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Corollary 9.27 A restriction of an arbitrary Möbius measure to the chart \( {FC}{F}_{1, l} \) is proportional to \( {\mu }_{{\omega }_{l}} \) . | Proof The statement follows from the proportionality of any two Möbius measures. | No |
Proposition 9.28 The form \( {\omega }_{l}\left( {{x}_{l},{y}_{l}}\right) \) is extendible to some form \( {\omega }_{1} \) of \( {FC}{F}_{1} \) . In coordinates \( \left( {{\varphi }_{1},{\varphi }_{2}}\right) \), the form \( {\omega }_{1} \) can be written as follows:\n\n\[ \n{\omega }_{1} = \frac{1}{4}{\cot }^{2}\le... | The proof of Proposition 9.28 is left as an exercise for the reader. | No |
Proposition 9.30 For every positive integer \( k \), the following holds:\n\n\[ \n{\mu }_{1}\left( {{}^{\prime \prime }{k}^{\prime \prime }}\right) = \ln \left( {1 + \frac{1}{k\left( {k + 2}\right) }}\right) .\n\] | Proof Consider a particular representative of an integer-linear type of a length- \( k \) segment: the segment with vertices \( \left( {0,1}\right) \) and \( \left( {k,1}\right) \) . The one-dimensional continued fraction contains the segment as an edge if and only if one of the straight lines defining the fraction int... | Yes |
Proposition 10.5 Consider an arbitrary nonzero real number \( \alpha \) . Then all its con-vergents are best approximations for \( \alpha \) . | Proof Let \( {p}_{k}/{q}_{k} \) be the \( k \) -convergent to \( \alpha \) (we assume that the integers \( {p}_{k} \) and \( {q}_{k} \) are relatively prime). We prove the proposition by induction on \( k \) .\n\nBase of induction. If the denominator \( {q}_{k} \) equals 1, then we have either \( \lfloor \alpha \rfloor... | Yes |
Lemma 10.6 Consider a nonzero real number \( \alpha > 0 \) with regular continued fraction \( \left\lbrack {{a}_{0};{a}_{1} : \cdots }\right\rbrack \) . Let \( {p}_{i}/{q}_{i} \) be its \( i \) -convergent for \( i = 0,1,2,\ldots \) . Then for every \( k \geq 1 \) every best approximation \( p/q \) of \( \alpha \) sati... | Proof Let \( A = \left( {{p}_{k - 1},{q}_{k - 1}}\right) \) and \( B = \left( {{p}_{k},{q}_{k}}\right) \) . Suppose that the lines \( {l}_{1} \) and \( {l}_{2} \) are as above (see Fig. 10.1).\n\nConsider \( E\left( {p, q}\right) \) . If \( \operatorname{ld}\left( {E,{OA}}\right) > 1 \), then the triangle \( \bigtriang... | Yes |
Corollary 10.7 All best approximations of \( \alpha = \left\lbrack {{a}_{0};{a}_{1} : \cdots }\right\rbrack \) satisfying\n\n\[ \n{q}_{k} < q \leq {q}_{k + 1} \]\n\nare of the form \( \left\lbrack {{a}_{0};\cdots : {a}_{k} : m}\right\rbrack \), where \( m \) is a positive integer not greater than \( {a}_{k} \) . | Proof From Lemma 10.6 we know that all best approximations satisfying the condition of the corollary are on a line \( {l}_{2} \) . All integer points of this line are of the form \( B + {mOA} \) for an integer \( m \), or in other words,\n\n\[ \n\left( {{p}_{k},{q}_{k}}\right) + m\left( {{p}_{k - 1},{q}_{k - 1}}\right)... | Yes |
Proposition 10.8 Every strong best approximation of a real number \( \alpha \) is a best approximation of \( \alpha \) . | Proof Consider a strong best approximation \( p/q \) of \( \alpha \) . Then for every rational number \( {p}^{\prime }/{q}^{\prime } \) with \( 0 < q < {q}^{\prime } \) we get\n\n\[ \left| {{q}^{\prime }\alpha - {p}^{\prime }}\right| > \left| {{q\alpha } - p}\right| \]\n\nSince \( 0 < {q}^{\prime } < q \), we get\n\n\[... | Yes |
Theorem 10.9 The set of all strong best approximations of a real number \( \alpha \) coincides with the set of all convergents of \( \alpha \) . | Proof Let \( {p}_{i}/{q}_{i} \) denote the \( i \) -convergent of \( \alpha = \left\lbrack {{a}_{0};{a}_{1} : {a}_{2} : \cdots }\right\rbrack \) .\n\nFrom Proposition 10.8 it follows that every strong best approximation is a best approximation. By Theorem 10.3 every best approximation is on a sail of the corresponding ... | Yes |
Example 10.11 Consider the example of \( \alpha = \frac{47}{21} = \left\lbrack {2;4 : 5}\right\rbrack \) . All best approximations and strong best approximations are shown in the following table. | <table><thead><tr><th>Approximations</th><th>[2]</th><th>\( \left\lbrack {2;3}\right\rbrack \)</th><th>\( \left\lbrack {2;4}\right\rbrack \)</th><th>\( \left\lbrack {2;4 : 3}\right\rbrack \)</th><th>\( \left\lbrack {2;4 : 4}\right\rbrack \)</th><th>\( \left\lbrack {2;4 : 5}\right\rbrack \)</th></tr></thead><tr><td>Best... | Yes |
Let us write the Markov-Davenport form for the arrangement of eigenlines of the Fibonacci operator\n\n\\[ \n\\left( \\begin{array}{ll} 1 & 1 \\\\ 1 & 0 \\end{array}\\right)\n\\]\n\nThe Fibonacci operator has two eigenlines,\n\n\\[ \ny = - {\\theta x}\\;\\text{ and }\\;y = {\\theta }^{-1}x,\n\\]\n\nwhere \\( \\theta \\)... | The Markov-Davenport form of the Fibonacci operator is\n\n\\[ \n\\frac{\\left( {y + {\\theta x}}\\right) \\left( {y - {\\theta }^{-1}x}\\right) }{-\\theta - {\\theta }^{-1}} = \\frac{1}{\\sqrt{5}}\\left( {{x}^{2} - {xy} - {y}^{2}}\\right) .\n\\] | Yes |
Theorem 10.19 Consider an arrangement \( R \) of the lines \( y = {\alpha }_{1}x \) and \( y = {\alpha }_{2}x \) . Let \( {\alpha }_{1} \) and \( {\alpha }_{2} \) be real numbers having infinite continued fractions with bounded elements. Then there exist constants \( {C}_{1},{C}_{2} > 0 \) such that for every integer \... | This theorem is a reformulation of one of the results of [101]. We begin the proof with the following two technical lemmas.\n\nLet \( {R}_{{\delta }_{1},{\delta }_{2}} \) denote the arrangement of lines \( y = \left( {{\alpha }_{i} + {\delta }_{i}}\right) x \) for \( i = 1,2 \) . | No |
Lemma 10.21 Let \( \varepsilon \) be a positive real number. Suppose that real numbers \( {\delta }_{1} \) and \( {\delta }_{2} \) satisfy \( \left| {\delta }_{1}\right| < \varepsilon \) and \( \left| {\delta }_{2}\right| < \varepsilon \) . Then the following estimate holds:\n\n\[ \rho \left( {R,{R}_{{\delta }_{1},{\de... | Proof The statement of the lemma follows directly from the estimate for \( R - {R}_{{\delta }_{1},{\delta }_{2}} \) , shown in (10.3). For the coefficient at \( {x}^{2} \) we have\n\n\[ \left| \frac{{\alpha }_{1}^{2}{\delta }_{2} - {\alpha }_{2}^{2}{\delta }_{1} + {\alpha }_{1}{\delta }_{1}{\delta }_{2} - {\alpha }_{2}... | Yes |
