Computable Absolutely Normal Numbers and Discrepancies

TL;DR

This paper examines algorithms for generating absolutely normal numbers, focusing on their convergence quality and complexity trade-offs, and compares classical methods by Sierpinski, Turing, and Schmidt.

Contribution

It provides a comparative analysis of explicit algorithms for normal numbers, highlighting the trade-offs between computational complexity and convergence speed.

Findings

Algorithms exhibit a trade-off between complexity and convergence rate.

Explicit variants of classical algorithms are analyzed in terms of discrepancy.

The study offers insights into the efficiency of different normal number generation methods.

Abstract

We analyze algorithms that output absolutely normal numbers digit-by-digit with respect to quality of convergence to normality of the output, measured by the discrepancy. We consider explicit variants of algorithms by Sierpinski, by Turing and an adaption of constructive work on normal numbers by Schmidt. There seems to be a trade-off between the complexity of the algorithm and the speed of convergence to normality of the output.