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arXiv · 1007.5249

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

Laurent Bienvenu·Adam Day·Mathieu Hoyrup·Ilya Mezhirov·Alexander Shen
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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.

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Laurent Bienvenu, Adam Day, Mathieu Hoyrup, Ilya Mezhirov, Alexander Shen. 2010-07-29. A constructive version of Birkhoff's ergodic theorem for Martin-L\"of random points. https://arxiv.org/abs/1007.5249

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