{"id":"c2cd5833-e7b8-48d1-a85f-6dcacca21206","arxiv_id":"2412.19956","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If a function lies in L^p and its Fourier transform is supported on a small fractal set on the moment curve, then for p up to a sharp threshold (d^2+d+2α)/(2α) for d≥3, and 4/α for d=2, the function is identically zero.","lead":"For functions whose Fourier transform is confined to a fractal subset of the moment curve, the paper finds the exact range of L^p spaces where the function must be zero. It proves the range is optimal by constructing random Cantor sets, and applies the result to restriction estimates and Wiener's Tauberian theorem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reader's rotating-frame concern about Lemma 2.3 does not land, since the f_{ε,i} live in vertical slabs; the real proof gap is that φ^{(2)}=1 only on B(0,2), which misses the moment-curve tail for d≥6, so f=Σ f_{ε,i} fails as written.","rationale":"I read Lemma 2.3 carefully. The functions f_{ε,i} are inverse Fourier transforms of fhat φ_i; φ_i is a product of a first-coordinate interval bump and a fixed cutoff in the orthogonal coordinates. Hence, for each fixed x', the Fourier transform of f_{ε,i}(·,x') is supported in a common interval I_i of length ≈ε. Rubio de Francia's theorem applies fiberwise, yielding Lemma 2.3; rotating Frenet boxes never enter at this stage (they appear only after convolution with η_{ε,i}, which is not part of Lemma 2.3). So the reader's weakest assumption is not the bottleneck. The genuine gap I found is the normalization φ^{(2)}=1 on B(0,2), insufficient for d≥6. This is a concrete, easily checked issue and it affects the central reduction f=Σ f_{ε,i}. Because the fix is a constant enlargement of the support, I do not think the mathematical claim is wrong; the verdict remains conditional with a precise repair.","tokens_in":29849,"tokens_out":36092,"duration_ms":335232,"concrete_test":"Fix d=6, E=Γ_6, α=1 and t=1; compute (t^2,...,t^6)=(1,...,1), whose Euclidean norm is √5>2, so φ^{(2)}=1 on B(0,2) fails on E and Σ_i φ_i≠1. Then symbolically replace B(0,2) by B(0,2d) in the definition of φ^{(2)} and check that the identities f=Σ f_{ε,i}, Lemma 2.3, and Lemma 2.2 go through unchanged. If they do, the concern is a constant-size fix; if not, the proof needs a more substantive revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 2, after (2.1), the authors define f_{ε,i}=F^{-1}(fhat φ_i) with φ_i(x)=φ_i^{(1)}(x_1)φ^{(2)}(x'), where φ^{(2)}=1 on B(0,2)⊂R^{d-1}. The identity f=Σ_i f_{ε,i} and the RHS of Lemma 2.3 are used throughout, but for d≥6 the moment curve γ(t)=(t,...,t^d), t∈[0,1], has |(t^2,...,t^d)|=√(d−1)>2 at t=1, so φ^{(2)} is not identically 1 on E=Γ_d. Hence Σ_i φ_i≠1 on the Fourier support, and f−Σ_i f_{ε,i} is a nonzero term with Fourier support on E; all subsequent estimates (Lemmas 2.1–2.4, weak convergence) control the wrong object. This is load-bearing for the proof as written, although it is repairable by taking φ^{(2)}=1 on B(0,C_d) with C_d≥√(d−1). The reader's worry about rotating Frenet frames in Lemma 2.3 is not the issue: for fixed x', the Fourier transform of f_{ε,i}(·,x') is supported in the first-coordinate interval I_i, so Rubio de Francia applies fiberwise without controlling rotations.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the L^p-integrability of functions whose Fourier transform is supported on a fractal subset E of the d-dimensional moment curve, under a uniform covering-number bound N(E,ε)≲ε^{-α}. Theorem 1.1 claims that any such f∈L^p with p≤p_α is identically zero, where p_α=(d^2+d+2α)/(2α) for d≥3 and p_α=4/α for d=2. Theorem 1.2 claims optimality of this range via a random Cantor-set construction. The paper also derives consequences for failure of restriction/extension estimates on the moment curve and for the Wiener Tauberian theorem. The proof for d≥3 follows the strategy of Guo–Iosevich–Zhang–Zorin-Kranich and Senthil Raani, using boxes adapted to the Frenet frame of the curve, Rubio de Francia's arbitrary-interval Littlewood-Paley inequality, and the Arkhipov–Chubarikov–Karatsuba oscillatory-integral bound; the optimality proof uses martingale/Cantor measures and Hoeffding's inequality. The d=2 statement is attributed to Senthil Raani's theorem and the d=2 optimality to Ryou's IMRN paper. I agree with the