Siegel-Veech Measures of Convex Flat Cone Spheres

Abstract

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of $\mathbb{R}^n$. Motivated by this result, Veech introduced an analogue for translation surfaces, known as the Siegel-Veech formula, which describes the average number of saddle connections of bounded length on the moduli space of translation surfaces. However, no such formula is known for flat surfaces with cone angles that are irrational multiples of $\pi$. A convex flat cone sphere is a Riemann sphere equipped with a conformal flat metric with conical singularities, all of whose cone angles lie in the interval $(0, 2\pi)$. In this paper, we extend the Siegel-Veech formula to this setting. We define a generalized Siegel-Veech transform and prove that it belongs to $L^\infty$ on the moduli space. This leads to the definition of a Siegel-Veech measure on $\mathbb{R}_{>0}$, obtained by integrating the Siegel-Veech transform over the moduli space. This measure can be viewed as a generalization of the classical Siegel-Veech formula. We show that it is absolutely continuous and piecewise real analytic. Finally, we study the asymptotic behavior of this measure on small intervals $(0,\varepsilon)$ as $\varepsilon \to 0$, providing an analogue of Siegel-Veech constants for convex flat cone spheres.