Expansions in Cantor real bases

Abstract

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

21 pages, 3 figures