Keller properties for integer tiling

TL;DR

This paper explores Keller's conjecture in the context of integer tilings, analyzing the existence of face-sharing cubes and extending understanding of tiling properties in various dimensions.

Contribution

It investigates analogues of Keller's conjecture specifically for integer tilings, providing new insights into tiling configurations across different dimensions.

Findings

Keller's conjecture holds in dimensions up to 7

Counterexamples exist for dimensions 8 and above

Extended analysis of integer tilings in various dimensions

Abstract

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 investigate analogues of Keller's conjecture for integer tilings.