Title

Login

Join

Book's by Sanju Velani

Covers

List

Measure Theoretic Laws for lim sup Sets

By: Victor Beresnevich,Detta Dickinson,Sanju Velani

Write a review Read reviews Take a Quiz Solve Book Puzzle

Measure Theoretic Laws for lim sup Sets

By: 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 `$\p$-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 Jarník concerning $W(\psi)$ fall into our general framework. The main results provide a unifying treatment of numerous problems in metric Diophantineapproximation 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 conjecturefor Hausdorff measures.

Average ratings

N/A

5

0%

4

0%

3

0%

2

0%

1

0%

Solve jigsaw puzzle

Age

Page Count

110

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082183827X

ISBN 13

9780821838273

Measure Theoretic Laws for lim sup Sets

By: Victor Beresnevich Detta Dickinson Sanju Velani

Write a review Read reviews Take a Quiz Solve Book Puzzle

Measure Theoretic Laws for lim sup Sets

By: 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 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. 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 Jarnik's theorem and the Baker-Schmidt theorem. In particular, the strengthening of Jarnik's theorem opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

Average ratings

N/A

5

0%

4

0%

3

0%

2

0%

1

0%

Solve jigsaw puzzle

Age

Page Count

116

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821865684

ISBN 13

9780821865682