Jean-Louis Verger-Gaugry
email: jlverger@ujf-grenoble.fr

On Self-Similar Finitely Generated Sphere Packings

Keywords: 2000 Mathematics Subject Classification: 52C17, 52C23 Delone set, Meyer set, mathematical quasicrystal, self-similarity, algebraic integer, Euclidean minimum, inhomogeneous minimum, hole, density, sphere packing

This paper is first a brief survey on links between Geometry of Numbers and aperiodic crystals in Physics, viewed from the mathematical side. In a second part, we prove the existence of a canonical cut-and-project scheme above a (ssfgud set) self-similar finitely generated packing of (equal) spheres Λ in R n and investigate its consequences, in particular the role played by the Euclidean and inhomogeneous minima of the algebraic number field generated by the self-similarity on the Delone constant of the sphere packing. We discuss the isolation phenomenon. The degree d of this field divides the Z-rank of Z[Λ -Λ]. We give a lower bound of the Delone constant of a k-thin ssfgud sphere packing which arises from a model set or a Meyer set when d is large enough.

Introduction

The mathematics of uniformly discrete point sets and Delone sets developped recently has at least four different origins: (i) the experimental evidence of nonperiodic states of matter in condensed matter physics, so-called aperiodic crystals, like quasicrystals [START_REF]Beyond Quasicrystals[END_REF] [51] [START_REF]Lectures on Quasicrystals[END_REF] [63] [START_REF] Shechtman | Metallic Phase with Long-Range Orientational Order and No Translational Symmetry[END_REF] incommensurate modulated crystals phases [62] [START_REF] Janssen | From Quasiperiodic to more Complex Systems[END_REF] and their geometric modelization (cf Appendix), (ii) some works of Delone (Delone) [START_REF] Delone | Neue Darstellung des Geometrische Kristallographie[END_REF] [START_REF] Delone | Sur la Sphère Vide[END_REF] [START_REF] Fadeev | Boris Nikolaevich Delone (On his Life and Creative Work)[END_REF] [START_REF] Ryshkov | Density of an (r, R)-system[END_REF] on geometric crystallography (comparatively, see [START_REF]International Tables for Crystallography[END_REF] [START_REF] Michel | N -Dimensional Crystallography[END_REF] [84] [START_REF] Schwarzenberger | N -Dimensional Crystallography[END_REF] for a classical mathematical approach of periodic crystals), (iii) some works of Meyer on now called cut-and-project sets and Meyer sets [START_REF] Meyer | Nombres de Pisot, Nombres de Salem et Analyse Harmonique[END_REF] [START_REF] Meyer | Algebraic Numbers and Harmonic Analysis[END_REF] [77] [START_REF] Patera | Quasicrystals and Discrete Geometry[END_REF] (for a modern language of Meyer sets in locally compact Abelian groups: [START_REF] Moody | Meyer sets and their Duals[END_REF]), (iv) the theory of self-similar tilings [START_REF] Bandt | Self-similar tilings and patterns described by mappings[END_REF] [70] [START_REF] Solomyak | Conjugates of beta-numbers and the zero-free domain for a class of analytic functions[END_REF] and the use of ergodic theory to understand diffractivity [START_REF] Baake | Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra[END_REF] [92] [START_REF] Solomyak | Conjugates of beta-numbers and the zero-free domain for a class of analytic functions[END_REF]. In particular, the impact on mathematics of the discovery of quasicrystals in 1984 [START_REF] Shechtman | Metallic Phase with Long-Range Orientational Order and No Translational Symmetry[END_REF], as long-range ordered phases, was outlined by Lagarias [START_REF] Lagarias | The Impact of Aperiodic Order on Mathematics[END_REF]. The term mathematical quasicrystals [START_REF]Directions in Mathematical Quasicrystals[END_REF] [START_REF] Lagarias | Mathematical Quasicrystals and the Problem of Diffraction[END_REF] was proposed to name these Delone sets which are used as discrete geometrical models of these new states of matter which have particular spectral or diffraction properties; in particular crystals those for which the spectrum is essentially pure point (see [START_REF]International Union of Crystallography, Report of the Executive Committee for 1991[END_REF] [96] and the Appendix for the new definition of what is a crystal, and [START_REF] Cowley | Diffraction Physics[END_REF] [53] [START_REF] Hof | Diffraction by aperiodic structures[END_REF] for spectral/diffraction theory). Delone sets are conceived as natural generalizations of lattices in modern crystallography.

In this note we will briefly review these notions (Section 2) and will consider more generally uniformly discrete sets of R n , in particular (ssfgud) self-similar finitely generated uniformly discrete sets (Definition 2. [START_REF] Böröczky | Finite packing and covering[END_REF]. A uniformly discrete set of R n of constant r > 0 is a packing of (equal) spheres of R n of (common) radius r/2. There are several advantages to consider uniformly discrete sets instead of Delone sets only: their R-spans may take arbitrary dimensions between 0 and n, while that of a Delone set is only n, they can be finite sets which is forbidden for Delone sets, they may exhibit (spherical) holes of arbitrary size at infinity whereas the size of holes in Delone sets is limited by the Delone constant. A classification of uniformly discrete sets, hence of Delone sets, which extends that given in [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF], is proposed in Subsection 2.3. Finitely generated uniformly discrete sets of R n constitute the largest class on which an address map (Subsection 2.3) can be defined.

The theory of ssfgud sets complements that of lattice packings of (equal) spheres of R n [START_REF] Böröczky | Finite packing and covering[END_REF] [START_REF] Cassels | An introduction to the Geometry of Numbers[END_REF] [26] [START_REF] Gruber | Geometry of Numbers[END_REF] [START_REF] Martinet | Les Réseaux Parfaits des Espaces Euclidiens[END_REF] [107] since a lattice is already a ssfgud set itself (integers are self-similarities), where lattices are or not O F -lattices for F an algebraic number field with involution [START_REF] Bayer-Fluckiger | Lattices and number fields[END_REF] [START_REF] Coulangeon | Voronoi theory over algebraic number fields, in Réseaux Euclidiens, designs sphériques et formes modulaires[END_REF], and makes use of algebraic integers of certain types (Subsection 2.4). Self-similar Meyer sets only admit self-similarities which are Pisot or Salem numbers [START_REF] Meyer | Nombres de Pisot, Nombres de Salem et Analyse Harmonique[END_REF], while self-similar finitely generated Delone sets only provide Perron or Lind numbers as selfsimilarities [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF]. It is an open problem to find a criterium which ensures that a given uniformly discrete set admits at least one self-similarity. For a general Delone set symmetries and in particular inflation symmetries are expected to be rare, especially when the dimension of the ambient space is large, probably more frequently than for lattices; a fortiori for uniformly discrete sets. For lattices, Bannai [START_REF] Bannai | Positive definite unimodular lattice with trivial automorphism groups[END_REF] has shown the existence of many unimodular Z-lattices with trivial automorphism group in a given genus of positive definite unimodular Z-lattices of sufficiently large rank (see also [START_REF] Collinet | Homologie rationelle du groupe On(Z[ 1 2 ) et automorphismes des réseaux unimodulaires[END_REF]).

The existence of cut-and-project schemes above Delone sets is useful to characterize the set of its self-similarities, inflation centers, local clustering, etc [START_REF] Cotfas | G-Model Sets and Their Self-Similarities[END_REF] [START_REF] Cotfas | On the Self-Similarities of a Model Set[END_REF] [49] [START_REF] Masákovà | Inflation Centers of the Cut-and-Project Quasicrystals[END_REF]. Given an arbitrary uniformly discrete set it is an open problem whether a cut-and-project scheme lies above it (Subsection 2.1.2). Theorem 1.1 answers in full generality to this problem for a given ssfgud set Λ ⊂ R n (with Λ ∈ UD f g , see Subsection 2.3) with self-similarity λ. The constructions use the Archimedean embeddings of the number field K := Q(λ) generated by the self-similarity λ (Section 3) in a vectorial way, as a product of copies of the étale R-algebra K R := K ⊗ Q R. Denote by Σ : K → K R the canonical map. The structure of the lattice in the cut-and-project scheme above Λ arises as a consequence of the Jordan invariants of R rk Λ as a K[X]module (from (ii) in Theorem 1.1).

Theorem 1.1. Let Λ ⊂ R n , n ≥ 1, be a uniformly discrete set such that m := rk Z[Λ -Λ] < +∞ with m ≥ 1. Let λ > 1 be a (affine) self-similarity of Λ, i.e. a real number > 1 such that λ(Λc) ⊂ Λc for a certain c ∈ R n . Then (i) λ is a real algebraic integer of degree d ≥ 1 and d divides m, (ii) there exist r = m/d Q-linearly independent vectors w 1 , w 2 , . . . , w r in the Q(λ)-vector space Q[Λ -Λ] such that Z[Λ -Λ] is a rank m Zsubmodule of the Z-module: Z[w 1 , λw 1 , . . . , λ d-1 w 1 , w 2 , λw 2 , . . . , λ d-1 w 2 , . . . , w r , λw r , . . . , λ d-1 w r ],

(iii) for every Z-basis {v 1 , v 2 , . . . , v m } of Z[Λ -Λ], a matrix relation: λV = M V holds, where V = t [v 1 , . . . , v m ] and M is an invertible integral m × m matrix with characteristic polynomial det(X I -M ) = (ϕ(X)) m/d in which ϕ(X) is the minimal polynomial of λ; in particular, det M = N K/Q (λ) m/d , where N K/Q (λ) is the algebraic norm of λ, (iv) there exists a cut-and-project scheme above Λ:

r i=1 K R w i w i ≃ H × R[Λ], L, π, pr 1
where the lattice L = r i=1 Σ Z[λ] wi wi is such that pr 1 (L) ⊃ Z[Λ -Λ], whose internal space H is the product of two spaces:

H = R K \ R[Λ] × G where R K is the image of R[Λ] in r i=1 K R wi
wi by the real and imaginary embeddings of K, and G the closure in r i=1 K R wi wi of the image by Σ of the space of relations over K between the generators w 1 , . . . , w r . The space R K \ R[Λ] is called the shadow space of Λ. This cut-and-project scheme is endowed with an Euclidean structure given by a real Trace-like symmetric bilinear form for which R K and G are orthogonal.

The central cluster of the basis (λ j w i ) i=1,...,r,j=0,...,d-1 is by definition the set {w 1 , w 2 , . . . , w r }. Note that some vectors in a central cluster may be Rlinearly dependent. When w 1 , w 2 , . . . , w r have identical norms and constitute orbits (i.e. F -clusters) under the action of a finite group, say F , constructions in (iv) in Theorem 1.1 can be deduced from [START_REF] Cotfas | A Mathematical Construction of n-dimensional Quasicrystals Starting from G-clusters[END_REF]. It is easy to check that

r = 1 in Theorem 1.1 =⇒ the R-span of Λ is one-dimensional .
The converse is generally wrong and Subsection 2.5 gives the example of sets Z β of beta-integers [START_REF] Barat | Directions in Numeration[END_REF] on the line for which open problems exist.

Corollary 1.2. If Λ is a self-similar finitely generated sphere packing in R n , with self-similarity λ, such that r = 1, i.e. for which the degree d of λ equals the rank m of Z[Λ -Λ], then Z[Λ -Λ] is the projection of a sublattice of finite index of an Arakelov divisor of K = Q(λ) in the cut-and-project scheme above Λ. This index is an integer multiple of (O K : Z[λ]).

Theorem 1.1 gives a framework for constructing aperiodic (equal) sphere packings B(Λ) for which local arrangements, for instance like t-designs [START_REF] Bachoc | Modular forms, lattices and spherical designs, Réseaux Euclidiens, designs sphériques et formes modulaires[END_REF], can be computed from a lattice in higher dimension above Λ.