stress-test note that the rotating-frame concern about Lemma 2.3 is not the real obstruction; the genuine gap in the present text is that the cutoff φ^(2) is not identically 1 on the full moment curve for d≥6, so the identity f=Σ_i f_{ε,i} and the subsequent weak-convergence argument fail as written.","tokens_in":30180,"tokens_out":9858,"duration_ms":105669,"significance":"If the proof is repaired, the results constitute a meaningful extension of the known L^p-integrability thresholds for Fourier supports on curves: they add the endpoint, allow fractal sets with a covering-number condition, and prove optimality through random Cantor measures. The applications to restriction estimates and Wiener Tauberian theorems are natural and potentially useful. The paper is generally careful and makes good use of standard external tools; the probabilistic construction is checkable and follows well-established sources. The main deficiency is a localized but load-bearing gap in Section 2, which is repairable within the manuscript's scope. The reliance of the d=2 case and optimality on a paper by one of the authors is not itself a flaw, but it should be transparent to the reader.","major_comments":[{"comment":"The identity f=Σ_i f_{ε,i} is asserted immediately after the definition f_{ε,i}=F^{-1}(fhat φ_i), where φ_i(x)=φ_i^{(1)}(x_1)φ^{(2)}(x') and φ^{(2)}=1 only on B(0,2)⊂R^{d-1}. For the moment curve γ_d(t)=(t,t^2,...,t^d), the non-first coordinates satisfy |(t^2,...,t^d)|≥√(d−1), which is greater than 2 for d≥6. Hence φ^{(2)} is not identically 1 on Γ_d, so Σ_i φ_i is not identically 1 on the Fourier support for a general E satisfying only N(E,ε)≲ε^{-α}; such an E may contain points with t close to 1. Consequently f−Σ_i f_{ε,i} is a nonzero term with Fourier support on E, and Lemmas 2.1–2.4 together with the dominated-convergence argument in the proof of Theorem 1.1 control the wrong object. This is load-bearing. The gap is local and fixable: take φ^{(2)}=1 on B(0,C_d) with C_d≥√(d−1) and adjust the constants in the boxes and in Lemma 2.1 accordingly, or introduce an additional localization in the t-variable that is guaranteed by a partition of unity on the parametrizing interval.","section":"Section 2, paragraph after (2.1)"},{"comment":"I do not agree with the concern that the rotated Frenet boxes invalidate the application of Rubio de Francia's inequality. Because f_{ε,i}=F^{-1}(fhat φ_i) with φ_i(ξ)=φ_i^{(1)}(ξ_1)φ^{(2)}(ξ'), the Fourier transform of f_{ε,i}(·,x') in the first variable is supported in an interval of length ≈ε on a fixed axis, independent of the rotation of the boxes B_{ε,i}. Thus Rubio de Francia's one-dimensional inequality can be applied fiberwise, as the paper states. This part becomes sound once the missing support issue for φ^{(2)} in the previous comment is fixed.","section":"Section 2, Lemma 2.3"}],"minor_comments":[{"comment":"The display defining the weight w appears to mix the variables x_1 and x': the condition is written with both |x_1| and |x^1| in what should be a single condition on the transverse variable. Please make the definition unambiguous, since the A_{p/2} verification depends on w(·,x') being constant in x_1 for fixed x'.","section":"Section 2, proof of Lemma 2.4"},{"comment":"There is an index inconsistency in the displayed estimates after the definition of C_2(E_j,s_1,s_2): some factors use M_j and others use β_{j+1}, and the proof appears to switch between M_j and M_{j+1}. Please recheck these exponents and make the indices uniform.","section":"Section 3, proof of Lemma 3.10"},{"comment":"The exponent in the bound for I in (2.6) is printed in a way that is difficult to parse; please re-typeset it and verify the exponent, since the positivity of the exponent is needed for the limit.","section":"Section 2, proof of Lemma 2.2"},{"comment":"The symbol N is used both for the covering number N(E,ε) and for the large decay parameter in a_j=min{2^{-jN},1}. This overloaded notation is confusing; please use a different letter for the decay parameter.","section":"Section 2.1, proof of (2.2)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely correct and the gap identified in the report is fixable, so I do not recommend rejection. The d=2 case and the optimality example depend on Senthil Raani's theorem and on Ryou's IMRN paper by one of the present authors; this is acceptable, but the editors may wish to satisfy themselves that the cited results are solid and that the dependence is properly disclosed. The manuscript would also benefit from a careful proofreading pass for the index and typographical issues noted in the minor comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is very likely true and this is a genuine advance. The proof as written has a localized, repairable gap for d≥6 that the reader's report missed; the rotating-frame worry in the report is a red herring.