A sufficient condition for having big pieces of bilipschitz images of subsets of euclidean space in Heisenberg groups

TL;DR

This paper extends a Euclidean geometric measure theory result to the Heisenberg group, providing a condition under which subsets contain large bilipschitz images of Euclidean spaces.

Contribution

It introduces a Carleson-type condition for Heisenberg groups that guarantees big pieces of bilipschitz images, extending prior Euclidean results.

Findings

Established a sufficient condition for bilipschitz images in Heisenberg groups

Generalized Euclidean geometric measure theory to non-commutative groups

Provided a framework for approximating sets by Heisenberg k-planes

Abstract

In this article we extend a euclidean result of David and Semmes to the Heisenberg group by giving a sufficient condition for a $k$-Ahlfors-regular subset to have big pieces of bilipschitz images of subsets of $\R^k$. This Carleson type condition measures how well the set can be approximated by the Heisenberg $k$-planes at different scales and locations. The proof given here follow the paper of David and Semmes.