Normal number constructions for Cantor series with slowly growing bases

TL;DR

This paper develops methods to construct numbers with normality properties in Cantor series expansions with slowly growing bases, including computable examples with atypical normality features.

Contribution

It introduces a construction for Q-normal and Q-distribution normal numbers in Cantor series with slowly growing bases, extending to computable numbers under certain conditions.

Findings

Constructed numbers are both Q-normal and Q-distribution normal.

Provided a method for computable constructions of such numbers.

Extended normality constructions to numbers with atypical properties.

Abstract

Let $Q=(q_n)_{n=1}^\infty$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties.