Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

TL;DR

This paper proves Fuglede's conjecture for certain groups by analyzing the divisibility of mask polynomials with cyclotomic polynomials, introducing a new technique combined with the mod-$p$-method.

Contribution

It develops a novel technique using cyclotomic polynomial divisibility and the mod-$p$-method to study Fuglede's conjecture in specific finite abelian groups.

Findings

Proof of Fuglede's conjecture for groups where $p^2q^2 \

Introduction of a new divisibility technique for mask polynomials,

Application of the mod-$p$-method to this problem.

Abstract

In this paper, we investigate Fuglede's conjecture for $\mathbb{Z}_{p^2q^2r}$ and provide a proof under the condition $p^2q^2 \leq r$. We develop a new technique by analyzing the divisibility of the mask polynomial of a given set by a system of cyclotomic polynomials. Combined with the so-called mod-$p$-method, this technique serves as a powerful tool for studying Fuglede's conjecture in this and related cases.