Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

Hui Zhu; Massimo Sorella; Nicolas Burq; Pierre Germain

REVIEW 3 major objections 5 minor 11 references

Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper characterizes which probability measures on the d-dimensional torus satisfy trace and observability inequalities for every Laplace eigenfunction: support dimension at least d−2 is necessary, and Fourier decay at rate…

desk verdict A substantial paper with a clean new necessary-condition threshold and a plausible Fourier-decay sufficiency theory, but the Sobolev theorem has a real λ-vs-ρ gap that needs repair before the full claims are accepted. read the letter →

arxiv 2507.16599 v1 pith:OEAPUAAH submitted 2025-07-22 math.AP math-phmath.CAmath.MPmath.NTmath.SP

Nicolas Burq , Pierre Germain , Massimo Sorella , Hui Zhu This is my paper · ORCID

classification math.APmath-phmath.CAmath.MPmath.NTmath.SP MSC 35P1058J5042B3711P21

keywords LaplaceeigenfunctionstorustraceinequalityobservabilityFourierdecayHausdorffdimensionlatticepointsonspheresdecoupling

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 asks which probability measures on the d-dimensional torus can act as a universal gauge for Laplace eigenfunctions: the trace inequality $\int |u|^2 d\mu \lesssim \int |u|^2 dx$ bounds the $\mu$-mass from above, and the observability inequality $\int |u|^2 dx \lesssim \int |u|^2 d\mu$ bounds it from below, uniformly over every eigenfunction $u$. The authors prove a sharp dimension threshold. If the trace inequality holds, then $\mu$ is upper $(d-2)$-regular, so its support has Hausdorff dimension at least $d-2$; if the observability inequality holds, the support has Minkowski dimension at least $d-2$. Conversely, if the Fourier coefficients of $\mu$ decay like $|k|^{-(d-2+\varepsilon)}$ for some $\varepsilon>0$, both inequalities hold uniformly, and the same holds for absolutely continuous measures with densities in the Sobolev space $W^{\varepsilon,(d+1)/2}(\mathbb{T}^d)$. The threshold $d-2$ mirrors the growth rate $\lambda^{d-2}$ of the number of lattice points on the sphere of radius $\lambda$, which the proofs exploit in both directions, and it extends classical two-dimensional results and earlier hypersurface restriction results to all dimensions.

What carries the argument

The load-bearing mechanism is the cluster structure of the lattice-point set $S_{\lambda}^{d-1} = \mathbb{Z}^d \cap \lambda S^{d-1}$ on the sphere of radius $\lambda$, codified in the clustering lemma: the set splits into clusters $\Omega_\alpha$, each contained in an affine subspace of dimension at most $d-1$, with mutual separation $\operatorname{dist}(\Omega_\alpha,\Omega_\beta) \gtrsim \lambda^{2/(d+1)!}$. This decomposition rewrites the quadratic form $\sum_{k,\ell} \widehat{\mu}_{k-\ell} \widehat{u}_k \widehat{u}_\ell$ as cluster-diagonal terms plus a small cross-cluster error, and it supports a mathematical induction on the affine dimension of the Fourier support, which is how the Fourier-decay and Sobolev sufficient conditions are proved. In the opposite direction, the necessary conditions are obtained from the concentrating eigenfunctions $\varphi_{\lambda,x_0}(x) = \sum_{k\in S_{\lambda}^{d-1}} e^{2\pi i k\cdot (x-x_0)}$, which attain value $N_d(\lambda) \sim \lambda^{d-2}$ at $x_0$ and force any trace-observing measure to give small balls at $x_0$ mass at most $r^{d-2}$. The $\ell^2$ decoupling theorem supplies the key estimate in the Sobolev-regularity proof.

What would settle it

Take $\mu$ to be the normalized surface measure on a smooth $(d-3)$-dimensional submanifold of $\mathbb{T}^d$ and compute the ratios $(\int |\varphi_{\lambda,x_0}|^2 d\mu)/\lambda^{d-2}$ for the concentrating eigenfunctions $\varphi_{\lambda,x_0}$; the necessary condition says these ratios must be unbounded along some sequence $\lambda\to\infty$, so finding a bounded subsequence would refute the necessary dimension threshold.

