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

Let $\{{\pi }_e\text{\hspace{0.17em}⁣}:\mathbb{H}\to {\mathbb{W}}_e:e\in S^1\}$ 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 $${\dim }_{\mathbb{H}}{\pi }_e(K)\ge {\dim }_{\mathbb{H}}K$$ 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 ${\dim }_{\mathbb{H}}K\in (5\mathrm{/}2,3)$.