Doubling constant
= Doubling constant
{title2=$|A+A|/|A|$}
The <doubling constant> of a nonempty <finite set> $S\subseteq G$ in an <abelian group> $G$ is
$$
K(S)=\frac{|S+S|}{|S|}.
$$
It 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>, $K(S)\geq1$: fixing $s_0\in S$ puts a translated copy $s_0+S$ inside $S+S$. The stronger <integer> bound gives $K(S)\geq2-1/|S|$ 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: $E(S)\geq|S|^3/K(S)$.