报告人：李文侠 教授（华东师范大学）

时间：2025年05月12日 10:00-

地点：理科楼LA104

摘要：Associated to the  L\"{u}roth expansion of an irrational number  $\bar{x}\in (0, 1)$ is a sequence $x=\{x_i\}\in {\mathbb N}^{\mathbb N}$. Then  for each $n$, one can get an infinite probability vector $\Pi(x|n)=(p_{i,n})_{i\in \mathbb N}$

where $p_{i,n}$ is the frequency of $i$ occurring in the prefix of $\{x_i\}$ of length $n$. Let $ A(\{\Pi(x|n)\}_{n\in \mathbb N})$ be the set of accumulation points of

the sequence $\{\Pi(x|n)\}_{n\in \mathbb N}$. Given a set $C$, let

$$

\Omega_{=C}=\left \{x \in {\mathbb N}^{\mathbb N}:

A(\{\Pi(x|n)\}_{n\in \mathbb N})= C\right \}\;\;\text{and}\;\;

\Omega_{\subseteq C}=\left\{x\in {\mathbb N}^{\mathbb N}:

A(\{\Pi(x|n)\}_{n\in \mathbb N})\subseteq C\right\}.

$$

In this talk, we introduce how to determine the Hausdorff dimensions of $\pi (\Omega_{=C})$ and $\pi (\Omega_{\subseteq C})$.

邀请人：数学研究中心

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