An improved bound on the Hausdorff dimension of sticky Kakeya sets in $\mathbb{R}^4$

Abstract

Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction. The Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered by Katz-Łaba-Tao in their 1999 breakthrough paper on the Kakeya problem. Then Wang-Zahl formalized the definition of a sticky Kakeya set, and proposed a special case of the Kakeya conjecture for such sets. Specifically this conjecture states that sticky Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$ and Wang-Zahl went on to prove the conjecture for $n=3$. A planebrush is a geometric object which is a higher dimensional analogue of Wolff's hairbrush. Using the planebrush argument, Katz-Zahl showed that Kakeya sets in $\mathbb{R}^4$ have Hausdorff dimension at least 3.059. If we restrict our attention to sticky Kakeya sets, we can improve upon this bound by combining the planebrush result with additional stickiness property. To be precise, we will show in this paper that sticky Kakeya sets in $\mathbb{R}^4$ have dimension at least 3.25.

Figures (8)

• Figure 1: Trilinear case
• Figure 2: "Weakly" plany case. Only two $\delta$-tubes shown for clarity.
• Figure 3: Tubes at scale $\rho$ are plany in this case.
• Figure 4: Discretizing sticky Kakeya sets
• Figure 5: Unit rescaling. For clarity, only a handful of cubes drawn.

Theorems & Definitions (16)

• Theorem 1.2
• Remark 2.4
• proof
• Remark 2.8
• Theorem 3.3
• Remark 3.5
• Remark 3.6
• proof
• Theorem 3.10
• proof : Proof of Theorem \ref{['sti2']} assuming Theorem \ref{['goal']}.