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

Abstract

In this short note 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 [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$. On the contrary we show that the conjecture does hold in $(\mathbb{Z}/2\mathbb{Z})^4$.