Strong marker sets for arbitrary generating sets of $\mathbb Z^n$

Jing Yu

Strong marker sets for arbitrary generating sets of $\mathbb Z^n$

T0 review · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read The support assumption is unnecessary for the strong clopen marker theorem on generating sets of Z^n.

desk verdict Short note removes the support assumption from Gao-Wang via a conjugacy reduction that holds up. read the letter →

arxiv 2606.06707 v1 pith:2QDXJGAI submitted 2026-06-04 math.CO math.LO

Jing Yu This is my paper · ORCID

classification math.COmath.LO

keywords markersetsZ^ngeneratingunimodulartransformationclopenconjugacylattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Gao and Wang proved a strong clopen marker theorem for finite generating sets of Z^n assuming each generator has support of size 1 or n. This paper demonstrates that the theorem holds without this support assumption. The key step is a unimodular change of coordinates that puts any finite set of nonzero lattice vectors into full-support position. The Gao and Wang theorem can then be applied, and the marker set conjugated back to the original coordinates. This extension makes the result applicable to arbitrary finite generating sets.

What carries the argument

Unimodular conjugacy to full-support position for applying the Gao-Wang marker theorem.

What would settle it

A concrete finite set of vectors in some Z^n for which no unimodular matrix makes all of them have full support in every coordinate.

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Extended reading notes

Core claim

Gao and Wang proved a strong clopen marker theorem for finite generating sets of Z^n under the assumption that each generator has support of size either 1 or n. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.

Load-bearing premise

There always exists a unimodular integer matrix transforming any finite collection of nonzero vectors in Z^n so that each has nonzero components in all n positions.

Editorial extensions

If this is right

Strong clopen marker sets exist for arbitrary finite generating sets of Z^n.
The support size restriction on generators is not required.
Marker sets for any finite set of nonzero vectors in Z^n can be obtained by conjugation.
The result simplifies the statement of the marker theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The conjugacy technique may apply to marker theorems in other discrete groups.
Applications in symbolic dynamics on lattices no longer need to restrict to special generators.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The paper shows that the support-size hypothesis in Gao and Wang can be dropped. After a unimodular change of coordinates, any finite set of nonzero vectors in Z^n can be made to have full support, the old theorem applies, and the marker set is conjugated back. That is the entire contribution.

The reduction is the new piece. It is a legitimate extension because the original work required the assumption and this removes it without adding new machinery. The argument stays short and uses only that the forbidden loci are a finite union of hyperplanes whose complement contains points that complete to a basis in GL(n,Z). The stress-test note confirms this existence step is standard and the conjugation preserves the marker property.

No major soft spots appear. The central claim reduces cleanly to the prior theorem, and nothing in the setup introduces circularity or unfalsifiable steps. Minor details like the explicit construction of the matrix A would need checking in the full text, but they are not load-bearing.

This is a narrow technical note aimed at people already working on marker sets or related tiling questions in Z^n. A reader inside that subfield gets a small, usable cleanup. It is worth sending to a serious referee because the claim is precise, the fix is correct, and the evidence is the reduction itself.

Referee Report

0 major / 0 minor

Summary. The manuscript extends Gao and Wang's strong clopen marker theorem for finite generating sets of Z^n by removing the assumption that each generator has support of size 1 or n. The proof is a short conjugacy argument: a unimodular change of coordinates places any finite set of nonzero lattice vectors in full-support position, the prior theorem is applied, and the resulting marker set is conjugated back.

Significance. If the result holds, it removes a restrictive hypothesis from an existing theorem, yielding a strictly more general statement on the existence of strong marker sets for arbitrary finite generating sets of the integer lattice. The argument is concise, invokes only the standard fact that the complement of finitely many hyperplanes in GL(n,Z) is nonempty, and preserves the marker property under lattice automorphisms. This strengthens the theorem's applicability without introducing new parameters or ad-hoc constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report accurately summarizes the contribution as a short conjugacy argument that removes the support-size hypothesis from Gao and Wang's theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; external theorem plus standard linear-algebra existence argument

full rationale

The derivation applies Gao-Wang's prior theorem after a coordinate change whose existence follows from avoiding a finite union of hyperplanes (standard fact about GL(n,Z) and linear inequalities over Z). This step is independent of the marker-set conclusion and does not reduce any quantity to a fitted input or self-referential definition. No self-citation is load-bearing; the cited result is treated as external. The paper is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The proof rests on standard facts from integer linear algebra; the key step is presented as always possible rather than constructed ad hoc.

assumptions (1)

standard math For any finite set of nonzero vectors in Z^n there exists a unimodular matrix A in GL(n,Z) such that every vector in the image set has full support.
Invoked as the enabling step that allows reduction to the Gao-Wang theorem.

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Cite this review

Pith. "Pith review of Strong marker sets for arbitrary generating sets of $\mathbb Z^n$." pith.science (2026). https://pith.science/paper/2QDXJGAI

@misc{pith2026260606707,
  author       = {Pith},
  title        = {Pith review of: Strong marker sets for arbitrary generating sets of $\mathbb Z^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QDXJGAI}},
  note         = {Machine review of arXiv:2606.06707}
}

abstract

Gao and Wang proved a strong clopen marker theorem for finite generating sets of $\mathbb Z^n$ under the assumption that each generator has support of size either $1$ or $n$. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

org/abs/2502.00598(preprint), 2025 (cit

[GW25] Su GaoandTianhao Wang.Strong marker sets and applications, https://arxiv. org/abs/2502.00598(preprint), 2025 (cit. on p. 1) 3

work page arXiv 2025

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

org/abs/2502.00598(preprint), 2025 (cit

[GW25] Su GaoandTianhao Wang.Strong marker sets and applications, https://arxiv. org/abs/2502.00598(preprint), 2025 (cit. on p. 1) 3

work page arXiv 2025