A group ring approach to Fuglede's conjecture in cyclic groups

Fuglede’s conjecture states that a subset $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles ${\mathbb{R}}^n$ by translation. The conjecture does not hold in both directions for ${\mathbb{R}}^n$, $n\ge 3$. However, this conjecture remains open in $\mathbb{R}$ and ${\mathbb{R}}^2$. Cyclic groups play important roles in the study of Fuglede’s conjecture in $\mathbb{R}$. In this paper, we introduce a new tool to study the spectral sets in cyclic groups. In particular, we prove that Fuglede’s conjecture holds in ${\mathbb{Z}}_{p^{n}qr}$.