On a theorem of Besicovitch and a problem in ergodic theory

Gwaltney, Ethan; Hagelstein, Paul; Herden, Daniel; King, Brian

doi:10.2140/involve.2019.12.961

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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

DOI: 10.2140/involve.2019.12.961

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 $T^{*} f$ 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

Milestones

Received: 26 June 2018

Revised: 26 February 2019

Accepted: 21 March 2019

Published: 3 August 2019

Communicated by Kenneth S. Berenhaut

Authors

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

Vol. 12, No. 6, 2019