Bounds on Rudin-Shapiro polynomials of arbitrary degree

TL;DR

This paper establishes sharp bounds on the magnitude of Rudin-Shapiro polynomials on the unit circle, confirming a longstanding conjecture and providing new insights into their growth and differences.

Contribution

It proves a sharp upper bound for Rudin-Shapiro polynomials' magnitude and refutes a previous conjecture about their differences, advancing understanding of their extremal properties.

Findings

Bound |P_{<n}(z)| ≤ √(6n−2)−1 is sharp for specific n and z.

Difference |P_{<n}(z)−P_{<m}(z)| ≤ √(10(n−m)) is asymptotically sharp.

Contradicts Montgomery's conjecture on polynomial differences.

Abstract

Let $P_{<n}(z)$ be the Rudin-Shapiro polynomial of degree $n-1$. We show that $|P_{<n}(z)|\le \sqrt{6n-2}-1$ for all $n\ge0$ and $|z|=1$, confirming a longstanding conjecture. This bound is sharp in the case when $n=(2\cdot 4^k+1)/3$ and $z=1$. We also show that for $n\ge m\ge0$, $|P_{<n}(z)-P_{<m}(z)|\le \sqrt{10(n-m)}$, which is asymptotically sharp in the sense that for any $\varepsilon>0$ there exists $n>m\ge0$ and $z$ with $|z|=1$ and $|P_{<n}(z)-P_{<m}(z)|\ge\sqrt{(10-\varepsilon)(n-m)}$, contradicting a conjecture of Montgomery.