  Let $Y_0$ be a not very well approximable $m\times n$ matrix, and let $M$ be
a connected analytic submanifold in the space of $m\times n$ matrices
containing $Y_0$. Then almost all $Y\in M$ are not very well approximable. This
and other similar statements are cast in terms of properties of certain orbits
on homogeneous spaces and deduced from quantitative nondivergence estimates for
`quasi-polynomial' flows on on the space of lattices.
