COUNTING PROBLEMS FROM THE VIEWPOINT OF ERGODIC THEORY: FROM PRIMITIVE INTEGER POINTS TO SIMPLE CLOSED CURVES

The uploaded paper (a survey) states Mirzakhani’s theorem s(X,L) ∼ s(X) L^{6g−6} for closed hyperbolic surfaces and develops the standard proof strategy via measured laminations, Thurston measure, ergodicity, and Weil–Petersson integration; it also identifies the orbitwise constants in the form c(γ)·B(X)/b_g and then sums over finitely many topological types. The candidate solution follows the same blueprint, citing the same ingredients and arriving at the same asymptotic with s(X) = (B(X)/b_{g,0}) Σ_γ c(γ). Minor notational differences (b_g vs b_{g,0}) aside, the arguments match; no substantive gaps beyond standard integrability and averaging steps the paper explicitly discusses.