  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.

Dynamical Systems