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Uniform distribution and algorithmic randomness

Published online by Cambridge University Press:  12 March 2014

Jeremy Avigad
Jeremy Avigad*
Affiliation:
Departments of Philosophy and Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA, E-mail: avigad@cmu.edu
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Abstract

A seminal theorem due to Weyl [14] states that if (an) is any sequence of distinct integers, then, for almost every x ∈ ℝ, the sequence (anx) is uniformly distributed modulo one. In particular, for almost every x in the unit interval, the sequence (anx) is uniformly distributed modulo one for every computable sequence (an) of distinct integers. Call such an x UD random. Here it is shown that every Schnorr random real is UD random, but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 334 - 344
Copyright
Copyright © Association for Symbolic Logic 2013

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