A paper of William Duke proves that integer points on the sphere are equidistributed:
$$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$
Up to reflections across the $x$, $y$ and $z$ axes, it seems intuitive these integer points should not accumulate in any octant.
The proof involves estimating averages over the points on the sphere, where a slow decay is obtained:
$$ \frac{1}{r_3(n)} \sum_{\xi \in \frac{1}{\sqrt{n}}V_n} u(\xi) \ll_{u,\epsilon} n^{-1/28+\epsilon},$$
where $u \in L^2(SO(3))$ is a spherical harmonic. This is enough to show the rescaled integer points become equidistributed.
If I understood the Siegel bound $r_3(n) \gg_\epsilon n^{1/2-\epsilon}$ and the difficult estimates of Ivaniec for holomorphic cusp forms, I would understand Duke's proof.
Is there really no more direct way of understanding this? It looks like we have taken a Riemann sum. We could try to measure the discrepancy
$$\left| \frac{1}{r_3(n)} \sum_{\xi \in \frac{1}{\sqrt{n}}V_n} f(\xi) - \int_{S^2} f(x) \, dS \right| \leq K_f D(V_n)$$
This is like the Erdős-Koksma-Hlawka inequality or something.