Consider elements $x,z \in \mathbb{Z}[i]$, the Gaussian integers. Let $S(T_1, T_2)$ be the subset of $\mathbb{Z}[i] \times \mathbb{Z}[i]$ consisting of those elements $(x,z)$ such that $T_1 < |x| \leq 2T_1, T_2 < |z| \leq 2T_2$. I want to estimate the cardinality of the set

$$\displaystyle S_k(T_1, T_2) =\{(x,z) \in S(T_1, T_2) : \left \lvert \Re(xz^k) \right \rvert \leq M \}, k \geq 2.$$

Note that if $M$ is close in size to $T_1 T_2^k$, then the restriction that $\lvert \Re(xz^k) \rvert \leq M$ is hardly a restriction at all and the obvious bound of $|S(T_1, T_2)|$ will give the correct order of magnitude. However when $M$ is very small compared to $T_1 T_2^k$ then $|S_k(T_1, T_2)|$ should be substantially smaller than $|S(T_1, T_2)|$.

If we write $x = r_1 e^{i \theta_1}$ and $z = r_2 e^{i \theta_2}$, then the condition $|\Re(xz^k)| \leq M$ can be written as $|r_1 r_2^k \cos(\theta_1 + k \theta_2)| \leq M$. If $M$ is small compared to $T_1 T_2^k$, then this is essentially saying that the angle $\theta_1 + k\theta_2 \pmod{\pi}$ is close to $\pi/2$.

Is there a good way to estimate the size of $S_k(T_1, T_2)$?