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Measure Theoretic Laws for lim sup Sets
Victor Beresnevich, Academy of Sciences of Belarus, Minsk, Belarus, Detta Dickinson, National University of Ireland, Kildare, Republic of Ireland, and Sanju Velani, University of York, England

Memoirs of the American Mathematical Society
2006; 91 pp; softcover
Volume: 179
ISBN-10: 0-8218-3827-X
ISBN-13: 978-0-8218-3827-3
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/179/846
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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

  • 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
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