The Cauchy-Davenport inequality on an additive group $G$ is: For all $A,B\subseteq G$,
$$|A+B|\ge \min\{|G|,|A|+|B|-1\}$$
The Cauchy-Davenport inequality fails in mod $n$ ($n$ composite) because we can take any subgroup $A$, then $|A+A|=|A|$. However what happens if we insist $|A|$ does not divide $n$ (and possibly $|B|$ does not divide $n$ too)? Is there a better bound than the trivial $|A+B|\ge\max\{|A|,|B|\}$?
(I am aware of the results of Kneser, although these are not relevant here as $n$ can be even)