Combinatorial properties of lazy expansions in Cantor real bases

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

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