Notes on the Szegő minimum problem. I. Measures with deep zeroes

The classical Szegő polynomial approximation theorem states that the polynomials are dense in the space L²(ρ), where ρ is a measure on the unit circle, if and only if the logarithmic integral of the measure ρ diverges. In this note we give a quantitative version of Szegő’s theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure ρ on a sufficiently rare subset of the circle.