Bi-Lipschitz Expansion of Measurable Sets
  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 $1-\gamma'$. 
Riddhipratim Basu, Vladas Sidoravicius, Allan Sly
math.AP math.PR