A constructive version of Birkhoffs ergodic theorem for Martin-L"of random points

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.

💡 Research Summary

The paper establishes a fully constructive version of Birkhoff’s ergodic theorem that applies specifically to Martin‑Löf random points. It begins by revisiting Kučera’s classical result, which states that for any Martin‑Löf random infinite binary sequence ω and any effectively open set A with measure μ(A)<1, some tail of ω eventually leaves A. The authors first give a streamlined proof of Kučera’s theorem and then extend its spirit to a broader class of statements concerning effectively open (Σ⁰₁) sets and random tails.

The core technical contribution is an effective weak form of Birkhoff’s ergodic theorem. The setting is a computable probability space (X, μ) equipped with a computable, μ‑preserving transformation T. For any L¹‑computable observable f, the time averages \