Liminf approximation sets for abstract rationals

Abstract

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for certain limsup sets. We discuss the analogous problem for liminf sets. Consider an infinite sequence of positive integers, $S=\{q_{n}\}_{n\in\mathbb{N}}$, exhibiting exponential growth. For a given $n$-tuple of functions denoted as $\Psi:=~(\psi_1, \ldots,\psi_n)$, each of the form $\psi_{i}(q)=q^{-\tau_{i}}$ for $(\tau_{1},\dots,\tau_{n})\in\mathbb{R}^{n}_{+}$, we calculate the Hausdorff dimension of the set of points that can be $\Psi$-approximated for all sufficiently large $q\in S$. We prove this result in the generalised setting of approximation by abstract rationals as recently introduced by Koivusalo, Fraser, and Ramirez (LMS, 2023). Some of the examples of this setting include the real weighted inhomogeneous approximation, $p$-adic weighted approximation, Diophantine approximation over complex numbers, and approximation on missing digit sets.