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Strictly Expansive Matrices

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Pith's one-line read An expansive integer matrix that is strictly diagonally dominant and strongly connected is strictly expansive; determinant-two expansive integer matrices escape symmetric, convex, and connected-boundary tiling sets.

desk verdict Two real results with patchable gaps: the strictly expansive theorem is likely correct but the proof of Lemma 2.7 needs a missing one-line argument, and the 2D classification has an unproved lemma. read the letter →

arxiv 2506.07538 v1 pith:JBQZ2C4D submitted 2025-06-09 math.CA

Darrin Speegle This is my paper
classification math.CA MSC 42C2015A1252C20
keywords strictlyexpansivematricesMeyer-typewaveletsdiagonaldominanceintegerdilationtranslationtilingslatticepackingsdeterminanttwostronglyconnectedgraphs
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

The paper establishes sufficient and negative conditions for an integer expansive matrix to be strictly expansive, meaning that a compact set $K$ tiles $\mathbb{R}^n$ by integer translations and $A(K^\circ)\supset K$. On the positive side, it proves that if $A$ is expansive, integer valued, and strictly diagonally dominant, and if the directed graph of its nonzero entries is strongly connected, then $A$ is strictly expansive. On the negative side, it proves that no expansive integer matrix with determinant two can be strictly expansive with respect to a tiling set that is symmetric, convex, or has connected boundary. These results matter because strict expansiveness is a known sufficient condition for the existence of a Meyer-type orthonormal $A$-wavelet: a wavelet basis whose Fourier transforms are smooth and compactly supported, with $|\det A|-1$ generators. The positive theorem gives a checkable matrix-entry condition for a large class of dilations, while the negative theorem shows that any strictly expansive set for determinant two would have to be quite irregular, leaving open whether such matrices are strictly expansive at all.

What carries the argument

The engines are: (i) the infinity-norm bound for diagonally dominant matrices, $\|A^{-1}\|_\infty \leq 1/\alpha$, which makes the $\alpha=1$ case reduce to a norm equal to 1; (ii) Lemma 2.7, which says strict expansiveness follows once the unit ball has exactly two boundary points $x$ with $\|Ax\| = 1$, because any third such point would produce a third preimage point and violate Lemma 2.6; (iii) the directed graph of nonzero entries, whose strong connectivity propagates signs across coordinates so that the set of norm-one vectors with $\|Ax\|_\infty = 1$ has exactly two elements; and (iv) for the negative results, an index-two sublattice $\Gamma$ of $\mathbb{Z}^n$ together with Lemma 2.13: if a tiling set $K$ contains two distinct points whose difference is a nonzero element of $\Gamma$, then no open set containing $K$ can pack by $\Gamma$-translations, which blocks $A(K^\circ) \supset K$ when $A\mathbb{Z}^n$ has index two.

What would settle it

Concretely, take $A=\begin{pmatrix}0&1\\2&0\end{pmatrix}$, an expansive integer matrix with determinant $-2$, and search for a compact convex set $K$ tiling $\mathbb{R}^2$ by $\mathbb{Z}^2$ translations with $A(K^\circ)\supset K$; finding one would disprove Theorem 2.16(2). Alternatively, for any compact convex tiling set $K$ and any index-two sublattice $\Gamma$ of $\mathbb{Z}^n$, count the boundary points of $K$ modulo $\Gamma$: if some such $K$ has no three distinct boundary points whose pairwise differences lie in $\Gamma$, then the asserted Lemma 2.5 route in Theorem 2.15(2) cannot be correct.

