DIMENSION DROP FOR DIAGONALIZABLE FLOWS ON HOMOGENEOUS SPACES

Let X = G /F, where G is a connected Lie group and F is a lattice in G . Let O be an open subset of X , and let F = {g(t ): 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, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X . This conjecture has been 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 Addiagonalizable 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 /F. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.