text: We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math 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 @math .
  @cite_0: for x 6= x′. A set Y ⊂ Rn is a net if there is an R such that d(x, Y ) 0 such that |y − y′| > ǫ > 0 for every pair y 6= y′ in Y . History. In 1965, J. Moser showed that any two positive, C∞ volume forms on a compact manifold with the same total mass are related by a diffeomorphism [Mos]. Extensions of this result to other smoothness classes such as Ck,α were given in [Rei1] and [DM]; see also [RY1], [RY2], and [Ye].
summary: The question of existence of bi-Lipschitz homeomorphisms between different subsets of @math , satisfying certain conditions, has been studied classically, notably by McMullen @cite_0 . It is shown in that paper that there exists a positive function @math on @math , @math , such that there does not exist a bi-Lipschitz homeomorphism @math with @math . It is also proved that there exists separated nets in @math ( @math ) which cannot be mapped into @math in a bi-Lipschitz way.