Abstract
It is well known due to Jarník that the set Bad R 1 of badly approximable numbers is of Hausdorff dimension 1. If Bad R 1 ( c ) denotes the subset of x ∈ Bad R 1 for which the approximation constant c ( x ) ⩾ c , then Jarník was in fact more precise and gave non-trivial lower and upper bounds on the Hausdorff dimension of Bad R 1 ( c ) in terms of the parameter c > 0 . Our aim is to determine simple conditions on a framework which allow one to extend ’Jarník's inequality’ to further examples. For many dynamical examples, these extensions are related to the Hausdorff dimension of the set of orbits that avoid a suitable given neighborhood of an obstacle. Among the applications, we discuss the set Bad R n r ¯ of badly approximable vectors in R n with weights r ¯ , the set of orbits in the Bernoulli shift that avoid a neighborhood of a periodic orbit, the set of geodesics in the hyperbolic space H n that avoid a suitable collection of convex sets, and the set of orbits of a toral endomorphism that avoid neighborhoods of a separated set.
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