Dense sphere packings of R n are of general interest [START_REF] Bezdek | Compact packings in the Euclidean space[END_REF] [22] [START_REF] Conway | Sphere packings, lattices and groups[END_REF] [50] [START_REF] Gruber | Geometry of Numbers[END_REF] [81] [START_REF] Zong | Sphere packings[END_REF]. For a sphere packing whose set of centers is a Delone set Λ which is a uniformly discrete of constant r > 0, of Delone constant R(Λ) := sup z∈R n inf µ∈Λ zµ , the density δ(B(Λ)) of B(Λ) satisfies [START_REF] Muraz | On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings[END_REF]: δ(B(Λ)) ≥ ( 2R(Λ)/r ) -n . If Λ is only a uniformly discrete set, no equivalent formula for bounding from below the density δ(B(Λ)) exists in general. However, it is not the case for Delone-like uniformly discrete sets of R n , which behave in some sense like Delone sets (Subsection 5.1).

Theorem 1.3. Let Λ be a uniformly discrete set of R n , of constant r > 0, which is Delone-like of pseudo-Delone constant R(Λ). Then the density of the sphere packing B(Λ) (of common radius r/2) whose set of centers is Λ satisfies:

δ(B(Λ)) ≥ 2R(Λ) r -n . (1.1)
Therefore it is crucial to obtain interesting lower bounds of the Delone constant (or pseudo-Delone constant) R(Λ) to control dense sphere packings, in particular, sphere packings whose set of centers is a ssfgud set.

In Section 6 we comment on the two origins of the (pseudo-) Delone constant of a sphere packing whose set of centers is a ssfgud set: the first one lies in the geometrical properties of the central cluster {w 1 , w 2 , . . . , w r } as given by Theorem 1.1 (ii), the second one is of purely arithmetical nature; it comes from the Euclidean and inhomogeneous minima associated with a sublattice of a product of ideal lattices [START_REF] Bayer-Fluckiger | Ideal Lattices, in A panorama of number theory or the view from Baker's garden[END_REF] [23] [START_REF] Cerri | Euclidean minima of totally real number fields[END_REF] in bijection with Z[Λ -Λ] in the cut-and-project scheme given by Theorem 1.1 (iv). Only the case r = 1 is reported in Section 6.

Theorem 1.4. Let Λ ⊂ R n , n ≥ 1, be a ssfgud set which is either a model set or a Meyer set in the cut-and-project scheme defined by Theorem 1.1 (iv) with r = 1, Ω as window and lattice L ′ such that pr

1 (L ′ ) = Z[Λ -Λ].
Assume that the self-similarity λ is of degree

d (≥ 3) large enough, that K = Q(λ) has a unit rank > 1 and is not a CM-field. Then, if Λ is k-thin, k ≥ 2, its Delone constant R(Λ) satisfies: R(Λ) ≥ √ d M (K) 2/d -M k (K) 2/d 1 2 > 0, (1.2)
where M (K), resp. M k (K), is the Euclidean minimum, resp. the k-th Euclidean minimum, of K.

The 75 ièmes Rencontres between Mathematicians and Physicists held at IRMA -Strasbourg on the Thema "Number Theory and Physics" have offered to the author the opportunity of writing this brief note, initially conceived as a short survey, on the relationships between sphere packings, the mathematics of aperiodic crystals, algebraic number theory and numeration in base an algebraic integer > 1.

2 Uniformly discrete sets and Delone sets

Definitions and Topology

Let us define uniformly discrete sets and Delone sets in two different contexts: in the metric case when the ambient space is a metric space which is σ-compact and locally compact, like R n , and when the ambient space is R n with a cutand-project scheme that lies above it with a locally compact abelian group as internal space.

Metric Case

Let (H, δ) be a σ-compact and locally compact metric space with infinite diameter (for δ). A discrete subset Λ of H is said to be uniformly discrete if there exists a real number r > 0 such that x, y ∈ Λ, x = y implies δ(x, y) ≥ r.

A uniformly discrete set is either the empty set, or a subset {x} of H reduced to one element, or, if it contains at least two points, they satisfy such an inequality. If r is equal to the minimal interpoint distance inf{δ(x, y) | x, y ∈ H, x = y} (when Card(Λ) ≥ 2) Λ is said to be a uniformly discrete set of constant r. The space of uniformly discrete sets of constant r > 0 of (H, δ) is denoted by UD(H, δ) r . It is the space SS(H, δ) r of systems of equal spheres (or space of sphere packings) of radius r/2 of (H, δ): Λ = (a i ) i∈N ∈ UD(H, δ) r is the set of sphere centers of

B(Λ) = {B(a i , r/2) | i ∈ N} ∈ SS(H, δ) r
where B(z, t) denotes generically the closed ball centered at z ∈ H of radius t > 0.

An element Λ ∈ ∪ r>0 UD(H, δ) r is said to be a Delone set if there exists R > 0 such that, for all z ∈ H, there exists an element λ ∈ Λ such that δ(z, λ) ≤ R (relative denseness property). Then a Delone set is never empty. If Λ is a Delone set, then

R(Λ) := sup z∈H inf λ∈Λ δ(z, λ) (2.1)
is called the Delone constant of Λ. In [START_REF] Muraz | On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings[END_REF] the range of values of the ratio R(Λ)/r in the case

H = R n , n ≥ 1, is shown to be the continuum √ 2 2 n n + 1 , +∞ . (2.2)
In the context of lattices which are O K -modules, with K a number field, the "ambient space" is obtained in a canonical way via the real and complex embeddings of K and the Delone constant is reminicent of the Euclidean minimum or inhomogeneous minimum, with possible isolated values instead of the continuum (2.2) (Section 6). The Delone constant of Λ is the maximal circumradius of all its Voronoi cells. If Λ ∈ UD(H, δ) r is a Delone set of Delone constant R, the discrete set Λ is also called a (r, R)-system [START_REF] Ryshkov | Density of an (r, R)-system[END_REF]. Let X(H, δ) r,R ⊂ UD(H, δ) r be the subset of uniformly discrete sets of constant r which are Delone sets of H of Delone constant ≤ R.

Theorem 2.1. Let (H, δ) be a σ-compact and locally compact metric space for which diam(H) is infinite. Then, for all r > 0, UD(H, δ) r can be endowed with a metric d such that the topological space (UD(H, δ) r , d) is compact and such that the Hausdorff metric on UD(H, δ) r,f is compatible with the restriction of the topology of (UD(H, δ) r , d) to UD(H, δ) r,f . For all R > 0 the subspace X(H, δ) r,R is closed.

In [START_REF] Muraz | On a Generalization of the Selection Theorem of Mahler[END_REF] several (classes of equivalent) metrics on H are constructed. In such constructions a base point, say α ∈ R n , is required. When H = R n , endowed with the Euclidean norm • , the topology on UD(R n , • ) r , r > 0, is expressed by "unique local pairings of points in big balls centered at the base point α", as follows (Proposition 3.6 in [START_REF] Muraz | On a Generalization of the Selection Theorem of Mahler[END_REF]). Let r = 1, the general case being the same.

Proposition 2.2. Let Λ, Λ ′ ∈ UD(R n , • ) 1 with Λ and Λ ′ nonempty. Let l = inf{ t -α | t ∈ Λ} < +∞ and ǫ ∈ (0, (1 + 2l) -1 ). Assume d(Λ, Λ ′ ) < ǫ. Then, for all λ ∈ Λ such that λ -α < 1-ǫ 2ǫ , (i) there exists a unique λ ′ ∈ Λ ′ such that λ -λ ′ < 1 2 , (ii) this pairing (λ, λ ′ ) satisfies the inequality: λ -λ ′ ≤ ( 1 2 + λ -α )ǫ.
2.1.2 Cut-and-Project Schemes Above Uniformly Discrete Sets A locally compact abelian (lca) group is an abelian group G endowed with a topology for which G is a Hausdorff space, each point admits a compact neighbourhood, and such that the mapping G × G → G, (x, y) → xy is continuous. In the sequel we will denote additively the additive law of G so that 0 is the neutral element of G.

Definition 2.3. Let G be a lca group. (i) A subset Λ of G is uniformly discrete if there exists an open neighbourhood W of 0 so that (Λ -Λ) ∩ W = {0}, (ii) a subset Λ of G is relatively dense if there exists a compact subset K of G such that G = Λ + K, (iii) a Delone set of G is a subset Λ of G which is relatively dense and uniformly discrete.

Definition 2.4. A lattice of R n , n ≥ 1, is a discrete Z-module of rank n.
A lattice in a lca group G is a subgroup L of G such that: (i) L is discrete, i.e. the topology on L induced by that of G is the discrete topology, (ii) L is cocompact, i.e. G/L is compact.

In the sequel we will only define cut-and-project schemes over uniformly discrete sets Λ which lie in finitely dimensional Euclidean spaces R n , leaving aside the general case where the ambient space of Λ is a lca group. Such more general constructions can be found in [START_REF] Meyer | Nombres de Pisot, Nombres de Salem et Analyse Harmonique[END_REF], Chap. II, and in [START_REF] Schreiber | Approximations diophantiennes et problèmes additifs dans les groupes abéliens localement compacts[END_REF]. Denote by L n the space of (affine) lattices of R n , n ≥ 1. Definition 2.5. A cut-and-project scheme (over R n ) is given by a 4-tuple (G × R n , L, π 1 , π 2 ) where:

(i) G × R n is the direct product of a lca group G and the n-dimensional

Euclidean space R n , n ≥ 1, (ii) L is a lattice in G × R n , such that the natural projections π 1 : G × R n → G and π 2 : G × R n → R n satisfy (1) the restriction π 2 | L of π 2 to L is a bijection from L to π 2 (L), ( 2
) the image π 1 (L) is dense in G.
G is called the internal space.

Definition 2.6. Let Λ be a uniformly discrete set in the n-dimensional Euclidean space R n , n ≥ 1. A cut-and-project scheme given by the 4-tuple

(G × R n , L, π 1 , π 2 ) is said to lie above Λ if there exists t ∈ R n such that Λ -t ⊂ π 2 (L).
Remark 2.7. The need to introduce the translation t in the last definition comes from the fact that the uniformly discrete set Λ does not necessarily contain the base point of the cut-and-project scheme, which is the origin of R n and at the same time the origin of G. Being a uniformly discrete set, or a Delone set, is an affine notion in the ambient space R n , while cut-andproject schemes priviledge a base point. For instance the lattice u + Z, u = 1/2, of R admits ({0} × R, Z, 0, Id) as cut-and-project scheme above it; the translation t being 1/2 in this case. If Λ is a Delone set, the translation t can be chosen such that: t ≤ R(Λ) the Delone constant of Λ.

In the last definition, the image π 2 (L) of the discrete subgroup L ⊂ G × R n is a Z-module in R n (the classical structure of Z-modules in R n is given for instance in [START_REF] Descombes | Eléments de Théorie des Nombres[END_REF], Theorem 2.3.7).

Cut-and-project sets, also called model sets, of R n form a particular class of Delone sets. Definition 2.8. A discrete subset Λ of R n , n ≥ 1, is a cut-and-project set, or model set, if there exists a cut-and-project scheme (G × R n , L, π 1 , π 2 ) over Λ, with G a lca group, and a relatively compact subset Ω of the internal space G, with nonempty interior, such that:

Λ -t = {π 2 (ω) | π 1 (ω) ∈ Ω},
for a certain t ∈ R n . The set Ω is called the window of the cut-and-project set Λ = Λ(Ω).

Model sets which arise from cut-and-schemes (G × R n , L, π 1 , π 2 ) with a lca group G as internal space do not differ too much from model sets that come from cut-and-project sets where the internal space is R m , for a certain m, by the following proposition. Proposition 2.9. Let Λ(Ω) be a cut-and-project set in the cut-and-project scheme (G×R n , L, π 1 , π 2 ) where G is a lca group. Then there exists a subgroup of G isomorphic to R m , for a certain m ≥ 0, and a model set

Λ ′ ⊂ R n having (R m × R n , L ′ , π 1 , π 2 )
as cut-and-project scheme above it such that Λ(Ω) is contained in a finite number of translates of Λ ′ . Proof. Proposition 2.7 in [START_REF] Moody | Meyer sets and their Duals[END_REF].

