On the construction of absolutely normal numbers

TL;DR

This paper constructs an absolutely normal number with a discrepancy order of O(N^{-1/2}) for all integer bases, which is smaller than typical for almost all numbers, demonstrating a new level of uniform distribution.

Contribution

It provides the first known explicit construction of an absolutely normal number with uniformly small discrepancy across all bases.

Findings

Constructed an absolutely normal number with discrepancy O(N^{-1/2})

Demonstrated existence of such numbers with smaller discrepancy than typical

Advances understanding of distribution properties of normal numbers

Abstract

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order $\mathcal{O}(N^{-1/2})$. This is below the order of discrepancy which holds for almost all real numbers. Even the existence of absolutely normal numbers having a discrepancy of such a small asymptotic order was not known before.