Preprint 35/2022

Pairs in discrete lattice orbits with applications to Veech surfaces

Claire Burrin, Samantha Fairchild, and Jon Chaika

Contact the author: Please use for correspondence this email.
Submission date: 29. Nov. 2022
Link to arXiv: See the arXiv entry of this preprint.

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

∑   ∑         f(x,y) x∈Λ1y∈Λ2

from which we derive new results for the set SM 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 SM with bounded determinant and on the number of pairs in SM 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 Fψt on any Veech surface M are disjoint.

01.06.2023, 02:22