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Extended reading notes

Core claim

The central claim is Theorem 2.8: an expansive, integer-valued, strictly diagonally dominant matrix $A$ whose associated directed graph is strongly connected is strictly expansive. The proof uses the infinity-norm bound for diagonally dominant matrices to get $\|A^{-1}\|_\infty \leq 1$; when equality holds, Lemma 2.7 reduces the task to showing that exactly two points $x$ on the boundary of the unit cube satisfy $\|Ax\|_\infty = 1$. Strong connectivity then forces every coordinate of such an $x$ to be determined up to the sign of one chosen coordinate, so there are exactly two such points, and Lemma 2.6 converts this fact into a compact tiling set $K$ with $A(K^\circ) \supset K$. The paper's second theorem, Theorem 2.16, asserts that if $|\det A| = 2$ then no compact tiling set $K$ that is centrally symmetric, convex, or connected in boundary can satisfy $A(K^\circ) \supset K$; the argument embeds the lattice $A\mathbb{Z}^n$ as an index-two sublattice of $\mathbb{Z}^n$ and shows that any $K$ of those types would force two points of $K$ to differ by a nonzero element of that sublattice, making $K$ unable to pack by the sublattice translations.

Load-bearing premise

The load-bearing premise for the determinant-two convex exclusion is that a compact convex set tiling $\mathbb{R}^n$ by $\mathbb{Z}^n$ translations has three distinct boundary points pairwise differing by elements of the index-two sublattice $\Gamma$, as asserted in Theorem 2.15(2); the proof cites Lemma 2.5 for this, but the lemma as proved yields only two such points, so the convex part of Theorem 2.16(2) depends on that step being repairable.

Editorial extensions

If this is right

  • Every expansive integer matrix that is strictly diagonally dominant with strongly connected nonzero-entry graph admits a Meyer-type orthonormal $A$-wavelet with $|\det A| - 1$ smooth, compactly supported Fourier-side generators.
  • The certificate $K$ for such matrices can be constructed explicitly: Lemma 2.6 starts from the unit cube and swaps small neighborhoods of two boundary points, producing a compact tiling set that satisfies $A(K^\circ) \supset K$.
  • An expansive integer matrix with determinant two, if it is strictly expansive at all, cannot have a strictly expansive set $K$ that is centrally symmetric, convex, or has connected boundary; any such $K$ would be irregular in all of these ways.
  • Proposition 2.18 upgrades the earlier measurable existence result: for every expansive integer matrix $A$ there is a compact tiling set $K$ with $0$ in its interior and $AK \supset K$, so the obstruction for determinant two is specific to the stronger containment $A(K^\circ) \supset K$ combined with set regularity.

Reading between the lines

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

  • The sign-propagation argument suggests a possible characterization: for strongly connected strictly diagonally dominant integer matrices, the equality case $\|A^{-1}\|_\infty = 1$ can occur only when the directed graph's structure forces the norm-one boundary set to be a single antipodal pair, so the strict expansiveness certificate may be readable directly from the graph's cycles.
  • The index-two sublattice obstruction likely has analogues for other prime determinants: for $|\det A| = p$ the relevant sublattice has index $p$, and a similar boundary-point counting condition would govern whether regular tiling sets can exist.
  • The open matrix $A=\begin{pmatrix}0&1\\2&0\end{pmatrix}$ is a concrete test case: the negative results say any strictly expansive $K$, if one exists, must be non-symmetric, non-convex, and have disconnected boundary, which makes a computer search over irregular compact tilers a feasible next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies strictly expansive integer matrices: expansive maps A for which there exists a compact set K tiling R^n by Z^n-translations such that A(K^circ) contains K. The main positive result, Theorem 2.8, claims that every expansive, integer-valued, strongly connected, (strictly) diagonally dominant matrix is strictly expansive. The main negative results, Theorem 2.16, state that an expansive integer matrix with determinant 2 cannot be strictly expansive with respect to a symmetric, convex, or connected-boundary tiling set. The paper also proves a compactness improvement, Proposition 2.18, of earlier existence results for K with AK containing K, and it sketches the two-dimensional classification of strict expansivity with respect to convex centrally symmetric sets. The arguments are mostly elementary and self-contained, relying on Varah's theorem, Minkowski's theorem, and prior wavelet results of Bownik and Speegle.