Proposition 2.10. Let Λ = Λ(Ω) be a model set in a cut-and-project scheme

(G × R n , L, π 1 , π 2 ) where G is a lca group. Then (i) Λ is Delone set of R n , (ii) if Ω ⊂ int(Ω) (adherence of its interior)
and Ω generates G as a group, the following equality holds:

Z[Λ -Λ] = π 2 (L).
Proof. Proposition 2.6 in [START_REF] Moody | Meyer sets and their Duals[END_REF].

Remark 2.11. In [START_REF] Moody | Meyer sets and their Duals[END_REF] and [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF], the origin implicitely belongs to the Delone set Λ, whereas in the present note we do not assume this minor fact. That is why we refer to Z[Λ -Λ] everywhere instead of Z[Λ], as in Theorem 1.1 or in Proposition 2.10 for instance.

Continuity of sphere packings arising from model sets

Let Λ be a model set in R n , viewed as set of centers of a sphere packing, and consider the cut-and-project scheme

(R k × R n , L, π 1 , π 2 ) above Λ which allows the construction of Λ by means of a window Ω ⊂ R k . Let us fix the direct product R k × R n . Let us write Λ = Λ L (Ω)
and consider how the model set Λ L (Ω) varies when Ω and L vary continuously. Let W(R k ) be the uniform space of nonempty open relatively compact subsets of R k (set of acceptance windows in R k ) whose affine hull is R k , endowed with the pseudo-metric

∆ W (Ω 1 , Ω 2 ) := ∆(Ω 1 , Ω 2 )
where ∆ is the Hausdorff metric on the space of nonempty closed subsets of R k . The space of lattices L n+k in R n+k is equipped with the quotient topology of GL(n + k, R)/GL(n + k, Z). The metric d built on r>0 UD(R n+k , • ) r with 0 as base point [START_REF] Muraz | On a Generalization of the Selection Theorem of Mahler[END_REF] is compatible with the quotient topology of GL(n + k, R)/GL(n + k, Z). Let UD = r>0 UD(R n , • ) r be the space of uniformly discrete subsets of R n , endowed with the metric d where here an arbitrary base point α ∈ R n is taken (see [START_REF] Muraz | On a Generalization of the Selection Theorem of Mahler[END_REF], Theorem 2.1 and Proposition 2.2). Denote by d α the metric d in this paragraph only. The two origins, of the cut-and-project scheme and of R n for the construction of the metric d α on UD, are taken a priori different.

Theorem 2.12. For any base point α ∈ R n , the mapping

W(R k ) × L n+k → (UD, d α ) : (Ω, L) → Λ L (Ω) is continuous. Proof. Let ǫ > 0. Let L 0 ∈ L n+k and Ω 0 ∈ W(R k ). Let us show the continuity at (Ω 0 , L 0 ). Let t = α + 1-ǫ/2 ǫ . Since Ω 0 is open, there exists η 1 > 0 such that all the sets {x ∈ L | π 1 (x) ∈ Ω 0 , π 2 (x) ≤ t} have the same cardinality if L belongs to the open set {L | d(L, L 0 ) < η 1 }. Since π 2 is continuous and π 2 | L 0
is assumed to be a bijection from L 0 onto π 2 (L 0 ), then π 2 | L is also a bijection from L onto π 2 (L) as soon as d(L, L 0 ) is small enough. Then, using Proposition 2.2 and invoking the continuity of π 2 , there exists

η ′ ≤ η 1 such that d(L, L 0 ) < η ′ implies d α (Λ L (Ω 0 ), Λ L0 (Ω 0 )) < ǫ/2. The subset {x ∈ L | π 1 (x) ∈ Ω, π 2 (x) ≤ t} of L is
such that its projection by π 1 is made of a finite collection of points which lie inside Ω (which is open), and its projection by π 2 is a finite subset of π 2 (L) which contains Λ L (Ω 0 ) ∩ B(α, 1-ǫ/2 ǫ ) (see Proposition 2.2). Since the projection mappings π 1 and π 2 are continuous and that π 2 | L is a bijection from L onto π 2 (L), the mapping

L → π 1 •(π 2 | L ) -1 is continuous on the open set {L | d(L, L 0 ) < η ′ }. Then there exists η ′′ > 0 such that ∆ W (Ω, Ω 0 ) < η ′′ implies d α (Λ L (Ω), Λ L (Ω 0 )) < ǫ/2 (
the value of t is chosen according to this last inequality and Proposition 2.2).

Then, as soon as ∆ W (Ω, Ω 0 ) < η ′′ and d(L, L 0 ) < η ′ hold, we have:

d α (Λ L (Ω), Λ L0 (Ω 0 )) ≤ d α (Λ L (Ω), Λ L (Ω 0 ))+d α (Λ L (Ω 0 ), Λ L0 (Ω 0 )) < ǫ 2 + ǫ 2 = ǫ.
We deduce the claim.

Note that the assumption "open" for windows in W(R k ) is essential to obtain the continuity in Theorem 2.12. If we consider a collection of model sets parametrized by a sequence of windows which are not necessarily open, but with nonempty interiors, then Theorem 2.12 should be applied with the collections of the interiors of the windows.

A Classification

Classes of Uniformly Discrete Sets and Delone sets in R n , and their relative inclusions, are given in Theorem 2.16, following Lagarias [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF] for Delone sets. We first define some classes of uniformly discrete sets intrinsically, i.e. without any cut-and-project scheme formalism above them. Then we indicate the definitions of point sets which invoke cut-and-project schemes. Definition 2.13. Let Λ be a nonempty uniformly discrete set of R n .

(i) Λ is finitely generated if the Z-module

Z[Λ -Λ] := finite α i (x i -y i ) | α i ∈ Z, x i , y i ∈ Λ is finitely generated, i.e. dim Q Q ⊗ Z[Λ -Λ] < +∞, (ii) Λ is of finite type if, for all t > 0, the intersection (Λ -Λ) ∩ B(0, t) is a finite set.
If Λ is a nonempty finitely generated uniformly discrete set, the rank of Λ, denoted by rk Λ, is by definition the dimension of the Q-vector space

Q ⊗ Z[Λ -Λ] = Q[Λ -Λ]. The rank rk Λ is an invariant of Λ. Let c ∈ R n . The rank of Z[Λ -c] varies with c and may be different of that of Z[Λ -Λ].
For instance, with c = 0 and Λ

= √ 2 + Z in R we have: rk Λ = 1 while the rank of Z[Λ] = Z[Λ -c] equals 2 (the notations rk Λ and rk Z[Λ] should not be confused); moreover Λ = Λ -c and Z[Λ -Λ] = Z are disjoint. Theorem 2.14 (Lagarias). Let Λ be a Delone set of finite type of R n , n ≥ 1. Then rk Λ ≤ Card ((Λ -Λ) ∩ B(0, 2R(Λ))) < + ∞ (2.3)
where R(Λ) is the Delone constant of Λ.

Proof. Theorem 2.1 in [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF].

Definition 2.15. Let Λ be a relatively dense discrete subset of R n . Λ is a Meyer set if one of the following equivalent assertions is satisfied:

(i) Λ -Λ is uniformly discrete,
(ii) Λ is a Delone set and there exists a finite set

F ⊂ R n such that Λ -Λ ⊂ Λ + F, (2.4)
(iii) Λ is a subset of a model set.

Proof. Theorem 9.1 and Proposition 9.2 in [START_REF] Moody | Meyer sets and their Duals[END_REF].

Conditions (i) and (ii) in the definition of Meyer sets are given independently of any "cut-and-project scheme above Λ" consideration while condition (iii) asserts the existence of such a cut-and-project scheme above it. In a similar way a (affine) lattice L ∈ L n in R n is intrinsically defined in R n , without any help of cut-and-project schemes, admits also ({0} × R n , L, 0, π 2 ) as cut-and-project scheme above it and is a model set in this cut-and-project scheme. The objectives of Theorem 1.1 consist in showing the existence of general constructions of cut-and-project schemes above ssfgud sets in R n .

Let n ≥ 1. Denote:

M (R)
:= { Model sets in R n arising from cut-and-project schemes having a m-dimensional Euclidean space R m as internal space } M (lcag) := { Model sets in R n arising from cut-and-project schemes having a lca group G as internal space } M (R) := { Meyer sets in R n arising from cut-and-project schemes having a m-dimensional Euclidean space R m as internal space } M (lcag) := { Meyer sets in R n arising from cut-and-project schemes having a lca group G as internal space

} UD := { Uniformly discrete sets in R n } UD fg := { Finitely generated uniformly discrete sets in R n } ⊂ ∪ r>0 UD(R n , • ) r UD ft := { Uniformly discrete sets of finite type in R n } X fg := { Finitely generated Delone sets in R n } X ft := { Delone sets of finite type in R n } Theorem 2.16.
The following inclusions hold:

UD ft ⊂ UD fg ∪ ∪ M (lcag) ⊃ M (lcag) ⊃ L n ⊂ M (R) ⊂ M (R) ⊂ X ft ⊂ X fg (2.
ϕ v i=1 m i e i = (m 1 , m 2 , . . . , m v ).
In Section 3 we will mainly use address maps of difference sets Λ -Λ for the elements Λ of UD f g .

Algebraic integers, inflation centers and self-similarities

Given Λ a uniformly discrete set of R n , a (affine) self-similarity of Λ is by definition a real number λ > 1 such that

λ(Λ -c) ⊂ Λ -c (2.6)
for a certain point c in R n (note that c belongs or not to Λ). A point c ∈ R n for which (2.6) occurs for a certain λ > 1 is called an inflation center of Λ. The concept of self-similarity is an affine notion and λ depends upon c. Denote by

C(Λ) := {c | ∃λ > 1 such that λ(Λ -c) ⊂ Λ -c} (2.7)
the set of inflation centers of Λ and by

S(c) := {λ > 1 | λ(Λ -c) ⊂ Λ -c} , for c ∈ C(Λ), (2.8)
the set of self-similarities associated with the point c.

Proposition 2.18. Let Λ = t+Z be a (affine) lattice of R of period 1 with t ∈ [0, 1). Then

(i) C(Λ) = t + Q, (ii) S(c) =    N \ {0} if c ∈ t + Z, (1 -qZ) ∩ N \ {0} if c ∈ C(Λ), c = t + p q , with p, q relatively prime (p ∈ Z, q ≥ 2).
The set of inflation centers of Λ of given (point) density 1/(2q) of selfsimilarities, with q an integer ≥ 2, is exactly the uniformly discrete set

t + { p q | p ∈ Z, gcd(p, q) = ±1}.
Proof. Routine, with the following definition of the (point) density of selfsimilarities of an inflation center c ∈ C(Λ):

dens(S(c)) = lim sup t→∞ 1 2t # (S(c) ∩ (1, t]) .
(2.9)

In the general case, given a uniformly discrete set, the characterization of C(Λ) and S(c) with c ∈ C(Λ) remains an open problem, even for Delone sets; see [START_REF] Cotfas | G-Model Sets and Their Self-Similarities[END_REF], [START_REF] Cotfas | On the Self-Similarities of a Model Set[END_REF] and [START_REF] Masákovà | Inflation Centers of the Cut-and-Project Quasicrystals[END_REF] for Penrose tilings and sets Z β of β-integers [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF], with β a quadratic Pisot number. At least, C(Λ) is expected to be far from being everywhere dense as in Proposition 2.18. Definition 2.19. A uniformly discrete of R n , n ≥ 1, is a (ssfgud set) selfsimilar finitely generated uniformly discrete set if it is finitely generated and admits at least one (affine) self-similarity.

Although a uniformly discrete set of R n , n ≥ 1, may be finite, let us observe that a nonempty (ssfgud) self-similar finitely generated uniformly discrete set in R n , n ≥ 1, is always infinite. Definition 2.20. Let λ > 1 be a real algebraic integer. Denote by λ (i) its conjugates. We say that λ is (i) a Pisot number if all its conjugates λ (i) satisfy |λ (i) | < 1, (ii) a Salem number if all its conjugates λ (i) satisfy |λ (i) | ≤ 1, with at least one on the unit circle, (iii) a Perron number if all its conjugates λ (i) satisfy |λ (i) | < λ, (iv) a Lind number if all its conjugates λ (i) satisfy |λ (i) | ≤ λ, with at least one on the circle {|z| = λ}.

