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For any positive number $c$ and positive integer $k$, show that there are infinitely many positive integer $n$ that satisfy

$$|\sin(n^k)| < c$$

I proved the case when $k=1$, but I can't solve when $k\geq2$.

choco_addicted
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김동하
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1 Answers1

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Due to Weyl's inequality and the equidistribution theorem, the irrationality of $\pi$ implies that $e^{i f(n)}$ is equidistributed (hence dense) on the unit circle for any $f(x)\in\mathbb{Z}[x]$. Since the map $z\to \text{Im}(z)$ is continuous it preserves density and the claim is a straightforward consequence.

Jack D'Aurizio
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