PATTERSON MEASURE AND UBIQUITY

Abstract

Let L be a closed subset of R , with Hausdorff dimension δ , 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 . 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 lim sup 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. The equivalent geometric result in terms of geodesic excursions on the quotient manifold is also obtained. These results are counterparts of Jarnik’s theorem on simultaneous diophantine approximation.