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  "body": "\n## Statement\n\nFor a real number $x$, let $\\{x\\} = x - \\floor{x}$ be its fractional part, where $\\floor{x}$ is the [[floor|08GR54CKEA]] of $x$.\nLet $k \\ge 1$ and let $\\theta_1, \\ldots, \\theta_k$ be real numbers such that $1, \\theta_1, \\ldots, \\theta_k$ are [[linearly independent|0EVHJFTG9Z]] over $\\Q$.\nThen the set\n$$\\set{(\\{n\\theta_1\\}, \\ldots, \\{n\\theta_k\\}) : n \\ge 1 \\text{ an integer}}$$\nis [[dense|0D198GPMTJ]] in $[0, 1]^k$.\nFor $k = 1$: if $\\theta$ is an [[irrational number|080T2TXVQV]], then the fractional parts $\\{n\\theta\\}$, $n \\ge 1$, are dense in $[0, 1]$.\n\n## Remarks\n\nFor $k = 1$, fix $\\varepsilon \u003e 0$.\nThe pigeonhole principle, as in [[Dirichlet's approximation theorem|04JAY9DGB7]], gives an $m \\ge 1$ with $0 \u003c \\{m\\theta\\} \u003c \\varepsilon$ or $1 - \\varepsilon \u003c \\{m\\theta\\} \u003c 1$.\nThe value $\\{m\\theta\\}$ is never $0$, since $\\theta$ is irrational.\nThe multiples of $m$ then move around the circle in steps shorter than $\\varepsilon$, so they come within $\\varepsilon$ of every point.\nIn general dimension, the theorem follows from the $k$-dimensional form of Weyl's criterion, which gives even equidistribution in $[0,1]^k$.\nThe independence hypothesis is necessary.\nSuppose $a_0 + a_1 \\theta_1 + \\cdots + a_k \\theta_k = 0$ with integers $a_i$ not all $0$; then some $a_i$ with $i \\ge 1$ is nonzero.\nAll points then satisfy $a_1 \\{n\\theta_1\\} + \\cdots + a_k \\{n\\theta_k\\} \\in \\Z$, so they lie on finitely many hyperplanes and are not dense.\nExample: $1, \\sqrt{2}, \\sqrt{3}$ are linearly independent over $\\Q$, so the points $(\\{n\\sqrt{2}\\}, \\{n\\sqrt{3}\\})$ are dense in the unit square.\n\n## Explanations\n\n### L1 human\n\nWalk around a circle of length $1$ in steps of an irrational length $\\theta$.\nYou never come back to the start, and in the long run you pass as close as you like to every point of the circle.\nKronecker's theorem says the same for several walks at once, as long as the step lengths have no rational relation between them.\n\n### L2 human\n\nThe theorem is the qualitative form of the [[equidistribution theorem|NCCGNTBEYV]] and its higher-dimensional versions.\nIt is a statement about the orbit of a rotation of the torus $\\R^k / \\Z^k$: under the independence condition, the orbit of $0$ is dense.\nIts higher-dimensional form is used in dynamics and in the theory of almost periodic functions.\n",
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