Optimally small sumsets in finite abelian groups. (Q1398954)

Let \(G\) be a finite Abelian group of order \(g\) and for the sum set \[ A+B:= \{a+b\mid a\in A,\,b\in B\}\;(A,B\subset G) \] denote \[ \mu_G(r,s):= \min\{| A+B|\mid | A|= r,\,| B|= s\}. \] In the introduction the authors recall earlier results on \(\mu_G(r,s)\). The purpose of this paper is the following theorem: For all \(r,s\in \{1,\dots, g\}\) one has \[ \mu_G(r,s)= \min_{d| g}\,\Biggl\{\Biggl(\Biggl\lceil{r\over d}\Biggr\rceil+ \Biggl\lceil{s\over d}\Biggr\rceil- 1\Biggr) d\Biggr\}. \]