M. Levin's construction of absolutely normal numbers with very low discrepancy

TL;DR

This paper analyzes Levin's 1979 construction of absolutely normal numbers, demonstrating its computability and complexity, and explores variants that balance discrepancy and computational effort.

Contribution

It provides a detailed analysis of Levin's construction in terms of computability and complexity, and proposes variants to optimize these aspects.

Findings

The construction yields a computable absolutely normal number.

Approximation error decreases exponentially with n.

Computational complexity is double exponential in n.

Abstract

Among the currently known constructions of absolutely normal numbers, the one given by Mordechay Levin in 1979 achieves the lowest discrepancy bound. In this work we analyze this construction in terms of computability and computational complexity. We show that, under basic assumptions, it yields a computable real number. The construction does not give the digits of the fractional expansion explicitly, but it gives a sequence of increasing approximations whose limit is the announced absolutely normal number. The $n$-th approximation has an error less than $2^{2^{-n}}$. To obtain the $n$-th approximation the construction requires, in the worst case, a number of mathematical operations that is double exponential in $n$. We consider variants on the construction that reduce the computational complexity at the expense of an increment in discrepancy.