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A Matrix Analogue of Rational Number Systems

This paper establishes sufficient conditions for the existence of digit systems with finiteness properties for matrix analogues of rational number systems and utilizes finite automata to construct systems possessing both finiteness and uniqueness properties in two dimensions, while also deriving vector expansions via expansion trees.

Original authors: Anjelo Gabriel R. Cruz, Manuel Joseph C. Loquias

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anjelo Gabriel R. Cruz, Manuel Joseph C. Loquias

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 system, but instead of counting with just one base like 10 (our usual decimal system) or 2 (binary), you are counting with a matrix. Think of a matrix not as a boring grid of numbers, but as a complex, multi-dimensional machine that stretches, twists, and rotates space.

This paper, written by Anjelo Gabriel R. Cruz and Manuel Joseph C. Loquias, asks a big question: Can we build a number system where the "base" is this twisting machine, and we can write down any vector (an arrow pointing in space) using a finite list of simple "digits"?

The Big Idea: The Matrix Machine

In our normal world, if you want to write the number 13 in base 10, you say "one ten and three ones." You are breaking the number down into powers of 10.

The authors are trying to do the same thing, but with arrows in space. They have two special machines, called matrices P and Q. They combine them to create a new machine, Q⁻¹P (think of this as "Q inverse times P"). This machine is the "base" of their new world.

The goal is to take any arrow x and write it as a sum:
x = (Machine)⁰ × (Digit) + (Machine)¹ × (Digit) + (Machine)² × (Digit) ...

The "digits" here aren't just 0 through 9; they are small arrows chosen from a specific, limited set called D.

The Two Golden Rules: Finiteness and Uniqueness

The authors are looking for a system that follows two strict rules:

The Finiteness Property: You must be able to write down any arrow in your system using only a finite number of digits. No infinite lists of digits allowed! If you keep adding digits forever, the system fails.
The Uniqueness Property: There should be only one way to write a specific arrow. If you can write the same arrow in two different ways, the system is messy and confusing.

The paper proves that if you choose your machines P and Q carefully (specifically, if they are "coprime" and the machine Q⁻¹P is "expanding"—meaning it stretches space so much that things fly apart), you can find a set of digits D that makes the system work.

The Secret Weapon: Finite Automata (The Magic Translators)

How do they prove this? They use something called finite automata. Imagine a tiny, super-fast robot that reads a string of digits and translates them.

In Section 3, the authors build these robots for 2-dimensional space (flat arrows on a piece of paper). They create a "transducer"—a machine that takes an input (like adding a tiny step to the right) and outputs the new string of digits.

They draw maps (called transition diagrams) showing how the robot moves from one state to another.
They found that for certain types of machines P and Q, these robots always settle down. They don't get stuck in an infinite loop of chaos; they eventually stop. This proves that the "Finiteness Property" holds.

The Expansion Tree: A Labyrinth of Paths

In Section 4, the authors introduce a concept called an Expansion Tree. Imagine a giant, branching tree where the root is the zero vector (the center of the universe).

Every branch represents adding a digit.
Every node (a spot on the tree) represents a specific arrow you can reach.
The path from the root to a node is the "code" or the expansion of that arrow.

They proved something fascinating about this tree:

It's not a simple loop: The tree is so complex that no simple computer program (a "regular language") can predict all the paths. It's infinitely intricate.
The only repeating pattern is zero: If you walk down the tree and see a pattern that repeats forever, the only pattern that works is a string of zeros. Any other repeating pattern leads to a dead end or a contradiction.

What About Real Numbers? (The Open Mystery)

So far, we've been talking about arrows made of whole numbers (integers). But what about real numbers (like 3.14)?

The authors suggest that if you let your digits go on forever to the right of a "decimal point" (using negative powers of the machine), you can represent real vectors.

They ran simulations (computer approximations) to see what these real vectors look like.
The Result: The set of all representable real vectors looks like a weird, jagged shape that tiles the plane (like a puzzle).
The Caveat: They do not say they have solved the problem of which real vectors can be represented. They explicitly state that determining this is "substantially more difficult" than the integer case. They offer a conjecture (a strong guess) that these shapes tile the entire space without overlapping, but this is based on their simulations and visual approximations, not a final proof.

