Pith. sign in

REVIEW 1 cited by

Quantitative flatness and obstructions in Fourier analysis

T0 review · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Quantifying flat parts of measures provides explicit obstructions to Fourier restriction, L^p-improving, and Fourier decay estimates.

desk verdict Fraser unifies obstruction methods for restriction, L^p-improving, and decay via quantifiable flatness, then applies it to get concrete Fourier dimension bounds for curves, surfaces, and several fractal measures. read the letter →

arxiv 2606.13170 v1 pith:5ETDFRTD submitted 2026-06-11 math.CA math.DGmath.DSmath.FAmath.MG

Jonathan M. Fraser This is my paper · ORCID
classification math.CAmath.DGmath.DSmath.FAmath.MG
keywords FourierrestrictiondecayL^p-improvingflatnessdimensionsurfacemeasuresPatterson-Sullivanself-affinesets
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 develops a unified abstract framework that uses quantitative measurements of flat regions in a measure to derive negative results for three key problems in Fourier analysis. These problems concern how measures interact with Fourier transforms in terms of restriction, improving integrability, and decay rates. By extending the idea of Knapp examples, the framework applies to various concrete measures from geometry and dynamics, yielding bounds on Fourier dimensions and connections to other fractal properties.

What carries the argument

Quantifying flat parts of the measure, detected via analytic and fractal geometric concepts to force obstructions to the desired estimates.

What would settle it

A surface measure on a C^2 surface where the Fourier dimension exceeds the smallest ambient rank of any point on the surface would show that detected flat parts do not always produce the claimed obstruction.

Watch

Extended reading notes

Core claim

The paper establishes a unified framework for providing negative results for the Fourier restriction problem, the L^p-improving problem, and the Fourier decay problem by quantifying flat parts of the measure in the spirit of the well-known Knapp examples from harmonic analysis. This framework applies generally and unifies and extends various parts of the literature through applications to surface measures, curves, Patterson-Sullivan measures, and ergodic measures on self-affine sets.

Load-bearing premise

The concrete measures arising in the applications possess quantifiable flat parts that can be detected and used to force the desired obstructions via the abstract framework.

Editorial extensions

If this is right

  • The Fourier dimension of the surface measure on a compact C^2 surface is bounded above by the smallest ambient rank of a point on the surface.
  • The Fourier dimension of a smooth curve in R^d is at most 4/(d+1), so such curves cannot be Salem for d >= 4.
  • Explicit upper bounds hold for the Fourier dimension of the Patterson-Sullivan measure for parabolic Kleinian group actions and ergodic measures on self-affine sets.
  • Fourier restriction and decay connect to the Assouad spectrum of projections and slices and to a strong form of tube-nullity.

Reading between the lines

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

  • The same flatness detection might yield obstructions for other Fourier-type problems involving different operators or transforms.
  • If flat parts can be quantified in non-Euclidean or infinite-dimensional settings, the framework could apply there to bound analogous dimensions.
  • Random or dynamically generated measures without obvious flat parts could be tested to see if the obstruction mechanism still activates indirectly.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 0 minor

Summary. The paper develops a unified abstract framework for deriving explicit obstructions to the Fourier restriction, L^p-improving, and Fourier decay problems by quantifying flat parts of a measure in the spirit of Knapp examples. The framework is applied to concrete measures (surface measures on C^2 surfaces, smooth curves in R^d, Patterson-Sullivan measures for parabolic Kleinian groups, and ergodic measures on self-affine sets) using analytic and fractal-geometric tools to obtain upper bounds on Fourier dimension (e.g., ≤ 4/(d+1) for smooth curves in R^d, and bounds in terms of ambient rank for surfaces). Additional results include connections to the Assouad spectrum of projections/slices and a strong form of tube-nullity, plus an auxiliary characterization of L^2-flattening in terms of the Fourier spectrum.

