{"id":"992ce701-3ab9-4a81-8633-30135cbc1036","arxiv_id":"2507.23494","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a specially constructed log-correlated field on T^d, the GMC measure almost surely has Fourier dimension d-γ^2 when γ<√(2d)/2 and (√(2d)-γ)^2 when √(2d)/2<=γ<√(2d), for all d>=1.","lead":"For every dimension d>=1, the paper constructs a log-correlated Gaussian field on the d-dimensional torus whose Gaussian multiplicative chaos measure has Fourier dimension exactly equal to its correlation dimension. This settles an open problem left by earlier work, which only handled low dimensions and restricted parameter ranges.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reader-identified gap is real: Proposition 3.1(P4) only controls derivatives up to order d, while Lemma 4.6 needs estimates up to order 2d; the gap is likely patchable but, as written, Lemma 4.11 and hence Proposition 4.4 are not justified.","rationale":"The central theorem is an existence result for a specially constructed log-correlated field, not for all log-correlated fields on the torus. The proof plausibly achieves the claimed Fourier dimension by combining a smooth mollified decomposition with a multi-resolution martingale argument, provided the high-order derivative estimates hold. The reader identified the precise weak point: Proposition 3.1(P4) is stated and verified only for |α| ≤ d, while Lemma 4.6 claims the analogous bound for all α, and Lemma 4.11 requires |α| = 2d. This is not a matter of disagreement with a known consensus; it is an internal gap in the written chain of implications. The gap is local and likely repairable because the explicit mollifier construction has enough decay: the scaling of D^α P_j contains ε_j^{-(d+|α|)}, and the support volume contributes ε_j^d, so after normalization by j^{2|α|}2^{j|α|} the remaining constant is independent of j. Thus I do not regard the result as wrong, only as requiring a completed verification. I also checked the abstract's stronger wording against Theorem 1.1; the theorem is an existence statement, so the abstract is overbroad, but this is a presentation issue and not load-bearing for the mathematical claim. The main concern is fully captured by the reader's weakest assumption, so the conditional verdict stands and no further change is needed.","tokens_in":29151,"tokens_out":5305,"duration_ms":55633,"concrete_test":"Reprove Lemma 3.9 without the restriction |α| ≤ d: for every multi-index α with |α| = 2d, show directly that sup_{j,z} E[|D^α ψ_j(z)|^p] / (j^{2|α|p} 2^{j|α|p}) < ∞, tracking the mollifier constant as ||D^α P_j||_{L∞} and the support volume |B_j|. Then substitute the resulting bound into the Faà di Bruno proof of Lemma 4.6 and confirm that Lemma 4.11 with A=2d still yields the factor k^{4dp} 2^{k(-dp + τp/2 + dp/q)} in Proposition 4.4. If the j-dependence changes, the final series convergence in §4.1 must be recomputed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.4 requires control of derivatives of the random factors X_j up to order 2d. Lemma 4.11 is applied with A=2d, and its proof invokes Lemma 4.6 for each derivative factor appearing in the Leibniz expansion of ∂^α X_φ_I. Lemma 4.6 in turn is stated for arbitrary multi-indices α, and its proof via the multivariate Faà di Bruno formula applies Proposition 3.1(P4) to every multi-index βℓ occurring in the partition of α. However, property (P4) in Proposition 3.1 is stated only for |α| ≤ d, and the verification in §3.2, specifically Lemma 3.9, is also restricted to |α| ≤ d. Since a partition of a multi-index of length 2d can contain a block of length 2d, the required estimate sup_j sup_z E[|D^β X_j(z)|^p] / (j^{2|β|p} 2^{j|β|p}) < ∞ for |β|=2d is not supplied by the manuscript as written. This is load-bearing because repeated integration by parts in Lemma 4.10 and the multinomial resummation in Lemma 4.11 depend exactly on being able to differentiate the product k times up to order 2d with j-dependent normalization j^{2|α|}2^{j|α|}. The gap is probably repairable: the mollifier construction has extra room, since ||D^α P_j||_{L∞} scales like j^{2(d+|α|)}2^{j(d+|α|)} and the support volume scales like j^{-2d}2^{-jd}, leaving the required factor j^{2|α|}2^{j|α|}; but this extension is not written in the paper, and without it the proof of Proposition 4.4 is incomplete.