Integer sequences with big gaps and the pointwise ergodic theorem

First, we show that there exists a sequence (a,) of integers which is a goo d averaging sequence in L-2 for the pointwise ergodic theorem and satisfies a(n)+1/a(n) > e((log n)-1-epsilon) for n > n (epsilon). This should be contrasted with an earlier result of ou rs which says that if a sequence (a(n)) of integers (or real numbers) satis fies a(n)+1/a(n) >e((log n)-1/2+epsilon) for some positive epsilon, then it is a bad averaging sequence in L-2 for t he pointwise ergodic theorem. Another result of the paper says that if we select each integer n with prob ability 1/n into a random sequence, then, with probability 1, the random se quence is a bad averaging sequence for the mean ergodic theorem. This resul t 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 [GRAPHICS] then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.