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 \(\mathbb {Z}_{p^{n}qr}\).

Fuglede 猜想指出，具有正且有限勒贝格测度的子集\(\Omega \subseteq \mathbb {R}^{n}\)是一个谱集当且仅当它平铺\(\mathbb {R}^{n} \)通过翻译。然而，对于\(\mathbb {R}^n\)、\(n\ge 3\) ，这个猜想在两个方向上都不成立。虽然该猜想在\(\mathbb {R}\)和\(\mathbb {R}^2\)中仍未解决，但循环群在\(\mathbb {R}\)中的研究中发挥了重要作用。本文介绍了一种研究循环群中谱集的新工具，特别是证明了 Fuglede 猜想在\(\mathbb {Z}_{p^{n}qr}\)中成立。