论文标题
在循环基团中的fuglede猜想的组环方法
A group ring approach to Fuglede's conjecture in cyclic groups
论文作者
论文摘要
Fuglede的猜想指出，一个$ω\ subseteq \ mathbb {r}^{n} $正面和有限的lebesgue度量是一个频谱集，并且仅当它通过翻译通过translation transpation tiles $ \ mathbb {r}^r}^{n} $。猜想并不能以$ \ mathbb {r}^n $，$ n \ ge3 $的方式沿两个方向。但是，此猜想仍然在$ \ mathbb {r} $和$ \ mathbb {r}^2 $中打开。循环群在研究fuglede的猜想中起着重要作用。在本文中，我们介绍了一种新工具，以研究循环基团的光谱集。特别是，我们证明Fuglede的猜想在$ \ mathbb {z} _ {p^{n} qr} $中。
Fuglede's conjecture states that a subset $Ω\subseteq\mathbb{R}^{n}$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles $\mathbb{R}^{n}$ by translation. The conjecture does not hold in both directions for $\mathbb{R}^n$, $n\ge3$. However, this conjecture remains open in $\mathbb{R}$ and $\mathbb{R}^2$. Cyclic groups play important roles in the study of Fuglede's conjecture in $\mathbb{R}$. In this paper, we introduce a new tool to study the spectral sets in cyclic groups. In particular, we prove that Fuglede's conjecture holds in $\mathbb{Z}_{p^{n}qr}$.
