Riesz Energy on the Torus: Regularity of Minimizers

We study sets of $N$ points on the $d-$dimensional torus ${\mathbb{T}}^d$ minimizing interaction functionals of the type $$\sum _{\genfrac{}{}{0px}{}{i,j=1}{i\mathrm{\ne }j}}^{N}f(x_i-x_j)\mathrm{.}$$ The main result states that for a class of functions $f$ that behave like Riesz energies $f(x)\sim \mathrm{\parallel }x{\mathrm{\parallel }}^{-s}$ 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 $f(x)=\exp (-N^{\frac{2}{d}}\mathrm{\parallel }x{\mathrm{\parallel }}^2)$.