Slope gap distribution of saddle connections on the <inline-formula><tex-math id="M1">$ 2n $</tex-math></inline-formula>-gon

Abstract

<p style='text-indent:20px;'>We explicitly compute the limiting slope gap distribution for saddle connections on any <inline-formula><tex-math id="M2">\begin{document}$ 2n $\end{document}</tex-math></inline-formula>-gon for <inline-formula><tex-math id="M3">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Sanchez-Wang.</p>