Normality preserving operations for Cantor series expansions and associated fractals, I

Abstract

It is well known that rational multiplication preserves normality in base $b$. We study related normality preserving operations for the $Q$-Cantor series expansions. In particular, we show that while integer multiplication preserves $Q$-distribution normality, it fails to preserve $Q$-normality in a particularly strong manner. We also show that $Q$-distribution normality is not preserved by non-integer rational multiplication on a set of zero measure and full Hausdorff dimension.