Origami slope gaps and the Hall distribution

TL;DR

This paper analyzes the slope gap distribution of a specific 10-tile origami, revealing it cannot be expressed as a sum of scaled Hall distributions, thus challenging previous assumptions about origami slope gap distributions.

Contribution

It derives the slope gap distribution for a particular origami and shows it differs from known distributions, highlighting new complexities in translation surface coverings.

Findings

The distribution is not a sum of scaled Hall distributions.

Demonstrates that branched coverings can have unique slope gap distributions.

Challenges previous models assuming additive distributions for origamis.

Abstract

In this paper we review the theory of slope gap distributions of translation surfaces and summarize the state-of-the-art for calculating slope gap distributions of Veech surfaces. We then derive the slope gap distribution of a particular 10-tile origami by considering the origami's return times to a Poincar\'e section under the horocycle flow on the moduli space associated with the origami. We show that the resulting distribution is not a sum of scaled Hall distributions, unlike all previously published origami slope gap distributions. More generally, this demonstrates that the slope gap distribution of a branched covering of a translation surface cannot necessarily be represented as a sum of scaled copies of the slope gap distribution of the base surface.