## Ubiquity and a general logarithm law for geodesics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Title of host publication | Dynamical Systems and Diophantine Approximation. Seminaires et Congres |
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Date | Published - 2009 |
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Pages | 21-36 |
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Number of pages | 16 |
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Volume | 19 |
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Original language | English |
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Name | Seminaires et Congres |
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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 tememoirs. 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 tememoirs.

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