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and Lindenstrauss is carried out. It is not immediately clear whether this work implies that LCFF and [MATH] –LC can only fail on a set of Hausdorff dimension zero. If true, this would provide a counterpart to the above–mentioned results by Einsiedler, Katok and Lindenstrauss |
and by Einsiedler and Kleinbock on the sets of exceptions to the Littlewood Conjecture and the [MATH] –adic Littlewood Conjecture, respectively. |
The aim of the present work is to fill up this gap. More precisely, the following theorem shows that [MATH] –LC fails over the field with three elements [MATH] . It provides an explicit counterexample defined from the Paper–Folding sequence [MATH] . Among many other ways, this sequence (also known as the Dragon Curve S... |
[EQUATION] where [MATH] is a non–zero integer . For accounts on some of the properties enjoyed by this sequence see , §6.5] and Theorem 1.1 |
The [MATH] –adic Littlewood Conjecture fails over [MATH] . Indeed, the Laurent series [EQUATION] where [MATH] is the Paper–Folding sequence seen as a sequence defined over [MATH] , is such that |
[EQUATION] If [MATH] is a power series in [MATH] , it can also be seen as an element in [MATH] , where [MATH] with [MATH] . Furthermore, it follows from \tagform@ 1.4 that its absolute value over the latter field equals its absolute value over the former field. Combined with the result of De Mathan and Teulié |
for the case when the ground field is infinite, these two observations lead one to the following corollary: Corollary 1.2 The [MATH] –adic Littlewood Conjecture fails over any ground field with characteristic 3. |
Reduction of the Problem to the Vanishing of certain Hankel Determinants The results established in this section are valid over any ground field [MATH] . Let [MATH] . Given the formulation of [MATH] –LC, one may restrict oneself without loss of generality to the case when the polynomial part of [MATH] vanishes. Write |
[EQUATION] where [MATH] is a sequence in [MATH] which, in what follows, is identified with the power series [MATH] itself. Define the infinite Hankel matrix [MATH] formed from the power series [MATH] as the matrix [MATH] ; that is, as |
[EQUATION] Given indices [MATH] and [MATH] , let [MATH] denote the finite rectangular [MATH] truncation of the previous matrix with top–left entry [MATH] ; that is, |
[EQUATION] (this representation corresponds to the case that [MATH] and is easily adapted to the case that [MATH] ). When [MATH] , one will more conveniently set |
[EQUATION] Definition 2.1 Let [MATH] be an integer. The sequence [MATH] of elements in [MATH] is said to have deficiency [MATH] if there exists integers [MATH] and [MATH] such that the [MATH] matrices |
[EQUATION] are singular but such that in any sequence of [MATH] matrices of the form [EQUATION] (where [MATH] and [MATH] ), at least one of them is non–singular. |
If, for any [MATH] and [MATH] , none of the matrices [MATH] is singular, the sequence [MATH] is said to have deficiency 1. It is said to have unbounded deficiency if for any integer [MATH] , there exist indices [MATH] and [MATH] such that all the matrices \tagform@ 2.2 are singular. |
Say that two finite square submatrices of [MATH] (corresponding to consecutive row and column indices) are nested if they share the same top left entry and if one can be obtained from the other by the addition of a row at the bottom and a column to the right. With this terminology, the sequence [MATH] having deficiency... |
The following theorem reduces [MATH] –LC to considerations of deficiency of sequences in [MATH] Theorem 2.2 Let [MATH] be a sequence in [MATH] identified with the power series [MATH] as in \tagform@ 2.1 . Then [MATH] satisfies equation \tagform@ 1.5 (that is, the [MATH] –adic Littlewood Conjecture over [MATH] is true f... |
Furthermore, [MATH] has deficiency [MATH] if and only if [EQUATION] Proof. Let [MATH] be a non–zero polynomial of degree [MATH] with coefficients in [MATH] . Let [MATH] and [MATH] be integers. Clearly, |
