A class of Littlewood polynomials that are not $L^{\alpha }$-flat

Abstract

We exhibit a class of Littlewood polynomials that are not $L^{\alpha }$-flat for any $\alpha \ge 0$. Indeed, it is shown that the sequence of Littlewood polynomials is not $L^{\alpha }$-flat, $\alpha \ge 0$, when the frequency of $-1$ is not in the interval $]\frac{1}{4},\frac{3}{4}[$. We further obtain a generalization of Jensen-Jensen-Hoholdt's result by establishing that the sequence of Littlewood polynomials is not $L^{\alpha }$-flat for any $\alpha >2$ if the frequency of $-1$ is not $\frac{1}{2}$. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not $L^{\alpha }$-flat for any $\alpha \ge 0$.

arXiv comment15 pages. The appendix is written jointly with M, G. Nadkarni. In this new version few misprints are corrected and Jensen-Jensen-Hoholdt's theorem is generalized