For a smooth map Φ\Phi satisfying Hörmander's nondegeneracy condition, a Φ\Phi-Kakeya set contains one curve from the associated 2(n−1)2(n-1)-parameter family for every direction, and the Kakeya index of Φ\Phi is the infimum of the Hausdorff dimensions of its Kakeya sets. Guo, Guth, Nadjimzadah, Shen and Zhang (arXiv:2607.27629), in a framework relating curved Kakeya problems to the projective geometry of paths, ask whether this index is always an integer when Φ\Phi is analytic and semi-algebraic, and in particular whether in R3\mathbb{R}^3 the index is always 22 or 33. By their Theorem 1.34, the latter is equivalent to asking whether a semi-algebraic non-degenerate curve family in R3\mathbb{R}^3 that fails the worst compression condition — the rank condition characterizing compression of the family into a surface — must have every associated Kakeya set of full Hausdorff dimension 33; their gap theorem only gives dimension at least 2+κ2+\kappa. No result addresses the integrality question or the dichotomy. The problem is open.

Background

From classical to curved Kakeya problems. A Kakeya set in Rn\mathbb{R}^n is a set containing a unit line segment in every direction; the Kakeya set conjecture asserts that every such set has full Hausdorff dimension nn (the Hausdorff dimension of a set is the exponent governing how the number of δ\delta-balls needed to cover it grows as δ→0\delta\to 0). Wang and Zahl (2025) resolved the conjecture in R3\mathbb{R}^3. In a curved Kakeya problem the lines are replaced by a 2(n−1)2(n-1)-parameter family of curve segments ℓξ,w\ell_{\xi,w} in Rn\mathbb{R}^n, where ξ∈Rn−1\xi\in\mathbb{R}^{n-1} is a direction parameter and w∈Rn−1w\in\mathbb{R}^{n-1} a position parameter, and a curved Kakeya set is a set containing one curve ℓξ,w(ξ)\ell_{\xi,w(\xi)} for each direction ξ\xi. The family is assumed non-degenerate, so that locally each point and tangent direction determines a unique curve; the model case of straight lines recovers the classical problem. Such families arise from Hörmander-type oscillatory integral operators Tλf(x)=∫eiλϕ(x,y)a(x,y)f(y) dyT_\lambda f(x)=\int e^{i\lambda\phi(x,y)}a(x,y)f(y)\,dy, whose geometry is carried by the characteristic curves of the phase ϕ\phi.

The map formulation and Bourgain's condition. Guo, Guth, Nadjimzadah, Shen and Zhang (2026) encode a curve family by a smooth map Φ(ω,t;ξ)=(X(ω,t;ξ),t)\Phi(\omega,t;\xi)=(X(\omega,t;\xi),t) with tt the curve parameter, and impose Hörmander's condition: the n×nn\times n matrix ∇(ω,t)Φ\nabla_{(\omega,t)}\Phi and the 2n×2n2n\times 2n matrix

[0∇(ω,t)Φ∇ξΦ∂tΦ∇(ω,t)∂tΦ∇ξ∂tΦ]\begin{bmatrix}0&\nabla_{(\omega,t)}\Phi&\nabla_{\xi}\Phi\\ \partial_{t}\Phi&\nabla_{(\omega,t)}\partial_{t}\Phi&\nabla_{\xi}\partial_{t}\Phi\end{bmatrix}

are both nonsingular. Already Bourgain (1991) showed that the curved situation in R3\mathbb{R}^3 differs sharply from the linear one: a Hörmander-type phase can have a curved Kakeya set of Hausdorff dimension only 22, the smallest value possible for a non-degenerate family; his compression example is the prototype of worst-case behavior. Subsequent work isolated Bourgain's condition — for a map in normal form, the requirement that ∇ξ∂t2X\nabla_{\xi}\partial_t^2 X be a pointwise scalar multiple of ∇ξ∂tX\nabla_{\xi}\partial_t X — as, conjecturally, exactly the class of phases that behave like the extremal Bochner–Riesz phase ϕ(x,t;ξ)=∣(x,t)−(ξ,φ(ξ))∣\phi(x,t;\xi)=|(x,t)-(\xi,\varphi(\xi))|: a line of work culminating in Guth–Hickman–Iliopoulou (2019) and Guo–Wang–Zhang (2024) established that the condition governs whether the optimal LpL^p bounds for the associated oscillatory integral operators are those of the classical restriction problem or can be improved, and the condition is conjectured to be necessary and sufficient for optimal curved Kakeya maximal function estimates. Bourgain's compression example fails the condition.

