  We 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 $L^1$. This contributes yet another counterexample for the conjecture
of Rosenblatt and Wierdl from 1991 asserting the failure of pointwise
convergence on $L^1$ 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 $L^p$, $p>1$, which is
derived by establishing uniform oscillation estimates and certain vector-valued
maximal estimates.
