Asymptotic counting of geometric objects has a long history. In the case of flat surfaces, various works by Masur and Veech showed the quadratic asymptotic growth of the number of saddle connections of bounded length. In this spirit, Athreya, Fairchild, and Masur showed that, for almost any flat surface, the number of pairs of saddle connection of bounded length and bounded virtual area increases quadratically with the constraint on length. In this case the « almost all » is with respect to the so-called Masur-Veech measure.

To demonstrate this, they use tools of ergodic theory (hence the result is true almost everywhere). This result can be extended in several ways, giving an error term or extending it to almost any SL(2,R)-invariant measure. We will present several useful tools for tackling these questions.