Journal of Symbolic Logic

Uniform distribution and algorithmic randomness

Jeremy Avigad

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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.

Article information

J. Symbolic Logic, Volume 78, Issue 1 (2013), 334-344.

First available in Project Euclid: 23 January 2013

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03D32, 11K06


Avigad, Jeremy. Uniform distribution and algorithmic randomness. J. Symbolic Logic 78 (2013), no. 1, 334--344. doi:10.2178/jsl.7801230.

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