Abstract
AbstractWe establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets B. As a corollary we obtain the corresponding pointwise convergence result on $$L^1$$
L
1
. This contributes yet another counterexample for the conjecture of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise convergence on $$L^1$$
L
1
of ergodic averages along arithmetic sets with zero Banach density. The second main result is a multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund along B on $$L^p$$
L
p
, $$p>1$$
p
>
1
, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.
Publisher
Springer Science and Business Media LLC
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