Given a strictly positive integer $d$ and $k$ square-matrices $A_1,\dots,A_k$ of size $d$, we define $S(A_1,\dots,A_k)=\lbrace n^{1/d}A_1^nA_2^n\cdots A_k^nV\ \vert\ n\in \mathbb N\rbrace$ where $V=(1,0,0,\dots,0)^t$.

Given $d\geq 1$ does there always exist an integer $k$ and $k$ orthogonal matrices $A_1,\dots,A_k$ such that $S(A_1,\dots,A_k)$ is quasi-isometric to the Euclidean space $R^d$? (Quasi-isometry means simply that there exists a real number $B$ such that balls of radius $B$ centered at all points of $S(A_1,\dots,A_k)$ cover $R^d$.)

The answer is yes for $d=1,2$ with $k=1$ (take $A_1=-1$ for $d=1$ and a rotation of angle $2\pi/\tau$ where $\tau=(1+\sqrt{5})/2$ or any irrational number with bounded continued fractions developpement for $d=2$).

$d=3$ is the first dimension where I am not sure of the answer. ($k=1$ is clearly impossible for $d\geq 3$.)