  Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in
$G$, and let $U$ be a subset of $X$ whose complement is compact. We use the
exponential mixing results for diagonalizable flows on $X$ to give upper
estimates for the Hausdorff dimension of the set of points whose trajectories
miss $U$. This extends a recent result of Kadyrov and produces new applications
to Diophantine approximation, such as an upper bound for the Hausdorff
dimension of the set of weighted uniformly badly approximable systems of linear
forms, generalizing an estimate due to Broderick and Kleinbock.
