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

TL;DR

This paper proves that in the Heisenberg group, sets with dimension greater than 3 almost surely have vertical projections of positive area, using advanced incidence and Kakeya inequalities.

Contribution

It introduces a novel application of point-plate incidence and Kakeya inequalities to analyze projections in the Heisenberg group for high-dimensional sets.

Findings

Vertical projections of sets with dimension > 3 have positive area almost surely.

The proof combines incidence methods with Kakeya inequalities in a novel way.

Related results on intersections with horizontal lines are also provided.

Abstract

It is shown that vertical projections in the Heisenberg group of sets of dimension strictly greater than 3 almost surely have positive area. The proof uses the point-plate incidence method introduced by F\"assler and Orponen, and also uses a similar approach to a recent maximal inequality of Zahl for fractal families of tubes. It relies on the endpoint trilinear Kakeya inequality in $\mathbb{R}^3$. Some related results are given on generic intersections with horizontal lines.