# Model theory and metric convergence II: Averages of unitary polynomial actions

## TL;DR

This paper proves pointwise convergence of averages of polynomial sequences of unitary transformations in Hilbert spaces using model theory, extending to general abelian group actions and including a case study of the lamplighter group.

## Contribution

It introduces a model-theoretic approach to establish convergence of polynomial unitary actions, covering Leibman sequences and generalizations to abelian groups.

## Key findings

- Proves pointwise convergence with uniform metastability rate for polynomial unitary averages.
- Extends convergence results to arbitrary Leibman sequences and actions of any abelian group.
- Demonstrates realization of the lamplighter group as a quadratic Leibman sequence.

## Abstract

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 $\mathbb{Z}$.