What They Explicitly Rule Out

The paper is very clear about what doesn't work or what requires extra conditions:

Uniqueness isn't automatic: Just because you have a finite set of digits doesn't mean the representation is unique. You have to pick the right set of digits (a "complete residue system") to get uniqueness.
Not all matrices work: The machine Q⁻¹P must be "expanding" (all its eigenvalues must have a modulus greater than 1). If the machine shrinks space instead of stretching it, the system breaks down.
Integer vs. Real: The methods that work perfectly for integer vectors (proven with the automata) do not automatically solve the problem for real vectors. The real vector case remains an open area of research, with the authors only suggesting a tiling property based on their visual models.

The Bottom Line

Cruz and Loquias have successfully built a bridge between the world of rational numbers (fractions) and the world of matrices. They showed that if you pick your machines right, you can write any integer vector using a finite, unique code. They built the "robots" (automata) to prove it and drew the "trees" to visualize it.

However, when it comes to the messy, infinite world of real numbers, they have only taken the first step. They have drawn a beautiful map of the territory and guessed that it covers the whole world, but they admit the full proof is still waiting to be discovered.

Technical Summary: A Matrix Analogue of Rational Number Systems

Problem Statement
This paper addresses the generalization of rational number base systems (specifically the $p/q$ -expansion introduced by Akiyama et al.) to the domain of matrix digit systems. While standard matrix digit systems typically utilize an integer base matrix $A$ and a digit set $D \subseteq \mathbb{Z}^d$ , this work investigates systems where the base is a rational matrix of the form $Q^{-1}P$ , with $P, Q \in \mathbb{Z}^{d \times d}$ being invertible coprime matrices. The central problem is to determine conditions under which every vector in a specific $\mathbb{Z}$ -module, $\mathbb{Z}^d[Q^{-1}P]$ , admits a finite expansion of the form:
$x = \sum_{i=0}^{\ell-1} (Q^{-1}P)^i Q^{-1} d_i$
where the digits $d_i$ belong to a finite set $D$ . The authors seek to establish sufficient conditions for the finiteness property (existence of such a finite expansion) and the uniqueness property (uniqueness of the expansion for a minimal length $\ell$ ).

Methodology
The authors employ a combination of algebraic number theory, dynamical systems, and automata theory:

Algebraic Framework: The study defines $\mathbb{Z}^d[Q^{-1}P]$ as the smallest non-trivial $Q^{-1}P$ -invariant $\mathbb{Z}$ -module containing $\mathbb{Z}^d$ . The authors utilize the concept of coprime matrices ( $P\mathbb{Z}^d + Q\mathbb{Z}^d = \mathbb{Z}^d$ ) to define a quotient group structure. They establish that the digit set $D$ must contain a complete residue system of $(\mathbb{Z}^d + P\mathbb{Z}^d[Q^{-1}P]) / P\mathbb{Z}^d[Q^{-1}P]$ .
Dynamical Systems: A dynamical system $\Phi$ (the quotient function) is defined to mimic the division algorithm. The existence of finite expansions is linked to the finiteness of the attractor $A_\Phi$ of this dynamical system. If the attractor is finite, the system possesses the finiteness property.
Restricted Division for Integer Vectors: To handle expansions of vectors in $\mathbb{Z}^d$ specifically, the authors introduce a "restricted digit function" $d_r$ mapping to residues of $\mathbb{Z}^d/P\mathbb{Z}^d$ . This leads to a restricted dynamical system $\Phi_r$ . A key lemma demonstrates that if $\Phi_r$ has a finite attractor, the extended system on $\mathbb{Z}^d[Q^{-1}P]$ also has a finite attractor.
Automata Theory (Dimension $d=2$ ): For the specific case of $2 \times 2$ matrices, the authors construct finite letter-to-letter transducers. These automata model the addition of basis vectors (e.g., $(1,0)^\top$ ) to a given digit expansion. By analyzing the transition diagrams and cycles (specifically self-loops and finite paths) of these transducers, the authors determine the structure of the attractor for various characteristic polynomial coefficients.
Expansion Trees: The paper constructs an "expansion tree" $T_{P/Q}$ to represent vectors in $\mathbb{R}^d$ . This tree generalizes the $p/q$ -expansion tree for real numbers, defining path labels as $P/Q$ -expansions and investigating the language properties of these paths.

