Do flat skew-reciprocal Littlewood polynomials exist?
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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.
