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I apologise if this is trivial or well known to be impossible:

Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$ such that for the function defined as $$ f_{a_1,\ldots,a_m}(x)=\max\left\{\left|\frac{\sin (a_1 x)}{\sin x}\right| ,\left|\frac{\sin (a_2 x)}{\sin x}\right| , \cdots,\left|\frac{\sin (a_m x)}{\sin x}\right| \right\} $$ the minimum satisfies $$ \min\{f_{a_1,\ldots,a_m}(x):0\leq ~x~\leq 2\pi\}>1? $$

Edit I can restrict to $x$ not equal to an odd multiple of $\pi/2$ as well.

More generally can the lower bound be made larger? Say larger than a small positive integer? Or a slowly growing function of $m$?

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It looks like this is impossible. Correct me if I am wrong.

For example, $f_{a_1,\cdots,a_m}(\pi/2)\leq 1$, so the minimum cannot exceed 1.

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  • $\begingroup$ Please see edit $\endgroup$ – kodlu Jul 3 '18 at 9:27
  • $\begingroup$ By considering $x_0=\frac \pi 2+\epsilon$ with $\epsilon$ sufficiently small, it still follows that $f_{a_1,\cdots,a_m}(x_0)<1$. $\endgroup$ – Cherng-tiao Perng Jul 3 '18 at 14:07

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