Asymptotic distribution of integral points on the three-dimensional sphere

Let $Q(X)=x_1^2+x_2^2+x_3^2$, $X=(x_1,x_2,x_3)$; $r(n)$ be the number of integral solutions of the equation
\begin{equation} Q(X)=n. \tag{1} \end{equation}
The following theorem is proved: $n=1,2,3,5,6\, (\operatorname{mod}8)$ and let $r(n,\Omega)$ be the number of integral solutions of equation (1) such that $Y=X/\sqrt{n}\in\Omega$ where $\Omega$ is an arbitrary convex domain with a piecewise smooth boundary on the unit sphere $S$: $Q(Y)=1$. Then
$$ r(n,\Omega)=\mu(\Omega)r(n)+O(n^{1/2-1/336+\varepsilon}),\qquad n\to\infty, $$
where $\mu(\Omega)$ is a measure, normalized by the condition $\mu(S)=1$. A similar result is obtained for the three-dimensional ellipsoid of general form. The mentioned theorem, in combination with the classical Guass–Siegel results on $r(n)$, yields the uniform distribution of the integral points on the three-dimensional sphere (1).