Let $0<k<n$ be positive integers. I want to find a subset of $k$ distinct elements of the set of $n$-th roots of unity which is "uniformly spread over the unit circle".
I do not have a precise definition for "uniformly spread over the unit circle", so I provisionally change the problem to minimizing the module of the sum:
Problem Let $\zeta_n=e^{2i\pi n}$ and $0<k<n$. Find distinct $0\leq i_1<\ldots<i_k<n$ that minimize: $$ \left\vert{\sum_{j=1}^k\zeta_p^{i_k}}\right\vert. $$
Question What is the solution to the problem above? Is there always a solution of the form: $$ \{i_j\}_{j=1,\ldots,k}=\{hj\bmod n\}_{j=1,\ldots,k} $$ for some integer $h$? (up to the order of the $i_j$'s)
Extra question What are some better definitions of "uniformly spread"? With my definition, if both $k$ and $n$ are even then any subset symmetric with respect to the origin has sum equal to zero, even if it is not necessarily "uniformly spread".
