On embedding of repetitive Meyer multiple sets into model multiple sets

Abstract

Abstract. Model sets are always Meyer sets but the converse is generally not true. In this work we show that for a repetitive Meyer multiple sets of R d with associated dynamical system (X, R d ), the property of being a model multiple set is equivalent for (X, R d ) to be almost automorphic. We deduce this by showing that a repetitive Meyer multiple set can always be embedded into a repetitive model multiple set having a smaller group of topological eigenvalues. OutlineIn this paper we address a study of particular point patterns of an Euclidean space. From a general point of view, a point pattern is a collection of points inside some space R d (or some locally compact Abelian group), which obeys some discreteness and relative density properties. In this work we will be turned onto the slight generalization of multiple point patterns, that is, a finite collection of point patterns which may overlap. Thanks to this minor generalization our results become true for symbolic sequences and arrays as well. We precise that the concept of point patterns can be much further generalized, by considering for instance weighted Dirac combs as done in [6], or the even more general concept of translation-bounded measures [2].In the topic of point patterns, probably the most understood and studied objects are the so-called model sets, or model multiple set according to our point of view. Formally, a model multiple set is a point set of an Euclidean space that arises as follows: we consider a 'total space' to be the product of R d (the ambient space) with a locally compact Abelian group H (the internal group), together with a latticeΓ in this product. A model multiple set is then a finite family (Λ i ) i∈I of patterns, each Λ i being obtained as projections on R d of the points of the lattice whose projection in H falls into some compact and topologically regular subset W i of H. The data of the group H together with the latticeΓ is called a cut & project scheme and the family {W i } i∈I is called a window. Any coding of a rotation is a model multiple set: such arrays may be defined as the model multiple sets associated with cut & project schemes and window {W i } i∈I obeying the following three conditions: (i) The internal space H is compact (ii) The window {W i } i∈I covers H (iii) The sets W i have pairwise disjoint interiors.It is also often required that the window admits a trivial redundancy subgroup. Famous examples of such arrays are the Toeplitz sequences and arrays [10], also called elsewhere limit-periodic point sets [7], which are uniquely characterized as being the coding of a rotation over an odometer H.

Pure point/continuous decomposition of translation-bounded measures and diffraction

“…In addition, compact topologically regular subset W of H, that is, a compact set which is the closure of its interiorW in H, will be called a window. The hard part of this Theorem is piq ñ piiq, which is sufficient to prove when S is itself a Meyer set, and which was proved by Meyer [35, Chapter II, Section 5, Proposition 4], later on followed by several works [25,36,5,2]. Assuming this let us then provide a short proof of the remaining statements:…”

Spectral notions of aperiodic order

“…In this case, particular attention has been paid to the case that Λ is a Meyer set. In this case, the corresponding parts of [1,11,42] can be summarised as giving that these three regimes correspond exactly to the situation that Λ is crystallographic, a regular model set, a model set respectively. We refrain from giving precise definitions or proofs but rather refer the reader to [2] for a recent discussion; see [40] as well.…”

Toeplitz flows and model sets

“…Then, define U ℓ = w∈A(ℓ, 1) [w] and V ℓ = w∈A(ℓ,0) [w]. Clearly, U ℓ ⊂ U ℓ+1 and V ℓ ⊂ V ℓ+1 hold for any ℓ.…”

“…With a focus on proper (but not necessarily regular) model sets their investigation has become a cornerstone of the theory; see, for example, the survey papers [19,20]. Recently, also the more general classes of repetitive Meyer sets [1,15] (see the survey [2] as well) and of weak model sets [4,16] have been studied in some detail.…”