Combinatorial properties of lazy expansions in Cantor real bases

The lazy algorithm for a real base $\beta$ is generalized to the setting of Cantor bases $\boldsymbol{\beta}=(\beta_n)_{n\in \mathbb{N}}$ introduced recently by Charlier and the author. To do so, let $x_{\boldsymbol{\beta}}$ be the greatest real number that has a $\boldsymbol{\beta}$-representation $a_0a_1a_2\cdots$ such that each letter $a_n$ belongs to $\{0,\ldots,\lceil \beta_n \rceil -1\}$. This paper is concerned with the combinatorial properties of the lazy $\boldsymbol{\beta}$-expansions, which are defined when $x_{\boldsymbol{\beta}}<+\infty$. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of $x_{\boldsymbol{\beta}}$ is proved. First, it is shown that the lazy $\boldsymbol{\beta}$-expansions are obtained by "flipping" the digits of the greedy $\boldsymbol{\beta}$-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy $\boldsymbol{\beta}$-expansions of some real number in $(x_{\boldsymbol{\beta}}-1,x_{\boldsymbol{\beta}}]$ is proved. Moreover, the lazy $\boldsymbol{\beta}$-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy $\boldsymbol{\beta}$-shift is sofic if and only if all quasi-lazy $\boldsymbol{\beta}^{(i)}$-expansions of $x_{\boldsymbol{\beta}^{(i)}}-1$ are ultimately periodic, where $\boldsymbol{\beta}^{(i)}$ is the $i$-th shift of the alternate base $\boldsymbol{\beta}$.