[EQUATION] The fractional part on the left–hand side of the latter inequality can be expanded as follows: [EQUATION] The second inequality in \tagform@ 2.4 means that the coefficients of [MATH] in the above expansion all vanish. Defining [MATH] as the transpose of the row vector [MATH] , this can be restated as follows... |
[EQUATION] Note that [MATH] is a rectangular matrix with dimensions [MATH] . Therefore, equation \tagform@ 2.5 holds for some non–zero vector [MATH] if and only if this matrix does not have maximal rank [MATH] . This shows that |
[EQUATION] The remainder of the proof is split into two parts in order to establish the following claim: the first inequality in \tagform@ 2.4 holds for some integers [MATH] and [MATH] and some non–zero polynomial [MATH] if and only if the sequence [MATH] has deficiency at least [MATH] . The statements in Theorem 2.2 t... |
To begin with, assume that the first inequality in \tagform@ 2.4 holds for some integers [MATH] and [MATH] and some non–zero polynomial [MATH] of degree [MATH] . The argument to prove that the sequence [MATH] has deficiency at least [MATH] is straightforward: the rank condition \tagform@ 2.6 means the [MATH] columns of... |
Conversely, assume that the sequence [MATH] has deficiency at least [MATH] . The goal is to prove that the first inequality in \tagform@ 2.4 holds for some integer [MATH] and some non–zero polynomial [MATH] . This amounts to proving the existence of integers [MATH] such that the rank condition \tagform@ 2.6 holds for t... |
Lemma 2.3 Let [MATH] be an [MATH] Hankel matrix with entries in [MATH] . Assume that the first [MATH] columns of [MATH] are linearly independent but that the first [MATH] columns are linearly dependent (here, [MATH] ). Then the principal minor of order [MATH] , that is, [MATH] , does not vanish. |
Proof. Denote by [MATH] the columns of the matrix [MATH] under consideration. By assumption, [MATH] are linearly independent whereas [MATH] can be expressed as a linear combination of the latter: |
[EQUATION] where [MATH] are coefficients in the field [MATH] . From the Hankel structure of the matrix, this implies that the entries of the matrix [MATH] satisfy the recurrence relation |
[EQUATION] valid for all [MATH] . Write [EQUATION] which is a matrix of rank [MATH] by assumption. It follows from the recurrence relations \tagform@ 2.7 that each of the rows of this matrix depends linearly on the preceding [MATH] rows, hence on the first [MATH] ones. Since the matrix has rank [MATH] , this implies th... |
Since it is assumed that the sequence [MATH] has deficiency at least [MATH] , its infinite Hankel matrix [MATH] contains [MATH] nested singular submatrices, say [MATH] , where [MATH] and where [MATH] . Since the largest of these matrices, viz. [MATH] , is singular, its [MATH] columns are linearly dependent. Since the s... |
This completes the proof of Theorem 2.2 Remark 2.4 It is worthwhile to mention another shorter but less self–contained proof of Theorem 2.2 . A best approximation degree of [MATH] is an integer [MATH] such that there exists a polynomial [MATH] of degree [MATH] with |
[EQUATION] Let [MATH] be the sequence of best approximation degrees of [MATH] and let [MATH] be the associated best approximation polynomials. It is known 40 , §9] that |
[EQUATION] Therefore, [EQUATION] Given [MATH] , let [MATH] be the [MATH] best approximation degree of [MATH] . Then [EQUATION] where |
[EQUATION] On the other hand, a normal index for [MATH] is an integer [MATH] such that the matrix [MATH] is invertible. It is a standard fact in the theory of Padé approximation that easily follows from \tagform@ 2.8 that [MATH] is a normal index if and only if [MATH] is a best approximation degree (see, e.g., 24 , Pro... |
Remark 2.5 Continuing on the previous remark, note that [MATH] is precisely the degree of the rational fraction [MATH] , which is also the degree of the [MATH] partial quotient in the continued fraction expansion of [MATH] (see 40 , §9] for an account on the theory of continued fractions in [MATH] ). Theorem 2.2 may th... |
