## Journal of Symbolic Logic

### Uniform distribution and algorithmic randomness

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.jsl/1358951118

Digital Object Identifier
doi:10.2178/jsl.7801230

Mathematical Reviews number (MathSciNet)
MR3087080

Zentralblatt MATH identifier
1275.03133

Subjects
Primary: 03D32, 11K06

#### Citation

Avigad, Jeremy. Uniform distribution and algorithmic randomness. J. Symbolic Logic 78 (2013), no. 1, 334--344. doi:10.2178/jsl.7801230. https://projecteuclid.org/euclid.jsl/1358951118