Let $\Lambda\subset \mathbb R^d$ a discrete subgroup, up to diminishing $d$ we assume it is of the form $A\mathbb Z^d$ with $A\in GL(d)$. Up to dilation we assume that the shortest vector in $\Lambda\setminus\{0\}$ has length $1$.
I would like to call this $\Lambda$ "self-similar" if for every $p\in \Lambda\setminus\{0\}$ one can complete $p$ to a sublattice $\Lambda'\subset \Lambda$ of the form $\Lambda'=\lambda R\Lambda$ with $\lambda=|p|$ and $R\in O(d)$ (i.e. $\Lambda'$ is a rotated dilated copy of $\Lambda$ such that $p$ is one of the shortest nonzero vectors in $\Lambda'$).
The motivation was for me that lattices $\mathbb Z^2$ and $A_2$ (the equilateral triangular lattice) in $d=2$ are self-similar, and I was wondering how rare is this property: What are other/all examples of self-similar lattices in other dimensions?
Pointers to possibly related concepts are very welcome.
