Shrinking rates of horizontal gaps for generic translation surfaces

Jon Chaika and Samantha Fairchild

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Submission date: 14. Jul. 2022
Pages: 27
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Link to arXiv: See the arXiv entry of this preprint.

Abstract:
A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R →∞ for the difference in angle between two almost horizontal saddle connections.