  In this paper we study the dimension of a family of sets arising in open
dynamics. We use exponential mixing results for diagonalizable flows in compact
homogeneous spaces $X$ to show that the Hausdorff dimension of set of points
that lie on trajectories missing a particular open ball of radius $r$ is at
most $$\dim X + C\frac{r^{\dim X}}{\log r},$$ where $C>0$ is a constant
independent of $r>0$. Meanwhile, we also describe a general method for
computing the least cardinality of open covers of dynamical sets using volume
estimates.
