Suppose $\{x_n\}_{n \geq 1}$ is a real valued sequence such that for every $a, r \in \mathbb Z_+$, we have that
$$\lim_{N \to \infty} \frac{1}{N} \sum_{i = 0}^{N-1} x_{a + ir}$$
exists and equals $L$ for some $L \in \mathbb R$.
Question: Is it true that $x_n \to L$ along a subsequence of natural density $1$? That is, does there exist a subsequence $n_k$ with
$$\lim_{M \to \infty} \frac{|\{k \in \mathbb N \, | \, n_k \leq M\}|}{M} = 1$$
such that $x_{n_k} \to L$?
Best Answer
No, just take $x_n = e^{2\pi i \alpha}$ for your favorite irrational $\alpha$. Then for any $a,r$, $x_{an+r} = e^{2\pi i r\alpha} (e^{2\pi i a\alpha)})^n$, and so $\sum_{n=0}^{N-1} x_{an+r} = e^{2\pi i r\alpha} \frac{1 - (e^{2\pi i a\alpha})^N}{1 - e^{2\pi i a\alpha}}$, with modulus bounded from above by a constant. So all of your Cesaro averages converge to 0, but the sequence itself always has modulus 1 and so can't converge to 0 along any subsequence.
In fact, more generally, for any totally ergodic system $(X, T, \mu)$ and any $f \in L^2$, almost every point $x$ will induce a sequence $x_n = f(T^n x)$ with all Cesaro averages along infinite arithmetic progressions converging to $\int f d\mu$, but very rarely would this sequence itself converge.