On the discrete Fuglede and Pompeiu problems

Kiss, Gergely; Malikiosis, Romanos Diogenes; Somlai, Gábor; Vizer, Máté

doi:10.2140/apde.2020.13.765

Cited by 27 publications

(43 citation statements)

References 31 publications

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“…An alternative proof that (i) implies (ii) (without using Theorem 4.7; compare [16, 42]) is as follows. By classic results on vanishing sums of roots of unity [3, 28, 32, 35, 36, 38], if and only if is a linear combination of the polynomials , where p runs over all prime divisors of N , with integer (but not necessarily nonnegative) coefficients.…”

Section: Cuboidsmentioning

confidence: 99%

“…Iosevich, Katz and Tao [14] proved in 2003 that Fuglede’s conjecture holds for convex sets in ; an analogous result in higher dimensions was proved only recently, by Greenfeld and Lev [12] for 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: 99%

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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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Section: Cuboidsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Combinatorial and harmonic-analytic methods for integer tilings

Łaba

Londner

2022

Forum of Mathematics, Pi

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“…It is proved in [10,16] that Fuglede's conjecture holds for every proper subgroup of Z 2 p × Z q . Hence we may assume neither S nor Λ are contained in a proper subgroup of Z 2 p × Z q , see Lemma 4.3 and Lemma 4.4 in [11].…”

Section: Spectral-tilementioning

confidence: 99%

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

Kiss

Somlai

2021

Proc. Amer. Math. Soc.

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The study of Fuglede’s conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For Z p 3 \mathbb {Z}_p^3 the problem is still open if p p is prime and p ≥ 11 p\ge 11 . In connection we prove that Fuglede’s conjecture holds on Z p 2 × Z q \mathbb {Z}_{p}^2\times \mathbb {Z}_q by developing a method based on ideas from discrete geometry.

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“…Remark that in some other works (see, e.g., [8], [9], [10], [11], [12], [13]) similar reformulations of tilings in terms of polynomials are introduced, and, moreover, similarly to this work, cyclotomic polynomials are applied. However, in such works the condition (T 1), first introduced in [8], is considered and used only for tilings of multiplicity 1.…”

Section: Introductionmentioning

confidence: 99%