Let $n$ be an integer, $n\ge2$. Let $m$ be a positive integer, $m\le n$, having no common factor with $n$. How can we select $m$ distinct complex $n$th roots of unity in such a way as to minimise the modulus of their sum?
The condition that $m$ and $n$ be coprime is included because if $m$ and $n$ have a common factor, it is easy to see that we can always find a sum which vanishes.
The obvious answer is that the roots chosen should be spread around the unit circle as evenly as possible. This does not appear to be correct: for example, if $n=9$, $m=4$ and if we write $\alpha=e^{2\pi i/9}$, then $$|1+\alpha^2+\alpha^4+\alpha^6|=2\cos\frac{2\pi}{9}-1=0.5321\ ,$$ while $$|1+\alpha+\alpha^4+\alpha^6|=2\cos\frac{\pi}{9}-2\cos\frac{2\pi}{9} =0.3473\ .$$