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A Normality Conjecture on Rational Base Number Systems

This paper conjectures that minimal and maximal words in rational base number systems are normal over appropriate subalphabets, supporting this hypothesis with extensive numerical experiments and discussing its potential implications for resolving long-standing open problems such as the existence of Z-numbers and the Collatz-inspired "4/3 problem."

Original authors: Mélodie Andrieu, Shalom Eliahou, Léo Vivion

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mélodie Andrieu, Shalom Eliahou, Léo Vivion

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical number machine. In our everyday life, we use base 10 (digits 0–9) to count. If you want to write the number 100, you just write "100." This is a very predictable, orderly system.

But what if you built a machine that counts in fractions, like "base 7/3"? This is the world of Rational Base Number Systems. It's a strange, chaotic place where the rules of counting twist and turn in ways that mathematicians are still trying to understand.

This paper is about a bold guess (a "conjecture") made by three mathematicians regarding the behavior of numbers in these weird fractional bases. Here is the story of their discovery, explained simply.

1. The Two Extreme Paths: The "Min" and "Max" Words

In this fractional world, every number has a unique "address" (a sequence of digits). But here's the twist: you can keep extending these addresses forever, creating infinite strings of digits.

The authors focus on two specific types of infinite strings:

The "Minimal" Word: Imagine you are walking through a maze. At every fork, you always choose the leftmost path (the smallest digit possible). If you keep doing this forever, you create a "Minimal Word."
The "Maximal" Word: Now, imagine you always choose the rightmost path (the biggest digit possible). This creates a "Maximal Word."

The Analogy: Think of a tree growing in a forest.

The Minimal Word is the branch that always leans as far left as gravity allows.
The Maximal Word is the branch that always leans as far right as possible.

2. The Big Guess: "Are They Random?"

For a long time, mathematicians thought these extreme branches might be chaotic or follow a secret, hidden pattern.

The Conjecture: The authors propose that these infinite strings are actually perfectly random.
In math, a "normal" number is one where every digit, and every combination of digits, appears with perfect fairness. For example, in a normal binary string (0s and 1s), you shouldn't see "00000" way more often than "11111." Over a long enough time, everything balances out perfectly.

The Claim: The authors believe that if you take any "Minimal Word" or "Maximal Word" from this fractional world, it will look exactly like a string of digits generated by a perfectly fair coin flip. There is no hidden order; it is pure, beautiful chaos.

3. Why Does This Matter? (The "Butterfly Effect")

You might ask, "Who cares if a weird number string looks random?" The authors show that if their guess is true, it solves four massive, unsolved mysteries in mathematics that have been stuck for decades:

The "Z-Number" Mystery: A famous problem from 1968 asks if there are any numbers that, when multiplied by 1.5 over and over, always land in a specific small range. The authors say: "If our random guess is true, the answer is NO. These numbers don't exist."
The "Triple Expansion" Mystery: Can a single number have three different addresses in this system? The authors say: "If our guess is true, the answer is NO. A number can have at most two addresses."
The "4/3 Problem": A puzzle about whether certain number games always end. The authors say: "If our guess is true, the game always ends."
The Collatz Connection: This is the most famous unsolved math problem (the "3x+1" problem). While the authors don't solve Collatz, they show their work is deeply related to it, suggesting that the chaotic behavior of these fractional numbers might hold the key to understanding why Collatz is so hard.

4. The Evidence: The Great Computer Experiment

Since they can't prove it with a simple pencil-and-paper formula yet, they did something very modern: They ran a massive simulation.

They used computers to generate millions of these "Minimal Words" and checked them for randomness.

The Test: They looked at how long it took for every possible combination of digits to appear (the "Richness Threshold").
The Result: They compared the fractional words to:
- Real random numbers (like rolling dice).
- Famous "normal" numbers like Pi ( $\pi$ ).
- The "Champernowne constant" (a number made by writing 1, 2, 3, 4, 5... all together).

The Finding: The fractional words behaved indistinguishably from the random dice rolls. They were just as "rich" and "fair" as the best random numbers we know. They were not like the Champernowne constant, which has a very predictable, structured pattern.

The Takeaway

The authors are essentially saying:

"We have a hunch that these strange, fractional number systems produce infinite strings that are perfectly random. We have tested this with millions of examples, and the data screams 'YES.' If we are right, we don't just understand these numbers better; we unlock the doors to four other giant mysteries in math."

It's a bit like finding out that the pattern of leaves falling from a tree isn't random wind, but actually follows a perfect, hidden law of nature that explains why the weather works the way it does. If they are right, the "chaos" of these numbers is actually the key to order in the mathematical universe.

1. Problem Statement and Context

Rational Base Number Systems:
The paper investigates number systems with a rational base $p/q$ (where $p > q \ge 1$ are coprime integers), a generalization of classical integer bases. Unlike integer bases, rational bases exhibit complex, non-regular behavior. The expansion of an integer $n$ in base $p/q$ , denoted $rep_{p/q}(n)$ , is computed via a "Modified Division Algorithm" (MD algorithm) involving Euclidean division of $qn$ by $p$ .

Minimal and Maximal Words:
Two specific families of infinite words emerge naturally from these systems:

Minimal words ( $w_{min}$ ): Infinite words where every prefix is the lexicographically smallest extension of a seed integer $u$ that remains a valid integer expansion.
Maximal words ( $w_{max}$ ): Infinite words where every prefix is the lexicographically largest such extension.

