Beurling–Landau densities of weighted Fekete sets and correlation kernel estimates

Let Q be a suitable real function on C. An n-Fekete set corresponding to Q is a subset {zn1,…,znn} of C which maximizes the expression . It is well known that, under reasonable conditions on Q, there is a compact set S known as the “droplet” such that the measures μn=n−1(δzn1+⋯+δznn) converges to the equilibrium measure as n→∞. In this note we prove that Fekete sets are, in a sense, maximally spread out with respect to the equilibrium measure. In general, our results apply only to a part of the Fekete set, which is at a certain distance away from the boundary of the droplet. However, for the potential Q=|z|2 we obtain results which hold globally, and we conjecture that such global results are true for a wide range of potentials.