Cube packings, second moment and holes

We consider tilings and packings of R by integral translates of cubes [0, 2[, which are 4Z-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension d ≤ 4. For higher dimension, we use random methods for generating some examples. Such a 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 − 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. Given such a cube packing and z ∈ Z, we denote by Nz the number of cubes inside the 4-cube z +[0, 4[ and call second moment the average of N2 z . We prove that the regular tiling by cubes has maximal second moment and give a lower bound on the second moment of a cube packing in terms of its density and dimension.