A counterexample to Fuglede's conjecture in $(\mathbb{Z}/p\mathbb{Z})^4$ for all odd primes

Abstract

We construct a spectral, non-tiling set of size $2p$ in $\mathbb Z/p\mathbb Z{4}$, $p$ odd prime. This example complements a previous counterexample in [C. Aten et al., \textit{Tiling sets and spectral sets over finite fields}, arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$. On the contrary we show that the conjecture does hold in $(\mathbb Z/2\mathbb Z)^4$.