Keller properties for integer tiling

Keller’s conjecture on cube tilings asserted that, in any tiling of ${\mathbb{R}}^d$ by unit cubes, there must exist two cubes that share a $(d-1)$-dimensional face. This is now known to be true in dimensions $d\le 7$ and false for $d\ge 8$. In this article, we investigate analogues of Keller’s conjecture for integer tilings.