Uniform distribution and algorithmic randomness

Journal of Symbolic Logic 78 (1):334-344 (2013)
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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

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Jeremy Avigad
Carnegie Mellon University

Citations of this work

Computable randomness and betting for computable probability spaces.Jason Rute - 2016 - Mathematical Logic Quarterly 62 (4-5):335-366.
The Equivalence of Definitions of Algorithmic Randomness.Christopher Porter - 2021 - Philosophia Mathematica 29 (2):153–194.

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References found in this work

Hermann WEYL.[author unknown] - 1957 - Revue Philosophique de la France Et de l'Etranger 147:133-133.

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