Additive energy

Remarks

Let $r(x)$ be the number of pairs $(a, b) \in A \times B$ with $a + b = x$. Then $E(A, B) = \sum_{x} r(x)^2$, summed over the sumset $A + B$. For $A = \set{0, 1, 2}$ in $\Z$, the values of $r$ on $0, 1, 2, 3, 4$ are $1, 2, 3, 2, 1$, so $E(A) = 1 + 4 + 9 + 4 + 1 = 19$. A quadruple with $a_1 + b_1 \ne a_2 + b_2$ is not counted; for example $(0, 1, 0, 0)$ is not counted for $A = B = \set{0, 1, 2}$, since $0 + 0 \ne 1 + 0$. The trivial quadruples with $a_1 = a_2$ and $b_1 = b_2$ always count, so $E(A, B) \ge \abs{A}\abs{B}$. Also $E(A, B) \le \min(\abs{A}^2\abs{B}, \abs{A}\abs{B}^2)$. The Cauchy–Schwarz inequality gives $E(A, B) \ge \abs{A}^2 \abs{B}^2 / \abs{A + B}$.