Riesz Energy on the Torus: Regularity of Minimizers

TL;DR

This paper investigates the regularity of point configurations on the torus that minimize interaction energies similar to Riesz energies, showing they are optimal for certain quadrature problems and extending to various interaction functions.

Contribution

It establishes the regularity properties of energy-minimizing point sets on the torus for a class of functions including Riesz energies, with implications for quadrature and distribution irregularities.

Findings

Minimizers exhibit optimal regularity in Fourier-analytic sense.

Configurations are optimal quadrature points for trigonometric polynomials.

Results extend to less singular functions like Gaussian interactions.

Abstract

We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for $0< s < d$, the minimizing configuration of points has optimal regularity w.r.t. a Fourier-analytic regularity measure that arises in the study of irregularities of distribution. A particular consequence is that they are optimal quadrature points in the space of trigonometric polynomials up to a certain degree. The proof extends to other settings and also covers less singular functions such as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.