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We have some lattice $$\Gamma=\bigoplus_{i=1}^n\mathbb{Z}v_i$$where the length of each basis vector, as well as the angle between every two vectors, is bounded. Is there an upper bound for $|\Gamma\bigcap [0,1]^n|?$

Isomorphism
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  • How many vertices on an $n$-dimensional cube? If it's an integer lattice, then vertices are only possible places of intersection. – Doug Jul 29 '22 at 16:40
  • I guess there might be some misunderstanding. For instance, take $n=1, v_1=1/2$, then there are three vectors $0,v_1,2v_1$ in the intersection, in which $v_1$ is not a vertex. – Isomorphism Jul 29 '22 at 17:20
  • Ok. I assumed that the lattice vectors were integer-valued. – Doug Jul 29 '22 at 17:28

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