Significance. If Theorem 2.8 is correct, it provides a broad and easily checkable sufficient condition for an integer expansive matrix to admit a Meyer-type orthonormal wavelet with |det A|-1 generators via the prior result [2]. The determinant-two obstruction in Theorem 2.16 is a useful complement, and the paper explicitly leaves open an interesting borderline case. The manuscript is careful to separate positive and negative results and includes several fully proved auxiliary lemmas. However, the current proof of the central positive theorem relies on a lemma that is not valid as stated, and the proof of the convex case in Theorem 2.15 contains an unsupported assertion; both gaps are repairable but must be addressed before the results are established.

major comments (4)
  1. [Section 2, Lemma 2.7 and Theorem 2.8] Lemma 2.7 is not a consequence of Lemma 2.6. The proof only rules out a third boundary point z with A^{-1}z in ∂K, but Lemma 2.6 requires exactly two points w in K with ||A^{-1}w||_K=1. If z is any boundary point with ||Az||_K<1, then w=Az lies in K^circ and satisfies ||A^{-1}w||_K=||z||_K=1, so the hypothesis of Lemma 2.6 is not met. Thus the bridge from the two-point boundary condition to strict expansivity is missing. In the specific setting of Theorem 2.8 the gap is closed by noting, as in Claim 2.9, that for z in ∂B with ||Az||_infty ≤ 1, the inequality 1 ≥ |(Az)_m| ≥ |a_mm| - Σ_{j≠m}|a_mj| ≥ 1 forces |(Az)_m|=1 for any m with |z_m|=1, so Az is on ∂B. This argument should be inserted into the proof of Theorem 2.8, and Lemma 2.7 should either be reformulated with the additional hypothesis that A maps boundary points with image in K to the boundary, or be deleted.
  2. [Theorem 2.15(2)] The proof of the convex case asserts: 'By Lemma 2.5, there exist distinct y1,y2,y3 in ∂K with y_i - y_j ∈ Γ.' Lemma 2.5(2) does not provide this; it supplies, for an extreme point x, two distinct boundary points y1,y2 with x - y_i ∈ Z^n. The intended argument is nonetheless sound: applying Lemma 2.13 directly to the triple x,y1,y2, whose pairwise differences lie in Z^n, proves that no open set containing K can pack by Γ-translations. This repair should be made explicit, as the current one-line proof is not valid as written.
  3. [Theorem 2.15(3)] The final step of the connected-boundary case says 'The result then follows from Lemma 2.11.' Lemma 2.11 requires a difference in Γ, but the construction produces points x, x+k1, x+k2 in ∂K with k1-k2 in Z^n, not necessarily in Γ. The correct tool is Lemma 2.13, whose hypotheses are satisfied by the triple x, x+k1, x+k2. As written, the connected-boundary case is not proved; this is a local citation error, but it must be corrected.
  4. [Theorems 2.15 and 2.16] Both theorems are stated for all n, but the proof of Theorem 2.15(1) explicitly assumes n>1, and the statements are false for n=1. For example, take K=[-1/2,1/2] and Γ=2Z; an open neighborhood of K packs by 2Z-translations, contradicting Theorem 2.15. Similarly, A=[2] is an expansive integer matrix with determinant 2, and K=[-1/2,1/2] satisfies A(K^circ) containing K, contradicting Theorem 2.16(1). Add the hypothesis n≥2 to both theorems, or treat the one-dimensional case separately.
minor comments (5)
  1. [Theorem 2.8 statement] The theorem says 'diagonally dominant', but the proof and the introduction require strict diagonal dominance with integer margin at least 1; otherwise the inequality chain in Claim 2.9 gives only 1 ≥ |(Az)_m| ≥ 0, not 1 ≥ |(Az)_m| ≥ 1. The hypothesis should be stated as 'strictly diagonally dominant'.
  2. [Proposition 2.18] The definition of \tilde E_m contains a likely typo: the union inside the parentheses uses E_{m-1} rather than E_i, and later 'E_{i+1}' should presumably be 'E_{m0+1}'. Please clarify the intended formula.
  3. [Proposition 2.18] Before applying Lemma 2.17, \tilde E_m must be compact; as written, \tilde E_m is the difference of a compact set and a closed set, which need not be closed. Either prove compactness or take a compact subset with the same union a.e. as guaranteed by Lemma 2.17.
  4. [Proposition 2.18] The proof that K is a finite union of the E_i should justify that the smallest J with A^{-J}x in Q is uniformly bounded for x in a fundamental domain; this follows from compactness and the expansivity of A, but it is not stated.
  5. [Theorem 2.2 and Corollary 2.3] Varah's theorem is normally stated for strictly diagonally dominant matrices; the paper's wording 'diagonally dominant' should be tightened to 'strictly diagonally dominant' to match the constant α>0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: main theorem is derived from Varah's bound and self-contained lemmas; self-citations are contextual only.

