Hilbert transforms and the equidistribution of zeros of polynomials
arXiv:2104.00105 · doi:10.1016/j.jfa.2021.109199
Abstract
We improve the current bounds for an inequality of Erdős and Turán 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 Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992).
18 pages, 3 figures, updated to incorporate referee's suggestions