Theorem 2.21 (Meyer). Let Λ ⊂ R n , n ≥ 1, be a Meyer set. If Λ is a ssfgud set, then all its self-similarities are Pisot or Salem numbers.

If β > 1 is a Pisot number of a Parry number, then the sets Z β are Meyer sets that admit by construction the self-similarity β [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF]. However, there exist many Meyer sets which have no self-similarities at all. Theorem 2.22 (Lagarias). Let Λ ⊂ R n , n ≥ 1, be a Delone set. If Λ is a ssfgud set, then all its (affine) self-similarities λ are algebraic integers such that degree(λ) divides rk Λ. Moreover, if Λ is of finite type, then all the selfsimilarities are Perron or Lind numbers.

The concept of (affine) self-similarity is extended as follows in a natural way [START_REF] Moody | Meyer sets and their Duals[END_REF].

Definition 2.23. Let Λ be a nonempty uniformly discrete set of R n . A selfsimilarity of Λ is given by a triple (c, λ, Q) where λ is a real number > 1, Q an element of the orthogonal group O(n, R) such that

λQ(Λ -c) ⊂ Λ -c (2.10)
for a certain c ∈ R n (note that c belongs or not to Λ).

A point c for which (2.10) occurs for certain couple (λ, Q) is called an inflation center of Λ, as in the affine case. Problems on self-similar sets are reported in [START_REF] Peres | Problems on self-similar sets and self-affine sets: An update[END_REF].

Sets Z β of beta-integers and Rauzy fractals

Meyer sets Z β ⊂ R of β-integers with β a Pisot number, and their vectorial extension to R n -so-called β-grids -, are useful tools for modeling quasicrystals in physics [START_REF] Duneau | Approximant phases of quasicrystals[END_REF] [37] [47] [START_REF] Gazeau | Counting Systems with Irrational Basis for Quasicrystals[END_REF]. Indeed, Penrose tilings in the plane and in space play a fundamental role in this modelling process, with suitable positioning of atoms in the tiles. Gazeau [START_REF] Gazeau | Counting Systems with Irrational Basis for Quasicrystals[END_REF] has observed that Penrose tilings can easily be deduced from τ -grids, where τ = 1+ √ 5 2

is the golden mean, quadratic Pisot number. Therefore it is natural to extend the constructions of Penrose tilings and β-grids with Pisot numbers β (or more generally with algebraic integers) of higher degree which could be used in the objective of providing possibly new models of aperiodic crystals in crystallography to physicists.

Let us recall the mathematical construction of Z β on the line and its properties when β > 1 is a real number, in a general way, and some open questions related to them when β is in particular an algebraic integer. We refer to [START_REF] Frougny | Number Representation and Finite Automata[END_REF] [45] [START_REF] Parry | On the β-expansions of real numbers[END_REF] [89] and [START_REF] Barat | Directions in Numeration[END_REF] for an overview on recent studies on Numeration and its applications.

Construction and Properties

For a real number x ∈ R, the integer part of x will be denoted by ⌊x⌋ and its fractional part by {x} = x -⌊x⌋. The smallest integer larger than or equal to x will be denoted by ⌈x⌉. For β > 1 a real number and z ∈ [0, 1] we denote by T β (z) = βz (mod 1) the β-transform on [0, 1] associated with β, and iteratively, for all integers j ≥ 0,

T j+1 β (z) := T β (T j β (z))
, where by convention T 0 β = Id. Let β > 1 be a real number. A beta-representation (or β-representation, or representation in base β ) of a real number x ≥ 0 is given by an infinite sequence (x i ) i≥0 and an integer k ∈ Z such that x = +∞ i=0 x i β -i+k , where the digits x i belong to a given alphabet (⊂ N). Among all the betarepresentations of a real number x ≥ 0, x = 1, there exists a particular one called Rényi β-expansion, which is obtained through the greedy algorithm [44] [45]: in this case, k satisfies β k ≤ x < β k+1 and the digits

x i := ⌊βT i β ( x β k+1 )⌋ i = 0, 1, 2, . . . , (2.11)
belong to the finite canonical alphabet A β := {0, 1, 2, . . . , ⌈β -1⌉}. If β is an integer, then A β := {0, 1, 2, . . . , β -1}; if β is not an integer, then A β := {0, 1, 2, . . . , ⌊β⌋}. We denote by

x β := x 0 x 1 x 2 . . . x k . x k+1 x k+2 . . . (2.12)
the couple formed by the string of digits x 0 x 1 x 2 . . . x k x k+1 x k+2 . . . and the position of the dot, which is at the kth position (between x k and x k+1 ). By definition the integer part (in base β) of x is k i=0 x i β -i+k and its fractional part (in base β) is +∞ i=k+1 x i β -i+k . If a Rényi β-expansion ends in infinitely many zeros, it is said to be finite and the ending zeros are omitted. If it is periodic after a certain rank, it is said to be eventually periodic (the period is the smallest finite string of digits possible, assumed not to be a string of zeros).

There is a particular Rényi β-expansion which plays an important role in the theory, which is the Rényi β-expansion of 1, denoted by d β (1) and defined as follows: since β 0 ≤ 1 < β, the value T β (1/β) is here set (by convention) to 1. Then using (2.11) for all i ≥ 1, we obtain: t 1 = ⌊β⌋, t 2 = ⌊β{β}⌋, t 3 = ⌊β{β{β}}⌋, etc. The equality d β (1) = 0.t 1 t 2 t 3 . . . corresponds to 1 = +∞ i=1 t i β -i . By definition, a real number β > 1 such that d β (1) is finite or eventually periodic is called a beta-number or more recently a Parry number (this new name appears in [START_REF] Elkharrat | Symmetry groups for beta-lattices[END_REF]). In particular, it is called a simple beta-number or a simple Parry number (after [START_REF] Elkharrat | Symmetry groups for beta-lattices[END_REF]) when d β (1) is finite. Betanumbers (Parry numbers) are algebraic integers [START_REF] Parry | On the β-expansions of real numbers[END_REF] and all their conjugates lie within a compact subset which looks like a fractal in the complex plane [START_REF] Flatto | The zeta function of the beta transformation[END_REF] [102]. The conjugates of Parry numbers are all bounded above in modulus by the golden mean 1 2

(1 + √ 5) [41] [102].
Definition 2.24. The set

Z β := {x ∈ R | |x| is equal to its integer part in base β}
is called set of beta-integers, or set of β-integers, or set of integers in base β.

By construction, the set Z β is discrete, relatively dense and locally finite (its intersection with any interval of the line is finite), self-similar, with β as self-similarity (with inflation center the origin), and symmetrical with respect to the origin: βZ β ⊂ Z β , Z β = -Z β . Its complete set of self-similarities is unknown. Thurston [START_REF] Thurston | Groups, Tilings and Finite State Automata[END_REF] has shown that it is uniformly discrete, hence a Delone set, when β is a Pisot number. From [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF], for β a Pisot number, the relatively dense set Z β ∩ R + is finitely generated over N.

Theorem 2.25. If β is a Pisot number, the Delone set Z β is a Meyer set which is a ssfgud set in R.

Proof. [START_REF] Burdik | beta-integers as natural counting systems for quasicrystals[END_REF], [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF]. For the definition of a ssfgud set, see Definition 2.19. It is a ssfgud set since βZ β ⊂ Z β .

Open problem (P 1 ).-What is the class of real numbers β > 1 for which Z β is uniformly discrete, equivalently a Delone set ?

We know that this class contains Pisot numbers (see [START_REF] Blanchard | β-expansions and Symbolic Dynamics[END_REF] [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF]) and Parry numbers. Whether it contains Salem numbers or Perron numbers, except a few cases [START_REF] Boyd | The beta expansion for Salem numbers[END_REF], is unknown [START_REF] Verger-Gaugry | On lacunary Rényi β-expansions of 1 with β > 1 a real algebraic number, Perron numbers and a classification problem[END_REF]. Problem (P 1 ) is equivalent to knowing whether the β-shift is specified [START_REF] Blanchard | β-expansions and Symbolic Dynamics[END_REF] [49] [START_REF] Verger-Gaugry | On lacunary Rényi β-expansions of 1 with β > 1 a real algebraic number, Perron numbers and a classification problem[END_REF].

The set Z β contains {0, ±1} and all the polynomials in β for which the coefficients are given by the equations (2.11). Parry [START_REF] Parry | On the β-expansions of real numbers[END_REF] has shown that the knowledge of d β (1) suffices to exhaust all the possibilities of such polynomials by the so-called "Conditions of Parry (CP β )". Let us recall them. Let (c i ) i≥1 ∈ A N β be the following sequence:

c 1 c 2 c 3 • • • =    t 1 t 2 t 3 • • • if d β (1) = 0.t 1 t 2 • • • is infinite, (t 1 t 2 • • • t m-1 (t m -1)) ω if d β (1) is finite and equal to 0.t 1 t 2 • • • t m , (2.13
) where ( ) ω means that the word within ( ) is indefinitely repeated. When the degree of β is ≥ 2 , we have c 1 = t 1 = ⌊β⌋ . Then the polynomial v i=0 y i β -i+v ≥ 0, with v ≥ 0, y i ∈ Z arbitrary, belongs to Z + β := Z β ∩ R + if and only if y i ∈ A β and the following v + 1 inequalities are satisfied:

(CP β ): (y j , y j+1 , y j+2 , . . . , y v-1 , y v , 0, 0, 0, . . .) ≺ (c 1 , c 2 , c 3 , . . .), for allj = 0, 1, 2, . . . , v, (2.14)
where "≺" means lexicographical smaller. For a negative polynomial, we consider the above criterium applied to its opposite. Conditions of Parry (CP β ) sieve the elements of the ring Z[β] in the number field Q(β).

The set Z β can be viewed as the set of vertices of the tiling T β of the real line for which the tiles are the closed intervals whose extremities are two successive β-integers. When β is a Pisot number, the number of (noncongruent) tiles in T β is finite [START_REF] Thurston | Groups, Tilings and Finite State Automata[END_REF]. If V is a tile of T β we denote by l(V ) its length. If β is a Pisot number and d β (1) is finite, say d β (1) = 0.t 1 t 2 . . . t m , then the set of the lengths of the tiles of T β is exactly

{T i β (1) | 0 ≤ i ≤ m -1} = {1, β -t 1 , β 2 -t 1 β -t 2 , . . . , β m-1 -t 1 β m-2 - t 2 β m-3 -. . . -t m-1 }. If β is a Pisot number with d β (1) eventually periodic, say d β (1) = 0.t 1 t 2 . . . t m (t m+1 t m+2 . . . t m+p ) ω , then the set of the lengths of the tiles of T β is exactly {T i β (1) | 0 ≤ i ≤ m + p -1} = {1, β -t 1 , β 2 -t 1 β - t 2 , . . . , β m-1 -t 1 β m-2 -t 2 β m-3 -. . . -t m-1 , β m -t 1 β m-1 -t 2 β m-2 -. . . - t m , . . . , β m+p-1 -t 1 β m+p-2 -t 2 β m+p-3 -. . . -t m+p-1 }. Hence, when β is a Pisot number, the set Z β is a Delone set of (sharp) constants (r, R) with r = min{l(V ) | V ∈ T β } > 0 and R = 1 2 max{l(V ) | V ∈ T β } = 1 2
. The tiling T β can be obtained directly from a substitution system on a finite alphabet which is associated to β in a canonical way [START_REF] Fabre | Substitutions et β-systèmes de numération[END_REF] [42] [START_REF] Frougny | Numeration Systems, Chapter 7 in Algebraic Combinatorics on Words[END_REF].

