kq_k}
(e^F- e^{F^-_k})
+ {\bf 1}_{\{X_k=-1\}}
\sqrt{p_kq_k}
(e^{F^+_k}- e^F) \\ & = &
{\bf 1}_{\{X_k=1\}}
\sqrt{p_kq_k}
e^F (1- e^{-\frac{1}{\sqrt{p_kq_k}}D_kF})
+ {\bf 1}_{\{X_k=-1\}}
\sqrt{p_kq_k} e^F (e^{\frac{1}{\sqrt{p_kq_k}} D_kF}- 1) \\ & = &
- X_k \sqrt{p_kq_k}
e^F (e^{-\frac{X_k}{\sqrt{p_kq_k}}D_kF}- 1) , \end{eqnarray*}
hence \begin{equation} \label{prd.0} D_k e^F =
X_k \sqrt{p_kq_k}e^F
(1-e^{-\frac{X_k}{\sqrt{p_kq_k}} D_kF}), \end{equation}
and since the function $x \mapsto (e^x-1)/x$ is positive
and increasing on ${\mathord{\mathbb R}}$ we have: $$\frac{e^{-sF}D_ke^{sF}}{D_kF}
= -\frac{X_k\sqrt{p_kq_k} }{D_kF}
\left(
e^{- s \frac{X_k}{\sqrt{p_kq_k}} D_kF} -1 \right)
\leq
\frac{e^{sK}-1}{K},$$
or in other terms: $$
\frac{e^{-sF}D_ke^{sF}}{D_kF}
= {\bf 1}_{\{X_k=1 \}}
\frac{e^{s ( F_k^--F_k^+) }-1}{F_k^--F_k^+}
+ {\bf 1}_{\{X_k= -1 \}}
\frac{e^{s ( F_k^+-F_k^-) }-1}{F_k^+-F_k^-}
\leq
\frac{e^{sK}-1}{K}.$$
We first assume that $F$ is a bounded random variable with $\mathbb{E}[F]=0$.
From Lemma~\ref{bbdd} applied to $DF$, we have \begin{eqnarray*} \mathbb{E}[Fe^{sF}] & =& \mathop{\hbox{\rm Cov}}\nolimits (F,e^{sF}) \\
& = & \mathbb{E}\left[\int_0^\infty
e^{-v}
\sum_{k=0}^\infty D_k e^{sF}
P_v D_k F dv \right] \\ & \leq &
\left\| \frac{e^{-sF}
De^{sF}}{DF} \right\|_{\infty}
\mathbb{E}\left[e^{sF}
\int_0^\infty e^{-v}
\Vert DFP_v DF \Vert_{\ell^1({\mathord{\mathbb N}} )} dv
\right] \\ & \leq &
\frac{e^{sK}-1}{K}
\mathbb{E}\left[e^{sF}
\Vert DF \Vert_{\ell^2 ({\mathord{\mathbb N}} )}
\int_0^\infty e^{-v}
\Vert P_v DF \Vert_{\ell^2({\mathord{\mathbb N}} )} dv
\right] \\ & \leq &
\frac{e^{sK}-1}{K}
\mathbb{E}\left[e^{sF} \right]
\Vert DF \Vert_{L^\infty (\Omega , \ell^2({\mathord{\mathbb N}} ))}^2
\int_0^\infty e^{-v} dv \\ & \leq &
\frac{e^{sK}-1}{K}
\mathbb{E}\left[e^{sF} \right]
\Vert DF \Vert_{L^\infty(\Omega ,\ell^2({\mathord{\mathbb N}} ))}^2. \end{eqnarray*}
In the general case, letting $L(s) = \mathbb{E}[e^{s(F-\mathbb{E}[F])}]$, we have \begin{eqnarray*} \log (\mathbb{E}[e^{t(F-\mathbb{E}[F])}])
& = &
\int_0^t \frac{L'(s)}{ L(s)}
ds \\ & = &
\int_0^t
\frac{\mathbb{E}[(F-\mathbb{E}[F]) e^{s(F-\mathbb{E}[F])}]}{\mathbb{E}[e^{s(F-\mathbb{E}[F])}]}
ds \\ & \leq &
\frac{1}{K} \Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2
\int_0^t (e^{sK}-1) ds \\ & = &
\frac{1}{K^2} (e^{tK}-tK-1)
\Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2 , \end{eqnarray*} $t\geq 0$.
We have for all $x\geq 0$ and $t\geq 0$: \begin{eqnarray*} \mathbb{P}(F-\mathbb{E}[F]\geq x) & \leq & e^{-tx} \mathbb{E}[e^{t(F-\mathbb{E}[F])}] \\ & \leq &
\exp \left(
\frac{1}{K^2} (e^{tK}-tK-1)
\Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2
-tx \right) , \end{eqnarray*}
The minimum in $t\geq 0$ in the above expression is attained with $$t = \frac{1}{K}
\log \left( 1 + \frac{xK}{ \Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2 }\right), $$
hence \begin{eqnarray*} \lefteqn{ \mathbb{P}(F-\mathbb{E}[F]\geq x) } \\ & \leq & \exp \left(
-\frac{1}{K} \left(
\left(
x+ \frac{1}{K} \Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2 \right)
\log \left( 1+ x K
\Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^{-2} \right)
- x\right) \right) \\
& \leq &
\exp \left(
-\frac{x}{2K}
\log \left( 1+ x K
\Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^{-2}
\right) \right) , \end{eqnarray*}
where we used the inequality
$(1+u)\log (1+u)-u \geq \frac{u}{2}\log (1+u)$.
If $K=0$, the above proof is still valid by replacing all terms
by their limits as $K\to 0$.
If $F$ is not bounded the conclusion
holds for $F_n = \max (-n , \min ( F, n))$, $n\geq 1$,
and $(F_n)_{n\in{\mathord{\mathbb N}}}$,
$(DF_n)_{n\in{\mathord{\mathbb N}}}$, converge respectively almost surely
and in $L^2(\Omega \times {\mathord{\mathbb N}} )$ to $F$ and $DF$, with
$\Vert DF_n \Vert_{L^\infty (\Omega ,L^2 ({\mathord{\mathbb N}} ))}^2
\leq \Vert DF \Vert_{L^\infty (\Omega ,L^2 ({\mathord{\mathbb N}} ))}^2$. \end{Proof}
In case $p_k=p$ for all $k\in{\mathord{\mathbb N}}$,
the conditions $$\frac{1}{\sqrt{pq}} \vert D_k F\vert \leq \beta, \quad
k\in{\mathord{\mathbb N}},
\quad
\mbox{and}
\quad
\Vert DF\Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2 \leq \alpha^2 , $$
give $$ \mathbb{P}(F-\mathbb{E}[F]\geq x) \leq \exp \left(
- \frac{\alpha^2 pq}{\beta^2}
g\left( \frac{x \beta}{\alpha^2 pq}\right)
\right)
\leq \exp \left(
- \frac{x}{2 \beta}
\log \left(
1 + \frac{x \beta}{\alpha^2 pq}
\right)\right), $$
which is Relation (13) in \cite{bobkov}.
In particular if $F$ is ${\cal F}_N$-measurable, then $$ \mathbb{P}(F-\mathbb{E}[F]\geq x) \leq \exp \left(
-Ng\left( \frac{x}{\beta N}\right) \right)
\leq \exp \left(
-\frac{x}{\beta }
\left(
\log\left(
1+\frac{x}{\beta N}\right)
-1\right) \right). $$
Finally we show a Gaussian concentration inequality
for functionals of $(S_n)_{n\in{\mathord{\mathbb N}}}$, using the covariance
identity \eqref{covclark}.
We refer to \cite{bobkovsyk}, \cite{bht2}, \cite{ht2},
\cite{ledouxesaim}, for other versions of this inequality. \begin{prop} \label{gsian}
Let $F: \Omega \rightarrow {\mathord{\mathbb R}}$ be such that $$
\left\|
\sum_{k=0}^\infty
\frac{1}{2(p_k\wedge q_k)}
\vert D_k F \vert
\Vert D_k F \Vert_\infty
\right\|_\infty
\leq K^2. $$
Then \begin{equation} \label{bdd} \mathbb{P}(F-\mathbb{E}[F]\geq x) \leq \exp \left(
- \frac{x^2}{2
K^2 }
\right), \qquad x\geq 0. \end{equation} \end{prop} \begin{Proof}
Again, we assume that $F$ is a bounded random variable with $\mathbb{E}[F] = 0$.
Using the inequality \begin{equation} \label{1..} \vert e^{tx}-e^{ty} \vert
\leq \frac{t}{2}
\vert x-y \vert (e^{tx}+e^{ty}),
\qquad x, y\in {\mathord{\mathbb R}}, \end{equation}
we have \begin{eqnarray}
\nonumber \vert D_k e^{tF} \vert & = &
\sqrt{p_kq_k}
\vert e^{tF_k^+}- e^{tF_k^-} \vert \\
\nonumber & \leq & \frac{1}{2}
\sqrt{p_kq_k}
t \vert F_k^+ - F_k^- \vert
(e^{tF_k^+} + e^{tF_k^-} ) \\
\nonumber & = & \frac{1}{2}
t \vert D_k F \vert
(e^{tF_k^+} + e^{tF_k^-} ) \\ \label{djkddsad}
& \leq & \frac{t}{2(p_k\wedge q_k)}
\vert D_k F \vert \mathbb{E}\left[
e^{tF}
\mid X_i, \ i\not=k
\right] \\
\nonumber
& = &
\frac{1}{2(p_k\wedge q_k)}
t \mathbb{E}\left[
e^{tF}
\vert D_k F \vert
\mid X_i, \ i\not=k
\right], \end{eqnarray}
where in \eqref{djkddsad}
the inequality is due to the absence of chain
rule of derivation for the operator $D_k$.
Now, Proposition~\ref{covclark0} yields \begin{eqnarray*} \mathbb{E}[Fe^{tF}] & = & \mathop{\hbox{\rm Cov}}\nolimits (F,e^{s F}) \\
&=&
\sum_{k=0}^\infty
\mathbb{E}[\mathbb{E}[D_k F \mid {\cal F}_{k-1} ]
D_k e^{tF} ] \\ & \leq & \sum_{k=0}^\infty
\Vert D_k F \Vert_\infty
\mathbb{E}\left[\vert D_k e^{tF} \vert \right] \\
& \leq & \frac{t}{2} \sum_{k=0}^\infty
\frac{1}{p_k\wedge q_k}
\Vert D_k F \Vert_\infty
\mathbb{E}\left[
\mathbb{E}\left[
e^{tF}
\vert D_k F \vert
\mid X_i, \ i\not=k
\right] \right] \\
& = & \frac{t}{2}
\mathbb{E}\left[
e^{tF}
\sum_{k=0}^\infty
\frac{1}{p_k\wedge q_k}
\Vert D_k F \Vert_\infty
\vert D_k F \vert
\right] \\
& \leq & \frac{t}{2}
\mathbb{E}[ e^{tF} ]
\left\|
\sum_{k=0}^\infty
\frac{1}{p_k\wedge q_k}
\vert D_k F \vert
\Vert D_k F \Vert_\infty
\right\|_\infty. \end{eqnarray*}
This shows that \begin{eqnarray*}
\log (\mathbb{E}[e^{t(F-\mathbb{E}[F])}])
& = & \int_0^t \frac{\mathbb{E}[(F-\mathbb{E}[F])e^{s(F-\mathbb{E}[F])}]}{\mathbb{E}[e^{s(F-\mathbb{E}[F])}]} ds \\ & \leq & K^2 \int_0^t s ds \\
& = & \frac{t^2}{2} K^2, \end{eqnarray*}
hence \begin{eqnarray*}
e^{x} \mathbb{P}(F-\mathbb{E}[F]\geq x)
& \leq & \mathbb{E}[e^{t(F-\mathbb{E}[F])}] \\ & \leq & e^{t^2 K^2 / 2 }, \qquad t\geq 0, \end{eqnarray*}
and $$
\mathbb{P}(F-\mathbb{E}[F]\geq x)
\leq e^{\frac{t^2}{2} K^2-tx}, \quad t\geq 0. $$
The best inequality is obtained for $t=x/K^2$.
If $F$ is not bounded the conclusion
holds for $F_n = \max (-n , \min ( F, n))$, $n\geq 0$,
and $(F_n)_{n\in{\mathord{\mathbb N}}}$,
$(DF_n)_{n\in{\mathord{\mathbb N}}}$, converge respectively to $F$ and $DF$
in $L^2(\Omega )$, resp. $L^2(\Omega \times {\mathord{\mathbb N}} )$, with
$\Vert DF_n \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2
\leq \Vert DF \Vert_{L^\infty (\Omega ,\ell^2 ({\mathord{\mathbb N}} ))}^2$. \end{Proof}
The bound \eqref{bdd} implies
$\mathbb{E}[e^{\alpha \vert F\vert}]<\infty$ for all $\alpha >0$,
and $\mathbb{E}[e^{\alpha F^2}]<\infty$ for all $\alpha <
1/(2K^2)$.
In case $p_k=p$, $k\in{\mathord{\mathbb N}}$, we obtain $$\mathbb{P}(F-\mathbb{E}[F]\geq x) \leq \exp \left(
- \frac{px^2}{\Vert DF \Vert_{
\ell^2({\mathord{\mathbb N}},L^\infty (\Omega ))}^2}
\right). $$ \section{Logarithmic Sobolev Inequalities}\index{logarithmic Sobolev inequalities!discrete time} \label{lsid}
The logarithmic Sobolev inequalities on Gaussian space
provide an infinite dimensional analog of Sobolev inequalities,
cf. e.g. \cite{ledouxmarkov}.
On Riemannian path space \cite{capitaine} and on Poisson space \cite{ane},
\cite{wuls2}, martingale methods have been successfully applied to
the proof of logarithmic Sobolev inequalities.
Here, discrete time martingale methods are used
along with the Clark predictable representation formula
\eqref{clk} as in \cite{gaopri},
to provide a proof of logarithmic Sobolev inequalities
for Bernoulli measures.
Here we are only concerned with modified logarithmic
Sobolev inequalities, and we refer to
\cite{saloff}, Theorem~2.2.8 and references
therein, for the standard version of the
logarithmic Sobolev inequality on the hypercube
under Bernoulli measures.
The entropy of a random variable
$F>0$ is defined by $${\mathrm{{\rm Ent \ \!}}} [F] = \mathbb{E} [F \log F ] -
\mathbb{E} [F] \log \mathbb{E} [F],$$
for sufficiently integrable $F$. \begin{lemma} \label{tensorlemma}
The entropy has the tensorization
property, i.e. if $F,G$ are sufficiently integrable
independent random variables we have \begin{equation} \label{tensop} {\mathrm{{\rm Ent \ \!}}} [ FG ]
= \mathbb{E}[F{\mathrm{{\rm Ent \ \!}}} [ G ] ] + \mathbb{E}[G{\mathrm{{\rm Ent \ \!}}} [F] ] . \end{equation} \end{lemma} \begin{Proof}
We have \begin{eqnarray*} \nonumber {\mathrm{{\rm Ent \ \!}}} [ FG ] & = & \mathbb{E}[FG\log (FG)] - \mathbb{E}[FG]\log \mathbb{E}[FG] \\ \nonumber & = & \mathbb{E}[FG ( \log F + \log G) ] - \mathbb{E}[F]\mathbb{E}[G] ( \log \mathbb{E}[F] + \log \mathbb{E}[G] ) \\ \nonumber & = & \mathbb{E}[G] \mathbb{E}[F \log F ] + \mathbb{E}[F] \mathbb{E}[ G \log G) ] - \mathbb{E}[F]\mathbb{E}[G] ( \log \mathbb{E}[F] + \log \mathbb{E}[G] ) \\
& = & \mathbb{E}[F{\mathrm{{\rm Ent \ \!}}} [ G ] ] + \mathbb{E}[G{\mathrm{{\rm Ent \ \!}}} [F] ] . \end{eqnarray*} \end{Proof}
In the next proposition we recover the
modified logarithmic Sobolev inequality of \cite{bobkov}
using the Clark representation formula in discrete time. \begin{theorem} \label{lsaprop}
Let $F\in {\mathrm{{\rm Dom \ \!}}} (D)$
with $F>\eta$ a.s. for some $\eta>0$.
We have \begin{equation} \label{lsa} {\mathrm{{\rm Ent \ \!}}} [F]
\leq \mathbb{E}
\left[\frac{1}{F} \Vert DF\Vert_{\ell^2({\mathord{\mathbb N}} )}^2
\right]. \end{equation} \end{theorem} \begin{Proof}
Assume that $F$ is ${\cal F}_N$-measurable and
let $M_n = \mathbb{E} [F\mid {\cal F}_n]$, $0\leq n\leq N$.
Using Corollary~\ref{cormes} and the Clark formula \eqref{clk}
we have $$M_n = M_{-1} + \sum_{k=0}^n
u_k Y_k,
\qquad
0 \leq n \leq N , $$
with $u_k = \mathbb{E} [D_k F \mid {\cal F}_{k-1}]$,
$0\leq k \leq n \leq N$,
and $M_{-1}=\mathbb{E}[F]$.
Letting $f(x) = x\log x$ and using the bound \begin{eqnarray*}
f(x+y)-f(x) & = & y\log x + (x+y)\log \left(
1+\frac{y}{x} \right) \\ & \leq & y(1+\log x) + \frac{y^2}{x}, \end{eqnarray*}
we have: \begin{eqnarray*}
{\mathrm{{\rm Ent \ \!}}} [F]
& = &
\mathbb{E}[f(M_N)]-\mathbb{E}[f(M_{-1})] \\ & = &
\mathbb{E} \left[ \sum_{k=0}^N f(M_k ) -f(M_{k-1}) \right] \\ & = & \mathbb{E} \left[ \sum_{k=0}^N f\left( M_{k-1} + Y_k u_k \right) -f(M_{k-1}) \right] \\ & \leq &
\mathbb{E} \left[ \sum_{k=0}^N
Y_k u_k (1+\log M_{k-1} ) + \frac{Y^2_k u^2_k}{M_{k-1}} \right] \\
& = &
\mathbb{E} \left[ \sum_{k=0}^N \frac{1}{\mathbb{E}[F\mid {\cal F}_{k-1}]}
(\mathbb{E}[D_kF\mid {\cal F}_{k-1}])^2 \right] \\
& \leq &
\mathbb{E} \left[ \sum_{k=0}^N
\mathbb{E}\left[\frac{1}{F}
| D_kF |^2 \mid {\cal F}_{k-1}\right] \right] \\
& = &
\mathbb{E} \left[ \frac{1}{F}
\sum_{k=0}^N
| D_kF |^2 \right]. \end{eqnarray*}
where we used the Jensen inequality
and the convexity of $(u,v)\mapsto v^2/u$ on $(0,\infty)\times {\mathord{\mathbb R}}$,
or the Schwarz inequality applied to $1/\sqrt{F}$
and $(D_k F/\sqrt{F})_{k\in {\mathord{\mathbb N}}}$,
as in the Wiener and Poisson cases \cite{capitaine} and \cite{ane}.
This inequality is extended by density to $F\in {\mathrm{{\rm Dom \ \!}}} (D)$. \end{Proof}
Theorem~\ref{lsaprop} can also be recovered by
the tensorization Lemma~\ref{tensorlemma} and the following
one-variable argument:
letting $p+q=1$, $p,q>0$, $f:\{ -1,1\} \to (0,\infty )$,
$\mathbb{E}[f]=pf(1)+qf(-1)$, and ${\mathord{{\rm d}}} f = f(1)-f(-1)$ we have: \begin{eqnarray*} \lefteqn{ {\mathrm{{\rm Ent \ \!}}} [ f ] = pf(1)\log f(1) + qf(-1)\log f(-1)
- \mathbb{E}[f]\log \mathbb{E}[f] } \\ & = & pf(1)\log (\mathbb{E}[f] + q{\mathord{{\rm d}}} f)
+ qf(-1)\log (\mathbb{E}[f] -p {\mathord{{\rm d}}} f)
- (pf(1)+qf(-1))\log \mathbb{E}[f] \\ & = & pf(1)\log \left( 1 + q \frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}\right)
+ qf(-1)\log \left( 1 - p \frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}\right) \\ & \leq & pqf(1) \frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}
- pqf(-1) \frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}
= pq \frac{ | {\mathord{{\rm d}}} f |^2}{\mathbb{E}[f]} \\
& \leq & pq \mathbb{E}\left[ \frac{1}{f} | {\mathord{{\rm d}}} f |^2
\right] . \end{eqnarray*}
Similarly we have \begin{eqnarray*} {\mathrm{{\rm Ent \ \!}}} [ f ] & = & pf(1)\log f(1) + qf(-1)\log f(-1)
- \mathbb{E}[f]\log \mathbb{E}[f] \\ & = & p (\mathbb{E}[f] + q{\mathord{{\rm d}}} f) \log (\mathbb{E}[f] + q{\mathord{{\rm d}}} f) \\ & & + q(\mathbb{E}[f] -p {\mathord{{\rm d}}} f) \log (\mathbb{E}[f] -p {\mathord{{\rm d}}} f)
- (pf(1)+qf(-1))\log \mathbb{E}[f] \\ & = & p \mathbb{E}[f] \log \left(1 + q\frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}\right)
+ p q{\mathord{{\rm d}}} f \log f(1) \\ & & + q\mathbb{E}[f] \log \left(1 -p \frac{{\mathord{{\rm d}}} f}{\mathbb{E}[f]}\right)
- qp {\mathord{{\rm d}}} f \log f(-1) \\ & \leq &
p q{\mathord{{\rm d}}} f \log f(1)
- pq {\mathord{{\rm d}}} f \log f(-1) \\ & = & pq \mathbb{E}\left[ {\mathord{{\rm d}}} f {\mathord{{\rm d}}} \log f \right] , \end{eqnarray*}
which, by tensorization, recovers the following $L^1$ inequality of
\cite{gao}, \cite{daipra}, and proved in \cite{wuls2} in the Poisson
case.
In the next proposition we state and prove
this inequality in the multidimensional case,
using the Clark representation formula, similarly
to Theorem~\ref{lsaprop}. \begin{theorem} \label{lsal1}
Let $F>0$ be ${\cal F}_N$-measurable.
We have \begin{equation} \label{lsa3} {\mathrm{{\rm Ent \ \!}}} [F]
\leq \mathbb{E}
\left[\sum_{k=0}^N
D_kF D_k \log F \right] . \end{equation} \end{theorem} \begin{Proof}
Let $f(x) = x\log x$ and $$\Psi (x,y) = (x+y)\log (x+y) - x\log x - (1+\log x)y,
\quad x, \ x+y >0. $$
From the relation \begin{eqnarray*} \lefteqn{ Y_ku_k = Y_k \mathbb{E} [ D_k F \mid {\cal F}_{k-1} ] } \\ & = & q_k {\bf 1}_{\{X_k=1\}}
\mathbb{E}[(F_k^+-F_k^-)\mid {\cal F}_{k-1}]
+ p_k {\bf 1}_{\{X_k=-1\}}
\mathbb{E}[(F_k^- - F_k^+)\mid {\cal F}_{k-1}] \\ & = & {\bf 1}_{\{X_k=1\}} \mathbb{E}[(F_k^+-F_k^-){\bf 1}_{\{X_k=-1\}}
\mid {\cal F}_{k-1} ] + {\bf 1}_{\{X_k= -1\}}
\mathbb{E}[(F_k^--F_k^+){\bf 1}_{\{X_k=1\}}
\mid {\cal F}_{k-1} ], \end{eqnarray*}
we have, using the convexity of $\Psi$: \begin{eqnarray*} \lefteqn{ {\mathrm{{\rm Ent \ \!}}} [F] = \mathbb{E} \left[ \sum_{k=0}^N f\left( M_{k-1} + Y_k u_k \right) -f(M_{k-1}) \right] } \\ & = &
\mathbb{E} \left[ \sum_{k=0}^N \Psi ( M_{k-1}, Y_k u_k ) + Y_k u_k (1+\log M_{k-1} ) \right] \\ & = &
\mathbb{E} \left[ \sum_{k=0}^N \Psi ( M_{k-1}, Y_k u_k ) \right] \\ & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k
\Psi
\left(
\mathbb{E}[F\mid {\cal F}_{k-1}]
,
\mathbb{E}[(F_k^+-F_k^-){\bf 1}_{\{X_k=-1\}}
\mid {\cal F}_{k-1} ]
\right) \right. \\ & & \left.
+
q_k
\Psi \left( \mathbb{E}[F\mid {\cal F}_{k-1}]
,
\mathbb{E}[(F_k^--F_k^+){\bf 1}_{\{X_k=1\}} \mid
{\cal F}_{k-1} ]
\right)
\right] \\ & \leq &
\mathbb{E} \left[
\sum_{k=0}^N
\mathbb{E}\left[
p_k
\Psi
\left(
F
,
(F_k^+-F_k^-){\bf 1}_{\{X_k=-1\}}
\right)
+
q_k
\Psi \left( F
,
(F_k^--F_k^+){\bf 1}_{\{X_k=1\}}
\right)
\mid {\cal F}_{k-1}
\right]\right] \\ & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k{\bf 1}_{\{X_k=-1\}}
\Psi
\left(
F_k^-
,
F_k^+-F_k^-
\right)
+
q_k{\bf 1}_{\{X_k=1\}}
\Psi \left( F_k^+
,
F_k^--F_k^+
\right)
\right] \\ & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k q_k
\Psi(F_k^-
,
F_k^+ - F_k^-
)
+
p_kq_k
\Psi(F_k^+
,
F_k^- - F_k^+
)
\right] \\ & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k q_k
(\log F_k^+ - \log F_k^- )
(F_k^+ - F_k^- ) \right] \\ & = &
\mathbb{E} \left[
\sum_{k=0}^N
D_k F D_k \log F \right] . \end{eqnarray*} \end{Proof}
The proof of Theorem~\ref{lsal1} can also be obtained
by first using the bound $$f(x+y)-f(x) = y\log x + (x+y)\log \left(
1+\frac{y}{x} \right)
\leq y(1+\log x) + y\log (x+y), $$
and then the convexity of $(u,v)\to v(\log (u+v)-\log u)$: \begin{eqnarray*} \lefteqn{
{\mathrm{{\rm Ent \ \!}}} [F]
=
\mathbb{E} \left[ \sum_{k=0}^N f\left( M_{k-1} + Y_k u_k \right) -f(M_{k-1}) \right] } \\ & \leq &
\mathbb{E} \left[ \sum_{k=0}^N
Y_k u_k (1+\log M_{k-1} ) +
Y_k u_k
\log (
M_{k-1} + Y_k u_k ) \right] \\
& = &
\mathbb{E} \left[ \sum_{k=0}^N
Y_k u_k
(\log (
M_{k-1} + Y_k u_k )
-
\log M_{k-1} ) \right] \\
& = &
\mathbb{E} \left[ \sum_{k=0}^N
\sqrt{p_kq_k}
\mathbb{E}[D_kF\mid {\cal F}_{k-1}]
(\log
\mathbb{E}[F + (F_k^+-F_k^-){\bf 1}_{\{X_k=-1\}}
\mid {\cal F}_{k-1} ]
-
\log \mathbb{E}[F\mid {\cal F}_{k-1}] ) \right. \\ & & \left.
- \sqrt{p_kq_k}
\mathbb{E}[D_kF\mid {\cal F}_{k-1}]
(\log
\mathbb{E}[F +(F_k^--F_k^+){\bf 1}_{\{X_k=-1\}} \mid {\cal F}_{k-1} ]
-
\log \mathbb{E}[F\mid {\cal F}_{k-1} ])\right] \\
& \leq &
\mathbb{E} \left[ \sum_{k=0}^N
\mathbb{E} \left[
\sqrt{p_kq_k}
D_kF
(\log (
F + (F_k^+-F_k^-){\bf 1}_{\{X_k=-1\}}
) -
\log F ) \right. \right. \\ & & \left. \left.
- \sqrt{p_kq_k}
D_kF
(\log (
F +(F_k^--F_k^+){\bf 1}_{\{X_k=1\}}
)
-
\log F)
\mid {\cal F}_{k-1} \right]\right] \\
& = &
\mathbb{E} \left[ \sum_{k=0}^N
\sqrt{p_kq_k}
D_kF {\bf 1}_{\{X_k=-1\}}
(\log F_k^+ -
\log F_k^- )
\right. \\
& &
\left.
- \sqrt{p_kq_k}
D_kF {\bf 1}_{\{X_k=1\}}
(\log F_k^- - \log F_k^+ ) \right] \\
& \leq &
\mathbb{E} \left[ \sum_{k=0}^N
\sqrt{p_kq_k}
q_k D_kF
(\log F_k^+
- \log F_k^- )
- \sqrt{p_kq_k}
p_k D_kF
(\log F_k^- - \log F_k^+ ) \right] \\ & = &
\mathbb{E} \left[ \sum_{k=0}^N
D_k F D_k \log F
\right] . \end{eqnarray*}
The application of Theorem~\ref{lsal1} to $e^F$ gives
the following inequality for $F>0$, ${\cal F}_N$-measurable: \begin{eqnarray} \nonumber {\mathrm{{\rm Ent \ \!}}}[e^F] & \leq & \mathbb{E} \left[
\sum_{k=0}^N
D_k F D_k e^F \right] \\ \nonumber & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k q_k
\Psi(e^{F_k^-}
,
e^{F_k^+} - e^{F_k^-}
)
+
p_k q_k
\Psi( e^{F_k^+}
,
e^{F_k^-} - e^{F_k^+}
)
\right] \\ \nonumber & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k q_k
e^{F_k^-}
((F_k^+-F_k^-)e^{F_k^+-F_k^-}
- e^{F_k^+-F_k^-}
+ 1 ) \right. \\ \nonumber
& & \left. +
p_k q_k
e^{F_k^+}
((F_k^- -F_k^+ )e^{F_k^- -F_k^+}
- e^{F_k^--F_k^+}
+ 1 ) \right] \\ \nonumber & = &
\mathbb{E} \left[
\sum_{k=0}^N
p_k {\bf 1}_{\{X_k=-1\}}
e^{F_k^-}
((F_k^+-F_k^-)e^{F_k^+-F_k^-}
- e^{F_k^+-F_k^-}
+ 1 ) \right. \\ \nonumber
& & \left. +
q_k {\bf 1}_{\{X_k=1\}}
e^{F_k^+}
((F_k^- -F_k^+ )e^{F_k^- -F_k^+}
- e^{F_k^--F_k^+}
+ 1 ) \right] \\ \label{lsa3.0} & = &
\mathbb{E} \left[
e^{F}
\sum_{k=0}^N
\sqrt{p_kq_k}
\vert Y_k \vert
( \nabla_k F e^{ \nabla_k F }
- e^{\nabla_k F} +1)
\right] . \end{eqnarray}
This implies \begin{equation} \label{lll}
{\mathrm{{\rm Ent \ \!}}} [e^F]
\leq
\mathbb{E} \left[
e^{F}
\sum_{k=0}^N
( \nabla_k F e^{ \nabla_k F }
- e^{\nabla_k F} +1)
\right]. \end{equation} \noindent
As already noted in \cite{daipra},
\eqref{lsa3} and the Poisson limit theorem
yield the $L^1$ inequality of \cite{wuls2}.
Let $M_n = (n + X_1 +\cdots +X_n)/2$,
$F = \varphi ( M_n )$,
and $p_k = \lambda /n$, $k\in {\mathord{\mathbb N}}$, $\lambda >0$.
Then \begin{eqnarray*} \lefteqn{ \sum_{k=0}^n
D_kF D_k \log F } \\
& = & \frac{\lambda}{n}
\left(1-\frac{\lambda}{n} \right)
(n-M_n) (\varphi (M_n + 1) - \varphi (M_n))
\log (\varphi (M_n + 1) - \varphi (M_n)) \\ & & +
\frac{\lambda}{n}\left(1-\frac{\lambda}{n}\right)
M_n (\varphi (M_n ) - \varphi (M_n - 1))
\log (\varphi (M_n ) - \varphi (M_n - 1)). \end{eqnarray*}
In the limit we obtain $${\mathrm{{\rm Ent \ \!}}} [\varphi (U) ] \leq \lambda \mathbb{E}
[ (\varphi (U+1)-\varphi (U))(\log \varphi (U+1)-\log \varphi (U))], $$
where $U$ is a Poisson random variable with parameter $\lambda$.
In one variable we have, still letting ${\mathord{{\rm d}}} f = f(1)-f(-1)$, \begin{eqnarray*} {\mathrm{{\rm Ent \ \!}}} [ e^f ] & \leq & pq \mathbb{E}\left[ {\mathord{{\rm d}}} e^f {\mathord{{\rm d}}} \log e^f \right] \\ & = & p q(e^{f(1)}-e^{f(-1)}) (f(1)-f(-1)) \\ & = & p qe^{f(-1)}(
(f(1)-f(-1))e^{f(1)-f(-1)}
- e^{f(1)-f(-1)}
+ 1 ) \\ & & +
p qe^{f(1)}(
(f(-1)-f(1))e^{f(-1)-f(1)}
- e^{f(-1)-f(1)}
+ 1 ) \\ & \leq & qe^{f(-1)}(
(f(1)-f(-1))e^{f(1)-f(-1)}
- e^{f(1)-f(-1)}
+ 1 ) \\ & & +
p e^{f(1)}(
(f(-1)-f(1))e^{f(-1)-f(1)}
- e^{f(-1)-f(1)}
+ 1 ) \\ & = & \mathbb{E}[e^f(
\nabla f e^{\nabla f}
- e^{\nabla f}
+ 1 ) ] , \end{eqnarray*}
where $\nabla_k$ is the gradient operator defined in
\eqref{mod2}.
This last inequality is not comparable to the optimal constant inequality \begin{equation} \label{sh} {\mathrm{{\rm Ent \ \!}}} [e^F]
\leq
\mathbb{E} \left[
e^{F}
\sum_{k=0}^N
p_kq_k ( \vert \nabla_k F \vert e^{\vert \nabla_k F \vert }
- e^{\vert \nabla_k F \vert} +1)
\right] , \end{equation}
of \cite{bobkov} since when $F_k^+-F_k^-\geq 0$ the right-hand side of
\eqref{sh} grows as $F_k^+e^{2F_k^+}$,
instead of $F_k^+e^{F_k^+}$ in \eqref{lll}.
In fact we can prove the following inequality which improves
\eqref{lsa}, \eqref{lsa3} and \eqref{sh}. \begin{theorem} \label{lsal1.1}
Let $F$ be ${\cal F}_N$-measurable. We have \begin{equation} \label{lsa3.01}
{\mathrm{{\rm Ent \ \!}}} [e^F]
\leq
\mathbb{E} \left[ e^{F}
\sum_{k=0}^N
p_kq_k
( \nabla_k F e^{ \nabla_k F }
- e^{\nabla_k F} +1)
\right] . \end{equation} \end{theorem}
Clearly, \eqref{lsa3.01} is better than
\eqref{sh}, \eqref{lsa3.0} and \eqref{lsa3}.
It also improves \eqref{lsa} from the bound $$xe^x-e^x+1\leq ( e^x-1 )^2, \quad x\in {\mathord{\mathbb R}} , $$
which implies $$e^F ( \nabla F e^{\nabla F} -e^{\nabla F} + 1 ) \leq e^F ( e^{\nabla F}-1 )^2
=
e^{-F} | \nabla e^{F} |^2 . $$
By the tensorization property \eqref{tensop}, the proof of
\eqref{lsa3.01} reduces to the following
one dimensional lemma. \begin{lemma} For any $0\leq p\leq 1$, $t\in{\mathbb R}$, $a \in{\mathbb R}$, $q=1-p$, \begin{eqnarray*} \lefteqn{ pt e^t+qa e^a-\left(p e^t+q e^a\right)\log\left( p e^t+ q e^a\right) } \\ &\leq & p q \left( q e^a\left((t-a)e^{t-a}-e^{t-a}+1\right) +pe^t\left((a-t)e^{a-t}-e^{a-t}+1\right)\right). \end{eqnarray*} \end{lemma} \begin{Proof} Set \begin{eqnarray*} g(t) &= &pq \left(qe^a\left((t-a)e^{t-a}-e^{t-a}+1\right) +pe^t\left((a-t)e^{a-t}-e^{a-t}+1\right)\right)\\ & & - pt e^t-qa e^a+\left(p e^t+qe^a\right)\log\left( p e^t+qe^a\right). \end{eqnarray*}
Then $$g'(t)= pq\left(q e^a(t - a)e^{t - a} + pe^t\left(-e^{a-t} + 1\right)\right)
- pte^t+pe^t\log(pe^t + q e^a) $$ and $g''(t)=p e^t h(t)$, where $$ h(t) = - a - 2pt - p + 2pa + p^2t - p^2a
+ \log( pe^t + q e^a ) + \frac{pe^t}{p e^t + q e^a }. $$
Now, \begin{eqnarray*} h'(t) & =&- 2p + p^2+ \frac{ 2 pe^t}{pe^t + qe^a} - \frac{p^2e^{2t}}{(p e^t + qe^a )^2}\\ &=&\frac{pq^2(e^t-e^a)(pe^t+(q+1)e^a )}{(pe^t+qe^a)^2}, \end{eqnarray*} which implies that $h'(a)=0$, $h'(t)<0 $ for any $t<a$ and $h'(t)>0 $ for any $t>a$.
Hence, for any $t\not= a$, $h(t)>h(a)=0$, and so $g''(t)\geq 0$ for any $t\in {\mathbb R}$ and $g''(t)=0$ if and only if $t=a$. Therefore, $g'$ is strictly increasing. Finally, since $t=a$ is the unique root of $g'=0$, we have that $g(t)\geq g(a)=0$ for all $t\in {\mathord{\mathbb R}}$. \end{Proof} \noindent
This inequality improves \eqref{lsa}, \eqref{lsa3}, and \eqref{sh},
as illustrated in one dimension in Figure~\ref{g5.0}, where
the entropy is represented as a function of $p\in [0,1]$
with $f(1)=1$ and $f(-1)=3.5$.
The inequality \eqref{lsa3.01} is a discrete analog of the sharp inequality
on Poisson space of \cite{wuls2}.
In the symmetric case $p_k=q_k=1/2$, $k\in {\mathord{\mathbb N}}$,
we have \begin{eqnarray*}
{\mathrm{{\rm Ent \ \!}}} [e^F]
& \leq &
\mathbb{E}\left[
e^F
\sum_{k=0}^N
p_kq_k
( \nabla_k F e^{ \nabla_k F }
- \nabla_k F +1) \right] \\
& = &
\frac{1}{8}
\mathbb{E} \left[
\sum_{k=0}^N
e^{F_k^-}
((F_k^+-F_k^-)e^{F_k^+-F_k^-}
- e^{F_k^+-F_k^-}
+ 1 ) \right. \\
& & \left. +
e^{F_k^+}
((F_k^- -F_k^+ )e^{F_k^- -F_k^+}
- e^{F_k^--F_k^+}
+ 1 ) \right] \\
& = &
\frac{1}{8}
\mathbb{E} \left[
\sum_{k=0}^N
(e^{F_k^+}-e^{F_k^-})
(F_k^+ -F_k^- ) \right] \\ & = &
\frac{1}{2}
\mathbb{E}\left[
\sum_{k=0}^N
D_k F D_k e^ F
\right], \end{eqnarray*}
which improves on \eqref{lsa3}. \\
\begin{figure}
\caption{\small
Graph of the entropy as a function of $p$. }
\label{g5.0}
\end{figure}
Letting $F=\varphi (M_n)$ we have \begin{eqnarray*} \lefteqn{
\mathbb{E}\left[
e^F
\sum_{k=0}^N
p_kq_k
( \nabla_k F e^{ \nabla_k F }
- \nabla_k F +1)
\right] } \\ & = &
\frac{\lambda}{n}
\left(1-\frac{\lambda}{n} \right)
\mathbb{E}\left[
M_n
e^{\varphi (M_n )} \right. \\
& &
\times
\left.
((\varphi (M_n )
- \varphi (M_n - 1)
)
e^{\varphi (M_n )
- \varphi (M_n - 1)
}
- e^{\varphi (M_n )
- \varphi (M_n - 1)
}
+ 1 )
\right] \\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \!
+
\frac{\lambda}{n}
\left(1-\frac{\lambda}{n} \right)
\mathbb{E}\left[
(n-M_n) e^{\varphi (M_n )}
\right. \\
& &
\times
\left.
((\varphi (M_n +1) - \varphi (M_n )
)
e^{\varphi (M_n +1 )
- \varphi (M_n )}
- e^{\varphi (M_n +1 )
- \varphi (M_n )
} + 1 ) \right] , \end{eqnarray*}
and in the limit as $n$ goes to infinity we obtain $${\mathrm{{\rm Ent \ \!}}} [ e^{\varphi (U)} ] \leq \lambda \mathbb{E}
[ e^{\varphi (U)} (
(\varphi (U+1)-\varphi (U)) e^{\varphi (U+1)-\varphi (U)}
- e^{\varphi (U+1)-\varphi (U)}
+1
) ], $$
where $U$ is a Poisson random variable with parameter $\lambda$.
This corresponds to the sharp inequality of \cite{wuls2}. \section{Change of Variable Formula} \label{s11}
In this section we state a discrete-time analog of It\^o's change
of variable formula which will be useful for the predictable
representation of random variables and for option hedging. \begin{prop} \label{it}
Let $(M_n)_{n\in {\mathord{\mathbb N}}}$ be a square-integrable martingale
and $f : {\mathord{\mathbb R}}\times {\mathord{\mathbb N}} \to {\mathord{\mathbb R}}$.
We have \begin{eqnarray} \label{note00} \lefteqn{
f(M_n,n) } \\ \nonumber
& = & f(M_{-1},-1)
+ \sum_{k=0}^n
D_k f(M_k,k)
Y_k
+ \sum_{k=0}^n
\mathbb{E}[f(M_k,k) -f(M_{k-1},k-1) \mid {\cal F}_{k-1}] . \end{eqnarray} \end{prop} \begin{Proof}
By Proposition~\ref{martrepr} there exists square-integrable
process $(u_k)_{k\in{\mathord{\mathbb N}}}$ such that $$M_n = M_{-1} + \sum_{k=0}^n u_k Y_k,
\qquad
n\in{\mathord{\mathbb N}}. $$
We write \begin{eqnarray*}
f(M_n,n) - f(M_{-1},-1) & = & \sum_{k=0}^n
f(M_k,k)-f(M_{k-1},k-1) \\ & = & \sum_{k=0}^n
f(M_k,k)-f(M_{k-1},k) + f(M_{k-1},k)-f(M_{k-1},k-1) \\ & = &
\sum_{k=0}^n
\sqrt{\frac{p_k}{q_k}} \left( f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}}, k \right) -f(M_{k-1},k) \right) Y_k \\ & & + \frac{p_k}{q_k} {\bf 1}_{\{X_k=- 1 \}} \left( f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}},k \right) -f(M_{k-1},k) \right) \\ & & + {\bf 1}_{\{X_k=- 1 \}} \left( f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} , k \right) -f(M_{k-1},k) \right) \\
& & +
\sum_{k=0}^n
f(M_{k-1},k)-f(M_{k-1},k-1) \\
& = &
\sum_{k=0}^n \sqrt{\frac{p_k}{q_k}} \left( f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}},k \right) -f(M_{k-1},k) \right) Y_k \\ \nonumber & & + \sum_{k=0}^n \frac{1}{q_k} {\bf 1}_{\{ X_k=-1 \}} \mathbb{E}[f(M_k,k)-f(M_{k-1},k) \mid {\cal F}_{k-1}] \\ \nonumber & & + \sum_{k=0}^n f(M_{k-1},k)-f(M_{k-1},k-1) . \end{eqnarray*}
Similarly we have \begin{eqnarray*} f(M_n,n) & = & f(M_{-1},-1) - \sum_{k=0}^n \sqrt{\frac{q_k}{p_k}} \left( f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} ,k \right) -f(M_{k-1} , k ) \right) Y_k \\ & & + \sum_{k=0}^n \frac{1}{p_k} {\bf 1}_{\{X_k= 1 \}} \mathbb{E}[f(M_k,k)-f(M_{k-1},k) \mid {\cal F}_{k-1}] \\
& & +
\sum_{k=0}^n
f(M_{k-1},k)-f(M_{k-1},k-1) , \end{eqnarray*}
Multiplying each increment in the above formulas
respectively by $q_k$ and $p_k$ and summing on $k$ we get \begin{eqnarray*} \lefteqn{ f(M_n,n) = f(M_{-1},-1) } \\ & &+ \sum_{k=0}^n \sqrt{p_kq_k} \left( f\left( _{k-1} + u_k \sqrt{\frac{q_k}{p_k}} ,k \right) - f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} ,k \right) \right) Y_k \\ & & + \sum_{k=0}^n \mathbb{E}[f(M_k,k) \mid {\cal F}_{k-1}]
-f(M_{k-1},k ) . \end{eqnarray*} \end{Proof}
Note that in \eqref{note00} we have $$ D_k f(M_k,k) = \sqrt{p_kq_k} \left( f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}} ,k \right) - f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} ,k \right) \right) , \quad k\in {\mathord{\mathbb N}}. $$
\noindent
On the other hand, the term $$ \mathbb{E}[f(M_k,k) -f(M_{k-1},k-1) \mid {\cal F}_{k-1}] $$
is analog to the generator part in the continuous time
It\^o formula, and can be written as $$
p_k f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}} ,k \right)
+
q_k f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} ,k \right)
- f\left( M_{k-1} ,k-1 \right) . $$
When $p_n=q_n=1/2$, $n\in{\mathord{\mathbb N}}$, we have \begin{eqnarray*}
f(M_n,n) & = & f(M_{-1},-1) + \sum_{k=0}^n \frac{f\left( M_{k-1} + u_k ,k \right) - f\left( M_{k-1} - u_k ,k \right) } {2} Y_k \\
& &
+ \sum_{k=0}^n \frac{f\left( M_{k-1} + u_k ,k \right) + f\left( M_{k-1} - u_k ,k \right)
- 2 f\left( M_{k-1} ,k-1 \right) } {2} . \end{eqnarray*}
The above proposition also provides an explicit version of
the Doob decomposition for supermartingales.
Naturally if $(f(M_n,n))_{n\in{\mathord{\mathbb N}}}$ is a martingale we have \begin{eqnarray*} \lefteqn{ f(M_n,n) = f(M_{-1},-1) } \\
& & + \sum_{k=0}^n \sqrt{p_kq_k} \left( f\left( M_{k-1} + u_k \sqrt{\frac{q_k}{p_k}} , k \right) - f\left( M_{k-1} - u_k \sqrt{\frac{p_k}{q_k}} , k \right) \right) Y_k \\
& = & f(M_{-1},-1)
+ \sum_{k=0}^n
D_k f(M_k,k) Y_k . \end{eqnarray*}
In this case the Clark formula, the martingale representation
formula Proposition~\ref{martrepr} and the change of variable formula all coincide.
In this case, we have in particular $$D_k f(M_k,k)
= \mathbb{E}[D_k f(M_n,n) \mid {\cal F}_{k-1}]
= \mathbb{E}[D_k f(M_k,k) \mid {\cal F}_{k-1}] , \qquad k\in {\mathord{\mathbb N}} . $$
If $F$ is an ${\cal F}_N$-measurable
random variable and $f$ is a function such that $$ \mathbb{E}[F \mid {\cal F}_n] = f(M_n,n), \qquad -1 \leq n \leq N, $$
we have $F = f(M_N,N)$, $\mathbb{E}[F]= f(M_{-1},-1)$ and \begin{eqnarray*}
F & = & \mathbb{E}[F]
+ \sum_{k=0}^n
\mathbb{E}[D_k f(M_N,N) \mid {\cal F}_{k-1}]
Y_k \\ & = & \mathbb{E}[F]
+ \sum_{k=0}^n
D_k f(M_k,k)
Y_k \\ & = & \mathbb{E}[F]
+ \sum_{k=0}^n
D_k \mathbb{E}[f(M_N,N) \mid {\cal F}_k ]
Y_k . \end{eqnarray*}
Such a function $f$ exists if $(M_n)_{n\in{\mathord{\mathbb N}}}$ is Markov and
$F=h(M_N)$.
In this case, consider the semi-group $(P_{k,n})_{0\leq k < n \leq N}$
associated to $(M_n)_{n\in{\mathord{\mathbb N}}}$ and defined by $$[P_{k,n} h](x) = \mathbb{E}[h(M_n) \mid M_k = x] . $$
Letting $f(x,n) = [P_{n,N}h](x)$ we can write $$
F = \mathbb{E}[F]
+ \sum_{k=0}^n
\mathbb{E}[D_k h(M_N) \mid {\cal F}_{k-1}]
Y_k
= \mathbb{E}[F]
+ \sum_{k=0}^n
D_k [P_{k,N}h (M_k) ] Y_k . $$ \section{Option Hedging in Discrete Time}\index{option hedging!discrete time} \label{hdg}
In this section we give a presentation of the Black-Scholes formula
in discrete time, or Cox-Ross-Rubinstein model, see e.g.
\cite{follmerschied},
\cite{lamberton}, $\S$15-1 of \cite{williams}, or \cite{ruiz},
as an application of the Clark formula. \\
In order to be consistent with the notation of the previous
sections we choose to use the time scale ${\mathord{\mathbb N}}$, hence the
index $0$ is that of the first random value of any stochastic
process, while the index $-1$ corresponds to its deterministic
initial value. \\
Let $(A_k)_{k\in {\mathord{\mathbb N}}}$ be a riskless asset with initial value
$A_{-1}$, and defined by $$ A_n = A_{-1} \prod_{k=0}^n (1+r_k), \qquad n\in {\mathord{\mathbb N}}, $$
where $(r_k)_{k\in {\mathord{\mathbb N}}}$, is a sequence
of deterministic numbers such that $r_k>-1$, $k\in{\mathord{\mathbb N}}$.
Consider a stock price with initial value $S_{-1}$, given
in discrete time as $$
S_n = \left\{
\begin{array}{ll} (1+b_n)S_{n-1}, & X_n=1, \\ \\ (1+a_n)S_{n-1}, & X_n=-1, \quad n\in {\mathord{\mathbb N}} , \end{array} \right. $$
where $(a_k)_{k\in {\mathord{\mathbb N}}}$ and $(b_k)_{k\in {\mathord{\mathbb N}}}$ are sequences
of deterministic numbers such that $$
-1 < a_k < r_k < b_k,
\qquad
k\in {\mathord{\mathbb N}}. $$
We have $$S_n = S_{-1} \prod_{k=0}^n
\sqrt{(1+b_k)(1+a_k)}
\left(
\frac{1+b_k}{1+a_k}\right)^{X_k/2}, \quad n\in {\mathord{\mathbb N}} . $$
Consider now the discounted stock price given as \begin{eqnarray*}
\tilde{S}_n & = &
S_n \prod_{k=0}^n
(1+r_k)^{-1} \\ & = &
S_{-1}
\prod_{k=0}^n
\left(
\frac{1}{1+r_k}
\sqrt{(1+b_k)(1+a_k)}
\left(
\frac{1+b_k}{1+a_k}\right)^{X_k/2}
\right) , \quad n\in {\mathord{\mathbb N}} . \end{eqnarray*}
If $-1 < a_k < r_k < b_k$, $k\in {\mathord{\mathbb N}}$, then
$(\tilde{S}_n)_{n\in{\mathord{\mathbb N}}}$ is a martingale with respect to
$({\cal F}_n)_{n\geq -1}$ under the probability $\mathbb{P}^*$ given by $$p_k = (r_k-a_k)/(b_k-a_k),
\quad
q_k = (b_k-r_k)/(b_k-a_k), \quad
k\in{\mathord{\mathbb N}}. $$
In other terms, under $\mathbb{P}^*$ we have $$
\mathbb{E}^* [ S_{n+1} \mid {\cal F}_n ]
= (1+r_{n+1}) S_n,
\qquad
n\geq -1 , $$
where $\mathbb{E}^*$ denotes the expectation under
$\mathbb{P}^*$.
Recall that under this probability measure there is
absence of arbitrage and the market is complete.
From the change of variable formula Proposition~\ref{it}
or from the Clark formula \eqref{clk}
we have the martingale representation $$\tilde{S}_n = S_{-1} + \sum_{k=0}^n
Y_k D_k \tilde{S}_k
= S_{-1} + \sum_{k=0}^n
\tilde{S}_{k-1}
\sqrt{p_kq_k}\frac{b_k-a_k}{1+r_k}
Y_k . $$ \begin{definition}
A portfolio strategy is a pair of predictable processes
$(\eta_k)_{k\in{\mathord{\mathbb N}}}$ and $(\zeta_k)_{k\in{\mathord{\mathbb N}}}$
where $\eta_k$, resp. $\zeta_k$
represents the numbers of units invested over the
time period $(k,k+1]$ in the asset $S_k$, resp. $A_k$,
with $k\geq 0$. \end{definition}
The value at time $k \geq -1$
of the portfolio $(\eta_k,\zeta_k)_{0\leq k \leq N}$
is defined as \begin{equation} \label{e43.0}
V_k = \zeta_{k+1} A_k + \eta_{k+1} S_k, \qquad
k \geq -1 , \end{equation}
and its discounted value is defined as \begin{equation} \label{plm} \tilde{V}_n = V_n \displaystyle\prod_{k=0}^n (1+r_k)^{-1},
\qquad
n\geq -1 . \end{equation} \begin{definition}
A portfolio $(\eta_k,\zeta_k)_{k \in {\mathord{\mathbb N}}}$ is said to be
self-financing if $$A_n(\zeta_{n+1}-\zeta_{n})+ S_n(\eta_{n+1}-\eta_{n} ) =0,
\qquad n\geq 0. $$ \end{definition}
Note that the self-financing condition implies $$
V_n= \zeta_n A_n + \eta_n S_n, \qquad n\geq 0 . $$
Our goal is to hedge an arbitrary claim on $\Omega$,
i.e. given an ${\cal F}_N$-measurable
random variable $F$ we search for a portfolio
$(\eta_k,\zeta_k)_{ 0 \leq k \leq n}$ such that the equality \begin{equation} \label{prblm1} F = V_N = \zeta_N A_N + \eta_N S_N \end{equation}
holds at time $N\in {\mathord{\mathbb N}}$. \begin{prop}
Assume that the portfolio
$(\eta_k,\zeta_k)_{0\leq k \leq N}$ is self-financing.
Then we have the decomposition \begin{equation} \label{e440.00}
V_n = V_{-1} \prod_{k=0}^n (1+r_k)
+ \sum_{i=0}^n
\eta_i S_{i-1}
\sqrt{p_iq_i} (b_i-a_i) Y_i
\prod_{k=i+1}^n (1+r_k) . \end{equation} \end{prop} \begin{Proof}
Under the self-financing assumption we have \begin{eqnarray*} \label{cll.0}
V_i-V_{i-1} & = & \zeta_i (A_i-A_{i-1}) + \eta_i (S_i-S_{i-1}) \\
& = & r_i \zeta_i A_{i-1} + (a_i{\bf 1}_{\{X_i=-1\}} + b_i{\bf 1}_{\{X_i=1\}}) \eta_i S_{i-1} \\
& = & \eta_i S_{i-1} (a_i{\bf 1}_{\{X_i= -1\}} + b_i {\bf 1}_{\{X_i=1\}} -r_i) + r_i V_{i-1} \\
& = & \eta_i S_{i-1} \sqrt{p_iq_i} (b_i-a_i) Y_i + r_i V_{i-1} , \qquad i\in {\mathord{\mathbb N}}, \end{eqnarray*}
hence for the discounted portfolio we get: \begin{eqnarray*}
\tilde{V}_i-\tilde{V}_{i-1}
& = &
\displaystyle\prod_{k=1}^i (1+r_k)^{-1}
{V}_i
-
\displaystyle\prod_{k=1}^{i-1} (1+r_k)^{-1}
{V}_{i-1} \\
& = &
\displaystyle\prod_{k=1}^{i} (1+r_k)^{-1}
(
{V}_i
-
{V}_{i-1}
- r_i {V}_{i-1}
) \\
& = & \eta_i S_{i-1}
\sqrt{p_iq_i} (b_i-a_i) Y_i
\displaystyle\prod_{k=1}^i (1+r_k)^{-1} , \qquad i\in {\mathord{\mathbb N}}, \end{eqnarray*}
which successively yields \eqref{e440.1.1} and \eqref{e440.00}. \end{Proof}
As a consequence of \eqref{e440.00} and \eqref{plm}
we immediately obtain \begin{equation} \label{e440.1.1}
\tilde{V}_n = \tilde{V}_{-1}
+ \sum_{i=0}^n
\eta_i S_{i-1}
\sqrt{p_iq_i} (b_i-a_i) Y_i
\displaystyle\prod_{k=0}^i (1+r_k)^{-1} ,
\qquad n\geq -1 . \end{equation}
The next proposition provides a solution to the hedging
problem under the constraint \eqref{prblm1}. \begin{prop}
Given $F\in L^2(\Omega , {\cal F}_N )$, let \begin{equation} \label{eta}
\eta_n = \frac{1}{S_{n-1} \sqrt{p_nq_n} (b_n-a_n) } \mathbb{E}^*
[D_n F \mid {\cal F}_{n-1} ]
\prod_{k=n+1}^N (1+r_k)^{-1} , \qquad 0\leq n \leq N , \end{equation}
and \begin{equation} \label{zeta} \zeta_n =
A_n^{-1}
\left(
\displaystyle\prod_{k=n+1}^N (1+r_k)^{-1}
\mathbb{E}^* [F \mid {\cal F}_n ]
-\eta_n S_n\right) ,
\qquad 0\leq n \leq N . \end{equation}
Then the portfolio $( \eta_k , \zeta_k )_{0\leq k \leq n}$
is self financing and satisfies $$
\zeta_{n} A_n + \eta_{n} S_n
=
\prod_{k=n+1}^N (1+r_k)^{-1}
\mathbb{E}^* [F \mid {\cal F}_n ] , \qquad 0 \leq n \leq N , $$
in particular we have $V_N = F$,
hence $(\eta_k ,\zeta_k )_{0 \leq k \leq N}$
is a hedging strategy leading to $F$. \end{prop} \begin{Proof}
Let $(\eta_k)_{-1\leq k \leq N}$ be defined by \eqref{eta} and
$\eta_{-1}=0$, and consider the process $(\zeta_n)_{0\leq n \leq N}$ defined by $$\zeta_{-1} = \frac{\mathbb{E}^* [F ]}{S_{-1}} \prod_{k=0}^N (1+r_k)^{-1}
\quad
\mbox{and}
\quad
\zeta_{k+1} = \zeta_k - \frac{(\eta_{k+1}-\eta_k)S_k}{A_k},
\qquad
k=-1,\ldots , N-1 . $$
Then $(\eta_k , \zeta_k )_{-1\leq k \leq N}$ satisfies
the self-financing condition $$A_k(\zeta_{k+1}-\zeta_k)+ S_k(\eta_{k+1}-\eta_k ) =0,
\qquad -1 \leq k \leq N-1 . $$
Let now $$V_{-1} =
\mathbb{E}^* [F ]
\prod_{k=0}^N (1+r_k)^{-1} ,
\quad
\mbox{and}
\quad
V_n = \zeta_{n } A_n + \eta_n S_n, \qquad 0 \leq n \leq N, $$
and $$\tilde{V}_n = V_n \displaystyle\prod_{k=0}^n (1+r_k)^{-1},
\qquad -1\leq n \leq N . $$
Since $(\eta_k,\zeta_k)_{-1 \leq k \leq N}$ is self-financing,
Relation~\eqref{e440.1.1} shows that \begin{equation} \label{tv}
\tilde{V}_n =
\tilde{V}_{-1}
+ \sum_{i=0}^n
Y_i
\eta_i S_{i-1}
\sqrt{p_iq_i} (b_i-a_i)
\prod_{k=1}^i (1+r_k)^{-1} ,
\qquad
-1\leq n \leq N . \end{equation}
On the other hand, from the Clark formula \eqref{clk} and the
definition of $(\eta_k)_{-1\leq k \leq N}$ we have \begin{eqnarray*} \lefteqn{
\mathbb{E}^* [ F \mid {\cal F}_n ]
\prod_{k=0}^N (1+r_k)^{-1} } \\
& = &
\mathbb{E}^* \left[
\mathbb{E}^* [F ] \prod_{k=0}^N (1+r_k)^{-1}
+
\sum_{i=0}^N
Y_i
\mathbb{E}^* [D_i F \mid {\cal F}_{i-1} ]
\prod_{k=0}^N (1+r_k)^{-1}
\Big|
{\cal F}_n
\right] \\
& = &
\mathbb{E}^* [F ] \prod_{k=0}^N (1+r_k)^{-1}
+
\sum_{i=0}^n
Y_i
\mathbb{E}^* [D_i F \mid {\cal F}_{i-1} ]
\prod_{k=0}^N (1+r_k)^{-1} \\
& = &
\mathbb{E}^* [F ] \prod_{k=0}^N (1+r_k)^{-1}
+ \sum_{i=0}^n
Y_i
\eta_i S_{i-1}
\sqrt{p_iq_i} (b_i-a_i)
\prod_{k=1}^i (1+r_k)^{-1} \\
& = &
\tilde{V}_n \end{eqnarray*}
from \eqref{tv}.
Hence $$
\tilde{V}_n
=
\mathbb{E}^* [ F \mid {\cal F}_n ]
\prod_{k=0}^N (1+r_k)^{-1} ,
\qquad -1 \leq n \leq N, $$
and $$
V_{n} =
\mathbb{E}^* [ F \mid {\cal F}_n ]
\prod_{k=n+1}^N (1+r_k)^{-1} , \qquad -1 \leq n \leq N. $$
In particular we have $V_N = F$.
To conclude the proof we note that from the
relation $V_n = \zeta_n A_n + \eta_n S_n$, $0\leq n \leq N$,
the process $(\zeta_n)_{0\leq n \leq N}$ coincides
with
$(\zeta_n)_{0\leq n \leq N}$ defined
by \eqref{zeta}. \end{Proof} \noindent
Note that we also have $$
\zeta_{n+1} A_n + \eta_{n+1} S_n
=
\mathbb{E}^* [F \mid {\cal F}_n ]
\prod_{k=n+1}^N (1+r_k)^{-1} , \qquad -1 \leq n \leq N . $$
The above proposition shows that there always exists a hedging strategy starting from $$\tilde{V}_{-1} = \mathbb{E}^* [F] \prod_{k=0}^N (1+r_k)^{-1}. $$
Conversely, if there exists a hedging strategy leading to $$\tilde{V}_N = F \prod_{k=0}^N (1+r_k)^{-1},$$
then $(\tilde{V}_n)_{-1\leq n \leq N}$ is necessarily a martingale with
initial value $$\tilde{V}_{-1} = \mathbb{E}^* [\tilde{V}_N ] = \mathbb{E}^* [F] \prod_{k=0}^N (1+r_k)^{-1}. $$
When $F=h(\tilde{S}_N)$, we have
$\mathbb{E}^* [h(\tilde{S}_N) \mid {\cal F}_{k}] = f(\tilde{S}_k,k)$ with $$f(x,k ) = \mathbb{E}^* \left[
h\left( x
\prod_{i=k+1}^n
\frac{\sqrt{(1+b_k)(1+a_k)}}{1+r_k}
\left(
\frac{1+b_k}{1+a_k}\right)^{X_k/2}
\right)
\right] . $$
The hedging strategy is given by \begin{eqnarray*}
\eta_k & = &
\frac{1}{S_{k-1} \sqrt{p_kq_k} (b_k-a_k) }
D_k f(\tilde{S}_k,k)
\prod_{i=k+1}^N (1+r_i)^{-1} \\
& = &
\frac{\prod_{i=k+1}^N (1+r_i)^{-1} }{S_{k-1} (b_k-a_k) }
\left(
f\left(\tilde{S}_{k-1}
\frac{1+b_k}{1+r_k} , k\right)
-
f\left(\tilde{S}_{k-1}
\frac{1+a_k}{1+r_k}
, k\right)
\right) , \qquad k\geq -1. \end{eqnarray*}
Note that $\eta_k$ is non-negative (i.e. there is no short-selling)
when $f$ is an increasing function, e.g. in the case of European options
we have $f(x) = (x-K)^+$.
\footnotesize
\def$'${$'$} \def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def$'${$'$}
\end{document} |
\begin{document}
\title{Could Gaussian regenerative stations act as quantum repeaters?}
\author{Ryo Namiki$^1$, Oleg Gittsovich$^{1}$, Saikat Guha$^2$, and Norbert L{\" u}tkenhaus$^1$} \affiliation{$^1$Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo ON, Canada N2L 3G1\\ $^2$Quantum Information Processing group, Raytheon BBN Technologies, Cambridge MA, USA 02138}
\begin{abstract}
Higher transmission loss diminishes the performance of optical communication---be it the rate at which classical or quantum data can be sent reliably, or the secure key generation rate of quantum key distribution (QKD). Loss compounds with distance---exponentially in an optical fiber, and inverse-square with distance for a free-space channel. In order to boost classical communication rates over long distances, it is customary to introduce regenerative relays at intermediate points along the channel. It is therefore natural to speculate whether untended regenerative stations, such as phase-insensitive or phase-sensitive optical amplifiers, could serve as repeaters for long-distance QKD. The primary result of this paper rules out all bosonic Gaussian channels
to be useful as QKD repeaters, which include phase-insensitive and phase-sensitive amplifiers as special cases, for any QKD protocol. We also delineate the conditions under which a Gaussian relay renders a lossy channel {\em entanglement breaking}, which in turn makes the channel useless for QKD. \end{abstract}
\pacs{03.67.Dd, 42.50.Lc} \keywords{quantum cryptography, quantum repeater, bosonic channel} \maketitle
\section{Introduction}
In recent years, performance of various communication tasks over an optical channel---when limited only by the fundamental noise of quantum mechanical origin---have been extensively studied. A few examples are: finding the communication capacities of the lossy optical channel for transmitting classical information~\cite{Gio04}, quantum information~\cite{Wol98}, and that for transmitting both classical and quantum information simultaneously in the presence of a limited amount of pre-shared entanglement~\cite{Wil12}. One of the biggest breakthroughs in optical communication using quantum effects was the invention of quantum key distribution (QKD), which is a suite of protocols that can generate information-theoretically-secure shared secret keys~\cite{Scarani_Renner_2008} between two distant parties Alice and Bob over a lossy-noisy optical channel, with the assistance of a two-way authenticated public classical channel. Security of QKD leverages quantum properties of light to ensure the generated shared keys are secure from the most powerful adversary that is physically consistent with the channel noise collectively estimated by Alice and Bob (despite the fact that much of that noise may actually be caused by non-adversarial or natural causes). Various QKD protocols have been proposed in the last three decades~\cite{SBCDLP09}, some of which have been transitioning to practice~\cite{DARPA02,SECOQC09,TOKYO_QKD}.
For all the communication protocols discussed above, the rates decrease rapidly with channel loss. For the task of classical communication over an ideal pure-loss channel (modeled by a beamsplitter of transmittance $\eta$), at any given value of the channel transmittance $\eta$, no matter how small, the data rate can in principle be increased without bound by increasing the input power~\cite{footnote1}. For QKD, this is not the case. For several well-known QKD protocols (such as BB84~\cite{BB84} with single photons and BB84 with weak laser light encoding and decoy states, E91~\cite{E91} with an ideal entanglement source, and CV-QKD with Gaussian modulation~\cite{Sil02, Gro03}), the secret key rate $R$ decays linearly with channel transmittance $\eta$ in the high-loss ($\eta \ll 1$) regime~\cite{footnote}. Recently, it was shown that this linear rate-transmittance scaling over the lossy bosonic channel---for secure-key generation with two-way public classical communication assistance---is impossible to improve upon, no matter how one may design a QKD protocol, or how much input power is used~\cite{Tak13}. To be specific, the secret key rate of any QKD protocol must be upper bounded by $R_{\rm UB}$ measured in bits/mode and given by \begin{equation} \label{TGWbound} R_{\rm UB} = \log_2\frac{1+\eta}{1-\eta} , \end{equation} which equals $R_{\rm UB} \approx 2.88\eta$, for $\eta \ll 1$. This fundamental rate-loss upper bound also applies to the following related tasks: quantum communication (sending qubits noiselessly over a lossy channel), direct secure communication~\cite{footnote3}, and entanglement generation (where each task may also use assistance of a separate authenticated two-way classical communication channel, in addition to transmissions over the quantum lossy bosonic channel itself)~\cite{Tak13}.
As we discussed above, for classical communication over an ideal lossy channel, one could in principle increase the input power without bound as the loss increases, to maintain a required data rate. However, an unbounded input power is impractical both from the point of view of the availability of a laser that is powerful enough, and also to avoid hitting up against the fiber's non-linearity-driven peak power constraint. This is why traditionally, electrical regenerators have been used to compensate for loss in long-haul optical fiber communications, which help restore the signal-to-noise ratio (SNR) of the digitally-modulated signals by periodically detecting and regenerating clean optical pulses. Over the last few decades, all-optical amplifiers, such as erbium-doped fiber amplifiers (EDFAs), have become popular in lieu of electrical regenerators, both due to their greater speeds as well as the low noise of modern EDFAs. Caves analyzed the fundamental quantum limits on the noise performance of optical amplifiers~\cite{Cav82}, for both phase-insensitive (PIA) and phase-sensitive amplifiers (PSA). Loudon analyzed the fundamental limitations on the overall SNR to `chains' of loss segments and optical amplifiers, both in the context of phase-sensitive (coherent detection) receivers, as well as direct detection receivers~\cite{Lou85}.
For QKD, one way to beat the linear rate-transmittance scaling is to break up the channel into low-loss segments by introducing physically-secured center stations; in this approach the overall key rate is still upper bounded by $R \le \log_2[(1+\eta^\prime)/(1-\eta^\prime)]$ bits/mode, but $\eta^\prime$ is the transmittance of the longest (lossiest) segment. Quantum repeaters are conceptual devices~\cite{SST_2011, LST_2009}, which if supplied at these intermediate stations, can beat the linear rate-transmission scaling without having to physically secure them. There is an approach to build a quantum repeater using one-way communication only~\cite{Mur14}, so they can act as passive untended devices. However, such structured implementations of those devices require quantum error correction codes operating on blocks of multiple qubits. A recently-proposed repeater protocol~\cite{Azu13} even eliminates the requirement of a quantum memory, but utilizes photonic cluster states. Building a functional quantum repeater is subject to intensive fundamental research, but is currently far from being a deployable technology. The natural question that thus arises---in analogy to Loudon's setup for classical optical communication~\cite{Lou85}---is whether all-optical amplifiers (PIAs or PSAs), left untended and inserted at regular intervals, might act to some degree as quantum repeaters and thereby help boost the distances over which QKD can be performed over a lossy channel.
{ The remainder of the paper is organized as follows. First, in Section~\ref{sec:MainResults} we summarize the main results derived in this article to put it into perspective. A central finding is a decomposition of a lossy quantum channel with intermediate bosonic Gaussian channel stations into another form without any insertion of middle stations as depicted in Fig.~\ref{fig:summary}. We then continue into the technical part.
In Section~\ref{sec:Gaussian}, we give an overview of bosonic Gaussian states and channels. In Section~\ref{sec:GaussianStations}, we analyze the scenario when a general multi-mode Gaussian channel is inserted between two pure-loss segments, and show how one can collect the entire pure loss in the center of the channel by appropriate modifications to the transmitter and the receiver. In Section~\ref{sec:EBconditions}, we consider single-mode Gaussian stations, and delineate the conditions for when the Gaussian center station renders the concatenation with the losses on its two sides, an entanglement-breaking channel. The quantum limited stations, the PSA and the PIA, are addressed as special cases. We conclude in Section~\ref{sec:conclusions} with a summary of the main results, and thoughts for future work. } \begin{figure}
\caption{(a) Any $n$-mode Gaussian channel ${\cal N}_G$ sandwiched between two pure-loss channel segments ${\cal A}_{\eta_1}^{\otimes n}$ and ${\cal A}_{\eta_2}^{\otimes n}$, respectively, can be decomposed into a single lossy channel ${\cal A}_{\eta}^{\otimes n}$ sandwiched by a pair of Gaussian channels, ${\cal N}_G^1$ and ${\cal N}_G^2$. The net loss in the channel is the sum (in dB) of the losses of the two individual lossy segments, i.e., $\eta = \eta_1\eta_2$ and the Gaussian channel at the receiver end ${\cal N}_G^2$ is a Gaussian unitary map. (b) Using this transformation recursively, one can `push' a collection of general Gaussian center stations interspersed through a lossy channel (b.1) to a single Gaussian operation at the input, and a single Gaussian operation at the output (b.2) of the entire loss accumulated in the center.}
\label{fig:summary}
\end{figure}
\section{Outline of main results} \label{sec:MainResults}
In this paper, we show that such is not possible when those all-optical amplifiers are limited to Gaussian channels. Note that by using the word ``channel'' we automatically imply the action to be trace-preserving. Examples of such channels involve beamsplitters, phase-shifters and squeezers~\cite{Bra05}. We prove our claim by transforming a concatenation of two lossy channel segments with a Gaussian channel in the middle, as a pair of Gaussian channels at the two ends, with the total loss collected in the middle (see Fig.~\ref{fig:summary}). The implication of our no-go result is that if Gaussian channels are employed in center station(s) placed along a lossy channel, the overall QKD key rate, for any QKD protocol, must be upper bounded by $R_{\rm UB} = \log_2[(1+\eta)/(1-\eta)]$ bits/mode, with $\eta$ being the total end-to-end channel transmittance.
{Simple protocols such as laser-decoy-based BB84, or Gaussian-modulated laser-based CV protocols, with no repeaters, can already attain key rates that have the optimal (linear) rate-transmittance scaling and are only a small constant factor below the general upper bound~\cite{SBCDLP09}.}
For any optical communication protocol over a lossy channel interspersed with Gaussian stations, our result shows there exists {\em another} protocol with the same performance that does not use any intermediate station, which can be derived from the original protocol by suitably amending the transmitted signals and the receiver measurement. Our result does {\em not} preclude a Gaussian channel in the middle of the lossy channel to improve the performance of a {\em given} protocol, {\em if} the transmitter and receiver are held to be the same. Nor does it preclude the existence of scenarios where it might be technologically easier to implement a protocol with such intermediate stations, as opposed to modifying the transmitter and the receiver per the prescription generated by our analysis. An example of such improvement is the increased range of a QKD protocol with a given level of detector noise (although, any increase in range must be consistent with the $R \sim \eta$ rate-transmission scaling).
{
Given that the overall rate-transmission scaling can not be changed, the question remains whether there might be other implementation advantages of Gaussian center stations. It turns out that there are strict conditions on such a scenario. To demonstrate this, }we delineate the conditions under which a Gaussian center station causes a lossy channel to become entanglement breaking (EB) \cite{Horo03,Hol07,Hol08}. It is well known that QKD is not possible on an EB channel, since the output of an EB channel can be simulated quantitatively correctly using a measure-and-prepare scheme~\cite{Cur04}. The pure lossy channel is not EB by itself for any non-zero transmittance, $\eta >0$.
Let us illustrate our reasoning for the better known case of classical communication over pure-loss bosonic channels. The channel capacity of the lossy bosonic channel (described by single photon transmittance $\eta$) using signals with mean photon number $\bar{n}$ per mode, is given by $g(\eta \; {\bar n}) = (1+\eta \; {\bar n})\log_2(1+\eta\; {\bar n}) - \eta \; {\bar n}\log_2 \eta \;{\bar n}$ bits per mode~\cite{Gio04}. We see, that increasing the mean photon number increases the classical communication rate. In practice, it is impractical to keep increasing the mean photon number due to non-linear effects in the fiber that limit the input power and can distort the signals. For these reasons, one limits the input power and builds optical amplifiers (phase sensitive or phase insensitive) into the fiber. According to our theorems, for the ideal loss-only bosonic channel, the setup of lossy segments with intermediate amplifiers is equivalent to a new transmitter consisting of the old transmitter combined with a very strong amplifier, followed by a transfer through the full distance of the lossy bosonic channel, and then a receiver consisting of a combination of another amplifier and on the original
receiver.
This replacement protocol corresponds to the situation of using a large input mean photon number, and realizes the classical capacity of the lossy bosonic channel. What we learn is that the intermediate amplifiers do not increase the channel capacity of the lossy bosonic channel, but realize an equivalent protocol that keeps the optical signals---throughout the communication channel---within a peak power level that is sufficiently below the level where non-linear effects would be encountered.
In QKD, the secrecy capacity of the lossy bosonic channel does not increase unboundedly with the input power of the signals, thus using strong signal pulses pushing into the non-linear domain of fibers is not important for QKD protocols: the use of equivalent replacement schemes utilizing optical amplifiers would not give any advantage. To the contrary, amplifiers will add additional noise which will eventually be detrimental to the performance of the QKD protocol, with the exception of effects of noisy pre-processing that can increase the secret key rate compared to protocols not using this approach~\cite{renner05b}. Note however that noisy pre-processing cannot improve on the fundamental secrecy capacity of the lossy bosonic channel, which is solely a function of the channel's end-to-end loss.
Our main result adds to the list of no-go results for Gaussian operations in quantum information protocols, i.e., those that cannot be performed with Gaussian operations and classical processing alone. Some examples are universal quantum computing~\cite{Bartlett2002}, entanglement distillation of Gaussian states~\cite{Eisert2002,Fiurasek2002,Giedke2002}, optimal cloning of coherent states~\cite{Cerf2005}, optimal discrimination of coherent states~\cite{Takeoka2008,Tsujino2011,Wittmann2010-1,Wittmann2010-2}, Gaussian quantum error correction~\cite{Niset2009}, and building a joint-detection receiver for classical communication~\cite{Tak14}.
\section{Gaussian states and channels} \label{sec:Gaussian} In this section, we will provide a basic introduction to the mathematics of Gaussian states and channels, sufficient to develop the results in this paper. For a more detailed account, see Ref.~\cite{Giedke2002}. A quantum state $\rho$ of an $n$-mode bosonic system is uniquely described by its characteristic function \begin{equation} \label{eq:characteristic_function} \chi (\mu) = {\rm Tr} \left[ \rho \mathcal{W} (\mu) \right] , \end{equation} where the Weyl operator, $\mathcal{W} (\mu) = \exp \left[ - i \mu^T R \right]$, with $R=[ \hat{x}_1, \cdots , \hat{x}_n , \hat{p}_1, \cdots , \hat{p}_n]^T$ consisting of field quadrature operators of the $n$ modes satisfying the commutation relations $[\hat{x}_k, \hat{p}_l]=i \delta_{kl}$, with $\mu =[ \mu_1, \cdots , \mu_{2n} ]$ a $2n$-length real vector. The characteristic function of a Gaussian state $\rho$ is given by, \begin{equation} \label{eq:chi_gaussian} \chi_\rho (\mu) = \exp\left[ - \frac{1}{4} \mu^T \gamma \mu + i d^T \mu \right] , \end{equation} where the $2n \times 2n$ matrix $\gamma$ is the covariance matrix (CM) and the $2n$-length vector $d:= (\ave{\hat x} , \ave{\hat p} )^T$ is the mean, or the displacement vector (DV), of $\rho$. The Gaussian state $\rho$ can thus be described uniquely by the pair $(\gamma, d)$. Due to the canonical uncertainty relation, any CM of physical states has to satisfy \begin{eqnarray} \gamma \ge \frac{i}{2} \sigma , \label{PhysConState} \end{eqnarray} where \begin{equation} \label{eq:S_decomposition} \sigma : = \left( \begin{array}{cc} 0 & \openone _ n \\ -\openone _ n & 0 \end{array} \right). \end{equation}
A Gaussian unitary operation $U_G$ transforms a Gaussian state $(\gamma, d)$ to a Gaussian state $(\gamma^\prime, d^\prime)$ as \begin{equation} \gamma^\prime = M^T \gamma M, \quad d^\prime = M^T d, \label{eq:symplectic} \end{equation} where $M$ is a symplectic matrix that satisfies \begin{eqnarray} M ^T \sigma M = \sigma . \label{Guni} \end{eqnarray}
A Gaussian channel ${\cal E}$ can be described by a triplet $(K,m,\alpha)$~\cite{Hol08}. It transforms a state $(\gamma, d)$ to the state $(\gamma^\prime, d^\prime)$ as \begin{eqnarray} \gamma^\prime = K^T \gamma K + \alpha, \quad d^\prime = K^T d +m. \end{eqnarray} From the regularity of CMs in Eq.~\eqref{PhysConState} the physical condition for the pair $(K,\alpha)$ is given by \begin{eqnarray} \alpha \ge \frac{i}{2}(\sigma - K^T \sigma K ) \label{PhysCon}. \end{eqnarray} Composition of two Gaussian channels $\mathcal E_1$ and $\mathcal E_2$ yields another Gaussian channel $\mathcal E_{12} = \mathcal E_2 \circ \mathcal E_1$, where \begin{eqnarray} K_{12} &=& K_1 K_2 \nonumber, \\ m_{12} &=& K_2^T m_1 + m_2, \,{\text{and}}\nonumber \\ \alpha_{12} &=& K_2^T \alpha _1 K_2 + \alpha_2. \label{CompoRule} \end{eqnarray} In this paper, we will focus on Gaussian channels with $m=0$.
In Appendix~\ref{app:displacement} we show an explicit calculation demonstrating how mean displacement terms can be separated out in any concatenation of Gaussian channels.
In the following subsections, we will delve a little deeper into properties of single-mode Gaussian channels that we use later on.
\subsection{Decomposing a Gaussian unitary operation} The symplectic matrix $M$ in Eq.~\eqref{eq:symplectic} of a Gaussian unitary can always be decomposed as \begin{equation} \label{eq:S_decomposition} M = B \left( \begin{array}{cc} \Lambda & 0 \\ 0 & \Lambda^{-1} \end{array} \right) B' , \end{equation} where $\Lambda$ is a positive diagonal matrix, and $B$, $B'$ are orthogonal symplectic matrices ($B^T = B^{-1}$)~\cite{Bra05}. This implies that any $n$-mode Gaussian unitary operation $U_G$ can be realized by a passive linear optic circuit $B$ (a circuit involving only beamsplitters and phase-shifters~\cite{Rec94}), followed by $n$ parallel (tensor-product) single-mode squeezers, followed by another $n$-mode passive linear optic circuit $B'$~\cite{Bra05}. Therefore a general Gaussian unitary operation can always be decomposed into passive linear optics (beamsplitters and phase shifters), single-mode squeezing and single-mode displacement operations.
Therefore, up to a displacement, a single-mode Gaussian unitary (described by its symplectic matrix $M$) can be decomposed as \begin{eqnarray} M = R_\theta S_{G} R_\phi, \end{eqnarray} where \begin{eqnarray} R_\theta &=& \left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin\theta & \cos \theta \\
\end{array} \right) \end{eqnarray} is the symplectic matrix of a single-mode phase rotation, and \begin{eqnarray} S_{G} &=& \left(
\begin{array}{cc}
\sqrt G +\sqrt{G -1} & 0 \\
0 & \sqrt G -\sqrt{G -1} \\
\end{array} \right) \end{eqnarray} is the symplectic matrix of a single-mode (phase-quadrature) squeezer. Note that it is sufficient to restrict the above decomposition to a phase-quadrature squeezer, is because one can absorb any additional phase in the squeezing operation into $R_\theta$ and $R_\phi$. This is because the symplectic matrix of a single-mode squeezer with gain $G$ and squeezing angle $\theta'$ can be expressed as \begin{eqnarray} S_{G, \theta'} = R_{\theta'} S_{G} R_{\theta'} ^\dagger. \end{eqnarray}
\subsection{Entanglement breaking channels}
An {\em entanglement breaking} (EB) channel is one whose action on one half of an entangled state (with an identity map on the other half) always yields a {\em separable} state. An EB channel can always be written in a {\em measure-and-prepare} form~\cite{Horo03, Hol08}. (See also Eq.~(\ref{eq:measureprepare}) below.) Any concatenation of $n$ (not necessarily Gaussian) channels, ${\cal E}_n \circ \ldots \circ {\cal E}_2 \circ {\cal E}_1$ is EB if one of channels ${\cal E}_i$ is EB. It is instructive to see the argument explicitly for $n= 3$. Consider the serially-concatenated channel, ${\cal E}_t= {\cal E}_3 \circ {\cal E}_2 \circ {\cal E}_1$, where the center station ${\cal E}_2 $ is EB. Supposing its measure-and-prepare form is given by ${\cal E}_2 (\rho ) = \sum_k \tr(M_k \rho )\sigma_k$, with $M_k \ge 0$ and $\sigma_k \ge 0 $, we can write ${\cal E}_t$ in a measure-and-prepare form, \begin{equation} \label{eq:measureprepare} {\cal E}_t (\rho) = \sum_k \tr[M_k {\mathcal E}_1 (\rho) ] {\mathcal E}_3(\sigma_k) = \sum_k \tr[M'_k \rho ] \sigma_k ', \end{equation} where $ \sigma_k '= {\mathcal E}_3(\sigma_k) \ge 0 $, and it is straightforward to show that $M'_k = \sum_i A_i^\dagger M _k A_i \ge 0$, where $\{A_i\}$ represent Kraus operators of $\mathcal E_1$, i.e., ${\mathcal E}_1(\rho) = \sum_i A_i \rho A_i^\dagger$.
The measure-and-prepare representation of an EB channel implies that the channel's quantum transmission can be seen as transmission of the (probabilistic) classical information obtained as a result of a hard quantum measurement made on the channel's input. This is the intuition behind why such a channel has zero secret-key capacity, and thus cannot be useful for QKD~\cite{Cur04}. Because of this reason, when we analyze concatenations of several Gaussian center stations for potential use as repeaters, we will limit our discussion to the case when all the channels $\mathcal E_i$ in the concatenation are non EB (since this is a necessary condition for QKD). Note however that when interspersed with loss segments, even when all center stations are non-EB, the overall input-output map can become EB---a topic that we will discuss in more detail later in Section~\ref{sec:EBconditions}.
\subsection{Unitary-equivalence classification for single-mode Gaussian channels}\label{sec:classification} Our analysis of general one-mode Gaussian operations will be based on the standard forms of such operations obtained from the unitary equivalence classification of quantum channels developed by Holevo~\cite{Hol07, Hol08}. We say that two quantum channels $\Phi$ and $\Phi_S$ are \textit{unitary equivalent} if there exist unitary operators $U_V, U_W$ such that, \begin{eqnarray} \Phi_S (\rho) = U_W \Phi (U_V \rho U_V ^\dagger) U_W^\dagger. \label{eq:unitary_equivalence} \end{eqnarray} If $U_V$ and $U_W$ above are Gaussian, we say $\Phi$ and $\Phi_S$ are {\em Gaussian unitary equivalent}. If a single-mode Gaussian channel ${\cal E} \triangleq (K, m, \alpha)$ is not an EB channel, it must be Gaussian unitary equivalent to a channel belonging to one of the following two classes: \\
\noindent {\bf (i) Phase insensitive channel (PIC)}: This class of channels is described by the triplet $(K, 0, \alpha)$, with \begin{eqnarray} K &=& \sqrt \kappa \openone_2, \,{\text{and}}\nonumber \\
\alpha &=&(|1-\kappa|/2 + N )\openone_2 , \label{i} \end{eqnarray} where $N \ge 0$ is the excess noise parameter and $\kappa \ge 0$ is a gain parameter. We will denote
this channel as ${\cal A}_\kappa^N$. It acts on the canonical quadratures phase-insensitively. When the gain $\kappa \ge 1$, we call it the {\em phase-insensitive amplifier} (PIA). When $\kappa < 1$, we call it the lossy bosonic channel (with excess thermal noise $N$). In this case, $\kappa$ is the channel's transmittance, the fraction of the input photons that appear at the channel's output. We will use the shorthand notation, ${\cal A}_\kappa \equiv {\cal A}_\kappa^0$ for a quantum-limited phase-insensitive amplifier, or a pure-loss channel, for $\kappa \ge 1$ and $\kappa < 1$, respectively.
It is known that the PIC is EB if and only if~\cite{Hol08}, \begin{eqnarray} N \ge \min (1, \kappa). \label{38} \end{eqnarray} In our analysis we will assume that the PIC is not EB, i.e., $N \in [0, \min(1, \kappa))$. Furthermore, using the composition rule of Eq. (\ref{CompoRule}), it is easy to see that any single-mode rotation (unitary) ${\cal R}$ commutes with a PIC, i.e., ${\cal R} \circ {\cal A}_\kappa^N = {\cal A}_\kappa^N \circ {\cal R}$. \\
\noindent {\bf (ii) Additive noise channel (ANC)}: This is a class of {\em phase-sensitive} Gaussian channels that adds rank-1 noise to the input state, and is described by the triplet $(K, 0, \alpha)$, with \begin{eqnarray} K &=& \openone_2, \,{\text{and}} \nonumber \\ \alpha&= & \frac{1}{2}\textrm{diag} (0,\epsilon), \label{ii} \end{eqnarray} where the noise parameter $\epsilon > 0$. We will denote this channel as $\mathcal I^{\epsilon}$, and will call it the additive noise channel (ANC).
\section{Gaussian regenerative stations in a lossy channel}\label{sec:GaussianStations} In this section we investigate lossy bosonic channels that have intermediate Gaussian channels inserted at some intervals. We will show that such an arrangement is still equivalent (up to Gaussian operations at the entrance and the exit) to a lossy bosonic channel with the total loss of the original loss segments. As a consequence, insertion of Gaussian channels
cannot increase the secrecy capacity of the lossy bosonic channel.
The setup for the main result of this paper is schematically depicted in Fig.~\ref{fig:summary}. Consider a pure-loss optical channel ${\cal A}_\eta$ with a given amount of total end-to-end ($A$ to $B$) transmittance $\eta \in (0, 1]$. Let us place a Gaussian center station---a quantum channel, or a trace-preserving completely positive map, ${\cal N}_G^{C_1 \to C_2}$---somewhere in the middle, thereby splitting ${\cal A}_\eta$ into two pure-loss segments: a pure-loss channel with transmittance $\eta_1$, ${\cal A}_{\eta_1}$ ($A$ to $C_1$), and a pure-loss channel with transmittance $\eta_2$, ${\cal A}_{\eta_2}$ ($C_2$ to $B$), such that $\eta_1\eta_2 = \eta$. We show that the overall channel action from $A$ to $B$ is unaffected by the transformation shown in Fig.~\ref{fig:summary}(a), which replaces the Gaussian center station ${\cal N}_G^{C_1 \to C_2}$ by a Gaussian operation ${{\cal N}_G^1}^{A \to A_1}$ at the input of the channel and a Gaussian operation ${{\cal N}_G^2}^{B_1 \to B}$ at the output of the channel. By applying this transformation recursively, it is easy to see that one can replace any number of Gaussian center stations interspersed through the lossy channel ${\cal A}_\eta$ into two Gaussian operations, at the input and the output, respectively.
\eat{ Consider an $n$-mode lossy bosonic channel ${\cal A}_\eta ^{\otimes n } \triangleq (K_0, 0,\alpha_0)$ with $K_0 = \sqrt{\eta } \openone_{2n}$ and $\alpha_0 = \frac{1- \eta}{2} \openone_{2n}$. Let an $n$-mode Gaussian channel ${\cal N}_G \triangleq (K, 0, \alpha)$ be the candidate center station, which could act collectively on $n$ spatial and/or temporal modes of the field. The loss-sandwiched composition $\Phi_0 := {\cal A}_{\eta_2} ^{\otimes n } \circ{\cal N}_G \circ {\cal A}_{\eta_1} ^{\otimes n}$ is therefore given by $(K_t, 0, \alpha_t)$, with \begin{eqnarray} K_t &=& \sqrt{\eta_1 \eta_2}K \nonumber, \,{\text{and}} \\ \alpha_t &=& \eta_2\left(\frac{1- \eta_1}{2} K^T K + \alpha \right)+ \frac{1- \eta_2}{2} \openone_{2n} \label{DefAlpha_main}. \end{eqnarray} Next we construct two Gaussian channels ${\cal N}_G^1$ and ${\cal N}_G^2$ (the latter suffices to be a unitary), such that ${\cal A}_{\eta_2} ^{\otimes n } \circ{\cal N}_G \circ {\cal A}_{\eta_1} ^{\otimes n } = {\cal N}_G^2 \circ {\cal A}_{\eta_1 \eta_2} ^{\otimes n } \circ{\cal N}_G^1$. The first step is to argue that there must exist a symplectic matrix $M$ that satisfies $M^T \alpha_t M \ge \frac{1- \eta_1 \eta_2}{2} \openone_{2n} + \eta_1 \eta _2 M^T \alpha M$ (see Appendix~\ref{app:mainresult} for detailed proof). Then we consider the channel ${\cal N}_G^1 \triangleq (\tilde K, 0, \tilde \alpha)$, with \begin{eqnarray} \tilde K &=& K M \nonumber, \,{\text{and}} \\ \tilde \alpha &=& \frac{1}{\eta_1 \eta_2 } \left(M^T \alpha _t M - \frac{1- \eta_1 \eta_2}{2} \openone_{2n}\right). \end{eqnarray} It is simple to then show that $K_t M = \sqrt{ \eta_1 \eta _2} \tilde K$, and $M^T \alpha_t M = \eta_1 \eta _2 \tilde \alpha + \frac{1- \eta_1 \eta_2}{2} \openone_{2n}$, which in turn implies that the compositions ${\cal A}_{\eta_1 \eta_2} ^{\otimes n } \circ{\cal N}_G ^1$, and $U_G(M) \circ \Phi_0$ are identical. Therefore, $\Phi _0 = {\cal N}_G ^2 \circ {\cal A}_{\eta_1 \eta_2} ^{\otimes n } \circ{\cal N}_G ^1 $, where ${\cal N}_G^2 \triangleq (M^{-1},0,0)$ is a Gaussian unitary. }
Let us consider an $n$-mode lossy bosonic channel ${\cal A}_\eta ^{\otimes n } \triangleq (K_0, 0,\alpha_0) $ with \begin{eqnarray} K_0 &=& \sqrt{\eta } \openone_{2n} \nonumber, \\ \alpha_0 &=& \frac{1- \eta}{2} \openone_{2n}. \end{eqnarray} Let ${\cal N}_G \triangleq (K, 0, \alpha)$ denote an $n$-mode Gaussian channel, which we consider as the candidate for a center station. Note that this Gaussian center station could act collectively on $n$ spatial and/or temporal modes of the propagating field. The main result of this section is the proof of the following proposition, also depicted schematically in Fig.~\ref{fig:summary}.
\begin{proposition}\label{prop:n-mode} For any $n$-mode Gaussian channel ${\cal N}_G$ there exists a Gaussian channel ${\cal N}_G^1$ and a Gaussian unitary channel ${\cal N}_G^2$ that satisfy \begin{eqnarray} {\cal A}_{\eta_2} ^{\otimes n } \circ{\cal N}_G \circ {\cal A}_{\eta_1} ^{\otimes n } = {\cal N}_G^2 \circ {\cal A}_{\eta_1 \eta_2} ^{\otimes n } \circ{\cal N}_G^1. \end{eqnarray} \end{proposition}
\begin{proof} Our goal is to find a pair of Gaussian channels ${\cal N}_G^1$ and ${\cal N}_G^2$ that satisfies the physical condition Eq. (\ref{PhysCon}). From the composition rule of Eq.~\eqref{CompoRule} we find the total channel action $\Phi_t := {\cal A}_{\eta_2} ^{\otimes n } \circ{\cal N}_G \circ {\cal A}_{\eta_1} ^{\otimes n}$ can be described by $\Phi_t \triangleq (K_t, 0, \alpha_t)$ with \begin{eqnarray} K_t &=& \sqrt{\eta_1 \eta_2}K \nonumber, \\ \alpha_t &=& \eta_2\left(\frac{1- \eta_1}{2} K^T K + \alpha \right)+ \frac{1- \eta_2}{2} \openone_{2n} \label{DefAlpha}. \end{eqnarray}
We will prove the proposition by constructing the required Gaussian channels using a symplectic matrix denoted by $M$. The properties of this matrix and its existence
are the subject of the following theorem: \begin{theorem}\label{thm1} For a given $\alpha_t$ in Eq.~\eqref{DefAlpha}, there exists a CM matrix $\gamma^\prime$ and a symplectic matrix $M$ such that \begin{equation} \alpha_t = \eta_1 \eta_2 \alpha + (1-\eta_1 \eta_2) \gamma^\prime \end{equation} and \begin{equation} M^T \gamma ^\prime M \ge \frac{1}{2} \openone_{2n}. \end{equation} \end{theorem} \begin{proof} From the physical condition of a Gaussian channel in Eq.~\eqref{PhysCon}, we have \begin{eqnarray}
\frac{i \sigma }{2}& \le& \frac{i \sigma}{2} + \frac{1}{2} K^T \left( \openone_{2n} - i \sigma \right) K \\ & = & \frac{1}{2} \left( K^T K + i (\sigma - K^T \sigma K )\right) \\ & \le& \frac{1}{2} ( K^T K + 2 \alpha )
\end{eqnarray} where we used in the first line that the matrix $ \openone_{2n} - i \sigma$ is positive semi-definite and in the last line that ${\cal N}_G$ is a physical channel. Our calculation implies $ \gamma := \frac{1}{2} ( K^T K + 2 \alpha ) $ is a CM of an $n$-mode Gaussian state due to Eq.~\eqref{PhysConState}. Consider now the convex combination of this CM with the CM of the $n$-mode vacuum state $\gamma^\prime := p \; \gamma + (1- p)\; \frac{1}{2} \openone_{2n}$with mixing probability $p = \eta_2(1-\eta_1 ) /(1-\eta_1 \eta_2 ) \in [0,1]$. A straightforward calculation verifies that $\alpha_t = \eta_1 \eta_2 \alpha + (1-\eta_1 \eta_2) \gamma^\prime $. As $\gamma^\prime$ is a valid CM, there exists a symplectic matrix $M$ such that one obtains a diagonal form $M^T \gamma ^\prime M \ge \frac{1}{2} \openone_{2n}$, which corresponds to a product of thermal states. \end{proof}
We are now in a position to define the Gaussian channels ${\cal N}_G^1\triangleq (\tilde K, 0, \tilde \alpha)$ and ${\cal N}_G^2\triangleq (M^{-1}, 0, 0)$ with the help of \begin{eqnarray} \tilde K & = & \frac{1}{\sqrt{\eta_1 \eta_2}} K_t M \;\;\;\left( \equiv K M\right) \; ,\\ \tilde \alpha &=& \frac{1}{\eta_1 \eta_2 } \left(M^T \alpha _t M - \frac{1- \eta_1 \eta_2}{2} \openone_{2n}\right). \label{13siki} \end{eqnarray}
To show that the channels are proper physical channels, we can concentrate on ${\cal N}_G^1$ since ${\cal N}_G^2$ corresponds to a unitary Gaussian channel. To prove that ${\cal N}_G^1$ is physical, we use in a first step the results of theorem \ref{thm1}, and then the physicality constraints on the channel ${\cal N}_G$, followed by a rewriting of the variables. These steps allow us to obtain \begin{equation} \tilde \alpha \ge M^T \alpha M \ge M^T \frac{i}{2}(\sigma - K^T \sigma K) M = \frac{i}{2}(\sigma - {\tilde K}^T \sigma \tilde K), \end{equation} Hence, ${\cal N}_G^1 \triangleq (\tilde K, 0, \tilde \alpha) $ is a valid Gaussian channel. It is again straightforward to verify that $\Phi _t = {\cal N}_G ^2 \circ {\cal A}_{\eta_1 \eta_2} ^{\otimes n } \circ{\cal N}_G ^1 $. This proves proposition \ref{prop:n-mode}.
\end{proof}
Overall, we showed the equivalence of a
bosonic Gaussian channel sandwiched between two lossy bosonic channels to a single lossy bosonic channel, bearing the total loss of the the original bosonic channels, and now sandwiched between two Gaussian channels.
This corresponds to the conversion of $(a.1)$ into $(a.2)$ of Fig.~\ref{fig:summary}. A simple iteration of this result shows that any pattern of Gaussian channels interspersed
between loss segments can be rearranged into a lossy bosonic channel sandwiched between Gaussian channels [See Fig.~\ref{fig:summary}(b)].
As the initial Gaussian channel can be combined with the state preparation, and the final Gaussian channel can be combined with the detection setup, it is evident that the total secret key rate of this arrangement is still bound by $R_{\textrm{UB}}$ of Eq.~(\ref{TGWbound}).
\section{Entanglement-breaking conditions for single-mode center stations}\label{sec:EBconditions}
As discussed in the introduction, there might be practical reasons
one want to use interspersed intermediate stations, even if the resulting key rate is still limited by the bound of Eq.~(\ref{TGWbound}). In this sections we will demonstrate severe restrictions on the situations where such an advantage may exist. To do so, we will investigate when such a sequence of lossy channels and Gaussian center stations becomes entanglement breaking (EB) so that its secrecy capacity goes to zero \cite{Cur04}. In the following part of this article we execute the central first step of such an investigation and focus on
single-mode Gaussian channels.
A pure-loss channel is not EB by itself, but increasing loss could make the channel progressively more fragile and susceptible to being EB when concatenated with other Gaussian operations, such as amplifiers. In the following subsections, we show the explicit conditions on the parameters of a Gaussian non-EB center station ${\cal N}_G$, such that the composition $\Phi_0 \equiv {\cal A}_{\eta_2} \circ {\cal N}_G \circ {\cal A}_{\eta_1}$ is EB, and specialize the conditions to the cases when ${\cal N}_G$ is either a PSA or a PIA.
\subsection{General non-EB center stations}\label{sec:general_nonEB} \begin{figure*}
\caption{(Color online) A general single-mode non-EB Gaussian channel ${\cal N}_G$ is unitary-equivalent to a Gaussian channel ${\cal N}$, which can be one of two forms, a {\em phase-insensitive channel} (PIC), ${\cal A}_g^N$ , or a phase-sensitive {\em additive noise channel} (ANC), ${\cal I}^\epsilon$. For both cases of the center stations sandwiched between two lossy segmenets ${\cal A}_{\eta_1}$ and ${\cal A}_{\eta_2}$, the total channel action $\Phi_0 \equiv {\cal A}_{\eta_2} \circ {\cal N}_G \circ {\cal A}_{\eta_1}$ is shown to be unitary-equivalent to a PIC channel ${\cal A}_{\eta_s}^{N_s}$ as in (b)1.c and (c)2.c. Green-shaded boxes denote single-mode unitary (reversible) operations, whereas red-shaded boxes denote (in-general irreversible) actions of a single-mode quantum channel---a trace-preserving completely-positive map.}
\label{fig:onemodechannel_main}
\end{figure*}
There is no point in considering EB center stations ${\cal N}_G$ as they would trivially render $\Phi_0$ EB. Any single-mode Gaussian non-EB station ${\cal N}_G$ is
unitary-equivalent to either a phase insensitive channel (PIC) or an additive noise channel (ANC) (See Sec.~\ref{sec:classification}). In order to evaluate EB conditions, we go deeper into decomposing $\Phi_0$, as depicted in Fig.~\ref{fig:onemodechannel_main}. The two branches of Fig.~\ref{fig:onemodechannel_main} consider decompositions when ${\cal N}_G$ is unitary-equivalent to a PIC or an ANC, respectively.
\noindent {\em 1. ${\cal N}_G$ unitary equivalent to a PIC ${\cal A}_g^N$}---Since phase-rotations commute with PICs, it is straightforward to see that the concatenated channel $\Phi_0 \equiv {\cal A}_{\eta_2} \circ {\cal N}_G \circ {\cal A}_{\eta_1}$ is unitary-equivalent to a channel $\Phi \equiv \mathcal A_{\eta_2} \circ \mathcal S_{ G_2, \theta }\circ \mathcal A_{g}^{N} \circ \mathcal S_{G_1} \circ \mathcal A_{\eta_1}$ [see Fig.~\ref{fig:onemodechannel_main}(b), lines~1.a~and~1.b], where $\mathcal S_{G_1}$ denotes a phase-quadrature squeezer (PSA) with gain $G_1$, and $\mathcal S_{G_2}$ is another PSA with gain $G_2$ and squeezing angle $\theta$. It is easy to deduce the parameters for the channel $\Phi$ as \begin{align} K_{\rm PIC} =& \sqrt{g\eta _1 \eta_2} S_1 {R_\theta S_2 R_\theta^\dagger} \; ,\\ \alpha_{\rm PIC} =& \frac{1}{2}\left[\eta_2(1-\eta_1) g K_\theta^T S_1 ^2 K_\theta + \right. \nonumber\\
&\left. \eta_2(|g-1|+2N) K_\theta^T K_\theta + (1-\eta_2) \openone_2\right], \label{eq3333} \end{align}
respectively, where $K_\theta = R_\theta S_2 R_\theta^\dagger$, and \begin{eqnarray} S_i &=& \left(
\begin{array}{cc}
\sqrt G_i +\sqrt{G_i-1} & 0 \\
0 & \sqrt G_i -\sqrt{G_i-1} \\
\end{array} \label{Gainnnnn} \right) , \\ R_\theta &=& \left(
\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin\theta & \cos \theta \\
\end{array} \right). \end{eqnarray} The following theorem shows that the total unitary equivalent channel $\Phi$ is further unitary-equivalent to a PIC $\Phi_s$, as shown in line 1.c of Fig.~\ref{fig:onemodechannel_main}(b).
\begin{theorem}\label{thm:phi_phis} $\Phi$ is unitary-equivalent to a channel $\Phi_s$ that is a PIC ${\cal A}_{\eta_s}^{N_s}$, whose descriptive parameters $(K_s,\alpha_s)$ are given by \begin{eqnarray} K_s&=& V K W = \sqrt{g \eta_1 \eta_2 } \openone_2, \, {\text{and}}\nonumber \\ \alpha _s &=& W^T \alpha_{\rm PIC} W = \sqrt{\det(\alpha_{\rm PIC})} \openone_2, \label{st0_main} \end{eqnarray} where $V$ and $W$ are Gaussian unitaries. \end{theorem} \begin{proof} See Appendix~\ref{app:proof_thm:phi_phis}. \end{proof}
Comparing Eq.~\eqref{st0_main} with Eq.~\eqref{i}, it is easy to see that $\Phi_s$ is in fact a PIC, ${\cal A}_{\eta_s}^{N_s}$, with \begin{eqnarray} \eta_s &=& g \eta_1 \eta_2, \,{\text{and}} \nonumber \\
N_s &=& \sqrt{ \det (\alpha_{\rm PIC})} - \frac{|1- \eta_s|}{ 2}. \label{DefEff} \end{eqnarray}
The condition under which the Gaussian center station ${\cal N}_G $ causes the lossy channel to be an EB channel is determined by applying Eq.~(\ref{38}) to the parameters of $\Phi_s$ in Eq.~\eqref{DefEff} since it is unitary equivalent to a PIC. Therefore, the channel $\Phi_0$ is EB if \begin{eqnarray} \sqrt{\det(\alpha_{\rm PIC} )} \ge \frac{1}{2}(1+ g\eta_1\eta_2). \label{ebpic38} \end{eqnarray} \\
\noindent {\em 2. ${\cal N}_G$ unitary equivalent to an ANC ${\cal I}^\epsilon$}--- it is straightforward to deduce [see line 2.b in Fig.~\ref{fig:onemodechannel_main}(c)] that, $\Phi_0 \equiv {\cal A}_{\eta_2} \circ {\cal N}_G \circ {\cal A}_{\eta_1}$ is unitary-equivalent to a channel $\Phi = \mathcal A_{\eta_2} \circ \mathcal S_{ G_2, \theta }\circ (\mathcal R_{-\phi} \circ \mathcal I ^{\epsilon } \circ \mathcal R_\phi) \circ \mathcal S_{G_1} \circ \mathcal A_{\eta_1}$, whose parameters are given by
\begin{eqnarray} K_{\rm ANC} &=& \sqrt{ \eta _1 \eta_2} S_1 K_\theta, \,{\text{and}}\\ \alpha_{\rm ANC} &=& \frac{1}{2}\left[ \eta_2(1-\eta_1) K_\theta^T S_1 ^2 K_\theta + \eta_2 K_\theta^T \epsilon^\prime K_\theta \right. \nonumber \\ &&\left. + (1-\eta_2) \openone_2\right], \end{eqnarray} where ${\cal R}_\phi$ is a phase rotation, $K_\theta = R_\theta S_2 R_\theta^\dagger$, and \begin{eqnarray}
\epsilon ':= \frac{1}{2}R_{\phi} \left(
\begin{array}{cc}
0 & 0 \\
0 & \epsilon \\
\end{array} \right)
R_{-\phi}^\dagger.
\end{eqnarray} Next we prove that $\Phi$ is unitary-equivalent to a PIC $\Phi_s$ [see line 2.c of Fig.~\ref{fig:onemodechannel_main}(b)]. \begin{theorem}\label{thm:phi_anc} $\Phi$ is unitary-equivalent to a PIC $\Phi_s$ described by, \begin{eqnarray} K_s&=& \sqrt{\eta_1 \eta_2 } \openone_2 , {\text{and}}\nonumber \\ \alpha _s &=& \sqrt{\det(\alpha_{\rm ANC})} \openone_2. \label{st1} \end{eqnarray} The excess noise parameter is given by \begin{eqnarray}
N_s &=& \sqrt{ \det (\alpha_{\rm ANC})} - \frac{|1- \eta_1\eta_2 |}{ 2}. \label{ancns1} \end{eqnarray} \end{theorem} \begin{proof} As in the proof of Theorem~\ref{thm:phi_phis} (see Appendix~\ref{app:proof_thm:phi_phis}), we can simultaneously diagonalize $K$ and $\alpha$. This follows Eq.~\eqref{st1}. Comparing Eqs.~\eqref{st1} and~\eqref{i}, we can determine the parameter $N_s$ in Eq.~\eqref{ancns1}. \end{proof} The condition under which the Gaussian center channel ${\cal N}_G $ causes the lossy channel to be an EB channel
is determined by applying Eq. (\ref{38}) to the parameters of $\Phi_s$ in Eqs.~\eqref{st1} and~\eqref{ancns1} since it is unitary equivalent to a PIC. The condition for $\Phi_0$ to be EB, translates to \begin{equation} \sqrt{\det(\alpha_{\rm ANC} )} \ge \frac{1}{2}(1+ \eta_1\eta_2). \end{equation}
\subsection{Explicit examples of a Gaussian center station: optical amplifiers}\label{sec:PIAPSA} In this subsection we illustrate our results in Section~\ref{sec:general_nonEB} with the important example of optical amplifiers used as center stations. We will consider the cases of a phase-sensitive amplifier (PSA) and phase-insensitive amplifier (PIA). Detailed proofs of the results will be deferred to Appendix~\ref{sec:PSAPIA}. If the center station ${\cal N}_G$ is a PSA of gain $G_{\rm PSA}$, then the composition $\Phi_0 \equiv {\cal A}_{\eta_2} \circ {\cal N}_G \circ {\cal A}_{\eta_1}$ becomes EB if the gain $G_{\rm PSA} $ exceeds a threshold $G^{\rm thres}_{\rm PSA}$ as \begin{equation} G_{\rm PSA} \ge G^{\rm thres}_{\rm PSA} := 1 + \frac{\eta_1}{(1-\eta_1)(1-\eta_2)}. \end{equation} (See Appendix~\ref{sec:PSA_app} for proof.) If ${\cal N}_G$ is a PIA of gain $G_{\rm PIA}$, then $\Phi_0$ becomes EB if the Gain $G_{\rm PIA}$ exceeds a threshold value $G^{\rm thres}_{\rm PIA}$ (see Appendix~\ref{sec:PIA_app} for proof): \begin{equation} G_{\rm PIA} \ge G^{\rm thres}_{\rm PIA} := \frac{1}{1-\eta_1}. \end{equation} Note that the transmittance $\eta_2$ of the loss segment {\em after} the PIA does not play a role in determining when $\Phi_0$ becomes EB. The expression for the threshold shows that when the channel transmittance $\eta_1$ of a the initial is low, an amplifier with even a small amount of gain can render the lossy channel EB. Finally, the concatenation of a chain of PSA center stations, $\eta_1 \to {\rm PSA}(G_1) \to \eta_2 \to {\rm PSA}(G_2) \to \ldots \to {\rm PSA}(G_k) \to \eta_{k+1}$, can be decomposed as ${\cal N}_G^2 \circ {\cal A}_{\eta_1 \eta_2 \ldots \eta_{k+1}} \circ {\cal N}_G^1$, where ${\cal N}_G^1$ is a PSA at the channel input (of an appropriate gain and squeezing angle) followed by classical thermal noise addition, ${\cal A}_{\eta_1\eta_2\ldots\eta_{k+1}}$ is the entire channel loss collected in the middle, and ${\cal N}_G^2$ is a PSA at the channel output. For expressions of the gain and phase parameters of the PSAs at the transmitter and the receiver, see Appendix~\ref{sec:PSAchain}.
We note here that PIAs can improve the signal-to-noise ratio (SNR) of a sub-unity-efficiency optical heterodyne detection receiver, albeit up to 3 dB of the quantum limited SNR, when preceding the receiver. PSAs on the other hand have been proposed for use in optical imaging~\cite{Dut10} and secure-key generation~\cite{Zha14}, to boost the effective detection efficiency of homodyne detection receivers, in principle pushing the receiver's performance all the way to the quantum limited SNR, by preceding the receiver with a PSA whose gain quadrature is phase-matched to the homodyne detector's local oscillator. Despite these practical uses of optical amplifiers, our results in the earlier sections show that these amplifiers cannot increase the secret key capacity, and the results in the current section show that it is unlikely that they will help to realize the given secret key capacity in a practical implementation.
\section{Conclusions}\label{sec:conclusions} It was recently shown \cite{Tak13} that for QKD (secure key generation), along with a few other optical quantum communication tasks such as quantum (qubit) communication, entanglement generation, and direct-secure communication (each with two-way authenticated classical communication assistance), the rates are upper bounded by $R_{\rm UB} = \log_2[(1+\eta)/(1-\eta)]$ bits per mode over a pure-loss optical channel of transmittance $\eta$. This upper bound reads $R_{\rm UB} \approx 2.88\eta$ when $\eta \ll 1$ (high loss), which translates to an exponential decay of rate with distance $L$ in fiber ($\eta \propto e^{-\alpha L}$), and an inverse-square decay with $L$ in free-space ($\eta \propto 1/L^2$). Quantum repeaters are conceptual devices that, when inserted along the lossy channel, can help circumvent this rate-loss trade-off.
In this paper, we have proven the inefficacy of bosonic Gaussian channels---optical processes that can be assembled using passive linear optics (beamsplitters and phase-shifters) and squeezers (phase-sensitive amplifiers, and the interaction of parametric downconversion)---to be used as quantum repeaters. We prove this by showing that any concatenation of such untended Gaussian operations along a lossy channel can be simulated by one Gaussian operation at the channel input and one at the channel output, where the entire loss in the channel is collected in the middle. We thereby argue that any communication protocol that uses such a chain of Gaussian center stations can be replaced by another protocol of the same performance without those stations, the transmitter and receiver of which are slightly modified versions of those used by the original protocol. As a consequence, the upper bound $R_{\rm UB}$, as shown above, still applies. Note, however, that our formulation is entirely based on the property of Gaussian channels and does not preclude the possibility that a trace-decreasing Gaussian operation \cite{Giedke2002} could serve as a quantum repeater.
It would be possible that intermediate trace-preserving Gaussian operations could be of practical advantage, while the same performance of any protocol working with such middle stations is in principle achievable {\it without} middle stations.
In order to demonstrate practical restrictions for use of conventional Gaussian stations, we separately analyzed the case of a general single-mode Gaussian channel sandwiched between lossy channels. We derived the conditions that the center station renders the end to end lossy channel entanglement breaking, and hence useless for QKD. From special cases for quantum-limited optical amplifiers as center stations, we found that in a high-loss regime, even modest amplification gains will render the overall channel entanglement breaking.
\acknowledgments
RN, OG, and NL were supported by the DARPA Quiness program under prime contract number W31P4Q-12-1-0017. SG was supported by the DARPA Quiness program subaward contract number SP0020412-PROJ0005188, under prime contract number W31P4Q-13-1-0004.
\appendix \section{Extracting one mean displacement in a concatenated Gaussian operation}\label{app:displacement}
The action of the concatenation of $n$ Gaussian channels ${\cal E}_n \circ \ldots \circ {\cal E}_2 \circ {\cal E}_1$, where ${\cal E}_i \triangleq (K_i, m_i, \alpha_i)$ can always be mimicked by a concatenation ${\cal E}^\prime_n \circ \ldots \circ {\cal E}^\prime_2 \circ {\cal E}^\prime_1$, where all the displacement terms are pushed to the $n$-th channel, i.e., ${\cal E}^\prime_i \triangleq (K_i, 0, \alpha_i)$, $1 \le i \le n-1$, and ${\cal E}^\prime_n \triangleq (K_n, m_t, \alpha_n)$, where $m_t$ is a function of $\left\{m_1, \ldots, m_n\right\}$, and $\left\{K_1, \ldots, K_n\right\}$. To see this, consider the composition of $n$ Gaussian channels: \begin{eqnarray} \mathcal E_{123 ... n } &=& \mathcal E_n \circ \cdots \circ \mathcal E_3 \circ \mathcal E_2 \circ \mathcal E_1, \end{eqnarray} for which we may write, \begin{eqnarray} d_{123 ... n } &:=& K_{123...n}^T d + m_{12...n} \nonumber, \\
&=& K_n^T K_{123...n-1}^T d + K_n ^T m_{123 ...n-1} +m_n \nonumber \\
& \vdots& \nonumber \\
&=& K_{1 \to n}^T d + \sum_{j=2}^n K_{j \to n }^T m_{j -1} +m_n \nonumber \\ &= & K_t d + m_t \label{70} \end{eqnarray} where \begin{eqnarray}
K_t :&=& K_{1 \to n } \; ,\nonumber \\
m_t:&=& \sum_{j=2}^n K_{j \to n }^T m_{j -1} +m_n \; ,\\
K_{j \to n } :&=& \begin{cases}
K_j K_{j+1} K_{j+2} \cdots K_{n-1} K_n & j< n-1 \nonumber \\
K_n & j=n. \end{cases} \end{eqnarray} From Eq. (\ref{70}) we can confirm that the total change in the first moment $d$ is given by $K_t = K_{1 \to n} $ and a constant shift $m_t= \sum_{j=2}^n K_{j \to n }^T m_{j -1} +m_n $. Hence, we can obtain the same transformation of $d$, for example, by setting $m_1 = m_2 = \cdots = m_{n-1} =0 $ and $m_n = m_t$, while leaving the gain terms $K_j$ of each channel $\mathcal E_ j $ as they are. To be precise, the following two channels equivalently act on $(\gamma, d) $. \begin{eqnarray} E & := & \underbrace{\mathcal E_n}_{(K_n, m_n,\alpha_n)} \circ \cdots \circ \underbrace{\mathcal E_2}_{(K_2, m_2,\alpha_2)} \circ \underbrace{\mathcal E_1}_{(K_1, m_1,\alpha_1)},
\\ E' & : = & \underbrace{\mathcal E_n}_{(K_n, m_t,\alpha_n)} \circ \cdots \circ \underbrace{\mathcal E_2}_{(K_2, 0,\alpha_2)} \circ \underbrace{\mathcal E_1}_{(K_1, 0,\alpha_1)}. \end{eqnarray}
Therefore, mean displacement terms can be absorbed into the final Gaussian operation, and their effect can be treated separately in the analysis of the sequential channel action. In this manner,
one can usually discuss Gaussian channel properties by assuming $m_j=0$ for all $j$ without loss of generality, and taking into account the effect of $m$'s, if at all needed, at once.
\section{Proof of Theorem~\ref{thm:phi_phis}}\label{app:proof_thm:phi_phis}
We restate Theorem~\ref{thm:phi_phis} below for completeness:
\begin{theorem*} $\Phi$ is unitary-equivalent to a channel $\Phi_s$ that is a PIC ${\cal A}_{\eta_s}^{N_s}$, whose descriptive parameters $(K_s,\alpha_s)$ are given by \begin{eqnarray} K_s&=& V K W = \sqrt{g \eta_1 \eta_2 } \openone_2, \, {\text{and}}\nonumber \\ \alpha _s &=& W^T \alpha W = \sqrt{\det(\alpha)} \openone_2, \label{st0} \end{eqnarray} where $V$ and $W$ are Gaussian unitaries. \end{theorem*} \begin{proof} Let $W_0$ be an orthogonal matrix that diagonalizes $\alpha$ so that $W_0^T \alpha W_0= \textrm{diag} (\lambda_1, \lambda_2) $. We then have, $\det(\alpha ) = \lambda_1 \lambda_2 $. It is then easy to see that the expression for $\alpha_s$ in Eq. (\ref{st0}) can be obtained by choosing \begin{eqnarray} W = (\lambda_1 \lambda_2)^{-1/4}{W_0 \sqrt{\textrm{diag} (\lambda_2, \lambda_1) }} . \end{eqnarray} Given this $W$, we can choose $V= W^{-1} K_\theta^{-1} S_1^{-1}$ to obtain the expression for $K_s$ in Eq. (\ref{st0}). Decomposing $\Phi$ into $\Phi_s$, sandwiched between unitaries $V$ and $W$ is depicted in line 1.c of Fig.~\ref{fig:onemodechannel_main}(b). \end{proof}
\section{Analysis of optical amplifiers as regenerative stations}\label{sec:PSAPIA}
In this Appendix, we prove the entanglement breaking conditions stated in Section~\ref{sec:PIAPSA}, for when an optical amplifier, either phase-insensitive (PIA) or phase-sensitive (PSA), is used as a center station, sandwiched between two pure-loss channel segments. We will evaluate these conditions by applying our general results from Section~\ref{sec:general_nonEB}.
The decomposition shown in line 1.b of Fig.~\ref{fig:onemodechannel_main}(b), with the excess noise parameter $N$ of ${\cal A}_g^N$ set to zero, includes the quantum-noise-limited PSA and PIA as special cases. For $g \ge 1$ and $N=0$, we obtain a simple expression of the determinant of $\alpha$ in Eq.~\eqref{eq3333}: \begin{eqnarray} \det(\alpha) &=&\frac{1}{4}\bigg \{ (1-g \eta_1\eta_2)^2
-4 \eta_2 \bigg[G_2 (1 - g \eta_1) (1 - \eta_2) \nonumber \\ &&
+ g G_1 (1 - \eta_1) (1 - g \eta_2) \nonumber \\
&& - 2 g\sqrt{ {G_1} {G_2}}(1 - \eta_1) (1 - \eta_2) \nonumber \\
&& \times \left( \sqrt{ {G_1} {G_2}}
+ \sqrt{( G_1-1)(G_2-1) } \cos 2\theta \right)\bigg] \bigg\}. \nonumber \\ \label{deta} \end{eqnarray}
\subsection{PSA sandwiched by two lossy channels}\label{sec:PSA_app} For the case of the quantum-limited PSA, by setting ${G_2=g= 1}$ in Eq. (\ref{deta}) one obtains
\begin{align} & \det( \alpha) = \frac{ (1- \eta_1 \eta_2)^2}{4} + \eta_2 (G_1-1) (1 - \eta_1) (1 -\eta_2) . \end{align} From this relation and with $\eta_s = \eta_1 \eta_2$ the EB condition of Eq.~\eqref{ebpic38} reads \begin{eqnarray}
G_1 \ge 1+ \frac{ \eta_1}{(1- \eta_1)(1- \eta _2)}. \label{interest} \end{eqnarray}
\begin{remark} A pure loss channel ${\cal A}_\eta$, $\eta \le 1$, is not EB. A quantum-limited PSA (which is a squeezer, and hence a unitary) is not EB. This is consistent with the observation that setting either $\eta_1$ or $\eta_2$ close to $1$ requires the PSA gain $G_1$ to go to infinity in order for the composition (loss-PSA-loss) to be EB. It is interesting that even though pure-loss channels and quantum-limited PSA are not EB by themselves, composing them can yield an EB channel if the gain and transmittances satisfy the condition in Eq.~\eqref{interest}. \end{remark}
\subsection{PIA sandwiched by two lossy channels}\label{sec:PIA_app}
For the case of the quantum-limited PIA by setting ${G_1=G_2= 1}$ in Eq. (\ref{deta}) one obtains \begin{eqnarray} \det (\alpha)= \frac{(1 + 2 (g-1)\eta_2 - g \eta_1 \eta_2 )^2}{4}. \end{eqnarray} From this relation and with $\eta_s = g \eta_1 \eta_2$, the EB condition of Eq.~\eqref{ebpic38} now reads \begin{eqnarray}\label{eq:EB_PIA}
g \ge \frac{1}{1-\eta_1}. \end{eqnarray} \begin{remark} One notable point is that the transmittance $\eta_2$ of the lossy channel that appears {\em after} the PIA, does not play a role in the EB condition in Eq.~\eqref{eq:EB_PIA}. \end{remark}
\subsection{Analysis of a chain of PSA center stations}\label{sec:PSAchain} Let us consider a chain of PSA center stations, interspersed between the number of $k+1$ lossy segments with transmission $\{\eta_i\}_{i= 1,2, \cdots, k+1}$ as in Fig.~\ref{fig:psaseq}(a). The channel action is formally written as \begin{align} \Phi_0 =& \mathcal A_{\eta_{k+1}} \circ \mathcal S_{ G_k } \circ \mathcal A_{\eta_k} \circ \cdots \nonumber \\
&\cdots \circ \mathcal A_{\eta_3} \circ \mathcal S_{ G_2 } \circ \mathcal A_{\eta_2} \circ \mathcal S_{G_1} \circ \mathcal A_{\eta_1}, \label{eqpsas} \end{align} where the action of PSAs $ \mathcal S_{ G_i } $ with the amplification gain of $\{G_i\}_{i= 1,2, \cdots, k}$ can be described by Eq.~\eqref{Gainnnnn}.
\begin{figure}
\caption{(Color online) (a) A sequence of PSAs $S_i$ with $i= 1,2, \cdots, k$ connected by the lossy segments of transmission $\eta_i$ with $i= 1, 2, \cdots , k+1$. The total loss is given by $\eta = \prod_{i=1}^{k+1} \eta_i$. (b) The PSA-loss chain $\Phi_0$ can be transformed to the standard form of a PIC $\Phi_s $ by using squeezing unitary operations $W$ and $S_0$. (c) The standard form $\Phi_s$ is decomposed into a thermal noise channel ${\cal A}_1^{N/\eta}$ and a transmission-$\eta $ pure lossy segment. Then, the origin channel $\Phi_0$ can be simulated by adding unitary operators to cancel out the unitary operators in (b) at the input-end and output end. This turns $\Phi_0$ into the form with original loss sandwiched by the operation ${\cal N}_G^1$ and the optput-end operation ${\cal N}_G^1$, --- an explicit example of our main result explained in Fig.~\ref{fig:summary}.}
\label{fig:psaseq}
\end{figure}
By repeatedly using the composition rule of Eq.~(\ref{CompoRule}) we can write the channel parameters $\Phi_0 \triangleq (K, 0, \alpha)$ as follows: \begin{widetext} \begin{align} K&= \sqrt{ \eta_{k+1}\eta_{k} \eta_{k-1}\cdots \eta_{1}} S_1 S_2 \cdots S_k = \sqrt{\eta} S_0, & \\ \alpha& = \frac{1}{2}\left[ \bar \eta_{k+1} \openone_2 + \eta_{k+1} \left\{ \bar \eta_k S_k^T S_k + \eta _k \bar \eta_{k-1} S_{k}^T S_{k-1}^T S_{k-1}S_k + \eta _k \eta _{k-1} \bar \eta_{k-2} S_{k}^T S_{k-1}^T S_{k-2}^T S_{k-2} S_{k-1}S_k +\cdots \right\} \right] \nonumber \\ &=\frac{1}{2}\left[ \bar \eta_{k+1} \openone_2 + \eta_{k+1} \left\{\sum_{n=0}^{k-1} \frac{\bar \eta_{k-n}}{\eta_{k-n}} \prod_{j=0}^{n} \eta_{k-j} (S_{k-j})^2 \right\} \right] = \left(
\begin{array}{cc}
\alpha^{(+)} & 0 \\
0 & \alpha^{(-)} \\
\end{array} \right),
\end{align} where \begin{align} {\eta} &:= { \eta_{k+1}\eta_{k} \eta_{k-1}\cdots \eta_{1}} , & \\ \bar \eta_i &:= 1- \eta_i ,\\
S_0&:= S_1 S_2 \cdots S_k , & \\ \alpha^{(\pm)}& := \frac{1}{2}\left[ \bar \eta_{k+1} + \eta_{k+1} \left\{\sum_{n=0}^{k-1} \frac{\bar \eta_{k-n}}{\eta_{k-n}} \prod_{j=0}^{n} \eta_{k-j}\left (\sqrt{G_{k-j} } \pm \sqrt{G_{k-j}-1}\right )^2 \right\} \right]. \end{align} \end{widetext}
Let us define a squeezer \begin{align} W := \left(\alpha^{(+)} \alpha^{(-)} \right )^{-1/2} \sqrt{\textrm{diag}[ \alpha^{(-)}, \alpha^{(+)}] } \end{align}
that symmetrizes $\alpha $ as $W^T \alpha W \propto \openone_2$ and set $V= W^{-1}S_0^{-1}$ similarly to the proof of Theorem~\ref{thm:phi_phis} in Appendix~\ref{app:proof_thm:phi_phis} [See Fig.~\ref{fig:psaseq}(b)]. Then, we can convert $\Phi_0$ to the standard form of a PIC $\Phi _S \triangleq (K_s,0, \alpha_s ) $ with \begin{align} K_s &= \sqrt{\eta} \openone_2,\\ \alpha_s&= \sqrt{ \det(\alpha ) } \openone_2 = \left[ \frac{1}{2} (1-\eta ) +N \right] \openone_2. \end{align} This implies the EB condition due to Eq.~\eqref{38}:
\begin{align}
\sqrt{ \det(\alpha ) } \ge \frac{1}{2} (1+\eta ). \end{align}
Let us now explicitly show the decomposition of the PSA chain into a pair of Gaussian operations at the input and the output. See Fig.~\ref{fig:psaseq} for a pictorial depiction. From the standard form we can split out the pure lossy segment by using the relation $\Phi_s\equiv {\cal A}_\eta^{N} = {\cal A}_\eta \circ {\cal A }_1^{N/\eta }$, which can be confirmed easily from the composition rule of Eq.~(\ref{CompoRule}). We can retrieve the original channel $\Phi_0$ by canceling the unitary operators $W$, $S_0$, and $W^{-1}$ as in Fig.~\ref{fig:psaseq}(c). To be specific, we can write the original channel in the sandwiched form $\Phi _0 = {\cal N }_G^{2} \circ {\cal A}_ \eta \circ {\cal N }_G^{1} $ with the two Gaussian channels ${\cal N }_G^{1} = {\cal A}_{1}^{N/ \eta} \circ {\cal W} \circ {\cal S}_0^{-1}$ and ${\cal N }_G^{2} = {\cal W } ^{-1} $. Note that the channel at the receiver ${\cal N }_G^{2}$ is a squeezing unitary, which is consistent with Proposition~\ref{prop:n-mode}.
\eat{ \section{Conclusions}\label{sec:conclusions}
We have shown that when a multi-mode Gaussian center station is sandwiched between two pure-loss channel segments, all the transmission loss can be collected in the middle, with appropriate modifications to the transmitter and the receiver stations (section~\ref{sec:GaussianStations}). Using a simple recursive argument, one can extend this statement to the case (shown in FIG.~\ref{fig:summary}(b)) when several Gaussian center stations are interspersed through a lossy channel, and show that it is possible to replace all the center stations by a single Gaussian operation at the input and a single Gaussian operation at the output, with the total transmittance of the entire length of the original lossy channel in between. We have also developed unitary-equivalent decompositions for a non-EB single-mode Gaussian station sandwiched by lossy segments, and determined the condition under which the non-entanglement-breaking relay station renders the whole channel entanglement breaking (see Appendix~\ref{app:singlemodeEB}). The following are a few key implications of our results:
\begin{itemize} \item Regardless of how a lossy channel is modified---by placing untended Gaussian center stations along the length of the channel---the secret-key rate achievable by {\em any} QKD protocol over that channel is upper bounded by $\log_2[(1+\eta)/(1-\eta)]$ secret key bits per mode~\cite{Tak13}, where $\eta$ is the total transmittance of the lossy channel. This fundamental rate-loss tradeoff establishes the real need for (non-Gaussian) quantum repeaters to make high-rate QKD feasible over long distances.
\item The secret-key capacity of a quantum channel $\cal N$, $C_s({\cal N}) = 0$, when ${\cal N}$ is EB. This is due to the fact that the output of an EB channel can be simulated quantitatively correctly using a measure-and-prepare scheme~\cite{Cur04}. We delineated the conditions under which Gaussian center stations, specifically quantum-noise-limited optical amplifiers, interspersed along a lossy channel, renders the entire channel EB (despite the fact that the pure-loss segments and quantum-noise-limited PSAs are not EB by themselves). For highly lossy channels, even a tiny amount of gain of an optical-amplifier center station renders the overall channel EB [as shown in Eqs.~\eqref{interest} and~\eqref{eq:EB_PIA}]. This shows that, for optical QKD over long distances, it is futile to use Gaussian optical amplifiers. In fact, one is better off not using them at all.
\item For any protocol using a lossy channel interspersed with Gaussian stations, there is {\em another} protocol with the same performance that does not use any center station. This new protocol can be derived from the original protocol by suitably amending the transmitted signals and the receiver measurement. Our result does not preclude a Gaussian center station to improve the performance of a protocol, {\em if} the transmitter and receiver is kept the same. Nor does it preclude the existence of scenarios where it might be technologically easier to implement a protocol with such center stations, as opposed to modifying the transmitter and the receiver per our prescription. \end{itemize} }
\end{document} |
\begin{document}
\title{Near Optimal Algorithm for Fault Tolerant Distance Oracle and Single Source Replacement Path problem
}
\begin{abstract} In a graph $G$ with a source $s$, we design a distance oracle that can answer the following query: $\textsc{Query}(s,t,e)$ -- find the length of shortest path from a fixed source $s$ to any destination vertex $t$ while avoiding any edge $e$. We design a deterministic algorithm that builds such an oracle in $\widetilde{O}(m\sqrt n)$ time\footnote{$\widetilde{O}()$ hides poly$\log n$ factor }. Our oracle uses $\widetilde{O}(n\sqrt n)$ space and can answer queries in $\widetilde{O}(1)$ time. Our oracle is an improvement of the work of Bil\`{o} et al. (ESA 2021) in the preprocessing time, which constructs the first deterministic oracle for this problem in $\widetilde{O}(m\sqrt n+n^2)$ time.
Using our distance oracle, we also solve the {\em single source replacement path problem} ($\textsc{Ssrp}$ problem). Chechik and Cohen (SODA 2019) designed a randomized combinatorial algorithm to solve the $\textsc{Ssrp}$ problem. The running time of their algorithm is $\widetilde{O}(m\sqrt n + n^2)$. In this paper, we show that the $\textsc{Ssrp}$ problem can be solved in $\widetilde{O}(m\sqrt n + |\mathcal{R}|)$ time, where $\mathcal{R}$ is the output set of the $\textsc{Ssrp}$ problem in $G$. Our $\textsc{Ssrp}$ algorithm is optimal (upto polylogarithmic factor) as there is a conditional lower bound of $\Omega(m\sqrt n)$ for any combinatorial algorithm that solves this problem. \end{abstract}
\section{Introduction} Real-life graph networks are prone to failures, e.g., nodes or links can fail. Thus, algorithms developed for these networks must be resilient to failures. For example, there may be some edges or links which are not working in the network and we want to avoid them. In this paper, we present an algorithm to create an oracle for the single source shortest path problem in a fault-prone graph. Such algorithms are also called fault-tolerant algorithms.
Consider an undirected and unweighted graph $G$ with a source $s$. We want to build an oracle that can find the length of shortest path from $s$ to any other vertex in the presence of faulty edges -- such an oracle is also called a {\em fault-tolerant distance oracle}. Formally,
\begin{definition}
A fault-tolerant distance oracle answers the following query in a graph $G$:
\begin{center}
$\textsc{Query}(s,t,F)$: Find the length of shortest path from $s$ to $t$ avoiding the set $F$ of edges.
\end{center}
\end{definition}
The time it takes to answer a query is called the {\em query time}. If the query is always from a fixed source $s$ and $|F|\le f$, then the distance oracle is called a $f$-edge fault tolerant single source distance oracle, or $\textsc{Sdo}(f)$ in short. If all vertices can be sources, the oracle is called $f$-edge fault tolerant distance oracle, or $\textsc{Do}(f)$. We list some results related to distance oracles:
Demetrescu et al. \cite{Demetrescu2008} designed a $\textsc{Do}(1)$ with $\widetilde{O}(n^2)$ space and $O(1)$ query time. Bernstein and Karger\cite{Bernstein2009} showed that this oracle can be built in $\widetilde{O}(mn)$ time. Pettie and Duan \cite{DuanP10} extended the result of Demestrescu et al. to two faults. They designed a $\textsc{Do}(2)$ with $\widetilde{O}(n^2)$ space and $\widetilde{O}(1)$ query time. Gupta and Singh \cite{GuptaS18} designed a $\textsc{Sdo}(1)$ with $\widetilde{O}(n\sqrt n)$ space and $\widetilde{O}(1)$ query time. Recently Bil{\`{o}} et al. \cite{BiloCFS21} built the $\textsc{Sdo}(1)$ (described in \cite{GuptaS18}) in $\widetilde{O}(m\sqrt n + n^2)$ time. Many different aspects of distance oracles have been studied in literature \cite{Bernstein2008,Bernstein2009,Bilo2016,ChechikCFK17,Chechik2010,10.5555/1496770.1496826,10.1145/2438645.2438646}.
In this paper, we will focus our attention on building $\textsc{Sdo}(1)$. Due to Bil{\`{o}} et al. \cite{BiloCFS21}, the time to build $\textsc{Sdo}(1)$ is $\widetilde{O}( m\sqrt n + n^2)$. Chechik and Cohen \cite{ChechikC19} showed that, the first term in this running time is a conditional lower bound for $\textsc{Ssrp}$ problem. But it is not clear if the second term is necessary. In this paper, we build a $\textsc{Sdo}(1)$ in $\widetilde{O}(m\sqrt n)$ time -- this preprocessing algorithm has a better runtime than \cite{BiloCFS21} for sparse graphs, which is state of the art for this problem till now. Using our $\textsc{Sdo}(1)$ data structure, we are able to reduce the runtime of the algorithm solving $\textsc{Ssrp}$ problem too. Our distance oracle is quite different from the distance oracle of Gupta and Singh \cite{GuptaS18} -- though we use the main technical idea of \cite{GuptaS18} crucially in our paper too. The construction of this new oracle is the main technical result of this paper.
\begin{theorem}
\label{thm:sdo}
For undirected, unweighted graphs there is a deterministic algorithm that can build a $\textsc{Sdo}(1)$ of size $\widetilde{O}(n\sqrt n)$ and query time $\widetilde{O}(1)$ in $\widetilde{O}(m\sqrt n)$ time. \end{theorem}
\subsection{Application: Single Source Replacement Path Problem} Let us first look at the {\em replacement path problem}. In this problem, we are given a source $s$ and a destination $t$. We assume that there is a unique shortest path from $s$ to $t$, denoted by $st$. \begin{definition}
(Replacement Path Problem) Let $s$ be a source and vertex $t$ be the destination in $G$.
For each $e \in st$ path, output the length of the shortest path from $s$ to $t$ avoiding $e$. \end{definition}
The replacement path problem was first investigated due to its relation with auction theory \cite{Hershberger2001, NisanR01} and has been studied extensively. For an undirected graph with non-negative edge weights, the replacement path problem can be solved in $\widetilde{O}(m+n)$ time\cite{MalikMG89,Hershberger2001,NardelliPW03}. We look at the generalization of the replacement path problem -- the single source replacement path problem. \begin{definition} ($\textsc{Ssrp}$ problem)
Let $s$ be a source in a graph $G$ which is undirected and unweighted. For each vertex $t \in G$ and each $e \in st$ path, output the length of the shortest path from $s$ to $t$ avoiding $e$. \end{definition} Chechik and Cohen \cite{ChechikC19} designed a randomized combinatorial algorithm that solves the $\textsc{Ssrp}$ problem in $\widetilde{O}(m\sqrt n +n^2)$ time. They also showed a matching conditional lower bound via Boolean Matrix Multiplication. \begin{lemma} \cite{ChechikC19}
\label{lem:lowerbound}
Let $\textsc{Bmm}(n,n)$ be the time taken to multiply two $n \times n$ boolean matrices with a total of $m$ ones. Under the assumption that any combinatorial algorithm for $\textsc{Bmm}(n,n)$ requires $mn^{1-o(1)}$ time\footnote{In a RAM\ model with words of $O(\log n)$ bits.}, any combinatorial algorithm for $\textsc{Ssrp}$ problem requires $\Omega(m\sqrt n)$ time. \end{lemma}
It may seem that the algorithm of Chechik and Cohen \cite{ChechikC19} is nearly optimal. It is indeed the case if the output size is $O(n^2)$. However, for a low-diameter graph, this extra additive factor seems unnecessary. If the graph is dense ($m \ge n^{3/2}$), then the $n^2$ factor is subsumed by the first term $m\sqrt n$. Thus, when $m < n^{3/2}$ and the graph has a low diameter, can we improve the running time of the $\textsc{Ssrp}$ problem? For such a graph, the algorithm of Chechik and Cohen \cite{ChechikC19} is not optimal. Similar to \cite{ChechikC19}, Gupta et al. \cite{GuptaJM20} also designed an algorithm for the $\textsc{Ssrp}$ problem. Even this algorithm has the running time $\widetilde{O}(m\sqrt n+n^2)$ -- though it uses an entirely different approach compared to \cite{ChechikC19}. Thus, the main question is: {\em can we remove this extra additive factor of $n^2$ from the running time of the $\textsc{Ssrp}$ problem?} In this paper, we design such an algorithm:
\begin{theorem}
\label{thm:main}
There is a deterministic algorithm for $\textsc{Ssrp}$ problem with a running time of $\widetilde{O}(m\sqrt n+ |\mathcal{R}|)$ where $|\mathcal{R}|$ is the output size of $\textsc{Ssrp}$ problem in $G$. \end{theorem}
In the above theorem, $|\mathcal{R}|$ is the output size, thus an implicit lower bound on the $\textsc{Ssrp}$ problem. Using \Cref{lem:lowerbound}, we conclude that our algorithm is nearly optimal up to a polylogarithmic factor.
To build an algorithm for the $\textsc{Ssrp}$ problem, we first build a $\textsc{Sdo}(1)$. Then, for each $t \in G$ and each $e \in st$ path, we call $\textsc{Query}(s,t,e)$ and output the answer. Thus, we claim the following lemma:
\begin{lemma}
\label{lem:reduction}
If we can build a $\textsc{Sdo}(1)$ with query time $q$ in time $T$, then there is an algorithm for $\textsc{Ssrp}$ problem with a running time $O(T+q|\mathcal{R}|)$, where $|\mathcal{R}|$ is the output size in $G$. \end{lemma}
The above lemma, along with \Cref{thm:sdo} implies \Cref{thm:main}.
\subsection{Related Work}
Other related problems include the fault-tolerant subgraph problem. In this problem, we want to find a subgraph of $G$ such that the shortest path from $s$ is preserved in the subgraph after any edge deletion. Parter and Peleg \cite{Parter2013} designed an algorithm to compute a single fault-tolerant subgraph with $ O(n^{3/2})$ edges. They also showed that their result could be easily extended to multiple sources. This result was later extended to dual fault by Parter [16] with $ O(n^{5/3}) $ edges. Gupta and Khan \cite{GuptaK17} extended the above result to multiple sources. All the above results are optimal due to a result by Parter \cite{Parter2015} which states that a multiple source $f$-fault tolerant subgraph requires $\Omega\Big(n^{2-\frac{1}{f+1}}\Big)$ edges. Bodwin et al. \cite{BodwinGPW17} showed the existence of a $f$-fault tolerant subgraph of size $O\Big( fn^{2-\frac{1}{2^f}} \Big)$.
\section{Preliminaries} Let $G(V,E)$ be an undirected unweighted graph with a source $s$.
Given two vertices $u$ and $v$ in a graph $H$, unless otherwise stated, $(uv)_H$ denotes the shortest path from $u$ to $v$ in $H$. If $H = G$, we will remove the subscript and the brackets -- we will apply this policy for all the notations below. $|uv|_H$ denotes the length of the shortest path in $H$. Some of our graphs will be weighted, even though $G$ is unweighted. If $H$ is weighted, then we will abuse notation and use $|uv|_H$ to denote the weight of the shortest path from $u$ to $v$ in $H$.
The edges and vertices of $H$ will be denoted by $E_H$ and $V_H$, respectively.\ Additionally, $m_H$ and $n_H$ will denote the number of edges and vertices in $H$, respectively.
$\textsc{Spt}_H(s)$ denotes the shortest path tree from $s$ in $H$. We can view the $\textsc{Spt}_H(s)$ to be drawn from top to bottom with the top vertex being $s$. For any two vertices $u,v$ on the $st$ path (the path between $s$ and $t$) in $\textsc{Spt}_H(s)$, we say that $u$ is before / above $v$ if $|su|_H <|sv|_H$. Similarly, we say that $u$ is after/below $v$ if $|su|_H >|sv|_H$.
For an edge $e$ in a weighted graph $H$, $wt_H(e)$ will denote the weight of $e$. Given two paths $(uv)_H$ and $(vw)_H$, the path $(uv)_H +(vw)_H$ denotes their concatenation.
$P[u,v]$ denotes a contiguous subpath of $P$ starting at $u$ and ending at $v$. Sometimes, we may also write {\em the interval $[u,v]$ of $P$} to denote $P[u,v]$. We say {\em $u$ comes before $v$} on a path $R$ starting from $s$, if $|R[s,u]| < |R[s,v]|$. Similarly, we can define the term {\em $u$ comes after $v$} on path $R$.
A replacement path $R$ is the shortest path from $s$ to $t$ avoiding an edge $e$ on $st$ path. There can be many replacement paths of the same length avoiding $e$. To ensure uniqueness, we will use the following definition of replacement path\footnote{This was referred to as {\em preferred replacement path} in \cite{GuptaK17}.}.
\input{fig_detour} \begin{definition} \label{def:replacementpath} ( Replacement Path) A path $R$ from $s$ to $t$ avoiding $e$ is called a replacement path if (1) it diverges from and merges to the $st$ path just once (2) its divergence point from the $st$ path is as close to $s$ as possible. (3) it is the {\bf lexicographically smallest}\footnote{Let $P$ and $P'$ first diverge from each other to $x \in P$ and $x' \in P'$ respectively. If the index of $x$ is lower than $x'$, then $P$ is said to be lexicographically smaller than $P'$.} shortest path in $G$ satisfying (1) and (2).
\end{definition}
We now define some terms related to replacement paths. $(st \diamond e)_H$ denotes the replacement path from $s$ to $t$ avoiding the edge $e$ in $H$. We can generalize this notation to a replacement path that avoids a set of edges. Thus, $(st \diamond F)_H$ denotes the replacement path from $s$ to $t$ in $H$ avoiding a set $F$ of edges. In our algorithm, after we find the replacement path $(st \diamond e)_H$, we will store its length in $d_H(s,t,e)$. Sometimes, we also want to store the length of $(st \diamond F)_H$. In that case, we will store it in $d_H(s,t,F)$.
\begin{definition}
\label{def:detour}
(Detour and Detour point of a replacement path) \cite{ChechikM20}
Let $R = st \diamond e$. Then, the detour of $R$ is $R \setminus st$. That is, let us assume that $R$ leaves $st$ above $e$ at a vertex $u$, and merges back on $st$ at vertex $v$ after $e$, then detour of $R$ is $R[u,v]$. Also, the vertex at which the detour starts is called the detour point of $R$. So, $u$ is the detour point of $R$ or in short $\textsc{Dp}(R)=u$.
\end{definition}
Lastly, in our algorithm, we will need to find least common ancestor of any two vertices $u$ and $v$ in $\textsc{Spt}_H(s)$. Let $\textsc{Lca}_H(u,v)$ denotes the least common ancestor of $u$ and $v$ in $\textsc{Spt}_H(s)$.
To find the $\textsc{Lca}$, we will use the following result:
\begin{lemma} (See \cite{BenderF00} and its references) \label{lem:lca}
Given a tree $T$ on $n$ vertices, we can build a data structure of size $O(n)$ in $O(n)$ time such that the least common ancestor query can be answered in $O(1)$ time.
\end{lemma}
\section{Overview of our algorithm to build SDO(1)}
We will use divide and conquer approach to build $\textsc{Sdo}(1)$. This strategy has been previously used for directed graphs in \cite{GrandoniW12,ChechikM20}. However, simply using this strategy will not get us close to our desired bound of $\widetilde{O}(m\sqrt n)$. For that, we need to combine this divide and conquer strategy with an idea of Gupta and Singh \cite{GuptaS18}. This combination is one of the technical contributions of the paper.
Like \cite{GrandoniW12,ChechikM20}, we use the following separator lemma to divide the graph $G$. \begin{theorem}(Separator Lemma \cite{GrandoniW12,ChechikM20}) \label{lem:separator}
Given a tree $T$ with $n$ nodes rooted at a source $s$, one can find in $O(n)$ time a vertex $r$ that separates the tree $T$ into edge disjoint sub-trees $M,N$ such that $E_{M}\cup E_{N}=E_{T}, V_{M} \cap V_{N}=\{r\}$ and $\frac{n}{3 } \leq | V_{M}|, |V_{N}| \leq \frac{2n}{3}$.
\end{theorem}
Without loss of generality, we will assume that $s\in \VV_{\MM}$. Thus, $r$ is the root of $N$. Also, note that $s$ and $r$ may or may not be the same. Let $G_{M}$ and $G_{N}$ be the graph induced by the vertices of $M$ and $N$, respectively.
There is one more important term that we will use in our paper:
\begin{definition} (Primary Path $\mathcal{P}$) Using the separator lemma, $\SPT(s)$ can be divided into two sub-trees $M$ and $N$ with roots $s$ and $r$. The path from $s$ to $r$ is called the primary path and is denoted by $\mathcal{P}$. \end{definition}
We now describe our data structure that we will build recursively. We can view the data structure as a binary tree $\mathcal{T}$. The root contains data structure for the entire graph $G$. We will abuse notation and say the root is the graph $G$.
The left child of the root will contain the graph $\GG_{\MM}$ and some weighted edges -- we will describe the utility of these weighted edges in the next section. These weighted edges are not in $G$ but are added by our algorithm. We will then build a data structure for $\GG_{\MM}$ recursively. The right child of the root will contain the graph $\GG_{\NN}$, again with some weighted edges.
At the root of $\mathcal{T}$, we store the following data structures. For each $v \in G$, set $d(s,v)=|sv|$. Similarly, set $d(r,v)= |rv|$. For each $v \in \GG_{\NN}$, set $d(s,v,\GG_{\NN}) = |sv \diamond \GG_{\NN}|$. Similarly, for each $v \in \GG_{\MM}$, set $d(s,v, \GG_{\MM}) = |sv \diamond \GG_{\MM}|$. All these quantities can be computed using a single source shortest path algorithm in $\widetilde{O}(m+n)$ time. Additionally, we will find the length of all replacement paths from $s$ to $r$ avoiding edges on the primary path $\mathcal{P}$. This can be done in $\widetilde{O}(m+n)$ time using\footnote{This algorithm work for graphs with non-negative edge weights. And our graph may have weighted edges.} \cite{Hershberger2001}. We will set $d(s,r,e) = |sr \diamond e| $ for each edge $e \in \mathcal{P}$.
\begin{comment}
\begin{enumerate}
\item $d(s,v)$: We will build the shortest path tree $\textsc{Spt}(s)$ from $s$ in $G$. For all $v \in G$, we will store the shortest path from $s$ to $v$ in $d(s,v)$ such that it can be retrieved in $O(1)$ time. This step takes $O(m+n)$ time.
\item $d(r,v)$: We will build the shortest path tree $\textsc{Spt}(r)$ from $r$ in $G$. For all $v \in G$, we will store the shortest path from $r$ to $v$ in $d(r,v)$. This step takes $\widetilde{O}(m+n)$ time
\item For each $v \in \GG_{\NN}$, we will find the shortest path from $s$ to $v$ avoiding all the edges in $\GG_{\NN}$. We can find this quantity by removing all the edges of $\GG_{\NN}$ from $G$ and then build the shortest path tree from $s$. We will set, $d(s,v,\GG_{\NN}) = |sv \diamond \GG_{\NN}|$. This step also takes $\widetilde{O}(m+n)$ time.
\item For each $v \in \GG_{\MM}$, we will find the shortest path from $r$ to $v$ avoiding all the edges in $\GG_{\MM}$. We can find this quantity by removing all the edges of $\GG_{\MM}$ from $G$ and then build the shortest path tree from $r$. We will set, $d(r,v,\GG_{\MM}) = |rv \diamond \GG_{\MM}|$. This step also takes $\widetilde{O}(m+n)$ time.
\item We will find all replacement paths from $s$ to $r$ avoiding edges on $\mathcal{P}= sr$ path. This can be done in $\widetilde{O}(m+n)$ time using \cite{Hershberger2001}. Note that this algorithm work for graphs with non-negative edge weights. We will set $d(s,r,e) = |sr \diamond e| $ for each edge $e \in \mathcal{P}$.
\end{enumerate} \end{comment} We store the above data-structure in each node of $\mathcal{T}$. If a node of $\mathcal{T}$ contains graph $H$, then we can contruct the above data-structures in $\widetilde{O}(m_H + n_H)$ time. We now describe our algorithm that finds replacements paths using $\mathcal{T}$.
\begin{comment}
\subsection{Algorithm to build $\textsc{Sdo}(1)$}
Let us see how we find and store replacement paths at the root of $\mathcal{T}$, that is, for the graph $G$. At first, we look at the replacement path for an edge on the primary path, that is $e \in \mathcal{P}$. Let $R = st \diamond e$. We define $R$ to be either {\em jumping} or {\em departing} depending on whether it merges back to the primary path or not.
\begin{definition}(Jumping and Departing paths)
Let $R = st \diamond e$ where $e \in \mathcal{P}$ . $R$ is called a \textbf{\em jumping path} if it uses some vertex $u \in \mathcal{P}$ after $e$. If the path is not jumping then it is a \textbf{\em departing path}. See Figure \Cref{fig:departingjumping} for a visualization of these two kinds of paths.\end{definition}
\input{fig_departing} Remember that a jumping or departing path is defined for edges on the primary path only. Also, If a replacement path is departing, then the destination $t$ cannot lie in $\mathcal{P}$. We will show the following new technical result in \Cref{sec:departing}.
\begin{lemma} \label{lem:departing} For each $t \in G$, all departing replacement paths to $t$ can be found in deterministic $\widetilde{O}(m\sqrt n)$ time. Moreover, all such departing paths can be stored in a data structure of size $\widetilde{O}(n\sqrt n)$ and can be queried in $\widetilde{O}(1)$ time. \end{lemma}
Thus, we just need to concentrate on the replacement paths that are either jumping or the faulty edge $e \notin \mathcal{P}$. We now divide remaining replacement paths depending on where the destination $t$ and faulty edge $e$ lies. There are following cases: \input{fig_gmgn} \begin{enumerate} \item $e \in \GG_{\MM}$ and $t \in \GG_{\NN}$
This case itself can be divided into two cases depending on whether $e$ lies on the primary path or not.
\begin{enumerate} \item $e \in \mathcal{P}$ (See \Cref{fig:gmgn} (a))
If $st \diamond e$ is a departing replacement path, then we have already found it using \Cref{lem:departing}. So let us assume that $st \diamond e$ is jumping. Since $st \diamond e$ is jumping, it merges back to $\mathcal{P}$. Since $\mathcal{P}$ is the prefix of $st$ path, we claim that $st \diamond e = sr \diamond e + rt$. So, in order to find the length of $st \diamond e$, we need to find the length of $sr \diamond e$ and $rt$.
To this end, we will build two data-structure. In the first data structure, we will find all the replacement paths from $s$ to $r$. For our second data structure, we will build the shortest path tree from $r$, that is $\SPT(r)$. Once we have these two data structure[s], we can find the length of $st \diamond e$.
\item $e \notin \mathcal{P}$ (See Figure \Cref{fig:gmgn} (b))
In this case, we claim that $st \diamond e = st$. This is because the $st$ path has $\mathcal{P}$ as its prefix. Since $\mathcal{P}$ lies in $\GG_{\MM}$ and survives after the deletion of $e$, $st$ path remains intact. \end{enumerate}
\item $e \in \GG_{\MM}$ and $t \in \GG_{\MM}$ \input{fig_gmgmcase1}
Again there are following main cases: \begin{enumerate} \item $st \diamond e$ completely lies inside $\GG_{\MM}$
In this case, we can recurse on the subgraph $\GG_{\MM}$.
\item $st \diamond e$ uses some edges of $\GG_{\NN}$
There are two subcases depending on where the edge $e$ lies:
\begin{enumerate} \item $e \in \mathcal{P}$ (See Figure \Cref{fig:gmgm}(a))
If $st \diamond e$ is a departing replacement path, then we have already found it in Lemma \Cref{lem:departing}. So let us assume that $st \diamond e$ is jumping. In Section \Cref{subsec:MM}, we will show that $st \diamond e$ passes through $r$. Thus, $st \diamond e = sr \diamond e + rt$. Similar to point 1(a), we have already found the two quantities on the right-hand side of the above equality. Thus, these jumping paths can be found easily.
\item $e \notin \mathcal{P}$ (See Figure \Cref{fig:gmgm}(b))
We will show in section \Cref{subsec:MM} that $st \diamond e$ passes through $r$. Moreover, we will show that $st \diamond e$ can be found by recursing in $\GG_{\MM}$ -- though we have to add some weighted edges in $\GG_{\MM}$. Thus, we can club this case with point 2(a), and we need to recurse in $\GG_{\MM}$ just once -- after adding some weighted edges. This is very important as otherwise, we will not be able to bound the running time of our algorithm. \end{enumerate}
\end{enumerate}
\item $e \in \GG_{\NN}$ and $t \in \GG_{\MM}$
In this case, we claim that the $st$ path survives. Thus, $st \diamond e = st$.
\item $e \in \GG_{\NN}$ and $t \in \GG_{\NN}$
Like point 2(b)(ii), in this case, we will show that we can recurse in $\GG_{\NN}$ after adding some weighted edges. Thus, we will be able to find replacement paths for such cases via recursion.
\item One endpoint of $e$ is in $\GG_{\MM}$ and the other endpoint is in $\GG_{\NN}$ and $t \in G$.
Similar to point(3), we claim that $st$ path survives. Thus, $st \diamond e = st$. \end{enumerate}
This completes the description of our algorithm at the root of the binary tree $\mathcal{T}$. The reader can verify that we have covered all the cases. We will see that the time taken by the algorithm at the root is dominated by the time taken in Lemma \Cref{lem:departing}. This will imply that the total time taken by our algorithm is $\widetilde{O}(m\sqrt n)$. Similarly, the reader will see that the space taken by our algorithm at the root is dominated by the space taken by the data structure in Lemma \Cref{lem:departing}. This will imply that the total space taken by our algorithm is $\widetilde{O}(n \sqrt n)$.
\end{comment}
\input{fig_departing}
Let us see how we find and store lengths of the replacement paths at the root of $\mathcal{T}$, that contains graph $G$. First, we find the replacement paths for edges on the primary path. Let $R = st \diamond e$ where $e \in \mathcal{P}$. We define $R$ to be either {\em jumping} or {\em departing} depending on whether it merges back to the primary path or not.
\begin{definition}(Jumping and Departing paths)
Let $R = st \diamond e$ where $e \in \mathcal{P}$ . $R$ is called a \textbf{\em jumping path} if it uses some vertex $u \in \mathcal{P}$ after $e$. If the path is not jumping then it is a \textbf{\em departing path}. If a replacement path is jumping, then it is called {\em jumping replacement path}. Similarly, we define departing replacement path. See \Cref{fig:departingjumping} for a visualization of these two kinds of paths.\end{definition}
Note that, jumping or departing path is defined only when the edge fault is on the primary path. Also, if a replacement path is departing, then the destination $t$ cannot lie on $\mathcal{P}$. In \Cref{sec:detail}, we will find all jumping replacement paths.
In \Cref{sec:departing}, we design a new algorithm for finding and storing all departing replacement paths. To this end, we will use the main idea in the paper of Gupta and Singh \cite{GuptaS18}. In \cite{GuptaS18}, the authors sampled a set of vertices with probability of $O(\frac{1}{\sqrt{n}})$. Then, for a vertex $t \in G$, they find a sampled vertex near $t$ on the $st$ path. They call this vertex $t_s$. Then, they show the following important lemma, which is the main idea of their paper:
\begin{lemma} (Lemma 11 in \cite{GuptaS18}) \label{lem:guptaaditi} The number of replacement paths from $s$ to $t$ that avoid edges in $st_s$ path and also avoid $t_s$ is $O(\sqrt n)$. \end{lemma}
An astute reader can see that the definition of replacement paths in the above definition looks very similar to departing replacement paths. We prove that this is indeed the case. Thus, we can transfer the result in \Cref{lem:guptaaditi} to departing replacement paths. This is the main novelty of the paper. The main technical result of \Cref{sec:departing} is as follows:
\begin{lemma} \label{lem:departing} For each $t \in G$, all departing replacement paths to $t$ can be found in deterministic $\widetilde{O}(m\sqrt n)$ time. Moreover, the length of all such departing paths can be stored in a data structure of size $\widetilde{O}(n\sqrt n)$ and can be queried in $\widetilde{O}(1)$ time. \end{lemma}
\input{fig_gmgn} \section{Algorithm to build SDO(1): Replacement paths that are not departing} \label{sec:detail} In \Cref{sec:departing}, we will find all and store all departing replacement paths. Thus, we just need to concentrate on the replacement paths that are either jumping or the faulty edge $e \notin \mathcal{P}$. We now divide remaining replacement paths depending on where the destination $t$ and faulty edge $e$ lies. There are following cases:
\subsection{$e \in \GG_{\MM}$ and $t \in \GG_{\NN}$} \label{sec:gmgn}
This case itself can be divided into two cases depending on whether $e$ lies on the primary path or not. \begin{enumerate}
\item $e \in \mathcal{P}$ (See \Cref{fig:gmgn}(a))
Let $R = st \diamond e$. If $R$ is departing then we will see how to find it in \Cref{sec:departing}. So, assume that $R$ is jumping . This implies that $R$ merges back to $\mathcal{P}$ at a vertex, say $w$, after the edge $e$. Since $t \in \GG_{\NN}$, $st = sw + wr + rt$. Thus, after merging with $\mathcal{P}$ at $w$, the replacement path passes through $r$. In that case, $|st \diamond e| = |sr \diamond e| + |rt|$. We can easily find the right hand side of the above equality as we have stored $d(s,r,e) = |sr \diamond e|$ and $d(r,t) = |rt|$.
\item $e \notin \mathcal{P}$ (See \Cref{fig:gmgn}(b))
In this case, we claim that $st \diamond e = st$. The $st$ path has $\mathcal{P}$ as its prefix. Since $\mathcal{P}$ lies in $\GG_{\MM}$ and survives after the deletion of $e$, $st$ path remains intact.
\end{enumerate}
\subsection{$e \in \GG_{\MM}$ and $t \in \GG_{\MM}$} \label{subsec:MM} \input{fig_gmgmcase1}
Since both $e$ and $t$ lie in $\GG_{\MM}$, one may think that we can recurse our algorithm in $\GG_{\MM}$ to find $st \diamond e$. If $st \diamond e$ completely lies inside $\GG_{\MM}$, this is indeed the case. However, $st \diamond e$ may also use edges of $\GG_{\NN}$. To handle such cases, before recursing in $\GG_{\MM}$, we will add weighted edges to it. For each $v \in \GG_{\MM}$, we will add an edge from $r$ to $v$ with a weight $|rv \diamond \GG_{\MM}|$. We have already calculated this weight, it is stored in $d(r,v,\GG_{\MM})$. Let the set of weighted edges added to $\GG_{\MM}$ be called $X$. We now look at two cases, (1) $e \in \mathcal{P}$ and (2) $e \notin \mathcal{P}$.
\subsubsection{$e \in \mathcal{P}$ } \label{sec:gmgmp} Let $R = st \diamond e$ be a jumping replacement path.
We will show that $st \diamond e = sr \diamond e + rt$. As we have calculated the length of both the paths in the right-hand side of the above equality, there is no need to even recurse in this case. To prove the above equality, we first prove the following simple lemma:
\begin{lemma} \label{lem:passr} Let $e \in \mathcal{P}$, $t \in \GG_{\MM}$. Assume that the jumping replacement path $R =st \diamond e$ uses some edges of $\GG_{\NN}$. Then $st \diamond e$ passes through $r$. \end{lemma}
\begin{proof} Since $R$ is jumping, it merges with $\mathcal{P}$. There are two ways in which $R$ can merge with $\mathcal{P}$. \begin{enumerate}
\item $R$ merges with $\mathcal{P}$ and then visits the edges of $\GG_{\NN}$.
Let us assume that $u$ is the last vertex of $\GG_{\NN}$ in the path $R$ and $R$ merges with $\mathcal{P}$ at $w$. Since $R$ first merges with $\mathcal{P}$ and then visits the edges of $\GG_{\NN}$, $u$ comes after $w$ on $R$. Since $w$ is below $e$ on $\mathcal{P}$, we claim that the sub-path $wu$ of $su$ survives in $G\setminus e$ and is also the shortest path from $w$ to $u$. But $wu$ path passes through $r$. Thus, $st \diamond e$ passes through $r$ by construction in \Cref{lem:separator}.
\item $R$ visits an edge of $\GG_{\NN}$ and then merges with $\mathcal{P}$ (See \Cref{fig:gmgm}(a))
In this case, we will show that $R$ merges with $\mathcal{P}$ at $r$. For contradiction, let $w$ be the vertex at which $R$ merges with $\mathcal{P}$ such that $w \neq r$.
Let $u$ be the first vertex of $\GG_{\NN}$ visited by $R$.
Also, $w$ lies after $u$ on path $R$. Thus, the replacement path $R = R[s,u]+ R[u,w] + R[w,t]$. Since $w$ lies below $e$ on $\mathcal{P}$, $wu$ sub-path of $su$ does not contain $e$ and is also the shortest path from $w$ to $u$. Thus, $R[u,w] = uw$. But $uw$ path passes through $r$. This implies that $R$ merges with $\mathcal{P}$ at $r$ contradicting our assumption that $w \neq r$. \end{enumerate}
\end{proof}
We are now ready to prove the main lemma in this subsection.
\begin{lemma} \label{lem:ste}
Let $e \in \mathcal{P}$ and $t \in \GG_{\MM}$. Assume that the jumping replacement path $R =st \diamond e$ uses some edges of $\GG_{\NN}$. Then $|st \diamond e| = |sr \diamond e| + |rt|$. \end{lemma}
\begin{proof}
Using \Cref{lem:passr}, $R$ passes through $r$. So, we have $R = R[s,r]+ R[r,t]$. The first summand on the right hand side of the above equality represent a path from $s$ to $r$ avoiding $e$. Thus, $|R[s,r]| = |sr \diamond e|$.
We will now show that $|R[r,t]| = |rt|$. Clearly $|R[r,t]| \ge |rt|$ as the first path avoid $e$ and the second path may or may not. We will now show that the second path also avoids $e$ which will imply that both paths are of same length. For contradiction, assume that $rt$ passes through $e$. Let us assume that there is a vertex $w$ before the edge $e$ on path $\mathcal{P}$ such that $rt = rw + wt$. Thus, $rw$ passes through $e$ but $wt$ avoids $e$. But then, there is a path $R'$ such that $R' = sw + wt$ which avoids $e$. We claim that $|R'| < |R|$ contradicting the fact that $R$ is the replacement path from $s$ to $t$ avoiding $e$.
To this end, \begin{tabbing}
\hspace{30mm}$|R|$\= $= |sr \diamond e| + |R[r,t]|$\\
\>$\ge |sr \diamond e| + |rt|$\\
\>$= |sr \diamond e| + |rw| + |wt|$\\
\> $\ge |sr \diamond e|+|wt|$\\
Since $|sw| < |sr|$, $|sw| < |sr \diamond e|$\\
\>$> |sw| + |wt|$\\
\>$= |R'|$. \end{tabbing} \ This completes the proof of the lemma. \end{proof}
\subsubsection{$e \notin \mathcal{P}$ } \label{sec:gmgnnp} In this case, we will show a path in $\GG_{\MM} \cup X$ such that it avoids $e$ and has the same length as $st \diamond e$. Please see \Cref{fig:gmgm}(b) for a visualization of this case.
\begin{lemma} \label{lem:enotinP}
Let $e \in \GG_{\MM} \setminus \mathcal{P}$ and $t \in \GG_{\MM}$. Assume that the replacement path $R =st \diamond e$ uses some edges of $\GG_{\NN}$. Then there is a path in $\GG_{\MM} \cup X$ that avoids $e$ and has length $|st \diamond e|$.
\end{lemma} \begin{proof}
Let us first prove that $R$ can alternate between edges of $\GG_{\NN}$ and $\GG_{\MM}$ just once. To this end, let $u $ be the last vertex of $\GG_{\NN}$ visited by $R$. Thus, $R = R[s,u] + R[u,t]$. But the shortest path $su$ remains intact in $G \setminus e$ as $e \notin \mathcal{P}$ and the shortest $su$ path passes through $r$. This implies that $R = R[s,u] + R[u,t] = R[s,r] +\ R[r,u] +\ R[u,t]$. By construction, the first path $R[s,r] = sr$ and it completely lies in $\GG_{\MM}$. The second path $R[r,u]=ru$ and it completely lies in $\GG_{\NN}$. Let $v$ be the vertex just after $u$ in $R$. So, $v \in \GG_{\MM}$. So we have $R =sr +\ ru +\ (u,v)+R[v,t]$. By construction, $R[v,t]$ completely lies in $\GG_{\MM}$. Thus, $R$ alternates from edges of $\GG_{\NN}$ to $\GG_{\MM}$\ just once.
From the above discussion $R = R[s,r] + R[r,u] + (u,v) + R[v,t] = R[s,r] + R[r,v]+ R[v,t]$. The first and the last paths of the above equality completely lies in $\GG_{\MM}$. By construction, $R[r,v]$ does not contain any edge of $\GG_{\MM}$. Thus, $R[r,v]$ is the shortest path from $r$ to $v$ avoiding edges of $\GG_{\MM}$, that is $|R[r,v]| =|rv \diamond \GG_{\MM}| = d(r,v,\GG_{\MM})$.
Since we have added an edge from $r$ to $v$ with weight $d(r,v,\GG_{\MM})$, we will now show that there is a path in $\GG_{\MM} \cup X$ that avoids $e$ and has same weight as $R$. Consider the path $R' = R[s,r] + (r,v) + R[v,t]$. The reader can check all the three subpaths lie completely in $\GG_{\MM} \cup X$. Moreover, $(r,v) \in X$ is a weighted edge. Thus the weight of $|R'| =|R[s,r]| + wt_{(\GG_{\MM}\cup X)}(r,v)+|R[v,t]| =|R[s,r]| + d(r,v,\GG_{\MM})+|R[v,t]|=|R[s,r]| + |R[r,v]|+|R[v,t]|=|R|$.
This completes the proof.
\end{proof}
\subsection{$e \in \GG_{\NN}$ and $t \in \GG_{\MM}$} \label{sec:gngm}
In this case, $st$ path completely lies in $\GG_{\MM}$ and thus survives. Thus, $|st \diamond e| = |st| = d(s,t)$.
\subsection{$e \in \GG_{\NN}$ and $t \in \GG_{\NN}$} \label{subsec:NN}
In this case, we will recurse in $\GG_{\NN}$. However, $\GG_{\NN}$ may not contain the source $s$ if $s \neq r$. In that case, before recursing, we add a new source $s_{N}$ in $ \GG_{\NN}$. We also add some {\em weighted }edges to $\GG_{\NN}$. For each $v \in \GG_{\NN}$, we add an edge from $s_{N}$ to $v$ with a weight $d(s,v,\GG_{\NN}) = |sv \diamond \GG_{\NN}| $. Let this set of edges be called $Y$. Let us now show that we will find all the replacement paths if we recurse in $\GG_{\NN} \cup Y$.
\begin{lemma} \label{lem:4.4}
Let $e \in \GG_{\NN}$ and $t \in \GG_{\NN}$. There is a path from $s_{N}$ to $t$ in $\GG_{\NN} \cup Y$ that avoids $e$ and has weight $|st \diamond e| $. \end{lemma} \begin{proof} \label{GNPath}
Let $R =st \diamond e$. Let us first prove that once $R$ visits an edge of $\GG_{\NN}$, it cannot visit an edge of $\GG_{\MM}$ anymore. Let $u$ be the last vertex of $\GG_{\MM}$ visited by $R$ and $v$ be the vertex after $u$ in $R$. Thus, $v \in \GG_{\NN}$. Thus, $R = R[s,u] +(u,v)+ R[v,t]$. By construction, $R[v,t]$ completely lies in $\GG_{\NN}$. The shortest path $su$ remains intact in $G \setminus e$ as $e \in \GG_{\NN}$ and the path $su$ completely lies in $\GG_{\MM}$. Thus, $R[s,u] = su$ . So, the path $R$ first visits only edges of $\GG_{\MM}$(in $R[s,u])$, then goes to $\GG_{\NN}$ (by taking the edge $(u,v)$) and then remains in $\GG_{\NN}$ (in $R[v,t]$). Thus, $R$ does not visit any edge of $\GG_{\MM}$ once it visits an edge of $\GG_{\NN}$.
By the above discussion, $R = R[s,u] +\ (u,v) + R[v,t] = R[s,v] +R[v,t]$ such that $R[v,t]$ completely lies in $\GG_{\NN}$. Also, $R[s,v]$ completely lies in $\GG_{\MM}$ except the last edge which has one endpoint in $\GG_{\MM}$ and other endpoint $v \in \GG_{\NN}$.
Thus, $R[s,v] = |sv \diamond \GG_{\NN}|= d(s,v,\GG_{\NN})$ and we have $R = sv \diamond \GG_{\NN} +\ R[v,t]$.
Now consider a path $R'$ that avoids $e$ from $s_{N}$ to $t$ in $\GG_{\NN} \cup Y$. We construct this path as follows: we will first take the weighted edge $s_{N} \to v$ and then the path $R[v,t]$. The reader can check that these two subpaths completely lie in $\GG_{\NN} \cup Y$. The weight of this path is $|R'| = |sv \diamond \GG_{\NN}| + |R[v,t]| = d(s,v,\GG_{\NN}) + |R[v,t]|=|R[s,v] |\ +\ |R[v,t]|= |R| $.
\end{proof}
\subsection{One endpoint of $e$ is in $\GG_{\MM}$ and the other is in $\GG_{\NN}$ and $t \in G$} \label{sec:middle} This is an easy case as the $st$ path survives in $G \setminus e$ as $st$ does not contain $e$. Thus, $st \diamond e = st$
\section{Departing Replacement paths} \label{sec:departing}
In the previous section, we have already found out a replacement path if it is jumping or if the edge failure is not on the primary path. In this section, we will try to find the best departing path after an edge failure. To this end, we define the following: \begin{definition} (Candidate departing paths) Let $e \in \mathcal{P}$. $P$ is called the candidate departing path for $e$, if among all departing paths avoiding $e$, $P$ has the minimum length. In case there are many departing paths avoiding same edge with same length, we will break ties using Definition \ref{def:replacementpath}. \end{definition}
Note that, a candidate departing path may or may not be a replacement path. In case it is, we call it a departing replacement path. Also, $P$ may be a candidate departing path for all edges in an interval, say $yz \in \mathcal{P}$, but may be a replacement path for a sub-interval of $yz$. With this definition in hand, we will now find all candidate departing paths.
\subsection{Finding all candidate departing paths}
In the previous section, we added some weighted edges in the graph when we recurse. Thus, there might be two types of edges in the graph -- {\em weighted} and {\em unweighted}. The weighted edges represent paths in the graph $G$, and the unweighted edges are present in $G$. $G$ contains only unweighted edges. However, the graph at an internal node of $\mathcal{T}$, say graph ${\widehat{G}}$, may have weighted as well as unweighted edges.
In the ensuing discussion, let ${\widehat{G}}$ be the graph processed by our algorithm at some internal node of $\mathcal{T}$. In the graph ${\widehat{G}}$, we will assume that there is a source $s$. Let $\overline{G}$ be the parent of ${\widehat{G}}$. In our algorithm, we partition $\overline{G}$ into two disjoint graphs and then recurse on it. If ${\widehat{G}}$ is the left child of $\overline{G}$, then it includes a set $X$ of weighted edges added by us in \Cref{subsec:MM}. Similarly, if ${\widehat{G}}$ is the right child of $\overline{G}$ then it includes a set $Y$ of weighted edges added by us in \Cref{subsec:NN}. Using the separator lemma, we will find the vertex $r$ that partitions $\SPT_{{\widehat{G}}}(s)$ . Also, the primary path $\mathcal{P} = sr$.
We now give an overview of our method to find candidate departing paths in ${\widehat{G}}$. To this end, we will use the main idea in the paper of Gupta and Singh \cite{GuptaS18}. In \cite{GuptaS18}, the authors proved \Cref{lem:guptaaditi}. Though they did not mention it, the paths in the \Cref{lem:guptaaditi} look very much like the departing paths. Indeed, that lemma is more general than what the authors originally intended it to be. The authors show the above lemma for a specific vertex $t_s$, but a careful reading of the paper suggests that the above lemma is true for any vertex on the $st$ path. We now generalise this lemma. However, there is one problem. The above result holds only for an unweighted graph, whereas ${\widehat{G}}$ is weighted. Thus, we cannot prove the above lemma as it is. However, we will prove the following weaker lemma:
\begin{lemma} \label{lem:guptaadapted} Let ${\widehat{G}}$ be the graph at an internal node in the binary tree $\mathcal{T}$. Let $s$ be the source of ${\widehat{G}}$. For a destination $t \in {\widehat{G}}$, let $p$ be any vertex on $st$ path. The number of replacement paths from $s$ to $t$ that avoid edges on $sp$ path and also avoid $p$ is $O(\sqrt n)$. \end{lemma}
Some discussion is in order. In an ideal case, the number of replacement paths that avoid edges on $sp$ and also avoid $p$ should have been $O(\sqrt {n_{{\widehat{G}}}})$. This result would have been similar to \Cref{lem:guptaaditi}. Unfortunately, we cannot prove this result as ${\widehat{G}}$ is weighted. However, if we just expand the weighted edge in the graph ${\widehat{G}}$, then we will get an unweighted graph. By expanding, we mean that for each weighted edge, add the path that represents that weighted edge. However, this process may increase the number of vertices in the graph to $n$. Now, we can adapt \Cref{lem:guptaaditi}. For an unweighted graph with $n$ vertices, this lemma guarantees that the number of replacement paths avoiding $p$ will be at most $\sqrt n$. Indeed, this is our result in \Cref{lem:guptaadapted}. We prove this lemma in \Cref{adapted} as it is an extension of the proof in \cite{GuptaS18}.
\Cref{lem:guptaadapted} implies that there are only $O(\sqrt n)$ replacement paths that have some special properties which are similar to the properties of departing paths. So, our plan of action is as follows:
\begin{enumerate}
\item Show that there are only $O(\sqrt n)$ candidate departing paths to $t$ in ${\widehat{G}}$. This will be done by showing the similarity between the replacement paths in \Cref{lem:guptaadapted} and candidate departing paths.
\item Show that we can find the lengths of all the candidate departing paths in ${\widehat{G}}$ in $O(m_{{\widehat{G}}} \sqrt n)$ time. Additionally, we show that we can store the lengths in a compact data structure of size $O(n_{{\widehat{G}}}\sqrt n)$. Given any query $\textsc{Query}(s,t,e)$ to this data-structure such that $s,t \in {\widehat{G}}$ and $e$ is on the primary path of ${\widehat{G}}$, we can find the length of corresponding candidate departing path in $\widetilde{O}(1)$ time. \end{enumerate}
This completes the overview of our algorithm for finding candidate departing paths.
\subsection{Similarity between candidate departing paths and replacement paths in \Cref{lem:guptaadapted}}
Let us first prove some simple results that will help us prove this section's main idea. \begin{lemma} \label{lem:departingproperty} Let $t \in {\widehat{G}}$ and $p = \textsc{Lca}(t,r)$. All the candidate departing paths from $s$ to $t$ avoiding edges on $sp$ path also avoid $p$. For any edge $e$ on $pr$ path , $st \diamond e = st$. \end{lemma}
\begin{proof} Let $R = st \diamond e$ where $e \in sp$ and $R$ is departing. The detour of $R$ must start above $e$ on $\mathcal{P}$. Since $R$ is departing, it can not merge with the path $\mathcal{P}$ again. So, it avoids $p$ also.
The $st$ path passes through $p$ and does not use any vertex of $\mathcal{P}$ below $p$. So, removing any edge on $pr$ path does not disturb $st$ path. So, for any edge $e \in pr$ , $st \diamond e = st$.
\end{proof}
We now show that the candidate departing paths and the replacement paths in \Cref{lem:guptaadapted} have the same property.
\begin{lemma} \label{lem:numberofdeparted} For any vertex $t \in {\widehat{G}}$, there are $O(\sqrt{n})$ candidate departing paths to $t$. \end{lemma} \begin{proof}
\label{cdpnumbers}
Remember that the candidate departing path avoids edges on path $\mathcal{P}$ only. Let $p = \textsc{Lca}(r,t)$. Using \Cref{lem:departingproperty}, $st$ is the candidate departing path avoiding all edges in $pr$ path. Again, by \Cref{lem:departingproperty}, all the candidate departing paths avoiding edges in $sp$ path avoid $p$ too. But by \Cref{lem:guptaadapted}, the number of such paths is $O(\sqrt n)$. This implies that there are $O(\sqrt n)$ candidate departing paths to $t$.
\end{proof}
Given the above lemma, we need to store $O(\sqrt n)$ candidate departing paths to $t$ in ${\widehat{G}}$. Before designing an algorithm to find candidate departing paths, we first see how we plan to store these paths in a compact data structure.
\subsection{The data-structure at each node of $\mathcal{T}$} Let us first discuss a small technical detail that may be perceived as a problem but is not. Our graph ${\widehat{G}}$ is weighted. It stands to reason that even the primary path $\mathcal{P}$ in ${\widehat{G}}$ maybe weighted. Since candidate departing paths are only for the faults on the primary path, it may not be clear what happens if the edge on the primary path is weighted. To this end, we show that on any $st$ path in ${\widehat{G}}$, all edges except may be the first one, are unweighted.
\begin{lemma} \label{lem:dsatT} For $t \in {\widehat{G}}$, except may be the first edge of $(st)_{\widehat{G}}$, all other edges of $(st)_{\widehat{G}}$ are unweighted. \end{lemma}
\begin{proof}
\label{structure}
We will prove using induction on the nodes of $\mathcal{T}$. In the root of $\mathcal{T}$, we have the graph $G$. Clearly, the graph $G$ satisfies the property of the lemma. Let us assume using induction that the parent of the graph ${\widehat{G}}$ satisfies the property of the lemma. Let $\overline{G}$ be the parent of ${\widehat{G}}$. Let $\overline{s}$ be the source in $\overline{G}$. Thus, in the graph $\overline{G}$, using the separator lemma, we find a $\overline{r}$ that divides $\overline{G}$ into two parts. The path from $\overline{s}$ to $\overline{r}$, say $\overline{\mathcal{P}}$ is the primary path. By induction, only the first edge of the primary path may be weighted. There are two cases:
\begin{enumerate} \item ${\widehat{G}}$ is the left child of $\overline{G}$.
In this case, the source of ${\widehat{G}}$ is also $\overline{s}$. For each $t \in {\widehat{G}}$, it can be observed that the path $(\overline{s}t)_{\overline{G}} = (\overline{s}t)_{\widehat{G}}$. Using induction hypothesis, since $(\overline{s}t)_{\overline{G}}$ satisfies the statement of lemma, so does $(\overline{s}t)_{\widehat{G}}$. \item ${\widehat{G}}$ is the right child of $\overline{G}$.
There are two cases. When $\overline{s} = \overline{r}$, then we fall back in point(1). So, let us look at the case when $\overline{s} \neq \overline{r}$.
For a $t \in {\widehat{G}}$, by construction, $(\overline{s}t)_{\overline{G}}$ passes through $\overline{r}$. Thus, $(\overline{s}t)_{\overline{G}} = (\bar{s}\bar{r})_{\overline{G}} + (\overline{r}t)_{\overline{G}}$. In ${\widehat{G}}$, we add a new source $s$. Also, we add a weighted edge from $s$ to $\overline{r}$ in ${\widehat{G}}$. The weight of this edge is $|\bar{s}\bar{r} \diamond {\widehat{G}}|_{\overline{G}}$. Also, $(\overline{r}t)_{\overline{G}} = (\overline{r}t)_{\widehat{G}}$. This implies that there is path in ${\widehat{G}}$ from $s$ to $t$, $(s,\overline{r})+ (\overline{r}t)_{\widehat{G}}$. By induction, we claim that on this path except $(s,\overline{r})$, all the edges are unweighted. \end{enumerate}
\end{proof}
Using the above lemma, all except the first edge of the primary path are unweighted. The weighted edges represent edges for which we have already found candidate departing paths at the parent or an ancestor of ${\widehat{G}}$. Thus, we will only find candidate departing paths for unweighted edges in $\mathcal{P}$. In the ensuing discussion, whenever we mention a path avoiding an edge on the primary path, it will always refer to an unweighted edge.
We now prove some simple lemmas that will help us build data structures for candidate departing paths. \begin{lemma} \label{lem:reldeparting1}
Let $R$ and $R'$ be two different candidate departing paths from $s$ to $t$ avoiding edges $e$ and $e'$ respectively on the path $\mathcal{P}$. If $e$ lies above $ e'$ on $\mathcal{P}$, then $|R|>|R'|$. \end{lemma} \begin{proof}
The detour of the candidate departing $R$ starts before $e$, and once it departs, it does not merge with $\mathcal{P}$ again. As $e$ lies above $e'$ on $\mathcal{P}$, $R$ also avoids $e'$. If $|R| \leq |R'|$, then by \Cref{def:replacementpath}, $R$ must be the candidate departing path avoiding $e'$, leading to a contradiction. So, it must be the case that $|R|>|R'|$. \end{proof}
\begin{lemma} \label{lem:reldeparting2} Let $R$ be a candidate departing path. Let $yz$ be the maximal subpath of $\mathcal{P}$ such that $R$ is the candidate departing path for edges in $yz$. Then $\textsc{Dp}(R) = y$. \end{lemma}
\begin{proof}
\label{app:reldeparting2}
Since $R$ avoids the edges of $yz$, its detour starts at or above $y$ on the path $\mathcal{P}$. Let us assume that $\textsc{Dp}(R)$ lies above $y$ on path $\mathcal{P}$, say at $x$. This implies that $R$ also avoids all edges on $xy$ path. But $R$ is not the candidate departing path for edges in the path $xy$. Let $R'$ be the candidate departing path for an edge, say $e' \in xy$. Now, for an edge $e \in yz$, $R$ is the replacement path and $e'$ lies above $e$ on $\mathcal{P}$. Using \Cref{lem:reldeparting1}, $|R'| > |R|$. But in that case, $R$ should be the replacement path avoiding $e'$ too. This leads to a contradiction. Thus, our assumption that $\textsc{Dp}(R)$ lies above $y$ is false and $\textsc{Dp}(R) = y$. \end{proof}
\label{app:reldeparting3}
The above lemma states that if $R$ avoids edges in $yz$, then the detour of $R$ necessarily starts from $y$.
We will now prove the contrapositive of the \Cref{lem:reldeparting1}.
\begin{lemma}
\label{lem:reldeparting3}
Let $R$ and $R'$ be two different candidate departing paths from $s$ to $t$. If $|R|>|R'|$, then $\textsc{Dp}(R)$ lies above $\textsc{Dp}(R')$ on $\mathcal{P}$. \end{lemma}
\begin{proof}
Since $R'$ does not merge with $\mathcal{P}$, $R'$ avoids all edges below $\textsc{Dp}(R')$ in $\mathcal{P}$. Assume for contradiction, that $\textsc{Dp}(R')$ lies above $\textsc{Dp}(R)$. This will imply that $R'$ also avoids all the edges avoided by $R$. But $|R| > |R'|$. Then, $R'$ should be the candidate departing path for all the edges avoided by $R$ -- a contradiction. Thus, $\textsc{Dp}(R)$ lies above $\textsc{Dp}(R')$ on $\mathcal{P}$. \end{proof}
We will now use the above lemmas and our deduction to build a compact data-structure for all candidate departing paths. To this end, we will store an array $\textsc{Dep}(t)$ for each $t \in {\widehat{G}}$. $\textsc{Dep}(t)$ will store candidate departing paths from $s$ to $t$ in increasing order of their lengths. By \Cref{lem:numberofdeparted}, there are $O(\sqrt n)$ such paths. let us denote them by $R_1, R_2, \dots, R_k$ where $k = O(\sqrt n)$. For any two consecutive candidate departing paths $R_i$ and $R_{i+1}$, using \Cref{lem:reldeparting3} and \Cref{lem:reldeparting2}, we claim that $R_{i+1}$ is the candidate departing path avoiding edges in $[\textsc{Dp}(R_{i+1}),\textsc{Dp}(R_{i})]$ on the primary path $\mathcal{P}$. Since the size of $\textsc{Dep}(t) = O(\sqrt n)$, the total size of $\textsc{Dep}$ data-structure is bounded by $O(n_{\widehat{G}} \sqrt n)$.
\begin{lemma} $\sum_{t \in {\widehat{G}}} \text{size of $\textsc{Dep}(t)$} = O(n_{\widehat{G}}\sqrt n)$ \end{lemma} \subsection{Finding and storing all candidate departing paths efficiently} Let us first describe the setting that will be used throughout this section. At an internal node ${\widehat{G}}$ of $\mathcal{T}$, we are planning to find all candidate departing paths from the source $s$. To this end, we will find a vertex $r$ that will divide $\textsc{Spt}_{{\widehat{G}}}(s)$ roughly equally. Also, $\mathcal{P} =sr$.
To find all candidate departing path, we will build an auxiliary graph $G$ which we will build incrementally. The source in this graph is $(s)$. All other vertices in $G$ are tuples of the form $(v,|R|)$, where $v \in {\widehat{G}} \setminus \mathcal{P}$ and $R$ is a candidate departing path to $v$. During initialization, we will add $(v,|sv|_{{\widehat{G}}})$ in $G$ for each $v \in {\widehat{G}} \setminus \mathcal{P}$. Also, there will be an edge from $(s)$ to $(v,|sv|_{{\widehat{G}}})$ with weight $|sv|_{{\widehat{G}}}$. We will show the following property at the end of our analysis.
\begin{prop} \label{prop1}
Let $R$ be the candidate departing path to $v$ avoiding edge on the subpath $yz$ of $\mathcal{P}$. Then, there is a vertex $(v,|R|)$ in
$G$. Moreover the shortest path from $(s)$ to $(v,|R|)$ in the graph $G$ is of length
$|R|$, that is $|R|= |(s)(v,|R|)|_{G}$ \end{prop}
In \Cref{lem:propertytrue}, we will show that \Cref{prop1} is true for all the nodes added during initialization. Also, we will create $\textsc{Dep}(v)$ for each $v$ during intialization.
We will store candidate departing paths in $\textsc{Dep}(v)$ in increasing order of lengths. Given a departing path $R$, we will assume that we will store the following information about $R$ in $\textsc{Dep}()$.
\begin{enumerate} \item The endpoints of $R$. \item The weight of path $R$. \item The last edge of $R$ and its weight. \item $\textsc{Dp}(R)$. \end{enumerate}
After initialization, we will run a variant of Dijkstra's algorithm in $G$. To this end, we will construct a min-heap $H$ in which we will store all the departing paths that we have discovered at any point in the algorithm. We use the first two points of \Cref{def:detour} to select the minimum element from $H$,i.e., given two candidate departing paths $R$ and $R'$, $R$ is {\em smaller} than $R'$ if $|R| < |R'|$ or $|R| = |R'|$ and $\textsc{Dep}(R)$ is closer to $s$ than $\textsc{Dep}(R')$. If $\textsc{Dep}(R) = \textsc{Dep}(R')$, then we can break ties arbitrarily.
We now explain the adaptation of Dijkstra's algorithm. After initialization,for each $(v, |sv|)$, and for each neighbor neighbor $w$ of $v$, we add the departing path $sv + (v,w)$ in $H$ if $w \notin \mathcal{P}$.
Then, we go over all the elements of the heap till it is empty. Let us assume that $R$ is the minimum element of the heap and it ends at $v$ and $(u,v) \in {\widehat{G}}$ is the last edge of $R$. This implies that $R$ was added in $H$ while processing a candidate departing path for $u$. Let this path be $R_u$. Thus, there is a node $(u,|R_u|)$ in $G$. We now need to decide whether $R$ is a candidate departing path for $v$.
To this end, we look at the last candidate departing path added by us in $\textsc{Dep}(v)$, let it be $Q$. We then check if the $d_{\widehat{G}}(s,\textsc{Dp}(R))$ is less than $d_{\widehat{G}}(s,\textsc{Dp}(Q))$. If yes, then we have found a new candidate departing path to $v$ that avoids all edges in $[\textsc{Dp}(R),\textsc{Dp}(Q)]$ of $\mathcal{P}$. Thus, we will add the vertex $(v,|R|)$ in the graph with an edge of weight $wt_{\widehat{G}}(u,v)$ from $(u,|R_u|)$. Also, for each neighbor $w$ of $v$, we will add the departing path $R+(v,w)$ to the heap if $w \notin \mathcal{P}$. The pseudocode of the algorithm for finding all candidate departing path is given in algorithm 1.
\begin{algorithm} \label{algo:cdpalgo} \caption{Algorithm to construct $\textsc{Dep}()$}
create a vertex $(s)$\;
\For{$v \in {\widehat{G}} \setminus \mathcal{P}$} {
Let $R_v = |sv|_{\widehat{G}}$\; create a vertex $(v,|R_v|)$\;
add edge between ($s)$ and $(v,|R_v|)$ with weight $|sv|_{\widehat{G}}$\;
append $R_v$ in $\textsc{Dep}(v)$\;
\For{$(v,w) \in {\widehat{G}}$ and $w \notin \mathcal{P}$}
{
add the path $R_v+(v,w)$ in min-heap $H$ \;
} }
\While{min-heap is non-empty} {
remove $R$ from the top of min-heap\;
Let us assume that $R$ ends at $v$ and the last edge of $R$ is $(u,v)$\; Also assume that $R_u = R \setminus (u,v)$ is a candidate departing path to $u$\;
Let $Q$ be the last candidate departing path added by us in $\textsc{Dep}(v)$\;
\If{$d_{{\widehat{G}}}(s,\textsc{Dp}(R))<d_{\widehat{G}}(s,\textsc{Dp}(Q))$}
{
create a vertex $(v,|R|)$\;
add edge between $(u,|R_u|)$ and $(v,|R|)$ with weight $wt_{\widehat{G}}(u,v)$ \;
append the path $R$ in $\textsc{Dep}(v)$\;
\For{$(v,w) \in {\widehat{G}}$ and $w \notin \mathcal{P}$}
{
add the departing path $R+(v,w)$ in min heap $H$\;
}
}
}
\end{algorithm}
\subsection{Correctness and running time of the algorithm storing candidate departing paths} \label{subsection:correct}
We claim that, the time taken to construct the $\textsc{Dep}()$ data-structure at a node of the binary tree with graph ${\widehat{G}}$ is $O(m_{\widehat{G}}\sqrt n)$. Moreover, the size of the data-structure is $O(n_{\widehat{G}} \sqrt n)$. In this section, we prove that our algorithm stores correct lengths of all candidate departing paths.\begin{lemma}
\label{lem:propertytrue}
Let $R$ be a candidate departing path to $v$ where $v \in {\widehat{G}} \setminus \mathcal{P}$. Let $yz$ be the maximal subpath of $\mathcal{P}$ such that $R$ is the candidate departing path for edges in $yz$. Then $(v,|R|) \in G$ and satisfies \Cref{prop1}. \end{lemma}
\begin{proof}
We will prove the above lemma using induction on the weighted distance of a vertex from $(s)$ in $G$. During initialization, for each $v \in G \setminus \mathcal{P}$, we add a vertex $(v,|sv|_{\widehat{G}}) \in G$. Also, the weight of the edge from $(s)$ to $(v,|sv|_{\widehat{G}})$ is $|sv|_{\widehat{G}}$. We claim that the statement of the lemma is true for the smallest candidate departing path to $v$. Indeed, using \Cref{lem:departingproperty}, $(sv)_{\widehat{G}}$ is the a replacement path for the edges in subpath $pr$ on $\mathcal{P}$ where $p = \textsc{Lca}_{{\widehat{G}}}(v,r)$. Also, $(sv)_{\widehat{G}}$ is the smallest replacement path because it is the shortest path from $s$ to $v$ in ${\widehat{G}}$. Thus, the base case is true for all $v \in {\widehat{G}} \setminus \mathcal{P}$.
Let us now assume that the statement of the lemma is true for all candidate departing paths to $v$ with length $< |R|$. Let the second last vertex of $R$ be $u$. Since $R$ is a candidate departing path, even $R \setminus(u,v)$ is a candidate departing path. Since $R[s,u]$ has length less than $R$, by induction hypothesis, there is a vertex $(u,|R \setminus(u,v)|)$ in $G$. Also there is at least one replacement path in $\textsc{Dep}(v)$ of weight less than $|R|$ -- as we have added $(sv)_{\widehat{G}}$ in $\textsc{Dep}(v)$. Let $Q$ be a candidate departing path of largest weight less than the weight of $R$. Let us also assume that $Q$ avoids edge on subpath $zz' \in \mathcal{P}$. Thus, using \Cref{lem:reldeparting2}, $\textsc{Dp}(Q) = z$. Since $|Q| < |R|$, using \Cref{lem:reldeparting3}, $\textsc{Dp}(R)$ lies above $\textsc{Dp}(Q)$ on $\mathcal{P}$. Using the induction hypothesis, there is a vertex $(v,|Q|)$ in $G$.
We will now show that our algorithm will add $(v,|R|)$ in $G$. There are four cases:
\begin{enumerate}
\item Our algorithm does not add any vertex for $v$ after $(v,|Q|)$
But our algorithm does process the vertex $(u,|R \setminus(u,v)|)$. Thus, it will check each neighbour of $u$. When it checks the neighbor $v$, it will add the departing path $R$ in the heap $H$. Thus, we will add the vertex $(v,|R|)$ in $G$ while processing $R$, leading to a contradiction.
\item Our algorithm adds a vertex $(v,|R'|)$ where $\textsc{Dp}(R)$ lies above $\textsc{Dp}(R')$ on $\mathcal{P}$
We claim that the weight of $R'$ cannot be less than the weight of $R$ as then $R$ is not the candidate departing path for all the edges in $\textsc{Dp}(R')z$ subpath, contradicting the statement of the lemma. So let us assume that $|R| = |R'|$. But then $\textsc{Dp}(R)$ lies above $\textsc{Dp}(R')$ on $\mathcal{P}$. Thus, the min-heap will give preference to the replacement path $R$ first, and our algorithm will make the vertex $(v,|R|)$. Again a contradiction.
\item Our algorithm adds a vertex $(v,|R'|)$ where $\textsc{Dp}(R)$ lies below $\textsc{Dp}(R')$ on $\mathcal{P}$
Again, we claim that the weight of $R'$ cannot be less than the weight of $R$ as then it $R$ is not the replacement path for all edges in $yz$ subpath, contradicting the statement of the lemma. So let us assume that $|R| = |R'|$. But $\textsc{Dp}(R') $ is closer to $s$ than $\textsc{Dp}(R)$. Thus, $R'$ should be the candidate departing path avoiding edges of $yz$. This contradicts our assumption that $R$ is the candidate departing path for all edges in $yz$.
\item Our algorithm adds a vertex $(v,|R'|)$ where $\textsc{Dp}(R) = \textsc{Dp}(R')$
$|R'|$ cannot be less than $|R|$ as otherwise our algorithm will give preference to path $R$. But, if $|R| = |R'|$, then there is a vertex $(v, |R'|)= (v,|R|)$ in $G$.
\end{enumerate}
So, we add the node $(v,|R|)$ in the graph $G$. At that moment, we also adds an edge from $(u,R\setminus(u,v))$ to $(v,|R|)$ in $G$ with weight $wt_{{\widehat{G}}}(u,v)$. Using induction, $|(s)(u,R\setminus(u,v))|_G = |R\setminus(u,v)|$. Thus, $|(s)(v,|R|)|_G = |R\setminus(u,v)|\ +\ wt_{\widehat{G}}(u,v) = |R|$. This completes our proof.
\end{proof}
Let's determine the running time of our algorithm. Using \Cref{lem:numberofdeparted}, for each $v \in {\widehat{G}}$, we make $O(\sqrt n)$ entries in $\textsc{Dep}(v)$. In other words, we make $O(\sqrt n )$ nodes of type $(v ,\dot)$ in $G$. Whenever we add a node $(v,|R|)$ in $G$, we see all the edges of $v$. This implies that the total time taken to process all the vertices of $v$ in $G$ is $O( \sqrt n \deg_{\widehat{G}}(v))$. Summing it over all the vertices gives us the bound of $O(m_{\widehat{G}}\sqrt n)$. Using a similar calculation, the total size of our data-structure for ${\widehat{G}}$ is $O(n_{\widehat{G}} \sqrt n)$. Thus, we claim the following lemma:
\begin{lemma} \label{lem:depconstruction} The time taken to construct the $\textsc{Dep}()$ data-structure at a node of the binary tree with graph ${\widehat{G}}$ is $O(m_{\widehat{G}}\sqrt n)$. Moreover, the size of the data-structure is $O(n_{\widehat{G}} \sqrt n)$. \end{lemma}
\subsection{Querying for a candidate departing path} \label{query} In this section, we describe how to find a candidate departing path using our data-structure $\textsc{Dep}()$. Let $t \in {\widehat{G}} \setminus \mathcal{P}$ and $e \in \mathcal{P}$ be an edge on $st$ path. Let $st \diamond e$ be a candidate departing path, then we can find it using algorithm given in \Cref{app:queryCDP}.
In this algorithm, we perform a binary search in $\textsc{Dep}(t)$ to find two consecutive paths $R$ and $Q$ such that $e$ lies in the interval [$\textsc{Dep}(R), \textsc{Dep}(Q)]$ of $\mathcal{P}$. Using \Cref{lem:reldeparting3} and \Cref{lem:reldeparting2}, $R$ is the candidate departing path avoiding $e$. The pseudocode of the query algorithm to find the Candidate Departing path here. \label{app:queryCDP} \begin{algorithm} \caption{$\textsc{Query-DEP}(s,t,e)$}
$R \xleftarrow{}$ binary search in the array of $\textsc{Dep}(t)$ to find a two consecutive path $Q$ and $R$ such that $e$ lies in interval $[\textsc{Dp}(R),\textsc{Dp}(Q)]$ of $\mathcal{P}$ \; return{ $|R|$} \label{algo:queryd_1} \end{algorithm}
\begin{algorithm}[H] \caption{$\textsc{Query}(s,t,e,\mathcal{T})$}
$mindist \xleftarrow{} \infty $\;
\tcc{ \Cref{sec:gmgn} }
\If{ $e \in \GG_{\MM},t \in \GG_{\NN} $} {
\If{ $e \in \mathcal{P}$}
{
\tcc{if $st \diamond e$ happens to be departing}
$mindist \leftarrow \textsc{Query-DEP}(s,t,e)$ using $\textsc{Dep}()$ data-structure at $\mathcal{T}$\;
\tcc{if $st \diamond e$ happens to be jumping}
$mindist \xleftarrow{} \min\{mindist, |sr \diamond e|+|rt|\}$
}
\Else{
$mindist \leftarrow |st|$
} } \tcc{ \Cref{subsec:MM}}
\If{ $e \in \GG_{\MM},t \in \GG_{\MM} $} { \tcc{ \Cref{sec:gmgmp}} \If{ $e \in \mathcal{P} $} {
\tcc{if $st \diamond e$ happens to be departing}
$mindist \leftarrow \textsc{Query-DEP}(s,t,e)$ using $\textsc{Dep}()$ data-structure at $\mathcal{T}$\;
\tcc{if $st \diamond e$ happens to be jumping}
$mindist \xleftarrow{} \min (mindist, |sr \diamond e|+|rt| ) $
}
\tcc{ \Cref{sec:gmgnnp}}
\If{ $e \in \GG_{\MM} \setminus \mathcal{P} $}
{
$mindist \xleftarrow{} \textsc{Query}(s,t,e,\text{left child of $\mathcal{T}$})$
}
}
\tcc{ \Cref{sec:gngm}} \If{ $e \in \GG_{\NN}, t \in \GG_{\MM} $} {
$mindist \leftarrow |st|$\; }
\tcc{ \Cref{subsec:NN}} \If{$ e \in \GG_{\NN},t \in \GG_{\NN}, $} {
$mindist \leftarrow \textsc{Query}(s,t,e,\text{right child of $\mathcal{T}$}$)\;
} \tcc{ \Cref{sec:middle}} \If{one endpoint of $e$ is in $\GG_{\MM}$ and other in $\GG_{\NN}$} {
$mindist \leftarrow |st|$\; }
return $mindist$\; \label{alg:query} \end{algorithm}
\section{Construction time, size and query time of the SDO(1) }
In this section, we show that the construction time of our algorithm is $\widetilde{O}(m \sqrt n)$. We also bound the size of the data-structure of our algorithm by $\widetilde{O}(n\sqrt n)$. We also design a query algorithm with a query time $\widetilde{O}(1)$. This proves the main result of the paper.
At the root of $\mathcal{T}$, except for the recursions, we claim that constructing all other data-structures take $O(m \sqrt{n})$ time. This is beacuse, the construction time is dominated by the time to construct $\textsc{Dep}()$, which using \Cref{lem:depconstruction}, is $O(m\sqrt n)$. At the second level of the tree $\mathcal{T}$, we have two nodes. In the left child, we have the graph $\GG_{\MM} \cup X$. This graph has $m_{\GG_{\MM}} + n_{\GG_{\MM}}$ edges and $n_{\GG_{\MM}}$ vertices. Again applying \Cref{lem:depconstruction}, the time taken to construct all the data-structures in the left child of root is $(m_{\GG_{\MM}}+n_{\GG_{\MM}})\sqrt n$. In the right child of the root, we have the graph $\GG_{\NN} \cup Y$. This graph has $m_{\GG_{\NN}} + n_{\GG_{\NN}}$ edges and $n_{\GG_{\NN}}+1$ vertices. The $+1$ is for the new root in $\GG_{\NN}$. Again applying \Cref{lem:depconstruction}, the time taken to construct all the data-structures in the right child of root is $(m_{\GG_{\NN}}+n_{\GG_{\NN}})\sqrt n$. Thus, the total time taken at the second level of $\mathcal{T}$ is = $(m_{\GG_{\MM}}+n_{\GG_{\MM}}) \sqrt{n} + (m_{\GG_{\NN}}+n_{\GG_{\NN}}) \sqrt{n}$. Since $m_{\GG_{\MM}}+m_{\GG_{\NN}} \le m$ and $n_{\GG_{\MM}} + n_{\GG_{\NN}} = n+1$, the total time taken is $\le (m+n+1) \sqrt n$. Note that $n_{\GG_{\MM}} + n_{\GG_{\NN}} = n+1$ because $r$ is shared both by $\GG_{\MM}$\ and $\GG_{\NN}$. Since, the number of nodes in $\mathcal{T}$ is $O(n)$, we claim that the number of vertices shared by sibling graphs at any level of $\mathcal{T}$ is $O(n)$. Similar to the second level, we claim that the time taken at level $\ell$ is $\widetilde{O}((m +n +$ \#nodes shared at level $\ell$$)\sqrt n) = \widetilde{O}((m+n)\sqrt n)$. We can assume that our graph $G$ is connected as we need not even process a component that is not reachable from our source $s$. Thus, the previous running time bound is equal to $\widetilde{O}(m \sqrt n)$. Since the height of the tree is $\widetilde{O}(1)$, the total time taken to construct our data-structure is $\widetilde{O}(m\sqrt n)$.
Using the same argument, we can bound the size of the data-structure of our algorithm by $\widetilde{O}(n\sqrt n)$.
\subsection{The Query Algorithm}
In this section, we will design our query algorithm that will take $s,t,e$ as its parameter. Additionally, it also takes the root of the tree $\mathcal{T}$ as a parameter which contains data structures of the main graph $G$. The algorithm then basically goes over all the possible cases described in \Cref{sec:detail} (Please see Algorithm 3). Also, the algorithm compares that output with the best candidate departing path given by Algorithm 2 and returns the minimum among them.
The reader can see that the time taken by the algorithm (excluding recursion) is $\widetilde{O}(1)$. Since, at each step in this algorithm, we either go to the left child of a node in the tree $\mathcal{T}$ or to the right child, the number of recursive steps in this algorithm is $\widetilde{O}(1)$. This implies that the running time of the algorithm is $\widetilde{O}(1)$. Thus, we have proven the main theorem of the paper.
\appendix
\section{Proof of Lemma \ref{lem:guptaadapted}} \label{adapted}
In lemma \ref{lem:guptaadapted}, we have a graph ${\widehat{G}}$ at some internal node of $\mathcal{T}$. The source of ${\widehat{G}}$ is $s$, and the primary path is $\mathcal{P} =sr$. A weighted edge in ${\widehat{G}}$ represents a path in the parent of ${\widehat{G}}$, say $\overline{G}$. First of all, we recursively expand all weighted edges in ${\widehat{G}}$. By expanding, we mean that we will replace a weighted edge $e \in {\widehat{G}}$ with its corresponding path in $\overline{G}$. Similarly, a weighted edge $e'$ in $\overline{G}$ represents a path in its parent. So, we will again recursively replace $e'$ with a path. The reader can see that this process will lead us to a graph in which each edge $e$ of ${\widehat{G}}$ is replaced by a path that contains only unweighted edges. However, this transformed graph may have $n$ vertices. Henceforth, we will assume that we are working with this transformed unweighted version of ${\widehat{G}}$.
Even in the transformed version of ${\widehat{G}}$, the shortest $st$ path remains the same -- since we have just expanded weighted edges. Since we now have an unweighted graph, we will follow the proof of \cite{GuptaS18}. We first define some terms: \begin{itemize}
\item Let $\mathcal{R}$ be the set of all replacement paths that avoid edges on $sp$ path and avoid $p$ too. At the end of the proof, we will show that $|\mathcal{R}| = O(\sqrt n)$.
\item If $R \in \mathcal{R}$ is a replacement path from $s$ to $t$ avoiding some edge, then we define the sets $(<R)$ and $(>R)$ as the set of all replacement paths from $s$ to $t$ avoiding edges on the path $st$ which has length less than the length of $R$ and greater than the length of $R$ respectively.
\item \textsc{Detour} $(R)$ is the sub-path of $R$ from the vertex it leaves $st$ to the vertex where it merges back to $st$ path. Thus, $\textsc{Detour}(R) = R \setminus st$.
\item \textsc{Unique} $(R)$ is the prefix of $R$ which does not intersect with any detours in $\bigcup_{R' \in (>R)}$ \textsc{Detour} $(R')$. \end{itemize}
We now state the following lemma from \cite{GuptaS18} that will be useful later:
\begin{lemma}(Lemma 9 of \cite{GuptaS18}) \label{lem:lessthanR}
If $R\in \mathcal{R}$ is a replacement path from $s$ to $t$ avoiding $e$ such that $|R|=|st|+l$ where $l \geq 0$, then the size of the set $(<R)$ is $\leq l$.
\end{lemma}
Now, let us go through the $st$ replacement paths in $\mathcal{R}$ in decreasing order of lengths. We can observe that the number of replacement paths with $\textsc{Unique} \geq \sqrt{n}$ will be $\le \sqrt{n}$ as the vertices on the detours must be mutually disjoint in \textsc{Unique}. Now, let $R$ be the first such replacement path such that \textsc{Unique}$(R) < \sqrt{n}$. We show that number of replacement paths in the set $(<R)$ will be $O(\sqrt{n})$. Now, we have the following cases.
\begin{itemize}
\item \textbf{ \textsc{Detour}$(R)$ does not intersect with detour of any path in $(>R)$}
Let the detour of $R$ starts at $a$ and ends at $b$. Thus $\textsc{Unique}(R) =ab$ and $|ab| \le \sqrt n$. Using Lemma \ref{lem:reldeparting3}, the detour of every replacement path avoiding edges between $a$ and $p$ from the set $(<R)$ must go through $a$ -- since the detour of these paths start below $a$. Now, $ |R \setminus sa |=|ab|+|bt| \leq |ab|+|at| < |at|+\sqrt{n} $. So using lemma \ref{lem:lessthanR}, the number of paths in the set $ \{R' \setminus sa | R' \in (<R) \}$ is $\le \sqrt{n}$. So, number of replacement paths with length less than $R$ will be $\leq \sqrt{n}$.
\item \textbf{ \textsc{Detour}$(R)$ intersects with detour of a path in $(>R)$}
\input{fig_oracle}
Let, the first path in $(>R)$ which intersects $R$ be $R'$. Let the detour of $R$ starts at $a$ and ends at $b$. Let the detour of $R'$ starts at $a'$ and ends at $b'$. Let $R$ and $R'$ intersect at point $c$. This implies that $\textsc{Unique}(R)=ac$ and $|ac| \le \sqrt n$. (See Figure \ref{fig:singlesecondcase})
\begin{align*}
|sa'|+|a'c|+|cb'|+|b't| &\leq |sa'|+|a'c|+|ca|+|at| \\
|cb'|+|b't| &\leq |ca| +|at| \\
|ac|+|cb'|+|b't| &\leq 2|ca|+|at| \end{align*}
Now, the left hand side of the above inequality represents the path $R \setminus sa$. So $|R \setminus sa| = |ac|+|cb'|+|b't| $. Since $|ca| \leq \sqrt{n}$, $|R \setminus sa| \leq 2 \sqrt{n}+ |at|$. Again, using Lemma \ref{lem:lessthanR}, the cardinality of the set of replacement paths $ \{R' \setminus sa | R' \in (<R) \}$ will be $\le 2 \sqrt{n}$.
\end{itemize} Thus, we have shown that the number of replacement paths in $\mathcal{R}$ with $\textsc{Unique} \ge \sqrt n$ is $O(\sqrt n)$. Moreover, once we find a path in $\mathcal{R}$ with $\textsc{Unique} < \sqrt n$, then there are $O(\sqrt n)$ path in $R$ left to be processed. Thus, the number of replacement paths in $\mathcal{R}$ is $O(\sqrt n)$.
This completes the proof of the lemma.
\end{document} |
\begin{document}
\title{Unbounded discrepancy in Frobenius numbers}
\author{Jeffrey Shallit\\ School of Computer Science \\ University of Waterloo\\ Waterloo, ON N2L 3G1 \\ Canada\\ {\tt shallit@cs.uwaterloo.ca }\\ \and James Stankewicz\footnote{Partially supported by NSF VIGRE grant DMS-0738586}\\ Department of Mathematics \\ University of Georgia \\ Athens, GA 30602 \\ USA \\ {\tt stankewicz@gmail.com }}
\maketitle
\begin{abstract}
Let $g_j$ denote the largest integer that is represented exactly $j$ times as a non-negative integer linear combination of $\lbrace x_1, \ldots, x_n\rbrace$. We show that for any $k > 0$, and $n = 5$, the quantity $g_0 - g_k$ is unbounded. Furthermore, we provide examples with $g_0 > g_k$ for $n \geq 6$ and $g_0 > g_1$ for $n \geq 4$. \end{abstract}
\section{Introduction}
Let $X = \lbrace x_1, x_2, \ldots, x_n \rbrace$ be a set of distinct positive integers such that $\gcd(x_1, x_2, \ldots, x_n) = 1$. The {\sl Frobenius number} $g(x_1, x_2, \ldots, x_n)$ is defined to be the largest integer that cannot be expressed as a non-negative integer linear combination of the elements of $X$. For example, $g(6, 9, 20) = 43$.
The Frobenius number --- the name comes from the fact that Frobenius mentioned it in his lectures, although he apparently never wrote about it --- is the subject of a huge literature, which is admirably summarized in the book of Ram\'{\i}rez Alfons\'{\i}n \cite{ramirez}.
Recently, Brown et al.\ \cite{brown} considered a generalization of the Frobenius number, defined as follows: $g_j(x_1, x_2, \ldots, x_n)$ is largest integer having exactly $j$ representations as a non-negative integer linear combination of $x_1, x_2, \ldots, x_n$. (If no such integer exists, Brown et al.\ defined $g_j$ to be $0$, but for our purposes, it seems more reasonable to leave it undefined.) Thus $g_0$ is just $g$, the ordinary Frobenius number. They observed that, for a fixed $n$-tuple $(x_1, x_2, \ldots, x_n)$, the function $g_j(x_1, x_2, \ldots, x_n)$ need not be increasing (considered as a function of $j$). For example, they gave the example $g_{35} (4,7,19) = 181$ while $g_{36}(4,7,19) = 180$. They asked if there are examples for which $g_1 < g_0$. Although they did not say so, it makes sense to impose the condition that
no $x_i$ can be written as a non-negative integer linear combination of the others, (*)
\noindent for otherwise we have trivial examples such as $g_0(4, 5, 8, 10) = 11$ and $g_1 (4,5,8,10) = 9$. We call a tuple satisfying (*) a {\sl reasonable} tuple.
In this note we show that the answer to the question of Brown et al.\ is ``yes'', even for reasonable tuples. For example, it is easy to verify that $g_0 (8,9,11,14,15) = 21$, while $g_1 (8,9,11,14,15) = 20$. But we prove much more: we show that $$g_0 (2n-2, 2n-1, 2n, 3n-3, 3n) = n^2 - O(n),$$ while for any fixed $k \geq 1$ we have $g_k (2n-2, 2n-1, 2n, 3n-3, 3n) = O(n)$. It follows that for this parameterized $5$-tuple and all $k \geq 1$, we have $g_0 - g_k \rightarrow \infty$ as $n \rightarrow \infty$.
\section{The main result}
We define $X_n = \lbrace 2n-2, 2n-1, 2n, 3n-3, 3n \rbrace$. It is easy to see that this is a reasonable $5$-tuple for $n \geq 5$. If we can write $t$ as a non-negative linear combination of the elements of $X_n$, we say $t$ has a representation or is representable.
We define $R(j) $ to be the number of distinct representations of $j$ as a non-negative integer linear combination of the elements of $X_n$.
\begin{theorem} \begin{enumerate}[$($a$)$] \item $g_k (X_n) = (6k+3)n - 1$ for $n > 6k+3$, $k \geq 1$. \item $g_0(X_n) = n^2 - 3n +1$ for $n \geq 6$; \end{enumerate} \label{ref1} \end{theorem}
Before we prove Theorem~\ref{ref1}, we need some lemmas.
\begin{lemma} \begin{enumerate}[$($a$)$] \item $R( (6k+3)n - 1) \geq k$ for $n \geq 4$ and $k \geq 1$. \item $R( (6k+3)n - 1) = k$ for $n > 6k+3$ and $k \geq 1$. \end{enumerate} \label{lem1} \end{lemma}
\begin{proof} First, we note that \begin{equation} (6k+3)n - 1 = 1 \cdot (2n-1) + (3t-1) \cdot (2n) + (2(k-t)+1)\cdot (3n) \label{eq1} \end{equation} for any integer $t$ with $1 \leq t \leq k$. This provides at least $k$ distinct representations for $(6k+3)n - 1$ and proves (a). We call these $k$ representations {\sl special}.
To prove (b), we need to see that the $k$ special representations given by (\ref{eq1}) are, in fact, all representations that can occur.
Suppose that $(a,b,c,d,e)$ is a $5$-tuple of non-negative integers such that \begin{equation} a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n) = (6k+3) n - 1 . \label{eq2} \end{equation} Reducing this equation modulo $n$, we get $ -2a -b -3d \equiv -1 $ (mod $n$). Hence there exists an integer $m$ such that $2a + b + 3d = mn + 1$. Clearly $m$ is non-negative. There are two cases to consider: $m = 0$ and $m \geq 1$.
If $m = 0$, then $2a + b + 3d = 1$, which, by the non-negativity of the coefficients $a, b, d$ implies that $a=d=0$ and $b = 1$. Thus by (\ref{eq2}) we get $ 2n-1 + 2cn + 3en = (6k+3) n - 1$, or \begin{equation} 2c + 3e = 6k+1. \label{eq3} \end{equation} Taking both sides modulo $2$, we see that $e \equiv 1$ (mod $2$), while taking both sides modulo $3$, we see that $c \equiv 2$ (mod $3$). Thus we can write $e = 2r+1$, $c = 3s-1$, and substitute in (\ref{eq3}) to get $k = r + s $. Since $s \geq 1$, it follows that $0 \leq r \leq k-1$, and this gives our set of $k$ special representations in (\ref{eq1}).
If $m \geq 1$, then $n +1 \leq mn+1 = 2a+b + 3d$, so $n \leq 2a+b+3d-1$. However, we know that $(6k+3)n - 1 \geq a(2n-2) + b(2n-1) + d(3n-3) > (n-1) (2a+b+3d)$. Hence $(6k+3)n > (n-1)(2a+b+3d) + 1 > (n-1)(2a+b+3d-1) \geq (n-1)n$. Thus $6k+3 > n-1$. It follows that if $n > 6k+3$, then this case cannot occur, so all the representations of $(6k+3)n - 1$ are accounted for by the $k$ special representations given in (\ref{eq1}). \end{proof}
We are now ready to prove Theorem~\ref{ref1} (a).
\begin{proof} We already know from Lemma~\ref{lem1} that for $n > 6k+3$, the number $N := (6k+3)n - 1$ has exactly $k$ representations. It now suffices to show that if $t$ has exactly $k$ representations, for $k \geq 1$, then $t \leq N$.
We do this by assuming $t$ has at least one representation, say $t = a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n)$, for some $5$-tuple of non-negative integers $(a,b,c,d,e)$. Assuming these integers are large enough (it suffices to assume $a,b,c,d,e \geq 3$), we may take advantage of the internal symmetries of $X_n$ to obtain additional representations with the following swaps.
\begin{enumerate}[$($a$)$]
\item $3(2n) = 2(3n)$; hence $$a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n) $$ $$ = a(2n-2) + b(2n-1) + (c+3)(2n) + d(3n-3) + (e-2)(3n).$$
\item $3(2n-2) = 2(3n-3)$; hence $$ a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n)$$ $$= (a+3)(2n-2) + b(2n-1) + c(2n) + (d-2)(3n-3) + e(3n).$$
\item $2n-2 + 2n = 2(2n-1)$; hence $$ a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n)$$ $$= (a+1)(2n-2) + (b-2)(2n-1) + (c+1)(2n) + d(3n-3) + e(3n).$$
\item $2n-2 + 2n-1 + 2n = 3n-3 + 3n$; hence $$ a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n)$$ $$= (a+1)(2n-2) + (b+1)(2n-1) + (c+1)(2n) + (d-1)(3n-3) + (e-1)(3n).$$
\end{enumerate}
We now do two things for each possible swap: first, we show that the requirement
that $t$ have exactly $k$ representations imposes upper bounds on the size of the coefficients. Second, we swap until we have a representation which can be conveniently bounded in terms of $k$.
\begin{enumerate}[$($a$)$]
\item If $\lfloor {e \over 2} \rfloor + \lfloor {c \over 3} \rfloor \geq k$, we can find at least $k+1$ representations of $t$. Thus we can find a representation of $t$ with $c \leq 2$ and $e \leq 2k-1$.
\item Similarly, if $\lfloor {d \over 2} \rfloor + \lfloor {a \over 3} \rfloor \geq k$, we can find at least $k+1$ representations of $t$. Thus we can find a representation of $t$ with $d \leq 2k-1$ and $a \leq 2$. Combining this with (a), we can
find a representation with $a,c \leq 2$ and $d + e \leq 2k -1$.
\item If $\lfloor {b \over 2} \rfloor + \min\{ a,c \} \geq k$, we can find at least $k+1$ representations of $t$.
Thus we can find a representation of $t$ with $| b - \min\{ a,c \} | \leq 1$. If we start with the assumption $a, c \leq 2$, this ensures that $\min\{a,b,c\} \leq \lfloor {{a+b+c}\over 3} \rfloor \leq \min\{ a,b,c \} + 1$ and $\max\{a,b,c\} - \min\{a,b,c\} \leq 3$.
\item If $\min\{ a,b,c \} + \min\{ d,e \} \geq k$ we can find at least $k+1$ representations of $t$. When this swap is followed by (a) or (b) (if necessary) we can find a representation with $d + e \leq 2k -1$, $a+b+c \leq 3$ and $a,c \leq 2$.
\end{enumerate}
Putting this all together, we see that $t \leq (2n-1) + 2 (2n) + (2k-1)(3n) = (6k+3)n - 1$, as desired. \end{proof}
In order to prove Theorem~\ref{ref1} (b), we need a lemma.
\begin{lemma} The integers $k(n-1), k(n-1)+1, \ldots, kn$ are representable for $k = 2$ and $k \geq 4$ and for $n \geq 4$. \label{lem2} \end{lemma}
\begin{proof} We prove the result by induction on $k$. The base cases are $k = 2,4$, and we have the representations given below:
\begin{eqnarray*} 4n-4 &=& 2 (2n-2) \\ 4n-3 &=& (2n-2) + (2n-1) \\ 4n-2 &=& 2(2n-1) \\ 4n-1 &=& (2n-1) + (2n) \\ 4n &=& 2 (2n). \end{eqnarray*}
Now suppose $ln-m$ is representable for $4 \leq l < k$ and $0 \leq m \leq l$. We want to show that $kn-t$ is representable for $0 \leq t \leq k$. There are three cases, depending on $k$ (mod $3$).
If $k \equiv 0$ (mod 3), and $k \geq 4$, then $(k-2)n - t = kn - t - 2n$ is representable if $t \leq k-2$; otherwise $(k-2)n - t + 2 = kn - t - (2n-2)$ is representable. By adding $2n$ or $2n+2$, respectively, we get a representation for $kn-t$.
If $k \equiv 1$ (mod 3), and $k \geq 4$, or if $k \equiv 2$ (mod 3), then $(k-3)n - t = kn -t - 3n$ is representable if $t \leq k-3$; otherwise $(k-3)n -t + 3 = kn-t - (3n-3)$ is representable. By adding $3n$ or $3n+3$, respectively, we get a representation for $kn-t$. \end{proof}
Now we prove Theorem~\ref{ref1} (b).
\begin{proof} First, let's show that every integer $> n^2 - 3n+1$ is representable. Since if $t$ has a representation, so does $t+2n-2$, it suffices to show that the $2n-2$ numbers $n^2 -3n+2, n^2-3n+3, \ldots, n^2-n-1$ are representable.
We use Lemma~\ref{lem2} with $k = n-2$ to see that the numbers $(n-2)(n-1) = n^2-3n+2, \ldots, (n-2)n = n^2-2n$ are all representable. Now use Lemma~\ref{lem2} again with $k = n-1$ to see that the numbers $(n-1)(n-1) = n^2-2n+1, \ldots, (n-1)n = n^2-n$ are all representable. We therefore conclude that every integer $> n^2-3n+1$ has a representation.
Finally, we show that $n^2-3n+1$ does not have a representation. Suppose, to get a contradiction, that it does: $$ n^2 -3n+1 = a(2n-2) + b(2n-1) + c(2n) + d(3n-3) + e(3n).$$ Reducing modulo $n$ gives $1 \equiv -2a -b -3d $ (mod $n$), so there exists an integer $m$ such that $2a+b+3d = mn-1$. Since $a, b, d$ are non-negative, we must have $m \geq 1$.
Now $n^2-3n+1 \geq a(2n-2)+ b(2n-1) + d(3n-3) > (n-1)(2a+b+3d)$. Thus \begin{equation} n^2 -3n+1 \geq (n-1)(mn-1) = mn^2 -(m+1)n + 1. \label{eq5} \end{equation} If $m = 1$, we get $n^2 - 3n+1 \geq n^2 - 2n + 1$, a contradiction. Hence $m \geq 2$.
From (\ref{eq5}) we get $(m-1)n^2 - (m-2)n \leq 0$. Since $n \geq 1$, we get $(m-1)n - (m-2) \leq 0$, a contradiction. \end{proof}
\section{Additional remarks}
One might object to our examples because the numbers are not pairwise relatively prime. But there also exist reasonable $5$-tuples with $g_0 > g_1$ for which all pairs are relatively prime: for example, $g_0(9,10,11,13,17) = 25$, but $g_1(9,10,11,13,17) = 24$. More generally one can use the techniques in this paper to show that $g_0(10n-1, 15n-1, 20n-1, 25n, 30n-1) = 50n^2 -1$ and $g_1(10n-1, 15n-1, 20n-1, 25n, 30n-1) = 50n^2 - 5n$ for $n \geq 1$, so that $g_0 - g_1 \rightarrow \infty$ as $n \rightarrow \infty$.
For $k \geq 2$, let $f(k)$ be the least non-negative integer $i$ such that there exists a reasonable $k$-tuple $X$ with $g_i(X) > g_{i+1}(X)$. A priori $f(k)$ may not exist. For example, if $k = 2$, then we have $g_i (x_1, x_2) = (i+1)x_1x_2 - x_1 - x_2$, so $g_i (x_1, x_2) < g_{i+1} (x_1, x_2)$ for all $i$. Thus $f(2)$ does not exist. In this paper, we have shown that $f(5) = 0$.
This raises the obvious question of other values of $f$.
\begin{theorem} We have $f(i) = 0$ for $i \geq 4$. \end{theorem}
\begin{proof} As mentioned in the Introduction, the example $(8,9,11,14,15)$ shows that $f(5) = 0$.
For $i = 4$, we have the example $g_0(10,15,32,48) = 101$ and $g_1(10,15,32,48) = 99$, so $f(4) = 0$. (This is the reasonable quadruple with $g_0 > g_1$ that minimizes the largest element.)
We now provide a class of examples for $i \geq 6$. For $n \geq 6$ define $X_n$ as follows: $$X_n = (n+1, n+4, n+5, [n+7..2n+1], 2n+3, 2n+4),$$ where by $[a..b]$ we mean the list $a, a+1, a+2, \ldots, b$.
For example, $X_8 = (9, 12, 13, 15, 16, 17, 19, 20)$. Note that $X_n$ is of cardinality $n$. We make the following three claims for $n \geq 6$.
\begin{enumerate}[$($a$)$] \item $X_n$ is reasonable. \item $g_0 (X_n) = 2n+7$. \item $g_1 (X_n) = 2n+6$. \end{enumerate}
(a): To see that $X_n$ is reasonable, assume that some element $x$ is in the ${\mathbb N}$-span of the other elements. Then either $x = ky$ for some $k \geq 2$, where $y$ is the smallest element of $X_n$, or $x \geq y+z$, where $y, z$ are the two smallest elements of $X_n$. It is easy to see both of these lead to contradictions.
(b) and (c): Clearly $2n+7$ is not representable, and $2n+6$ has the single representation $(n+1) + (n+5)$. It now suffices to show that every integer $\geq 2n+8$ has at least two representations. And to show this, it suffices to show that all integers in the range $[2n+8..3n+8]$ have at least two representations.
Choosing $(n+4) + [n+7..2n+1]$ and $(n+5)+[n+7..2n+1]$ gives two distinct representations for all numbers in the interval $[2n+12..3n+5]$. So it suffices to handle the remaining cases $2n+8, 2n+9, 2n+10, 2n+11, 3n+6, 3n+7, 3n+8$. This is done as follows:
\begin{alignat*}{2} 2n+8 &= (n+1)+(n+7) & \ = & \ 2(n+4) \\ 2n+9 &= (n+4)+(n+5) & \ = & \
\begin{cases}
3(n+1), & \text{if $n = 6$}; \\
(n+1)+(n+8), & \text{if $n \geq 7$.}
\end{cases} \\ 2n+10 &= 2(n+5) & \ = & \
\begin{cases}
(n+1)+(2n+3), & \text{if $n = 6$}; \\
3(n+1), & \text{if $n = 7$}; \\
(n+1)+(n+9), & \text{if $n \geq 8$.}
\end{cases} \\ 2n+11 &= (n+4)+(n+7) & \ = & \
\begin{cases}
(n+1)+(2n+4), & \text{if $n = 6$}; \\
(n+1)+(2n+3), & \text{if $n = 7$}; \\
3(n+1) , & \text{if $n = 8$}; \\
(n+1)+(n+10), & \text{if $n \geq 9$.}
\end{cases} \\ 3n+6 &= 2(n+1) + (n+4) & \ = & \ (n+5) + (2n+1) \\ 3n+7 &= 2(n+1) + (n+5) & \ = & \ (n+4) + (2n+3) \\ 3n+8 &= (n+5) + (2n+3) & \ = & \ (n+4) + (2n+4). \end{alignat*}
\end{proof}
We do not know the value of $f(3)$. The example \begin{align*} g_{14}(8,9,15) &= 172 \\ g_{15}(8,9,15) &= 169 \end{align*} shows that $f(3) \leq 14$.
\begin{conjecture} $f(3) = 14$. \end{conjecture}
We have checked all triples with largest element $\leq 200$, but have not found any counterexamples.
\section{Acknowledgments}
We thank the referee for useful comments. Thanks also go to Dino Lorenzini who sent us a list of comments after we submitted this paper. Among them was an encouragement to make more use of the formula of Brown et al, which led to the example that shows that $f(4) = 0$.
\end{document} |
\begin{document}
\maketitle
\begin{center} \today \end{center}
\begin{abstract} We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and $\Gamma$-convergence. \end{abstract}
\section{Introduction}
Neglecting inertia, a nonlinear viscoelastic material in Kelvin's-Voigt's rheology obeys the following system of equations \begin{align}\label{eq:viscoel}-{\rm div}\Big(\partial_FW(\nabla y) + \partial_{\dot F}R(\nabla y,\partial_t \nabla y) \Big) = f\text{ in $ [0,T] \times \color{black} \Omega$.} \end{align} Here, $[0,T]$ is a process time interval with $T>0$, $\Omega\subset\Bbb R^d$ ($d=2$ or $d=3$) is a smooth bounded domain representing the reference configuration, and \color{black} $y:[0,T]\times \Omega\to\Bbb R^d$ is a deformation mapping with corresponding \color{black} deformation gradient $\nabla y$. Further, \color{black} $W:\Bbb R^{d\times d}\to [0,\infty]\color{black}$ is a stored energy density, which represents a potential of the first Piola-Kirchhoff stress tensor ${T^E}$, i.e., ${T^E}:=\partial_F W:=\partial W/\partial F$ and $F\in\Bbb R^{d\times d}$ is the placeholder of $\nabla y$. Finally, $R: \Bbb R^{d \times d} \times \Bbb R^{d \times d} \to [0,\infty) \color{black} $ denotes a (pseudo)potential of dissipative forces, where $\dot F \in \Bbb R^{d \times d}$ is the placeholder of $\partial_t \nabla y$, \color{black} and $f:[0,T]\times \Omega\to\Bbb R^d$ is a volume density of external forces acting on $\Omega$. In the present contribution, we consider a version of \eqref{eq:viscoel} for nonsimple materials where the elastic stored energy density depends also on the second gradient of $y$. In this case, we get
\begin{align}\label{eq:viscoel-nonsimple} -{\rm div}\Big( \partial_F W(\nabla y) + \varepsilon\mathcal{L}_{P}(\nabla^2 y) + \partial_{\dot{F}}R(\nabla y,\partial_t \nabla y) \Big) = f\text{ in $\color{black} [0,T] \times \color{black} \Omega$,} \end{align}
where $\varepsilon>0$ is small and $\mathcal{L}_{P}$ is a first \color{black} order differential operator which is associated to an additional term $\int_\Omega P(\nabla^2 y)$ in the stored elastic energy, e.g., for $P(G):= \frac{1}{2} \color{black} |G|^2$ with $G\in\Bbb R^{d\times d\times d}$, we get $-{\rm div}\mathcal{L}_{P}(\nabla^2 y)= \Delta^2 y$. We refer to \eqref{LP-def} for more details. Thus, we resort to the so-called nonsimple materials, the stored energy density (and the first Piola-Kirchhoff stress tensor, too) of which depends also on the second gradient of the deformation. This idea was first introduced by Toupin \cite{Toupin:62,Toupin:64} and proved to be useful in mathematical elasticity, \color{black} see e.g.~\cite{BCO,Batra, chen, MielkeRoubicek:16,MielkeRoubicek,Podio} \color{black} because it brings additional compactness to the problem. The first Piola-Kirchhoff stress tensor, ${T^E}$, then reads for all $i,j\in\{1,\ldots, d\}$ \begin{align*} {T^E}_{ij}(F,G):= \color{black} \partial_{F_{ij}} W(F) +\varepsilon \big(\mathcal{L}_{P}(G)\big)_{ij}\color{black} = \partial_{F_{ij}} W(F) -\varepsilon\sum_{k=1}^d \partial_k \big(\partial_{G_{ijk}}P(G)\big), \end{align*} where $G\in\Bbb R^{d\times d\times d}$ is the placeholder for the second gradient of $y$. The term $\varepsilon\partial_{G}P(G)$ is usually called hyperstress.
We standardly assume that $W$ as well as $P$ are frame-indifferent functions, i.e., that $W(F)=W(QF)$ and $P(G)=P(QG)$ for every proper rotation $Q\in{\rm SO}(d)$, every $F\in\Bbb R^{d\times d}$, and every $G\in\Bbb R^{d\times d\times d}$. This implies that $W$ depends on the right Cauchy-Green strain tensor $C:=F^\top F$, see e.g.~\cite{Ciarlet}. \color{black} We wish to emphasize that, in the case of nonsimple materials, \color{black} no convexity properties of $W$ are needed, in particular, we do not have to assume that $W$ is polyconvex \cite{Ball:77,Ciarlet}. Moreover, it is shown in \cite{HealeyKroemer:09} that if $W$ satisfies suitable and physically relevant growth conditions (as $W(F)\to\infty$ if ${\rm det}\, F\to 0$), then every minimizer of the elastic energy is a weak solution to the corresponding Euler-Lagrange equations.
The second term on the left-hand side of \eqref{eq:viscoel} is the \color{black} viscous \color{black} stress tensor ${S}(F,\dot F):= \partial_{\dot F} R(F,\dot F)$ which has its origin in viscous dissipative mechanisms of the material. Notice that its potential $R$ plays an analogous role as $W$ in the case of purely elastic, i.e., non-dissipative processes. Naturally, we require that $R(F,\dot F)\ge R(F,0)=0$. The viscous stress tensor must comply with the time-continuous frame-indifference principle meaning that for all $F$ \begin{align*} {S}(F,\dot F)=F\tilde{S}(C,\dot C) \end{align*} where $\tilde{S}$ is a symmetric matrix-valued function. This condition constraints $R$ so that \cite{Antmann, Antmann:04,MOS} \color{black} (see also \cite{Demoulini}) \color{black} \begin{align}\label{eq:frame indifference-R} R(F,\dot F)=\tilde R(C,\dot C)\ , \end{align} for some nonnegative function $\tilde R$. In other words, \color{black} $R$ must depend on the right Cauchy-Green strain tensor $C$ and its time derivative $\dot C$.
In this work, we are interested in the case of small strains, i.e., when $\nabla u:=\nabla y - \mathbf{Id}$ is of order $\delta$ for some small $\delta >0$. Here, $u:=y-\mathbf{id}$ is the displacement corresponding to $y$ with $\mathbf{id}$ and $\mathbf{Id}$ standing for the identity map and identity matrix, respectively. Such a property is certainly meaningful if one considers initial values $y_0$ with $\Vert \nabla y_0 - \mathbf{Id} \Vert_{L^2(\Omega)} \le \delta$. Therefore, it is convenient to define the rescaled displacement $u = \delta^{-1}(y - \mathbf{id})$. Introducing a proper scaling in the above equation we get \begin{align}\label{eq:viscoel-nonsimple-scaled} -{\rm div}\Big( \delta^{-1}\partial_F W(\mathbf{Id} + \delta \nabla u) + \tilde{\varepsilon}\mathcal{L}_{P}(\delta\nabla^2 u) + \delta^{-1}\partial_{\dot{F}}R(\mathbf{Id} + \delta \nabla u, \delta \partial_t \nabla u) \Big) = f\end{align} for $\tilde{\varepsilon}=\tilde{\varepsilon}(\delta)$ appropriate. \color{black} Note that to obtain \eqref{eq:viscoel-nonsimple-scaled} from \eqref{eq:viscoel} we write the latter equation for $f:=\delta f$ and then divide the whole equation by $\delta$. \color{black}
Formally, we can pass to the limit and obtain the equation (for $\tilde{\varepsilon} \to 0$ as $\delta \to 0$) \begin{align}\label{eq:viscoel-small} -{\rm div}\Big( \Bbb C_W e(u) + \Bbb C_D e(\partial_t u) \Big) = f,
\end{align}
where $\Bbb C_W:=\partial^2_{F^2}W(\mathbf{Id})$ is the tensor of elastic constants, $\Bbb C_D:= \partial^2_{\dot F^2}R(\mathbf{Id},0)$ is the tensor of viscosity coefficients, and $e(u):=(\nabla u+(\nabla u)^\top)/2$ denotes the linear strain tensor.
The goal of this contribution is twofold: we first show existence of solutions \color{black} to the nonlinear system of equations \eqref{eq:viscoel-nonsimple-scaled}. \color{black} Afterwards, \color{black} we make the limit passage rigorous, i.e., we show that solutions to the nonlinear equations converge to the unique \color{black} solution of the linear systems as $\delta\to 0$. Interestingly, although the nonlinear viscoelastisity systems is written for a nonsimple material, in the limit we obtain the standard \color{black} linear equations without spatial gradients of $e(u)$.
Our general strategy is to treat the system of quasistatic viscoelasticity in the abstract setting of metric gradient flows \cite{AGS} which was, to our best knowledge, formulated for the first time in \cite{MOS} for simple materials (i.e.~only the first gradient of $y$ is considered).
However, in their setting, \color{black} a passage from time-discrete problems to a continuous one is only possible in a specific one-dimensional case. See also \cite{BallSenguel:15} for a related approach in materials undergoing phase transition. This, in our opinion, also supports models of nonsimple materials as their linearization \color{black} leads to \color{black} the usual small-strain viscoelasticity model which seems unreachable (or at least rather difficult) in the case of simple materials.
\color{black} An abstract framework for the study of metric gradient flows along a sequence of energies and metric spaces has been developed in \cite{S1,S2}. In practice, for each specific problem the challenge lies in proving that the additional conditions needed to ensure convergence of gradient flows are satisfied (we refer to \cite{S2} for some examples in that direction). Our aim is to show that the passage of nonlinear to linearized viscoelasticity can be formulated in this setting. Let us also mention that a rigorous analysis of the static, purely elastic case without viscosity goes back to \cite{DalMasoNegriPercivale:02}. \color{black}
\color{black} Heuristically, the idea of gradient flows in metric spaces stems from the observation that, having a Hilbert space (equipped with the dot product $\langle\cdot,\cdot\rangle$), the inequality $$
|u'|^2 +2\langle u',\nabla\phi(u)\rangle+|\nabla\phi(u)|^2\ge 0 $$
becomes equality if and only if $$ u'=-\nabla \phi(u), $$
i.e., if $u$ solves the gradient flow equation. This approach can be extended to metric spaces provided we are able to find analogies to $|u'|$ and $|\nabla\phi|$ in metric spaces. These are called the metric derivative and the upper gradient (or slope), respectively. Precise definitions can be found in Section~\ref{sec: defs} below. \color{black}
The plan of the paper is as follows. In Section~\ref{sec:Model}, we introduce the nonlinear and linear systems of viscoelasticity \color{black} in more detail and state our main results. In particular, Theorem~\ref{maintheorem1} and Theorem~\ref{maintheorem2} show the existence of solutions to the nonlinear and linear problems, respectively. These solutions can be identified with so-called \emph{curves of maximal slope} \color{black} introduced in \cite{DGMT}. \color{black} Proofs of existence rely on semidiscretization in time, and on the theory of \emph{generalized minimizing movements} and gradient flows in metric spaces \cite{AGS}, where the underlying metric is given by a \emph{dissipation distance} suitably related to the potential $R$ (see \eqref{intro:R}). \color{black} Finally, Theorem~\ref{maintheorem3} shows the relationship between the two systems. Besides convergence of solutions of \eqref{eq:viscoel-nonsimple} to solutions of \eqref{eq:viscoel-small}, we also get analogous convergences for semidiscretized problems. Moreover, convergences for vanishing time step and $\delta\to 0$ commute, see Figure~\ref{diagram}. (For a related commutativity result in an abstract setting we refer to \cite{BCGS}.)
Section~\ref{sec3} is devoted to definitions of generalized minimizing movements (GMM) and curves of maximal slope. \color{black} Here we also collect the necessary existence results proved in \cite{AGS}. Moreover, we present a statement similar to \cite{Ortner,S2} about sequences of curves of maximal slope and their limits as well as a corresponding result for minimizing movements. \color{black}
Further, Section~\ref{sec:energy-dissipation} shows interesting properties of dissipation distances related to our viscous dissipation. It turns out that by frame indifference \eqref{eq:frame indifference-R} the dissipation distances are genuinely non-convex. However, due to the presence of the higher order gradient we are able to obtain sufficiently good convexity properties in order to apply the abstract theory \color{black} \cite{AGS, S2}. \color{black} Finally, proofs of our results can be found in Section~\ref{sec results}. In particular, we relate curves of maximal slope for the nonlinear system with limiting curves of maximal slope as $\delta\to 0$ and identify these configurations as weak solutions of \eqref{eq:viscoel-nonsimple} and \eqref{eq:viscoel-small}. \color{black}
In what follows, we use standard notation for Lebesgue spaces, $L^p(\Omega)$, which are measurable maps on $\Omega\subset\Bbb R^d$ integrable with the $p$-th power (if $1\le p<+\infty$) or essentially bounded (if $p=+\infty$). Sobolev spaces, i.e., $W^{k,p}(\Omega)$ denote the linear spaces of maps which, together with their derivatives up to the order $k\in\Bbb N$, belong to $L^p(\Omega)$. \color{black} Further, $W^{k,p}_0(\Omega)$ contains maps from $W^{k,p}(\Omega)$ having zero boundary conditions (in the sense of traces). \color{black} In order to emphasize its Hilbert structure, we write $H^1(\Omega):=W^{1,2}(\Omega)$. We also work with the dual space to $H^1_0(\Omega)$ denoted by $H^{-1}(\Omega)$. We refer to \cite{AdamsFournier:05} for more details on Sobolev spaces and their duals.
If $A\in\Bbb R^{d\times d\times d\times d}$ and $e\in\Bbb R^{d\times d}$ then $Ae\in\Bbb R^{d\times d}$ such that for $i,j\in\{1,\ldots, d\}$ we define $(Ae)_{ij}:=A_{ijkl}e_{kl}$ where we use Einstein's summation convention. An analogous convention is used in similar occasions, in the sequel. Finally, at many spots, we follow closely notation introduced in \cite{AGS} to ease readability of our work because the theory developed there is one of the main tools of our analysis.
\section{The model and main results}\label{sec:Model}
\subsection{The nonlinear setting}
We adopt the usual setting of nonlinear elasticity: consider $\Omega \subset \Bbb R^d$ open, bounded with Lipschitz boundary. Fix $\delta>0$ (small), $p>d$ and $0< \alpha<1$. The parameter $\tilde\varepsilon(\delta)$ introduced in \eqref{eq:viscoel-nonsimple-scaled} is defined as $\tilde\varepsilon(\delta):=\delta^{1-p\alpha}$.
\textbf{Stored elastic energy and body forces:} We introduce the nonlinear elastic energy $\phi_\delta: W^{2,p}(\Omega;\Bbb R^d) \to [0,\infty]$ by \begin{align}\label{nonlinear energy} \phi_\delta(y) = \frac{1}{\delta^2}\int_\Omega W(\nabla y(x))\, dx + \frac{1}{\delta^{p\alpha}}\int_\Omega P(\nabla^2 y(x)) \, dx - \frac{1}{\delta}\int_\Omega f(x)\cdot y(x) \, dx \end{align} for a \emph{deformation} $y: W^{2,p}(\Omega;\Bbb R^d) \to \Bbb R^d$. Here, $W: \Bbb R^{d \times d} \to [0,\infty]$ is a single well, frame indifferent stored energy functional with the usual assumptions in nonlinear elasticity. Altogether, we suppose that there exists $c>0$ such that \begin{align}\label{assumptions-W} \begin{split} (i)& \ \ W \text{ continuous and $C^3$ in a neighborhood of $SO(d)$},\\ (ii)& \ \ \text{Frame indifference: } W(QF) = W(F) \text{ for all } F \in \Bbb R^{d \times d}, Q \in SO(d),\\ (iii)& \ \ W(F) \ge c\operatorname{dist}^2(F,SO(d)), \ W(F) = 0 \text{ iff } F \in SO(d), \end{split} \end{align} where $SO(d) = \lbrace Q\in \Bbb R^{d \times d}: Q^\top Q = \mathbf{Id}, \, \det Q=1 \rbrace$. Moreover, $P: \Bbb R^{d\times d \times d} \to [0,\infty]$ denotes a higher order perturbation satisfying \begin{align}\label{assumptions-P} \begin{split} (i)& \ \ \text{frame indifference: } P(QG) = P(G) \text{ for all } G \in \Bbb R^{d \times d \times d}, Q \in SO(d),\\ (ii)& \ \ \text{$P$ is convex and $C^1$},\\
(iii)& \ \ \text{growth condition: For all $G \in \Bbb R^{d \times d \times d}$ we have } \\& \ \ \ \ \ \ c_1 |G|^p \le P(G) \le c_2 |G|^p, \ \ \ \ \ \ \color{black} |\partial_G P(G)| \color{black} \le c_2 |G|^{p-1} \color{black} \end{split} \end{align} for $0<c_1<c_2$. Finally, $f \in L^\infty(\Omega;\Bbb R^d)$ denotes a volume force. From now on we always drop the target space $\Bbb R^d$ for notational convenience when no confusion arises. \color{black} We remark that by minor adaptions of our arguments we can also treat potentials with additional dependence on the material point $x \in \Omega$. We scale the energy appropriately with a (small) positive parameter $\delta$ as we will eventually be interested in the behavior in the small strain limit $\delta \to 0$.
\textbf{Dissipation potential and viscous stress:} Consider a time dependent deformation $y: [0,T] \times \Omega \to \Bbb R^d$. Viscosity is not only related to the strain $\nabla y(t,x)$, but also to the strain rate $\partial_t \nabla y(t,x)$ and can be expressed in terms of a dissipation potential $R(\nabla y, \partial_t \nabla y)$, where $R: \Bbb R^{d \times d} \times \Bbb R^{d \times d} \to [0,\infty)$. An admissible potential has to satisfy frame indifference in the sense (see \cite{Antmann, MOS}) \begin{align}\label{R: frame indiff} R(F,\dot{F}) = R(QF,Q(\dot{F} + AF)) \ \ \ \forall Q \in SO(d), A \in {\rm Skew}(d) \end{align} for all $F \in GL_+(d)$ and $\dot{F} \in \Bbb R^{d \times d}$, where $GL_+(d) = \lbrace F \in \Bbb R^{d \times d}: \det F>0 \rbrace$ and ${\rm Skew}(d) = \lbrace A \in \Bbb R^{d \times d}: A=-A^\top \rbrace$.
Following the discussion in \cite[Section 2.2]{MOS}, from the point of modeling it is much more convenient to postulate the existence of a (smooth) global distance $D: GL_+(d) \times GL_+(d) \to [0,\infty)$ satisfying $D(F,F) = 0$ for all $F \in GL_+(d)$, from which an associated dissipation potential $R$ can be calculated by \begin{align}\label{intro:R} R(F,\dot{F}) := \lim_{\varepsilon \to 0} \frac{1}{2\varepsilon^2} D^2(F+\varepsilon\dot{F},F) = \frac{1}{4} \partial^2_{F_1^2} D^2(F,F) [\dot{F},\dot{F}] \end{align} for $F \in GL_+(d)$, $\dot{F} \in \Bbb R^{d \times d}$, where $\partial^2_{F_1^2} D^2(F_1,F_2)$ denotes the Hessian of $D^2$ in direction of $F_1$ at $(F_1,F_2)$, being a fourth order tensor. We have the following assumptions on $D$ for some $c>0$. \begin{align}\label{eq: assumptions-D} (i) & \ \ D(F_1,F_2)> 0 \text{ if } F_1^\top F_1 \neq F_2^\top F_2,\notag \\ (ii) & \ \ D(F_1,F_2) = D(F_2,F_1),\\ (iii) & \ \ D(F_1,F_3) \le D(F_1,F_2) + D(F_2,F_3),\notag \\ (iv) & \ \ \text{$D(\cdot,\cdot)$ is $C^3$ in a neigborhood of $SO(d) \times SO(d)$},\notag \\ (v)& \ \ \text{Separate frame indifference: } D(Q_1F_1,Q_2F_2) = D(F_1,F_2)\notag \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall Q_1,Q_2 \in SO(d), \ \forall F_1,F_2 \in GL_+(d),\notag\\ (vi) & \ \ \text{$D(F,\mathbf{Id}) \ge c\operatorname{dist}(F,SO(d))$ $\forall F \in \Bbb R^{d \times d}$ in a neighborhood of $SO(d)$}.\notag \end{align} Note that conditions (i),(iii) state that $D$ is a true distance when restricted to symmetric matrices. We can not expect more due to the separate frame indifference (v). We also note that (v) implies \eqref{R: frame indiff} as shown in \cite[Lemma 2.1]{MOS}. Note that in our model we do not require any conditions of polyconvexity neither for $W$ nor for $D$ \cite{Ball:77}. For examples of admissible dissipation distances we refer the reader to \cite[Section 2.3]{MOS}.
\textbf{Equations of nonlinear viscoelasticity:} We will impose the boundary conditions $y(t,x) = x$ for $(t,x) \in [0,T] \times \partial \Omega$ and for convenience we define the set $W^{2,p}_\mathbf{id}(\Omega)= \lbrace y = \mathbf{id} + u \in W^{2,p}(\Omega): u \in W^{2,p}_0(\Omega) \rbrace$, where $\mathbf{id}$ denotes the identity function on $\Omega$.
We remark that our results can be extended to more general Dirichlet boundary conditions, too, which we do not include here for the sake of maximizing simplicity rather than generality. We now introduce a differential operator associated to the perturbation $P$ (cf. \eqref{assumptions-P}). To this end, we use the notation $(\nabla y)_{ik} = \partial_k y_i$ and $(\nabla^2 y)_{ijk} = \partial^2_{jk} y_i$ for $i,j,k \in \lbrace 1,\ldots,d\rbrace$ \color{black} and define \begin{align}\label{LP-def} \big(\mathcal{L}_P(\nabla^2 y)\big)_{ij} = -\sum\nolimits_{k=1}^d \partial_k(\partial_GP(\nabla^2 y))_{ijk}, \ \ \ \ \color{black} i,j \color{black} \in \lbrace 1,\ldots, d\rbrace \end{align} for $y \in W^{2,p}_\mathbf{id}(\Omega)$, where the derivatives have to be understood in the sense of distributions. The equations of nonlinear viscoelasticity then read as (respecting the different scalings of the terms in \eqref{nonlinear energy}) \begin{align}\label{nonlinear equation} \begin{split} \begin{cases} - {\rm div}\Big( \partial_FW(\nabla y) + \delta^{2-p\alpha}\mathcal{L}_{P}(\nabla^2 y) + \partial_{\dot{F}}R(\nabla y,\partial_t \nabla y) \Big) = \delta f & \text{in } [0,\infty) \times \Omega \\ y(0,\cdot) = y_0 & \text{in } \Omega \\ y(t,\cdot) \in W^{2,p}_\mathbf{id}(\Omega) &\text{for } t\in [0,\infty) \end{cases} \end{split} \end{align} for some $y_0 \in W^{2,p}_\mathbf{id}(\Omega)$, where $\partial_FW(\nabla y(t,x))$ denotes the first \emph{Piola-Kirchhoff stress tensor} and $\partial_{\dot{F}}R(\nabla y(t,x),\partial_t \nabla y(t,x))$ the \emph{viscous stress} with $R$ as introduced in \eqref{intro:R}.
The first goal of the present contribution is to prove the existence of weak solutions to \eqref{nonlinear equation}. More precisely, we say that $y \in L^\infty([0,\infty);W^{2,p}_{\mathbf{id}}(\Omega)) \cap W^{1,2}([0,\infty);H^1(\Omega))$ is a \emph{weak solution} of \eqref{nonlinear equation} if $y(0,\cdot) = y_0$ and for a.e. $t \ge 0$
\begin{align}\label{nonlinear equation2}
\begin{split} & \int_\Omega \Big( \partial_FW(\nabla y(t,x)) + \partial_{\dot{F}}R(\nabla y(t,x),\partial_t \nabla y(t,x))\Big) : \nabla \varphi(x) \, dx \\ & \ \ \ \ \ \ \ \ \ + \int_\Omega\delta^{2-p\alpha} \partial_GP(\nabla^2 y(t,x)) :\nabla^2 \varphi(x) \, dx = \delta \int_\Omega f(x) \cdot \varphi(x) \, dx \end{split}
\end{align}
for all $\varphi \in W^{2,p}_0(\Omega)$. In particular, we note that the first term in the second line is well defined for a weak solution by \eqref{assumptions-P}(iii) and H\"older's inequality.
\color{black}
\subsection{The linear problem} After rescaling with $\delta^{-1}$ and introducing the rescaled displacement field $u(t,x) = \delta^{-1} (y(t,x)-x)$, the partial differential equation \eqref{nonlinear equation} can be written as $$-{\rm div}\Big( \delta^{-1}\partial_FW(\mathbf{id} + \delta \nabla u) + \delta^{1-p\alpha}\mathcal{L}_{P}(\delta\nabla^2 u) + \delta^{-1}\partial_{\dot{F}}R(\mathbf{id} + \delta \nabla u, \delta \partial_t \nabla u) \Big) = f$$ with an initial datum $u_0 = \delta^{-1}(y_0 - \mathbf{id})$. For $\alpha$ small, letting $\delta \to 0$ we obtain, at least formally, the equation \begin{align}\label{linear equation} \begin{split} \begin{cases} -{\rm div}\Big( \Bbb C_W e(u) + \Bbb C_D e(\partial_t u) \Big) = f & \text{in } [0,\infty) \times \Omega \\ u(0,\cdot) = u_0 & \text{in } \Omega \\ u(t,\cdot) \in H^1_{0}(\Omega) &\text{for } t\in [0,\infty), \end{cases} \end{split} \end{align} where $\Bbb C_W := \partial^2_{F^2} W(\mathbf{Id})$ and $\Bbb C_D := \frac{1}{2}\partial^2_{F_1^2} D^2(\mathbf{Id},\mathbf{Id}) $ (cf. \eqref{intro:R}). Note that the frame indifference of the energy and the dissipation (see \eqref{assumptions-W}(ii) and \eqref{eq: assumptions-D}(v), respectively) imply that the contributions only depend on the symmetric part of the strain $e(u) = \frac{1}{2}( \nabla u +(\nabla u)^\top)$ and the strain rate $e(\partial_t u) = \frac{1}{2}( \partial_t \nabla u + \partial_t (\nabla u)^\top)$. Let us also mention that the stress tensor is related to the linearized elastic energy $\bar{\phi}_0 : H_0^1(\Omega) \to [0,\infty)$ given by \begin{align}\label{linear energy} \bar{\phi}_0(u) = \int_\Omega \frac{1}{2}\Bbb C_W[e(u)(x), e(u)(x)] \, dx - \int_\Omega f(x) \cdot u(x) \,dx \end{align} for $u \in H^1_0(\Omega)$. The goal of this article is to show that the above reasoning can be made rigorous: we will prove that \eqref{linear equation} admits a unique weak solution and that solutions of \eqref{nonlinear equation} converge to the solution of \eqref{linear equation} in a suitable sense. Here, similarly as before, we say $u \in W^{1,2}([0,\infty); H^1_0(\Omega))$ is a \emph{weak solution} of \eqref{linear equation} if $u(0,\cdot) = u_0$ and for a.e. $t \ge 0$ and all $\varphi \in H^{1}_0(\Omega)$ we have $$\int_\Omega ( \Bbb C_W e(u) + \Bbb C_D e(\partial_t u) ) : \nabla \varphi = \int_\Omega f\cdot \varphi$$. \color{black}
\subsection{Main results} Let us introduce the \emph{global dissipation distance} between two deformations for the nonlinear and linear setting by \begin{align}\label{eq: D,D0} \begin{split}
&\mathcal{D}_\delta(y_0,y_1) = \delta^{-1}\Big(\int_\Omega D^2(\nabla y_0, \nabla y_1) \Big)^{1/2}, \\
& \bar{\mathcal{D}}_0(u_0,u_1) = \Big(\int_\Omega \Bbb C_D[\nabla u_0 - \nabla u_1,\nabla u_0 - \nabla u_1] \Big)^{1/2}
\end{split} \end{align} for $y_0,y_1 \in W^{2,p}_\mathbf{id}(\Omega)$ and $u_0,u_1 \in H^1_0(\Omega)$, respectively. (In many notations we include an overline to indicate that the notion is related to the linear setting.) We also define the sublevel sets $\mathscr{S}_\delta^M := \lbrace y\in W^{2,p}_\mathbf{id}(\Omega): \phi_\delta(y) \le M\rbrace$. (For convenience we do not include $\Omega$ in the notation.) Our general strategy will be to show that the spaces $(\mathscr{S}_\delta^M, \mathcal{D}_\delta)$ and $(H^1_0(\Omega), \bar{\mathcal{D}}_0)$ are complete metric spaces and to follow the approach in \cite{AGS} (see Theorem \ref{th: metric space} and Theorem \ref{th: metric space-lin} below).
In particular, to show existence of solutions to the problems \eqref{nonlinear equation} and \eqref{linear equation}, we will apply an approximation scheme solving suitable time-incremental minimization problems and show that \color{black} time-continuous \color{black} limits are curves of maximal slope for the elastic energies $\phi_\delta, \bar{\phi}_0$, respectively. Finally, using the property that in Hilbert spaces curves of maximal slope can be related to gradient flows, we find solutions to \eqref{nonlinear equation}, \eqref{linear equation}.
Moreover, to study the relation between the nonlinear and linear problem we will apply some results about the limit of sequences of curves of maximal slope proved in Section \ref{sec: auxi-proofs}.
For the main definitions and notation for discrete solutions, (generalized) minimizing movements (abbreviated by MM and GMM, see Definition~\ref{main def1}) and curves of maximal slope we refer to Section \ref{sec: defs}. In particular, we define $\Phi_\delta$ and $\bar{\Phi}_0$, respectively, as in \eqref{incremental} replacing $\phi, \mathcal{D}$ by $\phi_\delta,\mathcal{D}_\delta$ and $\bar{\phi}_0,\bar{\mathcal{D}}_0$, respectively. Moreover, we write $|\partial \phi_\delta|_{\mathcal{D}_\delta}$, $|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$ for the (local) slopes and $|y'|_{\mathcal{D}_\delta}$, $|u'|_{\bar{\mathcal{D}}_0}$ for the metric derivatives, respectively (see Definition \ref{main def2}). Finally, discrete solutions for time step $\tau > 0$ will be denoted by \color{black} $\tilde{Y}^\delta_\tau$ and $\tilde{U}^0_\tau$, respectively. \color{black}
Our first main result addresses the existence of solutions to the nonlinear problem.
\begin{theorem}[Solutions to the nonlinear problem]\label{maintheorem1} Let $M>0$ and $\mathscr{S}_\delta^M = \lbrace y \in W^{2,p}_\mathbf{id}(\Omega): \phi_\delta(y) \le M\rbrace$. Then for $\delta>0$ sufficiently small only depending on $M$ the following holds:
(i) (Existence of GMM) \color{black} $GMM(\Phi_\delta;y_0) \neq \emptyset$ for all $y_0 \in \mathscr{S}^M_\delta$.
(ii) (Curves of maximal slope) \color{black} For all $y_0 \in \mathscr{S}^M_\delta $ each $y \in GMM(\Phi_\delta;y_0)$ is a curve of maximal slope for $\phi_\delta$
with respect to the strong upper gradient $|\partial \phi_\delta|_{\mathcal{D}_\delta}$, in particular for all $T>0$
we have the energy identity
\begin{align}\label{slopesolution}
\frac{1}{2} \int_0^T |y'|_{\mathcal{D}_\delta}^2(t)\,dt + \frac{1}{2} \int_0^T |\partial \phi_\delta|^2_{\mathcal{D}_\delta}(y(t))\,dt + \phi_\delta(y(T)) = \phi_\delta(y_0).
\end{align}
(iii) (Relation to PDE) \color{black} For all $y_0 \in \mathscr{S}^M_\delta $ each $y \in GMM(\Phi_\delta;y_0)$ is a weak solution of the partial differential equations of nonlinear viscoelasticity \eqref{nonlinear equation} in the sense of \eqref{nonlinear equation2}. \color{black}
\end{theorem}
For the linearized model we obtain the following results.
\begin{theorem}[Solutions to the linear problem]\label{maintheorem2} The limiting linear problem has the following properties.
(i) (Existence/Uniqueness of MM) \color{black} For all $u_0 \in H_0^1(\Omega)$ there exists a unique $u \in MM(\bar{\Phi}_0;u_0)$.
(ii) (Curves of maximal slope) \color{black} For all $u_0 \in H_0^1(\Omega)$ the minimizing movement $u \in MM(\bar{\Phi}_0;u_0)$ is the unique curve of maximal slope for $\bar{\phi}_0$ with respect to the strong upper gradient $|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$.
(iii) (Relation to PDE) \color{black} For all $u_0 \in H_0^1(\Omega)$ the unique $u \in MM(\Phi_\delta;u_0)$ is a weak solution of the partial differential equations of linear viscoelasticity \eqref{linear equation}.
\end{theorem}
In contrast to Theorem \ref{maintheorem1}, we get that the weak solution to \eqref{linear equation} for given initial value $u_0 \in H^1_0(\Omega)$ is uniquely determined and a minimizing movement (and not simply a generalized one). Finally, we study the relation of the solutions to the equations \eqref{nonlinear equation} and \eqref{linear equation}.
\begin{theorem}[Relation between nonlinear and linear problems]\label{maintheorem3} Fix a null sequence $(\delta_k)_k$ and a sequence of initial data $(y_0^k)_{k\in \Bbb N} \subset W^{2,p}_\mathbf{id}(\Omega)$ such that $$\sup\nolimits_{k\in\Bbb N} \phi_{\delta_k}(y_0^k)<\infty, \ \ \ \ \delta_k^{-p\alpha}\int_\Omega P(\nabla^2 y_0^k) \to 0, \ \ \ \ \delta_k^{-1}(y^k_0 - \mathbf{id}) \to u_0 \in H_0^1(\Omega).$$
Let $u$ be the unique element of $MM(\bar{\Phi}_0;u_0)$. Then the following holds:
(i) (Convergence of discrete solutions) \color{black} For all $\tau>0$ and all discrete solutions \color{black} $\tilde{Y}_\tau^{\delta_k}$ \color{black} as in \eqref{ds} below there is a discrete solution \color{black} $\tilde{U}^0_\tau$ \color{black} for the linearized system such that \color{black} $\delta_k^{-1}(\tilde{Y}_\tau^{\delta_k}(t) -\mathbf{id}) \to \tilde{U}^0_\tau(t)$ strongly in $H^1(\Omega)$ for all $t \in [0,\infty)$. \color{black}
(ii) (Convergence of continuous solutions) \color{black} Each sequence $y_k \in GMM(\Phi_{\delta_k};y_0^k)$, $k \in \Bbb N$, satisfies $\delta_k^{-1}(y_k(t) -\mathbf{id}) \to u(t)$ strongly in $H^1(\Omega)$ for all $t \in [0,\infty)$.
(iii) (Convergence at specific scales) \color{black} For each null sequence $(\tau_k)_k$ and each sequence of discrete solutions \color{black} $\tilde{Y}_{\tau_k}^{\delta_k}$ as in \eqref{ds} we have $\delta_k^{-1}(\tilde{Y}_{\tau_k}^{\delta_k}(t) -\mathbf{id}) \to u(t)$ strongly in $H^1(\Omega)$ for all $t \in [0,\infty)$. \color{black}
\end{theorem}
We remark that, in the formulation of \color{black} \cite{Braides, BCGS}, \color{black} property (iii) states that the configuration $u$ is a minimizing movement along $\phi_{\delta_k}$ at scale $\tau_k$. Let us emphasize that the converge in Theorem \ref{maintheorem3} is with respect to the strong $H^1(\Omega)$-topology. From now on we set $f \equiv 0 $ for convenience. The general case indeed follows with minor modifications, which are standard.\color{black}
\begin{figure}
\caption{ Illustration of the commutativity result given in Theorem \ref{maintheorem1}-Theorem \ref{maintheorem3}. The horizontal arrows are addressed in Theorem \ref{maintheorem1} and Theorem \ref{maintheorem2}, respectively. For the vertical and diagonal arrows we refer to Theorem \ref{maintheorem3}.\color{black} }
\label{diagram}
\end{figure}
\section{Preliminaries: Generalized minimizing movements and curves of maximal slope}\label{sec3}
In this section we first recall the relevant definitions and also give a convergence result for discrete solutions to curves of maximal slope proved in \cite{AGS}. In Section \ref{sec: auxi-proofs} we then present a result about the limit of sequences of curves of maximal slope being a variant of results presented in \cite{CG, S2}.
\subsection{Definitions}\label{sec: defs}
We consider a complete metric space $(\mathscr{S},\mathcal{D})$. We say a curve $u: (a,b) \to \mathscr{S}$ is \emph{absolutely continuous} with respect to $\mathcal{D}$ if there exists $m \in L^1(a,b)$ such that \begin{align}\label{metric-deriv} \mathcal{D}(u(s),u(t)) \le \int_s^t m(r) \, dr \ \ \ \text{for all} \ a \le s \le t \le b. \end{align}
The smallest function $m$ with this property, denoted by $|u'|_{\mathcal{D}}$, is called \emph{metric derivative} of $u$ and satisfies for a.e. $t \in (a,b)$ (see \cite[Theorem 1.1.2]{AGS} for the existence proof)
$$|u'|_{\mathcal{D}}(t) := \lim_{s \to t} \frac{\mathcal{D}(u(s),u(t))}{|s-t|}.$$ We now define the notion of a \emph{curve of maximal slope}. We only give the basic definition here and refer to \cite[Section 1.2, 1.3]{AGS} for motivations and more details. By $h^+:=\max(h,0)$ we denote the positive part of a function $h$.
\begin{definition}[Upper gradients, slopes, curves of maximal slope]\label{main def2}
We consider a complete metric space $(\mathscr{S},\mathcal{D})$ with a functional $\phi: \mathscr{S} \to (-\infty,+\infty]$.
(i) A function $g: \mathscr{S} \to [0,\infty]$ is called a strong upper gradient for $\phi$ if for every absolutely continuous curve $v: (a,b) \to \mathscr{S}$ the function $g \circ v$ is Borel and
$$|\phi(v(t)) - \phi(v(s))| \le \int_s^t g(v(r)) |v'|_{\mathcal{D}}(r)\,dr \ \ \ \text{for all} \ a< s \le t < b.$$
(ii) For each $u \in \mathscr{S}$ the local slope of $\phi$ at $u$ is defined by
$$|\partial \phi|_{\mathcal{D}}(u): = \limsup_{w \to u} \frac{(\phi(u) - \phi(w))^+}{\mathcal{D}(u,w)}.$$
(iii) An absolutely continuous curve $u: (a,b) \to \mathscr{S}$ is called a curve of maximal slope for $\phi$ with respect to the strong upper gradient $g$ if for a.e. $t \in (a,b)$
$$\frac{\rm d}{ {\rm d} t} \phi(u(t)) \le - \frac{1}{2}|u'|^2_{\mathcal{D}}(t) - \frac{1}{2}g^2(u(t)).$$ \end{definition}
We now introduce minimizing movements. In the following we will use an approximation scheme solving suitable time-incremental minimization problems: Consider a fixed time step $\tau >0$ and suppose that an initial datum $U^0_\tau$ is given. Whenever, $U_\tau^0, \ldots, U^{n-1}_\tau$ are known, $U^n_\tau$ is defined as (if existent) \begin{align}\label{incremental} U_\tau^n = {\rm argmin}_{v \in \mathcal{S}} \Phi(\tau,U^{n-1}_\tau; v), \ \ \ \Phi(\tau,u; v):= \frac{1}{2\tau} \mathcal{D}(v,u)^2 + \phi(v). \end{align} Supposing that for a choice of $\tau$ a sequence $(U_\tau^n)_{n \in \Bbb N}$ solving \eqref{incremental} exists, we define the piecewise constant interpolation by \begin{align}\label{ds} \color{black} \tilde{U}_\tau(0) = U^0_\tau, \ \ \ \tilde{U}_\tau(t) = U^n_\tau \ \text{for} \ t \in ( (n-1)\tau,n\tau], \ n\ge 1. \color{black} \end{align} In the following, \color{black} $\tilde{U}_\tau$ \color{black} will be called a \emph{discrete solution}. Note that the existence of discrete solutions is usually guaranteed by the direct method of the calculus of variations under suitable compactness, coercivity, and lower semicontinuity assumptions. Finally, we introduce the \emph{modulus of the derivative} \begin{align*}
|{\tilde U'_{\tau}}|_{\mathcal{D}}(t) = \frac{\mathcal{D}(U_{\tau}^n, U_{\tau}^{n-1})}{\tau} \ \text{ for } t \in ( (n-1)\tau, n\tau], \ n\ge 1. \end{align*}
\begin{definition}[Minimizing movements]\label{main def1} (i) We say a curve $u: [0,\infty) \to \mathscr{S}$ is a minimizing movement for $\Phi$ as defined in \eqref{incremental}, starting from the initial datum $u_0 \in \mathscr{S}$, if for every sequence of timesteps $(\tau_k)_k$ with $\tau_k \to 0$ there exist discrete solutions defined in \eqref{ds} such that \begin{align}\label{MM} \begin{split} &\lim_{k\to \infty} \phi(U^0_{\tau_k}) = \phi(u_0), \ \ \ \ \limsup\nolimits_{k \to \infty} \mathcal{D}(U^0_{\tau_k},u_0) < \infty, \\& \lim_{k\to \infty} \color{black} \mathcal{D}(\tilde{U}_{\tau_k}(t),u(t)) = 0 \ \ \ \forall t \in [0,\infty). \color{black} \end{split} \end{align} By $MM(\Phi;u_0)$ we denote the collection of all minimizing movements for $\Phi$ starting from $u_0$.
(ii) Likewise, we say a curve $u: [0,\infty) \to \mathscr{S}$ is a generalized minimizing movement for $\Phi$ starting from $u_0 \in \mathscr{S}$ if there exists a sequence of timesteps $(\tau_k)_k$ with $\tau_k \to 0$ and corresponding discrete solutions such that \eqref{MM} holds. The collection of all such curves is denoted by $GMM(\Phi;u_0)$. \end{definition}
\subsection{Compactness of discrete solutions and convergence to curves of maximal slope}\label{sec: AGS-results}
Suppose again that $(\mathscr{S},\mathcal{D})$ is a complete metric space. As discussed in \cite[Remark 2.0.5]{AGS}, it is convenient to introduce a weaker topology on $\mathscr{S}$ to have more flexibility in the derivation of compactness properties. Assume that there is a Hausdorff topology $\sigma$ on $\mathscr{S}$, which is compatible with $\mathcal{D}$ in the sense that $\sigma$ is weaker than the topology induced by $\mathcal{D}$ and satisfies \begin{align}\label{compatibility2} u_n \stackrel{\sigma}{\to} u, \ \ v_n \stackrel{\sigma}{\to} v \ \ \ \Rightarrow \ \ \ \liminf_{n \to \infty} \mathcal{D}(u_n,v_n) \ge \mathcal{D}(u,v). \end{align} Consider a functional $\phi: \mathscr{S} \to [0,+\infty)$ with the following properties: \begin{align}\label{basic assumptions} \begin{split} (i) & \ \ \text{$u_n \stackrel{\sigma}{\to} u$, \ \ $\sup\nolimits_{n,m}\mathcal{D}(u_n,u_m)< \infty \ \ \Rightarrow \ \ \liminf_{n \to \infty}\phi(u_n) \ge \phi(u)$,} \\ (ii)& \ \ \text{for all $N \in\Bbb N$ there is a $\sigma$-sequentially compact set $K_N$ such that} \\ & \ \ \text{$\lbrace u \in \mathscr{S}: \phi(u) + \mathcal{D}(u,u_*) \le N \rbrace \subset K_N$ for some point $u_* \in \mathscr{S}$.} \end{split} \end{align} Note that nonnegativity of $\phi$ can be generalized to a suitable \emph{coerciveness} condition, see \cite[(2.1.2b)]{AGS}, which we do not include here for the sake of simplicity. From \cite[Proposition 2.2.3, Theorem 2.3.3, Remark 2.3.4(i)]{AGS} we obtain the following compactness and convergence result.
\begin{theorem}\label{th: auxiliary1}
Suppose that $\phi$ satisfies \eqref{basic assumptions} and $v \in \mathscr{S} \mapsto |\partial \phi|_{\mathcal D}(v)$ is a strong upper gradient for $\phi$ and $\sigma$-lower semicontinuous. Then the following holds:
(i) Suppose that there is a sequence of initial data $(U^0_{\tau_k})_{k \in \Bbb N}$ and $u_0 \in \mathscr{S}$ with $\sup_k \mathcal{D}(U^0_{\tau_k},u_0)<+\infty$, $U^0_{\tau_k} \stackrel{\sigma}{\to} u_0$, and $\phi(U^0_{\tau_k}) \to \phi(u_0)$. Then there is an absolutely continuous curve $u:[0,\infty) \to \mathscr{S}$ and a subsequence, \color{black} not relabeled, \color{black} of $(\tau_k)_{k \in \Bbb N}$ such that a sequence of discrete solutions \color{black} $(\tilde{U}_{\tau_k})_{k \in \Bbb N}$ \color{black} defined in \eqref{ds} satisfies \color{black} $\tilde{U}_{\tau_k}(t) \stackrel{\sigma}{\to} u(t)$ for all $t \in [0,\infty)$.\color{black}
(ii) Every $u \in GMM(\Phi;u_0)$ for each $u_0 \in \mathscr{S}$ is a curve of maximal slope for $\phi$ with respect to $|\partial \phi|_{\mathcal{D}}$ and in particular $u$ satisfies the energy identity \begin{align}\label{maximalslope}
\frac{1}{2} \int_0^T |u'|_{\mathcal{D}}^2(t) \, dt + \frac{1}{2} \int_0^T |\partial \phi|_{\mathcal{D}}^2(u(t)) \, dt + \phi(u(T)) = \phi(u_0) \ \ \forall T>0. \end{align} Moreover, for a sequence of discrete solutions \color{black} $(\tilde{U}_{\tau_k})_{k \in \Bbb N}$ \color{black} as in (i) we have \begin{align*} \begin{split} &\lim_{k \to \infty} \color{black} \phi(\tilde{U}_{\tau_k}(t)) = \phi(u(t))\color{black} \ \ \ \forall t \in [0,\infty),\\
& \lim_{k \to \infty} |\partial \phi|_{\mathcal{D}}({\tilde U_{\tau_k}}) = |\partial \phi|_{\mathcal{D}}(u)\ \ \text{in} \ \ L^2_{\rm loc}([0,\infty)),\\
& \lim_{k \to \infty} |{\tilde U'_{\tau_k}}|_{\mathcal{D}} = |u'|_{\mathcal{D}} \ \ \text{in} \ \ L^2_{\rm loc}([0,\infty)). \end{split} \end{align*} \end{theorem}
In particular, Theorem \ref{th: auxiliary1}(i) states that the limit $u$ is a generalized minimizing movement, provided that $\sigma$ coincides with the topology induced by $\mathcal{D}$. We remark that $GMM(\Phi;u_0)$ could also be defined with respect to the weaker topology $\sigma$, see \cite[Definition 2.0.6]{AGS}. For our purposes, however, a definition in terms of $\mathcal{D}$ is more convenient.
The result can be considerably improved if $\Phi$ satisfies suitable convexity properties (see \cite[Theorem 4.0.4 and Theorem 4.0.7]{AGS}).
\begin{theorem}\label{th: auxiliary2} Suppose that $\phi$ is $\mathcal{D}$-lower semicontinuous and $\phi \ge 0$. Moreover, assume that for all $\tau>0$ and for all $w,v_0,v_1 \in \mathscr{S}$ there exists a curve $(\gamma_t)_{t \in [0,1]} \subset \mathscr{S}$ with $\gamma_0 = v_0$ and $\gamma_1 = v_1$ such that $$\Phi(\tau,w;\gamma_t) \le (1-t)\Phi(\tau,w; v_0 ) + t\Phi(\tau,w; v_1 ) - \frac{t(1-t)}{2\tau} \mathcal{D}(v_0,v_1)^2 \ \ \ \forall t \in [0,1].$$ Then for each $u_0 \in \mathscr{S}$ there exists a unique $u \in MM(\Phi;u_0)$. Moreover, the assertion of Theorem \ref{th: auxiliary1} (with $\sigma$ being the topology induced by $\mathcal{D}$) holds and for a discrete solution $\tilde{U}_{\tau}$ with $U^0_\tau = u_0$ we have
$\mathcal{D}(\tilde{U}_\tau(t),u(t))^2 \le \frac{1}{2}\tau^2|\partial \phi|_{\mathcal{D}}^2(u_0)$ for all $t>0$. \end{theorem} Note that in contrast to Theorem \ref{th: auxiliary1}, Theorem \ref{th: auxiliary2} yields also a uniqueness result for minimizing movements. Observe that \eqref{basic assumptions}(ii) is not necessary for Theorem \ref{th: auxiliary2} since the solvability of the problem ${\rm argmin}_{v \in \mathscr{S}} \Phi(\tau,u; v)$ for $\tau>0$ and $u \in \mathscr{S}$ (cf. \eqref{incremental}) follows from a convexity argument. In this setting, much more refined results can be established and we refer to \cite[Section 4]{AGS} for more details.
\subsection{Limits of curves of maximal slopes}\label{sec: auxi-proofs}
We now consider a set $\mathscr{S}$ and a sequence of metrics $(\mathcal{D}_n)_n$ on $\mathscr{S}$ as well as a limiting metric $\mathcal{D}$. We again assume that all metric spaces are complete. Moreover, let $(\phi_n)_n$ be a sequence of functionals with $\phi_n: \mathscr{S} \to [0,\infty]$. Suppose that there is a Hausdorff topology $\sigma$ on $\mathscr{S}$ which is weaker than the topology induced by each $\mathcal{D}_n,\mathcal{D}$ and satisfies similarly to \eqref{compatibility2} \begin{align}\label{compatibility} \begin{split} u_n \stackrel{\sigma}{\to} u, &\ \ v_n \stackrel{\sigma}{\to} v \ \ \ \Rightarrow \ \ \ \liminf_{n \to \infty} \mathcal{D}_n(u_n,v_n) \ge \mathcal{D}(u,v). \end{split} \end{align} Moreover, assume that $(\phi_n)_n $ satisfy \eqref{basic assumptions}(ii), i.e., for all $N \in\Bbb N$ there is a $\sigma$-sequentially compact set $K_N$ and $u_* \in \mathscr{S}$ such that for all $n \in \Bbb N$ \begin{align}\label{basic assumptions2} \lbrace u \in \mathscr{S}: \phi_n(u) + \mathcal{D}_n(u,u_*) \le N \rbrace \subset K_N. \end{align} To ensure the existence of limiting curves of maximal slope, we will apply the following refined version of the Arzel\`{a} Ascoli theorem.
\begin{theorem}\label{th: auxiliary3} Let $T>0$, let metrics $\mathcal{D}_n$, $\mathcal{D}$ and functionals $(\phi_n)_n$ be given such that \eqref{compatibility} holds with respect to the topology $\sigma$. Let $K \subset \mathscr{S}$ be a $\sigma$-sequentially compact set. Let $u_n:[0,T]\to \mathscr{S}$ be curves such that \begin{align*} u_n(t) \in K \ \forall n \in \Bbb N, t \in [0,T], \ \ \ \ \limsup_{n \to \infty}\mathcal{D}_n(u_n(s),u_n(t)) \le \omega(s,t) \ \ \ \forall s,t \in [0,T] \end{align*} for a symmetric function $\omega: [0,T]^2 \to [0,\infty)$ with $$\lim_{(s,t) \to (r,r)} \omega(s,t) = 0 \ \ \ \ \forall r\in [0,T] \setminus \mathscr{C}, $$ where $\mathscr{C}$ is an at most countable subset of $[0,T]$. Then there exists a (not relabeled) subsequence and a limiting curve $u:[0,T] \to \mathscr{S}$ such that $$u_n(t) \stackrel{\sigma}{\to} u(t) \ \ \ \forall t\in [0,T], \ \ \ u \text{ is $\mathcal{D}$-continuous in $[0,T] \setminus \mathscr{C}$.} $$
\end{theorem}
\par\noindent{\em Proof. } We follow the proof of \cite[Proposition 3.3.1]{AGS} with the only difference that the lower semicontinuity condition for the metric is replaced by our condition \eqref{compatibility} along the sequence of metrics. \nopagebreak\hspace*{\fill}$\Box$
Now consider also a limiting functional $\phi: \mathscr{S} \to [0,\infty]$. We suppose lower semicontinuity of the functionals and the slopes in the following sense: For all $u \in \mathscr{S}$ and $(u_k)_k \subset \mathscr{S}$ we have \begin{align}\label{eq: implication} \begin{split}
u_k \stackrel{\sigma}{\to} u \ \ \ \Rightarrow \ \ \ \liminf_{k \to \infty} |\partial \phi_{k}|_{\mathcal{D}_{k}} (u_{k}) \ge |\partial \phi|_{\mathcal{D}} (u), \ \ \ \liminf_{k \to \infty} \phi_{k}(u_{k}) \ge \phi(u). \end{split} \end{align}
We now obtain the following result about limits of curves of maximal slope.
\begin{theorem}\label{th:abstract convergence 1}
Consider a set $\mathscr{S}$, metrics $(\mathcal{D}_n)_{n \in \Bbb N}$ and functionals $\phi_n: \mathscr{S} \to [0,\infty]$, $n \in \Bbb N$, as well as $\mathcal{D}$ and $\phi: \mathscr{S}\to [0,\infty]$. Suppose that there is a weaker topology $\sigma$ on $\mathscr{S}$ such that \eqref{compatibility}, \eqref{basic assumptions2}, and the implication \eqref{eq: implication} hold. Moreover, assume that $|\partial \phi_n|_{\mathcal{D}_n}$, $|\partial \phi|_{\mathcal{D}}$ are strong upper gradients for $\phi_n$, $\phi$ with respect to $\mathcal{D}_n$, $\mathcal{D}$, respectively.
Let $T>0$ and $\bar{u} \in \mathscr{S}$. For all $n \in \Bbb N$ let $u_n$ be a curve of maximal slope for $\phi_n$ with respect to $|\partial \phi_n|_{\mathcal{D}_n}$ such that \begin{align}\label{eq: abstract assumptions1} \begin{split} (i)& \ \ \sup_{n \in \Bbb N} \sup_{t \in [0,T]} \big( \phi_n(u_n(t)) + \mathcal{D}_n(u_n(t),\bar{u}) \big) < \infty, \\ (ii)& \ \ u_n(0) \stackrel{\sigma}{\to}\bar{u}, \ \ \ \phi_n(u_n(0)) \to \phi(\bar{u}). \end{split} \end{align} Then there exists a limiting function $u: [0,T] \to \mathscr{S}$ such that up to a subsequence, \color{black} not relabeled, \color{black} $$u_n(t) \stackrel{\sigma}{\to} u(t), \ \ \ \ \ \phi_n(u_n(t)) \to \phi(u(t)) \ \ \ \forall t \in [0,T]$$
as $n \to \infty$ and $u$ is a curve of maximal slope for $\phi$ with respect to $|\partial \phi|_{\mathcal{D}}$. \end{theorem}
The result is an adaption of a statement in \cite{S2} where condition \eqref{compatibility} is replaced by a lower bound condition on the metric derivatives along the sequence. We also refer to \cite{CG}, where a similar result is proved without the assumption that the slopes are \emph{strong} upper gradients (cf. \cite[Definition 1.2.1 and Definition 1.2.2]{AGS} for the definition of strong and weak upper gradients), which comes at the expense that a suitable continuity condition along $(\phi_k)_k$ for sequences $(u_k)_k$ converging with respect to the metric has to be imposed.
\par\noindent{\em Proof. } From the properties of a curve of maximal slope we have (cf. \eqref{maximalslope}) \begin{align}\label{abstract1}
\frac{1}{2} \int_0^t |u'_n|_{\mathcal{D}_n}^2(s) \, ds + \frac{1}{2} \int_0^t |\partial \phi_n|_{\mathcal{D}_n}^2(u_n(s)) \, ds + \phi_n(u_n(t)) = \phi_n(u_n(0)) \end{align}
for all $t \in [0,T]$. (Here, we have used that $|\partial \phi_n|_{\mathcal{D}_n}$ are strong upper gradients for $\phi_n$ with respect to $\mathcal{D}_n$.) From \eqref{abstract1} and the equiboundedness of $\phi_n(u_n(t))$ (see \eqref{eq: abstract assumptions1}(i)) we get \begin{align*}
\sup_{n \in \Bbb N} \int_0^T|u'_n|_{\mathcal{D}_n}^2(t) \, dt + \sup_{n \in \Bbb N} \int_0^T |\partial \phi_n|_{\mathcal{D}_n}^2(u_n(t)) \, dt < \infty. \end{align*}
Consequently, there is a function $A \in L^2((0,T))$ such that $|u_n'|_{{\mathcal{D}}_{n}} \rightharpoonup A$ weakly in $L^2((0,T))$ up to a subsequence, \color{black} not relabeled. \color{black} In particular, this yields \begin{align}\label{abstract2}
\limsup_{n \to \infty} {\mathcal{D}}_n(u_n(s),u_n(t)) \le \limsup_{n \to \infty}\int_s^t|u_n'|_{\mathcal{D}_n}\le \omega(s,t):=\int_s^t A(r)\, dr \end{align}
for all $0 \le s \le t \le T$ by \eqref{metric-deriv}. Using \eqref{basic assumptions2}, \eqref{eq: abstract assumptions1}(i), and \eqref{abstract2}, we can apply Theorem \ref{th: auxiliary3} and obtain an absolutely continuous curve $u: [0,T] \to \mathscr{S}$ as well as a further subsequence \color{black} (not relabeled) \color{black} such that $u_n(t) \stackrel{\sigma}{\to} u(t)$ for all $t \in [0,T] $. Moreover, recalling \eqref{compatibility} we get $\mathcal{D}(u(s),u(t)) \le \int_s^t A(r)\,dr$, which gives $|u'| \le A$. By \eqref{eq: implication} we get \begin{align*} \begin{split}
|\partial \phi|_{\mathcal{D}} (u(t)) \le \liminf_{n \to \infty} |\partial \phi_n|_{\mathcal{D}_n} (u_n(t)),\ \ \
\phi(u(t)) \le \liminf_{n \to \infty} \phi_n(u_n(t)) \end{split} \end{align*}
for $t\in [0,T]$. This together with the fact that $|u_n'|_{{\mathcal{D}}_{n}} \rightharpoonup A$ weakly in $L^2((0,T))$ and $|u'| \le A$ gives \begin{align*} \begin{split}
&\frac{1}{2} \int_0^t |u'|_{\mathcal{D}}^2(s) \, ds + \frac{1}{2} \int_0^t |\partial \phi|_{\mathcal{D}}^2(u(s)) \, ds + \phi(u(t)) \\
&\le \frac{1}{2} \int_0^t A^2(s) \, ds + \frac{1}{2} \int_0^t \liminf_{n \to \infty} |\partial \phi_n|_{\mathcal{D}_n}^2(u_n(s)) \, ds + \liminf_{n \to \infty} \phi_n(u_n(t)) \\
&\le \liminf_{n \to \infty} \Big( \frac{1}{2} \int_0^t |u'_n|_{\mathcal{D}_n}^2(s) \, ds + \frac{1}{2} \int_0^t |\partial \phi_n|_{\mathcal{D}_n}^2(u_n(s)) \, ds + \phi_n(u_n(t))\Big) \end{split} \end{align*}
for all $t \in [0,T]$, where in the second step we used Fatou's lemma. Using \eqref{eq: abstract assumptions1}(ii), \eqref{abstract1}, and $\bar{u} = u(0)$ we get
$$\frac{1}{2} \int_0^t |u'|_{\mathcal{D}}^2(s) \, ds + \frac{1}{2} \int_0^t |\partial \phi|_{\mathcal{D}}^2(u(s)) \, ds + \phi(u(t)) \le \liminf_{n \to \infty} \phi_n(u_n(0)) = \phi(u(0)).$$
On the other hand, as $|\partial \phi|_{\mathcal{D}}$ is a strong upper gradient for $\phi$ with respect to $\mathcal{D}$, we obtain (recall Definition \ref{main def2}) \begin{align*}
\phi(u(0)) \le \phi(u(t)) + \int_0^t |\partial \phi|_{\mathcal{D}}(u(s))|u'|_{\mathcal{D}}(s) \,ds. \end{align*}
Therefore, combining the previous estimates and using Young's inequality we derive $$
|u'|_{\mathcal{D}}(t) = |\partial \phi|_{\mathcal{D}}(u(t)), \ \ \ \phi(u(0))- \phi(u(t)) = \int_0^t |\partial \phi|_{\mathcal{D}}(u(s))|u'|_{\mathcal{D}}(s) \,ds $$ for a.e. $t\in[0,T]$ and $\lim_{n \to \infty}\phi_n(u_n(t)) = \phi(u(t))$ for all $t \in [0,T]$. It follows that $u$ is absolutely continuous and for a.e. $t \in [0,T]$ we have \begin{align*}
\frac{\rm d}{ {\rm d} t} \phi(u(t)) = - |\partial \phi|_{\mathcal{D}}(u(t))|u'|_{\mathcal{D}}(t). \end{align*}
This concludes the proof. \nopagebreak\hspace*{\fill}$\Box$
We now study discrete solutions along the sequence of functionals $(\phi_n)_n$.
\begin{theorem}\label{th:abstract convergence 2}
Consider a set $\mathscr{S}$, metrics $(\mathcal{D}_n)_{n \in \Bbb N}$ and functionals $\phi_n: \mathscr{S} \to [0,\infty)$, $n \in \Bbb N$, as well as $\mathcal{D}$ and $\phi: \mathscr{S}\to [0,\infty)$. Suppose that there is a weaker topology $\sigma$ on $\mathscr{S}$ such that \eqref{compatibility}, \eqref{basic assumptions2} and the implication \eqref{eq: implication} hold. Moreover, assume that $|\partial \phi|_{\mathcal{D}}$ is a strong upper gradient for $ \phi $ with respect to $\mathcal{D}$.
Let $T>0$. Consider a null sequence $(\tau_k)_k$ and initial data $(U^0_{\tau_k})_k$, $\bar{u}$ with \begin{align*} \sup\nolimits_k \mathcal{D}_k(U^0_{\tau_k},\bar{u}) < + \infty, \ \ \ \ \ U^0_{\tau_k} \stackrel{\sigma}{\to} \bar{u} , \ \ \ \ \ \phi_k(U^0_{\tau_k}) \to \phi(\bar{u}). \end{align*}
Then for each sequence of discrete solutions $(\tilde{U}_{\tau_k})_k$ starting from $(U^0_{\tau_k})_k$ there is a curve $u$ of maximal slope for $\phi$ with respect to $|\partial \phi|_\mathcal{D}$ such that up to a subsequence, \color{black} not relabeled, \color{black} $\tilde{U}_{\tau_k}(t) \stackrel{\sigma}{\to} u(t)$ and $\phi_k(\tilde{U}_{\tau_k}(t)) \to \phi(u(t))$ for $t \in [0,T]$.
\end{theorem}
\color{black} For the proof we refer to \cite[Section 2]{Ortner}. Let us also mention the recently obtained variant \cite{BCGS} where, similarly to \cite{CG}, the lower semicontinuity along the sequence $(\phi_n)_n$ (see \eqref{eq: implication}) is replaced by a continuity condition. Note that in their setting it is not necessary to require that $|\partial \phi|_{\mathcal{D}}$ is a strong upper gradient. \color{black}
\section{Properties of energies and dissipation distances}\label{sec:energy-dissipation}
In this section we prove several properties about the energies and dissipation distances. Let $\delta>0$, $0 < \alpha < 1$ and recall the definition of the nonlinear energy in \eqref{nonlinear energy}-\eqref{assumptions-P} as well as \eqref{eq: assumptions-D}. We \color{black} recall that \color{black} $\mathscr{S}_\delta^M = \lbrace y \in W^{2,p}_\mathbf{id}(\Omega): \phi_\delta(y) \le M \rbrace$. In the whole section, $C\ge 1$ and $ 0 < c \le 1$ indicate generic constants, which may vary from line to line and depend on $M$, $\Omega$, the exponent $p>d$ (see \eqref{assumptions-P}), and on the constants in \eqref{assumptions-W}, \eqref{assumptions-P}, \eqref{eq: assumptions-D}, but are always independent of the small parameter $\delta$.
\subsection{Basic properties}
We start with some properties about the Hessian of $W$ and $D$. By $\partial^2 D^2$ we denote the Hessian and by $\partial^2_{F_1^2} D^2, \partial^2_{F_2^2} D^2$ the Hessian in direction of the first or second entry of $D^2$, respectively. Moreover, we define ${\rm sym }(F) = \frac{F + F^\top}{2}$ for $F \in \Bbb R^{d \times d}$ and recall the definition of $\Bbb C_W,\Bbb C_D$ in \eqref{linear equation}. By $\mathbf{Id} \in \Bbb R^{d \times d}$ we again denote the identity matrix.
\begin{lemma}[Properties of Hessian]\label{D-lin} Let $F_1,F_2 \in \Bbb R^{d \times d}$ and $Y \in \Bbb R^{d \times d}$ in a neighborhood of $\mathbf{Id}$ such that $\partial^2 D^2(Y,Y)$ exists.
(i) We have $\partial^2 D^2(Y,Y)[(F_1,F_2),(F_1,F_2)] = \partial^2_{F_1^2}D^2(Y,Y)[F_1-F_2,F_1-F_2] = \partial^2_{F_2^2}D^2(Y,Y)[F_1-F_2,F_1-F_2]$.
(ii) We have $\partial^2 D^2(\mathbf{Id},\mathbf{Id})[(F_1,F_2),(F_1,F_2)] = \Bbb C_D[{\rm sym}(F_1-F_2), {\rm sym}(F_1-F_2)]$.
(iii) There is a constant $c>0$ independent of $F$ such that $\Bbb C_W[F,F] \ge c|{\rm sym}(F)|^2$, $\Bbb C_D[F,F] \ge c|{\rm sym}(F)|^2$. \end{lemma}
\par\noindent{\em Proof. } (i) Set $H= \partial^2 D^2(Y,Y)$ for brevity. By symmetry \eqref{eq: assumptions-D}(ii) we find two fourth order tensors $H_1,H_2 : \Bbb R^{d\times d} \times \Bbb R^{d\times d} \to \Bbb R$ such that $H[(F_1,F_2),(F_1,F_2)] = H_1[F_1,F_1] + 2H_2[F_1,F_2] + H_1[F_2,F_2]$ and $H_2[F_1,F_2] = H_2[F_2,F_1]$. Note that $H_1 = \partial^2_{F_1^2}D^2(Y,Y) =\partial^2_{F_2^2} D^2(Y,Y)$. As $D(F,F)=0$ for all $F \in GL_+(d)$, we get $H[(F,F),(F,F)] = 0$ for all $F \in \Bbb R^{d \times d}$. Thus, we obtain $H_1[F,F] = -H_2[F,F]$ for all $F \in \Bbb R^{d \times d}$ and we compute \begin{align*} H_1[F_1-F_2,&F_1-F_2]\\& = - H_2[F_1-F_2,F_1-F_2] = -H_2[F_1,F_1] + 2H_2[F_1,F_2] - H_2[F_2,F_2] \\ &= H_1[F_1,F_1] + 2H_2[F_1,F_2] + H_1[F_2,F_2] = H[(F_1,F_2),(F_1,F_2)]. \end{align*} Property (ii) follows from frame indifference \eqref{eq: assumptions-D}(v) by an elementary computation. Finally, the growth condition for $\Bbb C_W$ and $\Bbb C_D$ stated in (iii) follow from \eqref{assumptions-W}(iii) and \eqref{eq: assumptions-D}(vi), respectively. \nopagebreak\hspace*{\fill}$\Box$
In the following, by $\mathbf{id}$ we again denote the identity function.
\begin{lemma}[Rigidity]\label{lemma:rigidity} There is constant $C>1$ independent of $\delta$ such that for $\delta$ sufficiently small for all $y \in \mathscr{S}_\delta^M$ we have
\begin{itemize} \item[(i)] $\Vert y - \mathbf{id} \Vert_{H^1(\Omega)} \le C\Vert \operatorname{dist}(\nabla y,SO(d)) \Vert_{L^2(\Omega)}$, \item[(ii)] $\Vert \nabla y -\mathbf{Id} \Vert_{L^\infty(\Omega)}\le C\delta^{\alpha}$, \ \ \ $\Vert y -\mathbf{id} \Vert_{L^\infty(\Omega)}\le C\delta^{\alpha}$. \end{itemize} \end{lemma}
\par\noindent{\em Proof. } (i) is a typical geometric rigidity argument, see e.g. \cite{DalMasoNegriPercivale:02, FrieseckeJamesMueller:02}: By \cite[Theorem 3.1]{FrieseckeJamesMueller:02} and Poincar\'e's inequality we find a rotation $Q \in SO(d)$ and $b \in \Bbb R^d$ such that \begin{align}\label{rig1} \Vert y - (Q\cdot + b) \Vert_{H^1(\Omega)} \le C\Vert \operatorname{dist}(\nabla y,SO(d)) \Vert_{L^2(\Omega)}. \end{align}
Passing to a trace estimate and using $y = \mathbf{id}$ on $\partial \Omega$, we get $\Vert \mathbf{id} - (Q\cdot + b) \Vert_{L^2(\partial \Omega)} \le C\Vert \operatorname{dist}(\nabla y,SO(d)) \Vert_{L^2(\Omega)}$. Using \cite[Lemma 3.3]{DalMasoNegriPercivale:02} we then find $|b| + |Q- \mathbf{Id}| \le C\Vert \mathbf{id} - (Q\cdot + b) \Vert_{L^2(\partial \Omega)}$ for a constant only depending on $\Omega$. This together with \eqref{rig1} implies (i).
We now prove (ii). By the definition of $\phi_\delta$ and \eqref{assumptions-P}(iii) we get $\Vert \nabla^2 y \Vert^p_{L^p(\Omega)} \le CM\delta^{p\alpha}$ for all $y \in \mathscr{S}^M_\delta$. As $p>d$, Poincar\'e's inequality yields some $F \in \Bbb R^{d \times d}$ and $b \in \Bbb R^d$ such that \begin{align}\label{rig2} \Vert y - (F\cdot + b)\Vert_{W^{1,\infty}(\Omega)} \le C\delta^{\alpha} \end{align} for a constant additionally depending on $\Omega$, M, and $p$. Using $\phi_\delta(y) \le M$, \eqref{assumptions-W}(iii), and (i) we compute \begin{align*}
\Vert (F\cdot + b)- \mathbf{id} \Vert^2_{H^1(\Omega)} \le C\Vert \operatorname{dist}(\nabla y,SO(d)) \Vert^2_{L^2(\Omega)} + C|\Omega|\delta^{2\alpha}
\le C\delta^2M + C|\Omega|\delta^{2\alpha} . \end{align*}
Since $\alpha \le 1$, this gives $|b| + |F - \mathbf{Id}| \le C\delta^\alpha$, which together with \eqref{rig2} yields (ii). \nopagebreak\hspace*{\fill}$\Box$
In the following we set for shorthand $H_Y := \frac{1}{2}\partial^2_{F_1^2} D^2(Y,Y) = \frac{1}{2}\partial^2_{F_2^2} D^2(Y,Y)$ for $Y \in GL_+(d)$ and given a deformation $y \in W^{2,p}_\mathbf{id}(\Omega)$ we also introduce the mapping $H_{\nabla y}: \Omega \to \Bbb R^{d \times d \times d \times d}$ by $H_{\nabla y}(x) = H_{\nabla y(x)}$ for $x \in \Omega$. Recall the definition of $\mathcal{D}_\delta, \bar{\mathcal{D}}_0$ in \eqref{eq: D,D0} and $\Bbb C_W$ below \eqref{linear equation}.
\begin{lemma}[Dissipation and energy]\label{lemma: metric space-properties} There are constants $0<c<1$, $C>1$ independent of $\delta$ such that for all $y,y_0,y_1 \in \mathscr{S}_\delta^M$ for $\delta$ sufficiently small we have
\begin{itemize}
\item[(i)] $\big|\delta^2\mathcal{D}_\delta(y_0,y_1)^2 - \int_\Omega H_{\nabla y_0}[\nabla (y_1 - y_0),\nabla (y_1 - y_0) ]| \le C \Vert \nabla (y_1- y_0) \Vert^3_{L^3(\Omega)}$, \item[(ii)] $c \Vert y_1 - y_0 \Vert_{H^1(\Omega)} \le \delta\mathcal{D}_\delta(y_0,y_1) \le C\Vert y_1 - y_0 \Vert_{H^1(\Omega)}$,
\item[(iii)] $\big|\mathcal{D}_\delta(y_0,y_1)^2 - \bar{\mathcal{D}}_0(u_0,u_1)^2\big| \le C\delta^\alpha$,
\item[(iv)] $\big|\delta^{-2} \int_\Omega W(\nabla y) - \int_\Omega \frac{1}{2}\Bbb C_W[e (u),e (u) ]\big| \le C\delta^\alpha,$ \end{itemize} where $u = \delta^{-1}(y - \mathbf{id})$ and $u_i = \delta^{-1}(y_i - \mathbf{id})$, $i=0,1$. In particular, (ii) shows that the topologies induced by $\mathcal{D}_\delta$ and $\Vert \cdot \Vert_{H^1(\Omega)}$ coincide. \end{lemma}
\par\noindent{\em Proof. }
Recall that $D^2$ is $C^3$ in a neighborhood of $(\mathbf{Id},\mathbf{Id})$. In view of the uniform bound on $\nabla y_0, \nabla y_1 $ (see Lemma \ref{lemma:rigidity}(ii)) and a Taylor expansion of $D^2$ at $(\nabla y_0, \nabla y_0)$, we derive by Lemma \ref{D-lin} \begin{align*} \begin{split} \int_\Omega D^2(\nabla y_0,\nabla y_1) &= \int_\Omega H_{\nabla y_0}[\nabla (y_1 - y_0),\nabla (y_1 - y_0) ] + O( \Vert \nabla (y_1- y_0) \Vert^3_{L^3(\Omega)}). \end{split} \end{align*} This gives (i). We obtain $\Vert H_{\nabla y_0} - \Bbb C_D \Vert_{L^\infty(\Omega)} \le C\delta^\alpha$ by regularity of $D$ and Lemma \ref{lemma:rigidity}(ii). This together with (i), Lemma \ref{lemma:rigidity}(ii), and Lemma \ref{D-lin} yields \begin{align}\label{eq:NNN} \begin{split} \int_\Omega D^2(\nabla y_0,\nabla y_1)& = \int_\Omega \Bbb C_D[e(y_1)-e(y_0),e(y_1)-e(y_0)]\\& \ \ \ + O(\delta^\alpha \Vert \nabla y_1- \nabla y_0 \Vert^2_{L^2(\Omega)}). \end{split} \end{align} Now by \eqref{eq:NNN}, Lemma \ref{D-lin}(iii), and Korn's inequality we derive for $\delta$ small enough
\begin{align*} \int_\Omega D^2(\nabla y_0,\nabla y_1) & \ge c \Vert e(y_1)-e(y_0) \Vert^2_{L^2(\Omega)} + O( \delta^\alpha\Vert \nabla y_1- \nabla y_0 \Vert^2_{L^2(\Omega)}) \\& \ge c \Vert \nabla y_1- \nabla y_0 \Vert^2_{L^2(\Omega)}. \end{align*} Here we used that $y_1 - y_0 = 0$ on $\partial \Omega$. The first inequality in (ii) follows from Poincar\'e's inequality. The other inequality can be seen along similar lines. By Lemma \ref{lemma:rigidity}(i), \eqref{assumptions-W}(iii) and the fact that $y_0,y_1 \in \mathscr{S}_\delta^M$ we get \begin{align}\label{eq:remD} \Vert \nabla y_i - \mathbf{Id} \Vert^2_{L^2(\Omega)} \le C\Vert \operatorname{dist}(\nabla y_i,SO(d)) \Vert^2_{L^2(\Omega)} \le C\phi_\delta(y_i) \le CM\delta^2 \end{align} for $i=0,1$. Recalling the definition of $\mathcal{D}_\delta, \bar{\mathcal{D}}_0$, we now obtain (iii) by \eqref{eq:NNN}.
Finally, to see (iv), an argument very similar to (i), essentially relying on a Taylor expansion and Lemma \ref{lemma: metric space-properties}(ii), yields
$$\Big|\delta^{-2} \int_\Omega W(\nabla y) - \int_\Omega \frac{1}{2}\Bbb C_W[e (u),e (u) ]\Big| \le C\delta^{\alpha-2} \Vert \nabla y - \mathbf{Id} \Vert^2_{L^2(\Omega)},$$ which together with \eqref{eq:remD} implies the claim. \nopagebreak\hspace*{\fill}$\Box$
We close this section with proving differentiablity of $\int_\Omega W(\nabla y)$.
\begin{lemma}[Differentiablity of $\int_\Omega W(\nabla y)$]\label{lemma:C1} For $(y_n)_n \subset \mathscr{S}_\delta^M$ and $y \in \mathscr{S}_\delta^M$ with $\mathcal{D}_\delta(y_n,y) \to 0$, we have $$\lim_{n \to \infty} \frac{\int_\Omega W(\nabla y_n) - \int_\Omega W(\nabla y) - \int_\Omega \partial_FW(\nabla y) : (\nabla y_n - \nabla y)}{\mathcal{D}_\delta(y_n,y)} = 0.$$ \end{lemma}
\par\noindent{\em Proof. } By a Taylor expansion we find a universal constant $C'>0$ such that $|W(F_2) - W(F_1) - \partial_F W(F_1) : (F_2 - F_1)| \le C'|F_1 - F_2|^2$ for all $F_1,F_2$ with $|F_1 - \mathbf{Id}|,|F_2-\mathbf{Id}| \le C\delta^\alpha$, where $C$ is the constant in Lemma \ref{lemma:rigidity}(ii). This together with Lemma \ref{lemma:rigidity}(ii) and Lemma \ref{lemma: metric space-properties}(ii) gives the result. \nopagebreak\hspace*{\fill}$\Box$
\subsection{Metric spaces and convexity}\label{sec: metric}
In this section we show that $(\mathscr{S}^M_\delta, \mathcal{D}_\delta)$, $(H^1_0(\Omega),\bar{\mathcal{D}}_0)$ are complete metric spaces and derive convexity properties for the energies and dissipation distances.
\begin{theorem}[Properties of $(\mathscr{S}^M_\delta, \mathcal{D}_\delta)$ and $\phi_\delta$]\label{th: metric space} For $\delta>0$ small enough we have \begin{itemize} \item[(i)] $(\mathscr{S}^M_\delta, \mathcal{D}_\delta)$ is a complete metric space. \item[(ii)] Compactness: If $(y_n)_n \subset \mathscr{S}^M_\delta$, then $(y_n)_n$ admits a subsequence converging weakly in $W^{2,p}(\Omega)$, strongly in $W^{1,\infty}(\Omega)$, and with respect to $\mathcal{D}_\delta$. \item[(iii)] Lower semicontinuity: $\mathcal{D}_\delta(y_n,y) \to 0$ \ \ $\Rightarrow$ \ \ $\liminf_{n \to \infty} \phi_\delta(y_n) \ge \phi_\delta(y)$. \end{itemize} \end{theorem}
\par\noindent{\em Proof. } First, recalling \eqref{nonlinear energy} and \eqref{assumptions-P}(iii), we have $\Vert \nabla^2 y \Vert^p_{L^p(\Omega)} \le CM\delta^{p\alpha}$ for all $y \in \mathscr{S}^M_\delta$, which together with Lemma \ref{lemma:rigidity}(ii) shows $\sup_{y \in \mathscr{S}^M_\delta}\Vert y \Vert_{W^{2,p}(\Omega)} < \infty$. This implies (ii) recalling $p>d$ and also using Lemma \ref{lemma: metric space-properties}(ii). In particular, for a sequence $(y_n)_n$ converging to $y$ with respect to $\mathcal{D}_\delta$ we have $y_n \rightharpoonup y$ weakly in $W^{2,p}(\Omega)$ and $y_n \to y$ strongly in $W^{1,\infty}(\Omega)$. Then (iii) follows from Fatou's lemma and the fact that $\liminf_{n\to \infty} \int_\Omega P(\nabla^2 y_n) \ge \int_\Omega P(\nabla^2 y)$ by \eqref{assumptions-P}(ii).
We now finally show (i). Apart from the positivity, all properties of a metric follow directly from \eqref{eq: assumptions-D} and \eqref{eq: D,D0}. To show that if $\mathcal{D}_\delta(y_0,y_1)=0$ for $y_0, y_1 \in \mathscr{S}^M_\delta$, then $y_0=y_1$, we apply Lemma \ref{lemma: metric space-properties}(ii). Finally, it remains to show that $(\mathscr{S}^M_\delta, \mathcal{D}_\delta)$ is complete. Let $(y_k)_k$ be a Cauchy sequence with respect to $\mathcal{D}_\delta$. By (ii) we find $y \in W^{2,p}(\Omega)$ and a subsequence \color{black} (not relabeled) \color{black} such that $y_k \to y$ in $W^{1,\infty}(\Omega)$. Then also $\lim_{k\to \infty}\mathcal{D}_\delta(y_k,y) = 0$ by Lemma \ref{lemma: metric space-properties}(ii). By (iii) we get $y \in \mathscr{S}_\delta^M$. The fact that $(y_k)_k$ is a Cauchy sequence now implies that the whole sequence $y_k$ converges to $y$ with respect to $\mathcal{D}_\delta$. This concludes the proof. \nopagebreak\hspace*{\fill}$\Box$
Similar properties can be derived in the linear setting. Recall the definition of $\bar{\mathcal{D}}_0$ in \eqref{eq: D,D0}.
\begin{theorem}[Properties of $(H^1_0(\Omega), \bar{\mathcal{D}}_0)$ and $\bar{\phi}_0$]\label{th: metric space-lin} We have
\begin{itemize} \item[(i)] $(H^1_0(\Omega), \bar{\mathcal{D}}_0)$ is a complete metric space. \item[(ii)] Continuity: $\bar{\mathcal{D}}_0(u_n, u) \to 0$ \ \ $\Rightarrow$ \ \ $\lim_{n \to \infty} \bar{\phi}_0(u_n) = \bar{\phi}_0(u)$. \end{itemize} \end{theorem}
\par\noindent{\em Proof. } By Lemma \ref{D-lin}(iii) we find a constant $c>0$ such that $$\bar{\mathcal{D}}_0(u_0,u_1)^2 \ge c \Vert e(u_0) - e(u_1) \Vert_{L^2(\Omega)}^2 \ge \Vert u_0 - u_1 \Vert^2_{H^1(\Omega)}, $$ where the last step follows from Korn's and Poincare's inequality. This show that $(H^1_0(\Omega), \bar{\mathcal{D}}_0)$ is a complete metric space, where $\bar{\mathcal{D}}_0$ is equivalent to the metric induced by $\Vert \cdot \Vert_{H^1(\Omega)}$. Recalling \eqref{linear energy} we find that $\bar{\phi}_0$ is continuous with respect to $\bar{\mathcal{D}}_0$. \nopagebreak\hspace*{\fill}$\Box$
The following properties are crucial to use the theory in \cite{AGS}.
\begin{theorem}[Convexity and generalized geodesics in the nonlinear setting]\label{th: convexity} There is a constant $C \ge 1$ independent of $\delta$ such that for $\delta$ small and for all $y_0,y_1 \in \mathscr{S}^M_\delta$: \begin{align*} (i)& \ \ \mathcal{D}_\delta(y_s,y_0)^2 \le s^2\mathcal{D}_\delta(y_1,y_0)^2 (1 + C \Vert \nabla y_1 - \nabla y_0 \Vert_{L^\infty(\Omega)}), \\ (ii) & \ \ \phi_\delta(y_s) \le (1-s) \phi_\delta(y_0) + s\phi_\delta(y_1), \end{align*} where $y_s := (1-s) y_0 + sy_1$, $s \in [0,1]$.
\end{theorem}
Note that $y_s$ is not a geodesic in the sense of \cite[Definition 2.4.2]{AGS}, but $y_s$ can be understood as a generalized geodesic. We also refer to \cite[Section 3.2, Section 3.4]{MOS} for a discussion about generalized geodesics in a related setting.
\par\noindent{\em Proof. } Let $y_s = (1-s)y_0 + sy_1$. By Lemma \ref{lemma: metric space-properties}(i) we obtain \begin{align*}
\delta^2\mathcal{D}_\delta(y_1,y_0)^2 &\ge \int_\Omega H_{\nabla y_0}[\nabla (y_1-y_0),\nabla (y_1-y_0)] - C\int_\Omega|\nabla y_1 - \nabla y_0|^3. \end{align*} Likewise, we get
\begin{align*}
\delta^2\mathcal{D}_\delta(y_s,y_0)^2 & \le s^2\int_\Omega H_{\nabla y_0}[\nabla (y_1-y_0),\nabla (y_1-y_0)]+ Cs^3\int_\Omega|\nabla y_1 - \nabla y_0|^3. \end{align*} Combining the two estimates, we therefore obtain
\begin{align*} \mathcal{D}_\delta(y_s,y_0)^2 & \le s^2\big( \mathcal{D}_\delta(y_1,y_0)^2 + C\delta^{-2}\Vert \nabla y_1 - \nabla y_0 \Vert_{L^3(\Omega)}^3\big), \end{align*} which together with Lemma \ref{lemma: metric space-properties}(ii) shows (i). To see (ii), it suffices to show $\int_\Omega W(\nabla y_s) \le (1-s)\int_\Omega W(\nabla y_0) + s \int_\Omega W(\nabla y_1)$ since $P$ is convex (see \eqref{assumptions-P}(ii)). A Taylor expansion gives $\int_\Omega W(\nabla y) = \frac{1}{2}\int_\Omega \Bbb C_W[\nabla y,\nabla y] + \omega(\nabla y)$ for a (regular) function $\omega: \Bbb R^{d \times d} \to \Bbb R$ with $\partial_F \omega(0) = 0$ and $\partial^2_{F^2} \omega(0) =0$. We get \begin{align}\label{quadratic convexity} \begin{split}
\int_\Omega \Bbb C_W[\nabla y_s,\nabla y_s] &= (1-s) \int_\Omega \Bbb C_W[\nabla y_0,\nabla y_0] + s \int_\Omega \Bbb C_W[\nabla y_1,\nabla y_1] \\ & \ \ \ - s(1-s) \int_\Omega \Bbb C_W[\nabla (y_1 - y_0),\nabla (y_1 - y_0)].
\end{split}
\end{align}
Denote by $B_{2C\delta^\alpha}(\mathbf{Id}) \subset \Bbb R^{d \times d}$ the ball with center $\mathbf{Id}$ and radius $2C\delta^\alpha$ with the constant $C$ from Lemma \ref{lemma:rigidity}(ii). Since $F \mapsto \omega(F) + \frac{1}{2}\Vert \partial^2_{F^2} \omega \Vert_{L^\infty(B_{2C\delta^\alpha}(\mathbf{Id}))}|F|^2$ is convex on $B_{2C\delta^\alpha}(\mathbf{Id})$, we get by Lemma \ref{lemma:rigidity}(ii)
\begin{align*}
\int_\Omega \omega(\nabla y_s ) & \le s\int_\Omega \omega (\nabla y_0) + (1-s) \int_\Omega \omega(\nabla y_1 ) \\& \ \ \ + \frac{1}{2}s(1-s) \Vert \partial^2_{F^2} \omega\Vert_{L^\infty(B_{2C\delta^\alpha}(\mathbf{Id}))} \int_\Omega|\nabla y_1 - \nabla y_0|^2.\notag \end{align*} By the fact that $\partial^2_{F^2 }w(0) =0$ and the regularity of $\omega$ we find $\Vert \partial^2_{F^2 } \omega \Vert_{L^\infty(B_{2C\delta^\alpha}(\mathbf{Id}))} \le C\delta^\alpha$. Combining the previous three estimates and recalling that $\int_\Omega W(\nabla y) = \frac{1}{2}\int_\Omega \Bbb C_W[\nabla y,\nabla y] + \omega(\nabla y)$, we conclude
\begin{align*}
&\int_\Omega W(\nabla y_s) - (1-s)\int_\Omega W(\nabla y_0) - s\int_\Omega W(\nabla y_1) \\ & \le - s(1-s) \int_\Omega \Bbb C_W[\nabla (y_1 - y_0),\nabla (y_1 - y_0)] +\frac{1}{2}s(1-s) C\delta^\alpha \int_\Omega|\nabla (y_1 - y_0)|^2 \le 0 \end{align*} for $\delta$ small enough, where the last step follows from Lemma \ref{D-lin}(iii) and Korn's inequality. \nopagebreak\hspace*{\fill}$\Box$
We note without proof that by a similar reasoning as in (ii) one can show that for given $w \in \mathscr{S}^M_\delta$ $$\mathcal{D}_\delta(y_s,w)^2 \le (1-s)\mathcal{D}_\delta(y_0,w)^2 + s\mathcal{D}_\delta(y_1,w)^2 - s(1-s)(1 - C\delta^\alpha)\mathcal{D}_\delta(y_1,y_0)^2.$$ This implies that $\mathcal{D}_\delta$ is $2(1-C\delta^\alpha)$-convex in the sense of \cite[Assumption 4.0.1]{AGS}. Note that this property is not strong enough to apply directly the results in \cite[Section 2.4, Section 4]{AGS}. \color{black} Nevertheless, we will be able to derive representations and lower semicontinuity properties for the slopes by direct computations (see Lemma \ref{lemma: slopes}, Lemma \ref{lemma: lsc-slope} below.) \color{black} However, in the linear setting we obtain $2$-convexity as the following result shows.
\begin{lemma}[Convexity in the linear setting]\label{lemma: convexity2} For all $u_0,u_1 \in H^1_0(\Omega)$ and $v \in H^1_0(\Omega)$ with $u_s := (1-s) u_0 + su_1$ we have $$\bar{\mathcal{D}}_0(u_s,v)^2 \le (1-s) \bar{\mathcal{D}}_0(u_0,v)^2 + s \bar{\mathcal{D}}_0(u_1,v)^2 - s(1-s) \bar{\mathcal{D}}_0(u_1,u_0)^2.$$ \end{lemma}
\par\noindent{\em Proof. } The property follows from an elementary computation as in \eqref{quadratic convexity} taking into account that $\bar{\mathcal{D}}_0^2$ is quadratic. \nopagebreak\hspace*{\fill}$\Box$
\subsection{Properties of local slopes}\label{sec: slopes}
We now derive representations and properties of the slopes corresponding to $\phi_\delta$ and $\bar{\phi}_0$. Recall Definition \ref{main def2}.
\begin{lemma}[Slopes]\label{lemma: slopes} (i) For $\delta>0$ small enough the local slopes in the nonlinear setting admit the representation \begin{align*}
&|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) = \sup_{w \neq y} \ \frac{(\phi_\delta(y) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y,w) (1 + C \Vert \nabla y - \nabla w \Vert_{L^\infty(\Omega)})^{1/2}} \ \ \ \ \forall y \in \mathscr{S}_\delta^M, \end{align*}
where $C$ is the constant from Theorem \ref{th: convexity}. The slopes are lower semicontinuous with respect to both $H^1(\Omega)$ and $\mathcal{D}_\delta$ and are strong upper gradients for $\phi_\delta$.
(ii) The local slope for the linear energy $\bar{\phi}_0$ admits the representation
$$ |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u) = \sup_{v \neq u} \ \frac{(\bar{\phi}_0(u) - \bar{\phi}_0(v))^+}{\bar{\mathcal{D}}_0(u,v)},$$
and is a strong upper gradient for $\bar{\phi}_0$. \end{lemma}
\par\noindent{\em Proof. } Before we start with the actual proof, let us recall from \cite[Lemma 1.2.5]{AGS} that in a complete metric space $(\mathscr{S,\mathcal{D}})$ with energy $\phi$ one has that $|\partial \phi|_{\mathcal{D}}$ is a weak upper gradient for $\phi$ in the sense of \cite[Definition 1.2.2]{AGS}. We do not repeat the definition of weak upper gradients, but only mention that weak upper gradients are also strong upper gradients if for each absolutely continuous curve $z:(a,b) \to \mathscr{S}$ with $|\partial \phi|_{\mathcal{D}}(z)|z'|_{\mathcal{D}} \in L^1(a,b)$, the function $\phi \circ z$ is absolutely continuous.
Moreover, \cite[Lemma 1.2.5]{AGS} also states that, if $\phi$ is $\mathcal{D}$-lower semicontinuous, then the global slope \begin{align}\label{global slope} \color{black} \mathcal{S}_{\phi}(v) \color{black} := \sup_{w \neq v} \frac{(\phi(v) - \phi(w))^+}{\mathcal{D}(v,w)} \end{align} is a strong (and thus also weak) upper gradient for $\phi$.
We now give the proof of (i). We partially follow the proofs of Theorem 2.4.9 and Corollary 2.4.10 in \cite{AGS}. To confirm the representation of $|\partial \phi_\delta|_{\mathcal{D}_\delta}$, we use the definition of the local slope in Definition \ref{main def2} and obtain with $C$ being the constant from Theorem \ref{th: convexity}(i) \begin{align*}
|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) & = \limsup_{w \to y} \frac{(\phi_\delta(y) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y,w)} = \limsup_{w \to y} \frac{(\phi_\delta(y) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y,w) (1 + C \Vert \nabla y- \nabla w \Vert_{\infty})^{1/2}} \\ &\le \sup_{w \neq y} \ \frac{(\phi_\delta(y) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y,w) (1 + C \Vert \nabla y - \nabla w \Vert_{\infty})^{1/2}}, \end{align*} where in the second \color{black} equality \color{black} we used that $w \to y$ (with respect to $\mathcal{D}_\delta$) implies $\Vert \nabla w - \nabla y\Vert_{L^ \infty(\Omega)} \to 0$ by Theorem \ref{th: metric space}(ii). To see the other inequality, it is not restrictive to suppose that $y \neq w$ and \begin{align}\label{proof2.1} \phi_\delta(y) - \phi_\delta(w)>0. \end{align} By Theorem \ref{th: convexity}(ii) with $y_0 = y$ and $y_1 = w$ we get
\begin{align*} \frac{\phi_\delta(y) - \phi_\delta(y_s)}{\mathcal{D}_\delta(y,y_s)} \ge \frac{\phi_\delta(y) - \phi_\delta(w)}{\mathcal{D}_\delta(y,w)} \frac{s\mathcal{D}_\delta(y,w)}{\mathcal{D}_\delta(y,y_s)}
\end{align*} for all $s \in [0,1]$, where $y_s = (1-s)y + s w$. Then we derive by \eqref{proof2.1} and Theorem \ref{th: convexity}(i) \begin{align*}
|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) \ge \frac{\phi_\delta(y) - \phi_\delta(w)}{\mathcal{D}_\delta(y,w) (1+ C \Vert \nabla y - \nabla w \Vert_\infty)^{1/2}}. \end{align*} The claim now follows by taking the supremum with respect to $w$. To confirm the lower semicontinuity, we consider $y_h \to y$ in $\mathcal{D}_\delta$ or equivalently in $H^1(\Omega)$ (see Lemma \ref{lemma: metric space-properties}(ii)). If $w \neq y$, then $w \neq y_h$ for $h$ large enough and thus \begin{align*}
\liminf_{h \to \infty} |\partial \phi_\delta|_{\mathcal{D}_\delta}(y_h) & \ge \liminf_{h \to \infty} \frac{(\phi_\delta(y_h) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y_h,w) (1 + C\Vert \nabla y_h - \nabla w \Vert_{\infty})^{1/2}} \\ & \ge \frac{(\phi_\delta(y) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y,w) (1 + C\Vert \nabla y - \nabla w \Vert_{\infty})^{1/2}}, \end{align*} where we used Theorem \ref{th: metric space}(ii),(iii). By taking the supremum with respect to $w$ the lower semicontinuity follows.
It remains to show that $|\partial \phi_\delta|_{\mathcal{D}_\delta}$ is a strong upper gradient. With Lemma \ref{lemma:rigidity}(ii), for $\delta$ small enough we find $\mathcal{S}_{\phi_\delta}(y) \le 2 |\partial \phi_\delta|_{\mathcal{D}_\delta}(y)$ with $\mathcal{S}_{\phi_\delta}$ as introduced in \eqref{global slope}. Recalling the remarks at the beginning of the proof, to show that $|\partial \phi_\delta|_{\mathcal{D}_\delta}$ is a strong upper gradient we have to check that for all absolutely continuous $z:(a,b) \to \mathscr{S}^M_\delta$ with $|\partial \phi_\delta|_{\mathcal{D}_\delta}(z)|z'|_{{\mathcal D}_\delta} \in L^1(a,b)$, the function $\phi_\delta \circ z$ is absolutely continuous. First, it follows $\mathcal{S}_{\phi_\delta}(z)|z'|_{{\mathcal D}_\delta} \in L^1(a,b)$ as $\mathcal{S}_{\phi_\delta} \le 2 |\partial \phi_\delta|_{\mathcal{D}_\delta}$. Since $\phi_\delta$ is $\mathcal{D}_\delta$-lower semicontinous, $\mathcal{S}_{\phi_\delta}$ is a strong upper gradient. Thus, we indeed get that $\phi_\delta \circ z$ is absolutely continuous, see Definition \ref{main def2}.
We now concern ourselves with (ii). The representation of the local slope follows from the convexity property in Lemma \ref{lemma: convexity2} as was shown in \cite[Theorem 2.4.9]{AGS}. Therefore, $\mathcal{S}_{\bar{\phi}_0} = |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$, which is $\bar{\mathcal{D}}_0$ lower semicontinous by Lemma \ref{th: metric space-lin}(ii) and thus $|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$ is a strong upper gradient. \nopagebreak\hspace*{\fill}$\Box$
\section{Proof of the main results}\label{sec results}
In this section we give the proof of Theorem \ref{maintheorem1}-Theorem \ref{maintheorem3}.
\subsection{Existence of curves of maximal slope}
In this section we prove the first two parts of Theorem \ref{maintheorem1} and Theorem \ref{maintheorem2}, which essentially follow from the properties of the metric spaces established in Section \ref{sec: metric}, \ref{sec: slopes} by applying the general results recalled in Section \ref{sec: AGS-results}.
\begin{proof}[Proof of Theorem \ref{maintheorem1}(i),(ii)] First, we note that the assumptions of Theorem \ref{th: auxiliary1} are satisfied by Lemma \ref{lemma: slopes}(i) and Lemma \ref{th: metric space}(ii),(iii), where we let $\mathscr{S} = \mathscr{S}_\delta^M$ and let $\sigma$ be the topology induced by $\mathcal{D}_\delta$.
(i) Fix $y_0 \in \mathscr{S}^M_\delta$. Define the initial data $U^0_\tau = y_0$ for all $\tau>0$. Applying Theorem \ref{th: auxiliary1}(i) we find a curve $y$ which is the limit of a sequence of discrete solutions with $y(0) = y_0$. Thus, in view of Definition \ref{main def1}, $y \in GMM(\Phi_{\delta};y_0)$, which is therefore nonempty.
(ii) To see that generalized minimizing movements are curves of maximal slope, it suffices to apply Theorem \ref{th: auxiliary1}(ii). \end{proof}
\begin{proof}[Proof of Theorem \ref{maintheorem2}(i),(ii)]
In the linear setting the convexity property given in Lemma \ref{lemma: convexity2} holds and $\bar{\phi}_0$ is convex by \eqref{linear energy} and Lemma \ref{D-lin}(iii). Thus, Theorem \ref{th: auxiliary2} is applicable. Apart from uniqueness, the result then follows from Theorem \ref{th: auxiliary2}. It remains to show that the unique minimizing movement is also the unique curve of maximal slope for $\bar{\phi}_0$ with respect to the strong upper gradient $|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$. To this end, we follow an idea used, e.g., in \cite{Gigli}.
We first observe that the metric derivative $|u'|^2_{\bar{\mathcal{D}}_0}$ is convex. Indeed, let $u^1,u^2:[0,\infty) \to H^1_0(\Omega)$ be two curves. We get for $u^{3} = \frac{1}{2}(u^1 + u^2)$ by Young's inequality (define $v^i = u^{i}(s) - u^{i}(t)$, $i=1,2$, for brevity) \begin{align*} \bar{\mathcal{D}}_0&(u^{3}(s), u^{3}(t))^2 = \int_\Omega \Bbb C_D[e((v^1+v^2)/2), e((v^1+v^2)/2)]\\& = \sum\nolimits_{i=1,2} \frac{1}{4}\int_\Omega \Bbb C_D[e(v^i), e(v^i)] + \frac{1}{2}\int_\Omega \Bbb C_D[e(v^1), e(v^2)] \\ & \le \sum\nolimits_{i=1,2}\frac{1}{2}\int_\Omega \Bbb C_D[e(v^i), e(v^i)] = \frac{1}{2} \bar{\mathcal{D}}_0(u^{1}(s), u^{1}(t))^2 + \frac{1}{2} \bar{\mathcal{D}}_0(u^{2}(s), u^{2}(t))^2. \end{align*}
Dividing by $|s-t|^2$ and letting $s$ go to $t$ we obtain the claim. We also anticipate from Lemma \ref{lemma: lin-slope} below that $u \mapsto |\partial \bar{\phi}_0|^2_{\bar{\mathcal{D}}_0}(u)$ is convex.
Assume there were two different curves of maximal slope $u^1$, $u^2$ starting from $u_0$, i.e., we find some $T$ such that $e(u^1(T)) \neq e(u^2(T))$ since otherwise the curves would coincide by Korn's inequality. Set $u^{3} = \frac{1}{2}(u^1 + u^2)$ and compute by the strict convexity of $\Bbb C_W$ on $\Bbb R^{d \times d}_{\rm sym}$ (see Lemma \ref{D-lin}(iii)), the convexity properties of the slope and metric derivative, and \eqref{maximalslope} \begin{align*}
\bar{\phi}_0(u_0) &= \frac{1}{2}\sum_{i=1,2} \Big( \frac{1}{2} \int_0^T |(u^i)'|_{\bar{\mathcal{D}}_0}^2(t) \, dt + \frac{1}{2} \int_0^T |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}^2(u^i(t)) \, dt + \bar{\phi}_0(u^i(T)) \Big)\\
& > \frac{1}{2} \int_0^T |(u^{3})'|_{\bar{\mathcal{D}}_0}^2(t) \, dt + \frac{1}{2} \int_0^T |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}^2(u^{3}(t)) \, dt + \bar{\phi}_0(u^{3}(T)), \end{align*}
which contradicts the fact that $ |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}$ is an upper gradient (see Definition \ref{main def2}(i) and use Young's inequality). This contradiction establishes uniqueness and concludes the proof. \end{proof}
\subsection{$\Gamma$-convergence and lower semicontinuity}
As a preparation for the passage to the linear problem, we recall and prove $\Gamma$-convergence results for the energies and lower semicontinuity for the slopes. In the following it is convenient to express all quantities in terms of the linear setting. To this end, recalling \eqref{nonlinear energy} and \eqref{eq: D,D0}, for $u,v \in W^{2,p}_0(\Omega)$ and $\tau,\delta>0$ we define \begin{align*} & \bar{\phi}_{\delta}(u) = \phi_\delta(\mathbf{id} + \delta u), \ \ \bar{\phi}_{\delta,P}(u) = \delta^{-p\alpha}\int_\Omega P(\delta\nabla^2 u), \ \ \bar{\phi}_{\delta,W}(u) = \bar{\phi}_\delta(u) - \bar{\phi}_{\delta,P}(u), \\ &\bar{\mathcal{D}}_\delta(u,v) = {\mathcal{D}}_\delta(\mathbf{id}+ \delta u, \mathbf{id} + \delta v), \ \ \ \bar{\Phi}_\delta(\tau,v;u) = \bar{\phi}_\delta(u) + \frac{1}{2\tau}\bar{\mathcal{D}}_\delta(u,v)^2,\\
&|\partial \bar{\phi}_\delta|_{\bar{\mathcal{D}}_\delta}(u) = |\partial {\phi}_\delta|_{{\mathcal{D}}_\delta}(\mathbf{id} + \delta u). \end{align*} We extend $\bar{\phi}_\delta$ to a functional defined on $H^1_0(\Omega) $ by setting $\bar{\phi}_\delta(u) = + \infty$ for $u \in H^1_0(\Omega) \setminus W^{2,p}_0(\Omega) $. Likewise, we extend $\bar{\Phi}_\delta$. Moreover, we say $u \in \bar{\mathscr{S}}_\delta^M$ if $\mathbf{id} + \delta u \in \mathscr{S}_\delta^M$. We obtain the following $\Gamma$-convergence results. (For an exhaustive treatment of $\Gamma$-convergence we refer the reader to \cite{DalMaso:93}.)
\begin{theorem}[$\Gamma$-convergence]\label{th: Gamma} Let $(\delta_n)_n$ be a null sequence.
(i) The functionals $\bar{\phi}_{\delta_n}: H^1_0(\Omega) \to [0,\infty]$ $\Gamma$-converge to $\bar{\phi}_0$ in the weak $H^1(\Omega)$-topology.
(ii) For each $\tau>0$, $M>0$, and each sequence $(\bar{v}_n)_n$ with $\bar{v}_n \in \bar{\mathscr{S}}_{\delta_n}^M$ and $\bar{v}_n \to \bar{v}$ strongly in $H^1(\Omega)$, the functionals $\bar{\Phi}_{\delta_n}(\tau,\bar{v}_n;\cdot): H^1_0(\Omega) \to [0,\infty]$ $\Gamma$-converge to $\bar{\Phi}_0(\tau,\bar{v}; \cdot)$ in the weak $H^1(\Omega)$-topology.
\end{theorem}
\par\noindent{\em Proof. } (i) The result is essentially proved in the paper \cite{DalMasoNegriPercivale:02} and we only give a short sketch highlighting the relevant adaptions. Since $\bar{\phi}_{\delta_n,P} \ge 0$, for the lower bound it suffices to prove $\liminf_{n \to \infty} \bar{\phi}_{\delta_n,W}(u_n) \ge \bar{\phi}_0(u)$ whenever $u_n \rightharpoonup u$ weakly in $H^1(\Omega)$. This was proved under more general assumptions in \cite[Proposition 4.4]{DalMasoNegriPercivale:02}. In our setting it follows readily by using Lemma \ref{lemma: metric space-properties}(iv) and the lower semicontinuity of $\bar{\phi}_0$ (see Lemma \ref{D-lin}(iii)).
By a general approximation argument in the theory of $\Gamma$-convergence it suffices to establish the upper bound for smooth functions $u$, cf. \cite[Proposition 4.1]{DalMasoNegriPercivale:02}. For such a function, setting $u_n = u$, we find $\lim_n \bar{\phi}_{\delta_n,W}(u_n) = \bar{\phi}_0(u)$ (see Lemma \ref{lemma: metric space-properties}(iv) or \cite[Proposition 4.1]{DalMasoNegriPercivale:02}) and moreover it is not hard to see that $\bar{\phi}_{\delta_n,P}(u_n) \to 0$ by the growth of $P$ and the fact that $\alpha<1$. This concludes the proof of (i).
(ii) We first suppose that the sequence $(\bar{v}_n)_n$ is constantly $\bar{v}$. Then $\bar{\Phi}_{\delta_n}(\tau,\bar{v};\cdot)$ $\Gamma$-converges to $\bar{\Phi}_{0}(\tau,\bar{v};\cdot)$ repeating exactly the proof of (i), where, in addition to Lemma \ref{lemma: metric space-properties}(iv), we also use Lemma \ref{lemma: metric space-properties}(iii). To obtain the general case, it now suffices to prove that for every sequence $(u_n)_n$ uniformly bounded in $H_0^1(\Omega)$ and $u_n \in \bar{\mathscr{S}}_{\delta_n}^M$ for some $M$ large enough we obtain
$$\lim\nolimits_{n\to \infty} |\bar{\mathcal{D}}_{\delta_n}(u_n,\bar{v}_{n})^2 - \bar{\mathcal{D}}_{\delta_n}(u_n,\bar{v})^2| = 0.$$
In view of Lemma \ref{lemma: metric space-properties}(iii), it suffices to show $\lim\nolimits_{n\to \infty} |\bar{\mathcal{D}}_{0}(u_n,\bar{v}_{n})^2 - \bar{\mathcal{D}}_{0}(u_n,\bar{v})^2| = 0$. To this end, we note that (recall \eqref{eq: D,D0}) \begin{align*} \bar{\mathcal{D}}_{0}(u_n,\bar{v}_{n})^2 - \bar{\mathcal{D}}_{0}(u_n,\bar{v})^2 &= \int_\Omega \Bbb C_D[\nabla \bar{v}_{n}, \nabla \bar{v}_{n}] - \int_\Omega \Bbb C_D[\nabla \bar{v}, \nabla \bar{v}] \\& \ \ \ - 2\int_\Omega \Bbb C_D[\nabla u_n, \nabla \bar{v}_{n} - \nabla\bar{v}], \end{align*} which by the assumption on $(\bar{v}_n)_n$ and $(u_n)_n$ converges to zero. \nopagebreak\hspace*{\fill}$\Box$
We remark that by a general result in the theory of $\Gamma$-convergence we get that (almost) minimizers associated to the sequence of functionals converge to minimizers of the limiting functional. We obtain the following strong convergence result for recovery sequences which in various settings has been derived in, e.g., \cite{DalMasoNegriPercivale:02, FriedrichSchmidt:2011, Schmidt:08}.
\begin{lemma}[Strong convergence of recovery sequences]\label{lemma: energy} Suppose that the assumptions of Theorem \ref{th: Gamma} hold. Let $M>0$, let $(u_n)_n$ be a sequence with $u_n \in \bar{\mathscr{S}}_{\delta_n}^M$. Let $u \in H_0^1(\Omega)$ such that $u_n\rightharpoonup u$ weakly in $H^1(\Omega)$ and $$(i) \ \ \bar{\phi}_{\delta_n}(u_n) \to \bar{\phi}_0(u) \ \ \ \text{or} \ \ \ (ii) \ \ \bar{\Phi}_{\delta_n}(\tau,\bar{v}_n; u_n) \to\bar{\Phi}_0(\tau,\bar{v}; u).$$ Then $u_n \to u$ strongly in $H^1(\Omega)$. \end{lemma}
\par\noindent{\em Proof. } If $\bar{\phi}_{\delta_n}(u_n) \to \bar{\phi}_0(u)$, we find $\bar{\phi}_0(u_n) \to \bar{\phi}_0(u)$ by Lemma \ref{lemma: metric space-properties}(iv) and thus by Lemma \ref{D-lin}(iii) \begin{align*} \Vert & e(u_n-u) \Vert^2_{L^2(\Omega)} \le C\int_\Omega \Bbb C_W[e(u_n-u),e(u_n-u)] \\ & = C\Big(\int_\Omega \Bbb C_W[e(u_n),e(u_n)] + \int_\Omega \Bbb C_W[e(u),e(u)] -2\int_\Omega \Bbb C_W[e(u_n),e(u)]\Big) \to 0 \end{align*} as $n \to \infty$. The assertion of (i) follows from Korn's inequality. The proof of (ii) is similar, where one additionally takes Lemma \ref{lemma: metric space-properties}(iii) into account. \nopagebreak\hspace*{\fill}$\Box$
We close this section with a lower semicontinuity result for the slopes.
\begin{lemma}[Lower semicontinuity of slopes]\label{lemma: lsc-slope}
For each sequence $(u_n)_n\subset \bar{\mathscr{S}}_{\delta_n}^M$ with $u_n \rightharpoonup u$ weakly in $H^1(\Omega)$ we have $\liminf_{n \to \infty}|\partial \bar{\phi}_{\delta_n}|_{\bar{\mathcal{D}}_{\delta_n}}(u_n) \ge |\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u)$. \end{lemma}
\par\noindent{\em Proof. } For $\varepsilon>0$ fix $u' \in C^\infty_c(\Omega;\Bbb R^d)$ with $\Vert u' - u \Vert_{H^1(\Omega)} \le \varepsilon$. Fix $v \in C^\infty_c(\Omega;\Bbb R^d)$, $v \neq u',u$. We first note that with $w_n := u_n - u'+ v$ we have by Lemma \ref{lemma: slopes}(i) \begin{align*}
|\partial \bar{\phi}_{\delta_n}|_{\bar{\mathcal{D}}_{\delta_n}}(u_n) & = \sup_{w \neq u_n} \ \frac{(\bar{\phi}_{\delta_n}(u_n) - \bar{\phi}_{\delta_n}(w))^+}{\bar{\mathcal{D}}_{\delta_n}(u_n,w)(1 + C \Vert \mathbf{Id} + \delta_n \nabla u_n - (\mathbf{Id} + \delta_n \nabla w) \Vert_{L^\infty(\Omega)})^{1/2}} \\ &\ge \frac{(\bar{\phi}_{\delta_n}(u_n) - \bar{\phi}_{\delta_n}(w_n ))^+}{\bar{\mathcal{D}}_{\delta_n}(u_n,w_n)(1 + C_v\delta_n)^{1/2}}, \end{align*} where $C_v$ is a constant depending also on $v$ and $u'$. Note that, since $u',v$ are smooth, we indeed get $w_n = u_n - u'+ v \in \bar{\mathscr{S}}_{\delta_n}^M$ for $n$ large enough for some possibly larger $M>0$. Consequently, by Lemma \ref{lemma: metric space-properties}(iii),(iv) we get \begin{align}\label{lsc-slope1}
\liminf_{n \to \infty}|\partial \bar{\phi}_{\delta_n}|_{\bar{\mathcal{D}}_{\delta_n}}(u_n) \ge \liminf_{n \to \infty} \frac{(\bar{\phi}_{0}(u_n) - \bar{\phi}_{0}(w_n) + \bar{\phi}_{\delta_n,P}(u_n) - \bar{\phi}_{\delta_n,P}(w_n) )^+}{\bar{\mathcal{D}}_{0}(u_n,w_n)}. \end{align} Recalling \eqref{linear energy} (for $f \equiv 0 $) we obtain by a direct computation \begin{align} \lim_{n \to \infty} &\big(\bar{\phi}_0(u_n) - \bar{\phi}_0(u_n - u'+ v)\big) = \lim_{n \to \infty} \big(-\bar{\phi}_0(v-u') - 2\int_\Omega \Bbb C_W[e(u_n),e(v-u')] \big)\notag \\ &= -\bar{\phi}_0(v-u') - 2\int_\Omega \Bbb C_W[e(u),e(v-u')]\notag\\& = \bar{\phi}_0(u) - \bar{\phi}_0(v) - \bar{\phi}_0(u'-u) +2\int_\Omega \Bbb C_W[e(u'-u),e(v)]. \end{align} Moreover, by convexity of $P$ and the definition $w_n := u_n - u'+ v$ we find \begin{align}\label{lsc-slope3} \bar{\phi}_{\delta_n,P}(u_n) - \bar{\phi}_{\delta_n,P}(u_n - u'+ v) \ge \delta_n^{-p\alpha} \int_\Omega \partial_GP(\delta_n\nabla^2w_n) : \delta_n(\nabla^2 u'- \nabla^2 v), \end{align} which vanishes as $n \to \infty$ by \eqref{assumptions-P}(iii), H\"older's inequality, $1+ \alpha(p-1) - \alpha p >0$, and the fact that $\Vert \delta_n \nabla^2 w_n \Vert^p_{L^p(\Omega)} \le CM\delta_n^{p\alpha}$. (The latter follows from $w_n \in \bar{\mathscr{S}}_{\delta_n}^M$.) Combining \eqref{lsc-slope1}-\eqref{lsc-slope3}, using $\bar{\mathcal{D}}_{0}(u_n,w_n) = \bar{\mathcal{D}}_{0}(v,u')$, and recalling $u_n \rightharpoonup u$, we get after some calculations \begin{align*}
\liminf_{n \to \infty}|\partial \bar{\phi}_{\delta_n}|_{\bar{\mathcal{D}}_{\delta_n}}(u_n)& \ge \frac{(\bar{\phi}_0(u) - \bar{\phi}_0(v) - \bar{\phi}_0(u'-u) +2\int_\Omega \Bbb C_W[e(u'-u),e(v)])^+}{\bar{\mathcal{D}}_{0}(v,u')}\\ & \ge \frac{(\bar{\phi}_0(u) - \bar{\phi}_0(v))^+}{\bar{\mathcal{D}}_{0}(v,u)} -C\varepsilon \end{align*} for some $C>0$ depending only on $u$, $u'$ and $v$. Letting first $\varepsilon\to 0$ and taking then the supremum with respect to $v$ we get \begin{align*}
\liminf_{n \to \infty}|\partial \bar{\phi}_{\delta_n}|_{\bar{\mathcal{D}}_{\delta_n}}(u_n) \ge \sup_{v \in C_c^\infty(\Omega), v \neq u} & \frac{(\bar{\phi}_0(u) - \bar{\phi}_0(v))^+}{\bar{\mathcal{D}}_{0}(v,u)}. \end{align*} In view of Lemma \ref{lemma: slopes}(ii), the claim now follows by approximating each $v \in H^1_0(\Omega)$ by a sequence of smooth functions noting that the right hand side is continuous with respect to $H^1(\Omega)$-convergence. \nopagebreak\hspace*{\fill}$\Box$
\subsection{Passage from nonlinear to linear viscoelasticity}
In this section we now give the proof of Theorem \ref{maintheorem3}. For the whole section we fix a null sequence $(\delta_k)_k$ and sequence of initial data $(y_0^k)_{k\in \Bbb N} \subset W^{2,p}_\mathbf{id}(\Omega)$ such that $\delta_k^{-1}(y^k_0 - \mathbf{id}) \to u_0 \in H_0^1(\Omega)$. Moreover, we fix $M>0$ so large that $y_0^k \in \mathscr{S}_{\delta_k}^M$ for $k \in \Bbb N$.
\begin{proof}[Proof of Theorem \ref{maintheorem3}(i)] Let $\tau>0$ and let $\tilde{Y}_\tau^{\delta_k}$ as in \eqref{ds} be a discrete solution. For each $k \in \Bbb N$ we then have the sequence $(U^n_k)_{n \in \Bbb N}$ with $U^n_k= \delta_k^{-1}(\tilde{Y}_\tau^{\delta_k}(n\tau) - \mathbf{id}) \in \bar{\mathscr{S}}_{\delta_k}^M$ for $n \in \Bbb N$. We need to show that there exists a sequence $(U^n_0)_{n \in \Bbb N}$ with $U^0_0 = u_0$ such that
$$(i) \ \ U^n_0 = {\rm argmin}_{v \in H^1_0(\Omega)} \bar{\Phi}_0(\tau,U^{n-1}_0; v), \ \ \ \ (ii) \ \ \text{$U^n_k \to U^n_0$ strongly in $H^1(\Omega)$} $$ for all $n \in \Bbb N$. We show this property by induction.
Suppose $(U^i_0)_{i=0}^n$ have been found such that the above properties hold. In particular, we note that (ii) holds for $n=0$ by assumption. We now pass from step $n$ to $n+1$.
As $U^n_k \to U^n_0$ strongly in $H^1(\Omega)$ and thus by Theorem \ref{th: Gamma}(ii) $\bar{\Phi}_{\delta_k}(\tau,U^{n}_k; \cdot)$ $\Gamma$-converges to $\bar{\Phi}_0(\tau,U^{n}_0; \cdot)$, we derive by properties of $\Gamma$-convergence that the (unique) minimizer of $\bar{\Phi}_0(\tau,U^{n}_0; \cdot)$, denoted by $U^{n+1}_0$, is the limit of minimizers of $\bar{\Phi}_{\delta_k}(\tau,U^{n}_k; \cdot)$. Consequently, we obtain $U^{n+1}_k \rightharpoonup U^{n+1}_0$ weakly in $H^1(\Omega)$ and $\bar{\Phi}_{\delta_k}(\tau,U^{n}_k; U^{n+1}_k) \to \bar{\Phi}_0(\tau,U^{n}_0; U^{n+1}_0) $. Thus, Lemma \ref{lemma: energy} implies that the sequence even converges strongly in $H^1(\Omega)$. This concludes the induction step. \end{proof}
In the following let $u$ be the unique element of $MM(\bar{\Phi}_0;u_0)$.
\begin{proof}[Proof of Theorem \ref{maintheorem3}(ii)] We let $\sigma$ be the weak $H^1(\Omega)$-topology. We consider the sequence of metrics $\mathcal{D}_k = \bar{\mathcal{D}}_{\delta_k}$ on $H^1_0(\Omega)$ and the functionals $\phi_k = \bar{\phi}_{\delta_k}$ as well as the limiting objects $\bar{\mathcal{D}}_0$ and $\bar{\phi}_0$. We note that \eqref{compatibility} is satisfied due to Lemma \ref{lemma: metric space-properties}(iii) and the fact that $\bar{\mathcal{D}}_0$ is quadratic and convex (see Lemma \ref{D-lin}(iii)). Moreover, also \eqref{eq: implication} is satisfied by the $\Gamma$-liminf inequality in Lemma \ref{th: Gamma}(i) and Lemma \ref{lemma: lsc-slope}.
Finally, also \eqref{basic assumptions2} holds. In fact, by the rigidity estimate in Lemma \ref{lemma:rigidity}(i) and \eqref{nonlinear energy}, \eqref{assumptions-W}(iii) we find for all $k \in \Bbb N$ and $u \in \bar{\mathscr{S}}_{\delta_k}^M$ letting $y = \mathbf{id} + \delta_k u$ \begin{align}\label{go to limit1} \begin{split} \Vert u \Vert^2_{H^1(\Omega)} &= \delta_k^{-2} \Vert y-\mathbf{id} \Vert^2_{H^1(\Omega)} \le C\delta_k^{-2} \Vert \operatorname{dist}(\nabla y,SO(d)\Vert^2_{L^2(\Omega)} \\&\le C\delta_k^{-2} \phi_{\delta_k}(y) \le CM. \end{split} \end{align}
Now consider a sequence $(y_k)_k$ of generalized minimizing movements starting from $y_0^k$ with $\delta_k^{-1}(y_0^k-\mathbf{id}) \to u_0$ in $H^1(\Omega)$. For convenience we also introduce the curves $u_k = \delta_k^{-1}(y_k - \mathbf{id})$. Fix $M>0$ so large that $y_0^k \in \mathscr{S}_{\delta_k}^M$ for $k \in \Bbb N$. As $\bar{\phi}_{\delta_k}(u_k(t)) \le \phi_{\delta_k}(y_k^0)$ for all $t \ge 0$, we get $\sup_k\sup_t (\phi_{\delta_k}(u_k(t)) + \mathcal{D}_k(u_k(t),u_0)) < \infty$ by \eqref{go to limit1} and Lemma \ref{lemma: metric space-properties}(iii).
Consequently, also \eqref{eq: abstract assumptions1}(i) holds and \eqref{eq: abstract assumptions1}(ii) is satisfied by the assumption on the initial data and Lemma \ref{lemma: metric space-properties}(iv). Since the slopes are strong upper gradients by Lemma \ref{lemma: slopes}, we can apply Theorem \ref{th:abstract convergence 1} and the existence of a limiting curve of maximal slope follows. As this curve is uniquely given by $u$ (see Theorem \ref{maintheorem2}(ii)), we indeed obtain $u_k(t) \rightharpoonup u(t)$ weakly in $H^1(\Omega)$ for all $t \in [0,\infty)$ up to a subsequence. Since the limit is unique, we see that the whole sequence converges to $u$ by Urysohn's subsequence principle.
It remains to observe that the convergence is actually strong. This follows from the fact that $\lim_{k \to \infty}\bar{\phi}_{\delta_k}(u_k(t)) = \bar{\phi}_0 (u(t))$ for all $t \in [0,\infty)$ (see Theorem \ref{th:abstract convergence 1}) and Lemma \ref{lemma: energy}. \end{proof}
\begin{proof}[Proof of Theorem \ref{maintheorem3}(iii)] Proceeding as in the previous proof, we see that all assumptions of Theorem \ref{th:abstract convergence 2} are satisfied. Therefore, we get that for any sequence of discrete solutions there is a subsequence converging pointwise weakly in $H^1(\Omega)$ to a curve of maximal slope for $\bar{\phi}_0$ which can again be identified as $u$. The strong convergence as well as the convergence of the whole sequence follow exactly as in the previous proof. \end{proof}
\subsection{Fine representation of the slopes and solutions to the equations}
In this section we derive fine representations for the slopes which will allow us to relate curves of maximal slope with solutions to the equations \eqref{nonlinear equation} and \eqref{linear equation}.
Recall that $\Bbb C_D$ as defined in \eqref{linear equation} is a fourth order symmetric tensor inducing a quadratic form $(F_1,F_2) \mapsto \Bbb C_D[F_1,F_2]$ which is positive definite on $\Bbb R^{d \times d}_{\rm sym}$ (cf. Lemma \ref{D-lin}). Moreover, it maps $\Bbb R^{d \times d}$ to $\Bbb R^{d \times d}_{\rm sym}$, denoted by $F \mapsto \Bbb C_D F$ in the following. More precisely, the mapping $F \mapsto \Bbb C_D F$ from $\Bbb R^{d \times d}_{\rm sym}$ to $\Bbb R^{d \times d}_{\rm sym}$ is bijective. By $\sqrt{\Bbb C_D}$ we denote its (unique) root and by $\sqrt{\Bbb C_D}^{-1}$ the inverse of $\sqrt{\Bbb C_D}$, both mappings defined on $\Bbb R^{d \times d}_{\rm sym}$. We start with a fine representation of the slope in the linear setting.
\begin{lemma}[Slope in the linear setting]\label{lemma: lin-slope} There exists a linear differential operator $\mathcal{L}_0: H^1_0(\Omega;\Bbb R^{d}) \to L^2(\Omega;\Bbb R^{d \times d}_{\rm sym})$ satisfying ${\rm div} \mathcal{L}_0(u) = 0$ in $H^{-1}(\Omega;\Bbb R^d)$ such that for all $u \in H^1_0(\Omega)$ we have
$$|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u) = \Vert \sqrt{\Bbb C_D}^{-1}\big(\Bbb C_W e(u) + \mathcal{L}_0(u) \big) \Vert_{L^2(\Omega)}.$$
Particularly, we note that $|\partial \bar{\phi}_0|^2_{\bar{\mathcal{D}}_0}$ is convex on $H^1_0(\Omega)$. \end{lemma}
\par\noindent{\em Proof. } Recalling \eqref{linear energy} (for $f \equiv 0 $), \eqref{eq: D,D0}, Definition \ref{main def2}(ii), and Lemma \ref{D-lin} we have \begin{align}\label{lin-slope1}
|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u) &= \limsup_{v \to u} \frac{(\bar{\phi}_0(u) - \bar{\phi}_0(v))^+}{\bar{\mathcal{D}}_0(u,v)}\\ &= \limsup_{v \to u} \frac{ (\int_\Omega \Bbb C_W[e(u), e(u-v)] - \frac{1}{2}\Bbb C_W[e(v-u), e(v-u)])^+} {(\int_\Omega \Bbb C_D[e(u-v),e(u-v)])^{1/2}}\notag\\ &= \limsup_{v \to u} \frac{ \int_\Omega \Bbb C_W[e(u), e(u-v)]} {\Vert \sqrt{\Bbb C_D} e(u-v) \Vert_{L^2(\Omega)}} = \sup_{w \neq 0} \frac{ \int_\Omega \Bbb C_W[e(u), e(w)]} {\Vert \sqrt{\Bbb C_D} e(w) \Vert_{L^2(\Omega)}},\notag \end{align} where in the second step we used $\int_\Omega \Bbb C_W[e(v-u), e(v-u)] / \Vert \sqrt{\Bbb C_D} e(u-v) \Vert_{L^2(\Omega)} \to 0$ as $v \to u$. Let $\bar{w}$ be \color{black} the unique \color{black} solution to the minimization problem
$$\min_{v \in H^1_0(\Omega)} \int_\Omega \Big(\frac{1}{2}|\sqrt{\Bbb C_D} e(v)|^2 - \int_\Omega \Bbb C_W[e(u), e(v)]\Big)\ . $$
\color{black} Clearly, $\bar{w}$ necessarily satisfies \color{black}
$$\int_\Omega \big(\sqrt{\Bbb C_D} e(\bar{w}) : \sqrt{\Bbb C_D}e(\varphi) - \Bbb C_W [e(u),e(\varphi)]\big) = 0 $$
for all $\varphi \in H^1_0(\Omega)$. This condition can also be formulated as \begin{align}\label{lin-slope2} \mathcal{L}_0(u): e( \varphi) = 0 \ \ \forall \varphi \in H^1_0(\Omega), \ \ \text{where} \ \ \mathcal{L}_0(u): = \Bbb C_D e(\bar{w}) - \Bbb C_We(u). \end{align}
As the solution $\bar{w}$ depends linearly on $u$, we also get that $\mathcal{L}_0$ is a linear operator. By \eqref{lin-slope1} and the property of $\mathcal{L}_0$ we now find \begin{align*}
|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u) &= \sup_{w \neq 0} \frac{ \int_\Omega (\Bbb C_W e(u) + \mathcal{L}_0(u)) : e(w)} {\Vert \sqrt{\Bbb C_D} e(w) \Vert_{L^2(\Omega)}} \\&= \sup_{w \neq 0} \frac{ \int_\Omega \big(\sqrt{\Bbb C_D}^{-1}(\Bbb C_We(u) + \mathcal{L}_0(u)) \big) : \sqrt{\Bbb C_D}e(w)} {\Vert \sqrt{\Bbb C_D} e(w) \Vert_{L^2(\Omega)}}\\ & \le \Vert \sqrt{\Bbb C_D}^{-1}(\Bbb C_We(u) + \mathcal{L}_0(u)) \Vert_{L^2(\Omega)},
\end{align*}
where in the last step we used the Cauchy-Schwartz inequality. On the other hand, by definition of $\mathcal{L}_0$ in \eqref{lin-slope2}, we get \begin{align*}
|\partial \bar{\phi}_0|_{\bar{\mathcal{D}}_0}(u)& \ge
\frac{ \int_\Omega \big(\sqrt{\Bbb C_D}^{-1}(\Bbb C_We(u) + \mathcal{L}_0(u)) \big) : \sqrt{\Bbb C_D}e(\bar{w})} {\Vert \sqrt{\Bbb C_D} e(\bar{w}) \Vert_{L^2(\Omega)}} \\
& = \Vert \sqrt{\Bbb C_D} e(\bar{w}) \Vert_{L^2(\Omega)} = \Vert \sqrt{\Bbb C_D}^{-1}(\Bbb C_We(u) + \mathcal{L}_0(u)) \Vert_{L^2(\Omega)}. \end{align*}
This concludes the proof. \nopagebreak\hspace*{\fill}$\Box$
Recall the definition of the symmetric fourth order tensor $H_Y = \frac{1}{2}\partial^2_{F_1^2} D^2(Y,Y)$ for $Y \in GL_+(d)$ (see before Lemma \ref{lemma: metric space-properties}). Let $Y \in \Bbb R^{d \times d}$ be in a small neighborhood of $\mathbf{Id}$ such that $Y^{-1}$ exists. Similarly to the discussion before Lemma \ref{lemma: lin-slope}, we get that $H_Y$ induces a bijective mapping from $Y^{-\top}\Bbb R^{d \times d}_{\rm sym}$ to $Y\Bbb R^{d \times d}_{\rm sym}$ by using frame indifference \eqref{eq: assumptions-D}(v) and the growth assumption \eqref{eq: assumptions-D}(vi). We then introduce $\sqrt{H_Y}$ as a bijective mapping from $Y^{-\top}\Bbb R^{d \times d}_{\rm sym}$ to $Y\Bbb R^{d \times d}_{\rm sym}$. In a similar fashion, we introduce the inverse $\sqrt{H_Y}^{-1}$.
For a given deformation $y: \Omega \to \Bbb R^d$ we introduce a mapping $H_{\nabla y}: \Omega \to \Bbb R^{d \times d \times d \times d}$ by $H_{\nabla y}(x) = H_{\nabla y(x)}$ for $x \in \Omega$. We note by Lemma \ref{lemma:rigidity}(ii), the fact that $D \in C^3$, and a continuity argument that \begin{align}\label{continuity for H} \Vert \sqrt{H_\mathbf{Id}} - \sqrt{H_{\nabla y}} \Vert_{L^\infty(\Omega)} \le C\delta^\alpha \end{align}
for all $y \in \mathscr{S}_\delta^M$ for a sufficiently large constant $C>0$. Moreover, recall the definition of the operator $\mathcal{L}_P: \lbrace \nabla^2 u: u \in W^{2,p}_\mathbf{id}(\Omega)\rbrace \to W^{-1,\frac{p}{p-1}}(\Omega;\Bbb R^{d \times d})$ in \eqref{LP-def}. We write $\beta = \delta^{2-\alpha p}$ in the following for convenience. Note that $\int_\Omega \partial_GP(\nabla^2 y) : \nabla^2 \varphi = \mathcal{L}_P(y) : \nabla \varphi $ for all $y \in W^{2,p}_\mathbf{id}(\Omega)$ and $\varphi \in W^{2,p}_0(\Omega)$, where the boundary term vanishes due to $\nabla \varphi =0$ on $\partial \Omega$. We now obtain the following result.
\begin{lemma}[Slope in the nonlinear setting]\label{lemma: nonlin-slope} There exists a differential operator $\mathcal{L}^*_P: \lbrace y \in W^{2,p}_\mathbf{id}(\Omega): {\rm div}\mathcal{L}_P(\nabla^2 y) \in H^{-1}(\Omega; \Bbb R^{d}) \rbrace \to L^2(\Omega; \Bbb R^{d\times d})$ satisfying ${\rm div}\mathcal{L}^*_P(y) = {\rm div}\mathcal{L}_P(\nabla^2 y)$ in $H^{-1}(\Omega; \Bbb R^{d})$ such that for $\delta>0$ small enough and for all $y \in \mathscr{S}_\delta^M$ we have \begin{align*}
|\partial \phi_\delta|_{{\mathcal{D}}_\delta}(y) = \begin{cases}
\tfrac{1}{\delta}\Vert \sqrt{H_{\nabla y}}^{-1}\big(\partial_FW(\nabla y) + \beta\mathcal{L}^*_P (y) \big) \Vert_{L^2(\Omega)} & \text{if} \ {\rm div}\mathcal{L}_P(\nabla^2 y) \in H^{-1}(\Omega),\\
+ \infty & \text{else}.
\end{cases}
\end{align*} \end{lemma}
\begin{rem}\label{rem-slope} {\normalfont We remark that the expression is well defined in the following sense: If $\nabla y(x) = Y(x)$ in the above notation, then we indeed have $\partial_FW(\nabla y(x)) + \beta\mathcal{L}^*_P (y(x)) \in Y(x) \Bbb R^{d \times d}_{\rm sym}$ for a.e. $x \in \Omega$. } \end{rem}
\par\noindent{\em Proof. } We (i) first prove the lower bond in the case ${\rm div}\mathcal{L}_P(\nabla^2 y) \in H^{-1}(\Omega)$ and (ii) afterwards if ${\rm div}\mathcal{L}_P(\nabla^2 y) \notin H^{-1}(\Omega)$. Finally, (iii) we establish the upper bound.
(i) Suppose that ${\rm div}\mathcal{L}_P(\nabla^2 y) \in H^{-1}(\Omega)$. Consider the minimization problem
$$\min_{w \in H^1_0(\Omega)} \int_\Omega \Big(\frac{1}{2}|\sqrt{H_{\nabla y}}\nabla w|^2 - (\partial_FW(\nabla y)+ \beta\mathcal{L}_P(\nabla^2 y)\Big): \nabla w.$$ By \eqref{continuity for H}, the fact that $\sqrt{H_\mathbf{Id}} = \sqrt{\Bbb C_D}$, Lemma \ref{D-lin}(iii), and Korn's inequality we have \begin{align*} \Vert \sqrt{H_{\nabla y}}\nabla w\Vert_{L^2(\Omega)}^2& \ge \Vert \sqrt{H_\mathbf{Id}}\nabla w\Vert_{L^2(\Omega)}^2 - C\delta^{2\alpha} \Vert \nabla w\Vert_{L^2(\Omega)}^2 \\&\ge C\Vert e(w)\Vert_{L^2(\Omega)}^2 - C\delta^{2\alpha} \Vert \nabla w\Vert_{L^2(\Omega)}^2 \ge C\Vert \nabla w\Vert_{L^2(\Omega)}^2 \end{align*}
for $\delta$ sufficiently small for all $w \in H^1_0(\Omega)$. Moreover, we have $|\int_\Omega \mathcal{L}_P(\nabla^2 y): \nabla w| \le \Vert {\rm div}\mathcal{L}_P(\nabla^2 y) \Vert_{H^{-1}(\Omega)} \Vert w \Vert_{H^1(\Omega)}$ for all $w \in H^1_0(\Omega)$. Thus, the solution $\bar{w}$ of the problem exists, is unique, and satisfies
$$(\partial_FW(\nabla y) + \beta\mathcal{L}_P(\nabla^2 y) ) : \nabla \varphi = \sqrt{H_{\nabla y}} \nabla \bar{w} : \sqrt{H_{\nabla y}} \nabla \varphi = H_{\nabla y} \nabla \bar{w} : \nabla \varphi $$
for all $\varphi \in H^1_0(\Omega)$. Define $\mathcal{L}^*_P(y) := \beta^{-1}( H_{\nabla y} \nabla \bar{w} - \partial_FW(\nabla y))$ and note that \begin{align}\label{nonlin-slope2} \mathcal{L}^*_P(y) : \nabla \varphi = \mathcal{L}_P(\nabla^2 y) : \nabla \varphi \ \ \ \text{ for all $\varphi \in H^1_0(\Omega)$} \end{align}
as well as $\mathcal{L}^*_P(y) \in L^2(\Omega)$. Moreover, since $\beta\mathcal{L}^*_P(y) + \partial_FW(\nabla y)= H_{\nabla y} \nabla \bar{w}$, recalling the properties of $H_{\nabla y}$ we see that Remark \ref{rem-slope} applies. Fix $\varepsilon>0$ and choose $w_\varepsilon \in C_c^\infty(\Omega;\Bbb R^d)$ with $\Vert \bar{w} - w_\varepsilon \Vert_{H^1(\Omega)} \le \varepsilon$. Letting $w_n = y - \frac{1}{n}w_\varepsilon$ we get by a Taylor expansion
\begin{align*}
n\delta^2( \phi_\delta(w_n) - & \phi_\delta(y)) = n\int_\Omega \partial_FW(\nabla y) : (\nabla w_n - \nabla y) + n O(\Vert \nabla w_n- \nabla y \Vert^2_{L^2(\Omega)})\notag\\
& + n \beta \int_\Omega \partial_GP(\nabla^2 y): (\nabla^2w_n - \nabla^2 y)+ n\beta O(\Vert \nabla^2 w_n - \nabla^2 y \Vert^2_{L^2(\Omega)}) \notag\\
& \ \ \ \ \ \ = - \int_\Omega \partial_FW(\nabla y) : \nabla w_\varepsilon - \beta \partial_GP(\nabla^2 y): \nabla^2w_\varepsilon + O(1/n),
\end{align*}
where $O(1/n)$ depends on the choice of $w_\varepsilon$. Similarly, we get by Lemma \ref{lemma: metric space-properties}(i)
\begin{align*}
n^2\delta^2\mathcal{D}_\delta(y,w_n)^2 &= n^2 \int_\Omega H_{\nabla y}[\nabla (y - w_n), \nabla (y - w_n)] + n^2 O(\Vert \nabla w_n- \nabla y\Vert^3_{L^3(\Omega)}) \notag \\& = \Vert \sqrt{H_{\nabla y}} \nabla w_\varepsilon \Vert^2_{L^2(\Omega)}+ O(1/n).
\end{align*}
For brevity we introduce
$$\Phi(w) = \Big(\int_\Omega (\partial_FW(\nabla y)+\beta\mathcal{L}_P(\nabla^2 y)) : \nabla w \Big)\Vert \sqrt{H_{\nabla y}} \nabla w \Vert_{L^2(\Omega)}^{-1}. $$
Since $\mathcal{D}_\delta(y,w_n) \to 0$, we now obtain
\begin{align*}
\delta|\partial \phi_\delta|_{\mathcal{D}_\delta}(y)& \ge \limsup_{n \to \infty} \frac{\delta(\phi_\delta(y) - \phi_\delta(w_n))^+}{\mathcal{D}_\delta(y,w_n)} \\&\ge \frac{\int_\Omega \partial_FW(\nabla y) : \nabla w_\varepsilon + \int_\Omega\beta \partial_GP(\nabla^2 y) : \nabla^2 w_\varepsilon }{\Vert \sqrt{H}_{\nabla y} \nabla w_\varepsilon \Vert_{L^2(\Omega)} } = \Phi(w_\varepsilon) \end{align*} where in the last step we used the definition of $\mathcal{L}_P$ in \eqref{LP-def}. Recalling the definition of $\mathcal{L}^*_P$ and \eqref{nonlin-slope2} we now derive
\begin{align*}
\Phi(\bar{w}) - \Phi(w_\varepsilon) + \delta|\partial \phi_\delta|_{\mathcal{D}_\delta}(y)& \ge \Phi(\bar{w}) = \frac{\int_\Omega H_{\nabla y} \nabla \bar{w} : \nabla \bar{w} }{\Vert \sqrt{H_{\nabla y}} \nabla \bar{w} \Vert_{L^2(\Omega)} } \\& = \frac{\int_\Omega \sqrt{H_{\nabla y}} \nabla \bar{w} :\sqrt{H_{\nabla y}} \nabla \bar{w} }{\Vert \sqrt{H_{\nabla y}} \nabla \bar{w} \Vert_{L^2(\Omega)} } = \Vert \sqrt{H_{\nabla y}} \nabla \bar{w} \Vert_{L^2(\Omega)} \\ & = \Vert \sqrt{H_{\nabla y}}^{-1}\big(\partial_FW(\nabla y) + \beta\mathcal{L}^*_P (y) \big) \Vert_{L^2(\Omega)}. \end{align*}
By definition of $w_\varepsilon$ we get $|\Phi(\bar{w}) - \Phi(w_\varepsilon)| \to 0$ as $\varepsilon \to 0$ and the lower bound in the case ${\rm div}\mathcal{L}_P(\nabla^2 y) \in H^{-1}(\Omega)$ follows.
(ii) Now suppose that ${\rm div}\mathcal{L}_P(\nabla^2 y) \notin H^{-1}(\Omega)$. Let $(y_n)_n$ be a sequence of smooth functions converging to $y$ in $W^{2,p}(\Omega)$. Then $ \mathcal{L}^*_P(y_n)$ is not bounded in $L^2(\Omega)$. Indeed, otherwise we would get by the definition of $\mathcal{L}_P$, \eqref{assumptions-P}(iii), and \eqref{nonlin-slope2} that \begin{align*}
\Big|\int_\Omega \mathcal{L}_P(\nabla^2 y) : \nabla \varphi\Big| & = \Big|\int_\Omega \partial_G P(\nabla^2 y): \nabla^2 \varphi\Big| = \lim_{n \to \infty}\Big|\int_\Omega \partial_G P(\nabla^2 y_n): \nabla^2 \varphi\Big| \\
& = \lim_{n \to \infty} \Big|\int_\Omega \mathcal{L}^*_P(y_n) : \nabla\varphi\Big| \le C \Vert \nabla \varphi\Vert_{L^2(\Omega)} \end{align*} for all $\varphi \in W^{2,p}_0(\Omega)$. This, however, contradicts the assumption ${\rm div}\mathcal{L}_P(\nabla^2 y) \notin H^{-1}(\Omega)$. As energy and dissipation are $W^{2,p}(\Omega)$-continuous (see \eqref{assumptions-W},\eqref{assumptions-P}, Lemma \ref{lemma: metric space-properties}(ii)), we find for some fixed $\varepsilon>0$ and $n$ large enough by Lemma \ref{lemma: slopes}(i)
$$\varepsilon + |\partial \phi_\delta|_{{\mathcal{D}}_\delta}(y) \ge \sup_{w \neq y_n} \frac{(\phi_\delta(y_n) - \phi_\delta(w))^+}{\mathcal{D}_\delta(y_n,w)(1 + C \Vert \nabla y_n - \nabla w \Vert_{L^\infty(\Omega)})^{1/2}} =|\partial \phi_\delta|_{{\mathcal{D}}_\delta}(y_n).$$ By the representation of the slope at $y_n$ and the fact that $ \mathcal{L}^*_P(y_n)$ is not bounded in $L^2(\Omega)$, the right hand side tends to infinity for $n \to \infty$, as desired.
(iii) For the upper bound, we first use Lemma \ref{lemma: metric space-properties}(i),(ii), and Lemma \ref{th: metric space}(ii) to get \begin{align*} 1 & = \lim_{w \to v} \frac{\mathcal{D}_\delta(v,w)^2}{\mathcal{D}_\delta(v,w)^2} \ge \limsup_{w \to v} \frac{\Vert \sqrt{H_{\nabla v}} \nabla (w-v) \Vert^2_{L^2(\Omega)} - C\Vert \nabla v - \nabla w \Vert^3_{L^3(\Omega)}}{\delta^2\mathcal{D}_\delta(v,w)^2} \\ & \ge \limsup_{w \to v} \frac{\Vert \sqrt{H_{\nabla v}} \nabla (w-v) \Vert^2_{L^2(\Omega)} }{\delta^2\mathcal{D}_\delta(v,w)^2} - C\limsup_{w \to v} \Vert \nabla v - \nabla w \Vert_{L^\infty(\Omega)}\\ & = \limsup_{w \to v} \frac{\Vert \sqrt{H_{\nabla v}} \nabla (w-v) \Vert^2_{L^2(\Omega)} }{\delta^2\mathcal{D}_\delta(v,w)^2}. \end{align*} This together with Lemma \ref{lemma:C1} and the convexity of $P$ gives \begin{align*}
\delta|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) & = \limsup_{w \to y}\frac{\delta^2(\phi_\delta(y) - \phi_\delta(w))^+}{\delta\mathcal{D}_\delta(y,w)} \\&\le \limsup_{w \to y} \frac{\int_\Omega \partial_FW(\nabla y) : \nabla (y - w) + \int_\Omega \beta \partial_GP(\nabla^2 y): \nabla^2 (y -w)}{\Vert \sqrt{H_{\nabla y}} (\nabla w - \nabla y) \Vert_{L^2(\Omega)} }. \end{align*} Recalling the definition of $\mathcal{L}_P$ and using \eqref{nonlin-slope2} as in the lower bound, we get \begin{align*}
\delta|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) & \le \limsup_{w \to y} \frac{\int_\Omega (\partial_FW(\nabla y) + \beta\mathcal{L}^*_P(y) ) : \nabla (y - w)}{\Vert \sqrt{H_{\nabla y}} (\nabla w - \nabla y) \Vert_{L^2(\Omega)} }. \end{align*} Finally, the Cauchy Schwartz inequality gives \begin{align*}
\delta|\partial \phi_\delta|_{\mathcal{D}_\delta}(y) & \le \limsup_{w \to y} \frac{\int_\Omega \sqrt{H_{\nabla y}}^{-1}(\partial_FW(\nabla y) + \beta\mathcal{L}^*_P(y) ) : \sqrt{H_{\nabla y}}\nabla (y - w)}{\Vert \sqrt{H_{\nabla y}} (\nabla w - \nabla y) \Vert_{L^2(\Omega)} } \\ & \le \Vert \sqrt{H_{\nabla y}}^{-1}(\partial_FW(\nabla y) + \beta\mathcal{L}^*_P(y)) \Vert_{L^2(\Omega)}. \end{align*}
\nopagebreak\hspace*{\fill}$\Box$
Finally, following \cite[Section 1.4]{AGS} we relate curves of maximal slope with solutions to the equations \eqref{nonlinear equation} and \eqref{linear equation}. Similar to \cite[Corollary 1.4.5]{AGS}, this relies on the fact that the stored energy can be written as a sum of a convex functional and a $C^1$ functional on $H^1(\Omega)$.
\begin{proof}[Proof of Theorem \ref{maintheorem1}(iii) and Theorem \ref{maintheorem2}(iii)]
We only give the proof for the nonlinear equation. The proof for the linear equation is easier and can be seen along similar lines.
First, the fact that $\phi_\delta(y(t))$ is decreasing in time together with \eqref{nonlinear energy}-\eqref{assumptions-P} gives $y \in L^\infty([0,\infty ); W^{2,p}_{\mathbf{id}}(\Omega))$. Moreover, since $|y'|_{\mathcal{D}_\delta} \in L^2([0,\infty ))$ by \eqref{slopesolution} and $\mathcal{D}_\delta$ is equivalent to the $H^1(\Omega)$-norm (see Lemma \ref{lemma: metric space-properties}(ii)), we observe that $y$ is an absolutely continuous curve in the Hilbert space $H^1(\Omega)$. By \cite[Remark 1.1.3]{AGS} this implies that $y$ is differentiable for a.e. $t$ with $\partial_t \nabla y(t) \in L^2(\Omega)$ for a.e. $t$, that \begin{align}\label{eq: derivative in Hilbert} \nabla y(t) - \nabla y(s) = \int_s^t \partial_t \nabla y(r) \, dr \ \ \ \text{a.e. in $\Omega$ \ \ \ for all } 0 \le s < t \end{align} and that $y \in W^{1,2}([0,\infty);H^1(\Omega))$. More precisely, by Fatou's lemma and Lemma \ref{lemma: metric space-properties}(i) we get for a.e. $t$ \begin{align}\label{PDE1} \begin{split}
|y'|_{\mathcal{D}_\delta}(t) & = \lim_{s \to t}\frac{\mathcal{D}_\delta(y(s),y(t))}{|s-t|} = \lim_{s \to t} \delta^{-1}\Big(\frac{\delta^2\mathcal{D}_\delta(y(s),y(t))^2}{|s-t|^2} \Big)^{1/2} \\&\ge \delta^{-1} \Big( \int_\Omega \liminf_{s\to t} \Big( H_{\nabla y(t)} \Big[\frac{\nabla y(s) - \nabla y(t)}{|s-t|}, \frac{\nabla y(s) - \nabla y(t)}{|s-t|}\Big] \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + |s-t|^{-2} O(|\nabla y(t) - \nabla y(s)|^3) \Big) \Big)^{1/2} \\ & =\delta^{-1} \Big(\int_\Omega H_{\nabla y(t)}[\partial_t \nabla y(t),\partial_t \nabla y(t) ]\Big)^{1/2} = \delta^{-1}\Vert \sqrt{H_{\nabla y(t)}} \partial_t \nabla y(t)\Vert_{L^2(\Omega)}. \end{split} \end{align}
We now determine the derivative $\frac{{\rm d}}{{\rm d}t} \phi_\delta(y(t))$ of the absolutely continuous curve $\phi_\delta \circ y$. Fix $t$ such that $\lim_{s \to t}\frac{\mathcal{D}_\delta(y(s),y(t))}{|s-t|}$ exists, which holds for a.e. $t$. Then by Lemma \ref{lemma:C1} we find
$$\lim_{s \to \infty} \frac{\int_\Omega W(\nabla y(s)) - \int_\Omega W(\nabla y(t)) - \int_\Omega \partial_FW(\nabla y(t)) : (\nabla y(s) - \nabla y(t))}{s-t} = 0.$$ The previous estimate together with the convexity of $P$ yields \begin{align*} \frac{{\rm d}}{{\rm d}t} \phi_\delta(y(t)) & = \lim_{s \to t} \frac{\phi_\delta(y(s)) - \phi_\delta(y(t))}{s-t} \\&\ge \liminf_{s \to t} \frac{1}{\delta^2(s-t)}\int_\Omega \Big(\partial_FW(\nabla y(t)) : (\nabla y(s) - \nabla y(t)) \\& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \beta \partial_GP(\nabla^2 y(t)) : (\nabla^2 y(s) - \nabla^2 y(t)) \Big)\\ & = \liminf_{s \to t} \frac{1}{\delta^2(s-t)}\int_\Omega \big(\partial_FW(\nabla y(t))+ \beta\mathcal{L}^*_P(y(t)) \big) : (\nabla y(s) - \nabla y(t)), \end{align*} where as before $\beta = \delta^{2-\alpha p}$. In the last step we integrated by parts and used ${\rm div}(\mathcal{L}^*_P(y(t))) = {\rm div}(\mathcal{L}_P(\nabla^2 y(t)))$ by Lemma \ref{lemma: nonlin-slope}. Note that the last term is well defined as $\mathcal{L}^*_P(y(t)) \in L^2(\Omega)$ for a.e. $t$ by Lemma \ref{lemma: nonlin-slope} and \eqref{slopesolution}. Now \eqref{eq: derivative in Hilbert} implies \begin{align*} \frac{{\rm d}}{{\rm d}t} \phi_\delta(y(t)) & \ge \delta^{-2}\int_\Omega \sqrt{H_{\nabla y(t)}}^{-1}\big(\partial_FW(\nabla y(t)) + \beta\mathcal{L}^*_P(y(t)) \big): \sqrt{H_{\nabla y(t)}}\partial_t\nabla y(t). \end{align*} We find by Lemma \ref{lemma: nonlin-slope}, \eqref{PDE1}, and Young's inequality
$$ \frac{{\rm d}}{{\rm d}t} \phi_\delta(y(t)) \ge - \frac{1}{2} \big(|\partial \phi_\delta|^2_{\mathcal{D}_\delta}(y(t)) + |y'|^2_{\mathcal{D}_\delta}(t)\big) \ge \frac{{\rm d}}{{\rm d}t} \phi_\delta(y(t)),$$ where the last step is a consequence of the fact that $y$ is a curve of maximal slope with respect to $\phi_\delta$. Consequently, all inequalities employed in the proof are in fact equalities and we get $$ \sqrt{H_{\nabla y(t)}}^{-1}\big(\partial_FW(\nabla y(t)) + \beta\mathcal{L}^*_P(y(t)) \big) = -\sqrt{H_{\nabla y(t)}}\partial_t\nabla y(t)
$$
pointwise a.e. in $\Omega$. Equivalently, recalling $\partial_{\dot{F}}R(F,\dot{F}) = \frac{1}{2} \partial^2_{F_1^2} D^2(F,F)\dot{F} = H_{F}\dot{F}$ from \eqref{intro:R}, we obtain $$ \big(\partial_FW(\nabla y(t)) + \beta\mathcal{L}^*_P(y(t)) \big) + \partial_{\dot{F}}R(\nabla y(t),\partial_t \nabla y(t)) =0$$
pointwise a.e. in $\Omega$. Multiplying the equation with $\nabla \varphi$ for $\varphi \in W_0^{2,p}(\Omega)$, using again $\int_\Omega \mathcal{L}^*_P(y(t)) : \nabla \varphi = \int_\Omega\mathcal{L}_P(\nabla^2 y(t)) : \nabla \varphi$ by Lemma \ref{lemma: nonlin-slope}, and the definition of $\mathcal{L}_P(\nabla^2 y(t)) $, we conclude that $y$ is a weak solution (see \eqref{nonlinear equation2}).
\end{proof}
\noindent \textbf{Acknowledgements} This work has been funded by the Vienna Science and Technology Fund (WWTF) through Project MA14-009. M.F.~acknowledges support by the Alexander von Humboldt Stiftung and thanks for the warm hospitality at \'{U}TIA AV\v{C}R, where this project has been initiated. M.K.~acknowledges support by the GA\v{C}R-FWF project 16-34894L. Both authors were also supported by the M\v{S}MT \v{C}R mobility project 7AMB16AT015. We wish to thank Ulisse Stefanelli for turning \color{black} our \color{black} attention to this problem.
\typeout{References}
\end{document} |
\begin{document}
\title{
Cold Brew: Distilling Graph Node Representations with Incomplete or Missing Neighborhoods}
\begin{abstract} Graph Neural Networks (GNNs) have achieved state-of-the-art performance in node classification, regression, and recommendation tasks. GNNs work well when rich and high-quality connections are available. However, their effectiveness is often jeopardized in many real-world graphs in which node degrees have power-law distributions. The extreme case of this situation, where a node may have no neighbors, is called Strict Cold Start (SCS). SCS forces the prediction to rely completely on the node's own features. We propose \textbf{Cold Brew}, a teacher-student distillation approach to address the SCS and noisy-neighbor challenges for GNNs. We also introduce feature contribution ratio (FCR), a metric to quantify the behavior of inductive GNNs to solve SCS. We experimentally show that FCR disentangles the contributions of different graph data components and helps select the best architecture for SCS generalization. We further demonstrate the superior performance of Cold Brew on several public benchmark and proprietary e-commerce datasets, where many nodes have either very few or noisy connections. Our source code is available at \url{https://github.com/amazon-research/gnn-tail-generalization}.
\end{abstract}
\section{Introduction} \label{sec:intro}
\begin{wrapfigure}{r}{0.46\textwidth}
\subfigure{
\includegraphics[width=2.3in]{figs/motivation.png}
}
\subfigure{
\includegraphics[width=2.3in]{figs/viz_distri.pdf}
}
\caption{\small \textbf{Top:} Graph nodes may have a power-law (``long-tail'') connectivity distribution, with a large fraction of nodes (yellow) having few to no neighbors. \textbf{Bottom:} Long-tail distributions in real-world datasets, making modern GNNs fail to generalize to the tail/cold-start nodes.}
\label{fig:motivation} \end{wrapfigure}
Graph Neural Networks (GNNs) achieve state-of-the-art results across a wide range of tasks such as graph classification, node classification, link prediction, and recommendation~\citep{wu2020comprehensive,goyal2018graph,kherad2020recommendation,shaikh2017recommendation,silva2010graph,zhang2019star}. Most modern GNNs rely on the principle of message passing to aggregate each node's features from its (multi-hop) neighborhood \citep{gcn, GAT, graphsage, gin, sgc, klicpera2018predict}. Therefore, the success of GNNs relies on the presence of dense and high-quality connections.
Even inductive GNNs \citet{graphsage} learn a function of the node feature and the node neighborhood, which requires the neighborhood to be present during inference.
A practical barrier for widespread applicability of GNNs arises from the long-tail node-degree distribution existing in many large-scale real-world graphs. Specifically, the node degree distribution is power law in nature, with a majority of nodes having very few connections \citep{wsdm,ding2021zero, lam2008addressing,lu2020meta}. Figure~\ref{fig:motivation} (top) illustrates a long-tail distribution, accompanied with the statistics of several public datasets (bottom). Many information retrieval and recommendation applications face the scenario of \textit{Strict Cold Start} (\textbf{SCS}) \citep{li2019zero,ding2021zero}, wherein some nodes have no edges connected. Predicting for these nodes admittedly is even more challenging than the tail nodes in the graph.
In these cases, existing GNNs fail to perform well due to the sparsity or absence of the neighborhood.
In this paper, we develop GNN models that have {\it truly inductive} capabilities: one can learn effective node embeddings for ``orphaned'' nodes in a graph. This capability is important to fully realize the potential of large-scale GNN models on modern, industry-scale datasets with very long tails and many orphaned nodes. To this end, we adopt the teacher-student knowledge distillation procedure \citep{yang2021extract, chen2020self} and propose \textbf{Cold Brew} to distill the knowledge of a GNN teacher into a multilayer perceptron (MLP) student.
The Cold Brew framework addresses two key questions: (1) how we can efficiently distill the teacher's knowledge for the sake of tail and cold-start generalization, and (2) how can a student make use of this knowledge.
We answer these two questions by learning a latent node-wise embedding using knowledge distillation, which both avoids ``over-smoothness'' \citep{oono2020graph,li2018deeper,nt2019revisiting} and discovers latent neighborhoods, which are missing for the SCS nodes.
Note that in contrast to traditional knowledge distillation \citep{hinton2015distilling}, our aim is not to train a simpler student model to perform as well as the more complex teacher. Instead, we aim to train a student model that is better than the teacher in terms of generalizing to tail or SCS samples.
In addition, to help select the cold-start friendly model architectures, we develop a metric called \textit{Feature Contribution Ratio} (FCR) that quantifies the contribution of node features with respect to the adjacency structure in the dataset for a specific downstream task. FCR indicates the difficulty level in generalizing to tail and cold-start nodes and guides our principled selection of both teacher and student model architectures in Cold Brew. We summarize our key contributions as follows:
\begin{itemize}
\item To generalize better to tail and SCS nodes, we design the Cold Brew knowledge distillation framework: we enhance the teacher GNN by appending the node-wise Structural Embedding (SE) to strengthen the teacher's expressiveness, and design a novel mechanism for the MLP student to rediscover the missing ``latent/virtual neighborhoods,'' on which it can perform message passing.
\item We propose Feature Contribution Ratio (FCR), which
quantifies the difficulty in generalizing to tail and cold-start nodes. We leverage FCR in a principled ``screening process'' to select the best model architectures for both the GNN teacher and the MLP student.
\item As the existing GNN studies only evaluate on the entire graph and do not explicitly evaluate on head/tail/SCS, we uncover the hidden differences of head/tail/SCS by creating bespoke train/test splits. Extensive experiments on public and proprietary e-commerce graph datasets validate the effectiveness of Cold Brew in tail and cold-start generalization.
\end{itemize}
\subsection{Problem Setup}
\label{sec:pf} GNNs effectively learn node representations using two components in graph data: they process {\it node features} through distributed node-wise transformations and process {\it adjacency structure} through localized neighborhood aggregations. For the first component, GNNs apply shared feature transformations to all nodes regardless of the neighborhoods. For the second component, GNNs use permutation-invariant aggregators to collect neighborhood information.
We take the node classification problem in the sequel for the sake of simplicity. All our proposed methods can be easily adapted to other semi-supervised or unsupervised problem settings, which we show in Section \ref{sec:exps}. We denote the graph data of interest by $\mathcal{G}$ with node set $\mathcal{V}, ~\ |\mathcal{V}| = N$. Each node possesses a $d_{in}-$dimensional feature and a $d_{out}-$dimensional label (either $d_{out}$ classes or a continuous vector in the case of regression). Let ${\bf X}^{0} \in{\mathbb{R}}^{N\times d_{in}}$ and ${\bf Y}\in{\mathbb{R}}^{N\times d_{out}}$ be the matrices of node features and labels, respectively. Let $\mathcal{N}_i$ be the neighborhood of the $i$-th node, $0\leq i < N$. In large-scale graphs, $|\mathcal{N}_i|$ is often small for a (possibly substantial) portion of nodes. We refer to these nodes as \textit{tail nodes}. Some nodes may have $|\mathcal{N}_i| = 0$, and we refer to these extreme cold start cases as \textit{isolated nodes}.
A classical GNN learns representations for the $i^{th}$ node at the $l^{th}$ layer as a function of its representation and its neighborhood's representations at the $(l-1)^{th}$ layer: \begin{equation} \label{eq:gnn_standard} x_i^l := f\left( \{x_i^{l-1}\}, \{ x_j^{l-1}\}_{j\in\mathcal{N}_i} \right) \end{equation} where $f(\cdot)$ is a general function that applies node-wise transformation on node $x_i^{l-1}$ and aggregates information of its neighborhood $\{ x_j^{l-1}\}_{j\in\mathcal{N}_i}$ to obtain the final node representation. Given $i$'s input features $x^0_i$ and its neighborhood $\mathcal{N}_i$, one can use \eqref{eq:gnn_standard} to obtain its representation and predict $y_i$, making these models inductive.
We are interested in improving the performance of these GNNs on a set of tail and cold-start nodes, where $\mathcal{N}_i$ for node $i$ is either unreliable\footnote{For example, a user with only one movie watched or an item with too few purchases.} or absent. In these cases, applying \eqref{eq:gnn_standard} will yield a suboptimal node representation, since $\{ x_j^{l-1}\}_{j\in\mathcal{N}_i} $ will be unreliable or empty at inference time.
\section{Related Work}
GNNs learn by aggregating neighborhood information to learn node representations \citep{gcn, GAT, graphsage, gin, sgc, klicpera2018predict}. Inductive variants of GNNs such as GraphSAGE \citep{graphsage} require initial node features as well as the neighborhood information of each node to learn the representation. Most works on improving GNNs have focused on learning better aggregation functions, and methods that can work when the neighborhood is absent or noisy have not been sufficiently exploited, except two recent concurrent works \citep{hu2021graph, zhang2021graph}.
In the context of cold start, \citep{wsdm} and \citep{ding2021zero} employ a transfer learning approach. \citep{yang2021extract} proposes a knowledge distillation approach for GNN, while \citep{chen2020self} proposes a self-distillation approach. In all the above cases, the models need full knowledge of the neighbors of the cold-start nodes in question and do not address the case of noisy or missing neighborhoods. Another possible solution is to directly train an MLP that only takes node features. \citep{hu2021graph} proposes to learn graph embeddings with only node-wise MLP, while using contrastive loss to regularize the graph structure.
Some previous works have studied the relation between node feature similarity and edge connections and how that influences the selection of appropriate graph models. \citep{pei2020geom} proposed the homophily metric that categorizes graphs into assortative and disassortative classes. \citep{wang2021dissecting} dissected the feature propagation steps of linear GCNs from a perspective of continuous graph diffusion and analyzed why linear GCNs fail to benefit from more propagation steps. \citep{liu2020non} further studied the influence of homophily on model selection and proposed a non-local GNN.
\section{Strict Cold Start Generalization} \label{sec:scs}
We now address the problem of generalization to the tail and cold-start nodes, where the neighborhood information is missing/noisy (Section \ref{sec:intro}). A naive baseline is to train an MLP to map node features to labels. However, such a method would disregard all graph information, and we show via our Feature Contribution Ratio and other experimental results that for most assortative graph datasets, the node-wise MLP approach is suboptimal.
The key idea of our framework is the following: the GNN maps node features into a $d$-dimensional embedding space, and since the number of nodes $N$ is usually much bigger than the embedding dimensionality $d$, we end up with an overcomplete set for this space using the embeddings as the basis. This implies the possibility that any node representation can be cast as a linear combination of $K \ll N$ existing node representations. Our aim will be to train a student model that can accurately discover the combination of the best $K$ existing node embeddings of a target isolated node. We call this procedure \textit{latent/virtual neighborhood discovery}, which is equivalent to using MLPs to ``mimic'' the node representations learned by the teacher GNN.
We adopt the knowledge distillation procedure \citep{yang2021extract, chen2020self} to improve the quality of the learned embeddings for tail and cold-start nodes. We use a teacher GNN model to embed the nodes onto a low-dimensional manifold by utilizing the graph structure. Then, the goal of the student is to learn a mapping from the node features to this manifold without knowledge of the graph that the teacher has. We further aim to let the student model generalize to SCS cases where the teacher model fails, beyond just mimicking the teacher as standard knowledge distillation does.
\begin{figure}
\caption{\small \textbf{(a)}: The proposed Cold Brew framework.
In normal case (left upper), GNN relies on both node feature and adjacency structure to make prediction. In cold start case (left lower) when the adjacency structure is missing, the cold brew student model first estimate the adjacency structure, then use both node feature and adjacency structure to make prediction.
The ``SE'' (right) is the structural embedding learned by Cold Brew's teacher GNN.
\textbf{(b)}: Four atomic components deciding the GNN embeddings of node $i$. Our proposed FCR metric disentangles them into two models: the MLP that only considers Part 1 and Part 3, and label propagation that only considers Part 1 and Part 2.}
\label{fig:illu}
\end{figure}
\subsection{The Teacher Model of Cold Brew: Structural Embedding GNN} \label{sec:teacher model}
Consider a graph $\mathcal{G}$. For a Graph Convolutional Network with $L$ layers, the $l$-th layer transformation can be written as\footnote{Compared to Equation \eqref{eq:gnn_standard}, multiplication by ${\tilde{\bm{A}}}$ plays the role of aggregating both $\{x_i\}$ and $\{x_j\}_{i\in\mathcal{N}_i}$.}: ${\bf X}^{(l+1)}=\sigma({\tilde{\bm{A}}}{\bf X}^{(l)} {\bf W}^{(l)})$, where ${\tilde{\bm{A}}}$ is the normalized adjacency matrix, ${\tilde{\bm{A}}}={\bf D}^{-1/2}{\bf A}{\bf D}^{-1/2}$, ${\bf D}$ is the diagonal degree matrix, and ${\bf A}$ is the adjacency matrix. ${\bf X}^{(l)}\in{\mathbb{R}}^{N\times d_{1}}$ is the node representations in the $l$-th layer, ${\bf W}^{(l)}\in{\mathbb{R}}^{d_1\times d_2}$ is the feature transformation matrix, where the values of $d_1/d_2$ depend on layer $l$: $(d_1,d_2)=(d_{in},d_{hidden})$ for $l=0$, $(d_{hidden},d_{hidden})$ for $1\leq l\leq L-2$, and $(d_{hidden},n_{classes})$ for $l=L-1$. $\sigma(\cdot)=$ is the nonlinear functions applied to each layer, (e.g., ReLU). $Norm(\cdot)$ refers to an optional batch or layer normalization.
GNNs typically suffer from oversmoothing \citep{oono2020graph,li2018deeper,nt2019revisiting}, i.e., node representations become too similar to each other. Inspired by the positional encoding in Transformers \citep{vaswani2017attention}, we train the teacher GNN to learn an additional set of node embeddings that can be appended, which we term the {\bf Structural Embedding (SE)}. SE learns to incorporate extra information besides original node features (such as node labels in the case of semi-supervised learning) through gradient backpropagation. The existence of SE avoids the oversmoothing issue in GNNs: the transformations applied to different nodes are no longer the same for every node since the SE of each node is different and participates in the feature transformation. This can be of independent interest to GNN researchers.
Specifically, for each layer $l,0\leq l\leq L-1$, the Structural Embedding takes the form of a learnable matrix ${\bf E}^{(l)}$, and the SE-GNN layer forward pass can be written as: \begin{equation}\label{eq:SE} {\bf X}^{(l+1)}=\sigma\left(\tilde{\bm{A}}\left({\bf X}^{(l)} {\bf W}^{(l)}+{\bf E}^{(l)}\right)\right), {\bf X}^{(l)} \in{\mathbb{R}}^{N\times d_{1}}, {\bf W}^{(l)} \in{\mathbb{R}}^{d_1\times d_{2}}, {\bf E}^{(l)} \in{\mathbb{R}}^{N\times d_{2}} \end{equation} \paragraph{Remark 1:}Note that SE is not the same as the bias term in traditional feature transformation ${\bf X}^{(l+1)}=\sigma\left(\tilde{\bm{A}}\left({\bf X}^{(l)} {\bf W}^{(l)}+{\bf b}^{(l)}\right)\right)$; in the bias ${\bf b}\in{\mathbb{R}}^{N\times d_{2}}$, the rows are copied/shared across all nodes. In contrast, we have a different structural embedding for every node. \paragraph{Remark 2:} SE is also unlike traditional label propagation (LP)~\citep{iscen2019label,wang2020unifying,huang2020combining}. LP encodes label information through iterating ${\bf E}^{(t+1)}=(1-\alpha){\bf G}+\alpha{\tilde{\bm{A}}}{\bf E}^{(t)}$, where ${\bf G}$ is a one-hot encoding of ground truth for training node classes and zeros for test nodes, and $0<\alpha<1$ is the portion of mixture at each iteration.
SE-GNN enables node $i$ to learn to encode the self and neighbors' label information\footnote{This will be inferred in the case of missing labels.} into its own node embedding through ${\tilde{\bm{A}}}$. We use the Graph Convolutional Networks \citep{gcn}, combined with other building blocks proposed in recent literature including: (1) initial/dense/jumping connections, and (2) batch/pair/node/group normalization as the backbone of Cold Brew's teacher GNN. More details are described in Appendix \ref{appendix:search space}. We also apply a regularization term to the loss function, yielding the following loss function: \begin{equation} loss=CE({\bf X}^{(L)}_{train},{\bf Y}_{train})+\eta\Vert{\bf E}\Vert_2^2 \end{equation} where ${\bf X}^{(L)}_{train}$ is the model's embedding at the $L$-th layer, $CE({\bf X}^{(L)}_{train},{\bf Y}_{train})$ is the Cross Entropy loss between the model output ${\bf X}^{(L)}_{train}$ and the ground truth ${\bf Y}$ on the training set, and $\eta$ is a regularization coefficient (grid-searched for different datasets in practice). The Cross-Entropy loss can be replaced by any other appropriate loss depending on the task.
\subsection{The Student MLP Model of Cold Brew} \label{sec:student model}
We design the student to be composed of two MLP modules. Given a target node, the first MLP module imitates the node embeddings generated by the GNN teacher. Next, given any node, we find a set of virtual neighbors of that node from the graph. Finally, a second MLP attends to both the target node and the virtual neighborhood and transforms them into the embeddings of interest.
Suppose we would like to obtain the embedding of a potentially isolated target node $i$ given only its feature ${\bf x}_i$. From the teacher GNN, at each layer $l$, we have access to two sets of node embeddings: ${\bf X}^{(l)}{\bf W}^{(l)}$ and ${\bf E}^{(l)}$. Denote $\bar{{\bf E}}$ as the embeddings that the teacher GNN passes over to the student MLPs. We offer two options for $\bar{{\bf E}}$: it can be the final output of the teacher GNN (in this case, $\bar{{\bf E}}\in{\mathbb{R}}^{N\times d_{out}}:= {\bf X}^{(L)}$), or it can be the concatenation of all intermediate results of the teacher GNN, similar to \citep{romero2014fitnets}: $\bar{{\bf E}}\in{\mathbb{R}}^{N\times (d_{hidden}*(L-1)+d_{out})}:={\bf X}^{(L)}\bigcup_{l=0}^{L-1}({\bf E}^{(l)} + {\bf X}^{(l)}{\bf W}^{(l)})$, where $\bigcup$ is the concatenation of matrices at the feature dimension (second dimension). $\bar{{\bf E}}$ acts as the target for the first MLP and also the input to the second MLP.
The first MLP learns a mapping from the input node features ${\bf X}^{(0)}$ to $\bar{{\bf E}}$, i.e., for node $i$, $\hat{\bm{e}_i} = \xi_1({\bf x}^{(0)}_i)$, where $\hat{\bm{e}_i}$ is trained with supervised learning to reproduce $\bar{{\bf E}}[i,:]$. Then, we discover the virtual neighborhood by applying an attention-based aggregation of the existing embeddings in the graph before linearly combining them: \begin{equation}\label{eq:LEMLP} \tilde{\bm{e}_i}=softmax(\Theta_K(\hat{\bm{e}_i}\bar{{\bf E}}^\top))\bar{{\bf E}} \end{equation} where $\Theta_K(\cdot)$ is the top-$K$ hard thresholding operator: for $z\in{\mathbb{R}}^{1\times N}$: $[\Theta_K(z)]_j=z_j$ if $z_j$ is among the top-$K$ largest elements of $z$, and $\Theta_K(z)_j=-\infty$ otherwise. Finally, the second MLP learns a mapping $\xi_2: \left[{\bf x}_i, \tilde{\bm{e}_i}\right]\to{\bf y}_i$, where ${\bf y}_i={\bf Y}[i,:]$ is the ground truth for node $i$.
Equation \eqref{eq:LEMLP} first selects $K$ nodes from the $N$ nodes that the teacher GNN was trained on via the hard thresholding operator. $\tilde{\bm{e}_j}$ is then a linear combination of $K$ node\footnote{We abuse terminology here since $\bm{E}$ contains node and structural embeddings from multiple layers.} embeddings. Thus, every sample whether or not seen previously while training the GNN can be represented as a linear combination of these representations. The MLP $\xi_2 (\cdot)$ maps this representation to the final target of interest. Thus, we decompose every node embedding as a linear combination of an (overcomplete) basis.
The training of $\xi_1(\cdot)$ occurs by minimizing the mean squared error over the non-isolated nodes in the graph (mimicking the teacher's embeddings), and the training of $\xi_2(\cdot)$ occurs by minimizing the cross entropy (for the node classification task) or mean squared error (for the node regression task) on the training split of the tail and isolated part of the graph. An illustration of SE-MLP's inference procedure for the isolated nodes is shown in Figure~\ref{fig:illu}. When the number of nodes is large, the ranking procedure involved in $\Theta_K(\cdot)$ can be precomputed after training the first part and before training the second part.
\subsection{Model Interpretation From A Label Smoothing Perspective}
We quote Theorem 1 in \citep{wang2020unifying}: \textit{Suppose that the latent ground-truth mapping from node features to node labels is differentiable and L-Lipschitz. If the edge weights $a_{ij}$ approximately smooth ${\bf x}_i$ over its immediate neighbors with error $\epsilon_i$, i.e., ${\bf x}_i=\frac{1}{d_{ii}}\Sigma_{j\in\mathcal{N} }a_{ij}{\bf x}_j+\epsilon_i$, then the $a_{ij}$ also approximately smooth $y_i$ to bound within error $|y_i-\frac{1}{d_{ii}}\Sigma_{j\in\mathcal{N}_i }a_{ij} y_j| \leq L||\epsilon||_2+o(\mathrm{max}_{j\in\mathcal{N}_i}(||{\bf x}_j-{\bf x}_i||_2))$, where $o(\cdot)$ denotes a higher order infinitesimal.}
This theorem indicates that the errors of the label predictions are determined by the difference of the features after neighborhood aggregation: if $\epsilon_i$ is large, then the error in the label prediction is also large, and vice versa. However, with structural embeddings, each node $i$ also learns an independent embedding $\bar{{\bf E}}[:,i]$ during the aggregation, which changes $\frac{1}{d_{ii}}\Sigma_{j\in\mathcal{N} }a_{ij}{\bf x}_j+\epsilon_i$ into $\frac{1}{d_{ii}}\Sigma_{j\in\mathcal{N} }a_{ij}{\bf x}_j+\bar{{\bf E}}[:,i]+\epsilon_i$. Deduced from this theorem, the structural embedding $\bar{{\bf E}}$ is important for the teacher model: it allows higher flexibility and expressiveness in learning the residual difference between nodes, and hence the error $\epsilon_i$ can be lowered if $\bar{{\bf E}}$ is properly learned.
From this theorem, one can also see the necessity of introducing neighborhood aggregations like that of the Cold Brew student model. If one directly applies MLP models without neighborhood aggregation, the $\epsilon_i$ turns out to be non-negligible, leading to higher losses in the label predictions. However, Cold Brew introduces the neighborhood aggregation mechanism so that the second part of the student MLP takes over the aggregation of neighborhood generated by the first MLP. Therefore, Cold Brew eliminates the above residual error even in the absence of the actual neighborhood.
\section{Model Selection And Graph Component Disentanglement With Feature Contribution Ratio}
\label{sec:fir}
We now discuss Feature Contribution Ratio (FCR), a metric to quantify the difficulty of learning representations under the truly inductive cold-start case, and a hyperparameter optimization approach to select the best suitable model architecture that helps tail and cold-start generalization.
As conceptually illustrated in Figure~\ref{fig:illu}, there are four atomic components contributing to the learned embedding of node $i$ in the graph: 1. the label of $i$ ({\it self-label}); 2. the label of neighbors of $i$ ({\it neighbor-labels}); 3. the features of $i$ ({\it self-feature}); 4. the features of neighbors of $i$ ({\it neighbor-features}). To quantize the SCS generalization difficulty, we first divide these four components into two submodules to disentangle the contributions of the {\it node features} with respect to the {\it adjacency structure} of the graph dataset. Then, we quantize it based on the assumption that the SCS generalization difficulty is proportional to the contribution ratio of the {\it node features}.
We posit that a submodule that learns accurate node representations must include the node's (self) label, so that training can be performed via backpropagation. What remains is to use the label with other atomic components to construct two specialized models that each make use of only the node features or the adjacency structure. For the first submodule, we build an MLP that maps the self-features to self-labels, ignoring any neighborhood information present in the dataset. For the second submodule, we adopt a Label Propagation (LP) method \citep{CNS}\footnote{We ignore the node features and use the label logits as explained in \citep{CNS}.} to learn representations from self- and neighbor-labels. This model ignores any node feature information.
With the above two submodules, we introduce the Feature Contribution Ratio (FCR) that characterizes the relative importance of the node features and the graph structure. Specifically, for graph dataset ${\mathcal{G}}$, we define the contribution of a submodule to be the \textit{residual performance} of the submodule compared to a full-fledged GNN (e.g., Equation \eqref{eq:gnn_standard}) using both the node feature as well as the adjacency structure. Denote $z_{MLP}, z_{LP},$ and $z_{GNN}$ as the performance of the MLP submodule, LP submodule, and the full GNN on the test set, respectively. If $z_{MLP} \ll z_{GNN}$, then $FCR({\mathcal{G}})$ is small and the graph structure is important, and noisy or missing neighborhood information will hurt model performance. Based on this intuition, we build SCR as: \begin{small} \begin{subequations}\label{eq:fir}
\begin{align} \delta_{MLP}=&z_{GNN}-z_{MLP},\quad \delta_{LP} = z_{GNN}-z_{LP} \\ FCR({\mathcal{G}}) = & \begin{cases} \frac{\delta_{LP}}{\delta_{MLP}+\delta_{LP}} \times100\% & z_{MLP} \leq z_{GNN} \\
1 + \frac{|\delta_{MLP}|}{|\delta_{MLP}|+\delta_{LP}} \times100\% & z_{MLP} > z_{GNN} \end{cases}
\end{align}
\end{subequations} \end{small}
\textbf{Interpreting FCR values.} For a particular graph ${\mathcal{G}}$, if {$0\% \leq FCR({\mathcal{G}}) < 50\%$}, it means $z_{GNN} > z_{LP} > z_{MLP}$, and the neighborhood information in ${\mathcal{G}}$ is mainly responsible for the GNN achieving good performance. If {$50\% \leq FCR({\mathcal{G}}) < 100\%$}, then $z_{GNN} > z_{MLP} > z_{LP}$, and the node features contribute more to the GNN's performance. If {$FCR({\mathcal{G}}) \geq 100\%$}, then $z_{MLP} > z_{GNN} > z_{LP}$, and the node aggregation in GNNs can actually lead to reduced performance compared to pointwise models. This case usually happens for some disassortative graphs, where the majority of neighborhoods hold labels different from that of the center nodes (e.g., as observed by \citep{liu2020non}).
\textbf{Integrate FCR as a tool to design teacher and student models.} For some graph datasets and models, the SCS generalization can be challenging without neighborhood information (i.e., $z_{GNN} > z_{LP} > z_{MLP}$). We hence consider FCR as a principled ``screening process'' to select model architectures for both teacher and student that own the best inductive bias for SCS generalization.
To achieve this, during the computation of FCR, we perform exhaustive grid search of the architectures (residual connection types, normalization, hidden layers, etc.) for the MLP, LP, and GNN modules, and pick the best-performing variant. Detailed definition of the search space can be found in Appendix~\ref{appendix:search space}. We treat this grid search procedure as a special case of architecture selection and hyperparameter optimization for Cold Brew.
We observe that FCR is able to identify the GNN and MLP architectures that are particularly friendly for SCS generalization, improving our method design.
In experiments, we observe that different model configurations are favored by different datasets, and we use the found optimal teacher GNN and student MLP architectures to perform Cold Brew. More detailed discussions are presented in section~\ref{sec:5.3}.
\section{Experiments and Discussion} \label{sec:exps}
In this section, we first evaluate FCR by training GNNs on several commonly used graph datasets and observing how well they generalize to tail and cold-start nodes. We also compare it to the graph homophiliy metric $\beta$ proposed in \citep{pei2020geom}. Next, we apply Cold Brew to these datasets and compare its generalization ability to baseline graph-based and pointwise MLP models on these datasets. We also show results on proprietary industry datasets.
\subsection{Datasets and Splits} \label{sec:datasets}
We perform experiments on five open-source datasets and four proprietary datasets. The proprietary e-commerce datasets, ``E-comm 1/2/3/4'', refer to graphs subsampled from anonymized logs of an e-commerce store. They are sampled so as to not reflect the actual raw traffic distributions, and results are provided with respect to a baseline model for these datasets. The different number suffixes refer to different product subsets, and the labels indicate product categories that we wish to predict. Node features are text embeddings obtained from a fine-tuned BERT model. We show FCR values for the public datasets.
The statistics of the datasets are summarized in Table \ref{tab:datasets}.
We create training and test splits of the data in order to specifically study the generalization ability of Cold Brew to tail and cold-start nodes. In the following tables, the {\it head} data corresponds to the top $10\%$ highest-degree nodes in the graph and the subgraph that they induce. We take the data that corresponds to the bottom $10\%$ of the degree distribution, and artifically remove all the edges emanating from these nodes. We then refer to this set of nodes as the {\it isolation} data. The {\it tail} data corresponds to the $10\%$ nodes in the remaining graph with lowest (non-zero) degree and the subgraph that they induce. All the zero-degree nodes are in the {\it isolation} data. The {\it Overall} data refers to the training/test splits without distinguishing head/tail/isolation.
\begin{minipage}[t]{1\columnwidth}
\begin{center}
\setlength{\tabcolsep}{1pt}
\renewcommand{1.2}{1.2}
\resizebox{0.8\columnwidth}{!}{
\begin{tabular}{l|c|c|c|c|c|c|c|c|cc}
\toprule[1.5pt]
& Cora & Citeseer & Pubmed & Arxiv & Chameleon & E-comm1 & E-comm2 & E-comm3 & E-comm4 \\
\midrule
Num. of Nodes & 2708 & 3327 & 19717 & 169343 & 2277 & 4918 & 29352 & 319482 & 793194 \\
Num. of Edges & 13264 & 12431 & 108365 & 2315598 & 65019 & 104753 & 1415646 & 8689910 & 22368070 \\
Max Degree & 169 & 100 & 172 & 13161 & 733 & 277 & 1721 & 4925 & 12452 \\
Mean Degree & 4.90 & 3.74 & 5.50 & 13.67 & 28.55 & 21.30 & 48.23 & 27.20 & 28.19\\
Median Degree & 4 & 3 & 3 & 6 & 13 & 10 & 21 & 15 & 14 \\
Isolated Nodes $\%$ & 3\% & 3\% & 3\% & 3\% & 3\% & 6\% & 5\% & 5\% & 6\% \\
\bottomrule[1.5pt]
\end{tabular}}
\captionof{table}{\small The statistics of datasets selected for evaluation.}
\label{tab:datasets}
\end{center}
\end{minipage}
\subsection{FCR Evaluation}
\label{sec:beta} In Table~\ref{fig:heng}, the top part presents the FCR results together with the homophily metric $\beta$ from \citep{pei2020geom} (Equation \ref{eq:beta}). The bottom part shows the prediction accuracies for the head and the tail nodes. As can be seen from the table, FCR differs among datasets and is negatively correlated with the homophily metric (with Pearson correlation coefficient -0.76). The high absolute correlation value and its negative sign indicate that the more similar the nodes are to their neighborhoods, the more difficult it is to generalize with MLP based models. FCR is thus an indicator of the tail generalization difficulty. Evaluations on more datasets (including the datasets where FCR $>100\%$) are presented in Appendix ~\ref{appendix:all exps}. \begin{small} \begin{equation}\label{eq:beta}
\beta({\mathcal{G}})=\frac{1}{|{\mathcal{V}}|}\sum_{v\in{\mathcal{V}}}\frac{\textrm{the number of $v$'s direct neighbors that have the same labels as $v$}}{\textrm{the number of $v$'s directly connected neighbors}}\times100\%
\end{equation} \end{small}
\begin{table}[h]
\centering
\resizebox{0.6\columnwidth}{!}{
\begin{tabular}{c|c c c c c cccccc }
\toprule[1.5pt]
& Cora & Citeseer & Pubmed & Arxiv & Chameleon \\
\hline
GNN & 86.96& 72.44& 75.96& 71.54& 68.51& \\
MLP & 69.02& 56.59& 73.51& 54.89& 58.65& \\
Label Propagation & 78.18& 45.00 & 67.8 & 68.26& 41.01\\
FCR \% & 32.86 \% & 63.39 \% & 76.91\% & 16.45\% & 73.61\% \\
$\beta({\mathcal{G}})$ \% & 83\% & 71\% & 79\% & 68\% & 25\% & \\
\hline
$head-tail (GNN)$ & 4.44& 23.98& 11.71& 5.9 & 0.24\\
$head-isolation (GNN)$ & 31.01& 33.09& 15.21& 28.81& 1.55& \\
\bottomrule[1.5pt]
\end{tabular}
}
\caption{\small Top part: FCR and its components. The $\beta$ metric is added as a reference. Bottom part: the performance difference of GNN on the head/tail and head/isolation splits. Here, the ``tail/isolation'' means the 10\% least connected, and isolated nodes in the graph.}
\label{fig:heng}
\end{table}
\subsection{Experimental Results on Tail Generalization With Cold Brew} \label{sec:5.3}
In Table~\ref{tab:proposed}, we present the performance of Cold Brew together with baselines on the tail and the isolation splits, across several different datasets. All the models in the table are evaluated on the training data, and are evaluated on the tail or isolation splits discussed in section~\ref{sec:datasets}.
