Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $d\geq 3$.)

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.