Projective geometry of paths. The 2026 paper attaches to each curve family a spray — a geometric structure on the tangent bundle whose unparameterized geodesics are the curves of the family — and studies its projective invariants, such as the Douglas and Weyl curvatures; vanishing of these invariants means the path geometry is locally projectively flat, that is, equivalent to straight lines after a coordinate change. Two incidence-geometric conditions organize the theory: the totally geodesic condition (the curves meeting two given intersecting curves lie in a common surface — the structure Wolff's hairbrush argument requires) and the Katz–Wolff condition (for two intersecting curves, the direction parameters of the curves meeting both form a one-dimensional rather than a two-dimensional set). Within the projectively flat class, the authors prove that Bourgain's condition forces direction-equivalence to a Bochner–Riesz-type family of lines, so that the Wang–Zahl theorem applies, while its failure makes every totally geodesic surface support a two-dimensional Kakeya set.

The worst-case theorem and the Kakeya index. For the worst-case behavior, Theorem 1.34 of the paper states: for a non-degenerate family of curves in R3\mathbb{R}^3 whose parametrization is semi-algebraic (definable from finitely many polynomial equalities and inequalities), the family admits a curved Kakeya set of Hausdorff dimension 22 if and only if it admits a curved Kakeya set contained in a two-dimensional surface; if not, there exists κ>0\kappa>0, depending on the family, such that every associated Kakeya set has dimension at least 2+κ2+\kappa. A concrete rank condition, the worst compression condition, produces two-dimensional Kakeya sets, and by Theorem 1.34 it is, in the semi-algebraic world, essentially the only mechanism for them. The Kakeya index of Φ\Phi is then defined as the infimum of the Hausdorff dimensions of Φ\Phi-Kakeya sets; the infimum is essential because individual Kakeya sets can carry artificial fractional dimensions — translating copies of Bourgain's two-dimensional example produces sets of any dimension between 22 and 33, within the compressing regime — without changing the family's intrinsic behavior. Throughout, maps are taken analytic, and semi-algebraicity is the hypothesis powering the gap theorem through semialgebraic geometry. What the framework leaves open is whether this semi-algebraic dimension landscape is discrete.

Problem Statement

Let Φ(ω,t;ξ)=(X(ω,t;ξ),t)\Phi(\omega,t;\xi)=(X(\omega,t;\xi),t), defined on an open set Ω⊂Rn−1×R×Rn−1\Omega\subset\mathbb{R}^{n-1}\times\mathbb{R}\times\mathbb{R}^{n-1}, be an analytic, semi-algebraic map satisfying Hörmander's condition (the two nonsingularity conditions of Definition 1.13 of arXiv:2607.27629, displayed in the Background above), and let a Φ\Phi-Kakeya set be a set containing, for every direction parameter ξ\xi, a curve of the family associated with Φ\Phi. Define the Kakeya index of Φ\Phi by

ind⁡(Φ)=inf⁡{dim⁡HE: E is a Φ-Kakeya set}.\operatorname{ind}(\Phi)=\inf\{\dim_{\mathrm{H}}E:\ E\text{ is a }\Phi\text{-Kakeya set}\}.

The question, posed by Guo–Guth–Nadjimzadah–Shen–Zhang (2026, Section 1.9):

(1) Integrality. Is ind⁡(Φ)\operatorname{ind}(\Phi) necessarily an integer?

(2) The three-dimensional dichotomy. For n=3n=3, is ind⁡(Φ)\operatorname{ind}(\Phi) necessarily equal to 22 or 33? By Theorem 1.34 of the source paper this is equivalent to: if Φ\Phi fails the worst compression condition, must every Φ\Phi-Kakeya set have Hausdorff dimension exactly 33?

