Astala's Conjecture on Distortion of Hausdorff Measures under   Quasiconformal Maps in the Plane
  Let E be a compact set in the plane, g be a K-quasiconformal map, and let 0<t<2. Then   H^t (E) = 0 implies H^{t'} (g E) = 0, for t'=[2Kt]/[2+(K-1)t].   This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in his celebrated paper on area distortion for quasiconformal maps and answers in the positive a Conjecture of K. Astala in op. cit. 
Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero