Randomness and uniform distribution modulo one

TL;DR

This paper explores the relationship between Martin-Löf and Schnorr randomness for real numbers through the lens of uniform distribution of sequences, extending classical results to broader classes of sequences.

Contribution

It introduces new characterizations of Martin-Löf and Schnorr randomness using uniform distribution, extending previous results to Koksma sequences and computably enumerable sets.

Findings

Necessary condition for Schnorr randomness via classical uniform distribution.

Sufficient condition for Martin-Löf and Schnorr randomness using computably enumerable open sets.

Extension of Avigad's result to Koksma sequences.

Abstract

We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform distribution of sequences. This extends the result proved by Avigad for sequences of linear functions with integer coefficients to the wider classical class of Koksma sequences of functions. And, by requiring equidistribution with respect to every computably enumerable open set (respectively, computably enumerable open set with computable measure) in the unit interval, we give a sufficient condition for Martin-L\"of (respectively Schnorr) randomness.