The Kakeya maximal function conjecture in $\mathbb{R}^3$ (the sharp $K=3$ case of the Wang–Zahl tube-volume inequality)

In February 2025, Wang and Zahl proved that every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3, resolving the three-dimensional Kakeya set conjecture. The engine of their proof is a volume lower bound for shaded unions of $\delta$-tubes that holds with some large, unspecified exponent $K$; the authors state explicitly that they do not resolve the Kakeya maximal function conjecture, which is the same inequality with the sharp exponent $K=3$, equivalently the sharp $L^p$ bounds for the Kakeya maximal operator in $\mathbb{R}^3$. This record states that classical conjecture precisely and records its literature status as of September 2026 — open, with the density-loss obstruction identified in Zahl's ICM survey and no resolution in subsequent literature.

Background

A Kakeya set (or Besicovitch set) in $\mathbb{R}^n$ is a compact set that contains a unit line segment pointing in every direction. Constructions from the 1920s show that such sets can have Lebesgue measure zero, so the meaningful question is their fractal size: the Kakeya set conjecture asserts that every Kakeya set in $\mathbb{R}^n$ has Minkowski and Hausdorff dimension $n$ (the Minkowski dimension measures the growth rate, as $\delta\to0$, of the number of $\delta$-balls needed to cover the set; the Hausdorff dimension is a finer multi-scale notion). The planar case was settled in the 1970s; every dimension $n\ge3$ remained open until 2025. The problem is central to Euclidean harmonic analysis because these constructions underlie the classical counterexample to $L^p$-boundedness of the ball multiplier (the operator that multiplies a function's Fourier transform by the indicator of the unit ball) for $p\ne2$, and quantitative Kakeya estimates are the standard inputs — and the standard obstructions — for the Fourier restriction problem (bounding the $L^p$ norms of functions whose Fourier transform is supported on a curved surface such as the sphere) and the Bochner–Riesz problem (the $L^p$ boundedness of smoothed ball multipliers).

The quantitative formulation uses tubes. A $\delta$-tube in $\mathbb{R}^3$ is the $\delta$-neighbourhood of a unit line segment (volume $\sim\delta^2$), and a family of tubes points in $\delta$-separated directions if any two of their directions subtend an angle at least $\delta$; discretizing a Kakeya set at scale $\delta$ produces roughly $\delta^{-2}$ such tubes. For a family $\mathbb{T}$ of $\delta$-tubes in the unit ball and a measurable shading $Y(T)\subseteq T$ with $|Y(T)|\ge\lambda|T|$ (a $\lambda$-dense shading), the Kakeya maximal function conjecture in $\mathbb{R}^3$ is the volume lower bound

$$\Big|\bigcup_{T\in\mathbb{T}}Y(T)\Big|\ \ge\ \delta^{\varepsilon}\,\lambda^{3}\sum_{T\in\mathbb{T}}|T|\qquad(\ast)$$

for every $\varepsilon>0$ and all sufficiently small $\delta$ (the customary $\delta^{\varepsilon}$ loss is permitted). Equivalent standard forms are: $\|\sum_{T\in\mathbb{T}}\chi_T\|_{L^p(\mathbb{R}^3)}\le C_\varepsilon\delta^{-\varepsilon}$ for all direction-separated families and all $1\le p\le3/2$; and, for the Kakeya maximal operator $K_\delta f(e)=\sup_T|T|^{-1}\int_T|f|$, the supremum over $\delta$-tubes $T$ pointing in direction $e$, the bound $\|K_\delta f\|_{L^3(S^2)}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\mathbb{R}^3)}$. These are the sharp $L^p$ maximal inequalities of the title. Córdoba proved the two-dimensional case in 1977.

Known partial results. Wolff's 1995 hairbrush argument (an induction organized around the tubes meeting a fixed tube) yields the maximal estimate at dimension $(n+2)/2=5/2$ in $\mathbb{R}^3$; in the shading form this carries an additional polynomial factor $\delta^{1/2}$, so it does not imply $(\ast)$ for any finite exponent at all. Katz–Łaba–Tao (2000) improved the upper Minkowski dimension of Kakeya sets in $\mathbb{R}^3$ to $5/2+c_0$ using sums-versus-products estimates from additive combinatorics — a dimension bound, not a maximal estimate. Hickman–Rogers–Zhang improved the maximal bounds only in dimensions $n=5$ and $n\ge7$, and Katz–Zahl's planebrush method gives a maximal estimate in $\mathbb{R}^4$. Two near-miss constructions — the Heisenberg group example of Katz–Łaba–Tao and the $SL_2$ example of Katz–Zahl — realize the natural non-clustering hypotheses at dimension $5/2$ over $\mathbb{C}$ and over a finite ring, respectively, showing that arguments using only such hypotheses (the Wolff axioms: no convex set $W$ contains more than about $|W|\,\delta^{-2}$ tubes) are capped at $5/2$; a proof in $\mathbb{R}^3$ must use arithmetic specific to the real field. The real-field $SL_2$ analogue was settled by Katz–Wu–Zahl, who proved that Kakeya sets in $\mathbb{R}^3$ formed from $SL_2$ lines nevertheless have full Hausdorff dimension 3.

