{"id":"95a94b13-c785-437c-9765-ebb4fd41474a","arxiv_id":"2507.16599","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that on the d-torus, trace and observability inequalities for Laplace eigenfunctions require the measure's support to have dimension at least d-2, and gives Fourier-decay and Sobolev-regularity conditions under which they hold.","lead":"The authors characterize which Borel measures on the d-dimensional torus admit uniform trace and observability estimates for Laplace eigenfunctions. They prove that the support must have dimension at least d-2 and give sufficient conditions via Fourier decay or Sobolev regularity, generalizing Zygmund and Bourgain-Rudnick.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The decoupling step in Theorem 8.1 does not control λ versus ρ, so the Sobolev-regularity sufficiency direction is not proved as written.","rationale":"The reader's verdict is CONDITIONAL, and I agree that the proof of Theorem 8.1 needs repair; the reader mentioned this decoupling scaling issue in the rationale. I focus on it here rather than on the cluster-induction concern, because Lemma 3.6 is stated for arbitrary spheres embedded in R^n, so applying it to the n-sphere containing supp u appears legitimate, and the exponent discrepancy in Theorem 7.1 looks typographical. The decisive gap is the λ-versus-ρ scaling: for eigenfunctions whose Fourier support lies in a lower-dimensional affine sphere of radius ρ, the full-sphere decoupling constant λ^ε is not compensated by N^{-ε} when N is tied to ρ and λ can be arbitrarily large. The family u_h above is a concrete instance of this, and no argument in the paper rules it out. Since Theorem 1.3 and the necessary-condition direction remain supported, the overall verdict stays CONDITIONAL rather than changing to ACCEPT or REJECT: the regularity-based sufficiency theorem is currently unproved as written, but the Fourier-decay route and the necessary conditions are independent and appear solid.","tokens_in":969,"tokens_out":897,"duration_ms":269713,"concrete_test":"For d = 4, n = 2, fix ρ > r and consider the family u_h(x) = e^{2πi h x_4} Σ_{|ξ|=ρ} c_ξ e^{2πi ξ·(x_1,x_2)}, where h^2 + ρ^2 is a sum of four squares so that u_h ∈ E_λ with λ ≈ h. Substitute this family into the estimate (8.2) and the subsequent residual bound, taking N = C_n ρ^{2/(n+2)!}. Check whether the displayed error term contains an unbounded factor such as (λ/N)^{ε/2}; if the error is not o(1) as h → ∞ for fixed r, the claimed uniform bound fails for this family. Separately, verify whether any published decoupling theorem gives ||u_h||_{L^{2(d+1)/(d-1)}} ≲ ρ^ε ||u_h||_2 for this family, with the loss depending on ρ rather than λ; if not, the proof of Theorem 8.1 for d ≥ 4 lacks a valid estimate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the induction step of Theorem 8.1 (d ≥ 3), supp u is contained in an n-sphere of radius ρ, with ρ ≳ r, and clusters are separated by N = C_n ρ^{2/(n+2)!}; P_N is chosen precisely so the low-frequency term has no cross-cluster contribution. The residual term is then bounded via Hölder and Bourgain–Demeter decoupling on the full sphere S^{d-1}_λ. The displayed chain effectively gives a factor λ^ε N^{-ε/2} (or similar), where λ is the eigenvalue and N is fixed once r is fixed. But λ and ρ are independent: taking an eigenfunction supported in the affine hyperplane x_d = h gives λ^2 = h^2 + ρ^2, so for fixed ρ ≥ r one can send h, and hence λ, to infinity while N stays fixed. The resulting bound is not o(1) as r → ∞ uniformly in the eigenfunction. In other words, the proof requires an unstated relation such as λ ≲ ρ^{2/(n+2)!}, which is false for these hyperplane-supported families. This does not invalidate the Fourier-decay theorem (Theorem 7.1), whose error terms depend on ρ and the Fourier decay of μ, but it leaves the regularity-based sufficient condition in Theorem 8.1 unproved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":24607,"tokens_out":17675,"duration_ms":189531,"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":[{"comment":"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":"Theorem 8.1, proof for d \\ge 3, inequality after Eq. (8.2)"},{"comment":"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.","section":"Section 8, proof of Theorem 8.1 for d \\ge 3, first term of (8.2)"},{"comment":"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.","section":"Lemma 3.6 and its use in Section 7"}],"minor_comments":[{"comment":"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}.","section":"Section 5, Eq. (5.5)"},{"comment":"There is a typo: \"Consequencely\" should be \"Consequently\".","section":"Remark 6.3"},{"comment":"The phrase \"T race inequality\" has an unwanted space; also \"Cauchy-Schwartz\" appears in several places and should be \"Cauchy-Schwarz\".","section":"Section 4"},{"comment":"The reference [GMZ24] contains a typo in the title: \"Llaplacian\" should be \"Laplacian\".","section":"References"},{"comment":"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.","section":"Section 2.1.