Intersection of projections and slicing theorems for the isotropic Grassmannian and the Heisenberg group

Abstract

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of $\mathbb{R}^{2n}$, as well as dimension of intersections of sets with isotropic planes. It is shown that if $A$ and $B$ are Borel subsets of $\mathbb{R}^{2n}$ of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of $A$ and $B$ under orthogonal projections onto these planes have positive Hausdorff $m$-measure. In addition, if $A$ is a measurable set of Hausdorff dimension greater than $m$, then there is a set $B\subset\mathbb{R}^{2n}$ with $\dim B\leq m$ such that for all $x\in\mathbb{R}^{2n}\setminus B$ there is a positive measure set of isotropic m-planes for which the translate by $x$ of the orthogonal complement of each such plane, intersects $A$ on a set of dimension $\dim A-m$. These results are then applied to obtain analogous results on the $n^{th}$ Heisenberg group.