Definition 1.15. Given $A,B\subseteq G$ we define the additive energy between $A$ and $B$ to be

$$E(A,B)=|\{(a,a^{\prime },b,b^{\prime })\in A\times A\times B\times B:a+b=a^{\prime }+b^{\prime }\}|.$$

We refer to the quadruples $(a,a^{\prime },b,b^{\prime })$ such that $a+b=a^{\prime }+b^{\prime }$ as additive quadruples.