Best known results on lattice point counting inside an ellipse

Question

Consider a positive definite and primitive integral binary quadratic form

$\displaystyle f(x,y) = ax^2 + bxy + cy^2$

which is reduced in the Gaussian sense: that is, we have the inequalities

$\displaystyle |b| \leq a \leq c, a > 0.$

Consider the counting function

$\displaystyle N_f(X) = \# \{(u,v) \in \mathbb{Z}^2 : f(u,v) \leq X \}.$

It is elementary that

$\displaystyle N_f(X) \sim \frac{\pi X}{\sqrt{4ac - b^2}}$

Let

$\displaystyle E_f(X) = \left \lvert N_f(X) - \frac{\pi X}{\sqrt{4ac - b^2}} \right \rvert.$

It is equally elementary to show that

$E_f(X) = O \left(\frac{X^{1/2}}{a} \right).$

However, surely a better upper bound for $E_f(X)$ is known. For example, in the case wen $a = c = 1, b = 0$ one has the bound

$E_{x^2 + y^2}(X) = O \left(X^{\frac{131}{416}} \right)$

according to Wikipedia; the exponent being due to Huxley.

In general, what is known about the error term $E_f(X)$? Both the exponent of $X$ and the dependence on $a,b,c$ or $D = 4ac - b^2$ are of interest.

davidlowryduda · Accepted Answer · 2024-08-08 16:12:37Z

The broad harmonic analytic method of Van der Corput applies here, and shows that $$E_f(X) \ll X^{1/3}.$$ One version says that if $D$ is a strongly convex domain in $n$ dimensions, then the number of standard lattice points inside $\lambda D$ is $$ \lambda^n \mathrm{Vol}(D) + O(\lambda^{n - 2 + \frac{2}{n + 1}}).$$ (Note the different normalization between $\lambda$ and $X$).

It's possible to track the constants explicitly. For example, Krätzel proved a version in dimension $2$ with explicit constants in Lattice points in planar convex domains, Monatsh. Math., 2004; the constants depend on bounds for the maximum and minimum of the curvature. There is a short trail of papers with minor improvements on Krätzel that improve the constants.

When the boundary of $D$ is $C^3$ and when we're in dimension $2$, Huxley (Exponential sums and lattice points III, Proc. London Math. Soc. (3) 87, 2003) showed the error is at most $$ O(\lambda^{131/208} \log^{18637/8320} \lambda). \qquad \left(\tfrac{131}{208} \approx 0.629\ldots\right).$$ It seems much harder to track the dependence on the exact ellipse through Huxley's work, but perhaps I'm wrong.

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