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

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

“…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 $$\mathbb {R}^{3}$$.…”

“…\end{equation*}$$Thus, $$V_{y} = e_{y}^{\perp }$$. Moreover, the lines $$\ell _{y} := \operatorname{span}(e_{y})$$ are all contained in a $$45^{\circ }$$ rotated copy of the light cone $$$$\begin{equation*} \mathcal {C} := \lbrace (x,y,t) \in \mathbb {R}^{3} : t^{2} = x^{2} + y^{2}\rbrace , \end{equation*}$$$$see [2, Section 2.2] for the details. This implies that the projections $$\lbrace \pi _{V_{y}}\rbrace _{y \in \mathbb {R}}$$ satisfy the curvature condition [3, (1)].…”