Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers?

The existence of the limes inferior follows from Dirichlet's approximation theorem, but the problem is to prove that it is $0$.

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    $\begingroup$ That's equivalent to asking whether $n\pi$ comes within $o(1/n)$ of an integer, which is a well-known open problem; it's expected to be true (if $\pi$ is replaced by a random number then it's true with probability $1$) but well beyond what can be proved by known methods. (The Dirichlet result you quote gives $O(1/n)$ in place of the desired $o(1/n)$.) $\endgroup$ – Noam D. Elkies Oct 6 '13 at 5:55
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    $\begingroup$ This question appears to be off-topic because it is about a well-known open question. $\endgroup$ – Gerry Myerson Oct 6 '13 at 6:41
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    $\begingroup$ math.niu.edu/~rusin/known-math/99/dense_sine $\endgroup$ – Terry Tao Oct 6 '13 at 15:17
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    $\begingroup$ This question was closed because it asks about a well-known open question. Why has it been reopened? Has someone solved it? $\endgroup$ – Gerry Myerson Nov 10 '13 at 4:35
  • $\begingroup$ Related: math.stackexchange.com/questions/221018/… $\endgroup$ – Per Alexandersson Apr 22 '14 at 20:42

This is an open question. See the comments for more details.

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    $\begingroup$ Just a technical answer (community wiki) so that this question can be considered answered and so that exact duplicates can be linked it. $\endgroup$ – Joël Apr 22 '14 at 16:21

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