The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.

This sequence seems to imply that the least number of distinct $n-$th roots of unity summing to the smallest possible nonzero magnitude is growing with something like linear growth with $n.$

I have asked about Fourier coefficients of characteristic functions of sets with a geometric series structure, that is sets of the form $\{1,2,\ldots,2^{s-1}\}$ with $n\geq 2^{s-1}$ implying $s=O(\log n),$ see the link below.

Answering the question in the title would provide a lower bound to the linked question when$n$ is a prime.

http://mathoverflow.net/questions/257558/lower-bound-on-geometric-series-structured-sets-fourier-coefficients?noredirect=1&lq=1

Related more general question:

How small can a sum of a few roots of unity be?