Abelian-normal decimal expansions

TL;DR

This paper introduces the concept of abelian-normal numbers, constructs a Champernowne-like constant that is abelian-normal, and explores the properties of digit rearrangements and abelian complexity in normality.

Contribution

It defines abelian-normal numbers, constructs a non-normal abelian-normal constant, and analyzes the impact of rearrangements on normality within this new framework.

Findings

Constructed a non-normal abelian-normal constant $D_{10}$

Proved $D_{10}$ is abelian-normal with respect to a weighting function

Presented open problems related to abelian-normal constants

Abstract

Many research works have concerned normality-preserving selection rules and operations on the sequence of digits of a given normal number that maintain or violate normality. This leads us to introduce rearrangement operations on finite subwords appearing within the digit expansions of normal numbers, and this is inspired by the concept of an abelian complexity function in the field of combinatorics on words. We introduce the concept of an abelian-normal number, with respect to a given base and a given weighting/counting function on subwords, by analogy with normal numbers and with the use of the equivalence classes associated with abelian complexity functions. We then construct a non-normal analogue $D_{10}$ of Champernowne's constant $C_{10}$ and prove that $D_{10}$ is abelian-normal with respect to a given weighting function. We conclude with two open problems concerning our Champernowne-like constant $D_{10}$.