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every integer d≥1, a centered log-correlated Gaussian field on the d-dimensional torus with covariance log^+(1/d_Td(z,w))+g(z\\bar w) for a bounded continuous function g, and proves that for every subcritical γ∈(0,√(2d)) the associated GMC measure almost surely has Fourier dimension exactly D_{γ,d}=d−γ² for 0<γ<√(2d)/2 and D_{γ,d}=(√(2d)−γ)² for √(2d)/2≤γ<√(2d). The proof combines a new mollified decomposition of the field into smooth independent Gaussian processes (Proposition 3.1), a smooth dyadic partition of unity, repeated integration by parts via Green's identity to obtain sharp localized Fourier estimates (Proposition 4.4), and Pisier's martingale type inequality to sum these local estimates into a global Fourier–Lebesgue bound (Proposition 1.2). The upper bound dim_F≤dim_2=D_{γ,d} is quoted from prior work, so the main work is the matching lower bound.","tokens_in":29475,"tokens_out":13003,"duration_ms":130476,"significance":"If the proof is completed, the result resolves the open problem left in [LQT24, LQT25] for d≥3 and confirms the Garban–Vargas phenomenon—Fourier dimension equal to correlation dimension—for GMC measures on tori of every dimension. The paper contains several genuine contributions: a clean construction of a log-correlated Gaussian field whose decomposition has explicit derivative scale bounds, a smooth partition-of-unity localization that avoids spectral leakage from sharp cutoffs, and a transparent final optimization over the exponent p that yields exactly D_{γ,d}. The overall strategy is convincing and the martingale bookkeeping is coherent. However, the derivative estimates in §4 require a regularity order that the construction in §3 does not currently supply, and one key lemma statement is inconsistent with its proof; these issues need repair before the main theorem is fully established.","major_comments":[{"comment":"Property (P4), as stated in Proposition 3.1 and verified in Lemma 3.9, controls only derivatives of order |α|≤d, with the normalization j^{2|α|p}2^{j|α|p}. Lemma 4.6, however, is stated for arbitrary multi-indices α, and its proof via the multivariate Faà di Bruno formula applies (P4) to every block βℓ in a partition of α. Since Lemma 4.11 is later used with A=2d, a partition of a multi-index of length 2d can contain a block of length 2d, so the required estimate sup_j sup_z E[|D^β X_j(z)|^p]/(j^{4dp}2^{2jdp})<∞ for |β|=2d is not supplied by (3.1). This is load-bearing: Lemma 4.10 applies Δ^d, creating derivatives of order 2d, and Lemma 4.11 needs derivative control up to that order. The gap is probably repairable by extending (P4) to all orders using the extra room from the mollifier ε_j=j^{-2}2^{-j}, but that extension is not written in the manuscript and the proof of Proposition 4.4 is incomplete as it stands.","section":"§3, Proposition 3.1(P4) and §4.2, Lemma 4.6"},{"comment":"The statement of Lemma 4.11 bounds sup_{|α|=A} (E[|∂^α X^φ_I(t)|^p])^{1/p} by a factor involving ∏_{j=1}^k (E[|D^α X_j(z)|^p])^{1/p}, with the same multi-index α of length A on the right-hand side. The proof, however, concludes with ∏_{j=1}^k (E[|X_j(z)|^p])^{1/p}: the derivative factors E[|D^{α_j}X_j|^p] are bounded separately via Lemma 4.6 and absorbed into the normalization (k^2 2^k)^A, and the product of undifferentiated X_j terms is then bounded by sup_z. Consequently the displayed bound (4.6), which contains E[|D^α X_j|^p], cannot be combined with Lemma 4.5 to produce the factor 2^{kp(p−1)γ²/2} used in (4.2). The statement of Lemma 4.11 and the display (4.6) should refer to E[|X_j(z)|^p], matching the proof and the conclusion in §4.3.4.","section":"§4.3, Lemma 4.11 and Eq. (4.6)"}],"minor_comments":[{"comment":"Proposition 1.2 states the equality E∥µ∞∥^p_{FL}=sup_{m≥1}E∥µm∥^p_{FL}, but the proof only establishes finiteness of the supremum. The inequality E∥µ∞∥^p≤sup_m E∥µm∥^p follows from Fatou's lemma and weak convergence, and this weaker statement is all that is needed for Theorem 1.1. Please either prove the asserted equality (for example by L^p convergence of the approximate