Siegel-Veech transforms are in \(L^2\) (Q2000067)

A translation surface is a pair $(X,\omega)$ where $X$ is a compact Riemann surface and $\omega$ is a holomorphic $1$-form that defines a singular flat metric on the underlying surface with conical singularities at the zeroes of $\omega$. A geodesic arc between two zeroes of $\omega$ is called a \textit{saddle connection}. Associated to each saddle connection $\gamma$ is its holonomy vector $z_{\omega}=\int_{\gamma} \omega \in \mathbb{C}$. The set of holonomy vectors $\Lambda_{\omega}$ is a discrete subset of the plane. For any compactly supported bounded function on the plane $f$, the seminal work of \textit{W. A. Veech} [Ann. Math. (2) 148, No. 3, 895--944 (1998; Zbl 0922.22003)] introduced the \textit{Siegel-Veech transform} $\hat{f}$ that associates to any translation surface $(X,\omega)$ the sum $\sum_{\nu \in \Lambda_{\omega}} f(\nu)$. In particular, it proved that for the Lebesgue measure $\mu$ of any connected component $\mathcal{H}$ of a stratum of translation surfaces, $\hat{f} \in L^{1}(\mathcal{H},\mu)$. It also proved that for any $\operatorname{SL}(2,\mathbb{R})$-invariant ergodic finite measure $\lambda$ such that $\hat{f} \in L^{1}(\mathcal{H},\lambda)$, an integral formula holds: $\int_{\mathcal{H}} \hat{f}d \lambda=c \int_{\mathbb{R}^{2}} f dm$ where $c$ is a constant that depends only on $\lambda$. These Siegel-Veech constants $c$ are important numerical invariants of the $\operatorname{SL}(2,\mathbb{R})$-invariant measures. This paper proves that $\hat{f} \in L^{2}(\mathcal{H},\mu)$. The result has some applications concerning the problem of counting saddle connections of length smaller than $R$. It is known (see for example [\textit{H. Masur}, Ann. Math. (2) 115, 169--200 (1982; Zbl 0497.28012)]) that these numbers have quadratic growth for generic translation surfaces, the constant being a Siegel-Veech constant. One of the theorems of this paper implies bounds on error terms for this counting. By extending the Siegel-Veech transform to functions on $\mathbb{R}^{2} \times \mathbb{R}^{2}$, it defines new invariants associated to some $\operatorname{SL}(2,\mathbb{R})$-invariant measures.