Logarithm laws for unipotent flows, I

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July 2009, 3(3): 359-378. doi: 10.3934/jmd.2009.3.359

Logarithm laws for unipotent flows, I

Jayadev S. Athreya ^1, and Gregory A. Margulis ^1,

1.	Department of Mathematics, Yale University, New Haven, CT 06520-8283, United States, United States

Received November 2008 Revised May 2009 Published August 2009

Abstract
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We prove analogs of the logarithm laws of Sullivan and Kleinbock--Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n, $\R$)/SL(n, $\Z$). The key lemma for our results says the measure of the set of unimodular lattices in $\R^n$ that does not intersect a 'large' volume subset of $\R^n$ is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.

Keywords: Logarithm laws, geometry of numbers., unipotent flows, diophantine approximation.

Mathematics Subject Classification: Primary: 327A17; Secondary: 11H1.

Citation: Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, I. Journal of Modern Dynamics, 2009, 3 (3) : 359-378. doi: 10.3934/jmd.2009.3.359

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