The Wang–Zahl trilogy and what remains. Wang–Zahl first proved the sticky case (sticky: the family obeys the non-clustering axioms at every intermediate scale $\delta\le\rho\le1$, a multi-scale self-similarity), then showed that a hypothetical Kakeya set in $\mathbb{R}^3$ of Assouad dimension below 3 (the Assouad dimension is a strongest-scale variant of the Minkowski dimension) would have to be sticky-like, and in their 2025 paper proved the general volume estimate — Theorem 1.2: for every $\varepsilon>0$ there exists $K>1$, large and not optimized, such that $(\ast)$ with $\lambda^K$ in place of $\lambda^3$ holds for all tube families satisfying the non-clustering hypothesis that no $a\times b\times2$ rectangular prism contains more than about $ab\,\delta^{-2}$ of the tubes (a condition implied by direction separation; its convex-set analogue, the Katz–Tao Convex Wolff Axioms, is used in their structure theorems). This implies their Theorem 1.1: every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3, settling the Kakeya set conjecture in three dimensions. The authors state explicitly that they do not resolve the Kakeya maximal function conjecture, i.e. $(\ast)$ with the sharp $K=3$. Zahl's ICM survey records the concrete obstruction: Córdoba-type planar estimates available inside the argument show only that a $\lambda$-dense shaded union fills a rectangular prism to $\lambda^2$-density, and these losses compound to $\lambda^{2^N}$ through the induction on scales; whether the argument can be reorganized to avoid them is, in his words, an interesting open problem, and no near-miss to the maximal conjecture in $\mathbb{R}^3$ is currently known. The sharp quantitative core of the three-dimensional Kakeya problem thus remains open.

Problem Statement

Prove, or refute by explicit construction, the Kakeya maximal function conjecture in three dimensions, in any of the following equivalent standard forms.

(Shading form.) For every $\varepsilon>0$ there exists $\delta_0>0$ such that for all $0<\delta<\delta_0$: if $\mathbb{T}$ is a collection of $\delta$-tubes contained in the unit ball of $\mathbb{R}^3$ pointing in $\delta$-separated directions, and $Y(T)\subseteq T$ is measurable with $|Y(T)|\ge\lambda|T|$ for some $\delta\le\lambda\le1$, then

$$\Big|\bigcup_{T\in\mathbb{T}}Y(T)\Big|\ \ge\ \delta^{\varepsilon}\,\lambda^{3}\sum_{T\in\mathbb{T}}|T|,$$

i.e. inequality (1.1) of Wang–Zahl with the sharp exponent $K=3$.

(Multiplicity form.) $\|\sum_{T\in\mathbb{T}}\chi_T\|_{L^p(\mathbb{R}^3)}\le C_\varepsilon\delta^{-\varepsilon}$ for every direction-separated family $\mathbb{T}$ of $\delta$-tubes and every $1\le p\le3/2$.

(Operator form.) $\|K_\delta f\|_{L^3(S^2)}\le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\mathbb{R}^3)}$ for the Kakeya maximal operator $K_\delta f(e)=\sup_T|T|^{-1}\int_T|f|$, the supremum over $\delta$-tubes $T$ pointing in direction $e$.

Scope notes. Since direction separation implies each of these non-clustering hypotheses, the shading form under a non-clustering hypothesis weaker than direction separation — for instance the Katz–Tao Convex Wolff Axioms (no convex set $W$ contains more than about $|W|\,\delta^{-2}$ tubes of the family) or the prism hypothesis of Wang–Zahl's Theorem 1.2 (no $a\times b\times2$ rectangular prism contains more than about $ab\,\delta^{-2}$ tubes) — is a stronger statement than the form stated above. The customary $\delta^{\varepsilon}$ loss is permitted; removing it with explicit constants is stronger still. A refutation must be consistent with the already-proven Wang–Zahl Theorems 1.1 and 1.2 — in particular the regime $\lambda\approx1$ is settled — so any counterexample necessarily lives in the regime of shadings whose density $\lambda$ tends to 0 with $\delta$.

