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(Uniform) convergence of twisted ergodic averages

Published online by Cambridge University Press:  13 April 2015

TANJA EISNER  and
BEN KRAUSE
TANJA EISNER
Affiliation:
Institute of Mathematics, University of Leipzig, PO Box 100 920, 04009 Leipzig, Germany email [email protected]
BEN KRAUSE
Affiliation:
UCLA Math Sciences Building, Los Angeles, CA 90095-1555, USA email [email protected]
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Abstract

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,

$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$
and for ‘twisted’ polynomial ergodic averages,
$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$
 for certain classes of badly approximable $\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $\unicode[STIX]{x1D707}$-almost everywhere for $f\in L^{p}(X),p>1,$ and arbitrary $\unicode[STIX]{x1D703}\in [0,1]$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2172 - 2202
Copyright
© Cambridge University Press, 2015 

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