Covering and separation of Chebyshev points for non-integrable Riesz potentials

Abstract

For Riesz $s$-potentials $K(x,y)=|x-y|^{-s}$, $s>0$, we investigate separation and covering properties of $N$-point configurations $\omega^*_N=\{x_1, \ldots, x_N\}$ on a $d$-dimensional compact set $A\subset \mathbb{R}^\ell$ for which the minimum of $\sum_{j=1}^N K(x, x_j)$ is maximal. Such configurations are called $N$-point optimal Riesz $s$-polarization (or Chebyshev) configurations. For a large class of $d$-dimensional sets $A$ we show that for $s>d$ the configurations $\omega^*_N$ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as $N\to \infty$ of the best covering constant. For these purposes we compare best-covering configurations with optimal Riesz $s$-polarization configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty$) of the maximal $s$-polarization constants. In addition, we introduce the notion of "weak separation" for point configurations and prove this property for optimal Riesz $s$-polarization configurations on $A$ for $s>\text{dim}(A)$, and for $d-1\leqslant s < d$ on the sphere $\mathbb{S}^d$.