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Given $a$ and $b$ irrational numbers with $a/b$ also irrational, how do you prove that $( \{ na\} , \{ nb \})$ is dense in $[0,1] * [0,1]$ , where $n$ ranges over the integers?

$\{x\}$ is the fractional part of $x$ .

I'm also curious about the general case, with $n$ irrational numbers , linearly independent over $Q$ , resulting in density over $[0,1]^n$ .

flag	asked Nov 2 at 11:55 Zero 33●3

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You don't. Take $a=\sqrt 2$ and $b=1+\sqrt 2$. The correct condition is that $1,a,b$ are linearly independent over $\mathbb Q$. Weil's criterion does the trick in no time giving even the equidistribution property. – fedja Nov 2 at 12:47

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@fedja, you meant Weyl's... – Asaf Nov 2 at 12:52

Right! $\\ \ \ $ – fedja Nov 2 at 12:59

For the general (and correct) statement you can find a quick proof here: mathoverflow.net/questions/106819/… – GH Nov 2 at 18:38

M Khan · Answer 1 · 2012-11-02 12:19:02Z UTC

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Just google Kronecker's approximation theorem.

answered Nov 2 at 12:19

M Khan
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