Normality of different orders for Cantor series expansions

TL;DR

This paper constructs basic sequences for Cantor series where normality of certain orders is independent, contrasting with base expansions, and shows these sets have full Hausdorff dimension, including computable examples.

Contribution

It demonstrates the existence of basic sequences with prescribed normality properties for different orders, including computable sequences, and analyzes their Hausdorff dimension.

Findings

Sets of numbers normal of specific orders have full Hausdorff dimension.

Constructed sequences where normality of orders in set S is independent of others.

Block frequencies along arithmetic progressions can fail to converge, contrasting with base expansions.

Abstract

Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all orders in $S$ but not $Q$-normal of all orders not in $S$ has full Hausdorff dimension. If the function $k \mapsto 1_S(k)$ is computable, then there exist computable examples. For example, there exists a computable basic sequence $Q$ where the set of numbers normal of all even orders and not normal of all odd orders has full Hausdorff dimension. This is in strong constrast to the $b$-ary expansions where any real number that is normal of order $k$ must also be normal of all orders between $1$ and $k-1$. Additionally, all numbers we construct satisfy the unusual condition that block frequencies sampled along non-trivial arithmetic progressions don't converge to the expected value. This is also in strong contrast to the case of the $b$-ary expansions, but more similar to the case of the continued fraction expansion. As a corollary, the set of $Q$-normal numbers that are not normal when sampled along any non-trivial arithmetic progression has full Hausdorff dimension.