Suppose $U\subset\mathbb{Z}_{2k}$ with $|U|=k$. Let $U^c$ denote the complement of $U$.

Let $v\in \mathbb{Z}_{2k}^{\times}$. How much is it known about $U+vU^c$?

For example: When $U+v\complement U = \mathbb{Z}_{2k}$?

Yet another example: if $U$ is aperiodic (i. e., its isotropy group is trivial), Kemperman's theorem implies $U-\complement U = \mathbb{Z}_{2k}\setminus \{0\}$. Anything beyond that?

othercoset modulo U, write it e+U. So U - C U = U - (e+U) = - e + U = e + U. And this is the same for this U for any v. What you say should/could be true if you assume U to be aperiodic. Likely if U is periodic (ie there is a nontriv subgroup such that U = U + H) you could mod out by the stabilizer H (which is also stab of the complement) and so reduce to aperiodic. – user9072 Feb 16 '13 at 9:41