Integer sequences with big gaps and the pointwise ergodic theorem

Abstract

First, we show that there exists a sequence $(a_n)$ of integers which is a good averaging sequence in $L^2$ for the pointwise ergodic theorem and satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-1-\epsilon}} $$ for $n>n(\epsilon)$. This should be contrasted with an earlier result of ours which says that if a sequence $(a_n)$ of integers (or real numbers) satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-\frac{1}{2}+\epsilon}} $$ for some positive $\epsilon$, then it is a bad averaging sequence in $L^2$ for the pointwise ergodic theorem.Another result of the paper says that if we select each integer $n$ with probability $1/n$ into a random sequence, then, with probability 1, the random sequence is a bad averaging sequence for the mean ergodic theorem. This result should be contrasted with Bourgain's result which says that if we select each integer $n$ with probability $\sigma_n$ into a random sequence, where the sequence $(\sigma_n)$ is decreasing and satisfies $$ \lim_{t\to\infty}\frac{\sum_{n\le t}\sigma_n}{\log t}=\infty, $$ then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.