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 $r$.

Duke Mathematical Journal76.3 (1994): 741-750.PDF download of preliminary version. $\endgroup$