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

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.