Hausdorff measures of sets in Exact Diophantine approximation
Bo Tan, Chen Tian
TL;DR
The paper establishes a general framework for exact Diophantine approximation sets $E(Q,R,\phi)$ in a delta-regular metric space, showing that under a local-ubiquity type structure and a divergence condition on a tailored sum, the $s$-dimensional Hausdorff measure $\mathcal{H}^s(E(Q,R,\phi))$ is infinite. The authors prove a dichotomy via a Cantor-set construction and a carefully designed measure, handling both finite and infinite $G=\limsup g(u_n)$ cases, and derive concrete applications to classical simultaneous approximation and restricted Diophantine settings. The results unify and extend classical Exact$(\psi)$ results, providing a robust method to compare $\mathcal{H}^s(E(Q,R,\phi))$ with $\mathcal{H}^s(W(Q,R,\phi))$ under suitable distribution hypotheses. The approach hinges on local ubiquity, $\delta$-regularity, and a precise Cantor framework to transfer divergent series into Hausdorff-measure lower bounds with explicit $s$-scaling.
Abstract
Let $(X, d)$ be a compact metric space. Given a countable subset $Q \subset X$, a positive function $R: Q \to \mathbb{R}^+:ξ\mapsto R_ξ$, and a non-decreasing function $φ$, we consider the set $E(Q, R, φ)$ of points $x \in X$ for which there are infinitely many $ξ\in Q$ satisfying $$d(x, ξ) < φ(R_ξ),$$ yet for any $0<ε<1$, there are only finitely many $ξ\in Q$ such that $$d(x, ξ) < (1-ε)φ(R_ξ).$$ We provide sufficient conditions (which are also necessary under some mild assumptions) for the $s$-dimensional Hausdorff measure of $E(Q, R, φ)$ to be infinite. This framework generalizes not only the classical set $\mathrm{Exact}(ψ)$ of points exactly $ψ$-approximable by rationals (where $ψ$ is non-increasing), but also certain restricted Diophantine approximation sets.