Hausdorff measures of sets in Exact Diophantine approximation

Abstract

Let $(X,d)$ be a compact metric space, and let $Q\subset X$ be countable. Given functions $R:Q\to {\mathbb{R}}^{+}$ and $\ddot{I}•:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, we consider the set $E(Q,R,\ddot{I}•)$ of points $x\in X$ that ``hit'' the shrinking balls $B(\hat{I}¾,\ddot{I}•(R(\hat{I}¾)))$ for infinitely many $\hat{I}¾\in Q$, yet, for every $\hat{I}\mu \in (0,1)$, are eventually ``cleared out'' from the slightly smaller neighborhoods $B(\hat{I}¾,(1-\hat{I}\mu )\ddot{I}•(R(\hat{I}¾)))$, that is, they lie outside all but finitely many of these smaller balls. We give sufficient conditions (also necessary under mild assumptions) for $E(Q,R,\ddot{I}•)$ to have infinite Hausdorff $f$-measure. This setting generalizes both the classical set $\mathrm{Exact}(\ddot{I}ˆ)$ of exactly $\ddot{I}ˆ$-approximable points (with $\ddot{I}ˆ$ non-increasing) and certain types of restricted Diophantine approximation sets.