13
$\begingroup$

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 $0$.

$\endgroup$
5
  • 19
    $\begingroup$ 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 $o(1/n)$.) $\endgroup$ Oct 6, 2013 at 5:55
  • 3
    $\begingroup$ This question appears to be off-topic because it is about a well-known open question. $\endgroup$ Oct 6, 2013 at 6:41
  • 13
    $\begingroup$ math.niu.edu/~rusin/known-math/99/dense_sine $\endgroup$
    – Terry Tao
    Oct 6, 2013 at 15:17
  • 4
    $\begingroup$ This question was closed because it asks about a well-known open question. Why has it been reopened? Has someone solved it? $\endgroup$ Nov 10, 2013 at 4:35
  • $\begingroup$ Related: math.stackexchange.com/questions/221018/… $\endgroup$ Apr 22, 2014 at 20:42

1 Answer 1

7
$\begingroup$

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

$\endgroup$
1
  • 2
    $\begingroup$ 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, 2014 at 16:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.