Let $G$ be an abelian group. Let $A\subset G$ be a finite set. $\sum\_A$ is defined as: $$\left\{\sum\_{b\in B}b \mid B\subset A\right\}$$ Is there any result similar to Freiman's Theorem for $\sum\_A$? Can we say anything about $A$ if $\sum\_A$ is small?



Size of distinct sums in A
One way in which $\sum\_A$ can be small is if $A$ is a small subgroup of $G$. To exclude such examples, define $X$ to be *aperiodic*, if the only solution to $X+x=X$ is $x=0$. DeVos, Goddyn, Mohar and Šámal [proved](http://arxiv.org/pdf/math/0612045.pdf?origin=publication_detail) that if $\sum\_A$ is aperiodic, then $\sum\_A$ exhibits quadratic growth. 


**Theorem**. Let $A \subseteq G \setminus \{0\}$, where $G$ is an abelian group. If $\sum\_A$ is aperiodic, then $|\sum\_A| \geq \frac{|A|^2}{64}$.


