Let $\mathbf{e}\in\mathbb{R}^n$ be a unit vector, $\hat L:=\mathbb{R}\mathbf{e}$ be the line in the direction of $\mathbf{e}$, which projects down to a line $L$ on the torus $\mathbb{T}^n:=\mathbb{R}^n/\mathbb{Z}^n$.
Question 1. Is the following statement true?
- The line $L$ is dense on the torus if and only if the components of $e$ are $\mathbb{Q}$-independent.
More generally, we can define
the rank $r(\mathbf{e})$ of $\mathbf{e}$, which is the dimension of the closure $K(\mathbf{e})$ of the line $L$ in $\mathbb{T}^n$.
the co-rank $z(\mathbf{e})$ of $\mathbf{e}$, which is the $\mathbb{Q}$-dimensions of the space of rational vectors $Z(e)=\{\mathbf{r}\in \mathbb{Q}^n: \mathbf{r}\cdot \mathbf{e} =0\}$.
Question 2. Does it hold that $r(\mathbf{e})+z(\mathbf{e})=n$?
Thanks!
Consider the flow $\hat\phi_t(x)=x+te$ of $R^n$, it commutes with the action of $Z^n$ and define a flow $\phi_t$ on $R$. Let $p:R^n\rightarrow T^n$ be the covering map, $L$ is the orbit of $p(0)$ by $\phi_t$.
Now we can apply the results of Yves Carriere on Riemannian flow or the results of Marina Ratner on unipotent flows which says that there exists a closed subgroup $H$ of $T^n$ such that $p(H(0))$ is closed in $R^n$ and the orbit of $p(0)$ by $\phi_t$ is a dense subset of $p(H(0))$.
Suppose that $H$ is distinct of $R^n$ and its a $l$-vector subspace $l<n$, since $p(H(0))$ is closed, there exists $u_1,..,u_l\in H$ with integral coordinates such that $(u_1,..,u_l)$ is a base of $H$, we can write $e=a_1u_1+..a_lu_l$. Since the coordinates of $u_i$ are integers the coordinates of $x$ are elements of the $Q$-vector space generated by $a_1,..,a_l$ since $l<n$. We deduce that they are $Q$-linearly dependent. Contradiction.
https://en.wikipedia.org/wiki/Ratner%27s_theorems
Carriere Yves, Flot Riemanniens.
in "Structures transverses des feuilletages", Asterisque, 116 (1984),31-52