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A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$

Larry Guth, Hong Wang, Joshua Zahl

TL;DR

This work presents a streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$ by reducing the general problem to a sticky Kakeya framework. It develops a robust toolkit of convex-set shadings, uniform-tube decompositions, and maximal-density factoring to organize incidences, and it proves key implications between Katz–Tao and Frostman type bounds. The main results show that KT bounds imply Frostman bounds (Main Lemma 1) and, under both KT and Frostman hypotheses, yield a gain which improves KT bounds (Main Lemma 2), enabling an iterative bootstrap to the Kakeya conclusion. The approach leverages a multi-scale, incidence-geometric analysis of slabs and planks, together with a sticky/non-sticky dichotomy, to deduce large coverage by unit tubes and control multiplicities, ultimately establishing that Besicovitch sets in $\mathbb{R}^3$ have Minkowski and Hausdorff dimension $3$.

Abstract

We present a streamlined and simplified proof of the Kakeya set conjecture in $\mathbb{R}^3$.

A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$

TL;DR

This work presents a streamlined proof of the Kakeya set conjecture in by reducing the general problem to a sticky Kakeya framework. It develops a robust toolkit of convex-set shadings, uniform-tube decompositions, and maximal-density factoring to organize incidences, and it proves key implications between Katz–Tao and Frostman type bounds. The main results show that KT bounds imply Frostman bounds (Main Lemma 1) and, under both KT and Frostman hypotheses, yield a gain which improves KT bounds (Main Lemma 2), enabling an iterative bootstrap to the Kakeya conclusion. The approach leverages a multi-scale, incidence-geometric analysis of slabs and planks, together with a sticky/non-sticky dichotomy, to deduce large coverage by unit tubes and control multiplicities, ultimately establishing that Besicovitch sets in have Minkowski and Hausdorff dimension .

Abstract

We present a streamlined and simplified proof of the Kakeya set conjecture in .
Paper Structure (30 sections, 27 theorems, 263 equations)

This paper contains 30 sections, 27 theorems, 263 equations.

Table of Contents

  1. Introduction to the Kakeya problem
  2. Notation and Definitions
  3. Convex sets and shadings
  4. Pigeonholing and $\lessapprox$ notation
  5. Uniform sets of tubes
  6. Partial Kakeya estimates
  7. Multiplicity bounds for larger values of $\Delta_{\max}(\mathbb{T})$ or $C_F(\mathbb{T})$
  8. Organizing convex sets
  9. Multiplicities, refinements, and factoring
  10. Multiplicity estimates for slabs and planks
  11. Slabs
  12. Planks
  13. A Katz-Tao multiplicity estimate
  14. A Frostman multiplicity estimate
  15. Tubes that factor through planks
  16. ...and 15 more sections

Key Result

Theorem 1.1

For every $\beta > 0$, there is some $\eta > 0$ so that the following holds for all sufficiently small $\delta>0$. If $\mathbb{T}$ is a set of $\delta$-tubes in $\mathbb{R}^3$ with $\Delta_{max}(\mathbb{T}) \leq \delta^{-\eta}$, and $Y$ is a shading of $\mathbb{T}$ with $\lambda(\mathbb{T}, Y) \geq

Theorems & Definitions (77)

  • Theorem 1.1: Kakeya set conjecture in $\mathbb{R}^3$
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • ...and 67 more