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

## TL;DR

This paper provides a quantitative analysis of Szego's theorem focusing on measures with deep zeroes on the unit circle, revealing how such zeroes influence polynomial density in $L^2$ spaces.

## Contribution

It introduces a quantitative version of Szego's theorem specifically addressing measures with deep zeroes, expanding understanding of polynomial approximation under these conditions.

## Key findings

- Deep zeroes significantly affect polynomial density in $L^2(
ho)$
- Quantitative criteria for divergence caused by deep zeroes
- Enhanced understanding of measure properties influencing Szego's theorem

## Abstract

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.