The Jarník-Besicovitch theorem forgeometrically finite Kleinian groups

Abstract

Let $G$ be a geometrically finite Kleinian group with parabolic elements and let $p$ be any parabolic fixed point of $G$. For each positive real $\tau$, let ${\cal W}_{p}(\tau)$ denote the set of limit points of $G$ for which the inequality $$ | x-g(p) | \leq |g^\prime(0)|^{\tau}$$ is satisfied for infinitely many elements $g$ in $G$. This subset of the limit set is precisely the analogue of the set of $\tau$-well approximable numbers in the classical theory of metric Diophantine approximation. In this paper we consider the following question. What is the `size' of the set ${\cal W}_{p}(\tau)$ expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for $\tau \geq 1$,$$ \dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$$ where $\mbox{rk}(p)$ denotes the rank of the parabolic fixed point $p$.

1991 Mathematics Subject Classification: 11K55, 11K60, 11F99, 58D20.