Asymptotics of discrete Riesz $d$-polarization on subsets of $d$-dimensional manifolds

We prove a conjecture of T. Erd{e}lyi and E.B. Saff, concerning the form of the dominant term (as $Nto infty$) of the $N$-point Riesz $d$-polarization constant for an infinite compact subset $A$ of a $d$-dimensional $C^{1}$-manifold embedded in $mathbb{R}^{m}$ ($dleq m$). Moreover, if we assume further that the $d$-dimensional Hausdorff measure of $A$ is positive, we show that any asymptotically optimal sequence of $N$-point configurations for the $N$-point $d$-polarization problem on $A$ is asymptotically uniformly distributed with respect to $mathcal H_d|_A$.