一般函数下Beta-展式中极致例外集的剩余性质

The Residual Property of the Extremely Exceptional Set in the Beta-Expansion under General Function

摘要: 给定$\beta >1$ 和$x\in \left(0,1\right]$ 。对于任意的$y\in \left(0,1\right]$ ，关于$x$ 的run-length函数$r_x\left(y,n\right)$ 定义为在$y$ 的$\beta$ -展式的前$n$ 个数字中出现的$x$ 的$\beta$ -展式的前缀的最大长度。令非负函数$\phi \left(n\right)$ 满足$\underset{n\to \infty }{\lim }\frac{n}{\phi \left(n\right)}=+\infty$ ，我们对以下关于该函数的极致例外集$E_x^{\phi \left(n\right)}\left(0,+\infty \right)=\left\{x\in \left[0,1\right]:\underset{n\to \infty }{\lim \inf }\frac{r_x\left(y,n\right)}{\phi \left(n\right)}=0,\underset{n\to \infty }{\lim \sup }\frac{r_x\left(y,n\right)}{\phi \left(n\right)}=+\infty \right\}$ 进行研究，给出该集合的剩余性质。

Abstract: Let $\beta >1$ and $x\in \left(0,1\right]$ . For any $y\in \left(0,1\right]$ , the run-length function $r_x\left(y,n\right)$ (with respect to $x$ ) is defined to be the maximal length of the prefix of the beta-expansion of $x$ which appears in the first $n$ terms of the beta-expansion of $y$ . Let the non-negative function $\phi \left(n\right)$ satisfy $\underset{n\to \infty }{\lim }\frac{n}{\phi \left(n\right)}=+\infty$ . We study the extremely exceptional set given by $E_x^{\phi \left(n\right)}\left(0,+\infty \right)=\left\{x\in \left[0,1\right]:\underset{n\to \infty }{\lim \inf }\frac{r_x\left(y,n\right)}{\phi \left(n\right)}=0,\underset{n\to \infty }{\lim \sup }\frac{r_x\left(y,n\right)}{\phi \left(n\right)}=+\infty \right\}$ and establish its residual properties.