Pairs in discrete lattice orbits with applications to Veech surfaces

Abstract

Let ${\Lambda }_1$, ${\Lambda }_2$ be two discrete orbits under the linear action of a lattice $\Gamma <{\mathrm{SL}}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$\sum _{\mathbf{x}\in {\Lambda }_1}\sum _{\mathbf{y}\in {\Lambda }_2}f(\mathbf{x},\mathbf{y})$$ 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 translations flows $F_{\theta }^t$ and $F_{\psi }^t$ on any Veech surface $M$ are disjoint.

By Claire Burrin and Samantha Fairchild with an appendix by Jon Chaika. Final version accepted to Journal of the European Mathematical Society