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

TL;DR

This paper explores how rational operations influence normality in $Q$-Cantor series expansions, demonstrating the existence of full Hausdorff dimension sets with specific normality properties and providing explicit examples.

Contribution

It shows that certain normality properties are preserved under rational operations and constructs explicit computable examples with full Hausdorff dimension.

Findings

Existence of a $Q$ with full Hausdorff dimension set of $Q$-normal but not $Q$-distribution normal numbers

Such sets remain invariant under rational multiplication and addition

Constructs an explicit computable example of such a number

Abstract

We investigate how non-zero 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.