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[EQUATION] for every [MATH] Definition 4.8 Given a tiling or a rectangular pattern [MATH] , given an [MATH] –coding [MATH] and an integer [MATH] , the coding [MATH] is said to be consistent with respect to an overlap of [MATH] (or, for short, [MATH] –consistent ) if for every integer vector [MATH] , every index [MATH] ... |
[EQUATION] whenever [MATH] and [MATH] are in the domain of [MATH] . In this equation, [MATH] and [MATH] appear in the [MATH] coordinate, the equality is as functions of the remaining [MATH] variables, and [MATH] stands for the [MATH] vector of the standard basis of [MATH] |
Figure illustrates the concept of a coding with overlap. If [MATH] is a tiling over [MATH] and [MATH] an [MATH] –coding which is [MATH] –consistent, define an [MATH] –coding [MATH] by setting |
[EQUATION] for every [MATH] . The resulting tiling [MATH] will be referred to as a tiling with [MATH] –coding and overlap [MATH] |
For any [MATH] , let [MATH] be the [MATH] –coding defined for a given [MATH] as a map [EQUATION] such that [EQUATION] for all [MATH] |
Lemma 4.9 With the above notation and definitions, the tiling [MATH] satisfies the relation [EQUATION] for any integer vector [MATH] such that both of these quantities are well–defined. |
Proof. Decompose the vectors [MATH] and [MATH] as [EQUATION] where each of the components of the integer vectors [MATH] and [MATH] lies between 1 and [MATH] . Decompose also the vector [MATH] as |
[EQUATION] where here also each of the components of the vector [MATH] lies between 1 and [MATH] Then, [EQUATION] Lastly, it will be useful to extend the definition of a coding with overlap to rectangular patterns: |
Definition 4.10 Let [MATH] be a rectangular pattern on [MATH] and let [MATH] be an [MATH] –consistent [MATH] –coding (where [MATH] ). The image of [MATH] under the coding [MATH] is the rectangular pattern [MATH] defined on |
[EQUATION] by [EQUATION] Here, the [MATH] function is applied to vectors coordinate–wise. Note that if the integer vector [MATH] is decomposed uniquely as |
[EQUATION] with [MATH] , then formula \tagform@ 4.9 can be rewritten more compactly as [EQUATION] This definition is designed to satisfy the following property: |
Lemma 4.11 If [MATH] is a tiling and [MATH] is an [MATH] –coding which is [MATH] –consistent, then for any rectangular pattern [MATH] in [MATH] , it holds that [MATH] is contained in [MATH] . More precisely, if [MATH] is a pattern on [MATH] and [MATH] is an integer vector such that [MATH] for all [MATH] , then |
[EQUATION] for every [MATH] Proof. Decompose any given [MATH] as [EQUATION] with [MATH] and [EQUATION] Furthermore, let [EQUATION] |
with [MATH] and [MATH] for any [MATH] . Then [EQUATION] In these relations, the third equality follows upon applying identity \tagform@ 4.5 |
[MATH] times in the [MATH] coordinate for each [MATH] . The last equality follows from Definition 4.10 and a direct calculation upon using decomposition \tagform@ 4.10 in each coordinate where [MATH] and upon using decompositions \tagform@ 4.10 and \tagform@ 4.11 in each coordinate where [MATH] |
Lemma 4.11 enables one to state the first of the two fundamental results of this subsection: Lemma 4.12 Assume that [MATH] is a tiling over [MATH] and that [MATH] is an [MATH] –coding which is [MATH] –consistent. Then for every integer [MATH] , every [MATH] –pattern in [MATH] is contained in the image under the coding ... |
[EQUATION] Remark 4.13 One could simplify Definition 4.10 and the proofs of Lemmata 4.11 and 4.12 by replacing the domain \tagform@ 4.8 with a smaller one, namely, [MATH] . This would come at the cost of replacing \tagform@ 4.12 with the possibly bigger number [MATH] . For the purpose of the current work, this would on... |
Proof. Assume that [MATH] is an [MATH] –pattern in [MATH] and let [MATH] be an integer vector such that [EQUATION] for every [MATH] . Decompose [MATH] as |
[EQUATION] with [MATH] for every [MATH] . Let [MATH] be the [MATH] –pattern in [MATH] defined by [EQUATION] for all [MATH] It then follows from Lemma 4.11 (applied with the [MATH] -tuple [MATH] all of whose components are equal to [MATH] ) that |
