 Doubling constant

The doubling constant of a nonempty finite set $S\subseteq G$ in an abelian group $G$ isIt measures the cardinality of the sumset relative to the original set. For the integer examples, $K(A)=7/4$ and $K(B)=10/4=5/2$. A nonconstant arithmetic progression with $n$ members has doubling constant $(2n-1)/n$, which stays below two as $n$ grows. This is the sense in which its sumset has small growth.

In any abelian group, $K(S)\ge 1$: fixing $s_0\in S$ puts a translated copy $s_0+S$ inside $S+S$. The stronger integer bound gives $K(S)\ge 2-1/|S|$ only in that setting. A finite subgroup $H$ has $K(H)=1$, as does every coset $x+H$, since its sumset is $2x+H$. In particular $H=\{0,2,4\}\subseteq \mathbb{Z}/6\mathbb{Z}$ has doubling constant one.

Small doubling constant is useful because it constrains repeated growth, as in the Plünnecke-Ruzsa inequality. It signals structure such as arithmetic progressions and subgroups; precise descriptions depend on the ambient abelian group and are the subject of the Freiman-Ruzsa theorem. The additive energy bound below gives a simpler immediate consequence: $E(S)\ge |S{|}^3/K(S)$.