Combinatorial cube packings in the cube and the torus

We consider sequential random packing of cubes z + [ 0 , 1 ] n with z ∈ 1 N Z n into the cube [ 0 , 2 ] n and the torus R n / 2 Z n as N → ∞ . In the cube case, [ 0 , 2 ] n as N → ∞ , the random cube packings thus obtained are reduced to a single cube with probability 1 − O ( 1 N ) . In the torus case, the situation is different: for n ≤ 2 , sequential random cube packing yields cube tilings, but for n ≥ 3 with strictly positive probability, one obtains non-extensible cube packings. introduce the notion of combinatorial cube packing, which instead of depending on N depends on some parameters. We use them to derive an expansion of the packing density in powers of 1 N . The explicit computation is done in the cube case. In the torus case, the situation is more complicated and we restrict ourselves to the case N → ∞ of strictly positive probability. We prove the following results for torus combinatorial cube packings: • e a general Cartesian product construction. ve that the number of parameters is at least n ( n + 1 ) 2 and we conjecture it to be at most 2 n − 1 . ve that cube packings with at least 2 n − 3 cubes are extensible. d the minimal number of cubes in non-extensible cube packings for n odd and n ≤ 6 .