Beta - Integers as Natural Counting Systems

Recently, discrete sets of numbers, the-integers ZZ , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational > 1 in which the-integers are all real numbers with no fractional part. These-integers appear as being quite appropriate to describing some quasilattices relevant to quasicrystallography when precisely is equal to 1+ p 5 2 (golden mean), to 1 + p 2, or to 2 + p 3, i.e. when is one of the self-similarity ratios observed in quasicrystalline structures. As a matter of fact,-integers are natural candidates for coordinating quasicrystalline nodes, and also the Bragg peaks beyond a given intensity in corresponding diiraction patterns: they could play the same role as ordinary integers do in crystallography. In this paper, we prove interesting algebraic properties of the sets ZZ when is a quadratic unit PV numberr, a class of algebraic integers which includes the quasicrystallographic cases. We completely characterize their respective Meyer additive and multiplicative properties where F and G are nite sets, and also their respective Galois conjugate sets ZZ 0. These properties allow one to develop a notion of a quasiring ZZ. We hope in this way initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be module on a quasiringg in IR d : = P i ZZ e i. We give also some two-dimensional examples with = .