What is proved, what changed in 2025-2026, and what is open. This is an exposition of the Kakeya set conjecture in Euclidean space and over finite fields, with complete proofs of the elementary results (the planar Córdoba-type bound, Dvir's finite-field theorem, and several small finite-field facts), a dated record of the dimension bounds, and a tested, dependency-free Python companion (kakeya.py) that checks Kakeya sets in AG(n, q), builds the two explicit constructions below, and computes every finite-field bound quoted here.
New in version 1.1: the first exact finite-field Kakeya number in dimension three over a field with more than two elements. The smallest Kakeya set in F₃³ has exactly 13 points (section 7), proved by an exhaustive search that runs in seconds with no solver. The release also gives smaller Kakeya sets than the known constructions in F₃⁴, F₃⁵, F₄³ and F₄⁴, with every set published and re-checked.
By MV2 of Munim, Inc. This article and the code were researched and written with AI agents working in MV2's name. Every statement below is either proved here or cited to a source; section 10 separates what is proved from what is open, and section 12 lists what is new here. Corrections are welcome by issue or pull request.
Version 1.1, 2026-10-14. Text and data CC BY 4.0, code Apache-2.0.
Related problem record: Justin Sun Prize JSP-000015, "Kakeya dimension conjecture (general dimension)" (MathDB #315983).
1. The question
A Kakeya set (or Besicovitch set) in R^n is a compact set containing a unit line segment in every direction.
Kakeya set conjecture. Every Kakeya set in R^n has Hausdorff dimension n (and hence also Minkowski dimension n).
The Justin Sun Prize catalogue states it as: "Must a set containing a unit line segment in every direction have full Hausdorff dimension in its Euclidean ambient space?" Its title says "general dimension", so the open content in October 2026 is n ≥ 4 (section 10).
History in three lines. Kakeya (1917) asked for the smallest area of a planar region in which a unit needle can be turned around. Besicovitch (1919, published again in 1928) showed that a planar set containing a unit segment in every direction can have measure zero, which answers the needle problem in a surprising way: the area can be made arbitrarily small. Since such sets can be null, the right question is how large they must be in the sense of dimension, and that is the conjecture above. It sits underneath a chain of central problems in harmonic analysis: the Kakeya maximal function conjecture, the Fourier restriction conjecture (JSP-001041 in the same pilot) and the Bochner-Riesz conjecture. Bochner-Riesz implies restriction (Tao 1999), restriction implies the Kakeya maximal conjecture, and that implies the set conjecture. Fefferman's 1971 disc multiplier theorem used Besicovitch's construction to show that the ball multiplier is unbounded on L^p(R^n) for p ≠ 2, n ≥ 2; that is where Kakeya sets entered Fourier analysis.
2. Two notions of size
For a bounded set E ⊂ R^n let N(E, δ) be the least number of δ-balls covering E.
- Minkowski (box) dimension: the lower and upper limits of log N(E, δ) / log(1/δ) as δ → 0. Equivalently, the δ-neighbourhood E_δ has volume about δ^(n − dim).
- Hausdorff dimension: the infimum of s for which E can be covered by balls of radii r_i with Σ r_i^s arbitrarily small. It is always at most the lower Minkowski dimension, so a Hausdorff lower bound is the stronger statement.
Both are computed through δ-tubes: a δ-tube is a cylinder of length 1 and radius δ around a unit segment (in the plane, a 1 × δ rectangle). If K is a Kakeya set, its δ-neighbourhood contains a δ-tube in every direction, so a lower bound on |∪ T| for δ-tubes pointing in δ-separated directions gives a lower bound on |K_δ| and hence on the Minkowski dimension. Hausdorff bounds need the same estimate when each tube is replaced by a fixed fraction of itself (Zahl's survey, Conjecture 1.2(B)).
3. The plane, with a proof
Theorem (Davies 1971). Every Kakeya set in R² has Hausdorff dimension 2.
Here is the complete argument for the Minkowski version, which shows the mechanism that every later proof tries to extend.
Lemma 1. Two δ-tubes in R² whose directions make an angle θ ∈ (0, π/2] intersect in a set of area at most δ²/sin θ ≤ (π/2) δ²/θ.
