2025/12/01 by Sohail Farhangi, Farhangi, Sohail, Bill Mance +1
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #semigroups and automata theory #Computability, Logic, AI Algorithms
paper · pdf · doi:10.48550/arxiv.2512.01239
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, β-expansions, and Lüroth series expansions. Let Q=(qn)n ∈ ℕ be a sequence of integers greater than or equal to 2. The Q-Cantor series expansion of x ∈ [0,1) is the unique sum of the form x=∑n=1^∞ (xn)/(q1q2⋯ qn), where xn ≠ qn-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.