Let \( M \) be an arbitrary positive integer. Consider an arrangement \( R \) of the lines \( y = 0 \) and \( y = {\alpha x} \), where the continued fraction \( \left\lbrack {{a}_{0};{a}_{1} : \cdots }\right\rbrack \) for \( \alpha \) is defined inductively:\n\n- \( {a}_{0} = 1 \) ;\n\n- let \( {a}_{0},\ldots ,{a}_{k} ... | Proof For every \( i \) we have\n\n\[ {q}_{i + 1} \geq {a}_{i}{q}_{i} = {q}_{i}^{M - 1}{q}_{i} = {q}_{i}^{M}. \]\n\nTherefore, the magnitude of best approximations for \( k \) -convergents are as follows:\n\n\[ \left| {{\alpha }_{1} - \frac{{p}_{k}}{{q}_{k}}}\right| \geq \frac{1}{{q}_{k}\left( {{q}_{k + 1} + {q}_{k}}\r... | Yes |
Lemma 10.26 Let \( v \) be an integer point in the \( w \) -continued fraction of an algebraic arrangement \( R \) . Then\n\n\[ \left| {{\Phi }_{R}\left( v\right) }\right| \geq w{\alpha }_{R} \] | Proof The statement follows directly from the fact that the \( w \) -continued fraction of \( R \) is homothetic to the continued fraction of \( R \) . | No |
Theorem 10.27 Let \( R \) be an algebraic arrangement. Then there exists \( C > 0 \) such that for every \( N \in {\mathbb{Z}}_{ + } \) the following holds: each of the two lines of the best approximation arrangement \( {R}_{N} \) contains an integer point of some \( w \) -sail for \( w < C \) . | Proof Let \( R \) be an arrangement of lines \( y = {\alpha }_{1}x \) and \( y = {\alpha }_{2}x \) . Let also \( {v}_{1, N}\left( {{x}_{1, N},}\right. \) \( \left. {y}_{1, N}\right) \) and \( {v}_{2, N}\left( {{x}_{2, N},{y}_{2, N}}\right) \) be two integer vectors of unit integer length in the lines defining \( {R}_{N... | Yes |
Proposition 10.28 Let \( m \) and \( n \) be two integers. Supposing that \( \left| {{\alpha }_{1} - \frac{p}{q}}\right| < \delta \) (or \( \left. {\left| {{\alpha }_{2} - \frac{p}{q}}\right| < \delta \text{, respectively}}\right) \), then the following holds:\n\n\[ \left| {{\alpha }_{1} - \frac{p}{q}}\right| > \frac{\... | Proof We have\n\n\[ \left| {{\alpha }_{1} - \frac{p}{q}}\right| = \frac{1}{q}\left| {p - {\alpha }_{1}q}\right| = \frac{1}{q}\frac{\left| {p - {\alpha }_{1}q}\right| \left( {p - {\alpha }_{2}q}\right) }{p - {\alpha }_{2}q} \]\n\n\[ = \frac{\left| {\Phi }_{R}\left( p, q\right) \right| }{{q}^{2}}\frac{\left| {\alpha }_{1... | Yes |
Proposition 11.5 Let \( \left( {{a}_{0},{a}_{1},\ldots ,{a}_{2n}}\right) \) be a sequence of arbitrary nonzero elements. Consider any broken line \( {A}_{0}\ldots {A}_{n} \) constructed by this sequence as above. Then the following holds:\n\n\[ \n{a}_{2k} = \left| {O{A}_{k} \times O{A}_{k + 1}}\right| ,\;k = 0,\ldots, ... | Proof Let us prove this proposition by induction on \( k \) .\n\nBase of induction. Directly from the first step of the construction we get\n\n\[ \n\left| {O{A}_{0} \times O{A}_{1}}\right| = {a}_{0} \]\n\nInductive Step. Supposing that the statement holds for \( k - 1 \), let us prove it for \( k \) . We start with \( ... | Yes |
Proposition 11.8 Let a broken line be an O-broken line. Then its LLS sequence coincides with its LSLS sequence. | Since the proof of this proposition is a straightforward calculation, we leave it to the reader. | No |
Proposition 11.9 Consider two broken lines \( {A}_{0}\ldots {A}_{n} \) and \( {B}_{0}\ldots {B}_{n} \) whose LLS sequences are \( \left( {{a}_{0},\ldots ,{a}_{2n}}\right) \) and \( \left( {{b}_{0},\ldots ,{b}_{2n}}\right) \) respectively. Suppose that there exists a \( \mathrm{{GL}}\left( {2,\mathbb{R}}\right) \) -oper... | Proof The statement follows directly from formulas of Proposition 11.5, since the area of any parallelogram in the definition is multiplied by \( \lambda \) . | Yes |
Corollary 11.13 Consider two broken lines \( {A}_{0}\ldots {A}_{n} \) and \( {B}_{0}\ldots {B}_{m} \) with LLS sequences \( \left( {{a}_{0},{a}_{1},\ldots ,{a}_{2n}}\right) \) and \( \left( {{b}_{0},{b}_{1},\ldots ,{b}_{2m}}\right) \) respectively. Let \( {B}_{0} = {A}_{0} \) and the vector \( {A}_{0}{A}_{1} \) coincid... | Proof Consider a transformation in the group \( \mathrm{{SL}}\left( {2,\mathbb{R}}\right) \) taking the point \( {A}_{0} \) to \( \left( {1,0}\right) \) and \( {A}_{1} \) to some point on the line \( x = 1 \) . It takes both broken lines to some broken lines with \( {B}_{0} = {A}_{0} = \left( {1,0}\right) \) and the po... | Yes |
Corollary 11.14 (On necessary and sufficient conditions for broken lines to be closed) Consider a broken line \( {A}_{0}{A}_{1}\ldots {A}_{n} \) with the LLS sequence \( \left( {{a}_{0},{a}_{1}}\right. \) , \( \left. {\ldots ,{a}_{2n}}\right) \) . Let \( {A}_{0} = \left( {1,0}\right) \) and \( {A}_{0}{A}_{1} \) be on t... | The condition \( {P}_{{2n} + 1} = 0 \) is equivalent to the following one:\n\n\[ \n\left\lbrack {{a}_{0};{a}_{1} : \cdots : {a}_{2n}}\right\rbrack = 0.\n\] | No |
Example 11.15 Let us write the conditions for a broken line consisting of three edges to form a triangle. Suppose that the LLS sequence of this broken line is \( \left( {{a}_{0},{a}_{1},{a}_{2},{a}_{3},{a}_{4}}\right) \) . Then the conditions are introduced by the following system: | \[ \left\{ \begin{array}{l} {a}_{0}{a}_{1}{a}_{2}{a}_{3}{a}_{4} + {a}_{0}{a}_{1}{a}_{2} + {a}_{0}{a}_{1}{a}_{4} + {a}_{0}{a}_{3}{a}_{4} + {a}_{2}{a}_{3}{a}_{4} + {a}_{0} + {a}_{2} + {a}_{4} = 0, \\ {a}_{1}{a}_{2}{a}_{3}{a}_{4} + {a}_{1}{a}_{2} + {a}_{1}{a}_{4} + {a}_{3}{a}_{4} + 1 = 1. \end{array}\right. \] | Yes |
Problem 6 Consider a broken line \( L \) and a proper Euclidean transformation \( T \) (the origin may not be preserved by \( T \) ). Find the relation between the elements of the LLS sequences of the broken lines \( L \) and \( T\left( L\right) \) . | Due to Proposition 11.9 it is sufficient to solve this problem only for translations by a vector. | No |
Proposition 11.17 (Areal density and the Kepler's second law) Suppose that a body moves along the curve \( \gamma \) with velocity \( 1/A \) . Then the sector area velocity of the body is constant and equals 1 . | The proposition follows directly from the definition. | No |
Theorem 11.19 Suppose that we know the areal density \( A\left( t\right) \) smoothly depending on a parameter \( t \) in some neighborhood of \( {t}_{0} \), the starting position \( \gamma \left( {t}_{0}\right) \), and the origin \( O \) . - If \( \left| {A\left( {t}_{0}\right) }\right| > \left| {{O\gamma }\left( {t}_{... | Proof In polar coordinates \( \left( {r,\varphi }\right) \) with center at \( O \), the curve \( \gamma \) is defined by the following system of differential equations: \[ \left\{ \begin{array}{l} {r}^{2}\dot{\varphi } = A, \\ {\dot{r}}^{2} + {r}^{2}{\dot{\varphi }}^{2} = 1. \end{array}\right. \] This system is equival... | Yes |
