# On absolutely normal numbers and their discrepancy estimate

## TL;DR

This paper constructs a specific absolutely normal number in base 2 whose digit sequence exhibits discrepancy properties similar to almost all real numbers across all integer bases, advancing understanding of normality and distribution.

## Contribution

It provides an explicit construction of an absolutely normal number with discrepancy behavior matching that of typical real numbers across all bases.

## Key findings

- Constructed an explicit absolutely normal number in base 2.
- Demonstrated the discrepancy of the sequence matches typical behavior.
- Showed uniform distribution properties hold across all integer bases.

## Abstract

We construct the base $2$ expansion of an absolutely normal real number $x$ so that, for every integer $b$ greater than or equal to $2$, the discrepancy modulo $1$ of the sequence $(b^0 x, b^1 x, b^2 x , \ldots)$ is essentially the same as that realized by almost all real numbers.