Worked solution: The three-dimensional case: every Kakeya set in $\mathbb{R}^3$ has dimension 3 (Wang–Zahl, 2025)

Frequently asked questions

Does this settle the stronger Kakeya maximal function conjecture too?

No. The set (Hausdorff/Minkowski dimension) conjecture proved here is implied by, but strictly weaker than, the Kakeya maximal function conjecture in $L^3(\mathbb{R}^3)$ , which controls how large the maximal directional average of a function can be, not just the size of the union of tubes. Wang and Zahl's techniques give some new bounds toward the maximal function conjecture but do not resolve it; it remains open as of 2026.

Source

Hong Wang, Joshua Zahl, 2025

Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

arXiv:2502.17655v1

Preprint

Condensed summary

Step 5 of 5: Closing the induction: thick prisms and thin prisms

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In plain words

What remains is exactly the case where grains do not rescale into a Kakeya-like configuration — they instead pile up far too densely ('super-Kakeya') inside some larger convex prism $W$ of unspecified dimensions. Wang and Zahl split on whether $W$ is 'thick' (every side length $\gg\delta$ ) or 'thin' (thickness $\approx\delta$ ). If $W$ is thick, a delicate further induction-on-scales argument shows that $K(d)$ upgrades to an 'x-ray' estimate strong enough to exhibit the Frostman violation of Step 3 directly. If $W$ is thin, the classical planar $L^2$ argument together with the super-Kakeya density of grains forces $E$ to nearly fill up $W$ entirely — but then $W$ itself could have been taken as an even larger grain in Step 4, contradicting the maximality with which the grains were originally chosen. Every configuration therefore falls into a case that closes the induction, completing the proof that $K(3-\varepsilon)$ holds for every $\varepsilon>0$ , i.e. every Kakeya set in $\mathbb{R}^3$ has Hausdorff and Minkowski dimension exactly $3$ .

W = \text{convex prism containing a `super-Kakeya' bundle of grains}; \qquad \text{thick } (\min\dim W \gg \delta) \ \text{vs.}\ \text{thin } (\min\dim W \approx \delta)

Detailed analysis

This case split (Wang–Zahl 2025, §§6–8, following the outline in Tao's blog exposition) is where the bulk of the 127 pages lives. For thick prisms $W$ , the authors prove an 'x-ray estimate' — a variant of $K(d)$ that bounds not just the total volume of a tube union but how it intersects every affine plane, strong enough that a super-Kakeya bundle of grains inside $W$ directly forces the Frostman violation of Step 3. For thin prisms, an elementary planar $L^2$ argument (structurally the same computation as Step 3 of the $\mathbb{R}^2$ proof, applied inside $W$ 's near-planar cross-section) combined with the super-Kakeya count shows $|E\cap W|\gtrsim |W|^{1-o(1)}$ , so $E$ nearly fills $W$ ; since $W$ was assumed larger than the maximal grain size fixed in Step 3, this contradicts that maximality and rules the case out. No configuration escapes both alternatives, which completes the induction on scales and the proof of $K(3-\varepsilon)$ for every $\varepsilon>0$ .