Almost-everywhere convergence and polynomials

Pages: 465 - 470,
Issue 3,
July
2008 doi:10.3934/jmd.2008.2.465

Michael Boshernitzan - Department of Mathematics, Rice Unviersity, Houston, TX 77005, United States (email)

Máté Wierdl - Department of Mathematical Sciences, 373 Dunn Hall, University of Memphis, Memphis, TN 38152-3240, United States (email)

Abstract:
Denote by $\Gamma$ the set of pointwise good sequences: 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 $(f(ku+v))_{k\geq 1}\in\Gamma$ 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 $(f(ku))_{k\geq 1}\in\Gamma$ 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 $\Gamma$.

Keywords: Pointwise ergodic theorems along
subsequences, polynomials.

Mathematics Subject Classification: Primary: 37A10, Secondary: 37A30.

Received: November 2007;
Revised:
March 2008;
Available Online: April 2008.