A counterexample to Fuglede’s conjecture in (Z/pZ) ⁴ for all odd primes

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