Cube packings, second moment and holes
| dc.creator | Dutour, Mathieu | |
| dc.creator | Itoh, Yoshiaki | |
| dc.creator | Poyarkov, Alexei | |
| dc.date | 2005-09-05 | |
| dc.date.accessioned | 2026-07-07T05:22:58Z | |
| dc.date.available | 2026-07-07T05:22:58Z | |
| dc.description | We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq 4$. For higher dimension, we use random methods for generating some examples. Such a cube packing is called {\em non-extendible} if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that $d$-dimensional cube packings with more than $2^d-3$ cubes can be extended to cube tilings. We also give a lower bound on the number $N$ of cubes of non-extendible cube packings. Given such a cube packing and $z\in \ZZ^d$, we denote by $N_z$ the number of cubes inside the $\4t$-cube $z+[0,4[^d$ and call {\em second moment} the average of $N_z^2$. We prove that the regular tiling by cubes has maximal second moment and give a lower bound on the second moment of a cube packing in terms of its density and dimension. | |
| dc.description | 11 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0509100 | |
| dc.identifier | http://arxiv.org/abs/math/0509100 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76262 | |
| dc.subject | Combinatorics | |
| dc.title | Cube packings, second moment and holes | |
| dc.type | text |