实数的唯一基：单纯正规基、非正规基与重有理数
Univoque bases of real numbers: simply normal bases, irregular bases and multiple rationals
摘要：
给定正整数$M$和实数$x\in(0,1]$，若存在唯一的单正规序列$(d_i)\in\{0,1,\ldots,M\}^\mathbb N$，使得$x=\sum_{i=1}^\infty d_i q^{-i}$，则称$q\in(1,M+1]$为$x$的唯一单正规基.类似地，如果存在唯一序列$(d_i)\in\{0,1,\ldots, M\}^\mathbb N$使得$x=\sum_{i=1}^\infty d_i q^{-i}$和序列$(d_i)$没有数字频率，则碱基$q\in(1,M+1]$被称为$x$的唯一不规则碱基。设$\mathcal U_{SN}(x)$和$\mathcal U_{I_r}(x)$分别是$x$的单正规基和单非正规基的集合.在本文中，我们证明了任何$x\in(0,1]$的$\mathcal U_{SN}(x)$和$\mathcal U_{I_r}(x)$有完整的Hausdorff维数。此外，我们还证明了存在一个完整的Hausdorff维数集的$x_1, x_2, \ldots, x_n\in(0,1]$，使每个$x_i$有一个唯一的扩展基$q$。
英文摘要：
Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$. Similarly, a base $q\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\in\{0,1,\ldots, M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\in(0,1]$ both $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \ldots, x_n\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$.
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