Diophantine approximations for translation surfaces and planar resonant sets

TL;DR

This paper investigates the Hausdorff dimension of sets of parameters related to Teichmüller geodesics on translation surfaces, including those with bounded excursions, prescribed escape rates, and fast recurrence, using planar resonant sets.

Contribution

It introduces a new framework for analyzing Hausdorff dimensions of geodesic sets on translation surfaces via planar resonant sets and establishes metric properties for these sets.

Findings

Bounds for Hausdorff dimension of bounded geodesic parameters

Dimension calculations for geodesics with prescribed escape rates

Dimension of directions with fast recurrence in rational billiards

Abstract

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.