Logarithm laws for unipotent flows, I

TL;DR

This paper establishes logarithm laws for unipotent flows on the space of lattices, extending classical results and providing measure estimates related to lattice intersections with large volume sets.

Contribution

It introduces new logarithm laws for unipotent flows and a measure estimate analogous to Minkowski's theorem in the geometry of numbers.

Findings

Logarithm laws for unipotent flows are proven.

Measure of lattices avoiding large volume sets is small.

Results apply to one-parameter actions on lattice spaces.

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 $SL(n, \R)/SL(n, \Z)$. 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' analogue of the classical Minkowski theorem in the geometry of numbers.