On low-dimensional uniform rectifiability in Heisenberg groups

Refining an earlier result due to Hahlomaa, we provide a new Carleson-type condition for $k$-regular sets in the Heisenberg group ${\mathbb{H}}^n$ to have big pieces of Lipschitz images of subsets of ${\mathbb{R}}^k$ for $1\le k\le n$. Our approach passes via the corona decompositions by normed spaces, recently introduced by Bate, Hyde, and Schul. Along the way, we prove implications between several notions of quantitative rectifiability for low-dimensional sets in ${\mathbb{H}}^n$.