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 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 $\eta_2 > \eta_1 > 0$ and a sequence $(P_n)$ of Littlewood polynomials $P_n$ of degree $n$ such that $$\eta_1 \sqrt{n} \leq |P_n(z)| \leq \eta_2 \sqrt{n}\,, \qquad z \in \mathbb{C}\,, \, \, |z| = 1,,$$ 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.