Pairs in discrete lattice orbits with applications to Veech surfaces

Claire Burrin, Samantha Fairchild, and Jon Chaika

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Submission date: 29. Nov. 2022
Bibtex
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
Let Λ₁, Λ₂ be two discrete orbits under the linear action of a lattice Γ < SL₂(ℝ) on the Euclidean plane. We prove a Siegel–Veech-type integral formula for the averages

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 (?,ψ) ∈ S¹ × S¹ the translations flows F_?^t and F_ψ^t on any Veech surface M are disjoint.