Dense line on complex torus Let $L\subset \mathbb{C}^n$ be a lattice of rank $2n$ and $v\in \mathbb{C}^n$. Is there a characterization when the set
$$ \{ z v \in \mathbb{C}^n/L \ : \ z \in \mathbb{C} \} $$ 
lies dense in $\mathbb{C}^n/L$?
In the real case we have for $(w_1, \cdots, w_{2n})=w\in \mathbb{R}^{2n}$ that
$$ \{ x w \in \mathbb{R}^{2n}/\mathbb{Z}^{2n} \ : \ x \in \mathbb{R} \} $$
lies dense in $\mathbb{R}^{2n}/\mathbb{Z}^{2n}$ iff the $w_i$ do not fulfill a relation of the form
$$ \sum_{j=1}^{2n} m_j \cdot \omega_j \in \mathbb{Z} $$
for $(m_1, \dots, m_n)\in \mathbb{Z}^n \setminus \{ 0 \}$. By transforming the lattice, one gets for a general lattice $L\subset \mathbb{R}^{2n}$ that
$$ \{ x w \in \mathbb{R}^{2n}/L \ : \ x \in \mathbb{R} \} $$
lies dense in $\mathbb{R}^{2n}/L$ iff the $w_i$ do not fulfill a relation of the form
$$ \sum_{j=1}^{2n} m_j \cdot \omega_j \in L $$
for $(m_1, \dots, m_n)\in \mathbb{Z}^n \setminus \{ 0 \}$.
I was hoping that there is a similar statement in the complex case for a general lattice. I.e. is it true that 
$$ \{ z v \in \mathbb{C}^n/L \ : \ z \in \mathbb{C} \} $$ 
lies dense in $\mathbb{C}^n/L$ iff the $v_i$ do not fulfill a relation of the form
$$ \sum_{j=1}^{n} m_j \cdot v_j \in L $$
for $(m_1, \dots, m_n)\in \mathbb{Z}^n \setminus \{ 0 \}$?
In fact, I would already be glad if there was a characterization in the case, when the lattice is of the form
$$ L= \langle e_1, \dots, e_n, \tau_1, \dots, \tau_n \rangle $$
where, $e_i\in \mathbb{C}^n$ denotes the $i$th standard basis vector and the matrix
$$ \tau = \begin{pmatrix} \tau_1 \dots \tau_n 
\end{pmatrix} \in Mat(n\times n, \mathbb{C})$$
is symmetric with positive definite imaginary part.
 A: Your compact topological abelian group is $G=\mathbb{C}^n/L$. Want to know whether the closure $\bar M$ of a certain set $M$ ($M$ closed under addition) is the whole group $G$ (apriori $\bar M$ is just a semigroup, but because of the compactness of $G$ it is a subgroup). Here is the criterion: $\bar M= G$ if an only the only character of $G$ that is $1$ on $M$ is the trivial character. 
The characters of $G$ are $$\phi_k \colon x \mapsto \exp(2\pi i k \cdot x)$$ where $k\in \hat L$, the dual lattice of $L$. With the above criterion, we can say: $\mathbb{C} \cdot v$ is not closed in $\mathbb{C}/L$ if and only if there exists $k \in \hat L$, $k\ne 0$ so that 
$$k \cdot v = 0 \\ k \cdot (iv) = 0$$ ( $\cdot $ the usual dot product). 
Let's consider the particular case $L=\mathbb{Z}^n$. The above equalities mean: there exists $(u_1, \ldots, u_n)\in \mathbb{Z}[i]^n$ not all $0$ so that $\sum u_i v_i = 0$
So, for instance, the line $\mathbb{C}\cdot v$ is dense in $\mathbb{C}^2/\mathbb{Z}^4$ if and only if the ratio of the components of $v$ is not in $\mathbb{Q}(i)$. 
