Dimension drop for diagonalizable flows on homogeneous spaces

Abstract

Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let O be an open subset of X, and let F = {gt : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F -orbit misses O; 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, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.