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

Abstract

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 {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-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=SLm+n(R), Γ=SLm+n(Z) and gt=diag(ent,…,ent,e−mt,…,e−mt), 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 SLm+n(R)/SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.