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 \({\mathcal N}(I,R_k-n)\) 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 \(\frac{n|I|}{8\pi } - \frac{2}{\pi } (2n\log n)^{1/2} - 1 \le N(I,R_k-n) \le \frac{n|I|}{\pi } + \frac{8}{\pi }(2n\log n)^{1/2},\quad k \ge 2,\) for every interval \(I:= [\alpha ,\beta ] \subset [0,2\pi )\) , where \(|I| = \beta -\alpha \) denotes the length of the interval I.
