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On the Dimension of the CC-geodesic Kakeya sets in the first Heisenberg group

T0 review · 1 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read CC-geodesic Kakeya sets in the first Heisenberg group have sharp Heisenberg dimension 3, not the expected 4.

desk verdict Clean negative answer to Lukyanenko’s question: full CC-geodesic Kakeya sets in H¹ still only force Heisenberg dimension 3, even when compact, and the constructions are elementary and checkable. read the letter →

arxiv 2607.26906 v1 pith:TRUZPCE7 submitted 2026-07-29 math.CA math.MG

Marianna Chatzakou This is my paper · ORCID
classification math.CAmath.MG MSC 28A7853C1722E25
keywords KakeyasetsCC-geodesicsHeisenberggroupHausdorffdimensionCarnot–Carathéodorygeometrycurvature-κsub-Riemannian
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

In the first Heisenberg group, a natural Kakeya-type set is one that contains a left translate of every unit-speed Carnot–Carathéodory geodesic segment leaving the identity. The homogeneous dimension is 4, so the direct analogue of the classical Kakeya conjecture would demand that every such set has Heisenberg Hausdorff dimension 4. This paper shows that prediction is false: every CC-geodesic Kakeya set has dimension at least 3, and 3 is sharp—even when the set is required to be compact. The same lower bound was already known for the weaker horizontal-only problem; the new work shows that forcing in all nonzero-curvature geodesics does not raise the bound. The paper also builds a (non-compact) example of full dimension 4 with zero Lebesgue measure, and shows that if one freezes a single positive curvature then compact examples of dimension 1 exist.

What carries the argument

The surface Σ parametrized by Φ(r,φ)=(r cos φ, r sin φ, r²φ/2), together with the horizontal plane Π. Suitable left translates of every nonzero-curvature unit geodesic land inside Σ, while the zero-curvature segments sit in Π; their union is therefore a CC-geodesic Kakeya set of Heisenberg dimension 3.

What would settle it

Exhibit a CC-geodesic Kakeya set whose Heisenberg Hausdorff dimension is strictly less than 3, or show that the constructed surface Σ fails to contain a left translate of some nonzero-curvature unit geodesic.

Watch

Extended reading notes

Core claim

Every CC-geodesic Kakeya set in the first Heisenberg group has Heisenberg Hausdorff dimension at least 3, and this lower bound is attained by an explicit Borel set (and even by a compact one). Consequently the natural full-dimension-4 conjecture fails. Separately, for each fixed curvature κ in (0,2π] there exist compact curvature-κ Kakeya sets of Euclidean and Heisenberg dimension exactly 1.

Load-bearing premise

The dimension comparison that every smooth two-dimensional Euclidean surface inside the Heisenberg group has Heisenberg Hausdorff dimension exactly 3.

Editorial extensions

If this is right

  • The CC-geodesic Kakeya conjecture in H¹ is false at the level of Hausdorff dimension; the sharp lower bound is 3 rather than 4.
  • Sharpness holds already inside the compact class, so unboundedness is not needed to reach dimension 3.
  • Full dimension 4 is still compatible with zero Lebesgue measure for CC-geodesic Kakeya sets.
  • Fixing a single positive curvature collapses the problem dramatically: compact examples of dimension 1 exist.
  • Horizontal-only Kakeya lower bounds already control the full geodesic family; extra curvature does not improve the constant.

Reading between the lines

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

  • The fixed-curvature dimension-1 examples suggest that any attempt to restore a dimension-4 conjecture would need to quantify over a positive-measure set of curvatures, not a single κ.
  • The same surface-plus-plane construction may adapt to higher-step or higher-dimensional Heisenberg groups once the geodesic equations are known explicitly.
  • Because the sharpness example is a union of two Euclidean surfaces, quantitative Kakeya maximal inequalities in H¹ cannot improve beyond the horizontal case without new geometric input.
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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

1 major / 5 minor

Summary. The paper studies CC-geodesic Kakeya sets in the first Heisenberg group H¹: Borel sets containing a left translate of every unit-speed minimising CC-geodesic segment of length 1 issuing from the identity. The natural analogue of the Kakeya conjecture would require Heisenberg Hausdorff dimension 4. The authors prove that every such set has dim_H^{H¹} ≥ 3 (inherited from the horizontal case K_CC ⊂ K({0})), and that 3 is sharp, both in general (Theorem 1.1, via the explicit union Σ ∪ Π of two smooth Euclidean surfaces) and among compact sets (Theorem 1.2, via a compact packing E_c = D² ∪ ⋃_j E_j of geodesic arcs). They also produce a CC-geodesic Kakeya set of full dimension 4 and Lebesgue measure zero (Corollary 1.3), and, for each fixed curvature κ ∈ (0, 2π], a compact curvature-κ Kakeya set of Euclidean and Heisenberg Hausdorff dimension 1 (Theorem 1.4).

