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  "body": "\n## Summary\n\nA Bohr almost periodic function is a continuous function on the real line whose $\\varepsilon$-almost periods are spread evenly over the line, for every $\\varepsilon \u003e 0$.\nThis was Harald Bohr's original definition of an [[almost periodic function|GZBSV8DYGS]].\nThese functions are exactly the uniform limits of trigonometric polynomials with arbitrary real frequencies, by [[Bohr's approximation theorem|GWCY1C1KQV]].\nThey have a mean value and a Fourier series, and they form a closed subalgebra of the bounded continuous functions.\n\n## Explanations\n\n### L1 human\n\nA periodic function with period $p$ repeats exactly after every multiple of $p$.\nA Bohr almost periodic function repeats only approximately, up to an error $\\varepsilon$.\nThe shifts that achieve this are not equally spaced, but they never leave large gaps: every interval of some length $L$ contains one.\nFor a continuous periodic function one can take the multiples of the period, with $L$ equal to the period.\n",
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