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?



About a sumset in $\mathbb{Z}_{2k}$
The majority of facts I wanted to know about this situation are contained in Theorem 7.5 (p. 75) of David J. Grynkiewicz's Ph. D. thesis "Sumsets, Zero-Sums and Extremal Combinatorics".


Thank you very much to those who offered their remarks.


