  Lattices and Z-modules in Euclidean space possess an infinitude of subsets
that are images of the original set under similarity transformation. We
classify such self-similar images according to their indices for certain 4D
examples that are related to 4D root systems, both crystallographic and
non-crystallographic. We encapsulate their statistics in terms of Dirichlet
series generating functions and derive some of their asymptotic properties.
