Bi-Lipschitz Expansion of Measurable Sets

TL;DR

This paper demonstrates the existence of bi-Lipschitz maps that can expand measurable sets in the unit square to nearly full measure while remaining identity on the boundary, with uniformly bounded Lipschitz constants.

Contribution

It introduces a method to construct boundary-preserving bi-Lipschitz maps that significantly increase the measure of measurable sets within the unit square.

Findings

Existence of bi-Lipschitz maps with bounded Lipschitz constants

Maps can increase measure of sets to at least 1−γ'

Maps are identity on the boundary

Abstract

We show that for $0<\gamma, \gamma' <1$ and for measurable subsets of the unit square with Lebesgue measure $\gamma$ there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least $1-\gamma'$.