Uniform distribution and algorithmic randomness

TL;DR

This paper explores the relationship between various notions of algorithmic randomness and uniform distribution modulo one, establishing that Schnorr randomness implies UD randomness, while Kurtz randomness does not, and analyzing the effective null sets involved.

Contribution

It demonstrates that all Schnorr random reals are UD random and identifies the limitations of Kurtz randomness in this context, refining the understanding of randomness notions.

Findings

Every Schnorr random real is UD random.

There exist Kurtz random reals that are not UD random.

Weyl's theorem holds relative to certain effectively closed null sets.

Abstract

A seminal theorem due to Weyl states that if (a_n) is any sequence of distinct integers, then, for almost every real number x, 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.