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_5: A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of Winkler' type, that is, they ask about what can be achieved by a clairvoyant demon.
  @cite_2: We describe a decomposition result for Lebesgue negligible sets in the plane, and outline some applications to real analysis and geometric measure theory. These results are contained in (2).
summary: Shortly before uploading this paper we came across a very recent paper @cite_5 on arXiv which investigates related questions and proves a variant of Theorem for sets @math of sufficiently small measure (Proposition 11, @cite_5 ), as well as much more. Their proof uses a covering lemma based on a result of @cite_2 , which is no longer true in higher dimensions. Our proof is different and we believe can be adapted to work in higher dimensions as well. Moreover, the stretching result only for sufficiently small sets is not sufficient for our purposes in proving rough isometry. Our work was independent of @cite_5 .