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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

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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