# A Nonlocal Functional Promoting Low-Discrepancy Point Sets

## TL;DR

This paper introduces a nonlocal energy functional designed to arrange points on a torus with minimal discrepancy, and demonstrates its effectiveness through numerical experiments on various low-discrepancy point sets.

## Contribution

The paper proposes a novel nonlocal energy functional that promotes low-discrepancy point distributions and shows its potential to improve regularity in existing low-discrepancy sets.

## Key findings

- Lattices in 2D are critical points of the energy functional.
- Numerical experiments show the energy functional can improve point set regularity.
- Some lattices are strict local minima of the energy.

## Abstract

Let $X = \left\{x_1, \dots, x_N\right\} \subset \mathbb{T}^d \cong [0,1]^d$ be a set of $N$ points in the $d-$dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce a curious energy functional $$ E(X) = \sum_{1 \leq m,n \leq N \atop m \neq n} \prod_{k=1}^{d}{ (1 - \log{\left(2 \sin{ \left( \pi |x_{m,k} - y_{m,k} |\right)} \right)})}$$ and to suggest that moving a set $X$ into the direction $-\nabla E(X)$ may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in $d=2$ are critical points of the energy functional, some (possibly all) are strict local minima.