{"id":"63d38e06-1e8e-41d2-9c60-c150015bc438","arxiv_id":"2508.17938","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove the optimal constant in a Fourier uncertainty principle is positive for all parameter values, show extremizers exist, and explicitly identify them in several regimes, confirming Steinerberger's conjecture for the interval.","lead":"The paper proves that a Fourier uncertainty inequality holds with a positive sharp constant for all parameters and that extremizers always exist. It then identifies those extremizers explicitly in several cases, confirming a 2021 conjecture about the smoothest local average.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The Reader's conditional verdict rests on the claim that (3.5) is not established. However, the missing step is a simple sign observation: after the third integration by parts, the integrand is a product of a positive function (-g''') and a bracket whose sign is fixed by the parity of k. For even k, 1-sin θ ≥0; for odd k, -1-sin θ ≤0. Hence (-1)^k c_k >0 for all k, and the upper bound follows immediately from the integrability of g'''. This closes the supposed gap in Theorem 1.4(1). I also checked the existence proof (Theorem 1.2): the normalization, tightness, and Prokhorov argument are sound, and the positivity of C* follows from continuity of the Fourier transform. The radial symmetrization correctly preserves the total variation and moment while not increasing the weighted Fourier norm. The supporting lemmas (3.2, 6.1, 6.3) have no obvious hidden assumptions for the parameter ranges used. I therefore recommend accepting the paper without the conditional revision, while noting that the text could be made more explicit around (3.5) for readability.","tokens_in":21304,"tokens_out":40780,"duration_ms":373548,"concrete_test":"Run an independent check of (3.5): for α∈{2.1, 3, 5, 10} and k=0,...,100, evaluate c_k^{(α)}=∫_0^{1/2}(1-|2x|^α)cos(2π(k+1/2)x)dx numerically; verify (-1)^k c_k>0 and (k+1/2)^3|c_k|≤C(α). Also verify analytically that the integrand bracket in the third integration by parts has constant sign for each parity of k. If all pass, the conditional concern is fully resolved.","verdict_should_be":"ACCEPT","load_bearing_attack":"After a full review, I find no load-bearing gap in the central claims. The Reader's flagged assertion (3.5) is actually valid. For α>2, the third integration by parts gives c_k = (2πλ)^{-3} ∫_0^{1/2} -g'''(x)[(-1)^k - sin(2πλx)] dx with λ=k+1/2. Since -g'''(x)>0 on (0,1/2] and the bracket is sign-definite — nonnegative for even k (1-sin θ ≥0) and nonpositive for odd k (-1-sin θ ≤0) for all θ∈[0,π(k+1/2)] — we get (-1)^k c_k ≥0, with strict positivity because the bracket is not identically zero. The upper bound follows from |bracket|≤2 and ∫-g''' = -g''(1/2)<∞. Thus (3.5) holds and the hypotheses of Lemma 3.2, including (3.1) and condition (iii), are satisfied. Theorem 1.2's existence proof via tightness and Prokhorov is sound, and the radial reduction is valid. I could not identify another internal inconsistency in the proof of Theorem 1.4 or in the supporting lemmas.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a sharp Fourier uncertainty principle for finite non-negative Borel measures on R^d, with the quantity C_{α,β,d}(μ) = ||\\hat{μ}(ξ)|ξ|^β||_∞^α I_μ(|x|^α)^β / ||μ||_{TV}^{α+β}. Theorem 1.2 establishes that the optimal constant C^*_{α,β,d} is positive for all α,β>0 and that a radial extremizer exists. Theorem 1.4 explicitly identifies extremizers in four parameter regimes, including the characteristic function of an interval in d=β=1 for α≥2, thereby confirming a conjecture of Steinerberger. Theorem 1.5 gives a structural dichotomy for extremizers: either the Fourier transform attains its norm at arbitrarily large frequencies or the extremizer is a finite sum of spherical-shell measures; Corollary 1.6 derives the resulting Fourier decay rates. The proofs use compactness via Prokhorov's theorem, a linear-programming weak-duality criterion for extremality, and detailed trigonometric inequalities.","tokens_in":21543,"tokens_out":11246,"duration_ms":119107,"significance":"If