In Table~\ref{tab:proposed} {\it GCN} refers to the the best configuration found through FCR-guided grid search (check Appendix~\ref{appendix:search space} for details), without Structural Embedding. Correspondingly, {\it GCN + SE} refers to the best FCR-guided configuration with Structural Embedding, which is the default teacher model of Cold Brew. {\it GraphSAGE} refers to \citep{graphsage}, {\it Simple MLP} refers to a simple node-wise MLP that has two hidden layers with 128 hidden dimensions, and {\it GraphMLP} refers to \citep{hu2021graph}. The results for the e-commerce datasets are presented as relative improvements to the baseline (each value is the difference with respect to the value of the {\it GCN 2 layers} on same dataset of the same split). We do not disclose the absolute numbers due to proprietary reasons.
As shown in Table~\ref{tab:proposed}, Cold Brew's student MLP improves accuracy on isolated nodes by up to +11\% on the e-commerce datasets and +2.4\% on the open-source datasets. Cold Brew's student model handles isolated nodes better, and the teacher GNN also achieves better performance on the tail split compared to all other models. Especially when compared with GraphMLP, Cold Brew's student MLP consistently performs better. This can be explained from their different mechanisms: GraphMLP encodes graph knowledge implicitly in the learned weights, while Cold Brew explicitly attends to neighborhoods even when they are absent. More detailed comparisons can be found in Appendix \ref{appendix:all exps}.
\begin{table}[h]
\centering
\resizebox{0.85\textwidth}{!}{
\begin{tabular}{l|lc|ccccc|cccccccccc}
\toprule[1.5pt]
\multirow{2}{*}{Splits} & \multicolumn{2}{c}{\multirow{2}{*}{Metrics/Models}} & \multicolumn{5}{|c|}{\bf Open-Source Datasets} & \multicolumn{4}{c}{\bf Proprietary Datasets} & \\
\cmidrule(r){4-8} \cmidrule(r){9-12}
& & & Cora & Citeseer & Pubmed & Arxiv & Chameleon & E-comm1 & E-comm2 & E-comm3 & E-comm4 \\
\midrule
\multirow{7}{*}{Isolation}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 58.02 & 47.09 & 71.50 & 44.51 & 57.28 & $-$ & $-$ & $-$ & $-$\\
& & GraphSAGE & 66.02 & 51.46 & 69.87 & 47.32 & 59.83 & +3.89 & +4.81 & +5.24 & +0.52\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 68.40 & \textbf{53.26} & 65.84 & 51.03 & 60.76 & +5.89 & +9.85 & +5.83 & +6.42\\
& & GraphMLP & 65.00 & 52.82 & 71.22 & 51.10 & \textbf{63.54 } & +6.27 & +9.46 & \textbf{+5.99 } & +7.37\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers & 58.37 & 47.78 & \textbf{73.85} & 45.20 & 60.13 & +0.27 & +0.76 & -0.50 & +1.22\\
& & Student MLP & \textbf{69.62} & 53.17 & 72.33 & \textbf{52.36} & 62.28 & \textbf{+7.56} & \textbf{+11.09} & +5.64 & \textbf{+9.05}\\
\midrule
\multirow{7}{*}{Tail}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 84.54 & \textbf{56.51} & 74.95 & 67.74 & 58.33 & $-$ & $-$ & $-$ & $-$\\
& & GraphSAGE & 82.82 & 52.77 & 73.07 & 63.23 & 61.26 & -3.82 & -3.07 & -2.87 & -6.42\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 70.76 & 54.85 & 67.21 & 52.14 & 50.12 & -0.37 & +1.74 & -0.13 & -0.45\\
& & GraphMLP & 70.09 & 55.56 & 71.45 & 52.40 & 52.84 & -0.33 & +1.64 & \textbf{+1.27 }& +0.80\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers &\textbf{ 84.66 }& 56.32 & \textbf{75.33} & \textbf{68.11} & \textbf{60.80} & \textbf{+0.85} & +0.44 & -0.60 & +1.10\\
& & Student MLP & 71.80 & 54.88 & 72.54 & 53.24 & 51.36 & +0.32 & \textbf{+3.09 }& -0.18 & \textbf{+2.09}\\
\bottomrule[1.5pt]
\end{tabular}}
\caption{\small The performance comparisons on the isolation and tail splits of different datasets. The full comparisons on head/tail/isolation/overall data are in the Appendix \ref{appendix:all exps}. GCN+SE 2 layers is Cold Brew's teacher model. Cold Brew outperforms GNN and other MLP baselines, and achieves the best performance on the isolation splits as well as some tail splits.}
\label{tab:proposed}
\end{table}
\begin{center} \begin{minipage}[t]{0.43\columnwidth}
\resizebox{0.96\textwidth}{!}{
\begin{tabular}{l|l|cccccccccccccc}
\toprule[1.5pt]
\multirow{2.5}{*}{Splits} & \multicolumn{1}{c}{\multirow{2.5}{*}{Models}} & \multicolumn{4}{|c}{\bf Datasets}\\
\cmidrule(r){3-6}
& & Cora & Citeseer & Pubmed & E-comm1 \\
\midrule
\multirow{5}{*}{Isolation}
& GCN 2 layers & 34.10 & 50.41 & 51.52 & $-$ \\
\cmidrule(r){2-2}
& TailGCN & 36.13 & 51.48 & 51.19 & +2.18 \\
\cmidrule(r){2-2}
& Meta-Tail2Vec & 36.92 & 50.90 & 51.62 & +2.34 \\
\cmidrule(r){2-2}
& Cold Brew's MLP & \textbf{44.59} & \textbf{55.14} & \textbf{54.82} & \textbf{+5.39} \\
\bottomrule[1.5pt]
\end{tabular}}
\captionof{table}{\small Link prediction Mean Reciprocal Ranks (MRR) on the isolation data. Note that Cold Brew outperforms baselines specifically built for generalizing to the tail. }
\label{tab:nov linkp} \end{minipage} \begin{minipage}[t]{0.4\columnwidth} \end{minipage} \begin{minipage}[t]{0.43\columnwidth}
\resizebox{0.96\textwidth}{!}{
\begin{tabular}{l|l|cccccccccccccc}
\toprule[1.5pt]
\multirow{2.5}{*}{Splits} & \multicolumn{1}{c}{\multirow{2.5}{*}{Models}} & \multicolumn{4}{|c}{\bf Datasets}\\
\cmidrule(r){3-6}
& & Cora & Citeseer & Pubmed & E-comm1 \\
\midrule
\multirow{5}{*}{Isolation}
& GCN 2 layers & 58.02 & 47.09 & 71.50 & $-$ \\
\cmidrule(r){2-2}
& TailGCN & 62.04 & 51.87 & 72.10 & +3.14 \\
\cmidrule(r){2-2}
& Meta-Tail2Vec & 61.16 & 50.46 & 71.80 & +2.80 \\
\cmidrule(r){2-2}
& Cold Brew's MLP & \textbf{69.62} & \textbf{53.17} & \textbf{72.33} & \textbf{+7.56} \\
\bottomrule[1.5pt]
\end{tabular}}
\captionof{table}{\small Node classification accuracies with other baselines specifically created to generalize to the tail. Cold Brew outperforms these methods when edge data is absent in the graph. }
\label{tab:nov classi} \end{minipage} \end{center}
We also evaluated the link prediction performance by replacing the node classification loss with the link prediction loss. On the manually created isolation split, the model is asked to recover the ground truth edges which are manually removed. The results are shown in Table \ref{tab:nov linkp}. The baseline models shown in table are TailGCN \citep{vetter1991eta} and Meta-Tail2Vec \citep{liu2020towards}. A comparison over these models on the node classification on the isolation split is provided in Table \ref{tab:nov classi}. As observed from the table \ref{tab:nov linkp} and \ref{tab:nov classi}, Cold Brew outperformed TailGCN and Meta-Tail2Vec on the isolation split, since both TailGCN and Meta-Tail2Vec explicitly are not zero-shot methods and require explicit neighborhood nodes, hence their performance degrades when the neighborhood is empty and padded by zero vectors.
The full performance on other splits are listed in Table \ref{tab:fullexp} in the appendix as a reference. The results across all splits in Table \ref{tab:fullexp} provide evidence for a few phenomena, for example, the high FCR means that the graph structure does not add too much information for the task at hand, and that GNN type models tend to perform better on the head while MLP type models tend to perform better on the tail/isolation splits. On the other hand, the proposed Structural Embeddings imply a potential to alleviate the over-smoothness \citep{oono2020graph,li2018deeper,nt2019revisiting} and bottleneck \citep{alon2020bottleneck} issues observed in deep GCN models. As shown in table Table~\ref{tab:64}, Cold Brew's GCN (GCN + SE) significantly outperformed the traditional GCN on 64 layers: the former has 34\% test accuracy higher on Cora, 23\% higher on Citeseer, and similar on others.
Finally, the improvement over isolation and tail splits (especially the isolation split) comes with a cost: we observed a performance drop for the student MLP model on the head and several other datasets' tail splits, compared with the naive GCN model. However, Cold Brew specifically targets the challenging strict cold start issues, as a new compelling alternative for in these cases. Meanwhile in the non-cold-start cases, the traditional GCN models can still be used to obtain good performance. Note that even on the head splits, the proposed GNN teacher model of Cold Brew still outperformed traditional GNN models. We hence consider as promising future work to adaptively switch between using Cold Brew teacher and student models, based on the current node connectivity degree.
\begin{table}[h]
\centering
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{l|l|ccccc|cccccccccc}
\toprule[1.5pt]
\multirow{2}{*}{Splits} & \multirow{2}{*}{Metrics/Models} & \multicolumn{5}{c|}{\bf Open-Source Datasets} & \multicolumn{4}{c}{\bf Proprietary Datasets} & \\
\cmidrule(r){3-7} \cmidrule(r){8-11}
& & Cora & Citeseer & Pubmed & Arxiv & Chameleon & E-comm1 & E-comm2 & E-comm3 & E-comm4 \\
\midrule
\multirow{2}{*}{Overall}
& GCN 64 layers & 40.04 & 23.66 & 75.65 & 65.53 & 58.14 & -5.49 & -6.59 & -6.13 & -3.57\\
& GCN + SE 64 layers & \textbf{74.23} & \textbf{46.80} & \textbf{78.12} & \textbf{69.28} & \textbf{59.88} & \textbf{-1.71 }& \textbf{-2.92} & \textbf{-3.29} & \textbf{-0.06}\\
\midrule
\multirow{2}{*}{Head}
& GCN 64 layers & 46.46 & 49.84 & 85.89 & 67.53 & 67.16 & -5.60 & -6.24 & -6.05 & -3.16\\
& GCN + SE 64 layers & \textbf{87.38} &\textbf{ 71.18 }& \textbf{86.81} & \textbf{71.35} & \textbf{69.63} & \textbf{-1.78} &\textbf{ -2.17} &\textbf{ -2.79} & \textbf{-0.35} \\
\midrule
\multirow{2}{*}{Tail}
& GCN 64 layers & 45.14 & 24.42 & 71.89 & 63.91 & 56.48 & -3.85 & -3.62 & -3.84 &\textbf{ -1.14}\\
& GCN + SE 64 layers & \textbf{79.56} & \textbf{36.52} & \textbf{74.88} & \textbf{65.19} & \textbf{61.73} & \textbf{-2.42} & \textbf{-2.52} &\textbf{ -3.68} & -1.23\\
\midrule
\multirow{2}{*}{Isolation}
& GCN 64 layers & 39.97 & 22.12 & 68.57 & 40.03 & 57.60 & -4.66 & -4.63 & -4.93 & \textbf{-1.89} \\
& GCN + SE 64 layers & \textbf{40.33} & \textbf{24.53} & \textbf{71.22} &\textbf{ 41.18} & \textbf{60.13} & \textbf{-3.08} & \textbf{-3.02} & \textbf{-4.00} & -2.32\\
\bottomrule[1.5pt]
\end{tabular}}
\caption{\small The comparisons of Cold Brew's GCN and the traditional GCN for deep layers. When the number of layers is large, Cold Brew's GCN retains good performance while the traditional GCN without SE suffers from the ``over-smoothess'' and degrades. Even with shallow layers, Cold Brew's GCN is better than traditional GCN.}
\label{tab:64}
\end{table}
\section{Conclusion}
In this paper, we studied the problem of generalizing GNNs to the tail and strict cold start nodes, whose neighborhood information is either sparse/noisy or completely missing. We proposed a teacher-student knowledge distillation procedure to better generalize to the isolated nodes. We added an independent set of structural embeddings in GNN layers to alleviate node over-smoothness, and also proposed a virtual neighbor discovery step for the student model to attend to latent neighborhoods. We additionally present the FCR metric to quantify the difficulty of truly inductive representation learning and to optimize our model architecture design. Experiments demonstrated the consistently superior performance of our proposed framework on both public benchmarks and proprietary datasets.
\appendix
\section{More illustrations} \label{sec:illu}
The more detailed inference procedures for GNN and Cold Brew are illustrated in Figure~\ref{fig:flowillu}.
\begin{figure}
\caption{Inference procedure illustration for GNN and Cold Brew.}
\label{fig:flowillu}
\end{figure}
\section{Search Space Details} \label{appendix:search space}
In computing FCR, we include a search space of model hyperparameters for GNN, MLP, and LP in order to find the best suitable configurations for distillation.
For the GNN model, we take GCN as a backbone and performed grid search over the number of hidden layers, whether it has the structural embedding, the type of residual connection, and the type of normalization. For the number of hidden layers, we considered 2, 4, 8, 16, 32, and 64. For the types of residual connections, we include: (1) connection to the last layer~\citep{li2019deepgcns, li2018deeper}, (2) initial connection to the initial layer~\citep{chen2020simple, klicpera2018predict, zhang2020revisiting}, (3) dense connection to all preceding layers~\citep{li2019deepgcns, li2018deeper, li2020deepergcn, luan2019break}, and (4) jumping connection combining all the preceding layers only at the final graph convolutional layer~\citep{xu2018representation, liu2020towards}. For the types of normalizations, we grid search over batch normalization (BatchNorm)~\citep{ioffe2015batch}, pair normalization (PairNorm)~\citep{zhao2019pairnorm}, node normalization (NodeNorm)~\citep{zhou2020understanding}, mean normalization (MeanNorm)~\citep{yang2020revisiting}, and differentiable group normalization (GroupNorm)~\citep{zhou2020towards}. For types of graph dropout methods, we include Dropout~\citep{srivastava2014dropout}, DropEdge~\citep{rong2020dropedge}, DropNode~\citep{dropedge2}, and LADIES~\citep{NEURIPS2019_91ba4a44}.
For the architecture design for Cold Brew's MLP, we conducted hyperparameter search over the number of hidden layers, the existence of residual connection, the hidden dimensions, and the optimizers. The number of hidden layers is searched over 2, 8, 16, and 32. The number of hidden dimensions is searched over 128 and 256. The optimizer is searched over (Adam(lr=0.001) Adam(lr=0.005), Adam(lr=0.02), SGD(lr=0.005))
For Label Propagation, we conducted hyperparameter search over the number of propagations, the propagation matrix type, and the mixing coefficient $\alpha$ \citep{CNS}. The number of propagations is searched over 10, 20, 50, 100, and 200. The propagation matrix type is searched over adjacency matrix and normalized Laplacian matrix. The mixing coefficient $\alpha$ is searched over 0.01, 0.1, 0.5, 0.9, and 0.99.
The best GCN, MLP, and LP configurations are reported in Tables~\ref{tab:bgcn}, \ref{tab:bmlp}, and \ref{tab:blp}, respectively.
\begin{table}[t] \centering \resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c | c | c | c c } \toprule[1.5pt] \multirow{2}{*}{\textbf{\makecell[c]{Dataset}}} & \multicolumn{4}{c}{\textbf{\makecell{Best GCN}}} \\ \cmidrule(r){2-5}
& num layers & whether has SE & residual type & normalization type \\
\hline Cora & 2 layer & has structural embedding & no residual & PairNorm \\ Citeseer & 2 layer & has structural embedding & no residual & PairNorm \\ Pubmed & 16 layer & has structural embedding & initial connection & GroupNorm \\ Arxiv & 4 layer & has structural embedding & initial connection & GroupNorm \\ Chameleon & 2 layer & has structural embedding & initial connection & BatchNorm \\ \bottomrule[1.5pt] \end{tabular}} \caption{Best GCN configurations.} \label{tab:bgcn} \end{table}
\begin{table}[t] \centering \resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c | c | c | c c } \toprule[1.5pt] \multirow{2}{*}{\textbf{\makecell[c]{Dataset}}} & \multicolumn{4}{c}{\textbf{\makecell{Best MLP}}} \\ \cmidrule(r){2-5} & hidden layers & residual connection & hidden dimensions & optimizer \\
\hline Cora & 2 layer & no residual & 128 & Adam(lr=0.001) \\ Citeseer & 4 layer & no residual & 128 & Adam(lr=0.001)\\ Pubmed & 2 layer & no residual & 256 & Adam(lr=0.02)\\ Arxiv & 2 layer & no residual & 256 & Adam(lr=0.001)\\ Chameleon & 2 layer & no residual & 256 & Adam(lr=0.001)\\ \bottomrule[1.5pt] \end{tabular} } \caption{Best MLP configurations.} \label{tab:bmlp} \end{table}
\begin{table}[h] \centering \resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c | c | c c c } \toprule[1.5pt] \multirow{2}{*}{\textbf{\makecell[c]{Dataset}}} & \multicolumn{4}{c}{\textbf{\makecell{Best LP}}} \\ \cmidrule(r){2-4} & number of propagations & propagation matrix type & mixing coefficient & \\
\hline Cora & 50 & Laplacian matrix & 0.1 \\ Citeseer & 100 & Laplacian matrix & 0.01 \\ Pubmed & 50 & Adjacency matrix & 0.5 \\ Arxiv & 100 & Laplacian matrix & 0.5 \\ Chameleon & 50 & Laplacian matrix & 0.1 \\ \bottomrule[1.5pt] \end{tabular}} \caption{Best Label Propagation configurations.} \label{tab:blp} \end{table}
\section{The performance on all splits of the data} \label{appendix:all exps} The performance evaluations on all splits are listed in Table~\ref{tab:fullexp}. The FCR evaluation on more datasets are presented in Figure~\ref{fig:hengfull}. We hypothesize that a high FCR means that the graph does not add too much information for the task at hand. We indeed see evidence for this hypothesis in Table \ref{tab:fullexp}, where for the Pubmed dataset $(FCR \approx 77\% )$, the MLP-type models tend to outperform GNN-type models in all splits On the other hand, regardless of FCR, for almost all datasets, the MLP-type models outperform the GNN-type models on the isolation split, and a few on the tail split, while the GNN-type models are superior in other splits.
\begin{table}[h]
\centering
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{l|lc|ccccc|cccccccccc}
\toprule[1.5pt]
\multirow{2}{*}{Splits} & \multicolumn{2}{c}{\multirow{2}{*}{Metrics/Models}} & \multicolumn{5}{|c|}{\bf Open-Source Datasets} & \multicolumn{4}{c}{\bf Proprietary Datasets} & \\
\cmidrule(r){4-8} \cmidrule(r){9-12}
& & & Cora & Citeseer & Pubmed & Arxiv & Chameleon & E-comm1 & E-comm2 & E-comm3 & E-comm4 \\
\midrule
\multirow{6}{*}{Isolation}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 58.02 & 47.09 & 71.50 & 44.51 & 57.28 & $-$ & $-$ & $-$ & $-$\\
& & GraphSAGE & 66.02 & 51.46 & 69.87 & 47.32 & 59.83 & +3.89 & +4.81 & +5.24 & +0.52\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 68.40 & \textbf{53.26} & 65.84 & 51.03 & 60.76 & +5.89 & +9.85 & +5.83 & +6.42\\
& & GraphMLP & 65.00 & 52.82 & 71.22 & 51.10 & \textbf{63.54 } & +6.27 & +9.46 & \textbf{+5.99 } & +7.37\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers & 58.37 & 47.78 & \textbf{73.85} & 45.20 & 60.13 & +0.27 & +0.76 & -0.50 & +1.22\\
& & Student MLP & \textbf{69.62} & 53.17 & 72.33 & \textbf{52.36} & 62.28 & \textbf{+7.56} & \textbf{+11.09} & +5.64 & \textbf{+9.05}\\
\midrule
\multirow{6}{*}{Tail}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 84.54 & \textbf{56.51} & 74.95 & 67.74 & 58.33 & $-$ & $-$ & $-$ & $-$\\
& & GraphSAGE & 82.82 & 52.77 & 73.07 & 63.23 & 61.26 & -3.82 & -3.07 & -2.87 & -6.42\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 70.76 & 54.85 & 67.21 & 52.14 & 50.12 & -0.37 & +1.74 & -0.13 & -0.45\\
& & GraphMLP & 70.09 & 55.56 & 71.45 & 52.40 & 52.84 & -0.33 & +1.64 & \textbf{+1.27 }& +0.80\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers &\textbf{ 84.66 }& 56.32 & \textbf{75.33} & \textbf{68.11} & \textbf{60.80} & \textbf{+0.85} & +0.44 & -0.60 & +1.10\\
& & Student MLP & 71.80 & 54.88 & 72.54 & 53.24 & 51.36 & +0.32 & \textbf{+3.09 }& -0.18 & \textbf{+2.09}\\
\midrule
\multirow{6}{*}{Head}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 88.68 & 80.37 & 85.79 & 73.35 & 67.49 & $-$ & $-$ & \textbf{$-$ }& $-$\\
& & GraphSAGE & 87.75 & 74.81 & 86.94 & 70.85 & 62.08 & -4.26 & -4.17 & -3.50 & -7.46\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 74.33 & 72.00 & 89.00 & 56.34 & 60.82 & -16.74 & -18.10 & -16.73 & -16.51\\
& & GraphMLP & 72.45 & 69.83 & 89.00 & 56.65 & 62.44 & -15.96 & -18.08 & -15.33 & -15.41\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers & \textbf{89.39 } & \textbf{80.76} & 87.83 &\textbf{74.01} & 70.56 & \textbf{+1.11} & \textbf{+0.47 }& -0.39 & \textbf{+1.28}\\
& & Student MLP & 74.53 & 72.33 & \textbf{90.33} & 57.41 & 61.28 & -15.28 & -17.42 & -17.02 & -15.41\\
\midrule
\multirow{6}{*}{Overall}
& \multirow{2}{*}{GNNs} & GCN 2 layers & 84.89 & 70.38 & 78.18 & 71.50 & 59.30 & $-$ & \textbf{$-$} & \textbf{$-$} & $-$\\
& & GraphSAGE & 80.90 & 66.21 & 76.73 & 68.33 & 70.02 & -3.09 & -3.86 & -2.58 & -5.48\\
\cmidrule(r){2-2}
& \multirow{2}{*}{MLPs} & Simple MLP & 69.02& 56.59& 73.51& 54.89& 58.65 & -12.69 & -12.86 & -12.68 & -13.16\\
& & GraphMLP & 71.87 & 68.22 & 82.03 & 53.81 & 57.67 & -12.26 & -12.01 & -10.80 & -11.41\\
\cmidrule(r){2-2}
& \multirow{2}{*}{Cold Brew} & GCN + SE 2 layers & \textbf{86.96} & \textbf{72.44} & 79.03 & \textbf{71.92} & \textbf{68.51} & \textbf{+0.65} & -0.24 & -0.77 & \textbf{+1.43}\\
& & Student MLP & 72.36 & 67.54 &\textbf{ 82.00} & 54.94 & 59.07 & -11.25 & -11.51 & -11.55 & -11.21\\
\bottomrule[1.5pt]
\end{tabular}}
\caption{The performance comparisons on all splits of different datasets.}
\label{tab:fullexp}
\end{table}
\begin{table}
\centering
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{c|c c c c c cccccc }
\toprule[1.5pt]
& Cora & Citeseer & Pubmed & Arxiv & Cham. & Squ. & Actor & Cornell & Texas & Wisconsin \\
\hline
GNN & 86.96& 72.44& 75.96& 71.54& 68.51& 31.95& 59.79& 65.1 & 61.08& 81.62 \\
MLP & 69.02& 56.59& 73.51& 54.89& 58.65& 38.51& 37.93& 86.26& 83.33& 85.42 \\
Label Propagation & 78.18& 45.00 & 67.8 & 68.26& 41.01& 22.85& 29.69& 32.06& 52.08& 40.62 \\
FCR \% & 32.86 \% & 63.39 \% & 76.91\% & 16.45\% & 73.61\% & 141.91\% & 57.93\% & 139.04\% & 171.2 \% & 108.48 \% \\
$\beta({\mathcal{G}})$ \% & 83\% & 71\% & 79\% & 68\% & 25\% & 22\% & 24\% & 11\% & 6\% & 16\% \\
\hline
$head-tail (GNN)$ & 4.44& 23.98& 11.71& 5.9 & 0.24& -6.51& 2.22& -4.37& -11.26& -33.92 \\
$head-isolation (GNN)$ & 31.01& 33.09& 15.21& 28.81& 1.55& -4.85& 22.61& -18.68& -24.62& -29.23 \\
\bottomrule[1.5pt]
\end{tabular}
}
\caption{Top part: FCR and its components. The $\beta$ metric is added as a reference. Bottom part: the performance difference of GNN on the head/tail and head/isolation splits.}
\label{fig:hengfull}
\end{table}
\section{Visualizing the learned embeddings}
Figure~\ref{fig:viz} visualizes the last-layer embeddings of different models after t-SNE dimensionality reduction. In the figure, colors denotes node labels and all nodes are marked as dots, with isolation subset nodes additionally marked with \textit{x}'s and the tail subset additionally marked with triangles. Although the GCN model did a decent job in separating different classes, a significant portion of the tail and isolation nodes fall into wrong class clusters. Cold Brew's MLP is more discriminative in the tail and isolation splits.
\def2.4in{2.4in}
\begin{figure}
\caption{Top two subfigures: the last-layer embeddings of GCN and Simple MLP. Bottom two subfigures: the last-layer embeddings of GraphMLP and Cold Brew's student MLP. All embeddings are projected to 2D with t-SNE. Cold Brew's MLP has the fewest isolated nodes that are misplaced into wrong clusters.}
\label{fig:viz}
\end{figure}
\end{document} |
\begin{document}
\title{The Supersingularity of Hurwitz Curves} \author{Erin Dawson, Henry Frauenhoff, Michael Lynch, Amethyst Price,\\ Seamus Somerstep, Eric Work \\
Graduate Student Advisor: Dean Bisogno\\ Faculty Advisor: Rachel Pries} \maketitle \makebox[\linewidth]{\rule{12cm}{0.4pt}} \begin{abstract} We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\ell}: X^nY^\ell + Y^nZ^\ell + Z^nX^\ell = 0$, with $n$ and $\ell$ relatively prime, is supersingular over the finite field $\mathbb{F}_{p}$ if and only if there exists an integer $i$ such that $p^i \equiv -1 \bmod (n^2 - n\ell + \ell^2)$. If this holds, we prove that it is also true that the curve is maximal over $\mathbb{F}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.
\begin{center} \textit{Keywords}: \\Hurwitz Curve, Hasse-Weil Bound, Maximal Curve,\\ Minimal Curve, Fermat Curve, Supersingular Curve\\ \textit{Subject Classification}: \\Primary: 11G20, 11M38, 14H37, 14H45, 11E81; \\Secondary: 11G10, 14H40, 14K15 \end{center}
\end{abstract} \makebox[\linewidth]{\rule{12cm}{0.4pt}}
\section{Introduction} In 1941, Deuring defined the basic theory of supersingular elliptic curves. Supersingular curves are useful in error-correcting codes called Goppa codes. They also have potential applications to quantum resistant cryptosystems.
In this paper we determine a condition for supersingularity of Hurwitz curves $H_{n,\ell}$ when $n$ and $\ell$ are relatively prime. In particular we show that every supersingular Hurwitz curve $H_{n,\ell}$ is maximal over some finite field. We also provide a classification of supersingular Hurwitz curves with genus less than 5 over fields with characteristic less than 37 and find some restrictions on the genera of Hurwitz curves.
\section{Background information}
We first define the Hurwitz curve and the Fermat curve. Next we define the zeta function of a curve. From the zeta function we compute the normalized Weil numbers which we use to study supersingularity. We must also state the Hasse-Weil bound in order to define maximality and minimality.
\subsection{The Hurwitz curve and the Fermat curve}\label{sec:curves} Let $n$, $\ell$, and $d$ be positive integers. Let $F$ be a field.
\begin{definition}[Hurwitz curve $H_{n,\ell}$] The {\em Hurwitz curve} $H_{n,\ell}$ over $F$ is given by the projective equation \begin{equation*} H_{n,\ell}: X^nY^\ell+Y^nZ^\ell+Z^nX^\ell=0. \end{equation*} \end{definition} Throughout this paper, set $m=n^2-n\ell+\ell^2$. The Hurwitz curve $H_{n,\ell}$ has the following genus \begin{equation*} g=\frac{m+2-3\gcd{(n,\ell)}}{2} \end{equation*} and is smooth when the characteristic $p$ of $F$ is relatively prime to $m$.
\begin{definition}[Fermat curve $\mathcal{F}_{d}$] The {\em Fermat curve} of degree $d$ over $F$ is given by the projective equation \begin{equation*} \mathcal{F}_{d}: U^{d}+V^{d}+W^{d}=0. \end{equation*} \end{definition} The Fermat curve $\mathcal{F}_d$ has genus $\frac{(d-1)(d-2)}{2}$ and is smooth when the characteristic $p$ of $F$ does not divide $d$. Note that the Hurwitz curve $H_{n,\ell}$ is covered by the Fermat curve of degree $m = n^2 - n\ell+\ell^2$; see Section \ref{sec:covers} for more details.
\subsection{Zeta Function}
Let $\mathbb{F}_q$ be a finite field of cardinality $q$ where $q$ is a power of a prime $p$. For a curve $C$ defined over $\mathbb{F}_q$, denote the number of points on $C$ by $\#C(\mathbb{F}_q)$. For extensions of $\mathbb{F}_q$, define $N_s = \#C(\mathbb{F}_{q^s})$. \begin{definition}[Zeta function] The {\em zeta function} of a curve $C/\mathbb{F}_q$ is the series \begin{equation} \label{Eq:Zeta} {Z}(C/\mathbb{F}_q,T)=\text{exp}\bigg(\sum_{s=1}^{\infty}\frac{{N}_{s}T^s}{s}\bigg). \end{equation} \end{definition} Rationality of the zeta function for curves was proven by Weil \cite{MR0027151, MR0029393}. In particular, Weil showed that the zeta function can be written as \begin{equation} \label{Eq:Zeta-Weil} {Z}(C/\mathbb{F}_q,T) = \frac{ L(C/\mathbb{F}_q,T)}{(1-T)(1-qT)}. \end{equation} The $L$-polynomial, $L(C/\mathbb{F}_q,T) \in \mathbb{Z}[T]$, has degree $2g$ \cite[page 152]{IrelandRosen}, \begin{equation} \label{Eq:L-poly} L(C/\mathbb{F}_q,T)=1+C_1T+...+C_{2g}T^{2g}. \end{equation} The $L$-polynomial of a curve $C$ over $\mathbb{F}_q$ with genus $g$ factors in $\mathbb{C}[T]$ as \begin{equation*} L(C/\mathbb{F}_q,T) = \displaystyle\prod_{i=1}^{2g}(1-\alpha_iT). \end{equation*}
Furthermore, $|\alpha_i| = \sqrt{q}$ for each $1 \leq i \leq 2g$ \cite[page 155]{IrelandRosen}. The normalized Weil numbers (NWNs) are the normalized reciprocal roots of the $L$-polynomial. \begin{definition}[Normalized Weil Numbers] The {\em Weil numbers} of $C/\mathbb{F}_q$ are the reciprocal roots $\alpha_i$ of $L(C/\mathbb{F}_q,T)$ for $1 \leq i \leq 2g$. The {\em normalized Weil numbers} are the values $\alpha_i/\sqrt{q}$ for $1 \leq i \leq 2g$. \end{definition} \begin{note} \label{Rmk:NWN/Extentions} If $\{\alpha_{1}, \ldots ,\alpha_{2g}\}$ are the normalized Weil numbers over $\mathbb{F}_q$, then $\{\alpha_{1}^i, \ldots ,\alpha_{2g}^i\}$ are the normalized Weil numbers over $\mathbb{F}_{q^i}$. \end{note}
The coefficients of $L(C/\mathbb{F}_q,T)$ follow a pattern. For $k \in {\mathbb N}$, we denote the set of partitions of $k$ by $\text{par}(k)$ and the length of a partition $\gamma$ by $\text{len}(\gamma)$.
\begin{lemma} \label{Lem:ZetaCoeffs} In Equation (\ref{Eq:L-poly}) for $0 \leq k \leq 2g$, the coefficient $C_k$ has the form \begin{align*} C_k=\sum_{\gamma \in {\rm par}(k)} \frac{\displaystyle \prod_{j \in \gamma}\frac{N_j}{j}}{\rm len(\gamma)!}-\sum_{i=0}^{k-1}(C_i \sum_{\mu =0}^{k-i}q^\mu). \end{align*} \end{lemma}
\begin{proof} Equation \eqref{Eq:Zeta} can be expanded using the Taylor series of the exponential function \begin{align*} Z(C/\mathbb{F}_q,T)=\displaystyle \sum_{i=0}^{\infty}\frac{(N_1T+\frac{N_2}{2}T^2+\ldots+\frac{N_{2g}}{2g}T^{2g})^i}{i!}. \end{align*} Collecting terms up through $T^3$ gives a pattern to follow: \begin{equation}\label{eq:zetaside1} Z(C/\mathbb{F}_q,T)=1 + (N_1)T + \bigg(\frac{N_2}{2}+\frac{N_1^2}{2}\bigg)T^2 + \bigg(\frac{N_3}{3}+\frac{N_1N_2}{2}+\frac{N_1^3}{6}\bigg)T ^3 + \ldots. \end{equation}
The key step is to recognize that the subscripts on the $N_j$ are the partitions of $k$. The coefficient on $T^k$ can be written as \begin{align*} \sum_{\gamma \in \text{par}(k)} \frac{\displaystyle \prod_{j \in \gamma}\frac{N_j}{j}}{\text{len}(\gamma)!}. \end{align*}
Equation \eqref{Eq:Zeta-Weil} gives a simplified version of $Z(C/\mathbb{F}_q,T)$. Using the Taylor series for each of the denominator terms as well as equation \eqref{Eq:L-poly} yields the following expansion: \begin{equation}\label{eq:zetaside2} Z(C/\mathbb{F}_q,T)=(1+C_1T+...+C_{2g}T^{2g})(1+T+T^2+\ldots)(1+qT+q^2T^2+\ldots). \end{equation} Expanding and collecting terms, the coefficients on $T^k$ are given by \begin{equation*} \sum_{i=0}^{k-1}(C_i \sum_{j =0}^{k-i}q^j)+C_k. \end{equation*} Setting equation \eqref{eq:zetaside1} and equation \eqref{eq:zetaside2} equal and comparing coefficients gives a linear system allowing one to solve for $C_k$ in terms of the values of $N_s$. \end{proof}
\subsection{The Newton Polygon and Supersingularity} Fix a curve $C/\mathbb{F}_q$ with associated $L$-polynomial $L(C/\mathbb{F}_q,T)$.
\begin{definition}[Supersingularity] \label{Rmk:SS-RootsOfUnity} The curve $C$ is {\em supersingular} if all its normalized Weil numbers are roots of unity. \end{definition}
Another way to check if $C$ is supersingular is with its Newton polygon.
\begin{definition}[Normalized Valuation on $\mathbb{F}_{p^r}$]\label{defn:val} Let $n = p^lk$ be an integer with $p \nmid k$. We denote the normalized $\mathbb{F}_{p^r}$-valuation of $n$ by $\text{val}_{p^r}(n) = \frac{l}{r}$ and the prime-to-$p$ part of $n$ by $n_p = k$. If $n=0$, we say $\text{val}_{p^r}(0) = \infty$. \end{definition} \begin{definition}[Newton Polygon] Fix a curve $C/\mathbb{F}_{p^r}$ with $L$-polynomial in the form of equation \eqref{Eq:L-poly}. The {\em Newton polygon} of $C/\mathbb{F}_{p^r}$ is the lower convex hull of the points $\{(i,\text{val}_{p^r}(C_i)) \mid 0 \leq i \leq 2g \}$. \end{definition}
\begin{note}\label{rmk:newton} Because $C_0 = 1$ for every curve $C/\mathbb{F}_{p^r}$, the Newton polygon will always have initial point $(0,0)$. Likewise the final coefficient of $L(C/\mathbb{F}_{p^r}, T)$ is always $C_{2g} = p^{rg}$. For this reason the Newton polygon always has terminal point $(2g,g)$. \end{note} From Remark \ref{rmk:newton}, we can see that the Newton polygon of a curve $C$ over $\mathbb{F}_{p^r}$ is always a union of line segments on or below the line $y = \frac{1}{2}x$ with increasing slopes. \begin{note} A curve $C/\mathbb{F}_{q}$ is supersingular if and only if its Newton polygon is a line segment with slope $\frac{1}{2}$. \end{note}
\subsection{Minimality and Maximality} As a consequence of the Weil conjectures, the number of points on a curve $C/\mathbb{F}_q$ is controlled by the Hasse-Weil bound: \begin{equation*} 1+q-2g\sqrt{q} \leq \#C(\mathbb{F}_{q}) \leq 1+q+2g\sqrt{q}. \end{equation*} The Hasse-Weil bound for curves was proven by Weil \cite{MR0027151}.
\begin{definition}[Minimal] A curve $C/\mathbb{F}_q$ is {\em minimal} if \begin{equation*} \#C(\mathbb{F}_q) = 1+q-2g\sqrt{q}. \end{equation*} \end{definition} \begin{definition}[Maximal] A curve $C/\mathbb{F}_q$ is {\em maximal} if \begin{equation*} \#C(\mathbb{F}_q) = 1+q+2g\sqrt{q}. \end{equation*} \end{definition}
\begin{note}[{\cite[page 22]{MR0027151}}, {\cite[page 69]{MR0029522}}]\label{rmk:nwnlemma} The curve C is maximal over $\mathbb{F}_q$ (resp.\ minimal over $\mathbb{F}_q$) if and only if all its normalized Weil numbers are -1 (resp.\ 1) over $\mathbb{F}_q$. \end{note}
In the following remark, we use the notation that $\zeta_k$ is the primitive $k^{\text{th}}$ root of unity $e^{\frac{2\pi i}{k}}$. Notice that there is a power $s$ such that $\zeta^s_k = -1$ if and only if $k$ is even. \begin{lemma} Let $C$ be a supersingular curve over $\mathbb{F}_q$. Suppose the normalized Weil numbers of $C/\mathbb{F}_q$ are of the form $\zeta^{t_1}_{k_1}, \ldots, \zeta^{t_{2g}}_{k_{2g}}$. Assume $\gcd(k_i,t_i) = 1$. The curve $C$ is maximal over $\mathbb{F}_{q^{r}}$ if and only if \begin{itemize} \item there exists $s \geq 1$ and $b_i$ odd, such that $k_i = 2^s(b_i)$ \item and $r$ is an odd multiple of {}$2^{s-1}{\rm lcm}(b_1, \ldots, b_n)$. \end{itemize} \end{lemma}
\begin{proof} Assume $C$ is maximal over $\mathbb{F}_{q^r}$. By Remark \ref{rmk:nwnlemma}, the curve $C$ is maximal over $\mathbb{F}_{q^r}$ if and only if $\zeta^{rt_i}_{k_i} = -1$ for all $i$. Consequently, $k_i$ is even for all $i$. Thus $k_i = 2^{s_i} b_i$ for some positive integer $s_i$ and odd integer $b_i$. The condition $\zeta^{rt_i}_{k_i} = -1$ for all $i$ implies that there exists an $s$ such that $s=s_i$ for all $i$ and $r$ is an odd multiple of $2^{s-1}\text{lcm}(b_1, \ldots, b_n)$.
For the converse, the conditions imply that the normalized Weil numbers of $C$ over $\mathbb{F}_{q^r}$ are all $-1$. \end{proof}
\section{Curve maps and covers} \subsection{Aoki's Curve} Let $\alpha = (a,b,c) \in \mathbb{N}^3$ with $a+b+c = m$. Note that $S_3$, the symmetric group on three letters, acts on $\alpha$ by permuting the coordinates. For $\sigma \in S_3$ we denote the action by $\alpha^\sigma$. We say two triples $\alpha = (a_1,a_2,a_3)$ and $\beta = (b_1,b_2,b_3)$ are equivalent, denoted $\alpha \approx \beta$, if there exist elements $t \in (\mathbb{Z}/m)^*$ and $\sigma \in S_3$ such that \begin{align*} (a_1,a_2,a_3) \equiv (tb_{\sigma(1)},tb_{\sigma(2)},tb_{\sigma(3)}) \bmod m. \end{align*}
In \cite{Aoki1} and \cite{Aoki2}, Aoki studies curves of the form \begin{equation*} D_\alpha: v^{m}=(-1)^{c}u^{a}(1-u)^{b}. \end{equation*} He provides the following conditions for when $D_\alpha$ is supersingular. \begin{theorem}[{\cite[Theorem 1.1]{Aoki2}}] \label{Thm:Aoki1.1} The curve $D_\alpha$ is supersingular over $\mathbb{F}_{p^{r}}$ if and only if at least one of the following conditions holds: \begin{itemize} \item $p^{i} \equiv -1 \bmod m$ for some $i$. \item $\alpha =(a,b,c) \approx (1,-p^{i},p^{i}-1)$ for some integer $i$ such that $d = \gcd(p^{i}-1,m) > 1$ and $p^{j} \equiv -1 \bmod \frac{m}{d}$ for some integer $j$. \end{itemize}\end{theorem}
\subsection{Covers of $H_{n,\ell}$ by $\mathcal{F}_m$}\label{sec:covers} In Section \ref{sec:curves}, we noted that the Hurwitz curve $H_{n,\ell}$ is covered by the Fermat curve $\mathcal{F}_m$ where $m = n^2-n\ell+\ell^2$. On an affine patch the Fermat and Hurwitz curves are given by the following equations \begin{align*} \mathcal{F}_m: u^m + v^m + 1 &= 0 \\ H_{n,l}: x^n y^\ell + y^n + x^\ell &= 0. \\ \end{align*} Then the following covering map is provided by \cite[Lemma 4.1]{AKT} \begin{align*} \phi: \mathcal{F}_m &\to H_{n,\ell} \\ (u, v)&\mapsto (u^{n}v^{-l}, u^{l}v^{n-l}). \end{align*}
Furthermore, it is known that $\mathcal{F}_m$ is supersingular over $\mathbb{F}_p$ if and only if $p^i \equiv -1 \bmod m$ for some integer $i$ \cite[Prop.\ 3.10]{shiodakatsura}. See also \cite[Theorem 3.5]{Yui}. In \cite[Theorem 5]{Tafazolian} it is shown that $\mathcal{F}_m$ is maximal over $\mathbb{F}_{p^{2i}}$ if and only if $p^{i}\equiv -1 \bmod m$. \begin{note} \label{serre} If $X \to Y$ is a covering of curves defined over $\mathbb{F}_{p^r}$, then the normalized Weil numbers of $Y/\mathbb{F}_{p^r}$ are a subset of the normalized Weil numbers of $X/\mathbb{F}_{p^r}$, see \cite{serre}. \end{note} Thus when a covering curve is supersingular (or maximal or minimal) the curve it covers is as well.
\subsection{A Birational Transformations} In \cite{Carbonne}, Bennama and Carbonne show that $H_{n,\ell}$ is isomorphic to a curve with affine equation \begin{equation} \label{Eq:Carbonne Hurwitz} y'^m=x'^{\lambda}(x'-1) \end{equation} via the following variable change. Suppose $1 \leq \ell < n$ and $\gcd(n,\ell)=1$. Then there exist integers $\theta$ and $\delta$ such that $1 \leq \theta \leq \ell$, $1 \leq \delta \leq n-1$, and $n \theta -\delta \ell =1$. Let $\lambda = \delta n - \theta (n-\ell)$ and $m=n^2-n\ell+\ell^2$. The birational transformation is as follows \begin{equation*} \begin{gathered} \begin{cases}
x=(-x')^{-\delta}((-1)^{\lambda}y')^n\\
y=(-x')^{-\theta}((-1)^{\lambda}y')^\ell \end{cases} \end{gathered} \end{equation*} \centerline{\text{and}} \begin{equation*} \begin{gathered} \begin{cases}
x'=-x^\ell y^{-n}\\
y'=(-1)^{\lambda}x^{\theta}y^{-\delta}. \end{cases} \end{gathered} \end{equation*} Equation \eqref{Eq:Carbonne Hurwitz} is very similar to the equation for $D_\alpha$ that Aoki studies but there are small differences. The following argument shows that these can be reconciled. Consequently, this variable change can be used to apply Aoki's results to Hurwitz curves.
Notice that equation \eqref{Eq:Carbonne Hurwitz} is divisible by $(x'-1)$ while Aoki studies curves whose equation contains a $(1-x')$ factor. Aoki requires that $a+b+c=m$ so the exponent on the negative sign is important. Inspecting equation \eqref{Eq:Carbonne Hurwitz} we see that $m$ will always be odd since $(n,\ell)=1$. Consequently, this negative sign is not an issue. Since $m$ is always odd we can replace $v$ with $-v$. This choice allows us to pick $c = m-a-b$. Then $b = 1$ and $a = \lambda$.
\section{Supersingular Hurwitz Curves} We arrive at explicit conditions on supersingularity for $H_{n,\ell}$ when $n$ and $\ell$ are relatively prime. We use results from \cite{Carbonne} and \cite{Aoki1} to accomplish this. We will be using affine equations for the Hurwitz curve in this section. \begin{lemma} \label{Lem:SS iff 2 conditions} If $n$ and $\ell$ are relatively prime then $x^{n}y^{\ell}+y^{n}+x^{\ell} = 0$ is supersingular over $\mathbb{F}_{p}$ if and only if at least one of the following conditions holds. \begin{enumerate} \item There exists $i\in\mathbb{Z}_{>0}$ such that $p^{i} \equiv -1 \bmod m$.
(In this case the Fermat curve covering the Hurwitz curve is maximal over $\mathbb{F}_{p^{2i}}$.)
\item There exists $i\in\mathbb{Z}_{>0}$ with $d = (p^{i}-1,m) > 1$ such that \begin{align*} (\delta(n-\ell)+\ell\theta-1,1,-(\delta(n-\ell)+\ell\theta)) \approx (1,-p^{i},p^{i}-1) \end{align*} and $p^{j} \equiv -1 \bmod(\frac{m}{d})$ for some integer $j$. \end{enumerate} \end{lemma}
\begin{proof} We use the variable substitution from \cite{Carbonne} to apply Aoki's results to Hurwitz curves. We use the substitutions: \begin{itemize} \item $m=n^{2}-n\ell+\ell^{2}$, \item $a=\lambda=\delta(n-\ell)+\ell\theta-1$, \item $b=1$, \item $c=m-(\delta(n-\ell)+\ell\theta)$. \end{itemize} Combining these with Aoki's results completes the proof. \end{proof} \begin{note} \label{Lem:(n,l,m)=1} If $n$ and $\ell$ are relatively prime, then $n$ and $\ell$ are relatively prime to $n^{2}-n\ell+\ell^2$. \end{note} \begin{theorem} \label{Thm:ss/Fp} Suppose $n$ and $\ell$ are relatively prime and $m = n^2 - n\ell + \ell^2$. Then $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$ if and only if $p^{i} \equiv -1 \bmod m$ for some positive integer $i$. \end{theorem}
\begin{proof} If $p^i \equiv -1 \bmod m$ for some positive integer $i$, then $\mathcal{F}_m$ is supersingular over $\mathbb{F}_p$ by \cite[Prop.\ 3.10]{shiodakatsura}. Recall from section \ref{sec:covers} that $\mathcal{F}_m$ covers $H_{n,\ell}$. Thus $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$ by Remark \ref{serre}.
Suppose $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$. By Lemma \ref{Lem:SS iff 2 conditions} it is enough to show condition 2 in Lemma \ref{Lem:SS iff 2 conditions} can not happen. We begin by simplifying it using the substitution $\theta=\frac{1+\ell\delta}{n}$ and reducing modulo $m$ to show that condition 2 is equivalent to $(\frac{\ell}{n}-1,1,-\frac{\ell}{n})\approx(1,-p^{i},p^{i}-1)$ for some $i$ such that $d = (p^{i}-1,m) > 1$ and $p^{j} \equiv -1 \mod (\frac{m}{d})$ for some integer $j$. Recall that $\alpha \approx \alpha' \text{ if } \alpha=t\alpha'^\sigma$ for some $t \in (\mathbb{Z}/m)^{*}$ and $\sigma\in S_3$. We will show that $p^i-1$ and $m$ are relatively prime. We label the three coordinates of $(\frac{\ell}{n}-1,1,-\frac{\ell}{n})$ as $(a,b,c)$ and the three coordinates of $(1,-p^{i},p^{i}-1)$ as $(A,B,C)$.
The proof will address six cases accounting for the orbit of $(A,B,C)$ under the action of $S_3$. In each case we will show that $\gcd(p^{i}-1,m)=1$. Specifically, we show $d=1$ by taking these congruences modulo $d$. By Remark \ref{Lem:(n,l,m)=1} we know that $n^{-1}$ exists modulo $m$ and modulo $d$. Finally, note that $\frac{\ell}{n}$ is relatively prime to $d$.
\begin{itemize} \item $(a,b,c)\equiv t(A,B,C) \bmod m$: Comparing $c$ and $tC$ yields \begin{align*} -\frac{\ell}{n}\equiv t(p^{i}-1) \bmod m. \end{align*} Consequently, $\frac{\ell}{n}\equiv 0 \bmod d$. Therefore, $d = 1$.
\item {$(a,b,c)\equiv t(B,A,C) \bmod m$}: Comparing $a$ with $tB$ and $b$ with $tA$ yields \begin{align*} \frac{\ell}{n}-1 &\equiv -tp^i \bmod m \\ 1 &\equiv t \bmod m. \end{align*} Substituting we have $\frac{\ell}{n}\equiv p^{i}-1 \bmod m$. Reducing modulo $d$ produces $\frac{\ell}{n}\equiv 0 \bmod d$, thus $d = 1$. \item {$(a,b,c)\equiv t(A,C,B) \bmod m$}: Comparing $b$ and $tC$ yields \begin{align*} -\frac{\ell}{n}\equiv t(p^{i}-1) \bmod m. \end{align*} This is identical to the first case. \item {$(a,b,c)\equiv t(C,B,A) \bmod m$}: Comparing $a$ and $tC$ yields \begin{align*} \frac{\ell}{n} - 1 \equiv t(p^i -1) \bmod m. \end{align*}
Thus $\frac{\ell}{n} -1 \equiv 0 \bmod d$. Recall by the definition of $m$ and selection of $d$, we have $d \mid n^2 - n\ell + \ell^2$. Hence, $d$ divides $1 - \frac{\ell}{n} + (\frac{\ell}{n})^2$. We conclude $d|(\frac{\ell}{n})$, thus $d = 1$. \item {$(a,b,c)\equiv t(C,A,B) \bmod m$}: Comparing $b$ with $tA$ and $c$ with $tB$ yields \begin{align*} 1 &\equiv t \bmod m \\ \frac{\ell}{n} &\equiv tp^i \bmod m. \end{align*} This case is completed as in the previous case. \item {$(a,b,c)\equiv t(B,C,A) \bmod m$}: Comparing $b$ with $tC$ yields \begin{align*} 1 \equiv t(p^{i}-1) \bmod m. \end{align*} Modulo $d$ this reduces to $1\equiv 0 \bmod d$. Therefore, $d = 1$. \end{itemize} \end{proof}
\begin{note}\label{rmk:TT} There is a family of Hurwitz type curves with affine equations $\mathcal{C}_{a_1,a_2,n_1,n_2}: x^{n_1} y^{a_1} + y^{n_2} + x^{a_2} = 0$. Set $\delta = a_1 a_2 - a_2 n_2 + n_1 n_2$. When $q = p^r$ is coprime to $\delta$ then the curve $C_{a_1,a_2,n_1,n_2}$ is $\mathbb{F}_q$-covered by the Fermat curve $\mathcal{F}_\delta$ of degree $\delta$. In \cite[Theorem 2.9]{MR3562537} Tafazolian and Torres show that under certain numerical conditions the statements \begin{itemize}
\item the Fermat curve $\mathcal{F}_\delta$ is maximal over $\mathbb{F}_{q^2}$;
\item the Hurwitz type curve $\mathcal{C}_{1,a_2,n_1,n_2}$ is maximal over $\mathbb{F}_{q^2}$;
\item and $q+1 \equiv 0 \mod \delta$ \end{itemize} are all equivalent.
The Hurwitz type curve $\mathcal{C}_{\ell,\ell,n,n}$ is the Hurwitz curve $H_{n,\ell}$. Thus in the case that $\ell = a_1=a_2$ and $n = n_1 = n_2$, Theorem \ref{Thm:ss/Fp} generalizes \cite[Theorem~2.9]{MR3562537}. \end{note}
\begin{note} Consider the family of curves with affine equations \begin{align*} N_{a_1,a_2,n_1,n_2} : x^{n_1} y^{a_1} + k_1 y^{n_2} + k_2 x^{a_2} = 0 \end{align*} over $\mathbb{F}_{p^r}$ with $k_1,k_2 \in (\mathbb{F}_q)^{*}$, $n_1 \geq a_1$, $n_1 + a_1 > a_2$, $n_1 + a_1 > n_2$, if $n_1 = a_1$ then $n_2 \geq a_2$, and $p \nmid \gcd(a_1, a_2, n_1, n_2)$. Set $d = \gcd(a_1,a_2,n_1,n_2)$ and $\delta$ as in Remark \ref{rmk:TT}. Recall the definition of $n_p$ in Definition \ref{defn:val}. With these assumptions \cite[Theorem 4.12]{MR3475542} shows that if $(\delta/d)_p$ divides $q + 1$ then $N_{a_1,a_2,n_1,n_2}$ is maximal over $\mathbb{F}_q$ and if $N_{a_1,a_2,n_1,n_2}$ is maximal over $\mathbb{F}_{q^2}$ then $(\delta/d)_p$ divides $q^2 + 1$.
Note $N_{\ell,\ell,n,n} = H_{n,\ell}$. Thus Theorem \ref{Thm:ss/Fp} generalizes \cite[Theorem 4.12]{MR3475542} when $a_1 = a_2 = \ell$ and $n_1 = n_2 = n$. \end{note}
\begin{corollary} \label{Cor:ss/Fp-max} If $n$ and $\ell$ are relatively prime and $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$, then it will be maximal over $\mathbb{F}_{p^{2i}}$ where $i$ is the same as in Theorem \ref{Thm:ss/Fp}. \end{corollary} \begin{proof} By Theorem \ref{Thm:ss/Fp}, if $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$, then $p^{i} \equiv -1 \bmod m$ for some $i$. By the results of \cite{Tafazolian}, this implies $\mathcal{F}_m$ will be maximal over $\mathbb{F}_{p^{2i}}$. Since $\mathcal{F}_m$ covers $H_{n,\ell}$, this implies $H_{n,\ell}$ will also be maximal over $\mathbb{F}_{p^{2i}}$. \end{proof}
A priori, if $H_{n,\ell}$ is supersingular (or maximal or minimal) over $\mathbb{F}_p$ then $\mathcal{F}_m$ may not be because it has more normalized Weil numbers. \begin{corollary} \label{cor:SSHurwitz Covered by SSFermat} If $n$ and $\ell$ are relatively prime and $H_{n,\ell}$ is supersingular over $\mathbb{F}_p$, then $\mathcal{F}_m$ is supersingular over $\mathbb{F}_p$. \end{corollary} \begin{proof} If $H_{n,\ell}$ supersingular over $\mathbb{F}_p$ and $\gcd(n,\ell) = 1$, Theorem \ref{Thm:ss/Fp} shows the existence of positive integer $i$ such that $p^i \equiv -1 \bmod m$. Then by \cite[Proposition~3.10]{shiodakatsura}, $\mathcal{F}_m$ is supersingular over $\mathbb{F}_p$. \end{proof}
Partial results are known for when a Hurwitz curve is maximal. \begin{theorem}[{\cite[Theorem 3.1]{AKT}}] Let $\ell = 1$. The curve $H_{n,1}$ is maximal over $\mathbb{F}_{q^{2j}}$ if and only if $p^j \equiv -1 \bmod m$ for some positive integer $j$. \end{theorem} \begin{theorem}[{\cite[Theorem 4.5]{AKT}}] Assume that $\gcd(n,\ell) = 1$ and $m$ is prime. Then $H_{n,\ell}$ is maximal over $\mathbb{F}_{p^{2j}}$ if and only if $p^j \equiv -1 \bmod m$ for some positive integer $j$. \end{theorem} Note that the key property used in \cite{AKT} is the existence of some positive integer $j$ such that \begin{equation}\label{keyprop} p^j \equiv -1 \bmod m. \end{equation} \begin{note} Under the requirements $\ell = 1$, or $\gcd(n,\ell) = 1$ and $m$ prime, the results in \cite{AKT} and \cite[Theorem 5]{Tafazolian} show that $\mathcal{F}_m$ is maximal over $\mathbb{F}_{q^2}$ if and only if $H_{n,\ell}$ is maximal over $\mathbb{F}_{q^2}$. \end{note} We consider the case when $H_{n,\ell}$ and $\mathcal{F}_m$ are minimal. \begin{corollary} \label{Tmin} If $\ell=1$, or $n$ and $\ell$ are relatively prime and $m$ is prime, $H_{n,\ell}$ is minimal over $\mathbb{F}_{p^{4i}}$ if and only if $\mathcal{F}_m$ is minimal over $\mathbb{F}_{p^{4i}}$. \end{corollary} \begin{proof} First suppose $\mathcal{F}_m$ is minimal over $\mathbb{F}_{p^{4i}}$ with set $N$ of normalized Weil numbers. Then the normalized Weil numbers of $H_{n,\ell}$ are a subset of $N$. Thus $H_{n,\ell}$ will also be minimal over $\mathbb{F}_{p^{4i}}$.
Now assume $H_{n,\ell}$ is minimal over $\mathbb{F}_{p^{4i}}$. Minimality implies supersingularity, thus $H_{n,\ell}$ must also be supersingular. By Theorem \ref{Thm:ss/Fp} supersingularity of $H_{n,\ell}$ over $\mathbb{F}_p$ implies $p^j \equiv -1 \bmod m$ for some positive integer $j$. Choose a minimal such $j$. Then Corollary \ref{Cor:ss/Fp-max} shows $H_{n,\ell}$ is maximal over $\mathbb{F}_{p^{2j}}$ thus minimal over $\mathbb{F}_{p^{4j}}$. Minimality of $j$ implies that $\mathbb{F}_{p^{4j}}$ is a subfield of $\mathbb{F}_{p^{4i}}$. Consequently, $j\mid i$.
Now, by \cite{AKT} $p^j \equiv -1 \bmod m$ implies that $\mathcal{F}_m$ is maximal over $\mathbb{F}_{p^{2j}}$. Hence, $\mathcal{F}_m$ is minimal over $\mathbb{F}_{p^{4j}}$. Because $j \mid i$, $\mathcal{F}_m$ is minimal over $\mathcal{F}_{p^{4i}}$. \end{proof}
\begin{note} \label{Rmk:HurwitzMax/FermatNot} The curve $H_{3,3}$ is maximal over $\mathbb{F}_{5^{2}}$ but $\mathcal{F}_{9}$ is not. The above theorems show a supersingular Hurwitz curve and its covering Fermat curve will both be maximal over $\mathbb{F}_{p^{2i}}$. This does not imply that the Fermat curve will always be maximal over the same field extension that the Hurwitz curve is. The Hurwitz curve could also be maximal over $\mathbb{F}_{p^{2j}}$ where $j\mid i$ with $i/j$ odd. In this case the Fermat curve may not be maximal over this field because it has a higher genus. Unfortunately our example of this does not have $n$ and $\ell$ being relatively prime. It is difficult to find an example with $n$ and $\ell$ relatively prime, as the genera of Hurwitz curves grow quickly causing the point counts to become computationally expensive. \end{note}
\begin{figure}
\caption{Current results regarding supersingularity, minimality, and maximality of Hurwitz and Fermat curves.}
\label{fig:theory}
\end{figure} Figure \ref{fig:theory} illustrates how the current theory fits together. The straight, dotted arrows are under the conditions $\ell = 1$, or $\gcd(n,\ell) = 1$ and $m$ prime. The notation max/$\mathbb{F}_{q^2}$ means, for some power $q$ of $p$, the curve is maximal over $\mathbb{F}_{q^2}$. If a curve is maximal over $\mathbb{F}_{q^2}$ then it is minimal over $\mathbb{F}_{q^4}$. The curved arrows show that under appropriate conditions a Hurwitz or Fermat curve is supersingular if and only if it is minimal over some field extension. Corollaries~\ref{Cor:ss/Fp-max} and \ref{cor:SSHurwitz Covered by SSFermat} are under the condition that $\gcd(n,\ell) = 1$, while \cite{AKT} and Corollary \ref{Tmin} are under the condition that $\ell = 1$, or $\gcd(n,\ell) = 1$ and $m$ is prime.
\section{Which Genera Occur and Data} Here we provide information about which genera occur for Hurwitz curves and provide a classification of supersingular Hurwitz curves having genus less than $5$ defined over ${\mathbb F}_p$ when $p < 37$.
Recall that the genus of the Hurwitz curve $H_{n,\ell}$ has the following equation \begin{equation*} g=\dfrac{n^2-n\ell+\ell^2 - 3\gcd(n,\ell) + 2}{2}. \end{equation*}
From this, it can be seen that the genus is determined by the quadratic form $q(x,y)=x^2-xy+y^2$ and $\gcd(x,y)$.
In this section, we provide information about which genera can appear as a result of these equations.
\begin{theorem}[{\cite[Vol.\ II, pages 310-314]{fermat}}]\label{Thm:PrimeDecomp} The equation $m=x^2-xy+y^2$ has solutions $x,y \in \mathbb{Z}$ if and only if for every prime $p$ in the prime decomposition of $m$, either $p \equiv 0,1 \bmod 3$ or $p$ is raised to an even power. \end{theorem}
There is no restriction in Theorem \ref{Thm:PrimeDecomp} on what the values $x$ and $y$ are. However, for Hurwitz curves we require $n$ and $\ell$ to be positive. The question remains as to when the equation $m=q(x,y)$ has solutions in the positive integers. To solve this we study the following automorphisms of $q(x,y)=m.$ \begin{gather*} \begin{cases} f: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid f(x,y) \mapsto (y,x)\\ g: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid g(x,y) \mapsto (-x,-y)\\ \varphi: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid \varphi(x,y) \mapsto (x,x-y)\\ I: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid I(x,y) \mapsto (x,y) \end{cases} \end{gather*} To see that $\varphi(x,y)$ is an automorphism, compute the following \begin{align*} q\circ\varphi(x,y) &= x^2-x(x-y)+(x-y)^2 \\ &= x^2 -x^2+xy+x^2-2xy+y^2 \\ &= x^2-xy+y^2 \\ &= q(x,y). \end{align*} \begin{corollary} \label{comorollary} If the equation $m=q(x,y)$ has a solution $(x,y)\in \mathbb{Z}^2$ then there is a solution with $(x',y')\in \mathbb{N}^2$. \end{corollary} \begin{proof} We separate into cases, depending on the values of $x$ and $y$. \begin{enumerate} \item If both $x$ and $y$ are negative, then $g(x,y) = (-x,-y) \in \mathbb{N}^2$. \item If $y$ negative and $x$ positive, then $\varphi(x,y) = (x,x-y) \in \mathbb{N}^2$. \item If $x$ negative and $y$ positive, then $\varphi(f(x,y)) = (y,y-x) \in \mathbb{N}^2$. \item If $x$ is 0, then $\varphi\circ f (0,y) = (y,y)$ and if $y$ is 0, then $\varphi(y,0) = (y,y)$. \end{enumerate} \end{proof}
By counting points and using Lemma \ref{Lem:ZetaCoeffs} we computed, using \cite{sage}, the $L$-polynomials and normalized Weil numbers of many supersingular Hurwitz curves over $\mathbb{F}_p$. When $n$ and $\ell$ are not relatively prime, it is possible that certain points of the equation for $H_{n,\ell}$ are singular. Resolving these singularities requires taking a field extension of $\mathbb{F}_p$. To adjust for this we check if $q\equiv 1\bmod\gcd(n,\ell)$ and count the multiplicities of singular points. This gives the correct point counts to compute the $L$-polynomial of the normalization of the equation. The table has all supersingular Hurwitz curves $H_{n,\ell}$ of genus less than 5 for primes less than 37. The table also includes some curves of genus 6.
\begin{table}[!ht] \small
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\textbf{n} & \textbf{l} & \textbf{p} & \textbf{g} & \textbf{L-Polynomial} & \textbf{NWNs (multiplicity)}\\
\hline
2 & 1 & 5 & 1 & $5T^2 + 1$ & i, -i\\
2 & 1 & 11 & 1 & $11T^2 + 1$ & i, -i\\
2 & 1 & 17 & 1 & $17T^2 + 1$ & i, -i\\
2 & 1 & 23 & 1 & $23T^2 + 1$ & i, -i\\
2 & 1 & 29 & 1 & $29T^2 + 1$ & i, -i \\
\hline
3 & 3 & 5 & 1 & $5T^2 + 1$ & i, -i\\
3 & 3 & 11 & 1 & $11T^2 + 1$ & i, -i\\
3 & 3 & 17 & 1 & $17T^2 + 1$ & i, -i\\
3 & 3 & 23 & 1 & $23T^2 + 1$ & i, -i\\
3 & 3 & 29 & 1 & $29T^2 + 1$ & i, -i\\
\hline
3 & 1 & 3 & 3 & $27T^6 + 1$ & i,-i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 1 & 5 & 3 & $125T^6 + 1$ & i,-i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 1 & 13 & 3 & $2197T^6 + 507T^4 + 39T^2 + 1$ & i(3), -i(3)\\
3 & 1 & 17 & 3 & $4913T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 1 & 19 & 3 & $6859T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 1 & 31 & 3 & $29791T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
\hline
3 & 2 & 3 & 3 & $27T^6 + 1$ & i,-i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 2 & 5 & 3 & $125T^6 + 1$ & i,-i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 2 & 13 & 3 & $2197T^6 + 507T^4 + 39T^2 + 1$ & i(3), -i(3)\\
3 & 2 & 17 & 3 & $4913T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 2 & 19 & 3 & $6859T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$\\
3 & 2 & 31 & 3 & $29791T^6 + 1$ & i, -i, $\zeta_{12}, \zeta_{12}^5, \zeta_{12}^7, \zeta_{12}^{11}$ \\
\hline
4 & 2 & 5 & 4 & $625T^8 + 500T^6 + 150T^4 + 20T^2 + 1$ &i(4), -i(4)\\
4 & 2 & 17 & 4 & $83521T^8 + 19652T^6 + 1734T^4 + 68T^2 + 1$ &i(4), -i(4)\\
4 & 2 & 29 & 4 & $707281T^8 + 97556T^6 + 5046T^4 + 116T^2 + 1$ &i(4), -i(4)\\
\hline
4 & 1 & 5 & 6 & $15625T^{12} + 1875T^8 + 75T^4 + 1$ & $\zeta_{8}(3), \zeta_{8}^3(3), \zeta_{8}^5(3), \zeta_{8}^7(3)$\\
\hline
4 & 3 & 5 & 6 & $15625T^{12} + 1875T^8 + 75T^4 + 1$ & $\zeta_{8}(3), \zeta_{8}^3(3), \zeta_{8}^5(3), \zeta_{8}^7(3)$\\
\hline
5 & 5 & 3 & 6 & $729T^{12} + 243T^8 + 27T^4 + 1$ &$\zeta_{8}(3), \zeta_{8}^3(3), \zeta_{8}^5(3), \zeta_{8}^7(3)$\\
5 & 5 & 7 & 6 & $117649T^{12} + 7203T^8 + 147T^4 + 1$ & $\zeta_{8}(3), \zeta_{8}^3(3), \zeta_{8}^5(3), \zeta_{8}^7(3)$\\
5 & 5 & 13 & 6 & $4826809T^{12} + 85683T^8 + 507T^4 + 1$ & $\zeta_{8}(3), \zeta_{8}^3(3), \zeta_{8}^5(3), \zeta_{8}^7(3)$\\
\hline \end{tabular} \caption{Supersingular Hurwitz curves in characteristic $p< 37$ with genus $< 5$.} \end{table}
\end{document} |
\begin{document}
factor density: \[d(V,W) = \frac{\text{number copies of } V \text{ in } W }{ \text{number of length-}n \text{ substrings in } W} \]
(instance) density... formerly denoted $\delta(V,W)$: \[D(V,W) = \frac{\text{number of substrings of } W \text{ that are } V \text{-instances} }{ \text{number of substrings of } W } \]
expected (q-)density... formerly $\EE(\delta(V,W_n))$ for $W_n \in [q]^n$ chosen uniformly at random: \[ \DD_n(V,q) \]
asymptotic expected (q-)density... formerly $\lim_{n\rightarrow \infty} \EE(\delta(V,W_n))$: \[ \DD(V,q) \]
liminf (q-)density... formerly $\underline{\delta}(V,q)$:
\[ \mathbb(d)(V,q) = \liminf_{\substack{W \in [q]^n \\ n \rightarrow \infty}} D(V,W) \]
instance count: \[ I_n(V,W) =\]
\end{document} |
\begin{document}
\title[Homogeneous coordinates]
{Homogeneous coordinates \\
for algebraic varieties}
\author[F. Berchtold]{Florian Berchtold} \address{Fachbereich Mathematik und Statistik, Universit\"at Konstanz} \email{berchtof@fmi.uni-konstanz.de}
\author[J.~Hausen]{J\"urgen Hausen} \address{Fachbereich Mathematik und Statistik, Universit\"at Konstanz} \email{Juergen.Hausen@uni-konstanz.de}
\subjclass{13A02,13A50,14C20,14C22}
\begin{abstract} We associate to every divisorial (e.g.~smooth) variety $X$ with only constant invertible global functions and finitely generated Picard group a $\operatorname{Pic}(X)$-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties $X$ with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization. \end{abstract}
\maketitle
\section*{Introduction}
The principal use of homogeneous coordinates is that they relate the geometry of algebraic varieties to the theory of graded rings. The classical example is the projective $n$-space: its homogeneous coordinate ring is the polynomial ring in $n+1$ variables, graded by the usual degree. Cox~\cite{Co} and Kajiwara~\cite{Ka} introduced homogeneous coordinate rings for toric varieties. Cox's construction is meanwhile a standard instrument in toric geometry; for example, it is used in~\cite{BrVe} to prove an equivariant Riemann-Roch Theorem, and in~\cite{MuSmTs} for a description of $\mathcal{D}$-modules on toric varieties.
In this article, we construct homogeneous coordinates for a fairly general class of algebraic varieties: Let $X$ be a divisorial variety --- e.g.~$X$ is ${\mathbb Q}$-factorial or quasiprojective~\cite{Bo} --- such that $X$ has only constant globally invertible functions and the Picard group $\operatorname{Pic}(X)$ is finitely generated. If the (algebraically closed) ground field ${\mathbb K}$ is of characteristic $p > 0$, then we require that the multiplicative group ${\mathbb K}^{*}$ is of infinite rank over ${\mathbb Z}$, and that $\operatorname{Pic}(X)$ has no $p$-torsion. Examples of such varieties are complete smooth rational complex varieties. Moreover, all Calabi-Yau varieties fit into this framework.
To define the homogeneous coordinate ring of $X$, consider a family of line bundles $L$ on $X$ such that the classes $[L]$ generate $\operatorname{Pic}(X)$. Choosing a common trivializing cover $\mathfrak{U}$ for the bundles $L$, one can achieve that they form a finitely generated free abelian group $\Lambda$, which is isomorphic to a subgroup of the group of cocycles $H^{1}(\mathcal{O}^{*}, \mathfrak{U})$. The sheaves of sections $\mathcal{R}_{L}$, where $L \in \Lambda$, then fit together to a sheaf $\mathcal{R}$ of $\Lambda$-graded $\mathcal{O}_{X}$-algebras. Such sheaves $\mathcal{R}$ and their global sections $\mathcal{R}(X)$ are often studied. For example, in~\cite{HK} they have been used to characterize when Mori's program can be carried out, and in~\cite{Ha1} they are the starting point for quotient constructions in the spirit of Mumford's Geometric Invariant Theory.
A first important observation is that we can pass from the above $\Lambda$-graded $\mathcal{O}_{X}$-algebras $\mathcal{R}$ to a universal $\mathcal{O}_{X}$-algebra $\mathcal{A}$, which is graded by the Picard group $\operatorname{Pic}(X)$. This solves in particular the ambiguity problem mentioned in~\cite[Remark p.~341]{HK}. More precisely, we introduce in Section~\ref{section3} the concept of a {\em shifting family\/} for the $\mathcal{O}_{X}$-algebra $\mathcal{R}$. This enables us to identify in a systematic manner two homogeneous parts $\mathcal{R}_{L}$ and $\mathcal{R}_{L'}$ if $L$ and $L'$ define the same class in $\operatorname{Pic}(X)$. The result is a projection $\mathcal{R} \to \mathcal{A}$ onto a $\operatorname{Pic}(X)$-graded $\mathcal{O}_{X}$-algebra $\mathcal{A}$.
The {\em homogeneous coordinate ring\/} of $X$ then is a pair $(A,\mathfrak{A})$. The first part $A$ is the $\operatorname{Pic}(X)$-graded ${\mathbb K}$-algebra of global sections $\mathcal{A}(X)$. The meaning of the second part $\mathfrak{A}$ is roughly speaking the following: It turns out that $A$ is the algebra of functions of a quasiaffine variety $\rq{X}$. Such algebras need not of finite type over ${\mathbb K}$, and $\mathfrak{A}$ is a datum describing all the possible affine closures of $\rq{X}$. From the algebraic point of view, the homogeneous coordinate ring is a {\em freely graded quasiaffine algebra}; the category of such algebras is introduced and discussed in Sections~\ref{section1} and~\ref{section2}.
The first main result of this article is that the homogeneous coordinate ring is indeed functorial, that means that given a morphism $X \to Y$ of varieties, we obtain a morphism of the associated freely graded quasiaffine algebras, see Section~\ref{section5}. In fact, we prove much more, see Theorem~\ref{fullyfaithful}:
\begin{introthm} The assignment $X \mapsto (A,\mathfrak{A})$ is a fully faithful functor from the category of divisorial varieties $X$ with finitely generated Picard group and $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$ to the category of freely graded quasiaffine algebras. \end{introthm}
Note that this statement generalizes in particular the description of the set ${\rm Hom}(X,Y)$ of morphisms of two divisorial toric varieties $X$, $Y$ obtained by Kajiwara in~\cite[Cor.~4.9]{Ka}. In the toric situation, $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$ is a usual nondegeneracy assumption: it just means that $X$ has no torus factors.
Having proved Theorem~\ref{fullyfaithful}, the task is to translate geometric properties of a given variety $X$ to algebraic properties of its homogeneous coordinate ring $(A,\mathfrak{A})$. In Section~\ref{section6}, we do this for basic properties of $X$, like smoothness and normality. In the latter case, the ${\mathbb K}$-algebra $A$ is a normal Krull ring. Moreover, we discuss quasicoherent sheaves, and we give descriptions of affine morphisms and closed embeddings.
In our second main result, we restrict to normal divisorial varieties $X$ with finitely generated Picard group and $\mathcal{O}(X) = {\mathbb K}$. We call such varieties {\em tame}. The homogeneous coordinate ring $(A,\mathfrak{A})$ of a tame variety $X$ is {\em pointed\/} in the sense that $A$ is normal with $A_{0} = {\mathbb K}$ and $A^{*} = {\mathbb K}^{*}$. Moreover, $(A,\mathfrak{A})$ is {\em simple\/} in the sense that the corresponding quasiaffine variety $\rq{X}$ admits only trivial ``linearizable'' bundles, see Section~\ref{section7} for the precise definition. In Theorem~\ref{equivthm}, we show:
\begin{introthm} The assignment $X \mapsto (A,\mathfrak{A})$ defines an equivalence of the category of tame varieties with the category of simple pointed algebras. \end{introthm}
Specializing further to the case of a free Picard group gives the class of {\em very tame\/} varieties, see Section~\ref{section8}. Examples are the Grassmannians and all smooth complete toric varieties. For this class, we obtain a nice description of products in terms of homogeneous coordinate rings, see Proposition~\ref{products}. The possibly most remarkable observation is that very tame varieties open a geometric approach to unique factorization conditions for multigraded Krull rings, see Proposition~\ref{freefactorial}:
\begin{introprop} A very tame variety is locally factorial if and only if its homogeneous coordinate ring is a unique factorization domain. \end{introprop}
We conclude the article with an example underlining this principle: Let $X$ be the projective line with the points $0,1$ and $\infty$ doubled, that means that $X$ is nonseparated. Nevertheless, $X$ is very tame and its Picard group is isomorphic to ${\mathbb Z}^{4}$. As mentioned before, $A = \mathcal{A}(X)$ is a unique factorization domain. It turns even out to be a classical example of a factorial singularity, namely $$ A = {\mathbb K}[T_{1}, \ldots, T_{6}]/\bangle{T_{1}^{2}+ \ldots + T_{6}^{2}}. $$
The quasiaffine variety $\rq{X}$ corresponding to the homogeneous coordinate ring of $X$ is an open subset of ${\rm Spec}(A)$. The prevariety $X$ is a geometric quotient of $\rq{X}$ by a free action of a fourdimensional algebraic torus. In particular, $\rq{X}$ is locally isomorphic to the toric variety ${\mathbb K} \times ({\mathbb K}^{*})^{4}$. That means that $\rq{X}$ is toroidal, even with respect to the Zariski Topology, but not toric.
\tableofcontents
\section{Quasiaffine algebras and quasiaffine varieties}\label{section1}
Throughout the whole article we work in the category of algebraic varieties following the setup of~\cite{Ke}. In particular, we work over an algebraically closed field ${\mathbb K}$, and the word point always refers to a closed point. Note that in our setting a variety is reduced but it need neither be separated nor irreducible.
The purpose of this section is to provide an algebraic description of the category of quasiaffine varieties. The idea is very simple: Every quasiaffine variety $X$ is an open subset of an affine variety $X'$ and hence is described by the inclusion $\mathcal{O}(X') \subset \mathcal{O}(X)$ and the vanishing ideal of the complement $X' \setminus X$ in $\mathcal{O}(X')$.
However, in general the algebra of functions $\mathcal{O}(X)$ of a quasiaffine variety $X$ is not of finite type, see for example~\cite{Re}. Thus there is no canonical choice of an affine closure $X'$ for a given $X$. To overcome this ambiguity, we have to treat all possible affine closures at once.
We introduce the necessary algebraic notions. By a ${\mathbb K}$-algebra we always mean a reduced commutative algebra $A$ over ${\mathbb K}$ having a unit element. We write $\bangle{I}$ for the ideal generated by a subset $I \subset A$. The set of nonzerodivisors of a ${\mathbb K}$-algebra $A$ is denoted by ${\rm nzd}(A)$. Recall that we have a canonical inclusion $A \subset {\rm nzd}(A)^{-1}A$ into the algebra of fractions.
\goodbreak
\begin{defi}\label{closedsubalgebra} Let $A$ be a ${\mathbb K}$-algebra. \begin{enumerate} \item A {\em closing subalgebra\/} of $A$ is a pair $(A',I')$ where
$A' \subset A$ is a subalgebra of finite type over ${\mathbb K}$ and $I'
\subset A'$ is an ideal in $A'$ with $$ I' = \sqrt{\bangle{I' \cap {\rm nzd}(A)}}, \qquad A = \bigcap_{f \in I' \cap {\rm nzd}(A)} A_{f}, \qquad A'_{f} = A_{f} \text{ for all } f \in I'. $$ \item Two closing subalgebras $(A',I')$ and $(A'',I'')$ of $A$ are
called {\em equivalent\/} if there is a closing subalgebra
$(A''',I''')$ of $A$ such that $$ A' \cup A'' \subset A''',
\qquad
I''' = \sqrt{\bangle{I'}} = \sqrt{\bangle{I''}}. $$ \end{enumerate} \end{defi}
Note that \ref{closedsubalgebra}~(ii) does indeed define an equivalence relation. In terms of these notions, the algebraic data to describe quasiaffine varieties are the following:
\begin{defi}\label{quasiaffalgdef} \begin{enumerate} \item A {\em quasiaffine algebra\/} is a pair $(A,\mathfrak{A})$,
where $A$ is a ${\mathbb K}$-algebra and $\mathfrak{A}$ is the
equivalence class of a closing subalgebra $(A',I')$ of $A$. \item A {\em homomorphism\/} of quasiaffine algebras
$(B,\mathfrak{B})$ and $(A,\mathfrak{A})$ is a homomorphism $\mu
\colon B \to A$ such that there exist $(B',J') \in \mathfrak{B}$
and $(A',I') \in \mathfrak{A}$ with $$ \mu(B') \subset A', \qquad I' \subset \sqrt{\bangle{\mu(J')}}. $$ \end{enumerate} \end{defi}
We show now that the category of quasiaffine varieties is equivalent to the category of quasiaffine algebras by associating to every variety $X$ an equivalence class $\mathfrak{O}(X)$ of closing subalgebras of $\mathcal{O}(X)$. We use the following notation: Given a variety $X$ and a regular function $f \in \mathcal{O}(X)$, let $$ X_{f} := \{x \in X; \; f(x) \ne 0\}.$$
\begin{defi}\label{naturalpairdef} Let $X$ be a quasiaffine variety. Let $A' \subset \mathcal{O}(X)$ be a subalgebra of finite type and $I' \subset A'$ a radical ideal. We call $(A',I')$ a {\em natural pair\/} on $X$, if for every $f \in I'$ the set $X_{f}$ is affine with $\mathcal{O}(X_{f}) = A'_{f}$ and the sets $X_{f}$, $f \in I'$, cover $X$. We define $\mathfrak{O}(X)$ to be the collection of all natural pairs on $X$. \end{defi}
So, our first task is to verify that the collection $\mathfrak{O}(X)$ is in fact an equivalence class of closing subalgebras of $\mathcal{O}(X)$. This is done in two steps:
\begin{lemma}\label{naturalpairs} Let $X$ be a quasiaffine variety. Let $(A',I')$ be a natural pair on $X$, and set $X' := {\rm Spec}(A')$. \begin{enumerate} \item The morphism $X \to X'$ defined by $A' \subset \mathcal{O}(X)$
is an open embedding, $I'$ is the vanishing ideal of
$X' \setminus X$, and $(A',I')$ is a closing subalgebra of
$\mathcal{O}(X)$. \item For a subalgebra $A'' \subset \mathcal{O}(X)$ of finite type
with $A' \subset A''$, consider the ideal $I'' :=
\sqrt{\bangle{I'}}$ of $A''$.
Then $(A'',I'')$ is a natural pair on $X$. \end{enumerate} \end{lemma}
\proof Recall that for any $f \in \mathcal{O}(X)$ we have $\mathcal{O}(X_{f}) = \mathcal{O}(X)_{f}$. In particular, $X \to X'$ is locally given by isomorphisms $X_{f} \to X'_{f}$, $f \in I'$. This implies that $X \to X'$ is an open embedding and that $I' \subset A'$ is the vanishing ideal of $X' \setminus X$. Finally, $(A',I')$ is a closing subalgebra, because up to passing to the radical, $I'$ is generated by the $f \in I'$ that are nontrivial on each irreducible component of $X$.
We turn to assertion~(ii). Let $X'' := {\rm Spec}(A'')$. It suffices to verify that the morphism $X \to X''$ defined by $A'' \subset \mathcal{O}(X)$ is an open embedding and that $I'' \subset A''$ is the vanishing ideal of the complement $X'' \setminus X$. Again this holds, because for every $f \in I'$ the map $X \to X''$ restricts to an isomorphism $X_{f} \to X''_{f}$. \endproof
\begin{lemma}\label{quasiaff2closingsubalg} The collection $\mathfrak{O}(X)$ of all natural pairs on a quasiaffine variety $X$ is an equivalence class of closing subalgebras of $\mathcal{O}(X)$. \end{lemma}
\proof First note that there exist natural pairs $(A',I')$ on $X$, because for every affine closure $X \subset X'$ we obtain such a pair by setting $A' := \mathcal{O}(X')$ and defining $I' \subset A'$ to be the vanishing ideal of the complement $X' \setminus X$. Moreover, by Lemma~\ref{naturalpairs}~(i), we know that every natural pair is a closing subalgebra of $\mathcal{O}(X)$.
We show that any two natural pairs $(A',I')$ and $(A'',I'')$ on $X$ are equivalent closing subalgebras of $\mathcal{O}(X)$. Let $A''' \subset \mathcal{O}(X)$ be any subalgebra of finite type containing $A' \cup A''$. Define an ideal in $A'''$ by $I''' := \sqrt{\bangle{I'}}$. Then Lemma~\ref{naturalpairs} tells us that the pair $(A''', I''')$ is a closing subalgebra.
We have to show that $I'''$ equals $\sqrt{\bangle{I''}}$. By Lemma~\ref{naturalpairs}, the morphism $X \to X'''$ defined by the inclusion $A''' \subset \mathcal{O}(X)$ is an open embedding and $I''' \subset A'''$ is the vanishing ideal of $X''' \setminus X$. For every $f \in I''$, the map $X \to X'''$ restricts to an isomorphism $X_{f} \to X'''_{f}$. Hence the desired identity of ideals follows from $$ X = \bigcup_{f \in I''} X_{f}. $$
Finally, we show that if a closing subalgebra $(A'',I'')$ is equivalent to a natural pair $(A',I')$, then also $(A'',I'')$ is natural. Choose $(A''',I''')$ as in~\ref{closedsubalgebra}~(ii). By Lemma~\ref{naturalpairs}~(ii), the pair $(A''',I''')$ is natural. In particular, $X_{f}$ is affine for every $f \in I''$. Moreover, $X$ is covered by these $X_{f}$, because $I'''$ equals $\sqrt{\bangle{I''}}$. \endproof
We are ready for the main result of this section. Given a quasiaffine variety $X$, we denote as before by $\mathfrak{O}(X)$ the collection of all natural pairs on $X$. For a morphism $\varphi \colon X \to Y$ of varieties, we denote by $\varphi^{*} \colon \mathcal{O}(Y) \to \mathcal{O}(X)$ the pullback of functions.
\begin{prop}\label{quasiaffequiv} The assignments $X \mapsto (\mathcal{O}(X), \mathfrak{O}(X))$ and $\varphi \mapsto \varphi^{*}$ define a contravariant equivalence of the category of quasiaffine varieties with the category of quasiaffine algebras. \end{prop}
\proof First of all, we check that the above assignment is in fact well defined on morphisms. Let $\varphi \colon X \to Y$ be any morphism of quasiaffine varieties. Choose a closing subalgebra $(B',J')$ in $\mathfrak{O}(Y)$. By Lemma~\ref{naturalpairs}~(ii), we can construct a closing subalgebra $(A',I')$ in $\mathfrak{O}(X)$ such that $\varphi^{*}(B') \subset A'$.
Now, consider the affine closures $X' := {\rm Spec}(A')$ and $Y' := {\rm Spec}(B')$ of $X$ and~$Y$. The morphism $\varphi' \colon X' \to Y'$ defined by the restriction $\varphi^{*} \colon B' \to A'$ maps $X$ to~$Y$. Since $I'$ and $J'$ are precisely the vanishing ideals of the complements $X' \setminus X$ and $Y' \setminus Y$, we obtain the condition required in~\ref{quasiaffalgdef}~(ii): $$I' \subset \sqrt{\bangle{\varphi^{*}(J')}}. $$
Thus $\varphi \mapsto \varphi^{*}$ is in fact well defined. Moreover, $X \mapsto (\mathcal{O}(X), \mathfrak{O}(X))$ and $\varphi \mapsto \varphi^{*}$ clearly define a contravariant functor, and this functor is injective on morphisms.
For surjectivity, let $\mu \colon \mathcal{O}(Y) \to \mathcal{O}(X)$ be a homomorphism of quasiaffine algebras. Let $(A',I') \in \mathfrak{O}(X)$ and $(B',J') \in \mathfrak{O}(Y)$ as in~\ref{quasiaffalgdef}~(ii). Then $\mu$ defines a morphism $\varphi'$ from ${\rm Spec}(A')$ to ${\rm Spec}(B')$. The condition on the ideals and Lemma~\ref{naturalpairs}~(i) ensure that $\varphi'$ restricts to a morphism $\varphi \colon X \to Y$. Clearly, we have $\varphi^{*} = \mu$.
It remains to show that up to isomorphism, every quasiaffine algebra $(A,\mathfrak{A})$ arises from a quasiaffine variety. Let $(A',I') \in \mathfrak{A}$, set $X' := {\rm Spec}(A')$, and let $X \subset X'$ be the open subvariety obtained by removing the zero set of $I'$. Then $\mathcal{O}(X) = A$, and $(A',I')$ is a natural pair on $X$. Lemma~\ref{quasiaff2closingsubalg} gives $\mathfrak{O}(X) = \mathfrak{A}$. \endproof
We conclude this section with the observation, that restricted on the category of quasiaffine varieties $X$ with $\mathcal{O}(X)$ of finite type, our algebraic description collapses in a very convenient way:
\begin{rem}\label{collaps} For any quasiaffine algebra $(A, \mathfrak{A})$ we have \begin{enumerate} \item The algebra $A$ is of finite type over ${\mathbb K}$ if and only if
$(A,I) \in \mathfrak{A}$ holds with some radical ideal $I \subset
A$. \item The quasiaffine algebra $(A,\mathfrak{A})$ arises from an affine
variety if and only if $(A,A) \in \mathfrak{A}$ holds. \end{enumerate} \end{rem}
\section{Freely graded quasiaffine algebras}\label{section2}
In this section, we introduce the formal framework of homogeneous coordinate rings, namely freely graded quasiaffine algebras and their morphisms. The geometric interpretation of these notions amounts to an equivariant version of the equivalence of categories presented in the preceding section.
\begin{defi}\label{freegradalgdef} Let $(A,\mathfrak{A})$ be a quasiaffine algebra, and let $\Lambda$ be a finitely generated abelian group. We say that $(A,\mathfrak{A})$ is {\em freely graded\/} by $\Lambda$ (or {\em freely $\Lambda$-graded\/}) if there is a grading $$ A = \bigoplus_{L \in \Lambda} A_{L}, $$ and there exists a closing subalgebra $(A',I') \in \mathfrak{A}$ admitting homogeneous elements $f_{1}, \ldots, f_{r} \in I'$ such that $I'$ equals $\sqrt{\bangle{f_{1}, \ldots, f_{r}}}$ and every localization $A_{f_{i}}$ has in each degree $L \in \Lambda$ a homogeneous invertible element. \end{defi}
\begin{exam}\label{polring} For $n \ge 2$, the polynomial ring ${\mathbb K}[T_{1}, \ldots, T_{n}]$ together with the usual ${\mathbb Z}$-grading can be made into a freely graded quasiaffine algebra: Let $\mathfrak{A}$ be the class of $(A,I)$, where $I := \bangle{T_{1}, \ldots, T_{n}}$. \end{exam}
The {\em weight monoid\/} of an integral domain $A$ graded by a finitely generated abelian group $\Lambda$ is the submonoid $\Lambda^{*} \subset \Lambda$ consisting of all weights $L \in \Lambda$ with $A_{L} \ne \{0\}$. For the weight monoid of a freely graded quasiaffine algebra, we have:
\begin{rem}\label{pointedweightcone} Let $(A,\mathfrak{A})$ be a freely $\Lambda$-graded quasiaffine algebra. Then the weight monoid $\Lambda^{*} \subset \Lambda$ of $A$ generates $\Lambda$ as a group. \end{rem}
We turn to homomorphisms. The final notion of a morphism of freely graded quasiaffine algebras will be given below. First we have to consider homomorphisms that are compatible with the structure:
\begin{defi}\label{admissiblehom} Let the quasiaffine algebras $(A,\mathfrak{A})$ and $(B,\mathfrak{B})$ be freely graded by $\Lambda$ and $\Gamma$, respectively. A homomorphism $\mu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ of quasiaffine algebras is called {\em graded\/}, if there is a homomorphism $\t{\mu} \colon \Gamma \to \Lambda$ with \begin{equation} \label{gradedhomcond} \mu(B_{E})
\subset A_{\t{\mu}(E)} \quad \text{for all } E \in \Gamma. \end{equation} \end{defi}
By Remark~\ref{pointedweightcone}, a graded homomorphism $\mu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ of freely graded quasiaffine algebras uniquely determines its accompanying homomorphism $\t{\mu} \colon \Gamma \to \Lambda$. Moreover, the composition of two graded homomorphisms is again graded.
For the subsequent treatment of our homogeneous coordinate rings we need a coarser concept of a morphism of freely graded quasiaffine algebras than the notion of a graded homomorphism would yield. This is the following:
\begin{defi}\label{pointedmorphdef} Let the quasiaffine algebras $(A,\mathfrak{A})$ and $(B,\mathfrak{B})$ be freely graded by finitely generated abelian groups $\Lambda$ and $\Gamma$ respectively. \begin{enumerate} \item Two graded homomorphisms $\mu, \nu \colon (B,\mathfrak{B})
\to (A,\mathfrak{A})$ are called {\em equivalent\/} if there is a
homomorphism $c \colon \Gamma \to A_{0}^{*}$ such that for every $E
\in \Gamma$ and every $g \in B_{E}$ we have
$$ \nu(g) = c(E) \mu(g). $$ \item A {\em morphism\/} $(B,\mathfrak{B}) \to (A,\mathfrak{A})$ of
the freely graded quasiaffine algebras $(B,\mathfrak{B})$ and
$(A,\mathfrak{A})$ is the equivalence class $[\mu]$ of a graded
homomorphism $\mu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$. \end{enumerate} \end{defi}
In the setting of~(i) we shall say that $\mu$ and $\nu$ {\em differ by a character\/} $c \colon \Gamma \to A_{0}^{*}$. Since equivalence of graded homomorphisms is compatible with composition, this definition makes the freely graded quasiaffine algebras into a category.
We give now a geometric interpretation of the above notions. We assume for the rest of this section that if ${\mathbb K}$ is of characteristic $p > 0$, then our finitely generated abelian groups $\Lambda$ have no $p$-torsion, i.e.~$\Lambda$ contains no elements of order $p$. Under this assumption, each $\Lambda$ defines a diagonalizable algebraic group $$ H := {\rm Spec}({\mathbb K}[\Lambda]). $$
Recall that the characters of this group $H$ are precisely the canonical generators $\chi^{L}$, $L \in \Lambda$, of the group algebra ${\mathbb K}[\Lambda]$. In fact, the assignment $\Lambda \mapsto H$ defines a contravariant equivalence of categories, see for example~\cite[Section~III.~8]{Bor}.
Now, suppose that a diagonalizable group $H = {\rm Spec}({\mathbb K}[\Lambda])$ acts by means of a regular map $H \times X \to X$ on a (not necessarily affine) variety $X$. A function $f \in \mathcal{O}(X)$ is called {\em homogeneous\/} with respect to a character $\chi^{L} \colon H \to {\mathbb K}^{*}$ if for every $(t,x) \in H \times X$ we have $$ f(t \! \cdot \! x) = \chi^{L}(t) f(x). $$
For $L \in \Lambda$, let $\mathcal{O}(X)_{L} \subset \mathcal{O}(X)$ denote the subset of all $\chi^{L}$-homogeneous functions. It is well known, use for example \cite[p.~67~Lemma]{Kn}, that the action of $H$ on $X$ defines a grading $$ \mathcal{O}(X) = \bigoplus_{L \in \Lambda} \mathcal{O}(X)_{L}.$$
Recall that one obtains in this way a canonical correspondence between affine $H$-varieties and $\Lambda$-graded affine algebras (the arguments presented in~\cite[p.~11]{Do} for the case $\Lambda = {\mathbb Z}$ also work in the general case).
We are interested in free $H$-actions on quasiaffine varieties $X$, where {\em free\/} means that every orbit map $H \to H \! \cdot \! x$ is an isomorphism. In this situation, we have:
\begin{lemma}\label{gradedqavar2qaalg} Let the diagonalizable group $H = {\rm Spec}({\mathbb K}[\Lambda])$ act freely by means of a regular map $H \times X \to X$ on a quasiaffine variety $X$. Then the associated $\Lambda$-grading of $\mathcal{O}(X)$ makes $(\mathcal{O}(X),\mathfrak{O}(X))$ into a freely graded quasiaffine algebra. \end{lemma}
\proof Let $(A'',I'')$ be any natural pair on $X$, and let $g_{1}, \ldots, g_{s}$ be a system of generators of $A''$. Let $A' \subset \mathcal{O}(X)$ denote the subalgebra generated by all the homogeneous components of the $g_{j}$. Then $A'$ is graded, and according to Lemma~\ref{naturalpairs}~(ii), we obtain a natural pair $(A',I')$ on $X$ by defining $I' := \sqrt{\bangle{I''}}$.
Now, the $\Lambda$-grading of $A'$ comes from an $H$-action on $X' := {\rm Spec}(A')$. This $H$-action extends the initial $H$-action on $X$. In particular, the ideal $I' \subset A'$ is graded, because it is the vanishing ideal of the invariant set $X' \setminus X$, see Lemma~\ref{naturalpairs}~(i). This fact enables us to verify the condition of~\ref{freegradalgdef} for $I'$:
\goodbreak
Choose generators $L_{1}, \ldots, L_{k}$ of $\Lambda$. Consider $x \in X$, and choose a homogeneous $h \in I'$ with $h(x) \ne 0$. Since $H$ acts freely, the orbit map $H \to H \! \cdot \! x$ is an isomorphism. Thus we find for every $i$ a $\chi^{L_{i}}$-homogeneous regular function $h_{i}$ on $H \! \cdot \! x$ with $h_{i}(x) \ne 0$. Since $H \! \cdot \! x$ is closed in $X_{h}$, the $h_{i}$ extend to $\chi^{L_{i}}$-homogeneous regular functions on $X_{h}$.
For a suitable $r > 0$, the product $f := h^{r}h_{1} \ldots h_{k}$ is a regular function on $X'$ with $f \in \bangle{h}$ and hence $f \in I'$. By construction, $f$ is homogeneous, and we have $f(x) \ne 0$. Moreover, the Laurent monomials in $h_{1}, \ldots, h_{k}$ provide for each degree $L \in \Lambda$ a $\chi^{L}$-homogeneous invertible function on $X_{f}$. Since finitely many of the $X_{f}$ cover $X$, this gives the desired property on the ideal $I' \subset A'$. \endproof
In order to give the equivariant version of Proposition~\ref{quasiaffequiv}, we have to fix the notion of a morphism of quasiaffine varieties with an action of a diagonalizable group. This is the following:
\begin{defi} Let $G \times X \to X$ and $H \times Y \to Y$ be algebraic group actions. A morphism $\varphi \colon X \to Y$ is called {\em equivariant\/} if there is a homomorphism $\t{\varphi} \colon G \to H$ of algebraic groups such that for all $(g,x) \in G \times X$ we have $$\varphi(g \! \cdot \! x) = \t{\varphi}(g) \! \cdot \! \varphi(x).$$ \end{defi}
This notion of an equivariant morphism makes the quasiaffine varieties with a free action of a diagonalizable group into a category. We obtain the following equivariant version of Proposition~\ref{quasiaffequiv}:
\begin{prop}\label{equivquasiaffequiv} The assignments $X \mapsto (\mathcal{O}(X), \mathfrak{O}(X))$ and $\varphi \mapsto \varphi^{*}$ define a contravariant equivalence from the category of quasiaffine varieties with a free diagonalizable group action to the category of freely graded quasiaffine algebras and graded homomorphisms. \end{prop}
\proof By Lemma~\ref{gradedqavar2qaalg}, the assignment $X \mapsto (\mathcal{O}(X), \mathfrak{O}(X))$ is well defined. From Proposition~\ref{quasiaffequiv} and the observation that equivariant morphisms of quasiaffine varieties correspond to graded homomorphisms of quasiaffine algebras we infer functoriality and bijectivity on the level of morphisms.
In order to see that up to isomorphism any quasiaffine algebra $(A,\mathfrak{A})$ which is freely graded by some $\Lambda$ arises in the above manner from a quasiaffine variety with free diagonalizable group action, we repeat the corresponding part of the proof of Proposition~\ref{quasiaffequiv} in an equivariant manner:
Let $(A',I')$ be as in Definition~\ref{freegradalgdef}. Let $A'' \subset A$ be any graded subalgebra of finite type with $A' \subset A''$, and let $I'' := \sqrt{\bangle{I'}}$. Then $(A'',I'')$ belongs to $\mathfrak{A}$, and the ideal $I''$ still satisfies the condition of Definition~\ref{freegradalgdef}.
The affine variety $X'' := {\rm Spec}(A'')$ comes along with an action of the diagonalizable group $H := {\rm Spec}({\mathbb K}[\Lambda])$ such that the corresponding grading of $\mathcal{O}(X'') = A''$ gives back the original $\Lambda$-grading of the algebra $A''$. Removing the $H$-invariant zero set of $I''$ from $X''$, gives a quasiaffine $H$-variety $X$.
By construction, the $\Lambda$-graded algebras $\mathcal{O}(X)$ and $A$ coincide, and $(A'',I'')$ is a natural pair on $X$. Moreover, the local existence of invertible homogeneous functions in each degree implies that for every $x \in X$ the orbit map $H \mapsto H \! \cdot \! x$ is an isomorphism. In other words, the action of $H$ on $X$ is free. \endproof
\begin{exam} The standard ${\mathbb K}^{*}$-action on ${\mathbb K}^{n+1} \setminus \{0\}$ has $(A,\mathfrak{A})$ of Example~\ref{polring} as associated freely graded quasiaffine algebra. \end{exam}
The remaining task is to translate the notion of equivalence of graded homomorphisms. For this let $X$ and $Y$ be quasiaffine varieties with actions of diagonalizable groups $H := {\rm Spec}({\mathbb K}[\Lambda])$ and $G := {\rm Spec}({\mathbb K}[\Gamma])$. Denote by $(A,\mathfrak{A})$ and $(B,\mathfrak{B})$ the freely graded quasiaffine algebras associated to $X$ and $Y$.
\begin{rem}\label{equivgeom} Two graded homomorphisms $\mu, \nu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ are equivalent if and only if there is an $H$-invariant morphism $\gamma \colon X \to G$ such that the morphisms $\varphi, \psi \colon X \to Y$ corresponding to $\mu$ and $\nu$ always satisfy $\psi(x) = \gamma(x) \! \cdot \! \varphi(x)$. \end{rem}
\section{Picard graded sheaves of algebras}\label{section3}
Let $X$ be an algebraic variety and denote by $\operatorname{Pic}(X)$ its Picard group. In this section we prepare the definition of a graded ring structure on the vector space $$ \bigoplus_{[L] \in \operatorname{Pic}(X)} H^{0}(X,L). $$
More generally, we even need a ring structure for the corresponding sheaves of vector spaces. The problem is easy, if $\operatorname{Pic}(X)$ is free: Then we can realize it as a group $\Lambda$ of line bundles as in~\cite[Sec.~2]{Ha}, and we can work with the associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra $\mathcal{R}$.
If $\operatorname{Pic}(X)$ has torsion, then we can at most expect a surjection $\Lambda \to \operatorname{Pic}(X)$ with a free group $\Lambda$ of line bundles. Thus the problem is to identify in a suitable manner isomorphic homogeneous components of the $\Lambda$-graded $\mathcal{O}_{X}$-algebra $\mathcal{R}$. This is done by means of shifting families and their associated ideals $\mathcal{I} \subset \mathcal{R}$, see~\ref{shiftfamdef} and~\ref{associdealdef}. The quotient $\mathcal{A} := \mathcal{R} / \mathcal{I}$ then will realize the desired ring structure.
To begin, let us recall the necessary constructions from~\cite{Ha}. Consider an open cover $\mathfrak{U} = (U_{i})_{i \in I}$ of~$X$. This cover gives rise to an additive group $\Lambda(\mathfrak{U})$ of line bundles on~$X$: For each \v{C}ech cocycle $\xi \in Z^{1}(\mathfrak{U},\mathcal{O}_{X}^{*})$, let $L_{\xi}$ denote the line bundle obtained by gluing the products $U_{i} \times {\mathbb K}$ along the maps $$ (x,z) \mapsto (x, \xi_{ij}(x)z).$$ Define the sum $L_{\xi} + L_{\eta}$ of two such line bundles to be $L_{\xi\eta} = L_{\eta\xi}$. This makes the set $\Lambda(\mathfrak{U})$ consisting of all the bundles $L_{\xi}$ into an abelian group, which is isomorphic to the cocycle group $Z^{1}(\mathfrak{U},\mathcal{O}_{X}^{*})$.
When we speak of a {\em group of line bundles\/} on $X$, we think of a finitely generated free subgroup of some group $\Lambda(\mathfrak{U})$ as above. Note that for any such group $\Lambda$ of line bundles, we have a canonical homomorphism $\Lambda \to \operatorname{Pic}(X)$ to the Picard group.
We come to the graded $\mathcal{O}_{X}$-algebra associated to a group $\Lambda$ of line bundles on a variety $X$. For each line bundle $L \in \Lambda$, let $\mathcal{R}_{L}$ denote its sheaf of sections. In the sequel, we shall identify $\mathcal{R}_{0}$ with the structure sheaf $\mathcal{O}_{X}$. The {\em graded $\mathcal{O}_{X}$-algebra\/} associated to $\Lambda$ is the quasicoherent sheaf $$ \mathcal{R} := \bigoplus_{L \in \Lambda} \mathcal{R}_{L}, $$ where the multiplication is defined as follows: The sections of a bundle $L_{\xi} \in \Lambda$ over an open set $U \subset X$ are described by families $f_{i} \in \mathcal{O}_{X}(U \cap U_{i})$ that are compatible with the cocycle $\xi$. For any two sections $f \in \mathcal{R}_{L}(U)$ and $f' \in \mathcal{R}_{L'}(U)$, the product $(f_{i}f'_{i})$ of their defining families $(f_{i})$ and $(f'_{i})$ gives us a section $ff' \in \mathcal{R}_{L+L'}(U)$.
In the sequel, we fix an open cover $\mathfrak{U} = (U_{i})_{i \in I}$ of~$X$ and a group $\Lambda \subset \Lambda(\mathfrak{U})$ of line bundles. Let $\mathcal{R}$ denote the associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra. Here comes the notion of a shifting family for $\mathcal{R}$:
\begin{defi}\label{shiftfamdef} Let $\Lambda_{0} \subset \Lambda$ be any subgroup of the kernel of $\Lambda \to \operatorname{Pic}(X)$. By a {\em $\Lambda_{0}$-shifting family\/} for $\mathcal{R}$ we mean a family $\varrho = ( \varrho_{E} )$ of $\mathcal{O}_{X}$-module isomorphisms $\varrho_{E} \colon \mathcal{R} \to \mathcal{R}$, where $E \in \Lambda_{0}$, with the following properties: \begin{enumerate} \item for every $L \in \Lambda$ and every $E \in \Lambda_{0}$ the
isomorphism $\varrho_{E}$ maps $\mathcal{R}_{L}$ onto
$\mathcal{R}_{L+E}$, \item for any two $E_{1}, E_{2} \in \Lambda_{0}$ we have
$\varrho_{E_{1} + E_{2}} = \varrho_{E_{2}} \circ \varrho_{E_{1}}$, \item for any two homogeneous sections $f,g$ of $\mathcal{R}$ and
every $E \in \Lambda_{0}$ we have
$\varrho_{E}(fg) = f \varrho_{E}(g)$. \end{enumerate} If $\Lambda_{0}$ is the full kernel of $\Lambda \to \operatorname{Pic}(X)$, then we also speak of a {\em full shifting family\/} for $\mathcal{R}$ instead of a $\Lambda_{0}$-shifting family. \end{defi}
The first basic observation is existence of shifting families and a certain uniqueness statement:
\begin{lemma}\label{shiftfamprops} Let $\Lambda_{0} \subset \Lambda$ be a subgroup of the kernel of $\Lambda \to \operatorname{Pic}(X)$. Then there exist $\Lambda_{0}$-shifting families for $\mathcal{R}$, and any two such families $\varrho$, $\varrho'$ differ by a character $c \colon \Lambda_{0} \to \mathcal{O}^{*}(X)$ in the sense that $\varrho'_{E} = c(E)\varrho_{E}$ holds for all $E \in \Lambda_{0}$. \end{lemma}
\proof
For the existence statement, fix a ${\mathbb Z}$-basis of the subgroup $\Lambda_{0} \subset \Lambda$. For any member $E$ of this basis choose a bundle isomorphism $\alpha_{E} \colon 0 \to E$ from the trivial bundle $0 \in \Lambda$ onto $E \in \Lambda$. With respect to the cover $\mathcal{U}$, this isomorphism is fibrewise multiplication with certain $\alpha_{i} \in \mathcal{O}^{*}(U_{i})$; so, on $U_{i} \times {\mathbb K}$ it is of the form \begin{equation}\label{localdata} (x,z) \mapsto (x, \alpha_{i}(x)z). \end{equation}
If $\alpha_{E'} \colon 0 \to E'$ denotes the isomorphism for a further member of the basis of $\Lambda_{0}$, then the products $\alpha_{i}\alpha_{i}'$ of the corresponding local data define an isomorphism $\alpha_{E+E'} \colon 0 \to E + E'$. Similarly, by inverting local data, we obtain isomorphisms $\alpha_{-E} \colon 0 \to -E$. Proceeding this way, we obtain an isomorphism $\alpha_{E} \colon 0 \to E$ for every $E \in \Lambda_{0}$.
The local data $\alpha_{i}$ of an isomorphism $\alpha_{E} \colon 0 \to E$ as constructed above define as well an isomorphism $L \to L + E$ for any $L \in \Lambda$. By shifting homogeneous sections according to $f \mapsto \alpha_{E} \circ f$, one obtains $\mathcal{O}_{X}$-module isomorphisms $\varrho_{E} \colon \mathcal{R} \to \mathcal{R}$ mapping each $\mathcal{R}_{L}$ onto $\mathcal{R}_{L+E}$. The Properties~\ref{shiftfamdef}~(ii) and~(iii) are then clear by construction.
We turn to the uniqueness statement. Let $\varrho$, $\varrho'$ be two $\Lambda_{0}$-shifting families for $\mathcal{R}$. Using Property~\ref{shiftfamdef}~(iii) we see that for every $E \in \Lambda_{0}$ and every homogeneous section $f$ of $\mathcal{R}$, we have $$ \varrho_{E}^{-1} \circ \varrho'_{E} (f) = \varrho_{E}^{-1} \circ \varrho'_{E} (f \cdot 1) = f \cdot \varrho_{E}^{-1} \circ \varrho'_{E} (1). $$ Thus, setting $c(E) := \varrho_{E}^{-1} \circ \varrho'_{E} (1)$ we obtain a map $c \colon \Lambda_{0} \to {\mathcal{O}}^{*} (X)$ such that $\varrho'_{E}$ equals $c(E)\varrho_{E}$. Properties~\ref{shiftfamdef}~(ii) and~(iii) show that $c$ is a homomorphism: \begin{eqnarray*} c(E_{1}+E_{2}) & = & \varrho_{E_{1}+E_{2}}^{-1} \circ \varrho'_{E_{1}+E_{2}}(1) \\ & = & \varrho_{-E_{1}} \circ \varrho_{-E_{2}} \circ \varrho'_{E_{2}} \circ \varrho'_{E_{1}}(1) \\ & = & \varrho_{-E_{1}} \circ \varrho_{-E_{2}} \circ \varrho'_{E_{2}} (\varrho'_{E_{1}}(1) \! \cdot \! 1)\\ & = & \varrho_{-E_{1}} ( \varrho'_{E_{1}}(1) c(E_{2})) \\ & = & c(E_{1})c(E_{2}). \qquad \qed \end{eqnarray*}
\goodbreak
We shall now associate to any shifting family an ideal in the $\mathcal{O}_{X}$-algebra $\mathcal{R}$. First we remark that for any subgroup $\Lambda_{0} \subset \Lambda$ the algebra $\mathcal{R}$ becomes $\Lambda/\Lambda_{0}$-graded by defining the homogeneous component of a class $[L] \in \Lambda/\Lambda_{0}$ as $$ \mathcal{R}_{[L]} := \sum_{L' \in [L]} \mathcal{R}_{L'}.$$
\begin{lemma}\label{associdealprops} Let $\Lambda_{0}$ be a subgroup of the kernel of $\Lambda \to \operatorname{Pic}(X)$, and let $\varrho$ be a $\Lambda_{0}$-shifting family. For each given open subset $U \subset X$ consider the ideal $$ \mathfrak{I} (U) \; := \; \bangle{f - \varrho_{E}(f); \; f \in \mathcal{R}(U), \; E \in \Lambda_{0}} \; \subset \; \mathcal{R}(U). $$ Let $\mathcal{I}$ denote the sheaf associated to the presheaf $U \mapsto \mathfrak{I}(U)$. Then $\mathcal{I}$ is a quasicoherent ideal of $\mathcal{R}$, and we have: \begin{enumerate} \item Every $\mathcal{I}(U)$ is homogeneous with respect to the $\Lambda/\Lambda_{0}$-grading of $\mathcal{R}(U)$. \item For every $L \in \Lambda$ we have $ \mathcal{R}_{L}(U) \cap \mathcal{I}(U) = \{0\}$. \end{enumerate} \end{lemma}
\proof
First note that the ideal sheaf $\mathcal{I}$ is indeed quasicoherent, because it is a sum of images of quasicoherent sheaves.
We check~(i). Using Property~\ref{shiftfamdef}~(iii), we see that each ideal $\mathfrak{I}(U)$ is generated by the elements $1 - \varrho_{E}(1)$, where $E \in \Lambda_{0}$. Consequently, each stalk $\mathfrak{I}_{x}$ is a $\Lambda/\Lambda_{0}$-homogeneous ideal in $\mathcal{A}_{x}$. This implies that the associated sheaf $\mathcal{I}$ is a $\Lambda/\Lambda_{0}$-homogeneous ideal sheaf in $\mathcal{A}$.
We turn to~(ii). By construction, it suffices to consider local sections $f \in \mathcal{R}_{L}(U) \cap \mathfrak{I}(U)$. By the definition of $\mathfrak{I}(U)$ and Property~\ref{shiftfamdef}~(iii), there exist homogeneous elements $f_{i} \in \mathcal{R}_{L_{i}}(U)$ such that we can write $f$ as \begin{equation}\label{minimalrep} f = \sum_{i=1}^{r} f_{i} - \varrho_{E_{i}}(f_{i}). \end{equation}
Since $\mathfrak{I}(U)$ is $\Lambda/\Lambda_{0}$-graded, all the $L_{i}$ belong to the class $[L]$ in $\Lambda/\Lambda_{0}$. Moreover, we can achieve in the representation~(\ref{minimalrep}) of $f$ that all $f_{i}$ are of degree $L \in [L]$. Namely, we can use Property~\ref{shiftfamdef}~(ii) to write $f_{i} - \rho_{E_{i}} (f_{i})$ in the form \begin{eqnarray*} f_{i} - \varrho_{E_{i}} (f_{i}) & = & \varrho_{L-L_{i}} (f_{i}) - \varrho_{E_{i} + L_{i} - L} (\varrho_{L-L_{i}} (f_{i})) \\ & & + (- \varrho_{L-L_{i}} (f_{i})) - \varrho_{L_{i}-L} (-\varrho_{L - L_{i}}(f_{i})). \end{eqnarray*}
Moreover we can choose the representation~(\ref{minimalrep}) minimal in the sense that $r$ is minimal with the property that every $f_{i}$ is of degree $L$. Then the $E_{i}$ are pairwise different from each other, because otherwise we could shorten the representation by gathering. But this implies $\varrho_{E_{i}}(f_{i}) = 0$ for every $i$. Hence we obtain $f=0$. \endproof
\begin{defi}\label{associdealdef} Let $\Lambda_{0}$ be a subgroup of the kernel of $\Lambda \to \operatorname{Pic}(X)$, and let $\varrho$ be a $\Lambda_{0}$-shifting family for $\mathcal{R}$. The {\em ideal associated to $\varrho$} is the $\Lambda/\Lambda_{0}$-graded ideal sheaf $\mathcal{I}$ of $\mathcal{R}$ defined in~\ref{associdealprops}. \end{defi}
With the aid of the ideal associated to a shifting family, we can pass from $\mathcal{R}$ to more coarsely graded $\mathcal{O}_X$-algebras:
\begin{lemma}\label{gradproject} Let $\Lambda_0 \subset \Lambda$ be a subgroup, and let $\varrho$ be a $\Lambda_0$-shifting family with associated ideal $\mathcal{I}$. Set $\mathcal{A} := \mathcal{R}/\mathcal{I}$, and let $\pi \colon \mathcal{R} \to \mathcal{A}$ denote the projection. \begin{enumerate} \item The $\mathcal{O}_{X}$-algebra $\mathcal{A}$ is quasicoherent,
and it inherits a
$\Lambda/\Lambda_{0}$-grading from $\mathcal{R}$ as follows
$$ \mathcal{A}
= \bigoplus_{[L] \in \Lambda/\Lambda_{0}} \mathcal{A}_{[L]}
:= \bigoplus_{[L] \in \Lambda/\Lambda_{0}}
\pi(\mathcal{R}_{[L]}). $$ \item For any $L \in \Lambda$ the induced map $\pi_{L} \colon \mathcal{R}_{L} \to \mathcal{A}_{[L]}$ is an isomorphism of $\mathcal{O}_{X}$-modules. In particular, we obtain $$\mathcal{A}(X) \cong \mathcal{R}(X) / \mathcal{I}(X).$$ \item The $\mathcal{O}_{X}$-algebra $\mathcal{A}$ is locally generated
by finitely many invertible homogeneous elements. \end{enumerate} \end{lemma}
\proof
The first assertion follows directly from the fact that we have a commutative diagram where the lower arrow is an isomorphism of sheaves: $$ \xymatrix{
& {\mathcal{R}} \ar[ld]_{\pi} \ar[rd] & \\ {\mathcal{A}} \ar[rr] & & {\bigoplus_{[L] \in \Lambda/\Lambda_{0}}
\mathcal{R}_{[L]}/\mathcal{I}_{[L]}} } $$
To prove (ii), note that $\pi_{L} \colon \mathcal{R}_{L} \to \mathcal{A}_{[L]}$ is injective by Lemma~\ref{associdealprops}~(ii). For bijectivity, we have to show that $\pi_{L}$ is stalkwise surjective. Let $h$ be a local section of ${\mathcal{A}_{[L]}}$ near some $x \in X$. Since ${\mathcal{A}_{[L]}}$ equals $\pi(\mathcal{R}_{[L]})$, we may assume that $h = \pi(f)$ with a local section $f$ of $\mathcal{R}_{[L]}$ near $x$. Write $f$ as the sum of its $\Lambda$-homogeneous components: $$ f = \sum_{L' \in [L]} f_{L'}. $$
For every $L' \ne L$, we subtract $f_{L'} - \varrho_{L-L'}(f_{L'})$ from $f$. The result is a local section $g$ of $\mathcal{R}_{L}$ near $x$ which still projects onto $h$. This proves bijectivity of $\pi_{L} \colon \mathcal{R}_{L} \to \mathcal{A}_{[L]}$. The isomorphy on the level of global sections then is due to left exactness of the section functor.
To prove assertion~(iii), note that the analogous statement holds for $\mathcal{R}$. In fact, for small $U \subset X$, the algebra $\mathcal{R}(U)$ is even a Laurent monomial algebra over $\mathcal{O}(U)$. Together with assertion (ii), this observation gives statement~(iii).
\endproof
\begin{defi}\label{picgradalg} Let $\Lambda_{0} \subset \Lambda$ be a subgroup of the kernel $\Lambda \to \operatorname{Pic}(X)$, and let $\varrho$ be a $\Lambda_{0}$-shifting family for $\mathcal{R}$ with associated ideal $\mathcal{I}$. We call the $\Lambda/\Lambda_{0}$-graded $\mathcal{O}_{X}$-algebra $\mathcal{A} := \mathcal{R}/ \mathcal{I}$ of~\ref{gradproject} the {\em Picard graded algebra\/} associated to $\varrho$. \end{defi}
If every global invertible function on $X$ is constant, then the Picard graded algebras associated to different $\Lambda_{0}$-shifting families are isomorphic (a graded homomorphism of sheaves is defined by requiring~\ref{gradedhomcond} on the level of sections):
\begin{lemma}\label{shiftfamunique} Suppose $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$. Let $\Lambda_{0} \subset \Lambda$ be a subgroup, and let $\varrho$, $\varrho'$ be $\Lambda_{0}$-shifting families for $\mathcal{R}$ with associated ideals $\mathcal{I}$ and $\mathcal{I}'$. Then there is a graded automorphism of $\mathcal{R}$ having the identity of $\Lambda$ as accompanying homomorphism and mapping $\mathcal{I}$ onto $\mathcal{I}'$. \end{lemma}
\proof By Lemma~\ref{shiftfamprops}, there exists a homomorphism $c \colon E \to {\mathbb K}^{*}$ such that $\varrho'_{E} = c(E) \varrho_{E}$ holds. By Lemma~\ref{charext} stated below, this homomorphism extends to a homomorphism $c \colon \Lambda \to {\mathbb K}^{*}$. Thus we can define the desired automorphism $\mathcal{R} \to \mathcal{R}$ by mapping a section $f \in \mathcal{R}_{L}(U)$ to $c(L)f \in \mathcal{R}_{L}(U)$. \endproof
In the proof of this lemma, we made use of the following standard property of lattices:
\begin{lemma}\label{charext} Let $\Lambda_{0} \subset \Lambda$ be an inclusion of lattices. Then any homomorphism $\Lambda_{0} \to {\mathbb K}^{*}$ extends to a homomorphism $\Lambda \to {\mathbb K}^{*}$. \end{lemma}
Let us give a geometric interpretation of Picard graded algebras. Let $\Lambda$ be a group of line bundles on $X$ with associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra $\mathcal{R}$. Fix a subgroup $\Lambda_{0}$ of the kernel of $\Lambda \to \operatorname{Pic}(X)$ and a $\Lambda_{0}$-shifting family $\varrho$ for $\mathcal{R}$.
Similar to the preceding section, we assume for the rest of this section that in the case of a ground field ${\mathbb K}$ of characteristic $p> 0$, the group $\Lambda/\Lambda_{0}$ has no $p$-torsion. Under this hypothesis, we can show that the quotient $\mathcal{A} := \mathcal{R}/\mathcal{I}$ by the ideal associated to the shifting family $\varrho$ is reduced:
\begin{lemma}\label{reduced} For every open $U \subset X$, the ideal $\mathcal{I}(U)$ is a radical ideal in $\mathcal{R}(U)$. \end{lemma}
\proof First note that we may assume that $U$ is a small affine open set such that $\mathcal{R}(U)$ is of finite type. Consider the affine variety $Z := {\rm Spec}(\mathcal{R}(U))$. Then the $\Lambda/\Lambda_{0}$-grading of $\mathcal{R}(U) = \mathcal{O}(Z)$ defines an action of the diagonalizable group $H := {\rm Spec}({\mathbb K}[\Lambda/\Lambda_{0}])$ on $Z$. Let $Z_{0} \subset Z$ denote the zero set of the ideal $\mathcal{I}(U) \subset \mathcal{R}(U)$.
Now we can enter the proof of the assertion. Let $f \in \mathcal{O}(Z)$ with $f^{n} \in \mathcal{I}(U)$. We have to show that $f \in \mathcal{I}(U)$ holds. Consider the decomposition of $f$ into homogeneous parts: $$ f = \sum_{[L] \in \Lambda/\Lambda_{0}} f_{[L]}. $$ Since $f$ vanishes along the $H$-invariant zero set $Z_{0}$ of the $\Lambda/\Lambda_{0}$-graded ideal $\mathcal{I}(U)$, also every homogeneous component $f_{[L]}$ has to vanish along $Z_{0}$.
We show that every $f_{[L]}$ belongs to $\mathcal{I}(U)$. Since the $f_{[L]}$ vanish along $Z_{0}$, Hilbert's Nullstellensatz tells us that for every degree $[L]$ some power $f_{[L]}^{m}$ lies in $\mathcal{I}(U)$. Now consider $$ g := \sum_{L' \in [L]} f_{L'} - (f_{L'} - \varrho_{L-L'}(f_{L'})). $$
Then $g$ is $\Lambda$-homogeneous of degree $L$. Moreover, by explicit multiplication, we see $g^{m} \in \mathcal{I}(U)$. But any $\Lambda$-homogeneous element of $\mathcal{I}(U)$ is trivial. Thus $g^{m} = 0$. Hence $g=0$, which in turn implies $f_{[L]} \in \mathcal{I}(U)$. \endproof
In our geometric interpretation, we use the global ``${\rm Spec}$''-construction, see for example~\cite{Ht}. Moreover, for any homogeneous section $f \in \mathcal{A} (U)$, we denote its zero set in $X$ by $Z(f)$. This is well defined, because the components $\mathcal{A}_{[L]}$ are locally free due to Lemma~\ref{gradproject}~(ii).
\begin{prop}\label{geominterp} Let $\rq{X} := {\rm Spec}(\mathcal{A})$, and let $q \colon \rq{X} \to X$ be the canonical map. \begin{enumerate} \item $\rq{X}$ is a variety, $q \colon \rq{X} \to X$ is an
affine morphism, and we have $\mathcal{A} = q_{*}
\mathcal{O}_{\rq{X}}$. \item For a homogeneous section $f \in \mathcal{A}_{[L]}(X)$ we obtain
$q^{-1}(Z(f)) = V(\rq{X};f)$, where $V(\rq{X};f)$ is
the zero set of the function $f \in \mathcal{O}(\rq{X})$. \item If $f_{i} \in \mathcal{A}(X)$ are homogeneous sections such that
the sets $X \setminus Z(f_{i})$ are affine and cover $X$, then
$\rq{X}$ is a quasiaffine variety. \end{enumerate} \end{prop}
\proof To check~(i), note that $\rq{X}$ is indeed a variety, because by Lemmas~\ref{gradproject}~(iii) and~\ref{reduced}, the algebra $\mathcal{A}$ is reduced and locally of finite type. The rest of~(i) are standard properties of the global ``${\rm Spec}$''-construction for sheaves of $\mathcal{O}_{X}$-algebras.
Assertion~(ii) is clear in the case $[L] = 0$, because then we have $\mathcal{A}_{0} = \mathcal{O}_{X}$. For a general $[L]$, we may reduce to the previous case by multiplying $f$ locally with invertible sections of degree $-[L]$. Note that invertible sections exist locally by Lemma~\ref{gradproject}~(iii). \endproof
\section{The homogeneous coordinate ring}\label{section4}
In this section, we give the precise definition of the homogeneous coordinate ring of a given variety, see Definition~\ref{homcoorddef}. Moreover, we show in Proposition~\ref{uniquehomcoord} that the homogeneous coordinate ring is unique up to isomorphism.
In order to fix the setup, recall from~\cite{Bo} that a (neither necessarily separated nor irreducible) variety $X$, is said to be {\em divisorial\/} if every $x \in X$ admits an affine neighbourhood of the form $X \setminus Z(f)$ where $Z(f)$ is the zero set of a global section $f$ of some line bundle $L$ on~$X$.
\begin{rem} Every separated irreducible ${\mathbb Q}$-factorial variety is divisorial, and every quasiprojective variety is divisorial. \end{rem}
Here is the setup of this section: We assume that the multiplicative group ${\mathbb K}^{*}$ is of infinite rank over ${\mathbb Z}$, e.g.~${\mathbb K}$ is of characteristic zero or it is uncountable. The variety $X$ is divisorial and satisfies $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$. Moreover, $\operatorname{Pic}(X)$ is finitely generated and, if ${\mathbb K}$ is of characteristic $p > 0$, then $\operatorname{Pic}(X)$ has no $p$-torsion.
\begin{lemma}\label{ontopic} There exists a group $\Lambda$ of line bundles on $X$ mapping onto $\operatorname{Pic}(X)$. For any such $\Lambda$ the associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra $\mathcal{R}$ admits homogeneous global sections $h_{1}, \ldots, h_{r}$ such that the sets $X \setminus Z(h_{i})$ are affine and cover~$X$. \end{lemma}
\proof
Only for the first statement there is something to show. For this, we may assume that $\operatorname{Pic}(X)$ is not trivial. Write $\operatorname{Pic}(X)$ as a direct sum of cyclic groups $\Pi_{1}, \ldots, \Pi_{m}$ and fix a generator $P_{l}$ for each $\Pi_{l}$. Choose a finite open cover $\mathfrak{U}$ of $X$ such that each $P_{l}$ is represented by a cocycle $\xi^{(l)} \in Z^{1}(\mathfrak{U},\mathcal{O}^{*})$. Choose members $U_{i}, U_{j}$ of $\mathfrak{U}$ such that $U_{i} \ne U_{j}$ holds and there is a point $x_{0} \in U_{i} \cap U_{j}$.
We adjust the $\xi^{(l)}$ as follows: By the assumption on the ground field~${\mathbb K}$, we find $a_{1}, \ldots, a_{m} \in {\mathbb K}^{*}$ which are linearly independent over ${\mathbb Z}$. Define a locally constant cochain $\eta^{(l)}$ by setting $\eta^{(l)} := a_{l}/\xi^{(l)}_{ij}(x_{0})$ on $U_{i}$ and $\eta^{(l)} := 1$ on the $U_{k}$ different from $U_{i}$. Let $\zeta^{(l)} \in Z^{1}(\mathfrak{U},\mathcal{O}^{*})$ be the product of $\xi^{(l)}$ with the coboundary of $\eta^{(l)}$.
Let $\Lambda \subset \Lambda(\mathfrak{U})$ be the subgroup generated by the line bundles arising from $\zeta^{(1)}, \ldots, \zeta^{(m)}$. By construction $\Lambda$ maps onto $\operatorname{Pic}(X)$. Moreover, we have $$ \Bigl(\bigl(\zeta_{ij}^{(1)}\bigr)^{n_{1}} \ldots \bigl(\zeta_{ij}^{(m)}\bigr)^{n_{m}}\Bigr)(x_{o}) = a_{1}^{n_{1}} \ldots a_{m}^{n_{m}} $$ for the cocycle corresponding to a general element of $\Lambda$. By the choice of the $a_{l}$, this cocylce can only be trivial if all exponents $n_{l}$ vanish. It follows that $\Lambda$ is free.
\endproof
We fix a group $\Lambda$ of line bundles on $X$ as provided by Lemma~\ref{ontopic}, and a full shifting family $\varrho$ for the $\Lambda$-graded $\mathcal{O}_{X}$-algebra $\mathcal{R}$ associated to $\Lambda$. Let $\mathcal{I}$ denote the ideal associated to the shifting family $\varrho$. As seen in Lemma~\ref{gradproject}~(i), the $\mathcal{O}_{X}$-algebra $\mathcal{A} := \mathcal{R}/\mathcal{I}$ is graded by $\operatorname{Pic}(X)$. In particular, we have a grading $$ \mathcal{A}(X) = \bigoplus_{[L] \in \operatorname{Pic}(X)} \mathcal{A}_{[L]}(X). $$
According to Lemmas~\ref{gradproject}~(ii) and~\ref{ontopic}, there are homogeneous $f_{1}, \ldots, f_{r} \in \mathcal{A}(X)$ such that the sets $X \setminus Z(f_{i})$ are affine and cover $X$. Hence Proposition~\ref{geominterp}~(iii) tells us that the variety $\rq{X} := {\rm Spec}(\mathcal{A})$ is quasiaffine. Thus we have the collection $\mathfrak{A}(X)$ of natural pairs on $\rq{X}$ as closing subalgebras for $\mathcal{A}(X) = \mathcal{O}(\rq{X})$, see Lemma~\ref{quasiaff2closingsubalg}.
\begin{prop}\label{coringisqualg} The pair $(\mathcal{A}(X), \mathfrak{A}(X))$ is a freely graded quasiaffine algebra. \end{prop}
\proof
We have to show that there is a natural pair $(A',I') \in \mathfrak{A}(X)$ with the properties of Definition~\ref{freegradalgdef}.
Choose homogeneous $f_{1}, \ldots, f_{r} \in \mathcal{A}(X)$ such that the sets $X \setminus Z(f_{i})$ form an affine cover of $X$. Let $q \colon \rq{X} \to X$ be the canonical map. By Proposition~\ref{geominterp}~(ii), each $\rq{X}_{f_{i}}$ equals $q^{-1}(X \setminus Z(f_{i}))$ and thus is affine. Consequently the algebras $$ \mathcal{A}(X)_{f_{i}} = \mathcal{O}(\rq{X})_{f_{i}} = \mathcal{O}(\rq{X}_{f_{i}}) $$ are of finite type. Thus we find a subalgebra $A' \subset \mathcal{A}(X)$ of finite type satisfying $A'_{f_{i}} = \mathcal{A}(X)_{f_{i}}$ for every $i$. Then $\b{X} := {\rm Spec}(A')$ is an affine closure of $\rq{X}$, and the vanishing ideal $I' \subset A'$ of $\b{X} \setminus \rq{X}$ is the radical of the ideal generated by $f_{1}, \ldots, f_{r}$. It follows that $(A',I')$ is a natural pair on $\rq{X}$.
We verify the condition on the degrees. Given $x \in \rq{X}$, choose an $f_{i}$ with $q(x) \in U := X \setminus Z(f_{i})$. By Lemma~\ref{gradproject}~(iii), there is a small neighbourhood $U_{h} \subset U$ of $x$ defined by some $h \in \mathcal{O}(U)$ such that every $[L] \in \operatorname{Pic}(X)$ admits an invertible section in $\mathcal{A}_{[L]}(U_{h})$.
Now, $U_{h}$ equals $X \setminus Z(hf_{i}^{n})$ for some large positive integer $n$. Since finitely many of such $U_{h}$ cover $X$, we obtain the desired Property~\ref{freegradalgdef} with finitely many of the homogeneous sections $hf_{i}^{n} \in I'$.
\endproof
\begin{defi}\label{homcoorddef} We call $(\mathcal{A}(X), \mathfrak{A}(X))$ the {\em homogeneous coordinate ring\/} of $X$. \end{defi}
We show now that homogeneous coordinate rings are unique up to isomorphism. This amounts to comparing Picard graded algebras arising from different groups of line bundles on $X$. As we shall need it later, we do this in a slightly more general setting:
\begin{lemma}\label{differentcomps} Let $\Lambda$ and $\Gamma$ be groups of line bundles on $X$ with associated graded $\mathcal{O}_{X}$-algebras $\mathcal{R}$ and $\mathcal{S}$. Suppose that the image of $\Lambda \to \operatorname{Pic}(X)$ contains the image of $\Gamma \to \operatorname{Pic}(X)$, and let $\varrho$ be a full shifting family for $\mathcal{R}$. \begin{enumerate} \item There exist a graded homomorphism $\gamma \colon \mathcal{S}
\to \mathcal{R}$ with accompanying homomorphism $\t{\gamma} \colon
\Gamma \to \Lambda$ and a full shifting family $\sigma$ for $\Gamma$
such that for every $K \in \Gamma$ we have $K \cong \t{\gamma}(K)$,
and, given an $F$ from the kernel of $\Gamma \to \operatorname{Pic}(X)$, there is
a commutative diagram of $\mathcal{O}_{X}$-module isomorphisms $$ \xymatrix{ {\mathcal{S}_{K}} \ar[rr]^{\gamma_{K}} \ar[d]_{\sigma_{F}} & & {\mathcal{R}_{\t{\gamma}(K)}} \ar[d]^{\varrho_{\t{\gamma}(F)}} \\ {\mathcal{S}_{K+F}} \ar[rr]_{\gamma_{K+F}} & & {\mathcal{R}_{\t{\gamma}(K)+\t{\gamma}(F)}} } $$ \item Given data as in (i), let $\mathcal{B}$ and $\mathcal{A}$
denote the Picard graded algebras associated to $\sigma$ and
$\varrho$. Then one has a commutative diagram $$ \xymatrix{ {\mathcal{S}} \ar[r]^{\gamma} \ar[d] & {\mathcal{R}} \ar[d] \\ {\mathcal{B}} \ar[r]_{\b{\gamma}} & {\mathcal{A}} } $$
of graded $\mathcal{O}_{X}$-algebra homomorphisms. The lower row
is an isomorphism if $\Gamma$ and $\Lambda$ have the same image in
$\operatorname{Pic}(X)$. \end{enumerate} \end{lemma}
\proof Let $\Gamma \subset \Lambda(\mathfrak{V})$ and $\Lambda \subset \Lambda(\mathfrak{U})$. Then $\Lambda$ and $\Gamma$ embed canonically into $\Lambda(\mathfrak{W})$, where $\mathfrak{W}$ denotes any common refinement of the open covers $\mathfrak{U}$ and $\mathfrak{V}$. Hence we may assume that $\Lambda$ and $\Gamma$ arise from the same trivializing cover.
Let $K_{1}, \ldots, K_{m}$ be a basis of $\Gamma$ and choose $E_{1}, \ldots, E_{m} \in \Lambda$ in such a way that the isomorphism class of $E_{i}$ equals the class of $K_{i}$ in $\operatorname{Pic}(X)$. Furthermore let $\t{\gamma} \colon \Gamma \to \Lambda$ be the homomorphism sending $K_{i}$ to $E_{i}$. For each $i = 1, \ldots, m$, fix a bundle isomorphism $\beta_{K_{i}} \colon K_{i} \to E_{i}$.
By multiplying the local data of the these homomorphisms, we obtain as in the proof of Lemma~\ref{shiftfamprops} a bundle isomorphism $\beta_{K} \colon K \to \t{\gamma}(K)$ for every $K \in \Gamma$. Shifting sections via these $\beta_{K}$ defines $\mathcal{O}_{X}$-module isomorphisms $\gamma_{K} \colon \mathcal{S}_{K} \to \mathcal{R}_{\t{\gamma}(K)}$. By construction, the $\gamma_{K}$ fit together to a graded homomorphism $\gamma \colon \mathcal{S} \to \mathcal{R}$ of $\mathcal{O}_{X}$-algebras.
Now it is clear how to define the full shifting family $\sigma$: Take an $F$ from the kernel of $\Gamma \to \operatorname{Pic}(X)$. Define $\sigma_{F} \colon \mathcal{S} \to \mathcal{S}$ by prescribing on the homogeneous components the (unique) isomorphisms $\mathcal{S}_{K} \to \mathcal{S}_{K+F}$ that make the above diagrams commutative. It is then straightforward to verify the properties of a shifting family for the maps $\sigma_{F}$. This settles assertion~(i).
We prove~(ii). By the commutative diagram of~(i), the ideal associated to $\sigma$ is mapped into the ideal associated to $\varrho$. Hence, we obtain the desired homomorphism $\b{\gamma} \colon \mathcal{B} \to \mathcal{A}$ of Picard graded algebras.
Now, assume that the images of $\Gamma$ and $\Lambda$ in $\operatorname{Pic}(X)$ coincide. Since every $\gamma_{K} \colon \mathcal{S}_{K} \to \mathcal{R}_{\t{\gamma}(K)}$ is an isomorphism, we can use Lemma~\ref{gradproject}~(ii) to see that $\b{\gamma}$ is an isomorphism in every degree. By assumption the accompanying homomorphism of $\b{\gamma}$ is bijective, whence the assertion follows. \endproof
The uniqueness of homogeneous coordinate rings is a direct consequence of the Lemmas~\ref{shiftfamunique} and~\ref{differentcomps}:
\begin{prop}\label{uniquehomcoord} Different choices of the group of line bundles and the full shifting family define isomorphic freely graded quasiaffine algebras as homogeneous coordinate rings for~$X$. \end{prop}
\proof
Let $\Lambda$ and $\Gamma$ be two groups of line bundles mapping onto $\operatorname{Pic}(X)$ and let $\mathcal{A}$ and $\mathcal{B}$ denote the Picard graded algebras associated to choices of full shifting families for the corresponding $\Lambda$ and $\Gamma$-graded $\mathcal{O}_{X}$-algebras. From Lemmas~\ref{shiftfamunique} and~\ref{differentcomps} we infer the existence of a graded $\mathcal{O}_{X}$-algebra isomorphism $\mu \colon \mathcal{B} \to \mathcal{A}$. In particular, we have $\mathcal{B}(X) \cong \mathcal{A}(X)$.
We show that $\mu$ defines an isomorphism of quasiaffine algebras. Let $(B',J') \in \mathfrak{B}(X)$ as in~\ref{freegradalgdef}. Then Lemma~\ref{quasiaff2closingsubalg} ensures that $(B',J')$ is a natural pair on ${\rm Spec}(\mathcal{B})$. We have to show that $(\mu(B'), \mu(I'))$ is a natural pair on ${\rm Spec}(\mathcal{A})$. Since $\mu$ is an $\mathcal{O}_{X}$-module isomorphism in every degree, we have $Z(\mu(g)) = Z(g)$ for any homogeneous $g \in J'$. Thus Proposition~\ref{geominterp}~(ii) tells us that $(\mu(B'), \mu(I'))$ is a natural pair.
\endproof
\section{Functoriality of the homogeneous coordinate ring} \label{section5}
In this section, we present the first main result. It says that homogeneous coordinates are a fully faithful contravariant functor, see Theorem~\ref{fullyfaithful}. But first we have to define the homogeneous coordinate ring functor on morphisms. The basic tool for this definition are Picard graded pullbacks, see~\ref{picpulldef} and~\ref{picpullex}.
\goodbreak
As in the preceding section, we assume that ${\mathbb K}^{*}$ is of infinite rank over ${\mathbb Z}$. Moreover, in this section we assume all varieties to be divisorial and to have only constant invertible global functions. Finally, we require that any variety has a finitely generated Picard group, and, if ${\mathbb K}$ is of characteristic $p > 0$, this Picard group has no $p$-torsion.
For a variety $X$ fix a group $\Lambda$ of line bundles mapping onto $\operatorname{Pic}(X)$ and denote the associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra by $\mathcal{R}$. Moreover, we fix a full shifting family $\varrho$ for $\mathcal{R}$ and denote the resulting Picard graded algebra by $\mathcal{A}$. For a further variety $Y$ we denote the corresponding data by $\Gamma$, $\mathcal{S}$, $\sigma$, and $\mathcal{B}$. Let $\varphi \colon X \to Y$ be a morphism of the varieties $X$ and $Y$.
\begin{defi}\label{picpulldef} By a {\em Picard graded pullback for $\varphi \colon X \to Y$} we mean a graded homomorphism $\mathcal{B} \to \varphi_{*}\mathcal{A}$ of $\mathcal{O}_{Y}$-algebras having the pullback map $\varphi^{*} \colon \operatorname{Pic}(Y) \to \operatorname{Pic}(X)$ as its accompanying homomorphism. \end{defi}
Note that the property of being an $\mathcal{O}_{Y}$-algebra homomorphism means in particular, that in degree zero any Picard graded pullback is the usual pullback of functions. As a consequence, we remark:
\begin{lemma}\label{pullzero} Let $\mu \colon \mathcal{B} \to \varphi_{*}\mathcal{A}$ be a Picard graded pullback for $\varphi \colon X \to Y$, and let $g \in \mathcal{B}(Y)$ be homogeneous. Then the zero set $Z(\mu(g)) \subset X$ is the inverse image $\varphi^{-1}(Z(g))$ of the zero set $Z(g) \subset Y$. \end{lemma}
\proof
It suffices to prove the statement locally, over small open $V \subset Y$. But on such $V$, we may shift $g$ by multiplication with invertible elements into degree zero. This does not affect zero sets, whence the assertion follows.
\endproof
The basic step in the definition of the homogeneous coordinate ring functor on morphisms is to show existence of Picard graded pullbacks and to provide a certain uniqueness property:
\begin{prop}\label{picpullex} There exist Picard graded pullbacks for $\varphi \colon X \to Y$. Moreover, any two Picard graded pullbacks $\mu, \nu \colon \mathcal{B} \to \varphi_{*}\mathcal{A}$ for $\varphi$ differ by a character $c \colon \operatorname{Pic}(Y) \to {\mathbb K}^{*}$ in the sense that $\nu_{P} = c(P) \mu_{P}$ holds for all $P \in \operatorname{Pic}(Y)$. \end{prop}
The proof of this statement is based on two lemmas. The first one is an extension property for shifting families:
\begin{lemma}\label{shiftext} Let $\Pi$ be any group of line bundles on $X$, and let $\Pi_{0} \subset \Pi_{1}$ be two subgroups of the kernel of $\Pi \to \operatorname{Pic}(X)$. Then every $\Pi_{0}$-shifting family $\tau^{0}$ for the $\Pi$-graded $\mathcal{O}_{X}$-algebra $\mathcal{T}$ associated to $\Pi$ extends to a $\Pi_{1}$-shifting family $\tau^{1}$ for $\mathcal{T}$ in the sense that $\tau^{1}_{E} = \tau^{0}_{E}$ holds for all $E \in \Pi_{0}$. \end{lemma}
\proof Let $\vartheta$ be any $\Pi_{1}$-shifting family for $\mathcal{T}$. Then $\vartheta$ restricts to a $\Pi_{0}$-shifting family. By Lemma~\ref{shiftfamprops}, there is a character $c \colon \Pi_{0} \to \mathcal{O}^{*}(X)$ with $\tau^{0}_{E} = c(E) \vartheta_{E}$ for all $E \in \Pi_{0}$. As we assumed $\mathcal{O}^{*} (X) = {\mathbb K}^{*}$, Lemma~\ref{charext} tells us that $c$ extends to $\Pi_{1}$. Thus, setting $\tau^{1}_{E} := c(E) \vartheta_{E}$ for $E \in \Pi_{1}$ gives the desired extension. \endproof
The second lemma provides a pullback construction for shifting families. By pulling back cocycles, we obtain the (again free) pullback group $\varphi^{*}\Gamma$. We denote the associated $\varphi^{*}\Gamma$-graded $\mathcal{O}_{X}$-algebra by $\varphi^{*}\mathcal{S}$. Indeed $\varphi^{*}\mathcal{S}$ is canonically isomorphic to the ringed inverse image of $\mathcal{S}$. Observe that we have a canonical sheaf homomorphism $\mathcal{S} \to \varphi_{*}\varphi^{*} \mathcal{S}$.
\goodbreak
\begin{lemma}\label{pullshift} Let $\Gamma_{0} \subset \Gamma$ a subgroup, and let $\sigma$ be a $\Gamma_{0}$-shifting family for $\mathcal{S}$. \begin{enumerate} \item The $\mathcal{O}_{X}$-module homomorphisms
$\varphi^{*}\sigma_{F}$ define a
$\varphi^{*}\Gamma_{0}$-shifting family
$\varphi^{*}\sigma$ for $\varphi^{*}\mathcal{S}$. \item The ideal $\mathcal{J}^{*}$ associated to $\varphi^{*}\sigma$
equals the pullback $\varphi^{*}\mathcal{J}$ of the ideal
$\mathcal{J}$ associated to $\sigma$. \end{enumerate} \end{lemma}
\proof For~(i), note that the isomorphisms $\sigma_{F} \colon \mathcal{S}_{K} \to \mathcal{S}_{K+F}$ can be written as $g \mapsto \beta_{K,F}(g)$ with unique line bundle isomorphisms $\beta_{K,F} \colon K \to K + F$. The family $\varphi^{*} \sigma_{F}$ corresponds to the collection $\varphi^{*} \beta_{K,F} \colon \varphi^{*} K \to \varphi^{*}K + \varphi^{*}F$. The properties of a shifting family become clear by writing the $\varphi^{*} \beta_{K,F}$ in terms of local data as in~\ref{localdata}.
To prove~(ii), we just compare the stalks of the two sheaves in question. By Property~\ref{shiftfamprops}~(iii), we obtain for any $x \in X$: \begin{eqnarray*} \mathcal{J}^{*}_{x} & = & \bangle{1_{x} - \varphi^{*} \sigma_{F}(1_{x});
\; F \in \Gamma_{0}} \\ & = & \bangle{\varphi^{*}(1_{\varphi(x)}) -
\varphi^{*}(\sigma_{F}(1_{\varphi(x)}));
\; F \in \Gamma_{0}} \\ & = & (\varphi^{*}\mathcal{J})_{x}. \qquad \qed \end{eqnarray*}
\proof[Proof of Proposition~\ref{picpullex}] We establish the existence of Picard graded pullbacks: As usual, let $\mathcal{I}$ and $\mathcal{J}$ denote the respective ideals associated to the shifting families $\varrho$ for $\mathcal{R}$ and $\sigma$ for $\mathcal{S}$. Thus the corresponding Picard graded algebras are $\mathcal{A} = \mathcal{R}/\mathcal{I}$ and $\mathcal{B} = \mathcal{S}/\mathcal{J}$.
By Lemma~\ref{pullshift}, we have the $\varphi^{*}\Gamma_{0}$-shifting family $\varphi^{*}\sigma$ for $\varphi^{*}\mathcal{S}$. Lemma~\ref{shiftext} enables us to choose a full shifting family $\varphi^{\sharp}\sigma$ extending $\varphi^{*}\sigma$. We denote by $\varphi^{\sharp}\mathcal{J}$ the ideal associated to $\varphi^{\sharp}\sigma$, and write $\varphi^{\sharp}\mathcal{B} := \varphi^{*}\mathcal{S}/\varphi^{\sharp}\mathcal{J}$ for the quotient. In this notation, we have a commutative diagram of graded $\mathcal{O}_{Y}$-algebra homomorphisms such that the unlabelled arrows are isomorphisms in each degree: $$ \xymatrix{ {\varphi_{*}\mathcal{R}} \ar[d] & & {\varphi_{*}\varphi^{*}\mathcal{S}} \ar[dl] \ar[dr] \ar[ll] & & {\mathcal{S}} \ar[ll]_{\varphi^{*}} \ar[d] \\ {\varphi_{*}{\mathcal{A}}} & {\varphi_{*}\varphi^{\sharp}\mathcal{B}} \ar[l] & & {\varphi_{*}\varphi^{*}\mathcal{B}} \ar[ll] & {\mathcal{B}} \ar[l]^{\varphi^{*}} } $$
Indeed, the right square is standard. To obtain the middle triangle, we only have to show that $\varphi^{\sharp}\mathcal{J}$ contains the kernel of $\varphi^{*}\mathcal{S} \to \varphi^{*}\mathcal{B}$. But this follows from exactness of $\varphi^{*}$ and Lemma~\ref{pullshift}~(ii). Existence of the left square follows from combining Lemmas~\ref{shiftfamunique} and~\ref{differentcomps}. Now the desired Picard graded pullback of $\varphi \colon X \to Y$ is the composition of the lower horizontal arrows.
We turn to the uniqueness statement. Let $P \in \operatorname{Pic}(Y)$. Since $\mathcal{B}_{P}$ locally admits invertible sections, we can cover $Y$ by open $V \subset Y$ such that there exist invertible sections $h \in \mathcal{B}_{P} (V)$. We define $$c(P,V) := \nu(h)/\mu (h) \in \mathcal{A}_{0}^{*} (\varphi^{-1}(V)).$$
This does not depend on the choice of $h$: For a further invertible $ g\in \mathcal{B}_{P} (V)$, the section $g/h$ is of degree zero. But in degree zero any Picard graded pullback is the usual pullback of functions. Thus we have $\mu(h/g) = \nu(h/g)$. Consequently, $\nu(g)/\mu(g)$ equals $\nu(h)/\mu(h)$.
\goodbreak
Similarly we see that for two open $V, V' \subset Y$ as above, the corresponding sections $c(P,V)$ and $c(P,V')$ coincide on the intersection $\varphi^{-1}(V) \cap \varphi^{-1}(V')$. Thus, by gluing, we obtain a global section $c(P) \in \mathcal{A}_{0}^{*} (X) = \mathcal{O}^{*}(X)$. Then it is immediate to check, that $P \mapsto c(P)$ has the desired properties.
\endproof
With the help of Picard graded pullbacks we can now make the homogeneous coordinate ring into a functor. We fix for any morphism $\varphi \colon X \to Y$ a Picard graded pullback $\mu_{\varphi} \colon \mathcal{B} \to \varphi_{*}\mathcal{A}$, and denote the induced homomorphism on global sections again by $\mu_{\varphi} \colon\mathcal{B}(Y) \to \mathcal{A}(X)$.
For a graded homomorphism $\nu$ of freely graded quasiaffine algebras, we denote by $[\nu]$ its equivalence class in the sense of Definition~\ref{pointedmorphdef}.
\begin{prop}\label{var2alghom} The assignments $X \mapsto (\mathcal{A}(X), \mathfrak{A}(X))$ and $\varphi \mapsto [\mu_{\varphi}]$ define a contravariant functor into the category of freely graded quasiaffine algebras. \end{prop}
\proof By Proposition~\ref{coringisqualg}, the homogeneous coordinate rings $(\mathcal{A}(X),\mathfrak{A}(X))$ and $(\mathcal{B}(Y),\mathfrak{B}(Y))$ of $X$ and $Y$ are in fact freely graded quasiaffine algebras. The first task is to show that the homomorphism $\mu_{\varphi} \colon \mathcal{B}(Y) \to \mathcal{A}(X)$ associated to a morphism $\varphi \colon X \to Y$ is a graded homomorphism of freely graded quasiaffine algebras.
As a Picard graded pullback, $\mu_{\varphi}$ is graded and has as accompanying homomorphism the pullback map $\operatorname{Pic}(Y) \to \operatorname{Pic}(X)$. Thus we are left with checking the conditions of Definition~\ref{quasiaffalgdef}~(ii) for $\mu_{\varphi}$. This is done geometrically in terms of the constructions of Proposition~\ref{geominterp}: $$ \rq{X} := {\rm Spec}(\mathcal{A}), \qquad \rq{Y} := {\rm Spec}(\mathcal{B}), \qquad
q_{X} \colon \rq{X} \to X, \qquad q_{Y} \colon \rq{Y} \to Y. $$
Let $(B',J') \in \mathfrak{B}(Y)$ be a closing subalgebra as in Definition~\ref{freegradalgdef}. Then Lemma~\ref{naturalpairs} provides a closing subalgebra $(A',I') \in \mathfrak{A}(X)$ such that $\mu_{\varphi}(B') \subset A'$ holds. We have to verify the condition on the ideals $I'$ and $J'$ required in~\ref{quasiaffalgdef}~(ii). For this, consider the affine closures of $\rq{X}$ and $\rq{Y}$: $$\b{X} := {\rm Spec}(A'), \qquad \b{Y} := {\rm Spec}(B'). $$
Then the restricted homomorphism $\mu_{\varphi} \colon B' \to A'$ defines a morphism $\b{\varphi} \colon \b{X} \to \b{Y}$. Recall from Section~\ref{section1} that $I'$ and $J'$ are the vanishing ideals of the complements $\b{X} \setminus \rq{X}$ and $\b{Y} \setminus \rq{Y}$. Thus we have to show that $\b{\varphi}$ maps $\rq{X}$ to $\rq{Y}$. For this, let $g_{1}, \ldots, g_{s} \in J'$ be homogeneous sections as in~\ref{freegradalgdef}. Using Lemma~\ref{pullzero}, we obtain:
\goodbreak
\begin{eqnarray*} \rq{X} & = & \bigcup_{j=1}^{r} q_{X}^{-1}(\varphi^{-1}(Y \setminus Z(g_{j}))) \\ & = & \bigcup_{j=1}^{r} q_{X}^{-1} (X \setminus Z(\mu_{\varphi}(g_{j}))) \\ & \subset & \bigcup_{j=1}^{r} \b{X}_{\mu_{\varphi}(g_{j})} \\ & = & \b{\varphi}^{-1}(\rq{Y}). \end{eqnarray*}
\goodbreak
Finally, we check that $\varphi \mapsto [\mu_{\varphi}]$ is functorial. Note that by Proposition~\ref{picpullex}, the class $[\mu_{\varphi}]$ does not depend on the choice of the Picard graded pullback $\mu_{\varphi}$ of a given morphism.
From this we conclude that the identity morphism of a variety is mapped to the identity morphism of its homogeneous coordinate ring. Moreover, as the composition of two Picard graded pullbacks is a Picard graded pullback for the composition of the respective morphisms, the above assignment commutes with composition.
\endproof
In the sequel we shall speak of the homogeneous coordinate ring functor. We present the first main result of this article. It tells us that the morphisms of two varieties are in one-to-one correspondence with the morphisms of their coordinate rings:
\begin{thm}\label{fullyfaithful} The homogeneous coordinate ring functor $X \mapsto (\mathcal{A}(X), \mathfrak{A}(X))$ and $\varphi \mapsto [\mu_{\varphi}]$ is fully faithful. \end{thm}
\proof Let $X$, $Y$ be varieties with associated Picard graded algebras $\mathcal{A}$ and $\mathcal{B}$. We denote the respective homogeneous coordinate rings of $X$ and $Y$ for short by $(A,\mathfrak{A})$ and $(B,\mathfrak{B})$. We construct an inverse to $${\rm Mor}(X,Y) \to {\rm Mor}((B,\mathfrak{B}),(A,\mathfrak{A})), \qquad \varphi \mapsto [\mu_{\varphi}].$$
So, start with any graded homomorphism $\mu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ of quasiaffine algebras. Then Lemma~\ref{naturalpairs} provides closing subalgebras $(A',I') \in \mathfrak{A}$ and $(B',J') \in \mathfrak{B}$ such that $(B',J')$ is as in Definition~\ref{freegradalgdef} and we have $\mu(B') \subset A'$.
Consider the affine closures $\b{X} := {\rm Spec}(A')$ and $\b{Y} := {\rm Spec}(B')$ of $\rq{X} := {\rm Spec}(\mathcal{A})$ and $\rq{Y} := {\rm Spec}(\mathcal{B})$. Then $\mu$ gives rise to a morphism $\b{\varphi} \colon \b{X} \to \b{Y}$, and restricting this morphism to $\rq{X}$ yields a commutative diagram
$$ \xymatrix{ {\rq{X}} \ar[r]^{\rq{\varphi}} \ar[d]_{q_{X}} &
{\rq{Y}} \ar[d]^{q_{Y}} \\ X \ar[r]^{\varphi} & Y } $$
where $q_{X}$ and $q_{Y}$ denote the canonical maps, and the morphism $\varphi \colon X \to Y$ has as its pullbacks on the level of functions the maps obtained by restricting the localizations $\mu_{g} \colon B_{g} \to A_{\mu(g)}$ to degree zero over the affine sets $Y_{g}$ for homogeneous $g \in J'$.
Observe that applying the above procedure to a further graded homomorphism $\nu \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ yields the same induced morphism $X \to Y$ if and only if the homomorphisms $\mu$ and $\nu$ are equivalent; the ``only if'' part follows from the uniqueness statement of Proposition~\ref{picpullex} and the fact that $\mu$ and $\nu$ define Picard graded pullbacks via localizing. Thus $[\mu] \mapsto \varphi$ defines an injection $${\rm Mor}((B,\mathfrak{B}),(A,\mathfrak{A})) \to {\rm Mor}(X,Y).$$
We check that this map is inverse to the one defined by the homogeneous coordinate ring functor. Start with a morphism $\varphi \colon X \to Y$, and let $[\mu_{\varphi}] \colon B \to A$ be as before Proposition~\ref{var2alghom}. Write shortly $\mu := \mu_{\varphi}$. Consider a homogeneous $g \in B$ such that $V := Y \setminus Z(g)$ is affine and let $U := \varphi^{-1}(V)$. Using Lemma~\ref{pullzero}, we obtain a commutative diagram $$\xymatrix{ {A_{(\mu(g))}} \ar@{=}[d] & {B_{(g)}} \ar[l]_{\mu_{(g)}} \ar@{=}[d] \\ {\mathcal{O}_{X}(U)} & {\mathcal{O}_{Y}(V)} \ar[l]^{\varphi^{*}} } $$ where the above horizontal map is the map on degree zero induced by the localized map $\mu_{g} \colon B_{g} \to A_{\mu(g)}$. Since $Y$ is covered by open affine sets of the form $V = Y \setminus Z(g)$, we see that the morphism $X \to Y$ associated to $\mu = \mu_{\varphi}$ is again $\varphi$. \endproof
So far, our homogeneous coordinate ring functor depends on the choice of the homogeneous coordinate ring for a given variety. By passing to isomorphism classes, the whole construction can even be made canonical:
\begin{rem}\label{canonical} If one takes as target category the category of isomorphism classes of freely graded quasiaffine algebras, then the homogeneous coordinate functor $X \to (\mathcal{A}(X), \mathfrak{A}(X))$ and $\varphi \mapsto [\mu_{\varphi}]$ becomes unique. \end{rem}
\section{A first dictionary}\label{section6}
We present a little dictionary between geometric properties of a variety and algebraic properties of its homogeneous coordinate ring. We consider separatedness, normality and smoothness. Moreover, we treat quasicoherent sheaves, and we describe affine morphisms and closed embeddings.
The setup is the same as in Sections~\ref{section4} and~\ref{section5}: The multiplicative group ${\mathbb K}^{*}$ of the ground field ${\mathbb K}$ is supposed to be of infinite rank over ${\mathbb Z}$. Moreover, $X$ is a divisorial variety with $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$ and its Picard group is finitely generated and has no $p$-torsion if ${\mathbb K}$ is of characteristic $p>0$.
Denote by $(A,\mathfrak{A}) := (\mathcal{A}(X),\mathfrak{A}(X))$ the homogeneous coordinate ring of $X$. Recall that $A$ is the algebra of global sections of a suitable Picard graded $\mathcal{O}_{X}$-algebra~$\mathcal{A}$. In the subsequent proofs, we shall often use the geometric interpretation provided by Propositions~\ref{geominterp} and~\ref{equivquasiaffequiv}:
\begin{lemma}\label{geomquot} Consider $\rq{X} := {\rm Spec}(\mathcal{A})$, the canonical map $q \colon \rq{X} \to X$ and the diagonalizable group $H := {\rm Spec}({\mathbb K}[\operatorname{Pic}(X)])$. \begin{enumerate} \item There is a unique free action of $H$ on
$\rq{X}$ such that each $\mathcal{A}_{[L]}(U)$ consists precisely
of the $\chi^{[L]}$-homogeneous functions of $q^{-1}(U)$. \item The canonical map $q \colon \rq{X} \to X$ is a geometric
quotient for the above $H$-action on $X$. \end{enumerate} \end{lemma}
\proof The first statement follows from Propositions~\ref{geominterp} and~\ref{equivquasiaffequiv}. The second statement is due to the facts that $\mathcal{O}_{X} = q_{*}(\mathcal{A}_{0})$ is the sheaf of invariants and the action of $H$ is free. \endproof
We begin with the dictionary. It is quite easy to characterize separatedness in terms of the homogeneous coordinate ring:
\begin{prop} The variety $X$ is separated if and only if there exists a graded closing subalgebra $(A',I') \in \mathfrak{A}$ and homogeneous $f_{1}, \ldots, f_{r} \in I'$ as in~\ref{freegradalgdef} such that each of the maps $A_{(f_{i})} \otimes A_{(f_{j})} \to A_{(f_{i}f_{j})}$ is surjective. \end{prop}
\proof
First recall that the sets $X_{i} := X \setminus Z(f_{i})$ form an affine cover of $X$. The above condition means just that the canonical maps from $\mathcal{O}(X_{i}) \otimes \mathcal{O}(X_{j})$ to $\mathcal{O}(X_{i} \cap X_{j})$ is surjective. This is the usual separatedness criterion~\cite[Prop.~3.3.5]{Ke}.
\endproof
Next we show how normality of the variety $X$ is reflected in its homogeneous coordinate ring (for us, a normal variety is in particular irreducible):
\begin{prop}\label{normal2normal} The variety $X$ is normal if and only if $A$ is a normal ring. \end{prop}
\proof
We work in terms of the geometric data $q \colon \rq{X} \to X$ and $H$ discussed in Lemma~\ref{geomquot}. First suppose that $A = \mathcal{A}(X)$ is a normal ring. Then the quasiaffine variety $\rq{X}$ is normal. It is a basic property of geometric quotients that the variety $X$ inherits normality from $\rq{X}$, see e.g.~\cite[p.~39]{Do}.
Suppose conversely that $X$ is normal. Luna's Slice Theorem tells us that $q \colon \rq{X} \to X$ is an $H$-principal bundle in the \'etale topology, see~\cite{Lu}, and~\cite[Prop.~8.1]{BaRi}. Thus, up to \'etale maps, $\rq{X}$ looks locally like $X \times H$. Since normality of local rings is stable under \'etale maps~\cite[Prop.~I.3.17]{Mi}, we can conclude that all local rings of $\rq{X}$ are normal.
It remains to show that $\rq{X}$ is connected. Assume the contrary. Then there is a connected component $\rq{X}_{1} \subset \rq{X}$ with $q(\rq{X}_{1}) = X$. Let $H_{1} \subset H$ be the stabilizer of $\rq{X}_{1}$, that means that $H_{1}$ is the maximal subgroup of $H$ with $H_{1} \! \cdot \! \rq{X}_{1} = \rq{X}_{1}$. Note that we have $t \in H_{1}$ if $t \! \cdot \! x \in \rq{X}_{1}$ holds for at least one point $x \in \rq{X}_{1}$. In particular, $H_{1}$ is a proper subgroup of $H$.
We claim that restricting the canonical map $q \colon \rq{X} \to X$ to $\rq{X}_{1}$ yields a geometric quotient for the action of $H_{1}$ on $\rq{X}_{1}$. Indeed, $H_{1}$ acts freely on $\rq{X}_{1}$. Hence we have a geometric quotient $\rq{X}_{1} \to \rq{X}_{1}/H_{1}$ and a commutative diagram $$ \xymatrix{ {\rq{X}_{1}} \ar[r]^{\subset} \ar[d]_{/H_{1}} & {\rq{X}} \ar[d]^{/H}_{q} \\ {\rq{X}_{1}/H_{1}} \ar[r] & X } $$
The map $\rq{X}_{1}/H_{1} \to X$ is bijective, because the intersection of a $q$-fibre with~$\rq{X}_{1}$ always is a single $H_{1}$-orbit. Since $X$ is normal, we may apply Zariski's Main Theorem to conclude that $\rq{X}_{1}/H_{1} \to X$ is even an isomorphism. This verifies our claim.
Since $H_{1}$ is a proper subgroup of $H$, we find a nontrivial class $[L] \in \operatorname{Pic}(X)$ such that the corresponding character $\chi^{[L]}$ of $H$ is trivial on $H_{1}$. We construct a defining cocycle for the class $[L]$: Cover $X$ by small open sets $U_{i}$ admitting invertible sections $g_{i} \in \mathcal{A}_{[L]}(U_{i})$. Then the cocycle $g_{i}/g_{j}$ defines a bundle belonging to the class $[L]$.
On the other hand, the $g_{i}$ are $\chi^{[L]}$-homogeneous functions on $q^{-1}(U_{i})$. So they restrict to $H_{1}$-invariant functions on $q^{-1}(U_{i}) \cap \rq{X}_{1}$. As seen before, $X$ is the quotient of $\rq{X}_{1}$ by the action of $H_{1}$. Thus we conclude that the $g_{i}/g_{j}$ form in fact a coboundary on $X$. Consequently, the class $[L]$ must be trivial. This contradicts the choice of $[L]$.
\endproof
Thus we see that if $X$ is normal, then $A$ is the ring of global functions of a normal variety. That means that $A$ belongs to a intently studied class of rings:
\begin{coro} Let $X$ be normal. Then $A$ is a Krull ring. \endproof \end{coro}
As we did in Proposition~\ref{normal2normal} for normality, we can characterize smoothness in terms of the homogeneous coordinate ring:
\begin{prop}\label{smooth} $X$ is smooth if and only if there is a closing subalgebra $(A',I') \in \mathfrak{A}$ such that all localizations $A_{\mathfrak{m}}$ are regular, where $\mathfrak{m}$ runs through the maximal ideals with $I' \not \subset \mathfrak{m}$. \end{prop}
\proof
Let $\rq{X} := {\rm Spec}(\mathcal{A})$, and consider the affine closure $\b{X} := {\rm Spec}(A')$ defined by any closing subalgebra $(A',I')$ of $A$. Recall from Lemmas~\ref{naturalpairs} and~\ref{quasiaff2closingsubalg} that $I'$ is the vanishing ideal of the complement $\b{X} \setminus \rq{X}$. So, the regularity of the local rings $A_{\mathfrak{m}}$, where $I' \not \subset \mathfrak{m}$, just means smoothness of $\rq{X}$.
The rest is similar to the proof of Proposition~\ref{normal2normal}: The canonical map $q \colon \rq{X} \to X$ is an \'etale $H$-principal bundle for a diagonalizable group $H$. Thus, up to \'etale maps, $\rq{X}$ looks locally like $X \times H$. The assertion then follows from the fact that regularity of local rings is stable under \'etale maps, see~\cite[Prop.~I.3.17]{Mi}. \endproof
We give a description of quasicoherent sheaves. Consider a graded $A$-module $M$. Given $f_{1}, \ldots, f_{r} \in A$ as in~\ref{freegradalgdef}, set $\mathcal{M}_{i} := M_{(f_{i})}$. Then these modules glue together to a quasicoherent $\mathcal{O}_{X}$-module $\mathcal{M}$ on $X$. As in the toric case \cite[Section~4]{ACHaSc}, one obtains:
\begin{prop} The assignment $M \mapsto \mathcal{M}$ defines an essentially surjective functor from the category of graded $A$-modules to the category of quasicoherent $\mathcal{O}_{X}$-modules. \endproof \end{prop}
We come to properties of morphisms. Let $Y$ be a further variety like $X$, and denote its homogeneous coordinate ring by $(B,\mathfrak{B})$. Let $\varphi \colon X \to Y$ be any morphism. Denote by $[\mu] \colon (B,\mathfrak{B}) \to (A,\mathfrak{A})$ the corresponding morphism of freely graded quasiaffine algebras.
\begin{prop} The morphism $\varphi \colon X \to Y$ is affine if and only if there are graded closing subalgebras $(A',I') \in \mathfrak{A}$ and $(B',J') \in \mathfrak{B}$ satisfying~\ref{quasiaffalgdef} such that $$ \sqrt{I'} = \sqrt{\bangle{\mu(J')}}. $$ Moreover, $\varphi \colon X \to Y$ is a closed embedding if and only if it satisfies the above condition and, given $g_{1}, \ldots, g_{s} \in B$ as in \ref{freegradalgdef}, every induced map $B_{(g_{i})} \to A_{(\mu(g_{i}))}$ is surjective. \end{prop}
\proof Let $\mathcal{B}$ be a Picard graded $\mathcal{O}_{Y}$-algebra with $B = \mathcal{B}(Y)$. Consider the affine closures $\b{X} := {\rm Spec}(A')$ of $\rq{X} := {\rm Spec}(\mathcal{A})$ and $\b{Y} := {\rm Spec}(B')$ of $\rq{Y} := {\rm Spec}(\mathcal{B})$. Then $\mu \colon B \to A$ gives rise to a commutative diagram
$$ \xymatrix{ {\rq{X}} \ar[r]^{\rq{\varphi}} \ar[d]_{q_{X}} &
{\rq{Y}} \ar[d]^{q_{Y}} \\ X \ar[r]^{\varphi} & Y } $$
The morphism $\varphi$ is affine if and only if $\rq{\varphi}$ is affine. The latter is equivalent to the condition of $\sqrt{I'} = \sqrt{\bangle{\mu(J')}}$ of the assertion. The supplement on embeddings is obvious. \endproof
\section{Tame varieties}\label{section7}
In this section we shed some light on the question which freely graded quasiaffine algebras occur as homogeneous coordinate rings. As before, we assume that the multiplicative group ${\mathbb K}^{*}$ of the ground field is of infinite rank over ${\mathbb Z}$. We consider varieties of the following type:
\begin{defi} A {\em tame variety\/} is a normal divisorial variety $X$ with $\mathcal{O}(X) = {\mathbb K}$ and a finitely generated Picard group $\operatorname{Pic}(X)$ having no $p$-torsion if ${\mathbb K}$ is of characteristic $p>0$. \end{defi}
The prototype of a tame variety lives in characteristic zero, and is a smooth complete variety with finitely generated Picard group. Moreover, in characteristic zero, every Calabi-Yau variety is tame, and every ${\mathbb Q}$-factorial rational variety $X$ with $\mathcal{O}(X) = {\mathbb K}$ is tame. Finally, in characteristic zero every normal divisorial variety with finitely generated Picard group admits an open embedding into a tame variety.
In order to figure out the coordinate rings of tame varieties, we need some preparation. Suppose that an algebraic group $G$ acts on a variety $X$. Recall that a {\em $G$-linearization} of a line bundle $E \to X$ is a fibrewise linear $G$-action on $E$ making the projection equivariant. By a {\em simple $G$-variety} we mean a $G$-variety for which any $G$-linearizable line bundle is trivial.
\begin{defi} Let $\Lambda$ be a finitely generated abelian group, and let $(A,\mathfrak{A})$ be a freely $\Lambda$-graded quasiaffine algebra. \begin{enumerate} \item We say that $(A,\mathfrak{A})$ is {\em pointed\/} if
$A$ is a normal ring, $A_{0} = {\mathbb K}$ holds,
and the set $A^{*} \subset A$ of
invertible elements is just ${\mathbb K}^{*}$. \item We say that $(A,\mathfrak{A})$ is {\em simple\/}
if $\Lambda$ has no $p$-torsion if ${\mathbb K}$ is
of characteristic $p>0$, and the quasiaffine
${\rm Spec}({\mathbb K}[\Lambda])$-variety corresponding to
$(A,\mathfrak{A})$ is simple. \end{enumerate} \end{defi}
These two subclasses define full subcategories of the categories of divisorial varieties with finitely generated Picard group and freely graded quasiaffine algebras. The second main result of this article is the following:
\begin{thm}\label{equivthm} The homogeneous coordinate ring functor restricts to an equivalence from the category of tame varieties to the category of simple pointed algebras. \end{thm}
\proof Let $X$ be a tame variety with Picard group $\Pi := \operatorname{Pic}(X)$, and denote the associated homogeneous coordinate ring by $(A,\mathfrak{A})$. Then $A$ is the algebra of global sections of some Picard graded $\mathcal{O}_{X}$-algebra $\mathcal{A}$ on $X$. We shall use again the geometric data discussed in Lemma~\ref{geomquot}: $$ \rq{X} := {\rm Spec}(\mathcal{A}), \qquad q \colon \rq{X} \to X, \qquad H := {\rm Spec}({\mathbb K}[\Pi]). $$
The first task is to show that $(A,\mathfrak{A})$ is in fact pointed. From Proposition~\ref{normal2normal} we infer that $A$ is a normal ring. Since we assumed $\mathcal{O}(X) = {\mathbb K}$, and $\mathcal{O}(X)$ equals $A_{0}$, we have $A_{0} = {\mathbb K}$. So we have to verify $A^{*} = {\mathbb K}^{*}$. For this, consider an arbitrary element $f \in A^{*}$.
Choose a direct decomposition of $\Pi$ into a free part $\Pi_{0}$ and the torsion part $\Pi_{{\rm t}}$. This corresponds to a splitting $H = H_{0} \times H_{{\rm t}}$ with a torus $H_{0}$ and a finite group $H_{{\rm t}}$. As an invertible element of $\mathcal{O}(\rq{X})$, the function $f$ is necessarily $H_{0}$-homogeneous, see~e.g.~\cite[Prop.~1.1]{Ma}. Thus, there is a degree $P \in \Pi_{0}$ such that $$ f = \sum_{G \in \Pi_{t}} f_{P+G}, \qquad f^{-1} = \sum_{G \in \Pi_{t}} f^{-1}_{-P+G}. $$
From the identity $ff^{-1} = 1$ we infer that $f_{P+G}f^{-1}_{-P-G} \ne 0$ holds for at least one component $f_{P+G}$ of $f$. Since $\mathcal{O}(X)={\mathbb K}$ holds, we see that the homogeneous section $f_{P+G} \in A$ is invertible. Thus the homogeneous component $\mathcal{A}_{P+G}$ is isomorphic to~$\mathcal{O}_{X}$.
On the other hand we noted in~\ref{gradproject}~(ii) that $\mathcal{A}_{P+G}$ is isomorphic to the sheaf of sections of a bundle representing the class $P+G$ in $\Pi_{0} \oplus \Pi_{{\rm t}}$. Thus $P+G$ is trivial, and we obtain $P = 0$. Hence all homogeneous components of $f$ have torsion degree. By $\mathcal{O}(X) = {\mathbb K}$ this yields that $f_{G} =0$ if $G \ne 0$. Thus we have $f \in A_{0} = {\mathbb K}$.
The next task is to show that $\rq{X}$ is a simple $H$-variety. For this, let $\operatorname{Pic}_{H}(\rq{X})$ denote the group of equivariant isomorphy classes of $H$-linearized line bundles on~$\rq{X}$, compare~\cite[Sec.~2]{Kr}. Moreover, let $\Pic_{\rm lin}(\rq{X}) \subset \operatorname{Pic}(\rq{X})$ denote the subgroup of the classes of all $H$-linearizable bundles. We have to show that $\Pic_{\rm lin}(\rq{X})$ is trivial.
First, we consider the possible linearizations of the trivial bundle $\rq{X} \times {\mathbb K}$. Using $\mathcal{O}^{*}(\rq{X}) = {\mathbb K}^{*}$, as verified before, one directly checks that any linearization of the trivial bundle is given by a character $\chi$ of $H$ as follows: \begin{equation}\label{trivbdlelin} t \! \cdot \! (x,z) := (t \! \cdot \! x, \chi(t) z) \end{equation}
In particular, the character group $\operatorname{Char}(H)$ canonically embeds into the group $\operatorname{Pic}_{H}(X)$. Since we obtain in~\ref{trivbdlelin} indeed any linearization of the trivial bundle, the map $\operatorname{Char}(H) \to \operatorname{Pic}_{H}(\rq{X})$ and the forget map $\operatorname{Pic}_{H}(\rq{X}) \to \Pic_{\rm lin}(\rq{X})$ fit together to an exact sequence, compare also~\cite[Lemma~2.2]{Kr}: \begin{equation}\label{thesequence} \xymatrix{
0 \ar[r] & {\operatorname{Char}(H)} \ar[r]^{} & {\operatorname{Pic}_{H}(\rq{X})} \ar[r] & {\Pic_{\rm lin}(\rq{X})} \ar[r] &
0 } \end{equation}
Thus, to obtain $\Pic_{\rm lin}(\rq{X}) = 0$, it suffices to split the map $\operatorname{Char}(H) \to \operatorname{Pic}_{H}(\rq{X})$ into isomorphisms as follows: \begin{equation}\label{specialsetting} \vcenter{ \xymatrix{ {\operatorname{Char}(H)} \ar[rr]^{} \ar[dr]^{{\cong}}_{{\chi^{P} \mapsto P}}& & {\operatorname{Pic}_{H}(\rq{X})} \\ & {\Pi} \ar[ur]_{q^{*}}^{\cong} &
}} \end{equation}
But this is not hard: The fact that $q^{*}$ induces an isomorphism of $\Pi = \operatorname{Pic}(X)$ and $\operatorname{Pic}_{H}(\rq{X})$ is due to~\cite[Prop.~4.2]{Kr}. To obtain commutativity, consider $P \in \Pi$. Choose invertible sections $g_i \in \mathcal{A}_{P}(U_{i})$ for small open $U_{i}$ covering $X$. Then the class of $P$ is represented by the bundle $P_{\xi}$ arising from the cocycle \begin{equation}\label{char2pic} \xi_{ij} \; := \; \frac{g_{j}}{g_{i}}. \end{equation} So the pullback class $q^{*}(P) \in \operatorname{Pic}_{H}(\rq{X})$ is represented by the trivially linearized bundle $q^{*}(P_{\xi})$, which in turn arises from the cocycle \begin{equation}\label{char2pic2} q^{*}(\xi_{ij}) \; := \; q^{*}\left(\frac{g_{j}}{g_{i}}\right) \; = \; \frac{g_{j}}{g_{i}}. \end{equation} But on $\rq{X}$, the $g_{i}$ are ordinary invertible functions. So we obtain an isomorphism from the representing bundle $q^{*}(P_{\xi})$ onto the trivial bundle by locally multiplying with $g_{i}$. Obviously, the induced linearization on the trivial bundle is the linearization~\ref{trivbdlelin} for $\chi = \chi^{P}$.
Thus we proved that $(A,\mathfrak{A})$ is in fact a simple pointed algebra. In other words, the homogeneous coordinate ring functor restricts to the subcategories in consideration. It remains to show that up to isomorphism, every simple pointed algebra is the homogeneous coordinate ring of some tame variety $X$.
So, let $(A,\mathfrak{A})$ be a simple pointed algebra, graded by some finitely generated abelian group $\Pi$. According to Proposition~\ref{equivquasiaffequiv}, we may assume that $(A,\mathfrak{A})$ equals $(\mathcal{O}(\rq{X}),\mathfrak{O}(\rq{X}))$ for some normal quasiaffine variety $\rq{X}$ with a free action of a diagonalizable group $H = {\rm Spec}({\mathbb K}[\Pi])$.
The action of $H$ on $\rq{X}$ admits a geometric quotient $q \colon \rq{X} \to X$: First divide by the finite factor $H_{{\rm t}}$ of $H$ to obtain a normal quasiaffine variety $\rq{X}/H_{{\rm t}}$, and then divide by the induced action of the unit component $H_{0}$ of $H$ on $\rq{X}/H_{{\rm t}}$, see for example~\cite[Ex.~4.2]{Do} and~\cite[Cor.~3]{Su}.
The candidate for our tame variety is $X$. Since the structure sheaf $\mathcal{O}_{X}$ is the sheaf of invariants $q_{*} (\mathcal{O}_{\rq X})^{H}$ and $A = {\mathcal{O}}(\rq{X})$ is pointed, we have $\mathcal{O}(X) = {\mathbb K}$. Moreover, as a geometric quotient space of a normal quasiaffine variety by a free diagonalizable group action, $X$ is again normal and divisorial, for the latter see~\cite[Lemma~3.3]{Ha1}.
To conclude the proof, we have to realize the ($\Pi$-graded) direct image $\mathcal{A} := q_{*}(\mathcal{O}_{\rq{X}})$ as a Picard graded algebra on $X$. First note that we have again the exact sequence~\ref{thesequence}. Since we assumed $\Pic_{\rm lin}(\rq{X}) = 0$, the character group $\operatorname{Char}(H)$ maps isomorphically onto $\operatorname{Pic}_{H}(\rq{X})$.
Moreover, we have a canonical map $\Pi \to \operatorname{Pic}(X)$: For a degree $P \in \Pi$ choose invertible $\chi^{P}$-homogeneous functions $g_{i} \in \mathcal{O}(q^{-1}(U_{i}))$ with small open $U_{i} \subset X$ covering $X$, see Definition~\ref{freegradalgdef}. As in~\ref{char2pic}, such functions define a cocycle $\xi$ and hence we may map $P$ to the class of the bundle $P_{\xi}$. In conclusion, we arrive again at a commutative diagram as in~\ref{specialsetting}. In particular, $\Pi \to \operatorname{Pic}(X)$ is an isomorphism.
In fact, the construction~\ref{char2pic} allows us to define a group $\Lambda$ of line bundles on~$X$: As in the proof of Lemma~\ref{ontopic}, we may adjust the sections $g_{i}$ for a system of generators $P$ of $\Pi$, such that that the corresponding cocycles $\xi$ generate a finitely generated free abelian group. Let $\Lambda$ be the resulting group of line bundles, and denote the associated $\Lambda$-graded $\mathcal{O}_{X}$-algebra by $\mathcal{R}$.
We construct a graded $\mathcal{O}_{X}$-algebra homomorphsim $\mathcal{R} \to \mathcal{A}$. The accompanying homomorphism will be the canonical map $\Lambda \to \Pi$, associating to $L$ its class under the identification $\Pi \cong \operatorname{Pic}(X)$. Now, the sections of $\mathcal{R}_{L}$, where $L = P_{\xi}$, are given by families $(h_{i})$ satisfying $$ h_{j} = \xi_{j} h_{i} = \frac{g_{j}}{g_{i}} h_{i} .$$
This enables us to define a map $\mathcal{R}_{L} \to \mathcal{A}_{P}$ by sending $(h_{i})$ to the section obtained by patching together the $h_{i}g_{i}$. Note that this indeed yields a graded homomorphism $\mathcal{R} \to \mathcal{A}$. By construction, this homomorphism is an isomorphism in every degree. Thus we only have to show that its kernel is the ideal associated a shifting family for $\mathcal{R}$.
Let $\Lambda_{0} \subset \Lambda$ denote the kernel of the canonical map $\Lambda \to \Pi$. Then every bundle $E \in \Lambda_{0}$ admits a global trivialization. In terms of the defining cocycle $g_{i}/g_{j}$ of $E$ this means that there exist invertible local funtions $\t{g}_{i}$ on $X$ with $$ \frac{g_{j}}{g_{i}} = \frac{\t{g}_{j}}{\t{g}_{i}}. $$
\goodbreak
The functions $\t{g}_{i}$ can be used to define a shifting family: Let $L \in \Lambda$ and $E \in \Lambda_{0}$. Then the sections of $\mathcal{R}_{L}$ are given by families $(h_{i})$ of function that are compatible with the defining cocycle. Thus we obtain maps $$ \varrho_{E} \colon \mathcal{R}_{L} \to \mathcal{R}_{L+E}, \qquad (h_{i}) \mapsto \left(\frac{h_{i}}{\t{g}_{i}}\right). $$
By construction, the $\varrho_{E}$ are homomorphisms, and they fit together to a shifting family $\varrho$ for $\mathcal{R}$. It is straightforward to check that the ideal $\mathcal{I}$ associated to $\varrho$ is precisely the kernel of the homomorphism $\mathcal{R} \to \mathcal{A}$. \endproof
\section{Very tame varieties}\label{section8}
Finally, we take a closer look to the case of a free Picard group. The only assumption in this section is that the multiplicative group ${\mathbb K}^{*}$ is of infinite rank over ${\mathbb Z}$. But even this could be weakened, see the concluding Remark~\ref{verytame}.
\begin{defi} A {\em very tame\/} variety is a normal divisorial variety with finitely generated free Picard group and only constant functions. \end{defi}
Examples of very tame varieties are Grassmannians and all smooth complete toric varieties. On the algebraic side we work with the following notion:
\begin{defi} A {\em very simple\/} algebra is a freely $\Lambda$-graded quasiaffine algebra $(A,\mathfrak{A})$ such that \begin{enumerate} \item the grading group $\Lambda$ of $(A,\mathfrak{A})$ is free, \item $A$ is normal, and we have $A_{0} = {\mathbb K}$ and $A^{*} = {\mathbb K}^{*}$, \item the quasiaffine variety associated to $(A,\mathfrak{A})$ has trivial Picard group. \end{enumerate} \end{defi}
Again, very tame varieties and very simple algebras form subcategories, and we have an equivalence theorem:
\begin{thm} The homogeneous coordinate ring functor restricts to an equivalence of the category of very tame varieties with the category of very simple algebras. \end{thm}
\proof Let $X$ be a very tame variety. We only have to show is that the quasiaffine $H$-variety $\rq{X}$ corresponding to the homogeneous coordinate ring of $X$ has trivial Picard group. Since $\rq{X}$ is normal and $H$ is a torus, every line bundle on $\rq{X}$ is $H$-linearizable, see~\cite[Remark p.~67]{Kn}. But from Theorem~\ref{equivthm}, we know that every $H$-linearizable bundle on $\rq{X}$ is trivial. \endproof
In the setting of very tame varieties, we can go further with the dictionary presented in Section~\ref{section6}. The first remarkable statement is that very tame varieties produce unique factorization domains:
\begin{prop}\label{freefactorial} Let $X$ be a very tame variety with homogeneous coordinate ring $(A, \mathfrak{A})$. Then $X$ is locally factorial if and only if $A$ is a unique factorization domain. \end{prop}
\proof
Let $A = \mathcal{A}(X)$ with some Picard graded $\mathcal{O}_{X}$-algebra $\mathcal{A}$, and the geometric quotient $q \colon \rq{X} \to X$ provided by Lemma~\ref{geomquot}. Since $\operatorname{Pic}(X)$ is free we divide by a torus $H$. Thus $q \colon \rq{X} \to X$ is an $H$-principal bundle with respect to the Zariski topology. In particular, $X$ is locally factorial if and only if $\rq{X}$ is so. But $\rq{X}$ is locally factorial if and only if $A$ is a factorial ring, because we have $\operatorname{Pic}(\rq{X}) = 0$.
\endproof
Next we treat products. Let $X$ and $Y$ be very tame varieties with homogeneos coordinate rings $(A,\mathfrak{A})$ and $(B,\mathfrak{B})$. Fix closing subalgebras $(A',I') \in \mathfrak{A}$ and $(B',J') \in \mathfrak{B}$, as in~\ref{freegradalgdef}, and consider the algebra $$ A \boxtimes B := \bigcap_{f} (A' \otimes_{{\mathbb K}} B')_{f} = \bigcap_{f} (A \otimes_{{\mathbb K}} B)_{f}, $$ where the intersections are taken in the quotient field of $A' \otimes_{{\mathbb K}} B'$ and $f$ runs through the elements of the form $g \otimes h$ with homogeneous $g \in I'$ and $h \in J'$.
Now $A$ and $B$ are graded, say by $\Lambda$ and $\Gamma$. These gradings give rise to a $(\Lambda \times \Gamma)$-grading of $A \boxtimes B$. Moreover, $$(A' \otimes_{{\mathbb K}} B', \sqrt{I' \otimes_{{\mathbb K}} J'})$$ is a closing subalgebra of $A \boxtimes B$. Let $\mathfrak{A} \boxtimes \mathfrak{B}$ denote the equivalence class of this closing subalgebra. Then we obtain:
\begin{prop}\label{products} Let $X$ and $Y$ be locally factorial very tame varieties. Then $X \times Y$ is locally factorial and very tame with homogeneous coordinate ring $(A \boxtimes B, \mathfrak{A} \boxtimes \mathfrak{B})$. Moreover, if $A$ and $B$ are of finite type over ${\mathbb K}$, then $A \boxtimes B$ equals $A \otimes_{{\mathbb K}} B$. \end{prop}
\proof
First note that for any two quasiaffine varieties $\rq{X}$ and $\rq{Y}$ with free diagonalizable group actions, their product $\rq{X} \times \rq{Y}$ is again such a variety. Moreover, if $\rq{X}$ and $\rq{Y}$ have only constant invertible functions, then so does $\rq{X} \times \rq{Y}$. If $\rq{X}$ and $\rq{Y}$ are additionally locally factorial with trivial Picard groups, then the same holds for $\rq{X} \times \rq{Y}$, use e.g.~\cite[Prop.~1.1]{FI}.
Now, let $\rq{X} := {\rm Spec}(\mathcal{A})$ and $\rq{Y} := {\rm Spec}(\mathcal{B})$. By Proposition~\ref{freefactorial} both are locally factorial. By construction $(A \boxtimes B, \mathfrak{A} \boxtimes \mathfrak{B})$ is the freely graded quasiaffine algebra corresponding to the product $\rq{X} \times \rq{Y}$. Thus the above observations and Proposition~\ref{equivquasiaffequiv} tell us that it is a coproduct in the category of simple pointed algebras. Hence the assertion follows from Theorem~\ref{equivthm}. The second statement is an easy consequence of Remark~\ref{collaps}~(i).
\endproof
\begin{coro} Let $X$ and $Y$ be locally factorial very tame varieties. Then $\operatorname{Pic}(X \times Y)$ is isomorphic to $\operatorname{Pic}(X) \times \operatorname{Pic}(Y)$. \endproof \end{coro}
We give an explicit example emphasizing the role of Proposition~\ref{freefactorial}. We assume that the ground field ${\mathbb K}$ is not of characteristic two. Consider the prevariety $X$ obtained by gluing two copies of the projective line ${\mathbb P}_{1}$ along the common open subset ${\mathbb K}^{*} \setminus \{1\}$. We think of $X$ as the projective line with three doubled points, namely $$ 0, \; 0', \quad 1, \; 1', \quad \infty, \; \infty'.$$
Note that $X$ is smooth and divisorial. Moreover, $\operatorname{Pic}(X)$ is isomorphic to ${\mathbb Z}^{4}$. Thus we obtain in particular that $X$ is very tame. Let $(\mathcal{A}(X),\mathfrak{A}(X))$ denote the homogeneous coordinate ring of $X$. We show:
\begin{prop}\label{example} $\mathcal{A}(X) \cong {\mathbb K}[T_{1}, \ldots, T_{6}]/\bangle{T_{1}^{2} + \ldots + T_{6}^{2}}$. \end{prop}
Before giving the proof, let us remark that the ring $\mathcal{A}(X)$ is a classical example of a singular factorial affine algebra. In view of our results, factoriality is a consequence of Proposition~\ref{freefactorial}.
\proof[Proof of Proposition~\ref{example}] First observe that we may realize $\operatorname{Pic}(X)$ as well as a subgroup $\Lambda$ of the group of Cartier divisors of $X$. For example $\operatorname{Pic}(X)$ is isomorphic to the group $\Lambda$ generated by $$ D_{0} := \{0\}, \qquad D_{1} := \{1\}, \qquad D_{1'} := \{1'\}, \qquad D_{\infty} := \{\infty\}.$$
For any Cartier divisor $D$ on $X$, let $\mathcal{A}_{D}$ denote its sheaf of sections. Then the homogeneous coordinate ring $\mathcal{A}(X)$ is the direct sum of the $\mathcal{A}_{D}(X)$, where $D \in \Lambda$. Consider the following homogeneous elements of $\mathcal{A}(X)$: $$ \begin{array}{ll} f_{1} := 1 \in \mathcal{A}_{D_{0}} (X), & f_{2} := 1 \in \mathcal{A}_{D_{1}} (X), \cr f_{3} := 1 \in \mathcal{A}_{D_{1'}}(X), & f_{4} := 1 \in \mathcal{A}_{D_{\infty}}(X). \cr f_{5} := \bigl( \frac{1}{z-1} \bigr)
\in \mathcal{A}_{D_{1} + D_{1'} - D_{\infty}}(X), & f_{6} := \bigl( \frac{z}{z-1} \bigr)
\in \mathcal{A}_{D_{1} + D_{1'} - D_{0}}(X). \end{array} $$
Let $\varphi$ be the algebra homomorphism ${\mathbb K}[T_{1}, \dots, T_{6}] \to \mathcal{A}(X)$ sending $T_{i}$ to $f_{i}$. It is elementary to check that $\varphi$ is surjective. Since we assumed ${\mathbb K}$ not to be of characteristic two, it suffices to show that the kernel of $\varphi$ is the ideal generated by $$ Q := T_{2}T_{3} + T_{5} T_{4} - T_{6} T_{1}. $$
An explicit calculation shows that the $f_{i}$ fulfil the claimed relation, that means that $Q$ lies in the kernel of $\varphi$. Conversely, consider an arbitrary element $R$ of the kernel of $\varphi$. Then there are $r_{j} \in {\mathbb K}[T_{1}, \dots, T_{5}]$ such that $R$ is of the form $$ R = \sum_{j=0}^{s} r_{j} T_{6}^{j}. $$
We proceed by induction on $s$. For $s = 0$ the fact that $f_{1}, \dots, f_{5}$ are algebraically independent implies $R = 0$. For $s > 0$ note first that that $\varphi(r_{j})$ is nonnegative in $D_{0}$ in the sense that its component in a degree containing a multiple $nD_{0}$ is trivial for negative $n$.
Since $f_{6}$ is negative in $D_{0}$, and $f_{1}$ is the only generator of $\mathcal{A}(X)$ which is strictly positive in degree $D_{0}$, we can write $r_{j} = \t{r}_{j} T_{1}^{j}$. Hence we obtain a representation $$ R = \sum_{j=0}^{s} \t{r}_{j} T_{1}^{j}T_{6}^{j}.$$ The element $\t{r}_{s} ((T_{1}T_{6})^{s} - (T_{2}T_{3} + T_{4}T_{5})^{s})$ is a multiple of~$Q$. In particular, it belongs to the kernel of $\varphi$. Subtracting it from $R$, we obtain $$ R'= \sum_{j=0}^{s-1} r_{j}' T_{1}^{j} T_{6}^{j},$$ with $r_{j}' = \t{r}_{j}$ for $j>0$ and $r_{0}' = \t{r}_{0} + \t{r}_{s}(T_{2}T_{3} + T_{4}T_{5})^{j}$. Applying the induction hypothesis to $R'$ yields that $R$ is a multiple of~$Q$.
\endproof
Finally, let us note that all our statements on very tame varieties hold under more general assumptions. This is due to the fact that free Picard groups always can be realized by (free) groups of line bundles. Hence in this case we don't need shifting families to define the homogeneous coordinate ring. This means:
\begin{rem}\label{verytame} For very tame varieties $X$, the results of this article hold over any algebraically closed ground field ${\mathbb K}$, and one might weaken the assumption $\mathcal{O}(X) = {\mathbb K}$ to $\mathcal{O}^{*}(X) = {\mathbb K}^{*}$. \end{rem}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{Classical invariants of Legendrian knots in the 3-dimensional torus}
\author{PAUL A. SCHWEITZER, S.J.}
\address{Departamento de Matem\'atica\\ Pontif\'icia Universidade Cat\'olica do Rio de Janeiro, PUC-Rio, Brazil}
\author{F\'ABIO S. SOUZA}
\address{Faculdade de Forma\c{c}\~ao de Professores\\ Universidade do Estado do Rio de Janeiro, UERJ, Brazil}
\begin{abstract}
All knots in $\mathbb{R}^3$ possess Seifert surfaces, and so the classical Thurs\-ton-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $\mathbb{R}^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $\xi$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight oriented contact structure on $T^3$ using projections.
\end{abstract}
\begin{keyword}
Legendrian knots \sep Thurston-Bennequin invariant \sep Maslov invariant\sep contact structures, 3-torus $T^3$ \sep Seifert surfaces.
\MSC 57R17 \sep 57M27
\end{keyword}
\end{frontmatter}
\section{Introduction and Statement of Results} \label{introduction}
Let $\xi$ be an {\it oriented contact structure} on a smooth oriented $3$-manifold $M$, i.e., a $2$-plane field that locally is the kernel of a totally non-integrable $1$-form $\omega$, so that locally $\xi=\ker(\omega)$ with the induced orientation and $\omega\wedge d\omega$ is non-vanishing. A {\it knot} in a smooth $3$-manifold $M$ is a smooth embedding $\alpha: S^1\to M$. A knot in the contact manifold $(M,\xi)$ is {\it Legendrian} if $\alpha$ is everywhere tangent to $\xi$. Two Legendrian knots $\alpha_0$ and $\alpha_1$ are {\it Legendrian homotopic} if there is a smooth $1$-parameter family of Legendrian knots $\alpha_t$, $t\in [0,1],$ that connects them.
The Thurston-Bennequin and rotation (or Maslov) numbers are well-known classical invariants of null-homologous oriented Legendrian knots in $(M^3,\xi)$ that depend only on their Legendrian homotopy class and, for the rotation invariant, on a fixed Legendrian vector field \cite{Be, Et}. Given a Seifert surface $\Sigma$ for the Legendrian knot $\alpha$ in $(M^3,\xi)$, the {\it Thurston-Bennequin invariant} $\rm tb(\alpha)$ is defined to be the number of times the contact plane $\xi$ rotates relative to the tangent plane to $\Sigma$ in one circuit of $\alpha$. The {\it rotation invariant} $r(\alpha)$ is the number of times the tangent vector $\alpha'$ rotates in $\xi$ relative to a fixed Legendrian vector field $Z$ in a single circuit of $\alpha$. For the standard contact structure $\xi_{std}=\ker(dz-ydx)$ on $\mathbb{R}^3$, both invariants of a generic Legendrian knot can be calculated using the front and Lagrangian projections of $\mathbb{R}^3$ to $\mathbb{R}^2$ (see Sections \ref{tbproj} and \ref{maslovproj}).
On the $3$-dimensional torus $T^3$, we define generalized Seifert surfaces for knots that are not null-homologous (Definition \ref{defseifsurf}) and use them to extend the definition of the Thurston-Bennequin invariant to all Legendrian knots in an arbitrary contact structure $\xi$ on $T^3$. Let $\alpha: S^1=\mathbb{R}/\mathbb{Z}\to T^3$ be a knot in $T^3$ and let $\widetilde{\alpha}:\mathbb{R}\to\mathbb{R}^3$ be a lift of $\alpha$ to the universal cover $\widetilde{T^3} = \mathbb{R}^3$, where $T^3$ is identified with $\mathbb{R}^3/\mathbb{Z}^3$. If a component of $\widetilde\alpha$ is compact, then it is a knot on ${\mathbb R^3}$ and the usual definition of Seifert surfaces applies. Hence we usually assume the following General Hypothesis, and then prove the following Proposition.
\begin{genhyp} \it Each component of $\widetilde\alpha$ is assumed to be non-compact. (Equivalently, $\alpha$ is not contractible in $T^3$.) \label{genhyp} \end{genhyp}
Note that the components of $\widetilde\alpha$ are homeomorphic to each other, so if one component is non-compact, then all of them are. In this case $\widetilde\alpha$ will be periodic, say with smallest period $(p,q,r)\in \mathbb{Z}^3$. We define a connected oriented surface $\Sigma \subset {\mathbb R}^3 $ to be a {\it covering Seifert surface} for such a knot $\alpha$ in $T^3$ if $\Sigma$ is $(p,q,r)$-periodic, $\partial\Sigma$ is one component of $\widetilde\alpha$, and outside a tubular neighborhood of that component, $\Sigma$ coincides with an affine half-plane in ${\mathbb R}^3 $. Then we call $\hat\Sigma = \Sigma/\mathbb{Z}(p,q,r)$ a (generalized) {\it Seifert surface} for $\alpha$ (See Definition \ref{defseifsurf}). The arguments using covering Seifert surfaces $\Sigma\subset \mathbb{R}^3$ and Seifert surfaces $\hat\Sigma\subset \mathbb{R}^3/\mathbb{Z}(p,q,r)$ are parallel and equivalent, so some attention is needed to distinguish the two types of Seifert surfaces.
\begin{prop} Let $\alpha$ be a smooth knot on $T^3$ whose lift $\widetilde\alpha$ has all components non-compact and let $\hat\alpha = \widetilde\alpha/\mathbb{Z}(p,q,r)$. Then \begin{enumerate}
\item There is a covering Seifert surface $\Sigma$ with corresponding (generalized) Seifert surface $\hat\Sigma$ for the knot $\alpha$;
\item If $\Sigma_1$ and $\Sigma_2$ are both covering Seifert surfaces for the knot $\alpha$, then the relative rotation number $\rho(\Sigma_1,\Sigma_2)$, defined to be the number of times $X_2$ rotates relative to $X_1$ in the normal plane to $\widetilde\alpha$ in one circuit of $\alpha$, where $X_i\ (i=1,2)$ is a unit vector field tangent to $\Sigma_i$ and orthogonal to $\widetilde\alpha'$, is zero. \end{enumerate}\label{2Seif} \end{prop}
Using these Seifert surfaces, we extend the definition of the Thurston-Bennequin invariant to $(T^3, \xi)$, as follows. The Thurston-Bennequin invariant $\rm tb(\alpha)$ for a Legendrian knot $\alpha$ in $(T^3, \xi)$ is defined to be the rotation number of the contact plane $\xi$ with respect to a (generalized) Seifert surface $\hat\Sigma$ for $\alpha$ in one circuit of $\alpha$ (See Definition \ref{def:tb}).
\begin{thm} \begin{enumerate} \item If $\alpha$ is a Legendrian knot in $(T^3,\xi)$ satisfying the General Hypothesis \ref{genhyp}, then the Thurston-Bennequin invariant $\rm tb(\alpha)$ is well-defined. \item If $\alpha$ satisfies \ref{genhyp} but is null-homologous, then our definition of $\rm tb(\alpha)$ agrees with the standard definition of $\rm tb(\alpha)$ in $T^3$. \item If $\alpha$ does not satisfy \ref{genhyp}, so that $\alpha$ is contractible, then the standard definitions of $\rm tb(\alpha)$ and $\rm tb(\tilde\alpha)$ coincide, where $\tilde\alpha$ and $\tilde\xi$ are the lifts of $\alpha$ and $\xi$ to the universal cover $(\tilde T^3,\tilde\xi)$ of $(T^3,\xi)$. \end{enumerate} \label{thm1} \end{thm}
\noindent This shows that Definition \ref{def:tb} extends the usual definition in ${\mathbb R}^3 $. Recall that Kanda \cite{Ka} defines a Thurston-Bennequin invariant for {\it quasilinear Legendrian knots} on $T^3$, i.e., those that are isotopic to knots with constant slope, using an incompressible torus containing the knot to replace the Seifert surface. In the universal cover, the torus lifts to a plane, half of which is isotopic to our covering Seifert surface, so the following result holds.
\begin{prop} For quasilinear knots in $T^3$, our definition of $\rm tb(\alpha)$ agrees with the definition of Kanda \cite{Ka}. \end{prop}
We shall be especially interested in the tight contact structures $$\xi_n=\ker(\cos(2\pi nz)dx + \sin(2\pi nz)dy)$$ on $T^3=\mathbb{R}^3/\mathbb{Z}^3$, where $n$ is a positive integer. Kanda \cite{Ka} has shown that for every tight contact structure $\xi$ on $T^3$ there is a contactomorphism (i.e., a diffeomorphism that preserves the contact structure) from $\xi$ to $\xi_n$, for some $n>0$.
Define projections $p_{xy}, p_{xz}: T^3\to T^2$ by setting $p_{xy}(x,y,z)=(x,y)$ and $p_{xz}(x,y,z)=(x,z)$, where $x,y,z$ are the coordinates modulo $1$ in $T^3$ and in $T^2$. The projection $p_{xy}$ is called the {\it front projection}, for if we identify $T^3$ with the space of co-oriented contact elements on $T^2$, then the wave fronts of the propagation of a wave on $T^2$ are images under $p_{xy}$ of Legendrian curves in $T^3$. Then a knot in $T^3$ is {\it generic} relative to both projections $p_{xy}$ and $p_{xz}$ if its curvature vanishes only at isolated points and the only singularities of the projected knot are transverse double points and cusps. Note that every Legendrian knot can be made generic by an arbitrarily small Legendrian homotopy. The following Theorem shows how to calculate both invariants of generic Legendrian knots for $\xi_n$ using the projections $p_{xy}, p_{xz}: T^3\to T^2$.
\begin{thm} Let $\alpha$ be a generic oriented Legendrian knot in $(T^3,\xi_n)$. \begin{enumerate} \item For the projection $p_{xy}$ of $\alpha$, $\rm tb(\alpha)= P - N + C/2$, where $P$ and $N$ are the numbers of positive and negative crossings and $C$ is the number of cusps for $p_{xy}\circ\alpha$ in one circuit of $\alpha$; \item For the projection $p_{xz}$ of $\alpha$, $\rm tb(\alpha)= P - N$, where $P$ and $N$ are the numbers of positive and negative crossings for $p_{xz}\circ\alpha$ in one circuit of $\alpha$, and there are no cusps; \item For the projection $p_{xy}$ of $\alpha$, the rotation invariant relative to the Legendrian vector field $Z=\partial/\partial z$ is $r(\alpha)= 1/2(C_+ - C_-)$, where $C_+$ and $C_-$ are the numbers of positive and negative cusps of $p_{xy}\circ\alpha$ in one circuit of $\alpha$;
\item Let $V=\{t\in S^1\ |\ (x'(t),y'(t))=(0,0) \}$ and suppose that $2nz(t)\notin {\mathbb Z}$ for every $t\in V$. Then for the projection $p_{xz}$ of $\alpha$, the rotation invariant relative to $Z=\partial/\partial z$ is $r(\alpha)= 1/2\sum_{t\in V} a(t)b(t)$, where $a(t)=(-1)^{[2nz(t)]}$ and $b(t)=\pm 1$ according to whether $p_{xz}\circ\alpha'(t)$ is turning in the positive or negative direction in the $xz$-plane. \end{enumerate} \label{projectionthm}\end{thm}
Generalized Seifert surfaces for knots in $T^3$ will be defined and studied in \S \ref{SeifSurf}. The Thurston-Bennequin invariant ${\rm tb}(\alpha)$ and the rotation invariant $r(\alpha)$ for a generic oriented Legendrian knot in $T^3$ will be treated in \S \ref{tbinv} and \S \ref{s:maslov}, respectively. The proofs of the four assertions of Theorem \ref{projectionthm} are given in the proofs of the Propositions \ref{tbpxz}, \ref{tbpxy}, \ref{maslovpxz}, and \ref{maslovpxy}, respectively, in the Subsections \ref{tbproj} and \ref{maslovproj}. In the last Section \ref{s:tb-inequality} we calculate the invariants for quasilinear Legendrian knots in $(T^3,\xi_n)$ and observe that the Bennequin inequality for null-homologous Legendrian knots in a tight contact structure has to be modified in this case. Finally we make a conjecture about the extension of the Bennequin inequality for tight contact structures on $T^3$.
This paper is a continuation of the work of the second author in his masters thesis \cite{So} at the Pontif\'\i cia Universidade Cat\'olica of Rio de Janeiro under the direction of the first author.
\section{Seifert surfaces in $T^3$} \label{SeifSurf}
In this section we consider smooth knots in $T^3$ and their Seifert surfaces, without reference to any contact structure, as a preparation for studying Legendrian knots and their Thurston-Bennequin and rotation invariants in the next two sections.
\begin{figure}
\caption{Inserting a twisted strip at a crossing.}
\label{fig:strip2}
\end{figure}
Recall that a {\it Seifert surface} for an oriented knot (or link) $\alpha$ in $\mathbb{R}^3$ is a compact connected oriented surface $\Sigma$ whose boundary is $\alpha$ with the induced orientation. Every knot and link $\alpha$ has Seifert surfaces, and there is a well-known method of constructing one using a regular projection of $\alpha$ in the plane (\cite{BZ}, pp. 16-18). Each crossing is replaced by a non-crossing that respects the orientation, the resulting circles are capped off by disjoint embedded disks, and then a twisted interval is inserted at each crossing, as in Figure \ref{fig:strip2}. Finally the {\em Seifert circles}, the boundary of the resulting surface, are capped off by disjoint embedded disks.
Let $\alpha: S^1=\mathbb{R}/\mathbb{Z}\to T^3$ be a knot in $T^3$ and let $\widetilde{\alpha}:\mathbb{R}\to\mathbb{R}^3$ be a lift of $\alpha$ to the universal cover $\widetilde{T^3} = \mathbb{R}^3$, where $T^3$ is identified with $\mathbb{R}^3/\mathbb{Z}^3$. Let $(p,q,r)\in {\mathbb Z}^3$ be a generator of the cyclic group of translations that preserve $\widetilde{\alpha}$. Note that $(p,q,r)$ is determined up to multiplication by $\pm 1$ and we choose the sign so that $\widetilde\alpha(t+1)=\widetilde\alpha(t)+ (p,q,r)$. Any subset of ${\mathbb R}^3 $ that is invariant under this group will be said to be {\it $(p,q,r)$-periodic}. The following definition adapts the classical concept of Seifert surfaces for knots in ${\mathbb R}^3$ to the present context.
\begin{defi} {\rm Let $\alpha$ be a knot in $T^3$ whose lift $\widetilde\alpha$ has non-compact components and let $(p,q,r)$ generate the cyclic group of translations that preserve $\widetilde\alpha$. A smooth surface $\Sigma \subset \mathbb{R}^3$ is a {\it covering Seifert surface} for $\alpha$ if it satisfies the following conditions:} \begin{enumerate} {\rm \item $\Sigma$ is connected, orientable, properly embedded in ${\mathbb R}^3 $, and $(p,q,r)$-periodic;
\item $\partial\Sigma = \widetilde\alpha$; and
\item There is an affine half-plane $P_+\subset {\mathbb R}^3 $ with boundary a straight line $S$ such that $\Sigma$ coincides with $P_+$ outside a $\delta$-neighborhood $N$ of $S$, for some sufficiently large $\delta$. } \end{enumerate} {\rm In this case we say that $\hat\Sigma = \Sigma/\mathbb{Z}(p,q,r)$ is a (generalized) {\it Seifert surface} for $\alpha$. If the components of $\widetilde\alpha$ are compact, then a Seifert surface for one of the components can be translated by the action of ${\mathbb Z}^3$ to give a periodic covering Seifert surface.} \label{defseifsurf} \end{defi}
\begin{figure}
\caption{A $(p,q,r)$-periodic knot projected into an affine plane.}
\end{figure}
In this section we shall usually deal with covering Seifert surfaces, but the same properties could be developed for Seifert surfaces, and there is a complete correspondence.
Clearly the half-plane $P_+$ and its boundary $S$ are also $(p,q,r)$-periodic. It is convenient to choose $P_+$ to be disjoint from $\widetilde\alpha$ and such that $P_+\subset\Sigma$. Given a covering Seifert surface $\Sigma$ of $\alpha$, we define the {\em associated vector field} $X=X(\alpha,\Sigma)$ along $\widetilde\alpha$ in ${\mathbb R}^3 $ to be the $(p,q,r)$-periodic unit vector field along $\widetilde\alpha$ that is orthogonal to $\widetilde\alpha$, tangent to $\Sigma$, and directed towards the interior of $\Sigma$.
Given two covering Seifert surfaces $\Sigma_i$ with associated vector fields $X_i$, $i=1,2$, the {\em relative rotation number} $\rho(\Sigma_1,\Sigma_2)$ of $\Sigma_2$ with respect to $\Sigma_1$ is defined to be the number of revolutions that $X_2$ makes with respect to $X_1$ in the positive direction in the normal plane field to $\widetilde\alpha$ along $\widetilde\alpha$ from a point on $\widetilde\alpha$ to its first $(p,q,r)$-translate in the positive direction, i.e., in one circuit of $\alpha$. Note that this number is independent of the orientation of $\alpha$, since changing the orientation of $\alpha$ also changes the orientation of the normal plane.
\begin{proof}[Proof of Proposition \ref{2Seif}.] First we construct a covering Seifert surface for a knot $\alpha$ in $T^3$ with period $(p,q,r)$. Choose a $(p,q,r)$-periodic plane $P$ meeting the proper curve $\widetilde\alpha$ such that the orthogonal projection of $\widetilde\alpha$ onto $P$ is regular (i.e., the only singularities are transverse double points). Fix an orientation of $\widetilde\alpha$. The $(p,q,r)$-periodicity of both $\widetilde\alpha$ and the affine plane $P$ permits us to adapt the classical construction of the Seifert surface of a knot in ${\mathbb R}^3 $ (see \cite{BZ}, pp. 16-18) in a $(p,q,r)$-periodic fashion. At each crossing of the image of $\widetilde\alpha$ in $P$, which we call the the knot diagram, replace the crossing by two arcs, respecting the orientation of $\widetilde\alpha$, and insert a twisted strip, as in Figure \ref{fig:strip2}. Do this so that the resulting collection of ``Seifert curves'' is pairwise disjoint and $(p,q,r)$-periodic. The Seifert curves that are simple closed curves are capped off in a periodic fashion by mutually disjoint disks meeting $P$ only in their boundaries. Then there will be a number of non-compact proper $(p,q,r)$-periodic Seifert curves left over. It is easy to check that this number will be odd, say $2k+1$, with $k+1$ of them oriented in the positive direction of $\widetilde\alpha$ and the other $k$ in the opposite direction. (To see this, consider a plane perpendicular to the direction $(p,q,r)$ that meets $\widetilde\alpha$ transversely and examine the sign of the intersections of $\widetilde\alpha$ with this plane.) These curves can be capped off in pairs with opposite orientations by disjoint oriented periodic infinite strips, starting with a pair whose projections are adjacent in the plane $P$. This process will leave one $(p,q,r)$-periodic Seifert curve which can be joined to a half plane $P_+$ contained in $P$ by another infinite periodic strip so that the result is embedded. The whole construction is done so as to preserve $(p,q,r)$-periodicity. The result is a covering Seifert surface for $\alpha$ that coincides with $P_+$ outside a sufficiently large tubular neighborhood $N$ of the line $S$.
Now suppose that $\Sigma_1$ and $\Sigma_2$ are two covering Seifert surfaces for $\alpha$. Take a $(p,q,r)$-periodic line $S$ in $\Sigma_1$ and a tubular neighborhood $N$ of $S$ sufficiently large so that the parts of $\Sigma_1$ and $\Sigma_2$ outside $N$ are half-planes. Remove these half-planes and add an infinite periodic strip in $\partial \bar N$ to connect $\Sigma_1$ and $\Sigma_2$, if necessary. Thus we obtain a new proper $(p,q,r)$-periodic surface $\Sigma$ which agrees with the union of $\Sigma_1$ and $\Sigma_2$ inside $N$. This surface $\Sigma$ will be a proper immersed surface contained in $\bar N$. Note that $\Sigma$ projects onto a compact oriented immersed surface on $T^3$. The following lemma will complete the proof, since by definition $\rho(\Sigma_1,\Sigma_2)=\rho(X_1,X_2)$. \end{proof}
\begin{lem} Let $\Sigma$ be a properly immersed $(p,q,r)$-periodic oriented surface in ${\mathbb R}^3$ whose boundary has two components, one being $\widetilde\alpha$ with the positive orientation and the other $\widetilde\alpha$ with the negative orientation, and which projects to a compact surface in $\widehat T^3={\mathbb R}^3/{\mathbb Z}(p,q,r)$. Let $X_1$ and $X_2$ be the vector fields associated to the two boundary components of $\Sigma$. Then the rotation number $\rho(X_1,X_2)$ of $X_2$ relative to $X_1$ along $\widetilde\alpha$ is zero. \label{2boundaries} \end{lem}
\begin{proof}[Proof] The quotient mappings $\pi': {\mathbb R}^3\to \widehat T^3$ and $\pi: \widehat T^3\to T^3$ are projections of covering spaces. Note that $\widehat T^3$ is diffeomorphic to ${\mathbb R}^2\times S^1$. The curve $\widetilde\alpha$ projects under $\pi'$ to a compact knot $\hat\alpha$ in $\widehat T^3$. Let $V$ be a small closed tubular neighborhood of $\hat\alpha$ (so that $V$ is diffeomorphic to $S^1\times D^2$, where $D^2$ is the closed unit disk in the plane) and set $M=\widehat T^3\smallsetminus {\rm Int}\ V$. Let $\alpha_1$ and $\alpha_2$ be the loops on the torus $\partial V=\partial M$ obtained by isotoping $\alpha$ in the directions of the vector fields $X_1$ and $X_2$. We claim that their homology classes satisfy $[\alpha_1] = [\alpha_2]\in H_1(\partial V)$, which implies that the mutual rotation number $\rho(X_1,X_2)$ vanishes, as claimed.
To see this claim, note that there is a compact oriented surface $\widehat\Sigma$ immersed in $M$ obtained from the projection of $\Sigma$ into $\widehat T^3$ by a small isotopy so that its boundary $\partial \widehat\Sigma$ is the union of $\alpha_1$ and $\alpha_2$ with opposite orientations. Consequently $i''_*([\alpha_1]-[\alpha_2])=0\in H_1M$, where $i'':\partial V\to M$ is the inclusion. Now let $\ell$ and $m$ be the oriented longitude and meridian of $\partial V$, so that there are integers $n_1$ and $n_2$ such that the homology classes of $\alpha_1$ and $\alpha_2$ on $\partial V$ satisfy $[\alpha_r]=[\ell]+n_r[m], r=1,2$. Since $m$ is contractible on the solid torus $V$, $i'_*([\alpha_1]-[\alpha_2])=0\in H_1V$, where $i':\partial V\to V$ is the inclusion. In the Mayer-Vietoris exact sequence $$\cdots \to H_2\widehat T^3 \overset{\partial_2}\to H_1\partial V \xrightarrow[\approx]{i_*} H_1 V\oplus H_1M \overset{j_*}\to H_1\widehat T^3 \to \cdots$$
$i_*=i'_*+i''_*$ so $i_*([\alpha_1]-[\alpha_2])=0$. Since $H_2\widehat T^3\approx H_2({\mathbb R}^3/{\mathbb Z}(p,q,r))=0$, $i_*$ is injective, so $[\alpha_1]=[\alpha_2]$, as claimed. \end{proof}
\section{The Thurston-Bennequin invariant}\label{tbinv}
In this section, we extend the classical definition of the Thurston-Bennequin invariant $\rm tb(\alpha)$ to all Legendrian knots for an arbitrary contact structure on $T^3$, and we show how to compute the invariant $\rm tb(\alpha)$ of Legendrian knots in $(T^3,\xi_n)$ using projections.
\subsection{The Thurston-Bennequin invariant for null-homol\-ogous knots.}
First we recall the definition of ${\rm tb}(\alpha)$ for an oriented null-homologous Legendrian knot $\alpha$ relative to a contact structure $\xi$ on an oriented $3$-manifold $M^3$. Since $\alpha$ is null-homologous it has a Seifert surface $\Sigma$, which by definition is an oriented compact connected surface embedded in $M^3$ with oriented boundary $\alpha$. Let $X$ and $Y$ be unit vector fields orthogonal to $\alpha$ (with respect to a metric on $M$), with $X$ tangent to $\Sigma$ and $Y$ tangent to $\xi$. Then ${\rm tb}(\alpha)$ is defined to be the algebraic number of rotations of $Y$ relative to $X$ in the normal plane field $\alpha^\perp$, which is oriented by the orientations of $M^3$ and $\alpha$, as we make one circuit of $\alpha$ in the positive direction. If we let $\alpha^+$ be a knot obtained by pushing $\alpha$ a short distance in the direction $Y$, then it is easy to see that ${\rm tb}(\alpha)$ is the intersection number of $\alpha^+$ with $\Sigma$. This is just the linking number of $\alpha^+$ with $\alpha$ because $\Sigma$ is a compact oriented surface with boundary $\alpha$. An argument analogous to the proof of Lemma \ref{2boundaries}, taking $\Sigma$ to be the disjoint union of two Seifert surfaces $\Sigma_1$ and $\Sigma_2$ for $\alpha$ with opposite orientations, shows that ${\rm tb}(\alpha)$ is independent of the choice of the Seifert surface.
\subsection{The Thurston-Bennequin invariant in $T^3$.}
Now consider $T^3$ with an oriented contact structure $\xi$ and let $\alpha$ be a Legendrian knot in $T^3$ with a covering Seifert surface $\Sigma$ for $\alpha$ using the lift $\widetilde\alpha$ to the universal cover ${\mathbb R}^3$. We can define the rotation number of the lifted contact structure $\widetilde\xi$ with respect to $\Sigma$ to be the number of rotations of one of the two unit orthogonal vector fields $Y$ along $\widetilde\alpha$ that is tangent to $\widetilde\xi$ with respect to a unit orthogonal vector field $X$ along $\widetilde\alpha$ that is tangent to $\Sigma$, in one circuit of $\alpha$.
\begin{defi} {\rm The {\it Thurston-Bennequin invariant} ${\rm tb}(\alpha)$ for a Legendrian knot $\alpha$ in $T^3$ is the rotation number of the contact structure $\tilde\xi$ with respect to a covering Seifert surface $\Sigma$ for $\alpha$.} \label{def:tb} \end{defi} It is clear that, on $T^3=\mathbb{R}^3/\mathbb{Z}(p,q,r)$, ${\rm tb}(\alpha)$ is the rotation number of the induced contact structure $\hat\xi$ with respect to a (generalized) Seifert surface $\hat\Sigma$ for $\alpha$ in one circuit of $\hat\alpha$. As for knots in $\mathbb{R}^3$, ${\rm tb}(\alpha)$ will be the intersection number of $\widetilde\alpha^+$, the lifted knot $\widetilde\alpha$ pushed a short distance in a direction transverse to the lifted contact structure $\widetilde\xi$, with the covering Seifert surface $\Sigma$, in one circuit of $\alpha$. In this case, however, if $\alpha$ is not null-homologous, then ${\rm tb}(\alpha)$ is not a linking number, since $\Sigma$ will not be compact.
We note that Definition \ref{def:tb} is an extension of the above definition of ${\rm tb}(\alpha)$ for a null-homologous Legendrian knot $\alpha$ in an oriented $3$-manifold $M^3$ endowed with a contact structure $\xi$. As in the null-homologous case, the following holds.
\begin{lem} The Thurston-Bennequin invariant for a Legendrian knot $\alpha$ in $T^3$ is independent of the choice of the covering Seifert surface and the orientation of $\alpha$. \end{lem}
\begin{proof} According to Proposition \ref{2Seif}, the rotation number of one covering Seifert surface for the knot $\alpha$ with respect to another one is zero. Hence the rotation numbers of the two covering Seifert surfaces with respect to the contact structure coincide. Given an orientation of $\alpha$, we choose the orientation of the plane field orthogonal to $\alpha$ such that the orientations of $\alpha$ and the plane field determine the standard orientation of $T^3$, so reversing the orientation of $\alpha$ reverses the orientation of the plane orthogonal field as well and the rotation number does not change. \end{proof}
It is worth remarking that our extended definition of ${\rm tb}(\alpha)$ continues to satisfy the usual properties: it does not change if we replace the vector field $Y$ tangent to $\widetilde\xi$ by $-Y$ or by a vector field $Y^\pitchfork$ transverse to $\widetilde\xi$, or if we use $-X$ or a vector field $X^\pitchfork$ transverse to $\Sigma$ in place of $X$.
\subsection{Computation of tb using projections}\label{tbproj}
In this subsection we compute the Thurston-Bennequin invariant ${\rm tb}(\alpha)$ of an oriented Legendrian knot $\alpha$ in $T^3$ relative to Kanda's tight contact structure $\xi_n$, $n>0$, using the projections $p_{xy}, p_{xz}: T^3\to T^2$ defined in the Introduction and a covering Seifert surface $\Sigma$ for $\alpha$, as defined in Section \ref{SeifSurf}.
First, we recall how to do this for a generic oriented Legendrian knot $\alpha$ in $\mathbb{R}^3$ with the standard contact structure $\xi_{std}$ utilizing its front and Lagrangian projections. The {\it front ({\rm resp.,} Lagrangian) projection} of a Legendrian knot $\alpha$ in $(\mathbb{R}^3,\xi_{std})$ is the map $\bar\alpha=pr_F\circ \alpha$ (resp., $\bar\alpha=pr_L\circ \alpha$) where the map $pr_F:\mathbb{R}^3\to\mathbb{R}^2$ (resp., $pr_L:\mathbb{R}^3\to \mathbb{R}^2$) is defined by $pr_F(x,y,z)=(x,z)$ (resp., $pr_L(x,y,z)=(x,y)$).
\begin{figure}
\caption{Front and Lagrangian projections of a Legendrian unknot and trefoil for $\xi_{std}$ on $\mathbb{R}^3$.}
\label{fig:projections}
\end{figure}
\begin{figure}
\caption{Positive and negative crossings.}
\label{fig:crossings}
\end{figure}
\noindent{\bf Computation of tb using projections of $\mathbb{R}^3$.} Let $\bar\alpha=pr_{F}\circ \alpha$ be the front projection of $\alpha$. The vector field $Y=\partial/\partial z$ is transverse to $\xi_{std}=\ker(dz-ydx)$ along $\alpha$, and we let $\alpha^+$ be a knot obtained by shifting $\alpha$ slightly in the direction $Y$. Then, as observed above, ${\rm tb}(\alpha)$ is the intersection number of $\alpha^+$ with the Seifert surface $\Sigma$, and this is the definition of the linking number of $\alpha^+$ with $\alpha$. This linking number is known to be half of the algebraic number of crossings of $\bar\alpha$ and $\bar\alpha^+$, where a crossing is positive if it is right handed and negative if it is left handed (see Figure \ref{fig:crossings}). One can check this directly by observing the intersections of $\alpha^+$ and $\Sigma$, if $\alpha^+$ is chosen to be slightly above the $(x,y)$-plane, and the part of $\Sigma$ near to $\bar\alpha$ is chosen to be in this plane. Each crossing of $\bar\alpha$ will yield one intersection point and contribute $(+1)$ if the crossing is positive and $(-1)$ if it is negative.
\begin{figure}
\caption{The pieces of the curves $\bar\alpha$ and $\bar\alpha^+$.}
\label{fig:shiftingknot}
\end{figure}
The crossings and cusps of $\bar\alpha$ and $\bar\alpha^+$ in the front projection are shown in Figure \ref{fig:shiftingknot}, with $\bar\alpha$ in black and $\bar\alpha^+$ in gray. Each cusp of $\bar\alpha$ pointing to the left contributes $0$ to the intersection of $\alpha$ and $\Sigma$ since $\alpha^+$ does not meet $\Sigma$ near the cusp, and a cusp pointing to the right contributes $(-1)$, so two adjacent cusps contribute $(-1)$. Hence if $P$ and $N$ are the number of positive and negative crossings of the front projection $\bar\alpha$, respectively, and $C$ is the number of cusps, we conclude that \begin{equation}\label{eq:two} {\rm tb}(\alpha)= P - N - C/2. \end{equation}
In the Lagrangian projection $pr_L(x,y,z)=(x,y)$) the knots $\alpha$ and $\alpha^+$ project to the same diagram, since $\alpha^+$ is obtained by moving $\alpha$ a small distance in the $Y=\partial/\partial z$-direction. We can see that ${\rm tb}(\alpha)$, the linking number of $\alpha$ and $\alpha^+$, is the algebraic number of the positive and negative crossings of the Lagrangian projection of $\alpha$, ${\rm tb}(\alpha)= P - N$, as in \cite{Et}, p. 13.
\noindent{\bf Computation of tb for Legendrian knots in $(T^3,\xi_n)$.} Now consider an oriented Legendrian knot $\alpha$ in $(T^3,\xi_n)$ for a fixed $n>0$. Let $$\hat{p}_{xy}, \hat{p}_{xz}:T^2\times\mathbb{R}\to T^2$$ be the lifts to the covering space $T^2\times\mathbb{R}$ of the projections $p_{xy}, p_{xz}:T^3 = T^2\times S^1\to T^2$, where $\mathbb{R}\to S^1$ is the universal cover of the circle. We shall show how to compute ${\rm tb}(\alpha)$ using the lifted projections $\hat{p}_{xy}, \hat{p}_{xz}$ in a similar way to the case of $({\mathbb R}^3,\xi_{std})$ treated above. We cannot use a linking number here, since the lifted knot $\hat\alpha$ does not bound a compact surface, but we can use the intersection number of a perturbed lifted knot $\hat\alpha^+$ with the image $\hat\Sigma$ in $T^2\times\mathbb{R}$ of the covering Seifert surface of $\Sigma$ of $\alpha$ in ${\mathbb R}^3$ in one circuit of $\alpha$.
\begin{prop} For a generic Legendrian knot $\alpha$ in $(T^3,\xi_n)$ and the projection $p_{xy}$, $$\rm tb(\alpha)=P-N+C/2$$ where $P$ is the number of positive crossings, $N$ is the number of negative crossings, and $C$ is the number of cusps of $p_{xy}\circ\alpha$, which must be even, in one circuit of $\alpha$. \label{tbpxy} \end{prop}
As before, since $\alpha$ is generic, the only singularities are transverse double points and isolated cusps. To determine which crossings are positive and which are negative we use a single component of the lift $\hat\alpha$ of $\alpha$ to $T^2\times{\mathbb R}$ to see which strand of this component is above and which one is below; then following $\hat\alpha$ from the double point on one arc to the same double point on the other arc, the change in the vertical coordinate $z$ determines which arc is above the other, and hence whether the crossing is positive or negative (see Figure \ref{fig:crossings}). Note that each crossing in the projection corresponds to exactly one pair of strands in $\hat\alpha$, and crossings involving two different components do not contribute anything. The statement of the Proposition \ref{tbpxy} could just as well be formulated in terms of one period of the lifted curve $\hat\alpha$.
\begin{figure}
\caption{The curve $\hat\alpha^+$, in gray, intersects $\hat\Sigma$ positively and negatively close to positive and negative crossings, respectively.}
\label{fig:puncturedcrossings}
\end{figure}
\begin{proof}[Proof of Proposition \ref{tbpxy}.] Lift the contact structure $\xi_n$ to the contact structure $\hat{\xi}_n$ on $T^2\times{\mathbb R}$. The perpendicular vector field $\hat Y=(\cos 2\pi nz, \sin 2\pi nz, 0)$ determines the orientation of $\hat{\xi}_n$. Let $\hat\alpha^+$ be a copy of $\hat\alpha$ obtained by shifting $\hat\alpha$ slightly in the positive direction of $\hat Y$. By Definition \ref{def:tb} the Thurston-Bennequin invariant of $\alpha$ is equal to the signed intersection number of $\hat\alpha^+$ with a Seifert surface $\hat\Sigma$ of $\alpha$ in one circuit of $\hat\alpha$. It is convenient to choose the Seifert surface $\hat\Sigma$ to descend vertically near the Seifert curves, except near the cusp points on $T^2$, where the covering Seifert surface must move out horizontally for a small distance before descending. Then it is easy to check that the contribution of a crossing will be $(+1)$ for a positive crossing and $(-1)$ for a negative crossing, since the upper strand of $\hat\alpha^+$ near the crossing will not meet $\hat\Sigma$, and the lower strand will pierce $\hat\Sigma$ just once, with the appropriate orientation, as shown in Figure \ref{fig:puncturedcrossings} for certain typical values of $z$.
The contribution of a cusp point of $\hat p_{xy}\circ \hat\alpha$ is illustrated in Figure \ref{fig:piercing}, which shows $\hat\alpha$ and $\hat\alpha^+$ on $T^2\times\mathbb{R}$. The arrows show the direction in which the vertical coordinate $z$ increases. If the vector field $Y$ points to the left of $\hat\alpha$ as $\hat\alpha$ approaches the cusp point, as in Figure \ref{fig:piercing} (a), then the knot $\hat\alpha^+$ perturbed in the direction $Y$ will be above $\hat\Sigma$, so there is no intersection and the contribution will be $0$. If, on the other hand, $Y$ points toward the right as $\hat\alpha$ approaches the cusp point, then there will be a single intersection point $p$ where the perturbed knot $\hat\alpha^+$ pierces $\hat\Sigma$, and the contribution will be $+1$, as the orientations in Figure \ref{fig:piercing} (b) show.
\begin{figure}
\caption{In part (a), the perturbed (gray) curve $\hat\alpha^+$ does not intersect $\hat\Sigma$ and, in part (b), $\hat\alpha^+$ intersects $\hat\Sigma$ positively.}
\label{fig:piercing}
\end{figure}
The following lemma will complete the proof of the Proposition. \end{proof}
\begin{lem} The contributions of the cusps alternate between $+1$ and $0$, so the total contribution of the cusps is $C/2$, where $C$ is the number of cusps. \end{lem}
\begin{proof} If the direction of increasing $z$ is the same from a cusp with value $+1$ to the next cusp, then $Y$ will point to the left as the next cusp is approached and its contribution will be $0$. On the other hand, if the direction of increasing $z$ reverses, then again the contribution will be $0$, since the direction of increasing $z$ will be reversed, so $Y$ will point to the right leaving the next cusp in the direction of increasing $z$. In a similar manner, if a cusp has contribution $0$, the next cusp will contribute $+1$. \end{proof}
\noindent{\bf The projection onto the $xz$-plane.} Now we shall compute $\rm tb(\alpha)$ using the projection $p_{xz}$ of a generic Legendrian knot $\alpha$. Set $\alpha(t)=(x(t),y(t),z(t))$ and note that by a small Legendrian perturbation of $\alpha$ we can suppose that \begin{equation} \hskip 2cm 2nz(t)\notin \mathbb{Z} {\rm\ \ whenever}\ \ (x'(t),y'(t))= (0,0).\hskip 3cm \label{znonzero} \end{equation}
In other words, for these values of $t$ the vertical component of $\alpha'(t)$ is not zero. For other values of $t$, the plane $\xi_n$ of the contact structure projects onto the tangent plane of $T^2$ under $p_{xz}$, and so the image $p_{xz}\circ\alpha$ is a smooth non-singular curve.
The argument used for the projection $p_{xy}$ shows the following result. By analogy with the previous analysis, we let the covering Seifert surface move off the knot $\hat\alpha$ in the direction of the $y$-axis, instead of the $z$-axis. To determine whether a crossing is positive or negative, lift the knot to $S^1\times {\mathbb R}\times S^1$, where the order of the arcs passing through a double point is well defined.
\begin{prop} For a generic Legendrian knot $\alpha$ in $(T^3,\xi_n)$ that satisfies (\ref{znonzero}), $$\rm tb(\alpha)=P-N$$ where $P$ is the number of positive crossings and $N$ is the number of negative crossings of $p_{xz}\circ\alpha$ in one circuit of $\alpha$. \label{tbpxz} \end{prop} It is possible to calculate $\rm tb(\alpha)$ for a generic Legendrian knot $\alpha$ that does not satisfy (\ref{znonzero}) using its projection in the $xz$-plane, but the formula is more complicated, so we omit it.
These calculations prove the first two assertions of Theorem \ref{projectionthm}, which relate to the Thurston-Bennequin invariant.
\section{The Rotation Number}\label{s:maslov}
Recall that a null-homologous oriented Legendrian knot $\alpha$ in a $3$-manifold $M$ with an oriented contact structure $\xi$ has an invariant $r(\alpha)$, called the rotation (or Maslov) number, which depends on the choice of a non-vanishing section $Z$ of $\xi$. If $\alpha$ is null-homologous, then it has a covering Seifert surface $\Sigma$, and the vector field $Z$ can be determined (up to Legendrian homotopy) by requiring that it extend to a non-vanishing section of $\xi$ over $\Sigma$.
\begin{defn} {\rm The {\it rotation number} (or Maslov number) of the oriented Legendrian knot $\alpha$, $r(\alpha)$, is the algebraic number of rotations of the tangent vector $\alpha'$ with respect to $Z$ in the plane field $\xi$ in a single circuit of $\alpha$.} \end{defn}
\begin{prop} If $\alpha$ is a null homologous oriented Legendrian knot, the rotation number $r(\alpha)$ does not depend on the section $Z$. Furthermore, two Legendrian knots that are isotopic through Legendrian knots have the same rotation number with respect to the same section $Z$. \label{independ} \end{prop}
\begin{proof} The second affirmation in obvious, since the rotation number is an integer that varies continuously as the Legendrian knot varies.
Now let $Z'$ be another global section of $\xi$ and let $f:M\to S^1$ be the function which gives the angle from $Z$ to $Z'$. Since $\alpha$ is null homologous in $M$, the image $f_*[\alpha]$ of its homology class must vanish in $H_1(S^1)$, so the mutual rotation number of $Z'$ relative to $Z$ is $0$. Hence the rotation numbers are the same. \end{proof}
Consequently $r(\alpha)$ for a null-homologous knot $\alpha$ depends only on the orientations of $\alpha$ and $\xi$. Reversing one of these orientations changes the sign of $r(\alpha)$. As in the case of the definition of the Thurston-Bennequin invariant, the rotation number can also be defined for non-null homologous oriented Legendrian knots, but then it does depend on the choice of the section $Z$ of $\xi$.
This dependence holds, in particular, when $M=T^3$ (see \cite{Gh}), but for Kanda's tight contact structure
$\xi_n$, we can use the covering Seifert surface of $\alpha$ to determine $Z|_{\widetilde\alpha}$ up to Legendrian homotopy.
\begin{lem} Let $\alpha$ be an oriented Legendrian knot for the contact structure $\xi_n$ on $T^3$, and let $\Sigma\subset \mathbb{R}^3 = \tilde T^3$ be a covering Seifert surface for $\alpha$ containing an affine half-plane $P\subset\mathbb{R}^3$. Then there is a non-vanishing Legendrian vector field $Z$ in
$\xi_n|\Sigma$ whose restriction $Z|_P$ is a section of $\xi_n \cap P$. Furthermore along $\widetilde\alpha$, the restriction $Z|_{\widetilde\alpha}$ is unique up to periodic Legendrian homotopy. \label{chooseZ} \end{lem}
\begin{proof} If the vertical Legendrian vector field $\partial/\partial z$ is in $P$, then along $P$, $Z=\partial/\partial z$ is a section of $\xi_n\cap P$. Furthermore, $\xi_n$ and $P$ are transverse except along isolated values of $z$, so by continuity the section $Z$ is determined up to multiplication by a non-vanishing function. If $\partial/\partial z$ is not in $P$, then $\xi_n$ and $P$ are transverse, so again $Z$ is determined as a section of the line field $\xi_n\cap P$ on $P$. Now extend $Z$ arbitrarily to a periodic and non-vanishing vector field tangent to $\Sigma$. Clearly $Z$ is determined up to periodic Legendrian homotopy on $P$, and the usual argument shows that the restriction $Z|_{\widetilde\alpha}$ is also determined up to periodic Legendrian homotopy along $\widetilde\alpha$. \end{proof}
It is interesting to note that on $P$ the restriction $Z|_P$ is periodically Legendrian homotopic to the vertical Legendrian vector field $\partial/\partial z$, since, in the second case of the above proof, the angle between them is never $\pi$.
\begin{defn} {\rm The {\it rotation number} (or Maslov number) of an oriented Legendrian knot $\alpha$, $r(\alpha)$, on the contact manifold $(T^3,\xi_n)$ is the algebraic number of rotations in the plane field $\xi_n$ of the tangent vector $\alpha'$ with respect to the vector field $Z$ given by Lemma \ref{chooseZ}, in a single circuit of $\alpha$.} \label{r on xin} \end{defn}
\subsection{Computation of the rotation invariant using projections}\label{maslovproj}
\noindent {\bf Computation of $r$ for Legendrian knots in $(\mathbb{R}^3,\xi_{std})$.} Let $\alpha$ be an oriented generic Legendrian knot in the standard contact structure $\xi_{std}=\ker(dz-ydx)$ on $\mathbb{R}^3$. In order to calculate the rotation number $r(\alpha)$ we fix the Legendrian vector field $Y=\partial/\partial y$. Then $r(\alpha)$ is the algebraic number of times the field of tangent vectors $\alpha'$ rotates in $\xi_{std}$ relative to $Y$, so $r(\alpha)$ can be obtained by counting how many times $\alpha'$ and $\pm Y$ point in the same direction. The sign is determined by whether $\alpha'$ passes $\pm Y$ counterclockwise (+1) or clockwise (-1), and then we must divide by two, since in one rotation $\alpha'$ passes both $Y$ and $-Y$.
\begin{figure}
\caption{Up cusps and down cusps.}
\label{fig:cusps}
\end{figure}
If $\bar\alpha$ denotes the front projection of $\alpha$, the field of tangent vectors to $\bar\alpha$, $\bar\alpha'$, points in the direction of $\pm Y=\pm \partial/\partial y$ at the cusps, which are horizontal in the $xz$-plane. Let us analyze the upwards left-pointing cusp, the first of the four cusps in Figure \ref{fig:cusps}. The value of $y$ is just the slope of $\bar\alpha'$, so $y$ is negative before the cusp and becomes positive, and thus at the cusp $y(t)$ is increasing so $y'(t)$ is positive and $\alpha'$ passes $+Y$ at the cusp point. Before the cusp, $x$ is decreasing, so $x'(t)$ passes from negative to positive at the cusp. Thus the vector $\bar\alpha'(t)$ turns in the negative direction, and the contribution is $(-1)$. By a similar analysis of the other three cases, we see that a cusp going upwards (the first two cusps in the figure) contributes $(-1)$, while a cusp going downwards (the third and fourth cusps in the figure) contributes $(+1)$. Therefore we have shown that the rotation number of $\alpha$ in the front projection is $$r(\alpha)=1/2(C_d-C_u),$$ where $C_u$ is the number of up cusps and $C_d$ is the number of down cusps in the front projection of $\alpha$. Since $\alpha$ is null-homologous, $r(\alpha)$ does not depend on the choice of the vector field $Y$, as we observed above.
In the Lagrangian projection $pr_L(x,y,z)=(x,y)$, the vector field $Y$ projects to $\partial/\partial y$, thus the rotation number of $\alpha$ is simply the winding number of the field of tangent vectors of the Lagrangian projection $pr_L\circ \alpha$ of $\alpha$ in $\xi_{std}$, $$r(\alpha)={\rm winding} (pr_L(\alpha)).$$
\noindent {\bf Computation of $r$ for Legendrian knots in $(T^3,\xi_n)$.} Let $\alpha$ be an oriented generic Legendrian knot in $T^3$ for the tight contact structure $\xi_n$. We shall calculate the rotation invariant $r(\alpha)$ relative to the vertical vector field $Z=\partial/\partial z\in \xi_n$.
\begin{figure}
\caption{Values of $b(t)$ in the projection $p_{xz}$.}
\label{fig:last}
\end{figure}
\noindent{\bf The projection $p_{xz}$.} First we use the projection $p_{xz}: T^3\to T^2$. By a small Legendrian perturbation, if necessary, we guarantee that if $2nz(t)\in {\mathbb Z}$ then the tangent vector $\alpha'(t)$ is not vertical, i.e., $(x'(t),y'(t))\neq (0,0)$. The only contributions to the rotation number $r(\alpha)$ occur for points where $\alpha'(t)$ is vertical, and then since $2nz(t) \notin{\mathbb Z}$ the projection $p_{xz}$ takes $\xi_n$ onto the tangent plane to $T^2$. Near to where $\alpha'(t)$ is vertical the tangent vector, which must be non-zero, will be turning in either the positive direction with respect to the orientation of the $xz$-plane and pass the vertical line in the positive direction, and then we set $b(t)= +1$ (as in the last two cases of Figure \ref{fig:last}), or in the negative direction (as in the first two cases) where we set $b(t)=-1$. Let $a(t)=(-1)^{[2nz(t)]}$, where the brackets indicate the largest integer function, so that $a(t)$ is positive where the projection $p_{xz}$ of $\xi_n$ onto the tangent $xz$-plane preserves the orientation and negative where the orientation is reversed, except when $2nz(t)\in {\mathbb Z}$, but we have guaranteed that then $\alpha'$ will not be vertical. Thus the contribution of a point where $\alpha'$ is vertical is half the product of $a(t)$ and $b(t)$. We have shown the following.
\begin{prop} The rotation invariant $r(\alpha)$ of a generic oriented knot $\alpha$ in $T^3$ with respect to the projection $p_{xz}: T^3\to T^2$ is $$r(\alpha)=1/2\sum_{t\in V} a(t)b(t)$$
where $V=\{t\in S^1\ |\ (x'(t),y'(t))=(0,0) \}$ in one circuit of $\alpha$, provided that $(x'(t),y'(t))\neq (0,0)$ whenever $2nz(t)\in {\mathbb Z}$. \label{maslovpxy} \end{prop}
\noindent{\bf The projection $p_{xy}$.} For the projection $p_{xy}:T^3\to T^2$, we must count how many times the tangent field of $\bar{\alpha}=p_{xy}\circ\alpha$ and $Z=\partial/\partial z$ point in the same direction, and this will happen where $\bar\alpha$ has a cusp since the projection $\bar\alpha$ will have velocity $\bar\alpha'(t_0)=0$ at such a point. Observe that the horizontal normal vector $Y=(\cos 2\pi nz,\sin 2\pi nz,0)$, which determines the orientation of the perpendicular contact plane $\xi_n$, projects to a vector $\bar Y=p_{xy*}(Y)=(\cos 2\pi nz,\sin 2\pi nz)$ perpendicular to the line tangent to the cusp in the $xy$-plane. The slope of this line, determined by the value of $z(t_0)$ at the cusp, may have any value.
\begin{figure}
\caption{A cusp of $p_{xz}\circ\alpha$.}
\label{fig:cusp_orientation}
\end{figure}
Consider the orientation of $\alpha$ and the direction of $Y$ in Figure \ref{fig:cusp_orientation}. Since the tangent vector $\bar\alpha'(t)$ is turning in the positive direction in the $xy$-plane, $z'(t_0)>0$ at the cusp. Before the cusp $\bar\alpha'(t)$ is directed toward the cusp, and afterwards, it is directed away from the cusp. Hence it is clear that $\alpha'(t)$ passes the vertical vector $Z$ in the positive direction in the contact plane $\xi_n$, so in this case the contribution of the cusp is $+1$, and we call the cusp {\em positive}. In this case the projection of $\bar\alpha'(t)$ onto the line through $Y$ has the same direction at $Y$, both before and after the cusp point. The result is the same if the diagram in Figure \ref{fig:cusp_orientation} is rotated in the $xy$-plane.
\begin{figure}
\caption{Positive and negative cusps for the projection $p_{xz}$.}
\label{fig:typesofcusps}
\end{figure}
Now it is clear that if the orientation of $\alpha$ or the direction of $Y$ is reversed, the sign of the contribution of the cusp changes. It follows that in all four cases of the orientation of $\alpha$ and the perpendicular direction of $Y$, the contribution of the cusp is $+1$ and the cusp is {\em positive} if the projection of $\bar\alpha'(t)$ onto the line through $Y$ both before and after the cusp has the same direction as $Y$, and the cusp is {\em negative}, with contribution $-1$, if the direction is opposite to $Y$, as shown in Figure \ref{fig:typesofcusps}.
\noindent Thus we have shown the following.
\begin{prop} The rotation number of a generic oriented knot $\alpha$ in $(T^3,\xi_n)$ with respect to the projection $p_{xy}$ is $$r(\alpha)=1/2(C_+-C_-)$$ where $C_+$ is the number of positive cusps and $C_-$ is the number of negative cusps of $p_{xy}\circ\alpha$ in one circuit of $\alpha$. \label{maslovpxz} \end{prop} This completes the calculation of the rotation invariant using the projections $p_{xz}$ and $p_{xy}$ as in Theorem \ref{projectionthm}, so its proof is complete.
\section{Does a Bennequin inequality hold?}\label{s:tb-inequality} For a null-homologous Legendrian knot $\alpha$ on a tight contact $3$-manifold $(M,\xi)$ with Seifert surface $\Sigma$, the Thurston-Bennequin inequality
\begin{equation} tb(\alpha) + |r(\alpha)| \leq -\chi(\Sigma) \label{Ben_classical} \end{equation} gives an upper bound on $tb(\alpha)$, provided that $\chi(\Sigma)\leq 0$ \cite{E,Et}. It is natural to ask (and we thank the referee for suggesting this) whether this inequality remains valid for our extension of these invariants. The following proposition gives an example which shows that the inequality must be modified. It also motivates a conjecture as to what ought to hold. Recall that according to Kanda \cite{Ka}, a Legendrian knot $\alpha$ in $(T^3,\xi_n)$ is quasilinear if $\alpha$ is isotopic on $T^3$ to a knot which lifts to a straight line in the universal cover $\tilde T^3 = \mathbb{R}^3$.
\begin{prop} For any $(p,q,r)\in\mathbb{Z}^3\smallsetminus \{(0,0,0)\}$ there is a quasilinear Legendrian knot $\alpha$ for the tight contact structure $(T^3,\xi_n)$ with a $(p,q,r)$-periodic lift to $\mathbb{R}^3=\tilde T^3$ such that $\alpha$ satisfies the following equation: \begin{equation} tb(\alpha) + r(\alpha) = -\chi(\hat\Sigma) + rn. \label{Ben-eq} \end{equation} \end{prop}
\begin{proof} First, we note that for every $n>0$ and $(p,q,r)\in \mathbb{Z}^3$ there exists a $(p,q,r)$-periodic Legendrian knot $\alpha$ in $(T^3,\xi_n)$, i.e., such that $\tilde\alpha(t+1) = \tilde\alpha(t) + (p,q,r)$, where $\widetilde\alpha(t) = (x(t),y(t),z(t))$ is the lift of $\alpha$ to $\mathbb{R}^3$. Furthermore, we may construct $\alpha$ so that for all $t\in \mathbb{R}$ if $r>0$ (respectively, $r=0$ or $r<0$) we have $z'(t)>0$ (respectively, $z'(t)=0$ or $r'(t)<0$). We construct such a piecewise linear Legendrian knot and then smooth it out by a small isotopy. If $r>0$, take $t_0, t_1\in [0,1]$ with $t_0<t_1$ such that $\xi_n(t_0)$ is parallel to the $x$-axis and $\xi_n(t_1)$ is parallel to the $y$-axis. Then define a piecewise linear $(p,q,r)$-periodic Legendrian knot by letting $z(t)$ increase linearly on the intervals $[0,t_0]$, $[t_0+\epsilon,t_1]$, and $[t_1+\epsilon, 1]$ modulo $1$ (for sufficiently small $\epsilon>0$, with $x(t)$ and $y(t)$ both constant on these intervals, while on the interval $[t_0,t_0+\epsilon]$ $x(t)$ increases by $p$ and on $[t_1,t_1+\epsilon]$ $y(t)$ increases by $q$, with $z(t)$ constant. Next, by a small Legendrian isotopy, deform this PL knot to a smooth Legendrian knot $\alpha$ so as to preserve the property that $z'(t)>0$ for every $t$. The case $r<0$ is similar. For the case $r=0$ we may take $\alpha$ to be the linear Legendrian knot $\alpha(t)=(pt,qt,z_0), t\in [0,1],$ where $z_0$ is such that the vector $Y=(p,q,0)\in \xi_n(x,y,z_0)$ for every $x,y\in\mathbb{R}$.
For such a Legendrian knot $\alpha$ with $r>0$ let us calculate the invariants. Since $z'(t)>0$, the rotation number of the tangent vector $\alpha'$ with respect to the constant Legendrian vector field $Z=\partial/\partial z$ in $\xi_n$ is $r(\alpha)=0$. Next, take a vector $X\in \mathbb{R}^3$ orthogonal to $(p,q,r)$ and construct a Seifert surface $\hat\Sigma\subset \hat T^3$ such that along $\hat\alpha$ $X$ is tangent to $T\hat\Sigma$ and points inwards towards $\hat\Sigma$. Then it follows that $tb(\alpha)=rn$, since as $z(t)$ increases by $r$ the contact structure $\xi_n$ rotates exactly $rn$ times. The Seifert surface $\hat\Sigma$ can be taken to be homeomorphic to $S^1\times [0,\infty)$, so $\chi(\hat\Sigma) = 0$. Thus equation (\ref{Ben-eq}) holds in this case. For the case that $z<0$, consider $-\alpha$, the knot $\alpha$ with the reversed orientation, and apply the case $r>0$. The signs of $r(\alpha)$ and $r$ are reversed, while $tb(\alpha)$ and $\chi(\hat\Sigma)$ continue to vanish, so the same formula holds. The case $r=0$ is analogous, with $r(\alpha)=tb(\alpha)=0$. \end{proof} These examples motivate the following conjecture. \begin{conj} For any Legendrian knot $\alpha$ on $(T^3,\xi_n)$ that is $(p,q,r)$-periodic (in the positive direction of $\alpha$), there is a Bennequin inequality \begin{equation} tb(\alpha) + r(\alpha) \leq -\chi(\hat\Sigma) + rn. \label{Ben-ineq} \end{equation} \end{conj}
We note that if $\alpha$ is a null-homologous Legendrian knot in $(T^3,\xi_n)$, then $(p,q,r)=(0,0,0)$ and (\ref{Ben-ineq}) is equivalent to the classical Bennequin inequality (\ref{Ben_classical}). Furthermore, since every tight contact structure on $T^3$ is contactomorphic to some $\xi_n$, the conjecture implies that a similar ineqality should hold for Legendrian knots in any tight contact structure $\xi$ on $T^3$, provided $Z$ is taken to correspond to $\partial/\partial z$ under the contactomorphism.
\end{document} |
\begin{document}
\title{A cubical flat torus theorem and the bounded packing property} \author[D.~T.~Wise]{Daniel T. Wise}
\address{Dept. of Math. \& Stats.\\
McGill Univ. \\
Montreal, QC, Canada H3A 0B9 }
\email{wise@math.mcgill.ca} \author{Daniel J. Woodhouse} \email{daniel.woodhouse@mail.mcgill.ca} \subjclass[2010]{20F67, 20F65} \keywords{CAT(0) cube complexes, Bounded Packing, Flat Torus Theorem} \date{\today} \thanks{Research supported by NSERC and the second author by Hydro Quebec.} \maketitle
\begin{com} {\bf \normalsize COMMENTS\\} ARE\\ SHOWING!\\ \end{com} \begin{abstract} We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.\end{abstract}
\section{Introduction}
Let $G$ be a finitely generated group, and let $\Upsilon$ be its Cayley graph with respect to some finite generating set. A subgroup $H \leqslant G$ has \emph{bounded packing in $G$} if for each $r>0$ there exists $m=m(r)$ such that if $g_1H,\ldots, g_mH$ are distinct left cosets of $H$, then there exists $i,j$ such that $\textup{\textsf{d}}_\Upsilon(g_ih, g_jh')>r$ for all $h,h'\in H$.
The motivating goal of this article is to prove the following:
\begin{thmA} Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde X$. Let $A$ be an abelian subgroup of $G$. Then $A$ has bounded packing in $G$. \end{thmA}
Since Theorem~\ref{thm:bounded packing abelian cubical} is limited to the setting of CAT(0) cube complexes, it offers no direction towards resolving the following problems: \begin{prob} \hspace{1cm}
\newline \noindent \begin{enumerate} \item Let $G$ act properly and cocompactly on a CAT(0) space. Does each [cyclic] abelian subgroup $A\leqslant G$ have bounded packing?
\item Let $G$ be an aTmenable group. Does each abelian subgroup $A\leqslant G$ have bounded packing?
\item Find a finitely generated group $G$ with an infinite cyclic subgroup $A \leqslant G$ that does not have bounded packing.
\end{enumerate} \end{prob}
The \emph{rank} of a virtually abelian group $A$ is the rank of any finite index free-abelian subgroup of $A$. A virtually abelian subgroup $A \leqslant G$ is \emph{highest} if $A$ does not have a finite index subgroup that lies in a virtually abelian subgroup of higher rank. The particular feature of CAT(0) cube complexes used to prove Theorem~\ref{thm:bounded packing abelian cubical} is Theorem~\ref{thm:hullTheorem}, which is the crux of this paper. A neat consequence of it is the following Cubical Flat Torus Theorem which asserts that a highest abelian subgroup acts on a product of quasilines. A \emph{cubical quasiline} is a CAT(0) cube complex that is quasi-isometric to $\mathbb{R}$. For brevity we will simply refer to cubical quasilines as \emph{quasilines}. \begin{thmD} Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde X$. Let $A$ be a highest virtually abelian subgroup of $G$ and let $p=\rank(A)$. Then $A$ acts properly and cocompactly on a convex subcomplex $\widetilde Y \subseteq \widetilde X$ such that $\widetilde Y \cong \prod_{i=1}^p C_i$ where each $C_i$ is a quasiline. \end{thmD}
We also present the following application of Theorem~\ref{thm:bounded packing abelian cubical}:
\begin{thmC} Let $H$ be a finitely generated virtually $[$compact$]$ special group. Let $A\subset H$ be a highest abelian subgroup. Let $G=H*_{A^t=A}$ be the HNN extension, where $t$ is the stable letter commuting with $A$, then $G$ is virtually $[$compact$]$ special. \end{thmC}
A version of Theorem~\ref{thm:Central HNN virtually special} was proven in \cite{WiseIsraelHierarchy} under the additional hypothesis of relative hyperbolicity, but Theorem~\ref{thm:bounded packing abelian cubical} allows us to avoid this hypothesis. Example~\ref{exmp:4 Z2 subgroups} shows that $G$ can fail to have a virtually compact cubulation when $H$ is a f.g. 2-dimensional right-angled Artin group, but $A$ is not highest.
Section~\ref{sec:restricted intersection} uses Theorem~\ref{thm:cocompact cubical flat} to restrict how highest abelian subgroups intersect. The following amusing consequence of Corollary~\ref{cor:too many intersection directions} shows that generic multiple cyclic HNN extensions of $\ensuremath{\field{Z}}^p$ cannot be virtually compactly cubulated:
\begin{GenericExmp}
Let $\big\{\langle b_1\rangle,\ldots,\langle b_r\rangle,\langle c_1\rangle,\ldots,\langle c_r\rangle \big\}$
be a collection of pairwise incommensurable infinite cyclic subgroups of $\ensuremath{\field{Z}}^p$,
and suppose that $r>\frac{p}{2}$. Let $G$ be the following multiple HNN extension of $\ensuremath{\field{Z}}^p =\langle a_1,\ldots, a_p\rangle$: $$G = \langle a_1,\ldots, a_p, t_1,\ldots, t_r \ \mid \ [a_i,a_j]=1, \ b_k^{t_k}=c_k \ : \ 1\leq k \leq r\rangle$$ Then $G$ does not contain a finite index subgroup that acts properly and cocompactly on a CAT(0) cube complex. \end{GenericExmp}
This paper is structured as follows: In Section~\ref{FlatDual} we prove Theorem~\ref{thm:hullTheorem}. In Section~\ref{BPproperty} we collect existing results and explain how, along with Theorem~\ref{thm:hullTheorem}, they allow us to prove Theorem~\ref{thm:bounded packing abelian cubical}. In Section~\ref{sec:restricted intersection} we observe that highest free-abelian subgroups have restricted intersections with other free-abelian subgroups. In Section~\ref{CentralizingExtensions} we prove Theorem~\ref{thm:Central HNN virtually special}.
\noindent {\bf Acknowledgement:} We are grateful to Mathieu Carette for helpful suggestions. We thank the referees for their comments and corrections.
\section{The dual to a flat} \label{FlatDual}
The goal of this section is to prove Theorem~\ref{thm:hullTheorem}, which we state below.
A \emph{quasiline} is a CAT(0) cube complex quasi-isometric to $\mathbb{R}$, and a \emph{quasiray} is a CAT(0) cube complex quasi-isometric to $[0,\infty) \subseteq \mathbb{R}$ .
A hyperplane in a CAT(0) cube complex $\widetilde X$ will be denoted by $H$, and its left and right halfspaces
are denoted by $\overleftarrow{H}$ and $\overrightarrow{H}$. As it is convenient to work with subcomplexes, we define $\overleftarrow{H},\overrightarrow{H}$ to be the smallest subcomplexes containing the complementary components of $\widetilde X-H$. We will be using wallspaces and Sageev's dual cube complex construction \cite{Sageev95}. We point the reader to~\cite{HruskaWiseAxioms} for an account of the techniques. The proof of the following is given at the end of this section, after we have developed the required language and lemmas.
A subset $S \subseteq \widetilde X$ of a geodesic metric space is \emph{convex} if every geodesic with endpoints in $S$ is contained in $S$.
When $\widetilde X$ is a complete CAT(0) space, a complete connected subspace $Y$ is convex if its inclusion into $\widetilde X$ is a local isometry.
When $\widetilde X$ is a CAT(0) cube complex, and $Y$ is a subcomplex, there is a simple combinatorial criterion equivalent to being a local isometry: For each $0$-cube $v$ of $Y$ the inclusion $\link_Y(v) \hookrightarrow \link_{\widetilde X}(v)$ is a full subcomplex.
We refer to~\cite{BridsonHaefliger} for a comprehensive account of CAT(0) metric spaces.
For a subset $S$ of a CAT(0) cube complex $\widetilde X$, let $\text{\sf hull}(S)$ be the smallest nonempty convex subcomplex of $\widetilde X$ that contains $S$.
\begin{thm}\label{thm:hullTheorem}
Let $A \leqslant G$ be a virtually abelian subgroup of rank~$p$ that acts properly and cocompactly on a flat $E$ in a CAT(0) cube complex $\widetilde X$ with $\textrm{dim}(\widetilde X) < \infty$.
Then either: \begin{enumerate}
\item \label{poss2} $\text{\sf hull}(E)$ is $A$-cocompact and $\text{\sf hull}(E) \cong \prod^{p}_{i=1}C_i$, where each $C_i$ is a convex subcomplex that is a quasiline.
\item \label{poss1} There exists a finite index subgroup $B \leqslant A$ such that $\min(B) \cap \text{\sf hull}(E)$ is not $B$-cocompact. \end{enumerate} \end{thm}
\begin{exmp}
Consider the cyclic group $A$ generated by a diagonal glide reflection acting on the standard cubulation of the plane $\mathbb{R}^2$.
Then $\min(A)$ is a diagonal line while $\text{\sf hull}(E)$ is $\mathbb{R}^2$. \end{exmp}
Let $A \leqslant G$ be a virtually abelian subgroup of rank~$p$ that acts properly and cocompactly on a flat $E$ in a CAT(0) cube complex $\widetilde X$. By a result of Bieberbach \cite{RatcliffeBook}, there exists a finite index free-abelian subgroup $ A_t \leqslant A$ that acts by translations on $E$.
Let $P$ be the set of all hyperplanes intersecting $E$.
The hyperplanes $H_1, H_2$ are \emph{parallel in $E$} if $H_1 \cap E$ and $H_2 \cap E$ are parallel in $E$.
Being parallel in $E$ is an equivalence relation on the hyperplanes intersecting $E$.
There are finitely many parallelism classes of hyperplanes in $E$, denoted $P_i \subseteq P$ for $1 \leq i \leq p$.
\begin{lem}
There exists a finite index subgroup $B \leqslant A_t$ that acts \emph{disjointly} in the sense that distinct hyperplanes in the same $B$-orbit are disjoint.
\end{lem}
\begin{proof}
For each parallelism class $P_i$, choose $g_i \in A_t$ such that the axis of $g_i$ crosses $H \cap E$ for $H \in P_i$. There exists $n_i >0$ such that $\langle g_i^{n_i} \rangle$ acts disjointly on the hyperplanes in $P_i$, as otherwise $g_i^jH$ intersects $g_i^kH$ for all $j,k \in \mathbb{Z}$, contradicting that $\dimension(\widetilde X)<\infty$.
Indeed $m$ pairwise intersecting hyperplanes mutually intersect in an $m$-cube.
As a CAT(0) cube complex is dual to the wallspace associated to its collection of hyperplanes, a point in $\widetilde X$, together with such a collection of pairwise crossing hyperplanes determines an $m$-cube (see eg~\cite{HruskaWiseAxioms}).
Alternatively, one can reach a contradiction from Proposition~\ref{prop:helly} applied to the subdivision of $\widetilde X$.
For $\epsilon>0$ to be determined below, we let $E^\epsilon$ denote $\mathcal N_\epsilon(E)$ and for a hyperplane $H\in P_i$ we let $H^\epsilon = H\cap E^\epsilon$ and let $P_i^\epsilon = \{H^\epsilon: H\in P_i\}$. Thus $(E^\epsilon, P_i^\epsilon)$ is a wallspace for each $i$.
We choose $\epsilon$ so that for each $i$, for each pair $H,H' \in P_i$ we have $H\cap E = H'\cap E$ if and only if $H,H'$ cross within $E^\epsilon$. Cocompactness of $E$ ensures that there are finitely many intersection angles between hyperplanes and $E$, and this allows us to bound the number of orbits of codimension-2 hyperplanes intersecting $\mathcal N_1(E)$ but not intersecting $E$. We choose $\epsilon$ to be less than the minimal distance from $E$ to any such codimension-2 hyperplane.
The dual cube complex $C(E^\epsilon, P_i^\epsilon)$ is a quasiline with an $A_t$-action.
The kernel $K_i$ of this action is isomorphic to $\ensuremath{\field{Z}}^{p-1}$.
Adjoining $g_i^{n_i}$ to $K_i$, we obtain the finite index subgroup $B_i = \langle K_i , g_i^{n_i} \rangle \leqslant A_t$. For each $H\in P_i$ we have $B_iH = \langle g_i^{n_i} \rangle H$ and hence $B_i$ acts disjointly on hyperplanes in $P_i$. Finally, $B = \bigcap_i B_i$ acts disjointly on the hyperplanes in $P$.
\end{proof}
For each parallelism class $P_i$, let $Z_i \leqslant B$ be an infinite cyclic group stabilizing a line $R_i \subseteq E$, non parallel to $H \cap E$ for all $H \in P_i$.
Consider two hyperplanes $H,H' \in P_i$ with distinct $Z_i$-orbits:
$\{z_i^jH: j\in \ensuremath{\field{Z}}\}$ and $\{z_i^jH' : j\in \ensuremath{\field{Z}}\}$.
Observe that if $H$ intersects both $z_i^jH'$ and $z_i^kH'$, then $H$ intersects $z_i^\ell H'$ for $j\leq \ell \leq k$.
\begin{defn} We say $H,H' \in P$ represent \emph{crossing} orbits if either $H \in P_i$ and $H' \in P_j$ with $i \neq j$, or if $H,H' \in P_i$ and $z_i^{j}H$ crosses $z_i^kH'$ for all $j,k\in \ensuremath{\field{Z}}$. We say $H, H' \in P_i$ represent \emph{aligned} orbits if $H$ intersects only finitely many $Z_i$-translates of $H'$. We say $H,H' \in P_i$ represent \emph{semi-crossing} orbits if they do not represent crossing orbits, and there exists $m \in \ensuremath{\field{Z}}$ such that either $z_i^jH$ crosses $z_i^kH'$ for $j - k > m$, or $z_i^jH$ crosses $z_i^kH'$ for $k-j > m$. Any pair of hyperplanes in $P$ must represent either crossing, semicrossing, or aligned orbits. \end{defn}
\begin{figure}
\caption{A cubical halfplane with two semi-crossing hyperplane orbits.}
\label{diag:Halfplane}
\end{figure}
\begin{exmp} Let $\mathbb{Z}$ act on the cubical halfplane in Figure~\ref{diag:Halfplane}. The flat plane $E$ is the line homeomorphic to $\mathbb{R}$. Observe that there are two parallelism classes of orbits that semi-cross. However, the action is not cocompact. \end{exmp}
\begin{lem}
Alignment of $Z_i$-orbits is an equivalence relation. \end{lem}
\begin{proof}
Reflexivity and symmetry are immediate. Suppose $Z_iH$ is aligned to $Z_iH'$ and $Z_iH'$ is aligned to $Z_iH''$, but infinitely many hyperplanes in $Z_iH$ cross $H''$.
By their alignment, there exists $j,k$ such that $H'' \subset (z_i^j \overrightarrow{H}' \cap z_i^k \overleftarrow{H}')$.
Since infinitely many elements of $Z_iH$ intersect $H''$, infinitely many of these elements intersect $H''$
but do not intersect $z_i^j H'$ or $z_i^k H'$.
This is a contradiction as infinitely many hyperplanes cannot separate $z_i^j H'$ and $z_i^k H'$.
\end{proof}
\begin{lem} \label{partialOrdering} Semi-crossing of $Z_i$-orbits is a partial ordering, denoted by $>$, where $Z_iH > Z_iH'$ if $Z_iH$ and $Z_iH'$ are semi-crossing and there exists $m \in \ensuremath{\field{Z}}$ such that $z_i^jH$ crosses $z_i^kH'$ for $j - k > m$. \end{lem}
\begin{proof} Antisymmetry holds since, if $Z_iH > Z_iH'$ and $Z_iH' > Z_iH$ but $Z_iH \neq Z_iH'$ then $Z_iH$ and $Z_iH'$ are crossing orbits. However, distinct crossing orbits are not semi-crossing.
To prove transitivity we first prove the following claim: If $Z_iH_1$ and $Z_iH_2$ are aligned orbits and $Z_iH' > Z_iH_1$, then $Z_iH' > Z_iH_2$. By their alignment, there exists $p<q$ such that $H_2 \subset (z_i^p \overrightarrow{H}_1 \cap z_i^q \overleftarrow{H}_1)$.
Suppose $z_i^j H'$ crosses $z_i^kH_1$ precisely for $j-k > m$.
Then $H_2$ is crossed by $z_i^jH'$ for $j - q > m$ as $z_i^jH'$ crosses both $z_i^qH_1$ and $z_i^pH_1$. Similarly $H_2$ is not crossed by $z_i^jH'$ for $j-p \leq m$. This implies that $Z_iH' > Z_iH_2$.
Similarly $Z_iH' < Z_iH_1$ would imply $Z_iH' < Z_iH_2$.
Suppose that $Z_iH_3 > Z_iH_2 > Z_iH_1$.
By the claim, $Z_iH_1$ cannot be aligned to $Z_iH_3$.
Therefore we need only exclude the possibility that $Z_iH_1$ and $Z_iH_3$ are crossing orbits.
Since $Z_iH_2> Z_iH_1$ there exists $N_1$ such that $z_i^nH_2$ is disjoint from $H_1$ for all $n \leq N_1$. Assume that $H \subseteq z_i^{N_1}\overrightarrow{H}_2$.
Since $Z_iH_3>Z_iH_2$ there exists $N_2$ such that $z_i^nH_3$ is disjoint from $z_i^{N_1}H_2$ for all $n \leq N_2$.
Since $z_i$ acts by translation on $E$ we can deduce that $z_i^nH_3 \subseteq z_i^{N_1}\overleftarrow{H}_2$ for $n \leq N_2$.
Hence, $z_i^nH_3$ is disjoint from $H_1$ for all $n \leq N_2$ as they are separated by $z_i^{N_1}{H}_2$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:hullTheorem}] We first assume there are no semi-crossing orbits. In the next three steps we will show that $\text{\sf hull}(E) = \prod_{i=1}^p C_i$ where each $C_i$ is the quasiline dual to the family of hyperplanes corresponding to an alignment class.
First, we claim that $\text{\sf hull}(E)$ is isomorphic to $C(\widetilde X, P)$. Indeed, each $0$-cube $x$ in $\text{\sf hull}(E)$ corresponds to the $0$-cube $y$ in $C(\widetilde X, P)$ where each hyperplane in $P$ is oriented towards $x$. Conversely, a $0$-cube $y$ in $C(\widetilde X, P)$ corresponds to a $0$-cube $x$ in $\widetilde X$ by orienting the hyperplanes not in $P$ towards $E$.
Moreover, $x \in \text{\sf hull}(E)$ since $x$ lies in each halfspaces containing $E$. This bijection preserves adjacency.
Secondly, let $\{ A_i \}_{i=1}^p$ be an enumeration of the alignment classes.
Observe $C(\widetilde X, P) \cong \prod_{i=1}^m C(\widetilde X, A_i)$ as every hyperplane in $A_i$ intersects every hyperplane in $A_j$ for $i \neq j$.
Indeed, a $0$-cube of $C(\widetilde X, P)$ determines a $0$-cube of each of the factors by ignoring the orientations in the other alignment classes.
Conversely, a choice $0$-cubes in each of the factors determines a $0$-cube in $C(\widetilde X, P)$ since the hyperplanes cross each other.
Again, it is easy to see this bijection preserves adjacency.
Thirdly, let $G_i = \langle g_i \rangle$ be an infinite cyclic subgroup of $B$ acting freely on $C(\widetilde X, A_i)$.
We will show that $C(\widetilde X, A_i)$ is $G_i$-cocompact, and therefore quasiisometric to $\ensuremath{\field{R}}$.
Let $H_1, \ldots, H_k$ be representatives of the distinct $G_i$-orbits.
Note that the dimension of $C(\widetilde X, A_i)$ is bounded by $k$.
We now show that there are finitely many $G_i$-orbits of maximal cubes.
A maximal cube corresponds to a collection of pairwise intersecting hyperplanes $g_i^{\alpha_1}H_{j_1}, \ldots g_i^{\alpha_\ell}H_{j_\ell}$.
By translating we can assume that $\alpha_1 = 0$, and therefore there are finitely many such collections since only finitely many hyperplanes can intersect $H_{j_1}$.
Then $\text{\sf hull}(E) = \prod_{i=1}^p C_i$ where each $C_i$ is the quasiline dual to the family of hyperplanes corresponding to an alignment class.
Observe that $B$ acts by translations on $E$ with disjoint hyperplane-orbits and hence stabilizes each alignment class and thus preserves the factors of the product structure.
If $p= \rank(A)$ the action on $\text{\sf hull}(E)$ is cocompact, which implies that the set of hyperplanes orthogonal to each $R_i$ belong to a single alignment class.
Otherwise $p>\rank(A)$, and as $B$ acts metrically properly and cocompactly on $C_i$,
each $C_i$ contains an isometrically embedded $B$-invariant line $\ell_i$.
Thus $\prod \ell_i \subseteq \prod_{i=1}^p C_i$ is not cocompact,
but is contained in $\min(B) \cap \text{\sf hull}(E)$.
Suppose there are at least two semi-crossing orbits in some parallelism class, and let $Q$ be a maximal alignment class with respect to the partial ordering. For each parallelism class $P_i$ and orbit $Z_iH \subseteq P_i - Q$:
either $Z_iH$ crosses the orbits in $Q$, or $Q\subset P_i$ and $Z_iH' > Z_iH$ for all $H' \in Q$.
We define a sequence of $B$-equivariant cubical maps
$\{\phi_k: \text{\sf hull}(E) \rightarrow \text{\sf hull}(E)\}_{k\in \ensuremath{\field{N}}}$
using the partition $P = Q \sqcup Q^c$:
A $0$-cell $x$ in $\text{\sf hull}(E)$ corresponds uniquely to a choice of orientation for each hyperplane intersecting $E$.
Let $x[H] \in \big\{\overleftarrow{H}, \overrightarrow{H}\big\}$ denote the halfspace of $H$ containing $x$ in its interior. Its image $\phi_k(x)$ is specified by how $\phi_k(x)$ orients the hyperplanes intersecting $E$.
For $H \in Q^c$ let $\phi_k(x)[H] = x[H]$.
For $H \in Q \subseteq P_i$ let $\phi_k(x)[z_i^{k}H] = x[H]$.
This defines a $0$-cube in $\text{\sf hull}(E)$ since only finitely many hyperplanes have their orientations changed, and disjoint hyperplanes are not oriented away from each other by $\phi_k(x)$:
Let $H \subseteq P_i \subseteq Q^c$ represent a $Z_i$-orbit not crossing the $Z_i$-orbits in $Q \subseteq P_i$, then $Z_iH' > Z_iH$ for any $H' \in Q$. Therefore, if $H'$ crosses $H$ then $z_i^{k}H'$ also crosses $H$.
The injectivity of $\phi_k$ on $0$-cubes holds since if $x_1\neq x_2$ then there exists $H \in P$ such that $x_1[H] \neq x_2[H]$.
If $H \in Q^c$ then $\phi_k(x_1)[H] = x_1[H] \neq x_2[H] = \phi_k(x_2)[H]$ so $\phi_k(x_1) \neq \phi_k(x_2)$.
If $H \in Q$ then $\phi_k(x_1)[z_i^{-k}H] = x_1[H] \neq x_2[H] = \phi_k(x_2)[z_i^{-k}H]$ so $\phi_k(x_1) \neq \phi_k(x_2)$.
Therefore $\phi_k$ is injective on the $0$-skeleton.
Similar reasoning shows that $\phi_k$ sends adjacent $0$-cubes to adjacent $0$-cubes and so $\phi_k$ extends to the $1$-skeleton of $\text{\sf hull}(E)$.
Moreover injective maps on the 1-skeleton send squares to squares, hence the map also extends to the 2-skeleton.
Any map defined on the 2-skeleton of a cube complex extends uniquely to a cubical map on the entire complex.
Observe that $B$ acts on $E$ by translation and preserves each $Z_j$-orbit in each $P_j$.
Therefore, for each $b \in B$ there exists $\ell_i$, for $1 \leq i \leq p$, such that $b H = z_i^{\ell_i}H$ for each $H \in P_i$.
Therefore $\phi_k$ is $B$-equivariant since if $H \in P_i$ but $H \notin Q$ then \begin{equation*}\begin{split} (b\cdot \phi_k(x))[H] &= \phi_k(x)[b^{-1} H] = \phi_k(x)[z_i^{-\ell_i} H] \\
&= x[z_i^{-\ell_i}H] = x[b^{-1} H] = (b \cdot x)[H] = \phi_k(b\cdot x)[H].
\end{split}
\end{equation*}
\noindent Similarly, if $H \in Q \subseteq P_i$ then \begin{equation*}\begin{split} (b\cdot \phi_k(x))[H] &= \phi_k(x)[b^{-1} H] = \phi_k(x)[z_i^{-\ell_i} H] \\ &= x[z_i^{k-\ell_i}H] = x[b^{-1}z_i^{k} H] = (b \cdot x)[z_i^kH] = \phi_k(b\cdot x)[H]. \end{split}\end{equation*}
We now show that $\textup{\textsf{d}}(\phi_k(x),b\cdot x)\geq k$ for each $b \in B$ and $x$ a canonical $0$-cube $x$ associated to a point in $E$.
For each $Z_j$-orbit $Z_jH$ fix a representative $H$ such that $x \in \overrightarrow{H} \cap z_i \overleftarrow{H}$.
Let $H \in Q \subseteq P_i$ be such a representative, then $\phi_k$ changes the orientation of precisely $k$ hyperplanes in $Z_iH$, namely $z_iH , \ldots, z_i^kH$.
For any representative $H \in P_i$, however, translation by $b$ changes the orientation of $\ell_i$ hyperplanes, namely $z_iH, \ldots, z_i^{\ell_i}H$.
As there is at least one $Z_i$-orbit in $P_i \supseteq Q$ not in $Q$, we can deduce that at least $k$ hyperplanes have distinct orientations in $\phi_k(x)$ and $b\cdot x$.
Therefore the distance from $bx$ to $\phi_k(x)$ is at least $k$.
Observe that $\textup{\textsf{d}}(\phi_k(y_1),\phi_k(y_2)) \leq \textup{\textsf{d}}(y_1,y_2)$ for $y_1,y_2\in \text{\sf hull}(E)$. Indeed, the CAT(0) metric on $\text{\sf hull}(E)$ is defined to be the infimal length of piecewise Euclidean paths joining points, and the map preserves lengths of paths. The $B$-equivariance together with that $\phi_k$ is distance-nonincreasing implies that $\phi_k(e)\in \min(B)$ for each $e\in E$.
In conclusion, $\textup{\textsf{d}}(\phi_k(E), E)) \rightarrow \infty$ as $k\rightarrow \infty$. For $e \in E$, the orbit $Be \subseteq E$ is mapped by a distance-nonincreasing function to a new orbit at distance $ \geq k$ from $E$. Since the original flat was in $\min(B)$, the image of the image orbit is isometric to the original orbit. Since $k$ is unbounded, $\min(B) \cap \text{\sf hull}(E)$ is non-cocompact.
\end{proof}
\section{The bounded packing property} \label{BPproperty}
Let $G$ be a finitely generated group with Cayley graph $\Upsilon$. Suppose $G$ acts by isometries on a geodesic metric space $\widetilde{X}$ such that the map $g \mapsto gx_0$ is a quasi-isometric embedding for some $x_0 \in \widetilde{X}$. Then $H \leqslant G$ has bounded packing if and only if for each $r >0$ there exists $m = m(r)$ such that if $g_1H, \ldots , g_mH$ are distinct left cosets of $H$, then there exists $i,j$ such that $\textup{\textsf{d}}_{\widetilde{X}}(g_iHx_0, g_jHx_0) > r$. We refer to \cite{HruskaWisePacking} for more about bounded packing, and specifically to Cor~2.9, Lem~2.3, and Lem~2.4 for the following:
\begin{lem} \label{lem:finite index subnormal} Suppose $H$ has bounded packing in $G$. If $K \leqslant H$ is a normal subgroup of $H$, then $K$ has bounded packing in $G$. If $K \leqslant G$ is a subgroup such that $[H:K\cap H]<\infty$ and $[K:K\cap H]<\infty$, then $K$ has bounded packing. \end{lem}
A proof of the following well known fact follows from the median space structure of the 1-skeleton of a CAT(0) cube complex ~\cite[Thm~2.2]{RollerPocSets}. A proof using disk diagrams can be found in~\cite[Sec 2]{WiseIsraelHierarchy}.
\begin{prop}[Helly Property]\label{prop:helly} Let $Y_1,\ldots, Y_r$ be convex subcomplexes of a CAT(0) cube complex. If $Y_i\cap Y_j\neq \emptyset$ for each $i,j$, then $\cap_{i=1}^r Y_i \neq \emptyset$. \end{prop}
The following is obtained in \cite[Lem~13.15]{HaglundWiseSpecial}:
\begin{lem}\label{lem:r thickening} Let $\widetilde Y\subset \widetilde X$ be a convex subcomplex of the CAT(0) cube complex $\widetilde X$. For each $r\geq 0$ there exists a convex subcomplex $\widetilde Y^{+r}$ such that $\mathcal N_r(\widetilde Y)\subset \widetilde Y^{+r} \subset \mathcal N_s(\widetilde Y)$ for some $s\geq 0$. Here $\mathcal N_m(\widetilde Y)$ denotes the $m$-neighborhood of $\widetilde Y$. \end{lem}
We infer the following from the above results.
\begin{lem}\label{lem:convex BP} Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde X$. Let $H$ be a subgroup that cocompactly stabilizes a nonempty convex subcomplex $\widetilde Y\subset \widetilde X$. Then $H$ has bounded packing in $G$. \end{lem} \begin{proof} Let $x_o \in \widetilde Y$. Let $g_1, g_2, \ldots$ be an enumeration of the left coset representatives of $H$. Let $\widetilde Y^{+r}$ be as in Lemma~\ref{lem:r thickening}. Observe that if $\textup{\textsf{d}}(g_jHx_o, g_kHx_o)< r$ then $g_j \widetilde Y^{+r} \cap g_k \widetilde Y^{+r} \neq \emptyset$. Thus, to show that $H$ has bounded packing, it suffices to find an upper bound on the number of distinct cosets $g_iH$ such that $\{g_i\widetilde Y^{+r}\}$ pairwise intersect.
Moreover, by Proposition~\ref{prop:helly}, if $\{g_1\widetilde Y^{+r}, \ldots, g_p\widetilde Y^{+r} \}$ pairwise intersect then $\cap_{i=1}^p g_i\widetilde Y^{+r}\neq \emptyset$.
It thus suffices to show that there is an upper bound $m$ on the multiplicity of $\{g_i\widetilde Y^{+r} : g_iH \in G / H \}$. However this collection of sets is uniformly locally finite since
$\widetilde Y^{+r}$ is $H$-cocompact and $[\stabilizer(Y^{+r}):H]< \infty$. \end{proof}
If $A$ is an abelian group acting by isometries on a metric space $\widetilde{X}$, then $\min(A)$ is the set of all $\widetilde{x} \in \widetilde{X}$ such that $\textup{\textsf{d}}(a\widetilde{x}, \widetilde{x}) \leq \textup{\textsf{d}}(a\widetilde{y}, \widetilde{y})$ for all $a \in A$ and $\widetilde{y} \in \widetilde{X}$. We refer to \cite{BridsonHaefliger} for the following:
\begin{prop}[Flat torus theorem]\label{prop:FPT} Let $A$ be a virtually free-abelian group of rank~$n$ acting metrically properly and semisimply on a CAT(0) space $\widetilde X$. There exists a subspace $V\times F \subset \widetilde X$ with $F$ isometric to $\ensuremath{\field{E}}^n$ such that $A$ stabilizes $V\times F$ and acts as: \ $a(v,f)=(v,af)$ for all $(v,f)\in V\times F$ and $a \in A$. Moreover, if $A \cong \mathbb{Z}^n $ then $ \min(A)= V\times F$. \end{prop}
\begin{thm}[Cubical flat torus theorem]\label{thm:cocompact cubical flat} Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde X$. Let $A$ be a highest virtually abelian subgroup of $G$ and let $p=\rank(A)$. Then $A$ acts properly and cocompactly on a convex subcomplex $\widetilde Y \subseteq \widetilde X$ such that $\widetilde Y \cong \prod_{i=1}^p C_i$ where each $C_i$ is a quasiline. \end{thm}
\begin{proof} By Proposition~\ref{prop:FPT}, $A$ stabilizes $E\subset \widetilde X$, where $E$ is isometric to $\ensuremath{\field{E}}^p$. By Theorem~\ref{thm:hullTheorem}, either $\text{\sf hull}(E) \cong \prod_{i=1}^p C_i$ is $A$-cocompact where each $C_i$ is a quasiline, or there exists a finite index free-abelian subgroup $B \leqslant A$ such that $\min(B) \cap \text{\sf hull}(E)$ is not $B$-cocompact.
We shall show that the second possibility contradicts that $A$ is highest.
Applying Proposition~\ref{prop:FPT} again, let $\min(B) =V\times F$, where $\diameter(V) = \infty$. For $v\in V$ let $N(\{v\}\times F)$ denote the smallest $B$-invariant connected subcomplex of $\widetilde X$ containing $\{v\}\times F$. Since $\{v\}\times F$ is $B$-cocompact, so is $N(\{v\}\times F)$. Moreover, the number of $B$-orbits of cells in $N(\{v\}\times F)$ is bounded by a constant independent of $v\in V$. Indeed, by $B$-cocompactness, there is $m>0$ such that $F = B\mathcal N_m(f )$ for each $f \in F$. For each $v$ there is a $B$-equivariant isometry $F \rightarrow \{v\}\times F$, and so $\{v\}\times F$ is likewise covered by the $B$ translates of each $m$-ball. However, the number of cells intersecting an $m$-ball in $\widetilde X$ is finite by properness and cocompactness. So the number of $B$-orbits of cells in $N(\{v\}\times F)$ has the same upper bound.
It follows that there are finitely many $G$-orbits of subcomplexes $N(\{v\}\times F)$.
As $\diameter(V)=\infty$, there are points $v_1,v_2\in V$ and $g\in G$ such that $N(\{v_1\}\times F)\cap N(\{v_2\}\times F)=\emptyset $, but $gN(\{v_1\}\times F)=N(\{v_2\}\times F)$.
Both $B$ and $g$ stabilize $\sqcup g^n N(\{v_1\}\times F)$ which is quasi-isometric to $\ensuremath{\field{E}}^{n+1}$. Hence $\langle g, B\rangle$ is a higher rank virtually abelian subgroup. \end{proof}
\begin{thm}\label{thm:bounded packing abelian cubical} Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde X$. Let $A$ be an abelian subgroup of $G$. Then $A$ has bounded packing in $G$. \end{thm} \begin{proof} By Proposition~\ref{prop:FPT} and the assumption that $\widetilde{X}$ is finite dimensional we can find a highest virtually free-abelian group $A'$ that contains a finite index subgroup of $A$. The result now follows by combining Lemmas~\ref{lem:finite index subnormal},~\ref{lem:convex BP}, and Theorem~\ref{thm:cocompact cubical flat}. \end{proof}
\section{Subproduct intersections} \label{sec:restricted intersection} This section illustrates the following consequence of Theorem~\ref{thm:cocompact cubical flat}:
\begin{thm} \label{thm:highestIntersectionSubgroups}
Let $G$ act properly and cocompactly on a CAT(0) cube complex $\widetilde{X}$.
Let $A \leqslant G$ be a highest free-abelian subgroup, and let $p=\rank(A)$.
There is a set $S = \{\hat a_1, \ldots, \hat a_p\} \subseteq A$ such that the following holds:
For any highest free-abelian subgroup $A' \leqslant G$, the intersection
$A'\cap A$ is commensurable to a subgroup generated by a subset of $S$. \end{thm}
Proving Theorem~\ref{thm:highestIntersectionSubgroups} requires the following consequence of the flat torus theorem.
\begin{lem} \label{lem:actionDecomposition}
Let $A$ be a rank~$p$ virtually abelian group acting properly and cocompactly on a CAT(0) cube complex $\prod_{i=1}^p C_i$, where each $C_i$ is a quasiline.
Then there exists a finite index free-abelian subgroup $\hat{A} \leqslant A$ with basis $\{\hat{a}_1,\ldots, \hat{a}_p\}$ such that $\hat{a}_i \cdot (c_1, \ldots , c_i, \ldots, c_p) = (c_1 ,\ldots, \hat{a}_i \cdot c_i, \ldots c_p)$ for each $i$. \end{lem}
\begin{proof}
The action of $A$ on $\prod_{i=1}^p C_i$ permutes the factors in the product, yielding a homomorphism $A \rightarrow S_p$ to the degree~$p$ symmetric group.
Its kernel is a finite index subgroup $B \leqslant A$ such that the $B$-action on $\prod_{i=1}^p C_i$
is the product of $B$-actions on the factors.
For each $i$ there is a finite index subgroup $B_i\leqslant B$ that acts by translations on an invariant line $\ell_i \subset C_i$.
Let $\hat A = \bigcap_{i=1}^p B_i$.
Consider a homomorphism $\phi : \hat A \rightarrow \ensuremath{\field{Z}}^p$ induced by the action of $\hat A$ on $\prod_{i=1}^p \ell_i$.
Since $\hat A$ acts cocompactly on $\prod_{i=1}^p \ell_i$ we deduce that $[\ensuremath{\field{Z}}^p : \phi(\hat A)]<\infty$.
Therefore, there are $\hat{a}_i \in \hat A$ such that $\phi(\hat{a}_i) = (0, \ldots , 0 , m_i, 0 ,\ldots , 0)$, where $m_i \neq 0$ is the $i$-th entry. \end{proof}
We earlier defined the halfspaces $\overleftarrow{H}, \overrightarrow{H}$ associated to a hyperplane $H$ of $X$ to be the smallest subcomplexes containing the components $X - H$. The \emph{small halfspaces} are the largest subcomplexes contained in the two components of $X-H$.
Equivalently, the small halfspaces are the components of $X - N^o(H)$, where $N^o(H)$ is the union of open cubes intersecting $H$.
Note that each small halfspace is convex as each component of $\boundary N^o(H)$ is convex.
It is readily verified that a subcomplex of $X$ is convex if and only if it is the intersection of small halfspaces.
\begin{lem} \label{lem:convexSubproduct} Let $\widetilde X = \prod \widetilde X_i$ where each $\widetilde X_i$ is a connected CAT(0) cube complex. Then a convex subcomplex $\widetilde Y \subseteq \widetilde X$ is a product $\widetilde Y = \prod \widetilde Y_i$, where $\widetilde Y_i \subseteq \widetilde X_i$ is a convex subcomplex. \end{lem}
\begin{proof}
Let $\widetilde Y$ be a convex subcomplex of $\widetilde X$.
Each $1$-cube is the product of some $0$-cubes and a single $1$-cube in some factor $\widetilde X_i$.
An $\widetilde X_i$ hyperplane is a hyperplane which is dual to a $1$-cube arising from a factor $\widetilde X_i$.
Let $\widetilde Y_i$ be the intersection of all small halfspaces containing $\widetilde Y$ that are associated to $\widetilde X_i$ hyperplanes.
Then it is immediate that $\widetilde Y = \prod \widetilde Y_i$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:highestIntersectionSubgroups}]
By Theorem~\ref{thm:cocompact cubical flat},
$A$ acts properly and cocompactly on a convex subcomplex $\widetilde{Y} \cong \prod_{i=1}^p C_i \subseteq \widetilde{X}$.
A halfspace is \emph{shallow} if it lies in a finite neighborhood of its hyperplane,
and is \emph{deep} otherwise.
By passing to a smallest nonempty convex $A$-invariant subcomplex of $\widetilde Y$, we may assume
that no hyperplane in $\widetilde Y$ has both a shallow and a deep halfspace.
The convex hull of an $A$-invariant $p$-flat $F \subseteq \widetilde Y$ has this property.
Indeed, each hyperplane intersecting $F$ in a $(p-1)$-flat necessarily has a pair of deep halfspaces, and a hyperplane containing $F$ has two shallow halfspaces by cocompactness.
By Lemma~\ref{lem:actionDecomposition}, there is a finite index subgroup $\hat{A}=\prod_{i=1}^p \langle \hat a_i \rangle$ of $A$ such that $\langle \hat{a}_i \rangle$ acts cocompactly on $C_i$, and trivially on $C_j$ for $j \neq i$. Let $S = \{ \hat{a}_1, \ldots, \hat{a}_p \}$.
Similarly, $A'$ cocompactly stabilizes a convex subcomplex $\widetilde{Y}'\subseteq \widetilde{X}$ which has its own induced product decomposition, and there exists a corresponding finite index subgroup $\hat A' = \prod_{i=1}^{p'} \langle \hat a_i' \rangle$ that acts cocompactly on $\widetilde Y'$.
By Lemma~\ref{lem:r thickening} for each $r$ there exists a cubical $r$-thickening $(\widetilde Y')^{+r}$ containing $\mathcal N_r(\widetilde Y')$ and $(\widetilde Y')^{+r}$ is convex and $\hat{A}'$-cocompact. Choose $r$ so that $\widetilde Y \cap (\widetilde Y')^{+r} \neq \emptyset$ and note that $\widetilde Y \cap (\widetilde Y')^{+r}$ is also convex. Therefore, by Lemma~\ref{lem:convexSubproduct}, the intersection is a subproduct $\widetilde Y \cap (\widetilde Y')^{+r} \subseteq \prod D_i \subseteq \prod C_i$ where each $D_i\subset C_i$ is a convex subcomplex. Thus each factor is either a quasiline, a quasiray, or a compact convex subcomplex. Furthermore, the action of $\hat{A} \cap \hat{A}'$ on $\widetilde Y \cap (\widetilde Y')^{+r}$ is cocompact. Indeed, the intersection $\widetilde Y \cap (\widetilde Y')^{+r}$ is the universal cover of a component of the fiber product of $\hat{A} \backslash \widetilde Y \rightarrow G \backslash \widetilde{X}$ and $\hat{A}' \backslash (\widetilde Y')^{+r} \rightarrow G \backslash \widetilde X$.
For each $i$, if $D_i$ is a quasiline or compact then let $E_i=D_i$, and otherwise let $E_i$ be the compact, $\hat A \cap \hat{A}'$-invariant subcomplex contained in the intersection of all shallow halfspaces of $D_i$ that have deep complements.
Note that $D_i$ is nonempty since by finite dimensionality, $D_i$ is the intersection of finitely many shallow halfspaces whose associated hyperplanes intersect, and thus the Helly property implies the intersection is nonempty. Let $E=\prod E_i$.
If $E_i$ is a quasiline, then $\stabilizer_{\hat{A}}(E_i) = \langle \hat a_1 , \ldots, \hat a_i^{n_i}, \ldots \hat a_p \rangle$ for some $n_i > 0$ since $\hat A \cap \hat{A}'$ must act cocompactly on $E$.
Otherwise, if $E_i$ is compact, then $\stabilizer_{\hat{A}}(E_i) = \langle \hat a_1 ,\ldots , \hat a_{i-1}, \hat a_{i+1}, \ldots, \hat a_p \rangle$.
Let $S_o \subseteq S$ be the subset of $S$ such that $i \in S_o$ if $E_i$ is a quasiline.
Therefore $\stabilizer_{\hat{A}}(E)$ acts cocompactly on $E$, is commensurable to the subgroup generated by $S_o$, and contains $\hat{A} \cap \hat{A}'$.
Assume now that $r$ is large enough that $(\widetilde Y)^{+r}\cap \widetilde Y'\neq \emptyset$ and as before $(\widetilde Y)^{+r}\cap \widetilde Y'$ contains a convex subcomplex of the form $E'=\prod E_j'$ where each $E_j'$ is either a quasiline or compact, and $\stabilizer_{\hat{A}'}(E')$ acts cocompactly on $E'$ and contains $\hat{A} \cap \hat{A}'$.
Any quasiline in $E$ provides a bi-infinite sequence of nested hyperplanes. Every hyperplane in this sequence intersects $\widetilde Y'$. Indeed, if some hyperplane in the sequence intersects $(\widetilde Y')^{+r}$ but does not intersect $\widetilde Y'$, then one side of the sequence would yield hyperplanes arbitrarily far from $\widetilde Y'$,
and this contradicts that $(\widetilde Y')^{+r}$ lies within a uniform distance of $\widetilde Y'$.
We deduce that this quasiline corresponds to an entire quasiline of $\widetilde Y'$
and thus a quasiline of the subproduct $E'$.
Let $E''$ denote the subcomplex of $(\widetilde Y)^{+r} \cap (\widetilde Y')^{+r}$ obtained by intersecting it
with all halfspaces that contain $E\cup E'$. We now show that $E'' \subset \mathcal N_s(E)$ and $E''\subset \mathcal N_s(E')$ for some $s>0$.
Indeed, suppose $E''\not\subset \mathcal N_s(E)$ for each $s\geq 0$. Then for each $s$, there is a length~$s$ geodesic $\gamma_s$ in $E''$ that starts at a $0$-cube of $E$, and such that no hyperplane of $E$ intersects $\gamma_s$. Let $\{H_{si}\}_{i=1}^s$ denote the sequence of hyperplanes dual to $\gamma_s$ and let $\overrightarrow H_{si}$ denote the halfspaces containing $E$. By definition of $E''$, each $H_{si}$ either intersects $E'$ or separates $E,E'$. Note that the number of hyperplanes separating $E, E'$ equals $\textup{\textsf{d}}(E,E')$. Thus for each $s$, all but $\textup{\textsf{d}}(E,E')$ of the hyperplanes in $\{H_{si}\}_{i=1}^s$ intersect $E'$.
By finite dimensionality there is an upper bound on the number of pairwise crossing hyperplanes, and so by Ramsey's theorem, for each $t$ there exists $S(t)$, such that $\gamma_s$ is crossed by $t$ pairwise disjoint hyperplanes whenever $s\geq S(t)$. We thus obtain arbitrarily long subsequence of hyperplanes that all intersect one of the finitely many factors of $E'=\prod D'_i$. Since the factors of $E'$ are either finite or quasilines, we see that such a subsequence belongs to a quasiline of $E'$. Thus it belonged to a quasiline of $E$, as explained earlier. But all hyperplanes of a quasiline of $E$ must cross $E$, which contradicts that no $H_{si}$ crosses $E$.
We now show that $B=\stabilizer_{\hat{A}}( E)$ and $B'=\stabilizer_{\hat{A}'}( E')$
are commensurable within $G$. We have already shown that $ E'$ and $ E$ are coarsely equal,
since each is coarsely equal to $E''$. Let $\Upsilon$ denote the Cayley graph of $G$ with respect
to a finite generating set. A $G$-equivariant map $\Upsilon\rightarrow \widetilde X$ shows that $B,B'$ lie within finite neighborhoods of each other within $\Upsilon$. The right action of $B$ thus stabilizes a finite collection of right cosets of $B'$, and so $B,B'$ are commensurable.
Let $H = B \cap B'$ which is a finite index subgroup of both $B$ and $B'$. As $\hat{A}\cap \hat{A}'\leqslant B\leqslant \hat{A}$ and $\hat{A}\cap \hat{A}'\leqslant B' \leqslant \hat{A}'$, we have $H = \hat{A} \cap \hat{A}'$. Thus $\hat{A}\cap \hat{A}'$ is a finite index subgroup of $B$, hence acts cocompactly on $E$. The claim then follows from the fact that $B$ is commensurable with a subgroup generated by $S_o$, and that $A \cap A'$ is commensurable to $\hat{A} \cap \hat{A}'$. \end{proof}
A $\ensuremath{\field{Z}}^p$ subgroup with a chosen product structure has $p \choose q$ distinct commensurability classes of $\ensuremath{\field{Z}}^q$ factor subgroups. We thus have the following corollary to Theorem~\ref{thm:highestIntersectionSubgroups}: \begin{cor}\label{cor:too many intersection directions} Suppose $G$ contains a highest free-abelian subgroup $A\cong \ensuremath{\field{Z}}^p$. Suppose there are ${p\choose k} +1$ other highest free-abelian subgroups $A_1,\ldots, A_{{p\choose k} +1}$ such that the subgroups $A\cap A_i$ are pairwise non-commensurable and isomorphic to $\ensuremath{\field{Z}}^k$. Then $G$ cannot act properly and cocompactly on a CAT(0) cube complex. \end{cor}
We now illustrate Corollary~\ref{cor:too many intersection directions} in a few situations.
\begin{exmp}\label{exmp:4 Z2 subgroups} We describe an easy example of a group that acts properly on a finite dimensional CAT(0) cube complex but does not have a finite index subgroup that acts properly and cocompactly on a CAT(0) cube complex. Consider the group $G$ presented as follows: $$ G = \langle a,b,r,s,t \mid [a,b], [a,r], [b,s], [ab,t]\rangle$$ Regard $G$ as a multiple HNN extension of $\langle a,b\rangle$ with stable letters $r,s,t$, we see that $G$ is a ``tubular group'', and deduce that $G$ acts properly on a finite dimensional CAT(0) cube complex by utilizing the \emph{equitable set} $\{ a, b \}$ (see \cite{WiseGerstenRevisited} and \cite{WoodhouseTubularAction}). However, $G$ does not have a finite index subgroup $G'$ that acts properly and cocompactly on a CAT(0) cube complex. Indeed, consider the following highest free-abelian subgroups: $A=\langle a,b\rangle$, $R=\langle a,r\rangle$, $S=\langle b,s\rangle$ and $T=\langle ab, t\rangle$. The intersections $R\cap A$, $S\cap A$, and $T\cap A$ are three pairwise non-commensurable cyclic subgroups of $A$, contradicting Corollary~\ref{cor:too many intersection directions}.
Note that $G$ is a central HNN extension of the 2-dimensional right-angled Artin group $\langle a,b,r,s \mid [a,b], [a,r], [b,s]\rangle$, and so the virtually compact version of Theorem~\ref{thm:Central HNN virtually special} fails without the assumption that $H$ is highest. \end{exmp}
\begin{exmp}\label{exmp:generic}
Let $\big\{\langle b_1\rangle,\ldots,\langle b_r\rangle,\langle c_1\rangle,\ldots,\langle c_r\rangle \big\}$
be a collection of pairwise incommensurable infinite cyclic subgroups of $\ensuremath{\field{Z}}^p$,
and suppose that $r>\frac{p}{2}$. Let $G$ be the following multiple HNN extension of $\ensuremath{\field{Z}}^p =\langle a_1,\ldots, a_r\rangle$: $$G = \langle a_1,\ldots, a_p, t_1,\ldots, t_r \ \mid \ [a_i,a_j]=1, b_k^{t_k}=c_k : 1\leq k \leq r\rangle$$ Then $G$ does not contain a finite index subgroup that acts properly and cocompactly on a CAT(0) cube complex.
Indeed, the subgroups $(\ensuremath{\field{Z}}^p)^{t_i^{\pm1}}$ intersect $\ensuremath{\field{Z}}^p$ in the various subgroups
$\{\langle a_i\rangle, \langle b_i\rangle\}$ and so Corollary~\ref{cor:too many intersection directions} applies. \end{exmp}
\section{Central HNN extensions of maximal free-abelian subgroups are special} \label{CentralizingExtensions}
This section presumes familiarity with the notions of specialness and canonical completion and retraction. We refer to \cite{HaglundWiseSpecial}.
\begin{lem}\label{lemma:CentralizingCompact Y} Let $X$ be a virtually special cube complex. Let $f:Y\rightarrow X$ be a local isometry where $Y$ is a compact nonpositively curved cube complex. Let $(P, p)$ be a based graph. Let $Z=\big( X \sqcup (Y\times P) \big) / \big\{ (y,p)\sim f(y) : \forall y\in Y\big\}$. Then $Z$ is virtually special. Moreover, there is a finite special cover $\widehat Z\rightarrow Z$ such that the preimage of $X$ is connected. \end{lem}
\begin{proof}
Let $X' \rightarrow X$ be a finite degree special cover of $X$. Let $Y_i'\rightarrow X'$ be the finitely many elevations of $Y\rightarrow X$. For each $i$, let $\canon{Y_i'}{X'}$ be the canonical completion of $Y_i' \rightarrow X'$ and identify $Y_i'$ with its image in $\canon{Y_i'}{X'}$. The canonical retraction $\canon{Y_i'}{X'}\rightarrow Y_i'$ ensures that the maps $Y_i'\rightarrow \canon{Y_i'}{X'}$ are \emph{tidy} in the sense that they are injective and that no hyperplane $U$ in $\canon{Y_i'}{X'}$ \emph{interosculates} with $Y_i'$ in the sense that $U$ is dual to an edge in $Y_i'$ and is also dual to an edge that is not in $Y_i'$ but has an endpoint in $Y_i'$. Suppose that a hyperplane $U$ intersecting $Y_i'$ were dual to an edge $e$ not in $Y_i'$, but adjacent to a vertex $v \in Y_i'$. Then $v$ must be adjacent to another edge $e'$ in $Y_i'$ dual to $U$ since the retraction sends hyperplanes to hyperplanes. This implies a contradiction since the retraction must preserve the orientations of the dual edges, but $U$ cannot self-osculate.
Let $\widehat X$ be a finite degree regular cover of $X$ that factors through each $\canon{Y_i'}{X'}$. Observe that now all elevations of $Y$ to $\widehat X$ are tidy, since tidiness is stable under covers. Finally, for each elevation $\widehat Y_j \hookrightarrow \widehat X$ of $Y\rightarrow X$, we adjoin a copy of $\widehat Y_j\times P$. We thus obtain a cover $\widehat Z \rightarrow Z$. The specialness of $\widehat Z$ holds due to the tidy embeddings and a case-by-case analysis of its hyperplanes: each hyperplane $W \subset \widehat X$ has $(\widehat Y_j \cap W) \times P$ attached for each $\widehat Y_j$. Therefore no self-crossings, 1-sided hyperplanes, and no self-osculations are introduced. The tidiness of each $\canon{\widehat Y_j}{\widehat X}$ guarantees that the new hyperplanes dual to the $P$ factors cannot interosculate with any hyperplane in $\widehat X$, as each elevation factors through some $Y_i'$. \end{proof}
\begin{rem} Lemma~\ref{lemma:CentralizingCompact Y} can be generalised from the case where $P$ is a graph, to the case where $P$ is a special cube complex. \end{rem}
We will need the following technical result about right-angled Artin groups. A subgroup $A \leq G$ is \emph{isolated} if $g^p \in A$ implies that $g \in A$ for some $p \in \mathbb{Z}$.
\begin{lem} \label{lem:isolation_in_raags}
Let $M$ be an abelian subgroup of a right-angled Artin group $R$.
Suppose that $M$ is not properly contained in another abelian subgroup, then $M$ is isolated. \end{lem}
\begin{proof}
Right-angled Artin groups are biorderable \cite{DuchampThibon92}, therefore if $[g^p, h] = 1$ then $[g,h] =1$. Indeed, if $ghg^{-1}>h$ then $(ghg^{-1})^n > h^n$ for all $n$, and likewise for $ghg^{-1}<h$. We conclude that by maximality of $M$, if $g^p \in M$, then $g \in M$. \end{proof}
\begin{cor} \label{cor:maxRankImpliesHighest}
If $M$ is a maximal rank abelian subgroup of a right angled Artin group $R$, then it is a highest subgroup of $R$. \end{cor}
\begin{proof} If $M$ is virtually contained in a higher rank subgroup $M'$ of $R$, then there exists $g \in M - M'$ with $g^p \in M'$.
This contradicts the isolation of $M$, by Lemma~\ref{lem:isolation_in_raags}. \end{proof}
\begin{figure}
\caption{The local isometry $Z \rightarrow S$.}
\label{diag:Another_Diagram}
\end{figure}
\begin{thm}\label{thm:Central HNN virtually special} Let $H$ be a finitely generated virtually $[$compact$]$ special group. Let $A\subset H$ be a highest abelian subgroup. Let $G=H*_{A^t=A}$ be the HNN extension, where $t$ is the stable letter commuting with $A$, then $G$ is virtually $[$compact$]$ special. \end{thm} \begin{proof} Let $X'$ be a [compact] nonpositively curved special cube complex such that $\pi_1X'$ is isomorphic to a finite index subgroup of $H$. We may assume that $X'$ has finitely many hyperplanes since $H$ is finitely generated.
Consider the local isometry to the associated Salvetti complex $X' \looparrowright R$,
and note that $R=R(X')$ is compact since $X'$ has finitely many hyperplanes.
Let $\{g_i \}$ be a finite set of representatives of the double cosets $\{Ag\pi_1X'\}$.
Let $\{A_i\}$ be the finitely many distinct intersections $\pi_1X'\cap g_i^{-1}Ag_i$.
Each $A_i$ is highest in $H'$, since $A$ is highest in $H$.
The subgroup $A_i \hookrightarrow \pi_1R$ is contained in a maximal free-abelian group $\dot B_i \leqslant \pi_1R$, which is highest in $\pi_1R$ by Corollary~\ref{cor:maxRankImpliesHighest}. As $A_i$ is highest in $H'$ we have $[H'\cap \dot{B}_i:A_i]<\infty$. The quotient $p_i: \dot B_i / A_i \cong T\oplus \ensuremath{\field{Z}}^m$ where $|T|<\infty$. The finite index subgroup $B_i=p_i^{-1}(\ensuremath{\field{Z}}^m)$ of $\dot B_i$ is still highest in $\pi_1R$ and has the additional property that $H' \cap B_i = A_i$.
By Theorem~\ref{thm:cocompact cubical flat}, for each $i$ there exists a local isometry $F_i\rightarrow R$ with $F_i$ a compact nonpositively curved cube complex, such that $\pi_1F_i$ maps to $B_i$. For each $i$, let $Y_i'\rightarrow R$ be the fiber-product of $X'\rightarrow R$ and $F_i\rightarrow R$.
Note that by possibly replacing $F_i$ with a sufficient convex finite thickening as provided by Lemma~\ref{lem:r thickening}, we can assume that $Y_i'$ is nonempty, so that $\pi_1Y_i'=A_i$. Let $Z = X'\cup \bigcup(Y_i'\times P) \slash \sim$. Note that $\pi_1Z$ is isomorphic to a finite index subgroup of $G$ since the graph of groups for $\pi_1 Z$ covers the graph of groups of $G$. Let $S= R\cup \bigcup(F_i\times P) \slash \sim$ be the space obtained from $R$ by attaching the various $F_i\times P$ along $F_i\times \{a\}$ using the map $F_i\rightarrow R$. See Figure~\ref{diag:Another_Diagram}.
A multiple use of Lemma~\ref{lemma:CentralizingCompact Y} shows that $S$ is virtually special. There is a local isometry $Z\rightarrow S$ given by the local isometry of $X'$ into $R$ extended along the local isometry $Y_i' \times P \rightarrow F_i \times P$, and hence $Z$ is virtually special.
\end{proof}
\end{document} |
\begin{document}
\markboth{Z. Shen $\&$ J. Wei}{Spatiotemporal patterns in a delay-diffusion mussel-algae model}
\title{Spatiotemporal patterns near the Turing-Hopf bifurcation in a delay-diffusion mussel-algae model}
\author{ZUOLIN SHEN\footnote{Email: mathust\_lin@foxmail.com} ~and JUNJIE WEI\footnote{Corresponding author. Email: weijj@hit.edu.cn}\\ {\small Department of Mathematics, Harbin Institute of Technology,
{\ }}\\ {\small Harbin, Heilongjiang, 150001, P.R.China
{\ }}\\ }
\maketitle
\begin{abstract} The spatiotemporal patterns of a reaction diffusion mussel-algae system with a delay subject to Neumann boundary conditions is considered. The paper is a continuation of our previous studies on delay-diffusion mussel-algae model. The global existence and positivity of solutions are obtained. The stability of the positive constant steady state and existence of Hopf bifurcation and Turing bifurcation are discussed by analyzing the distribution of eigenvalues. Furthermore, the dynamic classifications near the Turing-Hopf bifurcation point are obtained in the dimensionless parameter space by calculating the normal form on the center manifold, and the spatiotemporal patterns consisting of spatially homogeneous periodic solutions, spatially inhomogeneous steady states, and spatially inhomogeneous periodic solutions are identified in this parameter space through some numerical simulations. Both theoretical and numerical results reveal that the Turing-Hopf bifurcation can enrich the diversity of spatial distribution of populations. {\bf Keywords}: mussel-algae system; reaction diffusion; global stability; Hopf bifurcation; delay. \end{abstract}
\section{Introduction}
Two typical features of biological systems are the complexity of their organization structure and the interactions of various factors. Mussel beds are a typical system for the study of pattern formation and patterns develop at two distinctly separate scales in mussel beds \cite{Liu-2014}, large-scale banded patterns, and small-scale net-shaped patterns. One of the models used to describe the process of large-scale patterns is \begin{subequations}\label{eq_ma} \begin{equation}\label{eq_01}
\begin{cases}
\cfrac{\partial }{\partial t}M(x,t)=D_{M}\Delta M(x,t) +e c M(x,t) \left(A(x,t) -d_{M}\cfrac{k_{M}}{k_{M}+M(x,t)}\right), \\
\cfrac{\partial }{\partial t}A(x,t)=D_{A}\Delta A(x,t)+(A_{up}-A(x,t))f-\cfrac{c}{H}M(x,t)A(x,t).\\ \end{cases} \end{equation} with the following initial data and Neumann boundary conditions \begin{equation}\label{eq_02} \begin{array}{l} \partial_{\nu}M=\partial_{\nu}A=0, ~x\in \partial\Omega, ~t>0,\\
M(x,0)=M_{0}(x)\geq 0,~A(x,0)=A_{0}(x)\geq 0, ~x\in\Omega. \end{array} \end{equation} \end{subequations} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with a smooth boundary $\partial{\Omega}$. $M(x,t)$ represents the mussel biomass density at location $x$ and time $t>0$ on the sediment, and $A(x,t)$ represents the algae concentration in the lower water layer overlying the mussel bed while $A_{up}$ describes the uniform concentration of algae in the upper reservoir water layer. Here, $e$ is a conversion constant relating ingested algae to mussel biomass production, $c$ is the consumption constant, $d_{M}$ is the maximal per capita mussel mortality rate, $k_{M}$ is the value of $M$ at which mortality is half maximal, and the mussel mortality is assumed to decrease when mussel density increases because of a reduction of dislodgment and predation in dense clumps. $f$ is the rate of exchange between the lower and upper water layers, $H$ is the height of the lower water layer, $D_{M}$ and $D_{A}$ are the diffusion coefficients of the mussel and algae, respectively. $\nu$ is the outward unit normal vector on $\partial \Omega$. The homogeneous Neumann boundary condition indicates that there is no biomass input and output at the boundary.
Such a mussel-algae model was first proposed by van de Koppel {\it et al.} \cite{Kop-2005} to investigate the importance of self-organization in affecting the emergent properties of nature systems of large spatial scales. One thing that's different from Koppel's original model is that there is no random Brownian dispersion term $D_{A}\Delta A$, but a unidirectional advection term $V\nabla A$ instead used to describe the affect of tidal current. Cangelosi {\it et al.} \cite{Can} modified the model to the way it is now, and the modification is an extension of original model which as a first approximation to the field experiment of van de Koppel {\it et al.} \cite{Kop-2008} and Liu {\it et al.} \cite{Liu-2013} and in exact accordance with their laboratory experiment. Both models (original and modified) have been discussed by scholars, see \cite{Liu-2012, Liu-2014, ShM, WLS}. Ghazaryan and Manukian \cite{GhM} have captured the nonlinear mechanisms of pattern and wave formation of Koppel's original model by applying the geometric singular perturbation theory. Sherratt and Mackenzie \cite{ShM} have considered the implications of the algae's advection for pattern formation with the advection oscillating with tidal flow. Based on the normal form method, Song {\it et al.} \cite{SJL} have studied the Turing-Hopf bifurcation of \eqref{eq_ma} with a Neumann boundary conditions, and obtained the explicit dynamical classification in the corresponding critical point.
As is well known that delay can lead to the periodic solutions \cite{ChY, XuW}, while diffusion can cause Turing patterns \cite{Kla, NiT, OuS, Tur}. An obvious idea is how their interaction will affect the dynamics of the system. In this paper, we mainly study the following delay-diffusion mussel-algae system \begin{equation}\label{eq_MA_tau}
\begin{cases}
\cfrac{\partial }{\partial t}M(x,t)=D_{M}\Delta M(x,t) +e c M(x,t) \left(A(x,t-\tau) -d_{M}\cfrac{k_{M}}{k_{M}+M(x,t-\tau)}\right), \\
\cfrac{\partial }{\partial t}A(x,t)=D_{A}\Delta A(x,t)+(A_{up}-A(x,t))f-\cfrac{c}{H}M(x,t)A(x,t).\\
\partial_{\nu}M=\partial_{\nu}A=0, ~x\in \partial\Omega, ~t>0,\\
M(x,t)=M_{0}(x,t)\geq 0,~A(x,t)=A_{0}(x,t)\geq 0, ~x\in\Omega,~-\tau\leq t\leq 0. \end{cases} \end{equation} where $\tau$ is the digestion period of mussel and the mortality of mussels depends on the state whether they have eaten in the past. By employing the rescaling $$ \begin{array}{l} m=\cfrac{M}{k_{M}},~a=\cfrac{A}{A_{up}},~\omega=\cfrac{c k_{M}}{H},~\hat{t}=d_{M}t,~\alpha=\cfrac{f}{\omega},\\ r=\cfrac{e c A_{up}}{d_{M}},~\gamma=\cfrac{d_{M}}{\omega},~d=\cfrac{D_{M}}{\gamma D_{A}},~\hat{x}=x\sqrt{\cfrac{\omega}{D_{A}}}.\\ \end{array} $$ we have \begin{equation}\label{eq_ma_tau}
\begin{cases}
\cfrac{\partial }{\partial t}m(x,t)=d\Delta m(x,t)+m(x,t)\Big(ra(x,t-\tau) -\cfrac{1}{1+m(x,t-\tau)}\Big), &x\in\Omega, ~t>0,\\
\gamma \cfrac{\partial}{\partial t} a(x,t)=\Delta a(x,t)+\alpha(1-a(x,t))-m(x,t)a(x,t),&x\in\Omega, ~t>0,\\ \partial_{\nu}m=\partial_{\nu}a=0, &x\in\partial\Omega, ~t>0,\\
m(x,t)=m_{0}(x,t)\geq 0,~a(x,t)=a_{0}(x,t)\geq 0, &x\in\Omega, ~-\tau\leq t\leq 0. \end{cases} \end{equation} For simplicity, we have drop the `~$\hat{}$~'.
This paper is a continuation of our previous studies on delay-diffusion mussel-algae model. We mainly concern the spatiotemporal dynamics of Eq.\eqref{eq_ma_tau} near the Turing-Hopf bifurcation point with $\tau$ and $d$ as the bifurcation parameters. The study of Turing-Hopf bifurcation is not a new topic \cite{BGF, DLDB, HaR, SoZ, YaS}. Most of the studies have focused on the emergence of spatiotemporal patterns or the non-degenerate cases, but not many have been done on degenerate cases (Hopf bifurcation and Turing instability occur simulta neously). Recently, An and Jiang \cite{AnJ} extend the normal form methods proposed by Faria \cite{Far} to Turing-Hopf singularity of a general two-components delayed reaction diffusion system, and present a detailed calculation formulas. Motivated by their work, we study the spatiotemporal dynamics of system \eqref{eq_ma_tau}. Compared with the work of \cite{ShW-2, SJL}, we focus more on the common effects of delay and diffusion. Hence, a basic assumption is that the positive constant steady state is locally asymptotically stable under a homogeneous perturbation when time delay is equal to zero, and this assmption allows us to identify the importance of delay and diffusion in the process of pattern formation. The main contribution of this article can be concluded as: first, the proof of wellposedness of system \eqref{eq_ma_tau}; second, a detail bifurcation analysis with $\tau$ and $d$ as the bifurcation parameters; third, we show a rational explanation of different spatiotemporal distribution of mussel beds from both theoretical results and numerical simulations.
The rest of this paper is organized as follows. In section 2, we firstly give the proof of wellposedness of solutions, then study the stability of positive constant steady state including the existence of the Hopf bifurcation, Turing instability, and Turing-Hopf interaction. We take $\tau$ and $d$ as the bifurcation parameters which can reflect their effect on the dynamics of the system. In section 3, we show a detailed formulas for calculating the normal form of system \eqref{eq_ma_tau} with the method proposed by \cite{AnJ}. In section 4, we discuss the dynamic classification and spatiotemporal patterns near the Turing-Hopf bifurcation point, and for each dynamic region, some numerical simulations are presented to illustrate our theoretical analysis. In section 5, we end this paper with conclusions and some discussions about the following work of this model. Throughout the paper, we denote $\mathbb{N}$ as the set of positive integers, and $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ as the set of non-negative integers.
\section{Existence and stability analysis}
\subsection{Existence and boundedness} In this subsection, we first state the wellposedness result of the solutions of the initial value problem \eqref{eq_ma_tau}, for more details of abstract theory, refer to \cite{Pao-1996, Tay}.
\begin{theorem}\label{th-exis} Suppose that $\alpha$, $\gamma$, $r$ and $d$ are all positive, the initial data satisfies $m_{0}(x,t)\geq 0, a_{0}(x,t)\geq 0$ for $(x,t)\in \overline{\Omega}\times[-\tau,0]$. Then the system \eqref{eq_ma_tau} has a unique solution $(m(x,t),a(x,t))$ satisfying
$$0\leq m(x,t),~~0\leq a(x,t)\leq \max\{\|a_0\|_{\infty}, 1\}~~ \text{for}~~ (x,t)\in \overline{\Omega}\times[0,+\infty).$$
where $\|\psi\|_{\infty}=\sup_{x\in\overline{\Omega},t\in[-\tau,0]} \psi(x,t)$. Moreover, if $m_{0}(x,0) \not\equiv 0, a_{0}(x,0) \not\equiv 0$, then $m(x,t)>0, a(x,t)>0$ for $(x,t)\in \overline{\Omega}\times(0,+\infty)$. \end{theorem}
\begin{proof} Define $F=\Big(f(m,a,m_{\tau},a_{\tau}),g(m,a,m_{\tau},a_{\tau})\Big)^T$ with $$ f(m,a,m_{\tau},a_{\tau})=rma_{\tau} -\cfrac{m}{1+m_{\tau}}, ~g(m,a,m_{\tau},a_{\tau})=\alpha(1-a)-ma. $$ where $m=m(x,t)$, $a=a(x,t)$, $m_{\tau}=m(x,t-\tau)$, $a_{\tau}=a(x,t-\tau)$.
It is easy to prove that $F$ possesses a mixed quasi-monotone property since $D_af=0, D_{m_{\tau}}f>0, D_{a_{\tau}}f>0$ and $D_mg<0, D_{m_{\tau}}g= D_{a_{\tau}}g=0$ for $(m,a,m_{\tau},a_{\tau})\in \mathbb{R}^4_+$.
Let $(m^*(t),~a^*(t))$ be the unique solution of the following ODE system \begin{equation}\label{eq_ode} \begin{cases}
\cfrac{\text{d}m}{\text{d}t}=rma -\cfrac{m}{1+m},\\
\gamma \cfrac{\text{d}a}{\text{d}t}=\alpha(1-a), \\
m(0)=\phi_1, a(0)=\phi_2. \end{cases} \end{equation} with $$
\phi_1=\sup_{x\in\overline{\Omega},t\in[-\tau,0]}m_0(x,t), ~~~~\phi_2=\sup_{x\in\overline{\Omega},t\in[-\tau,0]}a_0(x,t); $$ Denote $(\widetilde{m}, \widetilde{a})=(m^*(t),~a^*(t))$, $(\widehat{m}, \widehat{a})=(0, 0)$. Since $$ \cfrac{\partial \widetilde{m}}{\partial t}-d\Delta\widetilde{m}-r\widetilde{m}\widetilde{a}-\cfrac{\widetilde{m}}{1+\widetilde{m}}=0\geq 0= \cfrac{\partial \widehat{m}}{\partial t}-d\Delta\widehat{m}-r\widehat{m}\widehat{a}-\cfrac{\widehat{m}}{1+\widehat{m}}, $$ $$ \gamma\cfrac{\partial \widetilde{a}}{\partial t}-\Delta\widetilde{a}-\alpha (1-\widetilde{a})-\widehat{m}\widetilde{a}=0\geq -\alpha= \gamma\cfrac{\partial \widehat{a}}{\partial t}-\Delta\widehat{a}-\alpha(1-\widehat{a})-\widetilde{m}\widehat{a}. $$ and $$ 0\leq m_0(x,t)\leq \phi_1, ~0\leq a_0(x,t)\leq \phi_2 ~~\text{for}~~ (x,t)\in \Omega\times[-\tau,0]. $$
Then $(\widetilde{m}, \widetilde{a})$ and $(\widehat{m}, \widehat{a})$ are the coupled upper and lower solutions of system \eqref{eq_ma_tau}. Hence, from Theorem 2.1 in \cite{Pao-1996}, the system \eqref{eq_ma_tau} has a unique solution $(m(x,t), a(x,t))$ which satisfies $$ 0\leq m(x,t)\leq m^*(t),~~0\leq a(x,t)\leq a^*(t)$$
Applying the comparison principle to the second equation of system \eqref{eq_ma_tau}, we can easily get $a(x,t)\leq \max\{\|a_0\|_{\infty}, 1\}$. To prove the positivity, we set $t\in[0,\tau]$, then $m_{\tau}$, $a_{\tau}$ coincide with the initial data $m_0(x,t-\tau), a_0(x,t-\tau)$. Since $m_{0}(x,0)\not\equiv 0$, $a_{0}(x,0)\not\equiv 0$, then $m(x,t)>0, a(x,t)>0$ for $(x,t)\in \Omega\times (0,\tau]$ from the standard maximum principle for semilinear parabolic equations. Repeating this process, we can obtain that $m(x,t)>0$, $a(x,t)>0$ for $(x,t)\in \Omega\times (0,\infty)$. \end{proof}
\subsection{Stability analysis}\label{linear and Hopf}
For the convenience of further discussion, we first define the following real-value Sobolev space
$$ X:=\Big\{(u,v)\in H^2(\Omega)\times H^2(\Omega)|\partial_{\nu}u=\partial_{\nu}v=0, x\in \partial\Omega\Big\} $$
and its complexification, $X_{\mathbb{C}}:=X\oplus iX=\{x_1+i x_2|x_1,x_2 \in X\}$ with a complex-valued $L^2$ inner product $<\cdot,\cdot>$ which defined as $$ <U_1,U_2>=\int_{\Omega}(\bar{u}_1u_2+\bar{v}_1v_2)dx $$ with $U_i=(u_i, v_i)^T \in X_{\mathbb{C}}, i=1,2$.
The system \eqref{eq_ma_tau} always has a non-negative constant solution $E_0(0,1)$ which corresponds to the bare sediment biologically, and the system also has a positive equilibrium $E_*(m^*,a^*)$ with $m^*=\cfrac{\alpha (r-1)}{1-\alpha r}, a^*=\cfrac{1-\alpha r}{r(1-\alpha)}$ if the following assumption satisfies: $$ \textsc{(H1)}~~~~~~\qquad~~~ 0<\alpha<1<r<\alpha^{-1}.~~~~~~~~ $$
Suppose that the spatial domain $\Omega=(0,l\pi)$, that is $\Omega$ is an interval in one space dimension. Here let the phase space $\mathscr{C}:=C([-\tau,0],X_{\mathbb{C}})$. Our main focus is the stability of positive constant steady state $E_*(m^*,a^*)$ with respect to the model \eqref{eq_ma_tau}, and the results of the boundary steady state $E_0(0,1)$ can be seen in \cite{ShW-2}.
The linearization of system \eqref{eq_ma_tau} at $E_*(m^*,a^*)$ is given by \begin{equation}\label{eq_linear} \dot{U}(t)=D\Delta U(t)+L(U_t), \end{equation} where $D= \text{diag}(d, \gamma^{-1})$, and $L:\mathscr{C}\to X_{\mathbb{C}}$ is defined as $$ L(\phi)=L_1\phi(0)+L_2\phi(-\tau), $$ with $$\begin{array}{l} L_1=\gamma^{-1}\left( \begin{array}{cc}
0 ~&~ 0\\
-a^* ~&~ -(\alpha+m^*)
\end{array}\right),~~ \quad L_2=\left( \begin{array}{cc}
\cfrac{m^*}{(1+m^*)^2} ~&~ r m^* \\
0 ~&~ 0
\end{array}\right), \end{array}$$ $$ \phi(t)=\big(\phi_1(t),~\phi_2(t)\big)^{^T},~~\phi_t(\cdot)=\big(\phi_1(t+\cdot),~\phi_2(t+\cdot)\big)^{^T}. $$
It is well known that the eigenvalue problem $$ -\Delta \xi=\sigma \xi,~~x\in(0,l\pi),~~ \xi'(0)=\xi'(l\pi)=0 $$ has eigenvalues $\sigma_n=\frac{n^2}{l^2}$, $n\in\mathbb{N}_0$, with corresponding eigenfunctions $\xi_n(x)=\cos\frac{n}{l}x$. Let $U(x,t)=e^{\lambda t}\xi(x)$, we have that the corresponding characteristic equation of system \eqref{eq_linear} satisfies \begin{equation}\label{c-eq1}
\lambda \xi-D\Delta \xi-L(e^{\lambda\,\cdot}\xi)=0, \end{equation} Then \eqref{c-eq1} can be transform into $$\det\Big(\lambda \text{I}+D\cfrac{n^2}{l^2}-L_1-L_2 e^{-\lambda\tau}\Big)=0,~~n\in\mathbb{N}_0.$$ That is, there exists some $n\in \mathbb{N}_0$ such that $\lambda$ satisfies the following characteristic equation \begin{equation}\label{c-eq2}
E_n(\lambda,\tau,d):=\gamma\lambda^2+T_n\lambda+(B\lambda+M_n)e^{-\lambda\tau}+D_n=0, ~~n\in\mathbb{N}_0, \end{equation} where \begin{equation}\label{TDMB} \begin{array}{l}
T_n=\alpha+m^*+(1+\gamma d)\cfrac{n^2}{l^2}, \quad M_n=ra^*m^*(1-\alpha r-ra^*\cfrac{n^2}{l^2});\\
D_n=d(\alpha+m^*+\cfrac{n^2}{l^2})\cfrac{n^2}{l^2},\quad B=-\gamma r^2{a^*}^2m^*. \end{array} \end{equation}
In the following, we analyze the existence of Turing-Hopf bifurcation for the positive constant steady state. In order to understand how delay and diffusion coefficient affect the Turing-Hopf bifurcation, we choose $\mu=(\tau, d)$ as the bifurcation parameters since Turing-Hopf is a codimension-two bifurcation. For a general case, $\mu=(\mu_1,\mu_2)\in\mathbb{R}^2$, the conditions for the occurrence of Turing-Hopf bifurcation can be described as:
\textbf{(TH)} There exists a neighborhood $\mathscr{N}(\mu_0)$ of $\mu_0=(\mu_{10}, \mu_{20})$, and $n_1, n_2 \in \mathbb{N}_0$ such that characteristic equation \eqref{c-eq2} has a pair of complex simple conjugate eigenvalues $\beta_{n_1}(\mu)\pm i\omega_{n_1}(\mu)$ and a simple real eigenvalue $\alpha_{n_2}(\mu)$ for $\mu\in\mathscr{N}(\mu_0)$, both continuously differentiable in $\mu$, and satisfy $\beta(\mu_0)=0, \omega(\mu_0)=\omega_0>0, \frac{\partial}{\partial{\mu_1}}\beta(\mu_0)\neq0, \alpha(\mu_0)=0, \frac{\partial}{\partial{\mu_2}}\alpha(\mu_0)\neq 0$; all other eigenvalues have non-zero real parts.
In our previous paper \cite{ShW-2}, it has been proved that, the system \eqref{eq_ma} without diffusion can undergo Hopf bifurcation when parameters are chosen appropriately.
\begin{lemma}\cite{ShW-2}\label{hopf} Assume that $\textsc{(H1)}$ is satisfied. For system \eqref{eq_ma} without diffusion, \begin{enumerate}
\item If $\mathcal{H}_{0}^2(r)<\mathcal{P}_{0}(r)$, the positive equilibrium $E_*(m^*,a^*)$ of system \eqref{eq_ma} is locally asymptotically stable;
\item If $\mathcal{H}_{0}^2(r)>\mathcal{P}_{0}(r)$, the positive equilibrium $E_*(m^*,a^*)$ of system \eqref{eq_ma} is unstable;
\item If $r_{_H}\in S$ satisfies the equation $\mathcal{H}_{0}^2(r)=\mathcal{P}_{0}(r)$, the system \eqref{eq_ma} undergoes a Hopf bifurcation at $r=r_{_H}$ which corresponds to spatially homogeneous periodic solution; the critical curve of Hopf bifurcation is defined by $\mathcal{H}_{0}^2(r)=\mathcal{P}_{0}(r)$, where \\
$\mathcal{H}_{0}(r)=\cfrac{1-\alpha r}{1-\alpha}$, $\mathcal{P}_{0}(r)=\cfrac{r(1-\alpha)}{\gamma(r-1)}$.
\end{enumerate} \end{lemma}
Lemma \ref{hopf} indicated that the positive equilibrium $E_*(m^*,a^*)$ of system \eqref{eq_ma} is stable to homogeneous perturbations when $\mathcal{H}_{0}^2(r)<\mathcal{P}_{0}(r)$. Since system \eqref{eq_ma} is a special case when $\tau=0$ of system \eqref{eq_ma_tau}, our next work is to discuss the stability of $E_*(m^*,a^*)$ when $\tau>0$. To ensure our stability analysis valid, we make the following assumption: $$ \textsc{(H2)}~~~~~~\quad~~~ \mathcal{H}_{0}^2(r)<\mathcal{P}_{0}(r).~~~~~~~~ $$
Hence, we let $\pm i\omega(\omega>0)$ be solutions of Eq.\eqref{c-eq2}, then we have
$$-\gamma\omega^2+i T_n \omega+(i\omega B+M_n)e^{-i\omega\tau}+D_n=0.$$ Separating the real and imaginary parts, it follows that \begin{equation}\label{re_im} \begin{cases} M_n\cos\omega\tau +\omega B\sin\omega\tau= \gamma\omega^2-D_n,\\ M_n\sin\omega\tau -\omega B\cos\omega\tau =T_n\omega . \\ \end{cases} \end{equation} that is \begin{equation}\label{omega} \gamma^2\omega^4+(T_n^2-2\gamma D_n-B^2)\omega^2+D_n^2-M^2_n=0. \end{equation} Let $z=\omega^2$. Then \eqref{omega} can be converted to
\begin{equation}\label{z_eq} \gamma^2 z^2+(T_n^2-2\gamma D_n-B^2)z+D_n^2-M^2_n=0. \end{equation} where $T_n^2-2\gamma D_n-B^2>0$ can be deduced by \textsc{(H2)}. Solving Eq.\eqref{z_eq} for $z$, we have \begin{equation}\label{z_n} z_n=\cfrac{-(T_n^2-2\gamma D_n-B^2)+\sqrt{(T_n^2-2\gamma D_n-B^2)^2-4\gamma^2(D_n^2-M_n^2)}}{2\gamma^2}, \end{equation}
Clearly, $D_0+M_0>0$, $D_0-M_0<0$, then $z_0=\omega_0^2$ is always exists, and $(\omega_0, \tau^j_0)$ always satisfies the characteristic equation \eqref{c-eq2}. This corresponds to a spatially homogeneous Hopf bifurcation. In the following, we shall look for the spatially inhomogeneous Hopf bifurcation. Note that \begin{equation} D_n+M_n=d\cfrac{n^4}{l^4}+(d\cfrac{\alpha}{a^*}-r^2 a^{*2}m^*)\cfrac{n^2}{l^2}+ar(r-1)a^* \end{equation} we can always choose a set of parameters ${d, \alpha, r}$ appropriately such that $D_n+M_n>0$ for all $n\in \mathbb{N}, l>0$. Hence, denote \begin{equation*}
\Gamma=\left\{(d,\alpha,r)|~~ D_n+M_n>0~\text{for all}~n\in\mathbb{N}, l>0\right\} \end{equation*} Now, the existence of $z_n$ is determined by the signal of $D_n-M_n$ when $(d,\alpha,r)\in \Gamma$. If $D_n-M_n<0$, then the (n+1)th equation of \eqref{c-eq2} has a pair of simple pure imaginary $\pm i\omega_n$, and if $D_n-M_n>0$, the (n+1)th equation of \eqref{c-eq2} has no pure imaginary.
Define \begin{equation}\label{l_n} l_n=n\cfrac{1}{\sqrt{S(d,\alpha,r)}},~~~~ n\in\mathbb{N}. \end{equation} where \begin{equation*}
S(d, \alpha,r)= -\cfrac{1}{2}\left(\frac{\alpha}{a^*}+\cfrac{ r^2a^{*2}m^*}{d}\right)+ \cfrac{1}{2d}\sqrt{(d\frac{\alpha}{a^*}+r^2a^{*2}m^*)^2+4d\alpha r (r-1)a^*}. \end{equation*} Then for $l_n<l<l_{n+1}$, and $1\leq {n_1}\leq n$, we have $$ \cfrac{n_1^2}{l^2}<S(d, \alpha,r) $$ which yields to $D_{n_1}-M_{n_1}<0$. Hence, we can find a series of root $z_{{n_1}}$ of Eq.\eqref{z_n} and critical values $\tau_{{n_1}}^j$ satisfies
\begin{equation}\label{tau} \tau_{{n_1}}^j=\begin{cases} \cfrac{1}{\omega_{n_1}}\Big(\arccos\cfrac{(\gamma M_{{n_1}}-BT_{{n_1}} )\omega_{{n_1}}^{2}-M_{{n_1}}D_{n_1}}{M_{n_1}^2+\omega_{n_1}^2B^2}+2j\pi\Big),&\sin\omega_{n_1}\tau_{n_1}^j >0 \\ \cfrac{1}{\omega_{n_1}}\Big(-\arccos\cfrac{(\gamma M_{n_1}-BT_{n_1} )\omega_{n_1}^{2}-M_{n_1}D_{n_1}}{M_{n_1}^2+\omega_{n_1}^2B^2}+2(j+1)\pi\Big), &\sin\omega_{n_1}\tau_{n_1}^j <0
\end{cases}~~0\leq{n_1}\leq n,~j\in\mathbb{N}_0. \end{equation} such that Eq.\eqref{c-eq2} has a pair of purely imaginary roots $\pm i\omega_{n_1}$.
Following the work of \cite{Cooke}, it is easy to verify that the following transversality condition holds. \begin{lemma}\label{trans_tau} Suppose that \textsc{(H1)} and \textsc{(H2)} are satisfied, $(d,\alpha,r)\in \Gamma$, and $l\in(l_n, l_{n+1}]$ with $l_n$ is defined as in \eqref{l_n}. Then $$ \cfrac{\partial}{\partial\tau}\beta(\tau^j_{n_1},d)>0,~~for~0\leq {n_1}\leq n, j\in\mathbb{N}_0, $$ where $\beta(\tau,d)= ~\textrm{Re}~ \lambda(\tau,d)$. \end{lemma} \begin{proof} Substituting $\lambda(\tau,d)$ into Eq.\eqref{c-eq2} and taking the derivative with respect to $\tau$ on both side, we obtain that $$ \Big(2\gamma\lambda+T_{n_1}+Be^{-\lambda\tau}-\tau(B\lambda+M_{n_1})e^{-\lambda\tau}\Big)\cfrac{\text{d}\lambda}{\text{d}\tau}-\lambda(B\lambda+M_{n_1})e^{-\lambda\tau}=0. $$ Thus $$ \left(\cfrac{\text{d}\lambda}{\text{d}\tau}\right)^{-1}=\cfrac{2\gamma\lambda+T_{n_1}+Be^{-\lambda\tau}-\tau(B\lambda+M_{n_1})e^{-\lambda\tau}}{\lambda(B\lambda+M_{n_1})e^{-\lambda\tau}}. $$ By Eq.\eqref{c-eq2} and Eq.\eqref{re_im}, we have \begin{equation*} \begin{array}{ll}
\text{Re}\Big(\cfrac{\text{d}\lambda}{\text{d}\tau}\Big)^{-1}\Big|_{\tau =\tau_{n_1}^j} &=\text{Re}\Big[\cfrac{(2\gamma\lambda+T_{n_1})e^{\lambda\tau}}{\lambda(B\lambda+M_{n_1})}+\cfrac{B}{\lambda(B\lambda+M_{n_1})}\Big]_{\tau =\tau_{n_1}^j}\\ &=\text{Re}\Big[\cfrac{(2\gamma\lambda+T_{n_1})e^{\lambda\tau}}{-\lambda(\gamma\lambda^2+T_{n_1}\lambda+D_{n_1})}+\cfrac{B}{\lambda(B\lambda+M_{n_1})}\Big]_{\tau =\tau_{n_1}^j}\\ &=\cfrac{2\gamma^2\omega_{n_1}^2-2\gamma D_{n_1}+T^2_{n_1}}{(\gamma\omega_{n_1}^2-D_{n_1})^2+\omega_{n_1}^2 T_{n_1}}+\cfrac{-B^2}{B^2\omega_{n_1}^2+M_{n_1}^2}\\ &=\cfrac{\sqrt{(T_{n_1}^2-2\gamma D_{n_1}-B^2)^2-4\gamma^2(D_{n_1}^2-M_{n_1}^2)}}{B^2\omega_{n_1}^2+M_{n_1}^2}. \end{array} \end{equation*} Since $\text{Sign} ~\beta(\tau,d)=\text{Sign} ~\beta^{-1}(\tau,d)$, the lemma follows immediately. \end{proof}
Let $\tau_0$ be the smallest value of $\tau^j_{n_1}$, that is $$ \tau_0=\min\{\tau_{n_1}^j(l), 0\leq{n_1}\leq n, j\in \mathbb{N}_0, ~\text{and}~l\in(l_n, l_{n+1}] ~\text{is defined as}~ \eqref{l_n}\}. $$
Summarizing the above analysis, we have the following result. \begin{theorem} Suppose that \textsc{(H1)} and \textsc{(H1)} are satisfied, $(d,\alpha,r)\in \Gamma$, and $l_n$ is defined as in \eqref{l_n}. Then
\begin{enumerate}
\item If $l\in(l_n, l_{n+1}]$, there exists $n+1$ series of points $\{\tau_{n_1}^j\}$ such that the system \eqref{eq_ma_tau} undergoes a Hopf bifurcation at $\tau=\tau_{n_1}^j, 0\leq{n_1}\leq n$, $j\in\mathbb{N}_0$.
\item Moreover, all the roots of Eq.\eqref{c-eq2} have negative real parts for $\tau\in[0,\tau_0)$, and Eq.\eqref{c-eq2} has at least one pair of conjugate complex roots with positive real parts for $\tau>\tau_0$. Especially for $l\leq l_1$, the Hopf bifurcation only occurs when $\tau=\tau_{0}^j ,j\in\mathbb{N}_0$ which corresponds to a spatially homogeneous periodic solution.
\end{enumerate} \end{theorem}
\begin{remark}
The condition \textsc{(H2)} ensure that the positive spatially homogeneous steady state is stable to a linear homogeneous perturbation when $\tau=0$. That is, the Hopf bifurcation was entirely induced by delay $\tau$. Biologically, the population will have a periodic oscillation if the digestion period $\tau$ is greater than a critical value $\tau_0$. \end{remark}
For the Turing instability to be realized and the spatial patterns to form, the real part of eigenvalue $\lambda$ of \eqref{c-eq2} must be greater than zero for some $n\neq 0$, moreover, there exists a real eigenvalue $\lambda^T$ pass through the origin from the left side of the complex plane to the right side. That is, if the system undergoes a Turing bifurcation, then the characteristic equation has a simple zero eigenvalue. Hence, Eq.\eqref{c-eq2} can be written as \begin{equation}\label{DM} h(d,n^2):=D_n+M_n=0 \end{equation} Clearly, Eq.\eqref{DM} is a quadratic equation with $n^2$, the critical $n^2$ can be obtained by the following formula \begin{equation}\label{n_c} n_2^2=\cfrac{l^2}{2d}\left(\cfrac{m^*}{(1+m^*)^2}-\cfrac{d \alpha}{a^*}\right). \end{equation} and the steady state is marginally stable at $n=n_2$ when \begin{equation}\label{d_nc} h(d,n_2^2)=0 \end{equation} Solving \eqref{d_nc} for $d$, we can get that \begin{equation}\label{d_0} d_0(\alpha,r)=\cfrac{\alpha(r-1)(1-\alpha r)^2}{(1-\alpha)^3(2\sqrt{1-\alpha r}+2-\alpha r)} \end{equation}
Now we are in the position to investigate the Turing instability that driven by diffusion coefficient $d$. Using the similar method in Lemma \ref{trans_tau}, we can obtain the following transversality without difficulty. \begin{lemma}\label{trans_d} Suppose that \textsc{(H1)} and \textsc{(H2)} are satisfied. Then $$ \cfrac{\partial}{\partial d}\alpha(\tau, d_0)<0. $$ where $\alpha(\tau, d)$ is the real eigenvalue of the characteristic equation \eqref{c-eq2} . \end{lemma} \begin{lemma}\label{d_lemma} Suppose that \textsc{(H1)} and \textsc{(H2)} are satisfied. Then \begin{enumerate}
\item If $d>d_0(\alpha, r)$, there is no Turing instability;
\item If $d<d_0(\alpha, r)$, there exists at least one $n\in\mathbb{N}$ such that $h(d,n^2)>0$, and the system undergoes a Turing bifurcation at $d=d_0$. \end{enumerate} \end{lemma} \begin{proof} It is easy to see $h(d, n_2^2)>0$ from the definition \eqref{DM} and \eqref{d_nc} when $d>d_0$ and $h(d, n_2^2)<0$ when $d<d_0$. \end{proof}
\begin{remark}
Noting that $d$ is only the diffusion coefficient of mussel, while the diffusion coefficient of algae is rescaled to $\frac{1}{\gamma}$. Lemma \ref{d_lemma} indicates that if mussel diffusivity is sufficiently large, there is no spatial patterns, but if it less than the threshold, Turing instability will happenㄛ and we shall observe the spatial distribution of the two species. This result is also suitable for high dimensional space where the patterns are more complicated and interesting. \end{remark}
The following Turing-Hopf bifurcation theorem is a direct result of the previous analysis. \begin{theorem}\label{theorem_TH}
Assume that \textsc{(H1)} and \textsc{(H2)} are satisfied, and $l\in(l_n, l_{n+1}]$ with $l_n$ is defined as in \eqref{l_n}. Then \begin{enumerate}
\item the constant steady state $E_*(m^*,a^*)$ is locally asymptotically stable when $\tau<\tau_0$ and $d>d_0$.
\item the $(n_1+1)$th equation of \eqref{c-eq2} has a pair of simple pure imaginary roots $\pm i\omega_{n_1}$, the $(n_2+1)$th equation of \eqref{c-eq2} has a simple zero when $\tau=\tau_{n_1}^j, d=d_0$, $j\in\mathbb{N}_0$, with $d_0$ is defined by \eqref{d_0}, $n_2$ is defined by \eqref{n_c} and $n_1$ is define as $0\leq n_1\leq n$, if $n_2>n$ or $0\leq n_1\leq n$, $n_1\neq n_2$, if $n_2\leq n$, and all other eigenvalues have non-zero real parts.
\item the system \eqref{eq_ma_tau} undergoes a Turing-Hopf bifurcation at $(\tau_{n_1}^j, d_0)$, where $n_1$ is well defined in (2). \end{enumerate}
Moreover, if $l\leq l_1$, then the characteristic equation \eqref{c-eq2} only has a pair of imaginary roots $\pm i\omega_0$ with $n_1=0$ and a simple zero with $n_2>0$, and all other eigenvalues with $(\tau, d)=(\tau_0, d_0)$ have strictly negative real parts.
\end{theorem} \begin{remark} Since $\cfrac{\partial}{\partial \lambda}E_{n_2}(0,\tau,d_0)=T_{n_2}+b-\tau M_{n_2}>0$ for any $\tau>0$, then $\lambda(\tau,d_0)=0$ is a simple root of characteristic equation \eqref{c-eq2}. This is determined by the model, in other words, $0$ may be a eigenvalue with multiplicity two for some other models(see \cite{AnJ}), in that case, there might exist a $\tau^*$ such that $E_{n_2}(0,\tau^*,d_0)=0$ and $\cfrac{\partial}{\partial \lambda}E_{n_2}(0,\tau^*,d_0)=0$ , and if other eigenvalues have non-zero real part, the system will undergoes a Bogdanov-Takens bifurcation or even a Turing-Turing-Hopf bifurcation at $(\tau^*, d_0)$ . \end{remark}
\begin{figure}
\caption{(a) The Hopf bifurcation curves and stable region in $d-\tau$ plane, the solid lines are Hopf bifurcation curves with $n=0, 1, 2$ from bottom to top respectively, and values of parameters are chosen as follows: $\gamma=4, r=1.1, \alpha=0.65$; (b) The critical curve of Turing bifurcation and unstable region in $\alpha-d$ plane with $r=1.1$.}
\label{fig-H-T}
\end{figure}
Clearly, $\tau^0_{n_1}<\tau^j_{n_1}$ for all $j>0$, and through a mass of numerical simulations, we have observed the trend of $\tau^0_{n_1}$ as $n_1$ get bigger. The result reveals that the smallest value $\tau_0$ is always obtained when $n_1=0$. Fig.\ref{fig-H-T}(a) is the geometric interpretation under the set of parameters that we used to run the numerical solutions in Section \ref{dynamic classifiction}. Hence, the first Turing-Hopf bifurcation point $(\tau_0, d_0)$ is $(\tau^0_0, d_0)$, our results below is the detailed analysis about this point and its neighborhood.
\section{Normal form of Turing-Hopf bifurcation}
In this section, we shall study the spatiotemporal dynamics of system \eqref{eq_ma_tau} by using the center manifold reduction \cite{LSW, Wu} and normal form theory \cite{AnJ, Far, SJL}. The amplitude equations are finally obtained to describe to dynamics near the critical Turing-Hopf bifurcation point, the truncated normal form is exactly the same to that of the ODE system with Hopf-Hopf bifurcation. In what follows, we will give a specific process and some explicit calculation formulas.
Let $\widetilde{m}(x,t)=m(x,t)-m^*$, $\widetilde{a}(x,t)=a(x,t)-a^*$, and $t\mapsto t/\tau$, dropping the tilde, then we have \begin{equation}\label{eq_origin} \begin{cases} \cfrac{\partial m}{\partial t}=\tau[d\Delta m+r^2a^{*2}m^* m_t(-1)+r m^*a_t(-1)+f_1(m_t,a_t)], & x\in\Omega,~t>0,\\ \gamma \cfrac{\partial a}{\partial t}=\tau[\Delta a-a^*m-(\alpha+m^*)a+f_2(m_t,a_t)], & x\in\Omega,~t>0,\\ \cfrac{\partial m}{\partial n_1}=0,~\cfrac{\partial a}{\partial {n_1}}=0, & x\in\partial \Omega,~t>0,\\
m(x,t)=m_0(x,t)-m^*,~ a(x,t)=a_0(x,t)-a^*, & x\in\Omega,-1\leq t\leq 0, \end{cases} \end{equation} where $$ m_t(\theta)=m(x,t+\theta),~a_t(\theta)=a(x,t+\theta),~~\theta\in [-1,0], $$ and for $\phi_1, \phi_2\in \mathcal{C}:=C([-1,0],X_{\mathbb{C}})$ \begin{equation}\label{f1} \begin{array}{ll} f_1(\phi_1,\phi_2)=&r\phi_1(0)\phi_2(-1)-\cfrac{m^*}{(1+m^*)^3}\phi_1^2(-1)+\cfrac{1}{(1+m^*)^2}\phi_1(0)\phi_1(-1)\\ &+\cfrac{m^*}{(1+m^*)^4}\phi_1^3(-1)-\cfrac{1}{(1+m^*)^3}\phi_1(0)\phi_1^2(-1)+\mathcal{O}(4),\\ f_2(\phi_1,\phi_2)=&-\phi_1(0)\phi_2(0). \end{array} \end{equation}
In order to study the dynamics near the Turing-Hopf bifurcation, we need to extend the domain of solution operator to a space of some discontinuous: $$
\mathcal{BC}:=\big\{\psi:[-1,0]\rightarrow X_{\mathbb{C}}~|~\psi~\text{is continuous on}~ [-1, 0), \exists\lim_{\theta\rightarrow 0^-}\psi(\theta)\in X_{\mathbb{C}}\big\} $$
Let $\mu=\mu_0+\mu_{\varepsilon}$, where $\mu=(\tau, d)$, $\mu_0=(\tau_0, d_0)$, and $\mu_{\varepsilon}=(\tau_{\varepsilon}, d_{\varepsilon})$. Then system \eqref{eq_origin} undergoes a Turing-Hopf bifurcation at the equilibrium $(0,0)$ when $\mu_{\varepsilon}=(0,0)$ and we can rewrite system \eqref{eq_origin} in an abstract form in the space $\mathcal{BC}$ as \begin{equation}\label{eq_abs_origin} \cfrac{d}{dt}U(t)=A U_t+X_0\mathcal{F}(\mu_\varepsilon,U_t), \end{equation} where \begin{equation*}
X_0(\theta)=\begin{cases}
0 , &\theta\in[-1, 0),\\
I, \quad &\theta=0.
\end{cases} \end{equation*} and $A$ is a operator from $\mathcal{C}_0^1:=\big\{\varphi\in \mathcal{C}: \dot{\varphi}\in \mathcal{C}, \varphi(0)\in \text{dom}(\Delta)\big\}$ to $\mathcal{BC}$ \cite{Paz}, defined by $$ A\varphi=\dot{\varphi}+X_0\big[\tau_0 D_0\Delta \varphi(0)+\tau_0 L_0(\varphi)-\dot{\varphi}(0)\big] $$ with $D_0=D(\mu_0)$, $L_0:\mathcal{C}\rightarrow X_{\mathbb{C}}$ is a linear operator given by $L_0(\varphi)= L(\mu_0)(\varphi)$ with $$ L(\mu)(\varphi)=L_1\varphi(0)+L_2\varphi(-1)$$ and $\mathcal{F}:\mathbb{R}^2 \times \mathcal{C}\to X_{\mathbb{C}}$ is a nonlinear operator and defined by \begin{equation*} \mathcal{F}(\mu_\varepsilon,\phi)=(\tau_0+\tau_{\varepsilon})\big[D\Delta\varphi(0) +L(\mu)(\varphi)+F(\mu_\varepsilon,\varphi)\big]-A\varphi(0) \end{equation*} with \begin{equation}\label{F} F(\mu_\varepsilon,\varphi)=(\tau_0+\tau_\varepsilon)(f_1(\varphi_1,\varphi_2),~\gamma^{-1}f_2(\varphi_1,\varphi_2))^{^T}, \end{equation} where $f_1$ and $f_2$ are defined by \eqref{f1}.
We denote $$
b_n=\cfrac{\cos (nx/l)}{\|\cos(nx/l)\|},~ ~\beta_n=\{\beta_n^1, \beta_n^2\}=\{(b_n, 0)^{T}, (0, b_n)^{T}\}, $$ where $$
\|\cos(nx/l)\|=\left(\int_0^{l\pi}\cos^2(nx/l)\text{d}x\right)^{\frac{1}{2}}. $$
For $\phi=(\phi^{^{(1)}},\phi^{^{(2)}})^{T}\in\mathcal{C}$, denote $$ \phi_n=\langle \phi,\beta_n\rangle=\left(\langle \phi,\beta_n^1\rangle, \langle \phi,\beta_n^2\rangle\right)^{T}. $$ Define $ A_{\varepsilon, n}$ as \begin{equation}\label{An}
A_{\varepsilon, n}(\phi_n(\theta)\beta_n)=\begin{cases}
\dot{\phi}_n(\theta)\beta_n,& \theta\in[-1,0), \\
\int_{-1}^{0}\text{d}\eta_n(\mu_\varepsilon,\theta)\phi_n(\theta)\beta_n ,\qquad &\theta=0,
\end{cases} \end{equation} where $$ \int_{-1}^{0}\text{d}\eta_n(\mu_\varepsilon,\theta)\phi_n(\theta)=-\cfrac{n^2}{l^2} (\tau_0+\tau_\varepsilon)D\phi_n(0)+L_{\varepsilon,n}(\phi_n), $$ with $$ L_{\varepsilon, n}(\phi_n)=(\tau_0+\tau_\varepsilon)L_1\phi_n(0)+(\tau_0+\tau_\varepsilon)L_2\phi_n(-1), $$ and \begin{equation*}
\eta_n(\mu_\varepsilon,\theta)=\begin{cases}\begin{array}{ll}
-(\tau_0+\tau_\varepsilon)L_2, & \theta=-1,\\
0, & \theta\in(-1,0),\\
(\tau_0+\tau_\varepsilon)\left(L_1-\cfrac{n^2}{l^2}D\right), & \theta=0.
\end{array}\end{cases} \end{equation*}
Denote $A^*$ as the adjoint operator of $A$ on $\mathcal{C}^*:=C([0,1],X_{\mathbb{C}})$. \begin{equation*} A^*\psi(s)=\begin{cases}\begin{array}{ll} -\dot{\psi}(s),~& s\in(0,1],\\ \sum_{n=0}^\infty\int_{-1}^0\psi_n(-\theta)\text{d}\eta_n^{^T}(0,\theta)\beta_n,~&s=0. \end{array}\end{cases} \end{equation*} Now, we introduce the bilinear formal $(\cdot,\cdot)$ on $\mathcal{C}^*\times\mathcal{C}$ \begin{equation*} (\psi,\phi)=\sum_{j_1,j_2=0}^\infty(\psi_{j_1},\phi_{j_2})\int_\Omega b_{j_1}b_{j_2}\text{d}x, ~k=1,2 \end{equation*} where $$ \psi=\sum_{n=0}^\infty \psi_n \beta_n\in\mathcal{C}^*,~\phi=\sum_{n=0}^\infty \phi_n \beta_n\in\mathcal{C}, $$ and $$ \phi_n\in C:=C([-1,0],\mathbb{C}^2),~~\psi_n\in C^*:=C([0,1],\mathbb{C}^2). $$ Notice that
$$
\int_\Omega b_{j_1}b_{j_2}\text{d}x=0~~\mbox{for}~~j_1\neq j_2,
$$ we have \begin{equation*}
(\psi,\phi)=\sum_{n=0}^\infty(\psi_n,\phi_n)|b_n|^2:=\sum_{n=0}^\infty(\psi_n,\phi_n)_n|b_n|^2, \end{equation*} where $(\cdot,\cdot)_n$(or $(\cdot,\cdot)$) is the bilinear form defined on $C^*\times C$ \begin{equation*} (\psi_n,\phi_n)_n=\psi_n(0)\phi_n(0)-\int_{-1}^0\int_{\xi=0}^\theta\psi_n(\xi-\theta) \text{d}\eta_n(0,\theta)\phi_n(\xi)\text{d}\xi. \end{equation*}
Let $\{\phi_1(\theta)b_{n_1}, \phi_2(\theta)b_{n_2}\}$ and $\{\psi_1(s)b_{n_1}, \psi_2(s)b_{n_2}\}$ are the eigenfunctions of $A$ and its dual $A^*$ relative to $\Lambda=\{i\omega_0\tau_0, 0\}$ such that $\phi_1, \phi_2\in C$, $\psi_1, \psi_2\in C^*$ and $$ (\psi_1,\phi_1)_1=1,~~(\psi_1,\overline{\phi}_1)_1=0,~~(\psi_2,{\phi}_2)_2=1 $$ By a straight forward calculation, we have $$ \begin{array}{ll} \phi_1(\theta)=q(0)e^{i\omega_0\tau_0\theta},& \psi_1(s)=M_1 q^*(0)e^{-i\omega_0\tau_0 s},\\ \phi_2(\theta)=p(0),&\psi_2(s)=M_2p^*(0), \end{array}$$ where $q(0)=(1, q_1)^{^T}$, $q^*(0)=(q_2, 1)$, $p(0)=(1, p_1)^{^T}$, $p^*(0)=(p_2, 1)$ and $$\begin{array}{ll} &q_1=\cfrac{a^*}{i\gamma\omega_0+\alpha+m^*},~~q_2=\cfrac{i \gamma \omega_0+\alpha+m^*}{r m^*e^{-i\omega_0\tau_0}},~~p_1=\cfrac{-a^*}{\frac{n^2_2}{l^2}+\alpha+m^*},~~p_2=\cfrac{\frac{n^2_2}{l^2}+\alpha+m^*}{r m^*},\\ &M_1=\cfrac{1}{q_1+q_2+\tau_0q_2e^{-i\omega_0\tau_0}(r^2a^{*2}m^*+rm^*q_1)},~~M_2=\cfrac{1}{p_1+p_2+\tau_0r m^*p_2(ra^{*2}+p_1)}
\end{array}$$
Denote $\Phi_1=(\phi_1, \overline{\phi}_1)$, $\Psi_1=(\psi^{^T}_1, \bar{\psi}^{^T}_1)^{^T}$ and $\Phi_2=\phi_2$, $\Psi_2=\psi_2$. From the discussion above, we know that the phase space $\mathcal{BC}$ can be decomposed as $$ \mathcal{BC}=\mathcal{P}\bigoplus\text{Ker}\pi, $$ where $\mathcal{P}$ is the is the 3-dimensional center subspace spanned by the basis eigenfunctions of the linear operator $A$ associated with the eigenvalues $\{\pm i\omega_0\tau_0, 0\}$ and $\text{Ker}\pi$ is the complementary space of $\mathcal{P}$ with $\pi: \mathcal{BC}\rightarrow \mathcal{P}$ is the projection defined by $$ \pi\varphi=\sum^2_{k=1}\Phi_k(\Psi_k,<\varphi(\cdot),\beta_{n_k}>)_k \cdot \beta_{n_k} $$ with $c\cdot \beta_{n_k}=c_1 \beta_{n_k}^1+c_2 \beta_{n_k}^2$ for $c=(c_1, c_2)^{_T}\in C$.
Then $U_t\in \mathcal{C}_0^1$ can be decomposed as \begin{equation*} \begin{split} U_t(\theta)&=\sum^2_{k=1}\Phi_k(\theta)(\Psi_k,<U_t,\beta_{n_k}>)_k \beta_{n_k}+y(\theta)\\ &=\sum^2_{k=1}\Phi_k(\theta)\tilde{z}_k(t)\cdot \beta_{n_k}+y(\theta), \end{split} \end{equation*} with $\tilde{z}_1=(z_1, \bar{z}_1)$, $\tilde{z}_2=z_2$, and $y\in Q^1:=\mathcal{C}_0^1\bigcap\text{Ker}\pi$. Then system \eqref{eq_abs_origin} on $\mathcal{BC}$ is equivalent to the following system \begin{equation} \begin{split}
& \dot{z}=Bz+\Psi(0)\left(\begin{array}{l}
<\mathcal{F}(\mu_\varepsilon,\sum^2_{k=1}\Phi_k\tilde{z}_k(t)\cdot \beta_{n_k}+y), \beta_{n_1}>\\
<\mathcal{F}(\mu_\varepsilon,\sum^2_{k=1}\Phi_k\tilde{z}_k(t)\cdot \beta_{n_k}+y), \beta_{n_2}>
\end{array}\right),\\
& \cfrac{d}{dt}y=A_{Q^1}y-(I-\pi)X_0\mathcal{F}\big(\mu_\varepsilon,\sum^2_{k=1}\Phi_k\tilde{z}_k(t)\cdot \beta_{n_k}+y\big), \end{split} \end{equation} where $z=(z_1, \bar{z}_1, z_2)$, $B=\text{diag}(i\omega_0\tau_0, -i\omega_0\tau_0, 0)$, $\Psi=\text{diag}(\Phi_1, \Phi_2)$, and $A_{Q^1}$ is the restriction of $A$ as an operator from $Q^1$ to $\text{Ker}\pi$.
From the Theorem 3.2 in \cite{AnJ}, the normal forms of system \eqref{eq_ma_tau} up to three order near a Turing-Hopf singularity $\mu=\mu_0$ are obtained \begin{equation}\label{z_norm} \begin{array}{ll}
\dot{z}_1=~~i\omega_0\tau_0z_1+&\cfrac{1}{2}f^{11}_{11}\alpha_1z_1+\cfrac{1}{2}f^{11}_{21}\alpha_2z_1+\cfrac{1}{6}g^{11}_{210}z^2_1\bar{z}_1
+\cfrac{1}{6}g^{11}_{102}z_1z^2_2+h.o.t.\\
\dot{\bar{z}}_1=-i\omega_0\tau_0z_1+&\cfrac{1}{2}f^{12}_{11}\alpha_1\bar{z}_1+\cfrac{1}{2}f^{12}_{21}\alpha_2\bar{z}_1
+\cfrac{1}{6}g^{12}_{210}\bar{z}^2_1 z_1+\cfrac{1}{6}g^{12}_{102}\bar{z}_1z^2_2+h.o.t.\\
z_2=&\cfrac{1}{2}f^{13}_{12}\alpha_1z_2+\cfrac{1}{2}f^{13}_{22}\alpha_2z_2+\cfrac{1}{6}g^{13}_{111}z_1\bar{z}_1z_2
+\cfrac{1}{6}g^{13}_{003}z_2^3+h.o.t.
\end{array} \end{equation} with $(\alpha_1,\alpha_2)=(\tau_{\varepsilon}, d_{\varepsilon})$, $f^{12}_{mn}=\overline{f^{11}_{mn}}$, $g^{12}_{mnk}=\overline{g^{11}_{mnk}}$ and \begin{equation*}
\begin{array}{ll}
f^{11}_{11}&=2\psi_1(0)\Big[\cfrac{\partial}{\partial\tau}A(\mu_0)\phi_1(0)+\cfrac{\partial}{\partial\tau}B(\mu_0)\phi_1(-1)\Big],\\
f^{11}_{21}&=2\psi_1(0)\Big[\cfrac{\partial}{\partial d}A(\mu_0)\phi_1(0)+\cfrac{\partial}{\partial d}B(\mu_0)\phi_1(-1)\Big],\\
f^{13}_{12}&=2\psi_2(0)\Big[-\cfrac{n_2^2}{l^2}\cfrac{\partial}{\partial \tau}\widetilde{D}(\mu_0)\phi_2(0)+
\cfrac{\partial}{\partial \tau}A(\mu_0)\phi_2(0)+\cfrac{\partial}{\partial \tau}B(\mu_0)\phi_2(-1)\Big],\\
f^{13}_{22}&=2\psi_2(0)\Big[-\cfrac{n_2^2}{l^2}\cfrac{\partial}{\partial d}\widetilde{D}(\mu_0)\phi_2(0)+
\cfrac{\partial}{\partial d}A(\mu_0)\phi_2(0)+\cfrac{\partial}{\partial d}B(\mu_0)\phi_2(-1)\Big],\\ \end{array} \end{equation*} \begin{equation*} \begin{array}{ll}
g^{11}_{210}&=f^{11}_{210}+\cfrac{3}{2i\omega_0\tau_0}\Big(-f^{11}_{110}f^{11}_{200}+f^{11}_{110}f^{12}_{110}+\cfrac{2}{3}f^{11}_{020}f^{12}_{200}\Big)\\
&~~+\cfrac{3}{2}\psi_1(0)\Big[S_{yz_1}(<h_{110}(\theta)b_{n_1},b_{n_1}>)+S_{y\bar{z}_1}(<h_{200}(\theta)b_{n_1},b_{n_1}>)\Big],\\
g^{11}_{102}&=f^{11}_{102}+\cfrac{3}{2i\omega_0\tau_0}\Big(-2f^{11}_{002}f^{11}_{200}+f^{12}_{002}f^{11}_{110}+2f^{11}_{002}f^{13}_{101}\Big)\\
&~~+\cfrac{3}{2}\psi_1(0)\Big[S_{yz_1}(<h_{002}(\theta)b_{n_1},b_{n_1}>)+S_{yz_2}(<h_{101}(\theta)b_{n_2},b_{n_1}>)\Big],\\
g^{13}_{111}&=f^{13}_{111}+\cfrac{3}{2i\omega_0\tau_0}\Big(-2f^{13}_{101}f^{11}_{110}+f^{13}_{011}f^{12}_{110}\Big)\\
&~~+\cfrac{3}{2}\psi_2(0)\Big[S_{yz_1}(<h_{011}(\theta)b_{n_1},b_{n_2}>)+S_{y\bar{z}_1}(<h_{101}(\theta)b_{n_1},b_{n_2}>)
+S_{yz_2}(<h_{110}(\theta)b_{n_2},b_{n_2}>)\Big],\\
g^{13}_{003}&=f^{13}_{003}+\cfrac{3}{2i\omega_0\tau_0}\Big(-f^{11}_{002}f^{13}_{101}+f^{12}_{002}f^{13}_{011}\Big)
+\cfrac{3}{2}\psi_2(0)\Big[S_{yz_2}(<h_{002}(\theta)b_{n_2},b_{n_2}>)\Big],
\end{array} \end{equation*} where $\widetilde{D}=\tau D$, $A(\mu)=\tau L_1$, $B(\mu)=\tau L_2$, $f^{12}_{mnk}=\overline{f^{11}_{mnk}}$, and \begin{equation*}
\begin{array}{ll}
f^{11}_{mnk}=\cfrac{1}{\sqrt{l\pi}}\psi_1(0)F_{mnk},~~~ & f^{13}_{mnk}=\cfrac{1}{\sqrt{l\pi}}\psi_2(0)F_{mnk},~~\text{when}~~m+n+k=2,\\ f^{11}_{mnk}=\cfrac{1}{l\pi}\psi_1(0)F_{mnk},~~~ & f^{13}_{mnk}=\cfrac{1}{l\pi}\psi_2(0)F_{mnk},~~\text{when}~~m+n+k=3.\\
\end{array} \end{equation*}
\begin{equation*}
\begin{array}{ll}
<h_{200}(\theta)b_{n_1},b_{n_1}>=\cfrac{e^{2i\omega_0\tau_0\theta}}{l\pi}\Big[2i\omega_0\tau_0-\tau_0 L_0(e^{2i\omega_0\tau_0\cdot}Id)\Big]^{-1}F_{200}
-\cfrac{1}{i\omega_0\tau_0\sqrt{l\pi}}\Big[f^{11}_{200}\phi_1(\theta)+\cfrac{1}{3}f^{12}_{200}\overline{\phi}_1(\theta)\Big],\\
<h_{110}(\theta)b_{n_1},b_{n_1}>=-\cfrac{1}{\sqrt{l\pi}}\Big[\tau_0L_0(Id)\Big]^{-1}F_{110}+\cfrac{1}{i\omega_0\tau_0\sqrt{l\pi}}
\Big[f^{11}_{110}\phi_1(\theta)-f^{12}_{110}\overline{\phi}_1(\theta)\Big],\\
<h_{110}(\theta)b_{n_2},b_{n_2}>= <h_{110}(\theta)b_{n_1},b_{n_1}>,\\
<h_{101}(\theta)b_{n_2},b_{n_1}>=\cfrac{e^{i\omega_0\tau_0\theta}}{l\pi}\Big[i\omega_0\tau_0+\cfrac{n_2^2}{l^2}\widetilde{D}(\mu_0)
-\tau_0L_0(e^{i\omega_0\tau_0 \cdot}Id)\Big]^{-1}F_{101}-\cfrac{1}{i\omega_0\tau_0\sqrt{l\pi}}f^{13}_{101}\phi_2(0),\\
\end{array} \end{equation*} \begin{equation*}
\begin{array}{ll}
<h_{011}(\theta)b_{n_1},b_{n_2}>=\cfrac{e^{-i\omega_0\tau_0\theta}}{l\pi}\Big[-i\omega_0\tau_0+\cfrac{n_2^2}{l^2}\widetilde{D}(\mu_0)
-\tau_0L_0(e^{-i\omega_0\tau_0\cdot}Id)\Big]^{-1}F_{011}+\cfrac{1}{i\omega_0\tau_0\sqrt{l\pi}}f^{13}_{011}\phi_2(0),\\
<h_{002}(\theta)b_{n_1},b_{n_1}>=-\cfrac{1}{l\pi}\Big[\tau_0L_0(Id)\Big]^{-1}F_{002}+\cfrac{1}{i\omega_0\tau_0\sqrt{l\pi}}
\Big[f^{11}_{002}\phi_1(\theta)-f^{12}_{002}\overline{\phi}_1(\theta)\Big],\\
<h_{002}(\theta)b_{n_2},b_{n_2}>=\cfrac{1}{2l\pi}\Big[\cfrac{(2n_2)^2}{l^2}\widetilde{D}(\mu_0)-\tau_0L_0(Id)\Big]^{-1}F_{002}+ <h_{002}(\theta)b_{n_1},b_{n_1}>,
\end{array} \end{equation*} and $S_{yz_i}(i=1,2)$, $S_{y\bar{z}_1}$ are linear operators from $Q_1$ to $X_{\mathbb{C}}$ given by \begin{equation*}
\begin{array}{ll}
S_{yz_i}(\varphi)=(F_{y_1(0)z_i},~ F_{y_2(0)z_i})\varphi(0)+(F_{y_1(-1)z_i}, F_{y_2(-1)z_i})\varphi(-1),\\
S_{y\bar{z}_1}(\varphi)=(\overline{F_{y_1(0)z_1}}, \overline{F_{y_2(0)z_1}})\varphi(0)+(\overline{F_{y_1(-1)z_1}}, \overline{F_{y_2(-1)z_1}})\varphi(-1).\\
\end{array} \end{equation*} For specific expressions of formulas $F_{y_i(\cdot)z_{j}}, F_{mnk}$, please refer to Appendix.
With the cylindrical coordinate transformation: $$ z_1=\widetilde{\rho} e^{i \sigma}, ~\bar{z}_1=\widetilde{\rho} e^{-i\sigma},~ z_2=\widetilde{\eta} $$ and variable substitution: $$
\rho=\sqrt{\cfrac{|\text{Re}(g^{11}_{210})|}{6}}\widetilde{\rho}, ~~\eta=\sqrt{\cfrac{|g^{13}_{003}|}{6}}\widetilde{\eta}, ~~\varepsilon=\text{Sign}\big(\text{Re}(g^{11}_{210})\big), ~~\widetilde{t}=t/\varepsilon, $$ the amplitude equation \eqref{z_norm} can be rewritten as \begin{equation}\label{rho-eta} \begin{array}{ll}
\cfrac{\text{d}\rho}{\text{d}\widetilde{t}}=\rho\Big(\epsilon_1(\mu_{\varepsilon})+\rho^2+b\eta^2\Big),\\
\cfrac{\text{d}\eta}{\text{d}\widetilde{t}}=\eta\left(\epsilon_2(\mu_{\varepsilon})+c \rho^2+\hat{d}\eta^2\right),\\ \end{array} \end{equation} where \begin{equation*}
\begin{array}{ll}
\epsilon_1(\mu_{\varepsilon})=\cfrac{\varepsilon}{2}\left[\text{Re}(f^{11}_{11})\tau_{\varepsilon}+\text{Re}
(f^{11}_{21})d_{\varepsilon}\right],\\
\epsilon_2(\mu_{\varepsilon})=\cfrac{\varepsilon}{2}\left[f^{13}_{12}\tau_{\varepsilon}+
f^{13}_{22}d_{\varepsilon}\right],\\
b=\cfrac{\varepsilon\text{Re}(g^{11}_{102})}{|g^{13}_{003}|},~~c=\cfrac{\varepsilon g^{13}_{111}}{|\text{Re}(g^{11}_{210})|},
~~\hat{d}=\cfrac{\varepsilon g^{13}_{003}}{|g^{13}_{003}|}=\pm1.
\end{array} \end{equation*} Notice that $\rho\geq 0$, and $\eta$ is arbitrarily real number. Hence, system \eqref{rho-eta} always has a zero equilibrium $E_1(0, 0)$ for all $\epsilon_1, \epsilon_2$, and three boundary equilibria \begin{equation*} \begin{array}{ll}
E_2\left(\sqrt{-\epsilon_1}, 0\right), ~~~&\text{for} ~~~\epsilon_1<0\\
E_3^{\pm}\Big(0,\pm \sqrt{-\cfrac{\epsilon_2}{\hat{d}}}\Big), ~~~&\text{for}~~~\epsilon_2\hat{d}<0,\\
\end{array} \end{equation*} and two possible positive equilibria \begin{equation*}
\begin{array}{ll}
E_4^{\pm}=\Big(\sqrt{\cfrac{b\epsilon_2-\hat{d}\epsilon_1}{\hat{d}-bc}}, \pm \sqrt{\cfrac{c\epsilon_1-\epsilon_2}{\hat{d}-bc}}\Big),
~~~\text{for} ~~~\sqrt{\cfrac{b\epsilon_2-\hat{d}\epsilon_1}{\hat{d}-bc}}>0, \sqrt{\cfrac{c\epsilon_1-\epsilon_2}{\hat{d}-bc}}>0.
\end{array} \end{equation*} There are 12 distinct types of unfoldings \cite{GuH} according to the signs of coefficients $b, c, \hat{d}$ and $\hat{d}-bc$.
\section{Numerical Simulations}\label{dynamic classifiction} In this section, we choose a set of parameters. Under these parameters, the dynamic classification of the system \eqref{eq_ma_tau}
near the Turing-Hopf bifurcation point is given and some simulations are carried out. \subsection{Dynamic classification} In this subsection, we apply the normal form method and the theoretical results obtained in previous sections to the system \eqref{eq_ma_tau}. The bifurcation diagram of system \eqref{rho-eta} with certain parameters near the Turing-Hopf bifurcation point in the $\tau_{\varepsilon}-d_{\varepsilon}$ parameter plane is firstly shown to determine the existential area of solutions, the critical lines separate the plane into six regions, and for each region, we shall given a detail analysis.
Take $$ {\bf (A)}~~~~~~~~~~~~r=1.10,~~~\gamma=4,~~~\alpha=0.654,~~~l=6.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $$ Then $m^*=0.233073$, $a^*=0.737257$. From \eqref{tau} and \eqref{d_0}, we have $\tau_0=7.084102$ with $n_1=0$, $d_0=0.0531255$ with $n_2=6$, and by a simple calculation, we have \begin{equation*}
\begin{array}{ll}
\epsilon_1 = -1.20727\times 10^{-2}\tau_{\varepsilon};~~
\epsilon_2 = 1.629874\times 10^{-8}\tau_{\varepsilon} + 6.085844 d_{\varepsilon};\\
\varepsilon = -1; ~~~\hat{d} = 1;~~~b = 1.582903; ~~~c = 0.993790;~~~\hat{d}-bc = -0.573073.
\end{array} \end{equation*} Then \eqref{rho-eta} becomes \begin{equation}\label{28}
\begin{array}{ll}
\dot{\rho}=\rho\left(-1.20727\times 10^{-2}\tau_{\varepsilon}+\rho^2+1.582903\eta^2\right),\\
\dot{\eta}=\eta\left(1.629874\times 10^{-8}\tau_{\varepsilon} + 6.085844 d_{\varepsilon}+0.993790\rho^2+\eta^2\right).\\
\end{array} \end{equation} According to the classification for the planar vector field \eqref{rho-eta} in [Page 399, \cite{GuH}], Case Ia occurs under this set of parameters. The detailed bifurcation diagram and corresponding phase portraits are shown in Fig.\ref{fig-TH}, in which the two blue lines are two pitchfork bifurcation curves: \begin{equation*} \begin{array}{ll}
T_1:~~d_{\varepsilon}=-1.253237 \times 10^{-3}\tau_{\varepsilon};\\
T_2:~~d_{\varepsilon}=-1.971431 \times 10^{-3}\tau_{\varepsilon}, \end{array} \end{equation*} and the other two solid lines $L_1$ and $L_2$ are \begin{equation*}
L_1:~~\tau_{\varepsilon}=0;~~~~~~~L_2:~~~d_{\varepsilon}=-2.678139\times 10^{-9}\tau_{\varepsilon}. \end{equation*} Notice that, under the parameters {\bf (A)}, the dynamics of original system \eqref{eq_ma_tau} near the $(\tau, d)=(\tau_0, d_0)$ is topologically equivalent to that of normal form system \eqref{28} at $(\tau_{\varepsilon}, d_{\varepsilon})=(0, 0)$. For system \eqref{28}, the equilibrium in the $\rho-$axis $(E_2)$ identifies the characteristics of the solutions of \eqref{eq_ma_tau} in time, while equilibrium in the $\eta-$axis $(E_3^{\pm})$ identifies the characteristics in space. Moreover, the positive equilibrium in the $\rho-\eta$ plane $(E_4^{\pm})$ identifies the characteristics of solutions of system \eqref{eq_ma_tau} both in time and space.
From Fig.\ref{fig-TH}, we see that the solid lines $L_1, L_2, T_1$ and $ T_2$ divide the plane into six regions, and in different regions there are different dynamics which can be summarized as follows.
When $(\tau_{\varepsilon}, d_{\varepsilon})\in D_1$, the amplitude system \eqref{28} has a stable trivial equilibrium $E_1(0,0)$, which means the constant steady state $E_*(m^*, a^*)$ of original system \eqref{eq_ma_tau} is locally asymptotically stable;
When $(\tau_{\varepsilon}, d_{\varepsilon})$ passes through $L_1$ into $D_2$, the constant steady state $E_*(m^*, a^*)$ lost its stability with a new stable spatially homogeneous periodic solution bifurcating from $E_*(m^*, a^*)$.
When $(\tau_{\varepsilon}, d_{\varepsilon})$ enters $D_3$ from $D_2$, two unstable non-constant steady states newly appear since a Turing bifurcation occurs at $L_2$. Moreover, $E_1$ of system \eqref{28} becomes an unstable node from a saddle.
When $(\tau_{\varepsilon}, d_{\varepsilon}) \in D_4$, two unstable spatially inhomogeneous periodic solutions newly appear and do coexist. The non-constant steady states become stable compared with its stability in region $D_3$.
When $(\tau_{\varepsilon}, d_{\varepsilon})$ enters $D_5$ from $D_4$, the two unstable spatially inhomogeneous solutions disappear since the parameters pass through another Turing bifurcation curve $T_2$, and the spatially homogeneous periodic solution loses its stability.
When $(\tau_{\varepsilon}, d_{\varepsilon})$ finally enters region $D_6$, the spatially homogeneous periodic solution disappears with a Hopf bifurcation occuring at $L_1$. Moreover, $E_1$ of system \eqref{28} becomes a saddle from an unstable node, and it will regain its stability when $(\tau_{\varepsilon}, d_{\varepsilon})$ passes through $L_2$ into $D_1$.
\begin{figure}
\caption{Bifurcation diagram of system \eqref{rho-eta} in the $\tau_{\varepsilon}-d_{\varepsilon}$ plane and the corresponding phase portraits.}
\label{fig-TH}
\end{figure}
\subsection{Simulations}
Numerical simulations of dynamics for original system \eqref{eq_ma_tau} at the Turing-Hopf bifurcation point are carried out in this subsection. For each region in Fig.\ref{fig-TH}, we shall select a set of parameters $(\tau_{\varepsilon}, d_{\varepsilon})$, and for obvious contrast, the parameters are always selected from a rectangle, see Fig.\ref{fig-parameter}. The little pink circles represent the points that that we choose in each region. \begin{figure}
\caption{The selection of parameters in the $\tau_{\varepsilon}-d_{\varepsilon}$ plane.}
\label{fig-parameter}
\end{figure}
Fig.\ref{fig-stable-pattern} shows the stable patterns in region $D_1, D_3, D_5$. The stable patterns in region $D_2$ and $D_6$ are similar with that in $D_3$ and $D_5$, respectively. In region $D_1$, there exists a stable spatially homogeneous steady state; in $D_3$, a stable spatially homogeneous periodic solution exists, and in $D_5$, two stable spatially inhomogeneous steady states coexist. For $D_4$, the non-constant steady state and spatially homogeneous periodic solution both can be considered as the stable patterns, which is related to the initial values. In addition, some transitions that connecting two state can be observed in our numerical simulations, detailed results refer to Fig.\ref{fig-D3}--Fig.\ref{fig-D5}.
\begin{figure}
\caption{The stable patterns in regions $D_1$, $D_3$, $D_5$. The above graphs shows the dynamics of mussel while the belows, the algae. }
\label{fig-stable-pattern}
\end{figure}
\begin{figure}
\caption{Stable spatially homogeneous periodic solution in $D_3$ with
$(\tau_{\varepsilon}, d_{\varepsilon})=(0.5, -0.0005)\in D_3$, and the initial function are $(m^*+0.1+0.3\cos x, a^*-0.1-0.3\cos x)$. (A)-(C): The dynamics of mussel; (D)-(F): The dynamics of algae.}
\label{fig-D3}
\end{figure}
Fig.\ref{fig-D3} shows a stable spatially homogeneous periodic solution in $D_3$. Fig.\ref{fig-D5} shows a stable spatially homogeneous steady state in $D_5$. (A) and (D) represent the trends of pattern formation; (B) and (E) show the transformation process at the beginning; (C) and (F) show the final stable behavior.
Fig.\ref{fig-D4_1} and Fig.\ref{fig-D4_2} shows the different evolutionary process of system \eqref{eq_ma_tau} with the same parameters but slightly different initial functions. Fix $(\tau_{\varepsilon}, d_{\varepsilon})=(0.5, -0.0009)\in D_4$, one case is that we choose the initial function $(m^*+0.3+0.5\cos x, a^*-0.5\cos x)$, then after a period of time evolution, one can see a spatially inhomogeneous periodic solution appears (see graph (B) and (E) of Fig.\ref{fig-D4_1}), but this is not the final state, the spatially inhomogeneous periodic solution disappears as time going on, and finally reach its stable state, a spatially homogeneous periodic solution. The other case is just the opposite. We choose$(m^*+0.3+0.5\cos x, a^*-0.1-0.5\cos x)$ as the initial functions, and the simulation shows the solution can also evolve into a spatially inhomogeneous periodic solution, but it ultimately becomes a spatially inhomogeneous steady state when time is long enough.
\begin{figure}
\caption{Stable spatially homogeneous periodic solution and unstable spatially inhomogeneous periodic solutions in $D_4$ with
$(\tau_{\varepsilon}, d_{\varepsilon})=(0.5, -0.0009)\in D_4$, and the initial function are $(m^*+0.3+0.5\cos x, a^*-0.5\cos x)$. (A)-(C): The dynamics of mussel; (D)-(F): The dynamics of algae.}
\label{fig-D4_1}
\end{figure}
\begin{figure}
\caption{Stable non-constant steady states in $D_4$ and unstable spatially inhomogeneous periodic solutions with
$(\tau_{\varepsilon}, d_{\varepsilon})=(0.5, -0.0009)\in D_4$, and the initial function are $(m^*+0.3+0.5\cos x, a^*-0.1-0.5\cos x)$. (A)-(C): The dynamics of mussel; (D)-(F): The dynamics of algae.}
\label{fig-D4_2}
\end{figure}
\begin{figure}
\caption{Stable spatially homogeneous periodic solution in $D_5$ with
$(\tau_{\varepsilon}, d_{\varepsilon})=(0.5, -0.002)\in D_5$, and the initial function are $(m^*+0.1+0.3\cos x, a^*-0.1-0.3\cos x)$. (A)-(C): The dynamics of mussel; (D)-(F): The dynamics of algae.}
\label{fig-D5}
\end{figure}
\begin{figure}
\caption{The positive constant steady state $E_*(m^*, a^*)$ of system \eqref{eq_ma_tau} is locally asymptotically stable in $D_1$ with
$(\tau_{\varepsilon}, d_{\varepsilon})=(-0.5, 0.01)\in D_1$, and the initial function are $(m^*+0.1\cos x, a^*-0.1\cos x)$.}
\label{fig-D1}
\end{figure}
\section{Discussion and conclusion} In this paper, we investigate the spatiotemporal patterns induced by the Turing-Hopf bifurcation for a mussel-algae model with delay and diffusion.
We first show the global existence of solutions of system \eqref{eq_ma_tau}. But, the boundedness of mussel $m(x,t)$ is still unknown. A reason is that the death rate of mussel depends on the density of mussels themselves. If the mortality of mussel is a constant, then the estimate of mussel can be obtained without difficulty. Hence, a open mathematical question for this model is the global stability of the positive spatially homogeneous steady state.
Under the assumption \textsc{(H1)} and \textsc{(H2)}, the positive spatially homogeneous steady state is locally asymptotically stable under a linear homogeneous perturbation when $\tau=0$. But when $\tau$ get the critical value $\tau_0$, the positive spatially homogeneous steady state will lose its stability , at the same time, a positive spatially homogeneous periodic solution appears and the system undergoes a Hopf bifurcation which is induced by the delay.
To investigate the Turing instability of system \eqref{eq_ma_tau}, we discuss the effect of diffusion coefficient $d$. If $d>d_0$, there is no Turing instability; and if $d<d_0$, one can always find a wave number $k$ such that Turing instability occurs. It's nothing that $d$ is not the true diffusivity ratio, actually, it is only the diffusion coefficient of the predator, mussel, and $\frac{1}{\gamma}$ is another diffusion coefficient belongs to the prey, algae. For fixed $\gamma$, if $d$ is sufficiently large, which means the diffusivity ratio $d\gamma$ is sufficient large, and by our result, there is no Turing instability. According to the mechanism of pattern formation presented by Turing in \cite{Tur}, the mussel represents the ``activator" while algae, the``inhibitor". It is somewhat different from the general predator-prey model.
The dynamics near the Turing-Hopf bifurcation is discussed in detail by using the method of normal form for partial functional differential equations. We divide the $\tau_{\varepsilon}-d_{\varepsilon}$ plane into six regions with the phase portraits of each region are different. There are four types of patterns: spatially homogeneous / inhomogeneous steady state; spatially homogeneous / inhomogeneous periodic solutions. From the numerical simulations, one can easily see that the delay $\tau$ and diffusion coefficient $d$ could result in complex spatiotemporal dynamics.
The interaction between mussel and algae contains a wealth of information. Considering the mechanisms of flow motion \cite{ShM} and formation of mussel bed \cite{Liu-2014}, there are still many problems to be solved. For example, If the advection term is added, how will it affect the dynamics of system? when the space domain expand to 2-dimension, what are the effects of time delay, diffusion coefficient and the advection ? and how do they interact each other ?
Within restoration ecology, the mussel beds are typical and active research system \cite{Don, Liu-2014}. Also, because of the high edible and medicinal value, mussel fisheries plays an important role in fiscal revenue in many coastal areas. The formation of spatiotemporal patterns may affect both the resilience and productivity of mussel beds. Hence, studying the mussel-algae model and the formation of different patterns has important biological and economic significance and we need more realistic and detailed models to depict those behaviors in the following work.
\section{Appendix}
The coefficient vectors $F_{y_i(\theta)z_j}$, $F_{mnk}$ presented in normal form \eqref{z_norm} and therein can be obtained by using the following calculation formulas, where $F_{mm}=\cfrac{\partial^2}{\partial m^2}F(0,0)$, $F(\mu_{\varepsilon}, U_t)$ is defined by \eqref{F}, and others can be deduced by analogy.
\begin{equation*}
\begin{array}{ll}
F_{y_1(0)z_1}=2(F_{mm}+F_{ma}q_1+F_{mm_{\tau}}e^{-i\omega_0\tau_0}+F_{ma_{\tau}}q_1e^{-i\omega_0\tau_0}),\\
F_{y_1(-1)z_1}=2(F_{mm_{\tau}}+F_{m_{\tau}a}q_1+F_{m_{\tau}m_{\tau}}e^{-i\omega_0\tau_0}+F_{m_{\tau}a_{\tau}}q_1e^{-i\omega_0\tau_0}),\\
F_{y_2(0)z_1}=2(F_{ma}+F_{aa}q_1+F_{m_{\tau}a}e^{-i\omega_0\tau_0}+F_{aa_{\tau}}q_1e^{-i\omega_0\tau_0}),\\
F_{y_2(-1)z_1}=2(F_{ma_{\tau}}+F_{aa_{\tau}}q_1+F_{m_{\tau}a_{\tau}}e^{-i\omega_0\tau_0}+F_{a_{\tau}a_{\tau}}q_1e^{-i\omega_0\tau_0}),\\
F_{y_1(0)z_2}=2(F_{mm}+F_{mm_{\tau}}+F_{ma}p_1+F_{ma_{\tau}}p_1),\\
F_{y_1(-1)z_2}=2(F_{mm_{\tau}}+F_{m_{\tau}m_{\tau}}+F_{m_{\tau}a_{\tau}}p_1+F_{m_{\tau}a}p_1),\\
F_{y_2(0)z_2}=2(F_{ma}+F_{m_{\tau}a}+F_{aa}p_1+F_{aa_{\tau}}p_1,\\
F_{y_2(-1)z_2}=2(F_{ma_{\tau}}+F_{m_{\tau}a_{\tau}}+F_{aa_{\tau}}p_1+F_{a_{\tau}a_{\tau}}p_1),\\ \end{array} \end{equation*} and \begin{equation*}
\begin{array}{ll}
F_{200}=&F_{mm}+F_{aa}q_1^2+F_{m_{\tau}m_{\tau}}e^{-2i\omega_0\tau_0}+F_{a_{\tau}a_{\tau}}q_1^2e^{-2i\omega_0\tau_0}+2(F_{ma}q_1+
F_{mm_{\tau}}e^{-i\omega_0\tau_0}\\
&+F_{ma_{\tau}}q_1e^{-i\omega_0\tau_0}+F_{m_{\tau}a}q_1e^{-i\omega_0\tau_0}+F_{aa_{\tau}}q_1^2e^{-i\omega_0\tau_0}+F_{m_{\tau}a_{\tau}}
q_1e^{-2i\omega_0\tau_0}),\\
F_{110}=&2\big[F_{mm}+F_{aa}q_1\bar{q}_1+F_{m_{\tau}m_{\tau}}+F_{a_{\tau}a_{\tau}}q_1\bar{q}_1+F_{ma}(q_1+\bar{q}_1)
+F_{mm_{\tau}}(e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0})\\
&+F_{ma_{\tau}}(q_1e^{-i\omega_0\tau_0}+\bar{q}_1e^{i\omega_0\tau_0})+F_{m_{\tau}a}(q_1e^{i\omega_0\tau_0}+\bar{q}_1e^{-i\omega_0\tau_0})
+F_{aa_{\tau}}q_1\bar{q}_1(e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0})\\
&+F_{m_{\tau}a_{\tau}}(q_1+\bar{q}_1)\big],\\
F_{101}=&2\big[F_{mm}+F_{aa}q_1 p_1+F_{m_{\tau}m_{\tau}}e^{-i\omega_0\tau_0}+F_{a_{\tau}a_{\tau}}q_1p_1e^{-i\omega_0\tau_0}
+F_{ma}(q_1+p_1)+F_{mm_{\tau}}(1+e^{-i\omega_0\tau_0})\\
&+F_{ma_{\tau}}(p_1+q_1e^{-i\omega_0\tau_0})+F_{m_{\tau}a}(q_1+p_1e^{-i\omega_0\tau_0})
+F_{aa_{\tau}}q_1p_1(1+e^{-i\omega_0\tau_0})\\
&+F_{m_{\tau}a_{\tau}}(q_1+p_1)e^{-i\omega_0\tau_0}\big],\\
F_{002}=&F_{mm}+F_{aa}p_1^2+F_{m_{\tau}m_{\tau}}+F_{a_{\tau}a_{\tau}}p_1^2+2(F_{ma}p_1+
F_{mm_{\tau}}+F_{ma_{\tau}}p_1+F_{m_{\tau}a}p_1\\
&+F_{aa_{\tau}}p_1^2+F_{m_{\tau}a_{\tau}}p_1),\\
F_{020}=&\overline{F_{200}},\\
F_{011}=&\overline{F_{101}}.
\end{array} \end{equation*} and \begin{equation*}
\begin{array}{ll}
F_{210}=&3\big[F_{mmm}+F_{mma}(2q_1+\bar{q}_1)+F_{mmm_{\tau}}(2e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0})+F_{maa}q_1(2\bar{q}_1+q_1)
+F_{mma_{\tau}}(2q_1e^{-i\omega_0\tau_0}\\
&+\bar{q}_1e^{i\omega_0\tau_0})+2F_{mam_{\tau}}(q_1e^{-i\omega_0\tau_0}+q_1e^{i\omega_0\tau_0}+\bar{q}_1e^{-i\omega_0\tau_0})
+2F_{maa_{\tau}}q_1(q_1e^{-i\omega_0\tau_0}+\bar{q}_1e^{i\omega_0\tau_0}+\bar{q}_1e^{-i\omega_0\tau_0})\\
&+F_{mm_{\tau}m_{\tau}}(e^{-2i\omega_0\tau_0}+2)+2F_{mm_{\tau}a_{\tau}}(q_1+\bar{q}_1+q_1e^{-2i\omega_0\tau_0})
+F_{ma_{\tau}a_{\tau}}q_1(2\bar{q_1}+q_1e^{-2i\omega_0\tau_0})+F_{aaa}q_1^2\bar{q}_1\\
&+F_{aam_{\tau}}q_1(2\bar{q}_1e^{-i\omega_0\tau_0}+q_1e^{i\omega_0\tau_0})
+F_{aaa_{\tau}}q_1^2\bar{q}_1(2e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0})+F_{am_{\tau}m_{\tau}}(2q_1+\bar{q}_1e^{-2i\omega_0\tau_0})\\
&+2F_{am_{\tau}a_{\tau}}q_1(\bar{q}_1+\bar{q}_1e^{-2i\omega_0\tau_0}
+q_1)+F_{aa_{\tau}a_{\tau}}q_1^2\bar{q}_1(2+e^{-2i\omega_0\tau_0})+F_{m_{\tau}m_{\tau}m_{\tau}}e^{-i\omega_0\tau_0}\\
&+F_{m_{\tau}m_{\tau}a_{\tau}}(2q_1e^{-i\omega_0\tau_0}+\bar{q}_1e^{-i\omega_0\tau_0})
+F_{m_{\tau}a_{\tau}a_{\tau}}q_1e^{-i\omega_0\tau_0}(2\bar{q}_1+q_1)
+F_{a_{\tau}a_{\tau}a_{\tau}}q_1^2\bar{q}_1e^{-i\omega_0\tau_0}\big];\\
\end{array} \end{equation*} \begin{equation*}
\begin{array}{ll}
F_{102}=&3\big[F_{mmm}+F_{mma}(q_1+2p_1)+F_{mmm_{\tau}}(e^{-i\omega_0\tau_0}+2)+F_{maa}p_1(2q_1+p_1)
+F_{mma_{\tau}}(2p_1+q_1e^{-i\omega_0\tau_0})\\
&+2F_{mam_{\tau}}(q_1+p_1+p_1e^{-i\omega_0\tau_0})
+2F_{maa_{\tau}}p_1(q_1+p_1+q_1e^{-i\omega_0\tau_0})\\
&+F_{mm_{\tau}m_{\tau}}(2e^{-i\omega_0\tau_0}+1)+2F_{mm_{\tau}a_{\tau}}(p_1+p_1e^{-i\omega_0\tau_0}+q_1e^{-i\omega_0\tau_0})
+F_{ma_{\tau}a_{\tau}}p_1(p_1+2q_1e^{-i\omega_0\tau_0})\\
&+F_{aaa}q_1p^2_1+F_{aam_{\tau}}p_1(2q_1+p_1e^{-i\omega_0\tau_0})
+F_{aaa_{\tau}}q_1p^2_1(2+e^{-i\omega_0\tau_0})+F_{am_{\tau}m_{\tau}}(q_1+2p_1e^{-i\omega_0\tau_0})\\
&+2F_{am_{\tau}a_{\tau}}p_1(q_1+q_1e^{-i\omega_0\tau_0}
+p_1e^{-i\omega_0\tau_0})+F_{aa_{\tau}a_{\tau}}q_1p_1^2(1+2e^{-i\omega_0\tau_0})+F_{m_{\tau}m_{\tau}m_{\tau}}e^{-i\omega_0\tau_0}\\
&+F_{m_{\tau}m_{\tau}a_{\tau}}(q_1e^{-i\omega_0\tau_0}+2p_1e^{-i\omega_0\tau_0})
+F_{m_{\tau}a_{\tau}a_{\tau}}p_1e^{-i\omega_0\tau_0}(2q_1+p_1)
+F_{a_{\tau}a_{\tau}a_{\tau}}q_1p_1^2e^{-i\omega_0\tau_0}\big],
\end{array} \end{equation*} \begin{equation*}
\begin{array}{ll}
F_{111}=&6\Big\{F_{mmm}+F_{mma}(q_1+\bar{q}_1+p_1)+F_{mmm_{\tau}}(e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0}+1)
+F_{maa}(q_1\bar{q}_1+q_1p_1+p_1\bar{q}_1)\\
&+F_{mma_{\tau}}(q_1e^{-i\omega_0\tau_0}+\bar{q}_1e^{i\omega_0\tau_0}+p_1)
+F_{mam_{\tau}}\big[q_1(1+e^{i\omega_0\tau_0})+\bar{q}_1(1+e^{-i\omega_0\tau_0})+p_1(e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0})\big]\\
&+F_{maa_{\tau}}\big[q_1p_1(1+e^{-i\omega_0\tau_0})+q_1\bar{q}_1(e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0})
+\bar{q}_1p_1(1+e^{i\omega_0\tau_0})\big]
+F_{mm_{\tau}m_{\tau}}(e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0}+1)\\
&+F_{mm_{\tau}a_{\tau}}\big[q_1(1+e^{-i\omega_0\tau_0})+\bar{q}_1(1+e^{i\omega_0\tau_0})
+p_1(e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0})\big]\\
&+F_{ma_{\tau}a_{\tau}}(q_1\bar{q_1}+\bar{q}_1p_1e^{i\omega_0\tau_0}+q_1p_1e^{-i\omega_0\tau_0})
+F_{aaa}q_1\bar{q}_1p_1
+F_{aam_{\tau}}(q_1\bar{q}_1+\bar{q}_1p_1e^{-i\omega_0\tau_0}+q_1p_1e^{i\omega_0\tau_0})\\
&+F_{aaa_{\tau}}q_1\bar{q}_1p_1(1+e^{-i\omega_0\tau_0}+e^{i\omega_0\tau_0})
+F_{am_{\tau}m_{\tau}}(p_1+q_1e^{i\omega_0\tau_0}+\bar{q}_1e^{-i\omega_0\tau_0})\\
&+F_{am_{\tau}a_{\tau}}\big[(q_1\bar{q}_1(e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0})+q_1p_1(1+e^{i\omega_0\tau_0})
+\bar{q}_1p_1(1+e^{-i\omega_0\tau_0})\big]\\
&+F_{aa_{\tau}a_{\tau}}q_1\bar{q}_1p_1(1+e^{i\omega_0\tau_0}+e^{-i\omega_0\tau_0})+F_{m_{\tau}m_{\tau}m_{\tau}}
+F_{m_{\tau}m_{\tau}a_{\tau}}(q_1+\bar{q}_1+p_1)\\
&+F_{m_{\tau}a_{\tau}a_{\tau}}(q_1\bar{q}_1+q_1p_1+\bar{q}_1p_1)
+F_{a_{\tau}a_{\tau}a_{\tau}}q_1\bar{q}_1p_1\Big\};\\
\end{array}
\end{equation*} \begin{equation*} \begin{array}{ll} F_{003}=&F_{mmm}+3F_{mmm_{\tau}}+3F_{mm_{\tau}m_{\tau}}+F_{m_{\tau}m_{\tau}m_{\tau}}+3F_{mma}p_1+3F_{mma_{\tau}}p_1+6F_{mam_{\tau}}p_1
+6F_{mm_{\tau}a_{\tau}}p_1\\
&+3F_{m_{\tau}m_{\tau}a_{\tau}}p_1+3F_{am_{\tau}m_{\tau}}p_1+3F_{maa}p_1^2+6F_{maa_{\tau}}p_1^2+
3F_{ma_{\tau}a_{\tau}}p_1^2
+F_{aaa}p_1^3+3F_{m_{\tau}a_{\tau}a_{\tau}}p_1^2\\
&+3F_{aam_{\tau}}p_1^2+3F_{aaa_{\tau}}p_1^3+6F_{am_{\tau}a_{\tau}}p_1^2+
3F_{aa_{\tau}a_{\tau}}p_1^3+F_{a_{\tau}a_{\tau}a_{\tau}}p_1^3.
\end{array} \end{equation*}
\end{document} |
\begin{document}
\title{A weighted extremal function and equilibrium measure} \author{Len Bos, Norman Levenberg, Sione Ma`u and Federico Piazzon}
\maketitle
\begin{abstract} Let $K=\mathbb{R}^n\subset \mathbb{C}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2 = x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in $\mathbb{R}^n\subset \mathbb{C}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of $\mathbb{C}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$:
$$V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2})$$ where $z=(z_1,...,z_n)$ and $z^2 = z_1^2+\cdots +z_n^2$. As a corollary, we find that the Alexander capacity $T_{\omega}(\mathbb{R} \mathbb P^n)$ of $\mathbb{R} \mathbb P^n$ is $1/\sqrt 2$. We also compute the Monge-Amp\`ere measure of $V_{K,Q}$: $$(dd^cV_{K,Q})^n = n!\frac{1}{(1+x^2)^{\frac{n+1}{2}}}dx.$$ \end{abstract}
\section{Introduction} For $K\subset \mathbb{C}^n$ compact, define the usual Siciak-Zaharjuta {\it extremal function} \begin{equation}\label{vk}
V_K(z)
:= \max \left\{ 0 ,
\sup _p \left\{ \frac{1}{deg(p)} \log|p(z)|: p \ \hbox{poly.}, \ ||p||_K:=\max_{z\in K} |p(z)| \leq 1\right\}
\right\}, \end{equation} where the supremum is taken over (non-constant) holomorphic polynomials $p$, and let $V_K^*(z):= \limsup_{\zeta \to z} V_K(\zeta)$ be its uppersemicontinuous (usc) regularization. If $K\subset \mathbb{C}^n$ is closed, a nonnegative uppersemicontinuous function $w:K\to [0, \infty)$ with $\{z\in K: w(z)=0\}$ pluripolar is called a weight function on $K$ and $Q(z):=-\log w(z)$ is the {\it potential} of $w$. The associated {\it weighted extremal function} is
$$V_{K,Q}(z):=\sup \{\frac{1}{deg(p)}\log |p(z)|: p \ \hbox{poly.}, \ ||pe^{-deg(p)Q}||_K\leq 1\}.$$
Note $V_K=V_{K,0}$. For unbounded $K$, the potential $Q$ is required to grow at least like $\log |z|$. If, e.g,
$$\liminf_{z\in K, \ |z|\to +\infty}\bigl(Q(z)-\log |z|\bigr) > -\infty $$ (we call $Q$ {\it weakly admissible}), then the Monge-Amp\`ere measure $(dd^cV_{K,Q}^*)^n$ may or may not
have compact support. A priori these extremal functions may be defined in terms of upper envelopes of {\it Lelong class} functions: we write $L(\mathbb{C}^n)$ for the set of all plurisubharmonic (psh) functions $u$ on $\mathbb{C}^n$ with the property that $u(z) - \log |z| = 0(1), \ |z| \to \infty$ and
$$ L^+(\mathbb{C}^n):=\{u\in L(\mathbb{C}^n): u(z)\geq \log^+|z| + C\}$$ where $C$ is a constant depending on $u$. For $K$ compact, either $V_K^*\in L^+(\mathbb{C}^n)$ or $V_K^*\equiv \infty$, this latter case occurring when $K$ is pluripolar; i.e., there exists $u\not \equiv -\infty$ psh on a neighborhood of $K$ with $K\subset \{u=-\infty\}$. In the setting of weakly admissible $Q$ it is a result of \cite{bs} that, provided the function $$\sup \{u(z): u\in L(\mathbb{C}^n), \ u\leq Q \ \hbox{on} \ K\}$$ is continuous, it coincides with $V_{K,Q}(z)$.
If we let $X=\mathbb P^n$ with the usual K\"ahler form $\omega$ normalized so that $\int_{\mathbb P^n} \omega^n =1$, we can define the class of {\it $\omega-$psh functions} (cf., \cite{GZ}) $$PSH(X,\omega) :=\{\phi \in L^1(X): \phi \ \hbox{usc}, \ dd^c\phi +\omega \geq 0\}.$$ Let ${\bf z}:=[z_0:z_1:\cdots :z_n]$ be homogeneous coordinates on $X=\mathbb P^n$. Identifying $\mathbb{C}^n$ with the affine subset of $\mathbb P^n$ given by $\{[1:z_1:\cdots:z_n]\}$, we can identify the $\omega-$psh functions with the Lelong class $L(\mathbb{C}^n)$, i.e., $$PSH(X,\omega) \approx L(\mathbb{C}^n),$$ and the bounded (from below) $\omega-$psh functions coincide with the subclass $L^+(\mathbb{C}^n)$: if $\phi \in PSH(X,\omega)$, then
$$u(z)=u(z_1,...,z_n):= \phi ([1:z_1:\cdots:z_n])+\frac{1}{2}\log (1+|z|^2)\in L(\mathbb{C}^n);$$ if $u\in L(\mathbb{C}^n)$, define $\phi \in PSH(X,\omega)$ via
$$\phi ([1:z_1:\cdots:z_n])=u(z)-\frac{1}{2}\log (1+|z|^2) \ \hbox{and}$$
$$\phi ([0:z_1:\cdots:z_n])=\limsup_{|t|\to \infty, \ t\in \mathbb{C}}[u(tz)-\frac{1}{2}\log (1+|tz|^2)].$$
Abusing notation, we write $u= \phi +u_0$ where $u_0(z):=\frac{1}{2}\log (1+|z|^2)$. Given a closed subset $K\subset \mathbb P^n$ and a function $q$ on $K$, we can define a {\it weighted $\omega-$psh extremal function} $$v_{K,q}({\bf z}):=\sup \{ \phi({\bf z}): \phi \in PSH(X,\omega), \ \phi \leq q \ \hbox{on} \ K\}.$$ Thus if $K\subset \mathbb{C}^n \subset \mathbb P^n$, for $[1:z_1:\cdots:z_n]=[1:z]\in \mathbb{C}^n$ we have \begin{equation}\label{wtdrel} v_{K,q}([1:z])=\sup \{u(z): u\in L(\mathbb{C}^n), \ u\leq u_0 +q \ \hbox{on} \ K\} -u_0(z)=V_{K,u_0+q}(z)-u_0(z).\end{equation} If $q=0$, the {\it Alexander capacity} $T_{\omega}(K)$ of $K\subset \mathbb P^n$ was defined in \cite{GZ} as $$T_{\omega}(K):=\exp {[-\sup_{\mathbb P^n} v_{K,0}]}.$$ This notion has applications in complex dynamics; cf., \cite{DS}.
These extremal psh and $\omega-$psh functions $V_K, V_{K,Q}$ and $v_{K,0}, v_{K,q}$, as well as the homogeneous extremal psh function $H_E$ of $E\subset \mathbb{C}^n$ (whose definition we recall in the next section), are very difficult to compute explicitly. Even when an explicit formula exists, computation of the associated Monge-Amp\`ere measure is problematic. Our main goal in this paper is to utilize a novel approach to explicitly compute $V_{K,Q}$ and $(dd^cV_{K,Q})^n$ for the closed set $K=\mathbb{R}^n\subset \mathbb{C}^n$ and the weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$ where $z^2 =z_1^2 +\cdots + z_n^2$ (see (\ref{magic}) or Theorem \ref{magic1}, and (\ref{monge})). Note the potential $Q(z)$ in this case is the standard K\"ahler potential $u_0(z)$ restricted to $\mathbb{R}^n$. As an application we can calculate the Alexander capacity $T_{\omega}(\mathbb{R} \mathbb P^n)$ of $\mathbb{R} \mathbb P^n$ (Corollary \ref{magiccor}).
We offer several methods to explicitly compute $V_{K,Q}$. For the first one, we relate this weighted extremal function to: \begin{enumerate} \item the extremal function $V_{B_{n+1}}$ of the {\it real $(n+1)-$ball} $$B_{n+1}=\{(u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2\leq1\}$$ in $\mathbb{R}^{n+1}\subset \mathbb{C}^{n+1}$ as well as \item the extremal function $V_{\widetilde K}$ of the {\it real $n-$sphere} $$\widetilde K = \{ (u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2=1\}$$ in $\mathbb{R}^{n+1}$ {\it considered as a compact subset of the complexified $n-$sphere} $$A := \{ (W_0,...,W_n)\in \mathbb{C}^{n+1}: \sum_{j=0}^n W_j^2=1\}$$ in $\mathbb{C}^{n+1}$. This function is the {\it Grauert tube function} of $\widetilde K$ in $A$; cf., \cite{Z}. \end{enumerate}
A similar (perhaps simpler) idea is a relation between $V_{K,Q}$ and \begin{enumerate} \item the extremal function $V_{B_{n}}$ of the {\it real $n-$ball}
$$B_n:= \{(u_1,...,u_n)\in \mathbb{R}^{n}: \sum_{j=1}^n u_j^2\leq1\}$$
in $\mathbb{R}^n\subset \mathbb{C}^{n}$ and \item the homogeneous extremal function $H_S$ of the {\it real $n-$upper hemisphere} $$S:=\{(u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2\leq1, \ u_0 >0\}$$ in $\mathbb{R}^{n+1}$ considered as a subset of $A$ \end{enumerate} \noindent obtained by projecting $S$ onto $B_n$.
In both cases we appeal to two well-known and highly non-trivial results: \begin{enumerate} \item using Theorem \ref{blmthm} (or \cite{bar}) we have a foliation of $\mathbb{C}^n\setminus B_n$ (and $\mathbb{C}^{n+1} \setminus B_{n+1}$) by complex ellipses on which $V_{B_n}$ ($V_{B_{n+1}})$ is harmonic; and \item using Theorem \ref{sadthm} we have $V_{\widetilde K}$ (and $H_S$) is locally bounded on $A$ and is maximal on $A\setminus \widetilde K$ (on $A\setminus S$). \end{enumerate} \noindent See the next section for statements of Theorems \ref{blmthm} and \ref{sadthm} and section 4 for details of these relations.
Bloom (cf., \cite{BL} and \cite{Bloomtams}) introduced a technique to switch back and forth between certain pluripotential-theoretic notions in $\mathbb{C}^{n+1}$ and their weighted counterparts in $\mathbb{C}^n$; we recall this in the next section. In section 3, we discuss a modification of Bloom's technique suitable for special weights $w$ and we use this modification in section 4 to construct a formula for $V_{K,Q}$ on a neighborhood of $\mathbb{R}^n$ for the set $K=\mathbb{R}^n\subset \mathbb{C}^n$ and weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$. This formula gives an explicit candidate $u\in L(\mathbb{C}^n)$ for $V_{K,Q}$. In section 5 we give another ``geometric'' interpretation of $u$ by observing a relationship with the Lie ball
$$L_n :=\{z=(z_1,...,z_n)\in \mathbb{C}^n: |z|^2 +\{ |z|^4 - |z^2|^2\}^{1/2} \leq 1\}$$ which we use to explicitly compute that $(dd^cu)^n=0$ on $\mathbb{C}^n\setminus \mathbb{R}^n$, verifying that $u=V_{K,Q}$. As a corollary, we compute the Alexander capacity $T_{\omega}(\mathbb{R} \mathbb P^n)$ of $\mathbb{R} \mathbb P^n$. Finally, section 6 utilizes results from \cite{blmr} to compute an explicit formula for the Monge-Amp\`ere measure $(dd^cV_{K,Q})^n$.
\section{Known results on extremal functions} In this section, we list some results and connections about extremal functions, all of which will be utilized.
One particular situation where we know much information about $V_K$ is when $K$ is a convex body in $\mathbb{R}^n$; i.e., $K\subset \mathbb{R}^n$ is compact, convex and int$_{\mathbb{R}^n}K\not =\emptyset$. \begin{theorem}\label{blmthm} Let $K\subset \mathbb{R}^n$ be a convex body. Through every point $z\in\mathbb{C}^n\setminus K$ there is either a complex ellipse $E$ with $z\in E$ such that $V_K$ restricted to $E$ is harmonic on $E\setminus K$, or there is a complexified real line $L$ with $z\in L$ such that $V_K$ is harmonic on $L\setminus K$. For such $E$, $E\cap K$ is a real ellipse inscribed in $K$ with the property that for its given eccentricity and orientation, it is the ellipse with largest area completely contained in $K$; for such $L$, $L\cap K$ is the longest line segment (for its given direction) completely contained in $K$. \end{theorem}
We refer the reader to Theorem 5.2 and Section 6 of \cite{blm2}; see also \cite{blm}. The ellipses and lines in Theorem \ref{blmthm} have parametrizations of the form $$ F(\zeta) = a + c\zeta + \frac{\bar c}{\zeta}, $$
$a\in\mathbb{R}^n$, $c\in\mathbb{C}^n$, $\zeta \in \mathbb{C}$ with $V_K(F(\zeta))=\log^+|\zeta|$ ($\bar c$ denotes the component-wise complex conjugate of $c$). These are higher dimensional analogs of the classical Joukowski function $\zeta\mapsto \frac{1}{2}(\zeta + \frac{1}{\zeta})$. For $K = B_n$, the real unit ball in $\mathbb{R}^n \subset \mathbb{C}^n$, the real ellipses $E\cap B_n$ and lines $L\cap B_n$ in Theorem \ref{blmthm} are symmetric with respect to the origin and, other than great circles in the real boundary of $B_n$, each $E\cap B_n$ and $L\cap B_n$ hits this real boundary at exactly two antipodal points. Lundin proved \cite{lunpre}, \cite{bar} that \begin{equation}
\label{eq:realball} V_K(z) =\frac{1}{2} \log h(|z|^2 + |z^2 -
1|), \end{equation}
where $|z|^2 = \sum |z_j|^2, \ z^2 = \sum z_j^2,$ and $h$ is the inverse Joukowski map $h(\frac{1}{2}(t + \frac{1}{t})) = t$ for $1 \leq t \in \mathbb{R}$. In this example, the Monge-Amp\`ere measure $(dd^cV_K)^n$ has the explicit form
$$(dd^cV_K)^n = n! \ vol(K) \ \frac{dx}{(1-|x|^2)^{\frac{1}{2}}}:=n! \ vol(K) \ \frac{dx_1 \wedge \cdots \wedge dx_n}{(1- |x|^2)^{\frac{1}{2}}} $$ (see also (\ref{eq:monge})).