The Core Problem:
While it is known that these words are non-periodic and have high complexity, their statistical distribution of subwords is unknown. The authors propose a Normality Conjecture:

Conjecture 1.2: For $q > 1$ , every minimal word (excluding the specific case of the empty word seed) is normal over the subalphabet $\{0, \dots, q-1\}$ . Similarly, every maximal word is normal over $\{p-q, \dots, p-1\}$ .
Definition of Normality: An infinite word is normal over a $d$ -ary alphabet if every finite subword of length $l$ appears with a limiting frequency of $1/d^l$ .

2. Methodology

The authors employ a dual approach combining rigorous theoretical proofs and extensive numerical experimentation.

A. Theoretical Framework

Equivalence of Conjectures: The authors establish a link between the normality of minimal words and the equidistribution of an iterated operator.
- They define an operator $T_{p/q}(x) = \lceil \frac{p}{q}x \rceil$ .
- Theorem 1.7: Conjecture 1.2 (Normality) is equivalent to Conjecture 1.6, which states that the sequence of iterates $(T^l_{p/q}(n))_{l \in \mathbb{N}}$ is equidistributed in residue classes modulo $q^k$ for all $k$ .
Implications for Open Problems: The authors prove that if their Normality Conjecture is true, it implies the truth of four long-standing open problems:
- Mahler's Z-numbers (1968): Non-existence of real numbers $x$ such that $\{x(p/q)^n\} \in [0, 1/q)$ .
- Akiyama's Triple Expansion (2008): No real number admits three distinct expansions in rational base $p/q$ .
- Dubickas-Mossinghoff '4/3 Problem' (2009): Termination of specific iterated maps on integers.
- Collatz-like dynamics: Connections to the distribution of trajectories in the Collatz map.

B. Numerical Experiments

To support the conjecture, the authors conducted massive computational experiments on 139 minimal words (up to length $10^6$ ) and 40,000 minimal words (up to length $10^5$ ) across various bases ( $2 \le q < p \le 26$ ).

They measured two key metrics:

Richness Threshold ( $rt_w$ ): The length $L$ at which all possible subwords of length $l$ first appear in the word. A word is "rich" if this threshold is finite for all $l$ . Richness is a necessary condition for normality.
Deviation from Uniformity (Discrepancy): The maximum difference between the empirical frequency of any subword of length $l$ and the expected uniform frequency ( $1/q^l$ ) in a prefix of length $n$ .

The experimental data was compared against:

Random $q$ -ary words (known to be normal with probability 1).
Famous conjectured normal numbers (e.g., expansions of $\pi$ and $\sqrt{2}$ in base $q$ ).
Structured normal words (Champernowne and de Bruijn words) to highlight differences in behavior.

3. Key Contributions

Formulation of the Normality Conjecture: The paper formally conjectures that minimal and maximal words in rational bases are normal over specific subalphabets, a property previously unproven and unobserved for these specific classes of words.
Theoretical Unification: The proof of Theorem 1.7 provides a powerful equivalence between the combinatorial property of word normality and the number-theoretic property of equidistribution in residue classes.
Resolution of Implications: The authors demonstrate that their conjecture is a "stronger" statement that subsumes several famous conjectures (Mahler, Akiyama, Dubickas-Mossinghoff). For instance, they show that proving minimal words contain the digit '0' (a weak condition implied by normality) is sufficient to disprove the existence of triple expansions.
Extensive Empirical Evidence: The paper provides the first large-scale statistical analysis of rational base minimal words, showing their statistical properties are indistinguishable from random sequences.

4. Results

Theoretical Results

Equivalence: The normality of minimal words is mathematically equivalent to the equidistribution of the sequence generated by the ceiling operator $T_{p/q}$ .
Implication for Z-numbers: If minimal words are normal, then for $p < q^2$ , no $Z_{p/q}$ -numbers exist. The proof relies on the fact that a normal word must contain the subword $(q-1)(q-1)$ , which leads to a contradiction if a $Z_{p/q}$ -number existed.
Triple Expansions: Normality implies that no word can be simultaneously minimal and maximal (since they use disjoint alphabets, except for the letter 0 which minimal words must contain). This confirms Akiyama's conjecture that no real number has three expansions.

Numerical Results

Richness: The computed minimal words were found to be rich. The richness thresholds for minimal words fell within the statistical distribution of random $q$ -ary words. They did not exhibit the extreme delays seen in structured words like Champernowne words.
Discrepancy: The deviation from uniformity ( $D_{w,l}(n)$ ) for minimal words decreased as $n$ increased, following the asymptotic rate $O(\sqrt{\log \log n} / \sqrt{n})$ typical of random sequences.
Distribution Shape: The distribution of richness thresholds and discrepancies for minimal words closely matched the Poisson-like distributions observed in random words.
Differentiation: Unlike Champernowne words (which have very high richness thresholds for specific subwords) or de Bruijn words (which have deterministic structures), minimal words in rational bases behave statistically like random noise.

5. Significance

This paper represents a significant advancement in the theory of numeration systems and combinatorics on words:

Bridging Fields: It connects the study of rational base number systems with deep problems in analytic number theory (distribution of fractional parts) and dynamical systems (Collatz-like maps).
New Perspective on Normality: It suggests that "chaotic" systems like rational base expansions naturally generate normal sequences, challenging the intuition that normality requires artificial construction (like Champernowne's constant).
Resolution of Open Problems: By framing the problem as a normality conjecture, the authors provide a unified path to solving multiple independent, decades-old conjectures. If the conjecture is proven, it instantly resolves the existence of Z-numbers and triple expansions.
Computational Benchmark: The extensive dataset and methodology (richness thresholds, discrepancy analysis) set a new standard for analyzing the statistical properties of infinite words generated by deterministic algorithms.

In conclusion, the authors present a compelling case that minimal and maximal words in rational base systems are not just complex, but normal, behaving statistically like random sequences. This hypothesis, if proven, would unify and resolve several major open problems in mathematics.