Background: I'm reading a paper about Brownian Motion.
Problem: I have a square lattice of mesh size $\frac{1}{\lceil \epsilon^{-1} \rceil}$ where $\epsilon > 0$. Given a circle with radius $r$, how many points of the lattice are at most inside the circle?
Solution: The answer should be $2\epsilon^{-4}\pi r^2$ according to the paper.
Thoughts: In a square with length/height of $r$, there are at least $(\frac r \epsilon )^2$ many points of the lattice included. The area of a circle with radius $r$ is $\pi r^2$. Unfortunately, I have not had any better ideas.