On the real part of ultraflat sequences of unimodular polynomials: Consequences implied by the resolution of the phase problem

Abstract

Let P n (z)=∑ k=0 n a k,n z k ℂ[z] be a sequence of unimodular polynomials (|a k,n |=1 for all k, n) which is ultraflat in the sense of Kahane, i.e., We prove the following conjecture of Queffelec and Saffari, see (1.30) in [QS2]. If q(0,∞) and (P n ) is an ultraflat sequence of unimodular polynomials P n of degree n, then for f n (t):=Re(P n (e it )) we have and where Γ denotes the usual gamma function, and the ∼ symbol means that the ratio of the left and right hand sides converges to 1 as . To this end we use results from [Er1] where we studied the structure of ultraflat polynomials and proved several conjectures of Queffelec and Saffari.