Keller Properties for Integer Tilings

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 $d\leq 7$ and false for $d\geq 8$. In this article, we propose analogues of Keller's face-sharing property for integer tilings. We construct counterexamples to a ``strong" version of this property, and prove that a weaker version holds for integer tilings under appropriate additional assumptions.