Siegel-Veech transforms are in $L^2$

Abstract

Let $\mathcal{H}$ denote a connected component of a stratum of translation surfaces. We show that the Siegel-Veech transform of a bounded compactly supported function on ${\mathbb{R}}^2$ is in $L^2(\mathcal{H},\mu )$, where $\mu$ is Lebesgue measure on $\mathcal{H}$, and give applications to bounding error terms for counting problems for saddle connections. We also propose a new invariant associated to $SL(2,\mathbb{R})$-invariant measures on strata satisfying certain integrability conditions.

23 pages, added appendix by Jayadev Athreya and Rene Ruhr

References in corpus (2)

• Volumes of the sets of translation surfaces with small saddle connections in rank one affine submanifolds
• Counting saddle connections in a homology class modulo $q$

Cited by in corpus (1)

• Counting saddle connections in a homology class modulo $q$