On the dimension drop conjecture for diagonal flows on the space of lattices

Let X=G/Γ, where G is a Lie group and Γ lattice in G, let U be an open subset of X, {gt} one-parameter subgroup G. Consider the set points X whose gt-orbit misses U; it has measure zero if flow ergodic. It been conjectured that this Hausdorff dimension strictly smaller than X. This conjecture proved when compact or simple real rank 1. In paper we prove for case G=SLm+n(R), Γ=SLm+n(Z) gt=diag(ent,…,ent,e−mt,…,e−mt), fact providing effective estimate codimension. The proof uses exponential mixing together with method integral inequalities height functions on SLm+n(R)/SLm+n(Z). We also discuss application to problem improving Dirichlet's theorem simultaneous Diophantine approximation.