Rauzy fractals and Meyer sets of beta-integers for β a Pisot number Rauzy fractals were introduced by Rauzy [1]

[2], [START_REF] Fogg | Substitutions in Dynamics, Arithmetics and Combinatorics[END_REF] (Chapter 7), [START_REF] Messaoudi | Propriétés arithmétiques et dynamiques du fractal de Rauzy[END_REF] [88] [START_REF] Siegel | Représentations des systèmes dynamiques substitutifs non unimodulaires[END_REF] [99] to provide geometric interpretations and geometric representations of symbolic dynamical systems, in the general objective of understanding whether substitutive dynamical systems are isomorphic to already known dynamical systems or if they are new. Rauzy [START_REF] Rauzy | Nombres algébriques et substitutions[END_REF] generalized the dynamical properties of the Fibonacci substitution [START_REF] Fogg | Substitutions in Dynamics, Arithmetics and Combinatorics[END_REF] to a three-letter alphabet substitution, called Tribonacci substitution or Rauzy substitution, defined by: 1 → 12, 2 → 13, 3 → 1. The incidence matrix of this substitution is

  1 1 1 1 0 0 0 1 0   , its characteristic polynomial is X 3 -X 2 -X -1 with β > 1 a
Pisot number as dominant root, and two complex conjugates roots α and α in the unit disc. The incidence matrix admits as eigenspaces in R 3 an expanding one-dimensional direction and a contracting plane [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF].

Theorem 2.26 (Rauzy). The Rauzy fractal generates a self-similar periodic tiling of the plane. The symbolic dynamical system generated by the Tribonacci substitution is measure-theoretically isomorphic to a toral substitution. The Tribonacci substitutive dynamical system has a purely discrete spectrum. [START_REF] Sirvent | Self-affine tiling via substitution dynamical systems and Rauzy fractals[END_REF]. Gazeau and Verger-Gaugry [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF] proved that the set Z β of integers in base β are in close relation with the Rauzy fractal, within the framework of a canonical cut-and-project scheme (R 3 = G × E, L = Z 3 , π 1 , π 2 ) above Z β , where E is a line in R 3 and G the corresponding internal space (hyperplane): the Rauzy fractal is the adherence of the image of Z β by the map π 1 • (π 2 | L ) -1 in the internal space G. This situation is quite general for Pisot numbers β and the Rauzy fractal appears as a compact canonical window [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF]. However, all the points of L are not selected by this window and only some of them which satisfy the conditions of Parry are projected on E, Z β being a Meyer set [START_REF] Burdik | beta-integers as natural counting systems for quasicrystals[END_REF] [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF].

Properties of the Rauzy fractal (connectedness, interiors, boundary, etc) were obtained in [2] [32] [60] [74] [98] [99] [100]

For all Perron numbers β, the construction of the cut-and-project scheme

(R d = G × E, L = Z d , π 1 , π 2 ) over Z β ,
where d is the degree of β, E a line in R d and G the corresponding internal space (hyperplane), is canonical, and does not use the fact that Z β should be uniformly discrete [START_REF] Gazeau | Geometric Study of the Beta-integers with β a Perron Number and Mathematical Quasicrystals[END_REF]. If β is a Pisot number then the image of L by π 1 • (π 2 | L ) -1 is relatively compact and its adherence is the (geometric) Rauzy fractal. Whether this image is relatively compact for β a general Salem number is not known.

The Z-module Z[Z β -Z β ] is finitely generated for β a Pisot number, but it is not known whether it is the case for Perron numbers in general (which are not Pisot numbers).

Open problem (P 2 ).-What is the class of real numbers β > 1 for which Z β is uniformly discrete and is not finitely generated, i.e. for which

rank Z[Z β -Z β ] = +∞? 3 Proof of Theorem 1.1 (i) (same proof as [65] Theorem 4.1 (i)) Let s = dim R R[Λ]
be the dimension of the R-span of Λ (by R-span of Λ, we mean the intersection of all the real affine subspaces of R n which contain Λ). Then 1 ≤ s ≤ n and m := rk Λ ≥ s.

By definition the

Z-module Z[Λ -Λ] := { finite a i (x i -x j ) | a i ∈ Z, x i , x j ∈ Λ} admits a set of m generators, say {v 1 , v 2 , . . . , v m }, which are Q-linearly independent (nonzero) vectors of R n . Then Z[Λ -Λ] = Z[v 1 , v 2 , . . . , v m ]. If λ > 1 is a self-similarity of Λ then there exists c ∈ R n such that λ(Λ -c) ⊂ Λ -c. Since Λ -Λ = Λ -c -(Λ -c), this implies λZ[Λ -Λ] ⊂ Z[Λ -Λ]. (3.1)
We deduce that there exist integers a i,j ∈ Z such that

λv i = α i,1 v 1 + α i,2 v 2 + . . . + α i,m v m i = 1, 2, . . . , m
with M = α i,j i,j ∈ Mat m (Z) the space of m × m integral matrices. Hence

λV = M V , with V =    v 1 . . . v m    . (3.2)
Equivalently the transposed matrix t M is the matrix associated with the Qlinear map which sends {v 1 , v 2 , . . . , v m } to the system {λv 1 , λv 2 , . . . , λv m } with respect to the Q-free system {v 1 , v 2 , . . . , v m }. Since the polynomial h(X) := det(X I -M ) ∈ Z[X] is monic and cancels at λ, the real number λ is an algebraic integer of degree less than m.

Let d be the degree of λ and

ϕ(X) = X d + a 1 X d-1 + a 2 X d-2 + . . . + a d , with a i ∈ Z, a d = 0,
be the minimal polynomial of λ. From (3.2) we deduce λ j V = M j V for all j ∈ N. Hence, since ϕ(λ) = 0,

ϕ(M )V = (M d + a 1 M d-1 + a 2 M d-2 + . . . + a d )V = 0. (3.3)
Since ϕ(M ) ∈ Mat m (Z) and that the vectors v 1 , v 2 , . . . , v m are Q-linearly independent, we deduce ϕ(M ) = ϕ( t M ) = 0. Hence the minimal polynomial ψ(X) ∈ Z[X] of the matrix t M divides ϕ(X) in Z[X]. Since ϕ(X) is irreducible over Q, there is equality: ψ(X) = ϕ(X).

Denote by K the number field Q(λ). Equation (3.1) implies that Z[Λ -Λ] is a module over the ring Z[λ] and that Q[Λ -Λ] is a K-vector space. The ring Z[λ] is a subring of finite index of the ring of integers O K of K. The m × m integral matrix t M corresponds to an endomorphism of R m , say u, expressed in the canonical basis {e 1 , e 2 , . . . , e m }. Since v 1 , v 2 , . . . , v m are Q-linearly independent, the base {e 1 , e 2 , . . . , e m } of R m and the system {v 1 , v 2 , . . . , v m } of R n can be identified as well as the two Q-vector spaces ⊕ m i=1 Qe i and ⊕ m i=1 Qv i . There are two cases: m = 1 and m > 1. When m = 1, then necessarily λ is an integer > 1 and d = 1. When m > 1 and d = 1, then the matrix M is the diagonal matrix λI and λ is an integer > 1. This case occurs for instance for (affine) lattices Λ of R n . Now, if m > 1 and d ≥ 2, then the endomorphism u induces a Jordan decomposition of R m as K[X]-module as follows (for instance [START_REF] Frenkel | Géométrie pour l'élève-professeur[END_REF] pp 295-301). We assume d ≥ 2 in the sequel. Let

A(ψ) =          0 0 . . . 0 -a d 1 . . . 0 . . . . . . 0 1 . . . . . . . . . . . . 0 -a 2 0 . . . 1 -a 1          , resp. U d :=    0 . . . 1 . . . 0 . . . 0 . . . 0    ,
be the d × d integral matrix whose terms are 0 except the last column which is composed of the coefficients of ψ(X) (up to sign) and the diagonal under the main diagonal, which is composed with 1, respectively the d × d matrix whose terms are 0 except the term of the first row and the last column, which is 1 (this makes sense since d > 1). Then there exists a basis of R m , say {ǫ 1 , ǫ 2 , . . . , ǫ m }, in which the matrix of the endomorphism u takes the diagonal form

J :=       J i1 0 . . . 0 0 J i2 . . . . . . . . . . . . . . . 0 0 . . . 0 J ir       where i 1 + i 2 + . . . + i r = m, i 1 ≥ i 2 ≥ . . . ≥ i r ≥ d, and J iq , 1 ≤ q ≤ r
, is the i q × i q integral matrix given by

J iq :=       A(ψ) 0 . . . 0 U d A(ψ) . . . . . . . . . . . . 0 0 . . . U d A(ψ)      
with 0 everywhere except A(ψ) on the main diagonal and U d on the diagonal under the main diagonal. Since all the diagonal terms of the matrix J iq are A(ψ), they are identical, and therefore d divides i q . Consequently d divides r q=1 i q = m.

(ii) Let us transcript back this Jordan decomposition to the ambient space R n of the uniformly discrete set Λ, block by block. Let us consider the first block J i1 , the situation being the same for the others. The system {ǫ 1 , ǫ 2 , . . . , ǫ i1 } satisfies the following relations:

-for 1 ≤ β < d, 0 ≤ α ≤ i1 d -1, u(ǫ αd+β ) = ǫ αd+β+1 , (3.4) -for β = d, 0 ≤ α < i1 d -1, u(ǫ (α+1)d ) = ǫ (α+1)d+1 -a 1 ǫ (α+1)d -a 2 ǫ (α+1)d-1 -. . . -a d ǫ αd+1 , (3.5) -for β = d, α = i1 d -1, u(ǫ i1 ) = -a 1 ǫ i1 -a 2 ǫ i1-1 -. . . -a d ǫ i1-d+1 . (3.6)
Now the matrices t M and J have coefficients in Q and are such that there exist a m × m invertible matrix C ∈ GL(m, R) such that t M = CJC -1 . Then (Corollary 2 in [START_REF] Lang | Algebra[END_REF], Chap. XV, §3) there exists a m × m invertible matrix C ′ in Q such that:

t M = C ′ JC ′-1 .
The matrix C ′ is the matrix associated with the linear map which sends {e 1 , e 2 , . . . , e m } to {ǫ 1 , ǫ 2 , . . . , ǫ m } with respect to the basis {e 1 , e 2 , . . . , e m }.

Let      f 1 f 2 . . . f m      = t C ′ V , with V =      v 1 v 2 . . . v m      . (3.7)
The Q-free system {f 1 , f 2 , . . . , f m } of nonzero vectors of R n , identified with the basis {ǫ 1 , ǫ 2 , . . . , ǫ m } of R m , admits the following structure: from (3.2)

λV = M V = t (C ′ JC ′-1 )    v 1 . . . v m    ⇔ t C ′-1 (λ I -t J) t C ′    v 1 . . . v m    = 0; (3.8) hence λ    f 1 . . . f m    = t J    f 1 . . . f m    . (3.9)
From (3.9), considering the first block J i1 , the situation being the same with the other blocks J iq , 1 ≤ q ≤ r, we deduce (see (3.4), (3.5)):

-for 1

≤ β < d, 0 ≤ α ≤ i1 d -1, f αd+β+1 = λf αd+β = λ β f αd+1 , (3.10)
-for β = d, 0 ≤ α < i1 d -1, f (α+1)d+1 = λf (α+1)d + a 1 f (α+1)d + a 2 f (α+1)d-1 + . . . + a d f αd+1 . (3.11)
Let us show that the assumption d < i 1 leads to a contradiction. Assume d < i 1 . Then we would have f d+1 = 0 from (3.7) since C ′ is invertible. But, from (3.10) and (3.11), with α = 0,

f d+1 = (λ + a 1 ) f d + a 2 f d-1 + . . . + a d f 1 = (λ + a 1 ) λ d-1 f 1 + a 2 λ d-2 f 1 + . . . + a d f 1 = ϕ(λ) f 1 = 0.
Contradiction. Hence d = i 1 . Proceeding now with the other blocks in the same way leads to the equalities d = i 1 = i 2 = . . . = i r .