Significance. The work answers Lukyanenko’s open question and shows that the natural full-dimension prediction fails: nonzero-curvature geodesics do not raise the sharp lower bound above the horizontal value 3. Sharpness is obtained by fully explicit left-translation constructions (Lemma 3.1 and the screw-motion identity (4.3)) whose dimension upper bounds rest only on standard comparison theorems for C¹ surfaces and Lipschitz images in H¹. The fixed-curvature near-miss at dimension 1 is a clean and striking counterpart. The arguments are elementary, self-contained, and free of circularity; the contribution is solid and appropriate for a serious journal in geometric measure theory / harmonic analysis.

major comments (1)
  1. [Abstract] Abstract vs. Theorems 1.1–1.2: the abstract states that “their sharp lower bound is 4”, while the body (Introduction, Theorem 1.1, Theorem 1.2, and the constructions) consistently proves that the sharp lower bound is 3 and that the prediction of 4 fails. This is a load-bearing misstatement of the main result and must be corrected to “3” before publication; as written, the abstract contradicts the theorems.
minor comments (5)
  1. [Abstract] In the abstract, insert the missing space in “is4” (and likewise check “dimension4”, “measure0”, etc., for consistent spacing).
  2. [Section 1] Page 1 / Introduction: the sentence “A fact that at first glance supports this conjecture is that, if one uses only horizontal unit segments… then the sharp lower bound… is 3” is slightly awkward; a brief clarifying clause that horizontal segments are the k = 0 members of the family would help non-specialist readers.
  3. [Lemma 3.1] Lemma 3.1: the two cases k > 0 and k < 0 differ only by a constant phase shift of π in the t-coordinate; a one-line remark that the same computation covers both signs (or a unified formula) would shorten the write-up.
  4. [Section 4.3] Proof of Theorem 1.2: the removable-singularity claim for F_j at κ = 0 is correct by Taylor expansion, but citing the explicit limiting expressions (or noting that the coordinate functions extend C^∞) would make the Lipschitz bound immediate without appealing to a neighbourhood of the rectangle.
  5. [References] References: the arXiv number and year of the Wang–Zahl Kakeya paper [26] should be double-checked against the final published version if available at proof stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: explicit geometric constructions with external dimension comparisons

full rationale

This is a self-contained existence-and-dimension paper in geometric measure theory. The lower bound dim ≥ 3 for CC-geodesic Kakeya sets is inherited from the strictly weaker horizontal Kakeya class K({0}) via the inclusion K_CC ⊂ K({0}), which was already proved by Liu and by Fässler–Pinamonti–Wald (independent authors). Sharpness is obtained by writing down explicit left translates (Lemma 3.1 for the non-compact surface Σ ∪ Π; the screw-motion identity (4.3) and length check for the compact union E_c) and then invoking standard external comparison theorems (Balogh–Durand-Cartagena–Fässler–Mattila–Tyson; Falconer) that C¹ Euclidean surfaces and Lipschitz images of 2-dimensional parameter domains have Heisenberg Hausdorff dimension 3. Nothing is fitted, nothing is defined in terms of the target dimension, and there is no load-bearing self-citation chain. Ordinary dependence on the paper’s own definitions of K_CC and K({κ}) is definitional bookkeeping, not circular derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper is a pure geometric-measure-theory existence and dimension paper. It rests on standard Heisenberg geometry (group law, CC-geodesics, Haar measure), published horizontal Kakeya lower bounds, and standard Hausdorff-dimension comparison theorems. No free parameters are fitted. The only ‘new entities’ are definitional classes of sets.

