The Fuglede Conjecture Holds in

Abstract

The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by 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 ${\mathbb{Z}}_3^5$ by exhausting the search space. Recently Iosevich, Mayeli and 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 those results by Aten et al. to allow a computer to carry out the lengthy computations.

Keywords:

• Fuglede's conjecture; tiling; computer proof; finite fields