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  "body": "\n## Statement\n\nThere are absolute constants $c \u003e 0$ and $C \u003e 0$ with the following property.\nLet $G$ be an [[abelian group|082GHNPJ53]], let $K \\ge 1$, and let $A, B \\subseteq G$ be finite sets with $\\abs{A} = \\abs{B} = n$.\nSuppose that the [[additive energy|MAQKH72P6V]] satisfies $E(A, B) \\ge n^3 / K$.\nThen there are subsets $A' \\subseteq A$ and $B' \\subseteq B$ such that\n$$\\abs{A'} \\ge c K^{-C} n, \\qquad \\abs{B'} \\ge c K^{-C} n, \\qquad \\abs{A' + B'} \\le C K^{C} n.$$\n\n## Remarks\n\nThe converse direction is easy: if $\\abs{A + B} \\le Kn$, then Cauchy–Schwarz gives $E(A, B) \\ge n^4 / (Kn) = n^3/K$.\nLarge energy alone does not give a small sumset for the whole sets.\nFor example, let $A$ be the union of an arithmetic progression of length $n/2$ and $n/2$ random-looking integers.\nThen $E(A, A)$ is at least of order $n^3$, but $\\abs{A + A}$ is of order $n^2$; the theorem finds the progression part.\nThe proof is graph-theoretic.\nOne forms the bipartite graph on $A \\cup B$ with an edge $ab$ when $a + b$ is a popular sum.\nA dependent-random-choice argument then finds $A'$ and $B'$ joined by many paths of length three, which bounds $\\abs{A' + B'}$.\nBalog and Szemerédi proved the theorem first, and Gowers obtained polynomial dependence on $K$.\n\n## Explanations\n\n### L1 human\n\nAdditive energy counts how often sums of elements of two sets coincide.\nIf there are many coincidences, one would like to say that the sets are structured.\nThe theorem says that large pieces of the two sets are indeed structured, with few distinct sums.\n\n### L2 human\n\nThe theorem lets one pass from a statistical statement, large energy, to a combinatorial one, a small sumset for large subsets.\nAfter that, [[Freiman's theorem|KEPFYE6WYZ]] gives structure.\nGowers used this chain in his quantitative proof of [[Szemerédi's theorem|B23B0QEMN5]].\n",
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