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July  2008, 2(3): 465-470. doi: 10.3934/jmd.2008.2.465

Almost-everywhere convergence and polynomials

Michael Boshernitzan 1, and  Máté Wierdl 2,

1. 

Department of Mathematics, Rice Unviersity, Houston, TX 77005, United States
2. 

Department of Mathematical Sciences, 373 Dunn Hall, University of Memphis, Memphis, TN 38152-3240, United States

Received  November 2007 Revised  March 2008 Published  April 2008

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: polynomials., Pointwise ergodic theorems along subsequences.
Mathematics Subject Classification: Primary: 37A10, Secondary: 37A3.
Citation: Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
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