Logarithm laws for unipotent flows, I

Abstract

We prove analogues 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 $SL(n,\mathbb{R})/SL(n,\mathbb{Z})$. The key lemma for our results says the measure of the set of unimodular lattices in ${\mathbb{R}}^n$ that does not intersect a `large' volume subset of ${\mathbb{R}}^n$ is `small'. This can be considered as a `random' analogue of the classical Minkowski theorem in the geometry of numbers.

submitted to the Journal of Modern Dynamics; revised version, paper is now split into two pieces, this first half contains results on the space of lattices, the second part will contain results on general homogeneous spaces