Counting social interactions for discrete subsets of the plane

Given a discrete subset V in the plane, how many points would you expect there to be in a ball of radius 100? What if the radius is 10,000? Due to the results of Fairchild and forthcoming work with Burrin, when V arises as orbits of non-uniform lattice subgroups of SL(2,R), we can understand asymptotic growth rate with error terms of the number of points in V for a broad family of sets. A crucial aspect of these arguments and similar arguments is understanding how to count pairs of saddle connections with certain properties determining the interactions between them, like having a fixed determinant or having another point in V nearby. We will focus on a concrete case used to state the theorem and highlight the proof strategy. We will also discuss some ongoing work and ideas which advertise the generality and strength of this argument.