Sequences with equi-distributed extreme points in uniform polynomial approximation

Let E be a compact set in C with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E, we consider the distribution of extreme points of the error of best uniform polynomial approximation on E. Let Λ = (nj) be a subsequence of N such that nj+1/nj → 1. If, for n ∈ Λ, An(f) ⊆ ∂E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ′ of Λ such that the distribution of any (n+2)-th Fekete point set Fn+2 of An(f) tends weakly to the equilibrium distribution on E as n → ∞ in Λ′. Furthermore, we prove a discrepancy result for the distribution of the point sets Fn+2 if the boundary of E is smooth enough.