Proof of Saffari's near-orthogonality conjecture for ultraflat sequences of unimodular polynomials
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Let Pn(z)=k=0nak,nz kC[z] be a sequence of unimodular polynomials (ak,n=1 for all k, n) which is ultraflat in the sense of Kahane, i.e., limnmaxz=1(n+1)-1/2Pn(z)-1=0. We prove the following conjecture of Saffari (1991): k=0nak,nan-k,n=o(n) as n, that is, the polynomial Pn(z) and its "conjugate reciprocal" Pn*(z)=k=0na n-k,nzk 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. 2001 Acadmie des sciences/ditions scientifiques et mdicales Elsevier SAS.