The Kakeya maximal conjecture in three dimensions

Introduction

For $a \in\mathbb{R}^{3}$, $\omega\in S^{2}$, and $0 < \delta< 1$, let

Thus $T_{\delta}(a,\omega)$ has volume $\pi\delta^{2}$. Define

The measure on $S^{2}$ below is its usual surface measure.

Theorem 1.1. For every $\varepsilon>0$ there exists a finite constant $C_{\varepsilon}$ such that, for every $0<\delta<1$ and every $f\in L^{3}(\mathbb{R}^{3})$,

Theorem 1.1 resolves the three-dimensional Kakeya maximal conjecture affirmatively. Its essential quantitative feature is uniformity in the density of the part of each tube on which a function is large. We explain that feature before describing the proof.

The set problem and the maximal problem

A Kakeya set contains a unit line segment in every direction. The problem has its origins in the study of moving a needle inside a small planar region; the early convex formulation appears in Fujiwara and Kakeya [8], Section 10. Besicovitch’s constructions showed that a set with segments in every direction can have measure zero, and that continuous needle reversal is possible in regions of arbitrarily small area [3, 6]. These are different requirements. The dimension conjecture asks whether directional containment nevertheless forces full Hausdorff dimension. Davies proved the planar case [6]; Córdoba’s planar maximal estimates provided a quantitative analytic counterpart [5].

The distinction relevant here is already visible for characteristic functions. Let $\mathcal{T}$ be a family of unit $\delta$-tubes in a fixed bounded region, with $\delta$-separated directions, and let a shading assign a measurable subset $Y(T) \subset T$ to each tube. If $|Y(T)| \ge\lambda|T|$, the maximal problem asks for the cubic density power in the union estimate

The power loss in $\delta$ may be arbitrarily small, uniformly in $\lambda$. A set-dimension conclusion can follow from weaker dependence on $\lambda$. In particular, Wang and Zahl’s set theorem gives this form of estimate with a power $\lambda^{K(\varepsilon)}$; the need to replace it by $\lambda^3$ is stated explicitly after their theorem [20], Theorem 1.2. The final reduction in Section 10 retains the density power through the passage from shadings to arbitrary $L^3$ functions.

The same overlapping tube configurations also enter Fourier restriction and summability problems. Fefferman’s ball-multiplier counterexample made this connection concrete [7]. Bourgain developed higher-dimensional maximal estimates and their Fourier-analytic applications [4], and Wolff’s hairbrush argument gave maximal estimates implying the Hausdorff-dimension bound $(n+2)/2$ in $\mathbb{R}^n$ [23]. In three dimensions this is $5/2$. Katz, Łaba, and Tao obtained a strict improvement for upper Minkowski dimension and analyzed three recurring structures in nearly extremal tube families: nearby tubes remain grouped across scales (stickiness), tubes meeting near a point have nearly coplanar directions (planiness), and their union organizes into thin planar pieces (graininess) [14]. Katz and Zahl subsequently obtained a strict Hausdorff-dimension improvement over $5/2$ [13].

The three-dimensional set problem was resolved through a sequence of works of Wang and Zahl. Their sticky Kakeya theorem was followed by the full Assouad-dimension theorem and then the Hausdorff- and Minkowski-dimension theorem for arbitrary Kakeya sets [20, 21, 22]. The corresponding preprints appeared in 2022, 2024, and 2025. The streamlined proof of Guth, Wang, and Zahl supplies the set estimate used in this paper [11], Theorem 1.1. It applies under a convex-clustering bound and a shading-density threshold $\lambda\ge\delta^\eta$, with $\eta> 0$ chosen from the desired volume exponent. Convex clustering bounds how many complete tubes can lie in a convex body; it is a formulation that remains useful after the restrictions and changes of scale needed here. The task is to retain the sharper density dependence required by (1) through those operations. For a broader account of the maximal formulation and its relation to the set problem, see Zahl [24], Conjecture 1.3” and Section 3.3.

Two further geometric estimates enter the proof. The multilinear Kakeya theorem of Bennett, Carbery, and Tao [2], Theorem 1.15, sharpened to the endpoint by Guth [10], Theorem 1.3, controls three transverse tube families. The planar Furstenberg theorem of Ren and Wang [17], Theorem 4.1, in the equivalent shaded-tube formulation recorded by Wang and Wu [19], Theorem 0.6, controls planar incidence and projection configurations. We state these inputs and prove the weighted forms used here in Section 3.

