Is $|\sum_{k=1}^{N}e^{2\pi i x_{k}}|$ bounded (with respect to other conditions)?

Question

Suppose $\gamma$ is a real number such that $0 < \gamma \leq \frac{1}{2}$. Suppose $x_{1},x_{2},...$ is a sequence in $\mathbb{R}$ such that $x_{j+1}-x_{j} \in [\gamma , 1-\gamma]$.

Is it true that $\lim_{N\rightarrow \infty}\frac{1}{N}\sum_{k=1}^{N}e^{2\pi i x_{k}} = 0$ ?

Is it true that $|\sum_{k=1}^{N}e^{2\pi i x_{k}}|$ is a bounded sequence with respect to $N$ ?

Is it true that $|\sum_{k=1}^{N}e^{2\pi i x_{k}}| \leq \frac{1}{\sin(\pi \gamma)}$

For example if if we take the sequence $\gamma,2\gamma,...$then $|\sum_{k=1}^{N}e^{2\pi i k\gamma}| \leq \frac{1}{\sin(\pi \gamma)} < \infty$ by geometric summation.

Answer 1

Let $\gamma = 1/6$

Define $x_0 = 1/6$

$$x_{j+1} = \begin{cases}x_j+4/6 & \text{ if $j$ odd} \\x_j+2/6 & \text{ if $f$ even} \end{cases}$$

Then $x_1 = 5/6$, $x_2 =7/6$, $x_3 = 11/6$ and so on.

In the complex plane, the sequence oscillates between two points $e^{i\pi/3}$ and $e^{-i\pi/3}$.

In fine, the real part is always $\cos \pi/3$, the limit in the first question is not $0$ and the sum of the terms is not bounded.

Answer 2

If one puts $x_k = f(k)$ then there are some estimations (due to Van der Corput and Weyl) of the sum with respect to derivative of $f$ ( plus, as far as I can remember f has to be monotonic). In particular, there is so called "first derivative test" which states that if $f$ is monotonic and its derivative is bounded from below by $M$ then the sum is bounded by $1/M$.