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REVIEW 2 major objections 4 minor 28 references

$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.19956 v2 pith:MZL4QXPV submitted 2024-12-27 math.CA

Shengze Duan , Minh-Quy Pham , Donggeun Ryou This is my paper · ORCID
classification math.CA MSC 42B1042B2028A75
keywords momentcurvefractalFouriersupportcoveringnumberL^pintegrabilityrestrictionestimatesWienerTauberiantheoremrandomCantorsetssquare-functioninequality
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 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.

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 (2)
  1. [Section 2, paragraph after (2.1)] 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.
  2. [Section 2, Lemma 2.3] 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.
minor comments (4)
  1. [Section 2, proof of Lemma 2.4] 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'.
  2. [Section 3, proof of Lemma 3.10] 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.
  3. [Section 2, proof of Lemma 2.2] 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.
  4. [Section 2.1, proof of (2.2)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivation is self-contained and uses external theorems; the only self-citation is an independent published result, not a restatement of the paper's claim.

full rationale

The proof of Theorem 1.1 is a forward derivation. The function is decomposed using a partition of unity adapted to Frenet boxes, and Lemmas 2.1 and 2.2 estimate the resulting u_eps through Plancherel's theorem, Hölder's inequality, Rubio de Francia's Littlewood–Paley inequality, and a cancellation b_{j,eps}→0. No fitted parameter is introduced, and the threshold p_alpha is a computed quantity rather than an input to the estimates. The d=2 case is imported from Senthil Raani [23], an independent external theorem, while the d≥3 case uses the Arkhipov–Chubarikov–Karatsuba oscillatory integral bound and Rubio de Francia's inequality. The optimality direction constructs random Cantor measures with dimension controlled by β_j, and the Fourier decay estimates are proved using Hoeffding's inequality and the independent oscillatory integral bound; the desired L^p summability is not assumed. The only self-citation is [20] (Ryou) for the d=2 Fourier decay estimate, and that is a published, independent result used as an external theorem rather than a restatement of the present claim. No equation-level identity makes a predicted quantity equal to an input by construction. Potential technical gaps in the proof, such as the support property of φ^(2), would be correctness issues rather than circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof imports standard tools from Fourier analysis, probability, and restriction theory. No free parameters are fitted; the construction parameters m_j and β_j are determined by the dimension parameter α. The d=2 part of the main theorem and of the optimality construction relies on published results, one of which (Ryou [20]) is by an author of this paper but is an independent published theorem. The main d≥3 proof does not depend on the authors' own results.

assumptions (7)
  • standard math Rubio de Francia's Littlewood-Paley inequality for arbitrary intervals (Theorem 1.2 in [19]) and its weighted version (Theorem 6.1 in [19])
    Used in Lemma 2.3 and Lemma 2.4 to bound the L^p norm of the square function of the localized pieces f_{ε,i}; this is the core estimate in the proof of Theorem 1.1.
  • standard math Arkhipov-Chubarikov-Karatsuba oscillatory integral estimate (Theorem 1.1 in [2])
    Used as Lemma 3.11 to control individual oscillatory integrals over intervals in the random Cantor set construction for optimality.
  • standard math Senthil Raani's L^p theorem for Fourier supports with covering number condition (Theorem 2.3 in [23] as restated in Remark 1.2)
    Used to handle the d=2 case of Theorem 1.1, giving threshold 4/α.
  • standard math Ryou's Fourier decay estimate for random Cantor sets on the parabola (Theorem 1.2 in [20])
    Used in the proof of Proposition 3.1 for d=2 to establish optimality when p > 4/α.
  • standard math Hoeffding's inequality (Lemma 3.8)
    Used in the probabilistic estimates for the random Cantor measure Fourier transform differences.
  • domain assumption Martingale convergence and Ahlfors-David regularity properties of the random Cantor measures from Shmerkin-Suomala [24] and Laba-Wang [15]
    Used to define the limiting measure ν and to assert AD-regularity and covering number control for the support.
  • domain assumption The equivalence between the covering number condition N(E,ε) ≲ ε^{-α} and the existence of a finite-overlapping cover by Frenet boxes of side lengths ε,...,ε^d
    Used at the start of Section 2 to set up the localization; not proved explicitly but geometrically plausible for the nondegenerate moment curve.

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Pith. "Pith review of $L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve." pith.science (2026). https://pith.science/paper/MZL4QXPV

@misc{pith2026241219956,
  author       = {Pith},
  title        = {Pith review of: $L^p$-integrability of functions with Fourier supports on fractal sets on the moment curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZL4QXPV}},
  note         = {Machine review of arXiv:2412.19956}
}
abstract

For $0 < \alpha \leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-\alpha}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*} 1 \leq p\leq p_\alpha:= \begin{cases} \frac{d^2+d+2\alpha}{2\alpha} & d \geq 3, \frac{4}{\alpha} &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.

Figures

Figures reproduced from arXiv: 2412.19956 by the authors.

