The Phase Problem of Ultraflat Unimodular Polynomials: the Resolution of the Conjecture of Saffari

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := ( pn : pn(z) = n X k=0 akz , ak ∈ C , |ak| = 1 ) . The class Kn is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (εn) of positive numbers tending to 0, we say that a sequence (Pn) of unimodular polynomials Pn ∈ Kn is (εn)-ultraflat if (1 − εn) √ n + 1 ≤ |Pn(z)| ≤ (1 + εn) √ n + 1 , z ∈ ∂D , n ∈ N . The existence of ultraflat unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdős (Problem 22 in [Er1]) asserting that, for all Pn ∈ Kn with n ≥ 1, max z∈∂D |Pn(z)| ≥ (1 + ε) √ n + 1 , where ε > 0 is an absolute constant (independent of n). Yet, refining a method of Körner [Kö], Kahane [Ka] proved that there exists a sequence (Pn) with Pn ∈ Kn which is (εn)-ultraflat, where εn = O n−1/17 √ log n . Thus the Erdős 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]. We prove five closely related conjectures of Saffari [Sa] (see also [QS2]). Most importantly the following one. Uniform Distribution Conjecture for the Angular Speed. Let (Pn) be a εnultraflat sequence of unimodular polynomials Pn ∈ Kn. Let Pn(e ) = Rn(t)e iαn(t) , Rn(t) = |Pn(e)| . In the interval [0, 2π], the distribution of the normalized angular speed α′n(t)/n converges to the uniform distribution as n → ∞. More precisely, we have m{t ∈ [0, 2π] : 0 ≤ αn(t) ≤ nx} = 2πx + on(x) for every x ∈ [0, 1], where limn→∞ on(x) = 0 for every x ∈ [0, 1]. 1991 Mathematics Subject Classification. 41A17.