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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|082GHNPJ53]], and let $A$ and $B$ be finite nonempty subsets of $G$.\nLet $K \\ge 1$ be a real number.\nSuppose that the [[sumset|XNBN31HQWB]] satisfies $\\abs{A + B} \\le K \\abs{A}$.\nThen for all integers $m, n \\ge 0$ with $m + n \\ge 1$,\n$$\\abs{mB - nB} \\le K^{m + n} \\abs{A}.$$\nHere $mB - nB$ is the set of all $b_1 + \\dots + b_m - b'_1 - \\dots - b'_n$ with all $b_i, b'_j \\in B$.\n\n## Remarks\n\nPetridis gave a short proof.\nChoose a nonempty $X \\subseteq A$ that minimizes the ratio $\\abs{X + B} / \\abs{X}$, and call the minimum $K'$, so $K' \\le K$.\nHe showed that $\\abs{X + B + C} \\le K' \\abs{X + C}$ for every finite set $C$, and induction gives $\\abs{X + mB} \\le K^m \\abs{X}$.\nThe [[Ruzsa triangle inequality|R7E5A4W6FJ]] then handles the differences.\nWith $B = A$, a set with $\\abs{A + A} \\le K\\abs{A}$ satisfies $\\abs{mA - nA} \\le K^{m+n}\\abs{A}$, so small doubling controls all iterated sums.\nFor $A = \\set{1, \\dots, N}$ one has $\\abs{A + A} = 2N - 1 \\le 2\\abs{A}$, and indeed $\\abs{mA - nA} \\le (m + n)N \\le 2^{m+n} N$.\n\n## Explanations\n\n### L1 human\n\nIf adding a set $B$ to a set $A$ makes it only a little larger, then adding $B$ many times, or subtracting it, also cannot make it much larger.\nThe growth is at most a fixed power of the first growth factor.\nSo a set with few sums is controlled at every later step.\n\n### L2 human\n\nThe Plünnecke–Ruzsa inequality is a basic tool in [[additive combinatorics|3NM362A2JG]].\nIt is used in the proof of [[Freiman's theorem|KEPFYE6WYZ]].\nIt is often combined with the [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]].\nPlünnecke's original proof used a graph-theoretic argument with layered commutative graphs, and Ruzsa turned it into the form above.\n",
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