Diophantine extremality of the Patterson measure

Bernd O Stratmann, M Urbanski

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19 Citations (Scopus)

Abstract

We derive universal Diophantine properties for the Patterson measure mu(Gamma) associated with a convex cocompact Kleinian group Gamma acting on (n + 1)-dimensional hyperbolic space. We show that mu(Gamma) is always an S-friendly measure, for every (Gamma, mu(Gamma))-neglectable set S, and deduce that if Gamma is of non-Fuchsian type then mu(Gamma) is an absolutely friendly measure in the sense of Pollington and Velani. Consequently, by a result of Kleinbock, Lindenstrauss and Weiss, mu(Gamma), is strongly extremal which means that mu(Gamma)-almost every point is not very well multiplicatively approximable. This is remarkable, since by a well-known result in classical metric Diophantine analysis the set of very well multiplicatively approximable points is of n-dimensional Lebesgue measure zero but has Hausdorff dimension equal to n.

Original languageEnglish
Pages (from-to)297-304
Number of pages8
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume140
Issue numberPart 2
DOIs
Publication statusPublished - Mar 2006

Keywords

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