Vertical projections in the Heisenberg group via cinematic functions and point-plate incidences

TL;DR

This paper establishes dimension preservation under vertical projections in the Heisenberg group for sets with certain Hausdorff dimensions, using cinematic functions and incidence geometry techniques.

Contribution

It extends known results to higher dimensions and the case of dimension 3, introducing a duality and incidence approach for the Heisenberg group.

Findings

Dimension inequality holds for almost every projection direction.

Introduces a duality transforming the problem into a point-plate incidence question.

Partial results obtained for sets with Hausdorff dimension between 2.5 and 3.

Abstract

Let $\{\pi_{e} \colon \mathbb{H} \to \mathbb{W}_{e} : e \in S^{1}\}$ be the family of vertical projections in the first Heisenberg group $\mathbb{H}$. We prove that if $K \subset \mathbb{H}$ is a Borel set with Hausdorff dimension $\dim_{\mathbb{H}} K \in [0,2] \cup \{3\}$, then $$ \dim_{\mathbb{H}} \pi_{e}(K) \geq \dim_{\mathbb{H}} K $$ for $\mathcal{H}^{1}$ almost every $e \in S^{1}$. This was known earlier if $\dim_{\mathbb{H}} K \in [0,1]$. The proofs for $\dim_{\mathbb{H}} K \in [0,2]$ and $\dim_{\mathbb{H}} K = 3$ are based on different techniques. For $\dim_{\mathbb{H}} K \in [0,2]$, we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl. To handle the case $\dim_{\mathbb{H}} K = 3$, we introduce a point-line duality between horizontal lines and conical lines in $\mathbb{R}^{3}$. This allows us to transform the Heisenberg problem into a point-plate incidence question in $\mathbb{R}^{3}$. To solve the latter, we apply a Kakeya inequality for plates in $\mathbb{R}^{3}$, due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets $K \subset \mathbb{H}$ with $\dim_{\mathbb{H}} K \in (5/2,3)$.