Shrinking rates of horizontal gaps for generic translation surfaces

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→ ∞ R → ∞ for the difference in angle between two almost horizontal saddle connections.