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  "body": "\n## Definition\n\nLet $(x_n)_{n \\in \\Z}$ be a two-sided [[sequence|0G9EKM0E15]] of complex numbers.\nFor $\\varepsilon \u003e 0$, an integer $p$ is an *$\\varepsilon$-almost period* of the sequence if\n$$\\abs{x_{n+p} - x_n} \u003c \\varepsilon \\quad \\text{for all } n \\in \\Z.$$\nThe sequence is *almost periodic* if the following holds for every $\\varepsilon \u003e 0$.\nThere is an integer $L \\ge 1$ such that every set of $L$ consecutive integers contains an $\\varepsilon$-almost period.\n\n## Remarks\n\nExample: a [[periodic sequence|Q3B6K9HYRG]] with period $p$ is almost periodic.\nEvery multiple of $p$ is an $\\varepsilon$-almost period for every $\\varepsilon \u003e 0$, and every set of $p$ consecutive integers contains a multiple of $p$, so $L = p$ works.\n\nNon-example: let $x_n = 0$ for $n \u003c 0$ and $x_n = 1$ for $n \\ge 0$.\nFor $p \u003e 0$ take $n = -1$, and for $p \u003c 0$ take $n = 0$: in both cases $\\abs{x_{n+p} - x_n} = 1$.\nSo for $\\varepsilon = 1/2$ the only almost period is $0$, and no $L$ works.\n\nEquivalent forms: a bounded sequence is almost periodic if and only if the set of its translates $(x_{n+p})_{n \\in \\Z}$, $p \\in \\Z$, is relatively compact for the supremum norm.\nIt is also equivalent that the sequence is a uniform limit on $\\Z$ of trigonometric polynomials $\\sum_{j=1}^{m} c_j e^{i \\lambda_j n}$ with $c_j \\in \\C$ and $\\lambda_j \\in \\R$.\nAn almost periodic sequence is bounded.\n",
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