Expansions in Cantor real bases

Abstract

We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\varvec{\beta }=(\beta _n)_{n\in {\mathbb {N}}}$, which we call $\varvec{\beta }$-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of $\varvec{\beta }$-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 $\varvec{\beta }$-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 $\varvec{\beta }$-shift is sofic if and only if all quasi-greedy $\varvec{\beta }^{(i)}$-expansions of 1 are ultimately periodic, where $\varvec{\beta }^{(i)}$ is the i-th shift of the Cantor real base $\varvec{\beta }$.