Diophantine extremality of the Patterson measure

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 ${\cal S}$-friendly measure, for every $(\Gamma,\mu_{_{\Gamma }})$-neglectable set ${\cal S}$, 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 $n$.