Abstract
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.
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