The matrix C ′ belongs to GL(m, Q). If C ′ ∈ GL(m, Z) we take

w 1 = f 1 , w 2 = f d+1 , w 3 = f 2d+1 , . . . , w r = f (r-1)d+1 .
Then we deduce from (3.7) and (3.10) (and its analogs for the other blocks) that ⊕ m i=1 Z v i and ⊕ r q=1 ⊕ m i=1 Zλ i w q are isomorphic as Z-modules. We deduce the result in this case. If C ′ ∈ GL(m, Q) \ GL(m, Z), let us denote by µ the lcm of the m 2 denominators of the coefficients of C ′-1 and take

w 1 = f 1 /µ, w 2 = f d+1 /µ, w 3 = f 2d+1 /µ, . . . , w r = f (r-1)d+1 /µ.
The coefficients of D := µ t C ′-1 are in Z and relatively prime so that

D                  f 1 /µ λf 1 /µ . . . λ d-1 f 1 /µ . . . f r /µ λf r /µ . . . λ d-1 f r /µ                  =      v 1 v 2 . . . v m      . (3.12)
From (3.12), the Z-module ⊕ m i=1 Z v i is a Z-submodule of ⊕ r q=1 ⊕ d-1 i=0 Zλ i w q . Hence the result.

(iii) Since d = i 1 = i 2 = . . . = i r and r = m d by (ii), we deduce that J is the m × m diagonal matrix for which the diagonal terms are all identical and equal to A(ψ):

J =       A(ψ) 0 . . . 0 0 A(ψ) . . . . . . . . . . . . 0 . . . . . . A(ψ)       .
Since det M = det J , we obtain the characteristic polynomial of M :

det(X I m -M ) = det(X I d -A(ψ)) m/d = (ϕ(X)) m/d ; in particular, det M = (detA(ψ)) m/d = N (λ) m/d
, where N (λ) = (-1) d a d , product of the conjugates of λ, is the algebraic norm of λ. This formula is reminiscent of the algebraic norm of an element in a ring extension [START_REF] Lang | Algebra[END_REF].

(iv) There are two cases: either (iv-1) w 1 , w 2 , . . . , w r are K-linearly independent, or (iv-2) they are K-linearly dependent. In the first case the cutand-project scheme above Λ will admit an internal space reduced to its shadow space (see below and [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF]), while, in the second case, the internal space will come from the shadow space and the space of relations over K between the vectors w i .

First let us fix some notations. Denote by C the finite set {w 1 , w 2 , . . . , w r } and call it the central cluster of the basis ( λ j w i ) i=1,...,r,j=0,1,...,d-1 . For i = 1, 2, . . . , r, let w i := w i -1 w i . Let C := { w 1 , . . . , w r } the image of C on the unit sphere S n-1 of R n . We have Card( C) ≤ r. We assume that the signature of the field

K = Q(λ), of degree d, is (r 1 , r 2 ). Then d = r 1 + 2r 2 .
Denote by σ j , 1 ≤ j ≤ r 1 , the real embeddings of K in R, and by σ j , σ r2+j = σ j , where

r 1 + 1 ≤ j ≤ r 1 + r 2 , the imaginary embeddings of K in C. Assume σ 1 (λ) = λ. Let Σ be the embedding of K in R r1 × C 2r2 defined by ∀ξ ∈ K, Σ(ξ) = (σ 1 (ξ), σ 2 (ξ), . . . , σ r1+2r2 (ξ)) .
Let (g j ) 1≤j≤d be a Z-basis of the ring of integers O K of K. We identify the field K to Q d via the mapping Ψ defined by Ψ(z) =

d i=1 z i g i if z ∈ Q d , resp. O K to Z d if z ∈ Z d . The composed mapping Φ := Σ • Ψ is then extended in a continuous way from Q d to R d ,
and denoted in the same way:

∀z ∈ R d , Φ(z) := d i=1 z i σ 1 (g i ), d i=1 z i σ 2 (g i ), . . . , d i=1 z i σ r1+2r2 (g i ) .
Σ is an injective homorphism for the ring structures while Ψ is Q-vector space isomorphism. Thus Φ is a R-vector space isomorphism from R d onto the étale R-vector space

K R := K ⊗ Q R = R r1 × {z ∈ C 2r2 | z r2+j = z j for all j = 1, 2, . . . , r 2 }.
The R-subspace Σ(O K ) of K R is a lattice. Let us extend Φ to C d as a Cendomorphism, keeping the same notation, by (with I = √ -1):

Φ(x + I y) = Φ(x) + I Φ(y), for all x, y ∈ R d .
Let us construct the cut-and-project scheme above Λ. Let x ∈ R[Λ] ⊂ R n be in the R-span of Λ. For i = 1, 2, . . . , r, denote by p i (x) the orthogonal projection of x onto the line Rw i and w i := w i -1 w i . Then p i (x) can be written

p i (x) = x, w i w i , w i w i = x, w i w i ,
where •, • is the standard Euclidean inner product. The point x ∈ R n will be said K-rational if the r coefficients x, w i belongs to K. In this case, for all i = 1, . . . , r, x, w i = d j=1 α i,j (x)g j with all coefficients α i,j (x) ∈ Q. Let us define Σ i : K → K R w i by Σ i (ξ) = Σ(ξ) w i , for all i = 1, . . . , r, and Φ i = Σ i • Ψ the R-vector space isomorphism from R d onto the R-vector space

K R w i := (R w i ) r1 ×{{z j w i } | z = (z j ) ∈ C 2r2 , z r2+j = z j for all j = 1, 2, . . . , r 2 }. The R-subspace Σ i (O K ) of K R w i is a lattice. For any set A, denote by pr k : A d → A the k-th projection, so that pr k (K R w i ) = σ k (K) w i for all i = 1, . . . , r and k = 1, . . . , d. Since the mapping R[Λ] → r i=1 pr 1 (K R w i ) , x → ( x, w i ) i (3.13)
is injective, as R-morphism of vector spaces, the R-span R[Λ] of Λ is identified by (3.13) with a s-dimensional R-subspace of the first component

r i=1 K w i = r i=1 pr 1 (K R w i ). Denote by R K the subspace of r i=1 K R w i which is the closure of {(Σ i ( x, w i )) i | x ∈ R[Λ] is K-rational}. It is a product of r 1 copies of R[Λ] and r 2 copies of C[Λ], with pr 1 (R K ) = R[Λ].
The space r i=1 K R w i is a K-vector space, the external law being given by

K × r i=1 K R w i → r i=1 K R w i (3.14) (µ, u) Σ(µ) • u
with componentwise multiplication, where u = (u k ) k=1,...,d , so that the external law, on the k-th component, is given by:

K × r i=1 pr k (K R w i ) → r i=1 pr k (K R w i ) . (3.15) (µ, u k ) σ k (µ) • u k
The actions (3.14) and (3.15) are extended from K to R by continuity. Thus, by (3.15) and since the conjugate fields σ j (K), σ r2+j (K) are not subfields of R for j = 1, 2, . . . , r 2 (if r 2 = 0), the usual scalar product •, • on R n should be considered as the restriction to R n of the standard hermitian form on C n ; in particular, it is anti-linear for the second variable. We now construct a real positive definite symmetric bilinear form on

r i=1 K R w i . Since r i=1 K R w i = d k=1 r i=1 pr k (K R ) w i it suffices to construct it on the k-th component r i=1 σ k (K) w i . Let us define q k : r i=1 σ k (K) w i × r i=1 σ k (K) w i → R by q k (U, V ) := r i=1 u i v i k = 1, 2, . . . , r 1 , r i=1 (u i v i + u i v i ) k = r 1 + 1, . . . , r 1 + r 2 . (3.16)
where U = (u i w i ) i and V = (v i w i ) i . Then we define q :

r i=1 K R w i × r i=1 K R w i → R, q(U, V ) := r1+r2 k=1 q k (pr k (U ), pr k (V )). (3.17) Let G := {ϕ ∈ R C | ϕ(w) w ∈ K for all w ∈ C, r i=1 w i -1 ϕ(w i ) w i = 0 }.
We call G the space of relations over K between the generators w 1 , . . . , w r .

The space G can be identified with a subspace of K C ≃ K r , therefore with a subspace of r i=1 K w i . Denote by G its image by r i=1 Σ i in r i=1 K R w i and by G, resp. G, the closure of G, resp. of G. For all x ∈ R[Λ] and all ϕ ∈ G,

x, r i=1 w i -1 ϕ(w i ) w i = r i=1 w i -1 ϕ(w i ) x, w i = 0. (3.18)
(3.18) implies that q 1 ( (

w i -1 ϕ(w i ) w i ) i , ( x, w i w i ) i ) = 0.
Then the two subspaces R[λ] and G are orthogonal and complementary in r i=1 R w i ≃ R r , of respective dimensions s and rs.

Let us prove that R K and G are orthogonal and complementary, of respective dimension sd and (r -s)d, in

r i=1 K R w i . It suffices to prove q k (U, V ) = 0 with U = (σ k ( w i -1 ϕ(w i )) w i ) i and V = (σ k ( x, w i ) w i ) i , for all k = 2, 3, . . . , d, x ∈ R[Λ] K-rational and ϕ ∈ G. We have 1 2 q k (U, V ) = Re r i=1 σ k ( w i -1 ϕ(w i )) σ k ( x, w i ) = Re σ k r i=1 w i -1 ϕ(w i ) x, w i = Re σ k x, r i=1 w i -1 ϕ(w i ) w i = 0.
We deduce the claim. The cut-and-project scheme above Λ we have constructed is the following:

r i=1 K R w i ≃ G × R K ≃ H × R[Λ], L, π, pr 1 where L = r i=1 Σ i (O K ) is a lattice in G × R K and pr 1 such that pr 1 (R K ) = R[Λ], pr 1 (G) = 0. Because of the structure of the Z-module Z[Λ -Λ] given by (ii), it suffices to take L = r i=1 Σ i (Z[λ]
). The projection mapping π is Id-pr 1 . The internal space, say H, is G×(R K \R[Λ]). By construction, π(L) is dense in H and pr 1 is one-to-one on L, onto pr

1 (L) = Z[λ][w 1 , w 2 , . . . , w r ] ⊃ Z[Λ -Λ]. Note that Z[Λ -Λ] is not necessarily a free Z[λ]-module, but it is of finite index in pr 1 (L) = Z[λ][w 1 , w 2 , . . . , w r ].
The Euclidean structure on the cutand-project scheme given by q is such that R[Λ] and H are orthogonal. The component R K \ R[Λ] of the internal space H is called the shadow space in [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF].

If the vectors w 1 , w 2 , . . . , w r are K-linearly independent (case (iv-1)) then G is trivial and the internal space

H is R K \ R[Λ].
Now, this cut-and-project scheme lies above Λ since, for all ν ∈ Λ, Λν ⊂ Λ -Λ ⊂ Z[Λ -Λ] ⊂ pr 1 (L). We deduce the claim.

Ideal lattices and proof of Corollary 1.2

The objectives of this section are the following: (i) to recall some definitions concerning ideal lattices, referring to [11] [12] [13] [14] [START_REF] Bayer-Fluckiger | Ideal lattices over totally real number fields and Euclidean minima[END_REF], (ii) to show that the sublattice (L ′ , q) of (L, q) such that pr 1 (L ′ ) = Z[Λ -Λ] in the cut-and-project scheme above the ssfgud set Λ given by Theorem 1.1 (iv) is a sublattice of an ideal lattice.

It will suffice to show that the canonical bilinear form q defined by (3.17) has suitable properties.

The canonical involution (or complex conjugation) of the algebraic number field K generated by the self-similarity λ is the involution -: K R → K R that is the identity on R r1 and complex conjugation on C r2 . Let P := {α ∈ K R | α = α and all components of α are > 0}. Let us denote by Tr : K R → R the trace map, i.e. T r(x 1 , x 2 , . . . , x d ) = x 1 + . . . + x d .

A generalized ideal will be by definition a sub O K -module of K-rank one of K R . As examples, fractional ideals of K are generalized ideals; ideals of the type uI where I is an O K -ideal and u ∈ K R are also generalized ideals. An ideal lattice is a lattice (I, b) where I is a generalized ideal, and b : K R × K R → R which satisfies the equivalent conditions of Proposition 4.1 with α ∈ P. Ideal lattices with respect to the canonical involution correspond bijectively to Arakelov divisors of the number field K [START_REF] Bayer-Fluckiger | Upper bounds for Euclidean minima of algebraic number fields[END_REF] [START_REF] Bayer-Fluckiger | Ideal Lattices, in A panorama of number theory or the view from Baker's garden[END_REF].

