Do flat skew-reciprocal Littlewood polynomials exist?

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 Vallée Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants ${\eta }_2>{\eta }_1>0$ and a sequence $(P_n)$ of Littlewood polynomials $P_n$ of degree $n$ such that $${\eta }_1\sqrt{n}\le |P_n(z)|\le {\eta }_2\sqrt{n},z\in \mathbb{C},|z|=1,,$$ 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.