  We consider improvements of Dirichlet's Theorem on space of matrices
$M_{m,n}(R)$. It is shown that for a certain class of fractals $K\subset
[0,1]^{mn}\subset M_{m,n}(R)$ of local maximal dimension Dirichlet's Theorem
cannot be improved almost everywhere. This is shown using entropy and dynamics
on homogeneous spaces of Lie groups.
