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Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
3
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Intersection of two lattices
Shouldn't this happen as long as $\mathbf{a}$ and $\mathbf{b}$ are linearly dependent mod $p$? For if so, you are talking about points in $\mathbb{Z}^n$ which reduce mod $p$ in two some $n-2$ dimensio …