[EQUATION] whenever [EQUATION] This condition holds in the present case. Indeed, since [MATH] and [MATH] for every [MATH] [EQUATION] |
whence the claim from the definition of the quantity [MATH] This concludes the proof of the lemma. Lemma 4.12 will be used in the proof of Theorem 2.6 In order to apply it, one first needs to verify that a given coding is consistent. While this can be a hard task in general, the following statement provides an easy–to–... |
Lemma 4.14 Let [MATH] be a [MATH] –substitution tiling (where [MATH] ). Assume that there are integers [MATH] [MATH] ) such that [MATH] if [MATH] is an [MATH] –tiling and such that [MATH] if [MATH] is a [MATH] –tiling verifying the following condition: every [MATH] –pattern contained in [MATH] is already contained in [... |
Then every [MATH] –pattern in [MATH] is already contained in [MATH] Before proving the lemma, extend first the definition of a [MATH] –substitution [MATH] over a set of tiles [MATH] to rectangular patterns. This can be done in the natural way by setting |
[EQUATION] for all [MATH] Proof. Let [MATH] be the set of tiles and let [MATH] be the substitution defining [MATH] . Let [MATH] be the [MATH] –pattern in [MATH] defined by |
[EQUATION] for some integer vector [MATH] . Assume that [MATH] is contained in [MATH] for some [MATH] . It will be shown by induction on the integer [MATH] that [MATH] also lies in [MATH] . The claim being true by assumption when [MATH] , assume that [MATH] and consider the rectangular pattern |
[EQUATION] defined by [EQUATION] Then, [MATH] is contained in a 2–pattern. Indeed, decompose the [MATH] component [MATH] of [MATH] as |
[EQUATION] and note that [EQUATION] Furthermore, the pattern [MATH] is contained in the image under [MATH] of the pattern [MATH] . Indeed, on the one hand, it easily follows from \tagform@ 4.17 that for any [MATH] one has that |
[EQUATION] On the other hand, [EQUATION] whence the claim. By the induction assumption, the pattern [MATH] , which is contained in [MATH] from its definition, already appears in [MATH] . Its image under [MATH] is therefore contained in the image of [MATH] under [MATH] , which is clearly contained in [MATH] . Thus, from... |
This completes the proof of the lemma. Corollary 4.15 Let [MATH] be a tiling satisfying the assumptions of Lemma 4.14 . Assume that [MATH] is an [MATH] –coding satisfying this property: there exists an integer [MATH] , where [MATH] , such that the consistency condition \tagform@ 4.5 holds for every [MATH] with [MATH] |
Then the coding [MATH] is [MATH] –consistent for the tiling [MATH] Proof. It immediately follows from Lemma 4.14 that the consistency condition \tagform@ 4.5 only needs to be checked in the region [MATH] for it to hold everywhere. |
Remark 4.16 Properties of substitution tilings with overlaps are also studied in , where they are referred to as C3DEL systems — an acronym representing “Deterministic Lindenmayer system with constant width, inflation, encoding and context”. This terminology extends the one presented in , §7.12] |
Generating the Putative Number Wall of the Paper–Folding Sequence over [MATH] The goal of this section is to explain how to generate an automatic Number Wall that coincides with a large segment of the Paper–Folding Number Wall. Verifying that this automatic tiling is in fact equal to the Paper–Folding Number Wall, and ... |
5.1 Building a Finite Portion of a Number Wall The recursive formula in Corollary 3.6 gives a formula for [MATH] in terms of elements occupying previous rows. This might seem impractical at first glance due to infinitely many computations required to calculate each row. However, looking more carefully reveals that the ... |
[EQUATION] given the finite portion [MATH] of the initial sequence [MATH] . The discovery of this algorithm is discussed in . It is described in Figure below. |
