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Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions

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Pith's one-line read The Mockenhaupt-Mitsis-Bak-Seeger restriction theorem is sharp: for every dimension d and every 0<a,b<d there is a measure meeting the hypotheses whose extension estimate fails below the claimed exponent.

desk verdict The main construction is strong and likely correct, but the paper as written does not prove the advertised full-range sharpness for b>2a; the gap is real and fixable by using q close to 1 instead of the cited Mitsis proposition. read the letter →

arxiv 2505.19526 v1 pith:ZGOCZKQZ submitted 2025-05-26 math.CA math.NT

Robert Fraser , Kyle Hambrook , Donggeun Ryou This is my paper · ORCID
classification math.CAmath.NT MSC 42B1011R0428A78
keywords FourierrestrictionextensionestimatesMockenhaupt-Mitsis-Bak-SeegertheoremSalemsetsFrostmanconditionalgebraicnumbertheoryDiophantineapproximationsharpness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the exponent p*(a,b,d)=(4d-4a+2b)/b in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem cannot be improved, for every dimension d and every pair of parameters 0

What carries the argument

The engine is the limsup set E(K,B,tau), consisting of points x in R^d that lie within distance |N(I)|^{-(tau+1)/d} of the inverse fractional ideal $I^{{-1}}$ for infinitely many ideals I of the ring of integers in a degree-d number field K. The measure is the weak limit of products of periodic bump sums F_k built from inverse prime ideals, and a convolution stability lemma controls the Fourier transform of these products scale by scale, giving the required ball regularity and Fourier decay. Failure of the extension estimate comes from constructive interference on long grids: the inverse of a prime ideal plays the role of an arithmetic progression, and the test functions are bumps supported on one such grid. A separation lemma based on ideal norms prevents the bumps from overlapping and supplies the counting estimates needed for the ball condition.

What would settle it

Take a concrete number field, say an imaginary quadratic field with d=2, and compute at a sequence of scales k the ratio ||\widehat{\Phi_{J_k,\eta_k}\mu}||_{L^p}/||\Phi_{J_k,\eta_k}||_{L^q(\mu)} for p slightly below p(\tau,\rho,q,d): the claimed divergence is quantitative in M_k, so a bounded ratio at arbitrarily large scales would disprove the construction. Separately, the reduction for b>2a is falsifiable by exhibiting a measure that satisfies (A) and (R) with p<2d/a, contradicting the cited external proposition.

Watch

Extended reading notes

Core claim

The central claim is that the exponent p*(a,b,d) is best possible on the full parameter range 0<a,b<d. Theorem 1.2 constructs, for any -d<rho<d and tau>1, a probability measure supported on E(K,B,tau) that satisfies the ball condition with exponent a<2d-rho_-/(1+tau), satisfies Fourier decay with exponent b<2(d-rho_+)/(1+tau), and admits functions f_k for which the extension ratio tends to infinity whenever p<p(tau,rho,q,d). Remark 1.3 converts these parameters into arbitrary a,b with 0<a,b<d, using the cited reduction to the case b<=2a, so the constructed measure violates the extension estimate for every p below p*(a,b,d). The same construction shows that E(K,B,tau) is a Salem set of dimension 2d/(1+tau).

Load-bearing premise

The full-range result relies on a previously published lemma, not proved here, saying that any measure satisfying the ball condition and the extension estimate must have p at least 2d/a; if that lemma carries hidden extra assumptions, the paper's coverage when b>2a would not follow from its construction.