Geometry of the proof

A density-sensitive extremal problem. Work in a bounded coordinate chart in which each tube follows an affine graph $M_i(t)=b_i+tu_i$, with $b_i,u_i\in\mathbb{R}^2$. At resolution $N=\delta^{-1}$, its shading is a set of marked time bins. We keep the original index $i$ and its weight under restriction, even when two indices have the same affine graph. A full-time plank counts the indices whose complete traces it contains; marked bins instead determine the occupied cells and their multiplicity. This separation permits the set input to be used after rescaling without identifying different shadings on the same trace.

Total shading density alone does not describe what happens after a scale cut. A temporal deficit profile records, in logarithmic units, how the number of marked bins falls short of full branching inside each occupied block. We introduce a family of multiplicity inequalities whose temporal cost depends on this profile and whose clustering term depends on the two plank widths. At the parameter that gives the maximal theorem, the cost reduces to the total density penalty. The critical exponent is the least extra power of $N$ needed to make the inequality hold for every configuration. A positive critical exponent would produce configurations attaining equality in the limit. We study their geometry, allowing restrictions that lose only a subpower fraction of the weighted incidence.

From equality to compatible plates. An equality configuration cannot contain a local packet with excessive mass: rescaling such a packet would improve the critical exponent. The resulting local bounds, together with multilinear Kakeya, force the directions on typical short blocks to lie close to a common plane. The planar Furstenberg estimate supplies density inside plates containing the reference segments. The plate Nikodym estimate then controls the support cost of filling their missing times. Equality therefore forces full temporal branching on an initial interval of logarithmic scales. On those scales, the weighted set theorem captures saturated plates: their masses attain the local bound. An anisotropic rescaling of aligned plates rules out equality with arbitrarily small total time deficit.

These local planes and plates are the forms of planiness and graininess needed by the present extremal argument. Their compatibility across scales must still be proved. We vary the two parameters in the critical inequality to select saturated shapes, then compare two ways of cutting an equality configuration. An outer cut removes initial scales; an inner cut retains a fine-scale model after aligning its plates. A minimum-delay argument forces a canonical temporal profile

with a flat interval followed by constant deficit slope. Repeating the outer cut reproduces this profile and the same normalized packet geometry. This is the stationarity used below.

Four coordinates and two frames. A packet has a time interval and a nearly constant normal parameter $\vartheta$. In that frame, write

Thus $X,Y$ describe position and $U,V$ describe velocity; the first coordinate in each pair measures the normal component. Their admissible widths change at four prescribed rates, determined by the stationary time and angle scales. The masses of the packets prescribe how much information these four coordinates must carry. Conditional entropy, normalized by $\log N$, measures that information as a logarithmic cell count.

Now choose two marked events on the same original index. They give two frames for the same affine graph. If their time and normal differences are $\Delta t$ and $\Delta\vartheta$, then

Changing time adds velocity to position; changing the normal mixes the two spatial components. The resulting scalar projections constrain the coordinate entropy rates through the planar Furstenberg theorem. When the two middle speeds coincide, the scale increments directly supply only their joint rate. Freezing the middle coordinates leaves the coefficient $\Delta t\,\Delta\vartheta$ multiplying $V$ in $X'$; this is the product slope for which the pinning estimate is needed.

The pinning argument adapts the radial-projection bootstrap of Shmerkin and Wang [18], Section 5.4 and of Orponen, Shmerkin, and Wang [16], Section 1.3. On small cells, planar projection estimates improve the nonconcentration of rays joining parameter points; this improvement can then be iterated. Section 9 proves the finite-scale product-slope estimate used here, including control of both single-event marginals.

The two events share an index, so their matrix and slope are generally dependent. To apply a projection theorem, we compare their actual joint law with the product of its conditional matrix and label marginals. The actual law always lands in the second event’s recorded support. If a coordinate-rate constraint failed, the product law would land there with polynomially small probability. Such a discrepancy requires a positive amount of mutual information. An averaged short-gap estimate shows that the available information is smaller. The resulting projection constraints contradict the entropy demand of the stationary packets, excluding a positive critical exponent.

