Completely uniformly distributed sequences based on de Bruijn sequences

TL;DR

This paper introduces a simplified construction of completely uniformly distributed sequences based on de Bruijn sequences with linearly increasing alphabet sizes, providing elementary and alternative proofs of their uniform distribution.

Contribution

It presents a new, simpler method for constructing completely uniformly distributed sequences using linearly increasing alphabet sizes, with rigorous proofs of uniformity.

Findings

The sequence is completely uniformly distributed.

Elementary proof of uniform distribution.

Alternative proof using Weyl's criterion.

Abstract

We study a construction published by Donald Knuth in 1965 yielding a completely uniformly distributed sequence of real numbers. Knuth's work is based on de Bruijn sequences of increasing orders and alphabet sizes, which grow exponentially in each of the successive segments composing the generated sequence. In this work we present a similar albeit simpler construction using linearly increasing alphabet sizes, and give an elementary proof showing that the sequence it yields is also completely uniformly distributed. In addition, we present an alternative proof of the same result based on Weyl's criterion.