Equidistribution of zeros of polynomials

TL;DR

This paper provides a concise, Fourier analysis-based proof of the classical Erdos-Turan theorem, demonstrating that certain polynomials have zeros that are evenly distributed around the unit circle.

Contribution

The paper offers a new, simplified proof of the Erdos-Turan equidistribution theorem using Fourier analysis techniques.

Findings

Zeros of polynomials with small values on the unit circle are equidistributed in angle.

The proof is short, self-contained, and relies on Fourier analysis.

The result confirms the clustering of zeros near the unit circle under specified conditions.

Abstract

A classical result of Erdos and Turan states that if a monic polynomial has small size on the unit circle and its constant coefficient is not too small, then its zeros cluster near the unit circle and become equidistributed in angle. Using Fourier analysis we give a short and self-contained proof of this result.