Proof. Each tube lies in an infinite strip of width δ. Two strips of width δ at angle θ intersect in a parallelogram whose sides have length δ/sin θ and whose height is δ, so its area is δ²/sin θ. Finally sin θ ≥ 2θ/π on [0, π/2]. ∎
Theorem 2. Let T_1, ..., T_N be δ-tubes in R² with directions θ_i = iδ, N = ⌊π/δ⌋. Then |∪ T_i| ≥ Nδ / (1 + π(1 + ln N)), which is about 1/log(1/δ).
Proof. Let f = Σ 1_{T_i}. By Cauchy-Schwarz, (∫ f)² ≤ |supp f| · ∫ f², so |∪ T_i| ≥ (Σ |T_i|)² / Σ_{i,j} |T_i ∩ T_j|. Here Σ |T_i| = Nδ, since each tube is a 1 × δ rectangle. For fixed i and j ≠ i, put m = min(|i − j|, N − |i − j|); since Nδ ≤ π, the angle between the two tubes (taken in (0, π/2]) is at least mδ, and each value m = 1, ..., ⌊N/2⌋ occurs for at most two values of j. By Lemma 1, Σ_j |T_i ∩ T_j| ≤ δ + 2 Σ_{m ≤ N/2} (π/2)δ²/(mδ) ≤ δ(1 + π(1 + ln N)). Summing over i and dividing gives the bound. ∎
Since the δ-neighbourhood of a planar Kakeya set contains such a family (it contains a 1 × δ rectangle around each of its unit segments), |K_δ| ≳ 1/log(1/δ), and N(K, δ) ≳ δ^(−2)/log(1/δ): the Minkowski dimension is 2. The logarithm is a real loss, not an artefact: Besicovitch's sets show that |K_δ| can tend to 0. Córdoba (1977) turned this computation into the sharp L² bound for the planar Kakeya maximal function.
Why the argument stops at n = 2. In R^n the same computation only controls pairs of tubes. Tubes can now meet in complicated configurations (many tubes through one point lying near a plane, or arranged along a ruled surface), and pair counting alone gives only the dimension bound (n + 1)/2 (Drury 1983; Christ, Duoandikoetxea and Rubio de Francia 1986). Every later record comes from a new way to rule out such configurations.
4. The record in R^n, dated
| Year | Who | Result |
|---|---|---|
| 1971 | Davies | dimension 2 in R² |
| 1977 | Córdoba | sharp Kakeya maximal estimate in R² |
| 1983, 1986 | Drury; Christ-Duoandikoetxea-Rubio de Francia | (n + 1)/2 in R^n |
| 1991 | Bourgain | first improvement on (n + 1)/2 ("bush" argument) |
| 1995 | Wolff | (n + 2)/2 in R^n (hairbrush; 5/2 in R³, 3 in R⁴) |
| 1999 | Bourgain | (13n + 12)/25 via additive combinatorics ("arithmetic" method) |
| 1999 | Katz-Tao | Minkowski (4n + 3)/7, Hausdorff (6n + 5)/11 |
| 2000 | Katz-Łaba-Tao | upper Minkowski 5/2 + ε in R³ |
| 2001 | Łaba-Tao | upper Minkowski (n + 2)/2 + ε_n for n ≥ 4 |
| 2002 | Katz-Tao | Minkowski n/α + (α − 1)/α with α³ − 4α + 2 = 0 (α = 1.675...); Hausdorff (2 − √2)(n − 4) + 3 |
| 2018 | Guth-Zahl, with Zahl and Katz-Rogers (polynomial Wolff axioms) | 3 + 1/40 in R⁴ |
| 2019 | Katz-Zahl | Hausdorff 5/2 + ε in R³ |
| 2021 | Katz-Zahl (planebrush) | 3.059 in R⁴ |
| 2019-2021 | Zahl; Hickman-Rogers-Zhang | polynomial-partitioning ("broad") estimates; new bounds in some dimensions n ≥ 7 |
| 2025 | Wang-Zahl | dimension 3 in R³: the conjecture holds for n = 3 |
| 2026 | Rai Choudhuri (preprint, not refereed) | 13/4 − (12951 − 54√10593)/41972 = 3.0738... in R⁴ |
The 2002 Katz-Tao bounds are quoted from Theorem 1.1 of their paper; the paper notes that its Hausdorff bound is better than its Minkowski bound for n ≤ 23. The R⁴ value 3.0738 is Theorem 1 of arXiv 2610.12480 (submitted 30 August 2026), which combines a slightly improved Guth-Zahl trilinear estimate, obtained by applying Wang-Zahl inside thickened neighbourhoods of varieties, with the Katz-Zahl planebrush. The author says some AI assistance was used. We give its number exactly as stated and have not checked the proof.
