Completely uniformly distributed sequences based on de Bruijn sequences

09/24/2019

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by   Emilio Almansi, et al.

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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.