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

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 \sum_{{\bf x}\in\Lambda_1}\sum_{{\bf y}\in\Lambda_2}f({\bf x},{\bf 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 translation flows F_{\theta}^{t} and F_{\psi}^{t} on any Veech surface M are disjoint.