  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.
