Pairs of saddle connections of typical flat surfaces on fixed affine orbifolds

TL;DR

This paper establishes that for almost every translation surface, the number of saddle connection pairs with length less than L and bounded virtual area grows quadratically, using ergodic theory and Siegel-Veech transforms.

Contribution

It proves quadratic asymptotics for pairs of saddle connections on typical flat surfaces within fixed affine orbifolds, extending previous understanding in the field.

Findings

Quadratic growth rate of saddle connection pairs with bounded virtual area

Siegel-Veech transforms of bounded functions are in L^{2+κ} for invariant measures

Results hold for almost every translation surface under ergodic SL(2,R) measures

Abstract

We prove that the asymptotic number of pairs of saddle connections with length smaller than $L$ with bounded virtual area is quadratic for almost every translation surface with respect to any ergodic $SL(2,\mathbb{R})$-invariant measure. A key tool of the proof is that Siegel-Veech transforms of bounded functions with compact supports are in $L^{2+\kappa}$ for every $SL(2,\mathbb{R})$-invariant measure.