  The classical Szego polynomial approximation theorem states that the
polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on
the unit circle, if and only if the logarithmic integral of the measure $\rho$
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 $\rho$ on a sufficiently rare subset of the circle.
