Lattices and $\ZZ$-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.

MSC Classifications:

11S45, 11H05, 52C07 show english descriptions Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16Hxx, 16Kxx]
unknown classification 11H05
Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21] 11S45 - Algebras and orders, and their zeta functions [See also 11R52, 11R54, 16Hxx, 16Kxx]
11H05 - unknown classification 11H05
52C07 - Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]

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