Limits of zeros of orthogonal polynomials on the circle

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of P. Turán): namely, for n < N , one can freely prescribe the n-th polynomial and N − n zeros of the N -th one. We shall also describe all possible limit sets of zeros within the unit disk. 1 Results Let D be the open unit disk. We consider Borel measures dμ(t), t ∈ [−π, π) on the unit circle (identified with R/mod 2π) of infinite support, and for such a measure let Φn(μ, z) = z + n−1 ∑