Bounding exponential sums

Question

There is classical inequality that seems to often appear: $$\sum_{i=1}^N \exp(2i\pi n \theta) \leq \|\theta\|^{-1}$$ where $\|\theta\|$ denotes the distance to the closest integer. I do understand that it can be proven by writing a sum of geometric sequence, and then using Euler's formula to get that it is bounded by $2/\sin(\pi \theta)$, but then I do not see that easily the relation with $\|\theta\|$.

Answer 1

Note that the sum is even periodic function of $\theta$ with period 1. So it is enough to prove the estimate for $\theta \in [0,1/2].$ Over this interval $\sin(\pi \theta)/2$ majorizes the distance to the origin, i.e., $\| \theta \|:$

So, we have $$ \|\theta\| \leq \frac{\sin(\pi \theta)}{2}. $$ Now take reciprocals.