Rigidity and the chessboard theorem for cube packings

We are concerned with subsets of R d that can be tiled with translates of the half-open unit cube in a unique way. We call them rigid sets. We show that the set tiled with [ 0 , 1 ) d + s , s ∈ S , is rigid if for any pair of distinct vectors t , t ′ ∈ S the number | { i : | t i − t i ′ | = 1 } | is even whenever t − t ′ ∈ { − 1 , 0 , 1 } d . As a consequence, we obtain the chessboard theorem which reads that for each packing [ 0 , 1 ) d + s , s ∈ S , of R d , there is an explicitly defined partition { S 0 , S 1 } of S such that the sets tiled with the systems [ 0 , 1 ) d + s , s ∈ S i , where i = 0 , 1 , are rigid. The technique developed in the paper is also applied to demonstrate certain structural results concerning cube tilings of R d .