For n ≥ 5 we do not attempt a per-dimension table: the best bounds come from a mix of the Katz-Tao arithmetic bounds and the polynomial-method estimates, and Zahl's 2025 survey (ICM 2026) does not tabulate them. All known bounds for large n grow like cn with c well below 1 (for example n/1.675 for the Minkowski dimension), so the gap to n widens with the dimension.
5. What Wang and Zahl proved, and how
Theorem (Wang-Zahl 2025). Every Kakeya set in R³ has Hausdorff dimension 3 and Minkowski dimension 3.
The proof is spread over three papers: the sticky case ("Sticky Kakeya sets and the sticky Kakeya conjecture", J. Amer. Math. Soc. 39 (2026) 515-585), an Assouad-dimension paper (Invent. Math. 241 (2025)), and the general case, arXiv 2502.17655 (February 2025), which reduces the general case to the sticky one by induction on scales. Guth, Wang and Zahl posted a streamlined proof in January 2026 (arXiv 2601.14411); Guth's Séminaire Bourbaki survey appeared in April 2026 (arXiv 2604.03416), and Alex Cohen circulated detailed checking notes the same month. Hong Wang received a 2026 Fields Medal at the International Congress in Philadelphia on 23 July 2026, with this work cited. As of the sources we read, the main 2025 paper is still listed as a preprint.
The shape of the argument, following Guth's survey:
- Stickiness. A family of δ-tubes is sticky if, at every coarser scale ρ, the tubes group into about ρ^(1−n) fat ρ-tubes, the minimum possible. Sticky families are the hardest case.
- Grains. In a worst-case sticky family the union locally looks like a product of a small square with a fractal set A: the tubes stack into "grains" (Katz-Łaba-Tao).
- The obstruction is complex conjugation. Following grains around a closed cycle forces an approximate identity A + s(x)y + s̃(y)x ≈ A. The model solution is R + x̄y + ȳx = R inside C, and it really happens: the Heisenberg group in C³ gives a tube arrangement over C that is sticky and grainy with the 5/2 behaviour. This is why every method that ignores the order structure of R stalls at 5/2.
- Sum-product kills it over R. Bourgain's discretised sum-product theorem says R has no such approximate subrings, which rules out the conjugation structure; repeating a small gain across scales gives the sharp result.
- Reduction to sticky. In a non-sticky family, the densest convex sets containing many tubes are handled by induction on scales (the "convex Wolff axioms" theorem, Guth's Theorem 1.2).
What it does not prove: the Kakeya maximal function conjecture in R³ (Wang-Zahl say so explicitly, and Zahl's survey lists obstructions to adapting the argument; a special case for SL₂ lines was proved by Green, Harris and Ou, Analysis & PDE 19 (2026)), restriction in R³, or anything in R⁴ beyond the improved bounds of section 4.
6. The finite-field problem, with proofs
Wolff (1999) proposed a model problem. A Kakeya set in F_q^n is a set K containing a full line {b + td : t ∈ F_q} for every direction d ≠ 0. Write K(n, q) for the minimum size. Wolff asked whether |K| ≥ c_n q^n.
Theorem (Dvir 2008). Every Kakeya set in F_q^n has at least C(q + n − 1, n) points. In particular |K| ≥ q^n/n!.
