What is the minimum value of $S=\sum_{k=1}^n|1+z^{(2^k)}|, z\in\mathbb{C}$ in terms of $n$ ?
Experimenting on desmos and wolfram suggests the following claims:
$S$ is minimized when $|z|=1$ and $\arg{z}=\dfrac{\lfloor{\frac{2^n}{3}}+\frac{1}{2}\rfloor}{2^n}\pi$ (The numerator is the Jacobsthal sequence.)
$S_\text{min}=n-f(n)$ where $\lim_{n\to\infty}f(n)\approx0.747771497$
I do not know how to prove any of these claims.
It is easy to show that if $|z|=1$ then $S=2\sum_{k=0}^{n-1} {|\cos{(2^kx)}|}$ where $x=\arg{z}$.
This is related, but I'm not sure if it helps: $|\cos x|+|\cos 2x|+\cdots+|\cos 2^nx|\geq \dfrac{n}{2\sqrt{2}}$.
That's all I've been able to do.
(This question was inspired by similar questions such as this, this, this and this.)
