The question refers to the following paper by Jean Bourgain: http://arxiv.org/abs/math-ph/0011053

Specifically, I can't derive the following inequality in (1.20):

\begin{equation} \left|\sum_{|k|\geq1}\varphi(k)\left[\frac{1}{R}\sum_{1\leq r\leq R}e^{2\pi i r k\omega}\right]e^{2\pi i k \theta}\right|\leq C\sum_{1\leq|k|\leq K}\frac1{|k|}\frac1{R\|k\omega\|+1}+\left|\sum_{|k|>K}\varphi(k)\left[\frac{1}{R}\sum_{1\leq r\leq R}e^{2\pi i r k\omega}\right]e^{2\pi i k \theta}\right| \end{equation}

where $\varphi(k)=\mathcal O\left(\frac1{|k|}\right)$, $R<K$, $\omega\in\mathbb T$, $k\in\mathbb Z$ and $\|\cdot\|$ denotes the distance to the nearest integer.

What is certainly true is $\sum_{1\leq r\leq R}e^{2\pi i r k\omega}=\frac{\sin(\pi R k\omega)}{\sin(\pi k\omega)}e^{i\pi (R+1)k\omega}$ hence $\left|\sum_{1\leq r\leq R}e^{2\pi i r k\omega}\right|\leq\frac1{2\|k\omega\|}\leq\frac1{\|k\omega\|}$.

Either it's trivial and I don't see it or...well...

Thanks for your help!