# Tiling sets and spectral sets over finite fields

@article{Aten2015TilingSA, title={Tiling sets and spectral sets over finite fields}, author={Charlotte Aten and B. Ayachi and E. Bau and D. Fitzpatrick and Alex Iosevich and H. X. Liu and Adam Lott and I. Mackinnon and Shir Maimon and S. Nan and Jonathan Pakianathan and Giorgis Petridis and C. Rojas Mena and A. Sheikh and Tim Tribone and Jean-Christophe Weill and C. Yu}, journal={arXiv: Classical Analysis and ODEs}, year={2015} }

## 22 Citations

### Exponential Riesz bases, multi-tiling and condition numbers in finite abelian groups

- Mathematics
- 2019

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We show that that every…

### Riesz bases of exponentials and multi-tiling in finite abelian groups

- Mathematics
- 2019

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that…

### Addendum to "Fuglede's conjecture fails in 4 dimensions over odd prime fields"

- MathematicsDiscret. Math.
- 2020

### The Fuglede conjecture holds in $\mathbb{Z}^3_5$.

- Mathematics
- 2019

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in…

### Fuglede ’ s conjecture holds in Z p × Z p n

- Mathematics
- 2021

Fuglede’s conjecture states that for a subset Ω of a locally compact abelian group G with positive and finite Haar measure, there exists a subset of the dual group of G which is an orthogonal basis…

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

- Mathematics, Computer Science
- 2016

A notion of bandwidth of such functions is introduced and its connection with the decomposition of this function into wavelets is discussed and a finite field Heisenberg uncertainty principle for sets that relates their bandwidth dimension and spatial dimension is established.

### Fuglede's conjecture holds for cyclic groups of order $pqrs$

- Mathematics
- 2020

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from…

### On Gabor orthonormal bases over finite prime fields

- MathematicsBulletin of the London Mathematical Society
- 2020

We study Gabor orthonormal windows in L2(Zpd) for translation and modulation sets, where p is prime and d⩾2 . We prove that for a set E⊂Zpd , the indicator function 1E is a Gabor window if and only…

### Equi‐distributed property and spectral set conjecture on Zp2×Zp

- MathematicsJournal of the London Mathematical Society
- 2020

In this paper, we show an equi‐distributed property in 2‐dimensional finite abelian groups Zpn×Zpm , where p is a prime number. By using this equi‐distributed property, we prove that Fuglede's…

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- Mathematics
- 2004

Abstract We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their L 2 space…

### Tiling and spectral properties of near-cubic domains

- Mathematics
- 2001

We prove that if a measurable domain tiles R or R 2 by translations, and if it is \close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar…

### Spectra of certain types of polynomials and tiling of integers with translates of finite sets

- Mathematics
- 2002

### The Fuglede spectral conjecture holds for convex planar domains

- Mathematics
- 2003

Let Ω be a compact convex domain in the plane. We prove that L2(Ω) has an orthogonal basis of exponentials if and only if Ω tiles the plane by translation. 0. Introduction Let Ω be a domain in R,…

### The Fuglede conjecture holds in ℤp × ℤp

- Mathematics
- 2017

In the 70s Bent Fuglede conjectured that if Ω is bounded domain in R, then L(Ω) has an orthogonal basis of exponentials if and only if Ω tiles R by translation. This idea led to much activity and…

### Universal spectra, universal tiling sets and the spectral set conjecture

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A subset $\Omega$ of $\mathbf{R}^d$ with finite positive Lebesgue measure is called a spectral set if there exists a subset $\Lambda\subset\mathbf{R}$ such that ${\mathcal E}_\Lambda :=\{e^{i2\pi…

### On Directions Determined by Subsets of Vector Spaces over Finite Fields

- MathematicsIntegers
- 2011

Abstract We prove that if a subset of a d-dimensional vector space over a finite field with q elements has more than q d–1 elements, then it determines all the possible directions. We obtain a…

### On Fuglede's Conjecture and the Existence of Universal Spectra

- Mathematics
- 2006

Recent methods developed by Tao [18], Kolountzakis and Matolcsi [7] have led to counterexamples to Fugelde’s Spectral Set Conjecture in both directions. Namely, in ${\Bbb R}^5$ Tao produced a…

### Spectral and Tiling properties of the Unit Cube

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Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2\pi tx)$, $t\in\T$ form an othonormal…

### Tiles with no spectra in dimension 4

- Mathematics
- 2006

We show by a counterexample that the "tiling $\Rightarrow$ spectral" part of Fuglede's Spectral Set Conjecture fails already in $\mathsf{Z}^4$ and $\mathsf {R}^4$.