Measure Theoretic Laws for Lim Sup Sets

Given a compact metric space $(Omega, d)$ equipped with a non-atomic, probability measure $m$ and a positive decreasing function $psi$, we consider a natural class of lim sup subsets $Lambda(psi)$ of $Omega$. The classical lim sup set $W(psi)$ of '$p$-approximable' numbers in the theory of metric Diophantine approximation fall within this class. We establish sufficient conditions (which are also necessary under some natural assumptions) for the $m$-measure of $Lambda(psi)$ to be either positive or full in $Omega$ and for the Hausdorff $f$-measure to be infinite. The classical theorems of Khintchine-Groshev and Jarnik concerning $W(psi)$ fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantine approximation including those for real, complex and $p$-adic fields associated with both independent and dependent quantities