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

Let either $R_k(t):=\mathrm{\mid }{P}_k(e^{it}){\mathrm{\mid }}^2$ or $R_k(t):=\mathrm{\mid }{Q}_k(e^{it}){\mathrm{\mid }}^2$, 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 $\text{Cal}N(I,R_k-n)$ 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 $$\frac{n\mathrm{\mid }I\mathrm{\mid }}{8\pi }-\frac{2}{\pi }(2n\log n{)}^{1\mathrm{/}2}-1\le N(I,R_k-n)\le \frac{n\mathrm{\mid }I\mathrm{\mid }}{\pi }+\frac{8}{\pi }(2n\log n{)}^{1\mathrm{/}2},k\ge 2,$$ for every interval $I:=[\alpha ,\beta ]\subset [0,2\pi )$, where $\mathrm{\mid }I\mathrm{\mid }=\beta -\alpha$ denotes the length of the interval $I$.