The Jarník-Besicovitch Theorem for Geometrically Finite Kleinian Groups

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JournalProc. Lond. Math. Soc
DatePublished - Nov 1998
Issue number3
Volume77
Number of pages27
Pages (from-to)524-550
Original languageEnglish

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 t, let Wp t 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 t-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 Wp t expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for t = 1, $\dim {\cal W}_{p} (\tau) = \min \left\{ \frac{\delta + \mbox{rk}(p) (\tau - 1) }{2 \tau - 1}, \, \frac{\delta}{\tau}\right\},$ where rk (p) denotes the rank of the parabolic fixed point p. 1991

    Research areas

  • Diophantine approximation, Kleinian groups, Hausdorff dimension, geodesic excursions

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