On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges

TL;DR

This paper investigates the distribution of zeros of the modulus squared of Rudin-Shapiro polynomials, providing bounds on their density in arbitrary intervals, extending previous results limited to the full period.

Contribution

It extends earlier work by establishing bounds on the number of zeros of Rudin-Shapiro polynomial moduli in any subinterval, not just the entire period.

Findings

Bounds on zeros are proportional to the interval length and degree.

Zeros are densely distributed with quantifiable fluctuations.

Results improve understanding of polynomial oscillations in harmonic analysis.

Abstract

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of the trigonometric polynomials $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let ${\Cal N}(I,R_k-n)$ denote the number of zeros, counted with multiplicities, of the trigonometric polynomial $R_k-n$ in an interval $I := [\alpha,\beta] \subset [0,2\pi)$. Improving earlier results proved only for the interval $I := [0,2\pi)$, in this paper we show that $$\frac{n|I|}{8\pi} - \frac{2}{\pi} (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}{\pi} + \frac{8}{\pi}(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every interval $I := [\alpha,\beta] \subset [0,2\pi)$, where $|I| = \beta-\alpha$ denotes the length of the interval $I$.