A Group Ring Approach to Fuglede’s Conjecture in Cyclic Groups

Fuglede’s conjecture states that a subset \(\Omega \subseteq \mathbb {R}^{n}\) with positive and finite Lebesgue measure is a spectral set if and only if it tiles \(\mathbb {R}^{n}\) by translation. However, this conjecture does not hold in both directions for \(\mathbb {R}^n\), \(n\ge 3\). While the conjecture remains unsolved in \(\mathbb {R}\) and \(\mathbb {R}^2\), cyclic groups are instrumental in its study within \(\mathbb {R}\). This paper introduces a new tool to study spectral sets in cyclic groups and, in particular, proves that Fuglede’s conjecture holds in \(\mathbb {Z}_{p^{n}qr}\).