The asymptotic value of the Mahler measure of the Rudin-Shapiro polynomials

TL;DR

This paper proves that the Mahler measure of Rudin-Shapiro polynomials asymptotically approaches a specific value, confirming a conjecture from 1985, with implications for signal processing and low-autocorrelation sequences.

Contribution

It establishes the asymptotic behavior of the Mahler measure of Rudin-Shapiro polynomials, confirming a long-standing conjecture using recent advances in related conjectures.

Findings

Mahler measure of Rudin-Shapiro polynomials asymptotically equals (2n/e)^{1/2}

Confirms Saffari's 1985 conjecture on Mahler measure

Utilizes recent proofs of Saffari and Montgomery conjectures by B. Rodgers

Abstract

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we show that the Mahler measure of the Rudin-Shapiro polynomials of degree $n=2^k-1$ is asymptotically $(2n/e)^{1/2}$, as it was conjectured by B. Saffari in 1985. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers.