  It is well known due to Jarnik that the set Bad of badly approximable numbers
is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for
which the approximation constant c > c(x), then Jarnik was in fact more precise
and gave nontrivial lower and upper bounds of the Hausdorff-dimension of Bad(c)
in terms of the parameter c > 0. Our aim is to determine simple conditions on a
framework which allow to extend 'Jarnik's inequality' to further examples;
among the applications, we discuss the set of badly approximable vectors in
with weights and the set of geodesics in the hyperbolic space which avoid a
suitable collection of convex sets.
