Diophantine approximations, large intersections and geodesics in negative curvature

Abstract

Let Γ be a group acting freely and properly on a CAT(-1) space X and ergodically on its visual boundary. We study the connection between metric aspects of the Γ action on the visual boundary of X and the asymptotic behaviour of geodesics on X/Γ. Our results include a logarithm law for approximation by geodesics in negatively curved manifolds, significantly extending existing results on the 'shrinking target problem'. Several of our results in this direction are new also in the case of manifolds with constant negative sectional curvature. Further, we obtain Hausdorff dimension estimates for the finer spiraling phenomena of geodesics, extending work of several authors including Hersonsky and Paulin [30]. Our proof of the logarithm laws involves a new adaptation of the 'well distributed systems' of Melián and Pestana [46], itself an adaptation of the regular systems of Baker and Schmidt [5]. We believe our 'adjusted well distributed systems' to be of independent interest. Another major theme of this paper is the investigation of the large intersection property of Falconer in the context of negative curvature. In particular, we prove that the Γ action on the visual boundary of a proper, geodesic, hyperbolic space has the large intersection property, provided Γ acts geometrically on the space. Our logarithm law results as well as our large intersection results have applications to Diophantine approximation and to hyperbolic geometry.