Poisson disk or blue noise sampling consists of finding a random, maximal set of points $\{x_i\}$ in a domain $\Omega$ such that $\|x_i - x_j\| \ge r$ for some fixed radius $r$ and for all $i$, $j$. There are several algorithms for efficient blue noise sampling, for example Bridson's algorithm. Blue noise sampling is used in many fields, including computer graphics and mesh improvement.
Many papers use the empirical periodogram for evaluating the quality of the resulting blue noise samples. This got me thinking: Is there any published work on generating approximate blue noise samples through Fourier inversion of the ideal periodogram? The resulting samples might have some pairs $x_i$, $x_j$ of points such that $\|x_i - x_j\| < r$. But strict adherence to this exclusion condition isn't necessarily a hard requirement for a lot of the things people use blue noise for. My hunch is that some kind of Fourier or spectral-based approach might (1) make better use of fast DFT algorithms, instead of spending all its time traversing a bunch of linked lists, and (2) be implementable in a single laptop screen of code at 12-point font. Bridson's algorithm fails on both counts.
I found this paper which is kind of in the right direction -- they calculate that the power spectral density of blue noise is
$$P(k) = 1 - \rho\left(\frac{2\pi r}{k}\right)^{\frac{d}{2}}J_{d/2}(kr)$$
for samples of density $\rho$ in dimension $d$, where $J_\nu$ are the Bessel functions of the first kind. But this is used more as a means to evaluate algorithms rather than the basis for an algorithm as such.
