Dimension theory of Diophantine approximation related to $\beta$-transformations

Wanlou Wu, Lixuan Zheng

Let $T_{\beta }$ be the $\beta$-transformation on $[0,1)$ defined by $$T_{\beta }(x)=\beta x\text{ mod }1.$$ We study the Diophantine approximation of the orbit of a point $x$ under $T_{\beta }$. Precisely, for…