Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity $$ \max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1} \left| 1+\omega^k+\omega^{2k}+\cdots+\omega^{(\ell-1)k}\right|^2, $$ as $n$ goes to infinity. Note that the optimization simplifies to $$ \max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1} \left| \frac{\sin (\pi \ell k/n)}{\sin (\pi k/n)}\right|^2, $$ and for concreteness case I don't mind taking $f(n)=\log^2 n, g(n)=\sqrt{n}-$not the iterated log but square of log.
I know the minimum will in general can be very small (at least for arbitrary sums of roots of unity as in question by Terry Tao) but what if we are free to look at multiple lengths for the sum, and take consecutive powers, things must improve, but how much?
Edit: The related problem $$ \min_{0<k\leq n-1}\quad \max_{f(n)\leq \ell \leq g(n)} \left| \frac{\sin (\pi \ell k/n)}{\sin (\pi k/n)}\right|^2, $$ which may be easier is also of interest.
One can take $n$ prime, if it helps.
