Hausdorff measures of sets in Exact Diophantine approximation

Let $(X,d)$ be a compact metric space, and let $Q\subset X$ be countable. Given functions $R:Q\to {\mathbb{R}}^{+}$ and $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, we consider the set $E(Q,R,\phi )$ of points $x\in X$ that hit'' the shrinking balls $B(ξ,{φ(R(ξ))})$ for infinitely many $ξ\in Q$, yet, for every $ε\in (0,1)$, are eventually cleared out’’ from the slightly smaller neighborhoods $B(\xi ,(1-\epsilon )\phi (R(\xi )))$, that is, they lie outside all but finitely many of these smaller balls. We give sufficient conditions (also necessary under mild assumptions) for $E(Q,R,\phi )$ to have infinite Hausdorff $f$-measure. This setting generalizes both the classical set $\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}(\psi )$ of exactly $\psi$-approximable points (with $\psi$ non-increasing) and certain types of restricted Diophantine approximation sets.