Fuglede's conjecture holds for cyclic groups of order $pqrs$

TL;DR

This paper proves Fuglede's conjecture for cyclic groups whose order is the product of four distinct primes, advancing understanding in the spectral-tile aspect of the conjecture for such groups.

Contribution

It establishes the validity of Fuglede's conjecture for cyclic groups of order pqrs, where p, q, r, s are distinct primes, extending previous results to this case.

Findings

Fuglede's conjecture holds for cyclic groups of order pqrs

The spectral-tile direction is confirmed for these groups

Advances understanding of the conjecture in non-square-free cases

Abstract

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede's conjecture holds in this case.