On the structure of cube tilings and unextendible systems of cubes in low dimensions
In the paper under review, cube tilings where each tile is a translate of the unit cube are studied. This topic is related to Keller's conjecture that in any cube tiling there are two cubes which share a hyperface. For the 3-dimensional space the authors prove that any cube tiling either is layered or has the structure as they describe. Also they classify the so-called unextendible systems of cubes in \(\mathbb{R}^3\), which can be applied in quantum information theory. For the 4-dimensional space it is proved that each cylinder of a cube tiling contains a column, and it is shown by an example that the latter result does not hold in \(\mathbb{R}^5\).
scientific article; zbMATH DE number 5982486
scientific article