5.2 Tiling a Finite Table with a Substitution, Coding and Overlap Assume one is given a rectangular segment of an automatic tiling. The goal is to find a substitution and a coding which generate the entire tiling. If the number of tiles in the infinite tiling is bounded and if the given pattern is large enough, this is... |
via finite automata. By a well–known process described in 29 , Propositions 10.1.5 & 10.2.2] , their algorithm also yields a substitution and a coding which generate the entire sequence. It is possible to extend this approach to higher dimensions. However, such an algorithm might not be efficient enough for our purpose... |
A different approach will be used hereafter. It was initiated by the third–named author in order to provide an algorithm sufficiently efficient in the present context. The underlying method only allows one to find automatic tilings with an injective coding; however, this proves adequate for the Paper–Folding Number Wal... |
5.3 The Paper–Folding Number Wall The algorithms of the previous sections enable one to detect a structure in the Number Wall of the Paper–Folding sequence: |
Theorem 5.1 The finite portion of the Paper–Folding Number Wall over [MATH] [EQUATION] agrees with a tiling [MATH] . This tiling is the image under a coding [MATH] of a substitution tiling [MATH] . Here, |
1. [MATH] is a [MATH] –substitution over [MATH] for which tiles [MATH] and [MATH] are respectively [MATH] and [MATH] –prolongable; |
2. [MATH] is an [MATH] –coding defined with the help of a 13–coding [MATH] by the relation [EQUATION] for every [MATH] and every [MATH] |
3. [MATH] is a [MATH] –coding defined over [MATH] and taking values in [MATH] . This coding is [MATH] –consistent for the tiling [MATH] restricted to the region |
[EQUATION] Remark 5.2 The additional translation included in the definition of the coding [MATH] in \tagform@ 5.2 is an adjustment of a technical nature (see the Appendix for details). It enables one to match the zeroth row of the tiling with the finite portion of the Paper–Folding sequence used to build part of its Nu... |
Theorem 5.1 is obtained by application of the algorithm described in Figure with the parameters [EQUATION] The region [MATH] defined in \tagform@ 5.3 is obtained from Pass 3 of this algorithm: it corresponds to the range of indices [MATH] such that |
[EQUATION] and [EQUATION] Note that Pass 3 then guarantees that the tiling [MATH] satisfies the assumptions of Lemma 4.14 upon setting |
[EQUATION] The resulting conclusion is stated as a proposition: Proposition 5.3 With the notation of Theorem 5.1 , every 2–pattern in [MATH] already appears in [MATH] |
Proving that the tiling [MATH] in Theorem 5.1 is the Paper–Folding Number Wall and determining its deficiency require a fine analysis of the pattern of zeros appearing in the portion [MATH] . This, in turn, relies on some properties of the coding [MATH] and of the 2–substitution [MATH] . These properties are stated in ... |
[EQUATION] Proposition 5.4 (Properties of the 13–coding [MATH] The 13–coding [MATH] introduced in Theorem 5.1 satisfies the following property: if, for a given tile [MATH] , the [MATH] square [MATH] contains a 4–pattern comprising only zero entries, then [MATH] |
More precisely, under such an assumption, either [MATH] and [MATH] is identically zero; or else [MATH] , in which case the top 9 rows of [MATH] are identically zero while all entries in the tenth row equal 1. |
Furthermore, for every [MATH] , the 13–pattern [MATH] is contained in the rectangular pattern \tagform@ 5.1 The last claim in Proposition 5.4 follows from the construction of the coding [MATH] in Pass 1 of the algorithm described in Figure . The rest of the proposition is established by (computer) inspection of the cod... |
in the file dragon_codes_B.dat Proposition 5.5 (Properties of the 2–substitution [MATH] The 2–substitution [MATH] introduced in Theorem 5.1 satisfies the following properties: |
1. the image of tile 5 is the [MATH] square all of whose entries are 5; 2. for any tile [MATH] [EQUATION] 3. if the image [MATH] of a tile [MATH] contains a tile in [MATH] , then [MATH] |
