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Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation.

Then by the classical Birkhoff's ergodic theorem if $p\ge 1$, then for any $f\in L^p(X, \mu)$ the sequence $$ \mathcal{M}_N f(x):=\frac{1}{N}\sum_{n=0}^N f(T^n x) $$ converges for almost every $x\in X$.

$\textbf{Question:}$ Is it true that the sequence $$ \mathcal{A}_N f(x):=\frac{1}{N}\sum_{n=0}^N f(T^{n^2} x) $$ is convergent a.e. for $f\in L^p(X,\mu)$ and $p\ge 1$?

I'll be more than happy to see the answer to my naive question for a particular case when: $X=\mathbb{Z}$, $\mu$ is a counting measure and $T$ being a regular shift operator $Tf(x)=f(x+1)$. In this case $$ \mathcal{A}_N f(x):=\frac{1}{N}\sum_{n=0}^N f(x+n^2), \qquad x\in\mathbb{Z} $$ for $f\in \ell^p(\mathbb{Z})$. It seems that the problem reduces to the study of the boundedness of the maximal function: $$ f\mapsto\sup_N \mathcal{A}_N |f|. $$ Is there a smart way of getting this boundedness from the corresponding result in the continuous setting? I tried to apply some known transference principles, but it seems to me, that the fact that there are large gaps between squares, namely $(n+1)^2-n^2\simeq n$, causes some trouble. Please excuse me if I'm overlooking something obvious here.

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2 Answers 2

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No - the sequence of squares is universally bad which was proved by Buczolich and Mauldin. I will quote from Tom Ward's review of their paper Divergent square averages, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530.

A consequence of J. Bourgain's work [Inst. Hautes Études Sci. Publ. Math. No. 69 (1989), 5–45; MR1019960] is an ergodic theorem along squares, answering earlier questions of Bellow and Furstenberg: If $(X,\mathcal B,T,\mu)$ is a measure-preserving system, then the non-conventional ergodic averages $$ \frac1{N} \sum_{n=0}^{N-1} f(T^{n^2} x) $$ converge almost everywhere for $f\in L^p$ with $p>1$. Here a comprehensive - and negative - answer is given to his question of whether the result extends to all of $L^1$. The authors show that the sequence $(n^2)$ is universally bad: for any ergodic measure-preserving system there is a function $f\in L^1$ for which the above averages fail to converge as $N\to\infty$ for $x$ in a set of positive measure.

PS The Birkhoff theorem does not apply to your ``particular case'' as it requires the presence of a finite invariant measure.

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  • $\begingroup$ Thanks for the comprehensive answer! $\endgroup$
    – Tony419
    May 12, 2020 at 17:52
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If $X=\mathbb{Z}$, $\mu$ is the counting measure, and $T$ is the shift operator given by $Tf(x)=f(x+1)$, then for all real $p\ge1$, $f\in\ell^p(\mathbb{Z})$, and $x\in\mathbb{Z}$, by Hölder's inequality, $$ |\mathcal{A}_N f(x)|\le \frac1N\,\sum_{n=0}^N|f(x+n^2)| \le\frac1N\,\|f\|_p\,(N+1)^{1-1/p}\to0 $$ and hence $\mathcal{A}_N f(x)\to0$ as $N\to\infty$.

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  • $\begingroup$ Oh, that's right! So the problem of everywhere convergence trivializes on $\mathbb{Z}$. The question about the boundedness of the related maximal function seems to remain valid in this case though and it's answered for $p>1$ in Bourgain's paper. $\endgroup$
    – Tony419
    May 12, 2020 at 18:35
  • $\begingroup$ No need for anything so sophisticated as $\ell^p\subset c_0$ :) $\endgroup$
    – R W
    May 12, 2020 at 20:37
  • $\begingroup$ @RW : Do you think this one-line application of Hölder's inequality is sophisticated? Of course, as you suggest, you can also use $\ell^p\subset c_0$, but then you will also need to use the Cesàro mean theorem math.stackexchange.com/questions/155839/… . I am not sure which way is simpler. $\endgroup$ May 13, 2020 at 0:25

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