[Math] Lattice points on the boundary of an ellipse

analytic-number-theorydiscrete geometrygeometrynt.number-theorysums-of-squares

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). Alternatively, this appears to be very similar to the question of how many points can the boundary of any axis-parallel ellipse contain from an $r\times r$ section of the integer lattice.

Let us say that the ellipse is defined as $(ax-b)^2+(cy-d)^2=r^2$. If $a,b,c,d$ are integers, we can reduce the problem to asking for the number of integer solutions to of an equation of the form $ax^2+by^2=r^2$ with integers $a,b$. This is known to have at most $r^{\frac{c}{\lg \lg r}}$ solutions, for some constant $c$. Does a similar bound exist for every axis-parallel ellipse?

In case it helps, I mainly care about the case where $a,b,c,d$ are not much larger than $r$.

Best Answer

For general ellipses I doubt you can do much better than Bombieri-Pila sort of bounds:

MR1016893 (90j:11099) Reviewed 
Bombieri, E.; Pila, J.
The number of integral points on arcs and ovals. 
Duke Math. J. 59 (1989), no. 2, 337–357. 
11P21 (11D99)

(you should check out the many papers that cite this, as well, but most of them seem to be interested in higher degrees/dimensions).

For integer coefficients, the best seems to be

Lattice points on Ellipses Cilleruelo and Cordoba, DMJ 1994

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