Equidistribution of expanding measures with local maximal dimension and diophantine approximation

We consider improvements of Dirichlet’s Theorem on the space of matrices ${M_{m,n}(\mathbb R)}$ . It is shown that for a certain class of fractals ${K\subset 0,1]^{mn}\subset M_{m,n}(\mathbb R)}$ 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.