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Abstract We establish pointwise convergence for nonconventional ergodic averages taken along $$\lfloor p^{c}\rfloor $$, where $$p$$ is a prime number and $$c\in (1,4/3)$$ on $$L^{r}$$, $$r\in (1,\infty )$$. In fact, we consider averages along more general sequences $$\lfloor h(p)\rfloor $$, where $$h$$ belongs in a wide class of functions, the so-called $$c$$-regularly varying functions. We also establish uniform multiparameter oscillation estimates for our ergodic averages and the corresponding multiparameter pointwise ergodic theorem in the spirit of Dunford and Zygmund. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $$L^{1}$$ to not seem entirely out of reach. 

Abstract We establish weak-type (1, 1) bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic setsB. 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 alongBon$$L^p$$ $L^p$,$$p>1$$ $p>1$, which is derived by establishing uniform oscillation estimates and certain vector-valued maximal estimates.