On a theorem of Besicovitch and a problem in ergodic theory

Abstract

In 1935, Besicovitch proved a remarkable theorem indicating that an integrable function $f$ on ${ℝ}^2$ is strongly differentiable if and only if its associated strong maximal function $M_{S}f$ is finite a.e. We consider analogues of Besicovitch’s result in the context of ergodic theory, in particular discussing the problem of whether or not, given a (not necessarily integrable) measurable function $f$ on a nonatomic probability space and a measure-preserving transformation $T$ on that space, the ergodic averages of $f$ with respect to $T$ converge a.e. if and only if the associated ergodic maximal function $T^{\ast }f$ is finite a.e. Of particular relevance to this discussion will be recent results in the field of inhomogeneous diophantine approximation.