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

Abstract

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.

10 pages, 27 references and three quotations: <<Begin at the beginning, the King said gravely, "and go on till you come to the end: then stop,''>> by Lewis Carroll, <<Those who know do not speak; those who speak do not know,>> by Laozi and <<Everyone writes, nobody reads,>> by Erdős (which should be attributed to Fejér according to Erdős)