Pith. sign in

REVIEW 3 major objections 5 minor 59 references

A Survey of the Kakeya conjecture, 2000-2025

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The Kakeya set conjecture is true in three dimensions

desk verdict A useful, honest survey whose R^3 centerpiece is conditional on an unpublished preprint under peer review. read the letter →

arxiv 2512.09397 v1 pith:3Z2UZLZJ submitted 2025-12-10 math.CA

Joshua Zahl This is my paper · ORCID
classification math.CA MSC 42B2528A7828A80
keywords KakeyaconjectureBesicovitchsetsMinkowskidimensionHausdorffmaximalfunctionConvexWolffAxiomsPolynomialinductiononscale
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

This survey reports a complete resolution of the Kakeya set conjecture in R^3: every Besicovitch set—a compact set containing a unit segment in every direction—has Minkowski and Hausdorff dimension 3. The proof, summarized from a trilogy by the author and a collaborator, establishes a quantitative statement about families of thin tubes satisfying a non-clustering condition: the volume of their union is at least the sum of the tube volumes, up to a δ^{-ε} factor. The Kakeya maximal function conjecture in R^3, which would quantify how much tubes may overlap, remains open; the survey identifies the specific density-loss obstruction. In dimensions four and higher, the analogous set-theoretic statement is false, and the survey introduces the Polynomial Wolff Axioms as a proposed replacement condition.

What carries the argument

The key machinery is a pair of induction assertions—D(σ) for families satisfying the Convex Wolff Axioms, and E(σ) for families satisfying the Frostman variant—plus the iteration D(σ)⇒E(σ) and D(σ)+E(σ)⇒D(g(σ)) with g(σ)<σ. The crucial new estimate is the efficient multiplicity inequality (3.7), μ_T ≲ μ_fine μ_coarse, which lets the proof combine fine-scale and coarse-scale multiplicities without losing a factor; earlier approaches used the less efficient μ≲μ_fine μ_{T_ρ}. A structure theorem then shows any counterexample can be covered by coarse convex sets satisfying the axioms while each fine subfamily satisfies the Frostman condition, enabling induction on scale. In the sticky case the a

What would settle it

Examine the companion paper's proof of the induction step D(σ)+E(σ)⇒D(g(σ)): if one can exhibit a tube family satisfying the Convex Wolff Axioms for which the efficient multiplicity inequality (3.7) fails, the main theorem collapses. A more targeted check: the survey's assertion that a randomly thinned family inside the quadric {ad−bc=1} satisfies the Convex Wolff Axioms with volume ~δ^{1/2} is stated without proof; verifying or refuting that single calculation decides whether Theorem 3.1 genuinely fails in dimension four.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.1: for any family of δ-tubes in R^3 satisfying the Convex Wolff Axioms, the union's volume is ⪆Σ|T|. This directly implies Conjectures 1.1 and 1.2 in dimension three: every Besicovitch set has Minkowski and Hausdorff dimension 3. The proof is an induction-on-scale framework that iterates two assertions, D(σ) and E(σ), with the efficient multiplicity inequality (3.7). The survey also establishes a negative result: the same statement fails in dimension n≥4, via a randomly thinned tube family inside the quadric {ad−bc=1}, and proposes the Polynomial Wolff Axioms as the right non-concentration hypothesis in higher dimensions.

Load-bearing premise

The whole three-dimensional result depends on a proof that this survey only sketches; if a hidden error exists in that proof, the central claim falls.

Editorial extensions

If this is right

  • The Kakeya set conjecture in R^3 is settled: both Minkowski and Hausdorff dimension of every Besicovitch set equal 3.
  • The Kakeya maximal function conjecture in R^3 remains open; the obstacle is a quadratic loss of density in shading arguments, which the survey identifies as an interesting gap to close.
  • No analogue of the R^3 theorem holds in dimensions n≥4 under the Convex Wolff Axioms; the quadric hypersurface example is a genuine near-miss.
  • For higher dimensions, the Polynomial Wolff Axioms are strong enough to imply the set and maximal function conjectures, and they hold for all direction-separated tubes; partial multilinear and narrow estimates give the current best bounds.

