Transversality of isotropic projections, unrectifiability, and Heisenberg groups

Abstract

We show that the family of $m$-dimensional isotropic projections in $\mathbb{R}^{2n}$ is transversal. As an application we show that the Besicovitch–Federer projection theorem holds for isotropic projections. We also use transversality to obtain almost sure estimates on the Hausdorff dimension of isotropic projections of subsets $E \subset \mathbb{R}^{2n}$. These results may also be applied to gain information on the horizontal projections of the Heisenberg group $\mathbb{H}^n$.