欧氏空间上的集合在加倍测度下的胖性与瘦性

According to the Lebesgue measure of sets, we divide the subsets of Euclidean space into two main classes, fat sets and thin sets. Let E be a subset of Euclidean space. We say that E is thin if its Lebesgue measure is equal to zero. Otherwise, we say that the set E is fat. This classification of subsets of Euclidean space is so coarse that the geometrical features of subsets are not sufficiently described. How to classify these subsets according to their features is a basic problem in Geometric Measure Theorey. An important idea is to divide them into finer subclasses by doubling measures. And then, fat sets are divided into three subclasses, which are called very fat, fairly fat and minimally fat. Similarly, thin sets are divided into three subclasses, which are called very thin, fairly thin and minimally thin. So, we have six classes of subsets of Euclidean Space. In this project, we shall study fat and thin sets for doubling measures in Euclidean space. Because little has been known for these sets in higher dimensional Euclidean space, we shall focus on the fatness and thinness of sets in higher dimensional case.

欧氏空间上的集合按照其Lebesgue测度是否为零可以分为两类：胖集和瘦集,将Lebesgue测度大于零的集合称为胖集，将Lebesgue测度等于零的集合称为瘦集。但是这种分类方式比较粗略，集合的一些几何特征和性质体现不出来。如何按照集合的特征性质做进一步的分类是几何测度论中一个基本的问题。利用加倍测度将集合进行更为细致的分类是一个重要的思想。我们将胖集（或瘦集）按照其在加倍测度下的不同表现特征又分成了三个小类：非常胖（或瘦）、相当胖（或瘦）、有一点胖（或瘦）。这样，欧氏空间的子集就分成了六个小类。本项目主要研究的是欧氏空间上的集合在加倍测度下的胖性与瘦性。由于在高维欧氏空间上，集合在加倍测度下的胖性与瘦性还所知不多，因此我们尤其要探讨在高维空间中集合的胖性与瘦性。

欧氏空间的子集根据其在加倍测度下的表现形式可以分为六类：非常胖、相当胖、有一点胖、非常瘦、相当瘦和有一点瘦。本项目主要研究的是平面上的Sierpinski地毯的这种分类，并且我们给出了描述其分类的充要条件。