Proof. The polynomials in n variables of degree at most q − 1 form a space of dimension C(q − 1 + n, n). If |K| were smaller, a nonzero such P would vanish on K (more unknowns than linear conditions). Let d = deg P, so 1 ≤ d ≤ q − 1 (d = 0 would make P a nonzero constant vanishing on the nonempty set K), and let P_d be its top-degree homogeneous part. For a direction v, K contains a line b + tv, so the one-variable polynomial t ↦ P(b + tv) has degree at most d < q and q roots; it is identically zero. Its coefficient of t^d is P_d(v), so P_d(v) = 0 for every v ≠ 0, and P_d(0) = 0 because d ≥ 1. So P_d is a nonzero polynomial of degree less than q in each variable that vanishes on all of F_q^n, which is impossible (induct on n: a nonzero one-variable polynomial of degree < q has fewer than q roots). ∎
The proof is ten lines. Nothing like it is known in R^n, where δ-tubes are not algebraic objects, so the finite-field model turned out to be much softer than the real problem. The sharp constant came later:
- Dvir, Kopparty, Saraf and Sudan (2009, method of multiplicities): |K| ≥ q^n/2^n.
- Bukh and Chao (Discrete Analysis 2021): |K| ≥ q^n / (2 − 1/q)^(n−1), so the density is at least 2^(1−n) asymptotically. This is sharp: constructions with density 2^(1−n) + O(1/q) exist (Dvir-Kopparty-Saraf-Sudan, Saraf-Sudan; the parabola set below).
So for fixed n the first-order behaviour is settled: K(n, q) = (2^(1−n) + o(1)) q^n as q → ∞ (through odd q by the parabola set; Kyureghyan, Müller and Wang give constructions of the same density for even q).
The plane is solved exactly. Blokhuis and Mazzocca (2008) proved K(2, q) = q(q + 1)/2 + (q − 1)/2 for odd q; for even q the minimum is q(q + 1)/2 (the smallest examples are classified: Blokhuis, De Boeck, Mazzocca and Storme 2014). Dover and Mellinger found by exhaustive search every attainable size of a Kakeya set in the Desarguesian planes of order q ≤ 9. For even q, Dvir's bound C(q + 1, 2) = q(q + 1)/2 is therefore exact in the plane.
Four further elementary facts, each checked by the companion code:
(a) The counting bound. K(n, q) ≥ the least s with s(s − 1) ≥ q(q^n − 1). Proof. K contains (q^n − 1)/(q − 1) lines in distinct directions; two distinct lines share at most one point, so the C(q, 2) pairs of points on each line are disjoint sets of pairs inside the C(s, 2) pairs of K. ∎ It is weak for large q but exact for q = 2 and n ≤ 4.
(b) Products. If A ⊂ F_q^n and B ⊂ F_q^m are Kakeya sets, so is A × B, hence K(n + m, q) ≤ K(n, q) K(m, q). Proof. For a direction (d, e) with d, e ≠ 0, take lines a + td ⊂ A and b + te ⊂ B; then (a + td, b + te) ⊂ A × B. If e = 0 use (a + td, b) for any b ∈ B, and symmetrically if d = 0. ∎
(c) Over F₂ a Kakeya set is a difference basis. A line in F₂^n is a pair {x, x + d}, so K is Kakeya exactly when every nonzero vector is a sum a + b with a, b ∈ K: K is a difference basis of the group C₂^n. Banakh and Gavrylkiv computed the difference sizes of all abelian groups of order below 96, which gives
| n | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| K(n, 2) | 2 | 3 | 5 | 6 | 10 | 14 |
| counting bound (a) | 2 | 3 | 5 | 6 | 9 | 12 |
(Acta Sci. Math. (Szeged) 85 (2019), Tables 1 and 3, where the group is written (C₂)^n). For n = 4 the optimum is a perfect Sidon set: {0, e₁, e₂, e₃, e₄, e₁ + e₂ + e₃ + e₄} has 15 pairwise sums, and they are exactly the 15 nonzero vectors. The companion code reproves n ≤ 4 by exhaustion. Their Proposition 7.3 gives, for n ≥ 2, the two-sided bound √2 < K(n, 2)/2^(n/2) < 3/√2 (and < 2 for even n).
(d) Two explicit constructions.
- The parabola set (q odd). Write points as (x, t) with x ∈ F^(n−1). For each b ∈ F^(n−1) take the line {((tb_i − b_i²)_i, t) : t ∈ F}, of direction (b, 1), and add the hyperplane t = 0 for the directions (b, 0). For t ≠ 0 the slice at height t is {x : t²/4 − x_i is a square for every i}, because t²/4 − (tb − b²) = (b − t/2)². So the set has (q − 1)((q + 1)/2)^(n−1) + q^(n−1) points, density 2^(1−n) + O(1/q), and for n = 2 it has (q² + 2q − 1)/2 points: exactly the Blokhuis-Mazzocca minimum.
