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 $\boldsymbol{ \beta}$.
Categories: math.CO cs.DM