Dimension bounds for escape on average in homogeneous spaces

Let $X=G\mathrm{/}\mathrm{\Gamma }$, where $G$ is a Lie group and $\mathrm{\Gamma }$ 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 $0<\delta \le 1$.