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.

Partial answers are given to two questions. When does a lattice $\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is geometrically similar to $\Lambda$? When is the sublattice ``clean'', in the sense that the boundaries of the Voronoi cells for $\Lambda'$ do not intersect $\Lambda$?

Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon (d)})$ for some positive $\varepsilon(d)$. Here we give for general convex bodies the weaker estimate \[ \left| G(\lambda M) -V(\lambda M) \right | \le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1}) \] where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term $S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$. Further we deal with families $\{P_\lambda\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have \[ \left| G(P_\lambda)-V(P_\lambda) \right| \le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr) \] where the convex body $K$ satisfies some simple condition, $V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the surface area of $P_\lambda$.