Pointwise Ergodic Theorems Along Fractional Powers of Primes

We establish pointwise convergence for nonconventional ergodic averages taken along pᶜ, where p is a prime number and c∈ (1,4/3) on Lʳ, r∈ (1,∞ ). In fact, we consider averages along more general sequences h(p), 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 L¹ to not seem entirely out of reach.