# The Fuglede conjecture holds in $\mathbb{Z}^3_5$

## TL;DR

This paper confirms the Fuglede conjecture in the three-dimensional vector space over the finite field with five elements by exhaustive computer search, extending previous results to this specific case.

## Contribution

The authors develop a computer program that exhaustively verifies the Fuglede conjecture in ^3, filling a gap in the understanding of the conjecture over prime fields.

## Key findings

- Fuglede conjecture holds in ^3.
- Computer-assisted proof confirms the conjecture in this case.
- Extends previous results for prime fields to dimension 3.

## Abstract

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer program that determines that the Fuglede conjecture holds in $\mathbb{Z}_5^3$ by exhausting the search space. A. Iosevich, A. Mayeli and J. Pakianathan showed that the Fuglede conjecture holds over prime fields when the dimension does not exceed 2. The question for dimension 3 was previously addressed by Aten et al. for $p=3$. In this paper we build upon the results of their work to allow a computer to carry out the lengthy computations.