三素数整数平铺中的无支撑分圆因子
Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
- Academy of Mathematics and Systems Science Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- School of Mathematical Sciences Soochow University（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总结：
本文证明三素数整数平铺中存在无支撑分圆因子，刻画了平方周期情形，并给出最小模数及Apéry集构造。
AI中文摘要：
分圆可除性对整数平铺施加了强素数幂结构。我们研究无支撑分圆因子：即阶的素数幂分量均不整除掩码，但阶中的每个素数都整除平铺基数的混合阶因子。Kiss、Łaba、Marshal 和 Somlai 曾问及这种现象是否会在三素数情形中出现。我们证明无支撑分圆因子在周期具有三个不同素数因子时已经出现。对于素数 \\(p<q<r\\)，我们刻画了平方周期情形：无支撑因子 \\(\Phi_{pqr}\\) 出现在 \\(\ZZ_{(pqr)^2}\\) 的平铺中当且仅当 \\(r\in\langle p,q\rangle\\)，且每个这样的平铺都位于模 \\(r\\) 的单个剩余类中。在无支撑阶整除指定模数的循环平铺中，最小模数为 \\(180\\)；若阶具有三个不同素数因子，则为 \\(900\\)。一个 Apéry 集构造为周期 \\(p^2q^2r^3\\) 的每个三元组给出了例子。我们的结果证明结合了傅里叶刚性、三圆柱分解和整数质量障碍。
英文摘要：
Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes \(p<q<r\), we characterize the square-period case: an unsupported factor \(Φ_{pqr}\) occurs in a tiling of \(\ZZ_{(pqr)^2}\) if and only if \(r\in\langle p,q\rangle\), and every such tile lies in a single residue class modulo \(r\). Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is \(180\); if the order has three distinct prime factors, it is \(900\). An Apéry-set construction gives examples for every triple at period \(p^2q^2r^3\).The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.
