We use model theory of metric structures to prove the pointwise convergence, with a uniform metastability rate, of averages of a polynomial sequence $\{ T_n\}$ (in Leibman's sense) of unitary transformations of a Hilbert space. As a special case, this applies to unitary sequences $\{ U^{ p(n)}\}$ where $p$ is a polynomial $\mathbb{ Z}\to\mathbb{ Z}$ and $U$ a fixed unitary operator; however, our convergence results hold for arbitrary Leibman sequences. As a case study, we show that the non-nilpotent "lamplighter group" $\mathbb{ Z}\wr\mathbb{ Z}$ is realized as the range of a suitable quadratic Leibman sequence. We also indicate how these convergence results generalize to arbitrary Folner averages of unitary polynomial actions of any abelian group $\mathbb{ G}$ in place of $\mathbb{ Z}$.
Categories: math.DS math.LO