We may consider the class
$$H:=\{u \in L(\mathbb{C}^n): \ u(tz) =\log {|t|} +u(z), \ t\in \mathbb{C}, \ z \in \mathbb{C}^n \} $$ of {\it logarithmically homogeneous} psh functions and, for $E\subset \mathbb{C}^n$, the {\it homogeneous extremal function of $E$} denoted by $H_E^*$ where $$H_E(z):=\max [0,\sup \{u(z):u \in H, \ u\leq 0 \ \hbox{on} \
E\}]. $$ Note that $H_E(z)\leq V_E(z)$. If $E$ is compact, we have
$$H_E(z)=\max [0,\sup \{\frac{1}{deg (h)}\log {|h(z)|}: h \ \hbox{homogeneous polynomial}, \ ||h||_E\leq 1\}]. $$ The $H-$principle of Siciak (cf., \cite{Kl}) gives a one-to-one correspondence between \begin{enumerate} \item homogeneous polynomials $H_d(t,z)$ of a fixed degree $d$ in $\mathbb{C}_t\times \mathbb{C}^n_z$ and polynomials $p_d(z)=H_d(1,z)$ of degree $d$ in $\mathbb{C}^n_z$ via $$H_d(t,z):=t^dp_d(z/t); $$ \item psh functions $h(t,z)$ in $H(\mathbb{C}_t\times \mathbb{C}^n_z)$ and psh functions $u(z)=h(1,z)$ in $L(\mathbb{C}^n_z)$ via
$$h(t,z)=\log {|t|} +u(z/t) \ \hbox{if} \ t\not = 0; \ h(0,z):=\limsup_{(t,\zeta)\to (0,z)}h(t,z); $$ \item extremal functions $V_E$ of $E\subset \mathbb{C}^n_z$ and homogeneous extremal functions $H_{1\times E}$ via 2.; i.e., \begin{equation}\label{easyfcn}V_E(z)=H_{1\times E}(1,z). \end{equation} \end{enumerate}
\noindent To expand upon 3., given a compact set $E\subset \mathbb{C}^n$, if one forms the circled set ($S$ is circled means $z\in S \iff e^{i\theta}z\in S$)
$$Z(E):=\{(t,tz)\in \mathbb{C}^{n+1}: z\in E, \ |t|=1\} \subset \mathbb{C}^{n+1},$$ then $$H_{Z(E)}(1,z) = V_E(z);$$ indeed, for $t\not = 0$,
$$H_{Z(E)}(t,z) = V_E(z/t)+\log |t|.$$
Note that $Z(E)$ is the ``circling'' of the set $\{1\}\times E\subset \mathbb{C}^{n+1}$. In general, if $E\subset \mathbb{C}^n$, the set $$E_c:=\{e^{i\theta}z: z\in E, \ \theta \in \mathbb{R}\}$$ is the smallest circled set containing $E$. If $E$ is compact, then $\widehat E_c$, the polynomial hull of $E_c$, is given by
$$\widehat E_c=\{tz: \ z\in E, \ |t|\leq 1\}$$ which coincides with the {\it homogeneous polynomial hull} of $E$:
$$\widehat E_{hom}:=\{z\in \mathbb{C}^n: |p(z)|\leq ||p||_E \ \hbox{for all homogeneous polynomials} \ p\}.$$ We have $H_{E_c}=V_{E_c}$. For future use we remark that if $E\subset F$ with $H_E=H_F=V_F$, it is {\it not} necessarily true that $V_E=H_E$. As a simple example, we can take $E=B_n$, the real unit ball, and $F=\widehat E_c=\widehat E_{hom}$. Then $F=L_n$, the {\it Lie ball}
$$L_n =\{z=(z_1,...,z_n)\in \mathbb{C}^n: |z|^2 +\{ |z|^4 - |z^2|^2\}^{1/2} \leq 1\}$$ (see section 5). Here, $V_{B_n}\not = V_{L_n}$.
More generally, if $K\subset \mathbb{C}^n$ is closed and $w$ is a weight function on $K$, we can form the circled set
$$Z(K,Q):= \{(t,tz)\in \mathbb{C}^{n+1}: z\in E, \ |t|=w(z)\}$$ and then $$H_{Z(K,Q)} (1,z) = V_{K,Q}(z);$$ indeed, for $t\not = 0$,
$$H_{Z(K,Q)} (t,z) = V_{K,Q}(z/t)+\log |t|.$$ This is the device utilized by Bloom (cf., \cite{BL} and \cite{Bloomtams}) alluded to in the introduction.
Finally, we mention the following beautiful result of Sadullaev \cite{Sad}. \begin{theorem}\label{sadthm} Let $A$ be a pure $m-$dimensional, irreducible analytic subvariety of $\mathbb{C}^n$ where $1\leq m \leq n-1$. Then $A$ is algebraic if and only if for some (all) $K\subset A$ compact and nonpluripolar in $A$, $V_K$ in (\ref{vk}) is locally bounded on $A$.
\end{theorem}
\noindent Note that $A$ and hence $K$ is pluripolar in $\mathbb{C}^n$ so $V_K^*\equiv \infty$; moreover, $V_K=\infty$ on $\mathbb{C}^n\setminus A$. In this setting, $V_K|_A$ (precisely, its usc regularization in $A$) is maximal on the regular points $A^{reg}$ of $A$ outside of $K$; i.e., $(dd^cV_K|_A)^m=0$ there, and $V_K|_A \in L(A)$. Here $L(A)$ is the set of psh functions $u$ on $A$ ($u$ is psh on $A^{reg}$ and locally bounded above on $A$) with the property that $u(z) - \log |z| = 0(1)$ as $|z| \to \infty$ through points in $A$, see \cite{Sad}.
\section{Relating extremal functions} Let $K\subset \mathbb{C}^n$ be closed and let $f$ be holomorphic on a neighborhood $\Omega$ of $K$. We define $F:\Omega \subset \mathbb{C}^n\to \mathbb{C}^{n+1}$ as $$F(z):=(f(z),zf(z))=W=(W_0,W')=(W_0,W_1,...,W_n)$$ where $W'=(W_1,...,W_n)$. Thus $$W_0= f(z), \ W_1 = z_1f(z), ..., \ W_n=z_nf(z).$$ Moreover we assume there exists a polynomial $P=P(z_0,z)$ in $\mathbb{C}^{n+1}$ with $P(f(z),z)=0$ for $z\in \Omega$; i.e., $f$ is {\it algebraic}. Taking such a polynomial $P$ of minimal degree, let \begin{equation}\label{variety} A:=\{W\in \mathbb{C}^{n+1}:P(W_0,W'/W_0)=P(W_0,W_1/W_0,...,W_n/W_0)=0 \}.\end{equation} Note that writing $P(W_0,W'/W_0)=\widetilde P(W_0,W')/W_0^s$ where $\widetilde P$ is a polynomial in $\mathbb{C}^{n+1}$ and $s$ is the degree of $P(z_0,z)$ in $z$ we see that $A$ differs from the algebraic variety $$\widetilde A:=\{W\in \mathbb{C}^{n+1}:\widetilde P(W_0,W')=0\}$$ by at most the set of points in $A$ where $W_0=0$, which is pluripolar in $A$. Thus we can apply Sadullaev's Theorem \ref{sadthm} to nonpluripolar subsets of $A$. Now $P(f(z),z)=0$ for $z\in \Omega$ says that $$F(\Omega)=\{(f(z),zf(z)): z \in \Omega\}\subset A.$$
We can define a weight function $w(z):=|f(z)|$ which is well defined on all of $\Omega$ and in particular on $K$; as usual, we set
\begin{equation}\label{need2}Q(z):=-\log w(z) = -\log |f(z)|.\end{equation} We will need our potentials defined in (\ref{need2}) to satisfy
\begin{equation}\label{need}Q(z):=\max \{-\log |W_0|: W\in A, \ W'/W_0=z\}\end{equation} and we mention that (\ref{need}) can give an a priori definition of a potential for those $z\in \mathbb{C}^n$ at which there exist $W\in A$ with $W'/W_0=z$.
We observe that for $K\subset \Omega$, we have two natural associated subsets of $A$: \begin{enumerate} \item $\widetilde K:= \{W\in A: W'/W_0\in K\}$ and \item $F(K)=\{W=F(z)\in A: z \in K\}$. \end{enumerate} \noindent Note that $F(K)\subset \widetilde K$ and the inclusion can be strict.
\begin{proposition}\label{exfcn} Let $K\subset \mathbb{C}^n$ be closed with $Q$ in (\ref{need2}) satisfying (\ref{need}). If $F(K)$ is nonpluripolar in $A$, $$V_{K,Q}(z)-Q(z)\leq H_{F(K)}(W) \ \hbox{for} \ z\in \Omega \ \hbox{with} \ f(z)\not =0 $$ where the inequality is valid for $W=F(z)\in F(\Omega)$. \end{proposition}
\noindent This reduces to (\ref{easyfcn}) if $w(z)\equiv 1$ ($Q(z)\equiv 0$) in which case $F(K)= \{1\}\times K$.
\begin{remark} In general, Proposition \ref{exfcn} only gives estimates for $V_{K,Q}(z)$ if $z\in \Omega$ and $f(z)\not =0$. We will use this and Lemma \ref{lowerest} in the next section to get a formula for $V_{K,Q}(z)$ when $K=\mathbb{R}^n\subset \mathbb{C}^n$ and the weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$ for $z$ in a neighborhood $\Omega$ of $\mathbb{R}^n$ and in section 5 we will verify that this formula is valid on all of $\mathbb{C}^n$.
\end{remark} \begin{proof} First note that for $z\in K$ and $W=F(z)\in F(K)$, given a polynomial $p$ in $\mathbb{C}^n$,
$$|w(z)^{deg p}p(z)|=|f(z)|^{deg p}|p(z)|= |W_0^{deg p} p(W'/W_0)|=|\widetilde p(W)|$$ where $\widetilde p$ is the homogenization of $p$. Thus $||w^{deg p}p||_K\leq 1$ implies $|\widetilde p|\leq 1$ on $F(K)$. Now fix $z\in \Omega$ at which $f(z)\not =0$ (so $Q(z)<\infty$) and fix $\epsilon >0$. Choose a polynomial $p=p(z)$ with $||w^{deg p}p||_K\leq 1$ and
$$\frac{1}{deg p} \log |p(z)|\geq V_{K,Q}(z) -\epsilon.$$ Thus
$$V_{K,Q}(z) -\epsilon -Q(z)\leq \frac{1}{deg p} \log |p(z)|-Q(z).$$ For $W\in A$ with $W_0\not =0$ and $W'/W_0=z$, the above inequality reads:
$$V_{K,Q}(z) -\epsilon -Q(z)\leq \frac{1}{deg p} \log |p(W'/W_0)|-Q(W'/W_0)\leq \frac{1}{deg p} \log |p(W'/W_0)|+\log |W_0|$$ from (\ref{need}). But
$$\frac{1}{deg p} \log |p(W'/W_0)|+\log |W_0|=\frac{1}{deg p} \log |W_0^{deg p}p(W'/W_0)|=\frac{1}{deg \widetilde p} \log |\widetilde p(W)|.$$ This shows that
$$V_{K,Q}(z) -\epsilon -Q(z)\leq \sup \{\frac{1}{deg \widetilde p} \log |\widetilde p(W)|: |\widetilde p|\leq 1 \ \hbox{on} \ F(K)\}\leq H_{F(K)}(W).$$
\end{proof}
Next we prove a lower bound involving $\widetilde K$ which will be applicable in our special case.
\begin{definition} \label{three4} \rm Let $A\subset\mathbb{C}^{n+1}$ be an algebraic hypersurface. We say that $A$ is \emph{bounded on lines through the origin} if there exists a uniform constant $c\geq 1$ such that for all $W\in A$,
if $\alpha W\in A$ also holds for some $\alpha\in\mathbb{C}$, then $|\alpha|\leq c$. \end{definition}
\begin{example} \label{three5} \rm A simple example of a hypersurface bounded on lines through the origin is one given by an equation of the form $p(W)=1$, where $p$ is a homogeneous polynomial. In this case, if $\alpha W\in A$ then $$1=p(\alpha W)=\alpha^{deg p}p(W)=\alpha^{deg p},$$ so $\alpha$ must be a root of unity. Hence we may take $c=1$. \end{example}
In order to get a lower bound on $V_{K,Q}-Q$ we need to be able to extend $Q$ to a function in $L(\mathbb{C}^n)$.
\begin{lemma}\label{lowerest}
Let $K\subset\mathbb{C}^n$ and let $Q(z)=-\log |f(z)|$ with $f$ defined and holomorphic on $\Omega\supset K$. Define $A$ as in (\ref{variety}) and assume $Q$ satisfies (\ref{need}). We suppose $A$ is bounded on lines through the origin, $\widetilde K$ is a nonpluripolar subset of $A$, and that $Q$ has an extension to $\mathbb{C}^n$ (which we still call $Q$) satisfying (\ref{need}) such that $Q\in L(\mathbb{C}^n)$. Then given $z\in\mathbb{C}^n$, $$ H_{\widetilde K}(W)\leq V_{\widetilde K}(W)\leq V_{K,Q}(z)-Q(z) $$ for all $W=(W_0,W')\in A$ with $W'/W_0=z$. \end{lemma}
\begin{proof} The left-hand inequality $H_{\widetilde K}(W)\leq V_{\widetilde K}(W)$ is immediate. For the right-hand inequality, we first note that $V_{\widetilde K}(W)\in L(A)$ if $\widetilde K$ is nonpluripolar in $A$. Hence there exists a constant $C\in\mathbb{R}$ such that $$
V_{\widetilde K}(W) \leq \log|W| + C
= \log|W_0| + \frac{1}{2}\log(1+|W'/W_0|^2) + C $$
for all $W\in A$ with $W_0\not = 0$.
Define the function $$ U(z):= \max\{V_{\widetilde K}(W): W\in A, W'/W_0=z\} + Q(z). $$ Note that the right-hand side is a locally finite maximum since $A$ is an algebraic hypersurface. Away from the singular points $A^{sing}$ of $A$ one can write $V_{\widetilde K}(W)$ as a psh function in $z$ by composing it with a local inverse of the map $A\ni W\mapsto z=W'/W_0\in\mathbb{C}^n$. Hence $U$ is psh off the pluripolar set $$\{z\in\mathbb{C}^n: z=W'/W_0 \ \hbox{ for some } \ W\in A^{sing}\},$$ and hence psh everywhere since it is clearly locally bounded above on $\mathbb{C}^n$.
Also, since $V_{\widetilde K}=0$ on $\widetilde K$ it follows that $U\leq Q$ on $K$. We now verify that $U\in L(\mathbb{C}^n)$ by checking its growth. By the definitions of $U$ and $Q$ and (\ref{need}), given $z\in\mathbb{C}^n$ there exist $W,V\in A$, with $z=W'/W_0= V'/V_0$, such that $$
U(z) = V_{\widetilde K}(W) + Q(z) \ \hbox{ and } \ Q(z)=-\log|V_0|. $$
Note that $W=\alpha V$, and since $A$ is uniformly bounded on lines through the origin, there is a uniform constant $c$ (independent of $W,V$) such that $|\alpha|\leq c$. We then compute \begin{eqnarray*}
U(z) = V_{\widetilde K}(W) - \log|V_0|
&\leq& V_{\widetilde K}(W) - \log|W_0| + \log c \\
&\leq& \log|W| + C - \log|W_0| + \log c \\
&=& \log|W/W_0| + C +\log c = \tfrac{1}{2}\log(1+|z|^2) + C +\log c \end{eqnarray*} where $C>0$ exists since $V_{\widetilde K}\in L(A)$. Hence $U\in L(\mathbb{C}^n)$, and since $U\leq Q$ on $K$ this means that $U(z)\leq V_{K,Q}(z)$. By the definition of $U$, $$ V_{\widetilde K}(W)+Q(z)\leq V_{K,Q}(z) $$ for all $W\in A$ such that $W'/W_0=z$, which completes the proof. \end{proof}
The situation of Lemma \ref{lowerest} will be the setting of our example in the next section.
\section{The weight $w(z)=|\frac{1}{(1+z^2)^{1/2}}|$ and $K=\mathbb{R}^n$} We consider the closed set $K=\mathbb{R}^n\subset \mathbb{C}^n$ and the weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$ where $z^2 =z_1^2 +\cdots + z_n^2$. Note that $f(z)\not =0$ and we may extend $Q(z)=-\log |f(z)|$ to all of $\mathbb{C}^n$ as $Q(z)=\frac{1}{2}\log |1+z^2|\in L(\mathbb{C}^n)$. Since $$(1+z^2)\cdot f(z)^2 -1 =0,$$ we take $$P(z_0, z) = (1+z^2)z_0^2 -1.$$ Here,
$$A = \{W:P(W_0,W'/W_0)=(1+W'^2/W_0^2)W_0^2-1= W_0^2+W'^2-1=0\}$$
is the complexified sphere in $\mathbb{C}^{n+1}$. From Definition \ref{three4} and Example \ref{three5}, $A$ is bounded on lines through the origin. Note that $f$ is clearly holomorphic in a neighborhood of $\mathbb{R}^n$; thus we can take, e.g., $\Omega=\{z= x+iy \in \mathbb{C}^n: y^2=y_1^2 +\cdots + y_n^2 < s <1\}$ in Proposition \ref{exfcn} and Lemma \ref{lowerest} where $z_j=x_j+iy_j$. Condition (\ref{need}) holds for $Q(z)=\frac{1}{2}\log |1+z^2|\in L(\mathbb{C}^n)$ at $z\in \mathbb{C}^n$ for which there exist $W\in A$ with $W'/W_0=z$ since $W=(W_0,W')\in A$ implies $W_0=\pm \sqrt {1-(W')^2}$ so that $|W_0|$ is the same for each choice of $W_0$. We have
$$F(K)= \{(f(z),zf(z)): z=(z_1,...,z_n) \in K=\mathbb{R}^n\}=\{(\frac{1}{(1+x^2)^{1/2}},\frac{x}{(1+x^2)^{1/2}}):x\in \mathbb{R}^n\}.$$
Writing $u_j = {\rm Re} W_j$, we see that
$$F(K)=\{ (u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2=1, \ u_0 >0\}.$$
On the other hand,
$$\widetilde K = \{W\in A: W'/W_0\in K\}= \{ (u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2=1\}.$$ Clearly $\widetilde K$ is nonpluripolar in $A$ which completes the verification that Lemma \ref{lowerest} is applicable.
We also observe that since for any homogeneous polynomial $h=h(W_0,...,W_n)$ we have
$$|h(-u_0,u_1,...,u_n)|=|h(u_0,-u_1,...,-u_n)|,$$
the homogeneous polynomial hulls of $\widetilde K$ and $\overline {F(K)}$ in $\mathbb{C}^{n+1}$ coincide so that $H_{\widetilde K} = H_{\overline{F(K)}}$ in $A$. Since
$$\overline{F(K)}\setminus F(K)=\{ (u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2=1, \ u_0 =0\}\subset A\cap
\{ W_0=0\}$$
is a pluripolar subset of $A$,
\begin{equation}\label{hulleq} H_{\widetilde K} = H_{F(K)}\end{equation}
on $A\setminus P$ where $P\subset A$ is pluripolar in $A$. Combining (\ref{hulleq}) with Proposition \ref{exfcn} and Lemma \ref{lowerest}, we have
\begin{equation}\label{fullin} H_{\widetilde K} (W)=V_{\widetilde K}(W) = V_{K,Q}(z) -Q(z)= H_{F(K)}(W)\end{equation}
for $z\in \widetilde \Omega :=\Omega \setminus \widetilde P$ and $W=F(z)$ where $\widetilde P$ is pluripolar in $\mathbb{C}^n$.
To compute the extremal functions in this example, we first consider $V_{\widetilde K}$ in $A$. Let
$$B:=B_{n+1}=\{(u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2\leq1\}$$
be the real $(n+1)-$ball in $\mathbb{C}^{n+1}$.
\begin{proposition} We have
$$V_B(W)=V_{\widetilde K}(W)$$
for $W\in A$.
\end{proposition}
\begin{proof} Clearly $V_B|_A \leq V_{\widetilde K}$. To show equality holds, the idea is that if we consider the complexified extremal ellipses $L_{\alpha}$ as in Theorem \ref{blmthm} for $B$ whose real points $S_{\alpha}$ are great circles on $\widetilde K$, the boundary of $B$ in $\mathbb{R}^{n+1}$, then the union of these varieties fill out $A$: $\cup_{\alpha} L_{\alpha} =A$. Since $V_B|_{L_{\alpha}}$ is {\it harmonic}, we must have $V_B|_{L_{\alpha}} \geq V_{\widetilde K}|_{L_{\alpha}}$ so that $V_B|_A = V_{\widetilde K}$.
To see that $\cup_{\alpha} L_{\alpha} =A$, we first show $A\subset \cup_{\alpha} L_{\alpha}$. If $W\in A\setminus \widetilde K$, then $W$ lies on {\it some} complexified extremal ellipse $L$ whose real points $E$ are an inscribed ellipse in $B$ with boundary in $\widetilde K$ (and $V_B|_L$ is harmonic). If $L\not = L_{\alpha}$ for some $\alpha$, then $E\cap \widetilde K$ consists of two antipodal points $\pm p$. By rotating coordinates we may assume $\pm p = (\pm 1,0,...,0)$ and
$$E\subset \{(u_0,...,u_n): \ u_2=\cdots =u_n=0\}.$$ We have two cases:
\begin{enumerate}
\item $E=\{(u_0,...,u_n): |u_0|\leq 1, \ u_1=0, \ u_2=\cdots =u_n=0\}$, a real interval:
\noindent In this case
$$L=\{(W_0,0,...,0): W_0\in \mathbb{C} \}.$$
But then $L\cap A =\{(W_0,0,...,0): W_0=\pm 1 \}=\{\pm p\} \subset \widetilde K$, contradicting $W\in A\setminus \widetilde K$.
\item $E=\{(u_0,...,u_n): u_0^2+u_1^2/r^2=1, \ u_2=\cdots =u_n=0\}$ where $0<r<1$, a nondegenerate ellipse:
\noindent In this case,
$$L:=\{(W_0,...,W_n): W_0^2+W_1^2/r^2=1, \ W_2=\cdots =W_n=0\}.$$
But then if $W\in L\cap A$ we have
$$ W_0^2+W_1^2/r^2=1= W_0^2+W_1^2$$
so that $W_1=\cdots =W_n=0$ and $W_0^2=1$; i.e., $L\cap A =\{\pm p\} \subset \widetilde K$ which again contradicts $W\in A\setminus \widetilde K$.
\end{enumerate}
For the reverse inclusion, recall that the variety $A$ is defined by $\sum_{j=0}^nW_j^2=1$. If $W=u+iv$ with $u,v\in \mathbb{R}^{n+1}$, we have
$$\sum_{j=0}^nW_j^2 = \sum_{j=0}^n[u_j^2-v_j^2]+ 2i\sum_{j=0}^nu_jv_j.$$
Thus for $W=u+iv \in A$, we have
$$\sum_{j=0}^nu_jv_j=0.$$
If we take an orthogonal transformation $T$ on $\mathbb{R}^{n+1}$, then, by definition, $T$ preserves Euclidean lengths in $\mathbb{R}^{n+1}$; i.e., $\sum_{j=0}^n u_j^2=1=\sum_{j=0}^n (T(u)_j)^2=1$. Moreover, if $u,v$ are orthogonal; i.e., $\sum_{j=0}^nu_jv_j=0$, then $\sum_{j=0}^n(T(u))_j\cdot (T(v))_j =0$. Extending $T$ to a complex-linear map on $\mathbb{C}^{n+1}$ via
$$T(W)=T(u+iv):= T(u) +iT(v),$$
we see that if $W\in A$, then $\sum_{j=0}^n(T(u))_j\cdot (T(v))_j =0$ so that
$$\sum_{j=0}^n(T(W)_j)^2 = \sum_{j=0}^n[(T(u)_j)^2-(T(v)_j)^2]=\sum_{j=0}^n[u_j^2-v_j^2]=1.$$
Thus $T$ preserves $A$.
Clearly the ellipse
$$L_0:=\{(W_0,...,W_n): W_0^2+W_1^2=1, \ W_2=\cdots =W_n=0\}$$
corresponding to the great circle
$S_0:=\{(u_0,...,u_n): u_0^2+u_1^2=1, \ u_2=\cdots =u_n=0\}$ lies in $A$ and any other great circle $S_{\alpha}$ can be mapped to $S_0$ via an orthogonal transformation $T_{\alpha}$. From the previous paragraph, we conclude that $ \cup_{\alpha} L_{\alpha}\subset A$.
\end{proof}
\noindent We use the Lundin formula for $V_B$ in (\ref{eq:realball}):
$$V_B(W)=\frac{1}{2}\log h\bigl( |W|^2+|W^2-1|\bigr)$$
where $h(t)=t+\sqrt{t^2-1}$ for $t\in \mathbb{C}\setminus [-1,1]$. Now the formula for $V_{\widetilde K}$ can only be valid on $A$; and indeed, since $W^2=1$ on $A$, by the previous proposition we obtain
$$V_{\widetilde K}(W)=\frac{1}{2}\log h (|W|^2), \ W\in A.$$ Note that since the real sphere $\widetilde K$ and the complexified sphere $A$ are invariant under real rotations, the Monge-Amp\`ere measure
$$(dd^cV_{\widetilde K} (W))^n=(dd^c \frac{1}{2}\log h( |W|^2))^n$$ must be invariant under real rotations as well and hence is normalized surface area measure on the real sphere $\widetilde K$. This can also be seen as a consequence of $V_{\widetilde K}$ being the {\it Grauert tube function} for $\widetilde K$ in $A$ as $(dd^cV_{\widetilde K} (W))^n$ gives the volume form $dV_g$ on $\widetilde K$ corresponding to the standard Riemannian metric $g$ there (cf., \cite{Z}).
Getting back to the calculation of $V_{K,Q}$, note that since $W=(\frac{1}{(1+z^2)^{1/2}},\frac{z}{(1+z^2)^{1/2}})$,
$$|W|^2:=|W_0|^2+|W_1|^2+\cdots +|W_n|^2=\frac{1+|z|^2}{|1+z^2|}.$$
Plugging in to (\ref{fullin})
$$V_{\widetilde K}(W)= V_B(W) =V_{K,Q}(z)-Q(z)=V_{K,Q}(z)- \frac{1}{2}\log |1+z^2|$$
gives
\begin{equation}\label{magic}V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2}\bigr)\end{equation} for $z\in \widetilde \Omega$. We show in section 5 that this formula does indeed give us the extremal function $V_{K,Q}(z)$ for all $z\in \mathbb{C}^n$.
A similar observation leads to another derivation of the above formula. Consider $\overline {F(K)}$ as the upper hemisphere
$$S:=\{(u_0,...,u_n)\in \mathbb{R}^{n+1}: \sum_{j=0}^n u_j^2 =1, \ u_0 \geq 0\}$$
in $\mathbb{R}^{n+1}$ and let $\pi: \mathbb{R}^{n+1}\to \mathbb{R}^n$ be the projection $\pi(u_0,...,u_n)=(u_1,...,u_n)$ which we extend to $\pi: \mathbb{C}^{n+1}\to \mathbb{C}^n$ via $\pi(W_0,...,W_n)=(W_1,...,W_n)$. Then
$$\pi(S) = B_n:= \{(u_1,...,u_n)\in \mathbb{R}^{n}: \sum_{j=1}^n u_j^2\leq1\}$$
is the real $n-$ball in $\mathbb{C}^{n}$. Each great semicircle $C_{\alpha}$ in $S$ -- these are simply half of the $L_{\alpha}$'s from before -- projects to half of an inscribed ellipse $E_{\alpha}$ in $B_n$, while the other half of $E_{\alpha}$ is the projection of the great semicircle given by the negative $u_1,...,u_n$ coordinates of $C_{\alpha}$ (still in $F(K)$, i.e., with $u_0>0$). As before, the complexification $E^*_{\alpha}$ of the ellipses $E_{\alpha}$ correspond to complexifications of the great circles.
\begin{proposition} \label{semi} We have
$$H_{F(K)}(W_0,...,W_n)=V_{B_n}(\pi(W))=V_{B_n}(W_1,...,W_n)=V_{B_n}(W')\leq V_{\widetilde K}(W_0,...,W_n)$$
for $W=(W_0,...,W_n)=(W_0,W')\in A$.
\end{proposition}
\begin{proof} Clearly $V_{B_n}(\pi(W))\leq V_{\widetilde K}(W)$. For the inequality $H_{F(K)}(W)\leq V_{B_n}(\pi(W))$, note that for $W\in A$ with $W=(W_0,W')$, we have $\pi^{-1}(W')=(\pm W_0,W')\in A$ but the value of $H_{F(K)}$ is the same at both of these points. Thus $W'\to H_{F(K)}(\pi^{-1}(W'))$ is a well-defined function of $W'$ for $W\in A$ which is clearly in $L(\mathbb{C}^n)$ (in the $W'$ variables) and nonpositive if $W'\in B_n$; hence $H_{F(K)}(\pi^{-1}(W'))\leq V_{B_n}(W')$.
\end{proof}
From (\ref{fullin}),
$$ H_{\widetilde K} (W)=V_{\widetilde K}(W) = V_{K,Q}(z) -Q(z)= H_{F(K)}(W)$$ for $z\in \widetilde \Omega$ and $W=F(z)$ so that we have equality for such $W$ in Proposition \ref{semi} and an alternate way of computing $V_{K,Q}$. From the Lundin formula, for $(W_0,W')\in A$ we have $W_0^2+W'^2=1$ so
$$V_{B_n}(W')=\frac{1}{2}\log h\bigl( |W'|^2+|W'^2-1|\bigr)=\frac{1}{2}\log h( |W|^2).$$ and we get the same formula (\ref{magic})
\[V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2}\bigr)\]
for $z\in \widetilde \Omega$.
\begin{remark}\label{onevarem} Note that for $n=1$, it is easy to see that
\begin{equation}\label{onevar} V_{K,Q}(z)=\max [\log |z-i|, \log |z+i|]\end{equation} which agrees with formula (\ref{magic}).
\end{remark}
\section{Relation with Lie ball and maximality of $V_{K,Q}$} One way of describing the Lie ball $L_n\subset \mathbb{C}^n$ is that it is the homogeneous polynomiall hull $\widehat{(B_n)}_{hom}$ of the real ball $$B_n:=\{x=(x_1,...,x_n)\in \mathbb{R}^n: x^2 = x_1^2+\cdots +x_n^2\leq 1\}.$$ A formula for $L_n$ is given by
$$L_n =\{z=(z_1,...,z_n)\in \mathbb{C}^n: |z|^2 +\{ |z|^4 - |z^2|^2\}^{1/2} \leq 1\}.$$ Note that (by definition) $L_n$ is circled. Writing $Z:=(z_0,z)=(z_0,z_1,...,z_n)\in \mathbb{C}^{n+1}$,
$$L_{n+1}=\{Z\in \mathbb{C}^{n+1}: |Z|^2 +\{ |Z|^4 - |Z^2|^2\}^{1/2} \leq 1\}.$$ The (homogeneous) Siciak-Zaharjuta extremal function of this (circled) set is
$$H_{L_{n+1}}(Z)=V_{L_{n+1}}(Z)= \frac{1}{2}\log^+ \bigl(|Z|^2 +\{ |Z|^4 - |Z^2|^2\}^{1/2}\bigr).$$ Thus
$$V_{L_{n+1}}(1,z)= \frac{1}{2}\log \bigl([1+|z|^2] +\{ [1+|z|^2]^2 - |1+z^2|^2\}^{1/2}\bigr)$$ so that from (\ref{magic}) $$V_{K,Q}(z)= V_{L_{n+1}}(1,z)$$ for $z\in \widetilde \Omega$.
The extremal function $V_{L_{n+1}}(Z)$ for the Lie ball in $\mathbb{C}^{n+1}$ is maximal outside $L_{n+1}$ and, since
$$V_{L_{n+1}}(\lambda Z)= \log |\lambda| + V_{L_{n+1}}(Z)$$
for $Z\in \partial L_{n+1}$ and $\lambda \in \mathbb{C}$ with $|\lambda|>1$, we see that $V_{L_{n+1}}$ is harmonic on complex lines through the origin (in the complement of $L_{n+1}$). Thus for each $Z\not \in L_{n+1}$, the vector $Z$ is an eigenvector of the complex Hessian of $V_{L_{n+1}}$ at $Z$ with eigenvalue $0$. We will use this to show: {\sl for $z\not \in \mathbb{R}^n$, the vector ${\rm Im} z$ is an eigenvector of the complex Hessian of the function $V_{K,Q}(z)$ defined in (\ref{magic}) at $z$ with eigenvalue $0$}.
To this end, let $u:\mathbb{C}^n\to\mathbb{R}$ denote our candidate function for $V_{K,Q}$ where $K=\mathbb{R}^n\subset \mathbb{C}^n$ and the weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$, i.e., for $z\in \mathbb{C}^n$, define
\[u(z):=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2}\bigr).\] Let $U:\mathbb{C}^{n+1}\to \mathbb{R}$ denote its homogenization, i.e,
\[U(Z)=\frac{1}{2}\log \bigl(|Z|^2+ \{|Z|^4-|Z^2|^2\}\bigr)\] with $Z:=(z_0,z)\in \mathbb{C}^{n+1},$ so that $u(z)=U(1,z)$. From above, $\max[0,U(Z)]$ is the extremal function for the Lie ball $L_{n+1}$, and since $U(Z)$ is psh, so is $u(z)$. Also, $U$ is symmetric as a function of its arguments and has the property that $U(\overline{Z})=U(Z)$; in particular it follows that \[\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(\overline{Z})= \frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(Z).\]
Now, for any function $v$, let $H_v(z)$ denote the complex Hessian of $v$ evaluated at the point $z.$ For any fixed $Z\in\mathbb{C}^{n+1}$ and $\lambda\in\mathbb{C},$
\[U(\lambda Z)=U(Z)+\log|\lambda|,\] which is harmonic as a function of $\lambda$ for $\lambda \not = 0$. It follows that \begin{equation} \label{atz} H_U(Z)Z=0\in \mathbb{C}^{n+1},\quad \forall Z\in \mathbb{C}^{n+1}\setminus \{0\} \end{equation} and that \begin{equation} \label{atzbar} H_U(Z)\overline{Z}=H_U(\overline{Z})\overline{Z}= 0\in \mathbb{C}^{n+1},\quad \forall Z\in \mathbb{C}^{n+1}\setminus \{0\}. \end{equation}
Equivalently, equation \eqref{atz} says that, for $0\le j\le n,$ \[\sum_{k=0}^{n} \frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(Z) \times Z_k=0.\] But then, for $1\le j\le n,$ we have \[\sum_{k=1}^{n} \frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(Z) \times Z_k=-\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_{0}}(Z)\times Z_{0}.\] Evaluating at $Z=(1,z)$ we obtain \[\sum_{k=1}^{n} \frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(1,z) \times z_k=-\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_{0}}(1,z)\times 1,\] i.e., \[\sum_{k=1}^{n} \frac{\partial^2u}{\partial z_j\partial \overline{z}_k}(z) \times z_k=-\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_{0}}(1,z).\] Similarly, from \eqref{atzbar} we obtain, for $1\le j\le n,$ \[\sum_{k=1}^{n} \frac{\partial^2U}{\partial Z_j\partial \overline{Z}_k}(Z) \times \overline{Z}_k=-\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_{0}}(Z)\times \overline{Z}_{0}\] so that evaluating at $Z=(1,z)$ gives \[\sum_{k=1}^{n} \frac{\partial^2u}{\partial z_j\partial \overline{z}_k}(z) \times \overline{z}_k=-\frac{\partial^2U}{\partial Z_j\partial \overline{Z}_{0}}(1,z).\] Consequently, \[H_u(z)z=H_u(z)\overline{z}, \,\,\hbox{i.e.,}\,\, H_u(z)(z-\overline{z})=0.\] In particular, for $z\neq \overline{z},$ i.e., $z\notin \mathbb{R}^n,$ ${\rm det}(H_u(z))=0,$ i.e., $(dd^c u)^n=0$ (note as $u$ is psh, $H_u(z)$ is a positive semi-definite matrix).
Since the function $u$ is maximal on $\mathbb{C}^n\setminus \mathbb{R}^n$ and $u(x)= Q(x)=\frac{1}{2}\log (1+x^2)$ for $x\in \mathbb{R}^n$ we have proved the following:
\begin{theorem} \label{magic1} For $K=\mathbb{R}^n\subset \mathbb{C}^n$ and weight $w(z)=|f(z)|=|\frac{1}{(1+z^2)^{1/2}}|$,
$$V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2}\bigr), \ z\in \mathbb{C}^n.$$
\end{theorem}
Note that from (\ref{wtdrel}), since the K\"ahler potential $u_0(x)=Q(x)$ for $x\in K=\mathbb{R}^n$, $$V_{K,Q}(z)= u_0(z)+ v_{K,0}([1:z]).$$ Thus we have found a formula for the (unweighted) extremal function of $\mathbb{R} \mathbb P^n$, the real points of $\mathbb P^n$.
\begin{corollary} \label{magiccor}The unweighted $\omega-$psh extremal function of $\mathbb{R} \mathbb P^n$ is given by
$$v_{\mathbb{R} \mathbb P^n,0}([1:z])=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2}\bigr)-u_0(z)$$
\begin{equation}\label{extrwt} =\frac{1}{2}\log \bigl( 1+[1-\frac{|1+z^2|^2}{(1+|z|^2)^2}]^{1/2}\bigr)\end{equation} for $[1:z]\in \mathbb{C}^n$ and
\begin{equation}\label{extrwt2}v_{\mathbb{R} \mathbb P^n,0}([0:z])=\frac{1}{2}\log \bigl( 1+[1-\frac{|z^2|^2}{(|z|^2)^2}]^{1/2}\bigr).\end{equation}
\end{corollary}
Since $|1+z^2|\leq 1+|z|^2$ (and $|z^2|\leq |z|^2$), we see that, e.g., upon taking $z=i(1/\sqrt n,...,1/\sqrt n)$ in (\ref{extrwt}) or letting $z\to 0$ in (\ref{extrwt2}), $$\sup_{{\bf z}\in \mathbb P^n}v_{\mathbb{R} \mathbb P^n,0}({\bf z})= \frac{1}{2}\log 2.$$ This gives the exact value of the Alexander capacity $T_{\omega}(\mathbb{R} \mathbb P^n)$ of $\mathbb{R} \mathbb P^n$ in Example 5.12 of \cite{GZ}: $$ T_{\omega}(\mathbb{R} \mathbb P^n)=1/\sqrt 2.$$ We remark that Dinh and Sibony had observed that the value of the Alexander capacity $T_{\omega}(\mathbb{R} \mathbb P^n)$ was independent of $n$ (Proposition A.6 in \cite{DS}). \section{Calculation of $(dd^cV_{K,Q})^n$ with $V_{K,Q}$ in (\ref{magic})}
We will compute $(dd^cV_{K,Q})^n$ for $V_{K,Q}$ in (\ref{magic}) after discussing some differential geometry. Let $\delta(x;y)$ be a Finsler metric where $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$ is a tangent vector at $x$. The Busemann density associated to this Finsler metric is $$\omega(x)=\frac{vol(\hbox{Euclidean unit ball in $\mathbb{R}^n$})}{vol(B_x)}$$ where $$B_x:= \{y: \delta(x;y)\leq 1\}.$$ The Holmes-Thompson density associated to $\delta(x;y)$ is $$\widetilde \omega(x)=\frac{vol(B_x^*)}{vol(\hbox{Euclidean unit ball in $\mathbb{R}^n$})}$$ where $$B_x^*:=\{y: \delta(x;y)\leq 1\}^*=\{x:x\cdot y= x^ty\leq 1 \ \hbox{for all} \ y\in B_x\}$$ is the dual unit ball. Here $x^t$ denotes the transpose of the (vector) matrix $x$. Finsler metrics arise naturally in pluripotential theory in the following setting: if $K=\bar \Omega$ where $\Omega$ is a bounded domain in $\mathbb{R}^n\subset \mathbb{C}^n$, the quantity \begin{equation}\label{baran}\delta_B(x;y):=\limsup_{t\to 0^+}\frac{V_K(x+ity)}{t}=\limsup_{t\to 0^+}\frac{V_K(x+ity)-V_K(x)}{t} \end{equation} for $x\in K$ and $y\in \mathbb{R}^n$ defines a Finsler metric called the {\it Baran pseudometric} (cf., \cite{blw}). It is generally not Riemannian: such a situation yields more information on these densities.
\begin{proposition} \label{ball} Suppose $$\delta(x;y)^2=y^tG(x)y$$ is a Riemannian metric; i.e., $G(x)$ is a positive definite matrix. Then $$vol(B_x^*)\cdot vol(B_x)=1 \ \hbox{and} \ vol(B_x^*)=\sqrt {\mathop{\mathrm{det}}\nolimits G(x)}.$$
\end{proposition}
\begin{proof} Writing $G(x)=H^t(x)H(x)$, we have $$\delta(x;y)^2=y^tG(x)y=y^tH^t(x)H(x)y.$$
Letting $||\cdot||_2$ denote the standard Euclidean ($l^2$) norm, we then have
$$B_x=\{y\in \mathbb{R}^N: ||H(x)y||_2\leq 1\}=H^{-1}(x)\bigl( \hbox{unit ball in $l^2-$norm})$$ and $$B_x^*=H(x)^t\bigl( \hbox{unit ball in $l^2-$norm}).$$ Hence $vol(B_x^*)\cdot vol(B_x)=1$ and $$vol\bigl(\{y:\delta(x;y)\leq 1\}^*\bigr)=vol(B_x^*)=\mathop{\mathrm{det}}\nolimits H(x)=\sqrt {\mathop{\mathrm{det}}\nolimits G(x)}.$$ \end{proof}
Motivated by (\ref{baran}) and Theorem \ref{main} below, for $u(z)=V_{K,Q}(z)$ in (\ref{magic}), we will show that the limit $$\delta_u(x;y):=\lim_{t\to 0^+}\frac{u(x+ity)-u(x)}{t}$$ exists. Fixing $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^n$, let $$F(t):=u(x+ity)=\frac{1}{2}\log \{ (1+x^2+t^2y^2) + 2 [t^2y^2+t^2x^2y^2-(x\cdot ty)^2]^{1/2}\}$$ $$=\frac{1}{2}\log \{ (1+x^2+t^2y^2) + 2 t[y^2+x^2y^2-(x\cdot y)^2]^{1/2}\}.$$ It follows that $$\delta_u(x;y)=F'(0)=\frac{1}{2}\frac{2 [y^2+x^2y^2-(x\cdot y)^2]^{1/2}}{1+x^2}=\frac{ [y^2+x^2y^2-(x\cdot y)^2]^{1/2}}{1+x^2}.$$ We write $$\delta_u^2(x;y)=\frac{ y^2+x^2y^2-(x\cdot y)^2}{(1+x^2)^2}=y^tG(x)y$$ where $$G(x):=\frac{(1+x^2)I -xx^t}{(1+x^2)^2}.$$ Since this matrix is positive definite, $\delta_u(x;y)$ defines a Riemannian metric.
We analyze this further. The eigenvalues of the rank one matrix $xx^t\in \mathbb{R}^{n\times n}$ are $x^2,0,\ldots,0$ for $$(xx^t)x = x(x^tx) = x^2\cdot x;$$ and clearly $v\perp x$ implies $(xx^t)v=x(x^tv)=0$. The eigenvalues of $(1+x^2)I -xx^t$ are then $$(1+x^2)-x^2, \ (1+x^2)-0, \ldots, \ (1+x^2)-0 \ = \ 1,1+x^2,\ldots, 1+x^2$$ and the eigenvalues of $G(x)$ are $$\frac{1}{(1+x^2)^2}, \frac{1}{1+x^2},\ldots, \frac{1}{1+x^2}.$$ This shows $G(x)$ is, indeed, positive definite (it is clearly symmetric) and $$\mathop{\mathrm{det}}\nolimits G(x)=\frac{1}{(1+x^2)^{n+1}}.$$ From Proposition \ref{ball}, $$vol(B_x^*) = \sqrt {\mathop{\mathrm{det}}\nolimits G(x)}=\frac{1}{(1+x^2)^{\frac{n+1}{2}}}=\frac{1}{vol(B_x)}.$$ In particular, the Busemann and Holmes-Thompson densities associated to $\delta_u(x;y)$ are \begin{equation}\label{dense}\frac{1}{(1+x^2)^{\frac{n+1}{2}}}\end{equation} up to normalization. Note from (\ref{onevar}) in Remark \ref{onevarem} this agrees with the density of $\Delta V_{K,Q}$ with respect to Lebesgue measure $dx$ on $\mathbb{R}$ if $n=1$ and this will be the case for the density of $ (dd^cV_{K,Q})^n$ with respect to Lebesgue measure $dx$ on $\mathbb{R}^n$ for $n>1$ as well. For motivation, we recall the main result of \cite{blmr} (see \cite{bt} for the symmetric case $K=-K$):
\begin{theorem} \label{main} Let $K$ be a convex body and $V_K$ its Siciak-Zaharjuta extremal function. The limit \begin{equation} \label{dblim} \delta (x;y):=\lim_{t\to 0^+}{V_K(x+ity)\over t} \end{equation} exists for each $x\in {\rm int}_{\mathbb{R}^n}K$ and $y\in \mathbb{R}^n$ and \begin{equation}
\label{eq:monge}
(dd^cV_K)^n=\lambda(x)dx \ \hbox{where} \ \lambda(x)=n!vol (\{y: \delta (x;y)\leq 1\}^*)=n!vol(B_x^*).
\end{equation}
\end{theorem}
\noindent The conclusion of Theorem \ref{main} required Proposition 4.4 of \cite{blmr}:
\begin{proposition} \label{propbaran} Let $D\subset \mathbb{C}^n$ and let $\Omega :=D\cap \mathbb{R}^n$. Let $v$ be a nonnegative locally bounded psh function on $D$ which satisfies:
$\begin{array}{rl}
i. & \Omega=\{v=0\}; \\ ii. & (dd^cv)^n=0 \; {\mbox{on}} \; D\setminus \Omega; \\ iii. & (dd^cv)^n=\lambda(x)dx \; on \; \Omega; \\ iv. & for \; all \; x\in \Omega, \ y \in \mathbb{R}^n, \; the \; limit \end{array}$
$$h(x,y):=\lim_{t\to 0^+} {v(x+ity)\over t} \; exists \; and \; is \; continuous \; on \; \Omega \times i\mathbb{R}^n;$$
$\begin{array}{rl} v. & for \; all \; x\in \Omega, y\to h(x,y) \; is \; a \; norm. \end{array}$
Then $$\lambda(x)=n! {\rm vol} \{y:h(x,y)\leq 1\}^*$$ and $\lambda(x)$ is a continuous function on $\Omega$. \end{proposition}
\begin{theorem} For $V_{K,Q}$ in (\ref{magic}), \begin{equation} \label{monge} (dd^cV_{K,Q})^n=n! \frac{1}{(1+x^2)^{\frac{n+1}{2}}}dx.\end{equation} \end{theorem}
\begin{proof}
Recall we extended $Q(x)=\frac{1}{2}\log (1+x^2)$ on $\mathbb{R}^n$ to all of $\mathbb{C}^n$ as $$Q(z)=\frac{1}{2}\log |1+z^2|\in L(\mathbb{C}^n).$$ With this extension of $Q$, and writing $u:=V_{K,Q}$, we claim
\begin{enumerate}
\item $Q$ is pluriharmonic on $\mathbb{C}^n\setminus V$ where $V=\{z\in \mathbb{C}^n: 1+z^2=0\}$;
\item $u-Q\geq 0$ in $\mathbb{C}^n$; and $\mathbb{R}^n=\{z\in \mathbb{C}^n: u(z)-Q(z)=0\}$;
\item for each $x,y\in \mathbb{R}^n$
$$\lim_{t\to 0^+}\frac{Q(x+ity)-Q(x)}{t} =0.$$ \end{enumerate} Item 1. is clear; 2. may be verified by direct calculation (the inequality also follows from the observation that $Q\in L(\mathbb{C}^n)$ and $Q$ equals $u$ on $\mathbb{R}^n$); and for 3., observe that
$$|1+(x+ity)^2|^2 = (1+x^2-t^2y^2)^2+4t^2(x\cdot y)^2=(1+x^2)^2+0(t^2)$$ so that
$$Q(x+ity)-Q(x)=\frac{1}{2}\log |1+(x+ity)^2|-\frac{1}{2}\log (1+x^2)$$ $$=\frac{1}{4} \log \frac{(1+x^2)^2+0(t^2)}{(1+x^2)^2}\approx \frac{1}{4}\frac{0(t^2)}{(1+x^2)^2} \ \hbox{as} \ t\to 0. $$ Thus 1. and 2. imply that $v:=u-Q$ defines a nonnegative plurisubharmonic function in $\mathbb{C}^n\setminus V$, in particular, on a neighborhood $D\subset \mathbb{C}^n$ of $\mathbb{R}^n$; from 1., \begin{equation} \label{maeq} (dd^cv)^n = (dd^cu)^n \ \hbox{on} \ D;\end{equation} and from 3., for each $x,y\in \mathbb{R}^n$
$$\lim_{t\to 0^+}\frac{v(x+ity)-v(x)}{t} =\lim_{t\to 0^+}\frac{u(x+ity)-Q(x+ity)-u(x)+Q(x)}{t}$$ $$=\lim_{t\to 0^+}\frac{u(x+ity)-u(x)}{t}- \lim_{t\to 0^+}\frac{Q(x+ity)-Q(x)}{t}=\delta_u(x;y).$$ Then (\ref{maeq}), (\ref{dense}) and Proposition \ref{propbaran} give (\ref{monge}). \end{proof}
{\bf Authors:}\\[\baselineskip] L. Bos, leonardpeter.bos@univr.it\\ University of Verona, Verona, ITALY \\[\baselineskip] N. Levenberg, nlevenbe@indiana.edu\\ Indiana University, Bloomington, IN, USA\\ \\[\baselineskip] S. Ma`u, s.mau@auckland.ac.nz\\ University of Auckland, Auckland, NEW ZEALAND\\ \\[\baselineskip] F. Piazzon, fpiazzon@math.inipd.it\\ University of Padua, Padua, ITALY \\[\baselineskip]
\end{document} |
\begin{document}
\title{A class of functional identities associated to curves over finite fields hanks{This work has been produced as part of the PhD thesis of the author at the Department of Mathematics Guido Castelnuovo, Sapienza Università di Roma, under the supervision of Federico Pellarin.} \begin{abstract}
Given a geometrically irreducible projective curve $X$ over $\mathbb{F}_q$ and a point $\infty\in X(\mathbb{F}_q)$, a class of zeta functions was introduced by Pellarin, and related to the Anderson-Thakur function via a functional identity in the case $g(X)=0$. Anglès, Ngo Dac, and Tavares Ribeiro introduced so-called "special functions" as a generalization of the Anderson-Thakur function to arbitrary genus; Green and Papanikolas then found another functional identity relating those special functions to zeta functions à la Pellarin in the case $g(X)=1$.
The aim of this paper is to prove that a generalization of those functional identities hold in arbitrary genus. Our proof exploits the topological nature of divisors on the curve $X$, as well as the introduction of a "dual shtuka function". This allows us to reinterpret the zeta functions as dual versions of the special functions. \end{abstract}
\tableofcontents
\section{Introduction}
Let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $X$ be a projective, geometrically irreducible, smooth curve of genus $g$ over $\mathbb{F}_q$, with a point $\infty\in X(\mathbb{F}_q)$. We call $A:=\mathcal{O}_X(X\setminus\{\infty\})$, $H$ the Hilbert class field of $K:=\Q(A)$, ${K_\infty}$ the completion of $K$ at $\infty$, and ${\mathbb{C}_\infty}$ a completion of the algebraic closure $K_\infty^{ac}$. We fix an inclusion $H\subseteq{K_\infty}$ and a multiplicative sign function at $\infty$: $\sgn:{\mathbb{C}_\infty}(X)\to\mathbb{C}_\infty^\times$.
Throughout the paper we fix the datum $(X,\infty,\sgn)$, and we work with a Drinfeld module $\phi$ - i.e. a ring homomorphism $A\to{\mathbb{C}_\infty}\{\tau\}$ where $\tau c=c^q\tau$ for all $c\in{\mathbb{C}_\infty}$ - of rank $1$ and sign-normalized, also called a Drinfeld-Hayes module. Following \cite{Thakur}, we associate to $\phi$ a rational function $f$ over $X_H$ called \textit{shtuka function}, with $\sgn(f)=1$ and $\Div(f)=V^{(1)}-V+\Xi-\infty$ where $\Xi:\Spec({\mathbb{C}_\infty})\to\Spec(A)\hookrightarrow X$ is given by the canonical inclusion $A\hookrightarrow{\mathbb{C}_\infty}$, and $V$ is a certain divisor of degree $g$.
A natural tool to study $\phi$ is the Tate algebra $\mathbb{T}:={\mathbb{C}_\infty}\hat\otimes A$, on which $\phi$ acts $A$-linearly in the second coordinate. Anglès, Ngo Dac, and Tavares Ribeiro proved in \cite{ANDTR} that there is a class of functions $\Sf(\phi)\subseteq\mathbb{T}$, that they called \textit{special functions}, on which $\phi$ acts by scalar multiplication - i.e. for all $\omega\in\Sf(\phi)$ we have the following equalities: \[\phi_a(\omega)=(1\otimes a)\omega,\;\forall a\in A.\]
Another interesting class of functions are the \textit{partial zeta functions} "à la Pellarin"; the partial zeta relative to an ideal $I\unlhd A$ is defined as: \[\zeta_I:=\sum_{a\in I\setminus\{0\}}a^{-1}\otimes a\in\mathbb{T}.\] They were first introduced in the case $I=A=\mathbb{F}_q[T]$ by Pellarin, who proved - for a canonical special function $\omega$ - the identity $\zeta_A\omega(1\otimes T-T\otimes1)=\tilde{\pi}$, where $\tilde{\pi}$ is a fundamental period of the period lattice (see \cite{Pellarin2011}[Thm. 1]). Green and Papanikolas proved a similar result (\cite{Green}[Thm. 7.1]) in the much wider context of $g(X)=1$. Together with \cite{Pellarin2011}[Thm. 1], it can be stated as follows:
\begin{teo*}[Pellarin-Green-Papanikolas]
Fix a datum $(X,\infty,\sgn)$ with $g(X)\leq1$. Fix the unique Drinfeld-Hayes module $\phi$ such that the associated period lattice is $\tilde{\pi}A\subseteq{\mathbb{C}_\infty}$ for some $\tilde{\pi}\in\mathbb{C}_\infty^\times$. There is a rational function $\delta\in\Q(H\otimes A)$ such that:
\[\Sf(\phi)=\frac{\delta^{(1)}(\tilde{\pi}\otimes1)}{f\zeta_A}\cdot (\mathbb{F}_q\otimes A),\]
where $f$ is the shtuka function of $\phi$. \end{teo*}
Green and Papanikolas also described $\delta$ in terms of its divisor. Note that this result does not encompass the totality of the case $g(X)=1$, since for each ideal class in the group $Cl(A)$ there is exactly one Drinfeld-Hayes module whose lattice is in that class.
In the present paper, we prove the following generalization for any datum $(X,\infty,\sgn)$, where: $I\unlhd A$ is any ideal, with $a_I\in I$ an element of least degree; $\phi$ is a Drinfeld-Hayes module with shtuka function $f$, dependent on the ideal class of $I$ (see Lemma \ref{con}); $\tilde{\pi}\in{K_\infty}$ is a fundamental period of the lattice of $\phi$ (see Definition \ref{definition Lambda'}); $V_*$ is the unique effective divisor of degree $g$ with $V_*\sim2g\infty-V$, and $\delta$ is the only function in $H\otimes A$ with $\sgn(\delta)=1$ and divisor $V+V_*-2g\infty$. \begin{customteo}{A}[Theorem \ref{Sf_I}, complete version]\label{A} The following $A$-submodules of $\mathbb{T}$ coincide: \[\Sf(\phi)=\frac{\delta^{(1)}(\tilde{\pi}\otimes1)}{f(a_I\otimes1)\zeta_I}\cdot (\mathbb{F}_q\otimes I).\] \end{customteo} \begin{oss}
The elements $a_I\in I$ and $\tilde{\pi}\in{K_\infty}$ are uniquely determined up to a factor in $\mathbb{F}_q^\times$. Moreover, one can check that the right hand side only depends on the ideal class of $I$. \end{oss} It's worth noting that the techniques employed by Green and Papanikolas were tailored to the case $g(X)=1$ - for example, they used the equation of a generic elliptic curve to carry out explicit computations. In the present paper, computations are only needed in Section \ref{section duality} to find the scalar factor $\tilde{\pi}\otimes 1$; the remaining content of Theorem \ref{A}, expressed as a partial version thereof in Section \ref{section special functions}, is built on the purely theoretical results of Section \ref{section topology} and Section \ref{Frobenius and divisors}. \\* \\* If $\phi$ is a Drinfeld-Hayes module and $f$ is its shtuka function, an element $\omega\in\mathbb{T}$ is a special function if and only if $\omega^{(1)}=f\omega$. If $f\in\mathbb{T}^\times$ there is $\alpha\in{K_\infty}$ such that \[(\alpha\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f}\right)^{(i)}\] is a well-defined element of $\mathbb{T}^\times$, and it is a special function; on the other hand, as Gazda and Maurischat noticed in \cite{Gazda}[Cor. 3.22], if there is an invertible special function, then $Sf(\phi)\cong A$, and it is not known if the converse is true, or in other words how restrictive is the hypothesis $f\in\mathbb{T}^\times$. We circumvent this problem and prove the following result for any datum $(X,\infty,\sgn)$. \begin{customteo}{B}[Theorem \ref{omega infinite product}]
Fix a Drinfeld-Hayes module $\phi$ with shtuka function $f$. There is some $\alpha\in K_\infty^\times$ such that the following element of ${K_\infty}\hat\otimes K$ is well defined:
\[\omega:=(\alpha\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f}\right)^{(i)}.\]
Moreover, $\omega\in(\mathbb{F}_q\otimes K)\Sf(\phi)$, and $\omega$ does not depend on the choice of $\alpha$. \end{customteo}
In the present paper, we introduce a \textit{dual shtuka function} $f_*$ - the unique rational function over $X_{K_\infty}$ with divisor $V_*-V_*^{(1)}+\Xi-\infty$ and $\sgn(f_*)=1$. In analogy with Theorem \ref{omega infinite product}, we prove the following identity, for a certain ideal $I\unlhd A$ with $a_I\in I$ an element of least degree. \begin{customteo}{C}[Theorem \ref{functional identity}, complete version]\label{C}
The following functional identity is well posed and true in ${K_\infty}\hat\otimes K$:
\[\zeta_I=-(a_I^{-1}\otimes a_I)\prod_{i\geq0}\left((\tilde{\pi}^{1-q}\otimes1)f_*^{(1)}\right)^{(i)}.\] \end{customteo}
Equivalently, we prove the following identity (Proposition \ref{a_I/pi}): \[\frac{(a_I\tilde{\pi}^{-1}\otimes1)\zeta_I}{\left((a_I\tilde{\pi}^{-1}\otimes1)\zeta_I\right)^{(1)}}=f_*^{(1)},\] whose similarity to the identity $\omega^{(1)}=f\omega$ for a special function $\omega$ suggests some sort of duality between special functions and zeta functions. As for Theorem \ref{A}, the complete version of Theorem \ref{C} - with the explicitation of the constant $\tilde{\pi}$ - is contained in Section \ref{section duality}, where we explore this idea of duality.
As far as the rest of the paper goes, the structure is as follows.
In Section \ref{section topology} we describe a functorial way of assigning a compact topology to the ${K_\infty}$-points of a proper $\mathbb{F}_q$-scheme $Y$. We then discuss some results about divisors of curves in finite characteristic from \cite{Milne}, and deduce a homeomorphism between certain spaces of rational functions and the spaces of their divisors; this allows us to prove statements about the convergence of the former by looking at the latter. Green and Papanikolas had already conjectured that the Jacobian variety and the divisor of $V$ would play a role in the generalization of \cite{Green}[Thm. 7.1], and in Section \ref{section topology} and Section \ref{Frobenius and divisors} we explain concretely how they are used.
Finally, in Section \ref{Dedekind zeta}, we generalize another theorem of Green and Papanikolas about Dedekind-like zeta functions (\cite[Thm. 7.3]{Green}), which, in the notation of Section \ref{Dedekind zeta}, are defined as: \[\xi_{\bar{I}}:=\sum_{J\unlhd A}\frac{\chi_{\bar{I}}(J)}{\chi_{\bar{I}}(J)(\Xi)}\in {K_\infty}\hat\otimes K.\] For a certain function $h_{I,\bar{A}}\in\Q(H\otimes A)\subseteq{K_\infty}\hat\otimes K$ defined as in Lemma \ref{final lemma}, the result is as follows.
\begin{customteo}{D}[Theorem \ref{xi equation}]
The function $\xi_{\bar{I}}$ is well defined, and the following identity holds: \[h_{I,\bar{A}}\xi_{\bar{I}}=-\left(\sum_{\sigma\in\G(H/K)}h_{I,\bar{A}}^\sigma\right)\zeta_{A}.\] \end{customteo}
\section{Notations and fundamental concepts}
Recall the notations from the first paragraph of the introduction. We also introduce the following. \begin{itemize}
\item[-]The degree map $\deg:K\to\mathbb{Z}$ is defined as the opposite of the valuation at $\infty$, and for all $A$-modules $\Lambda\subseteq K$, for all integers $d$, we define $\Lambda(d)$ (resp. $\Lambda(\leq d)$) the set $\{x\in\Lambda|\deg(x)=d\}$ (resp. $\deg(x)\leq d$).
\item[-]For any finite field extension $L/{K_\infty}$, we denote by $(\mathcal{O}_L,\mathfrak{m}_L)$ the associated local ring of integers, and by $\mathbb{F}_L$ the residue field.
\item[-]Throughout this paper unlabeled tensor products of modules are assumed to be over $\mathbb{F}_q$, while for unlabeled fiber products of schemes the base ring should be clear from the context.
\item[-]If $Y$ is an $R$-scheme and $S$ is an $R$-algebra, we denote $Y(S)$ the set of morphism of $R$-schemes from $\Spec(S)$ to $Y$, and with $Y_S$ the base change $Y\times\Spec(S)$. If $Y_S$ is integral, we denote $S(Y)$ the field of rational functions of $Y_S$.
\item[-]For all complete normed fields $L$, for all $\mathbb{F}_q$-vector spaces $M$, the module $L\otimes M$ is always endowed with the sup norm induced by $L$; we denote its completion by $L\hat\otimes M$.
\item[-]In analogy to \cite{Goss}, we use the relation symbol $a\in^*\Lambda$ to signify $a\in\Lambda\setminus\{0\}$. \end{itemize} \begin{oss}
Let's describe explicitly the sign function for any $h\in X({\mathbb{C}_\infty})^\times$. Since $X({\mathbb{C}_\infty})$ is the field of fractions of ${\mathbb{C}_\infty}\otimes A$, and $\sgn$ is a multiplicative function, we can assume $h\in{\mathbb{C}_\infty}\otimes A\setminus\{0\}$. We can write $h=\sum_{i=0}^k c_i\otimes a_i$, with $(a_i)_i$ in $A$ of strictly increasing degree and $(c_i)_i$ in $\mathbb{C}_\infty^\times$, and we have:
\[\sgn(h)=\sgn\left(\sum_{i=0}^k c_i\otimes a_i\right)=\sgn(c_k\otimes a_k)=c_k\sgn(a_k).\] \end{oss}
In the rest of this section, we present some basic results that are described in great detail in \cite{Goss}.
Let ${\mathbb{C}_\infty}\{\tau\}$ and ${\mathbb{C}_\infty}\{\tau^{-1}\}$ be the rings of non-commutative polynomials over ${\mathbb{C}_\infty}$, with the relations $\tau c=c^q\tau$ and $\tau^{-1}c^q=c\tau^{-1}$ for all $c\in{\mathbb{C}_\infty}$. There is a $\mathbb{F}_q$-linear and bijective antihomomorphism ${\mathbb{C}_\infty}\{\tau\}\to{\mathbb{C}_\infty}\{\tau^{-1}\}$ sending $\varphi:=\sum_i c_i\tau^i$ to $\varphi^*:=\sum \tau^{-i} c_i$.
A \textit{Drinfeld module} of rank $r$ is a ring homomorphism $\phi:A\to{\mathbb{C}_\infty}\{\tau\}$ sending $a$ to $\phi_a:=\sum_{i\geq0}a_i\tau^i$ with the following properties for all $a\in^* A$: \[\deg_x\left(\sum_{i\geq0}a_i x^{q^i}\right)=q^{r\deg(a)};\;a_0=a.\] If moreover $r=1$ and $a_{\deg(a)}=\sgn(a)$ for all $a\in^* A$, we call $\phi$ a \textit{Drinfeld-Hayes module}.
Fix $\phi,\psi$ Drinfeld modules. An element $f=\sum_i c_i \tau^i\in{\mathbb{C}_\infty}\{\tau\}$ is said to be a morphism from $\phi$ to $\psi$ if $f\circ\phi_a=\psi_a\circ f$ for all $a\in A$. It is known that every Drinfeld module of rank $1$ is isomorphic to a Drinfeld-Hayes module.
The \textit{exponential map} relative to a discrete $A$-module (called \textit{period lattice}) $\Lambda\subseteq{\mathbb{C}_\infty}$ is the following analytic function from ${\mathbb{C}_\infty}$ to itself: \[\exp_\Lambda(x):=x\prod_{\lambda\in^*\Lambda}\left(1-\frac{x}{\lambda}\right)\in{\mathbb{C}_\infty}[[x]].\] We can write $\exp_\Lambda(x)=\sum_{i\geq0}e_ix^{q^i}$, and its (bilateral) compositional inverse exists in ${\mathbb{C}_\infty}[[x]]$, is denoted $\log_\Lambda=\sum_{i\geq0}l_i x^{q^i}$, and is called \textit{logarithmic map}.
Finally, we recall that there is an equivalence between the (small) category of discrete $A$-submodules of ${\mathbb{C}_\infty}$ of rank $r$, with isogenies as morphisms, and the (small) category of Drinfeld modules of rank $r$.
\section{Convergence of divisors of rational functions over $X$}\label{section topology}
Recall the notations from the first paragraph of the introduction. Consider the $d$-th symmetric power $X^{(d)}$ for some positive integer $d$ - for all finite field extensions $L/{K_\infty}$, $X^{(d)}(L)$ is the set of $L$-rational divisors over $X$ of degree $d$. The aim of this section is to endow $X^{(d)}(L)$ with a compact topology such that the following property holds. \begin{customprop}{\ref{convergence of functions and divisors}} Consider a sequence $(h_m)_m$ in $L\otimes A(\leq d)$.
If the sequence $(\Div(h_m)+d\infty)_m$ converges to $D\in X^{(d)}(L)$, there are $(\lambda_m)_m$ in $L^\times$ such that $(\lambda_m h_m)_m$ converges to some $h\in L\otimes A(\leq d)\setminus\{0\}$ with $\Div(h)=D-d\infty$.
If the sequence $(h_m)_m$ converges to $h\in L\otimes A(\leq d)\setminus\{0\}$, then the sequence $(\Div(h_m)+d\infty)_m$ converges to $\Div(h)+d\infty\in X^{(d)}(L)$. \end{customprop}
In the following sections we need several times a topology on the $L$-points of other projective $\mathbb{F}_q$-schemes (such as the powers $\{X^d\}_{d\geq1}$ and the Jacobian variety $\mathcal{A}$ of $X$). To ensure their good interaction we prove that the topology that we define is functorial in Proposition \ref{Frobenius functor}.
\subsection{Functorial compact topology on ${K_\infty}$-rational points of $\mathbb{F}_q$-schemes}\label{topology}
Through this subsection, $L$ is a finite field extension of ${K_\infty}$, and $Y$ is a proper $\mathcal{O}_L$-scheme. We aim to construct a functor from proper schemes over $\mathcal{O}_L$ to compact Hausdorff topological spaces, sending $Y$ to $Y(\mathcal{O}_L)=Y(L)$.
\subsubsection{Definition of the functor}
\begin{lemma}\label{limit points} The natural maps $\left(\red_{L,k}:Y(\mathcal{O}_L)\to Y(\mathcal{O}_L/\mathfrak{m}_L^k)\right)_{k\geq1}$ induce a bijection $Y(\mathcal{O}_L)\cong\varprojlim_k Y(\mathcal{O}_L/\mathfrak{m}_L^k)$. \end{lemma} \begin{proof} Since $\Spec(\mathcal{O}_L)\cong\varinjlim_k\Spec(\mathcal{O}_L/\mathfrak{m}_L^k)$, we have: \[Y(\mathcal{O}_L)\cong\Hom_{\mathcal{O}_L}\left(\varinjlim_k\Spec(\mathcal{O}_L/\mathfrak{m}_L^k),Y\right) \cong\varprojlim_k\Hom_{\mathcal{O}_L}\left(\Spec(\mathcal{O}_L/\mathfrak{m}_L^k),Y\right)\cong\varprojlim_k Y(\mathcal{O}_L/\mathfrak{m}_L^k).\tag*{\qedhere}\] \end{proof} \begin{oss}\label{limit topology} The limit topology induced on $\varprojlim_k Y(\mathcal{O}_L/\mathfrak{m}_L^k)$ - where the indexed spaces are endowed with the discrete topology - is Hausdorff.
Since $Y$ is finite-type over $\mathcal{O}_L$ and $\mathcal{O}_L/\mathfrak{m}_L^k$ is finite for all $k$, $Y(\mathcal{O}_L/\mathfrak{m}_L^k)$ is finite for all $k$, so $Y(L)$ is compact. Moreover, it can be endowed with an ultrametric distance $\bar{d}$ as follows:
\[\bar{d}(P,Q):=\min_{k\in\mathbb{Z}}\left\{\frac{1}{q^k}\bigg|\red_{L,k}(P)\neq\red_{L,k}(Q)\right\}.\] \end{oss} \begin{Def}\label{red} Fix an inclusion $\mathbb{F}_L\hookrightarrow\mathcal{O}_L$: it induces a section of $\red_{L,1}:Y(\mathcal{O}_L)\to Y(\mathbb{F}_L)$. We denote $\red_L:Y(\mathcal{O}_L)\to Y(\mathcal{O}_L)$ the composition of the two maps. \end{Def}
From this point onwards, unless otherwise stated, we interpret the set $Y(\mathcal{O}_L)=Y(L)$ as endowed with this topology, which we call \textit{compact topology}. Similarly, if $Y'$ is a proper $\mathbb{F}_q$-scheme, the set $Y'(L)=Y'_{\mathcal{O}_L}(L)$ is always endowed with the compact topology.
\begin{prop}\label{Frobenius functor} The map associating to a proper $\mathcal{O}_L$-scheme $Y$ the topological space $Y(\mathcal{O}_L)$ can be extended to a functor $F_L$. \end{prop} \begin{proof} For every morphism $\varphi:Z\to Y$ of proper $\mathcal{O}_L$-schemes, the map $\varphi(\mathcal{O}_L):Z(\mathcal{O}_L)\to Y(\mathcal{O}_L)$ is continuous because it induces a system of maps $(\varphi(\mathcal{O}_L/\mathfrak{m}_L^k):Z(\mathcal{O}_L/\mathfrak{m}_L^k)\to Y(\mathcal{O}_L/\mathfrak{m}_L^k))_k$ which commute with the transition maps of the projective systems $(Z(\mathcal{O}_L/\mathfrak{m}_L^k))_k$ and $(Y(\mathcal{O}_L/\mathfrak{m}_L^k))_k$.
If we set $F_L(\varphi):=\varphi(\mathcal{O}_L)$ for all morphisms, it's easy to check that $F_L$ sends the identity map to the identity map and preserves composition, hence it is a functor. \end{proof} \begin{oss}\label{functor for F_q-schemes}
We also obtain a functor from proper $\mathbb{F}_q$-schemes to topological spaces, sending a scheme $Y$ to $Y(\mathcal{O}_L)=Y(L)$, by precomposing $\mathbb{F}_L$ with the base change $Y\mapsto Y_{\mathcal{O}_L}$. \end{oss}
\begin{lemma}\label{local homeomorphism}
Let $f:Z\to Y$ be a morphism of proper $\mathcal{O}_L$-schemes. Fix a subset $V\subseteq Y(L)$ with preimage $U\subseteq Z(L)$, such that $F_L(f)|_U:U\to V$ is bijective. Then $F_L(f)|_U$ is a homeomorphism. \end{lemma} \begin{proof} The map $F_L(f):Z(L)\to Y(L)$ is closed, being a continuous map between compact Hausdorff spaces. Any closed set of $U$ can be written as $C\cap U$, with $C\subseteq Z(L)$ closed. We have: \[F_L(f)(C\cap U)=F_L(f)\left(C\cap F_L(f)^{-1}(V)\right)=F_L(f)(C)\cap V,\]
which is closed in $V$ because $F_L(f)(C)$ is closed in $Y(L)$. This means that $F_L(f)|_U$ is closed, and since it induces a bijection between $U$ and $V$, it is a homeomorphism. \end{proof}
\begin{oss}\label{projective space}
In the case of the projective space $\mathbb{P}^n$ of dimension $n$ over $\mathbb{F}_q$, the set $\mathbb{P}^n(L)$ is in bijection with $L^{n+1}\setminus\{0\}/L^\times$; since the latter has a natural topology induced by $L$, the former also does, and it's easy to check that it's the same as the compact topology we defined. \end{oss}
\subsubsection{Application to group schemes}
The following statements show that the functor $F_L$ sends group schemes to topological groups.
\begin{lemma}\label{continuous functor} The topological spaces $F_L(Y\times_{\mathcal{O}_L} Y)$ and $F_L(Y)\times F_L(Y)$ are naturally isomorphic. \end{lemma} \begin{proof} The projections $\pi_1,\pi_2:Y\times Y\to Y$ induce a natural continuous map $F_L(Y\times_{\mathcal{O}_L} Y)\to F_L(Y)\times F_L(Y)$. Since both spaces are compact and Hausdorff, the map is closed; since the underlying function is the natural bijection $(Y\times_{\mathcal{O}_L} Y)(L)\cong Y(L)\times Y(L)$, the map is a homeomorphism. \end{proof}
\begin{prop}\label{Frobenius group scheme} If $Y$ is a (commutative) group scheme over $\mathcal{O}_L$, the metric on $Y(L)$ is translation invariant, and makes it into a (commutative) topological group. \end{prop} \begin{proof} By Lemma \ref{continuous functor}, we identify $F_L(Y\times_{\mathcal{O}_L}Y)\cong F_L(Y)\times F_L(Y)$ via a natural homeomorphism.
Call $e$ the identity, $i$ the inverse, and $m$ the multiplication of $Y$. Then $F_L(Y)$ has a natural structure of topological group, with identity $F_L(e)$, inverse $F_L(i)$ and multiplication $F_L(m)$, because all the necessary diagrams commute by functoriality. For the same reason, if $Y$ is commutative, $Y(L)$ is also commutative.
To prove the invariance of the metric, we need to show that every translation is an isometry. Fix a morphism of $\mathcal{O}_L$-schemes $P:\Spec(\mathcal{O}_L)\to Y$ (i.e. $P\in Y(\mathcal{O}_L)$), and consider the following: \[l_P:Y\cong \Spec(\mathcal{O}_L)\times_{\mathcal{O}_L}Y\xrightarrow{P\times id_Y} Y\times_{\mathcal{O}_L}Y\xrightarrow{m}Y,\] so that $F_L(l_P):Y(L)\to Y(L)$ is the left translation by $P$. It's immediate to check that, if we call $-P$ the inverse of $P$ in $Y(\mathcal{O}_L)$, $l_{-P}$ is the two-sided inverse of $l_P$, therefore they are isomorphisms. In particular, for all positive integers $k$, $l_P$ induces bijections $Y(\mathcal{O}_L/\mathfrak{m}_L^k)\to Y(\mathcal{O}_L/\mathfrak{m}_L^k)$, whose limit is precisely $F_L(l_P)$, hence $F_L(l_P)$ is an isometry. The proof for right translations is essentially the same. \end{proof}
\begin{cor}\label{series convergence} Suppose that $Y$ is commutative. Denote with addition the group law on $Y(L)$ and with $0$ its identity element. If $(P_i)_{i\in \mathbb{N}}$ is a sequence in $Y(L)$ converging to $0$, then the series $\sum_i P_i$ is a well defined element of $Y(L)$ (i.e. the sequence of partial sums converge). \end{cor} \begin{proof} Call $\bar{d}$ the distance on $Y(L)$. Since $d$ is ultrametric, we just need $\lim_k \bar{d}(S_k,S_{k-1})=0$, with $S_k:=\sum_{i=0}^k P_i$. Since the metric is translation invariant, $\lim_k \bar{d}(S_k,S_{k-1})=\lim_k \bar{d}(P_k,0)$, which is zero by hypothesis. \end{proof}
\subsection{Topology of the space of divisors}
In this subsection we state some propositions about the symmetric powers of a curve and its Jacobian. Most results are stated and proved in \cite{Milne}.
Recall the definition of $X$; $S_d$ is the permutation group of $d$ elements. We have the following (see \cite{Milne}[Prop. 3.1, Prop. 3.2]).
\begin{prop}\label{symmetric power} Fix a positive integer $d$. Consider the natural right action of $S_d$ on $X^d$ and call its quotient $X^{(d)}$. Then $X^{(d)}$ is a proper smooth $\mathbb{F}_q$-scheme. \end{prop}
The following result (see \cite{Milne}[Thm. 3.13]) gives us the functorial interpretation of the symmetric power $X^{(d)}$.
\begin{teo} Consider the functor $\Div^d_X$ which sends an $\mathbb{F}_q$-algebra $R$ to the set of relative effective Cartier divisors of degree $d$ on $X_R$ over $R$ (i.e. effective Cartier divisors on $X_R$ which are finite and flat of rank $d$ over $R$). This functor is represented by $X^{(d)}$. \end{teo}
\begin{cor}\label{cor Milne} For every field $E/\mathbb{F}_q$, $X^{(d)}(E)$ is in bijection with the $E$-subschemes of $X_E$ of degree $d$. \end{cor}
Let's continue with the fundamental property of the Jacobian variety (see \cite{Milne}[Thm. 1.1]).
\begin{teo} The functor from $\mathbb{F}_q$-algebras to abelian groups
\[R\mapsto\{\mathcal{L}\in\Pic(X_R)|\deg(\mathcal{L}_t)=0\;\forall t\in \Spec(R)\}/\pi_R^*(\Pic(R))\] is represented by an abelian variety $\mathcal{A}$ over $\mathbb{F}_q$, called the Jacobian variety of $X$. \end{teo} The following result clarifies the relation between the symmetric powers of $X$ and $\mathcal{A}$ (see \cite{Milne}[Thm. 5.2]). \begin{teo}\label{J^d} For all $d\geq1$, the point $\infty\in X(\mathbb{F}_q)$ induces a natural morphism of $\mathbb{F}_q$-schemes $J^d:X^{(d)}\to\mathcal{A}$. Moreover, the morphism $J^g:X^{(g)}\to\mathcal{A}$ is birational and surjective. \end{teo} \begin{oss} At the level of $L$-points, the morphism sends an effective divisor $D$ of degree $d$ to the class of $D-d\infty$. \end{oss}
Finally, we present a result on the fibers of the map $J^d$ (see \cite{Milne}[Rmk. 5.6.(c)]).
\begin{prop}\label{fibers} Fix a field extension $E/\mathbb{F}_q$ and a point $D\in X^{(d)}(E)$, corresponding to a sheaf of $\mathcal{O}_{X_E}$-ideals $\mathcal{I}_D$, with image $P\in\mathcal{A}(E)$. Call $V$ the $E$-vector space of the global sections of the $\mathcal{O}_{X_E}$-sheaf $\mathcal{I}_D^{-1}$. The fiber $(J^d)^*P$ is naturally isomorphic as an $E$-scheme to $\mathbb{P}(V)$.
For any field extension $E'/E$, for all $f\in E'\otimes_E V$, the isomorphism sends the line $E'\cdot f\in\mathbb{P}(V)(E')$ to $\Div(f)+D\in X^{(d)}(E')$. \end{prop}
\begin{cor}\label{preimage of h^0=1} Let $D\in X^{(d)}(E^{ac})$ with $h^0(D)=1$. If $J^d\circ D\in\mathcal{A}(E^{ac})$ factors through some $P\in\mathcal{A}(E)$, $D$ factors through some $D'\in X^{(d)}(E)$. \end{cor}
\begin{proof} WLOG $D\in X^{(d)}(E')$ for some finite field extension $\Phi:\Spec(E')\to\Spec(E)$. By Proposition \ref{fibers}, the pullback of $P\circ\Phi\in\mathcal{A}(E')$ along $J^d$ is a morphism $\Spec(E')\to X^{(d)}$, hence it is exactly $D$. If $Z\to X^{(d)}$ is the pullback of $P$ along $J^d$, $Z\times_{\Spec(E)}\Spec(E')$ is isomorphic to $\Spec(E')$; we deduce $Z\cong\Spec(E)$, and $D$ factors through $Z\to X^{(d)}$. \end{proof}
With the following proposition we can finally switch between convergence of functions and convergence of divisors, which is essential to prove the functional identity of Theorem \ref{functional identity}.
\begin{prop}\label{convergence of functions and divisors} Fix a finite field extension $L/{K_\infty}$ and consider a sequence $(h_m)_m$ in $L\otimes A(\leq d)$.
If the sequence $(\Div(h_m)+d\infty)_m$ converges to $D\in X^{(d)}(L)$, there are $(\lambda_m)_m$ in $L^\times$ such that $(\lambda_m h_m)_m$ converges to some $h\in L\otimes A(\leq d)\setminus\{0\}$ with $\Div(h)=D-d\infty$.
If the sequence $(h_m)_m$ converges to $h\in L\otimes A(\leq d)\setminus\{0\}$, the sequence $(\Div(h_m)+d\infty)_m$ converges to $\Div(h)+d\infty\in X^{(d)}(L)$. \end{prop} \begin{proof} Call $V:=\Gamma(\mathcal{I}_{d\infty}^{-1},X)$ and call $Z_d$ the pullback of $0\in\mathcal{A}$ along $J^d:X^d\to\mathcal{A}$, so that $\Div(h_m)+d\infty\in Z_d(L)$ for all $m$. As we noted in Remark \ref{projective space}, $\mathbb{P}(V)(L)$ is homeomorphic to $(L\otimes A(\leq d)\setminus\{0\})/L^\times$ endowed with the quotient topology. On the other hand, by Proposition \ref{fibers} (setting $E=\mathbb{F}_q$ and $D=d\infty$), the $\mathbb{F}_q$-schemes $\mathbb{P}(V)$ and $Z_d$ are isomorphic; in particular, the natural map $\mathbb{P}(V)(L)\to Z_d(L)$ - sending a line $L\cdot f\in L\otimes A(\leq d)$ to $\Div(f)+d\infty$ - is a homeomorphism in the compact topology, by Remark \ref{functor for F_q-schemes}.
If the sequence $(\Div(h_m)+d\infty)_m$ converges to $D\in Z_d(L)$, this proves that the equivalence classes $([h_m])_m$ in $L\otimes A(\leq d)/L^\times$ do converge to an equivalence class $[h]$ whose divisor is $D-d\infty$. Since the projection is open, we can lift this convergence to $L\otimes A(\leq d)$ up to scalar multiplication.
The map $L\otimes A(\leq d)\setminus\{0\}\to Z(L)$ sending a function $f$ to the effective divisor $\Div(f)+d\infty$ is continuous. In particular, if the sequence $(h_m)_m$ converges to $h\in L\otimes A(\leq d)\setminus\{0\}$, the sequence $(\Div(h_m)+d\infty)_m$ converges to $\Div(h)+d\infty\in Z_d(L)$. \end{proof}
\section{Frobenius and divisors}\label{Frobenius and divisors}
Fix an ideal $I\unlhd A$, with ideal class $\bar{I}\in Cl(A)$; with slight abuse of notation, call $I$ also the corresponding effective divisor of $X$. Call $\Xi\in X(K)$ the morphism $\Spec(K)\to X\setminus\infty$ corresponding to the canonical inclusion $A\hookrightarrow K$.
In the first subsection, we recall the notion of Frobenius twist $P^{(1)}$ for a point $P\in X^{(d)}({K_\infty})$, and study its behavior with respect to the compact topology. The main result is Proposition \ref{red limit}, where we prove that the sequence $\{P^{(m)}\}$ converges to $\red_{{K_\infty}}{P}$.
In the second subsection, we study the divisor of a rational function $h$ with respect to its expansion $\sum_{i\geq k}c_i u^i$ as an element of ${K_\infty}\hat\otimes K\cong K((u))$. Among several useful results, the most significant is Proposition \ref{Div and red commute}, where we state the identity $\Div(c_k)=\red_{K_\infty}(\Div(h))$.
Finally, in the third subsection, we construct the divisors $\{V_{I,*,m}\}_{m>0}$ and $V_{I,*}$ in $X^{(g)}({K_\infty})$ (see Lemma \ref{con}), uniquely defined by the following properties for $m\gg0$: \begin{align*}
&\begin{cases}V_{I,*,m}-V_{I,*,m}^{(1)}\sim\Xi^{(m)}-\Xi^{(1)}\\\red_{K_\infty}(V_{I,*,m})\sim (\deg(I)+g)\infty-I\end{cases};
&\begin{cases}V_{I,*}-V_{I,*}^{(1)}\sim\infty-\Xi\\\red_{K_\infty}(V_{I,*})\sim (\deg(I)+g)\infty-I\end{cases}. \end{align*} The main result is the convergence of the sequence $(V_{I,*,m})_{m\gg0}$ to $V_{I,*}^{(1)}$ in $X^{(g)}({K_\infty})$ (Proposition \ref{V_{I,m}}).
\subsection{Frobenius twist} In this subsection we define the Frobenius twist for a (proper) $\mathbb{F}_q$-scheme $Y$ and study its behavior with respect to the topology of $Y({K_\infty})$. The fundamental results are Proposition \ref{red limit} and Lemma \ref{Twist and divisors}. \begin{Def}\label{def frobenius} Let $Y$ be an $\mathbb{F}_q$-scheme, and $R$ an $\mathbb{F}_q$-algebra. Consider the morphism $\Frob_R:\Spec(R)\to\Spec(R)$ of $\mathbb{F}_q$-schemes induced by raising to the $q^\text{th}$ power.
The morphism $\Frob_R$ and the identity on $Y$ induce an endomorphism of $Y_R$ over $\Spec(\mathbb{F}_q)$: we denote it $F_R^Y:Y_R\to Y_R$.
Call $\pi_Y:Y_R\to Y$ and $\pi_R:Y_R\to\Spec(R)$ the natural projection. For all $P\in Y(R)$, denote $\overline{P}$ the unique element of $\Hom_R(\Spec(R),Y_R)$ such that $P=\pi_Y\circ\overline{P}$. We call Frobenius twist of $P$, and denote $P^{(1)}\in Y(R)$ the only element such that $\overline{P^{(1)}}$ is the pullback of $\overline{P}$ along $F_R^Y$. The $n$-th iteration of the twist is denoted $P^{(n)}$ for all $n\in\mathbb{N}$. \end{Def}
\begin{lemma}\label{Frob}
In the notation of Definition \ref{def frobenius}, we have $P^{(1)}=P\circ\Frob_R$. \end{lemma} \begin{proof}
We have the following cartesian diagram: \[\begin{tikzcd}
{\Spec(R)} & {Y_R} & {\Spec(R)} \\
{\Spec(R)} & {Y_R} & {\Spec(R)}
\arrow["{\pi_R}", from=1-2, to=1-3]
\arrow["{\overline{P^{(1)}}}", from=1-1, to=1-2]
\arrow["{\Frob_R}"', from=1-1, to=2-1]
\arrow["{\overline{P}}"', from=2-1, to=2-2]
\arrow["{F_R^Y}"{anchor=west}, from=1-2, to=2-2]
\arrow["{\pi_R}"', from=2-2, to=2-3]
\arrow["{\Frob_R}", from=1-3, to=2-3]
\arrow["\square"{anchor=center, pos=0.5}, draw=none, from=1-2, to=2-3]
\arrow["\square"{anchor=center, pos=0.5}, shift left=1, draw=none, from=1-1, to=2-2]. \end{tikzcd}\]
Since $\pi_Y\circ F_R^Y=\pi_Y$, we get: $P^{(1)}=\pi_Y\circ\overline{P^{(1)}}=\pi_Y\circ F_R^Y\circ\overline{P^{(1)}}=\pi_Y\circ\overline{P}\circ\Frob_R=P\circ\Frob_R.$ \end{proof} \begin{oss}
In light of Lemma \ref{Frob}, if $\Frob_R$ is an isomorphism, for all $P\in Y(R)$ we can redefine $P^{(k)}\in Y(R)$ as $P\circ(\Frob_R)^k$ for all $k\in\mathbb{Z}$. \end{oss}
\begin{lemma}\label{Frobenius twist of a power} Fix a positive integer $d$, an $\mathbb{F}_q$-scheme $Y$, and an $\mathbb{F}_q$-algebra $R$, and consider a point $(P_1,\dots,P_d)\in Y^d(R)$. Its Frobenius twist is $(P_1^{(1)},\dots,P_d^{(1)})$. \end{lemma} \begin{proof} The $i$-th projection $\pi_i:Y^d\to Y$ is such that $\pi_i\circ(P_1,\dots,P_d)=P_i$. By Remark \ref{Frob}: \[\pi_i\circ\left((P_1,\dots,P_d)^{(1)}\right)=\pi_i\circ(P_1,\dots,P_d)\circ\Frob_R=P_i\circ\Frob_R=P_i^{(1)}.\tag*{\qedhere}\] \end{proof} \begin{oss} The analogous statement, with the same proof, is true for any product of $\mathbb{F}_q$-schemes. \end{oss}
Let $L/{K_\infty}$ be a finite field extension and $Y$ a proper $\mathcal{O}_L$-scheme. Recall the notation $\red_L$ of Definition \ref{red}.
\begin{prop}\label{red limit} Fix a point $P\in Y(L)$, and set $k_L$ such that $\#\mathbb{F}_L=q^{k_L}$. The sequence $(P^{(m k_L+r)})_m$ converges to $\red_L(P)^{(r)}$ in $Y(L)$. \end{prop} \begin{proof} Since $\Spec(\mathcal{O}_L)$ only has one closed point, we can choose an open affine subscheme $U\subseteq Y$ with $B:=\mathcal{O}_Y(U)$ such that $P\in U(\mathcal{O}_L)$: $P$ corresponds to a map of $\mathcal{O}_L$-algebras $\chi_P:B\to\mathcal{O}_L$; its reduction modulo $\mathfrak{m}_L$, composed with the immersion $\mathbb{F}_L\hookrightarrow\mathcal{O}_L$, is $\chi_{\red_L(P)}$ by Definition \ref{red}. For all $i$, $P^{(i)}$ corresponds to the map $(\cdot)^{q^i}\circ\chi_P$, which modulo $\mathfrak{m}_L^{q^i}$ is the same as $\chi_{\red_L(P)^{(i)}}$, hence the projections of $P^{(i)}$ and $\red_L(P)^{(i)}$ onto $Y(\mathcal{O}_L/\mathfrak{m}_L^{q^i})$ coincide. Since $\red_L(P)^{(m k_L+i)}=\red_L(P)^{(i)}$ for all $m\geq0$, this proves the convergence. \end{proof}
\begin{oss} For any effective divisor $D$ of the curve $X_L$ over $\Spec(L)$, we can define its twist $D^{(1)}$ as the pullback along $F_L^X$. Obviously, if $D=\sum P_i$ with $P_i\in X_L(L_i)$, $D^{(1)}=\sum P_i^{(1)}$. \end{oss}
\begin{Def} Let $h$ be a rational function on $X_L$, i.e. a morphism of $L$-schemes $X_L\to\mathbb{P}^1_L$. We define the Frobenius twist $h^{(1)}:=h\circ F_L^X$. \end{Def} \begin{oss}
The field rational functions of $X_L$ is $\Q(L\otimes A)$, and if $h=\sum_i l_i\otimes a_i\in L\otimes A$, $h^{(1)}=\sum_i l_i^q\otimes a_i$. \end{oss}
We show that the Frobenius twists of divisors and rational functions are compatible.
\begin{lemma}\label{Twist and divisors} Let $h\in L(X)$, and call $\Div(h)$ its divisor. Then, $\Div(h^{(1)})=(\Div(h))^{(1)}$. \end{lemma} \begin{proof} For any closed point $P\in\mathbb{P}^1_L$, $(h^{(1)})^*(P)=(F_L^X)^*\circ h^*(P)$; setting $P=0$ and $P=\infty$, since the Frobenius twist on the divisors is induced by the pullback via $F_L^X$, we get our thesis. \end{proof}
\subsection{Rational functions on $X_{K_\infty}$ as Laurent series} Fix a finite field extension $L/{K_\infty}$ and a uniformizer $u\in\mathcal{O}_L$. Call $K'$ the fraction field of $X_{\mathbb{F}_L}$, i.e. $\mathbb{F}_L K$, and $A':=\mathbb{F}_L A$. \begin{oss}
Since $L(X)$ is the fraction field of $\mathcal{O}_L\otimes K'$, which has $\mathfrak{m}_L K'=(u\otimes1)\mathcal{O}_L\otimes K'$ as a maximal ideal, we can endow $L(X)$ with the $\mathfrak{m}_L K'$-adic topology. \end{oss} \begin{lemma}\label{K((u))} The following statements hold. \begin{enumerate}
\item The completion of $L(X)$ is naturally isomorphic to $K'((u))$ as a topological field.
\item For all positive integers $d$ the inclusion $L\otimes_{\mathbb{F}_L}A'(\leq d)\cong L\otimes A(\leq d)\hookrightarrow L(X)$ is an isometry (with respect to the sup norm on the left and the $u$-adic norm on the right) and a closed immersion.
\item For all $h\in L(X)\subseteq K'((u))$, if we write $h=\sum_{j\geq m}c_j u^j$, then $h^{(1)}=\sum_{j\geq m}c_j^{(1)}
u^{qj}$.
\item There is an immersion $L\hat\otimes A\hookrightarrow K'((u))$ whose image is exactly $A'[[u]][u^{-1}]$, i.e. the subset of Laurent series whose coefficients are in $A'$. \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate}
\item The natural isomorphisms $(\mathcal{O}_L\otimes K'/\mathfrak{m}_L^n K'\cong K'[u]/u^n)_n$ pass to the limit and to fraction fields, giving a natural isomorphism between the completion of $L(X)$ and $K'((u))$ as topological fields.
\item Since the metric space $L\otimes A(\leq d)$ is complete, it suffices to prove that its inclusion into $L(X)\subseteq K'((u))$ is an isometry. Fix an $\mathbb{F}_L$-linear basis $\{b_1,\dots,b_k\}$ of $A'(\leq d)$: we want to show that for all $k$-uples $(a_1,\dots,a_k)\in L^k\setminus\{0\}$, their minimum valuation is equal to the degree of $\sum_j a_j b_j\in K'((u))$. Up to multiplying all $a_j$'s by the same factor in $L^\times$, we can assume their minimum valuation to be $0$, so that we can take their projections $(\bar{a_j})_j$ onto $\mathcal{O}_L/\mathfrak{m}_L\cong\mathbb{F}_L$. We have that $\sum_j a_j b_j\in K'[[u]]$, and its coefficient of degree $0$ is $\sum_j\bar{a_j}b_j$; since not all $\bar{a_j}$ are zero, and the $b_j$'s are $\mathbb{F}_L$-linearly independent, this coefficient is nonzero.
\item Since the Frobenius twist on $\mathcal{O}_L\otimes A$ comes from the identity of $A$ and the Frobenius endomorphism of $\mathcal{O}_L$, we can extend it to $K'((u))$ by sending a generic element $\sum_{j\geq m}c_j u^j$ to $\sum_{j\geq m}c_j^{(1)} u^{qj}$.
\item Since $L\otimes A=\bigcup_d L\otimes A(\leq d)$, its inclusion in $K'((u))$ is an isometry, and since its image is contained in $A'[[u]][u^{-1}]$ we have an immersion $L\hat\otimes A\hookrightarrow A'[[u]][u^{-1}]$. On the other hand, every element $h\in A'[[u]][u^{-1}]$ is limit of its truncated expansions, which are in the image of $L\otimes A$, hence $h$ is contained in the image of $L\hat\otimes A$.\qedhere \end{enumerate} \end{proof}
To better understand the usefulness of $K'((u))$, let's state a couple of propositions. First, we prove a very natural result, analogous to Lemma \ref{Twist and divisors} but with the reduction instead of the twist. \begin{Def}
For all nonzero $h\in L(X)\subseteq K'((u))$, write $h=\sum_{j\geq m}c_j u^j$ with $c_m\neq0$, and set $\red_u(h):=c_m$. \end{Def} \begin{prop}\label{Div and red commute} For all nonzero $h\in L(X)$, $\Div(\red_u(h))=\red_L(\Div(h))$, where both are $\mathbb{F}_L$-rational divisors of $X_{\mathbb{F}_L}$. \end{prop} \begin{proof} Since for any nonzero $h\in L(X)$ there is a positive integer $d$ and $h_+,h_-\in\mathcal{O}_L\otimes_{\mathbb{F}_L} A'(\leq d)$ such that $h=\frac{h_+}{h_-}$, we can assume $h\in\mathcal{O}_L\otimes_{\mathbb{F}_L} A'(\leq d)$. Up to a factor in $L^\times$, we can also assume $h=\sum_{i\geq0}c_i u^i\in K'[[u]]$ with $c_0\in A'(\leq d)\setminus\{0\}$. By Lemma \ref{K((u))}.3, the sequence $(h^{(m k_L)})_m$ is equal to $(\sum_{j\geq0}c_j u^{j m k_L})_m$, hence it converges to $c_0$ in $K'[[u]]$; by Lemma \ref{K((u))}.2 this convergence lifts to $L\otimes_{\mathbb{F}_L} A'(\leq d)$. The sequence of divisors $(\Div(h^{(m f_L)})+d\infty)_m$, by Proposition \ref{convergence of functions and divisors}, converges to $\Div(c_0)+d\infty$ in $X^{(d)}(L)$; on the other hand, by Proposition \ref{red limit}, it converges to $\red_L(\Div(h))+d\infty$, hence we have the desired equality. \end{proof}
We prove now that the immersion $L(X)\hookrightarrow K'((u))$ behaves reasonably well with evaluations. \begin{prop}\label{conv eval entire rational} Fix $h\in L(X)$ with its only poles at $\infty$ (i.e. $h\in L\otimes A$), of degree $d$, written as $\sum_i h_{(i)} u^i\in K'((u))$; fix $P\in X_{\mathbb{F}_L}(L^{ac})\setminus\{\infty\}$, corresponding to a $\mathbb{F}_L$-linear homomorphism $\chi_P:A'\to L^{ac}$. Then, $h_{(i)}\in A'$ for all $i$ and $h(P)=\sum_i \chi_P(h_{(i)})u^i$. \end{prop} \begin{proof} We can write $h=\sum_j \gamma_j a_j$, with $\gamma_j\in L=\mathbb{F}_L((u))$ and $a_j\in A'(\leq d)$, hence $h_{(i)}\in A'(\leq d)$ for all $i$. For all integers $m$ define $\gamma_{j,m}$ as the truncation of $\gamma_j\in\mathbb{F}_L((u))$ at the degree $m$, and define $h_m:=\sum_j \gamma_{j,m} a_j\in K'((u))$, so that $h_m=\sum_{i\leq m} h_{(i)} u^i$. We have the equalities: \begin{align*}
&h_m(P)=\chi_P\left(\sum_j \gamma_{j,m} a_j\right)=\sum_j\gamma_{j,m}\chi_P(a_j);
&h_m(P)=\chi_P\left(\sum_{i\leq m} h_{(i)} u^i\right)=\sum_{i\leq m}\chi_P(h_{(i)})u^i; \end{align*} where we used that both summations are finite. Since the sequence $(\gamma_{j,m})_m$ converges to $\gamma_j$ in $\mathbb{F}_L((u))$ for all $j$, the first equation tells us that the sequence $(h_m(P))_m$ converges to $h(P)$. On the other hand, from the second equation, it also converges to $\sum_i \chi_P(h_{(i)})u^i$. \end{proof}
\begin{prop}\label{conv eval Tate} Let $h=\sum_i h_{(i)} u^i\in A'[[u]][u^{-1}]$ be a rational function over $X_L$; for any finite field extension $E/L$, fix $P\in X_L(E)$ such that $\red_E(P)\neq\infty$, corresponding to a function $\chi_P:A'\to\mathcal{O}_E$. Then $P$ is not a pole of $h$, and $h(P)=\sum_i \chi_P(h_{(i)})u^i$. \end{prop} \begin{proof} For $N\gg0$, we have a strict inclusion $\mathcal{L}(N\infty-\Div_-(h)-P)\subsetneq\mathcal{L}(N\infty-\Div_-(h))$, so we can choose $h_-$ in their difference, and set $h_+:=h h_-$; by definition, $h_+,h_-\in L\otimes A'$.
If we write $h_+=\sum_i h_{+,(i)} u^i$ and $h_-=\sum_i h_{-,(i)} u^i$, we have for all integers $k$ the equation $h_{+,(k)}=\sum_{i+j=k}h_{(i)} h_{-,(j)}$, which commutes with evaluation, being a finite sum. Since $\chi_P$ has image in $\mathcal{O}_E$, the series $\sum_i \chi_P(h_{(i)})u^i$ converges, hence by Proposition \ref{conv eval entire rational} we get the following equation in $\mathcal{O}_E$: \begin{align*}
h_+(P)&=\sum_k \chi_P(h_{+,(k)}) u^k=\sum_k\sum_{i+j=k}\chi_P(h_{(i)})\chi_P(h_{-,(j)})u^k\\
&=\left(\sum_i\chi_P(h_{(i)})u^i\right)\left(\sum_j \chi_P(h_{-,(j)})u^j\right)=\left(\sum_i\chi_P(h_{(i)})u^i\right)h_-(P). \end{align*} Since $h_-\in\mathcal{L}(N\infty-\Div_-(h))\setminus\mathcal{L}(N\infty-\Div_-(h)-P)$, if $P$ is a pole of $h$, then $P$ is a zero of $h_-$ of the same order, and $h_+(P)\neq0$, hence we reach a contradiction by the previous equation. Since $P$ is not a pole of $h$, then $h_-(P)\neq0$, and since $h_+(P)=h(P)h_-(P)$ we get: \[h(P)=\frac{h_+(P)}{h_-(P)}=\sum_i\chi_P(h_{(i)})u^i.\tag*{\qedhere}\] \end{proof} \begin{cor}\label{A[[u]][u^{-1}]} Let $h=\sum_i h_{(i)}u^i\in K'[[u]]$ be a rational function over $X_L$. Then, $h$ is in $A'[[u]][u^{-1}]$ if and only if all its poles reduce to $\infty$. \end{cor} \begin{proof} By Proposition \ref{conv eval Tate}, if $h\in A'[[u]][u^{-1}]$, its poles reduce to $\infty$. Vice versa, if $h\not\in A'[[u]][u^{-1}]$, call $m$ the least integer such that $h_{(m)}\not\in A'$ and set $h':=\sum_{i<m} h_{(i)} u^i$. By Lemma \ref{Div and red commute}: \[\Div(h_{(m)})=\Div(\red_u(h-h'))=\red_L(\Div(h-h'))\] and, since $h_{(m)}\not\in A'$, $h-h'$ has a pole at a point $P$ which does not reduce to $\infty$; on the other hand, since $h'\in A'[[u]][u^{-1}]$, it does not have a pole at $P$, hence $P$ is a pole of $h=(h-h')+h'$. \end{proof}
\begin{cor}\label{cor conv eval Tate} Let $h=\sum_i h_{(i)}u^i\in A'[[u]][u^{-1}]$ be a nonzero rational function over $X_L$, and suppose that the coefficients $(h_{(i)})_i$ are all contained some prime ideal $P\unlhd A'$. Then $P$, as a closed point of $X_L$, is a zero of $h$. \end{cor} \begin{proof} If we define $\mathbb{F}_P:=A'/P$, we can take a point $Q\in X_L(\mathbb{F}_P L)$ with support at $P$. By Proposition \ref{conv eval Tate}, we get $h(Q)=\sum_i h_{(i)}(Q)u^i=0$, hence $P$ is a zero of $h$. \end{proof}
\subsection{Notable divisors and convergence results} As anticipated by Lemma \ref{Twist and divisors}, in this subsection we explore the relation between Frobenius twists, divisors, and the compact topology.
\begin{lemma}[Drinfeld's vanishing lemma]\label{Drinfeld} Let $E/K$ be a field extension, $W\in X^{(d)}(E)$ for some $d\leq g$, $P,Q\in X(E)$. Suppose that $[W-W^{(m)}]=[P-Q]$, where $P\neq Q^{(sm)}$ for $0\leq s+d\leq2g$; then $d=g$ and $h^0(W)=1$. \end{lemma} \begin{proof} Call $W_0:=W$ and set $W_{i+1}=W_i+Q^{(im)}$ for all $i\in\mathbb{Z}$. Note that, since $\deg(W_k)=d+k$, $h^0(W_{-d-1})=0$ and $h^0(W_{2g-d-1})=g$. For all $i$, $h^0(W_i)\leq h^0(W_{i+1})\leq h^0(W_i)+1$, so there is a least integer $k\in[-d,0]$ such that $h^0(W_k)=1$.
Let's prove that for all $i\in[-d,2g-d-1[$, if $h^0(W_i)\geq1$, then $h^0(W_{i+1})=h^0(W_i)+1$. We have two relations: \begin{align*}
W_{i+1}&=(W+Q+\dots+Q^{((i-1)m)})+Q^{(im)}=W_i+Q^{(im)},\\
W_{i+1}&=(W+Q^{(m)}+\dots+Q^{(im)})+Q=(W_i^{(m)}-W^{(m)}+W)+Q\sim W_i^{(m)}+P; \end{align*} they induce two inclusions of vector spaces - $\mathcal{L}(W_i^{(m)})\subseteq \mathcal{L}(W_{i+1})$ and $\mathcal{L}(W_i)\subseteq \mathcal{L}(W_{i+1})$. To prove that those inclusions are strict, we need that $\mathcal{L}(W_i)\neq \mathcal{L}(W_i^{(m)})$ as subspaces of $\mathcal{L}(W_{i+1})$; they have the same dimension (because the Frobenius twist induces an isomorphism between the two vector spaces), so we just need $W_i\not\sim W_i^{(m)}$, but: \[W_i\not\sim W_i^{(m)}\Leftrightarrow W-W^{(m)}\not\sim Q^{(im)}-Q\Leftrightarrow P\not\sim Q^{(im)}\Leftrightarrow P\neq Q^{(im)},\] which is implied by our hypothesis. In particular, since $W_{2g-d-1}$ has degree $2g-1$, we get that \[g=h^0(W_{2g-d-1})=h^0(W_k)+2g-d-1-k=2g-d-k,\] therefore $g=d+k$; but $k\leq0$ and $d\leq g$ implies $d=g$ and $k=0$, therefore $h^0(W)=1$. \end{proof}
The previous lemma ensures that if such a divisor $W$ exists, it has no other effective divisors in its same equivalence class. On the other hand, the existence of such $W$ is ensured by the following propositions.
As usual, let $L$ be a finite field extension of ${K_\infty}$, with residue field $\mathbb{F}_L$ and $q^{k_L}:=\#\mathbb{F}_L$
\begin{prop}\label{finite covering} Call $\mathcal{A}_0(L)$ the kernel of $\red_L:\mathcal{A}(L)\to\mathcal{A}(\mathbb{F}_L)$ (which is a continuous homomorphism). The map $\mathcal{A}(L)\to\mathcal{A}_0(L)\times\mathcal{A}(\mathbb{F}_L)$ sending a point $D$ to the couple $(D-D^{(k_L)},\red_L(D))$ is an isomorphism of topological groups. \end{prop} \begin{proof} The map is obviously a continuous group homomorphism. Since domain and codomain are both compact and Hausdorff, it's sufficient to prove bijectivity.
On one hand, to prove injectivity, if we suppose $D-D^{(k_L)}=0$ we have $D\in\mathcal{A}(\mathbb{F}_L)$, so if $\red_L(D)=0$ we can deduce that $D=0$.
On the other hand, to prove surjectivity, we fix $(D_0,\tilde{D})\in\mathcal{A}_0(L)\times\mathcal{A}(\mathbb{F}_L)$ and show that they are the image of some $D\in\mathcal{A}(L)$. By Proposition \ref{red limit} we have that the sequence $(D_0^{(i k_L)})_i$ converges to $\red_L(D_0)=0$, hence by Corollary \ref{series convergence} the series $\tilde{D}+\sum_{i\geq0}D_0^{(i k_L)}$ converges to some point $D\in\mathcal{A}(L)$. Since the Frobenius twist and the reduction $\red_L$ are continuous endomorphisms of $\mathcal{A}(L)$, we get the following equations: \begin{align*}
&D-D^{(k_L)}=\tilde{D}+\sum_{i\geq0}D_0^{(i k_L)}-\tilde{D}^{(k_L)}-\sum_{i\geq1}D_0^{(i k_L)}=D_0;&\red_L(D)=\red_L(\tilde{D})=\tilde{D}, \end{align*} hence the image of $D$ is $(D_0,\tilde{D})$. \end{proof}
From now on, given an effective divisor $W\in X^{(d)}(L)$, we denote with $J(W)$ its image via the morphism $J^d:X^{(d)}\to\mathcal{A}$, i.e. the equivalence class of $W-d\infty$ in the Jacobian. \begin{cor}\label{V}
Fix a point $D\in\mathcal{A}(\mathbb{F}_L)$, and let $P,Q\in X(L)$ such that $\red_L(P)=\red_L(Q)$, with $P\neq Q^{(s)}$ for $|s|<2g$.
Then, there is a unique effective divisor $W$ such that: $W-W^{(k_L)}\sim P-Q$, $\red_L(J(W))=D$, and $\deg(W)\leq g$. Moreover, a fortiori, $W\in X^{(g)}(L)$ and, if $R$ is a point in the support of $W$, $R\not\in X(\mathbb{F}_q^{ac})$. \end{cor} \begin{proof} By Proposition \ref{finite covering} there is an element $D'\in\mathcal{A}(L)$ such that $D'-D'^{(k_L)}=[P-Q]$ and $\red_L(D')=D$. Since the morphism $J^g$ is surjective, there is a divisor $W\in X^{(g)}(L^{ac})$ such that $J(W)=D'$. By Drinfeld's vanishing lemma, there is only one divisor of degree $\leq g$ with the requested properties, hence $h^0(W)=1$; by Corollary \ref{preimage of h^0=1}, $W$ is $L$-rational.
Now, call $W'\leq W$ a maximal $\mathbb{F}_q^{ac}$-rational effective divisor ($W'\in X^{(d)}(\mathbb{F}_q^{ac})$), and call $G$ the group of $L$-linear field automorphisms of $L^{ac}$, which acts naturally on $X(L^{ac})$. Since $W\in X^{(g)}(L)$, it is fixed by the induced action of $G$; moreover, this action sends $X(\mathbb{F}_q^{ac})$ to itself, hence $W'\leq W$ is also fixed by $G$: since $W'$ is both $L$-rational and $\mathbb{F}_q^{ac}$-rational, $W'\in X^{(d)}(\mathbb{F}_L)$. We have: \[(W-W')-(W-W')^{(k_L)}=(W-W^{(k_L)})+(W'-W'^{(k_L)})=W-W^{(k_L)}\sim P-Q,\] but $\deg(W-W')=g$ from Drinfeld's vanishing lemma, hence $d=deg(W')=0$. \end{proof}
Recall the notations of $I,\bar{I},\Xi$ from the start of this section.
\begin{lemma}\label{red(Xi)} We have the identity $\red_{K_\infty}(\Xi)=\infty$ in $X({K_\infty})$. \end{lemma} \begin{proof} Since the image of the canonical inclusion $A\hookrightarrow{K_\infty}$ is not contained in $\mathcal{O}_{K_\infty}$, the morphism $\Xi:\Spec({K_\infty})\to X\setminus\infty$ does not factor through $\Spec(\mathcal{O}_{K_\infty})$, which means that $\red_{K_\infty}(\Xi)\not\in X(\mathbb{F}_q)\setminus\infty$, so $\red_{K_\infty}(\Xi)=\infty$. \end{proof}
Next, we construct some notable divisors.
\begin{lemma}\label{con} The following effective divisors of $X_{K_\infty}$ exist and are unique: \begin{itemize}
\item a divisor $V_{\bar{I}}$ of degree at most $g$, such that $V_{\bar{I}}-V_{\bar{I}}^{(1)}\sim\Xi-\infty$ and $\red_{K_\infty}(J(V_{\bar{I}}))=J(I)$;
\item for $m\geq1$, a divisor $V_{\bar{I},m}$ of degree at most $g$, such that $V_{\bar{I},m}-V_{\bar{I},m}^{(1)}\sim\Xi^{(1)}-\Xi^{(m+1)}$ and $\red_{K_\infty}(J(V_{\bar{I}}))=J(I)$;
\item a divisor $V_{\bar{I},*}$ of degree at most $g$ such that $J(V_{\bar{I},*})+J(V_{\bar{I}})=0$;
\item for $m\gg0$, a divisor $V_{\bar{I},*,m}$ of degree at most $g$ such that $J(V_{\bar{I},*,m})+J(V_{\bar{I},m})=0$. \end{itemize} Moreover, they all are in $X^{(g)}({K_\infty})$. \end{lemma} \begin{proof} Let's first note that the divisors, if they exist, are well defined: since for all $a,b\in^* A$ $J(aI)=J(bI)$, the properties of the divisors we want to construct only depend on the ideal class $\bar{I}\in Cl(A)$ of $I$.
Since $\red_{K_\infty}(\Xi)=\infty$ by Lemma \ref{red(Xi)}, we can apply Corollary \ref{V} to $P=\Xi$ and $Q=\infty$ (resp. $P=\Xi^{(1)}$ and $Q=\Xi^{(m+1)}$ for $m\geq1$), so the divisor $V_{\bar{I}}$ (resp. $V_{\bar{I},m}$) exists, is unique, and is contained in $X^{(g)}({K_\infty})$.
Since $J^g(K_\infty^{ac}):X^{(g)}(K_\infty^{ac})\to\mathcal{A}(K_\infty^{ac})$ is surjective, there is at least one effective divisor $V_{\bar{I},*}$ of degree at most $g$ such that $J(V_{\bar{I},*})=-J(V_{\bar{I}})$. It has the following properties: \begin{align*}
&[V_{\bar{I},*}-V_{\bar{I},*}^{(1)}]=[V_{\bar{I}}^{(1)}-V_{\bar{I}}]=[\infty-\Xi];
&\red_{K_\infty}(J(V_{\bar{I},*}))=-\red_{K_\infty}(J(V_{\bar{I}}))=-J(I). \end{align*} By Corollary \ref{V} applied to $P=\infty$ and $Q=\Xi$, $V_{\bar{I},*}$ is unique, ${K_\infty}$-rational, and of degree $g$.
In the same way we can prove the existence and uniqueness of $V_{\bar{I},*,m}$ for $m\gg0$, since it has the following properties: $V_{\bar{I},*,m}-V_{\bar{I},*,m}^{(1)}\sim\Xi^{(m+1)}-\Xi^{(1)}$, and $\red_{K_\infty}(J(V_{\bar{I},*,m}))=-J(I)$. \end{proof} \begin{oss}\label{Hayes} Since we fixed an inclusion $H\subseteq{K_\infty}$, the divisors $\{V_{\bar{I}}\}_{\bar{I}}$ are actually $H$-rational, and the natural action of $\G(H/K)$ on this set is free and transitive (see \cite{Hayes}[Prop. 3.2, Thm. 8.5]). Call $\bar{I}^\sigma\in Cl(A)$ the element such that $V_{\bar{I}^\sigma}=V_{\bar{I}}^\sigma$. Since this action commutes with morphisms of schemes, for all $\sigma\in\G(H/K)$, for all $\bar{I}\in Cl(A)$, we have that \[[V_{\bar{I},*}^\sigma-g\infty]=[V_{\bar{I},*}-g\infty]^\sigma=[g\infty-V_{\bar{I}}]^\sigma=[g\infty-V_{\bar{I}}^\sigma]=[g\infty-V_{\bar{I}^\sigma}]=[V_{\bar{I}^\sigma,*}-g\infty];\] hence $V_{\bar{I},*}^\sigma=V_{\bar{I}^\sigma,*}$ by Lemma \ref{con} because of uniqueness. \end{oss}
Finally, we state the main result of this subsection, which is central to the proof of Theorem \ref{functional identity}.
\begin{prop}\label{V_{I,m}} The sequences $(V_{\bar{I},m})_m$ and $(V_{\bar{I},*,m})_m$ converge respectively to the divisors $V_{\bar{I}}^{(1)}$ and $V_{\bar{I},*}^{(1)}$ in $X^{(g)}({K_\infty})$. \end{prop}
\begin{proof}
Define $U:=\{D\in X^{(g)}({K_\infty})|h^0(D)=1\}$, so that the restriction $J^g({K_\infty})|_U$ induces a bijection of $U$ with its image in $\mathcal{A}({K_\infty})$, which by Lemma \ref{local homeomorphism} is a homeomorphism. By Lemma \ref{con} - for $m\gg0$ - $h^0(V_{\bar{I},m})=h^0(V_{\bar{I}}^{(1)})=1$, so $V_{\bar{I},m},V_{\bar{I}}^{(1)}\in U$, and it suffices to prove the convergence of their images in $\mathcal{A}({K_\infty})$.
If we identify $\mathcal{A}({K_\infty})$ and $\mathcal{A}(\mathbb{F}_q)\times\mathcal{A}_0({K_\infty})$ by Proposition \ref{finite covering}, we have: \begin{align*}
\lim_m V_{\bar{I},m}=&\lim_m\left(\red_{K_\infty}(J^g(V_{\bar{I},m})),[V_{\bar{I},m}-V_{\bar{I},m}^{(1)}]\right)=\lim_m\left(J(I),[\Xi^{(1)}-\Xi^{(m+1)}]\right)\\
=&\left(J(I),[\Xi^{(1)}-\infty]\right)=\left(\red_{K_\infty}(J^g(V_{\bar{I}}^{(1)})),[V_{\bar{I}}^{(1)}-V_{\bar{I}}^{(2)}]\right)=V_{\bar{I}}^{(1)}. \end{align*} Similarly, for the other statement, it suffices to prove that the sequence $(J(V_{\bar{I},m,*}))_m$ converges to $J(V_{\bar{I},*}^{(1)})$, which is obvious because $J(V_{\bar{I},m,*})=-J(V_{\bar{I},m})$ for all $m\gg0$ and $J(V_{\bar{I},*}^{(1)})=-J(V_{\bar{I}}^{(1)})$. \end{proof}
\section{Zeta functions}\label{zeta functions} Throughout this and all the next sections we fix a uniformizer $u\in{K_\infty}$ and an ideal $I\unlhd A$. With the same abuse of notation as before, we also call $I$ the corresponding closed subscheme of $X\setminus\infty$, and $d$ its degree.
The following definition is a generalization of the zeta functions introduced by Pellarin in \cite{Pellarin2011}.
\begin{Def}
The \textit{partial zeta function} relative to $I$ is defined as the series:
\[\zeta_I:=\sum_{a\in^* I}a^{-1}\otimes a\in{K_\infty}\hat\otimes A.\] \end{Def} In this section, we first define the rational approximations $\{\zeta_{I,m}\}$ of $\zeta_I$ and compute their divisors, in analogy to what already done by Chung, Ngo Dac and Pellarin in the case $I=A$ (see \cite{Pellarin2021}[Lemma 2.1]). Afterwards, we use Proposition \ref{convergence of functions and divisors} to prove a functional identity regarding $\zeta_I$ in the shape of an infinite product, i.e. the partial version of Theorem \ref{functional identity}.
The complete version of the theorem, stated in the introduction, is be proven at the end of Section \ref{section duality}.
\subsection{The approximations of $\zeta_I$ and their divisors}\label{subsection def zeta}
For $m\in\mathbb{N}$, call $j_m$ the least integer such that $\dim_{\mathbb{F}_q}(I(\leq j_m))=h^0(j_m\infty-I)=m+1$. We call $a_I\in I$ the nonzero element with least degree (i.e. $a_I\in I(j_0)$) and sign $1$. \begin{oss}\label{j_m inequality}
Since $\deg(j_m\infty-I)+1-g\leq h^0(j_m\infty-I)\leq\deg(j_m\infty-I)+1$, we get the inequality:
\[m+d\leq j_m\leq m+g+d.\]
Moreover, for $m\gg0$, the rightmost inequality becomes an equality. \end{oss}
\begin{Def}
We set for all $m\geq0$:
\[\zeta_{I,m}:=\sum_{a\in^* I(\leq j_m)}a^{-1}\otimes a\in{K_\infty}\otimes A.\] \end{Def}
\begin{oss} The sequence $\zeta_{I,m}$ converges to $\zeta_I$ in ${K_\infty}\hat\otimes A\cong A[[u]]$. \end{oss} \begin{prop}\label{divisor of zeta_{I,m}} The divisor of $\zeta_{I,m}$ is $\Xi^{(1)}+\dots+\Xi^{(m)}+I+W_m-j_m\infty$ for some effective divisor $W_m$ with $h^0(W_m)=1$. Moreover, for $m\gg0$, $j_m=m+g+d$ and $W_m=V_{\bar{I},*,m}$. \end{prop}
We use the following technical lemma (see \cite{Goss}[Lemma 8.8.1]). \begin{lemma}\label{Goss} Let $U\subseteq{\mathbb{C}_\infty}$ be a finite $\mathbb{F}_q$-vector space of dimension $\alpha$, let $a$ be a nonnegative integer, and denote $a_i$ - with $0\leq a_i<q$ - the $i$-th digit of the expansion of $a$ in base $q$. Then, if $\sum_i a_i<(q-1)\alpha$, the polynomial $\sum_{w\in U}(x+w)^a\in{\mathbb{C}_\infty}[x]$ is identically zero. In particular, this happens for $0\leq a<q^\alpha-1$. \end{lemma}
\begin{proof}[Proof of Proposition \ref{divisor of zeta_{I,m}}] Since $\zeta_{I,m}$ is sum of elements whose divisor contains $I$, it's obvious that $\Div^+(\zeta_{I,m})\geq I$. For any positive integer $k$ we have: \[\zeta_{I,m}(\Xi^{(k)})=\sum_{a\in I(\leq j_m)}a^{q^k-1},\] which by Lemma \ref{Goss} is zero when $k\leq\dim(I(\leq j_m))-1=m$. Since the only poles are at $\infty$, and have multiplicity at most $j_m$, $\Div(\zeta_{I,m})=\Xi^{(1)}+\dots+\Xi^{(m)}+I+W_m-j_m\infty$ for some effective divisor $W_m$. To study $h^0(W_m)$, call $D_n:=j_m\infty-I-\sum_{i=1}^n\Xi^{(i)}$ for all nonnegative integers $n$.
Note that, since $(j_m\infty-I)^{(1)}=j_m\infty-I$, we deduce for all $n\geq0$: \begin{align*}
&\mathcal{L}(D_{n+1})\subseteq\mathcal{L}(D_n), &\mathcal{L}(D_{n+1})\subseteq\mathcal{L}(D_n^{(1)}), &&\mathcal{L}(D_n)\cap\mathcal{L}(D_n^{(1)})=\mathcal{L}(D_{n+1}). \end{align*} Let's prove that, if $h^0(D_n)\geq1$, then $h^0(D_{n+1})=h^0(D_n)-1$. If this were not the case, since for $k\gg0$ $h^0(D_k)<\deg(D_k)<0$, we could fix the maximum $n$ such that $h^0(D_{n+1})=h^0(D_n)>0$, and we would get the following implications: \begin{align*} &\left(\mathcal{L}(D_{n+1}^{(1)})+\mathcal{L}(D_{n+1})\subseteq\mathcal{L}(D_n^{(1)})\right)\Rightarrow\left(\mathcal{L}(D_n)=\mathcal{L}(D_{n+1})=\mathcal{L}(D_{n+1}^{(1)})\right)\\ \Rightarrow&\left(\mathcal{L}(D_{n+1})=\mathcal{L}(D_{n+1})\cap\mathcal{L}(D_{n+1}^{(1)})=\mathcal{L}(D_{n+2})\right)\Rightarrow \left(h^0(D_{n+2})=h^0(D_{n+1})\right), \end{align*} contradicting the maximality hypothesis on $n$. In particular, since $h^0(W_m)\geq1$, we have: \[h^0(W_m)=h^0(D_m)=h^0(D_0)-m=h^0(j_m\infty-I)-m=1.\]
On one hand, $\deg(W_m)=\deg(j_m\infty-\Xi^{(1)}-\dots-\Xi^{(m)}-I)=j_m-m-d$, which is $\leq g$ by Remark \ref{j_m inequality}. On the other hand, by Lemma \ref{Twist and divisors} and Proposition \ref{Div and red commute} we have: \begin{align*}
&0\sim\Div(\zeta_{I,m})-\Div(\zeta_{I,m})^{(1)}=\Xi^{(1)}-\Xi^{(m+1)}+W_m-W_m^{(1)},\\
&0\sim\Div(\red_u(\zeta_{I,m}))\sim\red_{K_\infty}(\Div(\zeta_{I,m}))=I+\red_{K_\infty}(W_m)-(d+g)\infty; \end{align*} so $W_m-W_m^{(1)}\sim \Xi^{(m+1)}-\Xi^{(1)}$, and $\red_{K_\infty}(W_m-g\infty)\sim d\infty-I$. Therefore, for $m\gg0$, $W_m=V_{\bar{I},*,m}$ by Lemma \ref{con}. \end{proof}
\subsection{The function $\zeta_I$ as an infinite product}
\begin{prop}\label{functions} In $\mathcal{O}_{K_\infty}\otimes K\subseteq K[[u]]$ there are functions $f'_{\bar{I},*},f'_{\bar{I}}\in 1+uK[[u]]$, with divisors $V_{\bar{I},*}-V_{\bar{I},*}^{(1)}+\Xi-\infty$ and $V_{\bar{I}}^{(1)}-V_{\bar{I}}+\Xi-\infty$, respectively. Moreover, there is a rational function $\delta'_{\bar{I}}\in\mathcal{O}_{K_\infty}\otimes K$, with divisor $V_{\bar{I}}+V_{\bar{I},*}-2g\infty$, such that $\frac{{\delta'_{\bar{I}}}^{(1)}}{\delta'_{\bar{I}}}=\frac{f'_{\bar{I}}}{f'_{\bar{I},*}}$. \end{prop} \begin{proof} From the definition of $V_{\bar{I},*}$, the divisor $V_{\bar{I},*}-V_{\bar{I},*}^{(1)}+\Xi-\infty$ is principal, hence it comes from some $f'_{\bar{I},*}\in{K_\infty}(X)$. Moreover, by Lemma $\ref{Div and red commute}$: \[\Div(\red_u(f'_{\bar{I},*}))=\red_{K_\infty}(\Div(f'_{\bar{I},*}))=\red_{K_\infty}(V_{\bar{I},*})-\red_{K_\infty}(V_{\bar{I},*})^{(1)}=0,\] hence $\red_u(f'_{\bar{I},*})\in\mathbb{F}_q$, and WLOG it's equal to 1; up to scalar multiplication, we can assume $f'_{\bar{I},*}=1+O(u)$. The existence of $f'_{\bar{I}}$ can be proven in the same way.
Since $V_{\bar{I}}+V_{\bar{I},*}-2g\infty\sim0$, we can pick $\tilde\delta'_{\bar{I}}\in{K_\infty}\otimes A(\leq2g)$ with that divisor, and up to scalar multiplication we can assume $\tilde\delta'_{\bar{I}}=c_0+O(u)$ for some $c_0\in K$. We get: \[\Div(\tilde\delta'_{\bar{I}})^{(1)}-\Div(\tilde\delta'_{\bar{I}})=\Div(f'_{\bar{I}})-\Div(f'_{\bar{I},*})\Longrightarrow\frac{\tilde\delta'_{\bar{I}}{}^{(1)}}{\tilde\delta'_{\bar{I}}}=\lambda\frac{f'_{\bar{I}}}{f'_{\bar{I},*}}\] for some $\lambda\in{K_\infty}$; moreover, by considering the expansion in $K((u))$, $\lambda=1+O(u)$, hence it admits a $q-1$-th root $\mu\in\mathcal{O}_{K_\infty}$. If we set $\delta'_{\bar{I}}:=\mu\tilde\delta'_{\bar{I}}$ we obtain the desired equation. \end{proof} \begin{oss} The choices of $f'_{\bar{I}},f'_{\bar{I},*},\delta'_{\bar{I}}$ are not unique. \end{oss}
\begin{teo}[Partial version]\label{functional identity} The infinite product $(a_I^{-1}\otimes a_I)\prod_{i\geq1}{f'_{\bar{I},*}}^{(i)}$ exists in $\mathcal{O}_{K_\infty}\hat\otimes K$ and is equal, up to a factor $\lambda\otimes1\in^*\mathcal{O}_{K_\infty}\otimes\mathbb{F}_q$, to $\zeta_I$. We can also write: \[\zeta_I=-(a_I^{-1}\otimes a_I)\prod_{i\geq0}\left((\lambda\otimes1)^{1-q}f'_{\bar{I},*}{}^{(1)}\right)^{(i)}.\] \end{teo} \begin{proof} Let's immerse $\mathcal{O}_{K_\infty}\hat\otimes K$ into $K[[u]]$. By Proposition \ref{functions}, $f'_{\bar{I},*}=1+O(u)$, hence ${f'_{\bar{I},*}}^{(i)}=1+O(u^{q^i})$ for all $i\geq0$, and the convergence of the infinite product is obvious. For all $m\geq0$: \[\red_u(\zeta_{I,m})=\sum_{\mu\in\mathbb{F}_q^\times}(\mu a_I)^{-1}\otimes(\mu a_I)=-a_I^{-1}\otimes a_I.\] In particular, by Proposition \ref{Div and red commute}, for $m\gg0$ we have: \[\Div(1\otimes a_I)=\Div(\red_u(\zeta_{I,m}))=\red_{K_\infty}(\Div(\zeta_{I,m}))=I+\red_{K_\infty}(V_{\bar{I},*,m})-(g+d)\infty;\] since $\red_{K_\infty}:X^{(g)}({K_\infty})\to X^{(g)}(\mathbb{F}_q)$ is a continuous map, and the sequence $(V_{\bar{I},*,m})_m$ converges to $V_{\bar{I},*}$ in $X^{(g)}({K_\infty})$ by Lemma \ref{V_{I,m}}, the equality passes to the limit: \[\red_{K_\infty}(V_{\bar{I},*})=\Div(1\otimes a_I)+(g+d)\infty-I.\] Define the rational function $\alpha_m:={\delta'_{\bar{I}}}^{(1)}\frac{\zeta_{I,m}}{{f'_{\bar{I},*}}^{(1)}\cdots {f'_{\bar{I},*}}^{(m)}}$ for $m\gg0$ and look at its divisor: \[\Div(\alpha_m)=I+V_{\bar{I},*}^{(m+1)}+V_{\bar{I},*,m}+V_{\bar{I}}^{(1)}-(3g+d)\infty\Longrightarrow \alpha_m\in{K_\infty}\otimes A(\leq3g+d).\] By Lemma \ref{V_{I,m}}, the sequence $(\Div(\alpha_m)+(3g+d)\infty)_m$ converges to \[I+\red_{K_\infty}(V_{\bar{I},*})+V_{\bar{I},*}^{(1)}+V_{\bar{I}}^{(1)}=(\Div(1\otimes a_I)+(g+d)\infty)+(\Div({\delta'_{\bar{I}}}^{(1)})+2g\infty)\] in $X^{(3g+d)}({K_\infty})$. Moreover, since $(\alpha_m)_m$ converges in $K((u))$ to ${\delta'_{\bar{I}}}^{(1)}\zeta_I\left(\prod_{i\geq1}{f'_{\bar{I},*}}^{(i)}\right)^{-1}$, by Lemma \ref{K((u))}.2 the latter is an element of ${K_\infty}\otimes A(\leq3g+d)$. By Proposition \ref{convergence of functions and divisors}, we have: \[\Div\left({\delta'_{\bar{I}}}^{(1)}\frac{\zeta_I}{\prod_{i\geq1}{f'_{\bar{I},*}}^{(i)}}\right)=\Div(\lim_m\alpha_m)=\lim_m\Div(\alpha_m)=\Div(1\otimes a_I)+\Div(\delta_{\bar{I}}'^{(1)}).\] In particular, there is some $\lambda\in {K_\infty}$ (a fortiori in $\mathcal{O}_{K_\infty}$) such that: \[\zeta_I=(\lambda\otimes1)(a_I^{-1}\otimes a_I)\prod_{i\geq1}{f'_{\bar{I},*}}^{(i)}.\]
As elements of $K((u))$, $\zeta_I(a_I\otimes a_I^{-1})=-1+O(u)$, and ${f'_{\bar{I},*}}^{(i)}=1+O(u)$ for all $i\geq0$, hence $\lambda\otimes1=-1+u\mathbb{F}_q[[u]]\subseteq\mathbb{F}_q((u))$. In particular, the infinite product $\prod_{i\geq0}(\lambda^{1-q}\otimes1)^{q^i}$ converges in $\mathbb{F}_q[[u]]$ to $-\lambda\otimes1$, so we deduce the following rearrangement: \[\zeta_I=-(a_I^{-1}\otimes a_I)\prod_{i\geq0}\left((\lambda^{1-q}\otimes1)f'_{\bar{I},*}{}^{(1)}\right)^{(i)}.\tag*{\qedhere}\] \end{proof}
\begin{Def}\label{def f, f_*, delta}
Define the functions $f_{\bar{I}},f_{\bar{I},*},\delta_{\bar{I}}$ respectively as the unique scalar multiples of the functions $f'_{\bar{I}}.f'_{\bar{I},*},\delta'_{\bar{I}}$ such that $\sgn(f_{\bar{I}})=\sgn(f_{\bar{I},*})=\sgn(\delta_{\bar{I}})=1$.
We call $\{f_{\bar{I}}\}_{\bar{I}\in Cl(A)}$ the \textit{shtuka functions} and $\{f_{\bar{I},*}\}_{\bar{I}\in Cl(A)}$ the \textit{dual shtuka functions}. \end{Def} \begin{oss}\label{delta}
We have the equality $\frac{\delta_{\bar{I}}^{(1)}}{\delta_{\bar{I}}}=\frac{f_{\bar{I}}}{f_{\bar{I},*}}$, since both sides have the same divisor and the same sign. \end{oss} \begin{oss}\label{Hayes2}
The functions $\{f_{\bar{I}}\}_{\bar{I}\in Cl(A)},\{f_{\bar{I},*}\}_{\bar{I}\in Cl(A)},\{\delta_{\bar{I}}\}_{\bar{I}\in Cl(A)}$ all have sign equal to $1$, and their divisors are all $H$-rational by Remark \ref{Hayes}, so all these functions are in $\Q(H\otimes A)$.
From Remark \ref{Hayes} we also know that, for all $\bar{I}\in Cl(A),\sigma\in G(H/K)\cong Cl(A)$:
\[\Div(f_{\bar{I}}^\sigma)=\Div(f_{\bar{I}})^\sigma=\left(V_{\bar{I}}^{(1)}\right)^\sigma-V_{\bar{I}}^\sigma+\Xi-\infty=V_{\bar{I}^\sigma}^{(1)}-V_{\bar{I}^\sigma}+\Xi-\infty=\Div(f_{\bar{I}^\sigma}),\]
and since both function have sign equal to $1$ we get $f_{\bar{I}}^\sigma=f_{\bar{I}^\sigma}$. Similarly, $f_{\bar{I},*}^\sigma=f_{\bar{I}^\sigma,*}$ and $\delta_{\bar{I}}^\sigma=\delta_{\bar{I}^\sigma}$. \end{oss}
\begin{cor}\label{gamma_I} There is $\gamma_I\in {\mathbb{C}_\infty}$, unique up to a factor in $\mathbb{F}_q^\times$, such that $\frac{((\gamma_I\otimes1)\zeta_I)^{(-1)}}{(\gamma_I\otimes1)\zeta_I}=f_{\bar{I},*}$. \end{cor}
\section{The module of special functions}\label{section special functions}
Fix an ideal $I\unlhd A$. By the \textit{shtuka correspondence} (see \cite{Goss}[Section 6.2]), we can associate a Drinfeld-Hayes module $\phi$ to the shtuka function $f_{\bar{I}}$. In this section, we use Theorem \ref{functional identity} (in its partial version) to describe somewhat explicitly the module of special functions relative to $\phi$.
Let's quickly review the notion of special function.
\begin{Def}[Special functions]
Set $\mathbb{T}:={\mathbb{C}_\infty}\hat\otimes A$. The set of special functions relative to a shtuka function $f_{\bar{I}}$ is defined as $\Sf_{\bar{I}}:=\{\omega\in\mathbb{T}|\omega^{(1)}=f_{\bar{I}}\omega\}$. \end{Def} \begin{oss}
As proven by Anglès, Ngo Dac, and Tavares Ribeiro (see \cite{ANDTR}[Lemma 3.6] and \cite{ANDTR}[Rmk. 3.10]), for all $\omega\in\mathbb{T}$:
\[\omega\in\Sf_{\bar{I}}\Longleftrightarrow\forall a\in A\;\phi_a(\omega)=(a\otimes1)\omega.\] \end{oss} Set $\zeta:=(\gamma_I\otimes1)\zeta_I$, with $\gamma_I$ defined as in Corollary \ref{gamma_I}, so that $\zeta^{(-1)}=f_*\zeta$. \begin{teo}[Partial version]\label{Sf_I} The $A$-module $\Sf_{\bar{I}}$ coincides with $(\mathbb{F}_q\otimes I)\frac{\delta_{\bar{I}}}{\zeta^{(-1)}}$. \end{teo}
\begin{oss}
We refer the reader to the article \cite{Gazda} from Gazda and Maurischat, where they described the isomorphism class of the module of special functions in relation to the period lattice, in the wider generality of Anderson modules (\cite{Gazda}[Thm. 3.11]). \end{oss}
Before the proof of Theorem \ref{Sf_I}, let's state some preliminary results.
\begin{oss} By Lemma \ref{K((u))}.4, we know that ${K_\infty}\hat\otimes A\cong A[[u]][u^{-1}]$. A rational function over $X_{K_\infty}$ is in $\mathbb{T}$ if and only if it's contained in $A[[u]][u^{-1}]$, which from Corollary \ref{A[[u]][u^{-1}]} happens if and only if its poles all reduce to $\infty$. \end{oss}
\begin{lemma}\label{fixed points of the twist}
The subset of ${\mathbb{C}_\infty}\hat\otimes K$ fixed by the Frobenius twist is $\mathbb{F}_q\otimes K$. \end{lemma} \begin{proof}
Fix an $\mathbb{F}_q$-basis $\{b_i\}_i$ of $K$: any element $c\in{\mathbb{C}_\infty}\hat\otimes K$ can be written in a unique way as a possibly infinite sum $\sum_i a_i\otimes b_i$, with $a_i\in{\mathbb{C}_\infty}$ for all $i$. If $c=c^{(1)}$, we need to have for all $i$ the equality $a_i^q=a_i$, hence $a_i\in\mathbb{F}_q$ for all $i$. \end{proof}
\begin{proof}[Proof of Theorem \ref{Sf_I}] First, let's show that $(\mathbb{F}_q\otimes K)\Sf_{\bar{I}}=(\mathbb{F}_q\otimes K)\frac{\delta_{\bar{I}}}{\zeta^{(-1)}}$. Pick any $\omega\in \Sf_{\bar{I}}$; since $\omega^{(1)}=f_{\bar{I}}\omega$, $\delta_{\bar{I}}^{(1)}=\frac{f_{\bar{I}}}{f_{\bar{I},*}}\delta_{\bar{I}}$, and $\zeta=\frac{1}{f_{\bar{I},*}}\zeta^{(-1)}$, we have: \[\left(\frac{\omega\zeta^{(-1)}}{\delta_{\bar{I}}}\right)^{(1)}=\frac{\omega^{(1)}\zeta}{\delta_{\bar{I}}^{(1)}}=\frac{(f_{\bar{I}}\omega)(f_{\bar{I},*}^{-1}\zeta^{(-1)})}{f_{\bar{I}}f_{\bar{I},*}^{-1}\delta_{\bar{I}}}=\frac{\omega\zeta^{(-1)}}{\delta_{\bar{I}}},\] hence $\frac{\omega\zeta^{(-1)}}{\delta_{\bar{I}}}\in \mathbb{F}_q\otimes K$ by Lemma \ref{fixed points of the twist}, or equivalently $(\mathbb{F}_q\otimes K)\omega=(\mathbb{F}_q\otimes K)\frac{\delta_{\bar{I}}}{\zeta^{(-1)}}$.
We can twist everything and multiply by $\gamma_I\otimes1$ without loss of generality: the thesis is now that $(1\otimes\lambda)\frac{\delta_{\bar{I}}^{(1)}}{\zeta_I}\in A[[u]][u^{-1}]$ if and only if $\lambda\in I$. Suppose $\lambda\in I$, and consider the sequence $\left((1\otimes\lambda)\frac{\delta_{I,m}^{(1)}}{\zeta_{I,m}}\right)_m$ in $K((u))$, whose limit is $(1\otimes\lambda)\frac{\delta_{\bar{I}}^{(1)}}{\zeta_I}$. The divisor of the $m$-th element of the sequence (for $m\gg0$) is \[V_{\bar{I},m}^{(1)}-(\Xi^{(1)}+\dots+\Xi^{(m)})-I+(m+d-g)\infty+\Div(1\otimes\lambda);\] since $\lambda\in I$, the only poles of the function reduce to $\infty$, hence $(1\otimes\lambda)\frac{\delta_{I,m}^{(1)}}{\zeta_{I,m}}\in A[[u]][u^{-1}]$ by Corollary \ref{A[[u]][u^{-1}]}, and so does the limit.
Vice versa, suppose $(1\otimes\lambda)\frac{\delta_{\bar{I}}^{(1)}}{\zeta_I}\in A[[u]][u^{-1}]$. Since the coefficients of $(1\otimes\lambda^{-1})\zeta_I\in K((u))$ are all contained in $\lambda^{-1}I$, $\delta_{\bar{I}}^{(1)}=\left((1\otimes\lambda)\frac{\delta_{\bar{I}}^{(1)}}{\zeta_I}\right)\left((1\otimes\lambda^{-1})\zeta_I\right)$ has all coefficients in $\lambda^{-1}I$, so the same is true for $\delta_{\bar{I}}$. If by contradiction $\lambda\not\in I$, there is a prime ideal $P\unlhd A$ which divides the fractional ideal $\lambda^{-1}I$, hence all the coefficients of $\delta_{\bar{I}}$ are in $A\cap\lambda^{-1}I\subseteq P$, which by Corollary \ref{cor conv eval Tate} means that $P$ is a zero of $\delta_{\bar{I}}$. Since $P\in X(\mathbb{F}_q^{ac})$ and $\Div(\delta_{\bar{I}})=V_{\bar{I}}+V_{\bar{I},*}-2g\infty$, this is a contradiction because, by Corollary \ref{V}, neither $V_{\bar{I}}$ or $V_{\bar{I},*}$ have $\mathbb{F}_q^{ac}$-rational points in their support. \end{proof}
To end this section, let's include an analogous result to Theorem \ref{functional identity} for special functions. \begin{teo}\label{omega infinite product} There is some $\alpha\in K_\infty^\times$ such that the following element of ${K_\infty}\hat\otimes K$ is well defined: \[\omega:=(\alpha\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f_{\bar{I}}}\right)^{(i)}.\] Moreover, $\omega\in(\mathbb{F}_q\otimes K)\Sf_{\bar{I}}$, and $\omega$ does not depend on the choice of $\alpha$. \end{teo} \begin{proof}
Take $\alpha,\beta\in K_\infty^\times$ such that the following elements of ${K_\infty}\hat\otimes K\cong K((u))$ are well defined:
\begin{align*}
&\omega(\alpha):=(\alpha\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f_{\bar{I}}}\right)^{(i)},
&\omega(\beta):=(\beta\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\beta\otimes1}{f_{\bar{I}}}\right)^{(i)}.
\end{align*}
The infinite products converge only if $\frac{\alpha\otimes1}{f_{\bar{I}}},\frac{\beta\otimes1}{f_{\bar{I}}}=1+O(u)$ in $K((u))$; in particular, we have $\gamma:=\alpha\beta^{-1}=1+O(u)$ in $\mathbb{F}_q[[u]]$, therefore:
\[\omega(\alpha):=(\gamma\beta\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\gamma\beta\otimes1}{f_{\bar{I}}}\right)^{(i)}=\omega(\beta)(\gamma\otimes1)^\frac{1}{q-1}\prod_{i\geq0}(\gamma^{q^i}\otimes1)=\omega(\beta),\]
and this proves the uniqueness of $\omega:=\omega(\alpha)$. On the other hand by Proposition \ref{functions}, we can choose $f'_{\bar{I}}\in\mathcal{O}_{K_\infty}\hat\otimes K$ and $\alpha\in {K_\infty}\otimes\mathbb{F}_q$ such that $f'_{\bar{I}}=1+O(u)$ and $f_{\bar{I}}=(\alpha\otimes1) f'_{\bar{I}}$. Finally, we have:
\[\frac{\omega^{(1)}}{\omega}=\left((\alpha\otimes1)^\frac{1}{q-1}\right)^q\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f_{\bar{I}}}\right)^{(i+1)}\left((\alpha\otimes1)^\frac{1}{q-1}\prod_{i\geq0}\left(\frac{\alpha\otimes1}{f_{\bar{I}}}\right)^{(i)}\right)^{-1}=(\alpha\otimes 1)\frac{f_{\bar{I}}}{\alpha\otimes1}=f_{\bar{I}},\]
so $\omega\in(\mathbb{F}_q\otimes K)\Sf_{\bar{I}}$ by the same considerations expressed in the proof of Theorem \ref{Sf_I}. \end{proof}
\section{Relation between zeta functions and period lattices}\label{section duality}
The aim of this section is to compute more explicitly the constant $\gamma_I$ defined in Corollary \ref{gamma_I}. To do so, we first study more in depth the zeta function $\zeta_I$ and its coefficients as a series in $K[[u]]$; afterwards, we draw a correspondence between the dual shtuka function $f_{I,*}$ and a certain Drinfeld-Hayes module $\phi$, obtaining the following result. \begin{customprop}{\ref{Lambda=I}}
The period lattice of $\phi$ is $\gamma_I^{-1}I\subseteq{\mathbb{C}_\infty}$. \end{customprop}
Finally, we state a complete version of Theorem \ref{Sf_I} - which generalizes \cite{Green}[Thm. 7.1] - and Theorem \ref{functional identity}.
\subsection{Evaluations of the zeta function}
The aim of this subsection, expressed in the following proposition, is to show that there is a well behaved notion of evaluation for the Pellarin zeta function $\zeta_I$ at any point $P\in X({\mathbb{C}_\infty})\setminus\{0\}$.
From now on, for any series $s\in K[[u^\frac{1}{q^n}]][u^{-1}]$ for some $n$, we denote by $s_{(i)}$ the coefficient of $u^i$. We can extend the valuation $v$: we denote $v(s)$ the least element in $\frac{1}{q^n}\mathbb{Z}$ such that $s_{(v(s))}\neq0$.
\begin{prop}\label{conv eval zeta} For all points $P\in X({\mathbb{C}_\infty})\setminus\{\infty\}$, corresponding to maps $\chi_P:A\to{\mathbb{C}_\infty}$, the sequence $(\zeta_{I,m}(P))_m$ and the series $\sum_{i\geq0}\chi_P\left((\zeta_I)_{(i)}\right)u^i$ converge to the same element of ${\mathbb{C}_\infty}$. \end{prop} To prove the proposition, we first need some results on the coefficients $\left((\zeta_I)_{(i)}\right)_i$. \begin{lemma}\label{(zeta_{I,m})_{(i)}} For all integers $i\geq0$, we have $\deg((\zeta_{I,m})_{(i)})\leq\log_q(i+1)+g+\deg(I)+1$ for $m\geq0$. \end{lemma} \begin{proof} Recall the definition of $j_m$, and that $m+\deg(I)+1\leq j_m\leq m+g+\deg(I)$, from Remark \ref{j_m inequality}. The coefficients of $\zeta_{I,0}$ have degree $j_0\leq g+\deg(I)$, so the lemma holds for $m=0$. Since $v(\zeta_{I,m})=j_0$ for all $m\geq0$, the coefficient $(\zeta_{I,m})_{(0)}$, is nonzero if and only if $I=A$; in that case, it's equal to $\sum_{a\in\mathbb{F}_q^\times}a^{-1}\otimes a=-1$, and its valuation is $0$, so the lemma also holds for $i=0$. Let's prove the lemma for $i\geq1$, $m\geq1$.
We claim that it suffices to prove the following inequality, for all $m\geq1$: \[v\left(\sum_{a\in I(j_m)}a^{-1}\otimes a\right)=v(\zeta_{I,m}-\zeta_{I,m-1})\geq q^{m-1}.\] If the inequality is true for $m\geq1$, fix $i>0$, and set $n:=\lfloor\log_q(i)\rfloor+1$, so that $q^{n-1}\leq i<q^n$; then, for $m\geq n$: \[\deg((\zeta_{I,m})_{(i)})=\deg\left(\left(\sum_{k=0}^m\zeta_{I,k}-\zeta_{I,k-1}\right)_{(i)}\right)=\deg\left(\left(\sum_{k=0}^n\zeta_{I,k}-\zeta_{I,k-1}\right)_{(i)}\right)\leq j_n,\] which is at most $n+g+\deg(I)=\lfloor\log_q(i)\rfloor+g+\deg(I)+1\leq\log_q(i+1)+g+\deg(I)+1$.
For $m\geq1$, $\zeta_{I,m}(\Xi)-\zeta_{I,m-1}(\Xi)=1-1=0$. By Proposition \ref{divisor of zeta_{I,m}}, on one hand, $\zeta_{I,m}-\zeta_{I,m-1}$ has only one pole, of degree at most $j_m$, at $\infty$, and has $I$ and $\Xi^{(1)},\dots,\Xi^{(m-1)}$ among its zeroes; on the other hand, \[h^0(W_m)=h^0(j_m\infty-I-\Xi-\cdots-\Xi^{(m-1)})=h^0(j_m\infty-I-\Xi^{(1)}-\cdots-\Xi^{(m)})=1,\] hence the remaining set of zeroes is $W_m^{(-1)}$, and $\zeta_{I,m}-\zeta_{I,m-1}$ is a scalar multiple of $\zeta_{I,m}^{(-1)}$.
If we fix $b\in I(j_m)$ with $\sgn(b)=1$, we get the following: \begin{align*}
\left((\zeta_{I,m}-\zeta_{I,m-1})(\Xi^{(-1)})\right)^q&=\sum_{a\in I(j_m)}a^{1-q}=-\sum_{\substack{a\in I(j_m)\\\sgn(a)=1}}a^{1-q}=-\sum_{c\in I(<j_m)}(b+c)^{1-q}\\
&=-b^{1-q}\sum_{c\in I(<j_m)}\sum_{i\geq0}\binom{1-q}{i}\frac{c^i}{b^i}=-b^{1-q}\sum_{i\geq0}b^{-i}\binom{1-q}{i}\sum_{c\in I(<j_m)}c^i. \end{align*} On the other hand, by Lemma \ref{Goss}, we have: \[\sum_{c\in I(<j_m)}c^i=0,\;\;\;\forall i<q^{\dim(I(<j_m))}-1=q^m-1.\] As elements of $\mathcal{O}_{K_\infty}\cong\mathbb{F}_q[[u]]\subseteq K[[u]]$, $v\left(\frac{c}{b}\right)\geq1$ for all $c\in I(<j_m)$, and $v(b^{-1})=j_m$, so we get: \begin{align*}
q\cdot v\left((\zeta_{I,m}-\zeta_{I,m-1})(\Xi^{(-1)})\right)&=(1-q)v(b)+v\left(\sum_{i\geq q^m-1}\binom{1-q}{i}\sum_{c\in I(<j_m)}\left(\frac{c}{b}\right)^i\right)\\
&\geq (1-q)v(b)+\min_{\substack{c\in I(<j_m)\\i\geq q^m-1}}\left\{i\cdot v\left(\frac{c}{b}\right)\right\}\geq j_m(q-1)+q^m-1. \end{align*} Since $\left(\zeta_{I,m}^{(-1)}\right)(\Xi^{(-1)})=\left(\zeta_{I,m}(\Xi)\right)^\frac{1}{q}=-1$, we get: \[\sum_{a\in I(j_m)}a^{-1}\otimes a=\zeta_{I,m}-\zeta_{I,m-1}=-\zeta_{I,m}^{(-1)}\cdot(\zeta_{I,m}-\zeta_{I,m-1})(\Xi^{(-1)}).\] Its valuation, since $\zeta_{I,m}^{(-1)}\in K[[u^\frac{1}{q}]]$, is at least $j_m\frac{q-1}{q}+q^{m-1}-\frac{1}{q}\geq q^{m-1}$, for $m\geq1$. \end{proof}
\begin{cor}\label{coefficients of zetas} For all $k\geq 0$, for all $i\in\frac{1}{q^k}\mathbb{N}$, $\deg((\zeta_I^{(-k)})_{(i)})\leq\log_q(i+1)+k+g+\deg(I)+1$. For all points $P\in X({\mathbb{C}_\infty})\setminus\{\infty\}$, corresponding to maps $\chi_P:A\to{\mathbb{C}_\infty}$, for all $k\geq0$, the following series converges: \[\sum_{i\geq0}\chi_P\left((\zeta_I^{(-k)})_{(i)}\right)u^i.\] \end{cor} First, a simple lemma. \begin{lemma}\label{k_P} Fix a point $P\in X({\mathbb{C}_\infty})\setminus\{\infty\}$, corresponding to map $\chi_P:A\to{\mathbb{C}_\infty}$. There is a positive real constant $k_P$ such that $v(\chi_P(a))\geq -k_P\deg(a)$ for all $a\in A$. \end{lemma} \begin{proof} We can pick a finite set $\{a_1,\dots,a_n\}$ such that for all $a\in A\setminus\mathbb{F}_q$ there is a product $a'$ of $a_i$'s with $\deg(a')=\deg(a)$. Let's define $k_P:=\max\left\{\frac{-v(\chi_P(a_i))}{\deg(a_i)}\right\}_i$, so that $v(\chi_P(a_i))\geq -k_P\deg(a_i)$ for all $i$. We prove the lemma by induction on $\deg(a)$.
If $\deg(a)=0$ the claim is trivially true. If $\deg(a)>0$ there is a product $a':=\lambda\prod_i a_i^{e_i}$, with $\lambda\in\mathbb{F}_q$, of the same degree and sign, hence $\deg(a-a')<\deg(a)$. We have: \begin{align*}
&v(\chi_P(a-a'))\geq -k_P\deg(a-a')>-k_P\deg(a),\text{ by inductive hypothesis;}\\
&v(\chi_P(a'))=\sum_i e_i\cdot v(\chi_P(a_i))\geq-\sum_i k_P e_i\cdot\deg(a_i)=-k_P\deg(a')=-k_P\deg(a). \end{align*} Hence, $v(\chi_P(a))\geq\min\{v(\chi_P(a')),v(\chi_P(a-a'))\}\geq-k_P\deg(a)$. \end{proof} \begin{proof}[Proof of Corollary \ref{coefficients of zetas}] The first part of the statement for $k=0$ follows from the inequality of Lemma \ref{(zeta_{I,m})_{(i)}}, using the fact that for all $i$ the sequence $((\zeta_{I,m})_{(i)})_m$ is eventually equal to $(\zeta_I)_{(i)}$. For $k>0$ and $i\in\frac{1}{q^k}\mathbb{N}$, we get: \[\deg\left((\zeta_I^{(-k)})_{(i)}\right)=\deg\left((\zeta_I)_{(iq^k)}\right)\leq\log_q(iq^k+1)+g+\deg(I)+1\leq\log_q(i+1)+k+g+\deg(I)+1.\]
Let's define $k_P$ as in Lemma \ref{k_P}. Then, for all $i>0$ we have: \[v\left(\chi_P\left((\zeta_I^{(-k)})_{(i)}\right)u^i\right)\geq-k_P\deg\left((\zeta_I^{(-k)})_{(i)}\right)+i\geq i-k_P\log_q(i+1)-k_P(k+g+\deg(I)+1),\] which tends to infinity as $i$ tends to infinity, proving the convergence of $\sum_{i\geq0}\chi_P\left((\zeta_I^{(-k)})_{(i)}\right)u^i$. \end{proof}
Finally we can prove Proposition \ref{conv eval zeta}.
\begin{proof}[Proof of Proposition \ref{conv eval zeta}] Define $k_P$ as in Lemma \ref{k_P}. For $m\geq0$, by Lemma \ref{(zeta_{I,m})_{(i)}} we have: \[v(\zeta_I-\zeta_{I,m})=v\left(\sum_{m'\geq m}\zeta_{I,m'+1}-\zeta_{I,m'}\right)\geq\min_{m'\geq m}v\left(\zeta_{I,m'+1}-\zeta_{I,m'}\right)\geq q^m.\] For all $i\geq q^m$, by Corollary \ref{coefficients of zetas}, we have: \begin{align*} \deg\left((\zeta_I-\zeta_{I,m})_{(i)}\right)&\leq\max\left\{\deg\left((\zeta_I)_{(i)}\right),\deg\left((\zeta_{I,m})_{(i)}\right)\right\}\\ &\leq\max\{\log_q(i+1)+g+\deg(I)+1,j_m\}=\log_q(i+1)+g+\deg(I)+1, \end{align*} since $j_m\leq m+g+\deg(I)+1$ and $m\leq\log_q(i+1)$. In particular: \begin{align*}
v\left(\sum_i \chi_P\left((\zeta_I-\zeta_{I,m})_{(i)}\right)u^i\right)&=v\left(\sum_{i\geq q^m}\chi_P\left((\zeta_I-\zeta_{I,m})_{(i)}\right)u^i\right)\\
&\geq\min_{i\geq q^m}\left\{i-k_P\cdot\deg\left((\zeta_I-\zeta_{I,m})_{(i)}\right)\right\}\\
&\geq\min_{i\geq q^m}\left\{i-k_P(\log_q(i+1)+g+\deg(I)+1)\right\}, \end{align*} which tends to infinity as $m$ tends to infinity. By Proposition \ref{conv eval entire rational}, $\zeta_{I,m}(P)=\sum_i \chi_P\left((\zeta_{I,m})_{(i)}\right)u^i$, hence we get that \[\lim_m \zeta_{I,m}(P)-\sum_{i\geq0}\chi_P\left((\zeta_I)_{(i)}\right)u^i=\lim_m\left(\sum_i \chi_P\left((\zeta_{I,m}-\zeta)_{(i)}\right)u^i\right)=0.\tag*{\qedhere}\] \end{proof}
\begin{Def}\label{evaluating zetas} We define the evaluation of $\zeta_I$ at $P$ as $\zeta_I(P):=\sum_i (\zeta_I)_{(i)}(P)u^i$. \end{Def}
\begin{cor}\label{zeta zero} For all $i\geq 1$, we have $\zeta_I(\Xi^{(i)})=0$. Similarly, for all $k\geq0$, for all $i\geq1$, $\sum_j\chi_{\Xi^{(i-k)}}((\zeta_I^{(-k)})_{(j)})u^j=0$ (where $j$ varies among $\frac{1}{q^k}\mathbb{N}$). \end{cor} \begin{proof} For the first identity we use that, for all $i\geq1$, $\zeta_{I,m}(\Xi^{(i)})=0$ for $m\gg0$. For the second identity, note that \[\left(\sum_{j\in\frac{1}{q^k}\mathbb{N}}\chi_{\Xi^{(i-k)}}((\zeta_I^{(-k)})_{(j)})u^j\right)^{q^k}=\sum_{j\in\mathbb{N}}\chi_{\Xi}^{(i)}((\zeta_I)_{(j)})u^j=0.\tag*{\qedhere}\] \end{proof}
\subsection{Dual Drinfeld modules and dual shtuka functions} From now on, in this section we use the following notations: $V_*:=V_{\bar{I},*}$, $f_*:=f_{\bar{I},*}$, $\zeta:=(\gamma_I\otimes1)\zeta_I$, with $\gamma_I$ defined as in Corollary \ref{gamma_I}, so that $\zeta^{(-1)}=f_*\zeta$.
The following proposition shows a connection between dual Drinfeld modules, dual shtuka functions, and zeta functions, which is meant to mirror the correspondence between Drinfeld modules, shtuka functions, and special functions (see for example \cite{Thakur}[Eq.($**$)]).
\begin{prop}\label{dual Drinfeld module} Set $e_d:=\prod_{i=0}^{d-1}f_*^{(-i)}$ for all nonnegative integers $d$. The collection $\{e_d\}_{d\geq0}$ is a basis of the ${\mathbb{C}_\infty}\otimes1$-vector space $\mathcal{O}(V_*^{(1)}):=\bigcup_{d\geq0}\mathcal{L}(V_*^{(1)}+d\infty)$.
For all $a\in A$ we have the equality $1\otimes a=\sum_{i=0}^{\deg(a)}(a_i\otimes1)e_i$ with $a_i^{q^i}\in {K_\infty}$, and the function $\phi^*:A\to {\mathbb{C}_\infty}\{\tau^{-1}\}$ sending $a$ to $\sum_i a_i\tau^{-i}$ is the dual of a Drinfeld-Hayes module $\phi$.
Finally, for all $a,b\in A$, $\phi^*_{ab}(\zeta)=(\phi^*_a\circ\phi^*_b)(\zeta)$. \end{prop} \begin{proof} For the first part, we just need to prove that, for all $d\geq0$, $e_d\in\mathcal{L}(V_*^{(1)})$ and it has a pole of multiplicity exactly $d$ at $\infty$; using that $\Div(f_*^{(-i)})=V_*^{(-i)}-V_*^{(1-i)}+\Xi^{(-i)}-\infty$, we get: \[\Div(e_d)=\Div\left(\prod_{i=0}^{d-1}f_*^{(-i)}\right)=V_*^{(1-d)}-V_*^{(1)}+\sum_{i=0}^{d-1}\Xi^{(-i)}-d\infty.\] If we fix $a\in A$ of degree $d$, $1\otimes a\in\mathcal{L}(V_*^{(1)}+d\infty)$, hence it can be written as $\sum_{i=0}^d (a_i\otimes1)e_i$. Moreover, if we twist $k$ times and evaluate at $\Xi$ for all $0\leq k\leq d$ we get the following triangular system of equations in the variables $(a_i)_i$: \[\left\{a=\sum_{i=0}^k\left(a_i^{q^k}\prod_{j=k-i}^k f_*^{(j)}(\Xi)\right)\right\}_k\Longrightarrow\left\{a_k^{q^k}=\left(\prod_{j=0}^k f_*^{(j)}(\Xi)\right)^{-1}\left(a-\sum_{i=0}^{k-1}a_i^{q^k}\prod_{j=k-i}^k f_*^{(j)}(\Xi)\right)\right\}_k.\] From this system we can deduce that $a_0=a$ and, since $f_*^{(j)}(\Xi)\in {K_\infty}$ for all $j\geq0$, that $a_k^{q^k}\in {K_\infty}$ for all $k$. Finally, since $\deg(a)=\deg(e_d)$, and for all $i\geq0$ $\sgn(f_*^{(i)})=\sgn(f_*)=1$, we have: \[\sgn(a)=\sgn\left(\sum_{i=0}^d (a_i^{q^d}\otimes1)e_i^{(d)}\right)=\sgn((a_d^{q^d}\otimes1)e_d^{(d)})=a_d^{q^d}\sgn(e_d^{(d)})=a_d^{q^d}\sgn\left(\prod_{i=1}^{d}f_*^{(i)}\right)=a_d^{q^d},\] so $a_d=\sgn(a)$. For all $a\in A$, write $\phi^*_a:=\sum_i a_i\tau^{-i}$. Since for all $k\geq 0$ and for all $a\in A$ we have $\zeta e_k=\zeta^{(-k)}$ and $1\otimes a=(1\otimes a)^{(-k)}=\sum_i (a_i^\frac{1}{q^k}\otimes 1)e_i^{(-k)}$, we get the following equations for all $k\geq0$ and $a,b\in A$:
\begin{align*} (1\otimes a)\zeta^{(-k)}&=\sum_i (a_i^\frac{1}{q^k}\otimes 1)(e_i\zeta)^{(-k)}=\sum_i (a_i^\frac{1}{q^k}\otimes 1)\zeta^{(-k-i)}=\tau^{-k}\circ\phi_a^*(\zeta);\\
\phi_{ab}^*(\zeta)&=(1\otimes ab)\zeta=(1\otimes a)\left((1\otimes b)\zeta\right)=\sum_i(1\otimes a)\left((b_i\otimes 1)\zeta^{(-i)}\right)\\
&=\sum_i(b_i\otimes 1)\left((1\otimes a)\zeta^{(-i)}\right)=\sum_i(b_i\otimes 1)\left(\tau^{-i}\circ\phi_a^*(\zeta)\right)=\left(\phi_b^*\circ\phi_a^*\right)(\zeta). \end{align*} Since the elements $(\zeta^{(-i)})_{i\geq0}=(\zeta e_i)_{i\geq0}$ are all ${\mathbb{C}_\infty}\otimes1$-linearly independent, we have the equality $\phi_{ab}^*=\phi_b^*\circ\phi_a^*$. Together with the fact that $\deg(\phi^*_a)=\deg(a)$ and $a_{\deg(a)}=\sgn(a)$, this means that the function $\phi:=(\phi^*)^*:A\to {K_\infty}\{\tau\}$ is a Drinfeld-Hayes module. \end{proof}
From this point onwards, $\phi$ and $\phi^*$ are defined as in Proposition \ref{dual Drinfeld module}.
\begin{Def}\label{definition Lambda'}
We call $\Lambda'\subseteq{\mathbb{C}_\infty}$ the unique rank $1$ projective $A$-module such that, for all $a\in A$:
\begin{align*}
&\exp_{\Lambda'}\circ(a\tau^0)=\phi_a\circ\exp_{\Lambda'},
&a\exp^*_{\Lambda'}=\exp^*_{\Lambda'}\circ\phi^*_a.
\end{align*}
We define $\exp:=\exp_{\Lambda'}$ and $\log:=\log_{\Lambda'}$. We call $\Lambda\subseteq K$ the unique fractional ideal isogenous to $\Lambda'$ such that $\Lambda(\leq0)=\mathbb{F}_q$. We choose a nonzero element of least degree in $\Lambda'$ and call it $\tilde{\pi}_{\Lambda'}$ - for simplicity we denote it $\tilde{\pi}$ for the rest of the section. \end{Def}
\begin{oss} Since $\rk(\Lambda')=1$, a lattice isogenous to $\Lambda'$ is uniquely determined by its nonzero elements of greatest norm, hence the hypothesis that for $\Lambda$ they are $\mathbb{F}_q$ already implies $\Lambda\subseteq K$. Our choice of $\tilde{\pi}$ is up to a factor in $\mathbb{F}_q^\times$, and we have $\tilde{\pi}\Lambda=\Lambda'$. \end{oss}
\begin{oss}\label{c_k in K}
If we write $\exp:=\sum_i c_i\tau^i$ and $\phi_a:=\sum_j a_j\tau^j$ for some $a\in A$ (with $a_0=a$), the equation $\exp\circ (a\tau^0)=\phi_a\circ\exp$ becomes:
\[\sum_k (c_k a^{q^k})\tau^k=\sum_k\left(\sum_{i+j=k} a_j c_i^{q^j}\right)\tau^k\Rightarrow c_k(a^{q^k}-a)=\sum_{i=0}^{k-1}a_{k-i}c_i^{q^{k-i}}.\]
Since $c_0=1$, we get that $c_k\in {K_\infty}$ for all $k\geq0$ by induction. \end{oss}
\begin{Def} For any rank $1$ projective $A$-module $L\subseteq{\mathbb{C}_\infty}$, we define, for all $k\geq1$: \begin{align*}
&S_k(L):=\sum_{\substack{\lambda_1,\dots,\lambda_k\in^* L\\i\neq j\Rightarrow \lambda_i\neq \lambda_j}} (\lambda_1\cdots \lambda_k)^{-1};
&P_k(L):=\sum_{\lambda\in^* L} \lambda^{-k}. \end{align*} We also set $S_0(L):=1$ and $P_0(L):=-1$. \end{Def} \begin{oss}
By definition $\exp_L(x):=x\prod_{\lambda\in^*L}\left(1-\frac{x}{\lambda}\right)\in{\mathbb{C}_\infty}[[x]]$, and by absolute convergence we can expand the product and rearrange the terms of the series, so we get $\exp_L(x)=\sum_{i\geq0}S_i(L)x^i$; in particular, if $i+1$ is not a power of $q$, $S_i(L)=0$. \end{oss}
\begin{oss}\label{c_i valuation} Note that in the summation that defines $S_{q^i-1}(\tilde{\pi}\Lambda)$ there is a unique summand of greatest norm, given by the product of the $q^i-1$ nonzero elements of lower degree of $\tilde{\pi}\Lambda$. Since $\mathbb{F}_q^\times$ are the nonzero elements of lowest degree of $\Lambda$, this product has valuation at least: \[\sum_{j=0}^{i-1} (q^{j+1}-q^j)(j+v(\tilde{\pi}))=i q^i-\left(\sum_{j=0}^{i-1} q^j \right)+q^i v(\tilde{\pi})\geq(i-1+v(\tilde{\pi}))q^i.\]
In particular, since $\lim_i \frac{1}{q^i}v\left(S_{q^i-1}(\tilde{\pi}\Lambda)\right)=\infty$, $\exp(t):{\mathbb{C}_\infty}\to{\mathbb{C}_\infty}$ is an entire function with an infinite radius of convergence, and $\exp^*(t):{\mathbb{C}_\infty}\to{\mathbb{C}_\infty}$, while not being a power series, is continuous, converges everywhere, and sends $0$ to $0$; moreover, $\lim_{\|z\|\to0}\exp^*(z)z^{-1}=1$. \end{oss} \begin{oss} Since they are continuous $\mathbb{F}_q$-linear endomorphisms of ${\mathbb{C}_\infty}$, we can extend uniquely both $\exp$ and $\exp^*$ to continuous $\mathbb{F}_q\otimes K$-linear endomorphisms of ${\mathbb{C}_\infty}\hat\otimes K$. \end{oss}
\begin{lemma}\label{log_L} For all rank $1$ lattices $L\subseteq{\mathbb{C}_\infty}$, $\log_L=-\sum_i P_{q^i-1}(L)\tau^i$. \end{lemma} \begin{proof} Newton's identities for an infinite number of variables tell us that, for all $i\geq1$, $iS_i(L)=\sum_{j=0}^{i-1}S_j(L)P_{i-j}(L)$. Setting $i=q^k-1$ with $k\geq1$, since $S_j(L)=0$ if $j+1$ is not a power of $q$, we get: \[-S_{q^k-1}(L)=\sum_{j=0}^{k-1}S_{q^j-1}(L)P_{q^k-q^j}(L)=\sum_{j=0}^{k-1}S_{q^j-1}(L)(P_{q^{k-j}-1}(L))^{q^j}.\] In particular: \[\exp_L\circ\left(-\sum_{i\geq0}P_{q^i-1}(L)\tau^i\right)=\sum_{k\geq0}\left(-\sum_{j=0}^k S_{q^j-1}(L)(P_{q^{k-j}-1}(L))^{q^j}\right)\tau^k=-S_0(L) P_0(L)=1.\] The uniqueness of right inverses proves the thesis. \end{proof}
\begin{prop}\label{exp(zeta)} We have the following functional identity: \[\exp^*(\zeta)=0.\] \end{prop} \begin{proof} For all $a\in A$ we have $\exp^*\circ\phi_a^*=(a\otimes 1)\exp^*$ as endomorphisms of ${\mathbb{C}_\infty}\hat\otimes K$; by Proposition \ref{dual Drinfeld module}, $\phi^*_a(\zeta)=(1\otimes a)\zeta$ . Hence, for all $a\in A$: \[0=\exp^*(0)=\exp^*(\phi^*_a(\zeta)-(1\otimes a)\zeta)=(a\otimes1-1\otimes a)\exp^*(\zeta).\] For $a\not\in\mathbb{F}_q$, $a\otimes1-1\otimes a$ is invertible in ${\mathbb{C}_\infty}\hat\otimes K$ - with inverse $\sum_{i\geq0}a^{-i-1}\otimes a^i$ - so we get the thesis. \end{proof}
\subsection{The fundamental period $\tilde{\pi}$}
Finally, in this subsection we are able to link the zeta function $\zeta_I$ and the fundamental period $\tilde{\pi}$.
\begin{prop}\label{Lambda=I} Fix an element $a_I\in^*I$ of least degree. We have $a_I^{-1}I=\Lambda$. \end{prop} \begin{proof} Since the nonzero elements of least degree of both $a_I^{-1}I$ and $\Lambda$ are $\mathbb{F}_q^\times$, it suffices to show that $I$ and $\Lambda$ are isogenous.
For all $k\in\mathbb{N}$, define $c_k:=S_{q^k-1}(\tilde{\pi}\Lambda)^\frac{1}{q^k}$; by Remark \ref{c_k in K}, $c_k\in\mathbb{F}_q((u^\frac{1}{q^k}))$. By Corollary \ref{coefficients of zetas} and Remark \ref{c_i valuation} respectively, we have the following inequalities for all $i\in\frac{1}{q^k}\mathbb{N}$, for all $k\in\mathbb{N}$: \begin{align*} &\deg\left((\zeta^{(-k)})_{(i)}\right)\leq\log_q(i+1)+k+g+\deg(I)+1, &v\left(c_k\right)\geq k-1+v(\tilde{\pi})=:k'. \end{align*}
Fix a positive integer $n$. Since $\exp^*(\zeta)=0$ by Proposition \ref{exp(zeta)}, for any arbitrarily large $N$ we can choose a positive integer $m\geq n$ such that $v\left(\sum_{k=0}^m c_k\zeta^{(-k)}\right)\geq N$. For all $k\leq m$ we can write the following, where the index $i$ varies among $\frac{1}{q^m}\mathbb{Z}$: \[c_k=\sum_{i\geq k'}\lambda_{k,i} u^i\in\mathbb{F}_q[u^{\pm\frac{1}{q^m}}]\text{ with }\lambda_{k,i}\in\mathbb{F}_q.\]
Let's rearrange $\sum_{k=0}^m c_k\zeta^{(-k)}$, with the indexes $i$ and $j$ varying among $\frac{1}{q^m}\mathbb{Z}$: \[\sum_{k=0}^m\sum_{j\geq k'} \lambda_{k,j}\zeta^{(-k)}u^j=\sum_{k=0}^m\sum_{j\geq k'}\lambda_{k,j}\sum_{i\geq0}\left(\zeta^{(-k)}\right)_{(i)}u^{i+j}=\sum_{i\geq0}\left(\sum_{k=0}^m\sum_{j=k'}^i\lambda_{k,j}\left(\zeta^{(-k)}\right)_{(i-j)}\right)u^i.\] Since $v\left(\sum_{k=0}^m c_k\zeta^{(-k)}\right)\geq N$, we get that, for $i\in\frac{1}{q^m}\mathbb{Z}$ and $i<N$: \[\sum_{k=0}^m\sum_{j=k'}^i\lambda_{k,j}\left(\zeta^{(-k)}\right)_{(i-j)}=0.\] Using this result and Corollary \ref{coefficients of zetas}, the evaluation $\sum_{k=0}^m c_k\zeta^{(-k)}(\Xi^{(-n)})$ can be rearranged as follows: \begin{align*} &\sum_{k=0}^m\sum_{j\geq k'}\lambda_{k,j}\zeta^{(-k)}(\Xi^{(-n)})u^j =\sum_{k=0}^m\sum_{j\geq k'}\lambda_{k,j}\sum_{i\geq0}\left(\zeta^{(-k)}\right)_{(i)}(\Xi^{(-n)}) u^{i+j}\\ =&\sum_{i\geq0}\left(\sum_{k=0}^m\sum_{j=k'}^i\lambda_{k,j}\left(\zeta^{(-k)}\right)_{(i-j)}\right)(\Xi^{(-n)})u^i =\sum_{i\geq N}\left(\sum_{k=0}^m\sum_{j=k'}^i\lambda_{k,j}\left(\zeta^{(-k)}\right)_{(i-j)}\right)(\Xi^{(-n)})u^i. \end{align*} For $i-j,k\geq0$, since $j\geq k'\geq v(\tilde{\pi})-1$, and since $\log_q(x)\leq x$ for all $x>0$, we have: \[\deg\left((\zeta^{(-k)})_{(i-j)}\right)\leq \log_q(i-j+1)+k+g+\deg(I)+1\leq i+k+g+\deg(I)+3-v(\tilde{\pi})=:i+C,\] so each summand has valuation at least $i-\frac{i+C}{q^n}\geq N-\frac{N+C}{q^n}$, which tends to infinity as $N$ tends to infinity. Since $m=m(N)$ depends on $N$ and tends to infinity as $N$ does, we have: \[0=\lim_{N\to\infty}\sum_{k=0}^{m(N)} c_k\zeta^{(-k)}(\Xi^{(-n)})=\lim_{m\to\infty}\sum_{k=0}^m c_k\zeta^{(-k)}(\Xi^{(-n)})=\sum_{k=0}^n c_k\zeta^{(-k)}(\Xi^{(-n)}),\] where we used that $\zeta^{(-k)}(\Xi^{(-n)})=0$ for $k>n$ by Corollary \ref{zeta zero}. For $0\leq k\leq n$ we can rewrite $\zeta^{(-k)}(\Xi^{(-n)})$ in the following way: \[\zeta^{(-k)}(\Xi^{(-n)})=\gamma_I^\frac{1}{q^k}\sum_{a\in^* I}a^{\frac{1}{q^n}-\frac{1}{q^k}}=\left(\gamma_I\sum_{a\in^*I}\left(\frac{a}{\gamma_I}\right)^{1-q^{n-k}}\right)^\frac{1}{q^n}=\left(\gamma_I P_{q^{n-k}-1}(\gamma_I^{-1}I)\right)^\frac{1}{q^n}.\]
We extrapolate the following equality for all $n\geq1$:
\[\left(\sum_{k=0}^n S_{q^k-1}(\tilde{\pi}\Lambda)^{q^{n-k}}P_{q^{n-k}-1}(\gamma_I^{-1}I)\right)^\frac{1}{q^n}=\gamma_I^{-\frac{1}{q^n}}\sum_{k=0}^n c_k\zeta^{(-k)}(\Xi^{(-n)})=0,\] which by Lemma \ref{log_L} implies that $\log_{\gamma_I^{-1}I}\circ\exp=1$. In particular, $\exp=\exp_{\gamma_I^{-1}I}$, therefore their zero loci are the same, which means that $\tilde{\pi}\Lambda=\gamma_I^{-1}I$. \end{proof} \begin{prop}\label{a_I/pi} The following identity holds in ${\mathbb{C}_\infty}\hat\otimes K$: \[\frac{(a_I\tilde{\pi}^{-1}\otimes1)\zeta_I}{\left((a_I\tilde{\pi}^{-1}\otimes1)\zeta_I\right)^{(1)}}=f_*^{(1)},\] \end{prop} \begin{proof} From the definition of $\gamma_I$ we have $\frac{\zeta_I}{\zeta_I^{(1)}}=(\gamma_I\otimes1)^{q-1}f_*$. Since $\Lambda=a_I^{-1}I$ and $\tilde{\pi}\Lambda=\gamma_I^{-1}I$, we deduce $\gamma_I=\frac{a_I}{\tilde{\pi}}$ up to a factor in $\mathbb{F}_q^\times$. \end{proof}
We can finally state and prove more precise versions of Theorems \ref{functional identity} and \ref{Sf_I}.
\begin{customteo}{\ref{functional identity}}[Complete version]
The following functional identity is well posed and true in ${K_\infty}\hat\otimes K$:
\[\zeta_I=-(a_I^{-1}\otimes a_I)\prod_{i\geq0}\left((\tilde{\pi}^{1-q}\otimes1)f_*^{(1)}\right)^{(i)}.\] \end{customteo} \begin{proof}
From the partial version of this theorem, we have the following identity in $\mathcal{O}_{K_\infty}\hat\otimes K$:
\[\zeta_I=-(a_I^{-1}\otimes a_I)\prod_{i\geq0}\left((\lambda^{1-q}\otimes1)f'_{\bar{I},*}{}^{(1)}\right)^{(i)},\]
where $f'_{\bar{I},*}$ is a scalar multiple of $f_{\bar{I},*}$, and $\lambda\in\mathcal{O}_{K_\infty}$ is some constant. We deduce:
\[\frac{\zeta_I}{\zeta_I^{(1)}}=(a_I^{q-1}\otimes 1)(\lambda^{1-q}\otimes1)f'_{\bar{I},*}{}^{(1)}.\]
On the other hand, by Corollary \ref{a_I/pi}, we know that
\[\frac{\zeta_I}{\zeta_I^{(1)}}=\left(\frac{a_I}{\tilde{\pi}}\otimes1\right)^{q-1}f_*^{(1)},\]
hence $(\lambda^{1-q}\otimes1)f'_{I,*}{}^{(1)}=(\tilde{\pi}^{1-q}\otimes1)f_*^{(1)}$ and we get the desired identity. \end{proof}
Recall the notations as in Definition \ref{def f, f_*, delta}. We restate Theorem \ref{Sf_I} to make it a proper generalization of \cite{Green}[Thm. 7.1]. \begin{customteo}{\ref{Sf_I}}[Complete version] The following $A$-submodules of ${\mathbb{C}_\infty}\hat\otimes A$ coincide: \[\Sf_{\bar{I}}=(\mathbb{F}_q\otimes I)\frac{\delta_{\bar{I}}^{(1)}(\tilde{\pi}\otimes1)}{f_{\bar{I}}(a_I\otimes1)\zeta_I}.\] \end{customteo} \begin{proof}
From the partial version of this theorem, we have the $A$-module $\Sf_I\subseteq{\mathbb{C}_\infty}\hat\otimes A$ coincides with $(\mathbb{F}_q\otimes I)\frac{\delta_I}{\zeta}$ , where $\zeta:=(\gamma_I\otimes1)\zeta_I$.
Since $\gamma_I=\frac{a_I}{\tilde{\pi}}$ up to a factor in $\mathbb{F}_q^\times$ by Corollary \ref{a_I/pi}, and $\delta_{\bar{I}}^{(1)}f_{\bar{I},*}=\delta_{\bar{I}}f_{\bar{I}}$ by Remark \ref{delta}, we deduce the thesis. \end{proof}
\section{Dedekind-like zeta function}\label{Dedekind zeta}
In this section, we prove the generalization of \cite[Thm. 7.3]{Green} in the form of Theorem \ref{xi equation}. We define a "Dedekind-like" zeta function $\xi_{\bar{I}}$ relative to a class $\bar{I}\in Cl(A)$, and then relate it to the Pellarin zeta function $\zeta_A$.
Fix a divisor $V_{\bar{I}}$, with corresponding Drinfeld-Hayes module $\phi:A\to {K_\infty}\{\tau\}$ sending $a$ to $\phi_a$ of degree $\deg(a)$. We can extend $\phi$ to ideals, sending $J=(a,b)\unlhd A$ to the generator $\phi_J$ of the left ideal $(\phi_a,\phi_b)<{K_\infty}\{\tau\}$, following a construction of Hayes (see \cite{Hayes}).
\begin{Def} Fix $\omega\in \Sf_{\bar{I}}$, and for $J\unlhd A$ define $\chi_{\bar{I}}(J):=\frac{\phi_J(\omega)}{\omega}$. \end{Def} \begin{oss} The previous definition does not depend on the choice of $\omega$. For $a\in A$, $J\unlhd A$, since $\phi_{aJ}=\phi_J\circ\phi_a$ we have that \[\chi_{\bar{I}}(aJ)=\frac{\phi_{aJ}(\omega)}{\omega}=\frac{\phi_J\circ\phi_a(\omega)}{\omega}=\frac{\phi_J((1\otimes a)\omega)}{\omega}=(1\otimes a)\frac{\phi_J(\omega)}{\omega}=\chi_{\bar{I}}(a)\chi_{\bar{I}}(J).\] It's easy to check that we can extend $\chi$ to all fractional ideals in a unique way such that for all $a\in K$ and for all fractional ideals $J$ we have $\chi_{\bar{I}}(a)\chi_{\bar{I}}(J)=\chi_{\bar{I}}(aJ)$. \end{oss} \begin{prop} For all $J\unlhd A$, $\chi_{\bar{I}}(J)\in\Q({K_\infty}\otimes A)$, and $\Div(\chi_{\bar{I}}(J))=V_{\bar{I}-\bar{J}}+J-V_{\bar{I}}-\deg(J)\infty$. \end{prop} \begin{proof} Consider $\mathcal{O}(V_{\bar{I}}):=\bigcup_{n\geq0}\mathcal{L}(V_{\bar{I}}+n\infty)$, which admits as a flag base $\{f_{\bar{I}}\cdots f_{\bar{I}}^{(k)}\}_{k\geq-1}$. By definition, for $a\in A$, if $\phi_a=\sum_i a_i\tau^i$ we have $1\otimes a=\sum_{i\geq0} (a_i\otimes 1) f_{\bar{I}}\cdots f_{\bar{I}}^{(i)}$. If we multiply everything by some $\omega\in \Sf_{\bar{I}}$ we get that $(1\otimes a)\omega=\sum_i (a_i\otimes 1)\omega^{(i)}=\phi_a(\omega)$, hence $\chi_{\bar{I}}(a)=1\otimes a$. For a fixed non principal ideal $J=(a,b)\unlhd A$, if we write $\phi_J=\sum_{i=0}^{\deg(J)} (c_i\otimes1)\tau^i$, we get: \[\chi_{\bar{I}}(J)=\frac{\phi_J(\omega)}{\omega}=\sum_{i=0}^{\deg(J)} c_i f_{\bar{I}}\cdots f_{\bar{I}}^{(i-1)}\in\mathcal{L}(V_{\bar{I}}+\deg(J)\infty).\] Moreover, if we write $\phi_J=\psi_1\circ\phi_a+\psi_2\circ\phi_b$ for some $\psi_1,\psi_2\in {K_\infty}\{\tau\}$, we get: \[\chi_{\bar{I}}(J)=\frac{\phi_J(\omega)}{\omega}=\frac{\psi_1\circ\phi_a(\omega)+\psi_2\circ\phi_b(\omega)}{\omega}=(1\otimes a)\frac{\psi_1(\omega)}{\omega}+(1\otimes b)\frac{\psi_2(\omega)}{\omega}.\] Since $1\otimes a,1\otimes b\in\mathcal{O}(-J)$, $\frac{\psi_1(\omega)}{\omega},\frac{\psi_2(\omega)}{\omega}\in\mathcal{O}(V_{\bar{I}})$, and the degree of $\chi_{\bar{I}}(J)$ is $\deg(J)$, we get $\chi_{\bar{I}}(J)\in\mathcal{L}(V_{\bar{I}}-J+\deg(J)\infty)$. The divisor $D:=V_{\bar{I}}+\deg(J)\infty-J$ has degree $g$ and is such that: \begin{align*}
&D-D^{(1)}\sim V_{\bar{I}}-V_{\bar{I}}^{(1)}\sim\Xi-\infty&\red(D-g\infty)\sim I-J+(\deg(J)-\deg(I))\infty. \end{align*} By Lemma \ref{con}, $D\sim V_{\bar{I}-\bar{J}}$, and $h^0(V_{\bar{I}-\bar{J}})=1$, hence $\Div(\chi_{\bar{I}}(J))=V_{\bar{I}-\bar{J}}+J-V_{\bar{I}}-\deg(J)\infty$. Since $\chi_{\bar{I}}(J)(\Xi)=c_0\in {K_\infty}$ and $\Div^+(\chi_{\bar{I}}(J)),\Div^-(\chi_{\bar{I}}(J))$ are ${K_\infty}$-rational, $\chi_{\bar{I}}(J)\in \Q({K_\infty}\otimes A)$. \end{proof} Let's prove a lemma before the last proposition. \begin{lemma}\label{final lemma}
Fix an ideal $I\unlhd A$, with degree $d_I$. Then, for all ideal classes $\bar{J}\in Cl(A)$, there is some $H$-rational function $h_{I,\bar{J}}$ with divisor:
\[\Div(h_{I,\bar{J}})=V_{\bar{J},*}^{(1)}+V_{\bar{I}+\bar{J}}-I-\Xi-(2g-d_I-1)\infty.\]
Moreover, we can choose $\{h_{I,\bar{J}}\}_{\bar{J}\in Cl(A)}$ such that for all $\bar{J}\in Cl(A)$, $\frac{h_{I,\bar{A}}}{h_{I,\bar{J}}}(\Xi)=1$. \end{lemma} \begin{proof}
Fix some ideal $J\unlhd A$, call $d_J$ its degree, and define $D:=I+\Xi+(2g-d_I-1)\infty$. Consider the divisor $D-V_{\bar{J},*}^{(1)}$: we want to prove that it is equivalent to $V_{\bar{I}+\bar{J}}$. First of all, its degree is $g$, hence it is equivalent to some effective divisor. Moreover, we have the following equivalences:
\begin{align*}
\red_{K_\infty}(D-V_{\bar{J},*}^{(1)})&\sim\red_{K_\infty}(D)-\red_{K_\infty}(V_{\bar{J},*})\\
&\sim (I+(2g-d_I)\infty)-((d_J+g)\infty-J)\\
&\sim (I+J)+(g-d_J-d_I)\infty\sim \red_{K_\infty}(V_{\bar{I}+\bar{J}});\\
(D-V_{\bar{J},*}^{(1)})-(D-V_{\bar{J},*}^{(1)})^{(1)}&\sim(D-D^{(1)})-(V_{\bar{J},*}-V_{\bar{J},*}^{(1)})^{(1)}\\
&\sim(\Xi-\Xi^{(1)})-(\infty-\Xi)^{(1)}\sim\Xi-\infty\sim V_{\bar{I}+\bar{J}}-V_{\bar{I}+\bar{J}}^{(1)}.
\end{align*} By Lemma \ref{con}, the two conditions imply that $D-V_{\bar{J},*}^{(1)}\sim V_{\bar{I}+\bar{J}}$.
By Remark \ref{Hayes}, the divisors $\{V_{\bar{J},*}^{(1)}+V_{\bar{I}+\bar{J}}-D\}_{\bar{J}\in Cl(A)}$ are $H$-rational. Moreover, by the same reasoning as Remarks \ref{Hayes} and \ref{Hayes2}, they are all conjugated by the action of $\G(H/K)$; we can define $h'_{I,\bar{A}}$ as any function in $\Q(H\otimes A)$ with divisor $V_{\bar{A},*}^{(1)}+V_{\bar{I}}-D$, while for all $\bar{J}\in Cl(A)$, we set $h'_{I,\bar{J}}:=h'_{I,\bar{A}}{}^{\sigma_{\bar{J}}}$, where $\sigma_{\bar{J}}$ is the appropriate element of $\G(H/K)$.
Now, for all $\sigma\in\G(H/K)$, we set $c_\sigma:=\frac{h'_{\bar{A}}{}^\sigma}{h'_{\bar{A}}}(\Xi)\in H^\times$. For all $\sigma,\tau\in\G(H/K)$ we have $c_{\sigma\tau}=c_{\sigma}c_{\tau}^\sigma$, so by Hilbert 90 there is some $b\in H$ such that, for all $\sigma\in\G(H/K)$, $c_\sigma=\frac{b}{b^\sigma}$. Finally, we can set $h_{I,\bar{J}}:=h'_{I,\bar{J}}\cdot b^{\sigma_{\bar{J}}}$, so that both the conditions we were looking for are satisfied. \end{proof} \begin{oss}
The quotients $\frac{h_{I,\bar{A}}}{h_{I,\bar{J}}}$ only depends on the class $\bar{I}$. \end{oss} \begin{Def}
We denote $\xi_{\bar{I}}:=\sum_{J\unlhd A}\frac{\chi_{\bar{I}}(J)}{\chi_{\bar{I}}(J)(\Xi)}\in {K_\infty}\hat\otimes K\cong K((u))$. \end{Def} In the notations of Lemma \ref{final lemma}, we have the following result. \begin{teo}\label{xi equation} The function $\xi_{\bar{I}}$ is well defined, and the following identity holds: \[h_{I,\bar{A}}\xi_{\bar{I}}=-\left(\sum_{\sigma\in\G(H/K)}h_{I,\bar{A}}^\sigma\right)\zeta_A.\] \end{teo} \begin{proof} Let's fix representatives $J_i\unlhd A$ for each ideal class $\bar{J_i}\in Cl(A)$. To prove convergence we rearrange the terms: \begin{align*}
\sum_{J\unlhd A}\frac{\chi_{\bar{I}}(J)}{\chi_{\bar{I}}(J)(\Xi)}
&=\sum_i\sum_{\substack{J\unlhd A\\\bar{J}=\bar{J_i}}}\frac{\chi_{\bar{I}}(J)}{\chi_{\bar{I}}(J)(\Xi)}=\sum_i\sum_{\substack{a\in^* J_i^{-1}\\\sgn(a)=1}}\frac{\chi_{\bar{I}}(aJ_i)}{\chi_{\bar{I}}(aJ_i)(\Xi)}=-\sum_i\sum_{a\in^* J_i^{-1}}\frac{\chi_{\bar{I}}(aJ_i)}{\chi_{\bar{I}}(aJ_i)(\Xi)}\\
&=-\sum_i\left(\frac{\chi_{\bar{I}}(J_i)}{\chi_{\bar{I}}(J_i)(\Xi)}\sum_{a\in^* J_i^{-1}}\frac{\chi_{\bar{I}}(a)}{\chi_{\bar{I}}(a)(\Xi)}\right)=-\sum_i\frac{\chi_{\bar{I}}(J_i)}{\chi_{\bar{I}}(J_i)(\Xi)}\zeta_{J_i^{-1}}. \end{align*} We now express explicitly the quotient $\frac{\zeta_{J_i^{-1}}}{\zeta_{A}}$. In $K((u))$, it is the limit of the sequence $\left(\frac{\zeta_{J_i^{-1},m}}{\zeta_{A,m}}\right)_m$, whose divisor for $m\gg0$ is $V_{-\bar{J_i},*,m}-V_{\bar{A},*,m}-J_i+\deg(J_i)\infty$, which we can rearrange as: \[(V_{-\bar{J_i},*,m}+V_{-\bar{J_i},m}-2g\infty)-(V_{\bar{A},*,m}+V_{-\bar{J_i},m}+J_i-(2g+\deg(J_i))\infty).\] Both divisors in the parentheses are principal, and their positive components converge; by Proposition \ref{convergence of functions and divisors}, the function $\frac{\zeta_{J_i^{-1}}}{\zeta_A}$ is rational, with divisor: \[\lim_m \left(V_{-\bar{J_i},*,m}-V_{\bar{A},*,m}-J_i+\deg(J_i)\infty\right)=V_{-\bar{J_i},*}^{(1)}-V_{\bar{A},*}^{(1)}-J_i+\deg(J_i)\infty.\] If we multiply $\frac{\zeta_{J_i^{-1}}}{\zeta_A}$ by $\chi_{\bar{I}}(J_i)$, the resulting divisor is $V_{-\bar{J_i},*}^{(1)}+V_{\bar{I}-\bar{J_i}}-V_{\bar{A},*}^{(1)}-V_{\bar{I}}$, which is the divisor of $\frac{h_{I,-\bar{J_i}}}{h_{I,\bar{A}}}$ by Lemma \ref{final lemma}. Since $-\zeta_{J_i^{-1}}(\Xi)=-\zeta_A(\Xi)=\frac{h_{I,-\bar{J_i}}}{h_{I,\bar{A}}}(\Xi)=1$, we have the equality $\frac{\zeta_{J_i^{-1}}}{\zeta_A}\chi_{\bar{I}}(J)=\chi_{\bar{I}}(J)(\Xi)\frac{h_{I,-\bar{J_i}}}{h_{I,\bar{A}}}$, so we can rewrite the Dedekind zeta as: \[\xi_{\bar{I}}=-\zeta_A\sum_i\frac{h_{I,-\bar{J_i}}}{h_{I,\bar{A}}}=-\zeta_A\sum_{\bar{J}\in Cl(A)}\frac{h_{I,\bar{J}}}{h_{I,\bar{A}}}=-\zeta_A\sum_{\sigma\in\G(H/K)}\frac{h_{I,\bar{A}}^\sigma}{h_{I,\bar{A}}}.\tag*{\qedhere}\] \end{proof} \begin{oss} Evaluating at $\Xi$, we get - modulo the characteristic of $\mathbb{F}_q$ - $\xi_{\bar{I}}(\Xi)=\# Cl(A)$. \end{oss}
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\end{document} |
\begin{document}
\begin{frontmatter} \title{Nonparametric Identification of Kronecker Networks }
\author[Padova]{Mattia Zorzi}\ead{zorzimat@dei.unipd.it}
\address[Padova]{Dipartimento di Ingegneria dell'Informazione, Universit\`a degli studi di Padova, via Gradenigo 6/B, 35131 Padova, Italy}
\begin{keyword} Linear system identification; Sparsity inducing priors; Kernel-based methods; Gaussian processes. \end{keyword}
\begin{abstract} We address the problem to estimate a dynamic network whose edges describe Granger causality relations and whose topology has a Kronecker structure. Such a structure arises in many real networks and allows to understand the organization of complex networks. We proposed a kernel-based PEM method to learn such networks. Numerical examples show the effectiveness of the proposed method. \end{abstract} \end{frontmatter}
\section{Introduction} In many applications we collect high dimensional data which describe systems having a wide amount of variables. Thus, there is the need to understand how those variables are ``connected each other'', i.e. we also need to learn from data the topology of the graphical model \citep{LAURITZEN_1996} representing those interactions. It is worth noting that a graphical model is meaningful only if it is not complete, i.e. it has some degree of sparsity in terms of edges. It is clear that the network identification paradigm depends on the definition of ``interactions''. The latter can describe conditional dependence relations between the stochastic processes characterizing the variables of the system and whose resulting network is undirected \citep{LINDQUIST_TAC13,LATENTG,8976281,RECS,zorzi2019graphical}. In the case of directed graphical models there is an edge from node $h$ to node $j$ if we need the process at node $h$ in order to reconstruct the process at node $j$. It is worth noting that such reconstruction can be both acausal \citep{MATERASSI1,MATERASSI2,v2020topology,sepehr2019blind,doddi2020estimating} and causal \citep{CHIUSO_PILLONETTO_SPARSE_2012,BSL_CDC}. In the latter case, we say that edges describes Granger causality relations \citep{GRANGER_CAUSALITY}. Finally, it is worth noting that a closely related topic is the problem to learn modules of a network only using data from neighbors nodes, see \cite{HOF1,HOF2}.
In this we paper we focus our attention on Kronecker networks \citep{leskovec2007scalable} whose topology possesses some properties emerging in many real networks, e.g. heavy-tailed degree distribution, small diameter, see \cite{leskovec2010kronecker} for more details. Such a structure is also useful to understand the organization of complex networks \citep{leskovec2009networks}. \cite{tsiligkaridis2013convergence} consider a network identification problem whose Kronecker structure is not only imposed on the topology of the network but also on the parameters of the model. The latter constraint, however, could be restrictive in some applications, e.g. for spatio-temporal MEG/EEG modeling \citep{bijma2005spatiotemporal}. A possible way to overcome this restriction is to impose the structure ``sum of Kronecker products'' on the parameters of the model \citep{QUARKS,tsiligkaridis2013covariance} at the price that we loose a meaningful definition of interaction, i.e. the resulting network is fully connected. In \cite{KRON,CDC_KRON}, instead, it has been considered the problem to learn undirected dynamic networks having the Kronecker structure on the topology, but not on the parameters of the model. Such a framework, however, has not been extended to the case of directed dynamic networks yet.
The present paper considers the problem to learn directed dynamic networks whose edges characterize Granger causality relations and whose topology has a Kronecker structure.
A well established system identification paradigm is the so called prediction error method (PEM), see \cite{LJUNG_SYS_ID_1999,SODERSTROM_STOICA_1988}. Within this framework, candidate models are described in fixed parametric model structures, e.g. ARMAX. Thus, the main difficulty is the correct choice of the best model structure which is usually performed by AIC and BIC criteria \citep{AKAIKE_1974}. Kernel-based PEM methods aim to overcome this issue, see \cite{PILLONETTO_DENICOLAO2010,EST_TF_REVISITED_2012,CHIUSO201624,MU2018381,ljung2020shift,9143975}. More precisely, the candidate model, described through the predictor impulse responses, is searched in an infinite dimensional nonparametric model class; this is clearly an ill posed problem because we have a finite set of measured data. However, it can be made into a well posed one using a penalty term which favors models with specific features. In the Bayesian viewpoint, this is equivalent to introduce an a priori Gaussian probability (prior) on the unknown model. Hence, the prior distribution is characterized by the covariance (i.e. kernel) function.
The contribution of this paper is the introduction of a kernel-based PEM method to estimate Kronecker networks. The a priori information is that the impulse responses must be Bounded Input Bounded Output (BIBO) stable and the topology of the network respects the Kronecker structure. We derive the corresponding kernel functions by using the maximum entropy principle.
The kernel functions are characterized by the decay rate of the predictor impulse responses and by the number of edges in the network. These features are not known and characterized by the so called hyperparameters. We estimate them by minimizing the negative log-marginal likelihood of the measured data, see \cite{RASMUSSEN_WILLIAMNS_2006}. Moreover, we show that the negative log-marginal likelihood automatically sets some hyperparameters in such a way that the a priori information is that the network has a sparse Kronecker structure. The degree of sparsity depends on the variance of the noise process affecting each node.
Finally, we provide some numerical experiments on synthetic data in order to test the effectiveness of the proposed method. Moreover, we use the latter for learning Granger causality relations in a bike sharing system.
The paper is organized as follows. In Section \ref{section_kron} we introduce the Kronecker model corresponding to a network characterized in terms of Granger causality relations. In Section \ref{section_pb_formulation} we introduce the Kronecker network identification problem. In Section \ref{section_kernel} we derive the maximum entropy kernels inducing BIBO stability and a Kronecker structure for the topology of the network, while Section \ref{sec:opt_procedure} is devoted to the estimation of the hyperparameters. In Section \ref{section_simulation} we present some numerical examples. Finally, we draw the conclusions in Section \ref{section_conclusions}. \subsection*{Notation}
$ \mathbb{N}$ is the set of natural numbers. Given a finite set $ \mathcal{I}$,
$| \mathcal{I} |$ denotes its cardinality.
$ \mathbb{E}[\cdot]$ denotes the expectation, while $ \mathbb{E}[\cdot|\cdot]$ denotes the conditional mean. Given three (possibly infinite dimensional) random vectors $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ we say that $\mathrm{a}$ is conditionally independent of $\mathrm{b}$ given $\mathrm{c}$ if $ \mathbb{E}[\mathrm{a}|\mathrm{b},\mathrm{c}] = \mathbb{E}[\mathrm{a}|\mathrm{c}]$. Given $G\in \mathbb{R}^{n\times p}$, $[G]_{i,j}$ denotes the entry of $G$ in position $(i,j)$; $G>0$ ($G\geq 0$) means that $G$ is a positive (semi-)definite matrix. $ \mathcal{D}_p$ denotes the vector space of diagonal matrices of dimension $p$. $\ell_2( \mathbb{N})$ denotes the space of $ \mathbb{R}$-valued infinite length sequences, which we think as infinite dimensional column vectors $g:=[g_1 \; g_2 \; \ldots \; g_j \; \ldots]^\top$, $g_k\in \mathbb{R}$, $k\in \mathbb{N}$, such that
$\|g\|_2:=\sqrt{\sum_{k=1}^\infty |g_k|^2}<\infty$. $\ell_2^{p\times n}( \mathbb{N})$ is the space of matrices of sequences in $\ell_2( \mathbb{N})$ \al{\Phi=\left[\begin{array}{ccc} (\phi^{[11]})^\top & \ldots & (\phi^{[1n]})^\top \\ \vdots & \ddots & \vdots \\ (\phi^{[p1]})^\top & \ldots & (\phi^{[pn]})^\top\end{array}\right]\nonumber} where $\phi^{[ij]}\in\ell_2 ( \mathbb{N})$, $i=1\ldots p$ and $j=1\ldots n$. $\ell_1( \mathbb{N})$ denotes the space of $ \mathbb{R}$-valued infinite length sequences $g$
such that $\|g\|_1:=\sum_{k=1}^\infty |g_k|<\infty$. $\ell_1^{p\times n}( \mathbb{N})$ is defined in similar way. $ \mathcal{S}_2( \mathbb{N})$ denotes the space of symmetric infinite dimensional matrices $K$ such that
$\| K\|_2:=\sqrt{\sum_{i,j=1}^\infty | [K]_{i,j}|^2}<\infty$. $ \mathcal{S}_2^{p}( \mathbb{N})$ is the space of symmetric infinite dimensional matrices \al{K=\left[\begin{array}{cccc} K^{[11]} & K^{[12]} & \ldots & K^{[1p]} \\ K^{[12]} & K^{[22]} & & K^{[2p]} \\ \vdots & & \ddots & \vdots \\ K^{[1p]} & K^{[2p]} &\ldots & K^{[pp]}\end{array}\right]\nonumber}
where $K^{[ij]} \in \mathcal{S}_2( \mathbb{N})$, $i,j=1\ldots p$. Given $\Phi\in\ell_2^{p \times n}( \mathbb{N})$, $\Psi\in\ell_2^{m \times n}( \mathbb{N})$ and $K\in \mathcal{S}_2^{n}( \mathbb{N})$, the products $\Phi \Psi^\top$ and $\Phi K \Psi^\top$ are understood as $p \times m$ matrices whose entries are limits of infinite sequences \citep{INFINITE_MATRICES}. Given $g\in \ell_2^{n\times 1}( \mathbb{N})$ and $K\in \mathcal{S}_2^{n}( \mathbb{N})$, $\|g\|^2_{K^{-1}}:=g^\top K^{-1}g$. The definition is similar in the case that $g$ and $K$ have finite dimension. With some abuse of notation the symbol $z$ will denote both the complex variable as well as the shift operator $z^{-1} y(t) :=y(t-1)$. Given a stochastic process $y=\{y(t)\}_{t\in \mathbb{Z}}$, with some abuse of notation, $y(t)$ will both denote a random vector and its sample value. From now on the time $t$ will denote \emph{present} and we shall talk about \emph{past} and \emph{future} with respect to time $t$. With this convention in mind, \al{\mathrm{y}^- =\left[\begin{array}{ccc} y(t-1)^\top & y(t-2)^\top & \ldots \end{array}\right]^\top\nonumber} denotes the (infinite length) past data vector of $y$ at time $t$. We denote as $\mathrm{supp}(G(z))$ the support of the transfer matrix $G(z)$, i.e. the entries of $G(z)$ different from the null function correspond to entries equal to one in $\mathrm{supp}(G(z))$, otherwise the latter are equal to zero.
\section{Kronecker Models} \label{section_kron} We consider the following nonparametric model \al{ \label{OEmodel_start}y(t)=& G(z)y(t)+F(z)u(t)+e(t)} where $y(t)\in \mathbb{R}^{p_1p_2}$ is the output, $u(t)\in \mathbb{R}^m$ is the input, $e(t)$ is zero mean normalized white Gaussian noise (WGN) whose components have variance $\sigma_{hk}^2$ with $h=1\ldots p_1$ and $k=1\ldots p_2$; $G(z)=\sum_{t=1}^\infty G_t z^{-t}$ is a BIBO stable transfer matrix of dimension $p_1 p_2\times p_1 p_2$, while $F(z)=\sum_{t=1}^\infty F_t z^{-t}$ is a BIBO stable transfer matrix of dimension $p_1p_2\times m$.
The minimum variance one-step ahead predictor of $y(t)$ based on the past data $\mathrm{y}^-$, denoted by $\hat y(t|t-1)$, is given by
\al{ \hat y(t|t-1)&= G(z)y(t)+F(z)u(t).\nonumber} Thus, $e(t)$ is the one-step-ahead prediction error $$
e(t) = y(t) - \hat y(t|t-1). $$ We assume that \al{\label{cond_supp}\begin{aligned}\mathrm{supp}(G(z))&=E_1 \otimes E_2\\ \mathrm{supp}(F(z))&=A_1 \otimes A_2 \end{aligned}} where $E_1$, $E_2$, $A_1$ and $A_2$ denote the adjacency matrices of dimension $p_1\times p_1$, $p_2\times p_2$, $p_1\times 1$ and $p_2\times m$, respectively. Let $y_{hk}$ denote the scalar process of $y$ in position $(h-1)p_2+k$ with $h\in I_1:=\{1\ldots p_1\}$ and $k\in I_2:=\{1\ldots p_2\}$. In a similar way, $\mathrm{y}^-_{hk}$ denotes the past of $y_{hk}$ at time $t$. Finally, $u_i$ denotes the scalar component of $u$ in position $i\in I_u:=\{1\ldots m\}$ and $\mathrm{u}_i^-$ the $i$-th component of $\mathrm{u}^-$. Throughout the paper we assume that $E_1$, $E_2$, $A_1$ and $A_2$ are different from the null matrix.
It is possible to describe the structure of model (\ref{OEmodel_start})-(\ref{cond_supp}) using a Bayesian network \citep{LAURITZEN_1996}. The nodes correspond to the scalar processes $y_{hk}$'s, with $h\in I_1$ and $k\in I_2$, while $u_i$'s, with $i\in I_u$, represent exogenous variables affecting some nodes of the network. Then, the connections among the nodes obey the following rules: \begin{itemize} \item there is a directed link from node $y_{jl}$ to node $y_{hk}$ if \al{
\mathbb{E}[y_{hk}(t) | \mathrm{y}^-,\mathrm{u}^-] \neq \mathbb{E}[y_{hk}(t) |\mathrm{y}_{\tilde j\tilde l}^-,\tilde j\neq j, \tilde l\neq l,\mathrm{u}^-],\nonumber }
i.e. if $\mathrm{y}_{jl}^-$ is needed to predict $y_{hk}(t)$ given $\mathrm{u}^-$ and $\mathrm{y}_{\tilde j\tilde l}^-$, for any $\tilde j\in I_1\setminus \{j\}$ and $\tilde l\in I_2\setminus\{l\}$; In this case, we shall say $y_{jl}$ conditionally Granger causes $y_{hk}$.
\item there is a directed link from $u_{i}$ to node $y_{hk}$ if \al{
\mathbb{E}[y_{hk}(t) | \mathrm{y}^-,\mathrm{u}^-] \neq \mathbb{E}[y_{hk}(t) |\mathrm{y}^-,\mathrm{u}^-_{\tilde i}, \tilde i\neq i]\nonumber }
i.e. if $\mathrm{u}_{i}^-$ is needed to predict $y_{hk}(t)$ given $\mathrm{y}^-$ and $\mathrm{u}_{\tilde i}^-$, for any $\tilde i\in I_u\setminus \{i\}$; In this case, we shall say $u_{i}$ conditionally Granger causes $y_{hk}$.
\end{itemize} In such a network we can recognize $p_1$ modules containing $p_2$ nodes and sharing the same topology described by $E_2$, while the interaction among these $p_1$ modules is described by $E_1$. Finally, $A_1$ describes the topology among the input and the modules; $A_2$ describes the topology among the input components and the nodes in a module which is the same in each module. These facts are formalized by the next proposition.
\begin{prop}\label{propo_cond_gr} Consider model (\ref{OEmodel_start})-(\ref{cond_supp}). Let $y^\star_h$ be the process obtained by stacking $y_{hk}$ with $k\in I_2$ and $y^\dag_k$ be the process obtained by stacking $h\in I_1$. Then: \begin{itemize} \item $y_j^\star$ conditionally Granger causes $y^\star_h$ if and only if $[E_1]_{h,j}=1$; \item $y_l^\dag$ conditionally Granger causes $y^\dag_k$ if and only if $[E_2]_{k,l}=1$; \item $u$ conditionally Granger causes $y^\star_h$ if and only if $[A_1]_{h}=1$; \item $u_i$ conditionally Granger causes $y^\dag_k$ if and only if $[A_2]_{k,i}=1$. \end{itemize} \end{prop} \begin{proof} We only prove the first claim, the proof of the others is similar. Let $g^{[hk,jl]}\in\ell_1( \mathbb{N})$ denote the impulse response in $G(z)$ having input $y_{jl}$ and output $y_{hk}$. By (\ref{cond_supp}), we have
\al{\label{supp_1}g^{[hk,jl]}\neq 0, \, \hbox{ for some } k,l\in I_2\iff [E_1]_{h,j}=1.}
From (\ref{OEmodel_start}) we have that
\al{y_{hk}&(t)=\sum_{\substack{j\in I_1\\ l\in I_2}}
[G(z)]_{(h-1)p_2+k,(j-1)p_2+l}y_{jl}(t) \nonumber\\
&+\sum_{i\in I_u} [F(z)]_{(h-1)p_2+k,i}u_i(t) + e_{hk}(t).\nonumber} Thus, condition (\ref{supp_1}) is equivalent to
\al{\label{cond_prop1} \mathbb{E}[y_{hk}(t) | \mathrm{y}^-,\mathrm{u}^-] \neq \mathbb{E}[y_{hk}(t) |\mathrm{y}_{\tilde j\tilde l}^-,\tilde j\neq j, \tilde l\neq l,\mathrm{u}^-],\nonumber\\\hbox{for some } k,l\in I_2 \iff [E_1]_{h,j}=1. } By (\ref{cond_prop1}), we have \al{\label{supp_2} \mathbb{E}[y_{hk}(t)|\mathrm{u}^-,\mathrm{y}^-]\neq \mathbb{E}[y_{hk}(t)|\mathrm{y}^{\star -}_{\tilde j}, \tilde j\neq j,\mathrm{u}^-], \nonumber\\ \hbox{for some } k\in I_2 \iff [E_1]_{h,j}=1. } Staking condition (\ref{supp_2}) for any $k\in I_2$, we obtain
\al{& \mathbb{E}[y_{h}^\star(t)|\mathrm{u}^-,\mathrm{y}^-]\neq \mathbb{E}[y_{h}^\star(t)|\mathrm{u}^-,\mathrm{y}^{\star -}_{\tilde j}, \tilde j\neq j] \nonumber\\ &\hspace{4cm} \iff [E_1]_{h,j}=1\nonumber} which proves the claim.
\vrule height 7pt width 7pt depth 0pt
\end{proof}
\begin{figure}
\caption{A example of dynamic Kronecker network with $p_1=3$ modules (red, blue and green) composed by $p_2=4$ nodes and $m=1$ exogenous component.}
\label{graph_ex}
\end{figure} It is worth noting that condition (\ref{cond_supp}) is weaker than $G(z)$ and $F(z)$ admit a Kronecker decomposition, e.g. $G(z)=G_1(z)\otimes G_2(z)$. Therefore, we do not constrain the dynamics in each module to be the same. An example of Kronecker network is depicted in Figure \ref{graph_ex}. Here, we have $$y=[\, y_{11} \,y_{12} \,y_{13} \,y_{14} \,y_{21} \,y_{22} \,y_{23} \,y_{24} \,y_{31} \,y_{32} \,y_{33} \,y_{34} \,]^\top$$ and the adjacency matrices are: {\small \al{E_1&=\left[\begin{array}{ccc}0 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0\end{array}\right], \, E_2=\left[\begin{array}{cccc}1 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 1 & 0 &0 \\ 0 & 0 & 0 &1 \end{array}\right],\, A_1=\left[\begin{array}{c}1 \\1 \\1 \end{array}\right],\, A_2=\left[\begin{array}{c}0 \\1 \\0 \\ 0\end{array}\right].\nonumber}}
\subsection{Motivating examples}\label{sec_examples} {\em Dynamic spatio-temporal modeling.} Consider a nonparametric time-varying model of the form \al{ \label{mod_LTV}x(s)=\tilde G_s(q) x(s)+\tilde F_s(q) w(s)+v(s),\; \; s\in \mathbb{Z}} where $x(s)\in \mathbb{R}^{p2}$ is the output, $w(s)\in \mathbb{R}^{\tilde m}$ is the input, $v(t)$ is normalized WGN and $\{\tilde G_s(q),\, s\in \mathbb{Z}\}$, $\{\tilde F_s(q),\, s\in \mathbb{Z}\}$ are sequences of BIBO stable (strictly causal) transfer matrices of dimension $p_2\times p_2$ and $p_2\times \tilde m$, respectively, such that \al{\tilde G_s(q)= \tilde G_{s+p_1}(q), \; \; \tilde F_s(q)=\tilde F_{s+p_1}(q)} where $q^{-1}$ is the shift operator corresponding to time variable $s$, e.g. $q^{-1}x(s)=x(s-1)$. Then, if we define \al{ y(t)&=[\, x((t-1)p_1+1)^\top\, \ldots\; x(tp_1)^\top\,]^\top\in \mathbb{R}^{p_1p_2}\nonumber\\ u(t)&=[\, w((t-1)p_1+1)^\top\, \ldots\; w(tp_1)^\top\,]^\top\in \mathbb{R}^{p_1\tilde m}\nonumber\\ e(t)&=[\, v((t-1)p_1+1)^\top\, \ldots\; v(tp_1)^\top\,]^\top\in \mathbb{R}^{p_1p_2},\nonumber} we can write (\ref{mod_LTV}) as (\ref{OEmodel_start}) where $G(z)$ and $F(z)$ are BIBO stable transfer matrices of dimension $p_1p_2\times p_1p_2$ and $p_1p_2\times p_1\tilde m$, respectively. Moreover, {\scriptsize \al{G(z)=\left[\begin{array}{ccccc}G_{1,1}(z) & G_{1,2}(z) & \ldots &\ldots & G_{1,p_1}(z) \\ zG_{2,1}(z) & G_{2,2}(z) & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots &\vdots \\ \vdots & & \ddots & \ddots & G_{p_1-1,p_1}(z) \\ zG_{p_1,1}(z) & \ldots & \ldots & zG_{p_1,p_1-1}(z) & G_{p_1,p_1}(z)\end{array}\right]\nonumber}}where $G_{hj}(z)=\sum_{t=1}^\infty G_{hj,t}z^{-t}$ is a transfer matrix of dimension $p_2\times p_2$ and $z^{-1}$ is the shift operator corresponding to time variable $t$; $F(z) $ is defined in a similar way: it is composed by the transfer matrices $F_{hj}(z)=\sum_{t=1}^\infty F_{hj,t}z^{-t}$, $h,j=1\ldots p_1$, of dimension $p_2\times \tilde m$ and the blocks in the strictly lower block triangular part are multiplied by $z$. In plain words, we have written (\ref{mod_LTV}) as a time-invariant model through $y(t)$ and $u(t)$. It is interesting to note that the time variables $s$ and $t$ play a different role; indeed, there is a decimation relationship between them and the decimation factor is $p_1$. If $G(z)$ and $F(z)$ satisfy condition (\ref{cond_supp}), then by Proposition \ref{propo_cond_gr} we have that $E_2$ is the matrix describing the conditional Granger causality relations among the components of $x(s)$, while $A_2$ is the matrix describing the conditional Granger causality relations among the components of $w(s)$ and $x(s)$. Moreover, $E_1$ describes the conditional Granger causality relations of $x(s)$ over the period $1\ldots p_1$, while $A_1$ describes the conditional Granger causality relations among $x(s)$ and $w(s)$ over the period $1\ldots p_1$. We conclude that (\ref{OEmodel_start}) can be understood as a dynamic spatio-temporal model. In particular, if we take \al{\label{cond_special} G(z)=\bar G, \;\; F(z)= \bar F,} i.e. constant matrices, then (\ref{OEmodel_start}) describes a spatio-temporal model without dynamic. The latter has been used to model magnetoencephalography (MEG) measurements for mapping brain activity \citep{bijma2005spatiotemporal}: $z(k,t)=[\,x_k((t-1)p_1+1) \, \ldots \,x_k(tp_1)\,]^\top$ models the measurements corresponding to the $k$-th
brain region during the $t$-th trial of length $p_1$, while $s(k,t)=[\,w_k((t-1)p_1+1) \, \ldots \,w_k(tp_1)\,]^\top$ is the stimulus applied to the $k$-th region during the $t$-th trial. These trials are assumed independent. Moreover, they are identically distributed because in each trial the task performed by the patient is the same and the way that the patient reacts to the stimulus is assumed to be the same. In our framework we do not impose the restrictive condition (\ref{cond_special}) and thus we let dependence among the trials. This means that the proposed model takes into account the fact that the trials are taken in a sequential way, and thus the fact that the behaviors of the patient in the next trial can be influenced by the previous one. Finally, such a dynamic spatio-temporal model could be also used for modeling the conditional Granger causality relations among the number rental bikes in a bikesharing system and the corresponding weather information, see Section \ref{sec_bike}.
{\em Multi-task learning.} Consider a network composed by $p_2\in \mathbb{N}$ agents (i.e. each node correspond to an agent). One agent is described by a model having an input and output. Our aim is to model such a network under $p_1\in \mathbb{N}$ heterogeneous conditions (i.e. tasks). The $k$-th agent under the $h$-th task has input $u(t)$ and output $y_{hk(t)}$. Then, we could model the network through the $p_1$ models: \al{\label{eq_multitask}y_h^\star(t)=G_h(z) y_h^\star+ F_h(z)u(t)+e_h(t), \; \; h\in I_1 } where $y_h^\star(t)$ has been defined in Proposition \ref{propo_cond_gr}. A more flexible approach is to model all the models in (\ref{eq_multitask}) together in order to exploit commonalities and differences across the tasks, see \cite{yu2009large}. More precisely, we consider the model \al{y(t)=G(z)y(t)+F(z)u(t)\nonumber} where $y(t)$ is obtained by stacking $y_h^\star(t)$ with $h\in I_1$. Under the assumption that (\ref{cond_supp}) holds, and in view of Proposition \ref{propo_cond_gr}, we have that $E_1$ describes the conditional Granger causality relations among the tasks; $E_2$ describes the conditional Granger causality relations among the agents in each task; $A_1$ describes the conditional Granger causality relations from the input to the tasks; $A_2$ describes the conditional Granger causality relations from the input components to the agents in each task.
\section{Problem Formulation}\label{section_pb_formulation} Assume to measure the data $\{y(t),u(t)\}_{t=0,..,N}$ generated by (\ref{OEmodel_start}). In this section, we address the problem of estimating $G(z)$ and $F(z)$ which admit the Kronecker product decomposition in (\ref{cond_supp}).
The transfer matrices $G(z)$, $F(z)$ are parametrized in terms of their impulse response coefficients $G_t$ and $F_t$. In particular, defining
$g^{[hk,jl]}\in \ell_1( \mathbb{N})$ to be the impulse response from input $(j-1)p_2+l$ to output $(h-1)p_2+k$, we have:
\al{ g^{[hk,jl]}:= &\left[
\begin{array}{cc}
[G_1]_{(h-1)p_2+k,(j+1)p_2+l} & \ldots \end{array} \right. \nonumber\\
&\hspace{0.2cm} \left. \begin{array}{ccc} \ldots & [G_s]_{(h-1)p_2+k,(j+1)p_2+l}&\ldots \\
\end{array}
\right]^\top ; \nonumber} defining $f^{[hk,i]}\in \ell_1( \mathbb{N})$ to be the impulse response from input $u_i$ to output $(h-1)p_2+k$, we have
\al{ f^{[hk,i]}:= \left[
\begin{array}{cccc}
[F_1]_{(h-1)p_2+k,i} & \ldots & [F_s]_{(h-1)p_2+k,i,i}&\ldots \\
\end{array}
\right]^\top . \nonumber} The coefficient vectors $\theta_g^\top\in\ell_1^{1\times p_1^2p_2^2}( \mathbb{N})$ and $\theta_f^\top\in\ell_1^{1\times p_1p_2m}( \mathbb{N})$ are defined as follows: \al{ \theta_g^\top =& \left[
\begin{array}{ccc|c}
(\theta_g^{[11]})^\top & \ldots & (\theta_g^{[1p_2]})^\top & \ldots\\
\end{array}
\right.\nonumber\\
& \hspace{0.2cm} \left.
\begin{array}{c|ccc}
\ldots & (\theta_g^{[p_11]})^\top & \ldots &(\theta_g^{[p_1p_2]})^\top \\
\end{array}
\right]\nonumber} \al{ \theta_f^\top =& \left[
\begin{array}{ccc|c}
(\theta_f^{[11]})^\top & \ldots & (\theta_f^{[1p_2]})^\top & \ldots\\
\end{array}
\right.\nonumber\\
& \hspace{0.2cm} \left.
\begin{array}{c|ccc}
\ldots & (\theta_f^{[p_11]})^\top & \ldots &(\theta_f^{[p_1p_2]})^\top \\
\end{array}
\right]\nonumber} where \al{ \theta_g^{[hk]^\top} =& \left[
\begin{array}{ccc|c}
(g^{[hk,11]})^\top & \ldots & (g^{[hk,1p_2]})^\top & \ldots\\
\end{array}
\right.\nonumber\\
& \hspace{0.2cm} \left.
\begin{array}{c|ccc}
\ldots & (g^{[kh,p_11]})^\top & \ldots &(g^{[hk,p_1p_2]})^\top \\
\end{array}
\right]\nonumber\\
\theta_f^{[hk]^\top} =& \left[
\begin{array}{ccc}
(f^{[hk,1]})^\top & \ldots & (f^{[hk,m]})^\top \\
\end{array}
\right]\nonumber.
} The measured data $y(1)\ldots y(N)$ are stacked in the vector $\mathrm{y}^+$ as follows
\al{ \mathrm{y}^+ =&\left[
\begin{array}{ccc|c|ccc}
\mathrm{y}_{11}^{+\top} & \ldots & \mathrm{y}_{1p_2}^{+\top} & \ldots & \mathrm{y}_{p_11}^{+\top}& \ldots & \mathrm{y}_{p_1p_2}^{+\top} \\
\end{array}
\right]^\top\nonumber}
where
\al{\mathrm{y}^+_{hk} =&\left[
\begin{array}{ccc}
y_{hk}(N)^\top & \ldots & y_{hk}(1)^\top \\
\end{array}
\right].\nonumber
}
The vector $\mathrm{e}^+$ is defined analogously. Let us also introduce the {\em Toeplitz} matrices $\phi_{hk} \in\ell_2^{N\times 1}( \mathbb{N})$, $\psi_{i} \in\ell_2^{N\times 1}( \mathbb{N})$:
\al{ &[\phi_{hk}]_{sn}:=y_{hk}(N-s- n+1)\nonumber\\
&[\psi_{i}]_{sn}:=u_{i}(N-s- n+1)
\nonumber} with $s=1\ldots N$ and $n\in \mathbb{N}$. Then, we define the regression matrices $\Phi\in\ell_2^{p_1p_2N \times p_1^2p_2^2} ( \mathbb{N})$ and $\Psi\in\ell_2^{p_1p_2N \times p_1p_2m} ( \mathbb{N})$ as:
\al{ \Phi&=I_{p_1p_2} \otimes \left[
\begin{array}{ccccccc}
\phi_{11} & \ldots & \phi_{1p_2} & \ldots & \phi_{p_1 1} & \ldots & \phi_{p_1p_2}\\
\end{array}
\right]\nonumber\\
\Psi&=I_{p_1p_2} \otimes \left[
\begin{array}{ccccccc}
\psi_{1} & \ldots & \psi_{m}\\
\end{array}
\right]\nonumber} so that, from (\ref{OEmodel_start}) the vector $\mathrm{y}^+$ containing the measured output data satisfies the linear regression model \al{\label{LM} \mathrm{y}^+= \Phi\theta_g+\Psi\theta_f+\mathrm{e}^+ } and $\hat\mathrm{y}^+:= \Phi \theta_g+\Psi\theta_f$ is the one-step ahead predictor of $\mathrm{y}^+$. It is worth noting that to construct $\Phi$ and $\Psi$ we need the remote past of the output and the input which is not available. Thus, model \eqref{LM} has to be approximated truncating $\Phi$ and $\Psi$ (and thus $\theta_g$ and $\theta_f$). This is equivalent to impose zero initial conditions, which is a reasonable approximation given the decay, as a function of $t$, of the coefficients $G_t$, $F_t$ by the BIBO stability conditions.
\begin{rem} \label{remark_spatio}In the case that we consider the dynamic spatio-temporal model of Section \ref{sec_examples}, the regression matrices in (\ref{LM}) are different because the strictly lower block triangular parts of $G(z)$ and $F(z)$ do not contain a delay. More precisely, $\Phi$ and $\Psi$ are $p_1\times p_1$ block diagonal matrices where the block in position $(h,h)$, with $h\in I_1$, is, respectively, \al{&I_{p_2}\otimes [\,\tilde\phi_{11} \,\ldots\, \tilde\phi_{h-1,p_2}\, \phi_{h,1}\, \ldots \phi_{p_1,p_2} \,]\nonumber\\ &I_{p_2}\otimes [\,\tilde \psi_{1} \,\ldots\, \tilde \psi_{(h-1)\tilde m}\, \psi_{(h-1)\tilde m+1}\, \ldots \psi_{p_1\tilde m} \,]\nonumber} and the {\em Toeplitz} matrices $\tilde \phi_{hk} \in\ell_2^{N\times 1}( \mathbb{N})$, $\tilde \psi_{i} \in\ell_2^{N\times 1}( \mathbb{N})$ are:
\al{ &[\tilde \phi_{hk}]_{sn}:=y_{hk}(N-s- n+2)\nonumber\\
&[\tilde \psi_{i}]_{sn}:=u_{i}(N- s- n+2).
\nonumber} \end{rem}
Therefore, our Kronecker identification problem can be formulated in terms of PEM as follows.
\begin{prob} \label{problem} Assume to collect the measurements $\{y(t),u(t) \}_{t=0\ldots N}$ and that the dimensions $p_1$ and $p_2$ are known. Find $\theta_g^\top\in\ell_1^{1\times p_1^2p_2^2}( \mathbb{N})$ and $\theta_f^\top\in\ell_1^{1\times p_1p_2m}( \mathbb{N})$ corresponding to a Kronecker model minimizing the prediction error squared norm $\| \mathrm{y}^+-\Phi \theta\|^2_{\Sigma^{-1}\otimes I_N}$ with $\Sigma=\mathrm{diag}( \sigma_{11}^2 \ldots\sigma_{1p_2}^2 \ldots\sigma_{p_11}^2 \ldots\sigma_{p_1 p_2})$. \end{prob} Following the nonparametric Gaussian regression approach in \cite{PILLONETTO_2011_PREDICTION_ERROR},
we model $\theta_g$ and $\theta_f$ as zero-mean processes with kernels $K_g\in \mathcal{S}_2^{p_1^2p_2^2}( \mathbb{N})$ and $K_f\in \mathcal{S}_2^{p_1p_2m}( \mathbb{N})$, respectively. These kernels may depend upon some tuning parameters, usually called hyperparameters and denoted with $\zeta$ hereafter. As illustrated in Section \ref{section_kernel}, according to the maximum entropy principle $\theta_g$ and $\theta_f$ will be modeled as Gaussian and independent. In the following $ \mathcal{H}_{K}$ denotes the reproducing Hilbert space \citep{ARONSZAJN1950} of deterministic functions on $ \mathbb{N}$, associated with $K$ and with norm denoted by $\|\cdot \|_{K^{-1}}$. We assume that the past data $\mathrm{y}^-$ neither affects the {\em a priori} probability on $\theta_g$ and $\theta_f$ nor carries information on $\zeta$ and $\Sigma$ \citep{PILLONETTO_DENICOLAO2010}, that is
\al{ \label{approx_pb}\mathbf{p} & (\mathrm{y}^+,\theta_g,\theta_f,\mathrm{y}^-|\zeta,\Sigma)\nonumber\\
& =\mathbf{p}(\mathrm{y}^+|\theta_g,\theta_f,\mathrm{y}^-,\zeta,\Sigma) \mathbf{p}(\theta_g,\theta_f|\mathrm{y}^-,\zeta,\Sigma)\mathbf{p}(\mathrm{y}^-|\zeta,\Sigma)\nonumber\\ & \approx
\mathbf{p}(\mathrm{y}^+|\theta_g,\theta_f,\mathrm{y}^-,\zeta,\Sigma) \mathbf{p}(\theta_g,\theta_f|\zeta,\Sigma)\mathbf{p}(\mathrm{y}^-).}
Let $\hat \theta_g= \mathbb{E}[\theta_g|\mathrm{y}^+,\zeta,\Sigma]$ and $\hat \theta_f= \mathbb{E}[\theta_f|\mathrm{y}^+,\zeta,\Sigma]$ be the minimum variance estimator, respectively, of $\theta_g$ and $\theta_f$ given $\mathrm{y}^+$, $\zeta$ and $\Sigma$. It is well known that $\hat \theta_g$ and $\hat \theta_f$ are almost surely solution to the following {\em Tikhonov}-type variational problem
\al{ \underset{\substack{\theta_g\in \mathcal{H}_{K_g}\\\theta_f\in \mathcal{H}_{K_f}}}{\arg\min}\|\mathrm{y}^+-\Phi \theta_g -\Psi \theta_f\|^2_{\Sigma^{-1}\otimes I_N}
+\| \theta_g\|^2_{K_g^{-1}}+\| \theta_f\|^2_{K_f^{-1}}.\nonumber} Moreover, almost surely: \al{ \begin{aligned}\hat \theta_g &= K_g\Phi^\top (\Phi K_g\Phi^\top+\Psi K_f\Psi^\top+\Sigma \otimes I)^{-1}\mathrm{y}^+\\ \hat \theta_f&= K_f\Psi^\top (\Phi K_g\Phi^\top+\Psi K_f\Psi^\top+\Sigma \otimes I)^{-1}\mathrm{y}^+\end{aligned}.\nonumber} In what follows, $\hat G(z)$ and $\hat F(z)$ denote the transfer matrices corresponding to $\hat \theta_g$ and $\hat \theta_f$, respectively. The main task now is to design the kernels $K_g$ and $K_f$ in such a way that $\hat G(z)$ and $\hat F(z)$ are almost surely BIBO stable while favoring a sparse Kronecker decomposition of $\mathrm{sup}(\hat G(z))$ and $\mathrm{sup}(\hat F(z))$.
\section{Maximum entropy priors} \label{section_kernel}
The probability law for the joint process $\theta=[\,\theta_g^\top\; \theta_f^\top]^\top$ under desired constraints can be obtained by the maximum entropy principle. Indeed, maximum entropy solutions rely on ``information'' arguments which essentially state that the maximum entropy distribution is the one satisfying the given constraints and containing the largest amount of ``uncertainty'' \citep{1456693}. In plain words, this principle guarantees that this solution does not satisfy additional constraints that are undesired and unnecessary. We shall make the rather mild assumption that the process $\theta$ is zero-mean and absolutely continuous with respect to the Lebesgue measure. We will see that the optimal solution (i.e. maximizing the differential entropy) is a Gaussian process where $\theta_g$ and $\theta_f$ are independent. Then, we will also characterize the corresponding kernel functions.
We start with the constraints on $\theta_g$ inducing BIBO stability on $\hat G(z)$. Let $P\in \mathcal{S}_2( \mathbb{N})$ be a strictly positive definite kernel (in the sense of Moore) such that $[P]_{t,t}\leq \kappa t^\alpha e^{-\beta t}$, $t\in \mathbb{N}$, with $\kappa,\beta>0$ and $\alpha\in \mathbb{R}$. Let also $\vartheta$ be a zero-mean process which satisfies the moment constraint
\al{ \mathbb{E}[\| \vartheta\|_{P^{-1}}^2]\leq c\nonumber}
where $c\geq 0$. Then, for any $\varepsilon>1$ there exists $\bar \kappa_\varepsilon>0$ such that the covariance function (kernel) $K$ of $\vartheta$ satisfies \cite[Proposition 4]{BSL}:
\al{[K]_{t,t}\leq \bar \kappa_\varepsilon t^{\alpha+\varepsilon} e^{-\beta t},\;\; t\in \mathbb{N}.\nonumber}
It is not difficult to see that the condition above on $P$ is satisfied by the kernels usually employed in the identification of dynamical models (e.g. stable spline, tuned/correlated and so on, see \cite{KERNEL_METHODS_2014}).
Thus, we consider the constraints \al{ \label{constraint_p_s} \mathbb{E}[\| g^{[hk,jl]}\|^2_{P^{-1} }] \leq \tilde c_{hk,jl}, \; \forall\,h,j\in I_1,\;\forall\, k,l\in I_2} where $c_{hk,jl} \geq 0$ and $P\in \mathcal{S}_2( \mathbb{N})$ as above. Then, by (\ref{constraint_p_s}), the covariance of the $k$-th element of $g^{[hk,jl]}$ decays exponentially. Accordingly, its posterior mean is a BIBO stable transfer function.
\begin{rem} The kernel $P$ induces a prior about the decay rate of the impulse responses $g^{[hk,jl]}$. On the other hand, the hyperparameters of $P$ are estimated from the measured data, see Section 5. Therefore, although $P$ could assign high probability to a small subset of BIBO stable systems, the latter is a ``wise'' set because it has been extrapolated from the data.\end{rem}
It remains to derive the constraints inducing a sparse Kronecker decomposition as in (\ref{cond_supp}), i.e.: \al{\label{cond_sp_1or}&[E_1]_{h,j}=0 \iff g^{[hk,jl]}=0, \; \forall\, k,l\in I_2\\ \label{cond_sp_2or}&[E_2]_{k,l}=0\iff g^{[hk,jl]}=0, \; \forall\, h,j\in I_1.} We consider the constraints:
\al{\label{constr1} \sum_{k,l\in I_2} \mathbb{E}[\| g^{[hk,jl]}\|^2_{P^{-1} }] \leq c_{hj}^\prime, \; \forall\,h,j\in I_1;\\
\label{constr2} \sum_{h,j\in I_1} \mathbb{E}[\| g^{[hk,jl]}\|^2_{P^{-1} }] \leq c_{kl}^{\prime\prime}, \; \forall\,k,l\in I_2 } where $c_{hj}^\prime,c_{kl}^{\prime\prime} \geq 0$. First, notice that (\ref{constr1}) and (\ref{constr2}) imply (\ref{constraint_p_s}) provided that $\max\{c_{hj}^\prime,c_{kl}^{\prime\prime}\}\leq \tilde c_{hk,jl}$ for any $h,k,j,l$. Accordingly, the posterior mean of $G(z)$, under the constraints (\ref{constr1})-(\ref{constr2}), is BIBO stable. Let $c_{hj}^\prime=0$, by (\ref{constr1}) we have that $g^{[hk,jl]}$ $\forall\, k,l\in I_2$ are the null sequence in mean square and
so are their posterior mean, i.e. the latter satisfy (\ref{cond_sp_1or}). Let $c_{kl}^{\prime \prime}=0$, by (\ref{constr2}) we have that $g^{[hk,jl]}$ $\forall\, h,j\in I_1$ are the null sequence in mean square and
so are their posterior mean, i.e. the latter satisfy (\ref{cond_sp_2or}).
In a similar way, condition (\ref{cond_supp}) on $F(z)$ can be written as:
\al{ &[A_1]_{h}=0 \iff f^{[hk,i]}=0, \; \forall\, k\in I_2, \; \forall\, i\in I_u\nonumber\\
&[A_2]_{k,i}=0\iff f^{[hk,i]}=0, \; \forall\, h\in I_1\nonumber.} Thus, the constraints inducing BIBO stability and a sparse Kronecker decomposition are:
\al{\label{constr1f} &\sum_{\substack{k\in I_2 \\i\in I_u}} \mathbb{E}[\| f^{[hk,i]}\|^2_{R^{-1} }] \leq d_{h}^\prime, \; \forall\,h\in I_1;\\
&\label{constr2f} \sum_{h\in I_1} \mathbb{E}[\| f^{[hk,i]}\|^2_{R^{-1} }] \leq d_{ki}^{\prime\prime}, \; \forall\,k\in I_2,\; \forall \, i\in I_u } where $d_{h}^\prime,d_{ki}^{\prime\prime} \geq 0$ and $R\in \mathcal{S}_2( \mathbb{N})$ is a strictly positive kernel such that $[R]_{t,t}\leq \kappa t^\alpha e^{-\beta t}$, $t\in \mathbb{N}$, $\kappa,\beta>0$ and $\alpha\in \mathbb{R}$.
In order to build the desired prior distribution we make use of the Kolmogorov extension Theorem, see \cite{Oksendal}, and work with finite vectors extracted from process $\theta=[\, \theta_g^\top\;\theta_f^\top\,]^\top$. More precisely, consider a finite index set $ \mathcal{I}= \mathcal{I}_g\times \mathcal{I}_f$ and $ \mathcal{I}_g, \mathcal{I}_f\subset \mathbb{N}$. Let $\check \theta$ be the random vector whose components are extracted, from processes $\theta_g$ and $\theta_f$ according to the index sets $ \mathcal{I}_g$ and $ \mathcal{I}_f$, respectively. We denote by $\mathbf{p}_{ \mathcal{I}}(\check \theta)$ the probability density of $\check \theta$. In view of the Kolmogorov extension Theorem, process $\theta$ can be characterized by specifying the joint probability density $\mathbf{p}_{ \mathcal{I}}$ for all finite sets $ \mathcal{I} \subset \mathbb{N}\times \mathbb{N}$. Thus, the maximum entropy process $\theta$ can be constructed building all the marginals $\mathbf{p}_{ \mathcal{I}}$ using the maximum entropy principle, which can thus be extended by the Kolmogorov extension theorem.
Such principle states that among all the probability densities $\mathbf{p}_{ \mathcal{I}}$ satisfying the desired constraints, the optimal one should maximize the differential entropy \citep{COVER_THOMAS}. Let $ P_{ \mathcal{I}_g}$ and $R_{ \mathcal{I}_f}$ be the kernel matrices whose entries are extracted from $P$ and $R$ according to $ \mathcal{I}_g$ and $ \mathcal{I}_f$, respectively. Therefore, the corresponding maximum entropy problem is
\al{ \label{ME_problem}\underset{\mathbf{p}_{ \mathcal{I}}\in \mathcal{P}}{\max}& \mathbf{H} (\mathbf{p}_{ \mathcal{I}})\nonumber\\ \hbox{s.t. } & \text{ (\ref{constr1}) (\ref{constr2}) (\ref{constr1f}) (\ref{constr2f}) hold for $ \mathcal{I}$}}
where $ \mathcal{P}$ is the class of probability densities in $ \mathbb{R}^{| \mathcal{I}|}$ which are bounded and taking positive values and $\mathbf{H} (\mathbf{p}_{ \mathcal{I}})$ denotes the differential entropy of $\check \theta$. Moreover, the constraints in (\ref{ME_problem}) must be understood as follows: (\ref{constr1}), (\ref{constr2}), (\ref{constr1f}) and (\ref{constr2f}) hold with $\theta$, $ P$ and $R$ replaced by $\check \theta$, $ P_{ \mathcal{I}_g}$ and $R_{ \mathcal{I}_f}$, respectively.
\begin{thm}\label{teo_ME} Under the assumption that $c_{hj}^\prime,c_{kl}^{\prime\prime},d_{h}^\prime,d_{ki}^{\prime\prime}>0$, $h,j\in I_1$, $k,l\in I_2$ and $i\in I _u$, the unique optimal solution to the maximum entropy problem (\ref{ME_problem}) is such that $\check \theta=[\, \check \theta_g^\top \; \check \theta_f^\top\,]^\top$ is Gaussian with zero mean. Moreover, $\check \theta_g$ and $\check \theta_f$ are independent and with kernel matrix, respectively, \al{\label{optimal_K} \begin{aligned} \check K_g&= X_g\otimes P_{ \mathcal{I}_g}\\ \check K_f&= X_f\otimes R_{ \mathcal{I}_f}\\ X_g&=(\Lambda\otimes \Gamma)(\Lambda\otimes I_{p_2^2}+I_{p_1^2}\otimes \Gamma)^{-1}\\ X_f&=(\Pi\otimes \Omega)(\Pi\otimes I_{p_2m}+I_{p_1}\otimes \Omega)^{-1}\\ \Lambda &=\mathrm{diag}(\lambda_{11}\ldots \lambda_{1p_1} \ldots \lambda_{p_11} \ldots \lambda_{p_1 p_1})\\ \Gamma &=\mathrm{diag}(\gamma_{11}\ldots \gamma_{1p_2} \ldots \gamma_{p_21} \ldots \gamma_{p_2 p_2})\\ \Pi &=\mathrm{diag}(\pi_{1}\ldots \pi_{p_1} )\\ \Omega &=\mathrm{diag}(\omega_{11}\ldots \omega_{1m} \ldots \omega_{p_21} \ldots \omega_{p_2 m}) \end{aligned} }
where $\lambda_{hj},\gamma_{kl}, \pi_h,\omega_{ki}>0$.
\end{thm} \begin{proof} We prove the claim by using the duality theory. The Lagrangian is \al{\label{Lagr}L(\mathbf{p}_{ \mathcal{I}},&\tilde \Lambda,\tilde \Gamma,\tilde \Pi,\tilde\Omega)=\mathbf{H}(\mathbf{p}_{ \mathcal{I}})\nonumber\\ &+\frac{1}{2}\sum_{h,j\in I_1}\tilde \lambda_{hj}\left(c_{hj}^\prime-
\sum_{k,l\in I_2} \mathbb{E}[\| \check g^{[hk,jl]}\|^2_{P_{ \mathcal{I}_g}^{-1} }] \right)\nonumber\\ &+\frac{1}{2}\sum_{k,l\in I_2}\tilde \gamma_{kl}\left( c_{kl}^{\prime\prime}-\sum_{h,j\in I_1} \mathbb{E}[\| \check g^{[hk,jl]}\|^2_{P_{ \mathcal{I}_g}^{-1} }] \right)\nonumber\\ &+\frac{1}{2}\sum_{h\in I_1}\tilde \pi_{h}\left(d_{h}^\prime-
\sum_{\substack{k\in I_2\\ i \in I_u}} \mathbb{E}[\| \check f^{[hk,i]}\|^2_{R_{ \mathcal{I}_f}^{-1} }] \right)\nonumber\\
&+\frac{1}{2}\sum_{\substack{k\in I_2\\ i \in I_u}}\tilde \omega_{ki}\left( d_{ki}^{\prime\prime}-\sum_{h\in I_1} \mathbb{E}[\| \check f^{[hk,i]}\|^2_{R_{ \mathcal{I}_f}^{-1} }] \right)\nonumber\\ &= - \mathbb{E}[\log( \mathbf{p}_{ \mathcal{I}})]+\frac{1}{2}\sum_{h,j\in I_1}\tilde \lambda_{hj} c_{hj}^\prime+\frac{1}{2}\sum_{k,l\in I_2} \tilde \gamma_{kl} c_{kl}^ {\prime\prime}\nonumber\\
&-\frac{1}{2}\sum_{\substack{h,j\in I_1\\k,l\in I_2}}(\tilde \lambda_{hj}+\tilde\gamma_{kl} ) \mathbb{E}[\| \check g^{[hk,jl]}\|^2_{P_{ \mathcal{I}_g}^{-1} }]\nonumber\\ &+\frac{1}{2}\sum_{h\in I_1}\tilde \pi_{h} d_{h}^\prime+\frac{1}{2}\sum_{\substack{k\in I_2\\ i\in I_u}} \tilde \omega_{ki} d_{ki}^{\prime\prime}\nonumber\\
&-\frac{1}{2}\sum_{\substack{h\in I_1,k\in I_2\\ i\in I_u}}(\tilde \pi_{h}+\tilde\omega_{ki} ) \mathbb{E}[\|\check f^{[hk,i]}\|^2_{R_{ \mathcal{I}_f}^{-1} }]\nonumber\\ &= - \mathbb{E}[\log( \mathbf{p}_{ \mathcal{I}})]+\frac{1}{2}\sum_{h,j\in I_1}\tilde \lambda_{hj} c_{hj}^\prime+\frac{1}{2}\sum_{k,l\in I_2} \tilde \gamma_{kl} c_{kl}^{\prime\prime}\nonumber\\
&+\frac{1}{2}\sum_{h\in I_1}\tilde \pi_{h} d_{h}^\prime -\frac{1}{2} \mathbb{E}[\|\check \theta_g\|^2_{(\tilde \Lambda \otimes I+ I\otimes \tilde \Gamma )\otimes P_{ \mathcal{I}_g}^{-1} }]\nonumber\\
&+\frac{1}{2}\sum_{\substack{k\in I_2\\ i\in I_u}} \tilde \omega_{ki} d_{ki}^{\prime\prime}-\frac{1}{2} \mathbb{E}[\|\check \theta_f\|^2_{(\tilde \Pi \otimes I+ I\otimes \tilde \Omega )\otimes R_{ \mathcal{I}_f}^{-1} }] } where $\tilde \Lambda=\mathrm{diag}(\tilde \lambda_{11} \ldots \tilde \lambda_{1p_1} \ldots \tilde \lambda_{p_11 } \ldots \tilde \lambda_{p_1 p_1} )$, $\tilde \Gamma=\mathrm{diag}(\tilde \gamma_{11} \ldots\tilde \gamma_{1p_2} \ldots\tilde \gamma_{p_21} \ldots\tilde \gamma_{p_2 p_2} )$, $\tilde \Pi=\mathrm{diag}(\tilde \pi_{1} \ldots \ldots\tilde \pi_{p_1} )$, $\tilde \Omega=\mathrm{diag}(\tilde \gamma_{11} \ldots\tilde \gamma_{1m} \ldots\tilde \gamma_{p_21} \ldots\tilde \gamma_{p_2 m} )$; moreover, $\tilde \lambda_{h,j},\tilde\gamma_{kl}, \tilde \pi_h, \tilde \omega_{hi}\geq 0$ are the Lagrange multipliers corresponding to (\ref{constr1}), (\ref{constr2}), (\ref{constr1f}) and (\ref{constr2f}), respectively. It is not difficult to see that (\ref{Lagr}) is strictly concave in $ \mathcal{P}$. Moreover, its point of maximum exists under the assumptions that $\tilde \lambda_{hj}+\tilde \gamma_{kl}>0$ and $\tilde \pi_{h}+\tilde \omega_{ki}>0$, and it is Gaussian distributed: \al{\label{opt_pI}\mathbf{p}_{ \mathcal{I}}(\check\theta)= \mathbf{p}_{ \mathcal{I}_g}(\check\theta_g)\mathbf{p}_{ \mathcal{I}_f}(\check\theta_f)} where \al{
\mathbf{p}_{ \mathcal{I}_f}(\check \theta_g)&=\frac{1}{\sqrt{(2\pi)^{p_1^2p_2^2| \mathcal{I}_g|} |\check K_g|}}\exp(-\frac{1}{2}\check\theta_g^\top \check K_g^{-1}\check\theta_g)\nonumber\\
\mathbf{p}_{ \mathcal{I}_f}(\check\theta_f)&=\frac{1}{\sqrt{(2\pi)^{p_1p_2m | \mathcal{I}_f|} |\check K_f|}}\exp(-\frac{1}{2}\check\theta_f^\top \check K_f^{-1}\check\theta_f)\nonumber} and \al{\check K_g&=(\tilde \Lambda \otimes I+ I\otimes \tilde \Gamma )^{-1}\otimes P_{ \mathcal{I}_g}\nonumber\\ \check K_f&=(\tilde \Pi\otimes I+ I\otimes \tilde \Omega )^{-1}\otimes R_{ \mathcal{I}_f}.\nonumber} In the case that $\tilde \lambda_{hj}+\tilde \gamma_{kl}=0$ or $\tilde \pi_{h}+\tilde \omega_{ki}=0$ for some $h,k,j,l,i$, then the maximum cannot be attained in $ \mathcal{P}$. Substituting the optimal form of $\mathbf{p}_{ \mathcal{I}}$ in the Lagrangian we obtain the dual function (up to terms not depending on $\tilde \Lambda$, $\tilde \Gamma$, $\tilde \Pi$ and $\tilde \Omega$): \al{J(\tilde \Lambda,\tilde \Gamma, \tilde\Pi, \tilde\Omega )=J_f(\tilde \Lambda,\tilde \Gamma)+J_g(\tilde \Pi,\tilde \Omega)\nonumber} where
\al{J_f(\tilde\Lambda,\tilde \Gamma)=-&\frac{| \mathcal{I}_g|}{2}\sum_{\substack{h,j\in I_1\\ k,l\in I_2}} \log(\tilde \lambda_{hj}+\tilde \gamma_{kl})\nonumber\\ &\hspace{0.5cm}+\frac{1}{2}\sum_{h,j\in I_1}\tilde \lambda_{hj} c_{hj}^\prime+\frac{1}{2}\sum_{k,l\in I_2} \tilde \gamma_{kl} c_{kl}^{\prime\prime}. \nonumber\\
J_g(\tilde\Pi,\tilde \Omega)=-&\frac{| \mathcal{I}_f|}{2}\sum_{\substack{h\in I_1, \, k\in I_2\\ i\in I_u}} \log(\tilde \pi_{h}+\tilde \omega_{ki})\nonumber\\ &\hspace{0.5cm}+\frac{1}{2}\sum_{h\in I_1}\tilde \pi_{hj} d_{h}^\prime+\frac{1}{2}\sum_{\substack{k\in I_1\\ i\in I_u}} \tilde \omega_{ki} d_{ki}^{\prime\prime}. \nonumber} Hence, we can minimize $(\tilde \Lambda,\tilde\Gamma)$ and $(\tilde \Pi, \tilde \Omega)$ in an independent way. We start to minimize $J_f(\tilde \Lambda,\tilde \Gamma)$ over the open and unbounded set \al{ \mathcal{C}=\{\, (\tilde \Lambda,\tilde \Gamma)\in \mathcal{D} \hbox{ s.t. } \tilde \Lambda\geq0,\, \tilde \Gamma \geq0,\, \tilde \Lambda \otimes I+I\otimes\tilde \Gamma>0 \, \}\nonumber} where $ \mathcal{D}= \mathcal{D}_{p_1^2}\times \mathcal{D}_{p_2^2}$.
Next, we show that we can restrict the search of the minimum of $J_f$ over a compact set and thus by the Weierstrass' theorem, the dual problem admits solution. Take a sequence $(\tilde\Lambda^{(n)}, \tilde \Gamma^{(n)})\in \mathcal{C}$, $n\in \mathbb{N}$, such that $\|\tilde \Lambda^{(n)}\|\rightarrow \infty$ and/or $\|\tilde \Gamma^{(n)}\|\rightarrow \infty$ as $n\rightarrow \infty$. Then, $J_f(\tilde \Lambda^{(n)},\tilde \Gamma^{(n)})\rightarrow \infty$ because the linear terms dominate the logarithmic ones. Therefore, such a sequence cannot be an infimizing sequence. Accordingly, we can restrict the set $ \mathcal{C}$ as \al{ \mathcal{C}_1=\{\, (\tilde \Lambda&,\tilde \Gamma)\in \mathcal{D} \hbox{ s.t. } 0\leq \tilde\Lambda\leq \lambda_{MAX}I,\nonumber\\ & 0\leq\Gamma \leq\gamma_{MAX}I,\, \tilde \Lambda\otimes I+I\otimes \tilde \Gamma>0 \, \}\nonumber} where $\lambda_{MAX},\gamma_{MAX}>0$ are constants taken sufficiently large. Take a sequence $(\tilde\Lambda^{(n)}, \tilde \Gamma^{(n)})\in \mathcal{C}_1$, $n\in \mathbb{N}$, such that $\lambda^{(n)}_{hj}+\tilde \gamma_{kl}^{(n)}\rightarrow 0$ as $n\rightarrow \infty$ for some $h,j\in I_1$ and $k,l\in I_2$. Then, we have
\al{J_f(\tilde \Lambda^{(n)},\tilde\Gamma^{(n)})\geq -\frac{| \mathcal{I}_g|}{2}\sum_{\substack{h,j\in I_1\\ k,l\in I_2}} \log(\tilde \lambda_{hj}+\tilde \gamma_{kl})\rightarrow \infty.\nonumber} Accordingly, it cannot be an infimizing sequence, and thus we can restrict the search to the set \al{ \mathcal{C}_2=\{\, (\tilde \Lambda&,\tilde \Gamma)\in \mathcal{D} \hbox{ s.t. } 0\leq \tilde\Lambda\leq \lambda_{MAX}I,\nonumber\\ & 0\leq\Gamma \leq\gamma_{MAX}I,\, \tilde \Lambda\otimes I+I\otimes \tilde \Gamma\geq \varepsilon I \, \}\nonumber}
where $\epsilon$ is a constant taken sufficiently small. Thus, $ \mathcal{C}_2$ is closed and bounded and thus compact. We conclude that $J_f$ admits minimum over $ \mathcal{C}$. In a similar way, it is possible to prove that $J_g$ admits minimum over the set
\al{ \mathcal{C}=\{\, (\tilde \Pi,\tilde \Omega)\in \tilde{ \mathcal{D}} \hbox{ s.t. } \tilde \Pi\geq0,\, \tilde \Omega \geq0,\, \tilde \Pi\otimes I+I\otimes \tilde \Omega>0 \, \}\nonumber} with $\tilde{ \mathcal{D}}= \mathcal{D}_{p_1}\times \mathcal{D}_{p_2m}$. We conclude that the dual problem admits solution. Accordingly, (\ref{opt_pI}) is the solution to the maximum entropy problem (\ref{ME_problem}). Finally, by defining $\Lambda=\tilde \Lambda^{-1}$, $\Gamma=\tilde \Gamma^{-1}$, $\Pi=\tilde \Pi^{-1}$ and $\Omega=\tilde \Omega^{-1}$ we obtain (\ref{optimal_K}).
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\end{proof}
We assumed that constraints (\ref{constr1}), (\ref{constr2}), (\ref{constr1f}) and (\ref{constr2f}) are totally binding in problem (\ref{ME_problem}). However, we are interested in the limiting cases where $c_{hj}^\prime$, $c_{kl}^{\prime\prime}$, $d_{h}^\prime$ and $d_{ki}^{\prime\prime}$ might be equal to zero in order to obtain a posterior corresponding to a Kronecker model. To include these scenarios, we consider the limits as $c_{hj}^\prime \rightarrow 0 $, $c_{kl}^{\prime\prime}\rightarrow 0$, $d_{h}^\prime \rightarrow 0 $, $d_{ki}^{\prime\prime}\rightarrow 0$ and extend the maximum entropy solution by continuity.
\begin{prop} \label{extended_ME} Let $ \mathcal{E}_1:=\{(h,j) \hbox{ s.t. } c_{hj}^\prime>0\}$, $ \mathcal{E}_2:=\{(k,l) \hbox{ s.t. } c_{kl}^{\prime\prime}>0\}$. If the sets $ \mathcal{E}_1, \mathcal{E}_2$ are nonempty, then the maximum entropy solution extended by continuity is the probability density such that
$\check \theta_g$ is Gaussian, zero-mean and with kernel matrix as in (\ref{optimal_K}) where:
\begin{itemize}
\item $(i,j)\notin \mathcal{E}_1$ $\implies$ $\lambda_{hj}=0$,
\item $(k,l)\notin \mathcal{E}_2$ $\implies$ $\gamma_{kl}=0$.
\end{itemize} \end{prop} \begin{proof} First, recall that $\lambda_{hj}=\tilde \lambda_{hj}^{-1}$, $\gamma_{kl}=\tilde \gamma_{kl}^{-1}$ and $\tilde \lambda_{hj}$, $\tilde \gamma_{kl}$ are the original Lagrange multipliers. The maximum entropy solution $\check \theta_{ME}$ is such that, by Theorem \ref{teo_ME}, $\check g_{ME}^{[hk,jl]}$ is a zero-mean process with kernel matrix \al{\frac{1}{\tilde \lambda_{hj}+\tilde\gamma_{kl}}P_{ \mathcal{I}_g}.\nonumber} Moreover, if $\tilde \gamma_{kl}>0$ ($\lambda_{hj}>0$) then the maximum entropy solution $\check \theta_{ME}$ satisfies the corresponding constraint with equality. We consider two sequences $\{c_{hj}^{\prime(n)}\}_{n\in \mathbb{N}}$, $c_{hj}^{\prime (n)}>0$, and $\{c_{kl}^{\prime\prime(n)}\}_{n\in \mathbb{N}}$, $c_{kl}^{\prime\prime (n)}>0$, such that $c_{hj}^{\prime(n)}\rightarrow c_{hj}^{\prime}$ and $c_{kl}^{\prime\prime(n)}\rightarrow c_{kl}^{\prime\prime}$ as $n\rightarrow\infty$. For any $c_{hj}^{\prime(n)}$, we have
\al{\label{ineq1}c_{hj}^{\prime(n)}&\geq \sum_{k,l\in I_2} \mathbb{E}[\|\check g_{ME}^{[hk,jl]}\|^2_{P_{ \mathcal{I}_g}^{-1}}]\nonumber\\ &=\sum_{k,l\in I_2}\mathop{\rm tr}\left( \mathbb{E}[\check g_{ME}^{[hk,jl]}\check g_{ME}^{[hk,jl]\top}]P^{-1}_{ \mathcal{I}_g}\right)\nonumber\\
&= | \mathcal{I}_g| \sum_{k,l\in I_2}\frac{1}{\tilde \lambda_{hj}^{(n)}+\tilde \gamma_{kl}^{(n)}}>0.}In a similar way, we have
\al{\label{ineq2}c_{kl}^{\prime\prime(n)}&\geq | \mathcal{I}_g| \sum_{h,j\in I_1}\frac{1}{\tilde \lambda_{hj}^{(n)}+\tilde \gamma_{kl}^{(n)}}>0.} Let $(h,j)\notin \mathcal{E}_1$, then $c^{\prime(n)}_{hj}\rightarrow 0$. Then, we consider the arbitrary subsequences $\tilde \lambda^{(n_r)}_{hj}$ and $\tilde \gamma^{(n_r)}_{kl}$, with $r\in \mathbb{N}$, which admit limit. Taking into account (\ref{ineq1}), we may have two possible cases. {\em First case}. $\tilde\gamma_{kl}^{(n_r)}$ converges for some $k,l\in I_2$. By (\ref{ineq1}), this implies that $\tilde \lambda_{hj}^{(n_r)}\rightarrow \infty$ and thus $\lambda_{hj}^{(n_r)}\rightarrow 0$. {\em Second case}. $\tilde \gamma_{kl}^{(n_r)}\rightarrow \infty$ for any $k,l\in I_2$. This means that there exists $\bar r\in \mathbb{N}$ sufficiently large for which $\tilde \gamma_{kl}^{(n_r)}>0$ for any $k,l\in I_2$ and $r\geq \bar r$. Accordingly, the corresponding constraints in (\ref{ineq2}) are satisfied with equality for $r\geq \bar r$ and $n=n_r$. Hence $c_{kl}^{\prime \prime (n_r)}\rightarrow 0$ for any $k,l\in I_2$ and thus $ \mathcal{E}_2=\emptyset$. This is not possible because $ \mathcal{E}_2\neq \emptyset$ by assumption. We conclude that $c_{hj}^{\prime(n_r)}\rightarrow 0$ implies $\lambda_{hj}^{(n_r)}\rightarrow 0$ for any arbitrary subsequence which admits limit and thus $\lambda_{hj}^{(n)}\rightarrow 0$. Accordingly, in a similar way, it is possible to prove that $c_{kl}^{\prime\prime(n)}\rightarrow 0$ implies $\gamma_{kl}^{(n)}\rightarrow 0$.
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\end{proof}
\begin{prop} \label{extended_ME2} Let $ \mathcal{A}_1:=\{(h) \hbox{ s.t. } d_{h}^\prime>0\}$, $ \mathcal{A}_2:=\{(k,i) \hbox{ s.t. } d_{ki}^{\prime\prime}>0\}$. If the sets $ \mathcal{A}_1, \mathcal{A}_2$ are nonempty, then the maximum entropy solution extended by continuity is the probability density such that $\check \theta_f$ is Gaussian, zero-mean and with kernel matrix as in (\ref{optimal_K}) where \begin{itemize} \item $h\notin \mathcal{A}_1$ $\implies$ $\pi_{h}=0$, \item $(k,l)\notin \mathcal{A}_2$ $\implies$ $\omega_{ki}=0$. \end{itemize} \end{prop}
\begin{proof} The proof is similar to one of Proposition \ref{extended_ME}.
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\end{proof}
Finally, in view of the Kolmogorov extension Theorem, from the probability density of $\check \theta$ we can characterize the probability law of $\theta$ maximizing the differential entropy.
\begin{cor} Consider the zero-mean Gaussian process $\theta=[\, \theta_g^\top\;\,\theta_f^\top]^\top$ with kernel, respectively, \al{ \label{ME_kernel} \begin{aligned}
K_g&= X_g\otimes P\\
K_f&= X_f\otimes R\\
\end{aligned} } where $X_g$ and $X_f$ are defined as in (\ref{optimal_K}). For all finite sets $ \mathcal{I}\subset \mathbb{N}\times \mathbb{N}$, its joint probability density is the extended solution to the maximum entropy problem (\ref{ME_problem}).\end{cor}
It is worth noting that the maximum entropy solution is such that $g^{[hk,jl]}$ and $f^{[hk,i]}$ are zero-mean Gaussian processes with kernel function, respectively, \al{\label{structK}K_g^{[hk,jl]}:=\frac{\lambda_{hj}\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}P, \; \; K_f^{[hk,i]}:=\frac{\pi_{h}\omega_{ki}}{\pi_{h}+\omega_{ki}}R.} In the case that both $\lambda_{hj}$ and $\gamma_{kl}$
are equal to zero, we define by continuity $\lambda_{hj}\gamma_{kl}/(\lambda_{hj}+\gamma_{kl})=0$. Accordingly, $K_g^{[hk,jl]}=0$ if $\lambda_{hj}=0$ or $\gamma_{kl}=0$. Similarly, in the case that $\pi_{h}$ or $\omega_{ki}$
is equal to zero, we define $\pi_{h}\omega_{ki}/(\pi_{h}+\omega_{ki})=0$.
In \cite{CHIUSO_PILLONETTO_SPARSE_2012} the transfer matrices in model (\ref{OEmodel_start}) are modeled as zero mean Gaussian processes with kernels \al{\label{structPillo}K_g^{[hk,jl]}:=\lambda_{hjkl}P, \; \; K_f^{[hk,i]}:=\pi_{hki} R} where $\lambda_{hjkl},\pi_{hki}\geq 0$, with $h,j\in I_1$, $k,l\in I_2$ and $i\in I_u$, are the hyperparameters representing the so called scale factors. The prior in (\ref{structPillo}) has been used to learn sparse networks. Finally, it is worth noting that there are many more hyperparameters in (\ref{structPillo}) compared with (\ref{structK}). \begin{rem} It is worth noting that the constants $c_{hj}^\prime$'s, $c_{kl}^{\prime\prime}$'s, $d_{h}^\prime$'s and $c_{ki}^{\prime\prime}$'s were needed only to derive the kernel structure in (\ref{ME_kernel}). Indeed, we are not interested in their particular values because, as we will show in the next section, the hyperparameters of the kernel will be found through another paradigm which exploits the measured data. \end{rem}
{\em Hierarchical dynamic networks}. An interesting scenario included in this framework is the case of a system without input (i.e. $y(t)$ is a stationary stochastic process which is characterized by the transfer matrix $G(z)$) with $p_1=p_2$ and $E_1=E_2$. It is then natural to take \al{K_g=(\Lambda\otimes \Lambda)(\Lambda\otimes I_{p_1^2}+I_{p_1^2}\otimes \Lambda)^{-1}\otimes P\nonumber} and thus $g^{[hk,jl]}$ is modeled is a Gaussian process with kernel function \al{\label{kernel_hier}K_g^{[hk,jl]}:=\frac{\lambda_{hj}\lambda_{kl}}{\lambda_{hj}+\lambda_{kl}}P.} This model corresponds to a network which is hierarchically organized into communities (clusters), the communities then grow, creating miniature copies of themselves. This idea can be applied recursively at the price that the dimension of $G(z)$ increases exponentially. Thus, $\Lambda$ is the Kronecker initiator for the prior on $G(z)$. These models have been proposed in \cite{leskovec2010kronecker}: the main difference with respect to our approach is the fact that we do not impose the Kronecker decomposition on $G(z)$, but on $\mathrm{supp}(G(z))$; moreover $y(t)$ is not white noise (i.e. we consider a process with dynamics).
\section{Hyperparameters Estimation}\label{sec:opt_procedure}
In order to compute $\hat \theta_g$ and $\hat \theta_f$, we have to estimate $\Sigma$ and $\zeta$. The latter contains the hyperparameters of $P, R$ and $\xi:=\{\Lambda, \Gamma, \Pi,\Omega\}$. The noise covariance matrix $\Sigma$ can be estimated using a low-bias ARX-model as suggested in \citep{GOODWIN_1992}. To estimate the hyperparameters of $P$ and $R$ we consider a nonparametric unstructured model, i.e. we consider model (\ref{OEmodel_start}) but we do not require that $G(z)$ and $F(z)$ satisfy condition (\ref{cond_supp}). Then, the hyperparameters of $P$ can be estimated by minimizing the negative log-marginal likelihood of $\mathrm{y}^+$ computed using the unstructured model, see \cite{PILLONETTO_2011_PREDICTION_ERROR}. Throughout the paper, kernel $P$ is chosen as Stable Spline (SS) with a modulation factor \citep{ZORZI2018125}:{\small\al{ &[P]_{t,s}=\left[\frac{e^{-\beta(t+s)}e^{-\beta\max(t,s)}}{2}-\frac{e^{-3\beta\max(t,s)}}{6}\right]\cos(\omega_0 (t-s))
\nonumber}}where $\beta\in(0,1)$ and $\omega_0\in (0,\pi)$ tune, respectively, the a priori information about the decay rate and the frequency content of the impulse responses. Kernel $R$ is chosen likewise. As explained in \cite{PILLONETTO_2011_PREDICTION_ERROR}, kernels of this type represent a good prior in the case that the impulse responses of the predictor have ``fast dynamics''.
Then, an estimate of $\xi$ is given by minimizing the negative log-marginal likelihood $\ell$ of $\mathrm{y}^+$ under model (\ref{OEmodel_start}), (\ref{ME_kernel}) with $\Sigma$, $P$ and $R$ fixed as above. Under the assumptions in (\ref{approx_pb}), we have (up to constant terms)
\al{\label{loglik}
\ell (\mathrm{y}^+,\xi)&=\log \det V+\frac{1}{2}(\mathrm{y}^+) ^\top V^{-1}\mathrm{y}^+\nonumber\\
V&= \Phi K_g\Phi^\top+\Psi K_f\Psi^\top+\Sigma \otimes I } where $K_g$ and $K_f$ have been defined in \eqref{ME_kernel} and $\xi$ belongs to the compact set \al{\label{def_XI}\Xi=\{(\Lambda,\Gamma,\Pi,\Omega)\in \mathcal{D} \hbox{ s.t. } 0\leq \Lambda,\Gamma,\Pi,\Omega\leq \kappa I\}} where $ \mathcal{D}= \mathcal{D}_{p_1^2} \times \mathcal{D}_{p_2^2} \times \mathcal{D}_{p_1} \times \mathcal{D}_{p_2 m}$ and $\kappa >0$ is a constant taken sufficiently large. Notice that the choice of $\Xi$ is not restrictive. Indeed, if we take: $\Gamma\geq 0$ such that $\gamma_{kl}>0$ for some $k,l\in I_2$; $\Lambda \geq 0$ such that $\lambda_{hj}\rightarrow \infty$ for some $h,j\in I_1$. Then, $\lambda_{hj}\gamma_{kl}/(\lambda_{hj}+\gamma_{kl})\rightarrow \infty$ and thus $\ell(\mathrm{y}^+,\xi)\rightarrow \infty$. This means that the minimum cannot be attained for $\lambda_{hj}>\kappa$. In the case that $\Gamma=0$, then we have $\ell(\mathrm{y}^+,\{\Lambda,\Gamma,\Pi,\Omega\})=\ell(\mathrm{y}^+,\{\Lambda+\tilde \Lambda,\Gamma,\Pi,\Omega\})$ for any $\tilde\Lambda\in \mathcal{D}_{p_1^2}$ such that $\Lambda+\tilde \Lambda\geq 0$; accordingly, we can restrict the values of $\Lambda$ as in (\ref{def_XI}). A similar reasoning holds for $\Gamma$, $\Pi$ and $\Omega$. Since $\ell$ is a continuous function over the compact set $\Xi$, it follows that $\ell$ admits a point of minimum over $\Xi$.
The minimization of (\ref{loglik}) with respect to $\xi$ in the numerical examples presented in the next section is performed through the nonlinear optimization Matlab solver \textbf{fmincon.m} which uses a gradient based method and the gradient, i.e. $\partial\; \ell (\mathrm y^+,\xi)\slash \partial \xi$, is supplied. The computation of the gradient can be done in a similar way to that explained in \cite[Section 5]{CHEN20132213}. In the case one is interested to make more efficient the algorithm, it is possible to perform the minimization by means of the scaled gradient projection method proposed in \cite{BONETTINI_2014}. Finally, it is worth noting that only a local minimum can be found because $\ell(\mathrm y^+,\xi)$ is not convex with respect to $\xi$.
Next, we prove that the posterior mean of $\theta_g$ and $\theta_f$ under the prior defined through kernels in (\ref{ME_kernel}) leads to a sparse Kronecker network. The analysis is performed by drawing inspiration from \cite{aravkin2014convex} which deals with the case of a sparse Bayesian network \citep{CHIUSO_PILLONETTO_SPARSE_2012}, i.e. the prior is defined through kernels in (\ref{structPillo}), and indeed our prior is closely related to Auto Relevance Determination (ARD) method \citep{mackay1994bayesian}. First, notice that the estimates $\hat \theta_g$ and $\hat \theta_f$ describe a sparse Kronecker network only if $\hat \xi=\{\hat \Lambda,\hat \Gamma, \hat \Pi,\hat \Omega\}$ is such that $\hat \Lambda,\Gamma, \hat \Pi,\hat \Omega$ are spare matrices. We assume that the noise variance at each node is the same, i.e. $\sigma_{hk}^2=\sigma^2$ with $h\in I_1$ and $k\in I_2$, and that \al{\label{cond_regressor}&(I\otimes P^{1/2})\Phi^\top\Phi(I\otimes P^{1/2}) =NI,\nonumber\\ &(I\otimes P^{1/2})\Phi^\top\Psi (I\otimes R^{1/2})=0,\nonumber\\ &(I\otimes R^{1/2})\Psi^\top\Psi (I\otimes R^{1/2})=NI.} Moreover $\sigma^2$, $P$ are $R$ are fixed. First, we consider the case that $g^{[hk,jl]}$'s and $f^{[hk,i]}$'s are finite dimensional vectors, say $T$ their dimension. Accordingly, the corresponding transfer matrices $G(z)$ and $F(z)$ are finite impulse responses (FIR) and $T$ represents their practical length. Moreover, we assume that $N\geq T$. We define \al{\hat \theta_{g,LS}&=(\Phi^\top\Phi)^{-1}\Phi^\top \mathrm{y}^+=\frac{1}{N}(I\otimes P)\Phi^\top \mathrm{y}^+,\nonumber\\ \;\hat \theta_{f,LS}&=(\Psi^\top\Psi)^{-1}\Psi^\top \mathrm{y}^+=\frac{1}{N}(I\otimes R)\Psi^\top \mathrm{y}^+\nonumber} which are the least squares estimators of $\theta_g$ and $\theta_f$. Thus, we have \al{ \label{LS_sparse}\hat g^{[hk,jl]}_{LS}=\frac{1}{N}P\phi^\top_{jl}\mathrm{y}^+_{hk}, \; \; \hat f^{[hk,i]}_{LS}=\frac{1}{N}R\psi^\top_{i}\mathrm{y}^+_{hk}.}
\begin{prop} \label{prop_ARD} Let $\lambda_{hj}$, $\gamma_{k,l}$, $\pi_h$ and $\omega_{ki}$ be the entries in the main diagonal of $\Lambda$, $ \Gamma$, $\Pi$, $ \Omega$, respectively. The following facts hold: \begin{itemize}
\item if \al{\label{cond_ARD_1}\frac{T}{N}\sigma^2 -\|\hat g_{LS}^{[hk,jl]}\|^2_{ P^{-1}}\geq 0, \; \; \forall \, k,l\in I_2,} then the subset $$\Xi_1=\{(\Lambda,\Gamma,\Pi,\Omega) \hbox{ s.t. }\Lambda,\Gamma,\Pi,\Omega\geq 0, \; \lambda_{hj}=0\}$$ contains a local minimum for $\ell$;
\item if \al{\frac{T}{N}\sigma^2 -\|\hat g_{LS}^{[hk,jl]}\|^2_{ P^{-1}}\geq 0, \; \; \forall \, h,j\in I_1,\nonumber}then the subset $$\Xi_2=\{(\Lambda,\Gamma,\Pi,\Omega) \hbox{ s.t. }\Lambda,\Gamma,\Pi,\Omega\geq 0, \; \gamma_{kl}=0\}$$ contains a local minimum for $\ell$;
\item if \al{\frac{T}{N}\sigma^2 -\|\hat f_{LS}^{[hk,i]}\|^2_{ R^{-1}}\geq 0, \; \; \forall \, k\in I_2,\;i\in I_u,\nonumber} then the subset $$\Xi_3=\{(\Lambda,\Gamma,\Pi,\Omega) \hbox{ s.t. }\Lambda,\Gamma,\Pi,\Omega\geq 0, \; \pi_{h}=0\}$$ contains a local minimum for $\ell$;
\item if \al{\label{cond_ARD_fin}\frac{T}{N}\sigma^2 -\|\hat f_{LS}^{[hk,i]}\|^2_{ R^{-1}}\geq 0, \; \; \forall \, h\in I_1,} then the subset $$\Xi_4=\{(\Lambda,\Gamma,\Pi,\Omega) \hbox{ s.t. }\Lambda,\Gamma,\Pi,\Omega\geq 0, \; \omega_{ki}=0\}$$ contains a local minimum for $\ell$. \end{itemize}\end{prop} \begin{proof} Let \al{X:= \left[\begin{array}{cc}X_g\otimes I& 0 \\0 & X_f\otimes I\end{array}\right] , \; \; \tilde K:=\left[\begin{array}{cc}I\otimes P & 0 \\0 & I\otimes R\end{array}\right],\nonumber} thus \al{K:= \left[\begin{array}{cc}K_g& 0 \\0 & K_f\end{array}\right]=X \tilde K=\tilde K X= \tilde K^{1/2}X \tilde K^{1/2} \nonumber} and condition (\ref{cond_regressor}) can be written as \al{\label{cond_regressor2} \tilde K^{1/2} \left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] \tilde K^{1/2} =N I .} In what follows, the symbol $\equiv$ means ``equal to up to terms not depending on $\xi$''. Next, we rewrite $\ell (\mathrm{y}^+,\xi)$ using (\ref{LS_sparse}) and (\ref{cond_regressor2}). First, {\small \al{&\log\det(\Phi K_g \Phi^\top+\Psi K_f\Psi^\top+\sigma^2 I)\nonumber\\ &\equiv \log\det\left(\frac{1}{\sigma^{2}}\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] K\left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]+ I\right)\nonumber\\ &\equiv \log\det\left(\frac{1}{\sigma^{2}}\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] \tilde K^{1/2}X^{1/2}X^{1/2}\tilde K^{1/2}\left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]+ I\right)\nonumber\\ & \equiv \log\det\left(\frac{1}{\sigma^{2}} X^{1/2} \tilde K^{1/2} \left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] \tilde K^{1/2} X^{1/2}+ I\right)\nonumber\\
&= \log\det\left(\frac{N}{\sigma^{2}}\left[\begin{array}{cc}X_g\otimes I_T & 0 \\0 & X_f \otimes I_T\end{array}\right] + I\right)\nonumber\\
&=\log\det\left(\frac{N}{\sigma^{2}}X_g\otimes I_T+I\right)+\log\det\left(\frac{N}{\sigma^{2}}X_f\otimes I_T+I\right)\nonumber\\
& = T\sum_{\substack{h\in I_1\\ k\in I_2}}\left[ \sum_{\substack{j\in I_1\\ l\in I_2}}\log \det \left(\frac{N}{\sigma^2}\frac{\lambda_{hj}\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}+1\right)\right.\nonumber\\ & \left. \hspace{0.4cm}+\sum_{\substack{i\in I_u}}\log \det \left(\frac{N}{\sigma^2}\frac{\pi_{h}\omega_{ki}}{\pi_{h}+\omega_{ki}}+1\right) \right]\nonumber }} {\small \al{&(\mathrm{y}^+)^\top(\Phi K_g \Phi^\top+\Psi K_f\Psi^\top+\sigma^2 I)^{-1}\mathrm{y}^+ \nonumber\\ &=(\mathrm{y}^+)^\top\left(\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] K \left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]+\sigma^2 I\right)^{-1}\mathrm{y}^+ \nonumber\\
&\equiv-\frac{1}{\sigma^{2}}(\mathrm{y}^+)^\top\left[\begin{array}{cc}\Phi & \Psi \end{array}\right] \left(\left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]\left[\begin{array}{cc}\Phi & \Psi \end{array}\right]+\sigma^2K^{-1} \right)^{-1} \left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right]\mathrm{y}^+ \nonumber\\
&=-\frac{N^2}{\sigma^{2}}\left[\begin{array}{cc}\hat \theta_{g,LS}^\top & \hat \theta_{f,LS}^\top \end{array}\right] \tilde K^{-1/2}\left(\tilde K^{1/2} \left[\begin{array}{c}\Phi^\top \\ \Psi^\top\end{array}\right] \left[\begin{array}{cc}\Phi & \Psi \end{array}\right] \tilde K^{1/2} \right.\nonumber\\ &\left. \hspace{0.4cm}+\sigma^2 X^{-1} \right)^{-1} \tilde K^{-1/2}\left[\begin{array}{c}\hat \theta_{g,LS} \\ \hat \theta_{f,LS}\end{array}\right] \nonumber\\
&=-\frac{N^2}{\sigma^{2}}\left[\begin{array}{cc}\hat \theta_{g,LS}^\top & \hat \theta_{f,LS}^\top \end{array}\right] \tilde K^{-1/2} \left( NI +\sigma^2 X^{-1} \right)^{-1} \tilde K^{-1/2}\left[\begin{array}{c}\hat \theta_{g,LS} \\ \hat \theta_{f,LS}\end{array}\right] \nonumber\\
&=-\frac{N^2}{\sigma^{2}} \sum_{\substack{h\in I_1\\ k\in I_2}}\left[\sum_{\substack{j\in I_1\\ l\in I_2}} \frac{\|\hat g^{[hk,jl]}_{LS}\|_{P^{-1}}^2}{N+\sigma^2\frac{\lambda_{hj}+\gamma_{kl}}{\lambda_{hj}\gamma_{kl}}} +\sum_{i\in I_u}\frac{\|\hat f^{[hk,i]}_{LS}\|_{R^{-1}}^2}{N+\sigma^2\frac{\pi_{h}+\omega_{ki}}{\pi_{h}\omega_{ki}}} \right]\nonumber.
}} Thus,
{\small\al{ &\ell(\mathrm{y}^+,\xi)\equiv \sum_{\substack{h\in I_1\\ k\in I_2}}\left[ T\sum_{\substack{j\in I_1\\ l\in I_2}}\log \det \left(\frac{N}{\sigma^2}\frac{\lambda_{hj}\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}+1\right)\right.\nonumber\\ &
\hspace{0.4cm}+T\sum_{\substack{i\in I_u}}\log \det \left(\frac{N}{\sigma^2}\frac{\pi_{h}\omega_{ki}}{\pi_{h}+\omega_{ki}}+1\right) \nonumber\\ & \left.-\frac{N^2}{\sigma^{2}} \sum_{\substack{j\in I_1\\ l\in I_2}} \frac{\|\hat g^{[hk,jl]}_{LS}\|_{P^{-1}}^2}{N+\sigma^2\frac{\lambda_{hj}+\gamma_{kl}}{\lambda_{hj}\gamma_{kl}}} -\frac{N^2}{\sigma^{2}}\sum_{i\in I_u}\frac{\|\hat f^{[hk,i]}_{LS}\|_{R^{-1}}^2}{N+\sigma^2\frac{\pi_{h}+\omega_{ki}}{\pi_{h}\omega_{ki}}} \right].\nonumber}}Next, we prove the first statement regarding $\lambda_{hj}$. The expression of the partial derivative of $\ell$ with respect to $\lambda_{hj}$ depends on the value of $\gamma_{kl}$'s. Given $\gamma_{kl}\geq 0$ for $k,l\in I_2$, let $\tilde J$ denotes the subset of $ I_2\times I_2$ containing the indexes $(k,l)$ such that $\gamma_{kl}>0$. Then, we have: {\small \al{& \frac{\partial \ell(\mathrm{y}^+,\xi)}{\partial \lambda_{hj}}\nonumber\\ &=\sum_{\substack{(k, l)\in \tilde J}}\left(\frac{\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}\right)^2
\frac{T\left[\frac{\lambda_{hj}\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}+\frac{\sigma^2}{N}\right]-\| \hat g_{LS}^{[hk,jl]}\|^2_{P^{-1}}}{\left[\frac{\lambda_{hj}\gamma_{kl}}{\lambda_{hj}+\gamma_{kl}}+\frac{\sigma^2}{N}\right]^2},\nonumber }} thus
\al{\label{grad_ineq}\left.\frac{\partial \ell(\mathrm{y}^+,\xi)}{\partial \lambda_{hj}}\right|_{\substack{\lambda_{hj}=0 }}= \sum_{\substack{(k, l)\in \tilde J}}
\frac{\frac{T }{N}\sigma^2-\| \hat g_{LS}^{[hk,jl]}\|^2_{P^{-1}}}{\sigma^4/N^2}.}
If condition (\ref{cond_ARD_1}) holds, and in view of (\ref{grad_ineq}), then \al{\label{cond_der}\left.\frac{\partial \ell(\mathrm{y}^+,\xi)}{\partial \lambda_{hj}}\right |_{\lambda_{hj}=0}\geq 0.} Notice that $\ell$ is continuous in the compact set $\Xi_1$ and thus $\ell$ admits a point of minimum in $\Xi_1$. Therefore, if (\ref{cond_der}) holds, then $\Xi_1$ contains a local minimum for $\ell$. The remaining statements can be proved in a similar way.
\vrule height 7pt width 7pt depth 0pt
\end{proof}
\begin{rem} \label{cor_ARD} In the nonparametric case, i.e. the limit case $T\rightarrow \infty$, the above result holds provided that the length of the data diverges in the same way. More precisely, assume that $T\rightarrow \infty$, $N\rightarrow \infty$ and $T/N\rightarrow 1$, then Proposition \ref{prop_ARD} holds where conditions (\ref{cond_ARD_1})-(\ref{cond_ARD_fin}) are replaced by
\al{&\sigma^2 -\|\hat g_{LS}^{[hk,jl]}\|^2_{ P^{-1}}\geq 0, \; \; \forall \, k,l\in I_2\nonumber\\
&\sigma^2 -\|\hat g_{LS}^{[hk,jl]}\|^2_{ P^{-1}}\geq 0, \; \; \forall \, h,j\in I_1\nonumber\\
&\sigma^2 -\|\hat f_{LS}^{[hk,i]}\|^2_{ R^{-1}}\geq 0, \; \; \forall \, k\in I_2,\; i\in I_u\nonumber \\
&\sigma^2 -\|\hat f_{LS}^{[hk,i]}\|^2_{ R^{-1}}\geq 0, \; \; \forall \, h\in I_1.\nonumber}\end{rem}
Let $\hat G_{LS}(z)$ and $\hat F_{LS}(z)$ be the transfer matrices corresponding to the least squares estimates in (\ref{LS_sparse}). Then, the corresponding model is \al{ y(t)=\hat G_{LS}(z)y(t)+ \hat F_{LS}(z) +e(t)\nonumber} The first statement of Remark \ref{cor_ARD} means that if the $\ell_2$ norms of $\hat g_{LS}^{[hk,jl]}$'s describing the conditional Granger causality relations from $y^\star_j$ to $y^\star_h$ do not dominate the variance of the noise process, then $\hat \lambda_{hj}=0$\footnote{Here, we implicitly assume that the local minimum in $\Xi_1$ is chosen as optimal solution.} and thus the posterior mean of $g^{[hk,jl]}$'s belonging to the block $(h,j)$ are null functions. Similar reasonings hold for the other statements.
\section{Numerical experiments}\label{section_simulation} In this section we test the performance of the proposed estimator. \subsection{Synthetic examples} We consider five Monte Carlo studies of 100 runs. For each run we generate an ARMAX model of the form \al{\label{model_true}y(t)=G_{\mathrm{TR}}(z)y(t)+F_{\mathrm{TR}}(z) u(t)+e(t)} as follows: \begin{itemize} \item the rational transfer matrices $G_{\mathrm{TR}}(z)$ and $F_{\mathrm{TR}}(z)$ are strictly casual and generated through the MATLAB function \texttt{drmodel.m} where the order of the system is equal to $20$; moreover, their poles are restricted to have absolute value less than 0.95. \item some entries of $G_{\mathrm{TR}}(z)$ and $F_{\mathrm{TR}}(z)$ are set equal to the null function in such a way that they satisfy condition (\ref{cond_supp}); matrices $E_1$, $E_2$, $A_1$ and $A_2$ are randomly generated in such a way such that the fraction of nonnull entries is equal to 0.6. \end{itemize} For each run we generate a dataset of length $N=500$ if not otherwise specified; the input $u$ has uncorrelated components which are generated through the MATLAB function \texttt{idinput.m} as a realization from a random Gaussian noise with band [0, 0.4].
We compare the following estimators: \begin{itemize} \item \textbf{K}: this is the estimator based on model (\ref{OEmodel_start}) where the transfer matrices are modeled as zero mean Gaussian processes with kernels (\ref{ME_kernel}); \item \textbf{S}: this is the estimator based on model (\ref{OEmodel_start}) where the transfer matrices are modeled as zero mean Gaussian processes with kernels (\ref{structPillo}); \item \textbf{SS}: this is the estimator based on model (\ref{OEmodel_start}) where the transfer matrices are modeled as zero mean Gaussian processes with kernels $\lambda I_{p_1^2p_2^2}\otimes P$ and $\pi I_{p_1p_2m}\otimes R$ where the hyperparameters $\lambda,\pi\geq 0$ represent the scale factors. \end{itemize}
The impulse responses are truncated with length $T=50$, i.e. we make the approximation $G(z)= \sum_{t=1}^\infty G_t z^{-t} \approx \sum_{t=1}^T G_t z^{-t}$. Such a truncation does not affect the estimates because both $P$ and $R$ force the estimated impulse responses to decay exponentially, i.e. it does not involve any kind of trade-off between bias and variance \citep{PILLONETTO_DENICOLAO2010}. In practice, it is only important to take the practical length very large.
The performance of these estimators is assessed through the following indexes: \begin{itemize} \item Average impulse response fit \al{\mathrm{AIRF}= \frac{fit(G_{\mathrm{TR}},\hat G)}{2}+\frac{fit(F_{\mathrm{TR}},\hat F)}{2} \nonumber} where
\al{&fit(G_{\mathrm{TR}},\hat G)= 100\left(1-\sqrt{\frac{\frac{1}{T}\sum_{t=1}^T\|G_{\mathrm{TR},t}-\hat G_t\|^2}{\frac{1}{T}\sum_{t=1}^T\|G_{\mathrm{TR},t}-\bar G_{\mathrm{TR}}\|^2}} \right)\nonumber\\ &\bar G_{\mathrm{TR}}= \frac{1}{T}\sum_{t=1}^TG_{\mathrm{TR},t}\nonumber} and $\hat G(z)$, $\hat F(z)$ are the estimated transfer matrices;
\item Fraction of misspecified edges{\small \al{\mathrm{ERR}=\frac{\| E_1\otimes E_2 - \hat E_1\otimes \hat E_2\|_0}{2p_1^2p_2^2}+\frac{\| A_1\otimes A_2 - \hat A_1\otimes \hat A_2\|_0}{2p_1p_2m}\nonumber}} where $\mathrm{supp}(\hat G(z))=\hat E_1 \otimes \hat E_2$, $\mathrm{supp}(\hat F(z))=\hat A_1 \otimes \hat A_2$, while $E_1$, $E_2$, $A_1$ and $A_2$ correspond to the support of $G_{\mathrm{TR}}(z)$ and $F_{\mathrm{TR}}(z)$, respectively. \end{itemize}
{\em First Monte Carlo study.} We consider the case in which model (\ref{model_true}) is without input $u$, i.e. we have a dynamic network corresponding to an ARMA stochastic process. In this case we only have to estimate $G(z)$, accordingly $\mathrm{AIRF}=fit(G_{\mathrm{TR}},\hat G)$ and $\mathrm{ERR}=\| E_1\otimes E_2 - \hat E_1\otimes \hat E_2\|_0 /(p_1^2p_2^2)$. Besides the previous estimators, we also consider: \begin{itemize} \item \textbf{PEM}: this is the classical PEM approach that uses BIC for model order selection. The orders of the polynomials in the ARMA model ranges from 1 to 15 and are not allowed to be different from each other since this would lead to a combinatorial explosion of the number of competitive models. \item \textbf{PEM+OR}: this is the same as \textbf{PEM} with an additional oracle knowing which impulse responses are null functions. \end{itemize} The length of the dataset is $N=3900$. It is worth noting the latter has been chosen so large in such a way that \textbf{PEM} with polynomial orders equal to 15 has enough data to estimate all the model parameters whose amount is 3848. \begin{figure}
\caption{First Monte Carlo study ``ARMA case'' with $p_1=4$, $p_2=4$. {\em Left panel}. Average impulse response fit. {\em Right panel}. Fraction of misspecified edges.}
\label{MC4}
\end{figure} Figure \ref{MC4} shows the performance of the estimators. \textbf{K} outperforms all the other estimators in terms of $\mathrm{AIRF}$ while both \textbf{PEM+OR}, even though it owns the correct network topology, and \textbf{PEM} are the worst. Finally, \textbf{K} is better than \textbf{S} in terms of $\mathrm{ERR}$. Notice that we did not plot $\mathrm{ERR}$ for \textbf{PEM} and \textbf{SS} because they always provide a full network. In particular, in regard to \textbf{SS}, the entries of $G(z)$ share the same prior (and thus the same scale factor $\lambda$ whose estimated value is always strictly positive). We also considered Monte Carlo studies where $p_1$ and $p_2$ are different: the results, in terms of performance, do not change.
{\em Second Monte Carlo study}. The output dimensions are $p_1=p_2=3$, while the input dimension is $m=2$. \begin{figure}
\caption{Second Monte Carlo study with $p_1=3$, $p_2=3$ and $m=2$. {\em Left panel}. Average impulse response fit. {\em Right panel}. Fraction of misspecified edges.}
\label{MC1}
\end{figure} Figure \ref{MC1} shows the performance of the three estimators: \textbf{K} outperforms all the other estimators in terms of $\mathrm{AIRF}$, while \textbf{SS} is the worst. Finally, \textbf{K} is better than \textbf{S} in terms of $\mathrm{ERR}$.
{\em Third Monte Carlo study}. We consider the case in which $p_1$ and $p_2$ are different; more precisely, we have $p_1=3$, $p_2=4$ and $m=2$. \begin{figure}
\caption{Third Monte Carlo study with $p_1=3$, $p_2=4$ and $m=2$. {\em Left panel}. Average impulse response fit. {\em Right panel}. Fraction of misspecified edges.}
\label{MC2}
\end{figure} Figure \ref{MC2} shows the obtained results. Also in this case \textbf{K} is the best estimator both in terms of $\mathrm{AIRF}$ and $\mathrm{ERR}$, while \textbf{SS} is the worst.
{\em Fourth Monte Carlo study}. We consider $p_1=4$, $p_2=3$ and $m=2$; this Monte Carlo study differs from the last one because the output dimensions are swapped. \begin{figure}
\caption{Fourth Monte Carlo study with $p_1=4$, $p_2=3$ and $m=2$. {\em Left panel}. Average impulse response fit. {\em Right panel}. Fraction of misspecified edges.}
\label{MC3}
\end{figure} The situation substantially does not change, see Figure \ref{MC3}.
{\em Fifth Monte Carlo study}. We consider the case in which model (\ref{model_true}) is without input $u$ and corresponds to a hierarchical dynamic network (i.e. $E_1= E_2$). We set $p_1=p_2=4$. Besides the previous estimators, we also consider: \begin{itemize} \item \textbf{H}: this is the estimator based on model (\ref{OEmodel_start}), without input, where the transfer matrix in $G(z)$ is modeled as zero mean Gaussian process with kernel (\ref{kernel_hier}). \end{itemize} $\mathrm{AIRF}$ and $\mathrm{ERR}$ are defined as in the first Monte Carlo study. \begin{figure}
\caption{Fifth Monte Carlo study ``ARMA case - hierarchical network'' with $p_1=4$, $p_2=4$. {\em Left panel}. Average impulse response fit. {\em Right panel}. Fraction of misspecified edges.}
\label{MC5}
\end{figure} In Figure \ref{MC5} the performance of the estimators is depicted. \textbf{S} and \textbf{SS} are worse than \textbf{H} and \textbf{K}. The latter perform in a similar way in terms of $\mathrm{AIRF}$. \textbf{H}, however, outperforms \textbf{K} in terms of $\mathrm{ERR}$.
\begin{rem} The negative log-marginal likelihood function is not convex with respect to the hyperparameters even in the simplest case where the hyperparameters are only the scale factors, see \cite{6883125}. Accordingly, the kernel-based PEM methods
could suffer about local minima in the estimation of the hyperparameters. In particular, both \textbf{S} and \textbf{K}-\textbf{H} have the same issue. The main difference is the number of hyperparameters to estimate: \textbf{K}-\textbf{H} have to estimate very less hyperparameters than \textbf{S}. More precisely, in \textbf{K} we have to estimate $p_1^2+p_2^2+p_1+p_2m$ ($p_1^2+p_2^2$ in the case of systems without input) hyperparameters, in \textbf{H} we have to estimate $p_1^2$ hyperparameters (only applicable in the case of systems without input and $p_1=p_2$), while in \textbf{S} we have to estimate $p_1^2p_2^2+p_1p_2m$ ($p_1^2p_2^2$ in the case of systems without input) hyperparameters. Table \ref{tab:table1} summarizes the number of hyperparameters needed in the previous Monte Carlo studies. \begin{table}[h!]
\begin{center}\caption{Number of hyperparameters to estimate.}
\label{tab:table1}
\begin{tabular}{l|c|c|c|}
\textbf{M.C. study} & \textbf{S} & \textbf{K}& \textbf{H}\\
\hline
First & 99 & 27 & -\\
Second & 168 & 36& -\\
Third & 168 & 35 &-\\
Fourth & 256 & 32 &- \\ Fifth & 256 & 32 & 16 \\
\end{tabular}
\end{center} \end{table}
Therefore, \textbf{K} and \textbf{H} alleviate the aforementioned issue since the search space for optimizing the marginal likelihood is small in respect to the one of \textbf{S}.\end{rem}
\subsection{Learning a bike sharing system} \label{sec_bike} Bike sharing systems aim to make automatic bike rentals, in particular the rental and the return back. We consider the number of users and some variables describing the weather, that is wind speed, humidity and apparent temperature. These quantities have been collected in the period 1 January 2011 - 14 May 2012 (500 days in total) from the Capital Bike Sharing (CBS) system at Washington, D.C., USA. The sampling time is equal to 1 hour. For more details see \cite{fanaee2014event}. These data describe a four dimensional stochastic process $x(s)=[\,x_1(s) \; x_2(s) \; x_3(s) \; x_4(s) \, ]^\top$, where $x_1(s)$ is the number of users, $x_2(s)$ is the wind speed, $x_3(s)$ is the humidity and $x_4(s)$ is the apparent temperature. Process $x(s)$ is non-stationary during a day: for instance, we expect that the number of users during the night is smaller than the ones during the day. Accordingly, we consider the process \al{y(t)=[\,x(24(t-1)+1)^\top \; \ldots \; \, x(24t)^\top \, ]^\top\nonumber} taking values in $ \mathbb{R}^{96}$ and the corresponding sampling time is 1 day. Thus, the corresponding dataset has length $N=500$. After detrending it, we model $y$ as the dynamic spatio-temporal model of Section \ref{sec_examples} where $p_1=24$, $p_2=4$. Then, we estimate $G(z)$ by \textbf{K} and the regression matrices are the ones in Remark \ref{remark_spatio}. The estimated network describing the conditional Granger causality relations among the number of users, wind speed, humidity and apparent temperature is depicted in Figure \ref{bike_E2}. \begin{figure}
\caption{Estimated network describing conditional Granger causality relations among: the number of users (U), wind speed (W), humidity (H) and apparent temperature (T).}
\label{bike_E2}
\end{figure} As expected, the number of users does not conditionally Granger causes the weather variables. Moreover, the number of users is conditionally Granger caused by the apparent temperature. \begin{figure}
\caption{Weighted matrix describing the estimated Granger causality relations among hours.}
\label{fig_hours}
\end{figure} Figure \ref{fig_hours} shows the weighted matrix describing the conditional Granger causality relations among hours. More precisely, the entry in position $(h,j)$ is the $\ell_1$ norm of the $4\times 4$ impulse response, formed by $\hat g^{[hk,jl]}$ with $k,l\in \{1,2,3,4\}$, which describes the strength of the conditional Granger causality relation from hour $j$ to hour $h$. We can notice that all the hours are mainly conditionally Granger caused by the previous hour.
\section{Conclusions} \label{section_conclusions} We have considered the problem to estimate a dynamic network describing conditional Granger causality relations and having a Kronecker structure. We have proposed a kernel-based PEM method for estimating such a model from data. The kernel functions have been derived from the maximum entropy principle. Numerical simulation showed that the proposed estimator outperforms the one estimating dynamic networks without structure. Finally, we have used the proposed method to learn the conditional Granger causality relations in a bike sharing system.
\end{document} |
\begin{document}
\title{\textsc{ A Strong Law of Large Numbers for Positive Random Variables} \thanks{~ We are deeply grateful to J\'anos \textsc{Koml\'os}, who went over the entire manuscript with the magnifying glass and offered line-by-line criticism and wisdom. We thank Daniel \textsc{Ocone}, Albert \textsc{Shiryaev} for invaluable advice; and Richard \textsc{Groenewald}, Tomoyuki \textsc{Ichiba}, Kostas \textsc{Kardaras}, Tze-Leung \textsc{Lai}, Kasper \textsc{Larsen}, Ayeong \textsc{Lee}, Emily \textsc{Sergel}, Nathan \textsc{Soedjak} for careful readings and suggestions.} }
\author{ \textsc{Ioannis Karatzas} \thanks{~ Department of Mathematics, Columbia University, New York, NY 10027 (e-mail: {\it ik1@columbia.edu}). Support from the National Science Foundation under Grant DMS-20-04977 is gratefully acknowledged. }
\and \textsc{Walter Schachermayer} \thanks{~ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria (email: {\it walter.schachermayer@univie.ac.at}). Support from the Austrian Science Fund (FWF) under
grant P-28861, and by the Vienna Science and Technology Fund (WWTF) through project MA16-021, is gratefully acknowledged.
}
}
\maketitle \begin{abstract} \noindent In the spirit of the famous \textsc{Koml\'os} (1967) theorem, every sequence of nonnegative, measurable functions $\{ f_n \}_{n \in \mathbb N}$ on a probability space, contains a subsequence which---along with all its subsequences---converges a.e.\,\,in \textsc{Ces\`aro} mean to some measurable $f_* : \Omega \to [0, \infty]$. This result of \textsc{von\,Weizs\"acker} (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of \textsc{Delbaen \& Schachermayer} (1994), replacing general convex combinations by \textsc{Ces\`aro} means. \end{abstract}
\noindent
{\sl AMS 2020 Subject Classification:} Primary 60A10, 60F15; Secondary 60G42, 60G46.
\noindent
{\sl Keywords:} Strong law of large numbers, hereditary convergence, partition of unity
\section{Introduction} \label{sec1}
On a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, consider real-valued measurable functions $f_1, f_2, \cdots \,.$ If these are independent and have the same distribution with $\mathbb E ( | f_1|) < \infty\,$, the celebrated \textsc{Kolmogorov} strong law of large numbers (\cite{K2};\,\cite{KAN};\,\cite{Du}, p.\,73) states that the ``sample average" $ \, ( f_1 + \cdots + f_N) / N\,$ converges $\mathbb{P}-$a.e.\,to the ``ensemble average" $\,\mathbb E (f_1) = \int_\Omega f_1 \, \mathrm{d} \mathbb{P}\,,$ as $ N \to \infty$. More generally, if $f_n (\omega) = f \big( T^{n-1}(\omega) \big), \, n \ge 2, \, \omega \in \Omega$ are the images of an integrable function $f_1: \Omega \to \mathbb R$ along the orbit of successive actions of a measure-preserving transformation $T: \Omega \to \Omega\,,$ then the above sample average converges $\mathbb{P}-$a.e.\,to the conditional expectation $f_* = \mathbb E ( f_1 |{\cal I})$ of $f_1$ given the $\sigma-$algebra ${\cal I}$ of $T-$invariant sets, by the \textsc{Birkhoff} pointwise ergodic theorem (\cite{Du}, p.\,333).
A deep result of \textsc{Koml\'os} \cite{K}, already 55 years old but always very striking, says that such ``stabilization via averaging" occurs within {\it any} sequence $f_1, f_2, \cdots \,$ of measurable, real-valued functions with $\, \sup_{n \in \mathbb N} \mathbb E ( | f_n|) < \infty\,.$ More precisely, there exist then an integrable function $f_*$ and a subsequence $ \{ f_{n_k} \}_{k \in \mathbb N} $ such that $ \, ( f_{n_1} + \cdots + f_{n_K}) / K\,$ converges to $f_*\,$, $\,\mathbb{P}-$a.e.\,\,as $ K \to \infty$; and the same is true for any further subsequence of this $ \{ f_{n_k} \}_{k \in \mathbb N}\,.$
This result inspired further path-breaking work in probability theory (\cite{G},\,\cite{Ch2},\,\cite{Ch3}) culminating with \\ \textsc{Aldous}
(1977), where exchangeability plays a crucial r\^ole. It, and its ramifications \cite{DS1},\,\cite{DS2} involving forward convex combinations, have been very useful in the field of convex optimization; more generally, when one seeks objects with specific properties, and tries to ascertain
their existence using weak compactness arguments. Stochastic control, optimal stopping and hypothesis testing are examples of the former (e.g.,\,\cite{KS},\,\cite{KW},\,\cite{CK},\,\cite{KZ},\,\cite{LZ}); the \textsc{Doob-Meyer} and \textsc{Bichteler-Dellacherie} theorems in stochastic analysis provide instances of the latter (e.g.,\,\cite{J},\,\cite{BSV1},\,\cite{BSV2}).
We develop here a very simple argument for the \textsc{Koml\'os} theorem, in the important special case of nonnegative $f_1, f_2, \cdots \,$ treated by \textsc{von\,Weizs\"acker} (2004). The argument dispenses with boundedness in $ \mathbb{L}^1$, at the cost of allowing the function $f_*$ to take infinite values.
\section{Background} \label{sec2}
We place ourselves on a given, fixed probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and consider a sequence $f_1, f_2, \cdots \,$ of measurable, real-valued functions defined on it.
We say that this sequence {\it converges hereditarily in \textsc{Ces\`aro} mean} to some measurable $f_*:\Omega \to \mathbb{R} \cup \{\pm \infty\}$, and write $ f_n \xrightarrow[n \to \infty]{hC}f_*\,,~~ \mathbb{P} -\hbox{a.e.,} $ if, for {\it every} subsequence $\big\{f_{n_k} \big\}_{k \in \mathbb{N}}$ of the original sequence, we have \begin{equation} \label{1} \lim_{K\to \infty} \frac{1}{K} \sum_{k=1}^K f_{n_k} = f_*\,,\qquad \mathbb{P} -\hbox{a.e.} \end{equation}
Clearly then, every other such sequence $g_1, g_2, \cdots \,$ which is
{\it equivalent} to $f_1, f_2, \cdots \,,$ in the sense of $\, \sum_{n \in \mathbb N} \mathbb{P}( f_n \neq g_n) < \infty\,$ (cf.\,\cite{KAN}), also has this property.
In 1967, \textsc{Koml\'os} proved the following remarkable result. The argument in \cite{K} is very clear, but also long and quite involved. Simpler proofs and extensions have appeared since (e.g.,\,\cite{S},\,\cite{T};\,\cite{B}).
\begin{theorem} [\textsc{Koml\'os} (1967)] \label{Kom} If the sequence $\{f_n \}_{n \in \mathbb{N}}$ is bounded in $ \mathbb{L}^1,$ i.e.,
$\sup_{n \in \mathbb{N}} \mathbb{E} (|f_n|) < \infty\,$ holds, there exist an integrable $f_*:\Omega \to \mathbb{R}$ and a subsequence $\big\{f_{n_k}\big\}_{k \in \mathbb{N}}$ of $\{f_n \}_{n \in \mathbb{N}}\,,$ which converges hereditarily in \textsc{Ces\`aro} mean to $f_*\,:$ \begin{equation} \label{02} f_{n_k} \xrightarrow[k \to \infty]{hC}f_*\,, \qquad \mathbb{P}-\hbox{a.e.} \end{equation} \end{theorem}
This result was motivated by an earlier one, Theorem \ref{Rev} right below. For the convenience of the reader, we provide in \S \,\ref{sec5f} a simple proof (in the manner of\,\cite{Ch}, pp.\,137-141) of that precursor result, which proceeds by extracting a {\it martingale difference} subsequence. This crucial idea, which establishes a powerful link to martingale theory and simplifies the arguments, appears in this context for the first time in \cite{K} (for related results, see \cite{K1}).
\begin{theorem} [\textsc{R\'ev\'esz} (1965)] \label{Rev} If the sequence $ \{f_n \}_{n \in \mathbb{N}}$ satisfies $\,\sup_{n \in \mathbb{N}} \mathbb{E} (f_n^2) < \infty\,,$ there exist a function $g \in \mathbb{L}^2$ and a subsequence $ \{f_{n_k} \}_{k \in \mathbb{N}}\,,$ such that $ \, \sum_{k \in \mathbb{N}} a_k \big(f_{n_k} -g \big)\, $ converges $\,\mathbb{P}-$a.e., for any sequence $ \{a_k \}_{k \in \mathbb{N}} \subset \mathbb R$ with $\,\sum_{k \in \mathbb{N}} a^2_k < \infty$. \end{theorem}
It is clear that this property of the subsequence $ \{f_{n_k} \}_{k \in \mathbb{N}} $ is inherited by all {\it its} subsequences (just ``stretch out" the $a_k$'s accordingly, and fill out the gaps with zeroes).
In a related development, \textsc{Delbaen \& Schachermayer} (\cite{DS1},\,Lemma A1.1;\,\cite{DS2}) showed with very simple arguments that, from every sequence $\{f_n \}_{n \in \mathbb{N}}$ of nonnegative, measurable functions, a sequence of convex combinations $\,g_n \in \text{conv}(f_n, f_{n+1}, \cdots ), ~ n \in \mathbb N\,$ of its elements can be extracted, which converges $\mathbb{P}-$a.e.\,to a measurable $f_* : \Omega \to [0, \infty]$. This result was called ``a somewhat vulgar version of \textsc{Koml\'os}'s theorem" in \cite{DS2}, and is implied
by Theorem \ref{theorem3} below. Indeed,
convergence for \textsc{Ces\`aro} averages is much more precise than
for unspecified forward convex combinations.
In several contexts, including optimization treated via convex duality, nonnegativity is often no restriction at all, but rather the natural setting (e.g.,\,\cite{KS};\,\cite{LZ}; \cite{KS2};\,\cite{KK}, Chapter 3 and Appendix). Then, in the presence of convexity, Lemma A1.1 in \cite{DS1}, or Theorem \ref{theorem3} here, are very useful analogues of Theorem \ref{Kom}: they lead to limit functions $f_*$ in convex sets (such as the positive orthant in $ \mathbb{L}^0$, or the unit ball in $ \mathbb{L}^1$) which are not compact in the usual sense, but {\it are} ``convexly compact" as in \textsc{\v Zitkovi\'c} \cite{Z}.
\section{Result} \label{sec3}
The purpose of this note is to prove with new and elementary tools the following version of Theorem \ref{Kom}, due to \textsc{von\,Weizs\"acker} \cite{vW} and studied further in \cite{Ta}, \S\,5.2.3 of \cite{KS2}.
\begin{theorem} \label{theorem3} Given a sequence $ \{f_n \}_{n \in \mathbb{N}}$ of {\rm nonnegative}, measurable functions on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, there exist a measurable function $f_*:\Omega \to [0, \infty]$ and a subsequence $\big\{f_{n_k}\big\}_{k \in \mathbb{N}}$ of the original sequence, such that \eqref{02} holds. \end{theorem}
Our proof appears in Section \ref{sec5}; it is, we believe, not without methodological/pedagogical merit. We observe that the result imposes no restriction whatsoever on the functions $f_1, f_2, \cdots$, apart from measurability and nonnegativity. This comes at a price: the function $f_*\,$, constructed here carefully in \eqref{5}--\eqref{8} below, can take the value $+\infty$ on a set of positive measure.
\section{Preparation} \label{sec4}
We place ourselves in the setting of Theorem \ref{theorem3}. The arguments that follow often necessitate passing to subsequences, and to diagonal subsequences, of a given $ \{f_n \}_{n \in \mathbb{N}}$. To simplify typography, we denote frequently such subsequences by the same symbols, $ \{f_n \}_{n \in \mathbb{N}}$.
For each integer $k \in \mathbb{N}$, we introduce now the truncated functions \begin{equation} \label{3} f_n^{(k)}\, :=\, f_n \cdot \mathbf{ 1}_{ \{ k-1 \le f_n < k \} }\,, \qquad n \in \mathbb{N} \end{equation} and note the partition of unity $\, \sum_{k \in \mathbb{N}} f^{(k)}_n = f_n\,, ~ \forall ~ n \in \mathbb{N}\,. $
\begin{lemma} \label{lemma04} For the sequence of functions $ \{f_n \}_{n \in \mathbb{N}}$ in Theorem \ref{theorem3}, there exists a subsequence, denoted by the same symbols and such that, for every $k\in \mathbb{N}$, the functions of \eqref{3} converge to an appropriate measurable function $f^{(k)}:\Omega \to [0,\infty)\,,$ in the sense \begin{equation} \label{4} f_n^{(k)} \, \xrightarrow[n \to \infty]{hC} \, f^{(k)}, \qquad \mathbb{P}-\text{a.e.} \end{equation} For each fixed $k \in \mathbb N,$ this convergence holds also in $\mathbb{L}^1.$
\end{lemma}
\noindent
{\it Proof} (after \cite{Ch}, pp.\,145--146): For arbitrary, fixed $k\in \mathbb{N}\,,$ the sequence $ \big\{f_n^{(k)} \big\}_{n \in \mathbb{N}}$ of \eqref{3} is bounded in $\mathbb{L}^\infty,$ thus also in $\mathbb{L}^2$. Theorem \ref{Rev} provides a function $ f^{(k)} \in \mathbb{L}^2$ and a subsequence $ \{ f_{n_j}^{(k)} \}_{j \in \mathbb{N}}$ of $ \{f_n^{(k)} \}_{n \in \mathbb{N}}\,$, such that $\,\sum_{j \in \mathbb{N}} (f_{n_j}^{(k)} -f^{(k)} )/ j\,$ converges $\,\mathbb{P}-$a.e.; and as mentioned right after Theorem \ref{Rev}, this is inherited by all subsequences of $ \{ f_{n_j}^{(k)} \}_{j \in \mathbb{N}}$, and the \textsc{Kronecker} Lemma (\cite{Du}, p.\,81) gives
$$
0= \lim_{J\to \infty} \frac{1}{J} \sum_{j=1}^J \big( f_{n_j}^{(k)} - f^{(k)} \big)=\lim_{ J \to \infty} \frac{1}{J} \sum_{j=1}^J f_{n_j}^{(k)} - f^{(k)} ,\qquad \mathbb{P} -\hbox{a.e.}
$$ We pass now to a diagonal subsequence, denoted $ \big\{f_n \big\}_{n \in \mathbb{N}}\,$ again, and such that \eqref{4} holds for {\it every} $k \in \mathbb N\,.$ The last claim follows by the dominated convergence theorem. \qed
With these ingredients, we introduce the measurable function $f:\Omega \to [0,\infty]$ via \begin{equation} \label{5} f \,:=\, \sum_{k \in \mathbb{N}} f^{(k)}, \qquad \text{and consider the set} \quad A_\infty \,:= \,\{f=\infty\}. \end{equation} With the help of \textsc{Fatou}'s Lemma, and the notation of (\ref{3})--(\ref{5}), Lemma \ref{lemma04} gives then \begin{equation} \label{6} \varliminf_{N\to \infty} \frac{1}{N} \sum^{N}_{n=1} f_n \geq f\,, \qquad \mathbb{P} -\text{a.e.} \end{equation} \begin{equation} \label{7} \lim_{N\to \infty} \frac{1}{N} \sum^{N}_{n=1} f_n = \infty =f\,, \qquad \mathbb{P} -\text{a.e.} \quad \text{on} \quad A_\infty \end{equation}
for a suitable subsequence (denoted by the same symbols) of the original sequence $ \{f_n \}_{n \in \mathbb{N}}\,,$ and for all further subsequences of this subsequence.
The inequality in \eqref{6} can easily be strict. Consider, for instance, $f_n \equiv n\,,$ so that $f_n^{(k)} =0$ holds in \eqref{3} for every fixed $k \in \mathbb{N}$ and all $n \in \mathbb{N}$ sufficiently large. We obtain $f^{(k)} =0$ in \eqref{4}, thus $f=0$ in \eqref{5}; and yet $\frac{1}{N} \sum^N_{n=1} f_n \to \infty$ as $N\to \infty$.
This preparation allows us to formulate a more technical and precise version of Theorem \ref{theorem3}, Proposition \ref{prop05} below, which implies it. The convention $ \,\infty \cdot 0=0$ is employed here, and throughout.
\begin{proposition} \label{prop05} Fix a sequence $\{ f_n \}_{n \in \mathbb N}$ of nonnegative, measurable functions on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and recall the notation \eqref{3}--\eqref{5}. There exist then a subsequence, denoted again $\{f_n\}_{n\in \mathbb{N}}\,$, and a set $A \supseteq A_\infty\,,$ such that \begin{equation} \label{8} f_n \, \xrightarrow[n \to \infty]{hC} \,f_* := \max \big(f, \, \infty \cdot \mathbf{1}_{A } \big)\,, \qquad \mathbb{P} -\text{a.e.} \end{equation} We have $A =A_\infty \,, $ thus also $f_* \equiv f ,$ when $\,\lim_{K\to \infty} \varlimsup_{n \to \infty} \, \mathbb{P} \big(f_n \ge K , f < \infty\big) =0\,$. \end{proposition}
This last condition holds if $\, \big\{ f_n \,\mathbf{1}_{ \{ f < \infty\} } \big\}_{n \in \mathbb N}\,$ is bounded in $\mathbb{L}^0$, i.e., $\,\lim_{K\to \infty} \sup_{n \in \mathbb N} \, {\mathbb P} (f_n \ge K, f < \infty ) = 0\, .$ A bit more stringently, if not only $\{ f_n \}_{n \in \mathbb N}$ but also its solid, convex hull in $\mathbb{L}^0_+,$ is bounded in $\mathbb{L}^0, $ then $\{ f_n \}_{n \in \mathbb N}$ is bounded in $\mathbb{L} ^1(\mathbb{Q})$ under some probability measure $ \mathbb{Q} \sim \mathbb{P},$ and thus $\mathbb{P} ( f < \infty) =1$ (e.g., Proposition A.11 in \cite{KK}).
Whereas, if $\{ f_n \}_{n \in \mathbb N}$ is bounded in $\mathbb{L}^1(\mathbb{P})$, i.e., $ \kappa := \sup_{n \in \mathbb{N}} \mathbb{E} (f_n)< \infty , $ then $f$ in \eqref{5} is integrable, since $\,\mathbb{E} (f ) \le \kappa $ holds from \eqref{6} and \textsc{Fatou}.
\section{Proofs} \label{sec5}
We shall need a couple of auxiliary results. First, and always with the notation of (\ref{3})--(\ref{5}), we note the following consequence of monotone and dominated convergence.
\begin{lemma} \label{lemma_2} Suppose a set $\,D\subseteq \Omega \backslash A_\infty = \{f < \infty \}\,$ satisfies $\,\mathbb{E}\big(f \, \mathbf{1}_D\big) < \infty\,.$ Then, for any given $\varepsilon \in (0,1),$ there exist $K \in \mathbb N $ and a subsequence of the given sequence $\{ f_n \}_{n \in \mathbb N}$ such that for it, and for any of its subsequences $($denoted again $\{ f_n \}_{n \in \mathbb N}),$ we have for arbitrary integers $L >K:$ \begin{equation} \label{10} \lim_{n \to \infty} \mathbb{E} \big[\,f_n \, \mathbf{ 1}_{ \{ K \le f_n <L \}\cap D }\, \big]=:\lim_{n \to \infty} \mathbb{E} \Big[\,f_n^{\,[K,L)}\, \mathbf{ 1}_D\, \Big] < \varepsilon\,. \end{equation} \end{lemma}
We are using throughout the notation \begin{equation} \label{11} f_n^{\,[K,L)} \,:= \sum^L_{k=K+1} f_n^{(k)} = f_n \, \mathbf{1}_{[K,L)} (f_n)\,, \quad ~~~f_n^{\,[K,\infty)} \,:=\, \sum_{k \geq K+1} f_n^{(k)} = f_n \, \mathbf{1}_{[K, \infty)} (f_n) \,; \end{equation}
in an analogous manner $\, f^{\,[K,L)} := \sum^L_{k=K+1} f^{(k)}\,, ~ \,f^{\,[K,\infty)} := \sum_{k \geq K+1} f^{(k)} \,,$ and Lemma \ref{lemma04} gives \begin{equation} \label{11a}
f_n^{\,[K,L)} \, \xrightarrow[n \to \infty]{hC} \, f^{\,[K,L)}
\,, \qquad \hbox{both $\,\mathbb{P}-$a.e. and in $\,\mathbb{L}^1$.} \end{equation}
Secondly, we recall \eqref{7} and observe the following dichotomy.
\begin{lemma} \label{lemma_3} In the setting of Proposition \ref{prop05}, consider any measurable set $B \supseteq \{f=\infty\} $ such that the property $\,f_n \xrightarrow[n \to \infty]{hC} \infty\,$ of \eqref{7} holds $\,\mathbb{P}-$a.e.\,on $B$. Then, either
\noindent (i) there exist a set $\,C \supseteq B$ with $\mathbb{P} (C) > \mathbb{P} (B)$ and a subsequence, still denoted $ \{f_n \}_{n \in \mathbb{N}}\,,$ with \begin{equation} \label{9} f_n \xrightarrow[n \to \infty]{hC} \infty \qquad \text{valid} \quad \mathbb{P} -\text{a.e.} ~ \text{on} ~ \,C\,; \qquad \text{or,} \end{equation} (ii) the \textsc{Ces\`aro} convergence $~f_n \xrightarrow[n \to \infty]{hC} f < \infty ~~ \text{holds}~~ \mathbb{P} -a.e. ~ \text{on} ~\, \Omega \setminus B \subseteq \{f< \infty\} \,.$ \end{lemma}
Under {\it Case\,(ii),} the set $B\supseteq A_\infty = \{f = \infty \}$ is maximal for the $\,\mathbb{P} -$a.e.\,\,property $f_n \stackrel{hC}{\longrightarrow} \infty\,$: it cannot be ``inflated" to a set $C\supseteq B,$ which satisfies \eqref{9} and has bigger measure. This leads eventually to Proposition \ref{prop05}, and thence to Theorem \ref{theorem3}.
Before proving these two results, we dispense with the proof of Theorem \ref{Rev}; this is completely self-contained, and has nothing to do with Lemma \ref{lemma_2} or Lemma \ref{lemma_3}.
\subsection{Proof of Theorem \ref{Rev}} \label{sec5f}
Because $\{f_n\}_{n \in \mathbb N}$ is bounded in $\mathbb{L}^2$, we can extract a subsequence that converges to some $g \in \mathbb{L}^2$ weakly in $\mathbb{L}^2$. Thus, it suffices to prove the result for a sequence $\{g_n\}_{n \in \mathbb N}$ bounded in $\mathbb{L}^2$, and with $g_n \to 0$ weakly in $\mathbb{L}^2$. We take such a sequence, then, and approximate each $g_n$ by a {\it simple} function $h_n \in \mathbb{L}^2$ with $\|g_n -h_n \|_2 \le 2^{-n}, ~\forall \,n \in \mathbb N.$ This gives, in particular, \begin{equation} \label{R1}
\sum_{n \in \mathbb N} \, \big| g_n - h_n \big| < \infty\,, \quad \mathbb{P} -\hbox{a.e.\,;}\qquad h_n \to 0 \quad \hbox{weakly in } \mathbb{L}^2. \end{equation} We construct now, by induction, a sequence $1=n_1 < n_2 < \cdots\,$ of integers, such that \begin{equation} \label{R2}
\big| \vartheta_k \big| < 2^{-k} \quad \text{holds } ~ \mathbb{P} -\text{a.e., for } ~ ~\vartheta_k:= \mathbb E \big( h_{n_k} \, \big| \, h_{n_1} , \cdots, h_{n_{k-1}} \big)\,,~~k=2,3, \cdots , \end{equation}
as follows: The function $h_{n_1} = h_1$ is simple, thus so is $ \,\mathbb E ( h_n | h_1) = \sum_{j=1}^J \gamma^{(n)}_j \mathbf{ 1}_{A_j} \,$ with $A_1, \cdots, A_J$ a partition of the space, and ${\mathbb P} (A_j)>0$, $\gamma^{(n)}_j := \big(1 / {\mathbb P} (A_j) \big) \cdot \mathbb E \big( h_n \,\mathbf{ 1}_{A_j} \big)\,.$ This last expectation tends to zero as $n \to \infty$ from \eqref{R1}, for every fixed $j$; so we can choose $n_2 > n_1 =1$ with $ \big|\gamma^{(n_2)}_j \big| < 2^{-2},$ for
\noindent
$ j=1, \cdots, J$; i.e., $\big|\vartheta_2 \big|< 2^{-2},$ ${\mathbb P}-$a.e. Clearly, we can keep repeating this argument since, at each stage, $\big( h_{n_1} , \cdots, h_{n_{k-1}} \big)$ generates a finite partition of the space; and this way we arrive at \eqref{R2}.
The sequence $\{ h_n \}_{n \in \mathbb N}$ is bounded in $\mathbb{L}^2$, thus so is the martingale $ X_n := \sum_{k=0}^n a_k \big(h_{n_k} - \vartheta_k \big)\,, ~ n \in \mathbb N_0\,, $
for any $ \{a_n \}_{n \in \mathbb{N}_0} \subset \mathbb R $ with $\sum_{n \in \mathbb{N}} a^2_n < \infty$. Martingale convergence theory (\cite{Du}, p.\,236) shows that the series $\sum_{k \in \mathbb N} a_k \big(h_{n_k} - \vartheta_k \big)$ converges ${\mathbb P}-$a.e. But we have also $\sum_{k \in \mathbb N} \big( \big| \vartheta_k \big| + \big| g_{n_k} - h_{n_k} \big| \big)< \infty, $ $\,{\mathbb P}-$a.e.\,\,from \eqref{R1}--\eqref{R2}, and deduce that $
\sum_{k \in \mathbb{N}} a_k \,g_{n_k} $ converges $\,\mathbb{P}-$a.e., the claim of the theorem. \qed
\subsection{Proof of Lemma \ref{lemma_2}} \label{sec5a}
Let us call {\it ``Lemma \ref{lemma_2}$^{\,\dagger}$"} the same statement as that of Lemma \ref{lemma_2}, except that \eqref{10} is now replaced by \begin{equation} \label{10a} \forall~ L=K+1, K+2, \cdots \,:~~~ \mathbb{E} \Big[\,f_n^{\,[K,L)}\, \mathbf{ 1}_D\, \Big] < \varepsilon\,, ~\text{for all but finitely many}~ n \in \mathbb N . \end{equation} {\it Claim: Lemma \ref{lemma_2}$^{\,\dagger}$ implies Lemma \ref{lemma_2}.} Let a subsequence of the original $\{ f_n \}_{n \in \mathbb N}$ be given (denoted $\{ f_n \}_{n \in \mathbb N}$ again), along with arbitrary $\varepsilon \in (0,1)$. Lemma \ref{lemma_2}$^{\,\dagger}$ guarantees the existence of $K \in \mathbb N$, depending on $\varepsilon$ and the subsequence, such that \eqref{10a} holds for all integers $L \ge K+1$.
Choose $L=K+1$ first. From Lemma \ref{lemma_2}$^{\,\dagger}$ and \textsc{Bolzano-Weierstrass}, (the current) $\{ f_n \}_{n \in \mathbb N}$ has a subsequence for which the expectation in \eqref{10a} converges, with limit $\le \varepsilon / 2.$ Now choose $L=K+2$ and a subsequence of the last subsequence, for which the expectation in \eqref{10a} converges and has limit $\le \varepsilon / 2.$ Continuing in this manner, then diagonalizing, we obtain a subsequence that satisfies \eqref{10a}.
\noindent {\it Proof of Lemma \ref{lemma_2}$^{\,\dagger}$.} We argue by contradiction, assuming that $\{ f_n \}_{n \in \mathbb N}$ has a subsequence for which Lemma \ref{lemma_2}$^{\,\dagger}$ fails. Then there exists an $\varepsilon \in (0,1)$ with the property that, for every subsequence of $\{ f_n \}_{n \in \mathbb N}$ and every $K \in \mathbb{N}$, there exists an integer $L>K$ such that \begin{equation} \label{11b} \mathbb{E} \bigg[\sum^L_{k=K+1} f^{(k)}_n \, \mathbf{1}_D \bigg] = \mathbb{E} \Big(f_n^{[K,L)} \, \mathbf{1}_D \Big) \geq \varepsilon \end{equation} holds for infinitely many integers $n \in \mathbb{N}$. But this means that there is a subsequence, again denoted by $\{f_n\}_{n \in \mathbb{N}}\,,$ {\it along which we have \eqref{11b} for every $n \in \mathbb{N};$} and, as a result, also \begin{equation} \label{11c} \mathbb{E} \bigg[\sum^L_{k=K+1} \Big(\frac{1}{N} \sum^N_{n=1} f_n^{(k)}\Big) \, \mathbf{1}_D \bigg] \geq \varepsilon\,, \qquad \forall ~ n \in \mathbb N\,. \end{equation} Now all the truncated functions $f_n^{(k)}$ as in \eqref{3}, for $k=K+1, \dots, L$ and $n \in \mathbb{N}$, take values on the ``Procrustean bed" $\{ 0 \} \cup [K,L)$; and $\, \lim_{N\to\infty} \frac{1}{N} \sum^N_{n=1} f_n^{(k)} = f^{(k)}\,$ holds $\mathbb{P} -$a.e., for the selected subsequence and all its subsequences, on account of Lemma \ref{lemma04}. Thus, $\, \mathbb{E} \big[\sum^L_{k=K+1} f^{(k)} \, \mathbf{1}_D \big] \geq \varepsilon \,$ from bounded convergence and \eqref{11c}; and the nonnegativity of these $f^{(k)}$'s implies also \begin{equation}\label{11d} \mathbb{E} \bigg(\sum_{k\geq K+1} f^{(k)} \, \mathbf{1}_D \bigg) = \mathbb{E} \Big(f^{\,[K, \infty)} \, \mathbf{1}_D \Big) \geq \varepsilon\,, \qquad \forall ~~K \in \mathbb{N}\,. \end{equation}
The nonnegativity gives also $\,\lim_{K\to \infty} \uparrow \sum^K_{k=1} f^{(k)}\, \mathbf{1}_D =f\, \mathbf{1}_D\,$, both $\mathbb{P} -$a.e.\,and in $\mathbb{L}^1$. Since $ \mathbb{E}\big(f \, \mathbf{1}_D\big) < \infty $ by assumption, $ \mathbb{E} \big[f^{\,[K, \infty)} \, \mathbf{1}_D\big] < \varepsilon/2\,$ holds for all $K \in \mathbb{N}$ large enough. But this contradicts \eqref{11d}, and we are done. \qed
\subsection{Proof of Lemma \ref{lemma_3}} \label{sec5b}
We start by fixing $j \in \mathbb{N}$ and distinguishing two contingencies, with the definitions \begin{equation} \label{12} D_j := \{f \leq j\} \backslash B \,, \qquad E_n^{[K, \infty)} \,:=\, \big\{f_n^{\,[K,\infty)} \geq K \big\} \cap D_j \,=\, \big\{f_n \geq K \big\} \cap D_j \,, \end{equation} \begin{equation} \label{12too} \alpha \,:=\, \lim_{K\to \infty} \varlimsup_{n \to \infty} \mathbb{P} \big(E_n^{[K, \infty)}\big) \,: \end{equation}
\noindent
{\it Contingency ~I:} $~\alpha > 0\,.$
\noindent
{\it Contingency II:} $~\alpha = 0\,.$
\noindent $\bullet~$ Under {\bf Contingency\,I}\,, we pass to a subsequence $\{f_n\}_{n \in \mathbb{N}}$ with $\mathbb{P} \big(E_n^{\,[n^2, \infty)}\big) \geq \alpha /2 \,,$ $\forall ~n \in \mathbb{N}\,$; and consider indicators $\,g_n := \mathbf{1}_{ E_n^{\,[n^2, \infty)}}, ~n \in \mathbb{N}\,,$ all of them supported on the set $\Omega \setminus B$. Arguing as in Lemma \ref{lemma04} we obtain
a subsequence, still denoted $\{g_n\} _{n \in \mathbb{N}}$, with $ \, g_n \xrightarrow[n \to \infty]{hC} g\,,$ $ \mathbb{P} -\hbox{a.e.,} $ for some $g:\Omega \to [0,1]\,$ with $\{g >0\} \subseteq \Omega \backslash B \,$ and $\mathbb{E}(g) \geq \alpha /2$ by bounded convergence.
Thus, $f_n \xrightarrow[n \to \infty]{hC} \infty\, $ holds $\, \mathbb{P} -$a.e. on $\,\{g >0\}\,$. This set has $\,\mathbb{P} \big( g>0\big) = \mathbb{E} [\mathbf{1}_{\{g>0\}} ] \geq \mathbb{E}(g) \geq \alpha/2 \,;$ we are under {\it Case (i)} of Lemma \ref{lemma_3}, with $C := \{g >0\} \cup B$ and $\mathbb{P}(C) > \mathbb{P}(B)$.
\noindent $\bullet~$ Now we pass to {\bf Contingency\,II}\,. We fix $\varepsilon >0,$ $D_j=\{f \leq j\} \backslash B $, and apply Lemma \ref{lemma_2} with this $D_j$ to construct inductively a subsequence $\big\{ n_m \big\}_{m \in \mathbb N}\,,$ along with sequences $\big\{ K_m \big\}_{m \in \mathbb N}\,, $ $\big\{ L_m \big\}_{m \in \mathbb N}\, $ of integers increasing to infinity and such that \begin{equation} \label{A.8}
\mathbb{P} \big(E_{n_m}^{\,[L_{m }, \infty ) }\big) = \mathbb{P} \big(\big\{ f_{n_m} \ge L_m \big\} \cap D_j \big) < 2^{- m } \end{equation} \begin{equation} \label{A.9}
\mathbb{E} \Big[\, f_{n_p}^{\,[K_m, L_p)} \, \mathbf{ 1}_{D_j} \, \Big]
< 2^{- m }\,,\qquad \forall ~~p = m, m+1, \cdots \end{equation} hold for every $ m \in \mathbb N$. With the choice (\ref{A.8}), the sequences $\, \big\{ f_{n_m} \cdot \mathbf{ 1}_{D_j} \big\}_{m \in \mathbb N}\,$ and $\, \big\{ f_{n_m}^{\,[0, L_m)} \cdot \mathbf{ 1}_{D_j} \big\}_{m \in \mathbb N}\,$ are equivalent in the sense introduced in section \ref{sec2}, as the probability of their respective general terms being different is bounded from above by $2^{- m }$. We claim that \begin{equation} \label{A.14} f_{n_m} \cdot \mathbf{ 1}_{D_j}\, \xrightarrow[m \to \infty]{hC} \,f \cdot \mathbf{ 1}_{D_j}\,,\quad \mathbb{P} -\hbox{a.e.;} \end{equation} and in view of the previous statement, this amounts to \begin{equation} \label{A.10} f_{n_m}^{\,[0, L_m)} \cdot \mathbf{ 1}_{D_j}\, \xrightarrow[m \to \infty]{hC} \,f \cdot \mathbf{ 1}_{D_j}\,,\quad \mathbb{P} -\hbox{a.e.} \end{equation}
To prove (\ref{A.10}), we start by observing that the sequence $\, \big\{ f_{n_m}^{\,[0, L_m)} \cdot \mathbf{ 1}_{D_j} \big\}_{m \in \mathbb N}\,$ is {\it uniformly integrable}, thus bounded in $\mathbb{L}^1$, as $$ \sup_{p \in \mathbb N \atop p \ge m} \, \mathbb{E} \Big[\, f_{n_p}^{\,[0, L_p)} \, \mathbf{ 1}_{D_j} \cdot \mathbf{ 1}_{ \big\{ f_{n_p}^{\,[0, L_p)} \ge K_m \big\} }\, \Big]
< 2^{- m } $$ holds on account of (\ref{A.9}) for every $m \in \mathbb N\,$. Theorem \ref{Kom} gives an integrable function $h : \Omega \to [0, \infty)\,$ with \begin{equation} \label{A.11} f_{n_m}^{\,[0, L_m)} \cdot \mathbf{ 1}_{D_j}\, \xrightarrow[m \to \infty]{hC} \,h \cdot \mathbf{ 1}_{D_j}\,,\quad \mathbb{P} -\hbox{a.e.,} \end{equation}
and we need to argue that this $h$ agrees with $f$ from (\ref{5}), $\mathbb{P}-$a.e.\,on $D_j$.
Indeed, for every $K \in \mathbb N$ and all $m$ large enough, $\, \sum_{k=1}^K \, f_{n_m} \, \mathbf{ 1}_{ \{ k-1 \le f_{n_m} < k \} } \,=\, f_{n_m}^{\,[0,K)} \,\le \, f_{n_m}^{\,[0,L_m)}\,
$ holds, therefore $\, \sum_{k=1}^K f^{(k)} \cdot \mathbf{ 1}_{D_j} \le h \cdot \mathbf{ 1}_{D_j}\,$ by letting $m \to \infty$, on account of (\ref{A.11}) and Lemma \ref{lemma04}. Passing now to the limit as $K \to \infty$ and recalling (4.3), we arrive at \begin{equation} \label{A.13} f \cdot \mathbf{ 1}_{D_j}\, \le \, h \cdot \mathbf{ 1}_{D_j}\,,\quad \mathbb{P} -\hbox{a.e.} \end{equation}
To obtain the inequality in the reverse direction, we take expectations. From (\ref{A.11}) and uniform integrability, we have $\,
\mathbb{E} \Big[\, f_{n_m}^{\,[0, L_m)} \cdot \mathbf{ 1}_{D_j} \, \Big] \, \xrightarrow[m \to \infty]{hC}\, \mathbb{E} \big[\, h \cdot \mathbf{ 1}_{D_j} \, \big]\,,$ therefore also
$$
\mathbb{E} \big[\, h \cdot \mathbf{ 1}_{D_j} \, \big]\,= \lim_{M \to \infty \atop K \to \infty} \frac{1}{M}\, \mathbb{E} \bigg[\, \sum_{m=1}^M f_{n_m}^{\,[0, L_m \wedge K)} \cdot \mathbf{ 1}_{D_j} \, \bigg] \,= \lim_{M \to \infty \atop K \to \infty} \frac{1}{M}\, \mathbb{E} \bigg[\, \sum_{m=1}^M \sum_{k=1}^{L_m \wedge K} f_{n_m}^{(k)} \cdot \mathbf{ 1}_{D_j} \, \bigg]~~~~~~~~~~~~
$$
$$ ~~~~~~~~~~~~~~ \le \lim_{M \to \infty \atop K \to \infty} \frac{1}{M}\, \mathbb{E} \bigg[\, \sum_{m=1}^M \bigg( \sum_{k=1}^{ K} f_{n_m}^{(k)} \bigg) \cdot \mathbf{ 1}_{D_j} \, \bigg]\,=\,\lim_{K \to \infty } \, \mathbb{E} \bigg[\, \sum_{k=1}^{K } \, f^{(k)} \cdot \mathbf{ 1}_{D_j} \, \bigg]\,\le\, \mathbb{E} \big[\, f \cdot \mathbf{ 1}_{D_j} \, \big]
$$
\noindent from Lemma \ref{lemma04}. In conjunction with (\ref{A.13}), this shows $\,f \cdot \mathbf{ 1}_{D_j}\, =\, h \cdot \mathbf{ 1}_{D_j}\,,~~ \mathbb{P} -\hbox{a.e.},$ as claimed; and on account of (\ref{A.11}) it establishes (\ref{A.10}), thus (\ref{A.14}) as well.
The final step is to let $ \,j\to \infty\,$: we do this again by extracting subsequences, successively for each $j \in \mathbb N\,$, then passing to a diagonal subsequence. We obtain then \eqref{A.14} with $D_j$ replaced by the set $D:= \bigcup_{j \in \mathbb N}D_j =\{ f < \infty \} \backslash B ,$ and deduce that we are in {\it Case\,(ii)} of Lemma \ref{lemma_3}. \qed
\subsection{Proofs of Proposition \ref{prop05} and Theorem \ref{theorem3}} \label{sec5c}
On the strength of Lemma \ref{lemma_3} we construct, by exhaustion or transfinite induction arguments and as long as we are under the dispensation of its {\it Case\,(i),} an increasing sequence $ B\subseteq B_1 \subseteq B_2 \subseteq \dots$ of sets as postulated there, whose union $B_\infty := \bigcup_{j \in \mathbb{N}} B_j \supseteq B \supseteq \{f=\infty\}$ is maximal with the property \eqref{9} for an appropriate subsequence. But maximality means that, on the complement $\Omega \backslash B_\infty$ of this set, we must be in the realm of {\it Case\,(ii)} in Lemma \ref{lemma_3}. This establishes the first claim of Proposition \ref{prop05} with $A = B_\infty \supseteq \{f=\infty\} \,$, thus also Theorem \ref{theorem3}.
For the second claim of the Proposition, we note that equality holds right above, that is, $ B_\infty = \{f=\infty\} ,$ if we are under Contingency II (i.e., $\alpha =0$) in \S\,\ref{sec5b} (proof of Lemma \ref{lemma_3}) and with $B = \{f=\infty\}\,$ in (\ref{12}); a sufficient condition for this, is $\, \lim_{K\to \infty} \varlimsup_{n \to \infty} \mathbb{P} (f_n \ge K, f < \infty ) =0$. The claim now follows. \qed
\end{document} |
\begin{document}
\title[Brauer group of the Moduli spaces of ${\rm PGL}_r({\mathbb C})$--bundles]{Unramified Brauer group of the moduli spaces of ${\rm PGL}_r({\mathbb C})$--bundles over curves}
\author[I. Biswas]{Indranil Biswas}
\address{School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India}
\email{indranil@math.tifr.res.in}
\author[A. Hogadi]{Amit Hogadi}
\address{School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India}
\email{amit@math.tifr.res.in}
\author[Y. I. Holla]{Yogish I. Holla}
\address{School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India}
\email{yogi@math.tifr.res.in}
\subjclass[2000]{14H60, 14E08, 14F22}
\keywords{Semistable projective bundle, moduli space, rationality, Brauer group, Weil pairing}
\date{}
\begin{abstract} Let $X$ be an irreducible smooth complex projective curve of genus $g$, with $g\,\geq\, 2$. Let $N$ be a connected component of the moduli space of semistable principal ${\rm PGL}_r(\mathbb C)$--bundles over $X$; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of $N$ is trivial. \end{abstract}
\maketitle
\section{Introduction}
Let $X$ be an irreducible smooth complex projective curve, with $\text{genus}(X)\, =\, g\, \geq\, 2$. For a fixed line bundle $\mathcal L$ over $X$, let $M_X(r, {\mathcal L})$ be the coarse moduli space of semistable vector bundles over $X$ of rank $r$ and determinant $\mathcal L$. It is a normal unirational complex projective variety, and if $\text{degree}({\mathcal L})$ is coprime to $r$, then $M_X(r, {\mathcal L})$ is known to be rational \cite{Ne}, \cite{KS}. Apart from these coprime case, and the single case of $g\,=\, r\, =\,\text{degree}(\mathcal L)\,=\,2$ when $M_X(r, {\mathcal L})\,=\, {\mathbb P}^3_{{\mathbb C}}$, the rationality of $M_X(r, {\mathcal L})$ is an open question in every other case. See \cite{Ho} for rationality of some other types of moduli spaces associated to $X$.
We consider the coarse moduli space $N_X(r,d)$ of semistable principal ${\rm PGL}_r(\mathbb C)$--bundles of topological type $d$ over $X$. Recall that a ${\rm PGL}_r(\mathbb C)$ bundle $P/X$ is said to be of topological type $d$ if the associated ${\mathbb P}^{r-1}$-bundle is isomorphic to ${\mathbb P}{\rm roj}({\mathbb E})$ for some rank $r$ vector bundle $E$ whose degree is congruent to $d$ modulo $r$. This $N_X(r,d)$ is an irreducible normal unirational complex projective variety. This paper is a sequel to \cite{BHH}, where we investigate the Brauer group of desginularization of moduli spaces attached to curves. This Brauer group is a birational invariant of the space and its vanishing is a necessary condition for the space involved to be rational.
In this paper we prove that the Brauer group of a desingularization of $N_X(r,d)$ is zero (see Theorem \ref{theorem}).
When $g\,=\, 2$, the moduli space $N_X(2,0)$ is a quotient of ${\mathbb P}^3_{{\mathbb C}}$ by a faithful action of the abelian group $({\mathbb Z}/2{\mathbb Z})^4$. In this special case it follows the quotient is rational.
\section{Preliminaries}\label{pril}
We continue with the above set--up and notation. Let $N_X(r,d)$ denote the coarse moduli space of S--equivalence classes of all semistable principal $\text{PGL}_r(\mathbb C)$--bundles of topological type $d$ over $X$. For notational convenience, $N_X(r,d)$ will also be denoted by $N$.
Let ${M}_X(r,{\mathcal L}_X)$ be the coarse moduli space of S-equivalence classes of semistable vector bundles over $X$ of rank $r$ and determinant ${\mathcal L}_X$. Let $\Gamma$ be the group of all isomorphism classes of algebraic line bundles $\tau$ over $X$ such that $\tau^{\otimes r}\, =\, {\mathcal O}_X$. This group $\Gamma$ has the following natural action on ${M}_X(r,{\mathcal L}_X)$: the action of any $\tau\, \in\, \Gamma$ sends any $E\, \in\, {M}_X(r,{\mathcal L}_X)$ to $E\, \otimes \, \tau$. The moduli space $N$ is identified with the quotient variety ${M}_X(r,{\mathcal L}_X)/\Gamma$. Let \begin{equation}\label{4f} f\,:\,{M}_X(r,{\mathcal L}_X)\, \longrightarrow\, {M}_X(r,{\mathcal L}_X)/\Gamma\,=\, N \end{equation} be the quotient morphism.
For notational convenience, the moduli space $M_X(r,{\mathcal O}_X)$ will also be denoted by $M_{{\mathcal L}_X}$.
Let \begin{equation}\label{ress} M_{{\mathcal L}_X}^{\rm st}\,\subset\, M_{{\mathcal L}_X} ~\, ~ \text{ and }~\, ~ N^{\rm st}\,\subset\, N \end{equation} be the loci of stable bundles. The above action of $\Gamma$ on $M_{{\mathcal O}_X}$ preserves $M_{{\mathcal O}_X}^{\rm st}$, and $$ f(M_{{\mathcal O}_X}^{\rm st})\,=\, N^{\rm st}\, . $$
\section{The action of $\Gamma$}
Consider the action of $\Gamma$ on $M_{{\mathcal O}_X}$ defined in Section \ref{pril}. For any primitive $\tau\, \in\, \Gamma$, i.e. an element of order $r$, let \begin{equation}\label{is} M^\tau_{{\mathcal O}_X}\, =\, \{E\, \in\, M_{{\mathcal O}_X}\, \mid\, E\otimes \tau\, =\, E\}\, \subset\, M_{{\mathcal O}_X} \end{equation} be the fixed point locus.
Take any nontrivial line bundle $\tau\, \in\, \Gamma$ of order $r$. Let \begin{equation}\label{res-phi} \phi\, :\, Y \, \longrightarrow\, X \end{equation} be the \'etale cyclic covering of degree $r$ given by $\tau$. We recall the construction of $Y$ as the spectral cover associated to the equation $\tau ^r \cong {\mathcal O}_X$.
Let $$ \beta\, :\, Y\, \longrightarrow\, Y $$ be a nontrivial generator of the Galois group $\text{Gal}(\phi)\,=\, {\mathbb Z}/r {\mathbb Z}$ of the covering $\phi$. The homomorphism $\xi\, \longmapsto\, \beta^*\xi$ defines an action of $\text{Gal}(\phi)$ on $\text{Pic}^d(Y)$ for any $d$.
Let \begin{equation}\label{pull-back} \phi^*:\, {\rm Pic}^0(X) \, \longrightarrow \, {\rm Pic}^0(Y) \end{equation} be the pullback homomorphism $L\, \longmapsto\, \phi^*L$. Let $K$ denote the kernel of $\phi^*$; it is a group of order $r$ generated by $\tau$. Let \begin{equation}\label{norm0} {\rm Nm}: \, {\rm Pic}^d (Y) \, \longrightarrow\, {\rm Pic}^d(X) \end{equation}
and
\begin{equation}\label{norm} {\rm N}: \, {\rm Pic}^d (Y) \, \longrightarrow\, {\rm Pic}^d(X) \end{equation}
be the norm homomorphism and the twisted norm morphism. We recall that ${\rm Nm}$ takes a line bundle $\xi$ to the descent of $\otimes (\beta^{*i}{\xi})$ and
${\rm N}$ sends a line bundle $\xi$ to ${\rm Nm}(\xi)\otimes \tau ^{(r(r-1)/2}$.
The group $\Gamma$ has a natural action on ${\rm Pic}^d (Y)$; any $\sigma\, \in\, \Gamma$ acts as the automorphism $\xi\, \longmapsto\, \xi\otimes \phi^*\sigma$. Therefore, $\phi^*$ in \eqref{pull-back} is $\Gamma$--equivariant, and the kernel $K$ acts trivially on ${\rm Pic}^d(Y)$. The morphism ${\rm N}$ in \eqref{norm} factors through the quotient morphism ${\rm Pic}^d(Y)\,\longrightarrow\,{\rm Pic}^d(Y)/\Gamma$. The action of $\Gamma$ on ${\rm Pic}^d (Y)$ clearly commutes with the action of $\text{Gal}(\phi)$ defined earlier.
Let \begin{equation}\label{recu} {\mathcal U}_{{\mathcal L}_X}\, :=\,{\rm N}^{-1}({\mathcal L}_X)\setminus ({\rm N}^{-1}({\mathcal L}_X))^{{\rm Gal}(\phi)} \, \subset \, {\rm Nm}^{-1}({\mathcal L}_X) \end{equation} be the complement of the fixed point locus for the action of $\text{Gal}(\phi)$. It is a $\Gamma$--invariant open subscheme.
Now we state a well-known result (cf. \cite[Lemma 3.4]{NR}).
\begin{lem}\label{BNR} Take any primitive line bundle $\tau\, \in\, \Gamma$. The reduced closed subscheme $$(M^{\rm st}_{{\mathcal L}_X})^\tau\,:=\,M^{\rm st}_{{\mathcal L}_X}\cap M^{\tau}_{{\mathcal L}_X}\, \subset\, M^{\rm st}_{{\mathcal L}_X}$$ (see \eqref{is} and \eqref{ress}) is $\Gamma$--equivariantly isomorphic to the quotient scheme $$ {\mathcal U}_{{\mathcal L}_X}/{\rm Gal}(\phi)\, . $$ \end{lem}
\begin{lem}\label{HP} The norm map as defined in \eqref{norm} is surjective, and there is a bijection of the set of connected components $\pi_0({\rm N}^{-1}({\mathcal L}_X))$ with the Cartier dual $K^{\vee}\, :=\, {\rm Hom}(K,\, {\mathbb C}^*)\,=\, {\mathbb Z}/r\mathbb Z$, where $K\,:=\, {\rm kernel}(\phi^*)\, =\, {\mathbb Z}/r{\mathbb Z}$. \end{lem}
Lemma \ref{HP} is proved in \cite{NR} (see \cite[Proposition 3.5]{NR}).
Let $V_0$ be the connected components of ${\rm N}^{-1}( {\mathcal O}_X)$, with ${\mathcal O}_Y\,\in\, V_0$. Since ${\rm Nm}^{-1}({\mathcal O}_X)$ is smooth, both $V_0$. is irreducible.
\begin{lem}\label{gal} The action of ${\rm Gal}(\phi)$ on ${\rm N}^{-1}({\mathcal O}_X)$ preserves the connected component $V_0$. For the action of ${\rm Gal}(\phi)$ on ${\rm N}^{-1}({\mathcal L}_X)$ the quotient ${\rm N}^{-1}({\mathcal L}_X)/{\rm Gal}(\phi)$ has exactly $(r,d)$ components which are smooth, here $(r,d)$ is the greatest common divisor of $r$ and $d$. \end{lem}
\begin{proof} The point ${\mathcal O}_Y\, \in\, \text{Pic}^0(Y)$ is fixed by ${\rm Gal}(\phi)$; hence $V_0$ is fixed by ${\rm Gal}(\phi)$. Therefore, the other component, namely $V_1$, is also fixed by ${\rm Gal}(\phi)$. See \cite{NR}, Proposition 3.5 for the proof of the second statement. \end{proof}
\begin{lem}\label{nss} Let $r=2$. The set of all points in the complement $M_{{\mathcal O}_X}\setminus M^{\rm st}_{{\mathcal O}_X}$ (see \eqref{ress}) fixed by $\tau$ is finite. \end{lem}
\begin{proof} Take any point $x\,\in\, M_{{\mathcal O}_X}\setminus M^{\rm st}_{{\mathcal O}_X}$. Let $E\,= \, L\oplus L^*$, with $L\, \in\, \text{Pic}^0(X)$, be the unique polystable vector bundle representing the point $x\,\in\, M_{{\mathcal O}_X}$. The action of $\tau$ takes the point $x$ to the point represented by the polystable vector bundle $(L\otimes \tau) \oplus (L^*\otimes\tau)$.
Assume that $\tau\cdot x\,=\,x$. Then the two vector bundles $L\oplus L^*$ and $(L\otimes \tau) \oplus (L^*\otimes\tau)$ are isomorphic. This implies that \begin{equation} \label{equal} L\otimes\tau \, \cong \, L^* \end{equation} (recall that $\tau$ is nontrivial; so $L\,\not=\, L\otimes\tau$). From \eqref{equal} it follows that $L^{\otimes 2}\,=\, (L^{\otimes 2})^*$.
Consequently, isomorphism classes of all line bundles $L\, \in\, \text{Pic}^0(X)$ satisfying \eqref{equal}, for a given $\tau$, is a finite subset. Therefore, there are only finitely many points of $M_{{\mathcal O}_X}\setminus M^{\rm st}_{{\mathcal O}_X}$ that are fixed by $\tau$. \end{proof}
\begin{rmk}\label{g2} {\rm When genus of $X$ equals $2$, ${\rm dim}({\rm Pic}^0(Y))=3$. It follows from Lemma \ref{BNR}, that $(M^{st}_{{\mathcal O}_X})^{\tau}$ is one dimensional and hence Lemma \ref{nss} implies that $M_{{\mathcal O}_X}^{\tau}$ is of codimension two in $M_{{\mathcal O}_X} \cong {\mathbb P}^3_{{\mathbb C}}$. } \end{rmk}
Let $\sigma\, \in \Gamma$ be another primitve element such that $\sigma$ and $\tau$ are ${\mathbb Z}/r{\mathbb Z}$ linearly independent. The subgroup of $\Gamma$ generated $\sigma$ and $\tau$ will be denoted by $A$. So $A$ is isomorphic to $({\mathbb Z}/r{\mathbb Z})^{\oplus 2}$.
We note that $\Gamma\subset {\rm Pic}(X)$ is identified with $H^1(X,\, {\mathbb Z}/r{\mathbb Z})\subset H^1(X,{\mathbb G}_m)$ under the natural inclusion. Let \begin{equation}\label{rese} e\, :\, \Gamma\otimes \Gamma\, \longrightarrow\, {\mathbb Z}/r\mathbb Z \end{equation} be the pairing given by the cup product $$ H^1(X,\, {\mathbb Z}/r{\mathbb Z})\otimes H^1(X,\, {\mathbb Z}/r{\mathbb Z}) \, \stackrel{\cup}{\longrightarrow}\, H^2(X,\, {\mathbb Z}/r{\mathbb Z})\,=\, {\mathbb Z}/ r\mathbb Z\, . $$ It is known that this $e$ coincides with the Weil pairing (see \cite[p. 183]{Mu}).
\begin{prop}\label{fixed-points} Let $\sigma$ and $\tau$ be two primitive elements of $\Gamma$ such that they generate a subgroup $A=({\mathbb Z}/r{\mathbb Z})^{\oplus 2}$. If the pairing $e(\sigma\,, \,\tau)\, =\, 0$ then there exists a nonempty closed irreducible $A$--invariant subset of $M_{{\mathcal O}_X}^{\rm st}$ which is fixed pointwise by $\tau$. \end{prop}
\begin{proof} By \cite{BP2}, Proposition 4.5, it follows that under the condition $e(\sigma, \tau)=0$ there is a stable bundle $E$ of rank $r$ and determinant ${\mathcal O}_X$ such that $E\otimes \sigma=E\otimes \tau=E$.
The condition $E\otimes \tau=E$ implies the existence of a line bundle $\xi \in {\rm N}^{-1}({\mathcal O}_X)$ such that $ \phi _*(\xi)=E$. The condition $E\otimes \sigma =E$ implies that there is a $\beta \in \text{Gal}(\phi)$ such that $\xi \otimes \phi ^*\sigma= \beta ^*\xi$. Hence $\phi^*\sigma = (\beta^*\xi) \otimes \xi^{-1}$ lies in $V_0$, because $\beta^*\xi$ and $\xi^{-1}$ lie in the same component (see both parts of the Lemma \ref{gal}).
One observes that ${\rm N}^{-1}({\mathcal L}_X)$ is $\Gamma$ equivariantly isomorphic to the translate $L \cdot {\rm Nm}^{-1}({\mathcal O}_X)$ by any line bundle $L$ such that $N(L)={\mathcal L}_X$. This along with the above fact that $\phi^*\sigma \in V_0$ implies that the translation by $\phi^*\sigma$ preserves the connected components of ${\rm N}^{-1}({\mathcal L}_X)$.
Hence we conclude that any connected component of the quotient ${\mathcal U}_{{\mathcal L}_X}/{\rm Gal}(\phi)\, \subset\, (M^{\rm st}_{{\mathcal O}_X})^\tau$ is a closed irreducible $A$-invariant subscheme of $M_{{\mathcal O}_X}^{\rm st}$ which is fixed pointwise by $\tau$. This completes the proof of the proposition. \end{proof}
\section{Brauer group of a desingularization of $N$}
In this section we identify the second cohomology $H^2(\Gamma,\,{\mathbb C}^*)$ with the space of alternating bi-multiplicative maps from $\Gamma$ to ${\mathbb C}^*$ (see \cite[p. 215, Proposition 4.3]{Ra}); the group $H^2(\Gamma,\, {\mathbb C}^*)$ is isomorphic to the dual of the second exterior power of $({\mathbb Z}/2{\mathbb Z})^{2g}$.
Recall that under the identification of $\Gamma$ with $H^1(X, {\mathbb Z}/r{\mathbb Z})$, the Weil pairing coincides with the intersection pairing. Let $\{a_1,b_1,\cdots a_g,b_g\}$ be a symplectic basis for $H^1(X, {\mathbb Z}/r{\mathbb Z})$. in other words we have $e(a_i,a_j)=0=e(b_i,b_j)$ for all $i$ and $j$, and $e(a_i,b_j)=\delta _{i,j}$.
Let $G\, \subset\, H^2(\Gamma,\,{\mathbb C}^*)$ be defined by $$ G\,:= \, \{ b\, \in\, H^2(\Gamma,\,{\mathbb C}^*)\, \mid \, e(\sigma_1\, ,\,\sigma_2)\, =\, 0\, \Rightarrow\, b(\sigma_1\, ,\,\sigma_2)\,= \,0\}\, . $$ Let $H$ be the subgroup of $G$ of order two generated by the Weil pairing $e$.
\begin{lem}\label{mistake} The group $G$ coincides with the subgroup $H$. \end{lem} \begin{proof} Fix an $i$ and $j$ such that $i\neq j$ then one checks that $e(a_i+b_j,a_j-b_i)=e(a_j,b_j)-e(a_i,b_i)=0$ hence if $f \in G$ then $0=f(a_i+b_j,a_j-b_i)=f(a_j,b_j)-f(a_i,b_i)$. This implies that $f$ is a multiple of $e$. \end{proof}
Our main theorem is the following. \begin{thm}\label{theorem} Let ${\widehat N}$ be a desingularization of the moduli space $N$. Then the Brauer group ${\rm Br}({\widehat N})=0$ \end{thm}
\begin{proof} We first assume that either $g\,\geq \,3$ or when $g=2$ rank $r>2$. The case of $g\,=\,2$ and rank $r=2$ will be treated separately.
It is enough to prove the theorem for some desingularization $\widehat{N}$ of $N$ because the Brauer group is a birational invariant for the smooth projective varieties. We choose a $\Gamma$--equivariant desingularization \begin{equation}\label{p} p\,:\, {\widetilde M}_{{\mathcal O}_X}\,\longrightarrow \,M_{{\mathcal O}_X} \end{equation} which is an isomorphism over $M^{\rm st}$; so ${\widetilde M}_{{\mathcal L}_X}$ is equipped with an action of $\Gamma$ given by the action of $\Gamma$ on $M_{{\mathcal L}_X}$. Define $$ {\widetilde N}\, :=\, {\widetilde M}_{{\mathcal L}_X}/\Gamma\, . $$
Let $$ {\widehat N}\, \longrightarrow\, {\widetilde N} $$ be a desingularization of ${\widetilde N}$ which is an isomorphism over the smooth locus. So ${\widehat N}$ is also a desingularization of $N$.
A stable principal $\text{PGL}_r({\mathbb C})$--bundle $E$ on $X$ is called \textit{regularly stable} if $$ \text{Aut}(E)\,=\, e $$ (by $\text{Aut}(E)$ we denote the automorphisms of the principal bundle $E$ over the identity map of $X$). It is known that the locus of regularly stable bundles in $N$, which we will denote by $N^{\rm rst}$, coincides with the smooth locus of $N$ \cite[Corollary 3.4]{BHof}. Define $$ M^{\rm rst}\, :=\, f^{-1}(N^{\rm rst})\, , $$ where $f$ is the morphism in \eqref{4f}. We note that the action of $\Gamma$ on $M_{{\mathcal L}_X}$ preserves $M^{\rm rst}$, because $f$ is an invariant for the action of $\Gamma$. The action of $\Gamma$ on $M^{\rm rst}$ can be shown to be free. Indeed, if $E\, =\, E\otimes\tau$, where $\tau$ is nontrivial, any isomorphism of $E$ with $E\otimes\tau$ produces a nontrivial automorphism of ${\mathbb P}(E)$, because ${\mathbb P}(E\otimes\tau)\,=\, {\mathbb P}(E)$. Hence such a vector bundle $E$ cannot lie in $M^{\rm rst}$.
Consequently, the projection $f$ in \eqref{4f} defines a principal $\Gamma$--bundle \begin{equation}\label{gfb} M^{\rm rst}\, \stackrel{f}{\longrightarrow}\, N^{\rm rst}\, . \end{equation} Since $N$ is normal, and $N^{\rm rst}$ is its smooth locus, it follows that the codimension of the complement $N\setminus N^{\rm rst}$ is at least two. Therefore, the codimension of the complement of $M^{\rm rst} \,\subset \, M_{{\mathcal O}_X}$ is at least two. Hence $$ H^0(M^{\rm rst},{\mathbb G}_m)={\mathbb C}^* $$ The Serre spectral sequence for the above principal $\Gamma$--bundle gives an exact sequence $$ \text{Pic}(N^{\rm rst}) \,\stackrel{\delta}{\longrightarrow}\, \text{Pic}(M^{\rm rst})^\Gamma\, \longrightarrow\, H^2(\Gamma,\, {\mathbb C}^*)\, . $$ We have $\text{Pic}(M^{\rm rst})^\Gamma/{\rm image}(\delta) \,=\, {\mathbb Z}/l{\mathbb Z}$ \cite{BH} (see (3.5) in \cite{BH} and lines following it) where $l=(r,d)$. Hence we get an inclusion \begin{equation}\label{res-go2} {\mathbb Z}/l{\mathbb Z}\, \hookrightarrow\, H^2(\Gamma,\, {\mathbb C}^*)\, , \end{equation} where the generator of ${\mathbb Z}/l{\mathbb Z}$ maps to the Weil pairing $e$ (see the proof of Proposition 9.1 in \cite[p. 203]{BLS}).
For the chosen desingularization $\widehat{N} \to N$, we have \begin{equation}\label{resem} {\rm Br}({\widehat N}) \,\subset\, {\rm Br}(N^{\rm rst})\,=\, {\rm Br}(M^{\rm rst}/\Gamma) \end{equation} using the inclusion of $N^{\rm rst}$ in ${\widehat N}$. The Brauer group ${\rm Br}(N^{\rm rst})$ is computed in \cite{BH}.
The Serre spectral sequence for the principal $\Gamma$--bundle in \eqref{gfb} gives the following exact sequence: \begin{equation}\label{rho} H^2(\Gamma,\, {\mathbb C}^*)\,\stackrel{\rho}{\longrightarrow} \,H^2(M^{\rm rst}/\Gamma,\,{\mathbb G}_m)\,\longrightarrow\,H^2(M^{\rm rst},\,{\mathbb G}_m)\, . \end{equation} Let $\mathbb S$ be the set of all bicyclic subgroups $A\, \subset\, \Gamma$ of the form $({\mathbb Z}/r{\mathbb Z})^{\oplus 2}$ satisfying the condition that there is some closed irreducible subvariety $\mathcal Z$ of ${\widetilde M}_{{\mathcal O}_X}$ preserved be the action of $A$ such that a primitive element of $A$ fixes $\mathcal Z$.
Define the subgroup $$G'\, :=\, \bigcap_{A\in{\mathbb S}}{\rm kernel}(H^2(\Gamma,\, {\mathbb C}^*) \,\rightarrow\,H^2(A,\,{\mathbb C}^*))\, \subset \, H^2(\Gamma,\,{\mathbb C}^*)\, . $$ Using a theorem of Bogomolov, \cite[p. 288, Theorem 1.3]{Bo}, we have \begin{equation}\label{i} \rho^{-1}(H^2({\widehat N},\,{\mathbb G}_m))\,\subset\,G' \end{equation} (see \eqref{resem}).
We will show that $G'$ is a subgroup of $G$ in $H^2(\Gamma,\, {\mathbb C}^*)$. This will prove that the image $\rho( \rho^{-1}(H^2({\widehat N},\,{\mathbb G}_m)))=0$
$b\, \in\, G'$. We need to check that $b(\sigma\, ,\,\tau)\, =\, 0$ whenever $e(\sigma\, ,\,\tau)\, =\, 0$ for a pair of primitive elements generating the subgroup $A={\mathbb Z}/r{\mathbb Z}^{\oplus 2}$. Since $e(\sigma\, ,\,\tau)\, =\, 0$, by Proposition \ref{fixed-points}, there is an irreducible closed subscheme $Z\, \subset\, M_{{\mathcal O}_X}^{\rm st}$ which is $A$--invariant and fixed pointwise by $\tau$.
Since the $\Gamma$--equivariant desingularization $p$ in \eqref{p} is an isomorphism over $M_{{\mathcal O}_X}^{\rm st}$, we conclude that the closure of $Z$ in ${\widetilde M}_{{\mathcal O}_X}$ is an $A$-invariant closed irreducible subscheme which is fixed pointwise by $\tau$. Hence the action of $A$ on this closure is cyclic. This implies that $A\, \in\, {\mathbb S}$, and hence $b(\sigma\, ,\,\tau)\, =\, 0$. Therefore $G'\, \subset\, G$.
Consequently, we have shown that $ H^2({\widehat N},\,{\mathbb G}_m) \cap {\rm image}(\rho)=0$.
This proves that the composition \begin{equation}\label{c1} H^2({\widehat N},\, {\mathbb G}_m) \,\longrightarrow\, H^2(M^{\rm rst}/\Gamma, \, {\mathbb G}_m)\,\longrightarrow \,H^2(M^{\rm rst},\, {\mathbb G}_m) \end{equation} is injective. We will prove that this composition is zero (these homomorphisms are induced by the inclusion $M^{\rm rst}/\Gamma\, \hookrightarrow\, \widehat N$ and the quotient map to $M^{\rm rst}/\Gamma$).
Consider the diagram $$ \xymatrix{ {\widehat M} \ar[r]\ar[d] & {\widetilde M}_{{\mathcal L}_X} \ar[d] \\ {\widehat N}\ar[r] & {\widetilde N}} $$ where ${\widehat M}$ is a $\Gamma$--equivariant desingularization of the closure of $$ M^{\rm rst}\,=\, {\widehat N}\times _{N^{\rm rst}} M^{\rm rst}$$ in the fiber product ${\widehat N}\times_{{\widetilde N}}{\widetilde M}_{{\mathcal L}_X}$. This gives an action of $\Gamma$ on the smooth projective variety ${\widehat M}$ which has a $\Gamma$--invariant open subscheme $M^{\rm rst}$ with the quotient $M^{\rm rst}/\Gamma$ being the Zariski open subset $N^{\rm rst}$ of ${\widehat N}$. Using the commutativity of $\Gamma$--actions we obtain a commutative diagram of homomorphisms \begin{equation}\label{dg} \xymatrix{ H^2({\widehat N}, {\mathbb G}_m) \ar[r]\ar[d] & H^2(N^{\rm rst}, {\mathbb G}_m) \ar[d] \\ H^2({\widehat M}, {\mathbb G}_m)\ar[r] & H^2(M^{\rm rst}, {\mathbb G}_m)} \end{equation} Since ${\widehat M}$ is also a desingularization of $M$, we conclude by \cite[p. 309, Theorem 1]{Ni} (see also \cite[Theorem 1]{BHH}) that $$ H^2({\widehat M}, \,{\mathbb G}_m)\,=\, 0\, . $$ Hence from \eqref{dg} it follows that the image of $H^2({\widehat N}, \,{\mathbb G}_m)$ in $H^2(M^{\rm rst},\, {\mathbb G}_m)$ by the composition in \eqref{c1} is zero. This completes the proof when $g\,\geq\, 3$.
Now assume that $g\,=\,2$ and rank $r=2$. So $M_{{\mathcal L}_X}$ is already smooth. We take ${\widetilde M}_{{\mathcal L}_X}\,=\, M_{{\mathcal L}_X}$. Let $M^{\rm free} \,\subset\, M_{{\mathcal L}_X}$ be the largest Zariski open subset where the action of $\Gamma$ is free. It follows from Remark \ref{g2} that the complement of $M^{\rm free}$ is of codimension two. The entire argument above works in this case after replacing $M^{\rm rst}$ by $M^{\rm free}$ and $N^{\rm rst}$ by $M^{\rm free}/\Gamma$. \end{proof}
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\begin{document}
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\title{Generic polynomials for cyclic function field extensions over certain finite fields} \begin{abstract} In this paper, we find all the generic polynomials for geometric $\ell$-cyclic function field extensions over the finite fields $\mathbb{F}_q$ where $q= p^n$, $p$ prime integer such that $q \equiv -1 \mod \ell$ and $(\ell , p)=1$. \end{abstract} Throughout the paper we will adopt the following notations unless mentioned differently. $K$ denotes a one variable function field with field of constants $\mathbb{F}_q$ where $q= p^n$, $p$ prime integer. Given $x \in K$, we denote $\mathcal{O}_{K,x}$ is the integral closure of $\mathbb{F}_q[x]$ in $L$. Let $L/K$ be a cyclic Galois extension of degree $\ell$ with $(\ell , p)=1$. We write $\ell = \prod_{i=1}^s \ell_i^{f_i}$ where $\ell_i$ are distinct prime integers and $f_i$ positive integer. Let $\xi$ be a primitive $\ell^{th}$-root of unity. We suppose that $q\equiv -1 \mod \ell$. So that, by \cite[Theorem 2.10]{Con2}, $\mathbb{F}_q$ do not contains any primitive $\ell^{th}$ root of unity. More precisely, we have $[\mathbb{F}_q( \xi) : \mathbb{F}_q]=2$ and the minimal polynomial of $\xi$ over $ \mathbb{F}_q$ is $X^2 - (\xi + \xi^{-1} )X +1$ and $(\xi + \xi^{-1} )\in \mathbb{F}_q$. We denote by $\sigma$ the generator of $Gal(\mathbb{F}_q( \xi) ,\mathbb{F}_q)$, we have that $\sigma (\xi) = \xi^{-1}$.
In this paper, our main result finds all the generic polynomials for geometric cyclic function field extensions over those finite fields $\mathbb{F}_q$ (Theorem \ref{m}). We find a one parameter family of generic polynomials of a cyclic extensions when $\ell$ is odd and a two parameters family of generic polynomials when $\ell$ is even (Corollary \ref{cm}). We also classify cyclic extensions up to isomorphism over those finite field $\mathbb{F}_q$ (Lemma \ref{clas}). Note that, in particular, this permits to classify all the geometric cyclic extensions of degree $3$, $4$ and $6$ over any finite fields $\mathbb{F}_q$. We describe the Galois action on a generator with a minimal polynomial in our form (Corollary \ref{gal}). We end the paper with the study of the ramification in term of our generation (Theorem \ref{ram}). In particular, we find that under our assumptions the ramified places are of even degree.
\section{Generic polynomials for cyclic extensions} \begin{lemma} \label{1} Let $L/K$ be a cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. Suppose $w$ is a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is of the form $X^{\ell} -a$, $y:=\sigma (w) + w$ is a generator for $L/K$ and $u:=\sigma(w) w \in K$. We write $\ell= 2 \iota + r$, where $r=0,1$. Then, the minimal polynomial of $y$ over $K$ is of the form $$ P^{\ell}_{u , \alpha} (X)= X^\ell -\ell u X^{\ell-2} + \sum_{s=1}^{\iota} c_{s,\ell} u^s X^{\ell -2s} - \alpha$$ where $\alpha = a + \sigma (a)$ and $ u^\ell = a \sigma(a)$, the coefficients $c^r_{s,j} \in \mathbb{F}_p$ and are defined recursively as $$c_{s,j}^r= - \sum_{k=1}^{s} \binom{2j+r}{k} c_{s-k,j-1-k} \in \mathbb{F}_p$$ for $ 0 \leq s \leq j$ and $1 \leq j \leq \ell$ and $c_{0,j}^r=1$ for all $j \geq 0$. In particular, we have that $c_{1,j}^r= -(2j+r)$, $c_{j,j}^1= (-1)^{j}(2j+1)$ and $c_{j,j}^0= (-1)^j2 $.\\ Defining, $$ Q^{l}_{u } (X)= X^l -\ell u X^{l-2} + \sum_{s=1}^{\iota} c_{s,l} u^s X^{l-2s} $$ for any $l$ integer where the $c_{s,l}$ are define recursively as before, we have that if $\ell= \prod_{i=1}^t l_i$ where $l_i$ are not necessarily distinct factors of $\ell$, then $$P^\ell_{u, \alpha} (X) = P_{l_t, u^{l/l_t} , \alpha} ( \cdots ( Q_{l_2, u^{l_1} } ( Q_{l_1, u}(X))$$
\end{lemma} \begin{proof} Under the assumption of the theorem we determine the minimal polynomial of $y=w + \sigma (w)$. For if, $$\begin{array}{ccl} y^\ell &=& (w + \sigma (w))^\ell= \sum_{i=0}^\ell \binom{\ell}{i} w^i \sigma(w)^{\ell-i} \\ &=& w^\ell + \sigma(w)^\ell +\sum_{i=1}^{\ell-1} \binom{\ell}{i} w^i \sigma(w)^{\ell-i}\\ &=& a + \sigma (a ) +\sum_{i=1}^{\iota-(1-r)} \binom{\ell}{i} w^i \sigma(w)^{i}(\sigma(w)^{\ell-2i}+w^{\ell-2i}) + (1-r)\binom{\ell}{\iota} w^\iota \sigma (w)^\iota \\ &=& a + \sigma (a) +\sum_{i=1}^{\iota-(1-r)} \binom{\ell}{i}u^i (\sigma(w)^{\ell-2i}+w^{\ell-2i}) + (1-r)\binom{\ell}{\iota} u^\iota \\ \end{array}$$ Note that $\ell - 2i$ is the same parity of $\ell$.
We set $\omega_j^r= w^{2j+r}+ \sigma(w)^{2j+r} $ for $0 \leq j\leq \frac{\ell -1}{2}$ and $(j,r)\neq 0$ and $\omega_0^0=1$, then $$\omega_{j}^r=w^{2j+r}+ \sigma(w)^{2j+r} = y^{2j+r} - \sum_{k=1}^{j} \binom{2j+r}{k} u^k \omega_{j-k}^r $$ We have $$\omega_{0}^1 = y,$$ $$\omega_{1}^1 = y^3-3 u y,$$ $$\omega_{2}^1 = y^5-5 u y^3+5 u^2 y,$$ $$\omega_{1}^0 = y^2-2u$$ $$\omega_{2}^0 = y^4-4uy^2+2u^2$$ By induction in $j$ for $r=0,1$, we prove that $$\omega_j^r = \sum_{s=0}^j c_{s,j}^r u^s y^{2j+r-2s}$$ where the coefficients are obtained recursively as
$c_{s,j}^r=- \sum_{k=1}^{s} \binom{2j+r}{k} c_{s-k,j-k}^r $, $1\leq s \leq j$ and $j\geq 1$ and $c_{0,j}^r=1$ for all $j \geq 0$. We also prove that $c_{1,j}^r= -(2j+r)$, $c_{j,j}^1= (-1)^{j}(2j+1)$ and $c_{j,j}^0= (-1)^j 2$.\\
From the computation above we see that this property is true for $j=1$. Suppose that this property is true for $\omega_t^r$, $1\leq t \leq j$, for some fixed $j$. We want to prove it remain true for $w_{j+1}^r$. Using the induction assumption, we obtain $$\begin{array}{ccl} \omega_{j+1}&=& y^{2(j+1)+r} - \sum_{k=1}^{j+1} \binom{2(j+1)+r}{k} u^k \omega_{j+1-k}^r \\ &=& y^{2(j+1)+r} -\sum_{k=1}^{j+1} u^k \binom{2(j+1)+r}{k} \sum_{s=0}^{j+1-k} c_{s,j+1-k}^r u^s y^{2(j+1)+r-2(s+k)} \\
&=& y^{2(j+1)+r} - \sum_{k=1}^{j+1} \sum_{s'=k}^{j+1} \binom{2(j+1)+r}{k} u^{s'} c_{s'-k,j+1-k}^r y^{2(j+1)+r-2s'} \ \text{ where \ $s' := s+k$} \\ &=& y^{2(j+1)+r} + \sum_{s'=1}^{j+1} (-\sum_{k=1}^{s'} \binom{2(j+1)+r}{k} c_{s'-k,j+1-k}^r ) u^{s'} y^{2(j+1)+r-2s'} \end{array}$$
Thus, $c_{s,j+1}^r=- \sum_{k=1}^{s} \binom{2(j+1)+r}{k} c_{s-k,j+1-k}^r $, for $1\leq s \leq j+1$.
Clearly, $c_{1,j+1}^r= -(2(j+1)+r)$.
Note that \begin{equation}
\begin{array}{ccl} 0 &=& (1-1)^{2n} \\ &=& \sum_{k=0}^{2n} \binom{2n}{k} (-1)^k \\ &=& \sum_{k=0}^n \binom{2n}{k} (-1)^k + \sum_{k=n+1}^{2n} \binom{2n}{k} (-1)^k \\ &=& \sum_{k=0}^n \binom{2n}{k} (-1)^k + \sum_{k=0}^{n-1} \binom{2n}{k} (-1)^k \\ &=& 2 \sum_{k=0}^{n-1} \binom{2n}{k} (-1)^k + \binom{2n}{n} (-1)^n \end{array} \end{equation}
Using the induction assumption, we have, $$\begin{array}{ccl} c_{j+1, j+1}^0 & =& - \sum_{k=1}^{j+1} \binom{2(j+1)}{k} c_{j+1-k,j+1-k}^0\\ &=& -2 (-1)^{j+1} \sum_{k=1}^{j} \binom{2(j+1)}{k} (-1)^k - \binom{2(j+1)}{j+1} \\ &=& 2 (-1)^{j+1} -2 (-1)^{j+1} \sum_{k=0}^{j} \binom{2(j+1)}{k} (-1)^k - \binom{2(j+1)}{j+1} \\ &=& (-1)^{j+1} 2 \text{ using $(1)$} \end{array}$$ and $$\begin{array}{ccl} & & c_{j+1, j+1}^1\\ & =& - \sum_{k=1}^{j+1} \binom{2(j+1)+1}{k} c_{j+1-k,j+1-k}^0\\ &=&- (-1)^{j+1} \sum_{k=1}^{j+1} \binom{2(j+1)+1}{k} (-1)^{k}(2(j+1-k)+1) \\ &=& -(-1)^{j+1} \sum_{k=1}^{j+1} \binom{2(j+1)+1}{k} (-1)^{k}(2(j+1)+1) + 2(-1)^{j+1} \sum_{k=1}^{j} \binom{2(j+1)+1}{k} k (-1)^{k} \\ &=& -(-1)^{j+1} (2(j+1)+1)( \sum_{k=1}^{j+1} (\binom{2(j+1)+1}{k}- 2\binom{2(j+1)}{k-1}) (-1)^{k} ) \\ &=& -(-1)^{j+1} (2(j+1)+1)( \sum_{k=1}^{j} \binom{2(j+1)}{k} (-1)^{k} - \sum_{k=1}^{j+1} \binom{2(j+1)}{k-1}) (-1)^{k}) \\ &=& -(-1)^{j+1} (2(j+1)+1) (\sum_{k=1}^{j+1} \binom{2(j+1)}{k} (-1)^{k} + \sum_{k=0}^{j} \binom{2(j+1)}{k}) (-1)^{k}) \\ &=& (-1)^{j+1} (2(j+1)+1) \text{ using (1)} \end{array}$$ Moreover, note that when $\ell= \prod_{i=1}^t l_i$ where $l_i$ are non necessarily distinct factor of $\ell$, the minimal polynomial of $y$ over $L^{l_1}$ the fixed field of $L$ by the subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ l_1 \mathbb{Z}$ is $$P^{l_1}_{u, \alpha_1}(X)$$ where $\alpha_1 = w^{l_1} + \sigma(w^{l_1} )$.
We let $Q^{l_1}_{ u}= P^{l_1}_{ u, \alpha_1}+ \alpha_1$. Since $y$ is a generator for $L/K$, we have that $[ K(w^{l_1} + \sigma(w^{l_1} )) : K] =\ell/ l_1$ and $w^{l_1}$ is a Kummer generator for $[ K(\xi )(w^{l_1} ) : K(\xi)]$, thus the minimal of $y$ over $L^{l_1l_2}$ the fixed field of $L$ by the subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ l_1 l_2 \mathbb{Z}$ is $$P^{l_2}_{ u^{l_1} , \alpha_2 } ( Q_{l_1, u}(X))$$ where $\alpha_2 = w^{l_1l_2} + \sigma(w^{l_1l_2} )$. Recursively, we obtain that the minimal polynomial $y$ of $L/K$ is $$P^{l_t}_{ u^{l/l_t} , \alpha} ( \cdots ( Q^{l_2}_{ u^{l_1} } ( Q^{l_1}_{ u}(X))$$ where $Q^{l_k}_{ u}= P^{l_k}_{ u, \alpha_k}+ \alpha_k$ and $\alpha_k= w^{l_1l_2\cdots l_k} + \sigma(w^{l_1l_2\cdots l_k} )$.
By uniqueness of the minimal polynomial of $y$ over $K$, we obtain that $$P^\ell_{u, \alpha} (X)= P_{l_t, u^{l/l_t} , \alpha} ( \cdots ( Q_{l_2, u^{l_1} } ( Q_{l_1, u}(X))$$ \end{proof}
\begin{remarque} \begin{enumerate} \item The previous lemma hold for general field extensions over a field of positive characteristic.
\item Note that when $2|\ell$, the polynomial $P^{\ell}_{u , \alpha}(X)$ only involve even degree monomials and when $(\ell , 2)=1$, the polynomial $P^{\ell}_{u , \alpha}(X)$ only involve odd degree monomials. \item We list some of those polynomials
\begin{itemize} \item[$\cdot$] $P^{3}_{u , \alpha}(X)=X^3-3uX- \alpha$, \item[$\cdot$] $P^{5}_{u , \alpha}(X)=X^5-5u X^3+5u^2X - \alpha$, \item[$\cdot$] $P^{7}_{u , \alpha}(X)=X^7-7uX^5+14u^2 X^3-7u^3X-\alpha$, \item[$\cdot$] $P^{9}_{u , \alpha}(X)= X^9-9uX^7+27u^2X^5-30u^3X^3+9u^4X-\alpha$, \item[$\cdot$] $P^{11}_{u , \alpha}(X)=X^{11}-11uX^9+44u^2X^7-77u^3X^5+55u^4 X^3-11u^5 X- \alpha$ \item[$\cdot$] $P^{13}_{u , \alpha}(X)=X^{13}-13uX^{11}+65u^2X^9-156u^3X^7+182u^4X^5-91u^5X^3+13 u^6X - \alpha$ \item[$\cdot$] $P^{2}_{u , \alpha}(X)=X^2-2u-\alpha$, \item[$\cdot$] $P^4_{u , \alpha}(X)= X^4-4uX^2+2u^2- \alpha$ \item[$\cdot$] $P^6_{u , \alpha}(X)=X^6-6uX^4+9u^2X^2-2u^3 - \alpha$ \item[$\cdot$] $P^8_{u , \alpha}(X)= X^8-8uX^6+20u^2X^4-16u^3X^2+2u^4- \alpha$. \end{itemize} We recognize $P^{3}_{u , \alpha}(X)$ when $u=1$ being the generic polynomials obtained in \cite{MWcubic} and \cite{MWcubic2}. We will prove that those polynomials $P^{\ell}_{u , \alpha}(X)$ are generic polynomials for cyclic extensions of degree $\ell$ over $\mathbb{F}_q$ when $q \equiv -1 \ mod \ 3$. \end{enumerate} \end{remarque}
\begin{lemma}\label{2} Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. For any $z$ Kummer generator for $L(\xi)/ K(\xi)$, $z\sigma (z)\in K$.
\end{lemma} \begin{proof} By \cite[Theorem 5.8.5]{Vil}, we know that $L(\xi ) / K(\xi)$ is a Kummer extension thus there exists a Kummer generator $z$ whose minimal polynomial is $X^\ell -a$ with $a \in K(\xi )$. Note that $\sigma (z)$ is also a Kummer generator for $L(\xi) / K(\xi)$ with minimal polynomial $X^\ell - \sigma(a)$.
We can write $L(\xi) / K(\xi)$ as a tower of prime degree extension $$z^{\ell_1} = w_{1, 1}, \cdots , w_{1,f_1}^{\ell_2}= w_{2,1} , \cdots , w_{s,f_s}^{\ell_s} = a$$ and we get a tower of cyclic prime degree extension $$L(\xi)/K(\xi)(w_{1,1}) /\cdots \cdots / K(\xi)( w_{s,f_s})/ K(\xi ).$$ We set $w_{0,1}=z$, $w_{s+1, f_{s+1}}=a$ and $f_0= f_{s+1}=1$.
We want to prove that $z \sigma(z) \in K$. Since $\sigma (z \sigma (z) ) = z \sigma (z)$, $z\sigma ( z )\in L$ and we have $$(z\sigma(z))^{\ell_1} = w_{1, 1}\sigma(w_{1,1}), \cdots , (w_{1,f_1}\sigma (w_{1,f_1}))^{\ell_2}= w_{2,1} \sigma(w_{2,1}), \cdots , (w_{s,f_s}\sigma (w_{s,f_s}))^{\ell_s} = a\sigma (a)$$
Similarly, $w_{i,j} \sigma (w_{i,t_i}) \in K( w_{i,t_i})$, for any $1 \leq i \leq s$ and $1\leq t_i \leq f_i$.
We have two cases, either all the $\ell_i$ are odd or one of the $\ell_i $ is equal to $2$, in which case we can suppose without loss of generality that $\ell_s=2$. \begin{enumerate} \item {\sf Case 1: all $\ell_j$ are odd, for $1\leq j \leq s$.}
Since $(w_{i,j} \sigma (w_{i,j} )) ^{\ell_k} = w_{k, l}\sigma(w_{k,l})$, for any $0 \leq i \leq s$ and $1\leq t_i \leq f_i$ with \begin{itemize} \item[$\cdot$] $k= i+1$ and $l=1$, when $k=i$ and $j= f_i$, \item[$\cdot$] $l=j+1$ and $i=k$, if $ 0\leq j< f_i$, \end{itemize} then either $w_{i,j} \sigma (w_{i,j}) \in K(w_{k, l}\sigma(w_{k,l}))$ or $K(w_{i,j} \sigma (w_{i,j} ))/ K(w_{k, l}\sigma(w_{k,l}))$ is a extension of degree $\ell_k$ in $L/K$. But the later case is impossible since $K(w_{i,j} \sigma (w_{i,j} ))/ K(w_{k, l}\sigma(w_{k,l}))$ is a subextension in the cyclic extension $L/K$ so it would be a Kummer extension, which is contradictory to the assumption that $q \equiv -1 \ mod\ \ell$ implying that $q \equiv -1 \ mod \ \ell_k$ (see \cite[Theorem 5.8.5]{Vil}). From this, we can prove that $$[K(z \sigma (z))/K ] = [ K (w_{1,1} \sigma (w_{1,1}) )/ K] = \cdots = [K(w_{s,f_s}\sigma (w_{s,f_s}))/K]=1$$ And thus, $z\sigma (z) \in K$. \item {\sf Case 2: one of the $\ell_i $ is equal to $2$, in which case we can suppose without loss of generality that $\ell_s=2$.}
From {\sf Case 1}, we know that $$[K(z \sigma (z))/K ] = [K(w_{s,1}\sigma (w_{s,1}))/K]$$ \begin{enumerate} \item If $f_s=1$, we have $(w_{s,1}\sigma (w_{s,1}))^2= a\sigma (a)$.
Thus either $w_{s,j}\sigma (w_{s,j}) \in K(w_{s,j+1}\sigma (w_{s,j+1}))$ or $w_{s,j}\sigma (w_{s,j} )$ is a Kummer generator for $K(w_{s,1}\sigma (w_{s,1}))/K$ and thus for $K(\xi)(w_{s,1}\sigma (w_{s,1}))/K(\xi)$. But then $w_{s,1}\sigma (w_{s,1})$ and $w_{s,1}$ are two Kummer generators for $K(\xi)(w_{s,1}\sigma (w_{s,1}))/K(\xi)$. Hence, by \cite[Theorem 5.8.5]{Vil}, there is $d\in K(\xi)$ such that $w_{s,1}\sigma (w_{s,1})= d w_{s,1}$, thus $ w_{s,1} = \sigma(d) \in K(\xi)$ which is impossible since $[K(\xi)(w_{s,1})/K(\xi)]=2$ by assumption. Thus $w_{s,1}\sigma (w_{s,1}) \in K$ and $[K(z \sigma (z))/K ]=1$ and $z\sigma(z)\in K$. \item If $f_s>1$,
$$(w_{s,j}\sigma (w_{s,j}))^4 = w_{s,j+2}\sigma (w_{s,j+2})$$ for $1 \leq j \leq f_s-1$. Thus either $w_{s,j}\sigma (w_{s,j})$ is a Kummer generator of the degree $4$ subextension $K(w_{s,j}\sigma (w_{s,j}))/ K( w_{s,j+2}\sigma (w_{s,j+2}))$ of $L/K$ (since any subextension of a cyclic extension is cyclic) or $w_{s,j}\sigma (w_{s,j}) \in K(w_{s,j+1}\sigma (w_{s,j+1})$. But since $q \equiv -1 \ mod \ 2^{e_s}$ thus $q \equiv -1 \ mod \ 4$ and since $K$ does not contain a primitive $4^{th}$ root of unity $w_{s,j}\sigma (w_{s,j})$ cannot be a Kummer generator of degree $4$ and $w_{s,j}\sigma (w_{s,j}) \in K(w_{s,j+1}\sigma (w_{s,j+1})$. So that, we can prove that $$[K(z \sigma (z))/K ] = [K(w_{s,1}\sigma (w_{s,1}))/K]=[K(w_{s,2}\sigma (w_{s,2}))/K] = [K(w_{s,e_s}\sigma (w_{s,e_s}))/K]$$ with $$(w_{s,e_s} \sigma (w_{s,e_s}))^2 = a \sigma (a)$$ And using the same reasoning as in the case $(a)$ where $f_s=1$ we can deduce that $w_{s,e_s} \sigma (w_{s,e_s})\in K$ and finally, we have again $z \sigma (z)\in K$. \end{enumerate} \end{enumerate}
\end{proof}
\begin{lemma} \label{3} Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. For any $z$ Kummer generator for $L(\xi)/ K(\xi)$, there is $\eta \in \mathbb{F}_q(\xi)^*$ such that $\sigma (\eta z) + \eta z$ is a generator for $L/K$ and $\sigma (\eta ) \eta =1$. \end{lemma} \begin{proof} We write $z= z_1 + \xi z_2$, we have thus that $z_1, \ z_2 \in L$ and $z_2\neq 0$ since $z \notin L$ and not both $z_1$ and $z_2$ are in proper the subextension of $L/K$ over $K$ since $z$ is a generator for $L(\xi ) / K(\xi)$.
We have $\xi + \xi^{-1} \neq \pm 2$, since if this equality was satisfied, we would have $X^2 - (\xi + \xi^{-1} )X +1= X^2 \mp 2 X +1$ is not irreducible over $\mathbb{F}_q$ contradicting that $q \equiv -1 \ mod \ \ell$ and thus $[\mathbb{F}_q(\xi): \mathbb{F}_q]=2$.
Taking $\zeta = \frac{ \xi - \xi^{-1} }{( \xi + \xi^{-1} )^2-4 },$ $$ \zeta z + \sigma (\zeta z) = \frac{ \xi - \xi^{-1} }{ ( \xi + \xi^{-1} )^2-4 } (z_1 +\xi z_2) + \frac{ \xi^{-1} - \xi }{ ( \xi + \xi^{-1} )^2-4 } (z_1 +\xi^{-1} z_2)= z_2. $$
Taking $\zeta' = \frac{ \xi^{-2} - 1 }{( \xi + \xi^{-1} )^2-4 }$, we get $$ \zeta' z + \sigma (\zeta' z) = \frac{ \xi^{-2} - 1 }{ ( \xi + \xi^{-1} )^2-4 } (z_1 +\xi z_2) + \frac{ \xi^{2} - 1 }{ ( \xi + \xi^{-1} )^2-4 } (z_1 +\xi^{-1} z_2)= z_1. $$
For any $1 \leq k \leq f_i$ and any $1 \leq i \leq s$, we denote $L^{\ell_i^k}$ be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/\ell_i^{k} \mathbb{Z}$. Since $z$ is a generator for $L(\xi)$ over $K(\xi)$, then $z$ is also a generator for $L(\xi)$ over $L^{\ell_i^k}(\xi ) $, for any $1 \leq k \leq f_i$ and any $1 \leq i \leq s$. Given $ 1 \leq i \leq s$, either $z_1$ or $z_2$ is a generator of $L/L^{\ell_i^{f_i}}$, indeed if that was not the case then both $z_1$ and $z_2$ would belong to a proper subextension of $L/L^{\ell_i^{f_i}}$ but since any such proper extension is contained in $L^{\ell_i^{f_i-1}}/K$. Then $z_1$ and $z_2$ would both belong to $L^{\ell_i^{f_i-1}}$ and $z=z_1 + \xi z_2 \in L^{\ell_i^{f_i-1}}(\xi)$ contradicting that $z$ is a generator for $L(\xi ) / L^{\ell_i^{f_i}}(\xi )$.
That also implies in this case, that either $\mathfrak{z}_1=z + \sigma (z)$ or $\mathfrak{z}_2=\xi^{-1} z+ \sigma (\xi^{-1} z)$ is a generator for $L/L^{\ell_i^{f_i}}$. Indeed, $z + \sigma (z) = 2 z_1 + (\xi + \xi^{-1} ) z_2$ and $\xi^{-1} z + \sigma (\xi^{-1} z) = 2 z_2 + (\xi + \xi^{-1} ) z_1$. As before, if none of them was a generator for $L/L^{\ell_i^{f_i}}$, we would have that they both belong to $L^{\ell_i^{f_i-1}}$ but then the same would be true for $$\begin{array}{ccl} && (\xi + \xi^{-1} ) (\xi^{-1} z + \sigma (\xi^{-1} z) )- 2 (z + \sigma ( z) )\\ &=& [(\xi + \xi^{-1} )^2-4] z_2\\ &=& (\xi + \xi^{-1} -2) (\xi + \xi^{-1} +2) z_2 \end{array}$$ and for $$\begin{array}{ccl} && (\xi + \xi^{-1} ) (z + \sigma ( z) )-2(\xi^{-1} z + \sigma (\xi^{-1} z) ) \\ &=& [(\xi + \xi^{-1} )^2-4] z_1\\ &=& (\xi + \xi^{-1} -2) (\xi + \xi^{-1} +2) z_1 \end{array}$$ Contradicting that either $z_1$ or $z_2$ is a generator for $L/L^{\ell_i^{f_i}}$, since $\xi + \xi^{-1}\neq \pm 2$.\\
When $s=1$, the theorem is proven by the above.
Otherwise, when $s>1$, note that if we have $w\in L$ generator for $L$ over $L^{\ell_i^{f_i}}$ and $L^{\ell_j^{f_j}}$ for $i\neq j$. Then $[L: L^{\ell_i^{f_i}}]=[L^{\ell_i^{f_i}}(w) : L^{\ell_i^{f_i}}]= \ell_i^{f_i}$ and $[L: L^{\ell_j^{f_j}}]=[L^{\ell_j^{f_j}}(w) : L^{\ell_j^{f_j}}]= \ell_j^{f_j}$ and $ \ell_i^{f_i} \ell_j^{f_j}| [L^{\ell_i^{f_i}\ell_j^{f_j}}(w) : L^{\ell_i^{f_i}\ell_j^{f_j}}]$ where $L^{\ell_i^{f_i}\ell_j^{f_j}}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/(\ell_i^{f_i} \ell_j^{f_j})\mathbb{Z}$ and $L=L^{\ell_i^{f_i}\ell_j^{f_j}}(w)$. Thus, $\mathfrak{z}_1$ is a generator for $L/L^I$ and $\mathfrak{z}_2$ is a generator for $L/L^J$ where $I$ and $J$ are subsets of $\{ 1 , \cdots , s\}$ such that $I\cup J=\{ 1 , \cdots , s\}$ with $I$ and $J$ maximal subsets having this property and defining $L^I$ to be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ (\prod_{i \in I}\ell_i^{f_i}) \mathbb{Z}$ and $L^J$ to be the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ (\prod_{i \in J}\ell_i^{f_i} ) \mathbb{Z}$. \\
As a consequence, if $\mathfrak{z}_1 = k \mathfrak{z}_2$ for some $k \in K^*$ then $\mathfrak{z}_1=z + \sigma (z)$ and $\mathfrak{z}_2=\xi^{-1} z+ \sigma (\xi^{-1} z)$ are both generators for $L/K$ and the theorem is proven. \\
Also, if either $I$ or $J$ is the all $\{1,\cdots , s\}$ then either $\mathfrak{z}_1=z + \sigma (z)$ or $\mathfrak{z}_2=\xi^{-1} z+ \sigma (\xi^{-1} z)$ is a generator for $L$ over $K$ and the theorem is again proven.\\
Otherwise, there is $i_0 \in \{ 1, \cdots , s\} \backslash I$ and $i_1 \in \{ 1, \cdots , s\} \backslash J$ and $i_0\neq i_1$.
Note also that since by assumption $\ell= \prod_{i=1}^s \ell_i^{f_i} | q-1$ then $q-1\geq \ell > s$. Indeed, $\ell$ is bigger or equal to the product of the $s$ smallest prime numbers and the $s^{th}$ smallest prime number is already bigger than $s$. We denote $U_{q+1}$ the set of the $(q+1)^{th}$ roots of unity in $\mathbb{F}_q(\xi)^*$, we have $U_{q+1}= q+1>s+2$. Note that by assumption $\xi$ and $\xi^{-1}$ is a root of $R(X)=X^2 - (\xi + \xi^{-1}) X+1$ with $ (\xi + \xi^{-1}) \in \mathbb{F}_q$. Hence, we have also that $\xi^q$ is a root of $R(X)$ and thus either $\xi= \xi^q$ or $\xi^{-1} = \xi^q$ but since $\xi\notin \mathbb{F}_q$, we have $\xi^{-1} = \xi^q$. So that, for $\chi=c_1 + \xi c_2\in U_{q+1}$, we have $$\chi \sigma (\chi ) = (c_1+ \xi c_2) (c_1+ \xi^{-1} c_2)= (c_1+ \xi c_2) (c_1+ \xi^{q} c_2)= (c_1+ \xi c_2)^{q+1}=1$$
Moreover, $\mathfrak{z}_1 , \ \mathfrak{z}_2$ and $\mathfrak{z}_1 + \chi \mathfrak{z}_2$ are distinct elements in $L$, for $\chi \in \mathbb{F}_q(\xi)^*$ and $\xi^{-1} \chi +1\in U_{q+1}$, since by assumption, $\mathfrak{z}_1 \neq k \mathfrak{z}_2$, for any $k, l \in K$. \\
Let $\chi_1 \in \mathbb{F}_q(\xi)^*$ such that $1 + \chi_1 \xi^{-1} \in U_{q+1}$ and $1 \leq i \leq s$, then either $\mathfrak{z}_1$ or $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2$ is a generator for $L/ L^{\ell_i^{f_i}}$, indeed, otherwise as before $\mathfrak{z}_1$ and $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2$ would both belong to $ L^{\ell_i^{f_i-1}}$, and $\chi_1 \mathfrak{z}_2 = \mathfrak{z}_1 + \chi_1 \mathfrak{z}_2- \mathfrak{z}_1\in L^{\ell_i^{f_i-1}}$, then $\mathfrak{z}_1$ and $\mathfrak{z}_2$ would both belong to $L/ L^{\ell_i^{f_i-1}}$ again contradicting that $z$ generates $L$ over $K$ as proven above. Thus, $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2$ is a generator of $L/L^{J_1}$ where $J_1$ is a subset of $\{ 1 , \cdots , s\}$ such that $I\cup J_1=\{ 1 , \cdots , s\}$ with $J_1$ maximal subset having this property and $L^{J_1}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ (\prod_{i \in J_1}\ell_i^{f_i} )\mathbb{Z}$. Similarly, given $1\leq i \leq s$, we can prove that either $\mathfrak{z}_2$ or $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2$ is a generator for $L/ L^{\ell_i^{f_i}}$ and thus $J\cup J_1= \{1 ,\cdots , s\}$. \\
If $J_1$ is all $\{1 , \cdots , s\}$ then $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2 = \eta z + \sigma ( \eta z )$ is a generator $L/K$ where $\eta = 1 + \chi_1 \xi^{-1} \in U_{q+1} $ thus $\eta \sigma (\eta )=1$, proving the theorem. \\
Otherwise, there is $i_2 \in \{1 ,\cdots , s\} \backslash J_1$. But since $J\cup J_1= \{1 ,\cdots , s\}$ and $I\cup J_1= \{1 ,\cdots , s\}$ then $i_0$ and $i_1\in J_1$ thus $i_0$, $i_1$ and $i_2$ are distinct and $|J_1|\geq 2$.
We then let $\chi_2 \in \mathbb{F}_q(\xi)^*$ such that $1 + \chi_2 \xi^{-1} \in U_{q+1}$ and $\chi_1 \neq \chi_2$, as before we prove that $\mathfrak{z}_1 + \chi_2 \mathfrak{z}_2 $ is also a generator of $L/L^{J_2}$ where $J_2$ is a subset of $\{ 1 , \cdots , s\}$ such that $I\cup J_2=\{ 1 , \cdots , s\}$, $L^{J_2}$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ (\prod_{i \in J_2}\ell_i^{e_i} )\mathbb{Z}$ with $I\cup J_2= \{ 1, \cdots , s \}$ and $J \cup J_2= \{ 1, \cdots , s \}$ with $J_2$ maximal subset having this property. Given $1 \leq i \leq s$, we can also prove using the same argument as before that either $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2$ or $\mathfrak{z}_1 + \chi_2 \mathfrak{z}_2$ is a generator for $L/ L^{\ell_i^{e_i}}$ since $\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2 - (\mathfrak{z}_1 + \chi_2 \mathfrak{z}_2) = (\chi_1 - \chi_2) \mathfrak{z}_2$ and $\frac{1}{\chi_1} (\mathfrak{z}_1 + \chi_1 \mathfrak{z}_2) - \frac{1}{\chi_2} (\mathfrak{z}_1 + \chi_2 \mathfrak{z}_2)= \frac{\chi_2 - \chi_1}{\chi_1\chi_2}\mathfrak{z}_1$. So that $J_1 \cup J_2 = \{ 1 , \cdots , s \}$. \\
If $J_2$ is all $\{1 , \cdots , s\}$ then $\mathfrak{z}_1 + \chi_2 \mathfrak{z}_2 = \eta z + \sigma ( \eta z )$ is a generator $L/K$ where $\eta = 1 + \chi_2 \xi^{-1} $ proving the theorem. \\
Otherwise, there is $i_3 \in \{ 1, \cdots , s\} \backslash J_1 $, and since $I\cup J_2 = J\cup J_2 = J_1 \cup J_2 =\{ 1, \cdots , s\}$ we have $i_0 ,i_1 , i_2 \in J_2$ and thus $i_0 , i_1 , i_2, i_3$ are all distinct and $|J_2| \geq 3$. Reproducing this process, with $\chi_3 , \cdots \chi_{s-1} \in \mathbb{F}_q(\xi)^*$ such that $1 + \chi_i \xi^{-1} \in U_{q+1}$, for $1 \leq i \leq s-1$, all distinct and distinct from $\chi_1$ and $\chi_2$ which exist since $s< |U_{q+1}|$, we find that either $\mathfrak{z}_1 + \chi_k \mathfrak{z}_2$ is a generator for $L/K$ for some $k \in \{ 3, \cdots , s-2\}$ or $|J_{s-1}|\geq s$ and $\mathfrak{z}_1 + \chi_{s-1} \mathfrak{z}_2$ is a generator for $L/K$. In any case, we find $\eta \in U_{q+1}$ such that $\eta z + \sigma (\eta z)$ is a generator for $L/K$
\end{proof}
\begin{theoreme} \label{m} Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. There exists $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$ and $u:=\sigma(w) w \in K$ so that the minimal polynomial of $y$ over $K$ is $ P^{\ell}_{u , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$. Conversely, if $L/K$ is a geometric extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$ and a generator $y$ whose minimal polymomial is of form $ P^{\ell}_{u , \alpha} (X)$ where $u , \alpha \in K$ such that $u^\ell= a \sigma (a)$ and $\alpha = a+ \sigma (a)$, for some $a\in K(\xi)\backslash K$ then $L/K$ is a cyclic extension. \end{theoreme} \begin{proof} The first statement of the theorem is a direct consequence of Lemmas \ref{1}, \ref{2} and \ref{3}, noting that if $z$ is a Kummer generator for $L(\xi ) /K(\xi)$ then $\eta z$ is also a Kummer generator for $L(\xi ) / K(\xi )$, by \cite[Theorem 5.8.5]{Vil}. \\
Conversely, suppose that $L/K$ is an extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$ and a generator $y$ whose minimal polymomial is of form $ P^{\ell}_{u , \alpha} (X)$ where $u , \alpha \in K$ such that $u^\ell= a \sigma (a)$ and $\alpha = a+ \sigma (a)$, for some $a\in K(\xi)$. Let $z_1, z_2 \in L^{alg}$ where $L^{alg}$ is the algebraic closure of $L$ the two roots of the polynomial $ X^2 - y X + u =0$. Then $y = z_1 + z_2$ and $u = z_1 z_2$, by Lemma \ref{1}, we have that $$ z_1^\ell + z_2^\ell = P^{\ell}_{u, \alpha}(y) + \alpha = \alpha $$ And since $z_2 = \frac{u}{z_1}$, we have $$ z_1^\ell + \frac{u^\ell }{z_1^\ell }= \alpha $$ and $$z_1^{2\ell}- \alpha z_1^\ell + u^\ell$$ We have that $$\alpha = a + \sigma (a) \text{ and } u^\ell = a \sigma (a)$$ then $a$ and $\sigma (a)$ are root of the polynomial $$X^{2}- \alpha X+ u^\ell=0$$ Hence, up to reindexing, $z_1^\ell= a$ and $z_2^\ell = \sigma (a)$ and $\sigma (z_1) \neq z_1$ since $a \in K(\xi)\backslash K$. Moreover, the coefficient of $T(X)=X^2-yX+u$ are in $L$, so that if $z_1$ is a root of $T(X)$ then $\sigma (z_1)$ is a root of $T(X)$ and thus, $z_2= \sigma (z_1)$. Since $\sigma (z_1) \neq z_1$, we also obtain that $\xi \in L(z_1)$ from $[L(z_1): L]\leq 2$ we can conclude that $ L(z_1)= L(\xi)$.
Since $z_1^\ell= a\in K(\xi)$, $[ K(\xi) (z_1): K(\xi)]\leq \ell$. If $[ K(\xi) (z_1): K(\xi)]< \ell$, since $K(\xi)(z_1)= K(\xi) (\sigma (z_1))$, then $y:=z_1 + \sigma (z_1)$ would belong to a proper subextension of $L/K$, and $[K(y): K]<\ell$, which contradicts that $y$ is a generator for $L/K$. As a consequence, $z_1$ is a Kummer generator for $L(\xi)/K(\xi)$ and $L(\xi)/ K(\xi)$ is cyclic. The set of all the $\zeta z_1$ where $\zeta$ is a $\ell^{th}$ root of unity is the set of distinct roots for the polynomial $X^\ell - a$ and then $\zeta z_1 + \sigma( \zeta z_1)$ are the distinct roots of $P^\ell_{u, \alpha}(X)$. Indeed, $\zeta = \xi^i$ for some $0\leq i \leq \ell -1$, thus $\sigma (\zeta ) = \zeta^{-1}$, $ \zeta z_1 \sigma( \zeta z_1)= u$ and $ (\zeta z_1)^\ell + \sigma( \zeta z_1)^\ell = z_1^\ell + \sigma(z_1)^\ell= \alpha $. Moreover, if $\zeta_1 \neq \zeta_2 $ $\ell^{th}$ root of unity, such that $\zeta_1 z_1 + \sigma( \zeta_1 z_1)=\zeta_2 z_1 + \sigma( \zeta_2 z_1)$ then $\frac{ \zeta_1 - \zeta_2}{- \sigma (\zeta_1 -\zeta_2) }= \frac{\sigma (z_1)}{ z_1}= \frac{1}{u} \sigma (z_1)^2$, which contradicts that $L/K$ is a geometric extension. Proving that $\zeta z_1 + \sigma( \zeta z_1)$ are the distinct roots of $P^\ell_{u, \alpha}(X)$. This proves that $L/K$ is cyclic.
\end{proof}
\begin{lemma}\label{d}
Let $\sigma$ a generator of $Gal( K(\xi ) /K)$ with $\xi$ a primitive $\ell^{th}$ root of unity. Let $d \in K (\xi )$, $\sigma (d) d =1$.
There is $A, B \in \mathcal{O}_{K,x}$ such that $$d = \frac{ A + \xi B}{A + \xi^{-1} B}= \frac{ C + \xi }{C + \xi^{-1} },$$ where $C= \frac{A}{B}$.
When $K= \mathbb{F}_q(x)$, one can choose $(A, B)=1$. \end{lemma} \begin{proof} Let $\gamma \in K^*$. If we have $\gamma + d \sigma (\gamma) = 0$, then $ d= - \frac{\gamma}{\sigma (\gamma )}$. Noting that $\frac{ \xi - \xi^{-1}}{\sigma ( \xi - \xi^{-1})} =-1$. In this case, on can take $\theta = (\xi - \xi^{-1}) \gamma $ and obtain $$d = \frac{ \theta}{\sigma (\theta )}.$$
Otherwise, let $\theta = \gamma + d \sigma (\gamma) \neq 0$, so that, \begin{align*} d\sigma (\theta )&= d \sigma (\gamma + d \sigma (\gamma)) \\ &=d \sigma(\gamma ) + d \sigma (d) \gamma \\ &= \gamma + d \sigma(\gamma ) \\ &=\theta \end{align*} We write $ \theta = \theta_1 + \xi \theta_2$, where $\theta_1, \ \theta_2 \in K$. One can write $\theta_1 = \frac{A_1}{B_1}$ and $\theta_2= \frac{A_2}{B_2}$ where $A_1, B_1 , A_2 , B_2 \in \mathcal{O}_{K,x}$ so that $\theta = \frac{A_1}{B_1} + \xi \frac{A_2}{B_2} = \frac{A_1 B_2 + \xi A_2 B_1}{B_1 B_2}$ and $\sigma ( \theta ) = \frac{A_1 B_2 + \xi^{-1} A_2 B_1}{B_1 B_2}$, thus $$d = \frac{ \theta } {\sigma ( \theta )}= \frac{ A_1 B_2 + \xi A_2 B_1}{ A_1 B_2 + \xi^{-1} A_2 B_1}$$ Taking $A= A_1 B_2$ and $B = A_2 B_1$, we obtain the theorem for general $K$. \\ When $K= \mathbb{F}_q(x)$, one can take $A= \frac{A_1 B_2}{gcd(A_1 B_2, A_2 B_1)}$ and $B= \frac{A_2 B_1}{gcd(A_1 B_2, A_2 B_1)}$ and prove the theorem, in this case. \end{proof} \begin{corollaire} \label{sigm} Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. There exists $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$ and so that the minimal polynomial of $y$ over $K$ is $ P^{\ell}_{u , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$ and \begin{enumerate} \item when $\ell$ is an odd integer, $u= w \sigma (w) = a\sigma (a) =1$;
\item when $2 | \ell$, $\frac{a}{u^{\frac{\ell}{2}}} \sigma \big( \frac{a}{u^{\frac{\ell}{2}}} \big) =1$ where $u = w \sigma (w) \in K$ and $$P^{\ell}_{u , \alpha} (X)=P^{\frac{\ell}{2}}_{1 , \frac{ \alpha}{u^{\frac{\ell}{2}}}} \big( \frac{X^2-2u}{u}\big).$$ More precisely, $y$ generates the extension $L/L_2$ where $L_2$ is the fixed field of $L$ by the unique subgroup of $Gal(L/K)$ isomorphic to $\mathbb{Z}/ 2 \mathbb{Z}$ with generating equation: $$y^2 =2 u + z^2 + \sigma (z)^2= u\big( 2 + \mathfrak{y} \big)$$ where $\mathfrak{y}= \frac{\sigma (z)}{z}+\frac{z}{\sigma (z)}$ generates $L_2/K$ with minimal polynomial $$ P^{\frac{\ell}{2}}_{1 , \frac{a}{u^{\frac{\ell}{2}}} } (X)$$ \end{enumerate} \end{corollaire} \begin{proof} \begin{enumerate} \item {\sf Suppose $\ell$ is an odd integer.}
Let $z$ be a Kummer generator for $L(\xi ) / K(\xi )$ whose minimal polynomial is $X^\ell -c$, then $\frac{\sigma (z)}{z}$ is a Kummer generator $L(\xi ) / K(\xi )$. Indeed, if it was not then $v=\frac{\sigma (z)}{z}$ would belong to a proper subextension $L'$ of $L$ but since $z \sigma (z) \in K$ by Lemma \ref{2}, then $\frac{\sigma (z)}{z}z \sigma (z) = \sigma (z)^2\in L'$ but since $z$ is a Kummer generator for $L(\xi ) / K(\xi )$, so is $\sigma (z)$ and since $\ell$ is odd, $\sigma (z)^2$ is a Kummer generator of $L(\xi ) / K(\xi )$, by \cite[Proposition 5.8.7]{Vil} which leads to a contradiction. By Lemma \ref{3}, there exist $\eta \in \mathbb{F}_q(\xi)^*$ such that $\eta \sigma (\eta )=1$, $\eta v + \sigma ( \eta v)$ is a generator for $L/K$. Thus taking $w= \eta v$, we have $y:= w+ \sigma (w)$ is a generator for $L/K$ and $ w \sigma (w)=1$, $w^\ell= \eta^\ell \frac{\sigma (c)}{c}=:a$ and by Lemma \ref{1}, the minimal polynomial of $y$ is $P^{\ell}_{1 , \alpha}$ where $\alpha = a + \sigma (a )$.\\
\item {\sf Suppose $\ell$ is an even integer.} Thanks to Theorem \ref{m}, we can choose $w$ to be a Kummer generator for $L(\xi ) / K(\xi )$ whose minimal polynomial is $X^\ell -a$ such that $y := \sigma (w) + w$ is a generator for $L/K$ with minimal polynomial $P^\ell_{u, \alpha} (X)$ with $u = w \sigma (w)\in K$ and $\alpha = a+ \sigma (a)$. Note that
$$ \frac{\sigma ( \omega^\ell)}{ \omega^\ell } =\frac{ \sigma (a)^2}{u^\ell}$$
thus
$$ \frac{\sigma ( \omega^{\frac{\ell}{2}})}{ \omega^{\frac{\ell}{2}} } =\frac{ \sigma (a)}{u^{\frac{\ell}{2}}}$$
And
$$\frac{ a}{u^{\frac{\ell}{2}}}= \frac{ \omega^{\frac{\ell}{2}} }{\sigma ( \omega^{\frac{\ell}{2}})}= \frac{a^{\frac{\ell}{2}}}{ \sigma (a)} $$
$$\sigma \big( \frac{ a}{u^{\frac{\ell}{2}}} \big)= \frac{u^{\frac{\ell}{2}}}{ a}.$$
As a consequence,
$$ \frac{a}{u^{\frac{\ell}{2}}} \sigma \big( \frac{a}{u^{\frac{\ell}{2}}} \big) =1$$ Note that since $P^\ell_{u, \alpha} (X)$ only involves even monomial we can write $P^\ell_{u, \alpha} (X)= Q(X^2)$. where $Q(X)$ is a monic irreducible polynomial over $K$ and $deg (Q(X))=\frac{\ell}{2}$. Thus $Q(X)$ is the minimal polynomial of $y^2$, $[K(y^2): K]= \frac{\ell}{2}$ and $L_2 = K(y^2)$ and $\mathfrak{y}:=\frac{y^2-2u}{u} = \frac{ \sigma (w)}{w} + \frac{w}{\sigma (w)}$ is also a generator for $L_2 /K$ since $u \in K$. Moreover, $\frac{\sigma (w)}{w} $ Kummer generator for $L_2/K$ such that $\big( \frac{\sigma (w)}{w} \big)^{\frac{\ell}{2}} =\frac{a}{u^{\frac{\ell}{2}}}$, thus by Lemma \ref{1}, the minimal polynomial of $\mathfrak{y}$ is $P^{\frac{\ell}{2}}_{1, \frac{a}{u^{\frac{\ell}{2}}}}(X)$ and the the result. \end{enumerate}
\end{proof}
Combining together Theorem \ref{m}, Corollary \ref{sigm} and Lemma \ref{d}, we obtain: \begin{corollaire}\label{cm}
Let $L/K$ be a cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. There exists $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$ and $u :=\sigma(w) w \in K$ so that the minimal polynomial of $y$ over $K$ is \begin{enumerate}
\item $$ P^{\ell}_{1 , \alpha} (X)= X^\ell -\ell X^{\ell-2} + \sum_{s=1}^{q} c_{s,\ell} X^{\ell -2s} - \frac{2 A^2 + 2 (\xi + \xi^{-1}) AB + (\xi^2 + \xi^{-2}) B^2 }{A^2 +(\xi + \xi^{-1} ) AB + B^2 }$$ for some $A, B \in \mathcal{O}_{K,x}$ such that $a = \frac{\sigma (A+ \xi B)}{A+\xi B}$ and $u=1$, when $\ell$ is an odd integer; \item $$ P^{\ell}_{u , \alpha} (X)= X^\ell -\ell u X^{\ell-2} + \sum_{s=1}^{q} c_{s,\ell} u^s X^{\ell -2s} - u^{\frac{\ell }{2}}\frac{2 A^2 + 2 (\xi + \xi^{-1}) AB + (\xi^2 + \xi^{-2}) B^2 }{A^2 +(\xi + \xi^{-1} ) AB + B^2 }, $$ for some $A, B \in \mathcal{O}_{K,x}$ such that $a =u^{\frac{\ell}{2}} \frac{\sigma (A+ \xi B)}{A+\xi B}$, when $\ell$ is an even integer;
\end{enumerate}
where $c_{s, \ell}$ are defined in Lemma \ref{1}. \end{corollaire} \begin{remarque} Note that $$\frac{2 A^2 + 2 (\xi + \xi^{-1}) AB + (\xi^2 + \xi^{-2}) B^2 }{A^2 +(\xi + \xi^{-1} ) AB + B^2 }= \frac{2 C^2 + 2 (\xi + \xi^{-1}) C + (\xi^2 + \xi^{-2}) }{C^2 +(\xi + \xi^{-1} ) C + 1}$$ where $C = \frac{A}{B}$. \end{remarque} One can obtain the following result combining Theorem \ref{m} and Proposition 5.8.7. VS \begin{lemma}\label{clas} Let $L_1/K$ and $L_2/K$ be two geometric cyclic extensions of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. For $i=1,2$, there exists $w_i$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a_i$ such that $y_i:=\sigma (w_i) + w_i$ is a generator for $L/K$, so that the minimal polynomial of $y_i$ over $K$ is $ P^{\ell}_{u_i , \alpha_i} (X)$ as in lemma \ref{1} where $ \alpha_i = a_i + \sigma (a_i)$ and $u_i:=\sigma(w_i) w_i \in K$. Then, the following assertions are equivalent: \begin{enumerate} \item $L_1= L_2$ \item $L_1 (\xi ) = L_2 (\xi)$ \item $w_2 = c w_1^j $ so that $y_2 = c w_1^j + \sigma (c) \sigma (w_1) ^j $ for all $1 \leq j \leq n - 1$ such that $( j, n) = 1$ and $c \in K(\xi)$. \item $a_2 = c^\ell a_1^j $ so that $\alpha_2= c^\ell a_1^j + \sigma (c^\ell) \sigma (a_1) ^j $, for all $1 \leq j \leq n - 1$ such that $( j, n) = 1$ and $c \in K(\xi)$.
