Riesz Energy on the Torus: Regularity of Minimizers

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 $f(x)\sim \Vert x{\Vert }^{-s}$ 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 $f(x)=\exp (-N^{\frac{2}{d}}\Vert x{\Vert }^2)$.

We found a gap in the proof of v1. v2 gives the best we can obtain with the argument

References in corpus (2)

• Distributing many points on spheres: minimal energy and designs
• Geodesic distance Riesz energy on the sphere