Significance. If the derivations hold, the work supplies a general, reusable method for producing negative results across three central problems in Fourier analysis, unifying and extending scattered results in the literature. The explicit dimension bounds for curves and other measures, together with the new links to Assouad spectrum and tube-nullity, are concrete contributions. The auxiliary L^2-flattening characterization is a useful byproduct. These strengths are grounded in the abstract framework and its applications as described.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper introduces an abstract framework quantifying flat parts of measures (in the spirit of Knapp examples) to derive obstructions for Fourier restriction, L^p-improving, and Fourier decay problems. It then applies this framework to concrete cases (surface measures, curves, Patterson-Sullivan measures, self-affine sets) using independent analytic and fractal-geometric tools such as the Assouad spectrum. No equation or claim reduces by construction to a fitted input, self-citation chain, or renamed ansatz; the derivations rely on external geometric concepts that remain falsifiable outside the paper's fitted values. The central results are therefore self-contained against external benchmarks.

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

Only the abstract is provided; no explicit free parameters, axioms, or invented entities can be identified from the given text.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantitative flatness and obstructions in Fourier analysis." pith.science (2026). https://pith.science/paper/5ETDFRTD

@misc{pith2026260613170,
  author       = {Pith},
  title        = {Pith review of: Quantitative flatness and obstructions in Fourier analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ETDFRTD}},
  note         = {Machine review of arXiv:2606.13170}
}
abstract

Three important problems in Fourier analysis are the Fourier restriction problem, the $L^p$-improving problem, and the Fourier decay problem. Positive results for any of these problems require a quantitative understanding of various geometric properties of the given measure, including curvature and arithmetic resonance. In this paper we establish a unified framework for providing negative results for all three problems (that is, we provide explicit obstructions to a measure satisfying certain Fourier restriction, $L^p$-improving, or Fourier decay estimates) by quantifying flat parts of the measure in the spirit of the well-known Knapp examples from harmonic analysis. Our main interest is in the application of these abstract results in various concrete settings where we use analytic and fractal geometric concepts to force `flatness'. Our framework applies generally and this allows us to unify and extend various parts of the literature. Some representative applications include: (i) we bound the Fourier dimension of the surface measure on a compact $C^2$ surface of arbitrary dimension above by the smallest ambient rank of a point on the surface; (ii) we prove that the Fourier dimension of a smooth curve in $\mathbb{R}^d$ is at most $4/(d+1)$ and so such curves cannot be Salem for $d \geq 4$ with analogous results for higher dimensional submanifolds; (iii) we obtain explicit upper bounds for the Fourier dimension of the Patterson-Sullivan measure for parabolic Kleinian group actions, as well as ergodic measures on self-affine sets; (iv) we establish novel connections between Fourier restriction/decay and a priori unrelated concepts in fractal geometry including the Assouad spectrum of projections and slices, and a strong form of tube-nullity. We establish several auxiliary results along the way, including a precise characterisation of L^2-flattening in terms of the Fourier spectrum.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

    math.DS 2026-07 conditional novelty 7.0 of 10

    Patterson-Sullivan measures of convex co-compact Schottky groups of dimension δ>1/2 satisfy |μ̂(ξ)| ≲ |ξ|^{-δ(2δ-1)/((2δ+1)(3-δ))}.

Reference graph

Works this paper leans on

76 extracted references · 11 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aaronson, M

    J. Aaronson, M. Denker and M. Urba\'nski. Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. , 337 , 495--548, (1993)

    1993

  2. [2]

    Algom and O

    A. Algom and O. Khalil. L^2 -Flattening of self-similar measures on non-degenerate curves, preprint available at: arXiv:2507.07321

    work page arXiv

  3. [3]

    Uniformly perfect measures on strictly convex planar graphs arel 2-flattening.https://arxiv.org/pdf/2509.09354,

    A. Algom and T. Orponen. Uniformly perfect measures on strictly convex planar graphs are L^2 -flattening, J. Math. Pures App. (to appear), preprint available at: arXiv:2509.09354

    work page arXiv

  4. [4]

    J-G. Bak, D. M. Oberlin and A. Seeger. Restriction of Fourier transforms to curves and related oscillatory integrals, Amer. J. Math. , 131 , 277--311, (2009)

    2009

  5. [5]

    Bak and A

    J.-G. Bak and A. Seeger. Extensions of the Stein--Tomas Theorem, Math. Res. Lett., 18(4), 767--781, (2011)

    2011

  6. [6]

    Baker and A

    S. Baker and A. Banaji. Polynomial Fourier decay for fractal measures and their pushforwards, Math. Ann. , 392 , 209--261, (2025)

    2025

  7. [7]