\end{enumerate} \end{lemma} The following result is a direct corollary of the previous lemma. \begin{corollaire} \label{gal} Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. There exists $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $ P_{\ell,u , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$ and $u:=\sigma(w) w \in K$. Then the Galois group can be identified with the group of the $\ell^{th}$ root of unity, and the Galois action is given by $y \mapsto y_\zeta = \sigma (\zeta w) +\zeta w$ \end{corollaire} \section{Ramification} \begin{theoreme} \label{ram}
Let $L/K$ be a geometric cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. As in Theorem \ref{m}, we choose $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $ P_{\ell,u , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$ and $u:=\sigma(w) w \in K$. Let $\mathfrak{p}$ be a place of $K$ and $\mathfrak{P}$ a place of $L$ above $\mathfrak{p}$. We denote $e(\mathfrak{P} | \mathfrak{p})$ the index of ramification of $\mathfrak{P}| \mathfrak{p}$,
We have \begin{enumerate} \item[$\cdot$] when $v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )\geq0$ then \begin{itemize}
\item[$\cdot$] $v_{\mathfrak{p}} \big( \alpha \big) \neq 0$, $\mathfrak{p}$ is of even degree and $e(\mathfrak{P}| \mathfrak{p}) = \frac{ \ell}{(\ell , v_{\mathfrak{p}} (\alpha ) )}.$
\item[$\cdot$] $v_{\mathfrak{p}} \big( \alpha \big) = 0$, $e(\mathfrak{P}| \mathfrak{p}) = 1$. \end{itemize} \item[$\cdot$] when $v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )<0$ then \begin{itemize}
\item[$\cdot$] $e(\mathfrak{P}| \mathfrak{p}) = 2$, if $(v_\mathfrak{p} ( u),2)=1$.
\item[$\cdot$] $e(\mathfrak{P}| \mathfrak{p}) = 1$, otherwise. \end{itemize} \end{enumerate} When $\ell$ is odd, one can choose $y$ such that $ u=1$, $\mathfrak{p}$ is unramified if and only if $v_{\mathfrak{p}} (\alpha ) \geq 0$. Moreover, when $\mathfrak{p}$ is ramified ,$\mathfrak{p}$ is of even degree and
$$e(\mathfrak{P}| \mathfrak{p}) = \frac{ \ell}{(\ell , v_{\mathfrak{p}} (\alpha ) )}.$$ \end{theoreme} \begin{proof} Let $\mathfrak{p}$ be a place of $K$, $\mathfrak{p}_\xi$ a place of $K(\xi)$ above $\mathfrak{p}$ and $\mathfrak{P}$ a place of $L$ above $\mathfrak{p}$
From \cite[Theorem 5.8.12]{Vil}, since $K(\xi )/K$ is unramified and the index of ramification is multiplicative in tower, $\mathfrak{p}$ is unramified in $L$ if and only if $\mathfrak{p}_\xi$ is unramified in $L(\xi)$. That is, $\ell | v_{\mathfrak{p}_\xi} (a)$. Thus, in order to prove the theorem, we need to determine $v_{\mathfrak{p}_\xi} (a)$ in terms of $v_{\mathfrak{p}}(\alpha )$ and $v_{\mathfrak{p}}(u )$. For if, note that $a$ satisfies the equation $$a^2 - \alpha a + u^\ell =0$$ since $\alpha = a + \sigma (a)$ and $u^\ell = a \sigma (a)$.\\ Moreover $K(a) = K(\xi)$ since $a \in K(\xi)\backslash K$, thus $K(a) / K$ is unramified. \\ We have $$\big( \frac{a}{\alpha}\big) ^2 - \frac{ a}{\alpha} = - \frac{u^\ell }{\alpha^2 } $$
When $v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )<0$, we find by the triangular inequality for valuation that $v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big) <0$ and $2v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big) = v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )$ then $$v_{\mathfrak{p}_\xi} \big(a) = v_{\sigma (\mathfrak{p}_\xi )} \big(a) =\frac{\ell v_\mathfrak{p} ( u )}{2}.$$
When $v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )=0$, by the triangular inequality, we obtain that $v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big) =0$, thus $$v_{\mathfrak{p}_\xi} \big(a) = v_{\mathfrak{p}_\xi} \big(\alpha \big) = v_{\mathfrak{p}} \big(\alpha \big) $$
When $v_\mathfrak{p} ( \frac{u^\ell }{\alpha^2 } )>0$, then by Kummer theorem applied to the polynomial $X^2 - X + \frac{u^\ell}{ \alpha^2 }$, we know that $\mathfrak{p}$ splits in $K(\xi )$, therefore, by \cite[Theorem 6.2.1]{Vil}, we obtain that $ \mathfrak{p} $ is of even degree. Moreover, by the triangular inequality $v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big) \geq 0$, for any place $\mathfrak{p}_\xi$ of $K(\xi )$ over $\mathfrak{p}$. More precisely, either $v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big) = 0$ or $v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha}\big)= v_{\mathfrak{p}} \big( \frac{u^\ell }{\alpha^2} \big)$. Moreover, $$\frac{u^\ell }{\alpha^2} = \frac{a}{\alpha} \sigma \big( \frac{a}{\alpha}\big)$$ Thus, $$v_{\mathfrak{p}} \big( \frac{u^\ell }{\alpha^2} \big)= v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha} \big) + v_{\mathfrak{p}_\xi} \big( \sigma \big( \frac{a}{\alpha} \big) \big) = v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha} \big) + v_{\sigma (\mathfrak{p}_\xi)} \big( \frac{a}{\alpha} \big) $$ From this we deduce that either $$v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha} \big) = 0 \text{ and } v_{\sigma (\mathfrak{p}_\xi)} \big( \frac{a}{\alpha} \big) =v_{\mathfrak{p}} \big( \frac{u}{\alpha^2} \big) $$ that is, $$v_{\mathfrak{p}_\xi} \big( a \big) = v_{\mathfrak{p}} \big( \alpha \big) \text{ and } v_{\sigma (\mathfrak{p}_\xi)} \big(a \big) =\ell v_{\mathfrak{p}} \big(u \big)- v_{\mathfrak{p}} \big( \alpha \big) $$ or $$v_{\mathfrak{p}_\xi} \big( \frac{a}{\alpha} \big) = v_{\mathfrak{p}} \big( \frac{u^\ell }{\alpha^2} \big) \text{ and } v_{\sigma (\mathfrak{p}_\xi)} \big( \frac{a}{\alpha} \big) =0 $$ that is, $$ v_{\mathfrak{p}_\xi} \big(a \big) =\ell v_{\mathfrak{p}} \big(u \big)- v_{\mathfrak{p}} \big( \alpha \big) \text{ and } v_{\sigma (\mathfrak{p}_\xi)} \big( a \big) = v_{\mathfrak{p}} \big( \alpha \big) $$ This permitting to obtain the desire result. \end{proof}
\begin{remarque} When $\ell$ is odd, one can choose a single place $\mathfrak{P}_\infty$ at infinity in $K$ such that $v_{\mathfrak{P}_\infty}(a) \geq 0$ and thus $\mathfrak{P}_\infty$ is unramified. (see \cite[Corollary 3.12]{MWcubic2}) \end{remarque} \begin{lemma} Let $\ell$ be an odd integer. Let $L/K$ be a cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. Given $\mathfrak{p}$ a place of $K$. There is $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$, so that the minimal polynomial of $y$ over $K$ is $ P^{\ell}_{1 , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$, $\sigma(w) w =1$ and either $v_{\mathfrak{p}} (\alpha ) = - m$ where $0\leq m \leq \ell -1$ or $v_{\mathfrak{p}} (\alpha ) \geq 0$.
Moreover, when $v_{\mathfrak{p}} (\alpha ) =-m$ where $0\leq m \leq \ell -1$ then $\mathfrak{p}$ is ramified and $v_{\mathfrak{p}} (\alpha ) \geq 0 $ then $\mathfrak{p}$ is unramified. \end{lemma} \begin{proof} Let $\mathfrak{p}$ a place of $K$. By Corollary \ref{cm}, there is $v$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -c$ such that $z:=\sigma (v) + v$ is a generator for $L/K$ and $\sigma( v) v=1 $ so that the minimal polynomial of $z$ over $K$ is $ P^{\ell}_{1 , \beta} (X)$ as in lemma \ref{1} where $ \beta = c + \sigma (c)$. From the proof of the previous lemma \ref{ram}, we have that either $v_\mathfrak{p} (\beta ) \geq 0$ when $v_{\mathfrak{p}_\xi} (c )=0$ or $\mathfrak{p}$ split $K(\xi )$ and for one of the place $\mathfrak{p}_\xi$ above $\mathfrak{p}$, we have $$v_\mathfrak{p} (\alpha ) = v_{\mathfrak{p}_\xi} (c )= -v_{\sigma (\mathfrak{p}_\xi) } (c ) <0$$ By Lemma \ref{d}, there is $\gamma \in K(\xi)^*$ such that $$ a = \frac{\sigma (\gamma )}{\gamma}. $$ We write $v_{\mathfrak{p}_\xi} (\gamma) = \ell s +m$ where $0 \leq m \leq \ell -1$, using weak approximation theorem we choose $\lambda \in K(\xi)$ such that $v_{\mathfrak{p}_\xi} (\lambda ) = -s$. So that $\frac{\sigma (\lambda )}{\lambda} v$ is a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -\lambda^\ell c$ and $$v_{\mathfrak{p}_\xi} (\big( \frac{\sigma (\lambda )}{\lambda} \big)^\ell c )=-v_{\sigma (\mathfrak{p}_\xi)} (\big( \frac{\sigma (\lambda )}{\lambda} \big)^\ell c ) =- m.$$ By Lemma \ref{3}, there is $\eta\in \mathbb{F}_q (\xi)^*$ such that $\sigma (\eta ) \eta =1$, $w= \eta \frac{\sigma (\lambda )}{\lambda} v $ is a Kummer generator $L(\xi ) /K(\xi)$ with minimal polynomial $X^\ell - a$ where $a = \eta^\ell \big( \frac{\sigma (\lambda )}{\lambda} \big)^\ell c$ and $y:=w+ \sigma (w )$ is a generator for $L/K$ and its minimal polynomial is $P^\ell_{1, \alpha } (X)$ with $\alpha= a + \sigma (a)$ and $v_{\mathfrak{p}} ( \alpha ) = -m $.
\end{proof} \begin{lemma}\label{form} Let $\ell$ be an odd integer. Suppose $K= \mathbb{F}_q(x)$. Let $L/K$ be a cyclic extension of degree $\ell$ with $q \equiv -1 \ mod \ \ell$. There is $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -a$ such that $y:=\sigma (w) + w$ is a generator for $L/K$ so that the minimal polynomial of $y$ over $K$ is $ P^{\ell}_{1 , \alpha} (X)$ as in lemma \ref{1} where $ \alpha = a + \sigma (a)$ and $\sigma(w) w =1$ such that at any $\mathfrak{p}$ place of $K$ either $v_{\mathfrak{p}} (\alpha ) = - m$ where $0\leq m \leq \ell -1$ or $v_{\mathfrak{p}} (\alpha ) \geq 0$.
Moreover, when $v_{\mathfrak{p}} (\alpha ) =-m$ where $0\leq m \leq \ell -1$ then $\mathfrak{p}$ is ramified, $v_{\mathfrak{p}} (\alpha ) \geq 0 $ then $\mathfrak{p}$ is unramified and $v_{\mathfrak{p}_\infty } (\beta )\geq 0$ where $\mathfrak{p}_\infty $ is the pole divisor of $x$ so that $\mathfrak{p}_\infty $ is unramified. \end{lemma} \begin{proof} By Corollary \ref{cm}, there is $w$ a Kummer generator for $L(\xi)/ K(\xi)$ whose minimal polynomial is $X^\ell -c$ such that $z:=\sigma (v) + v$ is a generator for $L/K$ so that the minimal polynomial of $z$ over $K$ is $ P^{\ell}_{1 , \beta} (X)$ as in lemma \ref{1} where $ \beta = c + \sigma (c)$ and $\sigma( v) v=1 $. By Lemma \ref{d}, there is $\gamma \in F_q(\xi)[x]^*$ such that $(\sigma (\gamma ) , \gamma)=1$ and $$ c = \frac{\sigma (\gamma )}{\gamma}. $$ Since $F_q(\xi)[x]$ is a unique factorization domain and we write $$\gamma = \prod_{i=1}^t \gamma_i^{e_i}$$ where $\gamma_i$ are distincts irreducible polynomials in $\mathbb{F}_q[x]$ and $e_i= \ell s_i +m_i$ where $0 \leq m_i \leq \ell -1$, for $0\leq i \leq t$. Since $(\sigma (\gamma ) , \gamma)=1$ we have $\sigma (\gamma_i ) \neq \gamma_i$.
Let $\lambda = \prod_{i=1}^t \gamma_i^{-s_i}$ so that $\theta = \lambda^\ell \gamma = \prod_{i=1}^t \gamma_i^{m_i}$ and $$a:=\frac{\sigma (\theta )}{\theta }=\frac{\sigma (\lambda^\ell \gamma )}{\lambda^\ell\gamma}= \frac{\prod_{i=1}^t \sigma(\gamma_i )^{m_i}}{\gamma_i^{m_i}}$$
Moreover, $ \frac{\sigma (\lambda )}{\lambda} v$ is a Kummer generator for $L (\xi) \ K(\xi)$. By Lemma \ref{3}, there is $\eta\in \mathbb{F}_q (\xi)^*$ such that $\sigma (\eta ) \eta =1$, $w= \eta \frac{\sigma (\lambda )}{\lambda} v $ is a Kummer generator $L(\xi ) /K(\xi)$ with minimal polynomial $X^\ell - a$ where $a = \eta^\ell \big( \frac{\sigma (\lambda )}{\lambda} \big)^\ell c$ and $y:=w+ \sigma (w )$ is a generator for $L/K$ and its minimal polynomial is $P^\ell_{1, \alpha } (X)$ with $$\alpha= a + \sigma (a) = \frac{ \gamma^2 + \sigma (\gamma )^2}{\gamma \sigma (\gamma)}=\frac{ \theta^2 + \sigma (\theta )^2}{\prod_{i=1}^t (\gamma_i \sigma (\gamma_i))^{m_i}}$$
$\gamma_i \sigma (\gamma_i)$ is then an irreducible polynomial in $\mathbb{F}_q[x]$ and $ (\theta \sigma (\theta ), \theta^2 + \sigma (\theta )^2)=1$ so that if $\mathfrak{p}_i$ is the finite place of $K$ corresponding to $\gamma_i \sigma (\gamma_i)$, $v_{\mathfrak{p}_i} (a + \sigma (a) )= - m_i$. Clearly, at any other finite place, $ v_{\mathfrak{p}_i} (a + \sigma (a) )\geq 0$.
Finally, since $deg (\theta) = deg( \sigma (\theta ))$, we have $v_{\mathfrak{p}_\infty} ( a)=0$, so that $v_{\mathfrak{p}_\infty} (a + \sigma (a)) \geq 0$, concluding the proof of the lemma.
\end{proof}
\end{document} |
\begin{document}
\title[\tiny{Algebraic characterizations of homeomorphisms between algebraic varieties}]{Algebraic characterizations of homeomorphisms between algebraic varieties} \author{Fran\c cois Bernard, Goulwen Fichou, Jean-Philippe Monnier and Ronan Quarez} \thanks{The second author wishes to thank Olivier Wittenberg for fruitful discussions. The authors have received support from the Henri Lebesgue Center ANR-11-LABX-0020-01 and the project EnumGeom ANR-18-CE40-0009.}
\address{François Bernard\\
Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France} \email{francois.bernard@univ-angers.fr}
\address{Goulwen Fichou\\ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France} \email{goulwen.fichou@univ-rennes1.fr}
\address{Jean-Philippe Monnier\\
Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France} \email{jean-philippe.monnier@univ-angers.fr}
\address{Ronan Quarez\\ Univ Rennes\\ Campus de Beaulieu, 35042 Rennes Cedex, France} \email{ronan.quarez@univ-rennes1.fr} \date\today \subjclass[2020]{14A10,13B22,14P99} \keywords{homeomorphisms of algebraic varieties, weak normalization, seminormalization, saturation, regulous functions, real closed fields}
\begin{abstract} We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case, whereas it comes from the theory of real closed fields otherwise. \end{abstract}
\maketitle
\vskip 15mm
Let $k$ be an algebraically closed field. Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$ and $\pi_k:Y(k)\to X(k)$ be its restriction to the closed points. The main purpose of this paper is to find algebraic characterizations for topological conditions on $\pi$ or $\pi_k$. In this direction, we compare bijections, isomorphisms and homeomorphisms with respect to the Zariski topology, and to a strong topology on the closed points when char($k$)=0 (corresponding to the Euclidean topology when $k=\C$ is the field of complex numbers).
As first comparisons, recall from the Nullstellensatz that $\pi$ is a homeomorphism if and only if $\pi_k$ is a homeomorphism. Also, it is clear that if $\pi$ is an isomorphism, then it is a homeomorphism and in particular $\pi_k$ is a bijection. However, in general, having a bijection at the level of closed points does not induce a homeomorphism or an isomorphism between the varieties. To obtain results of this kind, one must add some conditions on the varieties. For example, if $\pi_k$ is a bijection and $X$,$Y$ are irreducible curves, then $\pi$ is a homeomorphism. In greater dimension, a bijection (or a birational bijection in positive characteristics) between irreducible varieties induces an isomorphism when the target variety is normal by Zariski Main Theorem.
Assuming now $\pi$ to be a homeomorphism, there are similar results involving the notions of seminormality and weak normality instead of normality. Andreotti and Bombieri \cite{AB} proved that $\pi$ is an isomorphism if $X$ is weakly normal and $\pi$ is finite. Vitulli \cite{V2} managed to remove the finiteness assumption on $\pi$, by requiring that $X$ does not have one-dimensional components. This dimensional condition is necessary, as illustrated by the normalization of a nodal curve with one of the preimage of the singular point removed : we obtain a homeomorphism onto a seminormal curve which is not an isomorphism. The dimensional condition of Vitulli guaranties in fact that the homeomorphism is a finite morphism.
The weak normality property appearing in the results of Andreotti, Bombieri and Vitulli is very closed to the notion of seminormality and they are both related to the notions of subintegral and weakly subintegral morphisms. A morphism $\pi:Y\to X$ is (resp. weakly) subintegral if it is integral, bijective and equiresidual (resp. residually purely inseparable) i.e. it induces isomorphisms (resp. purely inseparable extensions) between the residue fields. The seminormalization (resp. weak normalization) of $X$ in $Y$, introduced by Bombieri and Andreotti, is the maximal variety with a (resp. weakly) subintegral morphism onto $X$ which factorizes $\pi$. Note that when $\car(k)=0$, weakly subintegral means subintegral and weak normality means seminormality. The notions of weak normality and seminormality were first introduced by Andreotti and Norguet \cite{AN} for complex analytic varieties, then by Andreotti and Bombieri \cite{AB} for schemes and by Traverso for rings \cite{T}. It appears in the study of Picard groups \cite{T} or as singularities in the minimal model program \cite{KoKo}. Seminormalization, weak normalization and (weakly) subintegral morphisms are studied in section \ref{sect-semi}. The close notion of radicial morphism, as introduced by Grothendieck \cite{Gr1}, only requires that the (non necessarily integral) morphism is injective and residually purely inseparable. We study in section \ref{sect-sat} the saturation for varieties as the geometric counterpart of radiciality. The saturation appears first in the context of Lipschitz geometry with works of Pham and Teissier \cite{PT} in complex analytic geometry and Lipman \cite{Lip} for ring extensions. For integral morphisms, saturation and weak normalization coincide, providing a different approach to weak normality as proposed by Manaresi \cite{Mana}. However, it is not established that the saturation produces a variety, contrarily to the weak normalization and the seminormalization. A comparaison between all these notions is done in section \ref{comparaison}.
It will be crucial in our discussions to consider another topology than the Zariski topology. For an algebraic variety $X$ over the complex numbers, the strong topology on the closed points $X(\C)$ of $X$ comes from the isomorphism $\R[\sqrt{-1}]=\C$ that gives an identification $\C\simeq \R^2$ and the property of $\R$ to be a real closed field. Indeed, the Euclidean topology on $\R^n$ has a basis of open sets given by semialgebraic subsets of $\R^n$, i.e. given by real polynomial equalities and inequalities. The theory of semialgebraic sets provides an algebraic way to discuss about topological question in real algebraic geometry, as developed in \cite{BCR}. The great advantage of this approach is that it generalizes from $\R$ to any real closed field. A real closed field is an ordered field that does not admit any ordered algebraic extension. Equivalently, adding a square root of $-1$ to a real closed field gives an algebraically closed field of characteristic zero. Real closed fields have been initially studied by Artin and Schreier \cite{AS} in the way of Artin's proof of Hilbert XVIIth Problem \cite{A}. The most basic examples away from $\R$ are the field of algebraic real numbers, which is the real closure of $\QQ$, and the field of Puiseux series with real coefficients, which is the real closure of the field $\R((T))$ of Laurent series ordered by $T$ positive and infinitely small. There are many of them, as illustrated by the fact that any algebraically closed field $k$ of characteristic zero contains (infinitely many) real closed subfields $R\subset k$ with $k=R[\sqrt{-1}]$. Fixing such a choice of $R$ leads to an identification $k\simeq R^2$ and equips $k$ with an order topology, called the $R$-topology on $k$. Note that in general $R$ is not connected, and a closed and bounded interval is not compact. Anyway, for a given algebraic variety $X$ over an algebraically closed field $k$ of characteristic zero, the choice of a real closed field $R\subset k$ such that $k=R[\sqrt{-1}]$ makes $X(k)$ a topological space for the $R$-topology. Of course, if $k=\C$ and $R=\R$ then we recover the Euclidean topology. The $R$-topology on the closed points of algebraic varieties over an algebraically closed field $k$ of characteristic zero is studied in section \ref{regulous}. The introduction of the $R$-topology allows us to characterize finite morphisms which are homeomorphisms via subintegrality and radiciality. Concerning subintegrality, it generalizes to any algebraically closed field of characteristic zero a result of the first author \cite[Thm. 3.1]{Be} in complex geometry.
\begin{thmx}\label{thmA} Let $\pi:Y\to X$ be a finite morphism between algebraic varieties over an algebraically closed field $k$ of characteristic zero. Let $R\subset k$ be a real closed subfield such that $k=R[\sqrt{-1}]$. The following properties are equivalent : \begin{enumerate} \item[(i)] $\pi$ is a homeomorphism. \item[(ii)] $\pi_k$ is a homeomorphism for the $R$-topology. \item[(iii)] $\pi$ is subintegral. \item[(iv)] $\pi$ is radicial. \end{enumerate} \end{thmx}
We focus in section \ref{sect-homeo} on the relations between the four equivalent properties appearing in Theorem \ref{thmA} when we remove the finiteness hypothesis. Note that the first two properties are topological whereas the last ones are algebraic. The equivalence between the four above properties is no longer true without the finiteness hypothesis. In particular, a homeomorphism with respect to the Zariski topology need not be a homeomorphism with respect to the $R$-topology, even for irreducible affine curves. Our interpretation is that the relevant topology to associate to subintegrality is the $R$-topology, whereas Zariski topology is rather related to the notion of radiciality.
\begin{thmx}\label{thmB} Let $\pi:Y\to X$ be a morphism between algebraic varieties over an algebraically closed field $k$ of characteristic zero. Let $R\subset k$ be a real closed subfield such that $k=R[\sqrt{-1}]$. Then : \begin{enumerate} \item[(i)] $\pi_k$ is a homeomorphism for the $R$-topology if and only if $\pi$ is subintegral. \item[(ii)] If $\pi$ is a homeomorphism then $\pi$ is radicial. \end{enumerate} \end{thmx}
The key result we prove to get the first statement of Theorem \ref{thmB} is that a homeomorphism for the $R$-topology is always finite. Note moreover that, if the second part of Theorem \ref{thmB} does not refer to the $R$-topology, our proof is a consequence of the first result where the use of the $R$-topology is crucial. We prove along the way that a homeomorphism with respect to the $R$-topology is a homeomorphism, the converse being true except for curves. In section \ref{sect-carpos} we consider the situation where $k$ has positive characteristic, and then the $R$-topology does not exist anymore. We provide an alternative proof of the second statement of Theorem \ref{thmB} in this context. As consequences, we completely answer the question considered by Vitulli \cite{V2} of characterizing when a homeomorphism is an isomorphism.
\begin{thmx}\label{thmC} Let $\pi:Y\to X$ be a morphism between algebraic varieties over an algebraically closed field $k$ of characteristic zero. Let $R\subset k$ be a real closed subfield such that $k=R[\sqrt{-1}]$. \begin{enumerate} \item[(i)] Assume $\pi_k$ is a homeomorphism for the $R$-topology. Then, $\pi$ is an isomorphism if and only if $X$ is seminormal in $Y$. \item[(ii)] Assume $\pi$ is a homeomorphism. Then, $\pi$ is an isomorphism if and only if $X$ is saturated in $Y$. \end{enumerate} \end{thmx}
The study of seminormalization over non algebraically closed fields presents some difficulty, and the last three authors managed to define a sort of seminormalization for algebraic varieties over the field of real numbers \cite{FMQ}. This notion has to do with the central points of a real algebraic variety, that is the Euclidean closure of the set of regular points. This approach would not have been possible without the recent study of continuous rational functions in real algebraic geometry, as initiated by Kucharz \cite{Ku}, Koll\'ar and Nowak \cite{KN}, and further developed in \cite{FHMM} as regulous functions. These regulous functions happens to be related to seminormality for complex algebraic varieties too as studied by the first author \cite{Be}. It is this approach of seminormality via continuous functions with respect to the $R$-topology we develop at the end of section \ref{sect-CR}. In particular, we provide a full study of the relation between seminormality and the $R$-topology completely parallel to the complex case in \cite{Be}. We introduce the continuous rational functions over $k$ in section \ref{sect-CR}. A remarkable fact is that the continuity of a rational function defined on an algebraic variety $X$ over $k$ does not depend on the choice of the real closed field $R$. Even more, these continuous rational functions coincide with the regular functions on the seminormalization of $X$, cf. Theorem \ref{caractK0}. As a consequence, fixing a real closed field $R\subset k$ brings all the flexibility of semialgebraic geometry over $R$ to algebraic geometry over $k$, without loosing in generality. We prove notably that the seminormalization determines a variety up to biregulous equivalence. As a consequence, we obtain the following result, which goes in the direction of the problems considered by Koll\'ar in \cite{Ko} (or \cite{KMOS,Ce}).
\begin{thmx}\label{thmD} \label{thmD} Let $X$ and $Y$ be seminormal algebraic varieties over an algebraically closed field $k$ of characteristic zero. If $X(k)$ and $Y(k)$ are biregulously equivalent, then $X$ and $Y$ are isomorphic. \end{thmx}
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In the paper, $k$ denotes a field (sometimes algebraically closed) of any characteristic, and an algebraic variety over $k$ is a reduced and separated scheme of finite type over $k$.
\section{Subintegrality, weak normalization and seminormalization}\label{sect-semi}
After some reminders on integral extensions and normalization, we recall the notions of (weakly) subintegral extensions and the respective constructions of Traverso \cite{T}, Andreotti and Bombieri \cite{AB} of the seminormalization and the weak normalization for ring extensions and morphisms between algebraic varieties. \vskip 5mm
{\bf Notation and terminology.}
Let $A$ be a ring. The Zariski spectrum $\Sp A$ of $A$ is the set of prime ideals of $A$. It is a topological space for the topology whose closed sets are generated by the sets $\V(f)=\{\p\in\Sp A\mid f\in\p\}$ for $f\in A$. We denote by $\Max A$ the subspace of maximal ideals of $A$. For $\p\in\Sp A$, we denote by $k(\p)$ the residue field at $\p$.
Let $(X,\SO_X)$ be a variety over $k$. For $x\in X$ we denote by $k(x)$ the residue field at $x$; for an affine neighborhood $U$ of $x$ then $x$ corresponds to a prime ideal $\p_x$ of $\SO_X(U)$ and we have $k(x)=k(\p_x)$. In case $X$ is affine then we denote by $k[X]$ the coordinate ring of $X$ i.e $k[X]=\SO_X(X)$. Let $K$ be a field containing $k$. We denote by $X(K)$ the set $\Mor(\Sp K,X)$ of $K$-rational points. If $K=k$ then $X(k)$ is also the set of $k$-closed points of $X$, i.e the points of $X$ with residue field equal to $k$. We have thus an inclusion $$X(k)\hookrightarrow X$$ that makes $X(k)$ a topological space for the Zariski topology. We denote by $\SO_{X(k)}$ the sheaf of regular functions on $X(k)$, for $x\in X(k)$ we have $\SO_{X,x}=\SO_{X(k),x}$. In the case $k$ algebraically closed, for an open subset $U$ of $X$, we may identify $\Max \SO_X(U)$ with $U(k)$ by the Nullstellensatz, and similarly we identify the regular functions on $U$ with those on $U(k)$, namely $\SO_X(U)=\SO_{X(k)}(U(k))$. If $T$ is a subset of $X$ or $X(k)$ or $\Sp A$ then we will denote by $\overline{T}^Z$ the closure of $T$ for the Zariski topology.
A ring extension $i:A\to B$ induces a map $\Sp(i):\Sp B\to \Sp A$, given by $\p\mapsto (\p\cap A)=i^{-1}(\p)$. If $\pi:Y\rightarrow X$ is a morphism between algebraic varieties over $k$, with $\SO_X\to \pi_*\SO_Y$ the associated morphism of sheaves of rings on $X$, then for any open subset $U\subset X$ the ring morphism $\SO_X(U)\to \SO_Y(\pi^{-1}(U))$ is an extension if $\pi$ is dominant. For a field extension $k\to K$, we denote by $\pi_{K}:Y(K)\to X(K)$ the induced map. Remark that $\pi_k$ is also the restriction of $\pi$ to the $k$-closed points.
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In the sequel, $\K(A)$ (resp. $\K$) will denote the total ring of fractions of $A$ (resp. the sheaf of total ring of fractions on $X$).
\subsection{Reminder on integral extensions and normalization}
A ring extension $A\to B$ is said of finite type (resp. finite) if it makes $B$ a finitely generated $A$-algebra (resp. $A$-module). The extension $A\to B$ is birational if it induces an isomorphism between $\K(A)$ and $\K(B)$.
An element $b\in B$ is integral over $A$ if $b$ is the root of a monic polynomial with coefficients in $A$, which is equivalent for $A[b]$ to be a finite $A$-module by \cite[Prop. 5.1]{AM}. As a consequence
$$A_B'=\{b\in B|\,b\,\, {\rm is\,\,
integral\,\, over}\,\,A\}$$ is a ring called the integral closure of $A$ in $B$. The extension $A\to B$ is said to be integral if $A_B'=B$. In case $B=\K(A)$ then the ring $A_{\K(A)}'$ is denoted by $A'$ and is simply called the integral closure of $A$. The ring $A$ is called integrally closed (resp. in $B$) if $A=A'$ (resp. $A=A_B'$).
We recall that a dominant morphism $Y\rightarrow X$ between algebraic varieties over $k$ is said of finite type (resp. finite, birational, integral) if for any open subset $U\subset X$ the ring extension $\SO_X(U)\rightarrow \SO_Y(\pi^{-1}(U))$ is of finite type (resp. finite, birational, integral). In this paper, a morphism between algebraic varieties is always of finite type.
Let $X$ be an algebraic variety over $k$. The normalization of $X$, denoted by $X'$, is the algebraic variety over $k$ with a finite birational morphism $\pi':X'\rightarrow X$, called the normalization morphism such that for any open subset $U\subset X$ we have $\SO_{X'} (\pi'^{-1}(U))=\SO_X(U)'$. We say that $X$ is normal if $\pi'$ is an isomorphism. A point $x\in X$ is said normal if $\SO_{X,x}$ is integrally closed.
\vskip 2mm
We will use frequently that an integral extension of rings induces surjectivity at the spectrum level (See \cite[Thm. 9.3]{Ma} or \cite[Thm. 5.10, Cor. 5.8]{AM}). \begin{prop} \label{lying-over} Let $A\to B$ be an integral extension of rings. The maps $\Sp B\to \Sp A$ and $\Max B\to \Max A$ are surjective and closed. \end{prop}
As a consequence, if $\pi$ is a finite morphism between algebraic varieties over $k$ then Proposition \ref{lying-over} implies that $\pi$ and $\pi_{k}$ are surjective.
\subsection{Subintegral extensions, weak normalization and seminormalization}
We recall the concept of subintegral and weakly subintegral extensions introduced respectively by Traverso \cite{T}, Andreotti and Bombieri \cite{AB}.
\begin{defn} Let $A\to B$ be an extension of rings. \begin{enumerate} \item For $\p\in\Sp B$, we say that $\Sp B\to\Sp A$ is equiresidual (resp. residually purely inseparable) at $\p$ if the extension $k(\p\cap A)\to k(\p)$ is an isomorphism (resp. purely inseparable).\\ Let $W\subset \Sp B$, we say that $\Sp B\to\Sp A$ is equiresidual (resp. residually purely inseparable) by restriction to $W$ if for any $\p\in W$, $\Sp B\to\Sp A$ is equiresidual (resp. residually purely inseparable) at $\p$. If $W=\Sp B$ then we simply say equiresidual (resp. residually purely inseparable).\\ The extension $A\to B$ is said equiresidual (resp. residually purely inseparable) if $\Sp B\to \Sp A$ is. \item The extension $A\to B$ and the map $\Sp B\to \Sp A$ are said (resp. weakly) subintegral if the extension is integral and $\Sp B\to\Sp A$ is bijective and equiresidual (resp. residually purely inseparable). \end{enumerate} \end{defn}
Note that a field extension is equiresidual (resp. residually purely inseparable) if and only if it is an isomorphism (resp. purely inseparable). Remark that a subintegral extension is weakly subintegral and that the converse holds in characteristic zero. We extend these definitions to the geometric setting, adding moreover a notion of hereditarily birational morphism that have been introduced in the real setting in \cite{FMQ}.
\begin{defn} Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. \begin{enumerate} \item We say that $\pi$ is equiresidual (resp. residually purely inseparable) if for any $y\in Y$ then the field extension $k(\pi(y))\to k(y)$ is an isomorphism (resp. purely inseparable). \item We say that $\pi$ is (resp. weakly) subintegral if $\pi$ is integral, bijective and equiresidual (resp. residually purely inseparable). \item We say that $\pi:Y\to X$ is hereditarily birational if for any open subset $U\subset X$ and
for any irreducible algebraic subvariety $V=\V(\p)\simeq\Sp(\SO_Y(\pi^{-1}(U))/\p)$ in $\pi^{-1}(U)$, the morphism $$\pi_{|V}: V\to W=\V(\p\cap \SO_X(U))\simeq\Sp\big(\SO_X(U)/(\p\cap \SO_X(U))\big)$$ is birational. \end{enumerate} \end{defn}
Geometrically speaking, a dominant morphism $\pi:Y\to X$ is equiresidual if and only if it is hereditarily birational. Indeed, for any open subset $U\subset X$ and for any irreducible algebraic subvariety $V=\V(\p)\simeq\Sp(\SO_Y(\pi^{-1}(U))/\p)$ in $\pi^{-1}(U)$, the restricted morphism $$\pi_{|V}: V\to W=\V(\p\cap \SO_X(U))\simeq\Sp \big( \SO_X(U)/(\p\cap \SO_X(U))\big)$$ is birational if and only if the extension $k(\p\cap \SO_X(U))=\K(W)\to k(\p)=\K(V)$ is an isomorphism.
An hereditarily birational morphism is not necessarily bijective. However, adding an integrality assumption and using Proposition \ref{lying-over}, we get the following characterization.
\begin{lem}
Let $\pi:Y\to X$ be an integral morphism between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate}
\item $\pi$ is hereditarily birational and injective.
\item $\pi$ is subintegral. \end{enumerate} \end{lem}
The notion of (resp. weak) subintegral extension leads to the notion of seminormalization (resp. weak normalization), in a similar way that integral extensions lead to normalization.
In order to define the notions of seminormality and weak normalization, we need to consider sequences of ring extensions. A ring $C$ is said intermediate between the rings $A$ and $B$ if there exists a sequence of extensions $A\to C\to B$. In that case, we say that $A\to C$ and $C\to B$ are intermediate extensions of $A\to B$ and we say in addition that $A\to C$ is a subextension of $A\to B$.
Seminormal (resp. weakly normal) extensions are maximal (resp. weakly) subintegral extensions. \begin{defn} \label{defseminorm} Let $A\rightarrow C\to B$ be a sequence of two extensions of rings with $A\to C$ (resp. weakly) subintegral. We say that $C$ is seminormal (resp. weakly normal) between $A$ and $B$
if for every
intermediate ring
$D$ between $C$ and $B$, with $C$ different from $D$, then $A\to D$ is not (resp. weakly)
subintegral.
We say that $A$ is seminormal (resp. weakly normal) in $B$ if $A$ is
seminormal (resp. weakly normal) between $A$ and $B$. We say that $A$ is seminormal (resp. weakly normal) if $A$ is seminormal (resp. weakly normal) between $A$ and $A'$. \end{defn}
Recall that the characteristic exponent $e(K)$ of a field $K$ is 1 if $\car (K) = 0$ and is $p$ if $\car (K) = p > 0$. Given an extension of rings $A\to B$, Traverso \cite{T}, Andreotti and Bombieri \cite{AB} (see also \cite{V}) proved respectively there exists a unique intermediate ring which is seminormal (resp. weakly normal) between $A$ and $B$. To this purpose, they introduced the rings
$$A_B^+=\{b\in A_B'|\,\,\forall\p\in\Sp A,\,\,b_{\p}\in A_{\p}+\JRad((A_B')_{\p})\},$$
$$A_B^*=\{b\in A_B'|\,\,\forall\p\in\Sp A,\exists n\geq 0,\,\,b_{\p}^{e(k(\p))^n}\in A_{\p}+\JRad((A_B')_{\p})\},$$ where $\JRad$ stands for the Jacobson radical, namely the intersection of all the maximal ideals. The idea to build $A_B^+$ and $A_B^*$ is, for all $\p\in\Sp A$, to glue together all the prime ideals of $A_B'$ lying over $\p$ (see \cite{Mnew}).
\begin{thm}\label{thmT} \cite{T}, \cite{AB}\\ Let $A\to B$ be an extension of rings. Then $A_B^+$ (resp. $A_B^*$) is the unique ring which is seminormal (resp. weakly normal) between $A$ and $B$.
Moreover, for any intermediate ring $C$ between A and B, the extension $ A\to C$ is (resp. weakly) subintegral if and only if $C\subset A^+_B$ (resp. $C\subset A_B^*$). \end{thm}
The ring $A^+_B$ (resp. $A_B^*$) is called the seminormalization (resp. weak normalization) of the ring extension $ A\to B$ or the seminormalization (resp. weak normalization) of $A$ in $B$. Note that $$A\subset A_B^+\subset A_B^*\subset A_B^\prime\subset B.$$ The ring $A^+_{A'}$ (resp. $A^*_{A'}$) is called the seminormalization (resp. weak normalization) of $A$ and is simply denoted by $A^+$ (resp. $A^*$). Note that when $A$ and $B$ are domains, then $A$ and $A_B^+$ have in particular the same fraction field and that $\K(A)\to \K(A_B^*)$ is purely inseparable.
Note that the inclusion $A_B^+\subset A_B^*$ can be strict. \begin{ex} Let $K$ be a field of characteristic $2$, $x$ be an indeterminate, and consider the integral extension $A=K[x^2]\to B=K[x]$. It follows from a criterion of Hamman that $A_B^+=A$ (see \cite[Ex. 2.13]{V}). Since $x^2$ and $2x= 0$ are both in $A$ but $x$ is not in $A$, it follows from \cite[Prop. 3.10]{V} that $A$ is not weakly normal in $B$. \end{ex}
\subsection{Seminormalization and weak normalization of a morphism between algebraic varieties}
Andreotti and Bombieri \cite{AB} have introduced and built the seminormalization and the weak normalization of a scheme in another one. In this section, we provide a different and elementary construction of the seminormalization and the weak normalization of an affine algebraic variety in another one.
The seminormalization and the weak normalization answer the respective following questions. Let $Y\to X$ be a dominant morphism between algebraic varieties over $k$. Does there exist a biggest algebraic variety $Z$ such that $Y\to X$ factorizes through $Z$ and $Z\to X$ is (resp. weakly) subintegral~?
\vskip 2mm
We recall first the notion of normalization of a variety in another one. Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. The integral closure $(\SO_X)_{\pi_*(\SO_Y)}'$ of $\SO_X$ in $\pi_*(\SO_Y)$ is a coherent sheaf \cite[Lem. 52.15]{STPmorph} and by \cite[II Prop. 1.3.1]{Gr2} it is the structural sheaf of a variety over $k$. \begin{defn} \label{defnorm} Let $\pi:Y\to X$ be a dominant morphism of finite type between algebraic varieties over $k$. The variety with structural sheaf equal to the integral closure of $\SO_X$ in $\pi_*(\SO_Y)$ is called the normalization of $X$ in $Y$ and is denoted by $X_Y'$. \end{defn}
Be aware that the normalization of a variety in another one is not necessarily a normal variety, nor it admits a birational morphism onto the original variety.
For a dominant morphism $Y\to X$ between algebraic varieties over $k$, we say that an algebraic variety $Z$ over $k$ is intermediate between $X$ and $Y$ if $Y\to X$ factorizes through $Z$. For affine varieties, it is equivalent to say that $k[Z]$ is an intermediate ring between $k[X]$ and $k[Y]$. The normalization of a variety in another one satisfies the following property :
\begin{prop} \label{PUnormalization} Let $Y\to X$ be a dominant morphism between algebraic varieties over $k$. Let $Z$ be an intermediate variety between $X$ and $Y$. Then $Z\to X$ is finite if and only if it factorizes $X_Y'\to X$. \end{prop}
We describe now an elementary construction of the seminormalization and the weak normalization of an affine algebraic variety in another one. Let $Y\to X$ be a dominant morphism between affine algebraic varieties over $k$. We want to check that the rings $A_1=k[X]^{+}_{k[Y]}$ and $A_2=k[X]^*_{k[Y]}$ are coordinate rings of algebraic varieties. We know that the morphism $X_Y'\to X$ is finite by Proposition \ref{PUnormalization}, so we can apply Lemma \ref{lemintermed} below to the extensions $$k[X]\subset A_1\subset A_2 \subset k[X'_Y]=k[X]'_{k[Y]} \subset k[Y]$$ to conclude.
\begin{lem}
\label{lemintermed} Let $\pi:Y\to X$ be a finite morphism between affine algebraic varieties over $k$. Let $A$ be a ring such that $k[X]\subset A\subset k[Y]$. Then $A$ is the coordinate ring of a unique affine algebraic variety over $k$ and $\pi$ factorizes through this variety. \end{lem}
\begin{proof} Since $k[Y]$ is a finite module over the Noetherian ring $k[X]$ then it is a Noetherian $k[X]$-module. Thus the ring $A$ is a finite $k[X]$-module as a submodule of a Noetherian $k[X]$-module. It follows that $A$ is a finitely generated algebra over $k$ and the proof is done. \end{proof}
For general constructions of the seminormalization and the weak normalization, one needs to check that the seminormalization and the weak normalization of the local charts of an affine covering glue together to give a global variety. This is done by Andreotti and Bombieri \cite{AB} using Grothendieck criterion \cite[II Prop. 1.3.1]{Gr2} concerning the quasi-coherence of sheaves. It leads to the following definitions.
\begin{defn} Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. The seminormalization (resp. weak normalization) of $X$ in $Y$ is the algebraic variety $X^+_Y$ (resp. $X_Y^*$) over $k$ with structural sheaf equal to the seminormalization (resp. weak normalization) of $\SO_X$ in $\pi_*(\SO_Y)$.
We call $X^{+}$ (resp. $X^*$) the seminormalization (resp. weak normalization) of $X$ in its normalization $Y=X'$. We say that $X$ is seminormal in $Y$ (resp. seminormal) if $X=X^+_Y$ (resp. $X=X^+$). We say that $X$ is weakly normal in $Y$ (resp. weakly normal) if $X=X^*_Y$ (resp. $X=X^*$).
\end{defn}
\begin{rem} Note that the seminormalization of $X$ in $Y$ is birational to $X$, even if $Y\to X$ is not birational. It is not the case for the normalization of $X$ in $Y$ and also for the weak normalization of $X$ in $Y$. We have in general $$Y\to X'_Y\to X^*_Y\to X^+_Y\to X.$$ \end{rem}
The seminormalization and the weak normalization of a variety in another one satisfies the following universal properties~: \begin{prop}\label{propCSEPvariety} Let $Y\to Z\to X$ be a sequence of dominant morphisms between algebraic varieties over $k$. Then $Z\to X$ is (resp. weakly) subintegral if and only if $X^+_Y\to X$ (resp. $X_Y^*$) factorizes though $Z$. \end{prop}
\begin{proof} It is a reformulation of the second part of Theorem \ref{thmT}. \end{proof}
\section{saturation}\label{sect-sat}
In the classical study of the seminormalization, some basic properties such as the local nature happen to be not so straightforward to prove. A nice algebraic approach has been proposed by Manaresi \cite{Mana}, in the spirit of the relative Lipschitz saturation \cite{Lip}, via the saturation of a ring $A$ in another ring $B$ which is integral over $A$. The saturation coincides with the weak normalization when the ring extension is finite. We aim to study the properties of this saturation for more general extensions, and establish its universal properties.
\subsection{Universal property of the saturation}
We define the saturation of a ring extension analogously to \cite{Mana}, but for non-necessarily integral extensions.
\begin{defn} Let $A\to B$ be an extension of rings. The saturation of $A$ in $B$, denoted by $\widehat{A}_B$, is defined by $$\widehat{A}_B=\{b\in B\mid b\otimes_A1-1\otimes_A b\in {\rm{NilRad}}(\BAB) \}$$ where the nil radical $\NilRad$ denotes the ideal of nilpotent elements.
We say that $A$ is saturated in $B$ if $\widehat{A}_B=A$. The saturation of $A$ is its saturation in $A'$ and it is simply denoted by $\widehat{A}$. We say that $A$ is saturated if $\widehat{A}=A$. \end{defn}
Recall that the nilradical is the intersection of all prime ideals. In order to study the saturation, we need to understand better the relation between prime ideals in $A$ and $B$ and prime ideals in $\BAB$. For a ring extension $A\to B$, we introduce the notation $\varphi_1$, $\varphi_2$ for the ring morphisms $\varphi_i:B\to \BAB$ defined by \begin{equation}\label{eq-phi} \varphi_1(b)= b\otimes_A 1 \textrm{~~~~~and~~~~~}\varphi_2(b)= 1\otimes_A b. \end{equation}
The data of a prime ideal $\omega$ in $\BAB$, or more precisely the data of a morphism $g:\BAB\to k(\omega)$ with kernel $\omega$, is equivalent to the data of a $4$-tuple of prime ideals $$(\p_1,\p_2,\q,\p)\in\Sp B\times\Sp B\times\Sp A\times\Sp (k(\p_1)\otimes_{k(\q)}k(\p_2))$$ such that $$\p_1=\ker (g\circ \varphi_1),~~\p_2=\ker (g\circ \varphi_2),~~\q=\p_1\cap A=\p_2\cap A,~~k(\omega)=k(\p)$$ and such that the composition $$\BAB\to k(\p_1)\otimes_{k(\q)}k(\p_2)\to k(\p)$$ coincides with $g$.
The saturation is a ring, compatible with inclusion.
\begin{lem}\label{lem-elem} Let $A\to B$ be an extension of rings. \begin{enumerate} \item $\widehat{A}_B$ is a subring of $B$ containing $A$. \item If $A\subset C \subset B$, then $\widehat{A}_B \subset \widehat{C}_B$. \end{enumerate} \end{lem}
\begin{proof} \begin{enumerate} \item The set $\widehat{A}_B$ is an $A$-module as the kernel of the $A$-module morphism $$B\xrightarrow{\varphi_1-\varphi_2} \dfrac{\BAB}{{\rm{NilRad}}(\BAB)}.$$ The stability under product comes from the identity $$b_1b_2\otimes_A 1-1\otimes_A b_1b_2=(b_1\otimes_A 1)(b_2\otimes_A 1-1\otimes_A b_2)+(1\otimes_A b_2)(b_1\otimes_A 1-1\otimes_A b_1)$$ and the fact that ${\rm{NilRad}}(\BAB)$ is an ideal. \item The image of a nilpotent element by the ring morphism $\BAB \to B\otimes_C B$ remains nilpotent. \end{enumerate} \end{proof}
In order to give a universal property of the saturation, we recall the notion of radicial extension introduced by Grothendieck \cite[I, def. 3.7.2]{Gr1}. We also introduce a notion of radicial sequence of extensions similarly to \cite{Mnew}, due to the lack of integrality of the ring extensions.
\begin{defn} \begin{enumerate} \item An extension of rings $A\to B$ is said radicial if $\Sp B\to\Sp A$ is injective and residually purely inseparable. \item A sequence of extensions $A\to C\to B$ of rings is said radicial if the restriction of $\Sp C\to\Sp A$ to the image of $\Sp B\to\Sp C$ is injective and residually purely inseparable. \end{enumerate} \end{defn}
\begin{rem} An extension (resp. a sequence of extensions) of fields $K\to K'$ (resp. $K\to K'\to K''$) is radicial if and only if $K\to K'$ is purely inseparable. \end{rem}
\vskip 2mm
The saturation furnishes radicial sequences of extensions.
\begin{prop}\label{prop-sat} Let $A\to B$ be a ring extension. For any $C\subset \widehat A_B$, the sequence $A\to C\to B$ is radicial. \end{prop}
Before entering into the proof, we state an elementary result about field extensions. Remark that it gives a proof of (2) implies (3) of Theorem \ref{PU1saturation} in the special case of a sequence of field extensions.
\begin{lem}\label{lem-field} Let $K\to K'\to K''$ be a non radicial sequence of field extensions i.e such that $K\to K'$ is not purely inseparable. Then, there are two $K$-morphisms $K''\to L$ into a (algebraically closed) field $L$ whose compositions with $K'\to K''$ are distinct. \end{lem}
\begin{proof} Since $K\to K'$ is not purely inseparable then it follows from \cite[Prop. I.3.7.1]{Gr1} that there are two distinct $K$-morphisms $\psi_1, \psi_2 : K'\to L'$ into a field $L'$. The point is to extend them to $K''$.
For $i\in\{1,2\}$, one can embed the field extensions $K'\to K''$ and $\psi_i: K' \to L'$ into a common extension $K'\to L'_i$ by amalgamation \cite[Chap 5, §4, Prop. 2]{Bour}. Denote by $\psi_i' :K''\to L'_i$ the induced extension. By amalgamation of $L'_1$ and $L_2'$ over $K''$, one can assume that $\psi_1'$ and $\psi_2'$ take values in a common field $L$.
The morphisms $\psi_1',\psi_2': K'' \to L$ fulfil the requirements since the restriction of $\psi_i'$ to $K'$ coincides with $\psi_i$. \end{proof}
\begin{rem} It is classical that one can choose $L=L'$ in the proof of Lemma \ref{lem-field} if the extension $K'\to K''$ is moreover algebraic, and this is used in \cite{Lip} to prove that the Lipschitz saturation is stable under contraction : in the setting of Proposition \ref{prop-sat}, if $C\to B$ is integral, then the Lipschitz saturation of $A$ in $C$ is equal to the intersection of $C$ with the Lipschitz saturation of $A$ in $B$. In our context the extensions are not assumed to be integral, and this contraction property does not hold, as illustrated by Example \ref{exVitdetail}. \end{rem}
\begin{proof}[Proof of Proposition \ref{prop-sat}] Let $\p_1$, $\p_2$ be two prime ideals of $B$ lying over the same ideal $\q$ of $A$. A first step is to prove that $\p_1$ and $\p_2$ lye over the same ideal of $C$.
Let $\p$ be a prime ideal of $k(\p_1)\otimes_{k(\q)}k(\p_2)$, and $\omega=(\p_1,\p_2,\q,\p)\in\Sp (\BAB)$ be the corresponding element, coming with a morphism $g:\BAB\to k(\omega)$ with $\ker g=\omega$. For $c\in\p_1\cap C$, we have $g\circ \varphi_1(c)=g(c\otimes_A 1)=0$ by construction of $g$. The element $\cAc$ is nilpotent in $\BAB$ by assumption, so that
$$0=g(\cAc)=g\circ \varphi_1(c)-g\circ \varphi_2(c).$$ As a consequence $g\circ \varphi_2(c)=0$ and thus $c\in\p_2\cap C$. By symmetry we obtain $$\p_1\cap C= \p_2\cap C.$$
The second step is to prove that the extension $\phi:k(\q)\to k(\p_1\cap C)$ is purely inseparable, where $\q=\p_1\cap A$. Assume by contradiction that $\phi$ is not purely inseparable, and consider the composition $$k(\q)\xrightarrow{\phi} k(\p_1\cap C)\rightarrow k(\p_1).$$ By Lemma \ref{lem-field}, there are two distinct $k(\q)$-morphisms $\psi_1,\psi_2:k(\p_1)\to L$ into some field $L$, which remain distinct by restriction to $k(\p_1\cap C)$. Thus there exists $c\in C$ such that $\psi_1\circ\pi (c)\not=\psi_2\circ\pi (c)$, where $\pi:B\to k(\p_1)$ denote the natural morphism.
The morphisms $\psi_1$ and $\psi_2$ induce a morphism $\psi:k(\p_1)\otimes_{k(\q)}k(\p_1)\to L$ given by $$\psi(\pi(b_1)\otimes_{k(\q)} \pi(b_2))=\psi_1\circ \pi(b_1)\cdot\psi_2\circ \pi(b_2)$$ for $b_1,b_2\in B$. The kernel $\p$ of $\psi$ gives rise to a prime ideal $\omega=(\p_1,\p_2,\q,\p)$ of $\BAB$ coming with a morphism $$g:\BAB\to k(\omega)\to L.$$ By our choice of $c$, the element $$\psi(\pi(c)\otimes 1-1\otimes \pi(c))=\psi_1\circ\pi (c)-\psi_2\circ\pi (c)$$ is not zero, so that $\cAc$ does not belong to $\ker g=\omega$, contradicting the inclusion $C\subset \widehat{A}_B$. \end{proof}
Actually the converse of the preceding result holds true, and it gives rise to universal properties of the saturation, in terms of radicial sequences of extensions. This result is, up to our knowledge, not present in the literature.
\begin{thm} \label{PU1saturation} Let $A\xrightarrow{i} C\xrightarrow{j} B$ be a sequence of extensions of rings. The following properties are equivalent: \begin{enumerate} \item For any field $K$, the map $$\Sp(j)\circ (\Mor(\Sp K, \Sp B))\to \Mor(\Sp K, \Sp A)$$ $$(\Sp(j)\circ \alpha)\mapsto \Sp(i)\circ (\Sp(j)\circ \alpha)$$ is injective. \item For any field $K$, if $\psi_1:B\to K$ and $\psi_2:B\to K$ are two field morphisms distinct by composition with $j$, then they are distinct by composition with $j\circ i$. \item The sequence $A\to C\to B$ is radicial. \item $j(C)\subset \widehat{A}_B$. \item The kernel of the morphism $\CAC\to C$ defined by $c_1\otimes_A c_2\mapsto c_1c_2$ is included in the nilradical of $\BAB$. \end{enumerate} \end{thm}
\begin{proof} The equivalence between (1) and (2) is straightforward. Since $\ker (\CAC\to C)$ is generated by the elements of the form $c\otimes_A1-1\otimes_A c$ for $c\in C$, then (4) $\Leftrightarrow$ (5). Note that $(4)$ implies $(3)$ by Proposition \ref{prop-sat}.
\vskip 2mm
Let us prove that (3) implies (2) by contraposition. Let $\psi_1:B\to K$ and $\psi_2:B\to K$ be two morphisms in a field K such that $\psi_1\circ j\not=\psi_2\circ j$ and $\psi_1\circ j\circ i=\psi_2\circ j \circ i$. Let $\p_1$, $\p_2$ and $\q$ denote respectively the kernels of $\psi_1$, $\psi_2$ and $\psi_1\circ j\circ i:A\to K$. For $i=1,2$ we get the following commutative diagram: $$\begin{array}{ccccccc}
A&\xrightarrow{i} & C &\xrightarrow{j} & B & \xrightarrow{\psi_i} & K\\
\downarrow&&\downarrow&& \downarrow & \nearrow & \\
k(\q)&\rightarrow & k(\p_i\cap C) & \rightarrow& k(\p_i) & &\\ \end{array}$$ If $\p_1\cap C$ is not equal to $\p_2\cap C$, then $\Sp C\to \Sp A$ is not injective on the image of $\Sp B$.\\ If $\p_1\cap C$ is equal to $\p_2\cap C$, then $\psi_1$ and $\psi_2$ induce two different $k(\q)$-morphisms $\psi'_\iota: k(\p_1\cap C) \to K$ since $\psi_1\circ j\not=\psi_2\circ j$. From \cite[Prop. I.3.7.1]{Gr1}, the extension $k(\q)\to k(\p_1\cap C)$ cannot be purely inseparable.
In both cases, the extension $A\to C\to B$ is not radicial.
\vskip 2mm Finally we prove that (2) implies (4) by contraposition. By assumption there are $c\in C$ and $\omega \in \Sp \BAB$ such that $\cAc \notin \omega$. The ideal $\omega$ comes with a morphism $g:\BAB \to K$ with $\ker g=\omega$. Consider the composition of $g$ with the morphisms $\varphi_1$ and $\varphi_2$ defined in \eqref{eq-phi}. By construction $g\circ \varphi_1:B\to K$ coincides with $g\circ \varphi_2:B\to K$ when composed with $j\circ i$, but not when composed with $j$ because $g\circ \varphi_1 (j(c))\neq g\circ \varphi_2(j(c))$. It contradicts $(2)$. \end{proof}
If we focus on the particular case of radicial extensions rather that sequences, we recover \cite[Prop. I.3.7.1]{Gr1}, with an additional condition using the saturation. \begin{prop} \label{radicial} Let $i:A\to B$ be an extension of rings and $\Sp(i):\Sp B\to\Sp A$ be the associated map. The following properties are equivalent: \begin{enumerate} \item For any field $K$, the map $$\Mor(\Sp K, \Sp B)\to \Mor(\Sp K, \Sp A)$$ $$\alpha\mapsto \Sp(i)\circ\alpha$$ is injective. \item If $\psi_1:B\to K$ and $\psi_2:B\to K$ are two distinct morphisms then the compositions $\psi_1\circ i$ and $\psi_2\circ i$ are different. \item $i:A\to B$ is radicial. \item $B=\widehat{A}_B$. \item The kernel of the morphism $\BAB\to B$ defined by $b_1\otimes_A b_2\mapsto b_1b_2$ is included in the nilradical of $\BAB$. \end{enumerate} \end{prop}
\begin{proof} Direct consequence of Theorem \ref{PU1saturation}, using the fact that an extension $A\to B$ is radicial if and only if the sequence of extensions $A\to B\to B$ is so. \end{proof}
\subsection{Saturation for varieties}
We begin with the definition of radiciality and saturation for morphisms \cite[Chap. I,3.7.2]{Gr1}. Then, we extend these definitions to sequences of morphisms.
\begin{defn} \begin{enumerate} \item Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. We say that $\SO_X\to \pi_*\SO_Y$ is radicial if for any open subset $U\subset X$ the extension $\SO_X (U)\to\SO_Y (\pi^{-1}(U))$ is radicial. In this situation, we say that $\pi$ is radicial. \item Let $Y\stackrel{\phi}{\to} Z\stackrel{\psi}{\to} X$ be a sequence of dominant morphisms between algebraic varieties over $k$. We say that $\SO_X\to \psi_*\SO_Z\to (\psi\circ\phi)_*\SO_Y$ is radicial if for any open subset $U\subset X$ the sequence of extensions $\SO_X (U)\to \SO_Z (\psi^{-1}(U))\to\SO_Y ((\psi\circ\phi)^{-1} (U))$ is radicial. In this situation, we say that the sequence of morphisms $Y\to Z\to X$ is radicial. \item We say that $X$ is saturated in $Y$ if $\SO_X$ is saturated in $\pi_*\SO_Y$, i.e for any open subset $U\subset X$ then $\SO_X(U)$ is saturated in $\SO_Y(\pi^{-1}(U))$. \end{enumerate} \end{defn}
\begin{rem} \label{defsatvar} \begin{enumerate} \item Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. Then, $\pi$ is radicial if and only if $\pi$ is injective and residually purely inseparable. \item Let $Y\stackrel{\phi}{\to} Z\stackrel{\psi}{\to} X$ be a sequence of dominant morphisms between algebraic varieties over $k$. Then, $Y\to Z\to X$ is radicial if and only $\psi$ is injective and residually purely inseparable by restriction to the image of $\phi$. \end{enumerate} \end{rem}
In order to translate the universal property of the saturation in terms of varieties, we recall the notion of universal injectivity from \cite[Chap. I, 3.4.3]{Gr1}.
\begin{defn} \begin{enumerate} \item A morphism $\pi:Y\to X$ between algebraic varieties over $k$ is said universally injective if for any field extension $k\to K$, the map $\pi_K:Y(K)\to X(K)$ is injective. \item A sequence of morphisms $Y\to Z\to X$ between algebraic varieties over $k$ is said universally injective if for any field extension $k\to K$, the map $Z(K)\to X(K)$ is injective by restriction to the image of $Y(K)\to Z(K)$. \end{enumerate} \end{defn}
Grothendieck \cite[Prop. 3.7.1]{Gr1} proved that the notions of radicial and universally injective morphisms coincide. The universal property given in Theorem \ref{PU1saturation} implies that it is also the case if we consider sequences of morphisms rather than morphisms.
\begin{prop}\label{PU1saturationvar} Let $Y\stackrel{\phi}{\to} Z\stackrel{\psi}{\to} X$ be a sequence of dominant morphisms between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate} \item $Y\to Z\to X$ is universally injective. \item $Y\to Z\to X$ is radicial. \item $\psi_*\SO_Z\subset (\widehat{\SO_X})_{(\psi\circ\phi)_*\SO_Y}$ i.e for any open subset $U\subset X$ we have $\SO_Z (\psi^{-1}(U))\subset\widehat{\SO_X (U)}_{\SO_Y ((\psi\circ\phi)^{-1} (U))}$. \end{enumerate} \end{prop}
\begin{rem} Let $\pi:Y\to X$ be a dominant morphism between affine algebraic varieties over $k$. Contrarily to the seminormalization case it is not clear whether $\widehat{k[X]}_{k[Y]}$ is a finitely generated algebra over $k$ and thus lead to the existence of a variety. \end{rem}
As a consequence of the previous remark, the statement $\pi$ is subintegral if and only if $X^+_Y=Y$ has no equivalent when $\pi$ is radicial. Nevertheless we get: \begin{cor} \label{PU2saturationvar} Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. Then $\pi$ is radicial if and only if $(\widehat{\SO_X})_{\pi_*\SO_Y}=\pi_*\SO_Y$. \end{cor}
\section{Comparison between saturation, seminormalization and weak normalization} \label{comparaison}
In general, the seminormalization and the weak normalization are only included in the saturation.
\begin{lem} \label{satetsemi} Let $A\to B$ be an extension of rings. Then $$A^+_B\subset A^*_B\subset \widehat{A}_B.$$ \end{lem}
\begin{proof} We already know that $A^+_B\subset A^*_B$. Since $A\to A^*_B$ is weakly subintegral then $A\to A^*_B\to B$ is radicial. The inclusion $A^*_B\subset \widehat{A}_B$ follows by Theorem \ref{PU1saturation}. \end{proof}
If $A\to B$ is a purely inseparable extension of fields which is not an isomorphism then we get $A^+_B\subset A^*_B= \widehat{A}_B$ and the inclusion is strict. So in the sequel we focus in the comparison between weak normalization and saturation.
Note that there is no special relationship between the saturation and the relative normalization. However saturation and weak normalization coincide when we restrict to integral extensions. \begin{prop} \label{satetsemi2} Let $A\to B$ be an integral extension of rings. Then $$A^*_B= \widehat{A}_B.$$ \end{prop}
\begin{proof} The direct inclusion comes from Lemma \ref{satetsemi}.
The sequence $A\to \widehat{A}_B\to B$ is radicial by Theorem \ref{PU1saturation}. Since $ \widehat{A}_B\to B$ is integral then $\Sp B\to \Sp \widehat{A}_B$ is surjective by Proposition \ref{lying-over}. It follows that $A\to \widehat{A}_B$ is radicial and integral and thus is weakly subintegral. This forces $ \widehat{A}_B$ to be equal to the weak normalization $A^*_B$ of $A$ in $B$ by Theorem \ref{thmT}. \end{proof}
Finally we state the relations between saturation, weak normalization and seminormalization for varieties induced by Lemma \ref{satetsemi} and Proposition \ref{satetsemi2}.
\begin{prop} \label{sat=semivar} Let $\pi:Y\to X$ be a dominant morphism between varieties over $k$. \begin{enumerate} \item If $\pi$ is subintegral then $\pi$ is weakly subintegral. \item If $\pi$ is weakly subintegral then $\pi$ is radicial. \item If $X$ is saturated in $Y$, then $X$ is weakly normal and seminormal in $Y$. \item If $\pi$ is moreover integral, then $\pi$ is weakly subintegral if and only if $\pi$ is radicial, and $X$ is saturated in $Y$ if and only if $X$ is weakly normal in $Y$. \end{enumerate} \end{prop}
In the following proposition and examples $k$ is an algebraically closed field and $\car (k)=0$.
We compare the notions of seminormality and relative seminormality. \begin{prop} \label{semimprel} Let $\pi:Y\to X$ be a dominant morphism between varieties over $k$. If $X$ is seminormal then $X$ is seminormal in $Y$. \end{prop}
\begin{proof} Suppose $X$ is not seminormal in $Y$. From Proposition \ref{propCSEPvariety} the morphism $\varphi:X_Y^+\to X$ is subintegral, factorizes $\pi$ and is not an isomorphism. Since $\varphi$ is birational and finite then it follows that the normalization map $X'\to X$ factorizes throught $\varphi$. By subintegrality of $\varphi$ and seminormality of $X$ then we get a contradiction. \end{proof}
The converse of Proposition \ref{semimprel} is false, take $Y=X$ with $X$ not seminormal for example.
From \cite[Rem. 1.4]{C}, we know that the notions of relative weak normalization and relative saturation differ in any characteristic when we do not consider integral extensions of rings and integral morphisms of varieties. We end this section by providing explicit examples to illustrate that the notions of relative saturation and seminormalization do not coincide for varieties over an algebraically closed field of characteristic null, in any dimension. The examples are built on Example \ref{exVitdetail}, constructed from a nodal curve, for which we offer two arguments : a simple geometric one, and a direct computational one in order to construct the generalization in any dimension in Example \ref{exVitdetail2}.
\begin{ex} \label{exVitdetail} \begin{enumerate} \item Let $X$ be the nodal plane curve with coordinate ring $A=k[X]=k[x,y]/(y^2-x^2(x+1))$. Its normalization $X'$ has coordinate ring $A'=k[X']=k[x,z]/(z^2-(x+1))=A[y/x]$ and the inclusion $A\to A'$ is given by $(x,y)\mapsto (x,xz)$. Let $Y$ be defined by removing one of the two points $p=(0,1)$ and $q=(0,-1)$ of $X'(k)$ lying above the singular point of $X(k)$, say $p$. The coordinate ring of $Y$ is $$B=k[Y]=k[x,z,s]/(z^2-(x+1),s(z-1)-1)=A'[1/(z-1)]=A'[s]=A[y/x,s],$$ and we have a sequence of inclusions $A\to A'\to B$.
Then $A^+_B=A$ whereas $\widehat{A}_B=B$. To see the first point, since the variety $X$ is seminormal (see \cite{GT}) then $X$ is seminormal in $Y$ by Proposition \ref{semimprel}. For the second point, note that $A\to B$ is radicial because, for irreducible curves, the prime ideals correspond to the generic point and the closed points, and here $Y\to X$ is birational with $Y(k)\to X(k)$ bijective. As a consequence $\widehat{A}_B=B$ by Proposition \ref{radicial}.
\item We revisit the nodal curve example proving the equality $\widehat{A}_B=B$ using the very definition of the saturation. Keeping previous notation, set $\alpha=(z\otimes_A 1)-(1\otimes_A z)$ and $\beta=(s\otimes_A 1)-(1\otimes_A s)$. It suffices to prove that $\alpha$ and $\beta$ are nilpotent elements of $\BAB$. Indeed $\widehat{A}_B$ is a ring containing $x,z$ and $s$ so that $\widehat{A}_B=B$ in that case.
Note that $$\begin{array}{cccc} x \alpha & = & x \big((z+1)\otimes_A 1-1\otimes_A(z+1)\big) & \\ &= & \big(x(z+1)\otimes_A 1\big) - \big(1\otimes_A x(z+1)\big) &\\ &= & \big((y+x)\otimes_A 1\big) - \big(1\otimes_A (y+x)\big) & \textrm{~~since~~} y+x=x(z+1) \textrm{~~in~~} B\\ &= & 0 &\\ \end{array}$$ hence $$\begin{array}{cccc} \alpha^2 & = & \alpha \big((z+1)\otimes_A 1-1\otimes_A(z+1)\big) & \\
& = & \alpha (xs\otimes_A 1-1\otimes_Axs) & \textrm{~~since~~} xs=z+1 \textrm{~~in~~} B\\ &= & x\alpha (s\otimes_A 1 - 1\otimes_A s) &\\ &= & 0. &\\ \end{array}$$ Actually we even have $\alpha=0$ in $\BAB$, since a straightforward computation shows that $\alpha=\frac1{4}\alpha^3$. Finally, using the equality $\alpha= (z-1)\otimes_A 1-1\otimes_A(z-1)$ and the relation $s(z-1)=1$, we observe that $$0=(s\otimes_A 1)\alpha (1\otimes_A s) =-\beta.$$ \end{enumerate} \end{ex}
\begin{ex}\label{exVitdetail2} Consider the curves $X$ and $Y$ as in Example \ref{exVitdetail}. For $n\geq 1$, the variety $X\times\Af_k^n$ is seminormal in $Y\times\Af_k^n$, whereas the saturation of $X\times\Af_k^n$ in $Y\times\Af_k^n$ is $Y\times\Af_k^n$.
To see this, note that $X$ and $\Af_k^n$ are seminormal, so $X\times \Af_k^n$ is also seminormal \cite[Cor. 5.9]{GT} and thus $X\times \Af_k^n$ is seminormal in $Y\times\Af_k^n$ by Proposition \ref{semimprel}.
For the saturation, if the radiciality of $$k[X\times\Af_k^n]= A[t_1,\ldots,t_n] \to B[t_1,\ldots,t_n]=k[Y\times\Af_k^n]$$ is not so straightforward since we no longer work with curves as in Example \ref{exVitdetail} (1), the computations done in Example \ref{exVitdetail} (2) still prove that $\widehat{A[t_1,\cdots,t_n]}_{B[t_1,\cdots,t_n]}$ contains $x,z$ and $s$, and so is equal to $B[t_1,\ldots,t_n]$. \end{ex}
\section{Strong topology on the rational closed points and regulous functions}\label{regulous}
Continuous rational functions and regulous functions have been originally studied in real algebraic geometry \cite{Ku,KN,FHMM}, where the continuity is regarded with respect to the Euclidean topology, which can be studied algebraically via semialgebraic open sets \cite{BCR}. For an algebraic variety $X$ over $\C$, we can consider the Euclidean (or strong) topology of the complex points $X(\C)$ seen as a topological variety (see \cite{Sha2}). For example, if $X$ is affine then we have $X\subset \mathbb A_{\C}^n$ for some $n\in\N$ and the strong topology is induced by the natural inclusion $X(\C)\subset \R^{2n}$. With this point of view Bernard has developed in \cite{Be} the theory of regulous functions for algebraic varieties over $\C$.
Subintegral extensions and regulous functions are strongly related in real algebraic geometry as developed in \cite{FMQ2}. Working with the field of complex numbers, we know from the work of Bernard that the same holds true in the geometric case. The purpose of this section is first to generalize the work of Bernard to varieties over any algebraically closed field of characteristic zero, and then to relate it to the theory of relative seminormalization.
\vskip 2mm
In this section $k$ is an algebraically closed field and $\car (k)=0$.
\subsection{Generalizing the strong topology of $\C$}\label{sect-R}
Since there is a priori no natural strong topology on the $k$-rational points of a variety over $k$, we use the theory of real closed fields to define such a topology as in \cite{K,HK} (see also \cite{BW} for a recent cohomological use of this approach).
\vskip 2mm
From Artin Schreier theory \cite{AS}, we know the existence of (many) real closed subfields of $k$ with algebraic closure equal to $k$. Let $R\subset k$ denote one of these real closed fields. Then $k=R[\sqrt{-1}]$ and $R$ comes with a unique ordering. The ordering on $R$ gives rise to an order topology on the affine spaces $R^n$, in a similar way than the Euclidean topology on $\R^n$, even if the topological space $R$ is not connected (except in the case $R=\R$) or the closed interval $[0,1]$ is in general not compact.
\vskip 2mm We use this choice of $R$ to define a topology on the closed points of an algebraic variety over $k$. First, for an algebraic variety $X$ over $R$, choose an affine covering of $X$ by Zariski open subsets $U_i$, and endow each affine sets $U_i(R)$ with the order topology. These open sets glue together to define the order topology on $X(R)$, and this topology does not depend on the choice of the covering. This topological space can be endowed additionally with the structure of a semialgebraic space by considering the sheaf of continuous semialgebraic functions \cite{DK2,DK}, or even of a real algebraic variety with the sheaf of regular functions on the $R$-points \cite{BCR,Hui}.
Consider now the case of a quasi-projective algebraic variety $X$ over $k$. By Weil restriction \cite{W,GrFGA}, we associate to $X$ an algebraic variety $X_R$ over $R$ whose $R$-points are in bijection with the $k$-points of $X$.
We endow $X(k)$ with the topology induced by the order topology on $X_R(R)$, and we call it the $R$-topology on $X(k)$.
If $X$ is no longer quasi-projective, then the Weil restriction does not necessarily exist. Anyway choose an affine open covering $(U_i)_{i\in I}$ of $X$, endow the $R$-points of the Weil restrictions $(U_i)_R$ with the order topology, and note that these open sets glue together to define a topology on $X(k)$. This topology does not depend on the choice of the covering by \cite[Lemma 5.6.1]{Sc}, and we call it the $R$-topology on $X(k)$. Again one can consider $X(k)$ as a semialgebraic space in the sense of \cite{DK} or as a real algebraic variety in the sense of \cite{BCR}.
\vskip 2mm The $R$-topology on $X(k)$ has many good properties, for instance $X(k)$ is semialgebraically connected and of pure dimension twice the dimension of $X$ if $X$ is irreducible \cite{K}. For $k=\C$ and $R=\R$, the $\R$-topology is nothing more than the strong topology.
The choice of a different real closed field $R$ in $k$ will lead to different topologies on $X(k)$ (for instance the semialgebraic fundamental group does depend on the choice of $R$ \cite{K}). Already with $k=\C$, one can choose a real closed field different from $\R$, even for instance a non-Archimedean $R\subset \C$. We will see however that in our setting, the choice of the real closed field is transparent.
\subsubsection{Basics on the $R$-topology of $k$-varieties}
In this section we fix a real closed field $R$ with algebraic closure $k$.
Let $X$ be a quasi-projective algebraic variety over $k$. Recall that by Weil restriction \cite{GrFGA,Sc}~: \begin{enumerate} \item The variety $X_R$ is nonsingular if $X$ is nonsingular. More precisely, a $k$-point in $X$ is singular if and only if its corresponding $R$-point in $X_R$ is singular. \item A Zariski open subset $U\subset X$ induces a Zariski open subset $U_R\subset X_R$. \item A proper morphism $Y\to X$ between quasi-projective algebraic varieties over $k$ induces a proper morphism $Y_R\to X_R$. \item A finite morphism $Y\to X$ between quasi-projective algebraic varieties over $k$ induces a finite morphism $Y_R\to X_R$. \end{enumerate}
Let $X$ be an affine algebraic variety over $k$. A regular function on $X$ gives rise to a polynomial (and thus continuous) mapping $X_R(R)\to R^2$. Indeed the regular function is polynomial, and by Weil restriction a polynomial function to $k$ induced a polynomial mapping to $R^2$ by taking the real and imaginary parts. Finally a polynomial function is continuous with respect to the $R$-topology.
The topological properties of $R$-varieties coming from $k$-varieties are much more moderate than for general $R$-varieties. For instance, if complex irreducible varieties are locally of equal dimension, irreducible algebraic subsets of $\R^n$ may have isolated points. From \cite{BCR}, a real algebraic variety is called central if its subset of nonsingular points is dense with respect to the $R$-topology.
For a subset $A\subset X(k)$, we denote by $\overline{A}^R$ the closure of $A$ with respect to the $R$-topology. We denote by $\Reg(X(k))$ the set of nonsingular points of $X(k)$. \begin{prop}\label{prop-cent} Let $X$ be an irreducible algebraic variety over $k$. Then $X(k)$ is central : $$\overline{\Reg(X(k))}^R=X(k).$$ \end{prop}
\begin{proof} The question being local, it suffices to assume $X$ is affine, and in particular the Weil restriction of $X$ exists.
If $X$ is nonsingular, then so is $X_R$ by Weil restriction, and so $X_R(R)$ is central.
Otherwise, consider a resolution $\sigma :\tilde X \to X$ of the singularities of $X$ which exists by \cite{Hiro} since $k$ has characteristic zero. Then $\sigma_k$ is surjective since $k$ is algebraically closed, and one can assume that $\sigma_k$ induces a bijection $$\tilde U=\sigma_k^{-1}(\Reg(X(k))\to \Reg(X(k))=U.$$ Let $x\in X(k)$, and choose a preimage $\tilde x\in \sigma_k^{-1}(x)$. The centrality of $\tilde X(k)$ implies the existence of a continuous semialgebraic curve $\tilde \gamma :[0,1]\to \tilde X(k)$ with $\tilde \gamma(0)=\tilde x$ and $\tilde \gamma(t)\in \tilde U$ for $t\in (0,1]\subset R$ by the Curve Selection Lemma \cite[Theorem 2.5.5]{BCR}. Its composition $\gamma=\sigma_k \circ \tilde \gamma$ is a continuous semialgebraic curve from $[0,1]$ to $X(k)$ with $\gamma(0)=x$ and $\gamma(t)\in U$ for $t\in (0,1]\subset R$. As a consequence $x$ belongs to the closure with respect to the $R$-topology of $\Reg(X(k))$ in $X(k)$, and so $X(k)$ is central. \end{proof}
\begin{rem} In particular, if the irreducible algebraic variety $X$ over $k$ has dimension $d$, then the local semialgebraic dimension of $X(k)$ at any point $x\in X(k)$ is equal to $2d$. \end{rem}
The following result is not valid in general for algebraic varieties over $R$, and even for $R=\R$. \begin{lem}\label{lem-dense} Let $X$ be an irreducible algebraic variety over $k$. A non-empty Zariski open subset of $X(k)$ is dense with respect to the $R$-topology. \end{lem}
\begin{proof} The question being local, it suffices to assume $X$ is affine, and in particular the Weil restriction of $X$ exists.
A Zariski open subset remains Zariski open by Weil restriction. Combined with Proposition \ref{prop-cent}, it suffices to check that a non-empty Zariski open set $U$ in a central irreducible algebraic variety over $R$ is dense with respect to the $R$-topology. This last property is classical ; for instance, the complement is an algebraic subset of strictly smaller dimension, and a semialgebraic triangulation of $X(k)$ adapted to the complement shows that locally, a point in the complement is in the boundary of a semialgebraic simplex in $U(k)$. \end{proof}
Over a general real closed field, the notion of compact sets is advantageously replaced by closed and bounded semialgebraic sets. For instance, the image of a closed and bounded semialgebraic set by a continuous semialgebraic map is again closed and bounded (and semialgebraic) \cite[Theorem 2.5.8]{BCR}. A semialgebraic map is said to be proper with respect to the $R$-topology if the preimage of a closed and bounded semialgebraic set is closed and bounded.
\begin{lem}\label{lem-surj} Let $\sigma:\tilde X\to X$ be a proper morphism between irreducible varieties over $k$. Then $\sigma_k$ is proper with respect to the $R$-topology. If $\sigma$ is moreover birational, then $\sigma_k$ is surjective. \end{lem}
\begin{proof}
The notion of properness is local on the target, so that there is an affine covering of $X$ such that for any open affine subset $U$ in the covering, the restriction $\sigma'=\sigma_{|\sigma^{-1}(U)}$ of $\sigma$ to $\sigma^{-1}(U)$ is proper. The properness of $\sigma'$ is kept by Weil restriction, so that $\sigma'_k$ is proper with respect to the $R$-topology by \cite[Theorem 9.6]{DK2}. Finally $\sigma_k$ is proper with respect to the $R$-topology since that notion of properness is local on the target too by \cite[Proposition 5.7]{DK}.
If $\sigma$ is birational, there are Zariski open sets $\tilde U\subset \tilde X$ and $U\subset X$ such that $\sigma_{|\tilde U}$ is a bijection onto $U$. Then $$U(k)= \sigma_k({\tilde U(k)})\subset \sigma_k(\overline{\tilde U(k)}^R)=\sigma_k(\tilde X(k)),$$ the right hand side equality coming from Lemma \ref{lem-dense}. Finally $\sigma_k(\tilde X(k))$ is closed for the $R$-topology by properness of $\sigma_k$, so that taking the closure with respect to the $R$-topology gives the result by Lemma \ref{lem-dense}. \end{proof}
\begin{lem}\label{lem-fini} Let $\sigma:\tilde X\to X$ be a finite morphism between algebraic varieties over $k$. Then $\sigma_k$ is closed with respect to the $R$-topology. \end{lem}
\begin{proof} By \cite[Theorem 4.2]{DK2}, a finite morphism between algebraic varieties over $R$ is closed with respect to the $R$-topology. The result follows since finiteness is local and Weil restriction preserves finite morphisms.
Alternatively when $X$ and $\tilde X$ are irreducible, a finite morphism is proper, and apply Lemma \ref{lem-surj}. \end{proof}
\subsection{Subintegrality and homeomorphisms}
We are now in position to generalize the characterization of subintegrality via homeomorphisms, as in \cite[Thm. 3.1]{Be}, over any algebraically closed field of characteristic zero. We also add a radicial property in the equivalences.
\begin{thm}\label{thmFB1} Let $\pi:Y\to X$ be a finite morphism between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate} \item $\pi$ is subintegral. \item $\pi_{k}$ is bijective. \item $\pi_{k}$ is a homeomorphism for the $R$-topology. \item $\pi_{k}$ is a homeomorphism for the Zariski topology. \item $\pi$ is a homeomorphism. \item $\pi$ is radicial. \end{enumerate} \end{thm}
\begin{proof} The equivalence between (4) and (5) is given by the Nullstellensatz. Using the Nullstellensatz, by Proposition \ref{lying-over} and proceeding similarly to Bernard's proof of \cite[Thm. 3.1]{Be} then we get the equivalence between (1), (2) and (5).
It is clear that (3) implies (2). The proof that (2) implies (3) is a direct consequence of Lemma \ref{lem-fini}. Indeed assuming (2), the map $\pi_k$ admits an inverse, and this inverse is continuous with respect to the $R$-topology since $\pi_k$ is a closed map by Lemma \ref{lem-fini}.
The equivalence between (1) and (6) is given by Proposition \ref{sat=semivar}. \end{proof}
\subsection{Regulous functions on the $k$-rational points}\label{sect-CR}
We fix a real closed field $R$ such that $R[\sqrt{-1}]=k$.
\subsubsection{Characterization of continuous rational functions} The choice of $R$ induces a topology, hence a notion of continuity. We define continuous rational functions on an algebraic variety over $k$ as follows.
\begin{defn}\label{def-rat-cont} Let $X$ be an algebraic variety over $k$. Let $U\subset X$ be an open subset of $X$. A continuous rational function on $U(k)$ is a function from $U(k)$ to $k$ which is the continuous extension to $U(k)$ of a rational function on $X$, when $X$ is endowed with the $R$-topology. \end{defn}
This notion comes initially from real algebraic variety \cite{Ku,KN,FHMM}, and has been studied also in complex algebraic geometry \cite{Be}. In the setting of Definition \ref{def-rat-cont}, it depends a priori on the choice of $R$. A rational function that is continuous with respect to the $R$-topology can be characterized by the fact that it becomes regular after applying a relevant proper birational map.
\begin{prop} Let $X$ be an algebraic variety over $k$. Let $f:X(k)\to k$ be an everywhere defined function, and assume that $f$ coincides with a regular function on a Zariski open subset of $X(k)$.
Then, $f$ is continuous with respect to the $R$-topology if and only if there is a proper birational map $\sigma:\tilde X\to X$ such that $f\circ \sigma_k :\tilde X(k)\to k$ is regular. \end{prop}
\begin{proof} Arguing similarly to \cite[Lem. 4.4]{Be}, we may assume $X$ is irreducible. Assume $f$ to be continuous, and denote by $g$ the rational function on $X$ that coincides with $f$ on a Zariski open subset of $X(k)$. One can resolve the indeterminacy of the rational map $g$ by a sequence of blowings-up along nonsingular centers, giving rise to a proper birational morphism $\sigma:\tilde X\to X$ such that $g\circ \sigma_k :\tilde X(k)\to \PP^1(k)$ is regular. The functions $f\circ \sigma_k$ and $g\circ \sigma_k$ are equal on a Zariski dense subset of $\tilde X(k)$, so they are equal on a subset dense with respect to the $R$-topology by Lemma \ref{lem-dense}. Therefore they coincide on $\tilde X(k)$ by continuity. As a consequence the regular function $g\circ \sigma_k$ takes its values in $k$ rather than in $\PP^1(k)$.
Conversely, let $C\subset k$ be a closed subset with respect to the $R$-topology. The set $(f\circ \sigma_k)^{-1}(C)$ is closed by continuity of $f\circ \sigma_k$, and its image under $\sigma_k$ is equal to $f^{-1}(C)$ by surjectivity of $\sigma_k$ via Lemma \ref{lem-surj}. As a consequence $f^{-1}(C)$ is closed by properness of $\sigma_k$ with respect to the $R$-topology, thanks to Lemma \ref{lem-surj} again. \end{proof}
Note that the characterization of continuity given above, via a resolution of indeterminacy, does not refer to the choice of $R$. In particular, the continuity with respect to the $R$-topology of a rational function is independent of the choice of the real closed field $R$.
\vskip 2mm
Let $U\subset X$ be an open subset of $X$. The continuous rational functions on $U(k)$ are the sections of a presheaf of $k$-algebras on $X(k)$, denoted by $\K_{X(k)}^0$ in the sequel. Since $\K$ is a sheaf, and the presheaf of locally continuous functions on $X(k)$ for the $R$-topology is also a sheaf for the Zariski topology, then $\K_{X(k)}^0$ is a sheaf called the sheaf of continuous rational functions. It makes $(X(k),\K_{X(k)}^0)$ a ringed space. In case $X$ is affine then we simply denote by $\SR(X(k))$ the global sections of $\K_{X(k)}^0$ on $X(k)$. A dominant morphism $\pi:Y\to X$ between varieties over $k$ induces an extension $\K_{X(k)}^0\to (\pi_{k})_*\K_{Y(k)}^0$, hence a morphism $(Y(k),\K_{Y(k)}^0)\to (X(k),\K_{X(k)}^0)$ of ringed spaces.
An important fact is that continuous rational functions on a normal variety are regular. More precisely, next proposition says that continuous rational functions are integral on the regular ones and thus are already regular if the variety is normal.
The set of indeterminacy points of a continuous rational function is related to the normal locus of the ambient variety.
\begin{prop}\label{prop-lieunorm} Let $X$ be an algebraic variety over $k$. We have: \begin{enumerate} \item $\K_{X(k)}^0\subset (\pi_{k}^\prime)_* \SO_{X'(k)}$ where $\pi':X'\to X$ is the normalization map. \item If $x$ is a normal closed point of $X(k)$ then $\K_{X(k),x}^0=\SO_{X,x}$. \end{enumerate} \end{prop}
\begin{proof} The properties are local so we may assume $X$ is affine. The original proof in \cite[Proposition 4.7]{Be} uses only one argument related to the complex setting. It is the density with respect to the strong topology of a Zariski dense open set, that can be replaced by Lemma \ref{lem-dense}. Note that Hartogs Lemma used in the proof is valid over $k$ : if $X$ is normal then the restriction map $\SO(X(k))\to \SO(\Reg(X(k)))$ is surjective since $$\dim(X(k)\setminus \Reg(X(k)))\leq \dim (X(k))-2$$ by \cite[p. 124]{Iit}. \end{proof}
In the remaining of this section, we show how seminormalization and continuous rational functions are in relation for varieties over $k$. We begin with a description of regular functions on the relative seminormalization in terms of regular functions on the relative normalization.
\begin{prop} \label{constant} Let $Y\to X$ be a dominant morphism between algebraic varieties over $k$. Let $U$ be an open subset of $X$. Then $$\SO_{X^{+}_Y}((\pi^+)^{-1}(U))=\{ f\in \SO_{X'_Y}((\pi')^{-1}(U))\mid \,f {\rm \,is\,constant\, on \,the\,fibers\,of}\,\pi'_{k}\}$$ where $\pi':X'_Y\to X$ (resp. $\pi^+:X^+_Y\to Y$) is the relative normalization (resp. seminormalization) morphism. \end{prop}
\begin{proof} By \cite[Cor. 3.7]{Be} (which is written in the case $U$ is affine, but all section 3 there is valid for any open subset $U$), we have
$$\SO_{X^{+}_Y}((\pi^+)^{-1}(U))=\{f\in \SO_{X'_Y}((\pi')^{-1}(U))\mid \forall x\in U(k), \,\,f\in \SO_{X,x}+\JRad(\SO_{X'_Y,x})\}$$ The radical being an intersection of maximal ideals, we see that the functions in $\SO_{X^{+}_Y}((\pi^+)^{-1}(U))$ correspond to the elements of $ \SO_{X'_Y}((\pi')^{-1}(U))$ constant on the fibers of $\pi'_{k}$. \end{proof}
We give a characterization of the structural sheaf of the seminormalization of an algebraic variety over $k$ in another one with continuous rational functions generalizing the main result in \cite{Be} to the relative seminormalization and over any algebraically closed field of characteristic zero. In order to state the result, we use the fiber product of two sheaf extensions.
\begin{thm} \label{thmintclosregulu} Let $\pi:Y\to X$ be a dominant morphism between algebraic varieties over $k$. Then $$(\pi^+_{k})_*\SO_{X^{+}_Y(k)}=\K_{X(k)}^0\times_{(\pi_{k})_*\K_{Y(k)}^0}(\pi_{k})_*\SO_{Y(k)} $$ where $\pi^+:X^+_Y\to X$ is the relative seminormalization morphism. \end{thm}
\begin{proof} We may assume $X$ and $Y$ are affine and thus we want to prove that $$k[X^{+}_Y]=\SR( X(k))\times_{\SR(Y(k))}k[Y],$$ where the right hand side stands for the fiber product of the rings. Let $\pi':X_Y^\prime\to X$ be the relative normalization map.
We consider the following diagram $$\begin{array}{ccccc}
k[X]&\rightarrow & k[X_Y']&\rightarrow &k[Y]\\
\downarrow&&\downarrow&& \downarrow \\
\SR(X(k))&\rightarrow &\SR(X'_Y(k))& \rightarrow&\SR(Y(k))\\ \end{array}$$
where the horizontal maps from the top (resp. the bottom) are given by composition with respectively $\pi'$ and $Y\to X_Y^\prime$ (resp. $\pi_{k}^\prime$ and $Y(k)\to X_Y^\prime(k)$).
We have $k[X^{+}_Y]\subset k[Y]$ by definition. Since $\pi^+$ is subintegral then it follows from Theorem \ref{thmFB1} that $\pi^+_k$ is an homeomorphism with respect to the $R$-topology. Since $\pi^+$ is in addition birational, the composition by $\pi_k^+$ gives an isomorphism between $\K^0(X(k))$ and $\K^0(X_Y^+(k))$. Therefore we get $k[X^{+}_Y]\subset \K^0(X_Y^+(k))=\K^0(X(k))$. In particular $k[X^{+}_Y]\subset\SR( X(k))\times_{\SR(Y(k))}k[Y]$.
Let us prove the converse inclusion. Let $f\in\SR( X(k)))\times_{\SR(Y(k))}k[Y]$. The continuous rational function $f$ is integral over $k[X]$ by Proposition \ref{prop-lieunorm}, therefore $f\in k[X'_Y]$ since additionally $f\in k[Y]$. As a function on $X'_Y(k)$, the function $f$ is constant on the fibers of $X'_Y(k)\to X(k)$ since $f$ induces a continuous function on $X(k)$. By Proposition \ref{constant}, we obtain then $f\in k[X^+_Y]$. It gives the reverse inclusion $\SR( X(k))\times_{\SR(Y(k))}k[Y]\subset k[X^+_Y]$. \end{proof}
A continuous rational function on a normal algebraic variety over $k$ is a regular function by Proposition \ref{prop-lieunorm}. The fact that the normalization of $X$ in $Y$ is not necessarily normal imposes to take the fiber product with $\SO_{Y(k)}$ in Theorem \ref{thmintclosregulu}. We state as a theorem the particular case $Y=X'$, the statement becoming much simpler by Proposition \ref{prop-lieunorm}. It says that the ring of continuous rational functions is isomorphic to the ring of regular functions on the seminormalization, it generalizes the main result in \cite{Be} over any algebraically closed field of characteristic zero.
\begin{thm} \label{caractK0} Let $X$ be an algebraic variety over $k$. The ringed space $(X^+(k),\SO_{X^{+}(k)})$ is isomorphic to $(X(k),\K_{X(k)}^0)$. \end{thm}
\begin{rem} Since $X^+$ doesn't depend of the choice of the real closed field, Theorem \ref{caractK0} gives another way to check that the continuity property of a rational function does not depend on the chosen real closed field. \end{rem}
\subsubsection{Regulous functions and homeomorphisms}
A regulous function $f$ is a continuous rational function that satisfies the additional property that $f$ remains rational by restriction to any subvariety. The first author proved that it is always the case \cite[Prop. 4.14]{Be} for complex varieties, contrarily to the real case \cite{KN}.
The following result asserts that continuous rational functions are always regulous. \begin{cor}\label{cor-reg} Let $X$ be an algebraic variety over $k$ and let $f\in\K^0(X(k))$. For any Zariski closed subset $V$ of $X$, the restriction $f_{\mid V(k)}$ belongs to $\K^0(V(k))$. \end{cor}
\begin{proof} The proof of \cite[Proposition 4.14]{Be} works verbatim using Theorem \ref{caractK0}. \end{proof}
In the sequel, we also say regulous for continuous rational.
\begin{defn} Let $X$ and $Y$ be algebraic varieties over $k$. Let $E\subset X(k)$ and $F\subset Y(k)$ be two closed subsets for the $R$-topology. We say that a map $h:E\to F$ is biregulous if it is a homeomorphism for the $R$-topology which is birational i.e there exist two dense Zariski open subsets $U_1\subset \overline{E}^Z$, $U_2\subset \overline{F}^Z$, a birational map $g:\overline{E}^Z\to\overline{F}^Z$ with $g_{\mid U_1}:U_1\to U_2$ an isomorphism such that $g=h$ by restriction to $U_1\cap E$. In this situation, we say that $E$ and $F$ are biregulous or biregulously equivalent. \end{defn}
In this situation and if in addition $E=Y(k)$, $F=X(k)$ and $h$ is an homeomorphism for the Zariski topology, so that the morphism $\K_{X(k)}^0\to (h)_*\K_{Y(k)}^0$ is well-defined, then this morphism is an isomorphism and thus the ringed spaces $(Y(k),\K_{Y(k)}^0)$ and $(X(k),\K_{X(k)}^0)$ are isomorphic. The following theorem is an extended version of Theorem \ref{thmFB1} and it explains how subintegral extensions, continuous rational functions and biregulous morphisms are related. It is a generalization of \cite[Thm. 3.1, Prop. 4.12]{Be} with an additional radiciality property.
\begin{thm}\label{thmFB} Let $\pi:Y\to X$ be a finite morphism between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate} \item $\pi$ is subintegral. \item $\pi_{k}$ is bijective. \item $\pi_{k}$ is biregulous.
\item The ringed spaces $(Y(k),\K_{Y(k)}^0)$ and $(X(k),\K_{X(k)}^0)$ are isomorphic. \item $\pi_{k}$ is a homeomorphism for the strong topology. \item $\pi_{k}$ is a homeomorphism for the Zariski topology. \item $\pi$ is a homeomorphism. \item $\pi$ is radicial. \end{enumerate} \end{thm}
\begin{proof} We already have the equivalence of (1), (2), (5), (6), (7) and (8) by Theorem \ref{thmFB1}.
Assuming $\pi$ to be subintegral, then it follows from Proposition \ref{propCSEPvariety} that $X$ and $Y$ have the same seminormalization. So $\K_{X(k)}^0\to (\pi_k)_*\K_{Y(k)}^0$ is an isomorphism by Theorem \ref{caractK0}. It shows (1) implies (4).
We show (4) implies (1) by contradiction. Assume $\pi_k$ is not bijective. We can suppose $X$ and $Y$ are affine. We may separate two different points $y,y'$ in the fibre $\pi^{-1}_k(x)$ of some $x\in X(k)$ by a regular function $f$ on $Y(k)$. But such a function is continuous with respect to the $R$-topology, and does not belong to the image of $\K^0(X(k))\to \K^0(Y(k))$ (given by the composition by $\pi_k$) since it is not constant on the fibres of $\pi_k$.
A biregulous map is bijective so (3) implies (2), and conversely by (4) we know that the inverse of $\pi_k$ is continuous rational, so regulous by Corollary \ref{cor-reg}.
\end{proof}
Remark also that a morphism satisfying the conditions of Theorem \ref{thmFB} is (bijective thus) automatically birational.
\vskip 2mm
We prove now that the seminormalization determines a variety up to biregulous equivalence. This result goes in the direction of the problems considered by Koll\'ar \cite{Ko}, \cite{KMOS}, \cite{Ce}.
Before that we need to prove that a biregulous map between algebraic varieties over $k$ is a homeomorphism for the Zariski topology.
Let $X$ be an algebraic variety over $k$ and let $E\subset X(k)$. We denote by $\overline{E}$ the closure of $E$ for the $R$-topology. Recall that a locally closed subset of $X$ or $X(k)$ is the intersection of a Zariski open subset with a Zariski closed subset and that a Zariski constructible subset is a finite union of locally closed subsets.
\begin{prop} \label{biregZhomeo} Let $X$ and $Y$ be algebraic varieties over $k$ and let $h:Y(k)\to X(k)$ be a biregulous map. Then $h$ is a homeomorphism for the Zariski topology. \end{prop}
\begin{proof} Let $Z$ be an irreducible closed subset of $Y(k)$. By restriction we get a biregulous map $Z\to h(Z)$. We claim that $h(Z)$ is a Zariski closed subset of $X(k)$. Since $h$ is biregulous then $h(Z)$ is closed in $X(k)$ for the $R$-topology. To prove the claim then it is sufficient to assume that $\overline{h(Z)}^Z=X(k)$, $X$ is irreducible and thus we have to prove that $h:Z\to X(k)$ is surjective. Since a regulous function $f$ on $Z$ is still regulous (and thus rational) by restriction then there exists a stratification of $Z$ in locally closed strata such that the restriction of $f$ to each stratum is regular (see \cite{FHMM} and \cite{Mnew}). It follows that $h(Z)$ is a Zariski constructible set and thus $h(Z)=\cup_{i=1}^n V_i\cap U_i$ such that, for $i=1,\ldots,n$, $V_i$ (resp. $U_i$) is a Zariski closed (resp. open) subset of $X(k)$. Since $\overline{h(Z)}^Z=X(k)$ and $X$ is irreducible then we may assume $V_1=X(k)$. We have $\overline{U_1}\subset \overline{h(Z)}=h(Z)$ and thus $h:Z\to X(k)$ is surjective by Lemma \ref{lem-dense}.
It shows that $h^{-1}$ is continuous for the Zariski topology. Similarly, $h$ is continuous for the Zariski topology. \end{proof}
\begin{thm} \label{QKoC} Let $X$ and $Y$ be algebraic varieties over $k$. Then, $X(k)$ and $Y(k)$ are biregulously equivalent if and only if $X^+$ and $Y^+$ are isomorphic. \end{thm}
\begin{proof} Assume $h:Y(k)\to X(k)$ is a biregulous map. From Proposition \ref{biregZhomeo} $h$ is a homeomorphism for the Zariski topology and thus the ringed spaces $(Y(k),\K_{Y(k)}^0)$ and $(X(k),\K_{X(k)}^0)$ are isomorphic. By Theorem \ref{caractK0} it follows that $X^+$ and $Y^+$ are isomorphic.
Since $X^+(k)\to X(k)$ and $Y^+(k)\to Y(k)$ are biregulous morphisms then we easily get the converse implication. \end{proof}
\begin{rem} Note that in the statement of the previous theorem, we do not require that there is a morphism from $X$ to $Y$ nor from $Y$ to $X$. Consider a smooth irreducible curve $Y$ over $k$ without automorphisms (e.g $\PP^1_{k}$ minus sufficiently many points). Let $P_1,P_2$ be two distincts points of $Y$. Let $X_1$ (resp. $X_2$) be the curve obtained from $Y$ by creating a cusp at $P_1$ (resp. $P_2$) i.e associated to the module $2P_1$ (resp. $2P_2$) (see \cite{Se}). We have $Y=X_1^+=X_2^+$ and thus $X_1(k)$ is biregulously equivalent to $X_2(k)$. However, there is no morphism from $X_1$ to $X_2$ (nor from $X_2$ to $X_1$) because otherwise it would lift to a finite and birational morphism $Y\to Y$ sending $P_1$ to $P_2$ (see \cite[Ex. 5.5]{FMQ2}) and since $Y$ is normal then this morphism is an isomorphism, a contradiction. \end{rem}
\section{Homeomorphisms between algebraic varieties}\label{sect-homeo}
In this section $k$ is an algebraically closed field and $\car (k)=0$. We fix a real closed field $R$ such that $R[\sqrt{-1}]=k$.
\vskip 1cm
Given a morphism $\pi:Y\to X$ between algebraic varieties over $k$, we are looking for algebraic conditions on $\pi$ which are respectively equivalent to the topological property that $\pi_{k}$ is an homeomorphism for the $R$-topology and $\pi$ is an homeomorphism for the Zariski topology.
In case $\pi$ is finite then we already know the answer. Indeed, from Theorem \ref{thmFB1} the two topological properties stated above are equivalent to each other, and moreover equivalent to the two algebraic conditions that $\pi$ is subintegral or $\pi$ is radicial. \begin{thm}\label{thmFB2} Let $\pi:Y\to X$ be a finite morphism between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate} \item $\pi_{k}$ is a homeomorphism for the $R$-topology. \item $\pi$ is a homeomorphism. \item $\pi$ is subintegral. \item $\pi$ is radicial \end{enumerate} \end{thm}
In the sequel, we compare these four properties when we drop the finiteness hypothesis.
\vskip 2mm
After some generalities on the relation between homeomorphism with respect to Zariski topology, isomorphism and normality, we provide a complete solution to this problem for $R$-homeomorphisms. As consequences, we give a partial answer to the problem for Zariski homeomorphisms and we provide statements that explain exactly when $R$-homeomorphisms and Zariski homeomorphisms are isomorphisms in the same spirit of Vitulli's result \cite[Thm. 2.4]{V2}.
\subsection{Bijection, birationality, homeomorphism} We aim to compare the notions of bijection, birational morphism, homeomorphism with respect to Zariski topology at the spectrum level and homeomorphism with respect to Zariski topology at the level of closed points.
\vskip 2mm
To begin with, recall that it follows from the Nullstellensatz that the property for a morphism to be a homeomorphism with respect to the Zariski topology is already decided at the level of closed points. \begin{prop} \label{NS} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. Then $\pi$ is a homeomorphism if and only if $\pi_k$ is a homeomorphism with respect to the Zariski topology. \end{prop}
Note that the previous result is still true if $\car(k)>0$. It is clear that an homeomorphism induces a bijection at the level of $k$-rational points, the converse being false in general as illustrated by Example \ref{exVit2} below.
Restricting our attention to curves, note that the converse holds true in the case of a morphism between irreducible algebraic curves. The irreducibility of the source space is crucial here, consider for instance the disjoint union of a point with a line minus a point, sent to a line. However even for morphisms between irreducible curves, a birational homeomorphism need not be an isomorphism as illustrated by the normalization of the cuspidal curve with equation $y^2=x^3$.
An important contribution to these questions is the fact that the bijectivity at the level of closed points induces the birationality for irreducible varieties, by Zariski Main Theorem.
\begin{prop}\label{prop-wk} Let $X$ and $Y$ be irreducible varieties over $k$. Then a morphism from $Y$ to $X$ inducing a bijection at the level of $k$-rational points is quasi-finite and birational. If in addition $X$ is normal, it is an isomorphism. \end{prop}
The proof is classical, but we include it for the clarity of the exposition.
\begin{proof} First note that $Y\to X$ is quasi-finite by \cite[Lem. 20.10]{STPmorph} and the Nullstellensatz. By Grothendieck's form of Zariski Main Theorem, a quasi-finite morphism $\pi: Y\to X$ between irreducible algebraic varieties over $k$ factorizes into an open immersion $Y\to Z$ and a finite morphism $Z\to X$. So we identify $Y$ with an open subset of $Z$ and further assume that $Y$ is Zariski dense in $Z$.
Assume $\pi_{k}:Y(k)\to X(k)$ bijective, so that $X$, $Y$ and $Z$ have the same dimension. Recall that the degree of the extension $\K(X)\to \K(Z)=\K(Y)$ is the cardinal of a generic fiber of $Z(k)\to X(k)$ by \cite[Thm. 7]{Sha}. Such a generic fiber is in general in $Y(k)$, otherwise the dimension of $\dim (Z\setminus Y)$ would be greater than or equal to $\dim X$, in contradiction with the density of $Y$. Thus the finite morphism $Z\to X$ has necessarily degree one, so that $Z\to X$ is birational and thus also $\pi$.
Assuming in addition $X$ normal implies that $Z$ is isomorphic to $X$. The open immersion is surjective at the level of $k$-rational points and from the Nullstellensatz it follows that it is surjective, thus an isomorphism. \end{proof}
The following example shows that a morphism which gives a bijection at the level of $k$-rational points needs not be an homeomorphism. \begin{ex} \label{exVit2} \begin{enumerate} \item Consider the varieties of Example \ref{exVitdetail2}, for which we have an open immersion $\psi$ and a finite morphism $\phi$ as follows : $$\pi : Y\times \Af_k^n \xrightarrow{\psi} X'\times \Af_k^n \xrightarrow{\phi} X\times\Af_k^n.$$ Even if $\pi_k$ is still bijective, the morphism $\pi$ is no longer a homeomorphism when $n>0$.
To see it, it suffices to consider the case $n=1$.
Denote by $O\in (X\times \Af_k^1)(k)$ the origin, and by $P=(0,1,0)$ and $Q=(0,-1,0)$ the two points in the fiber $\phi^{-1}(O)$, where the coordinates are $(x,z,t_1)$ in the notation of Example \ref{exVitdetail2}. Let $C$ be the curve in $X'\times \Af_k^1$ given by intersection with the plane $x-z+t_1+1=0$ in $\Af_k^2\times\Af_k^1$. Note that $P\in C$, $Q\notin C$, $C\setminus \{P\}$ is not a closed subset of $X'\times \Af_k^1$ but it is a closed subset of $Y\times \Af_k^1$ since $Y=X'\setminus\{P\}$. If $\pi$ was a homeomorphism, then $\pi_{k}(C(k)\setminus \{P\})$ should be Zariski closed in $(X\times \Af_k^1)(k)$ by Proposition \ref{NS}. However $O$ is in the closure of $\pi_{k}(C(k)\setminus\{P\})$.
\item This example is also interesting to consider relatively to Grothendieck's notion of universal homeomorphism \cite[Defn. 3.8.1]{Gr2}. Recall that a morphism $Y\to X$ is a universal homeomorphism if $Y\times_X Z\to Z$ is a homeomorphism for any morphism $Z\to X$.
The morphism $Y\to X$ of Example \ref{exVitdetail} is an homeomorphism but not a universal homeomorphism. Indeed, let $Z=X\times \Af_k^1$ and consider the base change $Z\to X$ given by the first projection. We have already checked that $Y\times_X Z=Y\times \Af_k^1\to Z=X\times \Af_k^1$ is not closed. \end{enumerate} \end{ex}
\subsection{Main results}
We focus now on the four properties appearing in Theorem \ref{thmFB2}, namely given a morphism $\pi:Y\to X$ between algebraic varieties over $k$, we consider the properties :\\ - $\pi_k$ is a homeomorphism for the $R$-topology,\\ - $\pi$ is a homeomorphism,\\ - $\pi$ is subintegral,\\ - $\pi$ is radicial.\\
By Theorem \ref{thmFB2}, these four properties are equivalent when $\pi$ is finite. Considering the open immersion $\Af_{k}^1\to \PP_{k}^1$, it is radicial but not bijective. Since finite morphisms are surjective by Proposition \ref{lying-over} then if do not assume $\pi$ to be finite then we have to replace the last property above by:\\
- $\pi$ is radicial and surjective.\\
The equivalence between the four above properties is no longer true without the finiteness hypothesis. In particular, an homeomorphism with respect to the Zariski topology need not be a homeomorphism with respect to the $R$-topology, even for irreducible affine curves.
\begin{ex} \label{exVit3} \begin{enumerate} \item Consider the morphism $\pi:Y\to X$ from Example \ref{exVitdetail}. We have already shown that $\pi$ is an homeomorphism, $\pi$ is radicial but $\pi$ is not subintegral since it is not an integral morphism. The morphism $\pi_k$ is not a homeomorphism with respect to the $R$-topology. Indeed, consider a small open ball $B$ of $Y(k)$ containing the point that is sent to the singular point of $X(k)$ by $\pi_k$. Then the image of $Y(k)\setminus B$ is not closed. \item Let $n>0$ and consider the morphism $\pi_n: Y\times\Af_k^n\to X\times \Af_k^n$ from Example \ref{exVitdetail2}. We have already proved that $\pi$ is radicial and surjective, $ \pi$ is not subintegral nor an homeomorphism. Note that $(\pi_n)_k$ is not a homeomorphism with respect to the $R$-topology either, following the same proof as in (1). \end{enumerate} \end{ex}
In view of the foregoing explanation, the only equivalence which remains possible to obtain is between the properties for a morphism to be subintegral and an homeomorphism for the $R$-topology. It will be the subject of our main result.
The following two results measure the rigidity of the $R$-topology.
\begin{prop}\label{prop-fini} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. If $\pi_{k}$ is a homeomorphism with respect to the $R$-topology, then $\pi$ is finite. \end{prop}
\begin{proof} It is sufficient to assume that $Y$ and $X$ are irreducible. By Proposition \ref{prop-wk} then $\pi$ is quasi-finite. By Grothendieck's form of Zariski Main Theorem, $\pi$ factorizes into an open immersion $g:Y\to Z$ and a finite morphism $h:Z\to X$. We consider $Y(k)$ embedded as an open subset of $Z(k)$ for the $R$-topology. We also assume $Y$ to be Zariski dense in $Z$, and thus $Y(k)$ is dense in $Z(k)$ for the $R$-topology by Lemma \ref{lem-dense}.
Since $\pi_{k}$ is bijective, the finite morphism $h$ is birational by Proposition \ref{prop-wk}. Moreover $h_{k}$ is surjective by Proposition \ref{lying-over}. Let us prove that $h_{k}$ is also injective. If not, there exist $y\in Y(k)$ and $z\in Z(k)\setminus Y(k)$ with $h_{k}(y)=h_{k}(z)$. Denote this point by $x\in X(k)$. Let $V_y$ be a closed neighborhood of $y$ in $Y(k)$ for the $R$-topology, and $V_z$ a closed neighborhood of $z$ in $Z(k)$ for the $R$-topology disjoint from $V_y$. Then $V_x=h_{k}(V_y)$ is a closed neighborhood of $x$ in $X(k)$ for the $R$-topology by assumption on $\pi_{k}$.
By the Curve Selection Lemma \cite[Thm. 2.5.5]{BCR}, there is a continuous semialgebraic curve $\gamma :[0,1)\to Z(k)$ with $\gamma(0)=z$ and $\gamma(0,1)\subset V_z\cap Y(k)$. Then $h_{k}\circ \gamma :[0,1)\to X(k)$ is a continuous semialgebraic curve with $ h_{k} \circ \gamma(0)=x$, so it meets $V_x\setminus\{x\}=h_{k}(V_y\setminus \{y\})$. As a consequence $V_y$ and $V_z$ cannot be disjoint because $\pi_{k}$ is bijective.
Therefore $h_{k}$ is bijective and thus $g_{k}$ is also bijective. The Nullstellensatz forces $g$ to be a bijective open immersion, thus an isomorphism. As a consequence $\pi$ is finite like $h$. \end{proof}
\begin{cor}\label{cor-homeo} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. If $\pi_{k}$ is a homeomorphism with respect to the $R$-topology, then $\pi$ is a homeomorphism. \end{cor}
\begin{proof} The finiteness follows from Proposition \ref{prop-fini}. Being a bijection on the closed points, it is a homeomorphism by Theorem \ref{thmFB1}. \end{proof}
Corollary \ref{cor-homeo} admits a converse, for varieties of dimension at least two.
\begin{prop} \label{equivhomeo} Let $\pi:Y\to X$ be a morphism between irreducible algebraic varieties over $k$ of dimension at least two. If $\pi$ is a homeomorphism, then $\pi_{k}$ is a homeomorphism with respect to the $R$-topology. \end{prop}
\begin{proof} By \cite[Theorem 2.2]{V2}, the morphism $\pi$ is finite (it is also birational by Proposition \ref{prop-wk}), so we conclude using Theorem \ref{thmFB1}. \end{proof}
Proposition \ref{equivhomeo} is however not true for curves as illustrated by Example \ref{exVit3} (1).
\vskip 1cm
We are now able to get the main result of the paper. We prove that, for a given morphism between algebraic varieties over $k$, the topological property to be a homeomorphism for the $R$-topology does not depend of the chosen real closed field since it is equivalent to the algebraic property to be subintegral. \begin{thm} \label{Euclhomeo} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. The following properties are equivalent: \begin{enumerate} \item $\pi_{k}$ is a homeomorphism with respect to the $R$-topology. \item $\pi$ is subintegral. \item $X_Y^+=Y$. \end{enumerate} \end{thm}
\begin{proof} From Theorem \ref{thmFB2} and Proposition \ref{propCSEPvariety} we know that (2) and (3) are equivalent and they imply (1).
Assume $\pi_{k}$ is a homeomorphism with respect to the $R$-topology. Then $\pi$ is finite by Proposition \ref{prop-fini} and thus $\pi$ is subintegral again by Theorem \ref{thmFB2}. \end{proof}
Note that we cannot replace in Theorem \ref{Euclhomeo} the topological assumption (1) on $\pi_{k}$ by $\pi_{k}$ is bijective or even $\pi$ or $\pi_k$ is a homeomorphism, as illustrated by Examples \ref{exVitdetail} and \ref{exVit3}.
As illustrated by Proposition \ref{prop-wk}, the normality of the target space plays a role to upgrade a bijection into an isomorphism. Next result, which is a direct consequence of Theorem \ref{Euclhomeo}, shows that relative seminormality is the correct notion to associate to a homeomorphism with respect to the $R$-topology in order to obtain an isomorphism.
\begin{cor} \label{corEuclhomeo1} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$ such that $\pi_{k}$ is a homeomorphism with respect to the $R$-topology. Then $\pi$ is an isomorphism if and only if $X$ is seminormal in $Y$. \end{cor}
We obtain an alternative version of \cite[Thm. 2.4]{V2}, where we replace the Zariski topology by the $R$-topology. In particular our statement is valid without any restriction on dimension.
\begin{cor} \label{corEuclhomeo2} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. If $\pi_{k}$ is a homeomorphism with respect to the $R$-topology and $X$ is seminormal, then $\pi$ is an isomorphism. \end{cor}
\begin{proof} From Proposition \ref{semimprel} then $X$ is seminormal in $Y$ and the result follows from Corollary \ref{corEuclhomeo1}. \end{proof}
We end the section by some results with a slightly different flavour. Forgetting about the $R$-topology, we compare now homeomorphisms with radicial and sujective morphisms. So we compare the two properties quoted at the beginning of the section, properties that do not appear in the equivalence of Theorem \ref{Euclhomeo}. We may also wonder what is the correct assumption to add to an homeomorphism in order to obtain an isomorphism in the spirit of Corollary \ref{corEuclhomeo1}. We start by proving that an homeomorphism between varieties over $k$ is radicial.
\begin{prop} \label{Zhomeo} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. If $\pi$ is a homeomorphism then $\pi$ is radicial. \end{prop}
\begin{proof} We may assume $X$ and $Y$ irreducible, since the irreducible components of $X$ and $Y$ are homeomorphic one-by-one.
Assume first that the dimension of $X$ and $Y$ is at least two. Then $\pi_{k}$ is a homeomorphism with respect to the $R$-topology by Proposition \ref{equivhomeo}, so $\pi$ is finite by Proposition \ref{prop-fini}. By Theorem \ref{Euclhomeo} we get that $\pi$ is subintegral and thus radicial (Proposition \ref{sat=semivar}).
Assume $X$ and $Y$ are curves. Then $X$ and $Y$ are birational by Proposition \ref{prop-wk}. Since moreover $\pi_{k}$ is bijective then $\pi$ is bijective and equiresidual. By definition $\pi$ is radicial. \end{proof}
Contrary to Theorem \ref{Euclhomeo}, the converse implication in Proposition \ref{Zhomeo} is not valid as Examples \ref{exVit2} (2) and \ref{exVit3} (2) show. Anyway, we get the analogue of Corollary \ref{corEuclhomeo1} with respect to the Zariski topology. It gives a generalization of \cite[Thm. 2.4]{V2} in any dimension for varieties over $k$.
\begin{cor} \label{corZhomeo} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$ such that $\pi$ is a homeomorphism. Then $\pi$ is an isomorphism if and only if $X$ is saturated in $Y$. \end{cor}
\begin{proof} Assume $X$ is saturated in $Y$, the converse implication being trivial. By Proposition \ref{Zhomeo} then $\pi$ is radicial. Then $Y\stackrel{Id}{\to} Y\stackrel{\pi}{\to} X$ is a radicial sequence of morphisms. We conclude that $\pi$ is an isomorphism by Proposition \ref{PU1saturationvar}. \end{proof}
We get the analogue of Corollary \ref{corEuclhomeo2} with respect to the Zariski topology only in dimension $\geq 2$. It is a kind of reformulation of \cite[Theorem 2.4]{V2} in characteristic zero. \begin{cor} \label{corZhomeo2} Let $\pi:Y\to X$ be a morphism between irreducible algebraic varieties over $k$ of dimension at least $2$. If $\pi$ is a homeomorphism and $X$ is saturated, then $\pi$ is an isomorphism. \end{cor}
\begin{proof} From Propositions \ref{prop-fini}, \ref{equivhomeo} and \ref{Zhomeo} then $\pi$ is radicial and finite. It follows from Proposition \ref{sat=semivar} that $\pi$ is subintegral and $X$ is seminormal. We conclude $\pi$ is an isomorphism from Corollary \ref{corEuclhomeo1}. \end{proof}
\begin{rem} Example \ref{exVitdetail} shows that Corollary \ref{corZhomeo2} is false for curves. \end{rem}
Note that, even though the statements of Proposition \ref{Zhomeo} and Corollaries \ref{corZhomeo}, \ref{corZhomeo2} does not mention the $R$-topology, it plays a crucial role in our proofs.
\section{Homeomorphisms versus isomorphisms in positive characteristic} \label{sect-carpos}
In this section $k$ is an algebraically closed field and $\car (k)=p>0$.
\vskip 2mm
One can wonder which statements of the previous section can be generalized in positive characteristic. Of course the $R$-topology no longer exists and therefore one can only try to obtain versions of Proposition \ref{Zhomeo} and Corollaries \ref{corZhomeo}, \ref{corZhomeo2}. Note also that Proposition \ref{prop-wk} is no longer valid : indeed, the Frobenious map $\varphi:\Af_k^n\to\Af_k^n$ defined by $\varphi(x_1,\ldots,x_n)=(x_1^p,\ldots,x_n^p)$ is a homeomorphism but is not birational.
We start by giving a version of Proposition \ref{Zhomeo} in positive characteristic. \begin{prop} \label{Zhomeocarpos} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$. If $\pi$ is a homeomorphism then $\pi$ is radicial. \end{prop}
\begin{proof} Assume $\pi:Y\to X$ is a homeomorphism. We may assume $X$ and $Y$ irreducible, since the irreducible components of $X$ and $Y$ are homeomorphic one-by-one.
Assume first that the dimension of $X$ and $Y$ is at least two. By \cite[Thm. 2.2]{V} then $\pi$ is finite. It is clear that $\pi$ is bijective. Proceeding similarly to Bernard's proof of \cite[Thm. 3.1]{Be} then for any $y\in Y$ we get that the algebraic field extension $k(\pi(y))\to k(y)$ has degree equal to one. It follows that $\pi$ is residually purely inseparable. So $\pi$ is weakly subintegral and thus radicial (Proposition \ref{sat=semivar}).
Assume $X$ and $Y$ are curves. We already know that $\pi$ is bijective. Let $y\in Y(k)$ then since we get an extension $k(\pi(y))\to k(y)=k$ and since $k(\pi(y)$ contains $k$ then $\pi(y)\in X(k)$. Let $x\in X(k)$, by bijectivity of $\pi$ then there exists $y\in Y$ such that $\pi(y)=x$. By Zariski's Lemma then $k(x)=k\to k(y)$ is finite and thus $y\in Y(k)$. We have proved $\pi_k$ is bijective. We adapt now in our situation the beginning of the proof of Proposition \ref{prop-wk}. First note that $Y\to X$ is quasi-finite by \cite[Lem. 20.10]{STPmorph} and the Nullstellensatz. By Grothendieck's form of Zariski Main Theorem, a quasi-finite morphism $\pi: Y\to X$ between irreducible algebraic varieties over $k$ factorizes into an open immersion $Y\to Z$ and a finite morphism $Z\to X$. So we identify $Y$ with an open subset of $Z$ and further assume that $Y$ is Zariski dense in $Z$. It follows that $Y\to Z$ is birational. Since $\pi_{k}:Y(k)\to X(k)$ is bijective, so that $X$, $Y$ and $Z$ have the same dimension. Recall that the degree of the extension $\K(X)\to \K(Z)=\K(Y)$ is the cardinal of a generic fiber of $Z(k)\to X(k)$ by \cite[Thm. 7]{Sha}. Such a generic fiber is in general in $Y(k)$, otherwise the dimension of $\dim (Z\setminus Y)$ would be $\geq \dim X$, in contradiction with the density of $Y$. Since $\pi_{k}:Y(k)\to X(k)$ is bijective then the finite morphism $Z\to X$ has necessarily degree one, so that $\K(X)\to \K(Z)$ is purely inseparable. It follows that $\pi$ is bijective and residually purely inseparable and thus radicial and surjective. \end{proof}
We end the paper by versions of Corollaries \ref{corZhomeo}, \ref{corZhomeo2} in positive characteristic.
\begin{cor} \label{corZhomeocarpos} Let $\pi:Y\to X$ be a morphism between algebraic varieties over $k$ such that $\pi$ is a homeomorphism. Then $\pi$ is an isomorphism if and only if $X$ is saturated in $Y$. \end{cor}
\begin{proof} We copy the proof of Corollary \ref{corZhomeo} using Proposition \ref{Zhomeocarpos} instead of Proposition \ref{Zhomeo}. \end{proof}
\begin{cor} \label{corZhomeo2carpos} Let $\pi:Y\to X$ be a morphism between irreducible algebraic varieties over $k$ of dimension $\geq 2$. If $\pi$ is a homeomorphism and $X$ is saturated, then $\pi$ is an isomorphism. \end{cor}
\begin{proof} We copy the proof of Corollary \ref{corZhomeo2} using Proposition \ref{Zhomeocarpos}, \cite[Thm. 2.2]{V} and Corollary \ref{corZhomeocarpos} instead respectively of Propositions \ref{Zhomeo}, \ref{prop-fini} and Corollary \ref{corZhomeo}. \end{proof}
\end{document} |
\begin{document}
\begin{frontmatter}
\title{A New Kalman Filter Model for Nonlinear Systems Based on Ellipsoidal Bounding}
\author{Ligang Sun\fnref{fn1}}
\fntext[fn1]{Geodetic Institute, Leibniz Universit\"at Hannover\\ Email address: \{ligang.sun, alkhatib, kargoll, neumann\}@gih.uni-hannover.de}
\author{Hamza Alkhatib\fnref{fn1}}
\author{Boris Kargoll\fnref{fn1}}
\author{Vladik Kreinovich\fnref{fn2}} \fntext[fn2]{Department of Computer Science, University of Texas at El Paso\\ Email address: vladik@utep.edu}
\author{Ingo Neumann\fnref{fn1}}
\begin{abstract} In this paper, a new filter model called set-membership Kalman filter for nonlinear state estimation problems was designed, where both random and unknown but bounded uncertainties were considered simultaneously in the discrete-time system. The main loop of this algorithm includes one prediction step and one correction step with measurement information, and the key part in each loop is to solve an optimization problem. The solution of the optimization problem produces the optimal estimation for the state, which is bounded by ellipsoids. The new filter was applied on a highly nonlinear benchmark example and a two-dimensional simulated trajectory estimation problem, in which the new filter behaved better compared with extended Kalman filter results. Sensitivity of the algorithm was discussed in the end. \end{abstract}
\begin{keyword} Set-membership Kalman filter, State estimation, Ellipsoidal bounding, Nonlinear programming, Optimization methods \end{keyword}
\end{frontmatter}
\section{Introduction}
State estimation is applicable to virtually all areas of engineering and science. Any discipline that is concerned with the mathematical modeling of its systems is a likely candidate for state estimation. This includes electrical engineering, mechanical engineering, chemical engineering, aerospace engineering, robotics, dynamical systems' control and many others. Nonlinear filtering can be a difficult and complex subject in the field of state estimation. It is certainly not as mature, cohesive, or well understood as linear filtering. There is still a lot of room for advances and improvement in nonlinear estimation techniques.
The optimal state estimation problem can be summarized as follows: given a mathematical model of a real system, and allowing some state perturbations and noise corrupted measurements, the state of the real system has to be estimated \cite{le2013zonotopic}. The estimation usually bases on the solving of an optimization problem, the estimated result relies on the assumptions made on uncertainties. Developed in the past hundreds years, the stochastic state estimation techniques are most widely applied in the real world. This approach bases on the probabilistic assumptions of the uncertainties in the system, such as Kalman filter \cite{kalman1960new} and extended Kalman filter (EKF) \cite{smith1962application,mcelhoe1966assessment} where uncertain parts (usually noise) in the system are assumed to have certain probability distribution (usually Gaussian distribution).
However, in many cases these probability distributions could be questionable, especially when the real process generating the data are complex so that only simplified models can be practically used in the estimation process \cite{milanese1996optimal}. There is another interesting approach, referred to set-membership uncertainty state estimation. Developed since 1960s \cite{witsenhausen1968sets,schweppe1968recursive,schweppe1973uncertainty}, this approach assumes that the uncertainty is unknown but bounded (UBB). No further assumption was made except for its membership of a given bound. Under this assumption, the optimal estimated state, noisy measurements and uncertainty are in some compact sets, respectively. This new technique is more appropriate in many cases where the bounded description is more realistic than stochastic distributed hypothesis. Classified by the geometrical representations, there are four major methods to bound the uncertainty, which are polytopes \cite{vicino1996sequential,walter1989exact}, ellipsoids \cite{bertsekas1971minimax,polyak2004ellipsoidal,durieu2001multi}, zonotopes \cite{combastel2005state,alamo2005guaranteed,althoff2009safety,le2013zonotopic,schon2005using} and intervals \cite{kreinovich2013computational,ferson2007experimental,kutterer2011recursive}. Polytope can be used to obtain better estimated accuracy, however, one major drawback is its computation load in multi-dimensional nonlinear systems, especially to zonotope. Ellipsoid has been widely used due to its simplicity of propagation, but the Minkovski sum of two ellipsoids is not an ellipsoid anymore, therefore the prorogation of its related algorithm requires solving an optimization problem.
In this paper, a new filter model called set-membership Kalman filter (SKF) for nonlinear systems was designed, in which both random and set-membership uncertainties were considered at the same time. This work extends Benjamin Noack's previous work in his PhD dissertation \cite{noack2014state}, where the linear case was discussed sufficiently. The novel SKF takes UBB uncertainties into account in both process equation and measurement equation, therefore it has a better uncertainty measures. It also keeps the recursive framework of random uncertainties from Kalman filter, thus the advantages of KF are reserved during the prorogation process. A better estimation under these more reliable assumptions is calculated based on solving an optimization problem in each step.
Section 2 gives mathematics preliminaries and dynamical system which would be considered later. Section 3 shows the detailed derivation of this new filter model. Section 4 is the algorithm in a practical form. Section 5 demonstrates how this new filter model works and shows that the SKF behaves better than EKF in some cases. The last section is the conclusion and future work.
\section{Mathematical Model}
\subsection{Preliminaries} The following definitions, theorems and corollaries are required for the derivation of the new filter model. The detailed proofs were given in \cite{durieu2001multi}.
\begin{mydef}\label{def_ellip} Given $S$ a positive-definite matrix, denoted by $S>0$, a bounded ellipsoid $\mathcal{E}$ in $\mathds{R}^n$ with nonempty interior is defined as \begin{equation}
\mathcal{E}=\mathcal{E}(c,S)=\{x\in\mathds{R}^n|(x-c)^TS^{-1}(x-c)\leq 1, S>0\} \end{equation} where $c\in\mathds{R}^n$ is called the center of the ellipsoid $\mathcal{E}$, and $S$ is the shape matrix which is positive-definite and specifies the size and orientation of the ellipsoid. \end{mydef}
\begin{mydef}\label{def_min} In geometry, the Minkowski sum is an operation of two sets $A$ and $B$ in Euclidean space $\mathds{R}^n$, which is defined by adding each vector in $A$ to each vector in $B$, i.e., \begin{equation}
A\oplus B=\{a+b|a\in A, b\in B\}. \end{equation} \end{mydef}
Given $K$ ellipsoids of $\mathds{R}^n$ \begin{equation} \mathcal{E}_k=\mathcal{E}(c_k,S_k), (k=1,2,\dots,K) \end{equation} their Minkowski sum is \begin{equation} \mathcal{U}_K=\sum_{k=1}^{K}\mathcal{E}_k, \end{equation} which is not an ellipsoid anymore but still a convex set.
Denote the problem of finding the smallest ellipsoid (under the criterion of matrix trace) containing the Minkowski sum of the $K$ ellipsoids as Problem $\mathrm{T^+}$: \begin{equation} \mathcal{E}^*=\underset{\mathcal{U}_K\subset\mathcal{E}}{\arg\min}\,\mathrm{tr} S\quad\mathrm{(Problem\,T^+)}, \end{equation} and from \cite{durieu2001multi}, this ellipsoid $\mathcal{E}^*$ exists and is unique.
\begin{theorem}\label{thm4.1} The center of the optimal ellipsoid $\mathcal{E}^*$ for Problem $T^+$ is given by \begin{equation}\label{e_center} c^*=\sum_{k=1}^{K}c_k \end{equation} \end{theorem}
\begin{theorem}\label{thm4.2} Let $\mathcal{D}$ be the convex set of all vectors $\alpha\in\mathds{R}^K$ with all $\alpha_k>0$ and $\sum_{k=1}^{K}\alpha_k=1$. For any $\alpha\in\mathcal{D}$, the ellipsoid $\mathcal{E}_\alpha=\mathcal{E}^+(c^*, S_\alpha)$, with $c^*$ defined by \eqref{e_center} and \begin{equation} S_\alpha=\sum_{k=1}^{K}\alpha_k^{-1}S_k, \end{equation} contains $\mathcal{U}_K$. \end{theorem}
\begin{corollary}\label{cor4.2} Special case of Theorem (2.2). When $K=2$, we have $\alpha_1+\alpha_2=1$, the $S_\alpha$ can be rewritten as \begin{equation} S_{\alpha}=\frac{1}{\alpha_1}S_1+\frac{1}{\alpha_2}S_2=(1+\frac{1}{\beta})S_1+(1+\beta)S_2 \end{equation} where $\beta$ can be any nonnegative real number. \end{corollary} \begin{proof} Let $\alpha_2=\frac{1}{1+\beta}$, $\beta\geq 0$ one can easily get above result. \end{proof}
\begin{theorem}\label{thm4.4} In the family $\mathcal{E}_\alpha=\mathcal{E}^+(c^*,S_\alpha)$, the minimal-trace ellipsoid containing the sum of the ellipsoids $\mathcal{E}_k=\mathcal{E}^+(c_k,S_k), k=1,2,\dots,K$ is obtained for \begin{equation} S_{\alpha^*}=\left(\sum_{k=1}^{K}\sqrt{\mathrm{tr}S_k}\right)\left(\sum_{k=1}^{K}S_k\sqrt{\mathrm{tr}S_k}\right) \end{equation} \end{theorem}
\begin{corollary} Special case of Theorem (2.3). When $K=2$, we have \begin{equation} S_{\alpha^*}=(1+\frac{1}{\beta^*})S_1+(1+\beta^*)S_2 \end{equation} where $\beta^*=\sqrt{\frac{\mathrm{tr}S_1}{\mathrm{tr}S_2}}$. \end{corollary}
\subsection{Dynamical System}
Consider the following nonlinear dynamical system:
\begin{gather} x_{k+1}=f_k(x_k,u_k,w_k,a_{1,k},a_{2,k},...,a_{I,k}) \label{system1}\\ y_k=h_k(x_k,v_k,b_k) \label{system2} \end{gather} where $x_k$ is a $n$-dimensional state vector, $u_k$ is the known input vector, $w_k\sim \textrm{N}(0,C_k^u)$ is a Gaussian system noise with covariance matrix $C_k^u$, $a_{i,k}\in\mathcal{E}(0,S_{ik}^u)$ is the unknown but bounded perturbation with shape matrix $S_{ik}^u$. $i=1,2,\dots, I.$ denotes the $i$th set-membership perturbation in the prediction equation. $v_k\sim \textrm{N}(0,C_k^z)$ is the a Gaussian measurement noise with covariance matrix $C_k^z$, and $b_k\in \mathcal{E}(0, S_k^z)$ is the unknown but bounded perturbation with shape matrix $S_k^z$. In this literature, $u$ and $z$ in the parameters denote they are relative to system equation and measurement equation, respectively. All the notations above represent the information at time $k$.
The following Fig. \ref{diagram} shows an estimated schematic diagram via set-membership Kalman filter in 2D case \cite{althoff2009safety}. Different with standard Kalman filter, where the output is usually an gaussian distribution and the mean of the distribution was regarded as the estimated point, in set-membership Kalman filter, a set containing all the mean values of possible distributions was put out.
\begin{comment} \begin{subequations}\label{system0} \begin{align} \label{system1} x_{k+1}&=f_k(x_k,u_k,w_k,a_{1,k},a_{2,k},...,a_{I,k})\\ \label{system2} y_k&=h_k(x_k,v_k,b_k) \end{align} \end{subequations} \end{comment}
\begin{figure}
\caption{Schematic diagram of 2D estimated result under SKF}
\label{diagram}
\end{figure}
\subsection{Linearization}
Recall the process of EKF, linearization is the first step in estimation for nonlinear dynamical systems. Perform Taylor series expansion for system equation \eqref{system1} around the point $(x_k=\hat{x}_k^+,u_k=u_k,w_k=0,a_{i,k}=0, 1\leq i\leq I)$: \begin{equation} \begin{split}
x_{k+1}=&f_k(\hat{x}_k^+,u_k,0,0)+\frac{\partial f_k}{\partial x_k}\bigg|_{(\hat{x}_k^+,u_k,0,0)}(x_k-\hat{x}_k^+)+\frac{\partial f_k}{\partial w_k}\bigg|_{(\hat{x}_k^+,u_k,0,0)}w_k\\
&+\sum_{i=1}^{I}\frac{\partial f_k}{\partial a_i}\bigg|_{(\hat{x}_k^+,u_k,0,0)}a_{i,k}+\cdots\\ \approx& f_k(\hat{x}_k^+,u_k,0,0)+F_{x,k}(x_k-\hat{x}_k^+)+F_{w,k}w_k+\sum_{i=1}^{I}F_{ai,k}a_{i,k}\\ =& F_{x,k}x_k+[f_k(\hat{x}_k^+,u_k,0,0)-F_{x,k}\hat{x}_k^+]+F_{w,k}w_k+\sum_{i=1}^{I}F_{ai,k}a_{i,k}\\ =& F_{x,k}x_k+\tilde{u}_k+F_{w,k}w_k+\sum_{i=1}^{I}F_{ai,k}a_{i,k}. \end{split} \end{equation} Here $\tilde{u}_k=f_k(\hat{x}_k^+,u_k,0,0)-F_{x,k}\hat{x}_k^+$.
Take Taylor series expansion for measurement equation \eqref{system2} around point $(x_k=\hat{x}_k^-,v_k=0,b_k=0)$: \begin{equation}\label{zk} \begin{split}
y_k=&h_k(\hat{x}_k^-,0,0)+\frac{\partial h_k}{\partial x_k}\bigg|_{(\hat{x}_k^-,0,0)}(x_k-\hat{x}_k^-)+\frac{\partial h_k}{\partial v_k}\bigg|_{(\hat{x}_k^-,0,0)}v_k\\
&+\frac{\partial h_k}{\partial b_k}\bigg|_{(\hat{x}_k^-,0,0)}b_k+\cdots\\ \approx& h_k(\hat{x}_k^-,0,0)+H_{x,k}(x_k-\hat{x}_k^-)+H_{v,k}v_k+H_{b,k}b_k\\
=& H_{x,k}x_k+\tilde{z}_k+H_{v,k}v_k+H_{b,k}b_k. \end{split} \end{equation} Here $\tilde{z}_k=h_k(\hat{x}_k^-,0,0)-H_{x,k}\hat{x}_k^-$. $\tilde{z}_k=0$ if measurement equation is linear.
Then we get the a linearized system for the original system \eqref{system1} and \eqref{system2}. \begin{gather} x_{k+1}=F_{x,k}x_k+\tilde{u}_k+F_{w,k}w_k+A_k \label{lsystem1}\\ y_k=H_{x,k}x_k+\tilde{z}_k+H_{v,k}v_k+H_{b,k}b_k \label{lsystem2} \end{gather} where $A_k=\sum_{i=1}^{I}F_{ai,k}a_{i,k}$.
\begin{comment} \begin{subequations}\label{lsystem0} \begin{align} x_{k+1}=F_{x,k}x_k+\tilde{u}_k+F_{w,k}w_k+A_k\label{lsystem1}\\ y_k=H_{x,k}x_k+\tilde{z}_k+H_{v,k}v_k+H_{b,k}b_k\label{lsystem2} \end{align} \end{subequations} where $A_k=\sum_{i=1}^{I}F_{ai,k}a_{i,k}$. \end{comment}
Both priori estimation $\hat{x}_k^-$ and posteriori estimation $\hat{x}_k^+$ are random variables. Assume that the expectation and covariance matrix of priori estimation $\hat{x}_k^-$ are $\hat{\mu}_k^-$ and $C_k^-$, the expectation and covariance matrix of posteriori estimation $\hat{x}_k^+$ are $\hat{\mu}_k^+$ and $C_k^+$. All the priori expectations $\hat{\mu}_k^-$ form an ellipsoid centered at $\hat{x}_k^{c-}$ with shape matrix $S_k^-$, i.e., $\hat{\mu}_k^-\in\mathcal{E}(\hat{x}_k^{c-},S_k^-)$. Similarly to posteriori expectation we have $\hat{\mu}_k^+\in \mathcal{E}(\hat{x}_k^{c+}, S_k^+)$.
Our objective is to calculate the explicit expressions of $\hat{x}_k^{c-}$, $C_k^-$, $S_k^-$ and $\hat{x}_k^{c+}$, $C_k^+$, $S_k^+$.
\section{Derivation of Set-membership Kalman Filter}
After getting the linearized dynamical system \eqref{lsystem1} and \eqref{lsystem2}, in this section we derive the set-membership Kalman filter model. Conclusions from section 2.1 are required and the results of this section would be summarized into one algorithm in section 4.
\subsection{Prediction}
Assume that the difference between the true state $x_k$ and the posteriori estimations center $\hat{x}_k^{c+}$ contains two components, i.e., the random part and the UBB part: \begin{equation}\label{diff+} x_k-\hat{x}_k^{c+}=\tilde{x}_k^{r+}+\tilde{x}_k^{s+}. \end{equation} So from last section we can get $\tilde{x}_k^{r+}\sim\textrm{N}(0,C_k^+)$ and $\tilde{x}_k^{s+}\in\mathcal{E}(0,S_k^+)$. And the mean squared error of posteriori estimation is given by \begin{equation}\label{cov_xe} \begin{split} &\mathrm{E}[(x_k-\hat{x}_k^{c+})(x_k-\hat{x}_k^{c+})^T]=\mathrm{E}[(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})^T]\\ =&\mathrm{E}[\tilde{x}_k^{r+}\tilde{x}_k^{r+,T}]+\mathrm{E}[\tilde{x}_k^{s+}\tilde{x}_k^{s+,T}]=C_k^++\tilde{x}_k^{s+}\tilde{x}_k^{s+,T}. \end{split} \end{equation}
Recalling EKF we have \begin{equation} \hat{x}_{k+1}^-=F_{x,k}\hat{x}_k^++\tilde{u}_k+F_{w,k}w_k+A_k. \end{equation}
Notice that $F_{w,k}w_k\sim\mathrm{N}(0,F_{w,k}C_k^uF_{w,k}^T)$, so for a fixed posteriori estimation $\hat{\mu}_k^+\in \mathcal{E}(\hat{x}_k^{c+},S_k^+)$, the predicted state follows by \begin{equation}\label{pn}
\hat{x}_{k+1}^-=F_{x,k}\hat{\mu}_k^++\tilde{u}_k+A_k+F_{w,k}w_k\sim\mathrm{N}(F_{x,k}\hat{\mu}_k^++\tilde{u}_k+A_k,F_{w,k}C_k^uF_{w,k}^T\big|\hat{x}_k^+). \end{equation} Therefore the expectation of $\hat{x}_{k+1}^-$ would be \begin{equation}\label{mu-} \hat{\mu}_{k+1}^-=\mathrm{E}(\hat{x}_{k+1}^-)=F_{x,k}\hat{\mu}_k^++\tilde{u}_k+A_k, \end{equation} which forms a set $\mathcal{E}(\hat{x}_{k+1}^{c-},S_{k+1}^-)$ when $\hat{x}_k^+$ being ergodic in the set $\mathcal{E}(\hat{x}_k^{c+},S_k^+)$.
Without loss of generality we have \begin{equation} \hat{x}_{k+1}^{c-}=F_{x,k}\hat{x}_k^{c+}+\tilde{u}_k. \end{equation}
Then the difference between the true state and the priori estimation center would be \begin{equation} \begin{split} x_{k+1}-\hat{x}_{k+1}^{c-}&=F_{x,k}(x_k-\hat{x}_k^{c+})+F_ww_k+A_k\\ &=F_{x,k}(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})+F_ww_k+A_k. \end{split} \end{equation}
Consider its covariance matrix we have \begin{equation}\label{cov_predict} \begin{split} &\mathrm{E}[(x_{k+1}-\hat{x}_{k+1}^{c-})(x_{k+1}-\hat{x}_{k+1}^{c-})^T]\\ =&\mathrm{E}\{[F_{x,k}(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})+F_ww_k+A_k]\cdot[F_{x,k}(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})+F_ww_k+A_k]^T\}\\
=&F_{x,k}\mathrm{E}[(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})(\tilde{x}_k^{r+}+\tilde{x}_k^{s+})^T]F_{x,k}^T+F_{x,k}\tilde{x}_k^{s+}A_k^T\\ &+F_{w,k}\mathrm{E}(w_kw_k^T)F_{w,k}^T+A_k\hat{x}_k^{s+,T}+A_kA_k^T\\
=&F_{x,k}C_k^+F_{x,k}^T+F_{w,k}C_k^uF_{w,k}^T+(F_{x,k}\tilde{x}_k^{s+}+A_k)(F_{x,k}\tilde{x}_k^{s+}+A_k)^T. \end{split} \end{equation}
Compared to equation \eqref{cov_xe}, we find that the predicted random uncertainty can be represented by \begin{equation}\label{C-} C_{k+1}^-=F_{x,k}C_k^+F_{x,k}^T+F_{w,k}C_k^uF_{w,k}^T. \end{equation}
Notice that a possible posteriori mean value $\hat{x}_k^+\in\mathcal{E}(\hat{x}_k^{c+},S_k^+)$, and \begin{gather} A_k=\sum_{i=1}^{I}F_{ai}a_{i,k}, a_{i,k}\in\mathcal{E}(0,S_{i,k}^u)\\ F_{ai}a_{i,k}\in\mathcal{E}(0,F_{ai}S_{i,k}^uF_{ai}^T). \end{gather}
So \begin{equation} A_k\in\sum_{i=1}^I\mathcal{E}(0,F_{ai}S_{i,k}^uF_{ai}^T). \end{equation} i.e., $A_k$ is one fixed element of a convex set which is the Minkowski sum of $I$ ellipsoids.
Recalling \eqref{mu-} we have \begin{equation}\label{mu-1} \begin{split} \hat{\mu}_{k+1}^-=&\mathrm{E}(\hat{x}_{k+1}^-)=F_{x,k}\hat{\mu}_k^++\tilde{u}_k+A_k\\ \in&\mathcal{E}(F_{x,k}\hat{x}_k^{c+}+\tilde{u}_k,F_{x,k}S_k^+F_{x,k}^T)\oplus\sum_{i=1}^I\mathcal{E}(0,F_{ai}S_{i,k}^uF_{ai}^T) \end{split} \end{equation}
Recalling \ref{thm4.4}, there exists an optimal ellipsoid $\mathcal{E}(c_k^*,S_{\alpha^*,k})$ such that \begin{equation} \begin{split} F_{x,k}\hat{\mu}_k^++\tilde{u}_k+A_k&\in\mathcal{E}(F_{x,k}\hat{x}_k^{c+}+\tilde{u}_k,F_{x,k}S_k^+F_{x,k}^T)\oplus\sum_{i=1}^I\mathcal{E}(0,F_{ai}S_{i,k}^uF_{ai}^T)\\ &\subset\mathcal{E}(\hat{x}_{k+1}^{c-},S_{k+1}^-) \end{split} \end{equation}
From \ref{thm4.1} we can get the center of the ellipsoid: \begin{equation}\label{xc-} \hat{x}_{k+1}^{c-}=F_{x,k}\hat{x}_k^{c+}+\tilde{u}_k. \end{equation}
From \ref{thm4.4} we can calculate the shape matrix of the ellipsoid: \begin{equation}\label{S-} \begin{split} S_{k+1}^-=&(\sqrt{\mathrm{tr}(F_{x,k}S_k^+F_{x,k}^T)}+\sum_{i=1}^I\sqrt{\mathrm{tr}(F_{a,i}S_{i,k}^uF_{a,i}^T)})\\ &\cdot(\frac{F_{x,k}S_k^+F_{x,k}^T}{\sqrt{\mathrm{tr}(F_{x,k}S_k^+F_{x,k}^T)}}+\sum_{i=1}^I\frac{F_{a,i}S_{i,k}^uF_{a,i}^T}{\sqrt{\mathrm{tr}(F_{a,i}S_{i,k}^uF_{a,i}^T)}}) \end{split} \end{equation}
Equation \eqref{C-}, \eqref{xc-} and \eqref{S-} gave us the elicit expressions of $C_k^-$, $\hat{x}_k^{c-}$ and $S_k^-$ respectively.
\subsection{Filtering}
Similar with \eqref{diff+}, here we assume that \begin{equation} x_k-\hat{x}_k^{c-}=\tilde{x}_k^{r-}+\tilde{x}_k^{s-}. \end{equation} So from last section we can get $\tilde{x}_k^{r-}\sim\textrm{N}(0,C_k^-)$ and $\tilde{x}_k^{s-}\in\mathcal{E}(0,S_k^-)$. And the mean squared error of priori estimation is given by \begin{equation}\label{cov_xe} \begin{split} &\mathrm{E}[(x_k-\hat{x}_k^{c-})(x_k-\hat{x}_k^{c-})^T]=\mathrm{E}[(\tilde{x}_k^{r-}+\tilde{x}_k^{s-})(\tilde{x}_k^{r-}+\tilde{x}_k^{s-})^T]\\ =&\mathrm{E}[\tilde{x}_k^{r-}\tilde{x}_k^{r-,T}]+\mathrm{E}[\tilde{x}_k^{s-}\tilde{x}_k^{s-,T}]=C_k^-+\tilde{x}_k^{s-}\tilde{x}_k^{s-,T}. \end{split} \end{equation} \begin{gather} \tilde{z}_k=h_k(\hat{x}_k^-,0,0)-H_{x,k}\hat{x}_k^-\\ y_k-\tilde{z}_k=H_{x,k}x_k+H_{v,k}v_k+H_{b,k}b_k. \end{gather}
Therefore, recalling equations \eqref{zk}, \eqref{lsystem1}, \eqref{lsystem2} and the derivation process in EKF, we also assume \begin{equation} \begin{split} \hat{x}_k^+=&\hat{x}_k^-+K_k[y-h_k(\hat{x}_k^-,0,0)]=\hat{x}_k^{-}+K_k[y-\tilde{z}_k(\hat{x}_k^{-})-H_{x,k}\hat{x}_k^{-}]\\ =&(I-K_kH_{x,k})\hat{x}_k^-+K_k[y-\tilde{z}_k(\hat{x}_k^-)]. \end{split} \end{equation}
The expectations $\hat{\mu}_k^+$ of posteriori estimations $\hat{x}_k^+$ would be \begin{equation}\label{mu+} \hat{\mu}_k^+=\mathrm{E}(\hat{x}_k^+)=(I-K_kH_{x,k})\hat{\mu}_k^-+K_k[y-\tilde{z}_k(\hat{\mu}_k^-)]. \end{equation}
The center of the ellipsoid $\mathcal{E}(\hat{x}_k^{c+},S_k^+)$ would be \begin{equation} \begin{split}\label{xc+} \hat{x}_{k}^{c+}=&\hat{x}_k^{c-}+K_k[y-h_k(\hat{x}_k^{c-},0,0)]=\hat{x}_k^{c-}+K_k[y-\tilde{z}_k(\hat{x}_k^{c-})-H_{x,k}\tilde{x}_k^{c-}]\\ =&(I-K_kH_{x,k})\hat{x}_k^{c-}+K_k[y-\tilde{z}_k(\hat{x}_k^{c-})]. \end{split} \end{equation}
Subtract $\hat{x}_k^{c+}$ from the true state $x_k$ we get: \begin{equation} \begin{split} x_k-\hat{x}_k^{c+}=&x_k-(I-K_kH_{x,k})\hat{x}_k^{c-}-K_k[y_k-\tilde{z}_k(\hat{x}_k^{c-})]\\ =&x_k-(I-K_kH_{x,k})\hat{x}_k^{c-}-K_k(H_{x,k}x_k+H_{v,k}v_k+H_{b,k}b,k)\\
=&(I-K_{k}H_{x,k})(\tilde{x}_k^{r-}+\tilde{x}_k^{s-})-K_k(H_{v,k}v_k+H_{b,k}b_k). \end{split} \end{equation}
So the mean squared error of the posteriori estimation center would be \begin{equation}\label{38} \begin{split} &\mathrm{E}[(x_k-\hat{x}_k^{c+})(x_k-\hat{x}_k^{c+})^T]\\ =&\mathrm{E}\{[(I-K_kH_{x,k})(\tilde{x}_k^{r-}+\tilde{x}_k^{s-})+K_k(H_{v,k}v_k+H_{b,k}b_k)]\cdot\\ &[(I-K_kH_{x,k})(\tilde{x}_k^{r-}+\tilde{x}_k^{s-})+K_k(H_{v,k}v_k+H_{b,k}b_k)]^T\}\\
=&(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T+K_kH_{v,k}C_k^zH_{v,k}^TK_k^T\\ &+[(I-K_kH_{x,k})\tilde{x}_k^{s-}-K_kH_{b,k}b_k][(I-K_kH_{x,k})\tilde{x}_k^{s-}-K_kH_{b,k}b_k]^T. \end{split} \end{equation}
Compared with equation \eqref{cov_xe}, we get \begin{equation}\label{cov+} C_k^+=(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T+K_kH_{v,k}C_k^zH_{v,k}^TK_k^T. \end{equation} Similar to Kalman filter (KF), the covariance matrices in the SKF provide us with a measure for uncertainty in our predicted and filtering state estimate, which is a very important feature for various applications of filtering theory since we then know how much to trust our predictions and estimates.
Notice that \begin{equation} \hat{\mu}_k^-\in\mathcal{E}(\hat{x}_k^{c-},S_k^-) \end{equation} and \begin{equation} y_k-\tilde{z}_k=H_{x,k}x_k+H_{v,k}v_k+H_{b,k}b_k\in\mathcal{E}(H_{x,k}x_k+H_{v,k}v_k,H_{b,k}S_k^zH_{b,k}^T). \end{equation}
So back to equation \eqref{mu+} we have \begin{equation} \begin{split} \hat{\mu}_k^+&=(I-K_kH_{x,k})\hat{\mu}_k^-+K_k(y_k-\tilde{z}_k)\\ &\in (I-K_kH_{x,k})\mathcal{E}(\hat{x}_k^{c-},S_k^-)\oplus K_k\mathcal{E}(H_{x,k}x_k+H_{v,k}v_k,H_{b,k}S_k^zH_{b,k}^T)\\ &=\mathcal{E}[(I-K_kH_{x,k})\hat{x}_k^{c-},(I-K_kH_{x,k})S_k^-(I-K_kH_{x,k})^T]\\ &\oplus\mathcal{E}[K_k(H_{x,k}x_k+H_{v,k}v_k),K_kH_{b,k}S_k^zH_{b,k}^TK_k^T]\subset\mathcal{E}(\hat{x}^{c+}_k,S_k^+), \end{split} \end{equation} where the midpoint is exactly in accordance with our previous assumption \eqref{xc+}: \begin{equation} \hat{x}_k^{c+}=(I-K_kH_{x,k})\hat{x}_k^{c-}+K_k[y-\tilde{z}_k(\hat{x}_k^{c-})], \end{equation} and from Corollary \ref{cor4.2} we have \begin{equation}\label{shape+} S_k^+(\beta)=(1+\frac{1}{\beta})(I-K_kH_{x,k})S_k^-(I-K_kH_{x,k})^T+(1+\beta)K_kH_{b,k}S_k^zH_{b,k}^TK_k^T. \end{equation}
\subsection{Optimization Problem} Now comparing to its counterpart in EKF, the only thing left is to derive the new optimal Kalman gain, which should minimize the mean square error of the posteriori estimation.
Here we introduce another parameter $\eta\in[0,1]$ to balance the random uncertainty and set-membership in the dynamical system, and define the following cost function as: \begin{equation}\label{costf} J(\beta)=(1-\eta)\mathrm{tr}(C_k^+)+\eta\mathrm{tr}(S_k^+(\beta)) \end{equation} which represents the global uncertainty of the posteriori estimation. The new optimal Kalman gain to be found should be used to minimize this cost function in a comprehensive way.
Plugging \eqref{cov_xe} and \eqref{shape+} into \eqref{costf} we get: \begin{equation} \begin{split} J(\beta)=&(1-\eta)\mathrm{tr}[(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T]\\ &+(1-\eta)\mathrm{tr}[K_kH_{v,k}C_k^zH_{v,k}^TK_k^T]\\ &+\eta(1+\frac{1}{\beta})\mathrm{tr}[(I-K_kH_{x,k})S_k^-(I-K_kH_{x,k})^T]\\ &+\eta(1+\beta)\mathrm{tr}(K_kH_{b,k}S_k^zH_{b,k}^TK_k^T)\\ \triangleq &(1-\eta)\mathrm{tr}[(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T]\\ &+(1-\eta)\mathrm{tr}[K_kH_{v,k}C_k^zH_{v,k}^TK_k^T]+\eta(1+\frac{1}{\beta})M+\eta(1+\beta)N. \end{split} \end{equation} where $M$ and $N$ are defined directly from above.
Notice that the cost function $J$ relies on two variables $K_k$ and $\beta$. Firstly we minimize $J$ respect with $\beta$.
Since $M>0$ and $N>0$, therefore \begin{equation} (1+\frac{1}{\beta})M+(1+\beta)N=M+N+\frac{1}{\beta}M+\beta N\geq M+N+2\sqrt{MN}. \end{equation} When $\frac{1}{\beta}M=\beta N$, i.e., $M=\beta^2N,\beta=\beta_1=\sqrt{\frac{M}{N}}$, we have \begin{equation} (1+\frac{1}{\beta_1})M+(1+\beta_1)N=M+N+2\sqrt{MN}=(\sqrt{M}+\sqrt{N})^2 \end{equation} Therefore we can find the local minimum point of function $J$ with respect to $\beta$: \begin{equation}\label{G} \begin{split} J(\beta_1)=&(1-\eta)\mathrm{tr}[(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T]\\ &+(1-\eta)\mathrm{tr}[K_kH_{v,k}C_k^zH_{v,k}^TK_k^T]+\eta(\sqrt{M}+\sqrt{N})^2. \end{split} \end{equation}
Next we calculate the global minimum by taking $K_k$ into account.
Notice that \begin{equation} \begin{split} \frac{\partial\sqrt{M}}{\partial K}=&\frac{1}{2}M^{-\frac{1}{2}}\frac{\partial M}{\partial K}=-\frac{1}{2\sqrt{M}}(I-K_kH_{x,k})(S_k^{-,T}+S_k^-)H_{x,k}^T\\ =&-\frac{1}{\sqrt{M}}(I-K_kH_{x,k})S_k^-H_{x,k}^T \end{split} \end{equation} \begin{equation} \begin{split} \frac{\partial\sqrt{N}}{\partial K}=&\frac{1}{2}N^{-\frac{1}{2}}\frac{\partial N}{\partial K}=\frac{1}{2\sqrt{N}}K_kH_{b,k}(S_k^{z,T}+S_k^z)H_{b,k}^T\\ &=\frac{1}{\sqrt{N}}K_kH_{b,k}S_k^zH_{b,k}^T \end{split} \end{equation}
Then \begin{equation} \begin{split} \frac{\partial J}{\partial K_k}=&2(1-\eta)(K_kH_{x,k}-I)C_k^-H_{x,k}^T+2(1-\eta)K_kH_{v,k}C_k^zH_{v,k}^T\\ &+2\eta(1+\frac{1}{\beta})(K_kH_{x,k}-I)S_k^-H_{x,k}^T+2\eta(1+\beta)K_kH_{b,k}S_k^zH_{b,k}^T. \end{split} \end{equation}
Let$\frac{\partial G_1}{\partial K_k}=0$ and solve for $K_k$, we get an adaptive Kalman gain: \begin{equation}\label{KG} \begin{split} K_k=&[(1-\eta)C_k^-H_{x,k}^T+\eta(1+\frac{1}{\beta})S_k^-H_{x,k}^T]\cdot[(1-\eta)H_{x,k}C_k^-H_{x,k}^T\\ &+(1-\eta)H_{v,k}C_k^zH_{v,k}^T+\eta(1+\frac{1}{\beta})H_{x,k}S_k^-H_{x,k}^T+\eta(1+\beta)H_{b,k}S_k^zH_{b,k}^T]^{-1} \end{split} \end{equation}
Now we get the elicit expression of the cost function \eqref{costf} by collecting \eqref{cov+}, \eqref{shape+} and \eqref{KG}. All the procedures in this filtering step rely on the solution of the following optimization problem.
\begin{equation}\label{optimization} \begin{split} &\underset{\beta}{\min}\,\,\,J(\beta)\\ &\mathrm{s.t.} \,\,\,\beta\in [0,+\infty) \subset \mathds{R}^1 \end{split} \end{equation} where the cost function $J(\beta)$ was defined in \eqref{costf} and the solution of above optimization problem was denoted by $\beta^*$. Putting $\beta^*$ into \eqref{KG}, \eqref{cov+} and \eqref{shape+} and we finished the filtering step.
Here are three remarks about this optimization problem: \begin{enumerate}[(1)] \item Problem \eqref{optimization} is a nonlinear programming problem since the objective function \eqref{costf} is nonlinear. \item Problem \eqref{optimization} is a convex optimization problem \cite{noack2014state}. Therefore, any existing local minimum is a global minimum. \item Usually, it is hard to solve a nonlinear programming problem due to the constrained equations or inequalities. MATLAB function \emph{fminsearch} is an efficient way to solve the problem \eqref{optimization}. Further, an advanced toolbox \emph{INTLAB} can also be used \cite{Ru99a}. \end{enumerate}
The parameter $\eta$ was introduced to balance the random uncertainty and set-membership uncertainty. There are three very interesting cases need to be noticed \cite{durieu2001multi}.
When $\eta=\frac{1}{2}$, the stochastic uncertainty and set-membership uncertainty have the same weight and $K(p)$ contains no $\alpha$ in this case. This solution is recommended to users when there is no expert-based information about $\eta$ available.
When $\eta=0$, \begin{equation} K_k(\beta)=C_k^-H_{x,k}^T\cdot[H_{x,k}C_k^-H_{x,k}^T+H_{v,k}C_k^zH_{v,k}^T]^{-1} \end{equation} which is exactly the Kalman gain in the standard EKF \cite{simon2006optimal}.
When $\eta=1$, the model now only contains set-membership uncertainty. In this case, \begin{equation} K_k(\beta)=(1+\frac{1}{\beta})S_k^-H_{x,k}^T\cdot[(1+\frac{1}{\beta})H_{x,k}S_k^-H_{x,k}^T+(1+\beta)H_{b,k}S_k^zH_{b,k}^T]^{-1}. \end{equation}
\section{Algorithm}
An algorithm for SKF was summarized according to previous derivation.
\begin{breakablealgorithm}
\caption{Set-membership Kalman filter model}
\begin{algorithmic}[1]
\State \textbf{Initialization}:
\begin{enumerate}[(1)]
\item Initial state midpoint $\hat{x}_0^{c+}=x_0$.
\item Initial estimated random covariance matrix $C_0^+$.
\item Initial estimated set-membership shape matrix $S_0^+$.
\end{enumerate}
\For{k=1,2,\dots,K}
\State \textbf{Input of Prediction Step}:
\begin{enumerate}[(1)]
\item Point post-estimation $\hat{x}_k^+$, with estimated covariance $C_k^+$ and shape matrix $S_k^+$.
\item Nonlinear system model
\begin{equation}
x_{k+1}=f_k(x_k,u_k,w_k,a_{1,k},a_{2,k},...,a_{I,k}),
\end{equation}
where $w_k\sim\textrm{N}(0,C_k^u)$ and $a_{i,k}\in \mathcal{E}(0,S_{i,k}^u)$, $i=1,2,\dots, I.$
\item Control input $u_k$, random noise covariance $C_k^u$ and shape matrices $S_{i,k}^u$, $i=1,2,\dots, I.$ for set-membership uncertainty.
\end{enumerate}
\State \textbf{Calculation of Prediction Step}:
\begin{enumerate}[(1)] \item Computation of error covariance matrix $C_{k+1}^-$ according to \begin{equation} C_{k+1}^-=F_{x,k}C_k^+F_{x,k}^T+F_{w,k}C_k^uF_{w,k}^T. \end{equation} \item The center of the priori ellipsoid: \begin{equation} \hat{x}_{k+1}^{c-}=F_{x,k}\hat{x}_k^{c+}+\tilde{u}_k. \end{equation} \item The shape matrix of the priori ellipsoid: \begin{equation} \begin{split} S_{k+1}^-=&(\sqrt{\mathrm{tr}(F_{x,k}S_k^+F_{x,k}^T)}+\sum_{i=1}^I\sqrt{\mathrm{tr}(F_{a,i}S_{i,k}^uF_{a,i}^T)})\\ &\cdot(\frac{F_{x,k}S_k^+F_{x,k}^T}{\sqrt{\mathrm{tr}(F_{x,k}S_k^+F_{x,k}^T)}}+\sum_{i=1}^I\frac{F_{a,i}S_{i,k}^uF_{a,i}^T}{\sqrt{\mathrm{tr}(F_{a,i}S_{i,k}^uF_{a,i}^T)}}). \end{split} \end{equation} The predicted point estimate $x_{k+1}^-$ is characterized by the random error $C_{k+1}^-$ and the set-membership error by $\hat{x}_{k+1}^{c-}$ and $S_{k+1}^-$. \end{enumerate}
\State \textbf{Output of Prediction Step}:
Priori estimated state: $\hat{x}_k^{c-}$, $C_k^-$, and $S_k^-$.
\State \textbf{Input of Filtering Step}:
\begin{enumerate}[(1)] \item Priori or predicted estimate $\hat{x}_k^-$ with error covariance matrix $C_k^-$ and ellipsoid center $\hat{x}_k^{c-}$ and shape matrix $S_k^-$. \item Nonlinear measurement model: \begin{equation} y_k=h_k(x_k,v_k,b_k), \end{equation} where $v_k\sim\textrm{N}(0,C_k^z)$ and $b_k\in \mathcal{E}(0,S_k^z)$. \item Observation $y_k$, sensor noise with random covariance $C_k^z$ and set-membership shape matrix $S_k^z$. \item $\tilde{z}_k(\hat{x}_k^-)=h_k(\hat{x}_k^-,0,0)-H_{x,k}\hat{x}_k^-$. \item Weighting parameter $\eta$. \end{enumerate}
\State \textbf{Calculation of Filtering Step}:
\begin{enumerate}[(1)] \item For given weighting parameter $\eta$, the optimal gain factor $K_k$ is \begin{equation} \begin{split} K_k(\beta)=&[(1-\eta)C_k^-H_{x,k}^T+\eta(1+\frac{1}{\beta})S_k^-H_{x,k}^T]\cdot[(1-\eta)H_{x,k}C_k^-H_{x,k}^T\\ &+(1-\eta)H_{v,k}C_k^zH_{v,k}^T+\eta(1+\frac{1}{\beta})H_{x,k}S_k^-H_{x,k}^T\\ &+\eta(1+\beta)H_{b,k}S_k^zH_{b,k}^T]^{-1}. \end{split} \end{equation} \item Computation of the center of updated estimate $\hat{x}_k^+$ by means of \begin{equation} \hat{x}_k^{c+}=(I-K_kH_{x,k})\hat{x}_k^{c-}+K_k[y-\tilde{z}_k(\hat{x}_k^{c-})]. \end{equation} \item Computation of updated error covariance matrix $C_k^+$ by \begin{equation} C_k^+(\beta)=(I-K_kH_{x,k})C_k^-(I-K_kH_{x,k})^T+K_kH_{v,k}C_k^zH_{v,k}^TK_k^T. \end{equation} \item Update the shape matrix $S_k^+$ by \begin{equation} S_k^+(\beta)=(1+\frac{1}{\beta})(I-K_kH_{x,k})S_k^-(I-K_kH_{x,k})^T+(1+\beta)K_kH_{b,k}S_k^zH_{b,k}^TK_k^T. \end{equation} \item The optimal parameter $\beta^*$ can be solved by \begin{equation}\label{op} \beta^*=\arg \min\{(1-\eta)\mathrm{tr}[C_k^+(\beta)]+\eta\mathrm{tr}[S_k^+(\beta)]\}. \end{equation} The updated point estimate $\hat{x}_k^+$ is characterized by random error characteristic $C_k^+=C_k^+(\beta)$ and set-membership error description $S_k^+=S_k^+(\beta)$. Put $\beta^*$ into above 4 functions to get the optimal output. \end{enumerate}
\State \textbf{Output of Filtering Step}:
Posteriori estimated state: $\hat{x}_k^{c+}$, $C_k^+(\beta^*)$, and $S_k^+(\beta^*)$.
\EndFor
\end{algorithmic} \end{breakablealgorithm}
\section{Applications}
\subsection{Example 1: Highly Nonlinear Benchmark Example}
Consider the following example: \begin{gather} x_{k+1}=\frac{1}{2}x_k+\frac{25x_k}{1+x_k^2}+8\cos[1.2(k-1)]+w_k+a_k,\\ y_k=\frac{1}{20}x_k^2+v_k+b_k. \end{gather}
where $x_k$ is a scalar, $u_k=8\cos[1.2(k-1)]$ is the input vector, $w_k\sim \textrm{N}(0,1)$ is a Gaussian process noise, $a_{i,k}\in\mathcal{E}(0,9)$ is the unknown but bounded perturbation, in this 1-D case the ellipsoid is the interval $[-3,3]$. $v_k\sim \textrm{N}(0,1)$ is the a Gaussian measurement noise, and $b_k\in \mathcal{E}(0,4)$ is the unknown but bounded perturbation in the interval $[-2,2]$. Initial true state is $x_0=0.1$, initial state estimate as $\hat{x}_0=x_0$, initial estimation covariance matrix is $C_0^+=2$ and initial shape matrix is $S_0^+=1\times10^{-3}$. We used a simulation length of 50 time steps. Weight parameter $\eta=0.5$.
This system was regarded as a benchmark in the nonlinear estimation theory \cite{kitagawa1987non}\cite{gordon1993novel}, and it is usually used to demonstrate the drawbacks of EKF comparing with particle filter \cite{simon2006optimal}. The high degree of nonlinearity in both the process and measurement equations makes this system a very difficult state estimation problem for a Kalman filter. We use this example to show the new SKF behaves better than the traditional first order EKF when some set-membership uncertainties are included in the system.
We repeated this simulation for 100 times. And Fig. \ref{com4} shows the comparison results between SKF and EKF at time $k=25,50,75,100$.
\begin{figure}
\caption{Comparison results in 4 simulations}
\label{com4}
\end{figure}
In above figures, the red stars denote the true positions of the state, the black crosses represent the estimated positions via EKF, the blue lines give the estimated ellipsoids (in this 1D case they are intervals) via SKF, and the blue plus signs mark the centers of the output ellipsoids. The center of the ellipsoid given by the new SKF is different with the traditional estimation via EKF, as what we expected, the ellipsoids include the true positions sometimes.
To further compare this new method with EKF, we calculate the distance vectors $d_s, d_e$ of SKF and EKF with the true states, respectively. Each distance vector is $50\times1$ for the total $50$ steps in every simulation. Table \ref{table1} shows the detailed $l_2$ norm comparison of these two distance vectors in $10$ simulations ($k=1,11,...,91$). The second row headed by $\mathrm{S}$ shows the distance error via SKF, and the third row headed by $\mathrm{E}$ shows the counterpart via EKF. We use the $l_2$ norm here as a generic measure of the distance between the estimated data and the true data, but other norms like $l_1$ and $l_{\infty}$ are possible for use. Without loss of generality, we choose the midpoints of these ellipsoids for comparison.
\begin{table} \scriptsize
\begin{tabular}{|p{0.12cm}|c|c|c|c|c|c|c|c|c|c|}
\hline
k & 1 & 11 & 21 & 31 & 41 & 51 & 61 & 71 & 81 & 91 \\
\hline
S & 53.59 & 116.38 & 70.38 & 102.95 & 100.16 & 75.81 & 82.11 & 48.71 & 66.30 & 81.55 \\
\hline
E & 72.49 & 116.64 & 230.08 & 234.69 & 80.73 & 31.32 & 109.53 & 79.69 & 8.63 & 64.95 \\
\hline \end{tabular} \caption{Comparison of SKF and EKF in 10 simulations} \label{table1} \end{table}
In the whole 100-time simulation experiments, the overall $l_2$ norm of the distance vector under SKF is 148.70, with its counterpart in extended Kalman filter 192.29. The new SKF behaved much better than EKF in these 100 simulations. However, this does not mean the SKF is always a better algorithm comparing with EKF, since it is also possible to get opposite results when repeating this experiment.
\subsection{Example 2: Two-Dimensional Trajectory Estimation}
A vehicle moves on a plane with a curved trajectory \cite{alkhatib2008comparison}. The state vector $x=(x,y,v_x,v_y)$ contains positions and velocities of the target, in x-direction and y-direction, respectively. After linearization, we do not consider the acceleration process anymore, and the mathematical model of this vehicle was assumed as following: \begin{equation} \mathbf{x}_{k+1}=F_k\mathbf{x}_k+\mathbf{w}_k+\mathbf{a}_k \end{equation} where $\mathbf{x}_k=(x_k,y_k,v_{x,k},v_{y,k})$ is the state vector at time $t_k$. The transition matrix $F_k$ is designed by: \begin{equation} F_k=\left(
\begin{array}{cccc}
1 & 0 & dt & 0 \\
0 & 1 & 0 & dt \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right). \end{equation} $\mathbf{w_k}$ which representing random uncertainty is gaussian with covariance matrix $C_k^u$, and $\mathbf{a_k}$ is the unknown but bounded uncertainty, which was bounded by an ellipsoid with shape matrix $S_k^u$. In total 300 points were observed and time step $dt=0.1$ seconds. The units of time, distance, angle are second, meter and degree, respectively.
In this experiment, two observation stations $S_1=[s_{12}, s_{12}]$ and $S_2=[s_{21}, s_{22}]$ were arranged to make the measurements. Each station measured the distance and the direction angle of the vehicle. Here is the measurement equation: \begin{equation} \mathbf{y}_k=h_k(\mathbf{x}_k,v_k,b_k). \end{equation} \begin{equation} \mathbf{y}_k=\left(
\begin{array}{c}
d_1 \\
d_2 \\
\theta_1 \\
\theta_2 \\
\end{array} \right)=\left(
\begin{array}{c}
\sqrt{[x-s_{11}]^2+[y-s_{12}]^2} \\
\sqrt{[x-s_{21}]^2+[y-s_{22}]^2} \\
\arctan [(y-s_{12})/(x-s_{11})] \\
\arctan [(y-s_{22})/(x-s_{21})] \\
\end{array}
\right)+\mathbf{v}_k+\mathbf{b}_k \end{equation} $\mathbf{v_k}$ which representing random uncertainty is gaussian with covariance matrix $C_k^z$, and $\mathbf{b_k}$ is the unknown but bounded uncertainty, which was bounded by an ellipsoid with shape matrix $S_k^z$.
The initial state, estimated covariance matrix and shape matrix are given by: $x_0=(0,0,0,0)$, $C_0^+=\mathrm{diag}(0.01,0.01,0.01,0.01)$ and $C_0^+=\mathrm{diag}(1\times10^{-6},1\times10^{-6},1\times10^{-6},1\times10^{-6})$.
\begin{comment} \begin{equation}\label{initial} x_0=\left(
\begin{array}{c}
0 \\
0 \\
0 \\
0 \\
\end{array}
\right), C_0^+=\left(
\begin{array}{cccc}
0.01 & 0 & 0 & 0 \\
0 & 0.01 & 0 & 0 \\
0 & 0 & 0.01 & 0 \\
0 & 0 & 0 & 0.01 \\
\end{array}
\right), S^+_0=1\times10^{-6}\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right). \end{equation} \end{comment}
The initial covariance matrices in process equation and measurement equation are given by: \begin{comment} \begin{gather} C_0^u=[0.0033,0,0.005,0;0,0.0033,0,0.005;0.005,0,0.01,0;0,0.005,0,0.01]\\ C_0^z=\mathrm{diag}[0.005^2,0.005^2,0.005^2,0.005^2]. \end{gather} \end{comment} \begin{equation}\label{Q} C^u_0=\left(
\begin{array}{cccc}
0.0033 & 0 & 0.005 & 0 \\
0 & 0.0033 & 0 & 0.005 \\
0.005 & 0 & 0.01 & 0 \\
0 & 0.005 & 0 & 0.01 \\
\end{array}
\right), C^z_0=\left(
\begin{array}{cccc}
0.005^2 & 0 & 0 & 0 \\
0 & 0.005^2 & 0 & 0 \\
0 & 0 & 0.005^2 & 0 \\
0 & 0 & 0 & 0.005^2 \\
\end{array}
\right). \end{equation}
The initial shape matrices of set-membership uncertainties in process equation and measurement equation are setting by: \begin{equation}\label{set_membership_setting} S^u_0=\left(
\begin{array}{cccc}
1^2 & 0 & 0 & 0 \\
0 & 1^2 & 0 & 0 \\
0 & 0 & 0.5^2 & 0 \\
0 & 0 & 0 & 0.5^2 \\
\end{array}
\right), S^z_0=\left(
\begin{array}{cccc}
0.01^2 & 0 & 0 & 0 \\
0 & 0.01^2 & 0 & 0 \\
0 & 0 & (\frac{\pi}{180})^2 & 0 \\
0 & 0 & 0 & (\frac{\pi}{180})^2 \\
\end{array}
\right). \end{equation}
Weight parameter $\eta=0.5$.
Below eight different trajectories were estimated by EKF and SKF from eight different data sets. The following Fig. \ref{8trajectory} shows the estimation results. Red stars mark the true position according to the reference data, black crosses denotes the estimated position via EKF, and the blue plus signs are the geometry centers of the ellipsoids.
\begin{figure}
\caption{Trajectory 1}
\caption{Trajectory 2}
\caption{Trajectory 3}
\caption{Trajectory 4}
\end{figure}
\begin{figure}
\caption{Trajectory 5}
\caption{Trajectory 6}
\caption{Trajectory 7}
\caption{Trajectory 8}
\caption{Eight trajectories examples to compare SKF with EKF}
\label{8trajectory}
\end{figure}
One may notice that both EKF and SKF perform well in most part of each trajectory, except that the ellipsoids getting large in the interaction area between the trajectory and the straight line of two stations. Again, we calculate the $l_2$ norm of distance vectors to make further comparison in Table \ref{table2}.
\begin{table}[H] \centering \scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Trajectory & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
SKF & 3.22 & 2.88 & 4.36 & 2.47 & 3.70 & 12.37 & 2.83 & 1.91 \\
\hline
EKF & 5.37 & 3.62 & 27.81 & 3.82 & 3.67 & 7.79 & 13.31 & 3.27 \\
\hline \end{tabular} \caption{Comparison of SKF and EKF in 8 trajectories} \label{table2} \end{table}
To check the estimation errors, we chose Trajectory 5 to repeat for 100 times and then get the following error distribution.
\begin{figure}
\caption{Distance Error of SKF in 100 times}
\label{distance_error}
\end{figure}
From above Fig. \ref{distance_error} it is obvious to notice that the estimated error was getting larger when $k\in [200, 225]$, i.e., in Fig. \ref{8trajectory} (e) one may get worse estimation results in the intersection area of the line between the two observation stations and the trajectory of the vehicle. The following Fig. \ref{local_5} shows more local details in the interaction area of Trajectory 5, where the straight line connects the two observation stations. Not only the estimated ellipsoids getting larger in the interaction area, but the semi-major axes of the largest ellipsoid is perpendicular to the straight line.
\begin{figure}
\caption{Local estimation details near the interaction area}
\label{local_5}
\end{figure}
There exist two major reasons causing this phenomena.
Firstly, the angle set-membership uncertainty played a more significant role in the estimation. From \eqref{set_membership_setting} we notice that in Fig. \ref{8trajectory} Trajectory 5, the set-membership uncertainty of distance is $[-0.01,0.01]$ meter, and its counterpart in angle is $[-1,1]$ degree. The distance between the observation station and the interaction area is at least 85 meters, i.e., the uncertainty caused by angles would be $80\times\frac{\pi}{180}\mathrm{m}=1.4835\mathrm{m}$ (in the vertical direction of the straight line), which is greatly larger than the distance uncertainties $0.01 \mathrm{m}$ (in the parallel direction of the straight line).
Secondly, the criterion of the optimization problem in \eqref{op} in the SKF algorithm is the trace of a shape matrix. There are several minimum criterions to get one optimal ellipsoid given a shape matrix $S$, e.g., the trace of the shape matrix $\mathrm{tr}(S)$, the determinant of the shape matrix $|S|$, and the largest eigenvalue of the shape matrix $\lambda_M(S)$. Minimizing the largest eigenvalue $\lambda_M(S)$ smoothes the mean curvature and makes the ellipsoid more like a ball (circular in 2D case). Minimizing the trace or the determinant of the shape matrix produces an ellipsoid with small volume, but sometimes causes the ellipsoid getting oblate, i.e., more uncertainties in one certain direction in this example.
\begin{figure}
\caption{Trajectory 5.1}
\caption{Trajectory 5.2}
\caption{Trajectory 5.3}
\caption{Trajectory 5.4}
\caption{Output ellipsoids are highly related to the initial settings.}
\label{terrible}
\end{figure}
Fig. \ref{terrible} shows that the output estimated ellipsoids are highly related to the initial boundary of the set-membership uncertainties in both equations. The principle semi-axes in Trajectory 5.3 (Su=diag([$100^2,100^2,50^2,50^2$]),\\ Sz=diag([$1^2,1^2,(\pi/1.8)^2,(\pi/1.8)^2$])) is 100 times larger than their counterparts in Trajectory 5.1 (Su=diag([$1^2,1^2,0.5^2,0.5^2$]),\\ Sz=diag([$0.01^2,0.01^2,(\pi/180)^2,(\pi/180)^2$]) and 10 times larger than their counterparts in Trajectory 5.2 (Su=diag([$10^2,10^2,5^2,5^2$]),\\ Sz=diag([$0.1^2,0.1^2,(\pi/18)^2,(\pi/18)^2$])), the the outputs of the SKF are getting very large. Both the input (accuracy of the instruments) and the output (estimated positions) in Trajectory 5.2 and 5.3 are not realistic and one more realistic example was shown in Trajectory 5.4 with initial shape matrix Su=diag([$0.5^2,0.5^2,0.5^2,0.5^2$]),\\ Sz=diag([$0.01^2,0.01^2,(\pi/180)^2,(\pi/180)^2$].
\section{Conclusion and Future Work}
One cannot state that the new SKF is always better than the standard EKF, however, the performance of SKF is much more reliable than EKF in some cases (like in previous simulated experiments). To say the least, the SKF is one reasonable and applicable model when some unknown but bounded uncertainties were included in the nonlinear system. A difference with the standard Kalman filter is that, the estimated states are ellipsoids instead of single points, and every inner points of one ellipsoid have the same estimation status. But one still can choose a series of particular points in these ellipsoids if necessary. The output is reasonable considering the unknown but bounded uncertainties which were included in the original system, and extra information in the measurement equation was issued properly in the filtering step.
Like other filter models, there is also some space for this SKF to improve. For instance, the shape matrices of the set-membership uncertainties in both system and measurement equation must be given properly at the beginning, and also the weighting parameter should be decided by the user or experts.
The future work of our research includes deriving a similar algorithm for second order extend Kalman filter or unscented Kalman filter, using zonotopes or interval boxes to bound the unknown but bounded uncertainty, and minimizing the determinant or the largest eigenvalue of the shape matrix when solving the optimization problem. Last but not least, the stability of this algorithm should be carefully discussed considering that the state estimation problem is usually ill-posed as an inverse problem \cite{blank2007state}.
\nocite{candy2016bayesian} \nocite{klir2005uncertainty} \nocite{li2018kalman} \nocite{sarkka2006recursive} \nocite{kurzhanskiui1997ellipsoidal}
\end{document} |
\begin{document}
\begin{abstract} We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots. \end{abstract}
\maketitle
\section{Introduction} Thurston proved that any non-trivial knot in the 3-sphere is either a \emph{torus knot}, a \emph{satellite knot}, or has \emph{hyperbolic complement} \cite{Thurston}. These are called the \emph{geometric types} of the knot. A satellite knot is a knot whose complement has an essential torus; a torus knot is a knot whose complement has essential annulus but does not have any essential torus; finally, a hyperbolic knot is a knot that is neither a torus knot nor a satellite knot. Thus, the \emph{geometric classification} of knots is related to the study of certain types of surfaces in the knot complement in $S^3$.
The geometric classification of knots has been an important task in modern research in knot theory. \emph{Alexander's theorem} tells us that every knot can be represented as the closure of a braid \cite{Birman}. So, a lot of study has been done to understand the knot types of knots given by braids. For example, Birman and Menasco studied surfaces in closed braid complements \cite{finiteness}. They also classified the positions of essential tori in closed braid complements \cite{positions}. Los proved that the Nielsen-Thurston classification of braids and the geometric structure of knot complements are related \cite{Los}. Ito studied the topology and geometry of closed braid complements by examining the Dehornoy ordering \cite{Ito}, which is a left-invariant total order on the braid group, introcuded by Dehornoy \cite{Dehornoy}.
Some important families of knots have appeared naturally with projections given by braids, such as \emph{Lorenz links} and \emph{twisted torus knots}. Twisted torus knots were introduced by Dean in his doctoral thesis \cite{Thesis} to study Seifert fibered spaces obtained by Dehn fillings. Lorenz links are knotted closed periodic orbits in the flow of the Lorenz system. The Lorenz system is a system of three ordinary differential equations in $\mathbb{R}^3$ introduced by the meteorologist Edward Lorenz to predict weather patterns. Birman and Kofman proved that Lorenz links coincide with \emph{T-links} \cite{newtwis}, which are links given by certain positive braids.
There haven't been many concrete conditions on braids to ensure their geometric type. For example, not much is known about the geometric classification of T-knots. However, there has been some study on the geometric classification of twisted torus knots(see \cite{LeeTorusknotsobtained}, \cite{hyperbolicity}, \cite{Composite}) of which some form a small class of T-knots.
This work builds upon the previous works by finding some conditions on some braids so that they close to give hyperbolic knots.
Let $\sigma_1, \dots, \sigma_{n-1}$ be the standard generators of the braid group $B_n$.
\begin{definition} Let $i, j, r$ be positive integers with $i<j$. The $(i, j, r)$-torus braid, denoted by $B_{i,j}^r$, is defined by the braid$$(\sigma_i\dots \sigma_{j-1})^{r}.$$ \end{definition}
Our main theorem is the following.
\begin{theorem}\label{theorem1} Let $K$ be a knot given by a positive braid $B$ with $p$ strands. Suppose that $B$ has one $(1, p, q + pk)$-torus braid, where $p, q$ are positive coprime integers with $p>q$ and $q, k\geq 2$, but $B \neq B_{1, p}^{q + pk}$. Also, assume that the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$. Then, the knot $K$ is hyperbolic. \end{theorem}
To prove this result, we combine a result of Ito related to the positions of essential surfaces with respect to the braid axis of the closed braid with a result of Williams related to the braid index of generalized cablings.
This theorem has some interesting applications to the geometric classification of T-links and twisted torus knots. There is an important conjecture, which was proposed by Morton, related to the geometric classification of Lorenz knots(see \cite{dePaivaPurcell:SatellitesLorenz} for more details). It says the following: A Lorenz knot that is a satellite has companion a Lorenz knot. Its pattern, when embedded in an unknotted torus in $S^3$, is equivalent to a Lorenz knot in $S^3$. Theorem~\ref{theorem1} gives us the next corollary, which helps to reduce the cases we need to consider Morton's conjecture as well as to understand the geometric classification of T-knots.
\begin{corollary}\label{T-knot} Let $p, q$ be positive coprime integers. Consider $p>q\geq r_1>1$. Then, for $k\geq 2$, the T-knots $$T((r_n, s_n), \dots, (r_1, s_1), (p, kp + q))\textrm{ and }T((r_n, s_n), \dots, (r_1, s_1), (kp + q, p))$$ are hyperbolic. \end{corollary}
The geometric classification of twisted torus knots has been intensely studied but the knot types of twisted torus knots of the form $T(p, q; r,\pm 1)$ are not fully understood(see \cite{dePaiva:Unexpected} for more details). We obtain the following two corollaries for these open cases.
\begin{corollary}\label{positiveTTT} Let $p, q$ be positive coprime integers. Consider $p>q\geq r>1$. Then, for $k\geq 2$, the twisted torus knots $$T(p, kp + q; r, 1)\textrm{ and }T(kp + q, p; r, 1)$$ are hyperbolic. \end{corollary} \begin{corollary}\label{negativeTTT} Let $p, q$ be positive coprime integers with $p>q$. Consider $p-q\geq r>1$. Then, for $k\geq 3$, the twisted torus knots $$T(p, kp + q; r, -1)\textrm{ and }T(kp + q, p; r, -1)$$ are hyperbolic. \end{corollary}
\subsection{Acknowledgment}I am grateful to Jessica Purcell, my supervisor, and Sangyop Lee for helpful discussions.
\section{The last case of Williams' result}
In this section we'll find the braid indexes of generalized cablings over trivial knots. This case wasn't considered by Williams in \cite{Williams}, but it'll be helpful for this work.
Williams extended the notion of cable knots by defining generalized cabling in \cite{Williams} as follows:
\begin{definition}A generalized $q$-cabling of a link $L$ we mean a link $L'$ contained in the interior of a tubular neighbourhood $L\times D^2$ of $L$ such that \begin{enumerate} \item each fiber $D^2$ intersects $L'$ transversely in $q$ points; and
\item all strands of $L'$ are oriented in the same direction as $L$ itself. \end{enumerate} \end{definition}
He also proved the following theorem in the same paper.
\begin{theorem}\label{Williams} The braid index is multiplicative under generalized cabling. In detail, if $L$ is a link with each component a non-trivial knot and $L'$ is a generalized $q$-cabling of $L$ then $$\beta(L') = q\beta(L),$$ where $\beta(*)$ is the braid index of $*$. \end{theorem}
Now we consider the case where $L$ is a trivial knot.
\begin{lemma}\label{lemma1} Let $L'$ be a generalized $q$-cabling of the unknot $L$, with $L$ given by a positive braid with $n$ strands, where $n>1$. Also, assume the knot inside $L$ is given by a positive braid. Then, $L'$ has braid index equal to $q$. \end{lemma} \begin{figure}
\caption{This series of drawings illustrate how the generalized 2-cabling $L'$ obtains one positive full twist after removing the crossing $\sigma_1$ from $L$. In this case we remove the first crossing of the braid $B$ of $L$. Similar ideas can be applied to see that $L'$ also obtains one positive full twist when we remove the last crossing of $B$ using the type II move. }
\label{trivial}
\end{figure} \begin{proof} Denote by $B$ the positive braid of $L$. The Markov theorem says that we can apply two types of moves(type I and type II) to transform $B$ into the trivial braid with one strand \cite{Birman}. The type I move geometrically sends crossings around the braid closure to the top or bottom of the braid. Thus, the type I move doesn't change the knot inside $L$. The type II move removes strands which bound discs by shrinking and undoing the circles that they form. In addition, we see that the type II move adds a positive full twist to the knot inside $L$ as explained in Figure~\ref{trivial}. Since the braid $B$ has more than one strand, we need to apply at least one type II move to transform $B$ into the trivial braid with one strand. After all these Markov moves, the knot $L'$ is given by a positive braid with $q$ strands and at least one positive full twist on the $q$ strands. Now it follows from [\cite{Franks}, corollary 2.4] that the knot $L'$ has braid index equal to $q$. \end{proof}
\section{Half Twists: the ``generators" of the braid group}
In this section we define \emph{half twist} as in [\cite{dePaiva:Unexpected}, section 3]. We'll use them to calculate the braid indexes of some braids in the next section.
\begin{definition} Let $J$ be an unknot bounding a disc $D$ which transversely intersects a set of $j$ parallel straight segments in a plane $Y$. A \emph{positive half twist along $J$} is obtained by the following procedure: cut along $D$ into two discs $D_1$ and $D_2$, where $D_1$, $D_2$ is the top, bottom disc, respectively. Then, rotate $D_1$ by $180^\circ$ degrees in the clockwise direction and then glue it back to $D_2$. A \emph{negative half twist along $J$} is defined similarly, only the rotation is in the anti-clockwise direction. \end{definition}
A positive half twist transforms the trivial braid with $j$ strands into the braid $$(\sigma_1\sigma_2\dots \sigma_{j - 2}\sigma_{j - 1})(\sigma_1\sigma_2\dots \sigma_{j - 3}\sigma_{j - 2})\dots (\sigma_1\sigma_2)(\sigma_1).$$ On the other hand, a negative half twist transforms the $j$ parallel straight segments into the braid $$(\sigma_{j - 1}^{-1}\sigma_{j - 2}^{-1} \dots \sigma_{2}^{-1}\sigma_{1}^{-1})(\sigma_{j - 1}^{-1}\sigma_{j - 2}^{-1} \dots \sigma_{2}^{-1})\dots (\sigma_{j - 1}^{-1}\sigma_{j - 2}^{-1})(\sigma_{j - 1}^{-1}),$$ where $\sigma_{i}^{-1}$ is the inverse of $\sigma_{i}$ in the braid group $B_j$.
We know that the $(i, j, (j-i+1)r)$-torus braid is obtained by Dehn filling along a circle encircling $j-i+1$ strands. The next lemma says that when the $(i, j, r)$-torus braid is not obtained by full twists, we need to use half twists in order to obtain it.
\begin{lemma}\label{half-twist} The $(i, j, r)$-torus braid is obtained by full and half twists along three circles as follows: start with the trivial braid $B$ with $j$ strands. Then, denote by $J_{i,j}$ the unknot encircling from the $i$ strand to the $j$ strand of the braid $B$. Consider $t$ an integer such that $0<t<j-i+1$ and $r=t+k(j-i+1)$ for some non-negative integer $k$. Further, augment the braid by placing unknots $J_{i, j-t}$ and $J_{j-t+1, j}$ into the trivial braid, where $J_{i, j-t}$, $J_{j-t+1, j}$ is enclosing from the strand $i$, $j-t+1$ to the strand $j-t$, $j$, respectively. Perform a positive half twist along $J_{i,j}$, a negative half twist along $J_{i, j-t}$, and a positive half twist along $J_{j-t+1, j}$. Finally, perform $(1/k)$-Dehn filling along $J_{i,j}$ and followed by $(1/0)$-Dehn fillings along each of $J_{i, j-t}$ and $J_{j-t+1, j}$ to remove them from the diagram. \end{lemma}
\begin{proof}We start by applying a positive half twist along $J_{i,j}$ to obtain the braid \[(\sigma_i\sigma_{i+1}\dots\sigma_{j-1})(\sigma_i\dots\sigma_{j-2})\dots(\sigma_i),\] encircled by $J_{i,j}$. We push $J_{i,j}$ to lie above the braid. A negative half twist along $J_{i, j-t}$ yields the braid \[(\sigma_{j-t-1}^{-1}\dots\sigma_{i+1}^{-1}\sigma_i^{-1})(\sigma_{j-t-1}^{-1}\dots\sigma_{i+1}^{-1})\dots(\sigma_{j-t-1}^{-1}).\] Then, we slide the component $J_{i, j-t}$ to be below the braid of the resulting half twist. This negative half twist along $J_{i, j-t}$ is cancelled with the positive half twist along the first $j-t-i+1$ strands of the first braid, as illustrated in the first and second drawings of Figure \ref{HalfTwistBraid}. Finally, the positive half twist along $J_{j-t+1, j}$ concatenates a positive half twist along the last $j-(j-t+1)+1$ strands, giving the braid \[(\sigma_i\dots\sigma_{j-1})^{t},\] as shown in the second and third drawings of Figure \ref{HalfTwistBraid}. \begin{figure}
\caption{These drawings illustrate the procedure described in the proof of lemma~\ref{half-twist} to obtain the $(1, 7, 4)$-torus braid by half twists. We start with the trivial braid on 7 strands together with the circles $J_{1, 7}$, $J_{1, 3}$, and $J_{4, 7}$, where $J_{1, 7}$ is encircling all 7 strands and $J_{1, 3}, J_{4, 7}$ is encircling the first 3, last 4 strands, respectively, with $J_{1, 7}$ above $J_{1, 3}$ and $J_{4, 7}$. Then, we apply a positive half twist along each $J_{1, 7}$ and $J_{4, 7}$ and a negative half twist along $J_{1, 3}$ to obtain the leftmost drawing. The negative half twist along $J_{1, 3}$ is cancelled with a positive half twist along just the first $3$ strands as shown in the second drawing. Finally, the positive half twist along $J_{4, 7}$ joins the rest of the positive half twist along $J_{1, 7}$ to give the $(1, 7, 4)$-torus braid as illustrated by the last drawing.}
\label{HalfTwistBraid}
\end{figure}
Now we perform $(1/k)$-Dehn filling along $J_{i,j}$ to add an additional $k(j-i+1)$ overstrands($k$ full twists) into the braid to obtain the desired braid $B_{i, j}^r$, although it is still augmented by the unknots $J_{i, j-t}$ and $J_{j-t+1, j}$. To remove these additional components, we perform $(1/0)$-Dehn filling along each. \end{proof}
\section{Hyperbolic knots given by positive braids}
In this section we prove theorem~\ref{theorem1}.
\begin{proposition}\label{proposition1} Let $K$ be a knot given by a positive braid $B$ with $p$ strands. Suppose that $B$ has one $(1, p, q + pk)$-torus braid, where $p, q$ are positive coprime integers with $p>q$ and $q, k\geq 2$. Also, assume that the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$. Then, $S^3-K$ has no essential tori, and therefore $K$ is not a satellite knot. \end{proposition}
\begin{proof}Consider that the complement of $K$ in $S^3$ has an essential torus $T$. Since $B$ has at least two positive full twists on $p$ strands, by Ito [\cite{Ito}, theorem 1.2(3)], $T$ doesn't intersect the braid axis $C$ of $B$ and the knot inside $T$ is given by a braid. So, $K$ is a generalized $b$-cabling of the knot $L$, where $L$ is the core of the solid torus bounded by $T$, with $b>1$. Thus, $b$ must divide $p$. After $(-1/k)$-Dehn surgery along $C$, the knot $K$ becomes the knot $K'$ given by the closure of the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ and the torus $T$ becomes a new torus $T'$ which doesn't intersect the braid axis of the last braid. The torus $T'$ is trivial or knotted. If $T'$ is knotted, then, by theorem~\ref{Williams}, $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $b\beta(T')$, where $\beta(T')$ is the braid index of the core of the solid torus bounded by $T'$. Thus, since the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$, we have that $q = b\beta(T')$, which is a contradiction since $p$ and $q$ are coprime. If $T'$ is trivial, then, by lemma~\ref{lemma1}, $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $b$. Thus, $q = b$, a contradiction for the same reason. So, $T$ does not exist. Therefore, $S^3-K$ has no essential tori. \end{proof}
\begin{corollary} Let $K$ be a knot given by a positive braid $B$ with $p$ strands. Suppose that $B$ has one $(1, p, q + pk)$-torus braid, where $p, q$ are positive coprime integers with $p>q\geq 2$. Also, assume that the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$. Then, for $k = 1, 0$, if $S^3-K$ has an essential torus, it intersects the braid axis of the braid $B$. \end{corollary} \begin{proof} Suppose $S^3-K$ has an essential torus $T$ that does not intersect the braid axis $C$ of $B$. Then, by doing Dehn surgery along $C$ with large slope, we can assume that the closure of the braid $BB_{1, p}^{k'p}$ has an essential torus for at least one $k'>2$, which is a contradiction with the last proposition. \end{proof}
The next lemma was proved by Los [\cite{Los}, corollary 1.2]]. We'll use it in the next proposition and in corollary~\ref{negativeTTT}.
\begin{lemma}\label{Los} Consider $\beta_1$, $\beta_2$ two braids which have minimal braid index and are representations of the same torus knot. Suppose that these two closed braids travel around the same braid axis $C$. Then, there is an isotopy in the complement of $C$ that takes $\beta_1$ to $\beta_2$. \end{lemma}
This lemma can also be seen by considering the Markov moves as mentioned in the proof of lemma~\ref{lemma1}. Since $\beta_1$, $\beta_2$ have minimal braid index, we only use type I moves to transform $\beta_i$ into $\beta_j$. Thus, this isopoty certainly happens in the complement of the braid axis.
\begin{proposition}\label{proposition2} Let $K$ be a knot given by a positive braid $B$ with $p$ strands. Suppose that $B$ has one $(1, p, q + pk)$-torus braid, where $p, q$ are positive coprime integers with $p>q\geq 2$ and $k\geq 1$, but $B \neq B_{1, p}^{q + pk}$. Also, assume that the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$. Then, the knot $K$ is not a torus knot. \end{proposition}
\begin{proof}Consider that $K$ is a torus knot. Since the braid $B$ contains at least one full twist on $p$ strands, it follows from [\cite{Franks}, corollary 2.4] that the knot $K$ has braid index equal to $p$. So, $K$ is the $(p, q + pk + d)$-torus knot with $d>0$. After $(-1/k)$-Dehn surgery along the braid axis $C$ of $B$, the knot $K$ becomes the knot $K'$ given by the closure of the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$. So, the knot $K'$ is the $(p, q + d)$-torus knot because, by lemma~\ref{Los}, there is an isotopy in the complement of $C$ that takes $K$ to $T(p, q + pk + d)$. As $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$, $K$ is also the $(q, w)-$torus knot with $w>0$. So, $q$ is equal to $p$ or $q + d$. Since $p, q$ are coprime, $q$ is equal to $q + d$. Thus, $d=0$, a contradiction. Therefore, $K$ is not a torus knot. \end{proof}
\begin{named}{Theorem~\ref{theorem1}} Let $K$ be a knot given by a positive braid $B$ with $p$ strands. Suppose that $B$ has one $(1, p, q + pk)$-torus braid, where $p, q$ are positive coprime integers with $p>q$ and $q, k\geq 2$, but $B \neq B_{1, p}^{q + pk}$. Also, assume that the braid $B(\sigma_{p-1}^{-1}\dots \sigma_1^{-1})^{kp}$ has braid index equal to $q$. Then, the knot $K$ is hyperbolic. \end{named}
\begin{proof} It immediately follows from proposition~\ref{proposition1} and proposition~\ref{proposition2}. \end{proof}
\begin{proposition}\label{proposition10} Consider $p, q$ positive coprime integers with $p>q$ and $q, k\geq 2$. Let $K$ be a knot given by a positive braid $B$ with $p$ strands of the form $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, p}^{q+kp}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}$$ and different from $B_{0, p}^{q+kp}$ or with $q$ strands of the form $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, q+kp}^{p}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}$$ and different from $B_{0, q+kp}^{p}$ such that if $a_i = 0$ then $b_i\leq q$ or if $a_i \neq 0$ then $b_i-a_i+1\leq q$. Then, the knot $K$ is hyperbolic. \end{proposition}
\begin{proof}We will calculate the braid index of the knot $K'$ given by the closure of the following braid, denoted by $B'$, $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, p}^{q}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}.$$ \begin{figure}
\caption{The second drawing is obtained from the first one by rotating it around the diagonal connecting the upper left vertex to the lower right vertex.}
\label{HK3}
\end{figure}
We push all $(a_i, b_i, r_i)$-torus braids which are above the biggest torus braid $B_{0, p}^{q}$ around the braid closure to be below $B_{0, p}^{q}$.
From lemma~\ref{half-twist}, each $(a_i, b_i, r_i)$-torus braid can be obtained by full and/or half twists along three circles where each one is encircling at most $q$ strands. Thus, $B'$ is obtained from the $(0, p, q)$-torus braid together with all these circles by full and/or half twists along them. By using the isotopy which deforms the $(p, q)$-torus knot to the $(q, p)$-torus knot, we can deform the $(0, p, q)$-torus braid to the $(0, q, p)$-torus braid. This isotopy can be described as a rotation around a diagonal of the square that gives the torus in which the $(p, q)-$torus knot lies, as illustrated in the first and second drawings of Figure~\ref{HK3}. After this isotopy, the circles encircle the meridional(horizontal) lines of the $(0, q, p)$-torus braid. Furthermore, they enclose the same amount of strands as before, which are at most $q$ strands. Now we can push all these circles to lie below the $(0, q, p)$-torus braid and then apply the full and/or half twists as before. We can see that after this last step we obtain a knot equivalent to the initial one(given by the closure of $B'$) because these isotopes preserve half twists as the tangles(bunches of crossings) of the torus braids travel in the same direction all the time. The knot $K'$ is now given by a positive braid with the $(0, q, p)$-torus braid on the top. Since $p>q$, this braid has at least one full twist. Therefore, it follows from \cite{Franks} that the knot $K'$ has braid index equal to $q$.
Now it follows from theorem~\ref{theorem1} that the knot given by the closure of the braid $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, p}^{q+kp}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}$$ is hyperbolic since $B' = B(\sigma_{p-1}^{-1}\dots \sigma_{1}^{-1})^{kp}$.
With similar ideas, we can isotope the braid $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, q+kp}^{p}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}$$ to a positive braid $B''$ with $p$ strands containing the torus braid $B_{0, p}^{q+kp}$. As before, we can see that the braid $B''(\sigma_1^{-1}\dots\sigma_{p-1}^{-1})^{kp}$ has braid index equal to $q$. Then, from theorem~\ref{theorem1}, the knot given by the closure of the braid $$B_{a_1, b_1}^{r_1}B_{a_2, b_2}^{r_2}\dots B_{a_{m-1}, b_{m-1}}^{r_{m-1}}B_{0, q+kp}^{p}B_{a_{m+1}, b_{m+1}}^{r_{m+1}}\dots B_{a_n, b_n}^{r_n}$$ is also hyperbolic. \end{proof}
\section{Applications to the geometric classification of T-knots and twisted torus knots} In this section we apply the results of the last section to get some corollaries related to the geometric classification of T-knots and twisted torus knots.
T-links are defined as follows: for $2\leq r_1< \dots < r_k$, and all $s_i>0$, the T-link $T((r_1,s_1), \dots, (r_k,s_k))$ is defined to be the closure of the following braid \[ (\sigma_1\sigma_2\dots\sigma_{r_1-1})^{s_1}(\sigma_1\sigma_2\dots\sigma_{r_2-1})^{s_2}\dots(\sigma_1\sigma_2\dots\sigma_{r_k-1})^{s_k}.\] Birman and Kofman showed that they are equivalent to Lorenz links \cite{newtwis}.
The author and Purcell classified some T-knots obtained by full twists on a $(p, q)$-torus knot in \cite{dePaivaPurcell:SatellitesLorenz}. The following corollary also includes T-knots not obtained by full twists, which are more difficult to find their knot types since they are not obtained by Dehn filling.
\begin{named}{Corollary~\ref{T-knot}} Let $p, q$ be positive coprime integers. Consider $p>q\geq r_1>1$. Then, for $k\geq 2$, the T-knots $$T((r_n, s_n), \dots, (r_1, s_1), (p, kp + q))\textrm{ and }T((r_n, s_n), \dots, (r_1, s_1), (kp + q, p))$$ are hyperbolic. \end{named}
\begin{proof} It follows from proposition~\ref{proposition10} since the first T-knot is the closure of the braid $B_{0, r_n}^{s_n}\dots B_{0, r_{1}}^{s_{1}} B_{0, p}^{kp + q}$ and the second T-knot is the closure of the braid $B_{0, r_n}^{s_n}\dots B_{0, r_{1}}^{s_{1}} B_{0, kp + q}^{p}$. \end{proof}
The twisted torus knot $T(p, q; r, s)$ is obtained by twisting $r$ adjacent strands of the $(p, q)$-torus knot a total of $s$ full twists(see \cite{LeeThiago} for more details). If $r<p$, then the twisted torus knot $T(p, q; r, s)$ is given by the braid $$(\sigma_1\sigma_2\dots\sigma_{p-1})^{q}(\sigma_1\sigma_2\dots\sigma_{r-1})^{sr}$$ if $s>0$ and by $$(\sigma_1\sigma_2\dots\sigma_{p-1})^{q}(\sigma_{r-1}^{-1}\sigma_{r-2}^{-1}\dots\sigma_{1}^{-1})^{sr}$$ if $s<0$.
The following corollary addresses the conjecture 4.3 of \cite{dePaiva:Unexpected}.
\begin{named}{Corollary~\ref{positiveTTT}} Let $p, q$ be positive coprime integers. Consider $p>q\geq r>1$. Then, for $k\geq 2$, the twisted torus knots $$T(p, kp + q; r, 1)\textrm{ and }T(kp + q, p; r, 1)$$ are hyperbolic. \end{named}
\begin{proof} It follows from the last corollary that the twisted torus knot $T(p, kp + q; r, 1)$ is hyperbolic since $T(p, kp + q; r, 1) = T((r, r), (p, kp + q))$. From [\cite{Knottypes}, lemmas 1,3,4,5], the twisted torus knot $T(kp + q, p; r, 1)$ is equivalent to the twisted torus knot $T(p, kp + q; r, 1)$. Therefore, $T(kp + q, p; r, 1)$ is also hyperbolic. \end{proof}
The author and Lee classified some negative twisted torus knots which are torus knots in \cite{LeeThiago}. The next corollary classifies some negative twisted torus knots which are hyperbolic.
\begin{named}{Corollary~\ref{negativeTTT}} Let $p, q$ be positive coprime integers with $p>q$. Consider $p-q\geq r>1$. Then, for $k\geq 3$, the twisted torus knots $$T(p, kp + q; r, -1)\textrm{ and }T(kp + q, p; r, -1)$$ are hyperbolic. \end{named}
\begin{proof} Suppose that $S^3-T(p, kp + q; r, -1)$ has an essential torus $T$. By Ito \cite{Ito}(see example 5.7), $T$ doesn't intersect the braid axis $C$ of $T(p, kp + q; r, -1)$. So, $T(p, kp + q; r, -1)$ is a generalized $b$-cabling of a knot $L$ with $b>0$, where $L$ is the core of the solid torus bounded by $T$. Thus, $b$ must divide $p$. After $(-1/(k+3))$-Dehn surgery along $C$, the knot $T(p, kp + q; r, -1)$ becomes the knot $K'$ given by the closure of the braid $B_{0, p}^{2p + p -q}B_{0, r}^{r}$ and the torus $T$ becomes a new torus $T'$ which doesn't intersect the braid axis of the last braid. The torus $T$ can't be trivial otherwise $b = p$ by lemma~\ref{lemma1}, which is not possible as $T$ wouldn't be essential in $S^3-T(p, kp + q; r, -1)$. So, $T$ is knotted and thus essential in $S^3-K'$. But, by the last corollary, the twisted torus knot $T(p, 3p -q; r, 1)$, which is given by the closure of the braid $B_{0, p}^{3p -q}B_{0, r}^{r}$, is hyperbolic, contradiction. Therefore, $T$ does not exist.
Consider now that the twisted torus knot $T(p, kp + q; r, -1)$ is a torus knot. By sending the crossings of the braid $(\sigma_{r-1}^{-1}\dots \sigma_{1}^{-1})^{r}$ clockwise around the braid closure, they are cancelled with some crossings of the first $r$ horizontal lines of $(\sigma_{1}\dots \sigma_{p-1})^{kp + q}$. After that, the total braid becomes a positive braid and has at least two full twists on $p$ strands. Thus, it has braid index equal to $p$ \cite{Franks}. So, $T(p, kp + q; r, -1)$ is a $(p, d)$-torus knot with $d>0$. From lemma~\ref{Los}, there is an isotopy in the complement of $C$ that takes $T(p, kp + q; r, -1)$ to $T(p, d)$. Then, we do $(-1/(k+3))$-Dehn surgery along $C$ to transform $T(p, kp + q; r, -1)$ into $T(p, 3p-q; r, 1)$. But, then $T(p, 3p-q; r, 1)$ would also be a $(p, d')$-torus knot with $d'>0$. But, $T(p, 3p-q; r, 1)$ can't be a torus knot due to the existence of the last corollary.
Finally, the twisted torus knot $T(kp + q, p; r, -1)$ is equivalent to the twisted torus knot $T(p, kp + q; r, -1)$ \cite{Knottypes}. Therefore, $T(kp + q, p; r, -1)$ is hyperbolic as well. \end{proof}
\end{document} |
\begin{document}
\allowdisplaybreaks
\title{On some extension of Gauss' work and applications}
\footnote{ 2010 \textit{Mathematics Subject Classification}. Primary 11E16; Secondary 11F03, 11G15, 11R37.} \footnote{ \textit{Key words and phrases}. binary quadratic forms, class field theory, complex multiplication, modular functions.} \footnote{ \thanks{ The first (corresponding) author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MIST) (2016R1A5A1008055 and 2017R1C1B2010652). The third author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2017R1A2B1006578), and by Hankuk University of Foreign Studies Research Fund of 2018.} }
\begin{abstract} Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$, and let $\tau_K$ be an element of the complex upper half-plane so that $\mathcal{O}_K=[\tau_K,\,1]$. For a positive integer $N$, let $\mathcal{Q}_N(d_K)$ be the set of primitive positive definite binary quadratic forms of discriminant $d_K$ with leading coefficients relatively prime to $N$. Then, with any congruence subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$ one can define an equivalence relation $\sim_\Gamma$ on $\mathcal{Q}_N(d_K)$. Let $\mathcal{F}_{\Gamma,\,\mathbb{Q}}$ denote the field of meromorphic modular functions for $\Gamma$ with rational Fourier coefficients. We show that the set of equivalence classes $\mathcal{Q}_N(d_K)/\sim_\Gamma$ can be equipped with a group structure isomorphic to $\mathrm{Gal}(K\mathcal{F}_{\Gamma,\,\mathbb{Q}}(\tau_K)/K)$ for some $\Gamma$, which generalizes further into the classical theory of complex multiplication over ring class fields. \end{abstract}
\title{On some extension of Gauss' work and applications}
\section {Introduction}
For a negative integer $D$ such that $D\equiv0$ or $1\Mod{4}$, let $\mathcal{Q}(D)$ be the set of primitive positive definite binary quadratic forms $Q(x,\,y)=ax^2+bxy+cy^2\in\mathbb{Z}[x,\,y]$ of discriminant $b^2-4ac=D$. The modular group $\mathrm{SL}_2(\mathbb{Z})$ (or $\mathrm{PSL}_2(\mathbb{Z})$) acts on the set $\mathcal{Q}(D)$ from the right and defines the proper equivalence $\sim$ as \begin{equation*} Q\sim Q'\quad\Longleftrightarrow\quad Q'=Q^\gamma =Q\left(\gamma\begin{bmatrix}x\\y\end{bmatrix}\right)~ \textrm{for some}~\gamma\in\mathrm{SL}_2(\mathbb{Z}). \end{equation*} In his celebrated work Disquisitiones Arithmeticae of 1801 (\cite{Gauss}), Gauss introduced the beautiful law of composition of integral binary quadratic forms. And, it seems that he first understood the set of equivalence classes $\mathrm{C}(D)=\mathcal{Q}(D)/\sim$ as a group. However, his original proof of the group structure is long and complicated to work in practice. After 93 years later Dirichlet (\cite{Dirichlet}) presented a different approach to the study of composition and genus theory, which seemed to be influenced by Legendre. (See \cite[$\S$3]{Cox}.) On the other hand, in 2004 Bhargava (\cite{Bhargava}) derived a wonderful general law of composition on $2\times2\times2$ cubes of integers, from which he was able to obtain Gauss' composition law on binary quadratic forms as a simple special case. Now, in this paper we will make use of Dirichlet's composition law to proceed the arguments. \par Given the order $\mathcal{O}$ of discriminant $D$ in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $I(\mathcal{O})$ be the group of proper fractional $\mathcal{O}$-ideals and $P(\mathcal{O})$ be its subgroup of nonzero principal $\mathcal{O}$-ideals. When $Q=ax^2+bxy+cy^2$ is an element of $\mathcal{Q}(D)$, let $\omega_Q$ be the zero of the quadratic polynomial $Q(x,\,1)$ in $\mathbb{H}
=\{\tau\in\mathbb{C}~|~\mathrm{Im}(\tau)>0\}$, namely \begin{equation}\label{wQ} \omega_Q=\frac{-b+\sqrt{D}}{2a}. \end{equation} It is well known that $[\omega_Q,\,1]=\mathbb{Z}\omega_Q+\mathbb{Z}$ is a proper fractional $\mathcal{O}$-ideal and the form class group $\mathrm{C}(D)$ under the Dirichlet composition is isomorphic to the $\mathcal{O}$-ideal class group $\mathrm{C}(\mathcal{O})=I(\mathcal{O})/P(\mathcal{O})$ through the isomorphism \begin{equation}\label{CDCO} \mathrm{C}(D)\stackrel{\sim}{\rightarrow}\mathrm{C}(\mathcal{O}),\quad [Q]\mapsto[[\omega_Q,\,1]]. \end{equation} On the other hand, if we let $H_\mathcal{O}$ be the ring class field of order $\mathcal{O}$ and $j$ be the elliptic modular function on lattices in $\mathbb{C}$, then we attain the isomorphism \begin{equation}\label{COGHK} \mathrm{C}(\mathcal{O})\stackrel{\sim}{\rightarrow}\mathrm{Gal}(H_\mathcal{O}/K), \quad [\mathfrak{a}]\mapsto(j(\mathcal{O}) \mapsto j(\overline{\mathfrak{a}})) \end{equation} by the theory of complex multiplication (\cite[Theorem 11.1 and Corollary 11.37]{Cox} or \cite[Theorem 5 in Chapter 10]{Lang87}). Thus, composing two isomorphisms given in (\ref{CDCO}) and (\ref{COGHK}) yields the isomorphism \begin{equation}\label{CDGHK} \mathrm{C}(D)\stackrel{\sim}{\rightarrow}\mathrm{Gal}(H_\mathcal{O}/K), \quad[Q]\mapsto(j(\mathcal{O})\mapsto j([-\overline{\omega}_Q,\,1])). \end{equation} \par Now, let $K$ be an imaginary quadratic field of discriminant $d_K$ and $\mathcal{O}_K$ be its ring of integers. If we set \begin{equation}\label{tauK} \tau_K=\left\{\begin{array}{ll} \sqrt{d_K}/2 & \textrm{if}~d_K\equiv0\Mod{4},\\ (-1+\sqrt{d_K})/2 & \textrm{if}~d_K\equiv1\Mod{4}, \end{array}\right. \end{equation} then we get $\mathcal{O}_K=[\tau_K,\,1]$. For a positive integer $N$ and $\mathfrak{n}=N\mathcal{O}_K$, let $I_K(\mathfrak{n})$ be the group of fractional ideals of $K$ relatively prime to $\mathfrak{n}$ and $P_K(\mathfrak{n})$ be its subgroup of principal fractional ideals. Furthermore, let \begin{eqnarray*}
P_{K,\,\mathbb{Z}}(\mathfrak{n})&=&\{\nu\mathcal{O}_K~|~ \nu\in K^*~\textrm{such that}~\nu\equiv^*m\Mod{\mathfrak{n}} ~\textrm{for some integer $m$ prime to $N$}\},\\
P_{K,\,1}(\mathfrak{n})&=&\{\nu\mathcal{O}_K~|~ \nu\in K^*~\textrm{such that}~\nu\equiv^*1\Mod{\mathfrak{n}}\} \end{eqnarray*} which are subgroups of $P_K(\mathfrak{n})$. As for the multiplicative congruence $\equiv^*$ modulo $\mathfrak{n}$, we refer to \cite[$\S$IV.1]{Janusz}. Then the ring class field $H_\mathcal{O}$ of order $\mathcal{O}$ with conductor $N$ in $K$ and the ray class field $K_\mathfrak{n}$ modulo $\mathfrak{n}$ are defined to be the unique abelian extensions of $K$ for which the Artin map modulo $\mathfrak{n}$ induces the isomorphisms \begin{equation*} I_K(\mathfrak{n})/P_{K,\,\mathbb{Z}}(\mathfrak{n}) \simeq\mathrm{Gal}(H_\mathcal{O}/K) \quad\textrm{and}\quad I_K(\mathfrak{n})/P_{K,\,1}(\mathfrak{n}) \simeq\mathrm{Gal}(K_\mathfrak{n}/K), \end{equation*} respectively (\cite[$\S$8 and $\S$9]{Cox} and \cite[Chapter V]{Janusz}). And, for a congruence subgroup $\Gamma$ of level $N$ in $\mathrm{SL}_2(\mathbb{Z})$, let $\mathcal{F}_{\Gamma,\,\mathbb{Q}}$ be the field of meromorphic modular functions for $\Gamma$ whose Fourier expansions with respect to $q^{1/N}=e^{2\pi\mathrm{i}\tau/N}$ have rational coefficients and let \begin{equation*} K\mathcal{F}_{\Gamma,\,\mathbb{Q}}(\tau_K)
=K(h(\tau_K)~|~h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}~ \textrm{is finite at}~\tau_K). \end{equation*} Then it is a subfield of the maximal abelian extension $K^\mathrm{ab}$ of $K$ (\cite[Theorem 6.31 (i)]{Shimura}). In particular, for the congruence subgroups \begin{eqnarray*}
\Gamma_0(N)&=&\left\{\gamma\in\mathrm{SL}_2(\mathbb{Z})~|~ \gamma\equiv\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ 0&\mathrm{*}\end{bmatrix}\Mod{N M_2(\mathbb{Z})}\right\},\\
\Gamma_1(N)&=&\left\{\gamma\in\mathrm{SL}_2(\mathbb{Z})~|~ \gamma\equiv\begin{bmatrix}1&\mathrm{*}\\ 0&1\end{bmatrix}\Mod{N M_2(\mathbb{Z})}\right\}, \end{eqnarray*} we know that \begin{equation}\label{specialization} H_\mathcal{O}=K\mathcal{F}_{\Gamma_0(N),\,\mathbb{Q}}(\tau_K) \quad\textrm{and}\quad K_\mathfrak{n}=K\mathcal{F}_{\Gamma_1(N),\,\mathbb{Q}}(\tau_K) \end{equation} (\cite[Corollary 5.2]{C-K} and \cite[Theorem 3.4]{K-S13}). On the other hand, one can naturally defines an equivalence relation $\sim_\Gamma$ on the subset \begin{equation}\label{QNdK} \mathcal{Q}_N(d_K)=
\{ax^2+bxy+cy^2\in\mathcal{Q}(d_K)~|~\gcd(N,\,a)=1\} \end{equation} of $\mathcal{Q}(d_K)$ by \begin{equation}\label{simG} Q\sim_\Gamma Q'\quad\Longleftrightarrow\quad Q'=Q^\gamma~\textrm{for some}~\gamma\in\Gamma. \end{equation} Observe that $\Gamma$ may not act on $\mathcal{Q}_N(d_K)$. Here, by $Q^\gamma$ we mean the action of $\gamma$ as an element of $\mathrm{SL}_2(\mathbb{Z})$. \par For a subgroup $P$ of $I_K(\mathfrak{n})$ with $P_{K,\,1}(\mathfrak{n})\subseteq P\subseteq P_K(\mathfrak{n})$, let $K_P$ be the abelian extension of $K$ so that $I_K(\mathfrak{n})/P\simeq\mathrm{Gal}(K_P/K)$. In this paper, motivated by (\ref{CDGHK}) and (\ref{specialization}) we shall present several pairs of $P$ and $\Gamma$ for which \begin{itemize} \item[(i)] $K_P=K\mathcal{F}_{\Gamma,\,\mathbb{Q}}(\tau_K)$, \item[(ii)] $\mathcal{Q}_N(d_K)/\sim_\Gamma$ becomes a group isomorphic to $\mathrm{Gal}(K_P/K)$ via the isomorphism \begin{equation}\label{desiredisomorphism} \begin{array}{ccl} \mathcal{Q}_N(d_K)/\sim_\Gamma&\stackrel{\sim}{\rightarrow}& \mathrm{Gal}(K_P/K)\\
\left[Q\right]&\mapsto&(h(\tau_K)\mapsto h(-\overline{\omega}_Q)~|~ h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}~ \textrm{is finite at}~\tau_K) \end{array} \end{equation} \end{itemize} (Propositions \ref{KPKF}, \ref{satisfies} and Theorems \ref{formclassgroup}, \ref{Galoisgroups}). This result would be certain extension of Gauss' original work. We shall also develop an algorithm of finding distinct form classes in $\mathcal{Q}_N(d_K)/\sim_\Gamma$ and give a concrete example (Proposition \ref{algorithm} and Example \ref{example}). To this end, we shall apply Shimura's theory which links the class field theory for imaginary quadratic fields and the theory of modular functions (\cite[Chapter 6]{Shimura}). And, we shall not only use but also improve the ideas of our previous work \cite{E-K-S17}. See Remarks \ref{difference}.
\section {Extended form class groups as ideal class groups}\label{sect2}
Let $K$ be an imaginary quadratic field of discriminant $d_K$ and $\tau_K$ be as in (\ref{tauK}). And, let $N$ be a positive integer, $\mathfrak{n}=N\mathcal{O}_K$ and $P$ be a subgroup of $I_K(\mathfrak{n})$ satisfying $P_{K,\,1}(\mathfrak{n})\subseteq P\subseteq P_K(\mathfrak{n})$. Each subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$ defines an equivalence relation $\sim_\Gamma$ on the set $\mathcal{Q}_N(d_K)$ described in (\ref{QNdK}) in the same manner as in (\ref{simG}). In this section, we shall present a necessary and sufficient condition for $\Gamma$ in such a way that \begin{eqnarray*} \phi_\Gamma:\mathcal{Q}_N(d_K)/\sim_\Gamma&\rightarrow& I_K(\mathfrak{n})/P\\ \left[Q\right]~~~&\mapsto&[[\omega_Q,\,1]] \end{eqnarray*} becomes a well-defined bijection with $\omega_Q$ as in (\ref{wQ}). As mentioned in $\S$1, the lattice $[\omega_Q,\,1]=\mathbb{Z}\omega_Q+\mathbb{Z}$ is a fractional ideal of $K$. \par The modular group $\mathrm{SL}_2(\mathbb{Z})$ acts on $\mathbb{H}$ from the left by fractional linear transformations. For each $Q\in\mathcal{Q}(d_K)$, let $I_{\omega_Q}$ denote the isotropy subgroup of the point $\omega_Q$ in $\mathrm{SL}_2(\mathbb{Z})$. In particular, if we let $Q_0$ be the principal form in $\mathcal{Q}(d_K)$ (\cite[p. 31]{Cox}), then we have $\omega_{Q_0}=\tau_K$ and \begin{equation}\label{isotropy} I_{\omega_{Q_0}}=\left\{ \begin{array}{ll} \left\{\pm I_2\right\} & \textrm{if}~d_K\neq-4,\,-3,
\\ \left\{\pm I_2,\,\pm S\right\} & \textrm{if}~d_K=-4,
\\ \left\{\pm I_2,\,\pm ST,\, \pm(ST)^2\right\} & \textrm{if}~d_K=-3 \end{array} \right. \end{equation} where $S=\begin{bmatrix} 0&-1\\1&0\end{bmatrix}$ and $T=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}$. Furthermore, we see that \begin{equation}\label{otherisotropy} I_{\omega_Q}=\{\pm I_2\}\quad \textrm{if $\omega_Q$ is not equivalent to $\omega_{Q_0}$ under $\mathrm{SL}_2(\mathbb{Z})$} \end{equation} (\cite[Proposition 1.5 (c)]{Silverman}). For any $\gamma=\begin{bmatrix}a&b\\c&d\end{bmatrix}\in \mathrm{SL}_2(\mathbb{Z})$, let \begin{equation*} j(\gamma,\,\tau)=c\tau+d\quad(\tau\in\mathbb{H}). \end{equation*} One can readily check that if $Q'=Q^\gamma$, then \begin{equation*} \omega_Q=\gamma(\omega_{Q'})\quad\textrm{and} \quad[\omega_Q,\,1]=\frac{1}{j(\gamma,\,\omega_{Q'})} [\omega_{Q'},\,1]. \end{equation*}
\begin{lemma}\label{primein} Let $Q=ax^2+bxy+cy^2\in\mathcal{Q}(d_K)$. Then $\mathrm{N}_{K/\mathbb{Q}}([\omega_Q,\,1])=1/a$ and \begin{equation*} [\omega_Q,\,1]\in I_K(\mathfrak{n})\quad\Longleftrightarrow\quad Q\in\mathcal{Q}_N(d_K). \end{equation*} \end{lemma} \begin{proof} See \cite[Lemma 2.3 (iii)]{E-K-S17}. \end{proof}
\begin{lemma}\label{prime} Let $Q=ax^2+bxy+cy^2\in\mathcal{Q}_N(d_K)$. \begin{enumerate} \item[\textup{(i)}] For $u,\,v\in\mathbb{Z}$ not both zero, the fractional ideal $(u\omega_Q+v)\mathcal{O}_K$ is relatively prime to $\mathfrak{n}=N\mathcal{O}_K$ if and only if $\gcd(N,\,Q(v,\,-u))=1$. \item[\textup{(ii)}] If $C\in P_K(\mathfrak{n})/P$, then \begin{equation*} C=[(u\omega_Q+v)\mathcal{O}_K]\quad \textrm{for some}~u,\,v\in\mathbb{Z}~\textrm{not both zero such that}~ \gcd(N,\,Q(v,\,-u))=1. \end{equation*} \end{enumerate} \end{lemma} \begin{proof} \begin{enumerate} \item[(i)] See \cite[Lemma 4.1]{E-K-S17} \item[(ii)] Since $P_K(\mathfrak{n})/P$ is a finite group, one can take an integral ideal $\mathfrak{c}$ in the class $C$ (\cite[Lemma 2.3 in Chapter IV]{Janusz}). Furthermore, since $\mathcal{O}_K=[a\omega_Q,\,1]$, we may express $\mathfrak{c}$ as \begin{equation*} \mathfrak{c}=(ka\omega_Q+v)\mathcal{O}_K\quad \textrm{for some}~k,\,v\in\mathbb{Z}. \end{equation*} If we set $u=ka$, then we attain (ii) by (i). \end{enumerate} \end{proof}
\begin{proposition}\label{surjective} If the map $\phi_\Gamma$ is well defined, then it is surjective. \end{proposition} \begin{proof} Let \begin{equation*} \rho: I_K(\mathfrak{n})/P \rightarrow I_K(\mathcal{O}_K)/P_K(\mathcal{O}_K) \end{equation*} be the natural homomorphism. Since $I_K(\mathfrak{n})/P_K(\mathfrak{n})$ is isomorphic to $I_K(\mathcal{O}_K)/P_K(\mathcal{O}_K)$ (\cite[Proposition 1.5 in Chapter IV]{Janusz}), the homomorphism $\rho$ is surjective. Here, we refer to the following commutative diagram: \begin{figure}
\caption{A commutative diagram of ideal class groups}
\label{diagram}
\end{figure} \noindent Let \begin{equation*} Q_1,\,Q_2,\,\ldots,\,Q_h\quad(\in\mathcal{Q}(d_K)) \end{equation*} be reduced forms which represent all distinct classes in $\mathrm{C}(d_K)=\mathcal{Q}(d_K)/\sim$ (\cite[Theorem 2.8]{Cox}). Take $\gamma_1,\,\gamma_2,\,\ldots,\,\gamma_h\in\mathrm{SL}_2(\mathbb{Z})$ so that \begin{equation*} Q_i'=Q_i^{\gamma_i}\quad(i=1,\,2,\,\ldots,\,h) \end{equation*} belongs to $\mathcal{Q}_N(d_K)$ (\cite[Lemmas 2.3 and 2.25]{Cox}). Then we get \begin{equation*} I_K(\mathcal{O}_K)/P_K(\mathcal{O}_K)=
\{[\omega_{Q_i'},\,1]P_K(\mathcal{O}_K)~|~i=1,\,2,\,\ldots,\,h\} \quad\textrm{and}\quad[\omega_{Q_i'},\,1]\in I_K(\mathfrak{n}) \end{equation*} by the isomorphism given in (\ref{CDCO}) (when $D=d_K)$ and Lemma \ref{primein}. Moreover, since $\rho$ is a surjection with $\mathrm{Ker}(\rho)=P_K(\mathfrak{n})/P$, we obtain the decomposition \begin{equation}\label{decomp} I_K(\mathfrak{n})/P=(P_K(\mathfrak{n})/P)\cdot
\{[[\omega_{Q_i'},\,1]]\in I_K(\mathfrak{n})/P~|~i=1,\,2,\,\ldots,\,h\}. \end{equation} \par Now, let $C\in I_K(\mathfrak{n})/P$. By the decomposition (\ref{decomp}) and Lemma \ref{prime} (ii) we may express $C$ as \begin{equation}\label{C} C=\left[\frac{1}{u\omega_{Q_i'}+v}\,[\omega_{Q_i'},\,1]\right] \end{equation} for some $i\in\{1,\,2,\,\ldots,\,h\}$ and $u,\,v\in\mathbb{Z}$ not both zero with $\gcd(N,\,Q_i'(v,\,-u))=1$. Take any $\sigma=\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ \widetilde{u}&\widetilde{v}\end{bmatrix}\in\mathrm{SL}_2(\mathbb{Z})$ such that $\sigma\equiv\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ u&v\end{bmatrix}\Mod{N M_2(\mathbb{Z})}$. We then derive that \begin{eqnarray*} C&=&\left[\frac{u\omega_{Q_i'}+v}{\widetilde{u}\omega_{Q_i'}+\widetilde{v}} \,\mathcal{O}_K\right]C\quad \textrm{because}~ \frac{u\omega_{Q_i'}+v}{\widetilde{u}\omega_{Q_i'}+\widetilde{v}} \equiv^*1\Mod{\mathfrak{n}}~\textrm{and}~P_{K,\,1}(\mathfrak{n}) \subseteq P\\ &=&\left[\frac{1}{\widetilde{u}\omega_{Q_i'}+\widetilde{v}} \,[\omega_{Q_i'},\,1]\right]\quad\textrm{by (\ref{C})}\\ &=&\left[\frac{1}{j(\sigma,\,\omega_{Q_i'})}\,[\omega_{Q_i'},\,1] \right]\\ &=&[[\sigma(\omega_{Q_i'}),\,1]]. \end{eqnarray*} Thus, if we put $Q=Q_i'^{\sigma^{-1}}$, then we obtain \begin{equation*} C=[[\omega_Q,\,1]]=\phi_\Gamma([Q]). \end{equation*} This prove that $\phi_\Gamma$ is surjective. \end{proof}
\begin{proposition}\label{injective} The map $\phi_\Gamma$ is a well-defined injection if and only if $\Gamma$ satisfies the following property: \begin{equation}\label{P} \begin{array}{c} \textrm{Let $Q\in\mathcal{Q}_N(d_K)$ and $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ such that $Q^{\gamma^{-1}}\in\mathcal{Q}_N(d_K)$. Then,}\\ j(\gamma,\,\omega_Q)\mathcal{O}_K\in P~\Longleftrightarrow~ \gamma\in\Gamma\cdot I_{\omega_Q}. \end{array} \end{equation} \end{proposition} \begin{proof} Assume first that $\phi_\Gamma$ is a well-defined injection. Let $Q\in\mathcal{Q}_N(d_K)$ and $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ such that $Q^{\gamma^{-1}}\in\mathcal{Q}_N(d_K)$. If we set $Q'=Q^{\gamma^{-1}}$, then we have $Q=Q'^\gamma$ and so \begin{equation}\label{wjw1} [\omega_{Q'},\,1]=[\gamma(\omega_Q),\,1]=\frac{1}{j(\gamma,\,\omega_Q)}[\omega_Q,\,1]. \end{equation} And, we deduce that \begin{eqnarray*} j(\gamma,\,\omega_Q)\mathcal{O}_K\in P&\Longleftrightarrow& [[\omega_Q,\,1]]= [[\omega_{Q'},\,1]]~\textrm{in}~I_K(\mathfrak{n})/P~ \textrm{by Lemma \ref{primein} and (\ref{wjw1})}\\ &\Longleftrightarrow&\phi_\Gamma([Q])=\phi_\Gamma([Q'])~ \textrm{by the definition of $\phi_\Gamma$}\\ &\Longleftrightarrow&[Q]=[Q']~\textrm{in $\mathcal{Q}_N(d_K)/\sim_\Gamma$ since $\phi_\Gamma$ is injective}\\ &\Longleftrightarrow&Q'=Q^\alpha~\textrm{for some}~\alpha\in\Gamma\\ &\Longleftrightarrow&Q=Q^{\alpha\gamma}~\textrm{for some}~ \alpha\in\Gamma~\textrm{because}~Q'=Q^{\gamma^{-1}}\\ &\Longleftrightarrow&\alpha\gamma\in I_{\omega_Q}~ \textrm{for some}~\alpha\in\Gamma\\ &\Longleftrightarrow&\gamma\in\Gamma\cdot I_{\omega_Q}. \end{eqnarray*} Hence $\Gamma$ satisfies the property (\ref{P}). \par Conversely, assume that $\Gamma$ satisfies the property (\ref{P}). To show that $\phi_\Gamma$ is well defined, suppose that \begin{equation*} [Q]=[Q']\quad\textrm{in}~\mathcal{Q}_N(d_K)/\sim_\Gamma ~\textrm{for some}~Q,\,Q'\in\mathcal{Q}_N(d_K). \end{equation*} Then we attain $Q=Q'^\alpha$ for some $\alpha\in\Gamma$ so that \begin{equation}\label{wjw2} [\omega_{Q'},\,1]=[\alpha(\omega_Q),\,1]=\frac{1}{j(\alpha,\,\omega_Q)}[\omega_Q,\,1]. \end{equation} Now that $Q^{\alpha^{-1}}=Q'\in\mathcal{Q}_N(d_K)$ and $\alpha\in\Gamma\subseteq\Gamma\cdot I_{\omega_Q}$, we achieve by the property (\ref{P}) that $j(\alpha,\,\omega_{Q})\mathcal{O}_K\in P$. Thus we derive by Lemma \ref{primein} and (\ref{wjw2}) that \begin{equation*} [[\omega_Q,\,1]]=[[\omega_{Q'},\,1]] \quad\textrm{in}~I_K(\mathfrak{n})/P, \end{equation*} which claims that $\phi_\Gamma$ is well defined. \par On the other hand, in order to show that $\phi_\Gamma$ is injective, assume that \begin{equation*} \phi_\Gamma([Q])=\phi_\Gamma([Q'])\quad \textrm{for some}~Q,\,Q'\in\mathcal{Q}_N(d_K). \end{equation*} Then we get \begin{equation}\label{wlw} [\omega_Q,\,1]=\lambda[\omega_{Q'},\,1]\quad \textrm{for some}~\lambda\in K^*~\textrm{such that}~ \lambda\mathcal{O}_K\in P, \end{equation} from which it follows that \begin{equation}\label{QQb} Q=Q'^\gamma\quad\textrm{for some}~\gamma\in\mathrm{SL}_2(\mathbb{Z}) \end{equation} by the isomorphism in (\ref{CDCO}) when $D=d_K$. We then derive by (\ref{wlw}) and (\ref{QQb}) that \begin{equation*} [\omega_{Q'},\,1]=[\gamma(\omega_Q),\,1]= \frac{1}{j(\gamma,\,\omega_{Q})} [\omega_Q,\,1]=\frac{\lambda}{j(\gamma,\,\omega_Q)}[\omega_{Q'},\,1] \end{equation*} and so $\lambda/j(\gamma,\,\omega_Q)\in\mathcal{O}_K^*$. Therefore we attain \begin{equation*} j(\gamma,\,\omega_Q)\mathcal{O}_K=\lambda\mathcal{O}_K\in P, \end{equation*} and hence $\gamma\in\Gamma\cdot I_{\omega_Q}$ by the fact $Q^{\gamma^{-1}}=Q'\in\mathcal{Q}_N(d_K)$ and the property (\ref{P}). If we write \begin{equation*} \gamma=\alpha\beta\quad\textrm{for some}~\alpha\in\Gamma~ \textrm{and}~\beta\in I_{\omega_Q}, \end{equation*} then we see by (\ref{QQb}) that \begin{equation*} Q=Q^{\beta^{-1}}=Q^{\gamma^{-1}\alpha}=Q'^\alpha. \end{equation*} This shows that \begin{equation*} [Q]=[Q']\quad\textrm{in}~\mathcal{Q}_N(d_K)/\sim_\Gamma, \end{equation*} which proves the injectivity of $\phi_\Gamma$. \end{proof}
\begin{theorem}\label{formclassgroup} The map $\phi_\Gamma$ is a well-defined bijection if and only if $\Gamma$ satisfies the property \textup{(\ref{P})} stated in \textup{Proposition \ref{injective}}. In this case, we may regard the set $\mathcal{Q}_N(d_K)/\sim_\Gamma$ as a group isomorphic to the ideal class group $I_K(\mathfrak{n})/P$. \end{theorem} \begin{proof} We achieve the first assertion by Propositions \ref{surjective} and \ref{injective}. Thus, in this case, one can give a group structure on $\mathcal{Q}_N(d_K)/\sim_\Gamma$ through the bijection $\phi_\Gamma: \mathcal{Q}_N(d_K)/\sim_\Gamma\rightarrow I_K(\mathfrak{n})/P$. \end{proof}
\begin{remark} By using the isomorphism given in (\ref{CDCO}) (when $D=d_K$) and Theorem \ref{formclassgroup} we obtain the following commutative diagram: \begin{figure}
\caption{The natural map between form class groups}
\label{diagram2}
\end{figure}
\noindent Therefore the natural map $\mathcal{Q}_N(d_K)/\sim_\Gamma\rightarrow\mathrm{C}(d_K)$ is indeed a surjective homomorphism, which shows that the group structure of $\mathcal{Q}_N(d_K)/\sim\Gamma$ is not far from that of the classical form class group $\mathrm{C}(d_K)$. \end{remark}
\section {Class field theory over imaginary quadratic fields}
In this section, we shall briefly review the class field theory over imaginary quadratic fields established by Shimura. \par For an imaginary quadratic field $K$, let $\mathbb{I}_K^\mathrm{fin}$ be the group of finite ideles of $K$ given by the restricted product \begin{eqnarray*} \mathbb{I}_K^\mathrm{fin}&=&{\prod_{\mathfrak{p}}}^\prime K_\mathfrak{p}^*\quad \textrm{where $\mathfrak{p}$ runs over all prime ideals of $\mathcal{O}_K$}\\
&=&\left\{s=(s_\mathfrak{p})\in\prod_\mathfrak{p}K_\mathfrak{p}^*~|~ s_\mathfrak{p}\in\mathcal{O}_{K_\mathfrak{p}}^*~\textrm{for all but finitely many $\mathfrak{p}$}\right\}. \end{eqnarray*} As for the topology on $\mathbb{I}_K^\mathrm{fin}$ one can refer to \cite[p. 78]{Neukirch}. Then, the classical class field theory of $K$ is explained by the exact sequence \begin{equation*} 1\rightarrow K^*\rightarrow \mathbb{I}_K^\mathrm{fin}\rightarrow\mathrm{Gal}(K^\mathrm{ab}/K) \rightarrow 1 \end{equation*} where $K^*$ maps into $\mathbb{I}_K^\mathrm{fin}$ through the diagonal embedding $\nu\mapsto(\nu,\,\nu,\,\nu,\,\ldots)$ (\cite[Chapter IV]{Neukirch}). Setting \begin{equation*} \mathcal{O}_{K,\,p}=\mathcal{O}_K\otimes_\mathbb{Z} \mathbb{Z}_p\quad\textrm{for each prime $p$} \end{equation*} we have \begin{equation*} \mathcal{O}_{K,\,p}
\simeq\prod_{\mathfrak{p}\,|\,p} \mathcal{O}_{K_\mathfrak{p}} \end{equation*} (\cite[Proposition 4 in Chapter II]{Serre}). Furthermore, if we let $\widehat{K}=K\otimes_\mathbb{Z}\widehat{\mathbb{Z}}$ with $\widehat{\mathbb{Z}}=\prod_p\mathbb{Z}_p$, then \begin{eqnarray*} \widehat{K}^*&=&{\prod_{p}}^\prime(K\otimes_\mathbb{Z} \mathbb{Z}_p)^*\quad\textrm{where $p$ runs over all rational primes}\\ &=&\left\{s=(s_p)\in\prod_p(K\otimes_\mathbb{Z}
\mathbb{Z}_p)^*~|~ s_p\in\mathcal{O}_{K,\,p}^*~ \textrm{for all but finitely many $p$}\right\}\\ &\simeq&\mathbb{I}_K^\mathrm{fin} \end{eqnarray*} (\cite[Exercise 15.12]{Cox} and \cite[Chapter II]{Serre}). Thus we may use $\widehat{K}^*$ instead of $\mathbb{I}_K^\mathrm{fin}$ when we develop the class field theory of $K$.
\begin{proposition}\label{1-1} There is a one-to-one correspondence via the Artin map between closed subgroups $J$ of $\widehat{K}^*$ of finite index containing $K^*$ and finite abelian extensions $L$ of $K$ such that \begin{equation*} \widehat{K}^*/J\simeq\mathrm{Gal}(L/K). \end{equation*} \end{proposition} \begin{proof} See \cite[Chapter IV]{Neukirch}. \end{proof}
Let $N$ be a positive integer, $\mathfrak{n}=N\mathcal{O}_K$ and $s=(s_p)\in \widehat{K}^*$. For a prime $p$ and a prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ lying above $p$, let $n_\mathfrak{p}(s)$ be a unique integer such that $s_p\in \mathfrak{p}^{n_\mathfrak{p}(s)}\mathcal{O}_{K_\mathfrak{p}}^*$. We then regard $s\mathcal{O}_K$ as the fractional ideal \begin{equation*} s\mathcal{O}_K=
\prod_p\prod_{\mathfrak{p}\,|\,p} \mathfrak{p}^{n_\mathfrak{p}(s)}\in I_K(\mathcal{O}_K). \end{equation*} By the approximation theorem (\cite[Chapter IV]{Janusz}) one can take an element $\nu_s$ of $K^*$ such that \begin{equation}\label{k}
\nu_ss_p\in 1+N\mathcal{O}_{K,\,p}\quad\textrm{for all}~p\,|\,N. \end{equation}
\begin{proposition}\label{rayidele} We get a well-defined surjective homomorphism \begin{eqnarray*} \phi_\mathfrak{n}~:~\widehat{K}^*&\rightarrow& I_K(\mathfrak{n})/P_{K,\,1}(\mathfrak{n})\\ s~~&\mapsto&~~~[\nu_ss\mathcal{O}_K] \end{eqnarray*} with kernel \begin{equation*} J_\mathfrak{n}= K^*\left(
\prod_{p\,|\,N}(1+N\mathcal{O}_{K,\,p})\times \prod_{p\,\nmid\,N}\mathcal{O}_{K,\,p}^*\right). \end{equation*} Thus $J_\mathfrak{n}$ corresponds to the ray class field $K_\mathfrak{n}$. \end{proposition} \begin{proof} See \cite[Exercises 15.17 and 15.18]{Cox}. \end{proof}
Let $\mathcal{F}_N$ be the field of meromorphic modular functions of level $N$ whose Fourier expansions with respect to $q^{1/N}$ have coefficients in the $N$th cyclotomic field $\mathbb{Q}(\zeta_N)$ with $\zeta_N=e^{2\pi\mathrm{i}/N}$. Then $\mathcal{F}_N$ is a Galois extension of $\mathcal{F}_1$ with $\mathrm{Gal}(\mathcal{F}_N/\mathcal{F}_1)\simeq \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$ (\cite[Chapter 6]{Shimura}).
\begin{proposition}\label{Galoisdecomposition} There is a decomposition \begin{equation*} \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}
=\left\{\pm\begin{bmatrix}1&0\\0&d\end{bmatrix}~|~ d\in(\mathbb{Z}/N\mathbb{Z})^*\right\}/\{\pm I_2\} \cdot\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}. \end{equation*} Let $h(\tau)$ be an element of $\mathcal{F}_N$ whose Fourier expansion is given by \begin{equation*} h(\tau)=\sum_{n\gg-\infty}c_nq^{n/N}\quad(c_n\in\mathbb{Q}(\zeta_N)). \end{equation*} \begin{enumerate} \item[\textup{(i)}] If $\alpha=\begin{bmatrix}1&0\\0&d\end{bmatrix}$ with $d\in(\mathbb{Z}/N\mathbb{Z})^*$, then \begin{equation*} h(\tau)^\alpha= \sum_{n\gg-\infty}c_n^{\sigma_d}q^{n/N} \end{equation*} where $\sigma_d$ is the automorphism of $\mathbb{Q}(\zeta_N)$ defined by $\zeta_N^{\sigma_d}=\zeta_N^d$. \item[\textup{(ii)}] If $\beta\in\mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$, then \begin{equation*} h(\tau)^\beta=h(\gamma(\tau)) \end{equation*} where $\gamma$ is any element of $\mathrm{SL}_2(\mathbb{Z})$ which maps to $\beta$ through the reduction $\mathrm{SL}_2(\mathbb{Z})\rightarrow \mathrm{SL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$. \end{enumerate} \end{proposition} \begin{proof} See \cite[Proposition 6.21]{Shimura}. \end{proof}
If we let $\widehat{\mathbb{Q}}=\mathbb{Q}\otimes_\mathbb{Z}\widehat{\mathbb{Z}}$ and $\displaystyle\mathcal{F}=\bigcup_{N=1}^\infty\mathcal{F}_N$, then we attain the exact sequence \begin{equation*} 1\rightarrow\mathbb{Q}^*\rightarrow\mathrm{GL}_2(\widehat{\mathbb{Q}}) \rightarrow\mathrm{Gal}(\mathcal{F}/\mathbb{Q})\rightarrow1 \end{equation*} (\cite[Chaper 7]{Lang87} or \cite[Chapter 6]{Shimura}). Here, we note that \begin{eqnarray*} \mathrm{GL}_2(\widehat{\mathbb{Q}})&=& {\prod_p}^\prime\mathrm{GL}_2(\mathbb{Q}_p) \quad\textrm{where $p$ runs over all rational primes}\\ &=&
\{\gamma=(\gamma_p)\in\prod_p\mathrm{GL}_2(\mathbb{Q}_p)~|~ \gamma_p\in\mathrm{GL}_2(\mathbb{Z}_p)~ \textrm{for all but finitely many $p$}\} \end{eqnarray*} (\cite[Exercise 15.4]{Cox}) and $\mathbb{Q}^*$ maps into $\mathrm{GL}_2(\widehat{\mathbb{Q}})$ through the diagonal embedding. For $\omega\in K\cap\mathbb{H}$, we define a normalized embedding \begin{equation*} q_\omega:K^*\rightarrow\mathrm{GL}_2^+(\mathbb{Q}) \end{equation*} by the relation \begin{equation}\label{defq} \nu\begin{bmatrix}\tau_K\\1\end{bmatrix}= q_\omega(\nu)\begin{bmatrix}\tau_K\\1\end{bmatrix}\quad(\nu\in K^*). \end{equation} By continuity, $q_\omega$ can be extended to an embedding \begin{equation*} q_{\omega,\,p}:(K\otimes_\mathbb{Z}\mathbb{Z}_p)^*\rightarrow\mathrm{GL}_2(\mathbb{Q}_p) \quad\textrm{for each prime $p$} \end{equation*} and hence to an embedding \begin{equation*} q_\omega:\widehat{K}^*\rightarrow\mathrm{GL}_2(\widehat{\mathbb{Q}}). \end{equation*} Let $\min(\tau_K,\,\mathbb{Q})=x^2+b_Kx+c_K$ ($\in\mathbb{Z}[x]$). Since $K\otimes_\mathbb{Z}\mathbb{Z}_p=\mathbb{Q}_p\tau_K+\mathbb{Q}_p$ for each prime $p$, one can deduce that if $s=(s_p)\in\widehat{K}^*$ with $s_p=u_p\tau_K+v_p$ ($u_p,\,v_p\in\mathbb{Q}_p$), then \begin{equation}\label{gamma_p} q_{\tau_K}(s)=(\gamma_p)\quad\textrm{with}~\gamma_p= \begin{bmatrix}v_p-b_Ku_p & -c_Ku_p\\u_p&v_p\end{bmatrix}. \end{equation} \par By utilizing the concept of canonical models of modular curves, Shimura achieved the following remarkable results.
\begin{proposition}[Shimura's reciprocity law]\label{reciprocity} Let $s\in\widehat{K}^*$, $\omega\in K\cap\mathbb{H}$ and $h\in\mathcal{F}$ be finite at $\omega$. Then $h(\omega)$ lies in $K^\mathrm{ab}$ and satisfies \begin{equation*}
h(\omega)^{[s^{-1},\,K]}=h(\tau)^{q_\omega(s)}|_{\tau=\omega} \end{equation*} where $[\,\cdot,\,K]$ is the Artin map for $K$. \end{proposition} \begin{proof} See \cite[Theorem 6.31 (i)]{Shimura}. \end{proof}
\begin{proposition}\label{model} Let $S$ be an open subgroup of $\mathrm{GL}_2 (\widehat{\mathbb{Q}})$ containing $\mathbb{Q}^*$ such that $S/\mathbb{Q}^*$ is compact. Let \begin{eqnarray*} \Gamma_S&=&S\cap\mathrm{GL}_2^+(\mathbb{Q}),\\
\mathcal{F}_S&=&\{h\in\mathcal{F}~|~h^\gamma=h~\textrm{for all}~\gamma\in S\},\\
k_S&=&\{\nu\in\mathbb{Q}^\mathrm{ab}~|~ \nu^{[s,\,\mathbb{Q}]}=\nu~\textrm{for all}~ s\in\mathbb{Q}^*\det(S) \subseteq\widehat{\mathbb{Q}}^*\} \end{eqnarray*} where $[\,\cdot,\,\mathbb{Q}]$ is the Artin map for $\mathbb{Q}$. Then, \begin{enumerate} \item[\textup{(i)}] $\Gamma_S/\mathbb{Q}^*$ is a Fuchsian group of the first kind commensurable with $\mathrm{SL}_2(\mathbb{Z})/\{\pm I_2\}$. \item[\textup{(ii)}] $\mathbb{C}\mathcal{F}_S$ is the field of meromorphic modular functions for $\Gamma_S/\mathbb{Q}^*$. \item[\textup{(iii)}] $k_S$ is algebraically closed in $\mathcal{F}_S$. \item[\textup{(iv)}] If $\omega\in K\cap\mathbb{H}$, then the subgroup $K^*q_\omega^{-1}(S)$ of $\widehat{K}^*$ corresponds to the subfield \begin{equation*}
K\mathcal{F}_S(\omega)=K(h(\omega)~|~h\in\mathcal{F}_S~ \textrm{is finite at}~\omega) \end{equation*} of $K^\mathrm{ab}$ in view of \textup{Proposition \ref{1-1}}. \end{enumerate} \end{proposition} \begin{proof} See \cite[Propositions 6.27 and 6.33]{Shimura}. \end{proof}
\begin{remark}\label{overQ} In particular, if $k_S=\mathbb{Q}$, then $\mathcal{F}_S=\mathcal{F}_{\Gamma_S,\,\mathbb{Q}}$ (\cite[Exercise 6.26]{Shimura}). \end{remark}
\section {Construction of class invariants}\label{classinvariants}
Let $K$ be an imaginary quadratic field, $N$ be a positive integer and $\mathfrak{n}=N\mathcal{O}_K$. From now on, let $T$ be a subgroup of $(\mathbb{Z}/N\mathbb{Z})^*$ and $P$ be a subgroup of $P_K(\mathfrak{n})$ containing $P_{K,\,1}(\mathfrak{n})$ given by \begin{eqnarray*}
P&=&\langle\nu\mathcal{O}_K~|~ \nu\in\mathcal{O}_K-\{0\}~\textrm{such that}~\nu\equiv t\Mod{\mathfrak{n}}~ \textrm{for some}~t\in T\rangle\\
&=&\{\lambda\mathcal{O}_K~|~\lambda\in K^*~ \textrm{such that}~\lambda\equiv^*t\Mod{\mathfrak{n}}~ \textrm{for some}~t\in T\}.\nonumber \end{eqnarray*} Let $\mathrm{Cl}(P)$ denote the ideal class group \begin{equation*} \mathrm{Cl}(P)=I_K(\mathfrak{n})/P \end{equation*} and $K_P$ be its corresponding class field of $K$ with $\mathrm{Cl}(P)\simeq\mathrm{Gal}(K_P/K)$. Furthermore, let \begin{equation*}
\Gamma=\left\{\gamma\in\mathrm{SL}_2(\mathbb{Z})~|~ \gamma\equiv\begin{bmatrix}t^{-1}&\mathrm{*}\\ 0&t\end{bmatrix}\Mod{N M_2(\mathbb{Z})}~ \textrm{for some}~t\in T\right\} \end{equation*} where $t^{-1}$ stands for an integer such that $tt^{-1}\equiv1\Mod{N}$. In this section, for a given $h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}$ we shall define a class invariant $h(C)$ for each class $C\in I_K(\mathfrak{n})/P$.
\begin{lemma}\label{subgroup} The field $K_P$ corresponds to the subgroup \begin{equation*}
\bigcup_{t\in T}K^*\left(\prod_{p\,|\,N}(t+N\mathcal{O}_{K,\,p}) \times\prod_{p\,\nmid\,N}\mathcal{O}_{K,\,p}^*\right) \end{equation*} of $\widehat{K}^*$ in view of \textup{Proposition \ref{1-1}}. \end{lemma} \begin{proof} We adopt the notations in Proposition \ref{rayidele}. Given $t\in T$, let $t^{-1}$ be an integer such that $tt^{-1}\equiv1\Mod{N}$. Let $s=s(t)=(s_p)\in \widehat{K}^*$ be given by \begin{equation*} s_p=\left\{\begin{array}{ll}
t^{-1} & \textrm{if}~p\,|\,N,\\ 1 & \textrm{if}~p\nmid N. \end{array}\right. \end{equation*} Then one can take $\nu_s=t$ so as to have (\ref{k}), and hence \begin{equation}\label{st} \phi_\mathfrak{n}(s)=[ts\mathcal{O}_K]=[t\mathcal{O}_K]. \end{equation} Since $P$ contains $P_{K,\,1}(\mathfrak{n})$, we obtain $K_P\subseteq K_\mathfrak{n}$ and $\mathrm{Gal}(K_\mathfrak{n}/K_P)\simeq P/P_{K,\,1}(\mathfrak{n})$. Thus we achieve by Proposition \ref{rayidele} that the field $K_P$ corresponds to \begin{eqnarray*} \phi_\mathfrak{n}^{-1}(P/P_{K,\,1}(\mathfrak{n}))&=& \phi_\mathfrak{n}^{-1}\left(\bigcup_{t\in T}[t\mathcal{O}_K]\right)\quad\textrm{by the definitions of $P_{K,\,1}(\mathfrak{n})$ and $P$}\\ &=&\bigcup_{t\in T}s(t)J_\mathfrak{n} \quad\textrm{by (\ref{st}) and the fact $J_\mathfrak{n}=\mathrm{Ker}(\phi_\mathfrak{n})$}\\
&=&\bigcup_{t\in T}K^*\left(\prod_{p\,|\,N}(t^{-1}+N\mathcal{O}_{K,\,p}) \times\prod_{p\,\nmid\,N}\mathcal{O}_{K,\,p}^*\right)\\
&=&\bigcup_{t\in T}K^*\left(\prod_{p\,|\,N}(t+N\mathcal{O}_{K,\,p}) \times\prod_{p\,\nmid\,N}\mathcal{O}_{K,\,p}^*\right). \end{eqnarray*} \end{proof}
\begin{proposition}\label{KPKF} We have $K_P=K\mathcal{F}_{\Gamma,\,\mathbb{Q}}(\tau_K)$. \end{proposition} \begin{proof} Let $S=\mathbb{Q}^*W$ ($\subseteq\mathrm{GL}_2(\widehat{\mathbb{Q}})$) with \begin{equation*} W=\bigcup_{t\in T}\left\{
\gamma=(\gamma_p)\in\prod_p\mathrm{GL}_2(\mathbb{Z}_p)~|~ \gamma_p\equiv\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ 0&t\end{bmatrix}\Mod{N M_2(\mathbb{Z}_p)}~\textrm{for all $p$}\right\}. \end{equation*} Following the notations in Proposition \ref{model} one can readily show that \begin{equation*} \Gamma_S= \mathbb{Q}^*\left\{
\gamma\in\mathrm{SL}_2(\mathbb{Z})~|~\gamma \equiv\begin{bmatrix}\mathrm{*}&\mathrm{*}\\0&t\end{bmatrix} \Mod{N M_2(\mathbb{Z})}~\textrm{for some}~t\in T\right\}\quad \textrm{and}\quad \det(W)=\widehat{\mathbb{Z}}^*. \end{equation*} It then follows that $\Gamma_S/\mathbb{Q}^*\simeq\Gamma/\{\pm I_2\}$ and $k_S=\mathbb{Q}$, and hence \begin{equation}\label{F_S} \mathcal{F}_S=\mathcal{F}_{\Gamma,\,\mathbb{Q}} \end{equation} by Proposition \ref{model} (ii) and Remark \ref{overQ}. Furthermore, we deduce that \begin{eqnarray*}
K^*q_{\tau_K}^{-1}(S)&=&K^*\{s=(s_p)\in\widehat{K}^*~|~q_{\tau_K}(s)\in \mathbb{Q}^*W\}\\
&=&K^*\{s=(s_p)\in\widehat{K}^*~|~q_{\tau_K}(s)\in W\} \quad\textrm{since}~q_{\tau_K}(r)=rI_2~\textrm{for every}~r\in\mathbb{Q}^*~\textrm{by (\ref{defq})}\\
&=&K^*\{s=(s_p)\in\widehat{K}^*~|~ s_p=u_p\tau_K+v_p~\textrm{with}~u_p,\,v_p\in\mathbb{Q}_p ~\textrm{such that}\\ &&\hspace{3.4cm}\gamma_p=\left[\begin{smallmatrix}v_p-b_Ku_p & -c_Ku_p\\u_p&v_p\end{smallmatrix}\right] \in W~\textrm{for all $p$}\}\quad\textrm{by (\ref{gamma_p})}\\
&=&\bigcup_{t\in T}K^*\{s=(s_p)\in\widehat{K}^*~|~ s_p=u_p\tau_K+v_p~\textrm{with}~u_p,\,v_p\in\mathbb{Z}_p ~\textrm{such that}\\ &&\hspace{4cm}\gamma_p\in\mathrm{GL}_2(\mathbb{Z}_p) ~\textrm{and}~\gamma_p\equiv \left[\begin{smallmatrix}\mathrm{*}&\mathrm{*}\\0&t\end{smallmatrix}\right] \Mod{N M_2(\mathbb{Z}_p)}~\textrm{for all $p$}\}\\
&=&\bigcup_{t\in T}K^*\{s=(s_p)\in\widehat{K}^*~|~ s_p=u_p\tau_K+v_p~\textrm{with}~u_p,\,v_p\in\mathbb{Z}_p ~\textrm{such that}\\ &&\hspace{4cm}\det(\gamma_p)=(u_p\tau_K+v_p)(u_p\overline{\tau}_K+v_p)\in\mathbb{Z}_p^*,\\ &&\hspace{4cm}u_p\equiv0\Mod{N\mathbb{Z}_p}~ \textrm{and}~v_p\equiv t\Mod{N\mathbb{Z}_p}~\textrm{for all $p$}\}\\ &=&\bigcup_{t\in T}K^*\left(
\prod_{p\,|\,N}(t+N\mathcal{O}_{K,\,p})\times \prod_{p\,\nmid\,N}\mathcal{O}_{K,\,p}^*\right). \end{eqnarray*} Therefore we conclude by Proposition \ref{model} (iv), (\ref{F_S}) and Lemma \ref{subgroup} that \begin{equation*} K_P=K\mathcal{F}_{\Gamma,\,\mathbb{Q}}(\tau_K). \end{equation*} \end{proof}
Let $C\in\mathrm{Cl}(P)$. Take an integral ideal $\mathfrak{a}$ in the class $C$, and let $\xi_1$ and $\xi_2$ be elements of $K^*$ so that \begin{equation*} \mathfrak{a}^{-1}=[\xi_1,\,\xi_2] \quad \textrm{and}\quad \xi=\frac{\xi_1}{\xi_2}\in\mathbb{H}. \end{equation*} Since $\mathcal{O}_K=[\tau_K,\,1]\subseteq\mathfrak{a}^{-1}$ and $\xi\in\mathbb{H}$, one can express \begin{equation}\label{A} \begin{bmatrix}\tau_K\\1\end{bmatrix}=A\begin{bmatrix} \xi_1\\\xi_2\end{bmatrix}\quad\textrm{for some}~ A\in M_2^+(\mathbb{Z}). \end{equation} We then attain by taking determinant and squaring \begin{equation*} \begin{bmatrix} \tau_K&\overline{\tau}_K\\1&1\end{bmatrix} =A\begin{bmatrix}\xi_1&\overline{\xi}_1\\ \xi_2&\overline{\xi}_2\end{bmatrix} \end{equation*} that \begin{equation*} d_K=\det(A)^2\mathrm{N}_{K/\mathbb{Q}}(\mathfrak{a})^{-2}d_K \end{equation*} (\cite[Chapter III]{Lang94}). Hence, $\det(A)=\mathrm{N}_{K/\mathbb{Q}}(\mathfrak{a})$ which is relatively prime to $N$. For $\alpha\in M_2(\mathbb{Z})$ with $\gcd(N,\,\det(\alpha))=1$, we shall denote by $\widetilde{\alpha}$ its reduction onto $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$ ($\simeq\mathrm{Gal}(\mathcal{F}_N/\mathcal{F}_1)$).
\begin{definition}\label{invariant} Let $h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}$ \textup{($\subseteq \mathcal{F}_N$)}. With the notations as above, we define \begin{equation*}
h(C)=h(\tau)^{\widetilde{A}}|_{\tau=\xi} \end{equation*} if it is finite. \end{definition}
\begin{proposition} If $h(C)$ is finite, then it depends only on the class $C$ regardless of the choice of $\mathfrak{a}$, $\xi_1$ and $\xi_2$. \end{proposition} \begin{proof} Let $\mathfrak{a}'$ be also an integral ideal in $C$. Take any $\xi_1',\,\xi_2'\in K^*$ so that \begin{equation}\label{a'} \mathfrak{a}'^{-1}=[\xi_1',\,\xi_2']\quad \textrm{and}\quad \xi'=\frac{\xi_1'}{\xi_2'}\in\mathbb{H}. \end{equation} Since $\mathcal{O}_K\subseteq\mathfrak{a}'^{-1}$ and $\xi'\in\mathbb{H}$, we may write \begin{equation}\label{A'} \begin{bmatrix}\tau_K\\1\end{bmatrix}=A'\begin{bmatrix} \xi_1'\\\xi_2'\end{bmatrix} \quad\textrm{for some}~ A'\in M_2^+(\mathbb{Z}). \end{equation} Now that $[\mathfrak{a}]=[\mathfrak{a}']=C$, we have \begin{equation*} \mathfrak{a}'=\lambda\mathfrak{a}\quad \textrm{with}~\lambda\in K^*~\textrm{such that}~\lambda\equiv^*t\Mod{\mathfrak{n}}~ \textrm{for some}~t\in T. \end{equation*} Then it follows that \begin{equation}\label{a-l-a-} \mathfrak{a}'^{-1}=\lambda^{-1}\mathfrak{a}^{-1}= [\lambda^{-1}\xi_1,\,\lambda^{-1}\xi_2] \quad\textrm{and}\quad\frac{\lambda^{-1}\xi_1} {\lambda^{-1}\xi_2}=\xi. \end{equation} And, we obtain by (\ref{a'}) and (\ref{a-l-a-}) that \begin{equation}\label{B} \begin{bmatrix}\xi_1'\\\xi_2'\end{bmatrix} =B\begin{bmatrix}\lambda^{-1}\xi_1\\ \lambda^{-1}\xi_2\end{bmatrix}\quad \textrm{for some}~B\in\mathrm{SL}_2(\mathbb{Z}) \end{equation} and \begin{equation}\label{x'Bx} \xi'=B(\xi). \end{equation} On the other hand, consider $t$ as an integer whose reduction modulo $N$ belongs to $T$. Since $\mathfrak{a},\,\mathfrak{a}'=\lambda\mathfrak{a}\subseteq\mathcal{O}_K$, we see that $(\lambda-t)\mathfrak{a}$ is an integral ideal. Moreover, since $\lambda\equiv^*t\Mod{\mathfrak{n}}$ and $\mathfrak{a}$ is relatively prime to $\mathfrak{n}$, we get $(\lambda-t)\mathfrak{a}\subseteq \mathfrak{n}=N\mathcal{O}_K$, and hence \begin{equation*} (\lambda-t)\mathcal{O}_K\subseteq N\mathfrak{a}^{-1}. \end{equation*} Thus we attain by the facts $\mathcal{O}_K=[\tau_K,\,1]$ and $\mathfrak{a}^{-1}=[\xi_1,\,\xi_2]$ that \begin{equation}\label{A''} \begin{bmatrix} (\lambda-t)\tau_K\\\lambda-t \end{bmatrix} =A'' \begin{bmatrix}N\xi_1\\N\xi_2\end{bmatrix} \quad\textrm{for some}~A''\in M_2^+(\mathbb{Z}). \end{equation} We then derive that \begin{eqnarray*} NA''\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}&=& \lambda\begin{bmatrix}\tau_K\\1\end{bmatrix} -t\begin{bmatrix}\tau_K\\1\end{bmatrix}\quad\textrm{by (\ref{A''})}\\ &=&\lambda A'\begin{bmatrix}\xi_1'\\\xi_2'\end{bmatrix}- tA\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix} \quad\textrm{by (\ref{A}) and (\ref{A'})}\\ &=&A'B\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}- tA\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix} \quad\textrm{by (\ref{B})}. \end{eqnarray*} This yields $A'B-tA\equiv O\Mod{N M_2(\mathbb{Z})}$ and so \begin{equation}\label{AtAB} A'\equiv tAB^{-1}\Mod{N M_2(\mathbb{Z})}. \end{equation} Therefore we establish by Proposition \ref{Galoisdecomposition} that \begin{eqnarray*}
h(\tau)^{\widetilde{A'}}|_{\tau=\xi'}&=&
h(\tau)^{\widetilde{tAB^{-1}}}|_{\tau=\xi'}\quad \textrm{by (\ref{AtAB})}\\ &&\hspace{2.8cm}\textrm{where $\widetilde{~\cdot~}$ means the reduction onto $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$} \\ &=&h(\tau)^{\widetilde{ \left[\begin{smallmatrix}1&0\\0&t^2\end{smallmatrix}\right]} \widetilde{\left[\begin{smallmatrix}t&0\\0&t^{-1}\end{smallmatrix}\right]}
\widetilde{AB^{-1}}}|_{\tau=\xi'}\\ &=&h(\tau)^{\widetilde{ \left[\begin{smallmatrix}t&0\\0&t^{-1}\end{smallmatrix}\right]}
\widetilde{AB^{-1}}}|_{\tau=\xi'}\quad \textrm{because $h(\tau)$ has rational Fourier coefficients}\\
&=&h(\tau)^{\widetilde{AB^{-1}}}|_{\tau=\xi'}\quad \textrm{since $h(\tau)$ is modular for $\Gamma$}\\
&=&h(\tau)^{\widetilde{A}}|_{\tau=B^{-1}(\xi')}\\
&=&h(\tau)^{\widetilde{A}}|_{\tau=\xi}\quad \textrm{by (\ref{x'Bx})}. \end{eqnarray*} This prove that $h(C)$ depends only on the class $C$. \end{proof}
\begin{remark}\label{identityinvariant} If we let $C_0$ be the identity class in $\mathrm{Cl}(P)$, then we have $h(C_0)=h(\tau_K)$. \end{remark}
\begin{proposition}\label{transformation} Let $C\in\mathrm{Cl}(P)$ and $h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}$. If $h(C)$ is finite, then it belongs to $K_P$ and satisfies \begin{equation*} h(C)^{\sigma(C')}=h(CC')\quad\textrm{for all}~C'\in \mathrm{Cl}(P) \end{equation*} where $\sigma:\mathrm{Cl}(P)\rightarrow\mathrm{Gal}(K_P/K)$ is the isomorphism induced from the Artin map. \end{proposition} \begin{proof} Let $\mathfrak{a}$ be an integral ideal in $C$ and $\xi_1,\,\xi_2\in K^*$ such that \begin{equation}\label{ax} \mathfrak{a}^{-1}=[\xi_1,\,\xi_2]\quad \textrm{with}~\xi=\frac{\xi_1}{\xi_2}\in\mathbb{H}. \end{equation} Then we have \begin{equation}\label{tAx} \begin{bmatrix}\tau_K\\1\end{bmatrix}=A\begin{bmatrix} \xi_1\\\xi_2\end{bmatrix}\quad\textrm{for some}~A\in M_2^+(\mathbb{Z}). \end{equation} Furthermore, let $\mathfrak{a}'$ be an integral ideal in $C'$ and $\xi_1'',\,\xi_2''\in K^*$ such that \begin{equation}\label{aax} (\mathfrak{a}\mathfrak{a}')^{-1}=[\xi_1'',\,\xi_2''] \quad\textrm{with}~\xi''=\frac{\xi_1''}{\xi_2''}\in\mathbb{H}. \end{equation} Since $\mathfrak{a}^{-1}\subseteq (\mathfrak{a}\mathfrak{a}')^{-1}$ and $\xi''\in\mathbb{H}$, we get \begin{equation}\label{xBx} \begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}=B\begin{bmatrix} \xi_1''\\\xi_2''\end{bmatrix}\quad \textrm{for some}~B\in M_2^+(\mathbb{Z}), \end{equation} and so it follows from (\ref{tAx}) that \begin{equation}\label{tABx} \begin{bmatrix}\tau_K\\1\end{bmatrix}= AB\begin{bmatrix}\xi_1''\\\xi_2''\end{bmatrix}. \end{equation} Let $s=(s_p)$ be an idele in $\widehat{K}^*$ satisfying \begin{equation}\label{s1pN} \left\{\begin{array}{ll}
s_p=1 & \textrm{if}~p\,|\,N,\\ s_p\mathcal{O}_{K,\,p}=\mathfrak{a}_p' & \textrm{if}~ p\nmid N \end{array}\right. \end{equation} where $\mathfrak{a}_p'=\mathfrak{a}'\otimes_\mathbb{Z} \mathbb{Z}_p$. Since $\mathfrak{a}'$ is relatively prime to $\mathfrak{n}=N\mathcal{O}_K$, we obtain by (\ref{s1pN}) that \begin{equation}\label{sOa} s_p^{-1}\mathcal{O}_{K,\,p}=\mathfrak{a}_p'^{-1}\quad \textrm{for all $p$}. \end{equation} Now, we see that \begin{equation*} q_{\xi,\,p}(s_p^{-1}) \begin{bmatrix}\xi_1\\\xi_2\end{bmatrix} =\xi_2q_{\xi,\,p}(s_p^{-1}) \begin{bmatrix}\xi\\1\end{bmatrix} =\xi_2s_p^{-1}\begin{bmatrix}\xi\\1\end{bmatrix}= s_p^{-1}\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}, \end{equation*} which shows by (\ref{ax}) and (\ref{sOa}) that $q_{\xi,\,p}(s_p^{-1})\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}$ is a $\mathbb{Z}_p$-basis for $(\mathfrak{a}\mathfrak{a}')_p^{-1}$. Furthermore, $B^{-1}\begin{bmatrix}\xi_1\\\xi_2\end{bmatrix}$ is also a $\mathbb{Z}_p$-basis for $(\mathfrak{a}\mathfrak{a}')^{-1}_p$ by (\ref{aax}) and (\ref{xBx}). Thus we achieve \begin{equation}\label{qpsgB} q_{\xi,\,p}(s_p^{-1})=\gamma_pB^{-1}\quad \textrm{for some}~\gamma_p\in\mathrm{GL}_2(\mathbb{Z}_p). \end{equation} Letting $\gamma=(\gamma_p)\in\prod_p\mathrm{GL}_2(\mathbb{Z}_p)$ we get \begin{equation}\label{qsgB} q_\xi(s^{-1})=\gamma B^{-1}. \end{equation} We then deduce that \begin{eqnarray*}
h(C)^{[s,\,K]}&=&(h(\tau)^{\widetilde{A}}|_{\tau=\xi})^{[s,\,K]}\quad \textrm{by Definition \ref{invariant}}\\
&=&(h(\tau)^{\widetilde{A}})^{\sigma(q_\xi(s^{-1}))}|_{\tau=\xi} \quad\textrm{by Propositioin \ref{reciprocity}}\\
&=&(h(\tau)^{\widetilde{A}})^{\sigma(\gamma B^{-1})}|_{\tau=\xi} \quad\textrm{by (\ref{qsgB})}\\ &=&h(\tau)^{\widetilde{A}\widetilde{G}}
|_{\tau=B^{-1}(\xi)}\quad \textrm{where $G$ is a matrix in $M_2(\mathbb{Z})$ such that}\\ &&\hspace{3.1cm}G\equiv\gamma_p
\Mod{N M_2(\mathbb{Z}_p)}~\textrm{for all}~p\,|\,N\\
&=&h(\tau)^{\widetilde{A}\widetilde{B}}|_{\tau=\xi''}
\quad\textrm{by (\ref{xBx}) and the fact that for each $p\,|\,N$,}\\ &&\hspace{2.5cm}\textrm{$s_p=1$ and so $\gamma_p B^{-1}=I_2$ owing to (\ref{s1pN}) and (\ref{qpsgB})}\\ &=&h(CC')\quad\textrm{by Definition \ref{invariant} and (\ref{tABx})}. \end{eqnarray*} In particular, if we consider the case where $C'=C^{-1}$, then we derive that \begin{equation*} h(C)=h(CC')^{[s^{-1},\,K]}=h(C_0)^{[s^{-1},\,K]} =h(\tau_K)^{[s^{-1},\,K]}. \end{equation*} This implies that $h(C)$ belongs to $K_P$ by Proposition \ref{KPKF}. \par For each $p\,\nmid\,N$ and $\mathfrak{p}$ lying above $p$, we have by (\ref{s1pN}) that $\mathrm{ord}_\mathfrak{p}~ s_p=\mathrm{ord}_\mathfrak{p}~\mathfrak{a}'$, and hence \begin{equation*}
[s,\,K]|_{K_P}=\sigma(C'). \end{equation*} Therefore we conclude \begin{equation*} h(C)^{\sigma(C')}=h(CC'). \end{equation*} \end{proof}
\section {Extended form class groups as Galois groups}
With $P$, $K_P$ and $\Gamma$ as in $\S$\ref{classinvariants}, we shall prove our main theorem which asserts that $\mathcal{Q}_N(d_K)/\sim_\Gamma$ can be regarded as a group isomorphic to $\mathrm{Gal}(K_P/K)$ through the isomorphism described in (\ref{desiredisomorphism}).
\begin{lemma}\label{unit} If $Q\in\mathcal{Q}(d_K)$ and $\gamma\in I_{\omega_Q}$, then $j(\gamma,\,\omega_Q)\in\mathcal{O}_K^*$. \end{lemma} \begin{proof} We obtain from $Q=Q^\gamma$ that \begin{equation*} [\omega_Q,\,1]=[\gamma(\omega_Q),\,1] =\frac{1}{j(\gamma,\,\omega_Q)}[\omega_Q,\,1]. \end{equation*} This claims that $j(\gamma,\,\omega_Q)$ is a unit in $\mathcal{O}_K$. \end{proof}
\begin{remark} This lemma can be also justified by using (\ref{isotropy}), (\ref{otherisotropy}) and the property \begin{equation}\label{jabt} j(\alpha\beta,\,\tau)=j(\alpha,\,\beta(\tau))j(\beta,\,\tau) \quad(\alpha,\,\beta\in\mathrm{SL}_2(\mathbb{Z}),\,\tau\in\mathbb{H}) \end{equation} (\cite[(1.2.4)]{Shimura}). \end{remark}
\begin{proposition}\label{satisfies} For given $P$, the group $\Gamma$ satisfies the property \textup{(\ref{P})}. \end{proposition} \begin{proof} Let $Q=ax^2+bxy+cy^2\in\mathcal{Q}_N(d_K)$ and $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ such that $Q^{\gamma^{-1}}\in\mathcal{Q}_N(d_K)$. \par Assume that $j(\gamma,\,\omega_Q)\mathcal{O}_K\in P$. Then we have \begin{equation*} j(\gamma,\,\omega_Q)\mathcal{O}_K=\frac{\nu_1}{\nu_2} \mathcal{O}_K\quad\textrm{for some}~\nu_1,\,\nu_2\in\mathcal{O}_K-\{0\} \end{equation*} satisfying \begin{equation}\label{ntnt} \nu_1\equiv t_1,\,\nu_2\equiv t_2\Mod{\mathfrak{n}}~ \textrm{with}~t_1,\,t_2\in T \end{equation} and hence \begin{equation}\label{zjnn} \zeta j(\gamma,\,\omega_Q)=\frac{\nu_1}{\nu_2}\quad \textrm{for some}~\zeta\in\mathcal{O}_K^*. \end{equation} For convenience, let $j=j(\gamma,\,\omega_Q)$ and $Q'=Q^{\gamma^{-1}}$. Then we deduce \begin{equation}\label{newQQ} \gamma(\omega_Q)=\omega_{Q'} \end{equation} and \begin{equation*} [\omega_Q,\,1]=j[\gamma(\omega_Q),\,1]=j[\omega_{Q'},\,1]= \zeta j[\omega_{Q'},\,1]. \end{equation*} So there is $\alpha=\begin{bmatrix}r&s\\ u&v\end{bmatrix}\in\mathrm{GL}_2(\mathbb{Z})$ which yields \begin{equation}\label{zz} \begin{bmatrix}\zeta j\omega_{Q'}\\ \zeta j\end{bmatrix}=\alpha \begin{bmatrix}\omega_Q\\1\end{bmatrix}. \end{equation} Here, since $\zeta j\omega_{Q'}/\zeta j=\omega_{Q'},\, \omega_Q\in\mathbb{H}$, we get $\alpha\in\mathrm{SL}_2(\mathbb{Z})$ and \begin{equation}\label{newww} \omega_{Q'}=\alpha(\omega_Q). \end{equation} Thus we attain $\gamma(\omega_Q)=\omega_{Q'}=\alpha(\omega_Q)$ by (\ref{newQQ}) and (\ref{newww}), from which we get $\omega_Q=(\alpha^{-1}\gamma)(\omega_Q)$ and so \begin{equation}\label{gaI} \gamma\in\alpha\cdot I_{\omega_Q}. \end{equation} Now that $aj\in\mathcal{O}_K$, we see from (\ref{ntnt}), (\ref{zjnn}) and (\ref{zz}) that \begin{eqnarray*} &&a\nu_2(\zeta j)\equiv a\nu_1\equiv at_1\Mod{\mathfrak{n}},~ \textrm{and}\\ &&a\nu_2(\zeta j)\equiv a\nu_2(u\omega_Q+v)\equiv ut_2(a\omega_Q)+at_2v\Mod{\mathfrak{n}}. \end{eqnarray*} It then follows that \begin{equation*} at_1\equiv ut_2(a\omega_Q)+at_2v\Mod{\mathfrak{n}} \end{equation*} and hence \begin{equation*} ut_2(a\omega_Q)+a(t_2v-t_1)\equiv0\Mod{\mathfrak{n}}. \end{equation*} Since $\mathfrak{n}=N\mathcal{O}_K=N[a\omega_Q,\,1]$, we have \begin{equation*} ut_2\equiv0\Mod{N}\quad\textrm{and}\quad a(t_2v-t_1)\equiv0\Mod{N}. \end{equation*} Moreover, since $\gcd(N,\,t_1)=\gcd(N,\,t_2)=\gcd(N,\,a)=1$, we achieve that \begin{equation*} u\equiv0\Mod{N}\quad\textrm{and}\quad v\equiv t_1t_2^{-1}\Mod{N} \end{equation*} where $t_2^{-1}$ is an integer satisfying $t_2t_2^{-1}\equiv1\Mod{N}$. This, together with the facts $\det(\alpha)=1$ and $T$ is a subgroup of $(\mathbb{Z}/N\mathbb{Z})^*$, implies $\alpha=\begin{bmatrix}r&s\\u&v\end{bmatrix}\in\Gamma$. Therefore we conclude $\gamma\in \Gamma\cdot I_{\omega_Q}$ by (\ref{gaI}), as desired. \par Conversely, assume that $\gamma\in\Gamma\cdot I_{\omega_Q}$, and so \begin{equation*} \gamma=\alpha\beta\quad \textrm{for some}~\alpha= \begin{bmatrix}r&s\\u&v\end{bmatrix}\in\Gamma~ \textrm{and}~\beta\in I_{\omega_Q}. \end{equation*} Here we observe that \begin{equation}\label{u0vt} u\equiv0\Mod{N}\quad\textrm{and}\quad v\equiv t\Mod{N}~\textrm{for some}~t\in T. \end{equation} We then derive that \begin{eqnarray*} j(\gamma,\,\omega_Q)&=&j(\alpha\beta,\,\omega_Q)\\ &=&j(\alpha,\,\beta(\omega_Q))j(\beta,\,\omega_Q)\quad \textrm{by (\ref{jabt})}\\ &=&j(\alpha,\,\omega_Q)\zeta\quad\textrm{for some}~ \zeta\in\mathcal{O}_K^*~\textrm{by the fact $\beta\in I_{\omega_Q}$ and Lemma \ref{unit}}. \end{eqnarray*} Thus we attain \begin{equation*} \zeta^{-1}j(\gamma,\,\omega_Q)-v =j(\alpha,\,\omega_Q)-v =(u\omega_Q+v)-v =\frac{1}{a}\{u(a\omega_Q)\}. \end{equation*} And, it follows from the fact $\gcd(N,\,a)=1$ and (\ref{u0vt}) that \begin{equation*} \zeta^{-1}j(\gamma,\,\omega_Q)\equiv^*v \equiv^*t\Mod{\mathfrak{n}}. \end{equation*} This shows that $\zeta^{-1}j(\gamma,\,\omega_Q) \mathcal{O}_K\in P$, and hence $j(\gamma,\,\omega_Q)\mathcal{O}_K\in P$. \par Therefore, the group $\Gamma$ satisfies the property (\ref{P}) for $P$. \end{proof}
\begin{theorem}\label{Galoisgroups} We have an isomorphism \begin{equation}\label{Galoisisomorphism} \begin{array}{ccl} \mathcal{Q}_N(d_K)/\sim_\Gamma&\rightarrow& \mathrm{Gal}(K_P/K)\\ \left[Q\right]&\mapsto&
\left(h(\tau_K)\mapsto h(-\overline{\omega}_Q)~|~ h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}~ \textrm{is finite at}~\tau_K\right). \end{array} \end{equation} \end{theorem} \begin{proof} By Theorem \ref{formclassgroup} and Proposition \ref{satisfies} one may consider $\mathcal{Q}_N(d_K)/\sim_\Gamma$ as a group isomorphic to $I_K(\mathfrak{n})/P$ via the isomorphism $\phi_\Gamma$ in $\S$\ref{sect2}. Let $C\in\mathrm{Cl}(P)$ and so \begin{equation*} C=\phi_\Gamma([Q])=[[\omega_Q,\,1]] \quad\textrm{for some}~Q\in\mathcal{Q}_N(d_K)/\sim_\Gamma. \end{equation*} Note that $C$ contains an integral ideal $\mathfrak{a}=a^{\varphi(N)}[\omega_Q,\,1]$, where $\varphi$ is the Euler totient function. We establish by Lemma \ref{prime} and the definition (\ref{wQ}) that \begin{equation*} \mathfrak{a}^{-1}=\frac{1}{\mathrm{N}_{K/\mathbb{Q}}(\mathfrak{a})} \overline{\mathfrak{a}}=\frac{1}{a^{\varphi(N)-1}} [-\overline{\omega}_Q,\,1] \end{equation*} and \begin{equation*} \begin{bmatrix}\tau_K\\1\end{bmatrix}= \begin{bmatrix} a^{\varphi(N)} & -a^{\varphi(N)-1}(b+b_K)/2\\ 0&a^{\varphi(N)-1} \end{bmatrix} \begin{bmatrix} -\overline{\omega}_Q/a^{\varphi(N)-1}\\ 1/a^{\varphi(N)-1} \end{bmatrix} \end{equation*} where $\min(\tau_K,\,\mathbb{Q})=x^2+b_Kx+c_K$ ($\in\mathbb{Z}[x]$). We then derive by Proposition \ref{Galoisdecomposition} that if $h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}$ is finite at $\tau_K$, then \begin{eqnarray*} h(C)&=&h(\tau)^{\widetilde{\left[\begin{smallmatrix} a^{\varphi(N)}&-a^{\varphi(N)-1}(b+b_K)/2\\ 0&a^{\varphi(N)-1}
\end{smallmatrix}\right]}}|_{\tau=-\overline{\omega}_Q} \quad\textrm{by Definition \ref{invariant}}\\ &&\hspace{5cm}\textrm{where $\widetilde{~\cdot~}$ means the reduction onto $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})/\{\pm I_2\}$}\\ &=&h(\tau)^{\widetilde{\left[\begin{smallmatrix} 1 & -a^{-1}(b+b_K)/2\\ 0&a^{-1}
\end{smallmatrix}\right]}}|_{\tau=-\overline{\omega}_Q} \quad\textrm{since}~a^{\varphi(N)}\equiv1\Mod{N}\\ &&\hspace{5cm}\textrm{where $a^{-1}$ is an integer such that $aa^{-1}\equiv1\Mod{N}$}\\ &=&h(\tau)^{\widetilde{\left[\begin{smallmatrix} 1&0\\0&a^{-1} \end{smallmatrix}\right]} \widetilde{\left[\begin{smallmatrix} 1 &-a^{-1}(b+b_K)/2\\0&1
\end{smallmatrix}\right]}}|_{\tau=-\overline{\omega}_Q}\\ &=&h(\tau)^{\widetilde{\left[\begin{smallmatrix} 1 &-a^{-1}(b+b_K)/2\\0&1
\end{smallmatrix}\right]}}|_{\tau=-\overline{\omega}_Q} \quad\textrm{because $h(\tau)$ has rational Fourier coefficients}\\
&=&h(\tau)|_{\tau=-\overline{\omega}_Q} \quad\textrm{since $h(\tau)$ is modular for $\Gamma$}\\ &=&h(-\overline{\omega}_Q). \end{eqnarray*} Now, the isomorphism $\phi_\Gamma$ followed by the isomorphism \begin{equation*} \begin{array}{ccl} \mathrm{Cl}(P)&\rightarrow&\mathrm{Gal}(K_P/K)\\ C&\mapsto&\left(
h(\tau_K)=h(C_0)\mapsto h(C_0)^{\sigma(C)}=h(C)=h(-\overline{\omega}_Q)~|~ h\in\mathcal{F}_{\Gamma,\,\mathbb{Q}}~ \textrm{is finite at}~\tau_K\right) \end{array} \end{equation*} which is induced from Propositions \ref{KPKF}, \ref{transformation} and Remark \ref{identityinvariant}, yields the isomorphism stated in (\ref{Galoisisomorphism}), as desired. \end{proof}
\begin{remark}\label{difference} In \cite{E-K-S17} Eum, Koo and Shin considered only the case where $K\neq\mathbb{Q}(\sqrt{-1}),\,\mathbb{Q}(\sqrt{-3})$, $P=P_{K,\,1}(\mathfrak{n})$ and $\Gamma=\Gamma_1(N)$. As for the group operation of $\mathcal{Q}_N(d_K)/\sim_{\Gamma_1(N)}$ one can refer to \cite[Remark 2.10]{E-K-S17}. They established an isomorphism \begin{equation}\label{rayGaloisgroups} \begin{array}{lcl} \mathcal{Q}_N(d_K)/\sim_{\Gamma_1(N)}&\rightarrow&\mathrm{Gal}(K_\mathfrak{n}/K)\\ \left[Q\right]=\left[ax^2+bxy+cy^2\right]&\mapsto& \left( h(\tau_K)\mapsto h(\tau)^{\widetilde{\left[
\begin{smallmatrix}a&(b-b_K)/2\\0&1\end{smallmatrix}\right]}}|_{\tau=
\omega_Q}~|~h(\tau)\in\mathcal{F}_N~ \textrm{is finite at $\tau_K$}\right). \end{array} \end{equation} The difference between the isomorphisms described in (\ref{Galoisisomorphism}) and (\ref{rayGaloisgroups}) arises from the Definition \ref{invariant} of $h(C)$. The invariant $h_\mathfrak{n}(C)$ appeared in \cite[Definition 3.3]{E-K-S17} coincides with $h(C^{-1})$. \end{remark}
\section {Finding representatives of extended form classes}
In this last section, by improving the proof of Proposition \ref{surjective} further, we shall explain how to find all quadratic forms which represent distinct classes in $\mathcal{Q}_N(d_K)/\sim_\Gamma$. \par For a given $Q=ax^2+bxy+cy^2\in\mathcal{Q}_N(d_K)$ we define an equivalence relation $\equiv_Q$ on $M_{1,\,2}(\mathbb{Z})$ as follows: Let $\begin{bmatrix}r&s\end{bmatrix},\, \begin{bmatrix}u&v\end{bmatrix}\in M_{1,\,2}(\mathbb{Z})$. Then, $\begin{bmatrix}r&s\end{bmatrix}\equiv_Q\begin{bmatrix}u&v\end{bmatrix}$ if and only if \begin{equation*} \begin{bmatrix}r&s\end{bmatrix} \equiv\pm t\begin{bmatrix}u&v\end{bmatrix}\gamma\Mod{NM_{1,\,2}(\mathbb{Z})} \quad\textrm{for some}~t\in T~\textrm{and}~\gamma\in \Gamma_Q \end{equation*} where \begin{equation*} \Gamma_Q=\left\{\begin{array}{ll} \left\{\pm I_2\right\} & \textrm{if}~d_K\neq-4,\,-3,
\\ \left\{\pm I_2,\, \pm\begin{bmatrix} -b/2&-a^{-1}(b^2+4)/4)\\a&b/2 \end{bmatrix} \right\} & \textrm{if}~d_K=-4,
\\ \left\{\pm I_2,\, \pm\begin{bmatrix} -(b+1)/2&-a^{-1}(b^2+3)/4\\a&(b-1)/2 \end{bmatrix},\, \pm\begin{bmatrix} (b-1)/2&a^{-1}(b^2+3)/4\\-a&-(b+1)/2 \end{bmatrix} \right\} & \textrm{if}~d_K=-3. \end{array} \right. \end{equation*} Here, $a^{-1}$ is an integer satisfying $aa^{-1}\equiv1\Mod{N}$.
\begin{lemma}\label{equivGamma} Let $Q=ax^2+bxy+cy^2\in\mathcal{Q}_N(d_K)$ and $\begin{bmatrix}r&s\end{bmatrix},\, \begin{bmatrix}u&v\end{bmatrix}\in M_{1,\,2}(\mathbb{Z})$ such that $\gcd(N,\,Q(s,\,-r))=\gcd(N,\,Q(v,\,-u))=1$. Then, \begin{equation*} [(r\omega_Q+s)\mathcal{O}_K]=[(u\omega_Q+v)\mathcal{O}_K]~ \textrm{in}~P_K(\mathfrak{n})/P \quad \Longleftrightarrow\quad \begin{bmatrix}r&s\end{bmatrix}\equiv_Q \begin{bmatrix}u&v\end{bmatrix}. \end{equation*} \end{lemma} \begin{proof} Note that by Lemma \ref{prime} (i) the fractional ideals $(r\omega_Q+s)\mathcal{O}_K$ and $(u\omega_Q+v)\mathcal{O}_K$ belong to $P_K(\mathfrak{n})$. Furthermore, we know that \begin{equation}\label{OK*} \mathcal{O}_K^*= \left\{ \begin{array}{ll} \{\pm1\} & \textrm{if}~K\neq\mathbb{Q}(\sqrt{-1}),\,\mathbb{Q}(\sqrt{-3}),\\ \{\pm 1,\,\pm\tau_K\} & \textrm{if}~K=\mathbb{Q}(\sqrt{-1}),\\ \{\pm 1,\,\pm\tau_K,\, \pm\tau_K^2\} & \textrm{if}~K=\mathbb{Q}(\sqrt{-3}) \end{array}\right. \end{equation} (\cite[Exercise 5.9]{Cox}) and so \begin{equation}\label{UK}
U_K=\{(m,\,n)\in\mathbb{Z}^2~|~m\tau_K+n\in\mathcal{O}_K^*\} =\left\{\begin{array}{ll} \{\pm(0,\,1)\} & \textrm{if}~K\neq\mathbb{Q}(\sqrt{-1}),\,\mathbb{Q}(\sqrt{-3}),\\ \{\pm(0,\,1),\,\pm(1,\,0)\} & \textrm{if}~K=\mathbb{Q}(\sqrt{-1}),\\ \{\pm(0,\,1),\,\pm(1,\,0),\,\pm(1,\,1)\} & \textrm{if}~K=\mathbb{Q}(\sqrt{-3}). \end{array}\right. \end{equation} Then we achieve that \begin{eqnarray*} &&[(r\omega_Q+s)\mathcal{O}_K]=[(u\omega_Q+v)\mathcal{O}_K] \quad\textrm{in}~P_K(\mathfrak{n})/P\\ &\Longleftrightarrow& \left(\frac{r\omega_Q+s}{u\omega_Q+v}\right)\mathcal{O}_K\in P\\ &\Longleftrightarrow& \frac{r\omega_Q+s}{u\omega_Q+v}\equiv^*\zeta t\Mod{\mathfrak{n}} \quad\textrm{for some}~\zeta\in\mathcal{O}_K^*~\textrm{and}~t\in T\\ &\Longleftrightarrow& a(r\omega_Q+s)\equiv\zeta ta(u\omega_Q+v)\Mod{\mathfrak{n}} \quad\textrm{since $a(u\omega_Q+v)\mathcal{O}_K$ is relatively prime to $\mathfrak{n}$}\\ &&\hspace{6.5cm}\textrm{and}~a\omega_Q\in\mathcal{O}_K\\ &\Longleftrightarrow& r\left(\tau_K+\frac{b_K-b}{2}\right)+as\equiv (m\tau_K+n)t\left\{ u\left(\tau_K+\frac{b_K-b}{2}\right)+av\right\}\Mod{\mathfrak{n}}\\ &&\hspace{10.5cm}\textrm{for some}~(m,\,n)\in U_K\\ &\Longleftrightarrow& r\tau_K+\left(\frac{r(b_K-b)}{2}+as\right)\equiv t(-mub_K+mk+nu)\tau_K+t(-muc_K+nk) \Mod{\mathfrak{n}}\\ &&\hspace{3.5cm}\textrm{with}~ k=\frac{u(b_K-b)}{2}+av,~\textrm{where}~\min(\tau_K,\,\mathbb{Q}) =x^2+b_Kx+c_K\\ &\Longleftrightarrow& r\equiv t\left\{-\left(\frac{b_K+b}{2}\right)m+n\right\}u+tmav\Mod{N}\quad\textrm{and}\\ &&s\equiv ta^{-1}\left(\frac{b_K^2-b^2}{4}-c_K\right)mu+ t\left\{-\left(\frac{b_K-b}{2}\right)m+n\right\}v\Mod{N}\\ &&\hspace{10cm}\textrm{by the fact}~\mathfrak{n}=N[\tau_K,\,1]\\ &\Longleftrightarrow& \begin{bmatrix}r&s\end{bmatrix}\equiv_Q \begin{bmatrix}u&v\end{bmatrix}\quad\textrm{by (\ref{UK}) and the definition of $\equiv_Q$}. \end{eqnarray*} \end{proof}
For each $Q\in\mathcal{Q}_N(d_K)$, let \begin{equation*}
M_Q=\left\{\begin{bmatrix}u&v\end{bmatrix}\in M_{1,\,2}(\mathbb{Z})~|~ \gcd(N,\,Q(v,\,-u))=1\right\}. \end{equation*}
\begin{proposition}\label{algorithm} One can explicitly find quadratic forms representing all distinct classes in $\mathcal{Q}_N(d_K)/\sim_\Gamma$. \end{proposition} \begin{proof} We adopt the idea in the proof of Proposition \ref{surjective}. Let $Q_1',\,Q_2',\,\ldots,\,Q_h'$ be quadratic forms in $\mathcal{Q}_N(d_K)$ which represent all distinct classes in $\mathrm{C}(d_K)=\mathcal{Q}(d_K)/\sim$. Then we get by Lemma \ref{equivGamma} that for each $i=1,\,2,\,\ldots,\,h$ \begin{equation*} P_K(\mathfrak{n})/P=
\left\{[(u\omega_{Q_i'}+v)\mathcal{O}_K]~|~ \begin{bmatrix}u&v\end{bmatrix}\in M_{Q_i'}/\equiv_{Q_i'}\right\}
=\left\{\left[\frac{1}{u\omega_{Q_i'}+v}\mathcal{O}_K\right]~|~ \begin{bmatrix}u&v\end{bmatrix}\in M_{Q_i'}/\equiv_{Q_i'}\right\}. \end{equation*} Thus we obtain by (\ref{decomp}) that \begin{eqnarray*} I_K(\mathfrak{n})/P&=&(P_K(\mathfrak{n})/P)\cdot
\{[[\omega_{Q_i'},\,1]]\in I_K(\mathfrak{n})/P~|~i=1,\,2,\,\ldots,\,h\}\\ &=&\left\{\left[\frac{1}{u\omega_{Q_i'}+v}
[\omega_{Q_i'},\,1]\right]~|~i=1,\,2,\,\ldots,\,h~ \textrm{and}~\begin{bmatrix}u&v\end{bmatrix} \in M_{Q_i'}/\equiv_{Q_i'}\right\}\\ &=&\left\{ \left[\left[\begin{bmatrix}\mathrm{*}&\mathrm{*}\\
\widetilde{u}&\widetilde{v}\end{bmatrix}(\omega_{Q_i'}),\,1\right]\right]~|~ i=1,\,2,\,\ldots,\,h~ \textrm{and}~\begin{bmatrix}u&v\end{bmatrix} \in M_{Q_i'}/\equiv_{Q_i'}\right\} \end{eqnarray*} where $\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ \widetilde{u}&\widetilde{v}\end{bmatrix}$ is a matrix in $\mathrm{SL}_2(\mathbb{Z})$ such that $\begin{bmatrix}\mathrm{*}&\mathrm{*}\\ \widetilde{u}&\widetilde{v}\end{bmatrix}\equiv \begin{bmatrix}\mathrm{*}&\mathrm{*}\\u&v\end{bmatrix} \Mod{NM_2(\mathbb{Z})}$. Therefore we conclude \begin{equation*} \mathcal{Q}_N(d_K)/\sim_\Gamma=\left\{ \left[Q_i'^{\left[\begin{smallmatrix} \mathrm{*}&\mathrm{*}\\\widetilde{u}&\widetilde{v} \end{smallmatrix}\right]^{-1}}
\right]~|~i=1,\,2,\,\ldots,\,h~ \textrm{and}~\begin{bmatrix}u&v\end{bmatrix} \in M_{Q_i'}/\equiv_{Q_i'} \right\}. \end{equation*} \end{proof}
\begin{example}\label{example} Let $K=\mathbb{Q}(\sqrt{-5})$, $N=12$ and $T=(\mathbb{Z}/N\mathbb{Z})^*$. Then we get $P=P_{K,\,\mathbb{Z}}(\mathfrak{n})$ and $K_P=H_\mathcal{O}$, where $\mathfrak{n}=N\mathcal{O}_K$ and $\mathcal{O}$ is the order of conductor $N$ in $K$. There are two reduced forms of discriminant $d_K=-20$, namely \begin{equation*} Q_1=x^2+5y^2\quad\textrm{and}\quad Q_2=2x^2+2xy+3y^2. \end{equation*} Set \begin{equation*} Q_1'=Q_1\quad \textrm{and}\quad Q_2'=Q_2^{\left[\begin{smallmatrix}1&1\\1&2\end{smallmatrix}\right]} =7x^2+22xy+18y^2 \end{equation*} which belong to $\mathcal{Q}_N(d_K)$. We then see that \begin{equation*} M_{Q_1'}/\equiv_{Q_1'}=\left\{ \begin{bmatrix}0&1\end{bmatrix},\, \begin{bmatrix}1&0\end{bmatrix},\, \begin{bmatrix}1&6\end{bmatrix},\, \begin{bmatrix}2&3\end{bmatrix},\, \begin{bmatrix}3&2\end{bmatrix},\, \begin{bmatrix}3&4\end{bmatrix},\, \begin{bmatrix}4&3\end{bmatrix},\, \begin{bmatrix}6&1\end{bmatrix} \right\} \end{equation*} with corresponding matrices \begin{eqnarray*} \begin{bmatrix} 1&0\\0&1 \end{bmatrix},\, \begin{bmatrix} 0&-1\\1&0 \end{bmatrix},\, \begin{bmatrix} 0&-1\\1&6 \end{bmatrix},\, \begin{bmatrix} 1&1\\2&3 \end{bmatrix},\, \begin{bmatrix} -1&-1\\3&2 \end{bmatrix},\, \begin{bmatrix} 1&1\\3&4 \end{bmatrix},\, \begin{bmatrix} -1&-1\\4&3 \end{bmatrix},\, \begin{bmatrix} 1&0\\6&1 \end{bmatrix} \end{eqnarray*} and \begin{equation*} M_{Q_2'}/\equiv_{Q_2'}=\left\{ \begin{bmatrix}0&1\end{bmatrix},\, \begin{bmatrix}1&5\end{bmatrix},\, \begin{bmatrix}1&11\end{bmatrix},\, \begin{bmatrix}2&1\end{bmatrix},\, \begin{bmatrix}3&1\end{bmatrix},\, \begin{bmatrix}3&7\end{bmatrix},\, \begin{bmatrix}4&5\end{bmatrix},\, \begin{bmatrix}6&1\end{bmatrix} \right\} \end{equation*} with corresponding matrices \begin{eqnarray*} \begin{bmatrix} 1&0\\0&1 \end{bmatrix},\, \begin{bmatrix} 0&-1\\1&5 \end{bmatrix},\, \begin{bmatrix} 0&-1\\1&11 \end{bmatrix},\, \begin{bmatrix} -1&-1\\2&1 \end{bmatrix},\, \begin{bmatrix} 1&0\\3&1 \end{bmatrix},\, \begin{bmatrix} 1&2\\3&7 \end{bmatrix},\, \begin{bmatrix} 1&1\\4&5 \end{bmatrix},\, \begin{bmatrix} 1&0\\6&1 \end{bmatrix}. \end{eqnarray*} Hence there are $16$ quadratic forms \begin{equation*} \begin{array}{llll} x^2+5y^2,&5x^2+y^2,&41x^2+12xy+y^2,&29x^2-26xy+6y^2,\\ 49x^2+34xy+6y^2,&61x^2-38xy+6y^2,&89x^2+46xy+6y^2,&181x^2-60xy+5y^2,\\ 7x^2+22xy+18y^2,&83x^2+48xy+7y^2,&623x^2+132xy+7y^2,&35x^2+20xy+3y^2,\\ 103x^2-86xy+18y^2,&43x^2-18xy+2y^2,&23x^2-16xy+3y^2,&523x^2-194xy+18y^2 \end{array} \end{equation*} which represent all distinct classes in $\mathcal{Q}_N(d_K)/ \sim_\Gamma=\mathcal{Q}_{12}(-20)/\sim_{\Gamma_0(12)}$. \par On the other hand, for $\begin{bmatrix}r_1&r_2\end{bmatrix}\in M_{1,\,2}(\mathbb{Q}) \setminus M_{1,\,2}(\mathbb{Z})$ the \textit{Siegel function} $g_{\left[\begin{smallmatrix}r_1&r_2\end{smallmatrix}\right]}(\tau)$ is given by the infinite product \begin{eqnarray*} g_{\left[\begin{smallmatrix}r_1&r_2\end{smallmatrix}\right]}(\tau) &=&-e^{\pi\mathrm{i}r_2(r_1-1)} q^{(1/2)(r_1^2-r_1+1/6)} (1-q^{r_1}e^{2\pi\mathrm{i}r_2})\\ &&\times \prod_{n=1}^\infty(1-q^{n+r_1}e^{2\pi\mathrm{i}r_2}) (1-q^{n-r_1}e^{-2\pi\mathrm{i}r_2})\quad(\tau\in\mathbb{H}) \end{eqnarray*} which generalizes the Dedekind eta-function $\displaystyle q^{1/24}\prod_{n=1}^\infty (1-q^n)$. Then the function \begin{equation*} g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12\tau)^{12} =\left(\frac{\eta(6\tau)}{\eta(12\tau)}\right)^{24} \end{equation*} belongs to $\mathcal{F}_{\Gamma_0(12),\,\mathbb{Q}}$ (\cite[Theorem 1.64]{Ono} or \cite{K-L}), and the Galois conjugates of $g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12\tau_K)^{12}$ over $K$ are \begin{equation*} \begin{array}{llll} g_1=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12\sqrt{-5})^{12}, & g_2=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12\sqrt{-5}/5)^{12}, \\ g_3=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(6+\sqrt{-5})/41)^{12}, & g_4=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-13+\sqrt{-5})/29)^{12},\\ g_5=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(17+\sqrt{-5})/49)^{12}, & g_6=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-19+\sqrt{-5})/61)^{12}, \\ g_7=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(23+\sqrt{-5})/89)^{12}, & g_8=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-30+\sqrt{-5})/181)^{12},\\ g_9=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(11+\sqrt{-5})/7)^{12}, & g_{10}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(24+\sqrt{-5})/83)^{12}, \\ g_{11}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(66+\sqrt{-5})/623)^{12}, & g_{12}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(10+\sqrt{-5})/35)^{12},\\ g_{13}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-43+\sqrt{-5})/103)^{12}, & g_{14}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-9+\sqrt{-5})/43)^{12}, \\ g_{15}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-8+\sqrt{-5})/23)^{12}, & g_{16}=g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12(-97+\sqrt{-5})/523)^{12} \end{array} \end{equation*} possibly with some multiplicity. Now, we evaluate \begin{eqnarray*} &&\prod_{i=1}^{16}(x-g_i)\\&=& x^{16}+1251968x^{15}-14929949056x^{14}+1684515904384x^{13} -61912544374756x^{12}\\ &&+362333829428160x^{11} +32778846351721632x^{10}-845856631699319872x^9\\ &&+4605865492693542918x^8+91164259067285621248x^7 -124917935291699694528x^6\\ &&+180920285564131280640x^5 -3000295144057714916x^4+8871452719720384x^3\\ &&+458008762175904x^2 -1597177179712x+1 \end{eqnarray*} with nonzero discriminant. Thus $g_{\left[\begin{smallmatrix} 1/2&0\end{smallmatrix}\right]}(12\tau_K)^{12}$ generates $K_P=H_\mathcal{O}$ over $K$. \end{example}
\begin{remark} In \cite{Schertz} Schertz deals with various constructive problems on the theory of complex multiplication in terms of the Dedekind eta-function and Siegel function. See also \cite{K-L} and \cite{Ramachandra}. \end{remark}
\address{ Applied Algebra and Optimization Research Center\\ Sungkyunkwan University\\ Suwon-si, Gyeonggi-do 16419\\ Republic of Korea} {hoyunjung@skku.edu} \address{ Department of Mathematical Sciences \\ KAIST \\ Daejeon 34141\\ Republic of Korea} {jkkoo@math.kaist.ac.kr} \address{ Department of Mathematics\\ Hankuk University of Foreign Studies\\ Yongin-si, Gyeonggi-do 17035\\ Republic of Korea} {dhshin@hufs.ac.kr}
\end{document} |
\begin{document}
\title{Local H\"older regularity of minimizers for nonlocal variational problems}
\author{ Matteo Novaga\footnote{Universit\`a di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy. E-mail: {\tt matteo.novaga@unipi.it}}\and Fumihiko Onoue\footnote{Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. E-mail: {\tt fumihiko.onoue@sns.it}} }
\date{\today}
\maketitle
\begin{abstract}
We study the regularity of solutions to a nonlocal variational problem, which is related to the image denoising model, and we show that, in two dimensions, minimizers have the same H\"older regularity as the original image. More precisely, if the datum is (locally) $\beta$-H\"older continuous for some $\beta\in(1-s,\,1]$, where $s\in (0,1)$ is a parameter related to the nonlocal operator, we prove that the solution is also $\beta$-H\"older continuous. \end{abstract}
\tableofcontents
\section{Introduction} Let $K:\mathbb{R}^n\setminus\{0\} \to \mathbb{R}$ be a given function and $f:\mathbb{R}^n \to \mathbb{R}$ be a given datum. We study the minimization problem \begin{equation}\label{miniProb}
\min\left\{ \mathcal{F}_{K,f}(u) \mid u \in BV_K(\mathbb{R}^n) \cap L^2(\mathbb{R}^n) \right\} \end{equation} where the functional $\mathcal{F}_{K,f}$ is defined as \begin{equation}\label{funcDenoisingProb}
\mathcal{F}_{K,f}(u) \coloneqq \frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} K(x-y)\,|u(x) - u(y)|\,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^n}(u(x)-f(x))^2\,dx \end{equation}
for any measurable function $u:\mathbb{R}^n \to \mathbb{R}$, and the space $BV_K(\mathbb{R}^n)$ is the set of all functions such that the first term of \eqref{funcDenoisingProb} is finite (see Section \ref{preliminaries} for the detail). The function $K$ is a kernel singular at the origin, and a typical example is the function $x \mapsto |x|^{-(n+s)}$ with $s \in (0,\,1)$. If $K$ is non-negative and we understand that $\mathcal{F}_{K,\,f}(u) = +\infty$ when $u \in L^2(\mathbb{R}^n)\setminus BV_K(\mathbb{R}^n)$, then we observe that the functional $\mathcal{F}_{K,\,f}$ is strictly convex, lower semi-continuous, and coercive in $L^2(\mathbb{R}^n)$. Hence, from the general theory of functional analysis (see, for instance, \cite{Brezis}), we obtain existence and uniqueness of solutions to \eqref{miniProb}.
In this paper we focus on the specific kernel $K(x) = |x|^{-(n+s)}$, with $s\in (0,1)$, and we study the regularity of the minimizers of $\mathcal{F}_{K,f}$, under some suitable conditions on the datum $f$.
Our minimization problem is motivated by the classical variational problem \begin{equation}\label{classicalDenosignProb}
\min\left\{ \mathcal{F}_{f}(u) \mid u \in BV(\mathbb{R}^n) \cap L^2(\mathbb{R}^n) \right\} \end{equation} where $\mathcal{F}_{f}(u)$ is defined as \begin{equation}\label{classicalTotalVariFunc}
\mathcal{F}_{f}(u) \coloneqq \int_{\mathbb{R}^n}|\nabla u|\,dx + \frac{1}{2}\int_{\mathbb{R}^n} |u-f|^2\,dx. \end{equation} Th minimization problem \eqref{classicalDenosignProb} has been studied by many authors since the celebrated work by Rudin, Osher, and Fatemi \cite{ROF}, and plays an important role in image denoising and restoration (see for instance \cite{CCN, Brezis02}). From the perspective of image processing, the datum $f$ in the functional $\mathcal{F}_{f}$ indicates an observed image and, when the given image has poor quality, then the minimizers of $\mathcal{F}_{f}$ or solutions to the Euler-Lagrange equation associated with $\mathcal{F}_{f}$ correspond to regularized images. It is easy to show that the minimizer of \eqref{classicalTotalVariFunc} exists and is unique, as a result of strict convexity, lower semi-continuity and coercivity of the functional. Moreover, the minimizer turns out to be the solution, in a suitable sense, of the Euler-Lagrange equation \begin{equation}\label{classicalTotalVariEq}
-\mathrm{div}\,\left( \frac{\nabla u}{|\nabla u|}\right) + u - f = 0 \quad \text{ in } \mathbb{R}^n. \end{equation}
The regularity of minimizers of $\mathcal{F}_{f}$ have been studied by several authors. In particular, the global and local regularity was investigated in a series of papers by Caselles, Chambolle and Novaga (see \cite{CCN01, CCN02, CCN}), who proved that the solution of \eqref{classicalTotalVariEq} inherits the local H\"older or Lipschitz regularity of the datum $f$, when the space-dimension $n$ is less than or equal to 7. In addition, if $f$ is globally H\"older or Lipschitz in a convex domain $\Omega \subset \mathbb{R}^n$, the global regularity also holds for the solution of \eqref{classicalTotalVariEq} with homogeneous Neumann boundary condition. In the recent papers \cite{Mercier, Porretta}, some of these results were extended to general dimensions. In \cite{Mercier}, Mercier has proved that the continuity of $f$ implies the continuity of a solution $u$ and, in the case of convex domains, the modulus of continuity is also inherited globally by the solution. Eventually, in \cite{Porretta}, Porretta was able to remove the restriction on the space-dimension.
For the variational problems associated with the nonlocal total variation, Aubert and Kornprobst in \cite{AuKo} and Gilboa and Osher in \cite{GiOs, GiOs02} have proposed the methods for approximating the solutions to \eqref{classicalDenosignProb} with a sequence of nonlocal total variations associated with non-singular smooth kernels. However, as far as we know, there are no results on the regularity of minimizers of the functional $\mathcal{F}_{K,f}$. Thus, in this paper, we consider the local H\"older regularity of the minimizers of \eqref{funcDenoisingProb} in two dimension as an analogy of the regularity results shown in \cite{CCN01, CCN02}. Precisely, we prove the following result:
\begin{theorem}\label{mainTheorem}
Let $n=2$. Assume that $K(x) = |x|^{-(2+s)}$ with $s \in (0,\,1)$ and $f \in L^2(\mathbb{R}^2) \cap L^{\infty}(\mathbb{R}^2)$. If $f$ is locally $\beta$-H\"older continuous
with $\beta \in (1-s,\,1]$, then the minimizer $u$ of the functional $\mathcal{F}_{K,f}$ is also locally $\beta$-H\"older continuous in $\mathbb{R}^2$. \end{theorem}
We point out that we cannot show the regularity result in any dimension due to the appearance of singularities on the boundary of the levelsets of minimizers. However, the two-dimensional case is of particular interest for the application to image denoising.
As discussed in the case of the denoising problem in \cite{CCN01, CCN02, CCN}, our regularity result is based on the following observation: if $u$ is a minimizer of the functional $\mathcal{F}_{K,f}$, then the super-level set $\{u > t\}$ for each $t \in \mathbb{R}$ is also a minimizer of the functional associated with the prescribed nonlocal mean curvature problem \begin{equation}\label{prescribedNonlocalMCProb}
\min\left\{ \mathcal{E}_{K,f,t}(E) \mid \text{$E \subset \mathbb{R}^n$: measurable}\right\} \end{equation} where we define the functional $\mathcal{E}_{K,f,t}$ by \begin{equation}\nonumber
\mathcal{E}_{K,f,t}(E) \coloneqq P_K(E) + \int_{E}(t-f(x))\,dx \end{equation} for any measurable set $E \subset \mathbb{R}^n$ and $t\in\mathbb{R}$. Here $P_K$ is the \textit{nonlocal perimeter} associated with the kernel $K$, which is given as \begin{equation}\nonumber
P_K(E) \coloneqq \int_{E}\int_{E^c}K(x-y)\,dx\,dy \end{equation} for any measurable set $E \subset \mathbb{R}^n$ (see Section \ref{preliminaries} for the detail). If $E_t$ is a minimizer of $\mathcal{E}_{K,f,t}$ for each $t$ and $\partial E_t$ is smooth ($C^2$-regularity is sufficient), then we can obtain that the boundary $\partial E_t$ satisfies the following \textit{prescribed nonlocal mean curvature equation} \begin{equation}\nonumber
H^K_{E_t}(x) + t - f(x) = 0 \end{equation} for any $x \in \partial E_t$. Here $H^K_{E_t}$ is the so-called \textit{nonlocal mean curvature} defined by \begin{equation}\label{definitionNonlocalMC}
H^K_{E_t}(x) \coloneqq \text{p.v.}\int_{\mathbb{R}^n}K(x-y)(\chi_{E_t}(x) - \chi_{E_t}(y)) \,dy \end{equation}
for any $x \in \mathbb{R}^n$, where we mean by ``p.v." the Cauchy principal value. Note that, if $K(x) = |x|^{-(N+s)}$, then we denote the nonlocal mean curvature at $x \in \partial E$ associated with $K$ by $H^s_E(x)$. One may observe that, if $\partial E$ is of class $C^{1,\alpha}$ with $\alpha > s$, then $H^s_E$ is finite at each point of $\partial E$.
The idea to show the local H\"older regularity of a minimizer $u$ is based on the observation that the distance between the boundaries of the two super-level sets $\{u>t\}$ and $\{u>t'\}$ for $t,\,t' \in \mathbb{R}$ with $t \neq t'$ should not be too close. To observe this, we compare between the nonlocal mean curvatures of $\partial \{u>t\}$ and $\partial \{u>t'\}$ at the points where the smallest distance between the boundaries $\partial \{u>t\}$ and $\partial \{u>t'\}$ is attained. The comparison can be done thanks to the computations of the first variation of the nonlocal mean curvature shown in \cite{DdPW, JuLa}. Thus we are able to derive some local estimate to assert the local H\"older regularity with the assumption of the local H\"older regularity of $f$.
The organization of this paper is as follows: In Section \ref{preliminaries}, we will introduce the notation related to the nonlocal total variations. In Section \ref{secELeqforEnergy}, we will show the correspondence between the minimizers of $\mathcal{F}_{K,f}$ in \eqref{funcDenoisingProb} and the solutions to the nonlocal 1-Laplace equation. In Section \ref{secComparisonMini}, we will give a sort of comparison principle for the minimizers. As a result of this claim, we will show that, if a datum $f$ is bounded, then the minimizer of $\mathcal{F}_{K,f}$ is also bounded. In Section \ref{secCharacteriMinimizers}, we will show that each super-level set of a minimizer of $\mathcal{F}_{K,f}$ is also a minimizer of $\mathcal{E}_{K,f,t}$ for $t\in\mathbb{R}$. In Section \ref{secBoundednessSuperLeverlsets}, we will show the boundedness of each super-level set of the minimizer and, moreover, this set can be uniformly bounded whenever the minimizer is bounded from below. Finally, by using all the previous results, in Section \ref{secLocalHolderConti} we prove the main theorem in this paper on the H\"older regularity of minimizers in two dimensions.
\section{Notation}\label{preliminaries} In this section, we give several definitions and properties of the space of functions with finite nonlocal total variations. First of all, we define the space $BV_K(\Omega)$ of functions with nonlocal bounded variations associated with the kernel $K$ by \begin{equation}\label{functionBVwithK}
BV_K(\Omega) \coloneqq \left\{u \in L^1(\Omega) \mid [u]_{K}(\Omega) <\infty \right\} \end{equation} where we set, for any measurable function $u$, \begin{equation}\label{seminormNonolocalTV}
[u]_{K}(\Omega) \coloneqq \frac{1}{2}\int_{\Omega}\int_{\Omega} K(x-y)\,|u(x) - u(y)|\,dx\,dy. \end{equation}
We observe that the space $BV_K(\Omega)$ coincides with the fractional Sobolev space $W^{s,\,1}(\Omega)$ when the kernel $K$ is given as $K(x)= |x|^{-(n+s)}$ with $s \in (0,\,1)$ (see, for instance, \cite{NPV}).
Secondly, if we set $\Omega = \mathbb{R}^n$ and substitute a characteristic function $\chi_E$ of a set $E\subset\mathbb{R}^n$ in \eqref{seminormNonolocalTV}, then we obtain the so-called \textit{nonlocal perimeter}. Namely, we define the nonlocal perimeter of a set $E \subset \mathbb{R}^n$ associated with the kernel $K$ by \begin{equation}\label{definitionNonlocalPeri}
P_K(E) \coloneqq \int_{E}\int_{E^c}K(x-y)\,dx\,dy. \end{equation}
In the case that $K(x) = |x|^{-(n+s)}$ for $s \in (0,\,1)$, we call $P_K$ the \textit{$s$-fractional perimeter}, and we denote it by $P_s$. This notion was introduced by Caffarelli, Roquejoffre, and Savin in \cite{CRS}. The authors' work in \cite{CRS} is motivated by the structure of inter-phases when long-range correlations exist (see also \cite{CaSo, Imbert} for the study of interfaces with fractional mean curvatures). After their work, any problems involving not only the $s$-fractional perimeter but also the nonlocal perimeter with the kernel $K$ were studied by many authors. We leave here a short list of papers, which are related to our problems, for those who are interested in the variational problems involving the nonlocal perimeter \cite{AuKo, AGP, Brezis02, CeNo, DRM, MRT01, MRT02} and the references are therein.
Next we can consider a localized version of the nonlocal perimeter $P_K$ as follows: let $\Omega \subset \mathbb{R}^n$ be any domain. Then the nonlocal perimeter in $\Omega$ associated with the kernel $K$ is given by \begin{align}
P_K(E;\Omega) &\coloneqq
\int_{\Omega \cap E}\int_{\Omega \cap E^c} K(x-y)\,dx\,dy + \int_{\Omega \cap E}\int_{\Omega^c \cap E^c} K(x-y) \,dx\,dy \nonumber\\
&\qquad + \int_{\Omega \cap E^c}\int_{\Omega^c \cap E} K(x-y) \,dx\,dy \nonumber \end{align} for any $E\subset \mathbb{R}^n$. Secondly, we give the definition of solutions to the so-called \textit{nonlocal 1-Laplace equations} associated with the kernel $K$. \begin{definition}\label{defSolutionNonlocal1Lap}
Let $F : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ be a measurable function in $L^2(\mathbb{R}^n \times \mathbb{R})$. We say that $u \in BV_K \cap L^2(\mathbb{R}^n)$ is a solution to the nonlocal equation
\begin{equation}\label{nonlocalSemilinearEq}
-\Delta^K_1 u(x) = F(x,\,u(x)) \quad \text{for a.e. $x\in \mathbb{R}^n$}
\end{equation}
if there exists a function $z: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $|z| \leq 1$ a.e. in $\mathbb{R}^n \times \mathbb{R}^n$ and $z(x,\,y) = - z(y,\,x)$ for a.e. $(x,\,y) \in \mathbb{R}^n \times \mathbb{R}^n$ such that
\begin{equation}\label{ELeqforF}
\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z(x,\,y)(v(x) - v(y))\,dx\,dy = \int_{\mathbb{R}^n}F(x,\,u(x))\,v(x)\,dx
\end{equation}
for every $v \in C^{\infty}_c(\mathbb{R}^n)$ with $[v]_K(\mathbb{R}^n) < \infty$ and
\begin{equation}\nonumber
z(x,\,y) \in \mathrm{sgn}\,(u(y) - u(x)) \quad \text{for a.e. $(x,\,y) \in \mathbb{R}^n \times \mathbb{R}^n$}
\end{equation}
where $\mathrm{sgn}\,(x)$ is a generalized sign function satisfying that
\begin{equation}\nonumber
\mathrm{sgn}\,(x) \in [-1,\,1], \quad \mathrm{sgn}\,(x)\,x = |x| \quad \text{for any $x \in \mathbb{R}$}.
\end{equation} \end{definition} We mention that the authors in \cite{MRT01, MRT02} give a similar definition of the nonlocal 1-Laplacian associated with an integrable kernel.
In the present paper, we only consider the case that $F(x,\,u(x)) = u(x) - f(x)$ for a given datum $f$. The concept of the definition is motivated by the Euler-Lagrange equation of the functional \begin{equation}\nonumber
\mathcal{I}_K(u) \coloneqq \frac{1}{2}\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} K(x-y)|u(x)-u(y)|\,dx\,dy. \end{equation} Indeed, when we assume that $u$ is a minimizer of $\mathcal{I}_K$ and consider the first variation of the functional $\mathcal{I}_K$, namely, the quantity $\frac{d}{d\varepsilon}\lfloor_{\varepsilon=0}\mathcal{I}_K(u + \varepsilon\phi)$ for any suitable test function $\phi$, we can formally obtain \begin{equation}\nonumber
\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\frac{u(x) - u(y)}{|u(x) - u(y)|}\,(\phi(x) - \phi(y))\,dx\,dy = 0. \end{equation}
However, it is quite problematic for us to give a rigorous meaning to the ratio $\frac{u(x) - u(y)}{|u(x) - u(y)|}$. To overcome this difficulty, we may apply Definition \ref{defSolutionNonlocal1Lap} and this can be regarded as one of the proper treatments for this issue. Indeed, in Definition \ref{defSolutionNonlocal1Lap}, we may consider the condition that $z(x,\,y)(u(y) - u(x)) = |u(y) - u(x)|$ for a.e. $(x,\,y) \in \mathbb{R}^n \times \mathbb{R}^n$ with $u(x) \neq u(y)$ as a natural requirement. Note that the framework of solutions in the sense of Definition \ref{defSolutionNonlocal1Lap} has been originally developed by, for instance, Maz\'on, Rossi, and Toledo in \cite{MRT} and can be seen as a nonlocal counterpart of the framework given in \cite{ABCM}, \cite{ACM}, and \cite{MRD}.
\section{Preliminary results}
\subsection{Euler-Lagrange equation for $\mathcal{F}_{K,f}$}\label{secELeqforEnergy} In this section, we show the necessary and sufficient condition for the minimizers of the functional $\mathcal{F}_{K,f}$ in $\mathbb{R}^n$. Before stating the claim, we give some conditions on the kernel $K$ which we will assume in the sequel.
\begin{itemize}
\item[(K1)] $K:\mathbb{R}^n \setminus \{0\} \to \mathbb{R}$ is a non-negative measurable function.
\item[(K2)] $K$ is symmetric with respect to the origin, namely $K(-x) = K(x)$ for any $x \in \mathbb{R}^n\setminus \{0\}$. \end{itemize}
We observe that a typical example of the kernel $K$ is given as $K(x)=|x|^{-(n+s)}$ with $s\in(0,\,1)$ and this function satisfies all the above assumptions.
In the following lemma, we show that the minimizer of $\mathcal{F}_{K,f}$ satisfies a prescribed nonlocal mean curvature equation. This equation can be regarded as the Euler-Lagrange equation. Moreover, we show that the converse statement is also valid. \begin{lemma}\label{equivMinimizerAndSolution}
Assume that the kernel $K$ satisfies (K1) and (K2) and a given datum $f$ is $L^2(\mathbb{R}^2)$. If $u \in BV_K \cap L^2(\mathbb{R}^n)$ is a minimizer of the functional $\mathcal{F}_{K,f}$, then $u$ satisfies the equation
\begin{equation}\label{nonlocalEquation00}
-\Delta^K_1 u = u -f \quad \text{in $\mathbb{R}^n$}
\end{equation}
in the sense of Definition \ref{defSolutionNonlocal1Lap}. Conversely, if $u \in BV_K \cap L^2(\mathbb{R}^n)$ is a solution of the equation \eqref{nonlocalEquation00} in the sense of Definition \ref{defSolutionNonlocal1Lap}, then $u$ is a minimizer of $\mathcal{F}_{K,f}$. \end{lemma} \begin{proof}
First, we recall the definition of the functional $\mathcal{I}_{K}$ and the non-negativity of $K$ and thus, find that $\mathcal{I}_K$ is convex, lower semi-continuous, and positive homogeneous of degree one. Then, by using the same argument as in \cite{MRT01, MRT02}, we can show the characterization of the sub-differential of $\mathcal{I}_{K}(u)$ as follows:
\begin{align}\label{characteriSubdiffNonlocalfunc}
&\partial \mathcal{I}_{K}(u) \nonumber\\
& \quad = \left\{ v \in L^2(\mathbb{R}^n) \mid \text{$-\Delta^K_1 u = v$ in the sense of Definition \ref{defSolutionNonlocal1Lap}} \right\}.
\end{align}
Here we recall the definition of the sub-differential $\partial \mathcal{E}(u)$ for $u \in X$ of the functional $\mathcal{E}: X \to \mathbb{R}\cup\{+\infty\}$ where $X$ is the Hilbert space with the inner product $(\cdot,\cdot)_X$. We say that $v \in X$ belongs to $\partial \mathcal{E}(u)$ for each $u \in X$ if it holds that, for any $w \in X$,
\begin{equation}\nonumber
\mathcal{E}(w) - \mathcal{E}(u) \geq (w,\,v)_X.
\end{equation}
Note that $u \in X$ is a minimizer of $\mathcal{E}$ if and only if $0 \in \partial \mathcal{E}(u)$. Then, from the general theory on the sub-differential, we can also show the identity
\begin{equation}\label{characteriSubdifferentialEnergy}
\partial \mathcal{F}_{K,f}(u) = \partial \mathcal{I}_{K}(u) + u-f.
\end{equation}
for any $u \in L^2$. Indeed, if $v \in \partial \mathcal{F}_{K,f}(u)$, then we can compute the functional of $u$ as follows; for any $w \in L^2(\mathbb{R}^n)$,
\begin{align}\label{characteriSubdiffe01}
\mathcal{I}_{K}(w) - \mathcal{I}_{K}(u) &= \mathcal{F}_{K,f}(w) - \mathcal{F}_{K,f}(u) + \frac{1}{2}\int_{\mathbb{R}^n}(u-f)^2\,dx - \frac{1}{2}\int_{\mathbb{R}^n}(w-f)^2 \nonumber\\
&\geq \int_{\mathbb{R}^n}v(w-u)\,dx - \frac{1}{2}\int_{\mathbb{R}^n}(w-u)(w+u-2f)\,dx \nonumber\\
&= \int_{\mathbb{R}^n}(v-u+f)(w-u)\,dx + \int_{\mathbb{R}^n}(u-f)(w-u)\,dx \nonumber\\
&\qquad - \frac{1}{2}\int_{\mathbb{R}^n}(w-u)(w+u-2f)\,dx \nonumber\\
&= \int_{\mathbb{R}^n}(v-u+f)(w-u)\,dx +\frac{1}{2}\int_{\mathbb{R}^n}(w-u)^2\,dx \nonumber\\
&\geq \int_{\mathbb{R}^n}(v-u+f)(w-u)\,dx.
\end{align}
Therefore we obtain $v-u+f \in \partial \mathcal{I}_{K}(u)$. On the other hand, if $v \in \partial \mathcal{I}_{K}(u)+u-f$, then we can compute in the following manner; for any $w \in L^2(\mathbb{R}^n)$, we have
\begin{align}\label{characteriSubdiffe02}
\mathcal{F}_{K,f}(w) - \mathcal{F}_{K,f}(u) &= \mathcal{I}_{K}(w) - \mathcal{I}_{K}(u) + \frac{1}{2}\int_{\mathbb{R}^n}(w-f)^2\,dx - \frac{1}{2}\int_{\mathbb{R}^n}(u-f)^2 \\
&\geq \int_{\mathbb{R}^n} (v-u+f)(w-u)\,dx + \frac{1}{2}\int_{\mathbb{R}^n}(w-u)(w+u - 2f)\,dx \nonumber\\
&= \int_{\mathbb{R}^n}v(w-u)\,dx + \frac{1}{2}\int_{\mathbb{R}^n}(w-u)^2\,dx \nonumber\\
&\geq \int_{\mathbb{R}^n}v(w-u)\,dx,
\end{align}
and thus we have that $v \in \partial \mathcal{F}_{K,f}(u)$. Therefore, from the computations \eqref{characteriSubdiffe01} and \eqref{characteriSubdiffe02}, we conclude that the first part of the claim is valid. Then, from \eqref{characteriSubdifferentialEnergy}, we can easily obtain the equity
\begin{align}\label{identityOverall}
&\partial \mathcal{F}_{K,f}(u) \nonumber\\
&\quad = \left\{ v+u-f \in L^2(\mathbb{R}^n) \mid \text{$-\Delta^K_1 u = v$ in the sense of Definition \ref{defSolutionNonlocal1Lap}} \right\}.
\end{align}
We can readily see that $0 \in \partial \mathcal{F}_{K,f}(u)$ whenever $u$ is a minimizer of $\mathcal{F}_{K,f}$ Therefore, we conclude that, if $u$ is a minimizer of $\mathcal{F}_{K,f}$, then $u$ is a solution of the equation \eqref{nonlocalEquation00}.
Conversely, if $u$ is a solution of the equation \eqref{nonlocalEquation00}, then from \eqref{identityOverall} we have that $0$ belongs to the set in the right-hand side of \eqref{identityOverall}, and thus we obtain $0 \in \partial \mathcal{F}_{K,f}(u)$. \end{proof}
\subsection{Comparison between minimizers}\label{secComparisonMini} In this section, we prove a comparison principle for the minimizers of $\mathcal{F}_{K,f}$. We assume that $K$ satisfies the assumptions (K1) and (K2) shown in Section \ref{secELeqforEnergy} and the data $f_1$ and $f_2$ satisfy that $f_1 \leq f_2$. Then we show that the minimizers $u_1$ and $u_2$ associated with $f_1$ and $f_2$, respectively, preserves the inequality. Precisely, we prove the following result: \begin{lemma}\label{comparisonMini}
Let $f_i$ be in $L^2(\mathbb{R}^n)$ for each $i \in \{1,\,2\}$ and $u_i \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of $\mathcal{F}_{K,f_i}$ for each $i\in\{1,\,2\}$. Assume that the kernel $K : \mathbb{R}^n \setminus \{0\} \to \mathbb{R}$ satisfies (K1) and (K2). If $f_1 \leq f_2$ $\mathcal{L}^n$-a.e. in $\mathbb{R}^n$, then $u_1 \leq u_2$ $\mathcal{L}^n$-a.e. in $\mathbb{R}^n$. \end{lemma} \begin{proof}
Let $u_1,\,u_2 \in BV_K(\mathbb{R}^n)$ be minimizers of $\mathcal{F}_{K,f}$ associated with given data $f_1,\,f_2 \in L^{2}(\mathbb{R}^n)$, respectively. First of all, we prove the following inequality:
\begin{equation}\label{submodularNonlocalTotalVari}
[u_{+}]_{K}(\mathbb{R}^n) + [u_{-}]_{K}(\mathbb{R}^n) \leq [u_1]_{K}(\mathbb{R}^n) + [u_2]_{K}(\mathbb{R}^n).
\end{equation}
Indeed, setting
\begin{equation}\label{maxMin}
u_{+}(x) \coloneqq \max\{u_1(x),\,u_2(x)\}, \quad u_{-}(x) \coloneqq \min\{u_1(x),\,u_2(x)\}
\end{equation}
for any $x \in \mathbb{R}^n$ and by the co-area formula, we have that
\begin{align}\label{coareaFormulaTV}
[u_{i}]_{K}(\mathbb{R}^n) &= \int_{-\infty}^{\infty}\,\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,|\chi_{\{u_{i}>t\}}(x) - \chi_{\{u_{i}>t\}}(y)| \,dx\,dy\,dt \nonumber\\
&= \int_{-\infty}^{\infty} P_K(\{u_{i} > t\}) \,dt
\end{align}
for any $i \in \{1,\,2,\,+,\,-\}$. We recall that the nonlocal perimeter $P_K$ is sub-modular, namely, it holds that
\begin{equation}\label{submodularNonlocalPeri}
P_K(E \cup F) + P_K(E \cap F) \leq P_K(E) + P_K(F)
\end{equation}
for any $E,\,F\subset \mathbb{R}^n$. Therefore from \eqref{submodularNonlocalPeri} and the definitions of $u_{+}$ and $u_{-}$, we obtain the claim.
Now from the general theory of calculus of variations, the minimizer of $\mathcal{F}_{K,f}$ is unique in $L^2(\mathbb{R}^n)$ and thus, it is sufficient to prove that
\begin{equation}\nonumber
\mathcal{F}_{K,f_2}(u_{+}) \leq \mathcal{F}_{K,f_2}(u_2)
\end{equation}
where $u_{+}$ is defined in \eqref{maxMin} to obtain the lemma. From a simple computation, we can easily see that the inequality
\begin{equation}\label{computationMinimizers}
(u_{-}- f_1)^2 + (u_{+}- f_2)^2 \leq (u_1- f_1)^2 + (u_2- f_2)^2
\end{equation}
in $\mathbb{R}^n$. From the minimality of $u_i$ for $i\in\{1,\,2\}$, we have
\begin{equation}\label{estimate01}
\mathcal{F}_{K,f_1}(u_1) + \mathcal{F}_{K,f_2}(u_2) \leq \mathcal{F}_{K,f_1}(u_{-}) + \mathcal{F}_{K,f_2}(u_{+}).
\end{equation}
On the other hand, from \eqref{submodularNonlocalTotalVari} and \eqref{computationMinimizers}, we have
\begin{align}\label{estimate02}
&\mathcal{F}_{K,f_1}(u_{-}) + \mathcal{F}_{K,f_2}(u_{+}) \\
&\leq [u_{-}]_{K}(\mathbb{R}^n) + \frac{1}{2}\int_{\mathbb{R}^n}(u_{-}- f_1)^2\,dx + [u_{+}]_{K}(\mathbb{R}^n) + \frac{1}{2}\int_{\mathbb{R}^n}(u_{+}- f_2)^2\,dx \nonumber\\
&= [u_1]_{K}(\mathbb{R}^n) + \frac{1}{2}\int_{\mathbb{R}^n}(u_1- f_1)^2\,dx + [u_2]_{K}(\mathbb{R}^n) + \frac{1}{2}\int_{\mathbb{R}^n}(u_2- f_2)^2\,dx \nonumber\\
&\quad +\frac{1}{2}\int_{\mathbb{R}^n}(u_{-}- f_1)^2\,dx - \frac{1}{2}\int_{\mathbb{R}^n}(u_1- f_1)^2\,dx \nonumber\\
&\quad \quad + \frac{1}{2}\int_{\mathbb{R}^n}(u_{+}- f_2)^2\,dx - \frac{1}{2}\int_{\mathbb{R}^n}(u_2- f_2)^2\,dx \nonumber\\
&\leq \mathcal{F}_{K,f_1}(u_1) + \mathcal{F}_{K,f_2}(u_2).
\end{align}
Thus from \eqref{estimate01} and \eqref{estimate02}, we obtain
\begin{equation}\label{identityEnergies}
\mathcal{F}_{K,f_1}(u_1) + \mathcal{F}_{K,f_2}(u_2) = \mathcal{F}_{K,f_1}(u_{-}) + \mathcal{F}_{K,f_2}(u_{+})
\end{equation}
Now suppose by contradiction that $\mathcal{F}_{K,f_2}(u_{+}) > \mathcal{F}_{K,f_2}(u_2)$. Then from \eqref{identityEnergies} we have
\begin{equation}\nonumber
\mathcal{F}_{K,f_1}(u_1) > \mathcal{F}_{K,f_1}(u_{-})
\end{equation}
which contradicts the minimality of $u_1$. Thus we obtain the inequality $\mathcal{F}_{K,f_2}(u_{+}) \leq \mathcal{F}_{K,f_2}(u_2)$. Therefore, by the uniqueness of the minimizer of $\mathcal{F}_{K}$ in $L^2(\mathbb{R}^n)$, we obtain that $u_{+} = u_2$ a.e. in $\mathbb{R}^n$, which implies that $u_2 \geq u_1$ a.e. in $\mathbb{R}^n$. \end{proof}
\begin{corollary}\label{nonnegativityMini}
Assume that the kernel $K: \mathbb{R}^n \setminus \{0\} \to \mathbb{R}$ satisfies the assumptions (K1) and (K2) in Section \ref{secELeqforEnergy}. If a datum $f \in L^2(\mathbb{R}^n)$ is non-negative a.e. in $\mathbb{R}^n$, then the minimizer $u \in BV_K \cap L^2(\mathbb{R}^n)$ is also non-negative a.e. in $\mathbb{R}^n$. \end{corollary} \begin{proof}
Since it holds that
\begin{equation}\nonumber
\mathcal{F}_{K,0}(0) = 0 \leq \mathcal{F}_{K,0}(v)
\end{equation}
for every $v \in BV_K \cap L^2(\mathbb{R}^n)$, we have that the unique solution of the problem
\begin{equation}\nonumber
\inf\{\mathcal{F}_{K,0}(v) \mid v \in BV_K \cap L^2\}
\end{equation}
is $v=0$. Hence, by applying Lemma \ref{comparisonMini} to the case that $f_1=0$ and $f_2=f$, we obtain that $0 \leq u$ a.e. in $\mathbb{R}^n$. \end{proof}
Finally, we show a sort of comparison property of minimizers under the assumption that a datum $f$ is bounded in $\mathbb{R}^n$. We do not derive the following proposition directly from Lemma \ref{comparisonMini} but from a simple computation.
\begin{proposition}\label{comparisonLinfty}
Let $u \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of $\mathcal{F}_{K,f}$ with a datum $f \in L^2(\mathbb{R}^n)$. Assume that the kernel $K:\mathbb{R}^n \setminus \{0\} \to \mathbb{R}$ is non-negative measurable function. If there exists a constant $C>0$ such that $|f(x)| \leq C$ for a.e. $x \in \mathbb{R}^n$, then $|u(x)| \leq C$ for a.e. $x \in \mathbb{R}^n$ with the same constant $C$. \end{proposition} \begin{proof}
It is sufficient to show that, if $f \leq C$ a.e. in $\mathbb{R}^n$ with some constant $C>0$, then $u \leq C$ a.e. in $\mathbb{R}^n$ with the same constant $C$ because we only repeat the same argument as we show in this proof. We define $v(x) \coloneqq \min\{u(x),\,C\}$ for $x \in \mathbb{R}^n$. It is sufficient to show that $u = v$ for a.e. in $\mathbb{R}^n$. From the definition, we can show the claim that $|v(x) - v(y)| \leq |u(x) - u(y)|$ for $x,\,y \in \mathbb{R}^n$. Indeed, if $u(x) \leq C$ and $u(y) \leq C$ or $u(x) > C$ and $u(y) > C$, then we can readily obtain the claim. If $u(x) \leq C$ and $u(y) > C$, then we have
\begin{align}
|u(x) - u(y)|^2 - |v(x) - v(y)|^2 &= u^2(y) - C^2 - 2u(x)\,u(y) + 2u(x)\,C \nonumber\\
&= (u(y)-C)(u(y) + C - 2u(x)) \geq 0. \nonumber
\end{align}
In the same way, we can prove the claim if $u(x) > C$ and $u(y) \leq C$. Moreover, we can show that $(v-f)^2 \leq (u-f)^2$ in $\mathbb{R}^n$. Therefore we compute the functional associated with $v$ as follows:
\begin{align}
\mathcal{F}_{K,f}(v) &= \frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,|v(x)-v(y)|\,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^n}(v-f)^2 \,dx \nonumber\\
& \leq \frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,|u(x)-u(y)|\,dx\,dy + \frac{1}{2}\int_{\mathbb{R}^n}(u-f)^2 \,dx \nonumber\\
&= \mathcal{F}_{K,f}(u). \nonumber
\end{align}
Thus, from the uniqueness of the minimizer of $\mathcal{F}_{K,f}$ in $L^2(\mathbb{R}^n)$, we obtain $v = u$ a.e. in $\mathbb{R}^n$ and this concludes the proof. \end{proof}
\subsection{Characterization of minimizers for $\mathcal{F}_{K,f}$}\label{secCharacteriMinimizers} In this section, we show the following claim which gives a relation between the minimizers of $\mathcal{F}_{K,f}$ and $\mathcal{E}_{K,f,t}$ for $t \in \mathbb{R}$. Recall that $\mathcal{E}_{K,f,t}(E)$ as \begin{equation}\label{nonlocalPeriEnergy}
\mathcal{E}_{K,f,t}(E) \coloneqq P_K(E) + \int_{E}(t-f(x))\,dx \end{equation} for every measurable set $E \subset \mathbb{R}^n$ where we assume that $f \in L^2(\mathbb{R}^n)$ is a given datum and $t\in\mathbb{R}$ is any number.
\begin{lemma}\label{relationMiniTwoEnergies}
Assume that the kernel $K(x) = |x|^{-(n+s)}$ for $x \in \mathbb{R}^n \setminus \{0\}$ with $s \in (0,\,1)$ and a datum $f \in L^2 \cap L^{\infty}(\mathbb{R}^n)$. If $u \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of $\mathcal{F}_{K,f}$, then the set $\{x\in\mathbb{R}^n \mid u(x)>t\}$ is also a minimizer of $\mathcal{E}_{K,f,t}(E)$ for every $t\in\mathbb{R}$ among measurable sets $E \subset \mathbb{R}^n$. \end{lemma} \begin{proof}
Let $F \subset \mathbb{R}^n$ be any measurable set. We may assume that $P_K(F) < \infty$; otherwise this set cannot minimize the functional $\mathcal{E}_{K,f,t}$. Moreover, we may assume that $\|\chi_F\|_{L^1} = |F| <\infty$ because of the nonlocal isoperimeteric inequality. Then it suffices to show that the super-level set $\{u > t\}$ for each $t\in\mathbb{R}$ satisfies the inequality
\begin{equation}\label{minimalityInequality}
P_K(\{u > t\}) + \int_{\{u > t\}}(t-f(x))\,dx \leq P_K(F) + \int_{F}(t-f(x))\,dx.
\end{equation}
From Lemma \ref{equivMinimizerAndSolution} and the assumption that $u$ is a minimizer of the functional $\mathcal{F}_{K,f}$, we have that $u$ is also a solution of the equation
\begin{equation}\label{nonlocalEq}
-\Delta^K_1 u = u -f \quad \text{ in }\mathbb{R}^n.
\end{equation}
Thus, from Definition \ref{defSolutionNonlocal1Lap}, there exists a function $z_u\in L^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ with $|z_u| \leq 1$ and $z_u$ being antisymmetric such that
\begin{equation}\label{defSolNonlocalEq}
\frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,(w(x) - w(y))\,dx\,dy = \int_{\mathbb{R}^n}(u-f)\,w(x)\,dx
\end{equation}
for any $w \in C^{\infty}_c(\mathbb{R}^n)$ with $[w]_K(\mathbb{R}^n) < \infty$ and moreover
\begin{equation}\label{propSolNonlocalEq}
z_u(x,\,y)(u(y) - u(x)) = |u(y) - u(x)|
\end{equation}
for a.e. $x,\,y\in \mathbb{R}^n$. From the co-area formula, we have the following two identities:
\begin{equation}\label{coarea01}
|u(x) - u(y)| = \int_{-\infty}^{+\infty}|\chi_{\{u > t\}}(x) - \chi_{\{u > t\}}(y)|\,dt
\end{equation}
and
\begin{equation}\label{coarea02}
(u(x) - u(y)) = \int_{-\infty}^{+\infty}(\chi_{\{u > t\}}(x) - \chi_{\{u > t\}}(y))\,dt
\end{equation}
for any measurable $u:\mathbb{R}^n \to \mathbb{R}$ and a.e. $x,\,y \in \mathbb{R}^n$. Thus from \eqref{propSolNonlocalEq}, \eqref{coarea01}, and \eqref{coarea02}, we obtain
\begin{equation}\label{identityLevelset}
z_u(x,\,y)(\chi_{\{u > t\}}(y) - \chi_{\{u > t\}}(x)) = |\chi_{\{u > t\}}(y) - \chi_{\{u > t\}}(x)|
\end{equation}
for a.e. $t \in \mathbb{R}$. Now we fix $t \in \mathbb{R}$ such that \eqref{identityLevelset} holds. From the specific choice of $K(x) = |x|^{-(n+s)}$, the function space $BV_K(\mathbb{R}^n)$ coincides with the fractional Sobolev space $W^{s,1}(\mathbb{R}^n)$. Recall that the space $C^{\infty}_c(\mathbb{R}^n)$ of smooth functions with compact supports is dense in $W^{s,1}(\mathbb{R}^n)$ (see \cite{Adam} for the detail). Hence, from the fact that $P_K(\{u>t\})$ and $P_K(F)$ are finite, we can choose sequences $\{\eta_{l}^u\}_{l\in\mathbb{N}}$ and $\{\eta_{l}^F\}_{l\in\mathbb{N}}$ in $C^{\infty}_c(\mathbb{R}^n)$ such that
\begin{equation}\label{approximationCharacterFunc}
\eta_{l}^u \xrightarrow[l \to \infty]{} \chi_{\{u>t\}}, \quad \eta_{l}^F \xrightarrow[l \to \infty]{} \chi_{F} \quad \text{in $W^{s,1}(\mathbb{R}^n)$}.
\end{equation}
From the choice of the approximation, we notice that the difference function $\eta_{l}^u - \eta_{l}^F$ is also in $C^{\infty}_c(\mathbb{R}^n)$ and $[\eta_{l}^u - \eta_{l}^F]_K(\mathbb{R}^n) < \infty$ for each $l \in \mathbb{N}$. Hence, from the definition of solutions to the equation \eqref{nonlocalEq}, we obtain
\begin{align}\label{identitySubstitutionCompetitor}
&\int_{\mathbb{R}^n}(u-f)\,(\eta_{l}^u - \eta_{l}^F)\,dx \nonumber\\
&= -\frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,[(\eta_{l}^u - \eta_{l}^F)(y) - (\eta_{l}^u - \eta_{l}^F)(x)]\,dx\,dy \nonumber\\
&= -\frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,(\eta_{l}^u(y) - \eta_{l}^u(x))\,dx\,dy \nonumber\\
&\qquad + \frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,(\eta_{l}^F(y) - \eta_{l}^F(x))\,dx\,dy.
\end{align}
By applying Proposition \ref{comparisonLinfty} and from the assumption that $f \in L^{\infty}(\mathbb{R}^n)$, we have that the minimizer $u$ is also in $L^{\infty}(\mathbb{R}^n)$ and thus
\begin{equation}\label{convergenceLinearTerm}
\left|\int_{\mathbb{R}^n}(u-f)(\eta_{l}^u - \eta_{l}^F)\,dx - \int_{\mathbb{R}^n}(u-f)(\chi_{\{u>t\}} - \chi_{F})\,dx \right| \xrightarrow[l \to \infty]{} 0.
\end{equation}
Hence by applying the dominated convergence theorem and from \eqref{approximationCharacterFunc}, \eqref{identitySubstitutionCompetitor}, and \eqref{convergenceLinearTerm}, we obtain that
\begin{align}\label{identitySubstitutionLimit}
&\int_{\mathbb{R}^n}(u-f)(\chi_{\{u>t\}} - \chi_{F})\,dx \nonumber\\
&= \lim_{l \to \infty}\int_{\mathbb{R}^n}(u-f)\,(\eta_{l}^u - \eta_{l}^F) \,dx \nonumber\\
&= -\frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,(\chi_{\{u>t\}}(y) - \chi_{\{u>t\}}(x))\,dx\,dy \nonumber\\
&\qquad -\frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y) \,z_u(x,\,y)\,(\chi_{F}(y) - \chi_{F}(x))\,dx\,dy.
\end{align}
From the definition of $z_u$, we have
\begin{align}\label{perimeterAnySetF}
&\frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)\,z_u(x,\,y)\,(\chi_{F}(x) - \chi_{F}(y))\,dx\,dy \nonumber\\
&\leq \frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)|\chi_{F}(x) - \chi_{F}(y)|\,dx\,dy = P_K(F).
\end{align}
Taking into account \eqref{identityLevelset}, \eqref{identitySubstitutionLimit}, and \eqref{perimeterAnySetF}, we obtain
\begin{align}\label{periEstimateOverall01}
&\int_{\mathbb{R}^n}(u-f)\,(\chi_{\{u>t\}} - \chi_{F})\,dx \nonumber\\
&\leq - \frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)|\chi_{\{u>t\}}(x) - \chi_{\{u>t\}}(y)| \,dx\,dy + P_{K}(F).
\end{align}
Regarding the left-hand side of \eqref{periEstimateOverall01}, we have
\begin{align}\label{estimateLHS}
\int_{\mathbb{R}^n}(u-f)\,(\chi_{\{u>t\}} - \chi_{F}) \,dx &= \int_{\mathbb{R}^n}(u-t+t-f)\,(\chi_{\{u>t\}} - \chi_{F}) \,dx \nonumber\\
&\geq \int_{\{u > t\} \cap F^c} (t-f) \,dx - \int_{\{u \leq t\} \cap F} (u-f) \,dx \nonumber\\
&\geq \int_{\{u > t\} \cap F^c} (t-f) \,dx - \int_{\{u \leq t\} \cap F} (t-f) \,dx \nonumber\\
&= \int_{\mathbb{R}^n}(t-f)\,(\chi_{\{u>t\}} - \chi_{F})\,dx
\end{align}
for a.e. $t\in\mathbb{R}$. Hence, from \eqref{periEstimateOverall01} and \eqref{estimateLHS}, we have
\begin{align}\label{periEstimateOverall02}
&P_K(\{u>t\}) + \int_{\mathbb{R}^n}(t-f)\,\chi_{\{u > t\}}\,dx \nonumber\\
&= \frac{1}{2} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}K(x-y)|\chi_{\{u>t\}}(x) - \chi_{\{u>t\}}(y)|\,dx\,dy + \int_{\mathbb{R}^n}(t-f)\,\chi_{\{u > t\}}\,dx \nonumber\\
&\leq P_{K}(F) + \int_{\mathbb{R}^n} (t-f)\,\chi_{F}\,dx
\end{align}
for a.e. $t\in\mathbb{R}$. Therefore we conclude that the inequality \eqref{minimalityInequality} holds for a.e. $t\in\mathbb{R}$. Notice that, for any $t\in\mathbb{R}$ such that \eqref{identityLevelset} does not hold, we can choose a sequence $\{t_j\}_{j\in\mathbb{N}}$ such that $t_j \rightarrow t$ as $j \to \infty$ and \eqref{identityLevelset} holds for any $t_j$; otherwise we can choose a constant $\delta>0$ such that $B_{\delta}(t) \subset \{t\in\mathbb{R} \mid \text{\eqref{identityLevelset} is not true}\}$. Since the condition \eqref{identityLevelset} holds true for a.e. $t \in \mathbb{R}$, we have that
\begin{equation}\nonumber
0< 2\delta = |B_{\delta}(t)| \leq |\{t\in\mathbb{R} \mid \text{\eqref{identityLevelset} is not true}\}| = 0,
\end{equation}
which is a contradiction. Thus from the lower semi-continuity of $P_K$ and the continuity of the map $t \mapsto |\{u>t\}|$, we conclude that \eqref{minimalityInequality} holds for every $t\in\mathbb{R}$. \end{proof}
\subsection{Boundedness of super-level sets of minimizers}\label{secBoundednessSuperLeverlsets} Let $u \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of $\mathcal{F}_{K,f}$ with a datum $f \in L^p(\mathbb{R}^n)$ with $p \in (\frac{n}{s},\,\infty]$. In this section, we show that the super-level set $\{u>t\}$ for each $t \in \mathbb{R}$ is bounded up to negligible sets. Precisely, we prove \begin{lemma}\label{boundednessMinimizers}
Assume that the kernel $K(x) = |x|^{-(n+s)}$ for $x \in \mathbb{R}^n \setminus \{0\}$ with $s \in (0,\,1)$ and $f \in L^p(\mathbb{R}^n)$ with $p \in (\frac{n}{s},\,\infty]$. If $E_T$ is a minimizer of $\mathcal{E}_{K,f,T}$ among sets with finite volumes for any $T\in\mathbb{R}$, then there exists a constant $R_T >0$ such that $|E_T \setminus B_{R_T}|=0$. \end{lemma} \begin{proof}
We basically follow the proof shown in \cite[Proposition 3.2]{CeNo}. Suppose by contradiction that $|E_T \setminus B_r| > 0$ for any $r > 0$. By setting $\phi_T(r) \coloneqq |E_T \setminus B_r|$ for any $r >0$, we have
\begin{equation}\nonumber
(\phi_T)'(r) = - \mathcal{H}^{n-1}(E_T \cap \partial B_r)
\end{equation}
for a.e. $r>0$. We fix any $R > 1$. From the minimality of $E_T$, we have
\begin{equation}\label{inequalityByMinimality00}
\mathcal{E}_{K,f,T}(E_T) \leq \mathcal{E}_{K,f,T}(E_T \cap B_r).
\end{equation}
Since it holds that
\begin{equation}\nonumber
P_K(A \cup B) = P_K(A) + P_K(B) - 2 \int_{A}\int_{B} K(x-y)\,dx\,dy
\end{equation}
for sets $A,\,B \subset \mathbb{R}^n$ with $A\cap B = \emptyset$, we have
\begin{equation}\label{inequalityByMinimality}
P_K(E_T\setminus B_r) \leq 2\int_{ E_T \cap B_r}\int_{E_T\setminus B_r
} K(x-y)\,dx\,dy - \int_{E_T \setminus B_r}(T - f(x))\,dx.
\end{equation}
From the isoperimetric inequality for the nonlocal perimeter, we can have the following lower bound of the term of the left-hand side in \eqref{inequalityByMinimality} (see for instance \cite{FFMMM}):
\begin{equation}\label{isoperiNonlocalPeri}
P_K(E_T\setminus B_r) \geq \frac{P_K(B_1)}{|B_1|^{\frac{n-s}{n}}}\,|E_T\setminus B_r|^{\frac{n-s}{n}} = C(n,s)\,\phi_T^{\frac{n-s}{n}}(r)
\end{equation}
for $r \geq R$, where we set $C(n,s) \coloneqq |B_1|^{-\frac{n-s}{n}}\,P_K(B_1)$. Secondly, from Fubini-Tonelli's theorem and the co-area formula, we can compute the first term of the right-hand side in \eqref{inequalityByMinimality} as follows:
\begin{align}\label{estifromMini01}
\int_{E_T \cap B_r}\int_{E_T\setminus B_r} K(x-y)\,dx\,dy &\leq \int_{E_T\setminus B_r}\int_{B_{|y|-r}(y)} \frac{1}{|x-y|^{n+s}}\,dx\,dy \nonumber\\
&= \int_{E_T\setminus B_r}|\mathbb{S}^{n-1}|\int_{|y|-r}^{\infty} \frac{1}{r^{1+s}}\,dr\,dy \nonumber\\
&\leq \frac{|\mathbb{S}^{n-1}|}{s}\int_{E_T \setminus B_r}(|y|-r)^{-s}\,dy \nonumber\\
&= \frac{|\mathbb{S}^{n-1}|}{s} \int_{r}^{+\infty}\frac{\mathcal{H}^{n-1}(E_T \cap \partial B_{\sigma})}{(\sigma - r)^s}\,d\sigma \nonumber\\
&= -\frac{|\mathbb{S}^{n-1}|}{s} \int_{r}^{+\infty} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,d\sigma
\end{align}
for any $r \geq R$. Finally, regarding the second term of the right-hand side in \eqref{inequalityByMinimality}, from the assumption of $f$ and Cauchy-Schwartz inequality (if $p \neq \infty$), we have
\begin{align}\label{estifromMini02}
\int_{E_T \setminus B_r }(-T + f(x))\,dx &\leq T\,|E_T \setminus B_r| + \|f\|_{L^p(\mathbb{R}^n)}\,|E_T \setminus B_r|^{\frac{1}{q}} \nonumber\\
&= T\,\phi_T(r) + \|f\|_{L^p(\mathbb{R}^n)}\,\phi_T^{\frac{1}{q}}(r) < \infty
\end{align}
for any $r \geq R>1$ where $q \geq 1$ satisfies $p^{-1}+q^{-1}=1$. By combining all the computations \eqref{isoperiNonlocalPeri}, \eqref{estifromMini01}, and \eqref{estifromMini02} with \eqref{inequalityByMinimality}, we obtain
\begin{equation}\label{estiByMinimality}
C(n,s)\,\phi_T^{\frac{n-s}{n}}(r) \leq -C_1 \int_{r}^{+\infty} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,d\sigma + T\,\phi_T(r) + \|f\|_{L^p(\mathbb{R}^n)}\,\phi_T^{\frac{1}{q}}(r)
\end{equation}
for any $r \geq R$ where we set $C_1 \coloneqq \frac{|\mathbb{S}^{n-1}|}{s}$. Since $\phi_T(r)$ vanishes as $r \to \infty$ and $\frac{1}{q} > \frac{n-s}{n}$, we can have that
\begin{equation}\nonumber
2T\,\phi_T(r) + 2\|f\|_{L^p(\mathbb{R}^n)}\phi_T^{\frac{1}{q}}(r) \leq C(n,s)\,\phi_T^{\frac{n-s}{n}}(r)
\end{equation}
for sufficiently large $r \geq R$. Hence, by integrating the both sides of \eqref{estiByMinimality} over $r \in (R,\infty)$, we obtain
\begin{equation}\label{estiByMinimality01}
\frac{C(n,s)}{2}\int_{R}^{\infty}\phi_T^{\frac{n-s}{n}}(r)\,dr \leq -C_1 \int_{R}^{\infty}\int_{r}^{+\infty} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,d\sigma\,dr.
\end{equation}
By exchanging the order of the integration, we have
\begin{equation}\label{exchangeIntergration}
\int_{R}^{\infty}\int_{r}^{+\infty} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,d\sigma\,dr = \int_{R}^{\infty}\int_{R}^{\sigma} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,dr\,d\sigma.
\end{equation}
Then by employing the similar computation shown in \cite{CeNo}, we obtain
\begin{equation}\nonumber
\int_{R}^{\infty}\int_{R}^{\sigma} \frac{(\phi_T)'(\sigma)}{(\sigma - r)^s}\,dr\,d\sigma \geq -\frac{\phi_T(R)}{1-s} - \int_{R+1}^{\infty} \frac{\phi_T(r)}{(\sigma - R)^s}\,d\sigma.
\end{equation}
Therefore, from \eqref{estiByMinimality01}, we have
\begin{align}
\frac{C(n,s)}{2}\int_{R}^{\infty}\phi_T^{\frac{n-s}{n}}(r)\,dr &\leq C_1\frac{\phi_T(R)}{1-s} + C_1\int_{R+1}^{\infty} \frac{\phi_T(\sigma)}{(\sigma - R)^s}\,d\sigma \nonumber\\
&\leq C_1\frac{\phi_T(R)}{1-s} + C_1\int_{R+1}^{\infty}\phi_T(\sigma)\,d\sigma. \nonumber
\end{align}
Again, by choosing $R$ sufficiently large so that the inequality
\begin{equation}\nonumber
C_1\int_{R+1}^{\infty}\phi_T(r)\,dr \leq \frac{C(n,s)}{4}\int_{R}^{\infty}\phi_T^{\frac{n-s}{n}}(r)\,dr
\end{equation}
holds, we have
\begin{equation}\nonumber
\int_{R}^{\infty}\phi_T^{\frac{n-s}{n}}(r)\,dr \leq \frac{4C_1}{C(n,s)(1-s)}\,\phi_T(R).
\end{equation}
Then by applying the method shown in, for instance, \cite{CNRV, CeNo}, we obtain the contradiction to the assumption that $\phi_T(r)>0$ for any $r>0$. Therefore, we conclude the essential boundedness of the set $E_T$. \end{proof}
We assume that $u \in BV_K \cap L^2(\mathbb{R}^n)$ is a minimizer of the functional $\mathcal{F}_{K,f}$ and $u$ is bounded from below with the constant $c \in \mathbb{R}$. Then, since the super-level set $\{u > c\}$ is also a minimizer of $\mathcal{E}_{K,f,c}$, we may obtain from Lemma \ref{boundednessMinimizers} that there exists a constant $R_c>1$ such that $|\{u>c\} \setminus B_{R_c}| = 0$. In addition to this, we have the inclusion of the super-level sets that $\{u>t'\} \subset \{u>t\}$ for any $t' > t$. Thus, we conclude that the following corollary holds. \begin{corollary}\label{corUniformBoundednessMinimizer}
Assume that the kernel $K(x) = |x|^{-(n+s)}$ for $x \in \mathbb{R}^n \setminus \{0\}$ with $s \in (0,\,1)$. Let $u \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of $\mathcal{F}_{K,f}$. If a datum $f$ is in $L^p(\mathbb{R}^n)$ with $p \in (\frac{n}{s},\,\infty]$ and $u \geq c$ a.e. in $\mathbb{R}^n$ for some $c \in \mathbb{R}$, then the super-level set $\{u>t\}$ is uniformly bounded with respect to $t \geq c$. Namely, there exists $R_c>0$, independent of $t$, such that $\{u>t\} \subset B_{R_c}$ for any $t \geq c$. \end{corollary}
\section{H\"older regularity of minimizers}\label{secLocalHolderConti}
First of all, we observe that, if $u$ satisfies some equation associated with the Euler-Lagrange equations of $\mathcal{E}_{K,f,t}$ and the boundary of $\{u>t\}$ is regular, then $u$ is continuous.
\begin{proposition}\label{continuityMinimizersLemma}
Assume that $K(x) = |x|^{-(N+s)}$ for any $x \in \mathbb{R}^n$ with $s \in (0,\,1)$ and the datum $f$ is in $L^2 \cap L^{\infty}(\mathbb{R}^n)$.
Let $u \in BV_K \cap L^2(\mathbb{R}^n)$. Assume that $\partial \{u > t\}$ is of class $C^{1,\alpha}$ with $\alpha \in (s,\,1]$ and $u$ satisfies the equation
\begin{equation*}
H^s_{\{u>t\}}(x) + t - f(x) = 0
\end{equation*}
for any $x \in \partial \{u>t\}$ and $t\in\mathbb{R}$. Then $u$ is continuous in $\mathbb{R}^n$. \end{proposition} \begin{proof}
Suppose by contradiction that $u$ is not continuous in $\mathbb{R}^n$. Then there exist a point $x_0 \in \mathbb{R}^n$ and $-\infty< t' < t < \infty$ such that $x_0 \in \partial E_t \cap \partial E_{t'}$. Indeed, if $u$ is not continuous at $x_0$, then it holds that $t_{+} \coloneqq \limsup_{x \to x_0}u(x) > \liminf_{x \to x_0}u(x) \eqqcolon t_{-}$. Note that $t_{+} \geq u(x_0) \geq t_{-}$ by definition.
Setting $\delta \coloneqq t_{+} - t_{-}>0$ and the definition of $t_{+}$, we can choose a sequence $\{x_n\}_{n\in\mathbb{N}}$ such that $x_k \to x_0$ and $u(x_k) > t_{+} - \frac{\delta}{2^{k}}$ for any $k\in\mathbb{N}$ with $k \geq 1$. If $u(x_0) = t_{+}$, then we have that $x_k \in \{u > u(x_0) - \frac{\delta}{2}\}$ for large $k\in\mathbb{N}$. Thus we obtain that $x_0 \in \overline{\{u > u(x_0) - \frac{\delta}{2} \}}$. However, from the definition of $\delta$, $x_0$ cannot be a interior point of $\{u > u(x_0) - \frac{\delta}{2} \}$; otherwise we can choose a sequence $\{y_k\}_{k\in\mathbb{N}}$ such that
\begin{equation}
u(x_0) - \frac{\delta}{2} < u(y_k) < t_{-} + \frac{\delta}{2^{k}}
\end{equation}
for any large $k$. From the definition of $\delta$ and the fact that $u(x_0)=t_{+}$, we obtain a contradiction. Thus we may assume that $u(x_0) < t_{+}$. Setting $\tilde{\delta} \coloneqq t_{+} - u(x_0)>0$ and since $u(x_k) > t_{+} - \frac{\delta}{2^{k}}$ for any $k\in\mathbb{N}$, we have that $u(x_k) > u(x_0) + \frac{1}{2}\tilde{\delta}$ for any $k\in\mathbb{N}$ with $k \geq (2\delta)^{-1}\tilde{\delta}$ and that $x_k \in \{u > u(x_0) + \frac{1}{2}\tilde{\delta}\}$ for large $k\in\mathbb{N}$. Hence, recalling that $x_k \to x_0$ as $k\to\infty$, we obtain that $x_0 \in \partial \{u > u(x_0) + \frac{1}{2}\tilde{\delta} \}$. In the same way, we can show that $x_0 \in \partial \{u > u(x_0) + \frac{3}{4}\tilde{\delta} \}$. Therefore, we conclude that, if $u$ is not continuous at $x_0$, we can find distinct constants $t,\,t'\in\mathbb{R}$ such that $x_0 \in \partial \{u>t\} \cap \partial \{u>t'\}$.
From the assumptions, we obtain that the following equations hold:
\begin{equation}\label{eulerLagrange01}
H_{E_t}^s(x) + t - f(x) = 0
\end{equation}
and
\begin{equation}\label{eulerLagrange02}
H_{E_{t'}}^s(x) + t' - f(x) = 0
\end{equation}
for each $x \in \partial E_t \cap \partial E_{t'}$. Recall that the nonlocal mean curvature associated with $K(x) = |x|^{-(N+s)}$ is well-defined at each point on $\partial E$ if $\partial E$ is at least of class $C^{1,\alpha}$ with $\alpha>s$ (see, for instance, \cite[Corollary 3.5]{Cozzi}).
Now we can readily see that, if two sets $E$ and $F$ satisfy that $E \subset F$ and $\partial E \cap \partial F \not= \emptyset$, then it holds that $H^s_{E} \geq H^s_{F}$ on $\partial E \cap \partial F$. Indeed, by definition, we have
\begin{align}\label{nonlocalMeanCurvature}
H^s_{E}(x) - H^s_{F}(x) &= \text{P.V.}\,\int_{\mathbb{R}^n}\frac{\chi_{E}(x) - \chi_{E}(y)}{|x-y|^{N+s}}\,dy \nonumber\\
&\qquad - \text{p.v.}\,\int_{\mathbb{R}^n}\frac{\chi_{F}(x) - \chi_{F}(y)}{|x-y|^{N+s}}\,dy \nonumber\\
&= \text{p.v.}\,\int_{\mathbb{R}^n}\frac{\chi_{E}(x) - \chi_{F}(x) - \chi_{E}(y) + \chi_{F}(y)}{|x-y|^{N+s}}\,dy
\end{align}
for any $x \in \partial E \cap \partial F$. Since $E \subset F$, it holds $\chi_E \leq \chi_F$ in $\mathbb{R}^n$ and $\chi_E(x)=\chi_F(x)$ for any $x \in \partial E \cap \partial F$. Thus from \eqref{nonlocalMeanCurvature} and the non-negativity of $K$, we obtain the claim.
Therefore, from \eqref{eulerLagrange01}, \eqref{eulerLagrange02}, and the fact that $H^s_{E_{t'}} \geq H^s_{E_{t}}$, we obtain
\begin{equation}
t' - f(x_0) \geq t - f(x_0) \nonumber
\end{equation}
and it turns out that $t' \geq t$. This contradicts the fact that $t' < t$.
\end{proof}
\subsection{Regularity of boundaries of super-level sets for minimizers}
Now we show some regularity results of the boundary of the set $\{u>t\}$ for each $t$ under suitable assumptions on the datum $f$, where $u$ is a minimizer of $\mathcal{F}_{K,f}$ with $K(x)=|x|^{-(N+s)}$. From Proposition \ref{comparisonLinfty}, we have that $u \in L^{\infty}(\mathbb{R}^n)$ whenever $f \in L^2 \cap L^{\infty}(\mathbb{R}^n)$. Since $\{u>t\} = \mathbb{R}^n$ if $t < - \|u\|_{L^{\infty}}$ and $\{u>t\} = \emptyset$ if $t \geq \|u\|_{L^{\infty}}$, in the sequel, we focus on the set $\{u>t\}$ only for $t \in [-\|u\|_{L^{\infty}}, \, \|u\|_{L^{\infty}})$ if $f \in L^{\infty}(\mathbb{R}^n)$. Recall that, from Corollary \ref{corUniformBoundednessMinimizer}, the super-level set $\{u>t\}$ is bounded uniformly in $t \in [-\|u\|_{L^{\infty}}, \, \|u\|_{L^{\infty}})$.
To obtain our main result on the regularity of minimizers, we exploit the regularity results proved by Caputo and Guillen \cite{CaGu}; Figalli, Fusco, Maggi, Millot, and Morini \cite{FFMMM}; Savin and Valdinoci \cite{SaVa}; and Barrios, Figalli, and Valdinoci \cite{BFV}.
Before recalling the regularity results, we give the definition of ``almost'' minimizers of $P_s$ in the sense of Figalli, et al \cite{FFMMM}. Given $\Lambda>0$, we say that a measurable bounded set $E \subset \mathbb{R}^n$ is an {\it almost minimizer} of $P_s$ if \begin{equation}\label{definitionAlmostMinimizer}
P_s(E) \leq P_s(F) + \frac{\Lambda}{1-s}|E \Delta F| \end{equation} for any measurable bounded set $F \subset \mathbb{R}^n$. Note that the concept of the almost minimality of $P_s$ was also given by Caputo and Guillen \cite{CaGu} and their definition can include a wider variety of sets than the definition by Figalli, et al. \cite{FFMMM}. In this paper, it is sufficient to apply the definition given by Figalli, et al \cite{FFMMM} and thus we do not write the definition given by Caputo and Guillen \cite{CaGu} here.
First, we recall the regularity of almost minimizers of $P_s$ in the sense of \eqref{definitionAlmostMinimizer}, which was shown by Figalli, Fusco, Maggi, Millot, and Morini \cite[Corollary 3.5]{FFMMM} (see also \cite{CaGu}). This result is a nonlocal analogue of the theory of Tamanini \cite{Tamanini} on almost minimal surfaces.
\begin{theorem}[\cite{FFMMM}]\label{improvementFlatnessFFMMM}
If $n \geq 2$, $\Lambda > 0$, and $s_0 \in (0,\,1)$, then there exist positive constants $0 < \varepsilon_0 < 1$, $C_0>0$, and $\alpha < 1$, depending on $n$, $\Lambda$, and $s_0$ only, with the following property: if $E$ is an almost minimizer of $P_s$ with $s \in (s_0, \,1)$ in the sense of \eqref{definitionAlmostMinimizer}, then $\partial E$ is of class $C^{1,\alpha}$ for some $0 < \alpha < 1$ except a closed set of Hausdorff dimension $n-2$. \end{theorem}
Next we recall the regularity result of fractional minimal cones in $\mathbb{R}^2$ by Savin and Valdinoci \cite{SaVa}. \begin{theorem}[\cite{SaVa}] \label{thmNonlocalCone2d}
Assume that $E \subset \mathbb{R}^2$ is a $s$-fractional minimal cone, namely, $E$ satisfies that $E = t\,E$ for any $t >0$. Then $E$ is a half-plane. \end{theorem} In particular, by combining the blow-up and blow-down arguments in \cite{CRS}, one may obtain that $s$-fractional minimal surfaces in $\mathbb{R}^2$ are fully $C^{1,\alpha}$-regular for any $\alpha \in (0,\,s)$. \begin{corollary}[\cite{SaVa}] \label{corNonlocalCone2d}
If $E$ is an $s$-fractional minimal set in $\Omega \subset \mathbb{R}^2$, then $\partial E \cap \Omega'$ is a $C^{1,\alpha}$-curve for any $\Omega' \Subset \Omega$. \end{corollary}
Originally, the regularity of nonlocal(fractional) minimal surfaces, which are defined by the boundaries of sets minimizing the fractional perimeter, was obtained by Caffarelli, Roquejoffre, and Savin \cite{CRS}. Precisely they proved that every fractional minimal surface is locally $C^{1,\alpha}$ except a closed set of Hausdorff dimension $n-2$. Moreover, thanks to Corollary \ref{corNonlocalCone2d}, this closed set of fractional minimal surfaces has Hausdorff dimension at most $n-3$.
As a consequence of these regularity results, we obtain \begin{lemma}[$C^{1,\alpha}$-regularity of boundary of super-level set of minimizers]\label{holderRegualrityLemma}
Let $s \in (0,\,1)$ and let $f \in L^2 \cap L^{\infty}(\mathbb{R}^n)$. Assume that $K(x) = |x|^{-(n+s)}$ for $x \in \mathbb{R}^n \setminus \{0\}$ and $u \in BV_K \cap L^2(\mathbb{R}^n)$ is a minimizer of the functional $\mathcal{F}_{K,f}$. Then, for each $t \in \mathbb{R}$, the boundary of the super-level set of $u$ is of class $C^{1,\alpha}$ with some $0 < \alpha < 1$, except a closed set of Hausdorff dimension $n-3$. \end{lemma} \begin{proof}
We fix $t \in \mathbb{R}$. Let $x_0 \in \partial \{u>t\}$ and $r>0$ be any number. First, from the assumption on $f$ and Lemma \ref{boundednessMinimizers} in Section \ref{secComparisonMini}, $u$ is non-negative and there exists a constant $R_0>0$ such that $E_t \coloneqq \{u>t\} \subset B_{\frac{R_0}{2}}$ for any $t \geq -\|u\|_{L^{\infty}}$. In order to apply Theorem \ref{improvementFlatnessFFMMM} to our case, it is sufficient to show that each set $E_t$ is an almost minimizer in the sense of \eqref{definitionAlmostMinimizer}. From Lemma \ref{relationMiniTwoEnergies}, we know that $\{u>t\}$ is a solution to the problem
\begin{equation}\nonumber
\min\{\mathcal{E}_{K,f,t}(E) \mid |E| < \infty\}
\end{equation}
for each $t \in \mathbb{R}$. Hence, from the minimality and boundedness of $E_t$, we have that
\begin{equation}\label{minimalitySublevel}
\mathcal{E}_{K,f,t}(E_t) \leq \mathcal{E}_{K,f,t}(F)
\end{equation}
for any bounded measurable set $F \subset \mathbb{R}^n$. Hence, from \eqref{minimalitySublevel}, we can compute as follows: for any bounded measurable set $F$, we have
\begin{align}\label{estimateQuasiNonlocalMini}
P_{K}(E_t) - P_{K}(F) &= \mathcal{E}_{K,f,t}(E_t) - \int_{E_t}(t-f(x))\,dx \nonumber\\
&\qquad - \mathcal{E}_{K,f,t}(F) + \int_{F}(t-f(x)\,dx \nonumber\\
&\leq \int_{\mathbb{R}^n}|\chi_{E_t}- \chi_{F}|\,|t - f(x)|\,dx \nonumber\\
&\leq \int_{B_r(x_0)}|t - f(x)|\,dx.
\end{align}
Since we assume that $f \in L^{\infty}(\mathbb{R}^n)$, we have
\begin{equation}\label{estimateResidue01}
\int_{B_r(x_0)}|t - f(x)|\,dx \leq (t +\|f\|_{L^{\infty}(\mathbb{R}^n)})\,|E_t \Delta F|.
\end{equation}
Hence, from \eqref{estimateQuasiNonlocalMini} and \eqref{estimateResidue01}, we have
\begin{equation}\nonumber
P_K(E_t) \leq P_K(F) + (t + \|f\|_{L^{\infty}(\mathbb{R}^n)})\,|E_t \Delta F|.
\end{equation}
Therefore, we apply Theorem \ref{improvementFlatnessFFMMM} and Corollary \ref{corNonlocalCone2d} to conclude that the claim is valid. \end{proof}
In addition, we employ another result of the regularity of solutions to integro-differential equations via the bootstrap argument. This result is obtained by Barrios, Figalli, and Valdinoci \cite[Theorem 1.6]{BFV}. They proved the following regularity theorem on the solutions to integro-differential equations. For simplicity, we do not describe the whole statement. See \cite[Theorem 1.6]{BFV} for the full statement. \begin{theorem}\label{theoremBootstrapBFV}
Let $v \in L^{\infty}(\mathbb{R}^{n-1})$ be a solution (in the viscosity sense) to the integro-differential equation
\begin{equation}\nonumber
\int_{\mathbb{R}^{n-1}}A_r(x',\,y')\left( v(x'+y') + v(x'-y') - 2v(x') \right) \,dy' = F(x', v(x'))
\end{equation}
for any $x' \in B'_r(0) \subset \mathbb{R}^{n-1}$ where $A_r$ satisfies the following assumptions:
\begin{itemize}
\item[(A1)] There exist constants $a_0,\,r_0>0$ and $\eta \in (0,\frac{a_0}{4})$ such that
\begin{equation*}
\frac{(1-s)(a_0-\eta)}{|y'|^{n+s}} \leq A_r(x',\, y') \leq \frac{(1-s)(a_0+\eta)}{|y'|^{n+s}}
\end{equation*}
for any $x' \in B'_r(0)$ and $y' \in B'_{r_0}(0) \setminus \{0\}$.
\item[(A2)] There exists a constant $C_0>0$ such that
\begin{equation*}
\| A_r(\cdot,\,y') \|_{C^{0,\beta}(B'_1)} \leq \frac{C_0}{|y'|^{n+s}}
\end{equation*}
for any $y' \in B'_{r_0}(0) \setminus \{0\}$.
\end{itemize}
and $F \in C^{0,\beta}(B'_r(0))$ with $\beta \in (0,\,1]$. Then $v \in C^{1,s+\alpha}(B'_{\frac{r}{2}}(0))$ for any $\alpha < \beta$. \end{theorem}
Taking into account all the above arguments, we can obtain that the boundary of the super-level set of the minimizer of $\mathcal{F}_{K,f}$ has the $C^{2,s+\beta-1}$-regularity under the $\beta$-H\"older regularity of a given datum $f$ with $\beta \in (1-s,\,1]$. Precisely, we prove \begin{lemma}\label{improvedRegularity}
Assume that $K(x) = |x|^{-(n+s)}$ for $x \in \mathbb{R}^n \setminus \{0\}$ with $s\ in (0,\,1)$ and $f$ is in $L^2 \cap L^{\infty}(\mathbb{R}^n)$. Let $u \in BV_K \cap L^2(\mathbb{R}^n)$ be a minimizer of the functional $\mathcal{F}_{K,f}$. If a datum $f$ is in $C^{0,\beta}_{loc}(\mathbb{R}^n)$ with $\beta \in (1-s,\,1]$, then for each $t \in \mathbb{R}$, the boundary of the super-level set $\{u > t\}$ is of class $C^{2,s+\alpha-1}$ with $1-s < \alpha < \beta \leq 1$ except a closed set of Hausdorff dimension $n-3$. \end{lemma} \begin{proof}
From Lemma \ref{holderRegualrityLemma} and the assumption that $f \in C^{0,\beta}_{loc} \cap L^{\infty}(\mathbb{R}^n)$ with $\beta \in (1-s,\,1]$, the boundary of the set $\{u > t\}$ has full $C^{1,\alpha}$-regularity with some $\alpha \in (0,\,1)$ except a closed set $\Sigma$ of Hausdorff dimension $n-3$, and thus we can represent $\partial \{u>t\} \setminus \Sigma$ locally as a graph of a $C^{1,\alpha}$-function $v_t$ in a bounded domain $U' \subset \mathbb{R}^{n-1}$. By employing the computation shown in \cite{BFV}, we may have that $v_t$ satisfies the equation, in the viscosity sense,
\begin{align}
&\int_{\mathbb{R}^{n-1}} A_r(x',\,y') \frac{v_t(x'+y') + v_t(x'-y') - 2v(x')}{|y'-x'|^{(n-1)+(1+s)}}\,dy' \nonumber\\
&\quad = G(x',\,v(x')) + t - f(x',\,v_t(x')) \quad \text{for $x' \in U' \subset \mathbb{R}^{n-1}$} \nonumber
\end{align}
where $A_r$ satisfies (A1) and (A2) and $G$ is a smooth function (see \cite{BFV} for the detail). Then, since $f \in C^{0,\beta}_{loc}(\mathbb{R}^n)$, we now apply Theorem \ref{theoremBootstrapBFV} several times, if necessary, to conclude that the regularity of $v_t$ can be improved up to $C^{2,s+\alpha-1}$ with $1-s < \alpha < \beta \leq 1$. From the compactness of the boundary of $\{u>t\}$ and by the standard covering argument, we obtain the $C^{2,s+\alpha-1}$-regularity of $\partial \{u>t\}$ for any $\alpha \in (1-s,\,\beta)$. \end{proof}
\subsection{Proof of the main regularity result} By using Lemma \ref{improvedRegularity}, we are now ready to prove the main result of this paper.
Let us briefly explain the strategy of the proof of Theorem \ref{mainTheorem}. Let $t_1, \, t_2 \in [-\|u\|_{L^{\infty}},\,\|u\|_{L^{\infty}})$ with $t_1 < t_2$ and we set $E_1 \coloneqq \{u>t_1\}$ and $E_2 \coloneqq \{u>t_2\}$. Notice that $E_2 \subset E_1$ because $t_1 < t_2$. In order to show the H\"older regularity of $u$, it is sufficient to observe that the boundaries of $E_1$ and $E_2$ are not too close. Precisely, using the regularity of $f \in C^{0,\beta}$, we show the inequality that \begin{equation}\label{inequalityCrucialMainTheorem}
t_2 - t_1 \lesssim \left( \mathrm{dist}\,(\partial E_1, \partial E_2) \right)^{\beta}. \end{equation} To see this, we compare the nonlocal mean curvatures on the boundaries $\partial E_1$ and $\partial E_2$. Notice that one can compare the curvatures at points which the boundaries have in common. Thus, we slide $\partial E_1$ (denoted by $\partial E^{\nu}_1$) along the outer unit normal $\nu$ of $\partial E_1$ until $\partial E^{\nu}_1$ touches $\partial E_2$. At the touching point, we can now compare the curvatures between $\partial E^{\nu}_1$ and $\partial E_2$. Moreover, by employing the computation by D\'avila, del Pino, and Wei \cite{DdPW}, we can also compare the curvatures between $\partial E_1$ and $\partial E^{\nu}_1$.
\begin{proof}[Proof of Theorem \ref{mainTheorem}]
Let $d_{t} \coloneqq d_{E_t}$ for $t \in [0,\,\infty)$ be a signed distance function from $\partial \{u>t\}$, which is negative inside $\{u>t\}$. We set $E_t \coloneqq \{x\mid u(x)>t\}$ for any $t$. Since $n=2$, from Lemma \ref{improvedRegularity} it follows that all the points on $\partial E_t$ are regular points. Thus, the signed distance function $d_t$ is of class $C^{2,s+\alpha-1}$ in a neighborhood of $\partial E_t$ with $1-s<\alpha<\beta$ (see, for instance, \cite{Wl, DeZo01, DeZo02, Bellettini} for the relation between the distance function and regularity of surfaces).
Recall that, from the assumption on $f$ and Proposition \ref{comparisonLinfty}, we have that $\|u\|_{L^{\infty}} \leq \|f\|_{L^{\infty}} < \infty$. We now take any $t_1 \in [-\|u\|_{L^{\infty}},\,\|u\|_{L^{\infty}})$ and set $E_1 \coloneqq E_{t_1}$. Then we can choose a neighborhood $U_1 \subset \mathbb{R}^2$ of the boundary $\partial E_1$ such that $d_1 \coloneqq d_{t_1} \in C^{2,s+\alpha-1}(U_1)$. Moreover, we take any $t_2 \in (-\|u\|_{L^{\infty}},\,\|u\|_{L^{\infty}})$ with $t_2 > t_1$ and set $E_2 \coloneqq E_{t_2}$. Then, from Lemma \ref{corUniformBoundednessMinimizer}, we obtain that there exists a constant $R_c>0$ independent of $t_1$ and $t_2$ such that $E_2 \subset E_1 \subset B_{R_c}$. We can choose points $x_1 \in \partial E_1$ and $x_2 \in \partial E_2$ such that
\begin{equation}\nonumber
\tilde{\delta} \coloneqq \mathrm{dist}\,(\partial E_1,\,\partial E_2) = |x_1 - x_2|.
\end{equation}
Since we study the local H\"older regularity of $u$, it is sufficient to consider the case that $x_2 \in U_1$.
We first show that the following inequality holds:
\begin{equation}\label{inequalityHolderRegularity}
t_2 - t_1 \leq ([f]_{\beta} + C\,\tilde{\delta}^{1-\beta} )\,\tilde{\delta}^{\beta}
\end{equation}
where $\beta$ is as in Theorem \ref{mainTheorem} and $C>0$ is a constant depending only on $s$ and $d_1$.
Without loss of generality, we may assume that $\widetilde{\delta} > 0$. Indeed, if $\widetilde{\delta} = 0$, then, from the definition of $\widetilde{\delta}$, we can easily see that $t_2=t_1$. This implies that the inequality \eqref{inequalityHolderRegularity} is valid. Thus, in the sequel, we always assume that $\tilde{\delta} > 0$.
Now we define $E_1^{\delta}$ as
\begin{equation}\nonumber
E_1^{\delta} \coloneqq \{ x \in E_1 \mid \mathrm{dist}\,(x,\,\partial E_1) \leq \delta\}
\end{equation}
for any $\delta \in (0,\,\tilde{\delta}]$. Then, from the choice of $t_2$ and the definition of $\widetilde{\delta}$, the boundary of $E_1^{\delta}$ can be described as $\partial E_1^{\delta} = \{x-\delta \nabla d_1(x) \mid x \in \partial E_1 \}$ for any $\delta \in (0,\,\widetilde{\delta}]$, where $\nabla d_1$ coincides with the outer unit normal vector of $\partial E_1$. From the definition of the nonlocal mean curvature, we can easily show that the following comparison inequality holds:
\begin{equation}\label{comparisonE_1dAndE_2}
H^K_{E_1^{\tilde{\delta}}}(x_2) \leq H^K_{E_2}(x_2).
\end{equation}
From the choice of $x_1$ and $x_2$, we have $x_2 = x_1 - \delta\,\nabla d_1(x_1)$. Now we compare the two nonlocal curvatures $H^{K}_{E_1^\delta}(x_2)$ and $H^{K}_{E_1}(x_1)$. To do this, we employ the computation shown by D\'avila, del Pino, and Wei in \cite{DdPW} (see also \cite{Cozzi, JuLa}). This computation is on the variation of the nonlocal(fractional) mean curvature. Precisely, we have that, for any set $E \subset \mathbb{R}^2$ with a smooth boundary (at least $C^2$), it holds that
\begin{align}\label{variationNonlocalMC}
&-\left.\frac{d}{d\delta}\right|_{\delta=0} H^K_{E_{\delta h}}(x-\delta
h(x)\nabla d_{E}(x)) \nonumber\\
&= 2\int_{\partial E}\frac{h(y) - h(x)}{|y-x|^{2+s}}\,d\mathcal{H}^{n-1}(y) \nonumber \\
&\quad + 2\int_{\partial E}\frac{(\nabla d_E(y) - \nabla d_E(x)) \cdot \nabla d_E(x)}{|y-x|^{2+s}} \,d\mathcal{H}^{n-1}(y)
\end{align}
for $x \in \partial E$ where $h \in L^{\infty}(\partial E)$ and $h$ is as smooth as $\partial E$. Here we define $E_{\delta h}$ in such a way that its boundary is given by $\partial E_{\delta h} \coloneqq \{x-\delta\,h(x)\,\nabla d_E(x) \mid x\in \partial E\}$ for any $\delta>0$. Then from \eqref{variationNonlocalMC} and by some computation, we have the estimate of the variation of the nonlocal mean curvature $H^s_{E_1^{\delta}}$ for small $\delta>0$. Precisely we can obtain that there exist constants $C>0$ and $\delta_0>0$, which depends on the space-dimension $n=2$, $s$, and the $L^{\infty}$-norm of $\nabla^2 d_1$ (equivalently the second fundamental form of $\partial E_1$), such that
\begin{equation}\label{estimateVariationNonlocalMC}
-\frac{d}{d\delta} H^K_{E_1^{\delta}}(\Psi_{\delta}(x_1)) \leq C\,\int_{\partial E_1}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y)
\end{equation}
for any $\delta \in (0,\,\delta_0)$ where we set $\Psi_{\delta}(x_1) \coloneqq x-\delta \,\nabla d_1(x)$. Indeed, choosing any smooth cut-off function $\eta_{\varepsilon}$ such that $\mathrm{spt}\,\eta_{\varepsilon} \subset
B^c_{\varepsilon}(0)$, $\eta_{\varepsilon} \equiv 1$ in $B^c_{2\varepsilon}(0)$, and $0 \leq \eta_{\varepsilon} \leq 1$, we can write the nonlocal curvature as follows:
\begin{align}\label{nonlocalPeriSplit}
&-H^s_{E_1^{\delta}}(\Psi_{\delta}(x_1)) \nonumber\\
&= \int_{\mathbb{R}^2}\frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|y-\Psi_{\delta}(x_1)|^{2+s}} \eta_{\varepsilon}(y-\Psi_{\delta}(x_1))\,dy \nonumber\\
&\qquad + \int_{\mathbb{R}^2}\frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|y-\Psi_{\delta}(x_1)|^{2+s}} (1-\eta_{\varepsilon}(y-\Psi_{\delta}(x_1)))\,dy \nonumber\\
&\eqqcolon A_{\varepsilon}(\delta) + B_{\varepsilon}(\delta).
\end{align}
Then we can compute the derivative of $A_{\varepsilon}(\delta)$ in \eqref{nonlocalPeriSplit} for small $\delta>0$ in the following manner: setting $\widetilde{y}_{\delta} \coloneqq y-\Psi_{\delta}(x_1)$ for simplicity, we have
\begin{align}\label{variationNonlocalMC01}
& \frac{d}{d\delta}\int_{\mathbb{R}^2}\frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|\widetilde{y}_{\delta}|^{2+s}} \eta_{\varepsilon}(\widetilde{y}_{\delta})\,dy \nonumber\\
&= \int_{\partial E_1^{\delta}} \frac{\eta_{\varepsilon}(\widetilde{y}_{\delta})}{|\widetilde{y}_{\delta}|^{2+s}} \,d\mathcal{H}^{n-1}(y) + \int_{\partial (E_1^{\delta})^c} \frac{\eta_{\varepsilon}(\widetilde{y}_{\delta})}{|\widetilde{y}_{\delta}|^{2+s}} \,d\mathcal{H}^{n-1}(y) \nonumber\\
&\qquad - (2+s)\int_{\mathbb{R}^2} \frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|\widetilde{y}_{\delta}|^{4+s}} (y-x_1+ \delta \nabla d_1(x_1)) \cdot \nabla d_1(x_1) \,\eta_{\varepsilon}(\widetilde{y}_{\delta})\,dy \nonumber\\
&\qquad \quad + \int_{\mathbb{R}^2} \frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|\widetilde{y}_{\delta}|^{2+s}} \nabla \eta_{\varepsilon}(\widetilde{y}_{\delta}) \cdot \nabla d_1(x_1)\,dy
\end{align}
for any $\delta \in (0,\,1)$ with $\Psi_{\delta}(x_1) \in U_1$. Then by using the Gauss-Green theorem, we have
\begin{align}\label{variationNonlocalMC02}
&- (2+s)\int_{\mathbb{R}^2} \frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|\widetilde{y}_{\delta}|^{4+s}} (y-x_1+ \delta \nabla d_1(x_1)) \cdot \nabla d_1(x_1) \,\eta_{\varepsilon}(\widetilde{y}_{\delta})\,dy \nonumber\\
&= \int_{\mathbb{R}^2} (\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)) \nabla_y \left(\frac{1}{|\widetilde{y}_{\delta}|^{2+s}}\right) \cdot \nabla d_1(x_1) \,\eta_{\varepsilon}(\widetilde{y}_{\delta})\,dy \nonumber\\
&= \int_{\partial E_1^{\delta}} \frac{\nabla d_1(x_1) \cdot \nabla d_{E_1^{\delta}}(y)}{|\widetilde{y}_{\delta}|^{2+s}} \eta_{\varepsilon}(\widetilde{y}_{\delta}) \,d\mathcal{H}^{n-1} \nonumber\\
&\qquad - \int_{\partial (E_1^{\delta})^c} \frac{\nabla d_1(x_1) \cdot (-\nabla d_{E_1^{\delta}}(y))}{|\widetilde{y}_{\delta}|^{2+s}} \eta_{\varepsilon}(\widetilde{y}_{\delta}) \,d\mathcal{H}^{n-1} \nonumber\\
&\qquad \quad - \int_{\mathbb{R}^2} \frac{\chi_{E_1^{\delta}}(y) - \chi_{(E_1^{\delta})^c}(y)}{|\widetilde{y}_{\delta}|^{2+s}} \nabla \eta_{\varepsilon}(\widetilde{y}_{\delta}) \cdot \nabla d_1(x_1)\,dy.
\end{align}
Thus from \eqref{variationNonlocalMC01} and \eqref{variationNonlocalMC02}, we obtain
\begin{align}
\frac{d}{d\delta}A_{\varepsilon}(\delta)
&= \int_{\partial E_1^{\delta}} \frac{2- 2(\nabla d_1(x_1) \cdot \nabla d_{E_1^{\delta}}(y))}{|\widetilde{y}_{\delta}|^{2+s}}\eta_{\varepsilon}(\widetilde{y}_{\delta}) \,d\mathcal{H}^{n-1}(y) \nonumber\\
&= \int_{\partial E_1^{\delta}} \frac{|\nabla d_1(x_1) - \nabla d_{E_1^{\delta}}(y)|^2}{|\widetilde{y}_{\delta}|^{2+s}}\eta_{\varepsilon}(\widetilde{y}_{\delta}) \,d\mathcal{H}^{n-1}(y) \nonumber
\end{align}
for any small $\delta >0$ with $\Psi_{\delta}(x_1) \in U_1$. Hence from the change of variables, we obtain
\begin{align}
\frac{d}{d\delta}A_{\varepsilon}(\delta)
&= \int_{\partial E_1} \frac{|\nabla d_1(x_1) - \nabla d_{1}(y)|^2}{|\Psi_{\delta}(y)-\Psi_{\delta}(x_1)|^{2+s}}\eta_{\varepsilon}(\Psi_{\delta}(y)-\Psi_{\delta}(x_1)) \,J_{\partial E_1}\Psi_{\delta}(y)\,d\mathcal{H}^{n-1}(y) \nonumber
\end{align}
where $J_{\partial E_1}\Psi_{\delta}(y)$ is the tangential Jacobian of $\partial E_1$ at $y$. As is shown in \cite{DdPW}, we can have that there exist constants $c'>0$ and $\delta'>0$, depending on the space-dimension $n=2$ and $s$ but independent of $\varepsilon >0$, such that $|\frac{d}{d\delta}B_{\varepsilon}(\delta)| \leq c'\varepsilon^{1-s}$ for any $\delta \in (0,\,\delta')$ and $\varepsilon \in (0,\,1)$. Therefore, we conclude that
\begin{align}
-\frac{d}{d\delta}H^s_{E_1^{\delta}}(\Psi_{\delta}(x_1)) &= \lim_{\varepsilon \downarrow 0}\left(\frac{d}{d\delta}A_{\varepsilon}(\delta) + \frac{d}{d\delta} B_{\varepsilon}(\delta) \right) \nonumber\\
&= \int_{\partial E_1} \frac{|\nabla d_1(x_1) - \nabla d_{1}(y)|^2}{|\Psi_{\delta}(y)-\Psi_{\delta}(x_1)|^{2+s}} J_{\partial E_1}\Psi_{\delta}(y) \,d\mathcal{H}^{n-1}(y) \nonumber
\end{align}
for any $\delta \in (0,\,\delta'_0)$ where $\delta'_0>0$ is a constant depending on the space-dimension $n=2$, $s$, and the $L^{\infty}$-norm of $\nabla^2 d_1$. From the definition of $\Psi_{\delta}$, we have that there exists a constant $C_0>0$ depending on the space-dimension $n=2$, $s$, and the $L^{\infty}$-norm of $\nabla^2 d_1$, such that
\begin{equation}\nonumber
\frac{J_{\partial E_1}\Psi_{\delta}(y)}{|\Psi_{\delta}(y) -\Psi_{\delta}(x_1)|^{2+s}} \leq \frac{C_0}{|y-x_1|^{2+s}}
\end{equation}
for any $y \in \partial E_1$ and $\delta \in (0,\,\delta'_0)$. Therefore we obtain that there exist constants $C>0$ and $\delta_0>0$, depending on the space-dimension $n=2$, $s$, and the second derivative of $d_1$ but independent of $\delta$, such that the inequality \eqref{estimateVariationNonlocalMC} with the constant $C$ holds for any $\delta \in (0,\,\delta_0)$. Thus, from the fundamental theorem of calculus and \eqref{estimateVariationNonlocalMC}, we obtain that
\begin{align}\label{estimateNonlocalCurvatures}
&-H^K_{E_1^{\delta}}(x-\delta
\,\nabla d_1(x)) \nonumber\\
&= -H^K_{E_1}(x_1) - \delta\,\int_{0}^{1} \frac{d}{d\delta} H^K_{E_1^{\delta}}(x- \lambda\delta
\,\nabla d_1(x)) \,d\lambda \nonumber\\
&\leq -H^K_{E_1}(x_1) + C\,\delta\,\int_{\partial E_1}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y)
\end{align}
for any $\delta \in (0,\,\delta_0)$. Now we show that the integral
\begin{equation}\nonumber
\int_{\partial E_1}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y)
\end{equation}
is uniformly bounded for any $x_1 \in V$ and any open set $V \subsetneq U_1$. Indeed, we define the set $U^r_1 \coloneqq \{x \in U_1 \mid \mathrm{dist}\,(x,\,\partial U_1)>r\}$ for any $r>0$ satisfying that $B_{2r}(x) \subset U_1$ for any $x \in U_1$. Then we can compute the integral as follows: for any $x_1 \in U^r_1$, it holds that
\begin{align}\label{estimateFracNormNormalVec}
&\int_{\partial E_1}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y) \nonumber\\
&= \int_{\partial E_1 \cap B_r(x_1)}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y) \nonumber\\
&\qquad + \int_{\partial E_1 \cap B^c_r(x_1)}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y) \nonumber\\
&\leq \int_{\partial E_1 \cap B_r(x_1)}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^2} \frac{1}{|y-x_1|^{n-2+s}} \,d\mathcal{H}^{n-1}(y) \nonumber\\
&\qquad + \int_{\partial E_1 \cap B^c_r(x_1)}\frac{4}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y).
\end{align}
From the fundamental theorem of calculus and the fact that $B_r(x_1) \subset U_1$ for any $x_1 \in U^r_1$, we have that
\begin{equation}\label{estiGradientDistance}
\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^2} \leq \|\nabla^2 d_1\|_{L^{\infty}(B_r(x_1))}^2
\end{equation}
for any $y \in B_r(x_1)$. Thus from \eqref{estimateFracNormNormalVec} and \eqref{estiGradientDistance} and noticing that $x_1 \in U^r_1$ and $E_t \subset B_{R_c}$ holds uniformly in $t \geq c$ where $c \coloneqq - \|u\|_{L^{\infty}} > -\infty$, we obtain
\begin{align}\label{estimateGradDistanceonSurface}
\int_{\partial E_1}\frac{|\nabla d_1(y) - \nabla d_1(x_1)|^2}{|y-x_1|^{2+s}} \,d\mathcal{H}^{n-1}(y) &\leq c_1\, \|\nabla^2 d_1\|_{L^{\infty}(B_r(x_1))}^3\,r^{1-s} \nonumber\\
&\qquad + \frac{c_2\,\|\nabla^2 d_1\|_{L^{\infty}(U_1)}}{r^s}
\end{align}
where $c_1>0$ and $c_2>0$ are constants depending on the space-dimension $n=2$ and $s$. Since we choose any $r$ in such a way that $B_r(x_1) \subset U_1$, we conclude the claim is valid. Thus, from \eqref{estimateNonlocalCurvatures} and \eqref{estimateGradDistanceonSurface}, we finally obtain the inequality
\begin{equation}\label{comparisonNonlocalMCSmallDiff}
-H^K_{E_1^{\delta}}(x_1-\delta
\,\nabla d_1(x)) \leq -H^K_{E_1}(x_1) + C(n,s,R_c)\,\delta
\end{equation}
for any $\delta \in (0,\,\delta_0)$ where $C(n,s,R_c)>0$ ($n=2$ is the space-dimension) and $\delta_0>0$ are some constants, which also depend on the $L^{\infty}$-norm of $\nabla^2 d_1$. Note that the constant $\delta_0$ can be bounded by the inverse of the $L^{\infty}$-norm of $\nabla^2 d_1$. Thus from \eqref{comparisonE_1dAndE_2} and \eqref{comparisonNonlocalMCSmallDiff}, we have that, for any $\delta \in (0,\,\delta_0)$,
\begin{equation}\label{comparisonNonlocalCurvature}
-H^K_{E_2}(x_2) \leq -H^K_{E_1}(x_1) + C(n,s,R_c)\,\delta.
\end{equation}
Now we consider the following two cases:
\textit{Case 1}: $0< \tilde{\delta} < \delta_0$. In this case, we simply substitute $\delta = \tilde{\delta}$ with \eqref{comparisonNonlocalCurvature} and obtain
\begin{equation}\nonumber
-H^K_{E_2}(x_2) \leq -H^K_{E_1}(x_1) + C(n,s,R_c)\,\tilde{\delta}
\end{equation}
where $\tilde{\delta} = \mathrm{dist}\,(\partial E_1,\,\partial E_2)$.
\textit{Case 2}: $\tilde{\delta} \geq \delta_0$. In this case, there exists a number $N \in \mathbb{N}$ such that $ \frac{\tilde{\delta}}{N} < \|\nabla^2 d_1\|_{L^{\infty}(U_1)}^{-1}$. Then setting $\tilde{\delta}_k \coloneqq \frac{k}{N}\tilde{\delta}$ for each $k \in \{1,\cdots,\,N\}$ and taking into account all the above arguments, we obtain the inequality that
\begin{equation}\label{comparisonNonlocalCurvatureIteration}
-H^{K}_{E_1^{\tilde{\delta}_k}}(x^{\tilde{\delta}_k}_1) \leq -H^{K}_{E_1^{\tilde{\delta}_{k-1}}}(x^{\tilde{\delta}_{k-1}}_1) + C(n,s,R_c)\,\frac{\tilde{\delta}}{N}
\end{equation}
for each $k \in \{1,\cdots,\,N\}$ where we understand the notation $x^{\tilde{\delta}_0}_1 = x_1$ and $E_1^{\tilde{\delta}_0} = E_1$. Thus by summing the inequality \eqref{comparisonNonlocalCurvatureIteration} for all $i \in \{1,\cdots,\,N\}$, we obtain
\begin{align}
-H^{K}_{E_1^{\tilde{\delta}}}(x_2) &= -H^{K}_{E_1^{\tilde{\delta}_N}}(x^{\tilde{\delta}_N}_1) \nonumber\\
&\leq -H^{K}_{E_1^{\tilde{\delta}_0}}(x^{\tilde{\delta}_0}_1) + N\,C(n,s,R_c)\,\frac{\tilde{\delta}}{N} = -H^{K}_{E_1}(x_1) + C(n,s,R_c)\,\tilde{\delta} \nonumber
\end{align}
where $\tilde{\delta} = \mathrm{dist}\,(\partial E_1,\,\partial E_2)$. In both cases, we finally obtain the inequality
\begin{equation}\label{comparisonFractionalMeanCurvatures}
- H^K_{E_2}(x_2) \leq -H^{K}_{E_1}(x_1) + C(n,s,R_c)\,\tilde{\delta}.
\end{equation}
Thanks to Lemma \ref{improvedRegularity}, the Euler-Lagrange equation
\begin{equation}
H^s_{E_t}(x) + t - f(x) = 0
\end{equation}
holds for every $x \in \partial E_t$. Then, since $E_i$ is the minimizer of $\mathcal{E}_{K,f,t_i}$ for $i \in \{1,2\}$ and from \eqref{comparisonFractionalMeanCurvatures}, we obtain
\begin{equation}\nonumber
t_2 - t_1 \leq f(x_2) - f(x_1) + C(n,s,R_c)\,\tilde{\delta}.
\end{equation}
Recalling the definition of $x_2$, the H\"older continuity of $f$, and the fact that $E_t \subset B_{R_c}$ for any $t \geq c$, we conclude that
\begin{equation}\label{keyInequalityHolder}
t_2 - t_1 \leq ([f]_{\beta}(B_{R_c}) + C(n,s,R_c)\,\tilde{\delta}^{1-\beta})\,\tilde{\delta}^\beta
\end{equation}
where $[f]_{\beta}(B_{R_c})$ is the H\"older constant of $f$ in $B_{R_c}$ given as
\begin{equation}\nonumber
[f]_{\beta}(B_{R_c}) \coloneqq \sup_{x,\,y \in B_{R_c}, \, x \neq y}\frac{|f(x) - f(y)|}{|x-y|^{\beta}}
\end{equation}
and the constant $\tilde{\delta}$ is defined as $\tilde{\delta} \coloneqq \mathrm{dist}\,(\partial E_1,\,\partial E_2)$. Note that the constant $C(n,s,R_c)>0$ also depends on the $L^{\infty}$-norm of $\nabla^2 d_1$.
We are now ready to prove the local H\"older continuity of $u$. Let $B_{r_0}(x_0) \subset \mathbb{R}^2$ be any open ball of radius $r_0$ with $x_0 \in \{u = t_0 \}$ for a number $t_0 \geq c \coloneqq -\|u\|_{L^{\infty}}$. We take any points $x,\,y \in B_{r_0}(x_0)$ with $x \neq y$ and set $t_1,\,t_2 \in \mathbb{R}$ as $t_1 \coloneqq u(x)$ and $t_2 \coloneqq u(y)$. We may assume that $t_1 > t_2 \geq c$ because we only repeat the same argument in the case of $t_1 < t_2$. In addition to this, we also assume that $t_1 > t_0 > t_2$. Indeed, in the case of $t_1 > t_2 \geq t_0$ or $t_0 \geq t_1 > t_2$, it is sufficient to take another point $x_0' \in B_{r_0}(x_0)$ and $t_0' \in \mathbb{R}$ such that $x_0' \in \{ u = t_0' \}$ and $t_1 > t_0' > t_2$, and do the argument that we will show below. Moreover, since we only observe the local regularity of $u$, it is sufficient to consider the case that $B_{r_0}(x_0) \subset U_0$ where $U_0$ is a neighborhood of $\partial \{u > t_0\}$ such that the signed distance function from $\partial \{u > t_0\}$ is of class $C^{2, s+\alpha-1}(U_0)$ with $\alpha \in (1-s,\, 1)$. Indeed, if $x \in B_{r_0}(x_0) \setminus U_0$ and $y \in B_{r_0}(x_0)$, then, from the continuity of $u$, we can choose a point $z_0$ in $B_{r_0}(x_0)$ and close to $x$ such that the estimate $|u(x) - u(z_0)| \leq |x-y|^{\beta}$ holds and $t_1 = u(x) > u(z_0) \geq u(y) = t_2$. In the case of $z_0 \in U_0$, we just apply the argument that we will show below with \eqref{keyInequalityHolder} for $z_0$, $x_0$, and $y$; otherwise we can repeat the above argument until we have the point belonging to $U_0$.
Now we choose sufficiently small $\varepsilon > 0$ such that $t_1 - \varepsilon > t_0$ and $t_0 - \varepsilon > t_2$ and then we have that $x \in \{ u > t_1 -\varepsilon\}$, $y\in \{ u > t_2 - \varepsilon\}$, and $x_0 \in \{ u > t_0 - \varepsilon\}$. Hence, from \eqref{keyInequalityHolder} and the fact that $x,\,y \in B_{r_0}(x_0)$, we obtain the two inequalities
\begin{align}\label{estimateHolder01}
u(x) - u(x_0) = t_1 - \varepsilon - (t_0 - \varepsilon) &\leq ( [f]_{\beta}(B_{R_c}) + C(n,s,R_c)\,\tilde{\delta}_1^{1-\beta})\,\tilde{\delta}_1^{\beta} \nonumber\\
&\leq ( [f]_{\beta}(B_{R_c}) + C(n,s,R_c)\,r_0^{1-\beta})\,\tilde{\delta}_1^{\beta}.
\end{align}
and
\begin{align}\label{estimateHolder02}
u(x_0) - u(y) = t_0 - \varepsilon - (t_2 - \varepsilon) &\leq ( [f]_{\beta}(B_{R_c}) + C(n,s,R_c)\,\tilde{\delta}_2^{1-\beta})\,\tilde{\delta}_2^{\beta} \nonumber\\
&\leq ( [f]_{\beta}(B_{R_c}) + C(n,s,R_c)\,r_0^{1-\beta})\,\tilde{\delta}_2^{\beta}
\end{align}
where we set $\tilde{\delta}_1 \coloneqq \mathrm{dist}\,(\partial E_{t_0},\partial E_{t_1})$ and $\tilde{\delta}_2 \coloneqq \mathrm{dist}\,(\partial E_{t_0},\partial E_{t_2})$. Note that the constant $C(n,s,R_c)>0$ also depends on the $L^{\infty}$-norm of $\nabla^2 d_{t_0}$, which can be uniformly bounded in $B_{r_0}(x_0)$. Notice that the inequality
\begin{equation}\nonumber
\tilde{\delta}_1 + \tilde{\delta}_2 = \mathrm{dist}\,(\partial E_{t_0},\,\partial E_{t_1}) + \mathrm{dist}\,(\partial E_{t_0},\,\partial E_{t_2}) \leq \mathrm{dist}\,(\partial E_{t_1},\,\partial E_{t_2}) \leq |x-y|
\end{equation}
holds because of the fact that $E_{t_1} \subset E_{t_0} \subset E_{t_2}$. Therefore from \eqref{estimateHolder01} and \eqref{estimateHolder02}, we obtain that there exists a constant $C=C(n,s,f,R_c,r_0,x_0)>0$ (we have assumed that the space-dimension $n$ is two) such that
\begin{align}
|u(x) - u(y)| &= |u(x) - u(x_0) + u(x_0) - u(y)| \nonumber\\
&\leq C\,(\tilde{\delta}_1^{\beta} + \tilde{\delta}_2^{\beta}) \leq C\,2^{1-\beta}(\tilde{\delta}_1 + \tilde{\delta}_2)^{\beta} \leq 2^{1-\beta}C\, |x-y|^{\beta}. \nonumber
\end{align}
Here, in the second inequality, we have used the fact that $2^{1-\beta}(x+1)^{\beta} \geq x^{\beta} + 1$ for any $x \geq 1$ and $\beta \in (0,\,1)$ and applied this fact with $x=\widetilde{\delta}_1\,\widetilde{\delta}_2^{-1}$ if $\widetilde{\delta}_1 \geq \widetilde{\delta}_2$ or $x=\widetilde{\delta}_2\,\widetilde{\delta}_1^{-1}$ if $\widetilde{\delta}_1 < \widetilde{\delta}_1$. \end{proof}
\end{document} |
\begin{document}
\title{On graphs with a large chromatic number containing no small odd cycles}
\author{S.L. Berlov\thanks{The work was supported by RFBR grant No.~10-01-00096-A.}, Ilya I. Bogdanov\footnotemark[1]}
\maketitle
\begin{abstract}
In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles. \end{abstract}
\section{Introduction}
P.~Erd\H os~\cite{erdos} showed that for every integer $n>1$ and $p>2$, there exists a graph of girth~$g$ and chromatic number greater than~$n$ which contains not more than $n^{2g+1}$ vertices. Later, he conjectured~\cite{erdos3} that for every positive integer~$s$ there exists a constant $c_s$ such that for every graph~$G$ having $N$ vertices and containing no odd cycles of length less than $c_sN^{1/s}$, its chromatic number does not exceed $s+1$.
