Completely uniformly distributed sequences based on de Bruijn sequences

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 we give an elementary proof showing that the generated sequence is also completely uniformly distributed. In addition, we present an alternative proof of the same result based on Weyl's criterion.