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Journal of Modern Dynamics (JMD)
 

Logarithm laws for unipotent flows, I

Pages: 359 - 378, Issue 3, July 2009      doi:10.3934/jmd.2009.3.359

 
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Jayadev S. Athreya - Department of Mathematics, Yale University, New Haven, CT 06520-8283, United States (email)
Gregory A. Margulis - Department of Mathematics, Yale University, New Haven, CT 06520-8283, United States (email)

Abstract: 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, unipotent flows, diophantine approximation, geometry of numbers.
Mathematics Subject Classification:  Primary: 327A17; Secondary: 11H16.

Received: November 2008;      Revised: May 2009;      Available Online: August 2009.