Can a circle of a given radius $r$ always be placed (in $\mathbb{R}^2$) such that the number of points with integer coordinates inside the circle is equal to the nearest integer of the circle's area?
In other words, if we imagine a square grid with the centers of the grid squares at integer points (and thus corners at $(n+0.5,m+0.5)$ for $n,m \in \mathbb{N}$), we want to place the circle at a point where if we approximate it by squares whose center it includes, we get an approximation that with respect to area is as close as possible.
I found a few older papers dealing with the number of integer points on or inside a circle with center at the origin, but not quite what I am looking for. Seeing no good theoretical approach, I wrote a small brute-force program that tries to place circles as required by sampling centers $(x,y)$ with $0 \leq x \leq 0.5$ and $x \leq y \leq 0.5$ (sufficient due to invariance to translation by an integer vector and reflection through a diagonal), and it seems indeed that even for small $r$ (e.g. $r=\sqrt{2/\pi}$) not only what I describe above is possible, but also having the circle overlap exactly $\lfloor \pi r^2 \rceil-1$ or $\lfloor \pi r^2 \rceil+1$ integer points. Yet the size of the regions for the centers of appropriately placed circles get smaller and smaller as $r$ increases, and I cannot see a general patter of how to arrive at valid center coordinates by any other means.