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We can prove that $\exists \{c_i\}_{i=1}^{\infty} \subseteq \mathbb N$ and $c_1<c_2<c_3<...$ such that $\lim_{i\rightarrow \infty}\sin (c_i^{\alpha})=c$, $\forall c\in [-1,1]$,where $\alpha $ is a positive rational number.

But what about the limit points of { $\sin(n+n^2)\mid n\in N\} $? Is it also the closed interval $[-1,1]$?

Furthermore, what about $\sin \bigg(\sum_{i=1}^ma_in^{i}\bigg)$, where all coefficients $a_i$ are rational numbers?

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  • $\begingroup$ You may want to write $n+n^2=\frac{(2n+1)^2}4-\frac14$; not sure though if it helps... $\endgroup$ – punctured dusk Aug 16 '15 at 10:26
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After being puzzled by this question for most of the day I received a tip that this can be proved by using van der Corput's theorem (which I never heard about before), it clearly works for the $n^2+n$, but should also work for all polynomials with rational coefficients* according to my source (I guess it's done by consecutive applications).

  • As a matter of fact it's not necessary for the coefficients to be rational, a sufficient condition is that at least one coefficient is not a $q\pi$ for some rational $q$.
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    $\begingroup$ Nice answer. You might want to include the statement of van der Corput's theorem just to make it a little more self-contained. (+1) $\endgroup$ – Micah Aug 16 '15 at 20:08

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