{"id":"0b92c0b8-d21f-4ba7-a2ae-c53313df2c80","arxiv_id":"1908.05753","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New counterexample constructions yield the exact parabolic Fourier decay rate (d−1)α/d for α in [d−1,d).","lead":"This paper proves new upper bounds on how fast the Fourier transform of fractal measures decays when averaged over spheres or paraboloids. It also determines the exact decay rate for parabolic averages when the measure dimension is between d−1 and d, and it shows a standard route to Falconer's distance set conjecture cannot reach its target threshold.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Corollary 1.4 follows from a self-contained upper-bound construction and cited lower bounds; the unproved lattice-point bound in Section 2 affects only the spherical theorems.","rationale":"The reader correctly identified the Section 2 lattice-point bound as the weakest external input, and I agree it is a genuine omitted proof. However, that bound is load-bearing only for the spherical upper bounds of Theorem 1.1, not for the central parabolic exact-rate result in Corollary 1.4. The parabolic construction in Section 3 uses a rectangular Euclidean lattice, so its measures and c_α estimates are computed directly without invoking lattice points on spheres. The only external dependencies of Corollary 1.4 are the lower bounds from [4,6], which are published and cited transparently; no circularity or inconsistency appears. The manuscript's conclusion is therefore well supported, and the reader's ACCEPT verdict stands unchanged.","tokens_in":15413,"tokens_out":22203,"duration_ms":203279,"concrete_test":"Verify the cited lattice-point bound in the smallest case needed by Theorem 1.1: d=4, m=1, so s=d−m=3 and the claim is #Γ ≳ R^{κ}. Check, via the standard theorem on representations of integers as sums of three squares in Fricker's survey [9], that for a sequence of N one has r_3(N) ≳ N^{1/2}, which translates to R^{κ}; if instead the truth carries an epsilonic loss, equation (2.9) would need an extra factor R^{-ε} and the spherical upper bounds would have to be relaxed. Corollary 1.4 would remain unaffected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Corollary 1.4. Its upper bound comes from Theorem 1.2(a) with m=1, giving κ4(1)=(d−α)/(2d), and the proof of that case (Lemma 3.2, Case I) is self-contained: the measure is a product of boxes on rectangular Euclidean lattices, so all volume and c_α estimates are elementary and explicit. The lower bound is cited from [4,6] and matches the upper bound algebraically. The only unproved external input I could locate is the lattice-point bound #Γ ≳ R^{κ(d−m−2)} used in Section 2 for the spherical example, equation (2.9); a failure of that bound would weaken Theorem 1.1, but it does not enter the parabolic construction, where the lattice is a rectangular Euclidean lattice rather than integer points on a sphere. I find no internal inconsistency in the κ assignments or the c_α computations for α∈[d−1,d).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the average Fourier decay rate β_d(α,S), defined via the squared L2(S)-norm of μ̂(R·) bounded by c_α(μ)‖μ‖R^{-β} for all α-dimensional measures μ. It constructs explicit product-type 'intermediate dimension' measures and uses them to prove new upper bounds: Theorem 1.1 for the unit sphere and Theorem 1.2 for the truncated paraboloid. The main consequence is Corollary 1.4, which determines the exact value β_d(α,P^{d−1})=(d−1)α/d for d≥3 and d−1≤α<d, by matching the new upper bound from Theorem 1.2(a) with the lower bounds of [4,6]. The proofs are explicit calculations of volumes, c_α(μ) constants, and phase-stationarity; the parabolic construction is self-contained, while the spherical example uses a cited lattice-point bound.","tokens_in":15601,"tokens_out":25974,"duration_ms":219030,"significance":"If correct, the paper is a clear advance: it resolves the parabolic average Fourier decay rate on the interval d−1≤α<d for every dimension d≥3, and it improves the previously known upper bounds for both paraboloids and spheres throughout the intermediate range. The test-measure construction is simple and explicit, and the c_α(μ) computations are checkable line by line. The lower bounds used in Corollary 1.4 come from independent published theorems, so there is no circularity. The 'intermediate dimension' idea is likely to be useful for further problems in Fourier restriction and maximal estimates.","major_comments":[],"minor_comments":[{"comment":"The bound #Γ ≳ R^{κ(d−m−2)} is quoted as 'well known' with reference [9], but no precise theorem or page is given; since Theorem 1.1 depends on it, please add a precise statement with a reference or a short proof