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Any counterexample to the three-dimensional Kakeya conjecture must be a grainy set.
Without evidence. The same authors as the 2025 resolution paper, writing earlier, credit graininess to Guth's paper and planiness to Bennett, Carbery and Tao, without argument of their own at this point. The statement is by leading specialists and is consistent with their later paper, but the two papers are one voice on this point. Sep 15, 2026

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Abstract0 passages

1 Introduction0 passages

2 Discretization0 passages

3 Extremal families of tubes and multi-scale structure0 passages

4 Planiness and graininess0 passages

5 The global grains slope function is C 2 C^{2}0 passages

6 The slope function has large slope0 passages

7 A C 2 C^{2} projection theorem0 passages

8 Proof of Theorem3 passages

References0 passages

License: CC BY 4.01 passage

Sticky Kakeya sets and the sticky Kakeya conjecture1 passage

Abstract1 passage

1 Introduction60 passages · 1 marked

1.2 The sticky Kakeya conjecture versus the Kakeya conjecture

In this section, we will informally discuss what Theorem 1.1 tells us about the Kakeya conjecture. In particular, how close does Theorem 1.1 get us to proving the Kakeya conjecture in ℝ 3 \mathbb{R}^{3} ? Katz and Tao’s arguments from [ 43 ] begin by conjecturing that a (hypothetical) counter-example K ⊂ ℝ 3 K\subset\mathbb{R}^{3} to the Kakeya conjecture in ℝ 3 \mathbb{R}^{3} must have the structural properties stickiness, planiness, and graininess. The results from [ 1 ] and [ 22 ] show that such a counter-example must be plainy and grainy, but it is unclear whether K K must be sticky. To reformulate the remaining part of Katz and Tao’s conjecture in our language, we need the following definition.

Any counterexample to the three-dimensional Kakeya conjecture must be a grainy set.SupportedReading · this claim’s Steward

Without evidence. The same authors as the 2025 resolution paper, writing earlier, credit graininess to Guth's paper and planiness to Bennett, Carbery and Tao, without argument of their own at this point. The statement is by leading specialists and is consistent with their later paper, but the two papers are one voice on this point.

2 Discretization17 passages

Next, let38 passages

𝒬 k ≲ ( 2 k ) dim H K + ε / 2 . \#\mathcal{Q}_{k}\lesssim(2^{k})^{\dim_{H}K+\varepsilon/2}.1 passage

For each ℓ ∈ L ′ \ell\in L^{\prime} , we have37 passages

3 Extremal families of tubes and multi-scale structure29 passages

𝕋 ′ ( p ) ≤ ( ρ / δ ) σ + ε for all p ∈ ℝ n . \#\mathbb{T}^{\prime}(p)\leq(\rho/\delta)^{\sigma+\varepsilon}\quad\textrm{for all}\ p\in\mathbb{R}^{n}.10 passages

For every measurable X ⊂ ℝ n X\subset\mathbb{R}^{n} we have3 passages

By ( 3.4 ), we have6 passages

Property (c) is straightforward: by ( 3.4 ) we have23 passages

𝕋 4 [ T ~ ] ( p ) ≤ ( ρ / δ ) σ n + ε 2 / 3 , \#\mathbb{T}_{4}[\tilde{T}](p)\leq(\rho/\delta)^{\sigma_{n}+\varepsilon^{2}/3},11 passages

𝕋 4 [ T ~ ] ≲ δ − η ( ρ / δ ) n − 1 , \#\mathbb{T}_{4}[\tilde{T}]\lesssim\delta^{-\eta}(\rho/\delta)^{n-1},14 passages

Comparing ( 3.13 ) and ( 3.15 ), we have36 passages

{ T ∈ 𝕋 ( p ) : ∠ ( v , dir ( T ) < δ ε } ≤ δ − σ n + ε σ n / 2 . \#\{T\in\mathbb{T}(p)\colon\angle(v,\operatorname{dir}(T)<\delta^{\varepsilon}\}\leq\delta^{-\sigma_{n}+\varepsilon\sigma_{n}/2}.7 passages

4 Planiness and graininess36 passages

⁡ { T ∈ 𝕋 1 ( p ) : ∠ ⁡ ( v , dir ⁡ ( T ) ) ≤ δ ε / 4 } ≤ δ − σ + ( ε / 4 ) σ / 2 ≤ δ − σ + 3 η . \#\{T\in\mathbb{T}_{1}(p)\colon\angle(v,\operatorname{dir}(T))\leq\delta^{\varepsilon/4}\}\leq\delta^{-\sigma+(\varepsilon/4)\sigma/2}\leq\delta^{-\sigma+3\eta}.5 passages

⁡ { ( T 1 , T 2 , T 3 ) ∈ ( 𝕋 2 ( p ) ) 3 : | v 1 ∧ v 2 ∧ v 3 | ≥ δ σ − 10 η } ≥ δ − 3 σ + 3 η , \#\{(T_{1},T_{2},T_{3})\in(\mathbb{T}_{2}(p))^{3}\colon|v_{1}\wedge v_{2}\wedge v_{3}|\geq\delta^{\sigma-10\eta}\}\geq\delta^{-3\sigma+3\eta},2 passages

For each p ∈ E 𝕋 3 p\in E_{\mathbb{T}_{3}} , we have0 passages

⁡ { ( T 1 , T 2 , T 3 ) ∈ 𝕋 3 ( p ) 3 : | v 1 ∧ v 2 ∧ v 3 | < δ σ − 10 η } ≥ μ 3 − δ − 3 σ + 3 η ≥ 1 2 μ 3 . \#\{(T_{1},T_{2},T_{3})\in\mathbb{T}_{3}(p)^{3}\colon|v_{1}\wedge v_{2}\wedge v_{3}|<\delta^{\sigma-10\eta}\}\geq\mu^{3}-\delta^{-3\sigma+3\eta}\geq\frac{1}{2}\mu^{3}.2 passages

