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

## TL;DR

This paper constructs a spectral, non-tiling set in the 4-dimensional vector space over finite fields for all odd primes, providing a counterexample to Fuglede's conjecture, while confirming the conjecture for the case p=2.

## Contribution

It presents a new counterexample to Fuglede's conjecture in $(bZ/pbZ)^4$ for all odd primes, extending previous results and confirming the conjecture for p=2.

## Key findings

- Counterexample exists for all odd primes p
- Fuglede's conjecture holds in $(bZ/2bZ)^4$
- The constructed set has size 2p

## 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 $(\mathbb{Z}/2\mathbb{Z})^4$.