Dimension theory of Diophantine approximation related to $β$-transformations

Abstract: 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 given two positive functions $\psi_1,~\psi_2: \mathbb{N} \rightarrow \mathbb{R}^+$, define $$\mathcal{L}(\psi_1):=\left{x\in[0,1]:T_\beta^n x<\psi_1(n),\text{ for infinitely many $n\in\mathbb{N}$}\right},$$ $$\mathcal{U}(\psi_2):=\left{x\in [0,1]:\forall~N\gg1,~\exists~n\in[0,N],\ s.t.\ T^n_\beta x<\psi_2(N)\right},$$ where $\gg$ means large enough. We compute the Hausdorff dimension of the set $\mathcal{L}(\psi_1)\cap\mathcal{U}(\psi_2)$. As a corollary, we estimate the Hausdorff dimension of the set $\mathcal{U}(\psi_2)$.