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MiS Preprint
35/2022

Pairs in discrete lattice orbits with applications to Veech surfaces

Claire Burrin, Samantha Fairchild and Jon Chaika

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.

Received:
Nov 29, 2022
Published:
Nov 29, 2022

Related publications

Preprint
2022 Repository Open Access
Claire Burrin, Samantha Fairchild and Jon Chaika

Pairs in discrete lattice orbits with applications to Veech surfaces

(Journal of the European Mathematical Society)