  We study the asymptotic equidistribution of points with discrete energy close
to Robin's constant of a compact set in the plane. Our main tools are the
energy estimates from potential theory. We also consider the quantitative
aspects of this equidistribution. Applications include estimates of growth for
the Fekete and Leja polynomials associated with large classes of compact sets,
convergence rates of the discrete energy approximations to Robin's constant,
and problems on the means of zeros of polynomials with integer coefficients.