It is easy to check that Proposition 4.1 is satisfied by the real symmetric bilinear form q defined by (3.16) and (3.17) in Section 3, where α = (α i ) 1≤i≤d with α i = 1 for all i. We assume r = 1, i.e. that the degree of the self-similarity λ is equal to the rank rk Λ. The lattice L ′ given by Theorem 1.1 (ii) such that pr

1 (L ′ ) = Z[Λ -Λ] is of finite index in the O K -module Σ(O K ) w 1 , of K-rank one in K R w 1 .
We have:

q(µ U, V ) = q(U, µV ) forall U, V ∈ K R w 1 and all µ ∈ K R .
Then, by Proposition 4.1, the bilinear form q has the following expression:

q(U, V ) = Tr(α U V ) forall U, V ∈ K R w 1 . (4.1)
Therefore (L ′ , q) is a sublattice of finite index of an Arakelov divisor of K in bijection with Z[Λ -Λ] by the projection mapping pr 1 | L . We deduce Corollary 1.2.

The construction of the real bilinear form q in Theorem 1.1 (iv) which provides the Euclidean structure to the cut-and-project scheme is obtained with α = (1) 1≤i≤d in Proposition 4.1. Other choices of α are possible and the parametrization of the set of possible constants α = α( w 1 ) in q in (4.1) is studied for instance in Schoof ([93] and related works).

5 Lower bounds of densities and pseudo-Delone constants

Pseudo-Delone sphere packings

The following definition is inspired by the "empty sphere" method of Delone [START_REF] Delone | Sur la Sphère Vide[END_REF]. If A ⊂ R n is any nonempty subset of R n and Λ is a uniformly discrete set of constant r > 0, we define the density of B(Λ) in A by

δ A (B(Λ)) := lim sup t→+∞ vol( zi∈Λ, zi ≤t B(z i , r/2) ∩ A)
vol(B(0, t) ∩ A) .

(5.1)

We omit the subscript "R n " when A = R n .

Definition 5.1. A uniformly discrete set Λ of R n , n ≥ 1, of constant r > 0 is Delone-like of constant R ξ > 0 when there exists a sequence ξ := (x i , T i ) i where (x i ) i is a sequence of points of R n and (T i ) i a sequence of real numbers such that, with the notation

A ξ := R n \ i • B(x i , T i ): (i) ∀i, T i+1 ≥ T i , with T i ≥ r/2, (ii) ∀i, • B(x i , T i ) ∩ B(Λ) = ∅, (iii) ∀x ∈ A ξ , ∃λ ∈ Λ such that x -λ ≤ R ξ , (iv) lim T →+∞ vol(A ξ ∩B(0,T )) vol(B(0,T )) = 1.
If it is finite, the infimum inf{R ξ } over all possibilites of point sets (x i ) i in R n and collections of radii (T i ) i , such that (i) to (iv) are satisfied, is called the pseudo-Delone constant of Λ and denoted by R(Λ). Let us call optimal a collection ξ such that R ξ is equal to R(Λ).

By Zorn's Lemma, optimal collections exist. The portion of space ∪ i • B(x i , T i ) defined by an optimal collection is an invariant, independent of the optimal collection used for defining it. Definition 5.1 means that we can remove the portion of ambient space which does not intervene at infinity for the computation of the density of Λ (in R n ). Note that, for a ssfgud set of R n , this portion of space does not contribute to the determination of the generators w i in Theorem 1.1 since i • B(x i , T i ) contains no point of Λ, and it is legitimate to remove it.

Proof of Theorem 1.3

Let R c := inf{R(Λ) | Λ is uniformly discrete of R n of constant 1} be the infimum of possible Delone constants over sphere packings of common radius 1/2. R c is only a function of n. Then, for all r > 0, rR c is the infimum of Delone constants of uniformly discrete sets of constant r.

Let r > 0 and ω n be the volume of the unit ball of R n . Let R ≥ rR c and T > R be a real number. Let Λ be a uniformly discrete set of R n of constant r > 0 which is pseudo-Delone of pseudo-Delone constant R. Let ξ := (x i , T i ) i be an optimal sequence and A

ξ := R n \ ∪ i • B(x i , T i ) i . For all ǫ > 0, the pseudo-Delone constant of Λ in A ξ is smaller than R + ǫ. Then (B(0, R + ǫ) + Λ) ∩ B(0, T ) covers the set B(0, T -R -ǫ) ∩ A ξ . The number of elements of Λ ∩ B(0, T ) is equal to the number of elements of Λ ∩ B(0, T ) ∩ A ξ . This number is at least ω n (T -R -ǫ) n -vol((R n \ A ξ ) ∩ B(0, T -R -ǫ)) ω n (R + ǫ) n = (T -R -ǫ) n (R + ǫ) n 1 - vol((R n \ A ξ ) ∩ B(0, T -R -ǫ)) ω n (T -R -ǫ) n .
On the other hand, since all the balls of radius r/2 centered at the elements of Λ ∩ B(0, T ) lie within B(0, T + r/2) and also within A ξ , the proportion of space they occupy in B(0, T + r/2) ∩ A ξ is at least

T -R -ǫ R + ǫ n 1 - vol((R n \ A ξ ) ∩ B(0, T -R -ǫ)) ω n (T -R -ǫ) n vol(B(0, r/2)) vol(B(0, T + r/2) ∩ A ξ ) (5.2) But, for all ǫ > 0, lim T →+∞ vol((R n \ A ξ ) ∩ B(0, T -R -ǫ)) ω n (T -R -ǫ) n = 0 and lim T →+∞ vol(B(0, T + r/2)) vol(B(0, T + r/2) ∩ A ξ ) = 1.
Hence, if T is large enough, the quantity (5.2) is greater than

r(T -R -ǫ) 2(R + ǫ)(T + r/2) n .
When T tends to infinity, this quantity tends to (2(R + ǫ)/r) -n , for all ǫ > 0, which is a lower bound of δ(B(Λ)). We deduce the claim.

Lower bounds of the Delone constant of a ssfgud set

The field K = Q(λ) generated by the self-similarity λ of the ssfgud set Λ in Theorem 1.1 has its own Euclidean spectrum [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF] [24] which leads to specific geometric properties of the Voronoi cell of the lattice L ′ [START_REF] Coulangeon | Voronoi theory over algebraic number fields, in Réseaux Euclidiens, designs sphériques et formes modulaires[END_REF] of the cut-andproject scheme above Λ, where L ′ such that pr 1 (L ′ ) = Z[Λ -Λ]. By projection by pr 1 , in this cut-and-project scheme above Λ, the Delone (or pseudo-Delone) constant of Λ, whatever the occupation of the elements of Λ -Λ in Z[Λ -Λ], reflects the arithmetical features of K (Euclidean minimum, Euclidean spectrum ... [23] [24]) as well as the geometrical characteristics of the central cluster {w 1 , . . . , w r }; in particular if Λ is a model set (see Proposition 2.10) or a Meyer set (see Definition 2.15 (iii) ), since, in both cases, a window in the internal space controls the thickness of the band around the R-span of Λ which is used for selecting the points of the lattice L ′ . Recall that the Delone (or pseudo-Delone) constant R(Λ) of the ssfgud set Λ, if finite, "measures" the maximal size of (spherical) holes in Λ with respect to the portion of space where the density is computed (see §2.1.1, §5 and [START_REF] Muraz | On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings[END_REF]) In the sequel, we recall these notions and refer to [11] [14] [15] [23] [START_REF] Cerri | Euclidean minima of totally real number fields[END_REF] [30] [START_REF] Lemmermeyer | The Euclidean algorithm in algebraic number fields[END_REF].

In the following we will only consider the case r = 1, i.e. the case where the degree of the self-similarity λ is equal to the rank of Z[Λ -Λ], leaving aside the case r > 1. Theorem 6.1, resp. Theorem 6.2, Corollary 6.3 and Theorem 6.4, is a reformulation in the present context of Theorem 3, resp. Theorem 5, Corollary 6 and Theorem 4 (Remark 2), obtained by Cerri [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF]. Recall that, for all ξ ∈ K, Σ 1 (ξ) = Σ(ξ) w 1 , with w 1 = w 1 -1 w 1 the unit vector.

Euclidean and inhomogeneous spectra of the number field generated by the self-similarity

Let N K/Q be the norm defined on K by

∀ξ ∈ K, N K/Q (ξ) = d i=1 σ i (ξ) = r1 i=1 σ i (ξ) r1+r2 i=r1+1 |σ i (ξ)| 2 . (6.1)
The field K is said to be norm-Euclidean if:

∀ξ ∈ K, ∃y ∈ O K such that N K/Q (ξ -y) < 1.
Following the notations of Section 3 and [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF] [24], we extend N K/Q • Ψ from Q d to R d by the map denoted by N as follows: (6.

∀x ∈ R d , N (x) = d i=1   d j=1 x j σ i (g j )   . ( 6
3)

The mapping m K • Ψ defined on Q d is extended to R d and is denoted by m:

m(z) := inf{ |N (z -l)| | l ∈ Z d } for z ∈ R d .
The inhomogeneous minimum of K is denoted by M (K) and is defined by

M (K) := sup x∈R d m(x). (6.4)
The mapping m

K • Σ -1 1 is extended to K R w 1 and is denoted by m K : m K (U ) := inf d i=1 (U i -Z i ) | Z = (Z i ) i w 1 ∈ Σ(O k ) w 1 for U = (U i ) i w 1 ∈ K R w 1 .
The set of values of m K , resp. of m, is called the Euclidean spectrum, resp. the inhomogeneous spectrum of K. Successive minima are enumerated: the second inhomogeneous minimum of K is defined by

M 2 (K) := sup x∈R d m(x)<M(K) m(x),
and the second Euclidean minimum of K by

M 2 (K) := sup ξ∈K mK (ξ)<M(K) m K (ξ).
Iteratively we define (p ≥ 2):

M p+1 (K) := sup x∈R d m(x)<Mp(K) m(x), and
M p+1 (K) := sup ξ∈K mK (ξ)<Mp(K) m K (ξ).
The inhomogeneous minimum M (K) of K is said to be isolated if

M 2 (K) < M (K).
This isolation phenomenon has been conjectured for d = 2 and K totally real by Barnes and Swinnerton-Dyer. Corollary 6.3 below shows that it occurs frequently.

If the inhomogeneous minimum M (K) of K satisfies the following property:

∀x ∈ R d , ∃l ∈ Z d such that |N (x -l)| ≤ M (K), (6.5)
we will say that M (K) is attained. Note that (6.5) is not verified for the quadratic field K = Q( √ 13) [START_REF] Lemmermeyer | The Euclidean algorithm in algebraic number fields[END_REF].

Theorem 6.1. Assume that the degree d of the field K generated by the selfsimilarity λ of the ssfgud sphere packing Λ is ≥ 3 and is equal to the rank of

Z[Λ -Λ]. If the unit rank r 1 + r 2 -1 of K is > 1, in particular if K is totally real, then (
i) there exists ξ ∈ K such that M (K) = m K (Σ 1 (ξ)), (ii)
M (K) = M (K) ∈ Q.
Then, if the unit rank r 1 + r 2 -1 of K is strictly greater than 1 and K is norm-Euclidean, there exists t ∈ (0, 1) such that the number of copies of A t to be considered for obtaining the fundamental region of L ′ is equal to the index of L ′ in Σ 1 (O K ) w 1 , an integer multiple of (O K : Z[λ]). Recall that ([23] Proposition 4):

(i) M (K) < 1 =⇒ K is norm-Euclidean, (ii) M (K) > 1 =⇒ K is not norm-Euclidean, If M (K) = 1,
it is not possible to conclude except if there exists ξ ∈ K such that M (K) = m K (ξ); in this case K is not norm-Euclidean. See [START_REF] Cerri | Euclidean minima of totally real number fields[END_REF] for computations of M p (K), p ≥ 1, with the conventions:

M (K) = M 1 (K), M (K) = M 1 (K).

