Tiling sets and spectral sets over finite fields

TL;DR

This paper investigates the relationship between tiling and spectral sets in finite field vector spaces, providing new counterexamples to the Fuglede conjecture in higher dimensions over prime fields.

Contribution

It demonstrates the existence of spectral sets that do not tile in certain higher-dimensional prime field vector spaces, extending known counterexamples to new cases.

Findings

Spectral sets without tiling in  ext{Z}_p^5 for all odd primes p

Spectral sets without tiling in  ext{Z}_p^4 for primes p  3 mod 4

Counterexamples extend previous results to more general prime cases

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 \cite{IMP15}). 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 $\mathbb{Z}_p^5$ for all odd primes $p$ and $\mathbb{Z}_p^4$ for all odd primes $p$ such that $p \equiv 3 \text{ mod } 4$. Although counterexamples in low dimensional groups over cyclic rings $\mathbb{Z}_n$ 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.