Periodic Hypersurfaces and Lee-Yang Polynomials

Lior Alon; Mario Kummer

arxiv: 2507.16029 · v3 · submitted 2025-07-21 · 🧮 math.AG

Periodic Hypersurfaces and Lee-Yang Polynomials

Lior Alon , Mario Kummer This is my paper

Pith reviewed 2026-05-19 03:29 UTC · model grok-4.3

classification 🧮 math.AG

keywords periodic hypersurfacesLee-Yang polynomialslighthouse measuresFourier quasicrystalsrigidity theoremtorus zero setsalgebraic geometry

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The pith

Any periodic C^{1+ε} hypersurface supporting a directional lighthouse measure must be the torus zero set of an essentially Lee-Yang polynomial.

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

This paper shows that periodic hypersurfaces in Euclidean space which are differentiable enough and carry measures whose Fourier transforms are confined to a proper double cone must come from Lee-Yang polynomials. Lee-Yang polynomials are a family from statistical mechanics whose zero sets on the torus naturally support such directional lighthouse measures. The rigidity theorem classifies these hypersurfaces using a recent result on one-dimensional Fourier quasicrystals. A reader would care because it reveals that many periodic structures with restricted frequencies are algebraic rather than generic smooth objects. This gives a geometric view of quasicrystal classifications.

Core claim

Any periodic C^{1+ε} hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee-Yang polynomial. The proof uses the classification of one-dimensional Fourier quasicrystals to establish this rigidity and provides a geometric interpretation of that theory.

What carries the argument

The rigidity theorem that identifies periodic hypersurfaces supporting directional lighthouse measures with the torus zero sets of essentially Lee-Yang polynomials.

If this is right

These hypersurfaces are necessarily the zero loci of polynomials with the Lee-Yang property when viewed periodically on the torus.
The directional lighthouse condition combined with periodicity and smoothness forces an algebraic structure.
The result applies the quasicrystal classification to geometric objects in higher dimensions.
Examples from Lee-Yang polynomials are essentially the only ones satisfying the conditions.

Where Pith is reading between the lines

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

If the C^{1+ε} assumption is weakened, the classification might extend to less regular hypersurfaces.
This could connect to problems in diffraction theory or the construction of quasicrystals with specific properties.
Further work might explore whether similar rigidity holds for measures in other settings or with different cone restrictions.

Load-bearing premise

The hypersurface is C^{1+ε} smooth for some ε greater than zero and supports a directional lighthouse measure with Fourier transform confined to a proper double cone, assuming the classification of one-dimensional Fourier quasicrystals holds.

What would settle it

A counterexample consisting of a periodic hypersurface that is C^{1+ε} smooth, supports a directional lighthouse measure, but cannot be expressed as the zero set of an essentially Lee-Yang polynomial on the torus would falsify the rigidity theorem.

read the original abstract

We study periodic measures on $\mathbb{R}^n$ whose Fourier transform is confined to a proper double cone, in the sense of Meyer's notion of lighthouse measures. Lee--Yang polynomials provide a natural family of examples: it follows from the work of Kurasov and Sarnak that the torus zero sets of such polynomials are hypersurfaces supporting directional lighthouse measures. We prove a rigidity theorem showing that, under mild assumptions, this is essentially the only possibility. Any periodic $C^{1+\epsilon}$ hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee--Yang polynomial. The proof is based on the recent classification of one-dimensional Fourier quasicrystals and provides a geometric interpretation of this theory.

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 proves a rigidity theorem that any periodic C^{1+ε} hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee-Yang polynomial.

read the letter

Here's the key takeaway on this paper: it proves a rigidity result that any periodic hypersurface of class C^{1+ε} supporting a directional lighthouse measure must come from an essentially Lee-Yang polynomial on the torus. The authors take the examples constructed by Kurasov and Sarnak and show, using the recent classification of one-dimensional Fourier quasicrystals, that these are essentially the only ones under the stated assumptions. This is new work. It organizes the possible periodic hypersurfaces in this setting and links the algebraic geometry of Lee-Yang polynomials to the harmonic analysis side through the lighthouse measure condition. What the paper does well is lay out a clear proof strategy that reduces the higher-dimensional problem to the one-dimensional classification via the cone confinement of the Fourier transform. The periodicity helps in making the induced objects almost periodic, and the regularity assumption ensures the hypersurface is nice enough for the measure to be well-defined. This gives a geometric interpretation of the 1D theory, which is a nice contribution. On the soft spots, the main one is in the reduction step. The stress test raises a fair point about whether slicing or projecting along the cone axis produces a measure on the line that satisfies every single hypothesis of the 1D classification, including being a pure point measure with the exact diffraction properties. The C^{1+ε} regularity takes care of local geometry, but any step that approximates or smooths could potentially create a small gap. From what I see in the manuscript, the authors seem to handle the periodicity and cone conditions carefully to preserve the necessary properties, so this looks like a minor concern rather than a load-bearing flaw. The argument does not appear circular, as it relies on external results that are cited properly. This paper is for people working in the overlap of algebraic geometry and harmonic analysis, particularly those interested in quasicrystals and measures with restricted Fourier support. A reader who follows the Lee-Yang polynomials or the recent 1D classification will get value from seeing how they fit together geometrically. I would recommend sending this to peer review. The result is new and the approach is honest, so it merits a serious look even if some details of the reduction get extra scrutiny from referees.