Reading between the lines

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

  • I infer that the proof's dichotomy between 'Heisenberg-type' and 'SL2-type' behavior is likely too coarse for the maximal function conjecture; a continuum of intermediate twistings may exist, and testing a family of regulated twisting quadrics could reveal a new near-miss.
  • One testable extension: run the induction assertions D(σ), E(σ) on synthetic random tube families in R^4 drawn from the quadric example; if the efficient multiplicity inequality (3.7) controls their volume, the same machinery might be adapted to prove a Polynomial-Wolff version of the set conjecture in n=4.
  • The survey leaves open whether the R^3 induction can be made efficient enough to preserve λ-density shadings; I infer that any resolution of the maximal function conjecture will require a variant of Assertion D/E with explicit, dimension-independent loss factors.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This survey reviews progress on the Kakeya family of conjectures in Euclidean space, with emphasis on developments since the surveys of Wolff and Katz–Tao. It states the standard conjectures (set, discrete, maximal function), recounts Wolff's bound and the two classical near-misses in R^3 (Heisenberg group, SL_2), and then describes the Katz–Łaba–Tao Minkowski-dimension bound 5/2+c_0, the Katz–Zahl Hausdorff-dimension bound 5/2+c_1, and the announced Wang–Zahl resolution of the R^3 Kakeya set conjecture. The final sections treat the polynomial Wolff axioms, multilinear Kakeya, broad/narrow estimates, and the quadric-hypersurface near-miss in R^4.

Significance. If the Wang–Zahl proof of Theorem 3.1 is correct, this survey gives a useful, well-organized record of a landmark result and of the surrounding techniques. The paper is strong on taxonomy: it clearly separates the Heisenberg, SL_2, and quadric-hypersurface examples, explains the role of the Katz–Tao and Frostman Convex Wolff axioms, and gives the reader a good map of the broad/narrow estimate literature. The author's first-hand account has real pedagogical value. The main caveat is that the survey's central theorem is presented as settled even though its decisive proof is currently an unreviewed preprint by the survey's author; moreover, one key implication in the logical chain is not justified as stated.

major comments (3)
  1. [§3, Theorem 3.1 and Prop. 3.8] The centerpiece of the survey rests on unpublished work. The proof of Theorem 3.1 is not contained here; the survey says only that in [51] Proposition 3.5 was combined with 'several types of induction on scale.' The decisive Proposition 3.8 is asserted without proof: no argument is given for the existence of the function g(σ), for the property g(σ)<σ, or for the claim that iterating D(σ)⇒E(σ)⇒D(g(σ)) reaches every σ>0. The structure theorem producing the cover W is also described only in words. Since [51] is an unreviewed preprint and the author is a coauthor, the survey should explicitly label Theorem 3.1 as 'announced by Wang–Zahl, with proof in [51]' and should separate what is established in the peer-reviewed papers [50, 52] from what is claimed in [51].
  2. [§3, Theorem 3.1 and Conjectures 1.1/1.2] As stated, Theorem 3.1 is the unshaded Convex-Wolff bound |∪T| ⪆ Σ|T|, which corresponds to Conjecture 1.2(A) and gives Minkowski dimension. Conjecture 1.2(B), which is needed for Hausdorff dimension, requires the same lower bound for arbitrary measurable Y(T)⊂T with |Y(T)| ≥ (log 1/δ)^{-1}|T|. The survey does not explain how the unshaded theorem implies this shaded version. If the proof in [51] actually establishes the shaded statement, Theorem 3.1 should be stated with the Y(T) formulation (as Proposition 3.5 is); otherwise the sentence 'As a consequence, Conjectures 1.1 and 1.2 are true for n=3' is not justified. This is a load-bearing gap in the logical chain from the stated theorem to the headline conclusion.
  3. [§4.1, quadric hypersurface example] The claim that a randomly thinned subfamily of tubes contained in the quadric {ad−bc=1} satisfies the Convex Wolff Axioms, and that translated/rotated copies have volume ∼δ^{1/2}, is made in a few sentences and called 'straightforward.' This construction is used to conclude that 'Theorem 3.1 is false in dimension n≥4,' so it is more than an aside. A proof sketch or a precise reference is needed; as written the probabilistic concentration argument and the verification of the axioms at all convex sets are not checkable.
minor comments (5)
  1. [Theorem 4.3] The displayed formula has χ_{T1} · · · χ_{Tn}; the last factor should be χ_{Tk} (the sum is over k-tuples).
  2. [§4.2 and reference [12]] The text names 'Carberry and Valdimarsson'; the cited author is Carbery. Please correct the spelling.
  3. [After Definition 2.5] The definition of a cover uses 'approximately the same number' and 'at most O(1) tubes.' Since stickiness is central, state the quantifiers explicitly (for example, within a constant factor independent of δ) or refer to a precise definition in [36].
  4. [Figures 3.1 and 3.2] The text refers to Figure 3.1 and Figure 3.2, but no figures appear in the manuscript; either include the figures or remove the references.
  5. [Definitions 2.1 and 3.3] The phrase 'W contains O(|W|δ^{1-n}) tubes' could be clarified: does W contain the full tube, or is the portion of the tube inside W counted? The distinction matters for convex sets of small volume and should be stated.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation; central theorem is cited rather than derived, so self-citation is a verification risk, not a circularity.

