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.