Rigidity and the chessboard theorem for cube packings

We are concerned with subsets of RdRd 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[0,1)d+s, s∈Ss∈S, is rigid if for any pair of distinct vectors tt, t′∈St′∈S the number |{i:|ti−ti′|=1}| is even whenever t−t′∈{−1,0,1}dt−t′∈{−1,0,1}d. As a consequence, we obtain the chessboard theorem which reads that for each packing [0,1)d+s[0,1)d+s, s∈Ss∈S, of RdRd, there is an explicitly defined partition {S0,S1}{S0,S1} of SS such that the sets tiled with the systems [0,1)d+s[0,1)d+s, s∈Sis∈Si, where i=0,1i=0,1, are rigid. The technique developed in the paper is also applied to demonstrate certain structural results concerning cube tilings of RdRd.