I am looking for the solution of this optimization problem: $$ \min_{x} \max_{1 \leq r\leq N-1} \left|\frac{\sin\pi r M x}{\sin\pi r x}\right|^2$$ where $M \ll N$, $x \in \mathbb{R}$, $r \in \mathbb{N}$.
My attempts: For $M = 2$ it can be shown that as long as $x \leq \frac{1}{N-1}$, $x = 1 / N$ is the solution. Surprisingly, I found by simulations that as long as $x \leq \frac{1}{N-1}$, $x = 1/N$ is the solution for any $M \ll N$.
