Unmotivated ergodic averages

TL;DR

This paper extends classical ergodic theorems to weighted averages indexed by primes, where weights are trace functions from algebraic geometry, covering mean, pointwise, and topological cases.

Contribution

It introduces new ergodic theorems for prime-indexed averages with algebraic geometric weights, expanding the scope of ergodic theory.

Findings

Extended mean-ergodic theorem for prime weights

Established pointwise ergodic theorem in this setting

Provided results in the topological ergodic context

Abstract

We consider weighted ergodic averages indexed by primes, where the weight depends on the prime, and is a "trace function" coming from algebraic geometry. We obtain extensions the classical mean-ergodic and pointwise ergodic theorems, as well as some result in the topological setting, and raise some further problems.