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope for an analysis and PDE journal, and the necessary conditions and Fourier-decay suffiency results are valuable. The main issue is the proof of Theorem 8.1: the decoupling step appears to confuse the eigenfrequency lambda with the cluster separation N, and the induction hypothesis is applied to a measure for which it was not established. The authors should be given the opportunity to repair or replace this argument. If the missing decoupling estimate cannot be supplied, the claims relying on Theorem 8.1, including the W^{\\epsilon,(d+1)/2} application to characteristic functions in Appendix A, should be withdrawn or reformulated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe 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.\n\nThe 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.\n\nA 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.\n\nOverall, 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.\n\nRecommendation: engage. The core results deserve referee time even if the Sobolev theorem needs substantial revision.","headline":"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.","tokens_in":25196,"tokens_out":1698,"would_cite":true,"duration_ms":20626,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35P10","58J50","42B37","11P21"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Laplace eigenfunctions","torus","trace inequality","observability inequality","Fourier decay","Hausdorff dimension","lattice points on spheres","decoupling"],"falsifier":"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.","tokens_in":24169,"feed_emoji":"🎯","tokens_out":15550,"duration_ms":136976,"temperature":0.7,"pith_summary":"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.","feed_headline":"d−2 decides which measures observe toral eigenfunctions","feed_subtitle":"The same exponent d−2 is necessary for both inequalities and sufficient once Fourier decay kicks in.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the clustering lemma that splits lattice points on a sphere into well-separated low-dimensional clusters; the engine of the induction proving the Fourier-decay sufficiency direction.","marker":"[Con76]"},{"why":"Supplies the $\\ell^2$ decoupling theorem used to prove the Sobolev-regularity sufficiency result for densities in $W^{\\varepsilon,(d+1)/2}$.","marker":"[BD15]"},{"why":"Establishes the $d=2$ trace inequality for $L^2$ densities, serving as the base case for the arguments in the present paper.","marker":"[Zyg74]"},{"why":"Establishes the $d=2$ observability and Cantor–Lebesgue results that the present paper generalizes to higher dimensions.","marker":"[Zyg72]"},{"why":"Proves hypersurface restriction and observability in dimensions 2 and 3 for curved surfaces, the pattern that motivates the curvature and Fourier-decay conditions here.","marker":"[BR09]"},{"why":"Formulates the conjectured $L^p$ eigenfunction bounds that the trace inequality for $(d-2+\\varepsilon)$-regular measures would imply.","marker":"[Bou93]"},{"why":"Provides the counting estimate for lattice points on lower-dimensional spheres used inside the Sobolev-regularity proof and in the rational-hyperplane context.","marker":"[HZ21]"},{"why":"Gives the standard cardinality estimates for integer points on spheres, which fix the critical exponent $d-2$.","marker":"[Gro85]"},{"why":"Supplies the two-dimensional clustering fact (clusters of size at most 2 with separation $\\gtrsim \\lambda^{1/3}$) used in the $d=2$ proof.","marker":"[Jar26]"},{"why":"Supplies the Fourier-analysis-to-dimension theory used to deduce Hausdorff dimension from Fourier decay when applying the trace and observability machinery to measures satisfying the decay assumption.","marker":"[Mat15]"}],"fun_headline_variants":["Exponent d−2 marks the sharp divide for toral eigenfunction measures","Fourier decay past d−2 yields both trace and observability on tori","Sharp d−2 threshold for measuring toral eigenfunctions","d−2: the exact dimension cutoff for eigenfunction inequalities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exponent d−2 marks the sharp divide for toral eigenfunction measures","Fourier decay past d−2 yields both trace and observability on tori","Sharp d−2 threshold for measuring toral eigenfunctions","d−2: the exact dimension cutoff for eigenfunction inequalities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000214,"raw_usage":{"total_tokens":1480,"prompt_tokens":1057,"completion_tokens":423,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":673,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":673,"tokens_out":423,"duration_ms":4611,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:08:13.441569+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}