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

## 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 |
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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 |

Related URLs: | |

URI: | http://eprints.whiterose.ac.uk/id/eprint/7010 |