This conjecture was proved by Kierstead, Szemer\'edi, and Trotter~\cite{kierstead}; in fact, they have proved a more general result. In our case, their result states that the chromatic number of any graph on $N$ vertices containing no odd cycles of length at most $4sN^{1/s}+1$ does not exceed~$s+1$.
Basing on these results, we introduce the following notation.
\begin{Def}
Assume that $n,k>1$ are two integers. Denote by $f(n,k)$ the maximal integer $f$ satisfying the following property: If a graph $G=(V,E)$ contains no odd cycles of length at most $2k-1$, and $|V|\leq f$, then there exists a proper coloring of its vertices in $n$ colors. \end{Def}
Notice that a graph contains no odd cycles of length at most $2k-1$ if and only if it contains no simple odd cycles of the same lengths.
The results mentioned above imply that $f(n,k)<n^{4k+1}$ and $f({s+1},[2sN^{1/s}]+1)\geq N$. One can obtain that the latter inequality is equivalent to the bound \begin{equation}
f(n,k)\geq \left(\frac k{2(n-1)}\right)^{n-1}-1.
\label{lower-kst} \end{equation}
A different upper bound for $f(n,k)$ can be obtained from the following graph constructed by Schrijver~\cite{schrijver}. Let $m,d$ be some positive integers. Set $X=\{1,2,\dots,2m+d\}$, $V=\{{x\subset X}:\; |x|=m,\; 1<|i-j|<{2m+d-1} \mbox{ \ for all pairs of distinct\ }i,j\in x\}$, $E=\{{(x,y)\in V^2}:\; {x\cap y=\emp}\}$. The Schrijver graph $(V,E)$
is $(d+2)$-chromatic, whilst it does not contain odd cycles of length less than $\frac{2m+d}d$. Next, we have $|V|=\frac{2m+d}{m+d}\binom{m+d}{d}$; now it is easy to obtain that \begin{equation}
f(n,k)<\frac{(n-1)(2k-1)+2}{(n-1)k+1}\binom{(n-1)k+1}{n-1}.
\label{upper-schr} \end{equation} When we fix the value of $n$, the bounds~\eqref{lower-kst} and~\eqref{upper-schr} become the polynomials in $k$ of the same degree; hence, in some sense they are close to each other. On the contrary, when we fix the value of $k$ and consider the values $n>k/(2e)+1$, we see that the right-hand part of~\eqref{lower-kst} decreases (as a function in $n$). Hence for larger values of $n$ this estimate does not provide any additional information.
On the other hand, for $k=2$ the asymptotics of $f(n,2)$ is tightly connected with the asymptotics of Ramsey numbers $R_{n,3}$. In the papers of Ajtai, Koml\'os, and E. Szemer\'edi~\cite{aks} and Kim~\cite{kim} it is shown that $c_1\frac{n^2}{\log n}\leq R(n,3)\leq c_2\frac{n^2}{\log n}$ for some absolute constants $c_1,c_2$. One can check that these results imply the bounds $$
c_3 n^2\log n\leq f(n,2)\leq c_4 n^2 \log n $$ for some absolute constants $c_3,c_4$.
In the present paper, we find nontrivial lower bounds for $f(n,k)$ for all values of $n\geq 2$ and $k\geq 2$. In Section~2, we make some combinatorial considerations leading to the recurrent bounds for $f(n,k)$. In Section~3, we obtain explicit bounds following from those results. In particular, we show (see Theorem~\ref{estimate}) that $$
f(n,k)\geq \frac{(n+k)(n+k+1)\cdots(n+2k-1)}{2^{k-1}k^k} $$ for all $n\geq2$ and $k\geq 2$.
\section{Recurrent bounds}
Firstly, we introduce some notation.
Let $G=(V,E)$ be an (unoriented) graph. We denote the {\em distance} between the vertices $u,v\in V$ by $\mathop{\operatorfont dist}_G(u,v)$.
Consider a vertex $v\in V$, and let $r$ be a nonnegative integer. We denote by $U_r(v,G)=\{u\in V\mid \mathop{\operatorfont dist}_G(u,v)\le r\}$ the {\em ball} of radius $r$ with the center at~$v$, and by $S_r(v,G)=\{u\in V\mid \mathop{\operatorfont dist}_G(u,v)= r\}$ the {\em sphere} with the same radius and center. In particular, $S_0(v,G)=U_0(v,G)=\{v\}$. Denote also by $\dd^{\rm out}_G V_1=\{u\in V\setminus V_1\mid \exists v\in V_1: (u,v)\in E\}$ the {\em outer boundary} of a set $V_1\subseteq V$. In particular, $S_r(v,G)=\dd^{\rm out}_G U_{r-1}(v,G)$.
For a set $V_1\subseteq V$, we denote by $G(V_1)$ the induced subgraph on the set of vertices~$V_1$.
Let us fix some integers $n$ and $k$ which are greater than~1. We need the following easy proposition.
\begin{Prop}
Graph $G$ does not contain odd cycles of length not exceeding $2k-1$ if and only if for each vertex $v\in V$ and each positive integer $r<k$, the subgraph $G(S_r(v,G))$ contains no edges.
\label{sph-empty} \end{Prop}
\proof Assume that the subgraph $G(S_r(v,G))$ contains an edge $(u_1,u_2)$. Supplementing this edge by shortest paths from $v$ to $u_1$ and $u_2$, we obtain a cycle of length~$2r+1\leq 2k-1$.
Conversely, assume that $G$ contains a cycle of length $\leq 2k-1$. Consider such a cycle $C$ of the minimal length $2r+1$ (then $r<k$). Choose any its vertex~$v$, and let $u_1,u_2$ be two vertices of~$C$ such that $\mathop{\operatorfont dist}_C(v,u_1)=\mathop{\operatorfont dist}_C(v,u_2)=r$. In fact, we have $\mathop{\operatorfont dist}_G(u_1,v)=\mathop{\operatorfont dist}_G(u_2,v)=r$. Actually, assume that $\mathop{\operatorfont dist}_G(v,u_1)<r$, and choose a path~$P$ of the minimal length connecting $v$ and $u_1$. Then one can supplement it by one of the two subpaths of~$C$ connecting $u_1$ and $v$ to obtain an odd cycle $C'$. The length of $C'$ is smaller than $r+(r+1)=2r+1$, that contradicts the choice of~$C$.
Thus, $u_1,u_2\in S_r(v,G)$, and the graph $G(S_r(v,G))$ contains an edge. \qed
Now let us fix an arbitrary graph $G=(V,E)$ with a minimal number of vertices such that it contains no odd cycles of length not exceeding $2k-1$, and $\chi(G)>n$ (hence $|V|=f(n,k)+1$). By the minimality condition, the graph~$G$ is connected. Moreover, for every $v\in V$ and $0\leq r\leq k$, the sphere~$S_r(v,G)$ is nonempty. Otherwise we would have $G=\cup_{i=0}^{r-1} S_i(v,G)$, where all the graphs $G(S_i(v,G))$ contain no edges by Proposition~\ref{sph-empty}. Therefore, it is possible to color this graph properly in two colors: the vertices of the sets $S_i(v,G)$ with even~$i$ in color~1, while those for odd~$i$ --- in color~2 (the vertex~$v$ should be colored in color~1).
Let us introduce the number $d=\max_{v\in V} |U_{k-1}(v,G)|$.
\begin{Lemma} \label{big okr}
For every vertex $v\in V$, we have $|U_{k-1}(v,G)|\geq n(k-1)+1$. In particular, $d\geq n(k-1)+1$. \end{Lemma}
\proof Notice that $U_{k-1}(v,G)=\bigcup_{r=0}^{k-1}S_r(v,G)$. Assume that $|S_r(v,G)|\geq n$ for every $r=1,\dots,k-1$; then $$
|U_{k-1}(v,G)|=\sum_{r=0}^{k-1}|S_r(v,G)|\geq 1+(k-1)\cdot n, $$ as desired.
Assume now that $|S_r(v,G)|< n$ for some $1\leq r\leq k-1$. Consider a subgraph $G'=G\bigl(V\setminus U_{r-1}(v,G)\bigr)$. From the minimality condition, it can be colored properly in $n$ colors. Consider an arbitrary such proper coloring; then the vertices of $S_r(v,G)$ are colored in at most $n-1$ colors, so there exists a color (say, color~1) different from them. Let us now color the vertices of~$S_{r-1}(v,G)$ in color~1, and then color all the remaining vertices of the sets~$S_i(v,G)$ ($i<r-1$) alternately: we use colors~1 and~2 (here 2 is any remaining color) for even and odd values of~$i-(r-1)$, respectively. It follows from Proposition~\ref{sph-empty} that this coloring is proper. This contradicts the choice of~$G$. \qed
\begin{Lemma}
\label{Sergey}
$|V|\geq f(n-1,k)+d+1$. \end{Lemma}
\proof Choose a vertex~$v$ such that $d=|U_{k-1}(v,G)|$. Assume that $|V\setminus U_{k-1}(v,G)|\leq f({n-1},k)$; then one can color properly vertices of the set $V\setminus U_{k-1}(v,G)$ in $n-1$ colors. Now we can color the vertices of the set $U_{k-1}(v,G)$ in colors~1 and~$n$ (where $n$ is a new color, and 1 is any of the old colors) in the following way: we color all the vertices of~$S_r(v,G)$ in color~1 or~$n$, if $r-(k-1)$ is odd or even, respectively. By Proposition~\ref{sph-empty}, we obtain a proper coloring of~$G$ in $n$ colors which is impossible.
Thus, our assumption is wrong, so $|V\setminus U_{k-1}(v,G)|\geq f(n-1,k)+1$, and $$
|V|\geq f(n-1,k)+1+|U_{k-1}(v,G)|=f(n-1,k)+d+1.
\eqno\qed $$
\begin{Lemma}
\label{Ilya}
$\ds |V|\geq \frac{d^{1/(k-1)}}{d^{1/(k-1)}-1}\bigl(f(n-2,k)+1\bigr)$. \end{Lemma}
\proof We will construct inductively a sequence of partitions of~$V$ into nonintersecting parts, $$
V=U_1\sqcup U_2\sqcup\dots\sqcup U_s\sqcup N_s\sqcup V_s, $$ such that the following conditions are satisfied:
(i) for all $i=1,\dots,s$ we have $\dd^{\rm out}_G U_i\subseteq N_s$; moreover, $\dd^{\rm out}_G V_s\subseteq N_s$;
(ii) for every $i=1,2,\dots,s$ the graph $G(U_i)$ is bipartite (in fact, $U_i$ is a ball with radius not exceeding~$k-1$ in a certain subgraph of~$G$);
(iii) $(d^{1/(k-1)}-1)(|U_1|+\dots+|U_s|)\geq |N_s|$.
For the base case $s=0$, we may set $V_0=V$, $N_0=\emp$ (there are no sets~$U_i$ in this case).
For the induction step, suppose that the partition $V=U_1\sqcup U_2\sqcup\dots\sqcup U_{s-1}\sqcup N_{s-1}\sqcup V_{s-1}$ has been constructed, and assume that the set~$V_{s-1}$ is nonempty. Consider the graph $G_{s-1}=G(V_{s-1})$ and choose an arbitrary vertex $v\in V_{s-1}$. Now consider the sets $$
U_0(v,G_{s-1})=\{v\},\quad U_1(v,G_{s-1}), \quad \dots, \quad U_{k-1}(v,G_{s-1}). $$ One of the ratios $$
\frac{|U_1(v,G_{s-1})|}{|U_0(v,G_{s-1})|}, \quad \frac{|U_2(v,G_{s-1})|}{|U_1(v,G_{s-1})|}, \quad
\dots, \quad \frac{|U_{k-1}(v,G_{s-1})|}{|U_{k-2}(v,G_{s-1})|} $$ does not exceed~$d^{1/(k-1)}$, since the product of these ratios is $$
|U_{k-1}(v,G_{s-1})|\leq |U_{k-1}(v,G)|\leq d. $$ So, let us choose $1\leq m\leq k-1$ such that $$
\frac{|U_m(v,G_{s-1})|}{|U_{m-1}(v,G_{s-1})|}\leq d^{1/(k-1)}. $$
Now we set $$
U_s=U_{m-1}(v,G_{s-1}), \quad N_s=N_{s-1}\cup S_m(v,G_{s-1}),
\quad
V_s=V_{s-1}\setminus U_m(v,G_{s-1}). $$ Since the condition~(i) was satisfied on the previous step, we have $$
\dd^{\rm out}_G V_s\subseteq \dd^{\rm out}_G V_{s-1}\cup S_m(v,G_{s-1})\subseteq N_s $$ and $$
\dd^{\rm out}_G U_s\subseteq \dd^{\rm out}_G V_{s-1}\cup S_m(v,G_{s-1})\subseteq N_s, $$ so this condition also holds now. The condition~(ii) is satisfied by Proposition~\ref{sph-empty}. Finally, the choice of~$m$ and the condition~(iii) for the previous step imply that \begin{gather*}
d^{1/(k-1)} |U_s|=d^{1/(k-1)}|U_{m-1}(v,G_{s-1})|\geq |U_m(v,G_{s-1})|,\\
(d^{1/(k-1)}-1)(|U_1|+\dots+|U_{s-1}|)\geq |N_{s-1}| \end{gather*} and hence $$
(d^{1/(k-1)}-1)(|U_1|+\dots+|U_{s}|)\geq
|N_{s-1}|+|U_m(v,G_{s-1})|-|U_{m-1}(v,G_{s-1})|
=|N_s|. $$ Thus, the condition~(iii) also holds on this step.
Continuing the construction in this manner, we will eventually come to the partition with $V_s=\emp$ since the value of $|V_s|$ strictly decreases. As the result, we obtain the partition $V=U_1\sqcup U_2\sqcup\dots\sqcup U_s\sqcup N_s$ such that $|N_s|\leq
(d^{1/(k-1)}-1)(|U_1|+\dots+|U_s|)$. So, $$
d^{1/(k-1)}|N_s|\leq (d^{1/(k-1)}-1)(|U_1|+\dots+|U_s|)+(d^{1/(k-1)}-1)|N_s|=|V|(d^{1/(k-1)}-1), $$
or $\ds |N_s|\leq |V|\frac{d^{1/(k-1)}-1}{d^{1/(k-1)}}$.
Assume now that $|N_s|\leq f(n-2,k)$; then one may color the vertices of~$G(N_s)$ in $n-2$ colors, and then color the vertices of each bipartite graph $G(U_i)$ in two remaining colors. This coloring might be not proper only if some vertices of two subgraphs~$G(U_i)$ and~$G(U_j)$ ($i\neq j$) are adjacent, which is impossible by the condition~(i). So, $G$ is $n$-colorable which is wrong. Therefore, $|N_s|\geq f(n-2,k)+1$ and hence $\ds |V|\geq \frac{d^{1/(k-1)}}{d^{1/(k-1)}-1}|N_s|\geq \frac{d^{1/(k-1)}}{d^{1/(k-1)}-1}{(f(n-2,k)+1)} $, as desired. \qed
\Zam 1. In the statement of the Lemma above, one may use the number
$\ds d'=\max_{\emp\ne V'\subseteq V}\min_{u\in V'}|U_{k-1}(u,G(V'))|$ instead of~$d$. For reaching that, on each step it is sufficient to choose the vertex $v\in V_{s-1}$ such that $$
|U_{k-1}(v,G_{s-1})|=\min_{u\in V_{s-1}}|U_{k-1}(u,G_{s-1})|. $$ Clearly, we have $d'\leq d$.
\Zam2.
On the other hand, the number $d^{1/(k-1)}$ in the same statement can be replaced by $(f(n,k)+1)^{1/k}$. Now, in the proof one may deal with $k+1$ sets
$$
U_0(v,G_{s-1})=\{v\},\quad U_1(v,G_{s-1}), \quad \dots, \quad U_k(v,G_{s-1})
$$
and use the condition $|U_k(v,G_{s-1})|\leq |V|=f(n,k)+1$.
The next theorem follows immediately from the Lemmas~\ref{Sergey} and~\ref{Ilya}.
\begin{Theorem}
\label{recurr}
For all integer $n,k\geq 2$, we have
\begin{equation}
f(n,k)\geq
\min_{t\geq n(k-1)+1}\max\left\{f(n-1,k)+t,\frac{t^{1/(k-1)}}{t^{1/(k-1)}-1}(f(n-2,k)+1)-1\right\}.
\label{minmax}
\end{equation}
\label{theo} \end{Theorem}
\proof From the choice of~$G$ we have $f(n,k)=|G|-1$. From Lemmas~\ref{Sergey} and~\ref{Ilya} it follows that $$
|G|\geq \max\left\{f(n-1,k)+d,\frac{d^{1/(k-1)}}{d^{1/(k-1)}-1}(f(n-2,k)+1)-1\right\}+1. $$ Since $d\geq n(k-1)+1$ by Lemma~\ref{big okr}, the statement holds. \qed
\begin{Cor}
For every real $g>1$, we have
\begin{equation}
f(n,k)\geq
\min\left\{f(n-1,k)+g,
\frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}(f(n-2,k)+1)-1\right\}.
\label{min}
\end{equation}
\label{cor} \end{Cor}
\proof Let $t_0$ be the integer for which the minimum in~\eqref{minmax} is achieved. As $t>1$ increases, the value of $f(n-1,k)+t$ also increases, while the value of $\ds \frac{t^{1/(k-1)}}{t^{1/(k-1)}-1}(f(n-2,k)+1)-1$ decreases. Thus, if $g\leq t_0$, then we have $$
{f(n-1,k)+g}\leq {f(n-1,k)+t_0}\leq f(n,k). $$ Otherwise, we have $g>t_0$ and $$
\frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}{(f(n-2,k)+1)}-1\leq
\frac{t_0^{1/(k-1)}}{t_0^{1/(k-1)}-1}{(f(n-2,k)+1)}-1\leq f(n,k).
\eqno\qed $$
\section{Explicit bounds}
Now we present the explicit lower bounds for $f(n,k)$ following from the results of the previous section.
Notice that for every $k$ we have $f(1,k)=1$ and $f(2,k)=2k$. Lemma~\ref{Sergey} implies now the following statement.
\begin{Theorem}
For all integer $n\geq 1$ and $k\geq 2$ the inequality
$\ds f(n,k)\geq n+\frac{(k-1)(n-1)(n+2)}2$ holds.
\label{Sergey2} \end{Theorem}
\proof Induction on~$n$. In the base cases $n=1$ or $n=2$ the statement holds. Assume now that $n>2$. By Lemmas~\ref{big okr} and~\ref{Sergey} we have $f(n,k)\geq f(n-1,k)+n(k-1)+1$. Next, the hypothesis of the induction implies that $$
f(n-1,k)\geq (n-1)+\frac{(k-1)(n-2)(n+1)}2. $$ Therefore, $$ f(n,k)\geq f(n-1,k)+n(k-1)+1\geq n+\frac{(k-1)(n-1)(n+2)}2, $$ as desired. \qed
The next estimate uses the whole statement of the Theorem~\ref{theo}. For the convenience, we use the notation $n\^k=n(n+1)\dots(n+k-1)$.
\begin{Lemma}
Suppose that for some integer $n_0\geq 1$, integer $k\geq 2$, and real $a$, the inequality
\begin{equation}
f(m,k)\geq \frac{(m+a)\^k}{2^{k-1}k^k}
\label{asym1}
\end{equation}
holds for two values $m=n_0$ and $m=n_0+1$. Then the same estimate holds for all integer $m\geq n_0$.
\label{common} \end{Lemma}
\proof We prove by induction on $n\geq n_0$ that the estimate~\eqref{asym1} holds for $m=n$. The base cases $m=n_0$ and $m=n_0+1$ follow from the theorem assumptions.
For the induction step, suppose that $n\geq n_0+2$. Let $c=2^{1-k}k^{-k}$, $g=ck(n+a)\^{k-1}$. By the induction hypothesis, we have \begin{equation}
f(n-1,k)+g\geq c(n+a-1)\^k+ck(n+a)\^{k-1}
=c(n+a)\^{k-1}(n+a-1+k)=c(n+a)\^k.
\label{est-lm} \end{equation} Notice that Lemmas~\ref{big okr} and~\ref{Sergey} imply that $f(n,k)\geq f(n-1,k)+n(k-1)+1$. Hence, if $g\leq n(k-1)+1$, then $f(n,k)\geq f(n-1,k)+g\geq c(n+a)\^k$, as desired.
Thus we may deal only with the case $g>n(k-1)+1$; in particular, $g>1$. We intend to use Corollary~\ref{cor}; for this, let us estimate the second term in the right-hand part of~\eqref{min}.
From the AM--GM inequality we have $$
g^{1/(k-1)}=(ck)^{1/(k-1)}\left((n+a)\^{k-1}\right)^{1/(k-1)}
\leq \frac1{2k}\left(n+a+\frac k2-1\right). $$ Let $s=n+a+\frac k2-1$; then $s\geq 2kg^{1/(k-1)}>2k$. Therefore, \begin{multline*}
\frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}
\geq \frac{s}{s-2k}\geq \frac{s+k-1}{s-(k+1)}\geq \\
\geq \frac{s^2+s(k-1)+\frac{k(k-2)}4}{s^2-s(k+1)+\frac{k(k+2)}4}
=\frac{(n+a+k-2)(n+a+k-1)}{(n+a-2)(n+a-1)}. \end{multline*}
Finally, from the induction hypothesis we get \begin{multline}
\frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}(f(n-2,k)+1)-1
\geq \frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}f(n-2,k)\geq\\
\geq \frac{(n+a+k-2)(n+a+k-1)}{(n+a-2)(n+a-1)}\cdot
c(n+a-2)\^k=c(n+a)\^k.
\label{est-lm2} \end{multline} Thus, for the value of~$g$ chosen above, Corollary~\ref{cor} and the estimates~\eqref{est-lm} and~\eqref{est-lm2} provide that $$
f(n,k)\geq
\min\left\{f(n-1,k)+g,
\frac{g^{1/(k-1)}}{g^{1/(k-1)}-1}(f(n-2,k)+1)-1\right\}
\geq c(n+a)\^k, $$ as desired. \qed
Finally, let us show that the constant $a$ in the previous Lemma can be chosen relatively large.
\begin{Theorem}
For all $k\geq 2$ and $n\geq 2$, we have $\ds f(n,k)\geq \frac{\left(n+k\right)\^k}{2^{k-1}k^k}$.
\label{estimate} \end{Theorem}
\proof Set $a=k$. Let us check the inequality~\eqref{asym1} for $n=2$ and $n=3$. Recall that $f(2,k)=2k$. Now for $m=2$ we get $$
2^{k-1}k^kf(2,k)=2^kk^{k+1}=(2k)^{k-1}\cdot 2k^2\geq (k+2)(k+3) \dots
2k\cdot (2k+1)=(k+2)\^k. $$ For $m=3$, Theorem~\ref{Sergey2} yields $f(3,k)\geq 5k-2$, and the previous estimate now implies that $$
2^{k-1}k^kf(3,k)\geq 2^{k-1}k^k(5k-2)\geq 2^kk^kf(2,k)
\geq 2(k+2)\^k>(k+3)\^k. $$ Thus, the inequality~\eqref{asym1} holds for $m=2$ and $n=3$, and hence for all $n\geq 2$ by Lemma~\ref{common}. \qed
The authors are very grateful to the referees for the valuable comments.
\end{document} |
\begin{document}
\title{On plane curves given by separated polynomials and their automorphisms} \date{} \author{Matteo Bonini, Maria Montanucci, Giovanni Zini}
\maketitle
\begin{abstract} Let $\mathcal{C}$ be a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_p$ by a separated polynomial, that is $\mathcal{C}: A(y)=B(x)$, where $A(y)$ is an additive polynomial of degree $p^n$ and the degree $m$ of $B(X)$ is coprime with $p$. Plane curves given by separated polynomials are well-known and studied in the literature. However just few informations are known on their automorphism groups. In this paper we compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod {p^n}$ and $B(X)$ has just one root in $K$, that is $B(X)=b_m(X+b_{m-1}/mb_m)^m$ for some $b_m,b_{m-1} \in K$. Moreover, some sufficient conditions for the automorphism group of $\mathcal{C}$ to imply that $B(X)=b_m(X+b_{m-1}/mb_m)^m$ are provided. As a byproduct, the full automorphism group of the Norm-Trace curve $\mathcal{C}: x^{(q^r-1)/(q-1)}=y^{q^{r-1}}+y^{q^{r-2}}+\ldots+y$ is computed. Finally, these results are used to construct multi point AG codes with many automorphisms. \end{abstract}
{\bf Keywords: } Plane curve, separated polynomial, AG code, code automorphisms.
{\bf MSC Code: } 14H05, 14H37, 94B27.
\section{Introduction}
Deep results on automorphism groups of algebraic curves, defined over a field of characteristic zero, have been achieved after the work of Hurwitz who was the first to prove that complex curves, other than the rational and the elliptic ones, can only have a finite number of automorphisms. Afterwards, a proof of Hurwitz's result which is independent from the characteristic of the ground field was provided, increasing the interest of studying curves defined over fields of positive charactestic, as e.g. finite fields. This is especially comprehensible recalling that curves in positive characteristic may happen to have much larger $K$-automorphism group compared to their genus, as the Hurwitz bound $|G| \leq 84(g-1)$ for a $K$-automorphism $G$ of a curve of genus $g \geq 2$ fails whenever $|G|$ is divisible by the characteristic of the ground field. From previous results, see e.g \cite{henn}, we know infinite families of curves $\mathcal{C}$ with $|{\rm Aut}(\mathcal{C})| \sim cg^3$ or with $|{\rm Aut}(\mathcal{C})| \sim cg^2$. Although curves with large automorphism groups may have several different features, they seems to share a common property, namely their $p$-rank is equal to zero. This common property and this so different situation with respect to the zero characteristic case, raises the problem of constructing and studying curves of $p$-rank zero defined over finite fields with unusual properties that a complex curve cannot have. Artin-Schreier curves and, in particular, Hermitian curves are of this type. A family of such plane curves arises from separated polynomial. It consists of curves $\mathcal{X}: A(Y)-B(X)$ where $p \nmid m$ with $m = \deg B(X) \geq 2$ and $A(Y)$ is any additive separable polynomial. The main known properties of $\mathcal{C}$ are extracted from the local analysis of its unique singular point $P_\infty$; see \cite{Stichorig} and Section \ref{prel}. The exposition describes the genus, the Weierstrass gap sequence at $P_\infty$ and the ramification groups of its translation automorphism group fixing $P_\infty$. The full $K$-automorphism group of $\mathcal{C}$ fixes $P_\infty$ except in two cases, namely, when $\mathcal{C}$ is the Hermitian curve $Y^{p^n} -Y -X^{p^n}+1=0$ or the curve, $Y^{p^n} + Y -X^m=0$ with $m < p^n$, and $p^n \equiv -1 \pmod m$ but now other informations are known in the literature. For $p > 2$ and $m = 2$, the latter curve is hyperelliptic. Notably for $p > 2$, these hyperelliptic curves and the Hermitian curves are the only curves whose $K$-automorphism groups have order larger than $8g^3$; see \cite{henn}. Deligne-Lusztig curves provide other examples of significant curves over finite fields, namely the DLS curves of Suzuki type and the DLR curves of Ree type. They are characterised by their genera and $K$-automorphism groups. For $p = 2$, the Hermitian curves, the DLS curves, and the hyperelliptic curves $Y^2+Y +X^{2^h}+1=0$ are the only curves with $K$-automorphism groups of order larger than $8g^3$.
In this paper we compute the full automorphism group of $\mathcal{C}$ when $m \not\equiv 1 \pmod {p^n}$ and $B(X)$ has just one root in $K$, that is $B(X)=b_m(X+b_{m-1}/mb_m)^m$ for some $b_m,b_{m-1} \in K$. Moreover, some sufficient conditions for the automorphism group of $\mathcal{C}$ to imply that $B(X)=b_m(X+b_{m-1}/mb_m)^m$ are provided. As a byproduct, the full automorphism group of the Norm-Trace curve $\mathcal{C}: x^{(q^r-1)/(q-1)}=y^{q^{r-1}}+y^{q^{r-2}}+\ldots+y$ is computed.
An important application of curves over finite fields is the construction of certain linear codes, called Algebraic Geometric codes (AG codes for short). The parameters of an AG code constructed from a curve $\mathcal{X}$ strictly depend on the geometry of $\mathcal{X}$, and in particular on two fixed divisors on $\mathcal{X}$. The Norm-Trace curve was used in the literature to construct one-point or two-point AG codes; see \cite{BR2013,G2003,MTT2008}. In this paper we construct multi point AG codes on the Norm-Trace curve. Our construction starts from a divisor on $\mathcal{X}$ which is invariant under the whole automorphism group of the curve; hence, our codes turns out to inherit many automorphisms.
\section{Preliminary results} \label{prel}
\subsection{Curves given by separated polynomials} Throughout the paper, $\mathcal{C}$ is a plane curve defined over the algebraic closure $K$ of a prime finite field $\mathbb{F}_{p}$ by an equation \begin{equation}\label{EqSeparated} A(Y)=B(X), \end{equation} satisfying the following conditions: \begin{enumerate} \item $\deg(\mathcal{C}) \geq 4$; \item $A(Y)=a_n Y^{p^n} + a_{n-1} Y^{p^{n-1}}+\ldots+a_0 Y$, $a_j \in K$, $a_0,a_n \ne 0$; \item $B(X)=b_m X^m + b_{m-1} X^{m-1} + \ldots + b_1 X + b_0$, $b_j \in K$, $b_m \ne 0$; \item $m \not\equiv 0 \pmod p$; \item $n \geq 1$, $m \geq 2$. \end{enumerate} Note that $2$ occurs if and only if $A(Y+a)=A(Y) + A(a)$ for every $a \in K$, that is, the polynomial $A(Y)$ is additive. The basic properties of $\mathcal{C}$ are collected in the following lemmas; see \cite[Section 12.1]{HKT} and \cite{Stichorig}.
\begin{lemma} \label{lem1} The curve $\mathcal{C}$ is an irreducible plane curve with at most one singular point. \begin{itemize}
\item[\rm (i)] If $|m-p^n| =1$, then $\mathcal{C}$ is non-singular. \item[\rm (ii)] \begin{itemize} \item[\rm (a)] If $m > p^n+1$, then $P_\infty=(0,0,1)$ is an $(m-p^n)$-fold point of $\mathcal{C}$. \item[\rm (b)] If $p^n>m+1$, then $P_\infty=(0,1,0)$ is a $(p^n-m)$-fold point of $\mathcal{C}$. \item[\rm (c)] In both cases, $P_\infty$ is the centre of only one branch of $\mathcal{C}$; also, $P_\infty$ is the unique infinite point of $\mathcal{C}$. \end{itemize} \item[\rm (iii)] $\mathcal{C}$ has genus $g=\frac{(p^n-1)(m-1)}{2}$; \item[\rm (iv)] Let $K(x,y)$ with $A(y)=B(x)$ denote the function field of $\mathcal{C}$. \begin{itemize} \item[\rm (a)] A translation $(x,y) \mapsto (x,y+a)$ preserves $\mathcal{C}$ if and only if $A(a)=0$; \item[\rm (b)] these translations form an elementary abelian group of order $p^n$, and $Aut_K(K(x,y))$ contains an elementary abelian $p$-group $G$ of order $p^n$ that fixes a unique place $\mathcal{P}_\infty$ centered at $P_\infty$ and acts transitively on the zeros of $x$; \item[\rm (c)] the sequence of ramification groups of $G$ at $\mathcal{P}_\infty$ is $$G=G_{\mathcal{P}_\infty}^{(1)}=G_{\mathcal{P}_\infty}^{(2)}=\ldots=G_{\mathcal{P}_\infty}^{(m)}, \quad G_{\mathcal{P}_\infty}^{(m+1)}=\{1\};$$ \item[\rm (d)] $\{\mathcal{P}_\infty\}$ is the unique short orbit of $G$, and $$div(K(x,y) / K(x,y)^G)=(p^n-1)(m+1) \mathcal{P}_\infty;$$ \item[\rm (e)] $K(x,y)^G$ is rational, and $\mathcal{C}$ has $p$-rank zero. \end{itemize} \end{itemize} \end{lemma}
\begin{lemma} \label{lem2}
Let $M$ be a $K$-automorphism group of $\mathcal{C}$, and let $M_{\mathcal{P}_\infty}=M_{\mathcal{P}_\infty}^{(1)} \rtimes H$ where $p \nmid |H|$. Then \begin{itemize}
\item[\rm (i)] $|H|$ divides $m(p^n-1)$;
\item[\rm (ii)] $|M_{\mathcal{P}_\infty}^{(1)}| \leq p^n(m-1)^2=\frac{4p^n}{(p^n-1)^2}g^2$;
\item[\rm (iii)] $|M_{\mathcal{P}_\infty}^{(1)}|=p^n$ when $m \not\equiv 1 \pmod {p^n}$, and so $g \not\equiv 0 \pmod {p^n}$;
\item[\rm (iv)] $|M_{\mathcal{P}_\infty}^{(2)}|=p^n$ when $m \equiv 1 \pmod {p^n}$, and so $g \equiv 0 \pmod {p^n}$. \end{itemize} \end{lemma}
\begin{lemma} \label{lem3} The $K$-automorphism group $Aut_K(\mathcal{C})$ fixes the place $\mathcal{P}_\infty$ except in the following two cases. \begin{enumerate} \item
\begin{itemize} \item[\rm (a)] Up to a linear substitution on $X$ and $Y$, $\mathcal{C}$ is the curve $Y^{p^n}+Y = X^m$, with $m<p^n$, $p^n \equiv -1 \pmod m$; \item[\rm (b)] $Aut_K(\mathcal{C})$ contains a cyclic normal subgroup $C_m$ of order $m$ such that $Aut_K(\mathcal{C}) / C_m \cong PGL(2,p^n)$; \item[\rm (c)] $C_m$ fixes each of the $p^n+1$ places with the same Weierstrass semigroup as $\mathcal{P}_\infty$; \item[\rm (d)] $Aut_K(\mathcal{C}) / C_m$ acts on the set of such $p^n+1$ places as $PGL(2,p^n)$. \end{itemize} \item \begin{itemize} \item[\rm (a)] Up to a linear substitution on $X$ and $Y$, $\mathcal{C}$ is the Hermitian curve $\mathcal H_{p^n}:Y^{p^n}+Y = X^{p^n+1}$; \item[\rm (b)] $Aut_K(\mathcal{C}) \cong PGU(3,p^n)$; \item[\rm (c)] $Aut_K(\mathcal{C})$ acts on the set of all places with the same Weierstrass semigroup as $\mathcal{P}_\infty$; \item[\rm (d)] $Aut_K(\mathcal{C})$ acts on the set of such places as $PGU(3,q)$ on the Hermitian unital. \end{itemize} \end{enumerate} \end{lemma}
\subsection{Algebraic Geometric codes}
We introduce in this section some basic notions on AG codes. We refer to \cite{Sti} for a detailed introduction.
Let $\mathcal{X}$ be a curve of genus $g$ over $\mathbb F_q$, $\mathbb F_q(\mathcal{X})$ be the field of $\mathbb F_q$-rational functions on $\mathcal{X}$, $\mathcal{X}(\mathbb F_q)$ be the set of $\mathbb F_q$-rational places of $\mathcal{X}$. For an $\mathbb F_q$-rational divisor $D=\sum_{P\in\mathcal{X}(\mathbb{F}_q)}n_P P$ on $\mathcal{X}$, denote by $$ \mathcal{L}(D):=\{f\in\mathbb{F}_q(\mathcal{X})\setminus\{0\}\mid (f)+D\geq0\}\cup\{0\} $$ the Riemann-Roch space associated to $D$, whose dimension over $\mathbb{F}_q$ is denoted by $\ell(D)$. Consider a divisor $D=P_1+\cdots P_n$ where $P_i\in\mathcal{X}(\mathbb{F}_q)$ and $P_i\ne P_j$ for $i\ne j$, and a second $\mathbb{F}_q$-rational divisor $G$ whose support is disjoint from the support of $D$. The \emph{functional AG code} $C_{\mathcal{L}}(D,G)$ is defined as the image of the linear evaluation map $$\begin{array}{llll} e_D : & \mathcal{L} (G) &\to &\mathbb{F}_q^n\\
& f & \mapsto & e_D(f)=(f(P_1),f(P_2),\ldots, f(P_n))\\ \end{array}. $$ The code $C_{\mathcal{L}}(D,G)$ has length $n$, dimension $k=\ell(G)-\ell(G-D)$, and minimum distance $d\geq d^*=n-\deg(G)$; $d^*$ is called the \emph{designed minimum distance} (or Goppa minimum distance). If $n>\deg(G)$, then $e_D$ is injective and $k=\ell(G)$. If $\deg(G)>2g-2$, then $k=\deg(G)+1-g$. The \emph{differential code} $C_{\Omega}(D,G)$ is defined as $$C_{\Omega}(D,G)= \left\{ (res_{P_1}(\omega),res_{P_2}(\omega),\ldots, res_{P_n}(\omega) \mid \omega \in \Omega(G-D)\right\},$$ where $\Omega(G-D)= \{\omega \in \Omega(\mathcal X) \mid (\omega) \geq G-D\} \cup \{0\}.$ The linear code $C_{\Omega}(D,G)$ has dimension $n-\deg(G)+g-1$ and minimum distance at least $\deg(G)-2g+2$.
Now we define the automorphism group of $C_\mathcal{L}(D,G)$; see \cite{GK2,JK2006}. Let $\mathcal{M}_{n,q}\leq{\rm GL}(n,q)$ be the subgroup of matrices having exactly one non-zero element in each row and column. For $\gamma\in Aut(\mathbb{F}_q)$ and $M=(m_{i,j})_{i,j}\in{\rm GL}(n,q)$, let $M^\gamma$ be the matrix $(\gamma(m_{i,j}))_{i,j}$. Let $\mathcal{W}_{n,q}$ be the semidirect product $\mathcal M_{n,q}\rtimes Aut(\mathbb{F}_q)$ with multiplication $M_1\gamma_1\cdot M_2\gamma_2:= M_1M_2^\gamma\cdot\gamma_1\gamma_2$. The \emph{automorphism group} $Aut(C_\mathcal{L}(D,G))$ of $C_\mathcal{L}(D,G)$ is the subgroup of $\mathcal{W}_{n,q}$ preserving $C_\mathcal{L}(D,G)$, that is, $$ M\gamma(x_1,\ldots,x_n):=((x_1,\ldots,x_n)\cdot M)^\gamma \in C_\mathcal{L}(D,G) \;\;\textrm{for any}\;\; (x_1,\ldots,x_n)\in C_\mathcal{L}(D,G). $$ Let $Aut_{\mathbb{F}_q}(\mathcal{X})$ be the $\mathbb{F}_q$-automorphism group of $\mathcal{X}$ and $$ Aut_{\mathbb{F}_q,D,G}(\mathcal{X}):=\{ \sigma\in Aut_{\mathbb{F}_q}(\mathcal{X})\,\mid\, \sigma(D)=D,\,\sigma(G)\approx_D G \}, $$ where $G'\approx_D G$ if and only if there exists $u\in\mathbb{F}_q(\mathcal{X})$ such that $G'-G=(u)$ and $u(P_i)=1$ for $i=1,\ldots,n$; note that $\sigma(G)=G$ implies $\sigma(G)\approx_D G$. Then the following holds. \begin{proposition}{\rm (\cite[Proposition 2.3]{BMZ2017})}\label{tivoglioiniettivo} If any non-trivial element of $Aut_{\mathbb{F}_q}(\mathcal{X})$ fixes less than $n$ $\mathbb{F}_q$-rational places of $\mathcal{X}$, then $Aut(C_{\mathcal{L}}(D,G))$ contains a subgroup isomorphic to $$ ({\rm Aut}_{\mathbb{F}_q,D,G}(\mathcal{X})\rtimes{\rm Aut}(\mathbb{F}_q))\rtimes \mathbb{F}_q^*. $$ \end{proposition}
In the construction of AG codes, the condition ${\rm supp}(D) \cap {\rm supp}(G)=\emptyset$ can be removed as follows; see \cite[Sec. 3.1.1]{TV}. Let $P_1,\ldots,P_n$ be distinct $\mathbb{F}_q$-rational places of $\mathcal{X}$ and $D=P_1+\ldots +P_n$, $G=\sum n_P P$ be $\mathbb{F}_q$-rational divisors of $\mathcal{X}$. For any $i=1,\ldots,n$ let $t_i$ be a local parameter at $P_i$. The map $$\begin{array}{llll} e^{\prime}_{D} : & \mathcal{L} (G) &\to &\mathbb{F}_q^n\\
& f & \mapsto & e^\prime_{D}(f)=((t^{n_{P_1}}f)(P_1),(t^{n_{P_2}}f)(P_2),\ldots, (t^{n_{P_n}}f)(P_n))\\ \end{array} $$ is linear. We define the \emph{extended AG code} $C_{ext}(D,G):=e^{\prime}(\mathcal{L}(G))$. Note that $e^\prime_D$ is not well-defined since it depends on the choise of the local parameters; yet, different choices yield extended AG codes which are equivalent. The code $C_{ext}$ is a lengthening of $C_{\mathcal{L}}(\hat D,G)$, where $\hat D = \sum_{P_i\,:\,n_{P_i}=0}P_i$. The extended code $C_{ext}$ is an $[n,k,d]_q$-code for which the following properties still hold: \begin{itemize} \item $d\geq d^*:=n-\deg(G)$. \item $k=\ell(G)-\ell(G-D)$. \item If $n>\deg(G)$, then $k=\ell(G)$; if $n>\deg(G)>2g-2$, then $k=\deg(G)+1-g$. \end{itemize}
\section{On the automorphism group of $\mathcal{C}$}\label{Sec:Aut}
At first we consider the norm-trace curve $\mathcal{N}_{q,r}$ with affine equation $$ X^{\frac{q^r-1}{q-1}} = Y^{q^{r-1}}+Y^{q^{r-2}}+\cdots+Y, $$ where $q$ is a $p$-power and $r$ is a positive integer. For $r=2$, this is the $\mathbb F_{q^2}$-maximal Hermitian curve, with automorphism group isomorphic to $PGU(3,q)$. For $r>2$, we determine the automorphism group of $\mathcal N_{q,r}$.
\begin{theorem}\label{AutNormTrace} For $r\geq3$, $Aut_K(\mathcal N_{q,r})$ has order $q^{r-1}(q^r-1)$ and is a semidirect product $G\rtimes C$, where $$ G=\left\{ (x,y)\mapsto(x,y+a)\mid Tr_{q^r\mid q}(a)=0 \right\}, \quad C=\{(x,y)\mapsto(b x,b^{\frac{q^r-1}{q-1}}y)\mid b\in\mathbb F_{q^r}^*\}. $$ \end{theorem}
\begin{proof} Suppose that $\mathcal N_{q,r}\cong\mathcal{H}_{\bar q}$ for some $p$-power $\bar q$. From Lemma \ref{lem1} (iii), $g(\mathcal{N}_{q,r})=g(\mathcal{H}_{\bar q})$ reads $\frac{(\frac{q^r-1}{q-1}-1)(q^{r-1}-1)}{2} = \frac{\bar q(\bar q-1)}{2}$. This implies $\bar q=q$ and $r=2$, a contradiction to the assumption on $r$.
Now suppose that $\mathcal{N}_{q,r}$ is isomorphic to the curve $\mathcal{X}:X^s=Y^{\bar q}+Y$ for some $p$-power $\bar q$, with $s<\bar q$, $s\mid(\bar q+1)$. From Lemma \ref{lem2}(iii), the Sylow $p$-subgroups $Aut_K(\mathcal{N}_{q,r})_{\mathcal P_\infty}^{(1)}$ and $Aut_K(\mathcal{X})_{\mathcal P_\infty}^{(1)}$ of $Aut_K(\mathcal{N}_{q,r})_{\mathcal P_\infty}$ and $Aut_K(\mathcal{X})_{\mathcal P_\infty}$ have order $q^{r-1}$ and $\bar q$, respectively. From Lemma \ref{lem1}(e) $\mathcal{N}_{q,r}$ and $\mathcal{X}$ have zero $p$-rank. Hence, $Aut_K(\mathcal{N}_{q,r})_{\mathcal P_\infty}^{(1)}$ and $Aut_K(\mathcal{X})_{\mathcal P_\infty}^{(1)}$ are Sylow $p$-subgroups of $Aut_K(\mathcal{N}_{q,r})\cong Aut_K(\mathcal{X})$; see \cite[Lemma 11.129]{HKT}. Therefore $q^{r-1}=\bar q$. Then $g(\mathcal{N}_{q,r})=g(\mathcal{X})$ yields $s=\frac{q^r-1}{q-1}=\bar q+\cdots+q+1$, a contradiction to $s<\bar q$.
From Lemma \ref{lem3}, this proves that $Aut_K(\mathcal N_{q^r\mid q})$ fixes $\mathcal P_\infty$. By direct checking $Aut_K(\mathcal{N}_{q,r})$ contains the group $G\rtimes C$ defined in the statement of the theorem. From Lemma \ref{lem3}, $Aut_K(\mathcal{N}_{q,r})=G\rtimes H$, where $H$ is a cyclic group. From Schur-Zassenhaus theorem, $H$ contains $C$ up to conjugation. By Lemma \ref{lem1}(e) the quotient curve $\mathcal{N}_{q,r}/G$ is rational, and its function field is $K(x)$. Hence the automorphism group $\bar H\cong H$ of $\mathcal{N}_{q,r}/G$ induced by $H$ has exactly two fixed places and acts semiregularly elsewhere; see \cite[Hauptsatz 8.27]{Hup}. Since $C\leq H$, the two places fixed by $\bar H$ are the place $\bar {\mathcal P}_\infty$ under $\mathcal P_\infty$ and the zero $\bar P_0$ of $x$. Let $\Omega=\{P_{(0,0)}, P_{(0,a_2)}, \ldots, P_{(0,a_{q^{r-1}})}\}$ be the orbit of $G$ lying over $\bar P_0$, so that $Aut_K(\mathcal{N}_{q,r})$ acts on $\Omega$; we denote by $P_{(0,0)}\in\Omega$ the zero of $y$, centered at the origin $(0,0)$. The group $H$ has a fixed point in $\Omega$ by the Orbit-Stabilizer theorem, and $P_{(0,0)}$ is the only fixed place of $C$ other than $\mathcal P_{\infty}$; thus, $H$ fixes $P_{(0,0)}$.
Therefore, $H$ fixes the unique pole of $x$ and $y$, fixes the unique zero of $y$, and acts on the $q^{r-1}$ simple zeros of $x$. This implies that a generator $h$ of $H$ acts as $h(x)=\mu x$, $h(y)=\rho y$ for some $\mu,\rho\in K^*$. By direct computation, $h$ is an automorphism of $\mathcal{N}_{q,r}$ if and only if $\rho=\rho^q$ and $\mu^{\frac{q^r-1}{q-1}}=\rho$. Hence, $H=C$. \end{proof}
The following result generalizes Theorem \ref{AutNormTrace}.
\begin{theorem}\label{AutMonom} Suppose that $m\not\equiv1\pmod{p^n}$ and $B(X)$ has just one root in $K$, so that Equation \eqref{EqSeparated} reads $$b_m\left(X+\frac{b_{m-1}}{m b_m}\right)^m = A(Y).$$ Then one of the following two cases occurs. \begin{itemize} \item[(i)] $m$ divides $p^n+1$ and $A(Y)$ is $p^n$-linearized, that is, $A(Y)=a_n Y^{p^n}+a_0 Y$. In this case, $\mathcal{C}$ is projectively equivalent to the curve $\mathcal{Q}_m$ with equation $X^m=Y^{p^n}+Y$ described in Case {\rm 1} of Lemma {\rm \ref{lem3}}. \item[(ii)] $m$ does not divide $p^n+1$ or $A(Y)$ is not $p^n$-linearized. Let $d=\gcd\left(j\geq1 : a_j\ne0\right)$ be the largest integer such that $A(Y)$ is $p^d$-linearized. Then $Aut_K(\mathcal{C})$ has order $p^n m(p^d-1)$ and $Aut_K(\mathcal{C})=G\rtimes C$, where $G=\left\{(x,y)\mapsto(x,y+a)\mid A(a)=0\right\}$ and $$ C=\left\{(x,y)\mapsto\left(bx+\frac{(b-1)b_{m-1}}{mb_m},b^m y\right)\mid b^{m(p^d-1)}=1\right\}. $$ \end{itemize} \end{theorem}
\begin{proof} Let $S$ be the stabilizer of $\mathcal P_\infty$ in $Aut_K(\mathcal{C})$. By direct checking, $S$ contains the semidirect product $G\rtimes C$. By Lemma \ref{lem2}, $S=G\rtimes H$, where $H$ is a cyclic group of order coprime to $p$. By Schur-Zassenhaus theorem, $H$ contains $C$ up to conjugation. Arguing as in the proof of Theorem \ref{AutNormTrace}, $\mathcal{C}/G$ is rational, and any nontrivial of the induced automorphism group $\bar H\cong H\leq Aut_K(\mathcal{C}/G)$ fixes the pole $\bar{\mathcal P}_\infty$ of $x$ and the zero $\bar P$ of $x+\frac{b_{m-1}}{m b_m}$. Hence $H$ acts on the $p^n$ distinct places of $\mathcal{C}$ lying over $\bar P$, and $H$ fixes one of them by the Orbit-Stabilizer theorem. The only fixed place of $C$ different from $\mathcal P_{\infty}$ is the unique zero $P$ of $y$, centered at the affine point $(\frac{-b_{m-1}}{m b_m},0)$; thus, $H$ fixes $P$. Let $h$ be a generator of $H$. We have shown that $h$ fixes the zero and the pole of $y$, which implies $h(y)=\rho y$ for some $\rho\in K$. Also, $h$ fixes the pole and acts on the simple zeros of $x+\frac{b_{m-1}}{m b_m}$; this implies $h(x+\frac{b_{m-1}}{m b_m}) = \mu (x+\frac{b_{m-1}}{m b_m})$ for some $\mu\in K$, that is, $h(x)=\mu x + \frac{(\mu-1)b_{m-1}}{m b_m}$. By direct checking, $h$ normalizes $G$ if and only if $A(\mu a)=0$ for all $a\in K$ satisfying $A(a)=0$. As $A(Y)$ is separable, this happens if and only if $A(\mu Y)=A(Y)$. This is equivalent to $\mu\in\mathbb F_{p^d}^*$, with $d$ defined as in the statement of this theorem. Then, in order for $h$ to be an automorphism of $\mathcal{C}$, we have $\rho^m=\mu$. We have shown that $S=G\rtimes C$.
From Lemma \ref{lem3}, either $Aut_K(\mathcal{C})=G\rtimes C$ and Case {\it (ii)} holds, or $\mathcal{C}$ is isomorphic to the curve $\mathcal Q_s: X^s=Y^{\bar q}+Y$ with $s\mid(\bar q+1)$, $s<\bar q$. Suppose that $\mathcal{C}\cong \mathcal{Q}_s$. By Lemma \ref{lem2} the Sylow $p$-subgroups of $Aut_K(\mathcal{C})$ and $Aut_K(\mathcal{Q}_s)$ have size $p^n$ and $\bar q$ respectively, so that $\bar q=p^n$; as $g(\mathcal{C})=g(\mathcal{Q}_s)$, we have $s=m$. The normalizer in $Aut_K(\mathcal{Q}_m)$ of a Sylow $p$-subgroup contains a cyclic group of order $p^n-1$, by Lemma \ref{lem3}(b). Hence, the same holds in $Aut_K(\mathcal{C})$ and $d=n$; this means that $\mathcal{C}$ has equation \begin{equation}\label{EqTuttoLin} b_m\left(X+\frac{b_{m-1}}{m b_m}\right)^m = a_n Y^{p^n} + a_0 Y. \end{equation} Conversely, if $\mathcal{C}$ is defined by Equation \eqref{EqTuttoLin}, then $\mathcal{C}$ is isomorphic to $\mathcal{Q}_m$. In fact, define $\varphi:(x,y)\mapsto(x^\prime,y^\prime):=(\gamma x,\delta a_0 y)$ with $\delta^{p^n-1}=a b^{-p^n}$ and $\gamma^m=\delta$. Then $K(x,y)=K(x^\prime,y^\prime)$ and $\varphi(\mathcal{C})=\mathcal{Q}_m$. Now the proof is complete. \end{proof}
Next result provides a converse to Theorem \ref{AutMonom} and extends \cite[Theorem 12.8]{HKT}.
\begin{theorem}\label{MonomAut}
Let $d=\gcd\left(j\geq1 : a_j\ne0\right)$ be the largest integer such that $A(Y)$ is $p^d$-linearized. If $|Aut_K(\mathcal{C})_{P_\infty}|/|Aut_K(\mathcal{C})_{P_\infty}^{(1)}|\geq m(p^d-1)$, then $B(X)$ has a unique root in $K$, that is, $$ B(X)=b_m\left(X+\frac{b_{m-1}}{mb_m}\right)^m. $$ \end{theorem}
\begin{proof} Let $S$ be the stabilizer of $\mathcal P_\infty$ in $Aut_K(\mathcal{C})$, $H$ be a cyclic complement of $S^{(1)}$ in $S$, and $\alpha$ be a generator of $H$. From Lemma \ref{lem2}, $G=\{(x,y)\mapsto(x,y+a)\mid A(a)=0\}$ is normal in $S$. Hence, $\alpha$ is an automorphism of the quotient curve $\mathcal{C}/G$; by Lemma \ref{lem1}(e), $\mathcal{C}/G$ is rational with function field $K(x)$. From \cite[Haptsatz 8.27]{Hup}, $\alpha$ has two fixed places in $K(x)$ and acts semiregularly elsewhere. One of the two places is the pole of $x$, lying under $\mathcal{P}_\infty$; the other place is the zero of $x^\prime:=x+u$ for some $u\in K$. Thus $\alpha(x^\prime)=bx^\prime$, for some $b\in K^*$ of order $ord(b)=ord(\alpha)$. Since $\alpha$ fixes the unique pole $\mathcal{P}_\infty$ of $y$ and the Weierstrass semigroup $H(\mathcal{P}_\infty)$ is generated by $-v_{\mathcal{P}_\infty}(x)=p^n$ and $-v_{\mathcal{P}_\infty}(x)=m$, we have that $\alpha(y)=ay+Q(x)$, where $a\in K^*$ and $Q(X)$ is a polynomial satisfying either $Q(X)=0$ or $\deg(Q(X))\cdot p^n<m$.
Let $B^\prime(X^\prime):=B(X)=B(X^\prime-u)$ and $Q^\prime(X^\prime):=Q(X)=Q(X^\prime-u)$. Since $\alpha$ is an automorphism of $\mathcal{C}$, the polynomial $A(aY+Q^\prime(X^\prime))-B^\prime(b X^\prime)$ is a multiple of the polynomial $A(Y)-B^\prime(X^\prime)$, say \begin{equation}\label{Transf} A(aY+Q^\prime(X^\prime))-B^\prime(b X^\prime) = k_1(A(Y)-B^\prime(X^\prime)) \end{equation}
with $k_1\in K^*$. As $A$ is a separable polynomial, Equation \eqref{Transf} implies $A(aY)=kA(Y)$ and hence $k_1=a^{p^j}$ for any $j$ such that $a_j\ne0$; thus, $k_1=a$ and $a^{p^d-1}=1$. Equation \eqref{Transf} also implies $B^\prime(bX^\prime)=B^\prime(X^\prime)+A(Q^\prime(X^\prime))$ and hence $k_1=b^m$ from the comparison of monomials ${X^\prime}^m$; thus, $(b^m)^{p^d-1}=1$ which yields $|H|=m(p^d-1)$.
Let $\beta:=\alpha^{p^d-1}$, which has order $m$ acts as $\beta(x^\prime)=b^{p^d-1}x^\prime$, $\beta(y)=y+Q^\prime(b^{p^d-2}x^\prime)$. As $\beta\in Aut_K(\mathcal{C})$, we have $$ A(Y+Q^\prime(b^{p^d-2}X^\prime))-B^\prime(b^{p^d-1} X^\prime) = k_2(A(Y)-B^\prime(X^\prime)) $$ with $k_2\in K^*$. Then $k_2=1$ and \begin{equation}\label{Transf2} B^\prime(b^{p^d-1} X^\prime)=B^\prime(X^\prime)+A(Q^\prime(b^{p^d-2}X^\prime)). \end{equation} We want to show that $\beta(y)=y$. Suppose by contradiction that $Q^\prime(b^{p^d-2}X^\prime)\ne0$. If $Q^\prime(b^{p^d-2}X^\prime)$ is a nonzero constant, then the order of $\beta$ is a multiple of $p$, a contradiction to $ord(\beta)=m$. If $\deg(Q^\prime(b^{p^d-2}X^\prime))>1$, then in Equation \eqref{Transf2} the right-hand side has a non-vanishing term of degree $p^n\cdot\deg(Q^\prime(X^\prime))$ while the left-hand side has not, a contradiction. Therefore, $\beta(x^\prime)=b^{p^d-1}x^\prime$ and $\beta(y)=y$, with $ord(b)=m(p^d-1)$. Since $\beta$ is an automorphism of $\mathcal{C}$, $B^\prime(X^\prime)=\lambda {X^\prime}^m$ for some $\lambda\in K^*$, that is, $B(X)=b_m\left(X+\frac{b_{m-1}}{mb_m}\right)^m$. \end{proof}
Even if $B(X)$ is not a monomial, the argument of the proof of Theorem \ref{MonomAut} shows the following result.
\begin{proposition}\label{Condiz} Let $Aut_K(\mathcal{C})_{P_\infty}=Aut_K(\mathcal{C})_{P_\infty}^{(1)}\rtimes H$ with $H=\langle\alpha\rangle$, and let $d=\gcd(j\geq1:a_j\ne0)$ be the largest integer such that $A(Y)$ is $p^d$-linearized. Then $\alpha(x)=bx+c$ for some $b,c\in K$, and $\alpha(B(x))=a B(x)$ for some $a\in\mathbb F_{p^d}^*$. \end{proposition}
\begin{remark} Once that $B(X)$ is explicitely given, Proposition {\rm \ref{Condiz}} provides a method to find $H$. In fact, $H$ has one fixed affine place in $K(x)$ and acts semiregularly on the other affine places; also, $H$ acts on the zeros of $B(x)$ with the same multiplicity. For instance: \begin{itemize}
\item If $B(X)$ has more than one root, but only one root with fixed multiplicity $M>1$, then $|H|$ divides either $M$ or $M-1$. \item If $B(X)$ has more than one root, and all the root have the same multiplicity $M>1$, then $H$ is trivial and $Aut_K(\mathcal{C})$ is a $p$-group of order $p^n$. \end{itemize} \end{remark}
\section{Multi point AG codes on the norm-trace curves}
Let $\ell,r\in\mathbb N$ with $r\geq3$, and let $\mathcal{N}_{q,r}$ be the norm-trace curve as defined in Section \ref{Sec:Aut}. Let $\Omega=\{P_{(0,y_1)},\ldots,P_{(0,y_{q^{r-1}})}\}$ be the set of the $q^{r-1}$ $\mathbb F_{q^r}$-rational places of $\mathcal{N}_{q,r}$ which are the zeros of $x$; the place $P_{(a,b)}$ is centered at the affine point $(a,b)$ of $\mathcal{N}_{q,r}$. Let $\Theta:=\mathcal{N}_{q,r}(\mathbb F_{q^r})\setminus\Omega$; note that $\Theta$ contains the place at infinity $P_\infty$. As pointed out in the proof of Theorem \ref{AutNormTrace}, the principal divisors of the coordinate functions are the following: \begin{itemize} \item $(x)=\sum_{P\in \Omega}P - q^{r-1} P_\infty$ ; \item $(y)=\frac{q^r-1}{q-1} P_{(0,0)} - \frac{q^r-1}{q-1} P_\infty$ . \end{itemize}
Define the $\mathbb F_{q^r}$-divisors $$G:=\sum_{P\in\Omega}\ell P\quad\textrm{and}\quad D:=\sum_{P\in\Theta}P.$$
Since $|\mathcal{N}_{q,r}(\mathbb F_{q^r})|=q^{2r-1}+1$ (see \cite[Lemma 2]{G2003}), $G$ and $D$ have degree $\ell q^{r-1}$ and $q^{2r-1}+1-q^{r-1}$, respectively. Denote by $C:=C_{\mathcal L}(D,G)$ the associated functional AG code over $\mathbb F_{q^r}$ having length $n= q^{2r-1}+1-q^{r-1}$, dimension $k$, and minimum distance $d$. The designed minimum distance is $$ d^*=n-\deg (G) = q^{2r-1}+1-(\ell+1)q^{r-1}. $$ The designed minimum distance is attained by $C$.
\begin{proposition} Whenever $d^*>0$, $C$ attains the designed minimum distance $d^*$. \end{proposition}
\begin{proof} By direct computation, the assumption $d^*>0$ is equivalent to $\ell<q^r$. Take $\ell$ distinct elements $c_1,\ldots,c_\ell\in \mathbb F_{q^r}^*$ and let $$f:=\prod_{i=1}^{\ell} \left(\frac{x-c_i}{x}\right).$$ The pole divisor of $f$ is exactly $G$, so that $f\in\mathcal{L}(G)$. By the properties of the norm and trace maps, $f$ has exactly $\ell q^{r-1}$ distinct $\mathbb F_{q^r}$-rational zeros. Thus, the weigth of $e_D(f)$ is $n-\ell q^{r-1}=d^*$. \end{proof}
We compute the dimension of $C$.
\begin{proposition} If $\frac{q^r-1}{q-1}-2\leq\ell\leq q^r-1$, then $$k=\ell q^{r-1}+1-\frac{1}{2}\left(\frac{q^r-1}{q-1}-1\right)\left(q^{r-1}-1\right).$$ \end{proposition}
\begin{proof} Since $n>\deg(G)>2g-2$, $k=\deg(G)+1-g$ by the Riemann-Roch Theorem. \end{proof}
\begin{proposition} \label{monomiallyeq} The code $C$ is monomially equivalent to the extended one-point code $C_{ext}(D,G^\prime)$, where $G^\prime=\ell q^{r-1}\mathcal{P}_{\infty}$. \end{proposition}
\begin{proof} We have $G=G^\prime+(x^\ell)$ and hence $\mathcal{L}(G^\prime)=\{f\cdot x^\ell \mid f\in\mathcal{L}(G)\}$. The codeword of $C_{\mathcal{L}}(D,G^\prime)$ associated to $f\cdot x^\ell$ is obtained as $$ \big( (f x^\ell)(P_1),\ldots,(f x^\ell)(\mathcal{P}_{\infty}),\ldots,(f x^\ell)(P_n) \big) = \big( f(P_1),\ldots,f(\mathcal{P}_{\infty}),\ldots,f(P_n) \big) \cdot M, $$ where $M$ is the diagonal matrix with diagonal entries $x(P_1)^\ell,\ldots,(t^{\ell q^{r-1}} x)(\mathcal{P}_{\infty})^\ell,\ldots,x(P_n)^\ell \in\mathbb F_{q^r}$, with $t$ a local parameter at $\mathcal{P}_\infty$. This means that $M$ defines a monomial equivalence between $C$ and $C_{ext}(D,G^\prime)$. \end{proof}
The Weierstrass semigroup $H(P_\infty)$ at $P_\infty$ is known to be generated by $q^{r-1}$ and $\frac{q^r-1}{q-1}$; see \cite{BR2013}. Thus, Proposition \ref{monomiallyeq} allows us to compute the dimension of $C$ also in those cases for which the Riemann-Roch Theorem does not give a complete answer.
\begin{corollary} If $1\leq\ell\leq\frac{q^r-1}{q-1}-3$, then the dimension of $C$ is $$ k=\ell+1+\frac{(q-1)}{2}\bigg \lfloor \frac{\ell}{q} \bigg \rfloor \bigg (\bigg \lfloor \frac{\ell}{q} \bigg \rfloor+1 \bigg)+\frac{(q^2-3q+2)}{2}+\Delta,$$ where, $$\Delta= \frac{(q-1)^2}{2} \bigg(\frac{\ell}{q}-1 \bigg)^2 + \bigg( \frac{(q-3)(q-1)}{2}\bigg) \bigg( \frac{\ell}{q}-1\bigg)+\frac{q(q-1)}{2} \bigg( \frac{\ell}{q}-1\bigg),$$
if $\ell \equiv 0 \pmod q$; $$\Delta= \frac{(q-1)^2}{2} \bigg \lfloor \frac{\ell}{q}\bigg \rfloor^2+\bigg( \frac{(q-3)(q-1)}{2}\bigg)\bigg \lfloor \frac{\ell}{q} \bigg \rfloor +\frac{q(q-1)}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor,$$
if $q \equiv q-1 \pmod q$; $$\Delta= \frac{(q-1)}{2} \bigg [ \bigg( \ell- \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q \bigg) \bigg \lfloor \frac{\ell}{q} \bigg \rfloor^2+\bigg( q-\ell+\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q-1\bigg) \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)^2\bigg ]+\bigg( \frac{q-3}{2}\bigg) \bigg[ \bigg(\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg \lfloor \frac{\ell}{q} \bigg \rfloor$$ $$+\bigg(q-\ell+\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q -1\bigg) \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)\bigg]+\frac{1}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor \bigg( \ell - \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg( \ell - \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q +1\bigg)$$ $$ + \frac{1}{2} \bigg ( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor -1\bigg) \bigg(q-1-\ell+ \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg( q+\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg),$$
otherwise. \end{corollary}
\begin{proof} Let $c:=(q^r-1)/(q-1)$. By the assumption on $\ell$, $\deg(G)<n$; hence, $k=\ell(G)$. From Proposition \ref{monomiallyeq}, $k=\ell(G^\prime)$ with $G^\prime=\ell q^{r-1}\mathcal{P}_{\infty}$. This means that $k$ equals the number of non-gaps $h\in H(\mathcal{P}_\infty)$ at $\mathcal{P}_\infty$ satisfying $h\leq\ell q^{r-1}$. From \cite{G2003} (see also \cite{BR2013}), $k$ is the number of couples $(i,j)\in\mathbb N^2$ such that $$ 0\leq i<q^r,\quad 0\leq j<q^{r-1},\quad i q^{r-1}+j c \leq \ell q^{r-1}. $$ Since $\ell\leq c-3$, this implies $$ k=\sum_{i=0}^{\ell} \left(\left\lfloor\frac{(\ell-i)q^{r-1}}{c}\right\rfloor+1\right) =\ell+1+\sum_{s=0}^{\ell}\left\lfloor\frac{ s q^{r-1}}{c}\right\rfloor. $$ Write $s=aq+b$ with $a\geq0$ and $1\leq b\leq q$. The condition $s\leq\ell$ is equivalent to $a\leq \lfloor\frac{\ell-b}{q}\rfloor$ when $b<q$, and to $a\leq \lfloor\frac{\ell}{q}\rfloor-1$ when $b=q$. Hence, \begin{equation}\label{conto1}
k =\ell+1+\sum_{a=0}^{\lfloor\frac{\ell}{q}\rfloor-1}\left\lfloor\frac{(aq+q)q^{r-1}}{c}\right\rfloor + \sum_{b=1}^{q-1} \sum_{a=0}^{\lfloor\frac{\ell-b}{q}\rfloor} \left\lfloor\frac{(aq+b)q^{r-1}}{c}\right\rfloor. \end{equation} By direct computation, \begin{small} \begin{equation}\label{conto2} \sum_{a=0}^{\lfloor\frac{\ell}{q}\rfloor-1}\left\lfloor\frac{(aq+q)q^{r-1}}{c}\right\rfloor = \sum_{a=0}^{\lfloor\frac{\ell}{q}\rfloor-1}\left\lfloor(a+1)(q-1)+\frac{a+1}{c}\right\rfloor = \sum_{a=0}^{\lfloor\frac{\ell}{q}\rfloor-1}(a+1)(q-1) = \frac{1}{2}(q-1)\left\lfloor \frac{\ell}{q} \right\rfloor \left(\left\lfloor \frac{\ell}{q} \right\rfloor+1\right). \end{equation} \end{small} Also, $$ \frac{(aq+b)q^{r-1}}{c} = a(q-1)+b-1+ \frac{q^r-1+a(q-1)-b(q^{r-1}-1)}{q^r-1}. $$ Assume that $1\leq b\leq q-1$ and $0\leq a\leq \left\lfloor\frac{\ell-b}{q}\right\rfloor \leq \left\lfloor\frac{\ell}{q}\right\rfloor$. By the assumption on $\ell$ follows $a\leq q\frac{q^{r-2}-1}{q-1}$. Thus, $$ \frac{q^r-1+a(q-1)-b(q^{r-1}-1)}{q^r-1}>0, \quad \frac{q^r-1+a(q-1)-b(q^{r-1}-1)}{q^r-1}<1, $$ so that $\left\lfloor\frac{(aq+b)q^{r-1}}{c}\right\rfloor=a(q-1)+b-1$. Thus, $$ \sum_{b=1}^{q-1} \sum_{a=0}^{\lfloor\frac{\ell-b}{q}\rfloor} \left\lfloor\frac{(aq+b)q^{r-1}}{c}\right\rfloor = \sum_{b=1}^{q-1} \sum_{a=0}^{\lfloor\frac{\ell-b}{q}\rfloor}\left(a(q-1)+b-1\right) =$$ $$\frac{(q-1)}{2} \sum_{b=1}^{q-1} \left\lfloor\frac{\ell-b}{q}\right\rfloor^2 + \bigg( \frac{q-3}{2} \bigg) \sum_{b=1}^{q-1} \left\lfloor\frac{\ell-b}{q}\right\rfloor + \sum_{b=1}^{q-1} b \left\lfloor\frac{\ell-b}{q}\right\rfloor + \frac{q^2-3q+2}{2}.$$ Denote by, $$A=\frac{(q-1)}{2} \sum_{b=1}^{q-1} \left\lfloor\frac{\ell-b}{q}\right\rfloor^2 , \quad B=\bigg( \frac{q-3}{2} \bigg) \sum_{b=1}^{q-1} \left\lfloor\frac{\ell-b}{q}\right\rfloor, \quad C= \sum_{b=1}^{q-1} b \left\lfloor\frac{\ell-b}{q}\right\rfloor .$$ We note that for a given $b=1,\ldots,q-1$, holds that $\bigg \lfloor \frac{\ell-b}{q} \bigg \rfloor \ne \bigg \lfloor \frac{\ell-b-1}{q} \bigg \rfloor$ if and only if $\ell-b \equiv 0 \pmod q$.
Thus if $\ell \equiv 0 \pmod q$ then $\bigg \lfloor \frac{\ell-b}{q} \bigg \rfloor=\frac{\ell}{q} - \bigg \lceil \frac{b}{q} \bigg \rceil=\frac{\ell}{q}-1$, for every $b=1,\ldots,q-1$; if $\ell \equiv q-1 \pmod q$ then $\bigg \lfloor \frac{\ell-b}{q} \bigg \rfloor=\bigg \lfloor \frac{\ell}{q} \bigg \rfloor$; while $\bigg \lfloor \frac{\ell-b}{q} \bigg \rfloor=\bigg \lfloor \frac{\ell}{q} \bigg \rfloor$ for $b=1,\ldots,\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q$ and $\bigg \lfloor \frac{\ell-b}{q} \bigg \rfloor=\bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)$ for $b=\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q+1,\ldots,q-1$, if $\ell \not\equiv 0,q-1 \pmod q$. In particular this implies that
$$A= \frac{(q-1)}{2}\sum_{b=1}^{q-1} \bigg( \frac{\ell}{q}-1\bigg)^2= \frac{(q-1)^2}{2} \bigg( \frac{\ell}{q}-1\bigg)^2,$$
if $\ell \equiv 0 \pmod q$, $$A=\frac{(q-1)}{2} \sum_{b=1}^{q-1} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor^2=\frac{(q-1)^2}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor^2,$$ if $\ell \equiv q-1 \pmod q$, and $$A=\frac{(q-1)}{2}\sum_{b=1}^{\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor^2+ \frac{(q-1)}{2} \sum_{b=\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q+1}^{q-1} \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)^2=$$ $$\frac{(q-1)}{2} \bigg[ \bigg( \ell- \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q \bigg) \bigg \lfloor \frac{\ell}{q} \bigg \rfloor^2+\bigg( q-\ell+\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q-1\bigg) \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)^2\bigg],$$
otherwise. Analagously, $$B= \frac{(q-3)}{2}\sum_{b=1}^{q-1} \bigg( \frac{\ell}{q}-1\bigg)= \frac{(q-3)(q-1)}{2} \bigg( \frac{\ell}{q}-1\bigg),$$ if $\ell \equiv 0 \pmod q$, $$B= \frac{(q-3)}{2} \sum_{b=1}^{q-1} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor=\frac{(q-1)(q-3)}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor,$$
if $\ell \equiv q-1 \pmod q$, while $$B=\frac{(q-3)}{2}\sum_{b=1}^{\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor+ \frac{(q-3)}{2} \sum_{b=\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q+1}^{q-1} \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)=$$ $$\bigg( \frac{q-3}{2}\bigg) \bigg[ \bigg(\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg \lfloor \frac{\ell}{q} \bigg \rfloor+\bigg(q-\ell+\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q -1\bigg) \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)\bigg]$$ otherwise, and $$C=\sum_{b=1}^{q-1} b \bigg( \frac{\ell}{q}-1\bigg)=\frac{q(q-1)}{2}\bigg( \frac{\ell}{q}-1\bigg),$$ if $\ell \equiv 0 \pmod q$, $$C=\sum_{b=1}^{q-1} b \bigg \lfloor \frac{\ell}{q} \bigg \rfloor=\frac{q(q-1)}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor,$$ if $\ell \equiv q-1 \pmod q$ and $$C=\sum_{b=1}^{\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q} b \bigg \lfloor \frac{\ell}{q} \bigg \rfloor+\sum_{b=\ell-\big \lfloor \frac{\ell}{q} \big \rfloor q+1}^{q-1} b \bigg( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor-1\bigg)=$$ $$\frac{1}{2} \bigg \lfloor \frac{\ell}{q} \bigg \rfloor \bigg( \ell - \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg( \ell - \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q +1\bigg)+ \frac{1}{2} \bigg ( \bigg \lfloor \frac{\ell}{q} \bigg \rfloor -1\bigg) \bigg(q-1-\ell+ \bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg) \bigg( q+\ell-\bigg \lfloor \frac{\ell}{q} \bigg \rfloor q\bigg),$$ otherwise. The claim now follows writing $k=\ell+1+\frac{(q-1)}{2}\bigg \lfloor \frac{\ell}{q} \bigg \rfloor \bigg (\bigg \lfloor \frac{\ell}{q} \bigg \rfloor+1 \bigg)+\frac{(q^2-3q+2)}{2}+A+B+C$. \end{proof}
We show that the automorphism group of $\mathcal{N}_{q,r}$ is inherited by the code $C$.
\begin{proposition} The automorphism group of $C$ has a subgroup isomorphic to $$ (Aut_K(\mathcal{N}_{q,r})\rtimes Aut_K(\mathbb F_{q^r}))\rtimes \mathbb F_{q^r}^*. $$ \end{proposition}
\begin{proof} The group $Aut_K(\mathcal{N}_{q,r})$ is defined over $\mathbb F_{q^r}$, so that $Aut_{\mathbb F_{q^r}}(\mathcal{N}_{q,r})=Aut_K(\mathcal{N}_{q,r})$. The support $supp(G)$ of the divisor $G$ is an orbit of $Aut_K(\mathcal{N}_{q,r})$, and $Aut_K(\mathcal{N}_{q,r})$ acts on the support $supp(D)=\mathcal{N}_{q,r}(\mathbb F_{q^r})\setminus supp(G)$ of the divisor $D$. Also, all places contained in $supp(G)$ have the same weight in $G$, which implies $\sigma(G)=G$ for any $\sigma\in Aut_K(\mathcal{N}_{q,r})$; analogously, $\sigma(D)=D$. Therefore, $Aut_{\mathbb F_{q^r},D,G}(\mathcal{N}_{q,r})$ is isomorphic to $Aut_K(\mathcal{N}_{q,r})$.
From the proof of Theorem \ref{AutNormTrace} follows that $Aut_K(\mathcal{N}_{q,r})$ has just two short orbits on $\mathcal{N}_{q,r}$. Namely, one short orbit is the singleton $\{\mathcal{P}_\infty\}$, which is fixed by the whole group $Aut_K(\mathcal{N}_{q,r})$; the other short orbit is the set $\Omega$ of the zeros of $x$, which has size $q^{r-1}$ and is fixed pointwise by the complement $H$ of the $p$-group $G$. Hence, any non-trivial element $\sigma\in Aut_K(\mathcal{N}_{q,r})$ is fixes at most $N:=q^{r-1}+1$ places on $\mathcal{N}_{q,r}$. Since the length $n$ of $C$ is bigger than $N$, the claim follows from Proposition \ref{tivoglioiniettivo}. \end{proof}
\begin{flushleft} Matteo Bonini\\ Dipartimento di Matematica,\\ University of Trento,\\ e-mail: {\sf matteo.bonini@unitn.it} \end{flushleft}
\begin{flushleft} Maria Montanucci\\ Dipartimento di Matematica, Informatica ed Economia,\\ University of Basilicata,\\ e-mail: {\sf maria.montanucci@unibas.it} \end{flushleft}
\begin{flushleft} Giovanni Zini\\ Dipartimento di Matematica e Informatica,\\ University of Florence,\\ e-mail: {\sf gzini@math.unifi.it} \end{flushleft}
\end{document} |
\begin{document}
\title[]{Boundedness and unboundedness results for some maximal operators on functions of bounded variation}
\author{J. M. Aldaz and J. P\'erez L\'azaro} \address{Departamento de Matem\'aticas y Computaci\'on, Universidad de La Rioja, 26004 Logro\~no, La Rioja, Spain.} \email{aldaz@dmc.unirioja.es} \address{Departamento de Matem\'aticas e Inform\'atica, Universidad P\'ublica de Navarra, 31006 Pamplona, Navarra, Spain.} \email{francisco.perez@unavarra.es}
\thanks{2000 {\em Mathematical Subject Classification.} 42B25, 26A84}
\thanks{Both authors were partially supported by Grant BFM2003-06335-C03-03 of the D.G.I. of Spain}
\thanks{The second named author thanks the University of La Rioja for its hospitality.}
\begin{abstract} We characterize the space $BV(I)$ of functions of bounded variation on an arbitrary interval $I\subset \mathbb{R}$, in terms of a uniform boundedness condition satisfied by the local uncentered maximal operator $M_R$ from $BV(I)$ into the Sobolev space $W^{1,1}(I)$.
By restriction, the corresponding characterization holds for
$W^{1,1}(I)$. We also show that if $U$ is open in $\mathbb{R}^d, d >1$, then boundedness from $BV(U)$ into $W^{1,1}(U)$ fails for the local directional maximal operator $M_T^{v}$, the local strong maximal operator $M_T^S$, and the iterated local directional maximal operator $M_T^{d}\circ \dots\circ M_T^{1}$. Nevertheless, if $U$ satisfies a cone condition, then $M_T^S:BV(U)\to L^1(U)$ boundedly, and the same happens with $M_T^{v}$, $M_T^{d} \circ \dots\circ M_T^{1}$, and $M_R$. \end{abstract}
\maketitle
\section {Introduction.}
\markboth{J. M. Aldaz, J. P\'erez L\'azaro} {A characterization of $BV(I)$}
The {\em local} uncentered Hardy-Littlewood maximal operator $M_R$ is defined in the same way as the uncentered Hardy-Littlewood maximal operator $M$, save for the fact that the supremum is taken over balls of diameter bounded by $R$, rather than all balls. The terms {\em restricted} and {\em truncated} have also been used in the literature to designate $M_R$. We showed in \cite{AlPe} that if $I$ is a bounded interval, then $M:BV(I)\to W^{1,1}(I)$ boundedly (Corollary 2.9). Here we complement this result by proving that for every interval
$I$, including the case of infinite length, $M_R:BV(I)\to W^{1,1}(I)$ boundedly. Of course, no result of this kind can hold if we consider $M$ instead of $M_R$, since $\|Mf\|_1 =\infty$ whenever $f$ is nontrivial. We shall see that if $f\in BV(I)$, then
$\|M_Rf\|_{W^{1,1}(I)}\le \max \{3 (1+2\log^+R), 4\} \|f\|_{BV(I)}$ (Theorem \ref{bd}), and furthermore, the logarithmic order of growth of $c:= \max \{3 (1+2\log^+R), 4\}$ cannot be improved (cf. Remark \ref{log} below). Also, since $c$ is nondecreasing in $R$, it provides a uniform bound for $M_T$ whenever $T\le R$. This observation leads to the following converse: Let $f\ge 0$. If there exists an $R>0$ and a constant $c = c(f,R)$ such that for all
$T\in (0,R]$, $M_Tf\in W^{1,1}(I)$ and $\| M_Tf\|_{W^{1,1}(I)} \le c$, then $f\in BV(I)$. A fortiori, given a locally integrable $f\ge 0$, we have that $f\in BV(I)$ if and only if for every $R>0$, $M_Rf\in W^{1,1}(I)$ and there exists a constant $c = c(f,R)$ such that for all
$T\in (0,R]$, $\| M_Tf\|_{W^{1,1}(I)} \le c$. By restriction to the functions $f$ that are absolutely continuous on $I$, we obtain the corresponding characterization for $W^{1,1}(I)$. If $f$ is real valued rather than nonnegative, since $f\in BV(I)$ (respectively $f\in W^{1,1}(I)$) if and only if both its positive and negative parts $f^+, f^-\in BV(I)$ (respectively $f^+, f^-\in W^{1,1}(I)$), we simply apply the previous criterion to $M_Tf^+$ and $M_T f^-$.
It is natural to ask whether the uniform bound condition is necessary to ensure that $f\in BV(I)$, or whether it is sufficient just to require that for all $T\in\mathbb{R}$, $M_Tf\in W^{1,1}(I)$. Uniform bounds are in fact needed (see Example \ref{counter1d}).
In higher dimensions we show that boundedness fails for the local strong maximal operator (where the supremum is taken over rectangles with sides parallel to the axes and uniformly bounded diameters) and the local directional maximal operator (where the supremum is taken over uniformly bounded segments parallel to a fixed vector), cf. Theorem \ref{Strong} below. But it is an open question whether the standard local maximal operator is bounded when $d > 1$, i.e., whether given a ``sufficiently nice" open set $U\subset\Bbb R^d$, $M_R$ maps $BV(U)$ boundedly into $W^{1,1}(U)$, or even into $BV(U)$. On the other hand, the direction from uniform boundedness of $M_Tf^+$ and $M_Tf^-$ to $f\in BV( U)$ follows immediately from the Lebesgue theorem on differentiation of integrals, even in the cases of the strong and directional maximal functions (cf. Theorem \ref{trivialdir}). All the maximal operators mentioned above map $BV(U)$ boundedly into $L^1(U)$, provided $U$ satisfies a cone condition (Theorem \ref{L1bounds}), so the question of boundedness of $M_R$ on $BV(U)$ is reduced to finding out how $DM_R$ behaves.
Previous results on these topics include the following. In \cite{Ha}, Piotr Haj\l asz utilized the local centered
maximal operator to present a
characterization, unrelated to the one given here, of the Sobolev space $W^{1,1}(\Bbb R^d)$. The boundedness of the centered Hardy-Littlewood maximal operator on the Sobolev spaces $W^{1,p}(\Bbb R^{d})$, for $1<p\le\infty$, was proven by Juha Kinnunen in \cite{Ki}. A local version of this result, valid on $W^{1,p}(\Omega )$, $\Omega\subset\Bbb R^{d}$ open, appeared in \cite{KiLi}. Additional work within this line
of research includes the papers \cite{HaOn}, \cite{KiSa}, \cite{Lu}, \cite{Bu}, \cite{Ko1}, and \cite{Ko2}. Of course, the case $p=1$ is significantly different from the case $p>1$. Nevertheless, in dimension $d=1$, Hitoshi Tanaka showed (cf. \cite{Ta}) that if $f\in W^{1,1}(\Bbb R)$, then the uncentered maximal function $Mf$ is differentiable a.e. and
$\| DMf\|_1 \le 2\| Df\|_1$ (it is asked in \cite{HaOn}, Question 1, p. 169, whether an analogous result holds when $d > 1$).
In \cite{AlPe} we strengthened Tanaka's result, showing that if $f\in BV(I)$, then $Mf$ is absolutely continuous and $\| DMf\|_1 \le | Df|(I)$, cf. \cite{AlPe} Theorem 2.5.
Finally we mention that the local (centered and uncentered) maximal operator has been used in connection with inequalities involving derivatives, cf. \cite{MaSh} and \cite{AlPe}. Another instance of this type of
application is given below (see Theorem \ref{ineq}). \section{Definitions, boundedness, and unboundedness results.}
Let $I$ be an interval and let $\lambda$ ($\lambda^d$ if $d > 1$) be Lebesgue measure. Since functions of bounded variation always have lateral limits, we can go from $(a,b)$ to $[a,b]$ by extension, and viceversa by restriction. Thus, in what follows it does not matter whether $I$ is open, closed or neither, nor whether it is bounded or has infinite length. \begin{definition} We say that $f:I \to \Bbb R$ is of bounded variation if its distributional derivative
$Df$ is a Radon measure with $|Df|(I) <\infty$, where $|Df|$ denotes the total variation of $Df$. In higher dimensions the definition is the same, save for the fact that $Df$ is (co)vector valued rather than real valued. More precisely, if $U\subset \Bbb R^d$ is an open set and $f:U \to \Bbb R$ is of bounded variation, then $Df$ is the vector valued Radon measure that satisfies, first, $\int_U f \operatorname{div }\phi dx = - \int_U \phi\cdot dDf$ for all
$\phi\in C_c^1(U, \Bbb R^d)$, and second, $|Df|(U) <\infty$. \end{definition}
In addition to $|Df|(I) <\infty$, it is often required that $f\in L^1(I)$. We do so only when defining the space $BV(I)$, and likewise in higher dimensions. The next definition is given only for the one dimensional case, being entirely analogous when $d > 1$. \begin{definition} Given the interval $I$, $$
BV(I) := \{f:I\to \Bbb R| f\in L^1(I), Df \mbox{ is a Radon measure, and } |Df|(I) <\infty\}, $$ and $$
W^{1,1}(I) := \{f:I\to \Bbb R| f\in L^1(I), Df \mbox{ is a function, and } Df\in L^1(I)\}. $$ \end{definition}
It is obvious that $W^{1,1}(I) \subset BV(I)$ properly. The Banach space $BV(I)$ is endowed with the norm $\|f\|_{BV(I)}:= \|f\|_1 + |Df|(I)$, and $
W^{1,1}(I),$ with the restriction of the $BV$ norm, i.e., $\|f\|_{W^{1,1}(I)}:= \|f\|_1 + \|Df\|_1$.
\begin{definition} The canonical representative of $f$ is the function $$ \overline{f}(x) := \limsup_{\lambda (I)\to 0, x\in I}\frac{1}{\lambda (I)}\int_I f(y)dy. $$ \end{definition}
In dimension $d=1$, bounded variation admits an elementary, equivalent definition. Given $P=\{x_1,\dots ,x_L\}\subset I$ with $x_1 <\dots <x_L$, the variation of the {\em function} $f:I\to \Bbb R$ associated to the partition $P$ is defined as $
V(f, I, P):= \sum_{j=2}^{L} |f(x_j) - f(x_{j-1})|, $ and the variation of $f$ on $I$, as $ V(f, I):=\sup_P V(f, I,P), $ where the supremum is taken over all partitions $P$ of $I$. Then $f$ is of bounded variation if $V(f, I) <\infty$. As it stands this definition is not $L^p$ compatible, in the sense that modifying $f$ on a set of measure zero can change $V(f,I)$, and
even make $V(f,I) = \infty.$ To remove this defect one simply says that $f$ is of bounded variation if $V(\overline{f}, I) <\infty$. It is then well known that $|Df|(I) = V(\overline{f}, I)$.
\begin{definition} Let $f: I\rightarrow \mathbb{R}$ be measurable and finite a.e.. The non-increasing rearrangement $f^*$ of $f$ is defined for $0<t<\lambda(I)$ as \begin{equation*}
f^*(t) = \sup_{\lambda(E)=t} \inf_{y\in E}|f(y)|. \end{equation*} \end{definition}
The function $f^*$ is non-increasing and equimeasurable with
$|f|$. Furthermore, \begin{equation}\label{rear}
\int_I f(y)dy =\int_0^{\lambda(I)}f^*(t)dt. \end{equation} For these and other basic properties of rearrangements see \cite[Chapter 2]{BeSh}. We mention that the same definition can be used for general measure spaces.
In the next definition, $\operatorname{ diam } (A)$ denotes the diameter of a set $A$, $U\subset \Bbb R^d$ denotes an open set, and $B\subset \Bbb R^d$ a ball with respect to some fixed norm.
\begin{definition} Given a locally integrable function $f:U\to \Bbb R$, the {\em local} uncentered Hardy-Littlewood maximal function $M_R f$ is defined by $$ M_Rf(x) := \sup_{ x\in B\subset U, \operatorname{ diam } B \le R}
\frac{1}{\lambda^d (B)}\int_B |f(y)|dy. $$ Of course, if the bound $R$ is eliminated then we get the usual uncentered Hardy-Littlewood maximal function $Mf$. \end{definition} As noted in the introduction, the terms {\em restricted} and {\em truncated} have also been used in the literature to designate $M_R$, but we prefer {\em local} for the reasons detailed in Remark 2.4 of \cite{AlPe}. Next we recall the well known weak type (1,1) inequality satisfied by $M$ in dimension 1, with the sharp constant 2. For all $f \in L^1(I)$ and all $ t>0$, \begin{equation}\label{weak}
(Mf)^*(t)\le 2 \|f\|_1/t. \end{equation}
\begin{definition} Let $U\subset \Bbb R^d$ be an open set, and let
$f:U\to \Bbb R$ be a locally integrable function. By a rectangle $R$ we mean a rectangle with sides parallel to the axes. The local uncentered {\em strong} Hardy-Littlewood maximal function $M_T^S f$ is defined by $$
M_T^Sf(x) := \sup_{ x\in R\subset U, \operatorname{ diam } (R)\le T}\frac{1}{\lambda^d (R)}\int_R |f(y)|dy. $$ Next, let $v\in \Bbb R$ be a fixed vector, and let $J$ denote a (one dimensional) segment in $\Bbb R^d$ parallel to $v$. The local uncentered {\em directional} Hardy-Littlewood maximal function $M_T^v f$ is defined by $$
M_T^v f(x) := \sup_{ x\in J\subset U, \lambda (J)\le T}\frac{1}{\lambda (J)}\int_J |f(y)|dy. $$ If $v=e_i$, then we write $M_T^i$ instead of $M_T^{e_i}$. \end{definition} We shall also be interested in the composition $M_T^d\circ\dots\circ M_T^1$ of the $d$ local directional maximal operators in the directions of the coordinate axes, since such composition controls $M_T^S$ pointwise. But first, we deal with the one dimensional case. \begin{theorem}\label{bd}
If $|f|\in BV(I )$, then $M_Rf\in W^{1,1}(I)$ and furthermore,
$\|M_Rf\|_{W^{1,1}(I)}\le 3 (1+2\log^+ R)\|f\|_{L^1 (I)}
+4\left|D|f|\right| (I).$ Hence, $\|M_Rf\|_{W^{1,1}(I)}\le \max \{3
(1+2\log^+ R), 4\} \|f\|_{BV (I)}.$
\end{theorem}
\begin{proof} Note that for any interval $J$ and any $h\in BV(J)$ \begin{equation}\label{eq3}
\|h\|_{L^\infty (J)} \le \operatorname{ essinf } |h| + |Dh| (J) \le \frac{\|h\|_{L^1(J)}}{\lambda(J)}+|Dh| (J). \end{equation}
Now, given $f:I\to \Bbb R$, if $\left|D|f|\right|$ is a finite Radon measure on $I$, then $M_Rf$ is absolutely continuous on $I$ and $\|DM_Rf\|_{L^1(I)}\le \left|D|f|\right|(I)$ by
\cite{AlPe}, Theorem 2.5 (we mention that for this bound on the size of the derivative, the hypothesis $f\in L^1(I)$ is not needed). Thus, it is enough to prove that given $|f|\in BV(I)$, \begin{equation}\label{est}
\|M_Rf\|_{L^1(I)}\le 3(1+2\log^+R) \|f\|_{L^1(I)} + 3 |D|f||(I). \end{equation} We may assume that $0\le f=\bar{f}$, since this does not change any value of $M_Rf$. Given $k\in\mathbb{Z}$ we denote by $I_k$ and $J_k$
the (possibly empty) intervals $I\cap[kR,(k+1)R)$ and $I\cap[(k-1)R,(k+2)R)$ respectively. We also set $f_k := f|_{J_k}$. Fix $k$. Then
\begin{equation}\label{eq2}
\int_{I_k}M_Rf(x)dx =
\int_{I_k}M_Rf_k(x)dx \le
\int_{I_k}Mf_k(x)dx. \end{equation} Suppose first that $\lambda(I_k)\le 1$. From (\ref{eq3}) we get \begin{equation}\label{smalllamb}
\int_{I_k}Mf_k(x)dx \le \lambda (I_k) \|f_k\|_{L^\infty (J_k)} \le
\|f_k\|_{L^1(J_k)}+|Df_k| (J_k). \end{equation} And if $\lambda (I_k) > 1$, then from (\ref{rear}) and (\ref{weak}) we obtain \begin{equation}\label{biglamb}
\int_{I_k}Mf_k(x)dx = \int_0^{\lambda (I_k)} (Mf_k)^*(t)dt
= \int_0^1 +\int_1^{\lambda (I_k)} \end{equation} \begin{equation*}
\le\|f_k\|_{L^\infty (J_k)} +2 \|f_k\|_{L^1 (J_k)}\int_1^{\lambda (I_k)}t^{-1}dt \end{equation*} \begin{equation*}
\le (1+2\log R)\|f_k\|_{L^1 (J_k)} +|Df_k| (J_k). \end{equation*} Since the intervals $I_k$ are all disjoint, and each nonempty $I_k$ is contained in $J_{k-1}, J_k$ and $J_{k+1}$, having empty intersection with all the other $J_i$'s, the
estimates (\ref{smalllamb}) and (\ref{biglamb}) yield \begin{equation}\label{L1norm}
\|M_R f\|_{L^1(I)} = \sum_{-\infty}^{\infty} \int_{I_k}M_Rf(x)dx \end{equation} \begin{equation*}
\le \sum_{-\infty}^{\infty} \left((1+2\log^+ R)\|f_k\|_{L^1 (J_k)}
+|Df_k| (J_k)\right) \end{equation*} \begin{equation*}
= 3\sum_{-\infty}^{\infty} (1+2\log^+ R)\|f_k\|_{L^1 (I_k)}
+3\sum_{-\infty}^{\infty}|Df_k| (I_k) \end{equation*} \begin{equation*}
= 3 (1+2\log^+ R)\|f\|_{L^1 (I)} +3|Df| (I). \end{equation*} Thus, \begin{equation*}\label{Rest}
\|M_R f\|_{BV(I)} \le 3 (1+2\log^+ R)\|f\|_{L^1 (I)} +4|Df| (I) \le
\max \{3 (1+2\log^+ R), 4\} \|f\|_{BV (I)}. \end{equation*} \end{proof}
\begin{remark}\label{log} The example $f:\Bbb R \to \Bbb R$ given by $f:=\chi_{[0,1]}$ shows that the logarithmic order of growth in the preceding theorem is the correct one. Here all the relevant quantities can be easily computed: $\|f\|_{L^1 (\Bbb R)}= 1$,
$|Df|(\Bbb R) = 2$, $\|M_R f\|_{L^1 (\Bbb R)}= 1 + 1/ R + 2\log R$ for $R\ge 1$, and
$|DM_Rf|(\Bbb R) = 2$ (for all $R>0$). \end{remark}
As noted in \cite{AlPe}, this kind of bounds on the size of maximal functions and their derivatives can be used to obtain variants of the classical Poincar\'e inequality, as well as other inequalities involving derivatives, under less regularity,
by using $DM_R f$ (a function) instead of $Df$ (a Radon measure). Here we present another instance of the same idea, a Poincar\'e type inequality involving $\|M_Rf\|_1$; the argument is standard but short, so we include it for the reader's convenience.
\
Given a compactly supported function $f$, denote by $N(f,R):= \operatorname{ supp }f + [-R, R]\subset \Bbb R$ the closed $R$-neighborhood of its support, that is, the set of all points at distance less than or equal to $R$ from the support of $f$.
\begin{theorem}\label{ineq}
Let $f\in BV(\Bbb R )$ be compactly supported. Then for all $R> 0$, we have
$ \|f\|_2^2 $
\begin{equation*}\label{MRpoin}
\le \min\left\{ \frac{(3 (1+2\log^+R))^2}{\lambda (N(f,R))} \|f\|_{BV(\mathbb{R})}^2+
\left(\frac{ \left(\lambda(N(f,R))\right)^2}{2}\right) \|D M_Rf\|_2^2, \lambda(N(f,R))^2
\|D M_Rf\|_2^2, \right\}.
\end{equation*}
\end{theorem} \begin{proof} Let $x< y$ be points in $\Bbb R$. By the Fundamental Theorem of Calculus, \begin{equation*}
M_R f(y) - M_R f(x) = \int_x^y DM_Rf(t) dt \le \|D M_Rf\|_1.
\end{equation*} Squaring and integrating with respect to $x$ and $y$ over $N(f,R)^2$, we get \begin{equation*}
\|M_Rf\|_2^2 \le
\frac{\|M_Rf\|_1^2}{\lambda (N(f,R))}+ \|D M_Rf\|_1^2\left( \frac{\lambda (N(f,R))}{2}\right).
\end{equation*}
Since $ \|f\|_2^2 \le \|M_Rf\|_2^2$, using (\ref{L1norm}) and either Jensen or H\"older inequality we obtain
\begin{equation*}\label{b}
\|f\|_2^2 \le \frac{(3 (1+2\log^+R))^2}{\lambda (N(f,R))} \|f\|_{BV(\mathbb{R})}^2+
\left(\frac{ \left(\lambda(N(f,R))\right)^2}{2}\right) \|D M_Rf\|_2^2.
\end{equation*}
On the other hand, integrating $M_R f(y) = \int_{\infty}^y DM_Rf(t) dt \le \|D M_Rf\|_1$ and repeating the previous steps we get
\begin{equation*}\label{a}
\|f\|_2^2 \le \lambda(N(f,R))^2
\|D M_Rf\|_2^2.
\end{equation*} \end{proof}
\begin{remark} In connection with the preceding inequality, we point out that if
$1 < p < \infty$ and $f\in W^{1,p}(\Bbb R )$, then $\|D M_Rf\|_p\le c_p \|D f\|_p$, with $c_p$ independent of $R$. Of course, the interest of the result lies in the fact that we can have $\|D M_Rf\|_p < \infty$ even if $Df$ is not a function (standard example, $f=\chi_{[0,1]}$). The cases $p=1,\infty$ are handled in \cite{AlPe}, Theorems 2.5 and 5.6. There we have $\|D M_Rf\|_p\le \|D f\|_p$. To see why $\|D M_Rf\|_p\le c_p \|D f\|_p$ holds with $c_p$ independent of $R$, repeat the sublinearity argument from \cite{Ki}, Remark 2.2 (i) (cf. also \cite{HaOn}, Theorem 1) using $M_Rf\le Mf$ to remove the dependency of the constant on $R$. \end{remark}
We shall consider next the local strong, directional, and iterated directional maximal operators, proving boundedness from $BV(U)$ into $L^1(U)$ and lack of boundedness from $BV(U)$ into $BV(U)$. Of course, since the strong maximal operator dominates pointwise (up to a constant factor) the maximal operator associated to an arbitrary norm, we also obtain the boundedness of $M_R$ from $BV(U)$ into $L^1(U)$ .
\begin{remark}\label{alt} It is possible to define $BV(U)$, where $U$ is open in $\mathbb{R}^d$,
without knowing a priori that $|Df|$ is a Radon measure: Write \begin{equation}\label{defvar}
\int_U |Df| := \sup\left\{\int_U f \operatorname{div} g: g\in C^1_c (U,\mathbb{R}^d), \|g\|_\infty \le 1\right\}. \end{equation}
Then $f\in BV(U)$ if $f\in L^1(U)$ and $\int_U |Df| < \infty$ (cf., for instance, Definition 1.3, pg. 4 of \cite{Giu}, or Definition 3.4, pg. 119 and Proposition 3.6, pg. 120 of \cite{AFP}). Integration by parts immediately yields that if $f\in C^1(U)$, then \begin{equation*}
\int_U |Df| = \int_U |\nabla f|dx, \end{equation*} (this is Example 1.2 of \cite{Giu}). With this approach
one has the following semicontinuity and approximation results (cf. Theorems 1.9 and 1.17 of \cite{Giu}), without any reference to Radon measures. \end{remark}
\begin{theorem}\label{semi} If a sequence of functions $\{f_n\}$ in $BV(U)$ converges in $L^1_{loc}(U)$ to $f$, then $\int_U|Df|\le \liminf_n \int_U |Df_n|$. \end{theorem}
\begin{theorem}\label{approx} If $f\in BV(U)$, then there exists a sequence of functions $\{f_n\}$ in $BV(U)\cap C^\infty(U)$ such that $\lim_n \int_U |f -f_n| dx = 0$ and
$\int_U|Df|= \lim_n \int_U |Df_n|$. \end{theorem}
Note that by passing to a subsequence, we may also assume that $\{f_n\}$ converges to $f$ almost everywhere.
If one uses the definition of $BV(U)$ given in Remark \ref{alt}, the fact that $Df$ is a Radon measure is obtained a posteriori via the Riesz Representation Theorem. Then of course $\int_U |Df| = |Df|(U)$.
\begin{definition} A finite cone $C$ of height $r$, vertex at $0$, axis $v$, and aperture angle $\alpha$, is the subset of $B(0,r)$ consisting of all vectors $y$ such that the angle between $y$ and $v$ is less than or equal to $\alpha /2$.
A finite cone $C_x$ with vertex at $x$, is a set of the form $x + C$, where the vertex of $C$ is $0$. Finally, an open set $U$ satisfies a cone condition if there exists a fixed finite cone $C$ such that every $x\in U$ is the vertex of a cone obtained from $C$ by a rigid motion. \end{definition}
We shall assume a cone condition in order to have available the following special case of the Sobolev embedding theorem (see, for instance, Theorem 4.12, pg. 85 of \cite{AdFo}). Of course, other type of conditions which also ensure the existence of such an embedding could be used instead (e.g., $U$ is an extension domain). The next Theorem and its Corollary are well known and included here for the sake of readability.
\begin{theorem}\label{Sobemb} Let the open set $U\subset \mathbb{R}^d$ satisfy a cone condition. Then there exists a constant $c>0$, depending only on $U$, such that for all $f\in W^{1,1}(U)$,
$\|f\|_{L^{\frac{d}{d-1}}(U)}\le c \|f\|_{W^{1,1}(U)}$. \end{theorem}
\begin{corollary}\label{BVemb} Let the open set $U\subset \mathbb{R}^d$ satisfy a cone condition. Then there exists a constant $c>0$, depending only on $U$, such that for all $f\in BV(U)$,
$\|f\|_{L^{\frac{d}{d-1}}(U)}\le c \|f\|_{BV(U)}$. \end{corollary} \begin{proof} Let $\{f_n\}$ be a sequence of functions in $BV(U)\cap C^\infty(U)$ such that
$f_n\to f$ a.e., $\lim_n \int_U |f -f_n| dx = 0$, and
$\int_U|Df|= \lim_n \int_U |\nabla f_n|dx$. By Fatou's lemma and Theorem \ref{Sobemb},
$\|f\|_{L^{\frac{d}{d-1}}(U)}\le \liminf_n \|f_n\|_{L^{\frac{d}{d-1}}(U)}
\le \lim_n c \|f_n\|_{W^{1,1}(U)}= c \|f\|_{BV(U)}$. \end{proof}
The next definition and lemma are valid for an arbitrary set $E\subset \mathbb{R}^k$, with measure defined by the restriction of the Lebesgue outer measure to the $\sigma$-algebra of all intersections of Lebesgue sets with $E$.
\begin{definition}\label{llog} Let $E\subset \mathbb{R}^k$ and $r\ge 1$. A function $g$ belongs to the Banach space $L(\log^+L)^r (E)$ if for some $t > 0$ we have \begin{equation}\label{condllogl}
\int \frac{|g(x)|}{t} \left(\log^+ \frac{|g(x)|}{t}\right)^r dx< \infty. \end{equation}
In that case the Luxemburg norm of $g$ is \begin{equation*}
\|g\|_{L(\log^+L)^r} := \inf\left\{t > 0: \int \frac{|g(x)|}{t} \left(\log^+ \frac{|g(x)|}{t}\right)^r dx \le 1\right\}. \end{equation*} \end{definition}
Note that by monotone convergence the inequality \begin{equation*}
\int \frac{|g(x)|}{t} \left(\log^+ \frac{|g(x)|}{t}\right)^r dx \le 1 \end{equation*}
holds when $t = \|g\|_{L(\log^+L)^r}$.
We mention that on finite measure spaces, the condition of Definition \ref{llog} is equivalent to the seemingly stronger requirement that for all $t> 0$, (\ref{condllogl}) hold.
The next lemma must be well known, but we include it for the reader's convenience. While stated for all $r\ge 1$, we only need the cases $r=1$ (used in Remark \ref{better}), $r=d-1$ (used in Theorem \ref{trivialdir}) and $r=d$ (used in Theorem \ref{L1bounds}).
\begin{lemma}\label{logemb} Let $E\subset \mathbb{R}^d$, where $d\ge 2$, and let $r\ge 1$. If $g\in L^{\frac{d}{d-1}}(E)$, then $g\in L(\log^+L)^r(E)$ and $\|g\|_{L(\log^+L)^r(E)}\le \left(r(d - 1)\right)^{\frac{r(d-1)}d}\|g\|_{L^{\frac{d}{d-1}}(E)}.$ \end{lemma}
\begin{proof} Note that $\log^+ y \le y^\alpha/\alpha$ for all $y,\alpha>0$, so given $t>0$, if we set $y = \frac{|g(x)|}{t}$ and $\alpha = \frac{1}{r(d-1)}$, we get \begin{equation*}
\int \frac{|g(x)|}{t} \left(\log^+ \frac{|g(x)|}{t}\right)^r
dx\le
\left(r(d - 1)\right)^{r}\left\|\frac{g}{t}\right\|_{L^{\frac{d}{d-1}}(E)}^{\frac{d}{d-1}}. \end{equation*}
Now let $t_0 < \|g\|_{L(\log^+L)^r}$. Then
$1 < \left(r(d - 1)\right)^{r}\left\|\frac{g}{t_0}\right\|_{L^{\frac{d}{d-1}}(E)}^{\frac{d}{d-1}}$, from which it follows that $\|g\|_{L(\log^+L)^r(E)}\le \left(r(d - 1)\right)^{\frac{r(d-1)}d}\|g\|_{L^{\frac{d}{d-1}}(E)}.$ \end{proof}
The proof of the next result is similar to that of Theorem \ref{bd}. We indicate the main differences: 1) In Theorem \ref{bd}, since $d=1$, no cone condition appears and we give a fully explicit constant; 2) when $d = 1$, we use the trivial embedding of $BV(I)$ in $L^\infty$ given in (\ref{eq3}) instead of Corollary \ref{BVemb} and Lemma \ref{logemb}; 3) for $d > 1$, bounds on the distributional gradient of the corresponding maximal operator are either false or not known.
\begin{theorem}\label{L1bounds} Let the open set $U\subset \Bbb R^d$ satisfy a cone condition. For every $R>0$, the local iterated directional maximal operator $M_R^{d}\circ \dots\circ M_R^{1}$ and the
local strong maximal operator $M_R^S$ map $BV(U)$ into $L^1(U)$ boundedly. Hence, so do the following operators: The standard local uncentered maximal operator $M_R$ associated to an arbitrary norm, the local directional maximal operator $M_R^v$, and $M_R^{i_k}\circ \cdots\circ M_R^{i_1}$, where $1\le k < d$ and $i_1 < \dots <i_k$. In fact, if $S_R$ is any of the above maximal operators, then there exists a constant $c > 0$, which depends only on the open set $U$, such that for all $f\in BV(U)$, \begin{equation}\label{conc}
\|S_Rf\|_{L^1(U)} \le c \left(\|f\|_{BV(U)} + (\log^+R)^d \|f\|_{L^1(U)} \right). \end{equation}
\end{theorem} \begin{proof} By Corollary \ref{BVemb}, it is enough to show that \begin{equation}\label{LplusL}
\|S_Rf\|_{L^1(U)} \le c \left(\|f\|_{L^{d/(d-1)}(U)} + (\log^+R)^d
\|f\|_{L^1(U)} \right). \end{equation} Now we can assume that $U=\mathbb{R}^d$. Else, we extend
$f$ without changing the right hand side of (\ref{LplusL}), by setting $f= 0$ on $\mathbb{R}^d\setminus U$.
The reason we are interested in having $U=\mathbb{R}^d$ is that later on, we will use the pointwise equivalence on $\mathbb{R}^d$ of maximal functions associated to different norms.
By $\eta$ we denote a generic $d$-tuple of integers $(n_1,\ldots,n_d)\in \mathbb{Z}^d$. For $\eta\in\mathbb{Z}^d$, we define the cubes $I_{\eta}=[n_1 R,(n_1+1)R)\times\dots\times [n_d R,(n_d +1)R)$ and
$J_{\eta}=[(n_1-1)R,(n_1+2)R)\times\dots\times [(n_d-1)R,(n_d+2)R)$. Set $f_{\eta}=f|_{J_{\eta}}$.
We want to estimate \begin{equation*} \alpha_{\eta}:= \int_{I_{\eta}}M^d_R\circ\cdots\circ M^1_Rf(x)dx \end{equation*} \begin{equation*}
= \int_{I_{\eta}}M^d_R\circ\cdots\circ M^1_Rf_{\eta}(x)dx \le
\int_{I_{\eta}}M^d\circ\cdots\circ M^1f_{\eta}(x)dx. \end{equation*} From \cite[\S I. Theorem 1]{Fa}, we get \begin{equation}\label{weakt}
\lambda^d(\{M^d\circ\cdots\circ M^1f_{\eta}>4t\})\le C
\int_{J_{\eta}} \frac{|f_{\eta}(x)|}{t}
\left(\log^+\frac{|f_{\eta}(x)|}{t}\right)^{d-1} dx, \end{equation} where $C$ is a constant that depends only on $d$. Moreover, calling
$A= \|f_{\eta}\|_{L(\log^+L)^{d}}$ and using (\ref{weakt}) we obtain \begin{equation*}
\alpha_{\eta} = 4 \int_0^\infty \lambda^d(I_{\eta}\cap \{M^d_R\circ\cdots\circ M^1_Rf_{\eta}(x)>4t\})dt =4\int_0^{A/R^d} + 4 \int_{A/R^d}^\infty \end{equation*} \begin{equation}\label{tres} \le 4 A + 4C\int_{A/R^d}^\infty \int_{J_{\eta}}
\frac{|f_{\eta}(x)|}{t}
\left(\log^+\frac{|f_{\eta}(x)|}{t}\right)^{d-1} dx dt = 4A + B. \end{equation}
Let $\tilde{J}_{\eta}:=J_{\eta}\cap\{|f(x)|>A/R^d\}$. Applying the Fubini-Tonelli Theorem and the change of variable $y(t)
=\log\frac{|f_{\eta}(x)|}{t}$ we have \begin{equation*}
B = 4 C\int_{\tilde{J}_{\eta}} \int_{A/R^d}^{|f_{\eta}(x)|}
\frac{|f_{\eta}(x)|}{t}
\left(\log\frac{|f_{\eta}(x)|}{t}\right)^{d-1}dt dx \end{equation*} \begin{equation*} = 4 C\int_{\tilde{J}_{\eta}}
|f_{\eta}(x)|dx\int_0^{\log^+\frac{|f_{\eta}(x)|R^d}{A}}
y^{d-1}dy
\end{equation*}
\begin{equation*}= \frac{4 C}{d}\int_{\tilde{J}_{\eta}}
|f_{\eta}(x)|\left(\log\frac{|f_{\eta}(x)|}{A} + d \log R\right)^d dx \end{equation*}
\begin{equation*}\le\frac{4 C 2^d}{d}\int_{{J}_{\eta}}
|f_{\eta}(x)|\left(\left(\log^+\frac{|f_{\eta}(x)|}{A}\right)^d + d^d \left(\log^+ R\right)^d\right)dx \end{equation*} \begin{equation*} = \frac{4 C 2^d }{d} \left(A\int_{J_{\eta}}
\frac{ |f_{\eta}(x)|}{A}\left(\log^+ \frac{|f_{\eta}(x)|}{A}\right)^d
dx + d^d\|f_{\eta}\|_{L^1(J_{\eta})} (\log^+R)^d\right)
\end{equation*}
\begin{equation}\label{last}
\le \frac{4 C 2^d }{d}\left(A + d^d\|f_{\eta}\|_{L^1(J_{\eta})} (\log^+
R)^d\right). \end{equation} Putting together (\ref{tres}), (\ref{last}), and Lemma \ref{logemb}, we get \begin{equation*}
\alpha_{\eta}\le C^\prime \left(\|f_{\eta}\|_{L^{d/(d-1)}(J_{\eta})}+ \|f_{\eta}\|_{L^1(J_{\eta})}(\log^+R)^d\right). \end{equation*} Next we sum over all $d$-tuples $\eta\in \mathbb{Z}^d$. Since a point in $\mathbb{R}^d$ cannot be contained in more than $3^d$ different cubes of type $J$, we conclude that for some $c > 0$, \begin{equation}\label{logd}
\int_{\mathbb{R}^d} M_R^d\circ \cdots\circ M_R^1f(x) dx \le c \left( \|f\|_{L^{d/(d-1)}(\mathbb{R}^d)}+ \|f\|_{L^1(\mathbb{R}^d)}(\log^+R)^d\right).
\end{equation} Since $M_R^Sf(x) \le M_R^d\circ \cdots\circ M_R^1f(x)$ for almost all $x \in \mathbb{R}^d$, the same inequality holds for $M_R^Sf$. Likewise, $M_R^S$
dominates pointwise the maximal operator $M_R$ associated to the $l^\infty$ norm (i.e., to cubes), so (\ref{conc}) also holds for $M_R$. Since local maximal operators associated to different norms are pointwise comparable by the equivalence of all norms in $\mathbb{R}^d$, inequality (\ref{conc}) holds, perhaps with a different value of $c$, for the maximal operator $M_R$ defined by any given norm. Finally, if $1\le k < d$ and $i_1 < i_2 <\dots<i_k$, we have $M_R^{i_k}\circ \cdots\circ M_R^{i_1}f(x) \le M_R^d\circ \cdots\circ M_R^1f(x)$ for all $x\in \mathbb{R}^d$, and $M_R^v$ obviously satisfies the same bounds as $M^1_R$, so (\ref{conc}) holds for all the operators under consideration. \end{proof}
\begin{remark}\label{better} It is possible to obtain bounds for $M_R$ directly, using essentially the same proof as in the previous theorem, rather than deriving them from the corresponding bounds for $M^S_R$. In fact, a direct approach yields a lower order of growth, $O(\log R)$ instead of $O((\log R)^d)$. More precisely, replace in the proof $L(\log^+L)^d$ by $L(\log^+L)$, and
inequality (\ref{weakt}) by the following well known refinement (due to N. Wiener, cf. \cite[Theorem 4$^\prime$]{Wi}) of the weak type inequality: \begin{equation*}
\lambda^d(\{Mf>t\})\le \frac{C}{t}\int_{\{|f|>t/2\}}|f(x)|dx
\qquad \textnormal{for all }t>0. \end{equation*} Then argue as before, to get \begin{equation*}
\int_U M_R f(x) dx \le c \left( \|f\|_{BV(U)}+ \|f\|_{L^1(U)}\log^+R\right).
\end{equation*}
An analogous remark can be made with respect to the operators $M_R^{i_k}\circ \cdots\circ M_R^{i_1}$ and $M_R^v$, obtaining orders of growth $O(\log^k R)$ and $O(\log R)$ respectively. \end{remark}
\begin{theorem}\label{Strong} Let $d > 1$ and let $U\subset\mathbb{R}^d$ be open. Given any $R>0$, the following maximal operators are unbounded on $BV(U)$: The local directional maximal operator $M_R^v$, the local iterated directional maximal operator $M_R^{d}\circ \dots\circ M_R^{1}$, and the local strong maximal operator $M_R^S$.
\end{theorem} \begin{proof} We will show that if $S_R$ denotes any of the maximal operators considered in the statement of the theorem, then there exists a sequence of characteristic functions
$f_{1/n}$ such that $\lim_{n\to\infty}\|f_{1/n}\|_{BV(U)} = 0$ and $$\lim_{n\to\infty}\frac{|DS_R(f_{1/n})|(U)}{\|f_{1/n}\|_{BV(U)}} = \infty.$$
In fact, the same result holds for the corresponding nonlocal maximal operators, which can be included in the notation $S_R$ by allowing the possibility $R=\infty$, as we do in this proof. So we take $0<R\le\infty$. Actually it is enough to consider $2 < R\le \infty$, since the argument we give below adapts to smaller values for $R$ just by rescaling. Similarly it is enough to consider the case $U=\mathbb{R}^d$. We start with $M^v_R$. By a rotation we may assume that $v = e_1$. For notational simplicity, we will write the proof for the case $d = 2$ only. Fix $R$. Given $0<\delta< 1$, set $f_\delta(x):=\chi_{[0,\delta]^2}(x)$. Then \begin{equation*}
\|f_\delta\|_1=\delta^2 \end{equation*}
and, since $|Df_\delta|(\mathbb{R}^2)$ is just the perimeter of the square $[0,\delta]^2$ (cf., for instance, Exercise 3.10 pg. 209 of \cite{AFP}), \begin{equation*}
|Df_\delta|(\mathbb{R}^2)=4\delta. \end{equation*} Thus \begin{equation}\label{une}
\|f_\delta\|_{BV(\mathbb{R}^2)} = O(\delta) \textnormal{ when } \delta \rightarrow 0. \end{equation} Next, let $\delta\le x\le 1$, and $0\le y\le \delta$. It is then easy to check that \begin{equation*} M_R^1(f_\delta)(x,y)=\frac{\delta}{x}. \end{equation*} Given $\delta \le t < 1$, the level sets $E_t :=\{M_R^1 (f_\delta ) > t\}$ are rectangles, with perimeter
$$|D\chi_{E_t}|(\mathbb{R}^2) \ge 2\delta + \frac{2\delta}{t}.$$
By the coarea formula for BV functions (cf. Theorem 3.40, pg. 145 of \cite{AFP}), we have \begin{equation}\label{coar}
|DM^1_R f_\delta|(\mathbb{R}^2)=\int_{-\infty}^{\infty}
|D\chi_{E_t}|(\mathbb{R}^2) dt \ge \int_{\delta}^{1}
|D\chi_{E_t}|(\mathbb{R}^2) dt \ge 2\delta \int_{\delta}^{1}\left( 1 + \frac{1}{t}\right) dt = \Theta\left(\delta\log \frac1{\delta}\right). \end{equation} where $\Theta$ stands for the exact order of growth.
From (\ref{une}) and (\ref{coar}) we obtain \begin{equation}\label{unbound}
\frac{|DM_R^1(f_\delta)|(\mathbb{R}^2)}{\|f_\delta\|_{BV((\mathbb{R}^2)}}\rightarrow \infty \textnormal{ when }\delta\rightarrow 0, \end{equation} as was to be proven.
Note next that on $[0,1]\times [0,\delta]$ the three maximal functions $M^1_Rf_\delta$,
$M_R^2\circ M_R^1 f_\delta$ and $M_R^S f_\delta$ take the same values, from which
it easily follows that for $\delta \le t < 1$,
$$|D\chi_{\{M_R^2\circ M_R^1 (f_\delta ) > t\}}|(\mathbb{R}^2) \ge 2\delta + \frac{2\delta}{t}$$ and
$$|D\chi_{\{M_R^S (f_\delta ) > t\}}|(\mathbb{R}^2) \ge 2\delta + \frac{2\delta}{t}.$$ Thus, the analogous statement to (\ref{unbound}) holds for $M_R^2\circ M_R^1 f_\delta$ and $M_R^S f_\delta$ also. \end{proof}
A standard mollification argument shows that the preceding maximal operators are not bounded on $W^{1,1}(U)$ either.
\section{Converses and a one dimensional characterization.}
Recall that $f^+$ and $ f^-$ denote respectively the positive and negative parts of $f$. Now, for any open set $U\subset \Bbb R^d$, $f\in BV(U)$ if and only if both $f^+\in BV(U)$ and $f^-\in BV(U)$. This can be seen as follows: If $f\in BV(U)$, it is immediate from the definition
\ref{defvar} contained in Remark \ref{alt} that $\int_U |Df| \ge
\int_U |D(f^+)|$ and $\int_U |Df| \ge \int_U |D(f^-)|$, so $f^+, f^-\in BV(U)$.
On the other hand, if both $f^+, f^-\in BV(U)$, then there are sequences $\{g_n\}$ and $\{h_n\}$ of $C^\infty$ functions that approximate $f^+$ and $f^-$ respectively, in the sense of Theorem \ref{approx}. Since $g_n - h_n\to f$ in
$L^1(U)$, by semicontinuity $|Df|(U) \le \liminf_n \int_U |\nabla
(g_n - h_n)| dx \le \lim_n \int_U |\nabla g_n| dx +\lim_n \int_U
|\nabla h_n| dx = |D(f^+)|(U) + |D(f^-)(U)|.$ Hence $f\in BV(U)$.
\begin{theorem}\label{trivialdir}
Let $U\subset \Bbb R^d$ be an open set and let $f:U\to \Bbb R $ be locally integrable. Suppose that there exists a sequence $\{a_n\}_1^\infty$ with $\lim_n a_n = 0$ and a constant $c$ such that for all $n$,
$M_{a_n}f^+\in W^{1,1}(U)$, $M_{a_n}f^-\in W^{1,1}(U)$, $\| M_{a_n}f^+\|_{W^{1,1}(U)} \le c$, and $\| M_{a_n}f^-\|_{W^{1,1}(U)} \le c$. Then $f\in BV(U)$. The same happens if instead of $M_R$ we consider either the local directional maximal operator, or, under the additional hypothesis that $U$ satisfies a cone condition, the local strong
maximal operator. \end{theorem}
\begin{proof} Consider first $f^+$. By the Lebesgue Theorem on differentiation of integrals we have that $\lim_n M_{a_n} f^+ = f^+$ a.e., so by dominated convergence, $M_{a_n} f^+ \to f^+$ in
$L^1(U)$, and by Theorem \ref{semi}, $\int_U|Df^+| \le \liminf_n
\int_U |DM_{a_n} f^+| \le c <\infty$. Repeating the argument for $f^-$ we get
$|Df|(U) \le |Df^+|(U) + |Df^-|(U) <\infty$. The result for the local strong maximal operator follows from
the well known Theorem of Jessen, Marcinkiewicz and Zygmund (\cite{JMZ}) stating that basis of rectangles (with sides parallel to the axes)
differentiates $L(\log ^+ L)^{d-1}_{loc}(U)$, and hence $BV(U)$ (cf. Corollary \ref{BVemb} and Lemma \ref{logemb}; for the first embedding we use the cone condition). Finally, the weak type $(1,1)$ boundedness of $M_T^v$ (which is obtained from the one dimensional result and the Fubini-Tonelli Theorem) also entails, by the standard argument, the corresponding differentiation of integrals result, so $\lim_n M^v_{a_n} f^+ = f^+$ and $\lim_n M^v_{a_n} f^- = f^-$. \end{proof}
For intervals $I\subset \mathbb{R}$ we have the following characterization.
\begin{theorem}\label{charact}
Let $f:I\to \Bbb R $ be locally integrable. Then the following are equivalent:
a) $f\in BV(I )$.
b) $M_Rf^+\in W^{1,1}(I)$, $M_Rf^-\in W^{1,1}(I)$,
$\|M_Rf^+\|_{W^{1,1}(I)}\le 3(1+2\log^+(R)) \|f^+\|_{L^1(I)} + 4 |Df^+|(I),$ and $\|M_Rf^-\|_{W^{1,1}(I)}\le 3(1+2\log^+(R)) \|f^-\|_{L^1(I)} + 4 |Df^-|(I).$
c) There exists a sequence $\{a_n\}_1^\infty$ with $\lim_n a_n = 0$ and a constant
$c = c(f, \{a_n\}_1^\infty)$ such that for all $n$,
$M_{a_n}f^+\in W^{1,1}(I)$, $M_{a_n}f^-\in W^{1,1}(I)$, $\| M_{a_n}f^+\|_{W^{1,1}(I)} \le c$, and $\| M_{a_n}f^-\|_{W^{1,1}(I)} \le c$.
d) There exists an $R>0$ and a constant $c = c(f, R)$ such that for all
$T\in (0,R]$, $M_Tf^+\in W^{1,1}(I)$, $M_Tf^-\in W^{1,1}(I)$, $\| M_Tf^+\|_{W^{1,1}(I)} \le c$, and $\| M_Tf^-\|_{W^{1,1}(I)} \le c$.
e) For every $R>0$ there exists a constant $c = c(f,R)$ such that for all
$T\in (0,R]$, $M_Tf^+\in W^{1,1}(I)$, $M_Tf^-\in W^{1,1}(I)$, $\| M_Tf^+\|_{W^{1,1}(I)} \le c$, and $\| M_Tf^-\|_{W^{1,1}(I)} \le c$.
If $f:I\to \Bbb R $ is absolutely continuous, then
a') $f\in W^{1,1}(I)$ is equivalent to b), c), d) and e).
\end{theorem}
\begin{proof} The implications b) $\to $ e), e) $\to $ d) and d) $\to $ c) are obvious, and a) $\to$ b) is the content of Theorem \ref{bd}. Without loss of generality we may take $I$ to be open, so c) $\to $ a) is a special case of Theorem \ref{trivialdir}. Finally, the last claim follows from the fact that
$f\in W^{1,1}(I)$ if and only if $f$ is absolutely continuous and
$f\in BV(I)$. \end{proof}
Let $f:I\to \Bbb R $ be locally integrable. By Theorem \ref{bd}, if $|f|\in BV(I )$ then for every $R >0$,
$M_Rf\in W^{1,1}(I)$ boundedly, with bound depending on $R$. Thus it is natural to ask whether
the latter condition alone suffices to ensure that $|f|\in BV(I )$. In other words, we
are asking whether the uniform bound condition appearing in parts c), d) and e) of
Theorem \ref{charact} is really needed. The following example shows that the answer is positive.
\begin{example}\label{counter1d}{\em There exists a non-negative function $f\in L^1(\Bbb R)\setminus BV(\Bbb R)$ such that for all $R>0$, $M_Rf\in W^{1,1}(\Bbb R )$.}
\begin{proof} Let $A$ be the closed set $[-1000,0]\cup\left(\cup_{n=0}^\infty [2^{-n}, 2^{-n} + 2^{-n-1}]\right)$, and let $f$ be the upper semicontinuous function $\chi_A$. Fix $R>0$. Clearly $M_R f \ge f$ everywhere, so by Lemma 3.4 of \cite{AlPe}, $M_Rf$ is a continuous function. Also, $M_R f|_{\mathbb{R}\setminus (0,2^{-n})}$ is Lipschitz, with $\operatorname{Lip }(M_Rf)\le \max\{R^{-1}, 2^{n+1}\}$, by Lemma 3.8 of \cite{AlPe}. Hence, if $E\subset \Bbb R$ has measure zero, so does $M_Rf(E)$, being a countable union of sets of measure zero. Next we show that $|DM_R f|(\Bbb R) < \infty$. Let $n\ge 1$ . On intervals of the form $( 2^{-n} + 2^{-n-1}, 2^{-n+1})$, if $R > 2^{-n-2}$ then $M_Rf > f$, so by Lemma 3.6 of \cite{AlPe} there exists an $x_n\in ( 2^{-n} + 2^{-n-1}, 2^{-n+1})$ such that
$M_Rf$ is decreasing on $( 2^{-n} + 2^{-n-1},x_n)$ and increasing on $(x_n, 2^{-n+1})$. Taking this fact into account, it is easy to see
that $V(M_R f, \mathbb{R})$ is decreasing in $R$, so we may suppose $R \in (0,1)$. Select $N\in \Bbb N$ such that $2^{-N+1} < R$. Then for $n > N$, \begin{equation*}
V(M_Rf,( 2^{-n} + 2^{-n-1}, 2^{-n+1})) = 2\left(1-M_Rf(x_n)\right) \le 2\left(1-\frac{R- 2^{-n+1}}{R}\right)\le \frac{ 2^{-n+2}}{R}. \end{equation*}
Hence $|DM_R f|(\Bbb R ) \le 2 + 2(N+1) <\infty$. Since $M_Rf$ is continuous, of bounded variation, and maps measure zero sets into measure zero sets, by the Banach Zarecki Theorem it
is absolutely continuous, so $M_Rf\in W^{1,1}(\Bbb R )$. \end{proof} Of course, using $\Bbb R$ above is not necessary, the example can be easily adapted to any other interval $I$. \end{example}
\end{document} |
\begin{document}
\begin{abstract} For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of A.~Grothendieck states that $C_k(X)$ has the Dunford-Pettis property and the sequential Dunford--Pettis property. We extend Grothendieck's result by showing that $C_k(X)$ has both the Dunford-Pettis property and the sequential Dunford-Pettis property if $X$ satisfies one of the following conditions: (i) $X$ is a hemicompact space, (ii) $X$ is a cosmic space (=a continuous image of a separable metrizable space), (iii) $X$ is the ordinal space $[0,\kappa)$ for some ordinal $\kappa$, or (vi) $X$ is a locally compact paracompact space. We show that if $X$ is a cosmic space, then $C_k(X)$ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. We prove that $C_p(X)$ has the Dunford--Pettis property and the sequential Dunford-Pettis property for every Tychonoff space $X$, and $C_p(X) $ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. \end{abstract}
\maketitle
\section{Introduction}
The class of Banach spaces with the Dunford--Pettis property enjoying also the Grothendieck property plays an essential and important role in the general theory of Banach spaces (particularly of continuous functions) and vector measures with several remarkable applications, we refer the reader to \cite{Dales-Lau}, \cite{Diestel-DP}, \cite{Diestel} and \cite[Chapter~VI]{Diestel-Uhl}.
It is well known by a result of A.~Grothendieck, see \cite[Corollary~4.5.10]{Dales-Lau}, that for every injective compact space $K$, the Banach space $C(K)$ has the Grothendieck property. Consequently, this applies for each extremely disconnected compact space $K$. A.~Grothendieck also proved that the Lebesgue spaces $L^{1}(\mu)$, the spaces $\pounds^{\infty}$ and $\pounds^{1}$ have the Dunford--Pettis property.
This line of research was continued by J.~Bourgain in \cite{bourgain}, where he showed that the spaces $C_{L^{1}}$ and $L^{1}_{C}$ enjoy also the Dunford--Pettis property. Moreover, in \cite{bourgain1} J.~Bourgain provided interesting sufficient conditions for subspaces $L$ of the Banach space $C(K)$ to have the Dunford--Pettis property. This results have been used by J.A.~Cima and R.M.~Timoney \cite{Ci-Ti} to study the Dunford--Pettis property for $T$-ivariant algebras on $K$. F.~Bombal and I.~Villanueva characterized in \cite{BomVil} those compact spaces $K$ such that $C(K)\hat{\otimes} C(K)$ have the Dunford--Pettis property.
Quite recently this area of research around Dunford--Pettis property for Banach spaces has been extended to a more general setting including general theory of locally convex spaces. This approach enabled specialists to apply this work to concrete problems related with the mean ergodic operators in Fr\'{e}chet spaces; we refer to articles \cite{ABR}, \cite{ABR1}, \cite{ABR2}, \cite{albaneze} and \cite{Bonet-Ricker} .
The classical Dunford--Pettis theorem states that for any measure $\mu$ and each Banach space $Y$, if $T:L_1(\mu)\to Y$ is a weakly compact linear operator, then $T$ is completely continuous (i.e., $T$ takes weakly compact sets in $L_1(\mu)$ onto norm compact sets in $Y$). This result motivates Grothendieck to introduce the following property (for comments, see \cite[p.633-634]{Edwards}):
\begin{definition}[\cite{Grothen}] \label{def:DP} A locally convex space $E$ is said to have the {\em Dunford--Pettis property} ($(DP)$ {\em property} for short) if every continuous linear operator $T$ from $E$ into a quasi-complete locally convex space $F$, which transforms bounded sets of $E$ into relatively weakly compact subsets of $F$, also transforms absolutely convex weakly compact subsets of $E$ into relatively compact subsets of $F$. \end{definition} Actually, it suffices that $F$ runs over the class of Banach spaces, see \cite[p.633]{Edwards}.
A.~Grothendieck proved in \cite[Proposition~2]{Grothen} that a Banach space $E$ has the $(DP)$ property if and only if given weakly null sequences $\{ x_n\}_{n\in\mathbb{N}}$ and $\{ \chi_n\}_{n\in\mathbb{N}}$ in $E$ and the Banach dual $E'$ of $E$, respectively, then $\lim_n \chi_n(x_n)=0$. He used this result to show that every Banach space $C(K)$ has the $(DP)$ property, see \cite[Th\'{e}or\`{e}me~1]{Grothen}. Extending this result to locally convex spaces (lcs, for short) and following \cite{Gabr-free-resp}, we consider the following ``sequential'' version of the $(DP)$ property.
\begin{definition} \label{def:sDP} A locally convex space $E$ is said to have the {\em sequential Dunford--Pettis property} ($(sDP)$ {\em property}) if given weakly null sequences $\{ x_n\}_{n\in\mathbb{N}}$ and $\{ \chi_n\}_{n\in\mathbb{N}}$ in $E$ and the strong dual $E'_\beta$ of $E$, respectively, then $\lim_n \chi_n(x_n)=0$. \end{definition}
It turns out, as A.A.~Albanese, J.~Bonet and W.J.~Ricker proved in \cite[Corollary~3.4]{ABR}, that the $(DP)$ property and the $(sDP)$ property coincide for the much wider class of Fr\'{e}chet spaces (or, even more generally, for strict $(LF)$-spaces). In \cite[Proposition~3.3]{ABR} they showed that every barrelled quasi-complete space with the $(DP)$ property has also the $(sDP)$ property. For further results we refer the reader to \cite{ABR,Bonet-Lin-93,BFV,Diestel-DP} and reference therein.
For a Tychonoff (=completely regular and Hausdorff) space $X$, we denote by $C_k(X)$ and $C_p(X)$ the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. Being motivated by the aforementioned discussion and results it is natural to ask:
\begin{problem} \label{prob:Ck(X)-DPP} Characterize Tychonoff spaces $X$ for which $C_k(X)$ and $C_p(X)$ have the $(DP)$ property or the $(sDP)$ property. \end{problem}
A.~Grothendieck proved that the Banach space $C(\beta\mathbb{N})$, where $\beta\mathbb{N}$ is the Stone--\v{C}ech compactification of the natural numbers $\mathbb{N}$ endowed with the discrete topology, has the following property: Any weak-$\ast$ convergent sequence in the Banach dual of $C(\beta\mathbb{N})$ is also weakly convergent. This result motivates the following important property. \begin{definition} \label{def:Grothendieck} A locally convex space $E$ is said to have the {\em Grothendieck property} if every weak-$\ast$ convergent sequence in the strong dual $E'_\beta$ is weakly convergent. \end{definition} These results mentioned above motivate also the second general question considered in the paper.
\begin{problem} \label{prob:Ck(X)-Grothendieck} Characterize Tychonoff spaces $X$ for which $C_k(X)$ and $C_p(X)$ have the Grothendieck property. \end{problem}
For spaces $C_p(X)$ we obtain complete answers to Problems \ref{prob:Ck(X)-DPP} and \ref{prob:Ck(X)-Grothendieck}. Recall that a subset $A$ of a topological space $X$ is called {\em functionally bounded in $X$} if every $f\in C(X)$ is bounded on $A$.
\begin{theorem} \label{t:DP-Cp} Let $X$ be a Tychonoff space. Then: \begin{enumerate} \item[{\rm (i)}] $C_p(X)$ has the $(sDP)$ property; \item[{\rm (ii)}] $C_p(X)$ has the $(DP)$ property; \item[{\rm (iii)}] $C_p(X) $ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. \end{enumerate} \end{theorem} We prove Theorem \ref{t:DP-Cp} in Section \ref{sec:Cp-GDP}. In this section we also recall some general results related to the $(DP)$ property and the Grothendieck property which are essentially used in the article.
For spaces $C_k(X)$ the situation is much more complicated. Following E.~Michael \cite{Mich}, a Tychonoff space $X$ is a {\em cosmic space} if $X$ is a continuous image of a separable metrizable space. The following theorem is our second main result. \begin{theorem} \label{t:DP-Ck} Assume that a Tychonoff space $X$ satisfies one of the following conditions: \begin{enumerate} \item[{\rm (i)}] $X$ is a hemicompact space; \item[{\rm (ii)}] $X$ is a cosmic space; \item[{\rm (iii)}] $X$ is the ordinal space $[0,\kappa)$ for some ordinal $\kappa$; \item[{\rm (iv)}] $X$ is a locally compact paracompact space. \end{enumerate} Then $C_k(X)$ has the $(sDP)$ property and the $(DP)$ property. \end{theorem} Since every compact subset of a cosmic space is metrizable, it is easy to see that indeed all four classes (i)-(iv) of Tychonoff spaces are independent (in the sense that there are spaces which belong to one of the classes but do not belong to other classes). In particular, the spaces $C_k(\mathbb{N}^\mathbb{N})$ and $C_k(\mathbb{Q})$ are non-Fr\'{e}chet spaces with the $(DP)$ property and the $(sDP)$ property.
For the Grothendieck property, essentially using Theorem \ref{t:DP-Cp} we prove the following result (the definitions of $\mu$-spaces and sequential spaces are given before the proof of this result).
\begin{theorem} \label{t:Grothendieck-Ck} Let $X$ be a Tychonoff space. \begin{enumerate} \item[{\rm (i)}] If $X$ is a $\mu$-space whose compact subsets are sequential (for example, $X$ is cosmic), then $C_k(X)$ has the Grothendieck property if and only if every functionally bounded subset of $X$ is finite. \item[{\rm (ii)}] If $X$ is a sequential space, then $C_k(X)$ has the Grothendieck property if and only if $X$ is discrete. \end{enumerate} \end{theorem}
Note that the condition on $X$ of being a sequential space in (ii) of Theorem \ref{t:Grothendieck-Ck} cannot be replaced by the condition ``$X$ is a $k$-space'', as the compact space $\beta\mathbb{N}$ shows. We prove Theorems \ref{t:DP-Ck} and \ref{t:Grothendieck-Ck} in Section \ref{sec:Ck-GDP}.
\section{The $(DP)$ property and the Grothendieck property for $C_p(X)$} \label{sec:Cp-GDP}
In what follows we shall use the following result. \begin{theorem}[\protect{\cite[Theorem~9.3.4]{Edwards}}] \label{t:Edwards-DP} An lcs $E$ has the $(DP)$ property if and only if every absolutely convex, weakly compact subset of $E$ is precompact for the topology $\tau_{\Sigma'}$ of uniform convergence on the absolutely convex, equicontinuous, weakly compact subsets of $E'_\beta$. \end{theorem}
A subset $A$ of a topological space $X$ is called {\em sequentially compact} if every sequence in $A$ contains a subsequence which is convergent in $A$. Following \cite{Gabr-free-resp}, a Tychonoff space $X$ is {\em sequentially angelic} if a subset $K$ of $X$ is compact if and only if $K$ is sequentially compact. It is clear that every angelic space is sequentially angelic. An lcs $E$ is {\em weakly sequentially angelic} if the space $E_w$ is sequentially angelic, where $E_w$ denotes the space $E$ endowed with the weak topology. The following proposition is proved in (ii) of Proposition~3.3 of \cite{ABR} (the condition of being quasi-complete is not used in the proof of this clause).
\begin{proposition}[\cite{ABR}] \label{p:DP-c0-quasibarrelled} Assume that an lcs $E$ satisfies the following conditions: \begin{enumerate} \item[{\rm (i)}] $E$ has the $(sDP)$ property; \item[{\rm (ii)}] both $E$ and $E'_\beta$ are weakly sequentially angelic. \end{enumerate} Then $E$ has the $(DP)$ property. \end{proposition}
Also we shall use repeatedly the next assertion, see \cite[Corollary~3.4]{ABR}. \begin{proposition}[\cite{ABR}] \label{p:DP-LF-DF} Let $E$ be a complete $(LF)$-space. Then $E$ has the $(DP)$ property if and only if it has the $(sDP)$ property. \end{proposition}
\begin{proposition} \label{p:sDP-strong-dual-Schur} Let $E$ be a locally complete lcs whose every separable bounded set is metrizable. Assume that $E$ is weakly sequentially angelic and does not contain an isomorphic copy of $\ell_1$. Then $E$ has the $(sDP)$ property if and only if the strong dual $E'_\beta$ has the Schur property. \end{proposition}
\begin{proof} Assume that $E$ has the $(sDP)$ property. Let $\{ \chi_{n}\}_{n\in\mathbb{N}}$ be a $\sigma(E',E'')$-null sequence in $E'$. Then Proposition 3.3(i) of \cite{Gabr-free-resp} guarantees that $\chi_n \to 0$ in the Mackey topology $\mu(E',E)$ on $E'$. As $E$ does not contain an isomorphic copy of $\ell_1$, a result of Ruess \cite[Theorem~2.1]{ruess} asserts that $\chi_n \to 0$ in the strong topology. Thus $E'_\beta$ has the Schur property. Conversely, if $E'_\beta$ has the Schur property, then $E$ has the $(sDP)$ property by Proposition 3.1(ii) of \cite{Gabr-free-resp}. \end{proof}
If $E$ is a Fr\'{e}chet space, the necessity in the following corollary is Theorem 2.7 of \cite{albaneze}. \begin{corollary} \label{c:Frechet-DP-L1-Schur} Let $E$ be a strict $(LF)$-space not containing an isomorphic copy of $\ell_1$. Then $E$ has the $(DP)$ property if and only if the strong dual $E'_\beta$ has the Schur property. \end{corollary}
\begin{proof} The space $E$ is weakly angelic by a result of B.~Cascales and J.~Orihuela, see \cite[Proposition~11.3]{kak}. Also it is well known that $E$ is even complete whose every bounded set is metrizable, see \cite{bonet}. Taking into account that the $(DP)$ property is equivalent to the $(sDP)$ property in the class of strict $(LF)$-spaces by Proposition \ref{p:DP-LF-DF}, the assertion follows from Proposition \ref{p:sDP-strong-dual-Schur}. \end{proof}
Let $X$ be a Tychonoff space. It is well known that the dual space of $C_p(X)$ is (algebraically) the linear space $L(X)$ of all linear combinations $\chi=a_1 x_1+\cdots + a_n x_n$, where $a_1,\dots,a_n$ are real numbers and $x_1,\dots,x_n\in X$. So \[ \chi(f)=a_1 f(x_1)+\cdots + a_n f(x_n), \quad f\in C_p(X). \] If all the coefficients $a_1,\dots,a_n$ are nonzero and all $x_1,\dots,x_n$ are distinct, we set \[ \mathrm{supp}(\chi):=\{ x_1,\dots,x_n\} \mbox{ and } \chi(x_i) := a_i, \; i=1,\dots,n. \] If $x\not\in \mathrm{supp}(\chi)$ we set $\chi(x):=0$.
We need the following proposition for which the statement (i) is noticed on page 392 of \cite{FKS-feral} and the case (ii) immediately follows from Theorem 5 or Theorem 10 of \cite{kakol}. Nevertheless, we provide its complete and independent proof for the sake of completeness and reader convenience. Following \cite{FKS-feral}, an lcs $E$ is called {\em feral} if every infinite-dimensional subset of $E$ is unbounded. Recall that an lcs $E$ is called {\em $c_0$-barrelled} if every null sequence in the weak-$\ast$ dual of $E$ is equicontinuous, see \cite[Chapter~12]{Jar} or \cite[Chapter~8]{bonet}.
\begin{proposition} \label{p:Cp-strong-feral} \begin{enumerate} \item[{\rm (i)}] The strong dual space $L_\beta(X)$ of $C_p(X)$ is feral. \item[{\rm (ii)}] $C_p(X)$ is $c_0$-barrelled if and only if it is barrelled. \end{enumerate} \end{proposition}
\begin{proof} (i) Suppose for a contradiction that there is a bounded infinite-dimensional subset $B$ in $L_\beta(X)$. For $n=1$, fix arbitrarily a nonzero $\chi_1\in B$ and let $x_1 \in \mathrm{supp}(\chi_1)$ be such that $\chi_1(x_1)=a_1 \not= 0$. Since $B$ is infinite-dimensional, by induction, for every natural number $n>1$ there exists a $\chi_{n}\in B$ satisfying the following condition: there is an $x_{n}\in \mathrm{supp}(\chi_{n})$ such that \begin{equation} \label{equ:DP-01} x_{n} \not\in \bigcup_{i=1}^{n-1}\mathrm{supp}(\chi_i) \; \mbox{ and } \; \chi_{n}(x_{n})=a_{n} \not= 0. \end{equation} Clearly, all elements $x_n$ are distinct. Passing to a subsequence if needed we can assume that, for every $n\in\mathbb{N}$, there is an open neighborhood $U_n$ of $x_n$ such that \begin{equation} \label{equ:DP-02} U_n \cap \left( \big[\mathrm{supp}(\chi_n)\setminus\{ x_n\}\big] \cup \bigcup_{i=1}^{n-1} U_i \right) =\emptyset. \end{equation} Finally, for every $n\in\mathbb{N}$, take a function $f_n\in C(X)$ such that $\mathrm{supp}(f_n)\subseteq U_n$ and $f_n(x_n)=n/a_n$. It is easy to see $f_n \to 0$ in $C_p(X)$, and hence the sequence $S=\{ f_n: n\in\mathbb{N}\}$ is bounded in $C_p(X)$. The choice of $f_n$, (\ref{equ:DP-01}) and (\ref{equ:DP-02}) imply $\chi_n(f_n)=f_n(x_n)=n \to\infty$. Therefore the sequence $\{ \chi_n: n\in\mathbb{N}\}\subseteq B$ is unbounded, a contradiction.
(ii) If $C_p(X)$ is barrelled then trivially it is $c_0$-barrelled. Conversely, assume that $C_p(X)$ is $c_0$-barrelled. By the Buchwalter--Schmets theorem, it suffices to prove that every functionally bounded subset of $X$ is finite. Suppose for a contradiction that $X$ has a one-to-one sequence $\{ a_n:n \in\mathbb{N}\}\subseteq X$ which is functionally bounded in $X$. For every $n\in \mathbb{N}$, set $\chi_n= 2^{-n}\delta_{a_n}\in L(X)$, where $\delta_{a_n}$ is the Dirac measure at $a_n$. Then, for every $f\in C(X)$, we have \[
|\chi_n (f)| \leq 2^{-n}\cdot \sup\{ |f(a_n)|: n\in\mathbb{N}\} \to 0. \] Therefore $\chi_n $ is a weak-$\ast$ null sequence. Since $C_p(X)$ is $c_0$-barreled we obtain that the sequence $S=\{ \chi_n: n\in\mathbb{N}\}$ is equicontinuous. So there is a neighborhood $U$ of zero in $E$ such that $S\subseteq U^\circ$. Since, by the Alaoglu theorem, $U^\circ$ is a $\sigma(E',E)$-compact convex subset of $E'$, we obtain that $S$ is strongly bounded, see Theorem 11.11.5 of \cite{NaB}. Clearly, the sequence $S$ is infinite-dimensional and hence, by (i), $S$ is not bounded in the strong topology. This contradiction finishes the proof. \end{proof}
Now Theorem \ref{t:DP-Cp} follows immediately from Proposition \ref{p:Cp-strong-feral} and the next result below.
\begin{theorem} \label{t:Cp-not-DP} Let $E$ be an lcs whose strong dual is feral. Then: \begin{enumerate} \item[{\rm (i)}] $E$ has the $(sDP)$ property; \item[{\rm (ii)}] $E$ has the $(DP)$ property; \item[{\rm (iii)}] $E$ has the Grothendieck property if and only if it is $c_0$-barrelled. \end{enumerate} \end{theorem}
\begin{proof} (i) Let $S'=\{ \chi_n :n\in\mathbb{N}\}$ be a weakly null sequence in $E'_\beta$. As $E'_\beta$ is feral, the sequence $S'$ is finite-dimensional, and hence for every weakly null (even bounded) sequence $\{ x_n :n\in\mathbb{N}\}$ in $E$ we trivially have $\chi_n (x_n) \to 0$. Thus $E$ has the $(sDP)$ property.
(ii) We use Theorem \ref{t:Edwards-DP}. First we note that every weakly compact subset of $E'_\beta$ is finite-dimensional. Therefore, every polar $A^\circ$ of an absolutely convex, equicontinuous and weakly compact subset $A$ of $E'_\beta$ defines a weak neighborhood at zero of $E$. Therefore $\tau_{\Sigma'}$ coincides with the weak topology of $E$. Thus $E$ has the $(DP)$ property.
(iii) Assume that $E$ has the Grothendieck property. Suppose for a contradiction that $E$ is not $c_0$-barrelled. Then there exists a weak-$\ast$ null sequence $S$ in $E'$ which is not equicontinuous. Clearly, $S$ is infinite-dimensional. Since $E'_\beta$ is feral it follows that $S$ is not strongly bounded. Thus $S$ does not converge to zero in the weak topology of $E'_\beta$, and hence $E$ does not have the Grothendieck property. This contradiction shows that $E$ must be $c_0$-barrelled.
Conversely, assume that $E$ is $c_0$-barrelled and let $S=\{ \chi_n: n\in\mathbb{N}\}$ be a weak-$\ast$ null sequence in $E'$. Then $S$ is equicontinuous. So there is a neighborhood $U$ of zero in $E$ such that $S\subseteq U^\circ$. Since, by the Alaoglu theorem, $U^\circ$ is a weak-$\ast$ compact convex subset of $E'$, we obtain that $S$ is strongly bounded, see Theorem 11.11.5 of \cite{NaB}. Therefore $S$ is finite-dimensional because $E'_\beta$ is feral. Hence $S$ is also a weakly null sequence in $E'_\beta$. Thus $E$ has the Grothendieck property. \end{proof}
\section{ The $(DP)$ property and the Grothendieck property for $C_k(X)$} \label{sec:Ck-GDP}
For a Tychonoff space $X$, we denote by $M_c(X)$ the space of all finite real regular Borel measures on $X$ with compact support (which will be denoted by $\mu,\nu$ etc.). It is well known that $M_c(X)$ is the dual space of $C_k(X)$.
Let $K$ be a compact subspace of a Tychonoff space $X$. Denote by $M_K(X)$ the linear subspace of $M_c(X)$ of all measures with support in $K$. Denote by $J: M(K)\to M_K(X)$ the natural inclusion map defined by \[ J(\nu)(A):= \nu(A\cap K), \] where $A$ is a Borel subset of $X$.
\begin{lemma} \label{l:M(K)-M(X)} Let $K$ be a compact subspace of a Tychonoff space $X$. Then $J$ is a linear isomorphism of the Banach space $M(K)$ onto the subspace $M_K(X)$ of $M_c(X)_\beta$. \end{lemma}
\begin{proof} It is clear that $J$ is a linear isomorphism. We show that $J$ is a homeomorphism.
Denote by $S$ the restriction map from $C_k(X)$ to $C(K)$, i.e., $S(f):= f|_K$ for every $f\inC_k(X)$. Clearly, $S$ is a continuous linear operator. Therefore its adjoint map $S^\ast: M(K) \to M_c(X)_\beta$ is continuous, see \cite[Theorem~8.11.3]{NaB}. Noting that \[ S^\ast (\nu)(f)=\nu(S(f)), \quad \nu\in M(K), \; f\in C_k(X), \] we see that $J$ is a corestriction of $S^\ast$ to $M_K(X)$. Thus $J$ is continuous.
To show that $J$ is also open it is sufficient to prove that $J(B_{M(K)})$ contains a neighborhood of zero in $M_K(X)$, where $B_{M(K)}$ is the closed unit ball of the Banach space $M(K)$. Define \[
B:= \{ f\in C(X): \; |f(x)|\leq 1 \mbox{ for every } x\in X\}. \]
It is clear that $B$ is a bounded subset of $C_k(X)$. Therefore, $B^\circ \cap M_K(X)$ is a neighborhood of zero in $M_K(X)$. We show that $B^\circ \cap M_K(X) \subseteq J(B_{M(K)})$. Indeed, let $\mu\in B^\circ \cap M_K(X)$ and denote by $\nu$ the restriction of $\mu$ onto $K$; so $\nu\in M(K)$ and $J(\nu)=\mu$. We have to prove that $\nu\in B_{M(K)}$. Fix an arbitrary function $g\in B_{C(K)}$. By the Tietze--Urysohn theorem, choose an extension $\widetilde{g}\in C(X)$ of $g$ onto $X$ such that $|\widetilde{g}(x)|\leq 1$ for every $x\in X$. Then $\widetilde{g}\in B$, and since $\mu\in B^\circ$ we obtain \[
|\nu(g)|=\left| \int_K g(x) d \nu\right| =\left| \int_X \widetilde{g}(x)d \mu\right| \leq 1. \] Thus $\nu\in B_{M(K)}$. \end{proof}
Below we provide a quite general condition on a Tychonoff space $X$ for which the space $C_k(X)$ has the $(sDP)$ property. Recall that the sets \[
[K;\varepsilon] :=\{ f\in C(X): |f(x)|<\varepsilon \; \forall x\in K\}, \] where $K$ is a compact subset of $X$ and $\varepsilon>0$, form a base at zero of the compact-open topology $\tau_k$ of $C_k(X)$.
\begin{theorem} \label{t:sequential-DP-c0-quasibarrelled} Each $c_0$-barrelled space $C_k(X)$ has the $(sDP)$ property. \end{theorem}
\begin{proof} Let $\{ f_n\}_{n\in\mathbb{N}}$ and $\{ \mu_n\}_{n\in\mathbb{N}}$ be weakly null sequences in $C_k(X)$ and its strong dual $M_c(X)_\beta$, respectively. We have to show that $\lim_n \mu_n(f_n)=0$.
Observe that the weak topology of $M_c(X)_\beta$ is stronger than the weak-$\ast$ topology on $M_c(X)$. Therefore the $c_0$-barrelledness of $C_k(X)$ implies that the sequence $S=\{ \mu_n\}_{n\in\mathbb{N}}$ is equicontinuous. So there is a compact subset $K$ of $X$ and $\varepsilon>0$ such that $S\subseteq [K;\varepsilon]^\circ$. Since $X$ is Tychonoff, it follows that $\mathrm{supp}(\mu_n)\subseteq K$ for every $n\in\mathbb{N}$. Indeed, otherwise, there is a function $f\in C(X)$ with support in $X\setminus K$ such that $\mu(f)>0$. It is clear that $\lambda f\in [K;\varepsilon]$ for every $\lambda>0$, and hence $\mu(\lambda f)> 1$ for sufficient large $\lambda$, a contradiction.
For every $n\in\mathbb{N}$, denote by $\nu_n$ the restriction of $\mu_n$ onto $K$, i.e., $\nu_n(A):=\mu_n(A\cap K)$ for every Borel subset $A$ of $X$. By Lemma \ref{l:M(K)-M(X)}, $\nu_n\to 0$ in the weak topology of the Banach space $M(K)$. Observe that the sequence $\{ f_n|_K\}_{n\in\mathbb{N}}$ is weakly null in the Banach space $C(K)$ because the restriction map $S:C_k(X) \to C(K)$, $S(f):= f|_K$, is continuous and hence is weakly continuous. Since the support of $\mu_n$ is contained in $K$ we obtain \[
\nu_n(f|_K)=\int_K f|_K(x)d\nu_n = \int_X f(x)d\mu_n =\mu_n(f) \] for every $f\in C(X)$. Now this equality and the $(sDP)$ property of $C(K)$ imply $
\lim_n \mu_n(f_n)=\lim_n \nu_n\big(f_n|_K\big)=0. $ Thus $C_k(X)$ has the $(sDP)$ property. \end{proof}
Let $\alpha$ and $\kappa$ be ordinals such that $\alpha<\kappa$. Since $[0,\alpha]$ and $(\alpha,\kappa)$ are clopen subspaces of $[0,\kappa)$, we have \begin{equation} \label{equ:DP-1} C_k\big([0,\kappa)\big) = C\big([0,\alpha]\big) \oplus C_k\big((\alpha,\kappa)\big), \end{equation} and hence, for strong dual spaces, \begin{equation} \label{equ:DP-2} M_c\big([0,\kappa)\big)_\beta =M_c\big([0,\alpha]\big)_\beta\oplus M_c\big((\alpha,\kappa)\big)_\beta. \end{equation}
\begin{proposition} \label{p:Ck-ordinal-c0-barrelled} For every ordinal $\kappa$, the space $C_k \big([0,\kappa)\big)$ is $c_0$-barrelled. \end{proposition}
\begin{proof} If $\kappa$ is a successor ordinal or has countable cofinality, then $C_k \big([0,\kappa)\big)$ is a Banach space or a Fr\'{e}chet space, respectively. Therefore $C_k \big([0,\kappa)\big)$ is even a barrelled space. Assume now that the cofinality $\mathrm{cf}(\kappa)$ of $\kappa$ is uncountable. For simplicity, set $E:=C_k\big([0,\kappa)\big)$ and let $E'_\beta :=M_c\big([0,\kappa)\big)_\beta $ be the strong dual of $E$.
Let $A=\{ \mu_n\}_{n\in\mathbb{N}}$ be a weakly-$\ast$ null sequence in $E'_\beta$. For every $n\in \mathbb{N}$, the support of $\mu_n$ is compact and hence there is an ordinal $\alpha_n$, $\alpha_n< \kappa$, such that $\mathrm{supp}(\mu_n) \subseteq [0,\alpha_n]$. Set $\alpha:= \sup\{ \alpha_n: n\in\mathbb{N}\}$. Since $\mathrm{cf}(\kappa)>\omega$, we have $\alpha<\kappa$. For every $n\in \mathbb{N}$, denote by $\nu_n $ the restriction $ \mu_n |_{[0,\alpha]}$ of $\mu_n$ onto $[0,\alpha]$. Since the restriction map $T: E\to C\big([0,\alpha]\big), T(f)=f|_{[0,\alpha]}$, is surjective we obtain that the sequence $S=\{ \nu_n\}_{n\in\mathbb{N}}$ is a weakly-$\ast$ null sequence in the dual $M\big([0,\alpha]\big)$ of the Banach space $C\big([0,\alpha]\big)$. Thus $S$ is equicontinuous, and hence there is $\lambda>0$ such that \[
S \subseteq \lambda \tilde{B}_\alpha^\circ, \mbox{ where } \tilde{B}_\alpha :=\big\{ g\in C\big([0,\alpha]\big): |g(x)|\leq 1 \mbox{ for all } x\in [0,\alpha]\big\}. \] Set $B_\alpha:= \tilde{B}_\alpha \times C_k\big((\alpha,\kappa)\big)$. It follows from (\ref{equ:DP-1}) that $B_\alpha$ is a neighborhood of zero in $E$. Then, for every $f\in B_\alpha$ and each $\mu_n\in A$, we have \[
|\mu_n(f)|=\left| \int_{[0,\kappa)} f(x) d\mu_n \right| =\left| \int_{[0,\alpha]} f|_{[0,\alpha]} (x) d\nu_n \right| \leq \lambda. \]
Therefore $A\subseteq \lambda B_\alpha^\circ$ and hence $A$ is equicontinuous. Thus $E$ is $c_0$-barrelled. \end{proof} The Nachbin--Shirota theorem, see \cite{bonet}, implies that $C_k \big([0,\kappa)\big)$ is barrelled if and only if $\kappa$ is a successor ordinal or has countable cofinality. Therefore, if the cofinality $\mathrm{cf}(\kappa)$ of $\kappa$ is uncountable (for example, $\kappa=\omega_1$), then $C_k \big([0,\kappa)\big)$ is $c_0$-barrelled but not barrelled.
Below we prove Theorem \ref{t:DP-Ck}.
{\em Proof of Theorem \ref{t:DP-Ck}}. (i) Assume that $X$ is a hemicompact space. Then the space $C_k(X)$ has the $(sDP)$ property by Theorem \ref{t:sequential-DP-c0-quasibarrelled}. Applying Proposition \ref{p:DP-LF-DF} we obtain that the space $C_k(X)$ has also the $(DP)$-property.
(ii) Assume that $X$ is a cosmic space. First we recall that each cosmic space is Lindel\"{o}f and every its compact subset is metrizable, see \cite{Mich}. Therefore $X$ is a $\mu$-space, and hence $C_k(X)$ is barrelled. Proposition 10.5 of \cite{Mich} implies that $C_p(X)$ is a cosmic space, and therefore every compact subset of $C_p(X)$ is metrizable. Observe that the weak topology of $C_k(X)$ is finer than the pointwise topology of $C_p(X)$. Thus every weakly compact subset of $C_k(X)$ is metrizable, and hence is $C_k(X)$ is weakly sequentially angelic.
The space $C_k(X)$ has the $(sDP)$ property by Theorem \ref{t:sequential-DP-c0-quasibarrelled}. To prove that $C_k(X)$ has also the $(DP)$ property we apply Proposition \ref{p:DP-c0-quasibarrelled}. We proved that $C_k(X)$ is barrelled and weakly sequentially angelic. Therefore it remains to check that the strong dual $M_c(X)_\beta$ of $C_k(X)$ is weakly sequentially angelic. Observe that the space $C_p(X)$ being cosmic is separable, see \cite[p.994]{Mich}. Therefore, by Corollary 4.2.2 of \cite{mcoy}, also the space $C_k(X)$ is separable. It follows that $M_c(X)$ with the weak-$\ast$ topology admits a weaker metrizable locally convex vector topology. As the weak topology of the strong dual $M_c(X)_\beta$ of $C_k(X)$ is evidently stronger than the weak-$\ast$ topology on $M_c(X)$, we obtain that every weakly compact subsets of $M_c(X)_\beta$ is even metrizable, and therefore $M_c(X)_\beta$ is weakly sequentially angelic. Finally, Proposition \ref{p:DP-c0-quasibarrelled} implies that $C_k(X)$ has the $(DP)$ property.
(iii) Let $X=[0,\kappa)$ for some ordinal $\kappa$. If $\kappa$ is a successor ordinal or has countable cofinality, then $[0,\kappa)$ is hemicompact. Thus, by (i), $C_k \big([0,\kappa)\big)$ has the $(sDP)$ property and the $(DP)$ property. Assume now that the cofinality $\mathrm{cf}(\kappa)$ of $\kappa$ is uncountable.
Proposition \ref{p:Ck-ordinal-c0-barrelled} implies that $C_k\big([0,\kappa)\big)$ is $c_0$-barrelled. Thus, by Theorem \ref{t:sequential-DP-c0-quasibarrelled}, $C_k\big([0,\kappa)\big)$ has the $(sDP)$ property. We show below that $C_k\big([0,\kappa)\big)$ also has the $(DP)$ property.
Suppose for a contradiction that $C_k\big([0,\kappa)\big)$ does not have the $(DP)$ property. Then, by Theorem \ref{t:Edwards-DP}, there exists an absolutely convex, weakly compact subset $Q$ of $C_k\big([0,\kappa)\big)$ which is not precompact in the topology $\tau_{\Sigma'}$. Therefore, by Theorem 5 of \cite{BGP}, there are an absolutely convex, equicontinuous, weakly compact subset $W$ of $M_c\big([0,\kappa)\big)_\beta$ and a sequence $\{ f_n: n\in\mathbb{N}\}$ in $Q$ such that \begin{equation} \label{equ:DP-c0-quasibarrelled-3} f_n - f_m \not\in W^\circ \; \mbox{ for every distinct } n,m \in\mathbb{N}. \end{equation}
For every $n\in \mathbb{N}$, choose $\alpha_n< \kappa$ such that $f_n(x)=f_n(\alpha_n)$ for every $x>\alpha_n$, see \cite[Example~3.1.27]{Eng}. Set $\alpha:= \sup\{ \alpha_n: n\in\mathbb{N}\}$. Since $\mathrm{cf}(\kappa)>\omega$, we have $\alpha<\kappa$. For every $n\in \mathbb{N}$, set $g_n:= f_n|_{[0,\alpha]}$. Since the restriction operator $T: C_k\big([0,\kappa)\big)\to C\big([0,\alpha]\big)$ onto $[0,\alpha]$ is continuous, it is weakly continuous and hence $T(Q)$ is a weakly compact subset of the Banach space $C\big([0,\alpha]\big)$. By the Eberlein--\v{S}mulian theorem, $T(Q)$ is weakly sequentially compact. Therefore, passing to a subsequence if needed, we can assume that $g_n$ weakly converges to some $g\in C\big([0,\alpha]\big)$. In particular, $g_n(\alpha)=f_n(\alpha) \to g(\alpha)$. Define $f\in E$ by $f|_{[0,\alpha]}:= g$ and $f|_{(\alpha,\kappa)}:=g(\alpha)$. Taking into account that $f_n|_{(\alpha,\kappa)}=f_n(\alpha) \mathbf{1}_{(\alpha,\kappa)}$ (where $\mathbf{1}_A$ denotes the characteristic function of a subset $A$), we obtain that $f_n\to f$ in the weak topology of $E$. Note that $f_n - f_{n+1}$ weakly converges to zero.
Now (\ref{equ:DP-c0-quasibarrelled-3}) implies that there is a sequence $S=\{ \mu_n: n\in\mathbb{N}\}$ in $W$ such that \begin{equation} \label{equ:DP-c0-quasibarrelled-4} \mu_n (f_n - f_{n+1}) >1 \; \mbox{ for every } n \in\mathbb{N}. \end{equation} For every $n\in \mathbb{N}$, choose $\alpha\leq\beta_n< \kappa$ such that $\mathrm{supp}(\mu_n) \subseteq [0,\beta_n]$. Set $\beta:= \sup\{ \beta_n: n\in\mathbb{N}\}$. Since $\mathrm{cf}(\kappa)>\omega$, we have $\beta<\kappa$. By Lemma \ref{l:M(K)-M(X)}, the sequence $S$ is contained in the closed subspace $L:= M_{[0,\beta]} \big([0,\kappa)\big)$ of $M_c \big([0,\kappa)\big)_\beta$ and $L$ is topologically isomorphic to the Banach space $M([0,\beta])$. Therefore, the set $W_\beta := W\cap L$ is a weakly compact subset of $M([0,\beta])$. Once again applying the Eberlein--\v{S}mulian theorem and passing to a subsequence if needed, we can assume that $\{ \mu_n\}_{n\in\mathbb{N}}$ weakly converges to a measure $\mu\in L\subseteq M_c \big([0,\kappa)\big)_\beta$. Now (\ref{equ:DP-c0-quasibarrelled-4}) and the $(sDP)$ property of $C_k\big([0,\kappa)\big)$ proved above imply \[ 1< \mu_n (f_n - f_{n+1}) = (\mu_n -\mu) (f_n - f_{n+1}) + \mu (f_n - f_{n+1}) \to 0. \] This contradiction shows that $C_k(X)$ has the $(DP)$ property.
(iv) Assume that $X$ is a locally compact and paracompact space. Theorem 5.1.27 of \cite{Eng} states that $X=\bigoplus_{i\in I} X_i$ is the direct topological sum of a family $\{ X_i\}_{i\in I}$ of Lindel\"{o}f locally compact spaces. Since all $X_i$ are hemicompact by \cite[Ex.3.8.c]{Eng}, (i) implies that all spaces $C_k(X_i)$ have the $(DP)$ property and the $(sDP)$ property. Therefore the space $C_k(X)=\prod_{i\in I} C_k(X_i)$ has the $(DP)$ property by \cite[9.4.3(a)]{Edwards}. Now we check that $C_k(X)$ has also the $(sDP)$ property.
Let $\{ f_n\}_{n\in\mathbb{N}}$ and $\{ \mu_n\}_{n\in\mathbb{N}}$ be weakly null sequences in $C_k(X)$ and $M_c(X)_\beta$, respectively. Choose a countable subfamily $J$ of the index set $I$ such that \[ \bigcup_{n\in\mathbb{N}} \mathrm{supp}(\mu_n) \subseteq \bigcup_{j\in J} X_j, \; \mbox{ and set } \; Y:=\bigcup_{j\in J} X_j. \]
For every $n\in\mathbb{N}$, set $g_n:= f_n |_Y$ and let $\nu_n$ be the restriction of $\mu_n$ onto $Y$. By construction, $Y$ and $X\setminus Y$ are clopen subsets of $X$. Therefore $C_k(X)=C_k(Y)\times C_k(X\setminus Y)$ and hence $M_c(X)_\beta = M_c(Y)_\beta \times M_c(X\setminus Y)_\beta$. Since the projection onto the first summand is continuous, it is weakly continuous as well. Thus $ \{ g_n\}_{n\in\mathbb{N}}$ and $ \{ \nu_n\}_{n\in\mathbb{N}}$ are weakly null sequences in $C_k(Y)$ and $M_c(Y)_\beta$, respectively. As $Y$ is hemicompact, (i) implies $\mu_n(f_n)= \nu_n(g_n) \to 0$ as $n\to\infty$. Thus $C_k(X)$ has the $(sDP)$ property. \qed
Having in mind the $(DP)$ property one may ask: {\em Under which condition a null-sequence $\{ f_n: n\in\mathbb{N}\}$ in $C_{p}(X)$ weakly converges to zero in $C_k(X)$}? Recall the following known observation which is a consequence of the Lebesgue's dominated convergence theorem and the fact that every measure $\mu \in C_k(X)'=M_c(X)$ has compact support: \begin{fact}\label{fact:weakly-null-Ck} Let $X$ be a Tychonoff space and let $S=\{ f_n:n\in\mathbb{N}\}$ be a null-sequence in $C_p(X)$. If $S$ is bounded in $C_k(X)$, then $f_{n}\rightarrow 0$ in the space $C_k(X)_{w}$, which means the space $C_k(X)$ endowed with the weak topology of $C_k(X)$. \end{fact}
Using (i) of Theorem \ref{t:DP-Ck} and Fact \ref{fact:weakly-null-Ck} we obtain the following easy \begin{proposition} \label{p:bounded-sequence-in-Ck} Let $X$ be a hemicompact space and let $S=\{ f_n:n\in\mathbb{N}\}$ be a null-sequence in $C_p(X)$. Then the following assertions are equivalent: \begin{enumerate} \item[{\rm (i)}] $S$ is bounded in the space $C_k(X)$; \item[{\rm (ii)}] $\mu_n(f_n)\to 0$ for every weakly null-sequence $\{\mu_{n}: n\in\mathbb{N}\}$ in the strong dual of $C_k(X)$. \end{enumerate} \end{proposition}
\begin{proof} (i)$\Rightarrow$(ii) immediately follows from (i) of Theorem \ref{t:DP-Ck} and Fact \ref{fact:weakly-null-Ck}.
(ii)$\Rightarrow$(i) Let $\{ K_n:n\in\mathbb{N}\}$ be a fundamental (increasing) sequence of compact sets in $X$. If all the $K_n$ are finite, then $C_{p}(X)=C_k(X)$ and hence $S$ is bounded by the assumption $f_{n}\rightarrow 0$ in $C_{p}(X)$. So we shall assume that all $K_n$ are infinite.
Suppose for a contradiction that $S$ is not bounded in $C_k(X)$. Then there exists $K:=K_{m}$ such that $S$ is unbounded in the Banach space $C(K):=(C(K),\|.\|)$. Let $k_1<k_2<\cdots$ be a sequence in $\mathbb{N}$ such that $\|f_{k_n}\|\geq n$ for all $n\in \mathbb{N}$. For every $n\in\mathbb{N}$, pick $x_{n}\in K$ such that $|f_{k_n}(x_{n})|=\|f_{k_n}\|$ and set \[
\mu_{i}:= \|f_{k_n}\|^{-1}\delta_{x_{n}} \mbox{ if } i=k_n \mbox{ for some } n\in\mathbb{N}, \mbox{ and } \mu_{i}:=0 \mbox{ otherwise}. \] Then the sequence $M:=\{ \mu_{i}:i\in\mathbb{N}\}$ converges to zero in the norm dual of $C(K)$. It follows from Lemma \ref{l:M(K)-M(X)} that
$\mu_i \to 0$ in the strong dual of $C_k(X)$. Therefore $M$ is a weakly null-sequence in the strong dual of $C_k(X)$. But since $|\mu_{k_n}(f_{k_n})|=1$ for every $n\in M$, we obtain that (ii) does not hold. This contradiction shows that $S$ is bounded in $C_k(X)$. \end{proof}
\begin{remark} \label{r:Ck-Cp-bounded} {\em Let $X$ be a Tychonoff space containing an infinite compact subset $K$. Then $C_p(X)$ contains a null-sequence which is not bounded for $C_k(X)$. Indeed, since $K$ is infinite, there is an infinite discrete sequence $\{ x_n\}_{n\in\mathbb{N}}$ in $K$ with pairwise disjoint neighborhoods $V_n$ of $x_n$ in $X$, see \cite[Lemma 11.7.1]{Jar}. For every $n\in\mathbb{N}$, choose a function $f_n: X \to [0,n]$ with support in $V_n$ and $f_n(x_n)=n$. As $V_n$ are pairwise disjoint, $f_n \to 0$ in $C_p(X)$. It is clear that, for each $m\in\mathbb{N}$, we have \[
f_n \not\in\{ f\inC_k(X): |f(x)|\leq m \mbox{ for all } x\in K\}, \mbox{ for every } n>m. \] Thus $\{ f_{n}:n\in\mathbb{N}\}$ is not bounded in $C_k(X)$.} \qed \end{remark}
To prove Theorem \ref{t:Grothendieck-Ck} we need the following lemma which
(for compact spaces) actually is noticed in \cite[p.138]{Dales-Lau}. Recall that a sequence $\{ x_n\}_{n\in\mathbb{N}}$ in a topological space $X$ is called {\em trivial} if there is $m\in\mathbb{N}$ such that $x_n =x_m$ for every $n\geq m$.
\begin{lemma} \label{l:Grothendieck-sequence} Let $X$ be a Tychonoff space. If $C_k(X)$ has the Grothendieck property, then $X$ does not contain non-trivial convergent sequences. \end{lemma}
\begin{proof} Suppose for a contradiction that there is a sequence $S=\{ x_n\}_{n\in\mathbb{N}}$ converging to a point $x\in X\setminus S$, we shall assume that $S$ is one-to-one. Then the sequence $\{ \delta_{x_n}-\delta_x\}_{n\in\mathbb{N}}$ evidently converges weakly-$\ast$ to $0$ (as usual $\delta_z$ denotes the Dirac measure at $z\in X$). Set $K:=S\cup\{ x\}$, so $K$ is a compact subset of $X$. It is easy to see that the map \[ \mu \mapsto \sum_{n\in\mathbb{N}} \mu\big( \{ x_n\} \big), \quad \mu\in M(K), \] is a continuous linear functional of the Banach space $M(K)$. Then, by Lemma \ref{l:M(K)-M(X)} and the Hahn--Banach extension theorem, there exists an extension $\chi$ of this map to a continuous linear functional on $M_c(X)_\beta$. Since $\chi\big( \delta_{x_n}-\delta_x\big) =1$ for every $n\in\mathbb{N}$, we see that $\delta_{x_n}-\delta_x \not\to 0$ in the weak topology of $M_c(X)_\beta$. \end{proof}
For the convenience of the reader we recall some definitions used in Theorem \ref{t:Grothendieck-Ck}.
A topological space $X$ is a {\em $\mu$-space} if $X$ is Tychonoff and every functionally bounded subset of $X$ is relatively compact. The Nachbin--Shirota theorem states that a Tychonoff space $X$ is a $\mu$-space if and only if $C_k(X)$ is barrelled. A topological space $X$ is called {\em sequential} if for each non-closed subset $A\subseteq X$ there is a sequence $\{a_n\}_{n\in\omega}\subseteq A$ converging to some point $a\in \overline{A}\setminus A$. We note that a sequential space $X$ is discrete if and only if it does not contain non-trivial convergent sequences. (Indeed, if $z\in X$ is non-isolated, then the set $A:=X\setminus\{z \}$ is non-closed and hence there is a sequence (necessarily non-trivial) in $A$ converging to $z$.) Now we are ready to prove Theorem \ref{t:Grothendieck-Ck}.
{\em Proof of Theorem \ref{t:Grothendieck-Ck}}. (i) Let $X$ be a $\mu$-space whose compact subsets are sequential. Assume that $C_k(X)$ has the Grothendieck property. Let $A$ be a functionally bounded subset of $X$. Then its closure $\overline{A}$ is a compact subset of $X$. By Lemma \ref{l:Grothendieck-sequence}, $\overline{A}$ does not contain non-trivial convergent sequences. Since $\overline{A}$ is sequential we obtain that $A$ is finite. Conversely, if every functionally bounded subset of $X$ is finite, then $C_k(X)=C_p(X)$ and Theorem \ref{t:DP-Cp} applies.
(ii) Let $X$ be a sequential space. Assume that $C_k(X)$ has the Grothendieck property. Then, by Lemma \ref{l:Grothendieck-sequence}, the space $X$ does not contain non-trivial convergent sequences. Since $X$ is sequential we obtain that $X$ is discrete. Conversely, if $X$ is discrete, then $C_k(X)=C_p(X)$ and Theorem \ref{t:DP-Cp} applies. \qed
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\begin{document}
\def\Ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\sim$}}}{}} \def\ubf#1{{\baselineskip=0pt\vtop{\hbox{$#1$}\hbox{$\scriptscriptstyle\sim$}}}{}} \def{\Bbb R}{{\Bbb R}} \def{\Bbb V}{{\Bbb V}} \def{\Bbb N}{{\Bbb N}} \def{\Bbb Q}{{\Bbb Q}}
\title{Vaught's conjecture on analytic sets} \author{Greg Hjorth \footnote{Research partially supported by NSF grant DMS 96-22977}} \date{\today} \maketitle
{\bf $\S$0 Prehistory}
In rough historical these are the groups for which we know the topological Vaught conjecture:
{\bf 0.1 Theorem} (Folklore) All locally compact Polish groups satisfy Vaught's conjecture -- that is to say, if $G$ is a locally compact Polish group acting
continuously on a Polish space $X$ then either $|X/G|\leq\aleph_0$ or there is a
perfect set of points with different orbits (and hence $|X/G|\geq 2^{\aleph_0}$).
{\bf 0.2 Theorem} (Sami) Abelian Polish groups satisfy Vaught's conjecture.
{\bf 0.3 Theorem} (Hjorth-Solecki) Invariantly metrizable and nilpotent Polish groups satisfy Vaught's conjecture.
{\bf 0.4 Theorem} (Becker) Complete left invariant metric and solvable Polish groups satisfy Vaught's conjecture.
In each of these case the result was shortly or immediately after extended to analytic sets. For this purpose let us write TVC$(G,\Ubf{\Sigma}^1_1)$ if whenever $G$ acts continuously on a Polish space $X$ and $A\subset X$ is $\Ubf{\Sigma}^1_1$
(or {\it analytic}) then either $|A/G|\leq\aleph_0$ or there is a perfect set of orbit inequivalent points in $A$. Thus we have TVC$(G,\Ubf{\Sigma}^1_1)$ for each of the group in the class mentioned in 0.1-0.4 above.
On the other hand, and in contrast to the usual topological Vaught conjecture, that merely asserts that 0.1-0.4 hold for arbitrary Polish groups, it is known that TVC$(S_{\infty},\Ubf{\Sigma}^1_1)$ {\it fails}.
Here it is shown that the presence of $S_{\infty}$ is a necessary condition for TVC$(G,\Ubf{\Sigma}^1_1)$ to fail:
{\bf 0.5 Theorem} If $G$ is a Polish group on which the Vaught conjecture fails on analytic sets then there is a closed subgroup of $G$ that has $S_{\infty}$ as a continuous homomorphic image.
The converse of 0.5 is known and by now considered trivial in light of 2.3.5 of \cite{beckerkechris}. Thus we have an exact characterization of TVC$(G,\Ubf{\Sigma}^1_1)$. If as widely believed the Vaught conjecture should fail for $S_{\infty}$ then this would as well characterize the groups for which the topological Vaught conjecture holds.
{\bf $\S$1 Preliminaries}
All of this can be found in \cite{hjorthorbit}.
{\bf 1.1 Theorem} (Effros) Let $G$ be a Polish group acting continuously on a Polish space $X$ (in other words, let $X$ be a {\it Polish $G$-space}. For $x\in X$ we have $[x]_G\in \Ubf{\Pi}^0_2$ if and only if \[G\rightarrow [x]_G,\] \[g\mapsto g\cdot x\] is open.
{\bf 1.2 Corollary} Let $G$ be a Polish group and $X$ a Polish $G$-space. Suppose that $[x]_G$ is $\Ubf{\Pi}^0_2$.
Then for all $V$ containing the identity we may find open $U$ such that for all $ x'\in U\cap[x]_G$ and $U'\subset X$ open \[[x]_G\cap U'\cap U\neq\emptyset\] implies that there exists $g\in V$ such that \[g\cdot x'\in U'.\]
{\bf 1.3 Definition} Let $X$ be a Polish space and ${\cal B}$ a basis. Let ${\cal L}({\cal B})$ be the propositional language formed from the atomic propositions $\dot{x}\in U$, for $U\in{\cal B}$. Let ${\cal L}_{\infty,0}({\cal B})$ be the infinitary version, obtained by closing under negation and arbitrary disjunction and conjunction. $F\subset$ ${\cal L}_{\infty, 0}({\cal B})$ is a {\it fragment} if it is closed under subformulas and the finitary Boolean operations of negation and finite disjunction and finite conjunction.
For a point $x\in X$ and $\varphi\in$${\cal L}_{\infty 0}({\cal B})$, we can then define $x\models \varphi$ by induction in the usual fashion, starting with \[x\models \dot{x}\in U\] if in fact $x\in U$. In the case that $X$ is a Polish $G$-space and $V\subset G$ open we may also define $\varphi^{\Delta V}$ by induction on the logical complexity of $\varphi$ so that in any generic extension in which $\varphi$ is hereditarily countable \[x\models \varphi^{\Delta V}\] if and only if \[\exists ^* g\in V (g\cdot x\models\varphi)\] (where $\exists^*$ is the categoricity quantifier ``there exists non-meagerly many'').
{\bf 1.4 Lemma} Let $X$ be a Polish $G$-space. ${\Bbb P}$ a forcing notion, $p\in {\Bbb P}$ a condition, $\sigma$ a ${\Bbb P}$-term. Suppose that ${\cal B}$ is a countable basis for $X$ and ${\cal B}_0$ a countable basis for $G$. Suppose that $G_0$ is a countable dense subgroup of $G$ and ${\cal B}$ is closed under $G_0$ translation and that ${\cal B}_0$ is closed under left and right $G_0$ translation. Suppose that \[p\Vdash_{\Bbb P}\sigma[\dot{G}]\in X\] and that $p$ decides the equivalence class of $\sigma$ in the sense that \[(p,p)\Vdash_{{\Bbb P}\times{\Bbb P}}\sigma[\dot{G}_l]E_G\sigma[\dot{G}_r].\] Then there is a formula $\varphi_0$ and a fragment $F_0$ containing $\varphi_0$ so that:
\leftskip 0.5in
\noindent (i) $\{\{x\in X: x\models \psi^{\Delta V}\}: \psi\in F_0, V\in {\cal B}_0\}$ provides the basis for a topology $\tau_0(F_0)$, and in any generic extension in which $F_0$ becomes countable $(X,\tau_0(F_0))$ is a Polish $G$-space;
\noindent (ii) $\varphi_0$ describes the equvalence class indicated by the triple $({\Bbb P},p,\sigma)$, in the sense that \[p\Vdash_{\Bbb P}\forall x\in X(x E_G\sigma[\dot{G}]\Leftrightarrow x\models \varphi_0);\]
\noindent (iii) and (ii) perserveres through all further forcing extensions, in that if $H\subset {\Bbb P}$ is $V$-generic below $p$ and $x=\sigma[H]$, then for all forcing notions ${\Bbb P}'\in V[H]$ \[ V[H]\models {\Bbb P}'\Vdash \forall y\in X(y E_G x\Leftrightarrow y\models \varphi_0).\]
\leftskip 0in
{\bf 1.5 Lemma} Let $G$ be a Polish group, $X$ a Polish $G$-space, $A\subset X$ a $\Ubf{\Sigma}^1_1$ set displaying a counterexample to TVC$(G,\Ubf{\Sigma}^1_1)$ -- so that $A/G$ has uncountably many orbits, but no perfect set of $E_G$-inequivalent points.
Then for each ordinal $\delta$ there is a sequence $({\Bbb P}_{\alpha}, p_{\alpha},\sigma_{\alpha})_{\alpha <\delta}$ so that:
\leftskip 0.5in
\noindent (i) for each $\alpha<\delta$ \[(p_{\alpha},p_{\alpha})\Vdash_{{\Bbb P}_{\alpha}\times{\Bbb P}_{\alpha}}\sigma_{\alpha}[\dot{G}_l]E_G\sigma_{\alpha} [\dot{G}_r];\]
\noindent (ii) for each $\alpha<\beta< \delta$ \[(p_{\alpha},p_{\beta})\Vdash_{{\Bbb P}_{\alpha}\times{\Bbb P}_{\beta}}\neg(\sigma_{\alpha}[\dot{G}_l] E_G\sigma_{\beta} [\dot{G}_r]).\]
\leftskip 0in
{\bf $\S$2 Proof}
{\bf 2.1 Definition} $U$ is a {\it regular open} set if \[(\overline{U})^o=U\] -- $U$ equals the interior of its closure. For $A$ a set let $RO(A)=(\overline{A})^o$.
Note then that $RO(A)$ is always a regular open set.
{\bf 2.2 Lemma} Let $G$ be a Polish group. For $V_0, V_1\subset G$ regular open sets, \[\{g\in G: V_0\cdot g= V_1\}\] is a closed subset of $G$.
($\Box$)
I need that the reader is willing to allow that we may speak of an $\omega$-model of set theory containing a Polish space, group, action, Borel set, and so on, provided suitable codes exist in the well founded part. Illfounded $\omega$-models are essential to the arguments below.
In what follows let ZFC$^*$ be some large fragment of ZFC, at the very least strong enough to prove all the lemmas of $\S$1, but weak enough to admit a finite axiomatization.
{\bf 2.3 Lemma} Let $M$ be an $\omega$-model of ZFC$^*$. Let $X$, $G$, $G_0$, ${\Bbb P}$, $F_0$, and so on, be as in 1.4 inside $M$. Suppose \[\pi:M\cong M\] is an automorphism of $M$ fixing $X$, $G$, $G_0$, ${\Bbb P}$, $F_0$, $\varphi_0$, and all elements of ${\cal B}$ and ${\cal B}_0$. Suppose $H\subset$ Coll($\omega, F_0$) is $M$-generic and $x\in X^{M[H]}$ with \[x\models \varphi_0.\]
Then there exists $\bar{g}\in G$ so that for all $\psi \in F_0$ and $V\in{\cal B}_0$ \[RO(\{g\in G_0: M[H]\models(g\cdot x\models \psi^{\Delta V})\})\bar{g}^{-1}= RO(\{g\in G_0: M[H]\models(g\cdot x\models \pi(\psi)^{\Delta V})\}).\]
Proof. It suffices to find $g_0, g_1\in G$ so that \[RO(\{g\in G_0: M[H]\models(g\cdot x\models \psi^{\Delta V})\}){g}_0^{-1}= RO(\{g\in G_0: M[H]\models(g\cdot x\models \pi(\psi)^{\Delta V})\})g^{-1}_1\] for all $\psi$ and $V$.
Let ${{\Bbb P}_0} $ be the forcing notion Coll$(\omega, F_0)$. Fixing $d_G$ a complete metric on $G$ we also build $h_i, h_i'\in G_0$, $\psi_i, \psi_i'\in F_0$, $W_i, W_i'\in {\cal B}_0$ so that
\leftskip 0.5in
\noindent (i) each $W_i$ is an open neighbourhood of the identity, $W_{i+1}\subset W_i$, $d_G(W_i)<2^{-i}$;
\noindent (ii) $\pi(\psi_i)=\psi_i'$;
\noindent (iii) $h_{2i}=h_{2i+1}$; $\forall g\in W_{2i+1}h_{2i}(d_G(g, h_{2i})<2^{-i})$;
\noindent (iv) $h_{2i+1}'=h_{2i+2}'$; $\forall g\in W_{2i+2}h_{2i+1}'(d_G(g, h_{2i+1}')<2^{-i})$;
\noindent (v) $h_{i+1}\in W_ih_i$; $h_{i+1}'\in W_ih_i'$;
\noindent (vi) $M[H]\models (h_i\cdot x\models (\psi_i)^{\Delta V_i})$;
\noindent (vii) $M[H]\models (h_i'\cdot x\models (\psi_i')^{\Delta V_i})$;
\noindent (viii) $M^{{\Bbb P}_0} $ satisfies that for all $y_0, y_1\in X$ all $\psi\in F_0$, and all $V\in {\cal B}_0$, if \[y_0\models \varphi_0\wedge (\psi_i)^{\Delta V_i}\] \[y_1\models \varphi_0\wedge (\psi_i)^{\Delta V_i}\wedge \psi^{\Delta V}\] then \[y_0\models ((\psi_i)^{\Delta V_i}\wedge \psi^{\Delta V})^{\Delta W_i};\]
\noindent (ix) conversely $M^{{\Bbb P}_0} $ satisfies that for all $y_0, y_1\in X$, $\psi\in F_0$, $V\in {\cal B}_0$, if \[y_0\models \varphi_0\wedge (\psi_i')^{\Delta V_i}\] \[y_1\models \varphi_0\wedge (\psi_i')^{\Delta V_i}\wedge \psi^{\Delta V}\] then \[y_0\models ((\psi_i')^{\Delta V_i}\wedge \psi^{\Delta V})^{\Delta W_i}.\]
\leftskip 0in
\noindent Note that (ix) actually follows from (viii), (ii), and the elementarity of $\pi$.
Before verifying that we may produce $h_i, h_i'\in G_0$, $\psi_i, \psi_i'\in F_0$, $W_i, V_i\in {\cal B}_0$ as above, let us imagine that it is already completed and see how to finish. Using (iii) and (iv) we may obtain $g_0=$ lim$h_i$ and $g_1=$ lim$h_i'$. It suffices to check that for all \[ g\in RO(\{h\in G_0: M[H]\models(x\models \psi^{\Delta V})\}){g}_0^{-1}\] we have \[g\in \overline{(\{h\in G_0: M[H]\models(h\cdot x\models \pi(\psi)^{\Delta V})\})} g_1^{-1}\] (since the converse implication will be exactly symmetric).
Then for sufficiently large $i$ we may choose a sufficiently small open neighbourhood $W$ of the identity and $\hat{g}\in G_0$ sufficiently close to $g$ so that $W\hat{g} W_i$ is an arbitrarily small neighbourhood of $g$ and \[M[H]\models (\hat{g} h_i\cdot x\models \psi^{\Delta V})\] \[\therefore M[H]\models ( h_i\cdot x\models (\psi^{\Delta V})^{\Delta W\hat{g}})\] hence, as witnessed by $y=h_i\cdot x$ \[M^{{\Bbb P}_0}\models \exists y( y\models \varphi_0\wedge (\psi_i)^{\Delta V_i} \wedge (\psi^{\Delta V})^{\Delta W\hat{g}}),\] \[\therefore M^{{\Bbb P}_0}\models \exists y( y\models \varphi_0\wedge (\psi_i')^{\Delta V_i} \wedge (\pi(\psi)^{\Delta V})^{\Delta W\hat{g}}),\] by elementarity of $\pi$, \[\therefore M[H]\models (h_i'\cdot x\models (\pi(\psi)^{\Delta V_i})^{\Delta W\hat{g}}) ^{\Delta W_i})\] by (ix), and so there exists some $\bar{g}\in W\hat{g} W_i$ so that \[M[H]\models (\bar{g}h_i'\cdot x\models \pi(\psi)^{\Delta V}).\] By letting $d_G(W\hat{g} W_i)\rightarrow 0$ and $h_i'\rightarrow g_1$ we get \[g\in \overline{\{h\in G_0: M[H]\models(x\models \pi(\psi)^{\Delta V})\}} g_1^{-1},\] as required.
We are left to hammer out the sequence.
Suppose that we have $\psi_j, \psi'_j, W_j,V_j, h_j, h_j'$ for $j\leq 2i$. Immediately we may find $W_{2i+1}\subset W_{2i}$ giving (iii), and then by 1.2 and 1.4(i) we can produce $\psi_{2i+1}$, $V_{2i+1}$ satisfying (viii) and such that \[M[H]\models h_{2i}\cdot x=_{df} h_{2i+1}\cdot x\models (\psi_{2i+1})^{V_{2i+1}}.\] Then by considering that $\pi$ is elementary \[M^{{\Bbb P}_0} \models \exists y(y\models \varphi_0\wedge \pi(\psi_{2i})^{\Delta V_{2i}} \wedge \pi(\psi_{2i+1})^{\Delta V_{2i+1}}).\] Thus by (ix) we may find $h'\in G_0\cap W_{2i}$ so that \[M[H]\models (h'h_{2i}\cdot x\models \pi(\psi_{2i})^{\Delta V_{2i}} \wedge \pi(\psi_{2i+1})^{\Delta V_{2i+1}}).\] In other words, by (ii), if we let $\psi_{2i+1}'=\pi(\psi_{2i+1})$ then \[M[H]\models (h'h_{2i}\cdot x\models (\psi_{2i}')^{\Delta V_{2i}} \wedge (\psi_{2i+1}')^{\Delta V_{2i+1}}).\] Taking $h_{2i+1}'=h'h_{2i}'$ we complete the transition from $2i$ to $2i+1$.
The further step of producing $\psi_{2i+2}, \psi_{2i+2}', W_{2j+2}, h_{2j+2}$, $V_{2j+2}$ and $h_{2j+2}'$ is completely symmetrical.
$\Box$
{\bf 2.4 Definition} $S_{\infty}$ {\it divides} a Polish group $G$ if there is a closed subgroup $H<G$ and a continuous onto homomorphism \[\pi:H\twoheadrightarrow S_{\infty}.\] (By Pettis' lemma, any Borel homomorphism between Polish groups must be continuous.)
{\bf 2.5 Lemma} $S_{\infty}$ divides Aut$({{\Bbb Q}} , <)$, the automorphism group of the rationals equipped with the usual linear ordering.
($\Box$)
{\bf 2.6 Definition} For $X$, $G$, $F_0$, and so on, as in 1.4, ${\Bbb P}_0$= Coll$(\omega, F_0)$, $\psi_0, \psi_1\in F_0$, $V_0, V_1\in {\cal B}_0$, set \[(\psi_0, V_0)R (\psi_1, V_1)\] if in $V^{{\Bbb P}_0}$ for all $x\models \varphi_0$ \[RO(\{g\in G_0: g\cdot x\models (\psi_0)^{\Delta V_0}\})\cap RO(\{g\in G_0: g\cdot x\models (\psi_1)^{\Delta V_1}\}\neq\emptyset).\] For $V\in {\cal B}_0$ let ${\cal B}(V)$ be the set of pairs $(\varphi, W)$ such that for all $\psi\in F_0$ and $W'\in {\cal B}_0$ \[V^{{\Bbb P}_0}\models \forall x_0\models \varphi_0\wedge \varphi^{\Delta W} ((\exists x_1\models \varphi_0\wedge \varphi^{\Delta W}\wedge \psi^{\Delta W'}) \Rightarrow x_0\models (\varphi^{\Delta W}\wedge \psi^{\Delta W'})^{\Delta V}).\] In other words, ${\cal B}(V)$ corresponds to the basic open sets witnessing 1.2 for $V$ in the topology $\tau_0(F_0)$.
The next lemma states that if the equivalence class corresponding to $\varphi_0$ requires large forcing to be introduced then the formulas $\{\psi^{\Delta V}: \psi\in F_0, V\in{\cal B}_0\}$ have large $R$-discrete sets.
{\bf 2.7 Lemma} Let $X$, $G$, $F_0$, ${\Bbb P}$, $\varphi_0$, and so on, be as in 1.4. Let $R$ be as in 2.6. Let $\kappa$ be a cardinal. Suppose no forcing notion of size less than $\kappa$ introduces a point in $X$ satisfying $\varphi_0$.
Then there is no infinite $\delta<\kappa$ such that each ${\cal B}(V)$ for $V\in {\cal B}_0$ has a maximal $R$-discrete set of size $\leq \delta$.
Proof. Suppose otherwise and choose large $\theta>\kappa$ so that $V_{\theta}\models$ ZFC$^*$ and choose an elementary substructure \[A {\prec} V_{\theta}\] so that
\[|A|=\delta,\] \[\delta+1\subset A,\] and $X$, $G$, $F_0$, $\varphi_0$, and so on, in $A$. Let $N$ be the transitive collapse of $A$ and \[\pi: N\rightarrow V_{\theta}\] the inverse of the collapsing map. Set $\hat{{\Bbb P}}=\pi^{-1}({\Bbb P}_0)$ (where ${{\Bbb P}_0}$= Coll$(\omega, F_0)$), $ \hat{\varphi}_0=\pi^{-1}(\varphi_0)$, $\hat{F}_0=\pi^{-1}(F_0)$, choose \[\hat{H}\subset \hat{\Bbb P},\] \[H\subset {\Bbb P}_0\] to be $V$-generic, and choose $\hat{x}\in N[\hat{H}]$ and $x\in V[H]$ so that \[N[\hat{H}]\models (\hat{x}\models \hat{\varphi}_0),\] \[V[H]\models (x\models\varphi_0).\] It suffices to show \[\hat{x}E_Gx.\]
As in the proof of 2.3 find $h_i, h_i'\in G_0$, $\psi_i\in F_0$, $\psi_i'\in {\hat{F}}_0$, $V_i, V'_i \in{\cal B}_0$, $W_i\in {\cal B}_0$ and $U_i\subset X$ basic open so that :
\leftskip 0.5in
\noindent (i) $W_{i+1}\subset W_i$, $W_i=(W_i)^{-1}$, $d_G(W_i)<2^{-i},$ $1_G\in W_i$; $U_{i+1}\subset U_i$, $d_X(U_i)<2^{-i}$;
\noindent (ii) $\pi(\psi_i')=\psi_i$;
\noindent (iii) $\forall g\in (W_{2i+1})^3 h_{2i}(d_G(g, h_{2i})<2^{-i})$; $h_{2i+1}=h_{2i}$;
\noindent (iv) $\forall g\in (W_{2i+2})^3 h_{2i+1}'(d_G(g, h_{2i+1})<2^{-i})$; $h_{2i+2}'=h_{2i+1}'$;
\noindent (v) $h_{i+1}\in (W_i)^3 h_i$, $h_{i+1}'\in (W_i)^3 h_i'$;
\noindent (vi) $V[H]\models (h_i\cdot {x}\models (\psi_i)^{\Delta V_i})$;
\noindent (vii) $N[\hat{H}]\models (h_i'\cdot \hat{x}\models (\psi_i')^{\Delta V_i'})$;
\noindent (viii) $V^{{\Bbb P}_0}\models (\psi_i, V_i)\in {\cal B}(W_i)$;
\noindent (ix) $N^{{\Bbb P}_0}\models (\psi_i', V_i')\in {\cal B}(W_i)$;
\noindent (x) $(\pi(\psi_i'), V_i') R(\psi_i, V_i)$;
\noindent (xi) $h_i\cdot x, h_i'\cdot \hat{x} \in U_i$.
\leftskip 0in
Granting all this may be found we finish quickly. By (iii) and (iv) we get $g_0$= lim$h_i$ and $g_1$= lim$h_i'$, whence \[g_0\cdot x=g_1\cdot \hat{x}\] by (xi). This would contradict ${\hat{\Bbb P}}$ being too small to introduce a representative of $[x]_G$.
So instead suppose we have built $V_j, V_j', \psi_j$ and so on for $j\leq 2i$ and concentrate on trying to show that we may continue the construction up to $2i+2$.
First choose $W_{2i+1}\subset W_{2i}$ in accordance with (i) and (iii) and then for (xi) and (i) choose $U_{2i+1}\subset U_{2i}$ containing $h_{2i}\cdot x (=_{df} h_{2i+1}\cdot x)$ with $d_X(U_{2i+1})<2^{-2i-1}$. Then by 1.2 we may choose $V_{2i+1}, \psi_{2i+1}$ as indicated at (vi) and (viii).
On the $N$ side we use the assumption on $R$ to find $V_{2i+1}'$ and $\psi_{2i+1}'$ in $N$ so that \[N^{\hat{\Bbb P}_0}\models (\psi_{2i+1}', V_{2i+1}')\in {\cal B}(W_{2i+1})\] and \[(\pi(\psi_{2i+1}'), V_{2i+1}') R (\psi_{2i+1}, V_{2i+1}).\] Unwinding the definitions gives \[V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge \pi(\psi_{2i}')^{\Delta V_{2i}'}) \Rightarrow y\models ((\psi_{2i})^{\Delta V_{2i}} \wedge \pi(\psi_{2i}')^{\Delta V_{2i}'})^{\Delta W_{2i}},\] \[V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge (\psi_{2i})^{\Delta V_{2i}}) \Rightarrow y\models ((\psi_{2i})^{\Delta V_{2i}} \wedge (\psi_{2i+1})^{\Delta V_{2i+1}})^{\Delta W_{2i}},\] \[V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge (\psi_{2i+1})^{\Delta V_{2i+1}}) \Rightarrow y\models ((\psi_{2i+1})^{\Delta V_{2i+1}} \wedge \pi(\psi_{2i+1}')^{\Delta V_{2i+1}'})^{\Delta W_{2i+1}}.\] In particular, assuming without loss of generality that \[(\psi_{2i+1})^{\Delta V_{2i+1}}\Rightarrow \dot{x}\in U_{2i+1}\] we have \[V^{{\Bbb P}_0}\models (y\models \varphi_0 \wedge \pi(\psi_{2i}')^{\Delta V_{2i}'}) \Rightarrow y\models ((\psi_{2i+1}')^{\Delta V_{2i+1}{'}} \wedge \dot{x}\in U_{2i+1})^{\Delta (W_{2i})^3}.\] Thus by elementarity of $\pi$ we may find $h'\in (W_{2i})^3\cap G_0$ so that $h'h_{2i}'\cdot \hat{x}\in U_{2i+1}$ and \[N[\hat{H}]\models (h'h_{2i}'\cdot \hat{x}\models (\psi_{2i+1}')^{\Delta V_{2i+1}'}).\] Then setting $h_{2i+1}=h'h_{2i}$ completes the transition from $2i$ to $2i+1$.
The step from $2i+1$ to $2i+2$ is similar.
$\Box$
We need a fact from infinitary model theory.
{\bf 2.8 Theorem} Let $\varphi\in {\cal L}_{\omega_1,\omega}$ and suppose \[N\models \varphi\] and $P$ is a predicate in the language of $N$ with
\[|(P)^N|\geq \beth_{\aleph_1}.\] Then $\varphi$ has a model with generating indiscernibles in $P$.
More precisely there is a model $M$ with language ${\cal L}^*\supset {\cal L}(N)$, ${\cal L}^*$ having a new symbol $<$, along with new function symbols of the form $f_{\hat{\varphi}}$ for $\hat{\varphi}$ in the fragment of ${\cal L}(N)_{\omega_1,\omega}$ generated by $\varphi$, and distinguished elements $(c_i)_{i\in {\Bbb N}}$, so that:
\leftskip 0.5in
\noindent (i) $(<)^M$ linearly orders $(P)^M$;
\noindent (ii) each $f_{\hat{\varphi}}$ is a Skolem function for $\hat{\varphi}$;
\noindent (iii) $M$ is the Skolem hull of $\{c_i:i\in{\Bbb N}\}$ (under the functions of the form $f_{\hat{\varphi}}$);
\noindent (iv) each $c_i\in (P)^M$;
\noindent (v) for all $\psi$ in the fragment of ${\cal L}^*_{\omega_1,\omega}$ generated by $\varphi$ and $i_1<i_2<...<i_n$, $j_1<...<j_n$ in ${\Bbb N}$ \[M\models \psi(c_{i_1},c_{i_2},...,c_{i_n})\Leftrightarrow \psi(c_{j_1},c_{j_2},...,c_{j_n});\]
\noindent (vi) $M\models \varphi$.
\leftskip 0in
See \cite{keisler}.
($\Box$)
{\bf 2.9 Theorem} Let $G$ be a Polish group for which TVC($G,\Ubf{\Sigma}^1_1$) fails. Then $S_{\infty}$ divides $G$.
Proof. Choose some Polish $G$-space $X$ witnessing the failure of TVC($G,\Ubf{\Sigma}^1_1$). Following 1.5 we may find some $({\Bbb P}, p,\sigma)$ introducing an equivalence class as in 1.4 that may not be produced by a forcing notion of size less than $\beth_{\aleph_1}$. Fix $\varphi_0$, ${\cal B}$, ${\cal B}_0$, $F_0$, $G_0$, and so on, as in 1.4, so that in all generic extensions $V[H]$ of $V$ \[V[H]\models p\Vdash_{\Bbb P}\forall y\in X(yE_G\sigma[\dot{G}]\Leftrightarrow y\models \varphi_0).\]
Let $V_{\theta}$ be large enough to contain $X$, $G$, $\varphi_0$, and so on, and satisfy ZFC$^*$. By 2.7 choose $P\subset V_{\theta}$ to be of size $\beth_{\aleph_1}$ and $R$-discrete (or more precisely, so for all $(\psi, V)\neq(\psi',V')\in P$ we have for any $V$-generic $H\subset$ Coll$(\omega, F_0)$ that $V[H]\models \neg ((\psi, V)R(\psi',V'))$). Applying 2.8 to $N=(V_{\theta}; \in, P, X, G, G_0, \varphi_0,...)$ and we may obtain an $\omega$-model with indiscernibles $(\psi_q, V_q)_{q\in {\Bbb Q}}$ in $B^M$. Let $H\subset$ Coll($\omega, (F_0)^M$) be $M$-generic. Choose $x\in M[H]$ so that \[M[H]\models (x\models \varphi_0).\]
All this granted we may define $G_1$ to be the set of $\bar{g}\in G$ so that for all $q\in {\Bbb Q}$ there exists $r\in{\Bbb Q}$ with \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}^{-1}= RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\})\] and for $q\in {\Bbb Q}$ there exists $r\in{\Bbb Q}$ with \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}= RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\}).\]
$G_1$ is $\Ubf{\Pi}^0_2$ in $G$, by 2.2 and since $\bar{g}$ is in $G_1$ if and only if the following four conditions hold:
\leftskip 0.5in
\noindent (i) for all $q\in {\Bbb Q}$ there exists $r\in{\Bbb Q}$ with \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}^{-1}\cap RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\})\neq\emptyset,\]
\noindent (ii) for all $q,r\in {\Bbb Q}$ \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}^{-1}\cap RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\})\neq\emptyset\] implies \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}^{-1}= RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\});\]
\noindent (iii) for all $q\in {\Bbb Q}$ there exists $r\in{\Bbb Q}$ with \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}\cap RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\})\neq\emptyset,\]
\noindent (iv) for all $q,r\in {\Bbb Q}$ \[RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_q)^{\Delta V_q})\})\bar{g}\cap RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\})\neq\emptyset\] implies \[RO(\{g\in G_0: M[H]\models(g\cdot x\models \psi_q^{\Delta V_q})\})\bar{g}^{-1}= RO(\{g\in G_0: M[H]\models(g\cdot x\models (\psi_r)^{\Delta V_r})\}).\]
\leftskip 0in
\noindent Since $G_1$ is a $\Ubf{\Pi}^0_2$ subgroup of $G$ it must be closed.
For $g\in G_1$ we may define the permutation $\hat{\pi}(g)$ of ${\Bbb Q}$ by the specification that for all $q\in {\Bbb Q}$ \[(\hat{\pi}(g))(q)=r\] if and only if $r$ is as above in the definition of $G_1$. This is well defined by the $R$-discreteness of the set $(P)^M$.
Now let $G_2$ be the set of $g\in G_1$ such that $\hat{\pi}(g)$ defines an automorphism of the structure $({\Bbb Q}, <)$. $G_2$ is a closed subgroup of $G_1$ and hence $G$. Since every order preserving permutation of the indiscernibles induces an automorphism of $M$ the map \[\hat{\pi}:G_2\rightarrow {\rm Aut}({\Bbb Q}, <)\] is onto by 2.3. Then by 2.5 $S_{\infty}$ divides $G$.
$\Box$
{\bf 2.10 Conjecture} Assume AD$^{L({\Bbb R})}$. Let $G$ be a Polish group, $X$ a Polish $G$-space, $A\subset X$ in $\Ubf{\Sigma}^1_1$, and suppose in ${L({\Bbb R})}$ there is an injection \[i:A/G\hookrightarrow 2^{<\omega_1}.\] Then there is a Polish $S_{\infty}$-space $Y$ and a $\Ubf{\Sigma}^1_1$ set $B\subset Y$ and a bijection \[\pi:A/G\cong B/S_{\infty}.\]
6363 MSB
Mathematics
UCLA
CA90095-1555
greg@math.ucla.edu
\end{document} |
\begin{document}
\title{Lehmer problem and Drinfeld modules} \begin{abstract} We propose a lower bound estimate in Dobrowolski's form of the canonical height of a Drinfeld module having a positive density of supersingular primes. This estimate takes into account the inseparable case and it is given as a function of: the degree of the field of coefficients, the height of the module and its rank. We will show that the class of Drinfeld modules we consider includes all CM Drinfeld modules with rank either 1 or a prime number different from the field characteristic. \end{abstract}
\section{Introduction} We study the natural analogue of Lehmer problem on Drinfeld modules. We consider in particular a special class of such modules, satisfying congruence properties that, for a suitable positive real number $r$, we call RV($r$) or RV($r$)$^{*}$ (see Definition 4 and Definition 5 below). We will call $A:=\mathbb{F}_{q}[T]$ the polynomial ring in one variable $T$ defined over the finite field of $q$ elements, where $q$ is a power of a chosen prime number $p$. We also call $k$ the fraction field of $A$, and $k_{\infty}=\mathbb{F}_{q}((1/T))$ the completion of $k$ with respect to the place at infinity. Let us call $\mathcal{C}:=(\overline{k_{\infty}})_{\infty}$ the completion of a chosen algebraic closure of $k_{\infty}$. This field is therefore algebraically closed and complete.\\\\
The main result we propose in this work (Theorem 2) is the following. Given a Drinfeld module (see Definition 1) $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ satisfying suitable congruence properties involving the density of supersingular primes (see Definition 4 and Definition 5), we provide a lower bound estimate of the canonical height of a non-torsion point $x\in \mathbb{D}(\overline{k})$ with respectively algebraic degree and purely inseparable degree $D$ and $D_{p.i.}$ over $k$ in the following form:\[C\frac{(\log\log{D})^{\mu}}{DD_{p.i.}^{\lambda}(\log{D})^{\kappa}},\]where the positive constants $C$, $\kappa$, $\mu$ and $\lambda$ are explicitly computed as functions of the three arithmetic parameters attached to $\mathbb{D}$: the degree of the field of coefficients $k(\Phi)$, the height $h(\Phi)$ of the Drinfeld module, and the rank $d$.\\\\
Let:\[\tau:\mathcal{C}\to \mathcal{C}\]\[z\mapsto z^{q}\]be the Frobenius map and:\[\overline{k}\{\tau\}:=\{c_{0}+c_{1}\tau + ... + c_{n}\tau^{n},\texttt{ }c_{1}, ..., c_{n}\in \overline{k},\texttt{ }n\in \mathbb{N}\}\]be the \textsl{Ore algebra} of the $\mathbb{F}_{q}-$additive forms with coefficients in $\overline{k}$\footnote{We remark that such an algebra is not commutative as of course in general for $c\in \overline{k}$ one has $\tau c=c^{q}\tau\neq c\tau$.}. \begin{de} A \textbf{Drinfeld module} of \textbf{rank $d$} defined over $\overline{k}$ is a pair:\[\mathbb{D}=(\mathbb{G}_{a},\Phi),\]where $\mathbb{G}_{a}$ is the additive group of $\mathcal{C}$ and $\Phi$ is an injective $\mathbb{F}_{q}-$algebra homomorphism:\[\Phi:A\to \overline{k}\{\tau\},\]defined so that:\[\Phi(T)=\sum_{i=0}^{d}a_{i}\tau^{i}\]where $a_{0}, ..., a_{d}\in \overline{k}$ are such that:\[a_{0}(T)=T\texttt{ and }a_{d}(T)\neq 0.\]We call $k(\Phi):=k(a_{1}, ..., a_{d})$ the \textbf{field of coefficients} of $\mathbb{D}$ (alternatively, we say that $\mathbb{D}$ is defined over $k(\Phi)$). \end{de} We call \textbf{torsion point} of the Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ a point $x\in \overline{k}$ such that there exists $a\in A\setminus\{0\}$ for which we have:\[\Phi(a)(x)=0.\]In particular, we say that $x$ is a \textbf{$a-$torsion point} for this $a$. We also denote by $\Phi[a]$ the set (which is in particular a $\mathbb{F}_{q}-$vector space) of the $a-$torsion points of $\mathbb{D}$. We define:\[\mathbb{D}(\overline{k})_{NT}:=\overline{k}\setminus \bigcup_{a\in A\setminus\{0\}}\Phi[a]\]the set of non-torsion points of $\mathbb{D}$.\\\\ The \textbf{Carlitz module} $C=(\mathbb{G}_{a},\Phi)$ is defined so that:\[\Phi(T)=T+\tau\]and it is the simplest example of a Drinfeld module having rank $1$.\\\\ The \textbf{Lehmer conjecture} in its original form concerns the multiplicative group $\mathbb{G}_{m}(\overline{\mathbb{Q}})$ and predicts a bound taking this shape:\[h(x)>>\frac{1}{[\mathbb{Q}(x):\mathbb{Q}]}\]for all $x\in \mathbb{G}_{m}(\overline{\mathbb{Q}})$ which are not roots of unity.\\\\%(see the second paragraph). Different versions of this conjecture have been proposed, in particular stating lower bounds of the same shape for the Néron-Tate height of non-torsion points of an abelian variety. L. Denis conjectured in \cite{Denis} the following analogue for the canonical height (see the second paragraph for the definition) of the algebraic non-torsion points of a general Drinfeld module defined over $\overline{k}$: \begin{conj} There exists a constant $c>0$ only depending on the Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$, such that each point $x\in\mathbb{D}(\overline{k})_{NT}$ of degree $D$ over $k$ satisfies the following inequality:\[\widehat{h}_{\mathbb{D}}(x)\geq\frac{c}{D}.\] \end{conj} For the specific case of the Carlitz module L. Denis also obtained in the same paper the following: \begin{thm}[Denis] Let $\mathbb{D}$ be the Carlitz module. There exists $\eta>0$ depending on $q$ such that for each $x$ non-torsion algebraic and separable point with degree $\leq D$ over $k$:\[\widehat{h}_{\mathbb{D}}(x)\geq\frac{\eta}{D}(\frac{\log\log(qD)}{\log(qD)})^{3}.\] \end{thm}
D. Ghioca (see \cite{Gh}, Remark 5) showed moreover, with no conditions on the Drinfeld module but on a strong local condition on $x$, that there exists a number $k\geq 1$, depending only on the chosen Drinfeld module, such that:\[\widehat{h}_{\mathbb{D}}(x)>>\frac{1}{D^{k}}.\]\\\\
Another result has also been found recently by S. David and A. Pacheco (see \cite{Dav-Pach}) who showed the following lower bound estimate:\[\widehat{h}_{\mathbb{D}}(x)\geq c(\mathbb{D},K)\]for a Drinfeld module $\mathbb{D}$ defined over the field $K\subset \overline{k}$, where $c(\mathbb{D},K)>0$ is a positive constant only depending on $\mathbb{D}$ and $K$, and $x\in K^{ab.}$, where $x$ is non-torsion and $K^{ab.}$ is the abelian closure of $K$ in $\overline{k}$. Such a result is in analogy with the work of F. Amoroso and R. Dvornicich (see \cite{Am-Dv}) which provides an estimate of this form for the height of an element $x\in \mathbb{G}_{m}(\mathbb{Q}^{ab.})\setminus \mathbb{G}_{m}(\mathbb{Q}^{ab.})_{tors.}$.\\\\
We give now the fundamental notations about the logarithmic functions we will use:\[\log(.):=\log_{q}(.).\]Each logarithm will have always basis $q$ unless we specify differently.\[\log_{+}(.):=\max\{\log(.),1\}\]\[\log\log_{+}(.):=\max\{\log\log(.),1\}.\]We will indicate from now on the degree in $T$ of each polynomial $a\in A=\mathbb{F}_{q}[T]$ by $\deg_{T}(a)$.\\\\ We define:\[S(A):=\{l\in A, \texttt{ monic and irreducible}\}.\]We also define, given some $N\in \mathbb{N}\setminus\{0\}$ :\[P_{N}(A):=\{l\in S(A), \deg_{T}(l) =N\}.\]We will also say that $l\in$ $S(A)$ \textbf{satisfies the RV property}\footnote{The acronym "RV" has been suggested to the author by the french word \textsl{relèvement} he was using to describe the property 1 of the primes involved, which means that there is a "lifting" of the $d-$th power of the Frobenius automorphism by an endomorphism of $\Phi$.} with respect to $\Phi$ if:\begin{enumerate}
\item For each place $v$ dividing $v_{l}$ (the place associated to $l$ over $k$) in the extension $k(\Phi)/k$, the coefficients $a_{i}$ of $\Phi$ are such that $v(a_{i})\geq 0$ and:\[\Phi(l)(X)\equiv X^{q^{d\deg_{T}(l)}} \texttt{ mod }(v)\]where:\[\Phi(l)(X)\in \mathcal{O}_{v}[X],\] the ring $\mathcal{O}_{v}$ being the ring of $v-$integers in $k(\Phi)$;
\item All places extending $v_{l}$ in $k(\Phi)$ have inertial degree $1$. \end{enumerate} \begin{de} Let $r\in ]0,1]$ be a real number and $c_{1}$ a fixed positive constant. A Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ is called \textbf{RV($r,c_{1}$)},
if for each natural number $N>0$:\[|\{l\in P_{N}(A), l\texttt{ is }RV\}|\geq c_{1}\frac{q^{rN}}{N}.\] \end{de} \begin{de}
A Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ is \textbf{RV($r,c_{1}$)$^{*}$}, with $r\in ]0,1]$ and $c_{1}>0$ a fixed constant, if there exists $N(\Phi)\in \mathbb{N}\setminus\{0\}$ such that, for each $N\geq N(\Phi)$:\[|\{l\in P_{N}(A), l\texttt{ is }RV\}|\geq c_{1}\frac{q^{rN}}{N}.\] \end{de} We fix $c_{1}=1/2r$ to ease notations reducing the number of parameters. We leave to the reader a generalization (not quite relevant) of the proposed estimates to a more general $c_{1}$. This directly follows by a mechanical repetition of the same steps of our argument. The choice of $c_{1}=1/2r$ has been suggested by the fact that, as we will see soon (Proposition 3) the value $c_{1}=1/2$ is the maximal that one can choose if $r=1$. \begin{de} Let $r\in ]0,1]$ be a real number. A Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ is \textbf{RV($r$)} if it is RV($r,1/2r$). \end{de} \begin{de} Let $r\in]0,1]$ be a real number. A Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ is \textbf{RV($r$)$^{*}$} if it is RV($r,1/2r$)$^{*}$. \end{de} It is clear that the condition RV($r$)$^{*}$ is implied by the RV($r$) one for every $r\in ]0,1]$. We also remark that the Carlitz module is RV($1$). Indeed, one can prove (see \cite{Hayes}, Proposition 2.4) that each $l\in S(A)$ has supersingular reduction with respect to the Carlitz module. In particular, the Carlitz module will satisfy our Theorems.\\\\
A result extending the study to rank 2 Drinfeld modules was showed by C. David in \cite{C. David}: in average, a rank 2 Drinfeld module with coefficients in $k$ (note that under this condition the properties for a prime to be supersingular or RV are equivalent) satisfies the analogue of the Lang-Trotter conjecture (in simple terms, the growth of the number of supersingular reduction primes takes the shape:\[|\{l\in P_{N}(A),\texttt{ }l\textsl{ is supersingular}\}|\sim_{N\to +\infty}\frac{q^{N/2}}{N}).\] This provides a considerable number of examples, in rank $2$, satisfying the RV($r,c_{q}$)$^{*}$ condition, with $r=1/d=1/2$ for some constant $c_{q}>0$ only depending on $q$. We point out anyway that this conjecture for Drinfeld modules is \textbf{false} (yet remaining open in the "classic" case of the elliptic curves), for each possible value of the rank, as a consequence of the remarkable work of B. Poonen, \cite{B. Poonen}.\\\\ The methods that we will present in the Appendix will also show that the class of Drinfeld modules with complex multiplication having either rank $1$ or a prime number different from the characteristic of $k$ is contained in RV($1,1/2d$)$^{*}$.\\\\
We will use the following notation for the degree of the extension fields which will be involved:\[D=[k(x):k],\texttt{ }c(\Phi):=[k(\Phi):k],\texttt{ }D':=[k(\Phi)(x):k(\Phi)].\]We also call:\[D_{p.i.}:=[k(x):k]_{p.i.}\]the inseparable degree of $x$ over $k$,\[D'_{p.i.}:=[k(\Phi)(x):k(\Phi)]_{p.i.}\]the inseparable degree of $x$ over $k(\Phi)$ and:\[D'_{sep.}:=[k(\Phi)(x):k(\Phi)]_{sep.}\]the separable degree of $x$ over $k(\Phi)$. We have that:\[D'=D'_{sep.}D'_{p.i.}.\]We also call $h(\Phi)$ the height of our Drinfeld module (see Section 2 for the definition). We now state our main result in this work.
\begin{thm} Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module defined over $\overline{k}$ satisfying the hypothesis $RV(r)$ or $RV(r)^{*}$. Let:\[c_{0}:=35000dh(\Phi)^{3}c(\Phi)^{3}q^{d+rh(\Phi)c(\Phi)}\]and:\[C_{0}:=\min\{q^{-5d(2(d+1)h(\Phi)+1)((q^{q+d+1}-1)c(\Phi))^{2}},\frac{h(\Phi)}{384rq^{d}c_{0}^{\frac{4h(\Phi)c(\Phi) d}{r}+1}}\}.\]Then, there exists $C>0$ such that for all $x\in\mathbb{D}(\overline{k})_{NT}$ one has:\[\widehat{h}_{\mathbb{D}}(x)\geq C\frac{(\log\log_{+}{D})^{\mu}}{D{D_{p.i.}}^{\lambda}(\log_{+}{D})^{\kappa}}\]where:\begin{equation}\mu:=2+\frac{d}{r}h(\Phi)c(\Phi);\label{eq:5}\end{equation} \begin{equation}\kappa:=1+\frac{3d}{r}h(\Phi)c(\Phi);\label{eq:6}\end{equation}\begin{equation}\lambda:=1+\frac{2d}{r}h(\Phi)c(\Phi);\label{eq:7}\end{equation}and:\[C=C_{0}\texttt{ under the hypothesis }RV(r)\]while\[0<C\leq C_{0}\texttt{ under the hypothesis }RV(r)^{*}.\] \end{thm}
As $D_{p.i.}\leq D$ we conclude that: \begin{cor} Under the same hypotheses of Theorem 2 we have:\[\widehat{h}(x)\geq C\frac{(\log\log_{+}{D})^{\mu}}{D^{1+\lambda}(\log_{+}{D})^{\kappa}}.\] \end{cor}
We note (see for example \cite{P1}, Proposition 2) that any lower bound in Dobrowolski's form of the canonical height associated to a Drinfeld module (and in particular our bound as well) extends essentially to the whole isogeny class of such a module, up to slight modifications of the multiplicative constant, depending on the degree of the isogeny. In particular, there are no changes at all to the multiplicative constant between isomorphic Drinfeld modules (case in which the isogeny degree is $0$, see \cite{Goss}, Chapter 4).
\section{Preliminary results} Let $\mathbb{P}^{1}(\overline{k})$ be the projective line defined over $\overline{k}$. If we take a \textbf{place} $v$ over $k$, it is well known that it is associated to an irreducible element $l\in A\setminus \{0\}$ or to the point $\infty\in \mathbb{P}^{1}(k)$ so that in the first case we have:\[v(x):=\deg_{T}(l)v_{l}(x)\texttt{ }\forall x\in k,\]where $v_{l}(x)$ is the $l-$divisibility index of $x$; while in the other one:\[v(x):=v_{\infty}(x):=-\deg_{T}(x)\texttt{ }\forall x\in k.\]Each one of such places has finitely many extensions to a finite field extension $L$ of $k$.
Now, for each $x\in \overline{k}$ and each place $w$ over $k(x)/k$ restricting to $v$ in $k$, one defines:\[n_{w}:=[k(x)_{w}:k_{v}],\]where $k_{v}$ and $k(x)_{w}$ are respectively the completion of $k$ with respect to $v$ and the completion of $k(x)$ with respect to $w$. We recall the well-known facts that:\[[k(x):k]=\sum_{w|v}n_{w}\texttt{ and }n_{w}=e_{w}f_{w},\]where $e_{w}$ and $f_{w}$ are respectively the ramification index and the inertial degree of $w|v$. We note that:\[v(k^{*})\subseteq \mathbb{Z}\texttt{ and }w(k(x)^{*})\subseteq \frac{1}{e_{w}}\mathbb{Z}.\]Moreover, for every $\alpha\in k$ and every $w|v$, we have that $v(\alpha)=w(\alpha)$.
The \textbf{height} of $x$ is defined as follows:\[h(x)=\frac{1}{D}\sum_{w\texttt{ over }k(x)/k}n_{w}\max\{0,-w(x)\},\]where $D=[k(x):k]$. By writing "$w$ over $k(x)/k$" we mean that the sum is on all the places extending in $k(x)$ every place over $k$ as described above. The more general definition of height of $\overline{x}=(x_{1}, ..., x_{n})\in \overline{k}^{n}$ (for some $n>1$) is the following:\[h(\overline{x}):=\frac{1}{D}\sum_{w\texttt{ over } k(\overline{x})/k}n_{w}\max_{i=1, ..., n}\{0,-w(x_{i})\},\]where $D=[k(\overline{x}):k]$ and $k(\overline{x})=k(x_{1}, ..., x_{n})$.
We list the main properties of the logarithmic height over $\overline{k}^{n}$ which will be needed in our proof. We start by introducing the following notation we will use along the entire text. Let $\overline{a}=(a_{1}, ..., a_{n})$ and $\overline{b}=(b_{1}, ..., b_{n})$ be two vectors of $n$ components, for any fixed positive integer $n$. We introduce the following notation:\[\overline{a}*\overline{b}:=(a_{1}b_{1}, ..., a_{n}b_{n}).\] \begin{prop} \begin{enumerate}
\item Let $\overline{\alpha}, \overline{\beta}\in \overline{k}^{n}$. We have that:\[h(\overline{\alpha}+\overline{\beta})\leq h(\overline{\alpha})+h(\overline{\beta}).\]
\item Let $\overline{\alpha},\overline{\beta}\in \overline{k}^{n}$. Then we have:\begin{equation} h(\overline{\alpha}*\overline{\beta})\leq h(\overline{\alpha})+h(\overline{\beta}). \label{eq:2} \end{equation}
\item Let $\overline{\alpha},\overline{\beta}\in \overline{k}^{n}$. Let $(\overline{\alpha},\overline{\beta})\in \overline{k}^{2n}$ be the vector of $2n$ entries obtained by "glueing" $\overline{\alpha}$ with $\overline{\beta}$. Then: \begin{equation} h(\overline{\alpha}+\overline{\beta})\leq h(\overline{\alpha},\overline{\beta}). \label{eq:3} \end{equation} \end{enumerate} \end{prop} These properties are easily implied by the previous definitions.\\\\
The \textbf{height $h(\Phi)$ of a Drinfeld module} $\mathbb{D}=(\mathbb{G}_{a},\Phi)$, where $\Phi(T)$ has coefficients $T, a_{1}, ..., a_{d}\in k(\Phi)$, is:\[h(\Phi)=h(T, a_{1},...,a_{d}).\]One can easily see that $h(\Phi)\geq 1$. The \textbf{Néron-Tate height}, or \textbf{canonical height} of a Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ with rank $d$ has been introduced by L. Denis \cite{Denis} as follows:\[\widehat{h}_{\mathbb{D}}(x)=\lim_{n\to\infty}\frac{h(\Phi(T^{n})(x))}{q^{dn}}.\]We replace from now on the notation "$\widehat{h}_{\mathbb{D}}$" by simply "$\widehat{h}$" as in the entire text there will be no reference to other possible Drinfeld modules.
\begin{prop}
Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module of rank $d$, such that:\[\Phi(T)(\tau)=T+a_{1}(T)\tau+...+a_{d}(T)\tau^{d}.\]We set:\[\gamma(\Phi):=\sup_{x\in \overline{k}}|h(x)-\widehat{h}(x)|.\]Then:\[\gamma(\Phi)<2(d+1)h(\Phi).\] \end{prop} \begin{proof} See \cite{D}, Théorème 1.2.7. \end{proof}
We give now a first rough lower bound estimate of the canonical height of a Drinfeld module. \begin{lem}
For each $\chi \geq 1$, $D\geq 1$, we have:\[|\{x\in\overline{k}, [k(x):k]\leq D,h(x)\leq\chi\}|\leq q^{5D^{2}\chi}.\] \end{lem} \begin{proof} See \cite{D}, Lemme 1.2.9.
\end{proof}
\begin{lem} Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module with rank $d$. By taking $c_{2}=q^{5d(2(d+1)h(\Phi)+1)c(\Phi)^{2}}$ we have, for all $x\in \mathbb{D}(\overline{k})_{NT}$ of degree $D$ over $k$:\[\widehat{h}(x)\geq\frac{1}{c_{2}^{D^{2}}}.\] \end{lem} \begin{proof}
We proceed by contradiction. Assume that $\widehat{h}(x)<\frac{1}{c_{2}^{D^{2}}}=\frac{1}{q^{c_{3}D^{2}c(\Phi)^{2}}}$. Therefore for any $a\in A$ we have that:\[\widehat{h}(\Phi(a)(x))=q^{d\deg_{T}(a)}\widehat{h}(x)<\frac{q^{d\deg_{T}(a)}}{q^{c_{3}D^{2}c(\Phi)^{2}}}.\]Let us choose:\[\deg_{T}(a)\leq \frac{c_{3}D^{2}c(\Phi)^{2}}{d}.\]Therefore, $\widehat{h}(\Phi(a)(x))<1$ and thus, by Proposition 2:\[h(\Phi(a)(x))\leq 1+\gamma(\Phi)\leq 1+2(d+1)h(\Phi).\]Lemma 1 allows us to say that the number of elements $y$ algebraic with degree $\leq Dc(\Phi)$ over $k$ such that $h(y)\leq 1+2(d+1)h(\Phi)$, is at most $q^{5(1+2(d+1)h(\Phi))D^{2}c(\Phi)^{2}}$. Now all elements of the form $y=\Phi(a)(x)$ are algebraic of degree $[k(\Phi)(x):k]\leq Dc(\Phi)$. Moreover, since $x$ is non-torsion (so that if $a\neq b$ then $\Phi(a)(x)\neq\Phi(b)(x)$), we have for all positive integers $M$:\[\left|\bigcup_{a\in A,\deg_{T}(a)\leq M}\{\Phi(a)(x)\}\right|=q^{M+1}.\]Choosing $M=[\frac{1}{d}(c_{3}D^{2}c(\Phi)^{2})]$ we obtain $q^{M+1}$ distinct elements with degree over $k$ at most $Dc(\Phi)$ and height at most $1+2(d+1)h(\Phi)$. We also know that such a set contains at most $q^{5(1+2(d+1)h(\Phi))D^{2}c(\Phi)^{2}}$ elements. Thus we obtain :\[[\frac{c_{3}D^{2}c(\Phi)^{2}}{d}]+1\leq 5(1+2(d+1)h(\Phi))D^{2}c(\Phi)^{2}\]which yields a contradiction and proves the statement by our choice of $c_{3}$. \end{proof}
\begin{prop} Let $X$ be the number of monic, irreducible polynomials in $A$ with degree $N$, for $N\in\mathbb{N}\setminus\{0\}$. Then:\[\frac{1}{2}\frac{q^{N}}{N}\leq X\leq \frac{q^{N}}{N}.\] \end{prop} \begin{proof}
The exact value of $X$ as a function of $N$ is:\[X=1/N\sum_{d|N}\mu(N/d)q^{d}\]where $\mu$ is the Moebius function, see \cite{Irrose}, page 84. Therefore, for each $d|N$, $\mu(N/d)\leq1$ where\footnote{If $N=1$ we remark that $X=q$, which satisfies our statement.} $N\geq 2$:\[\left|1/N\sum_{d|N,d\neq N}\mu(\frac{N}{d})q^{d}\right|\leq\frac{1}{N}\sum_{i=1}^{[N/2]}q^{i}\leq\frac{1}{N}\frac{q}{q-1}(q^{N/2}-1)\leq\frac{1}{2}\frac{q^{N}}{N},\]as $q^{N/2}-1\leq\frac{q-1}{2q}q^{N}$ for each $q$ and $N$ as in the hypotheses.
Now, we have that:\[X=\left|\frac{q^{N}}{N}+\frac{1}{N}\sum_{d|N,d\neq N}\mu(\frac{N}{d})q^{d}\right|\]\[=\left|\frac{q^{N}}{N}-(-\frac{1}{N}\sum_{d|N,d\neq N}\mu(\frac{N}{d})q^{d})\right|\geq \left|\frac{q^{N}}{N}\right|-\left|1/N\sum_{d|N,d\neq N}\mu(\frac{N}{d})q^{d}\right|\geq\frac{q^{N}}{N}-\frac{1}{2}\frac{q^{N}}{N}=\frac{1}{2}\frac{q^{N}}{N},\]as a consequence of our previous estimate. To prove the other inequality we use an analogue of the factorization of the polynomial $T^{m}-1\in \mathbb{Q}[T]$ in cyclotomic polynomials with degree dividing $m$:\[T^{q^{N}}-T=\prod_{d|N}\phi_{d}(T)\]where $\phi_{d}(T)\in\mathbb{F}_{q}[T]$ is the product of the irreducible, monic polynomials with degree $d$. If we call $X_{d}$ the number of these ones, we have:\[\deg_{T}(\prod_{d|N}\phi_{d}(T))=NX+\sum_{d|N,d\neq N}dX_{d}=\deg_{T}(T^{q^{N}}-T).\]In particular, we have:\[X\leq\frac{q^{N}}{N}.\] \end{proof}
We remark that an immediate consequence of Proposition 3 is that the set of Drinfeld modules which are RV($r$) is empty if $r>1$.\\\\ We state now a key lemma, of primary importance for our argument, as we will see. This is the \textbf{Siegel Lemma}, and its proof is contained in \cite{Denis}: \begin{lem} Let $a_{j,i}$ ($1\leq i\leq N$, $1\leq j\leq M$) be elements of $\overline{k}$ generating a finite algebraic extension $\widetilde{k}/k$ having degree $D$. We assume that $N>MD$. Then there exist $x_{1}, ..., x_{N}\in A$, not all $0$, such that:\[\sum_{1\leq i\leq N}x_{i}a_{j,i}=0\]for each $1\leq j\leq M$, and such that\[\deg_{T}(x_{i})\leq\frac{D}{N-MD}\sum_{1\leq j\leq M}h(a_{j,1}, ..., a_{j,N})\]for each $1\leq i\leq N$. \end{lem} \begin{lem} Let $x\in\mathbb{D}(\overline{k})_{NT}$ with separable degree $D'_{sep.}$ over $k(\Phi)$, and let $\sigma_{1}, ..., \sigma_{D'_{sep.}}$ be the different embeddings of $k(\Phi)(x)$ in its algebraic closure in $\overline{k}$, fixing $k(\Phi)$. \begin{enumerate} \item For each pair $(a,b)\in A^{2}$ such that $a/b\notin\mathbb{F}_{q}$, we have that:\[\sigma_{i}(\Phi(a)(x))\neq\sigma_{j}(\Phi(b)(x))\]for each pair $(i,j)\in\{1, ..., D'_{sep.}\}^{2}$.
\item Let \textbf{M} be a subset of $A$ whose elements are pairwise coprime. Suppose that for each $a\in$ \textbf{M}, there exist $i\neq j$ in $\{1, ..., D'_{sep.}\}$ such that $\sigma_{i}(\Phi(a)(x))=\sigma_{j}(\Phi(a)(x))$. Then, the number of elements of \textbf{M} is less than $\log{D'_{sep.}}/\log{2}$. \end{enumerate} \end{lem} \begin{proof} \begin{enumerate}
\item We consider without loss of generality that $\{\sigma_{1}, ..., \sigma_{D'_{sep.}}\}\subseteq Aut(k_{x}/k(\Phi))$, where $k_{x}$ is the normal closure of $k(\Phi)(x)$ in $\overline{k}$. If $\sigma_{i}(\Phi(a)(x))=\sigma_{j}(\Phi(b)(x))$ for some pair $(i,j)\in\{1, ..., D'_{sep.}\}^{2}$ and some $a$ and $b$ such that $a/b\notin\mathbb{F}_{q}$, $\Phi(a)(x)$ and $\Phi(b)(x)$ are conjugated over $k(\Phi)$, hence there exists $\sigma\in Aut(k_{x}/k(\Phi))$
such that $\sigma(\Phi(a)(x))=\Phi(b)(x)$. As $Aut(k_{x}/k(\Phi))$ is a finite group, there exists $\mu\in \mathbb{N}\setminus\{0\}$ such that $\sigma^{\mu}=id_{k_{x}}$. We thus have that:\[\Phi(a^{\mu})(x)=\sigma^{\mu}(\Phi(a^{\mu})(x))=\Phi(a^{\mu-1})(\sigma^{\mu}(\Phi(a)(x)))=\Phi(a^{\mu-1})(\sigma^{\mu-1}(\Phi(b)(x)))\]\[=\Phi(a^{\mu-2})(\sigma^{\mu-1}\Phi(a)(\Phi(b)(x)))=\Phi(a^{\mu-2})(\sigma^{\mu-2}\Phi(b^{2})(x))=...=\Phi(b^{\mu})(x).\]Hence $\Phi(a^{\mu}-b^{\mu})(x)=0$. Since $x$ is not a torsion point, it follows that $a^{\mu}=b^{\mu}$, hence $a/b\in \mathbb{F}_{q}$. This contradicts the hypothesis.
\item We take $a\in A$ and $j$ between $1$ and $D'_{sep.}$. Let:\[I(a,j)=\{i\in\{1, ..., D'_{sep.}\}/\sigma_{i}(\Phi(a)(x))=\sigma_{j}(\Phi(a)(x))\}.\]We have the following properties:
\begin{enumerate}
\item $|I(a,j)|=|I(a,i)|$ for each pair $(i,j)\in\{1, ..., D'_{sep.}\}^{2}$ and two different sets of this form are disjoint.
\item If $a$ and $b$ are coprime, $|I(a,i)\cap I(b,j)|\leq1$.
\item If $a$ and $b$ are coprime, $|I(ab,j)|\geq |I(a,j)||I(b,j)|$. \end{enumerate}
We start by proving the first point. If $i=j$ the statement is obvious. Let us assume $i\neq j$. We remark that for every $r\in I(a,i)$ there exists a unique $s\in \{1, ..., D'_{sep.}\}$ such that $\sigma_{r}^{-1}\sigma_{i}=\sigma_{s}^{-1}\sigma_{j}$. As $\sigma_{r}^{-1}\sigma_{i}(\Phi(a)(x))=\Phi(a)(x)=\sigma_{s}^{-1}\sigma_{j}(\Phi(a)(x))$, it is clear that $s\in I(a,j)$. The function $I(a,i)\to I(a,j)$ we have just constructed is moreover bijective: indeed, its inverse is defined in the same way in the opposite direction on the whole set $I(a,j)$. This shows that $|I(a,i)|=|I(a,j)|$. Lastly, if $I(a,i)\cap I(a,j)\neq \emptyset$, this immediately implies that $\sigma_{i}$ and $\sigma_{j}$ coincide on $k(\Phi(a)(x))$. The map $I(a,i)\to I(a,j)$ defined above takes then values in $I(a,i)$: indeed, for every $r\in I(a,i)$, we have that $\sigma_{r}(\Phi(a)(x))=\sigma_{i}(\Phi(a)(x))=\sigma_{j}(\Phi(a)(x))=\sigma_{s}(\Phi(a)(x))$, hence $\sigma_{i}^{-1}\sigma_{s}(\Phi(a)(x))=\Phi(a)(x)$, which means that $s\in I(a,i)$. It follows that $I(a,j)\subset I(a,i)$, hence $I(a,j)=I(a,i)$ since these finite sets have the same number of elements.\\\\
We now prove the second point: if $l,m\in I(a,i)\cap I(b,j)$, $\sigma_{m}(\Phi(b)(x))=\sigma_{l}(\Phi(b)(x))$ and $\sigma_{m}(\Phi(a)(x))=\sigma_{l}(\Phi(a)(x))$, so by Bachet-Bézout Theorem, $\sigma_{m}(\Phi((a,b))(x))=\sigma_{l}(\Phi((a,b))(x))$ (where the notation $(a,b)$ is to indicate the greatest common divisor of $a$ and $b$ in $A$), and therefore, as $a$ and $b$ are coprimes, $\sigma_{m}(x)=\sigma_{l}(x)$, so $m=l$.\\\\
In order to prove the third point, we first notice that the following inequality holds: $|I(ab,j)|\geq|\cup_{i\in I(a,j)}I(b,i)|$. Indeed, let $i\in I(a,j)$ and $l\in I(b,i)$. We have $\sigma_{l}(\Phi(b)(x))=\sigma_{i}(\Phi(b)(x))$, so $\sigma_{l}(\Phi(ab)(x))=\sigma_{i}(\Phi(ab)(x))=\sigma_{j}(\Phi(ab)(x))$. This shows that:\[\cup_{i\in I(a,j)}I(b,i)\subset I(ab,j),\]hence the above inequality. The two previous points now imply the point c.\\\\
Now, if we take $\textbf{M}$ as in the hypotheses, it follows that, for each $a\in \textbf{M}$ we have $|I(a,i)|\geq 2$ for each $i\in \{1, ..., D'_{sep.}\}$. Therefore:\[2^{|\textbf{M}|}\leq\prod_{a\in\textbf{M}}|I(a,i)|\leq|I(\prod_{a\in\textbf{M}}a,i)|\leq D'_{sep.}\]and:\[|\textbf{M}|\leq\frac{\log{D'_{sep.}}}{\log{2}}.\] \end{enumerate} \end{proof}
\section{Proof of Theorem 2} We consider from now on a Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ which is RV($r$) for $r\in ]0,1]$. We first prove Theorem 2 assuming the RV($r$) hypothesis, then we will complete the proof to the RV($r$)$^{*}$ case by a slight modification of the last passages.\\\\%We shall consider here just the general (non separable) case, the passages being exactly the same in the separable situation, which is obviously less interesting. We just send the reader to \cite{D} for an exhaustive proof of Theorem 1 after having proved the Theorem 2, as such a proof follows the same steps.\\\\ \textbf{Note:} From now on we let $x$ be an element of $\mathbb{D}(\overline{k})_{NT}$. If $D<q^{q+d+1}$, then Lemma 2 yields Theorem 2. Indeed, as it is easy to check, by Lemma 2 we have $\widehat{h}(x)\geq \frac{1}{c_{2}^{(q^{q+d+1}-1)^{2}}}$ and as $C_{0}\leq c_{2}^{-(q^{q+d+1}-1)^{2}}$, this yields $\widehat{h}(x)\geq C_{0}\geq C$ for all $C$ as in Theorem 2. Now, as $r\leq 1$, it follows that $dh(\Phi)c(\Phi)/r$ is always $\geq 1$. Thus $\mu\leq \kappa$, where $\mu$ and $\kappa$ are as in Theorem 2. We then conclude that $\frac{(\log\log_{+}D)^{\mu}}{D(\log_{+}D)^{\kappa}}\leq 1$ for all $D<q^{q+d+1}$ (where $\log_{+}(\cdot)$ and $\log\log_{+}(\cdot)$ take values $\geq 1$). Hence in what follows we will assume $D\geq q^{q+d+1}$. This will considerably ease many of the technicalities in our computations, as we will see.\\\\ To prove Theorem 2, we will argue by contradiction. We start therefore by assuming the following hypothesis which we want to contradict: \begin{hyp}
We have:\[\widehat{h}(x)<C_{0}\frac{(\log\log{D})^{\mu}}{D{D}_{p.i.}^{\lambda}(\log{D})^{\kappa}}\]where $C_{0},\mu,\lambda,\kappa$ are defined as in Theorem 2. \end{hyp}
We will proceed by the following steps.
\begin{enumerate}
\item We build an \textsl{auxiliary polynomial} with coefficients in $k(\Phi)$, vanishing with a certain multiplicity $t$ in $x$. By Siegel Lemma we will be able to bound the coefficients in some explicit way.
\item We show that for some specific $h<t$, the auxiliary polynomial vanishes at $\Phi(l)(x)$ with multiplicity at least $h$, for every $l\in A\setminus \mathbb{F}_{q}$ monic irreducible of some specific degree satisfying the RV condition. The proof of this fact is the real heart of the whole section and will be accomplished arguing again by contradiction. Assuming that there is an $l\in S(A)$ as above and such that our polynomial vanishes at $\Phi(l)(x)$ with multiplicity $h'<h$, we show that this contradicts Siegel's Lemma.
\item We will thus have that the sum of multiplicities of the roots $\Phi(l)(x)$ of our auxiliary polynomial for all $l\in S(A)$ satisfying the previous conditions (plus the case $l=1$), is at least $h$ times the cardinality of such a subset of $S(A)$. This will imply by a suitable choice of $h$ and $t$ that this number exceeds the degree of the auxiliary polynomial, yielding a contradiction. \end{enumerate}
\textbf{For the rest of the whole section we will assume Hypothesis 1.}
\subsection{Step 1 - Construction of the auxiliary polynomial} \begin{de} Given a polynomial $f(X)\in K[X]$ with $K$ any field of characteristic $p$, we call \textbf{hyperderivative} of $f$ of order $h$ the polynomial $d^{(h)}f(X)\in K[X]$ obtained as the coefficient of the term $H^{h}$ of $f(X+H)\in K[X][H]$, for some new parameter $H$.
\end{de}
\begin{rem} Let $A(X)\in \overline{k}[X]$. An element $x\in \overline{k}$ is a root of $A(X)$ of multiplicity at least $h\geq 1$ if and only if $d^{(h')}A(x)=0$ for each $h'=0, ..., h-1$. \end{rem} \begin{proof} See \cite{D}, Remarque 1.3.4.
\end{proof} We call $p^{e}$ the purely inseparable degree $D'_{p.i.}$ of $k(\Phi)(x)$ over $k(\Phi)$.\\\\ The following Proposition provides the explicit construction of the auxiliary polynomial we will use in our proof. \begin{prop} Let $L,t',t\in \mathbb{N}$ such that:\[t'=tp^{e}\]and:\[L^{2}>tDc(\Phi).\]Let $N\in A$ be such that:\begin{equation}\deg_{T}(N)=\left[\frac{1}{d}\log{L}\right]+1.\label{eq:8} \end{equation}Then there exists a polynomial:\[G(X,Y)=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}X^{i}Y^{j} \in A[X,Y]\setminus\{0\}\]such that:\[G_{N}(X):=G(X,\Phi(N)(X))\in k(\Phi)[X]\]is not identically 0, vanishes at $x$ with multiplicity at least $t'$ and such that the coefficients $p_{ij}\in A$ of $G(X,Y)$ satisfy the following condition:\[\deg_{T}(p_{ij})\leq \frac{Dc(\Phi)}{L^{2}-tDc(\Phi)}\Sigma\]for each $0\leq i,j\leq L-1$, where $\Sigma$ is the sum of the heights of all vectors which are the lines of the coefficient matrix of the linear system:\begin{equation} d^{(hp^{e})}G_{N}(x)=0 \label{eq:n} \end{equation} for $h=0, ..., t-1$, whose unknowns are precisely the coefficients of $G(X,Y)$. \end{prop} \begin{proof} Let us write:\[G(X,Y)=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}X^{i}Y^{j}.\]We choose an element $N\in A\setminus\{0\}$ such that (\ref{eq:8}) holds. Therefore, as $q^{d\deg_{T}(N)}>L-1$, it follows that $G_{N}$ is not identically $0$ in $k(\Phi)[X]$ as the algebraic variety of equation $Y=\Phi(N)(X)$ in $\mathcal{C}^{2}$ is not contained in the zero locus of $G(X,Y)$. Indeed:\[Y-\Phi(N)(X)\nmid G(X,Y)\]in $k(\Phi)[X,Y]$.
Now, the requirement that $G_{N}(X)$ vanishes at $x$ with order $t'$ means that we have to take the coefficients of $G_{N}(X)$ in the space of solutions of the linear system of $L^{2}$ unknowns and $t'$ conditions, given by the vanishing of the hyperderivatives of $G_{N}(X)$ at $x$ with order less than $t'$. We now show that the number of such conditions may actually be taken to be at most $t$. Indeed, if $x$ is a root of $G_{N}(X)$ we have as a first condition that:\[G_{N}(x)=0.\]This is a linear equation of $L^{2}$ unknowns and implies that:\[\Delta(X)|G_{N}(X)\]where $\Delta(X)\in k(\Phi)[X]$ is the minimal polynomial of $x$ over $k(\Phi)$. As the purely inseparable degree of $x$ over $k(\Phi)$ is $D'_{p.i.}=p^{e}$, $x$ is a root of $\Delta(X)$ of order $p^{e}$. Therefore, it is also a root of $G_{N}(X)$ with at least the same multiplicity. The single vanishing condition of $G_{N}(X)$ at $x$ implies therefore that the other $p^{e}-1$ conditions:\[d^{(h)}G_{N}(x)=0\]with $h\leq p^{e}-1$, are satisfied as well. In the same way, intersecting such a space of solutions with the one given by the linear equation:\[d^{(p^{e})}G_{N}(x)=0\]means that:\[\Delta(X)|\frac{G_{N}(X)}{\Delta(X)}\]in $k(\Phi)[X]$ as, by Remark 1, such an intersection implies that $x$ is a root of $G_{N}(X)$ with order at least $p^{e}+1$, while the roots of $\Delta(X)$ have multiplicity $p^{e}$. As:\[\Delta(X)^{2}|G_{N}(X)\]one actually has that the $p^{e}-1$ conditions:\[d^{(h)}G_{N}(x)=0\]with $h=p^{e}+1, ..., 2p^{e}-1$ follow directly from the first one $d^{(p^{e})}G_{N}(x)=0$. Now, repeating the same passages for each condition $G_{N}(x)=0$, $d^{(p^{e})}G_{N}(x)=0$, ..., $d^{((t-1)p^{e})}G_{N}(x)=0$, the linear system we have actually to solve takes the shape:\[d^{(hp^{e})}G_{N}(x)=0\]for $h=0, ..., t-1$ and it is equivalent to that of the form $d^{(h)}G_{N}(x)=0$ for each $h=0, ..., t'-1$.
We therefore obtain a linear system with $t$ conditions, whose unknowns are the $L^{2}$ coefficients of $G$. Thus, if we set $L^{2}>Dc(\Phi)t$, where $Dc(\Phi)\geq[k(\Phi)(x):k]$, the conditions of Lemma 3 are satisfied. If we choose $\Sigma$ as in the hypothesis the statement is proved. \end{proof}
\subsection{Step 2 - Vanishing of $G_{N}(X)$ with desired multiplicity at $\Phi(l)(x)$ for $l$ RV in $S(A)$} From now on, we choose the parameters $L$, $t$ and $h$ as follows (where $c_{0}$ is as in Theorem 2)\footnote{Such choices of the parameters have been suggested by Hugues Bauchère (LMNO, Caen), \cite{13}.}: \begin{equation} L:=\left[c_{0}^{2}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\right]+1; \label{eq:14} \end{equation} \begin{equation} t:=\left[c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{3}}p^{e}\right]; \label{eq:15} \end{equation} \begin{equation} h:=\left[c_{0}\frac{D}{(\log\log{D})^{2}}\right]. \label{eq:16}\end{equation} Let us take, as before:\[t'=tp^{e}.\] \begin{lem} With the above choice of the parameters we have:\[L^{2}-tDc(\Phi)\geq \frac{1}{2}L^{2}.\]In particular, the hypothesis of Proposition 4 is satisfied. \end{lem} \begin{proof} We know that:\[c_{0}\geq c(\Phi).\]Now, we state that: $\frac{1}{2}L^{2}-tDc(\Phi)\geq 0$. Indeed, as:\[L^{2}\geq c_{0}^{4}\frac{D^{2}(\log{D})^{2}p^{2e}}{(\log\log{D})^{4}}\]and:\[tDc(\Phi)\leq c_{0}^{4}\frac{D^{2}\log{D}}{(\log\log{D})^{3}}p^{2e}\]we have that:\[\frac{1}{2}L^{2}-tDc(\Phi)\geq c_{0}^{4}D^{2}p^{2e}\log{D}\left(\frac{\frac{1}{2}\log{D}-\log\log{D}}{(\log\log{D})^{4}}\right).\]If $D\geq q^{q+d+1}$ the right-hand term of such an inequality is not negative if and only if:\[\frac{1}{2}\log{D}\geq \log\log{D}\]which is easy to see to be always verified when $D\geq q^{q+d+1}$. \end{proof} By Proposition 4, we can construct a polynomial:\[G_{N}(X)=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}X^{i}(\Phi(N)(X))^{j}.\]By Remark 1 we can say that $x$ is a root of multiplicity at least $t'-hp^{e}$ of $d^{(hp^{e})}G_{N}(X)$ for $h=0, ..., t-1$. For a general $h\leq t-1$ we thus have the following decomposition:\[d^{(hp^{e})}G_{N}(X)=\Delta(X)^{t-h}R_{h}(X)\]where $\Delta(X)\in k(\Phi)[X]$ is the minimal polynomial of $x$ over $k(\Phi)$.\\\\ The goal of this subsection is to prove the following proposition. \begin{prop} Let $l\in S(A)$ satisfying the RV property and such that: \begin{equation} \deg_{T}(l):=h(\Phi)c(\Phi)\left[\frac{1}{r}\log\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)\right]. \label{eq:17} \end{equation} Then we have:\[d^{(h'p^{e})}G_{N}(\Phi(l)(x))=0\]for each $0\leq h'\leq h-1$. \end{prop} We will argue by contradiction. Assuming the conclusion of Proposition 5 is false, we will follow three steps: \begin{enumerate}
\item We provide an upper bound for the logarithmic height of $d^{(h'p^{e})}G_{N}(\Phi(l)(x))$.
\item We prove a lower bound of the same quantity, using our assumption that $l$ satisfies the RV condition.
\item We show that these two inequalities yield a contradition. \end{enumerate}
Given $v_{l}$ a place of $k$ associated to an irreducible element $l\in A\setminus\{0\}$, we write $w|v_{l}$ to say that a place $w$ extends $v_{l}$ to $k(\Phi)(x)$. \begin{prop}
In order to prove Theorem 2, we may assume that for all $l\in S(A)$ satisfying the RV condition and for all $w|v_{l}$ we have $w(x)\geq 0$. \end{prop} \begin{proof}
We assume that there exists an $l\in S(A)$ which is RV and an extension $w_{0}|v_{l}$ such that $w_{0}(x)<0$. Let us write:\[\Phi(l)(x)=lx+\alpha_{1}x^{q}+...+\alpha_{d\deg_{T}(l)}x^{q^{d\deg_{T}(l)}}.\]The RV hypothesis on $l$ implies that $w_{0}(\alpha_{i})>0$ for each $i=0, ..., d\deg_{T}(l)-1$, while $w_{0}(\alpha_{d\deg_{T}(l)})=0$. Therefore, as $w_{0}(x)<0$, we have that:\[w_{0}(\alpha_{d\deg_{T}(l)}x^{q^{d\deg_{T}(l)}})=q^{d\deg_{T}(l)}w_{0}(x)< q^{i}w_{0}(x)< w_{0}(\alpha_{i})+w_{0}(x^{q^{i}})=w_{0}(\alpha_{i}x^{q^{i}})\]for each $i=0, ..., d\deg_{T}(l)-1$. The properties of a non-Archimedean valuation imply therefore that:\[w_{0}(\Phi(l)(x))=q^{d\deg_{T}(l)}w_{0}(x).\]Iterating until we replace $x$ by $\Phi(l^{n-1})(x)$, we get, for all $n\in \mathbb{N}$:\[w_{0}(\Phi(l^{n})(x))=q^{d\deg_{T}(l)n}w_{0}(x).\]We derive from this:\[h(\Phi(l^{n})(x))=\frac{1}{[k(\Phi)(x):k]}\sum_{w\texttt{ over }k(\Phi)(x)/k}n_{w}\max\{0, -w(\Phi(l^{n})(x))\}\]\[\geq \frac{n_{w_{0}}}{[k(\Phi)(x):k]}\max\{0, -w_{0}(\Phi(l^{n})(x))\}\]\[=\frac{n_{w_{0}}q^{d\deg_{T}(l)n}}{[k(\Phi)(x):k]}\max\{0, -w_{0}(x)\}.\]Since $-n_{w_{0}}w_{0}(x)\geq 1$, we immediately deduce that:\[\widehat{h}(x)=\lim_{n\to+\infty}q^{-d\deg_{T}(l)n}h(\Phi(l^{n})(x))\geq \frac{1}{Dc(\Phi)}\]since:\[[k(\Phi)(x):k]\leq Dc(\Phi)\](we recall that $D=[k(x):k]$). This immediately provides an even stronger statement than Theorem 2, so we can easily get rid of the assumption of $w_{0}(x)$ to be negative. \end{proof} \subsubsection{Upper bound for $h(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))$}
\begin{prop} Let $l\in S(A)$. For each integer $h'$ with $0\leq h'\leq t-1$, we have:\[h(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))\leq h(p_{ij})+L[h(\Phi(l)(x))+h(\Phi(Nl)(x))]+\deg_{T}(N) h(\Phi)h'p^{e}.\] \end{prop} \begin{proof} We recall that:\[G(X,Y)=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}X^{i}Y^{j}.\]By definition, the hyperderivative of $G_{N}(X)$ at $x$ having order $h'p^{e}$ is the coefficient of $H^{h'p^{e}}$, for a new indeterminate $H$, of the polynomial:\[G_{N}(X+H)=\]\[=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}(\sum_{a=0}^{i}\binom{i}{a}X^{i-a}H^{a})(\sum_{b=0}^{j}\binom{j}{b}(\Phi(N)(X))^{j-b}(\Phi(N)(H))^{b}).\]Let us write:\[\Phi(N)(H)=\sum_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}H^{q^{s}},\]where $\widetilde{a}_{s}\in k(\Phi)$ for $s=0, ..., d\deg_{T}(N)$. The hyperderivative $d^{(h'p^{e})}G_{N}(X)$ is a sum of a certain number of terms, which is the number of all the possible ways to obtain the power $H^{h'p^{e}}$ in the above expression of $G_{N}(X+H)$. Each of such terms takes the following shape:\[\sum_{i=a}^{L-1}\sum_{j=b}^{L-1}p_{ij}\binom{i}{a}\binom{j}{b}\binom{b}{n_{0}, ..., n_{d\deg_{T}(N)}}X^{i-a}(\Phi(N)(X))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\]\[=\sum_{i=a}^{L-1}\sum_{j=b}^{L-1}p_{ij}\binom{i}{a}\binom{j}{j-b,n_{0}, ..., n_{d\deg_{T}(N)}}X^{i-a}(\Phi(N)(X))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\]for each pair $(a,b)$ and each $(d\deg_{T}(N)+1)-$tuple $\overline{n}=(n_{0}, ..., n_{d\deg_{T}(N)})\in \mathbb{N}^{d\deg_{T}(N)+1}$ such that:\[b=\sum_{s=0}^{d\deg_{T}(N)}n_{s}\]and\[h'p^{e}=a+\sum_{s=0}^{d\deg_{T}(N)}n_{s}q^{s}.\]We thus obtain that $0\leq a\leq h'p^{e}$ and $0\leq \sum_{s=0}^{d\deg_{T}(N)}n_{s}q^{s}=h'p^{e}-a$.
For each pair $(i,j)\in\{0, ..., L-1\}^{2}$ the coefficient associated to $p_{ij}$ in the linear system (\ref{eq:n}) introduced in Proposition 4 is then:\[\sum_{(a,b,\overline{n})\in\mathcal{I}(i,j,h')}\binom{i}{a}\binom{j}{j-b,n_{0}, ..., n_{d\deg_{T}(N)}}x^{i-a}(\Phi(N)(x))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}},\] where we define the set $\mathcal{I}(i,j,h')$ as follows:\[\mathcal{I}(i,j,h'):=\{(a,b,n_{0}, ..., n_{d\deg_{T}(N)})\in \mathbb{N}^{d\deg_{T}(N)+3}, 0\leq a\leq \min\{i,h'p^{e}\},\]\[,a+\sum_{s=0}^{d\deg_{T}(N)}n_{s}q^{s}=h'p^{e}, \sum_{s=0}^{d\deg_{T}(N)}n_{s}=b\}.\]The height of the $h'-$th line $L_{h'}$ of the system (\ref{eq:n}) is thus:\[h(L_{h'}):=h(\{\sum_{(a,b,\overline{n})\in\mathcal{I}(i,j,h')}\binom{i}{a}\binom{j}{j-b,n_{0}, ..., n_{d\deg_{T}(N)}}x^{i-a}(\Phi(N)(x))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\}_{(i,j)}).\] The property (\ref{eq:3}) of Proposition 1 leads us to the following upper bound of $h(L_{h'})$:\[h(L_{h'})\leq h(\{x^{i-a}(\Phi(N)(x))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\}_{(i,j,a,b,\overline{n})})\]\[=\frac{1}{[k(\Phi)(x):k]}\sum_{v\texttt{ over }k(\Phi)(x)/k}n_{v}\max_{(i,j,a,b,\overline{n})}\{0,-v(x^{i-a}(\Phi(N)(x))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}})\}.\]In other words, for each choice of $i,j=0, ..., L-1$, $a=0, ..., \min\{i,h'p^{e}\}$ and $b$ such that $h'p^{e}=a+\sum_{s=0}^{d\deg_{T}(N)}n_{s}q^{s}$, we multiply $x^{i-a}(\Phi(N)(x))^{j-b}$ by each one of the elements $\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}$ appearing for each $\overline{n}$ such that $\sum_{s=0}^{d\deg_{T}(N)}n_{s}=b$ and $h'p^{e}=a+\sum_{s=0}^{d\deg_{T}(N)}n_{s}q^{s}$. Therefore, by calling $\overline{\alpha}:=\{x^{i-a}(\Phi(N)(x))^{j-b}\}_{(i,j,a,b,\overline{n})}$ (note that for each multi-index $(i,j,a,b)$ the entry $x^{i-a}(\Phi(N)(x))^{j-b}$ appears a number of times which is precisely the cardinality of the set of the $\overline{n}$ associated to $(i,j,a,b)$), and calling:\[\overline{\beta}:=\{(\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}})\}_{(i,j,a,b,\overline{n})}\]we have:\[\overline{\alpha}*\overline{\beta}=\{x^{i-a}(\Phi(N)(x))^{j-b}\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\}_{(i,j,a,b,\overline{n})},\]whose height we are analysing. The product law (\ref{eq:2}) provides thus the following inequality:\[h(L_{h'})\leq h(\{x^{i-a}(\Phi(N)(x))^{j-b}\})+h(\{\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\}).\]By the properties of the logarithmic height, the first term is bounded as follows:\[h(\{x^{i-a}(\Phi(N)(x))^{j-b}\})\leq L[h(x)+h(\Phi(N)(x))].\]We search now for an upper bound of the second term too. Writing $N(T)= \alpha_{0}+\alpha_{1}T+...+\alpha_{\deg_{T}(N)}T^{\deg_{T}(N)}\in A$, we have:\[\Phi(N(T))=N(\Phi(T))=\alpha_{0}+\alpha_{1}\Phi(T)+...+\alpha_{\deg_{T}(N)}\Phi(T)^{\deg_{T}(N)}.\]We now focus on the height of each monomial. For $0\leq\delta\leq\deg_{T}(N)$:\[\Phi(T)^{\delta}=\sum_{i=0}^{d\delta}(\sum_{\overline{j}\in\Delta_{\delta}(i)}\prod_{s=1}^{\delta}a_{j(s)}^{q^{\sum_{\nu=0}^{s-1}j(\nu)}})\tau^{i}\]where:\[\Delta_{\delta}(i):=\{(j(1), ..., j(\delta))\in \mathbb{N}^{\delta};\sum_{s=1}^{\delta}j(s)=i\}\]with $j(s)\in\{0, ..., d\}$, $j(0):=0$. Recalling that $\widetilde{a}_{i}$ is the coefficient of $\tau^{i}$ in the expression of $\Phi(N)$, we obtain, for all places $w$ of $k(\Phi)$:\[-w(\widetilde{a}_{i})\leq\max\{\sum_{s=1}^{\delta}-q^{\sum_{\nu=0}^{s-1}j(\nu)}w(a_{j(s)})\}\leq \delta q^{i}\max_{j=0, ..., d}\{-w(a_{j})\}.\]Therefore:\[h(\{\prod_{s=0}^{d\deg_{T}(N)}\widetilde{a}_{s}^{n_{s}}\}_{(i,j,a,b,\overline{n})})=\frac{1}{c(\Phi)}\sum_{w\texttt{ over }k(\Phi)/k}n_{w}\max\{0,-\sum_{s=0}^{d\deg_{T}(N)}n_{s}w(\widetilde{a}_{s})\}\]\[\leq\frac{1}{c(\Phi)}\sum_{w\texttt{ over }k(\Phi)/k}n_{w}\sum_{s=0}^{d\deg_{T}(N)}\max\{0,-n_{s}w(\widetilde{a}_{s})\}\]\[\leq \frac{1}{c(\Phi)}\sum_{w\texttt{ over }k(\Phi)/k}n_{w}\sum_{s=0}^{d\deg_{T}(N)}n_{s} \deg_{T}(N) q^{s}\max_{j=0, ..., d}\{0,-w(a_{j})\}\leq \deg_{T}(N) h(\Phi)h'p^{e}.\]
Finally, for each $h'=0, ..., t-1$: \begin{equation} h(L_{h'})\leq L[h(x)+h(\Phi(N)(x))]+\deg_{T}(N) h(\Phi)h'p^{e}. \label{eq:10} \end{equation} Now:\[d^{(h'p^{e})}G_{N}(\Phi(l)(x))\]\[=\sum_{i=0}^{L-1}\sum_{j=0}^{L-1}p_{ij}\sum_{(a,b,\overline{n})\in\mathcal{I}(i,j,h')}\binom{i}{a}\binom{j}{j-b,n_{0}, ..., n_{d\deg_{T}(N)}}(\Phi(l)(x))^{i-a}(\Phi(Nl)(x))^{j-b}\prod_{i=0}^{d\deg_{T}(N)}\widetilde{a}_{i}^{n_{i}}.\]From this formula and the previous computations (applied to $\Phi(l)(x)$ instead of $x$), we obtain the bound of the Proposition. \end{proof}
\subsubsection{Lower bound for $h(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))$} \begin{prop} Let $l\in S(A)$ satisfying the RV condition and $h'\in \mathbb{N}$ with $0\leq h'\leq t-1$. If $d^{(h'p^{e})}G_{N}(\Phi(l)(x))\neq 0$, then we have:\[h(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))\geq \deg_{T}(l)\frac{(t-h')}{c(\Phi)}.\] \end{prop} \begin{proof} We call $\zeta:=N_{k(\Phi)(x)/k}(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))$. Since $\zeta$ is the product of $[k(\Phi)(x):k]$ conjugates (possibly equal) of $d^{(h'p^{e})}G_{N}(\Phi(l)(x))$, we have :\[h(\zeta) \leq [k(\Phi)(x):k]h(d^{(h'p^{e})}G_{N}(\Phi(l)(x))).\]
Let $w$ be a place of $k(\Phi)(x)$ such that $w|v_{l}$, and let $v$ be its restriction to $k(\Phi)$. Denote by $\mathcal{O}_{v}\subset k(\Phi)$ the valuation ring of $v$. As $w(x)\geq 0$ for each $w|v_{l}$ (see Proposition 6), the minimal (monic) polynomial $\Delta(X)$ of $x$ over $k(\Phi)$ has coefficients in $\mathcal{O}_{v}$. We know that:\[d^{(h'p^{e})}G_{N}(\Phi(l)(x))=\Delta(\Phi(l)(x))^{t-h'}R_{h'}(\Phi(l)(x))\]where $G_{N}(X)$ and $R_{h'}(X)$ are in $\mathcal{O}_{v}[X]\setminus\{0\}$
(we remark that as $\Delta(X), \Phi(l)(X)\in \mathcal{O}_{v}[X]$, we have $R_{h'}(X)\in \mathcal{O}_{v}[X]$ too). Let us call $\mathfrak{l}$ the prime ideal of $\mathcal{O}_{v}$ dividing $l$ and corresponding to $v$. Using the RV hypothesis on $l$ we have:\[\Delta(\Phi(l)(x))\equiv\Delta(x^{q^{d\deg_{T}(l)}})\texttt{ mod }(\mathfrak{l}\mathcal{O}_{w}),\]where $\mathcal{O}_{w}$ is the valuation ring of $k(\Phi)(x)$ with respect to $w$. Let $f_{v}=[\mathcal{O}_{v}/v:A/l]$ be the inertia degree of $v$ over $v_{l}$. By the RV condition, we have $f_{v}=1$, hence $|\mathcal{O}_{v}/v|=|A/l|=q^{\deg_{T}(l)}$. It follows that the coefficients of $\Delta(X)$ are congruent to their powers to $q^{d\deg_{T}(l)}$ mod ($\mathfrak{l}$). For this reason, we have that:\[\Delta(x^{q^{d\deg_{T}(l)}})\equiv \Delta(x)^{q^{d\deg_{T}(l)}}\equiv 0\texttt{ mod }(\mathfrak{l}\mathcal{O}_{w}).\]If we call $e_{w|v}$ the ramification index of $w$ over $v$, we have therefore that $w(\Delta(\Phi(l)(x)))\geq \frac{e_{w|v}}{e_{w}}=\frac{1}{e_{v}}$, where $e_{v}$ is the ramification index of $v$ over $v_{l}$.
So we can conclude that: \begin{equation} w(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))\geq \frac{t-h'}{e_{v}} \label{eq:m} \end{equation} for each $w|v_{l}$. Now, assuming $d^{(h'p^{e})}G(\Phi(l)(x))\neq 0$, we have that $\zeta\neq 0$ and therefore $\zeta^{-1}$ exists.
Therefore:\[h(\zeta)=h(\zeta^{-1})\geq\max\{0,-\deg_{T}(l)v_{l}(\zeta^{-1})\}=\max\{0,\deg_{T}(l)v_{l}(\zeta)\}.\]By (\ref{eq:m}) we have that $v_{l}(\zeta)=\sum_{w|v_{l}}n_{w}w(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))\geq \sum_{w|v_{l}}n_{w}\frac{t-h'}{e_{v}}\geq \sum_{w|v_{l}}n_{w}\frac{t-h'}{c(\Phi)}=[k(\Phi)(x):k]\frac{(t-h')}{c(\Phi)}$. Therefore:\[h(\zeta)\geq [k(\Phi)(x):k]\deg_{T}(l)\frac{(t-h')}{c(\Phi)}.\]As:\[h(\zeta)\leq [k(\Phi)(x):k]h(d^{(h'p^{e})}G_{N}(\Phi(l)(x)))\]the statement follows. \end{proof} \subsubsection{Final contradiction} In this subsection, we will finally complete the proof of Proposition 5. Let $l\in S(A)$ be as in this proposition. Arguing by contradiction, let us assume that there exists $0\leq h'\leq h-1$ such that $d^{(h'p^{e})}G_{N}(\Phi(l)(x))\neq 0$. By the inequalities showed in Proposition 7 and Proposition 8, we have that:\[\deg_{T}(l)\frac{(t-h')}{c(\Phi)}\leq h(p_{ij})+L[h(\Phi(l)(x))+h(\Phi(Nl)(x))]+\deg_{T}(N)h(\Phi)h'p^{e}.\]And, as $h'\leq h-1$, we easily conclude that: \begin{equation} \deg_{T}(l)\frac{(t-h)}{c(\Phi)}\leq h(p_{ij})+L[h(\Phi(l)(x))+h(\Phi(Nl)(x))]+\deg_{T}(N)h(\Phi)hp^{e}. \label{eq:11} \end{equation} We now show that for $c_{0}$ as in Theorem 2 the choice of the parameters $L$, $t$ and $h$ contradicts the inequality (\ref{eq:11}). By Proposition 4 and inequality (\ref{eq:10}), we have:\[h(p_{ij})\leq \frac{Dc(\Phi)}{L^{2}-tDc(\Phi)}\sum_{0\leq h\leq t-1}(L[h(x)+h(\Phi(N)(x))]+\deg_{T}(N)h(\Phi)hp^{e}).\]Hence, by Lemma 5 and Proposition 2, we obtain:\[h(p_{ij})\leq 2\frac{Dc(\Phi)}{L^{2}}\sum_{0\leq h\leq t-1}(L[\widehat{h}(x)+2\gamma(\Phi)+\widehat{h}(\Phi(N)(x))]+\deg_{T}(N)h(\Phi)hp^{e})\]\[\leq 2\frac{Dtc(\Phi)}{L^{2}}(2Lq^{d\deg_{T}(N)}\widehat{h}(x)+4(d+1)Lh(\Phi)+\deg_{T}(N)h(\Phi)p^{e}t/2).\]Condition (\ref{eq:8}) now provides\footnote{As $\log{L}\geq d$ (which is a consequence of the inequality $D\geq q^{q+d+1}$) one has that $\deg_{T}(N)\leq \frac{2}{d}\log{L}$.} the inequality:\[h(p_{i,j})\leq 2\frac{Dtc(\Phi)}{L^{2}}(2q^{d}L^{2}\widehat{h}(x)+4(d+1)Lh(\Phi)+\frac{1}{d}\log{L}h(\Phi)tp^{e})\] \begin{equation} =4q^{d}c(\Phi)Dt\widehat{h}(x)+\frac{8(d+1)h(\Phi)c(\Phi)}{L}Dt+\frac{2h(\Phi)c(\Phi)}{d}\frac{Dt^{2}\log{L}}{L^{2}}p^{e}. \label{eq:19} \end{equation} \\\\ Now, by (\ref{eq:19}) and (\ref{eq:11}):\[\deg_{T}(l)\frac{(t-h)}{c(\Phi)}< 4q^{d}c(\Phi)Dt\widehat{h}(x)+\frac{8(d+1)h(\Phi)c(\Phi)}{L}Dt+\frac{2h(\Phi)c(\Phi)}{d}\frac{Dt^{2}\log{L}}{L^{2}}p^{e}\]\[+L[2q^{d(\deg_{T}(N)+\deg_{T}(l))}\widehat{h}(x)+4(d+1)h(\Phi)]+\deg_{T}(N)h(\Phi)hp^{e}\]\[\leq 4q^{d}c(\Phi)Dt\widehat{h}(x)+\frac{8(d+1)h(\Phi)c(\Phi)}{L}Dt+\frac{2h(\Phi)c(\Phi)}{d}\frac{Dt^{2}\log{L}}{L^{2}}p^{e}\]\[+2q^{d\deg_{T}(l)}q^{d}L^{2}\widehat{h}(x)+4(d+1)h(\Phi)L+\frac{2}{d}\log{L}h(\Phi)hp^{e}.\]The choices (\ref{eq:14}), (\ref{eq:15}) and (\ref{eq:16}) imply that $h\leq t/2$. So:\[\deg_{T}(l)\frac{t}{c(\Phi)}\leq 8q^{d}c(\Phi)Dt\widehat{h}(x)+\frac{16(d+1)h(\Phi)c(\Phi)}{L}Dt+\frac{4h(\Phi)c(\Phi)}{d}\frac{Dt^{2}\log{L}}{L^{2}}p^{e}\]\[+4q^{d\deg_{T}(l)}q^{d}L^{2}\widehat{h}(x)+8(d+1)h(\Phi)L+\frac{4}{d}\log{L}h(\Phi)hp^{e}.\]Knowing that (see Lemma 5) $tDc(\Phi)<L^{2}$, we obtain that:\begin{equation} \deg_{T}(l)t<c_{4}(L^{2}q^{d\deg_{T}(l)}\widehat{h}(x)+h(\Phi)L+\frac{h(\Phi)c(\Phi)}{d}\frac{Dt^{2}}{L^{2}}p^{e}\log{L}+\frac{h(\Phi)}{d}hp^{e}\log{L}); \label{eq:20} \end{equation} where we put:\[c_{4}:=24q^{d}c(\Phi).\] \\
We now find a lower bound for $\deg_{T}(l)t$.\\\\ For each $a,b\in \mathbb{R}^{+}$ such that $a,b\geq 4$, we have that $[a][b]\geq \frac{1}{2}ab$. Therefore, we pose:\[\alpha:=h(\Phi)c(\Phi)\]and we remark that:\[c_{0}\geq q^{d}.\]Such a condition implies, as $D\geq q^{q+d+1}$, that $t,\deg_{T}(l)\geq 4$. Therefore:\[\deg_{T}(l)t\geq\frac{1}{2}\frac{\alpha}{r}(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}}-\log\log\log{D})c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{3}}p^{e}\]\[\geq \frac{1}{2}\frac{\alpha}{r}(4\log{c_{0}}+2\log\log{D})c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{3}}p^{e}\] \begin{equation} \geq \frac{\alpha}{r}c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}. \label{eq:23} \end{equation}\\\\ We now prove that (\ref{eq:20}) cannot hold if we assume Hypothesis 1. This will follow from the following facts: \begin{equation} \deg_{T}(l)t\geq 4c_{4}h(\Phi)L; \label{eq:24} \end{equation} \begin{equation} \deg_{T}(l)t\geq 4c_{4}\frac{h(\Phi)c(\Phi)}{d}\frac{Dt^{2}}{L^{2}}p^{e}\log{L}; \label{eq:25} \end{equation} \begin{equation} \deg_{T}(l)t\geq 4c_{4}\frac{h(\Phi)}{d}hp^{e}\log{L}; \label{eq:26} \end{equation}\begin{equation}\deg_{T}(l)t\geq 4c_{4}L^{2}q^{d\deg_{T}(l)}\widehat{h}(x).\label{eq:101}\end{equation}Let us check (\ref{eq:24}) first. We have : \begin{equation} c_{0}^{2}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\leq L\leq 2c_{0}^{2}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e} \label{eq:27} \end{equation} by (\ref{eq:14}) and the hypothesis that $D\geq q^{q+d+1}$. The inequality (\ref{eq:24}) is thus by (\ref{eq:23}) a consequence of the following one:\[\frac{\alpha}{r}c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\geq 8c_{4}\alpha c_{0}^{2}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\]which is true since:\[c_{0}\geq 8rc_{4}=192rq^{d}c(\Phi).\]The inequality (\ref{eq:25}) is on the other hand by (\ref{eq:23}) and (\ref{eq:27}) a consequence of the following one:\[\frac{\alpha}{r}c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\geq 4c_{4}\frac{\alpha}{d}c_{0}^{6}\frac{D^{3}(\log{D})^{2}p^{2e}}{(\log\log{D})^{6}}\frac{(\log\log{D})^{4}}{c_{0}^{4}p^{2e}D^{2}(\log{D})^{2}}p^{e}\log{L}\]which follows from this condition:\[c_{0}\log{D}\geq \frac{4rc_{4}}{d}(2\log{c_{0}}+\log{D}+\log\log{D}-2\log\log\log{D}+\log{2}+\log{p^{e}})\]which is implied by the following inequality: \begin{equation} c_{0}\log{D}\geq\frac{4rc_{4}}{d}(2\log{c_{0}}+4\log{D}). \label{eq:29} \end{equation} Now, (\ref{eq:29}) follows from these two facts:\[c_{0}\geq \frac{32rc_{4}}{d}=\frac{768rq^{d}c(\Phi)}{d},\]and:\[c_{0}\log{D}\geq \frac{16rc_{4}}{d}\log{c_{0}}.\]These are a consequence of:\begin{equation}\frac{c_{0}}{\log{c_{0}}}\geq\frac{16rc_{4}}{d}=\frac{384rq^{d}c(\Phi)}{d}.\label{eq:100}\end{equation}To prove (\ref{eq:100}) we consider two cases. Suppose first that $r\leq \frac{1}{384}$. As the function $\frac{X}{\log{X}}$ increases for $X\geq e$, hence in particular for $X\geq 2q^{d}c(\Phi)^{2}$, we use the fact that:\[c_{0}\geq 2q^{d}c(\Phi)^{2};\]so that (\ref{eq:100}) is implied by the following one:\[\frac{2q^{d}c(\Phi)^{2}}{d+2\log{c(\Phi)}+\log{2}}\geq \frac{384rq^{d}c(\Phi)}{d}.\]This inequality is satisfied because $r\leq \frac{1}{384}$ and $\frac{2c(\Phi)}{d+2\log{c(\Phi)}+\log{2}}\geq \frac{1}{d}$. Let us now examine the case where:\[r>\frac{1}{384}.\]Since $c_{0}\geq 35000dq^{d}c(\Phi)^{2}$ and $r\leq 1$, there exists a real number $X_{0}\geq 91d$ such that:\[c_{0}=X_{0}384rq^{d}c(\Phi)^{2}.\]We now claim that:\[\frac{X_{0}c(\Phi)}{\log{X_{0}}+\log(384r)+d+2\log{c(\Phi)}}\geq \frac{1}{d},\]for all $d\geq 1$. Indeed, as $q\geq 2$ and $\log(384r)\leq \log_{2}{384}< 9$, this inequality follows by:\[\frac{X_{0}c(\Phi)}{\log{X_{0}}+9+d+2\log{c(\Phi)}}\geq \frac{1}{d},\]and such a fact is true for all $X_{0}\geq 91$, so in particular for $X_{0}\geq 91d$ as well, for $d\geq 1$. Thus:\[\frac{c_{0}}{\log{c_{0}}}\geq \frac{X_{0}384rq^{d}c(\Phi)^{2}}{\log{X_{0}}+9+d+2\log{c(\Phi)}}\geq \frac{384rq^{d}c(\Phi)}{d},\]which gives (\ref{eq:100}). This completes the proof of the inequality (\ref{eq:25}).
By (\ref{eq:23}) and (\ref{eq:27}) we have that condition (\ref{eq:26}) is a consequence of the following:\[\frac{\alpha}{r}c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\geq 4c_{4}\frac{\alpha}{d}c_{0}\frac{D}{(\log\log{D})^{2}}p^{e}(2\log{c_{0}}+4\log{D})\]and, therefore, of the following one: \begin{equation} c_{0}^{2}\log{D}\geq \frac{8rc_{4}}{d}(2\log{c_{0}}+4\log{D}). \label{eq:38} \end{equation} As (\ref{eq:38}) is implied by (\ref{eq:29}), we thus also have (\ref{eq:26}).\\ Lastly, (\ref{eq:101}) follows precisely by Hypothesis 1, which yields the contradiction we need to prove Proposition 5. Indeed, by Hypothesis 1 we have:\[\widehat{h}(x)<C_{0}\frac{(\log\log{D})^{2+\frac{d\alpha}{r}}}{D(p^{e})^{1+\frac{2d}{r}\alpha}(\log{D})^{1+\frac{3d}{r}\alpha}}\leq \frac{\alpha(\log\log{D})^{2+\frac{d}{r}\alpha}}{r384q^{d}c(\Phi)c_{0}^{\frac{4d}{r}\alpha+1}D(p^{e})^{1+\frac{2d}{r}\alpha}(\log{D})^{1+\frac{3d}{r}\alpha}}.\]By (\ref{eq:27}) and (\ref{eq:17}) one can now easily check that:\[4c_{4}L^{2}q^{d\deg_{T}(l)}\widehat{h}(x)<c_{0}^{3}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e}\frac{\alpha}{r},\]which by (\ref{eq:23}) yields (\ref{eq:101}). We have therefore proved that inequality (\ref{eq:20}) cannot hold. This contradiction completes the proof of Proposition 5.
$\square$ \subsection{Step 3 - Counting zeroes of $G_{N}(X)$} We have shown in subsection 3.2 that assuming Hypothesis 1 yields $G_{N}(\Phi(l)(x))=0$ for each $l$ which satisfies the RV property and $\deg_{T}(l)$ chosen as in (\ref{eq:17}). Now, by Galois Theory we know that for each of such $l$ all the conjugates of $\Phi(l)(x)$ over $k(\Phi)$ are also zeroes of $G_{N}(X)$ with the same multiplicity of $\Phi(l)(x)$. Using Lemma 4 we can now compute the number of zeroes of $G_{N}(X)$ with their multiplicity. As we assume the Drinfeld module to be RV($r$) the polynomial $G_{N}(X)$ turns out to have at least\[(\frac{q^{r\deg_{T}(l)}}{2r\deg_{T}(l)}-\frac{\log{D'_{sep.}}}{\log{2}})D'_{sep.}\]zeroes, with multiplicity at least $hp^{e}$, where $\deg_{T}(l)$ is defined as in (\ref{eq:17}). As $\log{D'_{sep.}}\leq \log{D_{sep.}}$ and $D=[k(x):k]\leq [k(\Phi)(x):k(\Phi)][k(\Phi):k]=D'c(\Phi)$, it follows that the sum of multiplicities of the roots of $G_{N}(X)$ is at least\[(\frac{q^{r\deg_{T}(l)}}{2r\deg_{T}(l)}-\frac{\log{D_{sep.}}}{\log{2}})\frac{D}{c(\Phi)}h.\]Knowing that:\[\deg_{X}(G_{N}(X))\leq 2(L-1)q^{d\deg_{T}(N)}< 2q^{d}L^{2}\]we will now prove that Hypothesis 1 is false by showing that: \begin{equation} (\frac{q^{r\deg_{T}(l)}}{2r\deg_{T}(l)}-\frac{\log{D_{sep.}}}{\log{2}})\frac{D}{c(\Phi)}h\geq 2q^{d}L^{2}> \deg_{X}(G_{N}(X)). \label{eq:40} \end{equation} Indeed, (\ref{eq:40}) would prove that the sum of multiplicities of the roots of $G_{N}(X)$ exceeds the degree, so $G_{N}(X)$ has to be identically $0$, which would contradict Proposition 4. \begin{prop} As $c_{0}\geq 2q$, we have that: \begin{equation} \frac{q^{r\deg_{T}(l)}}{2r\deg_{T}(l)}\geq 2\frac{\log{D_{sep.}}}{\log{2}}. \label{eq:41} \end{equation} \end{prop} \begin{proof} As (\ref{eq:17}) implies that:\[\alpha\log\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)\geq r\deg_{T}(l)\geq r\alpha\left(\frac{1}{r}\log\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)-1\right)\]it follows that:\[q^{r\deg_{T}(l)}\geq \frac{\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)^{\alpha}}{q^{r\alpha}}.\]Therefore: \begin{equation} \frac{q^{r\deg_{T}(l)}}{2r\deg_{T}(l)}\geq \frac{\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)^{\alpha}}{q^{r\alpha}2\alpha(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}}-\log\log\log{D})}. \label{eq:42} \end{equation} Condition (\ref{eq:41}) will thus be a consequence of the following one:\[\frac{\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)^{\alpha}}{q^{r\alpha}2\alpha(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}})}\geq 2\frac{\log{D_{sep.}}}{\log{2}}.\]We thus have to show that:\[c_{0}^{4\alpha}(\log{D})^{3\alpha-1}p^{2\alpha e}\geq \frac{q^{r\alpha}4\alpha}{\log{2}}(\log\log{D})^{\alpha}(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}}).\]Such an inequality is a consequence of the following three conditions: \begin{equation} c_{0}^{4\alpha}(\log{D})^{3\alpha-1}p^{2\alpha e}\geq \frac{48q^{r\alpha}\alpha}{\log{2}}(\log\log{D})^{\alpha}\log{c_{0}}, \label{eq:43} \end{equation} \begin{equation} c_{0}^{4\alpha}(\log{D})^{3\alpha-1}p^{2\alpha e} \geq \frac{36q^{r\alpha}\alpha}{\log{2}}(\log\log{D})^{\alpha+1}; \label{eq:44} \end{equation} \begin{equation} c_{0}^{4\alpha}(\log{D})^{3\alpha-1}p^{2\alpha e} \geq \frac{24q^{r\alpha}\alpha}{\log{2}}(\log\log{D})^{\alpha}\log{p^{e}}. \label{eq:45} \end{equation} As we are assuming that $D\geq q^{q+d+1}$, (\ref{eq:43}) is satisfied if: \begin{equation} \frac{c_{0}^{4\alpha}}{\log{c_{0}}}\geq \frac{48q^{r\alpha}\alpha}{\log{2}}. \label{eq:46} \end{equation} Now, taking $c_{0}$ as in Theorem 2 and, therefore, such that:\[c_{0}\geq 2q,\]we see that (\ref{eq:46}) is satisfied. Indeed, we have that:\[\frac{2^{4\alpha}q^{3}}{\log{2}+1}\geq \frac{48\alpha}{\log{2}}\]and:\[q^{4\alpha-3}\geq q^{r\alpha}\]which is always true for each $\alpha \geq 1$ and $r\leq 1$.\\\\ Now, (\ref{eq:44}) follows from this inequality:\[c_{0}^{4\alpha}p^{\alpha e}\geq \frac{36q^{r\alpha}\alpha}{\log{2}}\]which is satisfied by (\ref{eq:46}). (\ref{eq:45}) directly follows from (\ref{eq:46}).
We thus proved (\ref{eq:41}). \end{proof}
\textbf{Proof of Theorem 2}\\\\ Recall that:\[c_{0}=35000 d\alpha^{3}q^{d+r\alpha}.\]If we call:\[A:=35000\]we see that the following three inequalities hold:\[\frac{A}{\log{A}}\geq 2304\]\[\frac{Ad}{d+\log{d}}\geq 2304\]\[\frac{A\alpha}{3\log{\alpha}+\alpha}\geq 2304.\]As:\[c_{0}=Ad\alpha^{3}q^{d+r\alpha},\]we have by the three inequalities above that: \[\frac{c_{0}}{\log{c_{0}}}\geq \left(\frac{\log{A}+\log{d}+d+3\log{\alpha}+\alpha}{Ad\alpha^{3}q^{d+r\alpha}}\right)^{-1}\]\[\geq \left(\frac{1}{2304d\alpha^{3}q^{d+r\alpha}}+\frac{1}{2304\alpha^{3}q^{d+r\alpha}}+\frac{1}{2304d\alpha^{2}q^{d+r\alpha}}\right)^{-1}\geq \frac{2304\alpha^{2}q^{d+r\alpha}}{3}=768\alpha^{2}q^{d+r\alpha}.\]We now see, by calling:\[X:=\log{D}\]and remembering that $D\geq q^{q+d+1}$ that:\[\frac{768q^{d+r\alpha}\alpha^{2}(\log{X})^{\alpha-2}\log{c_{0}}}{c_{0}^{4\alpha-3}X^{3\alpha-2}}\leq 1,\]\[\frac{384q^{d+r\alpha}\alpha^{2}(\log{X})^{\alpha-2}}{c_{0}^{4\alpha-3}X^{3(\alpha-1)}}\leq 1,\]\[\frac{576q^{d+r\alpha}\alpha^{2}(\log{X})^{\alpha-1}}{c_{0}^{4\alpha-3}X^{3\alpha-2}}\leq 1.\] Hence we have: \[\frac{64q^{d+r\alpha}\alpha^{2}((4\log{c_{0}}+2\log{p^{e}})(\log{X})^{\alpha-2}+3(\log{X})^{\alpha-1})}{c_{0}^{4\alpha-3}X^{3\alpha-2}p^{2(\alpha-1)e}}\leq 1.\]Therefore, since $c(\Phi)\leq \alpha$:\[\frac{c_{0}^{4\alpha-3}(\log{D})^{3\alpha-2}p^{2(\alpha-1)e}}{64q^{r\alpha+d}\alpha(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}})c(\Phi)(\log\log{D})^{\alpha-2}}\geq 1.\]By (\ref{eq:42}) and by the fact that:\[h\geq \frac{1}{2}c_{0}\frac{D}{(\log\log{D})^{2}},\]and:\[L\leq 2c_{0}^{2}\frac{D\log{D}}{(\log\log{D})^{2}}p^{e},\](which are a consequence of the hypotheses $D\geq q^{q+d+1}$ and $c_{0}\geq 1$) we have that:\[\frac{q^{r\deg_{T}(l)}}{4r\deg_{T}(l)}\frac{D}{c(\Phi)}h\]\[\geq \frac{\left(c_{0}^{4}\frac{(\log{D})^{3}p^{2e}}{\log\log{D}}\right)^{\alpha}}{q^{r\alpha}4\alpha(4\log{c_{0}}+3\log\log{D}+2\log{p^{e}}-\log\log\log{D})}\frac{D}{c(\Phi)}\frac{1}{2}c_{0}\frac{D}{(\log\log{D})^{2}}\]\[\geq 2q^{d}4c_{0}^{4}\frac{D^{2}(\log{D})^{2}}{(\log\log{D})^{4}}p^{2e}\geq 2q^{d}L^{2}.\]
By Proposition 9 it is now easy to see that (\ref{eq:40}) immediately follows. As we have seen, this is a contradiction. This means that Hypothesis 1 is false, hence the first part of Theorem 2 is proved under the RV($r$) hypothesis.\\\\
We now prove the second statement of Theorem 2, involving the RV($r$)$^{*}$ hypothesis.\\\\ We thus assume that the condition RV($r$)$^{*}$ is satisfied by the Drinfeld module $\mathbb{D}=(\mathbb{G}_{a}, \Phi)$. In our new situation, the condition RV($r$)$^{*}$ is not anymore sufficient to ensure the existence of all elements of $P_{\deg_{T}(l)}(A)$ with the desired degree in $T$. We will have a sufficiently large number of elements of $P_{\deg_{T}(l)}(A)$ only assuming a value of $\deg_{T}(l)$ large enough. We therefore modify the previous proof as follows. Let $N(\Phi)$ be an integer satisfying the conditions of Definition 3. We choose a positive integer $D_{\Phi}$ with $D_{\Phi}>q^{q+d+1}$ such that, for all $D\geq D_{\Phi}$, we have:\[h(\Phi)c(\Phi)\left[\frac{1}{r}\log\left(c_{0}^{4}\frac{(\log{D})^{3}}{\log\log{D}}\right)\right]\geq N(\Phi).\]We have now two cases. If $D\geq D_{\Phi}$, then we repeat exactly the same proof as before (with the same choice for the parameters $L$, $t$, $h$, $\deg_{T}(l)$) and we obtain the bound of Theorem 2 with $C=C_{0}$. If now $D<D_{\Phi}$, then we obtain, by Lemma 2, the lower bound of the theorem with:\[C=\min\{q^{-5d(2(d+1)h(\Phi)+1)((D_{\Phi}-1)c(\Phi))^{2}},\frac{h(\Phi)}{384rq^{d}c_{0}^{\frac{4h(\Phi)c(\Phi) d}{r}+1}}\}\leq C_{0}.\]In both cases, we thus get the estimate of Theorem 2.
\subsection{Separable case} A little improvement of the value of $c_{0}$ may be obtained if we restrict to the hypothesis that $x$ is \textbf{separable}. More precisely we have the following statement. \begin{thm} Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module defined over $\overline{k}$ satisfying the hypothesis $RV(r)$ or $RV(r)^{*}$. Let:\[c_{0}:=6500dh(\Phi)^{3}c(\Phi)^{3}q^{d+rh(\Phi)c(\Phi)}\]and:\[C_{0}:=\min\{q^{-5d(2(d+1)h(\Phi)+1)((q^{q+d+1}-1)c(\Phi))^{2}},\frac{h(\Phi)}{768rq^{d}c_{0}^{1+\frac{4d}{r}h(\Phi)c(\Phi)}}\}.\]Then, there exists $C>0$ such that for all $x\in\mathbb{D}(\overline{k})_{NT}$ separable with degree $D$ over $k$, one has:\[\widehat{h}_{\mathbb{D}}(x)\geq C\frac{(\log\log_{+}{D})^{2+\frac{d}{r}h(\Phi)c(\Phi)}}{D(\log_{+}{D})^{1+\frac{2d}{r}h(\Phi)c(\Phi)}}\]where:\[C=C_{0}\texttt{ under the hypothesis }RV(r)\]while\[0<C\leq C_{0}\texttt{ under the hypothesis }RV(r)^{*}.\]
\end{thm} \begin{proof} The proof repeats exactly the same steps as in the inseparable case, just assuming $D_{p.i.}=1$. We send the reader to \cite{D} for the explicit passages. \end{proof} This result contains L. Denis' result (see Theorem 1) about Carlitz modules: \begin{cor} Under the same hypotheses of Theorem 3, for $\mathbb{D}$ taken as the Carlitz module (which is RV($1$)) one finds the estimate of L. Denis (Theorem 1). \end{cor}
\section{Appendix: Drinfeld modules and supersingular reduction primes} In this section we concretely produce examples of Drinfeld modules satisfying RV($r, c_{1}$)$^{*}$ properties, showing in particular (see Theorem 6) that all CM Drinfeld modules with coefficients in $k$ and having rank $1$ or a prime number different from the field characteristic essentially belong to one of these classes. Chantal David already showed remarkably (see \cite{C. David} Theorem 1.2) that "in average" a rank 2 Drinfeld module with coefficients in $k$ satisfies the RV($r,c_{q}$)$^{*}$ condition, with $r=1/d=1/2$ for $c_{q}>0$ a constant depending only on $q$.
\begin{de}
Let $r\in]0,1]$, $c_{1}\in \mathbb{R}_{>0}$ and $\eta\in \mathbb{N}\setminus\{0\}$ and let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module. We say that $\mathbb{D}$ is \textbf{RV$_{\eta}$($r,c_{1}$)$^{*}$} if there exists a positive integer $N(\Phi)$ only depending on the choice of $\mathbb{D}$, such that for each $N\in \mathbb{N}$ such that $N\geq N(\Phi)$ and $N\equiv 1$ mod ($\eta$), we have:\[|\{l\in P_{N}(A), l\texttt{ is }RV\}|\geq c_{1}\frac{q^{rN}}{N}.\] \end{de} As it is easy to see, a Drinfeld module RV($r,c_{1}$)$^{*}$ is also RV$_{\eta}$($r,c_{1}$)$^{*}$ for $\eta>1$ and the two classes coincide when $\eta=1$.\\\\% We thus propose a proof of the fact that a CM Drinfeld module with rank $d$ where $d$ is a prime number, always satisfies the RV$_{\eta}$($1,1/2d$)$^{*}$ condition, for a suitable $\eta$ which will be defined later.\\\\
Given a Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$, we call a monic irreducible element $p(T)\in$ $S(A)$ which satisfies the RV condition with respect to $\mathbb{D}$ a \textbf{supersingular reduction prime} of $\Phi$. Note that our definition of supersingular prime is stronger than the one which is commonly used, only requiring supersingular reduction of the chosen Drinfeld module at $p(T)$, while the RV property also claims that all primes over $p(T)$ in the field of coefficients have inertia degree 1 on $p(T)$. For this reason we will focus only on Drinfeld modules defined over $k$, so that in such a setting our special notion of supersingular reduction prime clearly coincides with the common one. Thus, there will be no more need to make a distinction between the two definitions.\\\\
Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module of any \textbf{characteristic}\footnote{See \cite{Goss}, Definition 4.4.1.}, defined over $k$. We set:\[End_{k}(\Phi):=\{P(\tau)\in k\{\tau\}, \Phi(a)P=P\Phi(a),\forall a\in A\}.\]This is an $A-$module with respect to the action of $\Phi$ and a subring of $k\{\tau\}$ as well. One can see that this is a free $A-$module. \begin{lem} Let $\mathbb{D}$ be a Drinfeld module of characteristic 0, defined over $k$, of rank $d$. The rank of the $A-$module $End_{k}(\Phi)$ divides $d$. \end{lem} \begin{proof} See \cite{D}, Lemma 1.4.3. \end{proof} \begin{de} A Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ defined over $k$ is called \textbf{CM} or \textbf{with complex multiplication} if the rank of $End_{k}(\Phi)$ as an $A-$module is $d$. \end{de} We remark that every Drinfeld module with rank 1 has complex multiplication.
\subsection{Extending $\Phi$ to $End_{k}(\Phi)$} Let $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module with rank $d$ and characteristic 0, defined over $k$. Reduction of $\Phi(T)$ modulo $p(T)$ gives for every $p(T)\in$ $S(A)$, except possibly finitely many, a \textbf{reduced} Drinfeld module $\mathbb{D}_{p}:=(\mathbb{G}_{a},\Phi^{v_{p}})$, where, by calling $s:=\deg_{T}(p(T))$:\[\Phi^{v_{p}}:A\to \mathbb{F}_{q^{s}}\{\tau\}\]is the $\mathbb{F}_{q}-$algebra homomorphism defined by the association:\[T\mapsto (\Phi(T))^{v_{p}},\]where $(\Phi(T))^{v_{p}}$ is the reduction modulo $p(T)$ of the twisted polynomial $\Phi(T)$ and has still degree $d$ (see \cite{Goss}, Definition 4.10.1).
One can moreover see (\cite{Goss}, chapter 4) that there exists an injective ring homomorphism:\[\Phi^{v_{p}}:A\hookrightarrow End_{\mathbb{F}_{q^{s}}}(\Phi^{v_{p}}).\] It can be extended in the following way:\[\widetilde{\Phi^{v_{p}}}:End_{k}(\Phi)\hookrightarrow End_{\mathbb{F}_{q^{s}}}(\Phi^{v_{p}})\]\[P(\tau)\mapsto P(\tau)^{v_{p}}\]where $P(\tau)^{v_{p}}$ is obtained by reducing modulo $p(T)$ the coefficients of $P(\tau)$. We know (see \cite{Goss}, Proposition 4.7.13) that any isogeny between two Drinfeld modules divides an element of $A\setminus\{0\}$. We thus tensorize over $A$ with $k$ the category of Drinfeld modules and isogenies. One therefore extends in a natural fashion the algebra homomorphism $\Phi^{v_{p}}$ to $k$. This provides a field embedding $End_{k}(\Phi)\otimes_{A}k\hookrightarrow End_{\mathbb{F}_{q^{s}}}(\Phi^{v_{p}})\otimes_{A}k$. We now call:\[D_{p}:=End_{\mathbb{F}_{q^{s}}}(\Phi^{v_{p}})\otimes_{A}k\]and:\[E_{p}:=k(\tau^{s})\subset D_{p}.\]
\subsection{Counting supersingular primes} We state now a Theorem which provides a criterion to describe the supersingular primes of $\Phi$. See \cite{Goss}, Proposition 4.12.17 for the complete statement. \begin{thm} Let $\mathbb{D}_{p}=(\mathbb{G}_{a},\Phi^{v_{p}})$ be the rank $d$ Drinfeld module obtained by reducing modulo $p(T)$ the characteristic 0 one $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ defined over $k$. We then have the following equivalences: \begin{enumerate}
\item $p(T)$ is a supersingular reduction prime of $\mathbb{D}$.
\item
There is only one place in $E_{p}$ dividing $p(T)$. \end{enumerate} \end{thm} \begin{proof} See \cite{Goss}, Proposition 4.12.17. \end{proof} We now apply to this description the \textbf{Chebotarev Effective Density Theorem} for function fields (see \cite{Fried-Jarden}, Proposition 6.4.8). We send the reader to such a reference for all details.\\\\ Given $L$ a finite and Galois extension of $k$ we call $G(L/k)$ its Galois group, and for every $p(T)\in S(A)$ which is unramified in $L$ we call $\left(\frac{L/k}{p}\right)$ the corresponding \textbf{Artin symbol}. We recall (see the Introduction) that $P_{N}(A)$ is the set of all monic and irreducible $p(T)\in A$ such that $\deg_{T}(p(T))=N$, for any given $N\in \mathbb{N}\setminus\{0\}$. For any such an $N$, given a conjugacy class $\mathscr{C}$ in $G(L/k)$ we define:\[C_{N}(L/k, \mathscr{C}):=\{p(T)\in P_{N}(A),\textsl{ }p(T)\textsl{ unramified in }L/k,\textsl{ }\left(\frac{L/k}{p}\right)=\mathscr{C}\}.\] \begin{thm} Let $L$ be a finite Galois extension of $k$. Let $\mathbb{F}_{q^{\eta}}$ be the algebraic closure of $\mathbb{F}_{q}$ in $L$, and let $\mu:=[L:k\mathbb{F}_{q^{\eta}}]$. Let $\mathscr{C}$ be a conjugacy class in the Galois group $G(L/k)$. Let $a\in \mathbb{N}$
be such that:\[\sigma|_{\mathbb{F}_{q^{\eta}}}=\tau^{a}|_{\mathbb{F}_{q^{\eta}}}\]for each $\sigma\in \mathscr{C}$. \begin{enumerate}
\item If $N\not\equiv a\texttt{ } (\eta)$, then $C_{N}(L/k, \mathscr{C})=\emptyset$.
\item If $N\equiv a\texttt{ } (\eta)$, then $|C_{N}(L/k, \mathscr{C})|\thicksim_{N\to +\infty} \frac{|\mathscr{C}|q^{N}}{N\mu}$. \end{enumerate} \end{thm} \begin{thm} If a Drinfeld module $\mathbb{D}=(\mathbb{G}_{a},\Phi$) with characteristic $0$ and coefficients in $k$ has rank $d=1$ or a prime number different from the field characteristic, and if it is CM, then it is either RV($1,1/2d$)$^{*}$ or RV$_{d}$($1,1/2$)$^{*}$. \end{thm} \begin{proof} Let us call $L:=End_{k}(\Phi)\otimes_{A}k$. It is not hard to prove that $L$ is actually a field (see for example \cite{D}, Proposition 1.4.14). Moreover, $L/k$ is normal: given $\sigma\in Aut(\overline{k}/k)$ and $P(\tau)\in End_{k}(\Phi)$, for each $a\in A$ we have $\sigma(\Phi_{a}P(\tau))=\sigma(P(\tau)\Phi_{a})=\sigma(P(\tau))\Phi_{a}=\Phi_{a}\sigma(P(\tau))$. Lastly, as $[L:k]=d$ is a prime number different from the field characteristic, it follows immediately that the field extension $L/k$ has to be separable. Hence $L/k$ is a Galois field extension and Theorem 5 applies to it.\\ We know that every maximal field in $D_{p}$, for each $p(T)\in$ $S(A)$, always contains $E_{p}$ (by \cite{Goss}, Theorem 4.12.7, $D_{p}$ is central over $E_{p}$), and that it has degree at most $d$ over $k$. As $\mathbb{D}$ is CM, the field extension $L/k$ has degree $d$ and coincides, embedded in $D_{p}$ via $\widetilde{\Phi^{v_{p}}}$, with a maximal field of $D_{p}$. For each $p(T)\in$ $S(A)$, $L$ will thus always contain $E_{p}$.
This implies that the primes of $A$ which remain inert in $L$ also remain inert in $E_{p}$. In order to obtain a lower bound of the supersingular primes of $\Phi$ it is therefore sufficient, by Theorem 4, to focus only on the primes of $A$ which remain inert in $L$.
Now, all the primes of $A$ which are inert in $L$ are exactly those whose decomposition group (cyclic because they do not ramify in $L$) is $G(L/k)=\langle \sigma\rangle$, by calling $\sigma$ the generator of such a group. Note that our hypothesis that $d$ is a prime number is crucial. In case the cardinality $d$ of $G(L/k)$ were not a prime number, this group might not be cyclic, which would make empty the set of primes of $A$ which remain inert in $L$. Hence, the primes we are looking for are precisely all $p(T)\in$ $S(A)$ such that $\left(\frac{L/k}{p}\right)=\{\sigma\}$. Following the same notations as in Theorem 5 we have:\[\xymatrix{k=\mathbb{F}_{q}(T)\ar@{-}[r]^{\eta}& \mathbb{F}_{q^{\eta}}(T)\ar@{-}[r]^{\mu}&L\simeq \mathbb{F}_{q^{\eta}}(T^{1/\mu})}\]with $L/k$ finite and Galois cyclic extension of degree $d=\eta\mu$, which implies that $G(L/\mathbb{F}_{q^{\eta}}(T))\simeq \mathbb{Z}/\mu\mathbb{Z}$. Clearly, as we are assuming that $d$ is a prime number, it follows that either $\eta=1$ either $\eta=d$. As such extensions are cyclic, the restriction of $\sigma$ to $\mathbb{F}_{q^{\eta}}$ is the generator of the cyclic subgroup $G(L/\mathbb{F}_{q^{\eta}}(T))\simeq \mathbb{Z}/\mu \mathbb{Z}$. The integer $a$ such that res$_{\mathbb{F}_{q^{\eta}}}\tau^{a}=$res$_{\mathbb{F}_{q^{\eta}}}\sigma$ is therefore always $1$.
If $\eta=1$:\[L\simeq \mathbb{F}_{q}(T^{1/d})\Longrightarrow\texttt{ }|C_{N}(L/k, \{\sigma\})|\thicksim_{N\to +\infty} \frac{q^{N}}{dN}.\]Hence $\mathbb{D}$ is RV($1,1/2d$)$^{*}$ in this case. If on the other hand $\eta=d$ (which means that $\mu=1$ and $End_{k}(\Phi)=\mathbb{F}_{q^{d}}[T]$), we will have that:\[N\not\equiv 1\texttt{ }(d)\Longrightarrow C_{N}(L/k, \{\sigma\})=\emptyset\]\[N\equiv 1\texttt{ }(d)\Longrightarrow |C_{N}(L/k, \{\sigma\})|\thicksim_{N\to +\infty} \frac{q^{N}}{N}.\]This shows that $\mathbb{D}$ is RV$_{d}$($1,1/2$)$^{*}$ in this case.
\end{proof} \textsl{Remark}: Let $\mathbb{D}$ be a Drinfeld module satisfying the same hypotheses as in Theorem 6. We have seen in the proof that for $N$ sufficiently large with $N\equiv 1$ mod $(\eta)$, the number of supersingular primes $p(T)\in S(A)$ of degree $N$ is at least $\frac{q^{N}}{2N\mu}$. It easily follows from this that, for $N$ sufficiently large (without any congruence condition), the number of supersingular primes of degree $\leq N$ is at least $c_{\mathbb{D}}\frac{q^{N}}{N}$, for a certain real number $c_{\mathbb{D}}>0$ depending on $\mathbb{D}$. This gives an estimate in the same spirit as in C. David's work \cite{C. David}.
\[\]\[\]\[\]\[\]2010 Mathematics Subject Classification Codes: 11 G 09 and 11 G 50
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\begin{document}
\global\long\def\mathbb{E}{\mathbb{E}} \global\long\def\mathbb{P}{\mathbb{P}} \global\long\def\mathbb{N}{\mathbb{N}} \global\long\def\mathbb{I}{\mathbb{I}}
\title{{\normalsize\tt
\jobname.tex}\\ On convergence of 1D Markov diffusions to heavy-tailed invariant density}
\author{O.A. Manita\footnote{Moscow State University, Moscow, Russia; email: oxana.manita @ gmail.com}, A.Yu. Veretennikov\footnote{University of Leeds, UK, \& National Research University Higher School of Economics, \& Institute for Information Transmission Problems, Moscow, Russia; email: a.veretennikov @ leeds.ac.uk. For this author the work has been funded by the Russian Academic Excellence Project '5-100' (the sections 1 -- 2, the setting in the section 3 and both Lemmata) and by the Russain Science Foundation project no. 17-11-01098 (steps 1, 3, 8 -- 9 of the proof of the Theorem 1).} } \maketitle \begin{abstract} Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data. \end{abstract} Key words: 1D diffusion; invariant distribution; heavy tails; fast convergence
\noindent MSC codes: 60H10, 60J60.
\section{Introduction}
A topical area of Markov Chain Monte Carlo (MCMC) in theoretical statistics is around the following problem: given a fixed ``target'' density or distribution known up to a constant multiplier -- a normalizing constant -- how to construct a (Markov) process which would have this density as a (unique) invariant one and which would converge to this invariant one with a rate that could be theoretically evaluated? In particular, a permanent great interest in recent decades was about dealing with ``heavy-tailed'' densities with a polynomial decay at infinity. With this problem in mind, let us consider a polynomially decreasing probability density $\pi$ on the line \(\mathbb R^1\); in the precise setting it will be restricted to the half-line \(\mathbb R^1_+\). The question under consideration in this paper is constructing a Markov diffusion process with invariant measure $\pi(x)dx$ such that this measure is invariant for the constructed process and, moreover, so that an exponential convergence in total variation to the invariant distribution holds.
This problem has certain deep relations to ergodicity and to the Perron -- Frobenius theorem for Markov chains with finite state space, to spectral gap for semigroup generators, to upper and lower bounds for convergence to stationarity; yet, a spectral gap in this paper is not used. The literature in this area is huge and we only mention a few important references related to the subject of the paper more or less directly (see \cite{AitSahalia}, \cite{Cattiaux}, \cite{Fort}, \cite{Kovchegov}, \cite{kulik-leonenko}, \cite{Eva}, \cite{MenshPopov}, \cite{MenshPopov2}, et al.; also, see further references theiren).
The paper consists of four sections, the first one being this Introduction. In the section 2 two known receipts of constructing an SDE with a given stationary measure are shown: one is an SDE with a unit diffusion coefficient while another one is an SDE with an affine drift. In the section 3 we state the main result of this paper, and in the section 4 its proof is provided. The construction is based on the first one of the standard receipts from the section 2 and on a random time change. The proof uses certain recurrence type hitting time moment bounds introduced earlier in \cite{ayv_grad_drift}.
\section{Quick review: two standard receipts on $\mathbb R^1$} \subsection{Receipt 1: SDE with a unit diffusion coefficient}
Suppose a continuous and differentiable strictly positive probability density $\pi$ on $\mathbb{R}^{1}$ decreases at infinity polynomially, i.e. there exist constants $c>0$ and $m>1$ such that for any $x$, \begin{equation}
c\left(1+\left|x\right|\right)^{-m}\leq
\pi\left(x\right)\leq
c^{-1}\left(1+\left|x\right|\right)^{-m}. \label{eq:dens} \end{equation} Here $m>1$ is required so that the function $\pi$ were integrable; for further claims a bit more restrictive condition \(m>3\) will be assumed in the sequel.
On a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ let us fix a standard Wiener process $W_t$ with its natural filtration $\mathcal{F}_{t}=\mathcal{F}_{t}^W$ (as usual, $\mathbb P$ -- completed). On this probability space consider a Langevin diffusion $Y_t$ given by an SDE \begin{equation}\label{eq:lang} dY_{t}=dW_{t}+b\left(Y_{t}\right)dt, \quad Y_0=\xi, \end{equation} with an arbitrary nonrandom initial value \(\xi\), where \begin{equation}\label{eqb} b(x)=\frac{1}{2}(\ln\pi(x))'. \end{equation} If there is no explosion then this equation possesses a strong solution \cite{Ve81}. Random initial values will be mentioned briefly in the section \ref{sec22} and in principle could have been allowed here, too. It is assumed that two derivatives \(\pi'\) and $\pi''$ exist and that the drift \(b\) is locally bounded; its global boundedness is not required because, as it turns out \cite{ayv_grad_drift2, ayv_grad_drift}, a no blow-up is guaranteed just by the assumption (\ref{eq:dens}) on the function \(\pi\) only (see below the details). Emphasize that despite the assumed two derivatives, the only quantitative assumption will be just on $\pi$ itself given in (\ref{eq:dens}); also there is a hypothesis that the assumption about $\pi''$ could be dropped, and this is the reason why we refer to \cite{Ve81} instead of more standard results under a local Lipschitz condition on the drift, while talking on strong solutions in the sequel.
~
The receipt (\ref{eq:lang})--(\ref{eqb}) is, actually, a continuous time analogue of (one of) a standard MCMC receipt(s) in discrete time after a suitable limit. We do not recall it because the paper does not rely upon this limiting procedure; however, this is likely to signify a possible link to MCMC algorithms in discrete time. Obviously $\pi(x)dx$ is the (unique) invariant distribution of the process $Y_t$; this can be shown explicitly by checking the Kolmogorov equation for the invariant distribution. The process $Y_t$ is ergodic and has a polynomial rate of convergence to the stationary distribution with density $\pi$ \cite[Theorem 1]{ayv_grad_drift2} (under a bit more restricted conditions see also \cite{ayv_grad_drift}), at least, if \(m\) is not too small. Note that unlike in most of other works on convergence or mixing rates, Lyapunov functions are not used here, as they were not used in \cite{ayv_grad_drift2, ayv_grad_drift}. For close results for some particular distributions and for homogeneous Markov processes under various assumptions (usually more restrictive because of explicit assumptions about the derivative \(\pi'\)) see also \cite{Abu-Ver09, Abu-Ver09b, kulik-leonenko}; for discrete time examples -- that is, actually, about MCMC algorithms -- see, e.g., \cite{MP, Fort, MenshPopov2} and further references therein. Emphasize that the assumptions in \cite[Theorem 1]{ayv_grad_drift2} as well as in \cite[Theorem 1]{ayv_grad_drift} are all on $\pi$ and not on~$\pi'$ except that the latter derivative exists and that \(b\) is locally bounded. It remains to be our goal to avoid any assumptions on $\pi'$ beyond its existence and local boundedness of \(b\) in the sequel.
Moreover, it is known that under the {\it additional assumption} about $\pi'$, \begin{equation}\label{binfty}
\liminf_{|x| \to \infty} x b(x) =
\liminf_{|x| \to \infty} \frac{x\pi'(x)}{2\pi(x)} = -r < - 3/2, \end{equation} the {\em beta-mixing rate} of $Y_{t}$ is {\em no faster than polynomial, $\ge C t^{-k}$ with any $k> r-1/2$ and some $C>0$} (see the definition and the details in \cite{2006}). The notion of beta-mixing -- which is neither defined nor discussed here in detail -- is rather close although not identical to the convergence in total variation. Hence, and also because of close results about lower bounds for convergence rates in \cite{Klokov_lb, MenshPopov}, it is likely that convergence of $Y$ to the stationary distribution of $Y_t$ under (\ref{eq:dens}) is also no faster than some polynomial. The assumption (\ref{binfty}) will not be used in the sequel but was shown just for information. Recall that our aim is a faster convergence, and that we want to avoid any conditions on the derivative $\pi'$ except for its existence and local boundedness. \\
Note that in the case if for large \(|x|\) the density equals {\em exaclty} \(c(1+|x|)^{-m}\), it apparently follows that we need \(m>3\) in order to have the inequality \(r>3/2\) in (\ref{binfty}). Yet, we will not use conditions in terms of \(\pi'\), assuming just (\ref{eq:dens}). Also, emphasize that the requirement \(m>3\) is for the quick reference on some existing earlier results. We do not claim that for \(m\le 3\) a similar analysis and asymptotics are not possible, but just that we are not aware of such asymptotics for \(m\le 3\).
Note that for the density on a half-line $\mathbb R^1_+$ a natural analogue of (\ref{eq:lang}) is the process satisfying an SDE with a non-sticky reflection at zero, \begin{equation}\label{eq:lang2} dY_{t}=dW_{t}+b\left(Y_{t}\right)dt + d\phi^Y_t, \quad Y_0=\xi. \end{equation} For the process satisfying (\ref{eq:lang2}) similar mixing and convergence bounds follow from the bounds and from the calculus quite similar to those in \cite{ayv_grad_drift} applied to the situation of the reflected SDE, or just from a consideration of an SDE (\ref{eq:lang}) with a symmetric (\(b(-x)=b(x)\)) drift.
\subsection{Receipt 2: SDE with an affine ``mean-reverted'' drift}\label{sec22} There is another receipt different from (\ref{eq:lang}) offered in \cite{Bibby2005} (see also the references therein concerning some other earlier constructions): in a slightly simplified form it suggests to consider an SDE on the line \begin{equation}\label{sde_bibby} dZ_t = -(Z_t-\mu) \,dt + \sqrt{v(Z_t)}\,dW_t, \end{equation} with an appropriate initial distribution (e.g., stationary $\pi$ as in the reference paper), with \begin{equation}\label{bibby2} v(z) = \pi(z)^{-1} \, \int_{-\infty}^z (\mu-s)\,\pi(s)\,ds, \quad \mu = \int s\,\pi(s)\,ds. \end{equation} It is, of course, assumed that $\mu$ is finite, and then it is easily proved that $v\ge 0$, so that the SDE (\ref{sde_bibby}) is well-defined. (Indeed, $\int_{-\infty}^\mu (\mu-s)\,\pi(s)\,ds \ge 0$ since $\mu-s \ge 0$ for $s\le \mu$; and for $s>\mu$ the values $\mu-s$ are negative but the whole integral $\int_{-\infty}^{+\infty} (\mu-s)\,\pi(s)\,ds = 0$, so that for any $z<+\infty$, $v(z)\ge 0$ as required.) However, of course, ``good'' ergodic properties of the solution of this equation depend on some features of the density $\pi$. The solution locally exists due to local Lipschitz property of $\sqrt{v}$ combined with the affine drift assumption, but no-explosion should be derived from other conditions. Some related papers are, for example, \cite{Abu-Ver09, Abu-Ver09b, kulik-leonenko} which tackle particular parametric families of target densities $\pi$ -- Student, reciprocal Gamma, and Fisher-Snedekor diffusions. In all three papers a quadratic Lyapunov function allows to show an exponential convergence in total variation which is non-uniform in the initial state $Z_0$. It is interesting that in \cite{Bibby2005} an exponential character of the stationary correlation function is established; yet, convergence of a {\em non-}stationary process to a stationary regime was not studied. In fact, the process under investigation in \cite{Bibby2005} is stated to be ``ergodic'', which ergodicity is understood in the sense of being stationary without any convergence statements. At the same time, the assumptions on the density in \cite{Bibby2005} involve the stationary density \(\pi\) (in our notations) but not on its derivative (possibly with some additional non-restrictive requirements in some theorems like continuity of the target density). Recall that in the present paper $C^2$-differentiability of \(\pi\) is assumed, but convergence rate bounds only depend on the asymptotic assumptions at infinity on the density \(\pi\) itself. It looks plausible that, in principle, it may be possible to work with ``weak'' definitions of the process via Dirichlet forms theory (\cite{Fuku, MaRo}), but we prefer to have a well-defined solution trajectory; in particular, we will be working with strong solutions due to \cite{Ve81}. The receipt 2 naturally rises the question whether it is possible to arrange even a faster convergence, let theoretically.
\section{The setting \& main result} Our primary goal is a density $\pi$ on $\mathbb R^1_+ = [0,\infty)$ satisfying \begin{equation}
c\left(1+x\right)^{-m}\leq
\pi\left(x\right)\leq
c^{-1}\left(1+x\right)^{-m}, \quad x\ge 0. \label{eq:dens1} \end{equation} Receipts I \& II in the previous section can be applied to this setting if we just extend the density in a symmetric way to the whole line, with a natural normalisation. As was said in the Introduction, we aim at constructing a diffusion process on $\mathbb R^1_+$ which converges towards $\pi$ with an exponential rate uniformly with respect to the initial data. It is likely that a similar result holds true for a {\it symmetric} density $\pi$ on the whole line $\mathbb R^1$ satisfying (\ref{eq:dens}), which we mention as a remark.
In order to achieve yet a better convergence than typically guaranteed by the receipts (\ref{eq:lang})--(\ref{eqb}) or even by (\ref{sde_bibby})--(\ref{bibby2}), and for yet a more general class of densities than in \cite{Abu-Ver09, Bibby2005, kulik-leonenko} and in quite a few other works, let us consider two diffusion processes $Y_t$ and $X_{t}$ on $\mathbb R_+$ satisfying, respectively, SDEs with a non-sticky reflection at zero (\ref{eq:lang2}) and \begin{equation}\label{feq} dX_{t}=f\left(X_{t}\right)dW_{t}+f^{2}\left(X_{t}\right) b\left(X_{t}\right)dt + d\phi^X_t, \quad X_0 = \xi, \end{equation} with a local time $\phi_t^X$ at zero and with a special auxiliary function $f$, \begin{equation}
f(z):=
\displaystyle \left(1+\int_{0}^{z}\frac{dy}{\pi(y)}\right)^{1/2}, \quad z\ge 0. \label{eq:accel}\end{equation} The generator of this process is given by \[
L=f^2 L_0,
\]
where $L_0$ is the generator of the reflected diffusion \eqref{eq:lang2}: \[ L_0v(x) = \frac12 v''(x) + b(x) v'(x), \; \forall \, x>0, \quad \& \quad L_0v(0) = v'(0+). \] Recall the requirements on the non-sticky solution and on its local time: $\phi^X$ is a monotonically non-decreasing function; for any $t>0$, \begin{align*} \phi^X_t = \int_0^t 1(X_s=0)d\phi^X_s; \quad \int_0^t 1(X_s=0)ds = 0 \; \mbox{a.s.} \end{align*}
~
Of course, a question about existence of solution of this equation (\ref{feq}) on the whole half-line \(t\ge 0\) arises here, and a positive answer to this question for the first sight may look doubtful given fast increasing coefficients. However, it will be justified with the help of a random time change and of the law of large numbers that such a (strong) solution exists on the whole line and does not explode. The main result is the following Theorem.
\begin{theorem}\label{thm1} Assume that for a strictly positive probability density $\pi\in C^2$ with two locally bounded derivatives the bounds \eqref{eq:dens1} hold with some $m>3$. Then the SDE (\ref{feq}) has a strong solution $X_{t}$ for all \(t\ge 0\) which is strongly (pathwise) unique and which possesses an exponential rate of convergence to the stationary distribution $\pi\left(x\right)dx$, \begin{equation}\label{eq:crate}
\|\mu_t^{\xi} - \mu\| _{TV} \le C \exp(-\lambda t), \quad t\ge 0, \end{equation} uniformly with respect to $\xi$, with some constants $\lambda$ and $C$ which both admit certain evaluation, where $\mu^\xi_t$ is a marginal measure of the process $X_t$ that starts from $\xi$ at $t=0$, and $\mu (dx) = \pi(x)dx$ is the (unique) invariant measure of the process.
\end{theorem} The right hand side in (\ref{eq:crate}) does not depend on the initial value $\xi$. Theoretical evaluations of both constants in the bound (\ref{eq:crate}) is likely to be not very efficient, yet possible which is clearly better than pure existence of such constants.
\section{Proof of Theorem \ref{thm1}} The proof will be split into several steps.
\noindent {\bf Step 1. Random change of time.} Define the function $f(z)$ on
$\mathbb{R}^1_+$ by \eqref{eq:accel}. Obviously there exists $0<a\le 1$ (namely, any $a\in [0,c^2]$, with $c$ from (\ref{eq:dens1})) such that \begin{equation}
a\left(1+z\right)^{m+1}\leq f^{2}(z)\leq a^{-1}\left(1+z\right)^{m+1}, \quad \forall z \in \mathbb{R}_+^1. \label{eq:est_f}\end{equation} Let us define a random time change (cf. \cite{GS, McKean}) by \begin{equation}
\chi_{t}:=\int_{0}^{t}f^{-2}\left(Y_{s}\right)ds, \quad \& \quad \beta_t:= \chi^{-1}_t \quad \mbox{(the inverse function).} \label{eq:time-change} \end{equation} In other words, \[ \beta'_t = f^2(Y_{\beta_t}), \] and \[ t = \int_{0}^{\beta_t}f^{-2}\left(Y_{s}\right)ds. \] This time change $t\mapsto \beta_t$ is non-degenerate, that is, the following two conditions hold:
$\rm{(i)}$ there is no blow up at finite time: \begin{equation}
\mathbb{P}(\chi_{t}|_{t\rightarrow T-0}\rightarrow+\infty) =0\quad \forall T\in(0,+\infty). \label{eq:tc-bounds-2}\end{equation}
$\rm{(ii)}$ $\chi_t$ is unbounded as $t\rightarrow +\infty$ (i.e. when "real" time goes to infinity): \begin{equation}
\chi_{t}\geq0,\quad\chi_{t}|_{t\rightarrow\infty}\rightarrow+\infty\quad\mathbb{P}-\mbox{a.s.} \label{eq:tc-bounds-1}\end{equation}
To prove \eqref{eq:tc-bounds-2}, it suffices to notice that, due to \eqref{eq:est_f}, for any \(s<t\), \[
0\le \chi_{t}-\chi_{s}\leq a^{-1}\int_{s}^{t}\left(1+\left|Y_{r}\right|\right)^{-m-1}dr, \] hence $$
\chi_{t}^{'}\leq a^{-1}\cdot\sup_{r\in\mathbb{R}^{1}}\left(1+|r|\right)^{-m-1}=a^{-1}, \quad \mathbb{P}-\mbox{a.s.} $$ Then (\ref{eq:tc-bounds-2}) immediately follows.
~
From here we find, \[ \inf_{t\ge 0}\beta'_t \ge a>0, \quad \mathbb{P}-\mbox{a.s.}, \] and \[ \mathbb P(\limsup_{t\to\infty} \beta_t < \infty) = 0. \]
~
The assertion \eqref{eq:tc-bounds-1} follows from the following Lemma. \begin{lemma}\label{erg} Let $m>3$, and let $g$ be a bounded continuous function on $\mathbb{R}^{1}$. Assume that the diffusion process $Y_{t}$ satisfies
(\ref{eq:lang}), and let $\mu_{inv}$ be its unique invariant measure. Then for any $\delta>0$ and $\varepsilon>0$ there exists $T_0>0$ such that \[
\mathbb{P}\left(\left|\frac{1}{t}\int_{0}^{t}g\left(Y_{s}\right)ds-\int g(x)d\mu_{inv}(x)\right|
>\varepsilon\right)<\delta\qquad\mbox{for any }t\geq T_{0}. \] \end{lemma}
The Lemma with $g(r)=\left(1+\left|r\right|\right)^{-m-1}$ yields the assertion (\ref{eq:tc-bounds-1}). Indeed, let us fix any $\delta\in(0,1)$ and \(\varepsilon = a_g/2\). Naturally, \(a_g > 0\). Then with $\mathbb{P}$-probability at least $1-\delta$ one has $\chi_{t}\geq\left(a_{g}-\varepsilon\right)t=a_g \, t/2$ for {\bf any} $t$ large enough. This means that with probability at least \(1-\delta\) the change of time mapping does not stop up to at least $a_g\, t/2$. Since \(\delta\in (0,1)\) is arbitrary, \eqref{eq:tc-bounds-1} holds.
Proof of Lemma \ref{erg}. First of all, we will refer to the mixing results for SDEs on the whole line; however, in the case of symmetric coefficients ($b(-x)=b(x)$ and similarly for the diffusion if it is not a constant) such results straightforward imply similar bounds and convergence rates for (non-sticky) reflected at zero diffusions, too. In other words,
The process \((Y_t)\) is Markov ergodic with a finite variance (and, in fact, with any moment $m'<m-2$; $g$ is bounded) with a polynomial beta-mixing rate as well as convergence in total variation \(\beta^\xi(t) + |\mu_t^\xi - \mu_\infty|_{TV} \le C_k(\xi)(1+t)^{-k}\) with some \(C(\xi)\) for any \(k<m-1\), to the stationary regime $\mu_\infty$, see \cite{ayv_grad_drift, ayv_grad_drift2}. Indeed, the assumptions of \cite{ayv_grad_drift} are met with \(p=m-1\) where \(p\) is the standing parameter in \cite{ayv_grad_drift}. The assumption $m>3$ implies $k>2$. Moreover, the function $g$ is bounded; hence, the process $\int_0^t g(Y_s)\,ds$ possesses all moments (including exponential with any constant, although, this is far too much for our goal). The beta-mixing coefficient dominates the alpha-mixing, while certain convergence rate to zero of the alpha coefficient is the standing assumption in the Theorem 18.5.4 of \cite{IbrLinnik}. Hence, for the stationary regime, the assertion of the Lemma -- LLN -- follows from the Central Limit Theorem \cite[Theorem 18.5.4]{IbrLinnik}. Indeed, splitting the integral from zero to $t$ into a sum $\sum_1^{[t]}$ plus $\int_{[t]}^t$, the claim follows. For a {\it nonstationary} regime the desired LLN follows again from the CLT for the {\it stationary} case, from the Markov property, and from the polynomial convergence of $\mbox{Law }(Y_{s})$ to $\mu_{inv}$ in total variation, similarly to the proof of `` non-stationary CLT'' in \cite[Theorem 4]{Ve2} with the help of the results from \cite[Theorem 1]{ayv_grad_drift} with $m>3$. After mixing bounds have been found, see also \cite{Ve_LN} for LLN (formally, in \cite{Ve_LN} mixing is exponential, but obviously any polynomial would do such that the related sums or integrals converge).
This finishes the proof of the Lemma~\ref{erg}.
{}$\square$
See also \cite{Eva} for close results under slightly different assumptions. As was already mentioned, the statement of the Lemma will be used straight away for our reflected diffusion~(\ref{eq:lang2}).
~
\noindent {\bf Step 2. Constructing the process $X_t$.}
On the probability space $\left(\Omega,\mathcal{F}, (\mathcal{F}_{t}),\mathbb{P}\right)$ with a solution $Y_t$ to the equation (\ref{eq:lang2}), let us introduce stochastic processes \[
X_{t}:=Y_{\beta_{t}}, \quad \phi^X_t = \phi^Y_{\beta_{t}}. \] Then due to the time change \cite[Theorem 3.15.5]{GS} it follows that the process $X_{t}$ satisfies an SDE \begin{equation}
dX_{t}=f\left(X_{t}\right)\, d W_{t}+f^{2}\left(X_{t}\right) b\left(X_{t}\right)dt + d \phi^X_t, \quad X_0 = \xi, \label{eq:lang-ac} \end{equation} with a new Wiener process \(\displaystyle \tilde W_{t} = \int_0^{\beta_t} f^{-1}(X_s) \, dW_s\), and with the local time at zero \( \phi^X_t\); recall that $f(0)=1$. Indeed, outside zero the ``main part'' here $$1(X_t>0)dX_{t}=1(X_t>0)\left[f\left(X_{t}\right)\, d\tilde W_{t}+f^{2}\left(X_{t}\right) b\left(X_{t}\right)dt\right]$$ follows straightforward from \cite[Theorem 3.15.5]{GS}, and $$ 1(X_t=0)dX_{t}=1(X_t=0)d \phi^X_t $$ is a direct consequence of the equation $$ 1(Y_t=0)dY_{t}\,=\,1(Y_t=0)d\phi^Y_t. $$ Also, we have, \begin{align*} \int_0^t 1(X_s=0)ds = 0, \;\; \int_0^t 1(X_s=0)d\phi^X_s = \phi^X_t. \end{align*} Finally, \[ X_t - \xi - \int_0^t f\left(X_{s}\right)\, d W_{s}+\int_0^t f^{2}\left(X_{s}\right) b\left(X_{s}\right)ds - \phi^X_t = 0, \quad {\mbox{a.s.}} \] Thus, $X$ is the solution of the equation (\ref{eq:lang-ac}) with a non-sticky reflection, as required.
~
The equation (\ref{eq:lang-ac}) can be also derived from the time change for the SDE (\ref{eq:lang}) on the whole line with a symmetric drift and symmetrically extended $f$ after the application of It\^o--Tanaka's formula to the modulus, \begin{equation}
d\bar X_{t}=\bar f\left(\bar X_{t}\right)\, d\tilde W_{t}+\bar f^{2}\left(\bar X_{t}\right) b\left(\bar X_{t}\right)dt, \quad \bar X_0 = \xi, \label{eq:lang-ac2} \end{equation} with \[
\bar b(x) = \mbox{sign}(x)b(|x|), \;\; \bar f(x) = f(|x|), \quad \forall \, x\in \mathbb R^1. \]
By construction, the processes $X_t$ and $ \phi^X_t$ are regular, i.e. are defined for all $t\geq 0$, and adapted to the filtration \(\tilde {\cal F}_t \equiv {\cal F}_{\beta_t}\), see \cite{GS}. Recall the well-known fact that the new filtration \({\cal F}_{\beta_t}\) is well-defined because of the fact that for any \(t\), the random variable \(\beta_t\) is a stopping time.
Emphasize that the process $X_t$ is well defined on the whole half-line $t\ge 0$, it does not explode, and it neither reaches infinity from zero, nor vice versa (zero from infinity) over a finite time, all of these because of the construction via the time change.
\noindent {\bf Step 3.} The solution $X_t$ is strong. Indeed, it is well-defined on \(t\ge 0\), and the diffusion coefficient is locally continuously differentiable, and locally bounded, and locally non-degenerate, while the drift coefficient is also locally bounded. Due to the results in \cite{Ve81},
this suffices for strong uniqueness via the stopping time arguments with the help of the strong Markov property -- see \cite{Krylov_selection}. This will be used in the sequel in the coupling procedure (although, probably could be done with weak solutions, too).
\noindent {\bf Step 4. Stationary distribution for $X_t$.}
Let us prove that the process \(X_t\) has a unique invariant distribution $\pi(x)dx$. The stationary distribution $\mu$ satisfies the stationary Kolmogorov equation $L^* (\mu)=0$ on $\mathbb R_+$ -- or, equivalently, \(L^*\pi = 0\) -- where \begin{equation}\label{L} L=\frac{f^2}{2} D_x^2 +(f^2 b)D_x \end{equation} is the generator of $X_t$ and ${}^*$ is the adjoint with respect to the Lebesgue measure. First of all, the Kolmogorov equation $L^* (\mu)=0$ has at most one probability solution due to \cite[Example 4.1.1]{Bo}. Next, the measure $\mu(dx)=\pi(x) dx$ satisfies this equation. Indeed, since \begin{equation}
\frac{1}{2}\pi^{'}-\left(b\pi\right)=0, \label{eq:pl}\end{equation} we have \begin{multline}
\frac{1}{2}\left((f^{2})\pi\right)^{''}-\left((f^{2})b\pi\right)^{'}=
\frac{1}{2}\left((f^{2})^{''}\pi+2(f^{2})^{'}\pi^{'}+(f^{2})\pi^{''}\right)-
(f^{2})^{'}(b\pi)-(f^{2})(b\pi)^{'}=\\
(f^{2})\left(\frac{1}{2}\pi^{''}-
\left(b\pi\right)^{'}\right)+(f^{2})^{'}\pi^{'}+
\frac{1}{2}(f^{2})^{''}\pi-(f^{2})^{'}(b\pi)\overset{\eqref{eq:pl}}{=}
\frac{1}{2}(f^{2})^{''}\pi+\frac{1}{2}(f^{2})^{'}\pi^{'}=
\frac{1}{2}\left((f^{2})^{'}\pi\right)^{'}. \label{eq:check}\end{multline}
But \begin{equation}\label{one}
((f^{2})^{'}\pi)(x)=
\displaystyle \left(1+\int_{0}^{x}\frac{dy}{\pi(y)}\right)^{'}\cdot\pi(x)=
\frac{\pi(x)}{\pi(x)}=1, \quad x\ge 0, \end{equation} where at zero derivative is understood as right one. Hence the expression in the right hand side of \eqref{eq:check} equals zero, i.e., $\pi$ is a stationary measure for the new process \(X\). Note that the same calculus with $f$ replaced by $1$ shows the invariance of $\pi$ for $Y_t$
As may be expected, (\ref{one}) implies the equality \begin{equation}\label{e-inv}
\mathbb{E}_\pi h(X_t) = \int h(y)\pi(y)\,dy, \end{equation} for any \(t>0\) and \(h\in C_b(\mathbb{R}^1_+)\). By virtue of the Lebesgue dominated convergence theorem we can take \(h\in C^\infty_0(\mathbb{R}^1_+)\) (continuous with a compact support), but more than that, it suffices to consider functions $h\in C^\infty_0(\mathbb{R}^1_+)$ with $h'(0+)=0$: indeed, the latter class -- denoted in the sequel as $C^\infty_{00}(\mathbb{R}^1_+)$ -- is clearly dense in $C^\infty_0(\mathbb{R}^1_+)$. We have, due to $h'(0+)=0$, \begin{align*}
dh(X_t) = Lh(X_t)\,dt + h'(X_t)f(X_t)\,dW_t + 1(X_t=0)h'(0+)d\phi^X_t
\\\\ = Lh(X_t)\,dt + h'(X_t)f(X_t)\,dW_t. \end{align*} Moreover, since \(h\) has a compact support, \(Lh\) and \(h'f\) are bounded. So, by rewriting in integral form and taking expectations we get, \[
\mathbb{E}_\pi h(X_t) - \mathbb{E}_\pi h(X_0) = \mathbb{E}_\pi \int_0^t Lh(X_s)\,ds \equiv \int_0^t \mathbb{E}_\pi Lh(X_s)\,ds, \] the last equality due to Fubini's theorem. Denote by $p_s(y,z)$ the transition density of the Markov process $X$; its existence follows, e.g., from \cite[Corollary 2.9 \& Remark 2.17]{Bogach}. In a ``good case'' with all appropriate derivatives, the standard {\it formal} calculus runs as follows: \begin{align*}
\mathbb{E}_\pi Lh(X_s) = \iint (L_zh(z))\pi(y)p_s(y,z)\,dzdy
\\\\ = \iint \pi(y) h(z) L^*_zp_s(y,z)dzdy = \iint \pi(y) h(z) \partial_s p_s(y,z)dzdy
\\\\ = -\iint \pi(y) h(z) L_y p_s(y,z)dzdy = -\int h(z) \left(\int p_s(y,z) L^*_y \pi(y)dy\right)dz = 0, \end{align*} due to forward and backward Kolmogorov's equations, and Fubini's theorem. Therefore, we conclude that \begin{equation}\label{esta} \mathbb{E}_\pi h(X_t) = \mathbb{E}_\pi h(X_0), \end{equation} which is equivalent to (\ref{e-inv}). A rigorous justification without additional assumptions follows from \cite{Bogach}.
Note that uniqueness of the invariant measure will follow from the convergence bound (\ref {eq:crate}) once it is established.
\noindent {\bf Step 5. Uniform exponential moment bound.}
\noindent Let us take {\it any} $K>0$, and define \begin{equation} \gamma_{X}^{\xi}\equiv\gamma_X^{\xi,K} \equiv\gamma :=\inf\left(t\geq0:\,\, X_{t}
\leq K,
\,\, X_{0}=\xi \ge 0\right). \label{eq:moments} \end{equation} Let us show that $\mathbb{E}\exp(\alpha\gamma_{X}^{\xi}) <+\infty$
for $\alpha>0$ small enough, uniformly with respect to the initial state of the process. Denote $v_q \left(\xi\right):= \mathbb{E}_{\xi}\gamma^{q}$, with a convention $v_{0}\equiv1$.
Obviously, \[
\mathbb{E}_{\xi}e^{\alpha\gamma} =\sum_{q=0}^{+\infty}\frac{\alpha^{q}\mathbb{E}_{\xi}\gamma^{q}}{q!} = \sum_{q=0}^{+\infty}\frac{\alpha^{q}v_{q}\left(\xi\right)}{q!} < \infty, \] provided all quantities \(v_{q}(\xi)\) are finite and grow not too fast in $q$.
\noindent {\bf Step 6. Auxiliary results for polynomial moments.}
\noindent In order to guarantee that the values $v_q$, indeed, may not grow too fast, let us find alternative representations for them. By virtue of the identity \[ \left(\int_{0}^{\gamma}1dt\right)^{q}=q\int_{0}^{\gamma}\left(\int_{t}^{\gamma}1ds\right)^{q-1}dt, \] which holds both for finite and infinite $\gamma$, we get, \[ v_{q}(\xi)=q\mathbb{E}_{\xi}\int_{0}^{\gamma}v_{q-1}(X_{t})dt \] at least, for any $q\ge 1$ such that $v_{q}\left(\xi\right)$ is finite. Indeed, due to the Fubini's theorem (iii) and the Markov property (iv), \begin{multline*} v_{q}\left(\xi\right)\equiv \mathbb{E}_{\xi}\gamma^{q} = q \mathbb{E}_{\xi}\int_{0}^{\gamma} \left(\int_{t}^{\gamma}1ds \right)^{q-1}dt = q\mathbb{E}_{\xi}\int_{0}^{\infty}1 \left(\gamma > t\right) \left(\int_{t}^{\gamma}1ds\right)^{q-1}dt =
\\ \overset{(iii)}{=}q\int_{0}^{\infty}\mathbb{E}_{\xi}
1 \left(\gamma > t\right)\left(\int_{t}^{\gamma} 1ds\right)^{q-1}dt =
q\int_{0}^{\infty}\mathbb{E}_{\xi}1 \left(\gamma\geq t\right) \mathbb{E}_{\xi}\left(\left(\int_{t}^{\gamma}
1ds\right)^{q-1} | {\cal F}^X_t\right) dt=
\\ =
q\int_{0}^{\infty}\mathbb{E}_{\xi}1 \left(\gamma > t\right) \mathbb{E}_{\xi}\left((\gamma - t)^{q-1} | X_t\right) dt =q\int_{0}^{\infty}\mathbb{E}_{\xi}1 \left(\gamma > t\right)
\mathbb{E}_{X_{t}}\gamma^{q-1}dt =
\\ \overset{(iv)}{=}q\int_{0}^{\infty}\mathbb{E}_{\xi}1 \left(\gamma > t\right)v_{q-1}(X_{t})dt\overset{(iii)}{=} q\mathbb{E}_{\xi}\int_{0}^{\gamma}v_{q-1}(X_{t})dt. \end{multline*} Hence, if both quantities are finite, then \[ v_{q}\left(\xi\right)=q\mathbb{E}_{\xi}\int_{0}^{\gamma}v_{q-1}(X_{t})dt,\quad v_0=1. \] Note that for each $q\geq1$, if $v_q$ is finite, then it satisfies \begin{equation}
Lv_{q}(x)=-qv_{q-1}(x), \;\; x\ge K, \label{eq:pois}\end{equation} by virtue of the probabilistic representation of solutions of the elliptic equation with Dirichlet boundary condition, or equivalently by Duhamel's formula. Obviously $v_{q}(K)=0$, as well as $v_{q}(x)=0, \, 0\le x\le K$. Also, it is known that if $v_q(\xi)<\infty$ for some $\xi$ then it is finite for any $\xi$. However, we are not going to use this equation directly since it lacks the ``second boundary condition'' normally required for the second order differential equation. Instead, we will find solutions to boundary problems that approximate $v_q$. In fact, what we shall need instead is the following Lemma.
Let $\hat L:\,(\hat Lu)(x)=a(x)u^{''}(x)+c(x)u^{'}(x)$ be the generator of the diffusion process $(\zeta_t,t\geq 0)$ with locally bounded coefficients $a>0$ (the diffusion) and $c$ (the drift), and such that $a$ is locally uniformly non-degenerate, and which process is a strong solution of a corresponding SDE. Let us fix a positive (non-negative) function $\psi$ on $\mathbb{R}^1$. Let \[
\tau_K:=\inf\left(t\geq 0:\,\,\zeta_{t}\leq K,
\,\, \zeta_{0}=\xi\right) \quad (\xi>0), \] and \[ v\left(\xi\right) =\mathbb{E}_{\xi}\int_{0}^{\tau_K} \psi(\zeta_{t})\,dt. \] \begin{lemma}\label{PDE_lemma} For any \(N>K>0\), consider the boundary problem \begin{equation}
\hat Lv^+_{N}=-\psi,\quad v^+_{N}\left(K\right)=0,\,\,\left(v^+_{N}\right)^{'}(N)=0, \label{eq:bvp-lem} \end{equation} Then the function $v^+_{N}(\xi)\uparrow v(\xi)$ as $N\uparrow\infty$, for every $\zeta_0 = \xi$ with $\xi \ge K$. \end{lemma}
\textbf{Proof of Lemma \ref{PDE_lemma}.} For any
$0\le K\le \xi\le N$, let us consider a family of stochastic processes $\zeta_{t}^{N}$, given by the SDE with reflection, \[
d\zeta_{t}^{N}=\sqrt{2a(\zeta_t ^N)}dw_{t}+c(\zeta_{t}^{N})dt +d\phi_{t}^{N}, \quad \zeta_0^N = \xi, \] with values on $[0,N]$, with a non-sticky reflection at $N$ and an absorbtion at zero, where $\phi_{t}^{N}$ is its local time at $N$. Applying It\^o's formula (or, in fact, more precisely It\^o--Krylov's formula if continuity of $\psi$, $a$, and $c$ is not assumed) to $v_{N}(\zeta_{t}^{N})$, we get the following representations: \[
v_{N}(x)=\mathbb{E}_{x}\int_{0}^{\tau_{K,N}}\psi(\zeta_{s}^{N})ds, \] where \(
\tau_{K,N} = \inf\left(t\geq0:\,\, |\zeta^N _{t}| \le K\right) \) is the moment when the process $\zeta^N _t$ first hits the interval $[0,K]$. Note that $\tau_{K,N}$ monotonically increases as $N$ increases. Also note that if $\tau_K<\infty$ then, obviously, \(\tau_{K,N} \uparrow \tau_K\); and if $\tau_K=\infty$ then still \(\tau_{K,N} \uparrow \infty =\tau_K\). These all follow from the comparison theorem for one-dimensional SDEs possessing strong solutions with the same coefficients and different initial data. This comparison theorem can be shown as follows. Consider two SDEs with the same initial value $\xi$ but with two different $N_1 < N_2$, say. Denote the corresponding solutions by $\zeta^{N_1}_t$ and $\zeta^{N_2}_t$. Assuming that $\xi\in [K,N_1]$, due to the strong uniqueness $\zeta^{N_1}_t = \zeta^{N_2}_t$ until $\hat \tau^{}:=\inf(t\ge 0: \, \zeta^{N_1}_t = K \; \mbox{or} \; N_1)$. If at this moment -- which is a stopping time -- $\zeta^{N_1}_t = \zeta^{N_2}_t = K$, then the claim is justified because $K$ is the absorbtion point. If, however, $\zeta^{N_1}_t = \zeta^{N_2}_t = N_1$, then the first process $\zeta^{N_1}_t$ will remain less than or equal to $N_1$ all the time, while the second will exceed this level $N_1$ with probability one on any right interval of $\hat \tau$. This follows easily from the ``reverse'' time change which makes diffusion back equal to one and from the Girsanov theorem about eliminating the drift via change of measure, because for the standard Wiener process this property is well-known (e.g., it follows from Khintchin's iterated logarithm law for WP \cite{ItMc} along with the strong Markov property. Thus, on any small right neighbourhood of the moment $\hat\tau$ we would have $\zeta^{N_1} \le \zeta^{N_2}$, with strict inequality at least at infinitely many moments of time arbitrarily close to $\hat \tau$. Yet, both solutions are strong Markov. So, if we now start two processes with the same generator a new at two distinct initial value $\xi_1 < \xi_2$, then due to continuity the two solutions will satisfy $1(t>\hat\tau)1(\zeta^{N_1}_t<\zeta^{N_2}_t) = 1(t>\hat\tau)$, at least, until they meet again, i.e., for all $t< \bar\tau:= \inf(s\ge \hat\tau: \zeta^{N_1}_s = \zeta^{N_2}_s)$ (here, of course, $\inf(\emptyset) = \infty$, and, in fact, they will never meet again). But then, if we assume that $\bar \tau < \infty$, they will again coincide until the next moment when they touch the level $N_1$, after which we have again $\zeta^{N_1}\le \zeta^{N_2}$, and the cycle can repeated indefinitely. This shows that $\zeta^{N_1}_t\le \zeta^{N_2}_t$ for all $t\ge 0$. Hence, we have \(\tau_{K,N} \uparrow \tau_K\), $N\uparrow \infty$, and so, the monotonic convergence Theorem yields the assertion of the Lemma, as required.
{}$\square$
\noindent Similar calculi in similar situations yielding various close claims can be found in \cite{Mao, ayv_grad_drift2, ayv_grad_drift}.
~
\noindent {\bf Step 7. Bounds for polynomial moments.}
\noindent Let us prove that \begin{equation}
v_{q}\left(\xi\right)\leq q!\cdot C^{q} \label{eq:major_est-1} \end{equation} for all $q\geq1$, with \[
C=\frac{a^{-1}}{m}\cdot A_{m},\quad A_{m}:=\int_{K}^{\infty}\left(1+w\right)^{-m}dw. \] does not depend on $q$. We argue by induction. Now our particular generator is $\hat L = L$ from~(\ref{L}).
\underbar{Base}: Let $q=1$. Fix $N>0$. Notice that $v^{0}=1$ and consider a boundary value problem, \begin{equation}
Lv_{N}^{1}=-1,\quad v_{N}^{1}\left(K\right)=0,\,\,\left(v_{N}^{1}\right)^{'}(N)=0. \label{eq:BVP-1} \end{equation} Since $L=f^{2}L_{0}$, where $L_{0}u=\frac{1}{2}u^{''}+\frac{1}{2}\nabla\ln\pi(x)u^{'}$, this problem admits a unique solution \begin{equation}
v_{N}^{1}(\xi)=2\int_{K}^{\xi}\pi^{-1}(w_{1})dw_{1}
\int_{w_{1}}^{N}\frac{\pi(w_{2})}{f^{2}\left(w_{2}\right)}dw_{2}, \label{eq:sol_pois__no_lim-1}\end{equation} and due to \eqref{eq:dens} and \eqref{eq:est_f} we estimate replacing \(N\) by infinity in the upper limit of the integral, \begin{multline*}
v_{N}^{1}(\xi)\leq2c\int_{K}^{\xi}(1+w_{1})^{m}dw_{1}\int_{w_{1}}^{\infty}c^{-1}\cdot
(1+w_{2})^{-m}\cdot a^{-1}\left(1+w_{2}\right)^{-m-1}dw_{2}\leq\\
\leq2a^{-1}\int_{K}^{\xi}(1+w_{1})^{m}dw_{1}\int_{w_{1}}^{\infty}(1+w_{2})^{-m}\left(1+w_{2}\right)^{-m-1}dw_{2}
=\\
=2a^{-1}\int_{K}^{\xi}(1+w_{1})^{m}\frac{(1+w_{1})^{-2m}}{2m}dw_{1}\leq
\frac{a^{-1}}{m}\int_{K}^{\infty}(1+w_{1})^{-m}dw_{1}=A_{m}\cdot\frac{a^{-1}}{m}=:C. \end{multline*} By virtue of Lemma \ref{PDE_lemma}, this implies $v_{1}\left(\xi\right)=\lim_{N\rightarrow\infty}v_{N}^{1}\left(\xi\right)\leq C$.
\underbar{Induction Step}: Note that if the right hand side in the equation (\ref{eq:BVP-1}) is multiplied by a constant \(R>0\), then, given the specific boundary conditions, the bound for the solution will be also multiplied by this \(R\), so that instead of the upper bound \(C\) there will be a new upper bound \(RC\).
Suppose that for some \(q\) and for $n=q-1$ we have \[
v_{n}(\xi)\leq C^{n}\, n! \] with the same constant \(C\) as above. Then, by the remark in the beginning of the induction step with \(R=C^{n}\, n! \times q \equiv C^{q-1}\,q!\), we immediately obtain \[
v_{q}(\xi)\leq C^{q-1}\, q!\times C = C^{q}\, q!, \] as required. Hence, the inequality (\ref{eq:major_est-1}) follows. Note that a similar simple argument with a reference to the induction method and without using explicitly the second barrier \(N\) can be found in \cite[Lemma 3.1]{Mao}; practically the same calculus, yet with unbounded growing in \(x\) moments was used in \cite{ayv_grad_drift}.
~
Now, take any $\alpha\in\left(0,C^{-1}\right)$. Then due to (\ref{eq:major_est-1}) one has \begin{equation} \mathbb{E}_{\xi}e^{\alpha\gamma}=\sum_{q=0}^{+\infty}\frac{\alpha^{q}\mathbb{E}_{\xi}\gamma^{q}}{q!}
\leq\sum_{q=0}^{+\infty}\alpha^{q}
C^{q}=\frac{1}{1-\alpha C}<\infty. \label{eq:exp} \end{equation} It may be argued now that the desired ``exponential coupling'' can be arranged via the exponential moment bound (\ref{eq:exp}) and due to the elliptic Harnack inequality for divergent type equations \cite[Theorem 8.20]{GT} in the way similar to \cite{ayv_grad_drift}, see the next step. Note that it is a ``common knowledge'' that the bound (\ref{eq:major_est-1}) suffices for the Theorem claim. The reader who knows the exact reference may skip the rest of the proof.
\noindent {\bf Step 8. Using Harnack inequality. } The usage of coupling method assumes that glueing or meeting of two versions of the process -- one stationary and another non-stationary -- can be arranged with a positive probability bounded away from zero on each period of this construction. Here it suffices to consider a ``symmetric'' SDE on the whole line. By ``period'' in our case any finite interval may be taken; e.g., it is convenient to use \([0,2]\) which will be split into two equal parts, \([0,1]\) and \([1,2]\) (their intersection at one single point is not important). On the first half, according to the inequality (\ref{eq:exp}) and Bienaym\'e -- Chebyshev -- Markov's inequality, both independent versions of the process will attain some (actually, any) bounded neighbourhood of zero. On the second half we want to glue them with a probability also bounded away from zero. Note that the standard 1D or finite state space idea just to wait until the two trajectories intersect here does not work straightforward as we would like it. Or, rather, it works but the bound obtained in such a way would use some bounds on the derivative \(\pi'\), which we want to avoid by all means.
There is a recent rather general tool based on regeneration period moments \cite{Zv}. Yet, to verify the mild condition (*) required for this tool is probably no easier than -- or, maybe, equivalent to -- what we suggest instead in the next paragraphs.
One more approach which does not involve any properties of \(\pi'\) uses classical inequalities for divergence form PDEs. Indeed, this step justifies that it is possible by virtue of Moser's Harnack inequality for divergent type elliptic equation (see \cite{GT}) (cf., e.g., \cite{Ve1}, \cite{Ve2} where a parabolic Harnack inequality was used for the same goal). Here it is convenient to return to an SDE on the whole line with symmetric coefficients (\ref{eq:lang-ac2}) which solution is denoted by $\bar X_t$. Its modulus satisfies the equation (\ref{eq:lang2}) with a new Wiener process.
We argue that an elliptic Harnack inequality \begin{equation}\label{ha} \mathbb E_x g(\bar X_{\bar\sigma}) \le C \mathbb E_{x'} g(\bar X_{\bar\sigma}) \end{equation}
for any non-negative function $g$ and any $|x|, |x'| \le 1$ with
$\bar \sigma := \inf\,(t\ge 0:\; |\bar X_t|\ge 2)$ follows from \cite[Theorem 8.20]{GT} due to the equation $\mbox{div} (\exp(2U(x) \nabla v(x)) = 0$, here $v(x) = \mathbb E_x g(\bar X_\sigma)$. This reasoning should be combined with the bound $\mathbb P_x(\bar\sigma>t) \le
C t^{-1}$ for \(t>0\) with some $C>0$ depending on the sup-norms of all coefficients in the ball \(B:= \{|x|\le 2\}\). The latter bound follows from \cite[Theorem 8.16]{GT} applied and from the Bienaym\'e -- Chebyshev--Markov inequality \(\mathbb P_x(\sigma>t) \le t^{-1}\,\mathbb E_x \sigma\) since the function \(v(x) := \mathbb E_x \bar \sigma\) is a solution to the equation \(\frac12\,\exp(-2U)\mbox{div}(\exp(2U) \nabla v) + 1 = 0, \; \& \; v|_{\partial B}=0\), or, equivalently, to the (Poisson) equation \[
\mbox{div}(\exp(2U) \nabla v) + 2\exp(2U) = 0, \quad v|_{\partial B}=0, \]
to which the Theorem 8.16 \cite{GT} is applicable stating that solution \(v(x)\) is bounded by a constant, say, \(N\) depending only on \(\sup_{|x|\le 2}|U(x)|\) (actually, even on some integral norm of \(\exp(2U)\)). This immediately implies that by choosing \(t\ge 3N\) we have that \(\mathbb P_x(\bar\sigma\le t) \ge \frac23\). The same estimate holds true for the process $X_t$, with the stopping time $\sigma = \inf\,(t\ge 0:\; X_t\ge 2)$, and with a non-negative function $g$ on $\mathbb R^1_+$, i.e., $$ \mathbb E_x g(X_{\sigma}) \le C \mathbb E_{x'} g(X_{\sigma}). $$
Along with (\ref{eq:exp}), this suffices for a successful exponential coupling for the process $X_t$ with its stationary version. Although it will not be used here, note that the obtained bound implies a stronger exponential inequality $\mathbb P_x(\sigma>t) \le C \exp(-\lambda t)$ with some $C,\lambda>0$ by the well-known property of homogeneous Markov processes and their exit times.
Finally, we can change the function
$U$ outside the ball $|x|\le 3$ so that it becomes bounded, -- the latter is possible without changing the process until $\sigma$.
\noindent {\bf Step 9. Exponential convergence.} Let us return to the half-line $\mathbb R_+$ and to the process $X_t$, and let us fix some $K>0$. It is known that -- modulo the conclusion of the previous step -- for the proof of the desired exponential convergence in total variation, it sufficies to show that $\mathbb{E}\exp(\alpha\gamma_{X}^{\xi})$ is finite for some $\alpha>0$, and for some -- {\it actually, for any} -- $K>0$ where $\gamma_{X}^{\xi}= \gamma_{X}^{\xi, K}$ was defined in (\ref{eq:moments}).
Let $X_{t}^{st}$ be the {\it independent} stationary version of the Markov process $X_{t}$, i.e., $X_{t}^{st}$ is the process with the same generator and initial distribution with the density \(\pi\), if necessary, on some extended probability space with another independent Wiener process. (However, we will not change our notations for $\mathbb P$ and $\mathbb E$.) Naturally, the couple $(X_{t}, X_{t}^{st})$ is considered on some extension of the original probability space. For $\xi > K$ let $\tau\equiv \tau^\xi$ be the moment of the first intersection of $X_t$ started from \(X_0=\xi\) with the stationary version $X_t ^{st}$, i.e., $$
\tau :=\inf\left(t\geq0:\,\, X_{t}=X_{t}^{st}\right). $$
As the random variable $\tau$ is a stopping time and $X_{t}$ has strong Markov property, we can define a new strong Markov process \begin{equation} \hat{X}_{t}:=X_{t}1\left(t<\tau\right) +X_{t}^{st}1\left(t\geq\tau\right), \label{eq:coupling}\end{equation} with the property \[ \mbox{Law }(\hat X_{t}^{})=\mbox{Law }\left(X_{t}\right). \] Obviously on $\{t>\tau\} \cap \{\tau < \infty\}$ the trajectories of $X_{t}^{st}$ and $\hat{X}_{t}$ ``after $\tau$'' coincide. Then for any Borel set $A$ one has (we drop the initial value $\xi$ since the final estimate is uniform in it) \begin{align*}
\left|\mathbb P\left(X_{t}\in A\right)-\mathbb P\left(X_{t}^{st}\in A\right)\right|\overset{\eqref{eq:coupling}}{=}
\left|\mathbb P\left(\hat{X}_{t}\in A\right)-\mathbb P\left(X_{t}^{st}\in A\right)\right|=
\\\\
=\left|\mathbb{E}\left(\left(1 \left(\hat X_{t}\in A\right)-1 \left(X_{t}^{st}\in A\right)\right)
\times\left(1 \left(t<\tau\right)+1 \left(t\geq\tau\right)\right)\right)\right|=
\\\\
=\left|\mathbb{E}\left(\left(1 \left(\hat X_{t}\in A\right)-1 \left(X_{t}^{st}\in A\right)\right)
\times1 \left(t<\tau\right)\right)\right|
\\\\
\leq\mathbb{E}\left|\left
(1 \left(\hat X_{t}\in A\right)-1 \left(X_{t}^{st}\in A\right)\right)\right|
\times1 \left(t<\tau\right). \end{align*}
Since $\left|\left(1 \left(\hat X_{t}\in A\right)-1 \left(X_{t}^{st}\in A\right)\right)\right|\leq1$, taking into account the exponential version of Bienaym\'e -- Chebyshev--Markov's inequality, we conclude that \[
\left|\mathbb{P}\left(X_{t}\in A\right)-\mathbb{P}\left(X_{t}^{st}\in A\right)\right|\leq \mathbb{E}1 \left(t<\tau\right)=\mathbb{P}\left(t<\tau\right)
\leq\exp\left(-\alpha t\right)\mathbb{E}\exp(\alpha\tau). \] Passing to the supremum over Borel sets $A$ we obtain due to (\ref{eq:exp}), \[
\left\Vert \mu_{t}-\mu_{st}\right\Vert _{TV}\leq2\exp\left(-\alpha t\right)\mathbb{E}\exp(\alpha\tau) \le \frac{2}{1-\alpha C}\,\exp\left(-\alpha t\right), \] where $\mu_{t}=Law\left(X_{t}\right)$ and $\mu_{st}(dx)=\pi(x)dx$. This bound does not depend on the initial value $\xi$ which was dropped in the notation $\mathbb E_\xi$. Hence, the proof of the Theorem is completed. $
\square$
~
\begin{remark}
It is likely that for the symmetric density $\pi$ on $\mathbb R^1$ and for the equation (\ref{eq:lang}) on $\mathbb R^1$ this method is applicable, too, with the function $f(y) \equiv f(|y|)$, and that it provides a similar convergence rate bound (\ref{eq:crate}) as in the Theorem \ref{thm1}. We leave it till further papers. \end{remark}
~
\begin{remark} The form of the process $X_t$ is a result of an educated guess. We were looking for the process $X_t$
whose generator $L$ would be the generator of the Langevin diffusion multiplied by a positive function $F=f^2$, which is needed for applying a random time change. At the same time, we wanted the new process $X_t$ to have the same invariant density $\pi$. From this condition the function $F$ is determined up to two constants $c_1, c_2>0$ \[
F(x)=
\displaystyle c_1 + \int_0 ^{x} \frac{c_2 dy}{\pi(y)}, \quad x\geq 0. \] It can be checked by an explicit computation that with this choice of the function \(F\), the new process $X_t$ would still have an exponential rate of convergence to the invariant measure (we choose $c_1=c_2=1$ but in fact any strictly positive $c_1$ and $c_2$ give the same result). \end{remark}
\begin{remark} Note that in the Theorem \ref{thm1} the property of continuity of the state space is important.
If the state space is discrete, the modification we consider (multiplication of the generator by a function)
typically does not affect the rate of convergence. Indeed, let us consider a birth-death process $Y_t,\,\,t\geq 0$
with
birth rates $\{\lambda_n,\, n\in \mathbb{N}_{\geq 0}\}$ and death rates
$\{\mu_n,\, n\in \mathbb{N}\}$:
\begin{equation}
\mathbb{P}\left(Y_{t+h}=m|Y_{t}=n\right)=\left\{ \begin{array}{lcl}
\lambda_{n}h+o(h), & & m=n+1\\
\mu_{n}h+o(h), & & m=n-1\\
1-\mu_{n}h-\lambda_{n}h+o(h), & & m=n\\
o(h), & & |m-n|>1\end{array}\right. \label{eq:BDP}\end{equation} The generator $A_0$ of the process $Y_t$ is given by \[
A_0 \varphi (n) = \lambda_n ( \varphi (n+1) - \varphi (n) ) + \mu_n (\varphi (n-1)-\varphi (n)). \] If we want to apply the same transformation as in the continuous case, i.e. to consider a birth-death process $X_t$ with the generator $A=f_n \cdot A_0$, then the new process $X_t$ will have the birth and death rates $\{\lambda_n ^{'}=f_n \cdot \lambda_n,\, n\in \mathbb{N}_{\geq 0}\}$ and $\{\mu_n ^{'}=f_n \cdot \mu_n,\, n\in \mathbb{N}\}$ respectively.
The invariant distribution $\pi$ of $Y_t$ for the can be computed explicitly and equals \[
\pi(\{ n\} )\equiv\pi_{n}=
\pi_{0}\frac{\lambda_{0}\dots\lambda_{n-1}}{\mu_{1}\dots\mu_{n}},
\quad\pi_{0}=\left(1+\sum_{n\geq1}
\frac{\lambda_{0}\dots\lambda_{n-1}}{\mu_{1}\dots\mu_{n}}\right)^{-1}. \] Hence \[
\frac{\lambda_{n-1}}{\mu_n} = \frac{\pi_{n-1}}{\pi_n}\quad \mbox{for each}\quad n\in\mathbb{N}. \] The assumption that $X_t$ has the same invariant distribution as $Y_t$ yields \[
\frac{\lambda_{n-1} ^{'}}{\mu_n ^{'}}=\frac{f_{n-1} \cdot \lambda_{n-1}}{f_n \cdot \mu_n}
= \frac{\pi_{n-1}}{\pi_n}\quad \mbox{for each}\quad n\in\mathbb{N}, \] hence $f_n = f_0 = \mbox{const}$ for all values of $n$. So, such a transformation is just changing the time scale by multiplying it by a positive constant which doesn't influence the rate of convergence qualitatively.
\end{remark}
\end{document} |
\begin{document}
\title[Dimension of Bergman spaces]{On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces}
\author{A.-K. Gallagher} \address{Gallagher Tool \& Instrument LLC, Redmond, WA} \email{anne.g@gallagherti.com} \author{P. Gupta} \address{Department of Mathematics, Indian Institute of Science, Bangalore} \email{purvigupta@iisc.ac.in} \author{L. Vivas} \address{Department of Mathematics, The Ohio State University, Columbus, OH} \email{vivas@math.osu.edu}
\dedicatory{Dedicated to the memory of Berit Stens{\o}nes}
\begin{abstract} Given a holomorphic vector bundle $E$ over a compact Riemann surface $M$, and an open set $D$ in $M$, we prove that the Bergman space of holomorphic sections of the restriction of $E$ to $D$ must either coincide with the space of global holomorphic sections of $E$, or be infinite dimensional. Moreover, we characterize the latter entirely in terms of potential-theoretic properties of $D$. \end{abstract}
\keywords{Bergman spaces, compact Riemann surfaces, holomorphic vector bundles} \subjclass{32A36, 32L05, 30F99} \maketitle
\section{Introduction}\label{S:Intro}
Let $(M,g)$ be a compact Riemann surface equipped with a Hermitian metric, and $(E,h)$ be a Hermitian holomorphic vector bundle over $M$. Given an open subset $D\subset M$, the Bergman space of $E$-valued holomorphic sections on $D$ is the reproducing kernel Hilbert space given by
\bes
A^2(D,g;E,h)=\left\{s\in\mathcal{O}(D;E):||s||_h:=\left(\int_D h(s,s)\,\omega_g\right)^{1/2}<\infty\right\},
\ees where $\mathcal{O}(D;E)$ is the space of holomorphic sections of $E$ over $D$, and $\omega_g$ is the volume form associated to $g$. As a vector space, $A^2(D;E)=A^2(D,g;E,h)$ is independent of the choice of $g$ and $h$. We prove the following theorem.
\begin{theorem}\label{T:main} Let $M$ be a compact Riemann surface and $E$ be a holomorphic vector bundle over $M$. Suppose $D\subset M$ is a nonempty open subset of $M$. Then the following are equivalent.
\begin{itemize} \item[$(a)$] $M\setminus D$ is nonpolar. \item[$(b)$] $\mathcal{O}(M;E)\subsetneq A^2(D;E)$. \item[$(c)$] $\dim A^2(D;E)=\infty$. \item[$(d)$] There exists a bounded subharmonic function $\psi\in\mathcal{C}^\infty(D)$ such that $i\partial\overline\partial\psi\geq \omega$ on $D$ for some volume form $\omega$ on $M$. \end{itemize} \end{theorem}
This settles the question posed by Sz{\H o}ke in \cite{Sz20} on the dimension of vector bundle-valued Bergman spaces on open subsets of a compact Riemann surface. Sz{\H o}ke's question is motivated by the well-understood case of open subsets of the complex plane. A complete analogue of Theorem ~\ref{T:main} for open subsets of $\C$ holds by the works of Carleson \cite{Ca67}, Wiegerinck \cite{Wi84}, Gallagher-Harz-Herbort \cite{GaHaHe17}, and Gallagher-Lebl-Ramachandran \cite{GaLeRa21}. Using an argument similar to Wiegerinck's, Sz{\H o}ke established the equivalence of $(b)$ and $(c)$ in Theorem \ref{T:main} for the special case of $M=\mathbb{P}^1$. A proof of the implication $(b)\Rightarrow (a)$ for a general $M$ is also given in \cite{Sz20}. The rest of the theorem for $M=\mathbb{P}^1$ is established by the authors in \cite{GaGuVi22}, using the analytic approach of \cite{GaHaHe17} and \cite{GaLeRa21}. In comparison to Sz{\H o}ke's argument, the latter approach is not as sensitive to the topological and algebraic structure of the underlying surface and bundle. Thus, it lends itself more easily to generalization to all compact Riemann surfaces.
Since $(c)$ trivially implies $(b)$, and it is proven in \cite{Sz20} that $(b)$ implies $(a)$, the proof of Theorem~\ref{T:main} reduces to proving that $(a)$ implies $(d)$, and $(d)$ implies $(c)$. For the former, the Green's function of the domain is suitably modified to obtain the desired strictly subharmonic function; see Section~\ref{SS:(a)=>(d)}. For the latter, one relies on a version of H{\"o}rmander's $L^2$-method for solving the Cauchy--Riemann equations for bundle-valued forms; see Theorem~\ref{T:HDtheorem}. This cannot be directly invoked for the given Hermitian metric $h$ on $E$, as its curvature may not be positive. We instead apply H{\"o}rmander's theorem to the twisted metric $e^{-\widetilde\psi}h$, where $\widetilde\psi$ is a nonsmooth correction of $\psi$ as in $(d)$, chosen so as to yield linearly independent holomorphic sections in the untwisted Bergman space. Note that the twisted metric is singular, but the $L^2$-methods cited here are for smooth metrics. Due to the benign nature of the singularities, this issue is resolved by using an extension trick that appears to be standard in this context; see \cite[Lemma 5.1.3]{Be10}, for instance.
\noindent{\bf Acknowledgements.} This material is partially based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in the Summer Resarch in Mathematics program hosted by the Mathematical Sciences Research Institute in Berkeley, California, in June 2022. The authors are grateful to MSRI for providing financial support and a stimulating environment. Gupta and Vivas were partially supported by an Infosys Young Investigator Award, and NSF Grant No. DMS-1800777, respectively.
\section{Background and preliminaries}\label{S:prelim}
\subsection{Holomorphic vector bundles on compact Riemann surfaces} Throughout this section, $X$ is a Riemann surface equipped with a Hermitian metric $g$. The holomorphic and antiholomorphic tangent bundles of $X$ are denoted by $T^{1,0}X$ and $T^{0,1}X$, respectively. For $p,q\in\{0,1\}$, ${T^*}^{p,q}X$ is the bundle of $(p,q)$-forms on $X$ and $\Omega^{p,q}(X)$ is the space of its smooth sections. The standard Dolbeault operators on $\Omega^{p,q}(X)$ are denoted by $\partial$ and $\overline\partial$.
The volume form associated to $g$ is denoted by $\omega_g$. Note that $g$ and $\overline g$ are Hermitian metrics on $T^{1,0}X$ and $T^{0,1}X$, respectively. The dual Hermitian metrics on ${T^*}^{1,0}X$ and ${T^*}^{0,1}X$ are denoted by $g^*_{1,0}$ and $g^*_{0,1}$, respectively. The complex bundle ${T^*}^{1,1}X$ is endowed with the metric $g^*_{1,1}$ given on ${T^*}^{1,1}_xX$ by
\bes
g^*_{1,1}\left(\sigma\wedge\tau,\sigma'\wedge\tau' \right)=g^*_{1,0}(\sigma,\sigma')g^*_{0,1}(\tau,\tau'),
\ees for $\sigma,\sigma'\in {T^*}^{1,0}_xX$ and $\tau,\tau'\in {T^*}^{0,1}_xX$, and extended to general multilinear forms by sesquilinearity. Lastly, $g^*_{0,0}$ is the metric on ${T^*}^{0,0}X\cong X\times\C$ given at each $x\in X$ by the standard Hermitian metric on $\C$. In local coordinates, if $g=\gamma\: dz\otimes d\overline z$, then $\omega_g=i\:\gamma\: dz\wedge d\overline z$, $g^*_{1,0}(adz,bdz)=\frac{a\overline b}{\gamma}$ and $g^*_{1,1}(adz\wedge d\overline z,bdz\wedge d\overline z)=\frac{a\overline b}{\gamma^2}$.
Given a holomorphic vector bundle $E$ over $X$, its spaces of smooth and holomorphic sections are denoted by $\Gamma(X;E)$ and $\mathcal{O}(X;E)$, respectively. An $E$-valued $(p,q)$-form is a section of ${T^*}^{p,q}X\otimes E$, and for convenience, $\Gamma(X;{T^*}^{p,q}X\otimes E)$ is denoted by $\Omega^{p,q}(X;E)$. In particular, $\Omega^{0,0}(X;E)=\Gamma(X;E)$. The space of smooth compactly supported $E$-valued $(p,q)$-forms is denoted by $\Omega_c^{p,q}(X;E)$. For $p\in\{0,1\}$, the operator $\overline\partial:\Omega^{p,0}(X;E)\rightarrow \Omega^{p,1}(X;E)$ is given in local frames by
\bes
\overline\partial:s=\sum_{j=1}^r \sigma_j\otimes \xi_j\mapsto \sum_{j=1}^r \overline\partial\sigma_j\otimes \xi_j,
\ees where $\sigma_1,...,\sigma_r\in \Omega^{p,0}(X)$ and $\xi_1,...,\xi_r$ is a choice of local frames of $E$. Given a Hermitian metric $h$ on $E$, ${T^*}^{p,q}X\otimes E$ is endowed with the Hermitian metric ${g^*}_{\!\!p,q}\otimes h$. This induces the following inner product structure on $\Omega^{p,q}(X;E)$:
\begin{eqnarray*}
\left<s,t\right>_{g^*_{p,q}\otimes h}
&=&\int_X(g^*_{p,q}\otimes h)(s,t)\:\omega_g,\qquad s,t\in\Omega^{p,q}(X;E).
\end{eqnarray*} For simplicity, we denote $\left<s,t\right>_{g^*_{p,q}\otimes h}$ by $\left<s,t\right>_{h}$ since $g$ is fixed throughout. The Hilbert space completion of the inner product space $(\Omega_c^{p,q}(X;E),\left<\cdot,\cdot\right>_h)$ is denoted by $L^2_{p,q}(X,g;E,h)$. One extends $\overline\partial$ to an operator on $L^2_{p,0}(X,g;E,h)$, $p\in\{0,1\}$, in the sense of currents via the maximal extension of $\overline\partial$; see \cite[Chapter VIII, \S 3]{De97}. Recall that the Bergman space of sections of $(E,h)$ is defined as $A^2(X,g;E,h)=\mathcal{O}(X;E)\cap L^2_{0,0}(X,g;E,h)$. If $X$ is compact and $D\subset X$ is an open subset, the vector space $A^2(D,g;E,h)$ is independent of the choice of Hermitian metrics on $X$ and $E$; see Lemma 2.2 in \cite{GaGuVi22}, for instance. Thus, $A^2(D,g;E,h)$ and $A^2(D,g;{T^*}^{1,0}X\otimes E,g^*_{1,0}\otimes h)$ are simply denoted by $A^2(D;E)$ and $A^2_{1,0}(D;E)$, respectively.
For the rest of this section, $X$ is assumed to be noncompact. Since every holomorphic vector bundle over a noncompact Riemann surface is trivial, see \cite[\S 8.2]{Varolin11}, we may fix a global holomorphic frame, say $\xi=\{\xi_1,...,\xi_r\}$, of $E$ for computational ease. Any $s\in \Omega^{p,q}(X;E)$ can then be written uniquely as
\bes
s=\sum_{j=1}^r \sigma_j\otimes \xi_j,
\ees for some $\sigma_1,...,\sigma_r\in\Omega^{p,q}(X)$. The Hermitian metric $h$ is represented by the positive-definite Hermitian matrix-valued smooth function $h=\left(h_{jk}\right)$ on $X$, where $h_{jk}=h(\xi_j,\xi_k)$, $j,k\in\{1,...,r\}$. The Chern curvature tensor of $(E,h)$ is the $\text{Herm}(E,E)$-valued $(1,1)$-form $\Theta(h)=\overline\partial\left(\overline h^{-1}\partial \overline h\right)$. It may be expressed as the matrix of $(1,1)$-forms given by
\bes
\Theta(h)=\overline\partial\left(h^{\ell j}\partial h_{k\ell} \right),
\ees where $(h^{jk})$ is the inverse of $h=(h_{jk})$. By further choosing a global holomorphic frame $Z$ of ${T^*}^{1,0}X$, we may also write $\Theta(h)=\left(\theta_{jk}\right) Z\wedge\overline Z$, for some Hermitian-matrix valued smooth function $\theta=\left(\theta_{jk}\right)$ on $X$. Given two $\text{Herm}(E,E)$-valued $(1,1)$-forms $A=\left(a_{jk}\right) Z\wedge\overline Z$ and $B=\left(b_{jk}\right) Z\wedge\overline Z$, we say that $A\geq B$, if $(a_{jk})-(b_{jk})$ is positive semi-definite. This notion is independent of the choice of the global holomorphic frame $Z$.
Finally, in order to produce holomorphic functions on a noncompact Riemann surface, we will use the following version of H{\"o}rmander's theorem for the $\overline\partial$-problem on bundle-valued forms.
\begin{theorem}\cite[Theorem 6.1]{De97}\label{T:HDtheorem} Let $X$ be a noncompact Riemann surface with Hermitian metric $g$ and volume form $\omega_g$. Let $(F,\boldsymbol h)$ be a Hermitian holomorphic vector bundle over $X$ such that \be\label{E:positive} i\Theta(\boldsymbol h)\geq c\omega_g\otimes\text{I}_F \ee for some $c>0$. Then, for every $\alpha\in L^2_{1,1}(X,g;F,\boldsymbol h)$, there exists a $u\in L^2_{1,0}(X,g;F,\boldsymbol h)$ such that $\overline\partial u=\alpha$ in the sense of distributions, and
\bes
\left<u,u\right>_{\boldsymbol h}\leq \frac{1}{c}\left<\alpha,\alpha\right>_{\boldsymbol h}.
\ees \end{theorem}
Note that Theorem 6.1 in \cite{De97} is stated more generally for an $m$-semi-positive vector bundle $(F,\boldsymbol h)$ over a complete K{\"a}hler $n$-dimensional manifold $(X,\hat g)$ equipped with a possibly noncomplete metric $g$. Moreover, $\Omega_c^{1,1}(X;F)$ is endowed with the inner product: $(\alpha,\beta)=\int_X g^*_{1,1}(A^{-1}\alpha,\beta)\omega_g$, where $A=i\Theta(\boldsymbol h)\wedge\Lambda$ and $\Lambda$ is given by $g^*_{0,0}(\Lambda\alpha,f)=g^*_{1,1}(\alpha,f\wedge\omega_g)$, $\alpha\in\Omega^{1,1}(X;F)$, $f\in\Omega^{0,0}(X;F)$. In our case, since $X$ is a noncompact Riemann surface, it admits a complete K{\"a}hler metric; see \cite[Theorem~5.2]{De97}. Furthermore, since $m=n=1$, \eqref{E:positive} is equivalent to both the $m$-positivity of $(F,\boldsymbol h)$ and the positive-definiteness of $A$. For the latter, observe that when $n=1$, the operator $A:\Omega^{1,1}(X;F)\rightarrow \Omega^{1,1}(X;F)$ acts as $\sigma\otimes\xi\mapsto \frac{1}{\gamma}\sigma\otimes \Theta(\boldsymbol h)(\xi)$, $\sigma\in\Omega^{1,1}(X)$, $\xi\in\Gamma(X;F)$, where we view $\Theta(\boldsymbol h)$ as an $\text{End}(F)$-valued $(1,1)$-form on $X$.
\subsection{Potential theory on Riemann surfaces} Let $X$ be a Riemann surface. An upper semicontinuous function $s:X\rightarrow [-\infty, \infty)$ with $s\nequiv -\infty$ on any connected component of $X$, is said to be {\em subharmonic on $X$} if, for every coordinate chart $(V,\varphi)$ on $X$, $s\circ\varphi^{-1}$ is subharmonic on $\varphi(V)\subset \C$. The notion of harmonicity is defined analogously via charts. Of particular importance are the so-called Green's functions on $X$. \begin{definition}\label{D:Greensfunction}
Given $x\in X$, a {\em Green's function on $X$ with singularity at $x$} is a subharmonic function $G_x:X\rightarrow[-\infty,0)$ such that
\begin{itemize}
\item [$(i)$] $G_x$ is harmonic on $X\setminus\{x\}$,
\item [$(ii)$] if $(U,\varphi)$ is a coordinate chart containing $x$, then $y\mapsto G_x(y)-\log|\varphi(x)-\varphi(y)|$ is harmonic on $U$,
\item [$(iii)$] if $H:X\rightarrow[-\infty,0)$ is a subharmonic function satisfying $(i)$ and $(ii)$, then $H\leq G_x$.
\end{itemize}
If $G_x$ exists for every $x\in X$, we call $G(x,y):=G_x(y)$, $x,y\in X$, the {\em Green's kernel} of $X$, and say that $X$ admits a Green's kernel. \end{definition}
For an open subset of a compact Riemann surface, the existence of the Green's kernel is completely characterized by the nonpolarity of its complement. Recall that a set $K\subset X$ is said to be {\em polar} if for each $\zeta\in X$, there is an open neighborhood $V_\zeta\subset X$ of $\zeta$ and a subharmonic function $s_\zeta$ on $V_\zeta$ such that $K\cap V_\zeta\subset\{x\in V_\zeta:s_\zeta(x)=-\infty\}$. We recall the following well-known result.
\begin{theorem}\label{T:Greenian} Let $X$ be a compact Riemann surface and $D\subset X$ be an open subset of $X$. Then the following are equivalent.
\begin{itemize} \item [$(a)$] $X\setminus D$ is nonpolar. \item [$(b)$] $D$ admits a Green's kernel. \item [$(c)$] $D$ is potential-theoretically hyperbolic, i.e., $D$ admits a nonconstant bounded subharmonic function. \end{itemize} \end{theorem} Since the above theorem is a combination of several known results, we give some references. That $(a)$ implies $(c)$ follows from Kellog's theorem, see \cite[Theorem 4.2.5]{Ra95}, and \cite[Lemma 7.1.5]{Varolin11}. For $(c)\Rightarrow(b)$; see Theorem 7.1.13 in \cite{Varolin11}. Finally, the removability of closed polar sets for bounded subharmonic functions gives that $(b)$ implies $(a)$.
\section{Proof of Theorem~\ref{T:main}}\label{S:proofs} Since $\mathcal{O}(M;E)$ is finite dimensional whenever $M$ is compact, $(c)\Rightarrow(b)$. For the proof of $(b)\Rightarrow(a)$; see \cite[Proposition 2.1]{Sz20}. In Sections~\ref{SS:(a)=>(d)} and \ref{SS:dbar}, we prove the implications $(a)\Rightarrow(d)$ and $(d)\Rightarrow(c)$, respectively.
\subsection{Proof of $\mathbf{(a)}$ implies $\mathbf{(d)}$}\label{SS:(a)=>(d)}
Suppose $M\setminus D$ is nonpolar. Then, by Theorem \ref{T:Greenian}, $D$ admits a Green's kernel $G$. Fix an $x_0\in D$, and let $G_{x_0}(y):=G(x_0,y)$ be the associated Green's function on $D$. Define $\psi_{0}:D\longrightarrow [0,1)$ by \[
\psi_0(y):=
\begin{cases}
e^{2G_{x_0}(y)}, &\text{if } y\neq x_0, \\
0, &\text{if } y=x_0.
\end{cases} \] We first show that $\psi_0\in\mathcal{C}^\infty(D)$ is a subharmonic function which is strictly subharmonic outside a discrete set $Z$. By definition of the Green's function, see $(i)$ of Definition \ref{D:Greensfunction}, smoothness of $\psi_0$ only needs to be verified at $y=x_0$. For that, let $(U,\varphi)$ be a coordinate chart of $D$ which includes $x_0$. Then, by $(ii)$ of Definition \ref{D:Greensfunction} there exists a harmonic function $h$ on $U$ such that
$$G_{x_0}(y)-\ln|\varphi(x_0)-\varphi(y)|=h(y)\qquad\forall y\in U.$$ Therefore, \begin{align}\label{E:smoothpsi_0}
\psi_0(y)=e^{2h(y)}|\varphi(x_0)-\varphi(y)|^2\qquad\forall y\in U. \end{align} It follows that $\psi_0\in\mathcal{C}^\infty(D)$. To show that $\psi_0$ is subharmonic on $D$, we compute for $y\in D\setminus\{x_0\}$ that \begin{align}\label{E:subharmonicpsi_0}
i\partial\overline{\partial}\psi_0(y)
&=e^{2G_{x_0}(y)}\left(4i\partial G_{x_0}(y)\wedge\overline{\partial}G_{x_0}(y)+2i\partial\overline{\partial}G_{x_0}(y)\right)\notag\\
&=e^{2G_{x_0}(y)}\left(4i\partial G_{x_0}(y)\wedge\overline{\partial}G_{x_0}(y)\right), \end{align} where the last step follows from the harmonicity of $G_{x_0}$ on $D\setminus\{x_0\}$. A similar computation, using identity \eqref{E:smoothpsi_0} and the holomorphicity of $\varphi$, yields \begin{align*}
i\partial\overline{\partial}\psi_0(x_0)=\left(e^{2h}i\partial\varphi\wedge\overline{\partial\varphi}\right)(x_0). \end{align*} Thus, $\psi_0$ is strictly subharmonic near $x_0$. It then follows from \eqref{E:subharmonicpsi_0}, that $\psi_0$ is strictly subharmonic on $D\setminus Z$ with $$Z=\{y\in D\setminus\{x_0\}:\partial G_{x_0}(y)=0 \}.$$ Note that the harmonicity of $G_{x_0}$ implies that $\partial G_{x_0}$ is a holomorphic $(1,0)$-form. Hence, $Z$ is a discrete set in $D\setminus\{x_0\}$. To accommodate for the lack of strict subharmonicity on $Z$ and uniform strict subharmonicity near $bD$, we construct a particular set of cut-off functions.
We may assume that $M\setminus D$ lies in a single coordinate chart. If not, let $(U,\varphi)$ be a coordinate chart such that $U\setminus D$ is nonpolar. Let $K\subset U\setminus D$ be a nonpolar compact subset. Then $\widetilde D:=M\setminus K$ is an open set in $M$ containing $D$. Thus, if $(d)$ of Theorem \ref{T:main} holds on $\widetilde D$, it holds on $D$.
Let $(U_1,\varphi_1)$ be the coordinate chart which contains $M\setminus D$, and $V_1\Subset U_1$ be an open neighborhood of $M\setminus D$. Then, $Z\setminus V_1$ is a finite set of points $\{x_2,\dots,x_m\}$. For each $j\in\{2,\dots,m\}$, choose a coordinate chart $(U_j,\varphi_j)$ compactly contained in $D$ and containing $x_j$, and an open neighborhood $V_j$ of $x_j$ such that $V_j\Subset U_j$. Next, for each $j\in\{1,\ldots,m\}$,
let $\chi_j\in\mathcal{C}^\infty_c(U_j)$ such that $\chi_j=1$ on $V_j$, and $\zeta_j(y):=|\varphi_j(y)|^2$ for $y\in U_j$. Then, set
$\psi_j$ equal to $\chi_j\cdot\zeta_j$ on $U_j$ and zero on $M\setminus U_j$. Hence, each $\psi_j$, $j\in\{1,\ldots,m\}$, is a bounded, smooth function on $D$.
Moreover, $i\partial\overline{\partial}\sum_{j=1}^m\psi_j\geq \omega$ on $\bigcup_{j=1}^mV_j$ for some volume form $\omega$ on $M$.
Furthermore, $i\partial\overline{\partial}\psi_0\geq \omega$
on $D\setminus\bigcup_{j=1}^mV_j$ for some volume form $\omega$ on $M$.
By the compactness of $\bigcup_{j=1}^m\overline U_j$ and the
smoothness of $\psi_j$, $j\in\{0,\ldots,m\}$, there exists an $\epsilon>0$ such that \begin{align*}
i\partial\overline{\partial}\left(\psi_0+\epsilon\sum_{j=1}^m\psi_j \right)\geq \omega\qquad{on}\;D \end{align*} for some volume form $\omega$ on $M$. Therefore, $\psi:=\psi_0+\epsilon\sum_{j=1}^m\psi_j$ satisfies $(d)$ on $D$.
\subsection{Proof of $\mathbf{(d)}$ implies $\mathbf{(c)}$}\label{SS:dbar} Let $(E,h)$ be a Hermitian holomorphic vector bundle over $M$, and set $(F,\mathfrak{h})=(T^{1,0}M\otimes E, g\otimes h)$. Then $(F,\mathfrak{h})$ is a Hermitian holomorphic vector bundle over $M$ such that \begin{align*}
A^2(D;E)\cong A^2(D,g;T^{*1,0}M\otimes T^{1,0}M
\otimes E, g_{1,0}^*\otimes g\otimes h)
\cong A^2_{1,0}(D;F). \end{align*} Hence, it suffices to show that $A_{1,0}^2(D;F)$ is infinite dimensional whenever ($d$) of Theorem \ref{T:main} holds.
For fixed $N\in\mathbb{N}$, let $z_1,z_2,\ldots,z_N$ be distinct points in $D$. Choose charts $\left\{\left(U_j,\varphi_j\right)\right\}_{j=1}^N$, such that $U_j\subset D$, $z_j\in U_j$, and $\varphi_j(z_j)=0$. After possibly shrinking each $U_j$, we may assume that $U_j\cap U_k=\emptyset$ for any $j\neq k$ and $\varphi_j(U_j)$ is contained in the open unit disk $\mathbb{D}$ for each $j\in\{1,\ldots,N\}$. Choose a function $\chi\in\mathcal{C}^\infty_c(\mathbb{D})$ which equals $1$ near the origin, and define $\ell_j\in\mathcal{C}^\infty_c(D)$, $j\in\{1,\ldots,N\}$, by
$$\ell_j(z)=\chi(\varphi_j(z))\ln|\varphi_j(z)|\;\;\text{for}\;\;z\in U_j$$ and zero otherwise. Next, we set $$
\Phi_K:=K\psi+2\textstyle\sum_{j=1}^N\ell_j, $$ where $\psi$ is the function supplied by the hypothesis, i.e., $\psi\in\mathcal{C}^\infty(D)$ is a bounded function which satisfies $i\partial\overline{\partial}\psi\geq\omega$ on $D$ for some volume form $\omega$ on $M$. It then follows that $\Phi_K\in\mathcal{C}^\infty(D_N)$ for $D_N:=D\setminus\{z_1,\ldots,z_N\}$ and any $K>0$. Furthermore, for any $\widetilde K>0$, there exists a $K>0$ such that $\left(i\partial\overline\partial\Phi_{K}\right)\geq \widetilde K\omega_g$ on $D_N$. Note that $\mathfrak{h} e^{-\Phi_K}$ is a smooth Hermitian metric on $F$ over $D_N$, whose curvature is \begin{align*}
i\Theta(\mathfrak{h} e^{-\Phi_K})=i\Theta(\mathfrak{h})
+i\partial\overline{\partial}\Phi_K\otimes \text{I}_F, \end{align*} on $D_N$, for any $K>0$. Since $(F,\mathfrak{h})$ is a Hermitian vector bundle over $M$, there exists a constanct $c>0$ such that $i\Theta(\mathfrak{h})\geq -c\omega_g I_F$. We may now choose a $K>0$ such that $\boldsymbol h:=\mathfrak h e^{-\Phi_K}$ satisfies $i\Theta(\boldsymbol h) \geq\omega_g I_{F}$ on $D_N$. It follows that we may apply Theorem \ref{T:HDtheorem} for $(F,\boldsymbol h)$ on $D_N$.
Next, let $\xi$ and $\eta$ be nonvanishing holomorphic sections of $F$ and $T^{1,0}M$, respectively, on $D$. Set $\alpha=\overline{\partial}f\otimes\eta\otimes\xi$, where $f(z)=\chi(\varphi_N(z))$ for $z\in U_N$ and zero otherwise. It follows that $\alpha$ is a smooth, compactly supported $F$-valued $(1,1)$-form on $D_N$. Hence, $\alpha\in L_{1,1}^2(D_N,g;F,\boldsymbol h)$. Thus, by Theorem \ref{T:HDtheorem}, there exists a $u\in L^2_{1,0}(D_N,g;F,\boldsymbol h)$ such that $\overline\partial u=\alpha$ on $D_N$ in the sense of distributions and \begin{align*}
\langle u,u\rangle_{\boldsymbol h}\leq
\langle\alpha,\alpha\rangle_{\boldsymbol h}. \end{align*} Then there exists a constant $c>0$ such that \begin{align*}
c\langle u,u\rangle_{\mathfrak{h}}
\leq\langle \alpha,\alpha\rangle_{\boldsymbol h}<\infty \end{align*} since $e^{-\Phi_K}$ has a positive lower bound on $D_N$. Set $s=f\otimes\eta\otimes\xi-u$ on $D_N$. Then $\overline\partial s=0$ on $D_N$ in the sense of distributions. Thus, $s\in\mathcal{O}(D_N; T^{1,0}M\otimes F)$. Moreover, since both $f\otimes\eta\otimes\xi$ and $u$ belong to
$ L^2_{1,0}(D_N,g;F,\mathfrak{h})$, so does $s$. It then follows that there exists an $\tilde s\in A^2_{1,0}(D,g;F,\mathfrak{h})$ such that $\tilde s|_{D_N}=s$, since $D\setminus D_N$ is a compact polar set; see \cite[Theorem 9.5]{Co95}. Therefore, $\tilde u:=f\otimes\eta\otimes\xi-\tilde s\in\Omega^{1,0}(D;F)$ is such that $\tilde{u}$ equals $u$ outside a set of measure zero, and $\langle \tilde{u},\tilde{u}\rangle_{\boldsymbol h}=\langle u,u\rangle_{\boldsymbol h}<\infty$. By the lack of local integrability of $e^{-\Phi_K}$ at $z_j$, we obtain that $\tilde{u}(z_j)=0$, $j\in\{1,\ldots,N\}$. Therefore, $\tilde s(z_j)=0$ for all $j\in\{1,\ldots, N-1\}$ and $\tilde s(z_N)=\eta\otimes\xi$. As $N\in\mathbb{N}$ was arbitrary, it follows that $A^2_{1,0}(D,g;F,\mathfrak{h})$ is infinite dimensional.
{}
\end{document} |
\begin{document}
\title{Time-Optimal Guidance to Intercept Moving Targets by Dubins Vehicles} \begin{abstract} This paper is concerned with a Minimum-Time Intercept Problem (MTIP), for which a Dubins vehicle is guided from a position with a prescribed initial orientation angle to intercept a moving target in minimum time. Some geometric properties for the solution of the MTIP are presented, showing that the solution path must lie in a sufficient family of 4 candidates. In addition, necessary and sufficient conditions for optimality of each candidate are established. When the target's velocity is constant, by employing the geometric properties, those 4 candidates are transformed to a class of sufficiently smooth real-valued functions. In order to compute all the 4 candidates, an efficient and robust algorithm to find all the zeros of sufficiently smooth real-valued functions is developed. Since the MTIP with a constant target's velocity is equivalent to the path planning problem of Dubins vehicle in a constant drift field, developing such an algorithm also enables efficiently finding the shortest Dubins path in a constant drift field. Finally, some numerical examples are presented, demonstrating and verifying the developments of the paper. \end{abstract}
\keywords{Dubins vehicle\and Minimum-time path\and Path planning\and Intercept guidance}
\section{Introduction}
Autonomously guiding a pursuer to intercept a target in minimum time is a fundamental problem in the field of guidance and path planning \cite{Isaacs:1965,Merz:1971}. In this paper, we study a Minimum-Time Intercept Problem (MTIP), for which the target's moving strategy is given and the pursuer is considered to be a typical nonholonomic vehicle which moves only forward at a constant speed with a minimum turning radius. Such a nonholonomic vehicle has been commonly dubbed Dubins vehicle in the literature. As the Dubins vehicle provides an ideal kinematic model for a large class of vehicles, such as fixed-wing unmanned aerial vehicles, autonomous underwater vehicles, unmanned ground vehicles, etc., the shortest paths of Dubins vehicle from a fixed initial configuration (a location and a heading orientation angle) to intercept a target have been widely studied in many fields \cite{Matveev:2012,Matveev:11}. It should be noted that for a Dubins vehicle the shortest path is equivalent to the minimum-time path as the speed is constant.
Assuming that the target is stationary and considering that the final impact angle is fixed, the MTIP degenerates to the well-known Dubins problem between two configurations. L. E. Dubins used geometric arguments in \cite{Dubins:57} to show that the shortest Dubins paths between two configurations lie in a sufficient family of $6$ candidates. By relaxing the constraint on the final impact angle, the shortest Dubins path from a configuration to a stationary target was studied in \cite{Boissonnat:1994} and this problem is now called Relaxed Dubins Problem (RDP); the solution of RDP lies in a sufficient family of 4 candidates. With the advent of geometric optimal control theory, the developments in \cite{Dubins:57,Boissonnat:1994} were all verified in \cite{Sussmann:94} by using Pontryagin's maximum principle \cite{Pontryagin}. Recently, the shortest path of Dubins vehicle with three consecutive points was studied in \cite{Chen:19automatica} by proposing a polynomial method to compute the solution path. The syntheses in \cite{Dubins:57,Boissonnat:1994,Sussmann:94,Chen:19automatica} allow computing the shortest Dubins paths within a constant time since only a finite number of candidate paths need to be checked. However, if the target is moving, the results developed in \cite{Dubins:57,Boissonnat:1994,Sussmann:94,Chen:19automatica} do not apply any more for the MTIP.
As a matter of fact, synthesizing the solution of the MTIP with a moving target is fundamentally important in pursuit-evasion engagements \cite{Isaacs:1965}. For this reason, some variants of the MTIP have been studied in the literature. Meyer, Isaiah, and Shima are probably the first ones studying the solution of the MTIP \cite{Mayer:2015}. Those authors established some sufficient conditions to ensure that the solution of the MTIP is the same as that of the RDP. Whereas, it is not clear what the solution of the MTIP is if the sufficient conditions are not met. Fixing the final impact angle, the solution of the MTIP was studied under an assumption that the distance between the initial position and the moving target kept at least 4 times longer than the minimum turning radius \cite{Gopalan:2016}. This assumption restricted the solution into a family of 4 simple candidate paths, allowing formulating some nonlinear equations so that the optimal path was related to the roots of the nonlinear equations. Those authors proposed using a Newton-type method or a bisection method to find the roots of the nonlinear equations. However, the two numerical methods may not find the desired roots, as shown by the numerical examples in Section \ref{SE:Examples}. More recently, considering that the target moves along a circle, the MTIP was studied in \cite{Manyam:2019,Park:2020}. In both \cite{Manyam:2019} and \cite{Park:2020}, a strict condition that the initial point of Dubins vehicle is at least 4 times minimum turning radius apart from the target circle was assumed to hold. This strict assumption enables using geometric arguments to synthesize the solution path.
In all the papers cited in the previous paragraph, the target's velocity can be changing. If the target's velocity is constant, it can be proven by a simple coordinate transformation that the MTIP is equivalent to planning the shortest Dubins paths from a configuration to a point in a constant drift field \cite{Techy:2009,Bakolas:2013,McGee:2007}. For this reason, studying the MTIP with a constant target's velocity is quite important in practical scenarios since the motions of fixed-wing unmanned aerial vehicles and autonomous underwater vehicles are usually affected by wind and ocean current, respectively.
Because of the equivalence between the MTIP with a constant target's velocity and the path planning problem in a constant drift field, the MTIP with a constant target's velocity was usually studied from the perspective of synthesizing the shortest Dubins paths in a constant drift field. McGee, Spry, and Hedrick in \cite{McGee:2005} synthesized the shortest Dubins path in wind and an iterative method was proposed to find the shortest Dubins path. Later on, those authors studied the same problem, showing that the solution lies in a sufficiently family of 8 types \cite[Theorem 1]{McGee:2007}. Analogously, a single implicit equation for the minimum intercept time was formulated and an iterative algorithm was proposed to find the optimal solution in \cite{Looker:2008}, where it was assumed that the distance between the pursuer and the target was always greater than 4 times the minimum turning radius. In all \cite{Looker:2008}, \cite{McGee:2005}, and \cite{McGee:2007}, it was proposed to compute the shortest Dubins paths in constant drift field by iteratively finding a specific zero of nonlinear equations. However, as stated in the previous paragraph, it is challenging to find a specific zero of a nonlinear equation. The reasons include that (1) iterative methods may not converge, and (2) a nonlinear equation usually has multiple zeros so that a zero found by an iterative method cannot be guaranteed to be the desired zero, as shown by the numerical examples in Section \ref{SE:Examples}. In addition to \cite{McGee:2005,McGee:2007}, the properties of shortest Dubins paths in a constant drift field were synthesized in \cite{Bakolas:2013} by using standard optimal control tools and means of discontinuous mapping. More recently, the controllability of Dubins problem in a constant drift field was studied in \cite{Caillau:2019}.
Although some variants of the MTIP have been studied in the literature, it is not exaggerate to say that this fundamental problem has not been well addressed, as shown by the counter example presented in \cite[Example 1]{Mayer:2015}. From practical point of view, it is often required to compute the solution path of the MTIP in real time or onboard, especially for scenarios when the control decisions have to be made in situ or, if not exactly, at least efficiently. However, to the authors' best knowledge, an efficient and robust algorithm for computing the solution of the MTIP does not exist in the literature.
Unlike the aforementioned papers, the solution of the MTIP is thoroughly investigated in this paper without any assumption on the distance between the initial point and the target. First, by introducing three functions in terms of the target's trajectory and studying the continuity properties of the three functions, some geometric properties of the solution path are presented. Using these geometric properties, it is proven that the solution path of the MTIP lies in a sufficient family of 4 candidates, and necessary and sufficient conditions for optimality of each candidate are established. In addition, when considering that the target's velocity is constant, these geometric properties are used to formulate some nonlinear equations so that the solution of the MTIP is determined by a specific zero of the nonlinear equations.
In general, a nonlinear equation may have multiple zeros but only one specific zero is related to the solution of the MTIP. Since the typical Newton-like iteration method and bisection method proposed in \cite{McGee:2005,McGee:2007} cannot be guaranteed to find a specific zero of a nonlinear equation, a new algorithm is developed in this paper to find all the zeros of a sufficiently smooth real-valued function. Applying this algorithm allows computing the solution of the MTIP within a constant time, which provides a potential for onboard applications. It is worth mentioning that developing such an algorithm also enables computing the shortest Dubins path in a constant drift field, as shown by the last numerical example in Section \ref{SE:Examples}.
The paper is organized as follows. In Section \ref{SE:Preliminary} the MTIP is formulated as an optimal control problem and its necessary conditions for optimality are presented according to Pontryagin's maximum principle. Section \ref{SE:Syntheses} is attributed to establishing geometric properties for the solution of the MTIP. Then, a robust and efficient algorithm is designed in Section \ref{SE:Algorithm} to find the solution of the MTIP. All the developments are finally demonstrated and verified by numerical examples in Section \ref{SE:Examples}.
\section{Preliminary}\label{SE:Preliminary}
In this section, the MTIP is formulated as an optimal control problem, and its necessary conditions for optimality are established according to Pontryagin's maximum principle.
\subsection{Problem Formulation}
Consider a 2-dimensional engagement involving a pursuer and a moving target. The pursuer is a Dubins vehicle that moves only forward at a constant speed with a minimum turning radius. Denote the state (or configuration) of the pursuer by $\boldsymbol{z}:=(x,y,\theta)\in \mathbb{R}^2\times \mathbb{S}^1$, which consists of a position vector $(x,y)\in \mathbb{R}^2$ and a heading orientation angle $\theta \in \mathbb{S}^1$. Then, by normalizing the position $(x,y)$ so that the pursuer's speed is one, the kinematics is expressed as \begin{align} \qquad \dot{\boldsymbol{z}}(t)=\left[ \begin{matrix}
\cos \theta(t)\\
\sin \theta(t)\\
u(t)/\rho \end{matrix} \right],\ \ \ \ \ \ \ \ \ \ \ u\in [-1,1] \label{Eq:problem1} \end{align} where $t\geq 0$ denotes time, the dot denotes the differentiation with respect to time, $\rho>0$ is the minimum turning radius, and $u$ is the control input representing the lateral acceleration of the pursuer. Without loss of generality, we assume that the configuration at initial time $t=0$ is $$\boldsymbol{z}_0 := (0,0,\pi/2).$$
Denote by $\boldsymbol{v} = (v_x,v_y)\in \mathbb{R}^2$ the velocity of the target. Let the position of the target at initial time $t=0$ be $(\hat{x}_{0},\hat{y}_{0})$. Then, the position of the target at any time $t\geq 0$ is given by \begin{align} E(t) = (\hat{x}_{0},\hat{y}_{0})+ \int_{0}^t \boldsymbol{v}(\tau) \mathrm{d}\tau. \label{EQ:E(t)} \end{align} Throughout the paper, we assume that the target's moving strategy is given so that the position $E(t)$ for any $t\geq 0$ is available to the pursuer.
The MTIP is an optimal control problem defined as below. \begin{problem}[MTIP]\label{problem1} The MTIP consists of finding a minimum time $t_m > 0$ so that the system in Eq.~(\ref{Eq:problem1}) is steered by a measurable control $u(\cdot)$ over the interval $[0,t_m]$ from a fixed initial configuration $\boldsymbol{z}_0$ at $t=0$ to intercept a moving target at $t_m$, i.e., $(x(t_m),y(t_m)) = E(t_m)$. \end{problem} If $\boldsymbol{v} \equiv 0$, the MTIP degenerates to the well-known RDP \cite{Boissonnat:1994}. If the velocity $\boldsymbol{v}$ is constant, by a simple coordinate transformation, it can be proven that the MTIP is equivalent to planning path for a Dubins vehicle from a fixed configuration to a fixed point \cite{Bakolas:2013}.
In \cite{Cockayne:1967}, it was shown that the solution of the MTIP exists when $ \|{\boldsymbol{v}}\|< 1$, where the notation $\|\cdot\|$ denotes the Euclidean norm. In this paper, we assume that the condition $\|{\boldsymbol{v}}\|<1$ stands so that the solution of the MTIP exists.
\subsection{Necessary Conditions}
Let $p_x,\ p_y$ and $p_{\theta}$ be the costate variables of $x,\ y$, and $\theta$, respectively. Then, the Hamiltonian of the MTIP is \begin{align} H=p_x \cos \theta +p_y \sin \theta +p_{\theta}u/\rho-1 \label{Eq:problem4} \end{align} According to Pontryagin's maximum principle \cite{Pontryagin}, we have \begin{align}
\dot{p}_x(t)&=-\frac{\partial H}{\partial x}=0\label{EQ:px}\\ \dot{p}_y(t)&=-\frac{\partial H}{\partial y}=0\label{EQ:py}\\ \dot{p}_{\theta}(t)&=-\frac{\partial H}{\partial \theta}=p_x(t)\sin \theta(t)-p_y(t) \cos \theta(t)
\label{Eq:problem5} \end{align} It is apparent from Eq.~\eqref{EQ:px} and Eq.~\eqref{EQ:py} that $p_x$ and $p_y$ are constant. By integrating Eq.~(\ref{Eq:problem5}), we have \begin{align} p_{\theta}(t)=p_xy(t)-p_yx(t)+c_0 \label{Eq:problem6} \end{align} where $c_0$ is a constant. In view of Eq.~(\ref{Eq:problem6}), if $p_{\theta}\equiv 0$ on a nonzero interval, the path of $(x,y)$ is a straight line segment on this interval. Note that $u\equiv 0$ along a straight line. Thus, we have $u\equiv0$ if $p_{\theta} \equiv 0$. As a result, the maximum principle indicates that the optimal control $u$ is totally determined by $p_{\theta}$, i.e., \begin{align} u=\begin{cases} 1,&p_{\theta}>0\\ 0,&p_{\theta}\equiv0\\ -1,&p_{\theta}<0 \end{cases} \label{Eq:problem9} \end{align} The path $t\mapsto [x(t),y(t)]$ is a circular arc with right (resp. left) turning direction if $u = -1$ (resp. $u=1$). Therefore, the switching conditions in Eq.~\eqref{Eq:problem9} imply that the solution path of the MTIP is a concatenation of circular arcs and straight line segments.
The necessary conditions from Eq.~\eqref{EQ:px} to Eq.~\eqref{Eq:problem9} will be used in the next section to synthesize the solution path of the MTIP. Before proceeding, some notations and definitions are defined in the following subsection.
\subsection{Notations and definitions}
Denote by \begin{align} F:\mathbb{R}^2\rightarrow [0,+\infty),\ \ (x,y)\mapsto F[x,y] \end{align}
the minimum time (equivalent to the length of the shortest path since the speed of the pursuer is $1$) for the pursuer to move from the initial configuration $\boldsymbol{z}_0$ to the point $(x,y)\in \mathbb{R}^2$. By definition, it is apparent that the value of $F[x,y]$ denotes the length of the RDP's path from $\boldsymbol{z}_0$ to $(x,y)$. According to \cite{Boissonnat:1994}, the solution of the RDP can be computed in a constant time by checking at most four candidate paths. Thus, given any $t\geq 0$, the value of $F[E(t)]$ is readily available.
Let $\mathcal{C}_r$ and $\mathcal{C}_l$ be circles of radius $\rho$, lying on the right and left side of initial configuration $\boldsymbol{z}_0$, respectively, i.e., \begin{align}
\mathcal{C}_r = \{(x,y)\in \mathbb{R}^2 | (x-\rho)^2 + y^2 = \rho^2\}\nonumber \end{align} and \begin{align}
\mathcal{C}_l = \{(x,y)\in \mathbb{R}^2| (x+\rho)^2 + y^2 = \rho^2\}\nonumber \end{align} Moreover, set \begin{align}
\mathcal{D}_r = \{(x,y)\in \mathbb{R}^2 | (x-\rho)^2 + y^2 \leq \rho^2\} \nonumber \end{align} and \begin{align}
\mathcal{D}_l = \{(x,y)\in \mathbb{R}^2 | (x+\rho)^2 + y^2 \leq \rho^2\}\nonumber \end{align} as the regions bounded by $\mathcal{C}_r$ and $\mathcal{C}_l$, respectively. Denote by \begin{align} \boldsymbol{c}_0^{r} := \left( \begin{array}{c} \rho \\ 0 \end{array} \right) \ \text{and}\ \boldsymbol{c}_0^{l} := \left( \begin{array}{c} -\rho \\ 0 \end{array} \right)\nonumber \end{align} the centers of $\mathcal{C}_r$ and $\mathcal{C}_l$, respectively. We define three subregions $\mathcal{R}_1$, $\mathcal{R}_2$, and $\mathcal{R}_3$ in the $2$-dimensional plane as follow: \begin{align} \mathcal{R}_2 =&\ \mathcal{D}_r \cup \mathcal{D}_l,\nonumber\\
\mathcal{R}_{3} = &\ \big \{(x,y)\in \mathbb{R}^2 \ |\ y > 0, (x -\rho)^2 + y^2 \leq 9\rho^2, (x+\rho)^2 + y^2 \leq 9 \rho^2\big\} -( \mathcal{R}_2\cap \{y>0\}), \nonumber\\ \mathcal{R}_{1} =&\ \mathbb{R}^2 - \mathcal{R}_2 - \mathcal{R}_3.\nonumber \end{align} The geometries for $\mathcal{C}_r$, $\mathcal{C}_l$, $\mathcal{R}_1$, $\mathcal{R}_2$, and $\mathcal{R}_3$ are all illustrated in Fig.~\ref{Fig:synthese6}. \begin{figure}
\caption{The geometry of the circles $\mathcal{C}_r$ and $\mathcal{C}_l$ and the subregions $\mathcal{R}_1$, $\mathcal{R}_2$, and $\mathcal{R}_3$ in the 2-dimensional plane.}
\label{Fig:synthese6}
\end{figure}
Denote by ``S'' and ``C'' a straight line segment and a circular arc with radius of $\rho$, respectively. If a circular arc C has a right (resp. left) turning direction, we represent it by R (resp. L). In addition, we denote by ``S$_d$'' a straight line segment with a length of $d\geq 0$, and denote by ``C$_{\alpha}$'' a circular arc with a radian of $\alpha\geq 0$. Accordingly, we denote by L$_{\alpha}$ (resp. R$_{\alpha}$) a left-turning (resp. right-turning) circular arc with its radian being $\alpha \geq 0$. \begin{definition}[Feasible Dubins path] Given any path in the $x-y$ plane, it is said a feasible Dubins path if the curvature along the path everywhere is not greater than $1/\rho$. \end{definition}
As the solution of the RDP is used in the following text, we summarize the solution types of the RDP by the following remark. \begin{remark}[J.-D. Boissonnat and X.-N. Bui \cite{Boissonnat:1994}]\label{RE:RDP} The solution path of the RDP belongs to either CS or CC or substrings thereof, where \begin{itemize} \item CC = \{RL, LR\} \item CS = \{RS, LS\} \end{itemize} \end{remark}
\section{Characterizing the solution of the MTIP}\label{SE:Syntheses}
In this section, some geometric properties for the solution path of the MTIP will be established by analyzing the function $F[E(t)]$. For simplicity of presentation, all the proofs for the theorems of this section are postponed to Appendix \ref{Appendix:A}.
By the following lemma, we first recall from \cite{Mayer:2015} the properties for the solution of the MTIP in the case that $F[E(t)]$ is continuous. \begin{lemma}[Meyer, Isaiah, and Shima \cite{Mayer:2015}]\label{LE:continuous} If the function $F[E(t)]$ is continuous for $t\in [0,+\infty)$, then the following two statements hold: \begin{description} \item (1) The minimum interception time $t_m>0$ is the minimum fixed point of $F[E(t)]$, i.e., \begin{align}
t_m = \mathrm{min} \{t > 0\ \big| \ t = F[E(t)]\}.\nonumber \end{align} \item (2) The solution path of the MTIP is the same as that of the RDP from the initial condition $\boldsymbol{z}_0$ to the interception point $E(t_m)$. \end{description} \end{lemma}
This lemma presents the relationship between $t_m$ and $F[E(t)]$ under the sufficient condition that the function $F[E(\cdot)]$ is continuous. However, this sufficient condition may not be met, as shown by the following lemma. \begin{lemma}\label{TH:continuity} The function $F[E(t)]$ is discontinuous at a time $\bar{t}>0$ if and only if \begin{align}
E(\bar{t}) \in \{(x,y)\in \mathcal{C}_r \cup \mathcal{C}_l\ \big|\ y>0 \}.\nonumber \end{align} \end{lemma}
By extending Lemma \ref{LE:continuous}, the following lemma presents the solution property of the MTIP without requiring the continuity of $F[E(t)]$. \begin{lemma}\label{LE:fixed_RDP} No matter the function $F[E(t)]$ is continuous or not, if the minimum intercept time $t_m>0$ is a fixed point of $F[E(t_m)]$, i.e., $t_m = F[E(t_m)]$, the solution path of the MTIP is the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$. \end{lemma}
In order to establish necessary and sufficient conditions for $t_m = F[E(t_m)]$, we consider the location of $E(t_m)$ in different subregions of $\mathcal{R}_1$, $\mathcal{R}_2$, and $\mathcal{R}_3$ in the sequel. \begin{theorem}\label{LE:Occurance_R1} If the final point $E(t_m)$ of the MTIP lies in the interior of $\mathcal{R}_1\cup \mathcal{R}_2$, i.e., $E(t_m) \in \mathrm{int}(\mathrm{R}_1\cup \mathcal{R}_2)$, then the minimum interception time $t_m$ is a fixed point of $F[E(\cdot)]$, i.e., $t_m = F[E(t_m)]$. \end{theorem}
Combining Lemma \ref{LE:fixed_RDP} and Theorem \ref{LE:Occurance_R1}, we have that, no matter the function $F[E(t)]$ is continuous or not, only if $E(t_m) \in \mathrm{int}(\mathrm{R}_1\cup \mathrm{R}_2)$, the solution path of the MTIP must be the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$. Therefore, according to Remark \ref{RE:RDP}, we immediately have the following result.
\begin{remark}\label{RE:R1_R2}
If $E(t_m) \in \mathrm{int}(\mathrm{R}_1\cup \mathcal{R}_2)$, the solution path of the MTIP belongs to CC or CS or substrings thereof.