sketch, and note explicitly that this input is only needed for the spherical theorems and not for Corollary 1.4.","section":"Section 2, Eq. (2.9)"},{"comment":"In the proof of Lemma 3.2 the notations c_α(μ,r) and C_α(μ,r) are used interchangeably; please unify the notation.","section":"Section 3.2"},{"comment":"In the summary for m≤α≤d−m, the term c_α(μ,R^{2κ−1}) arising from (3.44) is omitted from the displayed maximum; the domination by the other terms is true under the stated restrictions, but the 'combining' step should include one sentence justifying this omission.","section":"Section 3.2, Case I"},{"comment":"The arguments in κ4(m;d,α) and κ5(m;d,α) should be written in the same order as in (1.3), namely κ4(m;α,d) and κ5(m;α,d).","section":"Section 1, after (1.2)"},{"comment":"The notation B^d_r and I_r is introduced but used only locally; the proof would be easier to follow if the notation were either used consistently throughout or removed.","section":"Section 3, after (3.24)"}],"recommendation":"minor_revision","confidential_remarks":"I found no evidence of circularity or overclaiming. The only external input that is not proved in the paper is the standard lattice-point bound in Section 2, and it does not affect the main parabolic result. The paper is a good fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine advance, not a repackaging. Du takes the intermediate-dimension trick used for Schrödinger maximal estimates and applies it to the Fourier-average problem. The new upper bounds in Theorems 1.1 and 1.2 are strictly better than the previous best in the full stated ranges, and Corollary 1.4 gives the exact parabolic decay rate β_d(α,P^{d−1})=(d−1)α/d for α∈[d−1,d). That exact range was open, so the paper resolves a mini-Mattila problem in those dimensions.\n\nThe proofs are explicit and honest. The test measures are concrete products of boxes on rectangular lattices, and the c_α computations are done by hand in Lemmas 2.2 and 3.2. The case analysis is tedious but coherent; I checked the exponent balance in the main regimes and it works. The lower bound side of Corollary 1.4 is imported from Du–Zhang and Du–Guth–Ou–Wang–Wilson–Zhang, which are published independent theorems; citing them here is not circular because the upper bound is what is new.\n\nThe main soft spot is the spherical example in Section 2. It relies on a lattice point bound #Γ ≳ R^{κ(d−m−2)} for integer points on a sphere, which is cited to Fricker's book but not proved. That is standard, and I would not block a paper over it. Still, it means Theorem 1.1 inherits a mild external input. The parabolic result does not: the lattice there is rectangular, so the volume estimates are elementary. If you care about a completely self-contained proof, the spherical part is the only place with a gap.\n\nI also like Remark 1.6: it quantifies what Mattila's scheme can and cannot do for Falconer's conjecture. The numbers he gets for the best possible threshold via that route are worth having in the literature, even if they are not a breakthrough by themselves.\n\nWho is this for? Anyone working on Fourier decay of fractal measures, distance sets, or restriction estimates. The paper is clearly written and the numerology is laid out in small dimensions, which helps a reader check the statements. It deserves full peer review and, in my view, acceptance. The one thing I would ask a referee to scrutinize is the lattice point bound and the exactness of the volume lower bound in Section 2, plus the case analysis in part (b)–(f) of Theorem 1.2, which is easy to get wrong but appears correct.","headline":"Du improves the known upper bounds for average Fourier decay rates and pins down the exact parabolic rate on [d−1,d); the paper deserves a serious referee.","tokens_in":16126,"tokens_out":2012,"would_cite":true,"duration_ms":18279,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that fractal measures of dimension α in [d−1,d) have parabolic average Fourier decay exactly (d−1)α/d, and it improves all prior upper bounds on spheres by an intermediate-dimension construction.","keywords":["Fourier decay rates","fractal measures","spherical averages","parabolic averages","average decay","intermediate dimension","distance sets","restriction estimates"],"falsifier":"For the exact parabolic claim, take $d=3$, $\\alpha=2$, where the paper predicts $\\beta_3(\\alpha,P^2)=4/3$; a direct evaluation of $\\|\\hat\\mu(R\\,\\cdot)\\|^2_{L^2(P^2)}$ for the measure constructed in Section 3 along $R\\to\\infty$ would either stay comparable to $R^{-4/3}$ or grow relative to it, settling the formula. For the spherical upper bound, a direct number-theoretic count of the lattice-point set $\\Gamma$ for $d-m=2$ would test the external estimate