Pairs in discrete lattice orbits with applications to Veech surfaces

Let ${\mathrm{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$ be two discrete orbits under the linear action of a lattice $\mathrm{\Gamma }<{\mathrm{S}\mathrm{L}}_2(\mathbb{R})$ on the Euclidean plane. We prove a Siegel$-$Veech-type integral formula for the averages $$\sum _{\mathbf{x}\in {\mathrm{\Lambda }}_1}\sum _{\mathbf{y}\in {\mathrm{\Lambda }}_2}f(\mathbf{x},\mathbf{y})$$ 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.