Expansions in Cantor real bases

2021/02/15 by Charlier, Émilie, Cisternino, Célia · 3 citations
#11A63 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.07722

Abstract

We introduce and study series expansions of real numbers with an arbitrary Cantor real base \boldsymbolβ=(β_n)_n∈ℕ, which we call \boldsymbolβ-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of \boldsymbolβ-representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy \boldsymbolβ-representations of some real number in the interval [0,1). We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the \boldsymbolβ-shift is sofic if and only if all quasi-greedy \boldsymbolβ⁽ⁱ⁾-expansions of 1 are ultimately periodic, where \boldsymbolβ⁽ⁱ⁾ is the i-th shift of the Cantor real base \boldsymbolβ.

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• Non-periodicity of the sequence of the last nonzero digits of factorials and its applications to transcendence
• Computing Expansions in Infinitely Many Cantor Real Bases via a Single Transducer
• An analogue of Solomyak's theorem for periodic Cantor real expansions in alternate bases

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