on which the construction's volume lower bound rests.","tokens_in":15221,"feed_emoji":"📐","tokens_out":11020,"duration_ms":99424,"temperature":0.7,"pith_summary":"Fractal measures have Fourier transforms that typically decay as the frequency grows; this paper asks how fast the average of that transform over a sphere or a paraboloid must decay. For a measure of dimension $\\alpha$ in $\\mathbb{R}^d$, the answer is unknown in most of the range $d/2<\\alpha<d$. The paper proves new upper bounds by building measures concentrated near an intermediate-dimensional slice of the hypersurface, and for the paraboloid these bounds meet the best known lower bounds when $d-1\\le\\alpha<d$. The result is an exact formula, $\\beta_d(\\alpha,P^{d-1})=(d-1)\\alpha/d$, for every $d\\ge 3$. That matters because such average decay rates control geometric questions such as the size of distance sets of fractal subsets.","feed_headline":"Fractal measure decay on paraboloids pinned down exactly","feed_subtitle":"For dimensions α between d−1 and d, the average decay rate is (d−1)α/d, matching lower bounds in all d≥3.","key_machinery":"The load-bearing mechanism is the intermediate-dimension parameter $\\kappa$, chosen from five functions $\\kappa_1,\\dots,\\kappa_5$ of $m,\\alpha,d$. A cap $\\Omega$ is formed by an $m$-dimensional ball of radius $R^{-1/2}$ fibered over lattice points spaced $R^{\\kappa-1}$ on the hypersurface, and $\\Lambda$ is the matching set in physical space, with the normal coordinate quantized so that $R(x\\cdot\\xi)$ is nearly an integer multiple of $2\\pi$. The two lemmas do the real work: on $\\Lambda$ the phase is almost constant modulo $2\\pi$, giving $|Ef(Rx)|\\sim\\sigma(\\Omega)$, and the dimension constant $c_\\alpha(\\mu)$ is $\\sim R^{\\alpha-d}$ exactly when $\\kappa$ equals the prescribed formula. Balancing these two estimates reduces the upper bound to the exponent computation $\\beta\\le\\alpha-1+2\\kappa$; the piecewise definitions in Theorems 1.1 and 1.2 are exactly the choices of $m$ and $\\kappa$ that minimize this exponent for each interval of $\\alpha$.","core_discovery":"The paper establishes that for the truncated paraboloid $P^{d-1}\\subset\\mathbb{R}^d$ and any $\\alpha$-dimensional measure with $d-1\\le\\alpha<d$, $d\\ge3$, the average Fourier decay exponent is exactly $\\beta_d(\\alpha,P^{d-1})=(d-1)\\alpha/d$. The upper-bound half is new: for each $\\alpha$ and a chosen integer $m$ with $0<m<d/2$, one constructs a measure $\\mu=\\chi_\\Lambda\\,dx$ supported on a thin lattice-like set and a test function $f$ supported on a matching cap $\\Omega$ on the paraboloid; the phase of the extension operator is within $1/100$ of $2\\pi\\mathbb{Z}$ on $\\Lambda$, so $|Ef(Rx)|\\sim|\\Omega|$ there. Choosing the lattice spacing parameter $\\kappa$ so that $c_\\alpha(\\mu)\\sim R^{\\alpha-d}$ makes the failure of (2.5) as strong as possible, forcing $\\beta_d(\\alpha,P^{d-1})\\le\\alpha-1+2\\tilde\\kappa(\\alpha,d)$. Part (a) gives $\\tilde\\kappa=(d-\\alpha)/(2d)$, which together with the lower bounds yields the exact formula. The same construction on the sphere gives improved piecewise upper bounds $\\beta_d(\\alpha,S^{d-1})\\le\\alpha-1+2\\kappa(\\alpha,d)$, though these do not reach the lower bounds.","pith_inferences":["The same intermediate-dimension construction is local and depends only on nonvanishing curvature, so the exact parabolic rate at $\\alpha\\in[d-1,d)$ may transfer to any bounded curved hypersurface through the standard reduction; the paper only states it for the paraboloid.","The piecewise structure of $\\tilde\\kappa$ suggests the true rate for $\\alpha\\in((d-1)/2,d-1)$ may itself have several regimes; one could test sharpness numerically by running the Section 3 construction for the finitely many intervals in Theorem 1.2 at, say, $d=5$ or $d=6$.","A variational reading of the construction—choose $m$ to minimize the resulting exponent subject to $c_\\alpha(\\mu)\\sim R^{\\alpha-d}$—gives a general ansatz for upper bounds on other hypersurfaces; this is an editorial extrapolation, not a theorem in the paper."],"forward_implications":["For every $d\\ge3$ and $d-1\\le\\alpha<d$, the parabolic average Fourier decay rate is now known exactly, so any competing upper-bound construction in that range must agree with $\\beta_d(\\alpha,P^{d-1})=(d-1)\\alpha/d$.","In the full range $\\alpha\\in((d-1)/2,d)$ the new upper bounds are strictly better than all previous ones, for spheres when $d\\ge4$ and for paraboloids when $d\\ge3$, because the chosen $\\kappa$ is always smaller than the previous $\\kappa_1(0)$ or $\\kappa_3(0)$.","Since $\\tilde\\kappa(\\alpha,d)<\\kappa(\\alpha,d)$, the paraboloid admits examples with slower decay than the sphere at the