Let $A, B \subseteq G$ for abelian group $G$. It is obvious that $A+B \leq A \cdot B$. Is there an easy characterization of sets $A, B$ meeting this bound? I am especially interested in the case where $G = {\mathbb Z}/p{\mathbb Z}$, with $p$ prime.
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3$\begingroup$ Clearly, the bound is attained iff $(AA)\cap(BB)=\{0\}$. I doubt there is much more to say without some constraints on $A$ and $B$. $\endgroup$ – Emil Jeřábek Jun 14 '18 at 14:24

$\begingroup$ OK, maybe I can refine the question a bit: Fix integer $k$. When do there exist $A, B \subset G$ with $A=B=k$ and $A+B = k^2$? Alternately, for a given $G$ what is the largest $k$ for which there exist sets $A, B$ with $A=B=k$ and $A+B = k^2$? $\endgroup$ – user30706 Jun 14 '18 at 15:53

$\begingroup$ Trivially $k\le\lfloor\sqrt{G}\rfloor$. For cyclic $G$, $k=\lfloor\sqrt{G}\rfloor$ is attained by $A=\{0,1,\dots,k1\}$, $B=\{0,k,2k,\dots,k^2k\}$. I’m pretty sure you can get close to $\sqrt{G}$ for arbitrary finite abelian $G$ as well. $\endgroup$ – Emil Jeřábek Jun 14 '18 at 16:51

$\begingroup$ After the fact, I noticed that a moregeneral answer appears here: mathoverflow.net/questions/241996/… $\endgroup$ – user30706 Jun 14 '18 at 18:04

$\begingroup$ Here is a fun generalization and hard question : Given a finite set $A$ of integers, when is there $B$ with $A+B=\mathbb{Z}_n$ where $AB=n$? This happens exactly when $A+C=\mathbb{Z}$ with all the sums distinct. The answer is known when $A$ has at most two distinct prime divisors. $\endgroup$ – Aaron Meyerowitz Jun 14 '18 at 20:47