A Harmonic Analysis Approach to Problems of Tiling

摘要DMS-9705775 Kolountzakis Kolountzakis将利用调和分析研究欧几里德空间的平移切片的性质。当函数f(X)在A的位置平移时，如果它加起来等于几乎所有地方的常数w(平铺的“权重”)，则称函数f(X)用离散的“平铺集”A来“平铺”欧几里德空间。人们得到了发生这种情况的一个等价条件：在函数f的FT的零点加上点0，必须支持在A的每个点具有一个点质量的分布的傅里叶变换。Kolountzakis建议研究的问题包括：(A)具有非紧支撑的线的平铺的结构，(B)高维的周期平铺猜想，(C)与泊松求和公式唯一性有关的结果，(D)对Steinhaus问题的进一步估计，(E)一般阿贝尔群中的同时基本域问题，以及(F)通过非阿贝尔调和分析方法的空间的非平移使用一种被称为“瓷砖”的建筑材料来研究空间的瓦片与结晶学有着重要的联系。在最近发现非周期性铺砌(即，铺设的空间不是由它们在任意大但有限的空间部分中的外观“确定”)确实存在之后，人们对此重新产生了兴趣。在这一领域，最重要的问题是，空间平铺的可能方式是什么。拟议的研究将有助于从更高的维度理解这种情况。所提出的方法的另一个应用领域涉及小波理论

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