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  "body": "\n## Statement\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]].\nLet $A \\in \\mathcal{B}$ with $\\mu(A) \u003e 0$, and let $k \\ge 1$ be an integer.\nThen\n$$\\liminf_{N \\to \\infty} \\frac{1}{N} \\sum_{n=1}^{N} \\mu\\big(A \\cap T^{-n}A \\cap T^{-2n}A \\cap \\cdots \\cap T^{-kn}A\\big) \u003e 0.$$\nIn particular, there is an integer $n \\ge 1$ with $\\mu(A \\cap T^{-n}A \\cap \\cdots \\cap T^{-kn}A) \u003e 0$.\n\n## Remarks\n\nFor $k = 1$ the weaker conclusion is the [[Poincaré recurrence theorem|TVYJPBHD0S]], and the average form follows from the mean ergodic theorem.\nFor general $k$ the proof goes through a structure theorem.\nOne first reduces to ergodic systems.\nEvery ergodic system is built from a trivial system by a possibly transfinite tower of compact extensions, followed possibly by one weakly mixing extension.\nThe property holds for compact extensions, where orbits come back almost periodically.\nIt also holds for weakly mixing extensions, where the sets $T^{-in}A$ behave as if independent.\nThen it passes up the tower and to limits.\nExample: for an irrational rotation and an arc $A$ of length $\\ell$, take $n$ with $n\\alpha$ within $\\ell/(2k)$ of an integer.\nThen $A \\cap T^{-n}A \\cap \\cdots \\cap T^{-kn}A$ contains an arc of length at least $\\ell/2$.\nExample: for a Bernoulli scheme and a cylinder set $A$, independence gives $\\mu(A \\cap T^{-n}A \\cap \\cdots \\cap T^{-kn}A) = \\mu(A)^{k+1}$ once $n$ exceeds the length of the cylinder.\nThe hypothesis $\\mu(X) \u003c \\infty$ cannot be dropped: the translation $x \\mapsto x + 1$ on $\\R$ with Lebesgue measure and $A = [0, 1)$ gives $A \\cap T^{-n}A = \\emptyset$ for every $n \\ge 1$.\n\n## Explanations\n\n### L1 human\n\nIn a system that preserves a finite volume, a set of positive volume does not only come back to itself.\nIt comes back at evenly spaced times $n, 2n, \\dots, kn$ simultaneously, for some $n$ and for any length $k$ you ask for.\nThis is a dynamical form of finding long arithmetic progressions.\n\n### L2 human\n\nFurstenberg proved the theorem in 1977 to give an ergodic-theoretic proof of [[Szemerédi's theorem|B23B0QEMN5]].\nThe Furstenberg correspondence principle turns a set of integers of positive upper density into a set $A$ of positive measure in a measure-preserving system.\nIt does so in such a way that a positive value of $\\mu(A \\cap T^{-n}A \\cap \\cdots \\cap T^{-kn}A)$ yields an arithmetic progression $a, a+n, \\dots, a+kn$ in the original set.\nThe method grew into ergodic Ramsey theory, with extensions to several commuting transformations, which gives multidimensional Szemerédi theorems, and to polynomial times.\n",
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