# Integer sequences with big gaps and the pointwise ergodic theorem

## Abstract

First, we show that there exists a sequence (an) of integers which is a good averaging sequence in L2 for the pointwise ergodic theorem and satisfies an+1/an > e(log n)-1-∈ for n > n(∈). This should be contrasted with an earlier result of ours which says that if a sequence (an) of integers (or real numbers) satisfies an+1/an > e(log n)-1/2+∈ for some positive ∈, then it is a bad averaging sequence in L2 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 σn into a random sequence, where the sequence (σn) is decreasing and satisfies (formula presented) then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.

## Publication Title

Ergodic Theory and Dynamical Systems

## Recommended Citation

Jones, R., Lacey, M., & Wierdl, M.
(1999). Integer sequences with big gaps and the pointwise ergodic theorem.* Ergodic Theory and Dynamical Systems**, 19* (5), 1295-1308.
https://doi.org/10.1017/S0143385799146819