Theorem 11.20 Let \( \gamma \) be a \( {C}^{2} \) -curve with arc-length parameterization. Then the sequence of functions \( \left( {A}_{n}\right) \) converges pointwise to the function \( A \), and the sequence of functions \( \left( {B}_{n}\right) \) converges pointwise to the function \( B \) . | Proof This follows directly from the definition of density functions and the properties of LLS sequences shown in Proposition 11.5. | No |
Proposition 12.6 The revolution number of any extended angle is well defined. | Proof Consider an arbitrary \( V \) -broken line and the corresponding extended angle \( \angle \left( {V,{A}_{0}{A}_{1}\ldots {A}_{n}}\right) \) . Let\n\n\[ \n{r}_{ + } = \left\{ {V + {\lambda V}{A}_{0} \mid \lambda \geq 0}\right\} \;\mathrm{{and}}\;{r}_{ - } = \left\{ {V - {\lambda V}{A}_{0} \mid \lambda \geq 0}\righ... | No |
Proposition 12.8 Consider a closed integer \( V \) -broken line \( L \) and enumerate all of its integer points \( {A}_{1},\ldots ,{A}_{d} \) (not only vertices). In addition, set \( {A}_{0} = {A}_{d} \) and \( {A}_{d + 1} = {A}_{1} \) . Then we have\n\n\[ \n{\operatorname{Rot}}_{V}\left( L\right) = \frac{1}{4}\mathop{... | For the proof of this proposition we refer to [79] and [211]. | No |
Lemma 12.11 Consider integers \( m, k \geq 1 \), and \( {a}_{i} > 0 \) for \( i = 0,\ldots ,{2n} \) .\n\n(i) Suppose that the LSLS sequences for the extended angles \( {\Phi }_{1} \) and \( {\Phi }_{2} \) are respectively\n\n\[ \left( {{\left( 1, - 2,1, - 2\right) }^{k - 1},1, - 2,1, - 2,{a}_{0},\ldots ,{a}_{2n}}\right... | Proof We start the proof with the first statement of the lemma. Without loss of generality we assume that the vertices of the extended angles \( {\Phi }_{1} \) and \( {\Phi }_{2} \) are at the origin, say\n\n\[ {\Phi }_{1} = \angle \left( {O,{A}_{0}\ldots {A}_{{2k} + n + 1}}\right) , \]\n\n\[ {\Phi }_{2} = \angle \left... | Yes |
Example 12.18 Let \( \Phi = {0\pi } + \) larctan 1 . Then | \[ \Phi + {}_{-3}\Phi = \pi + 1\arctan 1 \] \[ \Phi + {}_{-2}\Phi = \pi + \operatorname{larctan}0, \] \[ \Phi + {}_{-1}\Phi = {0\pi } + \operatorname{larctan}1, \] \[ \Phi { + }_{0}\Phi = {0\pi } + \operatorname{larctan}2, \] \[ \Phi + {}_{1}\Phi = {0\pi } + \operatorname{larctan}\frac{3}{2} \] | No |
Proposition 12.19 The M-sum of extended angles is nonassociative. | Proof For example, let\n\n\[ \n{\Phi }_{1} = {0\pi } + \operatorname{larctan}2 \n\]\n\n\[ \n{\Phi }_{2} = {0\pi } + \operatorname{larctan}\frac{3}{2} \n\]\n\n\[ \n{\Phi }_{3} = {0\pi } + \operatorname{larctan}5 \n\]\n\nThen\n\n\[ \n{\Phi }_{1} + {}_{-1}{\Phi }_{2} + {}_{-1}{\Phi }_{3} = \pi + \operatorname{larctan}4, \... | Yes |
Proposition 12.20 The M-sum of extended angles is noncommutative. | Proof For example, let\n\n\[ \n{\Phi }_{1} = {0\pi } + \operatorname{larctan}1\;\text{ and }\;{\Phi }_{2} = {0\pi } + \operatorname{larctan}\frac{5}{2}. \n\] \n\nThen \n\n\[ \n{\Phi }_{1} + {}_{1}{\Phi }_{2} = {0\pi } + \operatorname{larctan}\frac{12}{7} \neq {0\pi } + \operatorname{larctan}\frac{13}{5} = {\Phi }_{2} +... | Yes |
Theorem 12.24 Consider an extended angle \( \Phi = \angle \left( {V,{A}_{0}{A}_{1}\ldots {A}_{n}}\right) \) . Suppose that the normal form for \( \Phi \) is \( {k\pi } + \varphi \) for some pair \( \left( {k,\varphi }\right) \) . Let \( \left( {{a}_{0},{a}_{1},\ldots ,{a}_{{2n} - 2}}\right) \) be the LSLS sequence for ... | Proof Without loss of generality we assume that \( V \) is the origin \( O,{A}_{0} = \left( {1,0}\right) \), and\n\n\[ {A}_{0} + \frac{1}{{a}_{0}}\operatorname{sgn}\left( {{A}_{0}{O}^{\prime }{A}_{1}}\right) {A}_{0}{A}_{1} = \left( {1,1}\right) . \]\n\n(One can get this after a certain integer affine transformation of ... | Yes |
Lemma 12.27 Let \( \\alpha ,\\beta \\), and \( \\gamma \\) be nonzero integer angles. Suppose that\n\n\[ \n\\bar{\\alpha } + _{u}\\bar{\\beta } + _{v}\\bar{\\gamma } = \\pi \n\]\n\nThen there exists a triangle with three consecutive integer angles proper integer congruent to \( \\alpha ,\\beta \\), and \( \\gamma \\) r... | Proof Let\n\n\[ \nO = \\left( {0,0}\\right) ,\\;A = \\left( {1,0}\\right) ,\\;\\text{ and }\\;D = \\left( {-1,0}\\right) .\n\]\n\nChoose the integer points\n\n\[ \nB = \\left( {{q}_{1},{p}_{1}}\\right) \\;\\text{ and }\\;C = \\left( {{q}_{2},{p}_{2}}\\right)\n\]\n\nwith integers \( {p}_{1},{p}_{2} \\) and positive inte... | Yes |
For every collection of integer angles \( {\alpha }_{i}\left( {i = 1,\ldots, n}\right) \) there exist an integer \( k \geq n - 1 \) and a \( k \) -tuple of integers \( M = \left( {{m}_{1},\ldots ,{m}_{k}}\right) \) such that | Proof Consider any collection of integer angles \( {\alpha }_{i}\left( {i = 1,\ldots, n}\right) \) and set\n\n\[ \Phi = \overline{{\alpha }_{1}}{ + }_{1}\overline{{\alpha }_{2}}{ + }_{1}\cdots { + }_{1}\overline{{\alpha }_{n}} \]\n\nIt is clear that one of the LSLS sequences for \( \Phi \) is obtained by adding the LLS... | Yes |
Example 13.14 Let us construct a projective toric surface having a unique toric singularity with the sail pair \( \left( {7/5,7/3}\right) \) . | Consider \( \alpha \cong \operatorname{larctan}\left( {7/5}\right) \) . First of all, we draw the angle \( \pi - \alpha \) and its adjacent angle \( \pi - \left( {\pi - \alpha }\right) \) (which is \( \alpha \) ). Further, we subdivide the half-plane in the complement to the union of \( \alpha \) and \( \pi - \alpha \)... | Yes |
Proposition 14.1 The index of a sublattice generated by integer vectors \( {v}_{1},\ldots ,{v}_{k} \) in an integer \( k \) -dimensional plane equals the number of all integer points \( P \) satisfying\n\n\[ \n{AP} = \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{v}_{i}\;\text{ with }0 \leq {\lambda }_{i} < 1, i \in... | Proof The proof of this statement is similar to the proof of the planar one. Let \( H \) be a subgroup of \( {\mathbb{Z}}^{2} \) generated by \( {v}_{1},\ldots ,{v}_{k} \) . Define\n\n\[ \n\operatorname{Par}\left( {{v}_{1},\ldots ,{v}_{k}}\right) = \left\{ {\mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{v}_{i} \mid ... | Yes |
