Truncated geometric progression on the complex unit circle - how to minimize the maximum real value

Question

Let $a = \text{e}^{i 2 \pi k}$, and let $n$ be a natural number. Then I have a set defined as follows:

$S = \{ \text{Re} (a), \text{Re} ( a^2 ), \ldots, \text{Re} (a^n) \}$

I want to minimize $T = \max(S)$. My question is, what are the necessary and sufficient constraints on $k$ to ensure that $T$ is minimized?

I suspect that $k$ needs to be of the form $\frac{m}{n+1}$, where $m$ is coprime to $n+1$. This results in a local minimum of $T$. I am not sure how to prove that $k$ must be of the form $\frac{m}{n+1}$ and cannot take any other form.

Answer 1

Instead of minimizing the maximum real part, we can equivalently maximize the minimum angular distance from an element of the geometric sequence to $1$.

The key fact you should convince yourself of is:

The angular distance from $a^k$ to $a^j$ is the same as the angular distance from $1$ to $a^{|k-j|}$.

If $a^k$ and $a^j$ are in the sequence with $k \neq j$, then $a^{|k-j|}$ is also in the sequence. So the angular distance from $1$ to the closest point in the progression can be no larger than the angular distance between two points in the progression.

It follows that this distance is maximized when $1$ and the points in the progression are all equally spaced around the complex unit circle, which happens precisely when $a$ is a primitive $(n+1)$st root of unity (i.e., $k=\frac{m}{n+1}$ where $\gcd(m,n+1)=1$).