Localized quantitative criteria for equidistribution

Let (xn)n=1 be a sequence on the torus T (normalized to length 1). We show that if there exists a sequence of positive real numbers (tn)n=1 converging to 0 such that lim N→∞ 1 N2 N ∑ m,n=1 1 √ tN exp ( − 1 tN (xm − xn) ) = √ π, then (xn)n=1 is uniformly distributed. This is especially interesting when tN ∼ N−2 since the size of the sum is then essentially determined exclusively by local gaps at scale ∼ N−1. This can be used to show equidistribution of sequences with Poissonian pair correlation, which recovers a recent result of Aistleitner, Lachmann & Pausinger and Grepstad & Larcher. The general form of the result is proven on arbitrary compact manifolds (M, g) where the role of the exponential function is played by the heat kernel et∆: for all x1, . . . , xN ∈M and all t > 0 1 N2 N ∑ m,n=1 [eδxm ](xn) ≥ 1 vol(M) and equality is attained as N →∞ if and only if (xn)n=1 equidistributes.