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I want to answer the following simple question:

Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^3$ are inside the ellipsoid?

The answer is given by the volume of the ellipsoid (of order $Z^{3/2}$) plus an error term, and I am interested in the strongest possible error term, which I hope would be $O(Z^{3/4})$ or better. The error term should also depend on $Q$ in some fashion which is explicitly described.

It seems like this question must have been extensively studied by now. But I was unable to find a suitable reference. Two references that approach what I'm looking for are:

  1. Schmidt (Lemma 1, Northcott's theorem on Heights II. The quadratic case, Acta Arith., 1995) proves that if $S$ is (for example) a convex body in $n$ dimensions, lying in a ball of radius $r$, and $\Lambda$ is any lattice of rank $r$, the number of lattice points of $\Lambda$ inside $S$ is $$\frac{\rm{Vol}(S)}{\det(\Lambda)} + O\left(\frac{\lambda_n r^{n - 1}}{\det(\Lambda)}\right),$$ where $\lambda_n$ is the largest successive minimum of $\Lambda$. After a change of variable this is equivalent to my question, and the answer is of the form I am looking for -- but I believe a better error term should be possible when $S$ is an ellipsoid.

  2. Bentkus and Götze (main theorem, On the lattice point problem for ellipsoids, Acta Arith., 1997) formulate the question in the same way that I did, and obtain a power savings of $Z$ in the error term (as opposed to $Z^{1/2}$, which is what can be deduced from Schmidt's paper or any similar geometry of numbers method). Writing the quadratic form as $\langle Qx, x \rangle$, the theorem specificies the dependence of the error term on the eigenvalues of $Q$. This is of the shape that I'm interested in, but the paper requires that the dimension be at least 9.

I skimmed through the likelier looking references in the latter paper, as well as the books on counting lattice points of Krätzel and Fricker, and I found nothing. The problem of counting lattice points in three-dimensional ellipsoids is addressed, but if the dependence of the error term on the ellipsoid is made explicit, then I missed it.

Finally I should mention that I know how to solve my own problem: write down the Epstein zeta function associated to $Q$, and estimate its partial sums using Perron's formula and the method of Landau and Chandrasekharan-Narasimhan. (The methods in the books above don't use Epstein zeta functions, but after a brief reading they seem like equivalent arguments that don't go through the usual zeta function machinery.) The dependence of the error term on $Q$ can be described in terms of the functional equation for the Epstein zeta function.

But I would prefer to avoid inventing the wheel if I can help it. Does anyone know if such a theorem has already been proved? Thank you very much.

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    $\begingroup$ suspect this is not what you want, however: zakuski.math.utsa.edu/~kap/Duke_Schulze_Pillot_1990.pdf $\endgroup$
    – Will Jagy
    Commented Feb 28, 2016 at 20:46
  • $\begingroup$ Not what I'm looking for, but definitely interesting! (Indeed, their result seems much harder than what I'm looking for.) $\endgroup$
    – Frank Thorne
    Commented Feb 28, 2016 at 20:58
  • $\begingroup$ Just checking, does your quadratic form have integer coefficients? $\endgroup$
    – Will Jagy
    Commented Feb 28, 2016 at 20:59
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    $\begingroup$ Frank, I think this might reveal something about the minimum size of the error term. Do your estimates as carefully as you can for the two forms $x^2 + 3 y^2 + 36 z^2$ and $3 x^2 + 4 y^2 + 9 z^2.$ The splitting integers for the genus are squares, for other positive integers the number of lattice points on the ellipsoid come out the same. The spinor exceptional integers are $w^2,$ where all prime factors $p$ of $w$ have $p \equiv 1 \pmod 3.$ My guess is that this shows the worst possible comparison for two forms with equal determinants. $\endgroup$
    – Will Jagy
    Commented Feb 28, 2016 at 22:41
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    $\begingroup$ @GerryMyerson: True, nobody can get the conjectured error term. But where "naive geometry of numbers" gets you an error term of order $X^{1/2}$, you can smooth the characteristic function of the circle, use Poisson summation, nontrivially analyze the resulting Bessel functions, and get an error term of $X^{1/3}$. (The Epstein zeta function machinery is roughly equivalent to this.) You can save a little bit more with effort, but I'd be very happy with an analogue of this $X^{1/3}$. $\endgroup$
    – Frank Thorne
    Commented Feb 29, 2016 at 13:40

1 Answer 1

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As far as I can tell, the best result (too painful to typeset) is in:

Jingwei Guo, On lattice points in large convex bodies, Acta Arith. 151 (2012), no. 1, 83--108.

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    $\begingroup$ Thanks for this. Unfortunately both this paper and the paper of Müller cited in the introduction (not to mention pretty much every other reference I came across) allow the constant to depend on the shape of the ellipsoid in an unspecified fashion. This dependence would be worked out with some effort, for example via my (3) above, but this is what I was hoping to avoid. But thank you anyway. $\endgroup$
    – Frank Thorne
    Commented Feb 29, 2016 at 13:35

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