Needles in every direction?

0:00Needles and tubes

In 1917, Fujiwara and Kakeya asked for the smallest region in which a needle of length one can be turned all the way around. A disc of diameter one works, with area about 0.79. A three-cusped curve, the deltoid, does better, at about 0.39. Among convex regions, the best is an equilateral triangle of height one, with area about 0.58. Then Besicovitch found a surprise: a set can contain a unit segment in every direction and still have area zero, and the needle can be turned in as little area as you like. One way to see it: slice a triangle into thin pieces and slide them so they overlap. Each direction the triangle held is still covered by a unit segment, but the area shrinks. With 32 pieces it falls to 30% of the triangle, and with 4,096 pieces, to 17%. Slowly, but as far as you like. A set with a unit segment in every direction is called a Kakeya set. In higher dimensions such sets can also have volume zero, but the Kakeya conjecture says they cannot be thin: their Hausdorff dimension should be full. Davies proved this in the plane in 1971. A stronger, quantitative form thickens each segment into a tube of radius δ. The maximal function records, for each direction, the largest average of a function over tubes pointing that way. The maximal conjecture says this operator is bounded from Lⁿ of space to Lⁿ of the sphere of directions, up to a loss of δ^(−ε), and it implies the dimension statement. Wolff’s hairbrush argument gave dimension at least (n + 2)/2: 5/2 in space, three in four dimensions. In 2025, Wang and Zahl proved the set conjecture in three dimensions, and Guth, Wang and Zahl later streamlined the proof. The maximal version in three dimensions stayed open. In four dimensions, the best Hausdorff bound was about 3.06, from Katz and Zahl.

2:06The claims

Result 074 is two OpenAI preprints. The first, 97 pages, dated September 23, claims the Kakeya maximal conjecture in three dimensions. The key is how the bound depends on density. Take tubes pointing in different directions, and shade a fraction λ of each. The union of the shaded parts must have volume at least about λ³ times the total tube volume. A bush shows the cube cannot be improved: tubes through one point, shaded only near it, have density λ, but all their shading fits in a ball of volume about λ³. Wang and Zahl’s theorem gives this bound only with a large power of λ. The second preprint, 175 pages, dated September 24, claims the set conjecture in four dimensions: every subset of four-dimensional space with a unit segment in every direction has Hausdorff dimension four. No measurability or compactness is assumed. Then bounded Kakeya sets in four dimensions have Minkowski dimension four, and, by projection, Kakeya sets in five dimensions have dimension at least four. OpenAI published no Lean proof of either theorem; one uncatalogued Lean file states the three-dimensional maximal conjecture and proves groundwork at a fixed scale.

3:27Three dimensions

The three-dimensional proof argues by contradiction. It sets up a family of inequalities for weighted tubes with marked times, and asks for the least extra power of 1/δ needed to make them all hold: the critical exponent. If that exponent were positive, some configurations would nearly attain equality, and their geometry is then forced step by step. Multilinear Kakeya makes directions on short pieces nearly coplanar. The planar Furstenberg theorem of Ren and Wang gives density inside plates, and the Guth, Wang and Zahl estimate captures saturated plates. Comparing two ways of cutting the configuration forces a canonical profile: flat at first, then with constant slope. In that stationary regime, inside each packet, every tube is described by four coordinates: two for position and two for velocity. Two marked events on the same tube give two frames, related by a pair of shear matrices. Changing the time adds velocity to position; changing the normal direction mixes the two spatial coordinates. One coefficient, the product of the two changes, multiplies the last velocity coordinate. A pinning estimate for that product slope, adapted from work of Shmerkin and Wang, and of Orponen, Shmerkin and Wang, together with an entropy count, shows that the packets cannot carry the information they must. So the critical exponent is zero, and the maximal estimate follows.

4:58Four dimensions

In four dimensions, the danger is lines crowding near low-degree algebraic surfaces. A saddle, z = xy, contains two whole families of lines. The proof’s basic object is a chart: line pieces in a moving box, over a time interval, on which one quadratic polynomial test stays small, with enough incidence mass to count. Charts can be nested. A hypothetical Kakeya set of dimension below four gives a weighted line problem in which every substantial system of terminal charts must use intervals of a definite, shrinking length. The proof shows that some chart system always does better. It follows nested charts along a critical path, with a time rate k and a narrowness rate ℓ, and splits into three regimes. When ℓ is zero, separated velocities determine an approximate polynomial field with a potential, and if the directions lie near a conic, the leftover cubic and quartic terms are folded into a quadratic correction. When ℓ is positive and finite, either wide intervals already give an improvement, or the lines are forced into a horizontal model whose boxes have width H and thickness H². When ℓ is infinite, a final projection gives charts whose time cost tends to zero. Each case contradicts the extremal choice. Inputs include multilinear Kakeya, a dot-product theorem of Wang and Zahl, and a lemma from the three-dimensional paper.

6:26Where it stands

Neither preprint is peer reviewed, and neither has been revised. OpenAI published no Lean proof of either theorem. OSVerify reviewed the four-dimensional paper and reported no major issues and two minor ones: a citation that points to Lemma 2.3 of the three-dimensional paper instead of Lemma 3.3, and a sentence that gives a Jacobian for the wrong direction of a coordinate map, although the displayed estimate is correct. Both are easy to fix, and OSVerify notes that they do not show the theorem is false. We have seen no expert review of the three-dimensional paper, which supplies one lemma the four-dimensional proof uses. That paper also notes that result 077, another release paper, claims the same maximal estimate through Fourier restriction. If they hold, the Kakeya maximal conjecture is settled in three dimensions, and the Kakeya set conjecture in four. The maximal conjecture from four dimensions up, and the set conjecture in five dimensions and above, remain open.