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Hilbert transforms and the equidistribution of zeros of polynomials. (English) Zbl 1472.42002

Summary: We improve the current bounds for an inequality of P. Erdős and P. Turán [Ann. Math. (2) 51, 105–119 (1950; Zbl 0036.01501)] related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of K. Soundararajan [Am. Math. Mon. 126, No. 3, 226–236 (2019; Zbl 1409.42003)], 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 [loc. cit.], refinements of the discrepancy inequality of Erdős and Turán had been obtained by T. Ganelius [Ark. Mat. 3, 1–50 (1954; Zbl 0055.06905)] and M. Mignotte [C. R. Acad. Sci., Paris, Sér. I 315, No. 8, 907–911 (1992; Zbl 0773.31002)].

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
42A50 Conjugate functions, conjugate series, singular integrals
11K38 Irregularities of distribution, discrepancy
12D05 Polynomials in real and complex fields: factorization
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References:

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