Abstract:

For $\beta>1$, let $T_\beta$ be the $\beta$-transformation defined on $[0, 1)$. We study the sets of points whose orbits of $T_\beta$ have uniform Diophantine approximation properties. Precisely, for two given positive functions $\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, define ${\cal L}(\psi_1):=\left\{x\in[0, 1]:T_\beta^n x<\psi_1(n), \mbox{ for infinitely many $n\in{\Bbb N}$}\right\}, $ ${\cal 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 calculate the Hausdorff dimension of the set ${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set ${\cal U}(\psi_2)$. Our work generalizes the results of ^[4] where only exponential functions $\psi_1, \ \psi_2$ were taken into consideration.

Key words: Beta-transformation, Uniform Diophantine approximation, Hausdorff dimension