BETA-INTEGERS AS NATURAL COUNTING SYSTEMS FOR QUASI-CRYSTALS

Recently, discrete sets of numbers, the beta-integers Z(beta), have be en proposed as numbering tools in quasicrystalline studies. Indeed, th ere exists a unique numeration system based on the irrational beta > 1 in which the beta-integers are all real numbers with no fractional pa rt. These beta-integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely bet a is equal to 1 + root 5/2 (golden mean tau), to 1 + root 2, or to 2 root 3, i.e, when beta is one of the self-similarity ratios observed in quasicrystalline structures. As a matter of fact, beta-integers are natural candidates for coordinating quasicrystalline nodes, and also the Bragg peaks beyond a given intensity in corresponding diffraction patterns: they could play the same role as ordinary integers do in cry stallography. In this paper, we prove interesting algebraic properties of the sets Z(beta) when beta is a 'quadratic unit PV number', a clas s of algebraic integers which includes the quasicrystallographic cases . We completely characterize their respective Meyer additive and multi plicative properties Z(beta) + Z(beta) subset of Z(beta) + F Z(beta)Z( beta) subset of Z(beta) + G where F and G are finite sets, and also th eir respective GaIois conjugate sets Z(beta)'. These properties allow one to develop a notion of a quasiring Z(beta). We hope that in this w ay we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be 'module on a quasiring' in R- d : Lambda(beta) = Sigma(i) Z(beta ei). We give also some two-dimensio nal examples with beta = tau.