On Lipschitz Mappings Onto a Square

Abstract

The following problem was posed by Laczkovich [5]: Let $E \subseteq \mathbb{R}{\mathbb{R}}^{d}$ (d ≥ 2) be a set with positive Lebesgue measure λ ^d(E) > 0. Does there exist a Lipschitz mapping $f : {\mathbb{R}}^{d} \rightarrow Q = {[0,1]}^{d}$, such that f(E) = Q? Preiss [6] answered this question affirmatively for d = 2:

This paper was originally published in 1997. The present version is a minor revision from 2012, which includes some references to newer developments.

It is my pleasure to thank University College of London for enabling my visit, during which this research was conducted. The visit was also supported by the Humboldt Foundation.

Preiss in fact proves a slightly stronger statement, namely that f can be taken such that $f({\mathbb{R}}^{2} \setminus E)$ is countably rectifiable (i.e. it can be covered by a countable set of Lipschitz curves). In order to keep this note technically simple, we will not prove this strengthening here, although our method also provides it with some extra care.