densities) or reformulate the proposition with the inequality.","section":"§4.1, Proposition 1.2"},{"comment":"In the product estimate after the display 'By (P4) in Proposition 3.1', the exponent in the factor 2^{j|α|mℓpℓ} should be 2^{j|βℓ|mℓpℓ}; the displayed formula as written would not give the subsequent product 2^{j|α|p} after summing Pℓ mℓ|βℓ|=|α|.","section":"§4.2, proof of Lemma 4.6"},{"comment":"The support condition in Lemma 2.3(2) is written as E([−2^{−k},2^k]^d), which appears to be a typo for E([−2^{−k},2^{−k}]^d) in view of the construction in Appendix A; please correct the notation.","section":"Lemma 2.3 and Appendix A"},{"comment":"There are several minor typos, including 'avarage' in §1.2.2 and 'Thereofre' in the proof of Lemma 4.11; a careful proofreading pass is recommended.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The derivative-order gap identified above is the kind of issue that is likely fixable within the manuscript's scope: the mollifier construction has enough decay (ε_j=j^{-2}2^{-j}) to extend (P4) to all derivative orders, and the proof of Lemma 4.11 already contains the corrected argument. I would ask the authors to add the missing derivative bounds and to align Lemma 4.11 with its proof before acceptance. The paper is otherwise well within the journal's scope and the main theorem is significant if the technical repair is carried out."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take on arXiv:2507.23494. The paper does two genuinely new things: it builds a log-correlated Gaussian field on T^d whose decomposition into smooth processes has derivative bounds at the right dyadic scale, and it uses a smooth dyadic partition of unity to kill the spectral leakage that blocked the cube argument in d≥3. The multi-resolution estimate with Pisier's martingale type inequality is well organized, and the final optimization over ζ(p) lands exactly on D_{γ,d}. If the technical gap below is fixed, this is a solid resolution of the high-dimensional torus case for a constructed field.\n\nThe soft spot the referee will hit is real. Proposition 3.1(P4) controls derivatives of ψ_j only up to order d. Lemma 4.6 needs control up to order 2d because Lemma 4.11 differentiates the product X_φ^I up to A=2d, and the Faà di Bruno expansion in Lemma 4.6 can produce a block β of length 2d. The proof of Lemma 4.6 cites (P4) for arbitrary α, but the verification in §3.2 (Lemma 3.9) stops at |α|≤d. So the repeated integration by parts in Lemma 4.10 and the bound on the high-frequency piece in Proposition 4.4 are not justified as written. The gap looks patchable: the mollifier P_j has derivatives of all orders, and the L∞ bound scales like j^{2(d+|α|)}2^{j(d+|α|)}; multiplying by the support volume j^{-2d}2^{-jd} leaves the required j^{2|α|}2^{j|α|}. But the extension is not in the paper, and without it the proof is incomplete.\n\nA smaller issue: the abstract says “we determine exact values of Fourier dimensions for GMC measures on the torus,” while Theorem 1.1 asserts existence of a field with a specified kernel. It’s the difference between “there exists a GMC with this property” and “for every GMC.” The title is fine, but the abstract should be aligned with the theorem.\n\nOn balance, this is a serious paper. The construction is reusable, the structure is coherent, and the flaw is a missing order-of-derivative estimate, not a conceptual error. I would send it to a good referee and ask them to verify whether Lemma 3.9 can be extended to all orders; likely it can.\n\nFor you: yes, worth a reading group slot once the patch is checked.","headline":"Strong paper with a real but likely patchable gap: derivative estimates for the smoothing construction are stated only up to order d, while the high-frequency argument needs order 2d.","tokens_in":30090,"tokens_out":2681,"would_cite":true,"duration_ms":26577,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G57","42A61","46B09","60G46"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on the d-dimensional torus there exists a log-correlated Gaussian field whose sub-critical Gaussian multiplicative chaos measure has Fourier dimension exactly d−γ² for small γ and (√(2d)−γ)² for large γ, for every…","keywords":["Gaussian