    Baker, O

    S. Baker, O. Khalil and T. Sahlsten. Fourier Decay from L^2 -Flattening, preprint available at: arXiv:2407.16699

    work page arXiv

  8. [8]

    Fourier transform of nonlinear images of self-similar measures: qualitative aspects

    A. Banaji and H. Yu. Fourier transform of nonlinear images of self-similar measures: qualitative aspects, preprint available at: https://arxiv.org/pdf/2606.09743

    work page Pith review arXiv

Show all 76 references
  1. [9]

    B\'ar\'any, K

    B. B\'ar\'any, K. Simon and B. Solomyak. Self-similar and self-affine sets and measures, Mathematical Surveys and Monographs, 276 , American Mathematical Society, (2023)

    2023

  2. [10]

    A. F. Beardon. The geometry of discrete groups , Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, (1983)

    1983

  3. [11]

    A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, (1990)

    1990

  4. [12]

    Bourgain, S

    J. Bourgain, S. Dyatlov. Fourier dimension and spectral gaps for hyperbolic surfaces, Geom. Funct. Anal. , 27 , 744–771, (2017)

    2017

  5. [13]

    B. H. Bowditch. Geometrical finiteness for hyperbolic groups, J. Funct. Anal. , 113 , 245--317, (1993)

    1993

  6. [14]

    Carbery, F

    A. Carbery, F. Soria and A. Vargas. Localisation and weighted inequalities for spherical Fourier means, J. Anal. Math., 103 , 133--156, (2007)

    2007

  7. [15]

    Carnovale, J

    M. Carnovale, J. M. Fraser and A. E. de Orellana. Obtaining the Fourier spectrum via Fourier coefficients, Proc. Amer. Math. Soc., 154 , 2005--2017, (2026)

    2005

  8. [16]

    Carnovale, J

    M. Carnovale, J. M. Fraser and A. E. de Orellana. L^2 restriction estimates from the Fourier Spectrum, preprint: arXiv:2412.14896 https://arxiv.org/abs/2412.14896 (2024)

    2024

  9. [17]

    X. Chen. A Fourier restriction theorem based on convolution powers, Proc. Amer. Math. Soc., 142(11), 3897--3901, (2014)

    2014

  10. [18]

    L. Daw. A uniform result for the dimension of fractional Brownian motion level sets, Statist. Probab. Lett., 169 , Paper No. 108984, (2021)

    2021

  11. [19]

    C. Demeter. Fourier restriction, decoupling, and applications, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (2019)

    2019

  12. [20]

    Denker and M

    M. Denker and M. Urba\'nski. Hausdorff and conformal measures on Julia sets with rationally indifferent periodic points, J. Lond. Math. Soc. , 2 , 107--118, (1991)

    1991

  13. [21]

    Ekstr\"om, T

    F. Ekstr\"om, T. Persson and J. Schmeling. On the Fourier dimension and a modification, J. Fractal Geom., 2 , 309--337, (2015)

    2015

  14. [22]

    K. J. Falconer. Fractal geometry: mathematical foundations and applications, John Wiley and Sons, Hoboken, NJ, 3rd. ed., (2014)

    2014

  15. [23]

    K. J. Falconer. Seventy Years of Fractal Projections, preprint

  16. [24]

    K. J. Falconer, J. M. Fraser and X. Jin. Sixty years of fractal projections, Fractal Geometry and Stochastics V, Springer International Publishing, 3--25, (2015)

    2015

  17. [25]

    K. J. Falconer, J. M. Fraser and A. K\"aenm\"aki. Attainable values for the Assouad dimension of projections, Proc. Amer. Math. Soc., 148 , (2020), 3393--3405

    2020

  18. [26]

    D.-J. Feng. Dimensions of invariant measures for affine iterated function systems, Duke. Math. J., 172 , 701--774, (2023)

    2023

  19. [27]

    J. M. Fraser. Assouad dimension and fractal geometry , Cambridge University Press, Tracts in Mathematics Series, (2021)

    2021

  20. [28]

    J. M. Fraser. The Fourier spectrum and sumset type problems, Math. Ann., 390 , 3891--3930, (2024)

    2024

  21. [29]

    J. M. Fraser. Fourier decay of product measures, Illinois. J. Math. (to appear), preprint: arXiv:2405.05878 https://arxiv.org/abs/2405.05878 (2024)

