The distribution of gaps for saddle connections on the octagon

Abstract

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincar\'e section for the horocycle flow on $SL(2,\mathbb{R})/SL(X,\omega)$ associated to an arbitrary Veech surface $(X, \omega)$. As a corollary, we show that the associated gap distribution is piecewise real analytic.