Computing Absolutely Normal Numbers in Nearly Linear Time

TL;DR

This paper introduces an explicit algorithm for generating absolutely normal numbers in binary, achieving nearly linear time complexity by combining martingale methods with Lempel-Ziv parsing across all bases.

Contribution

It presents the first explicit construction of an absolutely normal number with nearly linear time complexity in the number of bits generated.

Findings

Generates the binary expansion of an absolutely normal number in n polylog(n) steps.

Uses a novel combination of martingale strategies and Lempel-Ziv parsing.

Achieves nearly linear time complexity for normal number generation.

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.