Title: 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 $\mathbb{S}^d$.