Dimension bounds for escape on average in homogeneous spaces.

Let $ X = G/\Gamma $, where $ G $ is a Lie group and $ \Gamma $ is a uniform lattice in $ G $, and let $ O $ be an open subset of $ X $. We give an upper estimate for the Hausdorff dimension of the set of points whose trajectories escape $ O $ on average with frequency $ \delta $, where $ 0 < \delta \le 1 $. [ABSTRACT FROM AUTHOR]