the results are correct, this is a substantial advance on a problem that has been open at the level of global extremizers. The explicit certificates in Lemma 3.2 are a particular strength: they reduce extremality to checking a non-negative comparison function, and they give closed-form extremizers and constants in several nontrivial regimes. The confirmation of Steinerberger's conjecture for α≥2 in the one-dimensional β=1 case is a clean, falsifiable statement. Theorem 1.5 and Corollary 1.6 provide new structural information about all extremizers, including a dichotomy between shell-type measures and measures whose Fourier transform reaches the maximal allowed growth at arbitrarily large frequencies. The paper is carefully written and the main line of argument is convincing.","major_comments":[],"minor_comments":[{"comment":"The proof of (3.5) is too terse: the sentence \"(3.5) follows by noting that ... -g'''(x)>0\" does not by itself explain why the oscillatory integral has the stated sign. The argument is valid, but it should be written out: after the third integration by parts one has c_k = (2πλ)^{-3} ∫_0^{1/2} (-g'''(x))[(-1)^k - sin(2πλx)] dx with λ=k+1/2, and the bracketed factor is nonnegative for even k and nonpositive for odd k, while -g'''(x)>0. The upper bound |c_k| ≲ (k+1/2)^{-3} also follows from |(-1)^k - sin(2πλx)|≤2. Please include these two or three lines.","section":"§3, proof of Theorem 1.4(1), inequality (3.5)"},{"comment":"The statement of Lemma 6.1 begins \"Let β=1 and 2≥α≥α0\", while Theorem 1.4(4) requires the union of the regime α∈[α0,2] with β=1 (handled by Lemma 6.1) and the regime α≥2, 0<β≤1 (handled by Lemma 6.3). The partition of the parameter ranges should be stated explicitly so the reader sees which lemma covers which part of the theorem.","section":"§6, Lemma 6.1 and Theorem 1.4(4)"},{"comment":"In the proof that the zero set of H_0 has no accumulation point at t=0, if A-λ*≠0 the conclusion is immediate from continuity, and this case should be separated before considering the expansion with s∈{l,α,l+r}. In the remaining case A=λ*, the stated expansion does give the result, but the role of the constant term should be made explicit.","section":"§4, Lemma 4.2, Step 4"},{"comment":"There is a typo in the definition of δ_{λ S^{d-1}}: \"uniform uniform measure\" should be \"uniform measure\".","section":"Notation, Section 1"},{"comment":"The phrase \"up to scaling and dilation\" in Remark 1.7(i) is redundant; the intended meaning is \"up to dilation and normalization\", and the wording could be unified with the earlier uniqueness statement in Theorem 1.4(1).","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read on Saucedo–Tikhonov (arXiv:2508.17938). Short version: this is a good paper, the main results hold up, and the one point that might trip up a referee is a non-issue.\n\nWhat is new: they prove C*_{α,β,d}>0 for all α,β>0 and existence of extremizers (Theorem 1.2), which was only known for β>d/2. They also give explicit extremizers in four regimes: the interval in d=1, α≥2, β=1 (confirming Steinerberger's conjecture), the tent function, the inverse-radius measure in R^3, and the sphere measure for certain parameters. Theorem 1.5 is a useful structural dichotomy: either the Fourier transform attains its weighted sup norm at arbitrarily large frequencies, or the extremizer is a finite sum of spherical shell measures. The duality framework in Section 4 is clean and gives the asymptotics as well.\n\nThe proof of Theorem 1.2 is exactly as solid as the reader's report says: tightness, Prokhorov, radial averaging. The explicit cases are handled by Lemma 3.2, which is a weak-duality certificate argument. The flagged issue is (3.5), the sign of the Fourier coefficients c_k. The reader worried that the paper just says '-g'''>0' and stops. On reading the actual integral, the step is fine: after three integrations by parts, the coefficient is (2πλ)^{-3} ∫ -g'''(x)[(-1)^k - sin(2πλx)] dx. For even k the bracket is 1 - sin θ ≥ 0; for odd k it is -1 - sin θ ≤ 0. Since -g'''>0, the product has the sign of (-1)^k, and it is not identically zero. So (3.5) holds and the certificate is non-negative. The proof could be expanded by half a sentence, but there's no mathematical gap.