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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be a [[continuous function|0X3WGX95FE]].\nFor $\\varepsilon \u003e 0$, a number $\\tau \\in \\R$ is an *$\\varepsilon$-almost period* of $f$ if\n$$\\sup_{x \\in \\R} \\abs{f(x + \\tau) - f(x)} \\le \\varepsilon.$$\nA set $T \\subseteq \\R$ is *relatively dense* if there is $L \u003e 0$ such that every closed interval of length $L$ contains a point of $T$.\nThe function $f$ is *Bohr almost periodic* if for every $\\varepsilon \u003e 0$ the set of $\\varepsilon$-almost periods of $f$ is relatively dense.\n\n## Remarks\n\nEvery Bohr almost periodic function is bounded and uniformly continuous.\nBochner proved that $f$ is Bohr almost periodic if and only if every sequence of translates of $f$ has a uniformly convergent subsequence.\nSo the definition agrees with the general [[almost periodic function|GZBSV8DYGS]] on the group $\\R$.\n\nExample: let $f$ be continuous with period $p \u003e 0$.\nEvery multiple $kp$, $k \\in \\Z$, satisfies $f(x + kp) = f(x)$, so it is an $\\varepsilon$-almost period for every $\\varepsilon \u003e 0$.\nEvery closed interval of length $p$ contains a multiple of $p$, so $f$ is Bohr almost periodic.\n\nNon-example: $f(x) = \\arctan x$ is continuous, but not Bohr almost periodic.\nFor $\\tau \u003e 0$ the difference $\\arctan(x + \\tau) - \\arctan x$ is largest at $x = -\\tau/2$, where it equals $2 \\arctan(\\tau/2)$, and the case $\\tau \u003c 0$ is symmetric.\nSo for $0 \u003c \\varepsilon \u003c \\pi$ the $\\varepsilon$-almost periods are the $\\tau$ with $\\abs{\\tau} \\le 2\\tan(\\varepsilon/2)$.\nThis set is bounded, hence not relatively dense.\n",
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