Beta-integers as natural counting systems for quasicrystals

Abstract

Recently, discrete sets of numbers, the -integers , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational in which the -integers are all real numbers with no fractional part. These -integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to (golden mean ), to , or to , 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 diffraction patterns: they could play the same role as ordinary integers do in crystallography. In this paper, we prove interesting algebraic properties of the sets when is a `quadratic unit PV number', 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 finite sets, and also their respective Galois conjugate sets . These properties allow one to develop a notion of a quasiring . We hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be `module on a quasiring' in : . We give also some two-dimensional examples with .