Hausdorff dimension of divergent trajectories on homogeneous spaces

For a one-parameter subgroup action on a finite-volume homogeneous space, we consider the set of points admitting divergent-on-average trajectories. We show that the Hausdorff dimension of this set is strictly less than the manifold dimension of the homogeneous space. As a corollary we know that the Hausdorff dimension of the set of points admitting divergent trajectories is not full, which proves a conjecture of Cheung [Hausdorff dimension of the set of singular pairs, Ann. of Math. (2) 173 (2011), 127–167].