Sticky Kakeya sets and the sticky Kakeya conjecture

TL;DR

This paper proves that sticky Kakeya sets in R^3 must have Hausdorff and Minkowski dimension 3, advancing the Kakeya program by isolating a structurally rigid, self-similar class of extremal configurations. The authors develop a discretized, multi-scale framework (lines, tubes, shadings) and introduce σ_n to quantify near-extremal behavior, linking it to dimension bounds for sticky Kakeya sets. A central innovation is planiness and graininess, yielding a Lipschitz plane map and a grain-based decomposition of tube unions; the global grains slope function is shown to be C^2, enabling a strong projection theory via twisted projections and a C^2 projection theorem. By combining multilinear Kakeya, hypergraph-density arguments, and Kaufman-type projection estimates, the authors rule out near-extremal counterexamples, establishing σ_3=0 and confirming the sticky Kakeya conjecture in dimension 3 with potential implications for the broader Kakeya problem.

Abstract

A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate multi-scale self-similarity, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. We propose a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We prove this conjecture in three dimensions.

Figures (6)

• Figure 1: The unit rescaling of $(\mathbb{T},Y)_\delta$. For clarity, only a few cubes in $Y(T)$ have been drawn.
• Figure 2: $V(p)$ is almost orthogonal to $\operatorname{dir}(\tilde{T}_Q)$ and $\operatorname{dir}(\tilde{T}_Q')$.
• Figure 3: The set $E_{\mathbb{T}'} \cap Q$ (blue) at scale $\rho$, and the unit vector $V(Q)$
• Figure 4: The union of the sets $Q\cap \tilde{Y}_1(T_i),\ i=1,\ldots,k$ fill out most of the slab (blue) inside $Q$.
• Figure 5: The rectangular prisms $\{P_S\colon S\in\mathcal{S}\}$, and the tubes from $\mathbb{T}$ (blue lines) that intersect them. For clarity, only a few tubes (blue lines) have been drawn.

Theorems & Definitions (117)

• Definition 1.1
• Conjecture 1.1
• Theorem 1.1
• Definition 1.2
• Conjecture 1.2
• Remark 1.1
• Conjecture 1.3
• Definition 2.1
• Proposition 2.1
• Lemma 2.2