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arXiv · 2508.16186

Origami slope gaps and the Hall distribution

Taylor McAdam·Xiaoxing Yu
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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.

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Taylor McAdam, Xiaoxing Yu. 2025-08-22. Origami slope gaps and the Hall distribution. https://arxiv.org/abs/2508.16186

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