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March 2013 Uniform distribution and algorithmic randomness
Jeremy Avigad
J. Symbolic Logic 78(1): 334-344 (March 2013). DOI: 10.2178/jsl.7801230

Abstract

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ 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.

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Jeremy Avigad. "Uniform distribution and algorithmic randomness." J. Symbolic Logic 78 (1) 334 - 344, March 2013. https://doi.org/10.2178/jsl.7801230

Information

Published: March 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1275.03133
MathSciNet: MR3087080
Digital Object Identifier: 10.2178/jsl.7801230

Subjects:
Primary: 03D32, 11K06

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 1 • March 2013
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