论文标题
关于循环基团的光谱和平铺子集的结构
On the structure of spectral and tiling subsets of cyclic groups
论文作者
论文摘要
本文的目的是研究环状基团的光谱和平铺子集的特性，以一维的频谱设定构想的介绍，该概念指出，$ \ mathbb {r} $的有限可测量子集接受了指数的正交基础，仅在IT tiles $ \ mathbb {r ransprancations pranslions。该猜想与离散的对应物有很强的连接，即，在每个有限的循环群中，当且仅当它是瓷砖时，子集是光谱。本文介绍的工具是最近在循环组设置中使用的工具的改进。统一根的消失总和是整个文本中的普遍概念，也是整数的平铺子集的结构。当指数为$ \ leq6 $时，或者当$ p^{m-2} <q^4 $时，我们设法证明了$ p^mq^n $的循环组的猜想，并且还证明了$ p_1^mp_1^mp_2 \ dotsm p_n $ p_n $ p_1^mp_1^mp_1^mp_1^mp_1^mp_2 \ dotsm p_n $的平式子集。
The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb{R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers. We manage to prove the conjecture for cyclic groups of order $p^mq^n$, when one of the exponents is $\leq6$ or when $p^{m-2}<q^4$, and also prove that a tiling subset of a cyclic group of order $p_1^mp_2\dotsm p_n$ is spectral.
