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.