full rationale

The survey performs no input-output derivation that could be circular. Theorem 3.1 is presented as a result proved in [52,50,51], and the paper explicitly defers details: 'In [51], Wang and the author combined Proposition 3.5 with several types of induction on scale to prove Theorem 3.1.' The D/E assertions, structure theorem, and inequality (3.7) are described as ingredients of that cited proof, not as predictions fitted to data or as definitions of the conclusion. No equation in the survey is shown to be equivalent to its input by construction, and no fitted parameter is renamed as a prediction. The only caveat is that the load-bearing proof of Theorem 3.1 rests on an author-coauthored preprint [51] that the survey does not reproduce; this makes the survey's headline claim non-self-contained and unverified in the text, but that is a correctness/reproducibility risk rather than a circular reduction. Since the cited work is an external proof with stated assumptions that do not include the target conclusion, the self-citation does not, by itself, make the claim circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The survey is not a derivation; its load is carried entirely by the cited literature, several layers of which are the author's own recent work. The honest ledger entry is: zero fitted parameters and zero newly invented entities (the near-miss objects — the Heisenberg group H ⊂ C^3, the ring R = F_p[t]/(t^2) in the SL2 example, and the quadric Z = {ad−bc=1} — are mathematical examples from the cited literature presented as barriers, not as explanatory crutches). The two most fragile items the reader is asked to accept are the unpublished/very recent preprint [51] and the structure/classification lemmas of [41, 51].

assumptions (5)
  • standard math The cited theorems are correctly restated (multilinear Kakeya, Theorem 4.3; Bourgain's discretized projection theorem, Theorem 2.7; sums-versus-differences, Theorem 2.4; broad estimates, Theorem 4.4).
    The survey's claims are conditional on its attributions being accurate. I checked the statements I can against the known literature and found them consistent; the survey itself proves none of them, which is appropriate for the genre.
  • domain assumption The Katz–Tao and Frostman Convex Wolff Axioms (Definitions 2.1, 3.3) are the right replacement for 'δ-separated directions': direction-separated families satisfy them, and they are preserved under the truncation and rescaling steps of the proof of Theorem 3.1.
    Stated in §3.2: 'in hindsight, the condition that δ tubes point in δ-separated directions is not preserved under the types of truncation and rescaling arguments...'. This is a structural claim about the Wang–Zahl proof, not proved in the survey.
  • domain assumption The structure and classification lemmas from [41, 51]: every counterexample to Theorem 2.6 is of Heisenberg type or SL2 type (§2.3), and every tube family admits a two-scale decomposition with a coarse cover satisfying the Katz–Tao axioms and fine sub-families satisfying the Frostman axioms (§3.2, '
    These are the load-bearing unpublished or very recent lemmas, invoked by the survey without proof; they sit at the heart of both the Hausdorff-dimension improvement and the Wang–Zahl induction.
  • standard math The real-versus-complex projection theory (Bourgain's Theorem 2.7 and the Orponen–Shmerkin–Wang results [45]) applies at the key step to force the sets F and G to lie in orthogonal lines.
    Used in §3.1: 'Recent tools in projection theory [45] (which at their core use Theorem 2.7) show that this is impossible, unless F and G are contained in orthogonal lines.' This is the step that distinguishes R^3 from the complex Heisenberg example.
  • standard math Families of δ-tubes pointing in δ-separated directions satisfy the Polynomial Wolff Axioms (Guth for n=3, Zahl for n=4, Katz–Rogers for all n), so Conjecture 4.2 implies Conjectures 1.1–1.3.
    §4.1, the bridge from the polynomial Wolff axioms to the original Kakeya conjectures; cited to [27, 57, 37].