\end{remark}
Next, we shall establish the geometric properties of the solution of the MTIP for the rest case that $E(t_m) \in \mathrm{int}(\mathcal{R}_3)$. Before proceeding, we present a symmetric property by the following remark. \begin{remark}[Symmetric property \cite{Boissonnat:1994}]\label{RE:Symmetric} Given any feasible Dubins path from $\boldsymbol{z}_0$ to $(x,y)\in \mathbb{R}^2$, there exists a feasible Dubins path from $\boldsymbol{z}_0$ to $(-x,y)$ so that the two feasible Dubins paths are symmetric with respect to the $y$ axis. \end{remark}
Thanks to this remark, in the following paragraphs we just consider the solution of the MTIP with the terminal point $E(t_m)$ in the subregion $\mathcal{R}_3\cap \{x> 0\}$.
Notice that for any point $(x,y)\in \mathrm{int}(\mathcal{R}_3\cap \{x>0\})$, there exist two circles of radius $\rho$ that not only pass through $(x,y)$ but also are tangent to $\mathcal{C}_l$, as illustrated in Fig.~\ref{Fig:L1L2}.
\begin{figure}
\caption{The geometries of the feasible Dubins paths related to $F$, ${L}^-$, and ${L}^+$.}
\label{Fig:L1L2}
\end{figure} Circular arcs on the two circles together with circular arcs on $\mathcal{C}_l$ form two feasible Dubins paths of type LR from $\boldsymbol{z}_0$ to $(x,y)$, and we denote by L$_{\alpha^+}$R and L$_{\alpha^-}$R the types of the two feasible Dubins paths, as shown in Fig.~\ref{Fig:L1L2}.
For any $(x,y)\in \mathcal{R}_3\cap \{x> 0\}$, let $\xi>0$ denote the angle between the positive $x$-axis and the vector from $\boldsymbol{c}_0^l$ to $(x,y)$, as shown in Fig.~\ref{Fig:L1L2}. It is apparent from Fig.~\ref{Fig:L1L2} that $\alpha^+ \geq \xi$ and $\alpha^- \leq \xi$ \cite{Ding:2019}.
\begin{definition}
We denote by $L^+[x,y]$ and $L^-[x,y]$ the lengths of the feasible Dubins paths from $\boldsymbol{z}_0$ to $(x,y)\in \mathcal{R}_3\cap\{x>0\}$ with types $L_{\alpha^+}R$ and $L_{\alpha^-}R$, respectively.
\end{definition}
The following lemma shows the continuity properties of $L^+[E(t)]$ and $L^-[E(t)]$.
\begin{lemma}\label{LE:continuity} Let $\bar{t}>0 $ be a time so that $E(\bar{t}) \in \mathcal{R}_3\cap \{x>0\}$. Then, both $L^-[E(t)]$ and $L^+[E(t)]$ are continuous at $\bar{t}$.
\end{lemma}
We recall from \cite{Ding:2019} the relationship of the three functions $F$, $L^-$, and $L^+$ by the following lemma. \begin{lemma}[Ding, Xin, and Chen \cite{Ding:2019}]\label{LE:Ding} Given any point $(x,y)$ in $\mathrm{int}(\mathcal{R}_3\cap \{x>0\})$, the following three statements hold: \begin{description} \item (1) $F[x,y] < L^-[x,y]<L^+[x,y]$. \item (2) For any $L>0$ in the open interval $ (L^-[x,y],L^+[x,y])$, there is not a feasible Dubins path with a length of $L$ from $\boldsymbol{z}_0$ to $(x,y)$. \item (3) For any $L>0$ in the closed interval $[F[x,y],L^-[x,y]]$ or in the semi-open interval $ [L^+[x,y],+\infty)$, there exists a feasible Dubins path with a length of $L$ from $\boldsymbol{z}_0$ to $(x,y)$. \end{description} \end{lemma} This lemma is a direct result of \cite[Theorem 2]{Ding:2019}. Thanks to Lemmas \ref{LE:continuity} and \ref{LE:Ding}, we have the following result. \begin{theorem}\label{LE:R3} If the minimum-time intercept point $E(t_m)$ between the pursuer and the target occurs in the interior of $\mathcal{R}_3\cap\{x>0\}$, i.e., $E(t_m)\in \mathrm{int}(\mathcal{R}_3\cap \{x>0\})$, we then have \begin{align} \begin{split}
t_m =& \min \{ t>0 \big| t = F[E(t)],\ t ={L}^-[E(t)],\ \text{or}
t={L}^+[E(t)]\}. \end{split} \label{EQ:tm_R3} \end{align} \end{theorem} It can be seen from Theorem \ref{LE:R3} that the minimum intercept time $t_m$ may not be the fixed point of $F[E(t)]$. Up to present, it has not been clear what the solution type of the MTIP is if $t_m \neq F[E(t_m)]$.
By the following remark, we shall present the solution property of the MTIP for the cases of $t_m = L^-[E(t_m)]$ and $t_m = L^+[E(t_m)]$. \begin{remark}\label{LE:type_L} If the minimum-time intercept point $E(t_m)$ lies in $\mathrm{int}(\mathcal{R}_3)\cap \{x>0\}$, the following statements hold: \begin{description} \item (1) If $t_m = L^-[E(t_m)]$, the solution of the MTIP is of type $L_{\alpha^-}R$. \item (2) If $t_m = L^+[E(t_m)]$, the solution of the MTIP is of type $L_{\alpha^+}R$. \end{description} \end{remark}
Note that the two types $L_{\alpha^-}R$ and $L_{\alpha^+}R$ belong to CC. Thus, combining Remark \ref{RE:R1_R2} with Remark \ref{LE:type_L}, we have the following result. \begin{remark} The solution of the MTIP lies in a sufficient family of 4 types in $\{RL,LR,LS,RS\}$. \end{remark}
Theorems \ref{LE:Occurance_R1} and \ref{LE:R3} indicate that finding the minimum intercept time $t_m$ is amount to computing the fixed points of $F[E(\cdot)]$, ${L}^-[E(\cdot)]$, and ${L}^+[E(\cdot)]$ over some specific intervals. Once $t_m$ is found, the final point $E(t_m)$ is available, and we can use the geometric properties in Remarks \ref{RE:R1_R2} and \ref{LE:type_L} to determine the solution path of the MTIP. In the next section, an algorithm will be developed to find those fixed points so that the minimum interception time $t_m$ can be obtained efficiently.
\section{Algorithm for the solution of MTIP}\label{SE:Algorithm}
In this section, we first present a robust and efficient algorithm to find the zeros of sufficiently smooth real-valued function in Subsection \ref{Subse:algorithm}, which will be employed in Subsection \ref{SE:procedure} to establish numerical methods to find the solution of the MTIP for the case that the target's velocity $\boldsymbol{v}$ is constant.
For simplicity of presentation, the proofs for all the lemmas of this section are postponed to Appendix \ref{Appendix:B}.
\subsection{Algorithm for finding zeros of sufficiently smooth real-valued functions}\label{Subse:algorithm}
Before proceeding, we first present a lemma regarding the relationship between extremas and zeros of a sufficiently smooth real-valued function. \begin{lemma}\label{LE:extreme_to_zero} Given a sufficiently smooth function $G(t)$ so that its number of zeros over an interval $[a,b]$ is finite, denote by $t_1$, $t_2,$, $\ldots$, $t_n$ in $[a,b]$ the zeros of the differentiation of $G(t)$ with respect to time, i.e., $G^{\prime}(t_i) = 0$ for $i=1,\ldots,n$. If $a\leq t_1 < t_2 < \ldots < t_n \leq b$, the following two statements hold: \begin{description} \item (1) if $G(t_i) \times G(t_{i+1}) >0$, the function $G(t)$ on the interval $[t_i,t_{i+1}]$ does not have a zero; \item (2) if $G(t_i) \times G(t_{i+1}) <0$, the function $G(t)$ on the interval $[t_i,t_{i+1}]$ has only one zero.
\end{description} \end{lemma} Thanks to this lemma, if $G(t_i)\times G(t_{i+1})< 0$, we can use a simple bisection method to find the only zero in the interval $(t_i,t_{i+1})$. For notational simplicity, if $G(t_i)\times G(t_{i+1})< 0$, we denote by $$z = \textbf{B}[G(t),t_{i},t_{i+1}]$$ the bisection method to find the zero $z$ of $G(t)$ in the interval $[t_i,t_{i+1}]$. Then, we can use Algorithm \ref{algorithm1} to compute all the zeros of $G(\cdot)$ in a constant time.
\begin{algorithm} \caption{ }\label{algorithm1} Given a sufficiently smooth function $G(t)$ so that its number of zeros is finite over $[a,b]$, let $t_1<t_2<\ldots<t_n$ be all the zeros of the differentiation of $G(t)$ with respect to $t$, i.e., $G^{\prime}(t_i) = 0$ for $i=1,2,\ldots,n$. Then, all the zeros of $G(t)$ over $[a,b]$ can be found by the following procedures: \begin{description} \item 1. set $i=0$, $t_0 = a$, $t_{n+1} = b$, and $Z=\varnothing$ \item 2. \textbf{while} $i<n$ \item 3. \qquad \textbf{if} $G(t_i) = 0$
\item 4. \qquad \qquad $Z = Z\cup \{t_i\}$
\item 5. \qquad \textbf{elseif} $G(t_i)\times G(t_{i+1}) < 0$
\item 6. \qquad \qquad $z =\textbf{B}[G(t),t_i,t_{i+1}]$ \item 7. \qquad \qquad $Z = Z\cup \{z\}$
\item 8. \qquad \textbf{endif} \item 9. \qquad $i=i+1$ \item 10. \textbf{endwhile} \end{description} \end{algorithm} Let us gather a few words to explain the pseudo codes in Algorithm \ref{algorithm1}. For any given $i\in \{0,1,\ldots,n+1\}$, if $G(t_i) = 0$, we have that $t_i$ is a zero of $G(t)$; thus, we add $t_i$ into the set $Z$ of zeros at step 4. If $G(t_i)\times G(t_{i+1}) < 0$, according to Lemma \ref{LE:extreme_to_zero} we have that the function $G(t)$ on the interval $(t_i,t_{i+1})$ has only one zero. Thus, a typical bisection method can be used to find that zero, as shown by step 6, and the zero is added to the set $Z$ at step 7. If $G(t_i)\times G(t_{i+1}) > 0$, none zero exists between $t_i$ and $t_{i+1}$ according to the first statement of Lemma \ref{LE:extreme_to_zero}. Thus, nothing is done in the while loop if $G(t_i)\times G(t_{i+1}) > 0$. As a result, after the while loop, the set $Z$ defined in Algorithm \ref{algorithm1} contains all the zeros of $G(t)$. It should be noted that the bisection method in step 6 can be completed within a constant time. To this end, given any sufficiently smooth real-valued function $G(t)$, if the number of zeros is finite over the interval $[a,b]$, we can use Algorithm \ref{algorithm1} to find all the zeros within a constant time.
In the following subsection, Algorithm \ref{algorithm1} will be applied to computing the solution of the MTIP.
\subsection{Computing the solution of the MTIP}\label{SE:procedure}
In this subsection, the geometric properties revealed in Section \ref{SE:Syntheses} will be employed to establish some nonlinear equations so that the solution length of the MTIP is determined by a specific zero of the nonlinear equations. As a result, Algorithm \ref{algorithm1} can be applied to finding the solution of the MTIP.
Due to the symmetric property presented in Remark \ref{RE:Symmetric}, we will only consider the scenario that the final point $E(t_m)$ lies on the right plane $\{x>0\}$ in the following subsections.
\subsubsection{The case of $E(t_m) \in \mathrm{int}(\mathcal{R}_1)\cap\{x>0\}$} \label{subsection:R2}
According to Theorem \ref{LE:Occurance_R1}, if $E(t_m) \in \mathrm{int}(\mathrm{R}_1\cap\{x>0\})$, we have $t_m = F(t_m)$, indicating that the solution path is the same as that of RDP from $\boldsymbol{z}_0$ to $E(t_m)$ (cf. Lemma \ref{LE:fixed_RDP}). For any terminal point in $\mathcal{R}_1\cap \{x>0\}$, the solution path of RDP is of type RS \cite{Boissonnat:1994}, as shown by Fig.~\ref{Fig:Types_Region} in Appendix \ref{Appendix:A}. By the following lemma, a nonlinear equation in terms of the parameters of the RS path will be established, so that the solution of the MTIP can be found by finding a specific zero of the nonlinear equation. \begin{lemma}\label{LE:analytic_CS} Assume that the target's velocity is constant. If the minimum-time intercept point $E(t_m)$ between the pursuer and the target occurs in the interior of $\mathcal{R}_1\cap\{x>0\}$, i.e., $E(t_m)\in \mathrm{int}(\mathcal{R}_1\cap\{x>0\})$, we have that the solution path of the MTIP is of type R$_{\alpha}$S and it holds \begin{align} G_{cs}(\alpha) \overset{\triangle}{=} \ & {A_1 \sin \alpha + A_2 \cos \alpha}+ \alpha (A_3 \cos \alpha + A_4 \sin \alpha ) + {A_5}= 0
\label{EQ:CS_alpha} \end{align} where $\alpha\in [0,2\pi]$ is the radian of the right-turning circular arc R$_{\alpha}$, and $A_1$--$A_5$ are constants in terms of $\rho$ and $\boldsymbol{z}_0$. \end{lemma}
The expressions of $A_1$--$A_5$ are given in the proof of this lemma in Appendix \ref{Appendix:B}.
Lemma \ref{LE:analytic_CS} shows that the radian $\alpha \in [0,2\pi]$ can be computed by finding the zeros of Eq.~\eqref{EQ:CS_alpha}. However, Eq.~\eqref{EQ:CS_alpha} is a transcendental equation that may have multiple zeros. Thus, the typical Newton-like iterative method or bisection method, proposed in \cite{McGee:2007}, may not find the desired zero that is related to the solution. In the following paragraph, a variant of $G_{cs}(\alpha)$ in Eq.~(\ref{EQ:CS_alpha}) will be presented so that Algorithm \ref{algorithm1} can be applied to finding the radian $\alpha$ of the right-turning circular arc.
By rearranging Eq.~\eqref{EQ:CS_alpha}, if $A_3 \cos \alpha +A_4 \sin \alpha \neq 0$ we have that $G_{cs}(\alpha)$ in Eq.~\eqref{EQ:CS_alpha} is equivalent to \begin{align}\label{EQ:bar_Gcs} \bar{G}_{cs}(\alpha) \overset{\triangle}{ =} \alpha + \frac{A_1 \sin \alpha + A_2 \cos \alpha + A_5}{A_3 \cos \alpha + A_4 \sin \alpha} \end{align} Differentiating this equation with respect to $\alpha$ leads to \begin{align}\label{EQ:dG_da} \frac{\mathrm{d} \bar{G}_{cs}(\alpha)}{\mathrm{d} \alpha} = 1&+ \big[(A_1 \cos \alpha - A_2 \sin \alpha )(A_3 \cos \alpha
+A_4 \sin \alpha) \notag\\
&- (A_1 \sin \alpha + A_2 \cos \alpha + A_5)
\times ( A_4 \cos \alpha - A_3 \sin \alpha )\big]/(A_3 \cos \alpha
+ A_4 \sin \alpha)^2 \end{align} By substituting the half-angle formulas \begin{align}\label{EQ:tan_x} \sin \alpha = \frac{2 \tan \frac{\alpha}{2}}{ 1 + \tan^2 \frac{\alpha}{2}}\ \text{and}\ \cos \alpha = \frac{1 - \tan^2 \frac{\alpha}{2}}{ 1 + \tan^2 \frac{\alpha}{2}} \end{align} into Eq.~\eqref{EQ:dG_da}, we have that $\tan(\alpha/2)$ is a zero of the following quartic polynomial: \begin{align} B_1 x^4 + B_2 x^3 + B_3 x^2 + B_4 x + B_5 = 0 \label{EQ:polynomial_quartic} \end{align} where \begin{align} \begin{split} B_1 & =-A_3+A_1A_3+A_4A_5-A_2A_4 \\ B_2 & = 2A_4+2A_3A_5\\ B_3 &=2A_1A_3-3A_2A_4+A_4A_5+A_2-A_5\\ B_4 & = 2A_4+2A_3A_5\\ B_5 & =A_3+A_1A_3-A_2A_4-A_4A_5 \end{split}\nonumber \end{align} The roots of any quartic polynomial can be readily obtained either by radicals or by standard polynomial solvers.
By the following remark, we shall show how to apply Algorithm \ref{algorithm1} to finding the shortest path of type R$_{\alpha}$S from the zeros of the quartic polynomial in Eq.~(\ref{EQ:polynomial_quartic}).
\begin{remark}\label{RE:procedure} If $E(t_m)\in \mathrm{int}(\mathcal{R}_1\cap \{x>0\})$ so that the solution path of the MTIP is of type $R_{\alpha}S$, then we can use the following procedure to find all the zeros of Eq.~\eqref{EQ:bar_Gcs}: \begin{description} \item (1) find all the real zeros of the quartic polynomial in Eq.~\eqref{EQ:polynomial_quartic} by either radicals or by a standard polynomial solver; \item (2) by combining Eq.~\eqref{EQ:tan_x} and the real zeros of Eq.~\eqref{EQ:polynomial_quartic}, we can find all the real zeros of Eq.~\eqref{EQ:dG_da}; \item (3) because all the real zeros of Eq.~\eqref{EQ:dG_da} are the extremas of Eq.~\eqref{EQ:bar_Gcs} and the function $\bar{G}_{cs}(\alpha)$ satisfies the conditions of Lemma \ref{LE:extreme_to_zero}, it follows that Algorithm \ref{algorithm1} can be used to find all the zeros of Eq.~\eqref{EQ:bar_Gcs} efficiently; \item (4) if we denote by $\alpha_1$, $\alpha_2$, $\ldots$, $\alpha_m$ the zeros of Eq.~\eqref{EQ:bar_Gcs}, any path of type R$_{\alpha_i}$S for $i=1,2,\ldots,m$ can be computed by a simple geometric analysis, and the shortest path of the type R$_{\alpha_i}$S is the solution of the MTIP. \end{description} \end{remark}
\subsubsection{The case of $E(t_m) \in \mathrm{int}(\mathcal{R}_2\cap\{x>0\})$}\label{Subsection:R1}
In view of Theorem \ref{LE:Occurance_R1}, if $E(t_m) \in \mathrm{int}(\mathrm{R}_2)$, we have $t_m = F(t_m)$, indicating that the solution path is the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$ (cf. Lemma \ref{LE:fixed_RDP}). For any terminal point in $\mathcal{R}_2\cap\{x>0\}$, the solution path of the RDP is of type LR \cite{Boissonnat:1994}, as shown by Fig.~\ref{Fig:Types_Region} in Appendix \ref{Appendix:A}.
By the following lemma, an equation in terms of the parameters of LR will be established. \begin{lemma}\label{LE:analytic_CC} If the minimum-time intercept point $E(t_m)$ occurs in $\mathrm{int}(\mathcal{R}_2\cap\{x>0\})$, then the solution of the MTIP is of type $L_{\beta}R_{\gamma}$ and it holds \begin{align}\label{EQ:G_cc} G_{cc}(\eta) \overset{\triangle}{=} \ & B_1 \eta^4 + B_2 \eta^3 + B_3 \eta^2 + B_4 \eta + B_5 + B_6 \cos \eta + B_7 \sin \eta + \eta (B_8 \cos \eta + B_9 \sin \eta) = 0 \end{align} where $\eta = \beta + \gamma$ is the sum of the radians of the left-turning circular arc $L_{\beta}$ and the right-turning circular arc $R_{\gamma}$, and $B_1$--$B_9$ are constants in terms of $\rho$ and $\boldsymbol{z}_0$. \end{lemma} The expressions of $B_1$--$B_9$ are given in the proof of this lemma in Appendix \ref{Appendix:B}.
The fourth derivative of $G_{cc}$ is expressed as \begin{align}\label{EQ:G_cc4} G_{cc}^{(4)}(\eta) =&\ (B_7 + 4B_8) \sin \eta + (B_6 - 4 B_9) \cos \eta +\eta (B_8 \cos \eta + B_9 \sin \eta) + 24 B_1. \end{align} Note that the form of $G_{cc}^{(4)}(\eta)$ is the same as that of $G_{cs}(\alpha)$ in Eq.~\eqref{EQ:CS_alpha}. Thus, all the zeros of $G_{cc}^{(4)}(\eta)$ can be obtained in a constant time by the procedure in Remark \ref{RE:procedure}. According to Lemma \ref{LE:extreme_to_zero}, Algorithm \ref{algorithm1} can be applied to finding the zeros of $G^{(i)}_{cc}(\eta)$ from the zeros of $G^{(i+1)}_{cc}(\eta)$. Hence, we are able to find all the real zeros of $G_{cc}(\eta)$ in Eq.~\eqref{EQ:G_cc} by applying Algorithm \ref{algorithm1} four times. Note that we have $t_m = \rho \eta$. Thus, the minimum positive zero of Eq.~\eqref{EQ:G_cc} gives the minimum interception time $t_m$.
\subsubsection{The case of $E(t_m) \in \mathrm{int}(\mathcal{R}_3)\cap \{x> 0\}$}\label{Subse:R3}
If $E(t_m) \in \mathrm{int}(\mathcal{R}_3\cap \{x>0\})$, not only do we need to check the fixed points of $F[E(t)]$ but also the fixed points of $L^-[E(t)]$ and $L^+[E(t)]$ according to Theorem \ref{LE:R3}. Since the path related to $F[E(t)]$ in $\mathrm{int}(\mathrm{R}_3\cap\{x>0\})$ is of type $RS$ \cite{Boissonnat:1994}, it follows that the fixed points of $F[E(t)]$ in this case can be obtained by the procedure in Remark \ref{RE:procedure}.
Even though the paths related to ${L}^-[E(t)]$ and ${L}^+[E(t)]$ are not the same as the solution of RDP from $\boldsymbol{z}_0$ to $E(t_m)$, their types are LR (cf. Lemma \ref{LE:type_L}). If we denote by $\eta> 0$ the sum of the radians of L and R, we have that Eq.~\eqref{EQ:G_cc} holds for both $t_m = L^+[E(t_m)]$ and $t_m=L^-[E(t_m)]$ according to the proof of Lemma \ref{LE:analytic_CC}. Therefore, the fixed points of ${L}^-[E(t)]$ and ${L}^+[E(t)]$ can be obtained by the same procedure as presented in Subsection \ref{Subsection:R1}.
\section{Numerical examples}\label{SE:Examples}
In this section, we present some examples to demonstrate the developments of the paper. In Subsection \ref{Subse:MTIP}, four examples of the MTIP are simulated, and in Subsection \ref{Subse:drift} an example is present to show how to apply the above algorithm to planning Dubins path in a constant drift field.
Note that the position vector $(x,y)$ is normalized so that the speed of pursuer is one in Section \ref{SE:Preliminary}. Thus, the units of variables in this section are omitted.
\subsection{Examples of the MTIP}\label{Subse:MTIP}
\subsubsection{Case A}
For case A, we present an example for which the function $F[E(t)]$ is continuous. We set
$\rho = 1$, $(\hat{x}_{0},\hat{y}_{0}) = (5,2)$, and $\boldsymbol{v} = (0.55,-0.55)$ to ensure that the half line $E(\cdot):[0,+\infty]$ does not intersect the two half circles $\{(x,y)\in C_r \cup C_l\ |\ y>0 \}$. According to Lemma \ref{TH:continuity}, these parameters guarantee that the function $F[E(t)]$ is continuous, as shown in Fig.~\ref{Fig:CaseAA1}. \begin{figure}
\caption{Case A: The profile of $F[E(t)]$ with respect to time.}
\label{Fig:CaseAA1}
\end{figure}
In such a case, the minimum intercept time $t_m$ is the minimum fixed point of $F[E(t)]$ according to Lemma \ref{LE:continuous}. By the developments in Subsections \ref{subsection:R2} and \ref{Subsection:R1}, all the fixed points of $F[E(t)]$ can be computed in a constant time. The time to compute the solution of case A is tested by MATLAB on a desktop with Intel(R) Core(TM)i3-4130U CPU@0.725GHz, showing that the solution is computed within $10^{-4}$ seconds. The minimum fixed point is computed as $18.45$, indicating that the minimum intercept time is $t_m = 18.45$. Since the target's velocity is constant, we have $E(t_m) = (\hat{x}_{0},\hat{y}_{0}) + \boldsymbol{v} t_m = (15.15,-8.15)$. In view of Lemma \ref{LE:fixed_RDP}, the solution path of the MTIP is the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m) = ((15.15,-8.15))$. Thus, the solution of the MTIP for case A is readily available by geometric analysis \cite{Boissonnat:1994}, and the solution path is shown in Fig.~\ref{Fig:CaseAA2}.
\begin{figure}
\caption{Case A: The solution path of the MTIP.}
\label{Fig:CaseAA2}
\end{figure}
\subsubsection{Case B}
The initial conditions of case A are set so that the minimum-time interception point $E(t_m)$ between the pursuer and the target occurs in $\mathcal{R}_1$. For case B, we choose $\rho = 1$, $(\hat{x}_{0},\hat{y}_{0}) = (1.2,0)$, and $\boldsymbol{v} = (-0.1,-0.1)$. These initial parameters are designed so that the minimum-time interception happens in $\mathcal{R}_2$. In addition, these initial parameters ensure that the half line $E(\cdot):[0,+\infty]$ does not intersect the two half circles $\{(x,y)\in C_r \cup C_l\ |\ y>0 \}$. Thus, according to Lemma \ref{TH:continuity}, the function $F[E(t)]$ for $t\geq 0$ is continuous, and it is plotted in Fig.~\ref{Fig:CaseBB1}
\begin{figure}
\caption{Case B: The profile of $F[E(t)]$ with respect to time.}
\label{Fig:CaseBB1}
\end{figure} According to Lemma \ref{LE:continuous}, the minimum fixed point of $F[E(t)]$ is the minimum time for the interception between the pursuer and the target, and it is computed as $5.43$. Then, the interception point is $E(t_m) = (\hat{x}_{0},\hat{y}_{0}) + \boldsymbol{v} t_m = (0.66,-0.54)$. With this terminal point, the solution path of the MTIP is the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$, and the solution path is readily available according to \cite{Boissonnat:1994}, as shown in Fig.~\ref{Fig:CaseBB2}.
\begin{figure}
\caption{Case B: The solution path of the MTIP.}
\label{Fig:CaseBB2}
\end{figure} The time to compute the solution of case B is tested by MATLAB on a desktop with Intel(R) Core(TM)i3-4130U CPU@0.725GHz, showing that the solution is computed within $10^{-4}$ seconds.
\subsubsection{Case C}
For case C, we set $\rho = 1$, $(\hat{x}_{0},\hat{y}_{0}) = (-3,0.8)$, and $\boldsymbol{v} = (0.15,0)$. These initial parameters are tailored so that the half line $E(\cdot):[0,t_f]$ intersects the two half circles $\{(x,y)\in C_r \cup C_l\ |\ y>0 \}$. In such a case, the function $F[E(t)]$ is not continuous (cf. Lemma \ref{TH:continuity}), as plotted in Fig.~\ref{Fig:CaseC1}. \begin{figure}
\caption{Case C: The profiles of $F[E(t)]$, $L^-[E(t)]$, and $L^+[E(t)]$ with respect to time.}
\label{Fig:CaseC1}
\end{figure} It is seen from Fig.~\ref{Fig:CaseC1} that the functions $L^-[E(t)]$ and $L^+[E(t)]$ exist over some intervals. This is reasonable because the target's path passes the subregion $\mathcal{R}_3$ over different intervals.
In the case that the function $F[E(t)]$ is discontinuous, checking the fixed points of $F[E(t)]$ is not enough to find the solution of the MTIP. One should also find the fixed points of $L^-[E(t)]$ and $L^+[E(t)]$. The procedures in Subsection \ref{SE:procedure} are applied, showing that the two functions $L^-[E(t)]$ and $L^+[E(t)]$ do not have a fixed point over their domains of definition. And, there is only one fixed point for the function $F[E(t)]$, which is computed as 3.15, i.e., $t_m = 3.15$.
Then, the interception point is $E(t_m) = (\hat{x}_{0},\hat{y}_{0}) + \boldsymbol{v} t_m = (-2.55,0.80)$. With this terminal point, the solution path of the MTIP is the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$ according to Lemma \ref{LE:fixed_RDP}.The solution path is of type LS, as shown in Fig.~\ref{Fig:CaseC2}. \begin{figure}
\caption{Case C: The solution path of the MTIP.}
\label{Fig:CaseC2}
\end{figure} The time to compute the solution of case C is tested by MATLAB on a desktop with Intel(R) Core(TM)i3-4130U CPU@0.725GHz, showing that the solution is computed within $10^{-4}$ seconds.
\subsubsection{Case D}
All the solution paths in the above three cases are the same as those of RDP from $\boldsymbol{z}_0$ to $E(t_m)$. In \cite{Mayer:2015}, an example was presented to show that the solution of the MTIP was not the same as that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$. However, it was not shown how to find the solution path. Thanks to the developments in Sections \ref{SE:Syntheses} and \ref{SE:Algorithm}, the example can be addressed efficiently as shown by the following paragraph.
For the example in \cite{Mayer:2015}, the minimum-turning radius is set as $\rho = 1$. The velocity vector and initial position of the target are given by $\boldsymbol{v} = (\frac{\sqrt{3}}{4\pi},0)$ and $(\hat{x}_{0},\hat{y}_{0}) = (-\frac{\sqrt{3}+1}{2},\frac{\sqrt{3}}{2})$, respectively.
The profiles of $F[E(t)]$, $L^-[E(t)]$, and $L^+[E(t)]$ are presented in Fig.~\ref{Fig:CaseA2}. \begin{figure}
\caption{Case D: The profiles of $F[E(t)]$, $L^-[E(t)]$, and $L^+[E(t)]$ with respect to time $t$.}
\label{Fig:CaseA2}
\end{figure} It is seen from Fig.~\ref{Fig:CaseA2} that the function $F[E(t)]$ is discontinuous. The procedures in Subsection \ref{SE:procedure} are directly applied to computing the solution of the MTIP, showing that the minimum intercept time $t_m$ is a fixed point of $L^+[E(t)]$. Then, according to Remark \ref{LE:type_L}, the solution path is of type L$_{\alpha^+}$R, which is consistent with the analysis in \cite{Mayer:2015}.
Having the value of $t_m$, the solution path of the MTIP for this example can be computed by applying the results in Subsection \ref{Subse:R3}, and it is presented in Fig.~\ref{Fig:CaseA1}. \begin{figure}
\caption{Case D: the solution path of the MTIP.}
\label{Fig:CaseA1}
\end{figure} It is apparent that the solution of RDP from $\boldsymbol{z}_0$ to $E(t_m)$ should be a single left turning circular arc, but the solution of the MTIP for case D is quite different. The time to compute the solution of case D is tested by MATLAB on a desktop with Intel(R) Core(TM)i3-4130U CPU@0.725GHz, showing that the solution is computed within $10^{-4}$ seconds.
\subsection{Path planning in constant drift field}\label{Subse:drift}
Consider an aerial/marine vehicle whose motion is described by \begin{align} \dot{\boldsymbol{z}}(t) = \left[ \begin{array}{c} \cos \theta(t) + w_x\\ \sin \theta(t) + w_y\\ u(t)/\rho \end{array} \right],\ \ \ \ \ u\in[-1,1] \label{EQ:problem_wind} \end{align} where $\boldsymbol{w}:=(w_x,w_y)$ is the constant drift field induced by local winds/currents, and all other variables have been defined in Eq.~(\ref{Eq:problem1}). We consider a path planning problem of steering Eq.~(\ref{EQ:problem_wind}) from $\boldsymbol{z}_0$ to a position $(x_f,y_f)\in \mathbb{R}^2$ so that the resulting path is the shortest. To the authors's best knowledge, there is not a robust algorithm in the literature to solve this path planning problem efficiently. In fact, such a problem is equivalent to the MTIP if $\boldsymbol{v} = -\boldsymbol{w}$ \cite{Bakolas:2013}. Thus, we can use the algorithm in Section \ref{SE:Algorithm} to find the shortest Dubins path for any terminal point $(x_f,y_f)$ and any constant drift field $\boldsymbol{w}$. Four examples of the shortest Dubins paths in constant drift field are presented in Fig.~\ref{Fig:CaseE}.
\begin{figure}
\caption{Shortest Dubins paths in constant drift fields.}
\label{Fig:CaseE}
\end{figure} It is apparent to see from Fig.~\ref{Fig:CaseE} that none-straight curves along the shortest Dubins paths in constant drift fields are not circular arcs any more, which is consistent to the results in \cite{Techy:2009}.
\section{Conclusions}
While the MTIP is a fundamental problem in pursuit-evasion engagements, it is not exaggerate to say that, without any assumption on the distance between the initial position of the pursuer and the target's trajectory, this problem has not been well addressed. In this paper, through introducing three functions $F[E(t)]$, $L^+[E(t)]$, and $L^-[E(t)]$ and analyzing their continuity properties, it was shown that the solution of the MTIP lies in a sufficient family of 4 candidates. Moreover, the geometric properties of each candidate path was established, indicating that each candidate is a circular arc followed by either a circular arc or a straight line segment. When the target's velocity is constant, the geometric properties enabled formulating some nonlinear equations so that the length of each candidate was determined by a specific zero of the nonlinear equations. An efficient and robust algorithm was developed to find all the zeros of sufficiently smooth functions. As a result, the solution of the MTIP can be computed within a constant time. Since the MTIP with a constant target's velocity is equivalent to the RDP in a constant drift field, the developments of this paper also allowed efficiently planning paths for aerial/marine vehicles in local winds/currents.
\appendix
\section{Proofs for the theorems in Section \ref{SE:Syntheses}} \label{Appendix:A}
Proof of Lemma \ref{TH:continuity}. Let us consider that the target's trajectory $E(t)$ intersects the half circle $\mathcal{C}_r\cap \{y> 0\}$ at a time $\bar{t}> 0$. Without loss of generality, assume that the target enters into the circle $\mathcal{C}_r$ at $\bar{t}$ from outside. In such a case, there exists a small $\varepsilon > 0$ so that $E(t+ \varepsilon) \in \mathrm{int} (\mathcal{D}_r)$ and $E(t- \varepsilon) \not \in \mathcal{D}_r$ where the notation $\mathrm{int}(\cdot)$ denotes the interior of a set. \begin{figure}
\caption{The regions for which the path of RDP terminates by different types. \cite{Boissonnat:1994}}
\label{Fig:Types_Region}
\end{figure} According to \cite{Boissonnat:1994}, the shortest paths of RDP from $\boldsymbol{z}_0$ to $E(\bar{t}-\varepsilon)$, $E(\bar{t})$, and $E(\bar{t}+\varepsilon)$ are of types RS$_{d(\varepsilon)}$, R, and LR$_{\alpha(\varepsilon)}$, respectively, as shown by Fig.~\ref{Fig:Types_Region}. \begin{figure}
\caption{The geometry for the path of RDP with terminal points $E(\bar{t}+\varepsilon)$, $E(\bar{t})$, and $E(\bar{t}-\varepsilon)$.}
\label{Fig:Intersection}
\end{figure} It is also known from \cite{Boissonnat:1994} that for any sufficiently small $\varepsilon>0$ we have $\alpha(\varepsilon)>\pi$, indicating that the function $F[E(t)]$ is discontinuous when the target moves from the circle $\mathcal{C}_r$ into its inside. For the case that the target's trajectory $E(t)$ intersects the left half circle $\mathcal{C}_l \cap \{y > 0\}$, it can be proven in the same way that the function $F[E(t)]$ is discontinuous at the intersection point.
From now on, we prove the necessity that the function $F[E(\bar{t})]$ is continuous if $E(\bar{t})\not \in \{(x,y)\in \mathcal{C}_r\cup \mathcal{C}_l \big | y> 0\}$. Let us choose a time $\bar{t}>0$ so that $E(\bar{t})\in \{(x,y)\in \mathbb{R}^2 \big | (x,y)\not \in \mathcal{D}_r\ \text{and}\ x>0\}$. According to \cite[Section 3.3]{Boissonnat:1994}, given a sufficiently small $\varepsilon > 0$, the path of RDP from $\boldsymbol{z}_0$ to $E(t+\delta )$ for any $\delta \in [-\varepsilon ,\varepsilon]$ is of type RS. Thus, the function $F[E(t)]$ is continuous at $\bar{t}$ if $E(\bar{t}) \in \{(x,y)\in \mathbb{R}^2 \big | (x,y)\not \in \mathcal{D}_r\ \text{and}\ x>0\}$.
We choose a time $\bar{t}>0$ so that $E(\bar{t})\in \mathrm{int}(\mathcal{D}_r)$. According to \cite[Section 3.3]{Boissonnat:1994}, given a sufficiently small $\varepsilon > 0$, the path of RDP from $\boldsymbol{z}_0$ to $E(t+\delta )$ for any $\delta \in [-\varepsilon ,\varepsilon]$ keeps being of type LR. Thus, the function $F[E(t)]$ is continuous at $\bar{t}$ if $E(\bar{t}) \in \mathrm{int}(\mathcal{D}_r)$.
Then, we consider the rest case that there exists a time $\bar{t}> 0 $ so that $E(\bar{t}) \in \mathcal{C}_r \cap \{y \leq 0\}$. Without loss of generality, let us assume that the target enters into the circle $\mathcal{C}_r$ at $\bar{t}$. Then, there exists a small $\varepsilon > 0$ so that $E(t+\varepsilon) \in \mathrm{int}(\mathcal{D}_r)$ and $E(t- \varepsilon )\not \in \mathrm{int}(\mathcal{D}_r)$. According to \cite[Section 3.3]{Boissonnat:1994}, the paths of RDP from $\boldsymbol{z}_0$ to $E(\bar{t}-\varepsilon )$, $E(\bar{t})$, and $E(\bar{t}+\varepsilon)$ are of types RS, R, and LR. Note that if $\varepsilon$ approaches to zero, the left turning circle $L$ and the straight line segment vanish. Thus, the function $F[E(t)]$ is continuous at $\bar{t}$ if $E(\bar{t}) \in \mathcal{C}_r \cap \{y \leq 0\}$.
If the time $\bar{t}>0$ is chosen so that $E(\bar{t}) \in \{x< 0\}$, the continuity of $F[E(t)]$ can be proven in the same way, completing the proof.
$\square$
\begin{lemma}\label{RE:1} The minimum time $t_m > 0$ for the pursuer to intercept the target is not smaller than $F[E(t_m)]$, i.e., $t_m \geq F[E(t_m)]$. \end{lemma} Proof of Lemma \ref{RE:1}. According to the definitions of $F[\cdot]$ and $E(\cdot)$, for any $t\geq 0$ the value $F[E(t)]$ denotes the minimum time for the Dubins vehicle to move from $\boldsymbol{z}_0$ to $E(t)$. Thus, the duration for the Dubins vehicle to move from $\boldsymbol{z}_0$ to $E(t_m)$ must be greater than $F[E(t_m)]$, completing the proof. $\square$
\iffalse
Proof of Lemma \ref{LE:continuous}. Consider a new function $\hat{F}(t) =t - F[E(t)]$, and we first prove that $t_m$ is a zero of $\hat{F}(t)$, i.e., $\hat{F}(t_m) = 0$. According to Lemma \ref{RE:1}, we must have $\hat{F}(t_m ) \geq 0 $. In order to prove that $\hat{F}(t_m) = 0$ holds, we by contradiction assume that $\hat{F}(t_m ) > 0 $. Note that $\hat{F}(0) < 0$ and $\hat{F}(t)$ is continuous. Thus, according to intermediate value theorem, there exists a time $\bar{t}\in (0,t_m)$ so that $\hat{F}(\bar{t}) = 0$, indicating $\bar{t} = F[E(\bar{t})]$. Thus, the target can be intercepted by the pursuer at a time $\bar{t}$ smaller than $t_m$. By contraposition, we have $t_m = F[E(t_m)]$. In case that the function $F[E(\cdot)]$ has multiple fixed points, the minimum interception time $t_m>0$ must be the minimum fixed point of $F[E(\cdot)]$, concluding the proof. $\square$
\fi
Proof of Lemma \ref{LE:fixed_RDP}. By contradiction, assume that the solution path of the MTIP does not follow that of the RDP from $\boldsymbol{z}_0$ to $E(t_m)$. Notice that $F[E(t_m)]$ denotes the time for the Dubins vehicle to follow the solution path of RDP from $\boldsymbol{z}_0$ to $E(t_m)$. Since the solution of RDP is the optimal path from $\boldsymbol{z}_0$ to $E(t_m)$, the contradicting assumption implies $t_m \neq F[E(t_m)]$. This contradicts the assumption of the lemma, completing the proof. $\square$
Proof of Theorem \ref{LE:Occurance_R1}. (1) By contradiction, assume that $t_m$ is not a fixed point of $F[E(\cdot)]$. According to Lemma \ref{RE:1}, this contradicting assumption implies $t_m > F[E(t_m)]$. In view of Lemma \ref{TH:continuity}, if $E(t_m) \in \mathrm{int}(\mathcal{R}_1\cup \mathcal{R}_2)$, we have that $F[E(\cdot)]$ is continuous around $t_m$. Thus, there exists a sufficiently small $\varepsilon>0$ so that \begin{align} t_m - \varepsilon > F[E(t_m - \varepsilon)]. \label{EQ:theorem_R1} \end{align} Since $E(t_m)\in \mathrm{int}(\mathcal{R}_1\cup \mathcal{R}_2)$ and $E(t)$ is continuous, if $\varepsilon > 0$ is sufficiently small, it holds that $E(t_m - \varepsilon)\in \mathrm{int}(\mathcal{R}_1\cup \mathcal{R}_2)$. Then, according to \cite[Theorem 2]{Ding:2019}, for any $t> F[E(t_m - \varepsilon)]$ there exists a feasible Dubins path with a duration of $t$ from initial condition $\boldsymbol{z}_0$ to the point $E(t_m-\varepsilon)$. For this reason, since $t_m - \varepsilon > F(t_m - \varepsilon )$ by Eq.~\eqref{EQ:theorem_R1}, it follows that there exists a feasible Dubins path with a duration of $t_m - \varepsilon$ to reach the point $E(t_m - \varepsilon)$. This means that the interception between the pursuer and the target can occur at a time $t_m - \varepsilon$, which is smaller than $t_m$. By contraposition, the proof of the first statement is completed.
(2) Combining the first statement and Lemma \ref{LE:fixed_RDP}, the second statement holds apparently. $\square$
Proof of Lemma \ref{LE:continuity}. Given a path of type L$_u$R$_v$, the terminal point is expressed by \cite{Boissonnat:1994} \begin{align} (x,y) = \left( \begin{array}{l} -\rho + 2\rho \cos u + \rho \cos (u+ \pi - v)\\ 2\rho \sin u + \rho \sin (u+ \pi - v) \end{array}\right) \nonumber \end{align} According to this formula, if the terminal point continuously changes, the values of $u$ and $v$ continuously change as well. Thus, the functions $L^-[E(t)]$ and $^+[E(t)]$ are continuous at $\bar{t}$, completing the proof. $\square$
Proof of Theorem \ref{LE:R3}. In view of the second statement of Lemma \ref{LE:Ding}, for any given $t$ in the open interval$ (L^-[E(t_m)],L^+[E(t_m)])$, any feasible Dubins path with a duration of $t$ cannot start from $\boldsymbol{z}_0$ and terminate at $E(t_m)$. Hence, in order to prove this theorem, we just need to prove that $t_m$ lies on the boundary of the interval $[F[E(t_m)],L^-[E(t_m)]]\cup [L^+[E(t_m)],+\infty)$.
By contradiction, assume $t_m \in (F[E(t_m)],L^-[E(t_m)])$. Note that $F[E(t)]$ and $L^-[E(t)]$ are continuous at $t_m$ according to Lemma \ref{TH:continuity} and Lemma \ref{LE:continuity}, respectively. Hence, there exists a sufficiently small $\varepsilon > 0$ so that $F[E(t_m - \varepsilon )] < t_m - \varepsilon < L^-[E(t_m - \varepsilon)]$. According to the third statement of Lemma \ref{LE:Ding}, for any $t$ in the interval $[F[E(t_m - \varepsilon)],L^-[E(t_m - \varepsilon)]]$, there exists a feasible Dubins path with a duration of $t$ from $\boldsymbol{z}_0$ to the point $E(t_m - \varepsilon)$. Thus, the target can be intercepted by the pursuer at the point $E(t_m - \varepsilon)$ by moving along a feasible Dubins path with a duration of $t_m - \varepsilon$, smaller than $t_m$. By contraposition, we have that $t_m$ does not lie in the interior of the interval $[F[E(t_m)],L^-[E(t_m)]]$, i.e., $t_m \not\in (F[E(t_m)],L^-[E(t_m)])$.
From now on, we assume that $t_m$ lies in the open interval $ (L^+[E(t_m)],+\infty)$ by contradiction. Note that $L^+[E(t)]$ is continuous at $t_m$ according to Lemma \ref{LE:continuity}. Therefore, there exists a sufficiently small $\varepsilon > 0$ so that $L^+(E(t_m - \varepsilon)) < t_m - \varepsilon$. Since $E(t_m)\in \mathrm{int}(\mathcal{R}_3\cap\{x>0\})$, if $\varepsilon>0$ is sufficiently small, it holds $E(t_m-\varepsilon)\in \mathrm{int}(\mathcal{R}_3\cap\{x>0\})$. Then, based on the third statement of Lemma \ref{LE:Ding}, for any $t$ in the semi-open interval $[L^+(E(t_m-\varepsilon)),+\infty)$, there exists a feasible Dubins path from $\boldsymbol{z}_0$ to the point $E(t_m - \varepsilon)$ with a duration of $t$, smaller than $t_m$. By contraposition, we have that $t_m$ does not lie in the open interval $(L^+[E(t_m)],+\infty)$.
To sum up, it is concluded that $t_m$ lies on the boundary of $[F[E(t_m)],L^-[E(t_m)]]\cup [L^+[E(t_m)],+\infty)$, indicating that $t_m \in \{t> 0 | t = F[E(t)], L^-[E(t)], or L^+[E(t)]\}$. Since $t_m$ is the minimum intercept time, we have that Eq.~\eqref{EQ:tm_R3} holds, completing the proof. $\square$
\section{Proofs for the lemmas in Section \ref{SE:Algorithm}}\label{Appendix:B}
Proof of Lemma \ref{LE:extreme_to_zero}. This lemma is a direct result of the intermediate value theorem. $\square$
Proof of Lemma \ref{LE:analytic_CS}. Let $\alpha \in [0,2\pi]$ be the radian of the right-turning circular arc R in the path of type RS, as presented in Fig.~\ref{Fig:LS1}. By geometric analysis, we have
\begin{align}
\boldsymbol{c}_0^r + \rho \left[ \begin{array}{c}
-\cos (\alpha )\\
\sin (\alpha ) \end{array} \right] + d\left[ \begin{array}{c} \cos(\frac{\pi}{2}- \alpha)\\ \sin (\frac{\pi}{2} - \alpha) \end{array} \right] =\left[ \begin{array}{c} x_f\\ y_f \end{array} \right]
\label{LS0} \end{align}
where $d>0$ is the length of the straight line segment and $(x_f,y_f)$ is the intercept point. \begin{figure}
\caption{The geometry of the solution path of RDP with a type of RS.}
\label{Fig:LS1}
\end{figure}
Because the pursuer and the target arrive at $(x_f,y_f)$ simultaneously, we have \begin{align} \rho \alpha + d = \frac{\sqrt{(x_f - \hat{x}_0)^2 + (y_f - \hat{y}_0)^2}}{\sqrt{v_x^2 + v_y^2}}\label{LS1} \end{align}
Note that we have \begin{align} \frac{y_f-\hat{y}_0}{x_f-\hat{x}_0}=\frac{v_y}{v_x}\nonumber \end{align} indicating \begin{align} y_f=\frac{v_y}{v_x}x_f-\frac{v_y}{v_x} \hat{x}_0+\hat{y}_0. \label{LS2} \end{align} Combining Eq.~(\ref{LS1}) and Eq.~(\ref{LS2}) yields \begin{align} d=\frac{x_f-\hat{x}_0}{v_x}-\rho \alpha \label{LS3} \end{align} Substituting Eq.~(\ref{LS2}) and Eq.~(\ref{LS3}) into Eq. ~\eqref{LS0} to eliminate $x_f$ and $d$, we have
\begin{align} \frac{a_1\!+\! a_2\sin \alpha\!-\!\rho \cos \alpha\!-\!\rho \alpha \sin \alpha}{a_3\!+\!a_2\cos \alpha\!+\!\rho \sin \alpha\!-\!\rho \alpha \cos \alpha} =\frac{v_x-\sin \alpha}{v_y-\cos \alpha} \label{LS4} \end{align}
where
\begin{align} \begin{cases} a_1=\rho\\ a_2=\frac{\hat{x}_0}{v_x}\\ a_3=-\hat{y}_0 + \frac{v_y}{v_x}\hat{x}_0 \end{cases}\nonumber \end{align}
Rearranging Eq.~(\ref{LS4}), we get \begin{align} A_1 \sin \alpha +A_2 \cos \alpha + \alpha(A_3 \cos \alpha +A_4 \sin \alpha) +A_5=0 \label{LS5} \end{align} where
\begin{align} \begin{cases} A_1=a_2 v_y + a_3 - \rho v_x\\ A_2=- a_1 - \rho v_y - a_2 v_x\\ A_3=\rho v_x \\ A_4=- \rho v_y \\ A_5=a_1 v_y - a_3 v_x + \rho \end{cases}\nonumber \end{align}
This concludes the proof of Lemma \ref{LE:analytic_CS}. $\square$
Proof of Lemma \ref{LE:analytic_CC}. Let $\alpha\in[0,2\pi]$ and $\beta\in[\pi,2\pi]$ be the radians of $L$ and $R$, respectively, as presented in Fig.~\ref{Fig:RL1}. \begin{figure}
\caption{The geometry of the solution path of RDP with a type of LR.}
\label{Fig:RL1}
\end{figure}
Note that $$\boldsymbol{c}_0^l+2\rho\left[ \begin{array}{c} \cos(\alpha)\\ \sin( \alpha) \end{array} \right] $$ and $$ \left[ \begin{array}{c} x_f\\ y_f \end{array} \right] + \rho \left[ \begin{array}{c} \cos(\alpha - \beta)\\ \sin( \alpha - \beta ) \end{array} \right]$$ are located at the same point (the center of the circle coinciding with R). Thus, we have \begin{align}
\boldsymbol{c}_0^l + 2\rho \left[ \begin{array}{c} \cos(\alpha)\\ \sin(\alpha ) \end{array} \right] = \left[ \begin{array}{c} x_f\\ y_f \end{array} \right] + \rho \left[ \begin{array}{c} \cos( \alpha - \beta )\\ \sin( \alpha - \beta) \end{array} \right] \label{RL2} \end{align}
As the pursuer and the target arrive at $(x_f,y_f)$ simultaneously, it follows that \begin{align} \rho(\alpha + \beta) = \frac{\sqrt{(x_f - \hat{x}_0)^2 + (y_f - \hat{y}_0)^2}}{\sqrt{v_x^2 + v_y^2}} \label{RL0} \end{align} We also have \begin{align} \frac{y_f-\hat{y}_0}{x_f-\hat{x}_0}=\frac{v_y}{v_x} \label{RL1} \end{align} Combining Eq.~\eqref{RL0} with Eq.~\eqref{RL1} leads to \begin{align} \begin{split}
\alpha + \beta=\frac{x_f-\hat{x}_0}{\rho v_x }= \frac{y_f-\hat{x}_0}{\rho vp_y}
\end{split} \label{RL11} \end{align} By the law of cosines, we also have \begin{align} 4 \rho^2+\rho^2-4\rho^2 \cos (2\pi-\beta)=s^2 \label{RL4} \end{align} which leads to \begin{align} \begin{cases} \cos \beta = \frac{5\rho^2-s^2}{4 \rho^2}\\ \sin \beta = \pm \sqrt{1 - \left[\frac{5\rho^2-s^2}{4 \rho^2}\right]^2} \end{cases} \label{RL5} \end{align} Denote by $\xi>0$ the angle between $x$-axis and the vector from $\boldsymbol{c}_0^l$ to $(x_f,y_f)$. Then, by using the law of sines, we have \begin{align} \frac{\sin (2\pi - \beta)}{s}=\frac{\sin (\alpha - \xi)}{\rho} \label{RL3} \end{align}
where $s = \| \boldsymbol{c}_0^l - (x_f,y_f)\|$ is the Euclidean distance between $\boldsymbol{c}_0^l$ and the interception point $(x_f,y_f)$. Taking into account the expression of $s$, from Eq.~(\ref{RL2}) we get \begin{align} 0 = s^2 - 4 \rho ( x_f + \rho) \cos \alpha - 4 \rho y_f \sin \alpha + 3 \rho^2 \label{RL7} \end{align} Note that $\cos \xi=\frac{x_f + \rho}{s}$ and $\sin \xi=\frac{y_f}{s}$. Then, according to Eq.~(\ref{RL3}), we have \begin{align} \sin \beta = \frac{y_f}{\rho} \cos \alpha - \frac{x_f + \rho}{\rho} \sin \alpha \label{RL9} \end{align} Substituting Eq.~(\ref{RL5}) and Eq.~(\ref{RL7}) into Eq.~(\ref{RL9}), we can get \begin{align} \begin{cases} \sin \alpha = \pm \frac{\rho ( x_f + \rho)\sqrt{1\!-\![\frac{s^2-5\rho^2}{4\rho^2}]^2}}{s^2} +y_f \frac{3\rho^2+s^2}{4\rho s^2}\\ \cos \alpha\!= \pm \!\frac{\rho y_f \sqrt{1\!-\![\frac{s^2-5\rho^2}{4\rho^2}]^2}}{s^2}\!+(x_ f + \rho)\frac{3\rho^2+s^2}{4\rho s^2} \end{cases} \label{RL10} \end{align}
Set $\eta = \alpha + \beta$. Then, we have \begin{align} \begin{split} \sin \eta = \sin \alpha \cos\beta + \cos \alpha \sin \beta\\ \cos \eta = \cos \alpha \cos \beta - \sin \alpha \sin \beta \end{split} \label{EQ:sin_cos_eta} \end{align} Substituting Eq.~(\ref{RL5}) and Eq.~(\ref{RL10}) into Eq.~\eqref{EQ:sin_cos_eta}, we have \begin{equation} \left\{ \begin{array}{l} \pm ( x_f +\rho )\frac{s^2-\rho^2}{2\rho s^2}\sqrt{1-[\frac{s^2-5\rho^2}{4\rho^2}]^2}=\sin \eta +\frac{(y_0^r - y_f )(s^4-6\rho^2s^2-3\rho^4)}{8\rho^3s^2}\\ \specialrule{0em}{0.5ex}{0.5ex} \pm y_f\frac{s^2-\rho^2}{2\rho s^2}\sqrt{1-[\frac{s^2-5\rho^2}{4\rho^2}]^2}
=\cos\eta+\frac{(x_0^r - x_f)(s^4-6\rho^2s^2-3\rho^4)}{8\rho^3s^2} \\ \end{array}\right. \label{RL13} \end{equation} where $(x_0^r,y_0^r) = \boldsymbol{c}_0^r$. Rewriting Eq.~(\ref{RL13}) yields \begin{align} y_f \sin\eta& + ( x_f +\rho) \cos \eta+\frac{s^4-6 \rho^2 s^2 - 3\rho^4}{8\rho^3} =0 \label{RL14} \end{align} According to Eq.~(\ref{RL11}), we have \begin{align} \begin{cases} x_f=\eta v_x \rho +\hat{x}_0\\ y_f=\eta v_y \rho +\hat{y}_0 \end{cases} \label{RL15} \end{align} Substituting Eq.~(\ref{RL15}) into Eq.~(\ref{RL14}), we eventually have \begin{align} F(\eta)& \overset{\triangle}{=}B_1\eta^4 + B_2\eta^3 + B_3\eta^2 + B_4\eta + B_5 +B_6\cos \eta + B_7 \sin \eta + \eta(B_8 \cos \eta + B_9 \sin \eta)= 0 \label{RL18} \end{align} where \begin{align} \begin{split} B_1&=C_a^2\\ B_2&=2C_aC_b\\ B_3&=C_b^2+2C_aC_c-6\rho^2C_a\\ B_4&=2C_cC_b-6\rho^2C_b\\ B_5&=C_c^2-6\rho^2C_c-3\rho^4\\ B_6&=8\rho^3(\rho + \hat{x}_0)\\ B_7&=8\rho^3\hat{y}_0\\ B_8&=8\rho^4V_x\\ B_9&=8\rho^4V_y \end{split}\nonumber
\end{align} with \begin{align} \begin{split} C_a &=V_x^2\rho^2+V_y^2\rho^2\\ C_b& =+2(\rho + \hat{x}_0)V_x\rho+2\hat{y}_0V_y\rho\\ C_c&=(\rho + \hat{x}_0)^2+\hat{y}_0^2 \end{split}\nonumber
\end{align}
\end{document} |
\begin{document}
\setcounter{page}{1}
\title[ The Stein restriction problem on the torus ]{A note on the Stein restriction conjecture and the restriction problem on the torus }
\author[D. Cardona]{Duv\'an Cardona} \address{
Duv\'an Cardona:
\endgraf
Department of Mathematics
\endgraf
Pontificia Universidad Javeriana.
\endgraf
Bogot\'a
\endgraf
Colombia
\endgraf
{\it E-mail address} {\rm duvanc306@gmail.com}
}
\subjclass[2010]{42B37.}
\keywords{Stein Restriction Conjecture, Fourier Analysis, Clifford's torus}
\begin{abstract} In this note revision we discuss the Stein restriction problem on arbitrary $n$-torus, $n\geq 2$. In contrast with the usual cases of the sphere, the parabola and the cone, we provide necessary and sufficient conditions on the Lebesgue indices, by finding conditions which are independent of the dimension $n$.
\end{abstract} \maketitle
\section{Introduction} This note is devoted to the Stein restriction problem on the torus $\mathbb{T}^n,$ $n\geq 2.$ In harmonic analysis, the Stein restriction problem for a smooth hypersurface $S\subset \mathbb{R}^n,$ asks for the conditions on $p$ and $q,$ $1\leq p,q<\infty,$ satisfying \begin{equation}\label{Stein}
\Vert \hat{f}|_{S}\Vert_{L^q({S},d\sigma)}:= \left(\int\limits_{S}|\hat{f}(\omega)|^qd\sigma(\omega)\right)^{\frac{1}{q}}\leq C\Vert f\Vert_{L^p(\mathbb{R}^n)},
\end{equation} where $d\sigma$ is a surface measure associated to $S,$ the constant $C>0$ is independent of $f,$ and $\widehat{f}|_{S}$ denotes the Fourier restriction of $f$ to $S,$ where \begin{equation}
\hat{f}(\xi)=\int\limits_{\mathbb{R}^n}e^{-i2\pi x\cdot \xi} f(x)dx, \end{equation} is the Fourier transform of $f.$ Let us note that for $p=1,$ the Riemann-Lebesgue theorem implies that $\hat{f}$ is a continuous function on $\mathbb{R}^n$ and we can restrict $\hat{f}$ to every subset $S\subset \mathbb{R}^n.$ On the other hand, if $f\in L^2(\mathbb{R}^n),$ the Plancherel theorem gives $\Vert f \Vert_{L^2(\mathbb{R}^n)}=\Vert \hat{f} \Vert_{L^2(\mathbb{R}^n)}$ and the Stein restriction problem is trivial by considering that every hypersurface is a subset in $\mathbb{R}^n$ with vanishing Lebesgue measure. So, for $1<p<2,$ a general problem is to find those hypersurfaces $S,$ where the Stein restriction problem has sense. However, the central problem in the restriction theory is the following conjecture (due to Stein). It is of particular interest because it is related to Bochner-Riesz multipliers and the Kakeya conjecture. \begin{conjecture}\label{ConjectureofStein}
Let $S=\mathbb{S}^{n-1}=\{x\in \mathbb{R}^n:|x|=1\}$ be the $(n-1)$-sphere and let $d\sigma$ be the corresponding surface measure. Then \eqref{Stein} holds true if and only if $1\leq p<\frac{2n}{n+1}$ and $q\leq p'\cdot \frac{n-1}{n+1},$ where $p'=p/p-1.$ \end{conjecture} That the inequalities $1\leq p<\frac{2n}{n+1}$ and $q\leq p'\cdot \frac{n-1}{n+1},$ are necessary conditions for Conjecture \ref{ConjectureofStein} is a well known fact. In this setting, a celebrated result by Tomas and Stein (see e.g. Tomas \cite{Tomas}) shows that \begin{equation}
\Vert \hat{f}|_{\mathbb{S}^{n-1}} \Vert_{L^2(\mathbb{S}^{n-1},d\sigma)}\leq C_{p,n}\Vert f\Vert_{L^{p}(\mathbb{R}^n)} \end{equation} holds true for every $1\leq p\leq \frac{2n+2}{n+3}.$ Surprisingly, a theorem due to Bourgain shows that the Stein restriction conjecture is true for $1<p<p_n$ where $p_n$ is defined inductively and $\frac{2n+2}{n+3}<p_n<\frac{2n}{n+1}.$ For instance, $p(3) = 31/23$. We refer the reader to Tao \cite{Tao2003} for a good introduction and some advances to the restriction theory.
In this paper we will consider the $n$-dimensional torus $\mathbb{T}^n=(\mathbb{S}^1)^n$ modelled on $\mathbb{R}^{2n},$ this means that \begin{equation}\label{DefiCliffordTorues}
\mathbb{T}^n=\{(x_{1,1},x_{1,2},x_{2,1},x_{2,2},\cdots,x_{n,1},x_{n,2}):x_{\ell,1}^2+x_{\ell,2}^2=1,\,1\leq \ell\leq n\}. \end{equation} In this case $$\mathbb{T}^n\subset {\sqrt{n}}\,\mathbb{S}^{2n-1} \subset \mathbb{R}^{2n}.$$ In order to illustrate our results, we will discuss the case $n=2,$ where \begin{equation}\label{DefiCliffordTorues2} \mathbb{T}^2\subset {\sqrt{2}}\,\mathbb{S}^{3} \subset \mathbb{R}^{4}.\end{equation} As it is well known, the $n$-dimensional torus can be understood of different ways. Topologically, $\mathbb{T}^n\sim \mathbb{S}^1\times \cdots \times \mathbb{S}^1,$ where the circle $\mathbb{S}^1$ can be identified with the unit interval $[0,1),$ where we have identified $0\sim 1.$ The case $n=2,$ implies that $\mathbb{T}^2\sim \mathbb{S}^1\times \mathbb{S}^1.$ From differential geometry, a stereographic projection $\pi$ from $\mathbb{S}^3\setminus \{N\}$ into $\mathbb{R}^3$ gives the following embedding of $(1/\sqrt{2})\mathbb{T}^2\subset \mathbb{S}^3,$ \begin{equation}\label{anothertorus}
\dot{\mathbb{T}}^2=\{((\sqrt{2}+\cos(\phi))\cos(\theta),(\sqrt{2}+\cos(\phi))\sin(\theta),\sin(\phi)\in \mathbb{R}^3: 0\leq \theta,\phi<2\pi \}, \end{equation} of the 2-torus in $\mathbb{R}^3.$ At the same time, the Fourier analysis and the geometry on the torus can be understood better by the description of the torus given in \eqref{DefiCliffordTorues}. So, we will investigate the restriction problem on the torus by using \eqref{DefiCliffordTorues2} instead of \eqref{anothertorus}. In this case, the Stein restriction conjecture for $S=\mathbb{S}^3$ assures that \eqref{Stein} holds true for every $1\leq p<\frac{8}{5}$ and $q\leq \frac{3}{5}p'.$ However, we will prove the following result, where we characterise the Stein restriction problem on $\mathbb{T}^2$. \begin{theorem}\label{ThrcardonaRest2018} Let $f\in L^{p}(\mathbb{R}^4).$ Then there exists $C>0,$ independent of $f$ and satisfying the estimate \begin{equation}\label{cardonaRest2018}
\Vert \hat{f}|_{\mathbb{T}^2}\Vert_{L^q({\mathbb{T}^2},d\sigma)}:= \left(\int\limits_{\mathbb{T}^2}|\hat{f}(\xi_1,\xi_2,\eta_1,\eta_2)|^qd\sigma(\xi_1,\xi_2,\eta_1,\eta_2)\right)^{\frac{1}{q}}\leq C\Vert f\Vert_{L^{p}(\mathbb{R}^4)}, \end{equation} if and only if $1\leq p< \frac{4}{3}$ and $q\leq p'/3.$ Here, $d\sigma(\xi_1,\xi_2,\eta_1,\eta_2)$ is the usual surface measure associated to $\mathbb{T}^2.$ \end{theorem} An important difference between the restriction problem on the $n$-torus, $n\geq 2,$ and the Stein-restriction conjecture come from the curvature notion. For example, the sphere $\mathbb{S}^2$, has Gaussian curvature non-vanishing, in contrast with the 2-torus $\mathbb{T}^2$ where the Gaussian curvature vanishes identically. In the general case, let us observe that the Stein conjecture for $S=\mathbb{S}^{2n-1}$ asserts that \eqref{Stein} holds true for all $1\leq p<\frac{ 4n}{2n+1}$ and $q\leq \frac{2n-1}{2n+1}p'.$ Curiously, the situation for the $n$-dimensional torus is very different, as we will see in the following theorem. \begin{theorem}\label{ThrcardonaRestTn2018} Let $f\in L^{p}(\mathbb{R}^{2n}),$ $n\geq 2.$ Then there exists $C>0,$ independent of $f$ and satisfying \begin{equation}\label{cardonaRestTn2018}
\Vert \hat{f}|_{\mathbb{T}^{n}}\Vert_{L^q({\mathbb{T}^{n}},d\sigma_n)}\leq C_n\Vert f\Vert_{L^{p}(\mathbb{R}^{2n})}, \end{equation} if and only if $1\leq p< \frac{4}{3}$ and $q\leq p'/3.$ Here, $d\sigma_n$ is the usual surface measure associated to $\mathbb{T}^{n}.$ \end{theorem} \begin{remark} By a duality argument we conclude the following fact: if
$F\in L^{q'}(\mathbb{T}^{n},d\sigma_n),$ then there exists $C>0,$ independent of $F$ and satisfying \begin{equation}
\Vert (Fd\sigma_n)^{\vee} \Vert_{L^{p'}(\mathbb{R}^{2n})}=\left\Vert \int\limits_{ \mathbb{T}^n } e^{i2\pi x\cdot \xi}F(\xi) d\sigma_n(\xi) \right\Vert_{L^{p'}({\mathbb{R}^{2n}})}\leq C_n\Vert F\Vert_{L^{q'}{(\mathbb{T}^n,d\sigma_n)}}, \end{equation} if and only if $p'>4$ and $q'\geq (p'/3)'.$ We have denoted by $(Fd\sigma_n)^{\vee}$ the inverse Fourier transform of the measure $\mu:=Fd\sigma_n.$ \end{remark}
We end this introduction by summarising the progress on the restriction conjecture as follows. Indeed, we refer the reader to,
\begin{itemize}
\item Fefferman \cite{Fefferman1970} and
Zygmund \cite{Zygmund74} for the proof of the restriction conjecture in the case $n=2$ (which is \eqref{cardonaRestTn2018} for $n=1$).
\item Stein \cite{Stein1986}, Tomas \cite{Tomas} and Strichartz \cite{Strichartz1977}, for the restriction problem in higher dimensions, with sharp $(L^q,L^2)$ results
for hypersurfaces with nonvanishing Gaussian curvature. Some more general classes of surfaces were treated by A. Greenleaf \cite{Greenlaf1981}. \item Bourgain \cite{Bourgain1991,Bourgain1995b}, Wolff \cite{Wolff1995}, Moyua, Vargas, Vega and Tao \cite{Moyua1996,Moyua1999,Tao1998} who established the so-called bilinear approach. \item Bourgain and Guth \cite{BoutgainGuth2011}, Bennett, Carbery and Tao \cite{BeCarTao2006}, by the progress on the case of nonvanishing curvature, by making use of multilinear restriction estimates. \item Finally, Buschenhenke, M\"uller and Vargas \cite{Muller}, for a complete list of references as well as the progress on the restriction theory on surfaces of finite type. \end{itemize} The main goal of this note is to give a simple proof of the restriction problem on the torus. This work is organised as follows. In Section \ref{Sec2} we prove Theorem \ref{ThrcardonaRest2018}. We end this note with the proof of Theorem \ref{ThrcardonaRestTn2018}. Sometimes we will use $(\mathscr{F}f)$ for the 2-dimensional Fourier transform of $f$ and $(\mathscr{F}_{\mathbb{R}^n}u)$ for the Fourier transform of a function $u$ defined on $\mathbb{R}^n.$
\section{Proof of Theorem \ref{ThrcardonaRest2018}}\label{Sec2} In this note we will use the standard notation used for the Fourier analysis on $\mathbb{R}^n$ and the torus (see e.g. Ruzhansky and Turunen \cite{Ruz}). Throughout this section we will consider the 2-torus $\mathbb{T}^2,$ \begin{equation}
\mathbb{T}^2=\{(x_{1},x_{2},y_{1},y_{2}):x_{1}^2+x_{2}^2=1,\,y_{1}^2+y_{2}^2=1 \}=\mathbb{S}^1_{(x_1,x_2)}\times \mathbb{S}^1_{(y_1,y_2)}\subset \mathbb{R}^4. \end{equation} Here, $\mathbb{T}^2$ will be endowed with the surface measure $$d\sigma(\xi_1,\xi_2,\eta_1,\eta_2)=d\sigma(\xi_1,\xi_2)d\sigma(\eta_1,\eta_2),$$ where $d\sigma(\xi_1,\xi_2)$ is the usual `surface measure' defined on $\mathbb{S}^1.$ Indeed, if $(\xi_1,\xi_2)\equiv (\xi_1(\varkappa),\xi_2(\varkappa))=(\cos(2\pi \varkappa),\sin(2\pi\varkappa)),$ $0\leq \varkappa< 1 ,$ then $d\varkappa=d\sigma(\xi_1,\xi_2).$
Conjecture \ref{ConjectureofStein} has been proved by Fefferman for $n=2,$ the corresponding announcement is the following (see Fefferman \cite{Fefferman1970} and Zygmund \cite{Zygmund1974}). \begin{theorem}[Fefferman restriction Theorem]\label{FeffermanStein}
Let $S=\mathbb{S}^{1}=\{x\in \mathbb{R}^2:|x|=1\}$ be the $1$-sphere and let $d\sigma$ be the corresponding `surface measure'. Then \eqref{Stein} holds true if and only if $1\leq p<\frac{4}{3}$ and $q\leq p'/3,$ where $p'=p/p-1.$ \end{theorem}
In order to prove Theorem \ref{ThrcardonaRest2018}, let us consider $1\leq p<\frac{4}{3}$, $q\leq p'/3$ and $f\in L^{p}(\mathbb{R}^4) .$ By the argument of density we can assume that $f\in C^\infty_{c}(\mathbb{R}^4).$ If $(\xi_1,\xi_2,\eta_1,\eta_2)\in \mathbb{T}^2,$ then \begin{equation}
\widehat{f}(\xi_1,\xi_2,\eta_1,\eta_2)=\int\limits_{\mathbb{R}^4}e^{-i2\pi (x\cdot\xi+y\cdot \eta)}f(x,y)dy\,dx,\,\,x=(x_1,x_2),\,y=(y_1,y_2). \end{equation} By the Fubini theorem we can write \begin{align*}
\widehat{f}(\xi_1,\xi_2,\eta_1,\eta_2)
=\int\limits_{\mathbb{R}^2}e^{-i2\pi x\cdot \xi}(\mathscr{F}_{y\rightarrow \eta}{f}(x,\cdot))(\eta)dx,\,\,\eta=(\eta_1,\eta_2), \end{align*} where $(\mathscr{F}_{y\rightarrow \eta}{f}(x,\cdot))(\eta)=\widehat{f}(x,\eta)$ is the 2-dimensional Fourier transform of the function $f(x,\cdot),$ for every $x\in \mathbb{R}^2.$ By writing \begin{equation}
\widehat{f}(\xi_1,\xi_2,\eta_1,\eta_2)=\mathscr{F}_{x\rightarrow \xi}(\mathscr{F}_{y\rightarrow \eta}{f}(x,\cdot))(\eta))(\xi), \end{equation} for $1\leq p<\frac{4}{3}$ and $q\leq p'/3,$ the Fefferman restriction theorem gives, \begin{equation}
\Vert \widehat{f}(\xi_1,\xi_2,\eta_1,\eta_2)\Vert_{L^q(\mathbb{S}^1,d\sigma(\xi))}\leq C\Vert \widehat{f}(x,\eta) \Vert_{L^p(\mathbb{R}^2_x)}. \end{equation} Now, let us observe that \begin{align*}
\Vert \widehat{f}|_{\mathbb{T}^2}\Vert_{L^q({\mathbb{T}^2},d\sigma)} &=\Vert\widehat{f}(\xi,\eta) \Vert_{L^q((\mathbb{S}^1,d\sigma(\eta));L^q(\mathbb{S}^1,d\sigma(\xi)))} \\
&\leq C\Vert \Vert \widehat{f}(x,\eta) \Vert_{L^p(\mathbb{R}^2_x)} \Vert_{L^q(\mathbb{S}^1,d\sigma(\eta))}=: C \Vert \widehat{f}(x,\eta) \Vert_{L^q((\mathbb{S}^1,d\sigma(\eta)); L^p(\mathbb{R}^2_x) )}\\
&:=I. \end{align*} Now, we will estimate the right hand side of the previous inequality. First, if we assume that $4/3\leq q<p'/3,$ then $p\leq q$ and the Minkowski integral inequality gives, \begin{align*}
I&=\left(\int\limits_{\mathbb{S}^1}\left(\int\limits_{\mathbb{R}^2}|\widehat{f}(x,\eta)|^pdx\right)^{\frac{q}{p}} d\sigma(\eta )\right)^{\frac{1}{q}} \leq \left(\int\limits_{\mathbb{R}^2}\left(\int\limits_{\mathbb{S}^1}|\widehat{f}(x,\eta)|^q d\sigma(\eta)\right)^\frac{p}{q} dx\right)^{\frac{1}{p}}\\
& \lesssim \left(\int\limits_{\mathbb{R}^2} \int\limits_{\mathbb{R}^2}|{f}(x,y)|^p dy dx\right)^{\frac{1}{q}}=\Vert f\Vert_{L^{p}(\mathbb{R}^4)}, \end{align*} where in the last inequality we have used the Fefferman restriction theorem. So we have proved that \eqref{cardonaRest2018} holds true for $4/3\leq q<p'/3.$ Now, if $q<\frac{4}{3},$ then we can use the finiteness of the measure $d\sigma(\xi,\eta)$ to deduce that \begin{equation}
\Vert \widehat{f}|_{\mathbb{T}^2}\Vert_{L^q({\mathbb{T}^2},d\sigma)}\lesssim \Vert \widehat{f}|_{\mathbb{T}^2}\Vert_{L^\frac{4}{3}({\mathbb{T}^2},d\sigma)} \leq C\Vert f\Vert_{L^{p}(\mathbb{R}^4)} \end{equation} holds true for $1\leq p<\frac{4}{3}.$ Now, we will prove the converse announcement. So, let us assume that $p$ and $q$ are Lebesgue exponents satisfying \eqref{cardonaRest2018} with a constant $C>0$ independent of $f\in L^p(\mathbb{R}^4).$ If $g\in C^\infty_{c}(\mathbb{R}^2),$ let us define the function $f$ by $f(x,y)=g(x)g(y).$ The inequality, \begin{equation}
\Vert \widehat{f}|_{\mathbb{T}^2}\Vert_{L^q({\mathbb{T}^2},d\sigma)}:= \left(\int\limits_{\mathbb{T}^2}|\widehat{f}(\xi_1,\xi_2,\eta_1,\eta_2)|^qd\sigma(\xi_1,\xi_2,\eta_1,\eta_2)\right)^{\frac{1}{q}}\leq C\Vert f\Vert_{L^{p}(\mathbb{R}^4)}, \end{equation} implies that \begin{equation}
\Vert \widehat{g}|_{\mathbb{S}^1}\Vert_{L^q({\mathbb{S}^1},d\sigma)}:= \left(\int\limits_{\mathbb{S}^1}|\widehat{g}(\xi_1,\xi_2)|^qd\sigma(\xi_1,\xi_2)\right)^{\frac{1}{q}}\leq C\Vert g\Vert_{L^{p}(\mathbb{R}^2)}. \end{equation}But, according with the Fefferman restriction theorem, the previous inequality only is possible for arbitrary $g\in C^\infty_{c}(\mathbb{R}^2),$ if $1\leq p<\frac{4}{3}$ and $q\leq p'/3.$
\section{Proof of Theorem \ref{ThrcardonaRestTn2018}}
Let us consider the $n$-dimensional torus \begin{equation}
\mathbb{T}^n=\{(x_{1,1},x_{1,2},x_{2,1},x_{2,2},\cdots,x_{n,1},x_{n,2}):x_{\ell,1}^2+x_{\ell,2}^2=1,\,1\leq \ell\leq n\}. \end{equation} We endow to $\mathbb{T}^n$ with the surface measure \begin{equation}
d\sigma_n (\xi_{1,1},\xi_{1,2},\xi_{2,1},\xi_{2,2},\cdots,\xi_{n,1},\xi_{n,2})=\bigotimes_{j=1}^n d\sigma(\xi_{j,1},\xi_{j,2}), \end{equation} where $d\sigma$ is the `surface measure' on $\mathbb{S}^1.$ In order to prove Theorem \ref{ThrcardonaRestTn2018} we will use induction on $n.$ The case $n=2$ is precisely Theorem \ref{ThrcardonaRest2018}. So, let us assume that for some $n\in\mathbb{N},$ there exists $C_n$ depending only on the dimension $n,$ such that \begin{equation}
\Vert (\mathscr{F}_{\mathbb{R}^n}{u})|_{\mathbb{T}^{n}}\Vert_{L^q({\mathbb{T}^{n}},d\sigma_n)}\leq C_n\Vert u\Vert_{L^{p}(\mathbb{R}^{2n})}, \end{equation} for every function $u\in L^{p}(\mathbb{R}^{2n}).$ If $f\in C^\infty_{c}(\mathbb{R}^{2n+2})\subset L^p(\mathbb{R}^{2n+2}),$ $1\leq p<\frac{4}{3}$ and $q\leq p'/3,$ by using the approach of the previous section, we can write \begin{align*}
\widehat{f}(\xi_1,\xi_2,\eta)
=\int\limits_{\mathbb{R}^2}e^{-i2\pi x\cdot \xi}(\mathscr{F}_{y\rightarrow \eta}{f}(x,\cdot))(\eta)dx,\,\,\eta\in \mathbb{R}^n. \end{align*} By applying the Fefferman restriction theorem we deduce \begin{equation}
\Vert\widehat{f}(\cdot,\cdot,\eta) \Vert_{L^q(\mathbb{S}^1,d\sigma(\xi))}\leq \Vert \mathscr{F}_{y\rightarrow \eta}{f}(x,\cdot))(\eta) \Vert_{L^p(\mathbb{R}^2_x)}. \end{equation} Now, by using that
\begin{align*}
\Vert \widehat{f}|_{\mathbb{T}^{n+1}}\Vert_{L^q({\mathbb{T}^{n+1}},d\sigma_{n+1})} &=\Vert\widehat{f}(\xi_1,\xi_2,\eta) \Vert_{L^q((\mathbb{T}^n,d\sigma_n(\eta));L^q(\mathbb{S}^1,d\sigma(\xi)))} \\
&\leq C\Vert \Vert \widehat{f}(x_{1,1},x_{1,2},\eta) \Vert_{L^p(\mathbb{R}^2_x)} \Vert_{L^q(\mathbb{T}^n,d\sigma_n(\eta))}\\
&=: C \Vert \widehat{f}(x,\eta) \Vert_{L^q((\mathbb{T}^n,d\sigma_n(\eta)); L^p(\mathbb{R}^2_x) )}\\
&:=II,
\end{align*} for $4/3\leq q<p'/3,$ $p\leq q,$ and the Minkowski integral inequality, we have \begin{align*}
II&=\left(\int\limits_{\mathbb{T}^n}\left(\int\limits_{\mathbb{R}^2}|\widehat{f}(x,\eta)|^pdx\right)^{\frac{q}{p}} d\sigma_n(\eta )\right)^{\frac{1}{q}} \leq \left(\int\limits_{\mathbb{R}^2}\left(\int\limits_{\mathbb{T}^n}|\widehat{f}(x,\eta)|^q d\sigma_n(\eta)\right)^\frac{p}{q} dx\right)^{\frac{1}{p}}\\
& \lesssim_n \left(\int\limits_{\mathbb{R}^{2}} \int\limits_{\mathbb{R}^{2n}}|{f}(x,y)|^p dy dx\right)^{\frac{1}{p}}=\Vert f\Vert_{L^{p}(\mathbb{R}^{2n+2})}, \end{align*} where in the last inequality we have used the induction hypothesis. So, we have proved Theorem \ref{ThrcardonaRestTn2018} for $4/3\leq q<p'/3.$ The case $q<\frac{4}{3}$ now follows from the finiteness of the measure $d\sigma_{n+1}$. That $1\leq p<\frac{4}{3}$ and $q\leq p'/3,$ are necessary conditions for \eqref{cardonaRestTn2018} can be proved if we replace $f$ in \eqref{cardonaRestTn2018} by a function of the form \begin{equation}
f(x_{1,1},x_{1,2},x_{2,1},x_{2,2},\cdots,x_{n,1},x_{n,2})=\prod_{j=1}^n g(x_{j,1},x_{j,2}),\,\,g \in C^\infty_c(\mathbb{R}^2). \end{equation} Indeed, we automatically have \begin{equation}
\Vert \widehat{g}|_{\mathbb{S}^1}\Vert_{L^q({\mathbb{S}^1},d\sigma)}:= \left(\int\limits_{\mathbb{S}^1}|\widehat{g}(\xi_1,\xi_2)|^qd\sigma(\xi_1,\xi_2)\right)^{\frac{1}{q}}\leq C\Vert g\Vert_{L^{p}(\mathbb{R}^2)}. \end{equation} Consequently, the Fefferman restriction theorem, shows that the previous inequality only is possible for arbitrary $g\in C^\infty_{c}(\mathbb{R}^2),$ if $1\leq p<\frac{4}{3}$ and $q\leq p'/3.$\\ An usual argument of duality applied to Theorem \ref{ThrcardonaRestTn2018}, allows us to deduce the following result. \begin{corollary}\label{ThrcardonaRestTn2018dual} Let $F\in L^{q'}(\mathbb{T}^{n},d\sigma_n).$ Then there exists $C>0,$ independent of $F$ and satisfying \begin{equation}\label{cardonaRestTn2018dual}
\Vert (Fd\sigma_n)^{\vee} \Vert_{L^{p'}(\mathbb{R}^{2n})}=\left\Vert \int\limits_{ \mathbb{T}^n } e^{i2\pi x\cdot \xi}F(\xi) d\sigma_n(\xi) \right\Vert_{L^{p'}({\mathbb{R}^{2n}})}\leq C_n\Vert F\Vert_{L^{q'}{(\mathbb{T}^n,d\sigma_n)}}, \end{equation} if and only if $p'>4$ and $q'\geq (p'/3)'.$ Here, $d\sigma_n$ is the usual surface measure associated to $\mathbb{T}^{n}$ and $r':=r/r-1.$ \end{corollary}
\noindent {\bf Acknowledgement}. I would like to thanks Felipe Ponce from \textit{Universidad Nacional de Colombia} who introduced me to the restriction problem.
\end{document} |
\begin{document}
\title{A heralded quantum gate between remote quantum memories} \author{P.~Maunz} \email{pmaunz@umd.edu} \author{S.~Olmschenk} \author{D.~Hayes} \author{D.~N.~Matsukevich} \affiliation{Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742} \author{L.-M.~Duan} \affiliation{FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, MI 48109} \author{C.~Monroe} \affiliation{Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742}
\begin{abstract}
We demonstrate a probabilistic entangling quantum gate between two
distant trapped ytterbium ions. The gate is implemented between the
hyperfine ``clock'' state atomic qubits and mediated by the
interference of two emitted photons carrying frequency encoded
qubits. Heralded by the coincidence detection of these two photons,
the gate has an average fidelity of $90\pm 2 \%$. This entangling gate
together with single qubit operations is sufficient to generate
large entangled cluster states for scalable quantum computing. \end{abstract} \pacs{03.67.Bg, 42.50.Ex, 03.67.Pp}
\maketitle
The conventional model of quantum computing, the quantum circuit model~\cite{deutsch_quantum_1989, blatt_entangled_2008}, consists of unitary quantum gate operations followed by measurements at the end of the computation process to read out the result. An equivalent model of quantum computation, which may prove easier to implement, is the ``one-way'' quantum computer~\cite{ briegel_persistent_2001,
raussendorf_one-way_2001, raussendorf_computational_2002}, where a highly entangled state of a large collection of qubits is prepared and local operations and projective measurements complete the quantum computation.
Experiments with entangled photon states have demonstrated basic quantum operations~\cite{walther_experimental_2005,
lu_experimental_2007} for one-way quantum computation. However, these experiments did not use quantum memories, and the photonic cluster states used as the resource for the computation are based on postselection and cannot easily be scaled~\cite{bodiya_scalable_2006}. In contrast, large entangled states of quantum memories can be generated using a photon-mediated quantum gate where the number of necessary operations asymptotically scales linearly with the number of nodes~\cite{duan_efficient_2005, duan_probabilistic_2006,
barrett_efficient_2005}. The successful operation of the gate is heralded by the coincidence detection of two photons. Because the entangling gate is mediated by photons, it can in principle be applied to a wide variety of quantum memories such as trapped ions, neutral atoms in cavities, atomic ensembles or quantum dots.
\begin{figure}
\caption{The experimental apparatus. Two $^{171}$Yb$^+$ ions are
trapped in identically constructed ion traps separated by one
meter. A magnetic field $B$ is applied perpendicular to the
excitation and observation axes to define the quantization
axis. About $2\%$ of the emitted light from each ion is collected by
objective lenses (OL) with numerical aperture of $0.23$ and coupled
into two single-mode fibers. Polarization control paddles are used
to adjust the fiber to maintain linear polarization. The output of
these fibers is directed to interfere on a polarization independent
$50\%$ beamsplitter. Polarizers (PBS) transmit only the
$\pi$-polarized light from the ions. The photons are detected by
single-photon counting photomultiplier tubes (PMT A and PMT
B). Detection of the atomic state is done independently for the two
traps with dedicated photomultiplier tubes (PMTs). }
\label{setup}
\end{figure}
In this Letter, we demonstrate this probabilistic, heralded entangling gate for two ytterbium ions confined in two independent traps separated by one meter. The gate is implemented between the long-lived hyperfine ``clock'' states and mediated by photons carrying frequency encoded qubits. Unlike the recent demonstration of teleportation between two ions~\cite{olmschenk_quantum_2009}, here we demonstrate and characterize the gate for arbitrary quantum states of both qubits, as required for scalable quantum computing. We perform the gate on a full set of input states for both qubits and measure an average fidelity of $90\pm 2 \%$. For the particular case that should result in the antisymmetric Bell state, we perform full tomography of the final state.
The gate has many favorable properties. First, the ions do not have to be localized to the Lamb-Dicke regime and the operation is not interferometrically sensitive to the optical path length difference. Because the qubits are encoded in the atomic hyperfine ``clock'' states and two well-separated photonic frequency states the system is highly insensitive to external influences. Finally, the operation of the gate between remote ions facilitates individual addressing for single qubit operations and measurement and there is no need to shuttle ions. While the success probability of the gate in the current experiment is very small ($2.2 \times 10^{-8}$), the scaling to large quantum networks is still efficient (polynomial instead of exponential)~\cite{duan_efficient_2005,
duan_probabilistic_2006}, furthermore, it should be possible to significantly improve this rate for practical applications.
We trap two single $^{171}$Yb$^+$ atoms in two identically constructed Paul traps, located in independent vacuum chambers separated by approximately one meter (Fig.~\ref{setup}).
An ion typically remains in the trap for several weeks. Doppler-cooling by laser light slightly red-detuned from the $\state{2}{S}{1/2} \leftrightarrow \state{2}{P}{1/2} $ transition at $369.5\unit{nm}$ localizes the ions to better than the diffraction limit of the imaging system but not to the Lamb-Dicke regime. With a probability of about $0.5\%$, the excited $\state{2}{P}{1/2}$ state decays to the meta-stable $\state{2}{D}{3/2}$ level. This level is depopulated with a laser near $935.2\unit{nm}$ to maintain efficient cooling and state detection. We apply an external magnetic field $B=5.2\unit{Gauss}$ to provide a quantization axis, break the degeneracy of the atomic states, and suppress coherent dark state trapping.
The atomic qubit is encoded in two $\state{2}{S}{1/2}$ ground-state hyperfine levels of the $^{171}$Yb$^+$ atom, with $\ket{0} := \ket{F=0,m_F=0}$ and $\ket{1} := \ket{F=1,m_F=0}$, which have a separation of $12.6\unit{GHz}$ (Fig.~\ref{entanglement_scheme} (a)). Here $F$ is the total angular momentum of the ion and $m_F$ its projection along the quantization axis. These hyperfine ``clock'' states are to first order insensitive to the magnetic field and thus form an excellent quantum memory~\cite{fisk_accurate_1997,
olmschenk_manipulation_2007}.
\begin{figure}\label{entanglement_scheme}
\end{figure}
The remote gate protocol is shown schematically in Fig.~\ref{entanglement_scheme}. We first initialize each ion in $\ket{0}$ with a $1\unit{\mu s}$ pulse of light resonant with the $\state{2}{S}{1/2} (F=1) \leftrightarrow \state{2}{P}{1/2} (F=1)$ transition. Then we independently prepare each ion ($i=1,2$) in any desired superposition state $\ket{\Psi_a}_i = \alpha_i\ket{0}_i + \beta_i\ket{1}_i$ by applying a resonant microwave pulse with controlled phase and duration $(0-16\unit{\mu s})$ directly to one of the electrodes of each trap.
Next, we use an ultrafast $\pi$-polarized resonant laser pulse to simultaneously transfer the superposition from the ground state qubit states to the $\state{2}{P}{1/2}$ hyperfine states $\ket{0^\prime} := \ket{F^\prime=1, m_F^\prime=0}$ and $\ket{1^\prime} := \ket{F^\prime=0, m_F^\prime=0}$ of each ion with near-unit efficiency. For $\pi$-polarized light the dipole selection rules allow only the transitions $\ket{0} \leftrightarrow \ket{0^\prime}$ and $\ket{1} \leftrightarrow \ket{1^{\prime}}$ which have equal transition strength. The two transitions are well-resolved, as their center frequencies are separated by $\Delta\nu = 14.7\unit{GHz}$, while the natural linewidth of the excited state is only about $20\unit{MHz}$. The bandwidth of the $1\unit{ps}$ pulse of about $300\unit{GHz}$ is broad compared to $\Delta\nu$ but small compared to the fine structure splitting in Yb$^+$ of about $100\unit{THz}$, allowing both transitions to be driven equally while the population of the excited $\state{2}{P}{3/2}$ state remains vanishingly small. Consequently, the qubit can be transferred coherently from the \state{2}{S}{1/2} ground state to the excited \state{2}{P}{1/2} state~\cite{madsen_ultrafast_2006}.
Following excitation, each ion will emit a single photon. Upon emission of a $\pi$-polarized $369.5\unit{nm}$ photon, the frequency-mode of the emitted photon and the state of the ion are in the entangled state \begin{equation}
\ket{\Psi_{ap}}_i = \alpha_i\ket{0}_i\ket{\nu_b}_i +
\beta_i\ket{1}_i\ket{\nu_r}_i, \end{equation} where $\ket{\nu_b}$ and $\ket{\nu_r}$ are the two possible frequency states of the emitted photon. The state of the total system is $\ket{\Psi_{apap}} = \ket{\Psi_{ap}}_1 \otimes \ket{\Psi_{ap}}_2$. For each ion, emitted photons are collected with a lens with a numerical aperture of $0.23$ and are coupled into a single-mode fiber. The output of the fiber from each ion is directed to interfere on a polarization-independent $50\%$ beamsplitter. Each output of the beamsplitter is directed through a linear polarizer and detected with a single-photon counting photomultiplier tube.
To ensure high contrast interference of the two photons from different ions the photons must be indistinguishable. To this end, we first carefully minimize the micromotion of the ions to prevent modulation of the emission frequency. Second, the geometrical mode from the two fibers is matched to better than $98\%$ as characterized with laser light. Third, the emitted photons are matched in their arrival time at the beamsplitter to better than $100\unit{ps}$. As a consequence of the quantum interference~\cite{hong_measurement_1987, shih_new_1988,
braunstein_measurement_1995}, two photons, each in a superposition of two frequency modes, can only emerge from different output ports of the beamsplitter if they are in the antisymmetric state $\ket{\psi^-_{pp}} = (\ket{\nu_b}_1\ket{\nu_r}_2 - \ket{\nu_r}_1\ket{\nu_b}_2)/\sqrt{2}$. Upon a coincidence detection event the ions are projected onto the state \begin{align}
\nonumber\ket{\Psi_{aa}} &= \frac{1}{\sqrt{2 \ensuremath{P_{|\psi^-\rangle}}}} \left(
\alpha_1\beta_2\ket{0}_1\ket{1}_2 -
\beta_1\alpha_2\ket{1}_1\ket{0}_2 \right) \\
&= \frac{1}{\sqrt{2 \ensuremath{P_{|\psi^-\rangle}}}}
\frac{Z_1(I-Z_1Z_2)}{2}\ket{\Psi_a}_1\ket{\Psi_a}_2, \end{align} where $Z$ is the Pauli-$z$ operator and $\ensuremath{P_{|\psi^-\rangle}} = (\alpha_1^2\beta_2^2+\beta_1^2\alpha_2^2)/2$ is the probability to find the photons in the antisymmetric Bell state. Thus the coincidence detection of two photons heralds the operation of the remote two-ion quantum gate $Z_1(I-Z_1Z_2)$. In contrast to a simple entanglement process~\cite{simon_robust_2003}, the final state depends on the initial states of both ions, and hence this scheme must keep track of the initial states and phases of the two qubits. This property of an entangling gate is essential for the generation of cluster states of more than two ions as the entangling gate has to preserve any entanglement which may already be present. Being a projection or measurement gate this process is not unitary. Indeed, for the input states $\ket{0}_1\ket{0}_2$ and $\ket{1}_1\ket{1}_2$ a heralding event should never occur, which would be calamitous in the circuit model. However, in the protocol to generate cluster states, the input qubits are always by design in a superposition of the two qubit states. In this case, the protocol succeeds with a nonvanishing probability and scales favorably~\cite{duan_probabilistic_2006}.
\begin{figure}\label{state_tomography}
\end{figure}
To verify the operation of the gate we first characterize the generation of the maximally entangled antisymmetric Bell state $\ket{\Psi_{aa}} = (\ket{0}\ket{1} - \ket{1}\ket{0})/\sqrt{2}$ by full state tomography. Both ions are prepared in $\ket{\Psi_a}_i = ( \ket{0}_i + \ket{1}_i )/\sqrt{2}$ and operated on by the gate. We then measure the state of each ion in three mutually unbiased bases. Detection of the quantum state of each ion in the $x$ ($y$) bases is done by first applying a resonant microwave $\pi/2$-pulse with a relative phase of $0$ ($\pi/2$) with respect to the initial microwave pulse. Standard fluorescence detection is then used to determine the quantum state of the ion; the ion scatters light if found in state \ket{1}, while it remains dark if found in \ket{0}~\cite{olmschenk_manipulation_2007}. Consequently, we get an answer in every attempt to measure the state and this answer is correct with more than $98\%$ probability. The density matrix (Fig.~\ref{state_tomography}) is obtained using a maximum likelihood algorithm~\cite{james_measurement_2001}. From this density matrix we calculate an entangled state fidelity of $F=0.87(2)$, a concurrence of $C=0.77(4)$ and an entanglement of formation $E_F=0.69(6)$. The entanglement of this state is considerably higher than in our previous experiments~\cite{moehring_entanglement_2007,
matsukevich_bell_2008} due to technical improvements and the superior coherence properties of the photonic frequency and atomic ``clock''state qubits.
\begin{table*}
\caption{Results of the remote quantum gate process. Listed in this
table are the input states of the two ions, the expected
output state of the gate, the measurement performed, the number of
heralding events, the obtained fidelity, the measured and ideal
probability for two photons to be in the antisymmetric Bell
state. The fidelity of the output states is obtained by parity
measurements in the appropriate bases. Here, the parity $\ensuremath{\mathcal{P}}_{xy}$
is the difference of the probabilities to find the two ions in the
same state and in opposite states when ion $1$, $2$ is measured in
the $x$, $y$ basis, respectively. The other parity values are
defined similarly. From these results we calculate the average
gate fidelity $\overline{F}=0.90(2)$. The success probability of
the gate is $P_{\text{gate}} = \ensuremath{P_{|\psi^-\rangle}} \times 8.5\times 10^{-8}$.} \label{process_fidelity} \begin{ruledtabular} \begin{tabular}{ccccccc}
input state & expected state & measurement & events & fidelity &
$\ensuremath{P_{|\psi^-\rangle}}$ (meas.) & \ensuremath{P_{|\psi^-\rangle}} (theo.) \\
$\ket{0+1}\otimes\ket{0+1}$&$\ket{0}\ket{1}-\ket{1}\ket{0}$&$\frac{1}{4}\left(
1-\ensuremath{\mathcal{P}}_{xx}-\ensuremath{\mathcal{P}}_{yy}-\ensuremath{\mathcal{P}}_{zz} \right)$ & $210$ & $0.89(2)$ & $0.26(1)$ & $1/4$ \\
$\ket{0+i}\otimes\ket{0+1}$&$\ket{0}\ket{1}-i\ket{1}\ket{0}$&$\frac{1}{4}\left(
1-\ensuremath{\mathcal{P}}_{xy}+\ensuremath{\mathcal{P}}_{yx}-\ensuremath{\mathcal{P}}_{zz} \right)$ & $179$ &$0.86(2)$ & $0.26(1)$ & $1/4$ \\
$\ket{0-1}\otimes\ket{0+1}$&$\ket{0}\ket{1}+\ket{1}\ket{0}$&$\frac{1}{4}\left(
1+\ensuremath{\mathcal{P}}_{xx}+\ensuremath{\mathcal{P}}_{yy}-\ensuremath{\mathcal{P}}_{zz} \right)$ & $178$ & $0.85(1)$ & $0.22(2)$ & $1/4$ \\
$\ket{0-i}\otimes\ket{0+1}$&$\ket{0}\ket{1}+i\ket{1}\ket{0}$&$\frac{1}{4}\left(
1+\ensuremath{\mathcal{P}}_{xy}-\ensuremath{\mathcal{P}}_{yx}-\ensuremath{\mathcal{P}}_{zz} \right)$ & $188$ & $0.81(2)$ & $0.27(2)$ & $1/4$\\
$\ket{0+1}\otimes\ket{1}$ & $\ket{0}\otimes\ket{1}$ & $\frac12 \left(
1+\ensuremath{\mathcal{P}}_{zz} \right) $ & $42$ & $0.86(5)$ & $0.24(4)$ & $1/4$\\
$\ket{0}\otimes\ket{0+1}$ & $\ket{0}\otimes\ket{1}$ & $\frac12 \left(
1+\ensuremath{\mathcal{P}}_{zz} \right) $ & $52$ & $0.90(4)$ & $0.20(3)$ & $1/4$\\
$\ket{0}\otimes\ket{1}$ & $\ket{0}\otimes\ket{1}$ & $\frac12 \left(
1+\ensuremath{\mathcal{P}}_{zz} \right) $ & $48$ & $0.98(2)$ & $0.39(6)$ & $1/2$\\
$\ket{0}\otimes\ket{0}$ & $0$ & &
$65$ & & $0.04(1)$ & $0$\\ \end{tabular} \end{ruledtabular} \end{table*}
To characterize the functionality of the gate for arbitrary input states we measure the fidelity of the output state for a representative set of input states as shown in Table~\ref{process_fidelity} and obtain an average fidelity of $\overline{F}=0.90(2)$.
We do not characterize the action of the gate on certain input states that differ only by global qubit rotations. Such states are identical to those considered up to an overall choice of basis, so the input states listed Table I are representative of a full set of unbiased qubit bases.
The observed entanglement and gate fidelity are consistent with known experimental imperfections. The primary error sources that reduce the gate fidelity are imperfect state detection $(3\%)$, geometrical mode mismatch on the beamsplitter $(6\%)$, and detection of $\sigma$-polarized light due to the non-zero solid angle and misalignment of the magnetic field $(<2\%)$. Micromotion at the rf-drive frequency of the ion trap, which alters the spectrum of the emitted photons and can degrade the quantum interference, is expected to contribute to the overall error less than $1\%$. Other error sources include imperfect state preparation, pulsed excitation to the wrong atomic state, dark counts of the PMT leading to false coincidence events, mismatch of the quantization and polarizer axes, and multiple excitation due to pulsed laser light leakage, and are each estimated to contribute much less than $1\%$ to the overall error.
The entangling gate demonstrated here is a heralded probabilistic process where the success probability is given by
\begin{equation*}
P_{\text{gate}} =
\ensuremath{P_{|\psi^-\rangle}} \left( p_\pi \frac{ \Delta\Omega}{4\pi}
T_{\text{fiber}} T_{\text{optics}} \eta \right) ^2 \approx \ensuremath{P_{|\psi^-\rangle}} 2.2
\times 10^{-8}. \end{equation*}
Here $p_{\pi} = 0.5$ is the probability that a collected $369.5\unit{nm}$ photon is $\pi$-polarized, $\Delta\Omega/4\pi = 0.02$ is the solid angle from which photons are collected, $T_{\text{fiber}}=0.2$ is the coupling and transmission efficiency of the collected light through the single-mode fiber, $T_{\text{optics}}=0.95$ is the transmission coefficient of the other optics and $\eta=0.15$ is the quantum efficiency of the photomultiplier tube. The probability to find two photons in the antisymmetric Bell state $\ensuremath{P_{|\psi^-\rangle}}$ is a function of the initial states and $0 \le \ensuremath{P_{|\psi^-\rangle}} \le 1/2$. As a coincidence detection of two photons is necessary to herald the operation of the gate, the success probability scales as the square of the photon detection probability.
In the current experiment photons are collected by a lens in free space which only covers a small solid angle. Increasing the collection solid angle, which can be possible by using parabolic mirrors~\cite{lindlein_new_2007} or microstructured lenses~\cite{streed_scalable_2008}, can significantly increase the success probability. Furthermore, the spontaneous emission into free space could be replaced by the induced emission into the small mode volume of a high finesse cavity~\cite{hennrich_vacuum-stimulated_2000,
mckeever_deterministic_2004} which can reach near unit efficiency. Even though the free spectral range of the cavity would have to be $14.7\unit{GHz}$ to simultaneously support both frequency modes, choosing a near-concentric design could still result in a small mode-volume and thus in a high emission probability into a well-defined Gaussian mode. Alternatively, a fast microwave $\pi$-pulse together with the photonic time-bin encoded qubit could be used~\cite{barrett_efficient_2005}. While the success probability in the current experiment is small, employing the aforementioned techniques to increase the photon collection probability may dramatically increase the success probability of the gate and could make the generation of large entangled cluster states feasible.
We have demonstrated a probabilistic, heralded entanglement gate between two remote matter qubits with an average fidelity of $\overline{F}=0.90(2)$. The remote entangling gate demonstrated here could be used to realize long-distance quantum repeaters and to demonstrate a loophole-free Bell-inequality violation. Furthermore, together with local operations, the entangling gate may be used to scalably generate cluster states for the realization of a one-way quantum computer~\cite{briegel_persistent_2001,
raussendorf_one-way_2001, raussendorf_computational_2002,
duan_efficient_2005, duan_probabilistic_2006}.
\begin{acknowledgments}
This work is supported by the Intelligence Advanced Research Project
Activity (IARPA) under Army Research Office contract, the National
Science Foundation Physics at the Information Frontier program, and
the NSF Physics Frontier Center at JQI. \end{acknowledgments}
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\begin{document}
\title{\LARGE \bf
Stochastic Strategies for Robotic Surveillance \\as Stackelberg Games
} \thispagestyle{empty} \pagestyle{empty}
\begin{abstract} This paper studies a stochastic robotic surveillance problem where a mobile robot moves randomly on a graph to capture a potential intruder that strategically attacks a location on the graph. The intruder is assumed to be omniscient: it knows the current location of the mobile agent and can learn the surveillance strategy. The goal for the mobile robot is to design a stochastic strategy so as to maximize the probability of capturing the intruder. We model the strategic interactions between the surveillance robot and the intruder as a Stackelberg game, and optimal and suboptimal Markov chain based surveillance strategies in star, complete and line graphs are studied. We first derive a universal upper bound on the capture probability, i.e., the performance limit for the surveillance agent. We show that this upper bound is tight in the complete graph and further provide suboptimality guarantees for a natural design. For the star and line graphs, we first characterize dominant strategies for the surveillance agent and the intruder. Then, we rigorously prove the optimal strategy for the surveillance agent. \end{abstract}
\begin{IEEEkeywords} Stochastic robotic surveillance, Markov chains, Stackelberg game, capture probabilities \end{IEEEkeywords}
\section{Introduction} \subsubsection*{Problem description and motivation} In a prototypical robotic surveillance scenario, mobiles robots patrol and move among locations in an environment (usually modeled by a graph) with the goal of capturing potential intruders; see Fig.~\ref{fig:surveillance} for an illustration. Here, we consider a similar setup, where the patrolling robot is, in addition, facing an omniscient intruder. We model the strategic interactions between the mobile robot and the intruder as a Stackelberg game, where the optimal strategy for the surveillance agent is constructed under the assumption that the intruder acts optimally against her. This formulation captures the worst-case scenario for the surveillance agent when playing against the strongest possible opponent. The corresponding Stackelberg solution is meaningful and practical when little or no information is known about the intruder. Similar models have appeared in \cite{NB-NG-FA:12} and \cite{ABA-SLS:16}, where heuristic algorithms without performance guarantees are provided. Instead, we analyze the problem from a mathematical perspective and obtain provably optimal or suboptimal solutions by considering three topologies: star, line and complete graph.
\begin{figure}
\caption{A surveillance scenario where a mobile robot patrols a graph with the goal of capturing potential intruders that attack certain locations.}
\label{fig:surveillance}
\end{figure}
\subsubsection*{Related work} There have been continuing efforts to study robotic surveillance problems under various settings, formulations, and assumptions. The early work on designing deterministic surveillance strategies started in~\cite{YC:04}. Recent deterministic extensions to cases where locations might have different importance, or to the coordination of multiple robots can be found in~\cite{YE-NA-GAK:09, fp-af-fb:09v, FP-JWD-FB:11h,
SA-EF-SLS:14, ABA-SLS-SS:19}. Unfortunately, deterministic strategies can be easily learned, and thus exploited in adversarial settings. In this respect, stochastic surveillance strategies are more appealing in that they are mostly unpredictable. One common approach to derive stochastic surveillance strategies is to model the motion of the surveillance agent as a first-order Markov chain~\cite{JG-JB:05, GC-AS:11,
NN-AR-JVH-VI:16}. Within this line of work, Patel \emph{et
al.}~\cite{RP-PA-FB:14b} studied minimum mean hitting time Markov chains for robotic surveillance with travel times on edges. They formulated a convex optimization problem by restricting attentions to the class of reversible Markov chains. More recently, George \emph{et
al.}~\cite{MG-SJ-FB:17b} and Duan \emph{et al.}~\cite{XD-MG-FB:17o} studied and quantified unpredictability of Markov chains and designed maxentropic surveillance strategies.
The aforementioned works do not explicitly describe the behavior of the malicious intruders. On the other hand, when these models are available, they can be leveraged to design improved surveillance strategies. In order to model the interplay between the surveillance agent and the intruder, surveillance problems have also been studied under a game-theoretic lens. Stackelberg security games, where the defenders and intruders are modeled as strategic players with sequential plays, have been successfully applied in various real-world scenarios \cite{AS-FF-BA-CK-MT:18} such as checkpoints placement and patrolling at airports \cite{JP-MJ-JM-FO-CP-MT-CW-PP-SK:08}, coast guard surveillance \cite{ES-BA-RY-MT-CB-JD-BM-GM:12} and wild life protection \cite{FF-PS-MT:15}. In these works, defenders allocate limited resources to a set of targets so as to optimize their objectives. The problems are formulated as matrix games and the topology of the environments are not explicitly taken into account. In~\cite{NB-NG-FA:12}, the authors introduce a patrolling game where a mobile robot moves on a graph to capture potential intruders who choose when and where to attack. Different intruder models were proposed and analyzed in \cite{ABA-SLS:16}, where the optimal Markov chain based stochastic strategy is computed via a pattern search algorithm. The authors in \cite{ABA-SLS:18} consider an intruder with limited observation time and design a strategy that is both hard to learn and hard to attack. An intruder model where the intruder decides where, when and for how long it attacks is studied in \cite{HY-ST-KSL-SL-JG:19}, and it is found that increasing the randomness of the strategy helps reduce the intruder's reward. The authors in \cite{JB-SK:17} study the impact of the graph topology on the maximization of the minimum expected hitting times. Finally, a sophisticated non-Markovian model was studied in \cite{TB-PH-AK-VR-MA:15} for the case of complete graphs.
While most of the existing works are concerned with the design of heuristics to compute suboptimal strategies without performance guarantees, in this paper, we derive provably optimal/suboptimal strategies. Towards this goal, we adopt the model of~\cite{ABA-SLS:16} and focus on graphs that correspond to prototypical robotic roadmaps.
\subsubsection*{Contributions} In this paper, we derive provably optimal and suboptimal strategies for surveillance agents in a Stackelberg game setting. We consider three prototypical robotic roadmaps are considered: star, complete and line graphs. Our problem formulation captures the worst-case scenario for the surveillance agent and thus the solution provides performance guarantees also in less pessimistic scenarios. Our main contributions are as follows. \begin{enumerate}
\item We derive a universal upper bound for the capture probability, i.e., the maximum achievable performance for the surveillance agent facing an omniscient intruder;
\item We show that this upper bound is tight in the case of the complete graph, and further provide suboptimality guarantees for a natural strategy often referred to as a random walk;
\item We study dominant strategies for both the intruder and the surveillance agent. Leveraging these insights, we obtain optimal strategies in the star and line graphs. \end{enumerate}
\subsubsection*{Paper organization} We provide preliminaries on Markov chains and formulate the Stackelberg game problem in Section~\ref{sec:prelim} and Section~\ref{sec:ProblemFormulation}, respectively. An upper bound on the capture probability and a suboptimal solution in the case of complete graph are obtained in Section~\ref{sec:upperboundcomplete}. We study dominant strategies for the players in Section~\ref{sec:dominancestarline}, and optimal strategies for star and line graphs are given in the same section.
\subsubsection*{Notation} Let $\mathbb{R}$, $\mathbb{Z}_{\geq0}$, and $\mathbb{Z}_{>0}$ denote the set of real numbers, nonnegative and positive integers, respectively. Let $\mathbb{1}_n$ and $\mathbb{0}_n$ denote column vectors in $\mathbb{R}^n$ with all entries being $1$ and $0$. $I_n\in \mathbb{R}^{n\times n}$ is the identity matrix. $\mathbb{e}_i$ denotes the $i$-th vector in the standard basis, whose dimension will be made clear when it appears. $[S]$ denotes a diagonal matrix with diagonal elements being $S$ if $S$ is a vector, or being the diagonal of $S$ if $S$ is a square matrix. Let $\otimes$ denote the Kronecker product. $\vecz(\cdot)$ is the vectorization operator that converts a matrix into a column vector.
\section{Preliminaries}\label{sec:prelim} \subsection{Markov chains}
We start by reviewing the basics of discrete-time Markov chains. A finite-state discrete-time homogeneous Markov chain with state space $\until{n}$ is a sequence of random variables $X_k$, $k\in\mathbb{Z}_{\geq0}$, taking values in $\until{n}$ and satisfying the Markov property, i.e., $X_k$ is such that $\mathbb{P}(X_{k+1}=j\,|\,X_{k}=i,\dots,X_{0}=i_0) =\mathbb{P}(X_{k+1}=j\,|\,X_{k}=i)=p_{ij},$ for all $i,j\in\until{n}$ and $k\in\mathbb{Z}_{\geq0}$, where $p_{ij}$ is the transition probability from state $i$ to state $j$, $P=\{p_{ij}\}\in\mathbb{R}^{n\times{n}}$ is the transition matrix satisfying $P\geq 0$, and $P\mathbb{1}_n=\mathbb{1}_n$; see \cite{JGK-JLS:76}, \cite{JRN:97}. A probability distribution $\bm{\pi}\in\mathbb{R}^n$ is \emph{stationary} for a Markov chain with transition matrix $P$ if it satisfies $\bm{\pi}\geq0$, $\bm{\pi}^\top\mathbb{1}_n=1$ and $\bm{\pi}^\top=\bm{\pi}^\top P$. The transition diagram of a Markov chain $P=\{p_{ij}\}\in\mathbb{R}^{n\times{n}}$ is a directed graph (digraph) $\mathcal{G}=(V,\mathcal{E})$ where $V=\{1,\dots,n\}$ and $\mathcal{E}=\{(i,j)\,|\,i,j\in V, p_{ij}>0\}$. A Markov chain is \emph{irreducible} if its transition diagram is strongly connected.
\subsection{Hitting time of Markov chains}\label{sec:returntime} In this paper, we consider a strongly connected digraph $\mathcal{G}=(V,\mathcal{E})$, where $V$ denotes the set of $n$ nodes $\until{n}$ and $\mathcal{E}\subset V\times V$ denotes the set of edges. Given the graph $\mathcal{G}=\{V,\mathcal{E}\}$, let $X_k\in \until{n}$ be the value of a Markov chain with transition diagram $\mathcal{G}$ and transition matrix $P$ at time $k \in \mathbb{Z}_{\geq0}$. For any pair of nodes $i,j\in V$, the \emph{first hitting time} from $i$ to $j$, denoted by $T_{ij}$, is the first time the Markov chain hits node $j$ starting from node $i$, that is \begin{equation*}
T_{ij}=\min\{k\,|\,X_0=i,X_k=j,k\geq1\}. \end{equation*} Note that $T_{ij}$ is itself a random variable. Let the $(i,j)$-th element of the \emph{first hitting time probability matrix} $F_k$ denote the probability that the Markov chain hits node $j$ for the first time in exactly $k$ time units starting from node $i$, i.e., $F_k(i,j)=\mathbb{P}(T_{ij}=k)$. It can be shown that the hitting time probabilities $F_k$ for $k\geq1$ satisfy the following recursive matrix equation \cite[Chapter 5, Eq. (2.4)]{EC:13} \begin{equation}\label{eq:Fk} F_{k+1}=P(F_k-\textup{diag}(F_k)), \end{equation} where $F_1=P$. The vectorized form of~\eqref{eq:Fk} can be written as \begin{equation}\label{eq:RecursiveInVec} \vecz(F_{k+1}) =(I_n\otimes P)(I_{n^2} - E)\vecz(F_{k}), \end{equation} where $E = \textup{diag}(\vecz(I_n))$. Note that~\eqref{eq:Fk} can be generalized to the case where there are times on the graph $\mathcal{G}$ \cite{XD-MG-FB:17o}.
\section{Problem formulation}\label{sec:ProblemFormulation} We consider a robotic surveillance problem where a mobile robot moves randomly between locations in a graph to perform surveillance tasks. Specifically, given a Markov chain strategy, the surveillance robot moves from the current node to a neighboring location according to the corresponding Markov chain transition matrix. An intruder attacks an unknown node in the graph by stationing at the given node for a certain period of time. The intruder is captured if the surveillance agent visits that node within the duration of the attack; see Fig.~\ref{fig:surveillance} for a pictorial representation.
In this work, we study how the mobile robot should move on the graph in order to maximize the probability of capturing the intruder.
We model the strategic interactions between the surveillance robot and the intruder as a two-player Stackelberg game with a leader and a follower. The game proceeds as follows: the leader commits to a strategy first, and then the follower, based on the knowledge of the leader's strategy, selects a strategy that optimizes her rewards. Having prior knowledge of the follower's best-response, the leader commits to a strategy that ultimately maximizes her own objective. In our problem setting, the surveillance agent is the leader who chooses a Markov chain as surveillance strategy. This strategy is observed and learned by the intruder (follower) who then chooses the best time and location to attack so as to minimize the probability of being captured. We describe the intruder and surveillance models in the following subsections.
\subsection{Intruder Model} An intruder aims to attack a location in the graph while the surveillance agent moves around with the goal of capturing her. The intruder requires $\tau$ units of time to complete the attack at any location in the graph. Once it commits to attacking a location, it stations at the location for that given period of time. We assume that the intruder is omniscient \cite{NB-NG-FA:12,ABA-SLS:16}, i.e., it knows or can learn the strategy (intended as a description of the Markov chain) as well as the current location of the surveillance agent perfectly. Given this information, the intruder decides when and where to attack so that it is least likely to be captured.
Given a Markov chain strategy for the surveillance agent, the intruder picks a pair of locations $i$ and $j$ so that the probability that the surveillance agent goes from location $i$ to location $j$ within the attack duration is minimized. Then, the intruder attacks location $j$ whenever the surveillance agent is at location $i$. Formally, for a surveillance strategy parameterized by a Markov transition matrix $P$, an optimal strategy for the intruder $(i^*,j^*)$ is given by \begin{equation}\label{eq:smartestintruder} (i^*,j^*)\in\argmin_{i,j}\{\mathbb{P}(T_{ij}(P)\leq \tau)\}. \end{equation} Note that optimal strategies in~\eqref{eq:smartestintruder} are not unique in general, and the intruder is free to pick any one of them.
\subsection{Surveillance model and problem formulation} The surveillance agent, knowing that the intruder selects an optimal strategy according to \eqref{eq:smartestintruder}, adopts a Markov chain $P^*$ that maximizes the probability of capturing the intruder, i.e., \begin{equation*} P^*=\argmax_{P}\min_{i,j}\{\mathbb{P}(T_{ij}(P)\leq \tau)\}. \end{equation*}
In this paper, we are interested in finding an optimal strategy for the surveillance agent when playing against the omniscient intruder just described. That is, we are interested in solving the following optimization problem. \begin{problem} \label{prob:optimalstrategy} Given a strongly connected digraph $\mathcal{G} = (V, \mathcal{E})$ and the attack duration $\tau\in\mathbb{Z}_{>0}$, find a Markov chain that conforms to the graph topology and maximizes the capture probability, i.e., solves the following optimization
problem: \begin{equation}\label{eq:originalopt} \begin{aligned} & \underset{P\in\mathbb{R}^{n\times n}}{\textup{maximize}} & &\min_{i,j}\{\mathbb{P}(T_{ij}(P)\leq \tau)\}\\ & \textup{subject to} && P\mathbb{1}_n=\mathbb{1}_n,\\ &&& p_{ij}\geq0,\quad\textup{for all }(i,j)\in\mathcal{E},\\ &&& p_{ij}=0,\quad\textup{for all }(i,j)\notin\mathcal{E}. \end{aligned} \end{equation} \end{problem} We let $\mathbb{V}\in[0,1]$ denote the optimal value of Problem~\ref{prob:optimalstrategy} and refer to it as the value of the game. Note that $\mathbb{V}$ represents the capture probability obtained when the surveillance agent utilizes an optimal strategy against the omniscent intruder. As a consequence, $\mathbb{V}$ lower bounds the capture probability also in less pessimistic circumstances, e.g., in the case of a nonstrategic intruder. \begin{remark}[Impact of the attack duration]\label{rk:attackduration} Problem~\ref{prob:optimalstrategy} is interesting only when the attack duration $\tau$ is in an appropriate range for a given graph topology. Specifically, \begin{enumerate}
\item the attack duration should be greater than or equal to the diameter $D$ of graph $\mathcal{G}$, i.e., $\tau\geq D$. Otherwise, the omniscient intruder always succeeds by attacking one end of the graph diameter when the surveillance agent is visiting the other. In this case, $\mathbb{V}=0$ no matter what strategy the surveillance agent uses;
\item the attack duration should be smaller than the length of any closed path on the graph $\mathcal{G}$ that has the same initial and final vertices and visits all locations at least once (e.g., a Hamiltonian tour of size $n$ if it exists). Otherwise, the surveillance agent does not benefit from using a Markov chain as a randomized strategy, and the capture is guaranteed by following the deterministic closed path. \end{enumerate} \end{remark} We assume hereafter that the attack duration $\tau$ takes a nontrivial value as described in Remark~\ref{rk:attackduration}. \begin{remark}[Irreducibility of optimal solutions]\label{remark:irreducibility} No irreducibility constraint is imposed on the Markov chain in Problem~\ref{prob:optimalstrategy}. However, if $\tau$ takes nontrivial values, an optimal Markov chain is necessarily irreducible. As the transition diagrams of any reducible Markov chains is not strongly connected, there must exist a pair of locations $i$ and $j$ such that $\mathbb{P}(T_{ij}\leq\tau)=0$, so that also $\mathbb{V}=0$. \end{remark}
\begin{remark}[Graph dimension]\label{rk:graphdim} Without loss of generality, we consider graphs with more than $2$ nodes in the rest of this paper, i.e., $n\geq3$. When $n=2$, the only meaningful case is a complete graph, and the optimal solution to Problem~\ref{prob:optimalstrategy} is $P=\frac{1}{2}\mathbb{1}_2\mathbb{1}_2^\top$ with $\mathbb{V}=\frac{1}{2}$ if $\tau=1$, and an irreducible permutation matrix (a Hamiltonian tour) with $\mathbb{V}=1$ if $\tau\geq2$. \end{remark}
\section{Upper bounds of value of the game and suboptimal solution in complete graphs}\label{sec:upperboundcomplete} In this section, we first derive a universal upper bound for the value of the game, which does not depend the graph topology. We then consider the case of complete graph and show that the upper bound can be tight. Further, we provide suboptimality guarantees for a a Markov chain whose corresponding transition matrix has identical entries.
\subsection{Upper bound of the value of the game} We introduce an auxiliary variable $\mu\in\mathbb{R}$ and exploit the iteration~\eqref{eq:Fk} to rewrite the optimization problem~\eqref{eq:originalopt} as follows \begin{equation}\label{eq:equivalentopt} \begin{aligned} & \underset{\mu\in\mathbb{R},P\in\mathbb{R}^{n\times n}}{\textup{maximize}} & &\mu\\ & \textup{subject to} && \mu\mathbb{1}_n\mathbb{1}_n^\top\leq\sum_{k=1}^{\tau} F_k,\\ &&&F_1 = P\\ &&& F_{k+1}=P(F_k-\textup{diag}(F_k)),\quad 1\leq k\leq\tau-1\\ &&& P\mathbb{1}_n=\mathbb{1}_n,\\ &&& p_{ij}\geq0,\quad\textup{for all }(i,j)\in\mathcal{E},\\ &&& p_{ij}=0,\quad\textup{for all }(i,j)\notin\mathcal{E}, \end{aligned} \end{equation} where the inequality in the first constraint is element-wise. Clearly, problem~\eqref{eq:equivalentopt} is equivalent to~\eqref{eq:originalopt}, and the optimal value $\mu^*$ is the value of the game. The following theorem shows how to obtain a universal upper bound on $\mu^*$.
\begin{theorem}[Upper bound for the value of the game]\label{thm:upperbound} Given a strongly connected digraph $\mathcal{G}=(V,\mathcal{E})$ with $n$ nodes and an attack duration $\tau$ that takes nontrivial values as in Remark~\ref{rk:attackduration},
the value of the game satisfies $\mathbb{V}\leq\frac{\tau}{n}$. \end{theorem} \begin{proof} By Remark~\ref{remark:irreducibility},
an optimal solution $P^*$ to problem~\eqref{eq:equivalentopt} is irreducible and thus has a unique stationary distribution $\bm\pi$. We multiply $\bm\pi^\top$ from the left on both sides of \eqref{eq:Fk} and obtain for $k\geq 1$, \begin{equation}\label{eq:multistationary} \bm\pi^\top F_{k+1}=\bm\pi^\top (F_k-\textup{diag}(F_k))\leq \bm\pi^\top F_k. \end{equation} By using \eqref{eq:multistationary} recursively, we have that for $k\geq1$, \begin{equation}\label{eq:uppderboundF} \bm\pi^\top F_k\leq \bm\pi^\top F_1= \bm\pi^\top P^*=\bm\pi^\top. \end{equation} Since $(\mu^*,P^*)$ is an optimal solution to problem~\eqref{eq:equivalentopt}, it satisfies the first constraint and thus \begin{equation}\label{eq:firstconst} \mu^*\mathbb{1}_n\mathbb{1}_n^\top\leq\sum_{k=1}^{\tau} F_k. \end{equation} Multiplying $\bm\pi^\top$ from the left on both sides of \eqref{eq:firstconst} and using \eqref{eq:uppderboundF}, we obtain \begin{equation*} \mu^*\mathbb{1}_n^\top\leq\sum_{k=1}^{\tau}\bm\pi^\top=\tau\bm\pi^\top. \end{equation*} Since $\bm\pi^\top\mathbb{1}_n=1$, we must have $\min\limits_{1\leq i\leq n}\pi_i\leq\frac{1}{n}$. Therefore, \begin{equation*} \mathbb{V}=\mu^*\leq\tau\min_{1\leq i\leq n}\pi_i=\frac{\tau}{n}. \end{equation*} \end{proof}
\subsection{Suboptimal solution in complete digraphs} Next, we show that for complete digraphs the upper bound $\mathbb{V}\le \frac{\tau}{n}$ can be achieved for certain combinations of $n$ and $\tau$.
\begin{lemma}[Optimal solution in special complete digraph]\label{lemma:completeOpt} Given a complete digraph $\mathcal{G}=(V,\mathcal{E})$ with $n$ nodes and the attack duration $\tau\leq n$, if $\tau$ divides $n$, then an optimal solution $P^*$ is given by \begin{equation*} P^*=\Pi_0\operatorname{\otimes}\frac{\tau}{n}\mathbb{1}_{\frac{n}{\tau}}\mathbb{1}_{\frac{n}{\tau}}^\top, \end{equation*}
where $\Pi_0\in\mathbb{R}^{\tau \times\tau}$ is any irreducible permutation matrix that represents a Hamiltonian tour in $\mathcal{G}$.
Moreover, the optimal strategy $P^*$ achieves the upper bound of the value of the game. \end{lemma} \begin{proof} By construction, the probability that a surveillance agent with $P^*$ starting from any location $i$ arrives at any location $j$ within $\tau$ time steps is $\frac{\tau}{n}$. \end{proof}
The previous result holds only when the combination of $n$ and $\tau$ is such that $\tau$ divides $n$. Nevertheless, leveraging Theorem~\ref{thm:upperbound} we are able to provide suboptimality guarantees for the natural choice of $P=\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top$.
\begin{lemma}[Constant factor optimality of random walk]\label{thm:complete} Given a complete digraph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes and the attack duration $\tau$, the random walk strategy $P=\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top$ achieves performance within \begin{equation*} \frac{n^\tau-(n-1)^\tau}{\tau n^{\tau-1}}\geq 1-\frac{1}{e} \end{equation*} of optimality, where $e$ is Euler's number. \end{lemma} \begin{proof} First, note that the capture probability for the random walk surveillance policy $P=\frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top$ is $1-(1-\frac{1}{n})^\tau$. Therefore, the random walk achieves performance within \begin{equation*} f(n,\tau)=\frac{n^\tau-(n-1)^\tau}{\tau n^{\tau-1}} \end{equation*} of optimality.
Let $n$ be fixed, and $g(\tau)=\frac{n^\tau-(n-1)^\tau}{\tau n^{\tau-1}}$. We relax $\tau$ to be a continuous variable and take derivative of $g(\tau)$ as \begin{align*} \frac{dg(\tau)}{d\tau}&=-\frac{n}{\tau^2}+\frac{n}{\tau^2}\big(1-\frac{1}{n}\big)^\tau-\frac{n}{\tau}\big(1-\frac{1}{n}\big)^\tau\log\big(1-\frac{1}{n}\big)\\ &=\frac{n}{\tau^2}\big((1-\tau\log(1-\frac{1}{n}))(1-\frac{1}{n})^\tau-1\big)\\ &\leq\frac{n}{\tau^2}\big((1-\tau(1-\frac{1}{1-\frac{1}{n}}))(1-\frac{1}{n})^\tau-1\big)\\ &=\frac{n}{\tau^2}\big((1+\frac{\tau}{n-1})(1-\frac{1}{n})^\tau-1\big)\leq0, \end{align*} where we used the upper bound $\log x\geq 1-\frac{1}{x}$ in the first inequality, and $\tau\leq n-1$ and $(1-\frac{1}{n})^\tau\leq\frac{1}{2}$ in the second inequality. Therefore, for fixed $n$, $f(n,\tau)$ is a decreasing function in $\tau$ and \begin{equation*} f(n,\tau)\geq f(n,n-1)=\frac{n}{n-1}-(1-\frac{1}{n})^{n-2}\triangleq h(n). \end{equation*} We relax $n$ to be a continuous variable in this case and compute the derivative of $h(n)$ as \begin{align*} \frac{dh(n)}{dn}&=-\frac{1}{(n-1)^2}+(1-\frac{1}{n})^{n-2}\big(\log\frac{n}{n-1}-\frac{n-2}{n^2-n}\big)\\ &\leq-\frac{1}{(n-1)^2}+(1-\frac{1}{n})^{n-2}(\frac{1}{n-1}-\frac{n-2}{n^2-n})\\ &=-\frac{1}{(n-1)^2}+(1-\frac{1}{n})^{n-1}\frac{2}{(n-1)^2}\\ &\leq-\frac{1}{(n-1)^2}+(1-\frac{1}{3})^{2}\frac{2}{(n-1)^2}\leq0,
\end{align*} where we used the upper bound $\log(1+x)\leq x$ and the proved that $(1-\frac{1}{n})^{n-1}$ is decreasing as $n$ increases. Therefore, we have that $h(n)$ is a decreasing function. In summary, we have \begin{multline*} f(n,\tau)\geq f(n,n-1)\\ \geq\lim_{n\rightarrow\infty}h(n)=\lim_{n\rightarrow\infty}\big(\frac{n}{n-1}-(1-\frac{1}{n})^{n-2}\big)=1-\frac{1}{e}. \end{multline*} \end{proof}
\section{Strategy dominance and optimal strategies in star and line graphs}\label{sec:dominancestarline} In this section, we first obtain dominated strategies for the intruder as well as the dominant strategies for the surveillance agent. Leveraging these result, we derive optimal strategies for the surveillance agent in star and line graphs. \subsection{Dominated strategies for the omniscient intruder} In this subsection, we present two lemmas characterizing dominated strategies for the omniscient intruder, i.e., strategies that intruder will never choose as they lead to a higher probability of being captured. \begin{lemma}[Dominated strategy for the intruder]\label{lemma:dominance_int} Given a strongly connected digraph $\mathcal{G}=(V,\mathcal{E})$ and an irreducible Markov chain strategy $P$ for the surveillance agent, attacking node $j\in V$ when the surveillance agent is at node $i\in V$, $i\neq j$, is not optimal for the omniscient intruder if: \begin{enumerate} \item\label{item:pre} there exists a node $k$ such that any path from node $k$ to node $j$ contains node $i$; or \item\label{item:post} there exists a node $k$ such that any path from node $i$ to node $k$ contains node $j$. \end{enumerate} Moreover, in case \ref{item:pre} (resp. in case \ref{item:post}) , attacking node $j$ when the surveillance agent is at node $k$ (resp. node $i$) is a better strategy for the omniscient intruder. \end{lemma} \begin{proof} The probability that the surveillance agent visiting node $i$ captures the intruder attacking node $j$ is \begin{equation*} \mathbb{P}(T_{ij}\leq \tau)=\sum_{ t=1}^{\tau} \mathbb{P}(T_{ij}= t). \end{equation*}
Regarding~\ref{item:pre}, we need to show that \begin{equation*} \mathbb{P}(T_{kj}\leq \tau)\leq\mathbb{P}(T_{ij}\leq \tau). \end{equation*} Since any path from node $k$ to node $j$ contains node $i$, by definition of probability, and the memoryless property of Markov chains, we have \begin{align*} \mathbb{P}(T_{kj}\leq\tau)&=\sum_{ t=1}^{\tau-1}\mathbb{P}(T_{ki}= t)\mathbb{P}(T_{ij}\leq \tau- t)\\ &\leq \max_{1\leq t\leq\tau-1}\mathbb{P}(T_{ij}\leq \tau- t)\\ &= \mathbb{P}(T_{ij}\leq \tau- 1)\\ &\leq \mathbb{P}(T_{ij}\leq \tau), \end{align*} where we used that $\mathbb{P}(T_{ij}\leq \tau- t)$ is decreasing with $t$.
The proof for~\ref{item:post} follows a similar argument as in~\ref{item:pre}. \end{proof}
\begin{lemma}[Dominated strategy on leaf nodes]\label{coro:intruder_dominance} Given a strongly connected digraph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes and an irreducible Markov chain strategy $P$ for the surveillance agent, if node $i\in V$ is a leaf node in $\mathcal{G}$, then attacking node $i$ when the surveillance agent just leaves node $i$ is not optimal for the omniscient intruder. \end{lemma} \begin{proof} The case of $\tau=1$ is uninteresting here because the surveillance agent fails with probability $1$ if the intruder attacks the leaf node $i$ when the surveillance agent visits other nodes than node $i$ and its neighbor. Therefore, we consider $\tau\geq2$ in the following. Let node $j$ be the neighbor node of the leaf node $i$, then \begin{equation}\label{eq:self} \mathbb{P}(T_{ii}\leq\tau)=p_{ii}+(1-p_{ii})\mathbb{P}(T_{ji}\leq\tau-1). \end{equation} Moreover, for $k\in V$, $k\neq i$ and $k\neq j$, \begin{align}\label{eq:ineq} \begin{split} \mathbb{P}(T_{ki}\leq\tau)&=\sum_{ t=1}^{\tau-1}\mathbb{P}(T_{kj}= t)\mathbb{P}(T_{ji}\leq \tau- t)\\ &\leq \max_{1\leq t\leq \tau-1}\mathbb{P}(T_{ji}\leq \tau- t)\\ &= \mathbb{P}(T_{ji}\leq\tau-1). \end{split} \end{align} Therefore, by \eqref{eq:self} and \eqref{eq:ineq} we have \begin{align*} \mathbb{P}(T_{ii}\leq\tau)&\geq p_{ii}+(1-p_{ii})\mathbb{P}(T_{ki}\leq\tau)\\ &\geq p_{ii}\mathbb{P}(T_{ki}\leq\tau)+(1-p_{ii})\mathbb{P}(T_{ki}\leq\tau)\\ &= \mathbb{P}(T_{ki}\leq\tau), \end{align*} which implies that attacking node $i$ when the surveillance agent is at node $k$ is a better strategy. \end{proof}
\subsection{Dominant strategies for the surveillance agent} In this subsection, we show that part of the optimal surveillance strategy can be determined readily when leaf nodes are present, where leaf nodes are nodes that have only one neighboring node.
\begin{lemma}[Dominant strategy on leaf nodes]\label{lemma:dominance_sur} Given a strongly connected digraph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes, if node $i\in V$ is a leaf node in $\mathcal{G}$ with node $j\in V$ as its only neighbor, then the optimal strategy $P^*$ satisfies \begin{equation*} P^*(i,i) = 0,\textup{ and } P^*(i,j) = 1. \end{equation*} \end{lemma} \begin{proof} Without loss of generality, suppose that node $1$ is a leaf node in $\mathcal{G}$ and node $2$ is its neighbor. Let $P$ be a strategy that is the same as $P^*$ except that $P(1,1)=p>0$ and $P(1,2)=1-p<1$. We prove that for all $i,j\in V$, the capture probability $\mathbb{P}(T_{ij}^*\leq\tau)$ for $P^*$ is greater than or equal to $\min_{\{i,j\in V\}}\mathbb{P}(T_{ij}\leq\tau)$ for $P$, which leads to $\min_{\{i,j\in V\}}\mathbb{P}(T_{ij}^*\leq\tau)\geq \min_{\{i,j\in V\}}\mathbb{P}(T_{ij}\leq\tau)$.
Since $P$ and $P^*$ differ by only the first row and node $1$ is a leaf node, we must have $\mathbb{P}(T_{i1}^*\leq \tau)=\mathbb{P}(T_{i1}\leq \tau)$ for all $i\in\{2,\dots,n\}$. Moreover, by Lemma~\ref{coro:intruder_dominance}, the strategy $(1,1)$ is a dominated strategy for the intruder, thus $\min_{\{i,j\in V\}}\mathbb{P}(T_{ij}^*\leq\tau)$ and $\min_{\{i,j\in V\}}\mathbb{P}(T_{ij}\leq\tau)$ do not attain minimum at $(1,1)$.
We next prove that $\mathbb{P}(T_{1j}^*\leq\tau)\geq\mathbb{P}(T_{1j}\leq\tau)$ for all $j\in\{2,\dots,n\}$ by induction. Let $d_{1j}$ be the length of the shortest paths from node $1$ to node $j$. The probabilities of these shortest paths are equal to the products of the edge probabilities along the paths, and for $P$ and $P^*$ they differ by a factor of $1-p$. Thus, we have that $\mathbb{P}(T_{1j}^*\leq d_{1j})>\mathbb{P}(T_{1j}\leq d_{1j})$. Suppose when $\tau\leq t$ for $t\geq d_{1j}$, we have $\mathbb{P}(T_{1j}^*\leq \tau)>\mathbb{P}(T_{1j}\leq \tau)$ and let $\ell_{ij}^t$ be the set of paths from node $i$ to node $j$ that do not contain node $1$ and have length less than or equal to $t$ for $i\in\{2,\dots,n\}$, then for $\tau=t+1$, \begin{align*} \mathbb{P}(T_{1j}^*\leq t+1)&=\mathbb{P}(T_{2j}^*\leq t)\\ &=\sum_{t_1=1}^t\mathbb{P}(T_{21}^*= t_1)\mathbb{P}(T_{1j}^*\leq t-t_1)+\mathbb{P}(\ell_{2j}^t)\\ &\geq \sum_{t_1=1}^t\mathbb{P}(T_{21}= t_1)\mathbb{P}(T_{1j}\leq t-t_1)+\mathbb{P}(\ell_{2j}^t)\\ &=\mathbb{P}(T_{2j}\leq t)\\ &\geq p\mathbb{P}(T_{1j}\leq t)+(1-p)\mathbb{P}(T_{2j}\leq t)\\ &=\mathbb{P}(T_{1j}\leq t+1), \end{align*} where the first inequality follows from the induction hypothesis and the second inequality follows from Lemma~\ref{lemma:dominance_int}.
Finally, for $i,j\in\{2,\dots,n\}$, we have \begin{align*} \mathbb{P}(T_{ij}^*\leq \tau)&=\sum_{t=1}^{\tau}\mathbb{P}(T_{i1}^*= t)\mathbb{P}(T_{1j}^*\leq \tau-t)+\mathbb{P}(\ell_{ij}^{\tau})\\ &\geq \sum_{t=1}^{\tau}\mathbb{P}(T_{i1}= t)\mathbb{P}(T_{1j}\leq \tau-t)+\mathbb{P}(\ell_{ij}^{\tau})\\ &=\mathbb{P}(T_{ij}\leq \tau). \end{align*}
In summary, we have that $\min_{\{i,j\in V\}}\mathbb{P}(T_{ij}^*\leq\tau)\geq \min_{\{i,j\in V\}}\mathbb{P}(T_{ij}\leq\tau)$, which completes the proof. \end{proof}
\subsection{Optimal solution for star graphs} In this subsection, we consider the star topology, which represents the abstraction of an environment where there is a corridor connecting multiple rooms. The optimal strategy for the surveillance agent is given in the following theorem.
\begin{theorem}[Optimal solution in star graph]\label{thm:star}
Given a directed star graph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes and node $1$ being the center, the optimal strategy $P^*$ for the surveillance agent is given by
\begin{equation}\label{eq:staroptimal}
P^*=\begin{bmatrix}
0&\frac{1}{n-1}&\frac{1}{n-1}&\cdots&\frac{1}{n-1}\\
1&0&0&\cdots&0\\
1&0&0&\cdots&0\\
\vdots&\vdots&\vdots&\cdots&\vdots\\
1&0&0&\cdots&0
\end{bmatrix}.
\end{equation}
Moreover, the value of the game $\mathbb{V}$ satisfies
\begin{equation*}
\mathbb{V}=\begin{cases}
1-(1-\frac{1}{n-1})^{\frac{\tau-1}{2}},&\quad\textup{if }\tau\geq2\textup{ is odd},\\
1-(1-\frac{1}{n-1})^{\frac{\tau}{2}},&\quad\textup{if }\tau\geq2\textup{ is even}.
\end{cases}
\end{equation*} \end{theorem} \begin{proof} For the directed star graph $\mathcal{G}$, the Markov chain $P$ corresponding to $\mathcal{G}$ has the following general structure, \begin{equation*} P=\begin{bmatrix} p_{11}&p_{12}&p_{13}&\cdots&p_{1n}\\ p_{21}&p_{22}&0&\cdots&0\\ p_{31}&0&p_{33}&\cdots&0\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ p_{n1}&0&0&\cdots&p_{nn} \end{bmatrix}. \end{equation*} Since node $2$ to node $n$ are leaf nodes, by Lemma~\ref{lemma:dominance_sur}, the optimal Markov chain does not have self loops at these nodes and we can reduce $P$ to \begin{equation}\label{eq:reducedP} P=\begin{bmatrix} p_{11}&p_{12}&p_{13}&\cdots&p_{1n}\\ 1&0&0&\cdots&0\\ 1&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ 1&0&0&\cdots&0 \end{bmatrix}. \end{equation} Note that the strategies $(1,j)$ are dominated for the intruder for all $j\in\{2,\dots,n\}$ by Lemma~\ref{lemma:dominance_int}, and the capture probabilities $\mathbb{P}(T_{i1}\leq \tau)=1$ for all $i\in\{1,\dots,n\}$ and $\tau\geq2$. Therefore, we only need to find a $P$ in~\eqref{eq:reducedP} that maximizes $\min_{i,j\in\{2,\dots,n\}}\mathbb{P}(T_{ij}\leq\tau)$. We divide the rest of the proof into two parts. In the first part, we show that there is no self loop at the center node for the optimal solution, i.e., $p_{11}=0$ in~\eqref{eq:reducedP}, and further reduce $P$; in the second part, we obtain the optimal solution.
\paragraph*{No self loop at center} We construct $P_1$ to be the same as $P$ in~\eqref{eq:reducedP} except for the first row where \begin{equation*} P_1(1,1)=0, ~P_1(1,j)=\frac{p_{1j}}{1-p_{11}}\textup{ for }j\in\{2,\dots,n\}. \end{equation*} We show that $P_1$ is a better strategy than $P$ in~\eqref{eq:reducedP} by induction on $\tau$, i.e., for all $i,j\in\{2,\dots,n\}$, the capture probabilities $\mathbb{P}(T_{ij}^1\leq\tau)$ for $P_1$ is greater than $\mathbb{P}(T_{ij}\leq\tau)$ of $P$. When $\tau=2$, we have $\mathbb{P}(T_{ij}^1\leq2)=\frac{p_{1j}}{1-p_{11}}>p_{1j}=\mathbb{P}(T_{ij}\leq2)$. Suppose when $\tau \leq t$, we have $\mathbb{P}(T_{ij}^1\leq\tau)>\mathbb{P}(T_{ij}\leq\tau)$, then for $\tau=t+1$, \begin{align*} \mathbb{P}(T_{ij}^1\leq t+1)&=\mathbb{P}(T_{1j}^1\leq t)\\ &=\frac{p_{1j}}{1-p_{11}}+\sum_{k\notin \{1,j\}}\frac{p_{1k}}{1-p_{11}}\mathbb{P}(T_{kj}^1\leq t-1)\\ &>\frac{p_{1j}}{1-p_{11}}+\sum_{k\notin \{1,j\}}\frac{p_{1k}}{1-p_{11}}\mathbb{P}(T_{kj}\leq t-1)\\ &=\frac{1}{1-p_{11}}(p_{1j}+\sum_{k\notin \{1,j\}}p_{1k}\mathbb{P}(T_{1j}\leq t-1))\\ &=\frac{1}{1-p_{11}}(\mathbb{P}(T_{1j}\leq t)-p_{11}\mathbb{P}(T_{1j}\leq t-1))\\ &\geq\frac{1}{1-p_{11}}(\mathbb{P}(T_{1j}\leq t)-p_{11}\mathbb{P}(T_{1j}\leq t))\\ &=\mathbb{P}(T_{1j}\leq t)\\ &=\mathbb{P}(T_{ij}\leq t+1), \end{align*} where the first inequality follows from the induction hypothesis and the second inequality follows from the fact that $\mathbb{P}(T_{1j}\leq t-1)\leq\mathbb{P}(T_{1j}\leq t)$. Therefore, we conclude that the optimal strategy does not have a self loop at the center node.
\paragraph*{Optimal solution} Note that for any $\tau\geq1$ and $j\in\{2,\dots,n\}$, we have that \begin{align*} \mathbb{P}(T_{1j}\leq\tau)&=p_{1j}+\sum_{ k\notin\{1,j\}}p_{1k}\mathbb{P}(T_{kj}\leq\tau-1)\\ &=p_{1j}+\sum_{ k\notin\{1,j\}}p_{1k}\mathbb{P}(T_{1j}\leq\tau-2)\\ &=p_{1j}+(1-p_{1j})\mathbb{P}(T_{1j}\leq\tau-2), \end{align*} with the initial condition $\mathbb{P}(T_{1j}\leq1)=\mathbb{P}(T_{1j}\leq2)=p_{1j}$. Therefore, the capture probability $\mathbb{P}(T_{1j}\leq\tau)$ satisfies \begin{equation}\label{eq:starcases} \mathbb{P}(T_{1j}\leq\tau)=\begin{cases} 1-(1-p_{1j})^{\frac{\tau-1}{2}},\quad&\textup{if }\tau\geq2\textup{ is odd},\\ 1-(1-p_{1j})^{\frac{\tau}{2}},\quad&\textup{if }\tau\geq2\textup{ is even}. \end{cases} \end{equation} By~\eqref{eq:starcases}, we have for odd $\tau\geq2$, \begin{align*} \min_{i,j\in\{2,\dots,n\}}\mathbb{P}(T_{ij}\leq\tau)&=\min_{j\in\{2,\dots,n\}}\mathbb{P}(T_{1j}\leq\tau-1)\\ &=\min_{j\in\{2,\dots,n\}}1-(1-p_{1j})^{\frac{\tau-1}{2}}\\ &=1-\max_{j\in\{2,\dots,n\}}(1-p_{1j})^{\frac{\tau-1}{2}}\\ &=1-(1-\min_{j\in\{2,\dots,n\}}p_{1j})^{\frac{\tau-1}{2}}, \end{align*} which along with the fact that $\sum_{j=2}^np_{1j}=1$ implies that~\eqref{eq:staroptimal} is the optimal solution for the directed star graph. \end{proof}
\subsection{Optimal solution for line graphs} In this subsection, we derive the optimal surveillance strategy for directed line graphs. In an $n$-node line graph, Problem~\ref{prob:optimalstrategy} is interesting only when the attack duration $\tau$ satisfies $n-1\leq\tau\leq 2n-3$. If $\tau<n-1$, the omniscient intruder always succeeds by attacking an end node when the surveillance agent is at the other end; if $\tau>2n-3$, the surveillance agent who walks back and forth between two ends of the line graph (a deterministic sweeping)
captures the omniscient intruder no matter how it attacks. Therefore, we consider only cases when $n-1\leq \tau\leq 2n-3$. Note that a sweeping strategy fails as long as $\tau< 2n-3$, because the intruder could attack an end node immediately after the surveillance agent just leaves that node and it succeeds with probability $1$. We label the nodes in an $n$-node line graph successively from left to right by $(1,\dots,n)$. We need the following conjecture to establish our main result.
\begin{conjecture}[Uniqueness of the optimal strategy]\label{conj:uniqueness}
Given a directed line graph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes, the optimal solution to Problem~\ref{prob:optimalstrategy} is unique. \end{conjecture}
We provide evidence for Conjecture~\ref{conj:uniqueness} in Remark~\ref{rk:uniqueness}.
\begin{theorem}[Optimal solution in line graph]
Given a directed line graph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes, if Conjecture~\ref{conj:uniqueness} holds true, then the optimal strategy $P^*$ is given by
\begin{equation}\label{eq:lineoptimal}
P^*=\begin{bmatrix}
0&1&0&\cdots&0\\
0.5&0&0.5&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&\cdots&0.5&0&0.5\\
0&\cdots&0&1&0\\
\end{bmatrix}.
\end{equation} \end{theorem} \begin{proof} We divide the proof into three parts. In the first part, we show that in the line graph, the optimal strategy for the intruder is to attack one end of the graph when the surveillance agent is at the other, in which case the objective function in Problem~\ref{prob:optimalstrategy} becomes $\min\{\mathbb{P}(T_{1n}\leq \tau),\mathbb{P}(T_{n1}\leq\tau)\}$. In the second part, we show that there are no self loops at any locations in the optimal surveillance strategy. In the last part, we obtain the optimal strategy by using Conjecture~\ref{conj:uniqueness} and a symmetry argument.
\paragraph*{Attack the end nodes} For $i,j\in V$ and $i< j$, by Lemma \ref{lemma:dominance_int}, we know that $\mathbb{P}(T_{ij}\leq\tau)\geq\mathbb{P}(T_{1j}\leq\tau)\geq\mathbb{P}(T_{1n}\leq\tau)$; on the other hand, for $i>j$, we have $\mathbb{P}(T_{ij}\leq\tau)\geq\mathbb{P}(T_{nj}\leq\tau)\geq\mathbb{P}(T_{n1}\leq\tau)$. Therefore, attacking any location in the middle while the surveillance agent is at another location in the middle is not optimal for the intruder. Since node $1$ and $n$ are leaf nodes, by Lemma~\ref{coro:intruder_dominance}, it is not optimal for the intruder to attack node $1$ or $n$ immediately after the surveillance agent leaves that node. Next, we show that attacking any node $i\in\{2,\dots,n-1\}$ immediately when the surveillance agent leaves node $i$ is dominated by attacking an end node when the surveillance agent is visiting the other. For $i\in\{2,\dots,n-1\}$, \begin{align*} \mathbb{P}(T_{ii}\leq \tau)&=p_{ii} + p_{i,i+1}\mathbb{P}(T_{i+1,i}\leq \tau-1) \\ &\quad+ p_{i,i-1}\mathbb{P}(T_{i-1,i}\leq \tau-1)\\ &\geq p_{ii} + p_{i,i+1}\mathbb{P}(T_{n1}\leq \tau) + p_{i,i-1}\mathbb{P}(T_{1n}\leq \tau)\\ &\geq \min\{1,\mathbb{P}(T_{n1}\leq \tau),\mathbb{P}(T_{1n}\leq \tau)\}\\ &= \min\{\mathbb{P}(T_{n1}\leq \tau),\mathbb{P}(T_{1n}\leq \tau)\}, \end{align*} where the first inequality follows from the facts that \begin{align*} &\quad\mathbb{P}(T_{n1}\leq \tau)\\&=\sum_{t_1,t_2}\mathbb{P}(T_{n,i+1}= t_1)\mathbb{P}(T_{i+1,i}\leq \tau-t_1-t_2)\mathbb{P}(T_{i1}=t_2)\\ &\leq\mathbb{P}(T_{i+1,i}\leq \tau-1), \end{align*} and \begin{align*} &\quad\mathbb{P}(T_{1n}\leq \tau)\\&=\sum_{t_1,t_2}\mathbb{P}(T_{1,i}= t_1)\mathbb{P}(T_{i,i+1}\leq \tau-t_1-t_2)\mathbb{P}(T_{i+1,n}=t_2)\\ &\leq\mathbb{P}(T_{i,i+1}\leq \tau-1). \end{align*} Therefore, the best strategy for the omniscient intruder is to attack an end node when the surveillance agent is at the other. Problem~\ref{prob:optimalstrategy} becomes $\max_{P}\min\{\mathbb{P}(T_{1n}\leq \tau),\mathbb{P}(T_{n1}\leq\tau)\}$.
\paragraph*{No self loop at any location} Fist, by Lemma~\ref{lemma:dominance_sur}, since nodes $1$ and $n$ are leaf nodes, the optimal strategy $P^*$ for the surveillance agent must satisfy $P^*(1,1)=0$, $P^*(1,2)=1$, $P^*(n,n)=0$ and $P^*(n,n-1)=1$. Next, we focus on $i\in\{2,\dots,n-1\}$. Let $P$ be any Markov chain strategy corresponding to the line graph with $p_{ii}>0$, and $P_1$ is the same as $P$ except for the $i$-th row where \begin{equation*} P_1(i,i)=0, ~P_1(i,i+1)=\frac{p_{i,i+1}}{1-p_{ii}},~P_1(i,i-1)=\frac{p_{i,i-1}}{1-p_{ii}}. \end{equation*} Note that \begin{align}\label{eq:linerecursion} \begin{split} \mathbb{P}(T_{1n}\leq\tau)&=\sum_{t_1=i-1}^{\tau}\mathbb{P}(T_{1i}= t_1)\mathbb{P}(T_{in}\leq \tau- t_1),\\ \mathbb{P}(T_{n1}\leq\tau)&=\sum_{t_1=n-i}^{\tau}\mathbb{P}(T_{ni}=t_1)\mathbb{P}(T_{i1}\leq \tau- t_1). \end{split} \end{align} Since $P$ and $P_1$ differ only by row $i$, their first hitting time probabilities satisfy $\mathbb{P}(T_{1i}= t_1)=\mathbb{P}(T^1_{1i}= t_1)$ and $\mathbb{P}(T_{ni}= t_1)=\mathbb{P}(T^1_{ni}= t_1)$ for all $t_1\geq1$. We first prove that $\mathbb{P}(T_{in}\leq \tau)\leq\mathbb{P}(T^1_{in}\leq \tau)$ for all $\tau$ by induction. When $\tau=n-i$, since $P_1(i,i+1)>p_{i,i+1}$, we have $\mathbb{P}(T_{in}\leq n-i)\leq\mathbb{P}(T^1_{in}\leq n-i)$. Suppose for all $\tau\leq t$, we have $\mathbb{P}(T_{in}\leq t)\leq\mathbb{P}(T^1_{in}\leq t)$. Then, when $\tau= t+1$,
\begin{align*} &\quad\mathbb{P}(T^1_{in}\leq t+1)\\ &=\frac{p_{i,i-1}}{1-p_{ii}}\mathbb{P}(T^1_{i-1,n}\leq t)+\frac{p_{i,i+1}}{1-p_{ii}}\mathbb{P}(T^1_{i+1,n}\leq t),\\ &\geq \frac{p_{i,i-1}}{1-p_{ii}}\mathbb{P}(T_{i-1,n}\leq t)+\frac{p_{i,i+1}}{1-p_{ii}}\mathbb{P}(T_{i+1,n}\leq t)\\ &=\frac{1}{1-p_{ii}}(\mathbb{P}(T_{in}\leq t+1)-p_{ii}\mathbb{P}(T_{in}\leq t))\\ &\geq\frac{1}{1-p_{ii}}(\mathbb{P}(T_{in}\leq t+1)-p_{ii}\mathbb{P}(T_{in}\leq t+1))\\ &=\mathbb{P}(T_{in}\leq t+1), \end{align*} where the first inequality follows from the hypothesis induction. A similar proof by induction shows that $\mathbb{P}(T_{ni}\leq \tau)\leq\mathbb{P}(T^1_{ni}\leq \tau)$ for all $\tau$. Then, by~\eqref{eq:linerecursion}, we have that $\mathbb{P}(T_{n1}\leq \tau)\leq\mathbb{P}(T^1_{n1}\leq \tau)$ and $\mathbb{P}(T_{1n}\leq \tau)\leq\mathbb{P}(T^1_{1n}\leq \tau)$ and therefore $P_1$ is a better strategy than $P$. In summary, we have that the optimal surveillance strategy does not have self loop at any location.
\paragraph*{Optimal solution} By the first two parts, we conclude that the optimal strategy for the surveillance agent has the following general structure, \begin{equation}\label{eq:generalP} P=\begin{bmatrix} 0&1&0&\cdots&0\\ 1-x_1&0&x_1&\cdots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ 0&\cdots&1-x_{n-2}&0&x_{n-2}\\ 0&\cdots&0&1&0\\ \end{bmatrix}, \end{equation} and thus the objective function in Problem~\ref{prob:optimalstrategy} can be parameterized by $0<x_1<1,\dots,0<x_{n-2}<1$. Let \begin{equation*} f(x_1,\dots,x_{n-2})=\min_{x_1,\dots,x_{n-2}}\{\mathbb{P}(T_{1n}\leq \tau),\mathbb{P}(T_{n1}\leq\tau)\}. \end{equation*} For the function $f$, following the proof in Appendix~\ref{appdix:proof1}, we have \begin{equation*} f(x_1,\dots,x_{n-2})=f(1-x_1,\dots,1-x_{n-2}). \end{equation*} If Conjecture~\ref{conj:uniqueness} holds, then at the optimal solution, we must have $x^*_i=1-x^*_{i}$ for all $i\in\{1,\dots,n-2\}$, which implies $x_{i}^*=\frac{1}{2}$. Therefore, the optimal solution is given by~\eqref{eq:lineoptimal}. \end{proof}
We prove a necessary condition for a strategy to be optimal in line graphs in Lemma~\ref{lemma:linenecessary} in Appendix~\ref{appendix:linenecessary}, which says that the optimal strategy must satisfy $\mathbb{P}(T^*_{1n}\leq\tau)=\mathbb{P}(T^*_{1n}\leq\tau)$. We provide evidence for Conjecture~\ref{conj:uniqueness} in the following remark.
\begin{remark}[Evidence for conjecture~\ref{conj:uniqueness}]\label{rk:uniqueness} We first consider two tractable cases: $n=3$ and $\tau=n-1$ or $\tau=n$. When $n=3$, the line graph is also a star graph, and by Theorem~\ref{thm:star}, we have that the optimal solution in unique. The case $\tau=n$ is the same as that of $\tau=n-1$ since the optimal Markov chain in the form~\eqref{eq:generalP} is periodic. For $\tau=n-1$, we have \begin{align*} &\mathbb{P}(T_{1n}\leq n-1)=p_{23}p_{34}\cdots p_{n-1,n},\\ &\mathbb{P}(T_{n1}\leq n-1)=p_{n-1,n-2}p_{n-2,n-3}\cdots p_{21}, \end{align*} and note that \begin{align}\label{eq:lineproduct} \begin{split} &\quad\mathbb{P}(T_{1n}\leq n-1)\mathbb{P}(T_{n1}\leq n-1)\\ &=(p_{21}p_{23})\cdots (p_{n-1,n-2}p_{n-1,n})\\ &\leq \big(\frac{p_{21}+p_{23}}{2}\big)^2\cdots\big(\frac{p_{n-1,n-2}+p_{n-1,n}}{2}\big)^2=\frac{1}{2^{2n-4}}. \end{split} \end{align} There is a unique strategy that achieves the upper bound in~\eqref{eq:lineproduct}: $p_{i,i+1}=p_{i,i-1}=\frac{1}{2}$ for all $i\in\{2,\dots,n-1\}$. Moreover, this strategy satisfies the necessary condition in Lemma~\ref{lemma:linenecessary}. Therefore, the optimal solution is unique.
As a second justification, following the Monte Carlo probability estimation method in~\cite[Remark V.1]{GN-FB:07z}, we randomly pick $27000$ different Markov chains with the structure~\eqref{eq:generalP} for $n=5$, $\tau=8$ and $n=6$, $\tau=8$. In all these cases, no chain has been found to have a better or same value as that of~\eqref{eq:lineoptimal}. Therefore, we have $99\%$ confidence that the probability of~\eqref{eq:lineoptimal} being optimal is at least $0.99$ for these cases. \end{remark}
\section{Conclusion}\label{sec:conclusion} In this paper, we studied a Stackelberg game formulation for the robotic surveillance problem, where the surveillance agent defends against an omniscient intruder who decides where and when to attack. We derived an upper bound on the performance of the surveillance agent and provided provably suboptimal solution in the complete graph. We derived dominant strategies and leveraged them to obtain optimal strategies for the star and the line topology. For future works, we will consider arbitrary graph topology and heterogeneous attack duration.
\begin{appendices} \section{Proof of a symmetry property}\label{appdix:proof1}
Let $P_1$ be a Markov chain of the form~\eqref{eq:generalP} and $P_2$ be
\begin{equation*}
\begin{bmatrix}
0&1&0&\cdots&0\\
x_1&0&1-x_1&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&\cdots&x_{n-2}&0&1-x_{n-2}\\
0&\cdots&0&1&0\\
\end{bmatrix},
\end{equation*}
which is, in terms of the objective function, equivalent to
\begin{equation}\label{eq:P22}
P_2=\begin{bmatrix}
0&1&0&\cdots&0\\
1-x_{n-2}&0&x_{n-2}&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&\cdots&1-x_{1}&0&x_{1}\\
0&\cdots&0&1&0\\
\end{bmatrix}.
\end{equation}
We work with $P_2$ in~\eqref{eq:P22} for the rest of the proof.
Let $\mathbb{P}(T_{1n}^1\leq\tau)$ and $\mathbb{P}(T_{1n}^2\leq\tau)$ be the capture probabilities for $P_1$ and $P_2$, respectively. We show that $\mathbb{P}(T_{1n}^1\leq\tau)=\mathbb{P}(T_{1n}^2\leq\tau)$, and a similar proof strategy works for the claim $\mathbb{P}(T_{n1}^1\leq\tau)=\mathbb{P}(T_{n1}^2\leq\tau)$. From~\eqref{eq:RecursiveInVec}, we have that
\begin{align*}
\mathbb{P}(T_{1n}^1\leq\tau)&=\mathbb{e}_1^\top\sum_{t=1}^\tau (P_1 - P_1\mathbb{e}_n\mathbb{e}_n^\top)^{t-1}P\mathbb{e}_n\\
&=\mathbb{e}_1^\top (I_n-(P_1 - P_1\mathbb{e}_n\mathbb{e}_n^\top)^{\tau})\\
&\quad\cdot(I_n-(P_1 - P_1\mathbb{e}_n\mathbb{e}_n^\top))^{-1}P\mathbb{e}_n\\
&=\mathbb{e}_1^\top(I_{n-1}-A^\tau)(I_{n-1}-A)^{-1}b,
\end{align*}
where $b=x_{n-2}\mathbb{e}_{n-1}$ and
\begin{equation*}
A=\begin{bmatrix}
0&1&0&\cdots&0\\
1-x_1&0&x_1&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&\cdots&1-x_{n-3}&0&x_{n-3}\\
0&\cdots&0&1-x_{n-2}&0\\
\end{bmatrix}.
\end{equation*}
Note that $(I_{n-1}-A)\mathbb{1}_{n-1}=b$. Therefore, we have
\begin{equation}\label{eq:captureP1}
\mathbb{P}(T_{1n}^1\leq\tau)=\mathbb{e}_1^\top(I_{n-1}-A^{\tau})\mathbb{1}_{n-1}.
\end{equation}
Similarly, for $P_2$, we have
\begin{equation}\label{eq:captureP2}
\mathbb{P}(T_{1n}^2\leq\tau)=\mathbb{e}_1^\top(I_{n-1}-B^{\tau})\mathbb{1}_{n-1},
\end{equation}
where
\begin{equation*}
B=\begin{bmatrix}
0&1&0&\cdots&0\\
1-x_{n-2}&0&x_{n-2}&\cdots&0\\
\vdots&\ddots&\ddots&\ddots&\vdots\\
0&\cdots&1-x_{2}&0&x_{2}\\
0&\cdots&0&1-x_{1}&0\\
\end{bmatrix}.
\end{equation*}
In order to show $\mathbb{P}(T_{1n}^1\leq\tau)=\mathbb{P}(T_{1n}^2\leq\tau)$, by~\eqref{eq:captureP1} and \eqref{eq:captureP2}, we only need to prove that for all $\tau\geq1$,
\begin{equation}\label{eq:powerequal}
\mathbb{e}_1^\top A^{\tau}\mathbb{1}_{n-1}=\mathbb{e}_1^\top B^{\tau}\mathbb{1}_{n-1}.
\end{equation}
First, note that for all $\tau\leq n-2$, we have
\begin{equation}\label{eq:powerequalsmall}
\mathbb{e}_1^\top A^{\tau}\mathbb{1}_{n-1}=\mathbb{e}_1^\top B^{\tau}\mathbb{1}_{n-1}=1.
\end{equation}
We next show that $A$ and $B$ have the same characteristic
polynomials, and then by the Cayley-Hamilton and
\eqref{eq:powerequalsmall}, we will
have~\eqref{eq:powerequal}. Since both $A$ and $B$ are tridiagonal
matrices, the characteristic polynomials of $A$ and $B$ can be
generated as follows \cite[equation~(2.3)]{MEAE:04}. Note that $A$
and $B$ are of order $n-1$. Let
$g_0^{n-1}(\lambda)=h_0^{n-1}(\lambda)=1$,
$g_1^{n-1}(\lambda)=h_1^{n-1}(\lambda)=\lambda$, where the
superscript indicates the order of the matrices, and for
$k=2,\dots,n-1$,
\begin{align}\label{eq:ghrecursive}
\begin{split}
g_k^{n-1}(\lambda)&=\lambda g_{k-1}^{n-1}(\lambda) - x_{k-2}(1-x_{k-1})g_{k-2}^{n-1}(\lambda),\\
h_k^{n-1}(\lambda)&=\lambda h_{k-1}^{n-1}(\lambda) - x_{k}(1-x_{k-1})h_{k-2}^{n-1}(\lambda),
\end{split}
\end{align}
where $x_0=x_{n-1}=1$ and we obtain the recurrence $h_{k}^{n-1}$ for $B$ starting from the bottom right of $B$ matrix. Then $g_{n-1}^{n-1}(\lambda)$ and $h_{n-1}^{n-1}(\lambda)$ are the characteristic polynomials for $A$ and $B$, respectively. Moreover, notice that
\begin{align*}
g_{n}^{n}&=\begin{vmatrix}
\lambda I_{n-1} -A &-x_{n-2}\mathbb{e}_{n-1}\\
-(1-x_{n-1})\mathbb{e}_{n-1}^\top&\lambda
\end{vmatrix}\\
&=\lambda g_{n-1}^{n-1}-x_{n-2}(1-x_{n-1})g_{n-2}^{n-1},
\end{align*}
and $g_{n-1}^n=g_{n-1}^{n-1}$. Thus,
\begin{align}\label{eq:gtog}
\begin{split}
\begin{bmatrix}
g_{n}^{n}(\lambda)\\
g_{n-1}^{n}(\lambda)
\end{bmatrix}&=
\begin{bmatrix}
\lambda&- x_{n-2}(1-x_{n-1})\\
1&0
\end{bmatrix}\begin{bmatrix}
g_{n-1}^{n-1}(\lambda)\\
g_{n-2}^{n-1}(\lambda)
\end{bmatrix}
\end{split}.
\end{align}
At the same time, note that
\begin{align}\label{eq:hrecursive}
\begin{split}
h_{n-1}^{n}&=|\lambda I_{n-1}-B+(1-x_{n-1})\mathbb{e}_{1}\mathbb{e}_{2}^\top|\\
&=\lambda h_{n-2}^{n-1}-(1-x_{n-2})x_{n-1}h_{n-3}^{n-1},\\
h_{n-2}^{n}&=h^{n-1}_{n-2}.
\end{split}
\end{align}
Thus,
\begin{align}\label{eq:htoh}
\begin{split}
\begin{bmatrix}
h_{n}^{n}(\lambda)\\
h_{n-1}^{n}(\lambda)
\end{bmatrix}&=
\begin{bmatrix}
\lambda&-(1-x_{n-1})\\
1&0
\end{bmatrix}\begin{bmatrix}
h_{n-1}^{n}(\lambda)\\
h_{n-2}^{n}(\lambda)
\end{bmatrix}\\
&=
\begin{bmatrix}
\lambda&-(1-x_{n-1})\\
1&0
\end{bmatrix}\\
&\quad\cdot\begin{bmatrix}
\lambda&-x_{n-1}(1-x_{n-2})\\
1&0
\end{bmatrix}\begin{bmatrix}
h_{n-2}^{n-1}(\lambda)\\
h_{n-3}^{n-1}(\lambda)
\end{bmatrix}\\
&=
\begin{bmatrix}
\lambda&-(1-x_{n-1})\\
1&0
\end{bmatrix}\begin{bmatrix}
\lambda&-x_{n-1}(1-x_{n-2})\\
1&0
\end{bmatrix}\\
&\quad\cdot\begin{bmatrix}
\lambda&-(1-x_{n-2})\\
1&0
\end{bmatrix}^{-1}\begin{bmatrix}
h_{n-1}^{n-1}(\lambda)\\
h_{n-1}^{n-2}(\lambda)
\end{bmatrix}\\
&=\begin{bmatrix}
\lambda x_{n-1}&(\lambda^2-1)(1-x_{n-1})\\
x_{n-1}&\lambda(1-x_{n-1})
\end{bmatrix}\begin{bmatrix}
h_{n-1}^{n-1}(\lambda)\\
h_{n-2}^{n-1}(\lambda)
\end{bmatrix},
\end{split}
\end{align}
where the second and third equalities follow from~\eqref{eq:hrecursive} and~\eqref{eq:ghrecursive}, respectively.
In the following, we prove that
\begin{align}\label{eq:byinductionmain}
\begin{split}
&g_{n-1}^{n-1}(\lambda)=h_{n-1}^{n-1}(\lambda),\\
&\lambda g_{n-1}^{n-1}(\lambda) = x_{n-2}g_{n-2}^{n-1}(\lambda) + (\lambda^2-1)h_{n-2}^{n-1},
\end{split}
\end{align} by induction on $n$. When $n=3$, by~\eqref{eq:ghrecursive}, we have
\begin{align*} &g_{2}^{2}(\lambda)=\lambda g_{1}^{2}(\lambda) - x_{0}(1-x_{1})g_{0}^{2}(\lambda)= \lambda^2-(1-x_1),\\ &h_2^{2}(\lambda)=\lambda h_{1}^{2}(\lambda) - x_{2}(1-x_{1})h_{0}^{2}(\lambda)=\lambda^2-(1-x_1),
\end{align*} and \begin{align*} \lambda g_{2}^{2}(\lambda)=\lambda^3-\lambda(1-x_1)=x_{1}g_{1}^{2}(\lambda) + (\lambda^2-1)h_{1}^{2}. \end{align*}
Suppose~\eqref{eq:byinductionmain} holds for $n=k$, then when $n=k+1$, \begin{align*}
&\quad g_{k}^{k}(\lambda)-h_{k}^{k}(\lambda)\\
&=\lambda g_{k-1}^{k-1}(\lambda)- x_{k-2}(1-x_{k-1})g_{k-2}^{k-1}(\lambda)\\
&\quad-\lambda x_{k-1} h_{k-1}^{k-1}(\lambda)-(\lambda^2-1)(1-x_{k-1})h_{k-2}^{k-1}(\lambda)\\
&=x_{k-2}x_{k-1}g_{k-2}^{k-1}(\lambda)-\lambda x_{k-1} h_{k-1}^{k-1}(\lambda)\\
&\quad+(\lambda^2-1)x_{k-1}h_{k-2}^{k-1}(\lambda)\\
&=\lambda x_{k-1} g_{k-1}^{k-1}(\lambda)-\lambda x_{k-1} h_{k-1}^{k-1}(\lambda)=0,
\end{align*}
where the first equality follows from~\eqref{eq:gtog} and~\eqref{eq:htoh}, the second and third follow from induction hypothesis. Moreover,
\begin{align*}
&\quad\lambda g_{k}^{k}(\lambda) - x_{k-1}g_{k-1}^{k}(\lambda) - (\lambda^2-1)h_{k-1}^{k}\\
&=(\lambda^2-x_{k-1})g_{k-1}^{k-1}(\lambda)-\lambda x_{k-2}(1-x_{k-1})g_{k-2}^{k-1}(\lambda)\\
&\quad-(\lambda^2-1)(x_{k-1}h_{k-1}^{k-1}(\lambda)+\lambda(1-x_{k-1})h_{k-2}^{k-1}(\lambda))\\
&=(1-x_{k-1})\lambda^2g_{k-1}^{k-1}(\lambda)-\lambda^2(1-x_{k-1})g_{k-1}^{k-1}(\lambda)=0,
\end{align*}
where the first equality follows from~\eqref{eq:gtog} and~\eqref{eq:htoh}, and the second follows from the induction hypothesis. The proof is completed.
\section{Necessary optimality condition in line graphs}\label{appendix:linenecessary} \begin{lemma}[Necessary optimality condition in line graph]\label{lemma:linenecessary}
Given a line graph $\mathcal{G}=(V,\mathcal{E})$ with $n\geq3$ nodes, if the Markov chain strategy $P^*$ is optimal for the surveillance agent, then it must satisfy
$\mathbb{P}(T^*_{1n}\leq\tau)=\mathbb{P}(T^*_{n1}\leq\tau)$. \end{lemma} \begin{proof}
We divide the proof into two parts. In the first part, we show a monotonicity property of the hitting times $\mathbb{P}(T_{1n}\leq\tau)$ and $\mathbb{P}(T_{n1}\leq\tau)$. In the second part, we obtain the necessary condition.
\paragraph*{Monotonicity of hitting times} We claim that $\mathbb{P}(T_{1n}\leq \tau)$ is monotonically increasing (decreasing) with $p_{i,i+1}$ ($p_{i,i-1}$), and $\mathbb{P}(T_{n1}\leq \tau)$ is monotonically decreasing (increasing) with $p_{i,i+1}$ ($p_{i,i-1}$). We prove that $\mathbb{P}(T_{1n}\leq \tau)$ is monotonically increasing with $p_{i,i+1}$, and a similar proof works for the other cases. For any $i\in\{2,\dots,n-1\}$, let $P^\epsilon$ be a Markov chain that is the same as $P$ except that $P^{\epsilon}(i,i+1)=P(i,i+1)+\epsilon$ and $P^{\epsilon}(i,i-1)=P(i,i-1)-\epsilon$, where $\epsilon>0$ is small enough such that $P^\epsilon$ remains irreducible. We first show that for $i\in\{2,\dots,n-1\}$, we have $\mathbb{P}(T^\epsilon_{in}\leq \tau)\geq \mathbb{P}(T_{in}\leq \tau)$ by induction. When $\tau=n-i$,
\begin{align*}
\mathbb{P}(T^\epsilon_{in}\leq n-i)&=(\epsilon+p_{i,i+1})p_{i+1,i+2}\cdots p_{n-1,n}\\
&>p_{i,i+1}p_{i+1,i+2}\cdots p_{n-1,n}\\
&= \mathbb{P}(T_{in}\leq n-i).
\end{align*}
Suppose $\mathbb{P}(T^\epsilon_{in}\leq \tau)\geq \mathbb{P}(T^\epsilon_{in}\leq \tau)$ holds for $\tau\leq t$. For $i\in\{2,\dots,n-2\}$, let $\ell_{i+1,n}^t$ be the set of paths from node $i+1$ to node $n$ that do not contain node $i$ and have length less than or equal to $t$. Then, when $\tau=t+1$, for $i\in\{2,\dots,n-2\}$,
\begin{align}\label{eq:linefromiton}
\begin{split}
&\mathbb{P}(T^\epsilon_{in}\leq t+1)\\
&=(p_{i,i-1}-\epsilon)\mathbb{P}(T^\epsilon_{i-1,n}\leq t)+ p_{i,i}\mathbb{P}(T^\epsilon_{in}\leq t) \\
&\quad+ (p_{i,i+1}+\epsilon)\mathbb{P}(T^\epsilon_{i+1,n}\leq t)\\
&=(p_{i,i-1}-\epsilon)\sum_{t_1=1}^{ t}\mathbb{P}(T^\epsilon_{i-1,i}=t_1)\mathbb{P}(T^\epsilon_{in}\leq t-t_1)\\
&\quad+p_{i,i}\mathbb{P}(T^\epsilon_{in}\leq t)+ (p_{i,i+1}+\epsilon)\mathbb{P}(\ell_{i+1,n}^t)\\
&\quad+(p_{i,i+1}+\epsilon)\sum_{t_1=1}^ t\mathbb{P}(T^\epsilon_{i+1,i}=t_1)\mathbb{P}(T^\epsilon_{in}\leq t-t_1)\\
&\geq (p_{i,i-1}-\epsilon)\mathbb{P}(T_{i-1,n}\leq t)+ p_{i,i}\mathbb{P}(T_{in}\leq t) \\
&\quad+ (p_{i,i+1}+\epsilon)\mathbb{P}(T_{i+1,n}\leq t)\\
&\geq\mathbb{P}(T_{in}\leq t+1),
\end{split}
\end{align}
where the first inequality follows from the induction hypothesis and the second inequality follows from the fact that $\mathbb{P}(T_{i+1,n}\leq t)\geq\mathbb{P}(T_{i-1,n}\leq t)$ by Lemma~\ref{lemma:dominance_int}. Moreover, when $i=n-1$, we equate $\mathbb{P}(T_{i+1,n}^\epsilon\leq t)=1$ in the third line of~\eqref{eq:linefromiton}, and a similar argument follows. Finally, we have
\begin{align*}
\mathbb{P}(T^\epsilon_{1n}\leq \tau)&=\sum_{t_1=1}^{\tau}\mathbb{P}(T^\epsilon_{1i}=t_1)\mathbb{P}(T^\epsilon_{in}\leq \tau-t_1)\\
&\geq\sum_{t_1=1}^{\tau}\mathbb{P}(T_{1i}=t_1)\mathbb{P}(T_{in}\leq \tau-t_1)\\
&= \mathbb{P}(T_{1n}\leq \tau),
\end{align*}
which completes the proof.
\paragraph*{Necessary condition for optimality} We prove by contradiction. Without loss of generality, suppose $P^*$ is optimal and $\mathbb{P}(T^*_{1n}\leq\tau)<\mathbb{P}(T^*_{n1}\leq \tau)$. By the monotonicity properties in the first part, for $i\in\{2,n-1\}$, if we increase $p_{i,i+1}^*$ and decrease $p_{i,i-1}^*$, then $\mathbb{P}(T^*_{1n}\leq\tau)$ increases and $\mathbb{P}(T^*_{n1}\leq \tau)$ decreases continuously, which leads to an increase in the objective function $\min\{\mathbb{P}(T^*_{1n}\leq\tau),\mathbb{P}(T^*_{n1}\leq \tau)\}$. Therefore, the strategy $P^*$ is not the optimal, which is a contradiction. \end{proof} \end{appendices}
\end{document} |
\begin{document}
\title{A single photoelectron transistor for quantum optical communications} \author{Hideo Kosaka$^{1\ast }$, Deepak S. Rao$^{1}$, Hans D. Robinson$^{1}$, Prabhakar Bandaru$^{1}$, Kikuo Makita$^{2}$, Eli Yablonovitch$^{1}$} \address{$^{1}$ Electrical Engineering Department, University of California Los Angeles, Los Angeles, CA, 90095-1594\\ $^{2}$ Photonics and Wireless Device Research Laboratories, NEC Corporation, \\ 34 Miyukigaoka, Tsukuba, Ibaraki 305-8501, Japan\\ $^{\ast }$ On leave from Fundamental Research Laboratories, NEC Corporation.} \date{\today}
\begin{abstract} A single photoelectron can be trapped and its photoelectric charge detected by a source/drain channel in a transistor. Such a transistor photodetector can be useful for flagging the safe arrival of a photon in a quantum repeater. The electron trap can be photo-ionized and repeatedly reset for the arrival of successive individual photons. This single photoelectron transistor (SPT) operating at the $\lambda$ = 1.3$\mu$m tele-communication band, was demonstrated by using a windowed-gate double-quantum-well InGaAs/InAlAs/InP heterostructure that was designed to provide near-zero electron g-factor. The g-factor engineering allows selection rules that would convert a photon's polarization to an electron spin polarization. The safe arrival of the photo-electric charge would trigger the commencement of the teleportation algorithm. \end{abstract}
\pacs{85.35.Gv, 73.50.Pz, 85.35.Be, 78.67.De}
\maketitle
Quantum information can take several different forms and it is beneficial to be able to convert among the different forms. One form is photon polarization, and another is electron spin polarization.
Photons are the most convenient medium for sharing quantum information between distant locations. Quantum key distribution \cite{Bennett92} has been demonstrated by sending photons through a conventional optical fiber up to distances over 80km \cite{Hiskett}. As the distance increases, the secure data rate decreases, owing to photon loss. To expand the distance dramatically, it is necessary to realize a quantum repeater, that is based on quantum teleportation \cite{Bennett93}. A quantum repeater requires quantum information storage \cite{van Enk}, and electron spin is a good candidate for such a quantum memory. We need to have a photodetector that converts from photon to electron, while transferring the quantum information from photon polarization to electron spin. This is sometimes called an entanglement preserving photodetector \cite{Vrijen}. In addition, the photodetector must provide a trigger signal to flag the arrival of a photo-electric charge, and to commence the teleportation algorithm.
A Field Effect Transistor (FET), and a Single Electron Transistor (SET) based on quantum dots, can both function as sensitive electrometers that can detect a single trapped electric charge. Our goal is to safely trap a photoelectron, so that its spin state can then be monitored. In this paper we demonstrate the trapping and manipulation of individual photoelectrons, but we have not yet measured the trapped electron's spin properties. Previous experiments have demonstrated interband photon absorption resulting in the trapping of photo{\it holes}; on self-assembled InAs quantum dots \cite{shields}, or on DX centers \cite{kosaka02}, near an FET source/drain channel. These produce positive photoconductivity, that is fairly common. The trapping of photo{\it electrons} is much more rarely observed, since it is accompanied by negative photoconductivity \cite{rose}.
\begin{figure}
\caption{ The energy band diagram of the Single Photoelectron Transistor (SPT) at zero bias simulated by using one-dimensional Poisson/Schr\"{o}dinger equation. Photo-induced transitions between the heavy hole band and the conduction band is shown with an arrow. Photo-ionization of donors by $ \lambda$ = 1.77 $\mu$m-light modulation dopes the channel. The tunneling time of trapped electrons in the top quantum well leaking to the bottom quantum well is estimated to be over 1{\nobreakspace}hour by WKB simulation.}
\label{fig1}
\end{figure}
Several kinds of photon effects on SETs made on modulation-doped semiconductors have been reported. Photon assisted tunneling is the most common effect. The tunneling takes place between an island and source-drain reservoir \cite{blick98,blick95}, between two adjacent islands \cite{vaart}, or between an inner island and an outer ring split into Landau levels by a magnetic field \cite{komiyama}. In all these cases, the rather long photon wavelengths are controlled by the electron sub-band energy difference, rather than by the fundamental bandgap as in our experiments.
These types of single photon detectors should be distinguished from avalanche photo-diodes, where the single photon sensitivity arises from avalanche gain. In the FET and SET photodetectors, a single trapped electric charge can influence the current of millions of electrons in the source/drain channel. This is indeed the mechanism of ``photoconductive gain'' \cite{rose} that is also sometimes called ``secondary photoconductivity'' \cite{rose}. But this form of gain can also be considered as arising from transistor action. Thus the name ''single photoelectron transistor'' (SPT) might be appropriate. Since the photoelectron is safely trapped, and is known to have a long spin lifetime in many semiconductors \cite{awschalom}, it can then be interrogated to determine its spin state. The initial goal is to monitor the photo-electric charge in such a way as to not disturb its spin state. Ultimately the goal is to measure its spin state as well.
\begin{figure}
\caption{ A single photoelectron transistor (SPT) with window-gate double-quantum-well modulation-doped heterostructure. (a), Top view of the window-gate part of the SPT. The center of the window gates is relatively positive to the surroundings when negative voltage is applied to the gates because of Fermi level pinning. The blue regions indicate the two-dimensional electron gas (2DEG) in the channel layer. (b{\bf )}, Cross-section view of the layers in the SPT. The upper quantum well (QW) functions as an absorption layer and lower QW serves as a 2DEG channel layer, which is connected to source and drain. The curve on the absorption layer illustrates the electron potential when negative voltage is applied to the gates. (c), Scanning electron micrograph (SEM) picture of the SPT. (d), Close-up SEM picture of the window gate part (circled part in c). The window diameter is 1 $\mu$m.}
\label{fig2}
\end{figure}
At least three requirements should be satisfied to make a photodetector for quantum repeaters: (1) The wavelength that should be in the 1.3$\mu $m or 1.55$\mu $m, the low-loss window of optical fibers. (2) The sign of the photoconductivity that should be negative, which means the trapped information carrier should be an electron instead of a hole. (3) The electron g$_{e}$-factor, which should be small, to make the up-and-down electron spin states as indistinguishable as possible \cite{Vrijen}. The 1st requirement suggests interband transition rather than intraband transition. The 2nd requirement suggests creation of a positively charged trap for an electron. The 3rd requirement is satisfied through g$_{e}$-factor engineering \cite{kosaka01,kiselev02}.
The single photoelectron transistor (SPT) that we present in this paper satisfies all of the above requirements. An InGaAs quantum well is used with a bandgap corresponding to $\lambda $ = 1.3 $\mu $m, as shown in Fig.{ \nobreakspace}1. In Fig.{\nobreakspace}2 is shown the window-shaped circular gates that are negatively biased above the two-dimensional electron gas (2DEG), leaving behind a relatively positive central island. The InGaAs absorption layer, which has a g$_{e}$-factor{\nobreakspace}={\nobreakspace} --4.5 in the bulk, is sandwiched between InP cladding layers, of g$_{e}$ -factor{\nobreakspace}={\nobreakspace}+1.2, to make the effective g$_{e}$ -factor in the absorption layer nearly zero. The measurements showed clear evidence for negative persistent photoconductivity steps. The abrupt drops in photoconductivity are strongly correlated with photon injection at the $ \lambda $ = 1.3 $\mu $m wavelength, leading to the conclusion that the SPT detects a single photon by sensing the charge of a safely trapped photoelectron in the absorption quantum well.
The photo-absorption layer is located above the source/drain channel layer, and both are made of In$_{0.53}$Ga$_{0.47}$As, separated by a high electron barrier layer made of In$_{0.52}$Al$_{0.48}$As to prevent leakage. The source/drain channel layer is modulation doped and formed into a 1-dimensional electron gas (1DEG) channel whose conductance is sensitive to the charge state of the island in the absorption layer above it. All layers were grown by gas-source molecular beam epitaxy on semi-insulating InP, and consisted of a nominally undoped InP buffer layer 100nm thick; an i-In$ _{0.52}$Al$_{0.48}$As buffer 1000nm thick; a Si-doped (5 $\times $ 10$^{17}$ /cm$^{3}$) n-In$_{0.52}$Al$_{0.48}$As doping layer 10nm thick; an i-In$ _{0.52}$Al$_{0.48}$As lower spacer layer 30nm thick; an i-In$_{0.53}$Ga$ _{0.47}$As channel layer 10nm thick; an i-In$_{0.52}$Al$_{0.48}$As barrier layer 20nm thick; an i-InP cladding layer 10nm thick; an i-In$_{0.53}$Ga$ _{0.47}$As absorption layer 4.5nm thick; an i-InP cladding layer 10nm thick; and an i-In$_{0.52}$Al$_{0.48}$As capping layer 60nm thick. The modulation-doped double-quantum well structure creates a 2DEG in the lower quantum well that is shaped into a 1DEG channel by the two split gates. The gates surround a circular window, 1 $\mu $m in diameter, that masks out unnecessary light exposure, and fixes the potential at the edges surrounding of the window. The Schottky gates, Al/Pt/Au, are fabricated using electron-beam lithography and electron-gun evaporation. The source/drain ohmic contacts are made of AuGe/Ni/Au. Scanning Electron Microscope pictures of the whole device and the window gates are shown in Figs.{\nobreakspace}2c{ \nobreakspace}and{\nobreakspace}2d, respectively. The energy band diagram at zero bias, simulated by one-dimensional Poisson/Schr\"{o}dinger equation, is shown in Fig.{\nobreakspace}1.
The sample is illuminated by monochromatic light through a large-core glass fiber, that is carefully shielded to block any photons from the outer jacket. The light is created by a tungsten lamp and then filtered by a monochromator, a long-pass filter passing wavelengths $\lambda${\tt >} 1000nm, and a 30dB neutral density filter. The optical power at the end of the fiber is measured by a InGaAs detector. The illumination area in the plane of the device is about 5mm in diameter owing to light diffraction from the end of the fiber. Given the small device active area of 7.9 $\times$ 10{ \-}$^{9}${\nobreakspace}cm$^{2}$, defined by the 1 $\mu$m diameter gate window, we estimate the actual light power in the active area to be 2.8 $ \times$ 10$^{-}$$^{8}$ times smaller than the total power (assuming a Gaussian profile). The incident photon number is estimated by multiplying this scaling factor by the measured power divided by the photon energy.
By applying a negative voltage to the split window gates, the source/drain current through the channel layer is pinched off. Simultaneously, the applied negative voltage creates a two-dimensional potential minimum in the window at the absorption layer. This is because the surface Fermi level in the circular area is pinned by the extrinsic surface states \cite{chou}. The electric field in the electrostatic potential well can separate an electron-hole pair created by a photon. The electron is attracted to the potential minimum at on center, and the hole is attracted to the negative gates as schematically shown in Fig.{\nobreakspace}2b.
\begin{figure}
\caption{ Negative persistent photoconductivity of the SPT to $\lambda $ = 1.3 $\mu $m light starting with finite conductance, and positive photoconductivity at $\lambda $ = 1.7 $\mu $m light starting with zero conductance. The source-drain current drops in discrete steps when the SPT is exposed to $\lambda $ = 1.3 $\mu $m. The inset shows the initial current{ \nobreakspace}-{\nobreakspace}gate voltage characteristics (I$_{sd}$-V$_{g}$ curves) and bias points for the $\lambda $ = 1.3 $\mu $m exposure and the $ \lambda $ = 1.7 $\mu $m exposure. The $\lambda $ = 1.3 $\mu $m photons create photoelectrons in the quantum well, which are trapped and pinch off the 2DEG, step by step. In contrast, the $\lambda $ = 1.7 $\mu $m photoionize the electrons and increase the 2DEG density. Photon number absorbed in the window area is 0.3 per second, on average.}
\label{fig3}
\end{figure}
The source/drain current is measured at a constant voltage drop (V$_{sd}$) of 0.5mV, at a temperature of 4.2{\nobreakspace}K. The interesting property of these photodetectors is that $\lambda $ = 1.77 $\mu $m light produces positive photoconductivity effectively doping the channel, and $\lambda $ = 1.3 $\mu $m light produces negative photoconductivity. We attribute the channel doping by $\lambda $ = 1.77 $\mu $m light to be due to photo-ionization of donors in the n-InAlAs doping layer. As a normal practice, we initially prepare the photodetectors for use by means of a deep soak in $\lambda $ = 1.77 $\mu $m light, to fully ionize the donors and to populate the source/drain channel. The pinch-off behavior in the source-drain conductance (I$_{sd}$-V$_{g}$ curve) is shown in the inset of Fig.{\nobreakspace}3. The left-most I-V curve in that inset corresponds to full modulation doping after a deep soak in $\lambda $ = 1.77 $\mu $m light.
After the deep soak in $\lambda$ = 1.77 $\mu$m light to produce full channel doping, the gate voltage is adjusted for a current around 0.6nAmp. The device is then exposed to a photon flux at a wavelength of $\lambda$ = 1.3 $ \mu$m (red curve labeled 1.3 $\mu$m in Fig.{\nobreakspace}3). The photon exposure at $\lambda$ = 1.3 $\mu$m causes current to drop inexorably, step-by-step, except for occasional upward spikes. Thus as a result of trapped photoelectrons, the current is again pinched off, and the I$_{sd}$-V$ _{g}$ curve was shifted toward positive gate voltages as shown in the right-most curve of the inset in Fig.{\nobreakspace}3. At this pinch-off condition, if the device was again exposed to $\lambda$ = 1.77 $\mu$m, the channel current would be restored (blue curve labeled 1.7 $\mu$m in Fig.{ \nobreakspace}3). The incident photon rate in the active window area for both wavelengths is about 100{\nobreakspace}photons/s. Since the absorptivity in the absorption layer is about 1\%, on average 1 photon/sec is absorbed in the window area. Thus the quantum efficiency for producing negative steps is estimated to be 1\%.
\begin{figure}
\caption{ Spectral dependence of the photoconductivity. The wavelength was swept from $\lambda$ = 1.0 $\mu$m to $\lambda$ = 1.8 $\mu$m while monitoring the source-drain current. From $\lambda$ = 1.0 $\mu$m to $\lambda$ = 1.3 $ \mu $m, the current monotonically decreases, which is the range of negative photoconductivity. On the contrary, from $\lambda$ = 1.3 $\mu$m to $\lambda$ = 1.8 $\mu$m, the current monotonically increases with increasing wavelength, which is the range of positive photoconductivity. The cross-over point, $\lambda$ = 1.3 $\mu$m, corresponds to the bandgap in InGaAs quantum wells.}
\label{fig4}
\end{figure}
The current drop for $\lambda $ = 1.3 $\mu $m means that net negative charge is trapped near the source/drain channel. The exposure to $\lambda $ = 1.77 $ \mu $m photons is energetically able to cause only photo-ionization, because the photon energy is smaller than any bandgaps. \newline Detailed examination of the spectral dependence is not straightforward since the channel conductance depends on the starting bias, and the full history of spectral exposure. In Fig.{\nobreakspace}4 we start with an unpinched channel, and sweep wavelength starting from $\lambda $ = 1 $\mu $m up to $ \lambda $ = 1.8 $\mu $m over an 80second time period. First the current monotonically decreases with increasing wavelength, corresponding to trapped electrons, with no further decrease at around $\lambda $ = 1.3 $\mu $m, the bandgap of the InGaAs quantum wells. Negative trapped charge at wavelengths shorter than $\lambda $ = 1.3 $\mu $m is caused by photon absorption in the absorption layer or the channel layer. The photoelectrons in the conducting channel are mobile, and thus cannot contribute to trapped charge. Thus the negative steps must originate from photoelectrons produced in the absorption layer.
\begin{figure}
\caption{ Bit wise current state switching near the cross-over from positive to negative photoconductivity. The photon source is gated to synchronize the current steps with the photons. The shutter was repeatedly opened for $\sim$ 10sec every 50sec. The negative and positive photoconductivity events (electron trapping and photo-neutralization), were balanced by incomplete soaking at $\lambda$ = 1.77 $\mu$m. The current alternates between a higher state and a lower state, the switching induced by optical pulses. In the dark, the state was stable for more than one hour. The photon number absorbed within the window area is 30 photons in 10seconds, on average.}
\label{fig5}
\end{figure}
By having an incomplete initial soak in $\lambda $ = 1.77 $\mu $m radiation, we can control the pinch off voltage in between -0.5V and +0.1V. Now, when the pinch-off voltage is set to nearly zero, the $\lambda $ = 1.3 $\mu $m photocurrent still shows steps but they are equally likely to be either positive or negative. The incomplete photo-ionization of donors in the initial state allows a balance between electron trapping and photo-ionization. To make this phenomenon clear, we periodically opened the optical shutter for 10 seconds in every 50 seconds, maintaining the SPT in a balanced condition biased at $\sim $0 Volts. The resulting current pulses are shown in Fig.{\nobreakspace}5. The optical shutter is open during the time slots labeled a, b, c, etc., and closed during the intervening periods. Successive optical pulses usually produced either electron trapping or photo-ionization, alternating, depending on the previous state. Sometimes multiple optical pulses were required before the state would alternate. Within the 10sec optical pulse there might be a transient thermal response, especially in time slot{\nobreakspace}g; but that returned to either of the two alternating states after the optical pulse. The photon number absorbed in the window area is{\nobreakspace}$\sim $30 on average within the 10sec pulse. The estimated quantum efficiency is consistent with that in Fig.{ \nobreakspace}3.
The switching behavior in Fig.{\nobreakspace}5 is due to photoelectron trapping/de-trapping located either in; (a){\nobreakspace}the shallow circular potential well between the window gates in the absorption layer, or (b){\nobreakspace}at donor sites. In case{\nobreakspace}(b) the donors that could contribute to trapping/de-trapping are the residual donors in the absorption layer rather than those in the modulation doped layer. The modulation-doped donors, which are located far below the channel, would only produce a smooth increase in conductivity by photo-ionization as was seen in Fig.{\nobreakspace}3 for $\lambda$ = 1.77 $\mu$m light. In either case{ \nobreakspace}(a){\nobreakspace}or{\nobreakspace}(b), there are two possible mechanisms for the positive steps in Fig.{\nobreakspace}5; photo-ionization of the trapped electron, or annihilation of the trapped electron by injected holes. The photo-ionization mechanism would require a specific photo-ionization cross-section to be consistent with the rough equality between trapping and de-trapping rates. On the other hand, annihilation by photo-holes would require a hole trapping rate that is roughly coincident with the electron trapping rate. Such an adjustment may have been made by the adjustment of potential wells through the pinch-off voltage requirement of Fig.{\nobreakspace}5.
In conclusion, we have trapped and safely stored single photoelectrons in a window-gate double-quantum-well transistor structure. This Single Photoelectron Transistor detector satisfies three key requirements for a quantum repeater photo-detector. It has a wavelength suitable for optical fibers, it safely traps and detects a single photoelectron, and the g$_{e}$ -factors can be designed to satisfy the requirements for an entanglement preserving photo-detector. The wavelength could be shifted to $\lambda$ = 1.55 $\mu$m, which is more preferable, by using strain engineered substrates [15]. The Single Photoelectron Transistor announces the arrival of the photoelectric charge, without disturbing the transfer of quantum information from photon polarization to the electron spin state. We have yet to prove the entanglement transfer, but we believe such a demonstration will be a breakthrough for realizing long-distance quantum key distribution or long distance teleportation.
The project is sponsored by the Defense Advanced Research Projects Agency \& Army Research Office Nos. MDA972-99-1-0017 and DAAD19-00-1-0172. The content of the information does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred.
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\end{document} |
\begin{document}
\title{Minimal invariant varieties and first integrals \ for algebraic foliations}
\begin{center} { \small Mathematisches Institut, Universit\"at Basel\\ Rheinsprung 21, Basel 4051, Switzerland\\ e-mail: bonnet@math-lab.unibas.ch} \end{center}
\begin{abstract} Let $X$ be an irreducible algebraic variety over $\mathbb C$, endowed with an algebraic foliation ${\cal{F}}$. In this paper, we introduce the notion of minimal invariant variety $V({\cal{F}},Y)$ with respect to $({\cal{F}},Y)$, where $Y$ is a subvariety of $X$. If $Y=\{x\}$ is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through $x$. First we prove that for very generic $x$, the varieties $V({\cal{F}},x)$ have the same dimension $p$. Second we generalize a result due to X.Gomez-Mont (see \cite{G-M}). More precisely, we prove the existence of a dominant rational map $F:X\rightarrow Z$, where $Z$ has dimension $(n-p)$, such that for very generic $x$, the Zariski closure of $F^{-1}(F(x))$ is one and only one minimal invariant variety of a point. We end up with an example illustrating both results. \end{abstract}
\section{Introduction}
Let $X$ be an affine irreducible variety over $\mathbb C$, and ${\cal{O}}_{X}$ its ring of regular functions. Let ${\cal{F}}$ be an algebraic foliation, i.e. a collection of algebraic vector fields on $X$ stable by Lie bracket. We consider the elements of ${\cal{F}}$ as $\mathbb C$-derivations on the ring ${\cal{O}}_{X}$. In this paper, we are going to extend the notion of algebraic solution for ${\cal{F}}$: this will be the minimal tangent varieties for ${\cal{F}}$. We will study some of their properties and relate them to the existence of rational first integrals for ${\cal{F}}$.
Recall that a subvariety $Y$ of $X$ is an algebraic solution of ${\cal{F}}$ if $Y$ is the closure (for the metric topology) of a leaf of ${\cal{F}}$. A non-constant rational function $f$ on $X$ is a first integral if $ \partial(f)=0$ for any $ \partial$ in ${\cal{F}}$. Since the works of Darboux, the existence of such varieties has been extensively studied in the case of codimension 1 foliations (see \cite{Jou},\cite{Gh},\cite{Bru}). In particular, from these works, we know that only two cases may occur for codimension 1 foliations:
\begin{itemize} \item{${\cal{F}}$ has finitely many algebraic solutions,} \item{${\cal{F}}$ has infinitely many algebraic solutions, and a rational first integral.} \end{itemize} So rational first integrals appear if and only if all leaves of ${\cal{F}}$ are algebraic solutions. In this case, the fibres of any rational first integral is a finite union of closures of leaves. This fact has been generalised by Gomez-Mont (see \cite{G-M}) in the following way.
\begin{thh} Let $X$ be a projective variety and ${\cal{F}}$ an algebraic foliation on $X$ such that all leaves are quasi-projective. Then there exists a rational map $F:X\to Y$ such that, for every generic point $y$ of $Y$, the Zariski closure of $F^{-1}(y)$ is the closure of a leaf of ${\cal{F}}$. \end{thh} We would like to find a version of this result that does not need all leaves to be algebraic. To that purpose, we need to give a correct definition to the algebraic object closest to a leaf. A good candidate would be the Zariski closure of a leaf, but this choice may rise difficulties due to the singularities of both $X$ and ${\cal{F}}$. We counterpass this problem by the following algebraic approach.
Let $Y$ be an algebraic subvariety of $X$ and $I_Y$ the ideal of vanishing functions on $Y$. Let ${\cal{J}}$ be the set of ideals $I$ in ${\cal{O}}_{X}$ satisfying the two conditions: $$ (i) \quad (0) \; \subseteq I \; \subseteq I_Y \quad \quad {\rm{and}} \quad \quad (ii) \quad \forall \; \partial \; \in \; {\cal{F}}, \quad \partial(I)\; \subseteq\; I $$ Since $(0)$ belongs to ${\cal{J}}$, ${\cal{J}}$ is non-empty and it is partially ordered by the inclusion. Since it is obviously inductive, ${\cal{J}}$ admits a maximal element $I$. If $J$ is any other ideal of ${\cal{J}}$, then $I+J$ enjoys the conditions $(i)$ and $(ii)$, hence it belongs to ${\cal{J}}$. By maximality, we have $I=I+J$ and $J$ is contained in $I$. Therefore $I$ is the unique maximal element of ${\cal{J}}$, which we denote by $I({\cal{F}},Y)$.
\begin{df} The minimal invariant variety $V({\cal{F}},Y)$ is the zero set of $I({\cal{F}},Y)$ in $X$. \end{df} From a geometric viewpoint, $V({\cal{F}},Y)$ can be seen as the smallest subvariety containing $Y$ and invariant by the flows of all elements of ${\cal{F}}$. In particular, if $x$ is a smooth point of $X$ where the foliation is regular, then $V({\cal{F}},x)$ is the Zariski closure of the leaf passing through $x$. In section \ref{order}, we show that $V({\cal{F}},Y)$ is irreducible if $Y$ is itself irreducible. \\
In this paper, we would like to study the behaviour of these invariant varieties, and relate it to the existence of first integrals. We analyze some properties of the function: $$ n_{{\cal{F}}} : X \longrightarrow \mathbb{N}, \quad x \longmapsto dim\; V({\cal{F}},x) $$ Let ${\cal{M}}$ be the $\sigma$-algebra generated by the Zariski topology on $X$. A function $f: X\rightarrow \mathbb{N}$ is {\em measurable for the Zariski topology} if $f^{-1}(p)$ belongs to ${\cal{M}}$ for any $p$. The space ${\cal{M}}$ contains in particular all countable intersections $\theta$ of Zariski open sets. A property ${\cal{P}}$ holds for {\em every very generic point $x$ in $X$} if ${\cal{P}}(x)$ is true for any point $x$ in such an intersection $\theta$.
\begin{thh} \label{mesure} Let $X$ be an affine irreductible variety over $\mathbb C$ and ${\cal{F}}$ an algebraic foliation on $X$. Then the function $n_{{\cal{F}}}$ is measurable for the Zariski topology. Moreover there exists an integer $p$ such that $(1)$ $n_{{\cal{F}}}(x)\leq p$ for any point $x$ in $X$ and $(2)$ $n_{{\cal{F}}}(x)=p$ for any very generic point $x$ in $X$. \end{thh} Set $p=max\; dim \; V({\cal{F}},x)$ and note that $p$ is achieved for every generic point of $X$. In the last section, we will produce an example of a foliation on $\mathbb C^4$ where the function $n_{{\cal{F}}}$ is measurable but not constructible for the Zariski topology. In this sense, theorem \ref{mesure} is the best result one can expect for any algebraic foliation.
Let $K_{{\cal{F}}}$ be the field generated by $\mathbb C$ and the rational first integrals of ${\cal{F}}$. By construction, the invariant varieties $V({\cal{F}},x)$ are defined set-theoretically, and they seem to appear randomly, i.e. with no link within each other. In fact there does exist some order among them, and we are going to see that they are "mostly" given as the fibres of a rational map. More precisely:
\begin{thh} \label{fibration} Let $X$ be an affine irreducible variety over $\mathbb C$ of dimension $n$ and ${\cal{F}}$ an algebraic foliation on $X$. Then there exists a dominant rational map $F: X\rightarrow Y$, where $Y$ is irreducible of dimension $(n-p)$, such that for every very generic point $x$ of $X$, the Zariski closure of $F^{-1}(F(x))$ is equal to $V({\cal{F}},x)$. In particular, the transcendence degree of $K_{{\cal{F}}}$ over $\mathbb C$ is equal to $(n-p)$. \end{thh} The idea of the proof is to construct enough rational first integrals. These will be the coordinate functions of the rational map $F$ given above. The construction consists in choosing a codimension $d$ irreducible variety $H$ in $X$. We show there exists an integer $r>0$ such that, for every very generic point $x$ of $X$, $V({\cal{F}},x)$ intersects $H$ in $r$ distinct points $y_1,...,y_r$. We then obtain a correspondence: $$ {\cal{H}}: x\longmapsto \{y_1,...,y_r\} $$ We can modify ${\cal{H}}$ so as to get a rational map $F$ that represents every $r$-uple $\{y_1,...,y_r\}$ by a single point. Since the image of $x$ only depends on the intersection of $V({\cal{F}},x)$ with $H$, the map $F$ will be invariant with respect to the elements of ${\cal{F}}$. \\
One question may arise after these two results. Does there exist an effective way of computing these minimal invariant varieties and detect the presence of rational first integrals? For instance, we may attempt to use the description of the ideals $I({\cal{F}},Y)$ given by lemma \ref{autre}. Unfortunately we cannot hope to compute them in a finite number of steps bounded, for instance, by the degrees of the components of the vector fields of ${\cal{F}}$. Indeed, consider the well-known derivation $ \partial$ on $\mathbb C^2$: $$ \partial=px\frac{ \partial}{ \partial x} + qy\frac{ \partial}{ \partial y} $$ For any couple of non-zero coprime integers $(p,q)$, this derivation will have $f(x,y)=x^qy^{-p}$ as a rational first integral, and we cannot find another one of smaller degree. The minimal invariant varieties of points will be given in general by the fibres of $f$. Therefore we cannot bound the degree of the generators of $I({\cal{F}},x)$ solely by the degree of $ \partial$.
However, we may find them by an inductive process. For one derivation, an approach is given in the paper of J.V.Pereira via the notion of extatic curves (see \cite{Pe}). The idea is to compute a series of Wronskians attached to the derivation. Then one of them vanishes identically if and only the derivation has a rational first integral. \\
Last thing to say is that the previous results carry over all algebraic irreducible varieties. Given an algebraic variety $X$ with an algebraic foliation, we choose a covering of $X$ by open affine sets $U_i$ and work on the $U_i$. For any algebraic subvariety $Y$ of $X$, we define the minimal invariant variety $V({\cal{F}},Y)$ by gluing together the Zariski closure of the varieties $V({\cal{F}},Y\cap U_i)$ in $X$.
\section{The contact order with respect to ${\cal{F}}$} \label{order}
In this section, we are going to show that the minimal invariant variety $V({\cal{F}},Y)$ is irreducible if $Y$ is irreducible. This result is already known when ${\cal{F}}$ consists of one derivation (see \cite{Ka}). We could reproduce the proof given in \cite{Ka} for any set of derivations, but we prefer to adopt another strategy. We will instead introduce a notion of contact order with respect to ${\cal{F}}$, and we will use it to show that $I({\cal{F}},Y)$ is prime if $I_Y$ is prime. Denote by $M_{{\cal{F}}}$ the ${\cal{O}}_{X}$-module spanned by the elements of ${\cal{F}}$. We start by giving the following characterisation of $I({\cal{F}},Y)$.
\begin{lem} \label{autre} $\displaystyle I({\cal{F}},Y)=\left\{ f \in I_Y, \; \forall \partial_1,..., \partial_k \in M_{{\cal{F}}}, \; \partial_1\circ ...\circ \partial_k(f)\in I_Y \right\}$ \end{lem} {\em Proof: } Let $f$ be an element of $I_Y$ such that $ \partial_1\circ ...\circ \partial_k(f)$ belongs to $I_Y$ for any $ \partial_1,..., \partial_k$ in $M_{{\cal{F}}}$. Then $ \partial_1\circ ...\circ \partial_k(f)$ belongs to $I_Y$ for any elements $ \partial_1,..., \partial_k$ of ${\cal{F}}$. Let $I$ be the ideal generated by $f$ and all the elements of the form $ \partial_1\circ ...\circ \partial_k(f)$, where every $ \partial_i$ lies in ${\cal{F}}$. By construction, this ideal is contained in $I_Y$, and is stable by every derivation of ${\cal{F}}$. Therefore $I$ is contained in $I({\cal{F}},Y)$, and a fortiori $f$ belongs to $I({\cal{F}},Y)$. We then have the inclusion: $$ \left\{ f \in I_Y, \; \forall \partial_1,..., \partial_k \in M_{{\cal{F}}}, \; \partial_1\circ ...\circ \partial_k(f)\in I_Y \right\}\subseteq I({\cal{F}},Y) $$ Conversely let $f$ be an element of $I({\cal{F}},Y)$. Since $I({\cal{F}},Y)$ is contained in $I_Y$ and is stable by every derivation of ${\cal{F}}$, $ \partial_1\circ ...\circ \partial_k(f)$ belongs to $I_Y$ for any elements $ \partial_1,..., \partial_k$ of ${\cal{F}}$. Since $M_{{\cal{F}}}$ is spanned by ${\cal{F}}$, $ \partial_1\circ ...\circ \partial_k(f)$ belongs to $I_Y$ for any $ \partial_1,..., \partial_k$ in $M_{{\cal{F}}}$ \begin{flushright} $\blacksquare$\end{flushright} Since the space of $\mathbb C$-derivations on ${\cal{O}}_{X}$ is an ${\cal{O}}_{X}$-module of finite type and ${\cal{O}}_{X}$ is noetherian, $M_{{\cal{F}}}$ is finitely generated as an ${\cal{O}}_{X}$-module.
Let $\{ \partial_1,..., \partial_r\}$ be a system of generators of $M_{{\cal{F}}}$. If $I=(i_1,...,i_n)$ belongs to $\{1,...,r\}^n$, we set $ \partial_I = \partial_{i_1}\circ ...\circ \partial_{i_n}$ and $|I|=n$. By convention $\{1,...,r\}^0=\{\emptyset\}$,
$|\emptyset|=0$ and $ \partial_{\emptyset}$ is the identity on ${\cal{O}}_{X}$. We introduce the following map: $$
ord_{{\cal{F}},Y}: {\cal{O}}_{X} \longrightarrow \mathbb{N} \cup \{+\infty\}, \quad f \longmapsto \inf\left\{|I|, \; \partial_I(f)\not\in I_Y\right\} $$
\begin{df} The map $ord_{{\cal{F}},Y}$ is the contact order with respect to $({\cal{F}},Y)$. \end{df} By lemma \ref{autre}, $f$ belongs to $I({\cal{F}},Y)$ if and only if $ord_{{\cal{F}},Y}(f)=+\infty$, and $f$ does not belong to $I_Y$ if and only if $ord_{{\cal{F}},Y}(f)=0$. A priori, the map $ord_{{\cal{F}},Y}$ depends on the set of generators chosen for $M_{{\cal{F}}}$. We are going to see that it only depends on ${\cal{F}}$. Let $\{d_1,...,d_s\}$ be another set of generators for $M_{{\cal{F}}}$, and define in an analogous way the map $ord_{{\cal{F}},Y} '$ corresponding to this set. By assumption there exist some elements $a_{i,j}$ of ${\cal{O}}_{X}$ such that: $$ \partial_i= \sum_{j=1} ^s a_{i,j} d_j $$
By Leibniz rule, it is easy to check via an induction on $|I|$ that there exist some elements $a_{I,J}$ in ${\cal{O}}_{X}$ such that: $$
\partial_I = \sum_{|J|\leq |I|} a_{I,J} d_J $$ Let $f$ be an element of ${\cal{O}}_{X}$ such that $ord_{{\cal{F}},Y}(f)=n$. Then there exists an index $I$ of length $n$ such that: $$
\partial_I(f) = \sum_{|J|\leq n} a_{I,J} d_J(f) \; \not\in \; I_Y $$ Since $I_Y$ is an ideal, this means there exists an index $J$ of length $\leq n$ such that $d_J(f)$ does not belong to $I_Y$. By definition we get that $ord_{{\cal{F}},Y} '(f)\leq n=ord_{{\cal{F}},Y}(f)$ for any $f$. By symmetry we find that $ord_{{\cal{F}},Y} '(f)=ord_{{\cal{F}},Y}(f)$ for any $f$, and the maps coincide.
\begin{prop} \label{contact} If $Y$ is irreducible, the contact order enjoys the following properties: \begin{itemize} \item{$ord_{{\cal{F}},Y}(f+g)\geq \inf\{ord_{{\cal{F}},Y}(f),ord_{{\cal{F}},Y}(g)\}$ with equality if $ord_{{\cal{F}},Y}(f)\not=ord_{{\cal{F}},Y}(g)$,} \item{$ord_{{\cal{F}},Y}(fg)=ord_{{\cal{F}},Y}(f)+ord_{{\cal{F}},Y}(g)$ for all $f,g$ in ${\cal{O}}_{X}$.} \end{itemize} \end{prop} {\it Proof of the first assertion}: If $ord_{{\cal{F}},Y}(f)=ord_{{\cal{F}},Y}(g)=+\infty$, then $f,g$ both belong to $I({\cal{F}},Y)$, $f+g$ belongs to $I({\cal{F}},Y)$ and the result follows. So assume that $ord_{{\cal{F}},Y}(f)$ is finite and for simplicity that $n=ord_{{\cal{F}},Y}(f)\leq ord_{{\cal{F}},Y}(g)$. For any index $I$ of length $<n$, $ \partial_{I}(f)$ and $ \partial_{I}(g)$ both belong to
$I_Y$. So $ \partial_{I}(f+g)$ belong to $I_Y$ for any $I$ with $|I|<n$, and $ord_{{\cal{F}},Y}(f+g)\geq n$. Therefore we have for all $f,g$: $$ ord_{{\cal{F}},Y}(f+g)\geq \inf\{ord_{{\cal{F}},Y}(f),ord_{{\cal{F}},Y}(g)\} $$ Assume now that $ord_{{\cal{F}},Y}(f)<ord_{{\cal{F}},Y}(g)$. Then there exists an index $I$ of length $n$ such that $ \partial_{I}(f)$
does not belong to $I_Y$. Since $|I|< ord_{{\cal{F}},Y}(g)$, $ \partial_{I}(g)$ belongs to $I_Y$. Therefore $ \partial_{I}(f+g)$ does not belong to $I_Y$ and $ord_{{\cal{F}},Y}(f+g)\leq n$, so that $ord_{{\cal{F}},Y}(f+g)= n$. \begin{flushright} $\blacksquare$\end{flushright} For the second assertion, we will need the following lemmas. The first one is easy to get via Leibniz rule, by an induction on the length of $I$.
\begin{lem} \label{calcul2} Let $ \partial_1,..., \partial_r$ a system of generators of $M_{{\cal{F}}}$. Then there exist some elements $\alpha_{I_1,I_2}$ of $\mathbb C$, depending on $I$ and such that for all $f,g$: $$
\partial_I (fg)= \sum_{|I_1|+|I_2|=|I|} \alpha_{I_1,I_2} \partial_{I_1} (f) \partial_{I_2}(g) $$ \end{lem}
\begin{lem} \label{calcul3} Let $f$ be an element of ${\cal{O}}_{X}$ such that $ord_{{\cal{F}},Y}(f)\geq n$. Let $I=(i_1,...,i_n)$ be any index. For any rearrangement $J=(j_1,...,j_n)$ of the $i_k$, $ \partial_J(f)- \partial_I(f)$ belongs to $I_Y$. \end{lem} {\em Proof: } Every rearrangement of the $i_k$ can be obtained after a composition of transpositions on two consecutive terms. So we only need to check the lemma in the case $J=(i_1,...,i_{l+1},i_l,...,i_n)$. If we denote by $I_1,I_2$ the indices $I_1=(i_1,...,i_{l-1})$ and $I_2=(i_{l+2},...,i_n)$, then we find: $$ \partial_J - \partial_I = \partial_{I_1} \circ [ \partial_{i_l}, \partial_{i_{l+1}}] \circ \partial_{I_2} $$ Since $M_{{\cal{F}}}$ is stable by Lie bracket, $d=[ \partial_{i_l}, \partial_{i_{l+1}}]$ belongs to $M_{{\cal{F}}}$. Then $ \partial_J - \partial_I$ is a composite of $(n-1)$ derivations that span $M_{{\cal{F}}}$. Since $ord_{{\cal{F}},Y}$ is independent of the set of generators and $ord_{{\cal{F}},Y}(f)=n$, $ \partial_J (f) - \partial_I(f)$ belongs to $I_Y$. \begin{flushright} $\blacksquare$\end{flushright} {\it Proof of the second assertion of Proposition \ref{contact}}: Let $f,g$ be a couple of elements of ${\cal{O}}_{X}$. If either $f$ or $g$ has infinite contact order, then one of them belongs to $I({\cal{F}},Y)$ and $fg$ belongs to $I({\cal{F}},Y)$, so that $ord_{{\cal{F}},Y}(fg)=+\infty = ord_{{\cal{F}},Y}(f) + ord_{{\cal{F}},Y}(g)$. Assume now that $ord_{{\cal{F}},Y}(f)=n$ and $ord_{{\cal{F}},Y}(g)=m$ are finite. By lemma \ref{calcul2}, we have: $$
\partial_I (fg)= \sum_{|I_1|+|I_2|=|I|} \alpha_{I_1,I_2} \partial_{I_1} (f) \partial_{I_2}(g) $$
Since $|I_1|+|I_2|<n+m$, either $|I_1|<n$ or $|I_2|<m$, and $ \partial_{I_1} (f) \partial_{I_2}(g)$ belongs to $I_Y$. So $ \partial_I (fg)$ belongs to $I_Y$ and we obtain: $$ord_{{\cal{F}},Y}(fg)\geq n+m$$ Conversely, consider the following polynomials $P,Q$ in the indeterminates $x,t_1,...,t_r$: $$ P(x,t_1,...,t_r)=(t_1 \partial_1 +...+ t_r \partial_r)^n(f)(x) \quad , \quad Q(x,t_1,...,t_r)=(t_1 \partial_1 +...+ t_r \partial_r)^m(g)(x) $$ By lemma \ref{calcul3}, we get that $ \partial_I(f)\equiv \partial_J(f)\; [I_Y]$ for any rearrangement $J$ of $I$ if $I$ has length $n$. Idem for $ \partial_I(g)$ and $ \partial_J(g)$ if $I$ has length $m$. Therefore in the expressions of $P,Q$, everything happens modulo $I_Y$ as if the derivations $ \partial_i$ commuted. We then obtain the following expansions modulo $I_Y$: $$ P\equiv \sum_{i_1+...+i_r=n} \frac{n!}{i_1 ! ...i_r!} t_1 ^{i_1}...t_r ^{i_r} \partial_1 ^{i_1} \circ ...\circ \partial_r ^{i_r}(f) \; [I_Y] $$ $$ Q\equiv \sum_{i_1+...+i_r=m} \frac{m!}{i_1 ! ...i_r!} t_1 ^{i_1}...t_r ^{i_r} \partial_1 ^{i_1} \circ ...\circ \partial_r ^{i_r}(g)\; [I_Y] $$ Since $ord_{{\cal{F}},Y}(f)=n$ and $ord_{{\cal{F}},Y}(g)=m$, both $P$ and $Q$ have at least one coefficient that does not belong to $I_Y$ by lemma \ref{calcul3}. So neither of them belong to the ideal $I_Y[t_1,...,t_r]$, which is prime because $I_Y$ is prime. So $PQ$ does not belong to $I_Y[t_1,...,t_r]$. If $ \partial=t_1 \partial_1+...+t_r \partial_r$, then we have by Leibniz rule: $$ \partial^{n+m}(fg)= \sum_{k=0} ^n C_{n+m} ^k \partial^k(f) \partial^{n+m-k} (g) $$ Since $ord_{{\cal{F}},Y}(f)=n$ and $ord_{{\cal{F}},Y}(g)=m$, $ \partial^k(f) \partial^{n+m-k} (g)$ belongs to $I_Y[t_1,...,t_r]$ except for $k=n$. So $ \partial^{n+m}(fg)= C_{n+m} ^{n}PQ$ does not belong to $I_Y[t_1,...,t_r]$. Choose a point $(y,z_1,...,z_r)$ in $Y\times \mathbb C^r$ such that $PQ(y,z_1,...,z_r)\not=0$ and set $d=z_1 \partial_1+...+z_r \partial_r$. By construction we have: $$ d^{n+m}(fg)(y)=C_{n+m} ^{n}PQ(y,z_1,...,z_r)\not=0 $$ So $d^{n+m}(fg)$ does not belong to $I_Y$ and $fg$ has contact order $\leq n+m$ with respect to the system of generators $\{ \partial_1,..., \partial_r,d\}$. Since the contact order does not depend on the system of generators, we find: $$ ord_{{\cal{F}},Y}(fg)=n+m=ord_{{\cal{F}},Y}(f) +ord_{{\cal{F}},Y}(g) $$ \begin{flushright} $\blacksquare$\end{flushright}
\begin{cor} \label{irreductible} Let $Y$ be an irreducible subvariety of $X$. Then the ideal $I({\cal{F}},Y)$ is prime. In particular, the minimal invariant variety $V({\cal{F}},Y)$ is irreducible. \end{cor} {\em Proof: } Let $f,g$ be two elements of ${\cal{O}}_{X}$ such that $fg$ belongs to $I({\cal{F}},Y)$. Then $fg$ has infinite contact order. By proposition \ref{contact}, either $f$ or $g$ has infinite contact order. So one of them belongs to $I({\cal{F}},Y)$, and this ideal is prime. \begin{flushright} $\blacksquare$\end{flushright}
\section{Behaviour of the function $n_{{\cal{F}}}$}
In this section we are going to establish theorem \ref{mesure} about the measurability of the function $n_{{\cal{F}}}$ for the Zariski topology. Recall that a function $f: X \rightarrow \mathbb{N}$ is lower semi-continuous for the Zariski topology if the set $f^{-1}([0,r])$ is closed for any $r$. Note that such a function is continuous for the constructible topology. We begin with the following lemma.
\begin{lem} \label{etape} Let $F$ be a finite dimensional vector subspace of ${\cal{O}}_{X}$. Then the map $\varphi_F: X \rightarrow \mathbb{N}, \; x\mapsto dim_{\mathbb C} \;F -dim_{\mathbb C} \; I({\cal{F}},x)\cap F$ is lower semi-continuous for the Zariski topology. \end{lem} {\em Proof: } For any fixed finite-dimensional vector space $F$, consider the affine algebraic set: $$ \Sigma_F=\left\{ (x,f) \in X\times F, \; \forall d_1,...,d_m \in M_{{\cal{F}}}, \; d_1\circ...\circ d_m(f)(x)=0\right\} $$ together with the projection $\Pi: \Sigma_F \longrightarrow X, \; (x,f)\longmapsto x$. Since $\Sigma_F$ is affine, there exists a finite collection of linear operators $\Delta_1,...,\Delta_r$, obtained by composition of elements of $M_{{\cal{F}}}$, such that: $$ \Sigma_F=\left\{ (x,f) \in X\times F, \; \Delta_1(f)(x)=...=\Delta_r(f)(x)=0\right\} $$ By lemma \ref{autre}, the fibre $\Pi^{-1}(x)$ is isomorphic to $I({\cal{F}},x)\cap F$ for any point $x$ of $X$. Since every $\Delta_i$ is linear, $\Delta_i$ can be considered as a linear form on $F$ with coefficients in ${\cal{O}}_{X}$. So the map $\Delta=(\Delta_1,...,\Delta_r)$ is represented by a matrix with entries in ${\cal{O}}_{X}$. We therefore have the equivalence: $$ f\in I({\cal{F}},x)\cap F \quad \Longleftrightarrow \quad f \in ker \; \Delta(x) $$ By the rank theorem, we have $\varphi_F(x)=rk \; \Delta(x)$. But the rank of this matrix is a lower semi-continuous function because it is given as the maximal size of the minors of $\Delta$ that do not vanish at $x$. Therefore $\varphi_F$ is lower semi-continuous for the Zariski topology. \begin{flushright} $\blacksquare$\end{flushright} {\it Proof of theorem \ref{mesure}}: Since $X$ is affine, we may assume that $X$ is embedded in $\mathbb C^k$ for some $k$. We provide $\mathbb C[x_1,...,x_k]$ with the filtration $\{F_n\}$ given by the polynomials of homogeneous degree $\leq n$. By Hilbert-Samuel theorem (see \cite{Ei}), for any ideal $I$ of $\mathbb C[x_1,...,x_k]$, the function: $$ h_I(n)=dim_{\mathbb C} \; F_n - dim_{\mathbb C} \; I\cap F_n $$ is equal to a polynomial for $n$ large enough, and the degree $p$ of this polynomial coincides with the dimension of the variety $V(I)$. It is therefore easy to show that: $$ p= \lim_{n\to +\infty} \frac{\log(h_I(n))}{n} $$ Let $\Pi: \mathbb C[x_1,...,x_k]\rightarrow {\cal{O}}_{X}$ be the morphism induced by the inclusion $X\hookrightarrow \mathbb C^k$, and set $\widetilde{F_n}=\Pi(F_n)$. For any ideal $I$ of ${\cal{O}}_{X}$, consider the function: $$ \widetilde{h_I}(n)=dim_{\mathbb C} \; \widetilde{F_n} - dim_{\mathbb C} \; I\cap \widetilde{F_n} $$ Since $\Pi$ is onto, we have $\widetilde{h_I}(n)=h_{\Pi^{-1}(I)}(n)$, so that $\widetilde{h_I}(n)$ coincides for $n$ large enough with a polynomial of degree $p$ equal to the dimension of $V(I)$. With the notation of lemma \ref{etape}, we obtain for $I=I({\cal{F}},x)$: $$ p=n_{{\cal{F}}}(x)= \lim_{n\to +\infty} \frac{\log(\widetilde{h_I}(n))}{n}=\lim_{n\to +\infty} \frac{\log(\varphi_{\widetilde{F_n}}(x))}{n} $$ By lemma \ref{etape}, every $\varphi_{\widetilde{F_n}}$ is lower semi-continuous for the Zariski topology, hence measurable. Since a pointwise limit of measurable functions is measurable, the function $n_{{\cal{F}}}$ is measurable for the Zariski topology. Moreover since $\varphi_{\widetilde{F_n}}$ is lower semi-continuous, there exist a real number $r_n$ and an open set $U_n$ on $X$ such that: \begin{itemize} \item{$\displaystyle \frac{\log(\varphi_{\widetilde{F_n}}(x))}{n}\leq r_n$ for any $x$ in $X$,} \item{$\displaystyle \frac{\log(\varphi_{\widetilde{F_n}}(x))}{n}=r_n$ for any $x$ in $U_n$.} \end{itemize} Denote by $U$ the intersection of all $U_n$. Since this intersection is not empty, there exists an $x$ in $X$ for which $\log(\varphi_{\widetilde{F_n}}(x))/n=r_n$ for any $n$, so that $r_n$ converges to a limit $p$. By passing to the limit, we obtain that: \begin{itemize} \item{$n_{{\cal{F}}}(x)\leq p$ for any $x$ in $X$,} \item{$n_{{\cal{F}}}(x)= p$ for any $x$ in $U$.} \end{itemize} Note that $p$ has to be an integer. The theorem is proved. \begin{flushright} $\blacksquare$\end{flushright}
\section{The family of minimal invariant varieties} \label{set}
In this section, we are going to study the set of minimal invariant varieties associated to the points of $X$. The result we will get will be the first step towards the proof of theorem \ref{fibration}. Let $M$ be the following set: $$ M=\left\{(x,y) \in X\times X, \; y \in V({\cal{F}},x)\right\} $$ together with the projection $\Pi: M\longrightarrow X, (x,y)\longmapsto x$. Note that for any $x$, the preimage $\Pi^{-1}(x)$ is isomorphic to $V({\cal{F}},x)$, so that the couple $(M,\Pi)$ parametrizes the set of all minimal invariant varieties. Our purpose is to show that:
\begin{prop} \label{ferm} The Zariski closure $\overline{M}$ is an irreducible affine set of dimension $dim\; X +p$, where $p$ is the maximum of the function $n_{{\cal{F}}}$. Moreover, for every very generic point $x$ in $X$, $\overline{M} \cap \Pi ^{-1}(x)$ is equal to $\{x\}\times V({\cal{F}},x)$. \end{prop} The proof of this proposition is a direct consequence of the following lemmas.
\begin{lem} The Zariski closure $\overline{M}$ is irreducible. \end{lem} {\em Proof: } For any $ \partial_i$ in ${\cal{F}}$, consider the new $\mathbb C$-derivation $\Delta_i$ on ${\cal{O}}_{X\times X}={\cal{O}}_{X} \otimes_{\mathbb C} {\cal{O}}_{X}$ given by the following formula: $$ \forall f,g \in {\cal{O}}_{X}, \quad \Delta_i(f(x)\otimes g(y))=f(x)\otimes \partial_i(g)(y) $$ It is easy to check that $\Delta_i$ is a well-defined derivation. Denote by ${\cal{G}}$ the collection of the $\Delta_i$, by $D$ the diagonal $\{(x,x), \; x \in X\}$ in $X\times X$ and set $M_0= V({\cal{G}},D)$. By corollary \ref{irreductible}, $M_0$ is irreducible. We are going to prove that $\overline{M}=M_0$.
First let us check that $M_0 \subseteq \overline{M}$. Let $f$ be a regular function on $X\times X$ that vanishes on $\overline{M}$. Then $f(x,y)=0$ for any couple $(x,y)$ where $y$ belongs to $V({\cal{F}},x)$. If $\varphi_t(y)$ is the flow of $ \partial_i$ at $y$, then $\psi_t(x,y)=(x,\varphi_t(y))$ is the flow of $\Delta_i$ at $(x,y)$. Since $y$ lies in $V({\cal{F}},x)$, $\varphi_t(y)$ belongs to $V({\cal{F}},x)$ for any small value of $t$, and we obtain: $$ f(\psi_t(x,y))=f(x,\varphi_t(y))=0 $$ By derivation with respect to $t$, we get that $\Delta_i(f)(x,y)=0$ for any $(x,y)$ in $M$. So $\Delta_i(f)$ vanishes along $\overline{M}$, and the ideal $I(\overline{M})$ is stable by the family ${\cal{G}}$. Since it is contained in $I(D)$, we have the inclusion: $$ I(\overline{M})\subseteq I({\cal{G}},D) $$ which implies that $M_0 \subseteq \overline{M}$.
Second let us show that $\overline{M}\subseteq M_0$. Let $f$ be a regular function that vanishes along $M_0$. Fix $x$ in $X$ and consider the function $f_x(y)=f(x,y)$ on $X$. Then for any $\Delta_1,...,\Delta_n$ in ${\cal{G}}$, we have: $$ \Delta_1 \circ ...\circ \Delta_n(f)(x,y)= \partial_1\circ ...\circ \partial_n(f_x)(y) $$ Since $M_0=V({\cal{G}},D)$, $D$ is contained in $M_0$ and $f_x(x)=0$. So $f(x,x)=0$ and for any $ \partial_1,..., \partial_n$ in ${\cal{F}}$ and any $x$ in $X$, we get that: $$ \partial_1\circ ...\circ \partial_n(f_x)(x)=0 $$ In particular, $f_x$ belongs to $I({\cal{F}},x)$ and $f_x$ vanishes along $V({\cal{F}},x)$. Thus $f$ vanishes on $\{x\}\times V({\cal{F}},x)=\Pi^{-1}(x)$ for any $x$ in $X$. This implies that $f$ is equal to zero on $M$ and on $\overline{M}$, so that $I({\cal{G}},D)\subseteq I(\overline{M})$. As a consequence, we find $\overline{M}\subseteq M_0$ and the result follows. \begin{flushright} $\blacksquare$\end{flushright}
\begin{lem} The variety $\overline{M}$ has dimension $\geq dim \; X+p$. \end{lem} {\em Proof: } Consider the projection $\Pi: \overline{M} \rightarrow X, \; (x,y)\mapsto x$. Since $M$ contains the diagonal $D$, the map $\Pi$ is onto. By the theorem on the dimension of fibres, there exists a non-empty Zariski open set $U$ in $X$ such that: $$ \forall x \in U, \quad dim \; \overline{M} = dim\; X + dim \; \Pi^{-1}(x) \cap \overline{M} $$ By theorem \ref{mesure}, there exists a countable intersection $\theta$ of Zariski open sets in $X$ such that $n_{{\cal{F}}}(x)=p$ for all $x$ in $X$. In particular, $U \cap \theta$ is non-empty. For any $x$ in $U \cap \theta$, $\Pi^{-1}(x) \cap \overline{M}$ contains the variety $V({\cal{F}},x)$ whose dimension is $p$, and this yields: $$ dim \; \overline{M} \geq dim\; X + p $$ \begin{flushright} $\blacksquare$\end{flushright}
\begin{lem} \label{note} The variety $\overline{M}$ has dimension $\leq dim \; X+p$. \end{lem} {\em Proof: } Let $\{F_n\}$ be a filtration of ${\cal{O}}_{X}$ by finite-dimensional $\mathbb C$-vector spaces, and set: $$ M_n =\left\{ (x,y) \in X\times X, \; \forall f \in I({\cal{F}},x)\cap F_n, \; f(y)=0 \right\} $$ The sequence $\{M_n\}$ is decreasing for the inclusion, and $M=\cap_{n \in \mathbb{N}}\; M_n$. Moreover every $M_n$ is constructible for the Zariski topology by Chevalley's theorem (see \cite{Ei}). Indeed its complement in $X\times X$ is the image of the constructible set: $$ \Sigma_n =\left\{ (x,y,f) \in X\times X \times F_n , \; \forall \partial_1,..., \partial_k \in {\cal{F}}, \; \partial_1 \circ ...\circ \partial_k(f)(y)=0 \; {\rm{and}} \; f(y)\not=0\right\} $$ under the projection $(x,y,f)\mapsto (x,y)$. Since $D$ is contained in every $M_n$, the projection $\Pi: M_n \rightarrow X$ is onto. By the theorem on the dimension of fibres applied to the irreducible components of $\overline{M_n}$, there exists a non-empty Zariski open set $U_n$ in $X$ such that: $$ \forall x \in U_n, \quad dim \; M_n \leq dim\; X + dim \; \Pi^{-1}(x) \cap M_n $$ Since $\overline{M}\subseteq \overline{M_n}$ for any $n$, and $\Pi^{-1}(x) \cap M_n\simeq V(I({\cal{F}},x)\cap F_n)$, we obtain: $$ \forall x \in U_n, \quad dim \; \overline{M} \leq dim\; X + dim \; V(I({\cal{F}},x)\cap F_n) $$ Since every $U_n$ is open, the intersection $\theta'=\cap_{n\in \mathbb{N}} \; U_n$ is non-empty. Let $\theta$ be an intersection of Zariski open sets of $X$ such that $n_{{\cal{F}}}(x)=p$ for any $x$ of $\theta$. For any fixed $x$ in $\theta \cap \theta'$, we have: $$ \forall n\in \mathbb{N}, \quad dim \; \overline{M} \leq dim\; X + dim \; V(I({\cal{F}},x)\cap F_n) $$ Since ${\cal{O}}_{X}$ is noetherian, there exists an order $n_0$ such that $I({\cal{F}},x)$ is generated by $I({\cal{F}},x)\cap F_n$ for any $n\geq n_0$. In this context, $V({\cal{F}},x)=V(I({\cal{F}},x)\cap F_n)$ for all $n\geq n_0$, and $V({\cal{F}},x)$ has dimension $p$, which implies that: $$ dim \; \overline{M} \leq dim\; X + p $$ \begin{flushright} $\blacksquare$\end{flushright}
\begin{lem} For every very generic point $x$ in $X$, $\overline{M} \cap \Pi ^{-1}(x)$ is equal to $\{x\}\times V({\cal{F}},x)$. \end{lem} {\em Proof: } Consider the constructible sets $M_n$ introduced in lemma \ref{note}. By construction their intersection is equal to $M$. The $\{\overline{M_n}\}$ form a decreasing sequence which converges to $\overline{M}$. Since these are algebraic sets, there exists an index $n_0$ such that for any $n\geq n_0$, we have $\overline{M_n}=\overline{M}$. We consider the sequence $\{M_n\}_{n\geq n_0}$ and denote by $G_n$ the Zariski closure of $\overline{M} - M_n$. By the theorem on the dimension of fibres, there exists a Zariski open set $V_n$ on $X$ such that for any $x$ in $V_n$, either $\Pi^{-1}(x)\cap G_n$ is empty or has dimension $<p$. Since $\Pi^{-1}(x)\cap M=\{x\}\times V({\cal{F}},x)$ for any $x$ in $X$, we have the following decomposition: $$ \Pi^{-1}(x)\cap \overline{M}= \{x\}\times V({\cal{F}},x) \cup \cup_{n\geq n_0} \Pi^{-1}(x)\cap G_n $$ For all $x$ in $\theta=\cap V_n$, the set $\Pi^{-1}(x)\cap G_n$ has dimension $<p$ for any $n\geq n_0$, hence its Hausdorff dimension is no greater than $(2p-2)$ (see \cite{Ch}). Consequently the countable union $\cup_{n\geq n_0} \Pi^{-1}(x)\cap G_n$ has an Hausdorff dimension $<2p$. Let $H_{i,x}$ be the irreducible components of $\Pi^{-1}(x)\cap \overline{M}$ distinct from $\{x\}\times V({\cal{F}},x)$. These $H_{i,x}$ are covered by the union $\cup_{n\geq n_0} \Pi^{-1}(x)\cap G_n$, hence their Hausdorff dimension does not exceed $(2p-2)$. Therefore the Krull dimension of $H_{i,x}$ is strictly less than $p$ for any $i$ and any $x$ in $\theta$. If $H_x$ denotes the union of the $H_{i,x}$, then we have for any $x$ in $\theta$: $$ \Pi^{-1}(x)\cap \overline{M}= \{x\}\times V({\cal{F}},x) \cup H_x \quad \mbox{and} \quad dim\; H_x<p $$ Now by Stein factorization theorem (see \cite{Ha}), the map $\Pi: \overline{M}\rightarrow X$ is a composite of a quasi-finite map with a map whose generic fibres are irreducible. In particular $\Pi^{-1}(x)\cap \overline{M}$ is equidimensionnal of dimension $p$ for generic $x$ in $X$. Therefore the variety $H_x$ should be contained in $\{x\}\times V({\cal{F}},x)$, and we have for any $x$ in $\theta$: $$ \Pi^{-1}(x)\cap \overline{M}= \{x\}\times V({\cal{F}},x) $$ \begin{flushright} $\blacksquare$\end{flushright}
\section{Proof of theorem \ref{fibration}}
Let $X$ be an irreducible affine variety over $\mathbb C$ of dimension $n$, endowed with an algebraic foliation ${\cal{F}}$. Let $p$ be the integer given by theorem \ref{mesure}. In this section we will establish theorem \ref{fibration}. We begin with a few lemmas.
\begin{lem} Let $F:X\rightarrow Y$ be a dominant morphism of irreducible affine varieties. Then for any Zariski open set $U$ in $X$, $F(U)$ is dense in $Y$. \end{lem} {\em Proof: } Suppose on the contrary that $F(U)$ is not dense in $Y$. Then there exists a non-zero regular function $f$ on $Y$ that vanishes along $\overline{F(U)}$. The function $f\circ F$ vanishes on $U$, hence on $X$ by density. So $F(X)$ is contained in $f^{-1}(0)$, which is impossible since this set is dense in $Y$. \begin{flushright} $\blacksquare$\end{flushright}
\begin{lem} \label{prep} Let $\overline{M}$ be the variety defined in section \ref{set}. Then there exists an irreducible variety $H$ in $X$ such that $\overline{M} \cap X\times H$ has dimension $n$ and the morphism $\Pi: \overline{M} \cap X\times H \rightarrow X$ induced by the projection is dominant. \end{lem} {\em Proof: } Let $(x,y)$ be a smooth point of $\overline{M}$ such that $x$ is a smooth point of $X$. By the generic smoothness theorem, we may assume that $d\Pi_{(x,y)}$ is onto. Consider the second projection $\Psi(x,y)=y$. Since the map $(\Pi,\Psi)$ defines an embedding of $\overline{M}$ into $X\times X$, and $d\Pi_{(x,y)}$ is onto, there exist some regular functions $g_1,...,g_p$ on $X$ such that $(d\Pi_{(x,y)},{dg_1}_{(y)},...,{dg_p}_{(y)})$ is an isomorphism from $T_{(x,y)} \overline{M}$ to $T_x X \oplus \mathbb C ^p$.
Let $G: \overline{M}\rightarrow \mathbb C^p$ be the map $(g_1,...,g_p)$, and denote by $E$ the set of points $(x,y)$ in $\overline{M}$ where either $\overline{M}$ is singular or $(\Pi,G)$ is not submersive. By construction $E$ is a closed set distinct from $\overline{M}$. Since $dG_{(y)}$ has rank $p$ on $T_{(x,y)} \overline{M}$, the map $G:\overline{M}\rightarrow \mathbb C^p$ is dominant. So its generic fibres have dimension $n$. Fix a fibre $G^{-1}(z)$ of dimension $n$ that is not contained in $E$. Then there exists a smooth point $(x,y)$ in $G^{-1}(z)$ such that $d(\Pi,G)_{(x,y)}$ is onto. The morphism $\Pi:G^{-1}(z)\rightarrow X$ is a submersion at $(x,y)$, hence it is dominant. Moreover $G^{-1}(z)$ is of the form $X\times F^{-1}(z)\cap \overline{M}$, where $F:X \rightarrow \mathbb C^p$ is the map $(g_1,...,g_p)$.
Choose an irreducible component $H$ of $F^{-1}(z)$ such that $\Pi: X\times H \cap \overline{M}\rightarrow X$ is dominant. By construction $X\times H \cap \overline{M}$ has dimension $\leq n$. Since the latter map is dominant, its dimension is exactly equal to $n$. \begin{flushright} $\blacksquare$\end{flushright} {\it Proof of theorem \ref{fibration}}: Let $H$ be an irreducible variety of codimension $p$ in $X$ satisfying the conditions of lemma \ref{prep}. Denote by $N$ the union of irreducible components of $\overline{M} \cap X\times H$ that are mapped dominantly on $X$ by $\Pi$. By construction $N$ has dimension $dim\; X$ and the morphism $\Pi: N\rightarrow X$ is quasi-finite. So there exists an open set $U$ in $X$ such that: $$ \widetilde{\Pi}: \Pi^{-1}(U)\cap N \longrightarrow U $$ is a finite unramified morphism. Let $r$ be the degree of this map. For any point $x$ in $U$, there exist $r$ points $y_1,...,y_r$ in $H$ such that ${\widetilde{\Pi}}^{-1}(x)=\{y_1,...,y_r\}$. Let $\mathfrak{S}_r$ act on $H^r$ by permutation of the coordinates, i.e $\sigma.(y_1,...,y_r)=(y_{\sigma(1)},...,y_{\sigma(r)})$. Since this action is algebraic and $\mathfrak{S}_r$ is finite, the algebraic quotient $H^r //\mathfrak{S}_r$ exists and is an irreducible affine variety (see \cite{Mu}). Let $Q: H\rightarrow H^r //\mathfrak{S}_r$ be the corresponding quotient morphism. Consider the mapping: $$ \varphi: U \longrightarrow H^r //\mathfrak{S}_r , \quad x \longmapsto Q(y_1,...,y_r) $$ Note that its graph is constructible in $U\times H^r //\mathfrak{S}_r$. Indeed it is given by the set: $$ \Sigma=\left\{ (x,y'), \; \exists (y_1,...,y_r) \in H^r, \;\forall i\not=j, \; y_i\not=y_j, \; (x,y_i) \in \overline{M} \; {\rm{and}} \; Q(y_1,...,y_r)=y' \right\} $$ By Serre's theorem (see \cite{Lo}), $\varphi$ is a rational map on $U$. Since $\widetilde{\Pi}$ is unramified, $\varphi$ is also holomorphic on $U$, hence it is regular on $U$. Denote by $Y$ the Zariski closure of $\varphi(U)$ in $H^r //\mathfrak{S}_r$. Since $U$ is irreducible, $Y$ is itself irreducible.
By construction, for any $x$ in $U$, $\{x\}\times \varphi^{-1}(\varphi(x))$ is equal to $\Pi^{-1}(x)\cap \overline{M}$. For every very generic point $x$ in $X$, $\Pi^{-1}(x)\cap \overline{M}$ corresponds to $\{x\}\times V({\cal{F}},x)$ by proposition \ref{ferm}. So $\varphi^{-1}(\varphi(x))=V({\cal{F}},x)$ for every generic point $x$ in $X$, hence it has dimension $p$. By the theorem on the dimension of fibres, $Y$ has dimension $(n - p)$.
Since $\varphi^{-1}(\varphi(x))=V({\cal{F}},x)$ for every generic point $x$ in $X$, this fibre is tangent to the foliation ${\cal{F}}$. Since tangency is a closed condition, all the fibres of $\varphi$ are tangent to ${\cal{F}}$. Let $f$ be a rational function on $Y$. In the neighborhood of any smooth point $x$ where ${\cal{F}}$ is regular and $f\circ \varphi$ is well-defined, the function $f\circ \varphi$ is constant on the leaves of ${\cal{F}}$. So $f\circ \varphi$ is a rational first integral of ${\cal{F}}$. Via the morphism $\varphi^*$ induced by $\varphi$, $K_{{\cal{F}}}$ is clearly isomorphic to $\mathbb C(Y)$ which has transcendence degree $(n-p)$ over $\mathbb C$. \begin{flushright} $\blacksquare$\end{flushright}
\section{An example}
In this last section, we introduce an example that illustrates both theorems \ref{mesure} and \ref{fibration}. Consider the affine space $\mathbb C^4$ with coordinates $(u,v,x,y)$, and the algebraic foliation ${\cal{F}}$ induced by the vector field: $$ \partial=ux\frac{ \partial}{ \partial x} + vy \frac{ \partial}{ \partial y} $$ For any $(\lambda,\mu)$ in $\mathbb C^2$, the plane $V(u-\lambda,v-\mu)$ is tangent to ${\cal{F}}$. Denote by $ \partial_{\lambda,\mu}$ the restriction of $ \partial$ to that plane parametrized by $(x,y)$. Then two cases may occur: \begin{itemize} \item{If $[\lambda;\mu]$ does not belong to $\mathbb{P}^1(\mathbb{Q})$, then $ \partial_{\lambda,\mu}$ has no rational first integrals. The only algebraic curves tangent to $ \partial_{\lambda,\mu}$ are the lines $x=0$ and $y=0$. There is only one singular point, namely $(0,0)$.} \item{If $[\lambda;\mu]$ belongs to $\mathbb{P}^1(\mathbb{Q})$, choose a couple of coprime integers $(p,q)\not=(0,0)$ such that $p\lambda+q\mu=0$. The function $f(x,y)=x^py^q$ is a rational first integral for $ \partial_{\lambda,\mu}$. The algebraic curves tangent to $ \partial_{\lambda,\mu}$ are the lines $x=0$, $y=0$ and the fibres $f^{-1}(z)$ for $z\not=0$. There is only one singular point, namely $(0,0)$.} \end{itemize} From those two cases, we can get the following values for the function $n_{{\cal{F}}}$:
\begin{itemize} \item{$n_{{\cal{F}}}(u,v,x,y)=2$ if $[\lambda;\mu] \not\in\mathbb{P}^1(\mathbb{Q})$ and $xy\not=0$,} \item{$n_{{\cal{F}}}(u,v,x,y)=0$ if $x=y=0$,} \item{$n_{{\cal{F}}}(u,v,x,y)=1$ otherwise.} \end{itemize} In particular, this function is measurable but not constructible for the Zariski topology, as can be easily seen from its fibre $n_{{\cal{F}}} ^{-1}(2)$. Moreover since $p=2$, its field $K_{{\cal{F}}}$ has transcendence degree 2 over $\mathbb C$. In fact it is easy to check that $K_{{\cal{F}}}=\mathbb C(u,v)$.
\end{document} |
\begin{document}
\title{Preprocessing to Reduce the Search Space: Antler Structures for Feedback Vertex Set}
\begin{abstract} The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic \textsc{Feedback Vertex Set} problem on undirected graphs, leading to a new type of graph structure called \emph{antler decomposition}, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated \emph{crown decomposition} which has been used for \textsc{Vertex Cover}. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve \textsc{Feedback Vertex Set}. \end{abstract}
\section{Introduction} \label{sec:intro} The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly~\cite{Fellows06,GuoN07a}. In a nutshell, this new direction can be summarized as: how can an algorithm identify part of an optimal solution in an efficient preprocessing phase? We explore this direction for the classic~\cite{Karp72} \textsc{Feedback Vertex Set} problem on undirected graphs, leading to a new graph structure called \emph{antler} which reveals vertices that belong to an optimal feedback vertex set.
We start by motivating the need for a new direction in the theoretical analysis of preprocessing. The use of preprocessing, often via the repeated application of reduction rules, has long been known~\cite{AchterbergBGRW16,Achterberg2013,Quine52} to speed up the solution of algorithmic tasks in practice. The introduction of the framework of parameterized complexity~\cite{DowneyF99} in the 1990s made it possible to also analyze the power of preprocessing \emph{theoretically}, through the notion of kernelization. It applies to \emph{parameterized decision problems}~$\Pi \subseteq \Sigma^* \times \mathbb{N}$, in which every instance~$x \in \Sigma^*$ has an associated integer parameter~$k$ which captures one dimension of its complexity. For \textsc{Feedback Vertex Set}, typical choices for the parameter include the size of the desired solution or structural measures of the complexity of the input graph. A kernelization for a parameterized problem~$\Pi$ is then a polynomial-time algorithm that reduces any instance with parameter value~$k$ to an equivalent instance, of the same problem, whose total size is bounded by~$f(k)$ for some computable function~$f$ of the parameter alone. The function~$f$ is the \emph{size} of the kernelization.
A substantial theoretical framework has been built around the definition of kernelization~\cite{CyganFKLMPPS15,DowneyF13,FlumG06,FominLSZ19,GuoN07a}. It includes deep techniques for obtaining kernelization algorithms~\cite{BodlaenderFLPST16,FominLMS12,KratschW12,PilipczukPSL18}, as well as tools for ruling out the existence of small kernelizations~\cite{BodlaenderJK14,DellM14,Drucker15,FortnowS11,HermelinKSWW15} under complexity-theoretic hypotheses. This body of work gives a good theoretical understanding of polynomial-time data compression for NP-hard problems.
However, we argue that these results on kernelization \emph{do not} explain the often exponential speed-ups (e.g.~\cite{AchterbergBGRW16},~\cite[Table 6]{AkibaI16}) caused by applying effective preprocessing steps to non-trivial algorithms. Why not? A kernelization algorithm guarantees that the input \emph{size} is reduced to a function of the parameter~$k$; but the running time of modern parameterized algorithms for NP-hard problems is not exponential in the total input size. Instead, fixed-parameter tractable (FPT) algorithms have a running time that scales polynomially with the input size, and which only depends exponentially on a problem parameter such as the solution size or treewidth. Hence an exponential speed-up of such algorithms cannot be explained by merely a decrease in input size, but only by a decrease in the \emph{parameter}!
We therefore propose the following novel research direction: to investigate how preprocessing algorithms can decrease the parameter value (and hence search space) of FPT algorithms, in a theoretically sound way. It is nontrivial to phrase meaningful formal questions in this direction. To illustrate this difficulty, note that strengthening the definition of kernelization to ``a preprocessing algorithm that is guaranteed to always output an equivalent instance of the same problem with a strictly smaller parameter'' is useless. Under minor technical assumptions, such an algorithm would allow the problem to be solved in polynomial time by repeatedly reducing the parameter, and solving the problem using an FPT or XP algorithm once the parameter value becomes constant. Hence NP-hard problems do not admit such parameter-decreasing algorithms. To formalize a meaningful line of inquiry, we take our inspiration from the \textsc{Vertex Cover} problem, the fruit fly of parameterized algorithms.
A rich body of theoretical and applied algorithmic research has been devoted to the exact solution of the \textsc{Vertex Cover} problem~\cite{AkibaI16,DzulfikarFH19,HespeLSS19,Hespe0S19}. A standard 2-way branching algorithm can test whether a graph~$G$ has a vertex cover of size~$k$ in time~$\ensuremath{\mathcal{O}}\xspace(2^k (n+m))$, which can be improved by more sophisticated techniques~\cite{ChenKX10}. The running time of the algorithm scales linearly with the input size, and exponentially with the size~$k$ of the desired solution. This running time suggests that to speed up the algorithm by a factor~$1000$, one either has to decrease the input size by a factor~$1000$, or decrease~$k$ by~$\log_2 (1000) \approx 10$.
It turns out that state-of-the-art preprocessing strategies for \textsc{Vertex Cover} indeed often \emph{succeed} in decreasing the size of the solution that the follow-up algorithm has to find, by means of crown-reduction~\cite{Abu-KhzamFLS07,ChorFJ04,Fellows03}, or the intimately related Nemhauser-Trotter reduction based on the linear-programming relaxation~\cite{NemhauserT75}. Recall that a vertex cover in a graph~$G$ is a set~$S \subseteq V(G)$ such that each edge has at least one endpoint in~$S$. Observe that if~$H \subseteq V(G)$ is a set of vertices with the property that there exists a minimum vertex cover of~$G$ containing all of~$H$, then~$G$ has a vertex cover of size~$k$ if and only if~$G - H$ has a vertex cover of size~$k - |H|$. Therefore, if a preprocessing algorithm can identify a set of vertices~$H$ which are guaranteed to belong to an optimal solution, then it can effectively reduce the parameter of the problem by restricting to a search for a solution of size~$k - |H|$ in~$G-S$.
A \emph{crown decomposition} (cf.~\cite{Abu-KhzamCFLSS04,ChorFJ04,Fellows03},~\cite[\S 2.3]{CyganFKLMPPS15},~\cite[\S 4]{FominLSZ19}) of a graph~$G$ serves exactly this purpose. It consists of two disjoint vertex sets~$(\ensuremath{\mathsf{head}}\xspace,\ensuremath{\mathsf{crown}}\xspace)$, such that~$\ensuremath{\mathsf{crown}}\xspace$ is a non-empty independent set whose neighborhood is contained in~$\ensuremath{\mathsf{head}}\xspace$, and such that the graph~$G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace]$ has a matching~$M$ of size~$|\ensuremath{\mathsf{head}}\xspace|$. As~$\ensuremath{\mathsf{crown}}\xspace$ is an independent set, the matching~$M$ assigns to each vertex of~$\ensuremath{\mathsf{head}}\xspace$ a private neighbor in~$\ensuremath{\mathsf{crown}}\xspace$. It certifies that any vertex cover in~$G$ contains at least~$|\ensuremath{\mathsf{head}}\xspace|$ vertices from~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace$, and as~$\ensuremath{\mathsf{crown}}\xspace$ is an independent set with~$N_G(\ensuremath{\mathsf{crown}}\xspace) \subseteq \ensuremath{\mathsf{head}}\xspace$, a simple exchange argument shows there is indeed an optimal vertex cover in~$G$ containing all of~$\ensuremath{\mathsf{head}}\xspace$ and none of~$\ensuremath{\mathsf{crown}}\xspace$. Since there is a polynomial-time algorithm to find a crown decomposition if one exists~\cite[Thm.~11--12]{Abu-KhzamFLS07}, this yields the following preprocessing guarantee for \textsc{Vertex Cover}: if the input instance~$(G,k)$ has a crown decomposition~$(\ensuremath{\mathsf{head}}\xspace,\ensuremath{\mathsf{crown}}\xspace)$, then a polynomial-time algorithm can reduce the problem to an equivalent one with parameter at most~$k - |\ensuremath{\mathsf{head}}\xspace|$, thereby giving a formal guarantee on reduction in the parameter based on the structure of the input.\footnote{The effect of the crown reduction rule can also be theoretically explained by the fact that interleaving basic 2-way branching with exhaustive crown reduction yields an algorithm whose running time is only exponential in the \emph{gap} between the size of a minimum vertex cover and the cost of an optimal solution to its linear-programming relaxation~\cite{LokshtanovNRRS14}. However, this type of result cannot be generalized to \textsc{Feedback Vertex Set} since it is already NP-complete to determine whether there is a feedback vertex set whose size matches the cost of the linear-programming relaxation (\cref{cor:lp-relax}).}
As the first step of our proposed research program into parameter reduction (and thereby, search space reduction) by a preprocessing phase, we present a graph decomposition for \textsc{Feedback Vertex Set} which can identify vertices~$S$ that belong to an optimal solution; and which therefore facilitate a reduction from finding a solution of size~$k$ in graph~$G$, to finding a solution of size~$k - |S|$ in~$G - S$. While there has been a significant amount of work on kernelization for \textsc{Feedback Vertex Set}~\cite{BodlaenderD10,BurrageEFLMR06,Iwata17,JansenRV14,Thomasse10}, the corresponding preprocessing algorithms do not succeed in finding vertices that belong to an optimal solution, other than those for which there is a self-loop or those which form the center a flower (consisting of~$k+1$ otherwise vertex-disjoint cycles~\cite{BodlaenderD10,BurrageEFLMR06,Thomasse10}, or a technical relaxation of this notion~\cite{Iwata17}). In particular, apart from the trivial self-loop rule, earlier preprocessing algorithms can only conclude a vertex~$v$ belongs to all optimal solutions (of a size~$k$ which must be given in advance) if they find a suitable packing of cycles witnessing that solutions without~$v$ must have size larger than~$k$. In contrast, our argumentation will be based on \emph{local} exchange arguments, which can be applied independently of the global solution size~$k$.
We therefore introduce a new graph decomposition for preprocessing \textsc{Feedback Vertex Set}. To motivate it, we distill the essential features of a crown decomposition. Effectively, a crown decomposition of~$G$ certifies that~$G$ has a minimum vertex cover containing all of~$\ensuremath{\mathsf{head}}\xspace$, because (i)~any vertex cover has to pick at least~$|\ensuremath{\mathsf{head}}\xspace|$ vertices from~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace$, as the matching~$M$ certifies that~$\mathrm{\textsc{vc}}(G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace]) \geq |\ensuremath{\mathsf{head}}\xspace|$, while (ii)~any minimum vertex cover~$S'$ in~$G - (\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace)$ yields a minimum vertex cover~$S' \cup \ensuremath{\mathsf{head}}\xspace$ in~$G$, since~$N_G(\ensuremath{\mathsf{crown}}\xspace) \subseteq \ensuremath{\mathsf{head}}\xspace$ and~$\ensuremath{\mathsf{crown}}\xspace$ is an independent set. To obtain similar guarantees for \textsc{Feedback Vertex Set}, we need a decomposition to supply disjoint vertex sets~$(\ensuremath{\mathsf{head}}\xspace,\ensuremath{\mathsf{antler}}\xspace)$ such that (i)~any minimum feedback vertex set contains at least~$|\ensuremath{\mathsf{head}}\xspace|$ vertices from~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace$, and (ii)~any minimum feedback vertex set~$S'$ in~$G - (\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace)$ yields a minimum feedback vertex set~$S' \cup \ensuremath{\mathsf{head}}\xspace$ in~$G$. To achieve (i), it suffices for~$G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace]$ to contain a set of~$|\ensuremath{\mathsf{head}}\xspace|$ vertex-disjoint cycles (implying that each cycle contains exactly one vertex of~$\ensuremath{\mathsf{head}}\xspace$); to achieve (ii), it suffices for~$G[\ensuremath{\mathsf{antler}}\xspace]$ to be acyclic, with each tree~$T$ of the forest~$G[\ensuremath{\mathsf{antler}}\xspace]$ connected to the remainder~$V(G) \setminus (\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace)$ by at most one edge (implying that all cycles through~$\ensuremath{\mathsf{antler}}\xspace$ intersect~$\ensuremath{\mathsf{head}}\xspace$). We call such a decomposition a 1-antler. Here \emph{antler} refers to the shape of the forest~$G[\ensuremath{\mathsf{antler}}\xspace]$, which no longer consists of isolated spikes of a crown (see \cref{fig:CrownAntler}). The prefix~$1$ indicates it is the simplest type of antler; we present a generalization later. An antler is \emph{non-empty} if~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{crown}}\xspace \neq \emptyset$, and the \emph{width} of the antler is defined to be~$|\ensuremath{\mathsf{head}}\xspace|$.
\begin{figure}
\caption{Graph structures showing there is an optimal solution containing all black vertices and no gray vertices, certified by the bold subgraph. Left: Crown decomposition for \textsc{Vertex Cover}. Right: Antler for \textsc{Feedback Vertex Set}. For legibility, the number of edges in the drawing has been restricted. It therefore has degree-2 vertices which make it reducible by standard reduction rules; but adding all possible edges between gray and black vertices leads to a structure of minimum degree at least three which is still a 1-antler.}
\label{fig:CrownAntler}
\end{figure}
Unfortunately, assuming \ensuremath{\mathsf{P}}~$\neq$~\ensuremath{\mathsf{NP}}\xspace there is no \emph{polynomial-time} algorithm that always outputs a non-empty 1-antler if one exists. We prove this in \cref{sec:np-hard}. However, for the purpose of making a preprocessing algorithm that reduces the search space, we can allow FPT time in a parameter such as~$|\ensuremath{\mathsf{head}}\xspace|$ to find a decomposition. Each fixed choice of~$|\ensuremath{\mathsf{head}}\xspace|$ would then correspond to a reduction rule which identifies a small ($|\ensuremath{\mathsf{head}}\xspace|$-sized) part of an optimal feedback vertex set, for which there is a simple certificate for it being part of an optimal solution. Such a reduction rule can then be iterated in the preprocessing phase, thereby potentially decreasing the target solution size (and search space) by an arbitrarily large amount. Hence we consider the parameterized complexity of testing whether a graph admits a non-empty 1-antler with~$|\ensuremath{\mathsf{head}}\xspace| \leq k$, parameterized by~$k$. On the one hand, we show this problem to be W[1]-hard in \cref{sec:w1-hard}. This hardness turns out to be a technicality based on the forced bound on~$|\ensuremath{\mathsf{head}}\xspace|$, though. We provide the following FPT algorithm which yields a search-space reducing preprocessing step.
\begin{theorem} \label{thm:1:antler} There is an algorithm that runs in~$2^{\ensuremath{\mathcal{O}}\xspace(k^5)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(1)}$ time that, given an undirected multigraph~$G$ on~$n$ vertices and integer~$k$, either correctly determines that~$G$ does not admit a non-empty 1-antler of width at most~$k$, or outputs a set~$S$ of at least~$k$ vertices such that there exists an optimal feedback vertex set in~$G$ containing all vertices of~$S$. \end{theorem}
Hence if the input graph admits a non-empty 1-antler of width at most~$k$, the algorithm is guaranteed to find at least~$k$ vertices that belong to an optimal feedback vertex set, thereby reducing the search space.
Based on this positive result, we go further and generalize our approach beyond 1-antlers. For a 1-antler~$(\ensuremath{\mathsf{head}}\xspace, \ensuremath{\mathsf{antler}}\xspace)$ in~$G$, the set of~$|\ensuremath{\mathsf{head}}\xspace|$ vertex-disjoint cycles in~$G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace]$ forms a very simple certificate that any feedback vertex set of~$G$ contains at least~$|\ensuremath{\mathsf{head}}\xspace|$ vertices from~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace$. We can generalize our approach to identify part of an optimal solution, by allowing more complex certificates of optimality. The following interpretation of a 1-antler is the basis of the generalization: for a 1-antler~$(\ensuremath{\mathsf{head}}\xspace, \ensuremath{\mathsf{antler}}\xspace)$ in~$G$, there is a subgraph~$G'$ of~$G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace]$ (formed by the~$|\ensuremath{\mathsf{head}}\xspace|$ vertex-disjoint cycles) such that~$V(G') \supseteq \ensuremath{\mathsf{head}}\xspace$ and~$\ensuremath{\mathsf{head}}\xspace$ is an optimal feedback vertex set of~$G'$; and furthermore this subgraph~$G'$ is simple because all its connected components, being cycles, have a feedback vertex set of size~$1$. For an arbitrary integer~$z$, we therefore define a $z$-antler in an undirected multigraph graph~$G$ as a pair of disjoint vertex sets~$(\ensuremath{\mathsf{head}}\xspace, \ensuremath{\mathsf{antler}}\xspace)$ such that (i)~any minimum feedback vertex set in~$G$ contains at least~$|\ensuremath{\mathsf{head}}\xspace|$ vertices from~$\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace$, as witnessed by the fact that~$G[\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace]$ has a subgraph~$G'$ for which~$\ensuremath{\mathsf{head}}\xspace$ is an optimal feedback vertex set and with each component of~$G'$ having a feedback vertex set of size at most~$z$; and~(ii) the graph~$G[\ensuremath{\mathsf{antler}}\xspace]$ is acyclic, with each tree~$T$ of the forest~$G[\ensuremath{\mathsf{antler}}\xspace]$ connected to the remainder~$V(G) \setminus (\ensuremath{\mathsf{head}}\xspace \cup \ensuremath{\mathsf{antler}}\xspace)$ by at most one edge. (So condition~(ii) is not changed compared to a 1-antler.) Our main result is the following.
\begin{theorem} \label{thm:z:antler} There is an algorithm that runs in~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time that, given an undirected multigraph~$G$ on~$n$ vertices and integers~$k \geq z \geq 0$, either correctly determines that~$G$ does not admit a non-empty $z$-antler of width at most~$k$, or outputs a set~$S$ of at least~$k$ vertices such that there exists an optimal feedback vertex set in~$G$ containing all vertices of~$S$. \end{theorem}
In fact, we prove a slightly stronger statement. If a graph~$G$ can be reduced to a graph~$G'$ by iteratively removing~$z$-antlers, each of width at most~$k$, and the sum of the widths of this sequence of antlers is~$t$, then we can find in time~$f(k,z) \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$ a subset of at least~$t$ vertices of~$G$ that belong to an optimal feedback vertex set. This implies that if a complete solution to \textsc{Feedback Vertex Set} can be assembled by iteratively combining $\ensuremath{\mathcal{O}}\xspace(1)$-antlers of width at most~$k$, then the entire solution can be found in time~$f'(k) \cdot n^{\ensuremath{\mathcal{O}}\xspace(1)}$. Hence our work uncovers a new parameterization in terms of the complexity of the solution structure, rather than its size, in which \textsc{Feedback Vertex Set} is fixed-parameter tractable.
Our algorithmic results are based on a combination of graph reduction and color coding. We use reduction steps inspired by the kernelization algorithms~\cite{BodlaenderD10,Thomasse10} for \textsc{Feedback Vertex Set} to bound the size of~$\ensuremath{\mathsf{antler}}\xspace$ in the size of~$\ensuremath{\mathsf{head}}\xspace$. After such reduction steps, we use color coding~\cite{AlonYZ95} to help identify antler structures. A significant amount of effort goes into proving that the reduction steps preserve antler structures and the optimal solution size.
\section{Preliminaries} \label{sec:prelims}
For any family of sets $X_1,\ldots,X_\ell$ indexed by $\{1,\ldots,\ell\}$ we define for all $1 \leq i \leq \ell$ the following $X_{<i} := \bigcup_{1 \leq j < i} X_j$, $X_{>i} := \bigcup_{i < j \leq \ell} X_j$, $X_{\leq i} := X_i \cup X_{<i}$ and $X_{\geq i} := X_i \cup X_{>i}$.
For a function~$f \colon A \to B$, let $\inv{f} \colon B \to 2^A$ denote the \emph{preimage function of $f$}, that is $\inv{f}(a) = \{b \in B \mid f(b) = a\}$.
For some set~$D$ of size~$n$ and integer~$k$ with~$n \geq k$ an~\emph{$(n,k)$-universal set for $D$} is a family $\mathcal{U}$ of subsets of $D$ such that for all $S \subseteq D$ of size at most~$k$ we have~$\{S \cap U \mid U \in \mathcal{U}\} = 2^S$.
\begin{theorem}[{\cite[Thm. 6]{NaorSS95}, cf.~\cite[Thm. 5.20]{CyganFKLMPPS15}}] \label{thm:universal-set}
For any set~$D$ and integers~$n$ and~$k$ with~$|D| = n \geq k$, an~$(n,k)$-universal set $\mathcal{U}$ for $D$ with $|\mathcal{U}| = 2^{\ensuremath{\mathcal{O}}\xspace(k)}\log n$ can be created in $2^{\ensuremath{\mathcal{O}}\xspace(k)} n \log n$ time. \end{theorem}
All graphs considered in this paper are undirected multigraphs, which may have loops. Based on the incidence representation of multigraphs (cf.~\cite[Example 4.9]{FlumG06}) we represent a multigraph $G$ by a vertex set $V(G)$, an edge set $E(G)$, and a function $\iota \colon E(G) \to 2^{V(G)}$ where $\iota(e)$ is the set of one or two vertices incident to $e$ for all $e \in E(G)$.
In the context of an algorithm with input graph~$G$ we use~$n = |V(G)|$ and~$m = |E(G)|$. We assume we can retrieve and update number of edges between two vertices in constant time, hence we can ensure in~$\ensuremath{\mathcal{O}}\xspace(n^2)$ time that there are at most two edges between any to vertices, meaning~$m \in \ensuremath{\mathcal{O}}\xspace(n^2)$. For a vertex set~$X \subseteq V(G)$ let $G[X]$ denote the subgraph of~$G$ induced by~$X$. For a set of vertices and edges~$Y \subseteq V(G) \cup E(G)$ let~$G-Y$ denote the graph obtained from $G[V(G)\setminus Y]$ by removing all edges in~$Y$. For a singleton set~$\{v\}$ we write~$G - v$ instead of~$G - \{v\}$. For two graphs~$G$ and~$H$ the graph~$G \cap H$ is the graph on vertex set~$V(G) \cap V(H)$ and edge set~$E(G) \cap E(H)$. For~$v \in V(G)$ the open neighborhood of~$v$ in~$G$ is~$N_G(v) := \{u \in V(G) \mid \exists e \in E(G) \colon \{u,v\} = \iota(e)\}$. For~$X \subseteq V(G)$ let~$N_G(X) := \bigcup_{v \in X} N_G(v) \setminus X$. The degree~$\deg_G(v)$ of a vertex $v$ in $G$ is the number of edge-endpoints incident to $v$, where a self-loop contributes two endpoints. For two disjoint vertex sets~$X,Y \subseteq V(G)$ the number of edges in~$G$ with one endpoint in~$X$ and another in~$Y$ is denoted by~$e(X,Y)$. To simplify the presentation, in expressions involving~$N_G(..)$ and~$e(.., ..)$ we sometimes use a subgraph~$H$ as argument as a shorthand for the vertex set~$V(H)$ that is formally needed. For a vertex~$v \in V(G)$ and an integer~$k$, a \emph{$v$-flower of order~$k$} is a collection of~$k$ cycles in~$G$ whose vertex sets only intersect in~$v$.
\begin{restatable}[{Cf.~\cite[Lemma 3.9]{RaymondT17}}]{lemma}{lemAltFVS} \label{lem:alt-fvs}
If~$v$ is a vertex in an undirected multigraph~$G$ such that~$v$ does not have a self-loop and~$G-v$ is acyclic, then we can find in $\ensuremath{\mathcal{O}}\xspace(n)$ time a set~$X \subseteq V(G)\setminus\{v\}$ such that~$G-X$ is acyclic and~$G$ contains a $v$-flower of order~$|X|$. \end{restatable} \begin{proof}
We prove the existence of such a set~$X$ and $v$-flower by induction on~$|V(G)|$. The inductive proof can easily be translated into a linear-time algorithm. If~$G$ is acyclic, output~$X = \emptyset$ and a $v$-flower of order~$0$. Otherwise, since~$v$ does not have a self-loop there is a tree~$T$ of the forest~$G - v$ such that~$G[V(T) \cup \{v\}]$ contains a cycle. Root~$T$ at an arbitrary vertex and consider a deepest node~$x$ in~$T$ for which the graph~$G[V(T_x) \cup \{v\}]$ contains a cycle~$C$. Then any feedback vertex set of~$G$ that does not contain~$v$, has to contain at least one vertex of~$T_x$; and the choice of~$x$ as a deepest vertex implies that~$x$ lies on all cycles of~$G$ that intersect~$T_x$. By induction on~$G' := G - V(T_x)$ and~$v$, there is a feedback vertex set~$X' \subseteq V(G') \setminus \{v\}$ of~$G'$ and a $v$-flower in~$G'$ of order~$|X'|$. We obtain a $v$-flower of order~$|X'| + 1$ in~$G$ by adding~$C$, while~$X := X' \cup \{x\} \subseteq V(G) \setminus \{v\}$ is a feedback vertex set of size~$|X'| + 1$. \end{proof}
\section{Feedback Vertex Cuts and Antlers} \label{sec:def:cuts:antlers}
In this section we present properties of antlers and related structures. A Feedback Vertex Set (FVS) in a graph~$G$ is a vertex set $X \subseteq V(G)$ such that $G-S$ is acyclic. The feedback vertex number of a graph~$G$, denoted by $\fvs(G)$, is the minimum size of a FVS in~$G$. A \emph{Feedback Vertex Cut} (FVC) in a graph~$G$ is a pair of disjoint vertex sets~$(C,F)$ such that $C,F \subseteq V(G)$, $G[F]$ is a forest, and for each tree~$T$ in~$G[F]$ we have $e(T,G-(C\cup F)) \leq 1$. The \emph{width} of a FVC~$(C,F)$ is~$|C|$, and~$(C,F)$ is \emph{empty} if~$|C \cup F| = 0$. \begin{observation} \label{obs:fvc-basics}
If $(C,F)$ is a FVC in~$G$ then any cycle in~$G$ containing a vertex from~$F$ also contains a vertex from~$C$. The set~$C$ is a FVS in $G[C \cup F]$, hence $|C| \geq \fvs(G[C \cup F])$. \end{observation} \begin{observation} \label{obs:subfvc}
If $(C,F)$ is a FVC in~$G$ then for any~$X \subseteq V(G)$ we have that~$(C\setminus X, F \setminus X)$ is a FVC in~$G-X$. Additionally, for any $Y \subseteq E(G)$ we have that~$(C,F)$ is a FVC in~$G-Y$. \end{observation}
We now present one of the main concepts for this work. An \emph{antler} in a graph~$G$ is a FVC~$(C,F)$ in $G$ such that~$|C| \leq \fvs(G[C \cup F])$. Then by \cref{obs:fvc-basics} the set~$C$ is a minimum FVS in~$G[C \cup F]$ and no cycle in~$G-C$ contains a vertex from~$F$. We observe: \begin{observation} \label{obs:remove-antler}
If~$(C,F)$ is an antler in~$G$, then~$\fvs(G) = |C| + \fvs(G-(C \cup F))$. \end{observation}
For a graph~$G$ and vertex set~$C \subseteq V(G)$, a \emph{$C$-certificate} is a subgraph~$H$ of~$G$ such that~$C$ is a minimum FVS in~$H$. We say a $C$-certificate has \emph{order}~$z$ if for each component~$H'$ of~$H$ we have~$\fvs(H') = |C \cap V(H')| \leq z$. For an integer~$z\geq0$, a $z$-antler in~$G$ is an antler~$(C,F)$ in~$G$ such that~$G[C \cup F]$ contains a $C$-certificate of order~$z$. Note that a $0$-antler has width~$0$.
\begin{observation} \label{obs:subantler}
If $(C,F)$ is a $z$-antler in $G$ for some $z\geq0$, then for any $X \subseteq C$, we have that $(C\setminus X,F)$ is a $z$-antler in $G - X$. \end{observation}
While antlers may intersect in non-trivial ways, the following proposition relates the sizes of the cross-intersections.
\begin{restatable}{proposition}{intersectionantlers} \label{prop:intersection_antlers}
If $(C_1,F_1)$ and $(C_2,F_2)$ are antlers in $G$, then $|C_1 \cap F_2| = |C_2 \cap F_1|$. \end{restatable} \begin{proof}
We show $\fvs(G[F_1 \cup F_2]) = |C_1 \cap F_2|$.
First we show $\fvs(G[F_1 \cup F_2]) \geq |C_1 \cap F_2|$ by showing $(C_1 \cap F_2, F_1)$ is an antler in $G[F_1 \cup F_2]$. Clearly $(C_1,F_1)$ is an antler in $G[F_1 \cup F_2 \cup C_1]$, so then by \cref{obs:subantler} $(C_1 \cap F_2,F_1)$ is an antler in $G[F_1 \cup F_2 \cup C_1] - (C_1 \setminus F_2) = G[F_1 \cup F_2]$.
Second we show $\fvs(G[F_1 \cup F_2]) \leq |C_1 \cap F_2|$ by showing $G[F_1 \cup F_2] - (C_1 \cap F_2)$ is acyclic. Note that $G[F_1 \cup F_2] - (C_1 \cap F_2) = G[F_1 \cup F_2] - C_1$.
Suppose $G[F_1 \cup F_2] - C_1$ contains a cycle. We know this cycle does not contain a vertex from $C_1$, however it does contain at least one vertex from $F_1$ since otherwise this cycle exists in $G[F_2]$ which is a forest.
We know from \cref{obs:fvc-basics} that any cycle in $G$ containing a vertex from $F_1$ also contains a vertex from $C_1$. Contradiction. The proof for~$\fvs(G[F_1 \cup F_2]) = |C_2 \cap F_1|$ is symmetric. It follows that~$|C_1 \cap F_2| = \fvs(G[F_1 \cup F_2]) = |C_2 \cup F_1|$. \end{proof}
\Cref{lem:antler_diff} shows that what remains of a $z$-antler~$(C_1,F_1)$ when removing a different antler~$(C_2,F_2)$, again forms a smaller $z$-antler. We will rely on this lemma repeatedly to ensure that after having found and removed an incomplete fragment of a width-$k$ $z$-antler, the remainder of that antler persists as a $z$-antler to be found later.
\begin{restatable}{lemma}{antlerdiff} \label{lem:antler_diff}
For any integer $z\geq0$, if a graph $G$ has a $z$-antler $(C_1,F_1)$ and another antler $(C_2,F_2)$, then $(C_1 \setminus (C_2 \cup F_2), F_1 \setminus (C_2 \cup F_2))$ is a $z$-antler in $G-(C_2 \cup F_2)$. \end{restatable}
Before we prove \cref{lem:antler_diff}, we prove a weaker claim:
\begin{restatable}{proposition}{intersectingantlers} \label{prop:intersecting_antlers}
If~$(C_1,F_1)$ and~$(C_2,F_2)$ are antlers in~$G$, then~$(C_1 \setminus (C_2 \cup F_2), F_1 \setminus (C_2 \cup F_2))$ is an antler in~$G - (C_2 \cup F_2)$. \end{restatable}
\begin{proof}
For brevity let $C_1' := C_1 \setminus (C_2 \cup F_2)$ and $F_1' := F_1 \setminus (C_2 \cup F_2)$ and $G' := G - (C_2 \cup F_2)$. First note that $(C_1', F_1')$ is a FVC in $G'$ by \cref{obs:subfvc}. We proceed to show that $\fvs(G'[C_1' \cup F_1']) \geq |C_1'|$. By \cref{obs:subantler} $(\emptyset,F_2)$ is an antler in $G-C_2$, so then by \cref{obs:subfvc} we have $(\emptyset, F_2 \cap (C_1 \cup F_1))$ is a FVC in $G[C_1 \cup F_1] - C_2$. Since a FVC of width $0$ is an antler we can apply \cref{obs:remove-antler} and obtain $\fvs(G[C_1 \cup F_1] - C_2) = \fvs(G[C_1 \cup F_1] - (C_2 \cup F_2)) = \fvs(G'[C_1' \cup F_1'])$. We derive \begin{align*} \fvs(G'[C_1' \cup F_1']) &= \fvs(G[C_1 \cup F_1] - C_2)\\
&\geq \fvs(G[C_1 \cup F_1]) - |C_2 \cap (C_1 \cup F_1)|\\
&= |C_1| - |C_2 \cap C_1| - |C_2 \cap F_1| &\text{Since $C_1 \cap F_1 = \emptyset$}\\
&= |C_1| - |C_2 \cap C_1| - |C_1 \cap F_2| &\text{By \cref{prop:intersection_antlers}}\\
&= |C_1| - |(C_2 \cap C_1) \cup (C_1 \cap F_2)| &\text{Since $C_2 \cap F_2 = \emptyset$}\\
&= |C_1 \setminus (C_2 \cup F_2)| = |C_1'|. \tag*{\qedhere} \end{align*} \end{proof}
We can now prove \cref{lem:antler_diff}. \begin{proof}
For brevity let $C_1' := C_1 \setminus (C_2 \cup F_2)$ and $F_1' := F_1 \setminus (C_2 \cup F_2)$ and $G' := G - (C_2 \cup F_2)$.
By \cref{prop:intersecting_antlers} we know $(C_1',F_1')$ is an antler, so it remains to show that $G'[C_1' \cup F_1']$ contains a $C_1'$-certificate of order $z$. Since $(C_1,F_1)$ is a $z$-antler in $G$, we have that $G[C_1 \cup F_1]$ contains a $C_1$-certificate of order $z$. Let $H$ denote this $C_1$-certificate and let $\overline{H}$ be the set of all edges and vertices in $G'[C_1' \cup F_1']$ that are not in $H$. Now $(C_1,F_1)$ is a $z$-antler in $G'' := G-\overline{H}$ since it is a FVC by \cref{obs:subfvc} and $G''[C_1 \cup F_1]$ contains a $C_1$-certificate of order $z$ since $H$ is a subgraph of $G''$. Note that $(C_2,F_2)$ is also an antler in $G''$ since $\overline{H}$ does not contain vertices or edges from $G[C_2 \cup F_2]$. It follows that $(C_1',F_1')$ is an antler in $G''$ by \cref{prop:intersecting_antlers}, so $G''[C_1' \cup F_1']$ is a $C_1'$-certificate in $G''$. Clearly this is a $C_1'$-certificate of order $z$ since $G''[C_1' \cup F_1']$ is a subgraph of $H$. Since $G''[C_1' \cup F_1']$ is a subgraph of $G'[C_1' \cup F_1']$ it follows that $G'[C_1' \cup F_1']$ contains a $C_1'$-certificate of order $z$. \end{proof}
\Cref{lem:antler_combine} shows that we can consider consecutive removal of two $z$-antlers as the removal of a single $z$-antler.
\begin{restatable}{lemma}{antlercombine} \label{lem:antler_combine}
For any integer~$z\geq0$, if a graph~$G$ has a $z$-antler~$(C_1,F_1)$ and~$G-(C_1 \cup F_1)$ has a $z$-antler~$(C_2,F_2)$ then~$(C_1 \cup C_2, F_1 \cup F_2)$ is a $z$-antler in~$G$. \end{restatable} \begin{proof}
Since~$(C_1,F_1)$ is a $z$-antler in~$G$ we know~$G[C_1 \cup F_1]$ contains a $C_1$-certificate of order~$z$, similarly~$(G-(C_1 \cup F_1))[C_2 \cup F_2]$ contains a $C_2$-certificate of order~$z$. The union of these certificate forms a $(C_1 \cup C_2)$-certificate of order~$z$ in~$G[C_1 \cup C_2 \cup F_1 \cup F_2]$. It remains to show that~$(C_1 \cup C_2, F_1 \cup F_2)$ is a FVC in~$G$.
First we show~$G[F_1 \cup F_2]$ is acyclic. Suppose for contradiction that~$G[F_1 \cup F_2]$ contains a cycle~$\mathcal{C}$. Since~$(C_1,F_1)$ is a FVC in~$G$, any cycle containing a vertex from~$F_1$ also contains a vertex from~$C_1$, hence~$\mathcal{C}$ does not contain vertices from~$F_1$. Therefore~$\mathcal{C}$ can only contain vertices from $F_2$. This is a contradiction with the fact that $G[F_2]$ is acyclic.
Finally we show that for each tree~$T$ in~$G[F_1 \cup F_2]$ we have $e(T,G-(C_1 \cup C_2 \cup F_1 \cup F_2)) \leq 1$. If~$V(T) \subseteq F_2$ this follows directly from the fact that~$(C_2,F_2)$ is a FVC in~$G-(C_1 \cup F_1)$. Similarly if~$V(T) \subseteq F_1$ this follows directly from the fact that~$(C_1,F_1)$ is a FVC in~$G$. So suppose~$T$ is a tree that contains vertices from both~$F_1$ and~$F_2$. Since $T$ is connected, each tree in~$T[F_1]$ contains a neighbor of a vertex in a tree in~$T[F_2]$. Hence no tree in~$T[F_1]$ contains a neighbor of~$V(G-(C_1 \cup C_2 \cup F_1 \cup F_2))$, so~$e(V(T) \cap F_1, G-(C_1 \cup C_2 \cup F_1 \cup F_2)) = 0$. To complete the proof we show~$e(V(T) \cap F_2, G-(C_1 \cup C_2 \cup F_1 \cup F_2)) \leq 1$. Recall each tree in~$G[F_2]$ has at most~$1$ edge to~$G-(C_1 \cup C_2 \cup F_1 \cup F_2)$, so it suffices to show that~$T[F_2]$ is connected. Suppose~$T[F_2]$ is not connected, then let $u,v \in F_2$ be vertices from different components of~$T[F_2]$. Since~$T$ is connected, there is a path from~$u$ to~$v$. This path must use a vertex~$w \in V(T-F_2) \subseteq F_1$. Let~$T'$ denote the tree in~$T[F_1]$ that contains this vertex. Since~$(C_1,F_1)$ is a FVC in~$G$ we have that~$e(T', F_2) \leq e(T',G-(C_1 \cup F_1)) \leq 1$ hence no vertex in~$T'$ can be part of a path from~$u$ to~$v$ in~$T$. This contradicts our choice of~$T'$. \end{proof}
The last structural property of antlers, given in \cref{col:few-fvcs}, derives a bound on the number of trees of a forest~$G[F]$ needed to witness that~$C$ is an optimal FVS of~$G[C \cup F]$. \Cref{col:few-fvcs} is a corollary to the following lemma.
\begin{lemma} \label{lem:small-cert}
If a graph~$G$ contains a $C$-certificate~$H$ of order~$z \geq 0$ for some~$C\subseteq V(G)$, then~$H$ contains a $C$-certificate $\hat{H}$ of order~$z$ such that~$\hat{H}-C$ has at most~$\frac{|C|}{2}(z^2+2z-1)$ trees. \end{lemma} \begin{proof}
Consider a tree $T$ in $H-C$, we show that $\fvs(H-V(T)) = \fvs(H)$ if
\begin{enumerate}
\item \label{item:vflower} for all $v \in C$ such that $H[V(T) \cup \{v\}]$ has a cycle, $H-V(T)$ contains an order-$z$ $v$-flower, and
\item \label{item:pumpkin} for all $\{u,v\} \in \binom{N_H(T)}{2}$ there are at least $z+1$ other trees in $H-C$ adjacent to $u$ and $v$.
\end{enumerate}
Consider the component $H'$ of $H$ that contains $T$. It suffices to show that $\fvs(H'-V(T)) = \fvs(H')$. Clearly $\fvs(H' - V(T)) \leq \fvs(H')$ so it remains to show that $\fvs(H' - V(T)) \geq \fvs(H')$. Assume $\fvs(H' - V(T)) < \fvs(H')$, then let $X$ be a FVS in $H'-V(T)$ with $|X| < \fvs(H') = |C \cap V(H')| \leq z$. For any $v \in C \cap V(H')$ such that $H[T \cup \{v\}]$ has a cycle we know from condition~\ref{item:vflower} that $H'-V(T)$ has $z > |X|$ cycles that intersect only in $v$, hence $v \in X$. By condition~\ref{item:pumpkin} we have that all but possibly one vertex in $N_G(T)$ must be contained in $X$, since if there are two vertices $x,y \in N_G(T)\setminus X$ then $H-V(T)-X$ has at least $z+1 - |X| \geq 2$ internally vertex-disjoint paths between $x$ and $y$ forming a cycle and contradicting our choice of $X$. Since there is at most one vertex $v \in N_G(T)\setminus X$ and $H[T \cup \{v\}]$ does not have a cycle, we have that $H'-X$ is acyclic, a contraction since~$|X| < \fvs(H')$.
The desired $C$-certificate $\hat{H}$ can be obtained from $H$ by iteratively removing trees from $H-C$ for which both conditions hold. We show that if no such tree exists, then $H-C$ has at most $\frac{|C|}{2}(z^2+2z-1)$ trees. Each tree $T$ for which condition~\ref{item:vflower} fails can be charged to a vertex $v \in C$ that witnesses this, i.e., $H[T \cup \{v\}]$ has a cycle and there are at most $z$ trees $T'$ such that $T' \cup v$ has a cycle. Clearly each vertex $v \in C$ can be charged at most $z$ times, hence there are at most $z \cdot |C|$ trees violating condition~\ref{item:vflower}.
Similarly each tree $T$ for which condition~\ref{item:pumpkin} fails can be charged to a pair of vertices $\{u,v\} \in \binom{N_H(T)}{2}$ for which at most~$z+1$ trees in $H-T$ are adjacent to $u$ and $v$. Clearly each pair of vertices can be charged at most~$z+1$ times. Additionally each pair consists of vertices from the same component of~$H$. Let $H_1,\ldots,H_\ell$ be the components in $H$, then there are at most~$\sum_{1 \leq i \leq \ell} \binom{|C \cap V(H_i)|}{2} = \sum_{1 \leq i \leq \ell} \frac{1}{2}|C \cap V(H_i)|(|C \cap V(H_i)|-1) \leq \sum_{1 \leq i \leq \ell} \frac{1}{2}|C \cap V(H_i)|(z-1) = \frac{|C|}{2}(z-1)$ such pairs. Thus $H-C$ has at most $z\cdot|C| + (z+1)\cdot\frac{|C|}{2}(z-1) = \frac{|C|}{2}(z^2+2z-1)$ trees violating condition~\ref{item:pumpkin}. \end{proof}
We can now give an upper bound on the number of trees in~$G[F]$ required for a~$z$-antler~$(C,F)$.
\begin{restatable}{lemma}{fewfvcs} \label{col:few-fvcs}
Let~$(C,F)$ be a $z$-antler in a graph~$G$ for some~$z \geq 0$. There exists an~$F' \subseteq F$ such that~$(C,F')$ is a $z$-antler in~$G$ and~$G[F']$ has at most~$\frac{|C|}{2}(z^2+2z-1)$ trees. \end{restatable} \begin{proof}
Since $(C,F)$ is a $z$-antler, $G[C \cup F]$ contains a $C$-certificate $H$ or order $z$ and by \cref{lem:small-cert} we know $H$ contains a $C$-certificate $\hat{H}$ of order $z$ such that $\hat{H}-C$ has at most~$\frac{|C|}{2}(z^2+2z-1)$ components. Take $F' := V(H-C)$ then $G[F']$ has at most $\frac{|C|}{2}(z^2+2z-1)$ components and $\hat{H}$ is a subgraph of $G[C \cup F']$, meaning $(C,F')$ is a $z$-antler. \end{proof}
\section{Finding Feedback Vertex Cuts} \label{sec:find-fvc}
As described in \cref{sec:intro}, our algorithm to identify vertices in antlers uses color coding. To allow a relatively small family of colorings to identify an entire antler structure~$(C,F)$ with~$|C| \leq k$, we need to bound~$|F|$ in terms of~$k$ as well. We therefore use several graph reduction steps. In this section, we show that if there is a width-$k$ antler whose forest~$F$ is significantly larger than~$k$, then we can identify a reducible structure in the graph. To identify a reducible structure we will also use color coding. In \cref{sec:fvc-kernel} we show how to reduce such a structure while preserving antlers and optimal feedback vertex sets.
Define the function~$f_r \colon \mathbb{N} \to \mathbb{N}$ as $f_r(x) = 2x^3+3x^2-x$. We say a FVC~$(C,F)$ is \emph{reducible} if~$|F| > f_r(|C|)$, and we say $(C,F)$ is a \emph{single-tree} FVC if~$G[F]$ is connected.
\begin{definition}
A FVC~$(C,F)$ is \emph{simple} if $|F| \leq 2f_r(|C|)$ and one of the following holds:~(a)~$G[F]$ is connected, or~(b)~all trees in $G[F]$ have a common neighbor $v$ and there exists a single-tree FVC $(C,F_2)$ with $v \in F_2 \setminus F$ and $F \subseteq F_2$. \end{definition}
The algorithm we will present can identify a reducible FVC when it is also simple. First we show that such a FVC always exists when the graph contains a single-tree reducible FVS.
\begin{restatable}{lemma}{smallfvc} \label{lem:small_fvc}
If a graph~$G$ contains a reducible single-tree FVC~$(C,F)$ there exists a simple reducible FVC~$(C,F')$ with~$F' \subseteq F$. \end{restatable}
\begin{proof}
We use induction on $|F|$. If $|F| \leq 2f_r(|C|)$ then $(C,F)$ is simple by condition~(a). Assume $|F| > 2f_r(|C|)$. Since $(C,F)$ is a FVC and $G[F]$ is connected there is at most one vertex $v \in F$ that has a neighbor in $V(G)\setminus(C \cup F)$. If no such vertex exists, take $v \in F$ to be any other vertex. Observe that $(C,F\setminus\{v\})$ is a FVC. Consider the following cases:
\begin{itemize}
\item All trees in $G[F] - v$ contain at most $f_r(|C|)$ vertices. Let $F'$ be the vertices of an inclusion minimal set of trees of $G[F]-v$ such that $|F'| > f_r(|C|)$. Clearly $|F'| \leq 2f_r(|C|)$ since otherwise the set is not inclusion minimal. Each tree in $G[F']$ contains a neighbor of $v$ and $F' \subseteq F$, hence $(C,F')$ is simple by condition~(b), and $(C,F')$ is reducible since $|F'| > f_r(|C|)$.
\item There is a tree $T$ in $G[F] - v$ that contains more than $f_r(|C|)$ vertices. Now $(C,V(T))$ is a single-tree reducible FVC with $|V(T)| < |F|$, so the induction hypothesis applies.
\qedhere
\end{itemize} \end{proof}
We proceed to show how a simple reducible FVC can be found using color coding. A vertex coloring of~$G$ is a function~$\chi \colon V(G) \to \{\ensuremath{\mathsf{\dot{C}}}\xspace,\ensuremath{\mathsf{\dot{F}}}\xspace\}$. We say a simple FVC~$(C,F)$ is \emph{properly colored} by a coloring~$\chi$ if~$F \subseteq \inv{\chi}(\ensuremath{\mathsf{\dot{F}}}\xspace)$ and~$C \cup N_G(F) \subseteq \inv{\chi}(\ensuremath{\mathsf{\dot{C}}}\xspace)$.
\begin{lemma} \label{lem:colorcoded-fvc} Given a graph~$G$ and coloring~$\chi$ of $G$ that properly colors a simple reducible FVC $(C,F)$, a reducible FVC $(C',F')$ can be found in $\ensuremath{\mathcal{O}}\xspace(n^3)$ time. \end{lemma} \begin{proof}
If~$(C,F)$ is simple by condition~(a), i.e,~$G[F]$ is connected, it is easily verified that we can find a reducible FVC in~$\ensuremath{\mathcal{O}}\xspace(n^3)$ time as follows: Consider the set~$\mathcal{T}$ of all trees in~$G[\inv{\chi}(\ensuremath{\mathsf{\dot{F}}}\xspace)]$. For each tree~$T \in \mathcal{T}$, if there is a vertex~$u \in N_G(T)$ such that~$e(\{u\},T) = 1$ take~$C' := N_G(T) \setminus \{u\}$, otherwise take~$C' := N_G(T)$. If~$|V(T)| > f_r(|C'|)$ return~$(C',V(T))$.
In the remainder of the proof we assume that~$(C,F)$ is simple by condition~(b).
\subparagraph*{Algorithm}
For each~$u \in \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ consider the set~$\mathcal{T}$ of all trees~$T$ in~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ such that~$e(\{u\},T) = 1$. Let~$C' \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \setminus \{u\}$ be the set of vertices (excluding $u$) with a neighbor in at least two trees in~$\mathcal{T}$ and let~$\mathcal{T}_1$ be the set of trees~$T \in \mathcal{T}$ for which~$N_G(T) \subseteq C' \cup \{u\}$. Now consider the set of trees~$\mathcal{T}_2 = \mathcal{T}\setminus\mathcal{T}_1$ as a set of objects for a 0-1 knapsack problem where we define for each~$T \in \mathcal{T}_2$ its weight as $|N_G(T)\setminus(C' \cup \{u\})|$ and its value as $|V(T)|$. Using the dynamic programming algorithm~\cite{Toth80} for the 0-1 knapsack problem we compute for all~$0 \leq b \leq |N_G(V(\mathcal{T}_2)) \setminus (C' \cup \{u\})|$ a set of trees~$\mathcal{T}_2^b \subseteq \mathcal{T}_2$ with a combined weight~$\sum_{T \in \mathcal{T}_2^b} |N_G(T) \setminus (C' \cup \{u\})| \leq b$ such that the combined value~$\sum_{T \in \mathcal{T}_2^b} |V(T)|$ is maximized. If for any such~$b$ we have~$|V(\mathcal{T}_1)|+ |V(\mathcal{T}_2^b)| > f_r(|C'| + b)$ then take~$\hat{C} := C' \cup N_G(V(\mathcal{T}_2^b)) \setminus \{u\}$ and~$\hat{F} := V(\mathcal{T}_1) \cup V(\mathcal{T}_2^b)$ and return~$(\hat{C},\hat{F})$.
\subparagraph*{Correctness}
To show that~$(\hat{C}, \hat{F})$ is a FVC, first note that~$G[\hat{F}]$ is a forest. For each tree~$T$ in this forest we have~$e(T,\{u\})=1$ and~$N_G(T) \subseteq C' \cup \{u\} \cup N_G(V(\mathcal{T}_2^b)) = \hat{C} \cup \{u\}$. It follows that~$e(T,G-(\hat{C} \cup \hat{F}))=e(T,\{u\}) = 1$. To show that~$(\hat{C},\hat{F})$ indeed reducible observe that~$\sum_{T \in \mathcal{T}_2^b} |N_G(T) \setminus (C' \cup \{u\})| = |\bigcup_{T \in \mathcal{T}_2^b} N_G(T) \setminus (C' \cup \{u\})|$ since if two trees~$T_1,T_2 \in \mathcal{T}_2^b$ have a common neighbor that is not~$u$, it must be in~$C'$ by definition, hence the neighborhoods only intersect on~$C' \cup \{u\}$. We can now deduce~$|\hat{F}| = |V(\mathcal{T}_1)| + |V(\mathcal{T}_2^b)| > f_r(|C'| + b) \geq f_r(|C'| + \sum_{T \in \mathcal{T}_2^b} |N_G(T) \setminus (C' \cup \{u\})|) = f_r(|C' \cup N_G(V(\mathcal{T}_2^b)) \setminus \{u\}|) = f_r(|\hat{C}|)$.
It remains to show that if~$\chi$ properly colors a simple reducible FVC~$(C,F)$ then for some~$u \in \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ there exists a~$b$ such that~$|V(\mathcal{T}_1)| + |V(\mathcal{T}_2^b)| \geq f_r(|C'| + b)$. Recall that we assumed~$(C,F)$ is simple by condition~(b), i.e., all trees in $G[F]$ have a common neighbor $v$ and there exists a single-tree FVC $(C,F_2)$ with $v \in F_2 \setminus F$ and $F \subseteq F_2$. Since~$(C,F)$ is properly colored we know~$v \in \inv{\chi}(\ensuremath{\mathsf{\dot{C}}}\xspace)$, so in some iteration we will have~$u = v$. Consider the sets~$\mathcal{T}$, $\mathcal{T}_1$, $\mathcal{T}_2$, and~$C'$ as defined in this iteration. We first show~$C' \subseteq C$. If~$w \in C'$ then~$w$ has a neighbor in two trees in~$\mathcal{T}$. This means there are two internally vertex disjoint paths between~$v$ and~$w$, forming a cycle. Since~$v \in F_2$ we have by \cref{obs:fvc-basics} for the FVC~$(C,F_2)$ that this cycle must contain a vertex in~$C$ which is therefore different from~$v$. Recall that~$(C,F)$ is properly colored, hence all vertices in~$C$ have color~\ensuremath{\mathsf{\dot{C}}}\xspace. Note that the internal vertices of these paths all have color~\ensuremath{\mathsf{\dot{F}}}\xspace because they are vertices from trees in~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$. Hence~$w \in C$ and therefore~$C' \subseteq C$. To complete the proof we show
\begin{restatable}{claim}{knapsack}
There exists a value~$b$ such that~$|V(\mathcal{T}_1)| + |V(\mathcal{T}_2^b)| \geq f_r(|C'| + b)$. \end{restatable}
\begin{claimproof}
Recall that we assumed existence of a properly colored FVC~$(C,F)$ that is reducible and simple by condition~(b) witnessed by the FVC~$(C,F_2)$. Consider the set~$\mathcal{T}'$ of trees in~$G[F]$. Note that any tree~$T'$ in $\mathcal{T}'$ is a tree in~$G[\inv{\chi}(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ since~$(C,F)$ is properly colored and note that~$T'$ contains a neighbor of~$v$. If~$e(T',\{v\}) > 1$ then~$G[F_2]$ contains a cycle, contradicting that~$(C,F_2)$ is a FVC in~$G$, hence~$e(T',\{v\})=1$. It follows that~$T' \in \mathcal{T}$, meaning~$\mathcal{T'} \subseteq \mathcal{T}$. Take~$\mathcal{T}'_2 = \mathcal{T}' \setminus \mathcal{T}_1 = \mathcal{T}' \cap \mathcal{T}_2$ and~$b = \sum_{T \in \mathcal{T}'_2} |N_G(V(T)) \setminus (C' \cup \{v\}|$. Clearly~$\mathcal{T}'_2$ is a candidate solution for the 0-1 knapsack problem with capacity~$b$, hence $|V(\mathcal{T}_2^b)| \geq |V(\mathcal{T}'_2)|$. We deduce
\allowdisplaybreaks \begin{align*}
|V(\mathcal{T}_1)| + |V(\mathcal{T}_2^b)|
&\geq |V(\mathcal{T}_1)| + |V(\mathcal{T}'_2)|
\geq |V(\mathcal{T}')|
= |F|\\
&> f_r(|C|) &&\text{since~$(C,F)$ is reducible}\\
&= f_r(|C' \cup C|) &&\text{since~$C' \subseteq C$}\\
&= f_r(|C' \cup (N_G(F) \setminus \{v\}) \cup C|) &&\text{since~$N_G(\mathcal{T}'_2) \setminus \{v\} \subseteq C$}\\
&\geq f_r(|C' \cup (N_G(\mathcal{T}'_2) \setminus \{v\})|) &&\text{since~$f_r$ is non-decreasing}\\
&= f_r(|C'| + |N_G(\mathcal{T}'_2) \setminus (C' \cup \{v\})|) &&\text{since~$|A \cup B| = |A| + |B \setminus A|$}\\
&> f_r(|C'| + b) &&\qedhere \end{align*} \end{claimproof}
\subparagraph*{Running time} For each~$u \in \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ we perform a number of~$\ensuremath{\mathcal{O}}\xspace(n+m)$ time operations and run the dynamic programming algorithm for a problem with~$\ensuremath{\mathcal{O}}\xspace(n)$ objects and a capacity of~$\ensuremath{\mathcal{O}}\xspace(n)$ yielding a run time of~$\ensuremath{\mathcal{O}}\xspace(n^2)$ for each~$u$ or~$\ensuremath{\mathcal{O}}\xspace(n^3)$ for the algorithm as a whole.
\end{proof}
It can be shown that whether a FVC of width~$k$ is properly colored is determined by at most~$1+k+2f_r(k) = \ensuremath{\mathcal{O}}\xspace(k^3)$ relevant vertices. By creating an~$(n,\ensuremath{\mathcal{O}}\xspace(k^3))$-universal set for $V(G)$ using \cref{thm:universal-set}, we can obtain in~$2^{\ensuremath{\mathcal{O}}\xspace(k^3)} \cdot n \log n$ time a set of~$2^{\ensuremath{\mathcal{O}}\xspace(k^3)} \cdot \log n$ colorings that contains a coloring for each possible assignment of colors for these relevant vertices. By applying \cref{lem:colorcoded-fvc} for each coloring we obtain the following lemma:
\begin{restatable}{lemma}{findfvc} \label{lem:find-fvc}
There exists an algorithm that, given a graph~$G$ and an integer~$k$, outputs a (possibly empty) FVC~$(C,F)$ in~$G$. If~$G$ contains a reducible single-tree FVC of width at most~$k$ then~$(C,F)$ is reducible. The algorithm runs in time~$2^{\ensuremath{\mathcal{O}}\xspace(k^3)} \cdot n^3 \log n$.
\end{restatable}
\begin{proof}
Take $s = 2f_r(k) + k + 1$. By \cref{thm:universal-set} an~$(n,s)$-universal set~$\mathcal{U}$ for $V(G)$ of size~$2^{\ensuremath{\mathcal{O}}\xspace(s)} \log n$ can be created in~$2^{\ensuremath{\mathcal{O}}\xspace(s)} n \log n$ time. For each~$Q \in \mathcal{U}$ let~$\chi_Q$ be the coloring of~$G$ with~$\inv{\chi}(\ensuremath{\mathsf{\dot{C}}}\xspace) = Q$. Run the algorithm from \cref{lem:colorcoded-fvc} on~$\chi_Q$ for every~$Q \in \mathcal{U}$ and return the first reducible FVC. If no reducible FVC was found return $(\emptyset,\emptyset)$.
We obtain an overall run time of~$2^{\ensuremath{\mathcal{O}}\xspace(s)} \cdot n^3 \log n = 2^{\ensuremath{\mathcal{O}}\xspace(k^3)} \cdot n^3 \log n$.
To prove correctness assume~$G$ contains a reducible single-tree FVC~$(C,F)$ with~$|C| \leq k$. By \cref{lem:small_fvc} we know~$G$ contains a simple reducible FVC~$(C,F')$. Coloring~$\chi$ properly colors~$(C,F')$ if all vertices in~$F' \cup C \cup N_G(F')$ are assigned the correct color. Hence at most~$|F'| + |C + N_G(F')| \leq 2f_r(k) + k + 1 = s$ vertices need to have the correct color. By construction of~$\mathcal{U}$, there is a~$Q \in \mathcal{U}$ such that~$\chi_Q$ assigns the correct colors to these vertices. Hence~$\chi_Q$ properly colors~$(C,F')$ and by \cref{lem:colorcoded-fvc} a reducible FVC is returned. \end{proof}
\section{Reducing Feedback Vertex Cuts} \label{sec:fvc-kernel}
We apply reduction operations inspired by \cite{BodlaenderD10, Thomasse10} on the subgraph~$G[C \cup F]$ for a FVC~$(C,F)$ in~$G$. We give 5 reduction operations and show at least one is applicable if~$|F| > f_r(|C|)$. The operations reduce the number of vertices~$v \in F$ with~$\deg_G(v) < 3$ or reduce~$e(C,F)$. The following lemma shows that this is sufficient to reduce the size of~$F$.
\begin{restatable}{lemma}{degreesum} \label{lem:degreesum}
Let~$G$ be a multigraph with minimum degree at least~$3$ and let~$(C,F)$ be a FVC in~$G$. We have~$|F| \leq e(C,F)$. \end{restatable} \begin{proof}
We first show that the claim holds if $G[F]$ is a tree. For all $i \geq 0$ let $V_i := \{v \in F \mid \degree_{G[F]}(v) = i\}$. Note that since $G[F]$ is connected, $V_0 \neq \emptyset$ if and only if $|F|=1$ and the claim is trivially true, so suppose $V_0 = \emptyset$. We first show $|V_{\geq 3}| < |V_1|$.
\begin{align*}
2|E(G[F])| &= \sum_{v \in F} \degree_{G[F]}(v) \geq |V_1| + 2|V_2| + 3|V_{\geq 3}|\\
2|E(G[F])| &= 2(|V(G[F])| - 1) = 2|V_1| + 2|V_2| + 2|V_{\geq 3}| - 2
\end{align*}
We obtain $|V_1| + 2|V_2| + 3|V_{\geq 3}| \leq 2|V_1| + 2|V_2| + 2|V_{\geq 3}| - 2$ hence $|V_{\geq 3}| < |V_1|$.
We know all vertices in $F$ have degree at least $3$ in $G$, so $e(V(G) \setminus F,F) \geq 2|V_1| + |V_2| > |V_1| + |V_2| + |V_{\geq 3}| = |F|$. By definition of FVC there is at most one vertex in $F$ that has an edge to $V(G) \setminus (C \cup F)$, all other edges must be between $C$ and $F$. We obtain $1+e(C,F) > |F|$.
If $G[F]$ is a forest, then let $F_1, \ldots, F_\ell$ be the vertex sets for each tree in $G[F]$. Since $(C,F_i)$ is a FVC in $G$ for all $1 \leq i \leq \ell$, we know $e(C,F_i) \geq |F_i|$ for all $1 \leq i \leq \ell$, and since $F_1, \ldots, F_\ell$ is a partition of $F$ we conclude $e(C,F) = \sum_{1 \leq i \leq \ell} e(C,F_i) \geq \sum_{1 \leq i \leq \ell} |F_i| = |F|$. \end{proof}
Next, we give the reduction operations. These operations apply to a graph $G$ and yield a new graph $G'$ and vertex set $S \subseteq V(G)\setminus V(G')$.
An operation is \emph{FVS-safe} if for any minimum feedback vertex set $S'$ of $G'$, the set $S \cup S'$ is a minimum feedback vertex set of~$G$.
An operation is \emph{antler-safe} if for all $z\geq0$ and any $z$-antler $(C,F)$ in $G$, there exists a $z$-antler $(C',F')$ in $G'$ with $C' \cup F' = (C \cup F) \cap V(G')$ and $|C'| = |C| - |(C \cup F) \cap S|$.
\begin{operation} \label{op:edge-multiplicity}
If $u,v \in V(G)$ are connected by more than two edges, remove all but two of these edges to obtain~$G'$ and take~$S := \emptyset$. \end{operation} \begin{operation} \label{op:contract-edge}
If $v \in V(G)$ has degree exactly $2$ and no self-loop, obtain $G'$ by removing $v$ from $G$ and adding an edge $e$ with $\iota(e) = N_G(v)$. Take~$S := \emptyset$.
\end{operation}
\Cref{op:edge-multiplicity,op:contract-edge} are well established and FVS-safe. Additionally \cref{op:edge-multiplicity} can easily be seen to be antler-safe. To see that \cref{op:contract-edge} is antler-safe, consider a $z$-antler~$(C,F)$ in~$G$ for some~$z\geq0$. If~$v \not\in C$ it is easily verified that~$(C,F\setminus\{v\})$ is a $z$-antler in~$G'$. If~$v \in C$ pick a vertex~$u \in N_G(v) \cap F$ and observe that~$(\{u\} \cup C \setminus \{v\}, F \setminus \{u\})$ is a $z$-antler in~$G'$.
\begin{operation} \label{op:remove-antler}
If $(C,F)$ is an antler in $G$, then $G' := G - (C \cup F)$ and $S := C$. \end{operation} \begin{lemma}
\Cref{op:remove-antler} is FVS-safe and antler-safe. \end{lemma} \begin{proof}
To show \cref{op:remove-antler} is FVS-safe, let $Z$ be a minimum FVS of $G'$. Now $(Z,V(G'-Z))$ is an antler in $G'=G-(C \cup F)$ so then $G$ contains the antler $(Z \cup C, V(G'-Z) \cup F) = (Z \cup S, V(G-(Z \cup S)))$ by \cref{lem:antler_combine}. It follows that~$Z \cup S$ is a minimum FVS of $G$.
To show \cref{op:remove-antler} is antler-safe, let $z\geq0$ and let $(\hat{C},\hat{F})$ be an arbitrary $z$-antler in $G$, then by \cref{lem:antler_diff} $(\hat{C}\setminus(C \cup F), \hat{F}\setminus(C \cup F)$ is a $z$-antler in $G' = G-(C \cup F)$. We deduce:
\begin{align*}
|\hat{C} \setminus (C \cup F)|
&= |\hat{C}| - |\hat{C} \cap C| - |\hat{C} \cap F|
&&\text{since $C \cap F = \emptyset$}\\
&= |\hat{C}| - |\hat{C} \cap C| - |C \cap \hat{F}|
&&\text{by \cref{prop:intersection_antlers}}\\
&= |\hat{C}| - |(\hat{C} \cap C) \cup (C \cap \hat{F})|
&&\text{since $\hat{C} \cap \hat{F} = \emptyset$}\\
&= |\hat{C}| - |(\hat{C} \cup \hat{F}) \cap C|
= |\hat{C}| - |(\hat{C} \cup \hat{F}) \cap \hat{S}'|.
\tag*{\qedhere}
\end{align*} \end{proof}
\begin{operation} \label{op:v-flower}
If $(C,F)$ is a FVC in $G$ and for some $v \in C$ the graph $G[F \cup \{v\}]$ contains a $v$-flower of order $|C|+1$, then $G' := G - v$ and $S := \{v\}$. \end{operation} \begin{lemma}
\Cref{op:v-flower} is FVS-safe and antler-safe. \end{lemma} \begin{proof}
We first show that any minimum FVS in $G$ contains $v$. Let $X$ be a minimum FVS in $G$. If $v \not\in X$ then $|F \cap X| > |C|$ since $G[F \cup \{v\}]$ contains a $v$-flower of order $|C|+1$. Take $X' := C \cup (X \setminus F)$, clearly $|X'| < |X|$ so $G-X'$ must contain a cycle since $X$ was minimum. This cycle must contain a vertex from $X \setminus X' \subseteq F$, so by \cref{obs:fvc-basics} this cycle must contain a vertex from $C$, but $C \subseteq X'$. Contradiction.
To show \cref{op:v-flower} is FVS-safe, suppose $Z$ is a minimum FVS of $G' = G-v$. Clearly~$Z \cup \{v\}$ is a FVS in~$G$. To show that~$Z \cup \{v\}$ is minimum suppose~$Z'$ is a smaller FVS in~$G$. We know $v \in Z$ so $Z'\setminus\{v\}$ is a FVS in $G-v$, but $|Z'\setminus\{v\}| < |Z|$ contradicting optimality of~$Z$.
To show \cref{op:v-flower} is antler-safe, suppose~$(\hat{C},\hat{F})$ is a $z$-antler in~$G$ for some~$z\geq0$. We show~$(\hat{C}\setminus\{v\}, \hat{F})$ is a $z$-antler in~$G'$. If~$v \in \hat{C}$ then this follows directly from \cref{obs:subantler}, so suppose~$v \not\in \hat{C}$. Note that~$v \in \hat{F}$ would contradict that any minimum FVS in~$G$ contains~$v$ by \cref{obs:remove-antler}. So~$G[\hat{C} \cup \hat{F}] = G'[\hat{C} \cup \hat{F}]$ and~$(\hat{C}\setminus\{v\}, \hat{F}) = (\hat{C},\hat{F})$ is a FVC in~$G' = G-v$ by \cref{obs:subfvc}, hence~$(\hat{C}\setminus\{v\}, \hat{F})$ is a $z$-antler in~$G'$.
\end{proof}
\begin{operation} \label{op:remove-edge}
If~$(C,F)$ is a FVC in~$G$, $v \in C$, and $X \subseteq F$ such that~$G[F \cup \{v\}] - X$ is acyclic, and if~$T$ is a tree in~$G[F]-X$ containing a vertex~$w \in N_G(v)$ such that for each~$u \in N_G(T)\setminus\{v\}$ there are more than~$|C|$ other trees~$T' \neq T$ in~$G[F]-X$ for which~$\{u,v\} \subseteq N_G(T')$,
then take~$S := \emptyset$ and obtain~$G'$ by removing the unique edge between~$v$ and~$w$, and adding double-edges between~$v$ and~$u$ for all~$u \in N_G(V(T))\setminus\{v\}$. \end{operation}
\begin{lemma}
\Cref{op:remove-edge} is FVS-safe and antler-safe. \end{lemma} \begin{proof}
Suppose $(\hat{C},\hat{F})$ is a $z$-antler in $G$ for some $z\geq0$. We first prove the following claim:
\begin{restatable}{claim}{disjointpaths} \label{claim:op4:disjointpaths}
For all $u \in N_G(T)\setminus\{v\}$ we have $v \in \hat{F} \Rightarrow u \in \hat{C}$ and $u \in \hat{F} \Rightarrow v \in \hat{C}$.
\end{restatable}
\begin{claimproof}
Each tree of $G[F]-X$ supplies a path between $u$ and $v$, hence there are more than $|C|+1$ internally vertex-disjoint paths between $u$ and $v$. Suppose $v \in \hat{F}$, we show $u \in \hat{C}$. The proof of the second implication is symmetric. Suppose for contradiction that $u \not\in \hat{C}$. All except possibly one of the disjoint paths between $u$ and $v$ must contain a vertex in $\hat{C}$ by \cref{obs:fvc-basics} since any two disjoint paths form a cycle containing a vertex from $\hat{F}$. Let $Y \subseteq \hat{C}$ be the set of vertices in $\hat{C}$ that are in a tree of $G[F]-X$ with neighbors of $u$ and $v$, so $|Y| > |C|$. Then $|C \cup \hat{C} \setminus Y| < |\hat{C}|$ we derive a contradiction by showing $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C} \setminus Y)$ is acyclic. We know $Y \subseteq F$, so any cycle in $G$ containing a vertex from $Y$ also contains a vertex from $C$ by \cref{obs:fvc-basics}. So if $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C} \setminus Y)$ contains a cycle, then so does $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C})$ which contradicts that $\hat{C}$ is a (minimum) FVS in~$G[\hat{C} \cup \hat{F}]$ since~$(\hat{C},\hat{F})$ is an antler in~$G$.
\end{claimproof}
To prove \cref{op:remove-edge} is antler-safe, we show that~$(\hat{C},\hat{F})$ is also a $z$-antler in $G'$. Suppose~$v \not\in \hat{C} \cup \hat{F}$, then~$G[\hat{C} \cup \hat{F}] = G'[\hat{C} \cup \hat{F}]$ as $G$ and $G'$ only differ on edges incident to $v$. It remains to show that for each tree $T'$ in $G'[\hat{F}]$ we have $e(T',G' - (\hat{C} \cup \hat{F})) \leq 1$. Suppose $T'$ is a tree in $G'[\hat{F}]$ with $e(T',G' - (\hat{C} \cup \hat{F})) > 1$. Since $e(T',G - (\hat{C} \cup \hat{F})) \leq 1$ we know that at least one of the edges added between $v$ and some $u \in N_G(T)$ has an endpoint in~$V(T') \subseteq \hat{F}$. Since $v \not\in \hat{F}$ we have $u \in \hat{F}$, so $v \in \hat{C}$ by \cref{claim:op4:disjointpaths} contradicting our assumption $v \not\in \hat{C} \cup \hat{F}$.
Suppose $v \in \hat{C} \cup \hat{F}$, we first show that $(\hat{C}, \hat{F})$ is a FVC in $G'$. If $v \in \hat{C}$ this is clearly the case, so suppose $v \in \hat{F}$. From \cref{claim:op4:disjointpaths} it follows that $N_G(T)\setminus\{v\} \subseteq \hat{C}$, so all edges added in~$G'$ are incident to vertices in $\hat{C}$ hence $(\hat{C},\hat{F})$ is still a FVC in $G'$.
We now show that~$G'[\hat{C} \cup \hat{F}]$ contains an $\hat{C}$-certificate of order $z$. We know $G[\hat{C} \cup \hat{F}]$ contains an $\hat{C}$-certificate of order $z$. Let $H$ be an arbitrary component of this certificate. Take~$Y := V(H) \cap \hat{C}$, so $Y$ is a minimum FVS in $H$. It suffices to show that $Y$ is also a minimum FVS of $G' \cap H$. This is easily seen to be true when $v \not\in V(H)$, so suppose $v \in V(H)$. First we argue that $(G' \cap H)-Y$ is acyclic. This is easily seen to be true when $v \in Y$ since $G$ and $G'$ only differ in edges incident to $v$, so suppose $v \not\in Y$. Then $v \not\in \hat{C}$ hence $v \in \hat{F}$ and by \cref{claim:op4:disjointpaths}~$N_G(T)\setminus\{v\} \subseteq \hat{C}$. It follows that $V(H) \cap (N_G(T)\setminus\{v\}) \subseteq Y$ so clearly~$(G' \cap H)-Y$ is acyclic since all edges in $G' \cap H$ that are not in $H$ are incident to a vertex in $Y$.
To show $Y$ is a minimum FVS, suppose there is a FVS $Y'$ of $G' \cap H$ with $|Y'| < |Y|$. Since $H-v$ is a subgraph of $G' \cap H$ we know $H-(Y' \cup \{v\})$ is acyclic, but since~$|Y'| < |Y| = \fvs(H)$ we also know $H-Y'$ contains a cycle. This cycle must contain the edge~$\{v,w\}$ since otherwise this cycle is also present in $(G' \cap H)- Y'$. Then there must be some~$u \in N_G(T)\setminus\{v\}$ on the cycle, so $u,v \not\in Y'$. But $G'$ contains a double-edge between~$u$ and~$v$ so~$(G' \cap H)-Y'$ contains a cycle, contradicting that $Y'$ is a FVS in $G' \cap H$.
We finally show \cref{op:remove-edge} is FVS-safe. Let $Z'$ be a minimum FVS in $G'$, and suppose~$Z'$ is not a FVS in $G$. Then $G-Z'$ contains a cycle. This cycle contains the edge $\{v,w\}$ since otherwise $G'-Z'$ also contains this cycle. Since $G'$ contains double-edges between $v$ and all $u \in N_G(T)\setminus\{v\}$ and $v \not\in Z'$, it follows that $N_G(T)\setminus\{v\} \subseteq Z'$, but then no cycle in $G-Z'$ can intersect $T$ and $\{v,w\}$ is not part of a cycle in $G-Z'$. We conclude by contradiction that $Z'$ is a FVS in $G$. To prove optimality, consider a minimum FVS~$Z$ in~$G$ and observe that~$(Z,V(G-Z))$ is an antler in~$G$. Since \cref{op:remove-edge} is antler-safe we know $G'$ contains an antler~$(C',F')$ with~$C' \cup F' = (Z \cup V(G-Z)) \cap V(G') = V(G')$ and~$|C'| = |Z| - |(Z \cup V(G-Z)) \cap S| = |Z|$. Since~$C' \cup F' = V(G')$ we know~$C'$ is a FVS in~$G'$, hence~$\fvs(G') \leq |C'| = |Z|$, hence~$\fvs(G) = |Z| \geq \fvs(G') = |Z'|$. \end{proof}
Finally we show that when we are given a reducible FVC~$(C,F)$ in~$G$, then we can find and apply an operation in $\ensuremath{\mathcal{O}}\xspace(n^2)$ time. With a more careful analysis better running time bounds can be shown, but this does not affect the final running time of the main algorithm.
\begin{lemma} \label{lem:fvc-kernel}
Given a graph~$G$ and a reducible FVC~$(C,F)$ in~$G$, we can find and apply an operation in~$\ensuremath{\mathcal{O}}\xspace(n^2)$ time. \end{lemma} \begin{proof}
Note that if a vertex $v \in V(G)$ has a self-loop then $(\{v\}, \emptyset)$ is an antler in $G$ and we can apply \cref{op:remove-antler}. If a vertex $v$ has degree 0 or 1 then $(\emptyset, \{v\})$ is an antler in $G$. Hence \Cref{op:edge-multiplicity,,op:contract-edge,,op:remove-antler} can always be applied if the graph contains a self-loop, a vertex with degree less than 3, or more than 2 edges between two vertices. So assume $G$ is a graph with no self-loops, minimum degree at least 3, and at most two edges between any pair of vertices.
By \cref{lem:degreesum} we have $e(C,F) \geq |F| > 2|C|^3+3|C|^2-|C|$ so then there must be a vertex $v$ in $C$ with more than $\frac{1}{|C|} \cdot (2|C|^3+3|C|^2-|C|) = 2|C|^2+3|C|-1$ edges to $F$. By \cref{lem:alt-fvs} we can find a set $X \subseteq F$ such that $G[F \cup \{v\}] - X$ is acyclic and $G[F \cup \{v\}]$ contains a $v$-flower of order $|X|$. Hence if $|X| \geq |C|+1$ \cref{op:v-flower} can be applied, so assume $|X| \leq |C|$. For each $u \in X \cup C \setminus \{v\}$ that is not connected to $v$ by a double-edge, mark up to $|C|+1$ trees $T'$ in $G[F]-X$ for which $\{u,v\} \in N_G(T)$. Note that we marked at most $(|C|+1) \cdot |X \cup C \setminus \{v\}| \leq (|C|+1)\cdot(2|C|-1) = 2|C|^2+|C|-1$ trees. Since $v$ has exactly one edge to each marked tree ($G[F \cup \{v\}]-X$ is acyclic) and at most two edges to each vertex in $X$, this accounts for at most $2|C|^2+3|C|-1$ edges from $v$ to $F$, so there must be at least one more edge from $v$ to a vertex $w \in F$, hence \cref{op:remove-edge} applies.
It can easily be verified that all operations described can be performed in~$\ensuremath{\mathcal{O}}\xspace(n^2)$ time.
\end{proof}
\section{Finding and Removing Antlers} \label{sec:find-antler} We will find antlers using color coding, using coloring functions of the form~$\chi \colon V(G) \cup E(G) \to \{\ensuremath{\mathsf{\dot{F}}}\xspace,\ensuremath{\mathsf{\dot{C}}}\xspace,\ensuremath{\mathsf{\dot{R}}}\xspace\}$. For all~$c \in \{\ensuremath{\mathsf{\dot{F}}}\xspace, \ensuremath{\mathsf{\dot{C}}}\xspace, \ensuremath{\mathsf{\dot{R}}}\xspace\}$ let~$\inv{\chi}_V(c) = \inv{\chi}(c) \cap V(G)$.
For any integer~$z\geq0$, a $z$-antler~$(C,F)$ in a graph~$G$ is \emph{$z$-properly colored} by a coloring~$\chi$ if all of the following hold: (i)~$F \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)$, (ii)~$C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$, (iii)~$N_G(F)\setminus C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{R}}}\xspace)$, and (iv)~$G[C \cup F] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is a $C$-certificate of order~$z$. Recall that~$\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ can contain edges as well as vertices so for any subgraph~$H$ of~$G$ the graph~$H - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is obtained from~$H$ by removing both vertices and edges. It can be seen that if~$(C,F)$ is a $z$-antler, then there exists a coloring that $z$-properly colors it. Consider for example a coloring where a vertex~$v$ is colored~\ensuremath{\mathsf{\dot{C}}}\xspace (resp.~\ensuremath{\mathsf{\dot{F}}}\xspace) if~$v \in C$ (resp.~$v \in F$), all other vertices are colored~\ensuremath{\mathsf{\dot{R}}}\xspace, and for some $C$-certificate~$H$ of order~$z$ in~$G[C \cup F]$ all edges in~$H$ have color~\ensuremath{\mathsf{\dot{F}}}\xspace and all other edges have color~\ensuremath{\mathsf{\dot{R}}}\xspace. The property in of a properly colored $z$-antler described in \cref{lem:propcolor} will be useful to prove correctness of the color coding algorithm.
\begin{lemma} \label{lem:propcolor}
For any $z\geq0$, if a $z$-antler $(C,F)$ in graph $G$ is $z$-properly colored by a coloring $\chi$ and $H$ is a component of $G[C \cup F] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ then each component $H'$ of $H-C$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ with $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$. \end{lemma} \begin{proof}
Note that since $C \cap \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace) = \emptyset$ we have that the statement $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$ implies that $N_{G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') = \emptyset$ and hence that $H'$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. We show $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$.
Suppose $v \in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ and let $u \in V(H')$ be a neighbor of $v$ in $G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. Since $V(H') \subseteq F$ we know $u \in F$. Since $(C,F)$ is $z$-properly colored we also have $N_G(F)\setminus C = \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, hence $N_G(u) \subseteq C \cup F \cup \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ so then $N_{G-\inv{\chi}}(u) \subseteq C \cup F$. By choice of $u$ we have $v \in N_{G-\inv{\chi}}(u) \subseteq C \cup F$. So since $u,v \in C \cup F$, and $u$ and $v$ are neighbors in $G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ we know $u$ and $v$ are in the same component of $G[C \cup F]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, hence $v \in V(H)$.
Suppose $v \not\in C$, so $v \in F$. Since also $u \in F$ we know that $u$ and $v$ are in the same component of $G[F]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, so $v \in H'$, but then $v \not\in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ contradicting our choice of $v$. It follows that $v \in C$ hence $v \in C \cap V(H)$. Since $v \in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ was arbitrary $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$. \end{proof}
We now show that a $z$-antler can be obtained from a suitable coloring of the graph.
\begin{restatable}[$\bigstar$]{lemma}{algfindantler} \label{lem:alg}
A~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time algorithm exists taking as input an integer~$z\geq0$, a graph~$G$, and a coloring~$\chi$ and producing as output a $z$-antler~$(C,F)$ in~$G$, such that for any $z$-antler~$(\hat{C},\hat{F})$ that is $z$-properly colored by~$\chi$ we have~$\hat{C} \subseteq C$ and~$\hat{F} \subseteq F$. \end{restatable}
\begin{proof}
We define a function~$W_\chi \colon 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)} \to 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)}$ as follows: for any~$C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ let~$W_\chi(C)$ denote the set of all vertices that are in a component~$H$ of~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ for which~$N_{G - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H) \subseteq C$.
The algorithm we describe updates the coloring~$\chi$ and recolors any vertex or edge that is not part of a $z$-properly colored antler to color~\ensuremath{\mathsf{\dot{R}}}\xspace.
\begin{enumerate}
\item \label{antleralg:init} Recolor all edges to color~\ensuremath{\mathsf{\dot{R}}}\xspace when one of its endpoints has color~\ensuremath{\mathsf{\dot{R}}}\xspace.
\item \label{antleralg:loop1} For each component~$H$ of~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ we recolor all vertices of~$H$ and their incident edges to color~\ensuremath{\mathsf{\dot{R}}}\xspace if~$H$ is not a tree or~$e(H,\inv{\chi}_V(\ensuremath{\mathsf{\dot{R}}}\xspace)) > 1$.
\item \label{antleralg:loop2} For each subset~$C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ of size at most~$z$, mark all vertices in~$C$ if $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C|$.
\item \label{antleralg:repeat} If~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ contains unmarked vertices we recolor them to color~\ensuremath{\mathsf{\dot{R}}}\xspace, remove markings made in step~\ref{antleralg:loop2} and repeat from step~\ref{antleralg:init}.
\item \label{antleralg:return} If all vertices in~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ are marked in step~\ref{antleralg:loop2}, return~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$.
\end{enumerate}
\subparagraph*{Running time}
The algorithm will terminate after at most~$n$ iterations since in every iteration the number of vertices in~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{R}}}\xspace)$ increases. Steps~\ref{antleralg:init}, \ref{antleralg:loop1}, \ref{antleralg:repeat}, and \ref{antleralg:return} can easily be seen to take no more than $\ensuremath{\mathcal{O}}\xspace(n^2)$ time. Step~\ref{antleralg:loop2} can be performed in $\ensuremath{\mathcal{O}}\xspace(4^z \cdot n^{z+1})$ time by checking for all~$\ensuremath{\mathcal{O}}\xspace(n^z)$ subsets~$C \in \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ of size at most~$z$ whether the graph~$G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ has feedback vertex number~$z$. This can be done in time~$\ensuremath{\mathcal{O}}\xspace(4^z \cdot n)$~\cite{IwataK19}.
Hence the algorithm runs in time~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
We show that any $z$-properly colored antler prior to executing the algorithm remains $z$-properly colored after termination and that in step~\ref{antleralg:return},~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a~$z$-antler in~$G$. Since~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ contains all properly colored antlers this proves correctness.
\begin{restatable}[$\bigstar$]{claim}{remainproperlycolored} \label{claim:remain}
All $z$-antlers $(\hat{C},\hat{F})$ that are $z$-properly colored by $\chi$ prior to executing the algorithm are also $z$-properly colored by $\chi$ after termination of the algorithm.
\end{restatable}
\begin{claimproof}
To show the algorithm preserves properness of the coloring, we show that every individual recoloring preserves properness, that is, if an arbitrary $z$-antler is $z$-properly colored prior to the recoloring, it is also $z$-properly colored after the recoloring.
Suppose an arbitrary $z$-antler $(\hat{C},\hat{F})$ is $z$-properly colored by $\chi$. An edge is only recolored when one of its endpoints has color~\ensuremath{\mathsf{\dot{R}}}\xspace. Since these edges are not in $G[\hat{C} \cup \hat{F}]$ its color does change whether $(\hat{C},\hat{F})$ is colored $z$-properly. All other operations done by the algorithm are recolorings of vertices to color~\ensuremath{\mathsf{\dot{R}}}\xspace. We show that any time a vertex~$v$ is recolored we have that $v \not\in \hat{C} \cup \hat{F}$, meaning $(\hat{C},\hat{F})$ remains colored $z$-properly.
Suppose $v$ is recolored in step~\ref{antleralg:loop1}, then we know $\chi(v) = \ensuremath{\mathsf{\dot{F}}}\xspace$, and $v$ is part of a component~$H$ of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$. Since $\chi$ $z$-properly colors $(\hat{C},\hat{F})$ we have $\hat{F} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)$ but $N_G(\hat{F}) \cap \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace) = \emptyset$, so since $H$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ we know either $V(H) \subseteq \hat{F}$ or $V(H) \cap \hat{F} = \emptyset$. If $V(H) \cap \hat{F} = \emptyset$ then clearly $v \not\in \hat{C} \cup \hat{F}$. So suppose $V(H) \subseteq \hat{F}$, then $H$ is a tree in $G[\hat{F}]$. Since $v$ was recolored and $H$ is a tree it must be that $e(H,\inv{\chi}_C(\ensuremath{\mathsf{\dot{R}}}\xspace)) > 1$ but this contradicts that $(\hat{C},\hat{F})$ is a FVC.
Suppose $v$ is recolored in step~\ref{antleralg:repeat}, then we know $v$ was not marked during step~\ref{antleralg:loop2} and $\chi(v) = \ensuremath{\mathsf{\dot{C}}}\xspace$, so $v \not\in \hat{F}$. Suppose that $v \in \hat{C}$. We derive a contradiction by showing that $v$ was marked in step~\ref{antleralg:loop2}.
Since $(\hat{C},\hat{F})$ is $z$-properly colored, we know that $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is a $\hat{C}$-certificate of order $z$, so if $H$ is the component of $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ containing $v$ then $\fvs(H) = |\hat{C} \cap V(H)| \leq z$. Since $\hat{C} \cap V(H) \subseteq \hat{C} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ we know that in some iteration in step~\ref{antleralg:loop2} we have $C = \hat{C} \cap V(H)$.
To show that $v$ was marked, we show that $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C|$. We know $G[W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is a forest since it is a subgraph of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ which is a forest by step~\ref{antleralg:loop1}, so we have that $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \leq |C|$.
To show $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \geq |C|$ we show that $H$ is a subgraph of $G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. By \cref{lem:propcolor} we have that each component $H'$ of $H-\hat{C}$ is also a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ with $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq \hat{C} \cap V(H) = C$. Hence $V(H-\hat{C}) = V(H-C) \subseteq W_\chi(C)$ so $H$ is a subgraph of $G[C \cup W_\chi(C)]$. Since $H$ is also a subgraph of $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ we conclude that $H$ is a subgraph of $G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ and therefore $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \geq \fvs(H) = |C|$.
\end{claimproof}
\begin{restatable}[$\bigstar$]{claim}{coloringisantler} \label{claim:coloredantler}
In step~\ref{antleralg:return}, $(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a $z$-antler in~$G$.
\end{restatable}
\begin{claimproof}
We know $(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a FVC in $G$ because each component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ is a tree and has at most one edge to a vertex not in $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ by step~\ref{antleralg:loop1}. It remains to show that $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ contains a $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$-certificate of order $z$.
Note that in step~\ref{antleralg:return} the coloring $\chi$ is the same as in the last execution of step~\ref{antleralg:loop2}. Let $\mathcal{C} \subseteq 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)}$ be the family of all subsets $C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ that have been considered in step~\ref{antleralg:loop2} and met the conditions for marking all vertices in $C$, i.e., $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C| \leq z$. Since all vertices in $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ have been marked during the last execution of step~\ref{antleralg:loop2} we know $\bigcup_{C \in \mathcal{C}} C = \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$.
Let $C_1,\ldots,C_{|\mathcal{C}|}$ be the sets in $\mathcal{C}$ in an arbitrary order and define $D_i := C_i \setminus C_{<i}$ for all $1 \leq i \leq |\mathcal{C}|$. Observe that $D_1, \ldots, D_{|\mathcal{C}|}$ is a partition of $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ with $|D_i| \leq z$ and $C_i \subseteq D_{\leq i}$ for all $1 \leq i \leq |\mathcal{C}|$. Note that $D_i$ may be empty for some $i$.
We now show that $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ contains a $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$-certificate of order $z$. We do this by showing there are $|\mathcal{C}|$ vertex disjoint subgraphs of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$, call them $G_1, \ldots, G_{|\mathcal{C}|}$, such that $\fvs(G_i) = |D_i| \leq z$ for each $1 \leq i \leq |\mathcal{C}|$.
Take $G_i := G[D_i \cup (W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i}))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ for all $1 \leq i \leq |\mathcal{C}|$. First we show that for any $i \neq j$ the graphs $G_i$ and $G_j$ are vertex disjoint. Clearly $D_i \cap D_j = \emptyset$. We can assume $i < j$, so $D_{\leq i} \subseteq D_{<j}$ and then $W_\chi(D_{\leq i}) \subseteq W_\chi(D_{<j})$. By successively dropping two terms, we deduce
\begin{align*}
(W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i})) \cap (W_\chi(D_{\leq j}) \setminus W_\chi(D_{<j}))
&\subseteq W_\chi(D_{\leq i}) \cap (W_\chi(D_{\leq j}) \setminus W_\chi(D_{<j})) \\
&\subseteq W_\chi(D_{\leq i}) \setminus W_\chi(D_{<j})
= \emptyset .
\end{align*}
We complete the proof by showing $\fvs(G_i) = |D_i|$ for all $1 \leq i \leq \ell$. Recall that $D_i = C_i \setminus C_{<i}$. Since $C_i \in \mathcal{C}$ we know $C_i$ is an optimal FVS in $G[C_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, so then clearly $D_i$ is an optimal FVS in $G[C_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace) - C_{<i} = G[D_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. We know that $C_i \subseteq D_{\leq i}$ so then also $W_\chi(C_i) \subseteq W_\chi(D_{\leq i})$. It follows that $D_i$ is an optimal FVS in $G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. In this graph, all vertices in $W_\chi(D_{<i})$ must be in a component that does not contain any vertices from $D_i$, so this component is a tree and we obtain $|D_i| = \fvs(G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = \fvs(G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace) - W_\chi(D_{<i})) = \fvs(G[D_i \cup (W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i}))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = \fvs(G_i)$.
\end{claimproof}
It can be seen from \cref{claim:remain} that for any $z$-properly colored antler~$(\hat{C},\hat{F})$ we have~$\hat{C} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ and~$\hat{F} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)$. \Cref{claim:coloredantler} completes the correctness argument. \end{proof}
If a graph~$G$ contains a reducible single-tree FVC of width at most~$k$ then we can find and apply an operation by \cref{lem:find-fvc} and \cref{lem:fvc-kernel}. If~$G$ does not contain such a FVC, but~$G$ does contain a non-empty $z$-antler~$(C,F)$ of width at most~$k$, then using \cref{col:few-fvcs} we can prove that whether~$(C,F)$ is $z$-properly colored is determined by the color of at most~$26 k^5 z^2$ relevant vertices and edges. Using two~$(n+m,26 k^5 z^2)$-universal sets, we can create a set of colorings that is guaranteed to contain a coloring that $z$-properly colors~$(C,F)$. Using \cref{lem:alg} we find a non-empty $z$-antler and apply \cref{op:remove-antler}. We obtain the following: \begin{restatable}[$\bigstar$]{lemma}{algfindop} \label{lem:find-op}
Given a graph $G$ and integers~$k \geq z \geq 0$. If~$G$ contains a non-empty $z$-antler of width at most $k$ we can find and apply an operation in~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$
time. \end{restatable} \begin{proof}
Consider the following algorithm:
Use \cref{lem:find-fvc} to obtain a FVC~$(C_1,F_1)$ in~$2^{\ensuremath{\mathcal{O}}\xspace(k^3)}\cdot n^3 \log n$ time. If~$(C_1,F_1)$ is reducible we can find and apply an operation in~$\ensuremath{\mathcal{O}}\xspace(n^2)$ time by \cref{lem:fvc-kernel} so assume~$(C_1,F_1)$ is not reducible. Create two~$(n+m, 26 k^5 z^2)$-universal sets~$\mathcal{U}_1$ and~$\mathcal{U}_2$ for~$V(G) \cup E(G)$ using \cref{thm:universal-set}. Define for each pair~$(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ the coloring~$\chi_{Q_1,Q_2}$ of~$G$ that assigns all vertices and edges in~$Q_1$ color~\ensuremath{\mathsf{\dot{C}}}\xspace, all vertices and edges in~$Q_2 \setminus Q_1$ color~\ensuremath{\mathsf{\dot{F}}}\xspace, and all vertices and edges not in~$Q_1 \cup Q_2$ color~\ensuremath{\mathsf{\dot{R}}}\xspace. For each~$(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ obtain in~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time a $z$-antler~$(C_2,F_2)$ by running the algorithm from \cref{lem:alg} on~$G$ and~$\chi_{Q_1,Q_2}$. If~$(C_2, F_2)$ is not empty, apply \cref{op:remove-antler} to remove~$(C_2,F_2)$, otherwise report~$G$ does not contain a $z$-antler of width at most~$k$.
\subparagraph*{Running time}
By \cref{thm:universal-set}, the sets~$\mathcal{U}_1$ and~$\mathcal{U}_2$ have size~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \log n$ and can be created in~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n \log n$ time. It follows that there are~$|\mathcal{U}_1 \times \mathcal{U}_2| = 2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \log^2 n$ colorings for which we apply the~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time algorithm from \cref{lem:alg}. We obtain an overall running time of~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$. Since a $z$-antler has width at least~$z$, we can assume~$k \geq z$, hence~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)} \leq 2^{\ensuremath{\mathcal{O}}\xspace(k^7)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
Suppose $G$ contains a $z$-antler $(C,F)$ of width at most $k$, we show the algorithm finds an operation to apply.
By \cref{col:few-fvcs} we know that there exists an~$F' \subseteq F$ such that~$(C,F')$ is a $z$-antler where $G[F']$ has at most $\frac{|C|}{2}(z^2+2z-1)$ trees. For each tree~$T$ in~$G[F']$ note that~$(C,V(T))$ is a single-tree FVC of width $|C| \leq k$. If for some tree~$T$ in~$G$ the FVC~$(C,V(T))$ is reducible, then $(C_1,F_1)$ is reducible by \cref{lem:find-fvc} and we find an operation using \cref{lem:fvc-kernel}, so suppose for all trees $T$ in $G[F']$ that $|V(T)| \leq f_r(|C|)$. So then $|F'| \leq \frac{|C|}{2}(z^2+2z-1) \cdot f_r(|C|)$. We show that in this case there exists a pair $(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ such that $\chi_{Q_1,Q_2}$ $z$-properly colors $(C,F')$.
Whether a coloring $z$-properly colors $(C,F')$ is only determined by the colors of $C \cup F' \cup N_G(F') \cup E(G[C \cup F'])$.
\begin{restatable}{claim}{antlercolorsize} \label{claim:antler-color-size}
$|C \cup F' \cup N_G(F') \cup E(G[C \cup F'])| \leq 26 k^5 z^2$.
\end{restatable}
\begin{claimproof}
Note that $|N_G(F') \setminus C| \leq \frac{|C|}{2}(z^2+2z-1)$ since no tree in $G[F']$ can have more than one neighbor outside $C$. Additionally we have
\begin{align*}
|E(G[C \cup F'])|
&\leq |E(G[C])| + |E(G[F'])| + e(C,F') \\
&\leq |E(G[C])| + |F'| + |C| \cdot |F'| &\text{since $G[F']$ is a forest} \\
&\leq |C|^2 + (|C|+1) \cdot |F'|\\
&\leq |C|^2 + (|C|+1) \cdot \frac{|C|}{2}(z^2+2z-1) \cdot f_r(|C|)\\
&\leq k^2 + (k+1) \cdot \frac{k}{2}(z^2+2z-1) \cdot (2k^3 + 3k^2 - k)\\
&\leq k^2 + \frac{z^2+2z-1}{2} \cdot (k^2+k) \cdot (2k^3 + 3k^2 - k)\\
&\leq k^2 + \frac{z^2+2z-1}{2} \cdot 2k^2 \cdot 5k^3 &\text{since $k=0$ or $k \geq 1$}\\
&\leq k^2 + \frac{3z^2}{2} \cdot 10k^5 &\text{since $z=0$ or $z \geq 1$}\\
&\leq k^2 + 15 k^5 z^2
\leq 16 k^5 z^2,
\end{align*}
hence
\begin{align*}
|C \cup F' \cup N_G(F') \cup E(G[C \cup F'])|\\
&\hspace{-5em}= |C| + |F'| + |N_G(F')\setminus C| + |E(G[C \cup F'])|\\
&\hspace{-5em}\leq |C|
+ \frac{z^2+2z+1}{2} f_r(|C|)
+ \frac{|C|}{2}(z^2+2z-1)
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3z^2}{2} f_r(k)
+ \frac{k}{2}(2z^2)
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3}{2}z^2 \cdot (2k^3 + 3k^2 - k)
+ z^2 \cdot k
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3}{2}z^2 \cdot 5k^3
+ z^2 \cdot k
+ 16 k^5 z^2
\leq 26 k^5 z^2. \qedhere
\end{align*}
\end{claimproof}
By construction of~$\mathcal{U}_1$ and~$\mathcal{U}_2$ there exist~$Q_1 \in \mathcal{U}_1$ and~$Q_2 \in \mathcal{U}_2$ such that~$\chi_{Q_1,Q_2}$ $z$-properly colors~$(C,F')$. Therefore the algorithm from \cref{lem:alg} returns a non-empty $z$-antler for~$\chi_{Q_1,Q_2}$ and \cref{op:remove-antler} can be executed.
\end{proof}
Note that applying an operation reduces the number of vertices or increases the number of double-edges. Hence by repeatedly using \cref{lem:find-op} to apply an operation we obtain, after at most~$\ensuremath{\mathcal{O}}\xspace(n^2)$ iterations, a graph in which no operation applies. By \cref{lem:find-op} this graph does not contain a non-empty $z$-antler of width at most~$k$. We show that this method reduces the solution size at least as much as iteratively removing $z$-antlers of width at most~$k$. We first describe the behavior of such a sequence of antlers. For integer~$k\geq0$ and~$z\geq0$, we say a sequence of disjoint vertex sets~$C_1, F_1,\ldots,C_\ell, F_\ell$ is a \emph{$z$-antler-sequence} for a graph~$G$ if for all~$1 \leq i \leq \ell$ the pair~$(C_i, F_i)$ is a $z$-antler in~$G - (C_{<i} \cup F_{<i})$. The \emph{width} of a $z$-antler-sequence is defined as $\max_{1 \leq i \leq \ell} |C_1|$.
\begin{proposition} \label{prop:antler-sequence}
If~$C_1, F_1,\ldots,C_\ell, F_\ell$ is a \emph{$z$-antler-sequence} for some graph~$G$, then the pair~$(C_{\leq i}, F_{\leq i})$ is a $z$-antler in~$G$ for any $1 \leq i \leq \ell$. \end{proposition} \begin{proof}
We use induction on~$i$. Clearly the statement holds for~$i = 1$, so suppose~$i > 1$. By induction~$(C_{<i},F_{<i})$ is a $z$-antler in~$G$, and since~$(C_i,F_i)$ is a $z$-antler in~$G-(C_{<i} \cup F_{<i})$ we have by \cref{lem:antler_combine} that~$(C_{<i} \cup C_i,F_{<i} \cup F_i) = (C_{\leq i},F_{\leq i})$ is a $z$-antler in~$G$. \end{proof}
The following theorem describes that repeatedly applying \cref{lem:find-op} reduces the solution size at least as much as repeatedly removing $z$-antlers of width at most~$k$. By taking~$t=1$ we obtain \cref{thm:z:antler}. \begin{restatable}{theorem}{maintheorem} \label{thm:main}
Given as input a graph $G$ and integers $k\geq z \geq 0$ we can find in $2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time a vertex set $S \subseteq V(G)$ such that
\begin{enumerate}
\item \label{thm:item:fvs-safe} there is a minimum FVS in~$G$ containing all vertices of~$S$, and
\item \label{thm:item:antlers} if~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler sequence of width at most~$k$
then~$|S| \geq |C_{\leq t}|$.
\end{enumerate} \end{restatable} \begin{proof}
We first describe the algorithm.
\subparagraph*{Algorithm}
We use \cref{lem:find-op} to apply an operation in $G$ and obtain the resulting graph $G'$ and vertex set $S$. If no applicable operation was found return an empty vertex set $S := \emptyset$. Otherwise we recursively call our algorithm on~$G'$ with integers~$z$ and~$k$ to obtain a vertex set~$S'$ and return the vertex set~$S \cup S'$.
\subparagraph*{Running time}
Note that since every operation reduces the number of vertices or increases the number of double-edges, after at most $\ensuremath{\mathcal{O}}\xspace(n^2)$ operations we obtain a graph where no operation can be applied. Therefore after at most $\ensuremath{\mathcal{O}}\xspace(n^2)$ recursive calls the algorithm terminates. We obtain a running time of $2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
We prove correctness by induction on the recursion depth, which is shown the be finite by the run time analysis.
First consider the case that no operation was found. Clearly condition~\ref{thm:item:fvs-safe} holds for~$G' := G$ and $S := \emptyset$. To show condition~\ref{thm:item:antlers} suppose~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler-sequence of width at most~$k$ for~$G$. The first non-empty antler in this sequence is a $z$-antler of width at most~$k$ in~$G$. Since no operation was found using \cref{lem:find-op} it follows that~$G$ does not contain a non-empty $z$-antler of width at most~$k$. Hence all antlers in the sequence must be empty and~$|C_{\leq t}| = 0$, so condition~\ref{thm:item:antlers} holds for~$G' := G$ and~$S := \emptyset$.
For the other case, suppose~$G'$ and~$S$ are obtained by applying an operation, then since this operation is FVS-safe we know for any minimum FVS~$S''$ of~$G'$ that~$S \cup S''$ is a minimum FVS in~$G$. Since $S'$ is obtained from a recursive call there is a minimum FVS in $G'$ containing all vertices of~$S'$. Let $S''$ be such a FVS in $G'$, so $S' \subseteq S''$ then we know $S \cup S''$ is a minimum FVS in $G$. It follows that there is a minimum FVS in $G$ containing all vertices of $S \cup S'$, proving condition~\ref{thm:item:fvs-safe}.
To prove condition~\ref{thm:item:antlers} suppose~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler-sequence of width at most~$k$ for~$G$. We first prove the following:
\begin{claim} \label{claim:antler-sequence}
There exists a $z$-antler-sequence~$C_1', F_1', \ldots, C_t', F_t'$ of width at most~$k$ for~$G'$
such that
\begin{enumerate}
\item \label{item:vsets} $C_{\leq t}' \cup F_{\leq t}' = V(G') \cap (C_{\leq t} \cup F_{\leq t})$ and
\item \label{item:cutsize} $|C_{\leq t}'| = \sum_{1 \leq i \leq t} |C_i| - |(C_i \cup F_i) \cap S|$.
\end{enumerate}
\end{claim}
\begin{claimproof}
We use induction on~$t$. Since $G'$ and $S$ are obtained through an antler-safe operation and $(C_1,F_1)$ is a $z$-antler in $G$, we know that $G'$ contains a $z$-antler $(C_1',F_1')$ such that $C_1' \cup F_1' = (C_1 \cup F_1) \cap V(G')$ and $|C_1'| = |C_1| - |(C_1 \cup F_1) \cap S|$. The claim holds for $t=1$.
For the induction step, consider~$t > 1$. By applying induction to the length-$(t-1)$ prefix of the sequence, there is a $z$-antler sequence~$C_1',F_1', \ldots, C_{t-1}',F_{t-1}'$ of width at most~$k$ for~$G'$ such that both conditions hold.
We have by \cref{prop:antler-sequence} that~$(C_{\leq t}, F_{\leq t})$ is a $z$-antler in~$G$. Since~$G'$ and~$S$ are obtained through an antler-safe operation from $G$ there is a $z$-antler~$(C',F')$ in~$G'$ such that~$C' \cup F' = V(G') \cap (C_{\leq t} \cup F_{\leq t})$ and~$|C'| = |C_{\leq t}| - |S \cap (C_{\leq t} \cup F_{\leq t})|$. Take~$C_{t}' := C' \setminus (C_{<t}' \cup F_{<t}')$ and~$F_{t}' := F' \setminus (C_{<t}' \cup F_{<t}')$. By \cref{lem:antler_diff} we have that~$(C_{t}',F_{t}')$ is a $z$-antler in~$G'-(C_{<t}' \cup F_{<t}')$, it follows that~$C_1',F_1',\ldots,C_{t}',F_{t}'$ is a $z$-antler-sequence for~$G'$. We first show condition~\ref{item:vsets}.
\begin{align*}
C_{\leq t}' \cup F_{\leq t}'
&= C_{t}' \cup F_{t}' \cup C_{<t}' \cup F_{<t}' \\
&= C' \cup F'
&&\text{by choice of $C_{t}'$ and $F_{t}'$}\\
&= V(G') \cap (C_{\leq t} \cup F_{\leq t})
&&\text{by choice of $C'$ and $F'$.}
\end{align*}
To prove condition~\ref{item:cutsize} and the $z$-antler-sequence~$C_1',F_1',\ldots,C_{t}',F_{t}'$ has width at most~$k$ we first show $|C_{t}'| = |C_t| - |(C_t \cup F_t) \cap S|$. For this observe that $(C_{\leq t}',F_{\leq t}')$ is an antler in $G'$ by \cref{prop:antler-sequence}.
\begin{align*}
|C_{t}'|
&= |C_{\leq t}'| - |C_{<t}'|
&&\hspace{-7em}\text{since $C_i' \cap C_j' = \emptyset$ for all $i \neq j$}\\
&= \fvs(G'[C_{\leq t}' \cup F_{\leq t}']) - |C_{<t}'|
&&\hspace{-7em}\text{by the above} \\
&= \fvs(G'[V(G') \cap (C_{\leq t} \cup F_{\leq t})]) - |C_{<t}'|
&&\hspace{-7em}\text{by condition~\ref{item:vsets}} \\
&= \fvs(G'[C' \cup F']) - |C_{<t}'|
&&\hspace{-7em}\text{by choice of $C'$ and $F'$}\\
&= |C'| - |C_{<t}'|
&&\hspace{-7em}\text{since $(C',F')$ is an antler in $G'$} \\
&= |C'| - \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)
&&\hspace{-7em}\text{by induction}\\
&= |C_{\leq t}| - | S \cap (C_{\leq t} \cup F_{\leq t})|
- \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)\\
&= \sum_{1 \leq i \leq t} (|C_i| - |S \cap (C_i \cup F_i)|)
- \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)\\
&&&\hspace{-11em}\text{since $C_1,F_1,\ldots,C_t,F_t$ are pairwise disjoint}\\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S|
\end{align*}
We know the $z$-antler-sequence~$C_1',F_1',\ldots,C_{t-1},F_{t-1}$ has width at most~$k$, so to show that this $z$-antler-sequence has width at most~$k$ it suffices to prove that~$|C_{t}'| \leq k$. Indeed~$|C_{t}'| = |C_{t}| - |(C_{t} \cup F_{t}) \cap S| \leq |C_{t}| \leq k$.
To complete the proof of \cref{claim:antler-sequence} we now derive condition~\ref{item:cutsize}:
\begin{align*}
|C_{\leq t}'|
&= |C_{t}'| + |C_{<t}'|
&&\text{since $C_{t}' \cap C_{<t}' = \emptyset$}\\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S| + |C_{<t}'| \\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S| + \sum_{1 \leq i \leq t-1} (|C_i| - |(C_i \cup F_i) \cap S|)
&&\text{by induction}\\
&= \sum_{1 \leq i \leq t} (|C_i| - |(C_i \cup F_i) \cap S|). &&\qedhere
\end{align*}
\end{claimproof}
To complete the proof of condition~\ref{thm:item:antlers} from \cref{thm:main} we show $|S \cup S'| \geq |C_{\leq t}|$. By \cref{claim:antler-sequence} we know a $z$-antler-sequence~$C_1',F_1',\ldots,C_t',F_t'$ of width at most~$k$ for $G'$ exists. Since $S'$ is obtained from a recursive call we have~$|S'| \geq |C_{\leq t}'|$, so then
\begin{align*}
|S \cup S'|
&= |S| + |S'| \\
&\geq |S| + |C_{\leq t}'| \\
&= |S| + \sum_{1 \leq i \leq t} ( |C_i| - |(C_i \cup F_i) \cap S| )
&&\text{by \cref{claim:antler-sequence}}\\
&= |S| + \sum_{1 \leq i \leq t} |C_i| - \sum_{1 \leq i \leq t} |(C_i \cup F_i) \cap S| \\
&= |S| + |C_{\leq t}| - |S \cap (C_{\leq t} \cup F_{\leq t})|
&&\text{since $C_1', F_1', \ldots, C_t', F_t'$ are disjoint}\\
&= |S| + |C_{\leq t}| - |S|\\
&\geq |C_{\leq t}|.
&&\qedhere
\end{align*} \end{proof}
As a corollary to this theorem, we obtain a new type of parameterized-tractability result for \textsc{Feedback Vertex Set}. For an integer~$z$, let the $z$-antler complexity of \textsc{fvs} on~$G$ be the minimum number~$k$ for which there exists a (potentially long) sequence~$C_1, F_1, \ldots, C_t, F_t$ of disjoint vertex sets such that for all~$1 \leq i \leq t$, the pair~$(C_{i}, F_{i})$ is a $z$-antler of width at most~$k$ in~$G - (C_{<i} \cup F_{<i})$, and such that~$G - (C_{\leq t} \cup F_{\leq t})$ is acyclic (implying that~$C_{\leq t}$ is a feedback vertex set in~$G$). If no such sequence exists, the $z$-antler complexity of~$G$ is~$+\infty$.
Intuitively, \cref{cor:main} states that optimal solutions can be found efficiently when they are composed out of small pieces, each of which has a low-complexity certificate for belonging to some optimal solution.
\begin{restatable}{corollary}{maincor} \label{cor:main} There is an algorithm that, given a graph~$G$, returns an optimal feedback vertex set in time~$f(k^*) \cdot n^{\ensuremath{\mathcal{O}}\xspace(z^*)}$, where~$(k^*,z^*)$ is any pair of integers such that the $z^*$-antler complexity of~$G$ is at most~$k^*$. \end{restatable} \begin{proof} Let~$(k^*,z^*)$ be such that the $z^*$-antler complexity of~$G$ is at most~$k^*$. Let~$p_1 \in \ensuremath{\mathcal{O}}\xspace(k^5 z^2), p_2 \in \ensuremath{\mathcal{O}}\xspace(z)$ be concrete functions such that the running time of \cref{thm:main} is bounded by~$2^{p_1(k,z)} \cdot n^{p_2(z)}$. Consider the pairs~$\{(k',z') \in \ensuremath{\mathbb{N}}\xspace^2 \mid 1 \leq z' \leq k' \leq n\}$ in order of increasing value of the running-time guarantee~$2^{p_1(k,z)} \cdot n^{p_2(z)}$. For each such pair~$(k',z')$, start from the graph~$G$ and invoke \cref{thm:main} to obtain a vertex set~$S$ which is guaranteed to be contained in an optimal solution. If~$G - S$ is acyclic, then~$S$ itself is an optimal solution and we return~$S$. Otherwise we proceed to the next pair~$(k',z')$.
\subparagraph*{Correctness}
The correctness of \cref{thm:main} and the definition of $z$-antler complexity ensure that for~$(k',z') = (k^*,z^*)$, the set~$S$ is an optimal solution. In particular, if~$C_1, F_1, \ldots, C_t, F_t$ is a sequence of vertex sets witnessing that the $z^*$-antler complexity of~$G$ is at most~$k^*$, then~\eqref{thm:item:antlers} of \cref{thm:main} is guaranteed to output a set~$S$ of size at least~$\sum _{1 \leq i \leq t} |C_i|$, which is equal to the size of an optimal solution on~$G$ by definition.
\subparagraph*{Running time} For a fixed choice of~$(k',z')$ the algorithm from \cref{thm:main} runs in time~$2^{\ensuremath{\mathcal{O}}\xspace((k')^5 (z')^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z')} \leq 2^{\ensuremath{\mathcal{O}}\xspace((k^*)^5 (z^*)^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z^*)}$ because we try pairs~$(k',z')$ in order of increasing running time. As we try at most~$n^2$ pairs before finding the solution, the corollary follows. \end{proof}
To conclude, we reflect on the running time of \cref{cor:main} compared to running times of the form~$2^{\ensuremath{\mathcal{O}}\xspace(\fvs(G))} \cdot n^{\ensuremath{\mathcal{O}}\xspace(1)}$ obtained by FPT algorithms for the parameterization by solution size. If we exhaustively apply \cref{lem:fvc-kernel} with the FVC~$(C,V(G)\setminus C)$, where~$C$ is obtained from a 2-approximation algorithm~\cite{BafnaBF99}, then this gives an \emph{antler-safe} kernelization: it reduces the graph as long as the graph is larger than~$f_r(|C|)$. This opening step reduces the instance size to~$\ensuremath{\mathcal{O}}\xspace(\fvs(G)^3)$ without increasing the antler complexity. As observed before, after applying~$\ensuremath{\mathcal{O}}\xspace(n^2)$ operations we obtain a graph in which no operations can be applied. This leads to a running time of~$\ensuremath{\mathcal{O}}\xspace(n^4)$ of the kernelization. Running \cref{thm:main} to solve the reduced instance yields a total running time of~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \fvs(G)^{\ensuremath{\mathcal{O}}\xspace(z)} + \ensuremath{\mathcal{O}}\xspace(n^4)$. This is asymptotically faster than~$2^{\ensuremath{\mathcal{O}}\xspace(\fvs(G))}$ when $z \leq k = o(\sqrt[7]{\fvs(G)})$ and $\fvs(G) = \omega(\log n)$, which captures the intuitive idea sketched above that our algorithmic approach has an advantage when there is an optimal solution that is large but composed of small pieces for which there are low-complexity certificates.
\section{Conclusion} We have taken the first steps of a research program to investigate how and when a preprocessing phase can guarantee to identify parts of an optimal solution to an NP-hard problem, thereby reducing the search space of the follow-up algorithm. Aside from the technical results concerning antler structures for \textsc{Feedback Vertex Set} and their algorithmic properties, we consider the conceptual message of this research program an important contribution of our theoretical work on understanding the power of preprocessing and the structure of solutions to NP-hard problems.
This line of investigation opens up a host of opportunities for future research. For combinatorial problems such as \textsc{Vertex Planarization}, \textsc{Odd Cycle Transversal}, and \textsc{Directed Feedback Vertex Set}, which kinds of substructures in inputs allow parts of an optimal solution to be identified by an efficient preprocessing phase? Is it possible to give preprocessing guarantees not in terms of the size of an optimal solution, but in terms of measures of the stability~\cite{AngelidakisABCD19,AwasthiBS12,DanielyLS12} of optimal solutions under small perturbations? Some questions also remain open concerning the concrete technical results in the paper. Can the running time of \cref{thm:z:antler} be improved to~$f(k) \cdot n^{\ensuremath{\mathcal{O}}\xspace(1)}$? We conjecture that it cannot, but have not been able to prove this. A related question applies to \textsc{Vertex Cover}: Is there an algorithm running in time~$f(k) \cdot n^{\ensuremath{\mathcal{O}}\xspace(1)}$ that, given a graph~$G$ which has disjoint vertex sets~$(C,H)$ such that~$N_G(C) \subseteq H$ and~$H$ of size~$k$ is an optimal vertex cover in~$G[C \cup H]$, outputs a set of size at least~$k$ that is part of an optimal vertex cover in~$G$? (Note that this is an easier target than computing such a decomposition of width~$k$ if one exists, which can be shown to be W[1]-hard.)
To apply the theoretical ideas on antlers in the practical settings that motivated their investigation, it would be interesting to determine which types of antler can be found in \emph{linear} time. One can show that a slight extension of the standard reduction rules~\cite[FVS.1--FVS.5]{CyganFKLMPPS15} for \textsc{Feedback Vertex Set} can be used to detect 1-antlers of width~$1$ in linear time. Can the running time of \cref{thm:1:antler} be improved to~$f(k) \cdot (n+m)$? It would also be interesting to investigate which types of antlers are present in practical inputs.
\appendix
\section{NP-hardness of finding 1-antlers} \label{sec:np-hard}
\begin{lemma}[{\cite[Lemma 2 for~$H = K_3$]{DonkersJ19}}] \label{lemma:reduction:disjointcycles} There is a polynomial-time algorithm that, given a CNF formula~$\Phi$, outputs a graph~$G$ and a collection~$\mathcal{H} = \{H_1, \ldots, H_\ell\}$ of vertex-disjoint cycles in~$G$, such that~$\Phi$ is satisfiable if and only if~$G$ has a feedback vertex set of size~$\ell$. \end{lemma}
\begin{definition}\label{def:1antler} A 1-antler in an undirected multigraph~$G$ is a pair of disjoint vertex sets~$(C,F)$ such that: \begin{enumerate}
\item $G[F]$ is acyclic,\label{def:1antler:forest}
\item each tree~$T$ of the forest~$G[F]$ is connected to~$V(G) \setminus (C \cup F)$ by at most one edge, and\label{def:1antler:edgeleaving}
\item the graph~$G[C \cup F]$ contains~$|C|$ vertex-disjoint cycles.\label{def:1antler:cycles} \end{enumerate} A 1-antler is called \emph{non-empty} if~$H \cup C \neq \emptyset$. \end{definition}
Will use the following easily verified consequence of this definition.
\begin{observation} \label{obs:1antler:cycleWithF} If~$(C,F)$ is a 1-antler in an undirected multigraph~$G$, then for each vertex~$c \in C$ the graph~$G[\{c\} \cup F]$ contains a cycle. \end{observation}
In the terminology of Section~\ref{sec:intro}, the cut~$C$ corresponds to~$\ensuremath{\mathsf{head}}\xspace$ and the forest~$F$ corresponds to~$\ensuremath{\mathsf{antler}}\xspace$. Observe that this self-contained definition of a 1-antler is equivalent to the general definition of $z$-antler from Section~\ref{sec:def:cuts:antlers} for~$z=1$.
\begin{corollary} Assuming \ensuremath{\mathsf{P}}~$\neq$~\ensuremath{\mathsf{NP}}\xspace, there is no polynomial-time algorithm that, given a graph~$G$, outputs a non-empty 1-antler in~$G$ or concludes that no non-empty 1-antler exists. \end{corollary} \begin{proof} We need the following simple claim.
\begin{claim} If~$G$ contains a packing of~$\ell \geq 1$ vertex-disjoint cycles and a feedback vertex set of size~$\ell$, then~$G$ admits a non-empty 1-antler. \end{claim} \begin{claimproof} Let~$C$ be a feedback vertex set in~$G$ of size~$\ell$ and let~$F := V(G) \setminus C$. \end{claimproof}
Now suppose there is a polynomial-time algorithm to find a non-empty 1-antler decomposition, if one exists. We use it to solve CNF-SAT in polynomial time. Given an input formula~$\Phi$ for CNF-SAT, use Lemma~\ref{lemma:reduction:disjointcycles} to produce in polynomial time a graph~$G$ and a packing~$\mathcal{H}$ of~$\ell$ vertex-disjoint cycles in~$G$, such that~$\Phi$ is satisfiable if and only if~$\fvs(G) = \ell$. The following recursive polynomial-time algorithm correctly tests, given a graph~$G$ and a packing~$\mathcal{H}$ of some~$\ell \geq 0$ vertex-disjoint cycles in~$G$, whether~$\fvs(G) = \ell$.
\begin{enumerate}
\item If~$\ell = 0$, then output \textsc{yes} if and only if~$G$ is acyclic.
\item If~$\ell > 0$, run the hypothetical algorithm to find a non-empty 1-antler~$(C,F)$.
\begin{enumerate}
\item If a non-empty 1-antler~$(C,F)$ is returned, then let~$\mathcal{H'}$ consist of those cycles in the packing not intersected by~$C$, and let~$\ell' := |\mathcal{H'}|$. Return the result of recursively running the algorithm on~$G' := G - (C \cup F)$ and~$\mathcal{H'}$ to test whether~$G'$ has a feedback vertex set of size~$|\mathcal{H'}|$.
\item Otherwise, return \textsc{no}.
\end{enumerate} \end{enumerate}
The claim shows that the algorithm is correct when it returns \textsc{no}. Observation~\ref{obs:remove-antler} shows that if we recurse, we have~$\fvs(G) = \ell$ if and only if~$\fvs(G - (C \cup H)) = \ell'$; hence the result of the recursion is the correct output. Since the number of vertices in the graph reduces by at least one in each iteration, the overall running time is polynomial assuming the hypothetical algorithm to compute a non-empty 1-antler. Hence using Lemma~\ref{lemma:reduction:disjointcycles} we can decide CNF-SAT in polynomial time. \end{proof}
\begin{corollary} \label{cor:lp-relax} It is NP-complete to determine whether~$\fvs(G) = \fvs LP(G)$. Here~$\fvs(G)$ denotes the minimum size of a feedback vertex set in~$G$, and~$\fvs LP(G)$ denotes the minimum cost of a solution to the linear programming relaxation of \textsc{Feedback Vertex Set} on~$G$. \end{corollary} \begin{proof} Membership in NP is trivial; we prove hardness. Suppose such an algorithm exists. As above, we use it to solve CNF-SAT in polynomial time. Given an input~$\Phi$ for SAT, use Lemma~\ref{lemma:reduction:disjointcycles} to produce in polynomial time a graph~$G$ and packing~$\mathcal{H}$ of~$\ell$ vertex-disjoint cycles in~$G$, such that~$\Phi$ is satisfiable if and only if~$\fvs(G) = \ell$.
Compute the cost~$c$ of an optimal solution to the linear programming relaxation of \textsc{Feedback Vertex Set} on~$G$, using the ellipsoid method. By the properties of a relaxation, if~$c > \ell$ then~$\fvs(G) > \ell$, and hence we can safely report that~$\Phi$ is unsatisfiable. If~$c \leq \ell$, then the existence of~$\ell$ vertex-disjoint cycles in~$G$ implies that~$c = \ell$. Run the hypothetical algorithm to test whether~$\fvs(G) = \fvs LP(G)$. If the answer is \textsc{yes}, then~$G$ has a feedback vertex set of size~$\ell$ and hence~$\Phi$ is satisfiable; if not, then~$\Phi$ is unsatisfiable. \end{proof}
\section{W[1]-hardness of finding bounded-width 1-antlers} \label{sec:w1-hard}
We consider the following parameterized problem.
\defparproblem{Bounded-Width 1-Antler Detection} {An undirected multigraph~$G$ and an integer~$k$.} {$k$.}
{Does~$G$ admit a non-empty 1-antler~$(C,F)$ with~$|C| \leq k$?}
We prove that \textsc{Bounded-Width 1-Antler Detection} is W[1]-hard by a reduction from \textsc{Multicolored Clique}, which is defined as follows.
\defparproblem{Multicolored Clique} {An undirected simple graph~$G$, an integer~$k$, and a partition of~$V(G)$ into sets~$V_1, \ldots, V_k$.} {$k$.}
{Is there a clique~$S$ in~$G$ such that for each~$1 \leq i \leq k$ we have~$|S \cap V_i| = 1$?}
The sets~$V_i$ are referred to as \emph{color classes}, and a solution clique~$S$ is called a \emph{multicolored clique}. It is well-known that \textsc{Multicolored Clique} is W[1]-hard (cf.~\cite[Thm.~13.25]{CyganFKLMPPS15}). Our reduction is inspired by the W[1]-hardness of detecting a Hall set~\cite[Exercise 13.28]{CyganFKLMPPS15}. In the proof, we use the following shorthand notation: for a positive integer~$n$, we denote by~$[n]$ the set~$\{1, \ldots, n\}$.
\begin{theorem} \label{thm:1antler:w1hard} \textsc{Bounded-Width 1-Antler Detection} is W[1]-hard. \end{theorem} \begin{proof}
We give a parameterized reduction~\cite[Def.~13.1]{CyganFKLMPPS15} from the \textsc{Multicolored Clique} problem. By inserting isolated vertices if needed, we may assume without loss of generality that for the input instance~$(G,k,V_1, \ldots, V_k)$ we have~$|V_1| = |V_2| = \ldots = |V_k| = n$ for some~$n \geq k(k-1) + 4$. For each~$i \in [k]$, fix an arbitrary labeling of the vertices in~$V_i$ as~$v_{i,j}$ for~$j \in [n]$. Given this instance, we construct an input~$(G', k')$ for \textsc{Bounded-Width 1-Antler Detection} as follows. \begin{enumerate}
\item For each~$i \in [k]$, for each~$j \in [n]$, create a set~$U_{i,j} = \{u_{i,j,\ell} \mid \ell \in [k] \setminus \{i\}\}$ of vertices in~$G'$ to represent~$v_{i,j}$. Intuitively, vertex~$u_{i,j,\ell}$ represents the connection that the~$j$th vertex from the~$i$th color class should have to the neighbor in the $\ell$th color class chosen in the solution clique.
\item Define~$\mathcal{U} := \bigcup _{i \in [k]} \bigcup _{j \in [n]} U_{i,j}$. Insert (single) edges to turn~$\mathcal{U}$ into a clique in~$G'$.
\item \label{w1hard:w} For each edge~$e$ in~$G$ between vertices of different color classes, let~$e = \{v_{i,j}, v_{i',j'}\}$ with~$i < i'$, and insert two vertices into~$G'$ to represent~$e$:
\begin{itemize}
\item Insert a vertex~$w_e$, add an edge from~$w_e$ to each vertex in~$U_{i,j} \cup U_{i', j'}$, and then add a second edge between~$w_e$ and~$u_{i,j,i'}$.
\item Insert a vertex~$w_{e'}$, add an edge from~$w_{e'}$ to each vertex in~$U_{i,j} \cup U_{i', j'}$, and then add a second edge between~$w_{e'}$ and~$u_{i',j',i}$.
\end{itemize}
Let~$W$ denote the set of vertices of the form~$w_e, w_{e'}$ inserted to represent an edge of~$G$. Observe that~$W$ is an independent set in~$G$.
\item Finally, insert a vertex~$u^*$ into~$G'$. Add a single edge from~$u^*$ to all other vertices of~$G'$, to make~$u^*$ into a universal vertex. \end{enumerate}
This concludes the construction of~$G'$. Note that~$G'$ contains double-edges, but no self-loops. We set~$k' := k(k-1)$, which is appropriately bounded for a parameterized reduction. It is easy to see that the reduction can be performed in polynomial time. It remains to show that~$G$ has a multicolored $k$-clique if and only if~$G'$ has a nonempty 1-antler of width at most~$k$. To illustrate the intended behavior of the reduction, we first prove the forward implication.
\begin{claim} \label{claim:w1hard:forward} If~$G$ has a multicolored clique of size~$k$, then~$G'$ has a non-empty 1-antler of width~$k'$. \end{claim} \begin{claimproof} Suppose~$S$ is a multicolored clique of size~$k$ in~$G$. Choose indices~$j_1, \ldots, j_k$ such that~$S \cap V_i = \{v_{i,j_i}\}$ for all~$i \in [k]$. Define a 1-antler~$(C,F)$ in~$G'$ as follows: \begin{itemize}
\item $C = \bigcup _{i \in [k]} U_{i, j_i}$.
\item $F = \{w_e, w_{e'} \mid e \text{~is an edge in~$G$ between distinct vertices of~$S$}\}$. \end{itemize}
Since each set~$U_{i,j_i}$ has size~$k-1$, it follows that~$|C| = k(k-1) = k'$. Since~$F \subseteq W$ is an independent set in~$G'$, it also follows that~$G'[F]$ is acyclic. Each tree~$T$ in the forest~$G'[F]$ consists of a single vertex~$w_e$ or~$w_{e'}$. By construction, there is exactly one edge between~$T$ and~$V(G') \setminus (C \cup F)$; this is the edge to the universal vertex~$u^*$. It remains to verify that~$G'[C \cup F]$ contains~$|C|$ vertex-disjoint cycles, each containing exactly one vertex of~$C$. Consider an arbitrary vertex~$u_{i, j_i, \ell}$ in~$C$; we show we can assign it a cycle in~$G'[C \cup F]$ so that all assigned cycles are vertex-disjoint. Since~$S$ is a clique, there is an edge~$e$ in~$G$ between~$v_{i, j_i}$ and~$v_{\ell, j_\ell}$, and the corresponding vertices~$w_e, w_{e'}$ are in~$F$. If~$i < \ell$, then~$w_e \in F$ and there are two edges between~$u_{i, j_i, \ell}$ and~$w_e$, forming a cycle on two vertices. If~$i > \ell$, then there is a cycle on two vertices~$u_{i, j_i, \ell}$ and~$w_{e'}$. Since for any vertex of the form~$w_e$ or~$w_{e'}$ there is a unique vertex of~$C$ that it has a double-edge to, the resulting cycles are indeed vertex-disjoint. This proves that~$(C,F)$ is a 1-antler of width~$k'$. \end{claimproof}
Before proving reverse implication, we establish some structural claims about the structure of 1-antlers in~$G'$.
\begin{claim} \label{claim:w1hard:antlers}
If~$(C,F)$ is a non-empty 1-antler in~$G'$ with~$|C| \leq k'$, then the following holds: \begin{enumerate}
\item $\mathcal{U} \cap F = \emptyset$.\label{w1:antler:u}
\item $u^* \notin C \cup F$.\label{w1:antler:ustar}
\item $W \cap C = \emptyset$.\label{w1:antler:w}
\item $C \subseteq \mathcal{U}$,~$F \subseteq W$, and each tree of the forest~$G'[F]$ consists of a single vertex.\label{w1:antler:subsets}
\item For each vertex~$w \in F$ we have~$N_{G'}(w) \cap \mathcal{U} \subseteq C$.\label{w1:antler:neighborhoodinC}
\item $F \neq \emptyset$.\label{w1:antler:fnonempty}
\end{enumerate} \end{claim} \begin{claimproof}
(\ref{w1:antler:u}) Assume for a contradiction that there is a vertex~$u_{i,j,\ell} \in \mathcal{U} \cap F$. Since~$G'[F]$ is a forest by Property~\eqref{def:1antler:forest} of Definition~\ref{def:1antler}, while~$\mathcal{U}$ is a clique in~$G'$, it follows that~$|F \cap \mathcal{U}| \leq 2$. By Property~\eqref{def:1antler:edgeleaving}, for a vertex in~$F$, there is at most one of its neighbors that belongs to neither~$F$ nor~$C$. Since~$|\mathcal{U}| \geq n \geq k(k-1) + 4$, and~$u_{i,j,\ell} \in F$ is adjacent to all other vertices of~$\mathcal{U}$ since that set forms a clique, the fact that~$|F \cap \mathcal{U}| \leq 2$ implies that~$|C \cap \mathcal{U}| \geq |\mathcal{U}| - 2 - 1 \geq k(k-1) + 4 - 3 > k'$. So~$|C| > k'$, which contradicts that~$(C,F)$ is a 1-antler with~$|C| \leq k'$.
(\ref{w1:antler:ustar}) Since~$u^*$ is a universal vertex in~$G'$, the set~$\mathcal{U} \cup \{u^*\}$ is a clique and the preceding argument shows that~$u^* \notin F$. To prove the claim we show that additionally,~$u^* \notin C$. Assume for a contradiction that~$u^* \in C$. By Observation~\ref{obs:1antler:cycleWithF}, the graph~$G'[\{u^*\} \cup F]$ contains a cycle. Since~$W$ is an independent set in~$G'$ and~$u^*$ is not incident on any double-edges, the graph~$G'[\{u^*\} \cup W]$ is acyclic. Hence to get a cycle in~$G'[\{u^*\} \cup F]$, the set~$F$ contains at least one vertex that is not in~$W$ and not~$u^*$; hence this vertex belongs to~$\mathcal{U}$. So~$\mathcal{U} \cap F \neq \emptyset$; but this contradicts Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:u}.
(\ref{w1:antler:w}) Assume for a contradiction that~$w \in W \cap C$. Again by Observation~\ref{obs:1antler:cycleWithF}, there is a cycle in~$G'[\{w\} \cup F]$, and since~$G'$ does not have any self-loops this implies~$N_{G'}(w) \cap F \neq \emptyset$. But by construction of~$G'$ we have~$N_{G'}(w) \subseteq \mathcal{U} \cup \{u^*\}$, so~$F$ contains a vertex of either~$\mathcal{U}$ or~$u^*$. But this contradict either Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:ustar} or Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:w}.
(\ref{w1:antler:subsets}) Since the sets~$\mathcal{U}, W, \{u^*\}$ form a partition of~$V(G')$, the preceding subclaims imply~$C \subseteq \mathcal{U}$ and~$F \subseteq W$. Since~$W$ is an independent set in~$G'$, this implies that each tree~$T$ of the forest~$G'[W]$ consists of a single vertex.
(\ref{w1:antler:neighborhoodinC}) Consider a vertex~$w \in F$, which by itself forms a tree in~$G'[F]$. Since~$u^* \notin C \cup F$, the edge between~$T$ and~$u^*$ is the unique edge connecting~$T$ to a vertex of~$V(G') \setminus (C \cup F)$, and therefore all neighbors of~$T$ other than~$u^*$ belong to~$C \cup F$. Since a vertex~$w \in W$ has~$N_{G'}(w) \subseteq \mathcal{U} \cup \{u^*\}$, it follows that~$N_{G'}(w) \cap \mathcal{U} \subseteq C$.
(\ref{w1:antler:fnonempty}) By the assumption that~$(C,F)$ is non-empty, we have~$C \cup F \neq \emptyset$. This implies that~$F \neq \emptyset$: if~$C$ would contain a vertex~$c$ while~$F = \emptyset$, then by Observation~\ref{obs:1antler:cycleWithF} the graph~$G'[\{c\} \cup F] = G'[\{c\}]$ would contain a cycle, which is not the case since~$G'$ has no self-loops. Hence~$F \neq \emptyset$. \end{claimproof}
With these structural insights, we can prove the remaining implication.
\begin{claim} \label{claim:w1hard:reverse}
If~$G'$ has a non-empty 1-antler~$(C,F)$ with~$|C| \leq k'$, then~$G$ has a multicolored clique of size~$k$. \end{claim} \begin{claimproof}
Let~$(C,F)$ be a non-empty 1-antler in~$G'$ with~$|C| \leq k'$. By Claim~\ref{claim:w1hard:antlers} we have~$C \subseteq \mathcal{U}$, while~$F \subseteq W$ and~$F \neq \emptyset$. Consider a fixed vertex~$w \in F$. Since~$F \subseteq W$, vertex~$w$ is of the form~$w_e$ or~$w_{e'}$ constructed in Step~\ref{w1hard:w} to represent some edge~$e$ of~$G$. Choose~$i^* \in [k], j^* \in [n]$ such that~$v_{i^*,j^*} \in V_{i^*}$ is an endpoint of edge~$e$ in~$G$. By construction we have~$U_{i^*,j^*} \subseteq N_{G'}(w)$ and therefore Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:neighborhoodinC} implies~$U_{i^*,j^*} \subseteq C$.
Consider an arbitrary~$\ell \in [k] \setminus \{i^*\}$. Then~$u_{i^*,j^*,\ell} \in U_{i^*,j^*} \subseteq C$, so by Observation~\ref{obs:1antler:cycleWithF} the graph~$G'[\{u_{i^*,j^*,\ell}\} \cup F]$ contains a cycle. Since~$G'[F]$ is an independent set and~$G'$ has no self-loops, this cycle consists of two vertices joined by a double-edge. By construction of~$G'$, such a cycle involving~$u_{i^*,j^*,\ell}$ exists only through vertices~$w_e$ or~$w_{e'}$ where~$e$ is an edge of~$G$ connecting~$v_{i^*,j^*}$ to a neighbor in class~$V_{\ell}$. Consequently,~$F$ contains a vertex~$w$ that represents such an edge~$e$. Let~$v_{\ell, j_\ell}$ denote the other endpoint of~$e$. Then~$N_{G'}(w) \supseteq U_{i^*,j^*} \cup U_{\ell, j_\ell}$, and by Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:neighborhoodinC} we therefore have~$U_{\ell, j_\ell} \subseteq C$.
Applying the previous argument for all~$\ell \in [k] \setminus \{i^*\}$, together with the fact that~$U_{i^*,j^*} \subseteq C$, we find that for each~$i \in [k]$ there exists a value~$j_i$ such that~$U_{i, j_i} \subseteq C$. Since~$|C| \leq k(k-1)$ while each such set~$U_{i, j_i}$ has size~$k-1$, it follows that the choice of~$j_i$ is uniquely determined for each~$i \in [k]$, and that there are no other vertices in~$C$. To complete the proof, we argue that the set~$S = \{ v_{i, j_i} \mid i \in [k] \}$ is a clique in~$G$.
Consider an arbitrary pair of distinct vertices~$v_{i, j_i}, v_{i', j_{i'}}$ in~$S$, and choose the indices such that~$i < i'$. We argue that~$G$ contains an edge~$e$ between these vertices, as follows. Since~$u_{i, j_i, i'} \in U_{i, j_i} \subseteq C$, by Observation~\ref{obs:1antler:cycleWithF} the graph~$G'[\{u_{i, j_i, i'}\} \cup F]$ contains a cycle. As argued above, the construction of~$G'$ and the fact that~$F \subseteq W$ ensure that this cycle consists of~$u_{i, j_i, i'}$ joined to a vertex in~$F$ by a double-edge. By Step~\ref{w1hard:w} and the fact that~$i < i'$, this vertex is of the form~$w_e$ for an edge~$e$ in~$G$ connecting~$v_{i, j_i}$ to a vertex~$v_{i',j'}$ in~$V_{i'}$. By construction of~$G'$ we have~$U_{i', j'} \subseteq N_{G'}(w_e)$, and then~$w_e \in F$ implies by Claim~\ref{claim:w1hard:antlers}\eqref{w1:antler:neighborhoodinC} that~$U_{i', j'} \subseteq C$. Since we argued above that for index~$i'$ there is a unique choice~$j_{i'}$ with~$U_{i', j_{i'}} \subseteq C$, we must have~$j' = j_{i'}$. Hence the vertex~$w_e$ contained in~$F$ represents the edge of~$G$ between~$v_{i, j_i}$ and~$v_{i', j_{i'}}$ in~$G$, which proves in particular that the edge exists. As the choice of vertices was arbitrary, this shows that~$S$ is a clique in~$G$. As it contains exactly one vertex from each color class, graph~$G$ has a multicolored clique of size~$k$. \end{claimproof}
Claims~\ref{claim:w1hard:forward} and~\ref{claim:w1hard:reverse} show that the instance~$(G,k)$ of \textsc{Multicolored Clique} is equivalent to the instance~$(G',k')$ of \textsc{Bounded-Width 1-Antler Detection}. This concludes the proof of Theorem~\ref{thm:1antler:w1hard}. \end{proof}
Observe that the proof of Theorem~\ref{thm:1antler:w1hard} shows that the variant of \textsc{Bounded-Width 1-Antler Detection} where we ask for the existence of a 1-antler of width \emph{exactly}~$k$, is also W[1]-hard.
\allowdisplaybreaks
\iffalse This section has been complete moved back into the main text
\section{Omitted proofs from \autoref{sec:prelims}} \lemAltFVS* \begin{proof}
We prove the existence of such a set~$X$ and $v$-flower by induction on~$|V(G)|$. The inductive proof can easily be translated into a linear-time algorithm. If~$G$ is acyclic, output~$X = \emptyset$ and a $v$-flower of order~$0$. Otherwise, since~$v$ does not have a self-loop there is a tree~$T$ of the forest~$G - v$ such that~$G[V(T) \cup \{v\}]$ contains a cycle. Root~$T$ at an arbitrary vertex and consider a deepest node~$x$ in~$T$ for which the graph~$G[V(T_x) \cup \{v\}]$ contains a cycle~$C$. Then any feedback vertex set of~$G$ that does not contain~$v$, has to contain at least one vertex of~$T_x$; and the choice of~$x$ as a deepest vertex implies that~$x$ lies on all cycles of~$G$ that intersect~$T_x$. By induction on~$G' := G - V(T_x)$ and~$v$, there is a feedback vertex set~$X' \subseteq V(G') \setminus \{v\}$ of~$G'$ and a $v$-flower in~$G'$ of order~$|X'|$. We obtain a $v$-flower of order~$|X'| + 1$ in~$G$ by adding~$C$, while~$X := X' \cup \{x\} \subseteq V(G) \setminus \{v\}$ is a feedback vertex set of size~$|X'| + 1$. \end{proof}
\fi
\iffalse This section has been complete moved back into the main text
\section{Omitted proofs from \autoref{sec:def:cuts:antlers}} \intersectionantlers* \begin{proof}
We show $\fvs(G[F_1 \cup F_2]) = |C_1 \cap F_2|$.
First we show $\fvs(G[F_1 \cup F_2]) \geq |C_1 \cap F_2|$ by showing $(C_1 \cap F_2, F_1)$ is an antler in $G[F_1 \cup F_2]$. Clearly $(C_1,F_1)$ is an antler in $G[F_1 \cup F_2 \cup C_1]$, so then by \cref{obs:subantler} $(C_1 \cap F_2,F_1)$ is an antler in $G[F_1 \cup F_2 \cup C_1] - (C_1 \setminus F_2) = G[F_1 \cup F_2]$.
Second we show $\fvs(G[F_1 \cup F_2]) \leq |C_1 \cap F_2|$ by showing $G[F_1 \cup F_2] - (C_1 \cap F_2)$ is acyclic. Note that $G[F_1 \cup F_2] - (C_1 \cap F_2) = G[F_1 \cup F_2] - C_1$.
Suppose $G[F_1 \cup F_2] - C_1$ contains a cycle. We know this cycle does not contain a vertex from $C_1$, however it does contain at least one vertex from $F_1$ since otherwise this cycle exists in $G[F_2]$ which is a forest.
We know from \cref{obs:fvc-basics} that any cycle in $G$ containing a vertex from $F_1$ also contains a vertex from $C_1$. Contradiction. The proof for~$\fvs(G[F_1 \cup F_2]) = |C_2 \cap F_1|$ is symmetric. It follows that~$|C_1 \cap F_2| = \fvs(G[F_1 \cup F_2]) = |C_2 \cup F_1|$. \end{proof}
The following proposition is needed for the proof of \cref{lem:antler_diff}.
\begin{restatable}{proposition}{intersectingantlers} \label{prop:intersecting_antlers}
If $(C_1,F_1)$ and $(C_2,F_2)$ are antlers in $G$, then $(C_1 \setminus (C_2 \cup F_2), F_1 \setminus (C_2 \cup F_2))$ is an antler in $G - (C_2 \cup F_2)$. \end{restatable}
\begin{proof}
For brevity let $C_1' := C_1 \setminus (C_2 \cup F_2)$ and $F_1' := F_1 \setminus (C_2 \cup F_2)$ and $G' := G - (C_2 \cup F_2)$. First note that $(C_1', F_1')$ is a FVC in $G'$ by \cref{obs:subfvc}. We proceed to show that $\fvs(G'[C_1' \cup F_1']) \geq |C_1'|$. By \cref{obs:subantler} $(\emptyset,F_2)$ is an antler in $G-C_2$, so then by \cref{obs:subfvc} we have $(\emptyset, F_2 \cap (C_1 \cup F_1))$ is a FVC in $G[C_1 \cup F_1] - C_2$. Since a FVC of width $0$ is an antler we can apply \cref{obs:remove-antler} and obtain $\fvs(G[C_1 \cup F_1] - C_2) = \fvs(G[C_1 \cup F_1] - (C_2 \cup F_2)) = \fvs(G'[C_1' \cup F_1'])$. We derive \begin{align*} \fvs(G'[C_1' \cup F_1']) &= \fvs(G[C_1 \cup F_1] - C_2)\\
&\geq \fvs(G[C_1 \cup F_1]) - |C_2 \cap (C_1 \cup F_1)|\\
&= |C_1| - |C_2 \cap C_1| - |C_2 \cap F_1| &\text{Since $C_1 \cap F_1 = \emptyset$}\\
&= |C_1| - |C_2 \cap C_1| - |C_1 \cap F_2| &\text{By \cref{prop:intersection_antlers}}\\
&= |C_1| - |(C_2 \cap C_1) \cup (C_1 \cap F_2)| &\text{Since $C_2 \cap F_2 = \emptyset$}\\
&= |C_1 \setminus (C_2 \cup F_2)| = |C_1'|. \tag*{\qedhere} \end{align*} \end{proof}
\antlerdiff* \begin{proof}
For brevity let $C_1' := C_1 \setminus (C_2 \cup F_2)$ and $F_1' := F_1 \setminus (C_2 \cup F_2)$ and $G' := G - (C_2 \cup F_2)$.
By \cref{prop:intersecting_antlers} we know $(C_1',F_1')$ is an antler, so it remains to show that $G'[C_1' \cup F_1']$ contains a $C_1'$-certificate of order $z$. Since $(C_1,F_1)$ is a $z$-antler in $G$, we have that $G[C_1 \cup F_1]$ contains a $C_1$-certificate of order $z$. Let $H$ denote this $C_1$-certificate and let $\overline{H}$ be the set of all edges and vertices in $G'[C_1' \cup F_1']$ that are not in $H$. Now $(C_1,F_1)$ is a $z$-antler in $G'' := G-\overline{H}$ since it is a FVC by \cref{obs:subfvc} and $G''[C_1 \cup F_1]$ contains a $C_1$-certificate of order $z$ since $H$ is a subgraph of $G''$. Note that $(C_2,F_2)$ is also an antler in $G''$ since $\overline{H}$ does not contain vertices or edges from $G[C_2 \cup F_2]$. It follows that $(C_1',F_1')$ is an antler in $G''$ by \cref{prop:intersecting_antlers}, so $G''[C_1' \cup F_1']$ is a $C_1'$-certificate in $G''$. Clearly this is a $C_1'$-certificate of order $z$ since $G''[C_1' \cup F_1']$ is a subgraph of $H$. Since $G''[C_1' \cup F_1']$ is a subgraph of $G'[C_1' \cup F_1']$ it follows that $G'[C_1' \cup F_1']$ contains a $C_1'$-certificate of order $z$. \end{proof}
\antlercombine* \begin{proof}
Since~$(C_1,F_1)$ is a $z$-antler in~$G$ we know~$G[C_1 \cup F_1]$ contains a $C_1$-certificate of order~$z$, similarly~$(G-(C_1 \cup F_1))[C_2 \cup F_2]$ contains a $C_2$-certificate of order~$z$. The union of these certificate forms a $(C_1 \cup C_2)$-certificate of order~$z$ in~$G[C_1 \cup C_2 \cup F_1 \cup F_2]$. It remains to show that~$(C_1 \cup C_2, F_1 \cup F_2)$ is a FVC in~$G$.
First we show~$G[F_1 \cup F_2]$ is acyclic. Suppose for contradiction that~$G[F_1 \cup F_2]$ contains a cycle~$\mathcal{C}$. Since~$(C_1,F_1)$ is a FVC in~$G$, any cycle containing a vertex from~$F_1$ also contains a vertex from~$C_1$, hence~$\mathcal{C}$ does not contain vertices from~$F_1$. Therefore~$\mathcal{C}$ can only contain vertices from $F_2$. This is a contradiction with the fact that $G[F_2]$ is acyclic.
Finally we show that for each tree~$T$ in~$G[F_1 \cup F_2]$ we have $e(T,G-(C_1 \cup C_2 \cup F_1 \cup F_2)) \leq 1$. If~$V(T) \subseteq F_2$ this follows directly from the fact that~$(C_2,F_2)$ is a FVC in~$G-(C_1 \cup F_1)$. Similarly if~$V(T) \subseteq F_1$ this follows directly from the fact that~$(C_1,F_1)$ is a FVC in~$G$. So suppose~$T$ is a tree that contains vertices from both~$F_1$ and~$F_2$. Since $T$ is connected, each tree in~$T[F_1]$ contains a neighbor of a vertex in a tree in~$T[F_2]$. Hence no tree in~$T[F_1]$ contains a neighbor of~$V(G-(C_1 \cup C_2 \cup F_1 \cup F_2))$, so~$e(V(T) \cap F_1, G-(C_1 \cup C_2 \cup F_1 \cup F_2)) = 0$. To complete the proof we show~$e(V(T) \cap F_2, G-(C_1 \cup C_2 \cup F_1 \cup F_2)) \leq 1$. Recall each tree in~$G[F_2]$ has at most~$1$ edge to~$G-(C_1 \cup C_2 \cup F_1 \cup F_2)$, so it suffices to show that~$T[F_2]$ is connected. Suppose~$T[F_2]$ is not connected, then let $u,v \in F_2$ be vertices from different components of~$T[F_2]$. Since~$T$ is connected, there is a path from~$u$ to~$v$. This path must use a vertex~$w \in V(T-F_2) \subseteq F_1$. Let~$T'$ denote the tree in~$T[F_1]$ that contains this vertex. Since~$(C_1,F_1)$ is a FVC in~$G$ we have that~$e(T', F_2) \leq e(T',G-(C_1 \cup F_1)) \leq 1$ hence no vertex in~$T'$ can be part of a path from~$u$ to~$v$ in~$T$. This contradicts our choice of~$T'$. \end{proof}
\Cref{col:few-fvcs} is a corollary to the following lemma.
\begin{lemma} \label{lem:small-cert}
If a graph $G$ contains a $C$-certificate $H$ of order $z \geq 0$ for some $C\subseteq V(G)$, then $H$ contains a $C$-certificate $\hat{H}$ of order $z$ such that $\hat{H}-C$ has at most $\frac{|C|}{2}(z^2+2z-1)$ trees. \end{lemma} \begin{proof}
Consider a tree $T$ in $H-C$, we show that $\fvs(H-V(T)) = \fvs(H)$ if
\begin{enumerate}
\item \label{item:vflower} for all $v \in C$ such that $H[V(T) \cup \{v\}]$ has a cycle, $H-V(T)$ contains an order-$z$ $v$-flower, and
\item \label{item:pumpkin} for all $\{u,v\} \in \binom{N_H(T)}{2}$ there are at least $z+1$ other trees in $H-C$ adjacent to $u$ and $v$.
\end{enumerate}
Consider the component $H'$ of $H$ that contains $T$. It suffices to show that $\fvs(H'-V(T)) = \fvs(H')$. Clearly $\fvs(H' - V(T)) \leq \fvs(H')$ so it remains to show that $\fvs(H' - V(T)) \geq \fvs(H')$. Assume $\fvs(H' - V(T)) < \fvs(H')$, then let $X$ be a FVS in $H'-V(T)$ with $|X| < \fvs(H') = |C \cap V(H')| \leq z$. For any $v \in C \cap V(H')$ such that $H[T \cup \{v\}]$ has a cycle we know from condition~\ref{item:vflower} that $H'-V(T)$ has $z > |X|$ cycles that intersect only in $v$, hence $v \in X$. By condition~\ref{item:pumpkin} we have that all but possibly one vertex in $N_G(T)$ must be contained in $X$, since if there are two vertices $x,y \in N_G(T)\setminus X$ then $H-V(T)-X$ has at least $z+1 - |X| \geq 2$ internally vertex-disjoint paths between $x$ and $y$ forming a cycle and contradicting our choice of $X$. Since there is at most one vertex $v \in N_G(T)\setminus X$ and $H[T \cup \{v\}]$ does not have a cycle, we have that $H'-X$ is acyclic, a contraction since~$|X| < \fvs(H')$.
The desired $C$-certificate $\hat{H}$ can be obtained from $H$ by iteratively removing trees from $H-C$ for which both conditions hold. We show that if no such tree exists, then $H-C$ has at most $\frac{|C|}{2}(z^2+2z-1)$ trees. Each tree $T$ for which condition~\ref{item:vflower} fails can be charged to a vertex $v \in C$ that witnesses this, i.e., $H[T \cup \{v\}]$ has a cycle and there are at most $z$ trees $T'$ such that $T' \cup v$ has a cycle. Clearly each vertex $v \in C$ can be charged at most $z$ times, hence there are at most $z \cdot |C|$ trees violating condition~\ref{item:vflower}.
Similarly each tree $T$ for which condition~\ref{item:pumpkin} fails can be charged to a pair of vertices $\{u,v\} \in \binom{N_H(T)}{2}$ for which at most~$z+1$ trees in $H-T$ are adjacent to $u$ and $v$. Clearly each pair of vertices can be charged at most~$z+1$ times. Additionally each pair consists of vertices from the same component of~$H$. Let $H_1,\ldots,H_\ell$ be the components in $H$, then there are at most~$\sum_{1 \leq i \leq \ell} \binom{|C \cap V(H_i)|}{2} = \sum_{1 \leq i \leq \ell} \frac{1}{2}|C \cap V(H_i)|(|C \cap V(H_i)|-1) \leq \sum_{1 \leq i \leq \ell} \frac{1}{2}|C \cap V(H_i)|(z-1) = \frac{|C|}{2}(z-1)$ such pairs. Thus $H-C$ has at most $z\cdot|C| + (z+1)\cdot\frac{|C|}{2}(z-1) = \frac{|C|}{2}(z^2+2z-1)$ trees violating condition~\ref{item:pumpkin}. \end{proof}
\fewfvcs* \begin{proof}
Since $(C,F)$ is a $z$-antler, $G[C \cup F]$ contains a $C$-certificate $H$ or order $z$ and by \cref{lem:small-cert} we know $H$ contains a $C$-certificate $\hat{H}$ of order $z$ such that $\hat{H}-C$ has at most $\frac{|C|}{2}(z^2+2z-1)$ components. Take $F' := V(H-C)$ then $G[F']$ has at most $\frac{|C|}{2}(z^2+2z-1)$ components and $\hat{H}$ is a subgraph of $G[C \cup F']$, meaning $(C,F')$ is a $z$-antler. \end{proof}
\fi
\iffalse This section has move into the main text
\section{Omitted proofs from \autoref{sec:find-fvc}} \label{ap:find-fvc}
\begin{restatable}[$\bigstar$]{lemma}{smallfvc} \label{lem:small_fvc}
If a graph~$G$ contains a reducible single-tree FVC~$(C,F)$ there exists a simple reducible FVC~$(C,F')$ with~$F' \subseteq F$. \end{restatable}
\begin{proof}
We use induction on $|F|$. If $|F| \leq 2f_r(|C|)$ then $(C,F)$ is simple by condition~(a). Assume $|F| > 2f_r(|C|)$. Since $(C,F)$ is a FVC and $G[F]$ is connected there is at most one vertex $v \in F$ that has a neighbor in $V(G)\setminus(C \cup F)$. If no such vertex exists, take $v \in F$ to be any other vertex. Observe that $(C,F\setminus\{v\})$ is a FVC. Consider the following cases:
\begin{itemize}
\item All trees in $G[F] - v$ contain at most $f_r(|C|)$ vertices. Let $F'$ be the vertices of an inclusion minimal set of trees of $G[F]-v$ such that $|F'| > f_r(|C|)$. Clearly $|F'| \leq 2f_r(|C|)$ since otherwise the set is not inclusion minimal. Each tree in $G[F']$ contains a neighbor of $v$ and $F' \subseteq F$, hence $(C,F')$ is simple by condition~(b), and $(C,F')$ is reducible since $|F'| > f_r(|C|)$.
\item There is a tree $T$ in $G[F] - v$ that contains more than $f_r(|C|)$ vertices. Now $(C,V(T))$ is a single-tree reducible FVC with $|V(T)| < |F|$, so the induction hypothesis applies.
\qedhere
\end{itemize} \end{proof}
\knapsack* \begin{claimproof}
Recall that we assumed existence of a properly colored FVC~$(C,F)$ that is reducible and simple by condition~(b) witnessed by the FVC~$(C,F_2)$. Consider the set~$\mathcal{T}'$ of trees in~$G[F]$. Note that any tree~$T'$ in $\mathcal{T}'$ is a tree in~$G[\inv{\chi}(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ since~$(C,F)$ is properly colored and note that~$T'$ contains a neighbor of~$v$. If~$e(T',\{v\}) > 1$ then~$G[F_2]$ contains a cycle, contradicting that~$(C,F_2)$ is a FVC in~$G$, hence~$e(T',\{v\})=1$. It follows that~$T' \in \mathcal{T}$, meaning~$\mathcal{T'} \subseteq \mathcal{T}$. Take~$\mathcal{T}'_2 = \mathcal{T}' \setminus \mathcal{T}_1 = \mathcal{T}' \cap \mathcal{T}_2$ and~$b = \sum_{T \in \mathcal{T}'_2} |N_G(V(T)) \setminus (C' \cup \{v\}|$. Clearly~$\mathcal{T}'_2$ is a candidate solution for the 0-1 knapsack problem with capacity~$b$, hence $|V(\mathcal{T}_2^b)| \geq |V(\mathcal{T}'_2)|$. We deduce
\begin{align*}
|V(\mathcal{T}_1)| + |V(\mathcal{T}_2^b)|
&\geq |V(\mathcal{T}_1)| + |V(\mathcal{T}'_2)|
\geq |V(\mathcal{T}')|
= |F|\\
&> f_r(|C|) &&\text{since~$(C,F)$ is reducible}\\
&= f_r(|C' \cup C|) &&\text{since~$C' \subseteq C$}\\
&= f_r(|C' \cup (N_G(F) \setminus \{v\}) \cup C|) &&\text{since~$N_G(\mathcal{T}'_2) \setminus \{v\} \subseteq C$}\\
&\geq f_r(|C' \cup (N_G(\mathcal{T}'_2) \setminus \{v\})|) &&\text{since~$f_r$ is non-decreasing}\\
&= f_r(|C'| + |N_G(\mathcal{T}'_2) \setminus (C' \cup \{v\})|) &&\text{since~$|A \cup B| = |A| + |B \setminus A|$}\\
&> f_r(|C'| + b) &&\qedhere \end{align*} \end{claimproof}
\message{replace this line by the findfvc* command} \begin{proof}
Take $s = 2f_r(k) + k + 1$. By \cref{thm:universal-set} an~$(n,s)$-universal set~$\mathcal{U}$ for $V(G)$ of size~$2^{\ensuremath{\mathcal{O}}\xspace(s)} \log n$ can be created in~$2^{\ensuremath{\mathcal{O}}\xspace(s)} n \log n$ time. For each~$Q \in \mathcal{U}$ let~$\chi_Q$ be the coloring of~$G$ with~$\inv{\chi}(\ensuremath{\mathsf{\dot{C}}}\xspace) = Q$. Run the algorithm from \cref{lem:colorcoded-fvc} on~$\chi_Q$ for every~$Q \in \mathcal{U}$ and return the first reducible FVC. If no reducible FVC was found return $(\emptyset,\emptyset)$.
We obtain an overall run time of~$2^{\ensuremath{\mathcal{O}}\xspace(s)} \cdot n^3 \log n = 2^{\ensuremath{\mathcal{O}}\xspace(k^3)} \cdot n^3 \log n$.
To prove correctness assume~$G$ contains a reducible single-tree FVC~$(C,F)$ with~$|C| \leq k$. By \cref{lem:small_fvc} we know~$G$ contains a simple reducible FVC~$(C,F')$. Coloring~$\chi$ properly colors~$(C,F')$ if all vertices in~$F' \cup C \cup N_G(F')$ are assigned the correct color. Hence at most~$|F'| + |C + N_G(F')| \leq 2f_r(k) + k + 1 = s$ vertices need to have the correct color. By construction of~$\mathcal{U}$, there is a~$Q \in \mathcal{U}$ such that~$\chi_Q$ assigns the correct colors to these vertices. Hence~$\chi_Q$ properly colors~$(C,F')$ and by \cref{lem:colorcoded-fvc} a reducible FVC is returned. \end{proof}
\fi
\iffalse This seciont has moved back into the main text
\section{Omitted proofs from \autoref{sec:fvc-kernel}} \degreesum* \begin{proof}
We first show that the claim holds if $G[F]$ is a tree. For all $i \geq 0$ let $V_i := \{v \in F \mid \degree_{G[F]}(v) = i\}$. Note that since $G[F]$ is connected, $V_0 \neq \emptyset$ if and only if $|F|=1$ and the claim is trivially true, so suppose $V_0 = \emptyset$. We first show $|V_{\geq 3}| < |V_1|$.
\begin{align*}
2|E(G[F])| &= \sum_{v \in F} \degree_{G[F]}(v) \geq |V_1| + 2|V_2| + 3|V_{\geq 3}|\\
2|E(G[F])| &= 2(|V(G[F])| - 1) = 2|V_1| + 2|V_2| + 2|V_{\geq 3}| - 2
\end{align*}
We obtain $|V_1| + 2|V_2| + 3|V_{\geq 3}| \leq 2|V_1| + 2|V_2| + 2|V_{\geq 3}| - 2$ hence $|V_{\geq 3}| < |V_1|$.
We know all vertices in $F$ have degree at least $3$ in $G$, so $e(V(G) \setminus F,F) \geq 2|V_1| + |V_2| > |V_1| + |V_2| + |V_{\geq 3}| = |F|$. By definition of FVC there is at most one vertex in $F$ that has an edge to $V(G) \setminus (C \cup F)$, all other edges must be between $C$ and $F$. We obtain $1+e(C,F) > |F|$.
If $G[F]$ is a forest, then let $F_1, \ldots, F_\ell$ be the vertex sets for each tree in $G[F]$. Since $(C,F_i)$ is a FVC in $G$ for all $1 \leq i \leq \ell$, we know $e(C,F_i) \geq |F_i|$ for all $1 \leq i \leq \ell$, and since $F_1, \ldots, F_\ell$ is a partition of $F$ we conclude $e(C,F) = \sum_{1 \leq i \leq \ell} e(C,F_i) \geq \sum_{1 \leq i \leq \ell} |F_i| = |F|$. \end{proof}
\disjointpaths*
\begin{claimproof}
Each tree of $G[F]-X$ supplies a path between $u$ and $v$, hence there are more than $|C|+1$ internally vertex-disjoint paths between $u$ and $v$. Suppose $v \in \hat{F}$, we show $u \in \hat{C}$. The proof of the second implication is symmetric. Suppose for contradiction that $u \not\in \hat{C}$. All except possibly one of the disjoint paths between $u$ and $v$ must contain a vertex in $\hat{C}$ by \cref{obs:fvc-basics} since any two disjoint paths form a cycle containing a vertex from $\hat{F}$. Let $Y \subseteq \hat{C}$ be the set of vertices in $\hat{C}$ that are in a tree of $G[F]-X$ with neighbors of $u$ and $v$, so $|Y| > |C|$. Then $|C \cup \hat{C} \setminus Y| < |\hat{C}|$ we derive a contradiction by showing $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C} \setminus Y)$ is acyclic. We know $Y \subseteq F$, so any cycle in $G$ containing a vertex from $Y$ also contains a vertex from $C$ by \cref{obs:fvc-basics}. So if $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C} \setminus Y)$ contains a cycle, then so does $G[\hat{C} \cup \hat{F}] - (C \cup \hat{C})$ which contradicts that $\hat{C}$ is a (minimum) FVS in~$G[\hat{C} \cup \hat{F}]$ since~$(\hat{C},\hat{F})$ is an antler in~$G$.
\end{claimproof}
\fi
\iffalse This section has moved to the main text
\section{Omitted proofs from \autoref{sec:find-antler}} \label{ap:find-antler} Before proving \cref{lem:alg} we prove the following lemma. \begin{lemma} \label{lem:propcolor}
For any $z\geq0$, if a $z$-antler $(C,F)$ in graph $G$ is $z$-properly colored by a coloring $\chi$ and $H$ is a component of $G[C \cup F] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ then each component $H'$ of $H-C$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ with $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$. \end{lemma} \begin{proof}
Note that since $C \cap \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace) = \emptyset$ we have that the statement $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$ implies that $N_{G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') = \emptyset$ and hence that $H'$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. We show $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$.
Suppose $v \in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ and let $u \in V(H')$ be a neighbor of $v$ in $G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. Since $V(H') \subseteq F$ we know $u \in F$. Since $(C,F)$ is $z$-properly colored we also have $N_G(F)\setminus C = \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, hence $N_G(u) \subseteq C \cup F \cup \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ so then $N_{G-\inv{\chi}}(u) \subseteq C \cup F$. By choice of $u$ we have $v \in N_{G-\inv{\chi}}(u) \subseteq C \cup F$. So since $u,v \in C \cup F$, and $u$ and $v$ are neighbors in $G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ we know $u$ and $v$ are in the same component of $G[C \cup F]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, hence $v \in V(H)$.
Suppose $v \not\in C$, so $v \in F$. Since also $u \in F$ we know that $u$ and $v$ are in the same component of $G[F]-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, so $v \in H'$, but then $v \not\in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ contradicting our choice of $v$. It follows that $v \in C$ hence $v \in C \cap V(H)$. Since $v \in N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H')$ was arbitrary $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq C \cap V(H)$. \end{proof}
\algfindantler* \begin{proof}
We define a function~$W_\chi \colon 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)} \to 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)}$ as follows: for any~$C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ let~$W_\chi(C)$ denote the set of all vertices that are in a component~$H$ of~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ for which~$N_{G - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H) \subseteq C$.
The algorithm we describe updates the coloring~$\chi$ and recolors any vertex or edge that is not part of a $z$-properly colored antler to color~\ensuremath{\mathsf{\dot{R}}}\xspace.
\begin{enumerate}
\item \label{antleralg:init} Recolor all edges to color~\ensuremath{\mathsf{\dot{R}}}\xspace when one of its endpoints has color~\ensuremath{\mathsf{\dot{R}}}\xspace.
\item \label{antleralg:loop1} For each component~$H$ of~$G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ we recolor all vertices of~$H$ and their incident edges to color~\ensuremath{\mathsf{\dot{R}}}\xspace if~$H$ is not a tree or~$e(H,\inv{\chi}_V(\ensuremath{\mathsf{\dot{R}}}\xspace)) > 1$.
\item \label{antleralg:loop2} For each subset~$C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ of size at most~$z$, mark all vertices in~$C$ if $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C|$.
\item \label{antleralg:repeat} If~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ contains unmarked vertices we recolor them to color~\ensuremath{\mathsf{\dot{R}}}\xspace, remove markings made in step~\ref{antleralg:loop2} and repeat from step~\ref{antleralg:init}.
\item \label{antleralg:return} If all vertices in~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ are marked in step~\ref{antleralg:loop2}, return~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$.
\end{enumerate}
\subparagraph*{Running time}
The algorithm will terminate after at most~$n$ iterations since in every iteration the number of vertices in~$\inv{\chi}_V(\ensuremath{\mathsf{\dot{R}}}\xspace)$ increases. Steps~\ref{antleralg:init}, \ref{antleralg:loop1}, \ref{antleralg:repeat}, and \ref{antleralg:return} can easily be seen to take no more than $\ensuremath{\mathcal{O}}\xspace(n^2)$ time. Step~\ref{antleralg:loop2} can be performed in $\ensuremath{\mathcal{O}}\xspace(4^z \cdot n^{z+1})$ time by checking for all~$\ensuremath{\mathcal{O}}\xspace(n^z)$ subsets~$C \in \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ of size at most~$z$ whether the graph~$G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ has feedback vertex number~$z$. This can be done in time~$\ensuremath{\mathcal{O}}\xspace(4^z \cdot n)$~\cite{IwataK19}.
Hence the algorithm runs in time~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
We show that any $z$-properly colored antler prior to executing the algorithm remains $z$-properly colored after termination and that in step~\ref{antleralg:return},~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a~$z$-antler in~$G$. Since~$(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ contains all properly colored antlers this proves correctness.
\begin{restatable}[$\bigstar$]{claim}{remainproperlycolored} \label{claim:remain}
All $z$-antlers $(\hat{C},\hat{F})$ that are $z$-properly colored by $\chi$ prior to executing the algorithm are also $z$-properly colored by $\chi$ after termination of the algorithm.
\end{restatable}
\begin{claimproof}
To show the algorithm preserves properness of the coloring, we show that every individual recoloring preserves properness, that is, if an arbitrary $z$-antler is $z$-properly colored prior to the recoloring, it is also $z$-properly colored after the recoloring.
Suppose an arbitrary $z$-antler $(\hat{C},\hat{F})$ is $z$-properly colored by $\chi$. An edge is only recolored when one of its endpoints has color~\ensuremath{\mathsf{\dot{R}}}\xspace. Since these edges are not in $G[\hat{C} \cup \hat{F}]$ its color does change whether $(\hat{C},\hat{F})$ is colored $z$-properly. All other operations done by the algorithm are recolorings of vertices to color~\ensuremath{\mathsf{\dot{R}}}\xspace. We show that any time a vertex~$v$ is recolored we have that $v \not\in \hat{C} \cup \hat{F}$, meaning $(\hat{C},\hat{F})$ remains colored $z$-properly.
Suppose $v$ is recolored in step~\ref{antleralg:loop1}, then we know $\chi(v) = \ensuremath{\mathsf{\dot{F}}}\xspace$, and $v$ is part of a component~$H$ of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$. Since $\chi$ $z$-properly colors $(\hat{C},\hat{F})$ we have $\hat{F} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)$ but $N_G(\hat{F}) \cap \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace) = \emptyset$, so since $H$ is a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ we know either $V(H) \subseteq \hat{F}$ or $V(H) \cap \hat{F} = \emptyset$. If $V(H) \cap \hat{F} = \emptyset$ then clearly $v \not\in \hat{C} \cup \hat{F}$. So suppose $V(H) \subseteq \hat{F}$, then $H$ is a tree in $G[\hat{F}]$. Since $v$ was recolored and $H$ is a tree it must be that $e(H,\inv{\chi}_C(\ensuremath{\mathsf{\dot{R}}}\xspace)) > 1$ but this contradicts that $(\hat{C},\hat{F})$ is a FVC.
Suppose $v$ is recolored in step~\ref{antleralg:repeat}, then we know $v$ was not marked during step~\ref{antleralg:loop2} and $\chi(v) = \ensuremath{\mathsf{\dot{C}}}\xspace$, so $v \not\in \hat{F}$. Suppose that $v \in \hat{C}$. We derive a contradiction by showing that $v$ was marked in step~\ref{antleralg:loop2}.
Since $(\hat{C},\hat{F})$ is $z$-properly colored, we know that $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is a $\hat{C}$-certificate of order $z$, so if $H$ is the component of $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ containing $v$ then $\fvs(H) = |\hat{C} \cap V(H)| \leq z$. Since $\hat{C} \cap V(H) \subseteq \hat{C} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ we know that in some iteration in step~\ref{antleralg:loop2} we have $C = \hat{C} \cap V(H)$.
To show that $v$ was marked, we show that $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C|$. We know $G[W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ is a forest since it is a subgraph of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ which is a forest by step~\ref{antleralg:loop1}, so we have that $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \leq |C|$.
To show $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \geq |C|$ we show that $H$ is a subgraph of $G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. By \cref{lem:propcolor} we have that each component $H'$ of $H-\hat{C}$ is also a component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ with $N_{G-\inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)}(H') \subseteq \hat{C} \cap V(H) = C$. Hence $V(H-\hat{C}) = V(H-C) \subseteq W_\chi(C)$ so $H$ is a subgraph of $G[C \cup W_\chi(C)]$. Since $H$ is also a subgraph of $G[\hat{C} \cup \hat{F}] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ we conclude that $H$ is a subgraph of $G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ and therefore $\fvs(G[C \cup W_\chi(C))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) \geq \fvs(H) = |C|$.
