Nondense orbits on homogeneous spaces and applications to geometry and number theory

Let $G$ be a Lie group, $\mathrm{\Gamma }\subset G$ a discrete subgroup, $X=G\mathrm{/}\mathrm{\Gamma }$, and $f$ an affine map from $X$ to itself. We give conditions on a submanifold $Z$ of $X$ guaranteeing that the set of points $x\in X$ with $f$-trajectories avoiding $Z$ is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on $X$. This has applications to constructing exceptional geodesics on locally symmetric spaces, and to non-density of the set of values of certain functions at integer points.