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

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The classical Szego 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 Szego'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.