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While trying to generalise a mechanics exercise with a friend, we came up with this question, in an attempt to understand wether sine curves with irrational period defined inside an annulus will end up 'filling up' the space in the annulus.

Mathematically, let $x \in \mathbb{R} \setminus \mathbb{Q}$:

Is $S_x=\{\mathrm{frac}(n x): n \in \mathbb{N}\}$ dense in $[0,1] \subset\mathbb{R}$?

This is equivalent to asking wether sequences of fractional parts of multiplies of one fixed irrational number can approximate every number in $[0,1]$.

My intuition says this may be true, but I see no reason why there shouldn't be some 'holes' in $S$. If that's the case, which kind of properties of $x$ determine the how $S_x$ is filled up?

Felipe Jacob
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  • You say this question was the outgrowth of trying to generalize a mechanics exercise? By that do you mean a classical mechanics problem? Do you happen to remember what the problem was? I realize this was asked a long time ago, but I am rather curious to know. – user193319 May 11 '17 at 12:02

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There can be no holes.

You can see that by running Euclidean algorithm on $1$ and $x$. If $x$ is irrational, it never stops and produces arbitrarily small number $\varepsilon=a+bx$ for $a,b\in\mathbb Z$.

Then $ka+kbx=k\varepsilon$ which means every multiple of $\varepsilon$ is covered. (You set $n\in\mathbb N$, not $\mathbb Z$, but that's just a technicality.)

user2345215
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