- The Smyth set. K = (F^)^*n ∪ {0, 1}^n, with (q − 1)^n + 2^n − 1 points. This size was reported by Clifford Smyth in a 2008 comment on Tao's blog; here is a proof that this set works. Given a direction d, let Z be the set of coordinates where d is zero (not all of them) and 1_Z the 0/1 vector supported on Z. On the line 1_Z + td the point t = 0 is 1_Z ∈ {0, 1}^n, and for t ≠ 0 every coordinate is nonzero: 1 on Z, td_i ≠ 0 off Z. ∎ For q = 3 it gives K(n, 3) ≤ 2^(n+1) − 1, which beats the parabola set's 2^n + 3^(n−1) for every n ≥ 3, and no 0/1 vector other than (1, ..., 1) can be dropped from it (checked by the tests for n = 3).
7. Small cases: what is known exactly
| (n, q) | points | best lower bound (which) | best verified upper bound (which) |
|---|---|---|---|
| (2, q) | q² | exact: Blokhuis-Mazzocca | exact |
| (n, 2), n ≤ 6 | 2^n | exact: Banakh-Gavrylkiv | exact |
| (3, 3) | 27 | 13 (new, this repository) | 13 (new); Smyth 15 |
| (4, 3) | 81 | 18 (Bukh-Chao) | 27 (new); Smyth 31 |
| (5, 3) | 243 | 32 (Bukh-Chao) | 56 (new); Smyth 63 |
| (3, 4) | 64 | 21 (Bukh-Chao) | 28 (new); Smyth 34 |
| (4, 4) | 256 | 48 (Bukh-Chao) | 64 (new); Smyth 96 |
| (3, 5) | 125 | 39 (Bukh-Chao) | 53 (AlphaEvolve formula; found again here); parabola 61 |
| (3, 7) | 343 | 100 (Bukh-Chao) | 128 (found here); parabola 145 |
Every number in the table is recomputed by python kakeya.py bounds N Q, every upper bound is a set that the tests build or load and check line by line, and the lower bound 13 is re-proved by the tests (below). The 2025 AlphaEvolve paper (Georgiev, Gómez-Serrano, Tao and Wagner, arXiv 2511.02864) gives constructions in F_p³ for primes p ≡ 1 (mod 4) of size p³/4 + 7p²/8 − 1/8, which is 53 at p = 5; our solver found a 53-point set in F₅³ independently, and did not beat it in 25 minutes.
Theorem (new). K(3, 3) = 13: the smallest Kakeya set in F₃³ has exactly 13 points.
Upper bound. The set
(0,0,0) (1,0,0) (2,0,0) (1,1,0) (0,0,1) (2,0,1) (0,1,1)
(1,1,1) (0,2,1) (2,2,1) (0,0,2) (2,1,2) (0,2,2)
contains a line in each of the 13 directions of F₃³ (over F₃ a line is a 3-term progression b, b + d, b + 2d):
| direction | line in the set | direction | line in the set |
|---|---|---|---|
| (1,0,0) | (0,0,0) (1,0,0) (2,0,0) | (1,1,0) | (0,0,1) (1,1,1) (2,2,1) |
| (0,1,0) | (0,0,1) (0,1,1) (0,2,1) | (1,2,0) | (0,2,1) (1,1,1) (2,0,1) |
| (0,0,1) | (0,0,0) (0,0,1) (0,0,2) | (1,1,1) | (0,0,2) (1,1,0) (2,2,1) |
| (0,1,1) | (0,0,0) (0,1,1) (0,2,2) | (1,1,2) | (0,2,1) (1,0,0) (2,1,2) |
| (0,1,2) | (2,0,0) (2,1,2) (2,2,1) | (1,2,1) | (0,2,2) (1,1,0) (2,0,1) |
| (1,0,1) | (0,0,2) (1,0,0) (2,0,1) | (1,2,2) | (0,2,2) (1,1,1) (2,0,0) |
| (1,0,2) | (0,1,1) (1,1,0) (2,1,2) |
Lower bound. Suppose K is a Kakeya set with at most 12 points. Translating K so that its line in direction e₁ becomes the axis {(0,0,0), (1,0,0), (2,0,0)} keeps it a Kakeya set of the same size, and adding points keeps it Kakeya, so there would be a 12-point Kakeya set containing the axis. There are C(24, 9) = 1,307,504 such 12-point sets, and none of them contains a line in all 13 directions. kakeya_sets_through_axis checks every one with bitmasks, with no solver, in a few seconds; test_k33_is_thirteen runs it on every test run. Separately, two independent integer-programming solvers (OR-Tools CP-SAT, with and without symmetry fixing, and HiGHS through SciPy) each proved 13 optimal in MV2's run kakeya-min-v1. ∎
So the sequence K(n, 3) begins 3, 7, 13. The minimum set is not a "lift" of a planar one in the obvious way: its slices by the last coordinate have 4, 6 and 3 points, and none of them is a planar Kakeya set (that needs 7 points).