Example 14.3 Consider a three-dimensional simplex \( S \subset {\mathbb{R}}^{4} \) with vertices\n\n\[ \n{s}_{1} = \left( {2,3,0,1}\right) ,\;{s}_{2} = \left( {1,4,2,4}\right) ,\;{s}_{3} = \left( {1,0,0,4}\right) ,\;{s}_{4} = \left( {1,0,0,1}\right) .\n\]\n\nThe integer volume of \( S \) is 6 . | We postpone the calculation of the volume of simplices for a while (see Example 14.32 below). | No |
Proposition 14.5 Consider two integer linear spaces \( {L}_{1} \) and \( {L}_{2} \) that are not contained one in another. Let the sets of independent integer vectors\n\n\[ \left\{ {{u}_{1},\ldots ,{u}_{k},{w}_{1},\ldots ,{w}_{m}}\right\} ,\;\left\{ {{v}_{1},\ldots ,{v}_{l},{w}_{1},\ldots ,{w}_{m}}\right\} ,\;\text{ an... | Proof First, let us change the basis \( \left( {w}_{i}\right) \) of \( {L}_{1} \cap {L}_{2} \) to the basis \( \left( {\bar{w}}_{i}\right) \) that generates the integer sublattice in \( {L}_{1} \cap {L}_{2} \) . The value of the formula stays unchanged, since the numerator and the denominator are both divided by \( {\m... | Yes |
Proposition 14.9 Consider two disjoint integer planes \( {\pi }_{1} \) and \( {\pi }_{2} \) . Let \( {L}_{1} \) and \( {L}_{2} \) be the spaces of vectors corresponding to \( {\pi }_{1} \) and \( {\pi }_{2} \), and let a be a vector with one integer endpoint in \( {\pi }_{1} \) and one integer endpoint in \( {\pi }_{2}... | The proof of Proposition 14.9 is similar to the proof of Proposition 14.5, so we skip it here. | No |
Proposition 14.11 Consider an integer simplex \( {A}_{0}{A}_{1}\ldots {A}_{n} \) in \( {\mathbb{R}}^{n} \) . Then the following statements are equivalent:\n\n(a) \( P\left( {{A}_{0};{A}_{1},\ldots ,{A}_{n}}\right) \) is empty;\n\n(b) \( \operatorname{lv}\left( {{A}_{0}{A}_{1}\ldots {A}_{n}}\right) = 1 \) ;\n\n(c) \( V\... | Proof of Proposition 14.11 (a) \( \Rightarrow \) (b). Let a parallelepiped \( P\left( {{A}_{0};{A}_{1},\ldots ,{A}_{n}}\right) \) be empty. Therefore, by Proposition 14.1 there is only one coset for the subgroup generated by vectors \( {A}_{1}{A}_{0},\ldots ,{A}_{n}{A}_{0} \) . Hence, the vectors \( {A}_{1}{A}_{0},\ldo... | Yes |
Proposition 14.13 For an arbitrary simplex \( S \) of full dimension we have the following formula:\n\n\[\n\operatorname{lV}\left( S\right) = V\left( {P\left( S\right) }\right)\n\] | Proof Let us prove this statement by induction on the integer volume of simplices.\n\nBase of induction. The statement for simplices of integer volume 1 follows directly from Proposition 14.11.\n\nStep of induction. Let the statement hold for all simplices of integer volume less then \( N\left( {N > 1}\right) \). Let u... | Yes |
Theorem 14.15 Let \( S \) be a \( k \) -dimensional integer simplex in \( {\mathbb{R}}^{n} \) and let \( L\left( S\right) \) denote the integer lattice of the \( k \) -dimensional integer plane containing \( S \) . Then we have\n\n\[ \operatorname{lv}\left( S\right) = \frac{V\left( {P\left( S\right) }\right) }{\det \le... | Proof Consider a linear map \( T \) sending \( L \) to \( {\mathbb{Z}}^{k} \subset {\mathbb{R}}^{k} \) . We have\n\n\[ \operatorname{lv}\left( {T\left( S\right) }\right) = \operatorname{lv}\left( S\right) ,\;V\left( {T\left( {P\left( S\right) }\right) }\right) = \frac{V\left( {P\left( S\right) }\right) }{\det \left( {L... | Yes |
Proposition 14.17 The integer volume of polyhedra is additive, i.e., if an integer polyhedron \( P \) is a disjoint union of integer polyhedra \( {P}_{1},\ldots ,{P}_{k} \) then\n\n\[ \operatorname{lV}\left( P\right) = \mathop{\sum }\limits_{{i = 1}}^{k}\operatorname{lV}\left( {P}_{k}\right) \] | Proof By definition the integer volumes of polyhedra of dimension \( n \) are proportional to their Euclidean volumes. Therefore, the additivity of Euclidean volume implies additivity of integer volumes. | No |
Theorem 14.18 For every convex integer polyhedron \( P \) (not contained in a hyperplane) there exists a decomposition into integer empty simplices. | Proof Let us give a sketch of the proof. The proof is based on induction on the number of integer points in \( P \in {\mathbb{R}}^{n} \) . Base of induction. If \( P \) has only \( n + 1 \) integer points inside, then it is an empty simplex. Step of induction. Suppose now that the statement holds for every \( k < m \) ... | Yes |
Theorem 14.19 For every convex integer polyhedron \( P \) in \( {\mathbb{R}}^{3} \) there exists a decomposition of \( {4P} \) into integer tetrahedra congruent to the basis tetrahedron. | Idea of the proof By Theorem 14.18 it is enough to find a proof for the list of empty tetrahedra, which is known due to White's theorem (see Corollary 15.3 below). It turns out that for every empty tetrahedron \( T \in {\mathbb{R}}^{3} \), the tetrahedron \( {4T} \) admits a decomposition into integer tetrahedra congru... | No |
The dimension of the space \( {\Lambda }^{2}\left( {\mathbb{R}}^{4}\right) \) is 6. | The lexicographic basis of \( {\Lambda }^{2}\left( {\mathbb{R}}^{4}\right) \) is as follows:\n\n\[ \left( {{e}_{1} \land {e}_{2},{e}_{1} \land {e}_{3},{e}_{1} \land {e}_{4},{e}_{2} \land {e}_{3},{e}_{2} \land {e}_{4},{e}_{3} \land {e}_{4}}\right) . \] | Yes |
Consider a two-dimensional linear subspace of \( {\mathbb{R}}^{3} \) generated by vectors \( {v}_{1} = \left( {1,1,1}\right) \) and \( {v}_{2} = \left( {1,2,5}\right) \). | \[ {v}_{1} \land {v}_{2} = \left( {{e}_{1} + {e}_{2} + {e}_{3}}\right) \land \left( {{e}_{1} + 2{e}_{2} + 5{e}_{3}}\right) \] \[ = \det \left( \begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right) {e}_{1} \land {e}_{2} + \det \left( \begin{array}{ll} 1 & 1 \\ 1 & 5 \end{array}\right) {e}_{1} \land {e}_{3} + \det \left( \... | Yes |
Theorem 14.30 Let \( {s}_{1}{s}_{2}\ldots {s}_{k + 1} \) be a simplex in \( {\mathbb{R}}^{n} \) and let \( \left( {{p}_{1},\ldots ,{p}_{N}}\right) \) be the lattice Plücker coordinates for the lattice generated by vectors \( {s}_{i}{s}_{k + 1} \) for \( i = 1,\ldots, k \) . Then\n\n\[\n\operatorname{lV}\left( {{s}_{1}{... | Denote by \( {v}_{i} \) the vector \( {s}_{i}{s}_{k + 1} \) for \( i = 1,\ldots, k \) . We begin with the following lemma. | No |
Lemma 14.31 The number \( \gcd \left( {{p}_{1},\ldots ,{p}_{N}}\right) \) is a \( \mathrm{{GL}}\left( {n,\mathbb{Z}}\right) \) -invariant. | Proof Let\n\n\[ \n{v}_{1} \land \cdots \land {v}_{k} = \mathop{\sum }\limits_{{i = 1}}^{N}{p}_{i}{\omega }_{i} \n\] \n\n(as usual, \( \left( {{\omega }_{1},\ldots ,{\omega }_{N}}\right) \) is the lexicographic basis associated to a chosen basis \( \left. \left( {{e}_{1},\ldots ,{e}_{n}}\right) \right) \) . Consider an ... | Yes |