full rationale

The derivation chain in this paper is essentially self-contained. Theorem 2.8 uses Varah's theorem (Theorem 2.2) as an external norm bound, Lemma 2.7 as a transfer statement about boundary points, and Lemma 2.6, whose proof directly constructs a modified tiling T and verifies A^{-1}T subset T^o. No parameter is fitted, no quantity is defined in terms of the result it is supposed to establish, and no uniqueness theorem from the authors' earlier work is invoked to force a choice. The determinant-two obstruction Theorem 2.16 is obtained from Theorem 2.15, whose three cases are proved from the elementary tiling Lemmas 2.11-2.13 and the classical Minkowski convex-tiling fact; again no input is renamed as a conclusion. Self-citations to [2] and [3] appear mainly in the two-dimensional summary and as motivation for the Meyer-type wavelet consequence, but the new results in this paper are supported by their own proofs, and the cited prior results are independent published statements rather than premises that already contain the present claims. Even where a proof gap may exist (e.g. the boundary-to-interior transfer in Lemma 2.7 or the three-boundary-point assertion in Theorem 2.15(2)), that is a correctness question, not circularity, since the disputed steps are not equivalent to the theorems they are used to prove by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The paper's dependency on standard results is normal; the notable ad hoc item is Lemma 3.7, which is unproved.

assumptions (5)
  • standard math Varah's inverse norm bound: a strictly diagonally dominant matrix A satisfies ||A^{-1}||∞ ≤ 1/α (Theorem 2.2, [19])
    Used in Corollary 2.3 and Theorem 2.8 to get ||A^{-1}||∞ ≤ 1 for integer strict diagonal dominance.
  • standard math Minkowski's theorem: a convex body that tiles by translations is centrally symmetric (cited via [16])
    Used in Theorem 2.15(2) to assert convex tilers are centrally symmetric.
  • standard math Existence of an ellipsoid Q with A Q^◦ ⊃ Q for expansive A (Proposition 2.1(1), [18])
    Basis for Proposition 2.18's construction.
  • standard math Classification of convex subsets of R×[-1/2,1/2] tiling by Z^2 as those in Lemma 3.5 (from [3])
    Used in Proposition 3.6 to reduce to checking equation (3.1).
  • ad hoc to paper Lemma 3.7: if a compact convex centrally symmetric set K has a point (0,z) with |z| ≥ 1/2 in its interior (or the other listed configurations), it does not tile R^2 by translations
    Presented without proof in Section 3; load-bearing for Proposition 3.8.

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Pith. "Pith review of Strictly Expansive Matrices." pith.science (2026). https://pith.science/paper/JBQZ2C4D

@misc{pith2026250607538,
  author       = {Pith},
  title        = {Pith review of: Strictly Expansive Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBQZ2C4D}},
  note         = {Machine review of arXiv:2506.07538}
}
abstract

If $A$ is an integer valued, strictly expansive matrix, then there exists an orthonormal $A$-wavelet whose Fourier transform is compactly supported and smooth. We show that strongly connected diagonally dominant integer matrices are strictly expansive, and that integer matrices with determinant two are not strictly expansive with respect to particularly nice sets.

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Works this paper leans on

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