multiplicative chaos","Fourier dimension","correlation dimension","log-correlated Gaussian field","Fourier–Lebesgue space","vector-valued martingale method","smooth partition of unity","torus"],"falsifier":"Compute, for the processes built in Proposition 3.1, $\\mathbb{E}[|D^\\alpha \\psi_j(z)|^p]$ for a fixed multi-index $\\alpha$ with $|\\alpha|=2d$ and check whether it stays bounded by a constant times $j^{2|\\alpha|p}2^{j|\\alpha|p}$. If it grows faster, Proposition 4.4 and hence the lower bound in Theorem 1.1 fail.","tokens_in":28870,"feed_emoji":"📐","tokens_out":5596,"duration_ms":55427,"temperature":0.7,"pith_summary":"The paper determines the exact Fourier dimension of a Gaussian multiplicative chaos (GMC) measure on the d-dimensional torus for every integer d≥1, a problem previously solved only in low dimensions or with partial parameter ranges. Fourier dimension measures how fast the Fourier coefficients of a measure decay, and the paper shows that for a suitably chosen log-correlated Gaussian field the GMC measure's Fourier dimension coincides exactly with its correlation dimension, given by $D_{\\gamma,d}=d-\\gamma^2$ when $\\gamma<\\sqrt{2d}/2$ and $D_{\\gamma,d}=(\\sqrt{2d}-\\gamma)^2$ otherwise. The advance is a new construction of the Gaussian field as a sum of smooth, practically independent processes, which lets the proof localize the measure at dyadic scales and apply repeated integration by parts without boundary artifacts. Combined with a global martingale estimate, this yields matching upper and lower bounds. If correct, the result closes the dimensional gap for torus GMC and provides a template for similar Fourier-type decay statements on other boundary-free spaces.","feed_headline":"Exact Fourier dimension found for torus Gaussian chaos","feed_subtitle":"A new smooth decomposition proves the random measure's Fourier dimension equals its correlation dimension in every dimension.","key_machinery":"The proof runs through three mechanisms. First, a field decomposition: the log-correlated kernel is written as the sum of kernels of independent stationary Gaussian processes whose paths are $C^\\infty$ and whose covariance has support within distance $3\\cdot 2^{-j}$; this is obtained by mollifying a dyadic field with carefully chosen smoothing parameters. Second, a smooth partition of unity adapted to dyadic cubes, which avoids the spectral leakage that sharp cutoffs would introduce. Third, a local estimate controlling the $\\ell^q$ Fourier–Lebesgue norm of each localized piece: repeated integration by parts through Green's identity uses the Laplacian eigenfunction equation $\\Delta z^n=-4\\pi^2|n|^2 z^n$ to trade high frequency $|n|$ for derivatives of the smooth random factors, and the martingale type inequality for vector-valued martingales is applied to sum the localized pieces globally. The precision comes from matching the derivative cost $(k^2 2^k)^A$ with the dyadic scale factor.","core_discovery":"The central claim is Theorem 1.1: for every integer $d\\ge 1$, there exists a centered log-correlated Gaussian field on $\\mathbb{T}^d$ with covariance $$K(z,w)=\\log^+\\left(\\frac{1}{d_{\\mathbb{T}^d}(z,w)}\\right)+g(z\\bar w),$$ where $g$ is bounded continuous, such that for every sub-critical $\\gamma\\in(0,\\sqrt{2d})$, the GMC measure $\\mathrm{GMC}^K_\\gamma$ almost surely has Fourier dimension $D_{\\gamma,d}$, with $D_{\\gamma,d}=d-\\gamma^2$ for $0<\\gamma<\\sqrt{2d}/2$ and $D_{\\gamma,d}=(\\sqrt{2d}-\\gamma)^2$ for $\\sqrt{2d}/2\\le\\gamma<\\sqrt{2d}$. The proof establishes a matching lower bound by showing that the GMC measure almost surely belongs to a weighted Fourier–Lebesgue space, which forces the desired polynomial decay of its Fourier coefficients.","pith_inferences":["The same smooth-decomposition scheme could be tested on other log-correlated fields, such as circular or spherical models, where the exact Fourier dimension is not yet settled; the torus construction indicates what derivative regularity the kernel decomposition must supply.","A quantitative version of the proof would give explicit constants in the bound on $\\mathbb{E}[\\|\\mu_\\infty\\|^p_{FL^{\\tau/2,q}}]$, yielding non-asymptotic Fourier-coefficient decay