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 is a lattice in G , let U be an open subset of X , and let { g t } be a one-parameter subgroup of G. Consider the set of points in X whose g t -orbit misses U ; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G = SL m + n (R) , Γ = SL m + n (Z) and g t = diag (e n t , ... , e n t , e − m t , ... , e − m t) , in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SL m + n (R) / SL m + n (Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation. [ABSTRACT FROM AUTHOR]