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, and divided differences, 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

$$\begin{aligned} \eta _1 \sqrt{n} \le |P_n(z)| \le \eta _2 \sqrt{n} , \qquad z \in {{\mathbb {C}}}, |z| = 1 , \end{aligned}$$

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.