On $L^{\alpha }$-flatness of Erdős-Littlewood's polynomials

It is shown that Erdös–Littlewood’s polynomials are not $L^{\alpha }$-flat when $\alpha >2$ is an even integer (and hence for any $\alpha \ge 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erdös–Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erdös–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. Révész.