\end{claimproof}
\begin{restatable}[$\bigstar$]{claim}{coloringisantler} \label{claim:coloredantler}
In step~\ref{antleralg:return}, $(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a $z$-antler in~$G$.
\end{restatable}
\begin{claimproof}
We know $(\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace),\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace))$ is a FVC in $G$ because each component of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ is a tree and has at most one edge to a vertex not in $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ by step~\ref{antleralg:loop1}. It remains to show that $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ contains a $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$-certificate of order $z$.
Note that in step~\ref{antleralg:return} the coloring $\chi$ is the same as in the last execution of step~\ref{antleralg:loop2}. Let $\mathcal{C} \subseteq 2^{\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)}$ be the family of all subsets $C \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ that have been considered in step~\ref{antleralg:loop2} and met the conditions for marking all vertices in $C$, i.e., $\fvs(G[C \cup W_\chi(C)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = |C| \leq z$. Since all vertices in $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ have been marked during the last execution of step~\ref{antleralg:loop2} we know $\bigcup_{C \in \mathcal{C}} C = \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$.
Let $C_1,\ldots,C_{|\mathcal{C}|}$ be the sets in $\mathcal{C}$ in an arbitrary order and define $D_i := C_i \setminus C_{<i}$ for all $1 \leq i \leq |\mathcal{C}|$. Observe that $D_1, \ldots, D_{|\mathcal{C}|}$ is a partition of $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ with $|D_i| \leq z$ and $C_i \subseteq D_{\leq i}$ for all $1 \leq i \leq |\mathcal{C}|$. Note that $D_i$ may be empty for some $i$.
We now show that $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$ contains a $\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$-certificate of order $z$. We do this by showing there are $|\mathcal{C}|$ vertex disjoint subgraphs of $G[\inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace) \cup \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)]$, call them $G_1, \ldots, G_{|\mathcal{C}|}$, such that $\fvs(G_i) = |D_i| \leq z$ for each $1 \leq i \leq |\mathcal{C}|$.
Take $G_i := G[D_i \cup (W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i}))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$ for all $1 \leq i \leq |\mathcal{C}|$. First we show that for any $i \neq j$ the graphs $G_i$ and $G_j$ are vertex disjoint. Clearly $D_i \cap D_j = \emptyset$. We can assume $i < j$, so $D_{\leq i} \subseteq D_{<j}$ and then $W_\chi(D_{\leq i}) \subseteq W_\chi(D_{<j})$. By successively dropping two terms, we deduce
\begin{align*}
(W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i})) \cap (W_\chi(D_{\leq j}) \setminus W_\chi(D_{<j}))
&\subseteq W_\chi(D_{\leq i}) \cap (W_\chi(D_{\leq j}) \setminus W_\chi(D_{<j})) \\
&\subseteq W_\chi(D_{\leq i}) \setminus W_\chi(D_{<j})
= \emptyset .
\end{align*}
We complete the proof by showing $\fvs(G_i) = |D_i|$ for all $1 \leq i \leq \ell$. Recall that $D_i = C_i \setminus C_{<i}$. Since $C_i \in \mathcal{C}$ we know $C_i$ is an optimal FVS in $G[C_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$, so then clearly $D_i$ is an optimal FVS in $G[C_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace) - C_{<i} = G[D_i \cup W_\chi(C_i)] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. We know that $C_i \subseteq D_{\leq i}$ so then also $W_\chi(C_i) \subseteq W_\chi(D_{\leq i})$. It follows that $D_i$ is an optimal FVS in $G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)$. In this graph, all vertices in $W_\chi(D_{<i})$ must be in a component that does not contain any vertices from $D_i$, so this component is a tree and we obtain $|D_i| = \fvs(G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = \fvs(G[D_i \cup W_\chi(D_{\leq i})] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace) - W_\chi(D_{<i})) = \fvs(G[D_i \cup (W_\chi(D_{\leq i}) \setminus W_\chi(D_{<i}))] - \inv{\chi}(\ensuremath{\mathsf{\dot{R}}}\xspace)) = \fvs(G_i)$.
\end{claimproof}
It can be seen from \cref{claim:remain} that for any $z$-properly colored antler~$(\hat{C},\hat{F})$ we have~$\hat{C} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{C}}}\xspace)$ and~$\hat{F} \subseteq \inv{\chi}_V(\ensuremath{\mathsf{\dot{F}}}\xspace)$. \Cref{claim:coloredantler} completes the correctness argument. \end{proof}
\algfindop* \begin{proof}
Consider the following algorithm:
Use \cref{lem:find-fvc} to obtain a FVC~$(C_1,F_1)$ in~$2^{\ensuremath{\mathcal{O}}\xspace(k^3)}\cdot n^3 \log n$ time. If~$(C_1,F_1)$ is reducible we can find and apply an operation in~$\ensuremath{\mathcal{O}}\xspace(n^2)$ time by \cref{lem:fvc-kernel} so assume~$(C_1,F_1)$ is not reducible. Create two~$(n+m, 26 k^5 z^2)$-universal sets~$\mathcal{U}_1$ and~$\mathcal{U}_2$ for~$V(G) \cup E(G)$ using \cref{thm:universal-set}. Define for each pair~$(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ the coloring~$\chi_{Q_1,Q_2}$ of~$G$ that assigns all vertices and edges in~$Q_1$ color~\ensuremath{\mathsf{\dot{C}}}\xspace, all vertices and edges in~$Q_2 \setminus Q_1$ color~\ensuremath{\mathsf{\dot{F}}}\xspace, and all vertices and edges not in~$Q_1 \cup Q_2$ color~\ensuremath{\mathsf{\dot{R}}}\xspace. For each~$(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ obtain in~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time a $z$-antler~$(C_2,F_2)$ by running the algorithm from \cref{lem:alg} on~$G$ and~$\chi_{Q_1,Q_2}$. If~$(C_2, F_2)$ is not empty, apply \cref{op:remove-antler} to remove~$(C_2,F_2)$, otherwise report~$G$ does not contain a $z$-antler of width at most~$k$.
\subparagraph*{Running time}
By \cref{thm:universal-set}, the sets~$\mathcal{U}_1$ and~$\mathcal{U}_2$ have size~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \log n$ and can be created in~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n \log n$ time. It follows that there are~$|\mathcal{U}_1 \times \mathcal{U}_2| = 2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \log^2 n$ colorings for which we apply the~$n^{\ensuremath{\mathcal{O}}\xspace(z)}$ time algorithm from \cref{lem:alg}. We obtain an overall running time of~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$. Since a $z$-antler has width at least~$z$, we can assume~$k \geq z$, hence~$2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)} \leq 2^{\ensuremath{\mathcal{O}}\xspace(k^7)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
Suppose $G$ contains a $z$-antler $(C,F)$ of width at most $k$, we show the algorithm finds an operation to apply.
By \cref{col:few-fvcs} we know that there exists an~$F' \subseteq F$ such that~$(C,F')$ is a $z$-antler where $G[F']$ has at most $\frac{|C|}{2}(z^2+2z-1)$ trees. For each tree~$T$ in~$G[F']$ note that~$(C,V(T))$ is a single-tree FVC of width $|C| \leq k$. If for some tree~$T$ in~$G$ the FVC~$(C,V(T))$ is reducible, then $(C_1,F_1)$ is reducible by \cref{lem:find-fvc} and we find an operation using \cref{lem:fvc-kernel}, so suppose for all trees $T$ in $G[F']$ that $|V(T)| \leq f_r(|C|)$. So then $|F'| \leq \frac{|C|}{2}(z^2+2z-1) \cdot f_r(|C|)$. We show that in this case there exists a pair $(Q_1,Q_2) \in \mathcal{U}_1 \times \mathcal{U}_2$ such that $\chi_{Q_1,Q_2}$ $z$-properly colors $(C,F')$.
Whether a coloring $z$-properly colors $(C,F')$ is only determined by the colors of $C \cup F' \cup N_G(F') \cup E(G[C \cup F'])$.
\begin{restatable}[$\bigstar$]{claim}{antlercolorsize} \label{claim:antler-color-size}
$|C \cup F' \cup N_G(F') \cup E(G[C \cup F'])| \leq 26 k^5 z^2$.
\end{restatable}
\begin{claimproof}
Note that $|N_G(F') \setminus C| \leq \frac{|C|}{2}(z^2+2z-1)$ since no tree in $G[F']$ can have more than one neighbor outside $C$. Additionally we have
\begin{align*}
|E(G[C \cup F'])|
&\leq |E(G[C])| + |E(G[F'])| + e(C,F') \\
&\leq |E(G[C])| + |F'| + |C| \cdot |F'| &\text{since $G[F']$ is a forest} \\
&\leq |C|^2 + (|C|+1) \cdot |F'|\\
&\leq |C|^2 + (|C|+1) \cdot \frac{|C|}{2}(z^2+2z-1) \cdot f_r(|C|)\\
&\leq k^2 + (k+1) \cdot \frac{k}{2}(z^2+2z-1) \cdot (2k^3 + 3k^2 - k)\\
&\leq k^2 + \frac{z^2+2z-1}{2} \cdot (k^2+k) \cdot (2k^3 + 3k^2 - k)\\
&\leq k^2 + \frac{z^2+2z-1}{2} \cdot 2k^2 \cdot 5k^3 &\text{since $k=0$ or $k \geq 1$}\\
&\leq k^2 + \frac{3z^2}{2} \cdot 10k^5 &\text{since $z=0$ or $z \geq 1$}\\
&\leq k^2 + 15 k^5 z^2
\leq 16 k^5 z^2,
\end{align*}
hence
\begin{align*}
|C \cup F' \cup N_G(F') \cup E(G[C \cup F'])|\\
&\hspace{-5em}= |C| + |F'| + |N_G(F')\setminus C| + |E(G[C \cup F'])|\\
&\hspace{-5em}\leq |C|
+ \frac{z^2+2z+1}{2} f_r(|C|)
+ \frac{|C|}{2}(z^2+2z-1)
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3z^2}{2} f_r(k)
+ \frac{k}{2}(2z^2)
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3}{2}z^2 \cdot (2k^3 + 3k^2 - k)
+ z^2 \cdot k
+ 16 k^5 z^2 \\
&\hspace{-5em}\leq k
+ \frac{3}{2}z^2 \cdot 5k^3
+ z^2 \cdot k
+ 16 k^5 z^2
\leq 26 k^5 z^2. \qedhere
\end{align*}
\end{claimproof}
By construction of~$\mathcal{U}_1$ and~$\mathcal{U}_2$ there exist~$Q_1 \in \mathcal{U}_1$ and~$Q_2 \in \mathcal{U}_2$ such that~$\chi_{Q_1,Q_2}$ $z$-properly colors~$(C,F')$. Therefore the algorithm from \cref{lem:alg} returns a non-empty $z$-antler for~$\chi_{Q_1,Q_2}$ and \cref{op:remove-antler} can be executed.
\end{proof}
Before we prove \cref{thm:main} we give the following definition and proposition:
\begin{definition}
If~$k\geq0$ and~$z\geq0$ are integers and~$G$ is a graph, then a sequence of disjoint vertex sets~$C_1, F_1,\ldots,C_\ell, F_\ell$ is called a \emph{$z$-antler-sequence} for~$G$ if for all~$1 \leq i \leq \ell$ the pair~$(C_i, F_i)$ is a $z$-antler in~$G - (C_{<i} \cup F_{<i})$. The \emph{width} of a $z$-antler-sequence is defined as $\max_{1 \leq i \leq \ell} |C_1|$. \end{definition}
\begin{proposition} \label{prop:antler-sequence}
If~$C_1, F_1,\ldots,C_\ell, F_\ell$ is a \emph{$z$-antler-sequence} for some graph~$G$, then the pair~$(C_{\leq i}, F_{\leq i})$ is a $z$-antler in~$G$ for any $1 \leq i \leq \ell$. \end{proposition} \begin{proof}
We use induction on~$i$. Clearly the statement holds for~$i = 1$, so suppose~$i > 1$. By induction~$(C_{<i},F_{<i})$ is a $z$-antler in~$G$, and since~$(C_i,F_i)$ is a $z$-antler in~$G-(C_{<i} \cup F_{<i})$ we have by \cref{lem:antler_combine} that~$(C_{<i} \cup C_i,F_{<i} \cup F_i) = (C_{\leq i},F_{\leq i})$ is a $z$-antler in~$G$. \end{proof}
\maintheorem* \begin{proof}
Note that using the definition of a $z$-antler-sequence, we can rephrase condition~\ref{thm:item:antlers} to: if~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler-sequence of width at most~$k$ then~$|S| \geq |C_{\leq t}|$.
\subparagraph*{Algorithm}
We use \cref{lem:find-op} to apply an operation in $G$ and obtain the resulting graph $G'$ and vertex set $S$. If no applicable operation was found return an empty vertex set $S := \emptyset$. Otherwise we recursively call our algorithm on~$G'$ with integers~$z$ and~$k$ to obtain a vertex set~$S'$ and return the vertex set~$S \cup S'$.
\subparagraph*{Running time}
Note that since every operation reduces the number of vertices or increases the number of double-edges, after at most $\ensuremath{\mathcal{O}}\xspace(n^2)$ operations we obtain a graph where no operation can be applied. Therefore after at most $\ensuremath{\mathcal{O}}\xspace(n^2)$ recursive calls the algorithm terminates. We obtain a running time of $2^{\ensuremath{\mathcal{O}}\xspace(k^5 z^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z)}$.
\subparagraph*{Correctness}
We prove correctness by induction on the recursion depth, which is shown the be finite by the run time analysis.
First consider the case that no operation was found. Clearly condition~\ref{thm:item:fvs-safe} holds for~$G' := G$ and $S := \emptyset$. To show condition~\ref{thm:item:antlers} suppose~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler-sequence of width at most~$k$ for~$G$. The first non-empty antler in this sequence is a $z$-antler of width at most~$k$ in~$G$. Since no operation was found using \cref{lem:find-op} it follows that~$G$ does not contain a non-empty $z$-antler of width at most~$k$. Hence all antlers in the sequence must be empty and~$|C_{\leq t}| = 0$, so condition~\ref{thm:item:antlers} holds for~$G' := G$ and~$S := \emptyset$.
For the other case, suppose~$G'$ and~$S$ are obtained by applying an operation, then since this operation is FVS-safe we know for any minimum FVS~$S''$ of~$G'$ that~$S \cup S''$ is a minimum FVS in~$G$. Since $S'$ is obtained from a recursive call there is a minimum FVS in $G'$ containing all vertices of~$S'$. Let $S''$ be such a FVS in $G'$, so $S' \subseteq S''$ then we know $S \cup S''$ is a minimum FVS in $G$. It follows that there is a minimum FVS in $G$ containing all vertices of $S \cup S'$, proving condition~\ref{thm:item:fvs-safe}.
To prove condition~\ref{thm:item:antlers} suppose~$C_1, F_1, \ldots, C_t, F_t$ is a $z$-antler-sequence of width at most~$k$ for~$G$. We first prove the following:
\begin{claim} \label{claim:antler-sequence}
There exists a $z$-antler-sequence~$C_1', F_1', \ldots, C_t', F_t'$ of width at most~$k$ for~$G'$
such that
\begin{enumerate}
\item \label{item:vsets} $C_{\leq t}' \cup F_{\leq t}' = V(G') \cap (C_{\leq t} \cup F_{\leq t})$ and
\item \label{item:cutsize} $|C_{\leq t}'| = \sum_{1 \leq i \leq t} |C_i| - |(C_i \cup F_i) \cap S|$.
\end{enumerate}
\end{claim}
\begin{claimproof}
We use induction on~$t$. Since $G'$ and $S$ are obtained through an antler-safe operation and $(C_1,F_1)$ is a $z$-antler in $G$, we know that $G'$ contains a $z$-antler $(C_1',F_1')$ such that $C_1' \cup F_1' = (C_1 \cup F_1) \cap V(G')$ and $|C_1'| = |C_1| - |(C_1 \cup F_1) \cap S|$. The claim holds for $t=1$.
For the induction step, consider~$t > 1$. By applying induction to the length-$(t-1)$ prefix of the sequence, there is a $z$-antler sequence~$C_1',F_1', \ldots, C_{t-1}',F_{t-1}'$ of width at most~$k$ for~$G'$ such that both conditions hold.
We have by \cref{prop:antler-sequence} that~$(C_{\leq t}, F_{\leq t})$ is a $z$-antler in~$G$. Since~$G'$ and~$S$ are obtained through an antler-safe operation from $G$ there is a $z$-antler~$(C',F')$ in~$G'$ such that~$C' \cup F' = V(G') \cap (C_{\leq t} \cup F_{\leq t})$ and~$|C'| = |C_{\leq t}| - |S \cap (C_{\leq t} \cup F_{\leq t})|$. Take~$C_{t}' := C' \setminus (C_{<t}' \cup F_{<t}')$ and~$F_{t}' := F' \setminus (C_{<t}' \cup F_{<t}')$. By \cref{lem:antler_diff} we have that~$(C_{t}',F_{t}')$ is a $z$-antler in~$G'-(C_{<t}' \cup F_{<t}')$, it follows that~$C_1',F_1',\ldots,C_{t}',F_{t}'$ is a $z$-antler-sequence for~$G'$. We first show condition~\ref{item:vsets}.
\begin{align*}
C_{\leq t}' \cup F_{\leq t}'
&= C_{t}' \cup F_{t}' \cup C_{<t}' \cup F_{<t}' \\
&= C' \cup F'
&&\text{by choice of $C_{t}'$ and $F_{t}'$}\\
&= V(G') \cap (C_{\leq t} \cup F_{\leq t})
&&\text{by choice of $C'$ and $F'$.}
\end{align*}
To prove condition~\ref{item:cutsize} and the $z$-antler-sequence~$C_1',F_1',\ldots,C_{t}',F_{t}'$ has width at most~$k$ we first show $|C_{t}'| = |C_t| - |(C_t \cup F_t) \cap S|$. For this observe that $(C_{\leq t}',F_{\leq t}')$ is an antler in $G'$ by \cref{prop:antler-sequence}.
\begin{align*}
|C_{t}'|
&= |C_{\leq t}'| - |C_{<t}'|
&&\hspace{-7em}\text{since $C_i' \cap C_j' = \emptyset$ for all $i \neq j$}\\
&= \fvs(G'[C_{\leq t}' \cup F_{\leq t}']) - |C_{<t}'|
&&\hspace{-7em}\text{by the above} \\
&= \fvs(G'[V(G') \cap (C_{\leq t} \cup F_{\leq t})]) - |C_{<t}'|
&&\hspace{-7em}\text{by condition~\ref{item:vsets}} \\
&= \fvs(G'[C' \cup F']) - |C_{<t}'|
&&\hspace{-7em}\text{by choice of $C'$ and $F'$}\\
&= |C'| - |C_{<t}'|
&&\hspace{-7em}\text{since $(C',F')$ is an antler in $G'$} \\
&= |C'| - \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)
&&\hspace{-7em}\text{by induction}\\
&= |C_{\leq t}| - | S \cap (C_{\leq t} \cup F_{\leq t})|
- \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)\\
&= \sum_{1 \leq i \leq t} (|C_i| - |S \cap (C_i \cup F_i)|)
- \sum_{1 \leq i < t} (|C_i| - |(C_i \cup F_i) \cap S|)\\
&&&\hspace{-11em}\text{since $C_1,F_1,\ldots,C_t,F_t$ are pairwise disjoint}\\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S|
\end{align*}
We know the $z$-antler-sequence~$C_1',F_1',\ldots,C_{t-1},F_{t-1}$ has width at most~$k$, so to show that this $z$-antler-sequence has width at most~$k$ it suffices to prove that~$|C_{t}'| \leq k$. Indeed~$|C_{t}'| = |C_{t}| - |(C_{t} \cup F_{t}) \cap S| \leq |C_{t}| \leq k$.
To complete the proof of \cref{claim:antler-sequence} we now derive condition~\ref{item:cutsize}:
\begin{align*}
|C_{\leq t}'|
&= |C_{t}'| + |C_{<t}'|
&&\text{since $C_{t}' \cap C_{<t}' = \emptyset$}\\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S| + |C_{<t}'| \\
&= |C_{t}| - |(C_{t} \cup F_{t}) \cap S| + \sum_{1 \leq i \leq t-1} (|C_i| - |(C_i \cup F_i) \cap S|)
&&\text{by induction}\\
&= \sum_{1 \leq i \leq t} (|C_i| - |(C_i \cup F_i) \cap S|). &&\qedhere
\end{align*}
\end{claimproof}
To complete the proof of condition~\ref{thm:item:antlers} from \cref{thm:main} we show $|S \cup S'| \geq |C_{\leq t}|$. By \cref{claim:antler-sequence} we know a $z$-antler-sequence~$C_1',F_1',\ldots,C_t',F_t'$ of width at most~$k$ for $G'$ exists. Since $S'$ is obtained from a recursive call we have~$|S'| \geq |C_{\leq t}'|$, so then
\begin{align*}
|S \cup S'|
&= |S| + |S'| \\
&\geq |S| + |C_{\leq t}'| \\
&= |S| + \sum_{1 \leq i \leq t} ( |C_i| - |(C_i \cup F_i) \cap S| )
&&\text{by \cref{claim:antler-sequence}}\\
&= |S| + \sum_{1 \leq i \leq t} |C_i| - \sum_{1 \leq i \leq t} |(C_i \cup F_i) \cap S| \\
&= |S| + |C_{\leq t}| - |S \cap (C_{\leq t} \cup F_{\leq t})|
&&\text{since $C_1', F_1', \ldots, C_t', F_t'$ are disjoint}\\
&= |S| + |C_{\leq t}| - |S|\\
&\geq |C_{\leq t}|.
&&\qedhere
\end{align*} \end{proof}
\maincor* \begin{proof} Let~$(k^*,z^*)$ be such that the $z^*$-antler complexity of~$G$ is at most~$k^*$. Let~$p_1 \in \ensuremath{\mathcal{O}}\xspace(k^5 z^2), p_2 \in \ensuremath{\mathcal{O}}\xspace(z)$ be concrete functions such that the running time of \cref{thm:main} is bounded by~$2^{p_1(k,z)} \cdot n^{p_2(z)}$. Consider the pairs~$\{(k',z') \in \ensuremath{\mathbb{N}}\xspace^2 \mid 1 \leq z' \leq k' \leq n\}$ in order of increasing value of the running-time guarantee~$2^{p_1(k,z)} \cdot n^{p_2(z)}$. For each such pair~$(k',z')$, start from the graph~$G$ and invoke \cref{thm:main} to obtain a vertex set~$S$ which is guaranteed to be contained in an optimal solution. If~$G - S$ is acyclic, then~$S$ itself is an optimal solution and we return~$S$. Otherwise we proceed to the next pair~$(k',z')$.
\subparagraph*{Correctness}
The correctness of \cref{thm:main} and the definition of $z$-antler complexity ensure that for~$(k',z') = (k^*,z^*)$, the set~$S$ is an optimal solution. In particular, if~$C_1, F_1, \ldots, C_t, F_t$ is a sequence of vertex sets witnessing that the $z^*$-antler complexity of~$G$ is at most~$k^*$, then~\eqref{thm:item:antlers} of \cref{thm:main} is guaranteed to output a set~$S$ of size at least~$\sum _{1 \leq i \leq t} |C_i|$, which is equal to the size of an optimal solution on~$G$ by definition.
\subparagraph*{Running time} For a fixed choice of~$(k',z')$ the algorithm from \cref{thm:main} runs in time~$2^{\ensuremath{\mathcal{O}}\xspace((k')^5 (z')^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z')} \leq 2^{\ensuremath{\mathcal{O}}\xspace((k^*)^5 (z^*)^2)} \cdot n^{\ensuremath{\mathcal{O}}\xspace(z^*)}$ because we try pairs~$(k',z')$ in order of increasing running time. As we try at most~$n^2$ pairs before finding the solution, the corollary follows. \end{proof}
\fi
\end{document} |
\begin{document}
\title[]{Sub shot-noise frequency estimation with bounded \textit{a priori} knowledge}
\author{Changhun Oh and Wonmin Son\footnote{Corresponding author: sonwm71@sogang.ac.kr}}
\address{Department of Physics, Sogang University, Mapo-gu, Shinsu-dong, Seoul 121-742, Korea}
\ead{sonwm71@sogang.ac.kr}
\begin{abstract} We analyze an efficient frequency estimation scheme that is applied to measure the unknown frequency of an atomic state in Ramsey spectroscopy. The scheme is employing appropriate combinations of uncorrelated probe atoms and Greenburgur-Horne-Zeilinger (GHZ) type correlated probe atoms to estimate its frequency. The estimation value of frequency is obtained through the Bayesian analysis of the final measurement outcomes. The proposed scheme allows us to obtain better precision than the scheme without quantum correlation and it also prevents us from ambiguity in the frequency estimation procedure with GHZ correlations only. We show that the scheme can beat the shot-noise limit and, in addition, it is found that there is the trade-off relation between the precision of the frequency estimation and the decoherence rate in the atomic states. \end{abstract}
\noindent{\it Keywords}:
\maketitle
\section{Introduction}
Measurement of a physical system with an arbitrary precision is an important task in many fields of experimental physics and technologies. The ultimate goal of the precision measurement is to attain the highest accuracy when a physical system is measured at a cost of given resources. In a real situation, the aim is needed to be achieved under the constraint of limited resources such as the finite number of trials and the limited duration time of the experiment. This situation is also true in the case of Ramsey spectroscopy experiment which is a scheme to measure the dynamical evolution of an atomic state and estimate the resonance frequency of the atomic state \cite{Ramsey}. Although precision in the measurements can be improved with the number of trials, it is also true that the limited resource restricts the possible number of repetitions of the measurement. Therefore, it is required to devise a method how to utilize the given resources as efficient as possible.
A possible method to achieve a higher accuracy in the precision measurement is to employ quantum entanglement. The examples are to use coherent-squeezed state in gravitational wave detector \cite{caves}, number entangled states of photons (so-called NOON states) in Mach-Zehnder interferometer \cite{noon} and GHZ states of atoms in frequency estimation with Ramsey spectroscopy \cite{wineland1994, wineland1996}. In contrast to the conventional measurement scheme that allows us to attain the shot-noise limit only, $\delta\theta\sim 1/\sqrt{N}$ \cite{Itano 1993}, the scheme with entangled states helps us to obtain the Heisenberg limit, $\delta\theta\sim 1/N$ \cite{giovannetti2006,maccone,davidovich2011} where $\theta$ is an unknown parameter to be measured and $N$ is the number of employed resources. However, the precision measurement scheme is sometimes restricted if a prior knowledge about the value of a parameter to be measured is not provided. In the case of phase estimations, there is recent work about an estimation scheme that is performed by using entangled coherent states without a prior knowledge about an unknown phase \cite{ecs}. The reason that the precise estimation of a completely unknown phase is possible is that without any prior knowledge about the unknown phase, we know that phases lie in $[0,2\pi)$. In general, the precision measurement scheme is only useful when we try to achieve a better precision on vaguely known values. If we do not have any knowledge on the measurement values, the scheme does not work well.
Especially, when there is periodicity of probabilities in each measurement, the measurement data cannot be used to specify the single value of phase estimation and results in more than one estimation value over the period. Thus, if we do not know the range of the parameter sharp enough, it is not possible to estimate the precise value of its phase and leaves ambiguity. Since the period of probabilities for the measurement with entangled states is even shorter than that of unentangled states, entanglement-based measurements give rise to a more serious ambiguity problem in the determination of an estimation value. It can be shown that in the scenario having a prior knowledge on the periodicity, the measurement scheme with highly entangled states merely attains an equal precision to that with uncorrelated states in the frequency estimation. Therefore, there will be no benefit of using entangled states in the case with limited {\it a priori} knowledge provided. Here, we propose measurement schemes that allow us to overcome the ambiguity of $2\pi$ periodicity and, at the same time, obtain a better precision beyond the shot-noise limit using quantum entanglement.
There have been prior discussions that, when highly entangled states are used, periodicity of probabilities in each measurement outcome sometimes prevents us from determining a unique estimation value and it can be overcome in idealized cases \cite{pezze2007, higgins, dunningham}. In the works, two different settings are considered (i) using GHZ states with particle numbers $1,2,4,...,2^{p-1}$, and (ii) using combinations of an uncorrelated state and GHZ states in order to avoid ambiguity. Each scheme has been discussed in interferometric systems for phase estimation and the introduced decoherence model was photon losses. In this paper, we consider more general situation with a different decoherence effect that occurs in trapped ions. We compare the results of the two schemes in the absence of decoherence and in the presence of decoherence, and conclude that while the first scheme allows us to achieve the Heisenberg limit in the ideal case, the second scheme becomes more advantageous as the decoherence rate increases and attains a sub shot-noise precision.
We organize this article in the following. In section 2, we review frequency estimation procedure in Ramsey spectroscopy when the atoms are uncorrelated and correlated under decoherence effects. Furthermore, we bring up an ambiguity problem that occurs in the frequency estimation. In section 3, we analyze two different schemes that overcome the ambiguity problem and, at the same time, improve precision by using entanglement, and compare the attainable precision of the proposed schemes with that of the conventional scheme using no entanglement. Finally, we summarize our works in section 4.
\section{Ramsey Spectroscopy} \subsection{Standard frequency estimation scheme under decoherence}
Ramsey spectroscopy is the measurement technique of transition frequency $\omega_0$ between the internal states of two-level atom \cite{Ramsey}. The system can only take one of two energy levels as they can denote the ground state $|g\rangle$ and the excited state $|e\rangle$. Standard Ramsey spectroscopy is operated as following. Initially, prepare $N$ ionized atoms confined in an ion trap \cite{wineland1998} and by optical pumping technique \cite{happer1970}, change the state of atoms into the same ground state, $|\psi_0\rangle=|g\rangle$. Then, the atoms are applied by $\pi/2$ pulse which leads each atom to be in the superposition state of the ground state and the excited state, $|\psi_1\rangle=(|g\rangle+|e\rangle)/\sqrt{2}$. After that, (classical) fields with frequency $\omega$ are applied to the atoms for interrogation time $t$ so that the state changes to $|\psi_2\rangle=(|g\rangle+e^{-i\Delta t}|e\rangle)/\sqrt{2}$ in a rotating frame, where $\Delta=\omega_0-\omega$ denotes the detuning between the atomic transition and classical driving field. Finally, the atoms are applied by the second $\pi/2$ pulse which changes the state to $|\psi_3\rangle=\sin(\Delta t/2)|g\rangle+\cos(\Delta t/2)|e\rangle$, and the internal state of each atom is measured by scattering light and detecting the fluorescence with a photomultipier (PMT). We estimate the transition frequency $\omega_0$ by counting the number of atoms in the excited state $|e\rangle$ and using the probability of detecting the excited state $|e\rangle$ in each atom which is given as \begin{equation}\label{p1}
P(e|\omega_0)=|\langle e|\psi_3\rangle|^2=\cos^2\bigg(\frac{\Delta t}{2}\bigg)=\frac{1+\cos\Delta t}{2}. \end{equation}
The probability of detecting the ground state is given as $P(g|\omega_0)=1-P(e|\omega_0)=\sin^2(\Delta t/2)$. The operation of Ramsey spectroscopy is shown in Fig \ref{ramseypicture}.
From its functional form, the probability $P(e|\omega_0)$ can be used to estimate the frequency $\omega_0$ of the atomic state within a certain precision. The statistical fluctuation in the estimation which is associated with the given probability $P(e|\omega_0)$ can be obtained by using Cram\'{e}r-Rao inequality \cite{cramer,Helstrom}, \begin{equation}\label{cramer} \delta\theta\geq\frac{1}{\sqrt{\nu F(\theta)}} \end{equation}
where $F(\theta)=\sum_i \frac{1}{P(i|\theta)}\big(\frac{\partial P(i|\theta)}{\partial \theta}\big)^2$ is the Fisher information and $\nu$ is the number of repetition of trials. $P(i|\theta)$ is the conditional probability of obtaining a result $i$ with a given parameter $\theta$ and the summation is taken over all possible results. The Fisher information is a quantity that measures how much information can be obtained when the parameter $\theta$ is changed infinitesimally. It is known that the lower bound of the Cram\'{e}r-Rao inequality can be achieved asymptotically by the maximum likelihood estimator \cite{fisher}. The Fisher information plays an important role in estimation theory in that it gives a lower bound of statistical fluctuation.
\begin{figure}
\caption{The operation of Ramsey spectroscopy using a single ion is shown. The probability of detecting $|g\rangle$ is $\sin^2(\Delta t/2)$ and the probability of detecting $|e\rangle$ is $\cos^2(\Delta t/2)$ in the ideal case.}
\label{ramseypicture}
\end{figure}
In Ramsey spectroscopy, the unknown parameter to be estimated is the transition frequency $\omega_0$ between two internal states $|g\rangle$ and $|e\rangle$. Using the fact that possible results are the excited state $|e\rangle$ with the probability $P(e|\omega_0)$ and the ground state $|g\rangle$ with the probability $P(g|\omega_0)$, the Cram\'{e}r-Rao lower bound is easily obtained as \begin{equation}\label{uncor}
|\delta \omega_0|=\frac{1}{\sqrt{\nu t^2}}=\frac{1}{\sqrt{NTt}}, \end{equation}
where $\nu=NT/t$ is the repetition of trials and $T$ is the total experimental time. Thus, the precision with $N$ atoms is proportional to $1/\sqrt{N}$, which is so-called the \textit{shot-noise limit} \cite{Itano 1993}. The shot-noise limit is originated from quantum projection noise. Note that the precision $|\delta \omega_0|$ is independent of the transition frequency $\omega_0$ with ignorance of decoherence effects.
So far, we have discussed in the ideal case. However, in a realistic experiment, it is necessary to include the effects of decoherence. As a realistic case, we introduce a model that was proposed in \cite{huelga1997}. In this model, dephasing of individual ions is the main decoherence, which is caused from collisions, stray fields, and laser instabilities. The effects of decoherence on a single ion $\rho$ can be described as the master equation in Lindblad form \cite{huelga1997,nielsen}: \begin{equation}\label{decoherence}
\dot{\rho}(t)=i\Delta(\rho|e\rangle\langle e|-|e\rangle\langle e|\rho)+\frac{\gamma}{2}(\sigma_z\rho\sigma_z-\rho) \end{equation}
where $\gamma$ is the decay rate and $\sigma_z=|g\rangle\langle g|-|e\rangle\langle e|$ denotes a Pauli spin operator. Eq. (\ref{decoherence}) is written in a rotating frame. The effect of decoherence is shown in the broaden signal (\ref{p1}) of a single ion: \begin{equation}\label{p1d}
P_\gamma (e|\omega_0)=\frac{1+\cos(\Delta t) e^{-\gamma t}}{2}, \end{equation} and again using Cram\'{e}r-Rao inequality (\ref{cramer}), the corresponding precision changes to \begin{equation}\label{deco}
|\delta\omega_0^{\rm{dec}}|=\sqrt{\frac{1-\cos^2(\Delta t)e^{-2\gamma t}}{NTte^{-2\gamma t}\sin^2(\Delta t)}}. \end{equation} It can be found that as the decay rate $\gamma$ increases, the uncertainty becomes larger. In contrast to the ideal case, a precision in the presence of decoherence is dependent to the frequency $\omega_0$. When $\Delta t=k\pi/2$ $(k$ {\rm{is odd}}$)$ and $t=1/2\gamma\leq T$ are satisfied, minimum uncertainty is obtained as \begin{equation}\label{minuncertainty}
|\delta \omega_0^{\rm{dec}}|_{\rm{min}}=\sqrt{\frac{2\gamma e}{NT}}. \end{equation}
Note that minimum uncertainty is attained when the probabilities of obtaining the measurement outcomes $|g\rangle$ and $|e\rangle$ are same.
\subsection{Improvement of precision with GHZ correlation} We have studied standard Ramsey spectroscopy using $N$ atoms, which allows us to obtain a shot-noise limit. It has been proposed that entanglement between atoms improves sensitivity of phase. Especially, one of the states that has been proposed to improve precision is maximally entangled multipartite state which is so-called Greenberger-Horne-Zeilinger (GHZ) state \cite{greenberger1990} \begin{equation}
|\psi_{\rm{GHZ}}\rangle=\frac{|g\rangle^{\otimes N}+|e\rangle^{\otimes N}}{\sqrt{2}} \end{equation}
where $|g\rangle^{\otimes N}\equiv|g\rangle |g\rangle \cdot\cdot\cdot |g\rangle$ and $|e\rangle^{\otimes N}\equiv|e\rangle |e\rangle \cdot\cdot\cdot |e\rangle$. The GHZ states accumulate $N$ times amplified phase information than without entanglement, which results in a better precision. Implementation of GHZ states among atoms has been demonstrated by Cirac \textit{et. al.} \cite{cirac} and it will be explained in the following.
The preparation procedure of GHZ states is, in principle, that after all $N$ atoms are prepared in the ground state $|g\rangle^{\otimes N}=|g\rangle|g\rangle\cdot\cdot\cdot|g\rangle$, the first ion is applied by $\pi/2$ pulse to create the state $(|g\rangle+|e\rangle)|g\rangle|g\rangle\cdot\cdot\cdot|g\rangle/\sqrt{2}$ and the ions are operated by a "controlled-NOT(CNOT)" gate, the first ion as a controlled qubit and the second ion as target qubits to entangle the first two atoms, which changes the state to $(|g\rangle|g\rangle+|e\rangle|e\rangle)|g\rangle\cdot\cdot\cdot|g\rangle/\sqrt{2}$. Continuing the operation of CNOT gates, the first ion as controlled qubit and other ions as target qubits, the final state becomes the GHZ state $(|g\rangle|g\rangle\cdot\cdot\cdot|g\rangle+|e\rangle|e\rangle\cdot\cdot\cdot|e\rangle)/\sqrt{2}=(|g\rangle^{\otimes N}+|e\rangle^{\otimes N})/\sqrt{2}$. Despite theoretical straightforwardness of generating GHZ states, it is known that large size GHZ states are extremely difficult to create in practice \cite{sorensen2000,leibfried2003,leibfried2004,leibfried2005,monz2011}.
After preparation of GHZ states, (classical) fields of frequency $\omega$ are applied to the atoms for interrogation time $t$, which changes the state to $(|g\rangle^{\otimes N}+e^{-iN\Delta t}|e\rangle^{\otimes N})/\sqrt{2}$. Finally, the atoms are disentangled by the second set of controlled-NOT gates and the internal state of the first ion is measured. Again, we estimate the true frequency by the measurement outcomes and the probability of detecting $|e\rangle$, \begin{equation}\label{pn}
P(e|N,\omega_0)=\frac{1+\cos N\Delta t}{2}=\cos^2\bigg(\frac{N\Delta t}{2}\bigg) \end{equation}
which denotes the probability of detecting all $N$ atoms in the excited state. The probability of detecting all $N$ atoms in the ground state is given as $P(g|N,\omega_0)=1-P(e|N,\omega_0)=\sin^2(N\Delta t/2)$. The principle that GHZ states allow us to obtain a better precision is originated from the fact that the phase shift of GHZ states in interrogation time is amplified $N$ times than without entanglement in the same period, which results in the change of the probability of detecting $|e\rangle$.
As the previous single ion case, using the Cram\'{e}r-Rao inequality (\ref{cramer}), we attain a $\sqrt{N}$ improved precision than that of uncorrelated $N$ atoms (\ref{uncor}), \begin{equation}\label{ghz}
|\delta\omega_0|=\frac{1}{\sqrt{\nu N^2 t^2}}=\frac{1}{N\sqrt{Tt}}, \end{equation} where $\nu=T/t$ is the repetition of trials. Now, uncertainy of frequency is proportional to $1/N$, which is referred to as the Heisenberg limit. As the previous case, uncertainty does not depend on the frequency in the ideal case.
Taking into account decoherence, similarly to uncorrelated states, we can easily see that the signal (\ref{pn}) of GHZ states with $N$ ions becomes \begin{equation}\label{pnd}
P_\gamma(e|N,\omega_0)=\frac{1+\cos(N\Delta t)e^{-N\gamma t}}{2}, \end{equation} and uncertainty of frequency becomes \begin{equation}\label{ghzdeco}
|\delta\omega_0^{\rm{dec}}|=\sqrt{\frac{1-\cos^2(N\Delta t)e^{-2N\gamma t}}{N^2 Tte^{-2N\gamma t}\sin^2(N\Delta t)}}. \end{equation}
The exponential part in the denominator indicates that if $N$ is too large, uncertainty $|\delta \omega_0^{\rm{dec}}|$ goes to infinity, which means that a large size of GHZ state is fragile against decoherence. Again, uncertainty depends on the frequency and minimum uncertainty is achieved when $\Delta t=k\pi/2N$ $(k$ {\rm{is odd}}$)$ and $t=1/2\gamma N\leq T$ are satisfied. Minimum uncertainty is obtained when the probabilities of obtaining each result $|g\rangle$ and $|e\rangle$ are same. As a consequence, minimum uncertainty is same with that of uncorrelated states (\ref{minuncertainty}). Therefore, in the presence of decoherence GHZ states do not help attaining a better precision than that of uncorrelated states \cite{huelga1997}.
\subsection{Ambiguity of $\pi$-period in the frequency estimation}
Let us consider a realistic experiment of frequency estimation with basic concepts of Ramsey spectroscopy. Basically, an estimation process is proceeded based on measurement outcomes which depend on the parameter to be estimated and the probability of obtaining the experimental data. In Ramsey spectroscopy, measurement outcomes are consisted of $|g\rangle$ and $|e\rangle$ only, and the probability of detecting the excited state $|e\rangle$ is given in (\ref{p1}) for a single atom. The probability (\ref{p1}) that we use in an estimation process has periodicity, which is shown in Fig \ref{periodicity}. A difficulty of frequency estimation comes from the periodicity because the periodicity prevents us from distinguishing a true estimate of frequency among a number of possible estimates. For example, let us suppose that the true frequency $\omega_0^{\rm{true}}$ is zero (we set $\omega=0$ for simplicity i.e. $\Delta=\omega_0$.), which leads to the measurement outcome $|e\rangle$ at all $N$ atoms. Then, our estimate obtained by the measurement outcomes is not $\omega_0=0$ but $\omega_0=2\pi m/t$ ($m$ an integer). Our estimation result is not unique. In other words, since two different frequencies $(\omega_0)_1=2\pi m/t$ and $(\omega_0)_2=2\pi n/t$ ($m$, $n$ integers) give the same experimental data, the data does not allow us to choose a single estimate of frequency. Hence, even after obtaining measurement outcomes, we are still required to determine a single estimation value. Determination of a unique estimate can be possible with the help of a prior knowledge about the true frequency $\omega_0$. If a provided prior knowledge about the true frequency is narrow enough to choose a unique estimate, the ambiguity problem is avoided. If a given prior knowledge is not enough narrow, we still have ambiguity in the estimation procedure. In short, because of periodicity, even when we try infinitely many measurements, it is not possible to determine a single estimation value without an appropriate prior knowledge. Thus, we are required to know about the true frequency initially with an enough degree of accuracy to choose a single estimation value so that we assume that an arbitrary prior knowledge about $\omega_0$ is always given in this paper.
\begin{figure}
\caption{Probability distribution of an excited state as a function of $\omega_0$ when $t=1$. (a) Periodicity of the probability distribution causes ambiguity in determining a single peak among the values in the periodic repetitions. If one knows that $\omega_0\in[-3\pi,3\pi)$, the estimate of $\omega_0$ can be taken from any values among $\pm2\pi$ and $0$. In order to avoid the ambiguity, we need to change the interaction time $t$ three times smaller than original. The new choice of interaction time allows us to discriminate the peak value of $\omega_0$ at $0$ from $\pm2\pi$. (b) Probability distibution of a single atom (blue) in an excited state and that of GHZ states with $3$ atoms (purple) are shown. If we know $\omega_0\in[-\pi,\pi)$ initially, the best choice of interrogation time is $t=1$. If we use a single atom, an estimation can be uniquely determined as $\widetilde{\omega_0}=0$. On the other hand, if we use GHZ states with $3$ atoms, the estimation becomes ambiguous because our estimation value is $\widetilde{\omega_0}=\pm 2\pi/3$ and $0$. To avoid the ambiguity, we need to choose $t$ three times smaller, which leads to the same uncertainty $|\delta \omega_0|$ with the scheme without GHZ correlation.}
\label{fig:combi}
\label{fig:combi}
\label{periodicity}
\end{figure}
In this situation, let us suppose that we are trying to improve precision. When we use $N$ atoms, the ultimate precision is given as (\ref{uncor}). Under a restriction of the limited number of atoms $N$ and a given total duration time $T$, a possible choice to improve preicision is increasing interrogation time $t$, since uncertainty of the frequency $|\delta \omega_0|$ decreases as the interrogation time $t$ increases. Hence, we can increase the time $t$ as possible in the only restriction $t\leq T$ where $T$ is sufficiently large. However, as the interrogation time $t$ increases, the period $2\pi/t$ of the signal (\ref{p1}) becomes shorter so that it causes ambiguity in determining a unique estimation value because a more number of frequencies become a possible estimation value as the period becomes shorter. In other words, the shorten period makes it harder to distinguish a true frequency from other possible estimates. Thus, while the longer interrogation time $t$ allows a better preicision, the increased interrogation time $t$ requires a narrower prior knowledge about the true frequency $\omega_0$ in order to determine a unique estimation value.
Let us suppose that we know that the true frequency $\omega_0$ lies in the interval $[0,\pi/L)$ where $L$ is a positive number. Under the prior knowledge, the largest possible value of interrogation time $t$ that allows us to determine a single peak is $L$. Then, the possible maximum interrogation time $t=L$ leads uncertainty to be \begin{equation}\label{noprior}
|\delta\omega_0|=\frac{1}{\sqrt{NTL}}. \end{equation} This is minimum uncertainty with the given prior knowledge $\omega_0\in [0, \pi/L)$. It implies that minimum uncertainty of frequency estimation depends on the accuracy of a prior knowledge about the true frequency. In other words, if we know the true frequency more accurately at the beginning, we can obtain a better precision.
As previously discussed, quantum entanglement between atoms help us to improve precision of frequency estimation. Let us consider introducing quantum entanglement between atoms by exploiting GHZ states under the same prior knowledge that $\omega_0\in [0, \pi/L)$. Similar to the unentangled state case, the signal of GHZ states with $N$ atoms has a period $2\pi/Nt$ from (\ref{pn}). Therefore, the largest possible $t$ that allows us to choose a single estimation value is $L/N$. Eventually, uncertainty obtained by using GHZ states is same with (\ref{noprior}). The reason that uncertainty with uncorrelated state and with GHZ state is same is that the period of signal with GHZ states is $N$ times shorter than that of uncorrelated state so that we need to use a shorter interrogation time to avoid ambiguity in the estimation process. Consequently, in the consideration of a prior knowledge about the frequency, exploiting GHZ states does not improve precision, which is shown in Fig \ref{periodicity}. However, it does not mean that GHZ states are useless in improving the precision of frequency because it obviously has a better sensitivity locally. We introduce useful schemes that utilize the potential power of GHZ states and improve precision, avoiding ambiguity.
\section{Schemes to overcome ambiguity of $\pi$-period} In this section, we analyze two measurement schemes to overcome ambiguity in frequency estimation of atomic states. Basic settings of the schemes are originally proposed by Pezze {\it et. al.} \cite{pezze2007} and Gkortsilas {\it et. al.}\cite{dunningham}. However, their schemes are limited either in the case of ideal measurement or in the case with photon losses. Here, we apply the schemes to frequency estimation with atomic GHZ states in the presence of decoherence. Decoherence effects on atomic states are different with photon losses in photonic states. Since photon losses can be recognized by comparing the number of detected photons with that of input photons, one is able to exclude measurement outcomes where photon losses occurred. Thus, one takes perfect measurement outcomes only in the estimation process. On the other hand, detection of internal states of atoms does not give any information whether decoherence had an effect on the atomic states. Therefore, all the measurement outcomes should be taken into account including atoms that were affected by decoherence. As such, decoherence in atomic states is different with photon losses.
The first scheme is using GHZ states with $1,2,2^2,...,2^{p-1}$ numbers of particles in order. It is known that this scheme allows phase estimation with the Heisenberg limit in the ideal case. The second scheme is using combinations of an uncorrelated state and GHZ states. The two schemes behave differently and give different precisions depending on the degree of decoherence effects. We analyze the two schemes in frequency estimation, considering decoherence effects on atomic states.
Before we introduce details of the schemes, let us review Bayesian analysis. Let us suppose that we know that the true frequency lies in $[0, \pi/L)$ again, and we choose $t=L$ which leads to the best sensitivity avoiding ambiguity for an uncorrelated state. According to the Bayes' theorem, the posterior probability distribution $P_p(\omega_0|N_T,X)$, given total resources $N_T$ and a data set of measurement outcomes $X=(x_1,x_2,...,x_{N_T})$, when $x_n\in \{g,e\}$, is given as \begin{equation}\label{bayes}
P_p(\omega_0|N_T,X)=\frac{P(X|N_T,\omega_0)P(\omega_0)}{P(N_T,X)}, \end{equation}
where $P(\omega_0)$ is a prior distribution of $\omega_0$ and is completely flat for an unknown frequency, $P(\omega_0)=L/\pi$, and $P(X)$ is treated as a normalization constant. The posterior probability $P(\omega_0|N_T,X)$ is a conditional probability of frequency $\omega_0$ based on a data set $X$ of measurement outcomes obtained by using $N_T$ resources. We choose the maximum of the distribution as estimator $\widetilde{\omega_0}$ and uncertainty $|\delta \omega_0|$ given by $\int_{\widetilde{\omega_0}-\delta \omega_0}^{\widetilde{\omega_0}+\delta \omega_0} d\omega_0 P_p(\omega_0|N_T,X)=0.6827$ when $P_p(\omega_0|N_T,X)$ goes to normal distribution. Bayesian analysis allows us to obtain the estimation value and associated uncertainty by using measurement outcomes.
\subsection{GHZ states with particle number $1,2,4,...2^{p-1}$}
\begin{figure}\label{GHZplot}
\end{figure}
Let us consider the first scheme : exploiting GHZ states with particle numbers $N_e=1,2,4,...,2^{p-1}$. Note that we have assumed that we know that $0\leq\omega_0^{\rm{true}}<\pi/L$ so that we have fixed the interrogation time $t=L$. Now let us suppose $\omega_0^{\rm{true}}=\pi/2L$ for simplicity. In this case, if we use a single atom, the probability of detecting the ground state and that of the excited state are same. Thus, the precision is maximum in the presence of decoherence. However, if we use GHZ states with even number of atoms, the measurement outcomes are either all $|g\rangle$ or all $|e\rangle$ where the precision is minimum for the realistic case, which is implied by (\ref{ghzdeco}). Thus, for GHZ states with the even number of atoms, we need to apply an additional phase shift $\pi/2$ in order to change the overall phase shift into one that gives minimum uncertainty where the probabilities of detecting $|g\rangle$ and $|e\rangle$ are same. Then, the effective frequency for GHZ states with the even number of atoms is $\omega_0+\pi/2N_e$.
Asymptotically, when we use GHZ states of $N$ atoms (If $N=1$, a single atom state) with the number of trials $\nu$ in frequency estimation, we obtain $\nu P(g|N,\omega_0^{\rm{eff}})$ number of the ground state and $\nu P(e|N,\omega_0^{\rm{eff}})$ number of the excited state. Due to the additional phase shift, the effective frequency $\omega_0^{\rm{eff}}$ is $\omega_0^{\rm{true}}+\pi/2N$ for GHZ states with even $N$ atoms and $\omega_0^{\rm{true}}$ for a single atom. Therefore, by using the Bayes' theorem (\ref{bayes}), when we use GHZ states of particle number $N_e=1,2,4,...,2^{p-1}$ with repetition $\nu=T/L$, the probability distribution in the ideal case becomes
\begin{eqnarray}\nonumber P_p(\omega_0|N_T=\sum_{k=0}^{p-1}2^k,X)\propto\prod_{k=0}^{p-1} P(\omega_0|2^k,e)^{\nu P(e|2^k,\omega_0^{\rm{eff}})}\times P(\omega_0|2^k,g)^{\nu P(g|2^k,\omega_0^{\rm{eff}})}\\= [\cos^2(\omega_0t/2) \sin^2(\omega_0t/2)]^{\nu/2}\prod_{k=1}^{p-1}[\cos^2(2^k\omega_0 t/2-\pi/4)\sin^2(2^k\omega_0 t/2-\pi/4)]^{\nu/2}\nonumber \\ \simeq e^{-\nu(4^p-1)(\omega_0 t-\frac{\pi}{2})^2/6} \simeq e^{-\nu N_T^2 (\omega_0 t-\frac{\pi}{2})^2/6} \end{eqnarray}
where $P(\omega_0|N_e,x)$ is defined by the Bayes' theorem (\ref{bayes}). We have used gaussian approximation $[\sin^2(\omega_0 t/2)]^{m}[\cos^2(\omega_0 t/2)]^{n}\sim e^{-\frac{m+n}{2}(\omega_0 t-\widetilde{\omega_0}t)^2}$ for large $m+n$, where $\widetilde{\omega_0}t=2\tan^{-1}(\sqrt{m/n})$ is the maximum point, and $N_T=\sum_{k=0}^{p-1} 2^k=2^p-1$. An advantage of using GHZ states with geometrically increasing number of atoms is cancellation of all peaks except the central one, which is shown in Fig \ref{GHZplot}. As a consequence, the probability distribution gives uncertainty of frequency which achieves the Heisenberg limit, \begin{equation}
|\delta\omega_0|=\frac{\sqrt{3}}{\sqrt{\nu N_T^2L^2}}=\frac{\sqrt{3}}{N_T\sqrt{TL}}. \end{equation}
In the realistic case, $P(x|N,\omega_0)$ are replaced by $P_\gamma(x|N,\omega_0)$. The results are shown in Fig \ref{optimization}. Since it achieves the Heisenberg limit in the ideal case, apparently this scheme is helpful to utilize GHZ states under a prior knowledge and to improve precision. Nevertheless, this scheme is extremely fragile against decoherence because it exploits a large size of GHZ states. The result of the first scheme in the presence of decoherence indicates that it is required to use a number of uncorrelated atoms that are more robust against decoherence than GHZ states. Therefore, we need to introduce another scheme that is more robust against decoherence in a realistic situation.
\subsection{Combination of different correlations} The second scheme is to use appropriate combinations of uncorrelated atoms $N_u$ and the $p$ copies of GHZ states with $N_e$ atoms. Here, the uncorrelated atoms play a role in suppressing other periods except one period where the true frequency exists, while GHZ states play a role in improving sensivity, which is shown in Fig \ref{schemes}. Similar to the previous scheme, the probability distribution with repetition $\nu$ is asymptotically written as
\begin{eqnarray} \nonumber P_p(\omega_0|N_T=N_u+p N_e,X)\propto \\ \qquad \qquad \qquad P(\omega_0|1, e)^{\nu N_u P(e|1,\omega_0^{\rm{true}})}\times P(\omega_0|1,g)^{\nu N_u P(g|1,\omega_0^{\rm{true}})}\nonumber
\\ \qquad \qquad \qquad \times P(\omega_0|N_e,e)^{\nu p P(e|N_e, \omega_0^{\rm{true}})}\times P(\omega_0|N_e,g)^{\nu p P(g|N_e,\omega_0^{\rm{true}})}.\label{asymp} \end{eqnarray} To minimize the standard deviation of the probability distribution, we need to choose optimal $N_u,N_e$ and $p$ numerically. For numerical optimization, first of all, we obtain the probability distribution in (\ref{asymp}) by using (\ref{p1}) and (\ref{pn}) in ideal case or by using (\ref{p1d}) and (\ref{pnd}) in realistic case. Then, for a proper value of repetition $\nu$ and fixed $N_T=N_u+p N_e$, by changing $N_u,N_e$ and $p$ appropriately, we find values of $N_u,N_e$ and $p$ that minimize the standard deviation of the probability distribution. Then, we iterate the same procedure for different values of $N_T$. After the numerical optimization, only one peak survives where the true frequency exists.
Let us consider the same situation with the first scheme, a prior knowledge that $\omega_0^{\rm{true}}\in[0,\pi/L)$, fixed $t=L$ and the true frequency $\omega_0^{\rm{true}}=\pi/2L$. In this case, we only use GHZ states with odd number of atoms because GHZ states with odd number of atoms attain minimum uncertainty at $\omega_0^{\rm{true}}=\pi/2L$ in the presence of decoherence. At $\omega_0^{\rm{true}}=\pi/2L$, the probability of detecting $|g\rangle$ and that of detecting $|e\rangle$ are $1/2$ for both uncorrelated states and GHZ states with odd number of atoms. Substituting $\omega_0^{\rm{true}}=\pi/2L$ into (\ref{asymp}), in the ideal case, the asymptotic probability distribution with the combination of $N_u$ uncorrelated atoms and $p$ copies of GHZ states of $N_e$ atoms and repetition $\nu=T/t$ can be obtained as \begin{eqnarray}
P_p(\omega_0|N_T=N_u+p N_e,X)\nonumber \\ =[\cos^2(\omega_0 t/2)\sin^2(\omega_0 t/2)]^{\nu N_u/2}[\cos^2(N_e \omega_0 t/2)\sin^2(N_e \omega_0 t/2)]^{\nu p/2}\nonumber \\ \simeq C e^{-(N_u+p N_e^2)(\omega_0 t-\pi/2)^2/2} \end{eqnarray} where $C$ is a normalization constant. The gaussian approximation is valid when the probability distribution is well-localized around $\pi/2$. If $N_u, N_e$ and $p$ are numerically optimized so that the probability distribution has a single peak, the approximation is valid. After the numerical optimization, uncertainty of frequency is obtained as \begin{equation}
|\delta \omega_0|=\sqrt{\frac{1}{\nu (N_u+p N_e^2)L^2}}=\sqrt{\frac{1}{(N_u+p N_e^2)TL}} \end{equation}
which is the consistent with the Cram\'{e}r-Rao bound. It implies that the Cram\'{e}r-Rao bound is saturated by Bayesian analysis if there is no ambiguity in the estimation process. Uncertainty obtained by the numerical optimization of $N_u, N_e$ and $p$ is shown in Fig \ref{optimization}. In the realistic case, $P(x|N,\omega_0)$ are replaced by $P_\gamma(x|N,\omega_0)$ in (\ref{asymp}).
\begin{figure}\label{fig:combi}
\label{fig:combi2}
\label{schemes}
\end{figure}
\begin{figure}\label{fig:noloss1}
\label{fig:loss1}
\label{fig:noloss1}
\label{fig:loss1}
\label{optimization}
\end{figure}
In the absence of decoherence, exploiting the GHZ states with $1,2,4,...,2^{p-1}$ atoms allows us to achieve the Heisenberg limit as known. A drawback of this scheme is that as the decoherence rate $\gamma$ increases, uncertainty becomes larger as shown in Fig \ref{optimization}. Especially, at $\gamma=0.1$ for large $N_T$, uncertainty is larger than that of uncorrelated states. In other words, since a large size of GHZ state is fragile against decoherence, this scheme is not practical when the decoherence rate is large. On the contrary, when we use appropriate combinations of an uncorrelated state and GHZ states, since we use a number of uncorrelated atoms, we can expect the robustness against decoherence. Indeed, Fig \ref{optimization} shows that the scheme with combinations of an uncorrelated state and GHZ states is robust against decoherence. More generally, it is noticed that the best scheme depends on decoherence rate $\gamma$. In the ideal case, the first scheme attains Heisenberg limit, which is the best sensitivity among our schemes. As the decoherence rate $\gamma$ increases, however, the second scheme becomes better. Thus, depending on the decoherence rate $\gamma$, we need to choose an optimal scheme.
In the second scheme, we have used combinations of uncorrelated atoms and GHZ states with fixed number of atoms for simplicity. However, it is possible to use combinations of uncorrelated states and various sizes of GHZ states. Indeed, the first scheme is one of the combinations of various GHZ states and the first scheme gives a better precision when the decoherence rate $\gamma$ is small. Therefore, it can be considered to use different various combinations depending on the decoherence rate $\gamma$.
We have assumed that the true frequency $\omega_0$ is $\pi/2L$ in the both schemes because the best sensitivity is attained at $\pi/2L$ for uncorrelated atoms in the realistic case. If the phase shift $\omega_0 L$ is not $\pi/2$, we need to modify the phase shift to be $\pi/2$ in order to attain minimum uncertainty, which is realizable by a feedback mechanism \cite{higgins, dunningham}. A feedback mechanism is implemented by the Bayesian analysis on the collected data. After each trial, we apply an additional phase shift which is determined by the Bayesian analysis to make the total phase shift to be $\pi/2$. For every trial, we repeat this process and then total phase shift becomes $\pi/2$ which gives minimum uncertainty of frequency.
One of the advantages in the second scheme is that it does not require a large size of GHZ state which is experimentally difficult to generate and has a short coherence time \cite{sorensen2000,leibfried2003,leibfried2004,leibfried2005,monz2011}. Indeed, the largest GHZ states that we have used in the numerical optimization are those of 5 atoms in both the ideal case and the realistic case. It indicates that our scheme is practical as well as useful. Nevertheless, since generating GHZ states requires extremely delicate experiment and the coherence time is very short, realization of our scheme may necessitate more advanced experiment devices and skills than current.
\section{Conclusion} Periodicity of probability distribution causes ambiguity in the frequency estimation process. Furthermore, in the consideration of a prior knowledge, exploiting GHZ states does not help improving precision of frequency estimation when one uses only GHZ states with the same number of atoms. In order to avoid the ambiguity and utilize GHZ states for improvement of precision, we implement two different schemes. The first scheme is employing GHZ states with $1,2,4,...,2^{p-1}$ atoms, which improves precision significantly in the ideal case. Nevertheless, since a large size of GHZ states is fragile against decoherence, the first scheme is no longer advantageous in the realistic case. The second scheme that is robust against decoherence is exploiting appropriate combinations of uncorrelated atoms and GHZ states. As the decoherence rate $\gamma$ increases, it is shown that the sensitivity of second scheme can be better than that of the first scheme. In addition, we conclude that the scheme with combinations of uncorrelated and GHZ states allows us to achieve a sub shot-noise precision in the presence of decoherence.
\section*{References}
\end{document} |
\begin{document}
\begin{abstract} In this note, we propose a new approach to solving the Calabi problem on manifolds with edge-cone singularities of prescribed angles along complex hypersurfaces. It is shown how the classical approach of Aubin-Yau in derving {\it a priori} estimates for the complex hessian can be made to work via adopting a \emph{good reference metric} and studying equivalent equations with different referrence metrics. This further allows extending much of the methods used in the smooth setting to the edge setting. These results generalise to the case of multiple hypersufaces with possibly normal crossing.
53C55, 35J60
\end{abstract}
\maketitle
\section{Introduction}
The study of problems around cone-edge singularities, in particular, the problems around finding K\"ahler-Einstein metrics prescribed edge-cone behaviour,
has received quite a bit of attention in the past years. One reason is the key r\^{o}le such metrics play in the approach taken by Chen, Donalsdon, and Sun in \cite{ch-do-su-1, ch-do-su-2, ch-do-su-3}, (and also other attempts for solving the same problem in \cite{ti}), in proving the relation between \(K\)-stability and the existence of K\"ahler-Einstein metrics on Fano manifolds. In the present work we shall show how one can use the geometry of edge-cone manifolds by constructing edge metrics with curvature bounded from below to obtain the estimates needed for solving the Calabi problem. Other than finding K\"ahler-Einstein metrics, this approach allows prescribing a wide class of Ricci forms. Since in most constructions and proofs it is straightforward to see how they should be modified for the case of divisors with possibly normal crossing, in the rest of this work, in order to keep the statements and proofs clearer, we confine ourselves to the case of one smooth hypersurface.
By K\"ahler metrics with \emph{edge} or \emph{edge-cone} singularities we mean a K\"ahler metric with conical singularity along a complex hypersurface, that is, a metric which asymptotically resembles a cone on \(\mathbb{C}\) of total angle \(2 \pi \tau\) in the directions normal to the hypersurface, and is smooth in the tangential directions. Examples of such metrics were already known as they arise as orbifold metrics. More generally, one may construct such metrics as follows. Let \((M^n,\omega_0)\) K\"ahler manifolds, where \(\omega_0\) is smooth. Assume that \(D^{n-1} \subset M^n\) is a complex hypersurface and that \(s\) is a holomorpic section of an hermitian line bundle \((L,h)\) which vanishes of order zero along \(D\). Then, the following metric \begin{equation} \label{cone-reference}
\omega_\tau := \omega_0 +\mathfrak{a} dd^c |s|_h^{2\tau} \end{equation} is an edge-cone metric along \(D\) of angle \(\tau\) when \(0<\mathfrak{a} \ll 1\). This statement, along with the rest of results can be generalised to the case wherein \(D\) consistes of a union of irreducible divisors, \(D_j\), with at most normal crossing. Indeed, to the best of the author's knowledge, in the works prior to the present work the metric \(\omega_\tau\), thus defined, has been taken as the conical background metric in order to do the analysis.
Various approaches to the study of the Calabi problem on manifolds with edge-cone singularities have proved effective. Ricci-flat edge-cone K\"ahler metrics were proved to exist under suitable topological conditions by Brendle in \cite{br} provided that \(\tau \leq {1 \over 2}\). There, the classical method used by Aubin and Yau in \cite{au,ya} was used. The fact that the classical calculations can be adapted to this situation is due to the fact that the reference metric \(\omega_\tau\) has bounded curvature away from the divisor \(D\) when \(\tau \leq {1 \over 2}\). More specifically, when \(\tau > {1 \over 2}\), the curvature of the reference metric \eqref{cone-reference} might become unbounded from below. This possible lack of lower bound in the curvature has been one main obstacle in deriving laplacian estimates and higher order regularity results since the approach of Aubin and Yau for deriving estimates on the complex hessian depends on the existence of a lower bound on the bisectional curvature of the referrence metric. Further, the lack of of a lower bound on the Ricci curvature of the reference metric means the lack of a lower bound on the laplacian of the Ricci potential, a yet another quantity the finiteness of whose lower bound is needed in deriving the laplacian estimate for the potential in the approach of Aubin-Yau.
Another work, which covers the case of larger angles as well is that of Jeffres, Mazzeo and Rubinstein \cite{je-ma-ru}. There, in order to derive the laplacian estimate in the absence of a lower bound on curvature of \(\omega_\tau\) a corollary of the Chern-Lu inequality has been used which, rather than the lower bound of the curvature of the background metric requires an upper bound on it. Along with the observation that the curvature of \(\omega_\tau\) is always bounded from above, the existence of K\"ahler-Einstein metrics are proved in \cite{je-ma-ru}. Using a different approach to the Chern-Lu inequality, X.-X. Chen \emph{et al} have also derived the laplacian estimate in the Fano case in \cite{ch-do-su-2}. This has been further refined by Yao in \cite{yao}.
In a different direction, in \cite{gu-pa} one finds a clever adaptation of the classical calculations to the edge-cone setting by using an auxiliary function to control the behaviour of the possibly unbounded curvature terms in the study of the complex Monge-Amp\`ere equation. The interested reader is referred to Chapter 7 of the survey article by Rubinstein \cite{rub}, wherein an extensve treatment of the laplacian estimates in the works mentioned above can be found.
However, one core idea in the present work is the observation that indeed the Pogorelov-Aubin-Yau approach can be used with very little modification if one observes that within the same edge-cone cohomology class there always exist metrics with lower bound on their bisectional curvature, and that with respect to this reference metric, the Ricci potential has the correct behaviour. This allows us to show the following.
\begin{theorem}
\label{thm-1-1}
(a) [Calabi's first problem] Let \(\tilde{\varrho} \in c_1(M) - (1- \tau) c_1(L) \in H^{1,1}(M,\mathbb{R}) \) be a closed
real (1,1)-form which is of class \(C^\alpha\) on \(M \sim D\),
such that \((\operatorname{tr}_{\omega_\tau} \tilde{\varrho})^+ = o(|s|^{-2\tau})\) for some \(\epsilon>0\), and in the vicinity that \(\tilde{\varrho}\) is generated by a local
potential of the H\"older class \(C^\alpha_\theta\) for some exponent \(\alpha\). Then, there is a potential \(\phi\), which belongs to the class \(C^{2,\theta}_\tau\) for some exponent \(\theta\), determined uniquely up to a constant, such that \begin{equation} \label{prescribe-ricci}
\varrho(\omega_\tau + dd^c \phi) = \tilde{\varrho} +2 \pi (1 - \tau) [D] \end{equation}
(b) [Calabi's second problem] Assume that \(c(M)- (1 - \tau)c_1(L) = \mu [\omega_0]\), where, either i) \(\mu \leq 0\), or ii) we have an \(L^\infty\) bound on the potential \(\phi\). Then, there is a potential, belonging to the class \(C^{2,\theta}_\tau\) for some exponent \(\theta\), such that \begin{equation} \label{twisted-ke}
\varrho( \omega_{\tau,\phi}) = \mu \omega_{\tau,\phi} + 2 \pi(1 - \tau) [D] \end{equation} for \(\omega_{\tau,\phi}=\omega_\tau + dd^c \phi\). Further, the potential \(\phi\) is unique up to addition of a constant when \(\mu = 0\), and is unique otherwise.
\end{theorem}
As in the smooth case, the solution of the Calabi problem relies on solving a complex Monge-Amp\`ere equation, but this time with an edge K\"ahler reference metric. This connection, which calls for a bit more careful handling than in the smooth case, will be clarified in \textsection \ref{mise-en-equation}.
\begin{theorem} \label{thm-1-2}
Assume that the metric \(\omega_\tau\) is as defined in \eqref{cone-reference}, and that for some exponent \(\alpha \in (0,1)\), \(f \in C^\alpha_\tau\)
is a function such that \( (\Delta f)^- = o( |s|_h^{-2 \tau})\).
Then, there exists a solution \(\phi\) to the following equation:
\begin{equation}\label{eq-1-4}
(\omega_{\tau} + dd^c \phi)^n = e^f \omega_{\tau}^n; \text{ } \int e^f \omega_{\tau}^n = \int \omega_{\tau}^n
\end{equation}
which is unique up to a constant and belongs to the edge-cone H\"older space \(C^{2,\theta}_\tau\) for some exponent \(\theta\). \end{theorem}
In order to solve the equation, we also use idea of approximating the edge-cone metric by a family of smooth metrics similar to what is done in \cite{ca-gu-pa, gu-pa}. An important observation that allows deriving estimates independent of the upper bound on the scalar curvature of the referrence metric is also borrowed from an earlier work of P\v{a}un \cite{pa}.
The proof of the theorems above relies on the following proposition which allows us to explicitly construct a good reference metric.
\begin{proposition} \label{good-metric-pro}
Let \(\omega_0\) be a smooth K\"ahler metric and let \(\tau \in (0,1)\) and \(\tau' \in (\tau,1)\) be two real numbers.
Then, for sufficiently small positive constant \(\mathfrak{c} > 0\), the following (1,1)-form
\begin{equation} \label{good-metric-eq-0}
\tilde{\omega} := \omega_\tau - \mathfrak{c} dd^c |s|_h^{2 \tau'}=
\omega_0 + \mathfrak{a} dd^c |s|_h^{2 \tau} - \mathfrak{c} dd^c |s|_h^{2 \tau'}
\end{equation}
is an edge-cone K\"ahler metric of cone angle \(\tau\), equivalent to the following edge-come metric:
\begin{equation}
\omega_{\tau}:= \omega_0 + \mathfrak{a} dd^c |s|_h^{2 \tau}
\end{equation}
in such a way that the curvature of \(\tilde{\omega}\) is bounded from below.
Further,
the parameter \(\mathfrak{c}\) can be chosen to be sufficiently small so that the metrics \(\omega_\tau\) and \(\tilde{\omega}_\tau\) are arbitrarily close
with respect to the H\"older norm \(C^\theta_\tau\) . \end{proposition}
Having proved the proposition above, we have to study how the equation and its components, in particular, the Ricci potential, transform under this change of the reference metric and then prove that we can indeed derive the estimates. The proof of this proposition and the study of how the equation transform under the change of the reference metric is the subject of \textsection \ref{good-metric-sec}.
Besides opening way for estimates previously known in the smooth case, which we shall hopefully explore elsewhere, one advantage of this approach is that this method is based on the geometry of the edge-cone metrics and we, therefore, hope that such observations about conical metrics will be of interest in their own right. Further, this approach allows a wide class of preassigned Ricci forms to be realised by edge-cone K\"ahler metrics.
\section{Edge-cone functional spaces, Linear elliptic theory }
In this section, we shall introduce the notation and basic concepts we shall frequently make use of in the rest of this note. For the definition of spaces we follow \cite{do}, where the linear elliptic theory for edge-cone metrics is developed.
Let us for the sake of clarity work on \(\mathbb{C}^n\) and assume that the edge-cone singularity occurs along the divisor \(\sum_{j=1}^k (1-\tau_j) [\mathfrak{z}_j = 0]\). The edge-cone K\"ahler form we consider as our model on \(\mathbb{C}^n\) is the following: \begin{equation} \label{cone-model}
\omega_{\mathfrak{K}} :=
dd^c \big(
\sum_{j=1}^k |\mathfrak{z}_j|^{2\tau_j} + \sum_{j=k+1}^n |\mathfrak{z}_j|^2
\big)
=
\sum_{j=1}^k \mathfrak{i} \tau_j^2 \vert \mathfrak{z}_j \vert^{2\tau_j - 2} d \mathfrak{z}_j \wedge d \bar{\mathfrak{z}}_j
+
\sum_{j=k+1}^n \mathfrak{i} d \mathfrak{z}_j \wedge d \bar{\mathfrak{z}}_j \end{equation}
To keep our notation simpler, in the following definitions we shall only state the case of a single divisor along \([\mathfrak{z}_1 = 0]\). The definitions can be extended to the case of multiple divisors with possible normal crossing in the obvious way.
\begin{definition} Consider the K\"ahler space \(\mathbb{C}_\tau \times \mathbb{C}^{n-1}\). For a given function \(f(\mathfrak{z}^1,..., \mathfrak{z}^n)\) define the associated function \(\tilde{f}(\xi, ... , \mathfrak{z}^n)\)
where \(\xi = |\mathfrak{z}_1|^{\tau - 1} \mathfrak{z}_1\). Then, \(f\) is said to be of class \(C^\alpha_\tau\) provided that \(\tilde{f} \in C^\alpha\). \end{definition}
There is another change of variable which we shall use which is compatible with the picture one has of in the case of conical angle \(\tau={1 \over p}\), for \(p \in \mathbb{N}\), on \(\mathbb{C}_{1 \over p}\), which can be \emph{uniformised} via the map \begin{eqnarray}
\mathbb{C}_{1 \over p} &\to& \mathbb{C} \nonumber \\
\mathfrak{z} &\mapsto& \mathfrak{z}^{1 \over p} =: \mathfrak{w} \end{eqnarray} The advantage of this transformation, unlike transformation \(\mathfrak{z} \mapsto \xi\), is that change of variable to \(\mathfrak{w}\) is a -local- bi-holomorphism, and hence we can calculate geometric quantities such as curvature in \(\zeta\). It is important to observe that \(\mathfrak{w}\) and and \(\xi\), defined above, only disagree in the angle variable. This, in particular, means that the pull-backs of the euclidean metric by the two transformations define equivalent distances. This allows us to define the H\"older spaces using either transformation.
Further, in either case, the function \(\vert \mathfrak{z} \vert^2\),
which is the K\"ahler potential of the flat metric on \(\mathbb{C}\), is sent to the function \(\vert \mathfrak{z} \vert^{2 \over p}\). We have defined the spaces \(C^{k,\alpha}_\tau\) in such a way that functions such as \(\vert \mathfrak{z} \vert^{2 \over p}\) will belong to them.
From this picture it should be clear that there is a more intrinsic way of defining the H\"older spaces \(C^{\alpha, \beta}\) which is the content of the following definition. We shall indeed make use of this equivalence later.
\begin{definition}
For a given function \(f\), define the semi-norm \([f]^\tau_{\alpha}\) as follows:
\[
[f]^\tau_{\alpha} := \sup_{x, y} {|f(x) - f(y)| \over d_\tau(x,y)^\alpha}
\]
Also, define the \(C^{\alpha}_\tau\) norm of the function \(f\) to be
\(\Vert f \Vert_{\alpha,\tau} = \Vert f \Vert_0 + [f]^\tau_{\alpha}\),
and let \(C^{\alpha}_\tau\) designate the space of all functions having finite \(\Vert . \Vert_{\alpha,\tau}\) bound. \end{definition}
Using the \(C^{\alpha}_\tau\) spaces for functions, we can now define \(C^{\alpha}_\tau\) forms and thereby define the space \(C^{2,\alpha}_\tau\).
\begin{definition}
Let \(\sigma\) be a \((1,0)\)-form. We say define:
\[
\Vert \sigma \Vert_{\alpha,\beta} := \sup
\] \end{definition}
Following Donaldson's definition of edge-cone H\"older spaces we recall the following definitions.
\begin{definition} We say that a function \(f\) is \(C^\alpha(\mathcal{S}_\tau)\) provided that \(f \) is \(\alpha\)-H\"older continuous on the the sector \(\arg \zeta \in [0,2 \pi \tau)\) with the two rays \(\tau =0\) and \(\tau = 2 \pi \tau\) identified outside of the origin. The spaces \(C^{k,\alpha}\) are defined similarly. \end{definition}
\section{Setting up the equation}
\label{mise-en-equation}
In this section, we shall see how to reduce the proof of Theorem \ref{thm-1-1} can be reduced to Theorem \ref{thm-1-2}. To this end, we study the behaviour of various components of the complex Monge-Amp\`ere equation written with respect to the reference metrics \eqref{cone-reference} and \eqref{good-metric-eq-0}. Here we will assume some of the results in \textsection \ref{good-metric-sec}.
In the case of smooth background K\"ahler metric \(\omega_0\) on a manifold which satisfies \(\mu \omega_0 \in c_1(M) \), when one tries to find a K\"ahler-Einstein metric with constant \(\mu \in \{-1, 0, 1\}\), the function \(f\) on the right hand side of the complex Monge-Amp\`ere equation \((\omega_0 + dd^c \phi)^n = e^{f - \mu\phi} \omega_0^n \) can be obtained by solving the following equation for \(f\): \[
\varrho(\omega_0) - \mu \omega_0 = dd^c f \]
However, as we shall see in \textsection \ref{good-metric-sec} the Ricci-form of the family of metrics \(\omega_\epsilon\) and \(\tilde{\omega}_\epsilon\) may not be bounded in general, the former from below and latter from above.
This lack of lower bound on the curvature of \(\omega_0 + dd^c |s|_h^{2 \tau}\) has already been well-known as a difficulty in the theory of edge-cone K\"ahler spaces as certain estimates depend on the lower bound of the curvature of the reference metric. But This, in particular, implies that we cannot assume a laplacian bound on \(f\).
To make this idea more quantitative, we recall that by a \emph{edge-cone K\"ahler-Einstein} metric, \(\omega\), of cone angle \(\tau\) we mean one that satisfies \(\varrho(\omega) = \mu \omega +2\pi (1 - \tau) [D]\). Similar to the smooth case, we are lead to the following equation for \(f\) \begin{equation} \label{ricci-potential}
dd^c f = \varrho(\omega_{\mathfrak{K}}) - \mu \omega_{\mathfrak{K}} - 2 \pi(1 - \tau) [D] \end{equation} wherein \(\omega_{\mathfrak{K}}\) is an arbitrary edge-cone background metric.
First, as we shall see in \textsection 4, the curvature of the reference metric \(\tilde{\omega}_\tau \)
becomes unbounded from above at the rate of \(|s|^{2\tau - 4}\), more precisely, we have:
\[
\tilde{R}(.,.,.,.) \geq C g_\tau \boxtimes g_\tau |s|^{2 \tau' -4\tau}
\] wherein \(\boxtimes\) denotes the Kalkurni-Nomizu product of tensors. As a result, the Ricci form of the metric also behaves asymptotically as:
\[
\varrho(\tilde{\omega}_\tau) \geq C g_{\tau} |s|^{2\tau' -4\tau} \]
Therefore, in terms of the asymptotic behaviour close to the divisor in \ref{ricci-potential} the term \(\varrho(\tilde{\omega}_\tau) \) dominates the metric. As a result, the form \(dd^c f \) is bounded from below by a multiple of the K\"ahler form \(\tilde{\omega}_\tau \), and the laplacian \(\Delta_\tau f\) is bounded from below.
Similarly, assume that instead of a K\"ahler-Einstein metric we seek to realise a prescribed Ricci form \(\tilde{\varrho} \). The potential \(f\) will then haveto satisfy the following: \[
dd^c f = \varrho(\tilde{\omega}_\tau) - \tilde{\varrho} \] Again, we see that so long as
\((\operatorname{tr}_{\omega_\tau} \tilde{\varrho})^+ = \mathcal{O}(|s|^{-2\tau })\) we can find a \(\tau' \in (\tau,1) \) so that the Ricci form \(\tilde{\varrho} (\tilde{\omega}_\tau) \) will have blow up at a higher rate and will, thereby, dominate the behaviour and will make \(dd^c f \) a current bounded from below. In particular, this means that \(\Delta_\tau f \) will be bounded from below which is what we need for the laplacian estimate.
Having made these observations, one can follow the usual way of reducing the statement of Theorem \ref{thm-1-1} to that of Theorem \ref{thm-1-2}.
\section{Choosing a good metric and approximation by smooth family} \label{good-metric-sec} In deriving the laplacian estimates, it is evident in the calculations of the Pogorelov-Aubin-Yau approach that the estimate depends on the lower bound of the bisectional curvature. However, once we add a potential of the form \(\vert s \vert^{2 \tau}_h\) to a smooth background metric in the vicinity of the zero locus of \(s\) the curvature might become unbounded from below , for \(\tau > {1 \over 2}\). Outside of the divisor, the curvature of such metric is always bounded from above, however, as it has been shown in the appendix of \cite{je-ma-ru}. \begin{comment}
Brendle has extended the solution of the Calabi problem to the case when the curvature is bounded from below, in particular, when \(\beta \leq {1 \over 2}\) \cite{br}. In \cite{je-ma-ru}, a different approach has been adopted based on the Chern-Lu inequality which allows to work with the case where curvature might be bounded from above and not below. In \cite{gu-pa} a different approach has been proposed for dealing with the lack of lower bounds on the curvature via introducing an auxiliary function in the calculations of the laplacian estimates. This allows using a modification of the Pogorelov-Aubin-Yau approach rather than the Chern-Lu inequality. \end{comment}
Here, we shall show how one could perturb the metric in the same -edge- cohomology class so that the curvature of the metric will become bounded from below. Indeed, the perturbed meric and the metric \(\omega_\tau\) are close in a suitable H\"older norm. Hence, bounding the laplacian with respect to one of them will suffice for deriving {\it a priori} bounds on the complex hessian.
The smallness of the parameter \(\mathfrak{c}\) guarantees that the (1,1)-form thus obtained is positive definite and therefore a K\"ahler metric. Obviously, we can always scale the metric \(h\) defined on the line bundle so that the expression \(\tilde{\omega}=\omega_0 + dd^c |s|_h^{2 \tau} - dd^c |s|_h^{2 \tau'}\) is positive definite. Indeed we shall assume this from now on and will drop the coefficients \(\mathfrak{a}\) and \(\mathfrak{c}\).
It is easy to observe in the proof that the correction \(|s|_h^{2 \tau'}\) to the potential does not change the curvature properties of the metric so much at the points away from the divisor.
Since we are going to construct the solution as the limit of a sequence of smooth solutions, each of which is obtained by solving with respect to a smooth reference metric, we need the following lemma which allows proving uniform estimate for a sequence of smooth approximations of \(\tilde{\omega}_\epsilon\).
\begin{lemma} \label{good-metric-lemma} The family \(\{ \omega_\epsilon \}_\epsilon\) defined as:
\begin{equation} \label{good-metric-eq}
\tilde{\omega}_\epsilon = \omega_0 + dd^c \big( (|s|_h^2 + \epsilon )^ \tau - (|s|_h^2 + \epsilon)^ {\tau'} \big) \end{equation} is a smooth family approximating \(\tilde{\omega}\). Further, elements of \(\tilde{\omega}_\epsilon\) have a uniform (in the parameter \(\epsilon\)) lower bound on their curvature
\end{lemma}
Indeed, what we shall prove is that in curvature of these metrics satisfy \(R^\epsilon(v, \bar{v}, w, \bar{w}) \geq C \vert v \vert_\tau \vert w \vert_\tau\) for some constant \(C\). The following proof essentially shows both Proposition \ref{good-metric-pro} and Lemma \ref{good-metric-lemma}. Indeed the lemma is just a small but useful observation about the content of Proposition \ref{good-metric-pro}. The rest of this section is dedicated to the proof of the proposition.
\subsection{Proof of Proposition \ref{good-metric-pro}}
Let us notice that a function of the form \(\eta = \vert s \vert_h^{2 \tau'}\), where \(\tau' \in ( \tau', 1)\), belongs to \(C^{2,\alpha}_\tau\) for some exponent \(\alpha\) in the sense defined before. In particular, \(\eta\) is a valid K\"ahler potential to be added to the metric. Let us first prove that
\(\tilde{\omega}\) is indeed a metric equivalent to \(\omega_\tau\). It will suffice to show that the complex hessian of the correction potential, \(dd^c |s|^{2 \tau'}\), is bounded, as a (1,1)-form, when measured with respect to the metric \(\omega_\tau\). Just as it is done already in \eqref{tilde-omega-good-coord}, by calculating the complex hessian of the potential \(|s|_h^{2 \tau'}\) in the special coordinates we see that the expression
\[ \Vert \tilde{\omega} - \omega_\tau \Vert_{\omega_\tau}^2 = g^{\kappa \bar{\lambda}}_\tau g^{\mu \bar{\nu}}_\tau \big( |s|_h^{2 \tau'} \big)_{,\mu \bar{\lambda}} \big( |s|_h^{2 \tau'} \big)_{,\kappa \bar{\nu}} \]
consists of terms of the form \(g^{\kappa \bar{\lambda}}_\tau g^{\mu \bar{\nu}}_\tau M_{,\kappa \bar{\nu}} M_{,\mu \bar{\lambda}} |\mathfrak{z}^1|^{4 \tau'} \), which are finite, and when all indices are equal to 1, other terms which are dominated by \(g^{1 \bar{1}}_\tau g^{1 \bar{1}}_\tau M^2 |\mathfrak{z}^1|^{4 \tau' - 4} \). Upon noticing that \(g^{1 \bar{1}} = \mathcal{O}( |\mathfrak{z}_1|^{2 - 2\tau})\), and that \(\tau' > \tau\), one concludes that \( \Vert \tilde{\omega} \Vert_{\omega_\tau} = \mathcal{O}(|\mathfrak{z}^1|^{2( \tau' - \tau)} ) \) which is finite and, indeed, tends to zero as the points approach the divisor.
\iffalse
Preliminary calculations: \begin{eqnarray}
\left[ (\epsilon + K |\mathfrak{z}^1|^2)^\beta \right]_{,i}
&=&
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta - 1} \left[ K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K \right] \nonumber \\
&=& \beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta - 1} \delta_{1i} \bar{\mathfrak{z}}_1 K \\
\left[ (\epsilon + K |\mathfrak{z}^1|^2)^\beta \right]_{,\bar{j}}
&=&
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta - 1} \left[ K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1j} \mathfrak{z}^1 K \right] \nonumber \\
&=&
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta - 1} \delta_{1j} \mathfrak{z}^1 K \\
{1 \over \beta} \left[ (\epsilon + K |\mathfrak{z}^1|^2)^\beta \right]_{,i \bar{j}} &=&
\nonumber \\
(\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
(\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1} &[& K_i |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K]_{\bar{j}}= \nonumber \\
(\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
(\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{i \bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K_i
+ \delta_{1i} \delta_{1j} K + \delta_{1i} \bar{\mathfrak{z}}_1 K_{\bar{j}}] \nonumber =\\
(\beta -1) (\epsilon + K|\mathfrak{z}^1|^2 )^{\beta-2} \delta_{1j} \delta_{1i} |\mathfrak{z}^1|^2 K^2
&+&
(\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1} (K_{,i\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1i} \delta_{1j} K) \end{eqnarray}
Now, let us calculate the components of the metric:
\begin{eqnarray}
g_{i \bar{j}}^\epsilon = g^0_{i \bar{j}} + \left[ (\epsilon + K |\mathfrak{z}^1|^2)^\beta \right]_{,i \bar{j}}
+
\left[ (\epsilon + K |\mathfrak{z}^1|^2)^\gamma \right]_{,i \bar{j}} \end{eqnarray}
which in the coordinates chosen properly assumes the following form:
\begin{eqnarray}
g_{i \bar{j}}^\epsilon &=& g^0_{i \bar{j}} + \beta (\beta -1) (\epsilon + K|\mathfrak{z}^1|^2 )^{\beta-2} \delta_{1j} \delta_{1i} |\mathfrak{z}^1|^2 K^2 +
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1} (K_{,i\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1i} \delta_{1j} K) \nonumber \\
&+&\gamma (\gamma -1) (\epsilon + K|\mathfrak{z}^1|^2 )^{\gamma-2} \delta_{1j} \delta_{1i} |\mathfrak{z}^1|^2 K^2 +
\gamma (\epsilon + K |\mathfrak{z}^1|^2)^{\gamma -1} (K_{,i\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1i} \delta_{1j} K)
\nonumber \end{eqnarray}
The first derivatives of the metric:
\begin{eqnarray}
g_{i \bar{j},k}^\epsilon &=& g^0_{i \bar{j},k} + \\
\beta (\beta -1) [(\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2}]_{,k} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K)_{,k} ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K)_{,k} \nonumber \\
+
\beta [(\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1} ]_{,k}
&[& K_{i \bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K_i
+ \delta_{1i} \delta_{1j} K + \delta_{1i} \bar{\mathfrak{z}}_1 K_{\bar{j}}] \nonumber \\
+
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{i \bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K_i
+ \delta_{1i} \delta_{1j} K + \delta_{1i} \bar{\mathfrak{z}}_1 K_{\bar{j}}]_{,k} \nonumber \\
&=& \nonumber \\
\beta (\beta -1)(\beta - 2) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-3} &(& K_{,k}|\mathfrak{z}^1|^2 + \delta_{1k} K \bar{\mathfrak{z}}_1) (K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j} k} |\mathfrak{z}^1|^2 + \delta_{1k} K_{\bar{j}} \bar{\mathfrak{z}}_1 + \delta_{1j} \delta_{1k} K + \delta_{1j} \mathfrak{z}^1 K_{,k} ) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,ik} |\mathfrak{z}^1|^2 + \delta_{1k} K_{,i} \bar{\mathfrak{z}}_1 + \delta_{1i} \bar{\mathfrak{z}}_1 K_{,k}) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -2} &(&K_{,k} |\mathfrak{z}^1|^2 + \delta_{1k} \bar{\mathfrak{z}}_1 K)
[ K_{i \bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K_i
+ \delta_{1i} \delta_{1j} K + \delta_{1i} \bar{\mathfrak{z}}_1 K_{\bar{j}}] \nonumber \\
+
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{,i \bar{j} k} |\mathfrak{z}^1|^2 + \delta_{ik} K_{i \bar{j}} \bar{\mathfrak{z}}_1 +
\delta_{1 j} \delta_{1k} K_i + \delta_{1j} \mathfrak{z}^1 K_{,ik}
+ \delta_{1i} \delta_{1j} K_{,k} + \delta_{1i} \bar{\mathfrak{z}}_1 K_{,\bar{j} k}] \nonumber \\
&=& \nonumber \\
\beta (\beta -1)(\beta - 2) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-3} &(& \delta_{1k} K \bar{\mathfrak{z}}_1) ( \delta_{1 j} \mathfrak{z}^1 K) ( \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(& \delta_{1j} \delta_{1k} K ) (\delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -2} &(& \delta_{1k} \bar{\mathfrak{z}}_1 K)
[ K_{i \bar{j}} |\mathfrak{z}^1|^2
+ \delta_{1i} \delta_{1j} K] \nonumber \\
+
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{,i \bar{j} k} |\mathfrak{z}^1|^2 + \delta_{ik} K_{i \bar{j}} \bar{\mathfrak{z}}_1 + + \delta_{1i} \bar{\mathfrak{z}}_1 K_{,\bar{j} k}] \nonumber \\ \end{eqnarray} [[The last ones are in special coord's]].\\
Similarly, in the good coordinate system, we have
\begin{eqnarray}
g^\epsilon_{i \bar{j}, \bar{l}} &=& g^0_{i \bar{j}, \bar{l}} + \\
\beta (\beta -1)(\beta - 2) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-3} &(& K_{,\bar{l}}|\mathfrak{z}^1|^2 + \delta_{1l} K \mathfrak{z}^1) (K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(& K_{,\bar{j} \bar{l}} |\mathfrak{z}^1|^2 + \delta_{1l} K_{,\bar{j}} \mathfrak{z}^1 + \delta_{1j} \mathfrak{z}^1 K_{,\bar{l}} ) ( K_{,i} |\mathfrak{z}^1|^2 + \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(&K_{,\bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i \bar{l}} |\mathfrak{z}^1|^2 + \delta_{1l} K_{,i} \mathfrak{z}^1 + \delta_{1i} \bar{\mathfrak{z}}_1 K_{,\bar{l}} + \delta_{1i} \delta_{1l} K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -2} &(&K_{,\bar{l}} |\mathfrak{z}^1|^2 + \delta_{1l} \mathfrak{z}^1 K)
[ K_{i \bar{j}} |\mathfrak{z}^1|^2 + \delta_{1 j} \mathfrak{z}^1 K_i
+ \delta_{1i} \delta_{1j} K + \delta_{1i} \bar{\mathfrak{z}}_1 K_{\bar{j}}] \nonumber \\
+
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{,i \bar{j} \bar{l}} |\mathfrak{z}^1|^2 + \delta_{il} K_{i \bar{j}} \mathfrak{z}^1 +
+
\delta_{1j} \mathfrak{z}^1 K_{,i \bar{l}}
+ \delta_{1i} \delta_{1j} K_{,\bar{l}} + \delta_{1i} \delta_{1l} K_{,\bar{j}}+ \delta_{1i} \bar{\mathfrak{z}}_1 K_{,\bar{j} \bar{l}}] \nonumber \\
&=&\nonumber \\
\beta (\beta -1)(\beta - 2) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-3} &(& \delta_{1l} K \mathfrak{z}^1) (+ \delta_{1 j} \mathfrak{z}^1 K) ( \delta_{1i} \bar{\mathfrak{z}}_1 K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta-2} &(& \delta_{1 j} \mathfrak{z}^1 K) ( K_{,i \bar{l}} |\mathfrak{z}^1|^2 + \delta_{1i} \delta_{1l} K) \nonumber \\
+
\beta (\beta -1) (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -2} &(&\delta_{1l} \mathfrak{z}^1 K)
[ K_{i \bar{j}} |\mathfrak{z}^1|^2
+ \delta_{1i} \delta_{1j} K ] \nonumber \\
+
\beta (\epsilon + K |\mathfrak{z}^1|^2)^{\beta -1}
&[& K_{,i \bar{j} \bar{l}} |\mathfrak{z}^1|^2 + \delta_{il} K_{i \bar{j}} \mathfrak{z}^1 +
+
\delta_{1j} \mathfrak{z}^1 K_{,i \bar{l}}
] \nonumber
\end{eqnarray}
We can now find the second derivative:
\fi
In order to study the curvature tensor we shall use the following lemma which will simplify our calculations. This observation seems to have been first stated in \cite{ti-ya}, in the proof of Lemma 4.3 in \cite{ti-ya}, and also used in the proof of the existence of an upper bound on the curvature of edge-cone metrics in \cite{je-ma-ru}. The version we state below as well as the proof may be found in \cite{ca-gu-pa}, where it appears as Lemma 4.1.
\begin{lemma}
\label{good-coord} \label{lemma-ti-ya}
Assume that for \(j \in \{1, ..., n_0\}\) the \(D_j\)'s are irreducible divisors with at most normal crossing with associated line bundles \(L_j\) with hermitian metrics \(h_j\) and defining sections of the corresponding line bundles \(s_j\). Let \(p \in \cap D_j \). Then, there is a neighbourhood of \(p_0\) in which for any point \(p\) there is a choice of local coordinates for the manifold \(M\) and trivialisations \(\theta_j\) for the line bundles \(L_j\) on an open set \(U\) so that
\begin{enumerate}
\item the hypersurfaces locally correspond to flat hyperplanes:
\(U \cap D_j = [z_j =0]\),
\item if the herimitian metric \(h_j\) is represented by \(e^{-\phi}\) in the trivialisation \(\theta_j\), then at the point \(p\) we have
\[
\phi_j(0) = 0, \quad d \phi_j(p) = 0, \quad
{\operatorname{\partial} \phi \over \operatorname{\partial} \mathfrak{z}^\alpha \operatorname{\partial} \mathfrak{z}^\beta}(p)= 0.
\]
\end{enumerate}
Further, all higher derivatives of \(\phi\) are bounded uniformly when the point varies on a compact subset of \(U\).
\end{lemma}
\begin{remark}
It is probably worth mentioning that the above lemma will be used to estimate the rate of blow-up of quantities in terms of \(\mathfrak{z}_1\)
However, in order for such an estimate to make sense one has to also notice that although the coordinates chosen do depend on the point \(p\), all the coordinates chosen, in particular \(\mathfrak{z}_1\), are uniformly equivalent as the point \(p\) varies on a compact set.
In particular, it is well-defined to speak of the rate of blow-up or the rate of vanishing in terms of powers of \(\mathfrak{z}^1\). \end{remark}
By bounds on the curvature we mean \(R(v, \bar{v}, w, \bar{w}) = R_{\alpha \bar{\beta} \gamma \bar{\delta}} v^\alpha \bar{v}^{\beta} w^\gamma \bar{w}^\delta\) when \(v= \operatorname{\partial}_\alpha v^\alpha\) and \(w = \operatorname{\partial}_\alpha w^\alpha\) are of unit norm with respect to the edge-cone metric \(\omega_{\mathfrak{K}}\). Since in our adopted coordinate system the metric satisfies \(g_{1 \bar{1}} \approx |\mathfrak{z}_1|^{2\tau - 2}\), we have that \(\vert v_1 \vert , \vert w_1 \vert \leq C |\mathfrak{z}_1|^{1 - \tau \over 2}\). Therefore, in studying the terms appearing in the curvature tensor, we shall consider only the once that persist as -potentially- infinite terms after multiplying \( |\mathfrak{z}^1|^{2 - 2\tau}\).
We have to prove that the family of metrics \(
\tilde{\omega}_\epsilon = \omega_0 + dd^c \big ( (\epsilon + |s|_h^2)^ {\tau} + {1 \over N} (\epsilon + |s|_h^2)^{\tau'} \big )\) has a uniform lower bound on the curvature tensor.
The coefficient \({1 \over N}\) is added to make sure that \(\tilde{\omega}_\epsilon\) stays positive definite. It is easy to see that the metric \(h\) on the line bundle \(L\) can always be scaled so that the positivity condition holds for the following family of metrics defined in \eqref{good-metric-eq}. Therefore, without loss of generality, we shall assume from now on that \(N=1\). One can find the curvature tensor \(R^\epsilon_{\alpha \bar{\beta} \gamma \bar{\delta}}\) by differentiating the metric.
However, one may note that since the components of the curvature tensor, written in coordinates, are combinations of various powers of \( \epsilon + |s|_h^2 \) and \(|s|_h\), it will suffice to show the claim when the parameter \(\epsilon \) is zero and that will prove the proposition for the entire range of the parameter \(\epsilon\).
In the coordinate system constructed in Lemma \ref{lemma-ti-ya} we can write \(|s|_h^{2 \tau} = a^\tau |\mathfrak{z}|^{2 \tau}\), and also \(|s|_h^{2 \tau'} = a^\gamma |\mathfrak{z}|^{2 \tau'}\). Let us keep make the following substitution in order to keep the notation simpler: \(K:= a^\tau\), \(M := a^{\tau'}\). Evidently, in the special coordinates of Lemms \ref{good-coord} we have at the point \(p\) that \(M(p)=K(p)=1\), \(dK(p)=dM(p)=0, K,_{\alpha \beta} = M,_{\alpha \beta} = 0\), for \( \alpha, \beta = 1 ... n\). Since we have taken \(\epsilon = 0\), we have: \[
\tilde{g}_{\alpha \bar{\beta}} =
g^0_{\alpha \bar{\beta}} + \big( K |\mathfrak{z}|^{2 \tau} - M |\mathfrak{z}|^{2 \tau'} \big)_{,\alpha \bar{\beta}} \] By differentiating directly we obtain: \begin{eqnarray} \label{tilde-omega}
\tilde{g}_{\alpha \bar{\beta}} &=& g^0_{\alpha \bar{\beta}}
+ K_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2 \tau}
+ \tau \delta_{1\beta} K_{,\alpha} |\mathfrak{z}|^{2 \tau -2} \mathfrak{z}
+ \tau \delta_{1 \alpha} K_{,\bar{\beta}} |\mathfrak{z}|^{2 \tau - 2} \bar{\mathfrak{z}}
+ \tau^2 \delta_{1 \alpha} \delta_{1 \beta} K |\mathfrak{z}|^{2 \tau -2} \nonumber \\
&-&
M_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2 \tau'}
- \tau' \delta_{1\beta} M_{,\alpha} |\mathfrak{z}|^{2 \tau' -2} \mathfrak{z}
- \tau' \delta_{1 \alpha} M_{,\bar{\beta}} |\mathfrak{z}|^{2 \tau' - 2} \bar{\mathfrak{z}}
- \tau'^2 \delta_{1 \alpha} \delta_{1 \beta} M |\mathfrak{z}|^{2 \tau' -2} \end{eqnarray} Which, in the coordinates of Lemma \ref{lemma-ti-ya} simplifies to the following \begin{eqnarray} \label{tilde-omega-good-coord}
\tilde{g}_{\alpha \bar{\beta}} &=& g^0_{\alpha \bar{\beta}}
+ K_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2 \tau}
+ \tau^2 \delta_{1 \alpha} \delta_{1 \beta} K |\mathfrak{z}|^{2 \tau -2}
-
M_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2 \tau'}
- \tau'^2 \delta_{1 \alpha} \delta_{1 \beta} M |\mathfrak{z}|^{2 \tau' -2} \nonumber \end{eqnarray} We now have for the first derivatives of the metric that \begin{eqnarray}
\tilde{g}_{\alpha \bar{\beta},\gamma} &=&
g^0_{\alpha \bar{\beta}, \gamma} + K_{,\alpha \bar{\beta} \gamma} |\mathfrak{z}|^{2 \tau}
+
\tau \delta_{1 \beta} K_{,\alpha \gamma} |\mathfrak{z}|^{2 \tau -2} \mathfrak{z}
+
\tau ( \delta_{1 \alpha} K_{,\bar{\beta} \gamma} + \delta_{1 \gamma} K_{,\alpha \bar{\beta}} ) |\mathfrak{z}|^{2\tau - 2} \bar{\mathfrak{z}} \nonumber \\
&+&
\tau^2 (\delta_{1 \beta} \delta_{1 \gamma} K_{,\alpha} + \delta_{1 \alpha} \delta_{1 \beta} K_{,\gamma}
+ \delta_{1 \alpha} \delta_{1 \gamma} K_{,\bar{\beta}} ) |\mathfrak{z}|^{2 \tau - 2}
+ \tau^2 ( \tau - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} K |\mathfrak{z}|^{2 \tau - 4} \bar{\mathfrak{z}} \nonumber \\
&-& M_{,\alpha \bar{\beta} \gamma} |\mathfrak{z}|^{2 \tau'}
-
\tau' \delta_{1 \beta} M_{,\alpha \gamma} |\mathfrak{z}|^{2 \tau' -2} \mathfrak{z}
-
\tau' ( \delta_{1 \alpha} M_{,\bar{\beta} \gamma} + \delta_{1 \gamma} M_{,\alpha \bar{\beta}} ) |\mathfrak{z}|^{2\tau' - 2} \bar{\mathfrak{z}} \nonumber \\
&-&
\tau'^2 (\delta_{1 \beta} \delta_{1 \gamma} M_{,\alpha} + \delta_{1 \alpha} \delta_{1 \beta} M_{,\gamma}
- \delta_{1 \alpha} \delta_{1 \gamma} M_{,\bar{\beta}} ) |\mathfrak{z}|^{2 \tau' - 2}
- \tau'^2 ( \tau' - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} M |\mathfrak{z}|^{2 \tau' - 4} \bar{\mathfrak{z}} \end{eqnarray} and in the coordinates of Lemma \ref{lemma-ti-ya} this simplifies to: \begin{eqnarray}
\tilde{g}_{\alpha \bar{\beta},\gamma} &=&
g^0_{\alpha \bar{\beta}, \gamma} + K_{,\alpha \bar{\beta} \gamma} |\mathfrak{z}|^{2 \tau}
+
\tau ( \delta_{1 \alpha} K_{,\bar{\beta} \gamma} + \delta_{1 \gamma} K_{,\alpha \bar{\beta}} ) |\mathfrak{z}|^{2\tau - 2} \bar{\mathfrak{z}}
+ \tau^2 ( \tau - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} K |\mathfrak{z}|^{2 \tau - 4} \bar{\mathfrak{z}} \nonumber \\
&-& M_{,\alpha \bar{\beta} \gamma} |\mathfrak{z}|^{2 \tau'}
-
\tau' ( \delta_{1 \alpha} M_{,\bar{\beta} \gamma} + \delta_{1 \gamma} M_{,\alpha \bar{\beta}} ) |\mathfrak{z}|^{2\tau' - 2} \bar{\mathfrak{z}}
- \tau'^2 ( \tau' - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} M |\mathfrak{z}|^{2 \tau' - 4} \bar{\mathfrak{z}} \end{eqnarray}
And similarly, the expression for \(g_{\alpha \bar{\beta}, \bar{\delta}}\) in the coordinates of Lemma \ref{lemma-ti-ya} is: \begin{eqnarray}
\tilde{g}_{\alpha \bar{\beta},\bar{\delta}} &=&
g^0_{\alpha \bar{\beta}, \bar{\delta}} + K_{,\alpha \bar{\beta} \bar{\delta}} |\mathfrak{z}|^{2 \tau}
+
\tau (\delta_{1 \beta} K_{,\alpha \bar{\delta}}
+
\delta_{1 \alpha} K_{,\bar{\beta} \bar{\delta}}) |\mathfrak{z}|^{2 \tau -2} \mathfrak{z} + \delta_{1 \delta} K_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2\tau - 2} \bar{\mathfrak{z}} \nonumber \\
&+&
\tau^2 (\delta_{1 \beta} \delta_{1 \delta} K_{,\alpha} + \delta_{1 \alpha} \delta_{1 \beta} K_{,\gamma}
+ \delta_{1 \alpha} \delta_{1 \delta} K_{,\bar{\beta}} ) |\mathfrak{z}|^{2 \tau - 2}
+ \tau^2 ( \tau - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \delta} K |\mathfrak{z}|^{2 \tau - 4} \mathfrak{z} \nonumber \\
&-& M_{,\alpha \bar{\beta} \bar{\delta}} |\mathfrak{z}|^{2 \tau'}
-
\tau' (\delta_{1 \beta} M_{,\alpha \bar{\delta}}
-
\delta_{1 \alpha} M_{,\bar{\beta} \bar{\delta}}) |\mathfrak{z}|^{2 \tau' -2} \mathfrak{z} +
\delta_{1 \delta} M_{,\alpha \bar{\beta}} |\mathfrak{z}|^{2\tau' - 2} \bar{\mathfrak{z}} \nonumber \\
&-&
\tau^2 (\delta_{1 \beta} \delta_{1 \delta} M_{,\alpha} + \delta_{1 \alpha} \delta_{1 \beta} M_{,\gamma}
- \delta_{1 \alpha} \delta_{1 \delta} M_{,\bar{\beta}} ) |\mathfrak{z}|^{2 \tau - 2}
- \tau'^2 ( \tau' - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \delta} M |\mathfrak{z}|^{2 \tau - 4} \mathfrak{z} \end{eqnarray}
Also, for the second derivative terms we have in the special coordinates that: \begin{eqnarray}
\tilde{g}_{\alpha \bar{\beta}, \gamma \bar{\delta}} \text{ } &=&
g^0_{\alpha \bar{\beta}, \gamma \bar{\delta}} + K_{,\alpha \bar{\beta} \gamma \bar{\delta}} |\mathfrak{z}^1|^{2 \tau}
+
\tau \big( \delta_{1 \beta} K_{,\alpha \gamma \bar{\delta}} + \delta_{1 \delta} K_{,\alpha \bar{\beta} \gamma} \big) |\mathfrak{z}^1|^{2 \tau - 2} \mathfrak{z}^1 \nonumber \\
&+&
\tau \big( \delta_{1 \gamma} K_{,\alpha \bar{\beta} \bar{\delta}} + \delta_{1 \alpha} K_{,\bar{\beta} \gamma \bar{\delta}} \big) |\mathfrak{z}^1|^{2\tau -2} \bar{\mathfrak{z}}_1 \nonumber \\
&+& \tau^2
\big ( \delta_{1 \alpha} \delta_{1\delta} K_{,\gamma \bar{\delta}}
+ \delta_{1 \gamma} \delta_{1 \delta} K_{,\alpha \bar{\beta}}
+ \delta_{1 \alpha} \delta_{1 \beta} K_{,\gamma \bar{\delta}}
+ \delta_{1 \beta} \delta_{1 \gamma} K_{,\alpha \bar{\delta}}
\big) |\mathfrak{z}^1|^{2\tau -2} \nonumber \\
&+& \tau (1 - \tau) \delta_{1 \beta} \delta_{1 \delta} K_{,\alpha \gamma} |\mathfrak{z}_1|^{2\tau - 4} \mathfrak{z}_1^2
+ \tau ( 1- \tau) \delta_{1 \alpha} \delta_{1 \gamma} K_{,\bar{\beta} \bar{\delta}} |\mathfrak{z}_1|^{2\tau - 4} \bar{\mathfrak{z}}_1^2
\big ) |\mathfrak{z}_1|^{2 \tau -2}
\nonumber \\
&+& \tau^2 ( 1 - \tau) \delta_{1 \alpha} \delta_{1 \gamma}
\big(
\delta_{1 \delta} K_{,\bar{\beta}}
+
\delta_{1 \beta} K_{,\bar{\delta}}
\big) |\mathfrak{z}^1|^{2 \tau -4} \bar{\mathfrak{z}}_1
+
\tau^2 (1 -\tau) \delta_{1 \beta} \delta_{1 \delta}
\big(
\delta_{1 \gamma} K_{,\alpha} + \delta_{1 \alpha} K_{,\gamma}
\big) |\mathfrak{z}^1|^{2 \tau - 4} \mathfrak{z}^1
\nonumber \\
&+& \tau^2 ( 1 - \tau)^2 \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} \delta_{1 \delta} K |\mathfrak{z}^1|^{2 \tau - 4} \nonumber \\
&-&
M_{,\alpha \bar{\beta} \gamma \bar{\delta}} |\mathfrak{z}^1|^{2 \tau'}
-
\tau' \big( \delta_{1 \beta} M_{,\alpha \gamma \bar{\delta}} + \delta_{1 \delta} K_{,\alpha \bar{\beta} \gamma} \big) |\mathfrak{z}^1|^{2 \tau' - 2} \mathfrak{z}^1
-
\tau' \big( \delta_{1 \gamma} M_{,\alpha \bar{\beta} \bar{\delta}} + \delta_{1 \alpha} M_{,\bar{\beta} \gamma \bar{\delta}} \big) |\mathfrak{z}^1|^{2\tau' -2} \bar{\mathfrak{z}}_1 \nonumber \\
&-& \tau'^2
\big ( \delta_{1 \alpha} \delta_{1\delta} M_{,\gamma \bar{\delta}}
- \delta_{1 \gamma} \delta_{1 \delta} M_{,\alpha \bar{\beta}}
- \delta_{1 \alpha} \delta_{1 \beta} M_{,\gamma \bar{\delta}}
- \delta_{1 \beta} \delta_{1 \gamma} M_{,\alpha \bar{\delta}}
\big) |\mathfrak{z}^1|^{2\tau' -2} \nonumber \\
&-& \tau' (1 - \tau') \delta_{1 \beta} \delta_{1 \delta} M_{,\alpha \gamma} |\mathfrak{z}^1|^{2\tau' - 4} \mathfrak{z}_1^2
- \tau' ( 1- \tau') \delta_{1 \alpha} \delta_{1 \gamma} M_{,\bar{\beta} \bar{\delta}} |\mathfrak{z}^1|^{2\tau' - 4} \bar{\mathfrak{z}}_1^2
\big ) |\mathfrak{z}^1|^{2 \tau' -2}
\nonumber \\
&-& \tau'^2 ( 1 - \tau) \delta_{1 \alpha} \delta_{1 \gamma}
\big(
\delta_{1 \delta} M_{,\bar{\beta}}
+
\delta_{1 \beta} M_{,\bar{\delta}}
\big) |\mathfrak{z}^1|^{2 \tau' -4} \bar{\mathfrak{z}}_1 \nonumber \\
&-&
\tau'^2 (1 -\tau') \delta_{1 \beta} \delta_{1 \delta}
\big(
\delta_{1 \gamma} M_{,\alpha} + \delta_{1 \alpha} M_{,\gamma}
\big) |\mathfrak{z}^1|^{2 \tau' - 4} \mathfrak{z}^1
- \tau'^2 ( 1 - \tau')^2 \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} \delta_{1 \delta} M |\mathfrak{z}^1|^{2 \tau' - 4} \end{eqnarray}
Which in the special coordinate system adopted at a given point becomes: \begin{eqnarray} \label{g-second-deriv}
\tilde{g}_{\alpha \bar{\beta}, \gamma \bar{\delta}} &=&
g^0_{\alpha \bar{\beta}, \gamma \bar{\delta}} + K_{,\alpha \bar{\beta} \gamma \bar{\delta}} |\mathfrak{z}^1|^{2 \tau}
+
\tau \big( \delta_{1 \beta} K_{,\alpha \gamma \bar{\delta}} + \delta_{1 \delta} K_{,\alpha \bar{\beta} \gamma} \big) |\mathfrak{z}^1|^{2 \tau - 2} \mathfrak{z}^1 \nonumber \\
&+&
\tau \big( \delta_{1 \gamma} K_{,\alpha \bar{\beta} \bar{\delta}} + \delta_{1 \alpha} K_{,\bar{\beta} \gamma \bar{\delta}} \big) |\mathfrak{z}^1|^{2\tau -2} \bar{\mathfrak{z}}_1 \nonumber \\
&+& \tau^2
\big ( \delta_{1 \alpha} \delta_{1\delta} K_{,\gamma \bar{\delta}}
+ \delta_{1 \gamma} \delta_{1 \delta} K_{,\alpha \bar{\beta}}
+ \delta_{1 \alpha} \delta_{1 \beta} K_{,\gamma \bar{\delta}}
+ \delta_{1 \beta} \delta_{1 \gamma} K_{,\alpha \bar{\delta}}
\big) |\mathfrak{z}^1|^{2\tau -2} \nonumber \\
&+& \tau^2 ( 1 - \tau)^2 \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} \delta_{1 \delta} K |\mathfrak{z}^1|^{2 \tau - 4} \nonumber \\
&-&
M_{,\alpha \bar{\beta} \gamma \bar{\delta}} |\mathfrak{z}^1|^{2 \tau'}
-
\tau \big( \delta_{1 \beta} M_{,\alpha \gamma \bar{\delta}} + \delta_{1 \delta} K_{,\alpha \bar{\beta} \gamma} \big) |\mathfrak{z}^1|^{2 \tau' - 2} \mathfrak{z}^1 \nonumber \\
&-&
\tau' \big( \delta_{1 \gamma} M_{,\alpha \bar{\beta} \bar{\delta}} + \delta_{1 \alpha} M_{,\bar{\beta} \gamma \bar{\delta}} \big) |\mathfrak{z}^1|^{2\tau' -2} \bar{\mathfrak{z}}_1 \nonumber \\
&-& \tau'^2
\big ( \delta_{1 \alpha} \delta_{1\delta} M_{,\gamma \bar{\delta}}
- \delta_{1 \gamma} \delta_{1 \delta} M_{,\alpha \bar{\beta}}
+\delta_{1 \alpha} \delta_{1 \beta} M_{,\gamma \bar{\delta}}
+ \delta_{1 \beta} \delta_{1 \gamma} M_{,\alpha \bar{\delta}}
\big) |\mathfrak{z}^1|^{2\tau' -2} \nonumber \\
&-& \tau'^2 ( 1 - \tau')^2 \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} \delta_{1 \delta} M |\mathfrak{z}^1|^{2 \tau' - 4} \end{eqnarray}
In order to study the behaviour of the terms with the inverse of the metric close to the divisor in its coordinate representation we shall need the following observation. \begin{lemma} \label{good-metric-inverse}
For the inverse matrix \(g^{\mu \bar{\nu}}\) we have:
\begin{itemize}
\item \(g^{1 \bar{1}} = {1 \over \tau^2} |\mathfrak{z}^1|^{2(1 - \tau)} \left(
1 + {\tau'^2 \over \tau^2} |\mathfrak{z}^1|^{2(\tau' - \tau)} + \mathcal{O} \big( |\mathfrak{z}^1|^{2(1 - \tau)}\big) \right) \),
\item \(g^{\mu \bar{1}} = \mathcal{O} \big( |\mathfrak{z}^1|^{2(1 - \tau)} \big)\), for \(\mu \neq 1\).
\end{itemize} \end{lemma}
\begin{proof}
We shall derive the aymptotic behaviour of the elements of \(g^{i \bar{j}}\) by studying the components \(g^{i\bar{j}}\) as quotients \({\det g^*_{i j} \over \det g} \), where \(g^*_{ij}\) is the minor obtained by removing the \(i\)-th row and \(j\)-th column.
Since in the coordinate systems introduced in Lemma \ref{good-coord} the only unbounded component of the metric tensor will be \(g^\epsilon_{1 \bar{1}} = 2 (\tau - 1) \mathcal{O}( |\mathfrak{z}^1|^{2(\tau -1)}) + \text{bounded terms}\), we see by expanding the determinant in the first row that:
\[
\det g = \big[
\tau^2|\mathfrak{z}^1|^{2(\tau -1)}
-
\tau'^2 |\mathfrak{z}^1|^{2(\tau' -1)}
\big] \det g^*_{11} + \text{bounded terms}
\]
and that \(\det g^*_{11}\) is bounded.
Therefore,
\({ \det g^*_{11} \over \det g} =
{\det g^*_{11} \over [ \tau^2 |\mathfrak{z}^1|^{2(\tau -1)}
-
\tau'^2 |\mathfrak{z}^1|^{2(\tau' -1)}] \det g^*_{11} + a_1}
\)
where \(a_j\)'s are bounded functions. One can now expand the quotient as
\begin{eqnarray}
\label{g-inverse-expansion}
g^{1 \bar{1}}={ \det g^*_{11} \over \det g} &=&
{1 \over \tau^2} |\mathfrak{z}^1|^{2(1 - \tau)} \Big[
1
+
{\tau'^2 \over \tau^2} |\mathfrak{z}^1|^{2(\tau' - \tau)}
+
{\tau'^2 \over \tau^2}{a_1 \over \det g_{11}^*}
K|\mathfrak{z}^1|^{2(1 - \tau)} \nonumber \\
&+& \mathcal{O} \big(
|\mathfrak{z}^1|^{4( \tau' -\tau )} \big)
\Big]
\end{eqnarray}
As it will become clear in the calculations of the curvature tensor, it is indeed the first order expansion of the fraction that is of importance for our purpose.
Indeed, when we do not have the extra term \( |\mathfrak{z}_1|^{2\tau}\) in the potential,
the term that could produce the negative infinity in the curvature is the term \( {-a_1 \over \det g^*_{11}} |\mathfrak{z}_1|^{2(1 - \tau)}\).
However, in our case this latter term is dominated by the term \( |\mathfrak{z}_1|^{2(\tau' - \tau)}\).
For components \(g^{i \bar{1}}, i \neq 1\) we can also derive a similar expression, the difference being that we do not need first order expansion, the signs of the terms appearing are not known, and do not play a r\^{o}le in the unboundedness of curvature tensor. \end{proof}
Having found the derivatives and the inverse of the metric (as a matrix in local coordinates) we can now turn to finding the curvature. Let us recall that we have the following formula for the curvature of a K\"ahler metric: \begin{equation}
\label{curv-form}
R_{\alpha \bar{\beta} \gamma \bar{\delta}} = - g_{\alpha \bar{\beta}, \gamma \bar{\delta}} + g^{\mu \bar{\nu}} g_{\alpha \bar{\nu}, \gamma} g_{\mu \bar{\beta}, \bar{\delta}} \end{equation}
It is not hard to see that the only terms in the second derivative, \(\tilde{g}_{\alpha \bar{\beta}, \gamma \bar{\delta}}\), which need to be considered are the following:
\( \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} \delta_{1 \delta} \big ( \tau^2 ( 1 - \tau)^2 K |\mathfrak{z}^1|^{2 \tau - 4} - \tau'^2 ( 1 - \tau')^2 M |\mathfrak{z}^1|^{2 \tau' - 4} \big )\). This term is non-zero only when all indices are equal to 1. In the first order expression \(g^{\mu \bar{\nu}} g_{\alpha \bar{\nu}, \gamma} g_{\mu \bar{\beta}, \bar{\delta}}\) one can first notice that \(g^{1 \bar{1}} = \mathcal{O}( |\mathfrak{z}^1|^{2 - 2\tau} ) \). So, the only term in the product \(g_{\alpha \bar{\nu}, \gamma} g_{\mu \bar{\beta}, \bar{\delta}}\) that might stay unbounded after multiplying into the relevant components of \(v\) and \(w\) is the following \begin{eqnarray}
\big(
\tau^2 ( \tau - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \delta} K |\mathfrak{z}|^{2 \tau - 4} \mathfrak{z}
&-&
\tau'^2 ( \tau' - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \delta} M |\mathfrak{z}|^{2 \tau' - 4} \mathfrak{z}
\big)
\big(
\tau^2 ( \tau - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} K |\mathfrak{z}|^{2 \tau - 4} \bar{\mathfrak{z}}
\nonumber \\
&-&
\tau'^2 ( \tau' - 1) \delta_{1 \alpha} \delta_{1 \beta} \delta_{1 \gamma} M |\mathfrak{z}|^{2 \tau' - 4} \bar{\mathfrak{z}}
\big) \end{eqnarray} and that only when \(\alpha = \beta = \gamma = \delta =1\). It takes a straightforward verification to see that after multiplying by relevant components of \(v\) and \(w\), the only potentially unbounded terms are in \(R_{1 \bar{1} 1 \bar{1}} \).
Let us study the terms in \(R_{\alpha \bar{\beta} \gamma \bar{\delta}} v^\alpha \bar{v}^{\beta} w^\gamma \bar{w}^\delta\) separately. First, we take \begin{equation} \label{curv-1}
-g_{\alpha \bar{\beta}, \gamma \bar{\delta}} v^\alpha \bar{v}^{\beta} w^\gamma \bar{w}^\delta \end{equation} By directly inspecting the terms in \eqref{g-second-deriv} we see that the only unbounded term in \eqref{curv-1} is the highest order term \[
\big( - \tau^2 ( 1 - \tau)^2 K |\mathfrak{z}^1|^{2 \tau - 4}
+ \tau'^2 ( 1 - \tau')^2 M |\mathfrak{z}^1|^{2 \tau' - 4}
\big) v^1 \bar{v}^1 w^1 \bar{w}^1. \]
Behaviour of the term \(g^{\mu \bar{\nu}} g_{\alpha \bar{\nu}, \gamma} g_{\mu \bar{\beta}, \bar{\delta}} v^\alpha \bar{v}^{\beta} w^\gamma \bar{w}^\delta\) can similarly be understood as follows: all the terms appearing in the product are bounded except the term
\begin{eqnarray}
g^{\mu \bar{\nu}}
\delta_{1 \mu} \delta_{1 \beta} \delta_{1 \delta}
\delta_{1 \alpha} \delta_{1 \nu} \delta_{1 \gamma}
\big(
\tau^2 ( \tau - 1) K |\mathfrak{z}^1|^{2 \tau - 4} \mathfrak{z}^1
&-&
\tau'^2 ( \tau' - 1) M |\mathfrak{z}^1|^{2 \tau' - 4} \mathfrak{z}^1
\big)
\big(
\tau^2 ( \tau - 1) K |\mathfrak{z}^1|^{2 \tau - 4} \bar{\mathfrak{z}}_1
\nonumber \\
&-&
\tau'^2 ( \tau' - 1) M |\mathfrak{z}^1|^{2 \tau' - 4} \bar{\mathfrak{z}}_1
\big) v^\alpha \bar{v}^\beta w^\gamma \bar{w}^\delta \nonumber \end{eqnarray} which, using \eqref{g-inverse-expansion} can be written as \begin{eqnarray}
{1 \over \tau^2} \Big(
1
&+&
{\tau'^2 \over \tau^2} |\mathfrak{z}^1|^{2(\tau' - \tau)}
+
\mathcal{O} (|\mathfrak{z}^1|^{2(1 - \tau)} )
\Big)
\big[
\tau^4 (1-\tau)^2 |\mathfrak{z}^1|^{2 \tau -4}
\nonumber \\
&-&
2 \tau^2 \tau'^2 (1 -\tau) (1-\tau') |\mathfrak{z}^1|^{4 \tau' - 2 \tau -4}
+
\tau'^4 ( 1- \tau')^2 |\mathfrak{z}^1|^{ 2 \tau' - 4}
\big]
v^1 \bar{v}^1 w^1 \bar{w}^1 \nonumber \\
&=&
\big( \tau^2 (1-\tau)^2 |\mathfrak{z}^1|^{2 \tau -4}
-
2 \tau'^2 (1-\tau)(1-\tau') |\mathfrak{z}^1|^{2 \tau' - 4}
+
{\tau'^4 \over \tau^2} ( 1 - \tau'^2) |\mathfrak{z}^1|^{4\tau' - 2 \tau - 4} \nonumber \\
&+&
\tau'^2 ( 1- \tau)^2 |\mathfrak{z}^1|^{2\tau' - 4}
- 2 {\tau'^4 \over \tau^2} ( 1 - \tau) (1 - \tau') |\mathfrak{z}^1|^{6 \tau' - 4\tau - 4}
+{\tau'^6 \over \tau^4}(1 - \tau')^2 |\mathfrak{z}^1|^{4\tau' - 2 \tau -4}
\nonumber \\
&+&
\mathcal{O}(|\mathfrak{z}^1|^{-2}) \big)
v^1 \bar{v}^1 w^1 \bar{w}^1 \nonumber \end{eqnarray}
Having obtained these expressions for the individual terms appearing in the components of the curvature tensor, we can now put them together to obtain the following expression for two unit vectors \(v, w\) \begin{eqnarray}
R_{1 \bar{1} 1 \bar{1}} v^1 \bar{v}^1 w^1 \bar{w}^1 &=&
\mathcal{O} (1) + \Big[
-
\tau^2 ( 1 - \tau)^2 |\mathfrak{z}^1|^{2 \tau - 4}
+
\tau'^2 ( 1 - \tau')^2 |\mathfrak{z}^1|^{2 \tau' - 4}
\nonumber \\
&+& \tau^2 (1-\tau)^2 |\mathfrak{z}^1|^{2 \tau -4}
-
2 \tau'^2 (1-\tau)(1-\tau') |\mathfrak{z}^1|^{2 \tau' - 4}
+
{\tau'^4 \over \tau^2} ( 1 - \tau'^2) |\mathfrak{z}^1|^{4\tau' - 2 \tau - 4} \nonumber \\
&+&
\tau'^2 ( 1- \tau)^2 |\mathfrak{z}^1|^{2\tau' - 4}
- 2 {\tau'^4 \over \tau^2} ( 1 - \tau) (1 - \tau') |\mathfrak{z}^1|^{6 \tau' - 4\tau - 4}
+{\tau'^6 \over \tau^4}(1 - \tau')^2 |\mathfrak{z}^1|^{4\tau' - 2 \tau -4}
\nonumber \\
&+&
\mathcal{O}(|\mathfrak{z}^1|^{-2})
\Big] v^1 \bar{v}^1 w^1 \bar{w}^1 \nonumber \end{eqnarray}
As one may observe, starting from the lowest power, the terms with \(|\mathfrak{z}^1|^{2 \tau -4}\), cancel, and the next lowest power, \(|\mathfrak{z}^1|^{2 \tau' - 4}\), has a positive coefficient: \(\tau'^2(1 - \tau')^2 + \tau'^2(1 - \tau)^2 - 2 \tau'^2 (1- \tau) (1 - \tau') = \tau'^2 (\tau' - \tau)^2 > 0 \).
This means we can disregard the terms with larger exponents altogether and the behaviour of the curvature is dominated by the positive term \( (\tau'- \tau)^2 |\mathfrak{z}^1|^{2 \tau' - 4}\). In particular, this means as \(|\mathfrak{z}^1| \to 0\), the \( R_{1 \bar{1} 1 \bar{1}} v^1 \bar{v}^2 w^1 \bar{w}^1\) blows up in the positive direction and is bounded from below.
Finally, it should be noted that the components of the curvature tensor other than the \(1 \bar{1} 1 \bar{1}\) component are bounded when multiplied by the corresponding elements of \(g^{\alpha \bar{\beta}}\). \qed \\
\begin{remark}
Althoug we have not detalied this here, but one can repeat similar calculations for a metric of the form
\(\omega_0 + dd^c \big( |s|_h^{2 \tau} \pm \mathfrak{c} |s|_h^{2\tau'} \big) \), \(\mathfrak{c}>0\), and observe that for such metric the curvature indeed blows up in the negative or the positive direction -depending on the sign before \(\mathfrak{c}\)- at the rate of \(|s|_h^{2\tau' - 4}\).
In particular, this gives an example of a smooth edge-cone metric whose curvature becomes unbounded from either below or above close to the divisor at a rate faster than \(|s|_h^{-2}\). This also means one can construct such edge-cone metrics even when \(\tau < {1 \over 2}\). We find it worthwhile to emphasise that this phenomenon is not exclusive to higher dimensions. In an identical fashion it is indeed possible to construct cone metrics on \(\mathbb{C}\) whose curvatures are unbounded from below or above. \end{remark}
\section{Proof of the main results-Solving the equation}
Having made the observations in \textsection 3 and 4, we can now prove Theorem \ref{thm-1-2}.
\begin{proof}[Proof of Theorem \ref{thm-1-2}] In order to do so, we shall approximate the equation by a family of equations with smooth components. In this section we first estiablish uniform
{\it a priori } estimates which will allow taking limit of the family of solutions.
As we have mentioned before, the way we solve the equation with edge reference metric is by approximating the edge metric by a family of smooth metrics and deriving estimates independent of the parameter of the sequence. We also take a family of smooth functions \(\{ f_\epsilon \}_\epsilon\) approximating the source term \(f\). That is, we solve the following family of equations \begin{equation}\label{5-1}
(\tilde{\omega}_\epsilon + dd^c \phi_\epsilon)^n = \tilde{\omega}_\epsilon^n e^{f_\epsilon} \end{equation}
It is not hard to see that the right hand side converges in \(L^p(\omega_0)\) for some \(p\) depending on the angle. The fundamental theorem of Ko\l odziej \cite{ko} on the stability in \(L^p\) of the Monge-Amp\`ere operator comes to our aid to guarantee that since the right hand side converges in \(L^p\), the potentials obtained as solutions converge to the unique H\"older continuous solution. This also takes care of the \(L^\infty\) estimate automatically.
Just as in the classical case, in order to derive estimates on the complex hessian, we derive an upper bound on the laplacian. This is the content of Theorem \ref{thm-5-1}.
Finally, we need to derive an estimate on the modulus of continuity of the second derivative, namely, the \(C^{2,\theta}_\tau\) estimates {\it \`a la} Evans and Krylov. This will be done in the following section. One can then repeat the usual method based on takign a sequence of solutions, \( \{\phi_{\epsilon_j} \}_j \), and if necessary pass to a subsequence and by evoking the uniform estimates prove that there is a solution as \(\epsilon \to 0^+\). This will thus conclude the proof of Theorem \ref{thm-1-2}. \end{proof}
Without mentioning, the functions and metric appearing below are assumed to be the ones corresponding to the \(\epsilon\)-approximation of the equation.
\begin{theorem}
\label{thm-5-1}
Let \(\phi\) be a \(C^3\) solution of Equation \ref{5-1}.
Then, we have the following {\it a priori} bound:
\begin{equation}
\Vert dd^c \phi_\epsilon \Vert_{\omega_\epsilon} \leq C
\end{equation}
where \(C= C \left (\Vert \phi \Vert_{L^\infty}, \inf R^\epsilon(.,.,.,.), \inf f_\epsilon, \inf (\Delta_{\omega_\epsilon} f_\epsilon)^- \right)\). \end{theorem}
\begin{proof} Other than a few observations, the argument is very similar to the classical argument. The reader can consult \textsection \cite{si}. We shall drop the subscript \(\epsilon\) in the rest of the proof. Let us set \(\Delta\) and \(\Delta'\) to be the laplacian associated to \(\tilde{\omega}\) and \(\tilde{\omega} + dd^c \phi\) respectively. In the rest of this section, we drop the subscript \(\epsilon\). Let us start with the following well-known inequality:
\begin{equation}
\Delta' \log ( n + \Delta \phi)
\geq
{1 \over n + \Delta \phi} \big[ \Delta f + \sum_{\alpha, \beta} \big(
- R_{\alpha \bar{\alpha} \beta \bar{\beta}} + R_{\alpha \bar{\alpha} \beta \bar{\beta}} {1 + \phi_{\alpha \bar{\alpha}} \over 1 + \phi_{\beta \bar{\beta}}}
\big)
\big]
\end{equation}
The reader can refer to \textsection 3.2 of \cite{si} for example. Using the symmetries of the curvature tensor, we now rewrite the expressions containing curvature terms as
\begin{eqnarray}
\sum_{\alpha, \beta} \big( - R_{\alpha \bar{\alpha} \beta \bar{\beta}} + R_{\alpha \bar{\alpha} \beta \bar{\beta}} {1 + \phi_{\alpha \bar{\alpha}} \over 1 + \phi_{\beta \bar{\beta}}} \big)
&=& 2\sum_{\alpha < \beta} \big(
{1 + \phi_{\alpha \bar{\alpha}} \over 1 + \phi_{\beta \bar{\beta}}}
+
{1 + \phi_{\beta \bar{\beta}} \over 1 + \phi_{\alpha \bar{\alpha}}}
-
2
\big)R_{\alpha \bar{\alpha} \beta \bar{\beta}} \nonumber \end{eqnarray}
In the original way the laplacian estimate was derived, it depended on the upper bound of the scalar curvature and the lower bound of the bi-sectional curvature.
This observation, which is already used in \cite{pa}, allows dropping the former requirement.
Let \(C\) be a positive constant so that \(R_{\alpha \bar{\alpha} \beta \bar{\beta}} u^\alpha \bar{u}^\alpha v^\beta \bar{v}^\beta \geq -C|u|_\omega|v|_\omega \).
Then, upon noticing that the terms
\({1 + \phi_{\alpha \bar{\alpha}} \over 1 + \phi_{\beta \bar{\beta}}} +
{1 + \phi_{\beta \bar{\beta}} \over 1 + \phi_{\alpha \bar{\alpha}}} -2 \)
are all non-negative, we deduce that
\begin{eqnarray}
\Delta' \log ( n + \Delta \phi - C_2 \phi )
&\geq&
{1 \over n + \Delta \phi} \big[ \Delta f - \sum_{\alpha, \beta} C \big( {1 + \phi_{\alpha \bar{\alpha}} \over 1 + \phi_{\beta \bar{\beta}}} +
{1 + \phi_{\beta \bar{\beta}} \over 1 + \phi_{\alpha \bar{\alpha}}} -2
\big)
\big] - C_2n + C_2 \sum_{\alpha} {1 \over 1 + \phi_{\alpha \bar{\alpha}}} \nonumber \\
&\geq&
{(\Delta f)^- \over n + \Delta \phi} - 2C \sum_\alpha {1 \over 1 + \phi_{\alpha \bar{\alpha}}} - 2 Cn^2
- C_2n + C_2 \sum_{\alpha} {1 \over 1 + \phi_{\alpha \bar{\alpha}}}
\end{eqnarray}
In the expression above, for a function \(v\), \(v^-:= \min \{ v, 0 \}\).
After choosing \(C_2\) to be sufficiently large, it will require a standard application of the maximum principle to conclude the argument. We deduce that the quantity \(\Delta \phi\) is bounded by a constant \(C_4=C_4((\Delta f)^-, \inf_{\alpha, \beta} R_{\alpha \bar{\alpha} \beta \bar{\beta}}, \Vert \phi \Vert_0)\).
All of these quantities are finite by the fact that we have chosen a reference metric whose curvature is bounded from below, along with the observations in Propositions \ref{good-metric-pro}, that guarantee the the laplacian of the Ricci potential is bounded from below. \end{proof}
Having proved the bounds on the complex hessian of the solution, we can now turn to proving the H\"older continuity of the second derivative using a version of the so-called Evans-Krylov theory to deduce that the solution \(\phi_0\), obtained as the limit of \(\{\phi_\epsilon \}_\epsilon \), belongs to the edge H\"older space \(C^{2,\theta}_\tau\) for some exponent \(\theta\). This will be the the subject of the next section.
\subsection{Change of the reference metric redux: Evans-Krylov theory} \label{ev-kr-sec}
The \(C^{2,\theta}_\tau\) estimates have been studied before. In \cite{ch-do-su-3} such an estimate has been derived for K\"ahler-Einstein metrics. Also, in \cite{ca-zh} the Evans-Krylov theory has been extended to the edge-cone setting with the restriction of \(\tau \leq {2 \over 3}\), and in \cite{je-ma-ru} for \(\tau \in (0,1)\). Later, an interesting argument based on approximating the cone angle by rational numbers and using the geometry of rational cone angles was developed in \cite{gu-pa}. We assume the validity Evans-Krylov theory on \(\mathbb{C}_\tau \times \mathbb{C}^{n-1}\), which is what has been established in \cite{ca-zh, je-ma-ru, chu, gu-pa}.
So, we first localise the equation as detailed in the next paragraph. In the backdrop there is indeed a local change of the background metric which allows considering the equation on the flat edge model. This is necessary since in the edge case, unlike the smooth case, the second derivatives of the reference metric might not be bounded, so taking derivatives of \eqref{eq-1-4} in its current form might not help with the proof.
In the case of the edge metrics, the equation that the metric satisfies is the following \[
\varrho(\omega_\phi) = \mu \omega_\phi +2 \pi (1- \tau) [D].\]
In the unit ball in \(\mathbb{C}_\tau \times \mathbb{C}^{n-1}\) the twisted K\"ahler-Einstein equation can be written as \begin{equation} \label{cone-local}
-dd^c \log (dd^c w)^n = dd^c \log |\mathfrak{z}_1|^{2 - 2 \tau} + \mu dd^c w \end{equation} Similar to the smooth case, we obtain the following equation for \(w\): \begin{equation} \label{cone-local-2}
\log \det (w_{,\alpha \bar{\beta}}) = \log |\mathfrak{z}_1|^{2\tau - 2} - \mu w + \mathfrak{H} \end{equation} for some pluri-harmonic function \(\mathfrak{H}\).
Of course since we have no boundary conditions, there are infinitely many choices of a pluri-harmonic function \(\mathfrak{H}\) which in general satisfy no uniformity of any sort. Noting the fact that this equation is satisfied locally by all K\"ahler-Einstein potentials, it comes as little surprise that with no boundary conditions prescribed and an undetermined source term, \(\mathfrak{H}\), Equation \ref{cone-local-2} has too many degrees of freedom. It might, at the face value, seem like by doing so we have lost a great deal of information. However, when one has readily obtained a bound on the complex hessian of the solution \(\phi\) on the manifold, it translates to the fact that \(dd^c w\) can be assumed to be bounded. Further, since \(\phi\), and hence \(w\), are bounded in \(L^\infty\), along with the fact that the form \(dd^c w\) is bounded, we see from the equation that \(\Vert \mathfrak{H} \Vert_{L^\infty}\) is {\em a priori} bounded. Then, since \(\mathfrak{H}\) is a pluri-harmonic, and in particular a harmonic function, it is a well-known fact \(\mathfrak{H}\) is analytic and that all the derivatives of \(\mathfrak{H}\) are controlled in terms of the oscillation of \(\mathfrak{H}\). (One way to prove this fact is by an application of the mean value theorem. For such a proof, see \cite{gi-tr}). Although, since we take \(\operatorname{\partial}_{\kappa \bar{\lambda}}\) derivatives, and \(\mathfrak{H}_{,\kappa \bar{\lambda}} =0\), we only need the bound on the first derivative of \(\mathfrak{H}\).
Now, in the global equation, \eqref{eq-1-4}, the Ricci potential, \(f\), is bounded, and in our local equation \(\omega_\tau\) is equivalent to the standard edge-cone metric, \(\omega_{\mathfrak{K}}\). This, along with the fact that the potential \(w\) is bounded allows us to immediately see that the oscillation of \(\mathfrak{H}\) is bounded, which, in turn, gives a bound on all higher derivatives of \(\mathfrak{H}\). We can summarise these observations as the following lemma:
Now one may apply the Evans-Krylov theory on the flat space to \eqref{cone-local-2} and establish the membership of \(w\), and hence \(\phi\), in \(C^{2,\theta}_\tau\) for some \(\theta \in (0,1)\).
In the case of prescribed Ricci form in Theorem \ref{thm-1-2} there is another consideration to be taken into account, namely the regularity of the Ricci potential when the problem is localied. It is not hard to see that the Ricci potential stays H\"older continuous after the change of the background metric and also after localising the problem. Assuming the H\"older continuity of the Ricci potential when the equation is localised with the falt reference metric, we can apply the result in \cite{chu}. Amongst the Evans-Krylov estimates derived in the edge-cone setting, the result in this reference is the only one which does not require differentiating the equation, and hence, does not require the boundedness of derivatives of the Ricci potential. We may summarise this as it is stated below.
\begin{theorem}
Assume that in Equation \ref{eq-1-4} the function \(\phi\) satisfies:
\[
\Vert \phi \Vert_{L^\infty}, \
\Vert \vert dd^c \phi \vert_{\omega_\tau} \Vert_{L^\infty},
\
\Vert f \Vert_{C^\alpha_\tau}
\leq C
\]
for some constant \(C>0\). \end{theorem}
We notice that the condition on the H\"older continuity of \(f\) is satisfied in particular when the prescribed Ricci form, \(\tilde{\varrho}\), can be locally realised by H\"older continuous potentials, which is one of the conditions is Theorem \ref{thm-1-1} part (a).
\end{document} |
\begin{document}
\title{ \bf{Change Point Detection for High-dimensional Linear Models: A General Tail-adaptive Approach}} \author{ Bin Liu\footnotemark[1],
Zhengling Qi\footnotemark[3],
Xinsheng Zhang\footnotemark[1],
Yufeng Liu\footnotemark[2]} \renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnotetext[1]{Department of Statistics and Data Science, School of Management at Fudan University, China; E-mail:{\tt liubin0145@gmail.com; xszhang@fudan.edu.cn }} \footnotetext[2]{Department of Statistics and Operations Research, Department of Genetics, and Department of Biostatistics, Carolina Center for Genome Sciences, Linberger Comprehensive Cancer Center, University of North Carolina at Chapel Hill, U.S.A; E-mail: {\tt yfliu@email.unc.edu }} \footnotetext[3]{Department of Decision Sciences, George Washington University, U.S.A; E-mail:{\tt qizhengling321@gmail.com }}
\date{} \maketitle
\begin{abstract}
We study the
change point detection problem for high-dimensional linear regression models. The existing literature mainly focused on the change point estimation with stringent sub-Gaussian assumptions on the errors. In practice, however, there is no prior knowledge about the existence of a change point or the tail structures of errors. To address these issues, in this paper, we propose a novel tail-adaptive approach for simultaneous change point testing and estimation. The method is built on a new loss function which is a weighted combination between the composite quantile and least squared losses, allowing us to borrow information of the possible change points from both the conditional mean and quantiles.
For the change point testing, based on the adjusted $L_2$-norm aggregation of a weighted score CUSUM process, we propose a family of individual testing statistics with different weights to account for the unknown tail structures. Combining the individual tests, a tail-adaptive test is further constructed that is powerful for sparse alternatives of regression coefficients' changes under various tail structures. For the change point estimation, a family of argmax-based individual estimators is proposed once a change point is detected. In theory, for both individual and tail-adaptive tests, the bootstrap procedures are proposed to approximate their limiting null distributions. Under some mild conditions, we justify the validity of the new tests in terms of size and power under the high-dimensional setup. The corresponding change point estimators are shown to be rate optimal up to a logarithm factor. Moreover, combined with the wild binary segmentation technique, a new algorithm is proposed to detect multiple change points in a tail-adaptive manner. Extensive numerical results are conducted to illustrate the appealing performance of the proposed method. \end{abstract}
\noindent {\bf Keywords:}
Efficient computation, Heavy tail, High dimensions, Structure change
\section{Introduction}
With the advances of data collection and storage capacity, large scale/high-dimensional data are ubiquitous in many scientific fields ranging from genomics, finance, to social science. Due to the complex data generation mechanism, the heterogeneity, also known as the structural break, has become a common phenomenon for high-dimensional data, where the underlying model of data generation changes and the identically distributed assumption may not hold anymore. Change point analysis is a powerful tool for handling structural changes since the seminal work by \cite{page1955control}. It received considerable attentions in recent years and has a lot of real applications in various fields including genomics \citep{Liu2019Unified}, social science \citep{Roy2017Change}, financial contagion \citep{pesaran2007econometric} in economy, and even for the recent COVID-19 pandemic studies \citep{jiang2021modelling}. Motivated by this, in this paper, we study the change point testing and estimation problem in the high-dimensional linear regression setting.
Specifically, we are interested in the following model and detect possible change points:
\begin{equation*}\label{equation: linear model}
Y=\bX^\top \bbeta+\epsilon,
\end{equation*} where $Y\in\RR$ is the response variable, $\bX=(X_1,\ldots,X_p)\in \RR^{p}$ is the covariate vector, $\bbeta=(\beta_1,\ldots,\beta_p)^\top$ is a $p\times1$ unknown vector of coefficients, and $\epsilon\in\RR$ is the error term.
Suppose we have $n$ independent but (time) ordered realizations $\{(Y_i,\bX_i),i=1,\ldots,n\}$ with $\bX_i=(X_{i1},\ldots,X_{ip})^\top$. For each time point $i$, consider the following regression model:
\begin{equation}\label{equation: linear models}
Y_i=\bX_i^\top \bbeta_i+\epsilon_i,
\end{equation} where $\bbeta_i=(\beta_{i1},\ldots,\beta_{ip})^\top$ is the regression coefficient vector for the $i$-th observation, and $\epsilon_i$ is the error term. For the above model, our first question is whether there is a change point. This can be formulated as the following hypothesis testing problem:
\begin{equation}\label{hypothesis: H0}
\small
\begin{array}{ll}
\Hb_0:\bbeta_1=\bbeta_2=\cdots=\bbeta_n~~\text{v.s.}~~
\Hb_1: \bbeta^{(1)}=\bbeta_1=\cdots=\bbeta_{k_1}\neq \bbeta_{k_1+1}=\cdots=\bbeta_{n}:=\bbeta^{(2)},
\end{array}
\end{equation} where $k_1$ is the possible but unknown change point location. In this article, we assume $k_1=\lfloor nt_1 \rfloor$ with $0<t_1<1$. According to (\ref{hypothesis: H0}), the linear regression structure between $Y$ and $\bX$ remains homogeneous if $\Hb_0$ holds, and otherwise there is a change point $k_1$ that divides the data into two segments with different regression coefficients $\bbeta^{(1)}$ and $\bbeta^{(2)}$. Our second goal of this paper is to identify the change point location if we reject $\Hb_{0}$ in (\ref{hypothesis: H0}). Note that the above two goals are referred as change point testing and estimation, respectively. In the practical use, both testing and estimation are important since practitioners typically have no prior knowledge about either the existence of a change point or its location. Therefore, it is very useful to consider simultaneous change point detection and estimation. {Furthermore, the tail structure of the error $\epsilon_i$ in Model \eqref{equation: linear models} is typically unknown, which could significantly affect the performance of the change point detection and estimation. In the existing literature, the performance guarantee of most methods on change point estimation relies on the assumption that the error $\epsilon_i$ follows a Gaussian/sub-Gaussian distribution. Such an assumption could be violated in practice when the data distribution is heavy-tailed or contaminated by outliers. While some robust methods can address these issues, they may lose efficiency when errors are indeed sub-Gaussian distributed. It is also very difficult to estimate the tail structures and construct a corresponding change point method based on that. Hence, it is of great interest to construct a tail-adaptive change point detection and estimation method for high-dimensional linear models. }
\subsection{Contribution}
{Motivated by our previous discussion, in this paper, under the high-dimensional setup with $p\gg n$, we propose a tail-adaptive procedure for simultaneous change point testing and estimation in linear regression models. The proposed method relies on a new loss function in our change point estimation procedure, which is a weighted combination between the composite quantile loss proposed in \cite{zou2008composite} and the least squared loss with the weight $\alpha\in[0,1]$ for balancing the efficiency and robustness. Thanks to this new loss function with different $\alpha$, we are able to borrow information related to the possible change point from both the conditional mean and quantiles in Model \eqref{equation: linear models}. Therefore, besides controlling the type I error to any desirable level when $\Hb_{0}$ holds, the proposed method simultaneously enjoys high power and accuracy for change point testing and identification across various underlying error distributions including both lighted and heavy-tailed errors when there exists a change point. By combining our single change point estimation method with the wild binary segmentation (WBS) technique \citep{fryzlewicz2014wild}, we also generalize our method for detecting multiple change points in Model \eqref{equation: linear models}.
{In terms of our theoretical contribution, for each given $\alpha$, a novel score-based $\RR^p$-dimensional individual CUSUM process $ \{{\bC}_\alpha(t),t\in[0,1]\}$ is proposed. Based on this, we construct a family of individual-based testing statistics $\{T_{\alpha},\alpha\in[0,1]\}$
via aggregating $\bC_\alpha(t)$ using the $\ell_2$-norm of its first $s_0$ largest order statistics, known as the $(s_0,2)$-norm proposed in \cite{Zhou2017An}.
A high-dimensional bootstrap procedure is introduced to approximate $T_{\alpha}$'s limiting null distributions (See Algorithm \ref{alg:first}). The proposed bootstrap method only requires mild conditions on the covariance structures of $\bX$ and the underlying error distribution $\epsilon$, and is free of tuning parameters and computationally efficient. Furthermore, combining the corresponding individual tests in $\{T_{\alpha},\alpha\in[0,1]\}$, we construct a tail-adaptive test statistic $T_{\rm ad}$ by taking the minimum $P$-values of $\{T_{\alpha},\alpha\in[0,1]\}$. The proposed tail-adaptive method $T_{\rm ad}$ chooses the best individual test according to the data and thus enjoys simultaneous high power across various tail structures. Theoretically, we adopt a low-cost bootstrap method for approximating the limiting distribution of $T_{\rm ad}$. In terms of size and power, for both individual and tail-adaptive tests, we prove that the corresponding test can control the type I error for any given significance level if $\Hb_0$ holds, and reject the null hypothesis with probability tending to one otherwise.
}
As for the change point estimation, once $\Hb_0$ is rejected by our test, based on each individual test statistic, we can estimate its location via taking argmax with respect to different candidate locations $t\in(0,1)$ for the $(s_0,2)$-norm aggregated process $\{\|\bC_\alpha(t)\|_{(s_0,2)},t\in[0,1]\}$. Under some regular conditions, for each individual based estimator $\{\hat{t}_{\alpha},\alpha\in[0,1]\}$, we can show that the estimation error is rate optimal up to a $\log(pn)$ factor. Hence, the proposed individual estimators for the change point location allow the signal jump size scale well with $(n,p)$ and are consistent as long as
$SNR(\alpha,\btau)\|\bSigma(\bbeta^{(2)}-\bbeta^{(1)})\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$ holds, where $SNR(\alpha,\btau)$ is the signal to noise constant related to the loss function and the underlying error distribution. It is worth noting that the computational cost for obtaining $\{\hat{t}_{\alpha},\alpha\in[0,1]\}$ is only $O(\rm Lasso(n,p))$ operations, where $\rm Lasso(n,p)$ is the cost to compute the Lasso estimator for the corresponding weighted loss. This can be much more efficient than the existing works with $O(\rm nLasso(n,p))$ operations in \cite{lee2016LASSO,lee2018oracle} and $O(\rm 2Lasso(n,p))$ operations in \cite{JMLR:v20:18-460}.
\begin{comment}
It is worth mentioning that our key condition for showing consistency of each individual test $T_{\alpha}$ is very mild in the sense that we essentially require the signal-to-noise ratio satisfies:
\begin{equation*}
SNR(\alpha,\btau)\times\|t_1(1-t_1)\bSigma(\bbeta^{(2)}-\bbeta^{(1)})\|_{(s_0,2)}\geq C\sqrt{{\log(pn)}/{n}},
\end{equation*}
for some $C>0$, where $SNR(\alpha,\btau)$ defined in (\ref{equation: theoretical SNR}) is referred as ``signal-to-noise-ratio'', which is only related to the choice of loss function. Here $\mathbf{\btau}$ refers to the quantile levels used in the loss function. Hence, the above inequality gives a unified condition for the consistency of change point testing for each individual test, such as the special cases when $\alpha=1$ and $\alpha=0$, which correspond to the squared error loss and composite quantile loss, respectively. More importantly, it shreds some light on why different loss functions perform differently for change point testing. We believe this is a novel contribution for change point testing in high-dimensional linear regression.
As for the change point estimation, once $\Hb_0$ is rejected by our test, based on each individual test statistic, we can estimate its location via taking argmax with respect to different candidate locations $t\in(0,1)$ for the $(s_0,2)$-norm aggregated process $\{\|\bC_\alpha(t)\|_{(s_0,2)},t\in[0,1]\}$. Under some regular conditions, for each individual based estimator $\{\hat{t}_{\alpha},\alpha\in[0,1]\}$, we can show that the estimation error is bounded by
\begin{equation*}
\big|\hat{t}_{\alpha}-t_1\big|=O_p\big(\dfrac{\log(pn)}{n}\times \dfrac{1}{SNR^2(\alpha,\tilde{\btau})\|\bSigma(\bbeta^{(2)}-\bbeta^{(1)})\|^2_{(s_0,2)}}\big),
\end{equation*}
which is rate optimal up to a $\log(pn)$ factor. Hence, the proposed individual estimators for the change point location allow the signal jump size scale well with $(n,p)$ and are consistent as long as
$SNR(\alpha,\btau)\|\bSigma(\bbeta^{(2)}-\bbeta^{(1)})\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$ holds.
\end{comment}
\subsection{Related literature}
For the low dimensional setting with a fixed $p$ and $p<n$, change point analysis for linear regression models has been well-studied. Specifically, \cite{quandt1958tests} considered testing (\ref{hypothesis: H0}) for a simple regression model with $p=2$. Other extensions include the maximum likelihood ratio tests \citep{horvath1995detecting}, partial sums of regression residuals \citep{gombay1994limit,bai1998estimating}, and the union intersection test \citep{horvath1995limit}. Other related methods include \cite{qu2008testing,zhang2014testing,oka2011estimating,lee2011testing} and among others.
Due to the curse of dimensionality, on the other hand, only a few papers studied high-dimensional change point analysis, which mainly focused on the change point estimation. See \cite{lee2016LASSO,JMLR:v20:18-460,lee2018oracle,leonardi2016computationally,zhang2015change-point,JMLR:v22:19-531,WLLZ2022}. However, none of the aforementioned papers develop hypothesis testing procedure, which is the prerequisite for the change point detection. Furthermore, most existing literature requires strong conditions on the underlying errors $\epsilon_i$ for deriving the desirable theoretical properties, which may not be applicable when the data are heavy-tailed or contaminated by outliers. One possible solution is to use the robust method such as median regression in \cite{lee2018oracle} for change point estimation. As discussed in \cite{zou2008composite,bradic2011penalized,2014A}, when making statistical inference for homogeneous linear models, the asymptotic relative efficiency of median regression-based estimators compared to least squared-based is only about 64\% in both low and high dimensions. In addition, inference based on quantile regression can have arbitrarily small relative efficiency compared to the least squared based regression. Our proposed tail-adaptive method is the first one that can perform simultaneous change point testing and estimation for high-dimensional linear regression models with different distributions. In addition to controlling the type I error to any desirable level, the proposed method enjoys simultaneously high power and accuracy for the change point testing and identification across various underlying error distributions when there exists a change point. Besides literature in the regression setting, change point analysis has also been carried out for the setting of high-dimensional mean vectors and some mean vector-based extensions. See \cite{Aue2009Break,Jirak2015Uniform,cho2016change,Liu2019Unified}) and many others.
\begin{comment}
Different from the low dimensional case, high-dimensional change point analysis for Model
\eqref{equation: linear models} is an important but more challenging problem due to the curse of dimensionality. Only a few papers exist until recently but are only focused on the change point estimation. For single change point estimation,
\cite{lee2016LASSO} considered a high-dimensional regression model with a possible change point due to a covariate threshold and proposed estimators for both regression coefficients as well as the threshold parameter. Based on the $L_1/L_0$ regularization, \cite{JMLR:v20:18-460} proposed a two-step algorithm for the detection and estimation of parameters in a high-dimensional change point regression model. In addition, assuming that at most one change point occurred, \cite{lee2018oracle} considered a three-step procedure for detecting and localizing a change point in high-dimensional quantile regression models, which can be regarded as a robust method for Model \eqref{equation: linear models}. As extensions to multiple structural breaks in high-dimensional linear models, \cite{leonardi2016computationally} proposed fast algorithms for multiple change point estimation based on the dynamic programming and binary search techniques. In addition,
\cite{zhang2015change-point} developed an approach for estimating multiple change points based on sparse group LASSO. Recently, \cite{JMLR:v22:19-531} proposed a projection-based algorithm for estimating multiple change points. In a different setting, \cite{WLLZ2022} extended the method by \cite{leonardi2016computationally} to high-dimensional generalized linear models for multiple change points estimation.
Even though the aforementioned papers have made important contributions for the change point analysis related to Model \eqref{equation: linear models}, many aspects of this topic still remain unsolved. First, most existing works in the high-dimensional setting mainly focus on the change point estimation, which is essentially built upon the penalized regression estimation. Few hypothesis testing methods for a change point in Model \eqref{equation: linear models} {in the high-dimensional setting} have been developed. Second, the most existing literature imposes strong conditions on the underlying errors $\epsilon_i$ for deriving the desirable theoretical properties. For example, \cite{lee2016LASSO,zhang2015change-point} required $\epsilon_i$ follow the Gaussian distributions, and \cite{JMLR:v20:18-460} assumed $\epsilon_i$ to be sub-Gaussian distributed. Those methods essentially used the least squared loss and may not be applicable when the data are heavy-tailed or contaminated by outliers. One possible solution is to use the robust method such as median regression in \cite{lee2018oracle} for change point estimation. As discussed in \cite{zou2008composite,bradic2011penalized,2014A}, when making statistical inference for homogeneous linear models, the asymptotic relative efficiency of median regression-based estimators compared to least squared-based is only about 64\% in both low and high dimensions. In addition, inference based on quantile regression can have arbitrarily small relative efficiency compared to the least squared based regression. This phenomenon can also been observed in change point analysis. To summarize, to the best of our knowledge, there is no change point testing and estimation methods in the literature that address both light-tailed and heavy-tailed distributions in the high-dimensional setting. Our proposed tail-adaptive method is the first one that can perform simultaneous change point testing and estimation for high-dimensional linear regression models with different distributions. In addition to controlling the type I error to any desirable level, the proposed method enjoys simultaneously high power and accuracy for the change point testing and identification across various underlying error distributions when there exists a change point.
{ Besides literature in the regression setting, change point analysis has also been carried out on the setting of high-dimensional mean vectors as well as some mean vector-based extensions. See \cite{Aue2009Break,Jirak2015Uniform,cho2016change,Liu2019Unified}) and many others.
In contrast, change point testing for Model \eqref{equation: linear models} is a much more challenging task and there are much less studies on that. The reasons are two folded. Firstly, there are no direct observations for $\bbeta_i$. Consequently,
there is no natural statistic compared with CUSUM process for mean vectors. One possible way is to use the LASSO estimators directly for constructing CUSUM. It is known that LASSO is biased and some de-biased procedure such as the one in \cite{van2014asymptotically} is typically needed. However, the direct extension of \cite{van2014asymptotically} is a non-trivial task, especially considering the general weighted loss we use in our procedure as well as the unknown change point in Model \eqref{equation: linear models}. To overcome this difficulty, we construct the score-based CUSUM process which can address this issue. Secondly, to control the size and demonstrate the power of testing methods, the analysis of penalized estimation such as LASSO properties under both $\Hb_0$ and $\Hb_1$ is generally required. This is significantly more challenging than the i.i.d setting. In this paper, we successfully overcome this difficulty and give the LASSO estimation error bounds for the general weighed loss under both $\Hb_0$ and $\Hb_1$, covering a wide range of loss functions. The non-smoothness of the composite quantile loss, as well as the heterogeneous observations make the proof non-trivial.}
\end{comment}
The rest of this paper is organized as follows. In Section \ref{section: methodology}, we introduce our new tail-adaptive methodology for detecting a single change point as well as multiple change points. In Section \ref{sec: theory}, we derive the theoretical results in terms of size and power as well as the change point estimation. In Section \ref{sec: summary of simulations}, we present our extensive simulation results. A real application to the S$\&$P 100 dataset is shown in Section \ref{section: real analysis}. The concluding remarks are provided in Section \ref{section: summary}. Detailed proofs and the full numerical results are given in the online supplementary materials.
{\bf{Notations:}} We end this section by introducing some notations. For $\bv=(v_1,\ldots,v_p)^\top \in \mathbb{R}^p$, we define its $\ell_p$ norm as $\|\bv\|_p=(\sum_{j=1}^d|v_j|^p)^{1/p}$ for $1\leq p\leq \infty$. For $p=\infty$, define $\|\bv\|_\infty=\max_{1\leq j\leq d}|v_j|$.
For any set $\cS$, denote its cardinality by $|\cS|$. For two real numbered sequences $a_n$ and $b_n$, we set $a_n=O(b_n)$ if there exits a constant $C$ such that $|a_n|\leq C|b_n|$ for a sufficiently large $n$; $a_n=o(b_n)$ if $a_n/b_n\rightarrow0$ as $n\rightarrow\infty$; $a_n\asymp b_n$ if there exists constants $c$ and $C$ such that $c|b_n|\leq|a_n|\leq C|b_n|$ for a sufficiently large $n$. For a sequence of random variables (r.v.s) $\{\xi_1,\xi_2,\ldots\}$, we set $\xi_n\xrightarrow{{\mathbb P}} \xi$ if $\xi_n$ converges to $\xi$ in probability as $n\rightarrow\infty$. We also denote $\xi_n=o_p(1)$ if $\xi_n\xrightarrow{{\mathbb P}} 0$. For a $n\times p$ matrix $\Xb$, denote $\Xb_i$ and $\Xb^j$ as its $i$-th row and $j$-th column respectively.
We define $\floor{x}$ as the largest integer less than or equal to $x$ for $x\geq 0$. Denote $(\mathcal{X},\cY)=\{(\bX_1,Y_1),\ldots,(\bX_n,Y_n)\}$.
\section{Methodology}\label{section: methodology}
In Section \ref{sec: method of single cpt}, we construct a family of oracle individual testing statistics $\{\tilde{T}_{\alpha},\alpha\in[0,1]\}$ for single change point detection. To estimate the unknown variance in $\tilde{T}_{\alpha}$, in Section \ref{sec: method of variance estimation}, a weighted variance estimation that is consistent under both $\Hb_0$ and $\Hb_1$ is proposed. With the estimated variance, a family of data-driven testing statistics $\{{T}_{\alpha},\alpha\in[0,1]\}$ is proposed for change point detection. To approximate the limiting null distribution of ${T}_{\alpha}$, we introduce a novel bootstrap procedure in Section \ref{sec: method of bootstrap}. In practice, the tail structure of the error term is typically unknown. Hence, it is desirable to combine the individual testing statistics $\{{T}_{\alpha},\alpha\in[0,1]\}$ for yielding a powerful tail-adaptive method. To solve this problem, in Section \ref{sec: method of tail-adaptive}, a tail-adaptive testing statistic $T_{\rm ad}$ is proposed. Lastly, we combine our testing procedure with the WBS technique for detecting multiple change points. \subsection{Single change point detection}\label{sec: method of single cpt} In this section, we introduce our new methodology for Problem \eqref{hypothesis: H0}. We first focus on detecting a single change point in Model \eqref{equation: linear models}.
In this case, Model \eqref{equation: linear models} reduces to:
\begin{equation}\label{equation: single cpt model}
Y_i=\bX_i^\top\bbeta^{(1)}\mathbf{1}\{i\leq k_1\}+\bX_i^\top\bbeta^{(2)}\mathbf{1}\{i> k_1\}+\epsilon_i,~\text{for}~i=1,\ldots,n.
\end{equation} In this paper, we assume $k_1=\floor{nt_1}$ for some constant $t_1\in(0,1)$. Note that $t_1$ is called the relative change point location. We assume the change point does not occur at the begining or end of data observations. Specifically, suppose there exists a constant $q_0\in(0,0.5)$ such that $q_0\leq t_1\leq 1-q_0$ holds, which is a common assumption in the literature \citep[e.g.,][]{dette2018relevant,Jirak2015Uniform}. For Model \eqref{equation: single cpt model}, given $\bX_i$, the conditional mean of $Y_i$ becomes:
\begin{equation}\label{model: mean}
{\mathbb E}[Y_i\, | \, \bX_i]=\bX_i^\top\bbeta^{(1)}\mathbf{1}\{i\leq k_1\}+\bX_i^\top\bbeta^{(2)}\mathbf{1}\{i> k_1\}.
\end{equation} Moreover, let $0<\tau_1<\ldots<\tau_K<1$ be $K$ candidate quantile indices. For each $\tau_k\in(0,1)$, let $b_k^{(0)}:=\inf\{t:{\mathbb P}(\epsilon\leq t)\geq \tau_k\}$ be the $\tau_k$-th theoretical quantile for the generic error term $\epsilon$ in Model \eqref{equation: single cpt model}. Then, conditional on $\bX_i$, the $\tau_k$-th quantile for $Y_i$ becomes:
\begin{equation}\label{model: quantile}
Q_{\tau_k}(Y_i|\bX_i)=b_k^{(0)}+\bX_i^\top\bbeta^{(1)}\mathbf{1}\{i\leq k_1\}+\bX_i^\top\bbeta^{(2)}\mathbf{1}\{i> k_1\}, ~k=1,\ldots,K,
\end{equation}
where $Q_{\tau_k}(Y_i|\bX_i):=\inf\{t:{\mathbb P}(Y_i\leq t|\bX_i)\geq \tau_k\}$. Hence, if there exists a change point in Model \eqref{equation: single cpt model}, both the conditional mean and the conditional quantile change after the change point. This suggests that we can make change point inference for $\bbeta^{(1)}$ and $\bbeta^{(2)}$ using either (\ref{model: mean}) or (\ref{model: quantile}). To propose our new testing statistic, we first introduce the following weighted composite loss function. In particular, let $\alpha\in[0,1]$ be some candidate weight. Define the weighted composite loss function as: \vspace*{-0.2in} \begin{equation}\label{equation: weighted loss function}
\ell_\alpha(\bx,y;\tilde{\btau},\bb,\bbeta):=(1-\alpha)\dfrac{1}{K}\sum_{k=1}^{K}\rho_{\tau_k}(y-b_k-\bx^\top\bbeta)+ \dfrac{\alpha}{2}(y-\bx^\top\bbeta)^2,
\vspace*{-0.2in} \end{equation} where $\rho_{\tau}(t):=t(\tau-\mathbf{1}\{t\leq 0\})$ is the check loss function \citep{koenker1978regression}, $\tilde{\btau}:=(\tau_1,\ldots,\tau_K)^\top$ are user-specified $K$ quantile levels, and $\bb=(b_1,\ldots,b_K)^\top\in \RR^K$ and $\bbeta=(\beta_1,\ldots,\beta_p)^\top\in \RR^p$. Note that we can regard $\ell_\alpha(\bx,y;\tilde{\btau},\bb,\bbeta)$ as a weighted loss function between the composite quantile loss and the squared error loss. For example, for $\alpha=1$, it reduces to the ordinary least squared-based loss function with $\ell_1(\bx,y)=(y-\bx^\top\bbeta)^2/2$. When $\alpha=0$, it is the composite quantile loss function $\ell_0(\bx,y)=\sum_{k=1}^{K}\rho_{\tau_k}(y-b_k-\bx^\top\bbeta)/K$ proposed in \cite{zou2008composite}. It is known that the least squared loss-based method has the best statistical efficiency when errors follow Gaussian distributions and the composite quantile loss is more robust when the error distribution is heavy-tailed or contaminated by outliers. As discussed before, in practice, it is challenging to obtain the tail structure of the error distribution and construct a corresponding change point testing method based on the error structure. Hence, we propose a weighted loss function by borrowing the information related to the possible change point from both the conditional mean and quantiles. More importantly, we use the weight $\alpha$ to balance the efficiency and robustness.
Our new testing statistic is based on a novel high-dimensional weighted score-based CUSUM process of the weighted composite loss function. In particular, for the composite loss function $\ell_\alpha(\bx,y;\tilde{\btau},\bb,\bbeta)$, define its score (subgradient) function ${\partial \ell_\alpha(\bx,y;\tilde{\btau},\bb,\bbeta) }/{\partial \bbeta}$ with respect to $\bbeta$ as:
\begin{equation}\label{equation: score loss}
\begin{array}{ll}
\bZ_{\alpha}(\bx,y;\tilde{\btau},\bb,\bbeta):=\Big[\dfrac{1-\alpha}{K}\sum\limits_{k=1}^{K}\bx\big(\mathbf{1}\{y-b_k-\bx^\top\bbeta\leq 0\}-\tau_k\big)\Big]-\alpha\big[\bx (y-\bx^\top\bbeta)\big].
\end{array}
\end{equation} Using $ \bZ_{\alpha}(\bx,y;\tilde{\btau},\bb,\bbeta)$, for each $\alpha\in[0,1]$ and $t\in(0,1)$, we first define the oracle score-based CUSUM as follows:
\begin{equation}\label{equation: score cusum}
\tilde{\bC}_\alpha(t;\tilde{\btau},\bb,\bbeta)
=\dfrac{1}{\sqrt{n}\sigma(\alpha,\tilde{\btau})}\big(\sum\limits_{i=1}^{\lfloor nt\rfloor}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\bb,\bbeta)-\dfrac{\lfloor nt \rfloor }{n}\sum\limits_{i=1}^{n}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\bb,\bbeta)\big),
\end{equation} where $\sigma^2(\alpha,\tilde{\btau}):=\text{Var}[(1-\alpha)e_i(\tilde{\btau})-\alpha\epsilon_i)]$ with $e_i(\tilde{\btau}):=K^{-1}\sum\limits_{k=1}^{K}(\mathbf{1}\{\epsilon_i\leq b_{k}^{(0)}\}-\tau_k)$.
Note that we call $\tilde{\bC}_\alpha(t;\tilde{\btau},\bb,\bbeta)$ oracle since we assume $\sigma^2(\alpha,\tilde{\btau})$ is known. In Section \ref{sec: method of variance estimation}, we will give the explicit form of $\sigma^2(\alpha,\tilde{\btau})$ under various combinations of $\alpha$ and $\tilde{\btau}$ and introduce its consistent estimator under both $\Hb_0$ and $\Hb_1$. {In the following, to motivate our test statistics, we study the behaviors of $\tilde{\bC}_\alpha(t;\tilde{\btau},\bb,\bbeta)$ under $\Hb_0$ and $\Hb_1$ respectively.}
First, under $\Hb_0$, if we replace $\bbeta=\bbeta^{(0)}$ and $\bb=\bb^{(0)}$ in (\ref{equation: score cusum}), the score based CUSUM becomes
\begin{equation*}
\begin{array}{ll} \tilde{\bC}_\alpha(t;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})
=\dfrac{1}{\sqrt{n}\sigma(\alpha,\tilde{\btau})}\Big(\sum\limits_{i=1}^{\lfloor nt\rfloor}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})-\dfrac{\lfloor nt \rfloor }{n}\sum\limits_{i=1}^{n}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})\Big).
\end{array}
\end{equation*} By noting that under $\Hb_0$, we have $Y_i=\bX_i^\top\bbeta^{(0)}+\epsilon_i$, the above CUSUM reduces to the following $\RR^p$--dimensional random noise based CUSUM process:
\begin{equation}\label{equation: random noise based CUSUM}
\begin{array}{ll}
\tilde{\bC}_\alpha(t;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})\\
=\dfrac{1}{\sqrt{n}\sigma(\alpha,\tilde{\btau})}\Big(\sum\limits_{i=1}^{\floor{nt}}\bX_i((1-\alpha) e_i(\tilde{\btau})-\alpha\epsilon_i)-\dfrac{\floor{nt}}{n}\sum\limits_{i=1}^n\bX_i((1-\alpha) e_i(\tilde{\btau})-\alpha\epsilon_i)\Big),\\
\end{array}
\end{equation} {whose asymptotic distribution can be easily characterized.} Since both $\bb^{(0)}$ and $\bbeta^{(0)}$ are unknown, we need some proper estimators that can approximate them well under $\Hb_0$. In this paper, for each $\alpha\in[0,1]$, we obtain the estimators by solving the following optimization problem with the $L_1$ penalty:
\begin{equation}\label{equation: lasso estimator}
\small
(\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)=\mathop{\text{\rm arg\,min}}_{\bb\in\RR^K,\atop\bbeta\in\RR^p}\Big[(1-\alpha)\dfrac{1}{n}\sum_{i=1}^{n}\dfrac{1}{K}\sum_{k=1}^{K}\rho_{\tau_k}(Y_i-b_i-\bX_i^\top\bbeta)+ \dfrac{\alpha}{2n}\sum_{i=1}^{n}(Y_i-\bX_i^\top \bbeta)^2+\lambda_{\alpha}\big\|\bbeta\big\|_1\Big],
\end{equation} where $\hat{\bb}_\alpha:=(\hat{b}_{\alpha,1},\cdots,\hat{b}_{\alpha,K})^\top$, $\hat{\bbeta}_{\alpha}:=(\hat{\beta}_{\alpha,1},\ldots,\hat{\beta}_{\alpha,p})^\top$, and $\lambda_{\alpha}$ is the non-negative tuning parameter controlling the overall sparsity of $\hat{\bbeta}_{\alpha}$. Note that the above estimators are obtained using all observations $\{(\bX_1,Y_1),\ldots,(\bX_n,Y_n)\}$. After obtaining $(\hat{\bb}_\alpha^\top,\hat{\bbeta}_\alpha^\top)$, we plug them into the score function in (\ref{equation: score cusum}) and obtain the final $\RR^p$--dimensional oracle score based CUSUM statistic as follows:
\begin{equation}\label{equation: score CUSUM}
\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha)=\big(\tilde{C}_{\alpha,1}(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha),\ldots,\tilde{C}_{\alpha,p}(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha)\big)^\top,~~t\in[q_0,1-q_0].
\end{equation} Using $(\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)$, we can prove that under $\Hb_0$, for each $\alpha\in[0,1]$, (\ref{equation: score CUSUM}) can approximate the random-noise based CUSUM process in (\ref{equation: random noise based CUSUM}) under some proper norm {aggregations}. {Next, we investigate the behavior of (\ref{equation: score CUSUM}) under $\Hb_1$. Observe that the score based CUSUM has the following decomposition:}
\begin{equation}
\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)=\underbrace{{\tilde{\bC}_\alpha(t;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})}}_{\rm \bf Random~ Noise}+\underbrace{{\bdelta_{\alpha}(t)}}_{\rm \bf Signal~Jump}+\underbrace{{\bR_{\alpha}(t;\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)}}_{\rm \bf Random ~Bias},
\end{equation} where $\tilde{\bC}_\alpha(t;\tilde{\btau},\bb^{(0)},\bbeta^{(0)})$ is the random noise based CUSUM process defined in (\ref{equation: random noise based CUSUM}), $\bR_{\alpha}(t;\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)$ is some random bias which has a very complicated form but can be controlled properly under $\Hb_1$, and $\bdelta_{\alpha}(t)$ is the signal jump function. More specifically, let
\begin{equation}\label{equation: theoretical SNR}
SNR(\alpha,\tilde{\btau}):=\dfrac{(1-\alpha) \big(\dfrac{1}{K}\sum\limits_{k=1}^{K}f_{\epsilon}(b_{k}^{(0)})\big)+\alpha}{\sigma(\alpha,\tilde{\btau})},
\end{equation} where $f_{\epsilon}(t)$ is the probability density function of $\epsilon$, and define the signal jump function
\begin{equation}
\bDelta(t;\bbeta^{(1)},\bbeta^{(2)}):=n^{-\frac{3}{2}} \times \left\{
\begin{array}{ll}
{\lfloor nt \rfloor (n-\floor {nt_1})}\bSigma\big(\bbeta^{(1)}-\bbeta^{(2)}\big),~\text{if}~t\leq t_1,\\
{\lfloor nt_1 \rfloor (n-\floor {nt})}\bSigma\big(\bbeta^{(1)}-\bbeta^{(2)}\big),~\text{if}~t> t_1.
\end{array} \right.
\end{equation} Then, the signal jump $\bdelta_{\alpha}(t)$ can be explicitly represented as the products of $SNR(\alpha,\tilde{\btau})$ and $\bDelta(t;\bbeta^{(1)},\bbeta^{(2)})$, which has the following explicit form:
\begin{equation}\label{equation: decomposition of the signal function}
\bdelta_{\alpha}(t):=SNR(\alpha,\tilde{\btau})\times \bDelta(t;\bbeta^{(1)},\bbeta^{(2)}).
\end{equation}
By (\ref{equation: decomposition of the signal function}), we can see that $\bdelta_{\alpha}(t)$ can be decomposed into a loss-function-dependent part $SNR(\alpha,\tilde{\btau})$ and a change-point-model-dependent part $\bDelta(t;\bbeta^{(1)},\bbeta^{(2)})$. More specifically, the first term $SNR(\alpha,\tilde{\btau})$ (short for the signal-to-noise-ratio) is only related to the parameters $\alpha,K,\bb^{(0)}$ as well as $\sigma(\alpha,\tilde{\btau})$, resulting from a user specified weighted loss function defined in (\ref{equation: weighted loss function}). In contrast, the second term $\{\bDelta(t;\bbeta^{(1)},\bbeta^{(2)}),t\in[0,1]\}$ is only related to Model \eqref{equation: single cpt model}, which is based on parameters $t_1$, $\bSigma$, $\bbeta^{(1)}$, and $\bbeta^{(2)}$ and is independent of the loss function. Moreover, for any weighted composite loss function, the process $\{\bDelta(t;\bbeta^{(1)},\bbeta^{(2)}),t\in[0,1]\}$ has the following properties: First, under $\Hb_1$, the non-zero elements of $\bDelta(t;\bbeta^{(1)},\bbeta^{(2)})$ are at most $(s^{(1)}+s^{(2)})$-sparse since we require sparse regression coefficients in the model; Second, we can see that $\|\bDelta(t;\bbeta^{(1)},\bbeta^{(2)})\|$ with $t\in[q_0,1-q_0]$ obtains its maximum value at the true change point location $t_1$, where $\|\cdot\|$ denotes some proper norm such as $\|\cdot\|_{\infty}$. Hence, in theory, the signal jump function $\bdelta_{\alpha}(t)$ also achieves its maximum value at $t_1$ under some proper norm. This is the key reason why using the score based CUSUM can correctly localize the true change point if $\bbeta^{(1)}-\bbeta^{(2)}$ is big enough. More importantly, for a given underlying error distribution $\epsilon$ in Model \eqref{equation: single cpt model}, we can use $SNR(\alpha,\tilde{\btau})$ to further amplify the magnitude of $\bdelta_{\alpha}(t)$ via choosing a proper combination of $\alpha$ and $\tilde{\btau}$. In particular, recall $\sigma^2(\alpha,\tilde{\btau}):=\text{Var}[(1-\alpha)e_i(\tilde{\btau})-\alpha\epsilon_i)]$. Then, we have
\begin{equation}\label{equation: orcale sigma square}
\sigma^2(\alpha,\tilde{\btau})=(1-\alpha)^2\text{Var}[e_i(\tilde{\btau})]+\alpha^2\sigma^2-2\alpha(1-\alpha)\text{Cov}(e_i(\tilde{\btau}),\epsilon_i),
\end{equation} where $\sigma^2:=\text{Var}(\epsilon)$. Using (\ref{equation: theoretical SNR}) and (\ref{equation: orcale sigma square}), $SNR(\alpha,\tilde{\btau})$ can be further simplified under some specific $\alpha$. For example, if $\alpha=1$, then $SNR(\alpha,\tilde{\btau})=1/\sigma$; If $\alpha=0$, then
\begin{equation*} SNR(\alpha,\tilde{\btau})=\dfrac{\sum\limits_{k=1}^{K}f_{\epsilon}(b_{k}^{(0)})}{\sqrt{\sum_{k_1=1}^{K}\sum_{k_2=1}^{K}\gamma_{k_1k_2}}}
\end{equation*}
with $\gamma_{k_1k_2}:=\min(\tau_{k_1},\tau_{k_2})-\tau_{k_1}\tau_{k_2}$; If we choose $\alpha\in(0,1)$, $K=1$ and $\tilde{\btau}=\tau$ for some $\tau\in(0,1)$. Then we have
\begin{equation}
SNR(\alpha,\tilde{\btau})=\dfrac{(1-\alpha)f_{\epsilon}(b_{\tau}^{(0)})+\alpha}{[(1-\alpha)^2\tau(1-\tau)+\alpha^2\sigma^2-2\alpha(1-\alpha)\text{Cov}(e({\tau}),\epsilon)]^{1/2}}.
\end{equation} Hence, for any underlying error distribution $\epsilon$ in Model \eqref{equation: single cpt model}, it is possible for us to choose a proper $\alpha$ and $\tilde{\btau}$ that makes $SNR(\alpha,\tilde{\btau})$ as large as possible for that distribution. See Figure \ref{figure: snr} for a direct illustration, where we plot the $SNR(\alpha,\tilde{\btau})$ under various error distributions for the weighted composite loss with $\tilde{\tau}=0.5$. Note that in this case, the loss function $\ell_{\alpha}(\bx,y)$ is a weighted combination between the absolute loss and the squared error loss. From Figure \ref{figure: snr}, $SNR(\alpha,\tilde{\btau})$ performs differently under various error distributions. For example, when $\epsilon\sim N(0,0.5)$, by choosing $\alpha=1$, $SNR(\alpha,0.5)$ achieves its maximum as expected. In contrast, when $\epsilon\sim \rm Laplace(0,1)$, $\alpha=0$ is the optimal in terms of $SNR(\alpha,0.5)$. Furthermore, we can see that for $\epsilon$ that follows the Student's $t_{v}$ distribution with a degree of freedom $v$, choosing $\alpha\in(0,1)$ has the highest $SNR$. In theory, we can prove that, for any distribution of $\epsilon$, the change point detection, as well as the estimating performance depend heavily on the choice of $\alpha$ and $\tilde{\btau}$ via $SNR(\alpha,\tilde{\btau})$. Hence, it motivates us to use $\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)$ to account for the tail properties for different $\epsilon$ using $\alpha$ and $\tilde{\btau}$. {Based on our investigation on our score based CUSUM process under $\Hb_0$ and $\Hb_1$, it is appealing to use $\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)$ to construct our testing statistics for adapting different tail structures in the data.}
\begin{figure}
\caption{$SNR(\alpha,\tilde{\btau})$ under various errors with different weights $\alpha\in\{0,0.1,\ldots,0.9,1\}$ for the weighted loss with $\tilde{\tau}=0.5$ and $K=1$.}
\label{figure: snr}
\end{figure} For change point detection, a natural question is how to aggregate the $\RR^p$--dimensional CUSUM process $\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)$. Note that for high-dimensional sparse linear models, there are at most $s=s^{(1)}+s^{(2)}$ coordinates in $\bbeta^{(1)}-\bbeta^{(2)}$ that can have a change point, which can be much smaller than the data dimension $p$, although we allow $s$ to grow with the sample size $n$. Motivated by this, in this paper, we aggregate the first $s_0$ largest statistics of $\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha)$. To that end, we introduce the $(s_0,2)$-norm as follows. Let $\bv=(v_1,\ldots,v_p)\in \RR^p$. For any $1\leq s_0\leq p$, define $\big\|\bv\big\|_{(s_0,2)}=(\sum_{j=1}^{s_0}|v_{(j)}|^2)^{1/2}$, where $|v_{(1)}\geq |v_{(2)}|\cdots\geq |v_{(p)}|$ are the order statistics of $\bv$. By definition, we can see that $\|\bv\|_{(s_0,2)}$ is the $L_2$-norm for the first $s_0$ largest order statistics of $(|v_1|,\ldots,|v_p|)^\top$, which can be regarded as an adjusted $L_2$-norm in high dimensions. Note that the $(s_0,2)$-norm is a special case of the $(s_0, \tilde p)$-norm proposed in \cite{Zhou2017An} by setting $\tilde p=2$. We also remark that taking the first $s_0$ largest order statistics can account for the sparsity structure of $\bbeta^{(1)}-\bbeta^{(2)}$.
Using the $(s_0,2)$-norm with a user-specified $s_0$ and known variance $\sigma^2(\alpha,\tilde{\tau})$, we introduce the oracle individual testing statistic with respect to a given $\alpha\in[0,1]$ as
\begin{equation*}
\tilde{T}_{\alpha}=\max_{q_0\leq t\leq 1-q_0}\Big\|\tilde{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_{a})\Big\|_{(s_0,2)},~~\text{with}~~\alpha\in[0,1].
\end{equation*}
By construction, $\tilde{T}_{\alpha}$ can capture the tail structure of the underlying regression errors by choosing a special $\alpha$ and $\tilde{\btau}$. Specifically, for $\alpha=1$, it equals to the least square loss-based method. In this case, since $\tilde{T}_{\alpha}$ only uses the moment information of the errors, it is powerful for detecting a change point with light-tailed errors such as Gaussian or sub-Gaussian distributions. For $\alpha=0$, $\tilde{T}_{\alpha}$ reduces to the composite quantile loss-based method, which only uses the information of ranks or quantiles. In this case, $\tilde{T}_{\alpha}$ is more robust to data with heavy tails such as Cauchy distributions. As a special case of $\alpha=0$, if we further choose $\tilde{\tau}=0.5$ and $K=1$, our testing statistic reduces to the median regression-based method. Moreover, if we choose a proper non-trivial weight $\alpha$, $\tilde{T}_{\alpha}$ enjoys satisfactory power performance for data with a moderate magnitude of tails such as the Student's $t_v$ or Laplace distributions. Hence, our proposed individual testing statistics can adequately capture the tail structures of the data by choosing a proper combination of $\alpha$ and $\tilde{\btau}$.
Another distinguishing feature for using $\tilde{T}_{\alpha}$ is that, we can establish a general and flexible framework for aggregating the score based CUSUM for high-dimensional sparse linear models. Instead of taking the $\ell_\infty$-norm as most papers adopted for making statistical inference of high-dimensional linear models \citep[e.g.,][]{xia2018two-sample}, we choose to aggregate them via using the $\ell_2$-norm of the first $s_0$ largest order statistics. Under this framework, the $\ell_\infty$-norm is a special case by taking $s_0=1$. Moreover, as shown by our extensive numerical studies, choosing $s_0>1$ is more powerful for detecting sparse alternatives, and can significantly improve the detection powers as well as the estimation accuracy for change points, compared to using $s_0=1$.
\subsection{Variance estimation under $\Hb_0$ and $\Hb_1$}\label{sec: method of variance estimation} Note that $\tilde{T}_{\alpha}$ is constructed using a known variance $\sigma^2(\alpha,\tilde{\btau})$ which is defined in (\ref{equation: orcale sigma square}). In practice, however, $\sigma^2(\alpha,\tilde{\btau})$ is typically unknown. Hence, to yield a powerful testing method, a consistent variance estimation is needed especially under the alternative hypothesis. For high-dimensional change point analysis, the main difficulty comes from the unknown change point $t_1$. To overcome this problem, we propose a weighted variance estimation. In particular, for each fixed $\alpha\in[0,1]$ and $t\in(0,1)$, define the score based CUSUM statistic without standardization as follows:
\begin{equation}\label{equation: score cusum without sd}
\breve{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_{\alpha},\hat{\bbeta}_\alpha)=\dfrac{1}{\sqrt{n}}\big(\sum\limits_{i=1}^{\lfloor nt\rfloor}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)-\dfrac{\lfloor nt \rfloor }{n}\sum\limits_{i=1}^{n}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)\big).
\end{equation} Then, for each $\alpha\in[0,1]$, we obtain the individual based estimation for the change point:
\begin{equation}\label{equation: individual single cpt estimator}
\hat{t}_{\alpha}=\mathop{\text{\rm arg\,max}}_{q_0\leq t\leq 1-q_0}\big\|\breve{\bC}_{\alpha}(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha)\big\|_{(s_0,2)}.
\end{equation}
In Theorem \ref{theorem: cpt estimation results}, we prove that under some regular conditions, if $\Hb_1$ holds, $\hat{t}_\alpha$ is a consistent estimator for $t_1$, e.g. $|n\hat{t}_{\alpha}-nt_1|=o_p(n)$.
This result enables us to propose a variance estimator which is consistent under both $\Hb_0$ and $\Hb_1$. Specifically, let $h\in(0,1)$ be a user specified constant, and define the samples $n_-=\{i: i\leq nh\hat{t}_{\alpha}\}$ and $n_+=\{i: \hat{t}_{\alpha}n+(1-h)(1-\hat{t}_{\alpha})n \leq i\leq n \}$. Let $((\hat{\bb}^{(1)}_\alpha)^\top,(\hat{\bbeta}^{(1)}_\alpha)^\top)$ and $((\hat{\bb}^{(2)}_\alpha)^\top,(\hat{\bbeta}^{(2)}_\alpha)^\top)$ be the estimators using the samples in $n_-$ and $n_+$.
For each $\alpha\in[0,1]$, we can calculate the regression residuals:
\begin{equation}\label{equation: regression residuals}
\hat{\epsilon}_i= (Y_i-\bX_i^\top\hat{\bbeta}_{\alpha}^{(1)})\mathbf{1}\{i \in n_-\}+(Y_i-\bX_i^\top\hat{\bbeta}_{\alpha}^{(2)})\mathbf{1}\{i\in n_+\}, ~\text{for}~ i\in n_-\cup n_+.
\end{equation} Moreover, compute $\hat{e}_i(\tilde{\btau})=K^{-1}\sum\limits_{k=1}^{K}\hat{e}_i(\tau_k)$ with $\hat{e}_i(\tau_k)$ defined as
\begin{equation}\label{equation: regression quantile residuals}
\small
\hat{e}_i(\tau_k):=(\mathbf{1}\{\hat{\epsilon}_i\leq \hat{b}_{\alpha,k}^{(1)}\}-\tau_k)\mathbf{1}\{i \in n_-\}+(\mathbf{1}\{\hat{\epsilon}_i\leq \hat{b}_{\alpha,k}^{(2)}\}-\tau_k)\mathbf{1}\{i\in n_+\},~\text{for}~ i\in n_-\cup n_+.
\end{equation} Lastly, based on $\hat{\epsilon}_i$ and $\hat{e}_i(\tilde{\btau})$, we can construct our weighted estimator for $\sigma^2(\alpha,\tilde{\btau})$ as
\begin{equation}\label{equation: variance estimator}
\hat{\sigma}^2(\alpha,\tilde{\btau})=\hat{t}_{\alpha}\times\hat{\sigma}_-^2(\alpha,\tilde{\btau})+(1-\hat{t}_{\alpha})\times\hat{\sigma}_+^2(\alpha,\tilde{\btau}),
\end{equation} where:
\begin{equation*}
\hat{\sigma}_-^2(\alpha,\tilde{\btau}):=\dfrac{1}{|n_-|}\sum_{i\in n_-}\big[(1-\alpha)\hat{e}_i(\tilde{\btau})-\alpha\hat{\epsilon}_i\big]^2,~~\hat{\sigma}_+^2(\alpha,\tilde{\btau}):=\dfrac{1}{|n_+|}\sum_{i\in n_+}\big[(1-\alpha)\hat{e}_i(\tilde{\btau})-\alpha\hat{\epsilon}_i\big]^2.
\end{equation*}
For the above variance estimation, we can prove that $|\hat{\sigma}^2(\alpha,\tilde{\btau})-{\sigma}^2(\alpha,\tilde{\btau})|=o_p(1)$ under either $\Hb_0$ or $\Hb_1$. As a result, the proposed variance estimator $\hat{\sigma}^2(\alpha,\tilde{\btau})$ avoids the problem of non-monotonic power performance as discussed in \cite{shao2010testing}, which is a serious issue in change point analysis. Hence, we replace ${\sigma}(\alpha,\tilde{\btau})$ in (\ref{equation: score CUSUM}) by $\hat{\sigma}(\alpha,\tilde{\btau})$ and define the data-driven score-based CUSUM process $\{{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha, \hat{\bbeta}_\alpha),t\in[q_0,1-q_0]\}$ with
\begin{equation}\label{equation: data-driven score CUSUM}
\begin{array}{ll}
{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,
\hat{\bbeta}_\alpha)
=\dfrac{1}{\sqrt{n}\widehat \sigma(\alpha,\tilde{\btau})}\big(\sum\limits_{i=1}^{\lfloor nt\rfloor}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)-\dfrac{\lfloor nt \rfloor }{n}\sum\limits_{i=1}^{n}\bZ_{\alpha}(\bX_i,Y_i;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_\alpha)\big).
\end{array}
\end{equation} Note that we call $ {\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha, \hat{\bbeta}_\alpha)$ data-driven since there are no unknown parameters in the testing statistic. For a user-specified $s_0\in\{1,\ldots,p\}$ and any $\alpha$, we define the final individual-based testing statistic as follows:
\begin{equation}\label{equation: final individual test statistic}
{T}_{\alpha}=\max_{q_0\leq t\leq 1-q_0}\Big\|{\bC}_\alpha(t;\tilde{\btau},\hat{\bb}_\alpha,\hat{\bbeta}_{a})\Big\|_{(s_0,2)},~~\text{with}~~\alpha\in[0,1].
\end{equation} Throughout this paper, we use $\{{T}_{\alpha},\alpha\in[0,1]\}$ as our individual-based testing statistics. \subsection{Bootstrap approximation for the individual testing statistic }\label{sec: method of bootstrap} In high dimensions, it is very difficult to obtain the limiting null distribution of $T_{\alpha}$. To overcome this problem, we propose a novel bootstrap procedure. In particular, suppose we implement the bootstrap procedure for $B$ times. Then, for each $b$-th bootstrap with $b=1,\ldots,B$, we generate i.i.d. random variables $e_1^b,\ldots,e_n^b$ with $e_i^b\sim N(0,1)$. Let $e_i^{b}(\tilde{\btau})=K^{-1}\sum\limits_{k=1}^{K}e_i^{b}(\tau_k)$ with $e_i^{b}(\tau_k):=\mathbf{1}\{\epsilon_i^b\leq \Phi^{-1}(\tau_k)\}-\tau_k$, where $\Phi(x)$ is the CDF for the standard normal distribution. Then, for each individual-based testing statistic $T_{\alpha}$, with a user specified $s_0$, we define its $b$-th bootstrap sample-based score CUSUM process as:
\begin{equation}\label{equation: bootstrap score CUSUM}
{\bC}^b_\alpha(t;\tilde{\btau})
=\dfrac{1}{\sqrt{n}v(\alpha,\tilde{\btau})}\big(\sum\limits_{i=1}^{\floor{nt}}\bX_i((1-\alpha) e^{b}_i(\tilde{\btau})-\alpha e^b_i)-\dfrac{\floor{nt}}{n}\sum\limits_{i=1}^n\bX_i((1-\alpha) e^{b}_i(\tilde{\btau})-\alpha e^b_i)\big),
\end{equation} where $v^2(\alpha,\tilde{\btau})$ is the corresponding variance for the bootstrap-based sample with
\begin{equation}\label{equation: oracle variance bootstrap}
v^2(\alpha,\tilde{\btau}):=(1-\alpha)^2\text{Var}[e_i^{b}(\tilde{\btau})]+\alpha^2-2\alpha(1-\alpha)\text{Cov}(e_i^{b}(\tilde{\btau}),e^b_i).
\end{equation} Note that for bootstrap, the calculation or estimation of $v^2(\alpha,\tilde{\btau})$ is not a difficult task since we use $N(0,1)$ as the error term. For example, when $\tilde{\tau}=0.5$, it has an explicit form of
\begin{equation*}
v^2(\alpha,\tilde{\btau})=(1-\alpha)^2\sum\limits_{k_1=1}^{K}\sum\limits_{k_2=1}^{K}\gamma_{k_1k_2}+\alpha^2-\alpha(1-\alpha)\sqrt{\dfrac{2}{\pi}}.
\end{equation*} Hence, for simplicity, we directly use the oracle variance $v^2(\alpha,\tilde{\btau})$ in (\ref{equation: bootstrap score CUSUM}). Using ${\bC}^b_\alpha(t;\tilde{\btau})$ and for a user specified $s_0$, we define the $b$-th bootstrap version of the individual-based testing statistic $T_{\alpha}$ as
\begin{equation}\label{equation: bootstrap individual test}
T_\alpha^b=\max_{q_0\leq t\leq 1-q_0}\Big\|\bC^b_{\alpha}(t;\tilde{\btau})\Big\|_{(s_0,2)},~~\text{with}~~\alpha\in[0,1].
\end{equation} Let $\gamma\in(0,0.5)$ be the significance level. For each individual-based testing statistic $T_\alpha$, let $F_{\alpha}={\mathbb P}(T_{\alpha}\leq t)$ be its theoretical CDF and $P_\alpha=1-F_{\alpha}(T_{\alpha})$ be its theoretical $p$-value. Using the bootstrap samples $\{T_{\alpha}^1,\ldots,T_{\alpha}^B\}$, we estimate $P_\alpha$ by
{
\begin{equation}\label{equation: p-value for individual test}
\hat{P}_{\alpha}=\dfrac{\sum_{b=1}^B\mathbf{1}\{T_{\alpha}^b> T_{\alpha}|\cX,\cY\}}{B+1},~~\text{with}~~\alpha\in[0,1].
\end{equation} } Given the significance level $\gamma$, we can construct the individual test as
\begin{equation}\label{equation: final individual test}
\Psi_{\gamma,\alpha}=\mathbf{1}\{ \hat{P}_{\alpha}\leq \gamma\}, ~~\text{with}~~\alpha\in[0,1].
\end{equation} For each $T_{\alpha}$, we reject $\Hb_0$ if and only if $\Psi_{\gamma,\alpha}=1$. Note that the above bootstrap procedure is easy to implement since it does not require any model fitting such as obtaining the LASSO estimators which is required by the data-based testing statistic $T_{\alpha}$. \subsection{Constructing the tail-adaptive testing procedure }\label{sec: method of tail-adaptive} \begin{algorithm}[!h]
\caption{: A bootstrap procedure to obtain the tail-adaptive testing statistic $T_{\rm ad}$ }\label{alg:first}
\begin{description}
\item[Input:] Given the data $(\mathcal{X},\cY)=\{(\bX_1,Y_1),\ldots,(\bX_n,Y_n)\}$, set the values for $\tilde{\btau}$, $s_0$, $q_0$, the bootstrap replication number $B$, and the candidate subset $\cA\subset[0,1]$.
\item[Step~1:] Calculate the individual-based testing stastisic $T_{\alpha}$ for $\alpha\in\cA$ as defined in (\ref{equation: final individual test statistic}).
\item[Step~2:] For each $\alpha\in \cA$, repeat the procedure (\ref{equation: bootstrap score CUSUM}) -- (\ref{equation: bootstrap individual test}) for $B$ times, and obtain the bootstrap samples $\{T_{\alpha}^1,\ldots,T_{\alpha}^B\}$ for $\alpha\in \cA$.
\item[Step~3:] Based on the bootstrap samples in Step~2, calculate empirical $P$-values $\hat{P}_{\alpha}$ for $\alpha\in\cA$ as defined in (\ref{equation: p-value for individual test}).
\item[Step~4:] Using $\hat{P}_{\alpha}$ with $\alpha\in\cA$, calculate the tail-adaptive testing statistic $T_{\rm ad}$ in (\ref{equation: tail-adaptive test statistic}).
\item[Output:] Algorithm \ref{alg:first} provides the bootstrap based samples $\{T_{\alpha}^1,\ldots,T_{\alpha}^B\}$ for $\alpha\in \cA$, and the tail-adaptive testing statistic $T_{\rm ad}$.
\end{description} \end{algorithm}
In Sections \ref{sec: method of single cpt} -- \ref{sec: method of bootstrap}, we propose a family of individual-based testing statistics $\{T_{\alpha},\alpha\in[0,1]\}$ and introduce a bootstrap-based procedure for approximating their theoretical $p$-values. As discussed in Sections \ref{sec: method of single cpt} and seen from Figure \ref{figure: snr}, $T_{\alpha}$ with different $\alpha$'s can have various power performance for a given underlying error distribution. For example, $T_{\alpha}$ with a larger $\alpha$ (e.g. $\alpha=1$) is more sensitive to change points with light-tailed error distributions by using more moment information. In contrast, $T_{\alpha}$ with a smaller $\alpha$ (e.g. $\alpha=0,0.1$) is more powerful for data with heavy tails such as Student's $t_v$ or even Cauchy distribution. In general, as shown in Figure \ref{figure: snr}, a properly chosen $\alpha$ can give the most satisfactory power performance for data with a particular magnitude of tails. In practice, however, the tail structures of data are typically unknown. Hence, it is desirable to construct a tail-adaptive method which is simultaneously powerful under various tail structures of data. One candidate method is to find $\alpha^*$ which maximizes the theoretical $SNR(\alpha,\tilde{\btau})$, i.e. $\alpha^*=\mathop{\text{\rm arg\,max}}_{\alpha}SNR(\alpha,\tilde{\btau})$, and constructs a corresponding individual testing statistic $T_{\alpha^*}$. Note that in theory, calculating $SNR(\alpha,\tilde{\btau})$ needs to know $\sigma(\alpha,\tilde{\btau})$ and $\{f_{\epsilon}(b_{k}^{(0)}),k=1,\ldots,K\}$, which could be difficult to estimate especially under the high-dimensional change point model. Instead, we construct our tail-adaptive method by combining all candidate individual tests for yielding a powerful one. In particular, as a small $p$-value leads to rejection of $\Hb_0$, for the individual tests $T_{\alpha}$ with $\alpha\in[0,1]$, we construct the tail-adaptive testing statistic as their minimum $p$-value, which is defined as follows:
\begin{equation}\label{equation: tail-adaptive test statistic}
T_{\rm ad}=\min_{\alpha\in\cA} \hat{P}_{\alpha},
\end{equation} where $\hat{P}_{\alpha}$ is defined in (\ref{equation: p-value for individual test}), and $\cA$ is a candidate subset of $\alpha$.
In this paper, we require $|\cA|$ to be finite, which is a theoretical requirement. Note that our tail-adaptive method is flexible and user-friendly. In practice, if the users have some prior knowledge about the tails of errors, we can choose $\cA$ accordingly. For example, we can choose $\mathcal{A}=\{0.9,1\}$ for light-tailed errors, and $\mathcal{A}=\{0\}$ for extreme heavy tails such as Cauchy distributions. However, if the tail structure is unknown, we can choose $\cA$ consisting both small and large values of $\alpha\in[0,1]$. For example, according to our theoretical analysis of $SNR(\alpha,\tilde{\btau})$, we find that $SNR(\alpha,\tilde{\btau})$ tends to be maximized near the boundary of $[0,1]$. Hence, we recommend to use $\cA=\{0,0.1,0.5,0.9,1\}$ in real applications, which is shown by our numerical studies to enjoy stable size performance as well as high powers across various error distributions. Algorithm \ref{alg:first} describes our procedure to construct $T_{\rm ad}$. Using Algorithm \ref{alg:first}, we construct the tail-adaptive testing statistic $T_{\rm ad}$. Let $F_{\rm ad}(x)$ be its theoretical distribution function. Note that $F_{\rm ad}(x)$ is unknown. Hence, we can not use $T_{\rm ad}$ directly for Problem \eqref{hypothesis: H0}. To approximate its theoretical $p$-value, we adopt the low-cost bootstrap method proposed by \cite{Zhou2017An}, which is also used in \cite{Liu2019Unified}. Let $\hat{P}_{\rm ad}$ be an estimation for the theoretical $p$-value of $T_{\rm ad}$ using the low-cost bootstrap. Given the significance level $\gamma\in(0,0.5)$, define the final tail-adaptive test as:
\begin{equation}\label{statistic: adaptive tests}
\begin{array}{lc}
\Psi_{\gamma,\rm ad}=\mathbf{1}\{\hat{P}_{\rm ad}\leq \gamma \}.
\end{array}
\end{equation}
For the tail-adaptive testing procedure, given $\gamma$, we reject $\Hb_0$ if $\Psi_{\gamma,\rm ad}=1$. If $\Hb_0$ is rejected by our tail-adaptive test, letting $\widehat{\alpha} = \mathop{\text{\rm arg\,min}}_{\alpha \in \mathcal{A}} \hat{P}_{\alpha}$, we return the change point estimation as $\hat{t}_{\rm ad} = \widehat{t}_{\widehat{\alpha}}$. As shown by our theory, under some conditions, we have $|\hat{t}_{\rm ad}-t_1 |=o_p(1)$.
\subsection{Multiple change point detection}\label{sec: method of multiple cpt}
\begin{algorithm}[!h]
\caption{: A WBS-typed tail-adaptive test for multiple change point detection}\label{alg:WBS}
\begin{description}
\item[Input:] Given the data $(\mathcal{X},\cY)=\{(\bX_1,Y_1),\ldots,(\bX_n,Y_n)\}$, set the values for $\tilde{\btau}$, the significance level $\gamma$, $s_0$, $q_0$, the bootstrap replication number $B$, the candidate subset $\cA\subset[0,1]$, and a set of random intervals $\{(s_\nu, e_\nu\}_{\nu = 1}^V$ with thresholds $v_0$ and $v_1$. Initialize an empty set $\cC$.
\item[Step~1:] For each $\nu = 1, \cdots, V$, compute $\hat{P}_{\rm ad}(s_\nu, e_\nu)$ following Section \ref{sec: method of tail-adaptive}.
\item[Step~2:] Perform the following function with $S=q_0$ and $E=1-q_0$.
\item[Function(S, E):] $S$ and $E$ are the starting and ending points for the change point detection.
\begin{enumerate}
\item[(a)] RETURN if $E-S \leq v_1$.
\item[(b)] Define $\cM = \{1 \leq \nu \leq V \, | \, [s_\nu, e_\nu] \subset [E, S]\}$.
\item[(c)] Compute the test statistics as $\overline{P}_{\rm ad} = \underset{\nu \in \cM, v_0 \leq e_{\nu} - s_\nu}{\min} \hat{P}_{\rm ad}(s_\nu, e_\nu)$ and the corresponding optimal solution $\nu^\ast$.
\item[(d)] If $\overline{P}_{\rm ad} \geq \gamma / V$, RETURN. Otherwise, add the corresponding change point estimator $\hat t_{\nu^\ast}$ to $\cC$, and perform Function(S, $\nu^\ast$) and Function($\nu^\ast$, E).
\end{enumerate}
\item[Output:] The set of multiple change points $\cC$.
\end{description} \end{algorithm}
In practical applications, it may exist multiple change points in describing the relationship between $\bX$ and $Y$. Therefore, it is essential to perform estimation of multiple change points if $\Hb_0$ is rejected by our powerful tail-adaptive test. In this section, we extend our single change point detection method by the idea of WBS proposed in \cite{fryzlewicz2014wild} to estimate the locations of all possible multiple change points.
Consider a single change point detection task in any interval $[s, e]$, where $0 \leq q_0 \leq s < e \leq 1-q_0$. Following Section \ref{sec: method of tail-adaptive}, we can compute the corresponding adaptive test statistics as $\hat{P}_{\rm ad}(s, e)$ using the subset of our data, i.e., $\{ \bX_{\floor{ns}}, \bX_{\floor{ns} + 1}, \cdots, \bX_{\floor{ne}} \}$ and $\{Y_{\floor{ns}}, Y_{\floor{ns} + 1}, \cdots, Y_{\floor{ne}}\}$. Following the idea of WBS, we first independently generate a series of random intervals by the uniform distribution. Denote the number of these random intervals as $V$. For each random interval $[s_\nu, e_\nu]$ among $\nu = 1, 2, \cdots, V$, we compute $\hat{P}_{\rm ad}(s_\nu, e_\nu)$ as long as $0 \leq q_0 \leq s_\nu < e_\nu \leq 1-q_0$ and $e_\nu - s_\nu \geq v_0$, where $v_0$ is the minimum length for implementing Section \ref{sec: method of tail-adaptive}. The threshold $v_0$ is used to reduce the variability of our algorithm for multiple change point detection. Based on the test statistics computed from the random intervals, we consider the final test statistics as $\overline{P}_{\rm ad} = \underset{1 \leq \nu \leq V, v_0 \leq e_{\nu} - s_\nu}{\min} \hat{P}_{\rm ad}(s_\nu, e_\nu)$, based on which we make decisions if there exists at least one change point among these intervals. We stop the algorithm if $\overline{P}_{\rm ad} \geq \bar c$, otherwise we report the change point estimation in $[s_{\nu^\ast}, e_{\nu^\ast}]$, where $\nu^\ast \in \underset{1 \leq \nu \leq V, v_0 \leq e_{\nu} - s_\nu}{\mathop{\text{\rm arg\,min}}} \hat{P}_{\rm ad}(s_\nu, e_\nu)$, and continue our algorithm. Given the first change point estimator denoted by $\hat t_{\nu^\ast}$, we split our data into two folds, i.e., before and after the estimated change point. Then we apply the previous procedure on each fold of the data using the same set of the random intervals as long as it satisfies the constraints. We repeat this step until the algorithm stops returning the change point estimation. For each step, we choose $\bar c = \gamma / V$, where $\gamma$ is the significance level used in each single change point detection algorithm. While we do not have the theoretical guarantee of using $\bar c$ in the proposed algorithm for controlling the size, the selection of this constant is based on the idea of Bonferroni correction, which is conservative. The numerical experiments in the appendix demonstrate the superiority of our proposed method in detecting multiple change points. Nevertheless, it is interesting to study the asymptotic property of $\overline{P}_{\rm ad}$, which we leave for the future work. The full algorithm of the multiple change point detection can be found in Algorithm \ref{alg:WBS}.
\section{Theoretical results}\label{sec: theory} In this section, we give theoretical results for our proposed methods. In Section \ref{section: basic assumptions}, we provide some basic model assumptions. In Section \ref{sec: Theoretical results of the individual test statistics}, we discuss the theoretical properties of the individual testing methods with a fixed $\alpha$ such as the size, power and change point estimation. In Section \ref{sec: Theoretical results of the adaptive test statistics}, we provide the theoretical results for the tail-adaptive method.
\subsection{Basic assumptions}\label{section: basic assumptions}
We introduce some basic assumptions for deriving our main theorems. Before that, we introduce some notations. Let $ e_i(\tilde{\btau}):=K^{-1}\sum\limits_{k=1}^{K}\big(\mathbf{1}\{\epsilon_i\leq b^{(0)}_k\}-\tau_k\big):=K^{-1}\sum\limits_{k=1}^{K}e_i(\tau_k)$. We set $\mathcal{V}_{s_0}:=\{\bv\in \mathbb{S}^{p}: \|\bv\|_0\leq s_0\}$, where $\mathbb{S}^{p}:=\{\bv\in\RR^p: \|\bv\|=1\}$. For each $\alpha\in[0,1]$, we introduce $\uwave{\bbeta}^*=((\bbeta^*)^\top,(\bb^*)^\top)^\top\in \RR^{p+K}$ with $\bbeta^*\in \RR^p$, $\bb^*=(b_1^*,\ldots,b_K^*)^\top\in \RR^K$, where
\begin{equation}
\uwave{\bbeta}^*:=\mathop{\text{\rm arg\,min}}_{\bbeta\in \RR^p,\bb\in \RR^K}{\mathbb E} \Big[(1-\alpha)\dfrac{1}{n}\sum_{i=1}^{n}\dfrac{1}{K}\sum_{k=1}^{K}\rho_{\tau_k}(Y_i-b_i-\bX_i^\top\bbeta)+ \dfrac{\alpha}{2n}\sum_{i=1}^{n}(Y_i-\bX_i^\top \bbeta)^2\Big].
\end{equation} Note that by definition, we can regard $\uwave{\bbeta}^*$ as the true parameters under the population level with pooled samples. We can prove that under $\Hb_0$, $\uwave{\bbeta}^*=((\bbeta^{(0)})^\top,(\bb^{(0)})^\top)^\top$ with $\bb^{(0)}=(b_{1}^{(0)},\ldots,b_{K}^{(0)})^\top$. Under $\Hb_1$, $\uwave{\bbeta}^*$ is generally a weighted combination of the parameters before the change point and those after the change point. For example, when $\alpha=1$, it has the explicit form of $ \uwave{\bbeta}^*=((t_1\bbeta^{(1)}+t_2\bbeta^{(2)})^\top,(\bb^{(0)})^\top)^\top$. More discussions about $\uwave{\bbeta}^*$ under our weighted composite loss function are provided in the appendix. With the above notations, we are ready to introduce our assumptions as follows:\\ {\bf{Assumption A (Design matrix):}} The design matrix $\Xb$ has i.i.d rows $\{\bX_i\}_{i=1}^n$.
(A.1) Assume that there are positive constants $\kappa_1$ and $\kappa_2$ such that $\lambda_{\rm min}(\bSigma)\geq\kappa_1>0$ and $\lambda_{\rm max}(\bSigma)\leq \kappa_2<\infty$ hold. (A.2) There exists some constant $M\geq 1$ such that $\max_{1\leq i\leq n}\max_{1\leq j\leq p}|X_{ij}|\leq M$ almost surely {for every $n$ and $p$.}\\ {\bf{Assumption B} (Error distribution):} The error terms $\{\epsilon_i\}_{i=1}^n$ are i.i.d. with mean zero and finite variance $\sigma_{\epsilon}^2$. There exist positive constants $c_{\epsilon}$ and $C_{\epsilon}$ such that $c^2_\epsilon\leq \text{Var}(\epsilon_i)\leq C^2_{\epsilon}$ hold. In addition, $\epsilon_i$ is independent with $\bX_i$ for $i=1,\ldots,n$.\\ {\bf{Assumption C (Moment constraints)}: } (C.1) There exists some constant $b>0$ such that ${\mathbb E}(\bv^\top\bX_i\epsilon_i)^{2}\geq b$ and ${\mathbb E}(\bv^\top\bX_ie_i(\tilde{\btau}))^{2}\geq b$, for $\bv\in \cV_{s_0}$ and all $i=1,\ldots,n$. Moreover, assume that $\inf_{i,j}{\mathbb E}[X^2_{ij}]\geq b$ holds.
(C.2) There exists a constant $K>0$ such that ${\mathbb E}|\epsilon_i|^{2+\ell}\leq K^{\ell}$, for $\ell=1,2$.\\ \ {\bf{Assumption D (Underlying distribution):}} The distribution function $\epsilon$ has a continuously differentiable density function $f_{\epsilon}(t)$ whose derivative is denoted by $f'_{\epsilon}(t)$. Furthermore, suppose there exist some constants $C_{+}$, $C_{-}$ and $C'_{+}$ such that
\begin{equation*}
\begin{array}{l}
\text{(D.1)} \sup\limits_{t\in\RR}f_{\epsilon}(t)\leq C_{+}; ~
\text{(D.2)} \inf\limits_{j=1,2}\inf\limits_{1\leq k\leq K} f_{\epsilon}(\bx^\top(\bbeta^*-\bbeta^{(j)})+b_k^*)\geq C_-;
\text{(D.3)} \sup\limits_{t\in\RR}|f'_{\epsilon}(t)|\leq C'_{+}.
\end{array}
\end{equation*} {\bf{Assumption E (Parameter space):}}
(E.1) We require $s_0^3\log(pn)=O(n^{\xi_1})$ for some $0<\xi_1<1/7$ and $s_0^4\log(pn)=O(n^{\xi_2})$ for some $0<\xi_2<\frac{1}{6}$. (E.2) Assume that $\dfrac{s_0^2s^3\log^3(pn)}{n}\rightarrow0$ as $(n,p)\rightarrow \infty$. (E.3) For $\bbeta^{(1)}$ and $\bbeta^{(2)}$, we require $\max (\|\bbeta^{(1)}\|_{\infty}, \|\bbeta^{(2)}\|_{\infty} )<C_{\bbeta}$ for some $C_{\bbeta}>0$. Moreover, we require $\|\bbeta^{(2)}-\bbeta^{(1)}\|_1\leq C_{\bDelta}$ for some constant $C_{\bDelta}>0$. (E.4) For the tuning parameters $\lambda_{\alpha}$ in (\ref{equation: lasso estimator}), we require $\lambda_{\alpha}=C_{\lambda}\sqrt{\log(pn)/n}$ for some $C_\lambda>0$.
Assumption A gives some conditions for the design matrix, requiring $\bX$ has a non-degenerate covariance matrix $\bSigma$ in terms of its eigenvalues. This is important for deriving the high-dimensional LASSO property with $\alpha\in[0,1]$ under both $\Hb_0$ and $\Hb_1$. Moreover, it also requires that $X_{ij}$ is bounded by some big constant $M>0$, which has been commonly used in the literature. Assumption B mainly requires the underlying error term $\epsilon_i$ has non-degenerate variance. Assumption C imposes some restrictions on the moments of the error terms as well as the design matrix. In particular, Assumption C.1 requires that $\bv^\top\bX\epsilon$, $\bv^\top\bX e(\tilde{\btau})$, as well as $X_{ij}$ have non-degenerate variances. Moreover, Assumption C.2 requires that the errors have at most fourth moments, which is much weaker than the commonly used Gaussian or sub-Gaussian assumptions. Both Assumptions C.1 and C.2 are basic moment conditions for bootstrap approximations for the individual-based tests. See Lemma C.6 in the appendix. Assumptions D.1 - D.3 are some regular conditions for the underlying distribution of the errors, requiring $\epsilon$ has a bounded density function as well as bounded derivatives. Assumption D.2 also requires the density function at $\bx^\top(\bbeta^*-\bbeta^{(j)})+b_k^*$ to be strictly bounded away from zero. Lastly, Assumption E imposes some conditions for the parameter spaces in terms of $(s_0,n,p,s,\bbeta^{(1)},\bbeta^{(2)})$. Specifically, Assumption E.1 scales the relationship between $s_0$, $n$, and $p$, which allows $s_0$ can grow with the sample size $n$. {This condition is mainly used to establish the high-dimensional Gaussian approximation for our individual tests.} Assumption E.2 also gives some restrictions on $(s_0,s,n,p)$. Note that both Assumptions E.1 and E.2 allow the data dimension $p$ to be much larger than the sample size $n$ as long as the required conditions hold. Assumption E.3 requires that the regression coefficients as well as signal jump in terms of its $\ell_1$-norm are bounded. Assumption E.4 imposes the regularization parameter $\lambda_\alpha=O(\sqrt{\log(pn)/n})$, which is important for deriving the desired error bound for the LASSO estimators under both $\Hb_0$ and $\Hb_1$ using our weighted composite loss function. See Lemmas C.9 - C.11 in the appendix.
\begin{remark}
Assumption C.2 with the finite fourth moment is mainly for the individual test with $\alpha=1$, while Assumption D without any moment constraints is for that with $\alpha=0$. Hence, our proposed individual-based change point method extends the high-dimensional linear models with sub-Gaussian distributed errors to those with only finite moments or without any moments, covering a wide range of errors with different tails. \end{remark}
\begin{remark}
Our proposed individual method with $\alpha=0$ needs to cover both cases with and without a change point. To obtain the desired error bound of LASSO estimation, we require that $\inf\limits_{j=1,2}\inf\limits_{1\leq k\leq K} f_{\epsilon}(\bx^\top(\bbeta^*-\bbeta^{(j)})+b_k^*)\geq C_-$. This is different from the classical assumption \citep{2014A} that $\inf\limits_{1\leq k\leq K} f_{\epsilon}(b_k^{(0)})\geq C_-$. Note that our assumption is quite mild since essentially, it only requires that the density function is non-generate at a neighborhood of $b_k^*$ which is shown to be satisfied under Assumptions A.2 and E.3. \end{remark}
\subsection{Theoretical results of the individual-based testing statistics}\label{sec: Theoretical results of the individual test statistics} \subsubsection{Validity of controlling the testing size} Before giving the size results, we first consider the variance estimation. Recall $\sigma^2(\alpha,\tilde{\btau})$ in (\ref{equation: orcale sigma square}) and $\hat{\sigma}^2(\alpha,\tilde{\btau})$ in (\ref{equation: variance estimator}). Theorem \ref{theorem: variance estimator under H0} shows that the pooled weighted variance estimator $\hat{\sigma}^2(\alpha,\tilde{\btau})$ is consistent under the null hypothesis, which is crucial for deriving the Gaussian approximation results as shown in Theorem \ref{theorem: size control for individual test} and shows that our testing method has satisfactory size performance. \begin{theorem}\label{theorem: variance estimator under H0}
For $\alpha=1$, suppose Assumptions A, B, C, E hold. For $\alpha=0$, suppose {Assumptions A, C.1, D, E} hold. For $\alpha\in(0,1)$, suppose { Assumptions A - E} hold.
Let $r_{\alpha}(n)=\sqrt{s\log(pn)/n}$ if $\alpha=1$ and $r_{\alpha}(n)=s\sqrt{\dfrac{\log(pn)}{n}}\vee s^{\frac{1}{2}}(\dfrac{\log(pn)}{n})^{\frac{3}{8}}$ if $\alpha\in[0,1)$.
Under $\Hb_0$, for each $\alpha\in[0,1]$, we have
\begin{equation*}\label{equation: variance estimation under H0}
|\hat{\sigma}^2(\alpha,\tilde{\btau})-{\sigma}^2(\alpha,\tilde{\btau})|=O_p(r_{\alpha}(n)).
\end{equation*} \end{theorem} Based on Theorem \ref{theorem: variance estimator under H0} as well as some other regularity conditions, the following Theorem \ref{theorem: size control for individual test} justifies the validity of our bootstrap-based procedure in Algorithm \ref{alg:first}. \begin{theorem}\label{theorem: size control for individual test}
For $\alpha=1$, suppose Assumptions A, B, C, E hold. For $\alpha=0$, suppose {Assumptions A, C.1, D, E} hold. For $\alpha\in(0,1)$, suppose { Assumptions A - E} hold. Then, under $\Hb_0$, for the individual test with $\alpha\in[0,1]$, we have
\begin{equation}\label{equation: size individual test }
\sup_{z\in (0,\infty)}\big|{\mathbb P}(T_{\alpha}\leq z)-{\mathbb P}(T_{\alpha}^{b}\leq z|\mathcal{X},\cY)\big|=o_p(1), ~\text{as}~ n,p\rightarrow\infty.
\end{equation} \end{theorem}
Theorem \ref{theorem: size control for individual test} demonstrates that we can uniformly approximate the distribution of $T_{\alpha}$ by that of $T_{\alpha}^b$. The following Corollary further shows that our proposed new test $\Psi_{\gamma,\alpha}$ can control the Type I error asymptotically for any given significant level $\gamma\in(0,1)$. \begin{corollary}\label{corollary: size}
Suppose the assumptions in Theorem \ref{theorem: size control for individual test} hold. Under $\Hb_0$, we have
\begin{equation*}
{\mathbb P}(\Psi_{\gamma,\alpha}=1)\rightarrow\gamma, ~ \text{as} ~n,p,B\rightarrow\infty.
\end{equation*} \end{corollary} \subsubsection{Change point estimation}
After analyzing the theoretical results under the null hypothesis, we next consider the performance of the individual test under $\Hb_1$. We first give some theoretical results on the change point estimation. To that end, some additional assumptions are needed. Recall $\Pi=\{j: \beta^{(1)}_j\neq \beta^{(2)}_j\}$ as the set with change points. For $j\in\{1,\ldots,p\}$, define the signal jump $\bDelta=(\Delta_1,\ldots,\Delta_p)^\top$ with $\Delta_j:=\beta^{(1)}_j-\beta^{(2)}_j$. Let $\Delta_{\min}=\min_{j\in\Pi}|\Delta_j|$ and $\Delta_{\max}=\max_{j\in \Pi}|\Delta_j|$. With the above notations, we introduce the following {{Assumption F}}.\\ {\bf{Assumption F}}. There exist constants $\underline{c}>0$ and $\overline{C}>0$ such that
\begin{equation}\label{inequality: ratio of maximum and minimum signal}
0<\underline{c}\leq \liminf\limits_{p\rightarrow\infty} \dfrac{\Delta_{\min}}{\Delta_{\max}} \leq\limsup\limits_{p\rightarrow\infty}\dfrac{\Delta_{\max}}{\Delta_{\min}}\leq \overline{C}<\infty.
\end{equation} Note that Assumption F is only a technical condition requiring that $\Delta_{\min}$ and $\Delta_{\max}$ are of the same order. With Assumption F as well as those of Assumptions A - E, Theorem \ref{theorem: cpt estimation results} below provides a non-asymptotic estimation error bound of the argmax-based individual change point estimator $\hat{t}_{\alpha}$ for $t_1$. To give a precise result, for change point estimation, we assume $s_0$ is fixed. \begin{theorem}\label{theorem: cpt estimation results}
Suppose $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$ and {Assumption F} hold. Moreover,
For $\alpha=1$, suppose { Assumptions A, B, C, E.2 - E.4 } as well as $n^{1/4}=o(s)$ hold;
For $\alpha=0$, suppose { Assumptions A, C.1, D, E.2 - E.4} as well as
\vspace*{-0.2in}
\begin{equation*}
\lim\limits_{n,p\rightarrow\infty}s_0^{1/2}s^2\sqrt{\log(p)/n}\|\bDelta\|_{(s_0,2)}=0
\vspace*{-0.2in}
\end{equation*}
hold;
For $\alpha\in(0,1)$, suppose { Assumptions A, B, C, D, E.2 - E.4 } as well as $n^{1/4}=o(s)$ \\ and $\lim\limits_{n,p\rightarrow\infty}s_0^{1/2}s^2\sqrt{\log(pn)/n}\|\bDelta\|_{(s_0,2)}=0$ hold.
Then, under $\Hb_1$, for each $\alpha\in[0,1]$, with probability tending to one, we have
\begin{equation}\label{inequality: estimation error bound of cpt}
\big|\hat{t}_{\alpha}-t_1\big|\leq C^*(s_0,\tilde{\btau},\alpha)\dfrac{\log(pn)}{nSNR^2(\alpha,\tilde{\btau})\|\bSigma\bDelta\|^2_{(s_0,2)}},
\end{equation}
where $C^*(s_0,\tilde{\btau},\alpha)>0$ is some universal constant only depending on $s_0, \tilde{\btau}$ and $\alpha$. \end{theorem}
Theorem \ref{theorem: cpt estimation results} shows that our individual estimators are consistent under the condition $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$. Moreover, according to \cite{rinaldo2021localizing}, for high-dimensional linear models, under Assumption F, if $\|\bDelta\|_{\infty}\gg 1/\sqrt{n} $, any change point estimator $\hat{t}$ has an estimation lower bound $|\hat{t}-t_1|\geq c_*\dfrac{1}{n\|\bDelta\|^2_{\infty}}$, for some constant $c_*>0$. Hence, considering (\ref{inequality: ratio of maximum and minimum signal}) and (\ref{inequality: estimation error bound of cpt}), with a fixed $s_0$, Theorem \ref{theorem: cpt estimation results} demonstrates that our individual-based estimators for the change point are rate optimal up to a $\log(pn)$ factor. More importantly, our proposed estimators $\{\hat{t}_{\alpha},\alpha\in[0,1]\}$ only require $O(\rm Lasso(n,p))$ operations to calculate, which is less computationally expensive than the grid search based method in \cite{lee2016LASSO}.
\subsubsection{Power performance} We discuss the power properties of the individual tests. Note that for the change point problem, variance estimation under the alternative is a difficult but important task. As pointed out in \cite{shao2010testing}, due to the unknown change point, any improper estimation may lead to nonmonotonic power performance. This distinguishes the change point problem substantially from one-sample or two-sample tests where homogenous data are used to construct consistent variance estimation. Hence, for yielding a powerful change point test, we need to guarantee a consistent variance estimation. Theorem \ref{theorem: variance estimator under H1} shows that the pooled weighted variance estimation is consistent under $\Hb_1$. This guarantees that our proposed testing method has reasonable power performance. \begin{theorem}\label{theorem: variance estimator under H1}
Suppose the assumptions in Theorem \ref{theorem: cpt estimation results} hold.
Let $r_{\alpha}(n)=\sqrt{s\log(pn)/n}$ if $\alpha=1$ and $r_{\alpha}(n)=s\sqrt{\dfrac{\log(pn)}{n}}\vee s^{\frac{1}{2}}(\dfrac{\log(pn)}{n})^{\frac{3}{8}}$ if $\alpha\in[0,1)$.
Under $\Hb_1$, for each $\alpha\in[0,1]$, we have
\begin{equation*}\label{equation: variance estimation under H1}
|\hat{\sigma}^2(\alpha,\tilde{\btau})-{\sigma}^2(\alpha,\tilde{\btau})|=O_p(r_{\alpha}(n)).
\end{equation*} \end{theorem}
According to the proof of Theorem \ref{theorem: variance estimator under H1}, several interesting observations can be drawn. Even if the signal strength is weak such as $\|\bDelta\|_{(s_0,2)}=O(\sqrt{\log(pn)/n})$, the pooled weighted variance estimator $\hat{\sigma}^2(\alpha,\tilde{\btau})$ can still be consistent for ${\sigma}^2(\alpha,\tilde{\btau})$. However, in this case, we can not guarantee that our change point estimator is consistent as $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$ is required in Theorem \ref{theorem: cpt estimation results} for consistency. In contrast, if the signal strength is strong enough such that $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\log(p)/n}$, then a consistent change point estimator such as the proposed $\hat{t}_{\alpha}$ is needed to guarantee Theorem \ref{theorem: variance estimator under H1}. These are insightful findings for variance estimation in change point analysis, which are not shown in the i.i.d. case.
Using the consistent variance estimation, we are able to discuss the power properties of the individual tests. To this end, we need some additional notations. Define the oracle signal to noise ratio vector $\bD=(D_1,\ldots,D_{p})^\top$ with
\begin{equation}\label{equation: signal to noise ration}
D_j:=\left\{\begin{array}{ll}
0,& \text{for}~~ j\in \Pi^c\\
SNR(\alpha,\tilde{\btau})\times\Big|{t_1(1-t_1)\big(\bSigma(\bbeta^{(1)}-\bbeta^{(2)})\big)_j}\Big|,& \text{for}~~j\in \Pi,
\end{array}\right.
\end{equation} where $SNR(\alpha,\tilde{\btau})$ is defined in (\ref{equation: theoretical SNR}). With the above notations and some regularity conditions, Theorem \ref{theorem: power control for individual test} stated below shows that we can reject the null hypothesis of no change point with probability tending to $1$. \begin{theorem}\label{theorem: power control for individual test}
Let $\epsilon_n:=O(s^{1/2}_0s\sqrt{{\log (pn)}/{n}}) \vee O({s_0^{1/2}s^2\sqrt{{\log (pn)}/{n}}\|\bDelta\|_{(s_0,2)}})$. For each $\alpha\in[0,1]$, assume the following conditions hold: When $\alpha=1$, suppose that { Assumptions A, B, C, E.2 - E.4 } hold; When $\alpha=0$, suppose that {Assumptions A, C.1, D, E.2 - E.4} as well as
\begin{equation}\label{euqation: condion for alpha=0} \lim\limits_{n,p\rightarrow\infty}s_0^{1/2}s^2\sqrt{\log(p)/n}\|\bDelta\|_{(s_0,2)}=0
\end{equation}
hold; When $\alpha\in(0,1)$, suppose that {Assumptions A - D, E.2 - E.4 } as well as (\ref{euqation: condion for alpha=0}) hold. Under $\Hb_1$, if $\bD$ in (\ref{equation: signal to noise ration}) satisfies
\begin{equation}\label{inequality: theoretical signal strengh}
\sqrt{n}\times \|\bD\|_{(s_0,2)}\geq \dfrac{C(\tilde{\btau},\alpha)}{1-\epsilon_n}s^{1/2}_0\big(\sqrt{\log(pn)}+\sqrt{\log(1/\gamma)}\big),
\end{equation}
then we have
\begin{equation*}
{\mathbb P}(\Phi_{\gamma,\alpha}=1)\rightarrow 1, \text{as}~n,p,B\rightarrow\infty,
\end{equation*}
where $C(\tilde{\btau},\alpha)$ is some universal positive constant only depending on $\tilde{\btau}$ and $\alpha$. \end{theorem}
Theorem \ref{theorem: power control for individual test} demonstrates that with probability tending to one, our proposed individual test with $\alpha\in[0,1]$ can detect the existence of a change point for high-dimensional linear models as long as the corresponding signal to noise ratio satisfies (\ref{inequality: theoretical signal strengh}). Combining (\ref{equation: signal to noise ration}) and (\ref{inequality: theoretical signal strengh}), for each individual test, we note that with a larger signal jump and a closer change point location $t_1$ to the middle of data observations, it is more likely to trigger a rejection of the null hypothesis. More importantly, considering $\epsilon_{n}=o(1)$, Theorem \ref{theorem: power control for individual test} illustrates that for consistently detecting a change point, we require the signal to noise ratio vector to be at least an order of $\|\bD\|_{(s_0,2)}\asymp s_0^{1/2}\sqrt{\log(pn)/n}$, which is particularly interesting to further discuss under several special cases. For example, if we choose $s_0=1$ and $\alpha=1$, our proposed individual test reduces to the least squared loss based testing statistic with the $\ell_\infty$-norm aggregation. In this case, we require $\|\bD\|_{\infty}\asymp \sqrt{\log(pn)/n}$ for detecting a change point. If we choose $\alpha=0$ with the composite quantile loss, the test is still consistent as long as $\|\bD\|_{\infty}\asymp \sqrt{\log(pn)/n}$. Note that the latter one is of special interest for the robust change point detection. Hence, our theorem provides the unified condition for detecting a change point under a general framework, which may be of independent interest. Moreover, Theorem \ref{theorem: power control for individual test} reveals that for detecting a change point, our individual-based method with $\alpha\in[0,1]$ can account for the tails of the data. For Model \eqref{equation: single cpt model} with a fixed signal jump $\bDelta$ and a change point location $t_1$, considering (\ref{equation: signal to noise ration}) and (\ref{inequality: theoretical signal strengh}), the individual test $T_{\alpha}$ is more powerful with a larger $SNR(\alpha, \tilde{\btau})$. This reveals why the individual tests with different weights $\alpha$ perform very differently under various error distributions.
Lastly, it is worth mentioning that the requirements for identifying and detecting a change point are different for Model \eqref{equation: single cpt model}. More specifically, for each individual-based method, Theorem \ref{theorem: cpt estimation results} demonstrates that, to consistently estimate the location of $t_1$, the signal strength should at least satisfy $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\log(pn)/n}$. In contrast, Theorem \ref{theorem: power control for individual test} shows that we can detect a change point if $\|\bDelta\|_{(s_0,2)}\geq C\sqrt{\log(pn)/n}$ holds. This reveals that we need more stringent conditions for localizing a change point than detecting its existence for high-dimensional linear models. To the best of our knowledge, this is still an open question on whether one can obtain consistent change point estimation if $\|\bDelta\|_{(s_0,2)}= O(\sqrt{\log(pn)/n})$.
\subsection{Theoretical results of the tail-adaptive testing statistics}\label{sec: Theoretical results of the adaptive test statistics}
In Section \ref{sec: Theoretical results of the individual test statistics}, we present the theoretical properties of the individual testing statistics $T_{\alpha}$ with $\alpha\in[0,1]$. In this section, we discuss the size and power properties of the tail-adaptive test $\Psi_{\gamma,\rm ad}$ defined in (\ref{statistic: adaptive tests}). To present the theorems, we need additional notations. Let $F_{T_\alpha}(x):={\mathbb P}(T_{\alpha}\leq x)$ be the CDF of $T_{\alpha}$. Then $\hat{P}_{\alpha}$ in (\ref{equation: p-value for individual test}) approximates the following individual tests' theoretical $P$-values defined as $P_{\alpha}:=1-F_{T_{\alpha}}(T_{\alpha})$. Hence, based on the above theoretical $P$-values, we can define the oracle tail-adaptive testing statistic
$\tilde{T}_{\rm ad}=\min_{\alpha\in\cA}P_{\alpha}.$
Let $\tilde{F}_{T,\rm ad}(x):={\mathbb P}(\tilde{T}_{\rm ad}\leq x)$ be the CDF of $\tilde{T}_{\rm ad}$. Then we can also define the theoretical tail-adaptive test's $P$-value as
$\tilde{P}_{\rm ad}:=\tilde{F}_{T,\rm ad}(\tilde{T}_{\rm ad})$.
Recall $\hat{P}_{\rm ad}$ be the low cost bootstrap $P$-value for $\tilde{P}_{\rm ad}$. In what follows, we show that $\hat{P}_{\rm ad}$ converges to $\tilde{P}_{\rm ad}$ in probability as $n,p,B\rightarrow\infty$.
We introduce Assumption $\mathbf{E.1'}$ to describe the scaling relationships among $n$, $p$, and $s_0$. Let $\bG_{i}=(G_{i1},\ldots,G_{ip})^\top$ with $\bG_i\sim N(\mathbf{0},\bSigma)$ being i.i.d. Gaussian random vectors, where $\bSigma:=\text{Cov}(\bX_1)$. Define
\begin{equation*}
\bC^{\bG}(t)=\dfrac{1}{\sqrt{n}}\big(\sum_{i=1}^{\floor{nt}}\bG_i-\dfrac{\floor{nt}}{n}\sum_{i=1}^n\bG_i\big) ~~\text{and}~~T^{\bG}=\max_{q_0\leq t\leq 1-q_0}\| \bC^{\bG}(t)\|_{(s_0,2)}.
\end{equation*} As shown in the proof of Theorem \ref{theorem: size control for individual test},
we use $T^{\bG}=\max\limits_{q_0\leq t\leq 1-q_0}\big\|\bC^{\bG}(t)\big\|_{(s_0,2)}$ to approximate $T_{\alpha}$. For $T^{\bG}$, let $f_{T^{\bG}}(x)$ and $c_{T^{\bG}}(\gamma)$ be the probability density function (pdf), and the $\gamma$-quantile of $T^{\bG}$, respectively. We then define $h(\epsilon)$ as $h(\epsilon)=\max_{x\in I(\epsilon)}f_{T^{\bG}}^{-1}(x)$, where $I(\epsilon):=[c_{T^{\bG}}(\epsilon),c_{T^{\bG}}(1-\epsilon)]$.
With the above definitions and notations, we now introduce Assumption $\mathbf{E.1'}$: \begin{description}
\item [(E.1)$'$] For any $0<\epsilon< 1$, we require
$ h^{0.6}(\epsilon)s_0^3\log(pn)=o(n^{1/10})$. \end{description}
Note that Assumption $\mathbf{E.1'}$ is more stringent than Assumption $\bf{E.1}$. The intuition of Assumption $\mathbf{E.1'}$ is that, we construct our tail-adaptive testing statistic by taking the minimum $P$-values of the individual tests. For analyzing the combinational tests, we need not only the uniform convergence of the distribution functions, but also the uniform convergence of their quantiles on $[\epsilon,1-\epsilon]$ for any $0<\epsilon<1$.
The following Theorem \ref{theorem: adaptive size} justifies the validity of the low-cost bootstrap procedure in Section \ref{sec: method of tail-adaptive}. It also shows that our tail-adaptive test has the asymptotic level of $\gamma$. \begin{theorem}\label{theorem: adaptive size}
For $T_{\rm ad}$, suppose {Assumptions A - D, E.1$'$, E.2 - E.4 } hold. Under $\Hb_0$, we have
\begin{equation*}\label{equation: adaptive size1}
\begin{array}{lc}
{\mathbb P}(\Psi_{\gamma,\rm ad}=1)\rightarrow \gamma,&\text{and}~~
\hat{P}_{\rm ad}-\tilde{P}_{\rm ad}\xrightarrow{{\mathbb P}}0,~\text{as}~n,p,B\rightarrow\infty.
\end{array}
\end{equation*} \end{theorem} After analyzing the size, we now discuss the power. Theorem \ref{theorem: adaptive power} shows that under some regularity conditions, our tail-adaptive test has its power converging to one. \begin{theorem}\label{theorem: adaptive power}
Let $\epsilon_n:=O(s^{1/2}_0s\sqrt{{\log (pn)}/{n}}) \vee O({s_0^{1/2}s^2\sqrt{{\log (pn)}/{n}}\|\bDelta\|_{(s_0,2)}})$.
Suppose {Assumptions A - D, E.2 - E.4 } as well as $\lim\limits_{n,p\rightarrow\infty}s_0^{1/2}s^2\sqrt{\log(p)/n}\|\bDelta\|_{(s_0,2)}=0$ hold.
If $\Hb_1$ holds with
\begin{equation}\label{inequality: theoretical signal strengh2}
\sqrt{n}\times \|\bD\|_{(s_0,2)}\geq \dfrac{C(\tilde{\btau},\alpha)}{1-\epsilon_n}s^{1/2}_0\big(\sqrt{\log(pn)}+\sqrt{\log(|\cA|/\gamma)}\big),
\end{equation}
then for $T_{\rm ad}$, we have
\begin{equation*}\label{equation: adaptive power }
\begin{array}{lc}
{\mathbb P}(\Psi_{\gamma,\rm ad}=1)\rightarrow 1 ~\text{as}~n,p,B\rightarrow\infty,
\end{array}
\end{equation*}
where $C(\tilde{\btau},\alpha)$ is some universal positive constant only depending on $\tilde{\btau}$ and $\alpha$. \end{theorem}
Note that based on the theoretical results obtained in Section \ref{sec: Theoretical results of the individual test statistics}, Theorems \ref{theorem: adaptive size} and \ref{theorem: adaptive power} can be proved using some modifications of the proofs of Theorems 3.5 and 3.7 in \cite{Zhou2017An}. Hence, we omit the detailed proofs for brevity. Lastly, recall the tail-adaptive based change point estimator $\hat{t}_{\rm ad} = \widehat{t}_{\widehat{\alpha}}$ with $\widehat{\alpha} = \mathop{\text{\rm arg\,min}}_{\alpha \in \mathcal{A}} \hat{P}_{\alpha}$. According to Theorem \ref{theorem: cpt estimation results}, the tail-adaptive estimator is consistent which is summarized as a corollary. \begin{corollary}\label{corollary: adaptive estimation}
Suppose {Assumptions A - D, E.2 - E.4, F } as well as $n^{1/4}=o(s)$ and \\ $\lim\limits_{n,p\rightarrow\infty}s_0^{1/2}s^2\sqrt{\log(pn)/n}\|\bDelta\|_{(s_0,2)}=0$ hold.
Suppose additionally $\|\bDelta\|_{(s_0,2)}\gg \sqrt{\dfrac{\log(pn)}{n}}$ holds.
Under $\Hb_1$, with probability tending to one, we have
\begin{equation*}
\big|\hat{t}_{\rm ad}-t_1\big|\leq C_{\rm ad}\dfrac{\log(pn)}{n\|\bSigma\bDelta\|^2_{(s_0,2)}},
\end{equation*}
where $C_{\rm ad}>0$ is some universal constant not depending on $n$ or $p$. \end{corollary}
\begin{figure}\label{figure: size}
\end{figure}
\begin{figure}\label{figure: power}
\end{figure}
\begin{figure}
\caption{Boxplots of the scaled Hausdorff distance of different methods for detecting multiple change points with $(n,p)=(1000,100)$ based on 100 replications. The three change points are at $(0.25,0.5,0.75)$. L\&B is the binary segmentation based technique in \cite{leonardi2016computationally} and SGL is the sparse graphical LASSO method in \cite{zhang2015change-point}. The constant $c$ represents the signal strength and a larger $c$ denotes stronger signal jump. }
\label{figure: boxplot multiple}
\end{figure}
\section{Simulation Studies}\label{sec: summary of simulations} We have carried out extensive numerical studies to examine the finite sample performance of our proposed new methods. To save space, we put the detailed model settings and results in Appendix A of the supplementary materials. The simulation results, including size, power and single and multiple change point estimation, can be summarized as follows:
1. As shown in Figure \ref{figure: size}, the proposed individual and tail-adaptive tests can control the size very well under various model settings with different tail structures including both lighted and heavy tails. The individual test with $\alpha=0$ can even control the size well for Student's $t_2$ and Cauchy distributions.
2. In terms of power performance, as shown in Figure \ref{figure: power}, the individual tests perform differently under various tail structures. However, the tail-adaptive method can have powers close to its best individual one whenever the errors are lighted or heavy-tailed.
3. For single and multiple change point estimation, similar to the power analysis, the performance of the individual estimators depends on the underlying error distributions. Figure \ref{figure: boxplot multiple} indicates that the tail-adaptive estimator can perform close to its best individual estimator. Moreover, compared with the existing techniques, the tail-adaptive method enjoys better performance for single and multiple change point detection.
4. For the choice of $s_0$, the size performance is stable across different choices of $s_0$ under $\Hb_0$. Moreover, it is shown that, under $\Hb_1$, choosing $s_0>1$ can have high powers and accuracies than only using $s_0=1$ for change point testing and estimation.
In summary, the numerical results are consistent with our theorems developed in Section \ref{sec: theory} and demonstrate the advantages of our tail-adaptive method over the existing methods. \section{Real data applications}\label{section: real analysis}
In this section, we apply our proposed methods to the S\&P 100 dataset to find multiple change points. We obtain the S$\&$P 100 index as well as the associated stocks from Yahoo! Finance (https://finance.yahoo.com/) including the largest and most established 100 companies in the S$\&$P 100. For this dataset, we collect the daily prices of 76 stocks that have remained in the S\&P 100 index consistently from January 3, 2007 to December 30, 2011. This covers the recent financial crisis beginning in 2008 and some other important events, resulting in a sample size $n=1259$.
In financial marketing, it is of great interest to predict the S\&P 100 index since it reveals the direction of the entire financial system. To this end, we use the daily prices of the 76 stocks to predict the S\&P100 index. Specifically, let $Y_t\in \RR^1$ be the S\&P 100 index for the $t$-th day and $\bX_t \in \RR^{76\times 2}$ be the stock prices with lag-1 and lag-3 for the $t$-th day. Our goal is to predict $Y_t$ using $\bX_t$ under the high dimensional linear regression models and detect multiple change points for the linear relationships between the S\&P100 index and the 76 stocks' prices. It is well known that the financial data are typically heavy-tailed and we have no prior-knowledge about the tail structure of the data. Hence, for this real data analysis, it seems very suitable to use our proposed tail-adaptive method. We combine our proposed tail-adaptive test with the WBS method (\cite{fryzlewicz2014wild}) to detect multiple change points, which is demonstrated in Algorithm \ref{alg:WBS}. To implement this algorithm, we set $\cA=\{0,0.1,0.5,0.9,1\}$, $s_0=5$, $B=100$, and $V=500$ (number of random intervals). Moreover, we consider the $L_1-L_2$ weighted loss by setting $\tilde{\btau}=0.5$ in (\ref{equation: weighted loss function}). The data are scaled to have mean zeros and variance ones before the change point detection. There are 14 change points detected which are reported in Table \ref{table: realdata}.
To further justify the meaningful findings of our proposed new methods, we refer to the T-bills and ED (TED) spread, which is short for the difference between the 3-month of London Inter-Bank Offer Rate (LIBOR), and the 3-month short-term U.S. government debt (T-bills). It is well-known that TED spread is an indicator of perceived risk in the general economy and an increased TED spread during the financial crisis reflects an increase in credit risk. Figure \ref{figure: Ted} shows the plot of TED where the red dotted lines correspond to the estimated change points. We can see that during the financial crisis from 2007 to 2009, the TED spread has experienced very dramatic fluctuations and the estimated change points can capture some big changes in the TED spread. In addition, the S\&P 100 index obtains its highest level during the financial crisis in October 2007 and then has a huge drop. Our method identifies October 29, 2007 as a change point. Moreover, the third detected change-point is January 10th, 2008. The National Bureau of Economic Research (NBER) identifies December of 2007 as the beginning of the great recession which is captured by our method. In addition, it is well known that affected by the 2008 financial crisis, Europe experienced a debt crisis from 2009 to 2012, with the Greek government debt crisis in October, 2009 serving as the starting point. Our method identifies October 5, 2009 as a change point after which S$\&$P 100 index began to experience a significant decline. Moreover, it is known that countries such as Italy and Spain were facing severe debt issues in July, 2011, raising fears about the stability of the Eurozone and the potential impact on global financial markets. As a result, there exists another huge drop for the S$\&$P 100 index in July 26, 2011, which can be successfully detected by our method.
\begin{figure}
\caption{Plots of the Ted spread (left) and the S\&P 100 index (right) with the estimated change-points (vertical lines) marked by $\#$ in Table \ref{table: realdata}. }
\label{figure: Ted}
\end{figure}
\begin{table}[!h]
\scriptsize
\caption{Multiple change poins detection for the S\&P 100 dataset.}\label{table: realdata}
\addtolength{\tabcolsep}{5pt}
\begin{center}
\begin{tabular}{cccccccc}
\toprule[2pt]
&&Change points &Date&Events&&& \\
\hline
&&117&2007/06/21& TED Spread$\#$&&& \\
& &207&2007/10/29&TED Spread$\#$&&& \\
& &257&2008/01/10&Global Financial Crisis (TED Spread)$\#$&&& \\
& &360&2008/06/09&TED Spread$\#$&&& \\
& &439&2008/09/30&TED Spread$\#$&&& \\
& &535&2009/02/18&Nadir of the crisis$\#$&&& \\
& &632&2009/07/08&&&& \\
& &694&2009/10/05&Greek debt crisis$\#$&&& \\
& &840&2010/05/05&Global stock markets fell due to fears of&&& \\
& &&& contagion of the European sovereign debt crisis$\#$ &&& \\
& &890&2010/07/16&&&& \\
& &992&2010/12/09&&&& \\
& &1074&2011/04/07&&&& \\
& &1149&2011/07/26&Spread of the European debt crisis to Spain and Italy$\#$&&& \\
& &1199&2011/10/05&&&& \\
\hline
\toprule[2pt]
\end{tabular}
\end{center}
\end{table}
\section{Summary}\label{section: summary} In this article, we propose a general tail-adaptive approach for simultaneous change point testing and estimation for high-dimensional linear regression models. The method is based on the observation that both the conditional mean and quantile change if the regression coefficients have a change point. Built on a weighted composite loss, we propose a family of individual testing statistics with different weights to account for the unknown tail structures. Then, we combine the individual tests to construct a tail-adaptive method, which is powerful against sparse alternatives under various tail structures. In theory, for both individual and tail-adaptive tests, we propose a bootstrap procedure to approximate the limiting null distributions. With mild conditions on the regression covariates and errors, we show the optimality of our methods theoretically in terms of size and power under the high-dimensional setting with $p\gg n$. For change point estimation, for each individual method, we propose an argmax-based estimator shown to be rate optimal up to a $\log (pn)$ factor. In the presence of multiple change points, we combine our tail-adaptive approach with the WBS technique to detect multiple change points. With extensive numerical studies, our proposed methods have better performance in terms of size, power, and change point estimation than the existing methods under various model setups.
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\begin{document}
\title{Harmonic models and spanning forests of residually finite groups} \author{Lewis Bowen} \author{Hanfeng Li} \address{\hskip-\parindent Lewis Bowen, Department of Mathematics, Texas A{\&}M University, College Station, TX 77843-3368, U.S.A.} \email{lpbowen@math.tamu.edu}
\address{\hskip-\parindent Hanfeng Li, Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, U.S.A.} \email{hfli@math.buffalo.edu}
\keywords{Harmonic model, algebraic dynamical system, Wired Spanning Forest, WSF, tree entropy}
\begin{abstract} We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions. \end{abstract}
\date{August 13, 2011}
\maketitle
\tableofcontents
\section{Introduction}
This paper is concerned with two related dynamical systems. The quickest way to explain the connections is to start with finite graphs. So consider a finite connected simple graph $G=(V,E)$. The graph Laplacian $\Delta_G$ is an operator on $\ell^2(V,{\mathbb R})$ given by $\Delta_G x(v) = \sum_{w: \{v,w\}\in E} (x(v)-x(w))$. Let $\det^*(\Delta_G)$ be the product of the non-zero eigenvalues of $\Delta_G$. By the Matrix-Tree Theorem (see, e.g., \cite{GR} Lemma 13.2.4), $|V|^{-1}\det^*(\Delta_G)$, is the number of spanning trees in $G$. Recall that a subgraph $H$ of $G$ is {\em spanning} if it contains every vertex. It is a {\em forest} if it has no cycles. A connected forest is a {\em tree}. The number of spanning trees in $G$ is denoted $\tau(G)$.
There is another interpretation for this determinant. Consider the space $({\mathbb R}/{\mathbb Z})^V$ of all functions $x:V \to {\mathbb R}/{\mathbb Z}$. The operator $\Delta_G$ acts on this space as well by the same formula. An element $x\in ({\mathbb R}/{\mathbb Z})^V$ is {\em harmonic mod $1$} if $\Delta_G x \in {\mathbb Z}^V$. The set of harmonic mod $1$ elements is an additive group $X_G < ({\mathbb R}/{\mathbb Z})^V$ containing the constants. The number of connected components of this group is denoted $|X_G|$. Let ${\mathbb Z}^V_0$ be the set of integer-valued functions $x:V \to {\mathbb Z}$ with zero sum: $\sum_{v\in V} x(v)=0$. Because $\Delta_G$ maps ${\mathbb Z}_0^V$ injectively into itself, $\det^*(\Delta_G) = |V||X_G|$. To our knowledge, this was first observed in \cite{So98}. We provide more details in \S \ref{S-approximation}.
Our main results generalize the equalities $\det^*(\Delta_G)|V|^{-1} = \tau(G) = |X_G|$ to Cayley graphs of finitely generated residually finite groups.
\subsection{Harmonic models and other algebraic systems} We begin with a discussion of the appropriate analogue of the group of harmonic mod $1$ points and, to provide further context, the more general setting of group actions by automorphisms of a compact group. This classical subject has long been studied when the acting group is ${\mathbb Z}$ or ${\mathbb Z}^d$ \cite{Schmidt} though not as much is known in the general case.
Let $\Gamma$ be a countable group and $f$ be an element in the integral group ring ${\mathbb Z}\Gamma$. The action of $\Gamma$ on the discrete abelian group ${\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ induces an action by automorphisms on the Pontryagin dual $X_f:=\widehat{{\mathbb Z}\Gamma/{\mathbb Z}\Gamma f}$, the compact abelian group of all homomorphisms from ${\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ to ${\mathbb T}$, the unit circle in ${\mathbb C}$. Call the latter action $\alpha_f$. The topological entropy and measure-theoretic entropy (with respect to the Haar probability measure) coincide when $\Gamma$ is amenable \cite{Berg, De06}. Denote this number by $h(\alpha_f)$.
\begin{example} If $S \subset \Gamma \setminus \{e\}$ is a finite symmetric generating set, where $e$ denotes the identity element of $\Gamma$, and $f$ is defined by $f_s=-1$ if $s\in S$, $f_e=|S|$ and $f_s=0$ for $s \notin S \cup \{e\}$ then $X_f$ is canonically identified with the group $\{x \in ({\mathbb R}/{\mathbb Z})^\Gamma:~ \sum_{s\in S} x_{ts} = |S|x_t, \forall t\in \Gamma\}$ of harmonic mod $1$ points of the associated Cayley graph. \end{example}
Yuzvinskii proved \cite{Yu65, Yu67} that if $\Gamma={\mathbb Z}$ then the entropy of $\alpha_f$ is calculable as follows. When $f=0$, $h(\alpha_f)=+\infty$. When $f\neq 0$, write $f$ as a Laurent polynomial $f = u^{-k}\sum_{j=0}^n c_j u^j$ by identifying $1 \in {\mathbb Z}=\Gamma$ with the indeterminate $u$ and requiring that $c_nc_0\ne 0, n\ge 0$. If $\lambda_1,\ldots, \lambda_n$ are the roots of $\sum_{j=0}^n c_j u^j$, then
$$h(\alpha_f) = \log |c_n| + \sum_{j=1}^n \log^+ |\lambda_j|$$ where $\log^+ t = \log \max(1,t)$. More generally, Yuzvinskii developed formulas for the entropy for any endomorphism of a compact metrizable group \cite{Yu67}.
When $\Gamma={\mathbb Z}^d$, we identify ${\mathbb Z}\Gamma$ with the Laurent polynomial ring ${\mathbb Z}[u_1^{\pm 1},\ldots, u_d^{\pm1}]$. Given a nonzero Laurent polynomial $f \in {\mathbb Z}\Gamma$, the Mahler measure of $f$ is defined by
$$\mathbb{M}(f)=\exp \left(\int_{{\mathbb T}^d} |f(s)|~ds\right)$$ where the integral is with respect to the Haar probability measure on the torus ${\mathbb T}^d$. In \cite{LSW90} it is shown that $h(\alpha_f) = \log \mathbb{M}(f)$. This is a key part of a more general procedure for computing the entropy of any action of ${\mathbb Z}^d$ by automorphisms of a compact metrizable group.
In \cite{FK}, Fuglede and Kadison introduced a determinant $\det_A f$ for elements $f$ of a von Neumann algebra $A$ with respect to a normal tracial state ${\rm tr}_A$. It has found widespread application in the study of $L^2$-invariants \cite{Luck}. We will apply it to the special case when $A$ is the group von Neumann algebra ${\mathcal N}\Gamma$ with respect to its natural trace ${\rm tr}_{{\mathcal N}\Gamma}$. Note that ${\mathbb Z}\Gamma$ is a sub-ring of ${\mathcal N}\Gamma$. These concepts are reviewed in \S \ref{S-notation}.
As explained in \cite[Example 3.13]{Luck}, if $\Gamma = {\mathbb Z}^d$ and $f \in {\mathbb Z}\Gamma$ is nonzero, then the Mahler measure of $f$ equals the logarithm of its Fuglede-Kadison determinant. So it was natural to wonder whether the equation $h(\alpha_f) = \log \det_{{\mathcal N}\Gamma} f$ holds whenever $\Gamma$ is amenable and $f$ is invertible in ${\mathcal N}\Gamma$. Some special cases were proven in \cite{De06} and \cite{DS} before the general case was solved in the affirmative in \cite{Li}.
Recall that $\Gamma$ is {\em residually finite} if the intersection of all finite-index subgroups of $\Gamma$ is the trivial subgroup. In this case, there exists a sequence of finite-index normal subgroups $\Gamma_n\lhd \Gamma$ such that $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$. Our main results concern residually finite groups; we do not, in general, require that $\Gamma$ is amenable.
A group $\Gamma$ is {\em sofic} if it admits a sequence of partial actions on finite sets which, asymptotically, are free actions. This large class of groups, introduced implicitly in \cite{Gr99} and developed further in \cite{We00, ES05, ES06}, contains all residually finite groups and all amenable groups. It is not known whether all countable groups are sofic.
Entropy was introduced in \cite{Ko58,Ko59} for actions of ${\mathbb Z}$. The definition and major results were later extended to all countable amenable groups \cite{Ki75,Ol85, OW87}. Until recently it appeared to many observers that entropy theory could not be extended beyond amenable groups. This changed when \cite{Bo10a} introduced an entropy invariant for actions of free groups. Soon afterwards, \cite{Bo10} introduced entropy for probability-measure-preserving sofic group actions. One disadvantage of the approach taken in \cite{Bo10} (and in \cite{Bo10a}) is that it only applies to actions with a finite-entropy generating partition. This requirement was removed in \cite{KL11a}. That paper also introduced topological sofic entropy for actions of sofic groups on compact metrizable spaces and proved a variational principle relating the two concepts analogous to the classical variational principle.
If $\Gamma$ is non-amenable then the definition of entropy of a $\Gamma$-action depends on a choice of sofic approximation. We will not need the full details here because we are only concerned with the special case in which $\Gamma$ is a residually finite group. A sequence $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ of finite-index normal subgroups of $\Gamma$ satisfying $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$ determines, in a canonical manner, a sofic approximation to $\Gamma$. Thus, we let $h_{\Sigma,\lambda}(\alpha_f)$ and $h_\Sigma(\alpha_f)$ denote the measure-theoretic sofic entropy and the topological sofic entropy of $\alpha_f$ with respect to $\Sigma$ respectively. Here $\lambda$ denotes the Haar probability measure on $X_f$. The precise definition of sofic entropy is given in \S \ref{sec:sofic} below.
In \cite{Bo11}, it was proven that if $f \in {\mathbb Z}\Gamma$ is invertible in $\ell^1(\Gamma)$ then $h_{\Sigma,\lambda}(\alpha_f) = \log \det_{{\mathcal N}\Gamma} f$ as expected. Also, if $f$ is invertible in the universal group $C^*$-algebra of $\Gamma$ then by \cite{KL11a}, $h_{\Sigma}(\alpha_f) = \log \det_{{\mathcal N}\Gamma} f$.
\begin{definition}\label{defn:well-balanced} We say that $f \in {\mathbb R}\Gamma$ is {\em well-balanced} if \begin{enumerate} \item $\sum_{s\in \Gamma}f_s=0$,
\item $f_s\le 0$ for every $s\in \Gamma\setminus \{e\}$,
\item $f=f^*$ (where $f^*$, the adjoint of $f$ is given by $f^*_s = f_{s^{-1}}$ for all $s\in \Gamma$).
\item the support of $f$ generates $\Gamma$. \end{enumerate} \end{definition} If $f \in {\mathbb Z}\Gamma$ is well-balanced then the dynamical system $\Gamma {\curvearrowright} X_f$ is called a {\em harmonic model} because $X_f$ is naturally identified with the set of all $x \in ({\mathbb R}/{\mathbb Z})^\Gamma$ such that $xf =0$, i.e., $x$ satisfies the harmonicity equation mod $1$: $$\sum_{s\in \Gamma} x_{ts}f_s = 0 ~\mod {\mathbb Z}$$ for all $t\in \Gamma$.
If $\Gamma={\mathbb Z}^d$ ($d\ge 2$), then much is known about the harmonic model: the entropy was computed in \cite{LSW90} in terms of Mahler measure, it follows from \cite[Theorem 7.2]{KS} that the periodic points are dense, \cite{SV} provides an explicit description of the homoclinic group in some cases, and \cite{LSV} contains a number of results on the homoclinic group, periodic points, the specification property and more in some cases. See also \cite{SV} for results relating the harmonic model to the abelian sandpile model.
Our first main result is: \begin{theorem}\label{thm:main1} Let $\Gamma$ be a countably infinite group and $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ a sequence of finite-index normal subgroups of $\Gamma$ satisfying $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$. Let $f \in {\mathbb Z}\Gamma$ be well-balanced. Then
$$h_{\Sigma,\lambda}(\alpha_f) = h_\Sigma(\alpha_f) = \log {\rm det}_{{\mathcal N}\Gamma} f = \lim_{n\to\infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|$$
where $|{\rm Fix}_{\Gamma_n}(X_f)|$ is the number of connected components of the set of fixed points of $\Gamma_n$ in $X_f$. \end{theorem}
In the appendix, we show that if $f$ is well-balanced then it is not invertible in $\ell^1(\Gamma)$ or even in the universal group $C^*$-algebra of $\Gamma$. Moreover, it is invertible in ${\mathcal N}\Gamma$ if and only if $\Gamma$ is non-amenable. Thus Theorem \ref{thm:main1} does not have much overlap with previous results.
In order to prove this theorem, we obtain several more results of independent interest. We show that if $\Gamma$ is not virtually isomorphic to ${\mathbb Z}$ or ${\mathbb Z}^2$ then the homoclinic subgroup of $X_f$ is dense in $X_f$. This implies $\Gamma {\curvearrowright} X_f$ is mixing of all orders (with respect to the Haar probability measure). Also, ${\rm Fix}_{\Gamma_n}(X_f)$ converges to $X_f$ in the Hausdorff topology as $n\to\infty$. As far we know, these results were unknown except in the case $\Gamma={\mathbb Z}^d$. Indeed, in the case $\Gamma={\mathbb Z}^d$, it follows from \cite[Theorem 7.2]{KS} that the periodic points are dense and it is known that the action is mixing of all orders if $d \ge 2$ (see Remark \ref{remark:homoclinic} for details).
\subsection{Uniform Spanning Forests} In this paper, all graphs are allowed to have multiple edges and loops. A subgraph of a graph is {\em spanning} if it contains every vertex. It is a {\em forest} if every connected component is simple connected (i.e., has no cycles). A connected forest is a {\em tree}. If $G$ is a finite connected graph then the {\em Uniform Spanning Tree} (UST) on $G$ is the random subgraph whose law is uniformly distributed on the collection of spanning trees. Motivated in part to develop an analogue of the UST for infinite graphs, R. Pemantle implicitly introduced in \cite{Pe91} the {\em Wired Spanning Forest} (WSF) on ${\mathbb Z}^d$. This model has been generalized to arbitrary locally finite graphs and studied intensively (see e.g., \cite{BLPS01}).
In order to define the WSF, let $G=(V,E)$ be a locally finite connected graph and $\{G_n\}_{n=1}^\infty$ an increasing sequence of finite subgraphs $G_n=(V_n,E_n) \subset G$ whose union is all of $G$. For each $n$, define the wired graph $G^w_n=(V^w_n,E^w_n)$ as follows. The vertex set $V^w_n= V_n \cup \{*\}$. The edge set $E_n^w$ of $G_n^w$ contains all edges in $G_n$. Also for every edge $e$ in $G$ with one endpoint $v$ in $G_n$ and the other endpoint $w$ not in $G_n$, let $e^*=\{v,*\}$. Then $G^w_n$ contains all edges of the form $e^*$. These are all of the edges of $G^w_n$. Let $\nu^w_n$ be the uniform probability measure on the set of spanning trees of $G^w_n$. We consider it as a probability measure on the set $2^{E_n^w}$ of all subsets of $E_n^w$. Because $E_n \subset E$, we can think of $2^{E_n}$ (which we identify with the set of all subsets of $E_n$) as a subset of $2^E$ (which we identify with the set of all subsets of $E$). The projection of $\nu^w_n$ to $2^{E_n} \subset 2^E$ converges (as $n\to\infty$) to a Borel probability measure $\nu_{WSF}$ on $2^E$ (in the weak* topology on the space of Borel probability measures of $2^E$). This measure does not depend on the choice of $\{G_n\}_{n=1}^\infty$. This was implicitly proven by Pemantle \cite{Pe91} (answering a question of R. Lyons). By definition, $\nu_{WSF}$ is the law of the Wired Spanning Forest on $G$. There is a related model, called the Free Spanning Forest (FSF) (obtained by using $G_n$ itself in place of $G^w_n$ above) which is frequently discussed in comparison with the WSF. However, we will not make any use of it here. For more background, the reader is referred to \cite{BLPS01}.
\begin{definition}\label{defn:Cayley}
Let $\Gamma$ be a countably infinite group and let $f \in {\mathbb Z}\Gamma$ be well-balanced (Definition~\ref{defn:well-balanced}). The Cayley graph $C(\Gamma,f)$ has vertex set $\Gamma$. For each $v \in \Gamma$ and $s \ne e$, there are $|f_s|$ edges between $v$ and $vs$. Let $E(\Gamma,f)$ denote the set of edges of $C(\Gamma,f)$. Similarly, for $\Gamma_n \triangleleft \Gamma$, we let $C_n^f=C(\Gamma/\Gamma_n,f)$ be the graph with vertex set $\Gamma/\Gamma_n$ such that for each $g\Gamma_n \in \Gamma/\Gamma_n$ and $s \in \Gamma$ there are $|f_s|$ edges between $g\Gamma_n$ and $gs\Gamma_n$. We let $E_n^f$ be the set of edges of $C^f_n$. \end{definition}
Let $\nu_{WSF}$ be the law of the WSF on $C(\Gamma,f)$. It is a probability measure on $2^{E(\Gamma,f)}$. Of course, $\Gamma$ acts on $E(\Gamma,f)$ which induces an action on $2^{E(\Gamma,f)}$ which preserves $\nu_{WSF}$.
In \cite{Lyons05}, R. Lyons introduced the {\em tree entropy} of a transitive weighted graph (and more generally, of a probability measure on weighted rooted graphs). In our case, the definition runs as follows. Let $\mu$ be the probability measure on $\Gamma$ defined by $\mu(s)=|f_s|/f_e$ for $s\in \Gamma \setminus \{e\}$. Then the tree entropy of $C(\Gamma,f)$ is $${\bf h}(C(\Gamma, f)):= \log f_e - \sum_{k\ge 1} k^{-1} \mu^k(e)$$ where $\mu^k$ denotes the $k$-fold convolution power of $\mu$. In probabilistic terms, $\mu^k(e)$ is the probability that a random walk with i.i.d. increments with law $\mu$ started from $e$ will return to $e$ at time $k$. In \cite{Lyons05}, R. Lyons proved that if $\Gamma$ is amenable then the measure-entropy of $\Gamma {\curvearrowright} (2^{E(\Gamma,f)}, \nu_{WSF})$ equals ${\bf h}(C(\Gamma,f))$. We extend this result to all finitely-generated residually finite groups (where entropy is taken with respect to a sequence $\Sigma$ of finite-index normal subgroups converging to the trivial subgroup).
In \cite[Theorem 3.1]{Lyons10}, it is shown that ${\bf h}(C(\Gamma,f)) = \log \det_{{\mathcal N}\Gamma} f$. We give another proof in \S \ref{S-approximation}. Thus by Theorem \ref{thm:main1}, the entropy of the WSF equals the entropy of the associated harmonic model.
In the case $\Gamma={\mathbb Z}^d$, the entropy of the harmonic model was computed in \cite{LSW90}. Then the topological entropy of the action of ${\mathbb Z}^d$ on the space of essential spanning forests was computed in \cite{BP93} and shown to coincide with the entropy of the harmonic model. This coincidence was mysterious until \cite{So98} explained how to derive this result without computing the entropy.
We also study a topological model related to the WSF (though not the same model as in \cite{BP93}). To describe it, we need to introduce some notation.
\begin{notation}\label{note:S} Let $S$ be the set of all $s\in \Gamma \setminus\{e\}$ with $f_s \ne 0$. Let $S_* \subset E(\Gamma,f)$ denote the set of edges adjacent to the identity element. Let $p:S_* \to S$ be the map which takes an edge with endpoints $\{e,g\}$ to $g$. Also, let $s_*^{-1}=s^{-1}s_* \in S_*$ where $p(s_*)=s$. Note that $p(s_*^{-1})= p(s_*)^{-1}$ and $(s_*^{-1})^{-1}=s_*$. \end{notation}
An element $y\in S_*^\Gamma$ defines, for each $g\in \Gamma$ a directed edge in $C(\Gamma,f)$ away from $g$ (namely, the edge $gs$ where $y_g=s$). Let ${\mathcal F}_f \subset S_*^\Gamma$ be the set of all elements $y \in S_*^\Gamma$ such that \begin{enumerate} \item edges are oriented consistently: if $y_g=s_*$ and $p(s_*)=s$ then $y_{gs} \ne s_*^{-1}$. \item there are no cycles: there does not exist $g_1,g_2,\ldots, g_n\in \Gamma$ and $s_1,\ldots, s_n \in S$ such that $g_is_i=g_{i+1}$, $g_1=g_n$ and $p(y_{g_i})=s_i$ for $1\le i \le n-1$. \end{enumerate} The space ${\mathcal F}_f \subset S_*^\Gamma$ is closed (where $S_*^\Gamma$ is given the product topology) and is therefore a compact metrizable space. Let $h_\Sigma({\mathcal F}_f,\Gamma)$ denote the topological sofic entropy of the action $\Gamma {\curvearrowright} {\mathcal F}_f$. Our second main result is:
\begin{theorem}\label{thm:WSF} If $\Gamma$ is a countably infinite group, $\Sigma=\{\Gamma_n\}_{n=1}^\infty$ is a sequence of finite-index normal subgroups of $\Gamma$, $\bigcap_{n=1}^\infty \bigcup_{i \ge n} \Gamma_i = \{e\}$ and $f \in {\mathbb Z}\Gamma$ is well-balanced then $$h_{\Sigma,\nu_{WSF}}(2^{E(\Gamma,f)},\Gamma) = h_{\Sigma}({\mathcal F}_f,\Gamma) = {\bf h}(C(\Gamma,f))=\log {\rm det}_{{\mathcal N}\Gamma} f = \lim_{n\to\infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f)$$ where $\tau(C_n^f)$ is the number of spanning trees of the Cayley graph $C(\Gamma/\Gamma_n,f)$.
\end{theorem}
{\bf Acknowledgements}: L. B. would like to thank Russ Lyons for asking the question, ``what is the entropy of the harmonic model on the free group?'' which started this project. L.B. would also like to thank Andreas Thom for suggesting the use of Property (T) to prove an upper bound on the sofic entropy of the harmonic model. This idea turned out to be a very useful entry way into the problem. Useful conversations occurred while L.B. visited the Banff International Research Station, the Mathematisches Forschungsinstitut Oberwolfach, and the Institut Henri Poincar\'e. L.B. was partially supported by NSF grants DMS-0968762 and DMS-0954606.
H. L. was partially supported by NSF Grants DMS-0701414 and DMS-1001625. This work was carried out while H.L. visited the Institut Henri Poincar\'{e} and the math departments of Fudan University and University of Science and Technology of China in the summer of 2011. He thanks Wen Huang, Shao Song, Yi-Jun Yao, and Xiangdong Ye for warm hospitality. H.L. is also grateful to Klaus Schmidt for sending him the manuscript \cite{LS}.
\section{Notation and preliminaries} \label{S-notation}
Throughout the paper, $\Gamma$ denotes a
countable group with identity element $e$. We denote $f \in \ell^p(\Gamma):=\ell^p(\Gamma, {\mathbb R})$ by $f=(f_s)_{s\in \Gamma}$. Given $g\in \ell^1(\Gamma)$ and $h\in \ell^\infty(\Gamma)$ we define the convolution $g h \in \ell^\infty(\Gamma)$ by
\begin{eqnarray*}
(gh)_w &:=& \sum_{s \in \Gamma} g_s h_{s^{-1}w} =\sum_{s \in \Gamma} g_{ws^{-1}} h_s, \quad \forall w\in \Gamma.
\end{eqnarray*}
More generally, whenever $g$ and $h$ are functions of $\Gamma$ we define $gh$ as above whenever this formula is well-defined. Let ${\mathbb Z}\Gamma\subset \ell^\infty(\Gamma)$ be the subring of all elements $f$ with $f_s \in {\mathbb Z}~\forall s \in \Gamma$ and $f_s=0$ for all but finitely many $s\in \Gamma$. Similarly ${\mathbb R}\Gamma\subset \ell^\infty(\Gamma)$ is the subring of all elements $f$ with $f_s \in {\mathbb R}~\forall s \in \Gamma$ and $f_s=0$ for all but finitely many $s\in \Gamma$. The element $1 \in {\mathbb Z}\Gamma$ is defined by $1_e=1, 1_s =0 ~\forall s\in\Gamma \setminus\{e\}$.
Given sets $A$ and $B$, $A^B$ denotes the set of all functions from $B$ to $A$. We frequently identify the unit circle ${\mathbb T}$ with ${\mathbb R}/{\mathbb Z}$. If $g\in {\mathbb Z}\Gamma$ and $h\in {\mathbb T}^\Gamma$ then define the convolutions $g h$ and $h g \in {\mathbb T}^\Gamma$ as above.
The {\em adjoint} of an element $g \in {\mathbb C}^\Gamma$ is defined by $g^*(s) := \overline{g(s^{-1})}$.
A probability measure $\mu$ on $\Gamma$ can be thought of as an element of $\ell^1(\Gamma)$. Thus we write $\mu_s=\mu(\{s\})$ for any $s\in \Gamma$, and $\mu^n$ denotes the $n$-th convolution power of $\mu$.
The group $\Gamma$ acts isometrically on the Hilbert space $\ell^2(\Gamma, {\mathbb C})$ from the left by $(sf)_t=f_{s^{-1}t}$ for all $f\in \ell^2(\Gamma, {\mathbb C})$ and $s, t\in \Gamma$. Denote by ${\mathcal B}(\ell^2(\Gamma, {\mathbb C}))$ the algebra of bounded linear operators from $\ell^2(\Gamma, {\mathbb C})$ to itself.
The group von Neumann algebra ${\mathcal N} \Gamma$ is the algebra of elements in ${\mathcal B}(\ell^2(\Gamma, {\mathbb C}))$ commuting with the left action of $\Gamma$ (see \cite[Section 2.5]{BO} for detail), and is complete under the operator norm $\|\cdot \|$.
For each $g\in {\mathbb R}\Gamma$, we denote by $R_g$ the operator $\ell^2(\Gamma, {\mathbb C})\rightarrow \ell^2(\Gamma, {\mathbb C})$ defined by
$R_g(x)=xg$ for $x\in \ell^2(\Gamma, {\mathbb C})$. It is easy to see that $R_g\in {\mathcal N}\Gamma$.
Then we have the injective ${\mathbb R}$-algebra homomorphism ${\mathbb R}\Gamma\rightarrow {\mathcal N}(\Gamma)$ sending $g$ to $R_{g^*}$. In this way we shall think of
${\mathbb R}\Gamma$ as a subalgebra of ${\mathcal N}\Gamma$.
For each $s\in \Gamma$, we also think of $s$ as the element in $\ell^2(\Gamma, {\mathbb C})$ being $1$ at $s$ and $0$ at $t\in \Gamma\setminus \{s\}$.
The canonical trace ${\rm tr}_{{\mathcal N}\Gamma}$ on ${\mathcal N}\Gamma$ is the linear functional ${\mathcal N}\Gamma\rightarrow {\mathbb C}$ defined by
$$ {\rm tr}_{{\mathcal N}\Gamma}(T)=\left< Te, e\right>.$$
An element $T\in {\mathcal N}\Gamma$ is called positive if $\left<Tx, x\right>\ge 0$ for all $x\in \ell^2(\Gamma, {\mathbb C})$. Let $T\in {\mathcal N}\Gamma$ be positive. The spectral measure of $T$, is the unique Borel probability measure $\mu$ on the interval $[0, \|T\|]$ satisfying \begin{align} \label{E-spectral measure}
\int_0^{\|T\|}p(t)\, d\mu(t)={\rm tr}_{{\mathcal N}\Gamma}(p(T)) \end{align} for every complex-coefficients polynomial $p$. If $\ker T=\{0\}$, then the Fuglede-Kadison determinant $\det_{{\mathcal N}\Gamma}(T)$ of $T$ \cite{FK} is defined as \begin{align} \label{E-determinant}
{\rm det}_{{\mathcal N}\Gamma}(T)=\exp\left(\int_{0+}^{\|T\|}\log t\, d\mu(t)\right). \end{align} If furthermore $f$ is invertible in ${\mathcal N}\Gamma$, then \begin{align} \label{E-determinant invertible}
{\rm det}_{{\mathcal N}\Gamma}(T)=\exp({\rm tr}_{{\mathcal N}\Gamma}\log T). \end{align}
For a locally compact abelian group $X$, we denote by $\widehat{X}$ its Pontryagin dual, i.e. the locally compact abelian group of continuous group homomorphisms $X\rightarrow {\mathbb R}/{\mathbb Z}$. For $f\in {\mathbb Z}\Gamma$, we set $X_f=\widehat{{\mathbb Z}\Gamma/{\mathbb Z}\Gamma f}$. Note that one can identify $\widehat{{\mathbb Z}\Gamma}$ with $({\mathbb R}/{\mathbb Z})^\Gamma$ naturally, with the pairing between $x\in ({\mathbb R}/{\mathbb Z})^\Gamma$ and $g\in {\mathbb Z}\Gamma$ given by $$ \left<x, g\right>=(xg^*)_e.$$ It follows that we may identify $X_f$ with $\{x\in ({\mathbb R}/{\mathbb Z})^\Gamma: xf^*=0\}$ naturally, with the pairing between $x\in X_f$ and $g+{\mathbb Z}\Gamma f\in {\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ for $g\in {\mathbb Z}\Gamma$ given by \begin{align} \label{E-pairing} \left<x, g+{\mathbb Z}\Gamma f\right>=(xg^*)_e. \end{align}
\section{Approximation by finite models}\label{S-approximation}
The purpose of this section is to prove: \begin{theorem}\label{T-det vs number of fixed point} Let $\Gamma$ be a finitely generated and residually finite infinite group. Let $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ be a sequence of finite-index normal subgroups of $\Gamma$ such that $\bigcap_{n\in {\mathbb N}}\bigcup_{i\ge n}\Gamma_i=\{e\}$. Let $f\in {\mathbb Z}\Gamma$ be well-balanced (Definition~\ref{defn:well-balanced}). Then
$$ \log {\rm det}_{{\mathcal N}\Gamma} f= {\bf h}(C(\Gamma,f)) = \lim_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f)= \lim_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|$$
where $\tau(C_n^f)$ is the number of spanning trees of the Cayley graph $C_n^f$ (Definition~\ref{defn:Cayley}), ${\rm Fix}_{\Gamma_n}(X_f)$ is the fixed point set of $\Gamma_n$ in $X_f$, and $|{\rm Fix}_{\Gamma_n}(X_f)|$ is the number of connected components of ${\rm Fix}_{\Gamma_n}(X_f)$. \end{theorem}
The following result is a special case of \cite[Theorem 3.1]{Lyons10}. For the convenience of the reader, we give a proof here.
\begin{proposition} \label{P-tree entropy} Let $\Gamma$ be a countable group and $f\in {\mathbb R}\Gamma$ be well-balanced. Set $\mu=-(f-f_e)/f_e$. We have $$ \log {\rm det}_{{\mathcal N}\Gamma}f
= \log f_e-\sum_{k=1}^\infty \frac{1}{k}(\mu^k)_e.$$ \end{proposition} \begin{proof} Note that $f$ is positive in ${\mathcal N}\Gamma$. Let $\varepsilon>0$. Then the norm of $\frac{f_e}{f_e+\varepsilon}\mu$ in ${\mathcal N}\Gamma$ is bounded above by
$\|\frac{f_e}{f_e+\varepsilon}\mu\|_1$, which in turn is strictly smaller than $1$. Thus $\varepsilon+f=(f_e+\varepsilon)(1-\frac{f_e}{f_e+\varepsilon}\mu)$ is positive and invertible in ${\mathcal N}\Gamma$.
Therefore in ${\mathcal N}\Gamma$ we have \begin{align*} \log (\varepsilon+f)= \log (f_e+\varepsilon)-\sum_{k=1}^\infty \frac{1}{k}\cdot \left(\frac{f_e}{f_e+\varepsilon}\mu\right)^k, \end{align*} and the right hand side converges in norm. Thus \begin{align*} \log {\rm det}_{{\mathcal N}\Gamma}(\varepsilon+f)&\overset{\eqref{E-determinant invertible}}={\rm tr}_{{\mathcal N}\Gamma} \log (\varepsilon+f)\\ &=\log (f_e+\varepsilon)-\sum_{k=1}^\infty \frac{(f_e)^k}{(f_e+\varepsilon)^kk}(\mu^k)_e. \end{align*} We have $\lim_{\varepsilon\to 0+}{\rm det}_{{\mathcal N}\Gamma}(f+\varepsilon)={\rm det}_{{\mathcal N}\Gamma}f$ \cite[Lemma 5]{FK}. Note that $(\mu^k)_e\ge 0$ for all $k\in {\mathbb N}$. Thus \begin{align*} \log {\rm det}_{{\mathcal N}\Gamma}f &=\lim_{\varepsilon\to 0+}\log {\rm det}_{{\mathcal N}\Gamma}(f+\varepsilon) \\ &= \log f_e- \sum_{k=1}^\infty \frac{1}{k}(\mu^k)_e. \end{align*}
\end{proof}
Recall that all graphs in this paper are allowed to have multiple edges and loops. \begin{lemma}\label{lem:matrix-tree}
Let $G=(V,E)$ be a finite connected graph and let $\Delta_G:{\mathbb R}^V \to {\mathbb R}^V$ be the graph Laplacian: $\Delta_G x(v) = \sum_{\{v,w\}\in E} (x(v)-x(w))$. Let $\det^*(\Delta_G)$ be the product of the nonzero eigenvalues of $\Delta_G$. Then $|V|^{-1}\det^* \Delta_G = \tau(G)$, the number of spanning trees in $G$. Moreover, if $v_0 \in V$ is any vertex, $V_0:=V \setminus \{v_0\}$, $P_0:{\mathbb R}^V \to {\mathbb R}^{V_0}$ is the projection map and $\Delta^0_G:{\mathbb R}^{V_0} \to {\mathbb R}^{V_0}$ is defined by $\Delta^0_G = P_0\Delta_G$, then
$$\det(\Delta^0_G) = |V|^{-1} {\rm det}^*(\Delta_G) = \tau(G).$$ \end{lemma}
\begin{proof} This is the Matrix-Tree Theorem (see e.g., \cite[Lemmas 13.2.3, 13.2.4]{GR}). \end{proof}
\begin{lemma}\label{lem:correspondence} Let $M$ be an $m\times m$ matrix with integral entries and inverse $M^{-1}$ with real entries. Let $\phi: {\mathbb R}^m \to {\mathbb R}^m$ be the corresponding linear transformation. Then the absolute value of the determinant of $M$ equals the number of integral points in $\phi([0,1)^m)$. \end{lemma}
\begin{proof} This is \cite[Lemma 4]{So98}. \end{proof}
\begin{lemma}\label{lem:harmonic points}
Let $G=(V,E)$ be a finite connected graph. Let $\Delta_G:{\mathbb R}^V \to {\mathbb R}^V$ be the graph Laplacian. We also consider $\Delta_G$ as a map from $({\mathbb R}/{\mathbb Z})^V$ to itself. Let $X_G < ({\mathbb R}/{\mathbb Z})^V$ be the subgroup consisting of all $x\in ({\mathbb R}/{\mathbb Z})^V$ with $\Delta_G x = {\mathbb Z}^V$. Let $|X_G|$ denote the number of connected components of $X_G$. Then
$$|X_G| = |V|^{-1}{\rm det}^*(\Delta_G)$$ where ${\rm det}^*(\Delta_G)$ is the product of the non-zero eigenvalues of $\Delta_G$. \end{lemma}
\begin{proof}
This lemma is an easy generalization of results in \cite{So98}. For completeness, we provide the details. As in Lemma \ref{lem:matrix-tree}, fix a vertex $v_0 \in V$, let $V_0:=V \setminus \{v_0\}$, think of ${\mathbb R}^{V_0}$ as a subspace of ${\mathbb R}^V$ in the obvious way, let $P_0:{\mathbb R}^V \to {\mathbb R}^{V_0}$ be the projection map and $\Delta^0_G:{\mathbb R}^{V_0} \to {\mathbb R}^{V_0}$ be defined by $\Delta^0_G = P_0\Delta_G$.
By Lemma \ref{lem:matrix-tree}, $\det \Delta_G^0 = |V|^{-1} \det^* \Delta_G$ is non-zero. Under the standard basis of ${\mathbb R}^{V_0}$ the linear map $\Delta_{G}^0$ is represented as a matrix with integral entries. Therefore, the previous lemma implies $\det \Delta_{G}^0$ equals the number of integral points in $\Delta_{G}^0( [0,1)^{V_0} )$.
If $x \in {\mathbb R}^V$ has $\| x\|_\infty < (2|E|)^{-1}$ and $\Delta_G x \in {\mathbb Z}^V$, then $\Delta_Gx=0$ and hence $x$ is constant. Therefore, each connected component of $X_G$ is a coset of the constants. Thus $|X_G|$ is the cardinality of $X_G/Z$ where $Z<({\mathbb R}/{\mathbb Z})^V$ denotes the constant functions.
Given $x \in X_G$, let $\tilde{x} \in [0,1)^V$ be the unique element with $\tilde{x} + {\mathbb Z}^V = x$. There is a unique element $x_0 \in x + Z$ such that $\tilde{x_0}(v_0)=0$. If we let $X^0_G$ be the set of all such $\tilde{x_0}$, then $X^0_G$ is a finite set with $|X^0_G|=|X_G|$.
We claim that $X^0_G$ is precisely the set of points $y \in [0,1)^{V_0}$ with $\Delta^0_G(y) \in {\mathbb Z}^{V_0}$. To see this, let $\tilde{x_0} \in X^0_G$. Then $\Delta_G^0 \tilde{x_0} = P_0\Delta_G \tilde{x_0} \in {\mathbb Z}^{V_0}$. To see the converse, let $S:{\mathbb R}^V \to {\mathbb R}$ denote the sum function: $S(y) = \sum_{v\in V} y(v)$. Note that $S(\Delta_G(y))=0$ for any $y\in {\mathbb R}^V$. Therefore, if $y \in [0,1)^{V_0}$ is any point with $\Delta_G^0(y) \in {\mathbb Z}^{V_0}$ then because $S(\Delta_G y)=0=\Delta_G y(v_0) + S(\Delta^0_G y)$, it must be that $\Delta_G y(v_0)$ is an integer and thus $\Delta_G y \in {\mathbb Z}^V$. Thus $y + {\mathbb Z}^V \in X_G$. Because $y(v_0)=0$, we must have $y \in X_G^0$ as claimed.
So $|X_G|$ equals $|X^0_G|$ which equals the number of points $y \in [0,1)^{V_0}$ with $\Delta_{G}^0(y) \in {\mathbb Z}^{V_0}$. We showed above that the later equals $\det\Delta_{G}^0 = |V|^{-1}\det^* \Delta_G$. \end{proof}
We are ready to prove Theorem~\ref{T-det vs number of fixed point}.
\begin{proof}[Proof of Theorem \ref{T-det vs number of fixed point}]
Define the Cayley graphs $C(\Gamma,f)$ and $C_n^f=C(\Gamma/\Gamma_n,f)$ as in Definition \ref{defn:Cayley}. Then the group of harmonic mod $1$ points on $C_n^f$ is canonically isomorphic with ${\rm Fix}_{\Gamma_n}(X_f)$ \cite[Lemma 7.4]{KL11a}. So Lemmas \ref{lem:matrix-tree} and \ref{lem:harmonic points} imply $|{\rm Fix}_{\Gamma_n}(X_f)| = \tau(C_n^f)$. By \cite[Theorem 3.2]{Lyons05}, $\lim_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f) = {\bf h}(C(\Gamma,f))$. By Proposition~\ref{P-tree entropy}, ${\bf h}(C(\Gamma,f)) = \log {\rm det}_{{\mathcal N}\Gamma}f$. This implies the result. \end{proof}
\section{Homoclinic Group} \label{S-homoclinic}
For an action of a countable group $\Gamma$ on a compact metrizable group $X$ by continuous automorphisms, a point $x\in X$ is called {\it homoclinic} if $sx$ converges to the identity element of $X$ as $\Gamma\ni s\to \infty$ \cite{LS99}. The set of all homoclinic points, denoted by $\Delta(X)$, is a $\Gamma$-invariant subgroup of $X$.
The main result of this section is the following
\begin{theorem} \label{T-dense homoclinic} Let $\Gamma$ be a countably infinite group such that $\Gamma$ is not virtually ${\mathbb Z}$ or ${\mathbb Z}^2$ (i.e., does not have any finite-index normal subgroup isomorphic to ${\mathbb Z}$ or ${\mathbb Z}^2$). Let $f\in {\mathbb Z}\Gamma$ be well-balanced. Then \begin{enumerate} \item The homoclinic group $\Delta(X_f)$ is dense in $X$.
\item $\alpha_f$ is mixing of all orders (with respect to the Haar probability measure of $X_f$). In particular, $\alpha_f$ is ergodic. \end{enumerate} \end{theorem}
\begin{remark}\label{remark:homoclinic}
When $\Gamma={\mathbb Z}^2$ and $f\in {\mathbb Z}\Gamma$ is well-balanced,
$\alpha_f$ is mixing of all orders. We are grateful to Doug Lind for explaining this to us. Indeed, let $\Gamma={\mathbb Z}^d$ for $d\in {\mathbb N}$ and $g\in {\mathbb Z}\Gamma$. If $\alpha_g$ has completely positive entropy (with respect to the Haar probability measure on $X_g$), then
$\alpha_g$ is mixing of all orders \cite{Kaminski} \cite[Theorem 20.14]{Schmidt}. By \cite[Theorems 20.8, 18.1, and 19.5]{Schmidt}, $\alpha_g$ has completely positive entropy exactly when $g$ has no factor in ${\mathbb Z}\Gamma={\mathbb Z}[u_1^{\pm}, \dots, u_d^{\pm}]$ as a generalized cyclotomic polynomial, i.e. $g$ has no factor of the form $u_1^{m_1}\dots u_d^{m_d}h(u_1^{n_1}\dots u_d^{n_d})$ for $m_1, \dots, m_d, n_1, \dots, n_d\in {\mathbb Z}$, not all $n_1, \dots, n_d$ being $0$, and $h$ being a cyclotomic polynomial in a single variable. When $d\ge 2$, if $g$ has a factor of such form, then $g(z_1, \dots, z_d)=0$ for some $z_1, \dots, z_d$ in the unit circle of the complex plane not being all $1$. On the other hand, when $f\in {\mathbb Z}\Gamma$ is well-balanced, if $f(z_1, \dots, z_d)=0$ for some $z_1, \dots, z_d$ in the unit circle of the complex plane, then $z_1=\dots=z_d=1$. \end{remark}
\begin{remark} \label{R-ergodic} Lind and Schmidt \cite{LS} showed that for any countably infinite amenable group $\Gamma$ which is not virtually ${\mathbb Z}$,
if $f\in {\mathbb Z}\Gamma$ is not a right zero-divisor in ${\mathbb Z}\Gamma$, then $\alpha_f$ is ergodic. In particular, if $\Gamma$ is virtually
${\mathbb Z}^2$ and $f\in {\mathbb Z}\Gamma$ is well-balanced, then $\alpha_f$ is ergodic. \end{remark}
\begin{remark} \label{R-integer not ergodic} When $\Gamma={\mathbb Z}$ and $f\in {\mathbb Z}\Gamma$ is well-balanced, $\alpha_f$ is not ergodic. This follows from \cite[Theorem 6.5.(1)]{Schmidt}, and can also be seen as follows. We may identify ${\mathbb Z}\Gamma$ with the Laurent polynomial ring ${\mathbb Z}[u^{\pm}]$. Since the sum of the coefficients of $f$ is $0$, one has $f=(1-u)g$ for some $g\in {\mathbb Z}[u^{\pm}]$. It follows that $g+{\mathbb Z}[u^{\pm}] f\in {\mathbb Z}[u^{\pm}]/{\mathbb Z}[u^{\pm}] f=\widehat{X_f}$ is fixed by the action of ${\mathbb Z}=\Gamma$ (i.e., it is fixed under multiplication by $u$). As ${\mathbb Z}[u^{\pm}]$ is an integral domain and $1-u$ is not invertible in ${\mathbb Z}[u^{\pm}]$, the element $g$ is not in ${\mathbb Z}[u^{\pm}]f$. Thinking of $g+{\mathbb Z}[u^{\pm}]f$ as a continuous ${\mathbb C}$-valued function on $X_f$, we find that $g+{\mathbb Z}[u^{\pm}]f$ has $L^2$-norm $1$ and is orthogonal to the constant functions with respect to the Haar probability measure. Thus $\alpha_f$ is not ergodic. \end{remark}
We recall first the definition of mixing of all orders.
\begin{lemma} \label{L-mixing of all order} Let $\Gamma$ be a countable group acting on a standard probability space $(X, {\mathcal B}, \lambda)$ by measure-preserving transformations. Let $n\in {\mathbb N}$ with $n\ge 2$. Let $s_1,\dots, s_n\in \Gamma$. The following conditions are equivalent: \begin{enumerate} \item for any $A_1, \dots, A_n\in {\mathcal B}$, one has $\lambda\big(\bigcap_{j=1}^ns_j^{-1}A_j\big )\to \prod_{j=1}^n\lambda(A_j)$ as $s_j^{-1}s_k\to \infty$ for all $1\le j<k\le n$.
\item for any $f_1, \dots, f_n$ in the space $L^{\infty}_{{\mathbb C}}(X, {\mathcal B}, \lambda)$ of essentially bounded ${\mathbb C}$-valued ${\mathcal B}$-measurable functions on $X$, one has $\lambda\big(\prod_{j=1}^ns_j(f_j)\big)\to \prod_{j=1}^n \lambda(f_j)$ as $s_j^{-1}s_k\to \infty$ for all $1\le j<k\le n$. \end{enumerate} If furthermore $X$ is a compact metrizable space, ${\mathcal B}$ is the $\sigma$-algebra of Borel subsets of $X$, then the above conditions are also equivalent to \begin{enumerate} \item[(3)] for any $f_1, \dots, f_n$ in the space $C_{\mathbb C}(X)$ of ${\mathbb C}$-valued continuous functions on $X$, one has $\lambda\big(\prod_{j=1}^ns_j(f_j)\big)\to \prod_{j=1}^n \lambda(f_j)$ as $s_j^{-1}s_k\to \infty$ for all $1\le j<k\le n$. \end{enumerate} If furthermore $X$ is a compact metrizable abelian group, ${\mathcal B}$ is the $\sigma$-algebra of Borel subsets of $X$, $\lambda$ is the Haar probability measure of $X$, and $\Gamma$ acts on $X$ by continuous automorphisms, then the above conditions are also equivalent to \begin{enumerate} \item[(4)] for any $f_1, \dots, f_n\in \widehat{X}$ not being all $0$, there is some finite subset $F$ of $\Gamma$ such that $\sum_{j=1}^ns_jf_j\neq 0$ for all $s_1, \dots, s_n\in \Gamma$ with
$s_j^{-1}s_k\not\in F$ for all $1\le j<k\le n$.
\item[(5)] for any $f_1, \dots, f_n\in \widehat{X}$ with $f_1\neq 0$, there is some finite subset $F$ of $\Gamma$ such that $f_1+\sum_{j=2}^ns_jf_j\neq 0$ for all $s_2, \dots, s_n\in \Gamma$ with
$s_j\not\in F$ for all $2\le j\le n$. \end{enumerate} \end{lemma}
\begin{proof} (1)$\Longleftrightarrow$(2) follows from the observation that (1) is exactly (2) when $f_j$ is the characteristic function of $A_j$ and the fact that the linear span of characteristic functions of elements in ${\mathcal B}$ is dense in $L^{\infty}_{\mathbb C}(X, {\mathcal B}, \lambda)$ under the essential supremum norm $\|\cdot \|_\infty$.
(2)$\Longleftrightarrow$(3) follows from the fact that for any $f\in L^{\infty}_{{\mathbb C}}(X, {\mathcal B}, \lambda)$ and $\varepsilon>0$ there exists $g\in C_{\mathbb C}(X)$ with
$\|g\|_\infty\le \|f\|_\infty$ and $\|f-g\|_2<\varepsilon$.
We identify ${\mathbb R}/{\mathbb Z}$ with the unit circle $\{z\in {\mathbb C}: |z|=1\}$ naturally. Then every $g\in \widehat{X}$ can be thought of as an element in $C_{\mathbb C}(X)$. Note that the identity element $0$ in $\widehat{X}$ is the element $1$ in $C_{\mathbb C}(X)$. Then (3)$\Longleftrightarrow$(4) follows from the observation that for any $g\in \widehat{X}$, $\lambda(g)=1$ or $0$ depending on whether $g=0$ in $\widehat{X}$ or not, and the fact that the linear span of elements in $\widehat{X}$ is dense in $C_{\mathbb C}(X)$ under the supremum norm.
(4)$\Longleftrightarrow$(5) is obvious. \end{proof}
When the condition (1) in Lemma~\ref{L-mixing of all order} is satisfied, we say that the action is {\it (left) mixing of order $n$}. We say that the action is {\it (left) mixing of all orders} if it is mixing of order $n$ for all $n\in {\mathbb N}$ with $n\ge 2$.
\begin{proposition} \label{P-homoclinic to mixing} Let a countable group $\Gamma$ act on a compact metrizable abelian group $X$ by continuous automorphisms. Suppose that the homoclinic group $\Delta(X)$ is dense in $X$. Then the action is mixing of all orders with respect to the Haar probability measure of $X$. \end{proposition} \begin{proof} We verify the condition (5) in Lemma~\ref{L-mixing of all order} by contradiction. Let $n\in {\mathbb N}$ with $n\ge 2$, and $f_1, \dots, f_n\in \widehat{X}$ with $f_1\neq 0$. Suppose that there is a sequence $\{(s_{m, 2}, \dots, s_{m, n})\}_{m\in {\mathbb N}}$ of $(n-1)$-tuples in $\Gamma$ such that $f_1+\sum_{j=2}^ns_{m,j} f_j=0$ for all $m\in {\mathbb N}$ and $s_{m, j}\to \infty$ as $m\to \infty$ for every $2\le j\le n$.
Let $x\in \Delta(X)$. Then $f_1(x)+\sum_{j=2}^nf_j(s_{m, j}^{-1}x)=(f_1+\sum_{j=2}^ns_{m,j}f_j)(x)=0$ in ${\mathbb R}/{\mathbb Z}$ for all $m\in {\mathbb N}$. Since $s_{m, j}^{-1}x$ converges to the identity element of $X$ as $m\to \infty$ for every $2\le j\le n$, we have $f_j(s_{m, j}^{-1}x)\to 0$ in ${\mathbb R}/{\mathbb Z}$ as $m\to \infty$ for every $2\le j\le n$. It follows that $f_1(x)=0$ in ${\mathbb R}/{\mathbb Z}$. Since $\Delta(X)$ is dense in $X$, we get $f_1=0$, a contradiction. \end{proof}
\begin{example} \label{E-invertible in von Neumann} For a countable group $\Gamma$, when $g\in {\mathbb Z} \Gamma$ is invertible in the group von Neumann algebra ${\mathcal N}\Gamma$, $\Delta(X_g)$ is dense in $X_g$ \cite[Lemma 5.4]{CL}, and hence by Proposition~\ref{P-homoclinic to mixing} the action $\alpha_g$ is mixing of all orders with respect to the Haar probability measure of $X_g$. \end{example}
Let $\Gamma$ be a finitely generated infinite group. Let $\mu$ be a finitely supported symmetric probability measure on $\Gamma$ such that the support of $\mu$ generates $\Gamma$. We shall think of $\mu$ as an element in ${\mathbb R} \Gamma$. We endow $\Gamma$ with the word length associated to the support of $\mu$.
By the Cauchy-Schwarz inequality for any $s\in \Gamma$ and $n\in {\mathbb N}$ one has $(\mu^{2n})_s=(\mu^n\cdot (\mu^n)^*)_s\le (\mu^n\cdot (\mu^n)^*)_e=(\mu^{2n})_e$. Also note that for any $s\in \Gamma$ and $n\in {\mathbb N}$ one has $(\mu^{2n+1})_s=(\mu^{2n}\cdot \mu)_s\le \|\mu^{2n}\|_\infty$. It follows that
$\sum_{k=0}^{\infty} (\mu^k)_e<+\infty$ if and only if $\sum_{k=0}^{\infty}\|\mu^k\|_\infty<+\infty$.
Now assume that $\Gamma$ is not virtually ${\mathbb Z}$ or ${\mathbb Z}^2$.
By a result of Varopoulos \cite{Varopoulos} \cite[Theorem 2.1]{Fu02}
\cite[Theorem 3.24]{Woess} one has $\sum_{k=0}^{\infty} (\mu^k)_e<+\infty$, and hence $\sum_{k=0}^{\infty}\|\mu^k\|_\infty<+\infty$. Thus we have the element $\sum_{k=0}^\infty \mu^k$ in $\ell^\infty(\Gamma)$. Let $\varepsilon>0$. Take $m\in {\mathbb N}$ such that
$\sum_{k=m+1}^\infty\|\mu^k\|_\infty<\varepsilon$. For each $s\in \Gamma$ with word length at least $m+1$, one has
$$|(\sum_{k=0}^\infty \mu^k)_s|=|(\sum_{k=m+1}^\infty \mu^k)_s|\le \sum_{k=m+1}^\infty\|\mu^k\|_\infty<\varepsilon.$$ Therefore $\sum_{k=0}^\infty \mu^k$ lies in the space $C_0(\Gamma)$ of ${\mathbb R}$-valued functions on $\Gamma$ vanishing at $\infty$. Since the support of $\mu$ is symmetric and generates $\Gamma$, one has $(\sum_{k=0}^\infty \mu^k)_s>0$ for every $s\in \Gamma$.
Now let $f\in {\mathbb R}\Gamma$ be well-balanced. Set $\mu=-(f-f_e)/f_e$. Then $\mu$ is a finitely supported symmetric probability measure on $\Gamma$ and the support of $\mu$ generates $\Gamma$. Set $\omega=f_e^{-1}\sum_{k=0}^\infty\mu^k\in C_0(\Gamma)$. We have $f=f_e(1-\mu)$, and hence $$ f\omega=(1-\mu)\sum_{k=0}^\infty\mu^k=1$$ and $$ \omega f=(\sum_{k=0}^\infty\mu^k)(1-\mu)=1.$$
Note that the space $C_0(\Gamma)$ is not closed under convolution. The above identities show that $\omega$ is a formal inverse of $f$ in $C_0(\Gamma)$. Now we show that $f$ has no other formal inverse in $C_0(\Gamma)$.
\begin{lemma} \label{L-associative} Let $g\in C_0(\Gamma)$ such that $gf\in \ell^1(\Gamma)$. Then $$ (gf)\omega=g.$$ \end{lemma} \begin{proof} In the Banach space $\ell^\infty(\Gamma)$ we have \begin{align*} (gf) \omega &= \lim_{m\to \infty}(gf)(f_e^{-1}\sum_{k=0}^m\mu^k) = \lim_{m\to \infty}g(f f_e^{-1}\sum_{k=0}^m\mu^k) = \lim_{m\to \infty}g((1-\mu)\sum_{k=0}^m\mu^k) \\ &= \lim_{m\to \infty}g(1-\mu^{m+1}) = g-\lim_{m\to \infty}g\mu^{m+1}. \end{align*}
Let $\varepsilon>0$. Take a finite set $F\subset\Gamma$ such that $\|g|_{\Gamma \setminus F}\|_\infty<\varepsilon$. Write $g$ as $u+v$ for $u, v\in \ell^\infty(\Gamma)$ such that $u$ has support contained in $F$ and $v$ has support contained in $\Gamma\setminus F$. For each $m\in {\mathbb N}$ we have \begin{align*}
\|g\mu^{m+1}\|_\infty &\le \|u\mu^{m+1}\|_\infty+\|v\mu^{m+1}\|_\infty \le \|u\|_1\|\mu^{m+1}\|_\infty+\|v\|_\infty\|\mu^{m+1}\|_1\\
&\le \|g\|_\infty \cdot |F|\cdot \|\mu^{m+1}\|_\infty+ \varepsilon. \end{align*}
Letting $m\to \infty$, we get $\limsup_{m\to \infty}\|g\mu^{m+1}\|_\infty\le \varepsilon$. Since $\varepsilon$ is an arbitrary positive number, we get
$\limsup_{m\to \infty}\|g\mu^{m+1}\|_\infty=0$ and hence $\lim_{m\to \infty}g\mu^{m+1}=0$. It follows that $(gf)\omega=g$ as desired. \end{proof}
\begin{corollary} \label{C-inverse} Let $g\in C_0(\Gamma)$ such that $gf=1$. Then $g=\omega$. \end{corollary} \begin{proof} By Lemma~\ref{L-associative} we have $$ g=(gf)\omega=\omega.$$ \end{proof}
Denote by $Q$ the natural quotient map $\ell^\infty(\Gamma)\rightarrow ({\mathbb R}/{\mathbb Z})^\Gamma$. We assume furthermore that $f\in {\mathbb Z}\Gamma$.
\begin{lemma} \label{L-homoclinic group} We have $$ \Delta(X_f)=Q({\mathbb Z}\Gamma \omega).$$ \end{lemma} \begin{proof} Let $h\in {\mathbb Z}\Gamma$. Then $$ (h\omega)f=h(\omega f)=h\in {\mathbb Z}\Gamma,$$ and hence $Q(h\omega)\in X_f$. Since $h\omega \in C_0(\Gamma)$, one has $Q(h\omega)\in \Delta(X_f)$. Thus $Q({\mathbb Z}\Gamma \omega)\subset\Delta(X_f)$.
Now let $x\in \Delta(X_f)$. Take $\tilde{x}\in C_0(\Gamma)$ such that $Q(\tilde{x})=x$. Then $\tilde{x} f\in C_0(\Gamma)\cap \ell^\infty(\Gamma, {\mathbb Z})={\mathbb Z}\Gamma$. Set $h=\tilde{x}f\in {\mathbb Z}\Gamma$. By Lemma~\ref{L-associative} we have $\tilde{x}=h\omega$. Therefore $x=Q(\tilde{x})=Q(h\omega)$, and hence $\Delta(X_f)\subset Q({\mathbb Z}\Gamma \omega)$ as desired. \end{proof}
\begin{corollary} \label{C-module} As a left ${\mathbb Z}\Gamma$-module, $\Delta(X_f)$ is isomorphic to ${\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$. \end{corollary} \begin{proof} By Lemma~\ref{L-homoclinic group} we have the surjective left ${\mathbb Z}\Gamma$-module map $\Phi: {\mathbb Z}\Gamma\rightarrow \Delta(X_f)$ sending $h$ to $Q(h\omega)$. Then it suffices to show $\ker \Phi={\mathbb Z}\Gamma f$.
If $h\in {\mathbb Z}\Gamma f$, say $h=gf$ for some $g\in {\mathbb Z}\Gamma$, then $$ Q(h\omega)=Q((gf)\omega)=Q(g(f\omega))=Q(g)=0.$$ Thus ${\mathbb Z}\Gamma \subset\ker \Phi$.
Let $h\in \ker \Phi$. Then $h\omega \in C_0(\Gamma)\cap \ell^\infty(\Gamma, {\mathbb Z})={\mathbb Z}\Gamma$. Set $g=h\omega \in {\mathbb Z}\Gamma$. Then $$ gf=(h\omega)f=h(\omega f)=h.$$ Thus $\ker\Phi\subset{\mathbb Z}\Gamma f$. \end{proof}
We are ready to prove Theorem~\ref{T-dense homoclinic}.
\begin{proof}[Proof of Theorem~\ref{T-dense homoclinic}] The assertion (2) follows from the assertion (1) and Proposition~\ref{P-homoclinic to mixing}.
To prove the assertion (1), by Pontryagin duality it suffices to show any $\varphi \in \widehat{X_f}$ vanishing on $\Delta(X_f)$ is $0$. Thus let $\varphi \in \widehat{X_f}={\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ vanishing on $\Delta(X_f)$. Say, $\varphi=g+{\mathbb Z}\Gamma f$ for some $g\in {\mathbb Z}\Gamma$. For each $h\in {\mathbb Z}\Gamma$, in ${\mathbb R}/{\mathbb Z}$ one has $$ 0=\left< Q(h\omega), \varphi\right>\overset{\eqref{E-pairing}}=((h\omega) g^*)_e+{\mathbb Z}=(h(\omega g^*))_e+{\mathbb Z},$$ and hence $(h(\omega g^*))_e\in {\mathbb Z}$. Taking $h=s$ for all $s\in \Gamma$, we conclude that $\omega g^*\in C_0(\Gamma)\cap \ell^\infty(\Gamma, {\mathbb Z})={\mathbb Z}\Gamma$. Set $v=\omega g^*\in {\mathbb Z}\Gamma$. Then $$ fv=f(\omega g^*)=(f\omega)g^*=g^*,$$ and hence $$ g=v^*f^*=v^*f\in {\mathbb Z}\Gamma f.$$ Therefore $\varphi=g+{\mathbb Z}\Gamma f=0$ as desired. \end{proof}
\section{Periodic Points} \label{S-periodic}
Throughout this section we let $\Gamma$ be a finitely generated residually finite infinite group, and let $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ be a sequence of finite-index normal subgroups of $\Gamma$ such that $\bigcap_{n\in {\mathbb N}}\bigcup_{i\ge n}\Gamma_i=\{e\}$.
For a compact metric space $(X, \rho)$, recall that the Hausdorff distance between two nonempty closed subsets $Y$ and $Z$ of $X$ is defined as $$ {\rm dist_H}(Y, Z):=\max(\max_{y\in Y}\min_{z\in Z}\rho(y, z), \max_{z\in Z}\min_{y\in Y}\rho(z, y)).$$
For $f\in {\mathbb Z}\Gamma$ we denote by ${\rm Fix}_{\Gamma_n}(X_f)$ the group of fixed points of $\Gamma_n$ in $X_f$. The main result of this section is
\begin{theorem} \label{T-dense periodic points}
Let $f\in {\mathbb Z}\Gamma$ be well-balanced. Let $\rho$ be a
compatible metric on $X_f$. We have ${\rm Fix}_{\Gamma_n}(X_f)\rightarrow X_f$ under the Hausdorff distance when $n\to \infty$. \end{theorem}
Denote by $\pi_n$ the natural ring homomorphism ${\mathbb R}\Gamma \rightarrow {\mathbb R}(\Gamma/\Gamma_n)$. Let $X_{\pi_n(f)}$ be $\widehat{{\mathbb Z}(\Gamma/\Gamma_n)/{\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)}$, which is the additive group of all maps $x:\Gamma/\Gamma_n \to {\mathbb R}/{\mathbb Z}$ satisfying $x\pi_n(f) =0$, i.e., $\sum_{s\in \Gamma} x_{ts\Gamma_n} f_{s^{-1}} = {\mathbb Z}$ for every $t\in \Gamma$.
\begin{lemma} \label{L-vanishing} Let $f\in {\mathbb Z}\Gamma$ and $\varphi=g+{\mathbb Z}\Gamma f\in {\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ for some $g\in {\mathbb Z}\Gamma$. Let $n\in {\mathbb N}$. Then $\varphi$ vanishes on ${\rm Fix}_{\Gamma_n}(X_f)$ if and only if $\pi_n(g)\in {\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$. \end{lemma} \begin{proof} By \cite[Lemma 7.4]{KL11a} we have a compact group isomorphism $\Phi_n:X_{\pi_n(f)} \rightarrow {\rm Fix}_{\Gamma_n}(X_f)$ defined by $(\Phi_n(x))_s=x_{s\Gamma_n}$ for all $x\in X_{\pi_n(f)}$ and $s\in \Gamma$.
For each $x\in X_{\pi_n(f)}$, in ${\mathbb R}/{\mathbb Z}$ we have \begin{align*} \left<\Phi_n(x), \varphi\right>&\overset{\eqref{E-pairing}}=(\Phi_n(x)g^*)_e =\sum_{s\in \Gamma} (\Phi_n(x))_sg_s =\sum_{s\in \Gamma} x_{s\Gamma_n}g_s \\ &= \sum_{s\Gamma_n \in \Gamma/\Gamma_n} x_{s\Gamma_n}\pi_n(g)_{s\Gamma_n} =\left<x, \pi_n(g)+{\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)\right>. \end{align*} Thus $\varphi$ vanishes on ${\rm Fix}_{\Gamma_n}(X_f)$ iff the element $\pi_n(g)+{\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$ in ${\mathbb Z}(\Gamma/\Gamma_n)/{\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$ vanishes on $X_{\pi_n(f)}$, iff $\pi_n(g)\in {\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$. \end{proof}
For the next lemma, recall that if $S \subset \Gamma \setminus \{e\}$ is a symmetric finite generating set then $C(\Gamma,S)$, the Cayley graph of $\Gamma$ with respect to $S$, has vertex set $\Gamma$ and edge set $\{ \{g,gs\} \}_{g\in \Gamma,s\in S}$. Similarly, $C(\Gamma/\Gamma_n, \pi_n(S))$ has vertex set $\Gamma/\Gamma_n$ and edge set $\{\{g\Gamma_n, gs\Gamma_n\}:~g\in \Gamma, s\in S\}$. A subset $A \subset \Gamma$ is identified with the induced subgraph of $C(\Gamma,S)$ which has vertex set $A$ and contains every edge of $C(\Gamma,S)$ with endpoints in $A$. Similarly, a subset $A \subset \Gamma/\Gamma_n$ induces a subgraph of $C(\Gamma/\Gamma_n,\pi_n(S))$. Thus we say that a subset $A \subset \Gamma$ (or $A \subset \Gamma/\Gamma_n$) is {\em connected} if its induced subgraph is connected.
\begin{lemma} \label{L-connectedness in Cayley graphs} Let $S\subset \Gamma\setminus \{e\}$ be a finite symmetric generating set of $\Gamma$. Let $A\subset\Gamma$ be finite. Then there exists a finite set $B\subset\Gamma$ containing $A$ such that when $n\in {\mathbb N}$ is large enough, in the Cayley graph $C(\Gamma/\Gamma_n, \pi_n(S))$ the set $(\Gamma/\Gamma_n)\setminus \pi_n(B)$ is connected.
\end{lemma}
\begin{proof} We claim that there exists a connected finite set $B \supset A \cup \{e\}$ such that every connected component of $C(\Gamma,S)\setminus B$ is infinite. To see this, let $A'\subset\Gamma$ be a finite connected set such that $A'\supset A\cup \{e\}$. For any connected component ${\mathcal C}$ of $C(\Gamma,S) \setminus A'$, taking a path in $C(\Gamma, S)$ from some point in ${\mathcal C}$ to some point in $A'$, we note that the last point of this path in ${\mathcal C}$ must lie in $A'S$, whence ${\mathcal C} \cap A'S\neq \emptyset$. It follows that $C(\Gamma,S) \setminus A'$ has only finitely many connected components. Denote by $B$ the union of $A'$ and all the finite connected components of $C(\Gamma,S) \setminus A'$. Then $B$ is finite, contains $A\cup \{e\}$, and is connected. Furthermore, the connected components of $C(\Gamma,S) \setminus B$ are exactly the infinite connected components of $C(\Gamma,S) \setminus A'$, whence are all infinite.
For each $t\in BS\setminus B$, since the connected component of $C(\Gamma,S) \setminus B$ containing $t$ is infinite, we can take a path $\gamma_t$ in $C(\Gamma,S) \setminus B$ from $t$ to some element in $\Gamma\setminus (B(\{e\}\cup S)^2B^{-1})$. Let $n \in {\mathbb N}$ be sufficiently large so that $\pi_n$ is injective on $B\cup (B(\{e\}\cup S)^2B^{-1}) \cup \bigcup_{t\in BS\setminus B}\gamma_t$.
An argument similar to that in the first paragraph of the proof shows that every connected component ${\mathcal C}$ of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus \pi_n(B)$ has nonempty intersection with $\pi_n(BS)$, and hence contains $\pi_n(t)$ for some $t\in BS\setminus B$. Then ${\mathcal C}$ contains $\pi_n(\gamma_t)$, whence contains $\pi_n(g)$ for some $g\in \gamma_t\setminus (B(\{e\}\cup S)^2B^{-1})$. Since $\pi_n$ is injective on $\gamma_t\cup (B(\{e\}\cup S)^2B^{-1})$, we have $\pi_n(g)\not \in \pi_n(B(\{e\}\cup S)^2B^{-1})$, equivalently, $g\pi_n(B\cup BS)\cap \pi_n(B\cup BS)=\emptyset$.
List all the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus \pi_n(B)$ as ${\mathcal C}_0,{\mathcal C}_1,\ldots, {\mathcal C}_k$.
We may assume that $|{\mathcal C}_0| = \min_{0\le j\le k} |{\mathcal C}_j|$. Take $g_0\in \Gamma$ with $\pi_n(g_0)\in {\mathcal C}_0$ and $g_0\pi_n(B\cup BS)\cap \pi_n(B\cup BS)=\emptyset$. Since $B$ is connected, $B\cup BS$ is connected. Thus $g_0\pi_n(B\cup BS)$ is connected and disjoint from $\pi_n(B)$.
Therefore $g_0\pi_n(B\cup BS)$ is contained in one of the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus \pi_n(B)$.
As $\pi_n(g_0)\in (g_0\pi_n(B\cup BS))\cap {\mathcal C}_0$, we get $g_0\pi_n(B\cup BS)\subset{\mathcal C}_0$.
Since $\Gamma$ acts on $C(\Gamma/\Gamma_n,\pi_n(S))$ by left translation, the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus g_0\pi_n(B)$ are $g_0{\mathcal C}_0, g_0{\mathcal C}_1, \dots, g_0{\mathcal C}_k$.
Suppose that $g_0{\mathcal C}_0 \cap \pi_n(B) = \emptyset$. Then $g_0{\mathcal C}_0$ must be contained in one of the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus \pi_n(B)$. Since ${\mathcal C}_0\cap \pi_n(BS)\neq \emptyset$, one has $g_0{\mathcal C}_0\cap g_0\pi_n(BS)\neq \emptyset$. Because $g_0\pi_n(BS)\subset{\mathcal C}_0$, we have
$g_0{\mathcal C}_0\cap {\mathcal C}_0\neq \emptyset$. Therefore $g_0{\mathcal C}_0\subset{\mathcal C}_0$. Because $|g_0{\mathcal C}_0|=|{\mathcal C}_0|$, this implies $g_0{\mathcal C}_0={\mathcal C}_0$. But this contradicts the fact that $g_0\pi_n(B) \subset{\mathcal C}_0$ but $g_0\pi_n(B) \cap g_0{\mathcal C}_0 = \emptyset$. So $g_0{\mathcal C}_0 \cap \pi_n(B) \ne \emptyset$.
Since $\pi_n(B\cup BS)$ is connected and disjoint from $g_0\pi_n(B)$, it is contained in one of the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus g_0\pi_n(B)$. Because $g_0{\mathcal C}_0 \cap \pi_n(B) \ne \emptyset$, we get $\pi_n(B\cup BS)\subset g_0{\mathcal C}_0$.
Suppose that $k>0$. Since ${\mathcal C}_k$ is disjoint from ${\mathcal C}_0$ and $g_0\pi_n(B)\subset{\mathcal C}_0$, ${\mathcal C}_k$ is disjoint from $g_0\pi_n(B)$. Thus ${\mathcal C}_k$ is contained in one of the connected components of $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus g_0\pi_n(B)$. Because ${\mathcal C}_k$ has nonempty intersection with $\pi_n(BS)$, and $\pi_n(B\cup BS)\subset g_0{\mathcal C}_0$, we get
${\mathcal C}_k\cup \pi_n(B) \subset g_0{\mathcal C}_0$. Therefore $|{\mathcal C}_0|=|g_0{\mathcal C}_0|>|{\mathcal C}_k|$
which contradicts the fact that $|{\mathcal C}_0| = \min_{0\le j\le k} |{\mathcal C}_j|$. Thus $k=0$; i.e., $C(\Gamma/\Gamma_n,\pi_n(S)) \setminus \pi_n(B)$ is connected. \end{proof}
\begin{lemma} \label{L-bounded} Let $f\in{\mathbb R}\Gamma$ be well-balanced and $g\in {\mathbb R}\Gamma$. Then there exists $C>0$ such that if $\pi_n(g)=h\pi_n(f)$ for some $n\in {\mathbb N}$ and $h\in {\mathbb R}(\Gamma/\Gamma_n)$, then $$\max_{}\{h_{s\Gamma_n}:~s\Gamma_n\in \Gamma/\Gamma_n\} -\min_{}\{h_{s\Gamma_n}:~s\Gamma_n\in \Gamma/\Gamma_n\}\le C.$$ \end{lemma} \begin{proof} Set $\mu=-(f-f_e)/f_e$. Then $\mu$ is a symmetric finitely supported probability measure on $\Gamma$. Denote by $S$ and $K$ the supports of $\mu$ and $g$ respectively. Set $a=\min_{s\in S} \mu_s$.
Suppose that $\pi_n(g)=h\pi_n(f)$ for some $n\in {\mathbb N}$ and $h\in {\mathbb R}(\Gamma/\Gamma_n)$. Denote by $W$ the set of $s\Gamma_n\in \Gamma/\Gamma_n$ satisfying $h_{s\Gamma_n}=\min_{}\{h_{t\Gamma_n}:~t\Gamma_n\in \Gamma/\Gamma_n\}$. If $s\Gamma_n\in W\setminus K\Gamma_n$, then from $$ 0=(\pi_n(g))_{s\Gamma_n}=(h\pi_n(f))_{s\Gamma_n}=f_e(h_{s\Gamma_n}-\sum_{t\in S}h_{st^{-1}\Gamma_n}\mu_t)$$ we conclude that $st\Gamma_n\in W$ for all $t\in S$. Since $S$ generates $\Gamma$, it follows that there exists some $s_{\min} \in K$ satisfying $h_{s_{\min}\Gamma_n}=\min_{}\{ h_{t\Gamma_n}:~ t\Gamma_n\in \Gamma/\Gamma_n\}$. Similarly, there exists some $s_{\max}\in K$ satisfying $h_{s_{\max}\Gamma_n}=\max_{}\{ h_{t\Gamma_n}:~ t\Gamma_n\in \Gamma/\Gamma_n\}$.
Now we show by induction on the word length $|t|$ of $t\in \Gamma$ with respect to $S$ that one has $h_{s_{\min}t\Gamma_n}\le h_{s_{\min}\Gamma_n}+|t|\cdot \frac{\|g\|_1}{f_ea^{|t|}}$ for all $t\in \Gamma$. This is clear when $|t|=0$, i.e. $t=e$. Suppose that this holds for all $t\in \Gamma$ with $|t|\le k$. Let $t\in \Gamma$ with $|t|=k+1$. Then
$t=t_1s_1$ for some $t_1\in \Gamma$ with $|t_1|=k$ and some $s_1\in S$. Note that \begin{align*}
-\|g\|_1&\le (\pi_n(g))_{s_{\min}t_1\Gamma_n}=(h\pi_n(f))_{s_{\min}t_1\Gamma_n}\\ &=f_e\left(h_{s_{\min}t_1\Gamma_n}-h_{s_{\min}t_1s_1\Gamma_n}\mu_{s_1^{-1}}-\sum_{s\in S\setminus \{s_1\}}h_{s_{\min}t_1s\Gamma_n}\mu_{s^{-1}}\right), \end{align*} and hence \begin{align*} h_{s_{\min}t\Gamma_n}&= h_{s_{\min}t_1s_1\Gamma_n}\\
&\le \left(\frac{\|g\|_1}{f_e}+h_{s_{\min}t_1\Gamma_n}-\sum_{s\in S\setminus \{s_1\}}h_{s_{\min}t_1s\Gamma_n}\mu_{s^{-1}}\right)/\mu_{s_1^{-1}}\\
&\le \left(\frac{\|g\|_1}{f_e}+h_{s_{\min}\Gamma_n}+k\cdot \frac{\|g\|_1}{f_ea^k}-\sum_{s\in S\setminus \{s_1\}}h_{s_{\min}\Gamma_n}\mu_{s^{-1}}\right)/\mu_{s_1^{-1}}\\
&= \left(\frac{\|g\|_1}{f_e}+h_{s_{\min}\Gamma_n}\mu_{s_1^{-1}}+k\cdot \frac{\|g\|_1}{f_ea^k}\right)/\mu_{s_1^{-1}}\le h_{s_{\min}\Gamma_n}+|t|\cdot \frac{\|g\|_1}{f_ea^{|t|}}. \end{align*} This finishes the induction.
Set $m=\max_{s\in K^{-1}K} |s|$. Taking $t=s_{\min}^{-1}s_{\max}$ in above we get \begin{align*}
h_{s_{\max}\Gamma_n}-h_{s_{\min}\Gamma_n}\le |s_{\min}^{-1}s_{\max}|\cdot \frac{\|g\|_1}{f_ea^{|s_{\min}^{-1}s_{\max}|}}\le \frac{m \|g\|_1}{f_e a^m}. \end{align*}
Now we may set $C=\frac{m \|g\|_1}{f_e a^m}$. \end{proof}
\begin{lemma} \label{L-connected component to left module} Let $f\in {\mathbb Z}\Gamma$ be well-balanced and $g\in {\mathbb Z}\Gamma$. Then $\pi_n(g)\in {\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$ for all $n\in {\mathbb N}$ if and only if $g\in {\mathbb Z}\Gamma f$. \end{lemma} \begin{proof} The ``if'' part is obvious. Suppose that $\pi_n(g)\in {\mathbb Z}(\Gamma/\Gamma_n)\pi_n(f)$ for all $n\in {\mathbb N}$. For each $n\in {\mathbb N}$, take $h_n\in {\mathbb Z}(\Gamma/\Gamma_n)$ such that $\pi_n(g)=h_n\pi_n(f)$. By Lemma~\ref{L-bounded} there exists some $C\in {\mathbb N}$ such that $$\max_{s\Gamma_n\in \Gamma/\Gamma_n}(h_n)_{s\Gamma_n}-\min_{s\Gamma_n\in \Gamma/\Gamma_n}(h_n)_{s\Gamma_n}\le C$$ for all $n\in {\mathbb N}$.
Let $S$ be the set of all $s\in \Gamma\setminus \{e\}$ with $f_s\neq 0$. Denote by $A_1$ the support of $g$. By Lemma~\ref{L-connectedness in Cayley graphs} we can find finite subsets $A_2, \dots, A_{C+1}$ of $\Gamma$ and $N\in {\mathbb N}$ such that for any $n\ge N$ and $1\le j\le C$, one has $A_j(\{e\}\cup S)\subset A_{j+1}$ and in the Cayley graph $C(\Gamma/\Gamma_n, \pi_n(S))$ the set
$(\Gamma/\Gamma_n)\setminus \pi_n(A_{j+1})$ is connected.
Let $n\ge N$. Set $a_j=\max_{}\{(h_n)_{s\Gamma_n}:~s\Gamma_n\in (\Gamma/\Gamma_n)\setminus \pi_n(A_j)\}$ and $b_j=\min_{}\{(h_n)_{s\Gamma_n}:~s\Gamma_n\in (\Gamma/\Gamma_n)\setminus \pi_n(A_j)\}$ for all $1\le j\le C+1$. We shall show by induction that $$ a_j-b_j\le C+1-j$$ for all $1\le j\le C+1$. This is trivial when $j=1$. Suppose that $a_j-b_j\le C+1-j$ for some $1\le j\le C$. If $a_{j+1}<a_j$,
then $a_{j+1}-b_{j+1}<a_j-b_j\le C+1-j$ and hence $a_{j+1}-b_{j+1}\le C+1-(j+1)$. Thus we may assume that $a_{j+1}=a_j$.
Denote by $W$ the set of elements in $(\Gamma/\Gamma_n)\setminus \pi_n(A_{j+1})$ at which $h_n$ takes the value $a_{j+1}$.
Let $t\Gamma_n\in W$. Since $\pi_n(g)$ takes value $0$ at $t\Gamma_n$, we have $$ f_e (\pi_n(h))_{t\Gamma_n}=\sum_{s\in S}(-f_s) (\pi_n(h))_{ts\Gamma_n}.$$ Note that $ts\Gamma_n\in (\Gamma/\Gamma_n)\setminus \pi_n(A_j)$ for all $s\in S$. Thus $(\pi_n(h))_{ts\Gamma_n}\le a_j=a_{j+1}=(\pi_n(h))_{t\Gamma_n}$ for all $s\in S$, and hence $(\pi_n(h))_{t\Gamma_n}=(\pi_n(h))_{ts\Gamma_n}$ for all $s\in S$. Therefore, if for some $s\in S$ one has $ts\Gamma_n\in (\Gamma/\Gamma_n)\setminus \pi_n(A_{j+1})$, then $ts\Gamma_n\in W$. Take $t_1\Gamma_n, t_2 \Gamma_n \in \Gamma/\Gamma_n$. By our choice of $A_{j+1}$ we have a path in $(\Gamma/\Gamma_n)\setminus \pi_n(A_{j+1})$ connecting $t_1\Gamma_n$ and $t_2\Gamma_n$. Therefore $t_1\Gamma_n\in W \Leftrightarrow t_2\Gamma_n \in W$, whence $a_{j+1}-b_{j+1}=0\le C+1-(j+1)$.
Now we have that $h_n$ is a constant function on $(\Gamma/\Gamma_n)\setminus \pi_n(A_{C+1})$. Replacing $h_n$ by the difference of $h_n$ and a suitable constant function, we may assume that $h_n$ is $0$ on $(\Gamma/\Gamma_n)\setminus \pi_n(A_{C+1})$. Then $\|h_n\|_\infty\le C$.
Passing to a subsequence of $\{\Gamma_n\}_{n\in {\mathbb N}}$ if necessary, we may assume that $h_n(s\Gamma_n)$ converges to some integer $h_s$ as $n\to \infty$ for every $s\in \Gamma$. Then $h_s=0$ for all $s\in \Gamma\setminus A_{C+1}$. Thus $h\in {\mathbb Z}\Gamma$. Note that $$ (hf)_s=\lim_{n\to \infty}(h_n\pi_n(f))_{s\Gamma_n}=\lim_{n\to \infty}(\pi_n(g))_{s\Gamma_n}=g_s$$ for each $s\in \Gamma$ and hence $g=hf\in {\mathbb Z}\Gamma f$. This proves the ``only if'' part. \end{proof}
We are ready to prove Theorem~\ref{T-dense periodic points}.
\begin{proof}[Proof of Theorem~\ref{T-dense periodic points}] Since $X_f$ is compact, the set of nonempty closed subsets of $X_f$ is a compact space under the Hausdorff distance \cite[Theorem 7.3.8]{BBI}. Thus, passing to a subsequence of $\{\Gamma_n\}_{n\in {\mathbb N}}$ if necessary, we may assume that ${\rm Fix}_{\Gamma_n}(X_f)$ converges to some nonempty closed subset $Y$ of $X_f$ under the Hausdorff distance as $n\to \infty$. A point $x\in X_f$ is in $Y$ exactly when $x=\lim_{n\to \infty} x_n$ for some $x_n\in {\rm Fix}_{\Gamma_n}(X_f)$ for each $n\in {\mathbb N}$. It follows easily that $Y$ is a closed subgroup of $X_f$. Thus, by Pontryagin duality it suffices to show that the only $\varphi\in \widehat{X_f}={\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ vanishing on $Y$ is $0$. Let $\varphi\in \widehat{X_f}={\mathbb Z}\Gamma/{\mathbb Z}\Gamma f$ vanishing on $Y$. Say, $\varphi=g+{\mathbb Z}\Gamma f$ for some $g\in {\mathbb Z}\Gamma$.
Let $U$ be a small neighborhood of $0$ in ${\mathbb R}/{\mathbb Z}$ such that the only subgroup of ${\mathbb R}/{\mathbb Z}$ contained in $U$ is $\{0\}$. Since ${\rm Fix}_{\Gamma_n}(X_f)$ converges to $Y$ under the Hausdorff distance, we have $\left<{\rm Fix}_{\Gamma_n}(X_f), \varphi\right>\subset U$ for all sufficiently large $n$. Note that $\left<{\rm Fix}_{\Gamma_n}(X_f), \varphi\right>$ is a subgroup of ${\mathbb R}/{\mathbb Z}$. By our choice of $U$, we see that $\varphi$ vanishes on ${\rm Fix}_{\Gamma_n}(X_f)$ for all sufficiently large $n$. Without loss of generality, we may assume that $\varphi$ vanishes on ${\rm Fix}_{\Gamma_n}(X_f)$ for all $n\in {\mathbb N}$. From Lemmas~\ref{L-vanishing} and \ref{L-connected component to left module} we get $g\in {\mathbb Z}\Gamma f$ and hence $\varphi=0$ as desired. \end{proof}
\section{Sofic Entropy}\label{sec:sofic} The purpose of this section is to review sofic entropy theory. To be precise, we use Definitions 2.2 and 3.3 of \cite{KL11b} to define entropy. So let $\Gamma$ act by homeomorphisms on a compact metrizable space $X$. Suppose the action preserves a Borel probability measure $\lambda$. Let $\Sigma:=\{\Gamma_n\}^\infty_{n=1}$ be a sequence of finite-index normal subgroups of $\Gamma$ such that $\bigcap_{n\in {\mathbb N}}\bigcup_{i\ge n}\Gamma_i=\{e\}$.
Let $\rho$ be a continuous pseudo-metric on $X$. For $n \in {\mathbb N}$, define, on the space ${\rm Map}(\Gamma/\Gamma_n,X)$ of all maps from $\Gamma/\Gamma_n$ to $X$ the pseudo-metrics \begin{eqnarray*} \rho_2(\phi,\psi)&:=& \left([\Gamma:\Gamma_n]^{-1}\sum_{s\Gamma_n \in \Gamma/\Gamma_n} \rho(\phi(s \Gamma_n), \psi(s\Gamma_n) )^2 \right)^{1/2},\\ \rho_\infty(\phi,\psi)&:=& \sup_{s\Gamma_n \in \Gamma/\Gamma_n} \rho(\phi(s \Gamma_n), \psi(s\Gamma_n)). \end{eqnarray*}
\begin{definition} Let $W\subset \Gamma$ be finite and nonempty and $\delta>0$. Define ${\rm Map}(W,\delta,\Gamma_n)$ to be the set of all maps $\phi: \Gamma/\Gamma_n \to X$ such that $\rho_2(\phi \circ s, s \circ \phi) \le \delta$ for all $s\in W$.
Given a finite set $L$ in the space $C(X)$ of continuous ${\mathbb R}$-valued functions on $X$, let ${\rm Map}_\lambda(W,L,\delta,\Gamma_n) \subset {\rm Map}(W,\delta,\Gamma_n)$ be the subset of maps $\phi: \Gamma/\Gamma_n \to X$ such that $|\phi_*U_n(p)-\lambda(p)| \le \delta$ for all $p\in L$, where $U_n$ denotes the uniform probability measure on $\Gamma/\Gamma_n$. \end{definition}
\begin{definition}\label{defn:separating} Let $(Z,\rho_Z)$ be a pseudo-metric space. A set $Y \subset Z$ is {\em $(\rho_Z,\epsilon)$-separating} if $\rho_Z(y_1,y_2) >\epsilon$ for every $y_1\ne y_2 \in Y$. If $\rho_Z$ is understood, then we simply say that $Y$ is {\em $\epsilon$-separating}. Let $N_\epsilon(Z,\rho_Z)$ denote the largest cardinality of a $(\rho_Z,\epsilon)$-separating subset of $Z$. \end{definition}
Define \begin{eqnarray*} h_{\Sigma,2}(\rho)&:=&\sup_{\epsilon>0} \inf_{W \subset \Gamma} \inf_{\delta>0} \limsup_{n\to\infty} [\Gamma:\Gamma_n]^{-1} \log N_\epsilon({\rm Map}(W,\delta,\Gamma_n),\rho_2)\\ h_{\Sigma,\lambda,2}(\rho)&:=&\sup_{\epsilon>0} \inf_{W \subset \Gamma} \inf_{L\subset C(X)} \inf_{\delta>0} \limsup_{n\to\infty} [\Gamma:\Gamma_n]^{-1} \log N_\epsilon({\rm Map}_\lambda(W,L,\delta,\Gamma_n),\rho_2). \end{eqnarray*} Similarly, define $h_{\Sigma,\infty}(\rho)$ and $h_{\Sigma,\lambda,\infty}(\rho)$ by replacing $\rho_2$ with $\rho_\infty$ in the formulae above.
The pseudo-metric $\rho$ is said to be {\em dynamically generating} if for any $x,y \in X$ with $x\ne y$, $\rho(sx,sy) >0$ for some $s \in \Gamma$. \begin{theorem}\label{thm:entropy} If $\rho$ is any dynamically generating continuous pseudo-metric on $X$ then $$h_{\Sigma,\lambda}(X,\Gamma)=h_{\Sigma,\lambda,2}(\rho) = h_{\Sigma,\lambda,\infty}(\rho),$$ $$h_{\Sigma}(X,\Gamma) = h_{\Sigma,2}(\rho) = h_{\Sigma,\infty}(\rho).$$ \end{theorem} \begin{proof} This follows from Propositions 2.4 and 3.4 of \cite{KL11b}. \end{proof}
\section{Entropy of the Harmonic Model}
In this section we prove Theorem~\ref{thm:main1}. Throughout this section we let $\Gamma$ be a countably infinite group, $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ be a sequence of finite-index normal subgroups of $\Gamma$ satisfying $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$, and $f \in {\mathbb Z}\Gamma$ be well-balanced.
\subsection{The lower bound}
Note that ${\rm Fix}_{\Gamma_n}(X_f) \subset X_f$ is a closed $\Gamma$-invariant subgroup. Let $\lambda_n$ be its Haar probability measure.
\begin{lemma} \label{L-measure convergence} The measure $\lambda_n$ converges in the weak* topology to the Haar probability measure $\lambda$ on $X_f$ as $n\to \infty$. \end{lemma}
\begin{proof} For $x\in X_f$, let $A_x:X_f \to X_f$ by the addition map $A_x(y)=x+y$. Each $A_x$ induces a map $(A_x)_*$ on the space $M(X_f)$ of all Borel probability measures on $X_f$. The map $X_f \times M(X_f) \to M(X_f)$ defined by $(x,\mu) \mapsto (A_x)_*\mu$ is continuous (with respect to the weak* topology on $M(X_f)$).
Choose an increasing sequence $\{n_i\}$ of natural numbers so that $\lim_{i\to\infty} \lambda_{n_i} = \lambda_\infty\in M(X_f)$ exists (this is possible by the Banach-Alaoglu Theorem). By the above if $x_i \in {\rm Fix}_{\Gamma_{n_i}}(X_f)$ and $\lim_{i\to\infty} x_i = x$ then $$\lim_{i\to\infty} (A_{x_i})_*\lambda_{n_i} = (A_x)_*\lambda_\infty.$$ Since $\lambda_{n_i}$ is the Haar probability measure on ${\rm Fix}_{\Gamma_{n_i}}(X_f)$, $(A_{x_i})_*\lambda_{n_i} = \lambda_{n_i}$, so the above implies $(A_x)_*\lambda_\infty = \lambda_\infty$. Because ${\rm Fix}_{\Gamma_{n_i}}(X_f)$ converges in the Hausdorff topology to $X_f$ by Theorem~\ref{T-dense periodic points}, we have that $(A_x)_*\lambda_\infty = \lambda_\infty$ for every $x\in X_f$ which, by uniqueness of the Haar probability measure, implies that $\lambda_\infty=\lambda$ as required. \end{proof}
\begin{definition}\label{defn:rho}
For $t \in {\mathbb R}/{\mathbb Z}$ let $t' \in [-1/2,1/2)$ be such that $t' + {\mathbb Z} = t$ and define $|t|: = |t'|$. Similarly, for $x\in ({\mathbb R}/{\mathbb Z})^\Gamma$, let $x' \in {\mathbb R}^\Gamma$ be the unique element satisfying $x'_s \in [-1/2,1/2)$ and $x'_s + {\mathbb Z} = x_s$ for all $s\in \Gamma$. Define $\|x\|_\infty = \|x'\|_\infty$. Let $\rho$ be the continuous pseudo-metric on $X_f$ defined by $\rho(x,y)=|(x-y)'_e|$. It is easy to check that $\rho$ is dynamically generating. \end{definition}
For $x\in {\rm Fix}_{\Gamma_n}(X_f)$, let $\phi_x:\Gamma/\Gamma_n \to X_f$ be the map defined by $\phi_x(s\Gamma_n) = sx$ for all $s\in \Gamma$. Let $W \subset \Gamma, L \subset C(X_f)$ be non-empty finite sets and $\delta>0$. Note that $\phi_x\in {\rm Map}(W,\delta,\Gamma_n)$ for all $x\in {\rm Fix}_{\Gamma_n}(X_f)$. Let ${\rm BAD}(W,L,\delta,\Gamma_n)$ be the set of all $x\in {\rm Fix}_{\Gamma_n}(X_f)$ such that $\phi_x \notin {\rm Map}_\lambda(W,L,\delta,\Gamma_n)$. Because $\phi_x\circ s = s \circ \phi_x$ for every $s\in \Gamma$, $\phi_x \notin {\rm Map}_\lambda(W,L,\delta,\Gamma_n)$ if and only if there exists $p\in L$ such that
$$\left| (\phi_x)_*U_n(p) - \lambda(p)\right| > \delta.$$
\begin{lemma}\label{lem:BAD1} Assume that $\Gamma$ is not virtually ${\mathbb Z}$ or ${\mathbb Z}^2$.
Then $$\lim_{n\to\infty} \lambda_n({\rm BAD}(W,L,\delta,\Gamma_n)) = 0.$$ \end{lemma} \begin{proof} The proof is similar to the proof of \cite[Theorem 3.1]{Bo11}. Let $n\in {\mathbb N}$. For each $\sigma \in \{-1,0,1\}^L$, let ${\rm BAD}_\sigma(W,L,\delta,\Gamma_n)$ be the set of all $x \in {\rm BAD}(W,L,\delta,\Gamma_n)$ such that for every $p \in L$, if $\sigma(p)\ne 0$ then $$\sigma(p)\left[(\phi_x)_*U_n(p)-\lambda(p)\right] =\sigma(p) \left[ -\lambda(p) + [\Gamma:\Gamma_n]^{-1}\sum_{s\Gamma_n\in \Gamma/\Gamma_n} p(sx) \right] > \delta$$
and if $\sigma(p)=0$ then $\left|(\phi_x)_*U_n(p)-\lambda(p)\right| \le \delta$.
Observe that for each $\sigma \in \{-1,0,1\}^L$, ${\rm BAD}_\sigma(W,L,\delta,\Gamma_n)$ is $\Gamma$-invariant. Moreover, $\{{\rm BAD}_\sigma(W,L,\delta,\Gamma_n):~\sigma \in \{-1,0,1\}^L\}$ is a partition of ${\rm BAD}(W,L,\delta,\Gamma_n)$.
Let $t_{n,\sigma} = \lambda_n({\rm BAD}_\sigma(W,L,\delta,\Gamma_n))$ and $t_{n,G} = 1- \lambda_n({\rm BAD}(W,L,\delta,\Gamma_n))$. So $t_{n,G} + \sum_\sigma t_{n,\sigma} = 1$. For each $\sigma \in \{-1,0,1\}^L$, define a Borel probability measure $\lambda_{n,\sigma}$ on $X_f$ by $$\lambda_{n,\sigma}(E) = \lambda_n(E \cap {\rm BAD}_\sigma(W,L,\delta,\Gamma_n)) t_{n,\sigma}^{-1},\quad \forall \mbox{ Borel } E \subset X_f$$ if $t_{n,\sigma} \ne 0$. Otherwise, define $\lambda_{n,\sigma}$ arbitrarily. Let $\lambda_{n,G}$ be the Borel probability measure on ${\rm Fix}_{\Gamma_n}(X_f)$ defined by $$\lambda_{n,G}(E) = \lambda_n( E \setminus {\rm BAD}(W,L,\delta,\Gamma_n) ) t_{n,G}^{-1},\quad \forall \mbox{ Borel } E \subset X_f$$ if $t_{n,G}\ne 0$. Otherwise, define $t_{n,G}$ arbitrarily. Observe that $$\lambda_n = t_{n,G}\lambda_{n,G} + \sum_{\sigma} t_{n,\sigma} \lambda_{n,\sigma}.$$ Because the space of Borel probability measures on $X_f$ is weak* sequentially compact (by the Banach-Alaoglu Theorem), there is a subsequence $\{n_i\}_{i=1}^\infty$ such that \begin{itemize} \item $\lambda_{n_i,G}$ converges in the weak* topology as $i\to\infty$ to a Borel probability measure $\lambda_{\infty,G}$ on $X_f$, \item each $\lambda_{n_i,\sigma}$ converges in the weak* topology as $i\to\infty$ to a Borel probability measure $\lambda_{\infty,\sigma}$ on $X_f$, \item the limits $\lim_{i\to\infty} t_{n_i,G} = t_{\infty,G}$ and $\lim_{i\to\infty} t_{n_i,\sigma} = t_{\infty,\sigma}$ exist for all $\sigma$. \end{itemize} By the previous lemma, $\lambda_n$ converges to $\lambda$ as $n\to\infty$. Therefore, $$\lambda = t_{\infty,G}\lambda_{\infty,G} + \sum_{\sigma} t_{\infty,\sigma} \lambda_{\infty,\sigma}.$$ Because weak* convergence preserves invariance, $\lambda_{\infty,G}$ and each of $\lambda_{\infty,\sigma}$ are $\Gamma$-invariant Borel probability measures on $X_f$. Because $\lambda$ is ergodic by Theorem~\ref{T-dense homoclinic}, this implies that for each $\sigma \in \{-1,0,1\}^L$ with $t_{\infty,\sigma} \ne 0$, $\lambda_{\infty,\sigma} = \lambda$. However, for any $p \in L$ with $\sigma(p)\ne 0$, $$\sigma(p)\left(\lambda_{\infty,\sigma}(p) - \lambda(p)\right)=\lim_{i\to\infty} \sigma(p)\left(\lambda_{n_i,\sigma}(p) -\lambda(p) \right) \ge \delta.$$ This contradiction implies $t_{\infty,\sigma} = 0$ for all $\sigma \in \{-1,0,1\}^L$ (if $\sigma$ is constantly $0$, then ${\rm BAD}_\sigma(W,L,\delta,\Gamma_n)$ is empty so $t_{\infty,\sigma}=0$). Thus $\lim_{n\to\infty} t_{n,\sigma} = 0$ for all $\sigma$ which, since $$\lambda_n({\rm BAD}(W,L,\delta,\Gamma_n)) = \sum_{\sigma} \lambda_n({\rm BAD}_\sigma(W,L,\delta,\Gamma_n) ) = \sum_{\sigma} t_{n,\sigma},$$ implies the lemma. \end{proof}
\begin{lemma}\label{lem:well-known} Let $x \in \ell^\infty(\Gamma)$ and suppose $xf = 0$. Suppose also that for some finite-index subgroup $\Gamma'<\Gamma$, $sx=x$ for all $s\in \Gamma'$. Then $x$ is constant. \end{lemma}
\begin{proof} Because $x$ is fixed by a finite-index subgroup, there is an element $s_0\in \Gamma$ such that $x_{s_0} = \min_{s\in \Gamma} x_s$. Because $xf=0$ and $f$ is well-balanced this implies that $x_{s_0t}=x_{s_0}$ for every $t$ in the support of $f$. By induction, $x_{s_0 t}=x_{s_0}$ for every $t$ in the semi-group generated by the support of $f$. By hypothesis, this semi-group is all of $\Gamma$. \end{proof}
\begin{lemma} \label{L-small suprenorm to constant}
There is a number $C>0$ such that if $x\in {\rm Fix}_{\Gamma_n}(X_f)$ for some $n\in {\mathbb N}$ satisfies $\|x\|_\infty< C$ then $x$ is constant. \end{lemma}
\begin{proof} Let $x'$ be as in Definition~\ref{defn:rho}.
Because $f$ has finite support, there is some number $C>0$ such that if $\|x'\|_\infty=\|x\|_\infty < C$ then $\| x'f\|_\infty < 1$. Since $x\in X_f$, $x'f \in \ell^\infty(\Gamma,{\mathbb Z})$. So $\|x'f\|_\infty < 1$ implies $x'f = 0$. Because $x'$ is fixed by a finite-index subgroup the previous lemma implies $x'$ is constant and hence $x$ is constant. \end{proof}
\begin{lemma}\label{lem:lower} Assume that $\Gamma$ is not virtually ${\mathbb Z}$ or ${\mathbb Z}^2$.
Then
$$h_{\Sigma,\lambda}(X_f,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|.$$ \end{lemma}
\begin{proof} Let $W \subset \Gamma, L \subset C(X_f)$ be non-empty finite sets and $\delta>0$. Let us identify ${\mathbb R}/{\mathbb Z}$ with the constant functions (on $\Gamma$) in $X_f$. It follows easily from Lemma~\ref{L-small suprenorm to constant} that the connected component of ${\rm Fix}_{\Gamma_n}(X_f)$ containing the identity element is exactly ${\mathbb R}/{\mathbb Z}$.
Choose a maximal set $Y_n \subset {\rm Fix}_{\Gamma_n}(X_f)$ such that $Y_n \cap {\rm BAD}(W,L, \delta, \Gamma_n) = \emptyset$ and for each $x \in Y_n$ and $t \in {\mathbb R}/{\mathbb Z}$ with $t\ne 0$, $x+t \notin Y_n$. By Lemma~\ref{lem:BAD1}, $\lim_{n\to \infty} |Y_n|^{-1} |{\rm Fix}_{\Gamma_n}(X_f)| = 1$. Let $C>0$ be the constant in Lemma~\ref{L-small suprenorm to constant}. By Lemma~\ref{L-small suprenorm to constant} if $x\ne y \in Y_n$ then $\|x-y\|_\infty \ge C$ which implies $\rho_\infty(\phi_x,\phi_y)\ge C$.
Therefore, if $0<\epsilon<C$ then $\{\phi_y:~y\in Y_n\}$ is $\epsilon$-separated with respect to $\rho_\infty$ which implies
$$ N_\epsilon( {\rm Map}_\lambda(W,L,\delta,\Gamma_n), \rho_\infty) \ge |Y_n|.$$
Because $\lim_{n\to \infty} |Y_n|^{-1} |{\rm Fix}_{\Gamma_n}(X_f)| = 1$, this implies
$$\limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)| \le h_{\Sigma,\lambda}(X_f,\Gamma).$$ \end{proof}
\begin{lemma} \label{L-amenable lower bound} Assume that $\Gamma$ is amenable. Then
$$h_{\Sigma,\lambda}(X_f,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|.$$ \end{lemma} \begin{proof} The argument in the proof of Lemma~\ref{lem:lower} shows that
$$ h_{\Sigma}(X_f,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|.$$ Note that $h_{\Sigma,\lambda}(X_f,\Gamma)$ coincides with the classical measure-theoretic entropy $h_\lambda(X_f, \Gamma)$ \cite[Theorem 1.2]{Bo12} \cite[Theorem 6.7]{KL11b}, and $h_{\Sigma}(X_f,\Gamma)$ coincides with the classical topological entropy $h(X_f, \Gamma)$ \cite[Theorem 5.3]{KL11b}. Since $\Gamma$ acts on $X_f$ by continuous group automorphism and $\lambda$ is the Haar probability measure of $X_f$, one has $h_\lambda(X_f, \Gamma)=h(X_f, \Gamma)$ \cite{Berg, De06}. Therefore
$$ h_{\Sigma,\lambda}(X_f,\Gamma)=h_\lambda(X_f, \Gamma)=h(X_f, \Gamma)=h_{\Sigma}(X_f,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|.$$ \end{proof}
\begin{lemma} \label{L-lower bound} We have
$$ h_{\Sigma,\lambda}(X_f,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)|.$$ \end{lemma} \begin{proof} If $\Gamma$ has a finite-index normal subgroup isomorphic to ${\mathbb Z}$ or ${\mathbb Z}^2$, then $\Gamma$ is amenable \cite[Theorem G.2.1 and Proposition G.2.2]{BHV}. Thus the assertion follows from Lemmas~\ref{lem:lower} and \ref{L-amenable lower bound}. \end{proof}
\subsection{The upper bound}
Let $U_n$ be the uniform probability measure on $\Gamma/\Gamma_n$. For $x,y\in {\mathbb R}^{\Gamma/\Gamma_n}$, let $\langle x,y\rangle_U$ be the inner product with respect to $U_n$ and $\|x\|_{p,U} := \left([\Gamma: \Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} |x_{s\Gamma_n}|^p\right)^{1/p}$ for $p\ge 1$.
For $x\in ({\mathbb R}/{\mathbb Z})^{\Gamma/\Gamma_n}$, let $x'$ be as in Definition~\ref{defn:rho}. Let $|x|=|x'|$ and $\|x\|_{p,U} = \|x'\|_{p,U}$ for $p\ge 1$.
We will use $\langle \cdot ,\cdot \rangle$ and $\|\cdot\|_p$ to denote the inner product and $\ell^p$-norm with respect to the counting measure.
\begin{lemma} \label{L-diffrent norms} For any $x\in ({\mathbb R}/{\mathbb Z})^{\Gamma/\Gamma_n}$,
$$\|x\|_{2,U}^2 \le \|x \|_{1,U} \le \|x\|_{2,U}\le 1/2.$$ \end{lemma}
\begin{proof}
This first inequality is immediate from $\|x\|_{\infty} \le 1/2$. Note \begin{eqnarray*}
\|x\|_1^2 &=& \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \sum_{t\Gamma_n \in \Gamma/\Gamma_n} |x_{s\Gamma_n}| |x_{t\Gamma_n}| = \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \sum_{t\Gamma_n \in \Gamma/\Gamma_n} |x_{s\Gamma_n}| |x_{ts\Gamma_n}| \\
&=& \sum_{t\Gamma_n \in \Gamma/\Gamma_n} \langle |x|, |x \circ t\Gamma_n|\rangle. \end{eqnarray*} By the Cauchy-Schwarz inequality, for any $t\in \Gamma$,
$$\langle |x|, |x \circ t\Gamma_n|\rangle \le \|x\|_2 \|x \circ t\Gamma_n\|_2 = \|x\|_2^2.$$ Hence
$$\|x\|_1^2 \le [\Gamma:\Gamma_n] \|x\|_2^2.$$
Since $\|x\|_{1,U} = [\Gamma:\Gamma_n]^{-1}\|x\|_1$ and $\|x\|_{2,U}^2 =[\Gamma:\Gamma_n]^{-1} \|x\|_2^2$, this implies the second inequality. The last one follows from $\|x\|_{2,U} \le \|x\|_\infty \le 1/2$. \end{proof}
Recall from Section~\ref{S-notation} that for a countable group $\Gamma'$ and $g\in {\mathbb R}\Gamma'$,
if $g$ is positive in ${\mathcal N}\Gamma'$, then we have the spectral measure of $g$ on
$[0, \|R_g\|]\subset [0, \|g\|_1]$ determined by \eqref{E-spectral measure}. For each $n\in {\mathbb N}$ we denote by $\pi_n$ the natural algebra homomorphism ${\mathbb R}\Gamma\rightarrow {\mathbb R}(\Gamma/\Gamma_n)$.
\begin{lemma} \label{L-bound preimage} Let $g\in {\mathbb R}\Gamma$ such that the kernel of $g$ on $\ell^2(\Gamma, {\mathbb C})$ is $\{0\}$,
and $\pi_n(g)$ is positive in ${\mathcal N}(\Gamma/\Gamma_n)$ for all $n\in {\mathbb N}$. For each $n\in {\mathbb N}$ and $\eta>0$ denote by $B_{n, \eta}$ the set of $x\in {\mathbb R}(\Gamma/\Gamma_n)$ satisfying
$\|x\pi_n(g)\|_{2, U}\le \eta$ and $\|P_n(x)\|_{2, U}\le 1$, where $P_n$ denotes the orthogonal projection from $\ell^2(\Gamma/\Gamma_n, {\mathbb C})$ onto $\ker \pi_n(g)$. For each $n\in {\mathbb N}$ denote by $\mu_n$ the spectral measure of $\pi_n(g)$ on $[0, \|g\|_1]$.
Let $\zeta>1$, $1>\varepsilon>0$, and $1/2>\kappa>0$. Then there exists $\eta>0$ such that when $n\in {\mathbb N}$ is large enough, one has
$$N_\varepsilon(B_{n, \eta}, \|\cdot\|_{2, U})<\zeta^{[\Gamma: \Gamma_n]}\exp(-[\Gamma: \Gamma_n]\int_{0+}^{\kappa}\log t \, d\mu_n(t))$$ where $N_\varepsilon(\cdot,\cdot)$ is as in Definition \ref{defn:separating}. \end{lemma} \begin{proof} Since $\pi_n(g)$ is positive in ${\mathcal N}(\Gamma/\Gamma_n)$, one has $(\pi_n(g))^*=\pi_n(g)$.
Let $Y_n$ be a maximal $(\|\cdot\|_{2, U},\varepsilon/6)$-separated subset of the closed unit ball of $\ker \pi_n(g)$ under $\|\cdot \|_{2, U}$. Then the open $\varepsilon/12$-balls centered at $y$ under $\|\cdot\|_{2, U}$ for all $y\in Y_n$ are pairwise disjoint, and their union is contained in the open $2$-ball of $\ker \pi_n(g)$ under $\|\cdot \|_{2, U}$. Comparing the volumes we obtain $|Y_n|\le (24/\varepsilon)^{\dim_{{\mathbb R}} \ker \pi_n(g)}$.
For each $n\in {\mathbb N}$ denote by $V_{n, \kappa}$ the linear span of the eigenvectors of $\pi_n(g)$ $\ell^2(\Gamma/\Gamma_n, {\mathbb C})$ with eigenvalue no bigger than $\kappa$. Note that $V_{n, 0}=\ker \pi_n(g)$. Since $\ker (g^*g)=\ker(g)=0$, by a result of L\"{u}ck \cite[Theorem 2.3]{Luck94} (it was assumed in \cite{Luck94} that $\Gamma_n\supset \Gamma_{n+1}$ for all $n\in {\mathbb N}$; but the argument there holds in general), one has
$\lim_{n\to \infty}[\Gamma: \Gamma_n]^{-1}\dim_{{\mathbb C}} \ker \pi_n(g)=\lim_{n\to \infty}[\Gamma: \Gamma_n]^{-1}\dim_{{\mathbb C}} \ker \pi_n(g^*g)=0$. It follows that $|Y_n|\le \zeta^{[\Gamma:\Gamma_n]}$ when $n$ is large enough.
Denote by $P_{n, \kappa}$ the orthogonal projection of $\ell^2(\Gamma/\Gamma_n, {\mathbb C})$ onto $V_{n, \kappa}$. Set $\eta=\min(\varepsilon/24, \kappa \varepsilon /12)$. Note that for each $x\in {\mathbb R}(\Gamma/\Gamma_n)$ one has
$$\|x\pi_n(g)\|_{2, U}^2=\|(P_{n, \kappa}(x))\pi_n(g)\|_{2, U}^2+\|(x-P_{n, \kappa}(x))\pi_n(g)\|_{2, U}^2\ge \kappa^2\|x-P_{n, \kappa}(x)\|_{2, U}^2.$$
Thus $\|x-P_{n, \kappa}(x)\|_{2, U}\le \eta/\kappa\le \varepsilon/12$ for every $x\in B_{n, \eta}$.
Then every two points in $({\rm Id}-P_{n, \kappa})(B_{n, \eta})$ have $\|\cdot \|_{2, U}$-distance at most $\varepsilon/6$. Let $X_n$ be a one-point subset of $({\rm Id}-P_{n, \kappa})(B_{n, \eta})$. Then $X_n$ is a maximal $(\varepsilon/6)$-separated subset of $({\rm Id}-P_{n, \kappa})(B_{n, \eta})$ under $\|\cdot \|_{2, U}$.
Denote by $E_{n, \kappa}$ the ordered set of all eigenvalues of $\pi_n(g)$ in $(0, \kappa]$ listed with multiplicity. Let $Z_n$ be a maximal $(\varepsilon/6)$-separated subset of $(P_{n, \kappa}-P_{n, 0})(B_{n, \eta})$ under $\|\cdot \|_{2, U}$. For each $z\in Z_n$ denote by $B_z$ the open ball centered at $z$ with radius $\varepsilon/12$ under $\|\cdot \|_{2, U}$. Note that $\|x\pi_n(g)\|_{2, U}\le \kappa\|x\|_{2, U}$ for all $x\in V_{n, \kappa}\ominus V_{n, 0}$. Thus every element in $(\bigcup_{z\in Z_n}B_z)\pi_n(g)$ has $\|\cdot \|_{2, U}$-norm at most $\eta+\kappa \varepsilon/12$. The volume of $(\bigcup_{z\in Z_n}B_z)\pi_n(g)$ is $\det(\pi_n(g)|_{V_{n, \kappa}\ominus V_{n, 0}})=\prod_{t\in E_{n, \kappa}}t$ times the volume of $\bigcup_{z\in Z_n}B_z$. It follows that
$$ |Z_n| \prod_{t\in E_{n, \kappa}} t\le \left(\frac{\eta+\kappa \varepsilon/12}{\varepsilon/12}\right)^{\dim_{\mathbb R} (V_{n, \kappa}\ominus V_{n, 0})}=\left(\frac{12\eta+\kappa \varepsilon}{\varepsilon}\right)^{\dim_{\mathbb R} (V_{n, \kappa}\ominus V_{n, 0})}\le 1.$$
Note that for every $t\in [0, \|g\|_1]$, the measure $\mu_n(\{t\})$ is exactly $[\Gamma:\Gamma_n]^{-1}$ times the multiplicity of $t$ as an eigenvalue of $\pi_n(g)$. When $n\in {\mathbb N}$ is sufficiently large, we have \begin{align*}
N_{\varepsilon}(B_{n, \eta}, \|\cdot \|_{2, U})&\le |X_n|\cdot |Y_n|\cdot |Z_n| \le \zeta^{[\Gamma: \Gamma_n]}\prod_{t\in E_{n, \kappa}}t^{-1} \\ &=\zeta^{[\Gamma: \Gamma_n]}\exp\left(-[\Gamma: \Gamma_n]\int_{0+}^{\kappa}\log t \, d\mu_n(t)\right) \end{align*} as desired. \end{proof}
\begin{lemma} \label{L-measure weak convergence} Let $g\in {\mathbb R}\Gamma$ be such that
$g$ is positive in ${\mathcal N}\Gamma$, and $\pi_n(g)$ is positive in ${\mathcal N}(\Gamma/\Gamma_n)$ for all $n\in {\mathbb N}$. Denote by $\mu$ the spectral measure of $g$ on $[0, \|g\|_1]$. For each $n\in {\mathbb N}$ denote by $\mu_n$ the spectral measure of $\pi_n(g)$ on $[0, \|g\|_1]$.
Let $\min(1, \|g\|_1)>\kappa>0$. Then
$$ \limsup_{n\to \infty}\int_{\kappa+}^{\|g\|_1}\log t \, d\mu_n(t)\le \int_{\kappa+}^{\|g\|_1}\log t \, d\mu(t).$$ \end{lemma} \begin{proof} It suffices to show
$$ \limsup_{n\to \infty}\int_{\kappa+}^{\|g\|_1}\log t \, d\mu_n(t)\le \eta(1+\|g\|_1)+\int_{\kappa+}^{\|g\|_1}\log t \, d\mu(t).$$ for every $\eta>0$. Let $\eta>0$.
For each sufficiently large $k\in {\mathbb N}$ define a real-valued continuous function $q_k$ on $[0, \|g\|_1]$ to be $0$ on $[0, \kappa]$,
$\log t$ at $t\in [\kappa+1/k, \|g\|_1]$, and linear on $[\kappa, \kappa+1/k]$. By the Lebesgue dominated convergence theorem one has $\int_{\kappa+}^{\|g\|_1} q_k(t)\, d\mu(t)\to \int_{\kappa+}^{\|g\|_1}\log t\, d\mu(t)$ as $k\to \infty$. Fix $k\in {\mathbb N}$ with $\int_{\kappa+}^{\|g\|_1} q_k(t)\, d\mu(t)\le \int_{\kappa+}^{\|g\|_1}\log t\, d\mu(t)+\eta$, and take
a real-coefficients polynomial $p$ such that $q_k+\eta \ge p\ge q_k$ on $[0, \|g\|_1]$. Then $p(t)\ge q_k(t)\ge \log t$ for all $t\in (0, \|g\|_1]$, and \begin{align*}
{\rm tr}_{{\mathcal N}\Gamma}(p(g))&\overset{\eqref{E-spectral measure}}= \int_{0}^{\|g\|_1}p(t) \, d\mu(t) \le \eta \|g\|_1 +\int_{0}^{\|g\|_1}q_k(t) \, d\mu(t) \\
&= \eta \|g\|_1 +\int_{\kappa+}^{\|g\|_1}q_k(t) \, d\mu(t) \le \eta(1+\|g\|_1)+\int_{\kappa+}^{\|g\|_1}\log t \, d\mu(t). \end{align*}
When $n\in {\mathbb N}$ is large enough, one has ${\rm tr}_{{\mathcal N}(\Gamma/\Gamma_n)}(p(\pi_n(g)))={\rm tr}_{{\mathcal N}\Gamma}(p(g))$ \cite[Lemma 2.6]{Luck94}, whence
$$ {\rm tr}_{{\mathcal N}\Gamma}(p(g))={\rm tr}_{{\mathcal N}(\Gamma/\Gamma_n)}(p(\pi_n(g)))\overset{\eqref{E-spectral measure}}=\int_{0}^{\|g\|_1}p(t) \, d\mu_n(t)\ge \int_{\kappa+}^{\|g\|_1}p(t) \, d\mu_n(t)\ge \int_{\kappa+}^{\|g\|_1}\log t \, d\mu_n(t) .$$ Thus \begin{align*}
\limsup_{n\to \infty}\int_{\kappa+}^{\|g\|_1}\log t \, d\mu_n(t)\le {\rm tr}_{{\mathcal N}\Gamma}(p(g))\le \eta(1+\|g\|_1)+\int_{\kappa+}^{\|g\|_1}\log t \, d\mu(t) \end{align*} as desired. \end{proof}
\begin{lemma} \label{L-upper bound} We have $$h_{\Sigma}(X_f,\Gamma) \le \log {\rm det}_{{\mathcal N}\Gamma} f.$$ \end{lemma} \begin{proof} Let $\rho$ be the pseudo-metric on $X_f$ defined as in Definition \ref{defn:rho}. Let $\zeta>1$ and $1/2>\kappa>0$. Let $1>\varepsilon>0$. Denote by $W$ the support of $f$. An argument similar to that in the proof of Lemma~\ref{lem:well-known} shows that the kernel of $f$ on $\ell^2(\Gamma, {\mathbb C})$ is $\{0\}$. Note that $f\ge 0$ in ${\mathcal N}\Gamma$ and $\pi_n(f)\ge 0$ in ${\mathcal N}(\Gamma/\Gamma_n)$ for all $n\in {\mathbb N}$. Take $\eta>0$ in Lemma~\ref{L-bound preimage} for $g=f$. Take $\delta>0$ such that
$2\|f\|_2^{1/2}|W|^{1/4}\delta^{1/2}<\eta$.
For $\phi \in {\rm Map}(W,\delta,\Gamma_n)$ define $y_\phi \in ({\mathbb R}/{\mathbb Z})^{\Gamma/\Gamma_n}$ by $y_\phi(s\Gamma_n) = \phi(s^{-1}\Gamma_n)_e$. Note that \begin{eqnarray*}
\|y_\phi \pi_n(f)\|_{2,U}^2&=&[\Gamma:\Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \left| \sum_{t\in W} y_\phi(st\Gamma_n) f(t^{-1})\right|^2 \\
&=& [\Gamma:\Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \left| \sum_{t\in W} \phi(t^{-1}s^{-1}\Gamma_n)_e f(t^{-1})\right|^2 \\
&=& [\Gamma:\Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \left| \sum_{t\in W} \left(\phi(t^{-1}s^{-1}\Gamma_n)_e - [t^{-1}\phi(s^{-1}\Gamma_n)]_e + [t^{-1}\phi(s^{-1}\Gamma_n)]_e\right) f(t^{-1})\right|^2 \\
&=& [\Gamma:\Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \left| \sum_{t\in W} \left(\phi(t^{-1}s^{-1}\Gamma_n)_e - [t^{-1}\phi(s^{-1}\Gamma_n)]_e \right) f(t^{-1})\right|^2 \\
&\le& [\Gamma:\Gamma_n]^{-1} \sum_{s\Gamma_n \in \Gamma/\Gamma_n} \left( \sum_{t\in W} \left|\phi(t^{-1}s^{-1}\Gamma_n)_e - [t^{-1}\phi(s^{-1}\Gamma_n)]_e \right|^2 \right) \|f \|_2^2 \\
&=& \|f\|_2^2 \sum_{t \in W} \rho_2(\phi \circ t^{-1}, t^{-1} \circ \phi)^2\\
&\le& \|f\|_2^2 |W| \delta^2. \end{eqnarray*}
For each $\phi\in {\rm Map}(W,\delta,\Gamma_n)$ take $\tilde{y}_\phi \in [-1/2, 1/2)^{\Gamma/\Gamma_n}$ such that $\tilde{y}_\phi + {\mathbb Z}^{\Gamma/\Gamma_n} =y_\phi$. Then there exists $z_\phi\in {\mathbb Z}^{\Gamma/\Gamma_n}$ such that $\tilde{y}_\phi\pi_n(f) - z_\phi \in [-1/2,1/2)^{\Gamma/\Gamma_n}$ which implies $\|\tilde{y}_\phi \pi_n(f)-z_\phi\|_{2,U} = \|y_\phi \pi_n(f)\|_{2,U}\le \|f\|_2 |W|^{1/2} \delta$.
Let $S_n:{\mathbb R}^{\Gamma/\Gamma_n} \to {\mathbb R}$ be the sum function: $S_n(y) = \sum_{s\Gamma_n \in \Gamma/\Gamma_n} y_{s\Gamma_n}$. Note that
$$|S_n(\tilde{y}_\phi \pi_n(f)-z_\phi)| \le \|y_\phi \pi_n(f)\|_1 \le (1/2) [\Gamma:\Gamma_n].$$
Note that $S_n(y\pi_n(f))=0$ for every $y \in {\mathbb R}^{\Gamma/\Gamma_n}$. In particular, $S_n(\tilde{y}_\phi\pi_n(f)-z_\phi)=-S_n(z_\phi) \in {\mathbb Z}$. So there exists $z'_\phi \in \{-1,0,1\}^{\Gamma/\Gamma_n}$ such that $S_n(\tilde{y}_\phi\pi_n(f)-z_\phi-z'_\phi)=0$ and $\|z'_\phi\|_1 \le \|y_\phi \pi_n(f)\|_1$. So by Lemma~\ref{L-diffrent norms} \begin{eqnarray*}
\| \tilde{y}_\phi\pi_n(f)-z_\phi-z'_\phi\|_{2,U} &\le& \|\tilde{y}_\phi \pi_n(f) - z_\phi\|_{2,U} + \|z'_\phi\|_{2,U} \\
&\le& \|\tilde{y}_\phi \pi_n(f) - z_\phi\|_{2,U} + \|z'_\phi\|_{1,U}^{1/2}\\
&\le& \|y_\phi \pi_n(f)\|_{2,U} +\|y_\phi \pi_n(f)\|_{2,U}^{1/2} \\
&\le& (1+2^{-1/2})\|y_\phi \pi_n(f)\|_{2,U}^{1/2} \\
&\le& 2\|f\|_2^{1/2}|W|^{1/4}\delta^{1/2}<\eta.
\end{eqnarray*}
Note that $\ker \pi_n(f)$ is the constants in $\ell^2(\Gamma/\Gamma_n, {\mathbb C})$. Denote by $\ell^2_0(\Gamma/\Gamma_n, {\mathbb C})$ the orthogonal complement of
the constants in $ \ell^2(\Gamma/\Gamma_n, {\mathbb C})$, and set $\ell^2_0(\Gamma/\Gamma_n, {\mathbb R})=\ell^2(\Gamma/\Gamma_n, {\mathbb R})\cap \ell^2_0(\Gamma/\Gamma_n, {\mathbb C})$.
Note that $y\in {\mathbb R}^{\Gamma/\Gamma_n}$ is in $\ell^2_0(\Gamma/\Gamma_n, {\mathbb R})$ exactly when $S_n(y)=0$. The operator $\pi_n(f)$ is invertible as an operator from $\ell_0^2(\Gamma/\Gamma_n, {\mathbb C})$ to itself. Since $\pi_n(f)$ preserves $\ell^2(\Gamma/\Gamma_n, {\mathbb R})$, it is also invertible from $\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$ to itself. Therefore there exists $\tilde{x}_\phi\in \ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$ such that $\tilde{x}_\phi\pi_n(f)=z_\phi+z'_\phi$. Let $P_n$ and $B_{n, \eta}$ be as in Lemma~\ref{L-bound preimage} for $g=f$. Then
$$ \|(\tilde{y}_\phi-\tilde{x}_\phi)\pi_n(f)\|_{2, U}=\| \tilde{y}_\phi\pi_n(f)-z_\phi-z'_\phi\|_{2,U}<\eta.$$
Note that $\|P_n(\tilde{y}_\phi-\tilde{x}_\phi)\|_{2, U}=\|P_n(\tilde{y}_\phi)\|_{2, U}\le \|\tilde{y}_\phi\|_{2, U}\le 1/2$. Therefore $\tilde{y}_\phi-\tilde{x}_\phi\in B_{n, \eta}$.
Let $\Phi_n$ be a $(\rho_2, 2\varepsilon)$-separated subset of ${\rm Map}(W,\delta,\Gamma_n)$ with $|\Phi_n|=N_{2\varepsilon}({\rm Map}(W,\delta,\Gamma_n), \rho_2)$.
Let $\phi \in \Phi_n$. Denote by $B_\phi$ the set of all $\psi\in \Phi_n$ satisfying
$\|(\tilde{y}_\phi-\tilde{x}_\phi)-(\tilde{y}_\psi-\tilde{x}_\psi)\|_{2, U}<\varepsilon$. Denote ${\mathbb Z}(\Gamma/\Gamma_n)\cap \ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$ by ${\mathbb Z}_0(\Gamma/\Gamma_n)$. We claim that the map $B_\phi\rightarrow {\mathbb Z}_0(\Gamma/\Gamma_n)/{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)$ sending $\psi$ to $z_\psi+z'_\psi+{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)$ is injective. Let $\psi, \varphi\in B_\phi$. Then \begin{align*}
\|(\tilde{y}_\psi-\tilde{x}_\psi)-(\tilde{y}_\varphi-\tilde{x}_\varphi)\|_{2, U}<2\varepsilon. \end{align*} Suppose that $z_\psi+z'_\psi+{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)=z_\varphi+z'_\varphi+{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)$. Then $\tilde{x}_\psi\pi_n(f)=\tilde{x}_\varphi\pi_n(f)+w\pi_n(f)$ for some $w\in {\mathbb Z}_0(\Gamma/\Gamma_n)$. Since the right multiplication by $\pi_n(f)$ is injective on $\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$, we get $\tilde{x}_\psi=\tilde{x}_\varphi+w$, which implies that
$$\rho_2(\psi, \varphi)=\|y_\psi-y_\varphi\|_{2, U}\le \|(\tilde{y}_\psi-\tilde{x}_\psi)-(\tilde{y}_\varphi-\tilde{x}_\varphi)\|_{2, U}<2\varepsilon,$$ and thus $\psi=\varphi$. This proves our claim.
Therefore $|B_\phi|\le |{\mathbb Z}_0(\Gamma/\Gamma_n)/{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)|$. Note that $\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$ is the linear span of ${\mathbb Z}_0(\Gamma/\Gamma_n)$. So any basis of ${\mathbb Z}_0(\Gamma/\Gamma_n)$ as a free abelian group is also a basis for $\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})$ as an ${\mathbb R}$-vector space.
Thus by Lemma~\ref{lem:correspondence}
one has
$|{\mathbb Z}_0(\Gamma/\Gamma_n)/{\mathbb Z}_0(\Gamma/\Gamma_n)\pi_n(f)|=|\det (\pi_n(f)|_{\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})})|$. Therefore
$$|B_\phi|\le |\det (\pi_n(f)|_{\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})})|.$$
Now we have \begin{align*}
|\Phi_n|\le N_\varepsilon(B_{n, \eta}, \|\cdot \|_{2, U}) \max_{\phi\in \Phi_n}|B_\phi|\le N_\varepsilon(B_{n, \eta}, \|\cdot \|_{2, U}) |\det (\pi_n(f)|_{\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})})|. \end{align*} Let $\mu$ and $\mu_n$ be as in Lemma~\ref{L-measure weak convergence} for $g=f$. When $n\in {\mathbb N}$ is large enough, by Lemma~\ref{L-bound preimage} we have
$$ N_\varepsilon(B_{n, \eta}, \|\cdot\|_{2, U})<\zeta^{[\Gamma: \Gamma_n]}\exp\left(-[\Gamma: \Gamma_n]\int_{0+}^{\kappa}\log t \, d\mu_n(t)\right),$$ whence \begin{align*}
N_{2\varepsilon}({\rm Map}(W,\delta,\Gamma_n), \rho_2)
&=|\Phi_n| \\
&\le \zeta^{[\Gamma: \Gamma_n]}\exp\left(-[\Gamma: \Gamma_n]\int_{0+}^{\kappa}\log t \, d\mu_n(t)\right)\left|\det (\pi_n(f)|_{\ell_0^2(\Gamma/\Gamma_n, {\mathbb R})})\right| \\
&=\zeta^{[\Gamma: \Gamma_n]}\exp\left([\Gamma: \Gamma_n]\int_{\kappa+}^{\|f\|_1}\log t \, d\mu_n(t)\right). \end{align*} It follows that \begin{align*}
\limsup_{n\to \infty}\frac{1}{[\Gamma: \Gamma_n]}\log N_{2\varepsilon}({\rm Map}(W,\delta,\Gamma_n), \rho)&\le \log \zeta+\limsup_{n\to \infty}\int_{\kappa+}^{\|f\|_1}\log t \, d\mu_n(t) \\
&\le \log \zeta+\int_{\kappa+}^{\|f\|_1}\log t \, d\mu(t), \end{align*} where the second inequality comes from Lemma~\ref{L-measure weak convergence}. Therefore
$$ h_{\Sigma}(X_f,\Gamma) \le \log \zeta+\int_{\kappa+}^{\|f\|_1}\log t \, d\mu(t).$$ Letting $\zeta\to 1+$ and $\kappa\to 0+$, we get
$$ h_{\Sigma}(X_f,\Gamma) \le \int_{0+}^{\|f\|_1}\log t \, d\mu(t)\overset{\eqref{E-determinant}}=\log {\rm det}_{{\mathcal N}\Gamma} f.$$ \end{proof}
We are ready to prove Theorem~\ref{thm:main1}.
\begin{proof}[Proof of Theorem~\ref{thm:main1}] From Lemmas \ref{L-lower bound} and \ref{L-upper bound} and Theorem~\ref{T-det vs number of fixed point} we obtain
$$h_{\Sigma}(X_f,\Gamma) \le \log {\rm det}_{{\mathcal N}\Gamma} f=\lim_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log |{\rm Fix}_{\Gamma_n}(X_f)| \le h_{\Sigma,\lambda}(X_f,\Gamma).$$ It follows immediately from Theorem \ref{thm:entropy} that $h_{\Sigma}(X_f,\Gamma) \ge h_{\Sigma,\lambda}(X_f,\Gamma).$ \end{proof}
\section{Entropy of the Wired Spanning Forest}
The purpose of this section is to prove Theorem \ref{thm:WSF}. To begin, let us set notation. Recall that $\Sigma=\{\Gamma_n\}^\infty_{n=1}$ a sequence of finite-index normal subgroups of $\Gamma$ satisfying $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$. All graphs in this paper are allowed to have multiple edges and loops. Let $f \in {\mathbb Z}\Gamma$ be well-balanced. The Cayley graph $C(\Gamma,f)$ has vertex set $\Gamma$. For each $v \in \Gamma$ and $s \ne e$, there are $|f_s|$ edges from $v$ to $vs$. Similarly, we let $C_n^f=C(\Gamma/\Gamma_n,f)$ be the graph with vertex set $\Gamma/\Gamma_n$ such that for each $g\Gamma_n \in \Gamma/\Gamma_n$ and $s \in \Gamma$ there are $|f_s|$ edges from $g\Gamma_n$ to $gs\Gamma_n$. For the sake of convenience we let $E=E(\Gamma,f)$ denote the edge set of $C(\Gamma,f)$ and $E_n=E_n^f$ denote the edge set of $C_n^f$. Recall the definition of $S$ and $S_*$ from Notation \ref{note:S}.
Let $\pi_n:\Gamma \to \Gamma/\Gamma_n$ denote the quotient map. We also denote by $\pi_n$ the induced map from ${\mathbb R}\Gamma$ to ${\mathbb R}\Gamma/\Gamma_n$ as well as the map from $E(\Gamma,f)$ (the edge set of $C(\Gamma,f)$) to $E_n^f$ (the edge set of $C_n^f$).
\subsection{The lower bound}
If ${\mathcal H} \subset C_n^f$ is a subgraph then its {\em lift} $\tilde{{\mathcal H}} \subset C(\Gamma,f)$ is the subgraph which contains an edge $gs$ (for $g\in \Gamma, s\in S_*$) if and only if ${\mathcal H}$ contains $\pi_n(gs)$. Let $2^E$ be the set of all spanning subgraphs of $C(\Gamma,f)$ and let $2^{E_n}$ be the set of all spanning subgraphs of $C_n^f$. Let $\nu_n$ be the probability measure on $2^{E_n}$ which is uniformly distributed on the collection of spanning trees of $C_n^f$. Let $\tilde{\nu}_n$ be its lift to $2^E$. To be precise, $\tilde{\nu}$ is uniformly distributed on the set of lifts $\tilde{{\mathcal T}}$ of spanning trees ${\mathcal T} \in 2^{E_n}$.
\begin{lemma}\label{lem:WSF2} $\tilde{\nu}_n$ converges in the weak* topology to $\nu_{WSF}$ as $n$ tends to infinity. \end{lemma}
\begin{remark} This lemma is a special case of \cite[Proposition 7.1]{AL07}. It is also contained in \cite[Theorem 4.3]{Bo04}. However, there is an error in the proof of \cite[Theorem 4.3]{Bo04} (namely, the fact that subspaces $S_i$ increase to $l^2_-(\Gamma)$ does not logically imply that $P_{S_i}(\star)$ converges to $\star$ in the strong operator topology). For the reader's convenience we provide another proof based on a negative correlations result of Feder and Mihail. \end{remark}
Let ${\mathcal G}=(V^{\mathcal G},E^{\mathcal G})$ be a finite connected graph. A collection ${\mathcal A}$ of spanning subgraphs is {\em increasing} if $x \subset y$ and $x\in {\mathcal A}$ implies $y\in {\mathcal A}$. We say that ${\mathcal A}$ {\em ignores} an edge ${\mathfrak e}$ if $x \setminus \{{\mathfrak e}\} = y \setminus \{{\mathfrak e}\}$ and $x \in {\mathcal A}$ implies $y \in {\mathcal A}$.
\begin{lemma} If ${\mathcal A}$ is increasing, ${\mathcal A}$ ignores ${\mathfrak e}$, ${\mathfrak e}$ is not a loop and $T$ denotes the uniform spanning tree on ${\mathcal G}$ then
$${\bf P}(T \in {\mathcal A}) \ge {\bf P}(T \in {\mathcal A}| {\mathfrak e} \in T) = \frac{{\bf P}(T \in {\mathcal A}, {\mathfrak e} \in T) }{{\bf P}({\mathfrak e} \in T) }$$
where ${\bf P}(\cdot)$ denotes probability. Equivalently, ${\bf P}(T \in {\mathcal A}) \le {\bf P}(T \in {\mathcal A}| {\mathfrak e} \notin T)$ whenever this is well-defined (i.e., whenever ${\bf P}({\mathfrak e} \notin T)>0$, or equivalently, whenever ${\mathcal G}\setminus \{{\mathfrak e}\}$ is connected). \end{lemma}
\begin{proof} This result is due to Feder and Mihail \cite{FM92}. The first statement is reproduced in \cite[Theorem 4.4]{BLPS01}. To see that the second inequality is equivalent to the first observe that
$$ {\bf P}(T \in {\mathcal A}| {\mathfrak e} \notin T) = \frac{ {\bf P}(T \in {\mathcal A}, {\mathfrak e}\notin T) }{{\bf P}({\mathfrak e} \notin T)} = \frac{ {\bf P}(T \in {\mathcal A}) - {\bf P}(T \in {\mathcal A}, {\mathfrak e}\in T) }{1-{\bf P}({\mathfrak e} \in T)}.$$
By multiplying denominators, we see that ${\bf P}(T \in {\mathcal A}) \le {\bf P}(T \in {\mathcal A}| {\mathfrak e} \notin T)$ if and only if $$ {\bf P}(T \in {\mathcal A}) - {\bf P}(T \in {\mathcal A}, {\mathfrak e}\in T) \ge {\bf P}(T \in {\mathcal A}) - {\bf P}(T \in {\mathcal A}){\bf P}({\mathfrak e} \in T)$$
which simplifies to ${\bf P}(T \in {\mathcal A}) \ge {\bf P}(T \in {\mathcal A}| {\mathfrak e} \in T)$. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:WSF2}]
For $n\ge 0$, let $B(n)$ denote the ball of radius $n$ centered at the identity element in $C(\Gamma,f)$. For each $n$, choose a non-negative integer $i_n$ so that the following hold. \begin{enumerate} \item $\lim_{n\to\infty} i_n \to \infty$. \item The quotient map $\pi_n$ restricted to $B(i_n)$ is injective but not surjective. Moreover, if $v,w$ are vertices in $B(i_n)$ then the number of edges in $C^f_n$ from $v\Gamma_n$ to $w\Gamma_n$ equals the number of edges in $B(i_n)$ from $v$ to $w$.
\end{enumerate} Because $\bigcap_{n=1}^\infty \bigcup_{i \ge n} \Gamma_i = \{e\}$, it is possible to find such a sequence.
Let $C^w_n$ denote the graph $C_n^f$ with all the vertices outside of $B(i_n)\Gamma_n$ contracted together. To be precise, $C^w_n$ has vertex set $B(i_n)\Gamma_n \cup \{*\}$. Every edge in $C_n^f$ with endpoints in $B(i_n)\Gamma_n$ is also in $C^w_n$. For every edge in $C_n^f$ with one endpoint $v$ in $B(i_n)\Gamma_n$ and the other endpoint not in $B(i_n)\Gamma_n$, there is an edge in $C^w_n$ from $v$ to $*$.
Similarly, let $D^w_n$ denote the graph $C(\Gamma,S)$ with all the vertices outside of $B(i_n)$ contracted together. By the choice of $i_n$, $D^w_n$ is isomorphic to $C^w_n$. Let $\nu^{C,w}_n$ be the law of the uniform spanning tree on $C^w_n$, $\nu^{D,w}_n$ be the law of the uniform spanning tree on $D^w_n$ and $\nu_n$ be the law of the uniform spanning tree on $C_n^f$.
Let ${\mathcal A} \subset 2^E$ be an increasing set which depends on only a finite number of edges (i.e., there is a finite subset $F \subset E$ such that if $x, y \in 2^E$ and $x \cap F = y \cap F$ then $x \in {\mathcal A} \Leftrightarrow y \in {\mathcal A}$). If $n$ is sufficiently large, then $F \subset B(i_n)$. So we define ${\mathcal A}_n \subset 2^{E_n}$ by: $x\in {\mathcal A}_n \Leftrightarrow \exists y \in {\mathcal A}$ such that $\pi_n(y \cap F) = x \cap \pi_n(F)$. By abuse of notation, we also consider ${\mathcal A}_n$ to be a subset of the set of edges of $C^w_n$.
Because $C^w_n$ is obtained from $C_n^f$ by adding some edges and contracting some edges, repeated applications of the previous lemma imply $\nu^{C,w}_n({\mathcal A}_n) \le \nu_n({\mathcal A}_n)$. By definition, $\nu^{C,w}_n({\mathcal A}_n) = \nu^{D,w}_n({\mathcal A})$ and $\nu_n({\mathcal A}_n) = \tilde{\nu}_n({\mathcal A})$. Thus, $$ \nu^{D,w}_n({\mathcal A}) \le \tilde{\nu}_n({\mathcal A}).$$
Let $E(i_n)$ denote the set of edges in the ball $B(i_n)$. We consider $2^{E(i_n)}$, the set of all subsets of $E(i_n)$, to be included in $2^E$, the set of all subsets of $E$, in the obvious way. By definition of the Wired Spanning Forest, the projection of $\nu^{D,w}_n$ to $2^{E(i_n)} \subset 2^E$ converges to $\nu_{WSF}$ in the weak* topology. So if $\tilde{\nu}_\infty$ is a weak* limit point of $\{\tilde{\nu}_n\}_{n=1}^\infty$ then we have $$\nu_{WSF}( {\mathcal A}) \le \tilde{\nu}_\infty( {\mathcal A})$$
for every increasing ${\mathcal A} \subset 2^E$ which depends on only a finite number of edges. This means that, for any finite subset $F \subset E$, the projection of $\nu_{WSF}$ to $2^F$, denoted $\nu_{WSF}|2^F$, is stochastically dominated by $\tilde{\nu}_\infty|2^F$. By Strassen's theorem \cite{St65}, there exists a probability measure $J_F$ on $$\{ (x,y) \in 2^F \times 2^F :~ x \subset y\}$$
with marginals $\nu_{WSF}|{2^F}$ and $\tilde{\nu}_\infty|{2^F}$ respectively. By taking a weak* limit point of $\{J_F\}_{F \subset E}$ as $F$ increases to $E$, we obtain the existence of a Borel probability measure $J$ on $$\{ (x,y) \in 2^E \times 2^E :~ x \subset y\}$$ with marginals $\nu_{WSF}$ and $\tilde{\nu}_\infty$ respectively.
Observe that the average degree of a vertex in the WSF is 2. To put it more formally, for every $g\in \Gamma$, let $\deg_g:2^E \to {\mathbb Z}$ be the map $\deg_g(x)$ equals the number of edges in $x$ adjacent to $g$. By \cite[Theorem 6.4]{BLPS01}, $\int \deg_g(x)~d\nu_{WSF}(x)=2$. Also $\int \deg_g(x)~d\tilde{\nu}_\infty(x)=2$. This can be seen as follows. Because $\tilde{\nu}_n$ is $\Gamma$-invariant, it follows that $\int \deg_g(x)~d\tilde{\nu}_{n}(x)$ is just the average degree of a vertex in a uniformly random spanning tree of $C_n^f$. However, each such tree has $[\Gamma:\Gamma_n]$ vertices and $[\Gamma:\Gamma_n]-1$ edges and therefore, the average degree is $2([\Gamma:\Gamma_n]-1)[\Gamma:\Gamma_n]^{-1}$ which converges to $2$ as $n\to\infty$.
Because $\int \deg_g(x)~d\nu_{WSF}(x)=\int \deg_g(x)~d\tilde{\nu}_\infty(x)$ for every $g\in \Gamma$, it follows that $J$ is supported on $\{(x,x):~x \in 2^E\}$. Thus $\tilde{\nu}_\infty = \nu_{WSF}$ as claimed. \end{proof}
For $x \in 2^E$, let $x_1$ denote the restriction of $x$ to the set of all edges containing the identity element. Let $\rho$ be the continuous pseudo-metric on $2^E$ defined by $\rho(x,y) = 1$ if $x_1\ne y_1$ and $\rho(x,y)=0$ otherwise. This pseudo-metric is dynamically generating.
For $x\in 2^{E_n}$, let $\phi_x: \Gamma/\Gamma_n \to 2^E$ be the map $\phi_x(g\Gamma_n) = \widetilde{gx}$. Let $W \subset \Gamma, L \subset C(2^E)$ be non-empty finite sets and $\delta>0$. Let ${\rm BAD}(W,L,\delta,\Gamma_n)$ be the set of all $x\in 2^{E_n}$ such that $\phi_x \notin {\rm Map}_{\nu_{WSF}}(W,L,\delta,\Gamma_n)$.
\begin{lemma}\label{lem:BAD2} $$\lim_{n\to\infty} \nu_n({\rm BAD}(W,L,\delta,\Gamma_n)) = 0.$$ \end{lemma} \begin{proof} The proof is similar to the proof of Lemma \ref{lem:BAD1}. It uses Lemma \ref{lem:WSF2} above and the fact that $\Gamma {\curvearrowright} (2^E,\nu_{WSF})$ is ergodic by \cite[Corollary 8.2]{BLPS01}. \end{proof}
\begin{lemma}\label{lem:lower2} $h_{\Sigma,\nu_{WSF}}(2^E,\Gamma) \ge \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f).$ \end{lemma}
\begin{proof} Let $W \subset \Gamma, L \subset C(2^E)$ be non-empty finite sets and $\delta>0$. Denote by $Y_n$ the set of spanning trees in $C_n^f$ not contained in ${\rm BAD}(W,L,\delta,\Gamma_n)$.
By the previous lemma, $\lim_{n\to \infty} |Y_n|^{-1} \tau(G_n) = 1$. By definition of $\rho_\infty$, if $x\ne y \in Y_n$ then $\rho_\infty(\phi_x,\phi_y)= 1$. Therefore, if $0<\epsilon<1$ then $\{\phi_y:~y\in Y_n\}$ is $\epsilon$-separated with respect to $\rho_\infty$ which implies
$$ N_\epsilon( {\rm Map}_{\nu_{WSF}}(W,L,\delta,\Gamma_n), \rho_\infty) \ge |Y_n|.$$
Because $\lim_{n\to \infty} |Y_n|^{-1}\tau(C_n^f) = 1$, this implies $$\limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f) \le h_{\Sigma,\nu_{WSF}}(2^E,\Gamma).$$ \end{proof}
\subsection{A topological model}
Recall the definition of ${\mathcal F}={\mathcal F}_f$ from the introduction. It is a closed $\Gamma$-invariant subset of $S_*^\Gamma$. We refer the reader to \cite[Chapter I.8]{BH} for background about the end space of a topological space.
\begin{lemma}\label{lem:model} If $\Gamma$ is not virtually ${\mathbb Z}$ then $h_{\Sigma,\nu_{WSF}}(2^E,\Gamma) \le h_\Sigma({\mathcal F},\Gamma)$. \end{lemma}
\begin{proof}
Because $\Gamma$ is not virtually cyclic, \cite[Theorem 10.1]{BLPS01} implies that for $\nu_{WSF}$-a.e. $x \in 2^E$, every component of $x$ is a 1-ended tree. Therefore, given such an $x$, for every $g\in \Gamma$ there is a unique edge $s_*\in S_*$ such that $gs_* \in x$ and if $C(x,g)$ is the connected component of $x$ containing $g$ then $C(x,g) \setminus \{gs_*\}$ has two components: a finite one containing $g$ and an infinite one containing $gs \in \Gamma$ (where $s=p(s_*) \in S$). Informally, $gs \in \Gamma$ is closer to the point at infinity of $C(x,g)$ than $g$ is. Let $\Phi(x)\in S_*^\Gamma$ be defined by $\Phi(x)_g=s_*$. Also let $\nu_{{\mathcal F}} = (\Phi_*)\nu_{WSF}$. Because $\nu_{WSF}$-a.e. $x\in 2^E$ is such that every component of $x$ is a $1$-ended tree, it follows that $\nu_{\mathcal F}$ is supported on ${\mathcal F}$. The random oriented subgraph with law $\nu_{\mathcal F}$ is called the {\em Oriented Wired Spanning Forest} (OWSF) in \cite{BLPS01}.
Note that $\Phi$ induces a measure-conjugacy from the action $\Gamma {\curvearrowright} (2^E, \nu_{WSF})$ to the action $\Gamma {\curvearrowright} ({\mathcal F},\nu_{\mathcal F})$. Thus $h_{\Sigma,\nu_{WSF}}(2^E,\Gamma) =h_{\Sigma,\nu_{{\mathcal F}}}({\mathcal F},\Gamma)$. Theorem \ref{thm:entropy} now implies the lemma. \end{proof}
\begin{lemma}\label{lem:model2} If $\Gamma$ is virtually ${\mathbb Z}$ then $h_{\Sigma,\nu_{WSF}}(2^E,\Gamma) \le h_\Sigma({\mathcal F},\Gamma)$. \end{lemma}
\begin{proof}
Let ${\rm{Ends}}(\Gamma)$ denote the space of ends of $C(\Gamma,f)$. Because $\Gamma$ is 2-ended, $|{\rm{Ends}}(\Gamma)|=2$. Let $x\in 2^E$ be connected and denote by ${\rm{Ends}}(x)$ the space of ends of $x$. Endow each edge in $C(\Gamma, f)$ with length $1$. Note that for any $g\in \Gamma$ and $r>0$ there exists $r'>0$ such that if $g'\in \Gamma$ has geodesic distance at least $r'$ from $g$ in $x$, then $g'$ has geodesic distance at least $r$ from $g$ in $C(\Gamma, f)$. Thus the argument in the proof of \cite[Proposition I.8.29]{BH} shows that the inclusion map of $x$ into $C(\Gamma,f)$ induces a map $\phi_x:{\rm{Ends}}(x) \to {\rm{Ends}}(\Gamma)$. We claim that this is a surjection.
Let $K \subset \Gamma$ be a finite set such that $C(\Gamma,f)\setminus K$ has two infinite components ${\mathcal C}_0,{\mathcal C}_1$ corresponding to the two ends $\eta_0,\eta_1$ of $C(\Gamma,f)$.
For each $i=0, 1$, define a subgraph $x|{\mathcal C}_i$ of $C(\Gamma, f)$ as follows: it has the same vertices as ${\mathcal C}_i$ does, and an edge $e$ in $E$ lies in $x|{\mathcal C}_i$ exactly when $e$ is in both $x$ and ${\mathcal C}_i$.
Because $x$ is connected, each component of $x|{{\mathcal C}_i}$ contains an element of $KS$. Since $KS$ is finite, this implies that at least one of the components of $x|{\mathcal C}_i$ must be infinite. Then any proper ray in this infinite component of $x|{\mathcal C}_i$ gives rise to an end $\omega_i$ of $x$, and also gives rise to an end of $C(\Gamma, f)$, which must be $\eta_i$.
It follows that $\phi_x(\omega_i)=\eta_i$. Because $i$ is arbitrary, $\phi_x$ is surjective as claimed.
By the claim, if $x \in 2^E$ is connected and 2-ended, we may identify ${\rm{Ends}}(x)$ with ${\rm{Ends}}(\Gamma)$ via the map $\phi_x$.
Given $(x,\eta) \in 2^E \times {\rm{Ends}}(\Gamma)$ with the property that $x$ is a 2-ended tree, we define $\Phi(x,\eta) \in {\mathcal F}$ as follows. For each $g \in \Gamma$, let $s_*\in S_*$ be the unique edge so that $x \setminus \{gs_*\}$ has two components: one containing $g$ (which is either finite or infinite with an end not equal to $\eta$), the other containing $gp(s_*)$ and having an end equal to $\eta$. Informally, $gp(s_*) \in \Gamma$ is ``closer'' to $\eta$ than $g$ is. Let $\Phi(x,\eta)\in S_*^\Gamma$ be defined by $\Phi(x,\eta)_g=s_*$. Clearly $\Phi(x, \eta)\in {\mathcal F}$.
Let $\zeta$ be the uniform probability measure on ${\rm{Ends}}(\Gamma)$. By \cite[Theorems 10.1 and 10.4]{BLPS01}, the WSF on $C(\Gamma,f)$ is a.s. a 2-ended tree. So $\nu_{{\mathcal F}} = (\Phi_*)(\nu_{WSF} \times \zeta)$ is well-defined.
The action of $\Gamma$ on $C(\Gamma,f)$ naturally extends to ${\rm{Ends}}(\Gamma)$. Note that $\Phi$ induces a measure-conjugacy from the action $\Gamma {\curvearrowright} (2^E \times {\rm{Ends}}(\Gamma), \nu_{WSF} \times \zeta)$ to the action $\Gamma {\curvearrowright} ({\mathcal F},\nu_{\mathcal F})$. By Theorem \ref{thm:entropy}, $$h_{\Sigma,\nu_{WSF} \times \zeta}(2^E \times {\rm{Ends}}(\Gamma),\Gamma) =h_{\Sigma,\nu_{{\mathcal F}}}({\mathcal F},\Gamma) \le h_\Sigma({\mathcal F},\Gamma).$$
Because $\Gamma$ is virtually ${\mathbb Z}$, it is amenable \cite[Theorem G.2.1 and Proposition G.2.2]{BHV}. Thus by \cite[Theorem 1.2]{Bo12} \cite[Theorem 6.7]{KL11b}, $h_{\Sigma,\nu_{WSF} \times \zeta}(2^E \times {\rm{Ends}}(\Gamma),\Gamma)$ is the classical entropy of the action, denoted by $h_{\nu_{WSF} \times \zeta}(2^E \times {\rm{Ends}}(\Gamma),\Gamma)$. It is well-known that classical entropy is additive under direct products. Thus, \begin{eqnarray*} h_{\nu_{WSF} \times \zeta}(2^E \times {\rm{Ends}}(\Gamma),\Gamma) &=& h_{\nu_{WSF}}(2^E,\Gamma) + h_{\zeta}({\rm{Ends}}(\Gamma),\Gamma)= h_{\nu_{WSF}}(2^E,\Gamma). \end{eqnarray*} The last equality holds because ${\rm{Ends}}(\Gamma)$ is a finite set. By \cite[Theorem 1.2]{Bo12} \cite[Theorem 6.7]{KL11b} again, $h_{\nu_{WSF}}(2^E,\Gamma) = h_{\Sigma,\nu_{WSF}}(2^E,\Gamma)$. Thus $$h_{\Sigma,\nu_{WSF}}(2^E,\Gamma)=h_{\Sigma,\nu_{WSF} \times \zeta}(2^E \times {\rm{Ends}}(\Gamma),\Gamma) \le h_\Sigma({\mathcal F},\Gamma).$$ \end{proof}
\subsection{The upper bound}
Given $s_* \in S_*$, let ${\vec{s}}_*$ denote the {\em oriented} edge from $e$ to $p(s_*)$. For $x,y \in {\mathcal F}$, let $\rho^{\mathcal F}(x,y) = 1$ if $x_e \ne y_e$. Let $\rho^{\mathcal F}(x,y)=0$ otherwise. This is a dynamically generating continuous pseudo-metric on ${\mathcal F}$.
For $\phi:\Gamma/\Gamma_n \to {\mathcal F}$, let $E(\phi)$ be the set of all edges of $C_n^f$ of the form $\pi_n(gs_*)$ where $\phi(g^{-1}\Gamma_n)_e=s_*$. Let ${\rm BAD}(\phi)\subset E(\phi)$ be those edges $\pi_n(gs_*)$ where $\phi(g^{-1}\Gamma_n)_e=s_*$ and $\phi( (gs)^{-1}\Gamma_n)_e = s_*^{-1}$ (where $s=p(s_*)$). Let ${\mathcal{G}}(\phi) = E(\phi) \setminus {\rm BAD}(\phi)$. Also let $\vec{{\mathcal{G}}}(\phi)$ be the set of all {\em oriented edges} of the form $\pi_n(g{\vec{s}}_*)$ where $\phi(g^{-1}\Gamma_n)_e=s_*$ and $\phi( (gs)^{-1}\Gamma_n)_e \neq s^{-1}_*$ (where $s=p(s_*)$, so the corresponding unoriented edge is in ${\mathcal{G}}(\phi)$). By abuse of notation, we will sometimes think of ${\mathcal{G}}(\phi)$ and ${\rm BAD}(\phi)$ as subgraphs of $C_n^f$ but not in the usual way. To be precise, we consider ${\mathcal{G}}(\phi)$ to be the smallest subgraph containing all the edges in ${\mathcal{G}}(\phi)$ (and similarly, for ${\rm BAD}(\phi)$). Thus ${\mathcal{G}}(\phi)$ and ${\rm BAD}(\phi)$ have no isolated vertices and, in general, are not spanning.
Because $\bigcap_{n=1}^\infty \bigcup_{i\ge n} \Gamma_i = \{e\}$, we may assume, without loss of generality, that for any $s_1\ne s_2 \in S \cup \{e\}$, $s_1\Gamma_n \ne s_2\Gamma_n$. An {\em oriented cycle} in $C_n^f$ is a sequence $g_0\Gamma_n,g_1\Gamma_n,\ldots, g_m \Gamma_n \in \Gamma/\Gamma_n$ such that $g_0\Gamma_n=g_m\Gamma_n$ and there exist $s_i \in S$ such that $\pi_n(g_is_i)= g_{i+1}\Gamma_n$. We consider two oriented cycles to be the same if they are equal up to a cyclic reordering of the vertices. Thus if $g_0\Gamma_n,g_1\Gamma_n,\ldots, g_m \Gamma_n$ is an oriented cycle then $g_i\Gamma_n, g_{i+1}\Gamma_n,\ldots, g_{m+i}\Gamma_n$ (indices $\mod m$) denotes the same oriented cycle. The cycle is {\em simple} if there does not exist $i,j$ with $0\le i < j<n$ such that $g_i\Gamma_n=g_j\Gamma_n$. By definition, $\pi_n(g_0s_0s_1\cdots s_{m-1}) = g_0\Gamma_n$. The cycle is {\em homotopically trivial} if $s_0s_1\cdots s_{m-1}$ is the identity element.
\begin{lemma} Let $W \subset \Gamma$ be a symmetric finite set containing $S$ and $\phi \in {\rm Map}(W,\delta,\Gamma_n)$ (where ${\rm Map}(W,\delta,\Gamma_n)$ is defined with respect to the pseudo-metric $\rho^{\mathcal F}$ above). Then \begin{enumerate}
\item $|{\rm BAD}(\phi)| \le \delta^2 |S_*|^2 [\Gamma:\Gamma_n]$ and the number of vertices in ${\rm BAD}(\phi)$ is at most $\delta^2|S| [\Gamma:\Gamma_n]$. \item each component of ${\mathcal{G}}(\phi)$ contains at most one cycle (i.e., each component is either a tree or is homotopic to a circle); \item if a component $c$ of ${\mathcal{G}}(\phi)$ is a tree, then there is a single vertex of $c$ incident with an edge in ${\rm BAD}(\phi)$;
\item for every integer $m>0$ there is an integer $N_m$ such that if $W \supset S^m\cup S$ and $n \ge N_m$ then the number of components of ${\mathcal{G}}(\phi)$ is at most $(\delta^2|W| + m^{-1})[\Gamma:\Gamma_n]$;
\item if $n\ge N_m$ and $W \supset S^m\cup S$ then there is a spanning tree $T_\phi$ of $C_n^f$ such that $|T_\phi \Delta {\mathcal{G}}(\phi)| \le (3\delta^2|W||S_*| + 2m^{-1})[\Gamma:\Gamma_n].$ \end{enumerate} \end{lemma}
\begin{proof}
Let ${\rm BAD}(W,\phi)$ be the set of all vertices $g\Gamma_n \in \Gamma/\Gamma_n$ such that $\rho^{\mathcal F}(w \circ \phi(g^{-1}\Gamma_n), \phi(wg^{-1}\Gamma_n)) =1$ for some $w\in W$. Because $\phi \in {\rm Map}(W,\delta,\Gamma_n)$, $|{\rm BAD}(W,\phi)| \le \delta^2 |W| [\Gamma:\Gamma_n]$.
We claim that the vertex set of ${\rm BAD}(\phi)$ is contained in ${\rm BAD}(W,\phi)$. So let $\pi_n(gs_*) \in {\rm BAD}(\phi)$ for some $g\in \Gamma$, $s_* \in S_*$. Let $s=p(s_*)$. By definition, $\phi(g^{-1}\Gamma_n)_e=s_*$ and $\phi((gs)^{-1}\Gamma_n)_e=s^{-1}_*$. Because $\phi(g^{-1}\Gamma_n) \in {\mathcal F}$, $\phi(g^{-1}\Gamma_n)_e=s_*$ implies $$(s^{-1} \phi(g^{-1}\Gamma_n))_e=\phi(g^{-1}\Gamma_n)_s \ne s^{-1}_* =\phi(s^{-1}g^{-1}\Gamma_n)_e.$$ Thus $\rho^{\mathcal F}(s^{-1} \phi(g^{-1}\Gamma_n), \phi(s^{-1}g^{-1}\Gamma_n))=1$ which implies $\pi_n(g) \in {\rm BAD}(W,\phi)$. By writing $\pi_n(gs_*)$ as $\pi_n(gs s_*^{-1})$ the same argument yields that $gs \in {\rm BAD}(W,\phi)$. So both endpoints of $\pi_n(gs_*)$ are in ${\rm BAD}(W,\phi)$. Because $\pi_n(gs_*) \in {\rm BAD}(\phi)$ is arbitrary, this implies the vertex set of ${\rm BAD}(\phi)$ is contained in ${\rm BAD}(W,\phi)$.
By choosing $W=S \subset \Gamma$ (by abuse of notation), we see that the number of vertices in ${\rm BAD}(\phi)$ is at most $\delta^2|S|[\Gamma:\Gamma_n]$. Since each vertex is incident to $|S_*|$ edges, $|{\rm BAD}(\phi)| \le \delta^2 |S_*|^2[\Gamma:\Gamma_n].$
Observe for every vertex $g\Gamma_n$ contained in ${\mathcal{G}}(\phi)$ either \begin{enumerate} \item $g\Gamma_n$ is contained in both ${\mathcal{G}}(\phi)$ and ${\rm BAD}(\phi)$ and there are no oriented edges of $\vec{{\mathcal{G}}}(\phi)$ with tail $g\Gamma_n$, or \item there is exactly one oriented edge $\pi_n(g{\vec{s}}_*) \in \vec{{\mathcal{G}}}(\phi)$ with tail $g\Gamma_n$. \end{enumerate}
For every vertex $g_0\Gamma_n$ of ${\mathcal{G}}(\phi)$, let $H(g_0\Gamma_n)$ be the set of all vertices ``ahead'' of $g_0\Gamma_n$. To be precise, this consists of all vertices $g_k\Gamma_n$ such that there exist oriented edges ${\mathfrak e}_0,{\mathfrak e}_1,\ldots, {\mathfrak e}_{k-1} \in \vec{{\mathcal{G}}}(\phi)$ with ${\mathfrak e}_i=(g_i\Gamma_n,g_{i+1}\Gamma_n)$. Then two vertices $g\Gamma_n,g'\Gamma_n$ are in the same component of ${\mathcal{G}}(\phi)$ if and only if $H(g\Gamma_n) \cap H(g'\Gamma_n) \ne \emptyset$ (one direction is obvious, the other can be shown by induction on the distance between $g\Gamma_n$ and $g'\Gamma_n$ in the component of ${\mathcal{G}}(\phi)$ containing both). Therefore, if $c$ is the collection of vertices in a connected component of ${\mathcal{G}}(\phi)$ then $$\bigcap_{g\Gamma_n \in c} H(g\Gamma_n)$$ is either a single vertex (contained in ${\rm BAD}(\phi)$) or a simple cycle. This implies items (2) and (3) in the statement of the lemma (since any cycle must be contained in $\bigcap_{g\Gamma_n \in c} H(g\Gamma_n)$).
Because $\bigcap_{n\in {\mathbb N}}\bigcup_{i\ge n}\Gamma_i=\{e\}$, there is an $N_m$ such that $n \ge N_m$ implies every homotopically nontrivial cycle in $C_n^f$ has length $> m$. Let us assume now that $W \supset S^m\cup S$ and $n\ge N_m$. We need to estimate the number of homotopically trivial oriented simple cycles of length $\le m$ in $\vec{{\mathcal{G}}}(\phi)$.
So suppose that $g_0\Gamma_n,g_1\Gamma_n,\ldots, g_k \Gamma_n = g_0\Gamma_n \in \Gamma/\Gamma_n$ is an oriented simple cycle in $\vec{{\mathcal{G}}}(\phi)$ and $k \le m$. By definition, $(g_i\Gamma_n,g_{i+1}\Gamma_n) \in \vec{{\mathcal{G}}}(\phi)$ for all $i$ (indices mod $k$). Thus if $s_{*,i} = \phi(g_i^{-1}\Gamma_n)_e$, $p(s_{*,i})=s_i$ and this cycle is homotopically trivial then $s_0s_1s_2\cdots s_{k-1}$ is the identity element.
By definition, ${\mathcal F}$ does not contain any simple cycles. Therefore, there is some $i \le k-1$ such $$((s_0\cdots s_i)^{-1}\phi(g_0^{-1}\Gamma_n))_e = \phi(g_0^{-1}\Gamma_n)_{s_0s_1\cdots s_i} \ne \phi( (g_0s_0s_1\cdots s_i)^{-1}\Gamma_n)_e.$$
Because $W \supset S^m$, $g_0\Gamma_n \in {\rm BAD}(W,\phi)$. Since $|{\rm BAD}(W,\phi)| \le \delta^2 |W| [\Gamma:\Gamma_n]$, this implies that the number of homotopically trivial simple cycles in ${\mathcal{G}}(\phi)$ of length at most $m$ is at most $\delta^2 |W| [\Gamma:\Gamma_n]$.
Since each component of ${\mathcal{G}}(\phi)$ either contains a vertex of ${\rm BAD}(W,\phi)$ or contains a simple cycle of length $>m$, it follows that the number of components is at most $(\delta^2|W| + m^{-1})[\Gamma:\Gamma_n]$.
Let $L \subset {\mathcal{G}}(\phi)$ be a set of edges such that each edge ${\mathfrak e} \in L$ is contained in a simple cycle of ${\mathcal{G}}(\phi)$ and no two distinct edges ${\mathfrak e}_1, {\mathfrak e}_2 \in L$ are contained in the same simple cycle. So $|L| \le (\delta^2|W| + m^{-1})[\Gamma:\Gamma_n]$ and ${\mathcal{G}}(\phi) \setminus L$ is a forest with at most $(\delta^2|W| + m^{-1})[\Gamma:\Gamma_n]$ connected components.
Note ${\mathcal{G}}(\phi) \setminus L$ contains every vertex in ${\mathcal{G}}(\phi)$ which contains at least $(1-\delta^2|S|)[\Gamma:\Gamma_n]$ vertices (because the number of vertices in ${\rm BAD}(\phi)$ is at most $\delta^2|S|[\Gamma:\Gamma_n]$).
An exercise shows that if ${\mathcal F}_n \subset E_n$ is any forest contained in $C_n^f$ with at most $c({\mathcal F}_n)$ components and at least $v({\mathcal F}_n)$ vertices then there is a spanning tree $T_n \subset E_n$ such that ${\mathcal F}_n \subset T_n$ and $|T_n \setminus {\mathcal F}_n| \le c({\mathcal F}_n)+ [\Gamma:\Gamma_n] - v({\mathcal F}_n)$. In particular, this implies the last statement of the lemma. \end{proof}
\begin{lemma}\label{lem:upper2} $$h_\Sigma({\mathcal F},\Gamma) \le \limsup_n [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f).$$ \end{lemma}
\begin{proof}
Let $1/2>\epsilon>0$. Let $m$ be a positive integer, $W \subset \Gamma$ be a finite set with $W \supset S^m\cup S$ and $\delta>0$. Let $n\ge N_m$ (where $N_m$ is as in the previous lemma). We assume $m$ is large enough and $\delta$ is small enough so that $\epsilon > 6\delta^2|W||S_*|^2 + 4m^{-1}$. By Stirling's Formula, there is a constant $C>1$ such that for every $k$ with $0\le k \le \epsilon [\Gamma:\Gamma_n]$,
$${ [\Gamma:\Gamma_n] \choose k } \le C \exp(H(\epsilon ) [\Gamma:\Gamma_n]), \quad { [\Gamma:\Gamma_n]|S_*|/2 \choose k } \le C \exp(H(\epsilon ) [\Gamma:\Gamma_n]|S_*|/2),$$ where $H(\epsilon) := -\epsilon \log(\epsilon) - (1-\epsilon)\log(1-\epsilon)$.
If $\phi,\psi \in {\rm Map}(W,\delta,\Gamma_n)$ are such that $\vec{{\mathcal{G}}}(\phi) = \vec{{\mathcal{G}}}(\psi)$ and ${\rm BAD}(\phi)={\rm BAD}(\psi)$, then $\rho^{\mathcal F}(\phi(g\Gamma_n),\psi(g\Gamma_n))=0$ for all $g\in \Gamma$. On the other hand, $|{\rm BAD}(\phi)| \le \delta^2 |S_*|^2 [\Gamma:\Gamma_n] \le \epsilon [\Gamma:\Gamma_n]$. Therefore, \begin{eqnarray*} &&N_0({\rm Map}(W,\delta,\Gamma_n), \rho^{\mathcal F}_2) \\
&\le& (\epsilon [\Gamma: \Gamma_n]+1) C \exp(H(\epsilon) [\Gamma:\Gamma_n]|S_*|/2) |\{\vec{{\mathcal{G}}}(\phi):~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}|. \end{eqnarray*}
Now suppose that $\phi, \psi \in {\rm Map}(W,\delta,\Gamma_n)$ are such that ${\mathcal{G}}(\phi)={\mathcal{G}}(\psi)$ and ${\rm Vert}({\rm BAD}(\phi))={\rm Vert}({\rm BAD}(\psi))$, where
${\rm Vert}({\rm BAD}(\phi))$ and ${\rm Vert}({\rm BAD}(\psi))$ denote the vertex sets of ${\rm BAD}(\phi)$ and ${\rm BAD}(\psi)$ respectively. Note that if ${\mathfrak e} \in {\mathcal{G}}(\phi)$ is not contained in a simple cycle then the orientation of ${\mathfrak e}$ in $\vec{{\mathcal{G}}}(\phi)$ is the same as the orientation of ${\mathfrak e}$ in $\vec{{\mathcal{G}}}(\psi)$. This is because either ${\mathfrak e}$ is contained in a component which contains a simple cycle (in which case, ${\mathfrak e}$ must be oriented towards the simple cycle), or ${\mathfrak e}$ is contained in a component which contains a vertex of ${\rm BAD}(\phi)$ (in which case ${\mathfrak e}$ must be oriented towards that vertex).
