Pointwise ergodic theorem along primes of the form $x^2 + ny^2$

TL;DR

This paper proves pointwise convergence of ergodic averages along primes of the form x^2 + ny^2 using the Hardy-Littlewood circle method, and shows the limits of such convergence for L^1 functions.

Contribution

It introduces novel major and minor arc estimates for primes of the form x^2 + ny^2 in the context of ergodic theory, extending Bourgain's work.

Findings

Established pointwise convergence for a class of polynomial primes.

Demonstrated the non-extendability of convergence results to L^1 functions.

Developed new estimates for prime ideals in the Hardy-Littlewood circle method.

Abstract

This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form $x^2 + ny^2$. Following the influential paper of Bourgain \cite{bourgain1989pointwise}, we employ the Hardy-Littlewood circle method where major arc and minor arc estimates for the set of prime ideals constitute the main novelty of the paper. We also prove that our convergence results cannot be extended to class of $L^1$ functions.