\n\nOne minor editorial flaw: the definition of c_k^{(α)} writes the integral from 0 to 1/2. As written, that gives half of the claimed value for α=2. It should be the integral over [-1/2,1/2] (or equivalently twice the given integral). The subsequent identities all use the correct convention, so it's just a typo, but a referee should flag it.\n\nBottom line: the paper deserves a serious referee. The main theorem is a genuine advance, the explicit extremizers are nontrivial, and the duality viewpoint is worth having. I would not hold the paper hostage to the (3.5) step. With the coefficient definition typo fixed, I'd expect it to go through.\n\nRegards.","headline":"Solid paper: proves the general positivity/existence result and confirms Steinerberger's conjecture; the one flagged sign inequality is actually fine, with just a minor typo in a coefficient definition.","tokens_in":22035,"tokens_out":15511,"would_cite":true,"duration_ms":120898,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the averaging uncertainty inequality has a sharp positive constant and an extremizer for every $\\alpha,\\beta>0$, and finds explicit extremizers, including the interval, in four regimes.","keywords":["Fourier uncertainty principle","extremizers","optimal averaging","radial measures","sharp constants","total variation","convex duality","trigonometric inequalities"],"falsifier":"Evaluate numerically the coefficients $c_k^{(\\alpha)}=\\int_0^{1/2}(1-|2x|^\\alpha)\\cos(2\\pi(k+\\tfrac12)x)\\,dx$ for, say, $\\alpha=3,4,5$ and $k=0,1,2,\\dots$; if any $k$ yields $c_k^{(\\alpha)}(-1)^k(k+\\frac12)^3\\le 0$, then the certificate $H$ in the interval case fails to be non-negative and the extremality proof via Lemma 3.2 collapses. A direct check of the positivity of $H(x)=|2x|^\\alpha+|2(x-1)|^\\alpha-2$ on $(1/2,3/2]$ for large $\\alpha$ would independently test the claim.","tokens_in":21128,"feed_emoji":"📐","tokens_out":9640,"duration_ms":88845,"temperature":0.7,"pith_summary":"This paper studies the averaging uncertainty inequality $\\|\\hat{\\mu}(\\xi)|\\xi|^\\beta\\|_\\infty^\\alpha (\\int |x|^\\alpha\\,d\\mu)^\\beta \\ge C \\|\\mu\\|_{TV}^{\\alpha+\\beta}$ for finite non-negative measures, which arose from an axiomatic approach to optimal averaging. The first main result is that the sharp constant $C_{\\alpha,\\beta,d}$ is strictly positive for every $\\alpha,\\beta>0$, and that a non-zero radial extremizer exists in every dimension. The second main result is a structural description of all radial extremizers: either their weighted Fourier transform reaches its supremum at arbitrarily large frequencies, or the measure is a finite sum of spherical-shell measures. In four concrete regimes the extremizers are identified explicitly — the interval characteristic function for $d=1$, $beta=1$, $\\alpha\\ge 2$ (confirming the conjecture that the interval gives the smoothest average), the tent function for $(\\alpha,\\beta,d)=(1,2,1)$, a radial $|x|^{-1}$ density in $\\mathbb{R}^3$, and the uniform sphere measure in $\\mathbb{R}^3$ — with exact sharp constants. A companion corollary fixes the decay rate of the Fourier transform of every extremizer, which was previously open.","feed_headline":"Interval average proven optimal in sharp Fourier bound","feed_subtitle":"Extremizers exist for every α, β > 0; for β = 1, the interval is extremal for all α ≥ 2.","key_machinery":"Lemma 3.2 is the engine of the paper. For a radial candidate $\\mu_*$ it asks for constants $C>0$, $D\\in\\mathbb{R}$ and a function $\\psi$ given by a sum of Fourier transforms of spherical measures, supported on the frequency set where $\\hat\\mu_*|\\xi|^\\beta$ is extremal, such that $H=\\psi+C|x|^\\alpha-D$ is non-negative and vanishes on the support of $\\mu_*$. When this holds, a Plancherel step shows that every competitor with equal total mass and equal weighted Fourier supremum has $I_\\mu(|x|^\\alpha)\\ge