  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 R\'enyi system with one real base. We focus
on the so-called alternate base $B$ given by a purely periodic sequence of real
numbers greater than 1. We answer an open question of Charlier et al. on the
set of numbers with eventually periodic $B$-expansions. We also investigate for
which bases all sufficiently small rationals have a purely periodic
$B$-expansion.