Key Contributions and Results

General Conditions for Finiteness: The paper proves that if $Q^{-1}P$ is an expanding matrix (all eigenvalues have modulus $>1$ ), the attractor of the associated dynamical system is finite. Consequently, a digit system $(P, Q, D)$ exists with the finiteness property, provided $D$ is a complete residue system of the relevant quotient group.
Uniqueness and Standard Systems: The authors clarify that if $D$ is a complete residue system and the expansion length is minimal, the uniqueness property holds. A system possessing both finiteness and uniqueness is termed "standard."
Explicit Attractor Classification for $d=2$ : For $2 \times 2$ $2 \times 2$ matrices $P$ $P$ and $Q$ $Q$ with coprime determinants and specific structural forms, the authors provide a complete classification of the attractor $A$ $A$ based on the coefficients $\alpha$ $α$ and $\beta$ $β$ of the characteristic polynomial of $Q^{-1}P$ $Q^{- 1} P$ ( $\chi(x) = x^2 + \alpha x + \beta$ $χ (x) = x^{2} + α x + β$ ).
- In cases where $0 \le \alpha \le \beta - 1$ or $0 < -\alpha \le 1$ , the attractor is trivial: $A = \{(0,0)^\top\}$ .
- In other cases (e.g., $1 < -\alpha \le \beta - 1$ ), the attractor is a finite set of vectors generated by specific linear combinations involving the matrix entries.
- These results are derived by analyzing the behavior of the constructed transducers, specifically identifying whether infinite periodic words (cycles in the automaton) correspond to non-zero attractor elements.
Language Properties: The paper establishes that the language $L_{P/Q}$ of finite $P/Q$ -expansions for integer vectors is prefix-closed but not regular. This is proven using the Myhill-Nerode theorem, showing that distinct vectors generate distinct equivalence classes.
Real Vector Representations: The authors define $P/Q$ -expansions for real vectors in $\mathbb{R}^d$ using infinite series with negative powers of the base. They introduce the concept of an expansion tree for real vectors and conjecture that the sets of vectors representable by these expansions form a tiling of $\mathbb{R}^d$ (a collection of sets whose union is $\mathbb{R}^d$ and whose pairwise intersections have Lebesgue measure zero).

Significance and Claims
The paper positions itself as a direct analogue to the rational base number systems studied by Akiyama et al., extending these concepts from scalar rational bases ( $p/q$ ) to matrix rational bases ( $Q^{-1}P$ ).

Theoretical Extension: The primary significance lies in bridging the gap between rational number systems and matrix digit systems. The authors demonstrate that the "modified division algorithm" used for $p/q$ bases can be successfully adapted to the matrix setting, provided the base matrix is expanding and the digit set is chosen as a residue system of a specific quotient group.
Constructive Approach: By utilizing finite automata (transducers) for the $d=2$ case, the paper moves beyond abstract existence proofs to provide constructive methods for determining the attractor and verifying the finiteness property.
Connection to Tiling: The paper connects these digit systems to the theory of self-affine tiles. The conjecture regarding the tiling of $\mathbb{R}^d$ by the sets $G_{P/Q}(v)$ suggests that these matrix expansions have geometric implications similar to those found in standard radix systems and self-affine tile theory.

The authors remain modest regarding the general case for $d > 2$ and the full characterization of the set of real vectors admitting such expansions, noting that determining which vectors in $\mathbb{R}^d$ admit a $Q^{-1}P$ -expansion is "substantially more difficult" than in the scalar case. The work serves as a foundational step, providing sufficient conditions and explicit constructions for the $2 \times 2$ case, while leaving the general geometric properties as an open area for future research.