Origami slope gaps and the Hall distribution

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.