Pairs in discrete lattice orbits with applications to Veech surfaces

TL;DR

This paper develops a new integral formula for counting pairs of vectors in discrete orbits related to Veech surfaces, leading to effective counting results and disjointness properties of translation flows.

Contribution

It introduces a Siegel–Veech-type integral formula for orbit pairs and applies it to derive counting estimates and flow disjointness results on Veech surfaces.

Findings

Effective count of saddle connection vectors in generic sets.

Upper bounds on pairs with bounded determinant or distance.

Disjointness of translation flows for almost every pair of directions.

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.