The Theory of Normality for Dynamically Generated Cantor Series Expansions

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 Lüroth 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 $x={\sum }_{n=1}^{\infty }\frac{x_n}{q_1{q}_2\cdots q_n}$, where $x_n\ne q_n-1$ 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.