标题： 折纸斜率间隙和Hall分布 显示英文标题

标题： Origami slope gaps and the Hall distribution

摘要： 在本文中，我们回顾了平移曲面的斜率间隙分布理论，并总结了计算Veech曲面斜率间隙分布的最新进展。 然后，我们通过考虑该10块折纸在与该折纸相关的模空间上的水平流的Poincaré截面的返回时间，推导出一个特定10块折纸的斜率间隙分布。 我们证明，所得的分布不是缩放后的Hall分布的和，这与所有之前发表的折纸斜率间隙分布不同。 更一般地，这表明分支覆盖的平移曲面的斜率间隙分布不一定能表示为基底曲面的斜率间隙分布的缩放副本的和。 显示英文摘要

摘要： 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.