On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

Abstract

In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we study the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle. We also show that the Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1$ with $n := 2^k$ have $o(n)$ zeros on the unit circle. This should be compared with a result of B. Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd primes $p$ the Fekete polynomials $f_p$ of degree $p-1$ have asymptotically $\kappa_0 p$ zeros on the unit circle, where $0.500813>\kappa_0>0.500668$. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that the $k$-th Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1 = 2^k-1$ have at least $c_2n$ zeros in the annulus $$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$