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

TL;DR

This paper proves that Rudin-Shapiro and Fekete polynomials, including their truncated and shifted variants, are not $L^eta$-flat for any $eta \,\geq\, 0$, highlighting limitations in their flatness properties.

Contribution

It establishes the non-$L^eta$-flatness of Rudin-Shapiro and Fekete polynomials, including truncated and shifted versions, for all $eta \,\geq\, 0$, providing new insights into their properties.

Findings

Rudin-Shapiro polynomials are not $L^eta$-flat for any $eta \geq 0$

Truncated Rudin-Shapiro sequences cannot generate $L^eta$-flat polynomials

Fekete polynomials and their variants are not $L^eta$-flat for any $eta \geq 0$

Abstract

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