Cantor实基中惰性展开式的组合性质
Combinatorial properties of lazy expansions in Cantor real bases
摘要：
本文将实基$β$的惰性算法推广到作者和Charlier最近提出的Cantor基$\boldsymbolβ=(β_n)_{n\in \mathbb{N}}$的情形。为此，令$x_{\boldsymbolβ}$为具有$\boldsymbolβ$表示$a_0a_1a_2\cdots$的最大实数，使得每个字母$a_n$属于$\{0,\ldots,\lceil β_n \rceil -1\}$。本文研究了当$\boldsymbolβ$.作为例子，研究了Thue-Morse序列的Cantor基，并证明了相应的$x_{\boldsymbolβ}$值公式.首先，它表明，懒惰的$\boldsymbolβ$ -扩展是通过“翻转”的贪婪的$\boldsymbolβ$ -扩展的数字。其次，证明了非负整数序列是$(x_{\boldsymbolβ}-1,x_{\boldsymbolβ}]$中某个实数的惰性$\boldsymbolβ$ -展开式的Parry型判别准则。进一步研究了$\boldsymbolβ$ -移位，在交替基即周期Cantor基的特殊情况下，证明了在惰性框架下Bertrand-Mathis定理的一个类似定理：$\boldsymbolβ$ -移位是sofic当且仅当$x_{\boldsymbolβ^{(i)}}-1$的所有准惰性$\boldsymbolβ^{(i)}$ -展开都是最终周期的，其中$\boldsymbolβ^{(i)}$是交替基$\boldsymbolβ$的第$i$次移位.
英文摘要：
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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