How far is an ultraflat sequence of unimodular polynomials from being conjugate-reciprocal?

Abstract

In this paper we study ultraflat sequences (Pn) of unimodular polynomials Pn ∈ Kn in general, not necessarily those produced by Kahane in his paper [Ka]. We examine how far is a sequence (Pn) of unimodular polynomials Pn ∈ Kn from being conjugate reciprocal. Our main results include the following. Theorem. Given a sequence (en) of positive numbers tending to 0, assume that (Pn) is a (en)-ultraflat sequence of unimodular polynomials Pn ∈ Kn. The coefficients of Pn are denoted by ak,n, that is, Pn(z) = n ∑ k=0 ak,nz k , , k = 0, 1, . . . , n, n = 1, 2, . . . . Then n ∑ k=0 k2|ak,n − an−k,n| 2 ≥ ( 1 3 + δn ) n , where (δn) is a sequence of real numbers converging to 0.