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.



Analog of the Birkhoff's ergodic theorem for the sequence of squares
No - the sequence of squares is **universally bad** which was proved by Buczolich and Mauldin. I will quote from Tom Ward's [review](https://mathscinet.ams.org/mathscinet-getitem?mr=2680392) 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. 


