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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.