Hilbert transforms and the equidistribution of zeros of polynomials

摘要

We improve the current bounds for an inequality of Erdos and Turan 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 Erdos and Turan had been obtained by Ganelius (1954) and Mignotte (1992). (C) 2021 Elsevier Inc. All rights reserved.