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Kakeya volume conjecture in R^d

Prove that for dimension d ≥ 2, any choice of translates x_θ of unit-radius, length-R tubes θ* pointing in all directions (θ running over 1/R caps on S^{d−1}) yields union volume at least C_{ε,d} R^{d − ε} for all ε > 0, independent of the translates.

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Background

The classical Kakeya problem asks whether one can pack one unit-length segment in every direction into a set of arbitrarily small measure; in higher dimensions the conjecture predicts essentially full dimension, reflected in near-maximal union volume.

This conjecture is deeply intertwined with Fourier restriction, oscillatory integrals, and decoupling, and it underlies Bourgain’s barrier connecting Dirichlet large-values to geometric measure theory.

References

Conjecture Fix a dimension d ≥ 2 and let θ and θ* be as above. For every ε > 0, there is a constant C_{ε, d} so that for every choice of x_θ,

| ⋃θ (θ* + xθ) | ≥ C_{ε,d} R{d − ε}.

— Large value estimates in number theory, harmonic analysis, and computer science (2503.07410 - Guth, 10 Mar 2025) in Section 9 (A barrier related to the Kakeya problem)