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

In this paper, we investigate Fuglede’s conjecture for ${\mathbb{Z}}_{p^2{q}^{2}r}$ and provide a proof under the condition $p^2{q}^2\le r$. 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.