On the dimension of Kakeya sets in the first Heisenberg group

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 { x o y } \{xoy\} -plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

“…Proof of Theorem Without loss of generality, we may assume that $$\mathcal {L} = \ell (P)$$ is a Borel set of lines, that is, $$P \subset \mathbb {R}^{3}$$ is a Borel set. For the full details of this reduction, see [7, Section 3] or [1, Theorem 7.9]. The idea is that we can first replace $$\cup \mathcal {L}$$ by a $$G_{\delta }$$‐set $$G \supset \cup \mathcal {L}$$ without affecting $$\dim _{\mathrm{H}}(\cup \mathcal {L})$$.…”

“…This interval would (be chosen to) consist of points $$y \in \mathbb {R}$$ with the property that the plane $$\mathbb {W}_{y}$$ intersects a family of segments corresponding to a $$(\dim _{\mathrm{H}}P - \epsilon )$$‐dimensional Borel subset $$P^{\prime } \subset P$$. We refer the reader to [7, Section 3] for a very similar argument. Clearly, (3.1) follows from (3.2) by the ‘Fubini inequality’ for Hausdorff measures (hence dimension), see [1, Theorem 5.8] or [8, Theorem 7.7].…”

“…The first one is a point‐line duality between horizontal lines and conical lines in $$\mathbb {R}^{3}$$, namely translates of lines contained in the light cone $$\lbrace (x,y,t) : t^{2} = x^{2} + y^{2}\rbrace$$. This duality was formalised in our paper [2], although it was already implicit in the work [7] of Liu. Using this point‐line duality, Kakeya‐type problems for horizontal lines can be transformed into projection problems in $$\mathbb {R}^{3}$$.…”

“…Without loss of generality, we may assume that  = 𝓁(𝑃) is a Borel set of lines, that is, 𝑃 ⊂ ℝ 3 is a Borel set. For the full details of this reduction, see[7, Section 3] or[1, Theorem 7.9]. The idea is that we can first replace ∪ by a 𝐺 𝛿 -set 𝐺 ⊃ ∪ without affecting dim H (∪).…”

Vertical projections in the Heisenberg group for sets of dimension greater than 3

“…They introduced a method of point-plate incidences, and proved this in the case dim A = 3 by using a square function estimate for the cone of Guth, Wang and Zhang [9] to control the average L 2 norms of pushforwards of discretised 3-dimensional measures. The point-line duality principle they used is due to Liu [10].…”