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

Diophantine approximations for translation surfaces and planar resonant sets