Eur. Phys. J. B 39, 287-294 (2004)
https://doi.org/10.1140/epjb/e2004-00193-x

Scale dependent partitioning of one-dimensional aperiodic set diffraction

Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Université Paris 7 - Denis Diderot, 2 Place Jussieu 75251 Paris Cedex 05, France

Corresponding author: ^a kharrat@ccr.jussieu.fr

Received: 23 December 2003
Revised: 24 March 2004
Published online: 12 July 2004

Abstract

We give a multiresolution partition of pure point parts of diffraction patterns of one-dimensional aperiodic sets. When an aperiodic set is related to the Golden Ratio, denoted by τ, it is well known that the pure point part of its diffractive measure is supported by the extension ring of τ, denoted by . The partition we give is based on the formalism of the so called τ-integers, denoted by . The set of τ-integers is a selfsimilar set obeying , . The pure point spectrum is then partitioned with respect to this “Russian doll" like sequence of subsets . Thus we deduce the partition of the pure point part of the diffractive measure of aperiodic sets.