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 $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the shrinking balls $B({\xi},{\phi(R(\xi))})$ for infinitely many $\xi \in Q$, yet, for every $\epsilon \in (0,1)$, 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 $\mathrm{Exact}(\psi)$ of exactly $\psi$-approximable points (with $\psi$ non-increasing) and certain types of restricted Diophantine approximation sets.