Measure Theoretic Laws for Lim Sup Sets

Measure Theoretic Laws for Lim Sup Sets

by Victor Beresnevich, Detta Dickinson, Sanju Velani
     
 

ISBN-10: 082183827X

ISBN-13: 9780821838273

Pub. Date: 12/01/2005

Publisher: American Mathematical Society

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

…  See more details below

Overview

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.

Read More

Product Details

ISBN-13:
9780821838273
Publisher:
American Mathematical Society
Publication date:
12/01/2005
Series:
Memoirs of the American Mathematical Society Series, #179
Pages:
91
Product dimensions:
(w) x (h) x 0.28(d)

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >