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USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS

Published online by Cambridge University Press:  10 October 2016

KENSHI MIYABE ,
ANDRÉ NIES  and
JING ZHANG
KENSHI MIYABE
Affiliation:
DEPARTMENT OF MATHEMATICS SCHOOL OF SCIENCE AND TECHNOLOGY MEIJI UNIVERSITY, JAPAN E-mail: kenshi.miyabe@gmail.com
ANDRÉ NIES
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF AUCKLAND, NEW ZEALAND E-mail: andre@cs.auckland.ac.nz
JING ZHANG
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY, USA E-mail: jingzhang@cmu.edu
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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. (1) all effectively closed classes containing z have density 1 at z.

  2. (2) all nondecreasing functions with uniformly left-c.e. increments are differentiable at z.

  3. (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 ${\rm{\Pi }}_n^0$ and ${\rm{\Sigma }}_1^1$ classes at a real.

Keywords

Lebesgue densityrandomnessergodic theoryalmost-everywhere theorem

Information

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 3 , September 2016 , pp. 305 - 331
Copyright
Copyright © The Association for Symbolic Logic 2016 

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