New York Journal of Mathematics
Volume 17 (2011) 601-617


Bill Mance

Typicality of normal numbers with respect to the Cantor series expansion

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Published: September 7, 2011
Keywords: Cantor series, normal numbers
Subject: 11K16, 11A63

Fix a sequence of integers Q={qn}n=1 such that qn is greater than or equal to 2 for all n. In this paper, we improve upon results by J. Galambos and F. Schweiger showing that almost every (in the sense of Lebesgue measure) real number in [0,1) is Q-normal with respect to the Q-Cantor series expansion for sequences Q that satisfy a certain condition. We also provide asymptotics describing the number of occurrences of blocks of digits in the Q-Cantor series expansion of a typical number. The notion of strong Q-normality, that satisfies a similar typicality result, is introduced. Both of these notions are equivalent for the b-ary expansion, but strong normality is stronger than normality for the Cantor series expansion. In order to show this, we provide an explicit construction of a sequence Q and a real number that is Q-normal, but not strongly Q-normal. We use the results in this paper to show that under a mild condition on the sequence Q, a set satisfying a weaker notion of normality, studied by A. Rényi, 1956, will be dense in [0,1).

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Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210-1174