Pointwise Ergodic Theorems Along Fractional Powers of Primes

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.