论文标题
光谱套装和薄弱的瓷砖
Spectral sets and weak tiling
论文作者
论文摘要
如果空间$ l^2（ω）$允许指数函数的正交基础，则$ω\ subset \ mathbb {r}^d $被认为是光谱。 Fuglede（1974）猜想$ω$是光谱，并且仅当它可以通过翻译铺平空间时。尽管该猜想是对一般集的反驳的，但最近证明了fuglede的猜想确实在$ \ mathbb {r}^d $中构成了凸面的类别。该证明是基于新的几何形状的光谱条件，称为“弱平地”。在本文中，我们研究了弱平铺概念的进一步特性，并将应用于凸形体，非凸层，乘积域和正值的cantor集合。
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $Ω$ is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in $\mathbb{R}^d$. The proof was based on a new geometric necessary condition for spectrality, called "weak tiling". In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.
