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The Jarník‐Besicovitch Theorem for Geometrically Finite Kleinian Groups
Author(s) -
Hill R,
Velani SL
Publication year - 1998
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611598000550
Subject(s) - mathematics , kleinian group , hausdorff dimension , combinatorics , diophantine approximation , limit set , limit point , limit (mathematics) , mathematics subject classification , hausdorff distance , dimension (graph theory) , rank (graph theory) , finite set , discrete mathematics , pure mathematics , diophantine equation , mathematical analysis
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 τ, let W p τ denote the set of limit points of G for which the inequality | x − g ( p ) | ⩽ | g ′ ( 0 ) | τis satisfied for infinitely many elements g in G . This subset of the limit set is precisely the analogue of the set of τ‐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 W p τ expressed in terms of its Hausdorff dimension? We provide a complete answer, namely that for τ ⩾ 1, dim W p ( τ ) = min { δ + rk ( p ) ( τ − 1 ) 2 τ − 1 ,δ τ } , where rk (p) denotes the rank of the parabolic fixed point p . 1991 Mathematics Subject Classification : 11K55, 11K60, 11F99, 58D20.