Universality Limits Involving Orthogonal Polynomials on an Arc of the Unit Circle

We establish universality limits for measures on a subarc of the unit circle. Assume that μ is a regular measure on such an arc, in the sense of Stahl, Totik, and Ullmann, and is absolutely continuous in an open arc containing some point z0 = e0 . Assume, moreover, that μ′ is positive and continuous at z0. Then universality for μ holds at z0, in the sense that the reproducing kernel Kn (z, t) for μ satisfies lim n→∞ Kn ( z0 exp ( 2πis n ) , z0 exp ( 2πit̄ n )) Kn (z0, z0) = eS ((s− t)T (θ0)) , uniformly for s, t in compact subsets of the plane, where S (z) = sinπz πz is the sinc kernel, and T/2π is the equilibrium density for the arc.