  We describe a framework for extending the asymptotic behavior of a
short-range interaction from the unit cube to general compact subsets of $
\mathbb R^d $. This framework allows us to give a unified treatment of
asymptotics of hypersingular Riesz energies and optimal quantizers. We further
obtain new results about the scale-invariant nearest neighbor interactions,
such as the $ k $-nearest neighbor truncated Riesz energy. Our generalized
approach has applications to methods for generating distributions with
prescribed density: strongly-repulsive Riesz energies, centroidal Voronoi
tessellations, and a popular meshing algorithm due to Persson and Strang.
