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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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    $\begingroup$ You seem to be using $X$ for both the measure space and limit of summation (I don't have the reputation to edit). $\endgroup$
    – Noah Stein
    Commented Oct 20, 2010 at 10:40
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    $\begingroup$ 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. $\endgroup$
    – Anthony Quas
    Commented Jul 19, 2014 at 3:45

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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.

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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

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