A Harmonic Analysis Approach to Problems of Tiling

摘要DMS-9705775 科隆扎基斯 Kolountzakis将研究平移平铺的性质， 欧氏空间使用调和分析。一个函数f（x）被称为 用离散的“瓦片集”A“瓦片”欧几里得空间，如果，当 在A的位置平移，它加起来是一个常数w（ “重”的瓷砖），几乎无处不在。人们得到一个等价物 发生这种情况的条件： 必须支持在A的每一点都有一个点质量的分布 在函数f的FT的零点处，加上点0。 Kolountzakis提出要研究的问题包括：（a）具有非紧瓦片的线的瓦片的结构 支持，（B）在高维周期平铺猜想，（c）结果有关的唯一性的泊松求和公式， (d)进一步估计的施泰因豪斯问题，（e）问题的 一般交换群中的同时基本域， (f)通过非阿贝尔方法的空间的非平移平铺 谐波分析 使用称为“瓦”的特定积木来研究空间的平铺与晶体学有着重要的联系。在相对较新的发现非周期性镶嵌（即，在自然界中确实存在着空间的铺路石，而这些铺路石并不取决于它们在空间的任意大但有限的部分中的外观。在这个领域中，最重要的问题是空间可以被平铺的可能方式是什么。 拟议的研究将有助于了解情况 在更高的维度。拟议的另一个应用领域 方法涉及波动理论

ABSTRACT DMS-9705775 Kolountzakis Kolountzakis will study properties of translational tilings of Euclidean Space using Harmonic Analysis. A function f(x) is said to "tile" the Euclidean space with the discrete "tile set" A if, when translated at the locations of A, it adds up to a constant w (the "weight" of the tiling), almost everywhere. One obtains an equivalent condition for this to happen: the Fourier transform of the distribution with one point mass at each point of A must be supported at the zeros of the FT of the function f, plus the point 0. Among the problems Kolountzakis proposes to study are: (a) the structure of tilings of the line with tiles of non-compact support, (b) the periodic tiling conjecture in higher dimension, (c) results related to the uniqueness of the Poisson Summation formula, (d) further estimates on the Steinhaus problem, (e) the problem of simultaneous fundamental domains in general abelian groups and (f) non-translational tilings of space via methods of non-abelian harmonic analysis. The study of tilings of space using a certain building block called the "tile" has significant connections with crystallography. Interest in this has been renewed after the, relatively recent, discovery that non-periodic tilings (i.e., pavings of space which are not "determined" by how they look in an arbitrarily large but finite part of the space) do exist in nature. The question of which are the possible ways that space can be tiled is the paramount one in this field. The proposed research will help to understand the situation in higher dimensions. Another domain of application of the proposed methods concern the theory of Wavelets