“…The key remaining part of the proof of Theorem 3.1 is to show that lim n→∞ K n (e i(θ−aσn) , e i(θ−aσn) ; µ) K n (e i(θ−aσn) , e i(θ−aσn) ; µ * ) = w(θ) and the convergence is uniform for a in compact subsets of C. Proving this is where we use the fact that the measure µ * is regular and involves interpreting the diagonal reproducing kernel in terms of Christoffel functions and a localization argument. The details appear in many places, such as [10,14,23], so we omit the lengthy calculations. Due to the possible presence of mass points in the gaps of supp(µ ac ), we mention that the Erdős-Turán criterion (see [31, page 101]) and [27,Theorem 11.3.2] imply that not only is µ * regular, but so is its restriction to supp(µ ac ).…”