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 $0$.
$\begingroup$
$\endgroup$
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 $o(1/n)$.) $\endgroup$– Noam D. ElkiesCommented Oct 6, 2013 at 5:55
-
3$\begingroup$ This question appears to be off-topic because it is about a well-known open question. $\endgroup$– Gerry MyersonCommented Oct 6, 2013 at 6:41
-
13$\begingroup$ math.niu.edu/~rusin/known-math/99/dense_sine $\endgroup$– Terry TaoCommented 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? $\endgroup$– Gerry MyersonCommented Nov 10, 2013 at 4:35
-
$\begingroup$ Related: math.stackexchange.com/questions/221018/… $\endgroup$– Per AlexanderssonCommented Apr 22, 2014 at 20:42
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
This is an open question. See the comments for more details.
-
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. $\endgroup$– JoëlCommented Apr 22, 2014 at 16:21