Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$

TL;DR

This paper proves that Fuglede's spectral set conjecture is true for cyclic groups of order the product of three distinct primes, confirming the conjecture in this specific algebraic setting.

Contribution

The paper establishes the validity of Fuglede's conjecture on cyclic groups of order pqr, where p, q, r are distinct primes, a case previously unresolved.

Findings

Fuglede's conjecture holds on cyclic groups $ ext{Z}_{pqr}$ with distinct primes p, q, r.

The proof confirms the conjecture in a new algebraic setting involving product of three primes.

Supports the conjecture's validity in specific finite abelian groups.

Abstract

Fuglede's spectral set conjecture states that a subset $\Omega$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of $L^2(\Omega)$. We prove that Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes.