Counting Problems for Invariant Point Processes

Using linear algebra and the ergodic theory of \(\mathrm {SL}(2,\mathbb {R})\) actions, we survey how to solve several natural asymptotic counting problems for discrete subsets of the plane using an axiomatic perspective. Applications include counting holonomies of saddle connections, lattice points, and fine scale distribution in various contexts. This is a perspective inspired by work of Veech (Ann. Math. (2) 148(3):895–944, 1998), and developed further by, among others, Eskin–Masur (Ergodic Theory Dyn. Syst. 21(2):443–478, 2001), Athreya–Ghosh (Enseign. Math. 64(1–2), 1–21, 2018), and Marklof (Lond. Math. Soc. Newsl. 493:42–49, 2021).