Equidistribution of expanding measures with local maximal dimension and Diophantine Approximation

摘要

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 Theoremcannot be improved almost everywhere. This is shown using entropy and dynamicson homogeneous spaces of Lie groups.