= Additive energy
{title2=$E(A)$}
{wiki}
The <additive energy> of a <finite set> $S\subseteq G$ in an <abelian group> $G$ counts <additive quadruples>, with order and repeated members allowed:
$$
E(S)=\bigl|\{(a,b,c,d)\in S^4:a+b=c+d\}\bigr|
=\sum_{s\in G}r_{S+S}(s)^2.
$$
At a fixed sum $s$, independently choosing two of its $r_{S+S}(s)$ <ordered pairs> gives $r_{S+S}(s)^2$ <additive quadruples>. Thus <additive energy> weights heavily represented sums more strongly than the <cardinality> of the <sumset> does.
For $A=\{0,1,2,3\}$ and $B=\{0,1,4,10\}$, the <sumset representation functions> give
$$
\begin{aligned}
E(A)&=1^2+2^2+3^2+4^2+3^2+2^2+1^2=44,\\
E(B)&=4\cdot1^2+6\cdot2^2=28.
\end{aligned}
$$
The <arithmetic progression> has additional coincidences such as $0+3=1+2$. The <set> $B$ still has coincidences forced by <commutativity>, including $0+1=1+0$, so its <additive energy> is not merely the sixteen identical-pair comparisons.
The same <additive energy> can be counted using the <difference set>. Define $r_{S-S}(t)=|\{(a,b)\in S\times S:a-b=t\}|$. The identity $a+b=c+d$ is equivalent to $a-c=d-b$, and the coordinate rearrangement $(a,b,c,d)\mapsto(a,c,d,b)$ is a <bijection>. Consequently
$$
E(S)=\sum_{t\in G}r_{S-S}(t)^2.
$$
For $A$, the difference counts from $-3$ through $3$ are again $1,2,3,4,3,2,1$. For $B$, the zero difference has four representations and each of the twelve nonzero differences has one, giving $4^2+12\cdot1^2=28$. The <sumset representation functions> for sums and differences can look different while their squared totals agree.
For every nonempty <finite set> $S\subseteq G$ in an <abelian group> $G$,
$$
\frac{|S|^4}{|S+S|}\leq E(S)\leq |S|^3.
$$
The lower bound follows from the <Cauchy-Schwarz inequality> and the total <sumset representation function> count:
$$
|S|^4=\left(\sum_{s\in S+S}r_{S+S}(s)\right)^2
\leq |S+S|\sum_{s\in S+S}r_{S+S}(s)^2.
$$
For the upper bound, any three members $a,b,c$ determine at most one fourth member $d=a+b-c$ of an <additive quadruple>. See https://ocw.mit.edu/courses/18-225-graph-theory-and-additive-combinatorics-fall-2023/MlYhHsq_tOU_transcript.pdf[Yufei Zhao's MIT lecture 25, pages 1-3, for representation counts and the energy bounds].
In the two <integer> examples, the lower bounds are $256/7$ and $256/10$, and the upper bound is $64$. For the <subgroup> $H=\{0,2,4\}$ of $\mathbb Z/6\mathbb Z$, each element of $H$ has three sum representations, so $E(H)=3\cdot3^2=27$ and both bounds are equalities.
\Image[/additive-energy-and-doubling.png]
{title=Sumset and difference-set sizes alongside additive energy and its bounds, including a cyclic subgroup}
For nonempty $S$, dividing <additive energy> by $|S|^3$ gives a relative measure between $1/K(S)$ and one. Small <doubling constant> therefore forces relatively large <additive energy>. A converse requires care: the <Balog-Szemerédi-Gowers theorem> extracts a large <subset> with small <doubling constant>, rather than asserting that all of $S$ has small <doubling constant>.
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