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  "body": "\n## Statement\n\nA generalized arithmetic progression of dimension $d$ in $\\Z$ is a set\n$$P = \\set{a + x_1 q_1 + \\dots + x_d q_d : x_i \\in \\Z, \\ 0 \\le x_i \u003c L_i \\text{ for } 1 \\le i \\le d},$$\nwhere $a, q_1, \\dots, q_d$ are integers and $L_1, \\dots, L_d$ are positive integers.\nLet $K \\ge 1$ be a real number.\nThen there are numbers $d(K)$ and $f(K)$, depending only on $K$, with the following property.\nEvery finite set $A \\subseteq \\Z$ with $\\abs{A + A} \\le K \\abs{A}$ is contained in a generalized arithmetic progression $P$ of dimension at most $d(K)$ with $\\abs{P} \\le f(K) \\abs{A}$.\n\n## Remarks\n\nHere $A + A$ is the [[sumset|XNBN31HQWB]] of $A$ with itself.\nRuzsa's proof has four steps.\nFirst, the [[Plünnecke–Ruzsa inequality|EBEVWSRD2E]] bounds $\\abs{2A - 2A}$.\nSecond, a large part of $A$ is modelled by a set in a cyclic group $\\Z/N\\Z$ with $N$ not much larger than $\\abs{A}$.\nThird, Bogolyubov's lemma finds a large Bohr set inside the set $2A' - 2A'$ of the model $A'$.\nFourth, geometry of numbers finds a progression inside the Bohr set, and a covering argument finishes.\nA converse holds: if $A$ is a large subset of a proper progression $P$ of dimension $d$, then $\\abs{A + A} \\le \\abs{P + P} \\le 2^d \\abs{P}$, so $A$ has small doubling.\nGreen and Ruzsa extended the theorem to all abelian groups, with coset progressions in place of progressions.\n\n## Explanations\n\n### L1 human\n\nSome sets of numbers have very few different pairwise sums compared with their size.\nArithmetic progressions like $3, 7, 11, 15, \\dots$ are the main example.\nFreiman's theorem says that every such set is a large piece of a \"multi-dimensional\" arithmetic progression.\nSo few sums means a lot of hidden regular structure.\n\n### L2 human\n\nFreiman's theorem is the central inverse theorem of [[additive combinatorics|3NM362A2JG]].\nGood bounds on $d(K)$ and $f(K)$ matter in applications, and finding the right ones is related to the polynomial Freiman–Ruzsa conjecture.\nIt is used in Gowers' proof of [[Szemerédi's theorem|B23B0QEMN5]], together with the [[Balog–Szemerédi–Gowers theorem|W5BZZ59CVM]].\n",
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