Mapping Borel sets onto balls and self-similar sets by Lipschitz and nearly Lipschitz maps

Abstract

If $X$ is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure $\mu$ on $X$ there is a set $N\subseteq X$ such that $\mu (N)>0$, an ultrametric space $Z$ and a Lipschitz bijection $\varphi :N\to Z$ whose inverse is nearly Lipschitz, i.e., $\beta$-Hölder for all $\beta <1$. As an application it is shown that a Borel set in a Euclidean space maps onto $[0,1{]}^n$ by a nearly Lipschitz map if and only if it cannot be covered by countably many sets of Hausdorff dimension strictly below $n$. The argument extends to analytic metric spaces satisfying the mild condition. Further generalization replaces cubes with self-similar sets, nearly Lipschitz maps with nearly Hölder maps and integer dimension with arbitrary finite dimension.