Bounds on autocorrelation coefficients of Rudin-Shapiro polynomials. II
The Rudin-Shapiro polynomials were discovered by \textit{H. S. Shapiro} [Extremal problems for polynomials and power series. Massachusetts Institute of Technology. (Ph.D. Thesis) (1953)]. These polynomials are also called the Shapiro--Rudin polynomials and constructed as follows: Let \(P_0(z) = 1\) and \(Q_0 (z) = 1\). For any integer \(n \geq 1\), define \[ P_n (z):= P_{n-1}(z) + z^{2^{n-1}} Q_{n-1}(z)\text{ and }Q_n (z) := P_{n-1} (z) - z^{2^{n-1}} Q_{n-1} (z). \] Polynomials \(P_n(z)\) and \(Q_n(z)\) are called Rudin-Shapiro polynomials. Note that \(P_n(z)\) and \(Q_n(z)\) are polynomials with \(\pm 1\) coefficients of degree \(L_n - 1\), where \(L_n = 2^n\). Also, for any \(t \in [0, 2 \pi )\), \( \left| P_n (e^{it}) \right|^2 + \left| Q_n (e^{it}) \right|^2 = L_{n+1} = 2^{n+1} \). By the Fourier coefficients of \(f\) at \(k\), we mean the coefficient for the term \(z^k\), or simply \[ \widehat{f}(k) = (f)^\wedge (k) := \frac{1}{2 \pi} \int_0^{2 \pi} f(e^{it}) e^{-ikt} dt. \] The author studies (periodic) autocorrelation coefficients of the Rudin--Shapiro polynomials and proves the following longstanding Saffari's conjecture. Theorem. If \(P_n\) and \(Q_n\) are the \(n\)-th Rudin-Shapiro polynomials and \[ |P_n (z)|^2=\sum_{k=-L_n + 1}^{L_n - 1} a_k z^k, \quad |z|=1 \] \((a_0 = L_n, a_k = a_{-k}, k \geq 1)\), then \[ 0.27771487 \cdots (1 + o(1))|\lambda|^n \leq \max_{1 \leq k \leq L_n - 1} |a_k| \leq (3.78207844 \cdots )|\lambda|^n, \] where \(\lambda \) is the real root of \(x^3 - x^2 - 2 x + 4 = 0\). Also, if \[ \left(\overline{ P_n} Q_n \right)(z)=\left(P_n \overline{Q_n}\right)( 1/z) =\sum_{k=-L_n + 1}^{L_n - 1} b_k z^k, \quad |z|=1 \] \((b_0 = 2 - L_n, b_k =\overline{b_{-k}}, k \geq 1)\), then \[ 0.46071984 \cdots (1 + o(1))|\lambda|^n \leq \max_{1 \leq k \leq L_n - 1} |b_k| \leq (3.78207844 \cdots )|\lambda|^n. \] This theorem improves the results obtained in the first part of this work by \textit{J. P. Allouche} et al. [Anal. Math. 45, No. 4, 705--726 (2019; Zbl 1449.42001)] and provides the upper bound of the correct order of infinity.
scientific article; zbMATH DE number 7187595