Let us calculate the volume of the three-dimensional tetrahedron \( S = {s}_{1}{s}_{2}{s}_{3}{s}_{4} \) of Example 14.3. Recall that\n\n\[ \n{s}_{1} = \left( {2,3,0,1}\right) ,\;{s}_{2} = \left( {1,4,2,4}\right) ,\;{s}_{3} = \left( {1,0,0,4}\right) ,\;{s}_{4} = \left( {1,0,0,1}\right) .\n\]\n\nThis tetrahedron defines ... | \[ \n= 0 \cdot {e}_{1} \land {e}_{2} \land {e}_{3} + \frac{1}{2} \cdot {e}_{1} \land {e}_{2} \land {e}_{4}\n\]\n\n\[ \n+ 6 \cdot {e}_{1} \land {e}_{3} \land {e}_{4} + {18} \cdot {e}_{2} \land {e}_{3} \land {e}_{4}\n\]\n\nThe lattice Plücker coordinates are \( \left( {0,{12},6,{18}}\right) \), and their greater common d... | Yes |
Theorem 14.36 (E. Ehrhart [54]) Consider an arbitrary convex polyhedron \( P \) in \( {\mathbb{R}}^{n} \) . Let us restrict the variable \( t \) to integers. Then the Ehrhart polynomial of \( P \) is a polynomial of degree \( d \) . | So for the integer variable \( t \) there exist rational numbers \( {a}_{0},\ldots ,{a}_{d} \) such that\n\n\[ L\left( {P, t}\right) = {a}_{d}{t}^{d} + {a}_{d - 1}{t}^{d - 1} + \cdots + {a}_{0}. \] | Yes |
Problem 10 Classify empty integer simplices in dimension \( n \) . | From Proposition 2.11 it follows that all empty triangles are integer congruent to the coordinate triangle \( \left( {n = 2}\right) \) . In Corollary 15.3 all distinct empty three-dimensional tetrahedra are listed \( \left( {n = 3}\right) \) . For the rest of the cases \( \left( {n \geq 4}\right) \), the problem is ope... | No |
Example 15.8 The width of every empty three-dimensional tetrahedron is 1. | The last example means that every empty tetrahedron is contained between two neighboring parallel integer planes (see Fig. 15.3). | No |
Lemma 15.12 Consider an integer parallelepiped \( P = {ABCD}{A}^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime } \) and a plane \( \pi \) parallel to \( {ABCD} \) . Let \( \pi \) intersect the parallelepiped (by some parallelogram). Then the following two statements hold.\n\n(i) The section of the parallelepiped \( P \... | We leave the proof of this lemma as a simple exercise for the reader. | No |
Corollary 15.13 All integer distances from the vertices of an integer (three-dimensional) tetrahedron with empty faces to the opposite faces are equal. | Proof Consider an integer tetrahedron \( {OABC} \) with empty faces. Suppose that\n\n\[ \operatorname{ld}\left( {A,{OBC}}\right) = n\text{.}\]\n\nLet us show that \( \operatorname{ld}\left( {B,{OAC}}\right) = \operatorname{ld}\left( {C,{OAB}}\right) = n \) .\n\nConsider the parallelepiped \( P\left( {OABC}\right) \) . ... | Yes |
Lemma 15.14 There is a unique integer node in the interior of the intersection of the plane \( x + y + z = r + 1 \) and the parallelepiped (here we restrict ourselves to the case \( r > 1 \) ). | Proof Since \( \operatorname{ld}\left( {A,{A}^{\prime }{B}^{\prime }{CD}}\right) = r \) and the parallelogram \( {A}^{\prime }{B}^{\prime }{CD} \) is contained in the plane \( x + y + z = r \), for every integer \( n \), the plane \( x + y + z = n \) is integer. So the plane \( x + y + z = r + 1 \) is also integer.\n\n... | Yes |
Proposition 16.15 Let \( u \in {\mathbb{R}}^{n} \) . Then for every \( \lambda \in \mathbb{R} \smallsetminus S \), where \( S \) is a countable set, the vector \( {\lambda u} \) is affinely irrational. | Proof The set \( S \) is formed by the intersections of the integer affine hyperplanes in \( {R}_{u} \) with the line \( \ell = \{ {\lambda u} \mid u \in \mathbb{R}\} \) . No such integer hyperplane \( \pi \) contains the line \( \ell \), since otherwise, \( \pi \) would contain \( {R}_{u} \) ; hence \( \pi \) intersec... | No |
Lemma 16.18 Consider \( u \in {\mathbb{R}}^{n} \) and let \( \dim {R}_{u} = d \) . Suppose that the integer lattice \( {L}_{u} = {R}_{u} \cap {\mathbb{Z}}^{n} \) has an integer basis \( \left( {{e}_{1},\ldots ,{e}_{d}}\right) \) .\n\n(i) For any nonzero \( \lambda \), we have \( {R}_{u} = {R}_{\lambda u} \) . | Proof Lemma 16.18(i) holds, since the linear spaces defined by \( u \) and \( {\lambda u} \) coincide. | No |
Theorem 16.16(i) and (ii) follow directly from Theorem 16.16(iii), and Theorem 16.16(iv) follows from Theorem 16.16(i). | All these prove the correctness of the step of induction and conclude the proof of the multidimensional Kronecker's approximation theorem. | No |
Proposition 16.19 Consider a simplicial \( n \) -dimensional cone \( C \) in \( {\mathbb{R}}^{n} \) with vertex at the origin. Let \( F \) be a face of \( C \) or else \( F = C \) . Then we have\n\n\[ \overline{\operatorname{A-hull}\left( C\right) } \cap F = \overline{\operatorname{A-hull}\left( F\right) } \oplus F. \] | If the dimension of the lattice contained in \( \operatorname{Span}F \) equals the dimension of \( F \), we have\n\n\[ \overline{\operatorname{A-hull}\left( F\right) } \oplus F = \overline{\operatorname{A-hull}\left( F\right) }.\] | No |
Lemma 16.22 Let \( C \) be an \( n \) -dimensional simplicial cone in \( {\mathbb{R}}^{n} \) and let \( x \) be an interior point of its A-hull. Then the shifted cone \( \{ x\} \oplus C \) is contained in the interior of \( \mathrm{A} \) -hull \( \left( C\right) \) . | Proof Since \( x \) is an interior point of A-hull \( \left( C\right) \), there exists a ball with center at \( x \) contained in A-hull \( \left( C\right) \) . Consider a point \( {x}_{1} \neq x \) in this ball such that \( x - {x}_{1} \in C \) . Then the cone \( \{ x\} \oplus C \) is in the interior of the cone \( \l... | Yes |
Theorem 16.24 Let a simplicial \( n \) -dimensional cone \( C \) be in general position. Then the set \( \operatorname{A-hull}\left( C\right) \) is closed. | We begin the proof of Theorem 16.24 with two lemmas. | No |
Lemma 16.25 Let a simplicial \( n \) -dimensional cone \( C \) be in general position. Then\n\n\[ \overline{\operatorname{A-hull}\left( C\right) } \cap \partial C = \varnothing \text{.} \] | Proof Consider an arbitrary face \( F \) of the cone \( C \) . From Proposition 16.19 we have\n\n\[ \overline{\operatorname{A-hull}\left( C\right) } \cap F = \overline{\operatorname{A-hull}\left( F\right) } \oplus F = \varnothing \oplus F = \varnothing . \]\n\nHence the intersection of \( \overline{\operatorname{A-hull... | Yes |
Theorem 16.29 Let a simplicial \( n \) -dimensional cone \( C \) be in general position. Then we have\n\n(i) all faces of the sail for \( C \) are compact integer polyhedra;\n\n(ii) the set of all vertices of the sail is discrete;\n\n(iii) the sail does not contain rays;\n\n(iv) each vertex of the sail is adjacent to o... | Proof We begin with Theorem 16.29(iii), proving it by contradiction. Let a cone \( C \) satisfy the conditions of the theorem and let its sail contain a ray \( r \) . Consider any support hyperplane of \( \overline{\operatorname{A-hull}\left( C\right) } \) containing this ray. This hyperplane divides the cone into two ... | Yes |
Corollary 16.32 Consider a simplicial \( n \) -dimensional cone \( C \) in \( {\mathbb{R}}^{n} \) . The closure of the A-hull of \( C \) is a quasipolyhedral set if every only if any face of \( C \) containing a nonzero integer point spans an integer affine space (i.e., a space with full-rank integer sublattice in it). | For more information we refer to [125]. | No |
Proposition 16.33 Every n-gon is realizable as a face of an \( m \) -dimensional continued fraction if \( m \geq n - 1 \) . | Proof It is clear that every integer \( n \) -gon \( P \) can be inscribed in an \( m \) -gon \( Q \), for \( m \geq n \), such that the convex hull of all integer points in \( Q \) coincides with \( P \) . We leave the details of the proof to the reader. | No |
Proposition 16.34 For arbitrary integers \( b \geq a \geq 1 \), the quadrangle with vertices \( \left( {-1,0}\right) ,\left( {-a - 1,1}\right) ,\left( {-1,2}\right) ,\left( {b - 1,1}\right) \) cannot be a face of a two-dimensional continued fraction. | Proof Suppose that the statement is false. Let there exist a two-dimensional continued fraction that has one of the compact faces (say \( F \) ) integer equivalent to a quadrangle with vertices \( \left( {-1,0}\right) ,\left( {-a - 1,1}\right) ,\left( {-1,2}\right) ,\left( {b - 1,1}\right) \) for some integers \( b \ge... | Yes |
Theorem 17.6 (Dirichlet's unit theorem) Let \( K \) be a field of algebraic numbers of degree \( n = s + {2t} \) . Consider an arbitrary order \( D \) in \( K \) . Then \( D \) contains units \( {\varepsilon }_{1},\ldots ,{\varepsilon }_{r} \) for \( r = s + t - 1 \) such that every unit \( \varepsilon \) in \( D \) ha... | We refer to [22] for a proof of this theorem. | No |
Proposition 17.8 Consider an arbitrary integer matrix \( A \) with characteristic polynomial irreducible over \( \mathbb{Q} \) . Let \( \xi \) be one of eigenvalues of \( A \) . Then the set \( {h}_{A,\xi }\left( {\Gamma \left( A\right) }\right) \) is an order in \( \mathbb{Q}\left( \xi \right) \) . | Proof First, the set \( {P}_{A} \) described above is closed under addition and multiplication, since if \( {p}_{1}\left( A\right) \) and \( {p}_{2}\left( A\right) \) are integer matrices, then \( \left( {{p}_{1} + {p}_{2}}\right) \left( A\right) \) and \( \left( {{p}_{1}{p}_{2}}\right) \left( A\right) \) are also inte... | Yes |
Theorem 17.9 Consider an arbitrary integer matrix \( A \) with characteristic polynomial irreducible over \( \mathbb{Q} \) . Let \( \xi \) be one of the eigenvalues of \( A \) . Then the Dirichlet group \( \Xi \left( A\right) \) is isomorphic to the multiplicative group of units in the order \( {h}_{A,\xi }\left( {\Gam... | Proof Recall that \( {h}_{A,\xi } \) is a one-to-one map between \( \Xi \left( A\right) \) and \( {h}_{A,\xi }\left( {\Xi \left( A\right) }\right) \) . Letting \( p\left( \xi \right) \in {h}_{A,\xi }\left( {\Gamma \left( A\right) }\right) \) be a unit, i.e., it is invertible, then there exists an element \( q \in P\lef... | Yes |
Proposition 17.11 Consider an arbitrary integer real spectrum matrix \( A \in \) \( \mathrm{{SL}}\left( {n,\mathbb{Z}}\right) \) with characteristic polynomial irreducible over \( \mathbb{Q} \) . Then \( {\Xi }_{ + }\left( A\right) = {\mathbb{Z}}^{n - 1} \) . | Proof Every generator of \( {\Xi }_{ + }\left( A\right) \) is a matrix with positive eigenvalues. Therefore, the operator is not cyclic. Hence \( {\Xi }_{ + }\left( A\right) \) is a free abelian group. Since for every \( B \in \Xi \left( A\right) \) we have \( {B}^{2} \in {\Xi }_{ + }\left( A\right) \), the rank of \( ... | Yes |
Proposition 17.12 For every integer irreducible real spectrum matrix \( A \) the set \( \Gamma \left( A\right) \) forms an additive group isomorphic to \( {\mathbb{Z}}^{n + 1} \) . | Proof In the diagonal basis, the group of all matrices commuting with \( A \) is isomorphic to \( {\mathbb{R}}^{n + 1} \) by addition. Hence the set of all integer matrices forms an integer lattice in this \( \left( {n + 1}\right) \) -dimensional subspace. So the group is isomorphic to \( {\mathbb{Z}}^{k} \) with \( k ... | Yes |
Corollary 17.13 The group \( \Gamma \left( A\right) \) is the intersection of the integer lattice \( {\mathbb{Z}}^{n \times n} \subset \) \( \operatorname{Mat}\left( {n,\mathbb{R}}\right) \) with the space \( \operatorname{Span}\left( {\operatorname{Id}, A,{A}^{2},\ldots ,{A}^{n}}\right) \) . | So in \( n + 1 \) steps we obtain a basis of \( \Gamma \left( A\right) \) that is the integer lattice in the space \( \operatorname{Span}\left( {\operatorname{Id}, A,{A}^{2},\ldots ,{A}^{n}}\right) \) according to Corollary 17.13. | No |
Theorem 17.16 (A.K. Lenstra, H.W. Lenstra, and L. Lovász [130]) Let \( {b}_{1},{b}_{2},\ldots \) , \( {b}_{n} \) be a reduced basis for a lattice \( L \) in \( {\mathbb{R}}^{n} \) . Then we have\n\n(i) \( {\left| {b}_{j}\right| }^{2} \leq {2}^{i - 1}{\left| {b}_{i}^{ * }\right| }^{2} \) for \( 1 \leq j \leq i \leq n \)... | Proof From the size reduced conditions and the Lovász conditions, we have\n\n\[ \n{\left| {b}_{i}^{ * }\right| }^{2} \geq \left( {\frac{3}{4} - {\mu }_{i, i - 1}^{2}}\right) \left| {b}_{i - 1}^{ * }\right| \geq \frac{1}{2}{\left| {b}_{i - 1}^{ * }\right| }^{2} \n\] \n\nfor all admissible \( i \), and therefore,\n\n\[ \... | Yes |
Theorem 17.17 (A.K. Lenstra, H.W. Lenstra, and L. Lovász [130]) Let \( L \subset {\mathbb{Z}}^{n} \) be a lattice with basis \( {b}_{1},{b}_{2},\ldots ,{b}_{n} \), and let \( B \in \mathbb{R}, B \geq 2 \), be such that \( {\left| {b}_{i}\right| }^{2} \leq B \) for \( 1 \leq i \leq n \) . Then the number of arithmetic o... | We are not going to give a proof here, since it is quite technical. The interested reader is referred to the original manuscript [130]. | No |
