Projection of a lattice onto a subspace Let $G$ be a $n \times n$ matrix with real entries and let $\Lambda = \{x^n \colon \exists i^n \in \mathbb{Z}^n \text{ such that } x^n = G \cdot i^n\}$ define a lattice. I am interested in projecting the lattice points onto a $k$-dimensional subspace $U$ with $k < n$. Let $A$ be a $n \times k$ matrix of vectors that span $U$. Then, the projection of $\Lambda$ onto $U$ has the generator matrix $P_A \cdot G$ where $P_A = A (A^TA)^{-1}A^T$ is the projection operator. Lets also assume that the entries of $G$ have no structure between them - i.e., they are generally chosen from the reals.
I am interested in finding out when the projection of $\Lambda$ onto $U$ (call it $\Lambda_U$) is also a $k$-dimensional lattice. It seems to me that in most cases, $\Lambda_U$ will fill out the $k$-dimensional space $U$ completely, i.e., the shortest distance between neighboring points in $\Lambda_U$ will be arbitrarily small (obs. 1). For certain subspaces $U$ that seem to be "aligned" correctly with $\Lambda$, we get a well-defined $\Lambda_U$. By well-defined, I mean that the spacing between the nearest neighbors of $\Lambda_U$ is bounded away from $0$. As far as I can tell, this happens when $A$ is chosen such that the columns of $P_A \cdot G$ are rational multiples of one another (obs. 2).
Questions:


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*Are the observations (obs. 1) and (obs. 2) correct?

*If so, is this a well-studied concept? I didn't have much success googling with the obvious keywords such as lattice "projection" or "matrix column rational multiples" etc.

*Assuming the observations are correct, given $G$, is there a way to choose $A$ such that the columns of $P_A \cdot G$ are rational multiples of one another?


The technique you suggest is sometimes used to construct quasi-crystaline structures or in mathematical parlance, aperiodic tilings. If you take a high dimensional lattice and you project on an appropriate subspace U, the result can be the lattice structure of a quasi-crystal. 
