The Theory of Normality for Dynamically Generated Cantor Series Expansions

TL;DR

This paper develops a unified theory of normality for a broad class of Cantor series expansions generated by dynamical systems, extending classical results from base expansions to more complex numeration systems.

Contribution

It introduces the class of dynamically generated Cantor series expansions, enabling classical normality theory to be applied to these systems, including examples like Thue-Morse and Champernowne numbers.

Findings

Normality and distribution normality coincide for bounded, zero entropy, ergodic-generated sequences.

The class includes many well-known sequences such as Thue-Morse and Champernowne.

Established a Hot Spot Theorem for these systems.

Abstract

The theory of normality for base $g$ expansions of real numbers in $[0,1)$ is rich and well developed. Similar theories have been developed for many other numeration systems, such as the regular continued fraction expansion, $\beta$-expansions, and L\"uroth series expansions. Let $Q=(q_n)_{n \in \mathbb{N}}$ be a sequence of integers greater than or equal to 2. The $Q$-Cantor series expansion of $x \in [0,1)$ is the unique sum of the form $x=\sum_{n=1}^\infty \frac{x_n}{q_1q_2\cdots q_n}$, where $x_n \neq q_n-1$ infinitely often. For the Cantor series expansions, most of the literature thus far considers $Q$ where the theory of normality differs drastically from that of the base $g$ expansions. We introduce the class of dynamically generated Cantor series expansions, which is a large class of Cantor series expansions for which much of the classical theory of base $g$ expansions can be developed in parallel. This class includes many examples such as the Thue-Morse sequence on $\{2,3\}$ and translated Champernowne numbers. A special case of our main results is that if $Q$ is a bounded basic sequence that is dynamically generated by an ergodic system having zero entropy, then normality base $Q$ coincides with distribution normality base $Q$, and $Q$ possesses a Hot Spot Theorem.