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  "body": "\n## Example\n\nThe *Gauss map* is\n$$G \\colon [0, 1) \\to [0, 1), \\qquad G(x) = \\frac{1}{x} - \\floor{\\frac{1}{x}} \\text{ for } x \\ne 0, \\qquad G(0) = 0.$$\nIf $x = [0; a_1, a_2, a_3, \\dots]$ is the [[continued fraction expansion|04VA28F3HB]] of an irrational $x \\in (0, 1)$, then $a_1 = \\floor{1/x}$ and $G(x) = [0; a_2, a_3, \\dots]$.\nSo $G$ shifts the continued fraction digits, as the doubling map shifts binary digits.\n\nThe *Gauss measure* is the probability measure\n$$\\mu(A) = \\frac{1}{\\log 2} \\int_A \\frac{dx}{1 + x}.$$\nIt is $G$-invariant; check on $A = [0, t)$ with $0 \u003c t \\le 1$.\nThe preimage $G^{-1}[0, t)$ is, up to the point $0$, the disjoint union over $n \\ge 1$ of the intervals $\\big(\\frac{1}{n + t}, \\frac{1}{n}\\big]$.\nThe measure of the $n$-th interval is $\\frac{1}{\\log 2}\\big(\\log\\frac{n+1}{n} - \\log\\frac{n+t+1}{n+t}\\big)$.\nThe partial sum up to $N$ telescopes to $\\frac{1}{\\log 2}\\big(\\log(N+1) - \\log(N+1+t) + \\log(1 + t)\\big)$, which tends to $\\frac{\\log(1+t)}{\\log 2} = \\mu([0, t))$.\n\nThe map $G$ is [[ergodic|AZRPHS478B]] for $\\mu$, and $\\mu$ is equivalent to [[Lebesgue measure|2Z2QVWZ262]].\nHence for Lebesgue almost every $x$, the digit $k$ appears among the $a_n$ with frequency $\\frac{1}{\\log 2}\\log\\frac{(k+1)^2}{k(k+2)}$, the Gauss–Kuzmin distribution.\nThe [[transfer operator|CNZV00JQRC]] of $G$ for Lebesgue measure is the Gauss–Kuzmin–Wirsing operator, $(\\mathcal{L}\\varphi)(x) = \\sum_{n \\ge 1} \\frac{1}{(x+n)^2} \\varphi\\big(\\frac{1}{x+n}\\big)$, and the density $\\frac{1}{(1+x)\\log 2}$ is its fixed point.\n\n## Explanations\n\n### L1 human\n\nThe Gauss map takes a number between $0$ and $1$, flips it to $1/x$, and keeps only the fractional part.\nDoing this again and again reads off the digits of the continued fraction of the number, one at a time.\nGauss found the measure that this map preserves, a density that is a bit larger near $0$ than near $1$.\nErgodic theory then tells how often each digit appears in the continued fraction of a typical number: the digit $1$ appears about $41.5\\%$ of the time.\n",
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