Here the worst compression condition is the following rank condition (Definition 1.23 of the source). Fix a position parameter w0w_{0} and two distinct direction parameters ξ′≠ξ′′\xi'\neq\xi''; for parameters t1,t2t_{1},t_{2}, let w(t1,t2)w(t_{1},t_{2}) and ξ(t1,t2)\xi(t_{1},t_{2}) denote the solution, unique when it exists, of the requirement that the curve {(X(w(t1,t2),t;ξ(t1,t2)),t)}\{(X(w(t_{1},t_{2}),t;\xi(t_{1},t_{2})),t)\} pass through the two points (X(w0,t1;ξ′),t1)(X(w_{0},t_{1};\xi'),t_{1}) and (X(w0,t2;ξ′′),t2)(X(w_{0},t_{2};\xi''),t_{2}). The map Φ\Phi satisfies the worst compression condition if such w0w_{0} and ξ′≠ξ′′\xi'\neq\xi'' can be found for which the union of these curves, ⋃t1,t2{(X(w(t1,t2),t;ξ(t1,t2)),t)}\bigcup_{t_{1},t_{2}}\{(X(w(t_{1},t_{2}),t;\xi(t_{1},t_{2})),t)\}, forms a two-dimensional surface and rank⁡[∂t1ξ(t1,t2),∂t2ξ(t1,t2)]=2\operatorname{rank}[\partial_{t_{1}}\xi(t_{1},t_{2}),\partial_{t_{2}}\xi(t_{1},t_{2})]=2 for some t1≠t2t_{1}\neq t_{2}; this is the mechanism by which a family compresses into a surface and admits a two-dimensional Φ\Phi-Kakeya set.

Note that Theorem 1.34 already excludes dimension 22 and guarantees dimension at least 2+κ2+\kappa for some κ>0\kappa>0 depending on the family when compression fails; the open content is whether a genuine intermediate dimension — or a non-integer index — can occur for an analytic semi-algebraic family. It suffices to resolve either question, by proof or by counterexample.

References

  1. Shaoming Guo, Larry Guth, Arian Nadjimzadah, Minxing Shen, Ruixiang Zhang (2026), 'Curved Kakeya problems and the projective geometry of paths', arXiv:2607.27629, DOI 10.48550/arXiv.2607.27629, https://arxiv.org/abs/2607.27629 (includes an appendix by Terence Tao)
  2. J. Bourgain (1991), 'L^p-Estimates for oscillatory integrals in several variables', Geometric and Functional Analysis 1(4), 321-374, DOI 10.1007/BF01895639, https://doi.org/10.1007/BF01895639
  3. Christopher Sogge (1999), 'Concerning Nikodym-type sets in 3-dimensional curved spaces', Journal of the American Mathematical Society 12(1), 1-31, DOI 10.1090/S0894-0347-99-00289-1, https://doi.org/10.1090/S0894-0347-99-00289-1
  4. L. Wisewell (2005), 'Kakeya sets of curves', Geometric and Functional Analysis 15(6), 1319-1362, DOI 10.1007/s00039-005-0540-9, https://doi.org/10.1007/s00039-005-0540-9
  5. Larry Guth, Jonathan Hickman, Marina Iliopoulou (2019), 'Sharp estimates for oscillatory integral operators via polynomial partitioning', Acta Mathematica 223(2), 251-376, DOI 10.4310/acta.2019.v223.n2.a2, https://doi.org/10.4310/acta.2019.v223.n2.a2
  6. Shaoming Guo, Hong Wang, Ruixiang Zhang (2024), 'A dichotomy for Hörmander-type oscillatory integral operators', Inventiones mathematicae 238(2), 503-584, DOI 10.1007/s00222-024-01288-8, https://doi.org/10.1007/s00222-024-01288-8
  7. Hong Wang, Joshua Zahl (2025), 'Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions', arXiv:2502.17655, https://arxiv.org/abs/2502.17655
  8. Arian Nadjimzadah (2025), 'New curved Kakeya estimates', arXiv:2503.15760, https://arxiv.org/abs/2503.15760
  9. Arian Nadjimzadah (2025), 'Bourgain's condition, sticky Kakeya, and new examples', arXiv:2511.10918, https://arxiv.org/abs/2511.10918
  10. Shaoming Guo, Diankun Liu, Yakun Xi (2025), 'Curved Kakeya sets for generic phases in odd dimensions', arXiv:2508.17706, https://arxiv.org/abs/2508.17706
  11. Yifan Jing, Shukun Wu (2026), 'Kakeya sets and dimension compression in compact Lie groups', arXiv:2608.11726, https://arxiv.org/abs/2608.11726