Averages over integer points of the sphere 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.
 A: First of all note that $n$ has to satisfy some trivial local conditions, otherwise $r_3(n)$ will be zero or too small. Let us assume that $n\equiv 1\pmod{4}$ for simplicity, then the theorem holds. Let me also remark that the best known exponent of $n$ in your second display is $-\frac{1}{12}+\epsilon$, see the work of Conrey-Iwaniec (Annals, 2000) and Matt Young (arXiv:1405.5457). The Grand Lindelöf Hypothesis (a consequence of the Grand Riemann Hypothesis) would yield the exponent $-\frac{1}{4}+\epsilon$.
As far as I know, Duke's proof (Inventiones, 1988) is still the only proof, and it was a real breakthrough at the time, based on the work of Iwaniec (Inventiones, 1987). More precisely, the Weyl-type sum in your second display is directly related to a central value of a twisted modular $L$-function, and now there are several methods and results about the subconvexity of these $L$-values (see the work of Duke, Friedlander, Iwaniec, Bykovskii, Conrey, Harcos, Blomer, Michel,  Sarnak, Cogdell, Piatetski-Shapiro, Venkatesh, Wu, Maga, Hoffstein, Hulse, Petrow, Young).
Linnik could prove the theorem under the condition that $n$ is a quadratic residue for any fixed odd prime, which of course implies (by throwing in more and more odd primes) that 100% of the $n$'s with $n\equiv 1\pmod{4}$ satisfy the equidistribution property. The work of Einsiedler, Lindenstrauss, Michel, Venkatesh builds on Linnik's ideas, and they develop them further, but I don't think they got rid of Linnik's congruence condition in the original Linnik problem. Correct me if I am wrong.
A: What you want to look at next is Duke and Schulze-Pillot, Invent. math. 99, pages 49-57, 1990, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. 
Much later, after Ellenberg-Venkatesh, Schulze-Pillot got involved and cleaned up the bounds and provided an elementary exposition for part of it. 
The distinction, really, is that in co-dimension 2, such as ternaries representing unaries (i.e. numbers) there is a contribution of the spinor genus, which caused no trouble in Linnik's original. In co-dimension 3 or larger, as in quaternaries representing numbers, there is no such contribution, the only obstacle is from anisotropy (which does not happen in codimension 4 or larger), and, morally, all lattices (positive forms) with sufficiently large minimum are represented.  Here is the arXiv preprint: http://arxiv.org/abs/0804.2158 where the summary is Theorem 11. 
