Key points are not available for this paper at this time.
Dimension-free bounds will be provided in maximal and r-variational inequalities on ᵖ (Zᵈ) corresponding to the discrete Hardy-Littlewood averaging operators defined over the cubes in Zᵈ. We will also construct an example of a symmetric convex body in Zᵈ for which maximal dimension-free bounds fail on ᵖ (Zᵈ) for all p (1, ). Finally, some applications in ergodic theory will be discussed.
Bourgain et al. (Tue,) studied this question.