  We show that a multiplicative form of Dirichlet's theorem on simultaneous
Diophantine approximation as formulated by Minkowski, cannot be improved for
almost all points on any analytic curve on R^k which is not contained in a
proper affine subspace. Such an investigation was initiated by Davenport and
Schmidt in the late sixties.
  The Diophantine problem is then settled by showing that certain sequence of
expanding translates of curves on the homogeneous space of unimodular lattices
in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on
unipotent flows, linearization techniques, and a new observation about
intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).
