Simple-looking bound on root of unity

Question

I am trying to prove some bound and stuck with the following:

If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$

($n,N$ are integers)

How can I prove it?

Answer 1

$\sqrt{2-2\cos \frac{2\pi n}{N}}\geq\frac{n}{N}$

Let $\alpha=n/N$. We want to prove $\sqrt{2(1-\cos (2\pi \alpha))}\geq\alpha$ for $-\frac{3}{4}\leq\alpha\leq\frac{3}{4}$ This is plainly true for $\alpha<0$, so we will only conider positive $\alpha$.

Note $\sqrt{2(1-\cos (2\pi\alpha))}\geq\alpha\iff 1\geq \frac{1}{2}\alpha^2+\cos(2\pi\alpha)$. Let's consider the derivative of $f(\alpha)=\frac{1}{2}\alpha^2+\cos(2\pi\alpha)$: $$ f'(\alpha)=\alpha-2\pi\sin(2\pi\alpha) $$

There is only one root in $0<\alpha<3/4$, and we may note $f''(\alpha)=1-4\pi^2\cos(2\pi\alpha)$, so it is reasonable to deduce that there will be a local minimum for $f(\alpha)$ in the desired range. This means we only need to check $f(0)$ and $f(3/4)$.

$$1\ge \frac{1}{2}\cdot0^2+\cos(2\pi\cdot 0)=0+1=1$$ $$ 1\ge \frac{1}{2}\left(\frac{3}{4}\right)^2+\cos\left(2\pi\frac{3}{4}\right)=\frac{9}{32}-0=\frac{9}{32} $$

So the inequality is satisfied for $|\alpha|\le \frac{3}{4}$