assumptions (5)
  • domain assumption Unit-speed minimizing CC-geodesics from the identity are exactly the family γ_{θ,k} for θ∈[0,2π), k∈[−2π,2π], with cut time 2π/|k| (Monti; Le Donne notes).
    Used throughout Definition 2.2 and all constructions; taken from the standard sub-Riemannian literature without reproof.
  • domain assumption Every E in the horizontal Kakeya class K({0}) (or Liu’s open-segment class) satisfies dim_{H¹}_H(E)≥3.
    Invoked via the inclusion K_CC ⊂ K({0}) to obtain the lower bound in Theorems 1.1–1.2 (Liu; Fässler–Pinamonti–Wald).
  • domain assumption Every C¹ Euclidean surface in H¹ has Heisenberg Hausdorff dimension 3; more generally the Balogh et al. dimension-comparison principle relating Euclidean and Heisenberg dimensions.
    Cited as [3, p. 414] and [3, Thm. 2.7]; load-bearing for sharpness upper bounds on Σ∪Π and E_c and for dim N=4.
  • standard math Hausdorff dimension is non-increasing under Lipschitz maps, and bi-Lipschitz equivalent metrics (d_K ~ d_CC) yield the same Hausdorff dimension.
    Used for dim K_κ≤1 and for identifying dim_{H¹}_H with either metric.
  • standard math L³ is a bi-invariant Haar measure on H¹ ≅ R³; smooth 2-surfaces and the constructed Cantor products are L³-null as claimed.
    Used for zero-measure statements in Theorem 1.1 and Corollary 1.3.
invented entities (2)
  • CC-geodesic Kakeya set (Definition 2.2 / class K_CC) independent evidence
    purpose: Formalizes Lukyanenko’s question: Borel sets containing a left translate of every unit-speed minimizing CC-geodesic segment from the identity.
    Definitional class; not a physical entity. Independent interest is the open question it encodes.
  • Curvature-κ Kakeya set (Definition 4.1) independent evidence
    purpose: Restricts the family to fixed curvature κ to produce dimension-1 near-misses.
    Definitional subclass suggested by Balogh (acknowledgements). Evidence is the explicit construction K_κ.

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Cite this review

Pith. "Pith review of On the Dimension of the CC-geodesic Kakeya sets in the first Heisenberg group." pith.science (2026). https://pith.science/paper/TRUZPCE7

@misc{pith2026260726906,
  author       = {Pith},
  title        = {Pith review of: On the Dimension of the CC-geodesic Kakeya sets in the first Heisenberg group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRUZPCE7}},
  note         = {Machine review of arXiv:2607.26906}
}
abstract

We study CC-geodesic Kakeya sets in the first Heisenberg group, defined by requiring a left translate of every unit-speed CC-geodesic segment issuing from the identity. The natural analogue of the Kakeya conjecture would predict full Heisenberg Hausdorff dimension 4 for such sets. We show that this prediction fails: their sharp lower bound is 4, and it remains sharp even among compact CC-geodesic Kakeya sets. We also construct a CC-geodesic Kakeya set of full Heisenberg Hausdorff dimension 4 and zero Lebesgue measure. Finally, for every $\kappa\in(0,2\pi]$, we construct a compact curvature-$\kappa$ Kakeya set of zero Lebesgue measure whose Euclidean and Heisenberg Hausdorff dimensions are both equal to 1.

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Works this paper leans on

28 extracted references · 2 linked inside Pith

  1. [1]

    Altaf, A one-dimensional planar Besicovitch-type set, arXiv:2412.11190, 2024

    I. Altaf, A one-dimensional planar Besicovitch-type set, arXiv:2412.11190, 2024

    arXiv 2024

  2. [2]

    Barilari, U

    D. Barilari, U. Boscain and R. W. Neel, Small-time heat kernel asymptotics at the sub-Riemannian cut locus,J. Differential Geom.92(2012), no. 3, 373–416

    2012

  3. [3]

    Z. M. Balogh, E. Durand-Cartagena, K. Fässler, P. Mattila and J. T. Tyson, The effect of projections on dimension in the Heisenberg group,Rev. Mat. Iberoam.29(2013), 381–432

    2013

  4. [4]

    A. S. Besicovitch, On Kakeya’s problem and a similar one,Math. Z.27(1928), 312–320

    1928

  5. [5]

    Bonfiglioli, E

    A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,Stratified Lie Groups and Potential Theory for Their Sub- Laplacians, Springer Monographs in Mathematics, Springer, Berlin, 2007

    2007

  6. [6]

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

    1971

  7. [7]

    K. J. Falconer,Fractal Geometry: Mathematical Foundations and Applications, second edition, John Wiley & Sons, Chichester, 2003

    2003

  8. [8]

    Fässler and T

    K. Fässler and T. Orponen, A note on Kakeya sets of horizontal andSL(2)lines,Bull. Lond. Math. Soc.55 (2023), no. 5, 2195–2204

    2023

Show all 28 references
  1. [9]

    Fässler, A

    K. Fässler, A. Pinamonti and P. Wald, A Kakeya maximal inequality in the Heisenberg group,Ann. Sc. Norm. Super. Pisa Cl. Sci.(5)26(2025), no. 3, 1451–1474