Proposition 5.5 is established by (computer) inspection of the substitution [MATH] , which is explicit and available at in the file dragon_ tetrads_B.dat . This file actually contains the 6721 2–patterns that can be found in the tiling [MATH] restricted to the region [MATH] . The first 2353 correspond to the successive... |
Together with the initial conditions [EQUATION] the 2–substitution [MATH] can be used to generate any finite portion of the tiling [MATH] . An example of such a portion is represented in Figure . Note the presence of the initial pattern |
[EQUATION] therein (cf. the second and third rows). Verifying the Properties of the Generated Number Wall Proving that the tiling described in the statement of Theorem 5.1 is the [MATH] –tiling given by the Paper–Folding Number Wall will be done by verifying that it satisfies the Frame Constraints. From Corollary 3.6 ,... |
Theorem 6.1 The coding [MATH] is 5–consistent for the tiling [MATH] Proof. The assumptions of Lemma \tagform@ 4.14 have been shown to hold in order to state Proposition 5.3 . Corollary 4.15 then implies that the consistency condition needs only be verified in [MATH] , where [MATH] and [MATH] are defined in \tagform@ 5.... |
Since [MATH] is a 13–coding which is 5 consistent, equation \tagform@ 5.2 implies that, with the notation of Section 4.3 [EQUATION] |
It then follows from Lemma 4.9 that the images of the tiling [MATH] under [MATH] and [MATH] are the same up to a shift of 3, both vertically and horizontally; that is, for any [MATH] |
[EQUATION] This equation enables one to interpret the squares and circles in Figure and to illustrate how the substitution, coding and overlap look like. The top circle marks the origin. The top–left [MATH] square is the image under [MATH] of tile [MATH] , with a circle in its center. The image of [MATH] under [MATH] i... |
[EQUATION] (this information is contained in Figure ). The images under [MATH] of these four tiles are represented by four squares with a circle around their respective centers. By definition, it then follows that [MATH] maps tiles 17, 35, 45, 47 and 58 to the top–left [MATH] squares sitting in their respective images ... |
Theorem 6.2 The zero entries with non–negative row indices in the tiling [MATH] appear in the form of squares with side lengths at most 3 and with horizontal and vertical edges. |
With the conventions and definitions adopted in Section 3.1 , Theorem 6.2 is thus saying that a window in the part of the tiling [MATH] with non–negative row indices has deficiency at most 4. |
The proof of Theorem 6.2 requires first the following lemma, which gives some properties of the tiling [MATH] Lemma 6.3 The tiling [MATH] is such that: |
1. all entries in rows with index at most -1 are identically equal to 5; 2. the zeroth row contains only tiles from [MATH] 3. the rows with positive indices contain no entry in [MATH] |
Here, the sets of tiles [MATH] and [MATH] are those defined in \tagform@ 5.5 Proof. Consider first the orthant comprising the point [MATH] with the associated initial condition [MATH] . Let [MATH] be a tile in the zeroth line of this orthant. The construction rule for the tiling [MATH] described in \tagform@ 4.2 implie... |
A similar reasoning applies to the orthant comprising the point [MATH] , which is mapped under [MATH] to tile 2, which is an element in [MATH] . This establishes the first two claims in the lemma. |
Consider now the orthant comprising the point [MATH] with the associated initial condition [MATH] . All elements in this orthant are obtained as entries in the image under [MATH] of a tile [MATH] lying in this orthant. As tile 4 is not in [MATH] , an easy induction based on Point 3 in Proposition 5.5 implies that no en... |
A similar reasoning applies to the orthant comprising the point [MATH] , which is mapped under [MATH] to tile 3, which is not an element in [MATH] . This establishes the last claim in the lemma. |
Proof of Theorem 6.2 The first step is to show that the part of the tiling [MATH] with non–negative row indices cannot contain any [MATH] zero window. |
