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 on the references (including the first such result by Akcoglu and del Junco in 1975) the Folner sequence which fails is actually $G_N=\{N,N+1,...,N+\lfloor N\rfloor\}$. 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.