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.
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