The Fuglede conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$

Abstract

In this paper we study subsets $E$ of ${\mathbb{Z}}_p^d$ such that any function $f:E\to ℂ$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede conjecture in ${\mathbb{Z}}_p^2$, which says in this context that $E\subset {\mathbb{Z}}_p^2$ is spectral if and only if $E$ tiles ${\mathbb{Z}}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof. The proof relies to a significant extent on the analysis of direction sets of Iosevich et al. (Integers 11 (2011), art. id. A39) and the tiling results of Haessig et al. (2011).