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Posted online: 2024-03-20 15:50:20Z by Stefan Steinerberger15

Cite as: P-240320.4

Is there a universal constant $c$ such that all translation and rotations of a cube $Q \subset \mathbb{R}^n$ with side-length $c$ always contains a point from the $\mathbb{Z}^n$?

The problem is that in high dimensions the lattice $\mathbb{Z}^n$ has gigantic holes and it is not so easy to rule out the possibility of a suitably rotated large cube hiding there. Henk-Tsintifas prove that $c=\sqrt{2}$ when $n=2$ or $n=3$.

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Created at: 2024-03-20 15:50:20Z

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