In this paper, we deal with the behavior of ergodic averages when it is sampled along a sparse sequence. It has been known for over twenty years that if we sample the ergodic averages along a lacunary sequence then in any aperiodic dynamical system we can find an indicator function for which the averages fail to converge almost everywhere. In this paper, we strengthen this result by showing that certain sublacunary sequences exhibit an extreme type of non-convergence behavior which is known as the strong sweeping out property. We also deduce similar conclusions for certain sparse random sequences.

The first author is supported by the National Science Foundation under grant number DMS-1855745.