Find $x$ that $w_n^{-l}+w_n^{-r}=w_n^{-x}$ in circulant matrix

Question

In the circulant matrix we have $w_{n}^{-l}=\exp\left\{ -\frac{2\pi i}{n}l\right\}$ where the $n$-th roots of unity and $i$ is the imaginary unit. I'm trying to figure if there is a closed form of the sum of two such values. In other words, I'm trying to find if there is $x$ so: $$ w_n^{-l}+w_n^{-r}=w_n^{-x} $$ I can't seem to figure it out. Is it possible?

Answer 1

Let $\omega:=\exp(\frac{2\pi i}{n})$.

You are asking whether two powers of $\omega$ can add to a third one so: $$ \omega^r +\omega^s = \omega^t. $$

By multiplying through by $\omega^{-t}$ we have $$ \omega^R +\omega^S = 1 $$ where $R=r-t,S=s-t$.

Now the sum of $\exp(i\theta)$ and $\exp(i\phi)$ is a multiple of $\exp(i\frac{\theta+\phi}{2})$. So we must have that $S=-R$.

Let $z:=\omega^R$; we now have that $z^2-z+1=0$, so that $z=\exp(\frac{2\pi i}{6})$.

For solutions to the original problem to exist you need to have $n=6R$ for some $R$, and the solutions are then $r=R+t$, $s=-R+t$, $t$ an arbritary integer.