Nilsystems and ergodic averages along primes

TL;DR

This paper proves that ergodic averages along primes converge everywhere for nilsystems and continuous functions, extending previous almost everywhere results using anti-correlation properties of the von Mangoldt function.

Contribution

It establishes everywhere convergence of ergodic averages along primes for nilsystems, leveraging Green and Tao's anti-correlation results, which is a novel extension of prior almost everywhere convergence results.

Findings

Ergodic averages along primes converge everywhere for nilsystems.

Utilizes Green and Tao's anti-correlation results for the von Mangoldt function.

Extends convergence results from almost everywhere to everywhere for specific systems.

Abstract

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao we observe everywhere convergence of such averages for nilsystems and continuous functions.