Consequences and organization

The theorem also gives Nikodym maximal estimates in $\mathbb{R}^3$ and locally on three-dimensional manifolds of constant sectional curvature, as well as local curved Kakeya estimates for a fixed nondegenerate translation-invariant phase satisfying Bourgain’s condition. These follow from the transfers of Gao, Liu, and Xi [9]; their full hypotheses and interpolation ranges appear in Section 11. An independent route to the same strong Kakeya maximal estimate through spherical Fourier extension appears in OpenAI [15], Corollary 1.3.

Section 2 defines the indexed model, explains the density reduction, and constructs the critical inequalities and equality sequences. Section 3 supplies the geometric inputs. Sections 4 and 5 establish the local geometry, canonical profile, and stationary packets. Section 6 proves angular nonconcentration and the conditional normal-position estimate.

Section 7 constructs the coordinate entropy rates, proves their lower demand, and obtains the velocity trace needed at the central offset. Section 8 derives the projection constraints from the two-frame experiment; Section 9 proves its auxiliary projection and pinning estimates. Section 10 closes the contradiction and passes from the density estimate to the full maximal theorem.

All limiting exponents arise from finite configurations. At each finite stage, the required scales, partitions, and conditional tests are imposed together before the resolution is increased. The regularization results in Section 2 give the losses and order of limits used in the later constructions.

Weighted configurations and critical exponents

The geometric problem is to bound the average multiplicity of shaded tubes in terms of their shading density and concentration in planks. We first express this problem for finite weighted, indexed families of affine lines. Restrictions and changes of scale preserve inherited weights, but may produce repeated line parameters, so the class includes those repetitions. We then define a family of critical inequalities and construct sequences that attain their optimal exponents. These sequences will be the input to the local geometric arguments.

The indexed model and logarithmic comparisons

Let $N = 2^j$ tend to infinity. An index $i$ consists of a positive weight $\omega_i$, a matrix $M_i = (b_i,u_i)$ in a fixed bounded subset of $\mathbb{R}^{2 \times2}$, and a nonempty subset $S_i$ of the $N$ dyadic time bins in $[0,1]$. Different indices may have identical matrices and different shadings. The spatial trace is

An event is a pair $(i,J)$ with $J \in S_i$. Its location is $(t_J,M_i(t_J))$, where $t_J$ is the bin center. Let $E$ be the union of the space-time grid cells of side $N^{-1}$ containing these locations. Write

Thus $k$ is initially a weighted mean, and $m$ is the raw average multiplicity. After regularization all $|S_i|$ will have the same exponent as $k$. Each event has mass $\omega_i$; probabilities on the event set always mean division by $\mathcal{I}$.

A full-time plank of widths $a,b$, where \$1 \leq a \leq b \leq N$, is a set with cross-sections

for an orthonormal spatial frame $(e_1,e_2)$ and affine center $b_0+tu_0$. Constant factors in these widths are permitted. An index belongs to a plank only if its entire trace over the indicated time interval is contained in it. We use the same definition on shorter dyadic time blocks; there the admissible widths are at most the block length in finest-cell units. In particular, membership is stronger than having an event in the plank. The mass $n_B$ of a plank is the sum of the inherited weights of its contained indices.

Fixed enlargements of cells, moving event locations a bounded number of fine cells, and replacing the bounded matrix patch by any other fixed-size patch do not affect the exponents below. For the last assertion, translate the matrix center and rescale space by a fixed constant. Both the old and new support grids admit bounded-overlap coverings by the other grid, and plank tests transfer after bounded enlargement and a fixed finite covering.

Convention 2.1 (Exponent notation). For positive quantities on a sequence with $N \to\infty$, write

A factor $N^{o(1)}$ has logarithm $o(\log N)$, uniformly over the objects being compared in that statement. Subsequence selection is allowed. Every contradiction will compare exponents separated by a fixed positive amount before the resolution tends to infinity. All these comparisons are homogeneous in the weights.

A refinement deletes events and then deletes empty indices; retained indices keep their original weights. It is extensive if its event mass is at least $N^{-o(1)}\mathcal{I}$. A restriction of loss exponent at most $e$ retains at least $N^{-e+o(1)}\mathcal{I}$.

Definition 2.2 (Uniform temporal profile). An admissible profile is a nondecreasing 1-Lipschitz function $F : [0,1] \to[0,1]$ with $F(0)=0$. A sequence of indexed configurations has uniform profile $F$ if, uniformly over every retained index and every occupied dyadic time block containing $K$ fine bins,

The uniform error also applies over the scale range $0 \le\kappa\le1$. In this case $k=\exp N^{1-\alpha}$, where $\alpha=F(1)$.