A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points

TL;DR

This paper develops an effective, constructive version of Birkhoff's ergodic theorem for Martin-Löf random points, extending previous results and providing a more general and stronger formulation.

Contribution

It introduces a new effective version of Birkhoff's ergodic theorem for Martin-Löf random points, improving upon prior work in the field.

Findings

Proves an effective version of a weak form of Birkhoff's ergodic theorem.

Generalizes results to a very broad effective ergodic theorem.

Enhances previous theorems by Yugin, Nandakumar, Hoyrup, and Rojas.

Abstract

A theorem of Ku\v{c}era states that given a Martin-L\"of random infinite binary sequence {\omega} and an effectively open set A of measure less than 1, some tail of {\omega} is not in A. We first prove several results in the same spirit and generalize them via an effective version of a weak form of Birkhoff's ergodic theorem. We then use this result to get a stronger form of it, namely a very general effective version of Birkhoff's ergodic theorem, which improves all the results previously obtained in this direction, in particular those of V'Yugin, Nandakumar and Hoyrup, Rojas.