Hilbert transforms and the equidistribution of zeros of polynomials
Classical Analysis and ODEs
2021-08-09 v4 Number Theory
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).
Keywords
Cite
@article{arxiv.2104.00105,
title = {Hilbert transforms and the equidistribution of zeros of polynomials},
author = {Emanuel Carneiro and Mithun Kumar Das and Alexandra Florea and Angel V. Kumchev and Amita Malik and Micah B. Milinovich and Caroline Turnage-Butterbaugh and Jiuya Wang},
journal= {arXiv preprint arXiv:2104.00105},
year = {2021}
}
Comments
18 pages, 3 figures, updated to incorporate referee's suggestions