On the growth of the polynomial entropy integrals for measures in the Szegő class

Let σσ be a probability Borel measure on the unit circle TT and {ϕn}{ϕn} be the orthonormal polynomials with respect to σσ. We say that σσ is a Szegő measure, if it has an arbitrary singular part σsσs, and ∫Tlogσ′dm>−∞∫Tlogσ′dm>−∞, where σ′σ′ is the density of the absolutely continuous part of σσ, mm being the normalized Lebesgue measure on TT. The entropy integrals for ϕnϕn are defined as ϵn=∫T|ϕn|2log|ϕn|dσ.ϵn=∫T|ϕn|2log|ϕn|dσ. It is not difficult to show that ϵn=o¯(n). In this paper, we construct a measure from the Szegő class for which this estimate is sharp (over a subsequence of nn’s).