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

## TL;DR

This paper investigates the Hausdorff dimension of intersections of isotropic projections of sets in Euclidean space and the Heisenberg group, revealing conditions under which these intersections have positive measure and specific dimensional properties.

## Contribution

It introduces new results on the dimensions of intersections under isotropic projections and extends these findings to the setting of the Heisenberg group.

## Key findings

- Positive measure of intersections for sets with dimension greater than m
- Existence of sets with controlled dimension intersecting under isotropic projections
- Extension of results to 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.