In view of Theorem 2.2 , Theorem 1.1 becomes an immediate corollary of the following statement, which will be established in the next sections: |
Theorem 2.6 The Paper–Folding sequence [MATH] has deficiency 4 over [MATH] Considerations of deficiency are ubiquitous in the literature due to their connections with linear recurrence sequences, Padé approximations and problems of irrationality. The results known in this topic are nevertheless rather limited. |
When [MATH] , it is not hard to construct a sequence with deficiency 1 by requiring that it should increase sufficiently fast. The situation turns out to be much more complicated in the case that one has to determine the deficiency of a given sequence. The fundamental work by Allouche, Peyrière, Wen and Wen |
establishes, with the help of sixteen recurrence relations, that the principal minors of the infinite Hankel matrix of the Thue–Morse sequence never vanish. Coons |
obtained a result with a similar flavour in the case of sequences defined from the sum of the reciprocals of the Fermat numbers. A combinatorial and simpler proof of both of these results was later provided by Bugeaud and Han |
. Coons and Vrbik also considered the case of the Paper–Folding sequence over [MATH] and established computationally that a large (finite) number of the principal minors of its Hankel matrix does not vanish. For further recent results on the non–vanishing of principal minors for classes of real sequences, the reader is... |
The problem is even less understood over finite fields. The only reasonably complete result seems to be , where it is shown that some sequences defined as Hankel determinants of the Thue–Morse sequence over [MATH] are 2–automatic. In the same paper are also considered the determinants of the successive nested submatric... |
A remarkable exception to the current poor understanding of the structure of determinants formed from square submatrices in the infinite Hankel matrix of given sequences is due to Kamae, Tamura and Wen |
. These authors consider the Fibonacci word over an alphabet [MATH] . Substituting [MATH] or [MATH] determines two real sequences and the corresponding infinite Hankel matrices. Rather explicit formulae are provided in |
for the determinant of any square submatrix sitting in these matrices. These formulae imply, in particular, that the Fibonacci sequences defined this way have unbounded deficiency. To the best of the authors’ knowledge, this, together with the results of the present paper, constitutes the only cases when the determinan... |
In what follows, Theorem 2.6 will be proved by introducing the concept of a Number Wall, which is an array containing information about the Hankel determinants formed from a given sequence. |
The Number Wall of a Sequence 3.1 Definition and Properties Let [MATH] be a doubly–infinite sequence defined over a field [MATH] . The Number Wall of this sequence is a two–dimensional array [MATH] defined as follows: for any [MATH] [MATH] is the Toeplitz determinant |
[EQUATION] when [MATH] and [MATH] (each diagonal contains the same entry); [MATH] when [MATH] and [MATH] , and [MATH] when [MATH] and [MATH] . In the Number Wall [MATH] |
rows and columns are indexed as in standard matrix notation; that is, the first index [MATH] records the row number and increases towards page bottom and the second index [MATH] records the column number and increases towards page right. This convention will be adopted throughout the paper for any array of numbers in t... |
A Number Wall records Toeplitz determinants of a sequence rather than its Hankel determinants. This enables one to express the properties of such a Wall (see below) in a symmetrical way (this insight is owed to John Conway). Since a Toeplitz determinant as above is obtained under reflection and sign change [MATH] from ... |
For the sake of simplicity of notation, set from now on [EQUATION] for any [MATH] . Properties of Number Walls have been extensively studied in |
. The most fundamental of them, which turns out to be a particular case of the Desnanot–Jacobi identity for determinants, can be stated as follows: |
