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Given an even positive integer $n$, I am interested in the behaviour of the finite sequence $C_n=\{\cos(k \pi/n)\},$ where $k=1, \ldots, n/2 -1$. In particular, I am looking for an approximation to the size $s_n$ of $C_n \cap (a,b)$ for an interval $(a,b) \subset (0,1)$ and $a,b \approx 1$. If, for instance, $|a-1|=|a-b|=O(1/n^2)$, can one prove something like $s_n=O(n)$ or $s_n=O(n^2)$?

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    In your specific case, $b-a=O(1/n^2)$... See my answer.2012-09-13

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In the example given in a comment, $a$ and $b$ depend on $n$ (this was not the setting of the question asked, but whatever), with $a_n=1-C/2n^2$ and $b_n=1-C/4n^2+o(1/n^2)$ for some fixed $C$.

Then for every fixed $k$, $\cos(k\pi/n)=1-k^2\pi^2/2n^2+o(1/n^2)$ hence $a_n\lt\cos(k\pi/n)\lt b_n$ roughly for $C/4\leqslant k^2\pi^2/2\leqslant C/2$, that is, $\sqrt{C/2}/\pi\leqslant k\leqslant\sqrt{C}/\pi$.

In particular, for every fixed $C$, $s_n$ stays bounded when $n\to\infty$ (that is, $s_n=O(1)$) and $s_n\to s_\infty(C)$. Furthermore, when $C\to\infty$, $s_\infty(C)\sim\lambda\sqrt{C}$, with $\lambda=(1-\sqrt2/2)/\pi$, hence $\lambda\approx0.09323$.