## 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 language | English |
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Pages (from-to) | 297-304 |

Number of pages | 8 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 140 |

Issue number | Part 2 |

DOIs | |

Publication status | Published - Mar 2006 |

## Keywords

- APPROXIMATION