Memoirs of the American Mathematical Society 2006; 91 pp; softcover Volume: 179 ISBN10: 082183827X ISBN13: 9780821838273 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/179/846
 Given a compact metric space \((\Omega,d)\) equipped with a nonatomic, 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 `\(\psi\)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 KhintchineGroshev and Jarník 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. Applications also include those to Kleinian groups and rational maps. Compared to previous works our framework allows us to successfully remove many unnecessary conditions and strengthen fundamental results such as Jarník's theorem and the BakerSchmidt theorem. In particular, the strengthening of Jarník's theorem opens up the DuffinSchaeffer conjecture for Hausdorff measures. Table of Contents  Introduction
 Ubiquity and conditions on the general setp
 The statements of the main theorems
 Remarks and corollaries to Theorem 1
 Remarks and corollaries to Theorem 2
 The classical results
 Hausdorff measures and dimension
 Positive and full \(m\)measure sets
 Proof of Theorem 1
 Proof of Theorem 2: \(0\leq G < \infty\)
 Proof of Theorem 2: \(G= \infty\)
 Applications
 Bibliography
