Bourgain has a nice paper, Pointwise ergodic theorems for arithmetic sets, (subsequently extended in various directions by other authors including my co-author, Máté Wierdl) on proving a version of the Birkhoff ergodic theorem where one averages along the sequence of square numbers, rather than the sequence of integers. That is: for a measure-preserving system $T\colon X\to X$ and a (square-integrable) function $f$, one considers convergence of the averages $$ \frac{1}{N}\sum_{j=1}^N f(T^{j^2}x). $$
Bourgain proves using analytic number theory techniques involving exponential sums that there is convergence almost everywhere, just as in the regular Birkhoff ergodic theorem (although not to the integral as in the regular case and the convergence fails for a typical $L^1$ function, unlike the regular case).
