Outside the Szegö condition: some proofs and disproofs

By the classical Szegö theorem the polynomials are dense in $L^2(\mu)$, where $\mu$ is a measure on the unit circle, if and only if the logarithmic integral of the density of $\mu$ diverges. We will discuss several quantitative versions of this theorem for the case of measure with a divergent logarithmic integral of its density. In particular, we will disprove one conjecture of Nevai. The talk is based on a joint work with A. Borichev and M. Sodin (arXiv:1902.00874, arXiv:1902.00872)