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

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 ${\mathbb{R}}^k$. 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.