Pointwise convergence of ergodic averages along cubes

Abstract

Let (X, B, µ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2 k − 1 bounded and measurable functions for all k. We show that this result can be derived from estimates about bounded sequences of real numbers and apply these estimates to establish the pointwise convergence of some weighted ergodic averages and ergodic averages along cubes for not necessarily commuting measure preserving transformations.