Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (2502.17655v1)

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.

Volume Estimates for Unions of Convex Sets, and the Kakeya Set Conjecture in Three Dimensions

This paper addresses the Kakeya set conjecture in three dimensions, focusing on volume estimates for unions of convex sets and their implications for the dimension of Kakeya sets. The authors, Hong Wang and Joshua Zahl, bring significant advancements in understanding the structural properties of these sets and their relationships to volume estimates, offering a resolution to the Kakeya set conjecture in $\mathbb{R}^3$.

Kakeya Set Conjecture Background

The Kakeya set conjecture posits that in $\mathbb{R}^n$, a Kakeya set—a compact set containing a unit line segment in every direction—must have full Hausdorff and Minkowski dimension $n$. While Davies resolved this conjecture for $n=2$, it remained open for three or more dimensions. This paper provides essential insights into the Kakeya conjecture specifically for $\mathbb{R}^3$, combining techniques from convex geometry and harmonic analysis.

Main Results and Techniques

The core contribution of the paper is a proof that every Kakeya set in $\mathbb{R}^3$ indeed possesses Minkowski and Hausdorff dimension 3. This is achieved through refined volume estimates for unions of $\delta$-tubes, which are small neighborhoods around unit line segments, under specific non-clustering conditions.

The authors establish a new breadth of analysis with the following key elements:

1. Non-Clustering Conditions: By ensuring that not many tubes can be contained in a common convex set, they show that the union must have almost maximal volume, leading to the desired dimensionality results for Kakeya sets.
2. Broadness and Grains Decomposition: Adapting Guth's grains decomposition, the authors develop a "two-scale" grains decomposition, examining how sets decompose into rectangular prisms—or "grains"—at varying scales and orientations. This decomposition plays a crucial role in understanding the density and volume distributions within the Kakeya set.
3. Multi-Scale Analysis: The interplay between scales allows for an exhaustive analysis that ultimately rules out counter-examples to the conjecture in three dimensions. If a Kakeya set failed to meet the conjectured dimension, it would imply an arrantly pathological structure at some scale, which the paper adeptly handles via recursive and iterative arguments.

Theoretical and Practical Implications

The resolution of the Kakeya set conjecture in $R^3$ has widespread implications in harmonic analysis, combinatorics, and related fields. The understanding of non-clustering in convex sets informs problems dealing with sphere packings and point distributions, which are pivotal in wireless communications, error-correcting codes, and more.

Further, the volume estimates established here lay foundational groundwork for analyzing problems of similar nature in higher dimensions or under additional geometric constraints.

Future Directions and Challenges

While the Kakeya conjecture for $\mathbb{R}^3$ is resolved, the extension to higher dimensions remains open and presents numerous challenges. The analytical techniques developed in this paper may provide pathways or structural heuristics that inspire new lines of attack in those settings.

Moreover, understanding precise volume estimates in more restricted or complex settings, such as those involving additional curvature or topological constraints, remains a frontier for future research. Building on the robust framework of geometric and analytic techniques, future inquiries could push the boundaries of what's achievable with current theoretical paradigms.

In summary, Wang and Zahl provide a substantial leap forward in resolving foundational conjectures in geometric analysis, and their work offers new tools and perspectives for tackling future problems both within and beyond geometric measure theory.