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

Abstract

Let $\text{P}{}_{\mathrm{n}}\text{(z)=∑}{}_{\mathrm{k=0}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{k,n}}\text{z}{}^{\mathrm{k}}\text{∈}\text{C}\text{[z]}$ be a sequence of unimodular polynomials (|a_k,n|=1 for all k, n) which is ultraflat in the sense of Kahane, i.e., $\text{lim}\text{n→∞}\text{max}\text{|z|=1}\text{|(n+1)}{}^{\mathrm{-1/2}}\text{|P}{}_{\mathrm{n}}\text{(z)|−1|=0.}$ We prove the following conjecture of Saffari (1991): ∑_k=0ⁿa_k,na_n−k,n=o(n) as n→∞, that is, the polynomial P_n(z) and its “conjugate reciprocal” $\text{P}{}_{\mathrm{n}}{}^{\mathrm{\ast }}\text{(z)=∑}{}_{\mathrm{k=0}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{n-k,n}}\text{z}{}^{\mathrm{k}}$ 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.