First  let me define *Difference multiset* for a set of integers 
$$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j
$$
as below:
$$
D = \{p_i-p_j \mod N ,\quad i \ne j\}
$$
I know that the minimum of $f(x)$ is same for all $P$'s having same difference multiset & and also the optimal $x$ (vector) for all of them is same for all of them up to permutation of elements, where $f(x)$ is as below:
$$\large
f(x) = {\sum_{j=1}^K \sum_{k=1}^K x_j x_k e^{\frac{i2\pi l(p_j-p_k)}N} \over \left( \sum_{i=1}^K x_i\right)^2}
$$
where $l$ is a constant integer

I achieved this result from simulations. I'm looking for a proof or even a justification which helps me prove it.