Annales Academić Scientiarum Fennicć
Mathematica
Volumen 38, 2013, 535-546

FAT AND THIN SETS FOR DOUBLING MEASURES IN EUCLIDEAN SPACE

Wen Wang, Shengyou Wen and Zhi-Ying Wen

Yunnan University, Department of Mathematics
Kunming 650091, P.R. China; sophia_84 'at' 126.com

Hubei University, Department of Mathematics
Wuhan 430062, P.R. China; sywen_65 'at' 163.com

Tsinghua University, Department of Mathematics
Beijing 100084, P.R. China; wenzy 'at' tsinghua.edu.cn

Abstract. According to the size of sets for doubling measures, subsets of Rⁿ 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.

2010 Mathematics Subject Classification: Primary 28A12; Secondary 30L10.

Key words: Uniform Cantor set, doubling measure, quasisymmetric mapping, fat set, thin set.

Reference to this article: W. Wang, S. Wen and Z.-Y. Wen: Fat and thin sets for doubling measures in Euclidean space. Ann. Acad. Sci. Fenn. Math. 38 (2013), 535-546.

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doi:10.5186/aasfm.2013.3827