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Kakeya maximal function conjecture in R^3

Prove that for all ε > 0, the inequality (1.1) holds with the exponent K = 3 in three dimensions: for every sufficiently small δ > 0, every set T of δ-tubes in R^3 satisfying the stated non-clustering condition, and measurable subsets Y(T) ⊂ T with |Y(T)| ≥ λ|T|, the union obeys |⋃_{T∈𝒯} Y(T)| ≥ C(ε) δ^{ε} λ^{3} ∑_{T∈𝒯} |T|.

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Background

The authors prove a weaker three-dimensional bound sufficient to establish the Kakeya set conjecture in R3, but explicitly state they do not resolve the Kakeya maximal function conjecture with the sharp exponent K=3. Cordoba proved the conjecture in R2; the optimal three-dimensional case remains open.

A complete resolution of the Kakeya maximal function conjecture in R3 would yield sharp bounds related to several central problems in harmonic analysis, including restriction, Bochner–Riesz, and oscillatory integral estimates.

References

The Kakeya maximal function conjecture asserts that for each $>0$, Inequality maximalFnBdpLamba is true for $K=3$. ... While we do not resolve the Kakeya maximal function conjecture in $R3$, the weaker statement given in Theorem \ref{maximalFnBdpThm} is nonetheless sufficient to obtain Theorem \ref{kakeyaSetThm}.

maximalFnBdpLamba:

$\Big|\bigcup_{T\inT}Y(T)\Big|\geq \delta^ \lambda^K \sum_{T\inT}|T|. $

— Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (2502.17655 - Wang et al., 24 Feb 2025) in Introduction, Section 1 (after Theorem 1.2)