[Paper Review] On the Dimension of Kakeya Sets in the First Heisenberg Group

Jiayin Liu|arXiv (Cornell University)|Jun 22, 2021

Advanced Harmonic Analysis Research4 citations

TL;DR

This paper establishes that every Kakeya set in the first Heisenberg group has Heisenberg Hausdorff dimension at least 3, using a duality principle that maps horizontal line segments to a subset in R⁴, followed by Marstrand-type projection theorems and co-area inequalities. The bound is sharp, as the {xoy}-plane achieves this dimension.

ABSTRACT

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the $\{xoy\}$-plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

Motivation & Objective

• To determine the infimum of the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group.
• To address the challenge that standard Euclidean projection arguments fail due to the non-Lipschitz nature of the Korányi metric.
• To establish a sharp lower bound on the dimension, showing 3 is optimal.
• To investigate whether Kakeya sets in the Heisenberg group can have zero 3-dimensional Heisenberg Hausdorff measure, posing an open problem.

Proposed method

• Define Kakeya sets in the Heisenberg group as sets containing a unit line segment in every direction within the horizontal plane.
• Encode each horizontal line segment in a Kakeya set E as a quadruple in R⁴, forming a subset L(E) ⊂ R⁴.
• Use a duality principle to relate intersections of E with vertical planes to projections of L(E) onto R³.
• Apply a Marstrand-type projection theorem in R³ (Käenmäki-Orponen-Venieri, 2014) to analyze the dimension of projected sets.
• Utilize the co-area inequality by Eilenberg-Harrold, Jr. to relate the Hausdorff dimension of slices to the overall dimension.
• Leverage the 1-Lipschitz property of the t-axis projection φ to transfer dimension bounds from R³ to the Heisenberg group.

Experimental results

Research questions

• RQ1 What is the minimal possible Heisenberg Hausdorff dimension of a Kakeya set in the first Heisenberg group?
• RQ2 Can the lower bound of 3 for the Heisenberg Hausdorff dimension of Kakeya sets be improved or is it sharp?
• RQ3 Is there a Kakeya set in the first Heisenberg group with zero 3-dimensional Heisenberg Hausdorff measure?
• RQ4 How does the non-Euclidean geometry of the Heisenberg group affect the dimension theory of Kakeya sets compared to Euclidean space?
• RQ5 To what extent do standard projection and slicing techniques from Euclidean geometric measure theory extend to the Heisenberg setting?

Key findings

• Every Kakeya set in the first Heisenberg group has Heisenberg Hausdorff dimension at least 3.
• The lower bound of 3 is sharp, as demonstrated by the {xoy}-plane, which is a Kakeya set with Heisenberg Hausdorff dimension exactly 3.
• For almost every c in a compact interval [c₀, c₀ + 1/4], the slice {x = c} ∩ E has Heisenberg Hausdorff dimension at least 2.
• For any α < 2, the α-dimensional Heisenberg Hausdorff measure of the slice {x = c} ∩ E is infinite.
• The 3-dimensional Heisenberg Hausdorff measure of the set F = E ∩ {(x,y,t) ∈ H | x ∈ [c₀, c₀ + 1/4]} is infinite, implying dimₕ(H)(F) ≥ 3.
• The proof relies on a duality between horizontal segments and projections in R⁴, combined with advanced tools from geometric measure theory in R³.