When is a lattice dense in a torus?

Question

Let $\pi: \mathbb{R}^n\rightarrow \mathbb{R}^n/\mathbb{Z}^n$. What (necessary and sufficient) criteria on $A\in GL_n(\mathbb{R})$ guarantee $\pi(A\mathbb{Z}^n)$ is dense?

Answer 1

One criterion comes from Kronecker's theorem: $\pi(A\mathbb{Z}^n)$ is dense if and only if $A^t(\mathbb{Z}^m\backslash \{0\})$ does not contain an integer vector. I.e. the rows of $A$ are linearly independent over $\mathbb{Q}$ from every rational vector.