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
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
![$\mathbb{Z}[\tau]$](/articles/epjb/abs/2004/11/b03705/b03705_tex_eq1.png)
. 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
![$\mathbb{Z}_\tau/\tau^{j-1}\subset\mathbb{Z}_\tau/\tau^j \subset \mathbb{Z}_\tau/\tau^{j+1} \subset\mathbb{Z}[\tau]$](/articles/epjb/abs/2004/11/b03705/b03705_tex_eq3.png)
, 
. 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.
PACS: 61.44.Br – Quasicrystals / 61.10.Dp – Theories of diffraction and scattering
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2004
