# A non-standard ergodic limit

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.

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You seem to be using $X$ for both the measure space and limit of summation (I don't have the reputation to edit). – Noah Stein Oct 20 '10 at 10:40
I don't think it's right that the answer is yes for an irrational rotation and the function $f$ is taken to be $L^1$. There is a theorem called the transference principle (I think of it as a photocopying machine) allowing you to transfer a counterexample that diverges for one measure-preserving transformation to any other aperiodic measure-preserving transformation. – Anthony Quas Jul 19 '14 at 3:45

Here is a partial answer: Mate Wierdl proved that the limit exists almost everywhere if $f \in L^r (\mu)$ for some $r>1$. See "Pointwise Ergodic Theorem along the Prime Numbers".
Also, there is a recent article by Trevor Wooley and Tamar Ziegler ("Multiple Recurrence and Convergence along the Primes") which proves $L^2$ convergence and multiple recurrence for more complicated ergodic averages of this kind.
Patrick LaVictoire shows that the answer is negative if you ask for all of $L^1$.