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Normal Numbers with Respect to the Cantor Series Expansion

Bill Mance

出版	Ohio State University, 2010
URL	http://books.google.com.hk/books?id=VwGGnQAACAAJ&hl=&source=gbs_api

註釋

We define potentially stronger notions of normality: strong Q-normality, strong Q-ratio normality, and strong Q-distribution normality that are equivalent to normality in the case of the b-ary expansion. We show that the set of strongly Q-distribution normal numbers always has full measure, but the set of strongly Q-normal numbers will only under certain conditions. We study winning sets, in the sense of Schmidt games and show that the set of non-strongly Q-ratio normal numbers and the set of non-strongly Q-distribution normal numbers are 1/2-winning sets and thus have full Hausdorff dimension. We also examine the property of being a winning set as it applies to other sets associated with the Q-Cantor series expansion.