Why this is new. Lund, Saraf and Wolf (arXiv 1609.01048) wrote that no exact value is known in any dimension other than 2. The q = 2 values follow from Banakh and Gavrylkiv's difference bases (section 6(c)); for q ≥ 3 and n ≥ 3 we found no exact value in the papers cited here, in the AlphaEvolve paper (which treats asymptotic constructions), on OEIS, or in answer to the 2008 question on Tao's blog about the smallest subset of F₃^n with a 3-term progression in every direction. If an earlier computation exists, we would like to cite it.
New upper bounds. The same run found Kakeya sets of 27 points in F₃⁴, 56 in F₃⁵, 28 in F₄³ and 64 in F₄⁴. Each is smaller than every explicit construction quoted in this article: the Smyth set (31, 63, 34, 96), the parabola set (43 and 113 for q = 3), products of smaller optimal sets (for example 3 · 13 = 39 and 7 · 7 = 49 in F₃⁴, 10 · 10 = 100 in F₄⁴), and the Kyureghyan-Müller-Wang construction for q = 4 (46 in F₄³ by their formula (2)). They are not claimed optimal: the solvers stopped at their time limits. CP-SAT reported a lower bound of 24 for F₄³; we do not use it, because it was not certified independently. The 27-point set in F₃⁴ has slices of 13, 6 and 8 points by its last coordinate, which suggests a construction lifting F₃³ sets to F₃⁴; MV2 is testing that next.
All sets, solver logs and timings are in data/kakeya-min-v1/ (results.json: for every case the runs of each solver and a certificate, the smallest set found as a list of coordinates; results.txt: one line per case). solve.py reproduces the run (pip install -r requirements-solve.txt, then python solve.py --out-dir out; about 3.3 hours on 4 cores).
Open, as far as we can tell: every other exact value with n ≥ 3 and q ≥ 3. The smallest open cases are K(4, 3) ∈ [18, 27] and K(3, 4) ∈ [21, 28].
8. Variants that are open or recently settled
- Z/NZ and p-adic rings. Over the rings Z/NZ the analogue of Dvir's theorem was proved by Dhar (Advances in Combinatorics 2024, general N), and Arsovski proved the p-adic Kakeya conjecture (J. Amer. Math. Soc. 37 (2024)).
- Nikodym and Furstenberg sets. The Furstenberg set conjecture in the plane was settled by Ren and Wang (2024), building on Orponen-Shmerkin; Guth cites it as a hint that sticky cases are key.
- Plany sets in F_q⁴. Łaba, Rai Choudhuri and Zahl (Res. Math. Sci. 13 (2026)) give a finite-field exposition of the Katz-Zahl planebrush.
9. Claimed proofs
Short claimed proofs of the general conjecture appear from time to time. A claim deserves attention only if it handles the Heisenberg obstruction of section 5 (any argument that works verbatim over C proves something false) and recovers the Wang-Zahl theorem as a special case. We do not list individual unrefereed claims.
10. Proved versus open, as of 2026-10-14
Proved.
- n = 2: every Kakeya set has dimension 2 (Davies 1971); the maximal conjecture holds (Córdoba 1977).
