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.


1 Answer 1


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


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