Sumset representation function ($r_{A+B}(s)$)

For finite sets $A,B\subseteq G$ in an abelian group $G$, the sumset representation function isIt counts ordered pairs in the Cartesian product, so $(a,b)$ and $(b,a)$ are different when $a\ne b$, even though commutativity makes their sums equal. It is zero outside the sumset, andFor $A=\{0,1,2,3\}$, its values at $s=0,1,\dots ,6$ are $1,2,3,4,3,2,1$. For $B=\{0,1,4,10\}$, its values at the increasing elements of $B+B$ are $1,2,1,2,2,1,2,2,2,1$. Both lists total sixteen, but the first concentrates more ordered pairs on fewer sums. This distinction is what additive energy quantifies.