It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences. 
The counter-example usually given is the Folner sequence $F\_N=\{N^2,N^2+1...,N^2+N\}$, however I've only seen it referenced to the first such result by Akcoglu and del Junco in 1975, where the Folner sequence which fails is actually $G\_N=\{N,N+1,...,N+\lfloor \sqrt{N}\rfloor\}$.
Since $\{F\_N\}$ is just a (quite sparse) subsequence of $\{G\_N\}$ it could happen that the ergodic averages converge pointwise along $\{F\_N\}$ even if they don't along $\{G\_N\}$.


Moreover, Lindenstrauss showed that every Folner sequence has a subsequence along which convergence holds.


So I'm just curious if there is a published proof (or if it is easy and I'm just overlooking something) that the sequence $\{F\_N\}$ is not good for pointwise ergodic theorem.



Failure of the Pointwise Ergodic Theorem
Bellow, Alexandra(1-NW); Jones, Roger(1-DPL); Rosenblatt, Joseph(1-OHS)
Convergence for moving averages.
Ergodic Theory Dynam. Systems 10 (1990), no. 1, 43–62. 


