Szego's theorem on Parreau-Widom sets

TL;DR

This paper extends Szego's theorem to infinite gap Parreau-Widom sets, including certain Cantor sets, by analyzing the equilibrium measure and employing advanced factorization and covering space techniques.

Contribution

It generalizes Szego's theorem to a broader class of sets, utilizing a canonical M-function factorization and covering space formalism.

Findings

Equilibrium measure is shown to be absolutely continuous.

Szego's condition is established for Parreau-Widom sets.

The approach unifies classical and fractal set cases.

Abstract

In this paper, we generalize Szego's theorem for orthogonal polynomials on the real line to infinite gap sets of Parreau-Widom type. This notion includes Cantor sets of positive measure. The Szego condition involves the equilibrium measure which is shown to be absolutely continuous. Our approach builds on a canonical factorization of the M-function and the covering space formalism of Sodin-Yuditskii.