Hilbert transforms and the equidistribution of zeros of polynomials

TL;DR

This paper improves bounds on the discrepancy of zeros of polynomials by connecting Erdős-Turán inequalities with Fourier analysis and Hilbert transforms, providing a complete solution to an extremal problem.

Contribution

It introduces a novel link between polynomial zero distribution inequalities and Fourier analysis, solving a key extremal problem involving Hilbert transforms.

Findings

Enhanced bounds for Erdős-Turán inequality.

Complete solution to an extremal Fourier analysis problem.

Improved understanding of zero distribution discrepancy.

Abstract

We improve the current bounds for an inequality of Erd\H{o}s and Tur\'an from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erd\H{o}s and Tur\'an had been obtained by Ganelius (1954) and Mignotte (1992).