Model sets

We now turn to a more precise notion about almost periodicity: model sets. We begin this section by motivating the introduction of these sets: we give an alternative construction of the setsbA(Zⁿ) in terms of model sets.

A pointx∈Zⁿbelongs tobA(Zⁿ) if and only if there existsy∈Zⁿsuch thatkx−Ayk_∞<

if we note p₁ and p₂ the projections of R²ⁿ on respectively the n firsts and then last coordinates, and if we set W =]−¹

2,¹₂]ⁿ, then bA(Zⁿ) =n

p₂(M_Av)|v∈Z²ⁿ, p₁(M_Av)∈Wo .

This notion is close to that of model set introduced by Y. Meyer in the early seventies [Mey72], but in our case the projection p₂ is not injective. Model sets are sometimes called “cut and project” sets in the literature.

Definition 7.32. LetΛbe a lattice ofR^m+n,p₁andp₂the projections ofR^m+non respec-tivelyR^m× {0}_Rn and {0}_Rm×Rⁿ, and W a Riemann integrable subset ofR^m. Themodel setmodelled on the latticeΛand thewindowW is (see Figure7.6)

Γ =n

p₂(λ)|λ∈Λ, p₁(λ)∈Wo .

Here, we will call model set every set of this type, even if the projectionp₂ is not injective. Indeed, this phenomenon is what is interesting for us, if it did not occur there would be no loss of injectivity when applying discretizations of linear maps. Notice that this definition, which could seem very restrictive for the setΓ, is in fact quite general: as stated by Y. Meyer in [Mey72], every Meyer set⁵is a subset of a model set. Conversely, model sets are Meyer sets (see [Mey95]).

Returning to our problem of images of the lattice Zⁿ by discretizations of linear maps, we have the following (trivial) result.

5. A setΓ is aMeyer setifΓ−Γ is a Delone set. It is equivalent to ask that there exists a finite set F such thatΓ−Γ⊂Γ+ F (see [Lag96]).

Proposition 7.33. LetA₁,· · ·,A_k ∈GL_n(R) bekinvertible maps, then the k-th imageΓ_k =

Remark7.34. This notion has the advantage that it builds the k-th image directly: the concept of time disappears, so we will be able “anticipate” the behaviour of successive images. The downside is the increasing of the dimension; thus it will be more difficult to have a geometric intuition. . .

In the sequel, we will only consider model sets whose window is regular.

Definition 7.35. Let W be a subset of Rⁿ. We say that W isregular if for every affine subspace V⊂Rⁿ, we have where Leb_Vdenotes the Lebesgue measure on V, and B_V

∂(V∩W),η

the set of points of V whose distance to∂(V∩W) is smaller thanη(of course, the boundary is also take in restriction to V).

The link with the previous sections is made by the following theorem.

Theorem 7.36(in collaboration with Y. Meyer). A model set modelled on a regular window is an almost periodic pattern.

In other words, for everyε>0, there exists R₀>0 and a relatively dense setN such that for everyv∈N and every R≥R₀, most of the points (i.e.a proportion greater than ε) of the model setΓ also belong tov+Γ (see Definition7.3).

We begin by proving a weak version of this theorem.

Lemma 7.37. A model set modelled on a window with nonempty interior is relatively dense.

Proof of Lemma7.37. We prove this lemma in the specific case where the window is B_η (recall that B_ηis the infinite ball of radiusηcentred at 0). We will use this lemma only in this case (and the general case can be treated the same way).

Let Γ be a model set modelled on a lattice Λ and a window B_η. We will use the fact that for any centrally symmetric convex set S⊂Rⁿ, if there exists a basis e₁,· · ·, e_n of Λsuch that for eachi,dn/2ee_i∈ S, then S contains a fundamental domain ofRⁿ/Λ, that is to say, for everyv ∈Rⁿ, we have (S +v)∩Λ,∅. This is due to the fact that the parallelepiped spanned by the vectorse_i is included into the simplex spanned by the vectorsdn/2ee_i.

and remark that imp₂= kerp₁⊂V, simply because for every vectorial line D⊂Rⁿ(and

We then use the remark made in the beginning of this proof and apply it to the linear space V⁰, the set S = this proves that the model set is relatively dense for the radius R.

Proof of Theorem7.36. LetΓ be a model set modelled on a latticeΛand a window W.

First of all, we decomposeΛinto three supplementary modules:Λ=Λ₁⊕Λ₂⊕Λ₃, such that (see [Bou98, Chap. VII, §1, 2]):

1. Λ₁= kerp₁∩Λ; consider-ing the dimensions in the decomposition (7.12), we get

dim span(Λ₃) = dim

kerp₁⊕span

p₁(Λ₃)

. (7.13)

The following matrix represents a basis ofΛ=Λ₂⊕Λ₃in a basis adapted to the decom-position (7.12).

We can see that the projection of the basis ofΛ₃on imp₂⊕span

p₁(Λ₃)

form a free family; by Equation (7.13), this is in fact a basis. Thus, span(Λ₃)⊃kerp₁= imp₂, so span

p₂(Λ₃)

= im(p₂).

Forη>0, letN(η) be the model set modelled onΛand B(0,η), that is N (η) ={p₂(λ₃)|λ₃∈Λ₃,kp₁(λ₃)k_∞≤η}.

Lemma7.37asserts thatN (η) is relatively dense in the space it spans, and the previous paragraph asserts that this space is imp₂. The next lemma, which obviously implies Theorem7.36, expresses that ifηis small enough, thenN (η) is the set of translations we look for.

Lemma 7.38. For every ε> 0, there exists η >0 and a regular model set Q(η) such that D⁺(Q(η))≤εand

v∈N(η)⇒(Γ +v)∆Γ ⊂Q(η).

We have now reduced the proof of Theorem7.36to that of Lemma7.38.

Proof of Lemma7.38. We begin by proving that (Γ +v)\Γ ⊂ Q(η) whenv ∈N (η). As v∈N(η), there existsλ₀∈Λ₃such thatp₂(λ₀) =vandkp₁(λ₀)k_∞≤η.

Ifx ∈Γ +v, thenx =p₂(λ₂+λ₃) +p₂(λ₀) =p₂(λ₂+λ₃+λ₀) whereλ₂∈Λ₂, λ₃ ∈Λ₃ and p₁(λ₂+λ₃) ∈ W. If moreover x < Γ, it implies thatp₁(λ₂+λ₃+λ₀) < W. Thus, p₁(λ₂+λ₃+λ₀)∈W_η, where (recall that V = span(p₁(Λ₃)))

W_η=n

k+w|k∈∂W, w∈V∩B_ηo .

We have proved thatΓ +v\Γ ⊂Q(η), where Q(η) =n

p₂(λ)|λ∈Λ, p₁(λ)∈W_ηo .

Let us stress that the model set Q(η) does not depend onv. We now observe that as W is regular, we have

X

λ₂∈Λ₂

Leb_V+p1(λ2)

W_η∩(V +p₁(λ₂))

−→

η→00.

Asp₁(Λ₃) is dense in V (thus, it is equidistributed), the uniform upper density of the model set Q(η) defined by the window W_ηcan be made smaller thanεby takingηsmall enough.

The treatment ofΓ \(Γ +v) is similar; this ends the proof of Lemma7.38.