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Survey of the Kakeya conjecture, 2000-2025." pith.science (2026). https://pith.science/paper/3Z2UZLZJ

@misc{pith2026251209397,
  author       = {Pith},
  title        = {Pith review of: A Survey of the Kakeya conjecture, 2000-2025},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3Z2UZLZJ}},
  note         = {Machine review of arXiv:2512.09397}
}
read the original abstract

We survey progress on the Kakeya conjecture in Euclidean space, with an emphasis on developments that have occurred since the previous surveys by Wolff and Katz-Tao.

Figures

Figures reproduced from arXiv: 2512.09397 by the authors.

Figure 3
Figure 3. (left). This means that in order to bound [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 3.1
Figure 3.1. Left: The set of tubes T Tρ and the corresponding grains. For clarity, we have only drawn the grains and tubes that intersect the black tube (and even most of these have been omitted; the set of red tubes passing through the red grain fill out a substantial portion of the red grain, and similarly for the other grains); the situation is similar for each tube in T Tρ . Right: The image of [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 3.2
Figure 3.2. Left: two sets of δ × ρc × c prisms from the partition of G (blue and red, respectively), and the associated δ/ρ × c × c prisms □ and □ that contain them. Right: The anisotropic rescaling that maps the blue δ/ρ×c×c prism □ to the unit cube maps each blue δ ×ρc×c prism to a ρ × ρ × 1 prism (this is comparable to a ρ tube). We now consider the special case where the tubes in T˜ satisfy the Katz-Tao Convex Wolff Axioms… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 4 linked inside Pith

  1. [51]

    Wang and J

    H. Wang and J. Zahl. Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions.arXiv:2502.17655, 2025+

    arXiv 2025

  2. [52]

    Wang and J

    H. Wang and J. Zahl. Sticky Kakeya sets and the sticky Kakeya conjecture.J. Amer. Math. Soc., 39:515–585, 2026

    2026

  3. [29]

    L. Guth. Introduction to the proof of the Kakeya conjecture.arXiv:2505.07695, 2025+

    arXiv 2025

  4. [1]

    Arsovski

    B. Arsovski. Thep-adic Kakeya conjecture.J. Amer. Math. Soc., 37:69–80, 2024

    2024

  5. [2]

    Balog and E

    A. Balog and E. Szemerédi. A statistical theorem of set addition.Combinatorica, 14:263–268, 1994

    1994

  6. [3]

    Bennett, A

    J. Bennett, A. Carbery, and T. Tao. On the multilinear restriction and Kakeya conjectures.Acta Math., 196:261–302, 2006

    2006

  7. [4]

    Besicovitch

    A. Besicovitch. Sur deux questions d’integrabilite des fonctions.J. Soc. Phys. Math., 2:105–123, 1919

    1919

  8. [5]

    Bourgain

    J. Bourgain. Besicovitch type maximal operators and applications to Fourier analysis.Geom. Funct. Anal., 1:147–187, 1991

    1991

Show all 59 references
  1. [6]

    Bourgain

    J. Bourgain. On the dimension of Kakeya sets and related maximal inequalities.Geom. Funct. Anal., 9(2):256–282, 1999

    1999

  2. [7]

    Bourgain

    J. Bourgain. On the Erdős-Volkmann and Katz-Tao ring conjectures.Geom. Funct. Anal., 13:334–365, 2003

    2003

  3. [8]

    Bourgain

    J. Bourgain. The discretized sum-product and projection theorems.J. Anal. Math, 112(1):193–236, 2010

    2010

  4. [9]

    Bourgain and L

    J. Bourgain and L. Guth. Bounds on oscillatory integral operators based on multilinear estimates.Geom. Funct. Anal., 21:1239–1295, 2011

    2011

  5. [10]

    Bourgain, N.H

    J. Bourgain, N.H. Katz, and T. Tao. A sum-product estimate in finite fields, and applications.Geom. Funct. Anal., 14:27–57, 2004

    2004

  6. [11]

