The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal

Abstract

Let K n ≔ { Q n : Q n ( z ) = ∑ k = 0 n a k z k , a k ∈ C , | a k | = 1 } . \begin{equation*} {\mathcal {K}}_n \coloneq \left \{Q_n: Q_n(z) = \sum _{k=0}^n{a_k z^k}, \quad a_k \in {\mathbb {C}}\,, \quad |a_k| = 1 \right \}\,. \end{equation*} A sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \!\in \! {\mathcal {K}}_n is called ultraflat if ( n + 1 ) − 1 / 2 | P n ( e i t ) | (n + 1)^{-1/2}|P_n(e^{it})| converge to 1 1 uniformly in t ∈ R t \!\in \! {\mathbb {R}} . In this paper we prove that 1 2 π ∫ 0 2 π | ( P n − P n ∗ ) ( e i t ) | q d t ∼ 2 q Γ ( q + 1 2 ) Γ ( q 2 + 1 ) π n q / 2 \begin{equation*} \frac {1}{2\pi } \int _0^{2\pi }{\left | (P_n - P_n^*)(e^{it}) \right |^q \, dt} \sim \frac {{2}^q \Gamma \left (\frac {q+1}{2} \right )}{\Gamma \left (\frac q2 + 1 \right ) \sqrt {\pi }} \,\, n^{q/2} \end{equation*} for every ultraflat sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \in {\mathcal {K}}_n and for every q ∈ ( 0 , ∞ ) q \in (0,\infty ) , where P n ∗ P_n^* is the conjugate reciprocal polynomial associated with P n P_n , Γ \Gamma is the usual gamma function, and the ∼ \sim symbol means that the ratio of the left and right hand sides converges to 1 1 as n → ∞ n \rightarrow \infty . Another highlight of the paper states that 1 2 π ∫ 0 2 π | ( P n ′ − P n ∗ ′ ) ( e i t ) | 2 d t ∼ 2 n 3 3 \begin{equation*} \frac {1}{2\pi }\int _0^{2\pi }{\left | (P_n^\prime - P_n^{*\prime })(e^{it}) \right |^2 \, dt} \sim \frac {2n^3}{3} \end{equation*} for every ultraflat sequence ( P n ) (P_n) of polynomials P n ∈ K n P_n \in {\mathcal {K}}_n . We prove a few other new results and reprove some interesting old results as well.