Meyer's concept of quasicrystal and quasiregular sets
Abstract
This paper relates two mathematical concepts of long-range order of a set of atoms Λ, each of which is based on restrictions on the set of interatomic distances Λ-Λ. A set Λ in &R;n is a Meyer set if Λ is a Delone set and there is a finite set F such thatΛ - Λ subseteq Λ + F.{text{ Y}}. Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept of quasiregular set, which is a set Λ such that both Λ and Λ-Λ are Delone sets. This paper shows that these two concepts are identical.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- August 1996
- DOI:
- 10.1007/BF02102593
- Bibcode:
- 1996CMaPh.179..365L