⁡ { ( T 1 , T 2 , T 3 ) ∈ 𝕋 3 ( p ) 3 : | v i ∧ v j | < δ ε / 4 } ≤ μ 2 δ − σ + 3 η ≤ 1 100 μ 3 . \#\{(T_{1},T_{2},T_{3})\in\mathbb{T}_{3}(p)^{3}\colon|v_{i}\wedge v_{j}|<\delta^{\varepsilon/4}\}\leq\mu^{2}\delta^{-\sigma+3\eta}\leq\frac{1}{100}\mu^{3}.19 passages

⁡ { T ~ ∈ T ~ 1 : T ~ associated to Q } ⪆ δ − σ + 2 η ( ρ / δ ) σ + ε 1 ⪆ δ 2 η + ε 1 ρ − σ . \#\{\tilde{T}\in\tilde{T}_{1}\colon\tilde{T}\ \textrm{associated to}\ Q\}\gtrapprox\frac{\delta^{-\sigma+2\eta}}{(\rho/\delta)^{\sigma+\varepsilon_{1}}}\gtrapprox\delta^{2\eta+\varepsilon_{1}}\rho^{-\sigma}.63 passages

This is illustrated in Figure 4 . Thus47 passages

Then by ( 4.24 ) and ( 4.25 ), we have74 passages

We will re-write ( 4.36 ) as4 passages

5 The global grains slope function is C 2 C^{2}50 passages

By ( 5.6 ), we have35 passages

𝒬 ≳ ρ − 5 / 2 + σ / 2 + 15 η . \#\mathcal{Q}\gtrsim\rho^{-5/2+\sigma/2+15\eta}.8 passages

Define20 passages

Define4 passages

ℋ = # 𝒲 † = # 𝒬 ′ ≳ ( ρ 1 / 2 ) 16 η − 3 . \#\mathcal{H}=\#\mathcal{W}^{{\dagger}}=\#\mathcal{Q}^{\prime}\gtrsim(\rho^{1/2})^{16\eta-3}.49 passages

Each interval I ∈ ℐ j I\in\mathcal{I}_{j} satisfies10 passages

A vertical trapezoid is a set of the form47 passages

Let4 passages

Define5 passages

We have9 passages

Thus7 passages

6 The slope function has large slope17 passages

{ S ∈ 𝒮 : S ∩ F ≠ ∅ } ≥ δ ε τ ρ − 1 / 2 , \#\{S\in\mathcal{S}\colon S\cap F\neq\emptyset\}\geq\delta^{\varepsilon}\tau\rho^{-1/2},40 passages

See Figure 55 passages

𝕋 S ≳ δ − 2 ( # 𝒮 ) − 1 ≳ δ 11 η − 2 ρ 1 − σ 2 , \#\mathbb{T}_{S}\gtrsim\delta^{-2}(\#\mathcal{S})^{-1}\gtrsim\delta^{11\eta-2}\rho^{\frac{1-\sigma}{2}},5 passages

{ T ∈ 𝕋 S : | dir ( T ) − v | ≤ δ } ≥ δ C 1 η ρ − σ / 2 ≥ δ C 1 η − σ ε , \#\{T\in\mathbb{T}_{S}\colon|\operatorname{dir}(T)-v|\leq\delta\}\geq\delta^{C_{1}\eta}\rho^{-\sigma/2}\geq\delta^{C_{1}\eta-\sigma\varepsilon},32 passages

7 A C 2 C^{2} projection theorem14 passages

For all f , g ∈ ℱ , f,g\in\mathcal{F}, we have4 passages

⁡ ( F ∩ B ) ≤ δ − ε ( r / δ ) for all balls B ⊂ C 2 ( I ) of diameter r . \#\big(F\cap B\big)\leq\delta^{-\varepsilon}(r/\delta)\quad\textrm{for all balls}\ B\subset C^{2}(I)\ \textrm{of diameter}\ r.1 passage

Then34 passages

8 Proof of Theorem 5.245 passages

n ℋ [ h I ] ≥ c ∏ i ∉ I # A i . \#n_{\mathcal{H}}[h_{I}]\geq c\prod_{i\not\in I}\#A_{i}.7 passages

n ℋ ′ [ h I ] < ε 2 k # ℋ ∏ i ∈ I # A i , \#n_{\mathcal{H}^{\prime}}[h_{I}]<\frac{\varepsilon}{2^{k}}\frac{\#\mathcal{H}}{\prod_{i\in I}\#A_{i}},55 passages

We compute76 passages

If ( 8.18 ) fails, then by Cauchy-Schwarz we have0 passages

𝒲 ≳ ( δ α | log ⁡ δ | ) 4 ( # G 1 ) 2 ( # G 2 ) \#\mathcal{W}\gtrsim\Big(\frac{\delta^{\alpha}}{|\log\delta|}\Big)^{4}(\#G_{1})^{2}(\#G_{2})18 passages

We will select76 passages

But54 passages

{ ( a , b 1 , b 2 ) ∈ ℋ : | a | ≥ δ 5 η , | b 1 − b 2 | ≥ δ 5 η } ≥ 1 2 # ℋ . \#\{(a,b_{1},b_{2})\in\mathcal{H}\colon|a|\geq\delta^{5\eta},\ |b_{1}-b_{2}|\geq\delta^{5\eta}\}\geq\frac{1}{2}\#\mathcal{H}.8 passages

References53 references

Instructions for reporting errors7 passages

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