# Maximum-size sum sets

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.

• Clearly, the bound is attained iff $(A-A)\cap(B-B)=\{0\}$. I doubt there is much more to say without some constraints on $A$ and $B$. – Emil Jeřábek Jun 14 '18 at 14:24
• 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 $|A+B| = k^2$? – user30706 Jun 14 '18 at 15:53
• Trivially $k\le\lfloor\sqrt{|G|}\rfloor$. For cyclic $G$, $k=\lfloor\sqrt{|G|}\rfloor$ is attained by $A=\{0,1,\dots,k-1\}$, $B=\{0,k,2k,\dots,k^2-k\}$. I’m pretty sure you can get close to $\sqrt{|G|}$ for arbitrary finite abelian $G$ as well. – Emil Jeřábek Jun 14 '18 at 16:51
• After the fact, I noticed that a more-general answer appears here: mathoverflow.net/questions/241996/… – user30706 Jun 14 '18 at 18:04
• 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 $|A||B|=n$? 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. – Aaron Meyerowitz Jun 14 '18 at 20:47