 

Bill Mance
Typicality of normal numbers with respect to the Cantor series expansion view print


Published: 
September 7, 2011

Keywords: 
Cantor series, normal numbers 
Subject: 
11K16, 11A63 


Abstract
Fix a sequence of integers Q={q_{n}}_{n=1}^{∞} such that q_{n} 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 Qnormal with respect to the QCantor 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 QCantor series expansion of a typical number. The notion of strong Qnormality, that satisfies a similar typicality result, is introduced. Both of these notions are equivalent for the bary 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 Qnormal, but not strongly Qnormal. 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).


Author information
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 432101174
mance@math.ohiostate.edu