Theorem 3.1 For any [MATH] [EQUATION] Proof. See 31 , p.8] It readily follows from this theorem that the entry [MATH] in row [MATH] in the Number Wall can be computed from entries in rows [MATH] and [MATH] |
provided that [MATH] does not vanish . In the case, however, that the this quantity vanishes, the above formula cannot be used anymore. A remarkable feature of Number Walls is that such zero entries can only occur in very specific shapes: |
Theorem 3.2 Zero entries in a Number Wall can only occur within windows ; that is, within square regions with horizontal and vertical edges. |
Proof. See 31 , p.9] For the sake of brevity, a window in a Number Wall containing only zero entries will from now on be referred to as a window . In what follows, it will be convenient to define (with a slight abuse of terminology) the deficiency of a window in a Number Wall as being equal to [MATH] if the window has ... |
Figure below depicts such window. The entries surrounding such a region (here corresponding to the sequences [MATH] and [MATH] ) will be referred to as the inner frame of the window. The entries surrounding the inner frame (here corresponding to the sequences [MATH] and [MATH] ) will be referred to as the outer frame o... |
Extending in the natural way the definition of the deficiency of a one–sided sequence (see Definition 2.1 ) to a doubly infinite sequence [MATH] , this concept can be reinterpreted in terms of some properties satisfied by the Number Wall [MATH] of [MATH] |
To this end, note first that, with the notation of Section , the determinant of the Hankel matrix [MATH] (where [MATH] and [MATH] ) is, up to a possible change of sign, the entry [MATH] of the Number Wall [MATH] . It thus follows from Definition 2.1 that [MATH] has deficiency [MATH] if and only if its Number Wall admit... |
Theorem 3.3 The inner frame of a window with finite deficiency [MATH] comprises four geometric sequences, along top, left, right and bottom edge with ratios [MATH] and [MATH] respectively, from origins at top–left and bottom–right corner (see Figure ). Furthermore, these ratios satisfy the relation |
[EQUATION] Proof. See 31 , p.11] Corollary 3.4 With the notation of Figure and Theorem 3.3 , the inner frame sequences (denoted by [MATH] [MATH] [MATH] and [MATH] ) satisfy the relation |
[EQUATION] for any [MATH] Proof. See 31 , p.11] Theorem 3.5 With the notation of Figure and Theorem 3.3 , the outer frame sequences (denoted by [MATH] [MATH] [MATH] and [MATH] ) lying immediately outside the inner frame sequences (denoted by [MATH] [MATH] [MATH] and [MATH] respectively) and aligned with them satisfy th... |
[EQUATION] for [MATH] Proof. See 31 , p.11] The relations above constitute the set of frame constraints of a Number Wall. They can be rephrased all together as follows: |
Corollary 3.6 (Frame Constraints) Given a doubly infinite sequence [MATH] over a ground field [MATH] , its Number Wall [MATH] is [MATH] in terms of the previous rows. More precisely, with the notation of Figure and Theorem 3.3 , given [MATH] |
[EQUATION] (In the last two equations above, the index [MATH] is determined in the natural way from [MATH] and [MATH] .) Conversely, an array satisfying this recurrence is the Number Wall of the sequence determined by [MATH] for all [MATH] |
Proof. This follows immediately from Theorem 3.3 , Corollary 3.4 and Theorem 3.5 . Note that all denominators are guaranteed to be non–zero. Also note that the parameters [MATH] [MATH] [MATH] and [MATH] required for these equations to be a recurrence relation are well–determined from previous rows when not within a win... |
The Frame Constraints expressed in Corollary 3.6 provide a necessary and sufficient condition for an infinite array to be the Number Wall of a sequence. They also give a Wall Builder Algorithm for generating finite segments of a given Number Wall as well as an alternative method for verifying its properties. This algor... |
3.2 On the Number Wall of the Paper–Folding Sequence over [MATH] Extend the Paper–Folding sequence to a doubly infinite sequence [MATH] by setting [MATH] and by defining [MATH] for [MATH] |
