Measure theoretic laws for lim–sup sets

Given a compact metric space (Ω, d) equipped with a non-atomic, probability measure m and a real, positive decreasing function ψ we consider a ‘natural’ class of lim sup subsets Λ(ψ) of Ω. The classical lim sup sets of ‘well approximable’ numbers in the theory of metric Diophantine approximation fall within this class. We show that m(Λ(ψ)) > 0 under a ‘global ubiquity’ hypothesis and the divergence of a certain m–volume sum. In fact, under a ‘local ubiquity’ hypothesis we show that Λ(ψ) has full measure; i.e. m(Λ(ψ)) = 1. This is the analogue of the divergent part of the classical Khintchine-Groshev theorem in number theory. Moreover, if the ’local ubiquity’ hypothesis is satisfied and a certain f -volume sum diverges then we are able to show that the Hausdorff f–measure of Λ(ψ) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of Λ(ψ) and various results concerning the dimension and measure of related ‘exact order’ sets. Essentially, the notion of ‘local ubiquity’ unexpectedly unifies ‘divergent’ type results for Λ(ψ) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical lim sup sets of ‘well approximable’ numbers, the framework strengthens the classical Hausdorff measure result of Jarńık and opens up the Duffin-Schaeffer conjecture for Hausdorff measures. This work has been partially supported by INTAS Project 00-429 Royal Society University Research Fellow