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Extended reading notes

Core claim

The central discovery is a sharp dimension cut-off for toral eigenfunctions. For the trace inequality $\int |u|^2 d\mu \lesssim \int |u|^2 dx$ to hold for every Laplace eigenfunction $u$ on $\mathbb{T}^d$, the measure $\mu$ must be upper $(d-2)$-regular, hence its support must have Hausdorff dimension at least $d-2$; for the observability inequality $\int |u|^2 dx \lesssim \int |u|^2 d\mu$, the support must have Minkowski dimension at least $d-2$. In the converse direction, if for some $\varepsilon>0$ the Fourier coefficients satisfy $|\widehat{\mu}_k| \lesssim |k|^{-(d-2+\varepsilon)}$, then both inequalities hold uniformly over all eigenfunctions; for $d\geq 5$ the decay condition can be relaxed to the weighted sum $\sum_{j\geq 0} 2^{j(d-2)} \sup_{|k|\in[2^j,2^{j+1}]} |\widehat{\mu}_k| < \infty$. Densities in $W^{\varepsilon,(d+1)/2}(\mathbb{T}^d)$ also give both inequalities. The exponent $d-2$ is therefore the exact boundary, forced by lattice-point counting and sufficient once the measure's Fourier spectrum decays past that rate.

Load-bearing premise

The sufficient-condition proofs apply the clustering lemma, which is stated for the whole lattice-point set of a sphere, to the lower-dimensional sub-spheres created during the induction, and this extension is asserted without proof; if that extension fails for some sphere, the trace and observability estimates in the sufficiency direction no longer follow from the argument as written.

Editorial extensions

If this is right

Cantor–Lebesgue theorems: the observability inequality for sets $E$ with $1_E \in W^{\varepsilon,(d+1)/2}$ implies that if spherical sums of an eigenfunction tend to zero in $L^2(E)$, then the coefficient sums tend to zero; Appendix A constructs fat Cantor sets with this Sobolev regularity.
Constraints on quantum limits: the uniform trace inequality for $W^{\varepsilon,(d+1)/2}$ densities gives a meaning to the pairing $\langle f, \mu\rangle$ between a quantum-measure density $f$ and the observing measure $\mu$, validating the chain of inequalities relating eigenfunction infima and suprema to $\int f\, d\mu$.
Schrödinger observability and control: the eigenfunction inequalities imply time-integrated observability for the Schrödinger propagator on $\mathbb{T}^d$, so the Hilbert Uniqueness Method yields exact controllability from any set whose characteristic function has the stated Sobolev regularity.
Sharp obstructions: no measure supported on a set of dimension $< d-2$ can satisfy trace or observability; in particular, observability fails for smooth submanifolds of codimension two or more, and trace fails for measures supported on rational linear subspaces of codimension two.
Open-question hierarchy: the trace inequality for general $(d-2+\varepsilon)$-regular measures would imply the hypersurface trace conjecture and the conjectured $L^p$ eigenfunction bounds, and the paper identifies the Fourier-decay measures of Theorem 7.1 as the only currently known family of such regular measures.

Reading between the lines

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

If the same $d-2$ threshold governs eigenfunction observation on general compact manifolds (the paper sketches this only for quasimodes via the spectral dimension count of eigenspaces), then observability from sets of dimension below $d-2$ would fail there too; this is an extension, not a result of the paper.
The clustering-lemma extension is the most fragile step; a direct proof or a replacement clustering theorem would strengthen the sufficiency results, and a counterexample to the extension would still leave the theorems' truth open but would indicate the current proof needs repair.
The sharpness at $\varepsilon=0$ suggests a phase-transition picture: measures with Fourier decay exactly $|k|^{-(d-2)}$ sit at the boundary, and a natural numerical test in low dimension is to track the trace constant as $\varepsilon\to 0^+$ for the family $f_\varepsilon(x)=|x|^{\varepsilon-2}$ and see whether the constant blows up only at $\varepsilon=0$.
The curvature-versus-dimension question raised by flat-hyperplane counterexamples invites a finer classification: among $(d-2)$-dimensional measures, those with positive Fourier dimension (like curved hypersurfaces) should satisfy observability, while flat ones should not—this is the natural Fourier-dimension refinement of the paper's necessary conditions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper is worth taking seriously. It proves genuinely new necessary conditions for trace and observability inequalities in d≥3 (dimension threshold d−2), and a Fourier-decay sufficient condition that generalizes Zygmund and Bourgain–Rudnick with the expected sharp rate. The d=2 part is a rehabilitation of known results, but Theorems 1.1, 1.2, 7.1, and 7.7 are new. The proof of the necessary conditions uses Bourgain's concentrated eigenfunctions with a pigeonhole lemma, and it is coherent. The observability necessity via Minkowski dimension and vanishing Fourier coefficient systems is also convincing.

