On the Oscillation of the Modulus of the Rudin–Shapiro Polynomials Around the Middle of Their Ranges

Let either Rₖ(t):= |Pₖ(eⁱᵗ)|² R k ( t ) : = | P k ( e it ) | 2 or Rₖ(t):= |Qₖ(eⁱᵗ)|² R k ( t ) : = | Q k ( e it ) | 2 , where Pₖ P k and Qₖ Q k are the usual Rudin–Shapiro polynomials of degree $$n-1$$ n - 1 with n=2ᵏ n = 2 k . The graphs of the trigonometric polynomials Rₖ R k on the period suggest many zeros of Rₖ(t)-n R k ( t ) - n in a dense fashion on the period. Let N(I,Rₖ-n) N ( I , R k - n ) denote the number of zeros, counted with multiplicities, of the trigonometric polynomial Rₖ-n R k - n in an interval $$I: