Abelian-normal decimal expansions

This paper introduces the concept of abelian-normal numbers by adapting abelian complexity functions to digit expansions, constructs a non-normal analogue of Champernowne's constant that satisfies this new property, and proposes open problems regarding its behavior.

The Gist

Here is an explanation of the paper "Abelian-normal decimal expansions" by John M. Campbell, translated into simple language with creative analogies.

The Big Idea: Sorting the Chaos

Imagine you have a giant, endless string of numbers, like a long necklace made of beads. In mathematics, a "normal number" is like a perfectly fair necklace. If you look at any section of it, every possible combination of beads (like "1-2-3" or "5-5-9") appears with the exact same frequency as every other combination of the same length.

The most famous example of this is Champernowne's Constant ($C_{10}$). It's created by just writing down all the counting numbers in a row:
0.123456789101112131415...

Mathematicians have known for a long time that this number is "normal." It's statistically perfect.

The Twist: What if we shuffle the beads?

The author of this paper asks a fun question: What happens if we take this perfect necklace and rearrange the beads, but only in a very specific way?

Imagine you have a section of the necklace that contains only the digits 0 and 1 (like a binary code). The author decides to take every single chunk of 0s and 1s in the number and sort them.

• If he finds a messy chunk like 101001, he turns it into 000111.
• He leaves all the other numbers (2 through 9) exactly where they are.

He creates a new number, let's call it $D_{10}$.

The Problem: It's No Longer "Normal"

If you look at the new number $D_{10}$, it is not a normal number anymore. Why?
Because the author forced every chunk of 0s and 1s to be sorted. This means you will never see the pattern 10 appear in the number! In a normal number, 10 should appear just as often as 01. Since 10 is banned, the number is "broken" or "abnormal" by the standard rules.

The Solution: A New Way to Count

Here is where the paper gets clever. The author says, "Okay, $D_{10}$ isn't normal by the old rules. But maybe it's normal by new rules."

He introduces a concept called "Abelian Normality."

The Analogy: The "Bag of Marbles" Rule
Imagine you are counting how often a specific pattern appears.

• Old Rule (Normality): You are a strict librarian. You only count the pattern 1-2-3. If you see 3-2-1, you ignore it.
• New Rule (Abelian Normality): You are a relaxed collector. You don't care about the order. You have a "bag of marbles." If you are looking for the pattern 1-2-3, you also count 3-2-1, 2-1-3, 1-3-2, etc. As long as the bag contains one 1, one 2, and one 3, it counts as a match.

The author proves that while $D_{10}$ is broken under the "Strict Librarian" rules, it is actually perfectly fair under the "Relaxed Collector" rules.

How Did He Prove It?

To prove this, the author had to invent a special "Weighting System."

Think of the number line as a busy highway.

• In the original number ($C_{10}$), cars (digits) drive in all sorts of random orders.
• In the new number ($D_{10}$), the author took every group of 0s and 1s and forced them to line up in a single file (sorted).

Because of this sorting, some patterns disappear, and new ones appear. To make the math work out to "fairness," the author had to assign a weight (or a score) to every pattern.

• If a pattern is very common in the original number but rare in the new one, he gives it a high weight.
• If a pattern is rare in the original but common in the new one, he gives it a low weight.

By carefully calculating these weights (using complex formulas involving factorials and limits), he showed that if you count the patterns in $D_{10}$ using his special weights, the number turns out to be Abelian-Normal.

The Takeaway

1. Normal Numbers: Perfectly random sequences where order matters.
2. Abelian-Normal Numbers: Sequences that might look "sorted" or "messy" if you look at the order, but are perfectly random if you only look at what is in the sequence, not where it is.
3. The Result: The author built a new number ($D_{10}$) that is "broken" by normal standards but "perfect" by these new "Abelian" standards.

Why Does This Matter?

This is like discovering a new type of randomness. Usually, we think randomness means "no pattern." This paper shows that you can have a number that looks like it has a pattern (because the 0s and 1s are sorted), but if you look at it through the right mathematical lens (ignoring the order), it is actually just as random as the original.

The paper ends with two open questions:

1. Is this new number $D_{10}$ a "transcendental" number (a number that isn't the solution to any simple algebraic equation)?
2. Can we find a number that is "Abelian-Normal" but fails the "Pure" version of this test?

In short, the paper takes a known mathematical object, breaks it, and then invents a new set of glasses that makes the broken object look perfect again.

Technical Summary

Here is a detailed technical summary of the paper "Abelian-normal decimal expansions" by John M. Campbell.

1. Problem Statement

The paper addresses a gap in the theory of normal numbers. A real number $\alpha$ is normal in base $B$ if every block of digits $E$ of length $\ell(E)$ appears with asymptotic frequency $B^{-\ell(E)}$. While many studies have explored operations (selection rules, insertions, deletions) that preserve or destroy normality, this paper investigates a new type of operation: rearranging subwords based on their abelian complexity.

The core problem is to determine if a number can be constructed that is not normal (in the standard sense) but satisfies a relaxed condition called abelian-normality. This condition requires that the frequency of a block $E$ and all its permutations (an abelian equivalence class) collectively satisfy the normality frequency, provided the count is normalized by a specific weighting function.

2. Methodology and Definitions

2.1 Abelian Normality

The author introduces the concept of an abelian-normal number by adapting the standard definition of normality using equivalence classes of words under permutation.