Lemma 6.4 Let [MATH] be a 4–pattern contained in [MATH] which is identically zero. Then [MATH] is contained in the portion of the tiling [MATH] with negative row indices. |
Proof. It is an immediate consequence of equation \tagform@ 6.1 that [MATH] also sits as a 4–pattern in the tiling [MATH] . Lemma 4.12 applied with the parameters [MATH] [MATH] and [MATH] then implies that [MATH] is contained in the image [MATH] of a 1–pattern [MATH] sitting in [MATH] . Let then [MATH] be such that [MA... |
Since the zero 4–pattern [MATH] is contained in [MATH] , Proposition 5.4 implies that [MATH] . Thus, from Lemma 6.3 , it holds that [MATH] and moreover that [MATH] when [MATH] and [MATH] when [MATH] |
Note then these two observations: on the one hand, [MATH] is the [MATH] subsquare in the [MATH] square [MATH] corresponding to the row and column indices between 4 and 11 (see equation \tagform@ 5.2 ). On the other hand, it follows from the way the tiling [MATH] is defined in \tagform@ 4.4 that the [MATH] square [MATH]... |
Since when [MATH] , the 4–pattern [MATH] cannot overlap with rows 10 to 13 in [MATH] (cf. Proposition 5.4 ), the above two observations enable one to conclude that the pattern [MATH] is contained in the region of the tiling [MATH] with row index at most -2. |
In order to complete the proof of Theorem 6.2 , consider a 4–pattern [MATH] in the region of the tiling [MATH] with non–negative row indices. From Lemma 6.4 [MATH] cannot be identically zero. Furthermore, Lemma 4.12 implies here also that [MATH] is contained in [MATH] for some [MATH] . From the last claim in Propositio... |
Since there cannot be a zero 4–pattern in the region [MATH] of the tiling [MATH] , the zeros in the pattern [MATH] take the shape of a rectangle [MATH] obtained as the intersection between a square [MATH] with side length at most 3 and the [MATH] square determined by [MATH] |
Assume that [MATH] is not entirely contained in the pattern [MATH] as there is otherwise nothing more to prove. The rectangle [MATH] then admits a point, say [MATH] , lying in the interior of the [MATH] square determined by [MATH] . Consider another 4–pattern [MATH] containing [MATH] , one of which corners coincides wi... |
This concludes the proof of Theorem 6.2 Theorem 6.5 The tiling [MATH] satisfies the Frame Constraints. Proof. By Theorem 6.2 the maximal side length of a window lying the in part of [MATH] with non–negative row indices is 3. Therefore, by Corollary 3.6 , the Frame Constraints are determined by patterns of size at most ... |
Also, from Theorem 5.1 , the image of [MATH] under [MATH] is, up to a translation by the vector [MATH] , the restriction of the Number Wall \tagform@ 5.1 to the region [MATH] . Note that from the values of [MATH] and [MATH] in \tagform@ 5.4 , this restriction followed by a translation by [MATH] is clearly contained in ... |
Theorems 6.2 and 6.5 can be rephrased as follows : Corollary 6.6 The tiling [MATH] is a Number Wall with deficiency 4. From the last claim in Corollary 3.6 [MATH] is the Number Wall of the sequence sitting in its zeroth row. This sequence is now determined: |
Theorem 6.7 The tiling [MATH] has the Paper–Folding sequence in its zeroth row. Proof. From Point 3 in Lemma 6.3 , only the eight tiles in [MATH] appear in the zeroth row of [MATH] . Figure tabulates these tiles, their images under the substitution [MATH] (in other words, the values of [MATH] and [MATH] in \tagform@ 5.... |
Using coded tiles, the zeroth row of the tiling [MATH] is thus the [MATH] –tiling [MATH] (this follows from the first two initial conditions in \tagform@ 5.7 , which are expressed in the language of non–coded tiles). |
Looking at the boxed segments in Figure that represent the coding [MATH] restricted to the zeroth line reveals that mapping the (coded) tiles [MATH] to [MATH] respectively in the substitution [MATH] is consistent with their definition on [MATH] (for instance, [MATH] and mapping 4 to 0 transforms the tiling [MATH] to [M... |
A further comparison with [MATH] and [MATH] introduced in Example 4.6 shows that, in fact, [EQUATION] for each [MATH] , and therefore for all [MATH] upon identifying tiles as above when needed. |
