  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.
