On $L^\alpha$-flatness of Erd\H{o}s-Littlewood's polynomials

Abstract

It is shown that Erd\"{o}s--Littlewood's polynomials are not $L^\alpha$-flat when $\alpha > 2$ is an even integer (and hence for any $\alpha \geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erd\"{o}s--Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erd\"{o}s--Littlewood polynomials. Our proof is short and simple. It relies on the classical lemma for $L^p$ norms of the Dirichlet kernel, the Marcinkiewicz--Zygmund interpolation inequalities, and the $p$-concentration theorem due to A. Bonami and S. R\'ev\'esz.