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  "body": "\n## Summary\n\nThe additive energy of two finite sets $A$ and $B$ in an abelian group counts the quadruples $(a_1, a_2, b_1, b_2)$ with $a_1 + b_1 = a_2 + b_2$.\nIt measures how often sums repeat, so it is large when the [[sumset|XNBN31HQWB]] $A + B$ is small.\nIt is a more robust measure of additive structure than the size of the sumset.\nThe [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]] turns large energy into a small sumset for large subsets.\n\n## Explanations\n\n### L1 human\n\nAdd each element of $A$ to each element of $B$ and look for coincidences among the sums.\nThe additive energy counts these coincidences, including trivial ones.\nFor an arithmetic progression many sums coincide, so the energy is large.\nFor a set like $\\set{1, 2, 4, 8}$, almost all sums are different, so the energy is as small as possible.\n",
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