Fuglede’s conjecture holds on ℤ_{𝕡}²×ℤ_{𝕢}

Kiss, Gergely; Somlai, Gábor

doi:10.1090/proc/15541

Cited by 15 publications

(7 citation statements)

References 15 publications

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“…While the conjecture has been disproved in its full generality in dimensions 3 and higher [44], [19], [20], [7], [32], [8], significant connections between tiling and spectrality do exist (see [4] for an overview of the problem in dimension 1), and there is a large body of work investigating such connections from many points of view. In higher dimensions, the conjecture has been proved for convex sets in R n , by Iosevich, Katz and Tao [13] for n = 2, Greenfeld and Lev [11] for n = 3, and by Lev and Matolsci [28] for general n. There have been many recent results on special cases of the finite abelian group analogue of the conjecture [14], [30], [5], [15], [16], [17], [29], [38], [39], [40], [6].…”

Section: Then the Tiling Conditionmentioning

confidence: 97%

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

Łaba¹,

Londner²

2021

Preprint

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It is well known that if a finite set A ⊂ Z tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A ⊕ B = Z M of a finite cyclic group. We are interested in characterizing all finite sets A ⊂ Z that have this property. Coven and Meyerowitz [3] proposed conditions (T1), (T2) that are sufficient for A to tile, and necessary when the cardinality of A has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of |A|.We prove that the Coven-Meyerowitz tiling condition (T2) holds for all integer tilings of period M = (p i p j p k ) 2 , where p i , p j , p k are distinct odd primes. The proof also provides a classification of all such tilings.

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Section: Then the Tiling Conditionmentioning

confidence: 97%

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

Łaba¹,

Londner²

2021

Preprint

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“…Nonetheless, many important cases remain open and continue to attract attention. Iosevich, Katz and Tao [14] proved in 2003 that Fuglede's conjecture holds for convex sets in R 2 ; an analogous result in higher dimensions was proved only recently, by Greenfeld and Lev [12] for 𝑛 = 3 and by Lev and Matolcsi [29] for general n. There has also been extensive work on the finite abelian group analogue of the conjecture [7,6,15,16,17,18,30,31,39,40,41,48].…”

Section: Introductionmentioning

confidence: 96%

Combinatorial and harmonic-analytic methods for integer tilings

Łaba

Londner

2022

Forum of Mathematics, Pi

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A finite set of integers A tiles the integers by translations if $\mathbb {Z}$ can be covered by pairwise disjoint translated copies of A. Restricting attention to one tiling period, we have $A\oplus B=\mathbb {Z}_M$ for some $M\in \mathbb {N}$ and $B\subset \mathbb {Z}$ . This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with A and B. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set A containing certain configurations can tile a cyclic group $\mathbb {Z}_M$ , or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period $(pqr)^2$ , where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

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“…The conjecture is known to be false, in its full generality, in dimensions 3 and higher [46], [20], [21], [8], [34], [9]. However, there are important special cases in which the conjecture was confirmed [14], [12], [30], and the finite abelian group analogue of the conjecture is currently a very active area of research [15], [32], [7], [16], [17], [18], [31], [40], [41], [42], [6], [49].…”

Section: Introductionmentioning

confidence: 99%

Splitting for integer tilings and the Coven-Meyerowitz tiling conditions

Łaba¹,

Londner²

2022

Preprint

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Suppose that a finite set A ⊂ Z tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization A ⊕ B = Z M of a finite cyclic group. We are interested in investigating the structure of such tilings, and in particular in a tentative characterization of finite tiles proposed by Coven and Meyerowitz [2]. Building on the work in [24,25], we prove that the Coven-Meyerowitz condition (T2) holds for all integer tilings of period M = (p i p j p k ) 2 , where p i , p j , p k are distinct primes. This extends the main result of [25], where we assumed that M is odd. We also provide a new combinatorial interpretation of some of the main tools of [24], [25], and introduce a new method based on splitting. In addition to applications in the proof of (T2) in the even case, we use splitting to prove weaker structural results in cases where (T2) is not yet available.

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Fuglede’s conjecture holds on ℤ_{𝕡}²×ℤ_{𝕢}

Cited by 15 publications

References 15 publications

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

Combinatorial and harmonic-analytic methods for integer tilings

Splitting for integer tilings and the Coven-Meyerowitz tiling conditions

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