Let
$\unicode[STIX]{x1D6FD}>1$ be a real number and define the
$\unicode[STIX]{x1D6FD}$-transformation on
$[0,1]$ by
$T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$. Further, define
$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$ and
$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$ where
$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$ is a positive function such that
$\unicode[STIX]{x1D6F9}(n)\rightarrow 0$ as
$n\rightarrow \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized 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.