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We study tiling and spectral sets in vector spaces over prime fields. Theclassical Fuglede conjecture in locally compact abelian groups says that a setis spectral if and only if it tiles by translation. This conjecture wasdisproved by T. Tao in Euclidean spaces of dimensions 5 and higher, usingconstructions over prime fields (in vector spaces over finite fields of primeorder) and lifting them to the Euclidean setting. Over prime fields, when thedimension of the vector space is less than or equal to $2$ it has recently beenproven that the Fuglede conjecture holds (see \cite{IMP15}). In this paper westudy this question in higher dimensions over prime fields and provide someresults and counterexamples. In particular we prove the existence of spectralsets 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 asmall, sporadic set of primes $p$ rather than general constructions. This paperis a result of a Research Experience for Undergraduates program ran at theUniversity of Rochester during the summer of 2015 by A. Iosevich, J.Pakianathan and G. Petridis.

Tiling sets and spectral sets over finite fields