Cube packings, second moment and holes

We consider tilings and packings of R d by integral translates of cubes [ 0 , 2 [ d , which are 4 Z d -periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimensions d ≤ 4 . For higher dimensions, we use random methods for generating some examples. cube packing is called non-extendible if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that d -dimensional cube packings with more than 2 d − 3 cubes can be extended to cube tilings. We also give a lower bound on the number N of cubes of non-extendible cube packings. such a cube packing and z ∈ Z d , we denote by N z the number of cubes inside the 4-cube z + [ 0 , 4 [ d and call the second moment the average of N z 2 . We prove that the regular tiling by cubes has maximal second moment and gives a lower bound on the second moment of a cube packing in terms of its density and dimension.