Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number

Dynamical directions in numeration

“…Section 6.3 deals with computer arithmetics and cryptography, and Section 6.4 is concerned with applications in physics, namely quasicrystals. Note that the current resarch on quasicrystals is very active, as shown in this volume by the contribution [153].…”

“…Meyer sets are proved to be subsets of model set of R k , for some k 1, that are relatively dense [255,256,257]. For more details, see for instance [38,138,153,233,229,301,332,333]. Note that there are close connections between such a generation process for quasicrystals and lattice tilings for Pisot unimodular substitutions (e.g., see [65,337,338]).…”

Noncommutative reading of the complex plane through Delone sequences

“…It is natural to ask in case of Parry numbers, whether the corresponding β-integers behave similarly to the integers. Some answers concerning the additive, multiplicative and diffractive properties of β-integers already exist (see [Gazeau(2006)] and references therein). In view of the "quasi"-periodic distribution of the β-integers on the real line, it is pertinent to investigate the extent to which they differ from ordinary integers in an asymptotic sense, according to the nature of β.…”

“…We underline that for quadratic numbers the notions of Parry and PV numbers coincide. The following proposition providing explicit formulae for β-integers for a quadratic PV unit β comes from [Gazeau(2006)].…”

Asymptotic Behavior of Beta-Integers

“…Let us mention that both of the previous asymptotic characteristics are known for β being a quadratic unit. The following proposition providing explicit formulae for β -integers for a quadratic unit β comes from [7]. Proposition 1.1.…”