Random sequences and pointwise convergence of multiple ergodic averages
We prove pointwise convergence, as N → ∞, for the multiple ergodic averages (1/N) Σn=1N f(Tnx) · g(San x), where T and S are commuting measure preserving transformations, and an is a random version of the sequence [nc] for some appropriate c > 1. We also prove similar mean convergence results for averages of the form (1/N) Σ n=1N (Tan x) · g(S an x), as well as pointwise results when T and S are powers of the same transformations. The deterministic versions of these results, where one replaces an with [nc], remain open, and we hope that our method will indicate a fruitful way to approach these problems as well.
Indiana University Mathematics Journal
Frantzikinakis, N., Lesigne, E., & Wierdl, M. (2012). Random sequences and pointwise convergence of multiple ergodic averages. Indiana University Mathematics Journal, 61 (2), 585-617. https://doi.org/10.1512/iumj.2012.61.4571