On the structure of cube tilings of and

A family of translates of the unit cube [ 0 , 1 ) d + T = { [ 0 , 1 ) d + t : t ∈ T } , T ⊂ R d , is called a cube tiling of R d if cubes from this family are pairwise disjoint and ⋃ t ∈ T [ 0 , 1 ) d + t = R d . A non-empty set B = B 1 × ⋯ × B d ⊆ R d is a block if there is a family of pairwise disjoint unit cubes [ 0 , 1 ) d + S , S ⊂ R d , such that B = ⋃ t ∈ S [ 0 , 1 ) d + t and for every t , t ′ ∈ S there is i ∈ { 1 , … , d } such that t i − t i ′ ∈ Z ∖ { 0 } . A cube tiling of R d 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 R 4 which, in contrast to cube tilings of R 3 , is not blockable. We give a new proof of the theorem saying that every cube tiling of R 3 is blockable.