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Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

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arxiv 2502.17655 v1 pith:MEYXHPM3 submitted 2025-02-24 math.CA math.MG

Hong Wang , Joshua Zahl This is my paper · ORCID
classification math.CAmath.MG
keywords tubesconvexkakeyamathbbsetsvolumealmostcommon
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abstract

We study sets of $\delta$ tubes in $\mathbb{R}^3$, with the property that not too many tubes can be contained inside a common convex set $V$. We show that the union of tubes from such a set must have almost maximal volume. As a consequence, we prove that every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Curved Kakeya problems and the projective geometry of paths

    math.CA 2026-07 conditional novelty 8.0 of 10

    Totally geodesic curved Kakeya families in R^3 are exactly projectively flat ones; under semi-algebraicity, dimension 2 occurs exactly under a compression condition.

  2. A smooth Alpert testing characterization of convolution type for the Fourier extension conjecture on paraboloids

    math.CA 2025-12 unverdicted novelty 8.0 of 10

    Proof of the Fourier extension conjecture on the paraboloid in d>2 by decomposing smooth Alpert projections, applying a bilinear reduction, and bounding the resulting oscillatory integral with periodic amplitude via l...

  3. On the Dimension of the CC-geodesic Kakeya sets in the first Heisenberg group

    math.CA 2026-07 accept novelty 7.0 of 10

    CC-geodesic Kakeya sets in the first Heisenberg group have sharp Heisenberg Hausdorff dimension 3 (even when compact), not the predicted 4.

  4. Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications

    math.CO 2026-03 accept novelty 7.0 of 10

    The refined-direction horizontal Kakeya operator on H₁(F_q) has exact ℓᵘ→ℓᵛ norm growth max{1/v, 1−1/u, 2/v−1/u, 1+2/v−3/u}, driven by a sharp, Fourier-proved ℓ²→ℓ² bound q^{1/2}.

  5. Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions

    math.NT 2025-07 conditional novelty 6.0 of 10

    A new zero-density estimate for Dirichlet L-functions with exponent 7/3, improving Huxley's 12/5, is proven by extending the Guth-Maynard large value method to character-twisted Dirichlet polynomials.

  6. A Survey of the Kakeya conjecture, 2000-2025

    math.CA 2025-12 conditional novelty 2.0 of 10

    An authoritative survey of the Kakeya conjecture (2000–2025), organized around the recent proof that Besicovitch sets in R^3 have full Hausdorff and Minkowski dimension 3.

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