On ε-escaping trajectories in homogeneous spaces

Let G/Γ be a finite volume homogeneous space of a semisimple Lie group G, and {exp(tD)} be a one-parameter Ad-diagonalizable subgroup inside a simple Lie subgroup G₀ of G. Denote by Z_ε,D the set of points x ∈ G/Γ whose {exp(tD)}-trajectory has an escape for at least an ε-portion of mass along some subsequence. We prove that the Hausdorff codimension of Z_ε,D is at least cε, where c depends only on G, G₀ and Γ.