For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed with respect to the normalised Lebesgue's mesure on the sphere as $N$ goes to infinity. In fact this is true if $N$ goes to infinity and affords values for which $\Omega_N$ is empty.
For $n=4$ the result is also true when the limit is taken such as the odd part of $N$ goes to infinity.
For $n=5$ this result is unconditionaly true.
Unfortunatly I can't find any references for the case $n=2$, and it doesn't seem trivial to me. I am asking whether the set $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^2~\text{and}~\|\alpha\|^2=x_1^2+x_2^2=N\right\}$$ is uniformly distributed with respect to the Lebesgue's mesure on $\textbf{S}^1$. I can guess that it will not be true for all $N$ thanks to the formula $$|\Omega_N|=4\sum_{\substack{d|N \\ d~\text{odd}}}{(-1)^{\frac{d-1}{2}}}. $$
Many thanks for your help!