Bi-Lipschitz pieces between manifolds (Q268244)

Let \(X\) and \(Y\) be Ahlfors \(s\)-regular, linearly locally contractible, complete, oriented, topological \(d\)-manifolds, \(s>0\), \(d\in \mathbb{N}\), and \(Y\) has \(d\)-manifold weak tangents. Let \(I\) be a dyadic \(0\)-cube in \(X\), and \(z: I\to Y\) a Lipschitz map. Then, for every \(\varepsilon>0\), there are measurable subsets \(E_{j}\subset I\), \(j=1, 2, \dots, n\), such that \(z|_{E_j}\) is bi-Lipschitz, and \(\left|z\left(I\setminus\bigcup_{j=1}^{n}E_{j}\right)\right|<\varepsilon|I|\).