Editorial extensions

If this is right

  • The Mockenhaupt-Mitsis-Bak-Seeger exponent cannot be lowered for any 0<a,b<d, so any sharper restriction estimate for measures satisfying (A) and (B) must impose additional structure beyond these two conditions.
  • The construction gives deterministic Salem sets of every dimension 2d/(1+tau) in every dimension d, complementing the usual random constructions.
  • Sharpness holds for general L^q extension estimates: for each 1<=q<infty, the extension estimate from L^q(mu) fails below p=q(dtau-rho)/((q-1)(d-rho_+)).
  • At the endpoint p=p*(a,b,d) the positive theorem still holds, so the result pins down the cutoff exactly rather than merely shrinking the possible range.
  • The same sharpness statement, with a modified construction, holds for the element-based set Eprin(K,B,tau), matching the classical Kaufman-type well-approximable sets in higher dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The full coverage of the range b>2a depends on a cited external proposition that the paper does not reprove; a reader checking the complete claim should verify that proposition's hypotheses apply exactly as stated.
  • The algebraic-number-field encoding suggests a portable template: other families of lattices with controlled norms and separation should yield sharpness results for other translation-invariant fractal restriction settings.
  • Because the counterexamples are deterministic, they may be usable where random Salem measures are not, such as in explicit constructions in additive combinatorics or in questions requiring a fixed measure with simultaneous Frostman and Fourier-decay control.
  • The dimension bound 2d/(1+tau) for E(K,B,tau) parallels classical Diophantine approximation exponents, so the Salem property established here can be read as an exact Fourier-analytic counterpart of those approximation results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper constructs, for each number field K of degree d and parameters τ > 1, -d < ρ < d, a deterministic Borel probability measure μ supported on the set E(K,B,τ) of points that are well approximable by inverse ideals. The measure is shown to satisfy a ball condition with exponent a < (2d - ρ₋)/(1 + τ), Fourier decay with exponent b < 2(d - ρ₊)/(1 + τ), and to violate the extension estimate for p below p(τ,ρ,q,d). By combining this with the parameter conversion in Remark 1.3, the authors claim that the exponent p*(a,b,d) in the Mockenhaupt-Mitsis-Bak-Seeger restriction theorem is best possible for every 0 < a,b < d and every dimension d, and that the construction also yields Salem sets.

Significance. If the full-range claim held, this would settle the sharpness of the MMBS restriction theorem in all dimensions and the full parameter range, with a deterministic construction. The paper contains substantial new technical work: a number-field generalization of Kaufman's construction, a convolution stability lemma, separation lemmas, and detailed Fourier estimates. The deterministic nature and the Salem-set corollary are notable strengths. However, the full-range claim depends on a reduction in Remark 1.3 that has a quantitative gap and on an external proposition that is not justified as stated. The portion of the argument covering b ≤ 2a appears internally consistent and is a significant contribution in its own right.

major comments (2)
  1. [Remark 1.3] The parameter reduction does not cover the range b > 2a. For such (a,b), p*(a,b,d) = 4(d-a)/b + 2 = 4d/b + (2 - 4a/b) > 4d/b. Any admissible choice a0,b0 in the remark must satisfy b0 ≤ 2a0 and b0 > b, so p*(a0,b0,d) = 4(d-a0)/b0 + 2 ≤ 4d/b0 < 4d/b < p*(a,b,d). Therefore no choice can satisfy p0 < p*(a0,b0,d) for p0 in the interval (4d/b, p*(a,b,d)). Consequently Theorem 1.2 supplies counterexamples only for p0 < 4d/b in the case b > 2a, and the claimed full-range optimality is not established.
  2. [Remark 1.3] The assertion that (A) and (B) imply b ≤ 2a via Mitsis [24, Proposition 3.1] is not justified and appears false as stated. The one-sided Frostman condition (A) gives an upper bound μ(B(x,r)) ≲ r^a, not the lower bound needed for the Knapp argument. For example, Lebesgue measure restricted to [0,1]^2 satisfies (A) with a = 1/2 and (B) with b = 2, and Theorem 1.1 with these parameters gives (R) for every p ≥ p*(1/2,2,2) = 5. The cited proposition, as stated, would force p ≥ 2d/a = 8, contradicting the theorem's consequence. Unless the proposition has additional hypotheses that are not reproduced in the manuscript, the derivation of b ≤ 2a is unsupported; independently of the parameter gap above, this affects the coverage of b > 2a.
minor comments (4)
  1. [Lemma 5.2] Lemma 5.2 is stated without proof and is used in Lemma 5.3 and in the proof of Proposition 8.1. Since the rest of the paper is detailed, the proof should be supplied or a precise explanation given of why it follows verbatim from the proof of Lemma 5.1.
  2. [Section 4.3] The sequence (M_k) is only required to grow 'sufficiently rapidly' or 'rapidly' without an explicit list of the growth conditions. Because these conditions are used in several inductive arguments (Lemmas 4.3, 5.1, 7.5, and 8.9), stating the quantitative conditions explicitly would improve verifiability.
  3. [Lemma 8.2] The proof of Lemma 8.2 establishes a lower bound for the truncated measures μ_l and then concludes the same bound for μ. The passage to the weak limit l → ∞ should be stated explicitly, since the estimates are uniform in l but the limit step is not written out.
  4. [Proof of Lemma 6.1] In the proof of Lemma 6.1, the citation for the Lévy continuity theorem repeats '[2, Section 26]' for both d = 1 and d ≥ 1; the second citation appears to be intended for [4].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is explicit and self-contained, and the sharpness reduction in Remark 1.3 is a parameter conversion rather than a fit or a self-citation chain.