On the other hand, for every simple cycle in ${\mathcal{G}}(\phi)$, there are two possible orientations it can have in $\vec{{\mathcal{G}}}(\psi)$. By the previous lemma, there are at most $(\delta^2|W| + m^{-1})[\Gamma:\Gamma_n] \le \epsilon[\Gamma:\Gamma_n]$ simple cycles. Therefore, \begin{eqnarray*}
&&|\{{\vec {\mathcal{G}}}(\phi):~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}| \\
&\le& 2^{\epsilon[\Gamma:\Gamma_n]} |\{({\mathcal{G}}(\phi), {\rm Vert}({\rm BAD}(\phi))):~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}|. \end{eqnarray*}
Because $|{\rm Vert}({\rm BAD}(\phi))| \le \delta^2 |S|[\Gamma:\Gamma_n] \le \epsilon [\Gamma:\Gamma_n]$, it follows that, \begin{eqnarray*}
&&|\{({\mathcal{G}}(\phi), {\rm Vert}({\rm BAD}(\phi))):~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}| \\
&\le& (\epsilon[\Gamma:\Gamma_n]+1)C\exp( H(\epsilon)[\Gamma:\Gamma_n]) |\{ {\mathcal{G}}(\phi) :~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}|. \end{eqnarray*}
Now suppose that $\phi, \psi \in {\rm Map}(W,\delta,\Gamma_n)$ are such that $T_\phi = T_\psi$ (where $T_\phi, T_\psi$ is a choice of spanning tree as in the previous lemma). Then $|{\mathcal{G}}(\phi) \Delta {\mathcal{G}}(\psi)| \le (6\delta^2|W||S_*| + 4m^{-1})[\Gamma:\Gamma_n] \le \epsilon [\Gamma:\Gamma_n]$. Therefore, \begin{eqnarray*}
&&|\{ {\mathcal{G}}(\phi) :~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}|\\
&\le& (\epsilon [\Gamma: \Gamma_n]+1) C\exp( H(\epsilon)[\Gamma:\Gamma_n] |S_*|/2) |\{ T_\phi :~ \phi \in {\rm Map}(W,\delta,\Gamma_n)\}|. \end{eqnarray*} We now have
$$N_0({\rm Map}(W,\delta,\Gamma_n),\rho^{\mathcal F}_2) \le (\epsilon [\Gamma: \Gamma_n]+1)^3 C^3 \exp\left( (\epsilon \log 2 + 2H(\epsilon) |S_*|) [\Gamma:\Gamma_n]\right) \tau(C_n^f).$$ Because $1/2>\epsilon>0$ is arbitrary and $H(\epsilon) \searrow 0$ as $\epsilon \searrow 0$, this implies the lemma. \end{proof}
We are ready to prove Theorem~\ref{thm:WSF}.
\begin{proof}[Proof of Theorem \ref{thm:WSF}] By Lemmas \ref{lem:lower2}, \ref{lem:model}, \ref{lem:model2} and \ref{lem:upper2}, \begin{eqnarray*} \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f) \le h_{\Sigma,\nu_{WSF}}(2^E,\Gamma) \le h_{\Sigma}({\mathcal F},\Gamma)\le \limsup_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f).
\end{eqnarray*} By Theorem 3.2 of \cite{Lyons05}, $ \lim_{n\to \infty} [\Gamma:\Gamma_n]^{-1} \log \tau(C_n^f) = {\bf h}(C(\Gamma,f))$. By Proposition~\ref{P-tree entropy}, $ {\bf h}(C(\Gamma,f)) = \log \det_{{\mathcal N}\Gamma} f$ so this proves the theorem. \end{proof}
\begin{question} There is another natural topological model for uniform spanning forests. Namely, let ${\mathcal F}_* \subset 2^E$ be the set of all subgraphs which are essential spanning forests. The word ``essential'' here means that every connected component of any $x\in {\mathcal F}_*$ is infinite. What is the topological sofic entropy of $\Gamma {\curvearrowright} {\mathcal F}_*$? Because $\nu_{WSF}$ can naturally be realized as an invariant measure on ${\mathcal F}_*$, the variational principle implies the topological sofic entropy of ${\mathcal F}_*$ is at least the measure-theoretic sofic entropy of $\Gamma {\curvearrowright} (2^E, \nu_{WSF})$. \end{question}
\appendix
\section{Non-invertibility}
\begin{theorem} Let $\Gamma$ be a countable group and $f\in {\mathbb R} \Gamma$ be well-balanced. Then $f$ is not invertible in the universal group $C^*$-algebra $C^*(\Gamma)$ (see \cite[Section 2.5]{BO}), or in $\ell^1(\Gamma)$. Moreover, $f$ is invertible in the group von Neumann algebra ${\mathcal N}\Gamma$ if and only if $\Gamma$ is non-amenable. \end{theorem}
\begin{proof} We note first that $f$ is not invertible in
$C^*(\Gamma)$. Indeed, the trivial representation of $\Gamma$ gives rise to a unital $C^*$-algebra homomorphism $\varphi: C^*(\Gamma)\rightarrow {\mathbb C}$ sending every $s\in \Gamma$ to $1$. Since $\varphi(f)=0$ is not invertible in ${\mathbb C}$, $f$ is not invertible in $C^*(\Gamma)$.
Next we note that $f$ is not invertible in $\ell^1(\Gamma)$. This follows from the natural unital algebra embedding $\ell^1(\Gamma)\hookrightarrow C^*(\Gamma)$ being the identity map on ${\mathbb R}\Gamma$ and the non-invertibility of $f$ in $C^*(\Gamma)$.
Finally we note that $f$ is invertible in
${\mathcal N}\Gamma$ if and only if $\Gamma$ is non-amenable. Suppose that $\Gamma$ is amenable. Let $\{F_n\}_{n\in {\mathbb N}}$ be a right F{\o}lner sequence of $\Gamma$. Then $|F_n|^{-1/2}\chi_{F_n}$ is a unit vector in $\ell^2(\Gamma, {\mathbb C})$ for each $n\in {\mathbb N}$, where $\chi_{F_n}$ denotes the characteristic function of $F_n$ in $\Gamma$, and $\lim_{n\to \infty}\||F_n|^{-1/2}\chi_{F_n}\cdot f\|_2=0$. Thus $f$ is not invertible in ${\mathcal N}\Gamma$. Now suppose that $\Gamma$ is non-amenable.
Set $\mu=-(f-f_e)/f_e\in {\mathbb R}\Gamma$ and denote by $\|\mu\|$ the operator norm of $\mu$ on $\ell^2(\Gamma, {\mathbb C})$, which is also the norm of $\mu$ in ${\mathcal N}\Gamma$.
Then $\mu$ is a symmetric probability measure on $\Gamma$ and the support of $\mu$ generates $\Gamma$. Since $\Gamma$ is non-amenable,
Kesten's theorem \cite[Theorem 4.20.ii]{Paterson} implies that $\|\mu\|<1$. Therefore $f=f_e(1-\mu)$ is invertible in the Banach algebra ${\mathcal N}\Gamma$. \end{proof}
\end{document} |
\begin{document}
\title[Hypergeometric Sequences with Quadratic Parameters]{The Membership Problem for Hypergeometric Sequences with Quadratic Parameters}
\author[G.~Kenison]{George Kenison} \address{George Kenison, Institute of Logic and Computation, TU Wien, Vienna, Austria} \email{george.kenison@tuwien.ac.at}
\author[K.~Nosan]{Klara Nosan} \address{Klara Nosan, Universit\'e Paris Cité, CNRS, IRIF, Paris, France} \email{nosan@irif.fr}
\author[M.~Shirmohammadi]{Mahsa Shirmohammadi} \address{Mahsa Shirmohammadi, Universit\'e Paris Cité, CNRS, IRIF, Paris, France} \email{mahsa@irif.fr}
\author[J.~Worrell]{James Worrell} \address{James Worrell, Department of Computer Science, University of Oxford, Oxford, UK} \email{jbw@cs.ox.ac.uk}
\thanks{ George Kenison gratefully acknowledges the support of ERC consolidator grant ARTIST 101002685 and WWTF grant ProbInG ICT19-018. Klara Nosan and Mahsa Shirmohammadi are supported by International Emerging Actions grant (IEA'22), by ANR grant VeSyAM (ANR-22-CE48-0005) and by the grant CyphAI (ANR-CREST-JST). James Worrell is supported by EPSRC fellowship EP/X033813/1. }
\maketitle
\DeclareRobustCommand{\gobblefive}[5]{} \DeclareRobustCommand{\gobblenine}[9]{} \newcommand*{\SkipTocEntry}{\addtocontents{toc}{\gobblenine}}
\begin{abstract} Hypergeometric sequences are rational-valued sequences that satisfy first-order linear recurrence relations with polynomial coefficients; that is, a hypergeometric sequence \(\seq[\infty]{u_n}{n=0}\) is one that satisfies a recurrence of the form $f(n)u_n = g(n)u_{n-1}$ where $f,g \in \mathbb{Z}[x]$.
In this paper, we consider the Membership Problem for hypergeometric sequences: given a hypergeometric sequence \(\seq[\infty]{u_n}{n=0}\) and a target value $t\in \mathbb{Q}$, determine whether $u_n=t$ for some index \(n\). We establish decidability of the Membership Problem under the assumption that either (i)~$f$ and $g$ have distinct splitting fields or (ii)~$f$ and $g$ are monic polynomials that both split over a quadratic extension of $\mathbb{Q}$. Our results are based on an analysis of the prime divisors of polynomial sequences $\langle f(n) \rangle_{n=1}^\infty$ and $\langle g(n) \rangle_{n=1}^\infty$ appearing in the recurrence relation. \end{abstract}
\maketitle
\section{Introduction}
\subsubsection*{Background and Motivation} Recursively defined sequences are ubiquitous in mathematics and computer science. A fundamental open problem in this context is the decidability of the \emph{Membership Problem}, which asks to determine whether a given value is an element of a given sequence. The Skolem Problem for \emph{C-finite} sequences (those sequences that satisfy a linear recurrence relation with constant coefficients) is the best known variant of the Membership Problem. The Skolem Problem asks to determine whether a given C-finite sequence vanishes at some index~\cite{everest2003recurrence}. Decidability of this problem is known for recurrences of order at most four \cite{mignotte1984distance,vereshchagin1985occurence} but is open in general. Proving decidability of the Skolem Problem would be equivalent to giving an effective proof of the celebrated Skolem--Mahler--Lech Theorem, which states that every non-degenerate C-finite sequence that is not identically zero has a finite set of zeros.
In this paper we consider the most basic case of the Membership Problem for a class of \emph{P-finite} sequences (those sequences that satisfy a linear recurrence with polynomial coefficients). Specifically, we consider the Membership Problem for the class of hypergeometric sequences. A rational-valued sequence $\langle u_n \rangle_{n=0}^\infty$ is \emph{hypergeometric} if it satisfies a recurrence relation of the form
\begin{equation}\label{eq:rel}
f(n)u_{n} - g(n)u_{n-1} = 0 \, , \end{equation}
where $f,g \in \mathbb{Z}[x]$ are polynomials, and $f(x)$
has no non-negative integer zeros.
By the latter assumption on~$f(x)$, the recurrence relation~\eqref{eq:rel}
uniquely defines an infinite sequence of rational numbers once the initial value $u_0\in\mathbb{Q}$ is specified. The term
\emph{hypergeometric} was introduced by John Wallis in the 17th century~\cite{wallis1655arithmetica} and
hypergeometric sequences and their
associated generating functions, the hypergeometric series,
have a long and illustrious history in the mathematics literature. In particular, hypergeometric series encompass many of the common mathematical functions
and have numerous applications in analytic combinatorics~\cite{FS09, Concrete11}.
The Membership Problem for hypergeometric sequences asks, given a recurrence~\eqref{eq:rel}, initial value $u_0\in \mathbb{Q}$, and target $t\in \mathbb{Q}$, whether $t$ lies in the sequence $\langle u_n \rangle_{n=0}^\infty$. At first glance, this problem may seem easy to decide. Without loss of generality we can assume that the sequence $\langle u_n \rangle_{n=0}^\infty$ either diverges to infinity or converges to a finite limit. If the sequence does not converge to~$t$ then one can compute a bound $B$ such that $u_n \neq t$ for all $n>B$. Such a bound can also be computed in case one is promised that $\langle u_n \rangle_{n=0}^\infty$ converges to $t$, by using the fact that the convergence to $t$ is ultimately monotonic. However the above case distinction does not suffice to show decidability of the Membership Problem! The problem is that it is not known how to decide whether a hypergeometric seqeuence converges to a given rational limit. The latter is related to deep conjectures about the gamma function (see the discussion below). In this paper we will take a different route to establish decidability of certain cases of the Membership Problem.
\subsubsection*{Contributions} We approach the Membership Problem by considering the prime divisors of the values of a hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$. The overall strategy is to exhibit an effective threshold $B$ such that for all $n>B$ there is a prime divisor of $u_n$ that is not a divisor of the target $t$. Our two main contributions are as follows: \begin{itemize}
\item The Membership Problem for hypergeometric sequences whose polynomial coefficients (as in \eqref{eq:rel}) have distinct splitting fields is decidable (\autoref{theo-distinctsplit}).
\item The Membership Problem for hypergeometric sequences whose polynomial coefficients are monic and split over a quadratic field is decidable (\autoref{theo-decide-quad}). \end{itemize}
The proofs of our main results involve two different implementations of our general strategy. The proof of \Cref{theo-distinctsplit} applies the Chebotarev density theorem to find a single prime $p\in \mathbb{Z}$ that does not divide the target $t$ but divides all members of an infinite tail of the sequence. Meanwhile, the proof of \Cref{theo-decide-quad} shows that for all sufficiently large $n$ there exists a prime $p$, that is allowed to depend on $n$, such that $p$ divides $u_n$ but not $t$. To find such a prime we rely on (a mild generalisation of) a result of~\cite{everest07} concerning prime divisors of the values of a quadratic polynomial.
\Cref{theo-distinctsplit} expands the class of sequences for which the Membership Problem can be solved and further isolates its hard instances. The paper~\cite{NPSW022} handles perhaps the easiest sub-case of the Membership Problem that does not fall under~\Cref{theo-distinctsplit}, namely when the polynomial coefficients both split over~$\mathbb{Q}$. The second main result of the present paper handles another naturally occurring sub-case: when the polynomial coefficients split over the ring of integers of a quadratic field $\mathbb{K}$. A common refinement of these two cases---that the polynomial coefficients split over $\mathbb{K}$---is the subject of current research. Generalisations of the results of~\cite{everest07} to higher-degree polynomials are a subject of ongoing research in number theory and potentially would allow us to extend our approach beyond the quadratic case.
\subsubsection*{Related Work} There is a growing body of work that addresses membership and threshold problems for sequences satisfying low-order polynomial recurrences. Here the \emph{Threshold Problem} asks to determine whether every term in a sequence lies above a given threshold, for example, whether every term is non-negative.
The recent preprint~\cite{kenison2022applications} establishes decidability results (some conditional on Schanuel's Conjecture) for both the Membership and Threshold Problems for hypergeometric sequences. The approach of~\cite{kenison2022applications} relies on transcendence theory for the gamma function (as well as underlying properties of modular functions established by Nesterenko~\cite{nesterenko1996modular}). By contrast, the algebraic techniques of the present paper seem appropriate only for the Membership Problem. We note that the approach of~\cite{kenison2022applications} requires certain restrictions, e.g., decidability is only unconditional when the parameters are drawn from imaginary quadratic fields.
The problem of deciding positivity of order-two P-finite sequences and of deciding the existence of zeros in such sequences is considered in~\cite{KauersP10,KenisonKLLMOW021,NeumannO021,PillweinS15}. These works all place syntactic restrictions on the degrees of the polynomial coefficients involved in the recurrences, and all four give algorithms that are not guaranteed to terminate for all initial values of a given recurrence. For example, in~\cite{KauersP10} the termination proof of the algorithm for determining positivity of order-two sequences requires that the characteristic roots of the recurrence be distinct and that one is working with a generic solution of the recurrence (in which the asymptotic rate of growth corresponds to the dominant characteristic root of the recurrence). Simple manipulations show that the Membership Problem considered in this paper is equivalent to the problem of finding a zero term in an order-two P-finite sequence $\langle u_n \rangle_{n=0}^\infty$ arising as a sum of two hypergeometric sequences.
Links between the Membership and Threshold Problems and the Rohrlich--Lang Conjecture appear in previous works \cite{kenison2020positivity, NPSW022}. Here the Rohrlich--Lang Conjecture concerns multiplicative relations for the gamma function evaluated at rational points.
The $p$-adic techniques used in the present paper bear many similarities with work on developing criteria for hypergeometric sequences to be integer valued. For example, work by Landau in 1900 \cite{landau1900factorielles} uses \(p\)-adic analysis to establish a necessary and sufficient condition for integrality in the so-called class of \emph{factorial} hypergeometric sequences. In more recent work, Hong and Wang \cite{hongarxiv2016} establish a criterion for the integrality of hypergeometric series with parameters from quadratic fields. We observe that some of the intermediate asymptotic results in Hong and Wang's note are close to \cite[Corollary~3.1]{Moll09} (\autoref{prop:moll} herein).
\subsubsection*{Structure}
The remainder of this paper is structured as follows. We briefly review preliminary material in \cref{sec-pre}, including some standard assumptions about instances of the Membership Problem that can be made without loss of generality. In \cref{sec-polyseq}, we recall useful technical results on the prime divisors of hypergeometric sequences that satisfy monic recurrence relations (see \eqref{eq:poly}). In \cref{sec-unequalsplitting}, we prove \cref{theo-distinctsplit}. The proof of \cref{theo-decide-quad} is given in \cref{sec-quad}. We discuss ideas for future research in \cref{sec:discussion}. The remaining appendices prove technical results omitted from the main text.
\section{Preliminaries} \label{sec-pre}
\subsubsection*{Hypergeometric Sequences}
A hypergeometric sequence $\langle u_n\rangle_{n=0}^\infty$ is a sequence of rational numbers that satisfies a recurrence of the form \eqref{eq:rel} where $f,g \in \mathbb{Z}[x]$ are polynomials, and $f(x)$ has no non-negative integer zeros. By the latter requirement on~$f(x)$, the recurrence~\eqref{eq:rel} uniquely defines an infinite sequence of rational numbers once the initial element $u_0$ is specified.
An instance of the Membership Problem for hypergeometric sequences consists of a recurrence~\eqref{eq:rel}, an initial value $u_0 \in \mathbb{Q}$, and a target $t \in \mathbb{Q}$. The problem asks to decide whether there exists \(n\in{\mathbb{N}}\) such that \(u_n = t\). We say that such an instance is in \emph{standard form} if~(S1) the initial condition is $u_0=1$; (S2)~the polynomial $g(x)$ has no positive integer root; (S3)~the target $t$ is non-zero; (S4)~the polynomials $f$ and $g$ have the same degree and leading coefficient.
For the purposes of deciding the Membership Problem, we can assume without loss of generality that all instances are in standard form. An arbitrary instance can be transformed into one satisfying Condition~(S1) by multiplying the sequence and target by a suitable constant. Instances of the Membership Problem that fail to satisfy Conditions~(S2) and (S3) are trivially solvable. The positive integer roots of $g$ can be computed and for any such root $n_0$, we have $u_n=0$ for all $n\geq n_0$. Finally, for recurrences that fail Condition~(S4) we have that \[ \frac{u_n}{u_{n-1}}=\frac{g(n)}{f(n)} \] either converges to $0$ or diverges in absolute value. Under the assumption that $t\neq 0$, in each case we can compute an effective threshold $n_0$ such that $u_n\neq t$ for all $n\geq n_0$.
\subsubsection*{The $p$-adic valuation} Let $p\in {\mathbb{N}}$ be a prime.
Denote by~$v_p:\mathbb{Q} \to \mathbb{Z} \cup\{\infty\}$ the $p$-adic valuation on~$\mathbb{Q}$. Recall that for a non-zero number~$x\in \mathbb{Q}$, $v_p(x)$ is the unique integer such that~$x$ can be written in the form \[x=p^{v_p(x)}\; \frac{a}{b}\] where $a,b\in \mathbb{Z}$ and $p$ divides neither $a$ nor $b$. The value $v_p(0)$ is defined to be $\infty$. The valuation possesses two important properties: \begin{enumerate}
\item[-]$v_p(x+y)\geq \min\{v_p(x),v_p(y)\}$ \, (\emph{strong triangle inequality}),
\item[-]$v_p(xy)=v_p(x)+v_p(y)$ \, (\emph{multiplicative property}). \end{enumerate}
\subsubsection*{Asymptotic estimates for series over primes} Given ${\sim} \in {\{<,=,> \}}$ and $x\in \mathbb{Q}$, we denote sums over primes \(p\in{\mathbb{N}}\) such that \(p \sim x\) by \(\sum_{p \sim x}\). Let \(\pi(x) := \sum_{p\le x} 1\) count the number of primes of size at most~\(x\). The following result is a consequence of the celebrated Prime Number Theorem.
\begin{theorem}\label{thm:pnt} For \(\pi(x)\) as above, we have
\begin{equation*}
\pi(x) = \frac{x}{\log x} + O\Bigl(\frac{x}{\log^2 x}\Bigr).
\end{equation*}
\end{theorem}
As an aside, an element \(a\in\mathbb{Z}\) is a \emph{square} modulo a prime \(p\in {\mathbb{N}}\) if there exists an \(x\in\mathbb{Z}\) such that \(x^2 \equiv a \pmod{p}\). An element \(a\in\mathbb{Z}\) is a \emph{quadratic residue} modulo \(p\) if \(a\) is both a square modulo \(p\), and furthermore \(a\) and \(p\) are co-prime. We denote by \(\mathcal{L}_p\) the set of quadratic residues modulo \(p\).
Recall the first of Mertens' three theorems \cite{mertens1874zahlentheorie} (see also \cite[Theorem~4.10]{apostol1998introduction}), \[
\sum_{p \leq x } \frac{\log p}{p} = \log x + O(1) \,.
\] In the sequel we shall make use of the following refinement of Mertens' theorem. \begin{proposition} \label{prop:apostol_primes_beta} Suppose that \(a \in \mathbb{Z}\) is not a perfect square. Then \begin{equation*} \sum_{p\leq x, \, a \in \mathcal{L}_p} \frac{\log p}{p} = \frac{1}{2}\log(x)+O(1). \end{equation*} \end{proposition} \autoref{prop:apostol_primes_beta} appears in work by Selberg \cite[Equation (3.3)]{selberg1950pnt-ap} on an elementary proof of Dirichlet's theorem in arithmetic progressions.
\section{Monic Recurrences} \label{sec-polyseq} In this section, we study hypergeometric sequences~$\langle u_n \rangle_{n=0}^\infty$, satisfying first-order recurrences of the special form \begin{equation}\label{eq:poly} u_n=f(n) u_{n-1} \quad \text{ and } \quad u_0=1, \end{equation} where $f\in \mathbb{Z}[x]$ has no non-negative integer roots. We call such a recurrence \emph{monic}. We analyse the prime divisors of sequences~$\langle u_n \rangle_{n=0}^\infty$ that satisfy such a monic recurrence. In particular, we recall two results that will serve as stepping stones toward our main decidability theorems in the subsequent sections. Following~\cite{Moll09}, for a fixed prime \(p\), the first result establishes an asymptotic estimate for the $p$-adic valuation $v_p(u_n)$ as $n$ tends to infinity. Next, following~\cite{everest07}, when $f$ is a quadratic polynomial we prove a result that yields asymptotic estimates on the size of the largest prime divisors of $u_n$ as $n$ tends to infinity. The restriction on the degree is necessary given the state of the art: estimates on large prime divisors constitute hard open problems in the theory of polynomials~\cite{hinz1996multiplicative,heathbrown2001largest}.
\subsection{Asymptotic growth of valuations} \label{subsec:asymval}
Let $p\in {\mathbb{N}}$ be prime. Consider a hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$, satisfying a monic recurrence~\eqref{eq:poly}. Since $u_n=\prod_{k=1}^n f(k)$, we have \begin{equation*}
v_p(u_n) = \sum_{k=1}^n v_p(f(k)). \end{equation*} In this section we recall the result of~\cite{Moll09} that characterises the asymptotic growth of $v_p(u_n)$ in terms of the number of roots of $f$ in $\mathbb{Z}/p\mathbb{Z}$. The key tool in this argument is Hensel's Lemma.
\begin{theorem}[Hensel's Lemma {\cite[Theorem 4.7.2]{gouvea2020padic}}] Let \(f(x)\in\mathbb{Z}[x]\) and assume that there exist polynomials \(g(x)\) and \(h(x)\) such that: i) \(g(x)\) is monic, ii) \(g(x)\) and \(h(x)\) are relatively prime modulo \(p\), and iii) \(f(x) = g(x)h(x) \pmod{p}\).
Then for all $e>0$ there exist polynomials \(g_1(x),h_1(x)\in\mathbb{Z}[x]\) such that: i) \(g_1(x)\) is monic, ii) \(g_1(x)\equiv g(x) \pmod{p}\) and \(h_1(x)\equiv h(x) \pmod{p}\), and \(f(x)=g_1(x)h_1(x) \pmod{p^e}\). \end{theorem}
Define a \emph{Hensel prime} for $f\in \mathbb{Z}[x]$ to be a prime that does not divide the discriminant of any irreducible factor of $f$. Since the discriminant of an irreducible polynomial is non-zero, all but finitely many primes are Hensel primes for a given polynomial.
Given a prime $p$, suppose that $f \in \mathbb{Z}[x]$ has $m$ roots in $\mathbb{Z}/p\mathbb{Z}$, i.e., suppose that $f$ factors as \[ f=(x-\alpha_1)^{m_1} \cdots (x-\alpha_\ell)^{m_\ell} g(x) \pmod{p},\] where $\alpha_1,\ldots,\alpha_\ell\in \mathbb{Z}$, $g\in \mathbb{Z}[x]$ has no root modulo $p$, and $m=m_1+\cdots+m_\ell$. In this case, if $p$ is a Hensel prime for $f$ then for all $e>0$ we can apply Hensel's Lemma to obtain a factorisation \[f(x)=(x-\beta_1)^{m_1} \cdots (x-\beta_\ell)^{m_\ell} h(x) \pmod{p^e}\] where $\beta_1,\ldots,\beta_\ell \in \mathbb{Z}$, and $h\in \mathbb{Z}[x]$ has no root modulo $p$. In other words, $f$ has exactly $m$ roots in the ring $\mathbb{Z}/p^e\mathbb{Z}$.
The following result is a reformulation of~\cite[Corollary 3.1]{Moll09}. For later use, we formulate the result so as to make explicit the dependence of the bounds for $v_p(u_n)$ on the prime $p$. The proof remains the same. \begin{proposition}[{\cite[Corollary~3.1]{Moll09}}] \label{prop:moll} Suppose that $\langle u_n \rangle_{n=0}^\infty$ satisfies the monic recurrence in Equation~\eqref{eq:poly} with polynomial coefficient $f \in \mathbb{Z}[x]$. Let $p$ be a Hensel prime of $f$ such that $f$ has $m$ roots modulo $p$. Then there exist effectively computable constants~$\varepsilon, n_0>0$ such that if $n>n_0$, \[m\Big(\frac{n}{p-1}- \frac{\varepsilon \log n}{\log p}\Big) \leq v_p(u_n) \leq m\Big(\frac{n}{p-1}+ \frac{\varepsilon \log n}{\log p}\Big)\] where $\varepsilon$ depends only on~$f$. \end{proposition} \begin{proof}
The function~$|f(x)|$ is eventually monotonically increasing on~${\mathbb{N}}$. There exists an effectively computable bound~$n_0$ such that for all $n \geq n_0$ and all $1 \leq k \leq n$, the inequality
$|f(k)| \leq |f(n)|$ holds.
Furthermore, there exists an effective constant $\varepsilon_0>0$, independent of $p$, such that for all $n\geq n_0$ and all $1\leq k \leq n$ we have
\[ |f(k)|< n^{\varepsilon_0} = p^{\varepsilon_0 \log n / \log p}.\]
Fix $n\geq n_0$ and define $e_{\max}$ to be the smallest power of~$p$ such that $p^{e_{\max}-1} \leq |f(n)| < p^{e_{\max}}$. Then \begin{equation}
\label{eq:emax} e_{\max} \leq \frac{\varepsilon_0 \log n}{\log p}. \end{equation}
Since $p$ is a Hensel prime, by Hensel's Lemma, there is a factorisation \[ f(x)=(x-\beta_1)^{m_1} \cdots (x-\beta_\ell)^{m_\ell} h(x) \pmod{p^{e_{\max}}}. \] where $m=m_1+\cdots+m_\ell$ and $h$ has no zero modulo~$p$.
Denote by $\mathbb{I}\{p^e \mid x\}$ the function such that \[ \mathbb{I}\{p^e \mid x \}:= \begin{cases}
1 & \text{ if }\ p^e \mid x, \\
0 & \text{ otherwise.} \end{cases} \] \color{black}
Since $v_p(f(k)) \leq e_{\max}$ for all~$k\leq n$, we have
\begin{align} v_p(u_n) =& \sum_{k=1}^n v_p(f(k))\notag \\
=& \sum_{k=1}^n \sum_{i=1}^\ell m_i \, v_p(k-\beta_i) \notag \\
=& \sum_{k=1}^n \sum_{i=1}^\ell \sum_{e=1}^{e_{\max}} m_i \, \mathbb{I}\{p^e \mid k - \beta_i\} \notag\\
=& \sum_{e=1}^{e_{\max}}\sum_{i=1}^\ell
\sum_{k=1}^n
m_i \mathbb{I}\{p^e \mid k - \beta_i\}. \label{eq:TAG} \end{align}
Now for all $1\leq e\leq e^{\max}$ the set $\{ k \in {\mathbb{N}} : p^e \mid k-\beta_i\}$ is an arithmetic progression with common difference $p^e$ and so \begin{equation} \label{eq-m-AP}
\ \frac{n}{p^e} -1 \leq \sum_{k=1}^n \mathbb{I}\{p^e \mid k-\beta_i \} \leq \frac{n}{p^e} +1 ,
\end{equation} Combining inequality~\eqref{eq-m-AP} with Equation~\eqref{eq:TAG} we obtain \begin{equation}
\label{eq-sum-III}
m \sum_{e=1}^{e_{\max}} \Big( \frac{n}{p^e} -1 \Big) \leq v_p(u_n) \leq m \sum_{e=1}^{e_{\max}} \Big( \frac{n}{p^e} +1 \Big). \end{equation}
Let $\varepsilon := \varepsilon_0+1$. The desired result follows by sandwiching the term $\sum_{e=1}^{e_{\max}} \frac{1}{p^e}$ in~\eqref{eq-sum-III} by \[
\frac{1-|f(n)|^{-1}}{p-1}\leq \frac{1-p^{-e_{\max}}}{p-1} =\sum_{e=1}^{e_{\max}} \frac{1}{p^e} \leq \, \frac{1}{p-1}\] in combination with the upper bound on $e_{\max}$ in~\eqref{eq:emax}. \end{proof}
\subsection{Asymptotic estimate for the largest prime divisor} Fix a polynomial $f(x):=x^2+\beta \in\mathbb{Z}[x]$. We assume that $-\beta$ is not a perfect square, which is equivalent to assuming that $f$ is irreducible. Let $a,b\in \mathbb{Q}$ be such that $0 \leq a <b$. Let $c,d \in {\mathbb{N}}$. For all $n\in {\mathbb{N}}$ we define \[I(n):=\{k \in {\mathbb{N}} : an \leq k\leq bn\} \cap (c{\mathbb{N}}+d) \] and \[F_n:= \prod_{k\in I(n)} f(k).\]
Informally speaking, the following theorem gives effective super-linear lower bounds on the growth of the function that maps $n$ to the greatest prime divisor of $F_n$. The result itself and the proof are a slight generalisation of~\cite[Theorem 5.1]{everest07}. The main difference is that we permit $I(n)$ to be the intersection of an interval and an arithmetic progression, whereas the work cited above considers unrefined intervals~$I(n)=\{1,\ldots,n\}$.
\begin{restatable}{theorem}{theobigprimespolyT} \label{theo:bigprimespolyT} Let $M \in {\mathbb{N}}$. There exists an effectively computable bound~$B\in {\mathbb{N}}$ such that for all \(n>B\) there exists a prime~$p>Mn$ that divides $F_n$. \end{restatable} \begin{proof} Given $n\in {\mathbb{N}}$, we have the prime factorisation $F_n = \prod_{p} p^{e_p}$ where $e_p:= v_p(F_n)$ for each prime $p$. Note that $e_p=0$ for all but finitely many $p$. Taking logarithms, we get \begin{equation*}
\log(F_n) =\sum_{p} e_p \log p . \end{equation*} Partitioning the above sum into a sub-sum over primes at most $Mn$ and a sub-sum over primes greater than $Mn$, we obtain \begin{equation} \label{eq:logunn}
\sum_{p >Mn} e_p \log p = \log(F_n) - \sum_{p \leq Mn} e_p
\log p . \end{equation}
The theorem at hand follows from a lower bound on the sum $\sum_{p >Mn} e_p \log p$ on the left-hand side of~\eqref{eq:logunn}. To this end we have two sub-goals: give a lower bound on $\log(F_n)$ and an upper bound on $\sum_{p \leq Mn} e_p \log p$.
Write $A:=\frac{b-a}{c}$. The following lower bound on $\log(F_n)$ is a consequence of Stirling's formula. The proof is in Appendix~\ref{sec:app-lowerbownun}. \begin{claim}
\label{claim:sizefunnyQ} We have the bound
$\log(F_n) \geq 2A (n\log n - n)$. \end{claim}
The next task is give an upper bound on $\sum_{p \leq Mn} e_p \log p$. Here we follow the approach in~\cite{everest07} and further partition the sum into those primes $p<n$ (treated in~\Cref{claim-small-prime-F}) and those primes \(n\le p\le Mn\) (treated in~\Cref{claim-average-prime}).
\begin{claim} \label{claim-small-prime-F} There exist positive constants \(\varepsilon, n_0>0\) such that if \(n>n_0\), then \begin{equation*}
\sum_{p <n} e_p \log p \leq An\log n + \varepsilon n. \end{equation*} \end{claim} \begin{proof} Let $S_n$ be the set of primes $p<n$ such that $p$ divides $F_n$ and $p$ is a Hensel prime for $f$. Observe that \[ \sum_{p < n} e_p \log p - \sum_{p\in S_n} e_p \log p \leq \varepsilon_0 \log n\] for an effective constant $\varepsilon_0$. Indeed, if $p<n$ is a prime divisor of ${F_n}$ that does not lie in $S_n$ then $p$ divides the discriminant of $f$---and there are finitely many such primes. Thus to prove the claim it will suffice to show the following bound for some effective constant $\varepsilon_1$: \begin{equation} \sum_{p \in S_n} e_p \log p \leq
An\log n + \varepsilon_1 n. \label{eq:SUM2} \end{equation}
For \(p\in S_n\), we establish an upper bound on $e_p$ which follows from \Cref{prop:moll}: \begin{equation}
e_p \leq \frac{2An}{p-1}+ \frac{\varepsilon_2 \log n}{\log p}. \label{eq:BOUND}
\end{equation}
Here the constant $\varepsilon_2$ is effective and independent of the prime $p$. The justification is given in~\Cref{sec:app-lowerbownun}.
We next argue that there exist effective constants $\varepsilon_3,\varepsilon_4,n_1>0$ such that the following chain of inequalities is valid
for all $n\geq n_1$.
We have that
\begin{eqnarray*}
\sum_{p\in S_n} e_p \log p & \leq & \sum_{p \in S_n} \left( \frac{2An }{p-1}+\varepsilon_2 \, \frac{\log n}{\log p} \right) \log p
\qquad\mbox{(by~\eqref{eq:BOUND})}\\
& \leq & \, 2An \sum_{p \in S_n} \frac{\log p }{p-1} + \varepsilon_2 \pi(n) \log n\\
&\leq & 2An \sum_{p \in S_n} \frac{\log p }{p-1} + \varepsilon_3 n
\qquad \mbox{(by \autoref{thm:pnt})}\\
&=& 2An \sum_{p \in S_n} \frac{\log p }{p} \left(1+\frac{1}{p-1}\right)+ \varepsilon_3 n\\
& \leq & 2An \sum_{p \in S_n} \frac{\log p }{p}+ \varepsilon_4.
\end{eqnarray*}
No prime in $S_n$ divides the discriminant of $f$. Since the latter
is equal to $-4\beta$, no prime in $S_n$ divides $\beta$.
In addition, every prime in $S_n$ is a divisor of $F_n$; i.e., a
divisor of $k^2+\beta$ for some $k\in I(n)$, we have that $\beta$ is
a quadratic residue modulo $p$ for every prime $p \in S_n$.
Thus, for sufficiently large \(n\), we have that \[ \sum_{p \in S_n} \frac{\log p}{p} \leq \frac{1}{2} \log n + \varepsilon_5\] (by \Cref{prop:apostol_primes_beta}) for some effective constant $\varepsilon_5$.
The desired bound~\eqref{eq:SUM2} follows by combining the previous two inequalities and fixing \(\varepsilon_1 \ge 2A\varepsilon_5 + \varepsilon_4\). \end{proof}
\begin{claim} \label{claim-average-prime}
There exist effectively computable constants \(n_0,\varepsilon>0\)
such that if \(n>n_0\), then \begin{equation*} \sum_{n\leq p \leq Mn} e_p \log p \leq \varepsilon n. \end{equation*} \end{claim}
\begin{proof}
Let $n \in {\mathbb{N}}$. Suppose that \(p> (b-a)n\) is a prime divisor of
\(F_n\). For such primes, we shall first show that
$e_p := v_p(F_n) \le 2$. Assume, for a contradiction, that there
are distinct integers $k_1 < k_2 < k_3$ in $I(n)$ such that $p$
divides $k_1^2+\beta$, $k_2^2+\beta$, and $k_3^2+\beta$. Then
$p \mid k_1^2 - k_2^2$.
Since \(p\) is prime, either
$p \mid k_1-k_2$ or \(p \mid k_1+k_2\). Since
\(0< k_2-k_1 < (b-a)n \le p\), we deduce that $p \mid k_1+k_2$. By
symmetric reasoning we have that $p \mid k_2+k_3$. Thus $p$ must
also divide $(k_2+k_3)-(k_1+k_2) = k_3 - k_1$. However, this leads
to a contradiction since $p \geq (b-a)n \geq k_3-k_1$. Hence for
each prime divisor \(p\mid F_n\) with \(p\ge (b-a)n\), we find that
$e_p = v_p(F_n) \le 2$.
Thus we bound the summation in the statement of the claim by \begin{equation*}
\sum_{n< p \leq Mn} e_p \log p
\le \sum_{p\le Mn} 2 \log p
\le 2 \log(Mn) \pi(Mn). \end{equation*} The desired result follows from the estimate on \(\pi(x)\) given by the Prime Number Theorem (\autoref{thm:pnt}). \end{proof}
We return to the proof of \autoref{theo:bigprimespolyT}. From Equation~\eqref{eq:logunn}, \cref{claim-small-prime-F}, and \cref{claim-average-prime}, there exist positive constants \(\varepsilon,n_0>0\) such that if \(n>n_0\) then \begin{equation*} \sum_{p > Mn} e_p \log p \geq An\log n - \varepsilon n. \end{equation*}
In turn, the above lower bound entails that for sufficiently large~\(n\), there exist prime divisors \(p \mid F_n\) such that \(p> Mn\). This concludes the proof. \end{proof}
\section{Decidability: different splitting fields} \label{sec-unequalsplitting} In this section we show decidability of the Membership Problem for recurrence sequences that satisfy a first-order relation of the form \eqref{eq:rel} subject to the condition that the polynomial coefficients $f,g\in \mathbb{Z}[x]$ have different splitting fields. To this end, it is useful to introduce the following terminology. Let $p$ be a Hensel prime for $fg$. We say that the recurrence~\eqref{eq:rel} is \emph{$p$-symmetric} if the two polynomials $f$ and $g$ have the same number of roots in $\mathbb{Z}/p\mathbb{Z}$. Otherwise we say that the recurrence is \emph{$p$-asymmetric}.
We first show decidability of the Membership Problem in the case of $p$-asymmetric recurrences and then we apply the Chebotarev Density Theorem to show that every recurrence in which $f$ and $g$ have different splitting fields is $p$-asymmetric for infinitely many primes $p$.
\begin{lemma}\label{lem-p-assym} There is a procedure to decide the Membership Problem for the class of hypergeometric sequences whose defining recurrences are $p$-asymmetric for some prime $p$. \end{lemma}
\begin{proof} Suppose that the hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$ satisfies the recurrence~\eqref{eq:rel} and moreover that there is a prime $p$ with respect to which the recurrence is $p$-asymmetric. We want to decide whether such a sequence reaches a given target value $t$.
Consider the sequences $\langle x_n \rangle_{n=0}^\infty$ and $\langle y_n \rangle_{n=0}^\infty$ respectively defined by the monic recurrences $x_n=g(n)x_{n-1}$, $y_n=f(n)y_{n-1}$, with $x_0=y_0=1$. Then $u_n = \frac{x_n}{y_n}$ and hence, for the aforementioned prime \(p\), \[v_p(u_n)=v_p(x_n)-v_p(y_n) = \sum_{\ell=1}^n (v_p(g(\ell)) - v_p(f(\ell)))\] by the multiplicative property.
Recall that $p$ is, by definition, a Hensel prime for both $f$ and $g$. Hence, by \Cref{prop:moll}, we obtain an asymptotic estimate of the form
\[|v_p(x_n)-v_p(y_n)| = \frac{|m_g-m_f|n}{p-1} + O(\log n)\] where \(m_f\) is the number of roots of \(f\) modulo \(p\) and \(m_g\) is defined similarly. Here the implied constant depends on \(fg\) and \(p\). The proof concludes by noting that $v_p(t)$ is a constant, whereas \(v_p(u_n)\) is bounded away from \(v_p(t)\) for sufficiently large \(n\) (note this threshold is computable). We deduce that \(u_n\neq t\), again, for sufficiently large \(n\), from which the desired result follows. \end{proof}
We now give a sufficient condition for a recurrence to be $p$-asymmetric. We use the following consequence of the Chebotarev Density Theorem. Let $\mathbb{K}$ be a Galois field of degree $d$ over $\mathbb{Q}$, and denote by ${\mathcal{O}}$ its ring of integers. Let ${\mathrm{Spl}}(\mathbb{K})$ be the set of rational primes~$p$ such that the ideal $p{\mathcal{O}}$ totally splits in ${\mathcal{O}}$, i.e., such that \(p{\mathcal{O}} = \mathfrak{p}_1 \cdots \mathfrak{p}_d\) where the $\mathfrak{p}_i$ are distinct prime ideals. The following result appears as \cite[Corollary 8.39]{milneANT} and \cite[Corollary 13.10]{neukirch1999algebraic}.
The latter reference attributes the result to Bauer. \begin{theorem} \label{theo-split-prime} Let $\mathbb{K}$ and ${\mathbb{L}}$ be Galois extensions of~$\mathbb{Q}$ such that $\mathbb{K} \neq {\mathbb{L}}$. Then ${\mathrm{Spl}}(\mathbb{K})$ and ${\mathrm{Spl}}({\mathbb{L}})$ differ in infinitely many primes. \end{theorem}
We state the main theorem of this section. \begin{restatable}{theorem}{theodistinctsplit} \label{theo-distinctsplit}
There is a procedure to decide the Membership Problem for the class of hypergeometric recurrences~\eqref{eq:rel} whose polynomial coefficients have different splitting fields. \end{restatable} \begin{proof}[Proof of \autoref{theo-distinctsplit}] Let $\langle u_n\rangle_{n=0}^\infty$ satisfy a recurrence~\eqref{eq:rel} for which the coefficients $f$ and $g$ have respective splitting fields $\mathbb{K}$ and ${\mathbb{L}}$, with $\mathbb{K}\neq {\mathbb{L}}$. Recall that there are only finitely many primes that are not Hensel primes for~$fg$. By Theorem~\ref{theo-split-prime}, there exists a Hensel prime for $fg$ that lies in exacly one of the two sets ${\mathrm{Spl}}(\mathbb{K})$ and ${\mathrm{Spl}}({\mathbb{L}})$. For such a prime $p$, the recurrence~\eqref{eq:rel} is $p$-asymmetric. Hence the result follows from~\Cref{lem-p-assym}. \end{proof}
We note that the recurrence~\eqref{eq:rel} can be $p$-asymmetric even when $f$ and $g$ have the same splitting field. We demonstrate this phenomenon with the following example.
\begin{example} Let $\langle u_n \rangle_{n=0}^\infty$ be the hypergeometric sequence defined by \begin{equation} \label{exam-rec-1} f(n)u_{n} - g(n)u_{n-1} = 0 \quad \text{and} \quad u_0=1,
\end{equation} where
\begin{equation*}
f(x) := (x^2+1)(x^2-2) \quad \text{ and } \quad
g(x) := x^4-2x^2+9.
\end{equation*}
It is easily checked that both $f$ and $g$ have splitting field $\mathbb{Q}(\sqrt{2},\iu)$. However we show that $f$ and $g$ have different numbers of roots in~$\mathbb{Z}/7\mathbb{Z}$, i.e., the recurrence~\eqref{exam-rec-1} is $7$-asymmetric.
It is straightforward to verify that $7$ is a Hensel prime for~$fg$ by noting that it does not divide the discriminants of the respective irreducible factors of $f$ and $g$. To show that the recurrence is $7$-asymmetric, observe first that $f$ factors as $(x+4)(x+3)(x^2+1)$ over $\mathbb{Z}/7\mathbb{Z}$, where~$x^2+1$ is irreducible; thus $f$ has two roots in $\mathbb{Z}/7\mathbb{Z}$. On the other hand, \(g\) factors into a pair of irreducible quadratic polynomials over $\mathbb{Z}/7\mathbb{Z}$ and hence has no roots.
We can now follow the argumentation of \autoref{lem-p-assym} to decide the Membership Problem for $\langle u_n\rangle_{n=0}^\infty$ with respect to any given target~$t \in \mathbb{Q}$. \color{black} Consider the monic recurrences \(x_n = g(n)x_{n-1}\) and \(y_n = f(n)y_{n-1}\), with initial conditions $x_0=y_0=1$. Note that $v_7(u_n)=v_7(y_n)-v_7(x_n)$. Since $g$ has no roots in $\mathbb{Z}/7\mathbb{Z}$, $v_p(g(k))=0$ for all integers $k> 0$. It follows that \(v_7(x_n) = \sum_{k=1}^n v_7(g(k)) = 0 \) and hence that $v_7(u_n)=v_7(y_n)$.
To obtain bounds on $v_7(y_n)$, note that \(|f(k)|\le n^4\) for all \(n\ge 2\) and \(1\le k\le n\). \autoref{prop:moll} gives the inequality
\begin{equation*}
\frac{2n}{6} - \frac{10\log n}{\log 7} \le v_7(y_n) .
\end{equation*}
For any target $t\in \mathbb{Q}$, the above bound allows us to compute a threshold $B$ such that for all $n>B$ we have Since \(v_7(u_n)=v_7(y_n)>v_7(t)\) and hence $u_n\neq t$. \end{example}
\section{Decidability: quadratic splitting fields} \label{sec-quad}
In this section, we focus on the decidability of the Membership Problem for recurrences \begin{equation} f(n)u_n - g(n)u_{n-1}=0, \qquad u_0=1 \tag{\ref{eq:rel}} \end{equation} in which both $f,g\in \mathbb{Z}[x]$ are monic and split completely over a quadratic (degree-two) extension $\mathbb{K}$ of $\mathbb{Q}$.
Recall that a number field \(\mathbb{K}\) is \emph{quadratic} if and only if there is a square-free integer \(\beta\) such that \(\mathbb{K} = \mathbb{Q}(\sqrt{\beta})\). The assumption that \(f\) and \(g\) are both monic ensures that the roots of both polynomials are algebraic integers in \(\mathbb{Q}(sqrt{\beta}) \). As shown in~\cite[Chapter 3]{stewart2016algebraic}, the following holds.
\begin{theorem} \label{thm:quadratic} Suppose that $\beta\in\mathbb{Z}$ is square-free. Then the ring of algebraic integers in $\mathbb{Q}(\sqrt{\beta})$ has the form $\mathbb{Z}[\theta]$, where \[\theta= \begin{cases}
\sqrt{\beta} & \text{if }\beta \not\equiv 1 \pmod{4},\\[3pt]
\textstyle\frac{\sqrt{\beta}-1}{2} & \text{if } \beta\equiv 1 \pmod{4}. \end{cases} \] \end{theorem}
The main result of the section is as follows.
\begin{theorem} \label{theo-decide-quad} The Membership Problem for recurrences of the form~\eqref{eq:rel} is decidable under the assumption that $f,g$ are both monic and both split over a quadratic extension $\mathbb{K}$ of $\mathbb{Q}$. \end{theorem}
The proof of Theorem~\ref{theo-decide-quad} is given in \Cref{subsec-partition-roots,subsec-thre,subsec-primediv,subsec-conclud}. The details differ slightly according to the two cases for the generator $\theta$ of the ring of integers of $\mathbb{K}$, as presented in Theorem~\ref{thm:quadratic}. In the subsections below, we treat the case for $\theta=\frac{\sqrt{\beta}-1}{2}$. The necessary adjustments for the case $\theta=\sqrt{\beta}$ are given in Appendix~\ref{app-sec-quad}. Henceforth we assume a normalised instance of the Membership Problem, given by the recurrence~\eqref{eq:rel} and target~$t\in \mathbb{Q}$. Our goal is to exhibit an effective bound~$B$ such that $u_n\neq t$ for all $n>B$. To this end, our strategy is to find $B$ such that for all~$n>B$ there exists a prime that divides $u_n$ but not~$t$. At the conclusion of the proof of \autoref{theo-decide-quad}, we demonstrate the argument and techniques with a worked example, namely \autoref{ex:example2} in \Cref{subsec-conclud}.
Let $\beta\equiv 1\pmod{4}$ be a square-free integer and $\mathbb{K}=\mathbb{Q}(\sqrt{\beta})$ a quadratic field over which the polynomials $f$ and $g$ in~\eqref{eq:rel} split completely. Let $\theta:=\frac{\sqrt{\beta}-1}{2}$ be such that $\mathbb{Z}[\theta]$ is the ring of integers of $\mathbb{K}$. Write $m_{\theta}(x):=x^2+x+\frac{1-\beta}{4} \in \mathbb{Z}[x]$ for the minimal polynomial of $\theta$.
\subsection{Partitioning the roots of \texorpdfstring{$fg$}{fg}} \label{subsec-partition-roots} Let $\mathcal R$ be the set of roots of $fg$. We partition \(\mathcal{R}\) into disjoint subsets (which we shall call the \emph{classes} of \(\mathcal{R}\)) with \(\alpha,\tilde{\alpha}\in \mathcal{R}\) in the same class if and only if \(\alpha-\tilde{\alpha}\in\mathbb{Z}\). We say that a subset of $\mathcal{S}\subseteq \mathcal {R}$ is \emph{balanced} if $f$ and $g$ have the same number of roots in $\mathcal S$, counting repeated roots according to their multiplicity. A subset is \emph{unbalanced} otherwise. The linchpin of the proof of \autoref{theo-decide-quad} is the balance of roots in the classes.
If each class (as above) is balanced then the roots of $f$ and $g$ can be placed in a bijection under which corresponding roots differ by an integer and have the same multiplicity in $f$ and $g$ respectively. In this case, by cancelling common factors in the expression $u_n = \prod_{k=1}^n \frac{g({k})}{f({k})}$, we see that for $n$ sufficiently large $u_n$ is a rational function in $n$. For such an instance, the Membership Problem reduces to the problem of deciding whether a univariate polynomial with rational integer coefficients has a positive integer root, which is straightforwardly decidable. A detailed account for this argument is given in~\cite[Appendix B]{NPSW022}.
Let us now consider the case where there is an unbalanced class~\(\mathcal{C}\). By the assumption that $f$ and $g$ have the same degree, there must, in fact, be at least two unbalanced classes. It follows that there is an unbalanced class that is not contained in $\mathbb{Z}$ (i.e., an unbalanced class of quadratic integers).
Here it is convenient to define the following linear ordering on~$\mathcal R$. Given elements $a\theta+b$ and $a'\theta+b'$ in $\mathcal R$ (where \(a,a',b,b'\in\mathbb{Z}\)), define $a\theta+b \prec a'\theta+b'$ if and only if one of the following four mutually exclusive conditions holds: \begin{enumerate} \item $a' \leq 0 < a$, \item $0 < a < a'$, \item $a<a' \leq 0$, \item $a=a'$ and $b<b'$. \end{enumerate}
Note that the classes in $\mathcal R$ are intervals with respect to the order~$\prec$. Thus the order lifts naturally to a linear order on classes. In particular, the \emph{least
unbalanced class}~$\mathcal C_0$ is well-defined. Let $\alpha_0=a_0\theta+b_0$ be the greatest element in $\mathcal C_0$. Then $ \{\alpha\in\mathcal{R}: \alpha \preccurlyeq \alpha_0 \} $ is unbalanced because this set is a disjoint union of balanced classes and~\(\mathcal{C}_0\). Further, $a_0> 0$ because the least unbalanced class is necessarily a subset of quadratic integers of the form \(a_0\theta + \mathbb{Z}\). Here we note that the image of an unbalanced class under the automorphism of~$\mathbb{K}$ that interchanges $\sqrt{\beta}$ and $-\sqrt{\beta}$ is likewise an unbalanced class and so $a_0>0$.
\colorlet{cyan}{purple} \begin{figure*}
\caption{ Image of \(\varphi\) on \(\mathbb{Z}\) as well as the positions of constants used in the proof of \autoref{thm:quadratic} to determine that \(v_p(u_k)\neq 0\) for \(k\) that satisfy \(a_0\theta'+\frac{1}{3}\theta' \le k \le a_0\theta'+\frac{2}{3}\theta'\). Note that the preimages \(\alpha\in\mathcal{R}\) such that \(1\le \varphi(\alpha)\le n\) are precisely those roots for which~$\alpha \preccurlyeq \alpha_0$.}
\label{fig:pretty-things}
\end{figure*}
\subsection{Threshold conditions} \label{subsec-thre}
Next we exhibit a threshold $B$ (defined in terms of the recurrence~\eqref{eq:rel}) such that for all $n>B$ there are rational integers $\theta'$ and $p$, with $p>n$ prime, satisfying the following conditions: \begin{enumerate} \item[(P1)] $m_\theta(\theta')\equiv 0\pmod{p}$; \item[(P2)] The function $\varphi:\mathcal{R}\rightarrow\mathbb{Z}$ defined by \[ \varphi(a\theta+b) = \begin{cases} a\theta' + b & \text{if } a> 0,\\ a\theta' + b + p & \text{if } a\leq 0 \end{cases}\] is an order embedding of $(\mathcal{R},\prec)$ in $(\{0,1,\ldots,p-1\},<)$. \item[(P3)] The set $\{ \alpha \in \mathcal{R} : 1 \leq \varphi(\alpha)
\leq n \}$ is unbalanced. \end{enumerate}
The definitions for $\theta'$ and $p$ follow. Consider the interval \begin{gather} I(n):= \left\{ k\in{\mathbb{N}} : \frac{6n}{3a_0+2}\leq k-1\leq
\frac{6n}{3a_0+1} \right\} \, .
\label{eq:INT}
\end{gather}
and let
$M$ be an upper bound on $\{|a|,|b|: a \theta+b\in \mathcal{R}\}$, and the height of the minimal polynomials of the elements of $\mathcal{R}$. By \autoref{theo:bigprimespolyT}, there is an effective threshold $B$, which we may assume to be greater than $3M(M+1)$, such that for all $n>B$ there exists a prime $p > 3Mn$ that divides the product \[ \prod_{\substack{k\in I(n)\\ k \in 2{\mathbb{N}}+1}} k^2-\beta.\] Furthermore, since \(p\) is prime, we deduce that there exists $k_0 \in I(n) \cap (2{\mathbb{N}}+1)$ such that $k_0^2 \equiv \beta \pmod{p}$. We define $\theta' \in {\mathbb{N}}$ to be the number such that $k_0=2\theta'+1$.
We will show that $\theta'$ and $p$ satisfy Conditions (P1)--(P3). Now \begin{equation*}
m_\theta(\theta') = m_\theta \biggl(\frac{k_0 - 1}{2} \biggr) \equiv k_0^2-\beta \equiv 0 \pmod{p}. \end{equation*} Thus $\theta'$ satisfies Condition (P1).
We turn next to establishing Condition (P2). Since $k_0\in I(n)$ and $k_0=2\theta'+1$, we have \begin{equation} (a_0+\textstyle\frac{1}{3})\theta' \leq n \leq (a_0+\textstyle\frac{2}{3})\theta'.
\label{eq:INEQ} \end{equation} Combining~\eqref{eq:INEQ} with the inequality $1\leq a_0 \leq M$ and rearranging terms gives $\frac{n}{M+1} \leq \theta' \leq \frac{3n}{4}$. Recalling that $p>3Mn$ and $n > B \geq 3M(M+1)$, we conclude that \begin{equation} 3M \leq \theta' \leq \frac{p}{4M} \, . \label{eq:INEQQ} \end{equation} The inequality $\theta'\leq\frac{p}{4M}$ in~\eqref{eq:INEQQ} implies that for all roots \(a\theta + b \in \mathcal{R}\), $\varphi(a\theta+b)$ is equal to
\begin{align*}
a\theta' + b \in &
\left\{0,\ldots,\frac{p-1}{2}\right\} \text{ if } a>0, \text{ and } \\
a\theta' + b + p \in & \left\{\frac{p-1}{2},\ldots,p\right\} \text{ if } a\leq 0
\end{align*} (for the latter, recall that $\mathcal{R}$ contains no positive integers).
Further, since $|b| \leq M<\theta'$ for all $a\theta+b\in \mathcal{R}$, we conclude that $\varphi$ is an order embedding of $(\mathcal{R},\prec)$ into $(\{0,\ldots,p-1\},<)$. This establishes~(P2).
Equation~\eqref{eq:INEQ} and the inequality $\theta' \leq 3M$ from~\eqref{eq:INEQQ} yields \[ \varphi(\alpha_0) < a_0\theta'+M < n < (a_0+1)\theta'-M \, .\] Hence \(\varphi(\alpha_0)\), the image of the greatest element in \(\mathcal{C}_0\) is upper bounded by \(n\). From the definition of the order \((\mathcal{R},\preccurlyeq)\), for $\alpha\in\mathcal R$ we have that $\alpha\preccurlyeq \alpha_0$ if and only if $\varphi(\alpha) \leq n$. Thus (P3) follows from the fact that the set $\{\alpha\in\mathcal{R}:\alpha \preccurlyeq \alpha_0\}$ is unbalanced.
\subsection{Prime divisors of \texorpdfstring{$u_n$}{un}} \label{subsec-primediv}
To conclude the proof, we now explain why properties (P1)--(P3) imply that $p$ divides $u_n$. Define $\psi:\mathbb{Z}[\theta]\rightarrow \mathbb{Z}/p\mathbb{Z}$ by \[ \psi(a\theta+b):=(a\theta'+b)\bmod{p}.\] Condition (P2) entails that $\psi$ and $\varphi$ agree on $\mathcal R$, while Condition (P1) entails that $\psi$ is a ring homomorphism. (We note in passing that the kernel of $\psi$ is a prime ideal~$\mathfrak{p}$ appearing in prime ideal factorisation of $p\mathbb{Z}[\theta]$.) Hence the polynomial $fg$ splits over $\mathbb{Z}/p\mathbb{Z}$ and $\varphi$ maps the roots of $fg$ in $\mathbb{K}$ to roots of $fg$ in $\mathbb{Z}/p\mathbb{Z}$.
Consider the decomposition of the \(p\)-adic valuation \begin{equation*}
v_p(u_n) = \sum_{k=1}^n (v_p(g(k))-v_p(f(k))). \end{equation*}
Let $h(x)$ be an irreducible factor of either $f$ or $g$. Then $h(x)$ is monic, of degree at most $2$ and height at most $M$. Since $p>3Mn$, we easily see that $|h(k)|<p^2$ for all $1\leq k \leq n$ and hence $v_p(h(k)) \in \{0,1\}$. It follows that $v_p(u_n)$ is equal to the number of roots of $g$ in $\mathbb{Z}/p\mathbb{Z}$ that lie in $\{1,\ldots,n\}$ minus the number of roots of $f$ in $\mathbb{Z}/p\mathbb{Z}$ that lie in $\{1,\ldots,n\}$, counting repeated roots according to their multiplicity. Observe that this count takes place on the set \(\{\alpha\in\mathcal{R} : 1\le \varphi(\alpha)\le n\}\). By Condition (P3), the aforementioned set is unbalanced and so it quickly follows that $v_p(u_n)\neq 0$.
\subsection{Concluding the proof of \autoref{theo-decide-quad}} \label{subsec-conclud} Finally, let us return to the decidability of the Membership Problem in the setting of \autoref{theo-decide-quad}. By our standing assumption that all instances of the problem are normalised we have that $t\neq 0$. We have exhibited a bound $B$ such that for all $n>B$ there exists a prime $p>3Mn$ such that $v_p(u_n)\neq 0$. This means that if $p_0$ is the largest prime such that $v_{p_0}(t)\neq 0$ then for $n>\max\left(B,\frac{p_0}{3M}\right)$ we have $u_n\neq t$. Thus we have reduced the Membership Problem in this setting to a finite search problem. This immediately establishes decidability and concludes our proof of \autoref{theo-decide-quad}.
We illustrate the construction underlying \autoref{theo-decide-quad} with a worked example.
\begin{example} \label{ex:example2} Let $\langle u_n \rangle_{n=0}^\infty$ be the hypergeometric sequence defined by the recurrence \begin{equation*} f(n)u_{n} - g(n)u_{n-1} = 0 \quad \text{with} \quad u_0=1,
\end{equation*} where
\(f(x) := x^2 -x -1\) and \(g(x) := x^2 + 2x - 4\).
The polynomials $f$ and $g$ both have splitting field $\mathbb{K} = \mathbb{Q}(\sqrt{5})$, with ring of integers $\mathbb{Z}[\frac{\sqrt{5} - 1}{2}]$. Define $\theta := \frac{\sqrt{5} - 1}{2}$, and write $m_\theta(x) = x^2 + x - 1$ for its minimal polynomial.
Since $f=(x-\theta-1)(x+\theta)$ and $g=(x-2\theta)(x+2\theta+2)$, the set of roots of $fg$ is $\mathcal{R} = \{\theta + 1, -\theta, 2\theta, -2\theta-2\}$. The definition of the linear ordering $\prec$ on $\mathcal{R}$
(see Section~\ref{subsec-partition-roots}) yields \[ \theta + 1 \prec 2\theta \prec -2\theta-2 \prec -\theta \, ,\]
with the least unbalanced class being $\mathcal{C}_0 := \{\theta + 1\}$. Define $M := 4$, which is an upper bound on $\{|a|,|b|: a \theta+b\in \mathcal{R}\}$ and the heights of $f$ and $g$ (which are the respective minimal polynomials of the elements of $\mathcal{R}$).
Write $p_0$ for the largest prime such that $v_{p_0}(t) \neq 0$.
By \Cref{theo:bigprimespolyT}, there is a bound $B> 3M(M+1)$ such that for all $n > \max(B, \frac{p_0}{3M})$, there is a prime $p$ with $v_p(u_n) \neq v_{p}(t)$. This permits us to reduce the Membership Problem for \(\langle {u_n}\rangle_{n=0}^{\infty}\) and $t$ to a finite search problem.
Given a target $t$ and sufficiently large $n$, the process in the proof of \Cref{theo-decide-quad} finds a prime $p$ with $v_p(u_n)\neq v_p(t)$. Below we illustrate the idea of the proof in the specific case $t=\frac{11}{59}$ and $n=61$. (Here we have $p_0=59$ and hence $n>3M(M+1)$ and $n>\frac{p_0}{3M}$, as required in the proof of Theorem~\ref{theo-decide-quad}.) We will establish the existence of a
prime $p$ such that \(v_p(u_{61})\neq 0\) \text{and} \(v_p(t)=0\), witnessing that $u_{61} \neq t$.
Guided by the proof of \Cref{theo:bigprimespolyT}, we observe that prime $p:=1481 > 3n M$ is a divisor of \[ \prod_{\substack{k\in I(61)\\ k \in 2{\mathbb{N}}+1}} k^2-\beta=(75^2 - 5)(77^2 - 5) \cdots(91^2-5).\]
In particular, we have $p|(77^2-5)$. Choosing $\theta':=\frac{77-1}{2}=38$, we observe that the pair $p$ and $\theta' $ satisfy conditions (P1)-(P3) in Section~\ref{subsec-thre}: \begin{enumerate}
\item[(P1)] $m_\theta(\theta')=38^2+38-1\equiv 0\pmod{p}$;
\item[(P2)] The map $\varphi:\mathcal{R}\rightarrow \mathbb{Z}/p\mathbb{Z}$ is an order embedding of $(\mathcal{R},\prec)$ into $(\{0,\ldots,p-1\},<)$, which can be seen by noting that \begin{align*}
\varphi(\theta + 1) = 39 \qquad & \qquad \varphi(2\theta) = 76\\
\varphi(-2\theta -2) = 1403 \qquad & \qquad \varphi(-\theta) = 1443 , \end{align*} whence $\varphi(\theta + 1) < \varphi(2\theta) < \varphi(-2\theta -2) < \varphi(-\theta)$.
\item[(P3)] The set $\{\alpha \in \mathcal{R}:1\leq \varphi(\alpha) \leq 61\} = \{\theta+1\}$ is unbalanced. \end{enumerate}
By the arguments above, in the equation \[ v_p(u_{61}) = \sum_{k=1}^{61} (v_p(g(k))-v_p(f(k))),\] the only non-zero term on the right-hand side is \[ v_p(f(\varphi(\theta + 1))) = v_p(f(39)) = v_p(1481) = 1.\] It follows that $v_p(u_{61}) = -1$, while $v_p(t)=0$.
\end{example}
\section{Discussion} \label{sec:discussion} In light of the results in~\Cref{sec-unequalsplitting} a clear direction for further research is to examine the decidability of the Membership Problem for recurrences whose polynomial coefficients share the same splitting field. We recall that previous work \cite{NPSW022} established decidability when the polynomial coefficients split over the rationals. The present work considers the case when the two polynomials split over the ring of integers of a quadratic field. In future work we will consider the more general case in which the all roots of the coefficient polynomials have degree at most two. As far as the authors are aware, the only known results in this direction are the (un)conditional decidability results for quadratic parameters in~\cite{kenison2022applications}. Extending the approach of the present paper to the case of polynomials with roots of degree more than two would require new results on large prime divisors on the values of such polynomials, which is an active area of research in number theory.
\appendix
\section{Proofs for Section~\ref{sec-polyseq}} \label{sec:app-lowerbownun}
\begin{proof}[Proof of \autoref{claim:sizefunnyQ}] First note that $A=\frac{b-a}{c}$. The claim states that \[ \log(F_n) \geq \frac{2(b-a)}{c} (n\log n - n) \, .\]
The proof is as follows. Given $y\in {\mathbb{N}}$, we first observe that \[\prod_{cx \leq y} (cx)^2 \geq c^{2y}\bigg(\bigg\lfloor\frac{y}{c} \bigg\rfloor!\bigg)^2.\] By Stirling's formula, the logarithm of the quantity above is at least \begin{equation} \label{eq:logQI} \frac{2y}{c}\log c+ \frac{2y}{c} \log y -\frac{2y}{c}. \end{equation}
Now \(F_n = \prod_{k\in I(n)} (k^2+\beta)\) is bounded from below by \begin{equation*}
F_n \geq \prod_{k\in I(n)} k^2
\geq \prod_{ an\leq cx \leq bn} (cx+d)^2
\geq \prod_{ an\leq cx \leq bn} (cx)^2.
\end{equation*} By the above, and Equation~\eqref{eq:logQI} we conclude that \(\log(F_n)\) is bounded from below by \begin{equation*}
\log \prod_{cx \leq bn} c^2x^2 - \log \prod_{cx \leq an} c^2x^2
\geq \frac{2(b-a)}{c}( n \log n -n), \end{equation*} as required. \end{proof}
We now prove the inequality~\eqref{eq:BOUND} from the proof of~\Cref{theo:bigprimespolyT}. Noting that $A=\frac{b-a}{c}$, the inequality states that \begin{equation}
e_p \leq \frac{2An}{p-1}+ \frac{\varepsilon_2 \log n}{\log p} \tag{\ref{eq:BOUND}} \end{equation}
\begin{proof}[Proof of Inequality \eqref{eq:BOUND}]
If $e_p=0$ then the bound trivially holds. Suppose $e_p>0$. Then the function $f$ has two roots in $\mathbb{Z}/p\mathbb{Z}$. Define $g\in\mathbb{Z}[x]$ by $g(x):=f(cx+d)$. In case $p>c$ then $g$ also has two roots in $\mathbb{Z}/p\mathbb{Z}$. For all $n\in{\mathbb{N}}$ define the products \begin{equation*}
G_n:=\prod_{k=1}^{\left\lfloor \frac{bn-d}{c} \right\rfloor} g(k)
\quad \text{and} \quad
H_n:=\prod_{k=1}^{\left\lceil \frac{an-d}{c} \right\rceil - 1} g(k) \end{equation*}
Then $F_n=\frac{G_n}{H_n}$ and hence
$e_p=v_p(F_n)=v_p(G_n)-v_p(H_n)$. Applying \Cref{prop:moll}, we get, for some constant $\varepsilon>0$, \begin{align*} v_p(G_n) \leq & \frac{2(bn-d)}{c(p-1)} + \frac{\varepsilon\log n}{\log p} \qquad {\text{ and } }\\ v_p(H_n) \geq & \frac{2(an-d-c)}{c(p-1)} - \frac{\varepsilon\log n}{\log p}. \end{align*} The upper bound in~\eqref{eq:BOUND} follows, for a suitable choice of the constant $\varepsilon_2$, by subtracting the upper bound for $v_p(G_n)$ from the lower bound for $v_p(H_n)$. \end{proof}
\section{Second Case in the proof of Theorem~\ref{theo-decide-quad}} \label{app-sec-quad}
Let $\beta \not\equiv 1 \pmod{4}$ be a square-free integer and $\mathbb{K}=\mathbb{Q}(\sqrt{\beta})$ a quadratic field over which the polynomials $f$ and $g$ in~\eqref{eq:rel} split completely. By Theorem~\ref{thm:quadratic}, the ring of integers of the field $\mathbb{K}$ is $\mathbb{Z}[\sqrt{\beta}]$. We define $\theta:=\sqrt{\beta}$, so that $m_{\theta}:=x^2-\beta$ is the minimal polynomial of~$\theta$.
Exactly as in Subsection~\ref{subsec-partition-roots}, we partition the set $\mathcal R$ of roots of $fg$ into classes, define the balanced and unbalanced classes, define the linear ordering~$\prec$ on $\mathcal R$, and consider the \emph{least unbalanced class} $C_0$. Let $a_0\theta+b_0$ be the greatest element in $C_0$ and note that $a_0\geq 1$ as before.
\subsection{Threshold conditions} Next we exhibit a threshold $B$ (defined in terms of the recurrence~\eqref{eq:rel}) such that for all $n>B$ there are rational integers $\theta'$ and $p$, with $p>n$ prime, satisfying the three conditions (P1)--(P3) as stated in Subsection~\ref{subsec-thre}.
The definitions for $\theta'$ and $p$ are as follows. Consider the interval \[ I(n):= \left\{ k\in{\mathbb{N}} : \frac{3n}{3a_0+2}\leq k \leq
\frac{3n}{3a_0+1} \right\} \]
and let
$M$ be an upper bound on $\{|a|,|b|: a \theta+b\in \mathcal{R}\}$, and the height of the minimal polynomials of the elements of $\mathcal{R}$. By \autoref{theo:bigprimespolyT}, there is an effective threshold $B$, which we may assume to be greater than $3M(M+1)$, such that for all $n>B$ there exists a prime $p > 3Mn$ that divides the product \[ \prod_{k\in I(n)} k^2-\beta.\] Further, since \(p>3Mn\) is prime, we deduce that for \(n>B\) there exists $\theta' \in I(n)$ such that $(\theta')^2 \equiv \beta \pmod{p}$.
We will show that $\theta'$ and $p$ satisfy Conditions (P1)--(P3). Now \begin{equation*}
m_\theta(\theta') \equiv (\theta')^2-\beta \equiv 0 \pmod{p}. \end{equation*} Thus $\theta'$ satisfies Condition (P1).
We turn next to establishing Condition (P2). Since $\theta'\in I(n)$, it is straightforward to show that \begin{equation} (a_0+\textstyle\frac{1}{3})\theta' \leq n \leq (a_0+\textstyle\frac{2}{3})\theta' \, .
\label{app-eq:INEQ} \end{equation} These
bounds are identical to those in~\eqref{eq:INEQ}.
In this case, Conditions (P2) and (P3) follow by an analogous argument to that given in Subsection~\ref{subsec-thre}.
The remaining part of the proof for the case $\beta\equiv 1 \pmod{4}$, as given in Subsection~\ref{subsec-primediv} and Subsection~~\ref{subsec-conclud}, carries over to the present case without change.
\end{document} |
\begin{document}
\title{ Convolution and square in abelian groups I } \author{Yves Benoist } \date{}
\maketitle
\begin{abstract} \noindent We prove that the functional equation $f\!\star\! f (2\,t) =\la f(t)^2$, for $t$ in $\m Z/d\m Z$ with $d$ odd, admits a non-zero solution $f$ if $\la=\sqrt{a}\!+\! i\sqrt{b}$ with $a,b$ positive integers such that $a\!+\! b\!=\!d$ and $a\!\equiv\frac{(d+1)^2}{4}$ mod $4$. The proof involves theta functions on elliptic curves with complex multiplication. \end{abstract}
\renewcommand{\arabic{footnote}}{\fnsymbol{footnote}} \footnotetext{\emph{2020 Math. subject class.} Primary 11F03~; Secondary 11F27} \footnotetext{\emph{Key words} Functional Equation, Convolution, Cyclic groups, Elliptic curves, Complex multiplication, Theta functions, Modular curve.} \renewcommand{\arabic{footnote}}{\arabic{footnote}}
\section{The functional equation}
It is well known that elliptic curves, theta functions and modular forms are very useful tools both in Algebraic Geometry and in Number Theory -see \cite{Farkas08}. In this paper, we focus on an elementary open problem raised by numerical experiments. The main surprise in this text is that the intriguing output of these numerical experiments cannot be understood without these tools.
\subsection{Introduction}
We will deal with a finite abelian group $G$ of odd order $d$, which, most of the time, will be the cyclic group $G=\m Z/d\m Z$, and with the functional equation \begin{equation} \label{eqntante} f*f\,(2t) =\la \, f^2(t)
\;\;\; \mbox{\rm for all $t$ in $G$,} \end{equation} where the unknown is a non-zero function $f: G\rightarrow \m C$ and where $\la\in \m C$ is a parameter. This equation expresses a proportionality condition between the ``convolution square'' of $f$ and its ``multiplication square''. A non-zero solution $f$ of this functional equation \eqref{eqntante} will be called a ``$\la$-critical function on $G$'' or, in short, a ``$\la$-critical function'', and a value $\la$ for which such a function $f$ exists will be called a ``critical value on $G$'', or a ``$d$-critical value'' when $G=\m Z/d\m Z$. Note that Equation \eqref{eqntante} has been chosen so that it is invariant by translation on the variable $t$. This equation \eqref{eqntante} can be rewritten as \begin{equation} \label{eqntante2} \textstyle \sum\limits_{\ell\in G}f(k\!+\!\ell)\,f(k\!-\!\ell) =\la \, f(k)^2
\;\;\; \mbox{\rm for all $k$ in $G$.} \end{equation}
The aim of this text is to point out the interest of this functional equation by gathering unexpected results and questions based on numerical experiments and by relating this apparently naive functional equation to elliptic curves with complex multiplication.
Indeed, our main result, Theorem \ref{thmrapirb}, gives explicit $d$-critical values. Eventhough the statement of this theorem is very short and purely elementary, surprisingly, our construction relies on the Jacobi theta functions $\theta(z,\tau)$ for special values of the parameter $\tau$, and the key point in the proof relies on modularity properties due to Hecke (Lemma \ref{lemtrafor}) of these theta functions.
I tried to keep this text as elementary and concrete as possible. In a second more technical paper \cite{CSAGII}, I will extend this construction of critical values to all finite abelian groups $G$ by using the Riemann theta functions on higher dimensional abelian varieties, and their modularity properties as functions on the Siegel upper half-space -- see Theorem \ref{thmmaivap}.
I thank G\'{e}rard Laumon, Samuel Leli\`evre and Emmanuel Ullmo for enlighting discussions on this project.
\subsection{Comments} \hspace*{1.5em}$\star$ Since $d$ is odd, the value $\la=0$ is not $d$-critical. Indeed the only function $f$ for which $f*f=0$ is $f=0$.
$\star$ By analogy, one might look at Equation \eqref{eqntante} on locally compact abelian groups $G$. When the group is $G=\m R$ or $G=\m Z$, the only $L^2$-solutions that I know are gaussian functions $f(t)=e^{at^2+bt+c}$ where $a,b,c \in \m C$, $Re(a)<0$, together with, when $G=\m Z$, their restrictions to subgroups. When the group is $G=\m R/\m Z$, the only $L^2$-solutions that I know are constant functions.
\subsection{Special critical values} \label{secspecri} \noindent
In this section, we list a few $d$-critical values that are easy to find. We call them special.
$\star$ If we choose $f(0)=1$ and $f(k)=\alpha} \def\be{\beta} \def\ga{\gamma$ to be constant for $k$ in $G\smallsetminus\{0\}$, we find four critical values: we find $\la = 1$ when $\alpha} \def\be{\beta} \def\ga{\gamma=0$, we find $\la=d$ when $\alpha} \def\be{\beta} \def\ga{\gamma=1$, and we find $\la=\frac{d-3+\eps\sqrt{D}}{2}$ when $\alpha} \def\be{\beta} \def\ga{\gamma=\frac{1-d-\eps\sqrt{D}}{2(d-1)}$, with $\eps=\pm 1$ and $D=(d\!-\!1)(d\!-\!9)$. Note that these $d$-critical values are real as soon as $d\geq 9$.
$\star$ If we choose $f$ to be a gaussian function $f(k):=\eta^{-k^2}$ with $\eta:=-e^{i\pi/d}$, we find the critical value $\la=\sqrt{d}$ when $d\equiv 1\,{\rm mod}\, 4$ and $\la= i\sqrt{d}$ when $d\equiv 3$ mod $4$. Moreover, its opposite $-\la$ is also often a critical value. This is the case when $d$ is not a square. But this is not always the case, for instance, when $d=9$, the value $-3$ is not $d$-critical.
\subsection{Induced critical values} \noindent
In this section we explain how to construct $d$-critical values when $d$ is a composite number starting from critical values for the factors of $d$. The method works for any abelian groups.
$\star$ Let $G_1\subset G$ be finite abelian groups and $\la_1$ be a critical value on $G_1$ with $\la_1$-critical function $f_1$. Then $\la_1$ is also a critical value on $G$ with $\la$-critical function $f:=f_1{\bf 1}_{G_1}$.
$\star$ Let $G_1\subset G$ be finite abelian groups, $d_1$ be the order of $G_1$ and $\la_2$ be a critical value on the quotient $G/G_1$ with $\la_2$-critical functions $f_2$. Then $\la:=d_1\la_2$ is a critical value on $G$ with $\la$-critical function $f:=f_2\circ \pi$ where $\pi:G\rightarrow G/G_1$ is the projection.
$\star$ Let $G=G_1\times G_2$ be the product of two finite abelian groups, let $\la_1$ be a critical value on $G_1$ with $\la_1$-critical function $f_1$ and $\la_2$ be a critical value on $G_2$ with $\la_2$-critical function $f_2$. Then the product $\la:=\la_1\la_2$ is a critical value on $G$ with $\la$-critical function $f:=f_1\otimes f_2$.
The most interesting critical values will be those that are not special and that are not obtained by these ``induction'' methods.
\subsection{Numerical experiments} \label{secnumexp}
\noindent The following lists of $d$-critical values rely on numerical experiments using the Buchberger's algorithm for computing Groebner basis (see \cite[Chap. 2]{CLO92}). We denote by $b_d$ the size of the list.
For $3\leq d\leq 9$ the complete lists of $d$-critical values are:
\noindent $\star$ \fbox{$d=3$} $b_3=4$.\; $\la=1$, $3$, and $\pm i\sqrt{3}$.
\noindent $\star$ \fbox{$d=5$} $b_5=6$.\; $\la=1$, $5$, $\pm \sqrt{5}$, and $1\pm 2i$.
\noindent $\star$ \fbox{$d=7$} $b_7=8$.\; $\la=1$, $7$, $\pm i \sqrt{7}$, and $\pm 2\pm i\sqrt{3}$.
\noindent $\star$ \fbox{$d=9$} $b_9\!= \!15$.\; $\la=1$, $9$, $3$, $\pm i\sqrt{3}$, $\pm 3i\sqrt{3}$, $\pm 1\pm 2i\sqrt{2}$, and $\pm \sqrt{5}\pm 2i$.
\noindent For $d=11$ and $ 13$, the lists below of $d$-critical values are still probably complete.
\noindent $\star$ \fbox{$d=11$} $b_{11}\!=\!20 $.\; $\la =1$ , $11$, $4\pm \sqrt{5}$, $\pm i\sqrt{11}$,\\ \hspace*{1em} $2\pm i\sqrt{7}$, $\pm 2\sqrt{2}\pm i\sqrt{3}$, and $\pm (1\!+\!\eps\sqrt{5})\pm i\sqrt{5\!-\!2\eps \sqrt{5}}$ with $\eps=\pm 1$.
\noindent $\star$ \fbox{$d=13$} $b_{13}\!=\! 18$.\; $\la =1$ , $13$, $5\pm 2\sqrt{3}$, $\pm i\sqrt{13}$,\\ \hspace*{1em} $\pm 1\pm 3i\sqrt{2}$, $\pm \sqrt{5}\pm 2i\sqrt{2}$, and $\pm 3\pm2i$.
\noindent For $d=15$ and $17$, here are just a few $d$-critical values. These values were obtained by the guess and check method. Looking at the critical values for $d\leq 13$ one can guess a few critical value for $d=15$ and $17$. The key point of the method is that the Buchberger's algorithm is much faster when checking if a given $\la$ is $d$-critical than when finding all the $d$-critical values.
\noindent $\star$ \fbox{$d=15$} $b_{15}\!=\! 60$.\; $\la=$ product of a $3$-critical and a $5$-critical value, and\\ \hspace*{1em} $\la=-3$, $-5$, $6\pm \sqrt{21}$, $\pm 2\pm i\sqrt{11}$, $\pm 2\sqrt{2}\pm i\sqrt{7}$, $\pm 2\sqrt{3}\pm i\sqrt{3}$, and\\ \hspace*{1em} $\pm \!2\sqrt{2\!-\!\eps\sqrt{3}} \pm (2\!+\eps\!\sqrt{3})i$ and $1\!+\!\eps\sqrt{5}\pm i\sqrt{9\!-\!2\eps \sqrt{5}}$ with $\eps=\pm 1$, and\\ \hspace*{1em} $\pm(\sqrt{3}\pm i\sqrt{2})(\sqrt{2}\pm i)$.
\noindent $\star$ \fbox{$d=17$} $b_{17}\!=\! 28$.\; $\la =1$ , $17$, $7\pm 4\sqrt{2}$, $\pm i\sqrt{17}$,\\ \hspace*{1em} $\pm 1\pm 4i$, $\pm \sqrt{5}\pm 2i\sqrt{3}$, $3\pm 2i\sqrt{2}$, $\pm \sqrt{13}\pm 2i$, and\\ \hspace*{1em} $\pm(1+2\eps\sqrt{2})\pm 2i\sqrt{2-\eps\sqrt{2}}$ with $\eps=\pm 1$.
\noindent For our numerical experiments we used SageMath and Maple softwares.
\section{ Critical values}
One of the motivations of Proposition \ref{procrival} and Theorem \ref{thmrapirb} below is to explain some of the intriguing patterns that occur in these experimental lists of critical values.
\subsection{Properties of critical values}
We first begin by a few properties of the critical values, that are valid on any finite abelian group.
\begin{Prop} \label{procrival} Let $G$ be a finite abelian group of odd order $d$, and $\la$ a critical value on $G$, then:\\ $(i)$ all the Galois conjugate of $\la$ are also critical values on $G$,\\
$(ii)$ one has $|\la|\leq d$ with equality if and only if $\la=d$,\\ $(iii)$ the ratio $d/\la$ is also a critical value on $G$,\\ $(iv)$ The ratio $\frac{\la -1}{2}$ is an algebraic integer. \end{Prop}
\begin{Cor} \label{corcrival} There exist only finitely many critical values on $G$. \end{Cor}
\begin{proof}[Proof of Corollary \ref{corcrival}] Since it is obtained by elimination, the set of critical values on $G$ is either finite or its complement in $\m C$ is finite. Since, by $(ii)$ it is bounded, it must be finite. \end{proof}
\begin{proof}[Proof of Proposition \ref{procrival}] $(i)$ Equations \eqref{eqntante2} have rational coefficients.
$(ii)$ This follows from Cauchy-Schwarz inequality. Indeed, setting\\
$\|f\|_\infty=\max\limits_{k\in G}|f(k)|$ and $\|f\|_2=(\sum_k|f(k)|^2)^{\frac12}$, one has $$
|\la|\|f\|^2_\infty=\| f\star f\|_\infty\leq \| f\|_2^2
\leq d\, \| f\|_\infty^2. $$
Hence $|\la|\leq d$. In case we have equality the function $f$ must have constant modulus, and must satisfy $f(k\!+\!\ell)f(k\!-\!\ell)=f(k)^2$, for all $k$, $\ell$. Hence $f$ is proportional to a character and one has $\la=d$.
$(iii)$ If $f$ is a $\la$-critical function on $G$, then its Fourier transform $\widehat{f}$, which is given by, for every character $\chi:G\rightarrow \m C^*$, $$ \textstyle \widehat{f}(\chi)= \sum\limits_{x\in G}f(x)\chi(x), $$ is a $d/\la$-critical function on the dual group $\widehat{G}$ which is isomorphic to $G$.
$(iv)$ Let $G_+$ be a subset of $G$ of cardinality $\frac{d-1}{2}$ such that for each non-zero $\ell\in G$ either $\ell$ or $-\ell$ is in $G_+$. The equations \eqref{eqntante2} can be rewritten as \begin{equation} \label{eqntante3} \tfrac{\la -1}{2} \, f(k)^2 = \textstyle\sum\limits_{\ell\in G_+}f(k\!+\!\ell)\,f(k\!-\!\ell)
\;\;\;\;\;\; \mbox{\rm for all $k$ in $G$} \end{equation} Let $K$ be the subfield of $\m C$ generated by the coefficients $f(k)$. To prove that $\la':=\tfrac{\la -1}{2}$ is an algebraic integer, it is enough to check that, for all non-archimedean absolute value
$|.|_v$ on $K$, one has $|\la'|_v\leq 1$.\\
We set $\| f\|_v:= \max\limits_{\ell\in G} |f(\ell)|_v$, we choose $k$ such that
$\|f\|_v=|f(k)|_v$, and we compute \begin{eqnarray*}
|\la'|_v \|f\|_v^2=|\la'f(k)^2|_v&=&|\textstyle\sum\limits_{\ell\in G_+} f(k\!+\!\ell)f(k\!-\!\ell)|_v\\
&\leq& \max\limits_{\ell\in G}\, |f(k\!+\!\ell)|_v|f(k\!-\!\ell)|_v
\;\leq\; \|f\|_v^2. \end{eqnarray*}
This proves that $|\la'|_v\leq 1$ as required. \end{proof}
\subsection{Construction of critical values}
From now on, $G$ will be the cyclic group $\m Z/d\m Z$. It is not clear from the definition that there does exist $d$-critical values that are non-induced and non-special. The following theorem tells us that this is always the case for $d\geq 5$.
\begin{Thm} \label{thmrapirb} Let $a$,$b$ be positive integers with $a\!+\!b\!=\!d$ and $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$. Then the complex number $\la:=\sqrt{a}+i\sqrt{b}$ is a $d$-critical value. \end{Thm}
\begin{Rem} The congruence assumption in Theorem \ref{thmrapirb} is equivalent to \begin{eqnarray} \label{eqnabmabm} a-b\equiv 1 \;\mbox{\rm mod}\; 4 &{\rm and}& ab \equiv 0\;\mbox{\rm mod}\; 4. \end{eqnarray} A more concrete way to state Theorem \ref{thmrapirb} is:
For $d\equiv 1$ mod $4$, the following values are $d$-critical:\\ $\sqrt{d}$ ,\; $\sqrt{d\!-\!4}\!+\!2i$ ,\; $\sqrt{d\!-\!8}\!+\! 2i\sqrt{2}$ ,\; $\sqrt{d\!-\!12}\!+\!2i\sqrt{3}$ , ...
For $d\equiv 3$ mod $4$, the following values are $d$-critical:\\ $i\sqrt{d}$ ,\; $2\!+\!i\sqrt{d\!-\!4}$ ,\; $2\sqrt{2}\!+\!i\sqrt{d\!-\!8} $ ,\; $2\sqrt{3}+i\sqrt{d\!-\!12}$ , ... \end{Rem}
More precisely, we will see that, surprisingly, for these values $\la$, the set of $\la$-critical functions has positive dimension. Indeed, we will construct a one-parameter family of $\la$-critical functions using a suitable Jacobi theta function.
Before that we discuss the above congruence condition on $a$.
\begin{Lem} \label{lemalgint} Let $a$,$b$ be positive integers with $a\!+\!b\!=\!d$ and let $\la:=\sqrt{a}+i\sqrt{b}$. The number $\frac{\la-1}{2}$ is an algebraic integer if and only if $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$. \end{Lem}
In particular, by Proposition \ref{procrival}.$iv$, when $a\!\not\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$, the complex number $\la=\sqrt{a}+i\sqrt{b}$ can not be a $d$-critical value.
\begin{Rem} Note that, for any algebraic number $\la$, one has the equivalence: \begin{eqnarray} \nu\!:=\!\tfrac{\la-1}{2}\; \mbox{\rm is an algebraic integer} \;\Longleftrightarrow\; \nu'\!:=\!\tfrac{\la^2-1}{4}\; \mbox{\rm is an algebraic integer}.&& \;\;\mbox{ } \end{eqnarray} Indeed, theses two elements $\nu$ and $\nu'$ are related by the equation $\nu^2+\nu=\nu'$. \end{Rem}
\begin{proof}[Proof of Lemma \ref{lemalgint}] The number $\nu'\!:=\!\tfrac{\la^2-1}{4}$ is equal to $\nu'=\tfrac{a-b-1}{4}+i\tfrac{\sqrt{ab}}{2}$. It is an algebraic integer if and only if $a-b\equiv 1 \;\mbox{\rm mod}\; 4$ \; {\rm and}\; $ab \equiv 0\;\mbox{\rm mod}\; 4$. As seen in \eqref{eqnabmabm}, this condition is equivalent to $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$. \end{proof}
\begin{Cor} \label{corrapirb2} Let $p$,$q$ be positive integers with $p$ odd and $q$ even and let $d:=p^2+q^2$. Then the complex number $\la:=p+iq$ is a $d$-critical value. \end{Cor}
\begin{proof} Condition \eqref{eqnabmabm} is true: $p^2-q^2\equiv 1$ mod $4$\; and\; $p^2q^2 \equiv 0$ mod $4$. \end{proof}
\subsection{More numerical experiments}
A reasonable aim in this topic would be to give, for each $d$ the list of the $d$-critical values $\la$, and for each $\la$ the description of the projective algebraic variety given by the $\la$-critical functions (number of connected components, their dimension,...).
Here are a few less ambitious questions supported by numerical experiments that suggest that some hidden structure has to be understood.
The first question deals with the properties of the $\la$-critical functions when $\la$ belongs to a real or imaginary quadratic number field.
\begin{Ques} \label{concrisym} Let $d$ be an odd integer and $\la$ a $d$-critical value which is quadratic. Does there exist an even $\la$-critical function? \end{Ques}
That is a $\la$-critical function $f$ such that $f(-k)=f(k)$ for all $k$.
We checked, using numerical experiments, that this is true for $d\leq 11$. Unfortunately this is not true when $\la$ is not quadratic as $2\sqrt{2}-1+2i\sqrt{2\!+\!\sqrt{2}}$.
The second question deals with the properties of the $d$-critical values.
\begin{Ques} \label{condslgal} Let $d$ be an odd prime and $\la$ a $d$-critical value.\\ Are $\la$ and $d/\la$ Galois conjugate, except for $\la=1$ and $\la=d$?
\end{Ques}
More generally, when $d$ is not prime, one might still expect a similar question to be true for {\it non-induced} $d$-critical values.
Note that we do not expect all the Galois conjugates in $\m C$ of the non-real critical values $\la$ to have absolute value equal to $\sqrt{d}$. We computed, using numerical experiments, an example of $d$-critical value $\la$ of degree $8$ over $\m Q$ with two real and six non-real Galois conjugates in $\m C$.
The third question deals with critical values that are real quadratic.
\begin{Ques} \label{conquacri} Let $d$ be an odd prime and $\la$ a $d$-critical value which is a real quadratic number. Is such a $\la$ special? \end{Ques}
We recall that special means that $\la=1$, $d$, $\pm \sqrt{d}$, or $\frac{d-3\pm\sqrt{(d-1)(d-9)}}{2}$.
Note that the $d$-critical values which are quadratic over $\m Q$ and non-real can be described up to sign thanks to Proposition \ref{procrival} and Theorem \ref{thmrapirb}.
We checked, using numerical experiments, that this is true for $d\leq 11$.
The last question deals with critical values that are quadratic over $\m Q$ but are non real. More precisely it deals with the sign of $Re(\la)$ in Theorem \ref{thmrapirb}.
\begin{Ques} \label{consgncri} Let $d$ be an odd integer and $a$,$b$ be positive integers with $a\!-\! b\!\equiv\! \frac{d^2+1}{2}\,{\rm mod}\, 8$ and $a\!+\!b\!=\!d$. If $d\not\equiv 2$ mod $3$, is the number $\la:=-\sqrt{a}+i\sqrt{b}$ a $d$-critical value? \end{Ques}
Moreover, for each $d\equiv 2$ mod $3$, one still expects the answer to this question to be true with at most one exception. Recall that the interesting case is when the critical value $\la$ is quadratic over $\m Q$ i.e. when the integer $a$ is a square (see Remark \ref{remmla}).
We checked, using numerical experiments, that $-1+2i$ and $-2+i\sqrt{7}$ are not critical values and that, for $d\leq 23$, the only other possible exceptions are $-3+2i\sqrt{2}$ and $-4+i\sqrt{7}$.
There is another similar question: let $d=a^2$ be the square of an odd integer $a\geq 3$. We know that this integer $a$ is a $d$-critical value. But when is its opposite $-a$ also $d$-critical? I checked, using numerical experiments, that $-3$ is not $9$-critical but that $-5$ is $25$-critical.
\section{Theta functions and elliptic curves} Our aim now is to prove Theorem \ref{thmrapirb}.
\subsection{Main result}
We recall the definition of the Jacobi theta function: $$\textstyle \theta_\tau (z)=\theta(z,\tau):=\sum\limits_{m\in \m Z}e^{i\pi\tau m^2}e^{2i\pi mz}, \;\;\mbox{\rm for $z\in \m C$ and $\tau\in \m H$,} $$ where $\m H$ is the upper half plane $\m H=\{ \tau\in \m C\mid Im(\tau)>0\}$. This function is $1$-periodic: $\theta_\tau(z+1)=\theta_\tau(z)$. We can now explain our construction of $\la$-critical functions on $\m Z/d\m Z$.
\begin{Def} \label{defrapirb} We will say that the function $\theta_{\tau}$ is $(\la,d)$-critical if, for all $z$ in $\m C$, the function $ f_{z,\tau}:\ell\mapsto \theta(z\!+\!\ell/d,\tau) $ is $\la$-critical on $\m Z/d\m Z$. \end{Def}
This means that, for all $z$ in $\m C$, $$\textstyle \sum\limits_{\ell\in\m Z/d\m Z} \theta(z+\ell/d,\tau)\,\theta(z-\ell/d,\tau) \;=\;\la\,\theta(z,\tau)^2. $$
Theorem \ref{thmrapirb} is a special case of the following Proposition \ref{prorapirb}.$(a)$. The whole Proposition \ref{prorapirb} tells us more. It tells us exactly for which parameters $d$, $\la$, $\tau$, the function $\theta_\tau$ is $(\la,d)$-critical.
\begin{Prop} \label{prorapirb} Let $a$,$b$ be positive integers with $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$ and $a\!+\!b\!=\!d$. Set $\la_0:=\sqrt{a}+i\sqrt{b}$ and \begin{eqnarray} \label{eqntau0} \tau_0&:=&\tfrac{1}{4d^2}(a-b-d^2+2i\sqrt{ab}). \end{eqnarray} $(a)$ The function $\theta_{\tau_0}$ is $(\la_0,d)$-critical.\\ $(b)$ Conversely, let $\tau\in \m H$ such that the function $\theta_\tau$ is $(\la,d)$-critical, then one has $\la=\pm\sqrt{a}\pm i\sqrt{b}$ with $a$, $b$ as above.\\ $(c)$ The function $\theta_\tau$ is $(\la,d)$-critical for $\la=\pm\la_0$ if and only if $\tau=(k+\tau_0)/p$ with $\tau_0$ as above, $k\in \m Z$ and $p>0$ a
divisor of the integer $N_k:=d^2\, |k\!+\! \tau_0|^2$.\\ $(d)$ The above sign $\eps=\pm$ is given by the Jacobi symbol $\eps=(\!\frac{p}{4k-1}\!)$. \end{Prop}
Remember that the Jacobi symbol $(\!\frac{\ga_0}{\delta} \def\eps{\varepsilon} \def\ze{\zeta_0}\!)=\pm 1$ is defined for two relatively prime integers $\ga_0$ and $\delta} \def\eps{\varepsilon} \def\ze{\zeta_0$ with $\delta} \def\eps{\varepsilon} \def\ze{\zeta_0$ odd, and that, by convention, when $\delta} \def\eps{\varepsilon} \def\ze{\zeta_0$ is negative, it is given by $(\!\frac{\ga_0}{\delta} \def\eps{\varepsilon} \def\ze{\zeta_0}\!)=(\!\frac{\ga_0}{-\delta} \def\eps{\varepsilon} \def\ze{\zeta_0}\!)$.
Note also that, in view of Point $(b)$, the assumption $\la=\pm \la_0$ in Point $(c)$ is not restrictive since one has the equivalence: $$\mbox{$\theta_\tau$ is $(\la,d)$-critical\; $\Longleftrightarrow$ \; $\theta_{-\ol{\tau}}$ is $(\ol{\la},d)$-critical.} $$
\begin{Rmq} The parameter $\tau_0$ will be called the {\it fundamental parameter} and the parameters $\tau_{k,p}:= (k+\tau_0)/p$ the {\it associated parameters}. These parameters $\tau_{k,p}$ and the integers $N_k$ can also be given by the simple formulas with $m_0:=\frac{a-b-d^2}{4}$ and $N_0:=\frac{(d+1)^2-4a}{16}$: $$ \fbox{$\tau_{k,p}=\frac{1}{d^2p}(d^2k+m_0+i\sqrt{ab/4})$} \;\; {\rm where}\;\;
\fbox{$p\, | \, N_k:=d^2k^2+2m_0k+N_0$}. $$
\end{Rmq}
\begin{Exa} To be very concrete, we give below the list of all values $\tau=\tau_{k,p}$ for which $\theta_\tau$ is $(\pm\la_0,d)$-critical with $k$ in $\m Z$ and $p$ divisor of $N_k$, when $d\leq 9$.
\noindent $\star$ $d=5$, $\la_0=1+2i$\;\; : $\tau_{k,p}=\tfrac{1}{25 p}(25k - 7+i)$\;\; \;\, where\;
$p\,|\, 25k^2\!-\! 14k\!+\! 2$.\\ $\star$ $d=7$, $\la_0=2+i\sqrt{3}$:
$\tau_{k,p}=\tfrac{1}{49 p}(49k\!-\! 12+i\sqrt{3})$ where\; $p\,|\, 49k^2\!-\! 24k\!+\! 3$.\\ $\star$ $d=9$, $\la_0=1\!+\! 2i\sqrt{2}$:
$\tau_{k,p}=\tfrac{1}{81 p}(81k\!-\! 22+i\sqrt{2})$ where\; $p\,|\, 81k^2\!-\! 44k\!+\! 6$.\\ $\star$ $d=9$, $\la_0=\sqrt{5}+2i$:
$\tau_{k,p}=\tfrac{1}{81 p}(81k\!-\! 20+i\sqrt{5})$ where\; $p\,|\, 81k^2\!-\! 40k\!+\! 5$. \end{Exa}
Given $d$ and $\la_0$, we have seen that it is always possible to choose $k$ and $p$ such that $\la=\eps\la_0$ with sign $\eps=(\!\frac{p}{4k-1}\!)$ equal to $+1$: we just choose $p=1$.
On the other hand, given $d$ and $\la_0$, it is sometimes possible to choose $k$ and $p$ such that $\la=\eps\la_0$ with sign $\eps=(\!\frac{p}{4k-1}\!)$ equal to $-1$. For instance,\\ $\star$ when $\la= -2-i\sqrt{3}$ with $\tau=\tfrac{1}{98}(37+i\sqrt{3})$, or \\ $\star$ when $\la=-\sqrt{5}-2i$ with $\tau=\tfrac{1}{162}(61+i\sqrt{5})$.\\ We mention the following corollary that tells us exactly when this is possible.
\begin{Cor} \label{corrapirb} Let $a$,$b$ be positive integers with $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$ and $a\!+\!b\!=\!d$. Set $\la:=-\sqrt{a}-i\sqrt{b}$. There exists $\tau$ in $\m H$ whose function $\theta_\tau$ is $(\la,d)$-critical if and only if either $a$ is not a square in $\m Z$ or $-b$ is not a square in $\m Z/2a\m Z$. \end{Cor}
Concretely, here are a few special cases of our criteria for such a function $\theta_\tau$ to exist, in which $a$ is a square in $\m Z$ and $b$ is either $4\ell$ or $4\ell-1$ with $\ell$ a positive integer:\\ $\star$ When $\la=-1-2i\sqrt{\ell}$ : never.\\ $\star$ When $\la=-3-2i\sqrt{\ell}$ : $\ell\equiv 1$ mod $3$.\\ $\star$ When $\la=-5-2i\sqrt{\ell}$ : $\ell\equiv 2$ or $3$ mod $5$.\\ $\star$ When $\la=-2-i\sqrt{4\ell-1}$: $\ell\equiv 1$ mod $2$.\\ $\star$ When $\la=-4-i\sqrt{4\ell-1}$: $\ell\equiv 1$ mod $2$.\\ $\star$ When $\la=-6-i\sqrt{4\ell-1}$: $\ell\equiv 1,2,3$ or $5$ mod $6$.
The proof of this corollary that relies on quadratic reciprocity is left to the reader. We will not use it below. The rest of the paper is devoted to the proof of Proposition \ref{prorapirb}.
\subsection{Preliminary formulas}
The proof of Proposition \ref{prorapirb}.$(a)$ relies on three classical formulas for the theta functions, the ``addition formula'', the ``isogeny formula'', the ``transformation formula''. We will only need special cases of these formulas that we state below.
We need to introduce the theta functions\footnote{\noindent With other ''classical'' notations for theta functions as in \cite{BiLa04}, one has the equalities,\\ $\theta_{[0]}(z,\tau)=\theta_{0,0}(2z,2\tau)= \theta \mbox{\scriptsize $\left[\!\!\! \begin{array}{c} 0 \\0\end{array}\!\!\! \right]$} (2z,2\tau)$ and $\theta_{[1]}(z,\tau)=\theta_{1,0}(2z,2\tau)= \theta \mbox{\scriptsize
$\left[\!\!\!\!
\begin{array}{c} 1/2\\0\end{array}\!\!\!\!
\right]$} (2z,2\tau).$ } \begin{eqnarray*} \theta_{[0]}(z )=\theta_{[0]}(z,\tau) &:=&\textstyle \sum\limits_{m\,\rm even}e^{i\pi\frac{\tau}{2} m^2}e^{2i\pi mz} \\ \theta_{[1]}(z )=\theta_{[1]}(z,\tau) &:=&\textstyle \sum\limits_{m\,\rm odd}e^{i\pi\frac{\tau}{2} m^2}e^{2i\pi mz}. \end{eqnarray*}
Note that one has the equalities: \begin{eqnarray} \label{eqnt0t1t2}\textstyle \theta_{[0]}(z,\tau)=\theta(2z,2\tau) &\;{\rm and}\;& \theta_{[0]}(z,\tau)+\theta_{[1]}(z,\tau)=\theta(z,\tau/2). \end{eqnarray}
Here is the first formula that we need.
\begin{Lem} \label{lemaddfor} {\bf Addition formula} For all $z,w$ in $\m C$, $\tau\in \m H$, one has \begin{eqnarray} \label{eqnaddfor} \theta(z+w,\tau)\theta(z-w,\tau)&=& \theta_{[0]}(w,\tau )\theta_{[0]}(z,\tau )+ \theta_{[1]}(w,\tau )\theta_{[1]}(z,\tau ). \end{eqnarray} \end{Lem}
\begin{proof} Just write the left-hand side as a double sum over $m$, $n$ in $\m Z$ and split this double sum according to the parity of $m\!-\! n$. \end{proof}
Here is the second formula which is simple but useful.
\begin{Lem} \label{lemisofor} {\bf Isogeny formula} For $\tau\in \m H$, $d$ odd positive integer, one has \begin{eqnarray*} \label{eqnisofor}\textstyle \sum\limits_{\ell \in \m Z/d\m Z}\theta_{[0]}(\ell /d,\tau)=d\,\theta_{[0]}(0,d^2\tau) &{\rm and}&\textstyle \sum\limits_{\ell \in \m Z/d\m Z}\theta_{[1]}(\ell /d,\tau)=d\,\theta_{[1]}(0,d^2\tau). \end{eqnarray*} \end{Lem}
\begin{proof}Just write the left-hand sides as a double sum over $m$ in $\m Z$ and $\ell $ in $\m Z/d\m Z$ and notice that $\sum_{\ell \in \m Z/d\m Z}e^{2i\pi \ell m/d}$ is equal to $d$ when $d$ divides $m$ and is equal to $0$ otherwise. \end{proof}
The last formula deals with an element $\sigma} \def\ta{\tau} \def\up{\upsilon =\mbox{ \scriptsize
$\left(\!
\begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be \\
\ga&\delta} \def\eps{\varepsilon} \def\ze{\zeta
\end{array}\!
\right)$} \in {\rm SL}(2,\m Z). $
\begin{Lem} \label{lemtrafor} {\bf Transformation formula} If $\sigma} \def\ta{\tau} \def\up{\upsilon\equiv \mathds{1}$ {\rm mod} $2$, and $\ga>0$, then \begin{eqnarray} \label{eqntrafor} \theta(0,\sigma} \def\ta{\tau} \def\up{\upsilon\tau) &=& i^\frac{\delta} \def\eps{\varepsilon} \def\ze{\zeta-1}{2}(\!\tfrac{\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)\; (\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)^{\frac12}\;\theta(0,\tau). \end{eqnarray} \end{Lem} In this formula, the ${\rm SL}(2,\m Z)$ action on the upper half plane $\m H$ is the standard action $\sigma} \def\ta{\tau} \def\up{\upsilon\tau =\frac{\alpha} \def\be{\beta} \def\ga{\gamma\tau+\be}{\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta}$, the number $z^{\frac12}$ is the square root of a complex number $z\in \m H$ whose real part is non negative, and the symbol $(\!\frac{\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)=\pm 1$ is still the Jacobi symbol.
Note that Formula \eqref{eqntrafor} can be equivalently rewritten as \begin{eqnarray*} \theta(0,\sigma} \def\ta{\tau} \def\up{\upsilon\tau) &=& \eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\,(\!\tfrac{2\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)\; (\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)^{\frac12}\;\theta(0,\tau), \end{eqnarray*} where $\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}= 1$ when $\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 1$ mod $4$, and $\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}=-i$ when $\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 3$ mod $4$.
\begin{proof} Up to sign, Formula \eqref{eqntrafor} follows from the following two formulas \begin{eqnarray*} \theta(0,\tau+2)&=&\theta(0,\tau),\\ \theta(0,-1/\tau)&=&(-i\tau)^\frac{1}{2}\, \theta(0,\tau). \end{eqnarray*} and from the fact that the map $(\sigma} \def\ta{\tau} \def\up{\upsilon,\tau)\mapsto \sigma} \def\ta{\tau} \def\up{\upsilon\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta$ is a cocycle on ${\rm SL}(2,\m Z)\times \m H$. The precise determination of the sign is a classical issue due to Hecke. It can be found for instance in \cite[p.181]{Rad73} in \cite[p.148]{Kob84} or in \cite[p.32]{Mum83}. \end{proof}
\begin{Rmq} \label{remmla} This precise determination of the sign is important for us because it will allow us to decide whether the critical value we will find is $\la=\sqrt{a}+i\sqrt{b}$ or its opposite. This is particularly important when $a$ is a square, because in this case $\la$ and $-\la$ are not Galois conjugate and one can not apply Proposition \ref{procrival}.i. \end{Rmq}
The following corollary of Lemma \ref{lemtrafor} will be very useful.
\begin{Cor} \label{cortrafor} If $\sigma} \def\ta{\tau} \def\up{\upsilon\equiv \pm\mathds{1}$ {\rm mod} $4$,
then, for all $\tau$ in $\m H$, one has \begin{eqnarray} \label{eqntrafor2} \frac{\theta_{[0]}(0,\sigma} \def\ta{\tau} \def\up{\upsilon\tau)}{\theta_{[0]}(0,\tau)}&=& \frac{\theta_{[1]}(0,\sigma} \def\ta{\tau} \def\up{\upsilon\tau)}{\theta_{[1]}(0,\tau)}\, . \end{eqnarray} \end{Cor}
\begin{proof} Let $$\sigma} \def\ta{\tau} \def\up{\upsilon' =\mbox{ $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be' \\ \ga'&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$} \;\;{\rm and}\;\; \sigma} \def\ta{\tau} \def\up{\upsilon'' =\mbox{ $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be'' \\ \ga''&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$}, $$ with $\be'=2\be$, $\ga'=\ga/2$ and $\be''=\be/2$, $\ga''=2\ga$, so that $$ \sigma} \def\ta{\tau} \def\up{\upsilon'(2\tau)=2\sigma} \def\ta{\tau} \def\up{\upsilon\tau\;\;{\rm and}\;\; \sigma} \def\ta{\tau} \def\up{\upsilon''(\tau/2)=\tfrac12 \sigma} \def\ta{\tau} \def\up{\upsilon\tau\, . $$ Since the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ is equal to $\pm\mathds{1}$ mod $4$, the two matrices $\sigma} \def\ta{\tau} \def\up{\upsilon'$ and $\sigma} \def\ta{\tau} \def\up{\upsilon''$ are equal to $\mathds{1}$ mod $2$. Therefore we can apply the transformation formula in Lemma \ref{lemtrafor} to both pairs $(\sigma} \def\ta{\tau} \def\up{\upsilon',2\tau)$ and $(\sigma} \def\ta{\tau} \def\up{\upsilon'',\tau/2)$. Using the multiplicativity properties of the Jacobi symbol, we see that the following two ratios are given by the same formula \begin{equation*} \frac{\theta(0, 2\sigma} \def\ta{\tau} \def\up{\upsilon\tau)}{\theta(0,2\tau)}= \frac{\theta(0,\tfrac12 \sigma} \def\ta{\tau} \def\up{\upsilon\tau)}{\theta(0,\tfrac12\tau)}. \end{equation*} We now conclude thanks to Equalities \eqref{eqnt0t1t2}. \end{proof}
\subsection{The condition on theta contants}
The first step in the proof of Proposition \ref{prorapirb} is the following criterion on $\la, \tau$ which ensures that the functions $f_{z,\tau}$ are $\la$-critical. This criterion is a relation between ``theta constants'', i.e. theta functions evaluated at $z=0$.
\begin{Lem} \label{lemrapirb} Let $\tau\in \m H$ and $\la\in \m C$. The function $\theta_\tau$ is $(\la,d)$-critical if and only if one has the equalities \begin{equation*} \label{eqnrapirb} \mbox{} \hspace{8em} \la= d\,\frac{\theta_{[0]}(0,d^2\tau)}{\theta_{[0]}(0,\tau)}= d\, \frac{\theta_{[1]}(0,d^2\tau)}{\theta_{[1]}(0,\tau)}\; . \hspace{6em} (T_{\la,\tau}) \end{equation*} \end{Lem}
\begin{proof} For $w$ in $\m C$ we introduce the function $$z\mapsto F_w(z)=F_w(z,\tau):=\theta(z+w,\tau)\,\theta(z-w,\tau).$$ We want to know when the two functions $\sum_\ell F_{\ell/d}$ and $F_0=\theta^2$ are proportional. The key point of the proof is that all these functions $F_w$ live in the same two-dimensional vector space and that this vector space has a very convenient basis: $(\theta_{[0]},\theta_{[1]})$. We only have to express that the coefficients of our two functions in this basis are proportional. These coefficients are given by the following calculation in which we apply successively the addition formula and the isogeny formula, \begin{eqnarray*} \sum_\ell F_{\ell /d}(z,\tau )&=& \sum_\ell \theta_{[0]}(\ell /d,\tau)\; \theta_{[0]}(z,\tau) + \sum_\ell \theta_{[1]}(\ell /d,\tau)\; \theta_{[1]}(z,\tau)\\ &=& d\,\theta_{[0]}(0,d^2\tau)\; \theta_{[0]}(z,\tau) +
d\, \theta_{[1]}(0,d^2\tau)\; \theta_{[1]}(z,\tau)
\;\;\; {\rm and} \end{eqnarray*} \begin{eqnarray*} \theta(z,\tau)^2&=& \theta_{[0]}(0,\tau)\; \theta_{[0]}(z,\tau) +
\theta_{[1]}(0,\tau)\; \theta_{[1]}(z,\tau).\hspace*{3em} \end{eqnarray*} These two functions are proportional with proportionality factor $\la$ if and only if one has \begin{equation*} \label{eqnlat0t1} \la= d\,\frac{\theta_{[0]}(0,d^2\tau)}{\theta_{[0]}(0,\tau)}= d\, \frac{\theta_{[1]}(0,d^2\tau)}{\theta_{[1]}(0,\tau)}\; . \end{equation*} This is the criterion $(T_{\la,\tau})$. \end{proof}
\subsection{The modular curve $X(4)$} \label{secmodcur}
In order to interpret the condition $(T_{\la,\tau})$, the following classical description of the modular curve $X(4)$ will be very useful.
For $m\geq 1$, we introduce the principal congruence subgroup of level $m$ \begin{eqnarray*} \label{eqnconsub} \Gamma(m)&:=&\{ \sigma} \def\ta{\tau} \def\up{\upsilon\in {\rm SL}(2,\m Z)\mid \sigma} \def\ta{\tau} \def\up{\upsilon\equiv \pm\mathds{1}\;\;{\rm mod}\; m \} \end{eqnarray*} and the modular curve of level $m$ \begin{eqnarray*} \label{eqnmodcur} X(m)&:=&\Gamma(m)\backslash \m H . \end{eqnarray*} Note that the element $-\mathds{1}\in {\rm SL}(2,\m Z)$ acts trivially on $\m H$. It is classical that $X(m)$ is a Riemann surface with finitely many cusps whose genus can be calculated thanks to Hurwitz formula. In this elementary paper we will only deal with $m=4$. In this case, $X(4)$ has genus zero and six cusps. The following lemma gives a nice interpretation of this fact.
We introduce the meromorphic function $\Phi$ on $\m H$ given by, for all $\tau$ in $\m H$, \begin{eqnarray*} \label{eqnmapPhi} \Phi(\tau):=\frac{\theta_{[1]}(0,\tau)}{\theta_{[0]}(0,\tau)}. \end{eqnarray*}
\begin{Lem} \label{lemmodcur} The map $\Phi$ induces a biholomorphism \begin{eqnarray*} \label{eqnmapphi} \varphi} \def\ch{\chi} \def\ps{\psi:X(4)&\longrightarrow &\m P^1\m C\smallsetminus\{ 0,\infty,\pm 1,\pm i\}. \end{eqnarray*} \end{Lem}
This lemma tells us that, as an hyperbolic surface, $X(4)$ is the ''regular ideal octahedron''.
The statement of this lemma is equivalent to the following four facts on the meromorphic map $\Phi$.\\ $(a)$ For all $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ and all $\tau$ in $\m H$ one has $\Phi(\sigma} \def\ta{\tau} \def\up{\upsilon\tau) =\Phi(\tau)$.\\ $(b)$ For all $\tau$ in $\m H$, one has $\Phi(\tau)\neq 0,\infty,\pm 1,\pm i$.\\ $(c)$ If $\Phi(\tau)=\Phi(\tau')$, there exists $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ such that $\tau'=\sigma} \def\ta{\tau} \def\up{\upsilon\tau$.\\ $(d)$ For all $z\neq 0,\infty,\pm 1,\pm i$, there exists $\tau$ in $\m H$ with $\Phi(\tau)=z$.
Note that the first fact $(a)$ is the most important one for us in order to prove Theorem \ref{thmrapirb} and that it is just a restatement of Corollary \ref{cortrafor}.
\begin{proof}[Proof of Lemma \ref{lemmodcur}] This lemma is classical for the experts. We will just relate it to the existing litterature. We will deduce these four facts from a very similar statement in Mumford's book \cite{Mum83}. In this book, Mumford uses the four Jacobi theta-functions $\theta_{a,b}$ with $a$, $b$ equal to $0$ or $1$, given by $$\textstyle \theta_{a,b}(z,\tau)=\sum\limits_{m\in \m Z} e^{i\pi\tau(m+\frac{a}{2})^2}e^{2i\pi(m+\frac{a}{2})(z+\frac{b}{2})} $$ It is proven in \cite[Theorem 10.1 p.51]{Mum83} that the map $\Psi$ given in homogeneous coordinates by $$ \Psi:\tau\mapsto [\theta_{0,0}^2(0,\tau),\theta_{0,1}^2(0,\tau),\theta_{1,0}^2(0,\tau)] $$ induces an biholomophism $\psi$ between the curve $X(4)$ and the curve $$ C:=\{[x_0,x_1,x_2]\in \m P^2\m C\mid x_0^2=x_1^2+x_2^2\; \mbox{\rm and all}\; x_i\neq 0\} $$ which is a conic with six points removed. By the addition formula \eqref{eqnaddfor}, these theta-constants $\theta_{a,b}$ are related to the theta-constants $\theta_{[0]}$ and $\theta_{[1]}$\, : \begin{eqnarray*} \theta_{0,0}^2(0,\tau)&=&\theta_{[0]}^2(0,\tau)+\theta_{[1]}^2(0,\tau),\\ \theta_{0,1}^2(0,\tau)&=&\theta_{[0]}^2(0,\tau)-\theta_{[1]}^2(0,\tau) ,\\ \theta_{1,0}^2(0,\tau)&=&2\,\theta_{[0]}(0,\tau)\,\theta_{[1]}(0,\tau). \end{eqnarray*} Hence one can express in a simple way the map $\Psi$ thanks to the function $\Phi$: $$ \Psi(\tau)=[1\!+\!\Phi^2(\tau)\, ,\, 1\!-\!\Phi^2(\tau)\, ,\, 2\Phi(\tau)], $$ for all $\tau$ in $\m H$. \end{proof}
\begin{Rmq} \label{remmodcur} Note that this identification of $X(4)$ is equivariant. More precisely, the finite group $G:={\rm PGL}(2,\m Z/4\m Z)$ has cardinality $48$ and acts by biholomorphisms or biantiholomorphisms on $X(4)$. The biholomorphism $\varphi} \def\ch{\chi} \def\ps{\psi$ identifies this group $G$ with the group of isometries of the octahedron. This follows from the identities $$ \Phi(-\ol{\tau})=\ol{\Phi(\tau)} \;\; ,\;\; \Phi(\tau+1)=\Phi(\tau) \;{\rm and}\;\; \Phi(-1/\tau)= \frac{-\Phi(\tau)+1}{\Phi(\tau+1}\, . $$ \end{Rmq}
\subsection{Elliptic curves with complex multiplication}
We can now go on the proof of Proposition \ref{prorapirb}, by explaining how we will check that a pair $(\la,\tau)$ satisfies Condition $(T_{\la,\tau})$.
For $\tau$ in $\m H$, we introduce the lattice $\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_\tau=\m Z\tau\oplus\m Z 1$ of $\m C$ so that the quotient $E_\tau:=\m C/\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_\tau$ is the elliptic curve associated to $\tau$. We will see that the values of $\la$ and $\tau=\tau_{k,p}$ in Theorem \ref{thmrapirb} have been chosen so that the elliptic curve $E_{\tau}$ has complex multiplication by $\mu:=\overline{\la}^2$. See \cite{Sche10} for more classical applications of complex multiplication. More precisely, they have been chosen so that $\mu\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_{\tau}=\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_{d^2\tau}$. This means that \begin{eqnarray} \label{eqncommul1} d^2\tau&=& \mu \; (\alpha} \def\be{\beta} \def\ga{\gamma\tau+\be),\\ \label{eqncommul2} 1&=& \mu\;(\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta), \end{eqnarray} for a matrix $\sigma} \def\ta{\tau} \def\up{\upsilon =\mbox{ $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be \\ \ga&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$} \in {\rm SL}(2,\m Z) $. We will be able to impose on $\sigma} \def\ta{\tau} \def\up{\upsilon$ the extra condition \begin{eqnarray} \label{eqng0mod4} \ga>0 &{\rm and}& \sigma} \def\ta{\tau} \def\up{\upsilon\equiv \pm\mathds{1}\;\;{\rm mod}\; 4. \end{eqnarray} We explain why such a choice is relevant in the following lemma.
\begin{Lem} \label{lemmulcom} Let $\tau\in \m H$ and $d$ an odd integer.\\ $(a)$ The function $\theta_\tau$ is $(\la,d)$-critical for some $\la$ in $\m C$ if and only if there exists $\sigma} \def\ta{\tau} \def\up{\upsilon =\mbox{ \scriptsize $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be \\ \ga&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$} \in {\rm SL}(2,\m Z) $ satisfying \eqref{eqng0mod4} such that $g\tau=d^2\tau$.\\ $(b)$ In this case, the critical value is given by, setting $\mu=(\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)^{-1}$, \begin{equation} \label{eqnlamuba} \la=\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\,(\!\tfrac{\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)\;\ol{\mu}^{1/2}. \end{equation} \end{Lem}
\noindent Recall that $\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}= 1$ when $\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 1$ mod $4$, and $\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}=-i$ when $\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 3$ mod $4$.
\begin{proof}[Proof of Lemma \ref{lemmulcom}] $(a)$ According to Lemma \ref{lemrapirb}, the function $\theta_\tau$ is $(\la,d)$-critical for some $\la$ in $\m C$ if and only if one has the equalities $\Phi(\tau)=\Phi(d^2\tau)$.
According to Lemma \ref{lemmodcur}, this equality is equivalent to the existence of an element $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ such that $\sigma} \def\ta{\tau} \def\up{\upsilon\tau=d^2\tau$. Note that this forces the lower-left coefficient $\ga$ to be non zero. Replacing $\sigma} \def\ta{\tau} \def\up{\upsilon$ by $-\sigma} \def\ta{\tau} \def\up{\upsilon$ if necessary, we can also assume that $\ga >0$.
$(b)$ To compute the critical value $\la$, we use again Lemma \ref{lemrapirb} combined with Formulas \eqref{eqnt0t1t2}. We obtain \begin{equation*} \la=d\,\frac{\theta_{[0]}(0,d^2 \tau)}{\theta_{[0]}(0,\tau)}=d\, \frac{\theta(0,d^2 2\tau)}{\theta(0,2\tau)} \; . \end{equation*} Let $\sigma} \def\ta{\tau} \def\up{\upsilon' =\mbox{ $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be' \\ \ga'&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$}, $ with $\be'=2\be$, $\ga'=\ga/2$, so that $ \sigma} \def\ta{\tau} \def\up{\upsilon'(2\tau)=d^2 2\tau. $ Since the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ equals $\pm\mathds{1}$ mod $4$, the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon'$ is equal to $\mathds{1}$ mod $2$. Hence we can apply the transformation formula in Lemma \ref{lemtrafor} to $\sigma} \def\ta{\tau} \def\up{\upsilon'$ and $\tau'$. We get \begin{equation} \label{eqnlaepde} \la =d\, \eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\,(\!\tfrac{\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)\;(\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)^\frac12. \end{equation}
By assumption, the number $\mu:=(\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)^{-1}$ is a complex multiplication for the lattice $\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_\tau$,
more precisely, one has $\mu\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_\tau=\Lambda} \def\Si{\Sigma} \def\Ph{\Phi_{d^2\tau}$. This gives the equality $\mu\ol{\mu}=d^2$, and Equation \eqref{eqncommul2} can be rewritten as $$ d^2\, (\ga\tau+\delta} \def\eps{\varepsilon} \def\ze{\zeta)=\ol{\mu}. $$ Now Equation \eqref{eqnlaepde} can be rewritten as $\la=\eps_{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\,(\!\frac{\ga}{\delta} \def\eps{\varepsilon} \def\ze{\zeta}\!)\,\ol{\mu}^\frac12$. \end{proof}
\subsection{Choosing the elliptic curve} \label{secchoell}
In order to end the proof of Proposition \ref{prorapirb}, it only remains to characterize those points $\tau$ in $\m H$ that satisfy the condition $\sigma} \def\ta{\tau} \def\up{\upsilon\tau=d^2\tau$ for some $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ and to express the $d$-critical value $\la$ given in Lemma \ref{lemmulcom} without using $\sigma} \def\ta{\tau} \def\up{\upsilon$.
We begin by recalling the notation of Proposition \ref{prorapirb}. Let $a$, $b$ be positive integers with $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$ and $a\!+\!b\!=\!d$. We introduced the ``fundamental parameter'' $\tau_0:=\tfrac{1}{4d^2}(a-b-d^2+2i\sqrt{ab})\in \m H$.
For $k$ in $\m Z$, we introduced the integer $N_k:=d^2|k+\tau_0|^2$. For $p\in \m N$ dividing $N_k$, we also introduced the ``associated parameters'' $\tau_{k,p}:=(k+\tau_0)/p\in \m H$.
\begin{Lem} \label{lemtaukg0} (a) Let $\tau$ in $\m H$ such that there exists $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ for which $\sigma} \def\ta{\tau} \def\up{\upsilon\tau=d^2\tau$, then there exists $k, p$ as above such that $\tau=\tau_{k,p}$ or $\tau=-\ol{\tau}_{k,p}$.\\ $(b)$ Conversely, for every $\tau=\tau_{k,p}$ as above, there exists $\sigma} \def\ta{\tau} \def\up{\upsilon$ in $\Ga(4)$ such that $\sigma} \def\ta{\tau} \def\up{\upsilon\tau=d^2\tau$.\\ $(c)$ This matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ can be chosen to be $\sigma} \def\ta{\tau} \def\up{\upsilon=\mbox{ \scriptsize $\left(\! \begin{array}{cc} 2(a\!-\!b)\!-\!d^2(1\!-\!4k)&-4 N_k/p \\ 4p&1-4k \end{array}\! \right)$} .$ \end{Lem}
\begin{proof} We first notice that, in this lemma, we can always add the extra conditions $\ga>0$ and $\sigma} \def\ta{\tau} \def\up{\upsilon\equiv \mathds{1}$ mod $4$. Indeed, if needed, we can always replace the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ by $-\sigma} \def\ta{\tau} \def\up{\upsilon$ without changing the point $\tau$. We can also replace $\sigma} \def\ta{\tau} \def\up{\upsilon$ by the matrix
$\mbox{ \scriptsize $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&-\be \\ -\ga&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$} $, the point $\tau$ is then replaced by $-\ol{\tau}$.
$(a)$ We set $\mu:=(\ga\tau\!+\!\delta} \def\eps{\varepsilon} \def\ze{\zeta)^{-1}$. Since the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon=\mbox{ \scriptsize $\left(\! \begin{array}{cc} \alpha} \def\be{\beta} \def\ga{\gamma&\be \\ \ga&\delta} \def\eps{\varepsilon} \def\ze{\zeta \end{array}\! \right)$} $ has deter\-minant $1$, the equations \eqref{eqncommul1} and \eqref{eqncommul2} can be rewritten as \begin{eqnarray} \label{eqncommul3} \;\; \ga^{-1}(\mu^{-1} -\delta} \def\eps{\varepsilon} \def\ze{\zeta)&=& \tau,\\ \label{eqncommul4} \mu^2- t_0\mu+d^2&=& 0,\;\; \end{eqnarray} with $t_0:=\alpha} \def\be{\beta} \def\ga{\gamma+d^2\delta} \def\eps{\varepsilon} \def\ze{\zeta$. We introduce $a$, $b$ such that $t_0=2(a-b)$ and $d=a+b$.\\ $\star$ Since $\alpha} \def\be{\beta} \def\ga{\gamma\equiv \delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 1$ mod $4$ and $d^2\equiv 1$ mod $4$, both numbers $a$ and $b$ are integers.\\
$\star$ Since $\mu$ is not a real number, one has $|t_0|<2d$ and the integers $a$ and $b$ are positive.\\ $\star$ Since $\sigma} \def\ta{\tau} \def\up{\upsilon\equiv \mathds{1}$ mod $4$, one has $\alpha} \def\be{\beta} \def\ga{\gamma\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 1$ mod $16$, and these integers satisfy $a-b\equiv \frac{1+d^2}{2}$ mod $8$, and hence $a\!\equiv\!\frac{(d+1)^2}{4}\;{\rm mod}\; 4$.\\ $\star$ Since $Im(\mu)<0$, solving Equation \eqref{eqncommul4}, one gets the equality \begin{eqnarray} \label{eqnmu0la0} \mu&:=\;(\sqrt{a}-i\sqrt{b})^2 \; =\; a-b-2i\sqrt{ab} \end{eqnarray} $\star$ We write $\delta} \def\eps{\varepsilon} \def\ze{\zeta=1-4k$ with $k\in \m Z$, and one computes \begin{eqnarray*}
N_k=d^2|k+\tau_0|^2&=& \tfrac{1}{16d^2}[((a-b)-(1-4k)d^2)^2+4ab]\\ &=&\tfrac{1}{16}[1-2(a-b)(1-4k)+d^2(1-4k)^2]\\ &=&(1-\alpha} \def\be{\beta} \def\ga{\gamma\delta} \def\eps{\varepsilon} \def\ze{\zeta)/16 \; =\; -\be\ga/16. \end{eqnarray*} $\star$ We write $\ga=4p$ so that this integer $p$ is a divisor of the integer $N_k$.\\ $\star$ Plugging these informations in \eqref{eqncommul3} gives $\tau=(k+\tau_0)/p$.
$(b)$ and $(c)$\; We assume now that $\tau=\tau_{k,p}$ as above and we want to construct the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$. We follow the same computation as in $(a)$ in opposite order. We set $\mu:=a-b-2i\sqrt{ab}$ and $t_0:=2(a-b)$ so that Equation \eqref{eqncommul4} is satisfied. We choose $\delta} \def\eps{\varepsilon} \def\ze{\zeta:=1-4k$ and $\alpha} \def\be{\beta} \def\ga{\gamma:=t_0-d^2\delta} \def\eps{\varepsilon} \def\ze{\zeta$. We first note that \begin{equation} \label{eqnadmod4} \alpha} \def\be{\beta} \def\ga{\gamma\delta} \def\eps{\varepsilon} \def\ze{\zeta\equiv 1\;\;{\rm mod}\;\; 16. \end{equation} To check \eqref{eqnadmod4}, just remember that one has $t_0\equiv 1+d^2$ mod $16$, and hence $$ \alpha} \def\be{\beta} \def\ga{\gamma\delta} \def\eps{\varepsilon} \def\ze{\zeta-1\equiv (\delta} \def\eps{\varepsilon} \def\ze{\zeta-1)(1-\delta} \def\eps{\varepsilon} \def\ze{\zeta d^2)\equiv 4^2\equiv 0\;\;\;{\rm mod}\; 16. $$ Then the same computation as above gives the equality $(1-\alpha} \def\be{\beta} \def\ga{\gamma\delta} \def\eps{\varepsilon} \def\ze{\zeta)/16=N_k$. Hence if we choose $\ga:=4p$ and $\be:=-4N_k/p$, the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ is in $\Ga(4)$. By construction these coefficients satisfy also Equality \eqref{eqncommul3}. Hence the matrix $\sigma} \def\ta{\tau} \def\up{\upsilon$ satisfies Equalities \eqref{eqncommul1} and \eqref{eqncommul2}. \end{proof}
\begin{proof}[Proof of Proposition \ref{prorapirb}] This proposition is now just a straightforward consequence of Lemmas \ref{lemmulcom} and \ref{lemtaukg0} combined with Formula \eqref{eqnmu0la0}. \end{proof}
\subsection{Conclusion and Perspective} \label{secconclusion}
The aim of this paper was to explain why the algebraic integers $\sqrt{a}+i\sqrt{b}$ that occur in the lists of section \ref{secnumexp} are indeed $d$-critical values when $a,b$ are positive integers with $a+b=d$ and $a\!\equiv\frac{(d+1)^2}{4}$ mod $4$. To keep this paper as elementary as possible, we have only discussed here these $d$-critical values.
However, in the lists of section \ref{secnumexp}, there are still remaining intriguing $d$-critical values. In a more technical forthcoming paper \cite{CSAGII}, we will see that these $d$-critical values belong to a wide class of critical values on finite abelian groups that can be explained by an extension of the construction of Proposition \ref{prorapirb}. This will be a nice application of the abelian varieties and their theta funtions, relying on works of Siegel, Stark, Igusa and of Taniyama-Shimura. See \cite{Beau13} for a recent paper surveying previous applications of these tools. Indeed we will prove in \cite{CSAGII}.
\begin{Thm} \label{thmmaivap} Let $A=\m C^g/\Lambda} \def\Si{\Sigma} \def\Ph{\Phi$ be a principally polarized abelian variety and $\nu$ be a unitary $\m Q$-endomorphism of $A$ preserving a theta structure of level $2$. Let $T_\nu$ be its tangent map, $G_\nu$ the group $\Lambda} \def\Si{\Sigma} \def\Ph{\Phi/(\Lambda} \def\Si{\Sigma} \def\Ph{\Phi\cap T_\nu\Lambda} \def\Si{\Sigma} \def\Ph{\Phi)$ and $d_\nu$ the order of $G_\nu$. Then there is a critical value $\la_\nu=\kappa} \def\la{\lambda} \def\rh{\rho_\nu\, d_\nu^{1/2}\,{\rm det}_{\m C}(T_\nu)^{1/2}$ on $G_\nu$ with $\kappa} \def\la{\lambda} \def\rh{\rho_\nu^4=1$. \end{Thm}
\noindent Note that $|\la_\nu|=d_\nu^{1/2}$ and that one can compute the fourth root of unity $\kappa} \def\la{\lambda} \def\rh{\rho_\nu$.
Note also that, any finite abelian group can occur as a group $G_\nu$ but, even when $g>1$, this group $G_\nu$ may be cyclic. We will construct concrete examples with $G_\nu$ cyclic when the abelian variety $A$ has complex multiplication. These constructions will explain all the intriguing $d$-critical values in our lists. For instance we will show in \cite{CSAGII} by using abelian surfaces.
\begin{Cor} \label{corparacri} Let $d_j\!=\!a_j\!+\!b_j$ with $d_1\wedge d_2=1$ and $a_j\!\equiv\! \frac{(d_j+1)^2}{4}\!+\! 2$ {\rm mod}~$4$ be positive integers. Then $\la\!=\!(\sqrt{a_1}\!+\! i\sqrt{b_1})(\sqrt{a_2}\!-\! i\sqrt{b_2}) $ is $d$-critical. \end{Cor}
\begin{Cor} \label{corrarbrc} Let $d=a\!+\! b\!+\!c$ be positive integers, with $b^2>4ac$, and $a\!\equiv\! b\!\equiv\! c\!\equiv\! 1$ mod $4$. Then $ \la= \sqrt{a}\!+\!\sqrt{c} + i\sqrt{b\!-\! 2 \sqrt{ac}} $ is $d$-critical. \end{Cor} \noindent A key remark in the proof of Corollary \ref{corrarbrc}, is an old factorization formula: \begin{equation*} \label{eqn1pr5pi} \textstyle \sqrt{a}\!+\!\sqrt{c}\!+\!i\sqrt{b\!-\!2\sqrt{ac}} \; =\; \sqrt{a}\left(\!1\!+\! i\sqrt{\tfrac{b+\sqrt{b^2-4ac}}{2a}}\right) \left(\!1\!-\! i\sqrt{\tfrac{b-\sqrt{b^2-4ac}}{2a}}\right). \end{equation*}
{\small
}
{\small \noindent Y. \textsc{Benoist}: CNRS, Universit\'e Paris-Saclay,
\texttt{yves.benoist@u-psud.fr}}
\end{document} |
\begin{document}
\title{The smallest hyperbolic 6-manifolds}
\author{Brent Everitt}
\address{Department of Mathematics, University of York, York YO10 5DD, England} \email{bje1@york.ac.uk}
\thanks{The first author is grateful to the Mathematics Department, Vanderbilt University for its hospitality during a stay when the results of this paper were obtained.}
\author{John Ratcliffe} \address{Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA} \email{ratclifj@math.vanderbilt.edu}
\author{Steven Tschantz} \address{Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA} \email{tschantz@math.vanderbilt.edu}
\subjclass{Primary 57M50}
\begin{abstract} By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic $6$-manifolds with Euler characteristic $-1$ are constructed. They are the first known examples of hyperbolic $6$-manifolds having the smallest possible volume. \end{abstract}
\maketitle
\section{Introduction}
The last few decades has seen a surge of activity in the study of finite volume hyperbolic manifolds--that is, complete Riemannian $n$-manifolds of constant sectional curvature $-1$. Not surprisingly for geometrical objects, volume has been, and continues to be, the most important invariant for understanding their sociology. The possible volumes in a fixed dimension forms a well-ordered subset of $\ams{R}$, indeed a discrete subset except in $3$-dimensions (where the orientable manifolds at least have ordinal type $\omega^\omega$). Thus it is a natural problem with a long history to construct examples of manifolds with minimal volume in a given dimension.
In $2$-dimensions the solution is classical, with the minimum volume in the compact orientable case achieved by a genus $2$ surface, and in the non-compact orientable case by a once-punctured torus or thrice-punctured sphere (the identities of the manifolds are of course also known in the non-orientable case). In $3$-dimensions the compact orientable case remains an open problem with the Matveev-Fomenko-Weeks manifold \cite{Matveev-Fomenko88,Weeks85} obtained via $(5,-2)$-Dehn surgery on the the sister of the figure-eight knot complement conjecturally the smallest. Amongst the non-compact orientable $3$-manifolds the figure-eight knot complement realizes the minimum volume \cite{Meyerhoff01} and the Gieseking manifold (obtained by identifying the sides of a regular hyperbolic tetrahedron as in \cite{Everitt02b,Prok98}) does so for the non-orientable ones \cite{Adams87}. One could also add ``arithmetic'' to our list of adjectives and so have eight optimization problems to play with (so that the Matveev-Fomenko-Weeks manifold is known to be the minimum volume orientable, arithmetic compact $3$-manifold, see \cite{Chinburg01}).
When $n\geq 4$ the picture is murkier, although in even dimensions we have recourse to the Gauss-Bonnet Theorem, so that in particular the minimal volume a $2m$-dimensional hyperbolic manifold could possibly have is when the
Euler characteristic $\chi$ satisfies $|\chi|=1$. The first examples of non-compact $4$-manifolds with $\chi=1$ were constructed in \cite{Ratcliffe00} (see also \cite{Everitt02}). The compact case remains a difficult unsolved problem, although if we restrict to arithmetic manifolds, then it is known [2,8] that a minimal volume arithmetic compact orientable 4-manifold $M$ has $\chi \leq 16$ and $M$ is isometric to the orbit space of a torsion-free subgroup of the hyperbolic Coxeter group $[5,3,3,3]$. The smallest compact hyperbolic $4$-manifold currently known to exist has $\chi=8$ and is constructed in \cite{Conder04}. Manifolds of very small volume have been constructed in $5$-dimensions \cite{Everitt02,Ratcliffe04}, but the smallest volume $6$-dimensional example hitherto known has $\chi=-16$ \cite{Everitt02}.
In this paper we announce the discovery of a number of non-compact non-orientable hyperbolic $6$-manifolds with Euler characteristic $\chi=-1$. The method of construction is classical in that the manifolds are obtained by identifying the sides of a $6$-dimensional hyperbolic Coxeter polytope.
\section{Coxeter polytopes}
Let $C$ be a convex (not necessarily bounded) polytope of finite volume in a simply connected space $X^n$ of constant curvature. Call $C$ a Coxeter polytope if the dihedral angle subtended by two intersecting $(n-1)$-dimensional sides is $\pi/m$ for some integer $m\geq 2$. When $X^n=S^n$ or the Euclidean space $E^n$, such polyhedra have been completely classified \cite{Coxeter34}, but in the hyperbolic space $H^n$, a complete classification remains a difficult problem (see for example \cite{Vinberg85} and the references there).
If $\Gamma$ is the group generated by reflections in the $(n-1)$-dimensional sides of $C$, then $\Gamma$ is a discrete cofinite subgroup of the Lie group $\text{Isom}\, X^n$, and every discrete cofinite reflection group in $\text{Isom}\, X^n$ arises in this way from some Coxeter polytope, which is uniquely defined up to isometry. The Coxeter symbol for $C$ (or $\Gamma$) has nodes indexed by the $(n-1)$-dimensional sides, and an edge labeled $m$ joining the nodes corresponding to sides that intersect with angle $\pi/m$ (label the edge joining the nodes of non-intersecting sides by $\infty$). In practice the labels $2$ and $3$ occur often, so that edges so labeled are respectively removed or left unlabeled.
Let $\Lambda$ be a $(n+1)$-dimensional Lorentzian lattice, that is, an $(n+1)$-dimensional free $\ams{Z}$-module equipped with a $\ams{Z}$-valued bilinear form of signature $(n,1)$. For each $n$, there is a unique such $\Lambda$, denoted $I_{n,1}$, that is odd and self-dual (see \cite[Theorem V.6]{Serre73}, or \cite{Milnor73, Neumaier83}). By \cite{Borel62}, the group $\text{O}_{n,1}\ams{Z}$ of automorphisms of $I_{n,1}$ acts discretely, cofinitely by isometries on the hyperbolic space $H^n$ obtained by projectivising the negative norm vectors in the Minkowski space-time $I_{n,1}\otimes\ams{R}$ (to get a faithful action one normally passes to the centerless version $\text{PO}_{n,1}\ams{Z}$).
Vinberg and Kaplinskaja showed \cite{Vinberg78,Vinberg72} that the subgroup $\text{Reflec}_n$ of $\text{PO}_{n,1}\ams{Z}$ generated by reflections in positive norm vectors has finite index if and only if $n\leq 19$, thus yielding a family of cofinite reflection groups and corresponding finite volume Coxeter polytopes in the hyperbolic spaces $H^n$ for $2\leq n\leq 19$. Indeed, Conway and Sloane have shown \cite[Chapter 28]{Conway93} or \cite{Conway82}, that for $n\leq 19$ the quotient of $\text{PO}_{n,1}\ams{Z}$ by $\text{Reflec}_n$ is a subgroup of the automorphism group of the Leech lattice. Borcherds \cite{Borcherds87} showed that the (non self-dual) even sublattice of $I_{21,1}$ also acts cofinitely, yielding the highest dimensional example known of a Coxeter group acting cofinitely on hyperbolic space.
When $4\leq n\leq 9$ the group $\Gamma=\text{Reflec}_n$ has Coxeter symbol, $$ \begin{pspicture}(0,0)(6,2)
\rput(3,1){\BoxedEPSF{announce1.eps scaled 800}} \rput(5.65,1.8){$4$} \rput(5,.25){$\Gamma_v$} \rput(3.8,.35){$\Gamma_e$} \rput(6.25,1.2){$F_1$} \rput(5.05,1.2){$F_2$} \end{pspicture} $$ with $n+1$ nodes and $C$ a non-compact, finite volume $n$-simplex $\Delta^n$ (when $n>9$, the polytope $C$ has a more complicated structure).
Let $v$ be the vertex of $\Delta^n$ opposite the side $F_1$ marked on the symbol, and let $\Gamma_v$ be the stabilizer in $\Gamma$ of this vertex. This stabilizer is also a reflection group with symbol as shown, and is finite for $4\leq n\leq 8$ (being the Weyl group of type $A_4,D_5,E_6,E_7$ and $E_8$ respectively) and infinite for $n=9$ (when it is the affine Weyl group of type $\widetilde{E}_8$). Let $$ P_n=\bigcup_{\gamma\in\Gamma_v} \gamma(\Delta^n), $$
a convex polytope obtained by gluing $|\Gamma_v|$ copies of the simplex $\Delta^n$ together. Thus, $P_n$ has finite volume precisely when $4\leq n\leq 8$, although it is non-compact, with a mixture of finite vertices in $H^n$ and cusped ones on $\partial H^n$. In any case, $P_n$ is an all right-angled Coxeter polytope: its sides meet with dihedral angle $\pi/2$ or are disjoint. This follows immediately from the observation that the sides of $P_n$ arise from the $\Gamma_v$-images of the side of $\Delta^n$ opposite $v$, and this side intersects the other sides of $\Delta^n$ in dihedral angles $\pi/2$ or $\pi/4$. Vinberg has conjectured that $n=8$ is the highest dimension in which finite volume all-right angled polytopes exist in hyperbolic space.
The volume of the polytope $P_n$ is given by
$$\text{vol}(P_n)=|\Gamma_v|\text{vol}(\Delta^n)=|\Gamma_v|[\text{PO}_{n,1}\ams{Z}:\Gamma]\text{covol}(\text{PO}_{n,1}\ams{Z}),$$
where $\text{covol}(\text{PO}_{n,1}\ams{Z})$ is the volume of a fundamental region for the action of $\text{PO}_{n,1}\ams{Z}$ on $H^n$ (and for $4\leq n\leq 9$ the index $[\text{PO}_{n,1}\ams{Z}:\Gamma]=1$). When $n$ is even, we have by \cite{Siegel36} and \cite{Ratcliffe97}, $$ \text{covol}(\text{PO}_{n,1}\ams{Z})=\frac{(2^{\frac{n}{2}}\pm 1)\pi^{\frac{n}{2}}}{n!}
\prod_{k=1}^{\frac{n}{2}} |B_{2k}|, $$ with $B_{2k}$ the $2k$-th Bernoulli number and with the plus sign if $n\equiv 0,2\mod 8$ and the minus sign otherwise.
Alternatively (when $n$ is even), we have recourse to the Gauss-Bonnet Theorem, so that $\text{vol}(P_n)=\kappa_n|\Gamma_v| \chi(\Gamma)$, where $\chi(\Gamma)$ is the Euler characteristic of the Coxeter group $\Gamma$ and $\kappa_n=2^n (n!)^{-1} (-\pi)^{n/2} (n/2)!$. The Euler characteristic of Coxeter groups can be easily computed from their symbol (see \cite{Chiswell92,Chiswell76} or \cite[Theorem 9]{Everitt02}). Indeed, when $n=6$, $\chi(\Gamma)=-1/\mathscr L$ where $\mathscr L=2^{10}\,3^4\,5$ and so
$\text{vol}(P_6)=8\pi^3|E_6|/15\mathscr L=\pi^3/15$.
The Coxeter symbol for $P_n$ has a nice description in terms of finite reflection groups. If $v'$ is the vertex of $\Delta^n$ opposite the side $F_2$, let $\Gamma_e$ be the pointwise stabilizer of $\{v,v'\}$: the elements thus stabilize the edge $e$ of $\Delta^n$ joining $v$ and $v'$.
Now consider the Cayley graph $\mathscr C_v$ for $\Gamma_v$ with respect to the generating reflections in the sides of the symbol for $\Gamma_v$. Thus, $\mathscr C_v$ has vertices in one to one correspondence with the elements of $\Gamma_v$ and for each generating reflection $s_{\alpha}$, an undirected edge labeled $s_{\alpha}$ connecting vertices $\gamma_1$ and $\gamma_2$ if and only if $\gamma_2=\gamma_1s_{\alpha}$ in $\Gamma_v$. In particular, $\mathscr C_v$ has $s_2$ labeled edges corresponding to the reflection in $F_2$. Removing these $s_2$-edges decomposes $\mathscr C_v$ into components, each of which is a copy of the Cayley graph $\mathscr C_e$ for $\Gamma_e$, with respect to the generating reflections.
Take as the nodes of the symbol for $P_n$ these connected components. If two components have an $s_2$-labeled edge running between any two of their vertices in $\mathscr C_v$, then leave the corresponding nodes unconnected, otherwise, connect them by an edge labeled $\infty$. The resulting symbol
(respectively the polytope $P_n$) thus has $|\Gamma_v|/|\Gamma_e|$ nodes (resp. sides). The number of sides of $P_n$ for $n=4,5,6,7,8$ is $10,16,27,56$ and $240$ respectively.
\section{Constructing the manifolds}
We now restrict our attention to the case $n=6$. We work in the hyperboloid model of hyperbolic 6-space
$$H^6=\{x\in \ams{R}^7: x_1^2+x_2^2+\cdots+x_6^2-x_7^2=-1\ \hbox{and}\ x_7>0\}$$
and represent the isometries of $H^6$ by Lorentzian $7\times 7$ matrices that preserve $H^6$. The right-angled polytope $P_6$ has 27 sides each congruent to $P_5$. We position $P_6$ in $H^6$ so that 6 of its sides are bounded by the 6 coordinate hyperplanes $x_i=0$ for $i=1,\ldots, 6$ and these 6 sides intersect at the center $e_7$ of $H^6$. Let $K_6$ be the group of 64 diagonal Lorentzian $7\times 7$ matrices ${\rm diag}(\pm 1,\ldots,\pm 1,1)$. The set $Q_6=K_6P_6$, which is the union of 64 copies of $P_6$, is a right-angled convex polytope with 252 sides. We construct hyperbolic 6-manifolds, with $\chi =-8$, by gluing together the sides of $Q_6$ by a proper side-pairing with side-pairing maps of the form $rk$ with $k$ in $K_6$ and $r$ a reflection in a side $S$ of $Q_6$. The side-pairing map $rk$ pairs the side $S'=kS$ to $S$ (see \S 11.1 and \S 11.2 of \cite{Ratcliffe94} for a discussion of proper side-pairings). We call such a side-pairing of $Q_6$ simple. We searched for simple side-pairings of $Q_6$ that yield a hyperbolic 6-manifold $M$ with a freely acting $\ams{Z}/8$ symmetry group that permutes the 64 copies of $P_6$ making up $M$ in such a way that the resulting quotient manifold is obtained by gluing together 8 copies of $P_6$. Such a quotient manifold has $\chi = -8/8=-1$. This is easier said than done, since the search space of all possible side-pairings of $Q_6$ is very large. We succeeded in finding desired side-pairings of $Q_6$ by employing a strategy that greatly reduces the search space. The strategy is to extend a side-pairing in dimension 5 with the desired properties to a side-pairing in dimension 6 with the desired properties.
Let $Q_5 = \{x\in Q_6: x_1 = 0\}$. Then $Q_5$ is a right-angled convex 5-dimensional polytope with 72 sides. Note that $Q_5$ is the union $K_5P_5$ of 32 copies of $P_5$ where $P_5= \{x\in P_6: x_1=0\}$ and $K_5$ is the group of 32 diagonal Lorentzian $7\times 7$ matrices ${\rm diag}(1,\pm 1,\ldots,\pm 1,1)$. A simple side-pairing of $Q_6$ that yields a hyperbolic 6-manifold $M$ restricts to a simple side-pairing of $Q_5$ that yields a hyperbolic 5-manifold which is a totally geodesic hypersurface of $M$. All the orientable hyperbolic 5-manifolds that are obtained by gluing together the sides of $Q_5$ by a simple side-pairing are classified in \cite{Ratcliffe04}.
We started with the hyperbolic 5-manifold $N$, numbered 27 in \cite{Ratcliffe04}, obtained by gluing together the sides of $Q_5$ by the simple side-pairing with side-pairing code {\tt 2B7JB47JG81}. The manifold $N$ has a freely acting $\ams{Z}/8$ symmetry group that permutes the 32 copies of $P_5$ making up $N$ in such a way that the resulting quotient manifold is obtained by gluing together 4 copies of $P_5$. A generator of the $\ams{Z}/8$ symmetry group of $N$ is represented by the Lorentzian $6\times 6$ matrix $$ \left(\begin{array}{cccccc}
\phantom{-}1 & 0 &\phantom{-}0 & \phantom{-}1 & 0 & -1 \\
\phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 1 & \phantom{-}0 \\
-1 & 0 & -1 & \phantom{-}0 & 0 & \phantom{-}1 \\
\phantom{-}0 & 1 &\phantom{-}0 & \phantom{-}0 & 0 & \phantom{-}0 \\
\phantom{-}0 & 0 & -1 & -1 & 0 & \phantom{-}1 \\
-1 & 0 & -1 & -1 & 0 & \phantom{-}2
\end{array} \right). $$ The strategy is to search for simple side-pairings of $Q_6$ that yield a hyperbolic 6-manifold with a freely acting $\ams{Z}/8$ symmetry group with generator represented by the following Lorentzian $7\times 7$ matrix that extends the above Lorentzian $6\times 6$ matrix. $$ \left(\begin{array}{ccccccc}
\phantom{-}1 &\phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 0 &\phantom{-}0 \\
\phantom{-}0 &\phantom{-}1 & 0 &\phantom{-}0 & \phantom{-}1 & 0 & -1 \\
\phantom{-}0 &\phantom{-}0 & 0 &\phantom{-}0 & \phantom{-}0 & 1 & \phantom{-}0 \\
\phantom{-}0 &-1 & 0 & -1 & \phantom{-}0 & 0 & \phantom{-}1 \\
\phantom{-}0 &\phantom{-}0 & 1 &\phantom{-}0 & \phantom{-}0 & 0 & \phantom{-}0 \\
\phantom{-}0 &\phantom{-}0 & 0 & -1 & -1 & 0 & \phantom{-}1 \\
\phantom{-}0 &-1 & 0 & -1 & -1 & 0 & \phantom{-}2
\end{array} \right) $$ For such a side-pairing the resulting quotient manifold can be obtained by gluing together 8 copies of $P_6$ by a proper side-pairing. By a computer search we found 14 proper side-pairings of 8 copies of $P_6$ in this way, and hence we found 14 hyperbolic $6$-manifolds with $\chi = -1$. Each of these 14 manifolds is noncompact with volume $8\text{vol}(P_6)=8\pi^3/15$ and five cusps. These 14 hyperbolic $6$-manifolds represent at least 7 different isometry types, since they represent 7 different homology types. Table 1 lists side-pairing codes for 7 simple side-pairings of $Q_6$ whose $\ams{Z}/8$ quotient manifold has homology groups isomorphic to $\ams{Z}^a\oplus(\ams{Z}/2)^b\oplus(\ams{Z}/4)^c\oplus(\ams{Z}/8)^d$ for nonnegative integers $a,b,c,d$ encoded by $abcd$ in the table. In particular, all 7 manifolds in Table 1 have a finite first homology group.
All of our examples, with $\chi = -1$, can be realized as the orbit space $H^6/\Gamma$ of a torsion-free subgroup $\Gamma$ of $\text{PO}_{6,1}\ams{Z}$ of minimal index. These manifolds are the first examples of hyperbolic 6-manifolds having the smallest possible volume. All these manifolds are nonorientable. In the near future, we hope to construct orientable examples of noncompact hyperbolic 6-manifolds having $\chi= -1$.
\begin{table} \begin{center} \begin{tabular}{lllllll} $N$&$SP$&\ \ $H_1$&\ \ $H_2$&\ \ $H_3$&\ \ $H_4$&\ \ $H_5$\\ &&$\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248& $\phantom{\mathbb Z}$0248&$\phantom{\mathbb Z}$0248\\ 1&{\tt GW8dNEEdN4ZJO1k2l1PIY}& \phantom{${\mathbb Z}$}0401& \phantom{${\mathbb Z}$}1910& \phantom{${\mathbb Z}$}4821& \phantom{${\mathbb Z}$}1500& \phantom{${\mathbb Z}$}0000\\ 2&{\tt HX9dNFEcM5aKU6f3f6UKa}& \phantom{${\mathbb Z}$}0401& \phantom{${\mathbb Z}$}1810& \phantom{${\mathbb Z}$}8710& \phantom{${\mathbb Z}$}5500& \phantom{${\mathbb Z}$}0000\\ 3&{\tt HX9dNFEcM5YIO1l3l1OIY}& \phantom{${\mathbb Z}$}0401& \phantom{${\mathbb Z}$}2900& \phantom{${\mathbb Z}$}7810& \phantom{${\mathbb Z}$}4400& \phantom{${\mathbb Z}$}1000\\ 4&{\tt HX9dNFEcM5YIO6l3l6OIY}& \phantom{${\mathbb Z}$}0401& \phantom{${\mathbb Z}$}2800& \phantom{${\mathbb Z}$}7910& \phantom{${\mathbb Z}$}4400& \phantom{${\mathbb Z}$}1000\\ 5&{\tt HX9dNFEcM5YIOxl3lyOIY}& \phantom{${\mathbb Z}$}0211& \phantom{${\mathbb Z}$}2800& \phantom{${\mathbb Z}$}4821& \phantom{${\mathbb Z}$}1400& \phantom{${\mathbb Z}$}1000\\ 6&{\tt HX9dNFEcM5YIOyl3lxOIY}& \phantom{${\mathbb Z}$}0211& \phantom{${\mathbb Z}$}2800& \phantom{${\mathbb Z}$}4930& \phantom{${\mathbb Z}$}1400& \phantom{${\mathbb Z}$}1000\\ 7&{\tt HX9dNFEcM5aKUxf3fyUKa}& \phantom{${\mathbb Z}$}0301& \phantom{${\mathbb Z}$}1900& \phantom{${\mathbb Z}$}5630& \phantom{${\mathbb Z}$}2500& \phantom{${\mathbb Z}$}0000\\ \end{tabular} \end{center} \caption{Side-pairing codes and homology groups of the seven examples} \end{table}
\end{document} |
\begin{document}
\title[Convolution Operators with Singular Measures of Fractional Type]{Convolution Operators with Singular Measures of Fractional Type on the Heisenberg Group} \author{Tom\'as Godoy and Pablo Rocha} \address{Universidad Nacional de C\'ordoba, FaMAF-UNC, C\'ordoba, 5000 C\'ordoba, Argentina} \email{godoy@famaf.unc.edu.ar, \ rp@famaf.unc.edu.ar} \thanks{\textbf{2.010 Math. Subject Classification}: 423A80, 42A38} \thanks{\textbf{Key words and phrases}: Singular measures, group Fourier transform, Heisenberg group, convolution operators} \thanks{Partially supported by Conicet and Secyt-UNC} \maketitle \begin{abstract} We consider the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n} \times \mathbb{R}$. Let $\mu_{\gamma}$ be the fractional Borel measure on $\mathbb{H}^{n}$ defined by
$$ \mu_{\gamma}(E) = \int_{\mathbb{C}^{n}}\chi_{E}\left(w,\varphi(w)\right) \prod_{j=1}^{n} \eta_j \left( |w_j|^{2} \right) | w_j |^{-\frac{\gamma}{n}}dw,$$ where $0 < \gamma < 2n$, $\varphi(w) = \sum\limits_{j=1}^{n} a_{j} \left\vert w_{j}\right\vert^{2}$, $w=(w_{1},...,w_{n}) \in \mathbb{C}^{n}$, $a_{j} \in \mathbb{R}$, and $\eta_j \in C_{c}^{\infty}(\mathbb{R})$. In this paper we study the set of pairs $(p,q)$ such that the right convolution operator with $\mu_{\gamma}$ is bounded from $L^{p}(\mathbb{H}^{n})$ into $L^{q}(\mathbb{H}^{n})$.
\end{abstract}
\section{Introduction}
Let $\mathbb{H}^{n} = \mathbb{C}^{n} \times \mathbb{R}$ be the Heisenberg group with group law $\left( z,t\right) \cdot \left(w,s\right) =\left( z+w,t+s+\left\langle z,w\right\rangle \right) $ where $\langle z,w \rangle = \frac{1}{2} Im(\sum \limits_{j=1}^{n}z_{j} \cdot \overline{w_{j}})$. For $x=(x_{1},...,x_{2n})\in \mathbb{R}^{2n}$, we write $x=(x^{\prime },x^{\prime \prime })$ with $x^{\prime }\in \mathbb{R}^{n}$, $x^{\prime \prime }\in \mathbb{R}^{n}$. So, $\mathbb{R}^{2n}$ can be identified with $\mathbb{C}^{n}$ via the map $\Psi (x^{\prime },x^{\prime \prime })=x^{\prime }+ix^{\prime \prime }$. In this setting the form $\langle z,w \rangle $ agrees with the standard symplectic form on $\mathbb{R}^{2n}$. Thus $\mathbb{H}^{n}$ can be viewed as $\mathbb{R}^{2n} \times \mathbb{R}$ endowed with the group law $$ \left( x,t\right) \cdot \left( y,s\right) =\left( x+y,t+s+\frac{1}{2} B(x,y) \right) $$ where the symplectic form $B$ is given by $B(x,y)=\sum \limits_{j=1}^{n}\left( y_{n+j}x_{j}-y_{j}x_{n+j}\right) $, with $x=(x_{1},...,x_{2n})$ and $y=(y_{1},...,y_{2n})$, with neutral element $(0,0)$, and with inverse $\left( x,t\right) ^{-1}=\left(-x,-t\right) $.
Let $\varphi :\mathbb{R}^{2n}\rightarrow \mathbb{R}$ be a measurable function, and let $\mu_{\gamma} $ be the fractional Borel measure on $\mathbb{H}^{n}=\mathbb{R}^{2n}\times \mathbb{R}$ supported on the graph of $\varphi$, given by \begin{equation} \left\langle \mu_{\gamma} ,f\right\rangle =\int\limits_{\mathbb{R}^{2n}}f\left( w,\varphi \left( w\right)
\right) \prod_{j=1}^{n} \eta_j \left( |w_j|^{2} \right) \left\vert w_j \right\vert ^{-\frac{\gamma}{n} }dw
\label{mu2} \end{equation} with $0< \gamma < 2n$, and where the $\eta _{j}$'s are functions in $C_{c}^{\infty }(\mathbb{R})$ such that $0\leq \eta _{j}\leq 1$, $\eta _{j}(t)\equiv 1$ if $t\in \lbrack -1,1]$ and $supp(\eta _{j})\subset (-2,2)$.
Let $T_{\mu_{\gamma} }$ be the right convolution operator by $\mu_{\gamma} $, defined by \begin{equation} T_{\mu_{\gamma} }f\left( x,t\right) =\left( f\ast \mu_{\gamma} \right) \left(x,t\right) =\int_{\mathbb{R}^{2n}}f\left( \left( x,t\right) \cdot
\left( w,\varphi \left( w\right) \right) ^{-1}\right) \prod_{j=1}^{n} \eta_j \left( |w_j|^{2} \right) \left\vert w_j \right\vert ^{-\frac{\gamma}{n}} dw. \label{tmu} \end{equation} We are interested in studying the type set $$ E_{\mu_{\gamma} }=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) \in \left[0,1\right] \times \left[ 0,1\right] :\left\Vert T_{\mu_{\gamma}}\right\Vert _{L^{p}-L^{q}}<\infty \right\} $$ where the $L^{p}$ - spaces are taken with respect to the Lebesgue measure on $\mathbb{R}^{2n+1}$. We say that that the measure $\mu_{\gamma}$ defined in (\ref{mu2}) is $L^{p}$-\textit{improving} if $E_{\mu_{\gamma}}$ does not reduce to the diagonal $1/p=1/q$.
This problem is well known if in (\ref{tmu}) we consider $\gamma =0$ and replace the Heisenberg group convolution with the ordinary convolution in $\mathbb{R}^{2n+1}$. If the graph of $\varphi$ has non-zero Gaussian curvature at each point, a theorem of Littman (see \cite{littman}) implies that $E_{\nu }$ is the closed triangle with vertices $(0,0)$, $(1,1)$, and $\left( \frac{2n+1}{2n+2},\frac{ 1}{2n+2}\right) $ (see \cite{oberlin}). A very interesting survey of results concerning the type set for convolution operators with singular measures can be found in \cite{ricci}.\\ Returning to our setting $\mathbb{H}^{n}$, in \cite{secco} and \cite{secco2} S. Secco obtains $L^{p}$-\textit{improving} properties of measures supported on curves in $\mathbb{H}^{1}$, under certain assumptions. In \cite{ricci2} F. Ricci and E. Stein showed that the type set of the measure given by (\ref{mu2}), for the case $\varphi(w)=0$, $\gamma =0$ and $n=1$, is the triangle with vertices $(0,0),$ $(1,1),$ and $\left( \frac{3}{4},\frac{1}{4}\right)$. In \cite{G-R}, the authors adapt the work of Ricci and Stein for the case of manifolds quadratic hypersurfaces in $\mathbb{R}^{2n+1}$, there we also give some examples of surfaces with degenerate curvature at the origin.
We observe that if $\left( \frac{1}{p},\frac{1}{q}\right) \in E_{\mu_{\gamma}}$ then \begin{equation} p\leq q, \,\,\,\,\,\,\,\,\,\,\,\, \frac{1}{q}\geq \frac{2n+1}{p}-2n, \,\,\,\,\,\,\,\,\,\,\,\, \frac{1}{q}\geq\frac{1}{(2n+1)p}. \label{restricciones} \end{equation}
Indeed, the first inequality follows from Lemma 3 in \cite{G-R}, replacing the sets $A_{\delta}$ and $F_{\delta, x}$ in the proof of Lemma 4 in \cite{G-R} by the sets $$ A'_{\delta }=\left\{ (x,t)\in \mathbb{R}^{2n}\times \mathbb{R}:x\in \widetilde{D}\wedge \left\vert t-\varphi (x)\right\vert \leq \frac{\delta }{4}\right\} $$ and $$ F'_{\delta ,x}=\left\{ y\in \widetilde{D}:\left\Vert x-y\right\Vert _{\mathbb{R} ^{2n}}\leq \frac{\delta }{4n(1+\left\Vert \nabla \varphi \mid _{supp(\eta )}\right\Vert _{\infty })}\right\} $$ where $\widetilde{D}$ is a closed disk in $\mathbb{R}^{2n}$ contained in the unit disk centered in the origin such that the origin not belongs to $\widetilde{D}$, we observe that the argument utilized in the proof of Lemma 4 in \cite{G-R} works in this setting so we get the others two inequalities.
Since $0 < \gamma < 2n$ it is clear that $\| T_{\mu_{\gamma}} f \|_{p} \leq c \|f\|_{p}$ for all Borel function $f \in L^{p}(\mathbb{H}^{n})$ and all $1 \leq p \leq \infty$, so $(\frac{1}{p}, \frac{1}{p}) \in E_{\mu_{\gamma}}$.
In Lemma 4, section 2 below, we obtain the following necessary condition for the pair $(\frac{1}{p}, \frac{1}{q})$ to be in $E_{\mu_{\gamma}}$: $$\frac{1}{q}\geq \frac{1}{p}-\frac{2n-\gamma }{2n+2}.$$
Let $D$ be the point of intersection, in the $(\frac{1}{p}, \frac{1}{q})$ plane, between the lines $\frac{1}{q}=\frac{2n+1}{p} -2n$, $\frac{1}{q}=\frac{1}{p}-\frac{2n-\gamma }{2n+2}$ and let $D^{\prime }$ be its symmetric with respect to the non principal diagonal. So \[ D=\left( \frac{4n^{2}+2n+\gamma }{2n(2n+2)},\frac{2n+(2n+1)\gamma }{2n(2n+2)} \right) =\left( \frac{1}{p_{D}},\frac{1}{q_{D}}\right) \text{ y\ }D^{\prime }=\left( 1-\frac{1}{q_{D}},1-\frac{1}{p_{D}}\right). \] Thus $E_{\mu_{\gamma}}$ is contained in the closed trapezoid with vertices $(0,0)$, $(1,1)$, $D$ and $D^{\prime }$.
Finally, let $C_{\gamma}$ be the point of intersection of the lines $\frac{1}{q}=1- \frac{1}{p}$ and $\frac{1}{q}= \frac{1}{p}- \frac{2n-\gamma}{2n+2}$, thus $C_{\gamma}= \left( \frac{4n+2-\gamma}{2(2n+2)}, \frac{2+\gamma}{2(2n+2)} \right)$.
In section 3 we prove the following results:
\begin{theorem} If $\mu_{\gamma}$ is the fractional Borel measure defined in (\ref{mu2}), supported on the graph of the function $\varphi (w)= \sum_{j=1}^{n} a_j \left\vert w_j \right\vert ^{2}$, with $n \in \mathbb{N}$, $a_j \in \mathbb{R}$ and $w_j \in \mathbb{R}^{2}$, then the interior of the type set $E_{\mu_{\gamma}}$ coincide with the interior of the trapezoid with vertices $(0,0)$, $(1,1)$, $D$ and $D^{\prime }$. Moreover the semi-open segments $\left[(1,1);(p_{D}^{-1},q_{D}^{-1}) \right)$ and $\left[(0,0);(1-q_{D}^{-1},1-p_{D}^{-1}) \right)$ are contained in $E_{\mu_{\gamma}}$. \end{theorem}
\begin{theorem} Let $\mu_{\gamma}$ be a fractional Borel measure as in Theorem 1. Then $C_{\gamma} \in E_{\mu_{\gamma}}$. \end{theorem}
Let $\widetilde{\mu}_{\gamma}$ be the Borel measure given by \begin{equation} \left\langle \widetilde{\mu}_{\gamma} ,f\right\rangle
=\int\limits_{\mathbb{R}^{2n}}f\left( w, |w|^{2m}
\right) \eta \left( |w|^{2} \right) | w |^{\gamma }dw,
\label{mu3} \end{equation} where $m \in \mathbb{N}_{\geq 2}$, $\gamma = \frac{2(m-1)}{(n+1)m}$, and $\eta$ is a function in $C_{c}^{\infty }(\mathbb{R})$ such that $0\leq \eta \leq 1$, $\eta(t)\equiv 1$ if $t\in \lbrack -1,1]$ and $supp(\eta)\subset (-2,2)$. \\
In a similar way we characterize the type set of the Borel measure $\widetilde{\mu}_{\gamma}$ supported on the graph of the function $\varphi(w) = |w|^{2m}$. In fact we prove
\begin{theorem} Let $\widetilde{\mu}_{\gamma}$ be the Borel measure defined in (\ref{mu3}) with $\gamma = \frac{2(m-1)}{(n+1)m}$, where $n \in \mathbb{N}$ and $m \in \mathbb{N}_{\geq 2}$. Then the type set $E_{\widetilde{\mu}_{\gamma}}$ is the closed triangle with vertices \[ A=\left( 0,0\right) ,\qquad B=\left( 1,1\right) ,\qquad C=\left( \frac{2n+1}{ 2n+2},\frac{1}{2n+2}\right). \] \end{theorem} This result improves to the one obtained in Theorem 2 in \cite{G-R}.
\qquad
Throughout this work, $c$ will denote a positive constant not necessarily the same at each occurrence.
\section{Auxiliary results}
\begin{lemma} Let $\mu_{\gamma}$ be the fractional Borel measure defined by (\ref{mu2}), where $\varphi(w) = \sum_{j=1}^{n} a_j |w_j|^{2}$ and $0 < \gamma < 2n$. If $\left(\frac{1}{p},\frac{1}{q}\right) \in E_{\mu_{\gamma}}$, then $\frac{1}{q}\geq \frac{1}{p}-\frac{2n-\gamma}{2n+2}$. \end{lemma}
\begin{proof} For $0<\delta \leq 1$, we define $Q_{\delta}=D_{\delta}\times \left[-(4M+n)\delta^{2},(4M+n)\delta^{2}\right]$, where $D_{\delta}=\left\{x \in \mathbb{R}^{2n}: \left\Vert x \right\Vert \leq \delta \right\}$ and $M=\max \left\{\left\vert \varphi(y)\right\vert:y \in D_{1} \right\}.$ We put $$A_{\delta} = \left\{(x,t) \in D_{\frac{\delta}{2}}\times \mathbb{R}: \left\vert t- \varphi(x) \right\vert \leq 2M\delta^{2}\right\}.$$ Let $f_{\delta} = \chi_{Q_{\delta}}$. We will prove first that $\left\vert (f_{\delta} \ast \mu_{\gamma})(x,t) \right\vert\geq c \delta^{2n-\gamma}$ for all $(x,t) \in A_{\delta}$, where $c$ is a constant independent of $\delta$. \\ If $(x,t) \in A_{\delta}$, we have that \begin{equation} (x,t)\cdot (y,\varphi(y))^{-1} \in Q_{\delta} \text{\textit{ for all }} y \in D_{\frac{\delta}{2}}; \label{qd} \end{equation} indeed, $(x,t)\cdot (y,\varphi(y))^{-1}=\left(x-y,t-\varphi(y)-\frac{1}{2} B(x,y) \right)$, from the homogeneity of $\varphi$ and since $\frac{1}{2}\left\vert B(x,y)\right\vert \leq n\left\Vert x\right\Vert _{ \mathbb{R}^{2n}}\left\Vert x-y\right\Vert _{\mathbb{R}^{2n}}$, (\ref{qd}) follows . So \[
\left\vert(f_{\delta} \ast \mu)(x,t) \right\vert = \int_{\mathbb{R}^{2n}} f_{\delta}\left((x,t)\cdot(y,\varphi(y))^{-1} \right) \prod_{j=1}^{n} \eta_j(|y_j|^{2}) \left\vert y_j \right\vert^{-\frac{\gamma}{n}} dy \] \[
\geq \int_{D_{\frac{\delta}{2}}} \left\vert y \right\vert^{-\gamma} \prod_{j=1}^{n} \eta_j(|y_j|^{2}) dy =\int_{D_{\frac{\delta}{2}}}\left\vert y \right\vert^{-\gamma} dy=c \delta^{2n-\gamma} \] for all $(x,t) \in A_{\delta}$ and all $0 < \delta < 1/2$. Thus \[ \left\Vert f_{\delta} \ast \mu_{\gamma} \right\Vert _{q}\geq \left(\int_{A_{\delta}} \left\vert f \ast \mu_{\gamma} \right\vert^{q} \right)^{\frac{1}{q}}\geq c\delta ^{2n-\gamma}\left\vert A_{\delta} \right\vert ^{\frac{1}{q}} = c \delta^{2n-\gamma+\frac{1}{q}(2n+2)}. \] On the other hand $\left(\frac{1}{p},\frac{1}{q}\right) \in E_{\mu}$ implies \[ \left\Vert f_{\delta} \ast \mu_{\gamma} \right\Vert _{q}\leq c \left\Vert f_{\delta} \right\Vert_{p}= c \delta^{\frac{1}{p}(2n+2)}, \] therefore $\delta^{2n-\gamma+\frac{1}{q}(2n+2)}\leq c \delta^{\frac{1}{p}(2n+2)}$ for all $0 < \delta < 1$ small enough, then $$\frac{1}{q}\geq \frac{1}{p} - \frac{2n-\gamma}{2n+2}.$$ \end{proof}
The following two lemmas deal on certain identities that involve to the Laguerre polynomials. We recall the definition of these polynomials: the Laguerre polynomials $L^{\alpha}_{n}(x)$ are defined by the formula \[ L^{\alpha}_{n}(x)= e^{x} \frac{x^{-\alpha}}{n!} \frac{d^{n}}{dx^{n}}(e^{-x} x^{n + \alpha}), \,\,\,\,\,\,\,\, n=0, 1, 2, ... \] for arbitrary real number $\alpha > -1$.
\begin{lemma} If $Re(\beta)>-1$, then $$\int\limits_{0}^{\infty }\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma =\frac{1}{k!} \left[\frac{d^{k}}{dr^{k}} \left( \frac{\Gamma (\beta +1)}{(1-r)^{n} \left( \frac{1}{2}+\frac{r}{1-r}+i\xi \right)^{\beta +1}}\right) \right]_{r=0},$$ for $n \in \mathbb{N}$ and $k \in \mathbb{N} \cup \{ 0 \}$. \end{lemma}
\begin{proof} Let $0<\epsilon <1$ be fixed. From the generating function identity (4.17.3) in \cite{Lebedev} p. 77, we have
\begin{equation} \sum\limits_{j\geq 0} \sigma ^{\beta }L_{j}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) } r^{j}=\frac{1}{(1-r)^{n}} \sigma ^{\beta }e^{-\sigma \left( \frac{1}{2}+\frac{r}{1-r}+i\xi
\right)}, \,\,\,\,\,\,\,\,\,\, |r|<1. \label{ident3} \end{equation} Since $\left\vert L_{j}^{n-1}\left( \sigma \right) e^{-\sigma \frac{1}{2}} \right\vert \leq \frac{(j+n-1)!}{j!(n-1)!}$ for all $\sigma > 0$ (see proposition 4.2 in \cite{thangavelu}), the series in (\ref{ident3}) is uniformly convergent on the interval $\left[\epsilon, \frac{1}{\epsilon} \right]$. Integrating on this interval we obtain $$ \sum\limits_{j\geq 0}\left( \int\limits_{\epsilon }^{\frac{1}{\epsilon } }\sigma ^{\beta }L_{j}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{ 2}+i\xi \right) }d\sigma \right) r^{j}=\frac{1}{(1-r)^{n}} \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma \left( \frac{1}{2}+\frac{r}{1-r}+i\xi \right) }d\sigma, $$ so \begin{equation} \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma =\frac{1}{k!}\left[ \frac{d^{k}}{dr^{k}}\left( \frac{1}{ (1-r)^{n}}\int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma \left( \frac{1}{2}+\frac{r}{1-r}+i\xi \right) }d\sigma \right)\right]_{r=0}. \label{ident4} \end{equation} Now let us computation $\left[ \frac{d^{k}}{dr^{k}} \left( \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma^{\beta} e^{-\sigma \left( \frac{1}{2}+\frac{r}{1-r}+i\xi \right) }d\sigma \right) \right]_{r=0}$. We start to compute first the derivatives of the function $u\rightarrow \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma u }d\sigma$, where $Re(u) > 0$. We define $\alpha_{\epsilon}(\sigma)= \sigma u$, $\sigma \in \left[\epsilon, \frac{1}{\epsilon} \right]$, so $$\int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma u }d\sigma = u^{-(\beta+1)} \int\limits_{\alpha_{\epsilon }} z^{\beta }e^{-z }dz$$ to apply the Cauchy's Theorem we have \begin{equation} \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma u }d\sigma = u^{-(\beta+1)} \left[\int\limits_{\epsilon }^{\frac{1}{\epsilon }} x^{\beta }e^{-x }dx + I_{1}(u,\epsilon) - I_{2}(u,\epsilon) \right] \label{ident5} \end{equation} where $$I_{1}(u,\epsilon)=\int\limits_{\left[\frac{1}{\epsilon},\frac{1}{\epsilon}u \right]} z^{\beta }e^{-z }dz$$ and $$I_{2}(u,\epsilon)=\int\limits_{\left[\epsilon,\epsilon u \right]} z^{\beta }e^{-z}dz$$ are line integrals on $\mathbb{C}$. Now we will prove that for each $u_{0} \in \mathbb{C}$ with $Re(u_{0})>0$ the following identity holds \begin{equation} \lim_{\epsilon \rightarrow 0} \left[ \frac{d^{k}}{du^{k}} I_{j}(u, \epsilon) \right]_{u=u_{0}} = 0 \label{ij} \end{equation} for $j=1,2$ and all $k \geq 0$. It is easy to check that $$I_{2}(u,\epsilon)=\epsilon ^{\beta+1} \int\limits_{\left[1, u \right]} z^{\beta }e^{-\epsilon z}dz.$$ Since $Re(\beta)>-1$ we have that $\lim_{\epsilon \rightarrow 0} I_{2}(u_{0}, \epsilon) = 0$. From the analyticity of the function $z \rightarrow z^{\beta }e^{-\epsilon z}$ on the region $\{ z: Re(z)>0 \}$ it follows for $k \geq 1$ $$\left[ \frac{d^{k}}{du^{k}} I_{2}(u, \epsilon) \right]_{u=u_{0}}=\epsilon ^{\beta+1} \left[ \frac{d^{k-1}}{du^{k-1}} u^{\beta} e^{-\epsilon u} \right]_{u=u_{0}},$$ then $\lim_{\epsilon \rightarrow 0} \left[ \frac{d^{k}}{du^{k}} I_{2}(u, \epsilon) \right]_{u=u_{0}} = 0$ for all $k\geq 0$. \\ Analogously and taking account the rapid decay of the function $z\rightarrow e^{-z}$ on the region $\{z : Re(z)>0 \}$ we obtain that $\lim_{\epsilon \rightarrow 0} \left[ \frac{d^{k}}{du^{k}} I_{1}(u, \epsilon) \right]_{u=u_{0}} =0$ for all $k\geq 0$, so (\ref{ij}) follows. To derive in (\ref{ident5}), from the Leibniz's formula and (\ref{ij}) it follows that \begin{equation} \lim_{\epsilon \rightarrow 0} \left[ \frac{d^{k}}{du^{k}} \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma u }d\sigma \right]_{u=u_{0}} =\Gamma(\beta+1)\left[ \frac{d^{k}}{du^{k}} u^{-(\beta+1)}\right]_{u=u_{0}}. \label{iu} \end{equation} Finally, from (\ref{iu}), to apply the chain rule to the function $r\rightarrow \int\limits_{\epsilon }^{\frac{1}{\epsilon }}\sigma ^{\beta }e^{-\sigma u(r) }d\sigma$ where $u(r)=\frac{1}{2}+\frac{r}{1-r}+i \xi$ and the Leibniz's formula give, to do $\epsilon \rightarrow 0$ in (\ref{ident4}), that $$\int\limits_{0}^{\infty }\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma =\lim_{{\epsilon \rightarrow 0}} \int\limits_{\epsilon }^{ \frac{1}{\epsilon }}\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma $$ $$=\frac{1}{k!} \left[\frac{d^{k}}{dr^{k}} \left( \frac{\Gamma (\beta +1)}{(1-r)^{n} \left( \frac{1}{2}+\frac{r}{1-r}+i\xi \right)^{\beta +1}}\right) \right]_{r=0}.$$ \end{proof}
\begin{lemma} If $Re(\beta) > -1$ and $w(\xi)= -\frac{\frac{1}{2}-i \xi}{\frac{1}{2}+i \xi}$ $(\xi \in \mathbb{R})$, then \[ \int\limits_{0}^{\infty }\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma =\frac{\Gamma(\beta+1)}{\left(\frac{1}{2}+i\xi \right)^{\beta+1}} \sum_{j+l=k} \frac{\Gamma(n-1-\beta+j)\Gamma(\beta+1+l)}{\Gamma(n-1-\beta)\Gamma(\beta+1)} \frac{w(\xi)^{l}}{j!l!}, \] for $n \in \mathbb{N}$ and $k \in \mathbb{N} \cup \{ 0 \}$. \end{lemma}
\begin{proof} We will start find the power series centered at $r=0$ of the following function
$$Q(r)=\frac{1}{(1-r)^{n} \left( \frac{1}{2}+ \frac{r}{1-r}+i \xi \right)^{\beta+1}}, \,\,\,\,\,\,\,\,\,\, |r| < 1.$$ We observe that $$Q(r)=\frac{1}{(1-r)^{n-\beta-1}\left(\frac{1}{2}+ i\xi+ r\left( \frac{1}{2}-i \xi \right) \right)^{\beta+1} },$$ doing $w=-\frac{\frac{1}{2}-i \xi}{\frac{1}{2}+i \xi}$, we obtain $$Q(r)=\frac{1}{ \left(\frac{1}{2}+i \xi \right)^{\beta+1} (1-r)^{n-1-\beta}(1-rw)^{\beta+1}}.$$ A simple computation gives $$(1-r)^{-n+\beta+1}=1+\sum_{j\geq 1} (n-1-\beta)(n-1-\beta+1)...(n-1-\beta+j-1) \frac{r^{j}}{j!}$$ \begin{equation} =\sum_{j\geq 0} \frac{\Gamma(n-1-\beta+j)}{\Gamma(n-1-\beta)} \frac{r^{j}}{j!}. \label{serie2} \end{equation} Analogously we have \[ (1-rw)^{-\beta-1}=\sum_{j\geq 0} \frac{\Gamma(\beta+1+j)}{\Gamma(\beta+1)} \frac{(rw)^{j}}{j!}. \] Thus $$Q(r)=\frac{1}{\left(\frac{1}{2}+i\xi \right)^{\beta+1}} \left( \sum_{j+l \geq 0} \frac{\Gamma(n-1-\beta+j)\Gamma(\beta+1+l)}{\Gamma(n-1-\beta)\Gamma(\beta+1)} \frac{r^{j+l}w^{l}}{j!l!} \right).$$ Finally, from Lemma 5 it follows $$\int\limits_{0}^{\infty }\sigma ^{\beta }L_{k}^{n-1}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma =\frac{\Gamma(\beta+1)}{\left(\frac{1}{2}+i\xi \right)^{\beta+1}} \sum_{j+l=k} \frac{\Gamma(n-1-\beta+j)\Gamma(\beta+1+l)}{\Gamma(n-1-\beta)\Gamma(\beta+1)} \frac{w^{l}}{j!l!}.$$ \end{proof}
\begin{lemma} If $Re(\beta) > -1$, then \[ \sum_{j+l=k} \frac{\Gamma(n-1-Re(\beta)+j)) \Gamma(Re(\beta)+1+l)}{\Gamma(n-1-Re(\beta))\Gamma(Re(\beta)+1)} \frac{1}{j!l!}=\frac{(n+k-1)!}{(n-1)!k!}, \] for $n \in \mathbb{N}$ and $k \in \mathbb{N} \cup \{ 0 \}$. \end{lemma}
\begin{proof} From $(\ref{serie2})$ it obtains $$\sum_{j+l=k} \frac{\Gamma(n-1-Re(\beta)+j)) \Gamma(Re(\beta)+1+l)}{\Gamma(n-1-Re(\beta))\Gamma(Re(\beta)+1)} \frac{1}{j!l!}$$ $$= \frac{1}{k!} \left[ \frac{d^{k}}{dr^{k}} (1-r)^{-n+Re(\beta)+1} (1-r)^{-Re(\beta)-1}\right]_{r=0}=\frac{1}{k!} \left[ \frac{d^{k}}{dr^{k}} (1-r)^{-n} \right]_{r=0}.$$ Since $$(1-r)^{-n} =\sum_{j\geq 0} \frac{\Gamma(n+j)}{\Gamma(n)} \frac{r^{j}}{j!}=\sum_{j\geq 0} \frac{(n+j-1)!}{(n-1)!j!} r^{j},$$ we have $$\sum_{j+l=k} \frac{\Gamma(n-1-Re(\beta)+j)) \Gamma(Re(\beta)+1+l)}{\Gamma(n-1-Re(\beta))\Gamma(Re(\beta)+1)} \frac{1}{j!l!}=\frac{(n+k-1)!}{(n-1)!k!}.$$ \end{proof}
\section{The main results}
To prove Theorem 1 we will decompose the operator $T_{\mu_{\gamma}}$ of the following way: we consider a family $\left\{T_{\mu_{k}} \right\}_{k \in \mathbb{N}}$ of operators such that $T_{\mu_{\gamma}} = \displaystyle{\sum_{k \in \mathbb{N}}} T_{\mu_{k}}$, $\left\Vert T_{\mu_{k}} \right\Vert_{1,1} \sim 2^{-k(2n-\gamma)}$ and $\left\Vert T_{\mu_{k}} \right\Vert_{p,q} \sim 2^{k\gamma}\left\Vert T_{\mu_{0}} \right\Vert_{p,q}$ where $T_{\mu_{0}}$ is the operator defined by (\ref{tmu}), taking there $\gamma =0$ and $\varphi(w)=\sum_{j=1}^{n} a_j \left\vert w_j\right\vert^{2}$. Then Theorem 1 will follow from Theorem 1 in \cite{G-R}, the Riesz-Thorin convexity Theorem and Lemma 4. \\ ${}$ \\ \textit{Proof of Theorem 1.} For each $k \in \mathbb{N}$ we define \[ A_{k}= \left\{ y=(y_1, ..., y_n) \in (\mathbb{R}^{2})^{n} : 2^{-k} < \left\vert y_j \right\vert \leq 2^{-k+1}, j=1, 2,..., n \right\} \] Let $\mu_{k}$ be the fracional Borel measure given by \[
\mu_{k}(E)= \int_{A_{k}} \chi_{E} \left(y, \varphi(y) \right) \prod_{j=1}^{n} \eta_j \left( |y_j|^{2} \right) | y_j |^{-\frac{\gamma}{n}} dy \] and let $T_{\mu_{k}}$ be its corresponding convolution operator, i.e: $T_{\mu_{k}}f=f \ast \mu_{k}$. Now, it is clear that $\mu_{\gamma}=\sum_{k}\mu_{k}$ and $\left\Vert T_{\mu_{\gamma}} \right\Vert_{p,q} \leq \sum_{k} \left\Vert T_{\mu_{k}} \right\Vert_{p,q}$. For $f\geq 0$ we have that $$
\int f(y,s) d\mu_{k}(y,s) \leq 2^{k\gamma}\int_{\mathbb{R}^{2n}} f\left(y, \varphi(y) \right) \prod_{j=1}^{n} \eta_j \left( |y_j|^{2} \right) dy.$$ Thus $\left\Vert T_{\mu_{k}} \right\Vert_{p,q} \leq c 2^{k\gamma}\left\Vert T_{\mu_{0}} \right\Vert_{p,q}$, from Theorem 1 in \cite{G-R} it follows that \[ \left\Vert T_{\mu_{k}} \right\Vert_{\frac{2n+2}{2n+1},2n+2} \leq c 2^{k\gamma}. \] It is easy to check that $\left\Vert T_{\mu_{k}} \right\Vert_{1,1}\leq \left\vert \mu_{k} (\mathbb{R}^{2n+1}) \right\vert \sim \int_{A_{k}} \left\vert y \right\vert^{-\gamma} dy= c 2^{-k(2n-\gamma)}.$ \\ For $0< \theta <1$, we define $$\left(\frac{1}{p_{\theta}}, \frac{1}{q_{\theta}} \right) = \left(\frac{2n+1}{2n+2}, \frac{1}{2n+2} \right) (1-\theta) + (1,1)\theta,$$ by the Riesz convexity Theorem we have $$ \left\Vert T_{\mu_{k}} \right\Vert_{p_{\theta},q_{\theta}} \leq c 2^{k\gamma(1-\theta)-k(2n-\gamma) \theta}$$ choosing $\theta$ such that $k\gamma(1-\theta)-k(2n-\gamma) \theta = 0$ result $\displaystyle{\sup_{k \in \mathbb{N}}} \left\Vert T_{\mu_{k}} \right\Vert_{p_{\theta},q_{\theta}} \leq c < \infty$. A simple computation gives $\theta=\frac{2n-\gamma}{2n}$, then $\left( \frac{1}{p_{\theta}}, \frac{1}{q_{\theta}} \right) = \left( \frac{1}{p_{D}}, \frac{1}{q_{D}} \right)$, so $\left\Vert T_{\mu_{k}} \right\Vert_{p_{D},q_{D}} \leq c$, where $c$ no depend on $k$. Interpolating once again, but now between the points $\left(\frac{1}{p_{D}}, \frac{1}{q_{D}} \right)$ and $(1,1)$ we obtain, for each $0< \tau <1$ fixed $$ \left\Vert T_{\mu_{k}} \right\Vert_{p_{\tau},q_{\tau}} \leq c 2^{-k(2n-\gamma) \tau},$$ since $\left\Vert T_{\mu_{\gamma}} \right\Vert_{p,q} \leq \sum_{k} \left\Vert T_{\mu_{k}} \right\Vert_{p,q}$ and $0< \gamma < 2n$, it follows that $$ \left\Vert T_{\mu_{\gamma}} \right\Vert_{p_{\tau},q_{\tau}} \leq c\sum_{k \in \mathbb{N}} 2^{-k(2n-\gamma) \tau} <\infty,$$ by duality we also have $$\left\Vert T_{\mu_{\gamma}} \right\Vert_{\frac{q_{\tau}}{q_{\tau}-1},\frac{p_{\tau}}{p_{\tau}-1}}\leq c_{\tau} <\infty.$$ Finally, the theorem follows from the Riesz convexity Theorem, the restrictions that appear in (\ref{restricciones}) and Lemma 4. $\square$
\qquad
To prove Theorem 2, we will consider an auxiliary operator $T_N$, with $N \in \mathbb{N}$ fixed, which will be embedded in an analytic family $T_{N,z}$ of operators on the strip $-\frac{2n-\gamma }{2+\gamma }\leq Re(z)\leq 1$ such that \begin{equation} \left\{ \begin{array}{c} \left\Vert T_{N,z}\left( f\right) \right\Vert _{L^{\infty }\left( \mathbb{H} ^{n}\right) }\leq A_{z}\left\Vert f\right\Vert _{L^{1}\left( \mathbb{H} ^{n}\right) }\qquad Re(z)=1 \\ \left\Vert T_{N,z}\left( f\right) \right\Vert _{L^{2}\left( \mathbb{H} ^{n}\right) }\leq A_{z}\left\Vert f\right\Vert _{L^{2}\left( \mathbb{H} ^{n}\right) }\qquad Re(z)=-\frac{2n-\gamma }{ 2+\gamma } \label{desig2} \end{array} \right. \end{equation} where $A_{z}$ will depend admissibly on the variable $z$ and it will not depend on $N$. We denote $T_N = T_{N,0}$. By Stein's theorem on complex interpolation, it will follow that the operator $T_{N}$ will be bounded from $L^{p_{\gamma}}(\mathbb{H}^{n})$ into $L^{p'_{\gamma}}(\mathbb{H}^{n})$, where $\left(\frac{1}{p_{\gamma}}, \frac{1}{p'_{\gamma}} \right)=C_{\gamma}$, uniformly in $N$. If we see that $T_{N}f(x,t) \rightarrow c T_{\mu_{\gamma}}f(x,t)$ a.e.$(x,t)$ as $N\rightarrow \infty$, then Theorem 2 will follow from Fatou's Lemma.
To prove the second inequality in (\ref{desig2}) we will see that such a family will admit the expression \[ T_{N,z}(f)(x,t)=\left( f\ast K_{N,z}\right) (x,t), \] where $K_{N,z}\in L^{1}(\mathbb{H}^{n})$, moreover it is a \textit{polyradial} function (i.e. the values of $K_{N,z}$ depend on $\left\vert w_{1}\right\vert ,$...$,\left\vert w_{n}\right\vert $ and $t$). Now our operator $T_{N,z}$ can be realized as a multiplication of operators via the group Fourier transform, i.e. \[ \widehat{T_{N,z}(f)}(\lambda )=\widehat{f}(\lambda )\widehat{K_{N,z}} (\lambda ) \] where, for each $\lambda \neq 0$, $\widehat{K_{N,z}}(\lambda )$ is an operator on the Hilbert space $L^{2}(\mathbb{R}^{n})$ given by \[ \widehat{K_{N,z}}(\lambda )g(\xi )=\int\limits_{\mathbb{H} ^{n}}K_{N,z}(\varsigma ,t)\pi _{\lambda }(\varsigma ,t)g(\xi )d\varsigma dt. \] It then follows from Plancherel's theorem for the group Fourier transform that \[ \left\Vert T_{N,z}f\right\Vert _{L^{2}(\mathbb{H}^{n})}\leq A_{z}\left\Vert f\right\Vert _{L^{2}(\mathbb{H}^{n})} \] if and only if \begin{equation} \left\Vert \widehat{K_{N,z}}(\lambda )\right\Vert _{op}\leq A_{z} \label{L21} \end{equation} uniformly over\textit{\ }$N$\textit{\ }and\textit{\ }$\lambda \neq 0.$ Since $K_{N,z}$ is a poliradial integrable function, then by a well known result of Geller (see Lemma 1.3, p. 213 in \cite{geller}), the operators $\widehat{ K_{N,z}}(\lambda ):L^{2}(\mathbb{H}^{n})\rightarrow L^{2}(\mathbb{H}^{n})$ are, for each $\lambda \neq 0$, diagonal with respect to a Hermite basis for $L^{2}(\mathbb{R}^{n})$. This is \[ \widehat{K_{N,z}}(\lambda )=C_{n}\left( \delta _{\gamma ,\alpha }\nu _{N,z}(\alpha ,\lambda )\right) _{\gamma ,\alpha \in \mathbb{N}_{0}^{n}} \] where $C_{n}=(2\pi )^{n}$, $\alpha =(\alpha _{1},...,\alpha _{n})$, $\delta _{\gamma ,\alpha }=1$ if $\gamma = \alpha$ and $\delta _{\gamma ,\alpha }=0$ if $\gamma \neq \alpha$, and the diagonal entries $\nu _{N,z}(\alpha _{1},...,\alpha _{n},\lambda )$ can be expressed explicitly in terms of the Laguerre transform. We have in fact \[ \nu_{N,z}(\alpha _{1},...,\alpha _{n},\lambda )=\int\limits_{0}^{\infty }\,...\,\int\limits_{0}^{\infty }\,K_{N,z}^{\lambda }(r_{1},...,r_{n})\prod_{j=1}^{n}\left( r_{j}L_{\alpha _{j}}^{0}(\frac{1}{2} \left\vert \lambda \right\vert r_{j}^{2})e^{-\frac{1}{4}\left\vert \lambda \right\vert r_{j}^{2}}\right) \,dr_{1}...dr_{n} \] where $L_{k}^{0}(s)$ are the Laguerre polynomials, i.e. $L_{k}^{0}(s)= \sum_{i=0}^{k}\left( \frac{k!}{(k-i)!i!}\right) \frac{(-s)^{i}}{i!}$ and $ K_{N,z}^{\lambda }(\varsigma )=\int\limits_{\mathbb{R}}K_{N,z}(\varsigma ,t)e^{i\lambda t}dt.$ Now (\ref{L21}) is equivalent to \[ \left\Vert T_{N,z}f\right\Vert _{L^{2}(\mathbb{H}^{n})}\leq A_{z}\left\Vert f\right\Vert _{L^{2}(\mathbb{H}^{n})} \] if and only if \begin{equation} \left\vert \nu_{N,z}(\alpha _{1},...,\alpha _{n},\lambda )\right\vert \leq A_{z} \label{L22} \end{equation} uniformly over\textit{\ }$N$, $\alpha _{j}$\textit{\ }and\textit{\ }$\lambda \neq 0.$ If $Re(z)=-\frac{2n-\gamma }{2+\gamma }$, in the proof of Theorem 2 we find that (\ref{L22}) holds with $A_{z}$ independent of $N$, $\lambda \neq 0$ and $\alpha _{j}$, and then we obtain the boundedness on $L^{2}(\mathbb{H}^{n})$ that is stated in (\ref{desig2}).
We consider the family $\{ I_{z} \}_{z \in \mathbb{C}}$ of distributions on $\mathbb{R}$ that arises by analytic continuation of the family $\{ I_{z} \}$ of functions, initially given when $Re(z)>0$ and $s\in \mathbb{R} \setminus\{ 0 \}$ by \begin{equation} I_{z}(s)=\frac{2^{-\frac{z}{2}}}{\Gamma \left( \frac{z}{2}\right) } \left\vert s\right\vert ^{z-1}. \label{iz} \end{equation} In particular, we have $ \widehat{I_{z}}=I_{1-z}$, also $I_{0}=c\delta $ where $\widehat{\cdot }$ denotes the Fourier transform on $\mathbb{R}$ and $\delta $ is the Dirac distribution at the origin on $\mathbb{R}$.
Let $H\in S(\mathbb{R)}$ such that $supp(\widehat{H})\subseteq \left( -1,1\right) $ and $\int \widehat{H}(t)dt=1$. Now we put $\phi _{N}(t)=H(\frac{t}{N})$ thus $\widehat{\phi _{N}}(\xi )=N\widehat{H}(N\xi )$ and $\widehat{\phi _{N}}\rightarrow \delta $ in the sense of the distribution, as $N\rightarrow \infty $.
For $z\mathbb{\in C}$ and $N\in \mathbb{N}$, we also define $J_{N,z}$ as the distribution on $\mathbb{H}^{n}$ given by the tensor products \begin{equation} J_{N,z}=\delta \otimes ...\otimes \delta \otimes \left( I_{z}\ast _{\mathbb{R }}\widehat{\phi _{N}}\right) \label{jz} \end{equation} where $\ast _{\mathbb{R}}$ denotes the usual convolution on $\mathbb{R}$ and $I_{z}$ is the fractional integration kernel given by (\ref{iz}). We observe that \begin{equation} J_{N,0}=\delta \otimes ...\otimes \delta \otimes c\widehat{\phi _{N}}\rightarrow \delta \otimes ...\otimes \delta \otimes c\delta \label{jz2} \end{equation} in the sense of the distribution as $N \rightarrow \infty $.
\qquad
\textit{Proof of Theorem 2.} Let $\left\{ T_{N,z}\right\} $ be the family operators on the strip $-\frac{2n-\gamma }{ 2+\gamma }\leq Re(z)\leq 1$, given by \[ T_{N,z}f=f\ast \mu_{\gamma, z}\ast J_{N,z}, \] where $J_{N,z}$ is given by (\ref{jz}) and $\mu _{\gamma, z}$ by \begin{equation} \mu_{\gamma, z}(E)=\int\limits_{\mathbb{R}^{2n}}\chi _{E}\left(
w,\varphi(w)\right) \prod_{j=1}^{n} \eta_j \left( |w_j|^{2} \right) \left\vert w_j \right\vert^{(z-1) \frac{\gamma}{n}}dw. \label{muz} \end{equation} Now (\ref{jz2}) implies that $T_{N, 0}f(x,t)\rightarrow c T_{\mu_{\gamma}}f(x,t)$ a.e.$(x,t)$ as $N\rightarrow \infty $.
\qquad
For $Re(z)=1$ we have
$$\mu _{\gamma, z}\ast J_{N,z}(x,t)= \left( I_{z}\ast _{\mathbb{R}}\widehat{\phi _{N}}\right) \left( t-\varphi(x)\right) \prod_{j=1}^{n}\eta_{j} \left( |x_j|^{2} \right) |x_j|^{iIm(z)\frac{\gamma}{n} },$$ so $\left\Vert \mu_{\gamma, z}\ast J_{N,z}\right\Vert _{\infty }\leq c\left\vert \Gamma \left( \frac{z}{2}\right) \right\vert ^{-1}$. Then, for $Re(z)=1$, we obtain $$ \left\Vert T_{N,z}f\right\Vert _{\infty }\leq \left\Vert f\ast \mu_{\gamma, z}\ast J_{N,z}\right\Vert _{\infty }\leq \left\Vert f\right\Vert _{1}\left\Vert \mu_{\gamma, z}\ast J_{N,z}\right\Vert _{\infty }\leq c\left\vert \Gamma \left( \frac{z}{2}\right) \right\vert ^{-1}\left\Vert f\right\Vert _{1} $$ where $c$ is a positive constant independent of $N$ and $z$.
\qquad
We put $K_{N, z} = \mu _{\gamma, z}\ast J_{N,z}$, for $Re(z)=-\frac{2n-\gamma }{2+\gamma }$ we have that $K_{N,z} \in L^{1}(\mathbb{H}^{n})$. Indeed \[
K_{N,z}(x,t)=\left( I_{z}\ast_{\mathbb{R}}\widehat{\phi _{N}}\right) \left( t-\varphi( x) \right) \prod_{j=1}^{n}\eta_{j} \left( |x_j|^{2} \right) |x_j|^{(z-1) \frac{\gamma}{n}}, \]
since $0<\gamma <2n$ it follows that $2+Re((z-1)\frac{\gamma}{n})=2-\frac{2n+2}{2+\gamma} \frac{\gamma}{n} > 0$ and so $\prod_{j=1}^{n}\eta_{j} \left( |x_j|^{2} \right) |x_j|^{(z-1) \frac{\gamma}{n}} \in L^{1}(\mathbb{R}^{2n})$, in the proof of Lemma 5 in \cite{G-R} it shows that $\left( I_{z}\ast_{\mathbb{R}}\widehat{\phi _{N}}\right) \in L^{1}(\mathbb{R})$. These two facts imply that $K_{N,z} \in L^{1}(\mathbb{H}^{n})$. In addition $K_{z,N}$ is a polyradial function. Thus the operator $\widehat{K_{z,N}}(\lambda )$ is diagonal with respect to a Hermite base for $L^{2}(\mathbb{R}^{n})$, and its diagonal entries $\nu_{z,N}(\alpha,\lambda )$, with $\alpha =(\alpha_1, ..., \alpha_n) \in \mathbb{N}_{0}^{n}$, are given by \[ \nu_{N, z}(\alpha ,\lambda )= \int\limits_{0}^{\infty }\,...\,\int\limits_{0}^{\infty }\,K_{N,z}^{\lambda
}(r_{1},...,r_{n})\prod_{j=1}^{n}\left( r_{j}L_{\alpha _{j}}^{0}(| \lambda| r_{j}^{2}/2)e^{-\frac{1}{4}\left\vert \lambda \right\vert r_{j}^{2}}\right) \,dr_{1}...dr_{n} \] \begin{equation} = I_{1-z}(-\lambda )\phi _{N}(\lambda ) \prod_{j=1}^{n} \int\limits_{0}^{\infty }\eta_{j}(r_j^{2})L_{\alpha_j}^{0}\left(
| \lambda |r_{j}^{2}/2\right)
e^{-\frac{1}{4}| \lambda | r_{j}^{2}}e^{i\lambda a_j r_{j}^{2}}r_j^{1+(z-1)\frac{\gamma}{n}} dr_j. \label{diag2} \end{equation} Thus, it is enough to study the integral \[
\int\limits_{0}^{\infty }\eta_{1}(r^{2})L_{\alpha_1}^{0}\left(|\lambda|r^{2}/2\right) e^{-\frac{1}{4} | \lambda|r^{2}} e^{i\lambda a_1 r^{2}} r^{1+(z-1)\frac{\gamma}{n}} dr, \]
where $a_1 \in \mathbb{R}$ and $\eta_1 \in C_{c}^{\infty}(\mathbb{R})$. We make the change of variable $\sigma = |\lambda|r^{2}/2$ in such an integral to obtain \[
\int\limits_{0}^{\infty }\eta_{1}(r^{2})L_{\alpha_1}^{0}\left(|\lambda|r^{2}/2\right) e^{-\frac{1}{4} | \lambda|r^{2}} e^{i\lambda a_1 r^{2}} r^{1+(z-1)\frac{\gamma}{n}} dr \] \[
= 2^{-\frac{(n+1) \gamma}{(2+\gamma)n}}| \lambda |^{-\left( 1+\frac{(z-1)\gamma}{2n}\right)} \int\limits_{0}^{\infty }\eta_{1}\left( \frac{2\sigma }{
| \lambda |}\right) L_{\alpha_1}^{0}\left( \sigma \right) e^{- \frac{\sigma }{2}}e^{i2sgn(\lambda ) a_1 \sigma }\sigma ^{\frac{(z-1)\gamma}{2n}}d\sigma \] \[
= 2^{-\frac{(n+1) \gamma}{(2+\gamma)n}} | \lambda |^{-\left( 1+\frac{(z-1)\gamma}{2n} \right) }\left( F_{\alpha_1 ,\beta}G_{\lambda }\right) \widehat{\left. {}\right. } (-2sgn(\lambda ) a_1) \] \[ = 2^{-\frac{(n+1) \gamma}{(2+\gamma)n}} \left\vert \lambda \right\vert ^{-\left( 1+\frac{(z-1)\gamma}{2n}\right) }(\widehat{F_{\alpha_1 ,\beta}}\ast \widehat{G_{\lambda }} )(-2sgn(\lambda ) a_1) \] where $$F_{\alpha_1, \beta}(\sigma ):=\chi _{(0,\infty )}(\sigma )L_{\alpha_1}^{0}\left( \sigma \right) e^{-\frac{\sigma }{2}}\sigma ^{\beta },$$ with $\beta =\frac{ (z-1) \gamma }{2n}$, and $$G_{\lambda }(\sigma ):=\eta _{1}\left( \frac{ 2\sigma }{\left\vert \lambda \right\vert }\right).$$ Now \begin{equation} \left\vert ( \widehat{F_{\alpha_1, \beta}}\ast \widehat{G_{\lambda }})(-2sgn(\lambda ) a_1)\right\vert \leq \left\Vert \widehat{F_{\alpha_1 ,\beta}}\ast \widehat{G_{\lambda }}\right\Vert _{\infty }\leq \left\Vert \widehat{F_{\alpha_1 ,\beta}}\right\Vert _{\infty }\left\Vert \widehat{ G_{\lambda }}\right\Vert _{1}=\left\Vert \widehat{F_{\alpha_1 ,\beta}}\right\Vert _{\infty }\left\Vert \widehat{\eta _{1}}\right\Vert _{1}. \label{fb} \end{equation} So it is enough to estimate $\left\Vert \widehat{F_{\alpha_1 ,\beta}}\right\Vert _{\infty }$. Since $$ \widehat{F_{\alpha_1 ,\beta}}(\xi )=\int\limits_{0}^{\infty }\sigma ^{\beta }L_{\alpha_1}^{0}\left( \sigma \right) e^{-\sigma \left( \frac{1}{2}+i\xi \right) }d\sigma, $$ from Lemma 6, with $n=1$, $k= \alpha_1$ and $\beta =\frac{ (z-1) \gamma }{2n}$ we obtain $$\widehat{F_{\alpha_1 ,\beta}}(\xi ) = \frac{\Gamma(\beta+1)}{\left(\frac{1}{2}+i\xi \right)^{\beta+1}} \sum_{j+l= \alpha_1} \frac{\Gamma(-\beta+j)\Gamma(\beta+1+l)}{\Gamma(-\beta)\Gamma(\beta+1)} \frac{w^{l}}{j!l!},$$ to take modulo in this expression and since $\left\vert w \right\vert = 1$ it follows that $$\left\vert \widehat{F_{\alpha_1 ,\beta}}(\xi ) \right\vert \leq \frac{\Gamma(-Re(\beta)) \Gamma(Re(\beta)+1)}{\left\vert \left(\frac{1}{2}+i\xi \right)^{\beta+1}\right\vert \left\vert \Gamma(-\beta) \right\vert} \sum_{j+l=\alpha_1} \frac{\Gamma(-Re(\beta)+j)) \Gamma(Re(\beta)+1+l)}{\Gamma(-Re(\beta))\Gamma(Re(\beta)+1)} \frac{1}{j!l!}.$$ From Lemma 7, with $n=1$ and $k = \alpha_1$ we have $$\sum_{j+l=\alpha_1} \frac{\Gamma(-Re(\beta)+j)) \Gamma(Re(\beta)+1+l)}{\Gamma(-Re(\beta))\Gamma(Re(\beta)+1)} \frac{1}{j!l!}=1.$$ So $$\left\vert \widehat{F_{\alpha_1 ,\beta}}(\xi ) \right\vert \leq \frac{\Gamma(-Re(\beta)) \Gamma(Re(\beta)+1)}{\left\vert \left(\frac{1}{2}+i\xi \right)^{\beta+1}\right\vert \left\vert \Gamma(-\beta) \right\vert} \leq \frac{\Gamma \left(\frac{(n+1) \gamma}{(2+\gamma)n} \right) \Gamma \left(\frac{2n-\gamma}{(2+\gamma)n}\right)}{(1/2)^{\frac{2n-\gamma}{(2+ \gamma)n}} e^{-\frac{Im(z) \pi \gamma}{4n}} \left\vert \Gamma\left(\frac{(1-z)\gamma}{2n} \right) \right\vert}.$$ Finally, for $Re(z)=-\frac{2n-\gamma}{2+\gamma}$, we obtain
$$| \nu_{z,N}(k,\lambda) |\leq c_{n, \gamma} |I_{1-z}(-\lambda) \phi_{N}(\lambda)| | \lambda |^{-\left( 1+\frac{(z-1)\gamma}{2n}\right)n} \prod_{j=1}^{n} \| \widehat{F_{\alpha_j, \beta}} \|_{\infty} \| \widehat{\eta_{j}} \|_{1}$$ $$ \leq \frac{c_{n, \gamma} \,\, e^{\frac {Im(z) \pi \gamma}{4}} \left\Vert H \right\Vert_{\infty} \left[ \Gamma \left(\frac{(n+1) \gamma}{(2+\gamma)n} \right) \right]^{n} \left[\Gamma \left(\frac{2n-\gamma}{(2+\gamma)n}\right) \right]^{n} \prod_{j=1}^{n} \left\Vert \widehat{\eta_{j}} \right\Vert_{1}} {\left\vert \Gamma \left(\frac{1-z}{2}\right) \right\vert \left\vert \Gamma\left(\frac{(1-z)\gamma}{2n} \right) \right\vert^{n}}.$$ By (\ref{L22}) it follows, for $Re(z)=-\frac{2n-\gamma}{2+\gamma}$, that \[ \left\Vert T_{N,z}f\right\Vert _{L^{2}(\mathbb{H}^{n})}\leq c_{n, \gamma} \,\, \frac{e^{\frac {Im(z) \pi \gamma}{4}}}{\left\vert \Gamma \left( \frac{1-z}{2}\right) \right\vert \left\vert \Gamma\left(\frac{(1-z)\gamma}{2n} \right) \right\vert^{n}} \left\Vert f\right\Vert _{L^{2}(\mathbb{H}^{n})} \] It is easy to see, with the aid of the Stirling formula (see \cite{stein4}, p. 326), that the family $\left\{ T_{N,z}\right\} $ satisfies, on the strip $ -\frac{2n-\gamma}{2+\gamma} \leq Re(z) \leq 1$, the hypothesis of the complex interpolation theorem (see \cite{stein2} p. 205) and so $T_{N,0}$ is bounded from $L^{p_{\gamma}}(\mathbb{H}^{n})$ into $L^{p_{\gamma}'}(\mathbb{H}^{n})$ uniformly on $N$, where $\left( \frac{1}{p_{\gamma}}, \frac{1}{p_{\gamma}'} \right) = C_{\gamma}$, then letting $N$ tend to infinity, we obtain that the operator $T_{\mu_{\gamma}}$ is bounded from $L^{p_{\gamma}}(\mathbb{H}^{n})$ into $L^{p_{\gamma}'}(\mathbb{H}^{n})$. $\square$
\qquad
We re-establish Theorem 1 and Theorem 2 in the following
\begin{theorem} Let $\mu_{\gamma}$ be a fractional Borel measure as in Theorem 1. Then the interior of $E_{\mu_{\gamma}}$ is the open trapezoidal region with vertices $(0,0)$, $(1,1)$, $D$ and $D'$. Moreover $C_{\gamma}$ and the closed segments joining $D'$ with $(0,0)$ and $D$ with $(1,1)$ except maybe $D$ and $D'$ are contained in $E_{\mu_{\gamma}}$. \end{theorem}
\qquad
\textit{Proof of Theorem 3.} We consider, for each $N \in \mathbb{N}$ fixed, the analytic family $\{ U_{N, z} \}$ of operators on the strip $-n \leq Re(z) \leq 1$, defined by $U_{N, z}f = f \ast \widetilde{\mu}_{(1-z)\gamma} \ast J_{N,z}$ where $\widetilde{\mu}_{(1-z)\gamma}$ i given by (\ref{mu3}), $J_{N, z}$ by (\ref{jz}) and $U_{N, 0}f(x,t) \rightarrow U_{\widetilde{\mu}_{\gamma}}f(x,t) := (f \ast \widetilde{\mu}_{\gamma})(x,t)$ a.e.$(x,t)$ as $N \rightarrow \infty$. Proceeding as in the proof of Theorem 2 it follows, for $Re(z) = 1$, that $\| U_{N, z} \|_{1, \infty} \leq c \left| \Gamma (z/2) \right|^{-1}$. Also it is clear that, for $Re(z) = -n$, the kernel $\widetilde{\mu}_{(1-z)\gamma} \ast J_{N,z} \in L^{1}(\mathbb{H}^{n})$ and it is a radial function. Now, our operator $\widehat{(\widetilde{\mu}_{(1-z)\gamma} \ast J_{N,z})}(\lambda)$ is diagonal, with diagonal entries $\nu_{N,z}(k, \lambda)$ given by \[
\nu_{z,N}(k,\lambda )=\frac{k!}{(k+n-1)!} \int\limits_{0}^{\infty} (\widetilde{\mu}_{(1-z)\gamma} \ast J_{N,z})(s, \widehat{-\lambda}) L_{k}^{n-1}(|\lambda|s^{2}/2) e^{-|\lambda|s^{2}/4} s^{2n-1} ds \] \[ =\frac{k!}{(k+n-1)!}I_{1-z}(-\lambda
)\phi _{N}(\lambda )\int\limits_{0}^{\infty }\eta(s^{2}) L_{k}^{n-1}(|\lambda|s^{2}/2) e^{-|\lambda|s^{2}/4} e^{i\lambda s^{2m}}s^{2n-1+(1-z)\gamma }ds. \]
Now we study this integral, the change of variable $\sigma = |\lambda| s^{2}/2$ gives \[
\int\limits_{0}^{\infty }\eta(s^{2}) L_{k}^{n-1}(|\lambda|s^{2}/2) e^{-|\lambda|s^{2}/4} e^{i\lambda s^{2m}}s^{2n-1+(1-z)\gamma }ds \] \[
=2^{n-1+ \frac{(1-z)\gamma}{2}} |\lambda|^{-\left(n + \frac{(1-z)\gamma}{2}\right)} \int\limits_{0}^{\infty }\eta\left( \frac{2\sigma}{ |\lambda|} \right) L^{n-1}_{k}(\sigma) e^{-\frac{\sigma}{2}} e^{i (2\sigma)^{m} |\lambda|^{1-m} sgn(\lambda)} \sigma^{n-1+ \frac{(1-z)\gamma}{2}} d\sigma. \] \[
=2^{n-1+ \frac{(1-z)\gamma}{2}} |\lambda|^{-\left(n + \frac{(1-z)\gamma}{2}\right)}\left( \widehat{F_{k,\beta}}\ast \left( \widehat{G_{\lambda }}\ast \widehat{R_{\lambda }}\right) \right) (0), \] where $$F_{k,\beta}(\sigma ):=\chi _{(0,\infty )}(\sigma )L_{k}^{n-1}\left( \sigma \right) e^{-\frac{\sigma }{2}}\sigma ^{\beta },$$ with $\beta =n-1+\frac{ (1-z)\gamma}{2}$,
$$G_{\lambda }(\sigma ):=\eta\left( 2\sigma / |\lambda|\right)$$ and $$R_{\lambda }(\sigma )=\chi _{(0,\left\vert \lambda \right\vert )}(\sigma )e^{i2^{m}sgn(\lambda )\left\vert \lambda \right\vert ^{1-m}\sigma ^{m}}.$$ Now \begin{equation} \left\Vert \widehat{F_{k,\beta}}\ast \left( \widehat{G_{\lambda }}\ast \widehat{ R_{\lambda }}\right) \right\Vert_{\infty } \leq \left\Vert \widehat{F_{k,\beta}} \right\Vert _{1}\left\Vert \widehat{G_{\lambda }}\right\Vert _{1}\left\Vert \widehat{R_{\lambda }}\right\Vert _{\infty }\label{fgr} \end{equation} So it is enough to estimate the right side of this inequality. From Lemma 6 and Lemma 7 we obtain $$\left\vert \widehat{F_{k,\beta}}(\xi ) \right\vert \leq \frac{\Gamma(n-1-Re(\beta)) \Gamma(Re(\beta)+1)}{\left\vert \left(\frac{1}{2}+i\xi \right)^{\beta+1}\right\vert \left\vert \Gamma(n-1-\beta) \right\vert} \frac{(n+k-1)!}{(n-1)!k!}.$$ Since $Re(z)=-n$, $\gamma = \frac{2(m-1)}{(n+1)m}$ and $\beta =n-1+\frac{ (1-z)\gamma}{2}$ we have $Re(\beta) = n- \frac{1}{m},$ thus it follows that \begin{equation} \left\Vert \widehat{F_{k,\beta_{z}}} \right\Vert_{1} \leq \frac{c \, e^{\frac{\vert Im(z) \vert \gamma \pi}{4}}} {\left\vert \Gamma \left(\frac{z-1}{2}\gamma \right) \right\vert} \, \frac{(n+k-1)!}{(n-1)!k!} \, \int\limits_{0}^{\infty} \frac{1}{ \left(\frac{1}{4}+\xi^{2} \right)^{\frac{1}{2} \left(n+1-\frac{1}{m} \right)}} d\xi, \label{estif} \end{equation} the last integral is finite for all $n \geq 1$ and all $m \geq 2$. It is clear that $\left\Vert \widehat{G_{\lambda }}\right\Vert_{1}=\left\Vert \widehat{\eta} \right\Vert_{1}$. Now, we estimate $\left\Vert \widehat{R_{\lambda }}\right\Vert _{\infty }$. Taking account of Proposition 2, p. 332, in \cite{stein3} we note that \begin{equation} \left\vert \widehat{R_{\lambda }}(\xi )\right\vert =\left\vert \int\limits_{0}^{\left\vert \lambda \right\vert }e^{i(2^{m}sgn(\lambda )\left\vert \lambda \right\vert ^{1-m}\sigma ^{m}-\xi \sigma )}d\sigma \right\vert \leq \frac{C_{m}}{\left\vert \lambda \right\vert ^{\frac{1-m}{m}}} \label{estir} \end{equation} where the constant $C_m$ does not depend on $\lambda$. Then, for $Re(z)=-n$, from (\ref{fgr}), (\ref{estif}) and (\ref{estir}) we obtain
$$|\nu_{N,z}(k, \lambda)| \leq c \, \frac{k!}{(k+n-1)!} \, |I_{1-z}(-\lambda) \phi_{N}(\lambda)| |\lambda|^{-\left(n + \frac{(1+n)\gamma}{2} \right)}\left\Vert \widehat{F_{k,\beta}}\ast \left( \widehat{G_{\lambda }}\ast \widehat{ R_{\lambda }}\right) \right\Vert_{\infty }$$ $$\leq c_{n,m} \Vert H \Vert_{\infty} \left\Vert \widehat{\eta} \right\Vert_{1} \frac{ e^{\frac{\vert Im(z) \vert \gamma \pi}{4}}} {\left\vert \Gamma \left(\frac{1-z}{2} \right) \right\vert \left\vert \Gamma \left(\frac{z-1}{2}\gamma \right) \right\vert} \int\limits_{0}^{\infty} \frac{1}{ \left(\frac{1}{4}+\xi^{2} \right)^{\frac{1}{2} \left(n+1-\frac{1}{m} \right)}} d\xi. $$ By (\ref{L22}) it follows, for $Re(z)=-n$, that \[
\| U_{N,z} f \|_{L^{2}(\mathbb{H})^{n}} \leq c_{n,m} \, \frac{ e^{\frac{\vert Im(z) \vert \gamma \pi}{4}}} {\left\vert \Gamma \left(\frac{1-z}{2} \right) \right\vert \left\vert \Gamma
\left(\frac{z-1}{2}\gamma \right) \right\vert} \|f\|_{L^{2}(\mathbb{H})^{n}}. \] It is clear that the family $\{ U_{N,z} \}$ satisfies, on the strip $-n \leq Re(z) \leq 1$, the hypothesis of the complex interpolation theorem. Thus $U_{N,0}$ is bounded from $L^{\frac{2n+2}{2n+1}}(\mathbb{H}^{n})$ into $L^{2n+2}(\mathbb{H}^{n})$ uniformly in $N$, and letting $N$ tend to infinity we conclude that the operator $U_{\widetilde{\mu}_{\gamma}}$ is bounded from $L^{\frac{2n+2}{2n+1}}(\mathbb{H}^{n})$ into $L^{2n+2}(\mathbb{H}^{n})$ for $n\in \mathbb{N}$. Finally, the theorem follows from the restrictions that appear in (\ref{restricciones}). $\square$
\end{document} |
\begin{document}
\title{A Comparative Code Study for Quantum Fault Tolerance}
\author{Andrew Cross \thanks{Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, 02139, USA and IBM Watson Research Center, P.O. Box 218, Yorktown Heights, NY, USA 10598. \texttt{andrew.w.cross@gmail.com}} \and David P. DiVincenzo \thanks{IBM Watson Research Center, P.O. Box 218, Yorktown Heights, NY, USA \texttt{divince@watson.ibm.com}} \and Barbara M. Terhal \thanks{IBM Watson Research Center, P.O. Box 218, Yorktown Heights, NY, USA 10598. \texttt{bterhal@gmail.com}}}
\maketitle \begin{abstract} We study a comprehensive list of quantum codes as candidates for codes used at the physical level in a fault-tolerant code architecture. Using the Aliferis-Gottesman-Preskill (AGP) ex-Rec method we calculate the pseudo-threshold for these codes against depolarizing noise at various levels of overhead. We estimate the logical noise rate as a function of overhead at a physical error rate of $p_0=1 \times 10^{-4}$. The Bacon-Shor codes and the Golay code are the best performers in our study.
\end{abstract}
\tableofcontents
\section{Introduction}
A great insight in the early history of quantum computing was that almost perfectly reliable quantum computation is possible with physical devices subjected to noise as long as the noise level is not too large. This observation has given us confidence that, ultimately, it will be possible to build a functioning quantum computer. However, the ``threshold theorem" that indicates how much noise can be tolerated has not otherwise given a very optimistic prognosis for progress. For example, the well-studied seven qubit Steane code [[7,1,3]] has a threshold against adversarial noise that is in the $O(10^{-5})$ range. This level of noise is far lower than anything that has been achieved in any laboratory -- it is actually a significantly lower error rate than many experiments are even capable of measuring.
In this paper we analyze both previously considered and new additional quantum codes and determine their thresholds, logical error rates and overheads using the ex-Rec method developed in \cite{AGP:ft}.
This allows us for the first time to compare the relative merit of many schemes. We will argue how the studied codes could serve as {\em bottom} (physical-level) codes in a fault-tolerant code architecture that minimizes the overall coding overhead.
In order to carry out this comparative code study we must make some simplifying assumptions. First of all, we assume that gates can be performed in a non-local geometry. It is likely that an ultimate quantum architecture will be largely restricted to local 1D, 2D or 3D geometries, hence the threshold numbers that we estimate will be affected by this architecture constraint. As was shown in \cite{STD:local,SR:localft}, the price for locality may be modest for small codes, but it will typically be worse for larger codes \cite{STD:local} since a bigger effort will have to be mounted to make the error-correction circuits local.
Secondly, the noise model for our study is a simple depolarizing noise model with equal probabilities for Pauli $X,Y$ and $Z$ errors. In particular, we assume that any single qubit location in the quantum circuit undergoes a Pauli $X$,$Y$ or $Z$ error with equal probability $p_0/3$. A two-qubit gate undergoes no error with probability $1-p_0$ and one of 15 errors with probability $p_0/15$. The reason for choosing this model is that it is the simplest unbiased choice for a noise model given that a comparative study of the performance of codes for general noise models (such as superoperator noise) is infeasible computationally. We do not expect the performance of these codes to differ greatly if we would choose a biased depolarizing noise model, since there are no intrinsic biases in the codes themselves. To deal with biased noise, it may be more advantageous to use specific biased code constructions such a non-square Bacon-Shor or surface code or the use of the repetition code against high-rate dephasing noise in \cite{AP:biased}.
We do not establish rigorous threshold lower-bounds by counting the number of malignant sets of faults as in \cite{AGP:ft}. Instead of counting or sampling of malignant faults, we simply simulate the depolarizing noise and keep track of when it leads to failures. We put error bars on our results such that a rigorous lower bound on the pseudo-threshold is within this statistical error bar. The level-1 pseudo-threshold is the value of $p_0$ where the level-1 encoded error rate $p_1=p_0$ \cite{SCCA:pseudo}. It is only the pseudo-threshold that is of interest in this study of bottom codes since we envision, see Section \ref{sec:coa}, that a different code would be used in the next level of encoding.
Our study is in some sense a continuation of the first comparative code study by Steane \cite{steane:overhead}.
Our analysis goes beyond these previous Monte Carlo studies of quantum fault-tolerance in that it includes more codes and focuses on a fault-tolerance analysis of the logical CNOT gate. One of the problems with comparing threshold estimates in the literature is that threshold numbers for different codes have been obtained by different methods, some more rigorous than other's. We believe that it would be advantageous to stick to one clear, rigorously motivated method. The AGP method has the advantage of being tied to a fully rigorous analysis \cite{AGP:ft} and the AGP method when used to (approximately) count the number of malignant sets of faults can give tight estimates of the (pseudo)-threshold.
With a few exceptions, we use standard Steane error-correction circuits (and sometimes omit possible code-specific optimizations) that allows us to compare codes directly. We discuss these possible code optimizations and our choices in Section \ref{sec:cc}. We do not separately analyze Knill's post-selected and Fibonacci schemes \cite{aliferis:fib, knill:nature} but we plot Knill's numbers in Figure \ref{fig:errorrate}. One scheme that is not included is the surface code scheme described in Ref. \cite{RHG:topo}.
However, we do study surface codes in the original setting of \cite{dennis+:top} using Shor error-correction circuits.
\subsection{The Code Architecture} \label{sec:coa}
The usefulness of error correction in computation is roughly measured by two parameters. First is the reduction in error rate that is obtained by using the code; this feature depends on the (pseudo)-threshold of the code for the particular noise model and the error correction circuits that are used. The second figure of merit is the smallness of the overhead that is incurred by coding. There is a tradeoff that one can expect between overhead and logical error-rate that mimics the trade-off between distance and rate of error-correction codes, see Figure \ref{fig:errorrate}. Since error levels of physical implementations are expected to be high, optimistically in the range from $O(10^{-2})$ to $O(10^{-6})$, it is clear that at the bottom level, optimizing the threshold has priority over optimizing overhead. This leads us to consider the following simple code architecture. At the physical level, we use a bottom code $C_{\rm bot}$ which is chosen to have a high noise threshold and a reasonable overhead. This paper will be devoted to a comparative study of such codes. We will pick some illustrative numbers to argue how one can envision completing the code architecture. We will see that one can find a bottom code that maps a base error rate of $p_0=O(10^{-4})$ onto a logical error rate of $p_1=O(10^{-7})$ (see Section \ref{sec:results}). To run a reasonable-sized factoring algorithm one may need an logical error rate of, say, $O(10^{-15})$ \footnote{An $n$-bit number can be factored using a circuit with space-time complexity of roughly $360n^4$ \cite{beckman:factor}, so RSA-1024 could be broken using a circuit with $O(10^{15})$ potential fault locations. Using different architectures, it may be possible to reduce this to $O(10^{11})$ see for example \cite{vanmeter:architecture}.}. Thus one needs a top code $C_{\rm top}$ that brings the error rate $O(10^{-7})$ to $O(10^{-15})$. The desirable features of the top code are roughly as follows (see also the Discussion at the end of the paper). The top code is a block code $[[n,k,d]]$ with good rate $k/n$ in order to minimize the overhead. The improvement in error-rate for a code which can correct $t$ errors is roughly \begin{equation} p_1 \approx p_0 \left(\frac{p_0}{p_{\rm th}}\right)^{t}, \label{eq:p1estim} \end{equation} where $p_0$ is the unencoded error-rate and $p_{\rm th}$ is the threshold error rate. Thus in order to get from $p_0=O(10^{-7})$ to $p_1=O(10^{-15})$ we could use a code which can correct $4$ errors and has a threshold of $O(10^{-5})$.
In \cite{steane:nature} Steane studied several block-codes which may meet these demands. The polynomial codes would also be an interesting family to study in this respect.
\section{Preliminaries} \label{sec:cc}
For our study it is necessary to select a subset of quantum codes. We focus on codes that are likely to have a good threshold, possibly at the cost of a sizeable but not gigantic overhead. To first approximation the threshold is determined by the equation \begin{equation} p_{\rm th}= N p_{\rm th}^{t+1} \Rightarrow p_{\rm th}=N^{-1/t}, \label{eq:thresh} \end{equation} where $t$ is the number of errors that the code can correct and $N$ is a combinatorial factor counting the sets of $t+1$ locations in an encoded gate that lead to the encoded gate failing. It is clearly desirable to minimize the number of locations and maximize $t$. This consideration has led us to primarily consider Calderbank-Shor-Steane (CSS) codes. Any stabilizer quantum code is CSS if and only if the CNOT gate is a transversal gate \cite{thesis:gottesman}. The advantage of a transversal CNOT is that it minimizes the size of the encoded CNOT; the bulk of the CNOT rectangle will be taken up by error correction. This is favorable for the noise threshold of $C_{\rm bot}$. Secondly, minimizing the error-rate of the encoded gate $C_{\rm bot}({\rm CNOT})$ will be useful at the next level of encoding, because CNOTs occur frequently in EC and their error rates play a large role in determining whether error rates are below the threshold (of $C_{\rm top}$).
However to demonstrate that this restriction to CSS codes is warranted we also consider the non-CSS 5-qubit code $[[5,1,3]]$ which is the smallest code that can correct a single error. We indeed find that this code performs {\em worse} than Steane's 7-qubit code $[[7,1,3]]$, see Section \ref{sec:results} and the Data Tables in Appendix \ref{app:data}.
\subsection{Approximate Threshold Scaling} \label{sec:scaling}
In this section we discuss the global behavior of the noise threshold as a function of block size $n$, distance, and other code properties. Let us consider Eq. (\ref{eq:thresh}) and see how we can get the best possible threshold. An upper-bound on $N$ is ${A \choose t+1}$ where $A$ is the total number of locations in the encoded gate (rectangle). Ideally, a code family has a distance that is linear in $n$, i.e. $t$ is linear in $n$. Let us assume for simplicity that only some fraction of all locations appears in the malignant fault sets of size $t+1$, i.e. we model $N \approx {A_{\rm mal} \choose t+1}$ where $A_{\rm mal} < A$. The locations in $A_{\rm mal}$ are in some sense the weak spots in the circuits; overall failure is most sensitive to failure at these locations. $A_{\rm mal}$ may be either linear or super-linear in the block size $n$. In case $A_{\rm mal}$ scales linearly with $n$, {\em and} $t=\delta n$ for some $\delta \leq 1/4$, the threshold in Eq. (\ref{eq:thresh}) {\em increases} as a function of $n$ and asymptotes in the limit of large $n$ to a finite value. Indeed, for $A_{\rm mal}=\alpha n$ and $\delta \ll \alpha$ (which is typically the case since $t \leq n/4$) we get, using Sterling's approximation, \begin{equation} p_{\rm th}=\lim_{n\rightarrow\infty}{\alpha n \choose \delta n+1}^{-1/(\delta n)}=\frac{\delta}{e \alpha}+ O\left(\frac{\delta^2}{\alpha^2}\right) \end{equation} Such monotonic increase of the threshold with block-size is clearly desirable. It is also clear that when $t$ is constant, for any polynomial $A_{\rm mal}={\rm poly}(n)$, the threshold $p_{\rm th}$ in Eq. (\ref{eq:thresh}) decreases as a function of $n$. When $A_{\rm mal}$ scales super-linearly with $n$ and $t$ is linear in $n$ we get the following behavior. First, the threshold increases with $n$ (the effect of larger $t$), then the threshold declines since the effect of a super-linear $A_{\rm mal}$ starts to dominate. For codes and EC circuits with this behavior, it is thus of interest to determine where this peak threshold performance occurs. We will see some evidence of these peaks in Figure \ref{fig:block} in Section \ref{sec:results}.
Now let us consider the scaling of $A$ (and $A_{\rm mal}$) in case we use Steane error correction. In Appendix \ref{app:css} we review how we can bound $A$ for a CSS code with Steane error correction, but a rough estimate is that \begin{equation} A=c_1 A_{\rm enc}+ c_2 A_{\rm ver}+c_3 n. \end{equation} Here $A_{\rm enc}$ is the number of locations in the encoding of the ancillas for error correction, and $A_{\rm ver}$ is the number of locations in the verification of the ancillas for error correction. The additional term linear in $n$ comes from the transversal encoded gates and the transversal syndrome extractions. For a CSS code and the standard encoding construction (see Appendix \ref{app:css}), $A_{\rm enc}$ typically scales as $O(w n)$ where $w$ is the maximum Hamming weight of the rows of the generator matrix of either $C_1$ or $C_2^{\perp}$ {\em in standard form}. However this standard construction may be sub-optimal, since by bringing the generator matrix in standard form one can increase the maximum weight of its rows.
For Steane-EC the full verification of the ancilla block requires other ancillas blocks; a fully fault-tolerant verification would give a pessimistic scaling of $A_{\rm ver}=O(w n t)$. However it is not necessarily desirable to have strict fault tolerance as long as the total probability of low-weight faults that produce faults with weight $t+1$ or more is low, see the discussion in Section \ref{sec:methods}. On the other hand for increasing $n$ the number of verification rounds should at least be increasing with $n$, perhaps $O(\log n)$ would be sufficient. If we assume that $A_{\rm mal}$ scales similarly as $A$, it follows that if we look for linear-scaling $A_{\rm mal}$ we need to look at code families which have simple encoders, scaling linearly with $n$. This seems only possible for stabilizer codes with constant weight stabilizers, such as quantum LDPC codes \cite{MMM:ldpc} and surface codes or for the Bacon-Shor codes (which has encoders that use $O(n)$ 2-qubit gates).
For the Bacon-Shor and surface codes the distance $t$ does not scale linearly with $n$ (but as $\sqrt{n}$). Nonetheless, the work in \cite{dennis+:top} shows that the effective distance for the surface codes does scale linearly with the block size, since there are very few incorrectible errors of weight $O(t)$. For the Bacon-Shor code family, where one has less syndrome information, this behavior has not been observed \cite{AC:bs} (see also Figure \ref{fig:perfectlonger}).
For code families with constant-weight stabilizers an interesting alternative to Steane-EC \cite{steane:active} is the use of Shor-EC \cite{NC:book} where the syndrome corresponding to each stabilizer is extracted using a cat state or simple unencoded qubit ancillas. As for ancilla verification in Steane-EC, the syndrome extraction needs to be repeated to make the circuits more fault-tolerant. In Section~\ref{sec:results} we will see the effect on the threshold of using Steane-EC versus Shor-EC for the surface codes \cite{BK:surface,FM:surface}, see Figure~\ref{fig:surfacethreshold} of Section~\ref{sec:results}. It is striking that the surface codes with Shor EC are the only known examples of a code family with a finite $n \rightarrow \infty$ threshold. This is despite the $O(n\sqrt{n})$ scaling of the total number of locations $A$ of the Shor error correction circuit.
\subsection{Choice of Codes}
The codes that we have studied are listed in Table~\ref{tab:codes}. All codes in this table are CSS codes with the exception of the [[5,1,3]] code. Some of these codes have been previously analyzed by Steane in Ref. \cite{steane:overhead}. There exist various families of binary CSS quantum codes; the families are the quantum Reed-Muller codes, the quantum Hamming codes, the quantum BCH codes, the surface codes and the sub-system Bacon-Shor codes. In our study we consider only a single member of the quantum Reed-Muller family, a $[[15,1,3]]$ code, since these codes typically don't have very good distance versus block-size \cite{steane:rm}. The $[[15,1,3]]$ code was first constructed in \cite{KLZ:faulttol} from a punctured Reed-Muller code ${\rm RM}(1,4)$ and its even sub-code. It is the smallest known distance-$3$ code with a transversal T gate.
We study various quantum Hamming and quantum BCH codes (see a complete list of quantum BCH codes of small block-size in \cite{grassl:bchcodes}) which are constructed from self-orthogonal classical Hamming and BCH codes respectively. We have chosen those codes that encode a single qubit and have maximum distance for a given block size. We have included the previously studied Bacon-Shor codes and the surface codes in our study. We have also included the concatenated 7-qubit code $[[49,1,9]]$ which we use in the way that was proposed by Reichardt in \cite{reichardt:concat7}, see the details in Section \ref{sec:FTECspecial}.
Another family of codes that has been proposed for fault tolerance \cite{AB:ftsiam} are the quantum Reed-Solomon codes or polynomial codes. These are codes that are naturally defined on qudits. In this study we consider them as candidates for bottom codes. An alternative use is to consider them as top codes where one uses a bottom code to map the qubits onto qudits. In our study we assume that quantum information is presented in the form of qubits and hence we will consider these codes as binary stabilizer codes.
We specifically chose to include the $[[21,3,5]]$ (a concatenated $[[7,1,4]]_8$) and the $[[60,4,10]]$ (a concatenated $[[15,1,8]]_{16}$) code from the family of dual-containing polynomial codes over $GF(2^m)$, because they are the smallest error-correcting polynomial codes in this family.
We find it impractical to simulate the encoded CNOT gate for BCH codes in this table which have block-size larger than $[[47,1,11]]$, see \ref{sec:software}. The threshold for these bigger codes will benefit considerably from the fact that $t/n$ is quite high. Some semi-analytical values for the thresholds of these codes have been given in \cite{steane:overhead}. Even with good thresholds, these bigger BCH codes have limited applicability due to their large overhead. The bottom code should be picked to obtain a logical error rate that is well below the threshold of some good block code but only at the price of a moderate overhead.
\begin{table} \begin{center}
\begin{tabular}{l|l|l} \textsc{Parameters} & \textsc{Notes} \\ \hline $[[5,1,3]]$ & non-CSS five qubit code \cite{laflamme+:5qubit}\\ $[[7,1,3]]$ & Steane's 7-qubit code (doubly-even dual-containing) \cite{steane:7qubit} \\ $[[9,1,3]],[[25,1,5]],[[49,1,7]],[[81,1,9]]$ & Bacon-Shor codes \cite{AC:bs} \\ $[[15,1,3]]$ & Quantum Reed-Muller code \cite{steane:rm,KLZ:faulttol} \\ $[[13,1,3]],[[41,1,5]],[[85,1,7]]$ & Surface codes \cite{BK:surface,FM:surface}\\ $[[21,3,5]]$ & Dual-containing polynomial code on $GF(2^3)$ \cite{grassl+:rscode} \\ $[[23,1,7]]$ & Doubly-even dual-containing Golay code (cyclic) \cite{thesis:reichardt} \\ $[[47,1,11]]$ & Doubly-even dual-containing quadratic-residue code (cyclic) \cite{grassl:bchcodes} \\ $[[49,1,9]]$ & Concatenated $[[7,1,3]]$ Hamming code \cite{reichardt:concat7} \\ $[[60,4,10]]$ & Dual-containing polynomial code on $GF(2^4)$ \cite{grassl+:rscode} \\ $[[79,1,15]],[[89,1,17]],[[103,1,19]],[[127,1,19]]$ & BCH codes, not analyzed \cite{grassl:bchcodes} \\ \end{tabular} \end{center} \caption{A list of the codes included in our study.} \label{tab:codes} \end{table}
\subsection{Universality} \label{sec:univ}
Universality for CSS codes can in principle be obtained using the technique of {\em injection-and-distillation} \cite{knill:nature,reichardt:distill, BK:magicdistill}. Let us briefly review how one may perform fault-tolerant computation for CSS codes for which, of the Clifford group gates, only the CNOT and Pauli operations are transversal. If one is able to perform any Clifford group gate transversally, including H and S, it is well known how to obtain a universal set of gates \cite{thesis:aliferis}. Note that a CSS code with only its transversal CNOT gives us the ability to fault-tolerantly prepare the states
$\{|\overline{+}\rangle,|\overline{0}\rangle\}$ and perform transversal $\overline{X}$ and $\overline{Z}$ measurements. However we do not necessarily have a fault-tolerant realization of the Hadamard gate H.
In this case the problem of constructing fault-tolerant single qubit Clifford gates can be reduced to the problem of preparing the encoded $|\overline{\textrm{+}i}\rangle\propto
|\overline{0}\rangle+i|\overline{1}\rangle$ ancilla \cite{preskill_aliferis:comm}. In particular, the gates
${\rm S}\propto\exp(-i\pi Z/4)$ and ${\rm Q}\propto\exp(+i\pi X/4)$ generate the single-qubit Clifford group and can be implemented given a $|\textrm{+}i\rangle$ ancilla, see Figure~\ref{fig:Sgate} and Figure~\ref{fig:Qgate}.
\begin{figure}
\caption{The S gate using a $|\textrm{+}i\rangle$ ancilla.}
\label{fig:Sgate}
\end{figure}
\begin{figure}
\caption{The Q gate using a $|\textrm{+}i\rangle$ ancilla.}
\label{fig:Qgate}
\end{figure}
An encoded $|\overline{\textrm{+}i}\rangle$ ancilla can be produced using the method of injection-and-distillation
\cite{BK:magicdistill,reichardt:distill}. The distillation procedure for distilling an unencoded $|\textrm{+}i\rangle$ from seven unencoded $|\textrm{-}i\rangle$ is shown in Figure~\ref{fig:Idistill}. In order to perform this distillation procedure in encoded form, one can generate an encoded noisy ancilla
$|\overline{\textrm{-}i}\rangle$ using Knill's idea of injecting a state in the code. The distillation circuit is then performed in encoded form.
\begin{figure}\label{fig:Idistill}
\end{figure}
\section{Error Correction Circuits} \label{sec:circ}
{\em Locations} in a quantum circuit are defined to be gates, single-qubit state preparations, measurement steps, or memory (wait) locations. After one level of encoding, every location (denoted as 0-Ga) is mapped onto a a rectangle or 1-rectangle (1-Rec), a space-time region in the encoded circuit, which consists of the encoded gate (1-Ga) followed by error correction (1-EC), as shown in Fig.~\ref{fig:1Rec}. For transversal gates, the 1-Ga consists of performing the 0-Ga's on each qubit in the block(s).
\begin{figure}
\caption{ A 1-rectangle (1-Rec), indicated by a dashed box, which replaces a single-qubit 0-Ga location. The 1-Rec consists of the encoded fault-tolerant implementation of the 0-Ga (1-Ga) followed by an error correction procedure (1-EC).}
\label{fig:1Rec}
\end{figure}
For the fault-tolerance analysis one also defines an extended 1-Rec or ex-Rec which consists of a rectangle along with its {\em preceding} 1-EC(s) on the input block(s). Let us now discuss the circuits for error correction.
\begin{figure}
\caption{Steane's error correction method for CSS codes involves coupling two encoded and verified ancilla's to a block of data qubits. The ancilla qubits are then measured in the $Z$- or $X$-basis and the syndrome $s$ is determined. From the syndrome $s$ the corresponding $Z$ or $X$ error is determined and the data qubits are corrected.}
\label{fig:SteFTEC}
\end{figure}
Steane error correction for CSS codes (Steane-EC) is schematically shown in Figure~\ref{fig:SteFTEC}. The $|\overline{0}\rangle$ and
$|\overline{+}\rangle$ ancilla blocks in Fig. \ref{fig:SteFTEC} can be prepared in the following way. First $n$ qubits are encoded using circuits derived from the generator matrix of a classical coset code of $C_2^\perp$, see Appendix \ref{app:css} for details and definitions. The memory locations in the encoder are determined using Steane's Latin rectangle method \cite{steane:fast}, discussed in more detail in Appendix \ref{app:latin}. Then the ancillas pass through a verification circuit. This error detection circuit measures the $X$ and $Z$ stabilizer generators of the encoded state some number $R$ of times. For a $\ket{\overline{0}}$ ancilla each round is given by the circuit in Figure~\ref{fig:singlestage}. For dual-containing codes, the Hadamard-conjugate of the circuit is used for a $\ket{\overline{+}}$ ancilla. If we detect any errors in any of the $R$ rounds, the encoded state is rejected. Otherwise the state is accepted and used for syndrome extraction. We will consider $L$ preparation attempts per ancilla and in our studies we will vary the parameters $R$ and $L$, giving rise to different overheads.
\begin{figure}\label{fig:singlestage}
\end{figure}
\subsection{Specific Code Considerations} \label{sec:FTECspecial}
Specific properties of a quantum code can often be used to simplify the error-correcting circuits. This section discusses each family of codes and the optimizations we have implemented or the code properties that have been used to modify the EC and 1-Ga circuits.
In general, we have opted to focus on the error-correcting properties of the codes rather than the possible simplifications to the Steane-EC network. One of the reasons for this approach is that it is not clear whether verification circuits that perform the minimal number of checks are superior to verification circuits that perform many thorough tests. Furthermore, changes to the network are difficult to parameterize and systematically study because there are many possible choices and few are clearly the best. In addition, we believe that the overall trends observed in this paper are not altered by omitting these optimizations.
Reichardt has suggested a generic optimization that uses different encoders for each logical ancilla in the verification circuit \cite{thesis:reichardt}. This optimization can reduce the number of necessary rounds of verification and possibly decrease the probability of correlated errors at the output of the verification circuit, conditioned on acceptance. We do not use this optimization for any of the codes in this study.
The Steane and Golay codes are constructed from perfect classical codes. Perfect codes have the property that every syndrome locates a unique error of weight $w\leq t$. As Ref. \cite{AGP:ft} observed, some parts of the error detection circuit can be removed for a CSS code constructed from perfect classical codes and the construction remains strictly fault-tolerant. Again we do not use this optimization.
For the Bacon-Shor codes we don't use Steane's Latin rectangle encoding method, but rather the simpler method described in \cite{AC:bs}. We do use the standard verification method for the bigger BS codes and not the simpler verification method in \cite{AC:bs}.
For the surface codes we consider both Steane-EC and Shor-EC to understand their effects on the threshold. We use Shor-EC using bare ancillas as in \cite{dennis+:top}. This is fault-tolerant for surface codes on a $5\times 5$ lattice or larger as long as the syndrome measurements are repeated enough times. The number of repeated measurements could in principle be varied, but we choose to repeat the measurements $\ell$ times for a $\ell\times\ell$ surface code, following \cite{dennis+:top}.
The $[[49,1,9]]$ concatenated Steane code is one of the CSS codes in our study whose network deviates from the construction described in the previous section. The preparations of $\ket{\overline{0}}$ and $\ket{\overline{+}}$ do not include a verification circuit. Instead each 7-qubit block has an error detection step after each [[7,1,3]]-encoded logical gate \cite{thesis:reichardt}. A $49$-qubit ancilla is rejected if any of these error-detections detects an error. This implies that any single fault will be detected, so the circuit is fault-tolerant. In fact, any pair of faults is also detected, so that a third order event is necessary to defeat the error-detections. This way of using $[[49,1,9]]$ is the one which Reichardt proposed. Unlike in his simulations we restrict ourselves to a finite number of ancilla preparation attempts $L$, since we care about the total overhead.
The polynomial codes that we consider are non-binary codes over $2^m$-dimensional qudits. We can choose the parameters of these codes so that when we consider each qudit as a block of $m$ qubits, the Fourier transform and controlled-SUM gates are implemented by bitwise application of Hadamard and CNOT, respectively. In this setting, the code is simply a binary CSS code encoding $m$ qubits which is constructed from a non-binary dual-containing classical code by concatenating using a self-dual basis of $GF(2^m)$. The advantage of such a construction is that we can decode the syndromes as if they were vectors over $GF(2^m)$, allowing us to correct more higher-order errors than we could otherwise correct as a binary code. To use this advantage, we do not need to change the way we construct the rectangles at all, only the way we interpret the classical measurement outcomes.
$[[5,1,3]]$ is the smallest distance-$3$ quantum error-correcting code and it is a perfect quantum code. Gottesman has shown how to compute fault-tolerantly with this code \cite{thesis:gottesman}, and there have been some numerical studies of logical error rates using Shor-EC \cite{fowler:thesis}. To our knowledge the threshold for this code has never been determined. Unfortunately, there are no two-qubit transversal gates for $[[5,1,3]]$, so it is necessary to construct a two-qubit gate from a three-qubit gate such as the ${\rm T_3}$ gate. The gate ${\rm T_3}$ is defined by the following action on Pauli operators: $XII \rightarrow XYZ, IXI \rightarrow YXZ, IIX \rightarrow XXX, ZII \rightarrow ZXY, IZI \rightarrow XZY, IIZ \rightarrow ZZZ$. This gate is a Clifford gate that can be combined with stabilizer-state preparations and transversal Pauli measurements to yield any gate in the Clifford group \cite{thesis:gottesman}. Specifically, CNOT, S, and Cyc gates (and their inverses) can be constructed from the ${\rm T_3}$ gate in this way. Here Cyc$={\rm SHSH}$ acts as $X\rightarrow Y\rightarrow Z\rightarrow X$. The fault-tolerant implementation of ${\rm T_3}$ is shown in Figure~\ref{fig:T3}.
\begin{figure}
\caption{The encoded implementation of ${\rm T_3}$ (with an additional permutation of the blocks $q_1, q_2, q_3$) using the gates CNOT, S, Cyc and Y (and inverses). Each gate in the circuit is applied transversally. The circuit is only a logical operation after completing all of the gates, i.e. CNOT and S are not valid transversal gates for $[[5,1,3]]$.}
\label{fig:T3}
\end{figure}
The $[[5,1,3]]$ construction also differs from other CSS constructions because we use Knill (or teleported) error correction (Knill-EC) \cite{knill:nature}. In our study we will simulate the extended ${\rm T_3}$-rectangle assuming that the logical Bell pairs of Knill-EC are perfect. We do this since it is simpler and shows that even using perfect logical Bell pairs the threshold is not very good, see Section \ref{sec:results}. For [[5,1,3]] the $R$ and $L$ parameters are replaced by $NC$ and $NB$, denoting the number of cat state preparation attempts per Pauli measurement and the number of logical Bell state preparation attempts per error correction, respectively. A circuit to prepare and verify encoded Bell pairs for Knill error correction for [[5,1,3]] is shown in Figure~\ref{fig:bell513} in Appendix \ref{app:decode}.
The construction for the $[[15,1,3]]$ Reed-Muller code is entirely standard. Since this code is not constructed from a dual-containing classical code, the $\ket{\overline{0}}$ and $\ket{\overline{+}}$ encoders are not simply related by a transversal Hadamard gate. For the same reason, the code can correct more $X$ errors than $Z$ errors. The most interesting feature of this code is that T is a transversal gate \cite{KLZ:faulttol,ZCC:trans}, but this does not enter directly into our analysis of the threshold.
\section{The Aliferis-Gottesman-Preskill (AGP) Method} \label{sec:methods}
According to \cite{AGP:ft} a rectangle is {\em correct} if the rectangle followed by an ideal decoder is equivalent to the ideal decoder followed by the ideal gate (0-Ga) that the rectangle simulates:
\begin{center} \begin{picture}(292,24)
\put(0,12){\line(1,0){10}} \put(10,0){\framebox(48,24){\shortstack{correct\\$1$-Rec}}}
\put(58,12){\line(1,0){10}} \put(68,0){\framebox(48,24){\shortstack{ideal\\$1$-decoder}}} \put(116,12){\line(1,0){10}}
\put(126,6){\makebox(20,12){=}}
\put(146,12){\line(1,0){10}} \put(156,0){\framebox(48,24){\shortstack{ideal\\$1$-decoder}}}
\put(204,12){\line(1,0){10}} \put(214,0){\framebox(48,24){\shortstack{ideal\\$0$-Ga}}} \put(262,12){\line(1,0){10}}
\put(272,6){\makebox(20,12){.}} \end{picture} \end{center}
As said before, an {\em extended rectangle} (ex-Rec) consists of a 1-Ga along with its leading and trailing error-corrections. The extended rectangles make an overlapping covering of the circuit. A set of locations inside an ex-Rec is called {\em benign} if the 1-Rec is correct for any set of faults occurring on these locations. If a set of locations is not benign, it is {\em malignant}. The design principles of {\em strict} fault-tolerance are described in pictures in Sec. 10 of \cite{AGP:ft}. If these properties hold for the 1-Ga and 1-EC, these gadgets
for a $[[n,1,d]]$ code with $t=\lfloor(d-1)/2\rfloor$ are called $t$-strictly fault-tolerant. The important consequence of these conditions is that for a $[[n,k,d]]$ code with $t$-strictly fault-tolerant constructions one can show that any set of $t$ or fewer locations in the ex-Rec is benign. A construction is called weakly fault-tolerant when, for a code that can correct $t$ errors, sets of $s < t$ locations can be malignant. Weak fault-tolerance is a useful concept in optimizing thresholds since weakly fault-tolerant circuits can be more compact than strictly fault-tolerant circuits, hence allowing for fewer fault locations and a potentially better threshold. On the negative side, weak fault-tolerance allows some low-weight faults to be malignant but if the number of such faults is small then the threshold is not much affected.
All our fault-tolerant schemes are $1$-strictly fault-tolerant implying that single faults can never be malignant. More precisely, any single fault in a 1-EC or a 1-Ga never propagates to become a weight-2 error in a block. In Steane-EC when we prepare ancillas with at least two attempts ($L\geq 2$) and one error detection stage ($R=1$), we eliminate malignant faults of weight $1$. For $R=1$ the EC is not 2-strictly fault-tolerant since there may be a pair of faults, one in each of the first two encoders, generating a high weight (possibly higher than $t$) error that passes the error detection circuit undetected. Since the number of these events is quite rare, they will not contribute much to the failure probability.
It is possible to show that $R\geq t$ and $L\geq t+1$ is necessary for $t$-strict fault-tolerance for a code that can correct $t$ errors by continuing the same argument\footnote{However, the standard verification stage would need additional error detections on the bottom ancilla pair for $R=t$ and $L=t+1$ to be both necessary and sufficient for $t$-strict fault-tolerance.}. For a specific $t$-error-correcting code the actual values required for $R$ and $L$ depend on how each encoding circuit propagates errors.
Let us review why the extended rectangle is the central object in a fault-tolerance analysis. An encoded circuit where the physical gates (0-Ga) have been replaced by rectangles can also be viewed as an encoded circuit with 0-Ga's with a different error model. This can be achieved simply by inserting perfect decoder-encoder pairs between the rectangles, see \cite{AGP:ft}. In an ex-Rec with malignant faults, the rectangle will correspond to a faulty 0-Ga, whereas for benign faults the rectangle will correspond to a perfect 0-Ga. The reason that one has to take into account an ex-Rec and not merely a Rec is that faults in the leading 1-EC can combine with faults in the 1-Rec to produce malignant faults. For example, a single error in a leading 1-EC does not cause a failure of the rectangle in which it is contained; this error however can combine with later errors to give rise to a logical error. At the same time the presence of a logical error in the leading 1-EC which maps one codeword onto another will not affect the failure rate of the rectangle that comes after it since the state which enters this rectangle is a codeword. Hence in the extended rectangle method there is no double-counting of errors. Instead, it is an efficient method to handle the effect of incoming errors and is likely to give a very tight estimate of the threshold if no other assumptions or simplifications are present.
In principle one may think that one would also need to be careful about the effect of incoming errors {\em into} the ex-Rec; perhaps an incoming error could combine with a seemingly benign fault in the ex-Rec and give rise to an incorrect rectangle. Thus perhaps one has to consider the malignancy of sets of faults given a possible {\em worst case} input to the extended rectangle.
However, one can argue for stabilizer codes and for {\em deterministic} (to be defined below) error-correction that malignancy does not depend on incoming errors to the ex-Rec. To show this, let us first review the formalism of stabilizer codes. A stabilizer code is the $+1$ eigenspace of an Abelian subgroup of the Pauli group $P_n$ which contains all $n$-qubit tensor-products of the Pauli operators $\{X,Y,Z,I\}$. The normalizer $N(S) \subseteq P_n $ of $S$ is defined as $N(S)=\{E| \forall s \in S, E s E^{\dagger} \in S \}$. For Pauli operators (which either commute or anti-commute with each other) $N(S)$ is the simply the group of Pauli operators that commute with any element in $S$. Any element of $N(S)\backslash S$ is a logical operator mapping codewords onto each other. All other Pauli operators $P \notin N(S)$ anti-commute with at least one element in $S$ and map a code word outside the code space indicated by a non-zero syndrome. Thus the Pauli group $P_n$ can be partitioned into cosets of $N(S)$ and each of these cosets is labeled by a different syndrome. The lowest-weight member of each coset is called the coset leader. Standard syndrome decoding finds, for each given syndrome, a coset leader with lowest weight and chooses this as the error correction. Thus the low-weight (non-degenerate) correctable errors correspond to distinct syndromes whose coset leader corrects the error. For high-weight errors $E_i$, all we can say is that $E_i E_{\rm correct} \in N(S)$ since $E_i$ and $E_{\rm correct}$ have the same syndrome.
Now let us consider the issue of incoming errors to an ex-Rec and assume the following properties of stabilizer error correction. First, we assume that the part of the 1-EC circuit which couples any ancillas to the incoming data is {\em deterministic}, i.e. does not depend on any incoming error on the data. The choice of which ancillas to couple may depend on some error detection or ancilla verification. This property holds for many but not all error-correction circuits; it does not hold, for example, when the number of repetitions of syndrome extraction depends on the value of these syndromes. This property {\em does} hold for the circuits used in this paper. Furthermore, given a stabilizer $S$ and the incoming error $E_{\rm in}$ on an encoded state, let the 1-EC be such that the syndrome of the 1-EC uniquely determines in which coset of $N(S)$ in the Pauli group the error $E_{\rm in}$ lies. In this sense the 1-EC must be complete error correction for the code that is used. For example, if for a CSS code the 1-EC only does $Z$ error correction whereas $X$ errors can map the state outside the code space, the syndrome information effectively partitions the Pauli group into cosets of $N(S_X) P_n(X)$. Here $P_n(X)$ is the subgroup of Pauli operators that only contain $X$ and $I$ and $N(S_X)$ is the normalizer of the stabilizer subgroup $S_X$ with only $X$ and $I$ Pauli operators. In this case the syndrome does not uniquely assign the incoming error to a coset of $N(S)$. Thirdly, upon any incoming error $E_{\rm in}$ a perfect 1-EC determines a syndrome that corrects $E_{\rm in}$ modulo a logical error (given by an element in $N(S)$). This is a basic property of stabilizer error correction as described above.
Let then the incoming state to an ex-Rec be a state in the code-space of the stabilizer with an additional error $E_{\rm in}$. We want to show that the state that comes out of the leading 1-EC is again some state in the code space with an additional error $E_{\rm out}$ that only depends on the errors inside the 1-EC, $E_{\rm ec}$, i.e $E_{\rm out}=f(E_{\rm ec})$ where $f$ is independent of $E_{\rm in}$. Any 1-EC circuit for stabilizer codes can be implemented with Clifford gates. Given an incoming error $E_{\rm in}$ and error inside the 1-EC $E_{\rm ec}$, it follows (because a 1-EC for any stabilizer code can be implemented with Clifford gates) that the 1-EC has syndrome $s(E_{\rm in} h_1(E_{\rm ec}))$ where $h_1$ is a function independent of $E_{\rm in}$. Based on the syndrome the correction step will be some $E_{\rm correct}= E_{\rm in} h_1(E_{\rm ec}) \mod N(S)$. Before error correction the data has error $h_2(E_{\rm ec}) E_{\rm in}$ where $h_2(E_{\rm ec})$ is the part of $E_{\rm ec}$ that has propagated to the data. After error correction the data thus has error $h_2(E_{\rm ec}) h_1(E_{\rm ec}) \mod N(S)$. We strip off the logical error in $N(S)$ and identify $E_{\rm out}=h_2(E_{\rm ec}) h_1(E_{\rm ec})$. Note that when the EC is not deterministic, the functions $h_1$ and $h_2$ can depend on $E_{\rm in}$. \qed
We discuss the explicit decoding of the error syndromes for each code in Appendix \ref{app:decode}.
\subsection{Monte-Carlo Implementation of Method} \label{sec:mcmethod}
Given the AGP method the numerical problem to be solved is whether a Rec is correct given a set of faults in the ex-Rec containing it. This set of faults is generated using depolarizing noise with error probability $p_0$ for each location in the circuit. We calculate the failure rate of the ex-Rec, i.e. the probability that the Rec is not correct, for fixed $R$ and $L$. This implies that sometimes there are no verified ancillas available for a 1-EC. If this happens for {\em any} of the 1-ECs inside the extended rectangle, we call this a failure of the extended rectangle. We do this for all codes except for Reichardt's use of [[49,1,9]]. The reason for this exception is that for Reichardt's method the failure rate of ancillas may be rather high. If we let failure of having verified ancillas in the leading 1-EC determine failure of the rectangle after this 1-EC, we are possibly double-counting errors. Hence in Reichardt's method we replace any failed ancillas in the leading 1-EC by a perfect ancilla and do not call failure. As the results show, even under this assumption, the [[49,1,9]] concatenated code with error-detection and finite resources is not a great performer.
In general, our assumption on the effect of failed ancilla preparations may make our estimates for the pseudo-threshold for the EC circuits slightly more pessimistic.
We will estimate the failure rate of a CNOT ex-Rec, since this is by far the biggest circuit among the Clifford ex-Recs. As we argued in Section \ref{sec:univ}, the non-Clifford (and possibly other Clifford) gates will be implemented via injection-and-distillation so that their implementation will not affect the threshold. Pauli gates are not applied within a Clifford ex-Rec because they can be stored in classical memory as the Pauli frame and applied only prior to the execution of non-Clifford gates.
Given a fixed $R$ and $L$, we will estimate the failure rate $p_1(p_0)=\frac{N_{\rm fail}}{N}$ where $N_{\rm fail}$ is the number of Monte-Carlo samples that fail (i.e. the number of times we simulate the extended rectangle with randomly generated faults and observe that the rectangle is incorrect) and $N$ is the total number of runs. With high probability this estimated $p_1$ lies within one standard deviation of the real $p_1$. In this way we collect data points $p_1(p_0)$ for different values of $p_0$. We then take these points as the mean of a normal distribution for each $p_0$. We sample from these normal distributions and for each set of samples we determine a small degree polynomial $p_1(p_0)$ fitting the samples.
The equation $p_1(p_0)=p_0$ gives us a sample of the threshold and we put an error bar on this result by calculating the standard deviation of the obtained threshold samples.
The way we test for correctness of a rectangle for a given pattern of faults in the ex-Rec is as follows: Let $E_{\rm out}$ be the outgoing error of the leading 1-EC. We use syndrome decoding to determine the coset leader $E_{\rm lead}$ corresponding to the coset of $N(S)$ in the Pauli group of this $E_{\rm out}$. We propagate this $E_{\rm lead}$ through the rectangle, let $f(E_{\rm lead})$ be the outgoing error on the data. We follow the rectangle by an ideal decoder and let $E_{\rm correct}$ be the correction suggested by the ideal decoder. Then we test whether $E_{\rm correct} f(E_{\rm lead})$ commutes with both $\overline{X}$ and $\overline{Z}$. If it does, we infer that no logical faults occurred, hence the rectangle was correct. Otherwise we call failure.
An alternative way of using the AGP correctness criterion is to count or sample malignant fault sets. This method is advantageous if one wants to estimate the threshold for worst-case adversarial noise. In such an application, one fixes the number of faults and counts or samples how many sets with this fixed number of faults are malignant. The failure rate $p_1$ is a polynomial in $p_0$ with factors that are determined using the malignant set counts (or estimates of these counts determined by sampling). For codes with large distance this method becomes cumbersome, as the total number of possible fault-sets of size $t+1$, ${A \choose t+1}$ becomes large. Here $A$ is the total number of locations in the ex-Rec. This sampling method is difficult but still possible for the Golay code, but it is not possible for codes of higher distance.
The advantage of the malignant set counting or sampling method is that one gets an upper bound on $p_1$ for arbitrary values of $p_0$. This makes it possible to estimate $p_1$ even for very small values of $p_0$. We use the Monte-Carlo simulation to estimate $p_1$, but the number of samples required becomes quite large if one wants to estimate $p_1$ with good relative error for small $p_0$. In such cases we extrapolate the values for $p_1$ obtained from larger values of $p_0$, see Section \ref{sec:results_pseudo}.
\subsection{Software and Computer Use} \label{sec:software}
On the website \cite{cross:tools} one can find a set of software tools that have been developed for this and other future fault-tolerance projects. The quantum circuits for the CNOT ex-Recs based on CSS codes are highly structured and can be mechanically assembled in $O(n^3)$ time for block-size $n$ given the classical codes $C_1$ and $C_2$. We have used MAGMA \cite{MAGMA} and/or GAP \cite{GAP} (using the GUAVA package \cite{GUAVA}) to construct quantum codes and compute their parameters. The code stabilizers are copied from the computer algebra programs into our circuit synthesis and simulation programs, where they are again verified to have the required commutation relations.
The simulation and circuit synthesis programs are implemented in C++ and use MPI \cite{MPI} for communication during embarrassingly parallel tasks. The project is entirely open source and makes use of preexisting open source libraries such as a Galois field implementation \cite{Arash} and a weighted matching algorithm \cite{thesis:rothberg}. Importantly, the same functions and procedures are used in the Monte-Carlo simulation. This gives us increased confidence in the simulation output.
The symmetries of the pair count matrix for some distance-3 code circuits
and the lack of single-location malignancies in all circuits strongly suggests that our circuit constructions are indeed fault-tolerant against single errors. Furthermore, we strictly check all input and intermediate results for consistency at runtime. The programs can be optimized and further improved, but we leave this to future work and encourage development by making the code publicly available \cite{cross:tools}.
The simulations were carried out on a relatively small allocation of Blue Gene L at the IBM T. J. Watson Research Center. Typically we used between 64 and 256 PowerPC 440 700 MHz CPUs. Each pair of CPUs had access to 512 Mb of local memory. Using 256 CPUs gave us roughly a factor of 50 speed-up over a typical single-processor desktop machine. The entire process of development and debugging took many months, but we estimate that all of the data could be retaken in several weeks with these computing resources.
\begin{figure}
\caption{Level-1 depolarizing pseudo-threshold for three families of codes with perfect ancillas for Steane-EC: surface codes, dual-containing codes, and Bacon-Shor codes. This plot indicates that under no circumstances can thresholds reach $1\%$ for the codes in our study. The data points are connected by lines merely as a guide to the eye. }
\label{fig:perfect}
\end{figure}
\section{Results} \label{sec:results}
Tables~\ref{table:complete1}, \ref{table:complete2}, \ref{table:complete3}, \ref{table:surfacedat}, and \ref{table:perfectdat} list the complete set of results of our studies. Our results are obtained assuming that all locations including memory locations suffer from noise at the same noise rate, unless specified otherwise.
\subsection{Perfect Ancillas}
In our first study, and only in this Section, we assume that ancillas for Steane error correction can be prepared flawlessly, see Figure \ref{fig:perfect}. In such a scenario, the threshold is largely determined by the error-correction properties of the code (see also the analysis in \cite{eastin}), in particular its (effective) distance. For families of quantum error-correcting codes in which the effective distance is linear in the block-size, we expect the threshold to be a monotonically increasing as a function of $n$, see Section \ref{sec:scaling}. In Figure~\ref{fig:perfect} and Figure \ref{fig:perfectlonger} we have plotted the pseudo-thresholds for three families of codes: surface codes, some dual-containing codes, and Bacon-Shor codes. The surface codes and Bacon-Shor codes apparently have fairly good distance properties, even though there is some decline in the BS code family for large $n$.
Figure~\ref{fig:perfect} shows we cannot expect a threshold over $1\%$ for the codes we have studied using Steane-EC -- introducing noise realistically into the ancilla preparation circuits cannot increase the pseudo-threshold. Note that if we do Shor-EC on the surface codes we cannot expect thresholds exceeding about $3\%$, see the arguments in \cite{dennis+:top}.
When we assume that the logical Bell pairs of Knill's circuit can be prepared flawlessly, the level-1 pseudo-threshold of the $[[5,1,3]]$'s ${\rm T_3}$ gate is $(2.0\pm 0.1)\times 10^{-4}$ \footnote{The pseudo-threshold in this case is the point at which the failure rate $p_1$ of a ${\rm T_3}$ ex-Rec is the same as the base error rate $p_0$ of all elementary gates in the ex-Rec.}. This is roughly an order of magnitude below the Steane code with perfect ancillas.
\begin{figure}
\caption{Level-1 depolarizing pseudo-threshold for surface codes and Bacon-Shor codes using perfect ancillas for Steane-EC.}
\label{fig:perfectlonger}
\end{figure}
\subsection{Pseudo-Thresholds} \label{sec:results_pseudo}
In Figure \ref{fig:block} we tabulate for each code the maximum pseudo-threshold over the various choices of $R$ and $L$. The maximum overall pseudo-threshold $(2.25\pm 0.03)\times 10^{-3}$ is attained by the Golay code with $L=30$ and $R=1$. The two code families, Bacon-Shor and surface, both attain a peak threshold and then decline when we use Steane-EC. The peak Bacon-Shor code is the $[[49,1,7]]$ at $(1.224\pm 0.005)\times 10^{-3}$ with $L=9$ and $R=1$. The peak surface code (using Steane error correction) is $[[41,1,5]]$ at $(1.008\pm 0.008)\times 10^{-3}$ at $L=30$ and $R=1$. Interestingly when we use Shor-EC for the surface codes the performance is quite different. Shor-EC does not do as well as Steane-EC for small block sizes, but for larger block size Shor-EC gives a threshold that asymptotes to a finite value, see Figure \ref{fig:surfacethreshold}. For small block size the thresholds of the surface codes are not as good as of some other codes such as the Golay code and the Bacon-Shor codes.
\begin{figure}
\caption{Level-1 depolarizing pseudo-threshold versus block size. The other codes are the $[[5,1,3]]$ non-CSS code, the $[[15,1,3]]$ Reed-Muller code, the $[[49,1,9]]$ (dual-containing) concatenated Steane code using $L=15$ attempts to prepare using error detection at level-1, and the $[[60,4,10]]$ (dual-containing) concatenated polynomial code using $L=20$ attempts to prepare ancillas.}
\label{fig:block}
\end{figure}
\begin{figure}
\caption{Surface code level-1 depolarizing pseudo-threshold versus $\ell$ for $\ell\times\ell$ surface code (the block-size $n=\ell^2+(\ell-1)^2$). The ex-Rec is a transversal CNOT gate with $\ell$ sequential Shor-EC steps per EC. The pseudo-threshold increases with $\ell$ and is expected to approach a constant value in the limit of large $\ell$, unlike the other codes in this study.}
\label{fig:surfacethreshold}
\end{figure}
It is clear from the data that the pseudo-threshold increases with increasing $L$. Our main interest in this study is in circuits with small overhead and hence with a relatively small number of preparation attempts $L$. In various cases the thresholds stated for finite $L$ will be thus be lower than the one in the $L \rightarrow \infty$ limit. Notably, this occurs for the $[[49,1,9]]$ code, where we expect thresholds approaching $1 \times 10^{-2}$ with many more ancilla preparation attempts \cite{reichardt:concat7}. In other cases one can take the perfect ancilla results in Figure \ref{fig:perfect} and the Tables as upper bounds on the $L \rightarrow \infty$ pseudo-threshold.
The use of weakly fault-tolerant circuits, i.e. small $R$ and small $L$ is meant to get improved threshold behavior for finite resources. The best performance of a code can be expected for $L \rightarrow \infty$, since one would always use ancillas which passed verification. When $L \rightarrow \infty$, it is clear that, --at least below threshold--, optimal performance is likely to be achieved when $R$ is taken as large as possible, since then an ancilla is maximally verified. However, at finite $L$, a larger $R$ will let more ancillas fail and hence increase the chance for an extended rectangle to fail (remember that we, pessimistically, call an extended rectangle failed if we don't have ancillas for EC). Hence for small $L$, small $R$, weakly fault-tolerant verification circuits can outperform circuits with the same $L$ and larger $R$.
\subsection{Influence of Storage Errors}
In Figure~\ref{fig:blockcompare} we replot the pseudo-threshold versus block-size when storage error rates (on memory locations) are zero. The peak pseudo-threshold increases to $(3.33\pm 0.02)\times 10^{-3}$. The Figure shows that storage errors do not influence the pseudo-threshold appreciably. The Bacon-Shor codes are least affected by storage errors because the encoding circuits are extremely simple. The non-CSS $[[5,1,3]]$ code is most greatly affected because storage errors can enter into the ${\rm T_3}$
gate sub-circuit, the $|\overline{0}\rangle$ encoders, and the cat-state encoders at many locations.
\begin{figure}
\caption{Pseudo-thresholds versus block size for Steane-EC and Knill-EC circuits, comparing the case where the memory failure rate equals the gate failure rate with the case where the memory failure rate is zero. Naturally the difference is smallest where we have taken advantage of simple encoders as those for the Bacon-Shor codes.}
\label{fig:blockcompare}
\end{figure}
\subsection{Logical Error Rate versus Overhead}
\begin{figure}\label{fig:errorrate_full}
\end{figure}
The threshold is an extremely important figure of merit for fault-tolerant circuit constructions. But practically speaking, we are also interested in how quickly the error rate decreases if the initial error rates are low enough for a given overhead. Figures \ref{fig:errorrate_full}, \ref{fig:errorrate}, and \ref{fig:surfaceerrorrate} plot the probability of failure of a CNOT ex-Rec (defined in Section \ref{sec:mcmethod}) versus the number of physical CNOTs in a rectangle at $p_0=10^{-4}$.
Even though there are other measures of overhead, such as total number of qubits involved in the rectangle or the depth of the rectangle circuit, we have chosen the number of CNOTs per rectangle as an estimate for the overhead since it approximately captures the total size, i.e. depth times width, of the rectangle.
The Golay code achieves the lowest logical error rate for codes with fewer than $O(10^{4})$ CNOT gates per rectangle, and that rate can be further reduced by increasing the number of verification rounds to $R=2$. There is a clear tradeoff between the number of physical CNOTs per rectangle and the logical error rate.
We note that given the lack of code specific optimizations, the achievable overheads for various codes may be somewhat less than what is estimated here. For the Golay code and the Bacon-Shor codes for example, the overhead may come down by at least a factor of 2 by using simplified verification circuits. Viewed an a log-scale such decrease in resources is relatively small. We also see in Figure \ref{fig:errorrate} that the approximate expression for the failure rate, Eq. (\ref{eq:p1estim}), gives a pretty good estimate of the actual failure rate.
\begin{figure}
\caption{Level-1 logical error rate (probability of failure of CNOT ex-Rec) versus the number of CNOTs per rectangle for the best performing codes. The subset of data plotted here was chosen so that the error rate decreases monotonically with the rectangle size and there is no code with lower error rate at a given rectangle size. The error rates are evaluated at a fixed $p_0=10^{-4}$.}
\label{fig:errorrate}
\end{figure}
\begin{figure}
\caption{Level-1 logical error rate versus the number of CNOTs per rectangle for the $\ell\times\ell$ surface codes, $\ell=5,7,9$. It is expected that the error rate decreases exponentially as $\ell$ increases for fixed $p_0=10^{-4}$. }
\label{fig:surfaceerrorrate}
\end{figure}
Some of the error rates plotted in Figure~\ref{fig:errorrate_full} were extrapolated from error rates at higher values of $p_0$ and may only be rough indications of the actual error rates. For small values of $p_0$ the logarithm of the error rate $p_1(p_0)$ is expected to be approximately linear in $p_0$. We extrapolate from a least-squares fit to this line.
Tables~\ref{table:complete1}, \ref{table:complete2}, and \ref{table:complete3} indicate these extrapolated rates by enclosing them in square brackets. The extrapolations are only plotted for the $5\times 5$ surface code and the $9\times 9$ Bacon-Shor code and are plotted without errorbars for these two points \footnote{Unfortunately, using the same method by which we obtain error estimates for our calculated pseudothresholds, we find error estimates that are an order of magnitude larger than these extrapolated values. In the few places where we make these extrapolations, the results should only be taken as rough indications of the actual error rates the codes can attain.}.
For the Golay code we have looked at the behavior of the threshold for $R=1, 2, 3$. One important empirical observation is the following. The pseudo-threshold can increase slightly while the logical error rate for $p_0=10^{-4}$ remains the same. This happens for the Golay code when $R=1$ and $L$ is increased from $10$ to $20$. Furthermore, the pseudo-threshold can decrease while the logical error rate decreases too. This also happens for the Golay code when $L=10$ and $R$ is increased from $1$ to $2$.
This suggests that the pseudo-threshold value is sensitive to higher order effects that quickly become negligible at lower error rates. Thus a desired logical error rate may be achievable with significantly fewer ancilla resources $L$ than are necessary to maximize the pseudo-threshold, provided the initial error rate $p_0$ is not too close to the pseudo-threshold.
In Figure \ref{fig:errorrate} we have also added Knill's $C_4/C_6$ Fibonacci scheme \cite{knill:nature} at 2 and 3 levels of encoding. These data points are derived from his paper \footnote{Note that his error model is slightly different from ours but we take the dominating physical CNOT error rate to be the same.}. At level 2 the detected error rate of the logical CNOT is $(1.06\pm 0.01)\times 10^{-5}$ and at level 3 the detected error-rate is $(2.18\pm 0.02 )\times 10^{-8}$.
The plot shows that [[9,1,3]] is still better than the $C_4/C_6$ scheme in terms of overhead, but the $C_4/C_6$ Fibonacci scheme definitely beats [[7,1,3]]. The next two Bacon-Shor codes fill a void between $C_4/C_6$ level 2 and $C_4/C_6$ level 3.
For the surface codes (see Fig. \ref{fig:surfaceerrorrate}) we note that the error rates are relatively high compared to other error-correcting codes with comparable numbers of CNOTs per rectangle. However one should remember that the circuits for the surface codes are already spatially local in two dimensions whereas the circuits for any of the other codes, for example, the Golay code, are not.
\section{Discussion}
In our study we have considered bottom codes and their performance in a bottom-top code architecture. Our best threshold around $2\times10^{-3}$ is seen for the Golay code, and many other codes both larger and smaller were studied and found to have much worse thresholds. An important figure of merit is the logical error rate versus overhead curve which shows that the Bacon-Shor codes are competitive with Knill's $C_4/C_6$ scheme at a base error rate of $10^{-4}$.
We have seen that the constraint of finite resources, i.e. limited $R$ and $L$ in Steane EC, can considerably and negatively impact noise thresholds. An example is Reichardt's estimate for $[[49,1,9]]$ when $L \rightarrow \infty$ versus our estimates for this construction at small $L$. For code families with low-weight stabilizers, Shor EC may give rise to thresholds which grow with block-size. For code families which do not have this property, e.g. general quantum BCH codes, the limit resource constraint on $R$ and $L$ and the complexity of the encoding circuits start pushing the thresholds down beyond some peak performance block-size.
In this landscape of codes and their performances, one of the missing players is the surface code scheme of \cite{RHG:topo} in which many qubits are encoded in one surface code and the CNOT gate is done in a topological manner. In principle, the possible advantage of this scheme is that if one uses enough space (meaning block size) one would reach the asymptotic threshold of a simple EC rectangle (no 1-Ga). We have in fact analyzed an ex-Rec where the Rec is only Shor-EC on a $\ell \times \ell$ surface and we find that this asymptotic memory threshold for $\ell \rightarrow \infty$ is about $3.5 \times 10^{-3}$. This is a factor of two lower than the number stated in \cite{RHG:topo}. For finite block size one could analyze a CNOT ex-Rec for this topological scheme just as for the other codes. Like all the other codes, the topological scheme will have a trade-off between overhead and logical error rate. It will be interesting to see whether topology and block coding provide an efficient way of using resources and how it compares to a local version of a bottom-top architecture discussed in this paper.
For a bottom-top architecture it will be important to study the performance of top codes in order to understand at what error rate one should switch from bottom to top code and what total overhead one can expect. Concerning a choice of top code we expect the following. First of all, given the constructions of \cite{SI:networks}, one can expect that a $[[n,k,d]]$ block code has a threshold comparable to a $[[n,1,d]]$ code. Secondly, the networks in \cite{SI:networks} show how to do logical gates on qubits inside the block codes using essentially gate-teleportation and Knill-EC. One issue of concern for block codes is the complexity of the encoding circuit as a function of block size. It would be highly desirable to consider block codes with EC circuits that are linear in $n$, otherwise one would expect the threshold to decline as a function of $n$.
There is another desirable property of top codes which relates to the transversality of gates. In order to minimize overhead, it is desirable that the T gate is transversal for the top code. The reason is as follows. In order to have maximal freedom in picking a bottom code we will only require that it has a transversal ${\rm CNOT}$. Thus all other gates, in particular ${\rm T}=e^{i \pi Z/8}$ and the phase gate S, should be either performed by more complicated fault-tolerant 1-Ga or be implemented by the injection-and-distillation scheme. If the fault-tolerant circuits for these non-transversal gates have poorer thresholds than the CNOT gate, then the injection-and-distillation scheme is the preferred solution. In the injection-and-distillation scheme, the obtained error rates of the encoded and distilled ancillas will be limited by the noise rates on the Clifford gates which distill the ancillas, since the Clifford distillation circuit is not fault-tolerant. Assume we teleport the ancillas into $C_{\rm top} \circ C_{\rm bot}$ and get Clifford gates with $O(10^{-15})$ error rate. Since a circuit such as Bravyi-Kitaev distillation uses $O(10^{3})$ gates,
the error rates of the distilled ancillas can be as high as $O(10^{-11})$. Thus by these schemes the, say, T error rate is always trailing the transversal gate error rates. But assume that the T gate is transversal for the top code and thus we only inject the T ancillas into $C_{\rm bot}$. Then even though the once encoded gate $C_{\rm bot}({\rm T})$ has an error rate of, say, $O(10^{-4})$, the twice-encoded gate $C_{\rm top} \circ C_{\rm bot}({\rm T})$ will mostly likely have an error rate similar to other Clifford gates since there are very few $C_{\rm bot}({\rm T})$ in the twice-encoded gate compared to the EC parts.
Of course, the top code will have other non-transversal gates; for example the [[15,1,3]] code has a transversal T gate but not a transversal Hadamard gate. If the bottom code has a transversal Hadamard gate, we can implement a fault-tolerant H in $C_{\rm top} \circ C_{\rm bot}$ by using the fault-tolerant non-transversal gadget for the Hadamard gate in $C_{\rm top}$ and implementing the resulting Clifford gates. This shows that there are possible constructions which would allow all gates needed for universality to be implemented with approximately the same, low, error rate, while the noise threshold of such scheme is determined by the noise threshold for the transversal Clifford gates.
\section{Various Aspects of Steane-EC} \label{app:css}
A binary $[[n,k,d]]$ CSS code $\textrm{CSS}(C_1,C_2)$ is constructed using two classical linear error correcting codes $C_2^\perp\subseteq C_1$ and has the codewords: \begin{equation}
\ket{\overline{a}} = \frac{1}{\sqrt{|C_2^\perp|}} \sum_{c\in C_2^\perp} \ket{c+a}\ \textrm{where}\ a\in C_1/C_2^\perp. \end{equation} Each row $r$ of the parity check matrix of $C_2$ gives the stabilizer generators $X(r)$, and each row $s$ of the parity check matrix of $C_1$ gives stabilizer generators $Z(s)$, where $U(r)=U_1^{r_1}\otimes\dots\otimes U_n^{r_n}$. It is easy to check that $\ket{\overline{a}}$ is a simultaneous eigenstate of these stabilizer generators: (1) a row $r$ of the parity check matrix of $C_2$ must be an element of $C_2^\perp$, so adding it to each codeword in the superposition $\ket{\overline{a}}$ leaves the state unchanged, and (2) every codeword in the superposition $\ket{\overline{a}}$ is an element of $C_1$, so it must pass the parity checks of $C_1$. A basis of the $2^k$ cosets of $C_2^\perp$ in $C_1$ corresponds to logical $X$ operations $\overline{X}(a)$ on the code space because $\overline{X}(a)\ket{\overline{0}}=\ket{\overline{a}}$. Similarly, a basis of the $2^k$ cosets of $C_1^\perp$ in $C_2$ corresponds to logical $Z$ operations $\overline{Z}(b)$ since $Z(b)\ket{\overline{a}}=(-1)^{b\cdot a}\ket{\overline{a}}$. We can choose these bases such that the logical operators obey the commutation relations of the $k$-qubit Pauli group.
A special case of the CSS construction occurs when $C_2^\perp=C_1^\perp$, in which case $C_1$ is a dual-containing code. The X and Z stabilizer generators have identical supports and the Hadamard H gate is transversal. If in addition the weight of each stabilizer generator is a multiple of $4$, $C_1^{\perp}$ is called doubly-even and the quantum code has a transversal S gate. The code does not have any transversal gates outside of the Clifford group \cite{rains:d2} in this case.
Steane error correction for a CSS code $CSS(C_1,C_2)$,
$C_2^\perp\subseteq C_1$ uses $|\overline{+}\rangle$ and
$|\overline{0}\rangle$ ancilla states. These states can be encoded directly from the generator matrices of $C_1$ and $C_2^\perp$, respectively, according to a well-known procedure. The generator matrix $G$ has $n$ columns and $k_1$ rows for $C_1$ or $n-k_2$ rows for $C_2^\perp$, and the quantum code encodes $k=k_1+k_2-n$
qubits. Gaussian elimination puts a generator matrix into standard form $G=(I |A)$ where $I$ is an identity matrix and $A$ is a binary matrix. The $i$th row of the generator matrix specifies the controls and targets of $w_i$ CNOT gates, where $w_i$ is the weight of the row minus one. In the next section we discuss how to implement this circuit in a way that minimizes the number of memory locations. The depth of the resulting CNOT circuit is $w=\textrm{max}\{w_i\}$, assuming equal cost for any pair of qubits to communicate. The number of fault locations in an encoder is summarized by the following expressions: \begin{align*} A_\textrm{enc}(n,k_1,k_2,w) & \leq n + w\textrm{max}(k_1,n-k_2),\ \textrm{no memory noise} \\ A_\textrm{enc}(n,k_1,k_2,w) & \leq n + w n,\ \textrm{memory noise}. \end{align*} For particular states, different scaling is possible. For example, for the Bacon-Shor codes one can make the encoded ancillas using $O(n)$ 2-qubit gates.
In general, any unitary stabilizer circuit has an equivalent circuit with $O(n^2)$ gates and $O(\log{n})$ depth \cite{AG:stabilizer}.
One method of verifying the encoded ancilla against low-weight correlated errors is to use transversal gates to perform error detection. One possible error detection method consumes three additional ancilla and uses 3 transversal CNOT gates and 3 transversal measurements.
The cost of verifying is: \begin{align*} A_\textrm{ver}(n,k_1,k_2,w,R) & \leq R(3A_\textrm{enc}(n,k_1,k_2,w) + 6n),\ \textrm{no memory noise} \\ A_\textrm{ver}(n,k_1,k_2,w,R) & \leq R(3A_\textrm{enc}(n,k_1,k_2,w) + 6n) + n,\ \textrm{memory noise}. \end{align*} Again, these expressions assume equal cost for any pair of qubits to communicate.
Finally, we can write expressions for the total number of fault locations in a CNOT extended rectangle using Steane error correction: \begin{equation*} A(n,k,w,R) \leq 8A_\textrm{enc}(n,k_1,k_2,w) + 8A_\textrm{ver}(n,k_1,k_2,w,R) + 17n. \end{equation*} If we set $R=t$ then the total number of fault locations is $A(n,k_1,k_2,w,R)=O(wnt)$ using this method of error correction and assuming equal communication costs between qubits. In the worst case this can be $O(n^3)$.
\subsection{Latin Rectangle Method for Optimizing Encoding Circuits} \label{app:latin}
There is a simple method for minimizing the number of memory locations in an ancilla encoding circuit due to Steane. Steane puts the generator matrix $G$ of a linear binary code into standard form $(I|A)$ using Gaussian elimination. An encoding circuit for the logical zero state can be constructing by looking at the $A$ matrix for the code $C_2^\perp$. Every $1$ in the $A$ matrix gives a CNOT gate in the encoder. The control qubits are the $1$s in the $I$ part of $G$ and the target qubits are the $1$s in the $A$ part of $G$.
For example, we have $G = (1010101,0110011,0001111)$ for the $[7,3,4]$ code, which is the $C_2^\perp$ for Steane's [[7,1,3]] code. Transposing columns 3 and 4 gives the standard form and an $A = (1101,1011,0111)$. This means there are 9 CNOT gates in the logical zero encoder. We can assign each CNOT a time-step so that no qubit is involved in two gates at once. That constraint makes a time-step assignment the same as finding a partial Latin rectangle. The Latin rectangle to complete is $\left[\begin{array}{cccc} ? & ? & & ? \\ ? & & ? & ? \\ & ? & ? & ? \end{array}\right]$ and one possible completion is $\left[\begin{array}{cccc} 1 & 2 & & 3 \\ 2 & & 3 & 1 \\
& 3 & 1 & 2 \end{array}\right]$. The circuit corresponding to the time-step assignment is: \begin{verbatim}
# time 1
cnot 1,4
cnot 2,7
cnot 3,6
# time 2
cnot 1,5
cnot 2,4
cnot 3,7
# time 3
cnot 1,7
cnot 2,6
cnot 3,5 \end{verbatim} We have to undo the qubit permutation that occurred in the Gauss elimination to standard form, so at the end we should switch back qubits 3 and 4.
This is the smallest depth $(3)$ that a circuit for $A$ can have. The smallest depth is the maximum row or column sum $w$ of $A$.
The problem of completing the Latin rectangle and therefore of computing the optimal time-step assignment for a matrix $A$ is equivalent to edge coloring a bipartite graph with the minimum number of colors. We construct the graph in the following way. The left set of vertices corresponds to the control qubits. The right set of vertices corresponds to the target qubits. A control and target vertex are connected by an edge if there is a CNOT between those two qubits. Assign a color to an edge to indicate what time-step we plan to do that CNOT gate. Since we cannot have two CNOT gates occur at the same time using the same qubit, all of the edges incident to a given vertex must have different colors. This means that a valid schedule corresponds to an edge coloring of this bipartite graph (bipartite because we have a set of control vertices that are only connected to target vertices, and a set of target vertices that are only connected to control vertices). By Hall's theorem \cite{steane:fast}, there is a coloring using $w$ colors, and $w$ colors is the minimum number of colors we can use. Here $w$ is maximum weight of the rows of $A$ minus 1. See \cite{KR:edgecolor} for an algorithm that finds an edge coloring with $w$ colors in time $O(n N_{{\rm CNOT}})$. Here $n$ is the number of qubits that are to be encoded (i.e. number of vertices) and $N_{\rm CNOT}$ is the number of CNOT gates in the encoder (i.e. number of edges).
We have tested several of the encoders produced by our Latin rectangle software tool with and without memory locations by simulating them using CHP\footnote{CHP stands for CNOT, Hadamard, Phase.} \cite{AG:stabilizer}.
\section{Syndrome Decoding} \label{app:decode}
General algorithms for constructing the classical circuits to decode measurement outcomes obtained in Steane error correction require exponential time and/or space. Therefore, we consider each code's syndrome decoder separately, essentially devising a special-purpose algorithm for each to make the decoding feasible.
\begin{table} \begin{center}
\begin{tabular}{l|l|l} \textsc{Code} & \textsc{Decoder} \\ \hline $[[5,1,3]]$ & Table Lookup\\ $[[7,1,3]]$ & Table Lookup (cyclic)\\ $[[9,1,3]],[[25,1,5]],[[49,1,7]],[[81,1,9]]$ & Majority \\ $[[15,1,3]]$ & Table Lookup \\ $[[13,1,3]],[[41,1,5]],[[85,1,7]]$ & Min. Wt. Matching \\ $[[21,3,5]]$ & Table Lookup (cyclic)\\ $[[23,1,7]]$ & Table Lookup (cyclic)\\ $[[47,1,11]]$ & Algebraic \cite{chen:decodingQR} \\ $[[49,1,9]]$ & Table Lookup with Message Passing \\ $[[60,4,10]]$ & Table Lookup (cyclic)\\ \end{tabular} \end{center} \caption{The decoders that we use for the codes in our study.} \label{tab:decoders} \end{table}
Table~\ref{tab:decoders} lists all of the codes we consider in this study and their syndrome decoders. There are six distinct decoding algorithms that we use to compute the error locations and type of error from the syndrome measurements: a generic table lookup algorithm, a table lookup algorithm for cyclic codes over arbitrary fields, a majority voting algorithm for Bacon-Shor codes, a minimum weight matching algorithm for surface codes, a simple message passing algorithm for the concatenated Hamming code, and an algebraic decoder for the $[[47,1,11]]$ quadratic residue code.
Rather than use a general table-lookup algorithm, we use a so-called Meggitt decoder which uses the fact that the polynomial codes and the Hamming, Golay, and quadratic residue (QR) codes are constructed from cyclic classical codes. Cyclic codes have a compact description in terms of a generating polynomial whose coefficients give one of the code words and whose cyclic shifts generate a basis for the code. The Meggitt decoding algorithm stores a table of syndromes and their associated error corrections \cite{book:HP}. For non-binary codes such as the polynomial codes, the table stores both error locations and error-type (the so-called amplitude). Only $n\choose w-1$ syndromes need to be stored for a weight $w$ error, since one of the coordinates can be fixed by the cyclic symmetry. Finding the appropriate recovery requires at most $n$ table lookups. If we fail to find a recovery in the table, a subroutine is triggered that applies some syndrome-dependent correction mapping the state back into the code space.
For cyclic codes with larger distance where table lookup is impractical, for example $[[47,1,11]]$, algebraic decoding techniques can be used. The generator polynomial's roots are used to compute a sequence of syndromes from which we can locate errors. BCH codes are easy to decode because their generator polynomials have a contiguous sequence of roots so the Berlekamp-Massey algorithm can find the error-locator polynomial whose roots give the error locations. Sometimes decoding up to the full minimum distance of the code is challenging because the generator polynomial may not have a long sequence of roots, so some syndromes are missing and the Berlekamp-Massey algorithm cannot be directly applied. In this case, unknown syndromes can sometimes be computed from algebraic equations involving the known syndromes. Algebraic decoding of the $[[47,1,11]]$ proceeds this way. For each error weight from zero to $t$, we compute any missing syndromes, construct a polynomial whose roots are the error locations, and find the roots of the polynomial. If the polynomial has enough roots, we correct those errors and stop. If we do not find enough roots for each of the locators, we return a ``failed'' result, triggering a subroutine that applies some syndrome-dependent correction that maps the state to some (possibly logically incorrect) state in the code space. The implementation details can be found in \cite{chen:decodingQR}.
The Bacon-Shor codes are essentially concatenated quantum repetition codes. Since the code stabilizer is preserved by bitwise Hadamard composed with a $90$ degree rotation of the square lattice, one syndrome decoder is sufficient for both $X$ and $Z$ error correction. Imagine a vector of $n^2$ syndrome bits placed on an $n$ by $n$ square lattice. Let $s_x$ be the syndrome vector for $X$ errors and $s_z$ be the syndrome vector for $Z$ errors. Let $R$ be the map on vectors of length $n^2$ that rotates them by $90$ degrees on the square lattice. The same syndrome decoder is applied to $s_x$ and $Rs_z$. The syndrome decoder decodes a variation on the classical repetition code on $n$ bits. First, the decoder computes the parity of each column of the lattice and stores each column parity as an element of a vector $p$. Next, the decoder computes the repetition code parity check $h=Hp$. This parity check $H$ is expressed in standard form $[I_{n-1}\ 1]$ where $1$ is the all ones column vector. Finally, the decoder infers the error locations from the parity check. If the weight of the parity check is greater than $t$, we must assume that the rightmost bit of $p$ was incorrect so that $h\oplus 1$ gives the error locations on the first $n-1$ bits of $p$. Otherwise, we infer that the rightmost bit of $p$ was correct so that $h$ gives the error locations on the first $n-1$ bits of $p$.
The surface code is decoded using Edmond's minimum-weight matching algorithm. The approach differs slightly depending on whether Steane-EC or Shor-EC is used but is essentially the same as \cite{dennis+:top}. Steane-EC gives a 2D matching problem whereas Shor-EC gives a 3D matching problem. The mapping from syndrome information to a matching problem is as follows.
Nonzero syndrome bits are called defects and are located somewhere in the $\ell \times \ell$ plane. We construct a complete weighted graph whose vertices represent defects and whose edge weights indicate the distance between defects. The surface code's syndrome may be such that there are lone defects which are not caused by error patterns connecting two defects, but by an error pattern connecting an edge-defect on the boundary to an inner defect. $X$ and $Z$ errors constitute separate matching problems and $X$-defects can be matched with, say, the horizontal boundaries and $Z$-defects with the vertical boundaries.
We can design an algorithm for decoding the surface code for, say, $Z$ errors, as follows: \begin{itemize} \item Imagine cutting the lattice vertically in two halves, left (L) and right (R). Let $N_{L/R}(i)$ be the number of defects in row $i$ of the left/right part of the lattice. For each row of the lattice, add $N_{L/R}(i)$ edge defects on the $i$th row on the left (right) boundary. \item Assign the weight of the edges between {\em any} edge defects as zero and assign the distance as the weight between edge defects and inner defects. \item Compute the minimum-weight perfect matching of the graph of defects. \item The recovery operation consists of applying phase flips on the qubits that are along the edges of each pair of matched vertices in the graph. \end{itemize}
Note that the algorithm enforces the property that the graph has an even number of vertices, so that every vertex can be matched.
\begin{figure}\label{fig:bell513}
\end{figure}
The concatenated [[7,1,3]] code, that is, the [[49,1,9]] code, can be decoded to distance $7$ if we treat it as a concatenated code. However, decoding the code to distance $9$ requires a slight modification of the algorithm so that a simple message is passed from level-1 to level-2.
Suppose the $49$ transversal measurement outcomes are organized into $7$ registers of $7$ bits each. We use these registers as temporary storage to compute the appropriate correction. First, we compute the level-1 syndromes for each register as we would normally do. These syndromes indicate errors $e_i$ in the $i$th level-1 register. We correct each level-1 register according to the $e_i$s and ``flag'' those registers for which $e_i\neq 0$. Next, we compute the level-2 (logical) syndrome of the resulting $49$ bit register, which now has trivial level-1 syndrome in each $7$ bit register. This level-2 syndrome indicates a logical correction $\bar{e}$ that is constant on each level-1 register (but two level-1 registers can take different values). The correction $c_1:=\left(\bigoplus_i e_i\right)\oplus\bar{e}$ corrects all errors of weight $4$ or less, except for one problem case. This case occurs when a pair of errors occurs in one level-1 register and another pair of errors occurs in a different level-1 register. The problem is overcome by comparing the register positions where $\bar{e}$ is $1$ with the positions of the flags whenever two flags are raised. If they disagree, apply the correction $c_2:=\left(\bigoplus_i e_i\right)\oplus\bar{f}$ where $\bar{f}$ is a logical correction on the flagged registers. Otherwise, apply the original correction $c_1$. This procedure corrects all errors of weight $4$ or less and returns the input to the codespace in all cases.
The classical decoding algorithms have been tested exhaustively for all of the codes in the paper except for the large Bacon-Shor and surface codes. The algorithms were found to correct all errors of weight $t$ or less. In the case of the polynomial codes, $t$ is the number of errors the underlying nonbinary code can correct. The same decoding algorithms that were tested exhaustively for small Bacon-Shor and surface codes were used for the larger codes in those families.
\section{Data Tables} \label{app:data}
\pagestyle{empty}
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\begin{landscape} \begin{table}[htp] \centering \begin{minipage}{\textwidth} \centering
\begin{tabular}{l|l|l|l|l|l|l|l} $[[n,k,d]]$ & \textsc{L}\footnote{for $[[5,1,3]]$ this parameter is \textsc{NB}.} & \textsc{R}\footnote{for $[[5,1,3]]$ this parameter is \textsc{NC}.} & \textsc{CX/Rec}\footnote{for $[[5,1,3]]$ this parameter is the number of CNOT gates in a $T_3$ rectangle} & $p_1(p_{\rm mem}=0,p_0=10^{-4})$ & $p_1(p_{\rm mem}=p_0=10^{-4})$ & $p_{\rm th}(p_{\rm mem}=0)$ & $p_{\rm th}(p_{\rm mem}=p_0)$ \\ \hline\hline
$[[5,1,3]]$ & 2 & 2 & 2,160 & -- & -- & \dat{3.9}{0.7}{5} & \dat{2.5}{0.4}{5} \\ $[[5,1,3]]$ & 3 & 3 & 5,117 & -- & -- & \dat{9.2}{0.5}{5} & \dat{3.7}{0.3}{5} \\ $[[5,1,3]]$ & 5 & 5 & 14,775 & -- & -- & \dat{9.2}{0.5}{5} & \dat{3.3}{0.6}{5} \\ $[[5,1,3]]$ & 10 & 3 & 18,536 & -- & -- & \dat{8.8}{0.5}{5} & \dat{4.3}{0.3}{5} \\ $[[5,1,3]]$ & 10 & 10 & 60,760 & -- & -- & \dat{8}{3}{5} & \dat{3.0}{0.6}{5} \\ \hline
$[[7,1,3]]$ & 2 & 1 & 519 & \dat{5.34}{0.07}{4} & \dat{7.05}{0.08}{4} & \dat{1.85}{0.05}{5} & \dat{1.46}{0.05}{5} \\ $[[7,1,3]]$ & 3 & 1 & 775 & \dat{2.3}{0.2}{5} & \dat{4.5}{0.2}{5} & \dat{3.11}{0.02}{4} & \dat{1.98}{0.01}{4} \\ $[[7,1,3]]$ & 4 & 1 & 1,031 & \dat{1.9}{0.1}{5} & \dat{3.7}{0.2}{5} & \dat{4.97}{0.07}{4} & \dat{2.56}{0.06}{4} \\ $[[7,1,3]]$ & 5 & 1 & 1,287 & \dat{1.8}{0.1}{5} & \dat{4.1}{0.2}{5} & \dat{5.3}{0.1}{4} & \dat{2.58}{0.06}{4} \\ \hline
$[[9,1,3]]$ & 1 & 1 & 69 & -- & \dat{4.90}{0.09}{5} & \dat{2.6}{0.1}{4} & \dat{2.06}{0.02}{4} \\ \hline
$[[13,1,3]]$ & 3 & 1 & 1,501 & -- & -- & \dat{1.59}{0.04}{4} & \dat{0.69}{0.03}{4} \\ $[[13,1,3]]$ & 4 & 1 & 1,997 & -- & -- & \dat{3.81}{0.07}{4} & \dat{1.95}{0.04}{4} \\ $[[13,1,3]]$ & 5 & 1 & 2,493 & -- & \dat{4.3}{0.2}{5} & \dat{4.9}{0.2}{4} & \dat{2.30}{0.08}{4} \\ $[[13,1,3]]$ & 10 & 1 & 4,973 & \dat{1.8}{0.1}{5} & \dat{3.9}{0.2}{5} & \dat{5.1}{0.2}{4} & \dat{2.54}{0.07}{4} \\ $[[13,1,3]]$ & 15 & 1 & 7,453 & \dat{1.9}{0.1}{5} & \dat{3.9}{0.2}{5} & \dat{4.9}{0.1}{4} & \dat{2.63}{0.07}{4} \\ \hline
$[[15,1,3]]$ & 3 & 1 & 2,127 & \dat{1.3}{0.2}{4} & \dat{4.5}{0.2}{4} & \dat{0.86}{0.03}{4} & \dat{0.33}{0.05}{4} \\ $[[15,1,3]]$ & 4 & 1 & 2,831 & \dat{4.9}{0.7}{5} & \dat{1.0}{0.1}{4} & \dat{1.5}{0.6}{4} & \dat{1.0}{0.2}{4} \\ $[[15,1,3]]$ & 5 & 1 & 3,535 & \dat{5.8}{0.8}{5} & \dat{1.0}{0.1}{4} & \dat{1.8}{0.2}{4} & \dat{1.0}{0.2}{4} \\ \hline
$[[23,1,7]]$ & 10 & 1 & 16,023 & \dat{1.1}{0.3}{7} & \dat{1.2}{0.6}{7} & \dat{1.14}{0.05}{3} & \dat{1.09}{0.01}{3} \\ $[[23,1,7]]$ & 20 & 1 & 32,023 & \dat{1.2}{0.4}{7} & \dat{9}{4}{8} & \dat{2.33}{0.02}{3} & \dat{1.97}{0.02}{3} \\ $[[23,1,7]]$ & 30 & 1 & 48,023 & -- & -- & \dat{2.98}{0.04}{3} & \dat{2.25}{0.03}{3} \\ $[[23,1,7]]$ & 40 & 1 & 64,023 & -- & -- & \dat{3.33}{0.02}{3} & \dat{2.19}{0.04}{3} \\ $[[23,1,7]]$ & 10 & 2 & 28,023 & \dat{4}{1}{8} & \dat{3}{2}{8} & \dat{5.76}{0.09}{4} & \dat{5.48}{0.09}{4} \\ $[[23,1,7]]$ & 20 & 2 & 56,023 & \dat{5}{1}{8} & $\approx <4\times 10^{-8}$ \footnote{one failure in $5\times 10^{7}$ samples} & \dat{1.23}{0.01}{3} & \dat{1.15}{0.01}{3} \\ $[[23,1,7]]$ & 30 & 2 & 84,023 & -- & -- & \dat{1.628}{0.006}{3} & \dat{1.487}{0.003}{3} \\ $[[23,1,7]]$ & 40 & 2 & 112,023 & -- & -- & \dat{1.95}{0.01}{3} & \dat{1.77}{0.02}{3} \\ \end{tabular} \end{minipage} \caption{Complete tabulation of code survey data, part 1} \label{table:complete1} \end{table} \end{landscape}
\begin{landscape} \begin{table}[htp] \centering \begin{minipage}{\textwidth} \centering
\begin{tabular}{l|l|l|l|l|l|l|l} $[[n,k,d]]$ & \textsc{L} & \textsc{R}\footnote{for $[[49,1,9]]$ this parameter is the number of preparation attempts for a 7-qubit encoded ancilla used in error detection} & \textsc{CX/Rec} & $p_1(p_{\rm mem}=0,p_0=10^{-4})$ & $p_1(p_{\rm mem}=p_0=10^{-4})$ & $p_{\rm th}(p_{\rm mem}=0)$ & $p_{\rm th}(p_{\rm mem}=p_0)$ \\
\hline\hline
$[[23,1,7]]$ & 10 & 3 & 40,023 & \dat{4}{2}{8} & \dat{3}{1}{8} & \dat{3.72}{0.05}{4} & \dat{3.45}{0.05}{4} \\ $[[23,1,7]]$ & 20 & 3 & 80,023 & -- & -- & \dat{8.03}{0.05}{4} & \dat{7.67}{0.05}{4} \\ $[[23,1,7]]$ & 30 & 3 & 120,023 & -- & -- & \dat{1.095}{0.003}{3} & \dat{1.036}{0.008}{3} \\ $[[23,1,7]]$ & 40 & 3 & 160,023 & -- & -- & \dat{1.366}{0.007}{3} & \dat{1.280}{0.009}{3} \\ \hline
$[[25,1,5]]$ & 4 & 1 & 1,465 & -- & \dat{1.2}{0.7}{6} & \dat{8.6}{0.2}{4} & \dat{7.44}{0.05}{4} \\ $[[25,1,5]]$ & 5 & 1 & 1,825 & -- & \dat{1.0}{0.1}{6} & \dat{1.13}{0.02}{3} & \dat{9.74}{0.07}{4} \\ $[[25,1,5]]$ & 6 & 1 & 2,185 & -- & \dat{1.08}{0.08}{6} & \dat{1.16}{0.02}{3} & \dat{1.034}{0.008}{3} \\ $[[25,1,5]]$ & 7 & 1 & 2,545 & -- & \dat{1.1}{0.2}{6} & \dat{1.17}{0.04}{3} & \dat{1.01}{0.04}{3} \\ \hline
$[[41,1,5]]$ & 5 & 1 & 11,321 & \dat{7.3}{0.8}{6} & \dat{2.39}{0.05}{4} & \dat{1.86}{0.02}{4} & \dat{7.9}{0.1}{5} \\ $[[41,1,5]]$ & 10 & 1 & 22,601 & $[3\times 10^{-7}]$\footnote{The values in square brackets are extrapolated from a linear least-squares fit to the logarithm of $p_1(p_0)$} & $[7\times 10^{-7}]$ & \dat{7.44}{0.03}{4} & \dat{3.44}{0.01}{4} \\ $[[41,1,5]]$ & 15 & 1 & 33,881 & -- & -- & \dat{1.224}{0.003}{3} & \dat{5.55}{0.02}{4} \\ $[[41,1,5]]$ & 20 & 1 & 45,161 & -- & -- & \dat{1.577}{0.004}{3} & \dat{7.61}{0.02}{4} \\ $[[41,1,5]]$ & 30 & 1 & 67,721 & -- & -- & \dat{2.06}{0.01}{3} & \dat{1.008}{0.008}{3} \\ \hline
$[[47,1,11]]$ & 10 & 1 & 52,527 & -- & -- & \dat{3.25}{0.04}{4} & \dat{2.15}{0.04}{4} \\ $[[47,1,11]]$ & 20 & 1 & 105,007 & -- & -- & \dat{6.89}{0.05}{4} & \dat{4.79}{0.03}{4} \\ $[[47,1,11]]$ & 30 & 1 & 157,487 & \dat{1.6}{0.9}{7} & -- & \dat{9.51}{0.04}{4} & \dat{6.45}{0.03}{4} \\ \hline
$[[49,1,9]]$ & 5 & 2 & 61,549 & -- & -- & \dat{1.02}{0.02}{4} & \dat{5.4}{0.1}{5} \\ $[[49,1,9]]$ & 10 & 2 & 123,049 & -- & -- & \dat{3.63}{0.08}{4} & \dat{2.23}{0.04}{4} \\ $[[49,1,9]]$ & 15 & 2 & 184,549 & -- & -- & \dat{4.0}{0.02}{4} & \dat{3.20}{0.08}{4} \\ $[[49,1,9]]$ & 20 & 2 & 246,049 & \dat{4}{1}{6} & -- & \dat{4.2}{0.3}{4} & -- \\ \hline
$[[49,1,7]]$ & 4 & 1 & 2,961 & -- & \dat{3.4}{0.2}{6} & \dat{4.73}{0.09}{4} & \dat{3.20}{0.02}{4} \\ $[[49,1,7]]$ & 6 & 1 & 4,417 & -- & \dat{2.8}{0.2}{7} & \dat{1.18}{0.01}{3} & \dat{8.7}{0.2}{4} \\ $[[49,1,7]]$ & 8 & 1 & 5,873 & -- & \dat{3.3}{0.6}{7} & \dat{1.41}{0.02}{3} & \dat{1.169}{0.005}{3} \\ $[[49,1,7]]$ & 9 & 1 & 6,601 & -- & \dat{2.2}{0.7}{7} & \dat{1.48}{0.02}{3} & \dat{1.224}{0.005}{3} \\ $[[49,1,7]]$ & 10 & 1 & 7,329 & -- & \dat{4.0}{0.9}{7} & \dat{1.42}{0.03}{3} & \dat{1.235}{0.005}{3} \\ $[[49,1,7]]$ & 11 & 1 & 8,057 & -- & \dat{2.5}{0.2}{7} & \dat{1.46}{0.03}{3} & \dat{1.241}{0.006}{3} \\ $[[49,1,7]]$ & 12 & 1 & 8,785 & -- & \dat{3}{2}{7} & \dat{1.46}{0.02}{3} & \dat{1.242}{0.006}{3} \\ \hline
$[[60,4,10]]$ & 10 & 1 & 86,460 & -- & -- & \dat{1.129}{0.004}{4} & -- \\ $[[60,4,10]]$ & 20 & 1 & 172,860 & -- & -- & \dat{3.91}{0.02}{4} & \dat{2.20}{0.04}{4} \end{tabular} \end{minipage} \caption{Complete tabulation of code survey data, part 2} \label{table:complete2} \end{table} \end{landscape}
\begin{landscape} \begin{table}[htp] \centering \begin{minipage}{\textwidth} \centering
\begin{tabular}{l|l|l|l|l|l|l|l} $[[n,k,d]]$ & \textsc{L} & \textsc{R} & \textsc{CX/Rec} & $p_1(p_{\rm mem}=0,p_0=10^{-4})$ & $p_1(p_{\rm mem}=p_0=10^{-4})$ & $p_{\rm th}(p_{\rm mem}=0)$ & $p_{\rm th}(p_{\rm mem}=p_0)$ \\ \hline\hline
$[[81,1,9]]$ & 4 & 1 & 4,977 & -- & \dat{4.4}{0.7}{5} & \dat{2.1}{0.2}{4} & \dat{1.407}{0.005}{4} \\ $[[81,1,9]]$ & 6 & 1 & 7,425 & -- & $[1\times 10^{-6}]$\footnote{The values in square brackets are extrapolated from a linear least-squares fit to the logarithm of $p_1(p_0)$} & \dat{7.1}{0.1}{4} & \dat{4.47}{0.03}{4} \\ $[[81,1,9]]$ & 10 & 1 & 12,321 & -- & $[7\times 10^{-7}]$ & \dat{1.25}{0.02}{3} & \dat{9.57}{0.03}{4} \\ $[[81,1,9]]$ & 11 & 1 & 13,545 & -- & -- & \dat{1.32}{0.02}{3} & \dat{1.029}{0.004}{3} \\ $[[81,1,9]]$ & 12 & 1 & 14,769 & -- & -- & \dat{1.29}{0.03}{3} & \dat{1.069}{0.006}{3} \\ $[[81,1,9]]$ & 18 & 1 & 22,113 & -- & -- & \dat{1.30}{0.03}{3} & \dat{1.113}{0.006}{3} \\ $[[81,1,9]]$ & 19 & 1 & 23,337 & -- & -- & \dat{1.34}{0.03}{3} & \dat{1.098}{0.006}{3} \\ $[[81,1,9]]$ & 20 & 1 & 24,561 & -- & -- & \dat{1.34}{0.02}{3} & \dat{1.112}{0.006}{3} \\ \hline
$[[85,1,7]]$ & 5 & 1 & 30,405 & -- & -- & \dat{5.7}{0.1}{5} & \dat{2.03}{0.07}{5} \\ $[[85,1,7]]$ & 10 & 1 & 60,725 & -- & -- & \dat{2.48}{0.01}{4} & \dat{1.03}{0.04}{4} \\ $[[85,1,7]]$ & 15 & 1 & 91,045 & -- & -- & \dat{4.18}{0.05}{4} & \dat{1.76}{0.02}{4} \\ $[[85,1,7]]$ & 20 & 1 & 121,365 & -- & $[2\times 10^{-7}]$ & \dat{5.59}{0.04}{4} & \dat{2.32}{0.02}{4} \end{tabular} \end{minipage} \caption{Complete tabulation of code survey data, part 3} \label{table:complete3} \end{table} \end{landscape}
\begin{landscape} \begin{table}[htp] \centering \begin{minipage}{\textwidth} \centering
\begin{tabular}{l|l|l|l} $[[n,k,d=\ell]]$ & \textsc{CX/Rec} & $p_1(p_{\rm mem}=p_0=10^{-4})$ & $p_{\rm th}(p_{\rm mem}=p_0)$ \\ \hline\hline $[[41,1,5]]$ & 1,481 & \dat{1.7}{0.1}{4} & \dat{6.8}{0.6}{5} \\ $[[85,1,7]]$ & 4,453 & \dat{5}{2}{5} & \dat{2.3}{0.2}{4} \\ $[[145,1,9]]$ & 9,937 & \dat{2}{1}{5} & \dat{4.5}{0.2}{4} \\ $[[221,1,11]]$ & 18,701 & $[8\times 10^{-6}]$\footnote{The values in square brackets are extrapolated from a linear least-squares fit to the logarithm of $p_1(p_0)$} & \dat{6.6}{0.2}{4} \\ $[[313,1,13]]$ & 31,513 & $[8\times 10^{-6}]$ & \dat{9.0}{0.4}{4} \end{tabular} \end{minipage} \caption{Surface code data using Shor-EC and a transversal CNOT as in \cite{dennis+:top}, taking $\ell$ syndromes for an $\ell\times\ell$ code EC.} \label{table:surfacedat} \end{table} \end{landscape}
\begin{landscape} \begin{table}[htp] \centering \begin{minipage}{\textwidth} \centering
\begin{tabular}{l|l|l} $[[n,k,d]]$ & family & $p_{\rm th}(\textrm{perfect ancilla})$ \\ \hline\hline $[[5,1,3]]$ & & \dat{2.0}{0.1}{4} \\ $[[7,1,3]]$ & doubly-even dual-containing & \dat{9.1}{0.2}{4} \\ $[[9,1,3]]$ & Bacon-Shor & \dat{6.0}{0.9}{4} \\ $[[13,1,3]]$ & surface & \dat{8.8}{0.1}{4} \\ $[[21,3,5]]$ & polynomial & $<10^{-5}$ \\ $[[23,1,7]]$ & dual-containing & \dat{5.34}{0.04}{3} \\ $[[25,1,5]]$ & Bacon-Shor & \dat{1.88}{0.04}{3} \\ $[[41,1,5]]$ & surface & \dat{3.8}{0.3}{3} \\ $[[47,1,11]$ & doubly-even dual-containing & \dat{7.67}{0.03}{3} \\ $[[49,1,7]]$ & Bacon-Shor & \dat{2.56}{0.05}{3} \\ $[[49,1,9]]$ & doubly-even dual-containing & \dat{4.8}{0.2}{3} \\ $[[60,4,10]]$ & polynomial & \dat{1.88}{0.04}{3} \\ $[[81,1,9]]$ & Bacon-Shor & \dat{2.88}{0.04}{3} \\ $[[85,1,7]]$ & surface & \dat{7.5}{0.3}{3} \\ $[[121,1,11]]$ & Bacon-Shor & \dat{2.83}{0.07}{3} \\ $[[145,1,9]]$ & surface & \dat{1.01}{0.02}{2} \\ $[[169,1,13]]$ & Bacon-Shor & \dat{2.97}{0.09}{3} \end{tabular} \end{minipage} \caption{Level-1 pseudo-thresholds for rectangles using Steane-EC with perfect (noiseless) ancilla, $n<200$.} \label{table:perfectdat} \end{table} \end{landscape}
\input{innercodes_refs.bbl}
\end{document} |
\begin{document}
\keywords{Axiom of Determinacy, forcing, Descriptive Set Theory} \subjclass{03E60,03E40,03E15}
\title[Preservation of AD via forcings]{Preservation of AD via forcings}
\author[D.\ Ikegami]{Daisuke Ikegami} \address[D.\ Ikegami]{College of Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-Ward, Saitama City, 337-8570, Saitama JAPAN}
\email[D.\ Ikegami]{\href{mailto:ikegami@shibaura-it.ac.jp}{ikegami@shibaura-it.ac.jp}}
\author[N.\ Trang]{Nam Trang} \address[N.\ Trang]{Department of Mathematics, University of North Texas, 1155 Union Circle 311430, Denton, TX 76203-5017, USA}
\email[N.\ Trang]{\href{mailto:nam.trang@unt.edu}{nam.trang@unt.edu}}
\thanks{The authors would like to thank W. Hugh Woodin for generously sharing his results and insight on the topic of this paper. They are also grateful to William Chan and Steve Jackson for their work in~\cite{MR4242147} that inspired the work in Section~\ref{sec:the-reals} of this paper. The first author would like to thank the Japan Society for the Promotion of Science (JSPS) for its generous support through the grant with JSPS KAKENHI Grant Number 19K03604. He is also grateful to the Sumitomo Foundation for its generous support through Grant for Basic Science Research. The second author is grateful to the National Science Foundation (NSF) for its generous support through CAREER Grant DMS-1945592.}
\begin{abstract} We show that assuming $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$, any poset which increases $\Theta$ does not preserve the truth of $\mathsf{AD}$. We also show that in $\mathsf{ZF} + \mathsf{AD}$, any non-trivial poset on $\mathbb{R}$ does not preserve the truth of $\mathsf{AD}$. This answers the question of Chan and Jackson~\cite[Question~5.7]{MR4242147}. Furthermore, we show that under the assumptions $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$'' $+ \lq\lq \Theta \text{ is regular}$'', there is a poset on $\Theta$ which adds a new subset of $\Theta$ while preserving the truth of $\mathsf{AD}$. This answers the question of Cunningham~\cite[Section~5]{Cunningham}.
\end{abstract}
\maketitle
\section{Introduction} In this paper, we discuss the relationship between forcing and the Axiom of Determinacy ($\mathsf{AD}$), especially on the question what kind of forcings preserve the truth of $\mathsf{AD}$.
Forcing was introduced by Cohen to prove the independence of the Continuum Hypothesis from Zermelo-Fraenkel set theory with the Axiom of Choice ($\mathsf{ZFC}$). Using forcing, he also proved that the Axiom of Choice is independent of Zermelo-Fraenkel set theory ($\mathsf{ZF}$). Since then, forcing has been a basic tool of constructing models of set theory, and it has been used to obtain various results on independence or unprovability of some mathematical statements from set theory as well as to analyze various kinds of models of set theory.
The Axiom of Determinacy ($\mathsf{AD}$) was introduced by Mycielski and Steinhaus to consider a situation where every set of reals has good properties that simple sets (such as Borel sets and analytic sets) enjoy, and examples of those good properties are Lebesgue measurability and the Baire property. While $\mathsf{AD}$ contradicts the Axiom of Choice in $\mathsf{ZF}$, it has many beautiful consequences on sets of reals. Furthermore, it has been shown that there are deep connections between models of $\mathsf{ZF} + \mathsf{AD}$ (or models of $\mathsf{ZF} + \mathsf{AD}^+$) and models of $\mathsf{ZFC}$ with Woodin cardinals, and $\mathsf{AD}$ has been playing an important role not only in descriptive set theory to analyze the properties of sets of reals, but also in the theory of large cardinals and inner model theory.
Let us mention how we got interested in the relationship between forcing and $\mathsf{AD}$, especially on the question what kind of forcings preserve the truth of $\mathsf{AD}$. By the result of Kunen, there is no non-trivial elementary embedding $j\colon V \to V$ such that $(V, \in , j)$ is a model of $\mathsf{ZFC}$.\footnote{Here $V$ is the class of all sets, $j$ in the structure $(V, \in , j)$ is considered as the interpretation of a binary predicate on the universe, and the structure $(V, \in , j)$ satisfies Comprehension and Replacement for first-order formulas with the binary predicates for $\in$ and $j$.} Furthermore, Hamkins, Kirmayer, and Perlmutter proved that for any set generic $G$ over $V$, there is no non-trivial elementary embedding $j \colon V \to V[G]$ such that $(V [G], \in , j)$ is a model of $\mathsf{ZFC}$.
One can then ask questions such as what if the structure $(V ,\in , j)$ or $(V[G] , \in , j)$ is a model of $\mathsf{ZF}$ or $\mathsf{ZF} + \mathsf{AD}$ instead of $\mathsf{ZFC}$. Using the method of symmetric models, Woodin proved that there are a set generic $G$ over $V$ and a non-trivial elementary embedding $j \colon V \to V[G]$ such that $(V[G], \in , j)$ is a model of $\mathsf{ZF} + \mathsf{AD}$. However, in his example, $j \upharpoonright \mathrm{Ord}$ is the identity map, so there is no critical point of $j$.
As far as we know, it is still open whether there are a set generic $G$ over $V$ and an elementary embedding $j \colon V \to V[G]$ such that $(V[G], \in , j)$ is a model of $\mathsf{ZF} + \mathsf{AD}$ and $j \upharpoonright \mathrm{Ord}$ is not the identity map. We are especially interested in the case when the critical point of $j$ is $\omega_1$ in $V$ because if the critical point of $j$ is $\omega_1$ in $V$, then the forcing to obtain $V[G]$ must add new reals to $V$ and $\mathsf{AD}$ has influence on reals and sets of reals. To obtain such a $j$, one needs to have a poset $\mathbb{P}$ to produce such a model $V[G]$, and the poset $\mathbb{P}$ must add new reals while preserve the truth of $\mathsf{AD}$ from $V$ to $V[G]$. Hence we have a test question: Is there any poset which adds a new real while preserving the truth of $\mathsf{AD}$? This is how we got interested in the relationship between forcing and $\mathsf{AD}$.
We still do not know if there is any poset which adds a new real while preserving the truth of $\mathsf{AD}$. Considering this question, we have observed that many forcings adding a new real do not preserve the truth of $\mathsf{AD}$. A typical example is Cohen forcing. It is well-known that if $G$ is $V$-generic for Cohen forcing, then in $V[G]$, the set of reals in $V$ does not have the Baire property. In particular, $\mathsf{AD}$ must fail in $V[G]$. On the other hand, there are posets which add a new set while preserving the truth of $\mathsf{AD}$. By the result of Woodin~\cite[Section~3]{MR736611}, if we assume $\mathsf{ZF} + \mathsf{AD} + \lq\lq V = \mathrm{L} (\mathbb{R})$'' and let $\kappa$ be a sufficiently big cardinal and $\mathbb{P}$ be the poset for adding a Cohen subset of $\kappa$ in $\mathrm{HOD}$, the class of hereditarily ordinal definable sets, then the poset $\mathbb{P}$ adds a new set while preserving the truth of $\mathsf{AD}$. Actually, the poset $\mathbb{P}$ does not add any set of reals to $V$ in this case.
We have been wondering what kind of forcings preserve the truth of $\mathsf{AD}$. Our intuition was such a poset $\mathbb{P}$ would not be able to change the structure of sets of reals drastically. The intuition was partially justified using the ordinal $\Theta$, the supremum of ordinals which are surjective images of $\mathbb{R}$, by the following theorem: \newtheorem*{sec3thm}{Theorem~\ref{thm:increasing-Theta-destruction}} \begin{sec3thm} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that a poset $\mathbb{P}$ increases $\Theta$, i.e., $\Theta^{V} < \Theta^{V[G]}$ for any $\mathbb{P}$-generic filter $G$ over $V$. Then $\mathsf{AD}$ fails in $V [G]$ for any $\mathbb{P}$-generic filter $G$ over $V$. \end{sec3thm}
However, the assumption $\lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$ is essential in Theorem~\ref{thm:increasing-Theta-destruction}: \newtheorem*{sec3ex}{Theorem~\ref{thm:increasing-Theta-example}} \begin{sec3ex} It is consistent relative to $\mathsf{ZF}+\mathsf{AD}_{\mathbb{R}}$ that $\mathsf{ZF}+\mathsf{AD}$ holds and there is a poset $\mathbb{P}$ increasing $\Theta$ while preserving $\mathsf{AD}$, i.e., for any $\mathbb{P}$-generic filter $G$ over $V$, we have $\Theta^{V} < \Theta^{V[G]}$ and that $\mathsf{AD}$ holds in $V[G]$.\footnote{For an expert on determinacy, the assumption $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}}$ is an overkill. The proof of Theorem~\ref{thm:increasing-Theta-example} shows that the assumption $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq \Theta > \Theta_0$'' is enough.} \end{sec3ex} In particular, there is a poset which adds a new set of reals (but does not add a new real) while preserving the truth of $\mathsf{AD}$.
After we announced Theorem~\ref{thm:increasing-Theta-destruction} and Theorem~\ref{thm:increasing-Theta-example}, Chan and Jackson~\cite{MR4242147} worked on the question what kind of forcings do not preserve the truth of AD. They proved that assuming $\mathsf{ZF} + \mathsf{AD}$, if a non-trivial poset $\mathbb{P}$ is a wellorderable forcing of cardinality less than $\Theta$, then $\mathbb{P}$ does not preserve the truth of $\mathsf{AD}$ (\cite[Theorem~3.2]{MR4242147}). They also proved that assuming $\mathsf{ZF} + \mathsf{AD} + \lq\lq \Theta \text{ is regular}$'', if a non-trivial poset $\mathbb{P}$ is a surjective image of $\mathbb{R}$, then $\mathbb{P}$ does not preserve the truth of $\mathsf{AD}$ (\cite[Theorem~5.5]{MR4242147}). Then they asked whether $\mathsf{ZF} + \mathsf{AD}$ only (i.e., without assuming the regularity of $\Theta$) implies that if a non-trivial poset $\mathbb{P}$ is a surjective image of $\mathbb{R}$, then $\mathbb{P}$ does not preserve the truth of $\mathsf{AD}$ (\cite[Question~5.7]{MR4242147}). We give a positive answer to their question: \newtheorem*{sec4thm}{Theorem~\ref{thm:the-reals}} \begin{sec4thm} Assume $\mathsf{ZF}+\mathsf{AD}$. Let $\mathbb{P}$ be any non-trivial poset which is a surjective image of $\mathbb{R}$ and $G$ be any $\mathbb{P}$-generic filter over $V$. Then $\mathsf{AD}$ fails in $V [G]$. \end{sec4thm}
We now turn to positive results on the question what kind of forcings preserve the truth of $\mathsf{AD}$. As was mentioned in a previous paragraph, By the result of Woodin~\cite[Section~3]{MR736611}, if we assume $\mathsf{ZF} + \mathsf{AD} + \lq\lq V = \mathrm{L} (\mathbb{R})$'' and let $\kappa$ be a sufficiently big cardinal and $\mathbb{P}$ be the poset for adding a Cohen subset of $\kappa$ in $\mathrm{HOD}$, the class of hereditarily ordinal definable sets, then the poset $\mathbb{P}$ adds a new set while preserving the truth of $\mathsf{AD}$. A natural question would be how small one can take $\kappa$ for this result. Cunningham~\cite{Cunningham} worked on this question. He proved that $\kappa$ can be taken as any regular cardinal larger than $\Theta^{+}$ (\cite[Subsection~4.1]{Cunningham}). Then he asked whether $\kappa$ can be taken as $\Theta$ (\cite[Section~5]{Cunningham}). We answer his question positively. In fact, we prove a more general theorem as follows: \newtheorem*{sec5thm}{Theorem~\ref{thm:subset-of-Theta-positive}} \begin{sec5thm} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that $\Theta$ is regular. Then there is a poset $\mathbb{P}$ on $\Theta$ which adds a subset of $\Theta$ while preserving $\mathsf{AD}$, i.e., for any $\mathbb{P}$-generic filter $G$ over $V$, there is a subset of $\Theta^{V}$ which belongs to $V[G] \setminus V$ and $\mathsf{AD}$ holds in $V[G]$.\footnote{The proof of Theorem~\ref{thm:subset-of-Theta-positive} shows that in both Case~\ref{case:ADRfail} and Case~\ref{case:ADR}, the poset $\mathbb{P}$ does not add any new set of reals to $V$. In particular, the poset $\mathbb{P}$ preserves the truth of $\mathsf{AD}^+$ as well.} \end{sec5thm}
Notice that $\mathsf{ZF} + \mathsf{AD} + \lq\lq V = \mathrm{L} (\mathbb{R})$'' implies the assumptions of Theorem~\ref{thm:subset-of-Theta-positive} including the regularity of $\Theta$. Also, in case of $\lq\lq V = \mathrm{L} (\mathbb{R})$'', the poset $\mathbb{P}$ is the one for adding a Cohen subset of $\Theta$ in $\mathrm{HOD}$ as in Case~\ref{case:ADRfail} in the proof of Theorem~\ref{thm:subset-of-Theta-positive}. Therefore, the arguments for Theorem~\ref{thm:subset-of-Theta-positive} answer the question of Cunningham~\cite[Section~5]{Cunningham}.
We also note that Theorem~\ref{thm:subset-of-Theta-positive} is optimal in the following two senses: In one sense, the size of the poset $\mathbb{P}$ cannot be smaller than $\Theta$. As was mentioned in a previous paragraph, by the result of Chan and Jackson~\cite[Theorem~3.2]{MR4242147}, any wellorderable forcing of cardinality less than $\Theta$ cannot preserve the truth of $\mathsf{AD}$. Also, by Theorem~\ref{thm:the-reals} (or by the result of Chan and Jackson~\cite[Theorem~5.5]{MR4242147} in case $\Theta$ is regular), any poset on $\mathbb{R}$ (or a surjective image of $\mathbb{R}$) cannot preserve the truth of $\mathsf{AD}$. In the other sense, unless the poset $\mathbb{P}$ adds a new real, the poset $\mathbb{P}$ cannot add a new bounded subset of $\Theta$ while preserving the truth of $\mathsf{AD}$. This is because if $\mathbb{P}$ does not add any real and both $V$ and $V[G]$ are models of $\mathsf{AD}$, then by the Moschovakis Coding Lemma, $V$ and $V[G]$ have the same bounded subsets of $\Theta$, leading to the situation that the poset $\mathbb{P}$ cannot add a bounded subset of $\Theta$.
After looking at Theorem~\ref{thm:subset-of-Theta-positive}, it is natural to ask whether the assumption $\lq\lq \Theta \text{ is regular}$'' is essential there. We do not know the answer to this question. However, in case $\Theta$ is singular, we have $\mathsf{AD}_{\mathbb{R}}$ under the assumptions of Theorem~\ref{thm:subset-of-Theta-positive}. In case $\mathsf{AD}_{\mathbb{R}}$ holds, which is Case~\ref{case:ADR} in the proof of Theorem~\ref{thm:subset-of-Theta-positive}, the poset $\mathbb{P}$ in Theorem~\ref{thm:subset-of-Theta-positive} is for adding a Cohen subset of $\Theta$ in $\mathrm{HOD}$. We show that this particular poset cannot preserve the truth of $\mathsf{AD}$ if $\Theta$ is singular: \newtheorem*{sec5ex}{Theorem~\ref{thm:subset-of-Theta-negative}} \begin{sec5ex} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that $\Theta$ is singular and let $\mathbb{P}$ be $\mathrm{Add} (\Theta, 1)$ in HOD, where $\mathrm{Add} (\Theta , 1) = \{ p \mid p \colon \gamma \to 2 \text{ for some $\gamma < \Theta$} \}$. Then $\mathsf{AD}$ fails in $V [G]$ for any $\mathbb{P}$-generic filter $G$ over $V$. \end{sec5ex}
\section{Basic definitions, theorems, and lemmas}\label{sec:basicnotions}
In this section, we introduce basic definitions, theorems, and lemmas we will use in later sections of the paper. We assume that readers are familiar with the basics of forcing and descriptive set theory. For basic definitions not given in this paper, see Jech~\cite{Jech} and Moschovakis~\cite{new_Moschovakis}. When we say \lq \lq reals", we mean elements of the Baire space $\omega^{\omega}$ or of the Cantor space $2^{\omega}$.
We start with some basic definitions which will be used throughout the paper: \begin{defn}
${}$
\begin{enumerate} \item The ordinal {\it $\Theta$} is the supremum of ordinals which are surjective images of $\mathbb{R}$.
\item A set $x$ is {\it OD from sets $y_1 , \ldots , y_n$} if $x$ is definable by a first-order formula with an ordinal and $y_1 , \ldots , y_n$ as parameters.
\item Let $Y$ be a set. We say a set $x$ is {\it hereditarily $\mathrm{OD}_Y$} if any element of the transitive closure of $\{ x \}$ is OD from some elements of $Y$.
\item For a set $Y$, we write {$\mathrm{HOD}_Y$} for the collection of sets which are hereditarily $\mathrm{OD}_Y$. When $Y$ is the empty set, we simply write $\mathrm{HOD}$ for $\mathrm{HOD}_Y$. \end{enumerate} \end{defn}
\begin{defn} Let $A$ and $B$ be sets of reals (or subsets of the Baire space $\omega^{\omega}$). We say $A$ is {\it Wadge reducible to $B$} if there is a continuous function $f \colon \omega^{\omega} \to \omega^{\omega}$ such that $A = f^{-1} (B)$. When $A$ is Wadge reducible to $B$, we write $A \le_{\text{W}} B$. The order $\le_{\text{W}}$ is called the {\it Wadge order on sets of reals}. \end{defn}
\begin{lem}[Wadge's Lemma] Assume $\mathsf{ZF} + \mathsf{AD}$. Then for any sets $A , B$ of reals, we have either $A \le_{\text{W}} B$ or $B \le_{\text{W}} \omega^{\omega} \setminus A$. \end{lem}
\begin{proof} See e.g., \cite[Lemma~2.1]{Wadge_van_Wesep}. \end{proof}
The following theorems will be used in Section~\ref{sec:increasing-Theta}.
\begin{thm}[Woodin]\label{thm:HODLA} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Then the model $\mathrm{HOD}$ is of the form $\mathrm{L} [Z]$ for some subset $Z$ of $\Theta$, and there are a poset $\mathbb{Q}$ in $\mathrm{HOD}$ and a $\mathbb{Q}$-generic filter $H$ over $\mathrm{HOD}$ such that $\mathrm{HOD} \subseteq V \subseteq \mathrm{HOD}[H]$. \end{thm}
\begin{proof} See e.g., \cite[Theorem~3.1.9]{NamThesis}. \end{proof}
\begin{thm}[Moschovakis]\label{thm:Moschovakis} Assume $\mathsf{ZF}+\mathsf{AD}$. Then $\Theta$ is a limit of measurable cardinals. \end{thm}
\begin{proof} For a proof without assuming $\mathsf{DC}_{\mathbb{R}}$, one could first prove that $\Theta$ is a limit of strong partition cardinals under $\mathsf{ZF} + \mathsf{AD}$ as in \cite{ADplus} and then verify that every strong partition cardinal is measurable under $\mathsf{ZF}$ as in \cite[28.10~Theorem]{MR1994835}. \end{proof}
\begin{thm}[Solovay]\label{thm:Solovay} Assume $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}}$. Then for any set $A$ of reals, there is a set $B$ of reals which is not $\mathrm{OD}$ from $A$ and any real. \end{thm}
\begin{proof} See \cite[Lemma~2.2]{Solovay_AD_R}. \end{proof}
The following theorem will be useed in Section~\ref{sec:the-reals}: \begin{thm}[Chan and Jackson]\label{thm:chan-Jackson} Assume $\mathsf{ZF}+\mathsf{AD}$ and $\Theta$ is regular. Then for any non-trivial poset $\mathbb{P}$ on $\mathbb{R}$ and any $\mathbb{P}$-generic filter $G$ over $V$, the axiom $\mathsf{AD}$ fails in $V [G]$. \end{thm}
\begin{proof} See \cite[Theorem~5.5]{MR4242147}. \end{proof}
The following theorems will be used in Section~\ref{sec:subset-of-Theta}: \begin{thm}[Moschovakis]\label{thm:CodingLemma} Assume $\mathsf{ZF} + \mathsf{AD}$. Then for any non-zero ordinal $\gamma < \Theta$, there is a set $A$ of reals such that there is a surjection from $\mathbb{R}$ to $\wp (\gamma)$ which is $\mathrm{OD}$ from $A$. \end{thm}
\begin{proof} For any surjection $\rho \colon \mathbb{R} \to \gamma$, the arguments in \cite[28.15~Theorem]{MR1994835} give us a surjection from $\mathbb{R}$ to $\wp (\gamma)$ which is OD from $\rho$. If $A$ is a prewellordering on $\mathbb{R}$ of length $\gamma$, then the surjection $\rho \colon \mathbb{R} \to \gamma$ induced from $A$ is clearly OD from $A$. Hence there is a surjection from $\mathbb{R}$ to $\wp (\gamma)$ which is OD from $A$, as desired. \end{proof}
\begin{thm}[Woodin]\label{thm:AD+ADRfail} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose also that $\mathsf{AD}_{\mathbb{R}}$ fails. Then there is a set $T$ of ordinals such that $V = \mathrm{L} (T, \mathbb{R})$. \end{thm}
\begin{proof} By the results of Woodin~\cite{ADplus}, the axiom $\mathsf{AD}^+$ and the failure of $\mathsf{AD}_{\mathbb{R}}$ imply that the set of Suslin cardinals is closed below $\Theta$ while not cofinal in $\Theta$. Hence there is a largest Suslin cardinal in $\Theta$. By the result of Woodin~\cite[Corollary~6]{ADplusreflection}, the assumptions $\mathsf{AD}^+$ and $\lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$ imply that the ultrapower $V^{\mathcal{D}} / \mu$ is well-founded where $\mathcal{D}$ is the set of Turing degrees and $\mu$ is the Martin measure on $\mathcal{D}$. Using the result of Woodin~\cite{ADplus}, it follows that there is a set $T$ of ordinals such that $\wp (\mathbb{R}) \subseteq \mathrm{L} (T, \mathbb{R})$. Since we assume $V = \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$, we have $V = \mathrm{L} (T, \mathbb{R})$, as desired. \end{proof}
\begin{thm}[Woodin]\label{thm:V=LTR} Assume $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V = \mathrm{L} (T, \mathbb{R})$'' for some set $T$ of ordinals. Then \begin{enumerate} \item for some subset $Z$ of $\Theta$, we have $\mathrm{HOD}_{\{ T \}} = \mathrm{L} [T, Z]$, and
\item for any real $x$, we have $\mathrm{HOD}_{\{T, x\}} = \mathrm{HOD}_{ \{ T \}} [ x]$. \end{enumerate} \end{thm}
\begin{proof} For (1), one can argue in the same way as in \cite[Corollary~7.21]{MR4132099}.
For (2), see \cite{ADplus}. \end{proof}
We next introduce Vop\v{e}nka algebras and their variants we will use in this paper: \begin{defn}\label{def:Vop} Let $\gamma$ be a non-zero ordinal and $T$ be a set of ordinals. \begin{enumerate} \item Let $n$ be a natural number with $n \ge 1$ and $\mathcal{O}_n$ be the collection of all nonempty subsets of $(\gamma^{\omega})^n$ which are OD from $T$. Fix a bijection $\pi_n \colon \eta \to \mathcal{O}_n$ which is OD from $T$, where $\eta$ is some ordinal. Let $\mathbb{Q}_n$ be the poset on $\eta$ such that for each $p , q$ in $\mathbb{Q}_n$, we have $p \le q$ if $\pi_n (p) \subseteq \pi_n (q)$. We call $\mathbb{Q}_n$ the{\it Vop\v{e}nka algebra for adding an element of $(\gamma^{\omega})^n$ in $\mathrm{HOD}_{\{ T \}}$}.
\item For all natural numbers $\ell$ and $m$ with $1 \le \ell \le m$, let $i_{\ell , m} \colon \mathbb{Q}_{\ell} \to \mathbb{Q}_m$ be the inclusion map induced from $\pi_{\ell}$ and $\pi_m$, i.e., for all $p \in \mathbb{Q}_{\ell}$, $\pi_m \bigl( i_{\ell , m} (p)\bigr) = \{ x \in (\gamma^{\omega})^m \mid x \upharpoonright \ell \in \pi_{\ell} (p) \}$. Then each $i_{\ell , m}$ is a complete embedding between posets. Let $\bigl(\mathbb{Q}_{\omega} , (i_n \colon \mathbb{Q}_n \to \mathbb{Q}_{\omega} \mid n < \omega) \bigr)$ be the direct limit of the system $( i_{\ell , m} \colon \mathbb{Q}_{\ell} \to \mathbb{Q}_m \mid 1 \le \ell \le m < \omega)$. We call $\mathbb{Q}_{\omega}$ the {\it finite support direct limit of Vop\v{e}nka algebras for adding an element of $\gamma^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$}. \end{enumerate}
\end{defn}
The following lemmas will be useful in Section~\ref{sec:subset-of-Theta}: \begin{lem}\label{lem:Qomega} Assume $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V = \mathrm{L} (T, \mathbb{R})$'' for some set $T$ of ordinals. \begin{enumerate} \item Let $\mathbb{Q}_1$ be the Vop\v{e}nka algebra for adding an element of $2^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$. Then the poset $\mathbb{Q}_1$ is of size at most $\Theta$ and $\mathbb{Q}_1$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$.
\item Let $\mathbb{Q}_{\omega}$ be the finite support limit of the Vop\v{e}nka algebras for adding an element of $2^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$. Then $\mathbb{Q}_{\omega}$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$.
\item (Woodin) There is a $\mathbb{Q}_{\omega}$-generic filter $H$ over $\mathrm{HOD}_{\{ T \}}$ such that $ V = \mathrm{L} (T, \mathbb{R}) \subseteq \mathrm{HOD}_{\{ T \}}[H]$ and the set $\mathbb{R}^V$ is countable in $\mathrm{HOD}_{\{ T \}} [H]$. \end{enumerate} \end{lem}
\begin{proof} For (1), we first show that the poset $\mathbb{Q}_1$ is of size at most $\Theta$ in $\mathrm{HOD}_{\{ T \}}$. Recall from Definition~\ref{def:Vop} that $\mathbb{Q}_1$ is a poset on some ordinal $\eta$ and $\pi_1$ is a bijection from $\eta$ to $\mathcal{O}_1$ which is OD from $T$, where $\mathcal{O}_1$ is the collection of all subsets of $2^{\omega}$ which are OD from $T$. We will argue that the ordinal $\eta$ is at most $\Theta$. For each $\alpha < \Theta$, let $W_{\alpha}$ be the collection of sets of reals in $\mathcal{O}_1$ of Wadge rank $\alpha$. Then we have $\mathcal{O}_1 = \bigcup_{\alpha < \Theta} W_{\alpha}$ and each $W_{\alpha}$ is a surjective image of $\mathbb{R}$. Since the set $\mathcal{O}_1$ is well-ordered, so is each $W_{\alpha}$ and there is a surjection from $\Theta$ to $W_{\alpha}$ which is OD from $T$. Hence there is a surjection from $\Theta \times \Theta$ to $\mathcal{O}_1$ which is OD form $T$, and therefore the set $\mathbb{Q}_1$ is of size at most $\Theta$ in $\mathrm{HOD}_{\{ T \}}$, as desired.
We next show that the poset $\mathbb{Q}_1$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$. To derive a contradiction, suppose that there is an antichain $(p_{\alpha} \mid \alpha < \Theta)$ in $\mathbb{Q}_1$ in $\mathrm{HOD}_{ \{ T \}}$. Then the family $\{ \pi_1 (p_{\alpha}) \mid \alpha < \Theta \}$ is a pairwise disjoint family of nonempty subsets of $2^{\omega}$, which would easily induce a surjection from $\mathbb{R}$ to $\Theta$, contradicting the definition of $\Theta$. Therefore, the poset $\mathbb{Q}_1$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$, as desired.
For (2), we first note that for all natural numbers $n$ with $n\ge 1$, the poset $\mathbb{Q}_n$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$ by the same argument as in (1). Then using the facts that $\Theta$ is regular in $V = \mathrm{L} (T, \mathbb{R})$ and that $\mathbb{Q}_{\omega}$ is the direct limit of $\mathbb{Q}_n$s, it follows that the poset $\mathbb{Q}_{\omega}$ has the $\Theta$-c.c. in $\mathrm{HOD}_{\{ T \}}$ as well.
For (3), one can argue in the same way as in \cite[Lemma~3.4 and Lemma~3.5]{MR2463615} by replacing $\mathcal{M}$ with $V$, and $\mathcal{H}$ with $\mathrm{HOD}_{\{ T \}}$. \end{proof}
\begin{lem}\label{lem:useful} Assume $\mathsf{ZFC}$. Let $\lambda$ be a regular uncountable cardinal, $\mathbb{P}$ be a $<$$\lambda$-closed poset, and $\mathbb{Q}$ be a $\lambda$-c.c. poset. Then for any $\mathbb{P}$-generic filter $G$ over $V$, the poset $\mathbb{Q}$ still has the $\lambda$-c.c. in $V[G]$. Furthermore, if $H$ is a $\mathbb{Q}$-generic filter over $V$, then $H$ is $\mathbb{Q}$-generic over $V[G]$ as well. \end{lem}
\begin{proof} Let $G$ be a $\mathbb{P}$-generic filter and $A$ be an antichain in $\mathbb{Q}$ in $V[G]$. We will show that $A$ is of size less than $\lambda$ in $V[G]$.
Towards a contradiction, we assume that $A$ is of size at least $\lambda$ in $V[G]$.
Let $\dot{A}$ be a $\mathbb{P}$-name with $\dot{A}^G = A$. Let $p$ be a condition in $G$ with $p \Vdash_{\mathbb{P}} \lq\lq \dot{A}$ is an antichain in $\check{\mathbb{Q}}$ of size at least $\check{\lambda} "$. Using the $<$$\lambda$-closure of $\mathbb{P}$ in $V$, one can construct a decreasing sequence $(p_{\alpha} \mid \alpha < \lambda)$ in $\mathbb{P}$ and a sequence $(a_{\alpha} \mid \alpha < \lambda)$ in $\mathbb{P}$ with the following properties: \begin{enumerate} \item $p_0 = p$,
\item for all $\alpha , \beta < \lambda$ with $\alpha \neq \beta$, we have $a_{\alpha} \neq a_{\beta}$, and
\item for all $\alpha < \lambda$, $p_{\alpha} \Vdash_{\mathbb{P}} \lq\lq \check{a}_{\alpha} \in \dot{A}"$. \end{enumerate}
Since $p \Vdash_{\mathbb{P}} \lq\lq \dot{A} \text{ is an antichain in } \check{\mathbb{Q}} "$, by (1), (2), and (3) above, for all $\alpha , \beta < \lambda$ with $\alpha < \beta$, the condition $p_{\beta}$ forces that $a_{\alpha}$ and $a_{\beta}$ are incompatible in $\mathbb{P}$. Therefore, the set $B = \{ a_{\alpha} \mid \alpha < \lambda \}$ is an antichain in $\mathbb{Q}$ of size $\lambda$ in $V$. This contradicts the assumption that $\mathbb{Q}$ has the $\lambda$-c.c. in $V$. Therefore, the antichain $A$ is of size less than $\lambda$ in $V[G]$, as desired.
Let $H$ be a $\mathbb{Q}$-generic filter over $V$. We will verify that $H$ is $\mathbb{Q}$-generic over $V[G]$ as well. Let $A$ be a maximal antichain in $\mathbb{Q}$ in $V[G]$. We will see that $H \cap A \neq \emptyset$. By the arguments in the previous paragraphs, $A$ is of size less than $\lambda$ in $V[G]$. Since $G$ is $\mathbb{P}$-generic over $V$ and $\mathbb{P}$ is $<$$\lambda$-closed in $V$ while $\mathbb{Q}$ is in $V$, there is no subset of $\mathbb{Q}$ of size less than $\lambda$ in $V[G] \setminus V$. Hence the antichain $A$ is in $V$ as well. By the genericity of $H$ over $V$, we have that $H \cap A \neq \emptyset$, as desired.
\end{proof}
\begin{lem}\label{thm:ADR-countable-sequences} Assume $\mathsf{ZF} + \mathsf{AD}^+ + \mathsf{AD}_{\mathbb{R}}$. Then for any set $C$ of reals, there is an $s \in \Theta^{\omega}$ such that $C$ is OD from $s$ and that $C$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$. \end{lem}
\begin{proof} Let $C$ be any set of reals. By the result of Woodin~\cite{ADplus}, under $\mathsf{ZF} + \mathsf{AD}^+ + \mathsf{AD}_{\mathbb{R}}$, every set of reals is Suslin. By the result of Martin and Steel~\cite{MR2463620}, every Suslin and co-Suslin set of reals is homogeneously Suslin. In particular, the complement $2^{\omega} \setminus C$ is homogeneously Suslin witnessed by the sequence $(\mu_u \mid u \in 2^{<\omega} )$ of measures on $\kappa^{<\omega}$ for some $\kappa < \Theta$. By the result of Kunen~\cite[28.21~Corollary]{MR1994835}, each measure $\mu_u$ is OD. Using the Moschovakis Coding Lemma and $\mathsf{AD}_{\mathbb{R}}$, one can show that each measure $\mu_u$ is definable from an ordinal below $\Theta$. Hence there is an $s \in \Theta^{\omega}$ such that the sequence $(\mu_u \mid u \in 2^{<\omega} )$ is definable from $s$. Now from the sequence $(\mu_u \mid u \in 2^{<\omega} )$, one can construct a Martin-Solovay tree $T$ such that $C = \text{p} [T]$. By the construction of $T$, it follows that $T$ is OD from $(\mu_u \mid u \in 2^{<\omega} )$. Hence $T$ is OD from $s$, which easily implies that the set $C$ is OD from $s$ and $C$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$, as desired. \end{proof}
\begin{lem}\label{lem:ADR-countable-sequences} Assume $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}} $. Let $\gamma < \Theta$ and $\mathbb{Q}_1$ be the Vop\v{e}nka algebra for adding an element of $\gamma^{\omega}$ in $\mathrm{HOD}$. Also let $\mathbb{Q}_{\omega}$ be the finite support limit of the Vop\v{e}nka algebras for adding an element of $\gamma^{\omega}$ in $\mathrm{HOD}$. \begin{enumerate} \item The posets $\mathbb{Q}_1$ and $\mathbb{Q}_{\omega}$ are of size less than $\Theta$ in $\mathrm{HOD}$.
\item Let $s \in \gamma^{\omega}$ and $h_s = \{ p \in \mathbb{Q}_1 \mid s \in \pi_1 (p) \}$, where $\pi_1 \colon \mathbb{Q}_1 \to \mathcal{O}_1$ is as in Definition~\ref{def:Vop}. Then the set $h_s$ is a $\mathbb{Q}_1$-generic filter over $\mathrm{HOD}$ such that $\mathrm{HOD} [h_s] = \mathrm{HOD}_{ \{ s \}}$.
\item (Woodin) There is a $\mathbb{Q}_{\omega}$-generic filter $H$ over $\mathrm{HOD}$ such that the set $(\gamma^{\omega})^V$ is countable in $\mathrm{HOD} [H]$. \end{enumerate} \end{lem}
\begin{proof} For (1), we first show that the poset $\mathbb{Q}_1$ is of size less than $\Theta$ in $\mathrm{HOD}$. Recall from Definition~\ref{def:Vop} that $\pi_1 \colon \mathbb{Q}_1 \to \mathcal{O}_1$ is a surjection which is OD, where $\mathcal{O}_1$ is the collection of all subsets of $\gamma^{\omega}$ which are OD. Since $\gamma < \Theta$, by Theorem~\ref{thm:CodingLemma}, there is a set $A$ of reals such that there is a surjection from $\mathbb{R}$ to $\wp (\gamma)$ which is OD from $A$. In particular, there is a surjection $\sigma \colon \mathbb{R} \to \gamma^{\omega}$ which is OD from $A$.
Hence for each $b \in \mathcal{O}_1$, the set $\sigma^{-1} (b)$ of reals is OD from $A$. Since we assume $\mathsf{AD}_{\mathbb{R}}$, by Theorem~\ref{thm:Solovay}, there is a set $B$ of reals which is not OD from $A$. By Wadge's Lemma under $\mathsf{ZF} + \mathsf{AD}$, for each $b \in \mathcal{O}_1$, the set $\sigma^{-1} (b)$ is Wadge reducible to $B$. In particular, there is a surjection from $\mathbb{R}$ to the family $\{ \sigma^{-1} (b) \mid b \in \mathcal{O}_1 \}$. Hence the family $\mathcal{O}_1$ is also a surjective image of $\mathbb{R}$ and the poset $\mathbb{Q}_1$ is of size less than $\Theta$ in $V$. Since $\Theta$ is a cardinal in $V$, it follows that the poset $\mathbb{Q}_1$ is of size less than $\Theta$ in $\mathrm{HOD}$ as well.
We next show that the poset $\mathbb{Q}_{\omega}$ is of size less than $\Theta$ in $\mathrm{HOD}$. Let $C = A \oplus B = \{ x \ast y \mid x \in A \text{ and } y \in B \}$, where $x \ast y (2\ell) = x(\ell)$ and $x \ast y (2\ell+1) = y (\ell)$ for all $\ell < \omega$. Then the argument in the last paragraph shows that there is a surjection from $\mathbb{R}$ to $\mathbb{Q}_1$ which is OD from $C$. Similarly, one can argue that for each natural numbers $n$ with $n \ge 1$, there is a surjection from $\mathbb{R}$ to $\mathbb{Q}_n$ which is OD from $C$. Since all such surjections are OD from $C$, one can pick a sequence $(\rho_n \colon \mathbb{R} \to \mathcal{O}_n \mid n \ge 1)$ of surjections, which would readily give us a surjection from $\mathbb{R}$ to $\mathbb{Q}_{\omega}$. Therefore, the poset $\mathbb{Q}_{\omega}$ is of size less than $\Theta$ in $V$. Since $\Theta$ is a cardinal in $V$, it follows that the poset $\mathbb{Q}_{\omega}$ is of size less than $\Theta$ in $\mathrm{HOD}$ as well.
For (2), for the $\mathbb{Q}_1$-genericity of $h_s$ over $\mathrm{HOD}$, see e.g., \cite[Theorem~15.46]{Jech}.
We will show the equality $\mathrm{HOD}[h_s] = \mathrm{HOD}_{\{s\}}$. The inclusion $\mathrm{HOD} [h_s] \subseteq \mathrm{HOD}_{\{ s \}}$ is easy because $h_s$ is OD from $s$ and $h_s$ is a set of ordinals. We will argue that $\mathrm{HOD}_{\{s \}} \subseteq \mathrm{HOD} [h_s]$. Since $\mathrm{HOD}_{\{ s \}}$ is a model of $\mathsf{ZFC}$, it is enough to see that every set of ordinals in $\mathrm{HOD}_{\{ s \} }$ is also in $\mathrm{HOD} [h_s]$. Let $X$ be any set of ordinals in $\mathrm{HOD}_{\{s \}}$. We will verify that $X$ is also in $\mathrm{HOD}[h_s]$. Let $\delta$ be an ordinal such that $X \subseteq \delta$. Since $X$ is in $\mathrm{HOD}_{\{s \} }$, the set $X$ is OD from $s$. So there is a formula $\phi$ such that for all $\alpha < \delta$, we have that $\alpha \in X$ if and only if $\phi [\alpha , s]$ holds. For each $\alpha < \delta$, let $b_{\alpha} = \{ x \in \gamma^{\omega} \mid \phi [\alpha , x]\}$. Then each set $b_{\alpha}$ is a subset of $\gamma^{\omega}$ which is OD. So each $b_{\alpha}$ is in $\mathcal{O}_1$. Now we have the following equivalences: For all $\alpha < \delta$, \begin{align*} \alpha \in X \iff \phi [\alpha , s] \iff s \in b_{\alpha} \iff \pi_1^{-1} (b_{\alpha}) \in h_s. \end{align*} Hence $X = \{ \alpha < \delta \mid \pi_1^{-1} (b_{\alpha}) \in h_s \}$. Since the sequence $\bigl(\pi_1^{-1} (b_{\alpha} ) \in \mathbb{Q}_1 \mid \alpha < \delta \bigr)$ is OD and $\mathbb{Q}_1$ is in $\mathrm{HOD}$, the sequence $\bigl(\pi_1^{-1} (b_{\alpha} ) \in \mathbb{Q}_1 \mid \alpha < \delta \bigr)$ belongs to $\mathrm{HOD}$. Hence the set $ \{ \alpha < \delta \mid \pi_1^{-1} (b_{\alpha}) \in h_s \}$ is in $\mathrm{HOD} [h_s]$. Therefore, the set $X$ is in $\mathrm{HOD} [h_s]$, as desired.
For (3), one can argue in the same way as in \cite[Lemma~3.4 and Lemma~3.5]{MR2463615} by replacing $\mathbb{R}$ with $\gamma^{\omega}$, $\mathcal{M}$ with $V$, and $\mathcal{H}$ with $\mathrm{HOD}$. \end{proof}
\section{On forcings increasing $\Theta$}\label{sec:increasing-Theta}
In this section, we prove the following theorems: \begin{thm}\label{thm:increasing-Theta-destruction} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that a poset $\mathbb{P}$ increases $\Theta$, i.e., $\Theta^{V} < \Theta^{V[G]}$ for any $\mathbb{P}$-generic filter $G$ over $V$. Then $\mathsf{AD}$ fails in $V [G]$ for any $\mathbb{P}$-generic filter $G$ over $V$. \end{thm}
\begin{thm}\label{thm:increasing-Theta-example} It is consistent relative to $\mathsf{ZF}+\mathsf{AD}_{\mathbb{R}}$ that $\mathsf{ZF}+\mathsf{AD}$ holds and there is a poset $\mathbb{P}$ increasing $\Theta$ while preserving $\mathsf{AD}$, i.e., for any $\mathbb{P}$-generic filter $G$ over $V$, we have $\Theta^{V} < \Theta^{V[G]}$ and that $\mathsf{AD}$ holds in $V[G]$. \end{thm}
\begin{proof}[Proof of Theorem~\ref{thm:increasing-Theta-destruction}]
Let $G$ be a $\mathbb{P}$-generic filter over $V$. We will show that $\mathsf{AD}$ fails in $V[G]$. Towards a contradiction, we assume that $\mathsf{AD}$ holds in $V[G]$.
Since we have $\mathsf{AD}^+$ and $V = \mathrm{L} \bigl (\wp (\mathbb{R}) \bigr)$, by Theorem~\ref{thm:HODLA}, the model HOD is of the form $\mathrm{L} [Z]$ for some subset $Z$ of $\Theta$, and there are a poset $\mathbb{Q}$ in HOD and a $\mathbb{Q}$-generic filter $H$ over HOD such that $\mathrm{HOD} \subseteq V \subseteq \mathrm{HOD}[H]$. In particular, $Z^{\#}$ does not exist in HOD. Since any poset does not add $Z^{\#}$, it follows that $Z^{\#}$ does not exist in $V$ either.
We will argue that $Z^{\#}$ exists in $V[G]$, which would contradict the fact that $Z^{\#}$ does not exist in $V$. Since $\mathbb{P}$ increases $\Theta$, we have $\Theta^V < \Theta^{V[G]}$. By assumption, we have $\mathsf{AD}$ in $V[G]$, so by Theorem~\ref{thm:Moschovakis}, it follows that $\Theta^{V[G]}$ is a limit of measurable cardinals in $V[G]$. In particular, there is a measurable cardinal $\kappa$ in $V[G]$ such that $\Theta^V < \kappa$. Let $U$ be a $<$$\kappa$-complete nonprincipal ultrafilter on $\kappa$ in $V[G]$. Then letting $M = \mathrm{L} [U, Z]$, the cardinal $\kappa$ is measurable also in $M$ witnessed by $U \cap M$. Since $M$ is a model of $\mathsf{ZFC}$ and $Z$ is a bounded subset of $\kappa$ in $M$, it follows that $Z^{\#}$ exists in $M$. By absolutness of $Z^{\#}$, we have $Z^{\#}$ in $V[G]$, contradicting the fact that $Z^{\#}$ does not exist in $V$.
Therefore, the assumption that $\mathsf{AD}$ holds in $V[G]$ was wrong, and $\mathsf{AD}$ fails in $V[G]$.
This completes the proof of Theorem~\ref{thm:increasing-Theta-destruction}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:increasing-Theta-example}]
We assume $\mathsf{ZF} + \mathsf{AD}_{\mathbb{R}}$ and will show that there is an inner model $M$ of $\mathsf{ZF} + \mathsf{AD}$ satisfying that there is a poset $\mathbb{P}$ increasing $\Theta$ while preserving $\mathsf{AD}$.
Let $M =\mathrm{HOD}_{\mathbb{R}}$, the class of all sets hereditarily ordinal definable from some real. We will show that $M$ is the desired inner model.
First notice that $M$ is a model of $\mathsf{ZF}$. Also since $M$ contains all the reals and $V$ satisfies $\mathsf{AD}$, we have that $M$ is a model of $\mathsf{AD}$ as well.
Since we have $\mathsf{AD}_{\mathbb{R}}$ in $V$, by Theorem~\ref{thm:Solovay}, there is a set $B$ of reals which is not definable from any ordinal and any real. Hence the set $B$ is not in $M$.
We will show that $M$ satisfies that there is a poset $\mathbb{P}$ increasing $\Theta$ while preserving $\mathsf{AD}$. The idea is to consider a variant of Vop\v{e}nka algebra in $M$ adding the set $B$ to $M$. Let $\mathcal{O} = \{ b \subseteq \wp (\mathbb{R}) \mid \text{$b$ is nonempty and OD from some real}\}$ ordered by inclusion. Then $\mathcal{O}$ is a poset which is OD. Let $\eta$ be a sufficiently large ordinal and let $\pi \colon \eta \times \mathbb{R} \to \mathcal{O}$ be a surjection which is OD such that if a set $b$ is in $\mathcal{O}$ and OD from a real $x$, then there is some $\alpha < \eta$ such that $\pi (\alpha , x) = b$. Let $\mathbb{P} = \pi^{-1} (\mathcal{O})$ and for $p_1, p_2\in \mathbb{P}$, we set $p_1 \le p_2$ if $\pi (p_1) \subseteq \pi (p_2)$. Then since $\pi$ is OD, the poset $\mathbb{P}$ is in $M$. For an $r $ in $\mathbb{P}$, let $\mathbb{P} \upharpoonright r = \{ p \in \mathbb{P} \mid p \le r\}$.
We will show that there is some $\mathbb{P}$-generic filter $G$ over $M$ such that $\Theta^M < \Theta^{M[G]}$ and $M[G]$ is a model of $\mathsf{AD}$. This is enough to end the arguments for the theorem because then there is some $r \in \mathbb{P}$ forcing the desired two statements for $M[G]$ over $M$, and the poset $\mathbb{P} \upharpoonright r$ is the desired poset in $M$.
Let $H = \{ b \in \mathcal{O} \mid B \in b\}$ and $G = \pi^{-1} (H)$. We will see that $G$ is the desired filter.
We first verify that $G$ is $\mathbb{P}$-generic over $M$. Let $D$ be a dense subset of $\mathbb{P}$ in $M$. We will argue that $G \cap D \neq \emptyset$. Let $E = \pi [D]$ and $b_E = \bigcup E$. By the definition of $\mathbb{P}$, the set $E$ is dense in $\mathcal{O}$. We claim that $b_E = \wp ( \mathbb{R} )$. Suppose not. Then since $D$ is in $M$ and $\pi$ is OD, the set $b_E$ is OD from some real. So $b_E$ is in $\mathcal{O}$. But then $\wp (\mathbb{R}) \setminus b_E$ is a nonempty set which is in $\mathcal{O}$ incompatible with any element of $E$, contradicting that $E$ is dense in $\mathcal{O}$. Hence $b_E = \wp (\mathbb{R})$. Since $B$ is in $\wp (\mathbb{R})$, we have that $B$ is in $b_E$, so there is a $b'$ in $E$ such that $B$ is in $b'$. By the definition of $H$, the condition $b'$ is also in $H$. Hence $H \cap E \neq \emptyset$. Since $G = \pi^{-1} (H)$ and $E = \pi [D]$, it follows that $G \cap D \neq \emptyset$, as desired. Therefore, $G$ is $\mathbb{P}$-generic over $M$.
We next verify that $M[G]$ is a model of $\mathsf{AD}$. Since $B$ is in $V$, $H = \{ b \in \mathcal{O} \mid B \in b\}$, and $G = \pi^{-1} (H)$, it follows that $G$ is in $V$ and $M[G]$ is a submodel of $V$. Since $M$ contains all the reals, so does $M[G]$. Finally, since $V$ is a model of $\mathsf{AD}$, it follows that $M[G]$ is also a model of $\mathsf{AD}$, as desired.
Finally, we verify that $\Theta^M < \Theta^{M[G]}$. Since both $M$ and $M[G]$ are models of $\mathsf{AD}$ containing all the reals, by the Wadge lemma under $\mathsf{ZF} + \mathsf{AD}$, it is enough to see that there is a set of reals in $M[G] \setminus M$. Since $B$ is not in $M$, it suffices to argue that $B$ is in $M[G]$. For each real $x$, let $b_x = \{ A \in \wp (\mathbb{R}) \mid x \in A\}$. Then $b_x$ is OD from $x$, so $b_x$ is in $\mathcal{O}$. By the choice of $\pi$, for each real $x$, there is an ordinal $\alpha$ such that $\pi (\alpha , x) = b_x$. For each real $x$, let $\alpha_x$ be the least ordinal with $\pi (\alpha_x , x) = b_x$. Then since $\pi$ and $\mathcal{O}$ are OD, the sequence $( \alpha_x \mid x \in \mathbb{R})$ is OD and is in $M = \mathrm{HOD}_{\mathbb{R}}$. From the sequence $(\alpha_x \mid x \in \mathbb{R})$ and $G$, one can compute the set $B$ as follows: for any real $x$, \begin{align*} x \in B \iff B \in b_x \iff b_x \in H \iff (\alpha_x , x) \in G. \end{align*} Therefore, the set $B$ is in $M[G]$, as desired.
We have verified that $G$ is the desired filter, and this completes the proof of Theorem~\ref{thm:increasing-Theta-example}. \end{proof}
\section{On forcings on the reals}\label{sec:the-reals}
In this section, we prove the following theorem which answers a question by Chan and Jackson~\cite[Question~5.7]{MR4242147}: \begin{thm}\label{thm:the-reals} Assume $\mathsf{ZF}+\mathsf{AD}$. Let $\mathbb{P}$ be any non-trivial poset which is a surjective image of $\mathbb{R}$ and $G$ be any $\mathbb{P}$-generic filter over $V$. Then $\mathsf{AD}$ fails in $V [G]$. \end{thm}
\begin{proof}[Proof of Theorem~\ref{thm:the-reals}]
Let $\mathbb{P}$ be any non-trivial poset which is a surjective image of $\mathbb{R}$ and $G$ be any $\mathbb{P}$-generic filter over $V$. We will show that $\mathsf{AD}$ fails in $V[G]$. Since $\mathbb{P}$ is a surjective image of $\mathbb{R}$, there is a poset on $\mathbb{R}$ which is forcing equivalent to $\mathbb{P}$. Hence we may assume $\mathbb{P}$ is a poset on $\mathbb{R}$.
Towards a contraction, we assume that $\mathsf{AD}$ holds in $V[G]$.
\begin{case}\label{RV-uncountable} When the set $\mathbb{R}^V$ is uncountable in $V[G]$. \end{case}
Here is the key point: \begin{claim}\label{key-claim} There is a real $r_0$ in $V[G]$ such that $\mathbb{R}^{V[G]} \subseteq \mathrm{L} (\mathbb{R}^V , r_0)$. \end{claim}
\begin{proof}[Proof of Claim~\ref{key-claim}] Since $V[G]$ satisfies $\mathsf{AD}$, the set $\mathbb{R}^V$ has the perfect set property in $V[G]$. Since $\mathbb{R}^V$ is uncountable in $V[G]$, the set $\mathbb{R}^V$ contains a perfect set $C$ in $V[G]$. Let $r_0$ code a perfect tree $T$ on $2 = \{0,1\}$ with $[T] = C$ in $V[G]$.
We will show that $\mathbb{R}^{V[G]} \subseteq \mathrm{L} (\mathbb{R}^V , r_0)$. Let $x$ be any element of $2^{\omega}$ in $V[G]$. We will see that $x$ is in $\mathrm{L} ( \mathbb{R}^V , r_0)$.
We say a node $t \in T$ is {\it splitting in $T$} if both $t^{\frown} \langle 0\rangle$ and $t^{\frown} \langle 1 \rangle $ are in $T$. Let $\{ t_s \in T \mid s \in 2^{<\omega} \}$ be the set of all splitting nodes in $T$ such that if $s_1 $ is a subsequence of $s_2$ in $2^{<\omega}$, then $t_{s_1}$ is a subsequence of $t_{s_2}$ in $T$. Let $y = \bigcup \{ t_{x \upharpoonright n} \mid n < \omega \}$. Then $y$ is in $[T]$. Since $[T] =C \subseteq \mathbb{R}^V$, the real $y$ is in $\mathbb{R}^V$. However, for all $n< \omega$ and $k \in 2 = \{ 0,1 \}$, \begin{align*} x(n) = k \iff t_{(x\upharpoonright n)^{\frown} \langle k \rangle} \subseteq y. \end{align*} Hence $x$ can be simply computed from $y$ and $T$. So $x \in \mathrm{L} [y, T]$. Since $\mathrm{L} [y, T] \subseteq \mathrm{L} [y, r_0] \subseteq \mathrm{L} (\mathbb{R}^V, r_0)$, the real $x$ is in $\mathrm{L} (\mathbb{R}^V , r_0)$, as desired.
This completes the proof of Claim~\ref{key-claim}. \end{proof}
Continuing to argue in Case~\ref{RV-uncountable}, let $r_0$ be a real in $V[G]$ such that $\mathbb{R}^{V[G]} \subseteq \mathrm{L} (\mathbb{R}^V , r_0)$ as in Claim~\ref{key-claim}.
Since the poset $\mathbb{P}$ is on $\mathbb{R}^V$, there is a $\mathbb{P}$-name $\dot{x}$ such that $\dot{x}^G = r_0$ and $\dot{x}$ is coded by some set $A$ of reals in $V$. Then setting $M = \mathrm{L} (\mathbb{R}^V , \mathbb{P} , A)$, we have that $M$ is an inner model of $V$ satisfying $\mathsf{AD}$ and the statement \lq\lq $\Theta$ is regular''. However, since $\mathbb{R}^{V[G]} \subseteq \mathrm{L} (\mathbb{R}^V , r_0) \subseteq M[G] \subset V[G]$ and we assumed that $V[G]$ satisfies $\mathsf{AD}$, the model $M[G]$ also satisfies $\mathsf{AD}$, contradicting Theorem~\ref{thm:chan-Jackson}. Therefore, the assumption that $V[G]$ satisfies $\mathsf{AD}$ was wrong and $\mathsf{AD}$ must fail in $V[G]$, as desired.
This finsihes the arguments for Theorem~\ref{thm:the-reals} in Case~\ref{RV-uncountable}.
\begin{case}\label{RV-countable} When the set $\mathbb{R}^V$ is countable in $V[G]$. \end{case}
Since $\mathbb{R}^V$ is countable in $V[G]$, any ordinal $\alpha$ below $\Theta^V$ is countable in $V[G]$ as well. Hence $\Theta^V \le \omega_1^{V[G]}$.
We will show that $\Theta^{V[G]} \le (\Theta^+)^V$, which would contradict the assumption that $\mathsf{AD}$ holds in $V[G]$, because $\mathsf{AD}$ in $V[G]$ would imply that $\Theta^{V[G]} > \omega_2^{V[G]} \ge (\Theta^+)^V$ since $\Theta^V \le \omega_1^{V[G]}$.
To see that $\Theta^{V[G]} \le (\Theta^+)^V$, let $f\colon \mathbb{R}^{V[G]} \to (\Theta^+)^V$ be any function in $V[G]$. We will show that $f$ is not surjective. As in the arguments in Case~\ref{RV-uncountable}, since $\mathbb{P}$ is on $\mathbb{R}$, any real in $V[G]$ can be coded by a set of reals in $V$. Hence we may assume that $f\colon \wp (\mathbb{R})^V \to (\Theta^+)^V$. Also, since $\mathbb{P}$ is on $\mathbb{R}^V$, there is a function $g \colon \wp (\mathbb{R})^V \times \mathbb{R}^V \to (\Theta^+)^V$ in $V$ such that $\mathrm{rng} (f) \subseteq \mathrm{rng} (g)$. Therefore, it is enough to see that $g$ is not surjective in $V$.
We now work in $V$. To see that $g$ is not surjective, for each $\alpha < \Theta$, let $W_{\alpha} = \{ B \in \wp (\mathbb{R}) \mid |B|_{\text{W}} = \alpha \}$, where $|B|_{\text{W}}$ is the Wadge ordinal of $B$. Then each $W_{\alpha}$ is a surjective image of $\mathbb{R}$ and so is the set $R_{\alpha} = \{ g(B, x) \mid B \in W_{\alpha}, x\in \mathbb{R} \}$. Hence, for every $\alpha < \Theta$, the order type of $R_{\alpha }$ is less than $\Theta$, and $\mathrm{rng} (g) = \bigcup_{\alpha < \Theta} R_{\alpha}$ is a surjective image of $\Theta \times \Theta$. Therefore, $\mathrm{rng} (g)$ is of cardinality at most $\Theta$ which is smaller than $\Theta^+$. Hence $g$ is not surjective in $V$, as desired.
This finishes the arguments for Theorem~\ref{thm:the-reals} in Case~\ref{RV-countable}.
This completes the proof of Theorem~\ref{thm:the-reals}. \end{proof}
\section{On forcings adding a subset of $\Theta$}\label{sec:subset-of-Theta}
In this section, we prove the following theorems: \begin{thm}\label{thm:subset-of-Theta-positive} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that $\Theta$ is regular. Then there is a poset $\mathbb{P}$ on $\Theta$ which adds a subset of $\Theta$ while presering $\mathsf{AD}$, i.e., for any $\mathbb{P}$-generic filter $G$ over $V$, there is a subset of $\Theta^{V}$ which belongs to $V[G] \setminus V$ and $\mathsf{AD}$ holds in $V[G]$. \end{thm}
\begin{thm}\label{thm:subset-of-Theta-negative} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that $\Theta$ is singular and let $\mathbb{P}$ be $\mathrm{Add} (\Theta, 1)$ in HOD, where $\mathrm{Add} (\Theta , 1) = \{ p \mid p \colon \gamma \to 2 \text{ for some $\gamma < \Theta$} \}$. Then $\mathsf{AD}$ fails in $V [G]$ for any $\mathbb{P}$-generic filter $G$ over $V$. \end{thm}
\begin{proof}[Proof of Theorem~\ref{thm:subset-of-Theta-positive}]
Throughout the proof of the theorem, we write $\lambda$ for $\Theta^V$.
We prove the theorem by considering the two cases whether $\mathsf{AD}_{\mathbb{R}}$ holds or not.
\setcounter{case}{0} \begin{case}\label{case:ADRfail} When $\mathsf{AD}_{\mathbb{R}}$ fails. \end{case}
Since $\mathsf{AD}_{\mathbb{R}}$ fails while we assume $\mathsf{AD}^+$ and $V = \mathrm{L}\bigl( \wp(\mathbb{R}) \bigr)$, by Theorem~\ref{thm:AD+ADRfail}, there is a set $T$ of ordinals such that $V = \mathrm{L}(T, \mathbb{R})$. We fix such a $T$ throughout the arguments for Case~\ref{case:ADRfail}.
Let $\mathbb{P}$ be $\mathrm{Add} (\lambda, 1)$ in $\mathrm{HOD}_{\{ T \}}$, where $\mathrm{Add} (\lambda , 1) = \{ p \mid p \colon \gamma \to 2 = \{ 0, 1 \} \text{ for some $\gamma < \lambda$} \}$.
Since $\mathbb{P}$ is computed in $\mathrm{HOD}_{\{ T \}}$ and $\lambda = \Theta^V$ is inaccessible in $\mathrm{HOD}_{\{ T \}}$, the poset $\mathbb{P}$ can be considered as a poset on $\lambda$.
We will show that $\mathbb{P}$ is the desired poset in Case~\ref{case:ADRfail}, i.e., $\mathbb{P}$ adds a subset of $\lambda = \Theta^V$ while preserving $\mathsf{AD}$ in Case~\ref{case:ADRfail}.
Let $G$ be any $\mathbb{P}$-generic filter over $V$. Then the function $\bigcup G \colon \lambda \to 2$ can be considered as a subset of $\lambda$ and by the genericity of $G$ over $V$, the subset is not in $V$. Hence the poset $\mathbb{P}$ adds a new subset of $\lambda$ to $V$.
We will show that $\mathsf{AD}$ holds in $V[G]$. We start with showing that the poset $\mathbb{P}$ does not add any bounded subset of $\lambda$: \setcounter{claim}{0} \begin{claim}\label{no-bounded-subset-in-vg-non-adr} For any $\gamma < \lambda$, we have $\wp (\gamma)^V = \wp (\gamma)^{V[G]}$. In particular, $\mathbb{R}^V = \mathbb{R}^{V[G]}$. \end{claim}
\begin{proof}[Proof of Claim~\ref{no-bounded-subset-in-vg-non-adr}]
Let $\gamma$ be an ordinal less than $\lambda$ and $f \colon \gamma \to 2$ in $V[G]$. We will show that $f$ is in $V$.
Let $\dot{f}$ be a $\mathbb{P}$-name with $\dot{f}^G = f$. Since $\mathbb{P}$ can be seen as a poset on $\lambda$ and $f \colon \gamma \to 2$, we may asssume that $\dot{f}$ is a subset of $\lambda \times \gamma \times 2$. Since $V = \mathrm{L} (T, \mathbb{R})$ and $\dot{f}$ is in $V$, we have that $\dot{f}$ is $\mathrm{OD}_{\{T,x\}}$ for some real $x$. Then since $\dot{f}$ is essentially a set of ordinals, $\dot{f}$ is in $\mathrm{HOD}_{\{T,x\}}$. By Theorem~\ref{thm:V=LTR}, we have that $\mathrm{HOD}_{\{T,x\}} = \mathrm{HOD}_{\{ T \}} [x]$ and $\mathrm{HOD}_{\{ T \}} = \mathrm{L} [T, Z]$ for some subset $Z$ of $\lambda$. Since $f = \dot{f}^G$, it follows that $f$ is in $\mathrm{HOD}_{\{T,x\}} [G] = \mathrm{HOD}_{\{ T \}} [x][G]$.
Let $\mathbb{Q}_1$ be the Vop\v{e}nka algebra for adding an element of $2^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$. Then the real $x$ induces a $\mathbb{Q}_1$-generic filter $h_x$ over $\mathrm{HOD}_{\{ T \}}$ such that $x \in \mathrm{HOD}_{\{ T \}}[h_x]$. Since $G$ was chosen to be $\mathbb{P}$-generic over $V$, it is also $\mathbb{P}$-generic over $\mathrm{HOD}_{\{ T \}} [h_x]$. Hence the filter $G \times h_x$ is $\mathbb{P} \times \mathbb{Q}_1$-generic over $\mathrm{HOD}_{\{ T \}}$ and $\mathrm{HOD}_{\{ T \}} [x][G] \subseteq HOD_{\{ T \}} [h_x][G] = \mathrm{HOD}_{\{ T \}} [G] [h_x]$.
Since $\dot{f}$ is in $\mathrm{HOD}_{\{ T \}} [x] \subseteq \mathrm{HOD}_{\{ T \}} [h_x]$, there is a $\mathbb{Q}_1$-name $\tau$ in $\mathrm{HOD}_{\{ T \}}$ such that $\tau^{h_x} = \dot{f}$. Let $\nu$ be a sufficiently big cardinal in $\mathrm{HOD}_{\{ T \}} [G]$ and let $N$ be $V_{\nu}$ in $\mathrm{HOD}_{\{ T \}} [G]$. By the $<$$\lambda$-closure of $\mathbb{P}$ in $\mathrm{HOD}_{\{ T \}}$, the ordinal $\lambda$ is regular in $\mathrm{HOD}_{\{ T \}}[G]$. Since $\mathrm{HOD}_{\{ T \}}[G]$ is a model of $\mathsf{ZFC}$, there is an elementary substructure $X$ of $N$ in $\mathrm{HOD}_{\{ T \}} [G]$ such that $\gamma + 1 \subseteq X$, $X \cap \lambda \in \lambda$, $X$ is of size less than $\lambda$, and $T, Z, G, \mathbb{P}, \mathbb{Q}_1 , \tau \in X$. Let $M$ be the transitive collapse of $X$ and let $\pi \colon M \to X$ be the inverse of the collapsing map. Then letting $\kappa = X \cap \lambda$, we have that $\kappa$ is the critical point of $\pi$ and $\pi (\kappa) = \lambda$.
For any $a \in X$, we write $\bar{a}$ for $\pi^{-1} (a)$, i.e., $\pi (\bar{a}) = a$.
We claim that $M$ is in $\mathrm{HOD}_{\{ T \}}$. Let $g = \bigcup G$. Then by the genericity of $G$, we have $g \colon \lambda \to 2$. Since $G$ is simply definable from $g$, we have $\mathrm{HOD}_{\{ T \}} [G] = \mathrm{HOD}_{\{ T \}} [g]$. Recall that $\mathrm{HOD}_{\{ T \}} = \mathrm{L} [T, Z]$, so $\mathrm{HOD}_{\{ T \}} [G] = \mathrm{L} [T, Z][G] = \mathrm{L} [T, Z][g]$. Hence the model $M$ is of the form $\mathrm{L}_{\mu} [\bar{T}, \bar{Z}][\bar{g}]$ for some $\mu$. Since $Z$ is a subset of $\lambda$, we have $\bar{Z} = Z \cap \kappa$ and hence $\bar{Z} \in \mathrm{HOD}_{\{ T \}}$.
Since $\bar{T}$ is a set of ordinals of size less than $\lambda$ in $\mathrm{HOD}_{\{ T \}} [G]$, by the $<$$\lambda$-closure of $\mathbb{P}$ in $\mathrm{HOD}_{\{ T \}}$, the set $\bar{T}$ is in $\mathrm{HOD}_{\{ T \}}$.
Since $g \colon \lambda \to 2$, we have $\bar{g} = g \upharpoonright \mu$, which is in $\mathbb{P}$. So $\bar{g}$ is in $\mathrm{HOD}_{\{ T\}}$. Since $M = \mathrm{L}_{\mu} [\bar{T}, \bar{Z} , \bar{g}]$, the model $M$ is in $\mathrm{HOD}_{\{ T \}}$, as desired.
Let $\bar{h}_x = \{ \bar{q} \mid q \in h_x \cap X\}$. We claim that $\bar{h}_x$ is $\bar{\mathbb{Q}}_1$-generic over $M$. Recall that $\mathbb{Q}_1$ is the Vop\v{e}nka algebra for adding an element of $2^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$. By Lemma~\ref{lem:Qomega}, we may assume that $\mathbb{Q}_1$ is on $\Theta^V = \lambda$ and $\mathbb{Q}_1$ has the $\lambda$-c.c. in $\mathrm{HOD}_{\{T\}}$. Let $A$ be a maximal antichain in $\bar{\mathbb{Q}}_1$ such that $A$ is in $M$. We will verify that $A \cap \bar{h}_x \neq \emptyset$. Since $\mathbb{P}$ is $<$$\lambda$-closed and $\mathbb{Q}_1$ has the $\lambda$-c.c. in $\mathrm{HOD}_{\{ T \}}$, by Lemma~\ref{lem:useful}, the poset $\mathbb{Q}_1$ still has the $\lambda$-c.c. in $\mathrm{HOD}_{\{ T \}} [G]$. By elementarity of $\pi$, the poset $\bar{\mathbb{Q}}_1$ has the $\kappa$-c.c. in $M$. In particular, the antichain $A$ is of size less than $\kappa$ in $M$. Since $\mathbb{Q}_1$ is on $\lambda$, the poset $\bar{\mathbb{Q}}_1$ is on $\kappa$ in $M$. So the antichain $A$ is a bounded subset of $\kappa$. Since $\kappa$ is the critical point of $\pi$, we have that $\pi (A) = A$.
By elementarity of $\pi$, the antichain $\pi (A) = A$ is maximal in $\mathbb{Q}_1$ in $\mathrm{HOD}_{\{ T \}} [G]$. Since $M$ is in $\mathrm{HOD}_{\{ T \}}$ and $A$ is in $M$, the antichain $A$ is maximal in $\mathbb{Q}_1$ in $\mathrm{HOD}_{\{ T \}}$ as well. By the genericity of $h_x$ over $\mathrm{HOD}_{\{ T \}}$, the set $A \cap h_x$ is nonempty. Let $q$ be an element of $A \cap h_x$. Since $A$ is in $M$ and $M$ is transitive, the condition $q$ is in $M$. Since $\mathbb{Q}_1$ is on $\lambda$, $\pi (\kappa) = \lambda$, and $q \in h_x \cap M$, we have that $\pi (q) = q$ and hence $q \in \bar{h}_x$.
Therefore, $q \in A \cap \bar{h}_x$ and the set $A \cap \bar{h}_x$ is nonempty, as desired.
Since the poset $\bar{\mathbb{Q}}_1$ has the $\kappa$-c.c. in $M$, by a standard argument, one can lift the embedding $\pi \colon M \to N$ to an elementary embedding $\hat{\pi} \colon M[\bar{h}_x] \to N [h_x]$ such that $\hat{\pi} (\bar{h}_x) = h_x$.
We now argue that the function $f$ is in $V$. It is enough to verify that $f$ is in $M [\bar{h}_x]$ because $M$ is in $\mathrm{HOD}_{\{ T \}}$, $\mathrm{HOD}_{\{ T \}} \subseteq V$, and $\bar{h}_x = \{ \bar{q} \mid q \in h_x \cap X \} = \{ q \mid q \in h_x \cap X \} = \bar{\mathbb{Q}}_1 \cap h_x$. Recall that $\tau$ is a $\mathbb{Q}_1$-name in $\mathrm{HOD}_{\{ T \}}$ such that $\tau^{h_x} = \dot{f}$ and that $\dot{f}$ is a $\mathbb{P}$-name in $\mathrm{HOD}_{\{ T \}} [h_x]$ such that $\dot{f}^G = f$. Since $\tau$ is in $X$, letting $\dot{g} = \bar{\tau}^{\bar{h}_x}$ and $g = \dot{g}^{\bar{G}}$, we have that $\hat{\pi} (g) = f$. By elementarity of $\hat{\pi}$, the set $g$ is a function from $\pi^{-1} (\gamma)$ to $2$. We now verify that $f=g$, which would imply that $f$ is in $M[\bar{h}_x]$ because $g$ is in $M[\bar{h}_x]$. Since $\gamma + 1 \subseteq X$, we have that $\pi \upharpoonright (\gamma + 1) = \mathrm{id}$. Hence $\pi^{-1} (\gamma) = \gamma$ and $g \colon \gamma \to 2$. Also, since $\hat{\pi} (g) = f$, by elementarity of $\hat{\pi}$, for any $\alpha < \gamma$ and $i < 2$, we have that $g (\alpha) = i$ if and only if $f( \alpha) = i$. Therefore, $f = g $, as desired.
This completes the proof of Claim~\ref{no-bounded-subset-in-vg-non-adr}. \end{proof}
By Claim~\ref{no-bounded-subset-in-vg-non-adr}, we know that $\mathbb{R}^V = \mathbb{R}^{V[G]}$. So we simply write $\mathbb{R}$ for $\mathbb{R}^V$ or $\mathbb{R}^{V[G]}$. Recall that we write $\lambda$ for $\Theta^V$.
We now show that $\mathbb{P}$ does not add any set of reals to $V$: \begin{claim}\label{no-set-of-reals-in-vg-non-adr} The equality $\wp (\mathbb{R})^V = \wp (\mathbb{R})^{V[G]}$ holds. \end{claim}
\begin{proof}[Proof of Claim~\ref{no-set-of-reals-in-vg-non-adr}]
Let $A$ be any set of reals in $V[G]$. We will show that $A$ is in $V$ as well.
We first claim that $\lambda$ is regular in $V[G]$ and $\lambda = \Theta^{V[G]}$. Let $\mathbb{Q}_{\omega}$ be the finite support direct limit of Vop\v{e}nka algebras for adding an element of $2^{\omega}$ in $\mathrm{HOD}_{\{ T \}}$. Then by Lemma~\ref{lem:Qomega}, the poset $\mathbb{Q}_{\omega}$ has the $\lambda$-c.c. in $\mathrm{HOD}_{\{ T \}}$ and there is a $\mathbb{Q}_{\omega}$-generic filter $H$ over $\mathrm{HOD}_{\{ T \}}$ such that $V = \mathrm{L} (T, \mathbb{R}) \subseteq \mathrm{HOD}_{\{ T \}} [H]$ and the set $\mathbb{R}$ is countable in $\mathrm{HOD}_{\{ T \}} [H]$. Since $\mathbb{P}$ is $<$$\lambda$-closed in $\mathrm{HOD}_{\{ T \}}$ and $\mathbb{Q}_{\omega}$ has the $\lambda$-c.c. in $\mathrm{HOD}_{\{ T \}}$, by Lemma~\ref{lem:useful}, the poset $\mathbb{Q}_{\omega}$ still has the $\lambda$-c.c. in $\mathrm{HOD}_{\{ T \}} [G]$ and the filter $H$ is $\mathbb{Q}_{\omega}$-generic over $\mathrm{HOD}_{\{ T \}} [G]$ as well. Hence $\lambda$ is still regular uncountable in $\mathrm{HOD}_{\{ T \}} [G][H]$, the filter $G \times H$ is $\mathbb{P} \times \mathbb{Q}_{\omega}$-generic over $\mathrm{HOD}_{\{ T \}}$, and $\mathrm{HOD}_{\{ T \}} [G][H] = \mathrm{HOD}_{\{ T \}} [H][G]$. Therefore, $\lambda$ is still regular uncountable in $\mathrm{HOD}_{\{ T \}} [H][G]$. Since $V[G] \subseteq \mathrm{HOD}_{\{ T \}} [H][G]$, the ordinal $\lambda$ is regular in $V[G]$ as well. Also, since $\mathbb{R}$ is countable in $\mathrm{HOD}_{\{ T \}} [H]$ and $\mathbb{R}^V = \mathbb{R}^{V[G]}$ while $V[G] \subseteq \mathrm{HOD}_{\{ T \}} [H][G]$, the ordinal $\Theta^{V[G]}$ is at most $\omega_1$ in $\mathrm{HOD}_{\{ T \}}[H][G]$. Since $\lambda$ is regular uncountable in $\mathrm{HOD}_{\{ T \}} [H][G]$, we have that $\Theta^{V[G]} \le \lambda$. Since $V \subseteq V[G]$ and $\lambda = \Theta^V$, the inequality $\lambda \le \Theta^{V[G]} $ also holds. Hence $\lambda = \Theta^{V[G]}$, as desired.
Let $\nu$ be a sufficiently large cardinal in $V[G]$ and let $N$ be $V_{\nu}$ in $V[G]$. Since $V = \mathrm{L} (T, \mathbb{R})$, the model $N$ is of the form $\mathrm{L}_{\nu} (T, \mathbb{R}) [G]$. Since every element of $N$ is definable from $T, G$, an ordinal, and some real while $\lambda$ is regular in $V[G]$ and $\lambda = \Theta^{V[G]}$, one can find an elementary substructure $X$ of $N$ in $V[G]$ such that $\mathbb{R} \subseteq X$, $\lambda \cap X \in \lambda$, the structure $X$ is a surjective image of $\mathbb{R}$, and $T, \mathbb{P} , G, A \in X$. Let $M$ be the transitive collapse of $X$ and let $\pi \colon M \to X$ be the inverse of the collapsing map. Then letting $\kappa = \lambda \cap X$, the critical point of $\pi$ is $\kappa$ and $\pi (\kappa) = \lambda$. For any $a$ in $X$, we write $\bar{a}$ for $\pi^{-1} (a)$, i.e., $\pi ( \bar {a}) = a$.
We will finish arguing that the set $A$ is in $V$. Since $\mathbb{R}$ is contained in $M$ and $\pi (\bar{A}) = A$, we have $\bar{A} = A$ and the set $A$ is in $M$. Hence it is enough to verify that the model $M$ is in $V$. Recall that $g = \bigcup G$ and $g \colon \lambda \to 2$. Since $G$ is simply definable from $g$, we have that $N = \mathrm{L}_{\nu} (T, \mathbb{R}) [G] = \mathrm{L}_{\nu} (T, \mathbb{R}) [g]$. Since $N$ is of the form $\mathrm{L}_{\nu} (T, \mathbb{R}) [g]$, $X$ is a surjective image of $\mathbb{R}$ in $V[G]$, and $\lambda = \Theta^{V[G]}$, it follows that $M$ is of the form $\mathrm{L}_{\mu} (\bar{T}, \mathbb{R}) [\bar{g}]$ for some ordinal $\mu < \lambda$. Since $\mu < \lambda$, the set $\bar{T}$ is a bounded subset of $\lambda$ in $V[G]$. By Claim~\ref{no-bounded-subset-in-vg-non-adr}, the set $\bar{T}$ is in $V$ as well.
Since $g \colon \lambda \to 2$ and $\pi (\kappa) = \lambda$, we have that $\bar{g} = g \upharpoonright \kappa$ and $\bar{g}$ is in $\mathbb{P}$. So $\bar{g}$ is in $V$ as well. Since $M = \mathrm{L}_{\mu} (\bar{T}, \mathbb{R}) [\bar{g}]$, the model $M$ is in $V$, and the set $A$ is in $V$, as desired.
This completes the proof of Claim~\ref{no-set-of-reals-in-vg-non-adr}. \end{proof}
By Claim~\ref{no-bounded-subset-in-vg-non-adr} and Claim~\ref{no-set-of-reals-in-vg-non-adr}, we have that $\mathbb{R}^V = \mathbb{R}^{V[G]}$ and $\wp (\mathbb{R})^V = \wp (\mathbb{R})^{V[G]}$. Since we assume $\mathsf{AD}$ in $V$, the axiom $\mathsf{AD}$ holds in $V[G]$ as well.
This finishes the arguments for Theorem~\ref{thm:subset-of-Theta-positive} in Case~\ref{case:ADRfail} when $\mathsf{AD}_{\mathbb{R}}$ fails.
\begin{case}\label{case:ADR} When $\mathsf{AD}_{\mathbb{R}}$ holds. \end{case}
Recall that we write $\lambda$ for $\Theta^V$. Let $\mathbb{P}$ be $\mathrm{Add} (\lambda , 1)$ in HOD, where $\mathrm{Add} (\lambda , 1) = \{ p \mid p \colon \gamma \to 2 = \{ 0,1 \} \text{ for some $\gamma < \lambda$}\}$. Since $\mathbb{P}$ is computed in HOD and $\lambda = \Theta^V$ is inaccessible in HOD, the set $\mathbb{P}$ can be considered as a poset on $\lambda$.
Let $G$ be a $\mathbb{P}$-generic filter over $V$. We will show that $\mathsf{AD}$ holds in $V[G]$.
\begin{claim}\label{no-new-set-of-reals-adr} The forcing $\mathbb{P}$ does not add any new set of reals, i.e., $\wp(\mathbb{R})^{V} = \wp(\mathbb{R})^{V[G]}$. \end{claim}
\begin{proof}[Proof of Claim~\ref{no-new-set-of-reals-adr}] We will show that for any $f \colon \mathbb{R}^V \to 2$ in $V[G]$, the function $f$ is also in $V$. Since any real in $V[G]$ can be simply coded as a subset of $\mathbb{R}^V$ in $V[G]$, this will show that $\mathbb{R}^V = \mathbb{R}^{V[G]}$ and $\wp (\mathbb{R})^V = \wp (\mathbb{R})^{V[G]}$ as well.
From now on, we write $\mathbb{R}$ for $\mathbb{R}^V$. \begin{subclaim}\label{subclaim} For some sequence $s \in \lambda^{\omega}$, the function $f$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$. \end{subclaim}
\begin{proof}[Proof of Subclaim~\ref{subclaim}] Let $\dot{f}$ be a $\mathbb{P}$-name with $\dot{f}^G= f$. Since $\mathbb{P}$ can be considered as a poset on $\lambda$, we may assume that $\dot{f}$ can be considered as a subset of $\lambda \times \mathbb{R} \times 2$. To make it simpler, we regard $\dot{f}$ as a subset of $\lambda \times \mathbb{R}$.
Since $V= \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$, there is a set $A$ of reals such that $\dot{f}$ is OD from $A$. For each $\alpha < \lambda$, let $X_{\alpha} = \{ x \in \mathbb{R} \mid (\alpha , x) \in \dot{f} \}$ and let $\xi_{\alpha}$ be the least ordinal $\xi < \lambda$ such that $X_{\xi}= X_{\alpha}$. For $\alpha , \beta < \lambda$, we write $\alpha \preceq \beta$ if $\xi_{\alpha} \le \xi_{\beta}$. Then the structure $(\lambda , \preceq)$ is a prewellordering. Let $\pi \colon (\lambda , \preceq ) \to (\gamma , \le)$ be the Mostowski collapsing map. For each $\delta < \gamma$, let $\eta_{\delta} = \min \pi^{-1} (\delta)$ and $Y_{\delta} = X_{\eta_{\delta} }$. Set $Y = ( Y_{\delta} \mid \delta < \gamma)$.
Since $\dot{f}$ is OD from $A$, so is $\pi$. Also $\pi$ is essentially a set of ordinals, so $\pi$ is in $\mathrm{HOD}_{\{ A \}}$.
We next verify that there is a set $B$ of reals such that $Y$ is in $\mathrm{L} (B, \mathbb{R})$. Since we have $\mathsf{AD}_{\mathbb{R}}$ in Case~\ref{case:ADR}, by Theorem~\ref{thm:Solovay}, there is a set $B_0$ of reals which is not OD from $A$ and any real. Since $\dot{f}$ is OD from $A$, so is $Y$. So each set $Y_{\delta}$ of reals is OD from $A$. By the Wadge Lemma under $\mathsf{ZF} + \mathsf{AD}$, each $Y_{\delta}$ is Wadge reducible to $B_0$. In particular, there is a surjection from $\mathbb{R}$ to $\{ Y_{\delta} \mid \delta < \gamma \}$ in $\mathrm{L} (B_0, \mathbb{R})$. Since the sequence $Y = ( Y_{\delta} \mid \delta < \gamma )$ is injective, there is a surjection $\rho \colon \mathbb{R} \to \gamma$ in $\mathrm{L} (B_0, \mathbb{R})$ as well. Let $B_1 = \{ x \ast y \mid x \in Y_{\rho (y)} \}$, where $x \ast y \, (2n) = x(n)$ and $x \ast y \, (2n+1) = y (n)$ for all $n \in \omega$. Then $Y$ is in $\mathrm{L} (\rho , B_1, \mathbb{R})$. So letting $B = B_0 \oplus B_1 = \{ x \ast y \mid x \in B_0 \text { and } y \in B_1 \}$, we have that $Y$ is in $\mathrm{L} (B, \mathbb{R})$, as desired.
We now argue that for some sequence $s \in \lambda^{\omega}$, the $\mathbb{P}$-name $\dot{f}$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$. Since $\pi$ is in $\mathrm{HOD}_{\{ A \}}$ and $Y$ is in $\mathrm{L} (B, \mathbb{R})$, letting $C = A \oplus B$, we have that $\pi$ is in $\mathrm{HOD}_{\{ C \}}$ and $Y$ is in $\mathrm{L} (C, \mathbb{R})$. Since we have $\mathsf{AD}_{\mathbb{R}}$ in Case~\ref{case:ADR}, by Lemma~\ref{thm:ADR-countable-sequences}, there is an $s \in (\Theta^V)^{\omega} = \lambda^{\omega}$ such that $C$ is OD from $s$ and that $C$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$. Hence both $\pi$ and $Y$ are in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$. Since $\dot{f}$ is simply definable from $\pi$ and $Y$, we have that $\dot{f}$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$, as desired.
Since $f = \dot{f}^G$ and $\dot{f}$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R})$, we have that $f$ is in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$.
This completes the proof of Subclaim~\ref{subclaim}. \end{proof}
Since we assume that $\lambda = \Theta^V$ is regular in $V$ and $s \in \lambda^{\omega} \cap V$, we can pick an ordinal $\gamma < \lambda$ such that $s \in \gamma^{\omega}$. Let $\mathbb{Q}_1$ be the Vop\v{e}nka algebra for adding an element of $\gamma^{\omega}$ in $\mathrm{HOD}$ and let $h_s = \{ p \in \mathbb{Q}_1 \mid s \in \pi_1 (p) \}$, where $\pi_1 \colon \mathbb{Q}_1 \to \mathcal{O}_1$ is as in Definition~\ref{def:Vop}. Then by Lemma~\ref{lem:ADR-countable-sequences}, we have that $h_s$ is a $\mathbb{Q}_1$-generic filter over $\mathrm{HOD}$ such that $ \mathrm{HOD}[h_s] = \mathrm{HOD}_{\{ s\}}$.
\begin{subclaim}\label{subclaim2} The ordinal $\lambda$ is regular in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ and there is no surjection from $\mathbb{R}^V$ to $\lambda$ in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$. \end{subclaim}
\begin{proof}[Proof of Subclaim~\ref{subclaim2}]
Let $\mathbb{Q}_{\omega}$ be the finite support limit of the Vop\v{e}nka algebras for adding an element of $\gamma^{\omega}$ in $\mathrm{HOD}$. Since $\mathbb{Q}_1$ is a complete suborder of $\mathbb{Q}_{\omega}$, by Lemma~\ref{lem:ADR-countable-sequences}, there is a $\mathbb{Q}_{\omega}$-generic filter $H$ over $\mathrm{HOD} $ such that $h_s \in \mathrm{HOD}[H]$ and that the set $(\gamma^{\omega})^V$ is countable in $\mathrm{HOD}[H]$. In particular, $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) \subseteq \mathrm{HOD} [h_s] (\mathbb{R}) \subseteq \mathrm{HOD} [H]$ and $\mathbb{R}^V$ is countable in $\mathrm{HOD}[H]$. Since we have $\mathsf{AD}_{\mathbb{R}}$ in Case~\ref{case:ADR}, by Lemma~\ref{lem:ADR-countable-sequences}, the poset $\mathbb{Q}_{\omega}$ is of size less than $\Theta^V = \lambda$. Since $\mathbb{P}$ is $<$$\lambda$-closed in $\mathrm{HOD}$ and $G$ is $\mathbb{P}$-generic over $\mathrm{HOD}$, we have that any subset of $\mathbb{Q}_{\omega}$ in $\mathrm{HOD} [G]$ is also in $\mathrm{HOD}$. Hence the filter $H$ is $\mathbb{Q}_{\omega}$-generic over $\mathrm{HOD} [G]$ as well.
We now argue that $\lambda$ is regular in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$. Since $\mathbb{P}$ is $<$$\lambda$-closed in $\mathrm{HOD}$ and $G$ is $\mathbb{P}$-generic over $\mathrm{HOD}$, the ordinal $\lambda$ is still regular in $\mathrm{HOD}[G]$. Also, since $\mathbb{Q}_{\omega}$ is of size less than $\lambda$ in $\mathrm{HOD}$ and $H$ is $\mathbb{Q}_{\omega}$-generic over $\mathrm{HOD} [G]$, the ordinal $\lambda$ is also regular in $\mathrm{HOD} [G][H]$. Since $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G] \subseteq \mathrm{HOD} [H][G] = \mathrm{HOD} [G][H]$, the ordinal $\lambda$ is regular in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$, as desired.
We next show that there is no surjection from $\mathbb{R}^V$ to $\lambda$ in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$. Since $\mathbb{R}^V$ is countable in $\mathrm{HOD}[H]$ while $\lambda$ is regular uncountable in $\mathrm{HOD}[H][G]$, there is no surjection from $\mathbb{R}^V$ to $\lambda$ in $\mathrm{HOD}[H][G]$. Since $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G] \subseteq \mathrm{HOD}[H][G]$, there is no surjection from $\mathbb{R}^V$ to $\lambda$ in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$, as desired.
This completes the proof of Subclaim~\ref{subclaim2}. \end{proof}
Recall that $\mathbb{Q}_1$ is the Vop\v{e}nka algebra for adding an element of $\gamma^{\omega}$ and $h_s$ is the $\mathbb{Q}_1$-generic filter over $\mathrm{HOD}$ derived from $s$ with $\mathrm{HOD} [h_s] = \mathrm{HOD}_{\{ s\}}$. Since we have $\mathsf{AD}_{\mathbb{R}}$ in Case~\ref{case:ADR}, by Lemma~\ref{lem:ADR-countable-sequences}, the poset $\mathbb{Q}_1$ is of size less than $\Theta^V = \lambda$ in $\mathrm{HOD}$. So the filter $h_s$ is essentially a bounded subset of $\lambda$. Since we assume $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V= \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$'', by Theorem~\ref{thm:HODLA}, there is a set $Z \subseteq \Theta^V = \lambda$ such that $\mathrm{HOD} = \mathrm{L} [Z]$. So the model $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ is of the form $\mathrm{L} (Z, h_s ,\mathbb{R}) [G]$ where $Z$ is a subset of $\lambda$ and $h_s$ is a bounded subset of $\lambda$. By Subclaim~\ref{subclaim}, the function $f$ is in the model $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$.
We are now ready to finish the arguments for Claim~\ref{no-new-set-of-reals-adr}, which are similar to those for Claim~\ref{no-set-of-reals-in-vg-non-adr}. Let $\nu$ be a sufficiently big cardinal in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ and let $N$ be $V_{\nu}$ in $V[G]$. Let $g = \bigcup G$. Then by the genericity of $G$, we have $g \colon \lambda \to 2$. Since $G$ is simply definable from $g$, we also have $\mathrm{HOD}_{\{ s \}} (\mathbb{R}) [G] = \mathrm{HOD}_{ \{ s \}} (\mathbb{R}) [g]$. Since $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) = \mathrm{L} (Z, h_s , \mathbb{R})$, the model $N$ is of the form $\mathrm{L}_{\nu} (Z , h_s, \mathbb{R}) [G] = \mathrm{L}_{\nu} (Z , h_s , \mathbb{R}) [g]$. Since every element of $N$ is definable from $Z, h_s, g$, an ordinal, and some real while $\lambda$ is regular in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ and there is no surjection from $\mathbb{R}$ to $\lambda$ in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ by Subclaim~\ref{subclaim2}, one can find an elementary substructure $X$ of $N$ in $\mathrm{HOD}_{\{ s\}} (\mathbb{R}) [G]$ such that $\mathbb{R} \subseteq X$, $\lambda \cap X \in \lambda$, the structure $X$ is a surjective image of $\mathbb{R}$, and $Z, \mathbb{P} , h_s, G, f \in X$. Let $M$ be the transitive collapse of $X$ and let $\pi \colon M \to X$ be the inverse of the collapsing map. Then letting $\kappa = \lambda \cap X$, the critical point of $\pi$ is $\kappa$ and $\pi (\kappa) = \lambda$. For any $a$ in $X$, we write $\bar{a}$ for $\pi^{-1} (a)$, i.e., $\pi ( \bar {a}) = a$.
We will finish arguing that the function $f$ is in $V$. Since $\mathbb{R}$ is contained in $M$ and $\pi (\bar{f}) = f$, we have $\bar{f} = f$ and the set $f$ is in $M$. Hence it is enough to verify that the model $M$ is in $V$. Since $N$ is of the form $\mathrm{L}_{\nu} (Z, h_s, \mathbb{R}) [g]$, the set $M$ is of the form $\mathrm{L}_{\mu} (\bar{Z}, \bar{H}_s, \mathbb{R}) [\bar{g}]$ for some ordinal $\mu$. Since $Z$ is a subset of $\lambda$, we have $\bar{Z} = Z \cap \kappa$, which is in $V$.
The filter $h_s$ is essentially bounded subset of $\lambda$ and $h_s$ is in $X$. So by elementarity of $X$, we have that $h_s \subseteq X$ and it follows that $\bar{h}_s = h_s$, which is also in $V$. Since $g \colon \lambda \to 2$, we have $\bar{g} = g \upharpoonright \kappa$ and so $\bar{g}$ is in $\mathbb{P}$. Hence we have $\bar{g} \in V$.
Since $M = \mathrm{L}_{\mu} (\bar{Z}, \bar{h}_s , \mathbb{R}) [\bar{g}]$, the model $M$ is in $V$, and hence the function $f$ is in $V$, as desired.
This completes the proof of Claim~\ref{no-new-set-of-reals-adr}. \end{proof}
By Claim~\ref{no-new-set-of-reals-adr}, we have $\wp(\mathbb{R})^{V[G]} = \wp(\mathbb{R})^{V}$. Since $\mathsf{AD}$ holds in $V$, so does in $V[G]$, as desired.
This finishes the arguments for Theorem~\ref{thm:subset-of-Theta-positive} in Case~\ref{case:ADR} when $\mathsf{AD}_{\mathbb{R}}$ holds.
This completes of the proof of Theorem~\ref{thm:subset-of-Theta-positive}. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:subset-of-Theta-negative}]
Let $G$ be any $\mathbb{P}$-generilc filter over $V$. We will show that $\mathsf{AD}$ fails in $V[G]$. To derive a contradiction, we assume $\mathsf{AD}$ in $V[G]$.
Since we have $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$'' in $V$, by Theorem~\ref{thm:increasing-Theta-destruction}, it is enough to show that $\mathbb{P}$ increases $\Theta$, i.e., $\Theta^V < \Theta^{V[G]}$.
Let $\gamma$ be the cofinality of $\Theta$ in $V$. Since $\Theta$ is singular in $V$, we have that $\gamma < \Theta$.
We will show that there is an injection from $\Theta^V$ to $\wp (\gamma)^{V[G]}$ in $V[G]$, which would imply $\Theta^V < \Theta^{V[G]}$ as follows: Since we assumed $\mathsf{AD}$ in $V[G]$, by Theorem~\ref{thm:CodingLemma}, there is a surjection from $\mathbb{R}^{V[G]}$ to $\wp (\gamma)^{V[G]}$ in $V[G]$. By the existence of an injection from $\Theta^V$ to $\wp (\gamma)^{V[G]}$, there would be a surjection from $\mathbb{R}^{V[G]}$ to $\Theta^V$ in $V[G]$. By the definition of $\Theta^{V[G]}$, we would have that $\Theta^V < \Theta^{V[G]}$, as desired.
We will construct a function $\iota \colon \Theta^V \to \wp (\gamma)^{V[G]}$ in $V[G]$ which is verified to be injective. Since $\mathbb{P} = \mathrm{Add} (\Theta , 1)$ in HOD and $G$ is $\mathbb{P}$-generic over $V$, the set $g = \bigcup G$ is a function from $\Theta^V$ to $2 = \{ 0, 1\}$. Since $\gamma$ is the cofinality of $\Theta$ in $V$, we can fix a cofinal increasing sequence $(\beta_{\alpha} \colon \alpha < \gamma )$ in $\Theta$ in $V$.
For each $\delta < \Theta^V$, let $a_{\delta}$ be the sequence $(\beta_{\alpha} + \delta \mid \alpha < \gamma)$ in $\Theta$ in $V$. Now let $\iota (\delta) = \{ \alpha < \gamma \mid g \bigl( a_{\delta} (\alpha) \bigr) =1 \}$. Then $\iota (\delta)$ is a subset of $\gamma$ for each $\delta < \Theta^V$.
We will verify that the function $\iota \colon \Theta^V \to \wp (\gamma)^{V[G]}$ is injective. Let $\delta , \epsilon$ be distinct ordinals less than $\Theta^V$. We will see that $\iota (\delta) \neq \iota (\epsilon)$. First notice that the functions $a_{\delta}$ and $a_{\epsilon}$ are different everywhere: For all $\alpha < \gamma$, we have $a_{\delta} (\alpha) = \beta_{\alpha} + \delta \neq \beta_{\alpha} + \epsilon = a_{\epsilon} (\alpha)$. Now since $a_{\delta}$ and $a_{\epsilon}$ are different everywhere and both are in $V$, by the genericity of $G$, there is an $\alpha < \gamma$ such that $g\bigl ( a_{\delta} (\alpha) \bigr) \neq g \bigl( a_{\epsilon} (\alpha) \bigr)$, and hence $\alpha \in \iota (\delta) \bigtriangleup \iota (\epsilon)$. Therefore, we have $\iota (\delta) \neq \iota (\epsilon)$, as desired.
This completes the proof of Theorem~\ref{thm:subset-of-Theta-negative}. \end{proof}
\section{Questions}\label{sec:Q}
We close this paper by raising two questions.
\begin{Q}\label{q1} Assume $\mathsf{ZF} + \mathsf{AD}$. Let $\mathbb{P}$ be a poset which adds a new real and let $G$ be $\mathbb{P}$-generic over $V$. Then must $\mathsf{AD}$ fail in $V[G]$? \end{Q}
To answer \lq No' to Question~\ref{q1}, one would need to find a poset which changes the structure of cardinals below $\Theta$ drastically as follows: Woodin proved that if there is a poset which adds a new real while preserving the truth of $\mathsf{AD}$, then the poset must collapse $\omega_1$. Also, by Theorem~\ref{thm:increasing-Theta-destruction}, if such a poset exists in a model of $\mathsf{ZF} + \mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl( \wp (\mathbb{R}) \bigr)$'', then the poset must preserve $\Theta$. Furthermore, by the arguments for \cite[Lemma~2.10]{MR4242147} by Chan and Jackson, if a poset adds a new real while preserving the truth of $\mathsf{AD}$, then any weak partition preperty of a cardinal in its generic extension cannot be witnessed by a club in the ground model. Hence, if such a poset preserves $\Theta$ as well, then for cofinaly many cardinals $\kappa$ below $\Theta$, the poset must shoot a club in $\kappa$ which does not contain any club in $\kappa$ in the ground model.
There are many things we do not know on forcings over $\mathsf{ZF} + \mathsf{AD}$ especially when $\Theta$ is singular. One of them is whether the assumption \lq\lq $\Theta$ is regular'' in Theorem~\ref{thm:subset-of-Theta-positive} is essential or not: \begin{Q} Assume $\mathsf{ZF}+\mathsf{AD}^+ + \lq\lq V = \mathrm{L} \bigl(\wp (\mathbb{R})\bigr)"$. Suppose that $\Theta$ is singular. Then is there any poset which adds a new subset of $\Theta$ while preserving $\mathsf{AD}$? \end{Q}
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