Let $\omega, \omega^2,\dots,\omega^{n-1},1$ be $n$-th roots of unity, and consider $$S_n(m):=\max_{1 \le i_1 < i_2 <\dots<i_m \le n}|\omega^{i_1}+\dots+\omega^{i_m}|$$ i.e. the maximal absolute value of the sum of $m$ different $n$-th roots of unity.
Based on computer calculations, I conjecture that $S_{2n+1}(m)=\phi_{n}(m)$, where $\phi_n(m)=\frac{\sin\left(\frac{m\pi}{2n+1}\right)}{\sin\left(\frac{\pi}{2n+1}\right)},2\le m\le n $ are the so-called 'golden numbers' which generalise the golden ratio. Geometrically, these are the ratios of the lengths of diagonals in a regular $(2n+1)$-gon to the side lengths.
Is anything known about this?