Jarnik-type inequalities for bounded orbits

It is well known due to Jarnik that the set Bad_R of badly approximable numbers is of Hausdorff-dimension one. If Bad_R(c) denotes the subset of x ∈ Bad_R for which the approximation constant c(x) > c, then Jarnik was in fact more precise and established non-trivial lower and upper bounds on the Hausdorff-dimension of Bad_R(c) in terms of c > 0. We extend ’Jarnik's inequality’ to further examples from Diophantine approximation and dynamical systems, where these extensions are related to the Hausdorff-dimension of the set of points whose orbits avoid a suitable given neighborhood of an obstacle. As main examples, we discuss the set Bad_Rn of badly approximable vectors in R^n and the set of geodesic rays in a (finite volume) hyperbolic manifold avoiding a neighborhood of an obstacle such as a cusp, a closed geodesic or a point.

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