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

In 1935, Besicovitch proved a remarkable theorem indicating that an integrable function f on 2 is strongly differentiable if and only if its associated strong maximal function MSf 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 Tf is finite a.e. Of particular relevance to this discussion will be recent results in the field of inhomogeneous diophantine approximation.

ergodic theory, maximal operators, Diophantine approximation
Mathematical Subject Classification 2010
Primary: 37A30, 42B25
Secondary: 11J20
Received: 26 June 2018
Revised: 26 February 2019
Accepted: 21 March 2019
Published: 3 August 2019

Communicated by Kenneth S. Berenhaut
Ethan Gwaltney
Department of Mathematics
Rice University
Houston, TX
United States
Paul Hagelstein
Department of Mathematics
Baylor University
Waco, TX
United States
Daniel Herden
Department of Mathematics
Baylor University
Waco, TX
United States
Brian King
Department of Statistics
Rice University
Houston, TX
United States