Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain

TL;DR

This paper proves that ergodic averages along integer-valued polynomials converge pointwise in measure-preserving systems and establishes the boundedness of their r-variation for r>2, extending Bourgain's theorem.

Contribution

It introduces new bounds on the r-variation of polynomial ergodic averages, showing convergence and sharpness of these bounds in L^2 spaces.

Findings

r-variation $ ext{V}^r(M_N(f))$ is bounded on L^2 for r>2

$ ext{V}^2(M_N(f))$ is unbounded on L^2

Pointwise convergence of polynomial ergodic averages

Abstract

Let $L^2(X,\Sigma,\mu,\tau)$ be a measure-preserving system, with $\tau$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} \tau^{P(n)} f \] converge pointwise for $f \in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.