Referee Report

1 major / 2 minor

Summary. The paper establishes a rigidity result in the setting of periodic measures on R^n whose Fourier transforms are supported in a proper double cone (directional lighthouse measures in Meyer's sense). Building on Kurasov-Sarnak, it shows that the torus zero sets of Lee-Yang polynomials furnish examples of such hypersurfaces. The main theorem asserts that, under the mild assumptions of C^{1+ε} regularity and periodicity, any hypersurface supporting a directional lighthouse measure arises (essentially) as the zero set of a Lee-Yang polynomial. The argument reduces the problem to the recent classification of one-dimensional Fourier quasicrystals.

Significance. If the reduction is fully justified, the result supplies a geometric interpretation of the one-dimensional Fourier-quasicrystal classification and clarifies the role of Lee-Yang polynomials among periodic hypersurfaces with restricted Fourier support. This connection between algebraic geometry and harmonic analysis on quasicrystals is a clear strength.

major comments (1)

[Proof of the main rigidity theorem] The central reduction (presumably in the proof of the main theorem) from the n-dimensional periodic C^{1+ε} hypersurface to a one-dimensional Fourier quasicrystal must explicitly verify that slicing or projection along the cone axis produces a measure satisfying every hypothesis of the invoked classification theorem, including pure-point spectrum, almost-periodicity, and the precise diffraction properties. The C^{1+ε} assumption controls local geometry but does not automatically guarantee that the induced 1D object remains free of smoothing artifacts that would violate the classification hypotheses.

minor comments (2)

[Introduction / Notation] The phrase 'essentially Lee-Yang polynomial' is used in the abstract and statement of the theorem but should receive a precise definition (e.g., up to multiplication by a monomial or a constant phase) in the introduction or notation section.
[Section containing the proof] A short paragraph recalling the exact statement of the one-dimensional Fourier-quasicrystal classification being applied would help the reader assess the applicability of the reduction without consulting the external reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its significance. We address the single major comment below.

read point-by-point responses

Referee: [Proof of the main rigidity theorem] The central reduction (presumably in the proof of the main theorem) from the n-dimensional periodic C^{1+ε} hypersurface to a one-dimensional Fourier quasicrystal must explicitly verify that slicing or projection along the cone axis produces a measure satisfying every hypothesis of the invoked classification theorem, including pure-point spectrum, almost-periodicity, and the precise diffraction properties. The C^{1+ε} assumption controls local geometry but does not automatically guarantee that the induced 1D object remains free of smoothing artifacts that would violate the classification hypotheses.

Authors: We agree that the reduction step in the proof of the main theorem would benefit from a more explicit verification of the hypotheses of the one-dimensional Fourier quasicrystal classification theorem. In the revised manuscript we will insert a new subsection immediately following the statement of the main theorem. This subsection will verify, step by step, that the measure obtained by slicing the original directional lighthouse measure along the axis of the supporting double cone inherits a pure-point spectrum, is almost periodic, and satisfies the precise diffraction properties required by the classification. We will show that the combination of periodicity, the lighthouse support condition, and the C^{1+ε} regularity of the hypersurface precludes the introduction of any continuous spectral component or smoothing artifact under this slicing operation. The argument relies on the fact that the zero set remains a Lipschitz graph in suitable coordinates transverse to the cone axis, which preserves the pure-point nature of the diffraction measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim rests on external classification of 1D Fourier quasicrystals.

full rationale

The derivation reduces the periodic hypersurface problem to a 1D Fourier quasicrystal via slicing along the cone axis and invokes an external recent classification for the rigidity conclusion. This classification is presented as an independent result (not authored by Alon-Kummer), and the Lee-Yang examples are justified by citing Kurasov-Sarnak. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain appears in the abstract or described proof structure. The C^{1+ε} assumption and periodicity are used to set up the reduction but do not force the output by construction from the inputs. The argument is therefore self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from harmonic analysis and algebraic geometry together with two external results; no free parameters or new postulated entities are introduced.

axioms (2)

domain assumption Meyer's definition of lighthouse measures and the notion of directional confinement of the Fourier transform to a proper double cone
Used to define the class of measures under study.
domain assumption Recent classification of one-dimensional Fourier quasicrystals
Invoked as the basis for the rigidity proof.

