The phase problem of ultraflat unimodular polynomials: The resolution of the conjecture of Saffari

where e > 0 is an absolute constant (independent of n). Yet, combining some probabilistic lemmas from Korner’s paper [Ko] with some constuctive methods (Gauss polynomials, etc.), which were completely unrelated to the deterministic part of Korner’s paper, Kahane [Ka] proved that there exists a sequence (Pn) with Pn ∈ Kn which is (en)-ultraflat, where en = O ( n−1/17 √ log n ) . Thus the Erdos conjecture was disproved for the classes Kn. 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].