OAI.WeakPinned.main

The theorem states that for every real s>0, the quantity F(n,s) tends to 0 as n→∞ through natural numbers. Here F(n,s), for n≥2, is the supremum, over all n-element finite sets P of points in the Euclidean plane ℝ², of the pair fraction of P at exponent s, and F(n,s)=0 for n<2. For a point set P, the pair fraction is the number of rich ordered pairs divided by n(n−1). An ordered pair (x,y) of distinct points of P is rich if k(P,x,y) ≥ n^s, where k(P,x,y) is the number of points z of P other than x with dist(z,x)=dist(y,x), that is, the number of points of P, excluding x itself, lying on the circle centered at x through y. So the statement is that, for any fixed positive exponent s, the largest possible proportion of ordered pairs of distinct points whose pinned-distance circle contains at least n^s points of the set goes to zero as the set size grows. The source states this with a sorry placeholder, so it is an admitted theorem.