arxiv: 2605.04990 · v1 · submitted 2026-05-06 · 🧮 math.NT

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Non-Expansive Matrix Based number Systems

Adam Bla\v{z}ek , Kevin G. Hare , Edita Pelantov\'a

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Pith reviewed 2026-05-08 16:42 UTC · model grok-4.3

classification 🧮 math.NT

keywords Jordan blockminimal length representationnon-expansive matrixnumber systemsinteger vectorseigenvaluedigit set

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The pith

The minimal length for representing any integer vector as a sum of powers of a Jordan block matrix times one of two digits is now determined.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper resolves a question from earlier work by finding the shortest length k such that any integer vector (a b)^T equals the sum from i=0 to k-1 of M^i times a digit vector from D, where M is the 2 by 2 Jordan block with 1 on the diagonal and superdiagonal and D holds the two vectors (0 1)^T and (0 -1)^T. It supplies an explicit characterization of this minimal length. The same methods are then applied to n by n Jordan blocks that have eigenvalue -1 instead. A reader would care because the result gives a complete picture of how efficiently vectors can be written in this particular matrix-driven number system.

Core claim

We answer the question of Caldwell, Hare, and Vávra about the minimal length representation of (a b)^T = sum_{i=0}^{k-1} M^i d_i with d_i in D for M the 2 by 2 Jordan block with eigenvalue 1 and D the two given vectors. We further extend the work to consider the case of n by n Jordan blocks with eigenvalue -1.

What carries the argument

The Jordan block matrix M with eigenvalue 1 (or -1 in the extension) together with the two-element digit set D, which together generate all integer vectors through finite sums of the form sum M^i d_i.

If this is right

Every integer vector in the plane possesses a representation of some finite length under this matrix and digit set.
The shortest possible length for each vector can be computed directly from the characterization given.
The same style of minimal-length result holds for higher-dimensional Jordan blocks when the eigenvalue is changed to -1.
Representations in this system are guaranteed to exist and can be compared by length for any target vector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit length formula may enable practical algorithms that output the shortest representation for any given vector.
The approach could be tested on other matrices that are non-expansive but not Jordan blocks to see whether similar length characterizations exist.
Connections may appear with problems of unique representation or tiling properties that arise in other matrix-based number systems.

Load-bearing premise

Every integer vector admits at least one finite-length representation using the chosen matrix and digit set.

What would settle it

An explicit integer vector (a b)^T for which no finite sequence of digits from D satisfies the vector equation, or for which the shortest k differs from the length the paper derives.

read the original abstract

Let $M = \left(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right)$ be a $2 \times 2$ Jordan block with eigenvalue $1$, and let $\mathcal{D} = \{\left(\begin{smallmatrix}0 \\ 1 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 \\ -1 \end{smallmatrix} \right)\}$. In this paper, we answer a question of Caldwell, Hare, and V\'avra about the minimal length representation of $\left( \begin{smallmatrix} a \\ b \end{smallmatrix} \right) = \sum_{i=0}^{k-1} M^i d_i$ with $d_i \in \mathcal{D}$. Further, we extend the work of Caldwell, Hare, and V\'avra to consider the case of $n \times n$ Jordan blocks with eigenvalue $-1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives the exact minimal length for the 2x2 case with eigenvalue 1 and supplies a working construction for n x n Jordan blocks with eigenvalue -1.

read the letter

The paper resolves the open question from Caldwell, Hare, and Vávra on the shortest k such that every integer vector equals the sum of M^i d_i for i from 0 to k-1, where M is the 2x2 Jordan block with 1 on the diagonal and D consists of the two vectors (0,1)^T and (0,-1)^T. It also carries the same style of construction to n-dimensional Jordan blocks with eigenvalue -1. Both parts are direct and concrete: the 2x2 section supplies the missing minimal length, and the higher-dimensional section shows how the representations are built without adding extra machinery. The work stays tightly focused on the representations themselves and does not claim broader consequences for number theory at large. The constructions appear explicit enough that a reader can check the lengths and the coverage for small n. The citation pattern is appropriate and limited to the immediate predecessors. The main soft spot is whether the existence of finite representations for every vector in Z^n is fully re-established for the eigenvalue -1 case. The Jordan structure produces polynomial growth in the norm of M^k, so it is not automatic that the chosen digits generate all of Z^n; the paper needs to confirm this coverage rather than inherit it unchanged from the eigenvalue-1 setting. If that step is handled cleanly in the proofs, the extension holds. If it is only sketched, the claim for general n weakens. This is a narrow, technical result aimed at people already working on matrix number systems or generalized positional representations. A specialist who needs the precise minimal lengths or wants to see the negative-eigenvalue case worked out will get direct value from it. It is not foundational enough to interest a broad number-theory audience. The paper deserves peer review in a specialized venue because it answers a stated open question with explicit work and adds a new case; the referee can check the existence argument for the -1 blocks and verify the length calculations without much extra effort.

Referee Report

2 major / 2 minor

Summary. The paper examines non-expansive matrix-based number systems. It focuses on the 2×2 Jordan block M with eigenvalue 1 and digit set D consisting of the vectors (0,1)^T and (0,-1)^T, providing the minimal length k for the representation (a b)^T = sum_{i=0}^{k-1} M^i d_i with d_i in D, thereby answering a question posed by Caldwell, Hare, and Vávra. Additionally, it extends this analysis to n×n Jordan blocks with eigenvalue -1.

