On the structure of cube tilings of R3 and R4

摘要

A 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.