Dickinson, D., Velani, S. and Beresnevich, V. (2006) Measure theoretic laws for lim sup sets. Memoirs of the American Mathematical Society, 179 ( 846). pp. 198. ISSN 00659266
Abstract
Given a compact metric space (X,d) equipped with a nonatomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable' numbers in the theory of metric Diophantine approximation fall within this class. We show that m(La(p))>0 under a `global ubiquity' hypothesis and the divergence of a certain mvolume sum. In fact, under a `local ubiquity' hypothesis we show that La(p) has full measure; i.e. m(La(p)) =1 . This is the analogue of the divergent part of the classical KhintchineGroshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain fvolume sum diverges then we are able to show that the Hausdorff fmeasure of La(p) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of La(p) and various results concerning the dimension and measure of related `exact order' sets. Essentially, the notion of `local ubiquity' unexpectedly unifies `divergent' type results for La(p) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical limsup sets of `well approximable' numbers, the framework strengthens the classical Hausdorff measure result of Jarnik and opens up the DuffinSchaeffer conjecture for Hausdorff measures.
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Institution:  The University of York 
Academic Units:  The University of York > Mathematics (York) 
Depositing User:  York RAE Import 
Date Deposited:  07 Apr 2009 16:09 
Last Modified:  07 Apr 2009 16:09 
Status:  Published 
Publisher:  American Mathematical Society 
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