FAT AND THIN SETS FOR DOUBLING MEASURES IN EUCLIDEAN SPACE
FAT AND THIN SETS FOR DOUBLING MEASURES IN EUCLIDEAN SPACE
复制标题
DOI：
10.5186/aasfm.2013.3827
复制
发表时间：
2013-06
期刊：
影响因子：
--
通讯作者：
Wen Wang;S. Wen;Z. Wen
中科院分区：
文献类型：
--
作者：
Wen Wang;S. Wen;Z. Wen
根据加倍度量的集合的大小，R n的子集可以分为六个类。这六个类中的集合被称为非常薄，相当薄，微薄，脂肪，相当脂肪且非常胖。在我们的主要结果中，我们证明，如果（0; 1）的准对称映射F（0; 1）映射到均匀的cantor套件到均匀的cantor套件f（e），则E仅在f（e）是这样的情况下进行正面的lebesgue测量。另外，我们证明，当且仅当每个因素都非常脂肪时，并且只有当一个因素之一是最小的脂肪时，且乘积的脂肪是最小的。
According to the size of sets for doubling measures, subsets of R n can be divided into six classes. Sets in these six classes are respectively called very thin, fairly thin, minimally thin, minimally fat, fairly fat, and very fat. In our main results, we prove that if a quasisymmetric mapping f of (0;1) maps a uniform Cantor set E onto a uniform Cantor set f(E), then E is of positive Lebesgue measure if and only if f(E) is so. Also, we prove that the product of n uniform Cantor sets is very fat if and only if each of the factors is very fat, and that the product is minimally fat if and only if one of the factors is minimally fat.