I_{\\mu_*}(|x|^\\alpha)$, so $\\mu_*$ is extremal; the same lemma also yields the uniqueness criteria. Section 4 upgrades the idea to a genuine Fenchel–Rockafellar duality for the variational problem, with slackness conditions that force the support of the dual variable onto the extremal frequency set. The dichotomy of Theorem 1.5 and the decay corollary both follow from this duality plus the classical asymptotics of Bessel functions.","core_discovery":"The central claim is that the averaging uncertainty principle is sharp in every parameter range and that its extremizers are not merely abstract: they obey a clean dichotomy and can be written down for several natural parameter choices. For radial measures the dichotomy states that an extremizer either has the property that $|\\hat\\mu(\\xi)||\\xi|^\\beta$ attains its essential supremum for arbitrarily large $|\\xi|$, or it is a finite non-negative combination of uniform spherical-shell measures. A certificate function $H(x)=\\psi(x)+C|x|^\\alpha-D$, with $\\psi$ built from Fourier transforms of spheres, is used to prove extremality through a Plancherel rearrangement: whenever $H\\ge 0$ and $H\\,d\\mu_*=0$, the candidate minimizes the $\\alpha$-th moment among all competitors with the same norm and weighted Fourier bound. The explicit extremizers found by this method include the interval characteristic function for $(\\alpha,1,1)$ with $\\alpha\\ge 2$, whose sharp constant is $1/((2\\pi)^\\alpha(\\alpha+1))$, settling the interval-optimality conjecture from the problem's origin.","pith_inferences":["The certificate method is likely to transfer to discrete and semi-discrete variants of the averaging problem, since it only requires a positive trigonometric (or spherical-harmonic) certificate and a Plancherel identity; testing it on the discrete averaging problem with $\\beta=2$ would be a direct next step.","The rigidity dichotomy suggests a computational strategy for finding extremizers: solve the dual convex program on a grid of spherical-shell atoms; if the answer converges to a sphere measure, Theorem 1.5(4) may hold beyond the stated ranges.","The interval-extremality conjecture could be made fully self-contained if the coefficient sign bound (3.5) were replaced by an exact evaluation of $c_k^{(\\alpha)}$ in terms of hypergeometric or gamma functions; such a formula would also simplify the proof.","If the lower bound in Corollary 1.6 could be upgraded to a limit rather than a limsup, the decay statement would identify the exact order of the Fourier transform at infinity, which would be a new result beyond the paper."],"forward_implications":["The sharp inequality holds for all parameters, so the optimal averaging function exists for every choice of $\\alpha,\\beta$ and every dimension, and the optimal constant is computable in principle as the minimum of an explicit convex problem.","In one dimension with $\\beta=1$ and $\\alpha\\ge 2$, the interval characteristic function is the unique extremizer up to dilation and normalization; this confirms that uniform averaging over an interval is the smoothest average in the sense of the original problem.","Every radial extremizer either oscillates to the maximum of its weighted Fourier transform out to infinity or is a finite sum of spherical shell measures; this reduces the search for optimal averaging measures to two structured families.","The Fourier transform of every extremizer decays like $|\\xi|^{-\\beta}$ (with the exponent $\\gamma$ replaced by $(d-1)/2$ when $\\beta<(d-1)/2$), with nonzero limsup, answering the previously open decay question.","For the tent function in one dimension, the same certificate proves global extremality among positive-definite functions for $(\\alpha,\\beta)=(2,2)$, resolving a question left open in earlier work."],"supporting_citations":[{"why":"introduced the averaging uncertainty inequality and conjectured the interval characteristic function is the extremizer for $d=\\beta=1$, $\\alpha\\ge 2$; also proved local extremality for $\\alpha=2,\\dots,6$.","marker":"[10]"},{"why":"proved the tent function is a local extremizer