Tiling sets and spectral sets over finite fields

Aten, Charlotte; Ayachi, B.; Bau, E.; Fitzpatrick, Daniel; Iosevich, Alex; Liu, H.; Lott, Adam; MacKinnon, I.; Maimon, Shir; Nan, S.; Pakianathan, Jonathan; Petridis, Giorgis; Mena, C. Rojas; Sheikh, A.; Tribone, T.; Weill, Jean-Christophe; Yu, C.

doi:10.48550/arxiv.1509.01090

Cited by 3 publications

(2 citation statements)

References 7 publications

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“…with respect to the basis of delta-functions, it is given by a p d × p d Hadamard matrix H of Butson type whose "(x, m)-entry" is χ(−x • m) (see for example [1] for details).…”

Section: Basis Of Eigenfunctions Of the Fourier Transformmentioning

confidence: 99%

Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields

Iosevich,

Liu,

Mayeli

et al. 2016

Preprint

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In this paper we study how zeros of the Fourier transform of a function f : Z d p → C are related to the structure of the function itself. In particular, we introduce a notion of bandwidth of such functions and discuss its connection with the decomposition of this function into wavelets. Connections of these concepts with the tomography principle and the Nyquist-Shannon sampling theorem are explored.We examine a variety of cases such as when the Fourier transform of the characteristic function of a set E vanishes on specific sets of points, affine subspaces, and algebraic curves. In each of these cases, we prove properties such as equidistribution of E across various surfaces and bounds on the size of E.We also establish a finite field Heisenberg uncertainty principle for sets that relates their bandwidth dimension and spatial dimension.

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“…with respect to the basis of delta-functions, it is given by a p d × p d Hadamard matrix H of Butson type whose "(x, m)-entry" is χ(−x • m) (see for example [1] for details).…”

Section: Basis Of Eigenfunctions Of the Fourier Transformmentioning

confidence: 99%

Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields

Iosevich,

Liu,

Mayeli

et al. 2016

Preprint

Self Cite

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“…However, the situation changes in higher dimension. In fact, it is proved in [1] that there exist spectral sets which are not tiles in F 5 p for all odd primes p and in F 4 p for all odd primes p such that p ≡ 3 mod 4, where F p is the prime field with p elements. This implies that there exist compact open spectral sets which are not tiles in Q 5 p for all odd primes p and in Q 4 p for all odd primes p such that p ≡ 3 mod 4.…”

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confidence: 99%

Fuglede's conjecture holds in $\mathbb{Q}_p$

Fan¹,

Fan²,

Liao³

et al. 2015

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Fuglede's conjecture in Q p is proved. That is to say, a Borel set of positive and finite Haar measure in Q p is a spectral set if and only if it tiles Q p by translation.Let p ≥ 2 be prime and Q p be the field of p-adic numbers. In the present paper, we will prove the following theorem that Fuglede's conjecture in Q p holds.Theorem 1.1. Assume that Ω ⊂ Q p is a Borel set of positive and finite Haar measure. Then Ω is a spectral set if and only if it is a tile of Q p . In this case, Ω must be an almost compact open set.

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A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

Mattheus¹

2019

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Tiling sets and spectral sets over finite fields

Cited by 3 publications

References 7 publications

Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields

Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields

Fuglede's conjecture holds in $\mathbb{Q}_p$

A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

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