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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.

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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.

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    $\begingroup$ Thanks! I couldn't find that paper online, but following citations I found what I was looking for in Butler, S. V.(1-IL); Rosenblatt, J. M.(1-IL) Moving averages. Colloq. Math. 113 (2008), no. 2, 251–266. $\endgroup$
    – Joel Moreira
    Commented Jan 15, 2013 at 17:52

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