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  "body": "\n## Definition\n\nAdditive combinatorics is the part of mathematics that studies finite subsets $A, B$ of an [[abelian group|082GHNPJ53]] $G$ through their additive structure.\nIts main objects are the [[sumsets|XNBN31HQWB]] $A + B$, the difference sets $A - B$, and the solutions of linear equations with all variables in $A$.\nIts typical results are of three kinds:\n\n1. Direct results bound the size of $A + B$ from below in terms of $\\abs{A}$ and $\\abs{B}$.\n2. Inverse results describe the structure of $A$ when $\\abs{A + A}$ is small compared with $\\abs{A}$.\n3. Density results show that large subsets of $G$ contain solutions of given linear equations.\n\n## Remarks\n\nThe [[Cauchy–Davenport theorem|F7K7XFACSG]] is a direct result: in $\\Z/p\\Z$ it gives $\\abs{A + B} \\ge \\min(p, \\abs{A} + \\abs{B} - 1)$.\n[[Freiman's theorem|KEPFYE6WYZ]] is an inverse result: sets of integers with small doubling lie in small generalized arithmetic progressions.\n[[Szemerédi's theorem|B23B0QEMN5]] is a density result about arithmetic progressions.\nA statement about sets without any addition, such as a bound on chains in a poset, is not a result of additive combinatorics.\nThe field is also called arithmetic combinatorics.\n",
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