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.