Expansions in Cantor real bases

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