The main soft spot is Theorem 8.1 for d≥3. In the induction step, the cluster separation N is chosen from the radius ρ of the lower-dimensional sphere, so N ≈ ρ^{2/(n+2)!}. But the error term from decoupling is controlled by a factor involving λ, the eigenvalue, and the displayed chain gives something like λ^ε N^{−ε/2}. Since λ and ρ are independent—hyperplane-supported eigenfunctions give λ^2 = h^2 + ρ^2 with h→∞ while N stays fixed—the error is not o(1) uniformly in the eigenfunction. Without an unstated relation such as λ ≲ ρ^{2/(n+2)!}, the proof of Theorem 8.1 does not close. This does not affect the Fourier-decay theorem, whose error terms depend on ρ and the Fourier decay of μ, but it leaves the Sobolev-regularity suffiency unproved as written. This is a load-bearing gap and should be flagged in a referee report.

A second, minor concern: the induction repeatedly applies Connes' clustering lemma to sub-clusters of the sphere of radius ρ, while Lemma 3.6 is stated for lattice points on the full sphere. The extension is plausible—the proof is volume-based and translation invariant—but it is not stated or proved. It probably follows from the same argument, but it needs checking, especially the constants when the center is not in Z^d.

Overall, the necessary conditions and the Fourier-decay sufficiency directions are a substantial contribution. The paper is honest about open questions and does not oversell the d=2 material. The Sobolev part is plausible but needs a different argument or a repaired estimate. I would send it to a serious referee, with instructions to focus on Section 8 and on the clustering lemma extension.

Recommendation: engage. The core results deserve referee time even if the Sobolev theorem needs substantial revision.

Referee Report

3 major / 5 minor

Summary. The paper studies uniform trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus with respect to arbitrary Borel probability measures. The main necessary conditions are that the trace inequality forces upper (d-2)-regularity of the measure (Theorem 1.1), and the observability inequality forces the support to have Minkowski dimension at least d-2 (Theorem 1.2). On the sufficient side, the paper proves trace and observability under pointwise Fourier decay |\hat\mu_k| \lesssim |k|^{-(d-2+\epsilon)} (Theorem 1.3 / 7.1), under a weighted l^1 Fourier condition for d \ge 5 (Theorem 1.4 / 7.7), and under Sobolev regularity of densities in W^{\epsilon,(d+1)/2} (Theorem 1.5 / 8.1). The proofs combine Connes' clustering of lattice points on spheres, cap-counting estimates, Bourgain-Demeter decoupling, and constructions of concentrated or vanishing eigenfunctions. Applications to Cantor-Lebesgue theorems, quantum limits, and Schr\"odinger observability are discussed.

Significance. If the main results stand, they represent a substantial advance: the necessary conditions give a clean dimensional obstruction, and the Fourier-decay sufficient condition generalizes classical results of Zygmund and Bourgain-Rudnick to arbitrary dimensions and measures. The paper is honest about the gap between its sufficient conditions and the conjectural (d-2+epsilon)-regularity threshold, and it provides explicit sharpness examples. The combinatorial arguments for the necessary conditions and the d=3 case of the Fourier-decay theorem are detailed and appear sound. However, the proof of the Sobolev-regularity theorem (Theorem 8.1) contains a decoupling error that is load-bearing for one of the paper's advertised sufficient conditions; the central claim of the paper is therefore only partially established as written.

major comments (3)

