A Density Theorem for Weighted Fekete Sets

We prove a result concerning the spreading of weighted Fekete points in the plane. Consider a system {zeta(j)}(n)(1) of identical point-charges in the complex plane C, subject to an external field nQ, where Q is a suitable real-valued function, which we call the "external potential." The energy of the system is taken to be H-n(zeta(1), ... , zeta(n)) = Sigma(j not equal k) log 1/vertical bar zeta(j) - zeta(k)vertical bar + n Sigma(n)(j=1) Q(zeta(j)). (0.1) A configurationF(n) = {zeta(j)}(n)(1) that minimizes H-n is called an n-Fekete set in external potential Q; the points of a Fekete set are called Fekete points. The analogue of H-n for continuous charge distributions (measures) mu is the weighted logarithmic energy in external potential Q, defined by I-Q[mu] = integral integral(C2) log 1/vertical bar zeta - eta vertical bar d mu(zeta)d mu(eta) + integral(C) Qd mu. (0.2) The equilibrium measure sigma minimizes I-Q among all Borel probability measures. This measure is absolutely continuous and has the form d sigma(zeta) = chi(S)(zeta)Delta Q(zeta) dA(zeta), (0.3) where the set S := supp sigma is called the droplet in external field Q. (See below for further notation.) A well-known "discretized" result asserts that that the normalized counting measures mu(n) = 1/n Sigma(n)(1) delta(zeta j) at Fekete points converge to the equilibrium measure sigma, as n -> infinity. (See e. g., [30], Section III. 1.) In this note, we study finer structure of the distribution of Fekete points. We shall prove that Fekete points are maximally spread out in S relative to the conformal metric ds(2) = Delta Q(zeta)vertical bar d zeta vertical bar(2) in the sense of Beurling-Landau densities. Our results generalize those of [2].