  The main goal of this work is to establish quantitative nondivergence
estimates for flows on homogeneous spaces of products of real and $p$-adic Lie
groups. These results have applications both to ergodic theory and to
Diophantine approximation. Namely, earlier results of Dani (finiteness of
locally finite ergodic unipotent-invariant measures on real homogeneous spaces)
and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of
$\Bbb R^n$) are generalized to the $S$-arithmetic setting.
