Siegel–Veech transforms are in \begin{document}$ \boldsymbol{L^2} $\end{document}(with an appendix by Jayadev S. Athreya and Rene Rühr)

Abstract

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