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

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 $\phi:N\to Z$ whose inverse is nearly Lipschitz, i.e., $\beta$-H\"older 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 H\"older maps and integer dimension with arbitrary finite dimension.

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