TITLE: A non-standard ergodic limit
QUESTION [7 upvotes]: 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.

REPLY [3 votes]: Patrick LaVictoire shows that the answer is negative if you ask for all of $L^1$.
His paper is Universally L^1-Bad Arithmetic Sequences (to appear in Journal d'Analyse Mathematique). The paper extends results of Buczolich and Mauldin (who showed a negative result if you sum along the squares). The paper can be found online at  http://arxiv.org/abs/0905.3865