rates; this is not stated in the paper.","Because the upper bound $\\dim_F\\le\\dim_2$ is general, the equality suggests that any log-correlated field on a boundary-free space with bounded continuous remainder should also have Fourier dimension equal to its correlation dimension; verifying this would be a natural next step.","The spectral-leakage argument suggests a rule of thumb: sharp spatial cutoffs degrade Fourier dimension by boundary terms, so exact Fourier dimensions should be formulated on boundary-free manifolds or with smooth windows."],"forward_implications":["The GMC measure's Fourier dimension equals its correlation dimension $D_{\\gamma,d}$ for the constructed field, confirming the general phenomenon on the torus.","All dimensions $d\\ge 1$ are covered by one argument, removing the earlier low-dimensional restriction.","Sharp cutoffs on the unit cube create $|n|^{-1}$ boundary decay; using smooth truncation transfers the torus result to the unit cube.","The method replaces the Fourier basis by Laplace–Beltrami eigenfunctions and should yield analogous abstract Fourier-type decay statements on general compact boundary-free Riemannian manifolds, as the paper announces as forthcoming work."],"supporting_citations":[{"why":"Supplies the earlier partial results for low dimensions, the correlation dimension computation, and the precise statement of the upper bound adapted from the cube setting.","marker":"[LQT25]"},{"why":"Introduces the vector-valued martingale method for one-dimensional GMC that the present proof extends to all dimensions.","marker":"[LQT24]"},{"why":"Formulates the conjecture that the Fourier dimension of GMC on the circle coincides with its correlation dimension, which the torus result confirms in a broader setting.","marker":"[GV23]"},{"why":"Provides the standard framework for Gaussian multiplicative chaos and the known correlation dimension of sub-critical GMC measures.","marker":"[RV14]"},{"why":"Supplies the result that the correlation dimension of the GMC measure is exactly $D_{\\gamma,d}$, used for the upper bound.","marker":"[Ber23]"},{"why":"Introduces the vector-valued martingale technique in the Mandelbrot cascade setting that the paper adapts to GMC.","marker":"[CHQW24]"},{"why":"States the martingale type inequality for $\\ell^q$-valued martingales, which is the global decoupling tool in the proof.","marker":"[Pis16]"},{"why":"Provides the T-martingale convergence theory that defines the GMC limit measure and guarantees its sub-critical non-degeneracy.","marker":"[Kah87]"}],"fun_headline_variants":["Fourier dimension found for torus GMC in all dimensions","Exact Fourier dimension for high-dimensional Gaussian chaos","New proof resolves GMC Fourier dimension open problem","Torus GMC Fourier dimension matches correlation dimension"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes, through property (P4), uniform $p$-th moment bounds for derivatives of the smooth approximating processes up to order $d$, yet Lemma 4.11 needs the same control for derivatives up to order $2d$; if that higher-order estimate is not derivable from the construction, the repeated integration by parts is not fully justified.","fun_headline_variants_meta":{"raw":{"variants":["Fourier dimension found for torus GMC in all dimensions","Exact Fourier dimension for high-dimensional Gaussian chaos","New proof resolves GMC Fourier dimension open problem","Torus GMC Fourier dimension matches correlation dimension"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000151,"raw_usage":{"total_tokens":1157,"prompt_tokens":861,"completion_tokens":296,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":233}},"tokens_in":477,"tokens_out":296,"duration_ms":3687,"temperature":1.0,"reasoning_tokens":233,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:42:33.685859+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the processes built in Proposition 3.1, $\\mathbb{E}[|D^\\alpha \\psi_j(z)|^p]$ for a fixed multi-index $\\alpha$ with $|\\alpha|=2d$ and check whether it stays bounded by a constant times $j^{2|\\alpha|p}2^{j|\\alpha|p}$. If it grows faster, Proposition 4.4 and hence the lower bound in Theorem 1.1 fail.","supporting_citations":[],"review_version":1}