\n\nWhat's new: Theorem 1.1 extends Guo–Iosevich–Zhang–Zorin-Kranich from the full moment curve to fractal subsets with N(E,ε)≲ε^{-α}, including the endpoint p=p_α. Theorem 1.2 supplies matching counterexamples via random Cantor sets for α<1 and d≥3, with d=2 handled by Senthil Raani and Ryou, and the authors disclose that dependency. The applications to restriction-estimate failure and Wiener Tauberian theorem are natural and correctly derived from Theorem 1.1. The proof structure is clear: Frenet boxes, an L^2 estimate, Rubio de Francia applied fiberwise, then martingale/Cantor estimates combined with the Arkhipov–Chubarikov–Karatsuba oscillatory lemma. The self-citation to Ryou's IMRN paper is appropriate here because that result is published and the authors flag exactly where it is used.\n\nThe real soft spot: in Section 2, φ^(2) is chosen to equal 1 only on B(0,2)⊂R^{d-1}. For d≥6, the moment curve γ(t)=(t,t^2,...,t^d), 0≤t≤1, has (t^2,...,t^d) with norm up to √(d−1)>2, so φ^(2) is not identically 1 on E. Consequently f=Σ_i f_{ε,i} is false as written, and Lemmas 2.1–2.4 plus the weak convergence argument control the wrong object. This is load-bearing, but it is also easy to repair: take φ^(2)=1 on B(0,C_d) with C_d≥√(d−1). Constants change, exponents do not.\n\nThe reader's concern about rotating Frenet frames in Lemma 2.3 does not land. For each fixed x', the one-dimensional Fourier transform of f_{ε,i}(·,x') is supported in the interval I_i, independent of the rotating frame, so Rubio de Francia's inequality applies fiberwise. There are minor typos, e.g., M_j versus M_{j+1} in Lemma 3.10, and the paper is long enough that these need a careful pass, but I did not find a second structural gap.\n\nThis paper deserves a serious referee. The main result is important, the strategy is sound, and the gap is localized and fixable. I would ask the referee to require the φ^(2) fix and a notation cleanup, then accept.","headline":"Strong, likely-correct sharp L^p threshold for fractal moment-curve sets, but the written proof has a localized support-cutoff bug for d≥6 that is easy to repair.","tokens_in":30739,"tokens_out":4515,"would_cite":true,"duration_ms":44674,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","42B20","28A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fourier support on a fractal moment-curve set forces $f$ to vanish identically in $L^p$ for $p\\le p_\\alpha$; random Cantor sets show the range is sharp.","keywords":["moment curve","fractal Fourier support","covering number","L^p integrability","restriction estimates","Wiener Tauberian theorem","random Cantor sets","square-function inequality"],"falsifier":"A direct refutation would be a nonzero $f\\in L^p(\\mathbb{R}^d)$ with $1\\le p\\le p_\\alpha$ and Fourier support contained in some $E\\subset\\Gamma_d$ with $N(E,\\varepsilon)\\lesssim\\varepsilon^{-\\alpha}$. A more local check: fix $d=3$, take an $\\varepsilon$-net of centres on $[0,1]$, form the Frenet boxes with side lengths $\\varepsilon,\\ldots,\\varepsilon^d$, and test numerically whether $\\|(\\sum_i |f_{\\varepsilon,i}|^2)^{1/2}\\|_p\\le C\\|f\\|_p$ holds uniformly in $\\varepsilon$; a visible failure at some scale would isolate the missing rotation estimate.","tokens_in":29644,"feed_emoji":"📐","tokens_out":17071,"duration_ms":153152,"temperature":0.7,"pith_summary":"The paper asks: if your function's Fourier transform is constrained to live on a thin fractal set, how integrable can the function be before it must be zero? For subsets of the moment curve $\\Gamma_d=\\{(t,t^2,\\ldots,t^d):t\\in[0,1]\\}$, the answer is a sharp exponent $p_\\alpha$. Theorem 1.1 says that if the set has $\\varepsilon$-covering number $N(E,\\varepsilon)\\lesssim\\varepsilon^{-\\alpha}$, then any $f\\in L^p(\\mathbb{R}^d)$ with support of $\\widehat{f}$ inside $E$ is identically