Siegel-Veech transforms are in $L^2$

TL;DR

This paper proves that the Siegel-Veech transform of certain functions on translation surfaces is square-integrable and explores applications to counting saddle connections, introducing a new invariant for specific measures.

Contribution

It establishes the $L^2$ integrability of Siegel-Veech transforms and introduces a novel invariant for $SL(2, eals)$-invariant measures under certain conditions.

Findings

Siegel-Veech transform is in $L^2$ for bounded compactly supported functions.

Provides bounds on error terms in saddle connection counting problems.

Introduces a new invariant for $SL(2, eals)$-invariant measures.

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.