Pairs in discrete lattice orbits with applications to Veech surfaces (with an appendix by Jon Chaika)

Abstract

Let ${\Lambda }_1$, ${\Lambda }_2$ be two discrete orbits under the linear action of a lattice $\Gamma <{\mathrm{SL}}_2(\mathbf{R})$ on the Euclidean plane. We prove a Siegel–Veech-type integral formula for the averages

from which we derive new results for the set $S_M$ of holonomy vectors of saddle connections of a Veech surface $M$. This includes an effective count for generic Borel sets with respect to linear transformations, and upper bounds on the number of pairs in $S_M$ with bounded determinant and on the number of pairs in $S_M$ with bounded distance. This last estimate is used in the appendix to prove that for almost every $(\theta ,\psi )\in S^1\times S^1$ the translation flows $F_{\theta }^t$ and $F_{\psi }^t$ on any Veech surface $M$ are disjoint.