Almost-Everywhere Convergence and Polynomials

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 $(f(ku+v){)}_{k\ge 1}\in \Gamma$ 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 $(f(ku){)}_{k\ge 1}\in \Gamma$ 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 $\Gamma$.

6 pages