Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about sequences of finite subsets $(F_n)_{n=1}^\infty$ of $\mathbb{Z}^2$ which are $L^p$-good for the pointwise ergodic theorem in the sense that for every $f \in L^p(X,\mu)$ the limit

$$ \lim_{n \to \infty} \frac{1}{|F_n|} \sum_{(m_1,m_2) \in F_n} f(S^{m_1} T^{m_2} x) $$

exists for almost every $x \in X$. If $(F_n)_{n=1}^\infty$ forms a tempered Folner sequence, it is well known that $(F_n)_{n=1}^\infty$ is $L^1$-good. I would like to hear what is known about analogs of Bourgain's subsequential ergodic theorems - i.e. the case where the $F_n$ are quite sparse. Specific examples of interest are $$F_n = \{(m_1^2,m_2^2): 0 \leq m_1,m_2 \leq n \},$$ more generally $$F_n = \{(p(m_1),q(m_2)): 0 \leq m_1,m_2 \leq n \}$$ for two polynomials $p$ and $q$ with integer coefficients, or $$ F_n = \{ (p_1,p_2): p_1 \mbox{ and } p_2 \mbox{ are among the first } n \mbox{ primes } \}.$$

There doesn't seem to be anything about this in Nevo's extensive survey 'Pointwise Ergodic Theorems for Actions of Groups' from the Handbook of Dynamical Systems.


For polynomials in several variables pointwise convergence has been recently proved by Mirek and Trojan: http://arxiv.org/abs/1405.5566


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.