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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.

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  • $\begingroup$ If you could estimate the left side of that display, it would give you a lower bound for the discrepancy. But to prove uniform distribution, you want an upper bound. $\endgroup$ Mar 6, 2015 at 5:51
  • $\begingroup$ The more direct way to understand this result is by associated equidistribution theorem about measures, which was pioneered by Linnik and Skubenko, and came to complete in a work by Einsiedler-Lindenstrauss-Michel-Venkatesh. Duke's proof is sharper somehow (one should be think about those estimates you've mentioned as analogues of Weyl sums), but need Iwaniec's estimate which is very difficult, and moreover can't be easily generalized. I guess the Michel-Venkatesh survey paper is a good place to start reading about that. $\endgroup$
    – Asaf
    Mar 6, 2015 at 6:44
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    $\begingroup$ In my modular forms course 2013-14 I did the much-easier analogous case of dimension 8n... which illustrates many of the principles without the delicacies and difficulties of the 3-D case. (math.umn.edu/~garrett/m/mfms) $\endgroup$ Mar 7, 2015 at 0:14

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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.

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  • $\begingroup$ ELMV indeed proved Duke's without Linnik's condition (which basically boils down to the fact that you have a split torus in some finite place), see ma.huji.ac.il/~elon/Publications/Erg-Duke.pdf. In subsequent works (Aka-Einsiedler and others), while investigating finer properties relating to Linnik's problems (such as joint equidistribution of lattice shapes) , in the low-dimensional settings, as the measure-classfication realys on diagonal dynamics results (Lindenstrauss et-al.) they need to generate positive entropy and occasionally the Linnik condition arises. $\endgroup$
    – Asaf
    Mar 7, 2015 at 6:11
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    $\begingroup$ @Asaf: What you say is not quite correct. There are several similar results that follow from the works of Iwaniec and Duke. Equidistribution of lattice points on the sphere, Heegner points, and equidistribution of closed geodesics. For the equidistribution of closed geodesics analog indeed ELMV give a proof based on Linnik's ideas, but for the problem discussed here GH from MO has described the situation carefully. $\endgroup$
    – Lucia
    Mar 7, 2015 at 6:30
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    $\begingroup$ @Asaf: I think the work of ELMV does not cover equidistribution on the sphere without the Linnik condition. They treat closed geodesics, hence in their situation there is a split torus at the real place automatically. My understanding is that the Linnik condition creates a split torus at a finite place for the group SO(3), making the problem similar to the variant on closed geodesics. I might be wrong, of course. $\endgroup$
    – GH from MO
    Mar 7, 2015 at 7:30
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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.

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