via formula \tagform@ 1.6 , where the integer [MATH] is then negative (in other words, [MATH] for any [MATH] ). Note that [MATH] . Consider then the [MATH] Hankel matrix formed from these three values together with [MATH] and [MATH] ; that is, consider the matrix |
[EQUATION] Clearly, all three principal minors of this matrix vanish, which shows that the Paper–Folding sequence has deficiency at least 4. The remainder of the proof will thus consist in proving that the Number Wall of this sequence contains no [MATH] zero block. |
Evidence for this conjecture is provided by Figure below, which represents the portion of the Number Wall [MATH] under consideration in the ranges of indices [MATH] and [MATH] (the squares and circles in the figure will be interpreted later). |
The Paper–Folding sequence is well–known to be 2–automatic. From a theorem by Cobham, this amounts to claiming that it is the image, under a coding, of a fixed point of a 2–substitution (see , §6.3] for proofs and definitions). The idea of the proof of Theorem 2.6 is that such a rich structure in the sequence should be... |
Build a suitably large segment of the Paper–Folding sequence and a large portion of its Number Wall; Construct a 2–dimensional substitution and a coding such that the generated tiles cover the portion of the Number Wall under consideration; |
Consider the infinite tiling obtained by these substitution and coding, and show that it is a valid Number Wall (in other words, that it satisfies the Frame Constraints); |
Check that the sequence generating the Number Wall thus obtained is the Paper–Folding sequence by showing that this sequence sits in row [MATH] |
Each of these steps will be detailed in the next sections. Beforehand, some lemmata related to the theory of tilings are proved. They will be needed to implement the above–described strategy. |
Tilings of [MATH] Let [MATH] be a set. Its elements will be referred to as tiles . Fix once and for all an integer [MATH] . A tiling of [MATH] (resp. of [MATH] ) over [MATH] is a function [MATH] (resp. a function [MATH] ). Referring to a tiling without referencing [MATH] or [MATH] will mean either. |
This section partially follows in order to recall a special type of tilings that are Several equivalent characterisations of such tilings are proved in |
. In particular, it is shown therein that this may be taken as a definition for an automatic tiling , some properties of which are established in this section. These properties will enable one to show in Section that the Number Wall of the Paper–Folding sequence is an automatic [MATH] –tiling, which will be used to est... |
Some notation is first introduced. Boldface letters such as [MATH] will denote vectors whose coordinates [MATH] are integers or positive integers, as should be clear from the context. Let [MATH] denote the ceiling of a real number [MATH] and set [MATH] . Given integers [MATH] and [MATH] , let [MATH] be the representati... |
[EQUATION] Also, the integer vector [MATH] (resp. [MATH] [MATH] [MATH] ) will stand for the vector all of whose components are equal to 0 (resp. equal to 1, to 2, to 3). Lastly, the notation [MATH] (resp. [MATH] [MATH] ) will be reserved to denote the vector all of whose components are equal to a given integer [MATH] (... |
4.1 Substitution Tilings Only tilings arising from a special type of substitutions will be required: Definition 4.1 Let [MATH] be a set of tiles and let [MATH] be an integer. A [MATH] –substitution is a map [MATH] . A uniform substitution is a [MATH] -substitution for some [MATH] |
In the above definition, the set [MATH] is the set of mappings from the set of [MATH] –tuples with integer entries between [MATH] to [MATH] to [MATH] . Thus, the [MATH] –substitution [MATH] maps each tile to a collection of [MATH] tiles which can be seen as being arranged in the shape of a [MATH] –dimensional hypercube... |
Given a substitution, one can construct a tiling by applying it again and again and “stacking up” shifts of the outcome. One way of doing so is introduced with the help of the following definition: |
