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  "body": "\n## Statement\n\nLet $\\alpha$ be an [[irrational number|0DZXAXEJQ9]].\nThen the sequence $(n\\alpha)_{n \\ge 1}$ is [[equidistributed modulo one|XM6DH57BWY]].\nThat is, for all $0 \\le a \u003c b \\le 1$, the proportion of $n \\le N$ with $n\\alpha - \\floor{n\\alpha} \\in [a, b)$ tends to $b - a$ as $N \\to \\infty$.\n\n## Remarks\n\nA short proof uses Weyl's criterion.\nFor a nonzero integer $h$, the number $e^{2\\pi i h \\alpha}$ is not $1$, since $h\\alpha \\notin \\Z$.\nThe sum $\\sum_{n=1}^{N} e^{2\\pi i h n \\alpha}$ is a geometric sum, so its absolute value is at most $2 / \\abs{1 - e^{2\\pi i h \\alpha}}$, a bound independent of $N$.\nDividing by $N$ gives a limit $0$, which is Weyl's condition.\nThe hypothesis that $\\alpha$ is irrational is necessary: if $\\alpha = p/q$ in lowest terms, then $n\\alpha$ takes only $q$ values modulo one.\nExample: $\\log_{10} 2$ is irrational, so the fractional parts of $n \\log_{10} 2$ are equidistributed.\nHence the leading digit of $2^n$ is $k$ for a proportion $\\log_{10}(1 + 1/k)$ of the exponents $n$; for instance the digit $1$ occurs about $30.1\\%$ of the time.\n\n## Explanations\n\n### L1 human\n\nWalk around a circle of circumference $1$ in steps of an irrational length $\\alpha$.\nYou never return to a point you have visited, and in the long run you visit every arc equally often.\nThis is a precise version of the idea that irrational rotations mix the circle well.\n\n### L2 human\n\nThe theorem is the simplest case of Weyl's results on polynomial sequences.\nIf a real polynomial $p$ has at least one irrational coefficient other than the constant term, then $(p(n))$ is equidistributed modulo one.\nIts proof shows the general pattern: equidistribution follows from cancellation in [[exponential sums|97PBMQCPPM]].\nIt strengthens Kronecker's theorem that $(n\\alpha)$ is dense modulo one, and it expresses the unique ergodicity of the irrational rotation of the circle.\n",
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