Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit

$\lim_{X\to\infty} \pi(X)^{-1}\sum_{p\leq X} f(T^{p}x)$

exist almost everywhere? Is it constant almost everywhere? Here the sum runs over primes, and $\pi(X)$ is the prime counting function.

When $X=\mathbb{R}/\mathbb{Z}$ with Lebesgue measure, and $T:x\to x+\theta$ is an irrational rotation, the answers to these questions are "yes" and "yes", by Vinogradov.