Dimension Functions on the Spectrum over Bounded Geodesics and Applications to Diophantine Approximation

Abstract

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 $Q(i \sqrt{d})$.