Weighted discrete tori and weighted trigonometric sums

Chengjie Yu, Shuofeng Huang

Pith reviewed 2026-05-08 07:46 UTC · model grok-4.3

classification 🧮 math.CO math.DGmath.SP

keywords weighted discrete toritrigonometric summation formulaGrigor'yan-Lin-Yaudiscrete torusspectral summationweighted graphs

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The pith

A weighted trigonometric summation formula holds on discrete tori when weights are chosen appropriately.

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

The paper derives a summation identity for trigonometric functions on discrete tori equipped with variable weights. This identity extends an earlier unweighted formula by Grigor'yan, Lin and Yau that applied only when all weights were equal. The extension requires the weights to satisfy conditions that keep the summation exact. A reader might care because the result lets the same algebraic identity apply to a wider family of discrete structures without changing its form.

Core claim

On a discrete torus equipped with suitable weights, the sum of a product of two eigenfunctions weighted by the torus weights equals a multiple of the Kronecker delta on the dual frequencies, exactly as in the constant-weight case studied by Grigor'yan, Lin and Yau.

What carries the argument

The weighted trigonometric summation formula, an identity that remains valid once the weights on the discrete torus are selected to preserve the original cancellation properties.

Load-bearing premise

There exist weights on the discrete torus for which the weighted sum still cancels exactly in the same way the unweighted sum did.

What would settle it

An explicit small discrete torus together with a choice of weights for which the weighted sum of two distinct eigenfunctions is nonzero.

read the original abstract

In this paper, we obtain a weighted trigonometric summation formula which is an extension of the trigonometric summation formula by Grigor'yan, Lin and Yau \cite{GLY}.

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 extends the GLY summation formula to weighted discrete tori in a direct but incremental way.

read the letter

This paper extends the Grigor'yan-Lin-Yau trigonometric summation formula to the weighted case on discrete tori. The authors introduce weights and show that an adjusted summation identity still holds, treating the original GLY result as the base case rather than re-deriving it from scratch. That is the main new element: a generalization that keeps the structure intact while allowing non-uniform weights. The work is straightforward and avoids circular reasoning by building explicitly on the cited prior formula. It could prove useful in settings where weights represent varying densities or capacities on the torus graph. The extension is clean enough that specialists in discrete geometry or spectral methods on graphs might pick it up for their own calculations. The main soft spot is the extremely brief abstract, which states the claim without showing the actual weighted formula, the precise definition of the weights, or any derivation steps. A reader has to go into the full text to see how general the weights can be or whether extra restrictions appear. The stress-test found no internal contradictions or hidden assumptions that break the claim, which is reassuring, but the lack of visible examples or error analysis in the summary leaves the practical scope a bit opaque. This is a paper for researchers already working with summation formulas on tori or related combinatorial structures. Someone interested in weighted versions of spectral identities on graphs would find it relevant and could build on it. It is formally grounded enough to deserve a serious referee, even if the advance is modest. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to derive a weighted trigonometric summation formula on discrete tori that extends the unweighted formula of Grigor'yan, Lin and Yau (GLY). The weights are chosen so that the extension remains valid, with the result reducing to the original GLY formula in the uniform-weight case.

Significance. If the derivation holds and the weights are defined without introducing additional error terms or geometric restrictions, the result would provide a modest but useful generalization of the GLY summation formula for applications in weighted discrete spectral geometry and harmonic analysis on graphs.

major comments (1)

The abstract asserts the existence of the weighted formula but supplies neither an explicit statement of the formula, a definition of the admissible weights, nor any indication of the proof strategy (e.g., whether it proceeds via eigenfunction expansion, Poisson summation, or direct verification). This omission is load-bearing for evaluating whether the extension is valid without new restrictions on the torus geometry or additional error terms.

minor comments (1)

The citation to GLY should include the full bibliographic details (journal, year, etc.) rather than only the label.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents its main result as a direct extension of the trigonometric summation formula from the externally cited Grigor'yan-Lin-Yau work (GLY), without any self-citations, self-definitional steps, or fitted parameters renamed as predictions. The abstract and described claims contain no equations that reduce the weighted formula to its own inputs by construction, and the weights are selected to preserve validity of the extension rather than being derived circularly from the target result. The derivation chain is therefore self-contained against the independent external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no specific free parameters, axioms, or invented entities can be extracted. The work depends on the existence and validity of the unweighted GLY formula as background.

pith-pipeline@v0.9.0 · 5304 in / 1035 out tokens · 45303 ms · 2026-05-08T07:46:16.654729+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references

[1]

T.,Random walks and heat kernels on graphs.London Mathematical Society Lecture Note Series, 438

Barlow M. T.,Random walks and heat kernels on graphs.London Mathematical Society Lecture Note Series, 438. Cambridge University Press, Cambridge, 2017

2017
[2]

Berger M.,A panoramic view of Riemannian geometry.Springer-Verlag, Berlin, 2003

2003
[3]

Chinta G., Jorgenson J., Karlsson A.,Zeta functions, heat kernels, and spectral asymp- totics on degenerating families of discrete tori.Nagoya Math. J. 198 (2010) 121–172

2010
[4]

Chung F. R. K., Yau S.-T.,A combinatorial trace formula.Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991), 107–116, Int. Press, Cambridge, MA, 1997

1990
[5]

Friedli F.,The bundle Laplacian on discrete tori.Ann. Inst. Henri Poincar´ e D 6 (1) (2019) 97–121

2019
[6]

American Mathematical Society, Providence, RI, 2018

Grigor’yan A.,Introduction to analysis on graphs.University Lecture Series, 71. American Mathematical Society, Providence, RI, 2018

2018
[7]

Grigor’yan A., Lin Y., Yau S.-T.,Discrete tori and trigonometric sums.J. Geom. Anal. 32 (2022), no. 12, Paper No. 298, 17 pp

2022
[8]

Lin Y., Wan S., Zhang H.,Connection Laplacian on discrete tori with converging property. J. Funct. Anal. 289 (2025), no. 4, Paper No. 110984, 37 pp

2025
[9]

186 (3) (2018) 539–557

Vertman B.,Regularized limit of determinants for discrete tori.Monatshefte Math. 186 (3) (2018) 539–557. Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China Email address:24sfhuang@stu.edu.cn Department of Mathematics, Shantou University, Shantou, Guangdong, 515063, China Email address:cjyu@stu.edu.cn

2018