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

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.