The discrepancy of the Champernowne constant

TL;DR

This paper provides a new, elementary proof for the discrepancy of the Champernowne constant, a well-known normal number, improving understanding of its distribution properties without relying on exponential sums.

Contribution

It offers a discrete, elementary proof of the discrepancy of the Champernowne constant, previously studied using exponential sums.

Findings

Elementary proof of discrepancy for Champernowne constant

Discrepancy quantifies how well the number approaches normality

Supports the understanding of digit distribution in normal numbers

Abstract

A number is normal in base $b$ if, in its base $b$ expansion, all blocks of digits of equal length have the same asymptotic frequency. The rate at which a number approaches normality is quantified by the classical notion of discrepancy, which indicates how far the scaling of the number by powers of $b$ is from being equidistributed modulo 1. This rate is known as the discrepancy of a normal number. The Champernowne constant $c_{10} = 0.12345678910111213141516\ldots$ is the most well-known example of a normal number. In 1986, Schiffer provided the discrepancy of numbers in a family that includes the Champernowne constant. His proof relies on exponential sums. Here, we present a discrete and elementary proof specifically for the discrepancy of the Champernowne constant.