  In this paper we discuss equidistribution results for weighted Fekete sets in
subsets of the plane. More precisely, we show that Fekete sets are maximally
spread out relative to a rescaled version of the Beurling--Landau density, in
the "droplet" corresponding to the given weight. Our method combines Landau's
idea to relate the density of a family of discrete sets to properties of the
spectrum of the concentration operator, with estimates for the correlation
kernel of the corresponding random normal matrix ensemble.
