Hausdorff measures of sets in Exact Diophantine approximation

TL;DR

This paper investigates the Hausdorff measure of specific sets in metric spaces related to Diophantine approximation, generalizing classical approximation sets and establishing conditions for infinite measure.

Contribution

It provides new sufficient and necessary conditions for the Hausdorff $f$-measure of these sets in a general metric space setting.

Findings

Conditions for infinite Hausdorff $f$-measure established

Generalizes classical Diophantine approximation sets

Includes necessary conditions under mild assumptions

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.