Consider the following set of complex numbers in the upper half plane: $$\{ic_n \pm \gamma_n: 0 \leq n \leq N, \hspace{1 mm} c_0=\gamma_0=0, c_n,\hspace{1 mm} \gamma_n>0\}.$$ Assume that this set has very small Fourier coefficients: $$\sum_{n=0}^{N} e^{im(ic_n \pm \gamma_n)}= 1+ 2\sum_{n=1}^{N} e^{-c_n m}\cos(\gamma_n m)= O(e^{-m}), \tag{$\star$}\label{star}$$ for $1 \leq m <M$ with $M \asymp N$.
I would like to show that $\gamma_n$ are uniformly distributed.
If $c_n=0, 1\leq n \leq N$ I can prove this using the Erdős–Turán inequality. This comes from the fact that we can approximate the characteristic function of an interval in terms of a trigonometric polynomial. For example we have $$\mathbb{1}_{(-\delta, \delta)}(\gamma) \simeq 2\delta + 2 \sum_{n=1}^{N} \frac{\sin(2\pi n \delta)}{\pi n} \cos(2 n \gamma).$$ If we sum the above over $\gamma_n$, assuming $c_n=0$, and using \eqref{star}, we get $\sum_{n}\mathbb{1}_{(-\delta, \delta)}(\gamma_n) \simeq 2\delta(2N+1)$.
We might try to imitate this and define a complex function $$I_{(-\delta, \delta)}(\gamma + ic) :=2\delta + 2 \sum_{n=1}^{N} \frac{\sin(2\pi n \delta)}{\pi n} e^{-2nc} \cos(2 n \gamma).$$ Summing over $\{ic_n \pm \gamma_n\}$ gives us a similar result. However, $I_{(-\delta, \delta)}(\gamma + ic)$, for large $c$ no longer approximates $\mathbb{1}_{(-\delta, \delta)}(\gamma)$. Is there a fix to this?
It is important to note that having $c_0=\gamma_0=0$ is essential. Otherwise assuming $c_n$ are large we can have \eqref{star} without any uniformity in the distribution of $\gamma_n$.
Also, \eqref{star} is a very strong condition, if $c_n=0$, we could have concluded uniform distribution of $\gamma_n$ from a less severe bound on the RHS of \eqref{star}. Therefore it is reasonable to expect that cancellation of order $e^{-m}$ on the $m$-th Fourier coefficient, should determine the distribution of the set.
If we consider this problem as finding the distribution in a box in $\mathbb{R}^2$, we need to know $$\sum_{n=0}^{N} e^{2\pi i\langle h\cdot(c_n, \pm \gamma_n)\rangle},$$ for many $h \in \mathbb{R}^2$, which is a information we do not have. Moreover, I am not sure if this is a right approach and we can get something meaningful in $\mathbb{R}^2$. My guess is $c_n$ very slowly goes from $0$ to $1$, as $n$ gets larger and $\gamma_n$ are uniformly distributed. For example $c_n= (\tfrac{n}{X})^2$ and $\gamma_n= \tfrac{n}{X}$ with large $X$ satisfy the condition.
