Sequences with equi-distributed extreme points in uniform polynomial approximation

Let E be a compact set in 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=(n_j) be a subsequence of such that n_j+1/n_j→1. If, for nΛ, A_n( 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 of A_n( 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 if the boundary of E is smooth enough.