References

1. Hong Wang and Joshua Zahl, 'Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions', arXiv:2502.17655 (2025), DOI 10.48550/arXiv.2502.17655, https://arxiv.org/abs/2502.17655
2. Hong Wang and Joshua Zahl, 'Sticky Kakeya sets and the sticky Kakeya conjecture', arXiv:2210.09581 (2022); Journal of the American Mathematical Society 39(2):515–585 (2026), DOI 10.1090/jams/1067, https://doi.org/10.1090/jams/1067
3. Hong Wang and Joshua Zahl, 'The Assouad dimension of Kakeya sets in $\mathbb{R}^3$', arXiv:2401.12337 (2024); Inventiones mathematicae 241(1):153–206 (2025), DOI 10.1007/s00222-025-01336-x, https://doi.org/10.1007/s00222-025-01336-x
4. Larry Guth, Hong Wang, and Joshua Zahl, 'A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$', arXiv:2601.14411 (2026), https://arxiv.org/abs/2601.14411
5. Larry Guth, 'The Kakeya conjecture, after Wang and Zahl', arXiv:2604.03416 (2026), https://arxiv.org/abs/2604.03416
6. Joshua Zahl, 'A Survey of the Kakeya conjecture, 2000-2025', arXiv:2512.09397 (2025), https://arxiv.org/abs/2512.09397
7. Jonathan Hickman, 'The Kakeya Conjecture: where does it come from and why is it important?', arXiv:2512.09842 (2025), https://arxiv.org/abs/2512.09842
8. Jonathan Hickman, Keith M. Rogers, and Ruixiang Zhang, 'Improved bounds for the Kakeya maximal conjecture in higher dimensions', arXiv:1908.05589; American Journal of Mathematics 144(6):1511-1560 (2022), DOI 10.1353/ajm.2022.0037, https://arxiv.org/abs/1908.05589
9. Nets Hawk Katz and Joshua Zahl, 'A Kakeya maximal function estimate in four dimensions using planebrushes', arXiv:1902.00989 (2019); Revista Matemática Iberoamericana 37(1):317–359 (2021), DOI 10.4171/rmi/1219, https://doi.org/10.4171/rmi/1219
10. Tainara Borges, Tiklung Chan, Mingfeng Chen, Diankun Liu, Yakun Xi, and Yufei Zhan, 'Restriction and Kakeya maximal estimates in $\mathbb{R}^4$', arXiv:2511.22824 (2025), https://arxiv.org/abs/2511.22824
11. Cristian Rios and Eric T. Sawyer, 'Equivalence of linear and trilinear Kakeya conjectures in three dimensions', arXiv:2506.21315 (2025), https://arxiv.org/abs/2506.21315
12. Zhixu Hua and Xiufan Yang, 'Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application', arXiv:2608.18870 (2026), https://arxiv.org/abs/2608.18870
13. Zhixu Hua, Xinshun Yao, and Xiufan Yang, 'Profile-Stable Buffered Multiplicity Factoring in Four Dimensions', arXiv:2608.18693 (2026), https://arxiv.org/abs/2608.18693
14. Thomas H. Wolff, 'An improved bound for Kakeya type maximal functions', Revista Matemática Iberoamericana 11(3):651-674 (1995), DOI 10.4171/RMI/188, https://doi.org/10.4171/RMI/188
15. Antonio Córdoba, 'The Kakeya Maximal Function and the Spherical Summation Multipliers', American Journal of Mathematics 99(1):1-22 (1977), DOI 10.2307/2374006, https://doi.org/10.2307/2374006
16. Nets Hawk Katz, Izabella Łaba, and Terence Tao, 'An Improved Bound on the Minkowski Dimension of Besicovitch Sets in $\mathbb{R}^3$', Annals of Mathematics (2) 152(2):383-446 (2000), DOI 10.2307/2661389, https://doi.org/10.2307/2661389
17. Nets Hawk Katz, Shukun Wu, and Joshua Zahl, 'Kakeya sets from lines in $SL_2$', arXiv:2211.05194 (2022); Ars Inveniendi Analytica (2023), Paper No. 6, DOI 10.15781/ep89-2217, https://doi.org/10.15781/ep89-2217
18. Nets Hawk Katz and Joshua Zahl, 'An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$', arXiv:1704.07210 (2017); Journal of the American Mathematical Society 32(1):195–259 (2019), DOI 10.1090/jams/907, https://doi.org/10.1090/jams/907