Proposition 18.2 Let \( A \) and \( B \) be matrices of \( \mathrm{{GL}}\left( {n + 1,\mathbb{R}}\right) \) with distinct real eigenvalues. The continued fractions associated to \( A \) and \( B \) are integer congruent if and only if there exists a matrix \( X \in \mathrm{{GL}}\left( {n + 1,\mathbb{Z}}\right) \) such ... | Proof Let \( A \) and \( B \) be matrices of \( \mathrm{{GL}}\left( {n + 1,\mathbb{R}}\right) \) with distinct real irrational eigenvalues and suppose that their continued fractions are integer congruent. Since the continued fractions are integer congruent, there exists a linear integer lattice-preserving transformatio... | Yes |
The second example was studied by A.D. Bryuno and V.I. Parus-nikov [27]. They constructed the continued fraction that is associated to the following matrix:\n\n\[ M = \left( \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 & 0 \\ 1 & 0 & 0 \end{matrix}\right) \] | The positive Dirichlet group \( {\Xi }_{ + }\left( M\right) \) is generated by the following two matrices:\n\n\[ X = {M}^{2},\;Y = 2\mathrm{{Id}} - {M}^{2}. \] | Yes |
The continued fractions of this series are associated to the following matrices for \( a \geq 0 \) :\n\n\[ \n{M}_{a} = \left( \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & - a - 5 \\ 0 & 1 & a + 6 \end{matrix}\right) \n\] | The positive Dirichlet group \( {\Xi }_{ + }\left( {M}_{a}\right) \) is generated by the following two matrices:\n\n\[ \n{X}_{a} = {M}_{a},\;{Y}_{a} = {\left( {M}_{a} - \mathrm{{Id}}\right) }^{2}. \n\]\n\nThe torus decomposition corresponding to \( {M}_{a} \) is homeomorphic to the following one:\n\n![1c600fd7-25c8-4af... | Yes |
Proposition 18.16 The continued fractions associated to the following two matrices are integer congruent (for integers \( a \geq 0 \) ):\n\n\[ \n{M}_{a,0} = \left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 + a & - a - 2 \end{matrix}\right) ,\;{M}_{a}^{\prime } = \left( \begin{matrix} 0 & 0 & 1 \\ 1 & 0 & - a - 5 \... | Proof The matrices \( {\left( \operatorname{Id} - {M}_{a,0}\right) }^{-1} \) and \( {M}_{a}^{\prime } \) are conjugate by the matrix \( X \) in the group \( \operatorname{SL}\left( {3,\mathbb{Z}}\right) \) :\n\n\[ \nX = \left( \begin{matrix} - 1 & - 1 & - 2 \\ 0 & 0 & - 1 \\ 1 & 0 & - 1 \end{matrix}\right)\n\]\n\n(i.e.... | Yes |
Example 18.25 Consider the torus decomposition consisting of one face with integer affine type of the simplest parallelogram with the vertices \( \left( {0,0}\right) ,\left( {0,1}\right) ,\left( {1,1}\right) \) , and \( \left( {1,0}\right) \) . | The integer distances to this face can be chosen arbitrarily. This decomposition is not realizable for periodic sails of cubic irrationalities. | Yes |
Example 18.26 The following two matrices having the same characteristic polynomial \( {x}^{3} + {11}{x}^{2} - {4x} - 1 \) (and hence the same cubic extension of \( \mathbb{Q} \) ) define integer noncongruent continued fractions: | \[ {\left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & - 2 \end{matrix}\right) }^{3},\;\left( \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 4 & - {11} \end{matrix}\right) \] | Yes |
Theorem 18.28 Let \( {\mathrm{{St}}}_{v} \) be a regular star. Then the set \( \Gamma \left( {\mathrm{{St}}}_{v}\right) \) is bounded. | Proof We prove the statement by induction on the dimension of the star.\n\nBase of induction. The statement clearly holds for every star (i.e., for one point) in \( {\mathbb{R}}^{1} \) .\n\nStep of induction. Let the statement hold for every star in \( {\mathbb{R}}^{d - 1} \) ; we prove the statement in \( {\mathbb{R}}... | Yes |
Corollary 19.6 A restriction of an arbitrary Möbius measure to the chart \( {FC}{F}_{n,\pi } \) is proportional to \( {\mu }_{{\omega }_{\pi }} \) . | Proof The statement follows from the proportionality of any two Möbius measures. | No |
Corollary 19.7 The form \( {\omega }_{\pi } \) is extendible to some form \( {\omega }_{n} \) of \( {FC}{F}_{n} \) . In coordinates \( {v}_{ij} \), the form \( {\omega }_{n} \) is as follows: | \[ {\omega }_{n} = \frac{{\left( -1\right) }^{\lfloor \left( {n + 3}\right) /4\rfloor }}{{2}^{n\left( {n + 1}\right) }}\left( {\mathop{\prod }\limits_{{i = 1}}^{{n + 1}}\mathop{\prod }\limits_{{j = i + 1}}^{{n + 1}}{\cot }^{2}\left( \frac{{\varphi }_{ij} - {\varphi }_{ji}}{2}\right) }\right) \cdot \left( {\mathop{\bigw... | Yes |
Proposition 19.13 In coordinates \( {a}_{1},{b}_{1},{a}_{2},{b}_{2},{a}_{3},{b}_{3} \) the form \( {\omega }_{\pi } \) can be written as follows: | \[ - \frac{{8d}{a}_{1} \land d{b}_{1} \land d{a}_{2} \land d{b}_{2} \land d{a}_{3} \land d{b}_{3}}{{\left( {a}_{3}{b}_{2} - {a}_{2}{b}_{3} + {a}_{1}{b}_{3} - {a}_{3}{b}_{1} + {a}_{2}{b}_{1} - {a}_{1}{b}_{2}\right) }^{3}}. \] | Yes |
Proposition 20.2 Consider an r-dimensional convex polytope \( {p}_{1}\ldots {p}_{s} \), and suppose it spans the plane \( \pi \) of dimension \( r \) . Consider a subset of indices \( {i}_{1} < \cdots < \) \( {i}_{r} \leq s \) . Let the tetrahedron \( T = {p}_{{i}_{1}}\ldots {p}_{{i}_{r}} \) have nonzero Euclidean (or ... | Proof The condition of the first item means exactly that all points are in one half-plane with respect to the hyperplane containing the tetrahedron \( {p}_{{i}_{1}}\ldots {p}_{{i}_{r}} \) .\n\nThe condition of the second item enumerates the points contained in the plane of \( T \) . | Yes |
Proposition 20.7 Let \( V \) be a vertex of the sail of the n-dimensional continued fraction of an \( \left( {n + 1}\right) \) -algebraic irrationality. Then there exists a fundamental domain of the sail such that all vertices of this domain are contained in the convex hull \( H \) of the origin and of the \( {2}^{n} \... | Proof Consider the polyhedral cone \( C \) with vertex at the origin and base at the convex polyhedron with vertices \( {V}_{{\varepsilon }_{1},\ldots ,{\varepsilon }_{n}} \) . We take the union of all images of this polyhedral cone under the actions of the operators with matrices\n\n\[ \n{A}_{{m}_{1},\ldots ,{m}_{n}} ... | Yes |
Theorem 20.15 Let the set of faces \( D \) satisfy the following conditions:\n\n(1) condition (i);\n\n(2) condition (ii);\n\n(3) positivity of all integer distances from the origin to the two-dimensional planes containing faces \( {F}_{i} \) ;\n\n(4) there are no integer points inside the pyramids with vertices at the ... | Let us prove that these seven stages are sufficient for the test.\n\n## 20.2.4.2 Lemma on the Injectivity of the Face Projection\n\nWe prove Theorem 20.15 in four lemmas.\n\nFirst let us give the necessary notation. Let the matrices \( {B}_{1} \) and \( {B}_{2} \) generate \( {\Xi }_{ + }\left( A\right) \) . For any in... | Yes |
