Logarithm laws for unipotent flows, I
Jayadev S. Athreya - 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.
Received: November 2008; Revised: May 2009; Available Online: August 2009. |