Quasi-linear transformations and discrete tilings

Tilings of the discrete plane Z2 generated by quasi-linear transformations (QLT) have been introduced by Nehlig [P. Nehlig, Applications quasi-affines: Pavages par images réciproques, Theoretical Computer Science 156 (1995) 1–38]. We studied these tilings and gave some results, such as periodicity and the number of neighbours of each of them [M.-A. Jacob-Da Col, Applications quasi-affines et pavages du plan discret, Theoretical Computer Science 259 (2001) 245–269. Also available in English: http://dpt-info.u-strasbg.fr/~jacob/articles/paving.pdf].The aim of this paper is to go on with this study in the discrete n-dimensional space Z2; we give a lower and an upper bound to the number of distinct tiles. We also give an algorithm to determine the points of a given tile, this algorithm will induce another algorithm to determine the number of distinct tiles associated to a QLT.