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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.
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