this seems intuitive to me but I'm struggling to prove it (is it false?).
Let $E$ be a subset of a lattice (free abelian group of finite rank) closed under addition, containing the origin and such that if $e \in E$ then $-e \notin E$ (it's some sort of "positive" cone, oh actually the correct term should probably be submonoid).
If I take any other element $x$ and consider the intersection of $E$ with $-(E +x)$, will it be finite (or empty)?
(All the properties considered are invariant under isomorphism, so we might as well consider the problem in $\mathbb{Z}^n$)
[please help me out for tags, it's not my usual cup of tea]