    2024

  22. [30]

    J. M. Fraser, K. E. Hare, K. G. Hare, S. Troscheit and H. Yu. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra, Ann. Acad. Sci. Fenn. Math., 44 , 379--387, (2019)

    2019

  23. [31]

    J. M. Fraser, T. L. J. Harris and N. G. Kroon. On the Fourier dimension of (d,k) -sets and Kakeya sets with restricted directions, Math. Zeit., 301 , 2497--2508, (2022)

    2022

  24. [32]

    J. M. Fraser and A. E. de Orellana. A Fourier analytic approach to exceptional set estimates for orthogonal projections, Indiana U. Math. J. (to appear), preprint: arXiv:2404.11179 https://arxiv.org/abs/2404.11179 (2024)

    2024

  25. [33]

    J. M. Fraser and A. E. de Orellana. Fourier restriction for the additive Brownian sheet, preprint: arXiv:2601.05802 https://arxiv.org/abs/2601.05802 (2026)

    2026

  26. [34]

    J. M. Fraser and T. Pham. On Fourier decay and the distance set problem, preprint: arXiv:2604.19486 https://arxiv.org/abs/2604.19486 (2026)

    2026 arXiv

  27. [35]

    J. M. Fraser and T. Sahlsten. On the Fourier analytic structure of the Brownian graph, Anal. PDE, 11(1), pp. 2730--2745, (2014)

    2014

  28. [36]

    J. M. Fraser and L. Stuart. Assouad type dimensions of parabolic Julia sets, Math. Zeit., 312 , Paper No. 108, (2026)

    2026

  29. [37]

    J. M. Fraser and H. Yu. New dimension spectra: Finer information on scaling and homogeneity, Adv. Math., 329, 273--328, (2018)

    2018

  30. [38]

    Grafakos

    L. Grafakos. Classical Fourier analysis , Graduate Texts in Mathematics, 249. Springer, New York, 2nd ed. (2008)

    2008

  31. [39]

    P. T. Gressman. On the Oberlin affine curvature condition, Duke Math. J., 168 , 2075--2126, (2019)

  32. [40]

    Hambrook and I

    K. Hambrook and I. aba. On the sharpness of Mockenhaupt's restriction theorem, Geom. Funct. Anal., 23, 1262--1277, (2013)

    2013

  33. [41]

    K. E. Hare amd M. Roginskaya. L^p -improving properties of measures of positive energy dimension, Colloq. Math. , 102 , 73--86, (2005)

    2005

  34. [42]

    P. Hartman. On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures, Trans. Amer. Math. Soc., 115 , 94--109, (1965)

    1965

  35. [43]

    Jordan, M

    T. Jordan, M. Pollicott and K. Simon. Hausdorff dimension for randomly perturbed self affine attractors, Comm. Math. Phys., 270 , 519--544, (2007)

    2007

  36. [44]

    Jordan and T

    T. Jordan and T. Sahlsten. Fourier transforms of Gibbs measures for the Gauss, Math. Ann., 364 , 983--1023, (2016)

    2016

  37. [45]

    J.-P. Kahane. Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, (1985)

    1985

  38. [46]

    O. Khalil. Exponential mixing via additive combinatorics, J. Amer. Math. Soc., 39 , 781--856, (2026)

    2026

  39. [47]

    C.-K Lai and C. Y. Lee. Fourier dimension of the graph of fractional Brownian motion with H 1/2 , preprint: arXiv:2502.09032 https://arxiv.org/pdf/2502.09032 (2025)

    2025 arXiv

  40. [48]

    G. Leclerc. Julia sets of hyperbolic rational maps have positive Fourier dimension, Comm. Math. Phys. , 397 , 503--546, (2023)

    2023

  41. [49]

    Lee and A

    S. Lee and A. Vargas. Restriction estimates for some surfaces with vanishing curvatures, J. Funct. Anal. , 258 , 2884--2909, (2010)

    2010

  42. [50]

    C. Y. Lee and S. Tindel. On the Fourier dimension of fractional Brownian graphs, preprint: arXiv:2510.09818 https://arxiv.org/pdf/2510.09818 (2025)

    2025

  43. [51]

    J. Li, F. Naud, and W. Pan. Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay, Duke Math. J. , 170 , 775--825, (2021)

