Mathematics – Number Theory
Scientific paper
2007-07-09
Mathematics
Number Theory
22 pages
Scientific paper
There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine's theorem and Jarnik's theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that they are both in fact a simple consequence of the notion of `local ubiquity'. The local ubiquity framework introduced here is a much simplified and more transparent version of that in \cite{memoirs}. Furthermore, it leads to a single local ubiquity theorem that unifies the Lebesgue and Hausdorff theories. As an application of our framework we consider the theory of metric Diophantine approximation on limit sets of Kleinian groups. In particular, we obtain a general Hausdorff measure version of Sullivan's logarithm law for geodesics -- an aspect overlooked in \cite{memoirs}.
Beresnevich Victor
Velani Sanju
No associations
LandOfFree
Ubiquity and a general logarithm law for geodesics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Ubiquity and a general logarithm law for geodesics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ubiquity and a general logarithm law for geodesics will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-159053