zero for $1\\le p\\le p_\\alpha$. Theorem 1.2 constructs random Cantor sets on the moment curve whose Fourier transforms lie in $L^p$ for every $p>p_\\alpha$, proving the range is optimal. The result extends the known full-curve theorem to all fractal subsets and includes the endpoint.","feed_headline":"Fractal moment-curve supports set a sharp zero-function threshold","feed_subtitle":"Covering size on the moment curve sets the exact L^p cutoff; larger p admits nonzero examples.","key_machinery":"The central object is a family of Frenet-adapted rectangular boxes $\\Gamma_{\\varepsilon,t}$ centred on the moment curve with side lengths $\\varepsilon,\\varepsilon^2,\\ldots,\\varepsilon^d$ along the Frenet frame at $\\gamma_d(t)$. These boxes cover a fractal $E$ with $\\lesssim\\varepsilon^{-\\alpha}$ members, and each Fourier piece $f_{\\varepsilon,i}$ is supported in one box; the proof then uses a square-function inequality for arbitrary intervals (through the boxes' one-dimensional projections) to control the square function $(\\sum_i |f_{\\varepsilon,i}|^2)^{1/2}$, and a weighted version of the same inequality to drive the tail terms $b_{j,\\varepsilon}$ to zero. The optimality construction is a random Cantor set on the moment curve, whose martingale structure controls sums of oscillatory integrals via a concentration inequality for bounded random variables and a trigonometric-sum estimate, yielding an almost-sure Fourier decay strong enough for $L^p$ summability.","core_discovery":"Let $d\\ge 2$ and $0<\\alpha\\le1$, and let $E\\subset \\Gamma_d=\\{(t,t^2,\\ldots,t^d):t\\in[0,1]\\}$ satisfy $N(E,\\varepsilon)\\lesssim \\varepsilon^{-\\alpha}$ for all $0<\\varepsilon<1$. Theorem 1.1 asserts that if $f\\in L^p(\\mathbb{R}^d)$ and $\\operatorname{spt}\\widehat{f}\\subset E$, then $f\\equiv0$ whenever $1\\le p\\le p_\\alpha$, where $p_\\alpha=(d^2+d+2\\alpha)/(2\\alpha)$ for $d\\ge3$ and $p_\\alpha=4/\\alpha$ for $d=2$. Theorem 1.2 says the range is optimal: for every $p>p_\\alpha$ there exists a nonzero $f\\in L^p(\\mathbb{R}^d)$ whose Fourier support is a subset of $\\Gamma_d$ with the same covering-number bound. Taken together, the two theorems identify a sharp dimension-dependent threshold separating forced vanishing from genuine existence.","pith_inferences":["A variable-coefficient or rotated-box square-function inequality, proved uniformly in $\\varepsilon$, would make the density step in Lemma 2.3 fully rigorous.","The formula $p_\\alpha=\\max\\{2d/\\alpha,(d^2+d+2\\alpha)/(2\\alpha)\\}$ suggests a general principle for curved submanifolds: the effective threshold is the larger of an isotropic dimension count and a curvature term; testing this on other polynomial curves would give a broader conjecture.","All optimality examples are almost-sure random Cantor measures; an explicit deterministic fractal subset of the moment curve with the same Fourier decay would show whether randomness is essential.","A weight adapted to the Frenet coordinates, replacing the scalar radial cut-off in the tail estimate, might streamline the weighted square-function step and is a natural testable refinement."],"forward_implications":["The endpoint $p=p_\\alpha$ is included, so the earlier full-curve result now holds for every fractal subset of the moment curve, not only for the curve itself.","For $d\\ge3$, $p_\\alpha>2d/\\alpha$, so the curvature of the moment curve forces a strictly stronger vanishing threshold than the Euclidean dimension-based one; for $d=2$ the known $2d/\\alpha$ threshold is recovered.","In restriction theory, any nonzero measure supported on a moment-curve fractal with covering exponent $\\alpha$ fails the extension estimate for $1\\le p\\le p_\\alpha$; with the additional ball-decay condition $\\mu(B(x,r))\\lesssim r^\\alpha$, failure also occurs for $p<q' d(d+1)/(2\\alpha)$.","In the Wiener Tauberian direction, if the zero set of $\\widehat{f}$ lies in such an $E$ and $p\\le p_\\alpha$, then the span of translates of $f$ is dense in $L^{p'}(\\mathbb{R}^d)$, improving the previously known range when $d\\ge3$.","For every $p>p_\\alpha$, the random Cantor construction yields a nonzero $L^p$ function whose Fourier support has covering number $\\varepsilon^{-\\alpha}$, so the threshold $p_\\alpha$ cannot be improved."],"supporting_citations":[{"why":"proves the full-moment-curve result whose