Take $n\geq 1$, and let $c_1,...,c_n$ be $n$ non-negative numbers.

For every $\phi_1,...,\phi_n$, the formulae $$v_k=\sum_{j=1}^n c_k \omega^{jk+\phi_k}$$ define a vector $v\in \mathbb{R}^n$, where $\omega=e^{2\pi \imath/n}$. Under what conditions is it possible to choose the $\phi_k$ so that $v_k\in \mathbb{Z}$ for every $1\leq k \leq n$, or $q v_k\in \mathbb{Z}$ for the smallest integer $q$ not depending on $k$ possible?

This problem is maybe not posed properly, it comes from the fact that I only know the auto-correlogram $\gamma_t=\sum_k v_k v_{k+t}$ of $v$, which gives me only access to the Fourier moduli of $v$.