Transversality of isotropic projections, unrectifiability and Heisenberg groups

TL;DR

This paper establishes the transversality of isotropic projections in Euclidean space, applies it to projection theorems, and derives Hausdorff dimension estimates, with implications for Heisenberg groups.

Contribution

It proves the transversality of isotropic projections and extends classical projection theorems to this setting, also providing dimension estimates for projections.

Findings

Isotropic projections in orm a transversal family.

The Besicovitch-Federer projection theorem applies to isotropic projections.

Almost sure Hausdorff dimension estimates for isotropic projections.

Abstract

We show that the family of $m$-dimensional isotropic projections in $\R^{2n}$ 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 $E \subset \R^{2n}$. These results may also be applied to gain information on the horizontal projections of the Heisenberg group $\H^n$.