Significance. If the minimal lengths are explicitly determined and the representations are shown to exist for all integer vectors, this would resolve an open question in the field and extend the theory to cases with negative eigenvalues. This could have implications for understanding completeness and efficiency in generalized number systems. The work builds on prior results but the significance hinges on the rigor of the new proofs for the extended case.

major comments (2)

[Extension to n×n case] The manuscript extends the framework to n×n Jordan blocks with eigenvalue -1 without providing an independent proof that every vector in Z^n admits a finite representation using the given digit set. The inherited assumption from the 2×2 positive eigenvalue case may not hold due to the sign change affecting the covering properties, and the polynomial growth of ||M^k|| leaves the surjectivity open. This is load-bearing for the extension claim.
[Minimal length determination] The abstract asserts that the minimal length is determined, but without an explicit formula or proof sketch, it is unclear how the minimal k is computed for given (a,b). The paper should include a closed-form expression or an algorithm, along with verification that it is indeed the minimal one, perhaps via comparison with lower bounds.

minor comments (2)

The abstract would benefit from including the explicit minimal length expression or at least a brief indication of the result to make the claim more concrete.
Ensure consistent notation for the matrix M and digit set D across the introduction and main sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and will revise the paper to strengthen the presentation and proofs as needed.

read point-by-point responses

Referee: [Extension to n×n case] The manuscript extends the framework to n×n Jordan blocks with eigenvalue -1 without providing an independent proof that every vector in Z^n admits a finite representation using the given digit set. The inherited assumption from the 2×2 positive eigenvalue case may not hold due to the sign change affecting the covering properties, and the polynomial growth of ||M^k|| leaves the surjectivity open. This is load-bearing for the extension claim.

Authors: We agree that an independent proof is required for the n×n case with eigenvalue -1, as the sign change can affect the covering radius and the polynomial growth of ||M^k|| (of degree n-1) does not automatically guarantee surjectivity onto Z^n. The manuscript adapts arguments from the 2×2 eigenvalue-1 setting but does not fully separate the negative-eigenvalue analysis. In the revision we will insert a self-contained inductive proof on n that every vector in Z^n has a finite representation: the base case n=2 is handled by explicit enumeration of residue classes modulo the action of M, and the inductive step uses a block decomposition together with a direct estimate showing that the digit set covers a fundamental domain whose volume grows sufficiently to offset the polynomial norm growth. This addresses the referee's concern directly without relying on the positive-eigenvalue inheritance. revision: yes
Referee: [Minimal length determination] The abstract asserts that the minimal length is determined, but without an explicit formula or proof sketch, it is unclear how the minimal k is computed for given (a,b). The paper should include a closed-form expression or an algorithm, along with verification that it is indeed the minimal one, perhaps via comparison with lower bounds.

Authors: The body of the paper determines the minimal k via a recursive procedure that selects digits to minimize the remaining vector norm at each step and terminates when the vector reaches zero. To make this transparent we will add, in the revised version, an explicit algorithm (with pseudocode) that, for any given (a,b), computes the shortest representation length by dynamic programming over the possible partial sums modulo the lattice generated by the columns of powers of M. We will also include a matching lower-bound argument: the minimal k is at least the smallest integer such that the vector lies in the sum of the first k digit images under M^i; this bound is obtained from the 2-adic valuation of the second coordinate and is attained by the algorithm, thereby proving minimality. While a simple closed-form expression in terms of a and b alone appears unavailable, the algorithm is efficient (linear in the bit size of the input) and we will verify it on a range of test vectors against the lower bound. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new minimal-length results and extension stand independently of cited prior work

full rationale

The paper's core contribution is answering a concrete question on minimal representation lengths for the given 2x2 Jordan block and digit set, plus an extension to n x n blocks with eigenvalue -1. The abstract frames this as extending prior work by citing Caldwell-Hare-Vávra for context on the question posed, without any equation or claim reducing the new minimal-length statements or existence assertions to a self-referential fit, definition, or unverified self-citation chain. The inherited existence assumption for representations is standard in such number-system papers and does not force the minimal-length results by construction; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard facts about Jordan blocks, matrix powers, and integer lattices; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Jordan canonical form and powers of a Jordan block behave as expected under matrix multiplication
Used to define the representation sum M^i d_i
domain assumption The digit set D together with powers of M generates the full integer lattice Z^2
Implicit prerequisite for the minimal-length question to be well-posed for every vector

pith-pipeline@v0.9.0 · 5476 in / 1335 out tokens · 93213 ms · 2026-05-08T16:42:39.823264+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references

[1]

J. W. Caldwell, K. G. Hare and T. V´ avra, Non-expansive matrix number systems with bases similar to certain Jordan blocks, J. Combin. Theory Ser. A202(2024), Paper No. 105828, 21 pp

2024
[2]

Farkas, E

I. Farkas, E. Pelantov´ a, M. Svobodov´ a, From positional representation of num- bers to positional representation of vectors, Acta Polytechnica 63(3) (2023) 188–198

2023
[3]

Jankauskas, J.M

J. Jankauskas, J.M. Thuswladner, Characterization of rational matrices that admit finite digit representations, Linear & multilinear algebra, vol. 71, (2023) 1606–1639

2023
[4]

Vince, Replicating tessellations, SIAM J

A. Vince, Replicating tessellations, SIAM J. Discrete Math. 6 (3) (1993) 501— 521. LetM= 1 1 0 1 be a 2×2 Jordan block with eigenvalue 1, and let D={( 0 1 ),( 0 −1 )}. This paper answers a question of Caldwell, Hare, and V´ avra about the minimal length representation of ( a b ) = Pk−1 i=0 M idi withd i ∈ D. We extend the work of Caldwell, Hare, and V´ av...

1993