Digit Matrices: Finite Expansions & Algorithms

• Digit matrices are integer or rational matrices that serve as bases in multidimensional positional number systems, generalizing classical numeral systems.
• They enable finite radix expansions through algorithmic construction of digit sets using block decomposition, Smith normal forms, and convex geometric methods.
• Applications range from algebraic number theory to symbolic dynamics and tiling theory, with finiteness determined by the eigenvalue spectral properties of the matrix.

A digit matrix is an integer or rational matrix that serves as the “base” in a higher-dimensional positional representation system, generalizing classical scalar base expansions to objects in $\mathbb{Z}^n$ or more generally, $\mathbb{Z}^n[A]$. The associated machinery enables finite "radix" expansions of lattice vectors analogous to positional number systems, utilizing a finite digit set as representatives. This theory unifies ideas from algebraic number systems, symbolic dynamics, and tiling theory, and allows for explicit, algorithmic constructions of expansions in both expanding and non-expanding cases.

1. Fundamental Definitions and Finiteness Characterization

Let $A \in M_n(\mathbb{Q})$ (or $M_n(\mathbb{Z})$ in the integral case). The minimal $A$-invariant $\mathbb{Z}$-module containing $\mathbb{Z}^n$ is defined as

$$\mathbb{Z}^n[A] = \bigcup_{k=1}^\infty \big(\mathbb{Z}^n + A\mathbb{Z}^n + \cdots + A^{k-1}\mathbb{Z}^n\big).$$

A digit system is a pair $(A, \mathcal{D})$ with $\mathcal{D}\subset\mathbb{Z}^n[A]$ finite. The system is said to have the finiteness property if every $z \in \mathbb{Z}^n[A]$ admits a finite expansion of the form

$$z = d_0 + A d_1 + \cdots + A^{k} d_{k},\qquad d_j \in \mathcal{D}.$$

The main criterion is: $(A, \mathcal{D})$ has the finiteness property for some finite $\mathcal{D}$ if and only if $A$ has no eigenvalue with absolute value $< 1$. In such a case, $\mathcal{D}$ can be taken in $\mathbb{Z}^n$ itself (Jankauskas et al., 2018, Jankauskas et al., 2021).

2. Algorithmic Construction and Explicit Digit Sets

The explicit construction of suitable digit sets involves canonical forms and convex geometry:

1. Block decomposition: Bring $A$ to rational or real (hyper-)companion form, decomposing into blocks associated to irreducible factors of its characteristic polynomial.
2. Polynomial analogue: For a block corresponding to $P(x)^{m}$, a finite set $N\subset\mathbb{Z}$ exists with $\mathbb{Z}[x]=N[x]+(P)$ iff all roots of $P$ satisfy $|\alpha|\ge1$.
3. Residue systems: Construct digit sets as coset representatives for $\mathbb{Z}^n[A]/A\mathbb{Z}^n[A]$. In expanding cases, any such system yields finiteness (Cruz et al., 8 Jul 2025, Jankauskas et al., 2021).
4. Convex hull conditions: For unimodular rotation blocks ($|\lambda|=1$), convex-geometric requirements on digits ensure all "angles" are covered so the algorithm converges to 0 (Jankauskas et al., 2021).
5. Block sum and twisting: Combine individual digit sets for each block by direct sum and apply necessary conjugations.
6. Optimization: Redundant digits may be pruned by analyzing the "repeller" set induced by the remainder-division map and using integer programming (Jankauskas et al., 2021).

A typical algorithm proceeds by: (i) block-decomposing $A$; (ii) computing Smith normal forms for residue classes; (iii) selecting/or optimizing digit representatives by convex-hull or contraction criteria; and (iv) constructing the global digit system.

3. Dynamical Systems Perspective and Attractors

Digit matrix systems can be interpreted via discrete dynamical systems. For a digit set $\mathcal{D}$ providing complete residue classes modulo $A\mathbb{Z}^n[A]$, define the digit function $\delta(x)$ and backward map (remainder-division or beta-map)

$\Phi(x) := A^{-1}(x - \delta(x)).$

If $A$ is expanding, $\Phi$ has a finite attractor $\mathcal{A}_\Phi$ that eventually absorbs all orbits. Each $x\in \mathbb{Z}^n[A]$ admits a unique representation

$$x = d_0 + A d_1 + \cdots + A^{k} d_{k} + A^{k+1}p, \quad p \in \mathcal{A}_\Phi.$$

Finiteness property corresponds to $\mathcal{A}_\Phi = \{0\}$. Otherwise, the attractor describes inherent periodic structures in the digit system, generalizing purely periodic expansions in non-integer bases (Cruz et al., 8 Jul 2025, Jankauskas et al., 2021).

