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Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

Published online by Cambridge University Press:  11 February 2015

IDRIS ASSANI ,
DAVID DUNCAN  and
RYO MOORE
IDRIS ASSANI
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA email assani@math.unc.edu, ryom@live.unc.edu
DAVID DUNCAN
Affiliation:
Department of Mathematics & Statistics, Coastal Carolina University, Conway, SC 29528, USA email dduncan@coastal.edu
RYO MOORE
Affiliation:
Department of Mathematics, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA email assani@math.unc.edu, ryom@live.unc.edu
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Abstract

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$
converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 4 , June 2016 , pp. 1037 - 1066
Copyright
© Cambridge University Press, 2015 

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