# Fuglede's conjecture fails in 4 dimensions over odd prime fields

## TL;DR

This paper disproves Fuglede's conjecture in four-dimensional vector spaces over odd prime fields using log-Hadamard matrices, provides proofs for some cases, and verifies the conjecture in others through computational methods.

## Contribution

The authors extend known counterexamples to four dimensions over odd primes and prove the conjecture in four dimensions over the field with two elements, using novel methods and computational verification.

## Key findings

- Fuglede's conjecture fails in $ ext{Z}_p^4$ for all odd primes p.
- The conjecture holds in $ ext{Z}_2^4$, resolving all four-dimensional cases over prime fields.
- Computational verification shows the conjecture holds in $ ext{Z}_2^5$ and $ ext{Z}_2^6$, but fails in $ ext{Z}_2^{10}$.

## Abstract

Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset $E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral, meaning any complex-valued function $f$ on $E$ can be written as a linear combination of characters orthogonal with respect to $E$. We disprove Fuglede's conjecture in $\mathbb{Z}_{p}^{4}$ for all odd primes $p$, by using log-Hadamard matrices to exhibit spectral sets of size $2p$ which do not tile, extending the result of Aten et al. that the conjecture fails in $\mathbb{Z}_{p}^{4}$ for primes $p \equiv 3 \pmod 4$ and in $\mathbb{Z}_{p}^{5}$ for all odd primes $p$. We show, however, that our method does not extend to $\mathbb{Z}_{p}^{3}$. We also prove the conjecture in $\mathbb{Z}_{2}^{4}$, resolving all cases of four-dimensional vector spaces over prime fields. Our simple proof method does not extend to higher dimensions. The authors, however, have written a computer program to verify that the conjecture holds in $\mathbb{Z}_{2}^{5}$ and $\mathbb{Z}_{2}^{6}$. Finally, we modify Terry Tao's counterexample to show that the conjecture fails in $\mathbb{Z}_{2}^{10}$. Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$ is now resolved in all cases except when $d=3$ and $p\geq 11$, or when $p=2$ and $d=7,8,9$.