  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
$f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.
