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 function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\hat{f}(k) \geq c|k|^{-2}$ and define a sequence via $$ x_n = \arg\min_x \sum_{k=1}^{n-1}{f(x-x_k)}.$$ Such sequences $(x_n)_{n=1}^{\infty}$ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval $J \subset [0,1]$ contains $\sim |J|n$ elements). We prove $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \leq \frac{c}{\sqrt{n}},$$ where $W_2$ is the 2-Wasserstein distance. Much stronger results seem to be true and it seems like an interesting problem to understand this dynamical system better. We obtain optimal results in dimension $d \geq 3$: using $G(x,y)$ to denote the Green's function of the Laplacian on a compact manifold, we show that $$ x_n = \arg\min_{x \in M} \sum_{k=1}^{n-1}{G(x,x_k)} \quad \mbox{satisfies} \quad W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}}.$$