Hilbert transforms and the equidistribution of zeros of polynomials

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).

[1]  David W. Boyd,et al.  Speculations Concerning the Range of Mahler's Measure , 1980, Canadian Mathematical Bulletin.

[2]  Sequences of analytic functions and their zeros , 1954 .

[3]  G. Pólya,et al.  Fonctions entières et intégrales de fourier multiples , 1936 .

[4]  Rima Alaifari,et al.  Lower bounds for the truncated Hilbert transform , 2013, 1311.6845.

[5]  Paul Erdös,et al.  On the Distribution of Roots of Polynomials , 1950 .

[6]  L. Grafakos.  Classical and modern Fourier analysis , 2003 .

[7]  G. Pólya,et al.  Fonctions entières et intégrales de Fourier multiples , 1936 .

[8]  F. G. Tricomi,et al.  ON THE FINITE HILBERT TRANSFORMATION , 1951 .

[9]  L. Debnath.  Tables of Integral Transforms , 2012 .

[10]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[11]  J. Vaaler.  SOME EXTREMAL FUNCTIONS IN FOURIER ANALYSIS , 2007 .

[12]  Burgess Davis.  On the weak type (1,1) inequality for conjugate functions , 1974 .

[13]  Namit Mahajan,et al.  THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS , 2000 .

[14]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[15]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[16]  R. Coifman,et al.  A Remark on the Arcsine Distribution and the Hilbert transform , 2018, The journal of fourier analysis and applications.

[17]  K. Soundararajan,et al.  Equidistribution of Zeros of Polynomials , 2018, Am. Math. Mon..

[18]  M. Mignotte.  Remarque sur une question relative à des fonctions conjuguées , 1992 .

[19]  S. K. Pichorides.  On the best values of the constants in the theorem of M. Riesz, Zygmund and Kolmogorov , 1972 .