same $\\alpha$; the curvature of the hypersurface measurably changes the optimal rate.","Through the standard scheme connecting Fourier decay to distance sets, the new parabolic bounds cap the dimension threshold that this scheme can reach at $d/2+1/(d+2)$ for even $d$ and $d/2+1/(d+3)$ for odd $d\\ge7$, with the small-dimensional values listed in Remark 1.6, so the scheme alone cannot settle the distance-set conjecture."],"supporting_citations":[{"why":"Supplies the previous best parabolic upper bound whose example is adapted through the intermediate-dimension trick.","marker":"[1]"},{"why":"Supplies the previous best spherical upper bound and the example the spherical construction adapts.","marker":"[11]"},{"why":"Provides the lower bound for d=3 that the new parabolic upper bound matches in the exact range.","marker":"[4]"},{"why":"Provides the lower bound for d≥4 that completes the exact parabolic formula.","marker":"[6]"},{"why":"Gives the earlier lower bounds and the d=2 exact rates that set the context for the new result.","marker":"[12]"},{"why":"Gives the d=2 exact decay rate that the higher-dimensional result extends.","marker":"[15]"},{"why":"Provides the previous upper bound α in the lower range and the Knapp-type example that the new construction improves.","marker":"[14]"},{"why":"Poses the problem of determining spherical average Fourier decay rates for fractal measures.","marker":"[13]"}],"fun_headline_variants":["Exact Fourier decay for fractal measures on paraboloids","Paraboloid Fourier decay rate now exact","Intermediate dimension trick achieves exact paraboloid decay","Upper bound matches lower for paraboloid Fourier decay","New exact rate for fractal Fourier decay on paraboloids"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the cited but unproved lattice-point estimate that $\\#\\{\\omega\\in S^{d-m-1}:R^\\kappa\\omega\\in2\\pi\\mathbb{Z}^{d-m}\\}\\gtrsim R^{\\kappa(d-m-2)}$ along a sequence $R\\to\\infty$, which the spherical volume estimate (2.9) needs and which the exact parabolic result does not use.","fun_headline_variants_meta":{"raw":{"variants":["Exact Fourier decay for fractal measures on paraboloids","Paraboloid Fourier decay rate now exact","Intermediate dimension trick achieves exact paraboloid decay","Upper bound matches lower for paraboloid Fourier decay","New exact rate for fractal Fourier decay on paraboloids"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000708,"raw_usage":{"total_tokens":3151,"prompt_tokens":865,"completion_tokens":2286,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":2211}},"tokens_in":481,"tokens_out":2286,"duration_ms":15683,"temperature":1.0,"reasoning_tokens":2211,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:06:35.861632+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the exact parabolic claim, take $d=3$, $\\alpha=2$, where the paper predicts $\\beta_3(\\alpha,P^2)=4/3$; a direct evaluation of $\\|\\hat\\mu(R\\,\\cdot)\\|^2_{L^2(P^2)}$ for the measure constructed in Section 3 along $R\\to\\infty$ would either stay comparable to $R^{-4/3}$ or grow relative to it, settling the formula. For the spherical upper bound, a direct number-theoretic count of the lattice-point set $\\Gamma$ for $d-m=2$ would test the external estimate on which the construction's volume lower bound rests.","supporting_citations":[{"cited_title":"Barcel´ o, J.M","cited_arxiv_id":null,"evidence_quote":"Supplies the previous best parabolic upper bound whose example is adapted through the intermediate-dimension trick."},{"cited_title":"Luc` a and K","cited_arxiv_id":null,"evidence_quote":"Supplies the previous best spherical upper bound and the example the spherical construction adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the lower bound for d=3 that the new parabolic upper bound matches in the exact range."},{"cited_title":"Du and R","cited_arxiv_id":null,"evidence_quote":"Provides the lower bound for d≥4 that completes the exact parabolic formula."},{"cited_title":"Mattila, Spherical averages of Fourier transforms of measures with fi nite energy; dimen- sions of intersections and distance sets , Mathematika 34 (1987), no","cited_arxiv_id":null,"evidence_quote":"Gives the earlier lower bounds and the d=2 exact rates that set the context for the new result."},{"cited_title":"W olff, Decay of circular means of Fourier transforms of measures , Int","cited_arxiv_id":null,"evidence_quote":"Gives the d=2 exact decay rate that the higher-dimensional result extends."},{"cited_title":"Mattila, Hausdorff dimension, projections, and the Fourier transfor m., Publ","cited_arxiv_id":null,"evidence_quote":"Poses the problem of determining spherical average Fourier decay rates for fractal measures."}],"review_version":1}