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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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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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