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

Abstract:

Let \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 $(P_n)$ of polynomials $P_n \!\in \! {\mathcal {K}}_n$ is called ultraflat if $(n + 1)^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t \!\in \! {\mathbb {R}}$. In this paper we prove that \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 $(P_n)$ of polynomials $P_n \in {\mathcal {K}}_n$ and for every $q \in (0,\infty )$, where $P_n^*$ is the conjugate reciprocal polynomial associated with $P_n$, $\Gamma$ is the usual gamma function, and the $\sim$ symbol means that the ratio of the left and right hand sides converges to $1$ as $n \rightarrow \infty$. Another highlight of the paper states that \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 $(P_n)$ of polynomials $P_n \in {\mathcal {K}}_n$. We prove a few other new results and reprove some interesting old results as well.