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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]] written additively, and let $A$ and $B$ be subsets of $G$.\nThe *sumset* of $A$ and $B$ is\n$$A + B = \\set{a + b : a \\in A, \\ b \\in B}.$$\nThe *difference set* is $A - B = \\set{a - b : a \\in A, \\ b \\in B}$.\nFor an integer $k \\ge 1$, the iterated sumset $kA$ is $A + \\dots + A$ with $k$ summands.\n\n## Remarks\n\nIn $\\Z$, take $A = \\set{0, 1, 2}$ and $B = \\set{0, 5}$.\nThen $A + B = \\set{0, 1, 2, 5, 6, 7}$, since each sum $a + b$ lies in this set and each element of it is such a sum.\nThe set $\\set{a + a : a \\in A} = \\set{0, 2, 4}$ is not the sumset $A + A$, because it misses sums of two different elements such as $0 + 1$.\nThe iterated sumset $2A = A + A$ is different from the dilate $2 \\cdot A = \\set{2a : a \\in A}$.\nFor finite nonempty $A, B \\subseteq \\Z$ one has $\\abs{A + B} \\ge \\abs{A} + \\abs{B} - 1$, with equality for arithmetic progressions of the same common difference.\nFor subsets of $\\R^n$ the sumset is called the Minkowski sum.\n",
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