    2021

  44. [52]

    L\"u and L.-F

    F. L\"u and L.-F. Xi. Quasi-Assouad dimension of fractals, J. Fractal Geom. , 3 , 187--215, (2016)

    2016

  45. [53]

    P. Mattila. Recent progress on dimensions of projections, Geometry and Analysis of Fractals, 283--301, (2014)

    2014

  46. [54]

    P. Mattila. Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (2015)

    2015

  47. [55]

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

    2002

  48. [56]

    Mockenhaupt

    G. Mockenhaupt. Salem sets and restriction properties of Fourier transforms, Geom. Funct. Anal., 10, 1579--1587 (2000)

    2000

  49. [57]

    M\"uller

    D. M\"uller. Problems of harmonic analysis related to finite-type hypersurfaces in R ^3 , and Newton polyhedra, Advances in analysis: the legacy of Elias M. Stein, 303--345, Princeton Math. Ser., 50, Princeton Univ. Press, Princeton, NJ, (2014)

    2014

  50. [58]

    D. M. Oberlin. Convolution with measures on hypersurfaces, Math. Proc. Cambridge Philos. Soc. , 129 , 517--526, (2000)

    2000

  51. [59]

    D. M. Oberlin. Affine dimension: measuring the vestiges of curvature, Michigan Math. J. , 51 , 13--26, (2003)

    2003

  52. [60]

    Peres and B

    Y. Peres and B. Solomyak. Existence of L^q -dimensions and entropy dimension for self-conformal measures, Indiana Univ. Math. J., 49 , 1603--1621, (2000)

    2000

  53. [61]

    Pyörälä, P

    A. Pyörälä, P. Shmerkin, V. Suomala and M. Wu. Covering the Sierpiński carpet with tubes, Israel J. Math. , 265 , 769–799, (2025)

    2025

  54. [62]

    Roos and A

    J. Roos and A. Seeger. Spherical maximal functions and fractal dimensions of dilation sets, American J. Math., 145 , 1077--1110, (2023)

    2023

  55. [63]

    Rossi and P

    E. Rossi and P. Shmerkin. On measures that improve L^q dimension under convolution, Rev. Mat. Iberoam. , 36 , 2217--2236, (2020)

    2020

  56. [64]

    C. D. Sogge. Fourier integrals in classical analysis. Second edition., Cambridge Tracts in Mathematics, 210 , Cambridge University Press, Cambridge, (2017)

    2017

  57. [65]

    E. M. Stein. Singular integrals and differentiability properties of functions. Princeton University Press, (1970)

    1970

  58. [66]

    E. M. Stein. Harmonic analysis on R ^n , Studies in Harmonic Analysis, MAA Studies in Mathematics, 13 , 97--135. Mathematical Association of America, Washington, DC, (1976)

    1976

  59. [67]

    E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, (1993)

    1993

  60. [68]

    B. O. Stratmann and M. Urba\'nski. The geometry of conformal measures for parabolic rational maps, Math. Proc. Camd. Philis. Soc. , 128 , 141--156, (2000)

    2000

  61. [69]

    B. O. Stratmann and S. L. Velani. The Patterson measure for geometrically finite groups with parabolic elements, new and old, Proc. Lond. Math. Soc. , 71 , 197--220, (1995)

    1995

  62. [70]

    Shmerkin and V

    P. Shmerkin and V. Suomala. Sets which are not tube null and intersection properties of random measures, J. London Math. Soc., 91 , 405--422, (2015)

    2015

  63. [71]

    Tao and J

    T. Tao and J. Wright. L^p improving bounds for averages along curves, J. Amer. Math. Soc., 16 , 605--638, (2003)

    2003

  64. [72]

    P. A. Tomas. A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc., 81, 705--714, (1975)

    1975

  65. [73]

    P. Tukia. The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math. , 152 , 127--140, (1984)

    1984

  66. [74]

    Y. Xiao. H\"older conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Theory Related Fields, 109 , 129--157, (1997)

    1997

  67. [75]

    J. Zhu. Fourier dimension of conical and cylindrical hypersurfaces, J. Geom. Anal., 35 , (2025)

    2025

  68. [76]

    J. Zhu. Fourier dimension of constant rank hypersurfaces, Math. Ann., 392 , 4831--4859, (2025)

    2025

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.