By a theorem proved by Erdos, Kunen and Mauldin, for any nonempty perfect set
P on the real line there exists a perfect set M of Lebesgue measure zero
such that P+M=R. We prove a stronger version of this theorem in
which the obtained perfect set M is a Dirichlet set. Using this result we
show that for a wide range of familes of subsets of the reals, all additive
sets are perfectly meager in transitive sense. We also prove that every proper
analytic subgroup G of the reals is contained in an F-sigma set F such that
F+G is a meager null set.Comment: 9 page