for $(\\alpha,\\beta,d)=(2,2,1)$ and raised the positive-definite question answered in Section 5.","marker":"[6]"},{"why":"previously established the cosine-coefficient sign estimate (3.5) that the interval-extremality proof relies on.","marker":"[3]"},{"why":"supplies the Fenchel–Rockafellar duality theorem used to prove strong duality and the slackness conditions in Lemma 4.1.","marker":"[11]"},{"why":"provides Prokhorov's theorem, which yields the compactness step in the proof that extremizers exist.","marker":"[1]"},{"why":"source for the radial Fourier transform representation and Bessel function asymptotics used to derive the decay corollary.","marker":"[4]"},{"why":"its Ingham-type trigonometric inequality gives the nonzero limsup in the decay corollary for spherical-shell sums.","marker":"[5]"}],"fun_headline_variants":["Fourier averaging bound is sharp; interval extremal for β=1, α≥2","Extremizers exist for all α,β>0; interval optimal when β=1","Sharp Fourier uncertainty: intervals beat all for averaging","Averaging Fourier inequality: extremizers found, interval for β=1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the interval function is optimal rests on the claimed sign bound $1\\ge c_k^{(\\alpha)}(-1)^k(k+1/2)^3>0$ for its cosine coefficients, which the paper justifies by integration by parts and monotonicity but whose final sign is not fully established for every $k$.","fun_headline_variants_meta":{"raw":{"variants":["Fourier averaging bound is sharp; interval extremal for β=1, α≥2","Extremizers exist for all α,β>0; interval optimal when β=1","Sharp Fourier uncertainty: intervals beat all for averaging","Averaging Fourier inequality: extremizers found, interval for β=1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000297,"raw_usage":{"total_tokens":1697,"prompt_tokens":897,"completion_tokens":800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":716}},"tokens_in":513,"tokens_out":800,"duration_ms":7349,"temperature":1.0,"reasoning_tokens":716,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:59:53.011125+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate numerically the coefficients $c_k^{(\\alpha)}=\\int_0^{1/2}(1-|2x|^\\alpha)\\cos(2\\pi(k+\\tfrac12)x)\\,dx$ for, say, $\\alpha=3,4,5$ and $k=0,1,2,\\dots$; if any $k$ yields $c_k^{(\\alpha)}(-1)^k(k+\\frac12)^3\\le 0$, then the certificate $H$ in the interval case fails to be non-negative and the extremality proof via Lemma 3.2 collapses. A direct check of the positivity of $H(x)=|2x|^\\alpha+|2(x-1)|^\\alpha-2$ on $(1/2,3/2]$ for large $\\alpha$ would independently test the claim.","supporting_citations":[{"cited_title":"Steinerberger","cited_arxiv_id":null,"evidence_quote":"introduced the averaging uncertainty inequality and conjectured the interval characteristic function is the extremizer for $d=\\beta=1$, $\\alpha\\ge 2$; also proved local extremality for $\\alpha=2,\\dots,6$."},{"cited_title":"Kravitz and S","cited_arxiv_id":null,"evidence_quote":"proved the tent function is a local extremizer for $(\\alpha,\\beta,d)=(2,2,1)$ and raised the positive-definite question answered in Section 5."},{"cited_title":"The zeros of certain Fourier transforms:Improvements of P\\'olya's results","cited_arxiv_id":"2110.01885","evidence_quote":"previously established the cosine-coefficient sign estimate (3.5) that the interval-extremality proof relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the Fenchel–Rockafellar duality theorem used to prove strong duality and the slackness conditions in Lemma 4.1."},{"cited_title":"Billingsley","cited_arxiv_id":null,"evidence_quote":"provides Prokhorov's theorem, which yields the compactness step in the proof that extremizers exist."},{"cited_title":"Grafakos","cited_arxiv_id":null,"evidence_quote":"source for the radial Fourier transform representation and Bessel function asymptotics used to derive the decay corollary."},{"cited_title":"Jaming and C","cited_arxiv_id":null,"evidence_quote":"its Ingham-type trigonometric inequality gives the nonzero limsup in the decay corollary for spherical-shell sums."}],"review_version":1}