Almost-Everywhere Convergence and Polynomials

TL;DR

This paper characterizes functions whose scaled sequences are pointwise good for ergodic averages, showing such functions must be polynomials on certain intervals, thus providing a converse to Bourgain's polynomial ergodic theorem.

Contribution

It proves that continuous or real analytic functions with all scaled sequences in mma are necessarily polynomials on some interval or the entire domain.

Findings

Functions with scaled sequences in mma are polynomials on an interval.

Real analytic functions with all scaled sequences in mma are polynomials on the whole domain.

Results serve as converses to Bourgain's polynomial ergodic theorem.

Abstract

Denote by $\Gamma$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages $\frac{1}{n} \sum_{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere. We prove the following two results. [1.] If $f: (0,\infty)\to\mathbb R$ is continuous and if $\big(f(ku+v)\big)_{k\geq 1}\in\Gamma$ for all $u, v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\infty)$ of positive length. [2.] If $f: [0,\infty)\to\mathbb R$ is real analytic and if $\big(f(ku)\big)_{k\geq 1}\in\Gamma$ for all $u>0$, then $f$ is a polynomial on the whole domain $[0,\infty)$. These results can be viewed as converses of Bourgain's polynomial ergodic theorem which claims that every polynomial sequence lies in $\Gamma$.