Slope gap distribution of the double heptagon and an algorithm for determining winning vectors

TL;DR

This paper introduces a method to find winning holonomy vectors on translation surfaces, enabling the explicit calculation of slope gap distributions, demonstrated on the double heptagon surface, extending previous work on related surfaces.

Contribution

The paper presents a new algorithm for identifying winning vectors, facilitating the computation of slope gap distributions for Veech surfaces, with explicit results for the double heptagon.

Findings

Explicit slope gap distribution for the double heptagon surface.

New algorithm for determining winning holonomy vectors.

Extension of previous work on the double pentagon surface.

Abstract

In this paper, we study the distribution of renormalized gaps between slopes of saddle connections on translation surfaces. Specifically, we describe a procedure for finding the "winning holonomy vectors" as defined by Kumanduri-Sanchez-Wang in arXiv:2102.10069, which constitutes a key step in calculating the slope gap distribution for an arbitrary Veech surface. We then apply this method to explicitly compute the gap distribution for the regular double heptagon translation surface. This extends work of Athreya-Chaika-Lelievre in arXiv:1308.4203 on the gap distribution for the "golden L" translation surface, which is equivalent to the regular double pentagon surface.