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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]], and let $A$ and $B$ be finite subsets of $G$.\nThe *additive energy* of $A$ and $B$ is the number\n$$E(A, B) = \\abs{\\set{(a_1, a_2, b_1, b_2) \\in A \\times A \\times B \\times B : a_1 + b_1 = a_2 + b_2}}.$$\nOne writes $E(A) = E(A, A)$.\n\n## Remarks\n\nLet $r(x)$ be the number of pairs $(a, b) \\in A \\times B$ with $a + b = x$.\nThen $E(A, B) = \\sum_{x} r(x)^2$, summed over the [[sumset|XNBN31HQWB]] $A + B$.\nFor $A = \\set{0, 1, 2}$ in $\\Z$, the values of $r$ on $0, 1, 2, 3, 4$ are $1, 2, 3, 2, 1$, so $E(A) = 1 + 4 + 9 + 4 + 1 = 19$.\nA 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 $0 + 0 \\ne 1 + 0$.\nThe trivial quadruples with $a_1 = a_2$ and $b_1 = b_2$ always count, so $E(A, B) \\ge \\abs{A}\\abs{B}$.\nAlso $E(A, B) \\le \\min(\\abs{A}^2\\abs{B}, \\abs{A}\\abs{B}^2)$.\nThe Cauchy–Schwarz inequality gives $E(A, B) \\ge \\abs{A}^2 \\abs{B}^2 / \\abs{A + B}$.\n",
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