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Abelian-normal decimal expansions

John M. Campbell

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 $D_{10}$.

Abelian-normal decimal expansions

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 of Champernowne's constant and prove that is abelian-normal with respect to a given weighting function. We conclude with two open problems concerning our Champernowne-like constant .
Paper Structure (6 sections, 2 theorems, 31 equations)

This paper contains 6 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Abelian normality
  3. An abelian-normal, non-normal number
  4. Mixed subwords
  5. A weighting function for a Champernowne-like constant
  6. Conclusion

Key Result

Lemma 1

For a word $v$ appearing in $C_{10}$ with at least one non-binary digit, we have that the relations among hold.

Theorems & Definitions (17)

  • Definition 1
  • Example 1
  • Definition 2
  • Example 2
  • Definition 3
  • Example 3
  • Example 4
  • Lemma 1
  • proof
  • Example 5
  • ...and 7 more