Pointwise ergodic theorems along fractional powers of primes

TL;DR

This paper proves pointwise convergence of ergodic averages along fractional powers of primes and general functions, using exponential sum estimates, with implications for harmonic analysis and number theory.

Contribution

It introduces new pointwise ergodic theorems along fractional prime powers and general functions, extending classical results and employing novel exponential sum techniques.

Findings

Established pointwise convergence for averages along fractional prime powers.

Developed uniform oscillation estimates for ergodic averages.

Connected exponential sum estimates to ergodic theory and number theory.

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.