Optimization Problem for Exponential Polynomials This question was asked on mathoverflow more than a year ago, with no answers
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.
 A: If $\omega$ is an $n^{th}$-root of unity, then so is $\omega^m$ for any integer $m$. So your minimum is just over all the $n^{th}$ roots of unity $\omega$ other than $1$. That is $$\min_{\underset {\omega \ne 1}{\omega^n = 1}}\left|1 + \dots +\omega^{\ell - 1}\right|^2$$
But the expression being minimized can be expressed as $$\left|\frac{\omega^{\ell} - 1}{\omega -1}\right|^2$$
This is minimized by sending the top to $0$ and maximized by sending the bottom to $0$. If $\omega = e^{i2k\pi/n}$, then the expression is minimized over $k$ by choosing $k$ as close to $\frac n\ell$ as possible. The result minimum is maximized over $\ell$ by choosing $\ell$ so that $\frac n\ell$ is as far from integer as possible.
And finally, we look at what happens as $n \to \infty$. As $n$ gets larger, $\omega$ divides the circle up into smaller and smaller parts, which means that we can find values for $\omega^\ell$ closer and closer to $1$, thus making the minimum ratio $k$ smaller, regardless of which $\ell$ we pick.
I.e., the limit you seek is $0$.