4. Expanding, Non-Expanding, and Contractive Cases

Case	Defining Feature	Digit System Behavior
Expanding	All $|\lambda|>1$	All vectors admit finite radix expansions; unique representations for each $x$ (Cruz et al., 8 Jul 2025, Curry, 2010)
Unimodular	All $|\lambda|=1$	Finiteness only with appropriately “balanced” or convex-enclosed digit sets; possible lack of uniqueness (Jankauskas et al., 2021, Pelantová et al., 2021)
Non-Expanding	Some $|\lambda|<1$	No digit system with finiteness property; expansions cannot reach all elements (Jankauskas et al., 2018)
Non-expansive Jordan block	$A \sim J_n(1)$	Expansions always possible with at most $n$ digits; full systems characterized by parity, residue, or block combinatorics (Caldwell et al., 2021)

This categorization clarifies why the spectral criterion $|\lambda|\ge 1$ is both necessary and sufficient for the finiteness property, whereas classic Euclidean division and its multidimensional analogues fail for matrices with contractive directions.

5. Notable Special Cases and Examples

Expanding Matrices

Diagonal $A = \mathrm{diag}(b, ..., b), b \geq 2$: Coordinate-wise digit sets $\{0, 1, ..., b-1\}^n$ yield classical base-$b$ expansion in $\mathbb{Z}^n$ (Curry, 2010).

Rational Rotations

$$A = \frac{1}{5}\begin{pmatrix}3 & -4\ 4 & 3\end{pmatrix}$$: A rotation in $\mathbb{Z}^2$ admits a digit set constructed by residue-system and convex-enclosure methods; the resulting system is finite after refining via repeller sets (Jankauskas et al., 2021).

Companion Matrices of Polynomials

For quadratic $p(x) = x^2+\alpha x + \beta \in \mathbb{Q}[x]$, with $$A = \begin{pmatrix}0 & -\beta\ 1 & -\alpha\end{pmatrix}$$, digits are coset representatives modulo $b = \mathrm{const}$. The system is finite if $0<\alpha\le\beta-1$. Attractors and automata describe all possible expansions and carries (Cruz et al., 8 Jul 2025).

Non-expansive Jordan Blocks

$J_n(1)$: For $n=2$, full expansion is characterized by explicit parity and gcd conditions on digits. For $n\ge3$, three digits suffice—canonical choices are $(0,\dots,0,\pm1)^T$, $(0,\dots,0,0)^T$; for similar matrices at most $n$ digits are necessary (Caldwell et al., 2021).

6. Generalizations and Connections

The digit matrix approach is a multidimensional generalization of classical positional numeration systems. It relates deeply to:

• The height-reducing property in algebraic number theory and digit representations in number fields (Jankauskas et al., 2018).
• Euclidean division algorithms generalized to modules and lattices (Curry, 2010).
• Symbolic dynamics: representation languages of zero are in general not context-free, but can be recognized in $\mathsf{LSPACE}$ by logarithmic-memory Turing machines (Caldwell et al., 2021).
• Algebraic tiling and self-affine tiles: attractors for higher-rank systems yield fractal boundaries and nontrivial topological features (Cruz et al., 8 Jul 2025).

7. Quantitative and Algorithmic Aspects

Quantitative bounds for expansion lengths and digit set sizes follow from block structure and polynomial bounds:

• For a companion matrix of $P(x) = a_d x^d + \cdots + a_0$, digit set cardinality can be $O(|a_0| + |a_d|)$.
• For matrices with spectral radius $\rho(A)>1$, expansion length is bounded above by $C \frac{\log \|z\|}{\log \rho(A)}$ for a suitable norm and constant $C$ (Jankauskas et al., 2018).
• Construction algorithms rely on Smith normal form, convex-geometry (for unimodular rotations), and integer programming for minimality (Jankauskas et al., 2021).

The interplay of spectral properties, lattice theory, and symbolic encoding enables systematic algorithmic construction and minimality improvement, integrating classical number-theoretic ideas with modern computational algebra and dynamics.