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Counting pairs of saddle connections
Jayadev Athreya, Samantha Fairchild and Howard Masur
We show that for almost every translation surface the number of pairs of saddle connections with bounded virtual area has asymptotic growth like $cR^2$ where the constant $c$ depends only on the area and the connected component of the stratum. The proof techniques combine classical results for counting saddle connections with the crucial result that the Siegel-Veech transform is in $L^2$. In order to capture information about pairs of saddle connections, we consider pairs with bounded virtual area since the set of such pairs can be approximated by a fibered set which is equivariant under geodesic flow. In the case of lattice surfaces, small virtual area is equivalent to counting parallel pairs of saddle connections, which also have a quadratic growth of $cR^2$ where $c$ depends in this case on the given lattice surface.