Using almost-everywhere theorems from analysis to study randomness

Kenshi Miyabe, André Nies, Jing Zhang

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10 Scopus citations

Abstract

We study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin-Löf (ML) randomness. We establish several equivalences. Given a ML-random real z, the additional randomness strengths needed for the following are equivalent. (1) all effectively closed classes containing z have density 1 at z. (2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z. (3) z is a Lebesgue point of each lower semicomputable integrable function. We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff's pointwise ergodic theorem. Lastly, we study randomness notions related to density of and classes at a real.

Original languageEnglish
Pages (from-to)305-331
Number of pages27
JournalBulletin of Symbolic Logic
Volume22
Issue number3
DOIs
StatePublished - 1 Sep 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 The Association for Symbolic Logic.

Funding

Miyabe was supported by JSPS. Nies was supported by the Marsden fund of New Zealand. Zhang was supported by a University of Auckland summer scholarship.

FundersFunder number
Marsden fund of New Zealand
University of Auckland
Japan Society for the Promotion of Science

    Keywords

    • Lebesgue density
    • almost-everywhere theorem
    • ergodic theory
    • randomness

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