发表机构
Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, Soochow University（中国科学院数学与系统科学研究院; 苏州大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了无平方因子阶循环群的Fuglede猜想，给出谱集与平铺集的完整结构刻画，核心方法是素数坐标消去定理及其迭代约化。
AI 中文摘要
我们证明了无平方因子阶有限循环群的Fuglede猜想。具体而言，若N是无平方因子的正整数，且A⊆Z_N是非空子集，|A|=m，则A是谱集当且仅当A平铺Z_N；此外，这些条件恰好当m整除N且A是模m的完全代表系时成立。这给出了Z_N中谱集和平铺集的完整结构描述。关键的新工具是一个素数坐标消去定理：若P是不整除nm的奇素数，则Z_n×Z_P中每个m点谱对都单射投影到Z_n中的谱对。其证明结合了特征矩阵正交性、分圆多项式系数比较以及相关Gram矩阵的P-adic可除性论证。迭代此约化方法可得到任意无平方因子N的结果，对其素数因子的个数或相对大小无任何限制。
英文摘要
We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if \(N\) is square-free and \(A\subseteq \mathbb Z_N\) is nonempty with \(|A|=m\), then \(A\) is spectral if and only if it tiles \(\mathbb Z_N\); moreover, these conditions hold precisely when \(m\mid N\) and \(A\) is a complete set of representatives modulo \(m\). This gives a complete structural description of spectral and tiling sets in \(\mathbb Z_N\). The key new ingredient is a prime-coordinate elimination theorem: if \(P\nmid nm\) is an odd prime, then every \(m\)-point spectral pair in \(\mathbb Z_n\times\mathbb Z_P\) projects injectively to a spectral pair in \(\mathbb Z_n\). Its proof combines character-matrix orthogonality, cyclotomic coefficient comparison, and a \(P\)-adic divisibility argument for associated Gram matrices. Iterating this reduction yields the result for arbitrary square-free \(N\), with no restriction on the number or relative sizes of its prime factors.
Comments12 pages