full rationale

The main construction in Theorem 1.2 is deterministic and proved directly: the measure is built from number-theoretic lattices, and the paper proves the support condition, the ball condition (A), the Fourier decay condition (B), and the failure of the extension estimate (R) for p below the stated threshold. The key estimates (Lemmas 4.3, 5.1, 7.5, 8.2 through 8.10, and the Convolution Stability Lemma 9.1) are proved in the paper rather than imported. Remark 1.3 is not circular: given b <= 2a, it selects a0 and b0 and converts them to tau and rho by explicit formulas so that the constructed measure has the desired exponents and its failure threshold equals p*(a0,b0,d); this is a parameter change, not an inversion of the conclusion. The reduction to the case b <= 2a uses the external Mitsis [24, Proposition 3.1] together with the forward direction of Theorem 1.1. That is a necessity argument and, in any case, it is an external citation, not a self-citation. If the external proposition or the coverage of the range b > 2a is incomplete, that is a correctness or completeness risk, not a circularity: no equation in the paper is defined in terms of the result it is used to prove, and no fitted quantity is renamed as a prediction. The self-citations to [9], [10], [12], and [13] are for technique and previous special cases, but the load-bearing estimates are established in this paper. Hence no identifiable circular step is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters: tau and rho are determined by the target exponents a0,b0, and the sequence (M_k) is chosen inductively rather than fitted. The construction relies on standard algebraic number theory and analytic tools, plus the cited Mitsis proposition for the reduction to b <= 2a.

assumptions (4)
  • standard math Landau prime ideal theorem: the number of prime ideals with norm in [M^d/2, M^d] is ~ M^d / log M
    Used in Section 4.2 to define the normalization c_k in (4.6) and to ensure Q(M_k) is non-empty.
  • standard math Dedekind zeta function zeta_K(z) converges for Re(z)>1
    Used in Proposition 10.1 to bound the Hausdorff dimension of E(K,B,tau).
  • domain assumption Mitsis [24, Proposition 3.1]: if (A) and (R) hold, then p >= 2d/a
    Used in Remark 1.3 to reduce the sharpness problem to the case b <= 2a, which is essential for the full-range claim.
  • standard math Levy continuity theorem and Poisson summation formula
    Used in Section 6 to pass to the weak limit measure mu and to extend Fourier decay estimates from integer frequencies to all frequencies.

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Cite this review

Pith. "Pith review of Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions." pith.science (2026). https://pith.science/paper/ZGOCZKQZ

@misc{pith2026250519526,
  author       = {Pith},
  title        = {Pith review of: Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGOCZKQZ}},
  note         = {Machine review of arXiv:2505.19526}
}
abstract

We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sharpness of convolution bounds for measures

    math.CA 2026-05 unverdicted novelty 8.0 of 10

    Sharp (p,q) ranges for convolution bounds with measures in P_{alpha,beta} are determined via constructions that also sharpen L^2 restriction estimates uniformly across regimes and dimensions.

  2. Fourier Frames on Salem Measures

    math.CA 2025-06 accept novelty 7.0 of 10

    For every 0<s≤1 there are s-dimensional Salem measures on the unit interval admitting no Fourier frame, and such measures appear in every known Salem construction type.

Reference graph

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