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

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