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

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.