Optimal embedding of Meyer sets into model sets

Abstract

Abstract. We give a constructive proof that a repetitive Meyer multiple set of R d admits a smallest model multiple set containing it colorwise.Meyer sets are objects of central importance in the mathematical theory of Quasicrystals developed in the last thirty years. A special kind of Meyer sets is the class of so-called model sets, which are point patterns for which a geometric picture is available, the latter highly desirable for the understanding of the point pattern as well as for the computation of its relevant quantities. It is known for a long time that a Meyer set always embeds into at least one model set, but it is also true that a huge collection of model sets will contains a common prescribed Meyer set. Hence a natural question is whether there exist a "smallest" model set containing a given Meyer set, and our aim in this note is to answer this question, in the special case where the Meyer set is repetitive, by the affirmative. The existential result is as follows:Theorem. For any repetitive Meyer set Λ of R d is associated a unique model set Λ such that whenever ∆ is a model set with Λ ⊆ ∆ then one hasWe provide a constructive proof of this result (see Theorem 3.1) by explicitly determining the CPS and window involved in the construction of Λ. The proof will be given in the slightly more general formalism of Meyer multiple set, as it for instance naturally appears in the setting of substitution point sets. Meyer sets and model sets of R dA subset Λ of R d is called a Delone set if it consists of a uniformly discrete collection of points, that is, it admits a uniform separation distance between any two of its points, and if it is relatively dense, meaning that any vector of R d fits at uniformly bounded distance to some point of Λ. A Delone set is called a Meyer set, or is said to have the Meyer property, if there is a finite subset F of R d such thatFinally we call a Meyer multiple set a finite collection Λ = (Λ i ) i∈I of Meyer sets where the support S(Λ) := ∪ i∈I Λ i is again a Meyer set. We will often call such quantity a point pattern along the text. There are several very interesting equivalent formulations of the Meyer property which can be found in [8]. The hull of a Meyer

Model sets with Euclidean internal space

“…In the CPS the space R d is called the physical space, the locally compact Abelian group H is called the internal space and the set L the lattice. Following [Auj16b], we have that a CPS can also be described as a triple (H, Γ, s H ) where H is a locally compact σ-compact Abelian group, Γ a countable subgroup of R d and s H : Γ → H a group morphism with range s H (Γ) dense in H such that the graph…”

“…We remark that Aujogue defined s H in [Auj16a] with a sign plus instead of a minus as we did. So, some results that we use from [Auj16a] and [Auj16b] look slightly different since we need to do a correction in the sign. From Proposition 5.1, the triple (H, Γ, s H ) is a CPS.…”

“…The CPS (H, Γ, s H ) and the window [Ξ Λ ] srp are called the optimal CPS and the optimal window for Λ, respectively. Indeed, in [Auj16b], the author proved that the model set that it defines,…”

“…Finally, we recall some results in [Auj16b] that we use in the proof of the Main Technical Lemma'. The first result allows to prove that a compact and irredundant set is a window.…”

Model sets with Euclidean internal space

“…Then we show that the maximal equicontinuous factor of the hull associated to the Lagarias CPS (for any window) is topologically conjugate to the address system. Then we use [Auj16b, Proposition 3.3] to prove that the closure of the projection in the internal space of the lifting of the Meyer set to the product space is a window for the Lagarias CPS. Finally, elaborating on the ideas in [Auj16a, Theorem 6.1], we show that if there is a point with a unique preimage under the maximal equicontinuous factor map of the hull of the Meyer set then the Meyer set is an inter-model set.…”