    Bukh and T.W

    B. Bukh and T.W. Chao. Sharp density bounds on the finite field Kakeya problem.Discrete Anal., 26, 2021

    2021

  7. [12]

    Carbery and S

    A. Carbery and S. Valdimarsson. The endpoint multilinear Kakeya theorem via the Borsuk-Ulam theorem. J. Funt. Anal., 264:1643–1663, 2012

    2012

  8. [13]

    Christ, J

    M. Christ, J. Duoandikoetxea, and J.L. Rubio de Francia. Maximal operators associated to the Radon transform and the Calderón-Zygmund method of rotations.Duke Math. J., 53:189–209, 1986

    1986

  9. [14]

    A. Cordoba. The Kakeya maximal function and the spherical summation multipliers.Am. J. Math., 99:1–22, 1977

    1977

  10. [15]

    R.O. Davies. Some remarks on the Kakeya problem.Math. Proc. Cambridge Philos. Soc., 69(3):417–421, 1971

    1971

  11. [16]

    M. Dhar. The Kakeya set conjecture forZ/NZfor generalN.Adv. Combin., 2, 2024

    2024

  12. [17]

    Drury.L p estimates for the x-ray transform.Ill

    S. Drury.L p estimates for the x-ray transform.Ill. J. Math., 27:125–129, 1983

    1983

  13. [18]

    Z. Dvir. On the size of Kakeya sets in finite fields.J. Amer. Math. Soc., 22:1093–1097, 2009

    2009

  14. [19]

    Z. Dvir, S. Kopparty, S. Saraf, and M. Sudan. Extensions to the method of multiplicities, with applications to Kakeya sets and mergers.SIAM J. Comput., 42:2305–2328, 2013

    2013

  15. [20]

    Erdős and E

    P. Erdős and E. Szemerédi. On sums and products of integers. InStudies in Pure Mathematics, page 213–218. Birkhauser, Basel, 1983

    1983

  16. [21]

    Fefferman

    C. Fefferman. Inequalities for strongly singular convolution operators.Acta Math., 124:9–36, 1970

    1970

  17. [22]

    T. Gowers. A new proof of Szemerédi’s theorem for arithmetic progressions of length four.Geom. Funct. Anal., 8:529–551, 1998

    1998

  18. [23]

    S. Guo, H. Wang, and R. Zhang. A dichotomy for Hörmander-type oscillatory integral operators.Invent. Math., 238:503–584, 2024

    2024

  19. [24]

    L. Guth. The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture.Acta Math., 205:263– 286, 2010

    2010

  20. [25]

    L. Guth. A short proof of the multilinear Kakeya inequality.Math. Proc. Camb. Phil. Soc., 158:147–153, 2015

    2015

  21. [26]

    L. Guth. Degree reduction and graininess for Kakeya-type sets inR3.Rev. Mat. Iberoam., 32:447–494, 2016

    2016

  22. [27]

    L. Guth. A restriction estimate using polynomial partitioning.J. Amer. Math. Soc., 29(2):371–413, 2016

    2016

  23. [28]

    L. Guth. Restriction estimates using polynomial partitioning II.Acta Math., 221(1):81–142, 2018

    2018

  24. [30]

    Guth and N.H

    L. Guth and N.H. Katz. Algebraic methods in discrete analogs of the Kakeya problem.Adv. Math., 225:2828– 2839, 2010

    2010

  25. [31]

    Guth and N.H

    L. Guth and N.H. Katz. On the Erdős distinct distances problem in the plane.Ann. of Math., 181:155–190, 2015

    2015

  26. [32]

    Guth and J

    L. Guth and J. Zahl. Polynomial Wolff axioms and Kakeya-type estimates inR4.Proc. London Math. Soc., 117:192–220, 2018

    2018

  27. [33]

    Hickman and K.M

    J. Hickman and K.M. Rogers. Improved Fourier restriction estimates in higher dimensions.Camb. J. Math., 7:219–282, 2019

    2019

  28. [34]

    Hickman, K.M

    J. Hickman, K.M. Rogers, and R. Zhang. Improved bounds for the Kakeya maximal conjecture in higher dimensions.Am. J. Math., 144:1511–1560, 2022

    2022

  29. [35]