Definition 4.2 Assume that [MATH] is a [MATH] –substitution on [MATH] (where [MATH] ). A tile [MATH] is said to be prolongable if [MATH] |
Assume that [MATH] is a [MATH] –substitution on [MATH] and that [MATH] is prolongable. Define an [MATH] –tiling by the recursive formula |
[EQUATION] and [EQUATION] for any [MATH] . This tiling will be denoted by [MATH] To describe a similar construction of substitution tilings in [MATH] , split first [MATH] into orthants : for any vector [MATH] , the [MATH] orthant of [MATH] is defined as |
[EQUATION] The following definition extends the above notion of prolongability within orthants of [MATH] Definition 4.3 Assume that [MATH] is a [MATH] –substitution on [MATH] (with [MATH] ). For any vector [MATH] , a tile [MATH] is said to be [MATH] –prolongable if [MATH] (recall that [MATH] ). |
Given a [MATH] –tuple [MATH] such that [MATH] is [MATH] –prolongable for [MATH] for any [MATH] , define recursively a [MATH] –tiling [MATH] |
with the help of \tagform@ 4.2 and of the following extension of the initial condition \tagform@ 4.1 : for every [MATH] let [EQUATION] |
The [MATH] –tiling thus obtained will be denoted by [MATH] Example 4.4 (The Thue–Morse [MATH] –tiling) To get used to the above definitions and notation, consider the tiling introduced in , p.20] . It is a substitution tiling of [MATH] over [MATH] given by the [MATH] –substitution |
[EQUATION] applied to the [MATH] –prolongable tile [MATH] A finite portion of the tiling [MATH] can be generated as follows: [EQUATION] |
Recall here the convention that the positive directions for the vertical and horizontal axes are as for matrices; that is, downwards and rightwards, respectively. |
4.2 Coding of a Tiling Given two sets of tiles [MATH] and [MATH] , a coding from [MATH] to [MATH] is a map [MATH] . It will be convenient to reinterpret this definition as a [MATH] –coding so as to fit a more general concept: |
Definition 4.5 Let [MATH] and [MATH] be sets and let [MATH] be an integer. An [MATH] –coding from [MATH] to [MATH] is a map [MATH] |
Given a tiling and a coding, another tiling can be [MATH] and [MATH] be sets. Let [MATH] be an [MATH] –coding from [MATH] to [MATH] , and let [MATH] be any tiling over [MATH] . The image of [MATH] under [MATH] is the tiling over [MATH] denoted by [MATH] and defined as follows: for any integer vector [MATH] in the domai... |
[EQUATION] A tiling which is the image of a uniform substitution tiling under a [MATH] –coding is said to be an automatic tiling . It can be shown that if [MATH] is a uniform substitution and [MATH] is any [MATH] –coding then [MATH] is automatic, i.e., there are a uniform substitution tiliing [MATH] and a [MATH] –codin... |
Example 4.6 The doubly–infinite Paper–Folding sequence introduced in §3.2 is well–known to be [MATH] –substitution [MATH] and the [MATH] –coding [MATH] defined in Figure below, applied to the [MATH] –prolongable and [MATH] –prolongable tiles [MATH] and [MATH] , respectively (see , Example 10.3.3] for further details). ... |
4.3 Consistent Overlaps In this subsection are proved two statements standing at the heart of our approach. To this end, some additional notation and definitions are first introduced. Given [MATH] , denote by |
[EQUATION] the rectangular shape with edges parallel to the coordinate axes and with opposite vertices [MATH] and [MATH] . It will also be convenient to introduce the following notation to exclude some border values of such a parallelepiped: |
[EQUATION] Finally, define the multiplication of this set by a positive integer [MATH] in the natural way: [EQUATION] Definition 4.7 |
Given a set of tiles [MATH] , a rectangular pattern over [MATH] is a map [MATH] for some [MATH] . In the case that [MATH] for some [MATH] , the map [MATH] is referred to as an [MATH] –pattern |
The rectangular pattern [MATH] is contained in [MATH] , which stands either for a tiling or for another rectangular pattern, if there exists [MATH] such that |
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