Perturbation of Sparse Ergodic Averages

TL;DR

This paper constructs sparse set sequences in groups where multidimensional averages fail to converge pointwise in some L^p spaces but succeed in others, revealing nuanced behaviors of ergodic averages.

Contribution

It introduces new examples of sparse set sequences in groups demonstrating varied convergence properties of ergodic averages across different L^p spaces.

Findings

Pointwise convergence fails in certain L^p spaces for the constructed sets.

Convergence occurs in other L^p spaces, showing a nuanced dependence on the space.

The method extends perturbation techniques to multidimensional and group settings.

Abstract

We provide examples of a nested sequences of sets {S_n}, suitably sparse, residing in a group G, for which multidimensional averages fail converge pointwise for f in certain L^p spaces, but do converge in others, for any free group action T. Our construction involves the method of perturbation pioneered by A. Bellow and applied in the integer cases by K. Reinhold and M. Wierdl.