On the structure of cube tilings of R3 and R4

AbstractA family of translates of the unit cube [0,1)d+T={[0,1)d+t:t∈T}, T⊂Rd, is called a cube tiling of Rd if cubes from this family are pairwise disjoint and ⋃t∈T[0,1)d+t=Rd. A non-empty set B=B1×⋯×Bd⊆Rd is a block if there is a family of pairwise disjoint unit cubes [0,1)d+S, S⊂Rd, such that B=⋃t∈S[0,1)d+t and for every t,t′∈S there is i∈{1,…,d} such that ti−ti′∈Z∖{0}. A cube tiling of Rd is blockable if there is a finite family of disjoint blocks B, |B|>1, with the property that every cube from the tiling is contained in exactly one block of the family B. We construct a cube tiling T of R4 which, in contrast to cube tilings of R3, is not blockable. We give a new proof of the theorem saying that every cube tiling of R3 is blockable