Slope Gap Distribution of Saddle Connections on the 2n-gon

TL;DR

This paper explicitly computes the limiting slope gap distribution for saddle connections on 2n-gons, revealing non-unimodal distributions and providing bounds on non-differentiability points, advancing understanding of translation surface dynamics.

Contribution

It provides the first explicit calculations of slope gap distributions for 2n-gons and establishes bounds on non-differentiability points, addressing open questions in the field.

Findings

Slope gap distribution is not always unimodal.

Linear bounds on non-differentiability points as n increases.

First example of non-trivial bounds on an infinite family of translation surfaces.

Abstract

We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as n grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.