论文标题
实际数字的单项基础：简单的基础，不规则基础和多个理性
Univoque bases of real numbers: simply normal bases, irregular bases and multiple rationals
论文作者
论文摘要
给定一个正整数$ m $和一个实际数字$ x \ in（0,1] $，我们称$ q \ in（1，m+1] $是univoque，如果存在独特的正常序列$（d_i）\ in \ in \ in \ {0,1，\ ldots，m \}，m \}^\ nath $ nater of $ x $，则为$ x $。 $ x = \ sum_ {i = 1}^\ infty d_i q^{ - i} $。 $ x = \ sum_ {i = 1}^\ infty d_i q^{ - i} $，序列$（d_i）$没有数字频率本文我们表明，对于任何$ x \ in（0,1] $均$ \ Mathcal u_ {sn}（x）$和$ \ Mathcal u_ {i_r}（x）$具有完整的Hausdorff尺寸。此外，给出了许多合理的$ x_1，x_2，x_2，x_2，x__n $ $ x_n $ a $ x_n $ a $ x_n $基本$ m+1 $的扩展，我们表明存在一个完整的Hausdorff尺寸集，$ q \ in（1，m+1] $，因此每个$ x_i $都具有基本$ Q $的独特扩展。
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 $q$.
