论文标题
Cantor Real Bases中懒惰扩展的组合特性
Combinatorial properties of lazy expansions in Cantor real bases
论文作者
论文摘要
真实基础$β$的懒算法被推广到Cantor Bases $ \BoldSymbolβ=（β_n）_ {n \ in \ Mathbb {n}} $最近推出的Charlier和作者最近引入的。 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 $\{0,\ldots,\lceil β_n \rceil -1\}$.本文涉及懒惰$ \boldsymbolβ$ - expansions的组合特性，当$ x _ {\boldsymbolβ} <+\ hyfty $时定义。作为例子，研究了thue-morse序列之后的cantor碱基，并证明了其相应值为$ x _ {\boldsymbolβ} $的公式。首先，证明懒惰$ \boldsymbolβ$ - 通过“翻转”贪婪的$ \boldsymbolβ$ -Expansions获得的数字获得。接下来，一个类似帕里的标准，表征了非阴性整数的序列，这些序列是$ \boldsymbolβ$ - 在$（x _ {x _ {\boldsymbolβ} -1，x _ {\boldsymbolβ}] $的$（x _ {\boldsymbolβ} -1）中的真实数字的extered。这就是周期性的基础，证明了Bertrand-Mathis定理的类似物：lazy $ \boldsymbolβ$ -shift是sofic的，只有当所有Quasi-Lazy lazy $ \boldsymbolβ^{（i）} $扩展$ x______________________________________________________________________________________________________________________________y-i。 $ \boldsymbolβ{（i）} $是$ i $ th的替代基础$ \boldsymbolβ$的转变。
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 $\boldsymbolβ$.
