Computing Absolutely Normal Numbers in Nearly Linear Time

Abstract: A real number $x$ is absolutely normal if, for every base $b\ge 2$, every two equally long strings of digits appear with equal asymptotic frequency in the base-$b$ expansion of $x$. This paper presents an explicit algorithm that generates the binary expansion of an absolutely normal number $x$, with the $n$th bit of $x$ appearing after $n$polylog$(n)$ computation steps. This speed is achieved by simultaneously computing and diagonalizing against a martingale that incorporates Lempel-Ziv parsing algorithms in all bases.