    2025

  2. [10]

    Fischer and M

    V. Fischer and M. Ruzhansky,Quantization on Nilpotent Lie Groups, Progress in Mathematics, vol. 314, Birkhäuser, Cham, 2016

    2016

  3. [11]

    G. B. Folland,Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989

    1989

  4. [12]

    Gaveau, Principe de moindre action et propagation de la chaleur pour le groupe d’Heisenberg,C

    B. Gaveau, Principe de moindre action et propagation de la chaleur pour le groupe d’Heisenberg,C. R. Acad. Sci. Paris Sér. A–B281(1975), A327–A328

    1975

  5. [13]

    Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents,Acta Math.139(1977), 95–153

    B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents,Acta Math.139(1977), 95–153

    1977

  6. [14]

    N. H. Katz, I. Łaba and T. Tao, An improved bound on the Minkowski dimension of Besicovitch sets inR3,Ann. of Math.(2)152(2000), no. 2, 383–446

    2000

  7. [15]

    N. H. Katz and T. Tao, Recent progress on the Kakeya conjecture,Publ. Mat., Extra Vol. (2002), 161–179. CC–GEODESIC KAKEYA SETS 18

    2002

  8. [16]

    N. H. Katz, S. Wu and J. Zahl, Kakeya sets from lines inSL2,Ars Inveniendi Analytica(2023), Paper No. 6, 23 pp

    2023

  9. [17]

    N. H. Katz and J. Zahl, An improved bound on the Hausdorff dimension of Besicovitch sets inR3,J. Amer. Math. Soc.32(2019), no. 1, 195–259

    2019

  10. [18]

    Le Donne,Lecture Notes on Sub-Riemannian Geometry: Carnot–Carathéodory Spaces from the Lie Group Viewpoint, lecture notes, 2023

    E. Le Donne,Lecture Notes on Sub-Riemannian Geometry: Carnot–Carathéodory Spaces from the Lie Group Viewpoint, lecture notes, 2023

    2023

  11. [19]

    Liu, On the dimension of Kakeya sets in the first Heisenberg group,Proc

    J. Liu, On the dimension of Kakeya sets in the first Heisenberg group,Proc. Amer. Math. Soc.150(2022), no. 8, 3445–3455

    2022

  12. [20]

    Lukyanenko,Research Questions,https://lukyanenko.net/questions.php

    A. Lukyanenko,Research Questions,https://lukyanenko.net/questions.php

  13. [21]

    Monti, Some properties of Carnot–Carathéodory balls in the Heisenberg group,Rend

    R. Monti, Some properties of Carnot–Carathéodory balls in the Heisenberg group,Rend. Lincei Mat. Appl.11 (2000), 155–167

    2000

  14. [22]

    E. M. Stein,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, with the assistance of T. S. Murphy, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993

    1993

  15. [23]

    Thangavelu,Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, vol

    S. Thangavelu,Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, vol. 159, Birkhäuser Boston, Boston, MA, 1998

    1998

  16. [24]

    Venieri, Heisenberg Hausdorff dimension of Besicovitch sets,Anal

    L. Venieri, Heisenberg Hausdorff dimension of Besicovitch sets,Anal. Geom. Metr. Spaces2(2014), no. 1, 319–327

    2014

  17. [25]

    Venieri,Dimension Estimates for Kakeya Sets Defined in an Axiomatic Setting, Ann

    L. Venieri,Dimension Estimates for Kakeya Sets Defined in an Axiomatic Setting, Ann. Acad. Sci. Fenn. Math. Diss.161(2017), 73 pp

    2017

  18. [26]

    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

    2025 arXiv

  19. [27]

    Wolff, Recent work connected with the Kakeya problem, inProspects in Mathematics(Princeton, NJ, 1996), H

    T. Wolff, Recent work connected with the Kakeya problem, inProspects in Mathematics(Princeton, NJ, 1996), H. Rossi (ed.), American Mathematical Society, Providence, RI, 1999, 129–162

    1996

  20. [28]

    Zahl, A survey of the Kakeya conjecture, 2000–2025, inProceedings of the International Congress of Math- ematicians 2026, Volume 4: Invited Lectures (Sections 5–8), S

    J. Zahl, A survey of the Kakeya conjecture, 2000–2025, inProceedings of the International Congress of Math- ematicians 2026, Volume 4: Invited Lectures (Sections 5–8), S. Friedlander and Y. Tschinkel (eds.), SIAM, Philadelphia, 2026, 270–287. Marianna Chatzakou: Department of ...

    2000

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