We 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 \(\varvec{\mathcal {B}}\) given by a purely periodic sequence \((\beta _n)_{n\ge 1}\) 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 \(\varvec{\mathcal {B}}\) -expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic \(\varvec{\mathcal {B}}\) -expansion. We show that a necessary condition for this phenomenon is that \(\delta =\prod _{n=1}^{p}\beta _n\) (where p is the period-length of \(\varvec{\mathcal {B}}\) ) 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 \(p=1\) . We provide a class of alternate bases in which all rational numbers in the interval [0, 1) have a purely periodic \(\varvec{\mathcal {B}}\) -expansion.
