Tiling sets and spectral sets over finite fields

Abstract

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6]). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z 5 p for all odd primes p and Z 4 p for all odd primes p such that p ≡ 3 mod 4. Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis.

Lattice sub-tilings and frames in LCA groups

“…However, in [10] the authors showed that the Fuglede Conjecture holds in twodimensional vector spaces over prime fields. Then, in [2] the authors prove that tiling implies spectral in Z 3 p , p prime, and Fuglede conjecture is true for Z 3 2 and Z 3 3 . Very recently, important developments in LCA groups, with crucial implications on sampling theory, have been developed by E. Agora, J. Antezana and C. Cabrelli in [1], where the authors could obtain a full characterization of Riesz bases for multi-tiling sets in LCA groups.…”

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

“…If A = 1 or p 3 , then A is a tile trivially. We suppose that 1 < A < p 3 By pigeon principle, we have (1,0)…”

“…Nevertheless, we either know few about for which group, the spectral set conjecture fails. We only know that the direction ‘Spectral sets Tiles’ fails on [15], [10], when prime and and when prime and [1]; the direction ‘Tiles Spectral sets ’ fails on [12], , where is prime with gcd [4] and [5]. Actually, these non‐tile spectral sets (respectively, non‐spectral tiles) in finite abelian groups lead to non‐tile spectral sets (respectively, non‐spectral tiles) in euclidean spaces.…”

“…(1, 0), (p, 0) ∈ Z B . By Lemmas 5.4 and 5.5, we obtain that B is a spectral set.Case 2: (1, 0) ∈ Z B and (p, 0) / ∈ Z B .…”

A characterization of isospectral measures on LCA groups

“…We could state a general Fuglede's conjecture for a locally compact Abelian group G, even for finite groups, and also called Fuglede's conjecture in G. In this general case, the generalized conjecture is far from being proved. For the works towards this general conjecture, we refer the readers to [4,5,6].…”