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Tiles and weak tiles in $\mathbb{Z}_{pq}$

T0 review · 0 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read In the cyclic group Z_pq with distinct primes p and q, every positive-definite weak tile is a classical translational tile.

desk verdict Clean, checkable case analysis that settles pd-tile = tile on Z_pq; the averaging worry is not load-bearing. read the letter →

arxiv 2607.02149 v2 pith:GOUPPQSC submitted 2026-07-02 math.CA

Mamateli Kadir , Kaibo Fan This is my paper
classification math.CA MSC 43A9905B4526E30
keywords weaktilepositive-definitefinitecyclicgroupDelsarteparametersCoven–MeyerowitzconditionsFourieranalysistranslationaltiling
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

This paper settles a natural question about how much one can relax the definition of tiling and still recover ordinary tiling, for the cyclic group of order pq. A positive-definite weak tile (pd-tile) is a set that tiles the group when convolved against some nonnegative, positive-definite density normalized at zero; ordinary tiles and spectral sets are both pd-tiles, but the converse can fail. The authors prove that, for Z_pq, the converse does not fail: every pd-tile is necessarily a classical tile. The argument classifies the possible Fourier supports of a candidate set, feeds those supports into Delsarte linear-programming parameters to force the set to have one of the sizes that arise from known tiles, and then checks the Coven–Meyerowitz algebraic conditions that characterize tiling when the cardinality has only two prime factors. The result enlarges the short list of groups where pd-tiling and tiling are known to coincide.

What carries the argument

Delsarte parameters D+(H) and D−(H) on unions of step classes H, together with their duality D+(H)D−(H′)=pq. These bounds force the cardinality of a pd-tile to equal one of the four sizes {1,p,q,pq} that arise from the six model tilings of Z_pq; the Coven–Meyerowitz conditions then finish the proof that the set tiles.

What would settle it

Exhibit a nonempty proper subset A of Z_pq whose Fourier support is one of the six allowed unions of step classes, that admits a nonnegative positive-definite weak tiling measure f, yet fails either Coven–Meyerowitz condition (T1) or (T2).

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Extended reading notes

Core claim

For distinct primes p and q, a subset A of the cyclic group Z_pq is a positive-definite weak tile if and only if it is a translational tile: there exists B such that A ⊕ B = Z_pq. Every classical tile is already a pd-tile; the new content is the converse.

Load-bearing premise

The claim rests on being able to average the Fourier-side test functions down to step functions without changing their total mass or feasibility; if that averaging step fails for some supports that only pd-tiles can have, the size bound no longer forces a classical tile.

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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

0 major / 5 minor

Summary. The paper proves that in the cyclic group Z_pq (p,q distinct primes), a subset A is a positive-definite weak tile (pd-tile) if and only if it is a classical translational tile. After recalling Fourier analysis on finite abelian groups, mask polynomials, step classes, and Delsarte parameters, the authors classify the possible Fourier supports H = supp(|1̂_A|^{2}). Two short lemmas eliminate the supports {0}∪R_1 and {0}∪R_p∪R_q, leaving exactly the six supports that arise from classical model tilings. Dual Delsarte bounds then force |A| ∈ {1,p,q,pq}. Because A and the corresponding model tile D share the same cyclotomic divisors, A satisfies the Coven–Meyerowitz conditions (T1)+(T2) and therefore tiles Z_pq.

Significance. The result cleanly extends the known list of groups on which pd-tiling coincides with translational tiling (previously Z_p, Z_p^{2}, Z_{p^n}) to the first square-free product of two distinct primes. The argument is a transparent case analysis that combines three standard tools—Fourier support classification, Delsarte duality, and Coven–Meyerowitz—without introducing new ad-hoc constructions. It supplies a concrete positive instance of the program outlined by Kiss–Matolcsi–Somlai and related works, and the open questions posed at the end (extensions to Z_{p^n q}, Z_{pqr}, structural decomposition of weak tiling measures) are natural and well-motivated.

