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 $f:E\to \mathbb{C}$ 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 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.

Cited by in corpus (8)

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• On the discrete Fuglede and Pompeiu problems
• Equi-distributed property and spectral set conjecture on ${\mathbb{Z}}_{p^2}\times {\mathbb{Z}}_p$
• Fuglede's conjecture holds for cyclic groups of order $pqrs$
• On the structure of spectral and tiling subsets of cyclic groups
• Convolution Idempotents with a given Zero-set
• Geometric structures in pseudo-random graphs
• On product spectral sets and functional tiles in ${\mathbb{Q}}_p^d$