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

Kaibo Fan; Mamateli Kadir

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

Tiles and weak tiles in mathbb{Z}_(pq)

Mamateli Kadir , Kaibo Fan This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-03 02:57 UTCgrok-4.3pith:GOUPPQSC record.json open to challenge →

classification math.CA

keywords weak tilestranslational tilescyclic groupsZ_pqFourier analysisCoven-Meyerowitz conditionsDelsarte parameters

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The pith

Weak tiles and translational tiles are equivalent in the cyclic group Z_pq.

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

The paper proves that in the finite cyclic group Z_pq a subset tiles by translation if and only if it is a weak tile. This unification of the two notions is achieved through Fourier analysis on the group, the Coven-Meyerowitz conditions, and Delsarte parameters. A reader might care because it means that the apparently weaker notion of weak tiling is in fact equivalent to the standard one in these specific groups. The result applies to groups of order pq where p and q are distinct primes.

Core claim

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.

What carries the argument

Fourier analysis combined with Coven-Meyerowitz conditions and Delsarte parameters to establish equivalence of tiling notions in Z_pq

If this is right

Tiling by translation and weak tiling are interchangeable concepts in Z_pq.
Any property proven for one type of tile applies to the other in this group.
The study of tiles in Z_pq can proceed using either definition without loss of generality.

Where Pith is reading between the lines

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

The method might extend to show equivalence in other cyclic groups.
Computational verification of tiling could be simplified by choosing the easier definition to check.
This equivalence highlights the special structure of groups with order a product of two primes.

Load-bearing premise

The Coven-Meyerowitz conditions and Delsarte parameters can be applied to Z_pq in combination with Fourier analysis to force the equivalence between weak tiles and translational tiles.

What would settle it

A counterexample subset in Z_pq that tiles weakly but does not tile translationally, or the reverse, would disprove the equivalence.

Figures

Figures reproduced from arXiv: 2607.02149 by Kaibo Fan, Mamateli Kadir.

**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 o… view at source ↗

read the original 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.

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 shows weak tiles equal translational tiles in Z_pq by combining Fourier analysis with Delsarte parameters and Coven-Meyerowitz conditions, but the result stays narrow.

read the letter

The main result is that weak tiles and translational tiles coincide in the cyclic group of order pq. The proof combines Fourier analysis on the group with Delsarte parameters and the Coven-Meyerowitz conditions to force the equivalence.

The work applies these standard tools directly to the pq case and obtains the claimed equivalence without apparent contradiction. That is the concrete contribution: a verification for this family of groups.

The argument looks routine once the conditions are in place, so the advance is incremental. The scope is narrow; nothing is said about other cyclic orders or about whether the same equivalence holds more generally. No new phenomena or counterexamples appear in the abstract, and the method does not extend the existing toolkit.

A reader already working on tiling problems in finite abelian groups will see the case filled in, but the paper will not shift broader questions in harmonic analysis. The citation pattern is light on recent related results, which is typical for a short note but leaves open whether earlier papers already covered this instance.

The paper is for specialists who track tiling equivalences in cyclic groups. It is coherent on its own terms and uses established external conditions, so it deserves referee time to check the details of the application.

Referee Report

0 major / 2 minor

Summary. The paper claims to prove that weak tiles and translational tiles are equivalent in the finite cyclic group Z_pq (with p, q distinct primes). The argument relies on Fourier analysis on the group, combined with Delsarte parameters and the Coven-Meyerowitz conditions to establish the equivalence.

Significance. If the central claim holds, the result would clarify the relationship between two notions of tiling in a standard family of cyclic groups, providing a concrete case where the notions coincide. The use of established tools (Fourier analysis, Delsarte parameters, Coven-Meyerowitz conditions) without introducing free parameters or ad-hoc entities is a strength that supports verifiability.

minor comments (2)

[Abstract] Abstract: the statement that the result holds 'in this group' would be clearer if it explicitly noted the assumption that p and q are distinct primes (standard for Z_pq but worth stating).
[Introduction] The manuscript would benefit from an explicit statement of the main theorem (e.g., 'Theorem 1.1: ...') early in the introduction, together with a brief outline of how the three tools are combined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The recommendation for minor revision is noted. No major comments were provided in the report, so there are no specific points requiring point-by-point rebuttal or revision at this stage. We remain available to address any minor issues or clarifications that may arise.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard tools

full rationale

The paper states it proves equivalence of weak tiles and translational tiles in Z_pq via Fourier analysis, Delsarte parameters, and Coven-Meyerowitz conditions. These are established external methods in the literature on tiling in cyclic groups, not defined or fitted within the paper itself. No equations, self-citations, or internal reductions are exhibited that would make any claimed result equivalent to its inputs by construction. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract supplies no information on free parameters, background axioms, or newly postulated entities.

pith-pipeline@v0.9.1-grok · 5555 in / 934 out tokens · 39327 ms · 2026-07-03T02:57:58.407171+00:00 · methodology