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Measure theoretic laws for lim sup sets

Dickinson, D., Velani, S. and Beresnevich, V. (2006) Measure theoretic laws for lim sup sets. Memoirs of the American Mathematical Society, 179 ( 846). pp. 1-98. ISSN 0065-9266

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Abstract

Given a compact metric space (X,d) equipped with a non-atomic, 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 m--volume 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 Khintchine-Groshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain f-volume sum diverges then we are able to show that the Hausdorff f--measure 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 Duffin-Schaeffer conjecture for Hausdorff measures.

Item Type: Article
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
Related URLs:
URI: http://eprints.whiterose.ac.uk/id/eprint/7010

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