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  "body": "\n## Statement\n\nLet $G$ be an [[abelian group|082GHNPJ53]], and let $A$, $B$ and $C$ be finite subsets of $G$.\nThen the difference sets satisfy\n$$\\abs{A} \\, \\abs{B - C} \\le \\abs{A - B} \\, \\abs{A - C}.$$\n\n## Remarks\n\nThe proof is an injection from $A \\times (B - C)$ into $(A - B) \\times (A - C)$.\nFor each $d \\in B - C$, fix one pair $(b_d, c_d)$ with $d = b_d - c_d$.\nSend $(a, d)$ to $(a - b_d, \\ a - c_d)$.\nThe difference of the two coordinates is $d$, which recovers $b_d$ and $c_d$, and then $a$.\nThe name comes from the distance $d(A, B) = \\log \\big( \\abs{A - B} / \\sqrt{\\abs{A}\\abs{B}} \\big)$ on nonempty finite sets.\nThe inequality is exactly $d(B, C) \\le d(B, A) + d(A, C)$.\nThis $d$ is not a metric, since $d(A, A) \u003e 0$ unless $A$ is a coset of a finite subgroup.\nFor example, with $A = B = C = \\set{0, 1}$ in $\\Z$, the inequality reads $2 \\cdot 3 \\le 3 \\cdot 3$.\n\n## Explanations\n\n### L1 human\n\nThe size of a difference set measures how well two sets fit together under subtraction.\nThe inequality says that if $A$ fits well with $B$ and with $C$, then $B$ and $C$ fit well with each other.\nIt behaves like the triangle inequality for distances.\n\n### L2 human\n\nThe Ruzsa triangle inequality is one of the first tools of [[additive combinatorics|3NM362A2JG]].\nTogether with the [[Plünnecke–Ruzsa inequality|EBEVWSRD2E]], it lets one pass between bounds on $A + A$, $A - A$ and longer sums.\nThe same injection proves a version in nonabelian groups, with products and inverses in place of sums and differences.\n",
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