# Measure Theoretic Laws for lim sup Sets

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*Victor Beresnevich; Detta Dickinson; Sanju Velani*

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 ‘\(\psi\)–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 Khintchine-Groshev 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 Baker-Schmidt theorem. In particular, the strengthening of Jarník's theorem opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

#### Table of Contents

# Table of Contents

## Measure Theoretic Laws for lim sup Sets

- Contents vii8 free
- Section 1. Introduction 112 free
- Section 2. Ubiquity and conditions on the general setup 819
- Section 3. The statements of t h e main theorems 1425
- Section 4. Remarks and corollaries t o Theorem 1 1627
- Section 5. Remarks and corollaries t o Theorem 2 1829
- Section 6. The classical results 2334
- Section 7. Hausdorff measures and dimension 2435
- Section 8. Positive and full m–measure sets 2637
- Section 9. Proof of Theorem 1 3041
- Section 10. Proof of Theorem 2: 0 ≤ G < ∞ 3748
- Section 11. Proof of Theorem 2: G = ∞ 6172
- Section 12. Applications 6576
- Bibliography 89100