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  "body": "\n## Definition\n\nFor a real number $x$, let $\\{x\\} = x - \\floor{x}$ be its fractional part, where $\\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.\nA sequence $(x_n)_{n \\ge 1}$ of real numbers is *equidistributed modulo one* if for all real $a, b$ with $0 \\le a \u003c b \\le 1$,\n$$\\lim_{N \\to \\infty} \\frac{\\#\\set{n : 1 \\le n \\le N, \\ \\{x_n\\} \\in [a, b)}}{N} = b - a.$$\n\n## Remarks\n\nExample: for irrational $\\alpha$, the sequence $x_n = n\\alpha$ is equidistributed modulo one; this is the [[equidistribution theorem|NCCGNTBEYV]].\nNon-example: $x_n = n/2$ has $\\{x_n\\} \\in \\set{0, 1/2}$, so for $[a, b) = [0.1, 0.4)$ the count is $0$ for every $N$, and the limit $0$ differs from $0.3$.\nWeyl's criterion gives an equivalent form: the sequence is equidistributed modulo one if and only if\n$$\\lim_{N \\to \\infty} \\frac{1}{N} \\sum_{n=1}^{N} e^{2\\pi i h x_n} = 0$$\nfor every nonzero integer $h$.\nAnother equivalent form: $\\frac{1}{N} \\sum_{n=1}^{N} f(\\{x_n\\}) \\to \\int_0^1 f(t)\\,dt$ for every continuous function $f$ on $[0, 1]$ with $f(0) = f(1)$.\nSome authors also say *uniformly distributed modulo one*.\n",
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