A Shrinking Target Problem with Target at Infinity in Rank One Homogeneous Spaces

In this paper, we give a definition of Diophantine points of type $\gamma$ for $\gamma\geq0$ in a homogeneous space $G/\Gamma$, and compute the Hausdorff dimension of the subset of points which are not Diophantine of type $\gamma$ when $G$ is a semisimple Lie group of real rank one. We also deduce a Jarn\'ik-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.