Mass Points of Measures and Orthogonal Polynomials on the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szegő recurrences. We assume that the reflection coefficients tend to some complex number a with 0 < |a| < 1. The orthogonality measure μ then lives essentially on the arc {e : α ≤ t ≤ 2π − α} where sin α 2 def = |a| with α ∈ (0, π). Under the certain rate of convergence it was proved in [6] that μ has no mass points inside this arc. We show that this result is sharp in a sense. We also examine the case of the whole unit circle and some examples of singular continuous measures given by their reflection coefficients.