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

TL;DR

This paper demonstrates that under mild conditions, Borel sets in certain metric spaces can be mapped onto ultrametric spaces or cubes via nearly Lipschitz maps, linking geometric measure properties with mapping behavior.

Contribution

It introduces a method to map Borel sets onto ultrametric spaces and cubes using nearly Lipschitz maps, extending previous results to more general metric spaces and self-similar sets.

Findings

Borel sets in analytic metric spaces can be mapped onto ultrametric spaces via Lipschitz bijections.

A Borel set in Euclidean space maps onto a cube if and only if it cannot be covered by sets of lower Hausdorff dimension.

The results extend to self-similar sets and nearly H"older maps, broadening the scope of geometric measure theory applications.

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 $\phi:N\to Z$ 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.