Abstract
In 1935, Besicovitch proved a remarkable theorem indicating that an integrable function on is strongly differentiable if and only if its associated strong maximal function 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 on a nonatomic probability space and a measure-preserving transformation on that space, the ergodic averages of with respect to converge a.e. if and only if the associated ergodic maximal function is finite a.e. Of particular relevance to this discussion will be recent results in the field of inhomogeneous diophantine approximation.
Citation
Ethan Gwaltney. Paul Hagelstein. Daniel Herden. Brian King. "On a theorem of Besicovitch and a problem in ergodic theory." Involve 12 (6) 961 - 968, 2019. https://doi.org/10.2140/involve.2019.12.961
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