  The set B of geodesic rays avoiding a suitable obstacle in a complete
negatively curved Riemannian manifold determines a spectrum S. While various
properties of this spectrum are known, we define and study dimension functions
on S in terms of the Hausdorff-dimension of suitable subsets of the set of
bounded geodesic rays. We establish estimates on the Hausdorff-dimension of
these subsets and thereby obtain non-trivial bounds for the dimension
functions. Moreover we discuss the property of B being an absolute winning set,
therefore satisfying a remarkable rigidity. Finally, we apply the obtained
results to the dimension functions on the spectrum of complex numbers badly
approximable by either an imaginary quadratic number field $Q(i \sqrt{d})$ or
by quadratic irrational numbers over $Q(i \sqrt{d})$.
