IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE

Let Rk(t) := |Pk(eit)|2 and Sk(t) := |Qk(eit)|2, where Pk and Qk are the usual Rudin-Shapiro polynomials of degree n - 1 with n = 2k. In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equations Rk(t) = (1 + eta)n and Sk(t) = (1 + eta)n have at least An0.5394282 distinct solutions in [0, 2 pi) whenever eta is real, |eta| < 2-8, and k is sufficiently large. In this paper we show that the equations Rk(t) = (1 + eta)n and Sk(t) = (1+ eta)n have at least (1/2-|eta|- epsilon)n/2 distinct solutions in [0, 2 pi) for every eta is an element of (-1/2, 1/2), epsilon > 0, and sufficiently large be k >= k eta,epsilon.