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  "body": "\n## Statement\n\nLet $(x_n)_{n \\ge 1}$ be a sequence of real numbers.\nThen $(x_n)$ is [[equidistributed modulo one|RT62ZAN3JB]] if and only if, for every nonzero integer $h$,\n$$\\lim_{N \\to \\infty} \\frac{1}{N} \\sum_{n=1}^{N} e^{2 \\pi i h x_n} = 0.$$\nHere $e^{2\\pi i t}$ is the [[complex exponential|07GJV36896]].\n\n## Remarks\n\nEquidistribution modulo one is equivalent to $\\frac{1}{N} \\sum_{n \\le N} f(x_n) \\to \\int_0^1 f(t) \\, dt$ for every continuous function $f \\colon \\R \\to \\C$ of period $1$.\nThe functions $t \\mapsto e^{2\\pi i h t}$ are such functions, with integral $0$ for $h \\ne 0$, which gives one direction.\nFor the other direction, the condition gives the limit for every trigonometric polynomial.\nUniform approximation by the Weierstrass theorem extends it to every continuous periodic $f$.\nFinally, squeezing the indicator of an interval between continuous functions gives the counting condition.\nExample: for irrational $\\alpha$ and $x_n = n \\alpha$, the sum is a geometric sum bounded by $2 / \\abs{1 - e^{2\\pi i h \\alpha}}$, so the averages tend to $0$; this proves the [[equidistribution theorem|NCCGNTBEYV]].\nFor a rational $\\alpha = p/q$ the test fails at $h = q$, where every term equals $1$.\nThe same criterion holds in $\\R^k$, with $h$ running over nonzero vectors of $\\Z^k$ and $h x_n$ replaced by the dot product $h \\cdot x_n$.\n\n## Explanations\n\n### L1 human\n\nTo check that a sequence spreads evenly around a circle, it is enough to test it against waves.\nFor each frequency $h$, average the points $e^{2\\pi i h x_n}$ on the unit circle.\nThe sequence is evenly spread exactly when all these averages tend to zero.\n\n### L2 human\n\nThe criterion turns a question about counting points in intervals into estimates of [[exponential sums|97PBMQCPPM]].\nIt gives a short proof that $(n\\alpha)$ is equidistributed for irrational $\\alpha$.\nWith Weyl differencing, it also shows that $p(n)$ is equidistributed for a real polynomial $p$ with an irrational coefficient other than the constant term.\nIts proof is a first example of approximating indicator functions by trigonometric polynomials, a basic method of Fourier analysis.\n",
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