## Journal of Symbolic Logic

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

### 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

**Permanent link to this document**

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