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

TL;DR

This paper identifies specific classes of Littlewood polynomials that are not $L^eta$-flat for any $eta \\geq 0$, based on the distribution of their coefficients, extending previous results and including palindromic cases.

Contribution

It generalizes known results by establishing non-flatness of Littlewood polynomials for various coefficient distributions and symmetries, including palindromic polynomials.

Findings

Littlewood polynomials are not $L^eta$-flat when the frequency of -1 is outside (1/4, 3/4)

They are not $L^eta$-flat for any $eta \\geq 0$ if the frequency of -1 is not 1/2, for $eta > 2$

Palindromic Littlewood polynomials with even degrees are not $L^eta$-flat for any $eta \\geq 0$

Abstract

We exhibit a class of Littlewood polynomials that are not $L^\alpha$-flat for any $\alpha \geq 0$. Indeed, it is shown that the sequence of Littlewood polynomials is not $L^\alpha$-flat, $\alpha \geq 0$, when the frequency of $-1$ is not in the interval $]\frac14,\frac34[$. 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 $\frac12$. Finally, we prove that the sequence of palindromic Littlewood polynomials with even degrees are not $L^\alpha$-flat for any $\alpha \geq 0$.