The behavior of time averages taken along subsequences of integers is the central question of subsequence ergodic theory. The existence of transference principles enables us to talk about convergence of operators, in the mean or pointwise, in a universal sense. For functions in $L^1$, however, the question of whether ergodic averages taken along subsequences converge pointwise is difficult. There are only a few examples of density zero sequences which are known to allow for pointwise convergence of averages for any function in $L^1$ and all aperiodic measure-preserving transformations.
We will discuss a few of the methods used to show that these sequences lead to pointwise convergence of the averages in the course of establishing related examples of good sequences of sets for $Z^d$ actions.
Joint work with P. LaVictoire (University of Wisconsin, Madison) and J. Rosenblatt (University of Illinois at Urbana-Champaign).