• Standard Count ($A_E$): The number of occurrences of a specific block $E$.
• Abelian Count ($B_E$): The number of occurrences of $E$ and any permutation of $E$ within the first $n$ digits.
• Weighting Function ($W$): To normalize the abelian count, a weighting function $W(E)$ is introduced. For a block $E$, $W(E)$ represents the size of the equivalence class (the number of distinct permutations of $E$).
• Definition: A number $\alpha$ is abelian-normal in base $B$ with respect to $W$ if:
  $$\lim_{n \to \infty} \frac{1}{W(E)} \frac{B_E(\alpha, n)}{n} = \frac{1}{B^{\ell(E)}}$$
  for all blocks $E$.

2.2 Construction of the Counter-Example ($D_{10}$)

The paper constructs a specific constant, $D_{10}$, which serves as a non-normal analogue to Champernowne's constant ($C_{10}$).

• Champernowne's Constant ($C_{10}$): Formed by concatenating positive integers: $0.123456789101112\dots$. It is known to be normal in base 10.
• The Transformation: The author defines a mapping $\sigma$ that operates on the decimal expansion of $C_{10}$.
  1. Identify maximal binary subwords (contiguous sequences of only digits 0 and 1) within the expansion.
  2. Sort each maximal binary subword lexicographically (i.e., all 0s come before all 1s).
  3. Replace the original subword with the sorted version.
• Resulting Constant ($D_{10}$): The resulting sequence is $D_{10}$.
  • Example: A segment like ...110100... (containing mixed 0s and 1s) becomes ...000111....
  • Consequence: By construction, $D_{10}$ never contains the subword 10 within a binary block context. Therefore, $D_{10}$ is not normal because the frequency of 10 does not match the expected $10^{-2}$.

2.3 Case Analysis and Weighting Derivation

To prove $D_{10}$ is abelian-normal, the author performs a rigorous case analysis on how permutations of words are counted in $C_{10}$ versus $D_{10}$.

• Case 1 ($\uparrow_C$): Occurrences in $C_{10}$ where a permutation $\pi(w)$ exists, but the corresponding sorted version in $D_{10}$ does not count as a permutation of $w$ due to boundary effects with non-binary digits.
• Case 2 ($\uparrow_D$): Occurrences in $D_{10}$ where a word $v$ (which is not a permutation of $w$ in $C_{10}$) maps to $w$ after sorting.
• Weighting Function Construction: The author defines a specific weighting function $W(w)$ that includes a correction term based on the asymptotic difference between the frequencies of these "mismatched" cases:
  $$W(w) = \frac{\ell(w)!}{\prod c_i!} + 10^{\ell(w)} \left( \lim_{n \to \infty} \frac{D_w(C_{10}, n)}{n} - \lim_{n \to \infty} \frac{C_w(C_{10}, n)}{n} \right)$$
  where $C_w$ and $D_w$ count the occurrences of the specific mismatch cases defined above.

3. Key Contributions and Results

3.1 Theorem 1: Abelian-Normality of $D_{10}$

The primary result is Theorem 1, which states that the constructed constant $D_{10}$ is abelian-normal with respect to the derived weighting function $W$.

• Proof Strategy: The proof decomposes the count of abelian occurrences in $D_{10}$ into:
  1. Permutations that remain unchanged by the sorting operation (non-binary boundaries).
  2. Corrections for the "loss" of counts due to sorting (Case 1).
  3. Corrections for the "gain" of counts from non-permutations becoming permutations (Case 2).
     By explicitly defining $W(w)$ to include the net difference of these limits, the author shows that the normalized frequency converges exactly to $10^{-\ell(w)}$.

3.2 Existence of Non-Normal Abelian-Normal Numbers

The paper successfully demonstrates the existence of a real number that is not normal (fails the standard definition) but is abelian-normal (satisfies the relaxed definition with the specific weighting). This resolves the question of whether the two concepts are distinct.

3.3 New Weighting Function

The paper introduces a novel weighting function that goes beyond simple factorial normalization (which counts permutations). It incorporates a dynamic correction term derived from the specific structural properties of the transformation applied to Champernowne's constant.

4. Significance and Future Directions

• Interdisciplinary Bridge: The work bridges Combinatorics on Words (specifically abelian complexity functions) and Number Theory (normal numbers). It suggests that abelian complexity can be used to define new classes of "random-like" numbers that are not strictly normal.
• Refining Randomness: It challenges the strictness of normality. $D_{10}$ exhibits a high degree of "randomness" in terms of digit distribution when order is ignored (abelian sense), despite having a deterministic, non-random structure (sorted binary blocks).
• Open Problems: The author concludes with two significant open questions:
  1. Transcendence: Can the methods used to prove $C_{10}$ is transcendental (Mahler's method) be adapted to prove $D_{10}$ is transcendental?
  2. Pure Abelian Normality: Does there exist a number that is "purely abelian-normal" (using the natural factorial weighting without correction terms) but not normal? This would imply a number where the abelian frequencies naturally balance out without artificial weighting adjustments.

Summary

John M. Campbell's paper introduces the concept of abelian-normal numbers and constructs a specific counter-example, $D_{10}$, derived from Champernowne's constant. By sorting binary subwords lexicographically, $D_{10}$ loses standard normality but retains a generalized form of normality when digit counts are weighted by a function that accounts for the specific permutations lost and gained during the sorting process. This work expands the theoretical framework of normal numbers by integrating concepts from combinatorics on words.