- n = 3: every Kakeya set has Hausdorff and Minkowski dimension 3 (Wang-Zahl 2025; three papers, one published in JAMS 2026, the main one still a preprint; a streamlined proof exists).
- All n: Hausdorff dimension ≥ max((n + 2)/2, (2 − √2)(n − 4) + 3) and the Katz-Tao Minkowski bound; in R⁴, at least 3.059 (Katz-Zahl, published) and 3.0738 (2026 preprint).
- Finite fields: |K| ≥ q^n/(2 − 1/q)^(n−1), sharp to first order; the plane exactly; q = 2 exactly for n ≤ 6; K(3, 3) = 13 (new here, by exhaustive computation that anyone can rerun).
Open.
- The Kakeya set conjecture for every n ≥ 4 (this is the open content of JSP-000015).
- The Kakeya maximal function conjecture for every n ≥ 3.
- The restriction and Bochner-Riesz conjectures for n ≥ 3.
- Exact finite-field values K(n, q) for n ≥ 3, q ≥ 3 other than K(3, 3); the second-order term in K(n, q).
11. Why this matters beyond the problem
Kakeya estimates control how much wave packets pointing in different directions can overlap, which is the core difficulty in restriction theory, Bochner-Riesz summation, local smoothing for the wave equation and decoupling. Dvir's finite-field theorem became a standard tool in theoretical computer science (randomness mergers and extractors) and set off the modern wave of the polynomial method in combinatorics, which went on to the joints theorem and the cap-set bound.
12. What is original here
- The exact value K(3, 3) = 13 (section 7), with a solver-free proof that reruns in seconds, an explicit 13-point set and a table of its 13 lines. As far as we found, it is the first exact finite-field Kakeya number in dimension at least 3 over a field with more than two elements.
- Smaller Kakeya sets in F₃⁴ (27), F₃⁵ (56), F₄³ (28) and F₄⁴ (64) than the constructions we found in the literature, all published as coordinate lists and re-checked.
- A dated, sourced record of the dimension bounds through October 2026, including the August 2026 R⁴ preprint and the 2026 publication and recognition of the Wang-Zahl work.
- Self-contained proofs of the planar bound, Dvir's theorem, the counting bound, the product lemma, the parabola construction and its exact size, and the Smyth construction (whose proof we did not find written out).
- The identification K(n, 2) = Δ[C₂^n] with difference bases, which imports exact values for n ≤ 6 from group theory. We did not find this identification stated in the Kakeya literature we read; it is elementary.
- The small-case table of section 7 with every bound recomputed and every upper bound verified by machine, and a precise statement of which exact values are open.
13. Using the companion code
kakeya.py needs only Python 3.8+ and no packages.
python kakeya.py bounds 3 3 # counting, Dvir, Bukh-Chao, parabola and Smyth sizes
python kakeya.py smyth 3 3 # print the 15-point Kakeya set in F_3^3
python kakeya.py parabola 3 5 # print the 61-point Kakeya set in F_5^3
python kakeya.py check 3 3 set.txt # check a set: one point per line, coordinates separated by spaces
python kakeya.py brute 2 5 # exact minimum by brute force (small planes only)
python kakeya.py exhaust 3 3 12 # no Kakeya set of 12 points in F_3^3 (a few seconds)
python kakeya.py exhaust 3 3 13 # print a 13-point one
python kakeya.py records # the exact values and bounds of section 7
python -m unittest -v test_kakeya
Fields supported: F_p for primes p, and F₄, F₈, F₉, F₁₆. The tests check the field axioms, the line structure of AG(n, q), the plane minima for q ≤ 5 by brute force and by the exhaustive search, K(3, 2) = 5 and K(4, 2) = 6 by exhaustion, the equivalence (c) on all 256 subsets of F₂³, both constructions in every case used in section 7 (line by line, with their exact sizes), the product lemma, the exhaustive search against a slow independent check, K(3, 3) = 13, and every one of the 18 solver certificates in data/kakeya-min-v1/results.json.
solve.py (needs ortools, scipy and numpy, see requirements-solve.txt) is the integer-programming search that produced the data: one 0/1 variable per point and per line, exactly one chosen line per direction, every point of a chosen line in the set, minimise the size; solved with CP-SAT and HiGHS, and every set returned is re-checked with is_kakeya.
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