Expansions in Cantor real bases

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

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