minor comments (5)
  1. Abstract and title speak of “weak tiles,” while the body (Definitions 1.1–1.2 and Theorem 1.3) works exclusively with positive-definite weak tiles (pd-tiles). A one-sentence clarification that the equivalence is proved for the pd-notion would remove any ambiguity.
  2. Remark 3.5 invokes the averaging procedure of [7] to replace the feasible functions u and v by step functions. Although the subsequent case analysis shows that only classical supports remain (so the equality |A|=D+(H′) is already forced by the model tilings), a brief parenthetical note that the averaging is used only for feasibility, not optimality, would make the logical dependence clearer.
  3. Figure 1 is helpful but the four question marks are never explained in the caption or surrounding text; a short legend indicating which arrows remain open would improve readability.
  4. Typographical inconsistencies: “Z pq” versus “Z_{pq}”, missing spaces after commas in several places (e.g., “p,qbe”), and the arXiv identifier appears as 2607.02149 (presumably a placeholder). Standard LaTeX cleanup would suffice.
  5. In the list of model tilings (pp. 10–11) the Fourier-support calculations are correct but terse; adding one intermediate line for the geometric-sum evaluation in case (1) would help non-specialist readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: |A| is forced by independent dual bounds on the six admissible Fourier supports, then C-M is verified elementary.

full rationale

The derivation of Theorem 1.3 is self-contained against external benchmarks and does not reduce the conclusion to its inputs by construction. After ruling out two impossible supports via Lemma 2.2 and Lemma 3.4, the only remaining H are exactly the six supports that arise from classical model tilings of Z_pq. For each such H the paper invokes the independent duality D+(H)D-(H')=M and the model-tiling evaluation D+(H')=|D| from [7, Prop. 3.3], then sandwiches |A| between a feasible lower bound from the normalized |1_A|^2 and a feasible upper bound from the normalized bf; equality forces |A| in {1,p,q,pq}. The subsequent verification that A and the model D share the same cyclotomic divisors (hence the same S_A) and that (T1)+(T2) hold is elementary polynomial arithmetic, independent of any fitted parameter or self-referential definition. The averaging remark (3.5) that replaces u,v by step functions is not load-bearing: the model tilings already supply explicit step-function witnesses attaining the dual values, so the sandwich survives without it. There is no self-citation by the present authors, no uniqueness theorem imported from their own prior work, and no quantity defined in terms of the claim being proved. Score 0 is therefore the correct assessment.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure-existence proof inside finite abelian harmonic analysis. It imports standard Fourier analysis, the definition of Delsarte parameters, the Coven–Meyerowitz conditions, and one averaging lemma from the literature. No free parameters are fitted; no new physical or combinatorial entities are postulated. The only non-standard ingredients are the six model supports (which are elementary) and the exclusion lemma proved in the paper itself.

assumptions (4)
  • standard math Fourier transform and inversion on finite abelian groups; convolution theorem.
    Used throughout §2 and §3 to translate the convolution equation 1_A * f = 1 into support conditions on Fourier transforms.
  • domain assumption Delsarte parameters D+(H), D−(H) and the duality D+(H) D−(H′) = |G| for step-function feasible sets (from [7]).
    Invoked in Step 2–3 of the proof to force |A| = D+(H′).
  • domain assumption Coven–Meyerowitz conditions (T1)+(T2) are necessary and sufficient for a subset of Z_pq to tile (known for |A| with at most two prime factors).
    Used in Step 4 to conclude that A tiles once |A| and the cyclotomic divisors match those of a model tile.
  • domain assumption Averaging any feasible function for the Delsarte programs to a step function preserves the objective value and feasibility ([7]).
    Remark 3.5; without it the functions u and v constructed from 1_A and f need not lie in A(M).

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Cite this review

Pith. "Pith review of Tiles and weak tiles in $\mathbb{Z}_{pq}$." pith.science (2026). https://pith.science/paper/GOUPPQSC

@misc{pith2026260702149,
  author       = {Pith},
  title        = {Pith review of: Tiles and weak tiles in $\mathbbZ_pq$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOUPPQSC}},
  note         = {Machine review of arXiv:2607.02149}
}
abstract

This paper investigates the relationship between tiles and weak tiles in the context of finite cyclic group $\mathbb{Z}_{pq}$. We prove that weak tiles and translational tiles are equivalent in this group. Our proof employs Fourier analysis, Delsarte parameters, and the Coven-Meyerowitz conditions.

Figures

Figures reproduced from arXiv: 2607.02149 by the authors.

Figure 1
Figure 1. The relationship between tiles, spectral sets, weak tiles, pd-tiles, and C-M conditions in cyclic groups. The remainder of the paper is organized as follows. In Section 2, we collect the necessary preliminaries, including the Fourier transform on finite abelian groups, mask polynomials and cyclotomic divisibility, step functions, and the Delsarte linear programming framework. Section 3 presents the proof of the main… view at source ↗

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Works this paper leans on

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