  We consider Teichm\"uller geodesics in strata of translation surfaces. We
prove lower and upper bounds for the Hausdorff dimension of the set of
parameters generating a geodesic bounded in some compact part of the stratum.
Then we compute the dimension of those parameters generating geodesics that
make excursions to infinity at a prescribed rate. Finally we compute the
dimension of the set of directions in a rational billiard having fast
recurrence, which corresponds to a dynamical version of a classical result of
Jarn\'ik and Besicovich. Our main tool are planar resonant sets arising from a
given translation surface, that is the countable set of directions of its
saddle connections or of its closed geodesics, filtered according to length. In
an abstract setting, and assuming specific metric properties on a general
planar resonant set, we prove a dichotomy for the Hausdorff measure of the set
of directions which are well approximable by directions in the resonant set,
and we give an estimate on the dimension of the set of badly approximable
directions. Then we prove that the resonant sets arising from a translation
surface satisfy the required metric properties.
