Positive-Definite Functions, Exponential Sums and the Greedy Algorithm: a Curious Phenomenon

Abstract

We describe a curious dynamical system that results in sequences of real numbers in [0,1] with seemingly remarkable properties. Let the even function f:T→R satisfy f̂(k)≥c|k|−2 and define a sequence via xn=argminx∑k=1n−1f(x−xk). Such sequences (xn)n=1∞ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval J⊂[0,1] contains ∼|J|n elements). We prove W2μ,ν≤cn,whereμ=1n∑k=1nδxk is the empirical distribution, ν=dx is the Lebesgue measure and W2(μ,ν) is the 2-Wasserstein distance between these two. Much stronger results seem to be true and it is an interesting problem to understand this dynamical system better. We obtain optimal results in dimension d≥3: using G(x,y) to denote the Green’s function of the Laplacian on a compact manifold, we show that xn=argminx∈M∑k=1n−1G(x,xk)satisfiesW21n∑k=1nδxk,dx≲1n1∕d.