Date of Original Version

7-2012

Type

Article

Rights Management

Copyright © Association for Symbolic Logic

Abstract or Description

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

DOI

http://dx.doi.org/10.2178/jsl.7801230

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Published In

The Journal of Symbolic Logic, 78, 1, 334-344.