Ergodic theorem and algorithmic randomness

TL;DR

This paper presents a constructive proof of Birkhoff's ergodic theorem, emphasizing the combinatorial aspects and raising questions about the effective ergodic theorem's broader implications.

Contribution

It provides a constructive version of the ergodic theorem with explicit combinatorial foundations and discusses related open questions.

Findings

Constructive proof of Birkhoff's ergodic theorem

Explicit combinatorial statement underlying the proof

Open questions on effective ergodic theorem

Abstract

We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the effective ergodic theorem in general).