Do flat skew-reciprocal Littlewood polynomials exist?

DOI10.1007/S00365-022-09575-4arXiv2001.08151MaRDI QIDQ6333325

Tamás Erdélyi

Publication date: 22 January 2020

Abstract: Polynomials with coefficients in $-1,1$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vall'ee Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollob'as, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants $et{a}_2>et{a}_1>0$ and a sequence $(P_n)$ of Littlewood polynomials $P_n$ of degree $n$ such that ${\displaystyle }$eta_1 sqrt{n} leq |P_n(z)| leq eta_2 sqrt{n},, qquad z in mathbb{C},, , , |z| = 1,,${\displaystyle }$ confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence $(P_n)$ of Littlewood polynomials $P_n$ is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of $P_n$ making the Littlewood polynomials $P_n$ close to skew-reciprocal.

Mathematics Subject Classification ID

Polynomials in number theory (11C08) Polynomials in real and complex fields: factorization (12D05) Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) (30C15) Inequalities in approximation (Bernstein, Jackson, Nikol'ski?-type inequalities) (41A17) Real polynomials: location of zeros (26C10)

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