Kakeya.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 4 contain sorry, 0 declared as axioms

Kakeya problem

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

def Kakeya.IsKakeya A set S in ℝⁿ is called a Kakeya set if it contains a unit line segment in every direction line 40 source
Source documentation
A set S in ℝⁿ is called a Kakeya set if it contains a unit line segment in every direction. For simplicity, we omit the compactness assumption here. For a discussion on the equivalence of definitions with and without compactness, see this paper.

L4DEL explanation
A set S in ℝⁿ is called a Kakeya set if it contains a unit line segment in every direction. For simplicity, we omit the compactness assumption here. For a discussion on the equivalence of definitions with and without compactness, see this paper.
Sources and attribution: Pinned Lean source (1646ca16afd6)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin
Proof uses: Eq, Norm.norm, OfNat.ofNat, ENNReal, Exists, LE.le, affineSegment, HAdd.hAdd
36 instances, coercions and auxiliary constants left out.
```
def IsKakeya {n : ℕ} (S : Set (ℝ^n)) : Prop :=
…
```
theorem Kakeya.isKakeya_closedBall A trivial example: the closed ball of radius 1 in ℝⁿ is a Kakeya set line 47 source
Source documentation
A trivial example: the closed ball of radius 1 in ℝⁿ is a Kakeya set.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Kakeya.IsKakeya, Metric.closedBall, EuclideanSpace, Real, Fin, OfNat.ofNat, ENNReal
Proof uses: Eq, Norm.norm, Exists.intro, LE.le, Set, affineSegment, HAdd.hAdd, Membership.mem, And, Set.Icc, AffineMap, AffineMap.lineMap, Eq.mpr, id, Eq.trans, Dist.dist, HSMul.hSMul, congrArg, congrFun', zero_add and 10 more
124 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem isKakeya_closedBall (n : ℕ) : IsKakeya (closedBall (0 : ℝ^n) 1) := by
…
```
Mentions: Kakeya.IsKakeya
def Kakeya.KakeyaSetConjectureDim The Kakeya set conjecture in dimension n: the statement that every Kakeya set in ℝⁿ has Hausdorff dimension n line 57 source
Source documentation
The Kakeya set conjecture in dimension n: the statement that every Kakeya set in ℝⁿ has Hausdorff dimension n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set, EuclideanSpace, Real, Fin, Kakeya.IsKakeya, Eq, ENNReal, dimH, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.
```
def KakeyaSetConjectureDim (n : ℕ) : Prop :=
…
```
theorem Kakeya.kakeya_set_conjecture The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$ sorry line 62 source
Source documentation
The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$.

depends on sorry what this means
Statement uses: Nat, GT.gt, OfNat.ofNat, Kakeya.KakeyaSetConjectureDim
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kakeya_set_conjecture (n : ℕ) (hn : n > 0) :
    KakeyaSetConjectureDim n := by
…
```
Mentions: Kakeya.KakeyaSetConjectureDim
theorem Kakeya.kakeya_2d The two-dimensional case, proved by Davies [Da71] sorry line 73 source
Source documentation
The two-dimensional case, proved by Davies [Da71].
[Da71] Davies, R. O., Some remarks on the Kakeya problem. Math. Proc. Cambridge Philos. Soc. 69 (1971), no. 3, 417–421.

depends on sorry what this means
Statement uses: Kakeya.KakeyaSetConjectureDim, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kakeya_2d : KakeyaSetConjectureDim 2 := by
…
```
Mentions: Kakeya.KakeyaSetConjectureDim
theorem Kakeya.kakeya_3d The three-dimensional case, proved by Wang, Zahl [WaZa25] sorry line 83 source
Source documentation
The three-dimensional case, proved by Wang, Zahl [WaZa25].
[WaZa25] Wang, H. and Zahl, J., Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. arXiv preprint, arXiv:2502.17655, 2025.

depends on sorry what this means
Statement uses: Kakeya.KakeyaSetConjectureDim, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kakeya_3d : KakeyaSetConjectureDim 3 := by
…
```
Mentions: Kakeya.KakeyaSetConjectureDim
def Kakeya.IsKakeyaFinite A finite field variant of the Kakeya problem considers subsets of 𝔽_qⁿ that contain a line in every direction line 91 source
Source documentation
A finite field variant of the Kakeya problem considers subsets of 𝔽_qⁿ that contain a line in every direction.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Fintype, Nat, Finset, Fin
Proof uses: Ne, OfNat.ofNat, Exists, Membership.mem, HAdd.hAdd, HSMul.hSMul
17 instances, coercions and auxiliary constants left out.
```
def IsKakeyaFinite {F : Type*} [Field F] [Fintype F] {n : ℕ} (S : Finset (Fin n → F)) : Prop :=
…
```
theorem Kakeya.kakeya_finite The finite field Kakeya conjecture asserts that any Kakeya set in 𝔽_qⁿ has size at least c_n · qⁿ for some constant c_n depending only on n sorry line 107 source
Source documentation
The finite field Kakeya conjecture asserts that any Kakeya set in 𝔽_qⁿ has size at least c_n · qⁿ for some constant c_n depending only on n. This was first proved by Dvir [Dv08]. The best known bound to date, due to Bukh and Chao [BuCh21], establishes that any Kakeya set in 𝔽_qⁿ has size at least qⁿ / (2 - 1/q)^(n - 1). The dimension is positive, as in kakeya_set_conjecture; for n = 0 the empty set is a Kakeya set and the bound would read 1 ≤ 0.
[Dv08] Dvir, Z., On the size of Kakeya sets in finite fields. Journal of the American Mathematical Society 22 (2009), no. 4, 1093–1097. [BuCh21] Bukh, B. and Chao, T.-W., Sharp density bounds on the finite field Kakeya problem. Discrete Analysis 26 (2021), 9 pp.

depends on sorry what this means
Statement uses: Field, Fintype, Nat, LT.lt, OfNat.ofNat, Finset, Fin, Kakeya.IsKakeyaFinite, LE.le, Rat, HDiv.hDiv, HPow.hPow, Fintype.card, HSub.hSub, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem kakeya_finite {F : Type*} [Field F] [Fintype F] {n : ℕ} (hn : 0 < n)
    (K : Finset (Fin n → F)) (hK : IsKakeyaFinite K) :
    card F ^ n / (2 - 1 / card F : ℚ) ^ (n - 1) ≤ K.card := by
…
```
Mentions: Kakeya.IsKakeyaFinite