Measure theoretic laws for lim sup sets
| dc.creator | Beresnevich, Victor | |
| dc.creator | Dickinson, Detta | |
| dc.creator | Velani, Sanju | |
| dc.date | 2004-01-12 | |
| dc.date | 2004-12-17 | |
| dc.date.accessioned | 2026-07-07T05:04:30Z | |
| dc.date.available | 2026-07-07T05:04:30Z | |
| dc.description | 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. | |
| dc.description | This is an updated version which has been accepted for publication by Memoirs of the AMS. Compared to earlier versions: Sections 1.1 and 1.2 have been moved around and various inaccuracies corrected | |
| dc.identifier | https://arxiv.org/abs/math/0401118 | |
| dc.identifier | http://arxiv.org/abs/math/0401118 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69826 | |
| dc.subject | Number Theory | |
| dc.subject | Dynamical Systems | |
| dc.title | Measure theoretic laws for lim sup sets | |
| dc.type | text |