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 $p\equiv 3(\mathrm{mod}4)$. On the contrary we show that the conjecture does hold in $(\mathbb{Z}/2\mathbb{Z}{)}^4$.

This result was found simultaneously and independently with [arXiv:1901.08734]. However, it contained an error which was fixed afterwards (again independently). We record it here as it gives a more geometrical approach to more or less the same construction