Patterson measure and ubiquity

Abstract

Let $L$ be a closed subset of $\bbfR\sp k$, with Hausdorff dimension $\delta$, which supports a probability measure $m$ for which the $m$- measure of a ball of radius $r$ and centred at a point in $L$ is comparable to $r\sp \delta$. By extending the notion of ubiquity from $k$-dimensional Lebesgue measure to $m$, a natural lower bound for the Hausdorff dimension of a fairly general class of $\limsup$ subsets of $L$ is obtained. This is applied to Patterson measure supported on the limit set of a convex co-compact group to obtain the Hausdorff dimension of the set of 'well-approximable' points associated with the limit set of a convex co-compact group. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of the Jarník's theorem on simultaneous diophantine approximation.