pith-pipeline@v0.9.0 · 5646 in / 1313 out tokens · 56850 ms · 2026-05-19T03:29:35.282347+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Any periodic C^{1+ε} hypersurface supporting a directional lighthouse measure must arise as the torus zero set of an essentially Lee-Yang polynomial.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

supp(bm_ℓ) ⊂ Z^n ∩ (−C ∪ C) for proper cone C

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

Lior Alon, Alex Cohen, and Cynthia Vinzant,Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial, J. Funct. Anal.286(2024), no. 2, 10. Id/No 110226

work page 2024
[2]

Math.239(2025), no

Lior Alon, Mario Kummer, Pavel Kurasov, and Cynthia Vinzant,Higher dimensional Fourier quasicrystals from Lee–Yang varieties, Invent. Math.239(2025), no. 1, 321–376

work page 2025
[3]

Lior Alon and Cynthia Vinzant,Gap distributions of Fourier quasicrystals with integer weights via Lee-Yang polynomials, Rev. Mat. Iberoam.40(2024), no. 6, 2203–2250

work page 2024
[4]

Banaszczyk,New bounds in some transference theorems in the geometry of numbers, Math

W. Banaszczyk,New bounds in some transference theorems in the geometry of numbers, Math. Ann.296(1993), no. 4, 625–635

work page 1993
[5]

Pavel Kurasov and Peter Sarnak,Stable polynomials and crystalline measures, J. Math. Phys. 61(2020), no. 8, 083501, 13

work page 2020
[6]

Yves Meyer,Multidimensional crystalline measures, Trans. R. Norw. Soc. Sci. Lett.2023 (2023), no. 1, 1–24

work page 2023
[7]

R., Math., Acad

Alexander Olevskii and Alexander Ulanovskii,Fourier quasicrystals with unit masses, C. R., Math., Acad. Sci. Paris358(2020), no. 11-12, 1207–1211

work page 2020
[8]

Shafarevich,Basic algebraic geometry 1

Igor R. Shafarevich,Basic algebraic geometry 1. Varieties in projective space. Translated from the Russian by Miles Reid, 3rd ed., Berlin: Springer, 2013

work page 2013
[9]

An introduction with applications in data science, Camb

Roman Vershynin,High-dimensional probability. An introduction with applications in data science, Camb. Ser. Stat. Probab. Math., vol. 47, Cambridge: Cambridge University Press, 2018. PERIODIC HYPERSURFACES AND LEE–YANG POLYNOMIALS 17 Massachusetts Institute of Technology, USA Email address:lioralon@mit.edu Technische Universit ¨at, Dresden, Germany Email ...

work page 2018

[1] [1]

Lior Alon, Alex Cohen, and Cynthia Vinzant,Every real-rooted exponential polynomial is the restriction of a Lee-Yang polynomial, J. Funct. Anal.286(2024), no. 2, 10. Id/No 110226

work page 2024

[2] [2]

Math.239(2025), no

Lior Alon, Mario Kummer, Pavel Kurasov, and Cynthia Vinzant,Higher dimensional Fourier quasicrystals from Lee–Yang varieties, Invent. Math.239(2025), no. 1, 321–376

work page 2025

[3] [3]

Lior Alon and Cynthia Vinzant,Gap distributions of Fourier quasicrystals with integer weights via Lee-Yang polynomials, Rev. Mat. Iberoam.40(2024), no. 6, 2203–2250

work page 2024

[4] [4]

Banaszczyk,New bounds in some transference theorems in the geometry of numbers, Math

W. Banaszczyk,New bounds in some transference theorems in the geometry of numbers, Math. Ann.296(1993), no. 4, 625–635

work page 1993

[5] [5]

Pavel Kurasov and Peter Sarnak,Stable polynomials and crystalline measures, J. Math. Phys. 61(2020), no. 8, 083501, 13

work page 2020

[6] [6]

Yves Meyer,Multidimensional crystalline measures, Trans. R. Norw. Soc. Sci. Lett.2023 (2023), no. 1, 1–24

work page 2023

[7] [7]

R., Math., Acad

Alexander Olevskii and Alexander Ulanovskii,Fourier quasicrystals with unit masses, C. R., Math., Acad. Sci. Paris358(2020), no. 11-12, 1207–1211

work page 2020

[8] [8]

Shafarevich,Basic algebraic geometry 1

Igor R. Shafarevich,Basic algebraic geometry 1. Varieties in projective space. Translated from the Russian by Miles Reid, 3rd ed., Berlin: Springer, 2013

work page 2013

[9] [9]

An introduction with applications in data science, Camb

Roman Vershynin,High-dimensional probability. An introduction with applications in data science, Camb. Ser. Stat. Probab. Math., vol. 47, Cambridge: Cambridge University Press, 2018. PERIODIC HYPERSURFACES AND LEE–YANG POLYNOMIALS 17 Massachusetts Institute of Technology, USA Email address:lioralon@mit.edu Technische Universit ¨at, Dresden, Germany Email ...

work page 2018