The Kakeya set conjecture in $\mathbb{R}^n$ for $n\ge 4$: Hausdorff and Minkowski dimension of Besicovitch sets

A Besicovitch (Kakeya) set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction; Besicovitch showed such sets can have Lebesgue measure zero, so the central question is dimensional. The Kakeya set conjecture asserts that every Besicovitch set in $\mathbb{R}^n$ has Hausdorff dimension and upper Minkowski dimension equal to $n$. The conjecture was proved for $n=2$ by Davies (1971) and, in the landmark 2025 breakthrough of Wang and Zahl, for $n=3$. This problem is the surviving open core: prove or refute the conjecture in every dimension $n\ge 4$. It remains open with best lower bounds near Wolff's $(n+2)/2$, and the 3D method provably fails in higher dimensions because tube families can cluster along infinitely ruled quadric hypersurfaces. A resolution would settle one of the central problems of Euclidean harmonic analysis and geometric measure theory, with direct consequences for the dimension theory of sets containing lines and major momentum for the Kakeya maximal function and Fourier restriction programs.

Background

A Besicovitch set (also called a Kakeya set) in $\mathbb{R}^n$ is a compact set that contains a unit line segment pointing in every direction. Besicovitch showed in 1919 that such sets can have Lebesgue measure zero already in the plane (a construction recounted in Zahl's 2025 survey), so the meaningful question is not about measure but about dimension. The Hausdorff dimension of a set is the infimum of exponents $s$ for which it can be covered by countably many balls of total $s$-dimensional cost arbitrarily small; the upper Minkowski (box-counting) dimension of a bounded set $K$ is $\limsup_{\delta\to 0}\log N_\delta(K)/\log(1/\delta)$, where $N_\delta(K)$ is the minimum number of $\delta$-balls needed to cover $K$. The two notions can differ for the same set, and the conjecture asserts both are maximal.

The Kakeya set conjecture states that every Besicovitch set in $\mathbb{R}^n$ has Hausdorff and upper Minkowski dimension equal to $n$: a set containing a segment in every direction cannot be too thin, no matter how cleverly the segments overlap. Davies proved the case $n=2$ in 1971. For $n\ge 3$ the problem became one of the central open questions of Euclidean harmonic analysis, because a stronger quantified statement — the Kakeya maximal function conjecture, about $L^p$ norms of averages of functions over thin tubes — implies the set conjecture, and both are intertwined with the Fourier restriction and Bochner–Riesz conjectures. Dvir's proof that Kakeya sets over a finite field with $q$ elements have size at least $C_n q^n$ cites the Euclidean conjecture explicitly as its motivation; its polynomial method resolves the finite-field analogue but does not transfer to Euclidean space.

Euclidean progress came through the discretized formulation: covering a Besicovitch set at scale $\delta$ yields a family of $\delta$-tubes ($\delta$-neighbourhoods of unit line segments) pointing in $\delta$-separated directions, and the conjecture becomes the assertion that the union of any such family has essentially maximal volume. Wolff (1995) proved that Besicovitch sets have dimension at least $(n+2)/2$ ($5/2$ in $\mathbb{R}^3$), a bound sharpened in $\mathbb{R}^3$ by Katz–Łaba–Tao (2000, upper Minkowski dimension $\ge 5/2+c$) and Katz–Zahl (2019, Hausdorff dimension $\ge 5/2+c$), with improvements in higher dimensions by Bourgain (1999), Katz–Tao (2002), and Hickman–Rogers–Zhang (2022). In February 2025, Wang and Zahl proved the full conjecture in three dimensions: every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3, via a volume estimate for unions of $\delta$-tubes satisfying the Convex Wolff Axioms, a non-clustering condition forbidding any convex set from containing far more tubes than its volume warrants.

Dimensions $n\ge 4$ are genuinely different, which is why the conjecture there remains open. The Wang–Zahl tube-volume theorem is false in every dimension $n\ge 4$: the quadric hypersurface $x_1^2+x_2^2+x_3^2=x_4^2$ contains a three-dimensional family of lines, so its $\delta$-neighbourhood hosts roughly $\delta^{-3}$ essentially distinct $\delta$-tubes that satisfy the Convex Wolff Axioms while occupying volume only about $\delta^{1/2}$, and combining translated and rotated copies produces a near-miss to the conjecture. The proposed repair — strengthening the non-concentration hypothesis so that clustering is also forbidden inside semi-algebraic sets (the Polynomial Wolff Axioms) — is itself an open program in every dimension $n\ge 4$. Thus, after the three-dimensional breakthrough, the conjecture in dimensions four and up stands as the unresolved core, with best lower bounds still close to $(n+2)/2$.

Problem Statement

Prove, or refute, the Kakeya set conjecture in dimensions four and higher: for every integer $n\ge 4$, every compact set $K\subset\mathbb{R}^n$ that contains a unit line segment in every direction has Hausdorff dimension $n$ and upper Minkowski dimension $n$.

Equivalently, in the standard discretized form: for every $0<\delta<1$ and every family $\mathbb{T}$ of $\delta$-tubes ($\delta$-neighbourhoods of unit line segments) in the unit ball of $\mathbb{R}^n$ pointing in $\delta$-separated directions, $|\bigcup_{T\in\mathbb{T}}T|\gtrapprox\sum_{T\in\mathbb{T}}|T|$ (the Minkowski-dimension version), and, if each tube carries a shaded subset $Y(T)\subset T$ with $|Y(T)|\ge(\log 1/\delta)^{-1}|T|$, then $|\bigcup_{T\in\mathbb{T}}Y(T)|\gtrapprox\sum_{T\in\mathbb{T}}|T|$ (the Hausdorff-dimension version); here $f\lessapprox g$ means $f\le C_\varepsilon\delta^{-\varepsilon}g$ for every $\varepsilon>0$, and $f\gtrapprox g$ means $g\lessapprox f$ (with $C_\varepsilon$ allowed to depend on the ambient dimension $n$).

The problem is already resolved for $n\le 3$ (Davies for $n=2$; Wang–Zahl for $n=3$), so the open problem is the full conjecture for all $n\ge 4$, whose first open instance is $n=4$. A resolution may proceed dimension by dimension — a proof for a single $n\ge 4$ settles that instance — but acceptance of the conjecture as stated requires all $n\ge 4$. A refutation consists of an explicit Besicovitch set in some $\mathbb{R}^n$ with $n\ge 4$ whose Hausdorff or upper Minkowski dimension is strictly less than $n$.

References

1. Z. Dvir, "On the size of Kakeya sets in finite fields", Journal of the American Mathematical Society 22 (2009) 1093–1097, DOI 10.1090/s0894-0347-08-00607-3, https://doi.org/10.1090/s0894-0347-08-00607-3 (also arXiv:0803.2336)
2. R. O. Davies, "Some remarks on the Kakeya problem", Mathematical Proceedings of the Cambridge Philosophical Society 69 (1971) 417–421, DOI 10.1017/S0305004100046867, https://doi.org/10.1017/S0305004100046867
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