Consider a disk of radius $r$ centered at $(x,y)$, where $(x,y)$ is chosen from the uniform distribution on $[0,1) \times [0,1)$, and let the random variable $N$ be the number of lattice points in the disk. The expected value of $N$ is clearly $\pi r^2$, but what is the variance of $N$, and why?

I have the impression that the solution (published by Kendall in the mid-20th century and perhaps found earlier by others) is straightforward, involving the Fourier transform of the indicator function of the disk (or rather a doubly-periodic union of disjoint copies of the disk), and using nothing more arcane that Parceval's identity and Bessel functions, but I haven't been able to find the details anywhere on the web, and I'm not enough of an analyst to work them out myself.