For any face of the polygonal surface \( U \), the map \( \pi \) is welldefined and injective on it. | Proof Consider any two-dimensional face \( F \) of the surface \( U \) . By condition 3, the distance from the origin to the plane containing \( F \) is greater than zero. Hence this plane does not contain the origin. Then \( \pi \) is welldefined and injective on \( F \) . | Yes |
Lemma 20.17 Let \( x \) be some point of the open invariant cone \( C \) . Then the union of all faces of \( D \) is contained in a finite union of solid angles of the type \( {B}_{n, m}\left( {N}_{x}\right) \) . | Proof By Dirichlet’s unit theorem it follows that for every interior point \( a \) of the open invariant cone \( C \) there exists an open neighborhood satisfying the following condition. The neighborhood can be covered by four solid angles of the type \( {B}_{n, m}\left( {N}_{x}\right) \) when \( a \) belongs to an ed... | Yes |
Corollary 20.18 Let \( x \) be contained in the open invariant cone \( C \) . Then the solid angle \( {N}_{x} \) contains only points from a finite number of fundamental domains of the type \( {B}_{n, m}\left( D\right) \) . | Proof From the last lemma it follows that \( D \) is contained in the finite union \( \mathop{\bigcup }\limits_{{k = 1}}^{l}{B}_{{n}_{k},{m}_{k}}\left( {N}_{x}\right) \) (for some positive \( l \) ). Then the solid angle \( {N}_{x} \) can contain only points of the fundamental domains \( {B}_{-{n}_{k}, - {m}_{k}}\left(... | Yes |
Corollary 21.7 Consider an \( \\mathrm{{SL}}\\left( {n,\\mathbb{Z}}\\right) \) operator \( A \) with matrix \( M \) and let \( B \\in \) \( \\Xi \\left( A\\right) \) . Then for an arbitrary \( v \) we have \( \\left( {M \\mid v}\\right) = \\left( {M \\mid B\\left( v\\right) }\\right) \) . | Proof Each step of the algorithm produces the same data for \( v \) and \( B\\left( v\\right) \), due to the fact that \( A \) and \( B \) commute. Therefore, \( \\left( {M \\mid v}\\right) = \\left( {M \\mid B\\left( v\\right) }\\right) \) . | Yes |
Proposition 21.9 Every Hessenberg matrix with positive Hessenberg complexity is identified by its Hessenberg type and characteristic polynomial. | Proof Let \( M = \left( {a}_{i, j}\right) \) be a Hessenberg matrix with positive Hessenberg complexity. The first \( n - 1 \) columns of \( \mathrm{M} \) are entirely defined by the Hessenberg type of \( M \) . The last column is uniquely defined from the characteristic polynomial of \( M \) ,\n\n\[ \n{x}^{n} + {c}_{n... | Yes |
Let us examine the Hessenberg type \( \langle 0,1 \mid 1,0,2\rangle \) . The set of all matrices of this Hessenberg type is a two-parameter family with parameters \( m \) and \( n \) : | \[ H\left( {\langle 0,1 \mid 1,0,2\rangle }\right) = \left\{ {\left. {\left( \begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 2 & 1 \end{array}\right) + m\left( \begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}\right) + n\left( \begin{array}{lll} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 2 \end{array}\right) }... | Yes |
Lemma 21.12 Consider an operator \( A \) with integer Hessenberg matrix \( M \) of type \( \Omega \) . Let \( v \) be the vector standing in the last column of \( M \) . Then \( M \in \mathrm{{GL}}\left( {n,\mathbb{Z}}\right) \) if and only if the following conditions hold:\n\n\( - 1\mathrm{\;V}\left( {\sigma \left( \O... | Proof Necessary condition. Consider an operator \( A \) with Hessenberg \( \mathrm{{GL}}\left( {n,\mathbb{Z}}\right) \) matrix \( M \) in some integer basis \( \left\{ {g}_{i}\right\} \) . Denote by \( {S}_{g}^{n - 1} \) the \( \left( {n - 1}\right) \) -dimensional simplex with vertices\n\n\[ O,{g}_{1},\ldots ,{g}_{n -... | Yes |
Example 21.22 Let us study an operator \( A \) with a Frobenius matrix \[ \left( \begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 3 \end{array}\right) \] This operator has one real and two complex conjugate eigenvalues. Therefore, the cone \( {\pi }_{ + } \) for \( A \) is a two-dimensional half-plane. In Fig. 21.2... | In Fig. 21.2b we show the cone \( {\pi }_{ + } \) . The invariant plane separates \( {\pi }_{ + } \) into two parts. The dots on \( {\pi }_{ + } \) are the \( \pi \) -integer points. The boundaries of the convex hulls in each part of \( {\pi }_{ + } \) are two factor-sails. One factor-sail is taken to another by the in... | No |
Proposition 21.27 Consider \( A \in \mathrm{{SL}}\left( {n,\mathbb{Z}}\right) \) and let \( B \in \Xi \left( A\right) \) . Then for an arbitrary point \( v \) we have\n\n\[{\Delta }_{A}\left( v\right) = {\Delta }_{A}\left( {B\left( v\right) }\right)\] | Proof of Proposition 21.27 Since \( B \in \Xi \left( A\right) \), we have the equality \( {A}^{n}B\left( v\right) = \) \( B{A}^{n}\left( v\right) \) . Hence the parallelepiped \( P\left( {A, B\left( v\right) }\right) \) coincides with \( B\left( {P\left( {A, v}\right) }\right) \) . Since \( B \in \mathrm{{SL}}\left( {n... | Yes |
Proposition 21.28 Let \( A \) be an \( \mathrm{{SL}}\left( {n,\mathbb{Z}}\right) \) operator whose characteristic polynomial has distinct roots. Then the MD-characteristic of \( A \) coincides with the absolute value of a form associated to \( A \) for a certain nonzero \( \alpha \) . | Proof Let us consider the formulas of the MD-characteristic of \( A \) in the eigenbasis. We assume that the coordinates in this eigenbasis are \( \left( {{t}_{1},\ldots ,{t}_{n}}\right) \) . Then for any vector \( v = \left( {{t}_{1},\ldots ,{t}_{n}}\right) \) we have\n\n\[ \n{A}^{j}\left( x\right) = \left( {{r}_{1}^{... | Yes |
Proposition 21.29 Consider an operator \( A \) with Hessenberg matrix \( M = \left( {a}_{i, j}\right) \) in some integer basis \( \left\{ {\widetilde{e}}_{i}\right\} \) . The Hessenberg complexity \( \varsigma \left( M\right) \) equals the value of the \( {MD} \) -characteristic \( {\Delta }_{A}\left( {\widetilde{e}}_{... | Proof Denote by \( {V}_{k} \) the plane spanned by vectors \( v, A\left( v\right) ,{A}^{2}\left( v\right) ,\ldots ,{A}^{k - 1}\left( v\right) \) . \n\nLet us inductively show that \n\n\[ \n{A}^{k}\left( {\widetilde{e}}_{1}\right) = \left( {\mathop{\prod }\limits_{{i = 1}}^{k}{a}_{i, i + 1}}\right) {\widetilde{e}}_{k + ... | Yes |
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