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.




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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](https://en.wikipedia.org/wiki/Low-discrepancy_sequence)


$$\left| \frac{1}{r\_3(n)} \sum\_{\xi \in \frac{1}{\sqrt{n}}V\_n} f(\xi) - \int\_{S^2} f(x) \, dS \right| \leq K\_f D(V\_n)$$


This is like the [Erdős-Koksma-Hlawka](https://en.wikipedia.org/wiki/Low-discrepancy_sequence#The_Erd.C5.91s.E2.80.93Tur.C3.A1n.E2.80.93Koksma_inequality) inequality or something.



Averages over integer points of the sphere
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.


