About symmetric rank-1 random matrices

co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Q)$ have a bounded basic province on $mathbb R^n$? Answer

Hello expensive customer to our community We will proffer you an answer to this query co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Q)$ have a bounded basic province on $mathbb R^n$? ,and the respond will breathe typical via documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Q)$ have a bounded basic province on $mathbb R^n$?

$DeclareMathOperatorGL{GL}$Let $G subset M_{ntimes n~}(mathbb Z)$ breathe a finitely generated subgroup of $GL(n,mathbb Q)$ (i.e. $gin G$ is an invertible matrix with entries in $mathbb Z$). Then $G$ acts on $mathbb R^n = mathbb Z^n otimes_{mathbb Z} mathbb R$ via $GL(n,mathbb Q)$.

Suppose that there’s a rational affine subspace $V subset mathbb R^n$ (by this, I denote that there’s a sub-lattice $L subset mathbb Z^n$ and $a in mathbb Z^n$ such that $V = a + (L otimes_{mathbb Z} mathbb R)$), and $V$ is invariant underneath the motion of $G$ (i.e. for any $vin V, gin G$, now we have $gcdot v in V$). Moreover, there exists $v in L$ (in truth, we are able to take $v=a$) such that
$$G cdot v = L.$$

Question: is there a bounded subset $P subset V$ such that $$bigcup_{g in G} gcdot P = V quad ? $$

Any suggestion on pertinent questions/references may be very welcome! Particularly, I do not know which bailiwick research such issues ….

Edit:

Example. Consider $(0,1)+L:=(mathbb Z,1) subset mathbb Z^2$, and $$G={commence{pmatrix} 1&okay&1end{pmatrix}mid kinmathbb Z}.$$ For $v=(0,1)$, now we have $G cdot v =(0,1)+L$. In this illustration, we are able to take $P$ to breathe the interval from $(0,1)$ to $(1,1)$.

we’ll proffer you the answer to co.combinatorics – When does a subgroup of $operatorname{GL}(n, mathbb Q)$ have a bounded basic province on $mathbb R^n$? query through our community which brings all of the solutions from a number of dependable sources.

Add comment