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arXiv · 2104.00105

Hilbert transforms and the equidistribution of zeros of polynomials

Emanuel Carneiro·Mithun Kumar Das·Alexandra Florea·Angel V. Kumchev·Amita Malik·Micah B. Milinovich·Caroline Turnage-Butterbaugh·Jiuya Wang
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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).

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Emanuel Carneiro, Mithun Kumar Das, Alexandra Florea, Angel V. Kumchev, Amita Malik, Micah B. Milinovich, Caroline Turnage-Butterbaugh, Jiuya Wang. 2021-03-31. Hilbert transforms and the equidistribution of zeros of polynomials. https://doi.org/10.1016/j.jfa.2021.109199

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