[Theorem 8.1, proof for d \ge 3, inequality after Eq. (8.2)] The estimate \|u\|_{L^{(d+1)/(d-1)}} \lesssim_{d,\epsilon} N^{\epsilon/2}\|u\|_{L^2} is not justified. The parameter N is the cluster separation N = C_n \rho^{2/(n+2)!}, whereas Bourgain-Demeter decoupling applied to the eigenfunction u, whose frequencies lie on S^{d-1}_\lambda, gives the constant \lambda^{\epsilon/2}. For a family with supp \hat u contained in the affine hyperplane k_d = h and \sum_{i<d} k_i^2 = \rho^2, one has \lambda^2 = \rho^2 + h^2 with \rho \ge r fixed and h \to \infty, so N stays bounded while \lambda tends to infinity. The resulting factor (\lambda/N)^{\epsilon/2} is not o(1) as r \to \infty uniformly over such eigenfunctions, so the Sobolev-regularity sufficiency theorem is not proved as written.
[Section 8, proof of Theorem 8.1 for d \ge 3, first term of (8.2)] The induction hypothesis of Theorem 7.1 is applied to the measure (P_N g) dx, but that hypothesis was proved only under the pointwise Fourier decay condition (7.1). The trigonometric polynomial P_N g has no uniform Fourier decay, and the constants in Theorem 7.1 depend on the measure in a way that is not tracked. The proof needs either a strengthened induction statement for W^{\epsilon,(d+1)/2} densities with constants controlled by the Sobolev norm, or a separate argument for finite Fourier series.
[Lemma 3.6 and its use in Section 7] The proof of Theorem 7.1 applies the clustering lemma to the lattice points of an n-sphere of radius \rho embedded in R^d, although Lemma 3.6 is stated for the full (d-1)-sphere in R^n. The manuscript asserts the extension because the proof uses only cap volume estimates, but gives no details. Since the induction step and the cross-cluster error estimate both rely on this extended statement, a proof or a precise citation covering the lower-dimensional case is required. The exponent is also inconsistent across the manuscript: \rho^{2/(n+1)!} in Section 7, s^{2/d!} in the error estimate, and \rho^{2/(n+2)!} in Section 8; the last value is the one matching Lemma 3.6 for an n-sphere.

minor comments (5)

[Section 5, Eq. (5.5)] The expression "4 \times 2\pi_n" should read "4 \times 2^{\pi_n}" to be consistent with the subsequent estimate N_2(\lambda) \gtrsim \lambda^{C'/\ln\ln\lambda}.
[Remark 6.3] There is a typo: "Consequencely" should be "Consequently".
[Section 4] The phrase "T race inequality" has an unwanted space; also "Cauchy-Schwartz" appears in several places and should be "Cauchy-Schwarz".
[References] The reference [GMZ24] contains a typo in the title: "Llaplacian" should be "Laplacian".
[Section 2.1.3] The claim that (2.1) is valid for X = W^{\epsilon,(d+1)/2} depends on Theorem 8.1, so it should be marked as conditional on the repair of that proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are proved from stated lemmas and external results, with no step reducing a conclusion to its own input.

full rationale

The paper's central claims are the necessary conditions (Theorems 1.1 and 1.2) and the sufficiency theorems (Theorems 1.3–1.5, 7.1, 7.7, 8.1). The necessary conditions are proved by testing the trace and observability inequalities against explicit Bourgain eigenfunctions concentrated at points or along subspaces; this is the standard contrapositive use of the defining inequality, not a definitional equivalence. The sufficiency proofs proceed from the lattice-point clustering lemma of Connes (Lemma 3.6), point-counting estimates (Lemmas 3.4–3.5), the Bourgain–Demeter ℓ2 decoupling theorem, and an inductive argument on affine dimension; none of these inputs is defined in terms of the trace or observability inequality being proved. No parameter is fitted to data and then renamed a prediction. The self-citations [GMZ24] and [BZ25] are contextual: [GMZ24] is cited in Proposition 5.2 for a line of argument that is fully reproduced in the present proof, and [BZ25] is cited for applications and open-question implications, not for any main theorem. Hence there is no load-bearing self-citation chain. A possible gap in Theorem 8.1 involving the relative sizes of λ and ρ (as flagged by a skeptic) is a correctness concern, not circularity, and does not change the circularity assessment.

