The Rudin-Shapiro polynomials and The Fekete polynomials are not $L^α$-flat

Abstract

We establish that the Rudin-Shapiro polynomials are not $L^α$-flat, for any $α\geq 0$. We further prove that the "truncated" Rudin-Shapiro sequence cannot generate a sequence of $L^α$-flat polynomials, for any $α\geq 0$. In the appendix, we present a simple proof of the fact that the Fekete polynomials and the modified or shifted Fekete polynomials are not $L^α$-flat, for any $α\geq 0$.

In this new version, minor misprints are corrected. The author further announce that he establishes a one-to-one correspondence between the L^2-normalized Littlewood polynomials and Bourgain-Newman polynomials. As a consequence he proved a criterion for L^1-flatness fro which he can deduce that the Littlewood polynomials are not L^α-flat if the frequency of 1 is not between 1/4 and 3/4