Logarithm laws for unipotent flows, I

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 ℝⁿ that does not intersect a 'large' volume subset of ℝⁿ is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.