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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
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Abstract |
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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. |
Keywords
ergodic theory, maximal operators, Diophantine
approximation
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Mathematical Subject Classification 2010
Primary: 37A30, 42B25
Secondary: 11J20
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Milestones
Received: 26 June 2018
Revised: 26 February 2019
Accepted: 21 March 2019
Published: 3 August 2019
Communicated by Kenneth S. Berenhaut
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Authors |
| Ethan Gwaltney | |
| Department of Mathematics Rice University Houston, TX United States |
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| Paul Hagelstein | |
| Department of Mathematics Baylor University Waco, TX United States |
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| Daniel Herden | |
| Department of Mathematics Baylor University Waco, TX United States |
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| Brian King | |
| Department of Statistics Rice University Houston, TX United States |
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