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.