It is elementary to verify that [EQUATION] Since the Paper–Folding sequence is [MATH] (cf. Example 4.6 ) , this concludes the proof that the zeroth row of the generated Number Wall is the Paper–Folding sequence. |
The [MATH] –adic Littlewood Conjecture in other Characteristics 7.1 Sequences with Small Deficiency over Finite Fields The aim of this section is to discuss to what extent the value of the deficiency appearing in Theorem 2.6 (viz. 4) can be improved and/or generalized to other characteristics. |
It is easily seen that one cannot have a Number Wall over [MATH] with no zero entry . This amounts to saying that any infinite Hankel matrix over [MATH] admits a singular connected minor (that is, a singular square submatrix whose row and column indices are consecutive). This prompts the following more general open pro... |
Question 7.1 Let [MATH] be a finite field with [MATH] elements. Does there exist an integer [MATH] such that any square matrix with dimensions [MATH] and with entries in [MATH] admits a singular connected minor (as defined above)? |
In other words, Question 7.1 amounts to asking how big a hyperinvertible matrix can be over a finite field, where hyperinvertibility of a matrix means that all connected square submatrices are invertible. Note that when the field is infinite, the well–known class of Cauchy matrices provides examples of arbitrarily larg... |
that there is no [MATH] matrix over [MATH] whose (non–necessarily connected) minors of order [MATH] and [MATH] are all different from zero. |
On another front, Theorem 2.6 raises the question so as to whether there exists a sequence with deficiency smaller than 4 over [MATH] and, possibly, with optimal value 2. This is indeed the case, and the discovery of a sequence generating a Number Wall with only isolated zeros, the Pagoda sequence |
[MATH] , was made in (see also for a fuller account; the origin of the name of the sequence is also explained in the latter reference). It is defined from the Paper–Folding sequence as follows: for any [MATH] |
[EQUATION] Tiling the Number Wall obtained from this sequence confirms the above claim: Theorem 7.2 The Number Wall of the Pagoda sequence over [MATH] has only isolated zero entries. |
The proof of Theorem 7.2 proceeds along the same lines as the proof of Theorem 5.1 and will not be detailed here. These details can be made explicit from the codes available at |
which deal, not only with the case of the Paper–Folding sequence, but also with that of the Pagoda sequence. The main missing ingredient is the verification of the analogue of Theorem 6.7 for the Pagoda sequence. This requires a comparison between two given automatic sequences, which in the case of the Paper–Folding se... |
[EQUATION] This equality corresponds to the “worst” possible case when [MATH] –LC fails over [MATH] As a matter of fact, a generalisation to other characteristics is suggested by computer evidence: |
Conjecture 7.3 The Paper–Folding and Pagoda sequences seen as sequences over a finite field [MATH] have bounded deficiency 4 and 2 respectively for all prime [MATH] , and unbounded deficiency for all other primes. |
Conjecture 7.3 has been checked extensively by computer inspection of the Number Walls of the sequences under consideration. For instance, in the case of the Paper–Folding sequence, it has been verified in finite [MATH] segments of its Number Walls over [MATH] for all [MATH] . The difficulty in validating this conjectu... |
If indeed true, Conjecture 7.3 would imply that the [MATH] –adic Littlewood Conjecture fails (at least) over any field with characteristic a prime congruent to 3 modulo 4. |
7.2 On the Laurent Series of the Paper–Folding Sequence in a Field with Characteristic 2 The Hankel matrices of the Paper–Folding and Pagoda sequences are much better understood over [MATH] . This follows from the fact that the Laurent series they define in [MATH] are both quadratic. Indeed, note that |
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