The distribution of gaps for saddle connection directions

TL;DR

This paper investigates the distribution of gaps in saddle connection directions on translation surfaces, revealing decay rates of the smallest gaps and characterizing lattice surfaces as exceptions.

Contribution

It provides new fine results on the distribution of saddle connection direction gaps and characterizes when the decay rate is not faster than quadratic.

Findings

Smallest gap decay faster than quadratic for almost all surfaces

Decay rate is not faster than quadratic only for lattice surfaces

Results apply to holomorphic differentials on genus g ≥ 2 surfaces

Abstract

Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential $\omega$ on a Riemann surface of genus $g \geq 2$ the smallest gap between saddle connection directions of length at most a fixed length decays faster than quadratically in the length. We also characterize the exceptional set: the decay rate is not faster than quadratic if and only if $\omega$ is a lattice surface.