range is extended here to fractal subsets and to the endpoint.","marker":"[11]"},{"why":"gives the fractal-support vanishing theorem with threshold 2d/alpha that handles the d=2 case and supplies the box-covering method adapted here.","marker":"[23]"},{"why":"provides the square-function inequality for arbitrary intervals and its weighted version used to control the pieces f_{epsilon,i}.","marker":"[19]"},{"why":"supplies the oscillatory-integral estimate used in Lemma 3.11 and the endpoint example for alpha=1.","marker":"[2]"},{"why":"gives the d=2 random Cantor Fourier-decay bound used to prove optimality in two dimensions.","marker":"[20]"},{"why":"supplies the almost-sure martingale and Fourier-decay properties of random Cantor measures used in the optimality construction.","marker":"[24]"},{"why":"provides the random translational Cantor-set construction used to build the optimality examples.","marker":"[15]"},{"why":"supplies the concentration inequality for sums of bounded random variables used to control sums of oscillatory integrals.","marker":"[13]"}],"fun_headline_variants":["Fractal moment curve: exact L^p zero threshold","Sharp vanishing theorem for Fourier support on moment curve","Moment-curve fractals: optimal range for L^p zero functions","Zero function forced by fractal support on moment curve","Precise L^p cutoff for moment-curve fractal supports"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that a square-function inequality proved for intervals on a fixed coordinate axis can be applied to the family of anisotropic boxes whose Frenet orientations rotate by $O(\\varepsilon)$ between neighbouring centres; the reduction to the fixed axis is asserted but not justified, and Lemma 2.1 depends on this step.","fun_headline_variants_meta":{"raw":{"variants":["Fractal moment curve: exact L^p zero threshold","Sharp vanishing theorem for Fourier support on moment curve","Moment-curve fractals: optimal range for L^p zero functions","Zero function forced by fractal support on moment curve","Precise L^p cutoff for moment-curve fractal supports"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000573,"raw_usage":{"total_tokens":2735,"prompt_tokens":1002,"completion_tokens":1733,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":1649}},"tokens_in":618,"tokens_out":1733,"duration_ms":11542,"temperature":1.0,"reasoning_tokens":1649,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:45:59.082033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct refutation would be a nonzero $f\\in L^p(\\mathbb{R}^d)$ with $1\\le p\\le p_\\alpha$ and Fourier support contained in some $E\\subset\\Gamma_d$ with $N(E,\\varepsilon)\\lesssim\\varepsilon^{-\\alpha}$. A more local check: fix $d=3$, take an $\\varepsilon$-net of centres on $[0,1]$, form the Frenet boxes with side lengths $\\varepsilon,\\ldots,\\varepsilon^d$, and test numerically whether $\\|(\\sum_i |f_{\\varepsilon,i}|^2)^{1/2}\\|_p\\le C\\|f\\|_p$ holds uniformly in $\\varepsilon$; a visible failure at some scale would isolate the missing rotation estimate.","supporting_citations":[{"cited_title":"$L^p$ integrability of functions with Fourier support on a smooth space curve","cited_arxiv_id":"2311.11529","evidence_quote":"proves the full-moment-curve result whose range is extended here to fractal subsets and to the endpoint."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the fractal-support vanishing theorem with threshold 2d/alpha that handles the d=2 case and supplies the box-covering method adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the square-function inequality for arbitrary intervals and its weighted version used to control the pieces f_{epsilon,i}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the oscillatory-integral estimate used in Lemma 3.11 and the endpoint example for alpha=1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the d=2 random Cantor Fourier-decay bound used to prove optimality in two dimensions."},{"cited_title":"Shmerkin and V","cited_arxiv_id":null,"evidence_quote":"supplies the almost-sure martingale and Fourier-decay properties of random Cantor measures used in the optimality construction."},{"cited_title":"L aba and H","cited_arxiv_id":null,"evidence_quote":"provides the random translational Cantor-set construction used to build the optimality examples."}],"review_version":1}