Periodicity and pure periodicity in alternate base systems

Abstract

AbstractWe study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base $$\varvec{\mathcal {B}}$$ B given by a purely periodic sequence $$(\beta _n)_{n\ge 1}$$ ( β n ) n ≥ 1 of real numbers greater than 1. We answer an open question of Charlier et al. (J Number Theory 254:184–198, 2024, https://doi.org/10.1016/j.jnt.2023.07.008) on the set of numbers with eventually periodic $$\varvec{\mathcal {B}}$$ B -expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic $$\varvec{\mathcal {B}}$$ B -expansion. We show that a necessary condition for this phenomenon is that $$\delta =\prod _{n=1}^{p}\beta _n$$ δ = ∏ n = 1 p β n (where p is the period-length of $$\varvec{\mathcal {B}}$$ B ) is a Pisot or a Salem unit. We also provide a sufficient condition. We thus generalize the results known for the Rényi numeration system, i.e. for the case when $$p=1$$ p = 1 . We provide a class of alternate bases in which all rational numbers in the interval [0, 1) have a purely periodic $$\varvec{\mathcal {B}}$$ B -expansion.