Polynomial averages and pointwise ergodic theorems on nilpotent groups

TL;DR

This paper proves pointwise convergence and maximal inequalities for polynomial ergodic averages in step-two nilpotent groups, confirming a conjecture and introducing a novel nilpotent circle method that extends classical harmonic analysis techniques.

Contribution

It establishes the first pointwise ergodic theorems for polynomial averages in nilpotent groups of step two and develops a new nilpotent circle method for non-commutative harmonic analysis.

Findings

Almost everywhere convergence of polynomial ergodic averages

Maximal inequalities on L^p spaces for 1<p≤∞

Variational inequalities on L^2 for 2<ρ<∞

Abstract

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $\rho$-variational inequalities on $L^2$ for $2<\rho<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.