A dichotomy law for the Diophantine properties in $\beta$-dynamical systems

Let $\beta>1$ be a real number and define the $\beta$-transformation on $[0,1]$ by $T_\beta:x\mapsto \beta x\bmod 1$. Further, define W_y(T_{\beta},\Psi):=\{x\in [0, 1]:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\} and W(T_{\beta},\Psi):=\{(x, y)\in [0, 1]^2:|T_\beta^nx-y|<\Psi(n) \mbox{ for infinitely many $n$}\}, where $\Psi:\mathbb{N}\to\mathbb{R}_{>0}$ is a positive function such that $\Psi(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarn\'ik-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.