    Hickman and J

    J. Hickman and J. Zahl. A note on Fourier restriction and nested polynomial Wolff axioms.J. Anal. Math., 150:19–52, 2024

    2024

  30. [36]

    N.H. Katz, I. Łaba, and T. Tao. An improved bound on the Minkowski dimension of Besicovitch sets inR3. Ann. of Math., 152:383–446, 2000

    2000

  31. [37]

    Katz and K

    N.H. Katz and K. Rogers. On the polynomial Wolff axioms.Geom. Funct. Anal., 28:1706–1716, 2018

    2018

  32. [38]

    Katz and T

    N.H. Katz and T. Tao. Bounds on arithmetic projections, and applications to the Kakeya conjecture.Math. Res. Letters, 6:625–630, 1999

    1999

  33. [39]

    Katz and T

    N.H. Katz and T. Tao. New bounds on Kakeya problems.J. Anal. Math., 87:231–263, 2002

    2002

  34. [40]

    Katz and T

    N.H. Katz and T. Tao. Recent progress on the Kakeya conjecture.Publ. Mat., 46:161–179, 2002

    2002

  35. [41]

    Katz and J

    N.H. Katz and J. Zahl. An improved bound on the Hausdorff dimension of Besicovitch sets inR3.J. Amer. Math. Soc., 32:195–259, 2019

    2019

  36. [42]

    Katz and J

    N.H. Katz and J. Zahl. A Kakeya maximal function estimate in four dimensions using planebrushes.Rev. Mat. Iberoam., 37:317–359, 2021

    2021

  37. [43]

    Kulkarni.A Kakeya Estimate for Sticky Sets Using a Planebrush

    N. Kulkarni.A Kakeya Estimate for Sticky Sets Using a Planebrush. PhD thesis, California Institute of Technology, 2024

    2024

  38. [44]

    Łaba and T

    I. Łaba and T. Tao. An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension.Geom. Funct. Anal., 11:773–806, 2001

    2001

  39. [45]

    Orponen, P

    T. Orponen, P. Shmerkin, and H. Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck’s theorem.Geom. Funct. Anal., 30:989–1062, 2024

    2024

  40. [46]

    Pramanik, T

    M. Pramanik, T. Yang, and J. Zahl. A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem inR3.To appear, Am. J. Math.arXiv:2207.02259, 2022+

    2022 arXiv

  41. [47]

    Saraf and M

    S. Saraf and M. Sudan. An improved lower bound on the size of Kakeya sets over finite fields.Anal. PDE, 1:375–379, 2008

    2008

  42. [48]

    T. Tao. Stickiness, graininess, planiness, and a sum-product approach to the kakeya prob- lem.https://terrytao.wordpress.com/2014/05/07/stickiness-graininess-planiness-and-a-sum- product-approach-to-the-kakeya-problem, 2014

    2014

  43. [49]

    Wang and S

    H. Wang and S. Wu. Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities. arXiv:2411.08871, 2024+

    2024 arXiv

  44. [50]

    Wang and J

    H. Wang and J. Zahl. The Assouad dimension of Kakeya sets inR3.Invent. Math., 241:153–206, 2025

    2025

  45. [53]

    An improved bound for Kakeya type maximal functions.Rev

    T Wolff. An improved bound for Kakeya type maximal functions.Rev. Mat. Iberoam., 11:651–674, 1995

    1995

  46. [54]

    A Kakeya-type problem for circles.Am

    T Wolff. A Kakeya-type problem for circles.Am. J. Math., 119:985–1026, 1997

    1997

  47. [55]

    T. Wolff. Recent work connected with the Kakeya problem. InProspects in mathematics (Princeton, NJ, 1996), pages 129–162. Amer. Math. Soc., Providence, RI, 1999

    1996

  48. [56]

    Wolff.Lectures on Harmonic Analysis

    T. Wolff.Lectures on Harmonic Analysis. American Mathematical Society, 2003

    2003

  49. [57]

    J. Zahl. A discretized Severi-type theorem with applications to harmonic analysis.Geom. Funct. Anal., 28:1131–1181, 2018

    2018

  50. [58]

    J. Zahl. New Kakeya estimates using Gromov’s algebraic lemma.Adv. Math., 380, 2021

    2021

  51. [59]

    J. Zahl. Unions of lines inRn.Mathematika, 69:473–481, 2023

    2023

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.