Positive-definite Functions, Exponential Sums and the Greedy Algorithm: a curious Phenomenon

TL;DR

This paper investigates a dynamical system generating sequences with remarkable uniform distribution properties, establishing optimal convergence rates in Wasserstein distance for these sequences on manifolds and the circle.

Contribution

It introduces a novel dynamical system based on positive-definite functions and exponential sums, proving optimal distribution convergence rates in Wasserstein distance.

Findings

Sequences are regularly distributed with favorable exponential sum estimates.

Wasserstein distance between empirical measures and uniform distribution decays as 1/√n.

Optimal convergence rates are established in higher dimensions using Green's functions.

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 $$ x_n = \arg\min_{x \in M} \sum_{k=1}^{n-1}{G(x,x_k)} \quad \mbox{satisfies} \quad W_2\left( \frac{1}{n} \sum_{k=1}^{n}{\delta_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}}.$$