Vol. 12, No. 6, 2019

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On a theorem of Besicovitch and a problem in ergodic theory

Ethan Gwaltney, Paul Hagelstein, Daniel Herden and Brian King

Vol. 12 (2019), No. 6, 961–968
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.

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Keywords
ergodic theory, maximal operators, Diophantine approximation
Mathematical Subject Classification 2010
Primary: 37A30, 42B25
Secondary: 11J20