Figure 1
Figure 1. Failure range of the extension estimate: qα “ # pα d ě 3 4 d “ 2. Also, when α “ 1, the arc length measure on Γd is an example such that (1.2) holds if and only if p ě q 1 dpd ` 1q 2α and p ą pα. (1.5) However, when 0 ă α ă 1, there is no such example known yet. When d “ 2, Ryou [20] constructed a measure such that (1.2) holds when p ą 6{α and q “ 2. Even with interpolation, this result only covers a partial range o… view at source ↗

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Works this paper leans on

28 extracted references · 25 canonical work pages

  1. [1]

    M. L. Agranovsky and E. K. Narayanan.L p-integrability, supports of Fourier transforms and unique- ness for convolution equations.J. Fourier Anal. Appl., 10(3):315–324, 2004

    work page 2004

  2. [2]

    G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba.Trigonometric sums in number theory and analysis, volume 39 ofDe Gruyter Expositions in Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin, 2004. Translated from the 1987 Russian original

    work page 2004

  3. [3]

    Beurling

    A. Beurling. On a closure problem.Ark. Mat., 1:301–303, 1951

    work page 1951

  4. [4]

    Brandolini, G

    L. Brandolini, G. Gigante, A. Greenleaf, A. Iosevich, A. Seeger, and G. Travaglini. Average decay estimates for Fourier transforms of measures supported on curves.J. Geom. Anal., 17(1):15–40, 2007

    work page 2007

  5. [5]

    Demeter.Fourier restriction, decoupling, and applications, volume 184 ofCambridge Studies in Advanced Mathematics

    C. Demeter.Fourier restriction, decoupling, and applications, volume 184 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2020

    work page 2020

  6. [6]

    N. P. Dobronravov. A measure with small support and p-summable fourier transform.arXiv preprint arXiv:2412.07314, 2024

    work page Pith review arXiv 2024

  7. [7]

    W. F. Donoghue, Jr.Distributions and Fourier transforms, volume 32 ofPure and Applied Mathe- matics. Academic Press, New York, 1969

    work page 1969

  8. [8]

    S. W. Drury. Restrictions of Fourier transforms to curves.Ann. Inst. Fourier (Grenoble), 35(1):117– 123, 1985

    work page 1985

Show all 28 references
  1. [9]

    G. A. Edgar and J. M. Rosenblatt. Difference equations over locally compact abelian groups.Trans. Amer. Math. Soc., 253:273–289, 1979

    1979

  2. [10]

    Grafakos.Classical Fourier Analysis

    L. Grafakos.Classical Fourier Analysis. Graduate Texts in Mathematics. Springer New York, NY, 2014. 35

    2014

  3. [11]

    S. Guo, A. Iosevich, R. Zhang, and P. Zorin-Kranich.L p integrability of functions with Fourier support on a smooth space curve.arXiv preprint arXiv:2311.11529, 2023

    2023 arXiv

  4. [12]

    C. S. Herz. A note on the span of translations inL p.Proc. Amer. Math. Soc., 8:724–727, 1957

    1957

  5. [13]

    Hoeffding

    W. Hoeffding. Probability inequalities for sums of bounded random variables.J. Amer. Statist. Assoc., 58:13–30, 1963

    1963

  6. [14]

    J.-P. Kahane. Dimension capacitaire et dimension de Hausdorff. InTh´ eorie du potentiel (Orsay, 1983), volume 1096 ofLecture Notes in Math., pages 393–400. Springer, Berlin, 1984

    1983

  7. [15]

    L aba and H

    I. L aba and H. Wang. Decoupling and near-optimal restriction estimates for Cantor sets.Int. Math. Res. Not. IMRN, 2018(9):2944–2966, 2018

    2018

  8. [16]

    Lev and A

    N. Lev and A. Olevskii. Wiener’s ‘closure of translates’ problem and Piatetski-Shapiro’s uniqueness phenomenon.Ann. of Math. (2), 174(1):519–541, 2011

    2011

  9. [17]

    T. Mitsis. A Stein-Tomas restriction theorem for general measures.Publ. Math. Debrecen, 60(1-2):89– 99, 2002

    2002

  10. [18]

    Prestini

    E. Prestini. A restriction theorem for space curves.Proc. Amer. Math. Soc., 70(1):8–10, 1978

    1978

  11. [19]

    J. L. Rubio de Francia. A Littlewood-Paley inequality for arbitrary intervals.Rev. Mat. Iberoameri- cana, 1(2):1–14, 1985

    1985

  12. [20]

    D. Ryou. Near-optimal restriction estimates for cantor sets on the parabola. Int. Math. Res. Notices,

  13. [21]

    R. Salem. On singular monotonic functions whose spectrum has a given Hausdorff dimension.Ark. Mat., 1:353–365, 1951

    1951

  14. [22]

    K. S. Senthil Raani. personal communication

  15. [23]

    K. S. Senthil Raani.L p-integrability, dimensions of supports of Fourier transforms and applications. J. Fourier Anal. Appl., 20(4):801–815, 2014

    2014

  16. [24]

    Shmerkin and V

    P. Shmerkin and V. Suomala. A class of random Cantor measures, with applications. InRecent developments in fractals and related fields, Trends Math., pages 233–260. Birkh¨ auser/Springer, Cham, 2017

    2017

  17. [25]

    Shmerkin and V

    P. Shmerkin and V. Suomala. Spatially independent martingales, intersections, and applications.Mem. Amer. Math. Soc., 251(1195):v+102, 2018

    2018

  18. [26]

    N. Wiener. Tauberian theorems.Ann. of Math. (2), 33(1):1–100, 1932

    1932

  19. [27]

    A. Zygmund. On Fourier coefficients and transforms of functions of two variables.Studia Math., 50:189–201, 1974. Department of Mathematics, University of Rochester, USA. Email address:sduan4@ur.rochester.edu Department of Mathematics, University of Rochester, USA. Email addres...

    1974

  20. [2023]

    doi: 10.1093/imrn/rnad223

    doi

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