报告题目:Diophantine analysis in Beta-dynamical systems and Lebesgue measures
报告摘要：Let $\varphi\colon \mathbb{N}\rightarrow(0,1]$ be a positive function and let $\{l_{n}\}_{n\geq 1}$ be a sequence of non-negative numbers. Using the Paley-Zygmund inequality, we prove that the set
\begin{equation*}
E(0, \varphi)=\{\beta>1\colon |T^{n}_{\beta}1-0|<\varphi(n) \textrm{ for infinitely many } n\in\mathbb{N}\}
\end{equation*}
is of zero or full Lebesgue measure in $(1,+\infty)$ according to $\sum\varphi(n)<+\infty$ or not, where $T_{\beta}$ is the $\beta$-transformation. We also determine the Lebesgue measure of the following set
\begin{equation*}
\mathfrak{E}(0, \{l_{n}\})=\{\beta>1\colon |T^{n}_{\beta}1-0|<\beta^{-l_{n}} \text{ for infinitely many } n\in\mathbb{N}\}.
\end{equation*}
报告人简介：吕凡，男，副教授。研究领域为分形几何与动力系统，目前主持国家自然科学基金青年基金一项，在Adv in Math等国际SCI刊物上已发表科研论文10余篇。
报告时间：2018年8月7日(星期二)下午3：30-4：30
