The Mahler measure of the Rudin-Shapiro polynomials

TL;DR

This paper investigates the Mahler measure of Rudin-Shapiro polynomials, revealing that it is comparable to their maximum modulus on the unit circle, providing new bounds and insights into their size properties.

Contribution

It establishes that the Mahler measure of Rudin-Shapiro polynomials matches their maximum modulus, offering the first nontrivial bounds for their Mahler measure.

Findings

Mahler measure and maximum modulus are of the same size for Rudin-Shapiro polynomials.

Mahler measure and maximum norm are comparable even on subarcs of the unit circle.

First nontrivial lower bounds for the Mahler measure of Rudin-Shapiro polynomials are provided.

Abstract

Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before.