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

The central claims rest on standard results in geometric measure theory, analytic number theory, and decoupling theory. No free parameters are fitted to data, and no new entities are postulated. The only novel ingredients are the combinatorial cluster arguments and the constructions of concentrating and vanishing eigenfunctions.

assumptions (5)

domain assumption Connes clustering lemma (Lemma 3.6)
Partitions lattice points on spheres into separated affine clusters; imported from [Con76] without proof and extended to subsets in the induction proofs of Sections 7 and 8.
domain assumption L2 decoupling theorem of Bourgain and Demeter
Used in Section 8 to bound L^p norms of eigenfunctions; cited as [BD15, Theorem 2.2]. The exact scaling used in the proof is not stated.
standard math Frostman's lemma and the energy criterion for Fourier dimension
Used in Proposition 5.1 and Remark 7.5 to pass from upper regularity or Fourier decay to lower bounds on support dimension.
standard math Prime number theorem for arithmetic progressions and Jacobi's formula for representations as sums of two squares
Used in Proposition 5.2 to build eigenfunctions on T^2 with many nonzero Fourier coefficients near a primorial frequency.
standard math Weyl law and standard spectral theory for toral eigenfunctions
Background for eigenspaces E_lambda and cluster decomposition; cited from [Sog17] and earlier works.

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Pith. "Pith review of Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus." pith.science (2026). https://pith.science/paper/OEAPUAAH

@misc{pith2026250716599,
  author       = {Pith},
  title        = {Pith review of: Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEAPUAAH}},
  note         = {Machine review of arXiv:2507.16599}
}

abstract

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int |u|^2 d \mu \lesssim \int |u|^2 d x \quad \text{(trace)}, \qquad \int |u|^2 d \mu \gtrsim \int |u|^2 d x \quad \text{(observability)}$$ hold uniformly for all eigenfunctions $u$ of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\mu$, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain--Rudnick to higher dimensions. Applications include results in the spirit of Cantor--Lebesgue theorems, constraints on quantum limits, and control theory for the Schr\"odinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.

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Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Semiclassical limits of eigenfunctions on flat n-dimensional tori

[Ais13] T. Aissiou. “Semiclassical limits of eigenfunctions on flat n-dimensional tori”. In: Canadian Mathematical Bulletin 56.1 (2013), pp. 3–12. [A W00] J. M. Ash and G. Wang. “Some spherical uniqueness theorems for multiple trigono- metric series”. In: Annals of Mathematics (2000), pp. 1–33. [A W07] J. M. Ash and G. Wang. “Uniqueness questions for mult...

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Elsevier, 2001, pp. 195–232. [BD15] J. Bourgain and C. Demeter. “The proof of the l 2 decoupling conjecture”. In: Annals of mathematics (2015), pp. 351–389. [BR09] J. Bourgain and Z. Rudnick. “Restriction of toral eigenfunctions to hypersurfaces”. In: Comptes Rendus Mathematique 347.21-22 (2009), pp. 1249–1253. [BR12] J. Bourgain and Z. Rudnick. “Restrict...

arXiv 2015

[3] [10]

2022, e24

Cambridge University Press. 2022, e24. [Gro85] E. Grosswald. Representations of Integers as Sums of Squares

work page 2022

[4] [129]

On the representations of a number as the sum of three squares

[Bat51] P. T. Bateman. “On the representations of a number as the sum of three squares”. In: Transactions of the American Mathematical Society 71.1 (1951), pp. 70–101. REFERENCES 21 [Bou93] J. Bourgain. “Eigenfunction bounds for the Laplacian on the n-torus”. In: Interna- tional Mathematics Research Notices 1993.3 (1993), pp. 61–66. [Bou96] J. Bourgain. “...

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work page 1984

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Trigonometric polynomials and lattice points

[CC92] J. Cilleruelo and A. Cordoba. “Trigonometric polynomials and lattice points”. In: Proc. Am. Math. Soc. 115.4 (1992), pp. 899–905. [Con76] B. Connes. “Sur les coefficients des s´ eries trigonom´ etriques convergentes sph´ eriquement”. In: CR Acad. Sci. Paris S´ er. AB283.4 (1976). [Coo79] R. Cooke. “The Cantor-Lebesgue theorem”. In: The American Mat...

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Explicit zero-free regions for Dirichlet L-functions

[McC84b] K. S. McCurley. “Explicit zero-free regions for Dirichlet L-functions”. In: J. Number Theory 19 (1984), pp. 7–32. [Sog17] C. D. Sogge. Fourier integrals in classical analysis . Vol

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[8] [1985]

Restriction of toral eigenfunctions to totally geodesic submanifolds

[HZ21] X. Huang and C. Zhang. “Restriction of toral eigenfunctions to totally geodesic submanifolds”. In: Analysis & PDE 14.3 (2021), pp. 861–880. [IK21] H. Iwaniec and E. Kowalski. Analytic number theory. Vol

work page 2021