If {c(n)} is an arbitrary sequence of irrational numbers converging to 0 then Q + c(n), the set obtained by adding c(n) to the set of rational numbers Q, is clearly disjoint from Q for each n.

Is there an uncountable dense set of the real numbers, say D, for which a sequence {c(n)} converging to 0 exists such that D + c(n) does not intersect D for all n?

If such a set exists, and is borel, then it must have measure 0 since D - D would contain an interval if D had a positive measure.