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.

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We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in this work we prove that if E ⊂ F d q is sufficiently large and ̟ is a non-degenerate multi-linear form then ̟ will attain all possible nonzero values as its arguments vary over E, under a certain quantitative assumption on the extent to which E is projective. We show that our bound is nontrivial in the case that n = 3 and d = 2 and construct examples of sets to which this applies. In particular, we give conditions under which every member of F * q

We introduce an n-dimensional analogue of the construction of tessellated surfaces from finite groups first described by Herman and Pakianathan. Our construction is functorial and associates to each n-ary alternating quasigroup both a smooth, flat Riemannian n-manifold which we dub the open serenation of the quasigroup in question, as well as a topological n-manifold (the serenation of the quasigroup) which is a subspace of the metric completion of the open serenation. We prove that every connected orientable smooth manifold is serene, in the sense that each such manifold is a component of the serenation of some quasigroup. We prove some basic results about the variety of alternating n-quasigroups and note connections between our construction and topics including Latin hypercubes, Johnson graphs, and Galois theory.

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