eBook ISBN:  9781470404475 
Product Code:  MEMO/179/846.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470404475 
Product Code:  MEMO/179/846.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 179; 2006; 91 ppMSC: Primary 11; 28
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

Chapters

1. Introduction

2. Ubiquity and conditions on the general setup

3. The statements of the main theorems

4. Remarks and corollaries to Theorem 1

5. Remarks and corollaries to Theorem 2

6. The classical results

7. Hausdorff measures and dimension

8. Positive and full $m$measure sets

9. Proof of Theorem 1

10. Proof of Theorem 2: $0 \leq G < \infty $

11. Proof of Theorem 2: $G = \infty $

12. Applications


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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.

Chapters

1. Introduction

2. Ubiquity and conditions on the general setup

3. The statements of the main theorems

4. Remarks and corollaries to Theorem 1

5. Remarks and corollaries to Theorem 2

6. The classical results

7. Hausdorff measures and dimension

8. Positive and full $m$measure sets

9. Proof of Theorem 1

10. Proof of Theorem 2: $0 \leq G < \infty $

11. Proof of Theorem 2: $G = \infty $

12. Applications