Small-large subgroups of the reals
  We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of R. 
Andrzej Roslanowski and Saharon Shelah
math.LO