Journal of Symbolic Logic

Uniform distribution and algorithmic randomness

Jeremy Avigad

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation