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

TL;DR

This paper investigates how certain operations affect normality in Cantor series expansions, revealing that integer multiplication preserves distribution normality but not normality, and non-integer rational multiplication generally does not preserve these properties.

Contribution

It introduces new results on the preservation of normality under multiplication in Cantor series expansions, extending understanding beyond base-$b$ systems.

Findings

Integer multiplication preserves $Q$-distribution normality.

Integer multiplication does not preserve $Q$-normality.

Non-integer rational multiplication generally does not preserve $Q$-distribution normality.

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.