I have seen the following question, where the equidistribution theorem can be generalized to 2 dimensions under certain conditions: Equidistribution in higher dimensions.
I was wondering if the same thing could be done for Vectors.2 Vectors in 2D - as long as not linearly dependent - span a parallelogram. Now if I were to have 4 Vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$, where ($\vec{a}$ and $\vec{b}$) and ($\vec{c}$ and $\vec{d}$) are linearly independent would I be able to span the whole parallelogram that is created by $\vec{c}$ and $\vec{d}$ with the series\begin{equation}n \cdot (\vec{a} + \vec{b}) \mod (\vec{c},\vec{d})\,?\end{equation}If no such modulo already exists, let us say that the modulo calculation $\mod (\vec{c},\vec{d})$ for vectors is defined in a way, that vectors $\vec{c}$ or $\vec{d}$ are subtracted whenever both entries in $n \cdot (\vec{a} + \vec{b})$ are big enough to do so. In the case where both Vectors could be subtracted, I am not sure of what would be mathematical more natural. What if I instead took a look at\begin{equation}n \vec{a} + m\cdot \vec{b} \mod (\vec{c},\vec{d})\,?\end{equation}where n and m are independent of each other?Is there a branch of mathematics, that can deal with questions like this? Would it be the same problem, if I do projections of the vectors $\vec{a}$ and $\vec{b}$ onto $\vec{c}$ and $\vec{d}$?
Thanks!
