The Discrepancy of the Lex-Least De Bruijn Sequence

TL;DR

This paper determines the discrepancy of the lexicographically-least binary de Bruijn sequence, showing it grows proportionally to 2^n log n divided by n, which answers a question posed by R. L. Graham.

Contribution

It provides a precise asymptotic estimate for the discrepancy of the lex-least binary de Bruijn sequence, a problem previously unresolved.

Findings

Discrepancy is Θ(2^n log n / n)

Quantifies imbalance in initial segments of the sequence

Addresses a question by R. L. Graham

Abstract

We answer the following question of R. L. Graham: What is the discrepancy of the lexicographically-least binary de Bruijn sequence? Here, "discrepancy" refers to the maximum (absolute) difference between the number of ones and the number of zeros in any initial segment of the sequence. We show that the answer is $\Theta(2^n \log n/n)$.