Proof of Theorem 1.4

Let us now deduce lower bounds of the Delone constant of the ssfgud set Λ ⊂ R n , n ≥ 1, that we will assume either a model set or a Meyer set, subset of a model set, defined by a window Ω in the internal space R K \ R[Λ] of the cut-and-project scheme

K R w 1 = ( R K \ R[Λ] ) × R[Λ], L ′ , π, pr 1 (6.6)
given by Theorem 1.1 (iv) with L ′ such that pr 1 (L ′ ) = Z[Λ -Λ] and the window Ω nonempty, open and relatively compact such that

Λ ⊂ ν + {pr 1 (U ) | U ∈ L ′ , π(U ) ∈ Ω} ⊂ ν + R w 1 with ν ∈ Λ. (6.7) Λ is a Delone set. The central cluster is {w 1 }. Denote by R[Λ] + Ω the band {P = (U, V ) ∈ K R w 1 | U = pr 1 (P ) ∈ R[Λ], V = π(P ) ∈ Ω} parallel to the one-dimensional R-span R[Λ] of Λ. Note that R[Λ] = ν + R w 1 in R n .
In the sequel we will use the notations of §6.1 and the assumptions of Theorem 6.2. Definition 6.5. Let k ≥ 2 be an integer. The self-similar finitely generated Delone set Λ defined by (6.6) and (6.7) is called (i) thin if the following condition on Ω and L ′ holds:

0 < π Σ 1 (x -t) < d M (K) 2/d , (6.8)
for all t ∈ O K such that Σ 1 (t) ∈ L ′ ∩ R[Λ]+Ω , and all x ∈ m -1 K (M (K)), (ii) k-thin if

d (M k+1 (K)) 2/d ≤ π Σ 1 (x -t) < d (M k (K)) 2/d (6.9) for all t ∈ O K such that Σ 1 (t) ∈ L ′ ∩ R[Λ]+Ω and x ∈ k-1 p=1 m -1 K (M p (K)).
This definition is consistent with the following facts: (a) the values of m constitute a strictly decreasing sequence of positive rational integers (Theorem 6.2 (ii))which are reached a finite number of times modulo L ′ (Theorem 6.2 (iii)), (b) the infinite (band) cylinder R[Λ] + Ω, parallel to the one-dimensional space R[Λ], can be made sufficiently narrow in order to avoid the set of points z of Q d such that m K • Ψ(z) ∈ k-1 p=1 M p (K), for all k ≥ 2, which is a finite union of translates of L ′ , (c) Assertion (b) is possible since free planes, a fortiori free lines, do exist in any lattice (equal) sphere packings in R d once d is large enough: Henk [START_REF] Henk | Free planes in lattice sphere packings[END_REF] has proved the existence of an d log 2 (d) -dimensional affine plane (called free plane) which does not meet any of the spheres in their interiors. Hence, provided d is large enough, free lines exist in the lattice sphere packing B(L ′ ) corresponding to L ′ in K R w 1 . Equivalently narrow bands, with section a nonempty open set, about free lines, exist in K R w 1 which do not intersect L ′ . By continuity, there exist narrow bands, with nonempty open cross-sections, about free lines, which do not intersect any finite union of translates of L ′ . For all x ∈ K, t ∈ O K , by the geometric mean inequality, we have: q Σ 1 (xt), Σ 1 (xt) d (6.10) where q Σ 1 (xt), Σ 1 (xt) = pr 1 (Σ 1 (xt)) 2 + π(Σ 1 (xt)) 2 with the notations of Section 3. We have: pr 1 (Σ 1 (xt)) 2 = (xt) w 1 2 = (ν + x w 1 ) -(ν + t w 1 ) 2 .

|N K/Q (x)| 2 = |N K/Q (x -t)| 2 = r1 i=1 |σ i (x -t)| 2
The set m -1

K (M (K)) = {x ∈ K | m K (x) = M (K) = M (K)} is such that Σ 1 m -1 K (M (K)) = {Σ 1 (x) | m K (x) = M (K) = M (K)
} is finite modulo L ′ by Theorem 6.1 and Theorem 6.2.

Let us take x in m -1 K (M (K)). Then, from (6.10), for all t ∈ O K ,

M (K) 2 ≤ 1 d (ν + x w 1 ) -(ν + t w 1 ) 2 + π(Σ 1 (x -t)) 2 d .
Hence, for all t ∈ O K such that Σ 1 (t) ∈ L ′ ∩ R[Λ] + Ω ,

(ν + x w 1 ) -(ν + t w 1 ) 2 ≥ d M (K) 2/d -π(Σ 1 (x -t)) 2 .
Since by hypothesis ν + x w 1 does not belong to Λ and that we consider the elements t ∈ O k for which ν + t w 1 belongs to Λ (with Σ 1 (t) ∈ L ′ ∩ (R[Λ] + Ω)), we have: R(Λ) ≥ (ν + x w 1 ) -(ν + t w 1 ) .

Hence R(Λ) 2 ≥ d M (K) 2/dsup π(Σ 1 (xt)) 2 > 0 (6.11)

where the supremum is taken over all x ∈ m -1 K (M (K)) and all t ∈ O k for which ν + t w 1 belongs to Λ, with Σ 1 (t) ∈ L ′ ∩ (R[Λ] + Ω).

Let us assume that Λ is k-thin and take x in k-1 p=1 m -1 K (M p (K)). Then the supremum in (6.11) is bounded from above by d M k (K) Using Theorem 1.3 or [START_REF] Muraz | On Lower Bounds of the Density of Delone Sets and Holes in Sequences of Sphere Packings[END_REF] we deduce a lower bound of the density of the ssfgud set Λ.

The isolation phenomenon which frequently occurs (Corollary 6.3 (ii) of Theorem 6.2) in higher dimension is likely to occur in R n as well by projection for Λ.

Appendix.-Crystallography of Aperiodic Crystals and Delone sets

New states of matter call for mathematical idealizations of packings of atoms and consequently a deep understanding of the mathematics which lies behind as far as they are characterized by experimental techniques described by a mathematical formalism which has to be mathematically settled: diffraction (X-rays, electrons, neutrons, synchrotron radiation, etc) and inverse problems (crystal reconstruction with satisfying local atom clustering, long-range order and self-similarities, etc). Indeed, the situation is well-known for (periodic) crystals [START_REF] Cowley | Diffraction Physics[END_REF] [53] [54] [63] [96] but fairly unknown, or at least badly understood for nonperiodic crystals. Quasicrystals and modulated crystals constitute exceptions since the use of cut-and-project sets allows periodization in higher dimension [3] [4] [51] [62] [64] [START_REF]Quasicrystals and Geometry[END_REF]. The parts of mathematics concerned with the crystallography of aperiodic crystals are mainly Geometry of Numbers and Discrete Geometry [START_REF] Cassels | An introduction to the Geometry of Numbers[END_REF] [52] [START_REF] Zong | Sphere packings[END_REF], N -dimensional crystallography when periodization in higher dimension is concerned [START_REF] Michel | N -Dimensional Crystallography[END_REF] [84] [START_REF] Schwarzenberger | N -Dimensional Crystallography[END_REF], Spectral Theory, Ergodic Theory and Fourier Transform of Delone sets as far as diffraction is concerned [START_REF] Baake | Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra[END_REF] [59], Harmonic Analysis as far as density is concerned (as an asymptotic measure). Atoms are viewed as hard spheres and aperiodic crys-tals as Delone sets (sphere centers). Implicitely this means that atoms behave like spheres, that is do have a spherical potential. This is far from covering the large variety of possibilities of chemical boundings between atomic species (see [START_REF] Weygand | Modèle d'un quasicristal covalent de Bore pur[END_REF] for quasicrystalline models of pure Boron for instance). This provides first-order crystalline models from which the computation of the electron density is made possible. Then comparison with experimental data (densities, physical properties, ...) leads to refine the models. Looking for a fine hierarchy of Delone sets, from the mathematical side, either from arithmetics [START_REF] Elkharrat | Symmetry groups for beta-lattices[END_REF] [47] [START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF] or from tiling theory [START_REF]Directions in Mathematical Quasicrystals[END_REF] [8] [START_REF] Moody | The Mathematics of Long-Range Aperiodic Order[END_REF] [102], leads to interesting and new questions concerning crystals, without knowing whether these crystals will exist or not. To finish up let us recall the new definition of a crystal ( in R 3 ) which was recently chosen by the International Union of Crystallography [START_REF]International Union of Crystallography, Report of the Executive Committee for 1991[END_REF] and the former one [START_REF] Schwarzenberger | N -Dimensional Crystallography[END_REF]. Definition 6.6 (former definition). Any solid for which the set of atom positions is a finite union of orbits under the action of a crystallographic group. Definition 6.7 (new definition). Any solid having an essentially discrete diffraction diagram. Definition 6.7 covers all cases of solids defined by Definition 6.6 by Poisson formula (see [START_REF] Lagarias | Mathematical Quasicrystals and the Problem of Diffraction[END_REF] for a proof).

Proposition 4 . 1 .

 41 Let b : K R × K R → R be a symmetric bilinear form. The following statements are equivalent:(i) there exists α ∈ K R with α = α such that b(x, y) = T r(α xy) for all x, y ∈ K R , (ii) the identity b(µx, y) = b(x, µy)holds for all x, y, µ ∈ K R .Proof.[START_REF] Bayer-Fluckiger | Upper bounds for Euclidean minima of algebraic number fields[END_REF] Proposition 2.1,[START_REF] Bayer-Fluckiger | Ideal Lattices, in A panorama of number theory or the view from Baker's garden[END_REF] Proposition 1.

. 2 )

 2 Let ξ ∈ K. The Euclidean minimum of ξ (relatively to the norm N K/Q ) is the real number m K (ξ) := inf{ |N K/Q (ξy)| | y ∈ O K }. The Euclidean minimum of K (for the norm N K/Q ) is denoted by M (K) and is by definition: M (K) := sup ξ∈K m K (ξ).

2 d

 2 which allows to deduce the claim.

5 )

 5 Proof. Theorem 9.1 in[START_REF] Moody | Meyer sets and their Duals[END_REF], Theorem 2.1 and Theorem 3.1 in[START_REF] Lagarias | Geometric Models for Quasicrystals I. Delone sets of Finite Type[END_REF].

Definition 2.17.

Let Λ ∈ UD f g . If {e 1 , e 2 , . . . , e v } is a Z-basis of Z[Λ], i.e. Z[Λ] = Z[e 1 ,

e 2 , . . . , e v ], then the address map ϕ : Z[Λ] → Z v of Λ associated to this basis is by definition

Proof. Theorem 3 in [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF].

The question whether ξ is unique under some assumptions is not clear [23] [24]. Theorem 6.2. Assume that the degree d of the field K generated by the selfsimilarity λ of the ssfgud sphere packing Λ is ≥ 3 and is equal to the rank of Z[Λ -Λ]. If the unit rank r 1 + r 2 -1 of K is > 1 and if K is not a CM-field, in particular if K is totally real, there exists a strictly decreasing sequence (y p ) p≥1 of positive rational integers, which satisfies:

Proof. Theorem 5 in [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF].

From the definitions, the inequality M (K) ≤ M (K) holds for an arbitrary number field, with equality if d = 2 (Barnes and Swinnerton-Dyer [START_REF] Lemmermeyer | The Euclidean algorithm in algebraic number fields[END_REF]). Recently Cerri [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF] (Corollary 3 of Theorem 3) proved that the equality M (K) = M (K) does hold true for every number field. Corollary 6.3. Under the same hypotheses M (K) is attained and

What are the possible fundamental regions of the sublattices

Theorem 6.4. Denote, for all t > 0,

Proof. Remark 2 after Theorem 4 in [START_REF] Cerri | Inhomogeneous and Euclidean spectra of number fields with unit rank greater than 1[END_REF].