论文标题
通过翻译函数的平铺：结果和开放问题
Tiling by translates of a function: results and open problems
论文作者
论文摘要
我们说，如果我们有$λ\ subset \ subset \ subset \ mathbb {r} $，则在l^1（\ mathbb {r}）$ w $中的功能$ f \，如果我们有$ \ sum_ {λ{λ\inλ} f（x-λ）= w $ a.e。在本文中，我们通过翻译功能调查了主要结果，并证明了几个新结果。讨论的现象包括有界密度和无限密度的瓷砖，翻译的均匀分布，周期性和非周期性的瓷砖以及零级的瓷砖。傅立叶分析在证明中起着重要作用。还给出了一些开放问题。
We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $Λ\subset \mathbb{R}$, if we have $\sum_{λ\in Λ} f(x-λ)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $\mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.
