  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.
