  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.

Number Theory;Dynamical Systems