Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the weighted power law potential $V=w(x,y)\left\vert x-y\right\vert^{-s}$, where $s>0$ is a fixed parameter and $w$ is an admissible weight. (In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.) Our main results concern the asymptotic behavior as $N\to \infty$ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution $\rho(x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our results provide a method for generating $N$-point configurations on $A$ that are ``well-separated'' and have asymptotic distribution $\rho (x)$ as $N\to \infty$.