Commentarii Mathematici Helvetici


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Volume 93, Issue 2, 2018, pp. 225–289
DOI: 10.4171/CMH/434

Published online: 2018-05-31

Diophantine approximations for translation surfaces and planar resonant sets

Luca Marchese[1], Rodrigo Treviño[2] and Steffen Weil[3]

(1) Université Paris 13, Villetaneuse, France
(2) New York University, USA
(3) Tel Aviv University, Israel

We consider Teichmüller 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 Jarník 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.

Keywords: Resonant sets, excursions of geodesics, Khinchin–Jarník theorem, Jarník inequality

Marchese Luca, Treviño Rodrigo, Weil Steffen: Diophantine approximations for translation surfaces and planar resonant sets. Comment. Math. Helv. 93 (2018), 225-289. doi: 10.4171/CMH/434