Proof of Saffari's near-orthogonality conjecture for ultraflat sequences of unimodular polynomials

Abstract

Let P n(z)=∑ k=0 na k,nz 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., lim n→∞ max |z|=1 |(n+1) −1/2|P n(z)|−1|=0. We prove the following conjecture of Saffari (1991): ∑ k=0 n a k, n a n− k, n =o( n) as n→∞, that is, the polynomial P n ( z) and its “conjugate reciprocal” P n ∗(z)=∑ k=0 n a n−k,nz k become “nearly orthogonal” as n→∞. To this end we use results from [3] where (as well as in [5]) we studied the structure of ultraflat polynomials and proved several conjectures of Saffari.