Dimension theory of Diophantine approximation related to $\beta$-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 ${\psi }_1,{\psi }_2:\mathbb{N}\to {\mathbb{R}}^{+}$, define $$\mathcal{L}({\psi }_1):=\left\{x\in [0,1]:T_{\beta }^{n}x<{\psi }_1(n),\text{ for infinitely many n∈N}\right\},$$ $$\mathcal{U}({\psi }_2):=\left\{x\in [0,1]:\forall N\gg 1,\exists n\in [0,N],s.t.T_{\beta }^{n}x<{\psi }_2(N)\right\},$$ 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 $\mathcal{U}({\psi }_2)$.