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 \(\varvec{\beta }\).

我们介绍并研究了具有任意康托尔实数基\(\varvec{\beta }=(\beta _n)_{n\in {\mathbb {N}}}\)的实数级数展开式，我们称之为\( \varvec{\beta }\) -表示。在这样做时，我们概括了实数在实数基和康托级数中的表示。我们展示了\(\varvec{\beta }\)表示的基本属性，每个表示都扩展了在实数基础上表示的现有结果。特别是，我们证明了 Parry 定理的推广，该定理表征了贪婪的非负整数序列\(\varvec{\beta }\)- 区间 [0, 1) 中某个实数的表示。我们特别注意周期性的康托实基，我们称之为交替基。在这种情况下，我们证明\(\varvec{\beta }\) -shift 是 sofic 当且仅当所有拟贪婪\(\varvec{\beta }^{(i)}\) -expansions of 1最终是周期性的，其中\(\varvec{\beta }^{(i)}\)是康托尔实数基数\(\varvec{\beta }\) 的第i个偏移。