New York Journal of Mathematics
Volume 21 (2015) 1311-1326


Dylan Airey, Bill Mance, and Joseph Vandehey

Normality preserving operations for Cantor series expansions and associated fractals. II

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Published: December 14, 2015
Keywords: Cantor series, Normal numbers, Hausdorff dimension
Subject: Primary 11K16, Secondary 11A63 and 28A80

We investigate how nonzero rational multiplication and rational addition affect normality with respect to Q-Cantor series expansions. In particular, we show that there exists a Q such that the set of real numbers which are Q-normal but not Q-distribution normal, and which still have this property when multiplied and added by rational numbers has full Hausdorff dimension. Moreover, we give such a number that is explicit in the sense that it is computable.


Research of the first and second authors is partially supported by the U.S. NSF grant DMS-0943870.

Author information

Dylan Airey:
Department of Mathematics, University of Texas at Austin, 2515 Speedway, Austin, TX 78712-1202, USA

Bill Mance:
Department of Mathematics, University of North Texas, General Academics Building 435, 1155 Union Circle, #311430, Denton, TX 76203-5017, USA

Joseph Vandehey:
Department of Mathematics, University of Georgia at Athens, Boyd graduate studies research center, Athens, GA 30606 USA